diff --git a/LeanPool.lean b/LeanPool.lean
index ddd609e9eb..7f0a0359e3 100644
--- a/LeanPool.lean
+++ b/LeanPool.lean
@@ -8092,6 +8092,216 @@ public import LeanPool.Zeta3Irrational.Equality
 public import LeanPool.Zeta3Irrational.Integral
 public import LeanPool.Zeta3Irrational.LegendrePoly
 public import LeanPool.Zeta3Irrational.LinearForm
+public import LeanPool.Zeta5Irrational
+public import LeanPool.Zeta5Irrational.Andreief
+public import LeanPool.Zeta5Irrational.ArcsineAtoms
+public import LeanPool.Zeta5Irrational.ArcsinePotential
+public import LeanPool.Zeta5Irrational.Arith.BasisChange
+public import LeanPool.Zeta5Irrational.Arith.BinomBasis
+public import LeanPool.Zeta5Irrational.Arith.ClassCount
+public import LeanPool.Zeta5Irrational.Arith.ClassFrame
+public import LeanPool.Zeta5Irrational.Arith.DirectBound
+public import LeanPool.Zeta5Irrational.Arith.DirectPoly
+public import LeanPool.Zeta5Irrational.Arith.InnerAlloc
+public import LeanPool.Zeta5Irrational.Arith.InnerWeights
+public import LeanPool.Zeta5Irrational.Arith.IntPoly
+public import LeanPool.Zeta5Irrational.Arith.LowRank
+public import LeanPool.Zeta5Irrational.Arith.MomentVal
+public import LeanPool.Zeta5Irrational.Arith.OuterBasis
+public import LeanPool.Zeta5Irrational.Arith.OuterEntries
+public import LeanPool.Zeta5Irrational.Arith.OuterEntry
+public import LeanPool.Zeta5Irrational.Arith.OuterEntryMain
+public import LeanPool.Zeta5Irrational.Arith.OuterFrame
+public import LeanPool.Zeta5Irrational.Arith.PoleFun
+public import LeanPool.Zeta5Irrational.Arith.PoleVal
+public import LeanPool.Zeta5Irrational.Arith.Pullback
+public import LeanPool.Zeta5Irrational.Arith.SmallPrime
+public import LeanPool.Zeta5Irrational.Arith.SmallPrimeF
+public import LeanPool.Zeta5Irrational.Arith.SmallPrimeRes
+public import LeanPool.Zeta5Irrational.Arith.SmallPrimeTau
+public import LeanPool.Zeta5Irrational.Arith.TEntry
+public import LeanPool.Zeta5Irrational.Arith.TauBound
+public import LeanPool.Zeta5Irrational.Arith.Unimodular
+public import LeanPool.Zeta5Irrational.Arith.Val
+public import LeanPool.Zeta5Irrational.Certificates
+public import LeanPool.Zeta5Irrational.CircleAtoms
+public import LeanPool.Zeta5Irrational.Constants
+public import LeanPool.Zeta5Irrational.Construction
+public import LeanPool.Zeta5Irrational.Criterion
+public import LeanPool.Zeta5Irrational.Degree
+public import LeanPool.Zeta5Irrational.DetIntegral
+public import LeanPool.Zeta5Irrational.Energy
+public import LeanPool.Zeta5Irrational.EnergyAtoms
+public import LeanPool.Zeta5Irrational.EnergyBlocks
+public import LeanPool.Zeta5Irrational.EnergyBound
+public import LeanPool.Zeta5Irrational.EnergyConst
+public import LeanPool.Zeta5Irrational.EnergyFinal
+public import LeanPool.Zeta5Irrational.EnergyLimit
+public import LeanPool.Zeta5Irrational.EnergyRaw
+public import LeanPool.Zeta5Irrational.Gaussian
+public import LeanPool.Zeta5Irrational.Growth
+public import LeanPool.Zeta5Irrational.Growth.Assembly1
+public import LeanPool.Zeta5Irrational.Growth.Assembly2
+public import LeanPool.Zeta5Irrational.Growth.Assembly3
+public import LeanPool.Zeta5Irrational.Growth.Assembly4
+public import LeanPool.Zeta5Irrational.Growth.ClassData
+public import LeanPool.Zeta5Irrational.Growth.ClassSum
+public import LeanPool.Zeta5Irrational.Growth.Constants
+public import LeanPool.Zeta5Irrational.Growth.InnerPrime
+public import LeanPool.Zeta5Irrational.Growth.InnerSum
+public import LeanPool.Zeta5Irrational.Growth.InnerTable
+public import LeanPool.Zeta5Irrational.Growth.OuterPrime
+public import LeanPool.Zeta5Irrational.Growth.OuterTable
+public import LeanPool.Zeta5Irrational.Growth.PieceIdx
+public import LeanPool.Zeta5Irrational.Growth.TableGen
+public import LeanPool.Zeta5Irrational.Growth.TableIntegrals
+public import LeanPool.Zeta5Irrational.Growth.TablePrime
+public import LeanPool.Zeta5Irrational.Growth.TailIntegral
+public import LeanPool.Zeta5Irrational.Growth.TailPrime
+public import LeanPool.Zeta5Irrational.Hermite
+public import LeanPool.Zeta5Irrational.HermiteFormula
+public import LeanPool.Zeta5Irrational.HermiteIBP
+public import LeanPool.Zeta5Irrational.HermiteIntegrals
+public import LeanPool.Zeta5Irrational.Kernel
+public import LeanPool.Zeta5Irrational.Legendre
+public import LeanPool.Zeta5Irrational.Main
+public import LeanPool.Zeta5Irrational.MainEstimate
+public import LeanPool.Zeta5Irrational.Measure
+public import LeanPool.Zeta5Irrational.Moments
+public import LeanPool.Zeta5Irrational.Norm
+public import LeanPool.Zeta5Irrational.PNT
+public import LeanPool.Zeta5Irrational.PartialFractions
+public import LeanPool.Zeta5Irrational.Positivity
+public import LeanPool.Zeta5Irrational.Potential
+public import LeanPool.Zeta5Irrational.PotentialV
+public import LeanPool.Zeta5Irrational.PrimeSum
+public import LeanPool.Zeta5Irrational.PrimeSumIntegral
+public import LeanPool.Zeta5Irrational.PrimeSumStep
+public import LeanPool.Zeta5Irrational.RealBound
+public import LeanPool.Zeta5Irrational.Stirling
+public import LeanPool.Zeta5Irrational.Table.Bracket
+public import LeanPool.Zeta5Irrational.Table.Common
+public import LeanPool.Zeta5Irrational.Table.Intervals
+public import LeanPool.Zeta5Irrational.Table.Tail
+public import LeanPool.Zeta5Irrational.Table.U00
+public import LeanPool.Zeta5Irrational.Table.U01
+public import LeanPool.Zeta5Irrational.Table.U02
+public import LeanPool.Zeta5Irrational.Table.U03
+public import LeanPool.Zeta5Irrational.Table.U04
+public import LeanPool.Zeta5Irrational.Table.U05
+public import LeanPool.Zeta5Irrational.Table.U06
+public import LeanPool.Zeta5Irrational.Table.U07
+public import LeanPool.Zeta5Irrational.Table.U08
+public import LeanPool.Zeta5Irrational.Table.U09
+public import LeanPool.Zeta5Irrational.Table.U10
+public import LeanPool.Zeta5Irrational.Table.U11
+public import LeanPool.Zeta5Irrational.Table.U12
+public import LeanPool.Zeta5Irrational.Table.U13
+public import LeanPool.Zeta5Irrational.Table.U14
+public import LeanPool.Zeta5Irrational.Table.U15
+public import LeanPool.Zeta5Irrational.Table.U16
+public import LeanPool.Zeta5Irrational.Table.U17
+public import LeanPool.Zeta5Irrational.Table.U18
+public import LeanPool.Zeta5Irrational.Table.U19
+public import LeanPool.Zeta5Irrational.Table.U20
+public import LeanPool.Zeta5Irrational.Table.U21
+public import LeanPool.Zeta5Irrational.Table.U22
+public import LeanPool.Zeta5Irrational.Table.U23
+public import LeanPool.Zeta5Irrational.Table.U24
+public import LeanPool.Zeta5Irrational.Table.U25
+public import LeanPool.Zeta5Irrational.Table.U26
+public import LeanPool.Zeta5Irrational.Table.U27
+public import LeanPool.Zeta5Irrational.Table.U28
+public import LeanPool.Zeta5Irrational.Table.U29
+public import LeanPool.Zeta5Irrational.Table.U30
+public import LeanPool.Zeta5Irrational.Table.U31
+public import LeanPool.Zeta5Irrational.Table.U32
+public import LeanPool.Zeta5Irrational.Table.U33
+public import LeanPool.Zeta5Irrational.Table.U34
+public import LeanPool.Zeta5Irrational.Table.U35
+public import LeanPool.Zeta5Irrational.Table.U36
+public import LeanPool.Zeta5Irrational.Table.U37
+public import LeanPool.Zeta5Irrational.Table.U38
+public import LeanPool.Zeta5Irrational.Table.U39
+public import LeanPool.Zeta5Irrational.Table.U40
+public import LeanPool.Zeta5Irrational.Table.U41
+public import LeanPool.Zeta5Irrational.Table.U42
+public import LeanPool.Zeta5Irrational.Table.U43
+public import LeanPool.Zeta5Irrational.Table.U44
+public import LeanPool.Zeta5Irrational.Table.U45
+public import LeanPool.Zeta5Irrational.Table.U46
+public import LeanPool.Zeta5Irrational.Table.U47
+public import LeanPool.Zeta5Irrational.Table.U48
+public import LeanPool.Zeta5Irrational.Table.U49
+public import LeanPool.Zeta5Irrational.Table.U50
+public import LeanPool.Zeta5Irrational.Table.U51
+public import LeanPool.Zeta5Irrational.Table.U52
+public import LeanPool.Zeta5Irrational.Table.U53
+public import LeanPool.Zeta5Irrational.Table.U54
+public import LeanPool.Zeta5Irrational.Table.U55
+public import LeanPool.Zeta5Irrational.Table.U56
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+public import LeanPool.Zeta5Irrational.Table.V00
+public import LeanPool.Zeta5Irrational.Table.V01
+public import LeanPool.Zeta5Irrational.Table.V02
+public import LeanPool.Zeta5Irrational.Table.V03
+public import LeanPool.Zeta5Irrational.Table.V04
+public import LeanPool.Zeta5Irrational.Table.V05
+public import LeanPool.Zeta5Irrational.Table.V06
+public import LeanPool.Zeta5Irrational.Table.V07
+public import LeanPool.Zeta5Irrational.Table.V08
+public import LeanPool.Zeta5Irrational.Table.V09
+public import LeanPool.Zeta5Irrational.Table.V10
+public import LeanPool.Zeta5Irrational.Table.V11
+public import LeanPool.Zeta5Irrational.Table.V12
+public import LeanPool.Zeta5Irrational.Table.V13
+public import LeanPool.Zeta5Irrational.Table.V14
+public import LeanPool.Zeta5Irrational.Table.V15
+public import LeanPool.Zeta5Irrational.Table.V16
+public import LeanPool.Zeta5Irrational.Table.V17
+public import LeanPool.Zeta5Irrational.Table.V18
+public import LeanPool.Zeta5Irrational.Table.V19
+public import LeanPool.Zeta5Irrational.Table.V20
+public import LeanPool.Zeta5Irrational.Table.V21
+public import LeanPool.Zeta5Irrational.Table.V22
+public import LeanPool.Zeta5Irrational.Table.V23
+public import LeanPool.Zeta5Irrational.Table.V24
+public import LeanPool.Zeta5Irrational.Table.V25
+public import LeanPool.Zeta5Irrational.Table.V26
+public import LeanPool.Zeta5Irrational.Table.V27
+public import LeanPool.Zeta5Irrational.Table.V28
+public import LeanPool.Zeta5Irrational.Table.V29
+public import LeanPool.Zeta5Irrational.Table.V30
+public import LeanPool.Zeta5Irrational.Table.V31
+public import LeanPool.Zeta5Irrational.Table.V32
+public import LeanPool.Zeta5Irrational.Table.V33
+public import LeanPool.Zeta5Irrational.Table.V34
+public import LeanPool.Zeta5Irrational.Table.V35
+public import LeanPool.Zeta5Irrational.Table.V36
+public import LeanPool.Zeta5Irrational.Table.V37
+public import LeanPool.Zeta5Irrational.Table.V38
+public import LeanPool.Zeta5Irrational.Table.V39
+public import LeanPool.Zeta5Irrational.Table.V40
+public import LeanPool.Zeta5Irrational.Table.V41
+public import LeanPool.Zeta5Irrational.Table.V42
+public import LeanPool.Zeta5Irrational.Table.V43
+public import LeanPool.Zeta5Irrational.Table.V44
+public import LeanPool.Zeta5Irrational.Table.V45
+public import LeanPool.Zeta5Irrational.Table.V46
+public import LeanPool.Zeta5Irrational.Table.V47
+public import LeanPool.Zeta5Irrational.Table.V48
+public import LeanPool.Zeta5Irrational.Table.V49
+public import LeanPool.Zeta5Irrational.Table.V50
+public import LeanPool.Zeta5Irrational.Table.V51
+public import LeanPool.Zeta5Irrational.Table.V52
+public import LeanPool.Zeta5Irrational.Table.V53
+public import LeanPool.Zeta5Irrational.Table.V54
+public import LeanPool.Zeta5Irrational.Table.V55
+public import LeanPool.Zeta5Irrational.Table.V56
+public import LeanPool.Zeta5Irrational.TableCheck
+public import LeanPool.Zeta5Irrational.Weight
+public import LeanPool.Zeta5Irrational.ZeroMass
+public import LeanPool.Zeta5Irrational.Zeta5
 public import LeanPool.ZetaH123
 public import LeanPool.ZetaH123.H1
 public import LeanPool.ZetaH123.H2
diff --git a/LeanPool/Zeta5Irrational.lean b/LeanPool/Zeta5Irrational.lean
new file mode 100644
index 0000000000..4de7ffb88d
--- /dev/null
+++ b/LeanPool/Zeta5Irrational.lean
@@ -0,0 +1,25 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Main
+public import LeanPool.Zeta5Irrational.MainEstimate
+public import LeanPool.Zeta5Irrational.Criterion
+public import LeanPool.Zeta5Irrational.Andreief
+public import LeanPool.Zeta5Irrational.Arith.Val
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-!
+# Irrationality of the Riemann zeta function at five
+
+Source: url:https://github.com/mo271/Zeta5/tree/f19a1960609f7d38e7b63fd2acb05e6f60a7b741
+Authors: Moritz Firsching
+Status: verified
+Main declarations: `Zeta5Irrational.irrational_five`
+Tags: irrationality, zeta-function, hankel-determinants, potential-theory, certified-computation
+MSC: 11J72, 11M06, 31A15
+-/
diff --git a/LeanPool/Zeta5Irrational/Andreief.lean b/LeanPool/Zeta5Irrational/Andreief.lean
new file mode 100644
index 0000000000..035eaf098e
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Andreief.lean
@@ -0,0 +1,148 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import Mathlib.Analysis.InnerProductSpace.Basic
+public import Mathlib.MeasureTheory.Integral.Bochner.Basic
+import Mathlib.MeasureTheory.Integral.Pi
+import Mathlib.Tactic.ENatToNat
+import Mathlib.Tactic.FieldSimp
+import Mathlib.Tactic.Positivity.Basic
+import Mathlib.Tactic.Ring.Basic
+
+/-! # Andréief's identity
+
+For functions `f i, g j : ℝ → ℝ` with `f i * g j` integrable,
+`det (∫ f i g j dμ) = (1/n!) ∫ det (f i (x j)) det (g i (x j)) dμⁿ(x)`.
+This is (6.10) of the paper in the form used for the Hankel determinant.
+-/
+
+@[expose] public section
+
+open MeasureTheory Finset Equiv
+
+namespace Zeta5Irrational
+
+variable {n : ℕ} {μ : Measure ℝ} [SigmaFinite μ]
+
+/-- The coordinate permutation `x ↦ (i ↦ x (τ i))` as a measurable equivalence. -/
+noncomputable def permEquiv (τ : Perm (Fin n)) : (Fin n → ℝ) ≃ᵐ (Fin n → ℝ) :=
+  MeasurableEquiv.arrowCongr' τ.symm (MeasurableEquiv.refl ℝ)
+
+lemma permEquiv_apply (τ : Perm (Fin n)) (x : Fin n → ℝ) (i : Fin n) :
+    permEquiv τ x i = x (τ i) := by
+  simp [permEquiv, MeasurableEquiv.arrowCongr', Equiv.arrowCongr']
+
+lemma measurePreserving_permEquiv (τ : Perm (Fin n)) :
+    MeasurePreserving (permEquiv τ) (Measure.pi fun _ : Fin n => μ)
+      (Measure.pi fun _ : Fin n => μ) :=
+  measurePreserving_arrowCongr' (fun _ : Fin n => μ) (fun _ : Fin n => μ) τ.symm
+    (MeasurableEquiv.refl ℝ) (fun _ => ⟨(MeasurableEquiv.refl ℝ).measurable, Measure.map_id⟩)
+
+/-- The basic integral `I(ρ) = ∫ ∏ᵢ f (ρ i) (xᵢ) g i (xᵢ)`. -/
+noncomputable def andreiefI (f g : Fin n → ℝ → ℝ) (ρ : Perm (Fin n)) : ℝ :=
+  ∫ x : Fin n → ℝ, ∏ i, f (ρ i) (x i) * g i (x i) ∂(Measure.pi fun _ : Fin n => μ)
+
+lemma integral_prod_prod_eq (f g : Fin n → ℝ → ℝ) (σ τ : Perm (Fin n)) :
+    ∫ x : Fin n → ℝ,
+        (∏ i, f (σ i) (x i)) * (∏ i, g (τ i) (x i)) ∂(Measure.pi fun _ : Fin n => μ) =
+      andreiefI (μ := μ) f g (σ * τ⁻¹) := by
+  unfold andreiefI
+  have hpt :
+    ∀ x : Fin n → ℝ,
+      (∏ i, f (σ i) (x i)) * (∏ i, g (τ i) (x i)) =
+        (fun y : Fin n → ℝ => ∏ k, f ((σ * τ⁻¹) k) (y k) * g k (y k)) (permEquiv τ⁻¹ x) := by
+    intro x
+    simp only [permEquiv_apply, ← Finset.prod_mul_distrib]
+    rw [← Equiv.prod_comp τ (fun k => f ((σ * τ⁻¹) k) (x (τ⁻¹ k)) * g k (x (τ⁻¹ k)))]
+    refine Finset.prod_congr rfl fun i _ => ?_
+    simp [Perm.mul_apply]
+  simp_rw [hpt]
+  exact
+    (measurePreserving_permEquiv (μ := μ) τ⁻¹).integral_comp (permEquiv τ⁻¹).measurableEmbedding
+      (fun y : Fin n → ℝ => ∏ k, f ((σ * τ⁻¹) k) (y k) * g k (y k))
+
+/-- **Andréief's identity**. -/
+theorem andreief (f g : Fin n → ℝ → ℝ) (hfg : ∀ i j, Integrable (fun x => f i x * g j x) μ) :
+    (Matrix.of fun i j => ∫ x, f i x * g j x ∂μ).det =
+      (1 / (n.factorial : ℝ)) *
+        ∫ x : Fin n → ℝ,
+          (Matrix.of fun i j => f i (x j)).det *
+            (Matrix.of fun i j => g i (x j)).det ∂(Measure.pi fun _ : Fin n => μ) := by
+  -- the left-hand side as a sum over permutations of basic integrals
+  have hL :
+    (Matrix.of fun i j => ∫ x, f i x * g j x ∂μ).det =
+      ∑ ρ : Perm (Fin n), (Perm.sign ρ : ℝ) * andreiefI (μ := μ) f g ρ := by
+    rw [Matrix.det_apply']
+    refine Finset.sum_congr rfl fun ρ _ => ?_
+    simp only [Matrix.of_apply, andreiefI]
+    rw [integral_fintype_prod_eq_prod (fun i x => f (ρ i) x * g i x)]
+      -- the right-hand side
+  have hint :
+    ∀ σ τ : Perm (Fin n),
+      Integrable
+        (fun x : Fin n → ℝ =>
+          (Perm.sign σ : ℝ) * (Perm.sign τ : ℝ) * ((∏ i, f (σ i) (x i)) * (∏ i, g (τ i) (x i))))
+        (Measure.pi fun _ : Fin n => μ) := by
+    intro σ τ
+    have :
+      (fun x : Fin n → ℝ => (∏ i, f (σ i) (x i)) * (∏ i, g (τ i) (x i))) = fun x =>
+        ∏ i, (fun i y => f (σ i) y * g (τ i) y) i (x i) :=
+      by funext x; simp only [← Finset.prod_mul_distrib]
+    refine Integrable.const_mul ?_ _
+    rw [this]
+    exact Integrable.fintype_prod fun i => hfg (σ i) (τ i)
+  have hR :
+    ∫ x : Fin n → ℝ,
+        (Matrix.of fun i j => f i (x j)).det *
+          (Matrix.of fun i j => g i (x j)).det ∂(Measure.pi fun _ : Fin n => μ) =
+      ∑ τ : Perm (Fin n),
+        ∑ σ : Perm (Fin n),
+          (Perm.sign σ : ℝ) * (Perm.sign τ : ℝ) * andreiefI (μ := μ) f g (σ * τ⁻¹) := by
+    have hpt :
+      ∀ x : Fin n → ℝ,
+        (Matrix.of fun i j => f i (x j)).det * (Matrix.of fun i j => g i (x j)).det =
+          ∑ τ : Perm (Fin n),
+            ∑ σ : Perm (Fin n),
+              (Perm.sign σ : ℝ) * (Perm.sign τ : ℝ) *
+                ((∏ i, f (σ i) (x i)) * (∏ i, g (τ i) (x i))) := by
+      intro x
+      rw [Matrix.det_apply', Matrix.det_apply', Finset.sum_mul_sum, Finset.sum_comm]
+      refine Finset.sum_congr rfl fun τ _ => Finset.sum_congr rfl fun σ _ => ?_
+      simp only [Matrix.of_apply]
+      ring
+    simp_rw [hpt]
+    rw [integral_finsetSum _ (fun τ _ => integrable_finsetSum _ (fun σ _ => hint σ τ))]
+    refine Finset.sum_congr rfl fun τ _ => ?_
+    rw [integral_finsetSum _ (fun σ _ => hint σ τ)]
+    refine Finset.sum_congr rfl fun σ _ => ?_
+    rw [integral_const_mul, integral_prod_prod_eq]
+      -- each inner sum is the left-hand side
+  have hsign :
+    ∀ σ τ : Perm (Fin n), (Perm.sign (σ * τ⁻¹) : ℝ) = (Perm.sign σ : ℝ) * (Perm.sign τ : ℝ) := by
+    intro σ τ
+    rw [Perm.sign_mul, Perm.sign_inv]
+    push_cast
+    ring
+  have hinner :
+    ∀ τ : Perm (Fin n),
+      ∑ σ : Perm (Fin n),
+          (Perm.sign σ : ℝ) * (Perm.sign τ : ℝ) * andreiefI (μ := μ) f g (σ * τ⁻¹) =
+        ∑ ρ : Perm (Fin n), (Perm.sign ρ : ℝ) * andreiefI (μ := μ) f g ρ := by
+    intro τ
+    conv_rhs => rw [← Equiv.sum_comp (Equiv.mulRight τ⁻¹)]
+    refine Finset.sum_congr rfl fun σ _ => ?_
+    simp only [Equiv.coe_mulRight]
+    rw [hsign]
+  rw [hR]
+  simp_rw [hinner]
+  rw [Finset.sum_const, Finset.card_univ, Fintype.card_perm, Fintype.card_fin, nsmul_eq_mul, hL]
+  have : (n.factorial : ℝ) ≠ 0 := by positivity
+  field_simp
+  try ring
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/ArcsineAtoms.lean b/LeanPool/Zeta5Irrational/ArcsineAtoms.lean
new file mode 100644
index 0000000000..e73144b514
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/ArcsineAtoms.lean
@@ -0,0 +1,178 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import Mathlib.Analysis.CStarAlgebra.Classes
+public import Mathlib.Analysis.SpecialFunctions.Complex.CircleMap
+public import Mathlib.Analysis.SpecialFunctions.Log.PosLog
+public import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
+import LeanPool.Zeta5Irrational.CircleAtoms
+import Mathlib.Tactic.ENatToNat
+import Mathlib.Tactic.Polynomial.Basic
+import Mathlib.Tactic.ReduceModChar
+import Mathlib.Tactic.FieldSimp
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.Positivity.Basic
+import Mathlib.Tactic.Ring.Basic
+
+/-! # The arcsine atoms: factorisation and the potential formula (A.1)
+
+For `w = e^{iθ}` and `y(θ) = m + r cos θ`, one has
+`z - y(θ) = -(r/(2w)) (w - q₁)(w - q₂)` where `q₁ + q₂ = 2ζ`, `q₁ q₂ = 1`, `ζ = (z - m)/r`.
+Hence `∫₀^{2π} log ‖z - y(θ)‖ dθ = 2π (log (r/2) + log⁺ ‖q₁‖ + log⁺ ‖q₂‖)` by Mathlib's
+circle-average identity, which gives the closed form `Uω` of (A.1) for real `z`.
+-/
+
+@[expose] public section
+
+open MeasureTheory Set Real intervalIntegral Filter Topology
+
+namespace Zeta5Irrational
+
+/-- The arcsine curve `θ ↦ m + r cos θ` (as a complex number). -/
+noncomputable def arcCurve (m r : ℝ) (θ : ℝ) : ℂ :=
+  ((m + r * Real.cos θ : ℝ) : ℂ)
+
+lemma continuous_arcCurve (m r : ℝ) : Continuous (arcCurve m r) := by unfold arcCurve; fun_prop
+
+lemma norm_arcCurve_le (m r θ : ℝ) : ‖arcCurve m r θ‖ ≤ |m| + |r| := by
+  unfold arcCurve
+  rw [Complex.norm_real, Real.norm_eq_abs]
+  calc
+    |m + r * Real.cos θ| ≤ |m| + |r * Real.cos θ| := abs_add_le _ _
+    _ ≤ |m| + |r| := by
+      rw [abs_mul]
+      have := Real.abs_cos_le_one θ
+      nlinarith [abs_nonneg r]
+
+/-- The factorisation of `z - y(θ)`. -/
+lemma sub_arcCurve_eq {z : ℂ} {m r : ℝ} (hr : r ≠ 0) (θ : ℝ) {sq : ℂ}
+    (hsq : sq ^ 2 = ((z - m) / r) ^ 2 - 1) :
+    z - arcCurve m r θ =
+      -(r / (2 * circleMap 0 1 θ)) *
+        ((circleMap 0 1 θ - ((z - m) / r + sq)) * (circleMap 0 1 θ - ((z - m) / r - sq))) := by
+  have hw : circleMap 0 1 θ ≠ 0 := by
+    intro h; have := norm_circleMap_zero_one θ; rw [h] at this;
+    simp at this
+  have hr' : (r : ℂ) ≠ 0 := by exact_mod_cast hr
+  have e1 :
+    (circleMap 0 1 θ - ((z - m) / r + sq)) * (circleMap 0 1 θ - ((z - m) / r - sq)) =
+      circleMap 0 1 θ ^ 2 - 2 * ((z - m) / r) * circleMap 0 1 θ + 1 :=
+    by linear_combination (-1 : ℂ) * hsq
+  rw [e1]
+  simp only [arcCurve, Complex.ofReal_add, Complex.ofReal_mul]
+  rw [cos_eq_half_add_inv]
+  field_simp
+  ring
+
+lemma norm_sub_arcCurve {z : ℂ} {m r : ℝ} (hr : 0 < r) (θ : ℝ) {sq : ℂ}
+    (hsq : sq ^ 2 = ((z - m) / r) ^ 2 - 1) :
+    ‖z - arcCurve m r θ‖ =
+      r / 2 * (‖circleMap 0 1 θ - ((z - m) / r + sq)‖ * ‖circleMap 0 1 θ - ((z - m) / r - sq)‖) :=
+  by
+  rw [sub_arcCurve_eq hr.ne' θ hsq, norm_mul, norm_neg, norm_div, norm_mul, norm_mul,
+    norm_circleMap_zero_one, Complex.norm_real, Real.norm_eq_abs, abs_of_pos hr]
+  simp
+
+/-- The exceptional set is countable. -/
+lemma arc_exceptional_countable (q₁ q₂ : ℂ) :
+    ({θ : ℝ | circleMap 0 1 θ = q₁ ∨ circleMap 0 1 θ = q₂}).Countable := by
+  have :
+    {θ : ℝ | circleMap 0 1 θ = q₁ ∨ circleMap 0 1 θ = q₂} =
+      circleMap 0 1 ⁻¹' { q₁ } ∪ circleMap 0 1 ⁻¹' { q₂ } :=
+    by ext θ; simp
+  rw [this]
+  exact
+    (circleMap_preimage_countable 0 one_ne_zero q₁).union
+      (circleMap_preimage_countable 0 one_ne_zero q₂)
+
+lemma arc_log_ae {z : ℂ} {m r : ℝ} (hr : 0 < r) {sq : ℂ} (hsq : sq ^ 2 = ((z - m) / r) ^ 2 - 1) :
+    ∀ᵐ θ ∂(volume : Measure ℝ),
+      Real.log ‖z - arcCurve m r θ‖ =
+        Real.log (r / 2) + Real.log ‖circleMap 0 1 θ - ((z - m) / r + sq)‖ +
+          Real.log ‖circleMap 0 1 θ - ((z - m) / r - sq)‖ := by
+  have h0 :=
+    (arc_exceptional_countable ((z - m) / r + sq) ((z - m) / r - sq)).measure_zero
+      (volume : Measure ℝ)
+  filter_upwards [compl_mem_ae_iff.mpr h0] with θ hθ
+  simp only [Set.mem_compl_iff, Set.mem_ofPred_eq, not_or] at hθ
+  rw [norm_sub_arcCurve hr θ hsq,
+    Real.log_mul (by positivity)
+      (by apply mul_ne_zero <;> exact norm_ne_zero_iff.mpr (sub_ne_zero.mpr (by tauto))),
+    Real.log_mul (norm_ne_zero_iff.mpr (sub_ne_zero.mpr hθ.1))
+      (norm_ne_zero_iff.mpr (sub_ne_zero.mpr hθ.2)),
+    add_assoc]
+
+lemma arc_log_integrable {z : ℂ} {m r : ℝ} (hr : 0 < r) :
+    IntervalIntegrable (fun θ => Real.log ‖z - arcCurve m r θ‖) volume 0 (2 * π) := by
+  set sq : ℂ := (((z - m) / r) ^ 2 - 1) ^ ((2 : ℕ) : ℂ)⁻¹ with hsq_def
+  have hsq : sq ^ 2 = ((z - m) / r) ^ 2 - 1 := Complex.cpow_nat_inv_pow _ two_ne_zero
+  have hg :
+    IntervalIntegrable
+      (fun θ =>
+        Real.log (r / 2) + Real.log ‖circleMap 0 1 θ - ((z - m) / r + sq)‖ +
+          Real.log ‖circleMap 0 1 θ - ((z - m) / r - sq)‖)
+      volume 0 (2 * π) :=
+    (intervalIntegrable_const.add (circle_log_integrable 0 _ 1)).add (circle_log_integrable 0 _ 1)
+  refine hg.congr_ae (ae_restrict_of_ae ?_)
+  filter_upwards [arc_log_ae hr hsq] with θ hθ
+  exact hθ.symm
+
+/-- The arcsine potential at a complex point: `∫₀^{2π} log ‖z - y(θ)‖ dθ`. -/
+lemma arc_log_integral {z : ℂ} {m r : ℝ} (hr : 0 < r) {sq : ℂ}
+    (hsq : sq ^ 2 = ((z - m) / r) ^ 2 - 1) :
+    ∫ θ in (0 : ℝ)..2 * π, Real.log ‖z - arcCurve m r θ‖ =
+      2 * π * (Real.log (r / 2) + log⁺ ‖(z - m) / r + sq‖ + log⁺ ‖(z - m) / r - sq‖) := by
+  rw [intervalIntegral.integral_congr_ae (g := fun θ =>
+      Real.log (r / 2) + Real.log ‖circleMap 0 1 θ - ((z - m) / r + sq)‖ +
+        Real.log ‖circleMap 0 1 θ - ((z - m) / r - sq)‖)
+      (by
+        filter_upwards [arc_log_ae hr hsq] with θ hθ _
+        exact hθ)]
+  rw [intervalIntegral.integral_add (intervalIntegrable_const.add (circle_log_integrable 0 _ 1))
+      (circle_log_integrable 0 _ 1),
+    intervalIntegral.integral_add intervalIntegrable_const (circle_log_integrable 0 _ 1),
+    intervalIntegral.integral_const, circle_log_integral_zero_one, circle_log_integral_zero_one]
+  simp only [sub_zero, smul_eq_mul]
+  ring
+
+/-- A uniform bound for `∫₀^{2π} |log ‖z - y(θ)‖| dθ`. -/
+lemma arc_abs_log_integral_le {z : ℂ} {m r : ℝ} (hr : 0 < r) {sq : ℂ}
+    (hsq : sq ^ 2 = ((z - m) / r) ^ 2 - 1) :
+    ∫ θ in (0 : ℝ)..2 * π, |Real.log ‖z - arcCurve m r θ‖| ≤
+      2 * π * |Real.log (r / 2)| + 4 * π * Real.log (1 + ‖(z - m) / r + sq‖) +
+        4 * π * Real.log (1 + ‖(z - m) / r - sq‖) := by
+  rw [intervalIntegral.integral_congr_ae (g := fun θ =>
+      |Real.log (r / 2) + Real.log ‖circleMap 0 1 θ - ((z - m) / r + sq)‖ +
+          Real.log ‖circleMap 0 1 θ - ((z - m) / r - sq)‖|)
+      (by
+        filter_upwards [arc_log_ae hr hsq] with θ hθ _
+        rw [hθ])]
+  have h1 := abs_log_circle_integral_le ((z - m) / r + sq)
+  have h2 := abs_log_circle_integral_le ((z - m) / r - sq)
+  have hi1 := (circle_log_integrable 0 ((z - m) / r + sq) 1).abs
+  have hi2 := (circle_log_integrable 0 ((z - m) / r - sq) 1).abs
+  have hmono :
+    (∫ θ in (0 : ℝ)..2 * π,
+        |Real.log (r / 2) + Real.log ‖circleMap 0 1 θ - ((z - m) / r + sq)‖ +
+            Real.log ‖circleMap 0 1 θ - ((z - m) / r - sq)‖|) ≤
+      ∫ θ in (0 : ℝ)..2 * π,
+        (|Real.log (r / 2)| + |Real.log ‖circleMap 0 1 θ - ((z - m) / r + sq)‖| +
+          |Real.log ‖circleMap 0 1 θ - ((z - m) / r - sq)‖|) := by
+    apply integral_mono_on (by positivity)
+    · exact
+        ((intervalIntegrable_const.add (circle_log_integrable 0 _ 1)).add
+            (circle_log_integrable 0 _ 1)).abs
+    · exact (intervalIntegrable_const.add hi1).add hi2
+    intro θ _
+    exact abs_add_three _ _ _
+  rw [intervalIntegral.integral_add (intervalIntegrable_const.add hi1) hi2,
+    intervalIntegral.integral_add intervalIntegrable_const hi1, intervalIntegral.integral_const,
+    sub_zero, smul_eq_mul] at hmono
+  linarith
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/ArcsinePotential.lean b/LeanPool/Zeta5Irrational/ArcsinePotential.lean
new file mode 100644
index 0000000000..3dad751de5
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/ArcsinePotential.lean
@@ -0,0 +1,242 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.ArcsineAtoms
+public import LeanPool.Zeta5Irrational.Potential
+import Mathlib.Tactic.ENatToNat
+import Mathlib.Tactic.Polynomial.Basic
+import Mathlib.Tactic.ReduceModChar
+import Mathlib.Tactic.FieldSimp
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.Positivity.Basic
+import Mathlib.Tactic.Ring.Basic
+import Mathlib.Tactic.Ring.RingNF
+
+/-! # The arcsine potential (A.1) and its modulus of continuity
+
+* `arc_potential_real` : for real `t`, `∫₀^{2π} log ‖t - y(θ)‖ dθ = 2π Uω a b t`;
+* `Gpot_sub_le` : the Hölder-`1/2` estimate for `log⁺ ‖ζ + s‖ + log⁺ ‖ζ - s‖` used for the circle
+  regularisation error.
+-/
+
+@[expose] public section
+
+open MeasureTheory Set Real intervalIntegral Filter Topology
+
+namespace Zeta5Irrational
+
+/-- A square root of `ζ² - 1` for real `ζ`. -/
+noncomputable def sqR (ζ : ℝ) : ℂ :=
+  if 1 ≤ ζ ^ 2 then ((Real.sqrt (ζ ^ 2 - 1) : ℝ) : ℂ)
+  else ((Real.sqrt (1 - ζ ^ 2) : ℝ) : ℂ) * Complex.I
+
+lemma sqR_sq (ζ : ℝ) : sqR ζ ^ 2 = (ζ : ℂ) ^ 2 - 1 := by
+  unfold sqR
+  split_ifs with h
+  · rw [← Complex.ofReal_pow, Real.sq_sqrt (by linarith)]; push_cast; ring
+  · rw [mul_pow, Complex.I_sq, ← Complex.ofReal_pow, Real.sq_sqrt (by linarith)]; push_cast; ring
+
+/-- The value `log⁺ ‖ζ + s‖ + log⁺ ‖ζ - s‖` for real `ζ`. -/
+lemma posLog_sum_real (ζ : ℝ) :
+    log⁺ ‖(ζ : ℂ) + sqR ζ‖ + log⁺ ‖(ζ : ℂ) - sqR ζ‖ =
+      if 1 ≤ |ζ| then Real.log (|ζ| + Real.sqrt (ζ ^ 2 - 1)) else 0 := by
+  unfold sqR
+  by_cases h : 1 ≤ ζ ^ 2
+  · rw [ite_eq_left h]
+    have h1 : 1 ≤ |ζ| := by nlinarith [abs_nonneg ζ, sq_abs ζ]
+    rw [ite_eq_left h1]
+    have hs0 : 0 ≤ Real.sqrt (ζ ^ 2 - 1) := Real.sqrt_nonneg _
+    have hs2 : Real.sqrt (ζ ^ 2 - 1) ^ 2 = ζ ^ 2 - 1 := Real.sq_sqrt (by linarith)
+    rw [← Complex.ofReal_add, ← Complex.ofReal_sub, Complex.norm_real, Complex.norm_real,
+      Real.norm_eq_abs, Real.norm_eq_abs]
+    rcases le_or_gt 0 ζ with hζ | hζ
+    · rw [abs_of_nonneg hζ] at h1 ⊢
+      set s := Real.sqrt (ζ ^ 2 - 1) with hs
+      have h2 : s ≤ ζ := Real.sqrt_le_iff.mpr ⟨hζ, by nlinarith⟩
+      have h3 : 0 < ζ - s := by
+        refine sub_pos.mpr (lt_of_le_of_ne h2 fun h => ?_)
+        rw [h] at hs2; linarith
+      have h4 : ζ - 1 ≤ s := (Real.le_sqrt (by linarith) (by nlinarith)).mpr (by nlinarith)
+      have hA : 0 < ζ + s := by linarith
+      rw [abs_of_pos hA, abs_of_pos h3, posLog_eq_log (by rw [abs_of_pos hA]; linarith),
+        (posLog_eq_zero_iff _).mpr (by rw [abs_of_pos h3]; linarith), add_zero]
+    · rw [abs_of_neg hζ] at h1 ⊢
+      set s := Real.sqrt (ζ ^ 2 - 1) with hs
+      have h2 : s ≤ -ζ := Real.sqrt_le_iff.mpr ⟨by linarith, by nlinarith⟩
+      have hA' : -1 - ζ ≤ s := (Real.le_sqrt (by linarith) (by nlinarith)).mpr (by nlinarith)
+      have hA : ζ + s ≤ 0 := by linarith
+      have hB : ζ - s < 0 := by linarith
+      have hB1 : 1 ≤ -(ζ - s) := by linarith
+      rw [abs_of_nonpos hA, abs_of_neg hB,
+        (posLog_eq_zero_iff _).mpr (by rw [abs_of_nonneg (by linarith)]; linarith),
+        posLog_eq_log (by rw [abs_of_nonneg (by linarith)]; exact hB1), zero_add]
+      ring_nf
+  · rw [ite_eq_right h]
+    have h1 : ¬1 ≤ |ζ| := by intro h1; apply h; nlinarith [abs_nonneg ζ, sq_abs ζ]
+    rw [ite_eq_right h1]
+    have hs2 : Real.sqrt (1 - ζ ^ 2) ^ 2 = 1 - ζ ^ 2 := Real.sq_sqrt (by linarith)
+    have e1 :
+      (ζ : ℂ) + ((Real.sqrt (1 - ζ ^ 2) : ℝ) : ℂ) * Complex.I =
+        (ζ : ℝ) + ((Real.sqrt (1 - ζ ^ 2) : ℝ) : ℂ) * Complex.I :=
+      rfl
+    have e2 :
+      (ζ : ℂ) - ((Real.sqrt (1 - ζ ^ 2) : ℝ) : ℂ) * Complex.I =
+        (ζ : ℝ) + ((-Real.sqrt (1 - ζ ^ 2) : ℝ) : ℂ) * Complex.I :=
+      by push_cast; ring
+    rw [e1, e2, Complex.norm_add_mul_I, Complex.norm_add_mul_I, neg_sq, hs2,
+      show ζ ^ 2 + (1 - ζ ^ 2) = 1 by ring, Real.sqrt_one]
+    simp
+
+/-- The closed form `Uω` in terms of `ζ = (t - m)/r`. -/
+lemma Uω_eq_log_add {a b t : ℝ} (hab : a < b) :
+    Uω a b t =
+      Real.log ((b - a) / 4) +
+        (if 1 ≤ |(t - (a + b) / 2) / ((b - a) / 2)| then
+          Real.log
+            (|(t - (a + b) / 2) / ((b - a) / 2)| +
+              Real.sqrt (((t - (a + b) / 2) / ((b - a) / 2)) ^ 2 - 1))
+        else 0) := by
+  set r := (b - a) / 2 with hr
+  set m := (a + b) / 2 with hm
+  have hr0 : 0 < r := by rw [hr]; linarith
+  have hζ : |(t - m) / r| = |t - m| / r := by rw [abs_div, abs_of_pos hr0]
+  by_cases h : a ≤ t ∧ t ≤ b
+  · rw [Uω_of_mem h.1 h.2]
+    have h1 : |t - m| ≤ r := by rw [abs_le]; constructor <;> linarith
+    have h2 : |(t - m) / r| ≤ 1 := by rw [hζ, div_le_one hr0]; exact h1
+    split_ifs with h3
+    · have h4 : |(t - m) / r| = 1 := le_antisymm h2 h3
+      rw [← sq_abs ((t - m) / r), h4, one_pow, sub_self, Real.sqrt_zero, add_zero, Real.log_one,
+        add_zero]
+    · ring
+  · have h1 : 1 < |(t - m) / r| := by
+      rw [hζ, one_lt_div hr0, lt_abs]
+      rcases not_and_or.mp h with h | h
+      · right; linarith [not_le.mp h]
+      · left; linarith [not_le.mp h]
+    rw [ite_eq_left h1.le]
+    unfold Uω
+    rw [ite_eq_right h]
+    have hpos : 0 < |t - m| / r + Real.sqrt (((t - m) / r) ^ 2 - 1) := by
+      have := Real.sqrt_nonneg (((t - m) / r) ^ 2 - 1)
+      rw [hζ] at h1; linarith
+    rw [hζ, ← Real.log_mul (by positivity) hpos.ne']
+    congr 1
+    have e : (t - a) * (t - b) = r ^ 2 * (((t - m) / r) ^ 2 - 1) := by
+      have hba : b - a ≠ 0 := by linarith
+      rw [hr, hm]; field_simp; ring
+    rw [e, Real.sqrt_mul (sq_nonneg r), Real.sqrt_sq hr0.le]
+    rw [show (b - a) / 4 = r / 2 by rw [hr]; ring, ← hm]
+    field_simp
+
+/-- **(A.1)**: the arcsine potential at a real point. -/
+lemma arc_potential_real {a b : ℝ} (hab : a < b) (t : ℝ) :
+    ∫ θ in (0 : ℝ)..2 * π, Real.log ‖(t : ℂ) - arcCurve ((a + b) / 2) ((b - a) / 2) θ‖ =
+      2 * π * Uω a b t := by
+  have hr0 : 0 < (b - a) / 2 := by linarith
+  have hsq :
+    sqR ((t - (a + b) / 2) / ((b - a) / 2)) ^ 2 =
+      (((t : ℂ) - ((a + b) / 2 : ℝ)) / ((b - a) / 2 : ℝ)) ^ 2 - 1 :=
+    by rw [sqR_sq]; push_cast; ring
+  rw [arc_log_integral hr0 hsq, Uω_eq_log_add hab]
+  congr 1
+  have e :
+    ((t : ℂ) - ((a + b) / 2 : ℝ)) / ((b - a) / 2 : ℝ) =
+      (((t - (a + b) / 2) / ((b - a) / 2) : ℝ) : ℂ) :=
+    by push_cast; ring
+  rw [e, add_assoc, posLog_sum_real, show (b - a) / 2 / 2 = (b - a) / 4 by ring]
+
+/-! ### Hölder continuity of the potential -/
+
+lemma norm_mul_roots {ζ sq : ℂ} (hsq : sq ^ 2 = ζ ^ 2 - 1) : ‖ζ + sq‖ * ‖ζ - sq‖ = 1 := by
+  rw [← norm_mul, show (ζ + sq) * (ζ - sq) = ζ ^ 2 - sq ^ 2 by ring, hsq]; simp
+
+/-- `log⁺ ‖ζ + s‖ + log⁺ ‖ζ - s‖ = log (max ‖ζ + s‖ ‖ζ - s‖)`. -/
+lemma posLog_sum_eq_log_max {ζ sq : ℂ} (hsq : sq ^ 2 = ζ ^ 2 - 1) :
+    log⁺ ‖ζ + sq‖ + log⁺ ‖ζ - sq‖ = Real.log (max ‖ζ + sq‖ ‖ζ - sq‖) := by
+  have h := norm_mul_roots hsq
+  have h1 := norm_nonneg (ζ + sq)
+  have h2 := norm_nonneg (ζ - sq)
+  rcases le_total 1 ‖ζ + sq‖ with hle | hle
+  · have h3 : ‖ζ - sq‖ ≤ 1 := by nlinarith
+    rw [max_eq_left (h3.trans hle),
+      posLog_eq_log (x := ‖ζ + sq‖) (by rw [abs_of_nonneg h1]; exact hle),
+      (posLog_eq_zero_iff ‖ζ - sq‖).mpr (by rw [abs_of_nonneg h2]; exact h3), add_zero]
+  · have h3 : 1 ≤ ‖ζ - sq‖ := by by_contra hc; rw [not_le] at hc; nlinarith
+    rw [max_eq_right (hle.trans h3),
+      posLog_eq_log (x := ‖ζ - sq‖) (by rw [abs_of_nonneg h2]; exact h3),
+      (posLog_eq_zero_iff ‖ζ + sq‖).mpr (by rw [abs_of_nonneg h1]; exact hle), zero_add]
+
+lemma one_le_max_roots {ζ sq : ℂ} (hsq : sq ^ 2 = ζ ^ 2 - 1) : 1 ≤ max ‖ζ + sq‖ ‖ζ - sq‖ := by
+  have h := norm_mul_roots hsq
+  have h1 := norm_nonneg (ζ + sq)
+  have h2 := norm_nonneg (ζ - sq)
+  by_contra hc
+  rw [not_le, max_lt_iff] at hc
+  nlinarith
+
+lemma log_sub_log_le_of_one_le {M M' : ℝ} (hM : 1 ≤ M) (hM' : 1 ≤ M') :
+    Real.log M - Real.log M' ≤ |M - M'| := by
+  rcases le_total M' M with h | h
+  · rw [abs_of_nonneg (by linarith), ← Real.log_div (by linarith) (by linarith)]
+    have := Real.log_le_sub_one_of_pos (show 0 < M / M' by positivity)
+    have h2 : M / M' - 1 = (M - M') / M' := by field_simp
+    have h3 : (M - M') / M' ≤ M - M' := by rw [div_le_iff₀ (by linarith)]; nlinarith
+    linarith
+  · have := Real.log_le_log (by linarith) h
+    linarith [abs_nonneg (M - M')]
+
+lemma abs_log_max_sub_le {M M' : ℝ} (hM : 1 ≤ M) (hM' : 1 ≤ M') :
+    |Real.log M - Real.log M'| ≤ |M - M'| := by
+  rw [abs_le]
+  constructor
+  · have := log_sub_log_le_of_one_le hM' hM; rw [abs_sub_comm] at this; linarith
+  · exact log_sub_log_le_of_one_le hM hM'
+
+lemma max_roots_sub_le_aux (ζ ζ' s s' : ℂ) :
+    |max ‖ζ + s‖ ‖ζ - s‖ - max ‖ζ' + s'‖ ‖ζ' - s'‖| ≤ ‖ζ - ζ'‖ + ‖s - s'‖ := by
+  refine (abs_max_sub_max_le_max _ _ _ _).trans (max_le ?_ ?_)
+  · calc
+      |‖ζ + s‖ - ‖ζ' + s'‖| ≤ ‖(ζ + s) - (ζ' + s')‖ := abs_norm_sub_norm_le _ _
+      _ = ‖(ζ - ζ') + (s - s')‖ := by ring_nf
+      _ ≤ ‖ζ - ζ'‖ + ‖s - s'‖ := norm_add_le _ _
+  · calc
+      |‖ζ - s‖ - ‖ζ' - s'‖| ≤ ‖(ζ - s) - (ζ' - s')‖ := abs_norm_sub_norm_le _ _
+      _ = ‖(ζ - ζ') - (s - s')‖ := by ring_nf
+      _ ≤ ‖ζ - ζ'‖ + ‖s - s'‖ := norm_sub_le _ _
+
+lemma max_roots_sub_le (ζ ζ' s s' : ℂ) :
+    |max ‖ζ + s‖ ‖ζ - s‖ - max ‖ζ' + s'‖ ‖ζ' - s'‖| ≤ ‖ζ - ζ'‖ + min ‖s - s'‖ ‖s + s'‖ := by
+  rcases le_total ‖s - s'‖ ‖s + s'‖ with h | h
+  · rw [min_eq_left h]; exact max_roots_sub_le_aux ζ ζ' s s'
+  · rw [min_eq_right h]
+    have := max_roots_sub_le_aux ζ ζ' s (-s')
+    rw [sub_neg_eq_add, sub_neg_eq_add, max_comm ‖ζ' + -s'‖,
+      show ζ' + -s' = ζ' - s' by ring] at this
+    exact this
+
+lemma min_roots_le_sqrt (s s' : ℂ) : min ‖s - s'‖ ‖s + s'‖ ≤ Real.sqrt ‖s ^ 2 - s' ^ 2‖ := by
+  rw [Real.le_sqrt (le_min (norm_nonneg _) (norm_nonneg _)) (norm_nonneg _)]
+  have e : s ^ 2 - s' ^ 2 = (s - s') * (s + s') := by ring
+  rw [e, norm_mul, sq]
+  exact
+    mul_le_mul (min_le_left _ _) (min_le_right _ _) (le_min (norm_nonneg _) (norm_nonneg _))
+      (norm_nonneg _)
+
+/-- The Hölder-`1/2` estimate for `G(ζ) = log⁺ ‖ζ + s‖ + log⁺ ‖ζ - s‖`. -/
+lemma Gpot_sub_le {ζ ζ' s s' : ℂ} (hs : s ^ 2 = ζ ^ 2 - 1) (hs' : s' ^ 2 = ζ' ^ 2 - 1) :
+    (log⁺ ‖ζ + s‖ + log⁺ ‖ζ - s‖) - (log⁺ ‖ζ' + s'‖ + log⁺ ‖ζ' - s'‖) ≤
+      ‖ζ - ζ'‖ + Real.sqrt (‖ζ - ζ'‖ * ‖ζ + ζ'‖) := by
+  rw [posLog_sum_eq_log_max hs, posLog_sum_eq_log_max hs']
+  have h1 := abs_log_max_sub_le (one_le_max_roots hs) (one_le_max_roots hs')
+  have h2 := max_roots_sub_le ζ ζ' s s'
+  have h3 := min_roots_le_sqrt s s'
+  rw [hs, hs', show ζ ^ 2 - 1 - (ζ' ^ 2 - 1) = (ζ - ζ') * (ζ + ζ') by ring, norm_mul] at h3
+  linarith [le_abs_self (Real.log (max ‖ζ + s‖ ‖ζ - s‖) - Real.log (max ‖ζ' + s'‖ ‖ζ' - s'‖))]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Arith/BasisChange.lean b/LeanPool/Zeta5Irrational/Arith/BasisChange.lean
new file mode 100644
index 0000000000..fd21e2cede
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Arith/BasisChange.lean
@@ -0,0 +1,115 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Construction
+import LeanPool.Zeta5Irrational.PartialFractions
+import Mathlib.RingTheory.PiTensorProduct
+import Mathlib.Tactic.ReduceModChar
+import Mathlib.Tactic.Ring.Basic
+
+/-! # Linearity of `μ_X` and change of basis in the Hankel determinant
+
+* `μX_add`, `μX_C_mul`, `μX_sum` : `μ_X` is `ℚ`-linear in the numerator;
+* `det_basis_change` : for polynomials `E a` of degree `< h`,
+  `det [μ_X(D_N⁶ E_a E_b / D_K)] = det(C)² Δ_K` where `C` is the coefficient matrix.
+-/
+
+@[expose] public section
+
+open Finset Polynomial
+
+namespace Zeta5Irrational
+
+lemma divByMonic_add' {q : ℚ[X]} (hq : q.Monic) (P Q : ℚ[X]) : (P + Q) /ₘ q = P /ₘ q + Q /ₘ q := by
+  refine ((div_modByMonic_unique (P /ₘ q + Q /ₘ q) (P %ₘ q + Q %ₘ q) hq ⟨?_, ?_⟩).1)
+  · have h1 := modByMonic_add_div P q
+    have h2 := modByMonic_add_div Q q
+    rw [mul_add]; linear_combination h1 + h2
+  · exact
+      (degree_add_le _ _).trans_lt
+        (max_lt (degree_modByMonic_lt _ hq) (degree_modByMonic_lt _ hq))
+
+lemma divByMonic_C_mul' {q : ℚ[X]} (hq : q.Monic) (c : ℚ) (P : ℚ[X]) :
+    (C c * P) /ₘ q = C c * (P /ₘ q) := by
+  refine ((div_modByMonic_unique (C c * (P /ₘ q)) (C c * (P %ₘ q)) hq ⟨?_, ?_⟩).1)
+  · have h1 := modByMonic_add_div P q
+    linear_combination C c * h1
+  · rw [← smul_eq_C_mul]; exact (degree_smul_le _ _).trans_lt (degree_modByMonic_lt _ hq)
+
+lemma μpoly_add (P Q : ℚ[X]) : μpoly (P + Q) = μpoly P + μpoly Q := by
+  unfold μpoly
+  exact Polynomial.sum_add_index _ _ _ (fun _ => by simp) (fun _ _ _ => by ring)
+
+lemma μpoly_C_mul (c : ℚ) (P : ℚ[X]) : μpoly (C c * P) = c * μpoly P := by
+  unfold μpoly
+  rw [← smul_eq_C_mul, Polynomial.sum_smul_index _ _ _ (fun _ => by simp), Polynomial.sum,
+    Polynomial.sum, Finset.mul_sum]
+  refine Finset.sum_congr rfl fun n _ => by ring
+
+lemma μX_add (K : ℕ) (P Q : ℚ[X]) : μX K (P + Q) = μX K P + μX K Q := by
+  unfold μX
+  rw [divByMonic_add' (D_monic K), μpoly_add, map_add]
+  simp only [eval_add, add_div, map_add, add_mul, Finset.sum_add_distrib]
+  ring
+
+lemma μX_C_mul (K : ℕ) (c : ℚ) (P : ℚ[X]) : μX K (C c * P) = C c * μX K P := by
+  unfold μX
+  rw [divByMonic_C_mul' (D_monic K), μpoly_C_mul, map_mul, mul_add, Finset.mul_sum]
+  congr 1
+  refine Finset.sum_congr rfl fun j _ => ?_
+  rw [eval_mul, eval_C, mul_div_assoc, map_mul]; ring
+
+lemma μX_sum {ι : Type*} (K : ℕ) (s : Finset ι) (f : ι → ℚ[X]) :
+    μX K (∑ i ∈ s, f i) = ∑ i ∈ s, μX K (f i) := by
+  classical
+    induction s using Finset.induction_on with
+  | empty =>
+    have := μX_C_mul K 0 0
+    simpa using this
+  | insert a s ha ih => rw [Finset.sum_insert ha, Finset.sum_insert ha, μX_add, ih]
+
+/-- The coefficient matrix of a family of polynomials. -/
+noncomputable def coeffMat {h : ℕ} (E : Fin h → ℚ[X]) : Matrix (Fin h) (Fin h) ℚ :=
+  Matrix.of fun a k => (E a).coeff k
+
+lemma sum_coeffMat {h : ℕ} (E : Fin h → ℚ[X]) (hE : ∀ a, (E a).natDegree < h) (a : Fin h) :
+    E a = ∑ k : Fin h, C (coeffMat E a k) * X ^ (k : ℕ) := by
+  conv_lhs => rw [as_sum_range' (E a) h (hE a)]
+  rw [Finset.sum_range (fun k => monomial k ((E a).coeff k))]
+  refine Finset.sum_congr rfl fun k _ => ?_
+  rw [coeffMat, Matrix.of_apply, C_mul_X_pow_eq_monomial]
+
+/-- **Change of basis**. -/
+theorem det_basis_change (n : ℕ) (E : Fin (37 * n) → ℚ[X]) (hE : ∀ a, (E a).natDegree < 37 * n) :
+    (Matrix.of fun a b => μX (40 * n) (D (3 * n) ^ 6 * E a * E b)).det =
+      C ((coeffMat E).det ^ 2) * Δ n := by
+  set Cm := (coeffMat E).map (C : ℚ →+* ℚ[X]) with hCm
+  have hM :
+    (Matrix.of fun a b => μX (40 * n) (D (3 * n) ^ 6 * E a * E b)) = Cm * G n * Cm.transpose := by
+    refine Matrix.ext fun a b => ?_
+    have hexp :
+      D (3 * n) ^ 6 * E a * E b =
+        ∑ k : Fin (37 * n),
+          ∑ l : Fin (37 * n),
+            C (coeffMat E a k * coeffMat E b l) * (D (3 * n) ^ 6 * X ^ ((k : ℕ) + (l : ℕ))) := by
+      rw [sum_coeffMat E hE a, sum_coeffMat E hE b]
+      simp only [Finset.mul_sum, Finset.sum_mul]
+      rw [Finset.sum_comm]
+      refine Finset.sum_congr rfl fun k _ => Finset.sum_congr rfl fun l _ => ?_
+      rw [C_mul, pow_add]; ring
+    rw [Matrix.of_apply, hexp, μX_sum]
+    simp only [μX_sum, μX_C_mul, Matrix.mul_apply, Matrix.transpose_apply, hCm, Matrix.map_apply,
+      G, Finset.sum_mul]
+    rw [Finset.sum_comm]
+    refine Finset.sum_congr rfl fun l _ => Finset.sum_congr rfl fun k _ => ?_
+    rw [C_mul]; ring
+  rw [hM, Matrix.det_mul, Matrix.det_mul, Matrix.det_transpose, hCm, ← RingHom.mapMatrix_apply, ←
+    RingHom.map_det, Δ, map_pow];
+  ring
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Arith/BinomBasis.lean b/LeanPool/Zeta5Irrational/Arith/BinomBasis.lean
new file mode 100644
index 0000000000..20b9142153
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Arith/BinomBasis.lean
@@ -0,0 +1,254 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import Mathlib.Algebra.Group.ForwardDiff
+public import Mathlib.NumberTheory.Bernoulli
+public import Mathlib.RingTheory.Polynomial.Pochhammer
+import Mathlib.Data.Rat.Star
+import Mathlib.Tactic.ENatToNat
+import Mathlib.Tactic.Polynomial.Basic
+import Mathlib.Tactic.ReduceModChar
+import Std.Tactic.BVDecide.Normalize.Prop
+import Mathlib.Tactic.FieldSimp
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.Ring.Basic
+
+/-! # The Bernoulli functional and the binomial basis
+
+* `Lb Q = ∑ Q_n B_n` (the Bernoulli umbral functional, `B₁ = -1/2`);
+* `Lb_shift` : `Lb (Q(x+1)) - Lb Q = Q'(0)`;
+* `bin k = x(x-1)⋯(x-k+1)/k!`, `bin_eval_nat : bin k (m) = choose m k`;
+* `Lb_bin` : `Lb (bin k) = (-1)^k/(k+1)`;
+* `derivative_bin` : `bin_k' = ∑_{j=1}^k (-1)^{j-1}/j · bin_{k-j}`;
+* `newton` : `P = ∑_{k ≤ d} Δ^k P(0) · bin k` for `deg P ≤ d`.
+-/
+
+@[expose] public section
+
+open Finset Polynomial
+
+namespace Zeta5Irrational
+
+/-- Polynomials over `ℚ` that agree at all natural numbers are equal. -/
+lemma poly_eq_of_nat {P Q : ℚ[X]} (h : ∀ n : ℕ, P.eval (n : ℚ) = Q.eval (n : ℚ)) : P = Q := by
+  apply Polynomial.eq_of_infinite_eval_eq
+  apply Set.infinite_of_injective_forall_mem (f := fun n : ℕ => (n : ℚ)) Nat.cast_injective
+  intro n
+  exact h n
+
+/-! ### The Bernoulli functional -/
+
+/-- `Lb Q = ∑_n Q_n B_n`. -/
+noncomputable def Lb (Q : ℚ[X]) : ℚ :=
+  Q.sum fun n a => a * _root_.bernoulli n
+
+lemma Lb_add (P Q : ℚ[X]) : Lb (P + Q) = Lb P + Lb Q := by
+  unfold Lb
+  exact Polynomial.sum_add_index _ _ _ (fun _ => by simp) (fun _ _ _ => by ring)
+
+lemma Lb_smul (c : ℚ) (Q : ℚ[X]) : Lb (c • Q) = c * Lb Q := by
+  unfold Lb
+  rw [Polynomial.sum_smul_index _ _ _ (fun _ => by simp), Polynomial.sum, Polynomial.sum,
+    Finset.mul_sum]
+  refine Finset.sum_congr rfl fun n _ => by ring
+
+lemma Lb_C_mul (c : ℚ) (Q : ℚ[X]) : Lb (C c * Q) = c * Lb Q := by rw [← smul_eq_C_mul, Lb_smul]
+
+lemma Lb_monomial (n : ℕ) (a : ℚ) : Lb (monomial n a) = a * _root_.bernoulli n := by
+  unfold Lb
+  rw [Polynomial.sum_monomial_index _ _ (by simp)]
+
+lemma Lb_X_pow (n : ℕ) : Lb (X ^ n) = _root_.bernoulli n := by
+  rw [← monomial_one_right_eq_X_pow, Lb_monomial, one_mul]
+
+lemma Lb_sum {ι : Type*} (s : Finset ι) (f : ι → ℚ[X]) : Lb (∑ i ∈ s, f i) = ∑ i ∈ s, Lb (f i) := by
+  classical
+    induction s using Finset.induction_on with
+  | empty => simp [Lb]
+  | insert a s ha ih => rw [Finset.sum_insert ha, Finset.sum_insert ha, Lb_add, ih]
+
+lemma Lb_neg (Q : ℚ[X]) : Lb (-Q) = -Lb Q := by rw [show -Q = (-1 : ℚ) • Q by simp, Lb_smul]; ring
+
+lemma Lb_sub (P Q : ℚ[X]) : Lb (P - Q) = Lb P - Lb Q := by
+  rw [sub_eq_add_neg, Lb_add, Lb_neg, sub_eq_add_neg]
+
+/-- **Shift identity**: `Lb (Q(x+1)) - Lb Q = Q'(0)`. -/
+theorem Lb_shift (Q : ℚ[X]) : Lb (Q.comp (X + 1)) - Lb Q = Q.derivative.eval 0 := by
+  induction Q using Polynomial.induction_on' with
+  | add P Q hP hQ => rw [add_comp, Lb_add, Lb_add, derivative_add, eval_add, ← hP, ← hQ]; ring
+  | monomial n
+    a =>
+    rw [← C_mul_X_pow_eq_monomial, mul_comp, C_comp, X_pow_comp, Lb_C_mul, Lb_C_mul, Lb_X_pow,
+      derivative_C_mul_X_pow, eval_C_mul, eval_pow, eval_X]
+    rw [add_pow, Lb_sum]
+    simp only [one_pow, mul_one]
+    have e :
+      ∀ k ∈ range (n + 1),
+        Lb (X ^ k * (n.choose k : ℚ[X])) = (n.choose k : ℚ) * _root_.bernoulli k := by
+      intro k _
+      rw [show (X ^ k * (n.choose k : ℚ[X])) = C (n.choose k : ℚ) * X ^ k by rw [mul_comm]; simp,
+        Lb_C_mul, Lb_X_pow]
+    rw [Finset.sum_congr rfl e, Finset.sum_range_succ, _root_.sum_bernoulli, Nat.choose_self]
+    rcases n with _ | n
+    · simp
+    · rcases n with _ | n
+      · simp; ring
+      · simp
+
+/-! ### The binomial basis -/
+
+/-- `bin k = x(x-1)⋯(x-k+1)/k!`. -/
+noncomputable def bin (k : ℕ) : ℚ[X] :=
+  C ((k.factorial : ℚ)⁻¹) * descPochhammer ℚ k
+
+lemma bin_eval_nat (k m : ℕ) : (bin k).eval (m : ℚ) = (m.choose k : ℚ) := by
+  unfold bin
+  rw [eval_mul, eval_C, descPochhammer_eval_eq_descFactorial,
+    Nat.descFactorial_eq_factorial_mul_choose]
+  push_cast
+  field_simp
+
+lemma bin_zero : bin 0 = 1 := by simp [bin]
+
+lemma bin_eval_zero {k : ℕ} (hk : 1 ≤ k) : (bin k).eval 0 = 0 := by
+  have := bin_eval_nat k 0
+  simp only [Nat.cast_zero] at this
+  rw [this, Nat.choose_eq_zero_of_lt hk]; simp
+
+/-- **Pascal**: `bin (k+1) (x+1) = bin (k+1) x + bin k x`. -/
+lemma bin_comp_add_one (k : ℕ) : (bin (k + 1)).comp (X + 1) = bin (k + 1) + bin k := by
+  apply poly_eq_of_nat
+  intro m
+  rw [eval_comp, eval_add, eval_X, eval_one, eval_add,
+    show (m : ℚ) + 1 = ((m + 1 : ℕ) : ℚ) by push_cast; ring, bin_eval_nat, bin_eval_nat,
+    bin_eval_nat, Nat.choose_succ_succ']
+  push_cast; ring
+
+lemma descPochhammer_eval_neg_one (n : ℕ) :
+    (descPochhammer ℚ n).eval (-1) = (-1) ^ n * n.factorial := by
+  induction n with
+  | zero => simp
+  | succ n
+    ih =>
+    rw [descPochhammer_succ_right, eval_mul, ih, eval_sub, eval_X, eval_natCast,
+      Nat.factorial_succ]
+    push_cast; ring
+
+/-- `bin_k'(0) = (-1)^{k-1}/k`. -/
+lemma bin_derivative_eval_zero (k : ℕ) :
+    (bin (k + 1)).derivative.eval 0 = (-1) ^ k / (k + 1 : ℚ) := by
+  unfold bin
+  rw [derivative_C_mul, eval_mul, eval_C, descPochhammer_succ_left, derivative_mul, derivative_X,
+    one_mul, eval_add, eval_mul, eval_X, zero_mul, add_zero, eval_comp, eval_sub, eval_X,
+    eval_one, zero_sub, descPochhammer_eval_neg_one, Nat.factorial_succ]
+  push_cast
+  field_simp
+
+/-- `Lb (bin k) = (-1)^k/(k+1)`. -/
+theorem Lb_bin (k : ℕ) : Lb (bin k) = (-1) ^ k / (k + 1 : ℚ) := by
+  have h := Lb_shift (bin (k + 1))
+  rw [bin_comp_add_one, Lb_add, bin_derivative_eval_zero] at h
+  linarith
+
+/-- A polynomial invariant under `x ↦ x + 1` is constant. -/
+lemma eq_C_of_comp_add_one {Q : ℚ[X]} (h : Q.comp (X + 1) = Q) : Q = C (Q.eval 0) := by
+  apply poly_eq_of_nat
+  intro m
+  rw [eval_C]
+  induction m with
+  | zero => simp
+  | succ m ih =>
+    have := congrArg (eval (m : ℚ)) h
+    rw [eval_comp, eval_add, eval_X, eval_one] at this
+    push_cast; rw [this, ih]
+
+/-- The coefficients of the derivative in the binomial basis. -/
+noncomputable def dcoef (j : ℕ) : ℚ :=
+  (-1) ^ (j - 1) / (j : ℚ)
+
+/-- **Derivative of the binomial polynomials**. -/
+theorem derivative_bin (k : ℕ) : (bin k).derivative = ∑ j ∈ Icc 1 k, C (dcoef j) * bin (k - j) := by
+  induction k with
+  | zero => simp [bin]
+  | succ k
+    ih =>
+    set D := (bin (k + 1)).derivative - ∑ j ∈ Icc 1 (k + 1), C (dcoef j) * bin (k + 1 - j) with hD
+    suffices hD0 : D = 0 by rw [hD] at hD0; exact sub_eq_zero.mp hD0
+    have hcomp : D.comp (X + 1) = D := by
+      rw [hD, sub_comp, Polynomial.sum_comp]
+      have h1 :
+        (bin (k + 1)).derivative.comp (X + 1) = (bin (k + 1)).derivative + (bin k).derivative := by
+        have := congrArg derivative (bin_comp_add_one k)
+        rw [derivative_comp, derivative_add, derivative_X, derivative_one, add_zero, one_mul,
+          derivative_add] at this
+        exact this
+      rw [h1, ih]
+      have h2 :
+        ∀ j ∈ Icc 1 (k + 1),
+          (C (dcoef j) * bin (k + 1 - j)).comp (X + 1) =
+            C (dcoef j) * bin (k + 1 - j) + (if j ≤ k then C (dcoef j) * bin (k - j) else 0) := by
+        intro j hj
+        rw [mul_comp, C_comp]
+        rw [Finset.mem_Icc] at hj
+        split_ifs with hjk
+        · obtain ⟨i, hi⟩ : ∃ i, k + 1 - j = i + 1 := ⟨k - j, by omega⟩
+          rw [hi, bin_comp_add_one, show k - j = i by omega]; ring
+        · have : k + 1 - j = 0 := by omega
+          rw [this, bin_zero, one_comp]; simp
+      rw [Finset.sum_congr rfl h2, Finset.sum_add_distrib]
+      have h3 :
+        ∑ j ∈ Icc 1 (k + 1), (if j ≤ k then C (dcoef j) * bin (k - j) else 0) =
+          ∑ j ∈ Icc 1 k, C (dcoef j) * bin (k - j) := by
+        rw [Finset.sum_ite, Finset.sum_const_zero, add_zero]
+        congr 1
+        ext j; simp only [Finset.mem_filter, Finset.mem_Icc]; omega
+      rw [h3]; ring
+    have h0 : D.eval 0 = 0 := by
+      rw [hD, eval_sub, eval_finsetSum, bin_derivative_eval_zero, ←
+        Finset.sum_erase_add _ _ (show k + 1 ∈ Icc 1 (k + 1) by simp)]
+      rw [Finset.sum_eq_zero]
+      · simp [bin_zero, dcoef]
+      · intro j hj
+        rw [Finset.mem_erase, Finset.mem_Icc] at hj
+        rw [eval_mul, bin_eval_zero (by omega), mul_zero]
+    rw [eq_C_of_comp_add_one hcomp, h0, C_0]
+
+/-! ### Newton expansion -/
+
+/-- **Newton expansion** of a polynomial in the binomial basis. -/
+theorem newton (P : ℚ[X]) {d : ℕ} (hd : P.natDegree ≤ d) :
+    P = ∑ k ∈ range (d + 1), C ((fwdDiff 1)^[k] P.eval 0) * bin k := by
+  apply poly_eq_of_nat
+  intro m
+  have h := shift_eq_sum_fwdDiff_iter (h := (1 : ℚ)) P.eval m 0
+  simp only [zero_add, nsmul_eq_mul, mul_one] at h
+  rw [h, eval_finsetSum]
+  simp only [eval_mul, eval_C, bin_eval_nat]
+  have hz : ∀ k, d < k → (fwdDiff 1)^[k] P.eval 0 = 0 := fun k hk => by
+    rw [Polynomial.fwdDiff_iter_eq_zero_of_degree_lt (by omega)]; rfl
+  have hz' : ∀ k, m < k → (m.choose k : ℚ) = 0 := fun k hk => by
+    rw [Nat.choose_eq_zero_of_lt hk];
+    simp
+  have e1 :
+    ∑ k ∈ range (m + 1), (m.choose k : ℚ) * (fwdDiff 1)^[k] P.eval 0 =
+      ∑ k ∈ range (m + d + 1), (m.choose k : ℚ) * (fwdDiff 1)^[k] P.eval 0 := by
+    apply Finset.sum_subset (Finset.range_subset_range.mpr (by omega))
+    intro k hk hk'
+    rw [Finset.mem_range] at hk hk'
+    rw [hz' k (by omega), zero_mul]
+  have e2 :
+    ∑ k ∈ range (d + 1), (fwdDiff 1)^[k] P.eval 0 * (m.choose k : ℚ) =
+      ∑ k ∈ range (m + d + 1), (fwdDiff 1)^[k] P.eval 0 * (m.choose k : ℚ) := by
+    apply Finset.sum_subset (Finset.range_subset_range.mpr (by omega))
+    intro k hk hk'
+    rw [Finset.mem_range] at hk hk'
+    rw [hz k (by omega), zero_mul]
+  rw [e1, e2]
+  exact Finset.sum_congr rfl fun k _ => mul_comm _ _
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Arith/ClassCount.lean b/LeanPool/Zeta5Irrational/Arith/ClassCount.lean
new file mode 100644
index 0000000000..9290182522
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Arith/ClassCount.lean
@@ -0,0 +1,261 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Arith.ClassFrame
+import Mathlib.Data.Rat.Star
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+import Mathlib.Tactic.Ring.Basic
+
+/-! # Class counting
+
+For `p = 2m + 1` every residue `c` mod `p` is `≡ ±a` for a unique `a ∈ {0, …, m}` (`ccls c`).
+We express the per-class counts `classCount` of the framework in terms of these classes.
+-/
+
+@[expose] public section
+
+open Finset Polynomial
+
+namespace Zeta5Irrational
+
+variable {p : ℕ} [hp : Fact p.Prime] {m : ℕ}
+
+/-- `μ(a, c) = [a ≡ c] + [a ≡ -c]`. -/
+def mu (p : ℕ) (a : ℕ) (c : ZMod p) : ℕ :=
+  (if ((a : ℤ) : ZMod p) = c then 1 else 0) + (if ((a : ℤ) : ZMod p) = -c then 1 else 0)
+
+/-- The class of a residue. -/
+def ccls (hm : 2 * m + 1 = p) (c : ZMod p) : Fin (m + 1) :=
+  if h : c.val ≤ m then ⟨c.val, by omega⟩ else ⟨p - c.val, by have := c.val_lt; omega⟩
+
+lemma ccls_spec (hm : 2 * m + 1 = p) (c : ZMod p) :
+    (((ccls hm c : ℕ) : ℤ) : ZMod p) = c ∨ (((ccls hm c : ℕ) : ℤ) : ZMod p) = -c := by
+  have hp0 : NeZero p := ⟨hp.out.ne_zero⟩
+  unfold ccls
+  split_ifs with h
+  · left; simp
+  · right
+    have hv := c.val_lt
+    simp only []
+    rw [show ((p - c.val : ℕ) : ℤ) = (p : ℤ) - (c.val : ℤ) by
+        push_cast [Nat.cast_sub hv.le];
+        ring]
+    push_cast
+    rw [ZMod.natCast_self, zero_sub, ZMod.natCast_zmod_val]
+
+lemma two_ne_zero_zmod (hm : 2 * m + 1 = p) : (2 : ZMod p) ≠ 0 := by
+  intro h
+  have : ((2 : ℕ) : ZMod p) = 0 := by exact_mod_cast h
+  rw [ZMod.natCast_eq_zero_iff] at this
+  have := Nat.le_of_dvd (by norm_num) this
+  rcases (Nat.le_succ_iff.mp this) with h1 | h1
+  · rcases Nat.le_one_iff_eq_zero_or_eq_one.mp h1 with h2 | h2
+    · exact hp.out.ne_zero h2
+    · exact hp.out.ne_one h2
+  · omega
+
+/-- Uniqueness of the class. -/
+lemma ccls_unique (hm : 2 * m + 1 = p) (c : ZMod p) (a : Fin (m + 1))
+    (h : (((a : ℕ) : ℤ) : ZMod p) = c ∨ (((a : ℕ) : ℤ) : ZMod p) = -c) : a = ccls hm c := by
+  have h1 := ccls_spec hm c
+  apply sq_injective hm
+  simp only
+  have e : ∀ x : ZMod p, x = c ∨ x = -c → x ^ 2 = c ^ 2 := by rintro x (rfl | rfl) <;> ring
+  have ha := e _ h
+  have hc := e _ h1
+  push_cast at ha hc
+  rw [ha, hc]
+
+lemma mu_eq (hm : 2 * m + 1 = p) (a : Fin (m + 1)) (c : ZMod p) :
+    mu p a c = if a = ccls hm c then (if c = 0 then 2 else 1) else 0 := by
+  unfold mu
+  by_cases ha : a = ccls hm c
+  · rw [ite_eq_left ha]
+    have hs := ccls_spec hm c
+    rw [← ha] at hs
+    by_cases hc : c = 0
+    · subst hc
+      rw [ite_eq_left rfl]
+      have : (((a : ℕ) : ℤ) : ZMod p) = 0 := by rcases hs with h | h <;> simpa using h
+      rw [ite_eq_left this, ite_eq_left (by rw [this, neg_zero])]
+    · rw [ite_eq_right hc]
+      have hnot : ¬((((a : ℕ) : ℤ) : ZMod p) = c ∧ (((a : ℕ) : ℤ) : ZMod p) = -c) := by
+        rintro ⟨h1, h2⟩
+        apply hc
+        have : (2 : ZMod p) * c = 0 := by have := h1.symm.trans h2; linear_combination this
+        exact (mul_eq_zero.mp this).resolve_left (two_ne_zero_zmod hm)
+      rcases hs with h | h
+      · rw [ite_eq_left h, ite_eq_right (fun h' => hnot ⟨h, h'⟩)]
+      · rw [ite_eq_right (fun h' => hnot ⟨h', h⟩), ite_eq_left h]
+  · rw [ite_eq_right ha]
+    rw [ite_eq_right (fun h => ha (ccls_unique hm c a (Or.inl h))),
+      ite_eq_right (fun h => ha (ccls_unique hm c a (Or.inr h)))]
+
+/-- `#{j ≤ X : j ≡ c} + #{j ≤ X : j ≡ -c}`. -/
+def SX (p X : ℕ) (c : ZMod p) : ℕ :=
+  ((Icc 1 X).filter fun j : ℕ => ((j : ℤ) : ZMod p) = c).card +
+    ((Icc 1 X).filter fun j : ℕ => ((j : ℤ) : ZMod p) = -c).card
+
+lemma SX_neg (X : ℕ) (c : ZMod p) : SX p X (-c) = SX p X c := by unfold SX; rw [neg_neg, add_comm]
+
+lemma SX_ccls (hm : 2 * m + 1 = p) (X : ℕ) (c : ZMod p) :
+    SX p X c = SX p X (((ccls hm c : ℕ) : ℤ) : ZMod p) := by
+  rcases ccls_spec hm c with h | h <;> rw [h]
+  rw [SX_neg]
+
+/-- The class constant `5[a = 0] + 6 S_N(a) - S_K(a)`. -/
+noncomputable def Bse (p K N : ℕ) (a : ℕ) : ℚ :=
+  (if a = 0 then 5 else 0) + 6 * (SX p N (((a : ℤ) : ZMod p)) : ℚ) - SX p K (((a : ℤ) : ZMod p))
+
+/-- The vanishing order of the row `s = (a, i)` at the class `b`. -/
+def nu {m : ℕ} (L : Fin (m + 1) → ℕ) (s : Σ a : Fin (m + 1), Fin (L a)) (b : Fin (m + 1)) : ℕ :=
+  if b = s.1 then s.2 else L b
+
+lemma card_filter_replicate (k : ℕ) (v : ℤ) (P : ℤ → Prop) [DecidablePred P] :
+    ((Multiset.replicate k v).filter P).card = if P v then k else 0 := by
+  split_ifs with h
+  · rw [Multiset.filter_eq_self.mpr fun a ha => by rw [Multiset.eq_of_mem_replicate ha]; exact h]
+    simp
+  · rw [Multiset.filter_eq_nil.mpr fun a ha => by rw [Multiset.eq_of_mem_replicate ha]; exact h]
+    simp
+
+lemma card_filter_sum_replicate {α : Type*} (s : Finset α) (L : α → ℕ) (v : α → ℤ) (P : ℤ → Prop)
+    [DecidablePred P] :
+    ((∑ c ∈ s, Multiset.replicate (L c) (v c)).filter P).card =
+      ∑ c ∈ s, if P (v c) then L c else 0 := by
+  classical
+    induction s using Finset.induction_on with
+  | empty => simp
+  | insert a s ha ih =>
+    rw [Finset.sum_insert ha, Multiset.filter_add, Multiset.card_add, ih, Finset.sum_insert ha,
+      card_filter_replicate]
+
+lemma count_classRoots (hm : 2 * m + 1 = p) (L : Fin (m + 1) → ℕ)
+    (s : Σ a : Fin (m + 1), Fin (L a)) (c : ZMod p) :
+    ((classRoots L s.1 s.2).filter fun γ : ℤ => (γ : ZMod p) = c).card +
+        ((classRoots L s.1 s.2).filter fun γ : ℤ => (γ : ZMod p) = -c).card =
+      (if c = 0 then 2 else 1) * nu L s (ccls hm c) := by
+  unfold classRoots
+  simp only [Multiset.filter_add, Multiset.card_add, card_filter_sum_replicate,
+    card_filter_replicate]
+  have key :
+    ∀ b : Fin (m + 1),
+      ∀ k : ℕ,
+        ((if (((b : ℕ) : ℤ) : ZMod p) = c then k else 0) +
+            (if (((b : ℕ) : ℤ) : ZMod p) = -c then k else 0)) =
+          k * mu p b c :=
+    by intro b k; unfold mu; split_ifs <;> ring
+  have e :
+    ∀ S : Finset (Fin (m + 1)),
+      (∑ b ∈ S, if (((b : ℕ) : ℤ) : ZMod p) = c then L b else 0) +
+          (∑ b ∈ S, if (((b : ℕ) : ℤ) : ZMod p) = -c then L b else 0) =
+        ∑ b ∈ S, L b * mu p b c :=
+    by intro S; rw [← Finset.sum_add_distrib]; exact Finset.sum_congr rfl fun b _ => key b (L b)
+  have e2 := key s.1 s.2
+  calc
+    _ =
+        ((∑ b ∈ Finset.univ.erase s.1, if (((b : ℕ) : ℤ) : ZMod p) = c then L b else 0) +
+            (∑ b ∈ Finset.univ.erase s.1, if (((b : ℕ) : ℤ) : ZMod p) = -c then L b else 0)) +
+          ((if (((s.1 : ℕ) : ℤ) : ZMod p) = c then (s.2 : ℕ) else 0) +
+            (if (((s.1 : ℕ) : ℤ) : ZMod p) = -c then (s.2 : ℕ) else 0)) :=
+      by ring
+    _ = ∑ b ∈ Finset.univ.erase s.1, L b * mu p b c + s.2 * mu p s.1 c := by rw [e, e2]
+    _ = _ := by
+      simp_rw [mu_eq hm]
+      unfold nu
+      by_cases h : ccls hm c = s.1
+      · rw [ite_eq_left h, ite_eq_left h.symm,
+          Finset.sum_eq_zero fun b hb => by
+            rw [ite_eq_right (fun h' => (Finset.mem_erase.mp hb).1 (h'.trans h)), mul_zero]]
+        ring
+      · rw [ite_eq_right h, ite_eq_right (Ne.symm h), mul_zero, add_zero,
+          Finset.sum_eq_single_of_mem (ccls hm c) (Finset.mem_erase.mpr ⟨h, Finset.mem_univ _⟩)
+            fun b _ hb => by rw [ite_eq_right hb, mul_zero]]
+        rw [ite_eq_left rfl]; ring
+
+lemma count_DN (N : ℕ) (c : ZMod p) :
+    ((6 • ((Icc 1 N).val.map fun j : ℕ => (j : ℤ))).filter fun γ : ℤ => (γ : ZMod p) = c).card +
+        ((6 • ((Icc 1 N).val.map fun j : ℕ => (j : ℤ))).filter fun γ : ℤ =>
+            (γ : ZMod p) = -c).card =
+      6 * SX p N c := by
+  unfold SX
+  simp only [Multiset.filter_nsmul, Multiset.card_nsmul, Multiset.filter_map, Multiset.card_map]
+  rw [mul_add]
+  rfl
+
+lemma card_plc_PlK (K : ℕ) (c : ZMod p) : (plc p (PlK K) c).card = SX p K c := by
+  unfold plc PlK SX
+  rw [Finset.filter_union,
+    Finset.card_union_of_disjoint (Finset.disjoint_filter_filter (PlK_disjoint K)),
+    Finset.filter_image, Finset.filter_image,
+    Finset.card_image_of_injective _ (fun a b h => by exact_mod_cast h),
+    Finset.card_image_of_injective _ (fun a b h => by simpa using h)]
+  congr 1
+  congr 1
+  ext j
+  simp only [Finset.mem_filter, Int.cast_neg]
+  constructor <;> rintro ⟨h1, h2⟩ <;> refine ⟨h1, ?_⟩ <;> [rw [← h2]; rw [h2]] <;> ring
+
+lemma ccls_eq_zero_iff (hm : 2 * m + 1 = p) (c : ZMod p) : ccls hm c = 0 ↔ c = 0 := by
+  constructor
+  · intro h
+    have hs := ccls_spec hm c
+    rw [h] at hs
+    simp only [Fin.val_zero, Nat.cast_zero, Int.cast_zero] at hs
+    rcases hs with h1 | h1
+    · exact h1.symm
+    · exact neg_eq_zero.mp h1.symm
+  · rintro rfl
+    unfold ccls
+    rw [dite_eq_left (by simp)]
+    ext; simp
+
+/-- **The class counts** in terms of `Bse` and the row vanishing orders. -/
+theorem classCount_eq (hm : 2 * m + 1 = p) (K N : ℕ) (L : Fin (m + 1) → ℕ)
+    (s t : Σ a : Fin (m + 1), Fin (L a)) (c : ZMod p) :
+    classCount p K (entryRoots N L s t) c =
+      Bse p K N (ccls hm c) +
+        (if ccls hm c = 0 then 2 else 1) * ((nu L s (ccls hm c) : ℚ) + nu L t (ccls hm c)) := by
+  unfold classCount entryRoots Bse
+  simp only [Multiset.filter_add, Multiset.card_add]
+  have h1 := count_DN (p := p) N c
+  have h2 := count_classRoots hm L s c
+  have h3 := count_classRoots hm L t c
+  rw [card_plc_PlK, ← SX_ccls hm N c, ← SX_ccls hm K c]
+  have hz : ((if c = 0 then 5 else 0 : ℕ) : ℚ) = if (ccls hm c : ℕ) = 0 then 5 else 0 := by
+    by_cases hc : c = 0
+    · rw [ite_eq_left hc, ite_eq_left (by rw [(ccls_eq_zero_iff hm c).mpr hc]; rfl)]; norm_num
+    · have hne : (ccls hm c : ℕ) ≠ 0 := fun h => hc ((ccls_eq_zero_iff hm c).mp (Fin.ext h))
+      simp [hc, hne]
+  have hz2 : ((if c = 0 then 2 else 1 : ℕ) : ℚ) = if ccls hm c = 0 then 2 else 1 := by
+    by_cases hc : c = 0
+    · rw [ite_eq_left hc, ite_eq_left ((ccls_eq_zero_iff hm c).mpr hc)]; norm_num
+    · rw [ite_eq_right hc, ite_eq_right (fun h => hc ((ccls_eq_zero_iff hm c).mp h))]; norm_num
+  have h1' :
+    (((Multiset.filter (fun γ : ℤ => (γ : ZMod p) = c)
+                (6 • ((Icc 1 N).val.map fun j : ℕ => (j : ℤ)))).card :
+            ℕ) :
+          ℚ) +
+        ((Multiset.filter (fun γ : ℤ => (γ : ZMod p) = -c)
+              (6 • ((Icc 1 N).val.map fun j : ℕ => (j : ℤ)))).card :
+          ℕ) =
+      6 * (SX p N c : ℚ) :=
+    by exact_mod_cast h1
+  have h2' := congrArg (fun x : ℕ => (x : ℚ)) h2
+  have h3' := congrArg (fun x : ℕ => (x : ℚ)) h3
+  simp only [Nat.cast_add, Nat.cast_mul] at h2' h3'
+  rw [hz2] at h2' h3'
+  push_cast
+  rw [← hz]
+  push_cast
+  linarith
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Arith/ClassFrame.lean b/LeanPool/Zeta5Irrational/Arith/ClassFrame.lean
new file mode 100644
index 0000000000..422e7f7093
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Arith/ClassFrame.lean
@@ -0,0 +1,197 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+import Mathlib.RingTheory.Int.Basic
+
+public import LeanPool.Zeta5Irrational.Arith.TEntry
+import LeanPool.Zeta5Irrational.Arith.Unimodular
+import Mathlib.Algebra.Ring.IsFormallyReal
+import Mathlib.Data.Rat.Star
+import Mathlib.Tactic.Polynomial.Basic
+import Mathlib.Tactic.ReduceModChar
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+import Mathlib.Tactic.Ring.Basic
+
+/-! # The class-basis framework
+
+Classes `a = 0, …, m` (with `p = 2m + 1`), dimensions `L a` with `∑ L = h`, basis polynomials
+`E_{a,i} = ∏_{c ≠ a} (t + c²)^{L_c} (t + a²)^i` (`i < L a`). If weights `w` satisfy the per-class
+entry inequalities, then `v_p^G(Δ) ≥ 2 ∑ w`.
+-/
+
+@[expose] public section
+
+open Finset Polynomial
+
+namespace Zeta5Irrational
+
+/-- The `t`-roots of the basis polynomial `E_{a,i}`. -/
+def classRoots {m : ℕ} (L : Fin (m + 1) → ℕ) (a : Fin (m + 1)) (i : ℕ) : Multiset ℤ :=
+  (∑ c ∈ Finset.univ.erase a, Multiset.replicate (L c) ((c : ℕ) : ℤ)) +
+    Multiset.replicate i ((a : ℕ) : ℤ)
+
+lemma card_classRoots {m : ℕ} (L : Fin (m + 1) → ℕ) (a : Fin (m + 1)) (i : ℕ) :
+    (classRoots L a i).card = ∑ c ∈ Finset.univ.erase a, L c + i := by
+  simp [classRoots, Multiset.card_sum]
+
+/-- `∏_{γ ∈ M} (t + γ²)` over `ℤ`. -/
+noncomputable def tpolZ (M : Multiset ℤ) : ℤ[X] :=
+  (M.map fun γ : ℤ => X + C (γ ^ 2)).prod
+
+lemma tpolZ_map_Q (M : Multiset ℤ) : (tpolZ M).map (Int.castRingHom ℚ) = tpol M := by
+  simp [tpolZ, tpol, Polynomial.map_multiset_prod, Multiset.map_map, Function.comp]
+
+lemma natDegree_tpolZ (M : Multiset ℤ) : (tpolZ M).natDegree ≤ M.card := by
+  unfold tpolZ
+  refine (natDegree_multiset_prod_le _).trans ?_
+  rw [Multiset.map_map]
+  have : ∀ γ ∈ M, (natDegree ∘ fun γ : ℤ => X + C (γ ^ 2)) γ = 1 := fun γ _ => natDegree_X_add_C _
+  rw [Multiset.map_congr rfl this, Multiset.map_const', Multiset.sum_replicate, smul_eq_mul,
+    mul_one]
+
+lemma prod_map_sum_replicate {α β : Type*} [CommMonoid β] (s : Finset α) (L : α → ℕ) (v : α → ℤ)
+    (g : ℤ → β) :
+    ((∑ c ∈ s, Multiset.replicate (L c) (v c)).map g).prod = ∏ c ∈ s, g (v c) ^ L c := by
+  classical
+    induction s using Finset.induction_on with
+  | empty => simp
+  | insert a s ha ih =>
+    rw [Finset.sum_insert ha, Multiset.map_add, Multiset.prod_add, ih, Finset.prod_insert ha,
+      Multiset.map_replicate, Multiset.prod_replicate]
+
+lemma tpolZ_map_ZMod (p : ℕ) {m : ℕ} (L : Fin (m + 1) → ℕ) (a : Fin (m + 1)) (i : ℕ) :
+    (tpolZ (classRoots L a i)).map (Int.castRingHom (ZMod p)) =
+      (∏ c ∈ Finset.univ.erase a, (X - C (-(((c : ℕ) : ZMod p) ^ 2))) ^ L c) *
+        (X - C (-(((a : ℕ) : ZMod p) ^ 2))) ^ i := by
+  unfold tpolZ classRoots
+  rw [Polynomial.map_multiset_prod, Multiset.map_map, Multiset.map_add, Multiset.prod_add,
+    prod_map_sum_replicate, Multiset.map_replicate, Multiset.prod_replicate]
+  simp [Function.comp, sub_eq_add_neg]
+
+/-- `D_m` as a `tpol`. -/
+lemma D_eq_tpol (m : ℕ) : D m = tpol ((Icc 1 m).val.map fun j : ℕ => (j : ℤ)) := by
+  unfold D tpol
+  rw [Multiset.map_map, Finset.prod_eq_multiset_prod]
+  congr 1
+
+lemma tpol_nsmul (k : ℕ) (M : Multiset ℤ) : tpol (k • M) = tpol M ^ k := by
+  induction k with
+  | zero => simp [tpol]
+  | succ k ih => rw [succ_nsmul, tpol_add, ih, pow_succ]
+
+/-- The squares of `0, …, m` are distinct modulo `p = 2m + 1`. -/
+lemma sq_injective {p m : ℕ} [hp : Fact p.Prime] (hm : 2 * m + 1 = p) :
+    Function.Injective fun c : Fin (m + 1) => -(((c : ℕ) : ZMod p) ^ 2) := by
+  intro c c' h
+  simp only [neg_inj] at h
+  have h2 : (p : ℤ) ∣ ((c : ℕ) - (c' : ℕ) : ℤ) * ((c : ℕ) + (c' : ℕ) : ℤ) := by
+    rw [← ZMod.intCast_zmod_eq_zero_iff_dvd]; push_cast
+    linear_combination h
+  have hc := c.isLt; have hc' := c'.isLt
+  rcases (Int.Prime.dvd_mul' hp.out h2) with h3 | h3
+  · have := Int.eq_zero_of_abs_lt_dvd h3 (by rw [abs_lt]; constructor <;> omega)
+    exact Fin.ext (by omega)
+  · have := Int.eq_zero_of_abs_lt_dvd h3 (by rw [abs_lt]; constructor <;> omega)
+    exact Fin.ext (by omega)
+
+/-- The `t`-roots of the numerator `D_N⁶ E_s E_t`. -/
+def entryRoots {m : ℕ} (N : ℕ) (L : Fin (m + 1) → ℕ) (s t : Σ a : Fin (m + 1), Fin (L a)) :
+    Multiset ℤ :=
+  6 • ((Icc 1 N).val.map fun j : ℕ => (j : ℤ)) + classRoots L s.1 s.2 + classRoots L t.1 t.2
+
+/-- The per-class count appearing in the entry bound. -/
+noncomputable def classCount (p : ℕ) (K : ℕ) (M : Multiset ℤ) (c : ZMod p) : ℚ :=
+  (((if c = 0 then 5 else 0) + (M.filter fun γ : ℤ => (γ : ZMod p) = c).card +
+          (M.filter fun γ : ℤ => (γ : ZMod p) = -c).card :
+        ℕ) :
+      ℚ) -
+    (plc p (PlK K) c).card
+
+/-- **The class-basis framework**: `v_p^G(Δ) ≥ 2 ∑ w`. -/
+theorem class_frame {p : ℕ} [hp : Fact p.Prime] (hp5 : 5 ≤ p) {m : ℕ} (hm : 2 * m + 1 = p) (n : ℕ)
+    (L : Fin (m + 1) → ℕ) (hL : ∑ c, L c = 37 * n) (w : (Σ a : Fin (m + 1), Fin (L a)) → ℚ)
+    (hsep : Sep p (PlK (40 * n))) (hK : 40 * n < p ^ 2)
+    (hdeg : 5 + 2 * (18 * n + 2 * (37 * n)) - 2 * (40 * n) + 1 < p ^ 2)
+    (hentry :
+      ∀ s t : (Σ a : Fin (m + 1), Fin (L a)),
+        ∀ c : ZMod p, w s + w t + 4 ≤ classCount p (40 * n) (entryRoots (3 * n) L s t) c) :
+    GV p (Δ n) (2 * ∑ s, w s) := by
+  classical
+  have hcard : Fintype.card (Σ a : Fin (m + 1), Fin (L a)) = 37 * n := by
+    rw [Fintype.card_sigma];
+    simpa using hL
+  set e : (Σ a : Fin (m + 1), Fin (L a)) ≃ Fin (37 * n) := Fintype.equivFinOfCardEq hcard
+  set R : Fin (37 * n) → Multiset ℤ := fun k => classRoots L (e.symm k).1 (e.symm k).2
+  have hRcard : ∀ k, (R k).card < 37 * n := by
+    intro k
+    rw [card_classRoots]
+    have h1 := Finset.add_sum_erase Finset.univ L (Finset.mem_univ (e.symm k).1)
+    have := (e.symm k).2.isLt
+    omega
+  set E : Fin (37 * n) → ℚ[X] := fun k => tpol (R k)
+  set Ez : Fin (37 * n) → ℤ[X] := fun k => tpolZ (R k)
+  have hEz : ∀ k, (Ez k).map (Int.castRingHom ℚ) = E k := fun k => tpolZ_map_Q _
+  have hdegZ : ∀ k, (Ez k).natDegree < 37 * n := fun k => (natDegree_tpolZ _).trans_lt (hRcard k)
+  have hdegE : ∀ k, (E k).natDegree < 37 * n := fun k => by
+    rw [← hEz];
+    exact
+      (natDegree_map_le).trans_lt
+        (hdegZ k)
+          -- unimodularity
+  have hunit :=
+    coeffMat_det_unit (p := p) Ez hdegZ (fun c : Fin (m + 1) => -(((c : ℕ) : ZMod p) ^ 2))
+      (sq_injective hm) L (fun k => (e.symm k).1) (fun k => (e.symm k).2)
+      (fun k => (e.symm k).2.isLt)
+      (fun k k' h1 h2 => by
+        apply e.symm.injective
+        exact Sigma.ext h1 (by rw [Fin.heq_ext_iff (by rw [h1])]; exact h2))
+      (fun k => tpolZ_map_ZMod p L _ _)
+  have hcoef : coeffMat E = coeffMat fun k => (Ez k).map (Int.castRingHom ℚ) := by simp only [hEz]
+  rw [← hcoef] at hunit
+  set dC := (coeffMat E).det with hdC
+    -- basis change
+  have hbc :=
+    det_basis_change n E
+      hdegE
+        -- entry bounds
+  have hentries :
+    ∀ k l,
+      GV p ((Matrix.of fun a b => μX (40 * n) (D (3 * n) ^ 6 * E a * E b)) k l)
+        (w (e.symm k) + w (e.symm l)) := by
+    intro k l
+    rw [Matrix.of_apply]
+    have hM : D (3 * n) ^ 6 * E k * E l = tpol (entryRoots (3 * n) L (e.symm k) (e.symm l)) := by
+      simp only [E, R, entryRoots, tpol_add, tpol_nsmul, D_eq_tpol]
+    rw [hM]
+    have hcardM : (entryRoots (3 * n) L (e.symm k) (e.symm l)).card ≤ 18 * n + 2 * (37 * n) := by
+      simp only [entryRoots, Multiset.card_add, Multiset.card_nsmul, Multiset.card_map,
+        Finset.card_val, Nat.card_Icc]
+      have h1 := hRcard k; have h2 := hRcard l
+      simp only [R] at h1 h2
+      omega
+    have :=
+      μX_tpol_bound hp5 (40 * n) (entryRoots (3 * n) L (e.symm k) (e.symm l)) hsep hK (by omega)
+        (w (e.symm k) + w (e.symm l) + 4) (fun c => hentry _ _ c)
+    simpa using this
+  have hdet := det_GV _ (fun k => w (e.symm k)) (fun k => w (e.symm k)) hentries
+  rw [hbc] at hdet
+  have hsum : ∑ k, w (e.symm k) + ∑ k, w (e.symm k) = 2 * ∑ s, w s := by
+    rw [Equiv.sum_comp e.symm w]; ring
+  rw [hsum] at hdet
+        -- divide by the unit `dC²`
+  have hΔ : Δ n = C ((dC ^ 2)⁻¹) * (C (dC ^ 2) * Δ n) := by
+    rw [← mul_assoc, ← C_mul, inv_mul_cancel₀ (pow_ne_zero _ hunit.1), C_1, one_mul]
+  rw [hΔ]
+  have hu : VG p ((dC ^ 2)⁻¹) 0 := by
+    refine (VG.inv (p := p) (r := 0) (pow_ne_zero _ hunit.1) ?_).mono (by norm_num)
+    rw [padicValRat.pow, hunit.2]; simp
+  simpa using GV.C_mul hu hdet
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Arith/DirectBound.lean b/LeanPool/Zeta5Irrational/Arith/DirectBound.lean
new file mode 100644
index 0000000000..b534fd1c38
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Arith/DirectBound.lean
@@ -0,0 +1,197 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Arith.PoleFun
+import Mathlib.Algebra.Ring.IsFormallyReal
+import Mathlib.Data.Rat.Star
+import Mathlib.Tactic.ENatToNat
+import Mathlib.Tactic.ReduceModChar
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.Ring.Basic
+
+/-! # The direct valuation bound for `τ_X`
+
+For `g = κ ∏_{ζ ∈ Z} (x - ζ) / ∏_{r ∈ Pl} (x - r)` with `p ≥ 5`, poles in the same residue class
+differing by exactly one power of `p`, harmonic indices `< p²` and polynomial part of degree
+`< p² - 1`:
+
+`v_p^G(τ_X(g)) ≥ min_c (e_c - ℓ_c) - 4`,
+
+where `e_c` (resp. `ℓ_c`) is the number of zeros (resp. poles) in the class `c` mod `p`.
+This replaces the `p`-adic distribution formula of the paper (Lemmas 3.1–3.2).
+-/
+
+@[expose] public section
+
+open Finset Polynomial
+
+namespace Zeta5Irrational
+
+variable {p : ℕ} [hp : Fact p.Prime]
+
+/-- The numerator `κ ∏_{ζ ∈ Z} (x - ζ)`. -/
+noncomputable def numOf (κ : ℚ) (Z : Multiset ℤ) : ℚ[X] :=
+  C κ * (Z.map fun ζ : ℤ => X - C (ζ : ℚ)).prod
+
+/-- Number of zeros in the class `c`. -/
+def cnt (p : ℕ) (Z : Multiset ℤ) (c : ZMod p) : ℕ :=
+  (Z.filter fun ζ : ℤ => (ζ : ZMod p) = c).card
+
+/-- Poles in the class `c`. -/
+def plc (p : ℕ) (Pl : Finset ℤ) (c : ZMod p) : Finset ℤ :=
+  Pl.filter fun r : ℤ => (r : ZMod p) = c
+
+/-! ### Integer valuations -/
+
+lemma zmod_eq_iff_dvd {a b : ℤ} : (a : ZMod p) = b ↔ (p : ℤ) ∣ a - b := by
+  rw [← ZMod.intCast_zmod_eq_zero_iff_dvd]; push_cast; exact sub_eq_zero.symm
+
+lemma VG_int_one {z : ℤ} (h : (p : ℤ) ∣ z) : VG p (z : ℚ) 1 := by
+  by_cases hz : z = 0
+  · left; simp [hz]
+  right
+  rw [padicValRat.of_int]
+  have := (padicValInt_dvd_iff (p := p) 1 z).mp (by simpa using h)
+  rcases this with h | h
+  · exact absurd h hz
+  · exact_mod_cast h
+
+lemma padicValRat_int_eq_zero {z : ℤ} (h : ¬(p : ℤ) ∣ z) : padicValRat p (z : ℚ) = 0 := by
+  rw [padicValRat.of_int]
+  have hz : z ≠ 0 := fun h' => h (h' ▸ dvd_zero _)
+  by_contra hne
+  have h1 : 1 ≤ padicValInt p z := Nat.one_le_iff_ne_zero.mpr (by exact_mod_cast hne)
+  exact h (by simpa using (padicValInt_dvd_iff (p := p) 1 z).mpr (Or.inr h1))
+
+lemma padicValRat_int_le_one {z : ℤ} (h2 : ¬(p : ℤ) ^ 2 ∣ z) : padicValRat p (z : ℚ) ≤ 1 := by
+  rw [padicValRat.of_int]
+  by_contra hne
+  rw [not_le] at hne
+  have : 2 ≤ padicValInt p z := by exact_mod_cast hne
+  have hz : z ≠ 0 := by rintro rfl; simp at hne
+  exact h2 ((padicValInt_dvd_iff (p := p) 2 z).mpr (Or.inr this))
+
+lemma padicValRat_finset_prod {ι : Type*} (s : Finset ι) (f : ι → ℚ) (hf : ∀ i ∈ s, f i ≠ 0) :
+    padicValRat p (∏ i ∈ s, f i) = ∑ i ∈ s, padicValRat p (f i) := by
+  classical
+    induction s using Finset.induction_on with
+  | empty => simp
+  | insert a s ha ih =>
+    rw [Finset.prod_insert ha, Finset.sum_insert ha,
+      padicValRat.mul (hf a (mem_insert_self _ _))
+        (Finset.prod_ne_zero_iff.mpr fun i hi => hf i (mem_insert_of_mem hi)),
+      ih fun i hi => hf i (mem_insert_of_mem hi)]
+
+lemma VG.multiset_prod (s : Multiset ℚ) (b : ℚ → ℚ) (h : ∀ q ∈ s, VG p q (b q)) :
+    VG p s.prod (s.map b).sum := by
+  induction s using Multiset.induction_on with
+  | empty => right; simp
+  | cons a s ih =>
+    rw [Multiset.prod_cons, Multiset.map_cons, Multiset.sum_cons]
+    exact
+      (h a (Multiset.mem_cons_self _ _)).mul (ih fun q hq => h q (Multiset.mem_cons_of_mem hq))
+
+/-! ### The harmonic numbers -/
+
+lemma VG_H5 {m : ℕ} (hm : m < p ^ 2) : VG p (H5 m) (-5) := by
+  unfold H5
+  refine VG.sum _ fun v hv => ?_
+  rw [Finset.mem_Icc] at hv
+  rw [one_div]
+  apply VG.inv (pow_ne_zero _ (by exact_mod_cast (by omega : v ≠ 0)))
+  rw [show ((v : ℚ) ^ 5) = ((v ^ 5 : ℕ) : ℚ) by push_cast; ring, padicValRat.of_nat,
+    padicValNat.pow (p := p) v 5]
+  have : padicValNat p v ≤ 1 := by
+    by_contra hne
+    rw [not_le] at hne
+    have h2 : p ^ 2 ∣ v := (padicValNat_dvd_iff_le (by omega)).mpr hne
+    exact absurd (Nat.le_of_dvd (by omega) h2) (by omega)
+  push_cast
+  have h3 : (padicValRat p (v : ℚ) : ℚ) ≤ 1 := by rw [padicValRat.of_nat]; exact_mod_cast this
+  linarith
+
+/-! ### Residues -/
+
+lemma eval_numOf (κ : ℚ) (Z : Multiset ℤ) (x : ℚ) :
+    (numOf κ Z).eval x = κ * (Z.map fun ζ : ℤ => x - (ζ : ℚ)).prod := by
+  unfold numOf
+  rw [eval_mul, eval_C, eval_multiset_prod, Multiset.map_map]
+  congr 2
+  exact Multiset.map_congr rfl fun ζ _ => by simp
+
+lemma VG_prod_sub (Z : Multiset ℤ) (x : ℤ) :
+    VG p ((Z.map fun ζ : ℤ => (x : ℚ) - (ζ : ℚ)).prod) (cnt p Z (x : ZMod p)) := by
+  induction Z using Multiset.induction_on with
+  | empty => right; simp [cnt]
+  | cons a Z ih =>
+    rw [Multiset.map_cons, Multiset.prod_cons]
+    unfold cnt at ih ⊢
+    rw [Multiset.filter_cons]
+    split_ifs with h
+    · rw [Multiset.card_add, Multiset.card_singleton]
+      have h1 : VG p ((x : ℚ) - (a : ℚ)) 1 := by
+        have := VG_int_one (p := p) ((zmod_eq_iff_dvd (p := p)).mp h.symm)
+        simpa using this
+      convert h1.mul ih using 1; push_cast; ring
+    · rw [zero_add]
+      have h1 : VG p ((x : ℚ) - (a : ℚ)) 0 := by simpa using VG.intCast (p := p) (x - a)
+      simpa using h1.mul ih
+
+lemma VG_eval_numOf (κ : ℚ) (hκ : VG p κ 0) (Z : Multiset ℤ) (x : ℤ) :
+    VG p ((numOf κ Z).eval (x : ℚ)) (cnt p Z (x : ZMod p)) := by
+  rw [eval_numOf]
+  simpa using hκ.mul (VG_prod_sub Z x)
+
+/-- Poles in the same class differ by exactly one power of `p`. -/
+def Sep (p : ℕ) (Pl : Finset ℤ) : Prop :=
+  ∀ r ∈ Pl, ∀ s ∈ Pl, r ≠ s → (r : ZMod p) = s → ¬(p : ℤ) ^ 2 ∣ r - s
+
+lemma padicValRat_denom_le (Pl : Finset ℤ) (hsep : Sep p Pl) {r : ℤ} (hr : r ∈ Pl) :
+    (padicValRat p (∏ s ∈ Pl.erase r, ((r : ℚ) - s)) : ℚ) ≤ (plc p Pl r).card - 1 := by
+  rw [padicValRat_finset_prod _ _
+      (fun s hs => by
+        rw [Finset.mem_erase] at hs
+        exact sub_ne_zero.mpr (by exact_mod_cast hs.1.symm))]
+  have hterm :
+    ∀ s ∈ Pl.erase r, (padicValRat p ((r : ℚ) - s) : ℚ) ≤ if (s : ZMod p) = r then 1 else 0 := by
+    intro s hs
+    rw [Finset.mem_erase] at hs
+    have e : ((r : ℚ) - s) = ((r - s : ℤ) : ℚ) := by push_cast; ring
+    rw [e]
+    split_ifs with h
+    · exact_mod_cast padicValRat_int_le_one (hsep r hr s hs.2 (Ne.symm hs.1) h.symm)
+    · rw [padicValRat_int_eq_zero
+          (fun hd =>
+            h ((zmod_eq_iff_dvd (p := p)).mpr (by rw [← neg_sub]; exact (dvd_neg).mpr hd)))]
+      simp
+  push_cast
+  refine (Finset.sum_le_sum hterm).trans ?_
+  rw [Finset.sum_ite, Finset.sum_const, Finset.sum_const_zero, add_zero, nsmul_eq_mul, mul_one]
+  have hcard : ((Pl.erase r).filter fun s : ℤ => (s : ZMod p) = r) = (plc p Pl r).erase r := by
+    unfold plc; rw [Finset.filter_erase]
+  rw [hcard, Finset.card_erase_of_mem (by unfold plc; simp [hr])]
+  have : 1 ≤ (plc p Pl r).card := Finset.card_pos.mpr ⟨r, by unfold plc; simp [hr]⟩
+  push_cast [this]
+  linarith
+
+/-- **Residue bound**: `v_p(res_r) ≥ e_c - ℓ_c + 1` for the class `c` of `r`. -/
+theorem VG_res (κ : ℚ) (hκ : VG p κ 0) (Z : Multiset ℤ) (Pl : Finset ℤ) (hsep : Sep p Pl) {r : ℤ}
+    (hr : r ∈ Pl) : VG p (resP (numOf κ Z) Pl r) ((cnt p Z r : ℚ) - (plc p Pl r).card + 1) := by
+  unfold resP
+  rw [div_eq_mul_inv]
+  have hD : ∏ s ∈ Pl.erase r, ((r : ℚ) - s) ≠ 0 :=
+    Finset.prod_ne_zero_iff.mpr fun s hs =>
+      by
+      rw [Finset.mem_erase] at hs
+      exact sub_ne_zero.mpr (by exact_mod_cast hs.1.symm)
+  have h1 := VG_eval_numOf κ hκ Z r
+  have h2 := VG.inv (p := p) hD (padicValRat_denom_le Pl hsep hr)
+  convert h1.mul h2 using 1
+  ring
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Arith/DirectPoly.lean b/LeanPool/Zeta5Irrational/Arith/DirectPoly.lean
new file mode 100644
index 0000000000..69254cdf03
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Arith/DirectPoly.lean
@@ -0,0 +1,333 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Arith.DirectBound
+import LeanPool.Zeta5Irrational.Arith.IntPoly
+import Mathlib.Data.Rat.Star
+import Mathlib.Tactic.ENatToNat
+import Mathlib.Tactic.Polynomial.Basic
+import Mathlib.Tactic.ReduceModChar
+import Mathlib.Tactic.FieldSimp
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+import Mathlib.Tactic.Ring.Basic
+import Mathlib.Tactic.Ring.RingNF
+
+import Mathlib.Tactic.NormNum.Prime
+
+/-! # The class-wise bound for the polynomial part
+
+For every integer `m`, `v_p(P(m)) ≥ β` where `P` is the polynomial part of
+`κ ∏ (x - ζ) / ∏ (x - r)` and `β ≤ e_c - ℓ_c` for every class `c`.
+-/
+
+@[expose] public section
+
+open Finset Polynomial
+
+namespace Zeta5Irrational
+
+variable {p : ℕ} [hp : Fact p.Prime]
+
+lemma prod_erase_split {f : ℤ → ℚ[X]} (S : Finset ℤ) (q : ℤ → Prop) [DecidablePred q] (r : ℤ) :
+    ∏ s ∈ S.erase r, f s =
+      (∏ s ∈ (S.filter q).erase r, f s) * ∏ s ∈ (S.filter (¬q ·)).erase r, f s :=
+  by rw [← Finset.filter_erase, ← Finset.filter_erase, Finset.prod_filter_mul_prod_filter_not]
+
+lemma GV_lin_comp_class (m ζ : ℤ) (h : (ζ : ZMod p) = m) :
+    GV p ((X - C (ζ : ℚ)).comp (C (m : ℚ) + C (p : ℚ) * X)) 1 := by
+  rw [sub_comp, X_comp, C_comp,
+    show C (m : ℚ) + C (p : ℚ) * X - C (ζ : ℚ) = C (((m - ζ : ℤ) : ℚ)) + C (p : ℚ) * X by
+      push_cast; rw [C_sub]; ring]
+  have h1 : VG p (((m - ζ : ℤ) : ℚ)) 1 := VG_int_one ((zmod_eq_iff_dvd (p := p)).mp h.symm)
+  have h2 : VG p (p : ℚ) 1 := by simpa using VG.primePow (p := p) 1
+  exact (GV.C h1).add (by simpa using GV.C_mul h2 GV.X)
+
+lemma GV_lin_comp (m ζ : ℤ) : GV p ((X - C (ζ : ℚ)).comp (C (m : ℚ) + C (p : ℚ) * X)) 0 := by
+  rw [sub_comp, X_comp, C_comp,
+    show C (m : ℚ) + C (p : ℚ) * X - C (ζ : ℚ) = C (((m - ζ : ℤ) : ℚ)) + C (p : ℚ) * X by
+      push_cast; rw [C_sub]; ring]
+  exact (GV.C (VG.intCast _)).add (by simpa using GV.C_mul (VG.natCast (p := p) p) GV.X)
+
+lemma GV_numOf_comp (κ : ℚ) (hκ : VG p κ 0) (Z : Multiset ℤ) (m : ℤ) :
+    GV p ((numOf κ Z).comp (C (m : ℚ) + C (p : ℚ) * X)) (cnt p Z (m : ZMod p)) := by
+  unfold numOf
+  rw [mul_comp, C_comp, multiset_prod_comp, Multiset.map_map]
+  suffices h :
+    GV p
+      ((Z.map ((fun q => q.comp (C (m : ℚ) + C (p : ℚ) * X)) ∘ fun ζ : ℤ => X - C (ζ : ℚ))).prod)
+      (cnt p Z (m : ZMod p))
+    by simpa using GV.C_mul hκ h
+  induction Z using Multiset.induction_on with
+  | empty => simpa [cnt] using GV.C (p := p) (VG.one (p := p))
+  | cons a Z ih =>
+    rw [Multiset.map_cons, Multiset.prod_cons]
+    unfold cnt at ih ⊢
+    rw [Multiset.filter_cons]
+    split_ifs with h
+    · rw [Multiset.card_add, Multiset.card_singleton]
+      have := (GV_lin_comp_class m a h).mul ih
+      convert this using 1
+      · simp [Function.comp]
+      · push_cast; ring
+    · rw [zero_add]
+      simpa using (GV_lin_comp m a).mul ih
+
+/-- `(x - r)(m + p z) = p (z - ρ)` with `ρ = (r - m)/p`, for `r ≡ m`. -/
+lemma lin_comp_eq {m r : ℤ} (h : (p : ℤ) ∣ r - m) :
+    (X - C (r : ℚ)).comp (C (m : ℚ) + C (p : ℚ) * X) =
+      C (p : ℚ) * (X - C (((r - m) / p : ℤ) : ℚ)) := by
+  have hp0 : (p : ℚ) ≠ 0 := by exact_mod_cast hp.out.ne_zero
+  have e : (p : ℚ) * (((r - m) / p : ℤ) : ℚ) = (r : ℚ) - m := by
+    have := Int.mul_ediv_cancel' h
+    exact_mod_cast this
+  rw [sub_comp, X_comp, C_comp, mul_sub, ← C_mul, e, C_sub]; ring
+
+/-- **Class-wise bound for the polynomial part.** -/
+theorem VG_polyPart_eval (κ : ℚ) (hκ : VG p κ 0) (Z : Multiset ℤ) (Pl : Finset ℤ)
+    (hsep : Sep p Pl) (β : ℚ) (hβ : ∀ c : ZMod p, β ≤ (cnt p Z c : ℚ) - (plc p Pl c).card)
+    (m : ℤ) : VG p ((polyPart (numOf κ Z) Pl).eval (m : ℚ)) β := by
+  classical
+  set A := numOf κ Z with hA
+  set P := polyPart A Pl with hP
+  set c : ZMod p := (m : ZMod p) with hc
+  set Plc := plc p Pl c with hPlc
+  set Plo := Pl.filter fun r : ℤ => ¬(r : ZMod p) = c with hPlo
+  set PiC := ∏ r ∈ Plc, (X - C (r : ℚ)) with hPiC
+  set PiO := ∏ r ∈ Plo, (X - C (r : ℚ)) with hPiO
+  have hPi : piPl Pl = PiC * PiO := by
+    unfold piPl;
+    rw [hPiC, hPiO, hPlc, plc, Finset.prod_filter_mul_prod_filter_not]
+  set H := P * PiO + ∑ r ∈ Plo, C (resP A Pl r) * ∏ s ∈ Plo.erase r, (X - C (s : ℚ)) with hH
+  set I := ∑ r ∈ Plc, C (resP A Pl r) * ∏ s ∈ Plc.erase r, (X - C (s : ℚ)) with hI
+    -- the splitting identity
+  have hAHI : A = PiC * H + PiO * I := by
+    have hpf := partial_fractionsP A Pl
+    rw [← Finset.sum_filter_add_sum_filter_not Pl (fun r : ℤ => (r : ZMod p) = c)] at hpf
+    have h1 :
+      ∀ r ∈ Pl.filter (fun r : ℤ => (r : ZMod p) = c),
+        C (resP A Pl r) * ∏ s ∈ Pl.erase r, (X - C (s : ℚ)) =
+          PiO * (C (resP A Pl r) * ∏ s ∈ Plc.erase r, (X - C (s : ℚ))) := by
+      intro r hr
+      rw [prod_erase_split Pl (fun s : ℤ => (s : ZMod p) = c) r]
+      have hro : r ∉ Plo := by rw [hPlo, Finset.mem_filter]; rw [Finset.mem_filter] at hr; tauto
+      rw [Finset.erase_eq_of_notMem hro]
+      rw [hPlc, plc]; ring
+    have h2 :
+      ∀ r ∈ Pl.filter (fun r : ℤ => ¬(r : ZMod p) = c),
+        C (resP A Pl r) * ∏ s ∈ Pl.erase r, (X - C (s : ℚ)) =
+          PiC * (C (resP A Pl r) * ∏ s ∈ Plo.erase r, (X - C (s : ℚ))) := by
+      intro r hr
+      rw [prod_erase_split Pl (fun s : ℤ => (s : ZMod p) = c) r]
+      have hrc : r ∉ Plc := by
+        rw [hPlc, plc, Finset.mem_filter]; rw [Finset.mem_filter] at hr;
+        tauto
+      rw [show (Pl.filter fun s : ℤ => (s : ZMod p) = c).erase r = Plc by
+          rw [hPlc, plc];
+          exact Finset.erase_eq_of_notMem hrc]
+      rw [hPiC]; ring
+    rw [Finset.sum_congr rfl h1, Finset.sum_congr rfl h2, ← Finset.mul_sum, ← Finset.mul_sum,
+      hPi] at hpf
+    have e1 : (Pl.filter fun r : ℤ => (r : ZMod p) = c) = Plc := by rw [hPlc, plc]
+    have e2 : (Pl.filter fun r : ℤ => ¬(r : ZMod p) = c) = Plo := by rw [hPlo]
+    rw [e1, e2] at hpf
+    conv_lhs => rw [hpf]
+    rw [hH, hI, hP];
+    ring
+      -- membership facts
+  have hmemc : ∀ r ∈ Plc, r ∈ Pl ∧ (r : ZMod p) = c := fun r hr => by
+    rw [hPlc, plc, Finset.mem_filter] at hr; exact hr
+  have hmemo : ∀ r ∈ Plo, r ∈ Pl ∧ ¬(r : ZMod p) = c := fun r hr => by
+    rw [hPlo, Finset.mem_filter] at hr; exact hr
+  have hdiv : ∀ r ∈ Plc, (p : ℤ) ∣ r - m := fun r hr =>
+    (zmod_eq_iff_dvd (p := p)).mp (by rw [(hmemc r hr).2, hc])
+  have hndiv : ∀ r ∈ Plo, ¬(p : ℤ) ∣ m - r := fun r hr hd =>
+    (hmemo r hr).2
+      (by rw [hc]; exact ((zmod_eq_iff_dvd (p := p)).mpr hd).symm)
+        -- Part A: `P(m) = H(m)/PiO(m) - ∑ res_r/(m - r)`
+  have hfac : ∀ r ∈ Plo, ((m : ℚ) - r) ≠ 0 := fun r hr h =>
+    hndiv r hr
+      (by
+        have : m - r = 0 := by exact_mod_cast h
+        rw [this]; exact dvd_zero _)
+  have hPiOm : PiO.eval (m : ℚ) = ∏ r ∈ Plo, ((m : ℚ) - r) := by rw [hPiO, eval_prod]; simp
+  have hPiOne : PiO.eval (m : ℚ) ≠ 0 := by rw [hPiOm]; exact Finset.prod_ne_zero_iff.mpr hfac
+  have hVGinv : ∀ r ∈ Plo, VG p ((m : ℚ) - r)⁻¹ 0 := fun r hr =>
+    by
+    refine (VG.inv (p := p) (r := 0) (hfac r hr) ?_).mono (by norm_num)
+    rw [show ((m : ℚ) - r) = ((m - r : ℤ) : ℚ) by push_cast; ring,
+      padicValRat_int_eq_zero (hndiv r hr)];
+    simp
+  have hVGinvPiO : VG p (PiO.eval (m : ℚ))⁻¹ 0 := by
+    refine (VG.inv (p := p) (r := 0) hPiOne ?_).mono (by norm_num)
+    rw [hPiOm, padicValRat_finset_prod _ _ hfac]
+    push_cast
+    rw [Finset.sum_eq_zero fun r hr => by
+        rw [show ((m : ℚ) - r) = ((m - r : ℤ) : ℚ) by push_cast; ring,
+          padicValRat_int_eq_zero (hndiv r hr)];
+        simp]
+  have hEo :
+    ∀ r ∈ Plo,
+      (∏ s ∈ Plo.erase r, (X - C (s : ℚ))).eval (m : ℚ) = PiO.eval (m : ℚ) * ((m : ℚ) - r)⁻¹ := by
+    intro r hr
+    have h := Finset.mul_prod_erase Plo (fun s : ℤ => X - C (s : ℚ)) hr
+    rw [← hPiO] at h
+    have h2 := congrArg (eval (m : ℚ)) h
+    rw [eval_mul, eval_sub, eval_X, eval_C] at h2
+    rw [← h2]; field_simp [hfac r hr]
+  have hPm :
+    P.eval (m : ℚ) =
+      H.eval (m : ℚ) * (PiO.eval (m : ℚ))⁻¹ - ∑ r ∈ Plo, resP A Pl r * ((m : ℚ) - r)⁻¹ := by
+    rw [hH, eval_add, eval_mul, eval_finsetSum]
+    simp only [eval_mul, eval_C]
+    rw [Finset.sum_congr rfl fun r hr => by rw [hEo r hr]]
+    rw [add_mul, Finset.sum_mul]
+    field_simp
+    ring_nf
+      -- Part B: the substitution `x = m + p z`
+  set G : ℚ[X] := C (m : ℚ) + C (p : ℚ) * X with hG
+  set e : ℕ := cnt p Z c with he
+  set ℓ : ℕ := Plc.card with hℓ
+  set ρ : ℤ → ℚ := fun r => (((r - m) / p : ℤ) : ℚ) with hρ
+  set T : ℚ[X] := ∏ r ∈ Plc, (X - C (ρ r)) with hT
+  have hp0 : (p : ℚ) ≠ 0 := by exact_mod_cast hp.out.ne_zero
+  have hPiCcomp : PiC.comp G = C ((p : ℚ) ^ ℓ) * T := by
+    rw [hPiC, Polynomial.prod_comp, Finset.prod_congr rfl fun r hr => lin_comp_eq (hdiv r hr),
+      Finset.prod_mul_distrib, Finset.prod_const, ← C_pow]
+  have hAcomp : A.comp G = PiC.comp G * H.comp G + PiO.comp G * I.comp G := by
+    conv_lhs => rw [hAHI]
+    rw [add_comp, mul_comp, mul_comp]
+  have hGA : GV p (A.comp G) e := by rw [hA, he, hc]; exact GV_numOf_comp κ hκ Z m
+  have hGPiO : GV p (PiO.comp G) 0 := by
+    rw [hPiO, Polynomial.prod_comp]
+    exact (GV.prod (p := p) Plo (r := fun _ => (0 : ℚ)) fun r _ => GV_lin_comp m r).mono (by simp)
+  have hGI : GV p (I.comp G) e := by
+    rw [hI, Polynomial.sum_comp]
+    refine GV.sum _ fun r hr => ?_
+    rw [mul_comp, C_comp, Polynomial.prod_comp]
+    have hprod : GV p (∏ s ∈ Plc.erase r, (X - C (s : ℚ)).comp G) ((Plc.erase r).card) := by
+      exact
+        (GV.prod (p := p) (Plc.erase r) (r := fun _ => (1 : ℚ)) fun s hs =>
+              GV_lin_comp_class m s (by rw [(hmemc s (Finset.mem_of_mem_erase hs)).2, hc])).mono
+          (by simp)
+    have hres := VG_res κ hκ Z Pl hsep (hmemc r hr).1
+    have hrc : (r : ZMod p) = c := (hmemc r hr).2
+    rw [hrc, ← hPlc, ← he] at hres
+    have := GV.C_mul hres hprod
+    refine this.mono (le_of_eq ?_)
+    rw [Finset.card_erase_of_mem hr]
+    have : 1 ≤ Plc.card := Finset.card_pos.mpr ⟨r, hr⟩
+    push_cast [this]; ring
+  have hRHS : GV p (A.comp G - PiO.comp G * I.comp G) e :=
+    hGA.sub ((hGPiO.mul hGI).mono (by simp))
+  set F : ℚ[X] := C ((p : ℚ) ^ (-(e : ℤ))) * (A.comp G - PiO.comp G * I.comp G) with hF
+  have hFint : GV p F 0 := by
+    have := GV.C_mul (VG.primePow (p := p) (-(e : ℤ))) hRHS
+    refine this.mono (le_of_eq ?_); push_cast; ring
+  have hFeq : F = T * (C ((p : ℚ) ^ ((ℓ : ℤ) - e)) * H.comp G) := by
+    rw [hF, show A.comp G - PiO.comp G * I.comp G = PiC.comp G * H.comp G by rw [hAcomp]; ring,
+      hPiCcomp]
+    have hz : (p : ℚ) ^ (-(e : ℤ)) * (p : ℚ) ^ ℓ = (p : ℚ) ^ ((ℓ : ℤ) - e) := by
+      rw [← zpow_natCast, ← zpow_add₀ hp0]; ring_nf
+    rw [← hz, C_mul]; ring
+  have hroot : ∀ x ∈ Plc.image ρ, F.eval x = 0 := by
+    intro x hx
+    rw [Finset.mem_image] at hx
+    obtain ⟨r, hr, rfl⟩ := hx
+    rw [hFeq, eval_mul, hT, eval_prod]
+    rw [Finset.prod_eq_zero hr (by simp), zero_mul]
+  have hρint : ∀ x ∈ Plc.image ρ, VG p x 0 := by
+    intro x hx
+    rw [Finset.mem_image] at hx
+    obtain ⟨r, _, rfl⟩ := hx
+    exact VG.intCast _
+  obtain ⟨S, hS, hFS⟩ := factor_roots (Plc.image ρ) hρint hFint hroot
+  have hinj : Set.InjOn ρ Plc := by
+    intro r hr s hs h
+    have h1 : ((r - m) / p : ℤ) = (s - m) / p := by
+      have h' : ((((r - m) / p : ℤ) : ℚ)) = (((s - m) / p : ℤ) : ℚ) := h
+      exact_mod_cast h'
+    have h2 := Int.ediv_mul_cancel (hdiv r hr)
+    have h3 := Int.ediv_mul_cancel (hdiv s hs)
+    have : r - m = s - m := by rw [← h2, ← h3, h1]
+    omega
+  rw [Finset.prod_image hinj, ← hT] at hFS
+  have hT0 : T ≠ 0 := (monic_prod_of_monic _ _ fun r _ => monic_X_sub_C _).ne_zero
+  have hcancel : C ((p : ℚ) ^ ((ℓ : ℤ) - e)) * H.comp G = S :=
+    mul_left_cancel₀ hT0 (hFeq.symm.trans hFS)
+  have hGm : G.eval 0 = (m : ℚ) := by rw [hG]; simp
+  have hHm : H.eval (m : ℚ) = (p : ℚ) ^ ((e : ℤ) - ℓ) * S.eval 0 := by
+    have := congrArg (eval 0) hcancel
+    rw [eval_mul, eval_C, eval_comp, hGm] at this
+    rw [← this, ← mul_assoc, ← zpow_add₀ hp0]; ring_nf; simp
+  have hVGH : VG p (H.eval (m : ℚ)) ((e : ℚ) - ℓ) := by
+    rw [hHm]
+    have := (VG.primePow (p := p) ((e : ℤ) - ℓ)).mul (VG.eval_zero hS)
+    simpa using
+      this
+        -- Part C: combine
+  rw [hPm]
+  have hβc : β ≤ (e : ℚ) - ℓ := hβ c
+  refine VG.sub ((hVGH.mul hVGinvPiO).mono (by linarith)) (VG.sum _ fun r hr => ?_)
+  have hres := VG_res κ hκ Z Pl hsep (hmemo r hr).1
+  have := hres.mul (hVGinv r hr)
+  refine this.mono ?_
+  have := hβ (r : ZMod p)
+  linarith
+
+lemma natDegree_numOf_le (κ : ℚ) (Z : Multiset ℤ) : (numOf κ Z).natDegree ≤ Z.card := by
+  unfold numOf
+  refine (natDegree_C_mul_le _ _).trans ?_
+  refine (natDegree_multiset_prod_le (t := Z.map fun ζ : ℤ => X - C (ζ : ℚ))).trans ?_
+  rw [Multiset.map_map]
+  have : ∀ ζ ∈ Z, ((natDegree ∘ fun ζ : ℤ => X - C (ζ : ℚ)) ζ) = 1 := fun ζ _ =>
+    natDegree_X_sub_C ((ζ : ℤ) : ℚ)
+  rw [Multiset.map_congr rfl this, Multiset.map_const', Multiset.sum_replicate, smul_eq_mul,
+    mul_one]
+
+lemma padicValNat_24 (hp5 : 5 ≤ p) : padicValNat p 24 = 0 := by
+  apply padicValNat.eq_zero_of_not_dvd
+  intro h
+  have h1 : p ≤ 24 := Nat.le_of_dvd (by norm_num) h
+  have hpr := hp.out
+  interval_cases p <;>
+    first
+    | omega
+    | (norm_num at hpr)
+
+
+/-- **The direct bound** `v_p^G(τ_X(g)) ≥ min_c (e_c - ℓ_c) - 4`. -/
+theorem tauX_direct (hp5 : 5 ≤ p) (κ : ℚ) (hκ : VG p κ 0) (Z : Multiset ℤ) (Pl : Finset ℤ)
+    (hsep : Sep p Pl) (hH : ∀ r ∈ Pl, dd r < p ^ 2) (hdeg : Z.card - Pl.card + 1 < p ^ 2) (β : ℚ)
+    (hβ : ∀ c : ZMod p, β ≤ (cnt p Z c : ℚ) - (plc p Pl c).card) :
+    GV p (tauX (numOf κ Z) Pl) (β - 4) := by
+  unfold tauX
+  refine GV.add (GV.C ?_) (GV.sum _ fun r hr => ?_)
+  · have hd : (polyPart (numOf κ Z) Pl).natDegree ≤ Z.card - Pl.card :=
+      (natDegree_polyPart_le _ _).trans (Nat.sub_le_sub_right (natDegree_numOf_le κ Z) _)
+    have h :=
+      tau_bound (p := p) hd (β := β)
+        (fun m _ => by
+          have := VG_polyPart_eval κ hκ Z Pl hsep β hβ (m : ℤ)
+          simpa using this)
+    have hlog : Nat.log p (Z.card - Pl.card + 1) ≤ 1 :=
+      Nat.lt_succ_iff.mp (Nat.log_lt_of_lt_pow (by omega) hdeg)
+    rw [padicValNat_24 hp5] at h
+    refine h.mono ?_
+    have : (Nat.log p (Z.card - Pl.card + 1) : ℚ) ≤ 1 := by exact_mod_cast hlog
+    push_cast; linarith
+  · have hres := VG_res κ hκ Z Pl hsep hr
+    have hH5 : GV p (C (H5 (dd r)) - X) (-5) :=
+      (GV.C (VG_H5 (hH r hr))).sub (GV.X.mono (by norm_num))
+    refine (GV.C_mul hres hH5).mono ?_
+    have := hβ (r : ZMod p)
+    linarith
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Arith/InnerAlloc.lean b/LeanPool/Zeta5Irrational/Arith/InnerAlloc.lean
new file mode 100644
index 0000000000..2f6824cd76
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Arith/InnerAlloc.lean
@@ -0,0 +1,351 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Arith.InnerWeights
+import Mathlib.Data.Rat.Star
+import Mathlib.Tactic.ENatToNat
+import Mathlib.Tactic.ReduceModChar
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.Positivity.Basic
+import Mathlib.Tactic.Ring.Basic
+
+/-! # The inner-range allocation (water filling)
+
+Classes `c ≠ 0` with integer offsets `β c` (the row weights of class `c` are `i + β c / 2`).
+At a level `k` (weights `< k/2`), class `c` receives `lrow k (β c) = ⌈(k - β c)/2⌉⁺` rows.
+We take the largest feasible level `kf ∈ [klo, ktop]` (at most `h - L0min` rows used) and put the
+remaining rows into the zero class. Then
+
+* the allocation is valid for `inner_bound_cap` (the zero-class condition),
+* `∑ wcap ≥ Φ(kf) - B₀`, with `Φ(k) = h k/2 - ∑_c ψ(k, β c)` concave in `k`,
+* `Φ(kf) ≥ Φ(k) - (k - klo) L0min/2` for every `k ∈ [klo, ktop]`.
+-/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+/-- The number of rows of offset `β` below the level `k`: `⌈(k - β)/2⌉⁺`. -/
+def lrow (k β : ℤ) : ℕ :=
+  ((k - β + 1) / 2).toNat
+
+lemma lrow_ge (k β : ℤ) : ((k : ℚ) - β) / 2 ≤ lrow k β := by
+  unfold lrow
+  have h1 : (k - β + 1) / 2 ≤ ((k - β + 1) / 2).toNat := Int.self_le_toNat _
+  have h2 : k - β ≤ 2 * ((k - β + 1) / 2) := by omega
+  have h3 : ((k - β : ℤ) : ℚ) ≤ 2 * (((k - β + 1) / 2).toNat : ℤ) := by
+    exact_mod_cast h2.trans (by omega)
+  push_cast at h3
+  linarith
+
+lemma lrow_lt {k β : ℤ} {i : ℕ} (hi : i < lrow k β) : 2 * (i : ℚ) + β < k := by
+  unfold lrow at hi
+  have : 2 * (i : ℤ) + β < k := by omega
+  exact_mod_cast this
+
+lemma lrow_le_max (k β : ℤ) : (lrow k β : ℚ) + β / 2 ≤ max (β : ℚ) (k + 1) / 2 := by
+  unfold lrow
+  rcases le_or_gt (k - β + 1) 0 with h | h
+  · have : ((k - β + 1) / 2).toNat = 0 := by omega
+    rw [this]; push_cast
+    have := le_max_left (β : ℚ) (k + 1); linarith
+  · have e : (((k - β + 1) / 2).toNat : ℤ) ≤ (k - β + 1) / 2 := by omega
+    have e2 : 2 * ((k - β + 1) / 2) ≤ k - β + 1 := by omega
+    have e3 : ((((k - β + 1) / 2).toNat : ℤ) : ℚ) * 2 ≤ ((k - β + 1 : ℤ) : ℚ) := by
+      exact_mod_cast (by omega : (((k - β + 1) / 2).toNat : ℤ) * 2 ≤ k - β + 1)
+    push_cast at e3
+    have := le_max_right (β : ℚ) (k + 1); linarith
+
+lemma lrow_mono {k k' : ℤ} (h : k ≤ k') (β : ℤ) : lrow k β ≤ lrow k' β := by unfold lrow; omega
+
+lemma lrow_succ (k β : ℤ) :
+    lrow (k + 1) β = lrow k β ∨ (lrow (k + 1) β = lrow k β + 1 ∧ 2 * (lrow k β : ℤ) + β = k) := by
+  unfold lrow; omega
+
+/-- `ψ(k, β) = ∑_{i < lrow} (k/2 - i - β/2)`. -/
+noncomputable def psiR (k β : ℤ) : ℚ :=
+  ∑ i ∈ range (lrow k β), (((k : ℚ) - 2 * i - β) / 2)
+
+lemma psiR_succ (k β : ℤ) : psiR (k + 1) β = psiR k β + (lrow (k + 1) β : ℚ) / 2 := by
+  unfold psiR
+  rcases lrow_succ k β with h | ⟨h, h2⟩
+  · rw [h, show (lrow k β : ℚ) / 2 = ∑ _i ∈ range (lrow k β), (1 / 2 : ℚ) by simp; ring, ←
+      Finset.sum_add_distrib]
+    refine Finset.sum_congr rfl fun i _ => ?_; push_cast; ring
+  · rw [h, Finset.sum_range_succ]
+    have h2' : 2 * ((lrow k β : ℕ) : ℚ) + β = k := by exact_mod_cast h2
+    rw [show ((lrow k β + 1 : ℕ) : ℚ) / 2 = ∑ _i ∈ range (lrow k β), (1 / 2 : ℚ) + 1 / 2 by
+        simp;
+        ring]
+    have :
+      ∑ i ∈ range (lrow k β), (((k + 1 : ℤ) : ℚ) - 2 * i - β) / 2 =
+        ∑ i ∈ range (lrow k β), ((((k : ℚ) - 2 * i - β) / 2) + 1 / 2) :=
+      by refine Finset.sum_congr rfl fun i _ => ?_; push_cast; ring
+    rw [this, Finset.sum_add_distrib]
+    push_cast
+    linarith
+
+variable {m : ℕ}
+
+/-- Rows used at level `k`. -/
+def rowsK (β : Fin (m + 1) → ℤ) (k : ℤ) : ℕ :=
+  ∑ c ∈ univ.erase (0 : Fin (m + 1)), lrow k (β c)
+
+/-- `Φ(k) = h k / 2 - ∑ ψ`. -/
+noncomputable def PhiK (β : Fin (m + 1) → ℤ) (h : ℕ) (k : ℤ) : ℚ :=
+  (h : ℚ) * k / 2 - ∑ c ∈ univ.erase (0 : Fin (m + 1)), psiR k (β c)
+
+lemma PhiK_succ (β : Fin (m + 1) → ℤ) (h : ℕ) (k : ℤ) :
+    PhiK β h (k + 1) = PhiK β h k + ((h : ℚ) - rowsK β (k + 1)) / 2 := by
+  unfold PhiK rowsK
+  simp only [psiR_succ, Finset.sum_add_distrib, ← Finset.sum_div]
+  push_cast
+  ring
+
+lemma rowsK_mono (β : Fin (m + 1) → ℤ) {k k' : ℤ} (h : k ≤ k') : rowsK β k ≤ rowsK β k' :=
+  Finset.sum_le_sum fun _c _ => lrow_mono h _
+
+/-- The chosen level. -/
+noncomputable def kfSel (β : Fin (m + 1) → ℤ) (h L0min : ℕ) (klo ktop : ℤ) : ℤ :=
+  klo + (Nat.findGreatest (fun j : ℕ => rowsK β (klo + j) + L0min ≤ h) (ktop - klo).toNat : ℤ)
+
+section
+
+variable (β : Fin (m + 1) → ℤ) (h L0min : ℕ) (klo ktop : ℤ)
+
+lemma kfSel_feasible (hlo : rowsK β klo + L0min ≤ h) :
+    rowsK β (kfSel β h L0min klo ktop) + L0min ≤ h := by
+  unfold kfSel
+  have :=
+    Nat.findGreatest_spec (P := fun j : ℕ => rowsK β (klo + j) + L0min ≤ h) (n :=
+      (ktop - klo).toNat) (Nat.zero_le _) (by simpa using hlo)
+  exact this
+
+lemma kfSel_ge : klo ≤ kfSel β h L0min klo ktop := by unfold kfSel; omega
+
+lemma kfSel_le (hk : klo ≤ ktop) : kfSel β h L0min klo ktop ≤ ktop := by
+  unfold kfSel
+  have :=
+    Nat.findGreatest_le (P := fun j : ℕ => rowsK β (klo + j) + L0min ≤ h) (n :=
+      (ktop - klo).toNat)
+  omega
+
+lemma kfSel_max {k : ℤ} (hk1 : kfSel β h L0min klo ktop < k) (hk2 : k ≤ ktop) :
+    h < rowsK β k + L0min := by
+  unfold kfSel at hk1
+  have hj :=
+    Nat.findGreatest_is_greatest (P := fun j : ℕ => rowsK β (klo + j) + L0min ≤ h) (n :=
+      (ktop - klo).toNat) (k := (k - klo).toNat) (by omega) (by omega)
+  simp only [not_le] at hj
+  rwa [show klo + ((k - klo).toNat : ℤ) = k by omega] at hj
+
+/-- **Concavity comparison**: `Φ(k) ≤ Φ(kf) + (k - klo) L0min / 2` on `[klo, ktop]`. -/
+theorem PhiK_le_kf (hlo : rowsK β klo + L0min ≤ h) {k : ℤ} (hk1 : klo ≤ k) (hk2 : k ≤ ktop) :
+    PhiK β h k ≤ PhiK β h (kfSel β h L0min klo ktop) + ((k - klo : ℤ) : ℚ) * L0min / 2 := by
+  have hfeas := kfSel_feasible β h L0min klo ktop hlo
+  have hge := kfSel_ge β h L0min klo ktop
+  have hmax : ∀ k', kfSel β h L0min klo ktop < k' → k' ≤ ktop → h < rowsK β k' + L0min :=
+    fun k' h1 h2 => kfSel_max β h L0min klo ktop h1 h2
+  generalize kfSel β h L0min klo ktop = kf at *
+  have hL0 : (0 : ℚ) ≤ L0min := by positivity
+  rcases le_or_gt k kf with hle | hgt
+  · -- below `kf` the steps are nonnegative
+    obtain ⟨t, rfl⟩ : ∃ t : ℕ, kf = k + t := ⟨(kf - k).toNat, by omega⟩
+    have : ∀ s : ℕ, s ≤ t → PhiK β h k ≤ PhiK β h (k + s) := by
+      intro s
+      induction s with
+      | zero => intro _; simp
+      | succ s ih =>
+        intro hs
+        have := ih (by omega)
+        rw [show k + ((s + 1 : ℕ) : ℤ) = (k + s) + 1 by push_cast; ring, PhiK_succ]
+        have hr : rowsK β (k + s + 1) ≤ rowsK β (k + t) := rowsK_mono β (by omega)
+        have : ((rowsK β (k + s + 1) : ℕ) : ℚ) ≤ h := by
+          exact_mod_cast (by omega : rowsK β (k + s + 1) ≤ h)
+        linarith
+    have h1 := this t le_rfl
+    have : (0 : ℚ) ≤ ((k - klo : ℤ) : ℚ) := by exact_mod_cast (by omega : (0 : ℤ) ≤ k - klo)
+    nlinarith
+  · -- above `kf` each step is less than `L0min / 2`
+    obtain ⟨t, rfl⟩ : ∃ t : ℕ, k = kf + t := ⟨(k - kf).toNat, by omega⟩
+    have : ∀ s : ℕ, s ≤ t → PhiK β h (kf + s) ≤ PhiK β h kf + (s : ℚ) * L0min / 2 := by
+      intro s
+      induction s with
+      | zero => intro _; simp
+      | succ s ih =>
+        intro hs
+        have := ih (by omega)
+        rw [show kf + ((s + 1 : ℕ) : ℤ) = (kf + s) + 1 by push_cast; ring, PhiK_succ]
+        have hr := hmax (kf + s + 1) (by omega) (by omega)
+        have : (h : ℚ) < rowsK β (kf + s + 1) + L0min := by exact_mod_cast hr
+        push_cast
+        linarith
+    have h1 := this t le_rfl
+    have : (t : ℚ) ≤ ((kf + t - klo : ℤ) : ℚ) := by
+      have : ((t : ℤ) : ℚ) ≤ ((kf + t - klo : ℤ) : ℚ) := by exact_mod_cast (by omega)
+      simpa using this
+    nlinarith
+
+end
+
+/-- The allocation. -/
+noncomputable def allocK (β : Fin (m + 1) → ℤ) (h : ℕ) (kf : ℤ) (c : Fin (m + 1)) : ℕ :=
+  if c = 0 then h - rowsK β kf else lrow kf (β c)
+
+lemma sum_allocK (β : Fin (m + 1) → ℤ) (h : ℕ) (kf : ℤ) (hkf : rowsK β kf ≤ h) :
+    ∑ c, allocK β h kf c = h := by
+  rw [← Finset.add_sum_erase _ _ (Finset.mem_univ (0 : Fin (m + 1)))]
+  unfold allocK
+  rw [ite_eq_left rfl, Finset.sum_congr rfl (fun c hc => ite_eq_right (Finset.mem_erase.mp hc).1)]
+  unfold rowsK at hkf ⊢
+  omega
+
+/-- The zero-class deficit: `∑_{i<L} (D - 2i)⁺ ≤ D⁺ (D⁺/2 + 1)`. -/
+lemma sum_deficit_le (D : ℚ) (L : ℕ) :
+    ∑ i ∈ range L, max 0 (D - 2 * i) ≤ max 0 D * (max 0 D / 2 + 1) := by
+  set E := max 0 D with hE
+  have hE0 : 0 ≤ E := le_max_left _ _
+  set N0 := ⌊E / 2⌋₊ + 1 with hN0
+  have hz : ∀ i, N0 ≤ i → max 0 (D - 2 * i) = 0 := by
+    intro i hi
+    have : E / 2 < (N0 : ℚ) := by rw [hN0]; push_cast; exact Nat.lt_floor_add_one _
+    have : (N0 : ℚ) ≤ i := by exact_mod_cast hi
+    apply max_eq_left
+    have := le_max_right 0 D; linarith
+  have h1 : ∑ i ∈ range L, max 0 (D - 2 * i) ≤ ∑ i ∈ range N0, max 0 (D - 2 * i) := by
+    rw [← Finset.sum_range_add_sum_Ico _ (show min L N0 ≤ L from min_le_left _ _), ←
+      Finset.sum_range_add_sum_Ico _ (show min L N0 ≤ N0 from min_le_right _ _)]
+    have hA : ∑ i ∈ Ico (min L N0) L, max 0 (D - 2 * (i : ℚ)) = 0 := by
+      refine Finset.sum_eq_zero fun i hi => hz i ?_
+      have := Finset.mem_Ico.mp hi; omega
+    have hB : 0 ≤ ∑ i ∈ Ico (min L N0) N0, max 0 (D - 2 * (i : ℚ)) :=
+      Finset.sum_nonneg fun _ _ => le_max_left _ _
+    linarith
+  have h2 : ∑ i ∈ range N0, max 0 (D - 2 * (i : ℚ)) ≤ ∑ _i ∈ range N0, E :=
+    Finset.sum_le_sum fun i _ =>
+      max_le hE0
+        (by
+          have h1 := le_max_right (0 : ℚ) D
+          have h2 : (0 : ℚ) ≤ (i : ℚ) := by positivity
+          rw [hE]; linarith)
+  rw [Finset.sum_const, Finset.card_range, nsmul_eq_mul] at h2
+  have h3 : (N0 : ℚ) ≤ E / 2 + 1 := by
+    rw [hN0]; push_cast;
+    have := Nat.floor_le (show 0 ≤ E / 2 by positivity); linarith
+  nlinarith
+
+section
+
+variable (hm1 : 1 ≤ m) (b : ℕ → ℚ) (β : Fin (m + 1) → ℤ)
+  (hβ : ∀ c : Fin (m + 1), c ≠ 0 → b c = (β c : ℚ) + 4)
+
+include hβ in
+lemma tmin_allocK_ge (h : ℕ) (kf : ℤ) : (kf : ℚ) / 2 ≤ tmin hm1 (allocK β h kf) b := by
+  unfold tmin
+  refine Finset.le_inf' _ _ fun c hc => ?_
+  have hc0 := (Finset.mem_erase.mp hc).1
+  unfold topw allocK
+  rw [ite_eq_right hc0, hβ c hc0]
+  have := lrow_ge kf (β c)
+  linarith
+
+include hβ in
+lemma tmin_allocK_le (h : ℕ) (kf : ℤ) :
+    tmin hm1 (allocK β h kf) b ≤ max (β ⟨1, by omega⟩ : ℚ) (kf + 1) / 2 := by
+  have hc : (⟨1, by omega⟩ : Fin (m + 1)) ≠ 0 := by simp [Fin.ext_iff]
+  refine (tmin_le hm1 _ b hc).trans ?_
+  unfold topw allocK
+  rw [ite_eq_right hc, hβ _ hc]
+  have := lrow_le_max kf (β ⟨1, by omega⟩)
+  linarith
+
+include hβ in
+/-- **The weight sum of the allocation**: `∑ wcap ≥ Φ(kf) - B₀`. -/
+theorem sum_wcap_allocK (h : ℕ) (kf : ℤ) (hkf : rowsK β kf ≤ h) :
+    PhiK β h kf -
+        max 0 ((kf : ℚ) / 2 - (b 0 - 4) / 2) * (max 0 ((kf : ℚ) / 2 - (b 0 - 4) / 2) / 2 + 1) ≤
+      ∑ s, wcap hm1 (allocK β h kf) b s := by
+  set L := allocK β h kf with hL
+  set T := tmin hm1 L b with hT
+  have hTge : (kf : ℚ) / 2 ≤ T := tmin_allocK_ge hm1 b β hβ h kf
+  rw [Fintype.sum_sigma]
+  rw [← Finset.add_sum_erase _ _ (Finset.mem_univ (0 : Fin (m + 1)))]
+    -- nonzero classes
+  have hnz :
+    ∀ c ∈ univ.erase (0 : Fin (m + 1)),
+      ∑ i : Fin (L c), wcap hm1 L b ⟨c, i⟩ =
+        ∑ i ∈ range (lrow kf (β c)), ((i : ℚ) + (β c : ℚ) / 2) := by
+    intro c hc
+    have hc0 := (Finset.mem_erase.mp hc).1
+    have hLc : L c = lrow kf (β c) := by rw [hL]; unfold allocK; rw [ite_eq_right hc0]
+    have hpt : ∀ i : Fin (L c), wcap hm1 L b ⟨c, i⟩ = ((i : ℕ) : ℚ) + (β c : ℚ) / 2 := by
+      intro i
+      unfold wcap wraw
+      simp only
+      rw [ite_eq_right hc0, hβ c hc0, show ((β c : ℚ) + 4 - 4) / 2 = (β c : ℚ) / 2 by ring]
+      apply min_eq_left
+      have hi : (i : ℕ) < lrow kf (β c) := hLc ▸ i.isLt
+      have := lrow_lt hi
+      linarith [hTge, hT]
+    rw [Finset.sum_congr rfl (fun i _ => hpt i),
+      Fin.sum_univ_eq_sum_range (fun i => (i : ℚ) + (β c : ℚ) / 2) (L c), hLc]
+      -- the zero class
+  have hz :
+    ((L 0 : ℕ) : ℚ) * kf / 2 -
+        max 0 ((kf : ℚ) / 2 - (b 0 - 4) / 2) * (max 0 ((kf : ℚ) / 2 - (b 0 - 4) / 2) / 2 + 1) ≤
+      ∑ i : Fin (L 0), wcap hm1 L b ⟨0, i⟩ := by
+    have hpt :
+      ∀ i : Fin (L 0),
+        (kf : ℚ) / 2 - max 0 ((kf : ℚ) / 2 - (b 0 - 4) / 2 - 2 * (i : ℕ)) ≤ wcap hm1 L b ⟨0, i⟩ :=
+  by
+      intro i
+      unfold wcap wraw
+      simp only [ite_true]
+      refine le_min ?_ ?_
+      · have := le_max_right (0 : ℚ) ((kf : ℚ) / 2 - (b 0 - 4) / 2 - 2 * (i : ℕ)); linarith
+      · have := le_max_left (0 : ℚ) ((kf : ℚ) / 2 - (b 0 - 4) / 2 - 2 * (i : ℕ));
+        linarith [hTge, hT]
+    refine le_trans ?_ (Finset.sum_le_sum fun i _ => hpt i)
+    rw [Fin.sum_univ_eq_sum_range
+        (fun i => (kf : ℚ) / 2 - max 0 ((kf : ℚ) / 2 - (b 0 - 4) / 2 - 2 * (i : ℚ))) (L 0),
+      Finset.sum_sub_distrib, Finset.sum_const, Finset.card_range, nsmul_eq_mul]
+    have := sum_deficit_le ((kf : ℚ) / 2 - (b 0 - 4) / 2) (L 0)
+    linarith
+  rw [Finset.sum_congr rfl hnz]
+  have hL0 : L 0 = h - rowsK β kf := by rw [hL]; unfold allocK; rw [ite_eq_left rfl]
+  have hL0q : ((L 0 : ℕ) : ℚ) = (h : ℚ) - rowsK β kf := by rw [hL0]; push_cast [hkf]; ring
+  have hpsi :
+    ∀ c ∈ univ.erase (0 : Fin (m + 1)),
+      psiR kf (β c) =
+        (lrow kf (β c) : ℚ) * kf / 2 - ∑ i ∈ range (lrow kf (β c)), ((i : ℚ) + (β c : ℚ) / 2) := by
+    intro c _
+    unfold psiR
+    rw [show (lrow kf (β c) : ℚ) * kf / 2 = ∑ _i ∈ range (lrow kf (β c)), (kf : ℚ) / 2 by
+        rw [Finset.sum_const, Finset.card_range, nsmul_eq_mul]; ring,
+      ← Finset.sum_sub_distrib]
+    refine Finset.sum_congr rfl fun i _ => ?_; ring
+  have hΦ :
+    PhiK β h kf =
+      (h : ℚ) * kf / 2 - ((rowsK β kf : ℕ) : ℚ) * kf / 2 +
+        ∑ c ∈ univ.erase (0 : Fin (m + 1)),
+          ∑ i ∈ range (lrow kf (β c)), ((i : ℚ) + (β c : ℚ) / 2) := by
+    unfold PhiK
+    rw [Finset.sum_congr rfl hpsi, Finset.sum_sub_distrib]
+    unfold rowsK
+    push_cast
+    rw [Finset.sum_mul, Finset.sum_div]
+    ring
+  rw [hΦ]
+  rw [hL0q] at hz
+  linarith
+
+end
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Arith/InnerWeights.lean b/LeanPool/Zeta5Irrational/Arith/InnerWeights.lean
new file mode 100644
index 0000000000..8ce315a0ae
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Arith/InnerWeights.lean
@@ -0,0 +1,216 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Arith.ClassCount
+import Mathlib.Data.Rat.Star
+import Mathlib.Tactic.ENatToNat
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.Ring.Basic
+
+/-! # The inner-range weights (4.6)–(4.7) and their validity
+
+With class constants `b a = Bse p K N a`, dimensions `L`, and `top a = L a + (b a - 4)/2`:
+* `w (a, i) = i + (b a - 4)/2` for `a ≠ 0`;
+* `w (0, i) = min (2 i + (b 0 - 4)/2) (min_{c ≠ 0} top c)`.
+If `top a - 1 ≤ top c` for all `a, c ≠ 0` (P) and `top a - 1 ≤ 2 L 0 + (b 0 - 4)/2` for `a ≠ 0`
+(Z), then `v_p^G(Δ) ≥ 2 ∑ w`.
+-/
+
+@[expose] public section
+
+open Finset Polynomial
+
+namespace Zeta5Irrational
+
+variable {m : ℕ}
+
+/-- `top a = L a + (b a - 4)/2`. -/
+noncomputable def topw (L : Fin (m + 1) → ℕ) (b : ℕ → ℚ) (a : Fin (m + 1)) : ℚ :=
+  L a + (b a - 4) / 2
+
+/-- The minimum of `top` over the ordinary classes. -/
+noncomputable def tmin (hm : 1 ≤ m) (L : Fin (m + 1) → ℕ) (b : ℕ → ℚ) : ℚ :=
+  (Finset.univ.erase (0 : Fin (m + 1))).inf' ⟨⟨1, by omega⟩, by simp [Fin.ext_iff]⟩ (topw L b)
+
+lemma tmin_le (hm : 1 ≤ m) (L : Fin (m + 1) → ℕ) (b : ℕ → ℚ) {c : Fin (m + 1)} (hc : c ≠ 0) :
+    tmin hm L b ≤ topw L b c :=
+  Finset.inf'_le _ (Finset.mem_erase.mpr ⟨hc, Finset.mem_univ _⟩)
+
+/-- The weights. -/
+noncomputable def winner (hm : 1 ≤ m) (L : Fin (m + 1) → ℕ) (b : ℕ → ℚ)
+    (s : Σ a : Fin (m + 1), Fin (L a)) : ℚ :=
+  if s.1 = 0 then min (2 * (s.2 : ℚ) + (b 0 - 4) / 2) (tmin hm L b)
+  else (s.2 : ℚ) + (b s.1 - 4) / 2
+
+/-- The abstract class inequality. -/
+theorem winner_ACI (hm : 1 ≤ m) (L : Fin (m + 1) → ℕ) (b : ℕ → ℚ)
+    (hP : ∀ a c : Fin (m + 1), a ≠ 0 → c ≠ 0 → topw L b a - 1 ≤ topw L b c)
+    (hZ : ∀ a : Fin (m + 1), a ≠ 0 → topw L b a - 1 ≤ 2 * L 0 + (b 0 - 4) / 2)
+    (s t : Σ a : Fin (m + 1), Fin (L a)) (a : Fin (m + 1)) :
+    winner hm L b s + winner hm L b t + 4 ≤
+      b a + (if a = 0 then 2 else 1) * ((nu L s a : ℚ) + nu L t a) := by
+  -- basic facts
+  have F1 : ∀ u : Σ a : Fin (m + 1), Fin (L a), u.1 ≠ 0 → winner hm L b u ≤ topw L b u.1 - 1 := by
+    intro u hu
+    unfold winner topw; rw [ite_eq_right hu]
+    have : (u.2 : ℚ) ≤ L u.1 - 1 := by
+      have := u.2.isLt; have : (u.2 : ℕ) + 1 ≤ L u.1 := this
+      have : ((u.2 : ℕ) : ℚ) + 1 ≤ L u.1 := by exact_mod_cast this
+      linarith
+    linarith
+  have F3 :
+    ∀ u : Σ a : Fin (m + 1), Fin (L a), ∀ c : Fin (m + 1), c ≠ 0 → winner hm L b u ≤ topw L b c :=
+  by
+    intro u c hc
+    by_cases hu : u.1 = 0
+    · unfold winner; rw [ite_eq_left hu]
+      exact (min_le_right _ _).trans (tmin_le hm L b hc)
+    · linarith [F1 u hu, hP u.1 c hu hc]
+  have Fz :
+    ∀ u : Σ a : Fin (m + 1), Fin (L a),
+      winner hm L b u ≤ (if u.1 = 0 then 2 * (u.2 : ℚ) else 2 * (L 0 : ℚ)) + (b 0 - 4) / 2 := by
+    intro u
+    by_cases hu : u.1 = 0
+    · unfold winner; rw [ite_eq_left hu, ite_eq_left hu]; exact min_le_left _ _
+    · rw [ite_eq_right hu]; linarith [F1 u hu, hZ u.1 hu]
+  unfold nu
+  by_cases ha : a = 0
+  · subst ha
+    rw [ite_eq_left rfl, show (((0 : Fin (m + 1)) : ℕ)) = 0 from rfl]
+    have h1 := Fz s; have h2 := Fz t
+    by_cases hs : s.1 = 0 <;> by_cases ht : t.1 = 0
+    · rw [ite_eq_left hs.symm, ite_eq_left ht.symm]; rw [ite_eq_left hs] at h1;
+      rw [ite_eq_left ht] at h2; linarith
+    · rw [ite_eq_left hs.symm, ite_eq_right (Ne.symm ht)]; rw [ite_eq_left hs] at h1;
+      rw [ite_eq_right ht] at h2
+      linarith
+    · rw [ite_eq_right (Ne.symm hs), ite_eq_left ht.symm]; rw [ite_eq_right hs] at h1;
+      rw [ite_eq_left ht] at h2
+      linarith
+    · rw [ite_eq_right (Ne.symm hs), ite_eq_right (Ne.symm ht)]; rw [ite_eq_right hs] at h1;
+      rw [ite_eq_right ht] at h2
+      linarith
+  · rw [ite_eq_right ha, one_mul]
+    have htop : b a - 4 = 2 * (topw L b a - L a) := by unfold topw; ring
+    by_cases hs : a = s.1 <;> by_cases ht : a = t.1
+    · rw [ite_eq_left hs, ite_eq_left ht]
+      have e1 : winner hm L b s = (s.2 : ℚ) + (b a - 4) / 2 := by
+        unfold winner;
+        rw [ite_eq_right (hs ▸ ha)]; simp only [hs]
+      have e2 : winner hm L b t = (t.2 : ℚ) + (b a - 4) / 2 := by
+        unfold winner;
+        rw [ite_eq_right (ht ▸ ha)]; simp only [ht]
+      rw [e1, e2]; linarith
+    · rw [ite_eq_left hs, ite_eq_right ht]
+      have e1 : winner hm L b s = (s.2 : ℚ) + (b a - 4) / 2 := by
+        unfold winner;
+        rw [ite_eq_right (hs ▸ ha)]; simp only [hs]
+      have := F3 t a ha
+      rw [e1]; unfold topw at this; linarith
+    · rw [ite_eq_right hs, ite_eq_left ht]
+      have e2 : winner hm L b t = (t.2 : ℚ) + (b a - 4) / 2 := by
+        unfold winner;
+        rw [ite_eq_right (ht ▸ ha)]; simp only [ht]
+      have := F3 s a ha
+      rw [e2]; unfold topw at this; linarith
+    · rw [ite_eq_right hs, ite_eq_right ht]
+      have h1 := F3 s a ha; have h2 := F3 t a ha
+      unfold topw at h1 h2; linarith
+
+/-- **Inner-range determinant bound** for an allocation satisfying (P) and (Z). -/
+theorem inner_bound {p : ℕ} [Fact p.Prime] (hp5 : 5 ≤ p) (hm : 2 * m + 1 = p) (hm1 : 1 ≤ m)
+    (n : ℕ) (L : Fin (m + 1) → ℕ) (hL : ∑ c, L c = 37 * n) (hsep : Sep p (PlK (40 * n)))
+    (hK : 40 * n < p ^ 2) (hdeg : 5 + 2 * (18 * n + 2 * (37 * n)) - 2 * (40 * n) + 1 < p ^ 2)
+    (hP :
+      ∀ a c : Fin (m + 1),
+        a ≠ 0 → c ≠ 0 → topw L (Bse p (40 * n) (3 * n)) a - 1 ≤ topw L (Bse p (40 * n) (3 * n)) c)
+    (hZ :
+      ∀ a : Fin (m + 1),
+        a ≠ 0 →
+          topw L (Bse p (40 * n) (3 * n)) a - 1 ≤ 2 * L 0 + (Bse p (40 * n) (3 * n) 0 - 4) / 2) :
+    GV p (Δ n) (2 * ∑ s, winner hm1 L (Bse p (40 * n) (3 * n)) s) := by
+  refine class_frame hp5 hm n L hL _ hsep hK hdeg fun s t c => ?_
+  rw [classCount_eq hm]
+  exact winner_ACI hm1 L _ hP hZ s t (ccls hm c)
+
+/-! ### Capped weights: validity needs only the zero-class condition -/
+
+/-- Uncapped weights. -/
+noncomputable def wraw (L : Fin (m + 1) → ℕ) (b : ℕ → ℚ) (s : Σ a : Fin (m + 1), Fin (L a)) : ℚ :=
+  if s.1 = 0 then 2 * (s.2 : ℚ) + (b 0 - 4) / 2 else (s.2 : ℚ) + (b s.1 - 4) / 2
+
+/-- Capped weights `min (wraw) (min_{c ≠ 0} top c)`. -/
+noncomputable def wcap (hm : 1 ≤ m) (L : Fin (m + 1) → ℕ) (b : ℕ → ℚ)
+    (s : Σ a : Fin (m + 1), Fin (L a)) : ℚ :=
+  min (wraw L b s) (tmin hm L b)
+
+theorem wcap_ACI (hm : 1 ≤ m) (L : Fin (m + 1) → ℕ) (b : ℕ → ℚ)
+    (hZ : tmin hm L b ≤ 2 * L 0 + (b 0 - 4) / 2) (s t : Σ a : Fin (m + 1), Fin (L a))
+    (a : Fin (m + 1)) :
+    wcap hm L b s + wcap hm L b t + 4 ≤
+      b a + (if a = 0 then 2 else 1) * ((nu L s a : ℚ) + nu L t a) := by
+  have F3 :
+    ∀ u : Σ a : Fin (m + 1), Fin (L a), ∀ c : Fin (m + 1), c ≠ 0 → wcap hm L b u ≤ topw L b c :=
+    fun u c hc => (min_le_right _ _).trans (tmin_le hm L b hc)
+  have Fraw : ∀ u : Σ a : Fin (m + 1), Fin (L a), wcap hm L b u ≤ wraw L b u := fun u =>
+    min_le_left _ _
+  have Fz :
+    ∀ u : Σ a : Fin (m + 1), Fin (L a),
+      wcap hm L b u ≤ (if u.1 = 0 then 2 * (u.2 : ℚ) else 2 * (L 0 : ℚ)) + (b 0 - 4) / 2 := by
+    intro u
+    by_cases hu : u.1 = 0
+    · rw [ite_eq_left hu]; have := Fraw u; unfold wraw at this; rw [ite_eq_left hu] at this;
+      exact this
+    · rw [ite_eq_right hu]; exact (min_le_right _ _).trans hZ
+  unfold nu
+  by_cases ha : a = 0
+  · subst ha
+    rw [ite_eq_left rfl, show (((0 : Fin (m + 1)) : ℕ)) = 0 from rfl]
+    have h1 := Fz s; have h2 := Fz t
+    by_cases hs : s.1 = 0 <;> by_cases ht : t.1 = 0
+    · rw [ite_eq_left hs.symm, ite_eq_left ht.symm]; rw [ite_eq_left hs] at h1;
+      rw [ite_eq_left ht] at h2; linarith
+    · rw [ite_eq_left hs.symm, ite_eq_right (Ne.symm ht)]; rw [ite_eq_left hs] at h1;
+      rw [ite_eq_right ht] at h2
+      linarith
+    · rw [ite_eq_right (Ne.symm hs), ite_eq_left ht.symm]; rw [ite_eq_right hs] at h1;
+      rw [ite_eq_left ht] at h2
+      linarith
+    · rw [ite_eq_right (Ne.symm hs), ite_eq_right (Ne.symm ht)]; rw [ite_eq_right hs] at h1;
+      rw [ite_eq_right ht] at h2
+      linarith
+  · rw [ite_eq_right ha, one_mul]
+    have hraw :
+      ∀ u : Σ a : Fin (m + 1), Fin (L a), a = u.1 → wcap hm L b u ≤ (u.2 : ℚ) + (b a - 4) / 2 := by
+      intro u hu
+      have := Fraw u
+      unfold wraw at this; rw [ite_eq_right (hu ▸ ha)] at this; simpa only [hu] using this
+    by_cases hs : a = s.1 <;> by_cases ht : a = t.1
+    · rw [ite_eq_left hs, ite_eq_left ht]
+      have := hraw s hs; have := hraw t ht; linarith
+    · rw [ite_eq_left hs, ite_eq_right ht]
+      have := hraw s hs; have h2 := F3 t a ha
+      unfold topw at h2; linarith
+    · rw [ite_eq_right hs, ite_eq_left ht]
+      have := hraw t ht; have h2 := F3 s a ha
+      unfold topw at h2; linarith
+    · rw [ite_eq_right hs, ite_eq_right ht]
+      have h1 := F3 s a ha; have h2 := F3 t a ha
+      unfold topw at h1 h2; linarith
+
+/-- **Inner-range determinant bound** with capped weights; only the zero-class condition. -/
+theorem inner_bound_cap {p : ℕ} [Fact p.Prime] (hp5 : 5 ≤ p) (hm : 2 * m + 1 = p) (hm1 : 1 ≤ m)
+    (n : ℕ) (L : Fin (m + 1) → ℕ) (hL : ∑ c, L c = 37 * n) (hsep : Sep p (PlK (40 * n)))
+    (hK : 40 * n < p ^ 2) (hdeg : 5 + 2 * (18 * n + 2 * (37 * n)) - 2 * (40 * n) + 1 < p ^ 2)
+    (hZ : tmin hm1 L (Bse p (40 * n) (3 * n)) ≤ 2 * L 0 + (Bse p (40 * n) (3 * n) 0 - 4) / 2) :
+    GV p (Δ n) (2 * ∑ s, wcap hm1 L (Bse p (40 * n) (3 * n)) s) := by
+  refine class_frame hp5 hm n L hL _ hsep hK hdeg fun s t c => ?_
+  rw [classCount_eq hm]
+  exact wcap_ACI hm1 L _ hZ s t (ccls hm c)
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Arith/IntPoly.lean b/LeanPool/Zeta5Irrational/Arith/IntPoly.lean
new file mode 100644
index 0000000000..8b2ecef0a2
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Arith/IntPoly.lean
@@ -0,0 +1,89 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Arith.Val
+public import Mathlib.Algebra.Polynomial.Div
+import Mathlib.Tactic.ReduceModChar
+import Mathlib.Tactic.Ring.Basic
+
+/-! # `p`-integral polynomials
+
+`GV p F 0` means that `F ∈ ℚ[X]` has `p`-integral coefficients. We show closure under powers,
+composition and evaluation, and division by `X - a` (for `p`-integral roots `a`).
+-/
+
+@[expose] public section
+
+open Finset Polynomial
+
+namespace Zeta5Irrational
+
+variable {p : ℕ} [hp : Fact p.Prime]
+
+lemma GV.pow {F : ℚ[X]} {r : ℚ} (h : GV p F r) (n : ℕ) : GV p (F ^ n) (n * r) := by
+  induction n with
+  | zero => simpa using GV.C (p := p) (VG.one (p := p))
+  | succ n ih => rw [pow_succ]; convert ih.mul h using 1; push_cast; ring
+
+lemma GV.comp {F G : ℚ[X]} (hF : GV p F 0) (hG : GV p G 0) : GV p (F.comp G) 0 := by
+  rw [comp_eq_sum_left, Polynomial.sum]
+  refine GV.sum _ fun n _ => ?_
+  have := GV.C_mul (hF n) (hG.pow n)
+  simpa using this
+
+lemma VG.eval {F : ℚ[X]} {z : ℚ} (hF : GV p F 0) (hz : VG p z 0) : VG p (F.eval z) 0 := by
+  rw [eval_eq_sum, Polynomial.sum]
+  refine VG.sum _ fun n _ => ?_
+  have := (hF n).mul (hz.pow n)
+  simpa using this
+
+omit hp in
+lemma VG.eval_zero {F : ℚ[X]} {r : ℚ} (hF : GV p F r) : VG p (F.eval 0) r := by
+  rw [← coeff_zero_eq_eval_zero]; exact hF 0
+
+lemma GV.divByMonic_X_sub_C {F : ℚ[X]} {a : ℚ} (hF : GV p F 0) (ha : VG p a 0) :
+    GV p (F /ₘ (Polynomial.X - Polynomial.C a)) 0 := fun n =>
+  by
+  rw [coeff_divByMonic_X_sub_C]
+  refine VG.sum _ fun i _ => ?_
+  have := (ha.pow (i - (n + 1))).mul (hF i)
+  simpa using this
+
+/-- An integral polynomial vanishing at `p`-integral distinct points is divisible by
+`∏ (X - ρ)` with an integral quotient. -/
+theorem factor_roots (ρs : Finset ℚ) (hρ : ∀ ρ ∈ ρs, VG p ρ 0) {F : ℚ[X]} (hF : GV p F 0)
+    (hroot : ∀ ρ ∈ ρs, F.eval ρ = 0) : ∃ S : ℚ[X], GV p S 0 ∧ F = (∏ ρ ∈ ρs, (X - C ρ)) * S := by
+  classical
+    induction ρs using Finset.induction_on generalizing F with
+  | empty => exact ⟨F, hF, by simp⟩
+  | insert a ρs ha
+    ih =>
+    have hra : F.IsRoot a := hroot a (Finset.mem_insert_self _ _)
+    set F1 := F /ₘ (X - C a) with hF1
+    have hFeq : F = (X - C a) * F1 := ((mul_divByMonic_eq_iff_isRoot).mpr hra).symm
+    have hF1int : GV p F1 0 := GV.divByMonic_X_sub_C hF (hρ a (Finset.mem_insert_self _ _))
+    have hF1root : ∀ ρ ∈ ρs, F1.eval ρ = 0 := by
+      intro ρ hρs
+      have h1 := hroot ρ (Finset.mem_insert_of_mem hρs)
+      rw [hFeq, eval_mul, eval_sub, eval_X, eval_C] at h1
+      have hne : ρ - a ≠ 0 := sub_ne_zero.mpr fun h => ha (h ▸ hρs)
+      exact (mul_eq_zero.mp h1).resolve_left hne
+    obtain ⟨S, hS, hSeq⟩ := ih (fun ρ h => hρ ρ (Finset.mem_insert_of_mem h)) hF1int hF1root
+    exact ⟨S, hS, by rw [hFeq, hSeq, Finset.prod_insert ha, mul_assoc]⟩
+
+/-- Multiset products. -/
+lemma GV.multiset_prod (s : Multiset ℚ[X]) (b : ℚ[X] → ℚ) (h : ∀ f ∈ s, GV p f (b f)) :
+    GV p s.prod (s.map b).sum := by
+  induction s using Multiset.induction_on with
+  | empty => simpa using GV.C (p := p) (VG.one (p := p))
+  | cons a s ih =>
+    rw [Multiset.prod_cons, Multiset.map_cons, Multiset.sum_cons]
+    exact
+      (h a (Multiset.mem_cons_self _ _)).mul (ih fun f hf => h f (Multiset.mem_cons_of_mem hf))
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Arith/LowRank.lean b/LeanPool/Zeta5Irrational/Arith/LowRank.lean
new file mode 100644
index 0000000000..32ff1b6892
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Arith/LowRank.lean
@@ -0,0 +1,80 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+import Mathlib.LinearAlgebra.Matrix.SchurComplement
+
+public import LeanPool.Zeta5Irrational.Arith.Val
+import Mathlib.Data.Rat.Star
+import Mathlib.Tactic.Polynomial.Basic
+import Mathlib.Tactic.ReduceModChar
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+import Mathlib.Tactic.Ring.Basic
+
+/-! # Lemma 4.2 (simplified): determinants with a low-rank `p⁻¹`-correction
+
+If `v(A_{st}) ≥ w_s + w_t` with `w ≤ 0`, and `U`, `V` are `p`-integral of inner dimension `r`, then
+`v_p^G(det(A + p⁻¹ U V)) ≥ 2 ∑ w - r` (via the Schur complement).
+-/
+
+@[expose] public section
+
+open Finset Polynomial Matrix
+
+namespace Zeta5Irrational
+
+variable {p : ℕ} [hp : Fact p.Prime]
+
+theorem det_lowrank_bound {h r : ℕ} (A : Matrix (Fin h) (Fin h) ℚ[X])
+    (U : Matrix (Fin h) (Fin r) ℚ) (V : Matrix (Fin r) (Fin h) ℚ) (w : Fin h → ℚ)
+    (hw : ∀ s, w s ≤ 0) (hA : ∀ s t, GV p (A s t) (w s + w t)) (hU : ∀ s u, VG p (U s u) 0)
+    (hV : ∀ u t, VG p (V u t) 0) :
+    GV p (A + (((p : ℚ)⁻¹ • (U * V)).map (C : ℚ →+* ℚ[X]))).det (2 * ∑ s, w s - r) := by
+  set B : Matrix (Fin h) (Fin r) ℚ[X] := (-U).map (C : ℚ →+* ℚ[X])
+  set Cm : Matrix (Fin r) (Fin h) ℚ[X] := ((p : ℚ)⁻¹ • V).map (C : ℚ →+* ℚ[X])
+  have hdet :
+    (A + (((p : ℚ)⁻¹ • (U * V)).map (C : ℚ →+* ℚ[X]))).det = (Matrix.fromBlocks A B Cm 1).det := by
+    rw [det_fromBlocks_one₂₂]
+    congr 1
+    refine Matrix.ext fun s t => ?_
+    simp only [B, Cm, Matrix.add_apply, Matrix.sub_apply, Matrix.mul_apply, Matrix.map_apply,
+      Matrix.neg_apply, Matrix.smul_apply, smul_eq_mul, map_sum, map_mul, map_neg, Finset.mul_sum]
+    rw [sub_eq_add_neg, ← Finset.sum_neg_distrib]
+    congr 1
+    refine Finset.sum_congr rfl fun u _ => ?_
+    ring
+  rw [hdet]
+  have hinv : VG p ((p : ℚ)⁻¹) (-1) := by
+    have := VG.primePow (p := p) (-1)
+    simpa using this
+  have key :=
+    det_GV (Matrix.fromBlocks A B Cm 1) (Sum.elim w fun _ => (-1 : ℚ))
+      (Sum.elim w fun _ => (0 : ℚ))
+      (by
+        intro i j
+        rcases i with s | u <;> rcases j with t | u'
+        · exact hA s t
+        · simp only [Matrix.fromBlocks_apply₁₂, B, Matrix.map_apply, Matrix.neg_apply,
+            Sum.elim_inl, Sum.elim_inr, add_zero]
+          exact (GV.C (hU s u').neg).mono (hw s)
+        · simp only [Matrix.fromBlocks_apply₂₁, Cm, Matrix.map_apply, Matrix.smul_apply,
+            smul_eq_mul, Sum.elim_inl, Sum.elim_inr]
+          refine (GV.C (hinv.mul (hV u t))).mono ?_
+          linarith [hw t]
+        · simp only [Matrix.fromBlocks_apply₂₂, Sum.elim_inr, add_zero]
+          rw [Matrix.one_apply]
+          split_ifs
+          · exact (GV.C (p := p) VG.one).mono (by norm_num)
+          · exact GV.zero _)
+  refine key.mono (le_of_eq ?_)
+  simp [Fintype.sum_sum_type]
+  ring
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Arith/MomentVal.lean b/LeanPool/Zeta5Irrational/Arith/MomentVal.lean
new file mode 100644
index 0000000000..4266d6c97f
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Arith/MomentVal.lean
@@ -0,0 +1,129 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Arith.Val
+public import LeanPool.Zeta5Irrational.Construction
+import LeanPool.Zeta5Irrational.Arith.DirectPoly
+import Mathlib.Data.Rat.Star
+import Mathlib.Tactic.ENatToNat
+import Mathlib.Tactic.ReduceModChar
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+import Mathlib.Tactic.Positivity.Basic
+import Mathlib.Tactic.Ring.Basic
+
+/-! # Valuations of the polynomial moments
+
+`μ(t^e) = (-1)^e B_{2e+2} (2e+3)(2e+4)(2e+5)/24` has `v_p ≥ -1` (`p ≥ 5`), and is `p`-integral
+for `e < 2p - 3` (von Staudt–Clausen).
+-/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+variable {p : ℕ} [hp : Fact p.Prime]
+
+lemma VG_bernoulli_even (k : ℕ) : VG p (_root_.bernoulli (2 * k)) (-1) := by
+  obtain ⟨z, hz⟩ := Bernoulli.vonStaudt_clausen k
+  have e :
+    _root_.bernoulli (2 * k) =
+      (z : ℚ) - ∑ q ∈ range (2 * k + 2) with q.Prime ∧ (q - 1) ∣ 2 * k, (1 : ℚ) / q :=
+    by rw [hz]; ring
+  rw [e]
+  refine ((VG.intCast (p := p) z).mono (by norm_num)).sub (VG.sum _ fun q hq => ?_)
+  rw [Finset.mem_filter] at hq
+  have hq0 : (q : ℚ) ≠ 0 := by exact_mod_cast hq.2.1.ne_zero
+  rw [one_div]
+  refine VG.inv hq0 ?_
+  rw [padicValRat.of_nat]
+  have : padicValNat p q ≤ 1 := by
+    by_cases hpq : p = q
+    · subst hpq; rw [padicValNat_self]
+    · have := Fact.mk hq.2.1
+      rw [padicValNat_primes hpq]; omega
+  exact_mod_cast this
+
+lemma VG_bernoulli_even_int (k : ℕ) (hk : ¬(p - 1) ∣ 2 * k) : VG p (_root_.bernoulli (2 * k)) 0 :=
+  by
+  obtain ⟨z, hz⟩ := Bernoulli.vonStaudt_clausen k
+  have e :
+    _root_.bernoulli (2 * k) =
+      (z : ℚ) - ∑ q ∈ range (2 * k + 2) with q.Prime ∧ (q - 1) ∣ 2 * k, (1 : ℚ) / q :=
+    by rw [hz]; ring
+  rw [e]
+  refine (VG.intCast (p := p) z).sub (VG.sum _ fun q hq => ?_)
+  rw [Finset.mem_filter] at hq
+  have hq0 : (q : ℚ) ≠ 0 := by exact_mod_cast hq.2.1.ne_zero
+  have hpq : p ≠ q := fun h => hk (h ▸ hq.2.2)
+  rw [one_div]
+  refine (VG.inv (r := 0) hq0 ?_).mono (by norm_num)
+  have := Fact.mk hq.2.1
+  rw [padicValRat.of_nat, padicValNat_primes hpq]; simp
+
+lemma VG_inv24 (hp5 : 5 ≤ p) : VG p ((24 : ℚ)⁻¹) 0 := by
+  refine (VG.inv (r := 0) (by norm_num) ?_).mono (by norm_num)
+  rw [show (24 : ℚ) = ((24 : ℕ) : ℚ) by norm_num, padicValRat.of_nat, padicValNat_24 hp5]; simp
+
+/-- `v_p(μ(t^e)) ≥ -1`. -/
+theorem VG_μmono (hp5 : 5 ≤ p) (e : ℕ) : VG p (μmono e) (-1) := by
+  unfold μmono
+  rw [show 2 * e + 2 = 2 * (e + 1) by ring, div_eq_mul_inv]
+  have h1 : VG p ((-1 : ℚ) ^ e) 0 := by
+    rcases neg_one_pow_eq_or ℚ e with h | h <;> rw [h]
+    · exact VG.one
+    · exact VG.one.neg
+  have h3 : VG p (((2 * e + 3) * (2 * e + 4) * (2 * e + 5) : ℕ) : ℚ) 0 := VG.natCast _
+  have := ((h1.mul (VG_bernoulli_even (e + 1))).mul h3).mul (VG_inv24 hp5)
+  push_cast at this
+  simpa using this
+
+/-- `μ(t^e)` is `p`-integral for `e < 2p - 3`. -/
+theorem VG_μmono_int (hp5 : 5 ≤ p) {e : ℕ} (he : e + 3 < 2 * p) : VG p (μmono e) 0 := by
+  unfold μmono
+  rw [show 2 * e + 2 = 2 * (e + 1) by ring, div_eq_mul_inv]
+  have h1 : VG p ((-1 : ℚ) ^ e) 0 := by
+    rcases neg_one_pow_eq_or ℚ e with h | h <;> rw [h]
+    · exact VG.one
+    · exact VG.one.neg
+  by_cases hd : (p - 1) ∣ 2 * (e + 1)
+  · -- `2e + 2 = j (p - 1)` with `j ∈ {1, 2, 3}`
+    obtain ⟨j, hj⟩ := hd
+    have hp1 : 1 ≤ p - 1 := by omega
+    have hj3 : j ≤ 3 := by
+      by_contra h; rw [not_le] at h
+      have : (p - 1) * 4 ≤ (p - 1) * j := Nat.mul_le_mul_left _ h
+      omega
+    have hj0 : 1 ≤ j := by
+      rcases Nat.eq_zero_or_pos j with h | h
+      · rw [h, mul_zero] at hj; omega
+      · exact h
+    have hprod : VG p (((2 * e + 3) * (2 * e + 4) * (2 * e + 5) : ℕ) : ℚ) 1 := by
+      apply VG.of_eq
+      intro hne
+      rw [padicValRat.of_nat]
+      have hdvd : p ∣ (2 * e + 3) * (2 * e + 4) * (2 * e + 5) := by
+        interval_cases j
+        · exact Dvd.dvd.mul_right (Dvd.dvd.mul_right ⟨1, by omega⟩ _) _
+        · exact Dvd.dvd.mul_right (Dvd.dvd.mul_left ⟨2, by omega⟩ _) _
+        · exact Dvd.dvd.mul_left ⟨3, by omega⟩ _
+      have hne' : (2 * e + 3) * (2 * e + 4) * (2 * e + 5) ≠ 0 := by positivity
+      have := one_le_padicValNat_of_dvd hne' hdvd
+      exact_mod_cast this
+    have := ((h1.mul (VG_bernoulli_even (e + 1))).mul hprod).mul (VG_inv24 hp5)
+    push_cast at this
+    simpa using this
+  · have h3 : VG p (((2 * e + 3) * (2 * e + 4) * (2 * e + 5) : ℕ) : ℚ) 0 := VG.natCast _
+    have := ((h1.mul (VG_bernoulli_even_int (e + 1) hd)).mul h3).mul (VG_inv24 hp5)
+    push_cast at this
+    simpa using this
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Arith/OuterBasis.lean b/LeanPool/Zeta5Irrational/Arith/OuterBasis.lean
new file mode 100644
index 0000000000..c3fbe5d171
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Arith/OuterBasis.lean
@@ -0,0 +1,181 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Arith.ClassCount
+import Mathlib.Tactic.Polynomial.Basic
+import Mathlib.Tactic.ReduceModChar
+import Mathlib.Tactic.Ring.Basic
+
+/-! # The outer-range basis
+
+For `p = 2m + 1` and the tail poles `j ∈ (N, K]`, grouped by class `jc j` (`j ≡ ±c`):
+row `(c, i)` (`i < Lo c`) is `tpol (rootsO c i)` with
+`rootsO c i = {tail poles of other classes} + (i < nb c ? i·{c} : big c + (i - nb c)·{c})`,
+where `big c` are the tail poles of class `c` above `p`.
+-/
+
+@[expose] public section
+
+open Finset Polynomial
+
+namespace Zeta5Irrational
+
+variable {p : ℕ} [hp : Fact p.Prime] {m : ℕ}
+
+/-- The class of a natural number. -/
+def jc (hm : 2 * m + 1 = p) (j : ℕ) : Fin (m + 1) :=
+  ccls hm (((j : ℤ) : ZMod p))
+
+lemma sq_eq_iff_ccls (hm : 2 * m + 1 = p) (x y : ZMod p) :
+    x ^ 2 = y ^ 2 ↔ ccls hm x = ccls hm y := by
+  constructor
+  · intro h
+    have h2 : (x - y) * (x + y) = 0 := by linear_combination h
+    rcases mul_eq_zero.mp h2 with h3 | h3
+    · rw [sub_eq_zero.mp h3]
+    · have : x = -y := by linear_combination h3
+      rw [this]
+      exact
+        (ccls_unique hm (-y) (ccls hm y)
+            (by
+              rcases ccls_spec hm y with h4 | h4
+              · right; rw [h4, neg_neg]
+              · left; rw [h4])).symm
+  · intro h
+    have e : ∀ z : ZMod p, z ^ 2 = ((((ccls hm z : ℕ) : ℤ) : ZMod p)) ^ 2 := by
+      intro z
+      rcases ccls_spec hm z with h4 | h4 <;> rw [h4]; ring
+    rw [e x, e y, h]
+
+variable (hm : 2 * m + 1 = p) (n : ℕ)
+
+/-- Tail poles `N < j ≤ K`. -/
+def tailO (n : ℕ) : Finset ℕ :=
+  Icc (3 * n + 1) (40 * n)
+
+/-- Outer pole indices belonging to the residue class `c`. -/
+def tailC (c : Fin (m + 1)) : Finset ℕ :=
+  (tailO n).filter fun j => jc hm j = c
+
+/-- Number of rows allocated to the outer residue class `c`. -/
+def LoC (c : Fin (m + 1)) : ℕ :=
+  (tailC hm n c).card
+
+/-- Outer pole indices in class `c` that exceed the prime `p`. -/
+def bigC (c : Fin (m + 1)) : Finset ℕ :=
+  (tailC hm n c).filter fun j => p < j
+
+/-- Number of outer poles in class `c` that exceed `p`. -/
+def nbC (c : Fin (m + 1)) : ℕ :=
+  (bigC hm n c).card
+
+/-- The roots of the outer row `(c, i)`. -/
+def rootsO (c : Fin (m + 1)) (i : ℕ) : Multiset ℤ :=
+  ((tailO n).filter fun j => jc hm j ≠ c).val.map (fun j : ℕ => (j : ℤ)) +
+    (if i < nbC hm n c then Multiset.replicate i ((c : ℕ) : ℤ)
+    else
+      (bigC hm n c).val.map (fun j : ℕ => (j : ℤ)) +
+        Multiset.replicate (i - nbC hm n c) ((c : ℕ) : ℤ))
+
+lemma sum_LoC : ∑ c, LoC hm n c = 37 * n := by
+  have :=
+    Finset.card_eq_sum_card_fiberwise (f := jc hm) (s := tailO n) (t := Finset.univ)
+      (fun j _ => Finset.mem_univ _)
+  unfold LoC tailC
+  rw [← this, tailO, Nat.card_Icc]; omega
+
+lemma card_rootsO (c : Fin (m + 1)) {i : ℕ} (hi : i < LoC hm n c) :
+    (rootsO hm n c i).card < 37 * n := by
+  unfold rootsO
+  have hsplit := Finset.card_filter_add_card_filter_not (s := tailO n) (fun j => jc hm j = c)
+  have htail : (tailO n).card = 37 * n := by rw [tailO, Nat.card_Icc]; omega
+  have hnb : nbC hm n c ≤ LoC hm n c := Finset.card_filter_le _ _
+  have hLo : LoC hm n c = ((tailO n).filter fun j => jc hm j = c).card := rfl
+  have hother : ((tailO n).filter fun j => jc hm j ≠ c).card = 37 * n - LoC hm n c := by
+    have :
+      ((tailO n).filter fun j => jc hm j ≠ c).card =
+        ((tailO n).filter fun a => ¬(jc hm a = c)).card :=
+      rfl
+    omega
+  split_ifs with h
+  · simp only [Multiset.card_add, Multiset.card_map, Finset.card_val, Multiset.card_replicate]
+    omega
+  · simp only [Multiset.card_add, Multiset.card_map, Finset.card_val, Multiset.card_replicate]
+    have : nbC hm n c = (bigC hm n c).card := rfl
+    omega
+
+/-- The class of a class representative is itself. -/
+lemma ccls_self (c : Fin (m + 1)) : ccls hm ((((c : ℕ) : ℤ) : ZMod p)) = c :=
+  (ccls_unique hm _ c (Or.inl rfl)).symm
+
+lemma sq_jc (j : ℕ) : (((j : ℤ) : ZMod p)) ^ 2 = ((((jc hm j : ℕ) : ℤ) : ZMod p)) ^ 2 := by
+  rw [sq_eq_iff_ccls hm, ccls_self]; rfl
+
+/-- `γ_c = -c²` in `𝔽_p`. -/
+def γO (c : Fin (m + 1)) : ZMod p :=
+  -((((c : ℕ) : ℤ) : ZMod p)) ^ 2
+
+lemma map_factor (γ : ℤ) :
+    (X + C (γ ^ 2) : ℤ[X]).map (Int.castRingHom (ZMod p)) = X - C (-((γ : ZMod p)) ^ 2) := by
+  simp [sub_eq_add_neg]
+
+lemma prod_map_const_pow {s : Finset ℕ} {f : ℕ → (ZMod p)[X]} {q : (ZMod p)[X]}
+    (h : ∀ j ∈ s, f j = q) : ∏ j ∈ s, f j = q ^ s.card := by
+  rw [Finset.prod_congr rfl h, Finset.prod_const]
+
+/-- **Reduction of the outer rows mod `p`.** -/
+theorem rootsO_map_ZMod (c : Fin (m + 1)) (i : ℕ) :
+    (tpolZ (rootsO hm n c i)).map (Int.castRingHom (ZMod p)) =
+      (∏ c' ∈ Finset.univ.erase c, (X - C (γO (p := p) c')) ^ LoC hm n c') *
+        (X - C (γO (p := p) c)) ^ i := by
+  have hfac :
+    ∀ j : ℕ,
+      ((X + C (((j : ℤ)) ^ 2) : ℤ[X]).map (Int.castRingHom (ZMod p))) =
+        X - C (γO (p := p) (jc hm j)) :=
+    by intro j; rw [map_factor, γO]; congr 2; exact congrArg Neg.neg (sq_jc hm j)
+  have hcc :
+    ((X + C ((((c : ℕ) : ℤ)) ^ 2) : ℤ[X]).map (Int.castRingHom (ZMod p))) =
+      X - C (γO (p := p) c) :=
+    by rw [map_factor]; rfl
+  unfold tpolZ rootsO
+  rw [Polynomial.map_multiset_prod, Multiset.map_map, Multiset.map_add, Multiset.prod_add]
+  have h1 :
+    ((((tailO n).filter fun j => jc hm j ≠ c).val.map (fun j : ℕ => (j : ℤ))).map
+          ((Polynomial.map (Int.castRingHom (ZMod p))) ∘ fun γ : ℤ => X + C (γ ^ 2))).prod =
+      ∏ c' ∈ Finset.univ.erase c, (X - C (γO (p := p) c')) ^ LoC hm n c' := by
+    rw [Multiset.map_map, ← Finset.prod_eq_multiset_prod]
+    simp only [Function.comp, hfac]
+    rw [←
+      Finset.prod_fiberwise_of_maps_to (g := jc hm) (t := Finset.univ.erase c)
+        (fun j hj => Finset.mem_erase.mpr ⟨(Finset.mem_filter.mp hj).2, Finset.mem_univ _⟩)]
+    refine Finset.prod_congr rfl fun c' hc' => ?_
+    rw [prod_map_const_pow (q := X - C (γO (p := p) c'))
+        (fun j hj => by rw [(Finset.mem_filter.mp hj).2])]
+    congr 1
+    unfold LoC tailC
+    congr 1
+    ext j
+    simp only [Finset.mem_filter]
+    constructor
+    · rintro ⟨⟨h1, _⟩, h3⟩; exact ⟨h1, h3⟩
+    · rintro ⟨h1, h3⟩; exact ⟨⟨h1, h3 ▸ (Finset.mem_erase.mp hc').1⟩, h3⟩
+  rw [h1]
+  congr 1
+  split_ifs with h
+  · rw [Multiset.map_replicate, Multiset.prod_replicate]; simp only [Function.comp, hcc]
+  · rw [Multiset.map_add, Multiset.prod_add, Multiset.map_map, Multiset.map_replicate,
+      Multiset.prod_replicate, ← Finset.prod_eq_multiset_prod]
+    simp only [Function.comp, hfac]
+    rw [show jc hm ((c : ℕ)) = c from ccls_self hm c]
+    rw [prod_map_const_pow (q := X - C (γO (p := p) c))
+        (fun j hj => by rw [(Finset.mem_filter.mp (Finset.mem_filter.mp hj).1).2]),
+      ← pow_add]
+    have : nbC hm n c = (bigC hm n c).card := rfl
+    congr 1; omega
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Arith/OuterEntries.lean b/LeanPool/Zeta5Irrational/Arith/OuterEntries.lean
new file mode 100644
index 0000000000..0dc14a0f50
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Arith/OuterEntries.lean
@@ -0,0 +1,324 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Arith.OuterBasis
+import LeanPool.Zeta5Irrational.Arith.IntPoly
+import LeanPool.Zeta5Irrational.Arith.OuterEntry
+import LeanPool.Zeta5Irrational.Arith.PoleVal
+import Mathlib.Data.Rat.Star
+import Mathlib.RingTheory.Int.Basic
+import Mathlib.Tactic.ENatToNat
+import Mathlib.Tactic.ReduceModChar
+import Std.Tactic.BVDecide.Normalize.Prop
+import Mathlib.Tactic.FieldSimp
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+import Mathlib.Tactic.Positivity.Basic
+import Mathlib.Tactic.Ring.Basic
+
+/-! # Outer range: entry bounds
+
+Helper lemmas (memberships, root counts, zero-class valuations).
+-/
+
+@[expose] public section
+
+open Finset Polynomial
+
+namespace Zeta5Irrational
+
+variable {p : ℕ} [hp : Fact p.Prime] {m : ℕ}
+
+/-- The entry roots `6·[1..N] + rows`. -/
+def entryO (hm : 2 * m + 1 = p) (n : ℕ) (cs : Fin (m + 1)) (is : ℕ) (ct : Fin (m + 1)) (it : ℕ) :
+    Multiset ℤ :=
+  6 • ((Icc 1 (3 * n)).val.map fun j : ℕ => (j : ℤ)) + rootsO hm n cs is + rootsO hm n ct it
+
+variable (hm : 2 * m + 1 = p) (n : ℕ)
+
+lemma mem_entryO_small {cs ct : Fin (m + 1)} {is it : ℕ} {k : ℕ} (hk : k ∈ Icc 1 (3 * n)) :
+    (k : ℤ) ∈ entryO hm n cs is ct it := by
+  unfold entryO
+  refine Multiset.mem_add.mpr (Or.inl (Multiset.mem_add.mpr (Or.inl ?_)))
+  rw [Multiset.mem_nsmul_of_ne_zero (by norm_num)]
+  exact Multiset.mem_map.mpr ⟨k, hk, rfl⟩
+
+lemma mem_rootsO_other {c : Fin (m + 1)} {i j : ℕ} (hj : j ∈ tailO n) (hc : jc hm j ≠ c) :
+    (j : ℤ) ∈ rootsO hm n c i := by
+  unfold rootsO
+  refine Multiset.mem_add.mpr (Or.inl (Multiset.mem_map.mpr ⟨j, ?_, rfl⟩))
+  exact Finset.mem_val.mpr (Finset.mem_filter.mpr ⟨hj, hc⟩)
+
+lemma mem_rootsO_big {c : Fin (m + 1)} {i j : ℕ} (hi : nbC hm n c ≤ i) (hj : j ∈ bigC hm n c) :
+    (j : ℤ) ∈ rootsO hm n c i := by
+  unfold rootsO
+  rw [ite_eq_right (by omega)]
+  refine Multiset.mem_add.mpr (Or.inr (Multiset.mem_add.mpr (Or.inl ?_)))
+  exact Multiset.mem_map.mpr ⟨j, Finset.mem_val.mpr hj, rfl⟩
+
+lemma mem_entryO_left {cs ct : Fin (m + 1)} {is it : ℕ} {γ : ℤ} (h : γ ∈ rootsO hm n cs is) :
+    γ ∈ entryO hm n cs is ct it := by
+  unfold entryO
+  exact Multiset.mem_add.mpr (Or.inl (Multiset.mem_add.mpr (Or.inr h)))
+
+lemma mem_entryO_right {cs ct : Fin (m + 1)} {is it : ℕ} {γ : ℤ} (h : γ ∈ rootsO hm n ct it) :
+    γ ∈ entryO hm n cs is ct it := by
+  unfold entryO
+  exact Multiset.mem_add.mpr (Or.inr h)
+
+/-- Count of roots in the class of `c` in a row of class `c`. -/
+lemma count_rootsO (c : Fin (m + 1)) (i : ℕ) (x : ZMod p)
+    (hx : x ^ 2 = ((((c : ℕ) : ℤ) : ZMod p)) ^ 2) :
+    i ≤ ((rootsO hm n c i).filter fun γ : ℤ => ((γ : ZMod p)) ^ 2 = x ^ 2).card := by
+  have hcx : ((((c : ℕ) : ℤ) : ZMod p)) ^ 2 = x ^ 2 := hx.symm
+  unfold rootsO
+  rw [Multiset.filter_add, Multiset.card_add]
+  split_ifs with h
+  · rw [card_filter_replicate, ite_eq_left hcx]; omega
+  · rw [Multiset.filter_add, Multiset.card_add, card_filter_replicate, ite_eq_left hcx]
+    have hbig :
+      ((bigC hm n c).val.map fun j : ℕ => (j : ℤ)).filter
+          (fun γ : ℤ => ((γ : ZMod p)) ^ 2 = x ^ 2) =
+        (bigC hm n c).val.map fun j : ℕ => (j : ℤ) := by
+      refine Multiset.filter_eq_self.mpr fun γ hγ => ?_
+      obtain ⟨j, hj, rfl⟩ := Multiset.mem_map.mp hγ
+      have hj' : j ∈ bigC hm n c := Finset.mem_val.mp hj
+      have hjc : jc hm j = c := (Finset.mem_filter.mp (Finset.mem_filter.mp hj').1).2
+      rw [hx, sq_jc hm j, hjc]
+    rw [hbig, Multiset.card_map, Finset.card_val]
+    have : nbC hm n c = (bigC hm n c).card := rfl
+    omega
+
+/-- The valuation of `∏ (γ² - j²)` for `p ∣ j`: two per root divisible by `p`. -/
+lemma VG_prod_sq_sub_zero (M : Multiset ℤ) {j : ℤ} (hj : (p : ℤ) ∣ j) :
+    VG p ((M.map fun γ : ℤ => (γ : ℚ) ^ 2 - (j : ℚ) ^ 2).prod)
+      (2 * (M.filter fun γ : ℤ => (p : ℤ) ∣ γ).card) := by
+  induction M using Multiset.induction_on with
+  | empty => right; simp
+  | cons a M ih =>
+    rw [Multiset.map_cons, Multiset.prod_cons, Multiset.filter_cons]
+    split_ifs with h
+    · rw [Multiset.card_add, Multiset.card_singleton]
+      have h1 : VG p ((a : ℚ) ^ 2 - (j : ℚ) ^ 2) 2 := by
+        rw [show (a : ℚ) ^ 2 - (j : ℚ) ^ 2 = ((a - j : ℤ) : ℚ) * ((a + j : ℤ) : ℚ) by
+            push_cast;
+            ring]
+        have := (VG_int_one (p := p) (dvd_sub h hj)).mul (VG_int_one (p := p) (dvd_add h hj))
+        simpa [one_add_one_eq_two] using this
+      convert h1.mul ih using 1; push_cast; ring
+    · rw [zero_add]
+      have h1 : VG p ((a : ℚ) ^ 2 - (j : ℚ) ^ 2) 0 := by
+        have := VG.intCast (p := p) (a ^ 2 - j ^ 2); push_cast at this; exact this
+      simpa using h1.mul ih
+
+/-- Denominator for `p ∣ j`: at most two per other multiple of `p`. -/
+lemma padicValRat_den_zero_le {K j : ℕ} (hj : j ∈ Icc 1 K) (hK : 2 * K < p ^ 2)
+    (hj0 : (p : ℤ) ∣ (j : ℤ)) :
+    (padicValRat p (∏ i ∈ (Icc 1 K).erase j, ((i : ℚ) ^ 2 - (j : ℚ) ^ 2)) : ℚ) ≤
+      2 * (((Icc 1 K).erase j).filter fun i : ℕ => (p : ℤ) ∣ (i : ℤ)).card := by
+  have hne : ∀ i ∈ (Icc 1 K).erase j, ((i : ℚ) ^ 2 - (j : ℚ) ^ 2) ≠ 0 := by
+    intro i hi
+    have hij := (Finset.mem_erase.mp hi).1
+    have : (i : ℚ) ≠ j := by exact_mod_cast hij
+    intro h
+    have h2 : ((i : ℚ) - j) * ((i : ℚ) + j) = 0 := by linear_combination h
+    rcases mul_eq_zero.mp h2 with h3 | h3
+    · exact this (by linarith)
+    · have : (0 : ℚ) ≤ i := by positivity
+      have : (1 : ℚ) ≤ j := by exact_mod_cast (Finset.mem_Icc.mp hj).1
+      linarith
+  rw [padicValRat_finset_prod _ _ hne]
+  push_cast
+  have hrhs :
+    (2 * ((((Icc 1 K).erase j).filter fun i : ℕ => (p : ℤ) ∣ (i : ℤ)).card : ℚ)) =
+      ∑ i ∈ (Icc 1 K).erase j, (if (p : ℤ) ∣ (i : ℤ) then (2 : ℚ) else 0) := by
+    rw [Finset.card_filter]; push_cast; rw [Finset.mul_sum]
+    exact Finset.sum_congr rfl fun i _ => by split_ifs <;> norm_num
+  rw [hrhs]
+  refine Finset.sum_le_sum fun i hi => ?_
+  have hi' := Finset.mem_erase.mp hi
+  have hiK := (Finset.mem_Icc.mp hi'.2)
+  have hjK := (Finset.mem_Icc.mp hj)
+  rw [show ((i : ℚ) ^ 2 - (j : ℚ) ^ 2) = (((i - j) * (i + j) : ℤ) : ℚ) by push_cast; ring]
+  have hz : ((i : ℤ) - j) * ((i : ℤ) + j) ≠ 0 := by
+    have := hne i hi
+    intro h; apply this;
+    rw [show ((i : ℚ) ^ 2 - (j : ℚ) ^ 2) = (((i - j) * (i + j) : ℤ) : ℚ) by
+    push_cast; ring, h];
+    simp
+  have hz1 := (mul_ne_zero_iff.mp hz).1
+  have hz2 := (mul_ne_zero_iff.mp hz).2
+  split_ifs with hc
+  · rw [padicValRat.of_int, padicValInt.mul hz1 hz2]
+    have hle : ∀ z : ℤ, z ≠ 0 → |z| ≤ 2 * K → padicValInt p z ≤ 1 := by
+      intro z hz0 hzK
+      have :=
+        padicValRat_int_le_one (p := p) (z := z)
+          (fun hd => by
+            have h1 := Int.natAbs_dvd_natAbs.mpr hd
+            rw [Int.natAbs_pow, Int.natAbs_natCast] at h1
+            have h2 := Nat.le_of_dvd (Int.natAbs_pos.mpr hz0) h1
+            have h3 : (z.natAbs : ℤ) ≤ 2 * K := by rw [Int.natCast_natAbs]; exact hzK
+            omega)
+      rw [padicValRat.of_int] at this; exact_mod_cast this
+    have hb1 := hle _ hz1 (by rw [abs_le]; constructor <;> omega)
+    have hb2 := hle _ hz2 (by rw [abs_le]; constructor <;> omega)
+    push_cast
+    have : ((padicValInt p ((i : ℤ) - j) : ℕ) : ℚ) ≤ 1 := by exact_mod_cast hb1
+    have : ((padicValInt p ((i : ℤ) + j) : ℕ) : ℚ) ≤ 1 := by exact_mod_cast hb2
+    linarith
+  · have hnd : ¬(p : ℤ) ∣ ((i : ℤ) - j) * ((i : ℤ) + j) := by
+      intro hd
+      rcases Int.Prime.dvd_mul' hp.out hd with h1 | h1
+      · exact hc (by have := dvd_add h1 hj0; simpa using this)
+      · exact hc (by have := dvd_sub h1 hj0; simpa using this)
+    rw [padicValRat_int_eq_zero hnd]; simp
+
+/-- Residue bound for a nonzero class. -/
+lemma VG_res_class (hp3 : 3 ≤ p) {K : ℕ} (hK : 2 * K < p ^ 2) (M : Multiset ℤ) {j : ℕ}
+    (hj : j ∈ Icc 1 K) (hj0 : ¬(p : ℤ) ∣ (j : ℤ)) :
+    VG p (res K (tpol M) j)
+      (((M.filter fun γ : ℤ => ((γ : ZMod p)) ^ 2 = ((j : ℤ) : ZMod p) ^ 2).card : ℚ) -
+        (((Icc 1 K).erase j).filter fun i : ℕ =>
+            ((i : ℤ) : ZMod p) ^ 2 = ((j : ℤ) : ZMod p) ^ 2).card) := by
+  rw [res_tpol hj, div_eq_mul_inv]
+  have hden : ∏ i ∈ (Icc 1 K).erase j, ((i : ℚ) ^ 2 - (j : ℚ) ^ 2) ≠ 0 := by
+    refine Finset.prod_ne_zero_iff.mpr fun i hi => ?_
+    have hij := (Finset.mem_erase.mp hi).1
+    have : (i : ℚ) ≠ j := by exact_mod_cast hij
+    intro h
+    have h2 : ((i : ℚ) - j) * ((i : ℚ) + j) = 0 := by linear_combination h
+    rcases mul_eq_zero.mp h2 with h3 | h3
+    · exact this (by linarith)
+    · have : (0 : ℚ) ≤ i := by positivity
+      have : (1 : ℚ) ≤ j := by exact_mod_cast (Finset.mem_Icc.mp hj).1
+      linarith
+  have h1 := VG_prod_sq_sub (p := p) M (j : ℤ)
+  have h2 := VG.inv (p := p) hden (padicValRat_den_le hp3 hj hK hj0)
+  push_cast at h1
+  simpa [sub_eq_add_neg] using h1.mul h2
+
+/-- Divided differences of integral polynomials are integral. -/
+lemma eval_sub_eval_int {S : ℚ[X]} (hS : GV p S 0) {u v : ℚ} (hu : VG p u 0) (hv : VG p v 0) :
+    ∃ T : ℚ, S.eval u - S.eval v = (u - v) * T ∧ VG p T 0 := by
+  refine
+    ⟨∑ k ∈ range (S.natDegree + 1), S.coeff k * ∑ i ∈ range k, u ^ i * v ^ (k - 1 - i), ?_, ?_⟩
+  · rw [eval_eq_sum_range, eval_eq_sum_range, ← Finset.sum_sub_distrib, Finset.mul_sum]
+    refine Finset.sum_congr rfl fun k _ => ?_
+    have := Commute.geom_sum₂_mul (Commute.all u v) k
+    rw [← mul_sub, ← this]; ring
+  · refine VG.sum _ fun k _ => ?_
+    have : VG p (∑ i ∈ range k, u ^ i * v ^ (k - 1 - i)) 0 :=
+      VG.sum _ fun i _ => by simpa using (hu.pow i).mul (hv.pow (k - 1 - i))
+    simpa using (hS k).mul this
+
+omit hp in
+lemma GV_tpol (M : Multiset ℤ) : GV p (tpol M) 0 := by
+  rw [← tpolZ_map_Q]; intro k;
+  rw [coeff_map]; exact VG.intCast _
+
+lemma sq_ne_of_ne {i j : ℕ} (h : i ≠ j) : (i : ℚ) ^ 2 - (j : ℚ) ^ 2 ≠ 0 := by
+  intro h2
+  have h3 : ((i : ℚ) - j) * ((i : ℚ) + j) = 0 := by linear_combination h2
+  rcases mul_eq_zero.mp h3 with h4 | h4
+  · exact h (by exact_mod_cast (sub_eq_zero.mp h4))
+  · have : (0 : ℚ) ≤ i := by positivity
+    have : (0 : ℚ) ≤ j := by positivity
+    have hi : (i : ℚ) = 0 := by linarith
+    have hj : (j : ℚ) = 0 := by linarith
+    exact h (by exact_mod_cast hi.trans hj.symm)
+
+/-- **The pairing lemma**: two remaining poles `a, p - a` give an integral contribution. -/
+theorem pair_GV (hp3 : 3 ≤ p) {K : ℕ} (M : Multiset ℤ) {a : ℕ} (ha1 : 1 ≤ a) (hap : a < p)
+    (hne : a ≠ p - a) (haK : a ≤ K) (hbK : p - a ≤ K)
+    (hroots : ∀ i ∈ Icc 1 K, i ≠ a → i ≠ p - a → (i : ℤ) ∈ M)
+    (hres : VG p (res K (tpol M) (p - a)) (-1)) :
+    GV p (C (res K (tpol M) a) * poleValue a + C (res K (tpol M) (p - a)) * poleValue (p - a))
+      0 := by
+  set b := p - a with hb
+  set W := ((Icc 1 K).erase a).erase b with hW
+  set ρ : ℕ → ℚ := fun i => -(i : ℚ) ^ 2 with hρ
+  have hinj : Set.InjOn ρ W := by
+    intro i _ j _ h
+    simp only [hρ, neg_inj] at h
+    have := (sq_ne_of_ne (i := i) (j := j))
+    by_contra hij; exact this hij (by rw [h]; ring)
+  have hroot : ∀ x ∈ W.image ρ, (tpol M).eval x = 0 := by
+    intro x hx
+    obtain ⟨i, hi, rfl⟩ := Finset.mem_image.mp hx
+    have hi' := Finset.mem_erase.mp hi
+    have hi'' := Finset.mem_erase.mp hi'.2
+    rw [eval_tpol]
+    apply Multiset.prod_eq_zero
+    exact Multiset.mem_map.mpr ⟨i, hroots i hi''.2 hi''.1 hi'.1, by simp [hρ]⟩
+  have hsqv : ∀ i : ℕ, VG p (-(i : ℚ) ^ 2) 0 := fun i => by
+    have := (VG.natCast (p := p) (i ^ 2)).neg; push_cast at this; exact this
+  obtain ⟨S, hS, hNum⟩ :=
+    factor_roots (W.image ρ)
+      (fun x hx => by
+        obtain ⟨i, _, rfl⟩ := Finset.mem_image.mp hx
+        exact hsqv i)
+      (GV_tpol M) hroot
+  rw [Finset.prod_image hinj] at hNum
+  have haIcc : a ∈ Icc 1 K := Finset.mem_Icc.mpr ⟨ha1, haK⟩
+  have hbIcc : b ∈ Icc 1 K := Finset.mem_Icc.mpr ⟨by omega, hbK⟩
+  have hPW : ∀ j : ℕ, j ∉ W → ∏ i ∈ W, ((i : ℚ) ^ 2 - (j : ℚ) ^ 2) ≠ 0 → True := fun _ _ _ =>
+    trivial
+  have hWne : ∀ j, j = a ∨ j = b → ∏ i ∈ W, ((i : ℚ) ^ 2 - (j : ℚ) ^ 2) ≠ 0 := by
+    intro j hj
+    refine Finset.prod_ne_zero_iff.mpr fun i hi => sq_ne_of_ne ?_
+    have hi' := Finset.mem_erase.mp hi
+    have hi'' := Finset.mem_erase.mp hi'.2
+    rcases hj with rfl | rfl
+    · exact hi''.1
+    · exact hi'.1
+  have hNumEval :
+    ∀ j : ℕ,
+      (tpol M).eval (-(j : ℚ) ^ 2) =
+        (∏ i ∈ W, ((i : ℚ) ^ 2 - (j : ℚ) ^ 2)) * S.eval (-(j : ℚ) ^ 2) := by
+    intro j
+    rw [hNum, eval_mul, eval_prod]
+    congr 1
+    refine Finset.prod_congr rfl fun i _ => ?_
+    simp [hρ]; ring
+  have hEa : (Icc 1 K).erase a = insert b W := by
+    rw [hW, Finset.insert_erase (Finset.mem_erase.mpr ⟨Ne.symm hne, hbIcc⟩)]
+  have hEb : (Icc 1 K).erase b = insert a W := by
+    rw [hW, Finset.erase_right_comm, Finset.insert_erase (Finset.mem_erase.mpr ⟨hne, haIcc⟩)]
+  have hbW : b ∉ W := by simp [hW]
+  have haW : a ∉ W := by simp [hW]
+  have hres_a : res K (tpol M) a = S.eval (-(a : ℚ) ^ 2) / ((b : ℚ) ^ 2 - (a : ℚ) ^ 2) := by
+    unfold res
+    rw [eval_derivative_D haIcc, hEa, Finset.prod_insert hbW, hNumEval]
+    field_simp [hWne a (Or.inl rfl), sq_ne_of_ne (Ne.symm hne)]
+  have hres_b : res K (tpol M) b = S.eval (-(b : ℚ) ^ 2) / ((a : ℚ) ^ 2 - (b : ℚ) ^ 2) := by
+    unfold res
+    rw [eval_derivative_D hbIcc, hEb, Finset.prod_insert haW, hNumEval]
+    field_simp [hWne b (Or.inr rfl), sq_ne_of_ne hne]
+  obtain ⟨T, hT, hTv⟩ :=
+    eval_sub_eval_int hS (u := -(a : ℚ) ^ 2) (v := -(b : ℚ) ^ 2) (hsqv a) (hsqv b)
+  have hsum : res K (tpol M) a + res K (tpol M) b = T := by
+    rw [hres_a, hres_b]
+    have hab : (b : ℚ) ^ 2 - (a : ℚ) ^ 2 ≠ 0 := sq_ne_of_ne (Ne.symm hne)
+    have hba : (a : ℚ) ^ 2 - (b : ℚ) ^ 2 ≠ 0 := sq_ne_of_ne hne
+    have key : S.eval (-(a : ℚ) ^ 2) - S.eval (-(b : ℚ) ^ 2) = ((b : ℚ) ^ 2 - (a : ℚ) ^ 2) * T :=
+      by rw [hT]; ring
+    rw [div_add_div _ _ hab hba, div_eq_iff (mul_ne_zero hab hba)]
+    linear_combination ((a : ℚ) ^ 2 - (b : ℚ) ^ 2) * key
+  have hsplit :
+    C (res K (tpol M) a) * poleValue a + C (res K (tpol M) b) * poleValue b =
+      C T * poleValue a + C (res K (tpol M) b) * (poleValue b - poleValue a) :=
+    by rw [← hsum, C_add]; ring
+  rw [hsplit]
+  have hPa := GV_poleValue_lt hp3 ha1 hap
+  have hcong := GV_poleValue_congr hp3 ha1 hap
+  exact ((GV.C_mul hTv hPa).mono (by norm_num)).add ((GV.C_mul hres hcong).mono (by norm_num))
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Arith/OuterEntry.lean b/LeanPool/Zeta5Irrational/Arith/OuterEntry.lean
new file mode 100644
index 0000000000..de78a1982b
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Arith/OuterEntry.lean
@@ -0,0 +1,164 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+import Mathlib.RingTheory.Int.Basic
+
+public import LeanPool.Zeta5Irrational.Arith.TEntry
+import Mathlib.Data.Int.Star
+import Mathlib.Data.Rat.Star
+import Mathlib.Tactic.ENatToNat
+import Mathlib.Tactic.ReduceModChar
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+import Mathlib.Tactic.Positivity.Basic
+import Mathlib.Tactic.Ring.Basic
+
+/-! # Outer range: general residue and pole-value lemmas
+
+For a numerator `tpol M = ∏_{γ ∈ M} (t + γ²)`:
+* `res_tpol` : `res K (tpol M) j = ∏_γ (γ² - j²) / ∏_{i ≠ j} (i² - j²)`;
+* valuation bounds for residues and pole values;
+* `PV_congr` : `poleValue (p - c) ≡ poleValue c (mod p)`.
+-/
+
+@[expose] public section
+
+open Finset Polynomial
+
+namespace Zeta5Irrational
+
+variable {p : ℕ} [hp : Fact p.Prime]
+
+lemma eval_tpol (M : Multiset ℤ) (x : ℚ) :
+    (tpol M).eval x = (M.map fun γ : ℤ => x + (γ : ℚ) ^ 2).prod := by
+  unfold tpol
+  rw [eval_multiset_prod, Multiset.map_map]
+  congr 1
+  exact Multiset.map_congr rfl fun γ _ => by simp
+
+lemma res_tpol {K j : ℕ} (hj : j ∈ Icc 1 K) (M : Multiset ℤ) :
+    res K (tpol M) j =
+      (M.map fun γ : ℤ => (γ : ℚ) ^ 2 - (j : ℚ) ^ 2).prod /
+        ∏ i ∈ (Icc 1 K).erase j, ((i : ℚ) ^ 2 - (j : ℚ) ^ 2) := by
+  unfold res
+  rw [eval_derivative_D hj, eval_tpol]
+  congr 1
+  congr 1
+  exact Multiset.map_congr rfl fun γ _ => by ring
+
+lemma res_tpol_eq_zero {K j : ℕ} (hj : j ∈ Icc 1 K) {M : Multiset ℤ} {γ : ℤ} (hγ : γ ∈ M)
+    (h : γ ^ 2 = (j : ℤ) ^ 2) : res K (tpol M) j = 0 := by
+  rw [res_tpol hj, Multiset.prod_eq_zero, zero_div]
+  rw [Multiset.mem_map]
+  exact ⟨γ, hγ, by rw [sub_eq_zero]; exact_mod_cast h⟩
+
+/-- The numerator valuation: at least the number of roots `γ` with `γ² ≡ j²`. -/
+lemma VG_prod_sq_sub (M : Multiset ℤ) (j : ℤ) :
+    VG p ((M.map fun γ : ℤ => (γ : ℚ) ^ 2 - (j : ℚ) ^ 2).prod)
+      ((M.filter fun γ : ℤ => ((γ : ZMod p)) ^ 2 = (j : ZMod p) ^ 2).card) := by
+  induction M using Multiset.induction_on with
+  | empty => right; simp
+  | cons a M ih =>
+    rw [Multiset.map_cons, Multiset.prod_cons, Multiset.filter_cons]
+    split_ifs with h
+    · rw [Multiset.card_add, Multiset.card_singleton]
+      have h1 : VG p ((a : ℚ) ^ 2 - (j : ℚ) ^ 2) 1 := by
+        have :=
+          VG_int_one (p := p) (z := a ^ 2 - j ^ 2)
+            (by rw [← ZMod.intCast_zmod_eq_zero_iff_dvd]; push_cast; rw [h, sub_self])
+        simpa using this
+      convert h1.mul ih using 1; push_cast; ring
+    · rw [zero_add]
+      have h1 : VG p ((a : ℚ) ^ 2 - (j : ℚ) ^ 2) 0 := by
+        have := VG.intCast (p := p) (a ^ 2 - j ^ 2); push_cast at this; exact this
+      simpa using h1.mul ih
+
+/-- The denominator valuation for a nonzero class: at most the number of other nodes in the
+class (each factor has valuation `≤ 1`). -/
+lemma padicValRat_den_le (hp3 : 3 ≤ p) {K j : ℕ} (hj : j ∈ Icc 1 K) (hK : 2 * K < p ^ 2)
+    (hj0 : ¬(p : ℤ) ∣ (j : ℤ)) :
+    (padicValRat p (∏ i ∈ (Icc 1 K).erase j, ((i : ℚ) ^ 2 - (j : ℚ) ^ 2)) : ℚ) ≤
+      (((Icc 1 K).erase j).filter fun i : ℕ =>
+          ((i : ℤ) : ZMod p) ^ 2 = ((j : ℤ) : ZMod p) ^ 2).card := by
+  have hne : ∀ i ∈ (Icc 1 K).erase j, ((i : ℚ) ^ 2 - (j : ℚ) ^ 2) ≠ 0 := by
+    intro i hi
+    have hij := (Finset.mem_erase.mp hi).1
+    have : (i : ℚ) ≠ j := by exact_mod_cast hij
+    intro h
+    have h2 : ((i : ℚ) - j) * ((i : ℚ) + j) = 0 := by linear_combination h
+    rcases mul_eq_zero.mp h2 with h3 | h3
+    · exact this (by linarith)
+    · have : (0 : ℚ) ≤ i := by positivity
+      have : (1 : ℚ) ≤ j := by exact_mod_cast (Finset.mem_Icc.mp hj).1
+      linarith
+  rw [padicValRat_finset_prod _ _ hne]
+  push_cast
+  rw [Finset.card_filter]
+  push_cast
+  refine Finset.sum_le_sum fun i hi => ?_
+  have hi' := Finset.mem_erase.mp hi
+  have hiK := (Finset.mem_Icc.mp hi'.2)
+  have hjK := (Finset.mem_Icc.mp hj)
+  rw [show ((i : ℚ) ^ 2 - (j : ℚ) ^ 2) = (((i - j) * (i + j) : ℤ) : ℚ) by push_cast; ring]
+  split_ifs with hc
+  · -- exactly one factor divisible by `p`, with valuation `≤ 1`
+    rw [padicValRat.of_int]
+    have hz : ((i : ℤ) - j) * ((i : ℤ) + j) ≠ 0 := by
+      have := hne i hi
+      intro h; apply this;
+      rw [show ((i : ℚ) ^ 2 - (j : ℚ) ^ 2) = (((i - j) * (i + j) : ℤ) : ℚ) by
+      push_cast; ring, h];
+      simp
+    have hz1 := (mul_ne_zero_iff.mp hz).1
+    have hz2 := (mul_ne_zero_iff.mp hz).2
+    rw [padicValInt.mul hz1 hz2]
+    have hle : ∀ z : ℤ, z ≠ 0 → |z| ≤ 2 * K → padicValInt p z ≤ 1 := by
+      intro z hz0 hzK
+      have :=
+        padicValRat_int_le_one (p := p) (z := z)
+          (fun hd => by
+            have h1 := Int.natAbs_dvd_natAbs.mpr hd
+            rw [Int.natAbs_pow, Int.natAbs_natCast] at h1
+            have h2 := Nat.le_of_dvd (Int.natAbs_pos.mpr hz0) h1
+            have h3 : (z.natAbs : ℤ) ≤ 2 * K := by rw [Int.natCast_natAbs]; exact hzK
+            omega)
+      rw [padicValRat.of_int] at this; exact_mod_cast this
+    have hb1 : padicValInt p ((i : ℤ) - j) ≤ 1 :=
+      hle _ hz1 (by rw [abs_le]; constructor <;> omega)
+    have hb2 : padicValInt p ((i : ℤ) + j) ≤ 1 :=
+      hle _ hz2 (by rw [abs_le]; constructor <;> omega)
+    have hp2 : ¬((p : ℤ) ∣ ((i : ℤ) - j) ∧ (p : ℤ) ∣ ((i : ℤ) + j)) := by
+      rintro ⟨h1, h2⟩
+      apply hj0
+      have h3 : (p : ℤ) ∣ 2 * (j : ℤ) := by
+        have := dvd_sub h2 h1;
+        rw [show (i : ℤ) + j - (i - j) = 2 * j by ring] at this; exact this
+      rcases (Int.Prime.dvd_mul' hp.out h3) with h4 | h4
+      · exfalso
+        have : (p : ℤ) ≤ 2 := Int.le_of_dvd (by norm_num) h4
+        omega
+      · exact h4
+    by_cases hd1 : (p : ℤ) ∣ ((i : ℤ) - j)
+    · have hd2 : ¬(p : ℤ) ∣ ((i : ℤ) + j) := fun h => hp2 ⟨hd1, h⟩
+      have : padicValInt p ((i : ℤ) + j) = 0 := padicValInt.eq_zero_of_not_dvd hd2
+      rw [this]; push_cast;
+      linarith [(show (padicValInt p ((i : ℤ) - j) : ℚ) ≤ 1 by exact_mod_cast hb1)]
+    · have : padicValInt p ((i : ℤ) - j) = 0 := padicValInt.eq_zero_of_not_dvd hd1
+      rw [this]; push_cast;
+      linarith [(show (padicValInt p ((i : ℤ) + j) : ℚ) ≤ 1 by exact_mod_cast hb2)]
+  · have hnd : ¬(p : ℤ) ∣ ((i : ℤ) - j) * ((i : ℤ) + j) := by
+      intro hd
+      apply hc
+      rw [← ZMod.intCast_zmod_eq_zero_iff_dvd] at hd
+      push_cast at hd
+      linear_combination hd
+    rw [padicValRat_int_eq_zero hnd]; simp
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Arith/OuterEntryMain.lean b/LeanPool/Zeta5Irrational/Arith/OuterEntryMain.lean
new file mode 100644
index 0000000000..552418b37d
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Arith/OuterEntryMain.lean
@@ -0,0 +1,588 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Arith.OuterEntries
+public import LeanPool.Zeta5Irrational.Arith.OuterFrame
+import LeanPool.Zeta5Irrational.Arith.MomentVal
+import LeanPool.Zeta5Irrational.Arith.OuterEntry
+import LeanPool.Zeta5Irrational.Arith.PoleVal
+import Mathlib.Data.Int.Star
+import Mathlib.Data.Rat.Star
+import Mathlib.Tactic.ReduceModChar
+import Std.Tactic.BVDecide.Normalize.Prop
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+import Mathlib.Tactic.Positivity.Basic
+import Mathlib.Tactic.Ring.Basic
+
+/-! # Outer range: the entry theorem
+-/
+
+@[expose] public section
+
+open Finset Polynomial
+
+namespace Zeta5Irrational
+
+variable {p : ℕ} [hp : Fact p.Prime] {m : ℕ}
+
+/-- The `μ`-part below `2p - 3` of an integral polynomial is integral. -/
+lemma VG_μpoly_sub_μGe (hp5 : 5 ≤ p) {Q : ℚ[X]} (hQ : GV p Q 0) : VG p (μpoly Q - μGe p Q) 0 := by
+  have e : μpoly Q = ∑ e ∈ range (Q.natDegree + 1), Q.coeff e * μmono e := by
+    unfold μpoly;
+    rw [Polynomial.sum_over_range]; intro n; simp
+  rw [e, μGe, ← Finset.sum_sub_distrib]
+  refine VG.sum _ fun e _ => ?_
+  split_ifs with h
+  · simp [VG.zero]
+  · simpa using (hQ e).mul (VG_μmono_int hp5 (by omega))
+
+omit hp in
+/-- `μ_X` minus its high part, expanded. -/
+lemma μX_sub_μGe (K : ℕ) (P : ℚ[X]) :
+    μX K P - C (μGe p (P /ₘ D K)) =
+      C (μpoly (P /ₘ D K) - μGe p (P /ₘ D K)) + ∑ j ∈ Icc 1 K, C (res K P j) * poleValue j :=
+  by unfold μX res; rw [map_sub]; ring
+
+variable (hm : 2 * m + 1 = p) (n : ℕ)
+
+lemma entryO_eq (cs : Fin (m + 1)) (is : ℕ) (ct : Fin (m + 1)) (it : ℕ) :
+    D (3 * n) ^ 6 * (tpolZ (rootsO hm n cs is)).map (Int.castRingHom ℚ) *
+        (tpolZ (rootsO hm n ct it)).map (Int.castRingHom ℚ) =
+      tpol (entryO hm n cs is ct it) :=
+  by rw [tpolZ_map_Q, tpolZ_map_Q, D_eq_tpol, ← tpol_nsmul, ← tpol_add, ← tpol_add]; rfl
+
+/-- The integral numerator quotient. -/
+lemma GV_entry_quot (cs : Fin (m + 1)) (is : ℕ) (ct : Fin (m + 1)) (it : ℕ) :
+    GV p (tpol (entryO hm n cs is ct it) /ₘ D (40 * n)) 0 := by
+  rw [← tpolZ_map_Q];
+  exact GV_divByMonic_D _ _
+
+/-- Residues vanish at poles `j ≤ N` and at tail poles outside the common class. -/
+lemma res_entry_zero {cs ct : Fin (m + 1)} {is it : ℕ} {j : ℕ} (hj : j ∈ Icc 1 (40 * n))
+    (h : j ≤ 3 * n ∨ jc hm j ≠ cs ∨ jc hm j ≠ ct) :
+    res (40 * n) (tpol (entryO hm n cs is ct it)) j = 0 := by
+  have hj' := Finset.mem_Icc.mp hj
+  rcases h with h | h | h
+  · exact res_tpol_eq_zero hj (mem_entryO_small hm n (Finset.mem_Icc.mpr ⟨hj'.1, h⟩)) rfl
+  · by_cases hN : j ≤ 3 * n
+    · exact res_tpol_eq_zero hj (mem_entryO_small hm n (Finset.mem_Icc.mpr ⟨hj'.1, hN⟩)) rfl
+    · exact
+        res_tpol_eq_zero hj
+          (mem_entryO_left hm n (mem_rootsO_other hm n (Finset.mem_Icc.mpr ⟨by omega, hj'.2⟩) h))
+          rfl
+  · by_cases hN : j ≤ 3 * n
+    · exact res_tpol_eq_zero hj (mem_entryO_small hm n (Finset.mem_Icc.mpr ⟨hj'.1, hN⟩)) rfl
+    · exact
+        res_tpol_eq_zero hj
+          (mem_entryO_right hm n (mem_rootsO_other hm n (Finset.mem_Icc.mpr ⟨by omega, hj'.2⟩) h))
+          rfl
+
+/-- The pole sum reduces to the tail poles of the common class. -/
+lemma pole_sum_reduce (c : Fin (m + 1)) (is it : ℕ) :
+    ∑ j ∈ Icc 1 (40 * n), C (res (40 * n) (tpol (entryO hm n c is c it)) j) * poleValue j =
+      ∑ j ∈ tailC hm n c, C (res (40 * n) (tpol (entryO hm n c is c it)) j) * poleValue j := by
+  symm
+  apply Finset.sum_subset
+  · intro j hj
+    have := (Finset.mem_filter.mp hj).1
+    unfold tailO at this
+    exact
+      Finset.mem_Icc.mpr
+        ⟨by have := (Finset.mem_Icc.mp this).1; omega, (Finset.mem_Icc.mp this).2⟩
+  · intro j hj hjn
+    rw [res_entry_zero hm n hj, C_0, zero_mul]
+    by_contra h
+    push Not at h
+    apply hjn
+    exact Finset.mem_filter.mpr ⟨Finset.mem_Icc.mpr ⟨by omega, (Finset.mem_Icc.mp hj).2⟩, h.2.1⟩
+
+lemma pole_sum_cross {cs ct : Fin (m + 1)} (hc : cs ≠ ct) (is it : ℕ) :
+    ∑ j ∈ Icc 1 (40 * n), C (res (40 * n) (tpol (entryO hm n cs is ct it)) j) * poleValue j = 0 :=
+  by
+  refine Finset.sum_eq_zero fun j hj => ?_
+  rw [res_entry_zero hm n hj, C_0, zero_mul]
+  by_cases h : jc hm j = cs
+  · right; right; rw [h]; exact hc
+  · right; left; exact h
+
+/-- `ℓ(c) = #{j ≤ K : j ≡ ±c}`. -/
+def ellC (c : Fin (m + 1)) : ℕ :=
+  ((Icc 1 (40 * n)).filter fun j => jc hm j = c).card
+
+/-- `δ(c) = [c ≤ N]`. -/
+def delC (c : Fin (m + 1)) : ℕ :=
+  if (c : ℕ) ≤ 3 * n then 1 else 0
+
+/-- The outer weights (4.12). -/
+noncomputable def wO (c : Fin (m + 1)) (i : ℕ) : ℚ :=
+  if c = 0 then (if i < nbC hm n 0 then -2 else if nbC hm n 0 = 0 then -1 / 2 else 0)
+  else
+    (if i < nbC hm n c then min 0 ((i : ℚ) + 3 * delC n c - ((ellC hm n c : ℚ) + 4) / 2) else 0)
+
+lemma wO_nonpos (c : Fin (m + 1)) (i : ℕ) : wO hm n c i ≤ 0 := by
+  unfold wO;
+  split_ifs <;>
+    first
+    | exact min_le_left _ _
+    | norm_num
+
+lemma jc_eq_zero_iff (j : ℕ) : jc hm j = 0 ↔ (p : ℤ) ∣ (j : ℤ) := by
+  unfold jc;
+  rw [ccls_eq_zero_iff, ZMod.intCast_zmod_eq_zero_iff_dvd]
+
+/-- Denominator count for a nonzero class. -/
+lemma den_count {c : Fin (m + 1)} {j : ℕ} (hj : j ∈ Icc 1 (40 * n)) (hjc : jc hm j = c) :
+    (((Icc 1 (40 * n)).erase j).filter fun i : ℕ =>
+          ((i : ℤ) : ZMod p) ^ 2 = ((j : ℤ) : ZMod p) ^ 2).card =
+      ellC hm n c - 1 := by
+  have e :
+    (((Icc 1 (40 * n)).erase j).filter fun i : ℕ =>
+        ((i : ℤ) : ZMod p) ^ 2 = ((j : ℤ) : ZMod p) ^ 2) =
+      ((Icc 1 (40 * n)).filter fun i => jc hm i = c).erase j := by
+    rw [Finset.filter_erase]
+    congr 1
+    ext i
+    simp only [Finset.mem_filter, sq_eq_iff_ccls hm]
+    constructor
+    · rintro ⟨h1, h2⟩; exact ⟨h1, h2.trans hjc⟩
+    · rintro ⟨h1, h2⟩; exact ⟨h1, h2.trans hjc.symm⟩
+  rw [e, Finset.card_erase_of_mem (Finset.mem_filter.mpr ⟨hj, hjc⟩)]
+  rfl
+
+/-- Numerator count for a nonzero class. -/
+lemma num_count {c : Fin (m + 1)} (hc0 : c ≠ 0) {is it : ℕ} {j : ℕ} (hjc : jc hm j = c) :
+    6 * delC n c + is + it ≤
+      ((entryO hm n c is c it).filter fun γ : ℤ =>
+          ((γ : ZMod p)) ^ 2 = ((j : ℤ) : ZMod p) ^ 2).card := by
+  have hx : ((j : ℤ) : ZMod p) ^ 2 = ((((c : ℕ) : ℤ) : ZMod p)) ^ 2 := by rw [sq_jc hm j, hjc]
+  unfold entryO
+  rw [Multiset.filter_add, Multiset.filter_add, Multiset.card_add, Multiset.card_add]
+  have h1 := count_rootsO hm n c is _ hx
+  have h2 := count_rootsO hm n c it _ hx
+  have h3 :
+    6 * delC n c ≤
+      ((6 • ((Icc 1 (3 * n)).val.map fun j : ℕ => (j : ℤ))).filter fun γ : ℤ =>
+          ((γ : ZMod p)) ^ 2 = ((j : ℤ) : ZMod p) ^ 2).card := by
+    unfold delC
+    split_ifs with hc
+    · rw [Multiset.filter_nsmul, Multiset.card_nsmul]
+      have :
+        1 ≤
+          (((Icc 1 (3 * n)).val.map fun j : ℕ => (j : ℤ)).filter fun γ : ℤ =>
+              ((γ : ZMod p)) ^ 2 = ((j : ℤ) : ZMod p) ^ 2).card := by
+        apply Multiset.card_pos_iff_exists_mem.mpr
+        refine
+          ⟨((c : ℕ) : ℤ), Multiset.mem_filter.mpr ⟨Multiset.mem_map.mpr ⟨c, ?_, rfl⟩, hx.symm⟩⟩
+        have : (c : ℕ) ≠ 0 := fun h => hc0 (Fin.ext h)
+        exact Finset.mem_val.mpr (Finset.mem_Icc.mpr ⟨by omega, hc⟩)
+      omega
+    · omega
+  omega
+
+lemma ellC_pos {c : Fin (m + 1)} {j : ℕ} (hj : j ∈ Icc 1 (40 * n)) (hjc : jc hm j = c) :
+    1 ≤ ellC hm n c :=
+  Finset.card_pos.mpr ⟨j, Finset.mem_filter.mpr ⟨hj, hjc⟩⟩
+
+lemma mem_Icc_of_tailC {c : Fin (m + 1)} {j : ℕ} (hj : j ∈ tailC hm n c) : j ∈ Icc 1 (40 * n) := by
+  have := Finset.mem_Icc.mp (Finset.mem_filter.mp hj).1
+  exact Finset.mem_Icc.mpr ⟨by omega, this.2⟩
+
+/-- Residue bound on a nonzero class. -/
+lemma VG_res_entry (hp3 : 3 ≤ p) (hK : 2 * (40 * n) < p ^ 2) {c : Fin (m + 1)} (hc0 : c ≠ 0)
+    {is it : ℕ} {j : ℕ} (hj : j ∈ tailC hm n c) :
+    VG p (res (40 * n) (tpol (entryO hm n c is c it)) j)
+      ((6 * delC n c + is + it : ℕ) - ((ellC hm n c : ℚ) - 1)) := by
+  have hjc : jc hm j = c := (Finset.mem_filter.mp hj).2
+  have hjI := mem_Icc_of_tailC hm n hj
+  have hj0 : ¬(p : ℤ) ∣ (j : ℤ) := fun h => hc0 (hjc ▸ (jc_eq_zero_iff hm j).mpr h)
+  have h := VG_res_class hp3 hK (entryO hm n c is c it) hjI hj0
+  rw [den_count hm n hjI hjc] at h
+  refine h.mono ?_
+  have h1 := num_count hm n hc0 (is := is) (it := it) hjc
+  have h2 := ellC_pos hm n hjI hjc
+  push_cast [h2]
+  have :
+    ((6 * delC n c + is + it : ℕ) : ℚ) ≤
+      ((entryO hm n c is c it).filter fun γ : ℤ =>
+          ((γ : ZMod p)) ^ 2 = ((j : ℤ) : ZMod p) ^ 2).card :=
+    by exact_mod_cast h1
+  push_cast at this
+  linarith
+
+/-- **Case (a)**: both rows below `nb`. -/
+lemma pole_case_a (hp3 : 3 ≤ p) (hK : 2 * (40 * n) < p ^ 2) {c : Fin (m + 1)} (hc0 : c ≠ 0)
+    {is it : ℕ} (his : is < nbC hm n c) (hit : it < nbC hm n c) :
+    GV p (∑ j ∈ tailC hm n c, C (res (40 * n) (tpol (entryO hm n c is c it)) j) * poleValue j)
+      (wO hm n c is + wO hm n c it) := by
+  refine GV.sum _ fun j hj => ?_
+  have hjI := mem_Icc_of_tailC hm n hj
+  have hjI' := Finset.mem_Icc.mp hjI
+  have hPV := GV_poleValue_sq hp3 hjI'.1 (by nlinarith)
+  refine (GV.C_mul (VG_res_entry hm n hp3 hK hc0 hj) hPV).mono ?_
+  unfold wO
+  rw [ite_eq_right hc0, ite_eq_right hc0, ite_eq_left his, ite_eq_left hit]
+  have e1 := min_le_right (0 : ℚ) ((is : ℚ) + 3 * delC n c - ((ellC hm n c : ℚ) + 4) / 2)
+  have e2 := min_le_right (0 : ℚ) ((it : ℚ) + 3 * delC n c - ((ellC hm n c : ℚ) + 4) / 2)
+  push_cast
+  linarith
+
+lemma count_rootsO_zero (i : ℕ) : i ≤ ((rootsO hm n 0 i).filter fun γ : ℤ => (p : ℤ) ∣ γ).card := by
+  unfold rootsO
+  rw [Multiset.filter_add, Multiset.card_add]
+  have hz : (p : ℤ) ∣ (((0 : Fin (m + 1)) : ℕ) : ℤ) := by simp
+  split_ifs with h
+  · rw [card_filter_replicate, ite_eq_left hz]; omega
+  · rw [Multiset.filter_add, Multiset.card_add, card_filter_replicate, ite_eq_left hz]
+    have hbig :
+      ((bigC hm n 0).val.map fun j : ℕ => (j : ℤ)).filter (fun γ : ℤ => (p : ℤ) ∣ γ) =
+        (bigC hm n 0).val.map fun j : ℕ => (j : ℤ) := by
+      refine Multiset.filter_eq_self.mpr fun γ hγ => ?_
+      obtain ⟨j, hj, rfl⟩ := Multiset.mem_map.mp hγ
+      have hj' : j ∈ bigC hm n 0 := Finset.mem_val.mp hj
+      exact (jc_eq_zero_iff hm j).mp (Finset.mem_filter.mp (Finset.mem_filter.mp hj').1).2
+    rw [hbig, Multiset.card_map, Finset.card_val]
+    have : nbC hm n 0 = (bigC hm n 0).card := rfl
+    omega
+
+/-- Multiples of `p` up to `K` are `p` and the big zero-class poles. -/
+lemma mult_count (hpN : 3 * n < p) :
+    ((Icc 1 (40 * n)).filter fun i : ℕ => (p : ℤ) ∣ (i : ℤ)).card ≤ nbC hm n 0 + 1 := by
+  have hsub : ((Icc 1 (40 * n)).filter fun i : ℕ => (p : ℤ) ∣ (i : ℤ)) ⊆ insert p (bigC hm n 0) :=
+  by
+    intro i hi
+    rw [Finset.mem_filter, Finset.mem_Icc] at hi
+    rw [Finset.mem_insert]
+    by_cases hip : i ≤ p
+    · left
+      have hd : p ∣ i := by exact_mod_cast hi.2
+      have := Nat.le_of_dvd (by omega) hd
+      omega
+    · right
+      refine
+        Finset.mem_filter.mpr
+          ⟨Finset.mem_filter.mpr ⟨?_, (jc_eq_zero_iff hm i).mpr hi.2⟩, by omega⟩
+      unfold tailO; exact Finset.mem_Icc.mpr ⟨by omega, hi.1.2⟩
+  have := Finset.card_le_card hsub
+  have h2 := Finset.card_insert_le p (bigC hm n 0)
+  have : nbC hm n 0 = (bigC hm n 0).card := rfl
+  omega
+
+lemma nbC_zero_le (hK3 : 40 * n < 3 * p) : nbC hm n 0 ≤ 1 := by
+  have hsub : bigC hm n 0 ⊆ {2 * p} := by
+    intro j hj
+    have h1 := Finset.mem_filter.mp hj
+    have h2 := Finset.mem_filter.mp h1.1
+    have h3 := Finset.mem_Icc.mp h2.1
+    have hd : p ∣ j := by exact_mod_cast (jc_eq_zero_iff hm j).mp h2.2
+    obtain ⟨k, rfl⟩ := hd
+    have hk1 : 1 < k := by by_contra h; rw [not_lt] at h; interval_cases k <;> omega
+    have hk3 : k < 3 := by
+      by_contra h; rw [not_lt] at h
+      have : p * 3 ≤ p * k := Nat.mul_le_mul_left _ h
+      omega
+    rw [Finset.mem_singleton]
+    interval_cases k; ring
+  have := Finset.card_le_card hsub
+  rw [Finset.card_singleton] at this
+  exact this
+
+/-- **Zero class.** -/
+lemma pole_case_zero (hp7 : 7 ≤ p) (hK3 : 40 * n < 3 * p) (hK : 2 * (40 * n) < p ^ 2)
+    (hpN : 3 * n < p) (is it : ℕ) :
+    GV p (∑ j ∈ tailC hm n 0, C (res (40 * n) (tpol (entryO hm n 0 is 0 it)) j) * poleValue j)
+      (wO hm n 0 is + wO hm n 0 it) := by
+  have hnb := nbC_zero_le hm n hK3
+  refine GV.sum _ fun j hj => ?_
+  have hjI := mem_Icc_of_tailC hm n hj
+  have hjI' := Finset.mem_Icc.mp hjI
+  have hjc : jc hm j = 0 := (Finset.mem_filter.mp hj).2
+  have hpj : (p : ℤ) ∣ (j : ℤ) := (jc_eq_zero_iff hm j).mp hjc
+  obtain ⟨k, hk⟩ : ∃ k, j = p * k := by exact_mod_cast hpj
+  have hk1 : 1 ≤ k := by
+    rcases Nat.eq_zero_or_pos k with h | h
+    · subst h; simp at hk; omega
+    · exact h
+  have hkp : k < p := by
+    by_contra h; rw [not_lt] at h
+    have : p * p ≤ p * k := Nat.mul_le_mul_left _ h
+    nlinarith
+  have hPV : GV p (poleValue j) (-1) := by
+    rw [hk];
+    exact
+      GV_poleValue_mul (by omega) hk1
+        hkp
+          -- residue bound
+  have hden : ∏ i ∈ (Icc 1 (40 * n)).erase j, ((i : ℚ) ^ 2 - (j : ℚ) ^ 2) ≠ 0 :=
+    Finset.prod_ne_zero_iff.mpr fun i hi => sq_ne_of_ne (Finset.mem_erase.mp hi).1
+  have hnum := VG_prod_sq_sub_zero (p := p) (entryO hm n 0 is 0 it) hpj
+  have hdv := VG.inv (p := p) hden (padicValRat_den_zero_le hjI hK hpj)
+  have hres :
+    VG p (res (40 * n) (tpol (entryO hm n 0 is 0 it)) j) (2 * (is + it : ℚ) - 2 * nbC hm n 0) := by
+    rw [res_tpol hjI, div_eq_mul_inv]
+    refine (hnum.mul hdv).mono ?_
+    have hc1 : (is + it : ℕ) ≤ ((entryO hm n 0 is 0 it).filter fun γ : ℤ => (p : ℤ) ∣ γ).card := by
+      unfold entryO
+      rw [Multiset.filter_add, Multiset.filter_add, Multiset.card_add, Multiset.card_add]
+      have := count_rootsO_zero hm n is
+      have := count_rootsO_zero hm n it
+      omega
+    have hc2 :
+      (((Icc 1 (40 * n)).erase j).filter fun i : ℕ => (p : ℤ) ∣ (i : ℤ)).card ≤ nbC hm n 0 := by
+      have h1 := mult_count hm n hpN
+      rw [Finset.filter_erase] at *
+      rw [Finset.card_erase_of_mem (Finset.mem_filter.mpr ⟨hjI, hpj⟩)]
+      omega
+    have e1 :
+      ((is + it : ℕ) : ℚ) ≤
+        (((entryO hm n 0 is 0 it).filter fun γ : ℤ => (p : ℤ) ∣ γ).card : ℚ) :=
+      by exact_mod_cast hc1
+    have e2 :
+      ((((Icc 1 (40 * n)).erase j).filter fun i : ℕ => (p : ℤ) ∣ (i : ℤ)).card : ℚ) ≤
+        nbC hm n 0 :=
+      by exact_mod_cast hc2
+    push_cast at e1 ⊢
+    linarith
+  refine (GV.C_mul hres hPV).mono ?_
+  unfold wO
+  rw [ite_eq_left rfl, ite_eq_left rfl]
+  interval_cases h : nbC hm n 0
+  · simp
+    have : (0 : ℚ) ≤ is := by positivity
+    have : (0 : ℚ) ≤ it := by positivity
+    linarith
+  · by_cases h1 : is < 1 <;> by_cases h2 : it < 1
+    · rw [ite_eq_left h1, ite_eq_left h2]; push_cast; have : is = 0 := by omega
+      have : it = 0 := by omega
+      subst_vars; norm_num
+    · rw [ite_eq_left h1, ite_eq_right h2, ite_eq_right (by norm_num)]
+      have : (1 : ℚ) ≤ it := by exact_mod_cast (by omega : 1 ≤ it)
+      have : (0 : ℚ) ≤ is := by positivity
+      push_cast; linarith
+    · rw [ite_eq_right h1, ite_eq_left h2, ite_eq_right (by norm_num)]
+      have : (1 : ℚ) ≤ is := by exact_mod_cast (by omega : 1 ≤ is)
+      have : (0 : ℚ) ≤ it := by positivity
+      push_cast; linarith
+    · rw [ite_eq_right h1, ite_eq_right h2, ite_eq_right (by norm_num)]
+      have : (1 : ℚ) ≤ is := by exact_mod_cast (by omega : 1 ≤ is)
+      have : (1 : ℚ) ≤ it := by exact_mod_cast (by omega : 1 ≤ it)
+      push_cast; linarith
+
+lemma jc_small {i : ℕ} (hi : i < p) :
+    ((jc hm i : ℕ) = i ∧ i ≤ m) ∨ ((jc hm i : ℕ) = p - i ∧ m < i) := by
+  have hv : (((i : ℤ) : ZMod p)).val = i := by
+    rw [Int.cast_natCast, ZMod.val_natCast, Nat.mod_eq_of_lt hi]
+  unfold jc ccls
+  split_ifs with h
+  · left; rw [hv] at h; exact ⟨by simp [Nat.mod_eq_of_lt hi], h⟩
+  · right; rw [hv] at h; exact ⟨by simp [Nat.mod_eq_of_lt hi], by omega⟩
+
+lemma jc_small_eq {c : Fin (m + 1)} {i : ℕ} (hi : i < p) (h : jc hm i = c) : i = c ∨ i = p - c := by
+  have hc := c.isLt
+  rcases jc_small hm hi with ⟨h1, _⟩ | ⟨h1, _⟩ <;> rw [h] at h1 <;> omega
+
+/-- Class members up to `K`: the big ones plus at most the two small ones. -/
+lemma ellC_le (hpN : 3 * n < p) {c : Fin (m + 1)} (hc0 : c ≠ 0) :
+    ellC hm n c ≤ nbC hm n c + ((Icc 1 (40 * n)).filter fun i => jc hm i = c ∧ i < p).card := by
+  have hsub :
+    ((Icc 1 (40 * n)).filter fun j => jc hm j = c) ⊆
+      bigC hm n c ∪ ((Icc 1 (40 * n)).filter fun i => jc hm i = c ∧ i < p) := by
+    intro i hi
+    obtain ⟨hiI, hic⟩ := Finset.mem_filter.mp hi
+    have hiI' := Finset.mem_Icc.mp hiI
+    rw [Finset.mem_union]
+    by_cases hip : i < p
+    · right; exact Finset.mem_filter.mpr ⟨hiI, hic, hip⟩
+    · left
+      have hne : i ≠ p := by
+        intro hh; rw [hh] at hic
+        exact hc0 (hic ▸ (jc_eq_zero_iff hm p).mpr dvd_rfl)
+      refine Finset.mem_filter.mpr ⟨Finset.mem_filter.mpr ⟨?_, hic⟩, by omega⟩
+      unfold tailO; exact Finset.mem_Icc.mpr ⟨by omega, hiI'.2⟩
+  have := Finset.card_le_card hsub
+  have :=
+    Finset.card_union_le (bigC hm n c) ((Icc 1 (40 * n)).filter fun i => jc hm i = c ∧ i < p)
+  unfold ellC nbC
+  omega
+
+lemma small_sub {c : Fin (m + 1)} :
+    ((Icc 1 (40 * n)).filter fun i => jc hm i = c ∧ i < p) ⊆ {(c : ℕ), p - c} := by
+  intro i hi
+  obtain ⟨_, hic, hip⟩ := Finset.mem_filter.mp hi
+  rcases jc_small_eq hm hip hic with h | h <;> simp [h]
+
+lemma small_card_le {c : Fin (m + 1)} :
+    ((Icc 1 (40 * n)).filter fun i => jc hm i = c ∧ i < p).card ≤ 2 :=
+  (Finset.card_le_card (small_sub hm n)).trans (Finset.card_le_two)
+
+/-- **Case (b)**: one row at or above `nb`; the pole sum is integral. -/
+lemma pole_case_b (hp3 : 3 ≤ p) (hK : 2 * (40 * n) < p ^ 2) (hpN : 3 * n < p) {c : Fin (m + 1)}
+    (hc0 : c ≠ 0) {is it : ℕ} (his : nbC hm n c ≤ is) :
+    GV p (∑ j ∈ tailC hm n c, C (res (40 * n) (tpol (entryO hm n c is c it)) j) * poleValue j)
+      0 := by
+  have hcm := c.isLt
+  have hc1 : 1 ≤ (c : ℕ) := by
+    rcases Nat.eq_zero_or_pos (c : ℕ) with h | h
+    · exact absurd (Fin.ext h) hc0
+    · exact h
+  have hℓ := ellC_le hm n hpN hc0
+  have hS2 := small_card_le hm n (c := c)
+  set Sm := (Icc 1 (40 * n)).filter fun i => jc hm i = c ∧ i < p with hSm
+    -- big poles have zero residue
+  have hbig : ∀ j ∈ tailC hm n c, ¬j < p → res (40 * n) (tpol (entryO hm n c is c it)) j = 0 := by
+    intro j hj hjp
+    have hjc : jc hm j = c := (Finset.mem_filter.mp hj).2
+    have hne : j ≠ p := by
+      intro hh; rw [hh] at hjc
+      exact hc0 (hjc ▸ (jc_eq_zero_iff hm p).mpr dvd_rfl)
+    have hjb : j ∈ bigC hm n c := Finset.mem_filter.mpr ⟨hj, by omega⟩
+    exact
+      res_tpol_eq_zero (mem_Icc_of_tailC hm n hj)
+        (mem_entryO_left hm n (mem_rootsO_big hm n his hjb)) rfl
+  rw [← Finset.sum_filter_add_sum_filter_not _ (fun j => j < p)]
+  rw [Finset.sum_eq_zero (s := (tailC hm n c).filter fun j => ¬j < p)
+      (fun j hj => by
+        have := Finset.mem_filter.mp hj
+        rw [hbig j this.1 this.2, C_0, zero_mul]),
+    add_zero]
+    -- individual small-pole bound
+  have hind :
+    ∀ j ∈ (tailC hm n c).filter (fun j => j < p),
+      (0 : ℚ) ≤ ((6 * delC n c + is + it : ℕ) : ℚ) - ((ellC hm n c : ℚ) - 1) →
+        GV p (C (res (40 * n) (tpol (entryO hm n c is c it)) j) * poleValue j) 0 := by
+    intro j hj h0
+    have hj' := Finset.mem_filter.mp hj
+    have hjI' := Finset.mem_Icc.mp (mem_Icc_of_tailC hm n hj'.1)
+    refine
+      (GV.C_mul (VG_res_entry hm n hp3 hK hc0 hj'.1) (GV_poleValue_lt hp3 hjI'.1 hj'.2)).mono ?_
+    linarith
+  by_cases hδ : delC n c = 1
+  · refine GV.sum _ fun j hj => hind j hj ?_
+    have : (ellC hm n c : ℚ) ≤ nbC hm n c + 2 := by
+      exact_mod_cast (by omega : ellC hm n c ≤ nbC hm n c + 2)
+    have : (nbC hm n c : ℚ) ≤ is := by exact_mod_cast his
+    rw [hδ]; push_cast; linarith
+  have hcN : 3 * n < (c : ℕ) := by unfold delC at hδ; split_ifs at hδ with h <;> omega
+  have hδ0 : delC n c = 0 := by
+    unfold delC;
+    rw [ite_eq_right (by omega)]
+      -- small class members all lie in the tail
+  have hSmT : Sm ⊆ (tailC hm n c).filter (fun j => j < p) := by
+    intro i hi
+    obtain ⟨hiI, hic, hip⟩ := Finset.mem_filter.mp hi
+    have hiI' := Finset.mem_Icc.mp hiI
+    have : 3 * n < i := by rcases jc_small_eq hm hip hic with h | h <;> omega
+    have hiT : i ∈ tailO n := by unfold tailO; exact Finset.mem_Icc.mpr ⟨by omega, hiI'.2⟩
+    exact Finset.mem_filter.mpr ⟨Finset.mem_filter.mpr ⟨hiT, hic⟩, hip⟩
+  have hTSm : (tailC hm n c).filter (fun j => j < p) ⊆ Sm := by
+    intro i hi
+    have h1 := Finset.mem_filter.mp hi
+    exact Finset.mem_filter.mpr ⟨mem_Icc_of_tailC hm n h1.1, (Finset.mem_filter.mp h1.1).2, h1.2⟩
+  have hTeq : (tailC hm n c).filter (fun j => j < p) = Sm := Finset.Subset.antisymm hTSm hSmT
+  by_cases hS1 : Sm.card ≤ 1
+  · refine GV.sum _ fun j hj => hind j hj ?_
+    have : (ellC hm n c : ℚ) ≤ nbC hm n c + 1 := by
+      exact_mod_cast (by omega : ellC hm n c ≤ nbC hm n c + 1)
+    have : (nbC hm n c : ℚ) ≤ is := by exact_mod_cast his
+    push_cast;
+    linarith
+      -- the pair case
+  have hSeq : Sm = {(c : ℕ), p - c} :=
+    Finset.eq_of_subset_of_card_le (small_sub hm n) ((Finset.card_le_two).trans (by omega))
+  have hcne : (c : ℕ) ≠ p - c := by omega
+  have hmemc : (c : ℕ) ∈ Sm := by rw [hSeq]; simp
+  have hmemb : p - (c : ℕ) ∈ Sm := by rw [hSeq]; simp
+  have hcK := (Finset.mem_Icc.mp (Finset.mem_filter.mp hmemc).1).2
+  have hbK := (Finset.mem_Icc.mp (Finset.mem_filter.mp hmemb).1).2
+  rw [hTeq, hSeq, Finset.sum_pair hcne]
+  refine pair_GV hp3 _ hc1 (by omega) hcne hcK hbK ?_ ?_
+  · intro i hi hia hib
+    have hi' := Finset.mem_Icc.mp hi
+    by_cases hiN : i ≤ 3 * n
+    · exact mem_entryO_small hm n (Finset.mem_Icc.mpr ⟨hi'.1, hiN⟩)
+    have hiT : i ∈ tailO n := by unfold tailO; exact Finset.mem_Icc.mpr ⟨by omega, hi'.2⟩
+    by_cases hic : jc hm i = c
+    · refine mem_entryO_left hm n (mem_rootsO_big hm n his ?_)
+      have hne : i ≠ p := by
+        intro hh; rw [hh] at hic
+        exact hc0 (hic ▸ (jc_eq_zero_iff hm p).mpr dvd_rfl)
+      have hip : ¬i < p := fun hip =>
+        by
+        rcases jc_small_eq hm hip hic with h | h
+        · exact hia h
+        · exact hib h
+      exact Finset.mem_filter.mpr ⟨Finset.mem_filter.mpr ⟨hiT, hic⟩, by omega⟩
+    · exact mem_entryO_left hm n (mem_rootsO_other hm n hiT hic)
+  · have hjb : p - (c : ℕ) ∈ tailC hm n c := (Finset.mem_filter.mp (hSmT hmemb)).1
+    refine (VG_res_entry hm n hp3 hK hc0 hjb).mono ?_
+    have : (ellC hm n c : ℚ) ≤ nbC hm n c + 2 := by
+      exact_mod_cast (by omega : ellC hm n c ≤ nbC hm n c + 2)
+    have : (nbC hm n c : ℚ) ≤ is := by exact_mod_cast his
+    push_cast; linarith
+
+lemma entryO_comm (c : Fin (m + 1)) (is it : ℕ) : entryO hm n c is c it = entryO hm n c it c is :=
+  by unfold entryO; rw [add_right_comm]
+
+/-- **The outer entry theorem.** -/
+theorem outer_entry (hp7 : 7 ≤ p) (hK3 : 40 * n < 3 * p) (hK : 2 * (40 * n) < p ^ 2)
+    (hpN : 3 * n < p) (cs : Fin (m + 1)) (is : ℕ) (ct : Fin (m + 1)) (it : ℕ) :
+    GV p
+      (μX (40 * n) (tpol (entryO hm n cs is ct it)) -
+        C (μGe p (tpol (entryO hm n cs is ct it) /ₘ D (40 * n))))
+      (wO hm n cs is + wO hm n ct it) := by
+  have hp3 : 3 ≤ p := by omega
+  have hw := add_nonpos (wO_nonpos hm n cs is) (wO_nonpos hm n ct it)
+  rw [μX_sub_μGe]
+  refine GV.add ((GV.C (VG_μpoly_sub_μGe (by omega) (GV_entry_quot hm n cs is ct it))).mono hw) ?_
+  by_cases hc : cs = ct
+  · subst hc
+    rw [pole_sum_reduce]
+    by_cases hc0 : cs = 0
+    · subst hc0; exact pole_case_zero hm n hp7 hK3 hK hpN is it
+    by_cases his : nbC hm n cs ≤ is
+    · exact (pole_case_b hm n hp3 hK hpN hc0 his).mono hw
+    by_cases hit : nbC hm n cs ≤ it
+    · rw [entryO_comm]; exact (pole_case_b hm n hp3 hK hpN hc0 hit).mono hw
+    exact pole_case_a hm n hp3 hK hc0 (by omega) (by omega)
+  · rw [pole_sum_cross hm n hc]; exact (GV.zero _).mono hw
+
+/-- **The outer-range bound** `v^G(Δ) ≥ 2 ∑ w − r`. -/
+theorem outer_bound (hp7 : 7 ≤ p) (hK3 : 40 * n < 3 * p) (hK : 2 * (40 * n) < p ^ 2)
+    (hpN : 3 * n < p) :
+    GV p (Δ n)
+      (2 * ∑ c, ∑ i ∈ range (LoC hm n c), wO hm n c i -
+        ((37 * n - ((2 * p + 22 * n - 3) - (37 * n - 1)) : ℕ) : ℚ)) := by
+  classical
+  have hcard : Fintype.card (Σ c : Fin (m + 1), Fin (LoC hm n c)) = 37 * n := by
+    rw [Fintype.card_sigma]; simpa using sum_LoC hm n
+  set e : (Σ c : Fin (m + 1), Fin (LoC hm n c)) ≃ Fin (37 * n) := Fintype.equivFinOfCardEq hcard
+  have h :=
+    outer_frame (p := p) (by omega) n (fun a => tpolZ (rootsO hm n (e.symm a).1 (e.symm a).2))
+      (fun a => (natDegree_tpolZ _).trans_lt (card_rootsO hm n _ (e.symm a).2.isLt))
+      (fun c : Fin (m + 1) => γO (p := p) c)
+      (fun a b hab => sq_injective hm (by simpa [γO] using hab)) (LoC hm n)
+      (fun a => (e.symm a).1) (fun a => (e.symm a).2) (fun a => (e.symm a).2.isLt)
+      (fun a b h1 h2 => by
+        apply e.symm.injective
+        exact Sigma.ext h1 (by rw [Fin.heq_ext_iff (by rw [h1])]; exact h2))
+      (fun a => rootsO_map_ZMod hm n _ _) (fun a => wO hm n (e.symm a).1 (e.symm a).2)
+      (fun a => wO_nonpos hm n _ _)
+      (fun a b => by
+        rw [entryO_eq]
+        exact outer_entry hm n hp7 hK3 hK hpN _ _ _ _)
+  have hsum :
+    ∑ a, wO hm n (e.symm a).1 (e.symm a).2 = ∑ c, ∑ i ∈ range (LoC hm n c), wO hm n c i := by
+    rw [Equiv.sum_comp e.symm (fun s => wO hm n s.1 s.2), Fintype.sum_sigma]
+    refine Finset.sum_congr rfl fun c _ => ?_
+    exact Fin.sum_univ_eq_sum_range (fun i => wO hm n c i) _
+  rw [hsum] at h
+  exact h
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Arith/OuterFrame.lean b/LeanPool/Zeta5Irrational/Arith/OuterFrame.lean
new file mode 100644
index 0000000000..c3450fce6c
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Arith/OuterFrame.lean
@@ -0,0 +1,260 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Arith.Val
+public import LeanPool.Zeta5Irrational.Construction
+import LeanPool.Zeta5Irrational.Arith.ClassFrame
+import LeanPool.Zeta5Irrational.Arith.LowRank
+import LeanPool.Zeta5Irrational.Arith.MomentVal
+import LeanPool.Zeta5Irrational.Arith.Unimodular
+import LeanPool.Zeta5Irrational.PartialFractions
+import Mathlib.Algebra.Ring.IsFormallyReal
+import Mathlib.Data.Rat.Star
+import Mathlib.RingTheory.PiTensorProduct
+import Mathlib.Tactic.ENatToNat
+import Mathlib.Tactic.ReduceModChar
+import Mathlib.Tactic.FieldSimp
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+import Mathlib.Tactic.Ring.Basic
+
+/-! # The outer-range framework
+
+`μ = μ_{<} + μ_{≥}` where `μ_{≥}` collects the moments `μ(t^e)` with `e ≥ 2p - 3` (which may have
+`v_p = -1`). In any basis the `μ_{≥}`-part of the Hankel matrix has rank `≤ r` and is `p⁻¹` times
+an integral product `U V`; with weights `w ≤ 0` for the integral part this gives
+`v_p^G(Δ) ≥ 2 ∑ w - r`.
+-/
+
+@[expose] public section
+
+open Finset Polynomial
+
+namespace Zeta5Irrational
+
+variable {p : ℕ} [hp : Fact p.Prime]
+
+/-- The moments `e ≥ 2p - 3` of a polynomial. -/
+noncomputable def μGe (p : ℕ) (Q : ℚ[X]) : ℚ :=
+  ∑ e ∈ range (Q.natDegree + 1), if 2 * p ≤ e + 3 then Q.coeff e * μmono e else 0
+
+lemma μGe_eq (p : ℕ) (Q : ℚ[X]) {N : ℕ} (hN : Q.natDegree < N) :
+    μGe p Q = ∑ e ∈ range N, if 2 * p ≤ e + 3 then Q.coeff e * μmono e else 0 := by
+  unfold μGe
+  apply Finset.sum_subset (Finset.range_subset_range.mpr (by omega))
+  intro e _ he
+  rw [Finset.mem_range] at he
+  rw [coeff_eq_zero_of_natDegree_lt (by omega)]; simp
+
+omit hp in
+lemma μGe_add (P Q : ℚ[X]) : μGe p (P + Q) = μGe p P + μGe p Q := by
+  set N := max (max P.natDegree Q.natDegree) (P + Q).natDegree + 1 with hN
+  rw [μGe_eq p (P + Q) (N := N) (by omega), μGe_eq p P (N := N) (by omega),
+    μGe_eq p Q (N := N) (by omega), ← Finset.sum_add_distrib]
+  refine Finset.sum_congr rfl fun e _ => ?_
+  split_ifs <;> simp [add_mul]
+
+omit hp in
+lemma μGe_C_mul (c : ℚ) (Q : ℚ[X]) : μGe p (C c * Q) = c * μGe p Q := by
+  set N := max Q.natDegree (C c * Q).natDegree + 1 with hN
+  rw [μGe_eq p (C c * Q) (N := N) (by omega), μGe_eq p Q (N := N) (by omega), Finset.mul_sum]
+  refine Finset.sum_congr rfl fun e _ => ?_
+  split_ifs <;> simp [coeff_C_mul]; ring
+
+omit hp in
+lemma μGe_eq_zero {Q : ℚ[X]} (hQ : Q.natDegree + 3 < 2 * p) : μGe p Q = 0 := by
+  unfold μGe
+  refine Finset.sum_eq_zero fun e he => ?_
+  rw [Finset.mem_range] at he
+  rw [ite_eq_right (by omega)]
+
+lemma VG_p_μGe (hp5 : 5 ≤ p) {Q : ℚ[X]} (hQ : GV p Q 0) : VG p ((p : ℚ) * μGe p Q) 0 := by
+  have h : VG p (μGe p Q) (-1) := by
+    unfold μGe
+    refine VG.sum _ fun e _ => ?_
+    split_ifs
+    · simpa using (hQ e).mul (VG_μmono hp5 e)
+    · exact VG.zero _
+  simpa using (VG.primePow (p := p) 1).mul h
+
+omit hp in
+/-- The quotient of an integer polynomial by `D_K` has integral coefficients. -/
+lemma GV_divByMonic_D (K : ℕ) (Pz : ℤ[X]) : GV p (Pz.map (Int.castRingHom ℚ) /ₘ D K) 0 := by
+  have hD : D K = (tpolZ ((Icc 1 K).val.map fun j : ℕ => (j : ℤ))).map (Int.castRingHom ℚ) := by
+    rw [tpolZ_map_Q, D_eq_tpol]
+  have hmon : (tpolZ ((Icc 1 K).val.map fun j : ℕ => (j : ℤ))).Monic := by
+    unfold tpolZ
+    exact monic_multiset_prod_of_monic _ _ fun γ _ => monic_X_add_C _
+  rw [hD, ← map_divByMonic _ hmon]
+  intro n
+  rw [coeff_map]
+  exact VG.intCast _
+
+omit hp in
+lemma μGe_sum {ι : Type*} (s : Finset ι) (f : ι → ℚ[X]) :
+    μGe p (∑ i ∈ s, f i) = ∑ i ∈ s, μGe p (f i) := by
+  classical
+    induction s using Finset.induction_on with
+  | empty => simp [μGe]
+  | insert a s ha ih => rw [Finset.sum_insert ha, Finset.sum_insert ha, μGe_add, ih]
+
+lemma divByMonic_sum' {q : ℚ[X]} (hq : q.Monic) {ι : Type*} (s : Finset ι) (f : ι → ℚ[X]) :
+    (∑ i ∈ s, f i) /ₘ q = ∑ i ∈ s, f i /ₘ q := by
+  classical
+    induction s using Finset.induction_on with
+  | empty => simp
+  | insert a s ha ih => rw [Finset.sum_insert ha, Finset.sum_insert ha, divByMonic_add' hq, ih]
+
+lemma D_pow_mul_X_pow_int (N e : ℕ) : ∃ Pz : ℤ[X], Pz.map (Int.castRingHom ℚ) = D N ^ 6 * X ^ e :=
+  by
+  refine ⟨tpolZ ((Icc 1 N).val.map fun j : ℕ => (j : ℤ)) ^ 6 * X ^ e, ?_⟩
+  rw [Polynomial.map_mul, Polynomial.map_pow, Polynomial.map_pow, Polynomial.map_X, tpolZ_map_Q,
+    D_eq_tpol]
+
+/-- **The outer-range framework.** -/
+theorem outer_frame (hp5 : 5 ≤ p) (n : ℕ) (Ez : Fin (37 * n) → ℤ[X])
+    (hdegZ : ∀ s, (Ez s).natDegree < 37 * n) {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (γ : ι → ZMod p) (hγ : Function.Injective γ) (Lc : ι → ℕ) (cls : Fin (37 * n) → ι)
+    (idx : Fin (37 * n) → ℕ) (hidx : ∀ a, idx a < Lc (cls a))
+    (hinj : ∀ a b, cls a = cls b → idx a = idx b → a = b)
+    (hred :
+      ∀ a,
+        (Ez a).map (Int.castRingHom (ZMod p)) =
+          (∏ c ∈ Finset.univ.erase (cls a), (X - C (γ c)) ^ Lc c) * (X - C (γ (cls a))) ^ idx a)
+    (w : Fin (37 * n) → ℚ) (hw : ∀ s, w s ≤ 0)
+    (hA :
+      ∀ s t,
+        GV p
+          (μX (40 * n)
+              (D (3 * n) ^ 6 * (Ez s).map (Int.castRingHom ℚ) * (Ez t).map (Int.castRingHom ℚ)) -
+            C
+              (μGe p
+                ((D (3 * n) ^ 6 * (Ez s).map (Int.castRingHom ℚ) *
+                    (Ez t).map (Int.castRingHom ℚ)) /ₘ
+                  D (40 * n))))
+          (w s + w t)) :
+    GV p (Δ n) (2 * ∑ s, w s - ((37 * n - ((2 * p + 22 * n - 3) - (37 * n - 1)) : ℕ) : ℚ)) := by
+  classical
+  set k0 := (2 * p + 22 * n - 3) - (37 * n - 1) with hk0
+  set r := 37 * n - k0 with hr
+  set E : Fin (37 * n) → ℚ[X] := fun s => (Ez s).map (Int.castRingHom ℚ) with hE
+  have hdegE : ∀ s, (E s).natDegree < 37 * n := fun s => (natDegree_map_le).trans_lt (hdegZ s)
+  have hunit := coeffMat_det_unit Ez hdegZ γ hγ Lc cls idx hidx hinj hred
+  set dC := (coeffMat E).det with hdC
+  have hbc := det_basis_change n E hdegE
+  set Cq := coeffMat E with hCq
+  have hCqint : ∀ s k, VG p (Cq s k) 0 := fun s k => by
+    simp only [hCq, coeffMat, Matrix.of_apply, hE, coeff_map]; exact VG.intCast _
+  set β0 : ℕ → ℚ := fun e => μGe p ((D (3 * n) ^ 6 * X ^ e) /ₘ D (40 * n)) with hβ0
+  have hβ0int : ∀ e, VG p ((p : ℚ) * β0 e) 0 := fun e =>
+    by
+    obtain ⟨Pz, hPz⟩ := D_pow_mul_X_pow_int (3 * n) e
+    have := GV_divByMonic_D (p := p) (40 * n) Pz
+    rw [hPz] at this
+    exact VG_p_μGe hp5 this
+  have hβ0zero : ∀ e, e + 18 * n < 2 * p - 3 + 40 * n → β0 e = 0 := by
+    intro e he
+    apply μGe_eq_zero
+    rw [natDegree_divByMonic _ (D_monic _),
+      natDegree_mul (pow_ne_zero _ (D_monic _).ne_zero) (pow_ne_zero _ X_ne_zero), natDegree_pow,
+      natDegree_D, natDegree_D, natDegree_X_pow]
+    omega
+      -- the correction as a double sum
+  have hβ :
+    ∀ s t,
+      μGe p ((D (3 * n) ^ 6 * E s * E t) /ₘ D (40 * n)) =
+        ∑ k : Fin (37 * n), ∑ l : Fin (37 * n), Cq s k * Cq t l * β0 (k + l) := by
+    intro s t
+    have hexp :
+      D (3 * n) ^ 6 * E s * E t =
+        ∑ k : Fin (37 * n),
+          ∑ l : Fin (37 * n), C (Cq s k * Cq t l) * (D (3 * n) ^ 6 * X ^ ((k : ℕ) + (l : ℕ))) := by
+      rw [sum_coeffMat E hdegE s, sum_coeffMat E hdegE t]
+      simp only [Finset.mul_sum, Finset.sum_mul]
+      rw [Finset.sum_comm]
+      refine Finset.sum_congr rfl fun k _ => Finset.sum_congr rfl fun l _ => ?_
+      rw [C_mul, pow_add]; ring
+    rw [hexp, divByMonic_sum' (D_monic _), μGe_sum]
+    refine Finset.sum_congr rfl fun k _ => ?_
+    rw [divByMonic_sum' (D_monic _), μGe_sum]
+    refine Finset.sum_congr rfl fun l _ => ?_
+    rw [divByMonic_C_mul' (D_monic _), μGe_C_mul]
+      -- the low-rank factorization
+  set U : Matrix (Fin (37 * n)) (Fin r) ℚ := fun s u => Cq s ⟨k0 + u, by omega⟩ with hU
+  set V : Matrix (Fin r) (Fin (37 * n)) ℚ := fun u t =>
+    (p : ℚ) * ∑ l : Fin (37 * n), β0 (k0 + u + l) * Cq t l with hV
+  have hp0 : (p : ℚ) ≠ 0 := by exact_mod_cast hp.out.ne_zero
+  have hUV :
+    ∀ s t, μGe p ((D (3 * n) ^ 6 * E s * E t) /ₘ D (40 * n)) = ((p : ℚ)⁻¹ • (U * V)) s t := by
+    intro s t
+    rw [hβ, Matrix.smul_apply, Matrix.mul_apply, smul_eq_mul, Finset.mul_sum]
+    set F : Fin (37 * n) → ℚ := fun k => ∑ l : Fin (37 * n), Cq s k * Cq t l * β0 (k + l)
+    have hF0 : ∀ k : Fin (37 * n), (k : ℕ) < k0 → F k = 0 := by
+      intro k hk
+      refine Finset.sum_eq_zero fun l _ => ?_
+      rw [hβ0zero _ (by have := l.isLt; omega), mul_zero]
+    set F' : ℕ → ℚ := fun k => if hk : k < 37 * n then F ⟨k, hk⟩ else 0 with hF'
+    have hL : ∑ k : Fin (37 * n), F k = ∑ k ∈ range (37 * n), F' k := by
+      rw [← Fin.sum_univ_eq_sum_range (fun k => F' k) (37 * n)]
+      refine Finset.sum_congr rfl fun k _ => ?_
+      simp [hF', k.isLt]
+    have hR : ∑ u : Fin r, (p : ℚ)⁻¹ * (U s u * V u t) = ∑ u ∈ range r, F' (k0 + u) := by
+      rw [← Fin.sum_univ_eq_sum_range (fun u => F' (k0 + u)) r]
+      refine Finset.sum_congr rfl fun u _ => ?_
+      have hu : k0 + (u : ℕ) < 37 * n := by omega
+      simp only [hF', dite_eq_left hu, F, hU, hV]
+      rw [Finset.mul_sum, Finset.mul_sum, Finset.mul_sum]
+      apply Finset.sum_congr rfl
+      intro l _
+      field_simp
+    have hzero : ∀ k, k < k0 → F' k = 0 := by
+      intro k hk
+      simp only [hF']
+      split_ifs with h
+      · exact hF0 ⟨k, h⟩ hk
+      · rfl
+    rw [hL, hR]
+    by_cases hk : k0 ≤ 37 * n
+    · rw [Finset.range_eq_Ico, ← Finset.sum_Ico_consecutive _ (Nat.zero_le k0) hk,
+        Finset.sum_eq_zero (fun k hk' => hzero k (Finset.mem_Ico.mp hk').2), zero_add,
+        Finset.sum_Ico_eq_sum_range]
+    · rw [show r = 0 by omega, Finset.sum_range_zero]
+      exact
+        Finset.sum_eq_zero fun k hk' =>
+          hzero k
+            (by rw [Finset.mem_range] at hk'; omega)
+              -- the matrix identity
+  set A' : Matrix (Fin (37 * n)) (Fin (37 * n)) ℚ[X] :=
+    Matrix.of fun s t =>
+      μX (40 * n) (D (3 * n) ^ 6 * E s * E t) -
+        C (μGe p ((D (3 * n) ^ 6 * E s * E t) /ₘ D (40 * n)))
+  have hG :
+    (Matrix.of fun a b => μX (40 * n) (D (3 * n) ^ 6 * E a * E b)) =
+      A' + (((p : ℚ)⁻¹ • (U * V)).map (C : ℚ →+* ℚ[X])) := by
+    refine Matrix.ext fun s t => ?_
+    simp only [A', Matrix.add_apply, Matrix.of_apply, Matrix.map_apply, ← hUV]
+    ring
+  have hlow :=
+    det_lowrank_bound A' U V w hw (fun s t => by simp only [A', Matrix.of_apply]; exact hA s t)
+      (fun s u => hCqint s _)
+      (fun u t => by
+        simp only [hV, Finset.mul_sum]
+        exact VG.sum _ fun l _ => by rw [← mul_assoc]; simpa using (hβ0int _).mul (hCqint t l))
+  rw [← hG, hbc] at hlow
+  have hΔ : Δ n = C ((dC ^ 2)⁻¹) * (C (dC ^ 2) * Δ n) := by
+    rw [← mul_assoc, ← C_mul, inv_mul_cancel₀ (pow_ne_zero _ hunit.1), C_1, one_mul]
+  rw [hΔ]
+  have hu : VG p ((dC ^ 2)⁻¹) 0 := by
+    refine (VG.inv (p := p) (r := 0) (pow_ne_zero _ hunit.1) ?_).mono (by norm_num)
+    rw [padicValRat.pow, hunit.2]; simp
+  have := GV.C_mul hu hlow
+  rw [zero_add] at this
+  exact this
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Arith/PoleFun.lean b/LeanPool/Zeta5Irrational/Arith/PoleFun.lean
new file mode 100644
index 0000000000..71089cba4d
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Arith/PoleFun.lean
@@ -0,0 +1,96 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Arith.TauBound
+public import LeanPool.Zeta5Irrational.Construction
+public import Mathlib.LinearAlgebra.Lagrange
+import Mathlib.Tactic.Polynomial.Basic
+import Mathlib.Tactic.ReduceModChar
+
+/-! # The functional `τ_X` on rational functions with simple integer poles
+
+For a numerator `A ∈ ℚ[x]` and a finite set `Pl ⊆ ℤ` of poles, `g = A / ∏_{r ∈ Pl} (x - r)`:
+
+* `polyPart A Pl = A /ₘ ∏ (x - r)` (the polynomial part);
+* `resP A Pl r = A(r) / ∏_{s ≠ r} (r - s)` (the residues);
+* `tauX A Pl = τ(polyPart) + ∑_r res_r (H⁽⁵⁾_{d(r)} - X) ∈ ℚ[X]`, with `d(r) = r` for `r ≥ 0` and
+  `d(r) = -r - 1` for `r < 0`.
+
+`partial_fractions` : `A = P Π + ∑_r res_r ∏_{s ≠ r} (x - s)`.
+-/
+
+@[expose] public section
+
+open Finset Polynomial
+
+namespace Zeta5Irrational
+
+/-- `∏_{r ∈ Pl} (x - r)`. -/
+noncomputable def piPl (Pl : Finset ℤ) : ℚ[X] :=
+  ∏ r ∈ Pl, (X - C (r : ℚ))
+
+lemma piPl_monic (Pl : Finset ℤ) : (piPl Pl).Monic :=
+  monic_prod_of_monic _ _ fun _r _ => monic_X_sub_C _
+
+lemma natDegree_piPl (Pl : Finset ℤ) : (piPl Pl).natDegree = Pl.card := by
+  unfold piPl
+  rw [natDegree_prod_of_monic _ _ fun r _ => monic_X_sub_C _]
+  rw [Finset.sum_congr rfl fun (r : ℤ) _ => natDegree_X_sub_C ((r : ℤ) : ℚ)]
+  simp
+
+/-- The polynomial part. -/
+noncomputable def polyPart (A : ℚ[X]) (Pl : Finset ℤ) : ℚ[X] :=
+  A /ₘ piPl Pl
+
+/-- The residue at `r`. -/
+noncomputable def resP (A : ℚ[X]) (Pl : Finset ℤ) (r : ℤ) : ℚ :=
+  A.eval (r : ℚ) / ∏ s ∈ Pl.erase r, ((r : ℚ) - s)
+
+/-- The harmonic index `d(r)`. -/
+def dd (r : ℤ) : ℕ :=
+  if 0 ≤ r then r.toNat else (-r - 1).toNat
+
+/-- The functional `τ_X`. -/
+noncomputable def tauX (A : ℚ[X]) (Pl : Finset ℤ) : ℚ[X] :=
+  C (tau (polyPart A Pl)) + ∑ r ∈ Pl, C (resP A Pl r) * (C (H5 (dd r)) - X)
+
+lemma lagrange_basis_eq (Pl : Finset ℤ) (r : ℤ) :
+    Lagrange.basis Pl (fun s : ℤ => (s : ℚ)) r =
+      C (∏ s ∈ Pl.erase r, ((r : ℚ) - s))⁻¹ * ∏ s ∈ Pl.erase r, (X - C (s : ℚ)) := by
+  unfold Lagrange.basis Lagrange.basisDivisor
+  rw [Finset.prod_mul_distrib, ← map_prod, Finset.prod_inv_distrib]
+
+/-- **Partial fractions**. -/
+theorem partial_fractionsP (A : ℚ[X]) (Pl : Finset ℤ) :
+    A = polyPart A Pl * piPl Pl + ∑ r ∈ Pl, C (resP A Pl r) * ∏ s ∈ Pl.erase r, (X - C (s : ℚ)) :=
+  by
+  have hinj : Set.InjOn (fun s : ℤ => (s : ℚ)) Pl := fun a _ b _ h => by simpa using h
+  have hmod : A %ₘ piPl Pl = ∑ r ∈ Pl, C (resP A Pl r) * ∏ s ∈ Pl.erase r, (X - C (s : ℚ)) := by
+    rw [Lagrange.eq_interpolate_of_eval_eq (f := A %ₘ piPl Pl) (fun r : ℤ => A.eval (r : ℚ)) hinj
+        (by
+          have := degree_modByMonic_lt A (piPl_monic Pl)
+          rwa [degree_eq_natDegree (piPl_monic Pl).ne_zero, natDegree_piPl] at this)
+        (fun r hr => by
+          have h1 := modByMonic_add_div A (piPl Pl)
+          have h2 : (piPl Pl).eval (r : ℚ) = 0 := by
+            unfold piPl; rw [eval_prod]
+            exact Finset.prod_eq_zero hr (by simp)
+          conv_rhs => rw [← h1]
+          rw [eval_add, eval_mul, h2, zero_mul, add_zero])]
+    simp only [Lagrange.interpolate_apply]
+    refine Finset.sum_congr rfl fun r _ => ?_
+    rw [lagrange_basis_eq, resP, div_eq_mul_inv, C_mul, mul_assoc]
+  conv_lhs => rw [← modByMonic_add_div A (piPl Pl)]
+  rw [hmod, polyPart, add_comm, mul_comm]
+
+lemma natDegree_polyPart_le (A : ℚ[X]) (Pl : Finset ℤ) :
+    (polyPart A Pl).natDegree ≤ A.natDegree - Pl.card := by
+  unfold polyPart
+  rw [natDegree_divByMonic A (piPl_monic Pl), natDegree_piPl]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Arith/PoleVal.lean b/LeanPool/Zeta5Irrational/Arith/PoleVal.lean
new file mode 100644
index 0000000000..2960ec3cd4
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Arith/PoleVal.lean
@@ -0,0 +1,274 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Arith.Val
+public import LeanPool.Zeta5Irrational.Construction
+import LeanPool.Zeta5Irrational.Arith.DirectBound
+import Mathlib.Algebra.Ring.IsFormallyReal
+import Mathlib.Data.Rat.Star
+import Mathlib.Tactic.ENatToNat
+import Mathlib.Tactic.Polynomial.Basic
+import Mathlib.Tactic.ReduceModChar
+import Mathlib.Tactic.FieldSimp
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+import Mathlib.Tactic.Positivity.Basic
+import Mathlib.Tactic.Ring.Basic
+import Mathlib.Tactic.Ring.RingNF
+
+/-! # Valuations of the pole values `μ_X(1/(t + j²)) = j⁴(X - H⁽⁵⁾_j) - 1/4 + 1/(2j)`
+-/
+
+@[expose] public section
+
+open Finset Polynomial
+
+namespace Zeta5Irrational
+
+variable {p : ℕ} [hp : Fact p.Prime]
+
+/-- `v_p(1/u) ≥ -e` if `v_p(u) ≤ e`. -/
+lemma VG_inv_nat_le {u : ℕ} (hu : u ≠ 0) {e : ℕ} (he : padicValNat p u ≤ e) :
+    VG p ((u : ℚ)⁻¹) (-(e : ℚ)) := by
+  refine VG.inv (by exact_mod_cast hu) ?_
+  rw [padicValRat.of_nat]; exact_mod_cast he
+
+omit hp in
+lemma padicValNat_lt_p {u : ℕ} (hu : 1 ≤ u) (hup : u < p) : padicValNat p u = 0 :=
+  padicValNat.eq_zero_of_not_dvd fun h => absurd (Nat.le_of_dvd (by omega) h) (by omega)
+
+lemma padicValNat_lt_sq {u : ℕ} (hu : 1 ≤ u) (hup : u < p ^ 2) : padicValNat p u ≤ 1 := by
+  by_contra h
+  rw [not_le] at h
+  have : p ^ 2 ∣ u := (padicValNat_dvd_iff_le (by omega)).mpr h
+  exact absurd (Nat.le_of_dvd (by omega) this) (by omega)
+
+lemma VG_H5_lt {j : ℕ} (hj : j < p) : VG p (H5 j) 0 := by
+  unfold H5
+  refine VG.sum _ fun u hu => ?_
+  have hu' := Finset.mem_Icc.mp hu
+  rw [one_div, show ((u : ℚ) ^ 5) = ((u ^ 5 : ℕ) : ℚ) by push_cast; ring]
+  refine (VG_inv_nat_le (e := 0) (pow_ne_zero _ (by omega)) ?_).mono (by norm_num)
+  rw [padicValNat.pow (p := p) u 5, padicValNat_lt_p hu'.1 (by omega)]
+
+lemma VG_quarter (hp3 : 3 ≤ p) : VG p (1 / 4 : ℚ) 0 := by
+  rw [one_div, show (4 : ℚ) = ((4 : ℕ) : ℚ) by norm_num]
+  refine (VG_inv_nat_le (e := 0) (by norm_num) ?_).mono (by norm_num)
+  rw [show (4 : ℕ) = 2 ^ 2 by norm_num, padicValNat.pow (p := p) 2 2]
+  have := Fact.mk Nat.prime_two
+  rw [padicValNat_primes (p := p) (q := 2) (by omega)]
+
+lemma VG_inv_2j (hp3 : 3 ≤ p) {j e : ℕ} (hj : 1 ≤ j) (he : padicValNat p j ≤ e) :
+    VG p (1 / (2 * (j : ℚ))) (-(e : ℚ)) := by
+  rw [one_div, show (2 * (j : ℚ)) = ((2 * j : ℕ) : ℚ) by push_cast; ring]
+  refine VG_inv_nat_le (by omega) ?_
+  have := Fact.mk Nat.prime_two
+  rw [padicValNat.mul (by norm_num) (by omega), padicValNat_primes (p := p) (q := 2) (by omega)]
+  omega
+
+/-- Integral pole values for `1 ≤ j < p`. -/
+lemma GV_poleValue_lt (hp3 : 3 ≤ p) {j : ℕ} (hj : 1 ≤ j) (hjp : j < p) : GV p (poleValue j) 0 := by
+  unfold poleValue
+  have h1 : GV p (C ((j : ℚ) ^ 4) * (X - C (H5 j))) 0 := by
+    have := GV.C_mul (VG.natCast (p := p) (j ^ 4)) (GV.X.sub (GV.C (VG_H5_lt hjp)))
+    simpa using this
+  refine (h1.sub (GV.C (VG_quarter hp3))).add (GV.C ?_)
+  have := VG_inv_2j hp3 hj (e := 0) (by rw [padicValNat_lt_p hj hjp])
+  simpa using this
+
+/-- Pole values `≥ -5` for `1 ≤ j < p²`. -/
+lemma GV_poleValue_sq (hp3 : 3 ≤ p) {j : ℕ} (hj : 1 ≤ j) (hjp : j < p ^ 2) :
+    GV p (poleValue j) (-5) := by
+  unfold poleValue
+  have h1 : GV p (C ((j : ℚ) ^ 4) * (X - C (H5 j))) (-5) := by
+    have :=
+      GV.C_mul (VG.natCast (p := p) (j ^ 4)) ((GV.X.mono (by norm_num)).sub (GV.C (VG_H5 hjp)))
+    simpa using this
+  refine (h1.sub ((GV.C (VG_quarter hp3)).mono (by norm_num))).add (GV.C ?_)
+  exact (VG_inv_2j hp3 hj (e := 1) (padicValNat_lt_sq hj hjp)).mono (by norm_num)
+
+/-- Pole values `≥ -1` for `j = p k`, `1 ≤ k < p`. -/
+lemma GV_poleValue_mul (hp3 : 3 ≤ p) {k : ℕ} (hk : 1 ≤ k) (hkp : k < p) :
+    GV p (poleValue (p * k)) (-1) := by
+  have hj : 1 ≤ p * k := Nat.one_le_iff_ne_zero.mpr (by positivity)
+  have hjp : p * k < p ^ 2 := by rw [sq]; exact Nat.mul_lt_mul_of_pos_left hkp (by omega)
+  unfold poleValue
+  have h4 : VG p (((p * k : ℕ) : ℚ) ^ 4) 4 := by
+    right
+    rw [padicValRat.pow, padicValRat.of_nat, padicValNat.mul (by omega) (by omega),
+      padicValNat_self, padicValNat_lt_p hk hkp];
+    norm_num
+  have h1 : GV p (C (((p * k : ℕ) : ℚ) ^ 4) * (X - C (H5 (p * k)))) (-1) := by
+    have := GV.C_mul h4 ((GV.X.mono (by norm_num)).sub (GV.C (VG_H5 hjp)))
+    refine this.mono (by norm_num)
+  refine (h1.sub ((GV.C (VG_quarter hp3)).mono (by norm_num))).add (GV.C ?_)
+  exact VG_inv_2j hp3 hj (e := 1) (padicValNat_lt_sq hj hjp)
+
+lemma VG_int_dvd_one {z : ℤ} (h : (p : ℤ) ∣ z) : VG p (z : ℚ) 1 :=
+  VG_int_one h
+
+/-- `1/(p-u)⁵ + 1/u⁵ ≡ 0 (mod p)`. -/
+lemma VG_pair (_hp3 : 3 ≤ p) {u : ℕ} (hu : 1 ≤ u) (hup : u < p) :
+    VG p (1 / ((p : ℚ) - u) ^ 5 + 1 / (u : ℚ) ^ 5) 1 := by
+  have hpu : ((p : ℚ) - u) = ((p - u : ℕ) : ℚ) := by push_cast [Nat.cast_sub hup.le]; ring
+  have hne1 : ((p - u : ℕ) : ℚ) ≠ 0 := by exact_mod_cast (by omega : p - u ≠ 0)
+  have hne2 : (u : ℚ) ≠ 0 := by exact_mod_cast (by omega : u ≠ 0)
+  rw [hpu, div_add_div _ _ (pow_ne_zero _ hne1) (pow_ne_zero _ hne2), div_eq_mul_inv]
+  have hnum : VG p (1 * (u : ℚ) ^ 5 + ((p - u : ℕ) : ℚ) ^ 5 * 1) 1 := by
+    have :=
+      VG_int_one (p := p) (z := (u : ℤ) ^ 5 + ((p : ℤ) - u) ^ 5)
+        (by
+          rw [← ZMod.intCast_zmod_eq_zero_iff_dvd]; push_cast
+          rw [ZMod.natCast_self]; ring)
+    convert this using 1
+    push_cast [Nat.cast_sub hup.le]; ring
+  have hden : VG p (((p - u : ℕ) : ℚ) ^ 5 * (u : ℚ) ^ 5)⁻¹ 0 := by
+    rw [show ((p - u : ℕ) : ℚ) ^ 5 * (u : ℚ) ^ 5 = (((p - u) ^ 5 * u ^ 5 : ℕ) : ℚ) by
+        push_cast;
+        ring]
+    have hA : (p - u) ^ 5 ≠ 0 := pow_ne_zero _ (by omega)
+    have hB : u ^ 5 ≠ 0 := pow_ne_zero _ (by omega)
+    refine (VG_inv_nat_le (e := 0) (mul_ne_zero hA hB) ?_).mono (by norm_num)
+    rw [padicValNat.mul hA hB, padicValNat.pow (p := p) _ 5, padicValNat.pow (p := p) _ 5,
+      padicValNat_lt_p (by omega) (by omega), padicValNat_lt_p hu hup]
+  simpa using hnum.mul hden
+
+/-- `H⁽⁵⁾_{p-1} ≡ 0 (mod p)`. -/
+lemma VG_H5_pm1 (hp3 : 3 ≤ p) : VG p (H5 (p - 1)) 1 := by
+  have h2 : 2 * H5 (p - 1) = ∑ u ∈ Icc 1 (p - 1), (1 / ((p : ℚ) - u) ^ 5 + 1 / (u : ℚ) ^ 5) := by
+    unfold H5
+    rw [Finset.sum_add_distrib, two_mul]
+    congr 1
+    apply Finset.sum_nbij' (fun u => p - u) (fun u => p - u)
+    · intro u hu; simp only [Finset.mem_Icc] at hu ⊢
+      omega
+    · intro u hu; simp only [Finset.mem_Icc] at hu ⊢
+      omega
+    · intro u hu; simp only [Finset.mem_Icc] at hu
+      omega
+    · intro u hu; simp only [Finset.mem_Icc] at hu
+      omega
+    · intro u hu
+      simp only [Finset.mem_Icc] at hu
+      push_cast [Nat.cast_sub (show u ≤ p by omega)]; ring_nf
+  have hsum : VG p (2 * H5 (p - 1)) 1 := by
+    rw [h2];
+    exact
+      VG.sum _ fun u hu =>
+        VG_pair hp3 (Finset.mem_Icc.mp hu).1 (by have := (Finset.mem_Icc.mp hu).2; omega)
+  have hhalf : VG p ((1 / 2 : ℚ)) 0 := by
+    rw [one_div, show (2 : ℚ) = ((2 : ℕ) : ℚ) by norm_num]
+    have := Fact.mk Nat.prime_two
+    refine (VG_inv_nat_le (e := 0) (by norm_num) ?_).mono (by norm_num)
+    rw [padicValNat_primes (p := p) (q := 2) (by omega)]
+  have := hhalf.mul hsum
+  rw [show 1 / 2 * (2 * H5 (p - 1)) = H5 (p - 1) by ring] at this
+  simpa using this
+
+lemma H5_succ' {j : ℕ} (hj : 1 ≤ j) : H5 j = H5 (j - 1) + 1 / (j : ℚ) ^ 5 := by
+  obtain ⟨i, rfl⟩ : ∃ i, j = i + 1 := ⟨j - 1, by omega⟩
+  unfold H5
+  rw [Finset.sum_Icc_succ_top (by omega), Nat.add_sub_cancel]
+
+omit hp in
+lemma H5_split {c : ℕ} (hc : 1 ≤ c) (hcp : c < p) :
+    H5 (p - 1) = H5 (p - c) + ∑ u ∈ Icc 1 (c - 1), 1 / ((p : ℚ) - u) ^ 5 := by
+  unfold H5
+  rw [show Icc 1 (p - 1) = Ioc 0 (p - 1) from Finset.Icc_add_one_left_eq_Ioc 0 _,
+    show Icc 1 (p - c) = Ioc 0 (p - c) from Finset.Icc_add_one_left_eq_Ioc 0 _]
+  have h :=
+    Finset.sum_Ioc_consecutive (fun k : ℕ => 1 / (k : ℚ) ^ 5) (Nat.zero_le (p - c))
+      (show p - c ≤ p - 1 by omega)
+  rw [← h]
+  congr 1
+  apply Finset.sum_nbij' (fun k => p - k) (fun u => p - u)
+  · intro k hk; simp only [Finset.mem_Ioc, Finset.mem_Icc] at hk ⊢; omega
+  · intro u hu; simp only [Finset.mem_Ioc, Finset.mem_Icc] at hu ⊢; omega
+  · intro k hk; simp only [Finset.mem_Ioc] at hk; omega
+  · intro u hu; simp only [Finset.mem_Icc] at hu; omega
+  · intro k hk
+    simp only [Finset.mem_Ioc] at hk
+    push_cast [Nat.cast_sub (show k ≤ p by omega)]
+    ring_nf
+
+/-- **The congruence** `poleValue (p - c) ≡ poleValue c (mod p)` for `1 ≤ c < p`. -/
+theorem GV_poleValue_congr (hp3 : 3 ≤ p) {c : ℕ} (hc : 1 ≤ c) (hcp : c < p) :
+    GV p (poleValue (p - c) - poleValue c) 1 := by
+  set d := p - c with hd
+  have hdq : (d : ℚ) = p - c := by rw [hd]; push_cast [Nat.cast_sub hcp.le]; ring
+  have hc0 : (c : ℚ) ≠ 0 := by exact_mod_cast (by omega : c ≠ 0)
+  have hd0 : (d : ℚ) ≠ 0 := by exact_mod_cast (by omega : d ≠ 0)
+  set S' := ∑ u ∈ Icc 1 (c - 1), 1 / ((p : ℚ) - u) ^ 5 with hS'
+  set Sc := ∑ u ∈ Icc 1 (c - 1), 1 / (u : ℚ) ^ 5 with hSc
+  have hHd : H5 d = H5 (p - 1) - S' := by rw [H5_split hc hcp]; ring
+  have hHc : H5 c = Sc + 1 / (c : ℚ) ^ 5 := by rw [H5_succ' hc]; rfl
+  set κ : ℚ :=
+    -(d : ℚ) ^ 4 * H5 d + (c : ℚ) ^ 4 * H5 c + (1 / (2 * (d : ℚ)) - 1 / (2 * (c : ℚ))) with hκ
+  have hκ' :
+    κ =
+      -(d : ℚ) ^ 4 * H5 (p - 1) + ((d : ℚ) ^ 4 - (c : ℚ) ^ 4) * S' +
+          (c : ℚ) ^ 4 * ∑ u ∈ Icc 1 (c - 1), (1 / ((p : ℚ) - u) ^ 5 + 1 / (u : ℚ) ^ 5) +
+        (1 / (2 * (c : ℚ)) + 1 / (2 * (d : ℚ))) := by
+    rw [hκ, hHd, hHc, Finset.sum_add_distrib, ← hS', ← hSc]
+    field_simp
+    ring
+  have hdiff : poleValue d - poleValue c = C ((d : ℚ) ^ 4 - (c : ℚ) ^ 4) * X + C κ := by
+    unfold poleValue
+    rw [hκ]
+    ext n
+    simp only [coeff_add, coeff_sub, coeff_C_mul, coeff_C, coeff_X, mul_sub]
+    split_ifs <;> ring
+  rw [hdiff]
+  have hd4 : VG p ((d : ℚ) ^ 4 - (c : ℚ) ^ 4) 1 := by
+    have :=
+      VG_int_one (p := p) (z := (d : ℤ) ^ 4 - (c : ℤ) ^ 4)
+        (by
+          rw [← ZMod.intCast_zmod_eq_zero_iff_dvd]; push_cast
+          rw [hd, Nat.cast_sub hcp.le, ZMod.natCast_self]; ring)
+    simpa using this
+  refine (by simpa using GV.C_mul hd4 (GV.X (p := p)) : GV p _ 1).add (GV.C ?_)
+  rw [hκ']
+  have t1 : VG p (-(d : ℚ) ^ 4 * H5 (p - 1)) 1 := by
+    have := ((VG.natCast (p := p) (d ^ 4)).mul (VG_H5_pm1 hp3)).neg
+    rw [show -(d : ℚ) ^ 4 * H5 (p - 1) = -(((d ^ 4 : ℕ) : ℚ) * H5 (p - 1)) by push_cast; ring]
+    simpa using this
+  refine ((t1.add ?_).add ?_).add ?_
+  · refine (hd4.mul (VG.sum (r := 0) _ fun u hu => ?_)).mono (by norm_num)
+    have hu' := Finset.mem_Icc.mp hu
+    have hu2 : u < p := by omega
+    rw [show ((p : ℚ) - u) = ((p - u : ℕ) : ℚ) by
+        push_cast [Nat.cast_sub (show u ≤ p by omega)];
+        ring,
+      one_div, show (((p - u : ℕ) : ℚ) ^ 5) = (((p - u) ^ 5 : ℕ) : ℚ) by push_cast; ring]
+    refine
+      (VG_inv_nat_le (e := 0) (pow_ne_zero _ (Nat.sub_ne_zero_of_lt hu2)) ?_).mono (by norm_num)
+    rw [padicValNat.pow (p := p) _ 5,
+      padicValNat_lt_p (Nat.one_le_iff_ne_zero.mpr (Nat.sub_ne_zero_of_lt hu2))
+        (Nat.sub_lt (by omega) (by omega))]
+  · have :=
+      (VG.natCast (p := p) (c ^ 4)).mul
+        (VG.sum (r := 1) _ fun u hu =>
+          VG_pair hp3 (Finset.mem_Icc.mp hu).1
+            (lt_of_le_of_lt (Finset.mem_Icc.mp hu).2 (lt_of_le_of_lt (Nat.sub_le c 1) hcp)))
+    simpa using this
+  · have e : 1 / (2 * (c : ℚ)) + 1 / (2 * (d : ℚ)) = (p : ℚ) * (1 / (2 * (c * d : ℕ) : ℚ)) := by
+      push_cast; rw [hdq]
+      have : ((p : ℚ) - c) ≠ 0 := by rw [← hdq]; exact hd0
+      field_simp; ring
+    rw [e]
+    have hp1 : VG p (p : ℚ) 1 := by simpa using VG.primePow (p := p) 1
+    refine
+      (hp1.mul
+            (VG_inv_2j (e := 0) hp3
+              (Nat.one_le_iff_ne_zero.mpr (Nat.mul_ne_zero (by omega) (by omega))) ?_)).mono
+        (by norm_num)
+    rw [padicValNat.mul (by omega) (by omega), padicValNat_lt_p hc hcp,
+      padicValNat_lt_p (by omega) (by omega)]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Arith/Pullback.lean b/LeanPool/Zeta5Irrational/Arith/Pullback.lean
new file mode 100644
index 0000000000..de20caca78
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Arith/Pullback.lean
@@ -0,0 +1,279 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Arith.PoleFun
+public import LeanPool.Zeta5Irrational.Degree
+import LeanPool.Zeta5Irrational.PartialFractions
+import Mathlib.Algebra.Ring.IsFormallyReal
+import Mathlib.Data.Int.Star
+import Mathlib.Order.Interval.Finset.Box
+import Mathlib.RingTheory.PiTensorProduct
+import Mathlib.Tactic.ENatToNat
+import Mathlib.Tactic.Polynomial.Basic
+import Mathlib.Tactic.ReduceModChar
+import Mathlib.Tactic.FieldSimp
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+import Mathlib.Tactic.Ring.Basic
+import Mathlib.Tactic.Ring.RingNF
+
+/-! # The pullback identity (3.1): `μ_X(R) = τ_X(x⁵ R(-x²))`
+
+With `PlK K = {±1, …, ±K}` and `pull K P = (-1)^K x⁵ P(-x²)` we have
+`μX K P = tauX (pull K P) (PlK K)` (`μX_eq_tauX`), since
+`x⁵ P(-x²) / D_K(-x²) = pull K P / ∏_{r ∈ PlK K} (x - r)`.
+-/
+
+@[expose] public section
+
+open Finset Polynomial
+
+namespace Zeta5Irrational
+
+/-! ### Linearity of `τ` -/
+
+lemma tau_add (P Q : ℚ[X]) : tau (P + Q) = tau P + tau Q := by
+  unfold tau
+  simp only [Function.iterate_succ, Function.iterate_zero, Function.comp_apply, id,
+    derivative_add, Lb_add]
+  ring
+
+lemma tau_C_mul (c : ℚ) (P : ℚ[X]) : tau (C c * P) = c * tau P := by
+  unfold tau
+  simp only [Function.iterate_succ, Function.iterate_zero, Function.comp_apply, id,
+    derivative_C_mul, Lb_C_mul]
+  ring
+
+lemma tau_neg (P : ℚ[X]) : tau (-P) = -tau P := by
+  rw [show -P = C (-1) * P by simp, tau_C_mul];
+  ring
+
+lemma tau_sum {ι : Type*} (s : Finset ι) (f : ι → ℚ[X]) :
+    tau (∑ i ∈ s, f i) = ∑ i ∈ s, tau (f i) := by
+  classical
+    induction s using Finset.induction_on with
+  | empty => simp [tau, Lb]
+  | insert a s ha ih => rw [Finset.sum_insert ha, Finset.sum_insert ha, tau_add, ih]
+
+lemma tau_X_pow (n : ℕ) : tau (X ^ n) = (n.descFactorial 3 : ℚ) * _root_.bernoulli (n - 3) / 24 :=
+  by
+  unfold tau
+  rw [iterate_derivative_X_pow_eq_smul, Lb_smul, Lb_X_pow]
+
+lemma tau_X_pow_odd (e : ℕ) : tau (C ((-1 : ℚ) ^ e) * X ^ (2 * e + 5)) = μmono e := by
+  rw [tau_C_mul, tau_X_pow, μmono]
+  rw [show 2 * e + 5 - 3 = 2 * e + 2 by omega,
+    show (2 * e + 5).descFactorial 3 = (2 * e + 5) * (2 * e + 4) * (2 * e + 3) by
+      simp [Nat.descFactorial]; ring]
+  push_cast
+  ring
+
+/-- `τ(x⁵ Q(-x²)) = μ(Q)`. -/
+lemma tau_pull_poly (Q : ℚ[X]) : tau (X ^ 5 * Q.comp (-X ^ 2)) = μpoly Q := by
+  induction Q using Polynomial.induction_on' with
+  | add P Q hP hQ =>
+    rw [add_comp, mul_add, tau_add, hP, hQ]
+    unfold μpoly
+    rw [Polynomial.sum_add_index _ _ _ (fun _ => by simp) (fun _ _ _ => by ring)]
+  | monomial e a =>
+    rw [← C_mul_X_pow_eq_monomial, mul_comp, C_comp, X_pow_comp]
+    have : X ^ 5 * (C a * (-X ^ 2) ^ e) = C a * (C ((-1 : ℚ) ^ e) * X ^ (2 * e + 5)) := by
+      rw [neg_pow, ← pow_mul, map_pow, map_neg, map_one]; ring
+    rw [this, tau_C_mul, tau_X_pow_odd]
+    unfold μpoly
+    rw [C_mul_X_pow_eq_monomial, Polynomial.sum_monomial_index _ _ (by simp)]
+
+lemma tau_cubic (a : ℚ) : tau (-X ^ 3 - C a * X) = -1 / 4 := by
+  rw [sub_eq_add_neg, tau_add, tau_neg, tau_neg, tau_C_mul, tau_X_pow,
+    show (X : ℚ[X]) = X ^ 1 by ring, tau_X_pow]
+  simp [Nat.descFactorial, _root_.bernoulli_zero]
+  norm_num
+
+/-! ### The pole set `{±1, …, ±K}` -/
+
+/-- The poles `±1, …, ±K` in the variable `x`. -/
+def PlK (K : ℕ) : Finset ℤ :=
+  (Icc 1 K).image (fun j : ℕ => (j : ℤ)) ∪ (Icc 1 K).image (fun j : ℕ => -(j : ℤ))
+
+lemma PlK_disjoint (K : ℕ) :
+    Disjoint ((Icc 1 K).image (fun j : ℕ => (j : ℤ))) ((Icc 1 K).image (fun j : ℕ => -(j : ℤ))) :=
+  by
+  rw [Finset.disjoint_left]
+  intro r h1 h2
+  rw [Finset.mem_image] at h1 h2
+  obtain ⟨a, ha, rfl⟩ := h1
+  obtain ⟨b, hb, hab⟩ := h2
+  rw [Finset.mem_Icc] at ha hb
+  omega
+
+lemma sum_PlK {M : Type*} [AddCommMonoid M] (K : ℕ) (f : ℤ → M) :
+    ∑ r ∈ PlK K, f r = ∑ j ∈ Icc 1 K, (f j + f (-j)) := by
+  rw [PlK, Finset.sum_union (PlK_disjoint K),
+    Finset.sum_image (fun a _ b _ h => by exact_mod_cast h),
+    Finset.sum_image (fun a _ b _ h => by simpa using h), Finset.sum_add_distrib]
+
+lemma prod_PlK {M : Type*} [CommMonoid M] (K : ℕ) (f : ℤ → M) :
+    ∏ r ∈ PlK K, f r = ∏ j ∈ Icc 1 K, (f j * f (-j)) := by
+  rw [PlK, Finset.prod_union (PlK_disjoint K),
+    Finset.prod_image (fun a _ b _ h => by exact_mod_cast h),
+    Finset.prod_image (fun a _ b _ h => by simpa using h), Finset.prod_mul_distrib]
+
+lemma X_add_C_comp_neg_sq (a : ℚ) : (X + C a).comp (-X ^ 2) = -X ^ 2 + C a := by simp
+
+lemma piPl_PlK (K : ℕ) : piPl (PlK K) = C ((-1 : ℚ) ^ K) * (D K).comp (-X ^ 2) := by
+  unfold piPl D
+  rw [prod_PlK, Polynomial.prod_comp,
+    show C ((-1 : ℚ) ^ K) = ∏ j ∈ Icc 1 K, C (-1 : ℚ) by
+      rw [Finset.prod_const, Nat.card_Icc, Nat.add_sub_cancel, map_pow],
+    ← Finset.prod_mul_distrib]
+  refine Finset.prod_congr rfl fun j _ => ?_
+  rw [X_add_C_comp_neg_sq]
+  push_cast
+  rw [C_neg, C_pow, map_neg, map_one]
+  ring
+
+/-- `pull K P = (-1)^K x⁵ P(-x²)`. -/
+noncomputable def pull (K : ℕ) (P : ℚ[X]) : ℚ[X] :=
+  C ((-1 : ℚ) ^ K) * X ^ 5 * P.comp (-X ^ 2)
+
+lemma card_PlK (K : ℕ) : (PlK K).card = 2 * K := by
+  rw [PlK, Finset.card_union_of_disjoint (PlK_disjoint K),
+    Finset.card_image_of_injective _ (fun a b h => by exact_mod_cast h),
+    Finset.card_image_of_injective _ (fun a b h => by simpa using h)]
+  simp; ring
+
+/-- The cofactor `B_j = (-1)^K ∏_{i ≠ j} (i² - x²)`. -/
+noncomputable def Bj (K j : ℕ) : ℚ[X] :=
+  C ((-1 : ℚ) ^ K) * (∏ i ∈ (Icc 1 K).erase j, (X + C ((i : ℚ) ^ 2))).comp (-X ^ 2)
+
+lemma piPl_eq_Bj {K j : ℕ} (hj : j ∈ Icc 1 K) :
+    piPl (PlK K) = (-X ^ 2 + C ((j : ℚ) ^ 2)) * Bj K j := by
+  rw [piPl_PlK, Bj, D, ← Finset.mul_prod_erase _ _ hj, mul_comp, X_add_C_comp_neg_sq]; ring
+
+lemma natDegree_Bj_le {K j : ℕ} (hj : j ∈ Icc 1 K) : (Bj K j).natDegree ≤ 2 * (K - 1) := by
+  unfold Bj
+  refine (natDegree_C_mul_le _ _).trans ?_
+  refine natDegree_comp_le.trans ?_
+  have h1 : (∏ i ∈ (Icc 1 K).erase j, (X + C ((i : ℚ) ^ 2))).natDegree ≤ K - 1 := by
+    refine (natDegree_prod_le _ _).trans ?_
+    refine
+      (Finset.sum_le_sum fun (i : ℕ) _ => (natDegree_X_add_C (((i : ℕ) : ℚ) ^ 2)).le).trans ?_
+    simp only [Finset.sum_const, smul_eq_mul, mul_one]
+    rw [Finset.card_erase_of_mem hj]; simp
+  have h2 : (-X ^ 2 : ℚ[X]).natDegree = 2 := by rw [natDegree_neg, natDegree_X_pow]
+  rw [h2]; omega
+
+/-- The polynomial part of the pulled-back function. -/
+theorem polyPart_pull (K : ℕ) (P : ℚ[X]) :
+    polyPart (pull K P) (PlK K) =
+      X ^ 5 * (P /ₘ D K).comp (-X ^ 2) +
+        ∑ j ∈ Icc 1 K, C (res K P j) * (-X ^ 3 - C ((j : ℚ) ^ 2) * X) := by
+  unfold polyPart
+  refine
+    (div_modByMonic_unique _ (∑ j ∈ Icc 1 K, C (res K P j) * (C ((j : ℚ) ^ 4) * X * Bj K j))
+        (piPl_monic _) ⟨?_, ?_⟩).1
+  · have hP := modByMonic_add_div P (D K)
+    rw [modByMonic_eq_sum_res] at hP
+    conv_rhs => rw [pull, ← hP]
+    rw [add_comp, mul_comp, Polynomial.sum_comp, piPl_PlK]
+    simp only [mul_comp, C_comp, mul_add, Finset.mul_sum]
+    have hj :
+      ∀ j ∈ Icc 1 K,
+        C ((-1 : ℚ) ^ K) * X ^ 5 *
+            (C (res K P j) * (∏ i ∈ (Icc 1 K).erase j, (X + C ((i : ℚ) ^ 2))).comp (-X ^ 2)) =
+          C (res K P j) * (C ((j : ℚ) ^ 4) * X * Bj K j) +
+            C ((-1 : ℚ) ^ K) * (D K).comp (-X ^ 2) *
+              (C (res K P j) * (-X ^ 3 - C ((j : ℚ) ^ 2) * X)) := by
+      intro j hjK
+      have := piPl_eq_Bj hjK
+      rw [piPl_PlK] at this
+      rw [this, Bj]
+      have e : (C ((j : ℚ) ^ 4) : ℚ[X]) = C ((j : ℚ) ^ 2) * C ((j : ℚ) ^ 2) := by
+        rw [← C_mul];
+        ring_nf
+      rw [e]; ring
+    rw [Finset.sum_congr rfl hj, Finset.sum_add_distrib]
+    ring
+  · rw [degree_eq_natDegree (piPl_monic _).ne_zero, natDegree_piPl, card_PlK]
+    rcases Nat.eq_zero_or_pos K with rfl | hK
+    · simp
+    · refine (degree_le_natDegree).trans_lt ?_
+      have :
+        (∑ j ∈ Icc 1 K, C (res K P j) * (C ((j : ℚ) ^ 4) * X * Bj K j)).natDegree ≤ 2 * K - 1 := by
+        refine natDegree_sum_le_of_forall_le _ _ fun j hjK => ?_
+        refine (natDegree_C_mul_le _ _).trans ?_
+        refine (natDegree_mul_le).trans ?_
+        refine
+          (Nat.add_le_add ((natDegree_C_mul_le _ _).trans natDegree_X_le)
+                (natDegree_Bj_le hjK)).trans
+            ?_
+        omega
+      exact_mod_cast (Nat.lt_of_le_of_lt this (by omega))
+
+lemma prod_erase_eq_derivative (Pl : Finset ℤ) {r : ℤ} (hr : r ∈ Pl) :
+    ∏ s ∈ Pl.erase r, ((r : ℚ) - s) = (derivative (piPl Pl)).eval (r : ℚ) := by
+  have h := Lagrange.eval_nodal_derivative_eval_node_eq (s := Pl) (v := fun s : ℤ => (s : ℚ)) hr
+  rw [show piPl Pl = Lagrange.nodal Pl (fun s : ℤ => (s : ℚ)) from rfl, h, Lagrange.eval_nodal]
+
+lemma resP_pull (K : ℕ) (P : ℚ[X]) {r : ℤ} (hr : r ∈ PlK K) (hr0 : r ≠ 0) :
+    resP (pull K P) (PlK K) r =
+      -((r : ℚ) ^ 4 / 2) * (P.eval (-(r : ℚ) ^ 2) / (derivative (D K)).eval (-(r : ℚ) ^ 2)) := by
+  unfold resP
+  rw [prod_erase_eq_derivative _ hr, piPl_PlK, derivative_C_mul, derivative_comp, eval_mul,
+    eval_C, eval_mul, eval_comp, pull, eval_mul, eval_mul, eval_C, eval_pow, eval_X, eval_comp]
+  simp only [derivative_neg, derivative_X_pow, eval_neg, eval_mul, eval_C, eval_pow, eval_X]
+  have hr0' : (r : ℚ) ≠ 0 := by exact_mod_cast hr0
+  have hK : ((-1 : ℚ) ^ K) ≠ 0 := pow_ne_zero _ (by norm_num)
+  by_cases hD : (derivative (D K)).eval (-(r : ℚ) ^ 2) = 0
+  · rw [hD]; simp
+  · field_simp
+    norm_num
+    ring
+
+lemma dd_nat (j : ℕ) : dd (j : ℤ) = j := by simp [dd]
+
+lemma dd_neg {j : ℕ} (hj : 1 ≤ j) : dd (-(j : ℤ)) = j - 1 := by
+  unfold dd
+  rw [ite_eq_right (by omega)]
+  omega
+
+lemma H5_succ_eq {j : ℕ} (hj : 1 ≤ j) : H5 j = H5 (j - 1) + 1 / (j : ℚ) ^ 5 := by
+  obtain ⟨i, rfl⟩ : ∃ i, j = i + 1 := ⟨j - 1, by omega⟩
+  unfold H5
+  rw [Finset.sum_Icc_succ_top (by omega), Nat.add_sub_cancel]
+
+/--
+**The pullback identity (3.1)**: `μ_X(P / D_K) = τ_X((-1)^K x⁵ P(-x²) / ∏_{r ∈ PlK} (x - r))`. -/
+theorem μX_eq_tauX (K : ℕ) (P : ℚ[X]) : μX K P = tauX (pull K P) (PlK K) := by
+  unfold μX tauX
+  rw [polyPart_pull, tau_add, tau_pull_poly, tau_sum, sum_PlK]
+  simp only [tau_C_mul, tau_cubic]
+  rw [map_add, map_sum, add_assoc, ← Finset.sum_add_distrib]
+  congr 1
+  refine Finset.sum_congr rfl fun j hj => ?_
+  have hj1 : 1 ≤ j := (Finset.mem_Icc.mp hj).1
+  have hjP : (j : ℤ) ∈ PlK K := by
+    rw [PlK, Finset.mem_union]; left;
+    exact Finset.mem_image_of_mem _ hj
+  have hjN : -(j : ℤ) ∈ PlK K := by
+    rw [PlK, Finset.mem_union]; right;
+    exact Finset.mem_image_of_mem _ hj
+  rw [resP_pull K P hjP (by omega), resP_pull K P hjN (by omega), dd_nat, dd_neg hj1]
+  have hj0 : (j : ℚ) ≠ 0 := by exact_mod_cast (by omega : j ≠ 0)
+  unfold poleValue res
+  rw [H5_succ_eq hj1]
+  push_cast
+  ext n
+  simp only [coeff_add, coeff_sub, coeff_C_mul, coeff_C, coeff_X]
+  split_ifs <;>
+    first
+    | omega
+    | (field_simp; ring)
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Arith/SmallPrime.lean b/LeanPool/Zeta5Irrational/Arith/SmallPrime.lean
new file mode 100644
index 0000000000..fc3ec824f2
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Arith/SmallPrime.lean
@@ -0,0 +1,117 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Arith.BinomBasis
+public import LeanPool.Zeta5Irrational.Arith.Val
+public import LeanPool.Zeta5Irrational.Construction
+import LeanPool.Zeta5Irrational.Arith.TauBound
+import Mathlib.Data.Rat.Star
+import Mathlib.Tactic.ENatToNat
+import Mathlib.Tactic.ReduceModChar
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.Ring.Basic
+
+/-! # Small primes: generic tools for the bound (3.12)
+
+* integer values of the binomial polynomials at all integers;
+* a polynomial of degree `≤ d` whose values at `d + 1` consecutive integers have valuation `≥ β`
+  has valuation `≥ β` at every integer (`VG_eval_int_of_window`);
+* Kummer's bound `v_p(C(n, k)) ≤ log_p n`;
+* `v_p(H^{(5)}_m) ≥ -5 log_p m`.
+-/
+
+@[expose] public section
+
+open Finset Polynomial
+
+namespace Zeta5Irrational
+
+variable {p : ℕ} [hp : Fact p.Prime]
+
+/-- Values of `bin k` at integers are integers. -/
+lemma bin_eval_int (k : ℕ) (z : ℤ) : ∃ w : ℤ, (bin k).eval (z : ℚ) = w := by
+  induction k generalizing z with
+  | zero => exact ⟨1, by simp [bin_zero]⟩
+  | succ k
+    ih =>
+    have hstep : ∀ y : ℚ, (bin (k + 1)).eval (y + 1) = (bin (k + 1)).eval y + (bin k).eval y := by
+      intro y
+      have := congrArg (eval y) (bin_comp_add_one k)
+      rw [eval_comp, eval_add, eval_X, eval_one, eval_add] at this
+      exact this
+    induction z using Int.induction_on with
+    | zero => exact ⟨0, by rw [Int.cast_zero, bin_eval_zero (by omega)]⟩
+    | succ i hi =>
+      obtain ⟨w, hw⟩ := hi
+      obtain ⟨w', hw'⟩ := ih (i : ℤ)
+      refine ⟨w + w', ?_⟩
+      push_cast at hw hw' ⊢
+      rw [hstep, hw, hw']
+    | pred i hi =>
+      obtain ⟨w, hw⟩ := hi
+      obtain ⟨w', hw'⟩ := ih (-(i : ℤ) - 1)
+      refine ⟨w - w', ?_⟩
+      have := hstep ((-(i : ℤ) - 1 : ℤ) : ℚ)
+      push_cast at this hw hw' ⊢
+      rw [show (-(i : ℚ) - 1 + 1) = -(i : ℚ) by ring, hw] at this
+      linarith
+
+omit hp in
+lemma VG_bin_eval_int (k : ℕ) (z : ℤ) : VG p ((bin k).eval (z : ℚ)) 0 := by
+  obtain ⟨w, hw⟩ := bin_eval_int k z
+  rw [hw]; exact VG.intCast w
+
+/-- **Window lemma**: values at `s, …, s + d` control the values at all integers. -/
+theorem VG_eval_int_of_window {P : ℚ[X]} {d : ℕ} (hd : P.natDegree ≤ d) (s : ℤ) {β : ℚ}
+    (hv : ∀ i ≤ d, VG p (P.eval ((s + i : ℤ) : ℚ)) β) (z : ℤ) : VG p (P.eval (z : ℚ)) β := by
+  set Q := P.comp (X + C (s : ℚ)) with hQ
+  have hQd : Q.natDegree ≤ d := by
+    rw [hQ]
+    refine (natDegree_comp_le).trans ?_
+    have : (X + C (s : ℚ)).natDegree ≤ 1 := by rw [natDegree_X_add_C]
+    calc
+      P.natDegree * (X + C (s : ℚ)).natDegree ≤ d * 1 := Nat.mul_le_mul hd this
+      _ = d := by ring
+  have hQv : ∀ m ≤ d, VG p (Q.eval (m : ℚ)) β := by
+    intro m hm
+    have := hv m hm
+    rw [hQ, eval_comp]
+    simp only [eval_add, eval_X, eval_C]
+    push_cast at this ⊢
+    rwa [add_comm]
+  obtain ⟨e, he, hev⟩ := binRep_of_values hQd hQv
+  have hPz : P.eval (z : ℚ) = Q.eval ((z - s : ℤ) : ℚ) := by rw [hQ, eval_comp]; simp
+  rw [hPz, he, eval_finsetSum]
+  refine VG.sum _ fun m hm => ?_
+  rw [eval_mul, eval_C]
+  have :=
+    (hev m (by have := Finset.mem_range.mp hm; omega)).mul (VG_bin_eval_int (p := p) m (z - s))
+  simpa using this
+
+/-- **Kummer**: `v_p(C(n, k)⁻¹) ≥ -log_p n`. -/
+lemma VG_choose_inv {n k : ℕ} (hk : k ≤ n) : VG p ((n.choose k : ℚ)⁻¹) (-(Nat.log p n : ℚ)) := by
+  apply VG.inv (by exact_mod_cast (Nat.choose_pos hk).ne')
+  rw [padicValRat.of_nat, ← Nat.factorization_def _ hp.out]
+  exact_mod_cast Nat.factorization_choose_le_log
+
+/-- `v_p(H^{(5)}_m) ≥ -5 log_p m`. -/
+lemma VG_H5_log (m : ℕ) : VG p (H5 m) (-5 * (Nat.log p m : ℚ)) := by
+  unfold H5
+  refine VG.sum _ fun v hv => ?_
+  have hv' := Finset.mem_Icc.mp hv
+  have h := (VG.inv_nat (p := p) hv'.1 hv'.2).pow 5
+  rw [inv_pow] at h
+  rw [one_div]
+  refine h.mono ?_
+  push_cast; linarith
+
+omit hp in
+lemma log_mono {a b : ℕ} (h : a ≤ b) : (Nat.log p a : ℚ) ≤ Nat.log p b := by
+  exact_mod_cast Nat.log_mono_right h
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Arith/SmallPrimeF.lean b/LeanPool/Zeta5Irrational/Arith/SmallPrimeF.lean
new file mode 100644
index 0000000000..8fcdd11e03
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Arith/SmallPrimeF.lean
@@ -0,0 +1,324 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Arith.BasisChange
+public import LeanPool.Zeta5Irrational.Arith.BinomBasis
+public import LeanPool.Zeta5Irrational.Arith.Val
+import LeanPool.Zeta5Irrational.Arith.SmallPrime
+import LeanPool.Zeta5Irrational.Arith.SmallPrimeTau
+import LeanPool.Zeta5Irrational.PartialFractions
+import Mathlib.Algebra.Ring.IsFormallyReal
+import Mathlib.Data.Int.Star
+import Mathlib.Data.Rat.Star
+import Mathlib.Tactic.ENatToNat
+import Mathlib.Tactic.Polynomial.Basic
+import Mathlib.Tactic.ReduceModChar
+import Mathlib.Tactic.FieldSimp
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+import Mathlib.Tactic.Positivity.Basic
+import Mathlib.Tactic.Ring.Basic
+
+/-! # Small primes: the bound (3.12)
+
+`v_p^G(F_K) ≥ -6h ⌊log_p (5K)⌋ - h v_p(24)` for every prime `p` (`small_prime_bound`), via the
+basis `q_0 = 1`, `q_i(t) = (-1)^i 2t D_{i-1}(t)/(2i)!` with `q_i(-z²) = C(z+i, 2i) + C(z+i-1, 2i)`.
+-/
+
+@[expose] public section
+
+open Finset Polynomial
+
+namespace Zeta5Irrational
+
+variable {p : ℕ} [hp : Fact p.Prime]
+
+/-- The basis `q_a`. -/
+noncomputable def qP (a : ℕ) : ℚ[X] :=
+  if a = 0 then 1 else C ((-1 : ℚ) ^ a * 2 / ((2 * a).factorial : ℚ)) * (X * D (a - 1))
+
+/-- Its leading coefficient. -/
+noncomputable def qlead (a : ℕ) : ℚ :=
+  if a = 0 then 1 else (-1 : ℚ) ^ a * 2 / ((2 * a).factorial : ℚ)
+
+lemma qlead_ne_zero (a : ℕ) : qlead a ≠ 0 := by
+  unfold qlead; split_ifs
+  · norm_num
+  · have : ((2 * a).factorial : ℚ) ≠ 0 := by positivity
+    exact div_ne_zero (mul_ne_zero (pow_ne_zero _ (by norm_num)) (by norm_num)) this
+
+lemma qP_eq (a : ℕ) : qP a = C (qlead a) * (if a = 0 then 1 else X * D (a - 1)) := by
+  unfold qP qlead; split_ifs <;> simp
+
+lemma monic_qbase (a : ℕ) : (if a = 0 then (1 : ℚ[X]) else X * D (a - 1)).Monic := by
+  split_ifs
+  · exact monic_one
+  · exact monic_X.mul (D_monic _)
+
+lemma natDegree_qbase (a : ℕ) : (if a = 0 then (1 : ℚ[X]) else X * D (a - 1)).natDegree = a := by
+  split_ifs with h
+  · simp [h]
+  · rw [natDegree_mul X_ne_zero (D_monic _).ne_zero, natDegree_X, natDegree_D]; omega
+
+lemma natDegree_qP (a : ℕ) : (qP a).natDegree = a := by
+  rw [qP_eq, natDegree_C_mul (qlead_ne_zero a), natDegree_qbase]
+
+lemma coeff_qP_self (a : ℕ) : (qP a).coeff a = qlead a := by
+  rw [qP_eq, coeff_C_mul]
+  have := (monic_qbase a).coeff_natDegree
+  rw [natDegree_qbase] at this
+  rw [this, mul_one]
+
+/-! ### Integer values -/
+
+lemma prod_Icc_eq_prod_range (N : ℕ) (g : ℕ → ℚ) :
+    ∏ j ∈ Icc 1 N, g j = ∏ i ∈ range N, g (i + 1) := by
+  induction N with
+  | zero => simp
+  | succ N ih => rw [Finset.prod_Icc_succ_top (by omega), ih, Finset.prod_range_succ]
+
+lemma bin_eval_eq (k : ℕ) (y : ℚ) :
+    (bin k).eval y = (∏ i ∈ range k, (y - i)) / (k.factorial : ℚ) := by
+  unfold bin
+  rw [eval_mul, eval_C, descPochhammer_eval_eq_prod_range]
+  ring
+
+/-- `∏_{i < 2c+1} (z + c - i) = z ∏_{j ≤ c} (z² - j²)`. -/
+lemma prod_window (z : ℚ) (c : ℕ) :
+    ∏ i ∈ range (2 * c + 1), (z + c - i) = z * ∏ j ∈ Icc 1 c, (z ^ 2 - (j : ℚ) ^ 2) := by
+  induction c with
+  | zero => simp
+  | succ c
+    ih =>
+    rw [show 2 * (c + 1) + 1 = (2 * c + 1) + 1 + 1 by ring, Finset.prod_range_succ',
+      Finset.prod_range_succ]
+    have e :
+      ∏ i ∈ range (2 * c + 1), (z + ((c + 1 : ℕ) : ℚ) - ((i + 1 : ℕ) : ℚ)) =
+        ∏ i ∈ range (2 * c + 1), (z + c - i) :=
+      by refine Finset.prod_congr rfl fun i _ => ?_; push_cast; ring
+    rw [e, ih, Finset.prod_Icc_succ_top (by omega)]
+    push_cast; ring
+
+/-- `q_a(-z²) = C(z+a, 2a) + C(z+a-1, 2a)` for `a ≥ 1`. -/
+lemma qP_eval_neg_sq {a : ℕ} (ha : 1 ≤ a) (z : ℚ) :
+    (qP a).eval (-z ^ 2) = (bin (2 * a)).eval (z + a) + (bin (2 * a)).eval (z + a - 1) := by
+  obtain ⟨c, rfl⟩ : ∃ c, a = c + 1 := ⟨a - 1, by omega⟩
+  rw [qP, ite_eq_right (by omega), bin_eval_eq, bin_eval_eq, show c + 1 - 1 = c by omega]
+  rw [eval_mul, eval_C, eval_mul, eval_X]
+  unfold D
+  rw [eval_prod]
+  simp only [eval_add, eval_X, eval_C]
+  have h1 :
+    ∏ i ∈ range (2 * (c + 1)), (z + ((c + 1 : ℕ) : ℚ) - i) =
+      (z + ((c + 1 : ℕ) : ℚ)) * (z * ∏ j ∈ Icc 1 c, (z ^ 2 - (j : ℚ) ^ 2)) := by
+    rw [show 2 * (c + 1) = (2 * c + 1) + 1 by ring, Finset.prod_range_succ']
+    have e :
+      ∏ i ∈ range (2 * c + 1), (z + ((c + 1 : ℕ) : ℚ) - ((i + 1 : ℕ) : ℚ)) =
+        ∏ i ∈ range (2 * c + 1), (z + c - i) :=
+      by refine Finset.prod_congr rfl fun i _ => ?_; push_cast; ring
+    rw [e, prod_window]; push_cast; ring
+  have h2 :
+    ∏ i ∈ range (2 * (c + 1)), (z + ((c + 1 : ℕ) : ℚ) - 1 - i) =
+      (z * ∏ j ∈ Icc 1 c, (z ^ 2 - (j : ℚ) ^ 2)) * (z - ((c + 1 : ℕ) : ℚ)) := by
+    rw [show 2 * (c + 1) = (2 * c + 1) + 1 by ring, Finset.prod_range_succ]
+    have e :
+      ∏ i ∈ range (2 * c + 1), (z + ((c + 1 : ℕ) : ℚ) - 1 - (i : ℚ)) =
+        ∏ i ∈ range (2 * c + 1), (z + c - i) :=
+      by refine Finset.prod_congr rfl fun i _ => ?_; push_cast; ring
+    rw [e, prod_window]; push_cast; ring
+  rw [h1, h2]
+  have h3 :
+    ∏ j ∈ Icc 1 c, (-z ^ 2 + (j : ℚ) ^ 2) = (-1) ^ c * ∏ j ∈ Icc 1 c, (z ^ 2 - (j : ℚ) ^ 2) := by
+    rw [show ((-1 : ℚ) ^ c) = ∏ _j ∈ Icc 1 c, (-1 : ℚ) by
+        rw [Finset.prod_const, Nat.card_Icc, Nat.add_sub_cancel],
+      ← Finset.prod_mul_distrib]
+    refine Finset.prod_congr rfl fun j _ => ?_; ring
+  rw [h3]
+  have hf : ((2 * (c + 1)).factorial : ℚ) ≠ 0 := by positivity
+  field_simp
+  push_cast
+  have hc : (-1 : ℚ) ^ (c * 2) = 1 := Even.neg_one_pow ⟨c, by ring⟩
+  linear_combination (2 * z ^ 2 * ∏ j ∈ Icc 1 c, (z ^ 2 - (j : ℚ) ^ 2)) * hc
+
+lemma VG_qP_eval (a : ℕ) (z : ℤ) : VG p ((qP a).eval (-(z : ℚ) ^ 2)) 0 := by
+  rcases Nat.eq_zero_or_pos a with rfl | ha
+  · simp only [qP, ↓reduceIte, eval_one]; exact VG.one
+  · rw [qP_eval_neg_sq ha]
+    refine VG.add ?_ ?_
+    · have := VG_bin_eval_int (p := p) (2 * a) (z + a); push_cast at this; exact this
+    · have := VG_bin_eval_int (p := p) (2 * a) (z + a - 1); push_cast at this; exact this
+
+/-- `D_N(-z²) = (N!)² C(N+z, N) C(N-z, N)`. -/
+lemma D_eval_neg_sq (N : ℕ) (z : ℚ) :
+    (D N).eval (-z ^ 2) = (N.factorial : ℚ) ^ 2 * ((bin N).eval (N + z) * (bin N).eval (N - z)) :=
+  by
+  rw [bin_eval_eq, bin_eval_eq]
+  unfold D
+  rw [eval_prod]
+  simp only [eval_add, eval_X, eval_C]
+  have hf : (N.factorial : ℚ) ≠ 0 := by positivity
+  have r1 : ∏ i ∈ range N, ((N : ℚ) + z - i) = ∏ i ∈ range N, (z + ((i + 1 : ℕ) : ℚ)) := by
+    rw [← Finset.prod_range_reflect (fun i => z + ((i + 1 : ℕ) : ℚ)) N]
+    refine Finset.prod_congr rfl fun i hi => ?_
+    have := Finset.mem_range.mp hi
+    push_cast [show i ≤ N - 1 by omega, show 1 ≤ N by omega]; ring
+  have r2 : ∏ i ∈ range N, ((N : ℚ) - z - i) = ∏ i ∈ range N, (((i + 1 : ℕ) : ℚ) - z) := by
+    rw [← Finset.prod_range_reflect (fun i => ((i + 1 : ℕ) : ℚ) - z) N]
+    refine Finset.prod_congr rfl fun i hi => ?_
+    have := Finset.mem_range.mp hi
+    push_cast [show i ≤ N - 1 by omega, show 1 ≤ N by omega]; ring
+  rw [r1, r2, prod_Icc_eq_prod_range N (fun j => -z ^ 2 + (j : ℚ) ^ 2)]
+  field_simp
+  rw [← Finset.prod_mul_distrib]
+  refine Finset.prod_congr rfl fun i _ => ?_
+  ring
+
+lemma VG_D_eval (N : ℕ) (z : ℤ) : VG p ((D N).eval (-(z : ℚ) ^ 2) / (N.factorial : ℚ) ^ 2) 0 := by
+  have hf : (N.factorial : ℚ) ≠ 0 := by positivity
+  rw [D_eval_neg_sq, mul_div_cancel_left₀ _ (pow_ne_zero _ hf)]
+  have h1 := VG_bin_eval_int (p := p) N (N + z)
+  have h2 := VG_bin_eval_int (p := p) N (N - z)
+  push_cast at h1 h2
+  simpa using h1.mul h2
+
+/-! ### Assembly -/
+
+lemma det_coeffMat_qP (h : ℕ) : (coeffMat (fun a : Fin h => qP a)).det = ∏ a : Fin h, qlead a := by
+  rw [Matrix.det_of_isLowerTriangular]
+  · refine Finset.prod_congr rfl fun a _ => ?_
+    simp only [coeffMat, Matrix.of_apply]
+    exact coeff_qP_self a
+  · intro i j hij
+    simp only [coeffMat, Matrix.of_apply]
+    apply coeff_eq_zero_of_natDegree_lt
+    rw [natDegree_qP]
+    exact hij
+
+lemma qlead_sq {a : ℕ} (ha : 1 ≤ a) : qlead a ^ 2 = 4 / (((2 * a).factorial : ℚ) ^ 2) := by
+  unfold qlead
+  rw [ite_eq_right (by omega), div_pow, mul_pow, ← pow_mul, show a * 2 = 2 * a by ring, pow_mul]
+  norm_num
+
+lemma prod_qlead_sq (h : ℕ) (hh : 1 ≤ h) :
+    (∏ a : Fin h, qlead a) ^ 2 =
+      4 ^ (h - 1) / ∏ i ∈ Icc 1 (h - 1), (((2 * i).factorial : ℚ) ^ 2) := by
+  rw [← Finset.prod_pow, Fin.prod_univ_eq_prod_range (fun a => qlead a ^ 2) h]
+  obtain ⟨h', rfl⟩ : ∃ h', h = h' + 1 := ⟨h - 1, by omega⟩
+  rw [Finset.prod_range_succ', show qlead 0 = 1 by simp [qlead], one_pow, mul_one, ←
+    prod_Icc_eq_prod_range h' (fun a => qlead a ^ 2), Nat.add_sub_cancel]
+  rw [Finset.prod_congr rfl (fun i hi => qlead_sq (Finset.mem_Icc.mp hi).1),
+    Finset.prod_div_distrib, Finset.prod_const, Nat.card_Icc, Nat.add_sub_cancel]
+
+/-- `S_K = ((K!)²/(N!)¹²)^h det(C)²`. -/
+lemma S_eq_qlead (n : ℕ) (hn : 1 ≤ n) :
+    S n =
+      (((40 * n).factorial : ℚ) ^ 2 / ((3 * n).factorial : ℚ) ^ 12) ^ (37 * n) *
+        (∏ a : Fin (37 * n), qlead a) ^ 2 := by
+  rw [prod_qlead_sq _ (by omega)]
+  unfold S
+  have h1 : ((3 * n).factorial : ℚ) ≠ 0 := by positivity
+  have h2 : ∏ i ∈ Icc 1 (37 * n - 1), (((2 * i).factorial : ℚ) ^ 2) ≠ 0 :=
+    Finset.prod_ne_zero_iff.mpr fun i _ => by positivity
+  rw [div_pow, ← pow_mul, ← pow_mul, show 12 * (37 * n) = 12 * (37 * n) by rfl]
+  field_simp
+
+/-- **The bound (3.12)** at every prime. -/
+theorem small_prime_bound {n : ℕ} (hn : 1 ≤ n) :
+    GV p (F n)
+      (-(6 * (37 * n : ℕ) * (Nat.log p (200 * n) : ℚ) + (37 * n : ℕ) * padicValNat p 24)) := by
+  set h := 37 * n with hh
+  set K := 40 * n with hK
+  set N := 3 * n with hN
+  set E : Fin h → ℚ[X] := fun a => qP a with hE
+  have hEdeg : ∀ a, (E a).natDegree < h := fun a => by
+    rw [hE]; simp only; rw [natDegree_qP];
+    exact a.isLt
+  have hbc := det_basis_change n E hEdeg
+  set c : ℚ := (K.factorial : ℚ) ^ 2 / (N.factorial : ℚ) ^ 12 with hc
+  set M := Matrix.of fun a b => μX (40 * n) (D (3 * n) ^ 6 * E a * E b) with hM
+  have hF : F n = ((C c) • M).det := by
+    rw [Matrix.det_smul, hbc, Fintype.card_fin, F, S_eq_qlead n hn, det_coeffMat_qP]
+    simp only [map_mul, map_pow]
+    ring
+  rw [hF]
+  set d := 5 + 12 * N + 4 * (h - 1) with hd
+  set Λ : ℚ := (Nat.log p (200 * n) : ℚ) with hΛ
+  have hd1 : d + 1 ≤ 200 * n := by omega
+  have hdK : 2 * K ≤ d := by omega
+  have hentry :
+    ∀ a b : Fin h,
+      GV p (((C c) • M) a b)
+        ((-3 * Λ - padicValNat p 24 / 2) + (-3 * Λ - padicValNat p 24 / 2)) := by
+    intro a b
+    rw [Matrix.smul_apply, hM, Matrix.of_apply, smul_eq_mul, ← μX_C_mul, μX_eq_tauX]
+    set Q := D (3 * n) ^ 6 * E a * E b with hQ
+    set A : ℚ[X] := C ((-1 : ℚ) ^ K / (N.factorial : ℚ) ^ 12) * X ^ 5 * Q.comp (-X ^ 2) with hA
+    have hpull : pull (40 * n) (C c * Q) = C ((K.factorial : ℚ) ^ 2) * A := by
+      have e1 : C c = C ((K.factorial : ℚ) ^ 2) * C (1 / (N.factorial : ℚ) ^ 12) := by
+        rw [← C_mul]; congr 1; rw [hc]; ring
+      have e2 :
+        C ((-1 : ℚ) ^ K / (N.factorial : ℚ) ^ 12) =
+          C ((-1 : ℚ) ^ (40 * n)) * C (1 / (N.factorial : ℚ) ^ 12) :=
+        by rw [← C_mul]; congr 1; rw [hK]; ring
+      rw [pull, hA, mul_comp, C_comp, e1, e2]; ring
+    rw [hpull]
+    have hQdeg : Q.natDegree ≤ 6 * N + 2 * (h - 1) := by
+      rw [hQ]
+      refine (natDegree_mul_le).trans ?_
+      have h1 : (D (3 * n) ^ 6 * E a).natDegree ≤ 6 * N + a := by
+        refine (natDegree_mul_le).trans ?_
+        rw [natDegree_pow, natDegree_D, hE]; simp only; rw [natDegree_qP]
+      have h2 : (E b).natDegree ≤ b := by rw [hE]; simp only; rw [natDegree_qP]
+      have := a.isLt; have := b.isLt
+      omega
+    have hAdeg : A.natDegree ≤ d := by
+      rw [hA]
+      refine (natDegree_mul_le).trans ?_
+      have h1 : (C ((-1 : ℚ) ^ K / (N.factorial : ℚ) ^ 12) * X ^ 5 : ℚ[X]).natDegree ≤ 5 :=
+        (natDegree_C_mul_le _ _).trans (by rw [natDegree_X_pow])
+      have h2 : (Q.comp (-X ^ 2 : ℚ[X])).natDegree ≤ 2 * (6 * N + 2 * (h - 1)) := by
+        rw [natDegree_comp]
+        have : (-X ^ 2 : ℚ[X]).natDegree = 2 := by rw [natDegree_neg, natDegree_X_pow]
+        rw [this]; linarith
+      omega
+    have hAint : ∀ z : ℤ, VG p (A.eval (z : ℚ)) 0 := by
+      intro z
+      have e :
+        A.eval (z : ℚ) =
+          (-1 : ℚ) ^ K * (z : ℚ) ^ 5 * ((D N).eval (-(z : ℚ) ^ 2) / (N.factorial : ℚ) ^ 2) ^ 6 *
+              (qP a).eval (-(z : ℚ) ^ 2) *
+            (qP b).eval (-(z : ℚ) ^ 2) := by
+        rw [hA, hQ, hE]
+        simp only [eval_mul, eval_C, eval_pow, eval_X, eval_comp, eval_neg]
+        have : (N.factorial : ℚ) ≠ 0 := by positivity
+        field_simp
+        rw [hN]; ring
+      rw [e]
+      have hs : VG p ((-1 : ℚ) ^ K) 0 := by
+        rcases neg_one_pow_eq_or ℚ K with h | h <;> rw [h]
+        · exact VG.one
+        · exact VG.one.neg
+      have hz : VG p ((z : ℚ) ^ 5) 0 := by simpa using (VG.intCast (p := p) z).pow 5
+      have :=
+        (((hs.mul hz).mul ((VG_D_eval (p := p) N z).pow 6)).mul (VG_qP_eval (p := p) a z)).mul
+          (VG_qP_eval (p := p) b z)
+      simpa using this
+    have hb := crude_tauX (p := p) (K := K) (d := d) (by omega) hdK A hAdeg hAint
+    refine hb.mono ?_
+    have : (Nat.log p (d + 1) : ℚ) ≤ Λ := log_mono hd1
+    linarith
+  have hdet :=
+    det_GV _ (fun _ => -3 * Λ - padicValNat p 24 / 2) (fun _ => -3 * Λ - padicValNat p 24 / 2)
+      hentry
+  refine hdet.mono ?_
+  simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul]
+  linarith
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Arith/SmallPrimeRes.lean b/LeanPool/Zeta5Irrational/Arith/SmallPrimeRes.lean
new file mode 100644
index 0000000000..28fb192d3b
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Arith/SmallPrimeRes.lean
@@ -0,0 +1,216 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Arith.Pullback
+import Mathlib.Tactic.Polynomial.Basic
+import Mathlib.Tactic.ReduceModChar
+import Std.Tactic.BVDecide.Normalize.Prop
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+import Mathlib.Tactic.Positivity.Basic
+import Mathlib.Tactic.Ring.Basic
+
+/-! # Small primes: products over the poles `±1, …, ±K`
+
+* `∏_{s ∈ PlK K} (m - s) = (K!)² C(m-1, K) C(m+K, K)` for `m > K`;
+* the residue denominators `∏_{s ≠ r} (r - s)` at `r = ±j`, and
+  `v_p((K!)² / ∏_{s ≠ r} (r - s)) ≥ -log_p (2K)`.
+-/
+
+@[expose] public section
+
+open Finset Polynomial
+
+namespace Zeta5Irrational
+
+variable {p : ℕ} [hp : Fact p.Prime]
+
+lemma prod_Icc_sub_mul_factorial (a : ℕ) :
+    ∀ c : ℕ,
+      c < a →
+        (∏ i ∈ Icc 1 c, ((a : ℚ) - i)) * ((a - 1 - c).factorial : ℚ) = ((a - 1).factorial : ℚ) := by
+  intro c
+  induction c with
+  | zero => intro _; simp
+  | succ c ih =>
+    intro hc
+    rw [Finset.prod_Icc_succ_top (by omega), ← ih (by omega)]
+    have e : a - 1 - c = (a - 1 - (c + 1)) + 1 := by omega
+    rw [e, Nat.factorial_succ]
+    push_cast [show c + 1 ≤ a by omega, show 1 ≤ a by omega]
+    rw [show ((a - 1 - (c + 1) : ℕ) : ℚ) = (a : ℚ) - 1 - (c + 1) by
+        push_cast [show c + 1 ≤ a - 1 by omega, show 1 ≤ a by omega]; ring]
+    ring
+
+lemma prod_Icc_add_mul_factorial (a c : ℕ) :
+    (∏ i ∈ Icc 1 c, ((a : ℚ) + i)) * (a.factorial : ℚ) = ((a + c).factorial : ℚ) := by
+  induction c with
+  | zero => simp
+  | succ c
+    ih =>
+    rw [Finset.prod_Icc_succ_top (by omega), mul_right_comm, ih, ← add_assoc, Nat.factorial_succ]
+    push_cast; ring
+
+lemma prod_Ioc_sub (j t : ℕ) :
+    ∏ i ∈ Ioc j (j + t), ((j : ℚ) - i) = (-1) ^ t * (t.factorial : ℚ) := by
+  induction t with
+  | zero => simp
+  | succ t
+    ih =>
+    rw [← add_assoc, Finset.prod_Ioc_succ_top (by omega), ih, Nat.factorial_succ, pow_succ]
+    push_cast; ring
+
+/-- `∏_{j ≤ K} (m - j) = K! C(m-1, K)` for `m > K`. -/
+lemma prod_Icc_sub_eq_choose {m K : ℕ} (h : K < m) :
+    ∏ j ∈ Icc 1 K, ((m : ℚ) - j) = (K.factorial : ℚ) * ((m - 1).choose K : ℚ) := by
+  have h1 := prod_Icc_sub_mul_factorial m K h
+  have h2 :
+    ((m - 1).choose K : ℚ) * (K.factorial : ℚ) * ((m - 1 - K).factorial : ℚ) =
+      ((m - 1).factorial : ℚ) :=
+    by exact_mod_cast Nat.choose_mul_factorial_mul_factorial (by omega : K ≤ m - 1)
+  have hne : ((m - 1 - K).factorial : ℚ) ≠ 0 := by positivity
+  apply mul_right_cancel₀ hne
+  rw [h1, ← h2]; ring
+
+/-- `∏_{j ≤ K} (m + j) = K! C(m+K, K)`. -/
+lemma prod_Icc_add_eq_choose (m K : ℕ) :
+    ∏ j ∈ Icc 1 K, ((m : ℚ) + j) = (K.factorial : ℚ) * ((m + K).choose K : ℚ) := by
+  have h1 := prod_Icc_add_mul_factorial m K
+  have h2 :
+    ((m + K).choose K : ℚ) * (K.factorial : ℚ) * (m.factorial : ℚ) = ((m + K).factorial : ℚ) := by
+    have := Nat.choose_mul_factorial_mul_factorial (by omega : K ≤ m + K)
+    rw [Nat.add_sub_cancel] at this
+    exact_mod_cast this
+  have hne : (m.factorial : ℚ) ≠ 0 := by positivity
+  apply mul_right_cancel₀ hne
+  rw [h1, ← h2]; ring
+
+/-- The pole product at a point `m > K`. -/
+lemma prod_PlK_sub {m K : ℕ} (h : K < m) :
+    ∏ s ∈ PlK K, ((m : ℚ) - s) =
+      (K.factorial : ℚ) ^ 2 * ((m - 1).choose K : ℚ) * ((m + K).choose K : ℚ) := by
+  rw [prod_PlK, Finset.prod_mul_distrib]
+  simp only [Int.cast_natCast, Int.cast_neg, sub_neg_eq_add]
+  rw [prod_Icc_sub_eq_choose h, prod_Icc_add_eq_choose]; ring
+
+/-- The product `∏_{i ≤ K, i ≠ j} (j - i)`. -/
+lemma prod_Icc_ite_sub {j K : ℕ} (hj1 : 1 ≤ j) (hjK : j ≤ K) :
+    ∏ i ∈ Icc 1 K, (if i = j then (1 : ℚ) else (j : ℚ) - i) =
+      (-1) ^ (K - j) * ((j - 1).factorial : ℚ) * ((K - j).factorial : ℚ) := by
+  have hsplit : Icc 1 K = Icc 1 j ∪ Ioc j K := by
+    ext i;
+    simp only [Finset.mem_union, Finset.mem_Icc, Finset.mem_Ioc]; omega
+  have hdisj : Disjoint (Icc 1 j) (Ioc j K) := by
+    rw [Finset.disjoint_left]; intro i h1 h2
+    simp only [Finset.mem_Icc, Finset.mem_Ioc] at h1 h2; omega
+  rw [hsplit, Finset.prod_union hdisj]
+  have hA : ∏ i ∈ Icc 1 j, (if i = j then (1 : ℚ) else (j : ℚ) - i) = ((j - 1).factorial : ℚ) := by
+    obtain ⟨c, rfl⟩ : ∃ c, j = c + 1 := ⟨j - 1, by omega⟩
+    rw [Finset.prod_Icc_succ_top (by omega), ite_eq_left rfl, mul_one]
+    rw [Finset.prod_congr rfl
+        (fun i hi => ite_eq_right (by have := Finset.mem_Icc.mp hi; omega) :
+          ∀ i ∈ Icc 1 c,
+            (if i = c + 1 then (1 : ℚ) else ((c + 1 : ℕ) : ℚ) - i) = ((c + 1 : ℕ) : ℚ) - i)]
+    have := prod_Icc_sub_mul_factorial (c + 1) c (by omega)
+    rw [show c + 1 - 1 - c = 0 by omega, Nat.factorial_zero, Nat.cast_one, mul_one] at this
+    rw [this]
+  have hB :
+    ∏ i ∈ Ioc j K, (if i = j then (1 : ℚ) else (j : ℚ) - i) =
+      (-1) ^ (K - j) * ((K - j).factorial : ℚ) := by
+    rw [Finset.prod_congr rfl
+        (fun i hi => ite_eq_right (by have := Finset.mem_Ioc.mp hi; omega) :
+          ∀ i ∈ Ioc j K, (if i = j then (1 : ℚ) else (j : ℚ) - i) = (j : ℚ) - i)]
+    have := prod_Ioc_sub j (K - j)
+    rwa [show j + (K - j) = K by omega] at this
+  rw [hA, hB]; ring
+
+/-- Residue denominator at `r = ±j`, up to sign. -/
+lemma prod_erase_PlK {j K : ℕ} (hj1 : 1 ≤ j) (hjK : j ≤ K) (ε : ℤ) (hε : ε = 1 ∨ ε = -1) :
+    ∃ σ : ℚ,
+      (σ = 1 ∨ σ = -1) ∧
+        ∏ s ∈ (PlK K).erase (ε * j), (((ε * j : ℤ) : ℚ) - s) =
+          σ * ((j - 1).factorial : ℚ) * ((K - j).factorial : ℚ) * ((K + j).factorial : ℚ) /
+            (j.factorial : ℚ) := by
+  classical
+  set r : ℤ := ε * j with hr
+  have hrmem : r ∈ PlK K := by
+    rcases hε with rfl | rfl
+    · simp only [hr, one_mul, PlK, Finset.mem_union, Finset.mem_image]
+      exact Or.inl ⟨j, Finset.mem_Icc.mpr ⟨hj1, hjK⟩, rfl⟩
+    · simp only [hr, neg_one_mul, PlK, Finset.mem_union, Finset.mem_image]
+      exact Or.inr ⟨j, Finset.mem_Icc.mpr ⟨hj1, hjK⟩, rfl⟩
+  set f : ℤ → ℚ := fun s => if s = r then 1 else (r : ℚ) - s with hf
+  have e1 : ∏ s ∈ (PlK K).erase r, ((r : ℚ) - s) = ∏ s ∈ PlK K, f s := by
+    rw [← Finset.prod_erase_mul _ _ hrmem]
+    simp only [hf, ite_eq_left rfl, mul_one]
+    refine Finset.prod_congr rfl fun s hs => ?_
+    rw [ite_eq_right (Finset.mem_erase.mp hs).1]
+  rw [e1, prod_PlK]
+  have hPj := prod_Icc_ite_sub hj1 hjK
+  have hjf : (j.factorial : ℚ) ≠ 0 := by positivity
+  have hA' : ∏ i ∈ Icc 1 K, ((j : ℚ) + i) = ((K + j).factorial : ℚ) / (j.factorial : ℚ) := by
+    rw [eq_div_iff hjf, prod_Icc_add_mul_factorial j K, add_comm]
+  rcases hε with hε | hε
+  · refine ⟨(-1) ^ (K - j), by rcases neg_one_pow_eq_or ℚ (K - j) with h | h <;> simp [h], ?_⟩
+    have e2 :
+      ∀ i ∈ Icc 1 K,
+        f (i : ℤ) * f (-(i : ℤ)) = (if i = j then (1 : ℚ) else (j : ℚ) - i) * ((j : ℚ) + i) := by
+      intro i hi
+      have hi' := Finset.mem_Icc.mp hi
+      simp only [hf, hr, hε, one_mul]
+      rw [ite_eq_right (by omega : -(i : ℤ) ≠ (j : ℤ))]
+      by_cases hij : i = j
+      · subst hij; simp
+      · rw [ite_eq_right (by exact_mod_cast hij), ite_eq_right hij]; push_cast; ring
+    rw [Finset.prod_congr rfl e2, Finset.prod_mul_distrib, hPj, hA']
+    ring
+  · refine ⟨(-1) ^ (K - j) * (-1) ^ (2 * K - 1), ?_, ?_⟩
+    · rcases neg_one_pow_eq_or ℚ (K - j) with h | h <;>
+          rcases neg_one_pow_eq_or ℚ (2 * K - 1) with h' | h' <;>
+        simp [h, h']
+    have e2 :
+      ∀ i ∈ Icc 1 K,
+        f (i : ℤ) * f (-(i : ℤ)) =
+          ((-1) * ((j : ℚ) + i)) * ((-1) * (if i = j then (-1 : ℚ) else (j : ℚ) - i)) := by
+      intro i hi
+      have hi' := Finset.mem_Icc.mp hi
+      simp only [hf, hr, hε, neg_one_mul]
+      rw [ite_eq_right (by omega : (i : ℤ) ≠ -(j : ℤ))]
+      by_cases hij : i = j
+      · subst hij; simp; ring
+      · rw [ite_eq_right (by intro h; exact hij (by exact_mod_cast neg_injective h)),
+          ite_eq_right hij]
+        push_cast; ring
+    rw [Finset.prod_congr rfl e2, Finset.prod_mul_distrib, Finset.prod_mul_distrib,
+      Finset.prod_mul_distrib, Finset.prod_const]
+    simp only [Nat.card_Icc, Nat.add_sub_cancel]
+    have hPj' :
+      ∏ i ∈ Icc 1 K, (if i = j then (-1 : ℚ) else (j : ℚ) - i) =
+        -((-1) ^ (K - j) * ((j - 1).factorial : ℚ) * ((K - j).factorial : ℚ)) := by
+      rw [← hPj, ← Finset.mul_prod_erase _ _ (Finset.mem_Icc.mpr ⟨hj1, hjK⟩), ←
+        Finset.mul_prod_erase (Icc 1 K) (fun i => if i = j then (1 : ℚ) else (j : ℚ) - i)
+          (Finset.mem_Icc.mpr ⟨hj1, hjK⟩),
+        ite_eq_left rfl, ite_eq_left rfl]
+      rw [Finset.prod_congr rfl (fun i hi => by rw [ite_eq_right (Finset.mem_erase.mp hi).1]),
+        Finset.prod_congr rfl
+          (fun i hi => by rw [ite_eq_right (Finset.mem_erase.mp hi).1] :
+            ∀ i ∈ (Icc 1 K).erase j, (if i = j then (1 : ℚ) else (j : ℚ) - i) = (j : ℚ) - i)]
+      ring
+    rw [hPj', hA']
+    have h2K : (-1 : ℚ) ^ (2 * K - 1) * (-1) = 1 := by
+      rw [← pow_succ, show 2 * K - 1 + 1 = 2 * K by omega, pow_mul]; norm_num
+    have hodd : (-1 : ℚ) ^ (2 * K - 1) = -1 := by linear_combination (-1 : ℚ) * h2K
+    have hK2 : (-1 : ℚ) ^ K * (-1) ^ K = 1 := by rw [← pow_add, ← two_mul, pow_mul]; norm_num
+    rw [hodd]
+    linear_combination
+      (-((-1 : ℚ) ^ (K - j) * ((j - 1).factorial : ℚ) * ((K - j).factorial : ℚ) *
+            (((K + j).factorial : ℚ) / (j.factorial : ℚ)))) *
+        hK2
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Arith/SmallPrimeTau.lean b/LeanPool/Zeta5Irrational/Arith/SmallPrimeTau.lean
new file mode 100644
index 0000000000..1bbc6a51ec
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Arith/SmallPrimeTau.lean
@@ -0,0 +1,229 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Arith.Pullback
+import LeanPool.Zeta5Irrational.Arith.SmallPrime
+import LeanPool.Zeta5Irrational.Arith.SmallPrimeRes
+import Mathlib.Data.Rat.Star
+import Mathlib.Tactic.ENatToNat
+import Mathlib.Tactic.ReduceModChar
+import Mathlib.Tactic.FieldSimp
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+import Mathlib.Tactic.Positivity.Basic
+import Mathlib.Tactic.Ring.Basic
+
+/-! # Small primes: the analogue of Lemma 3.3
+
+For `A` integer-valued on `ℤ` with `deg A ≤ d`, `2K ≤ d`:
+`v_p^G(τ_X((K!)² A / ∏_{0<|r|≤K}(x - r))) ≥ -6 log_p (d+1) - v_p(24)`.
+
+Instead of the paper's ball analysis we evaluate the polynomial part on the window
+`K+1, …, d-K+1` beyond all poles, where `(K!)²/∏(m - r) = 1/(C(m-1,K) C(m+K,K))`, and use the
+window lemma.
+-/
+
+@[expose] public section
+
+open Finset Polynomial
+
+namespace Zeta5Irrational
+
+variable {p : ℕ} [hp : Fact p.Prime]
+
+lemma mem_PlK_iff {K : ℕ} {r : ℤ} (hr : r ∈ PlK K) :
+    ∃ j ∈ Icc 1 K, ∃ ε : ℤ, (ε = 1 ∨ ε = -1) ∧ r = ε * j := by
+  simp only [PlK, Finset.mem_union, Finset.mem_image] at hr
+  rcases hr with ⟨j, hj, rfl⟩ | ⟨j, hj, rfl⟩
+  · exact ⟨j, hj, 1, Or.inl rfl, by ring⟩
+  · exact ⟨j, hj, -1, Or.inr rfl, by ring⟩
+
+/-- `(K!)² / ((K-j)! (K+j)!) = C(2K, K+j) / C(2K, K)`. -/
+lemma factorial_ratio {j K : ℕ} (hjK : j ≤ K) :
+    (K.factorial : ℚ) ^ 2 / (((K - j).factorial : ℚ) * ((K + j).factorial : ℚ)) =
+      ((2 * K).choose (K + j) : ℚ) / ((2 * K).choose K : ℚ) := by
+  have h1 :
+    ((2 * K).choose (K + j) : ℚ) * ((K + j).factorial : ℚ) * ((K - j).factorial : ℚ) =
+      ((2 * K).factorial : ℚ) := by
+    have := Nat.choose_mul_factorial_mul_factorial (by omega : K + j ≤ 2 * K)
+    rw [show 2 * K - (K + j) = K - j by omega] at this
+    exact_mod_cast this
+  have h2 :
+    ((2 * K).choose K : ℚ) * (K.factorial : ℚ) * (K.factorial : ℚ) = ((2 * K).factorial : ℚ) := by
+    have := Nat.choose_mul_factorial_mul_factorial (by omega : K ≤ 2 * K)
+    rw [show 2 * K - K = K by omega] at this
+    exact_mod_cast this
+  have hc : ((2 * K).choose K : ℚ) ≠ 0 := by exact_mod_cast (Nat.choose_pos (by omega)).ne'
+  have ha : ((K - j).factorial : ℚ) ≠ 0 := by positivity
+  have hb : ((K + j).factorial : ℚ) ≠ 0 := by positivity
+  rw [div_eq_div_iff (mul_ne_zero ha hb) hc]
+  linear_combination h2 - h1
+
+/-- The residue factor `(K!)² / ∏_{s ≠ r} (r - s)` has valuation `≥ -log_p (2K)`. -/
+lemma VG_res_factor {K : ℕ} {r : ℤ} (hr : r ∈ PlK K) :
+    VG p ((K.factorial : ℚ) ^ 2 / ∏ s ∈ (PlK K).erase r, ((r : ℚ) - s))
+      (-(Nat.log p (2 * K) : ℚ)) := by
+  obtain ⟨j, hj, ε, hε, rfl⟩ := mem_PlK_iff hr
+  have hj' := Finset.mem_Icc.mp hj
+  obtain ⟨σ, hσ, he⟩ := prod_erase_PlK hj'.1 hj'.2 ε hε
+  rw [he]
+  have hjf : (j.factorial : ℚ) = j * ((j - 1).factorial : ℚ) := by
+    rw [← Nat.mul_factorial_pred (by omega : j ≠ 0)]; push_cast; ring
+  have hσ2 : σ * σ = 1 := by rcases hσ with rfl | rfl <;> norm_num
+  have e :
+    (K.factorial : ℚ) ^ 2 /
+        (σ * ((j - 1).factorial : ℚ) * ((K - j).factorial : ℚ) * ((K + j).factorial : ℚ) /
+          (j.factorial : ℚ)) =
+      (σ * j) * (((2 * K).choose (K + j) : ℚ) * ((2 * K).choose K : ℚ)⁻¹) := by
+    rw [← div_eq_mul_inv, ← factorial_ratio hj'.2, hjf]
+    have hσ0 : σ ≠ 0 := by rcases hσ with rfl | rfl <;> norm_num
+    have h1 : ((j - 1).factorial : ℚ) ≠ 0 := by positivity
+    have h2 : ((K - j).factorial : ℚ) ≠ 0 := by positivity
+    have h3 : ((K + j).factorial : ℚ) ≠ 0 := by positivity
+    have hj0 : (j : ℚ) ≠ 0 := by exact_mod_cast (by omega : j ≠ 0)
+    field_simp
+    linear_combination (-1 : ℚ) * hσ2
+  rw [e]
+  have hσv : VG p (σ * j) 0 := by
+    rcases hσ with rfl | rfl
+    · simpa using VG.natCast (p := p) j
+    · simpa using (VG.natCast (p := p) j).neg
+  have :=
+    hσv.mul
+      ((VG.natCast (p := p) ((2 * K).choose (K + j))).mul
+        (VG_choose_inv (p := p) (by omega : K ≤ 2 * K)))
+  exact this.mono (by simp)
+
+/-- **The analogue of Lemma 3.3** for the poles `±1, …, ±K`. -/
+theorem crude_tauX {K d : ℕ} (hK : 1 ≤ K) (hdK : 2 * K ≤ d) (A : ℚ[X]) (hA : A.natDegree ≤ d)
+    (hint : ∀ z : ℤ, VG p (A.eval (z : ℚ)) 0) :
+    GV p (tauX (C ((K.factorial : ℚ) ^ 2) * A) (PlK K))
+      (-6 * (Nat.log p (d + 1) : ℚ) - padicValNat p 24) := by
+  set g := C ((K.factorial : ℚ) ^ 2) * A with hg
+  set Λ : ℚ := (Nat.log p (d + 1) : ℚ) with hΛ
+  have hΛ0 : (0 : ℚ) ≤ Λ := by positivity
+  have hlog2K : (Nat.log p (2 * K) : ℚ) ≤ Λ := log_mono (by omega)
+  have hv24 : (0 : ℚ) ≤ padicValNat p 24 := by
+    positivity
+      -- residues
+  have hres : ∀ r ∈ PlK K, VG p (resP g (PlK K) r) (-Λ) := by
+    intro r hr
+    unfold resP
+    rw [hg, eval_mul, eval_C, mul_comm, mul_div_assoc]
+    refine ((hint r).mul (VG_res_factor hr)).mono ?_
+    linarith
+  have hgdeg : g.natDegree ≤ d := (natDegree_C_mul_le _ _).trans hA
+  unfold tauX
+  refine GV.add (GV.C ?_) (GV.sum _ fun r hr => ?_)
+  · -- the polynomial part
+    set P := polyPart g (PlK K) with hP
+    have hPd : P.natDegree ≤ d - 2 * K := by
+      have := natDegree_polyPart_le g (PlK K)
+      rw [card_PlK, ← hP] at this
+      omega
+        -- values on the window
+    have hwin : ∀ i ≤ d - 2 * K, VG p (P.eval ((((K + 1 : ℕ) : ℤ) + i : ℤ) : ℚ)) (-2 * Λ) := by
+      intro i hi
+      set m : ℕ := K + 1 + i with hm
+      have hmK : K < m := by omega
+      have hmd : m + K ≤ d + 1 := by omega
+      have hcast : ((((K + 1 : ℕ) : ℤ) + i : ℤ) : ℚ) = (m : ℚ) := by rw [hm]; push_cast; ring
+      rw [hcast]
+      have hpf :
+        g.eval (m : ℚ) =
+          P.eval (m : ℚ) * (piPl (PlK K)).eval (m : ℚ) +
+            ∑ r ∈ PlK K,
+              (C (resP g (PlK K) r) * ∏ s ∈ (PlK K).erase r, (X - C (s : ℚ))).eval (m : ℚ) := by
+        conv_lhs => rw [partial_fractionsP g (PlK K)]
+        rw [eval_add, eval_mul, eval_finsetSum]
+      have hpi : (piPl (PlK K)).eval (m : ℚ) = ∏ s ∈ PlK K, ((m : ℚ) - s) := by
+        unfold piPl;
+        rw [eval_prod]; simp
+      have hpi' := prod_PlK_sub (K := K) hmK
+      have hne : ∀ s ∈ PlK K, (m : ℚ) - s ≠ 0 := by
+        intro s hs
+        obtain ⟨j, hj, ε, hε, rfl⟩ := mem_PlK_iff hs
+        have := (Finset.mem_Icc.mp hj).2
+        rcases hε with rfl | rfl <;> push_cast <;> intro h
+        · have : (m : ℚ) = j := by linarith
+          have : m = j := by exact_mod_cast this
+          omega
+        · have : (m : ℚ) + j = 0 := by linarith
+          have : m + j = 0 := by exact_mod_cast this
+          omega
+      have hPi0 : ∏ s ∈ PlK K, ((m : ℚ) - s) ≠ 0 := Finset.prod_ne_zero_iff.mpr hne
+      have hterm :
+        ∀ r ∈ PlK K,
+          (C (resP g (PlK K) r) * ∏ s ∈ (PlK K).erase r, (X - C (s : ℚ))).eval (m : ℚ) =
+            resP g (PlK K) r / ((m : ℚ) - r) * ∏ s ∈ PlK K, ((m : ℚ) - s) := by
+        intro r hr
+        rw [eval_mul, eval_C, eval_prod, ← Finset.mul_prod_erase _ _ hr]
+        simp only [eval_sub, eval_X, eval_C]
+        field_simp [hne r hr]
+      rw [Finset.sum_congr rfl hterm, hpi, ← Finset.sum_mul] at hpf
+      have hPm :
+        P.eval (m : ℚ) =
+          g.eval (m : ℚ) / ∏ s ∈ PlK K, ((m : ℚ) - s) -
+            ∑ r ∈ PlK K, resP g (PlK K) r / ((m : ℚ) - r) := by
+        rw [eq_sub_iff_add_eq, eq_div_iff hPi0, add_mul]
+        exact hpf.symm
+      rw [hPm]
+      refine VG.sub ?_ (VG.sum _ fun r hr => ?_)
+      · rw [hpi', hg, eval_mul, eval_C]
+        have hc1 : ((m - 1).choose K : ℚ) ≠ 0 := by exact_mod_cast (Nat.choose_pos (by omega)).ne'
+        have hc2 : ((m + K).choose K : ℚ) ≠ 0 := by exact_mod_cast (Nat.choose_pos (by omega)).ne'
+        have hKf : (K.factorial : ℚ) ≠ 0 := by positivity
+        have e :
+          (K.factorial : ℚ) ^ 2 * A.eval (m : ℚ) /
+              ((K.factorial : ℚ) ^ 2 * ((m - 1).choose K : ℚ) * ((m + K).choose K : ℚ)) =
+            A.eval (m : ℚ) * (((m - 1).choose K : ℚ)⁻¹ * ((m + K).choose K : ℚ)⁻¹) :=
+          by field_simp
+        rw [e]
+        have h1 := VG_choose_inv (p := p) (by omega : K ≤ m - 1)
+        have h2 := VG_choose_inv (p := p) (by omega : K ≤ m + K)
+        have l1 : (Nat.log p (m - 1) : ℚ) ≤ Λ := log_mono (by omega)
+        have l2 : (Nat.log p (m + K) : ℚ) ≤ Λ := log_mono (by omega)
+        have := (hint m).mul (h1.mul h2)
+        push_cast at this
+        exact this.mono (by linarith)
+      · obtain ⟨j, hj, ε, hε, rfl⟩ := mem_PlK_iff hr
+        have hj' := Finset.mem_Icc.mp hj
+        have hpos : ∃ u : ℕ, 1 ≤ u ∧ u ≤ d + 1 ∧ (m : ℚ) - ((ε * j : ℤ) : ℚ) = u := by
+          rcases hε with rfl | rfl
+          · exact ⟨m - j, by omega, by omega, by push_cast [show j ≤ m by omega]; ring⟩
+          · exact ⟨m + j, by omega, by omega, by push_cast; ring⟩
+        obtain ⟨u, hu1, hud, hu⟩ := hpos
+        rw [div_eq_mul_inv, hu]
+        have := (hres _ hr).mul (VG.inv_nat (p := p) hu1 hud)
+        exact this.mono (by rw [← hΛ]; linarith)
+    have hall := VG_eval_int_of_window hPd ((K + 1 : ℕ) : ℤ) hwin
+    have htau :=
+      tau_bound (p := p) hPd (β := -2 * Λ) (fun m _ => by have := hall m; simpa using this)
+    refine htau.mono ?_
+    have : (Nat.log p (d - 2 * K + 1) : ℚ) ≤ Λ := log_mono (by omega)
+    linarith
+  · -- the pole terms
+    obtain ⟨j, hj, ε, hε, hrj⟩ := mem_PlK_iff hr
+    have hj' := Finset.mem_Icc.mp hj
+    have hdd : dd r ≤ d + 1 := by
+      rw [hrj]; rcases hε with rfl | rfl
+      · rw [one_mul, dd_nat]; omega
+      · rw [neg_one_mul, dd_neg hj'.1]; omega
+    have hH : GV p (C (H5 (dd r)) - X) (-5 * Λ) :=
+      (GV.C
+            ((VG_H5_log (p := p) (dd r)).mono
+              (by
+                have : (Nat.log p (dd r) : ℚ) ≤ Λ := log_mono hdd
+                linarith))).sub
+        (GV.X.mono (by linarith))
+    refine (GV.C_mul (hres r hr) hH).mono ?_
+    linarith
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Arith/TEntry.lean b/LeanPool/Zeta5Irrational/Arith/TEntry.lean
new file mode 100644
index 0000000000..e852026dbc
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Arith/TEntry.lean
@@ -0,0 +1,108 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Arith.Pullback
+public import LeanPool.Zeta5Irrational.Arith.DirectBound
+import LeanPool.Zeta5Irrational.Arith.DirectPoly
+import Mathlib.Tactic.Ring.Basic
+
+/-! # Entry bounds for numerators `∏ (t + γ²)`
+
+For a multiset `M` of integers, `tpol M = ∏_{γ ∈ M} (t + γ²)`. Its pullback is
+`pull K (tpol M) = ± x⁵ ∏_{γ ∈ M} (x - γ)(x + γ)`, so the direct bound applies to
+`μ_X(tpol M / D_K)`.
+-/
+
+@[expose] public section
+
+open Finset Polynomial
+
+namespace Zeta5Irrational
+
+/-- `∏_{γ ∈ M} (t + γ²)`. -/
+noncomputable def tpol (M : Multiset ℤ) : ℚ[X] :=
+  (M.map fun γ : ℤ => X + C ((γ : ℚ) ^ 2)).prod
+
+lemma tpol_add (M M' : Multiset ℤ) : tpol (M + M') = tpol M * tpol M' := by
+  simp [tpol, Multiset.map_add, Multiset.prod_add]
+
+/-- The `x`-roots of the pullback of `tpol M`. -/
+def xroots (M : Multiset ℤ) : Multiset ℤ :=
+  Multiset.replicate 5 0 + M + M.map (fun γ => -γ)
+
+lemma pull_tpol (K : ℕ) (M : Multiset ℤ) :
+    pull K (tpol M) = numOf ((-1 : ℚ) ^ (K + M.card)) (xroots M) := by
+  unfold pull tpol numOf xroots
+  rw [multiset_prod_comp, Multiset.map_map]
+  simp only [Multiset.map_add, Multiset.prod_add, Multiset.map_replicate, Multiset.prod_replicate]
+  have h1 :
+    ∀ γ : ℤ,
+      ((fun q : ℚ[X] => q.comp (-X ^ 2)) ∘ fun γ : ℤ => X + C ((γ : ℚ) ^ 2)) γ =
+        C (-1 : ℚ) * ((X - C (γ : ℚ)) * (X - C ((-γ : ℤ) : ℚ))) := by
+    intro γ
+    simp only [Function.comp, add_comp, X_comp, C_comp]
+    push_cast
+    rw [map_neg, map_one, C_neg, C_pow]; ring
+  rw [Multiset.map_congr rfl fun γ _ => h1 γ, Multiset.prod_map_mul, Multiset.prod_map_mul]
+  simp only [Multiset.map_const', Multiset.prod_replicate, Multiset.map_map, Function.comp]
+  simp only [Int.cast_zero, map_zero, sub_zero]
+  rw [pow_add, map_mul, map_pow, map_pow]
+  ring
+
+/-- Number of `x`-roots in the class `c`. -/
+lemma cnt_xroots (p : ℕ) (M : Multiset ℤ) (c : ZMod p) :
+    cnt p (xroots M) c =
+      (if c = 0 then 5 else 0) + (M.filter fun γ : ℤ => (γ : ZMod p) = c).card +
+        (M.filter fun γ : ℤ => (γ : ZMod p) = -c).card := by
+  unfold cnt xroots
+  rw [Multiset.filter_add, Multiset.filter_add, Multiset.card_add, Multiset.card_add]
+  congr 1
+  · congr 1
+    by_cases h : (0 : ZMod p) = c
+    · rw [ite_eq_left h.symm,
+        Multiset.filter_eq_self.mpr
+          (fun a ha => by rw [Multiset.eq_of_mem_replicate ha]; simpa using h)]
+      simp
+    · rw [ite_eq_right (Ne.symm h),
+        Multiset.filter_eq_nil.mpr
+          (fun a ha => by rw [Multiset.eq_of_mem_replicate ha]; simpa using h)]
+      simp
+  · rw [Multiset.filter_map, Multiset.card_map]
+    congr 1
+    refine Multiset.filter_congr fun γ _ => ?_
+    simp only [Function.comp, Int.cast_neg]
+    constructor <;> intro h <;> [rw [← h]; rw [h]] <;> ring
+
+/-- **Entry bound** for `μ_X(tpol M / D_K)`. -/
+theorem μX_tpol_bound {p : ℕ} [Fact p.Prime] (hp5 : 5 ≤ p) (K : ℕ) (M : Multiset ℤ)
+    (hsep : Sep p (PlK K)) (hK : K < p ^ 2) (hdeg : 5 + 2 * M.card - 2 * K + 1 < p ^ 2) (β : ℚ)
+    (hβ :
+      ∀ c : ZMod p,
+        β ≤
+          ((if c = 0 then 5 else 0) + (M.filter fun γ : ℤ => (γ : ZMod p) = c).card +
+                (M.filter fun γ : ℤ => (γ : ZMod p) = -c).card :
+              ℕ) -
+            (plc p (PlK K) c).card) :
+    GV p (μX K (tpol M)) (β - 4) := by
+  rw [μX_eq_tauX, pull_tpol]
+  have hκ : VG p ((-1 : ℚ) ^ (K + M.card)) 0 := by
+    rcases neg_one_pow_eq_or ℚ (K + M.card) with h | h <;> rw [h]
+    · exact VG.one
+    · exact VG.one.neg
+  refine tauX_direct hp5 _ hκ _ _ hsep ?_ ?_ β ?_
+  · intro r hr
+    rw [PlK, Finset.mem_union, Finset.mem_image, Finset.mem_image] at hr
+    rcases hr with ⟨j, hj, rfl⟩ | ⟨j, hj, rfl⟩
+    · rw [dd_nat]; have := (Finset.mem_Icc.mp hj).2; omega
+    · rw [dd_neg (Finset.mem_Icc.mp hj).1]; have := (Finset.mem_Icc.mp hj).2; omega
+  · have : (xroots M).card = 5 + 2 * M.card := by simp [xroots]; ring
+    rw [this, card_PlK]; exact hdeg
+  · intro c
+    rw [cnt_xroots]; exact hβ c
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Arith/TauBound.lean b/LeanPool/Zeta5Irrational/Arith/TauBound.lean
new file mode 100644
index 0000000000..37daf112a5
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Arith/TauBound.lean
@@ -0,0 +1,153 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Arith.BinomBasis
+public import LeanPool.Zeta5Irrational.Arith.Val
+import Mathlib.Data.Rat.Star
+import Mathlib.Tactic.ReduceModChar
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+import Mathlib.Tactic.Ring.Basic
+
+/-! # The bound (3.9) for the polynomial functional `τ(P) = L(P''')/24`
+
+If `deg P ≤ d` and `v_p(P(m)) ≥ β` for `m = 0, …, d`, then
+`v_p(τ(P)) ≥ β - 4 ⌊log_p (d+1)⌋ - v_p(24)`.
+-/
+
+@[expose] public section
+
+open Finset Polynomial
+
+namespace Zeta5Irrational
+
+variable {p : ℕ} [hp : Fact p.Prime]
+
+/-- `τ(P) = L(P''')/24`. -/
+noncomputable def tau (P : ℚ[X]) : ℚ :=
+  Lb (derivative^[3] P) / 24
+
+/-- `Q` is a combination of `bin 0, …, bin k` with coefficients of valuation `≥ r`. -/
+def BinRep (p : ℕ) (k : ℕ) (r : ℚ) (Q : ℚ[X]) : Prop :=
+  ∃ e : ℕ → ℚ, Q = ∑ m ∈ range (k + 1), C (e m) * bin m ∧ ∀ m ≤ k, VG p (e m) r
+
+lemma derivative_bin' {k m : ℕ} (hm : m ≤ k) :
+    (bin m).derivative = ∑ n ∈ range (k + 1), C (if n < m then dcoef (m - n) else 0) * bin n := by
+  rw [derivative_bin]
+  symm
+  rw [← Finset.sum_filter_add_sum_filter_not (range (k + 1)) (fun n => n < m)]
+  rw [Finset.sum_eq_zero (s := (range (k + 1)).filter fun n => ¬n < m)
+      (fun n hn => by rw [Finset.mem_filter] at hn; rw [ite_eq_right hn.2, C_0, zero_mul]),
+    add_zero]
+  apply Finset.sum_nbij' (fun n => m - n) (fun j => m - j)
+  · intro n hn; simp only [Finset.mem_filter, Finset.mem_range] at hn ⊢;
+    simp only [Finset.mem_Icc]; omega
+  · intro j hj; simp only [Finset.mem_Icc] at hj ⊢
+    simp only [Finset.mem_filter, Finset.mem_range]; omega
+  · intro n hn; simp only [Finset.mem_filter, Finset.mem_range] at hn; omega
+  · intro j hj; simp only [Finset.mem_Icc] at hj; omega
+  · intro n hn
+    simp only [Finset.mem_filter, Finset.mem_range] at hn
+    rw [ite_eq_left hn.2, show m - (m - n) = n by omega]
+
+lemma VG_dcoef {j k : ℕ} (hj : 1 ≤ j) (hjk : j ≤ k) : VG p (dcoef j) (-(Nat.log p k : ℚ)) := by
+  unfold dcoef
+  rw [div_eq_mul_inv]
+  have h1 : VG p ((-1 : ℚ) ^ (j - 1)) 0 := by
+    rcases neg_one_pow_eq_or ℚ (j - 1) with h | h <;> rw [h]
+    · exact VG.one
+    · exact VG.one.neg
+  simpa using h1.mul (VG.inv_nat hj hjk)
+
+/-- Differentiation loses at most `⌊log_p k⌋` in the binomial basis. -/
+theorem BinRep.derivative {k : ℕ} {r : ℚ} {Q : ℚ[X]} (h : BinRep p k r Q) :
+    BinRep p k (r - Nat.log p k) Q.derivative := by
+  obtain ⟨e, rfl, he⟩ := h
+  refine ⟨fun n => ∑ m ∈ range (k + 1), e m * (if n < m then dcoef (m - n) else 0), ?_, ?_⟩
+  · rw [Polynomial.derivative_sum]
+    simp_rw [derivative_mul, derivative_C, zero_mul, zero_add]
+    rw [Finset.sum_congr rfl fun m hm => by
+        rw [derivative_bin' (Nat.lt_succ_iff.mp (Finset.mem_range.mp hm)), Finset.mul_sum]]
+    rw [Finset.sum_comm]
+    refine Finset.sum_congr rfl fun n _ => ?_
+    rw [map_sum, Finset.sum_mul]
+    refine Finset.sum_congr rfl fun m _ => ?_
+    rw [C_mul]; ring
+  · intro n _
+    apply VG.sum
+    intro m hm
+    have hmk : m ≤ k := Nat.lt_succ_iff.mp (Finset.mem_range.mp hm)
+    split_ifs with hnm
+    · have := (he m hmk).mul (VG_dcoef (p := p) (j := m - n) (k := k) (by omega) (by omega))
+      simpa [sub_eq_add_neg] using this
+    · rw [mul_zero]; exact VG.zero _
+
+/-- The Bernoulli functional loses at most `⌊log_p (k+1)⌋`. -/
+theorem BinRep.Lb {k : ℕ} {r : ℚ} {Q : ℚ[X]} (h : BinRep p k r Q) :
+    VG p (Lb Q) (r - Nat.log p (k + 1)) := by
+  obtain ⟨e, rfl, he⟩ := h
+  rw [Lb_sum]
+  apply VG.sum
+  intro m hm
+  have hmk : m ≤ k := Nat.lt_succ_iff.mp (Finset.mem_range.mp hm)
+  rw [Lb_C_mul, Lb_bin, div_eq_mul_inv]
+  have h1 : VG p ((-1 : ℚ) ^ m) 0 := by
+    rcases neg_one_pow_eq_or ℚ m with h | h <;> rw [h]
+    · exact VG.one
+    · exact VG.one.neg
+  have h2 : VG p (((m : ℚ) + 1)⁻¹) (-(Nat.log p (k + 1) : ℚ)) := by
+    have := VG.inv_nat (p := p) (j := m + 1) (n := k + 1) (by omega) (by omega)
+    simpa using this
+  have := (he m hmk).mul (h1.mul h2)
+  simpa [sub_eq_add_neg] using this
+
+omit hp in
+lemma BinRep.mono {k : ℕ} {r s : ℚ} {Q : ℚ[X]} (h : BinRep p k r Q) (hs : s ≤ r) :
+    BinRep p k s Q := by
+  obtain ⟨e, he, hv⟩ := h
+  exact ⟨e, he, fun m hm => (hv m hm).mono hs⟩
+
+/-- Values at `0, …, d` give a binomial representation. -/
+theorem binRep_of_values {P : ℚ[X]} {d : ℕ} (hd : P.natDegree ≤ d) {β : ℚ}
+    (hv : ∀ m ≤ d, VG p (P.eval (m : ℚ)) β) : BinRep p d β P := by
+  refine ⟨fun k => (fwdDiff 1)^[k] P.eval 0, newton P hd, fun k hk => ?_⟩
+  change VG p ((fwdDiff 1)^[k] P.eval 0) β
+  rw [fwdDiff_iter_eq_sum_shift]
+  apply VG.sum
+  intro j hj
+  have hjd : j ≤ d := by have := Finset.mem_range.mp hj; omega
+  rw [zero_add, zsmul_eq_mul, nsmul_eq_mul, mul_one]
+  have h1 : VG p (((-1 : ℤ) ^ (k - j) * (k.choose j : ℤ) : ℤ) : ℚ) 0 := VG.intCast _
+  have := h1.mul (hv j hjd)
+  simpa using this
+
+lemma VG_inv_24 : VG p ((24 : ℚ)⁻¹) (-(padicValNat p 24 : ℚ)) := by
+  apply VG.inv (by norm_num)
+  rw [show (24 : ℚ) = ((24 : ℕ) : ℚ) by norm_num, padicValRat.of_nat]
+  exact le_refl _
+
+/-- **(3.9)**: `v_p(τ(P)) ≥ β - 4 ⌊log_p (d+1)⌋ - v_p(24)` if `deg P ≤ d` and
+`v_p(P(m)) ≥ β` for `0 ≤ m ≤ d`. -/
+theorem tau_bound {P : ℚ[X]} {d : ℕ} (hd : P.natDegree ≤ d) {β : ℚ}
+    (hv : ∀ m ≤ d, VG p (P.eval (m : ℚ)) β) :
+    VG p (tau P) (β - 4 * Nat.log p (d + 1) - padicValNat p 24) := by
+  have h0 := binRep_of_values hd hv
+  have hl : (Nat.log p d : ℚ) ≤ Nat.log p (d + 1) := by
+    exact_mod_cast Nat.log_mono_right (Nat.le_succ d)
+  have h3 : BinRep p d (β - 3 * Nat.log p (d + 1)) (derivative^[3] P) := by
+    have := h0.derivative.derivative.derivative
+    simp only [Function.iterate_succ, Function.iterate_zero, Function.comp_apply, id]
+    exact this.mono (by linarith)
+  have := h3.Lb.mul (VG_inv_24 (p := p))
+  unfold tau
+  rw [div_eq_mul_inv]
+  exact this.mono (by linarith)
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Arith/Unimodular.lean b/LeanPool/Zeta5Irrational/Arith/Unimodular.lean
new file mode 100644
index 0000000000..e823b1352e
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Arith/Unimodular.lean
@@ -0,0 +1,176 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Arith.BasisChange
+import LeanPool.Zeta5Irrational.Arith.DirectBound
+import Mathlib.RingTheory.Henselian
+import Mathlib.RingTheory.RegularLocalRing.Defs
+import Mathlib.RingTheory.SimpleRing.Principal
+import Mathlib.Tactic.Polynomial.Basic
+import Mathlib.Tactic.ReduceModChar
+import Mathlib.Tactic.Ring.Basic
+
+/-! # Unimodularity of the class bases
+
+Let `E a ∈ ℤ[t]` (`a : Fin h`) be such that, modulo `p`,
+`E a ≡ ∏_{c ≠ cls a} (t - γ_c)^{L_c} · (t - γ_{cls a})^{idx a}`,
+where the `γ_c ∈ 𝔽_p` are distinct, `idx a < L (cls a)` and `(cls, idx)` is injective.
+Then the coefficient matrix of the `E a` has a determinant prime to `p`.
+-/
+
+@[expose] public section
+
+open Finset Polynomial
+
+namespace Zeta5Irrational
+
+variable {F : Type*} [Field F]
+
+/-- A polynomial `∑_{i ∈ s} λ_i (X - γ)^i` vanishes only if all `λ_i` vanish. -/
+lemma sum_pow_X_sub_C_eq_zero {ι : Type*} (s : Finset ι) (d : ι → ℕ) (hd : Set.InjOn d s)
+    (lam : ι → F) (γ : F) (h : ∑ a ∈ s, C (lam a) * (X - C γ) ^ d a = 0) : ∀ a ∈ s, lam a = 0 := by
+  intro a ha
+  have h2 := congrArg (fun f : F[X] => (f.comp (X + C γ)).coeff (d a)) h
+  simp only [Polynomial.sum_comp, mul_comp, C_comp, pow_comp, sub_comp, X_comp,
+    add_sub_cancel_right, zero_comp, coeff_zero, finsetSum_coeff, coeff_C_mul, coeff_X_pow] at h2
+  rw [Finset.sum_eq_single a] at h2
+  · simpa using h2
+  · intro b hb hba
+    rw [ite_eq_right (fun h => hba (hd hb ha h.symm)), mul_zero]
+  · intro h; exact absurd ha h
+
+/-- **Independence of the class bases over a field.** -/
+theorem classBasis_independent {h : ℕ} {ι : Type*} [Fintype ι] [DecidableEq ι] (γ : ι → F)
+    (hγ : Function.Injective γ) (L : ι → ℕ) (cls : Fin h → ι) (idx : Fin h → ℕ)
+    (hidx : ∀ a, idx a < L (cls a)) (hinj : ∀ a b, cls a = cls b → idx a = idx b → a = b)
+    (v : Fin h → F)
+    (hv :
+      ∑ a,
+          C (v a) *
+            ((∏ c ∈ Finset.univ.erase (cls a), (X - C (γ c)) ^ L c) *
+              (X - C (γ (cls a))) ^ idx a) =
+        0) :
+    v = 0 := by
+  funext a0
+  set c := cls a0 with hc
+  set U := ∏ c' ∈ Finset.univ.erase c, (X - C (γ c')) ^ L c' with hU
+  have hsplit :=
+    Finset.sum_filter_add_sum_filter_not Finset.univ (fun a => cls a = c)
+      (fun a =>
+        C (v a) *
+          ((∏ c' ∈ Finset.univ.erase (cls a), (X - C (γ c')) ^ L c') *
+            (X - C (γ (cls a))) ^ idx a))
+  rw [hv] at hsplit
+        -- the rows of other classes are divisible by `(X - γ_c)^{L_c}`
+  have hdvd_other :
+    (X - C (γ c)) ^ L c ∣
+      ∑ a ∈ Finset.univ.filter (fun a => ¬cls a = c),
+        C (v a) *
+          ((∏ c' ∈ Finset.univ.erase (cls a), (X - C (γ c')) ^ L c') *
+            (X - C (γ (cls a))) ^ idx a) := by
+    refine Finset.dvd_sum fun a ha => ?_
+    rw [Finset.mem_filter] at ha
+    refine Dvd.dvd.mul_left (Dvd.dvd.mul_right ?_ _) _
+    exact Finset.dvd_prod_of_mem _ (Finset.mem_erase.mpr ⟨Ne.symm ha.2, Finset.mem_univ _⟩)
+  have hsame :
+    ∑ a ∈ Finset.univ.filter (fun a => cls a = c),
+        C (v a) *
+          ((∏ c' ∈ Finset.univ.erase (cls a), (X - C (γ c')) ^ L c') *
+            (X - C (γ (cls a))) ^ idx a) =
+      U * ∑ a ∈ Finset.univ.filter (fun a => cls a = c), C (v a) * (X - C (γ c)) ^ idx a := by
+    rw [Finset.mul_sum]
+    refine Finset.sum_congr rfl fun a ha => ?_
+    rw [Finset.mem_filter] at ha
+    rw [ha.2]; ring
+  have hdvd :
+    (X - C (γ c)) ^ L c ∣
+      U * ∑ a ∈ Finset.univ.filter (fun a => cls a = c), C (v a) * (X - C (γ c)) ^ idx a := by
+    rw [← hsame]
+    have :
+      ∑ a ∈ Finset.univ.filter (fun a => cls a = c),
+          C (v a) *
+            ((∏ c' ∈ Finset.univ.erase (cls a), (X - C (γ c')) ^ L c') *
+              (X - C (γ (cls a))) ^ idx a) =
+        -(∑ a ∈ Finset.univ.filter (fun a => ¬cls a = c),
+            C (v a) *
+              ((∏ c' ∈ Finset.univ.erase (cls a), (X - C (γ c')) ^ L c') *
+                (X - C (γ (cls a))) ^ idx a)) :=
+      by rw [eq_neg_iff_add_eq_zero]; exact hsplit
+    rw [this]; exact (dvd_neg).mpr hdvd_other
+  have hcop : IsCoprime ((X - C (γ c)) ^ L c) U := by
+    rw [hU]
+    refine IsCoprime.prod_right fun c' hc' => ?_
+    refine IsCoprime.pow ?_
+    exact
+      isCoprime_X_sub_C_of_isUnit_sub
+        (sub_ne_zero.mpr fun h => (Finset.mem_erase.mp hc').1 (hγ h).symm).isUnit
+  have hdvd2 := hcop.dvd_of_dvd_mul_left hdvd
+  set f := ∑ a ∈ Finset.univ.filter (fun a => cls a = c), C (v a) * (X - C (γ c)) ^ idx a
+  have hdeg : f.natDegree < L c := by
+    rcases Nat.eq_zero_or_pos (L c) with h0 | hpos
+    · exact absurd (hidx a0) (by rw [← hc, h0]; omega)
+    refine
+      lt_of_le_of_lt (natDegree_sum_le_of_forall_le _ _ (n := L c - 1) fun a ha => ?_) (by omega)
+    rw [Finset.mem_filter] at ha
+    refine (natDegree_C_mul_le _ _).trans ?_
+    rw [natDegree_pow, natDegree_X_sub_C, mul_one]
+    have := hidx a; rw [ha.2] at this; omega
+  have hf0 : f = 0 := by
+    by_contra hne
+    have := natDegree_le_of_dvd hdvd2 hne
+    rw [natDegree_pow, natDegree_X_sub_C, mul_one] at this
+    omega
+  have :=
+    sum_pow_X_sub_C_eq_zero (Finset.univ.filter (fun a => cls a = c)) idx
+      (fun a ha b hb h => hinj a b (by rw [Finset.coe_filter] at ha hb; rw [ha.2, hb.2]) h) v
+      (γ c) hf0
+  exact this a0 (by simp [hc])
+
+/-- **Unimodularity**: the rational determinant of the coefficient matrix is a `p`-adic unit. -/
+theorem coeffMat_det_unit {h : ℕ} {p : ℕ} [hp : Fact p.Prime] {ι : Type*} [Fintype ι]
+    [DecidableEq ι] (Ez : Fin h → ℤ[X]) (hdeg : ∀ a, (Ez a).natDegree < h) (γ : ι → ZMod p)
+    (hγ : Function.Injective γ) (L : ι → ℕ) (cls : Fin h → ι) (idx : Fin h → ℕ)
+    (hidx : ∀ a, idx a < L (cls a)) (hinj : ∀ a b, cls a = cls b → idx a = idx b → a = b)
+    (hred :
+      ∀ a,
+        (Ez a).map (Int.castRingHom (ZMod p)) =
+          (∏ c ∈ Finset.univ.erase (cls a), (X - C (γ c)) ^ L c) * (X - C (γ (cls a))) ^ idx a) :
+    (coeffMat fun a => (Ez a).map (Int.castRingHom ℚ)).det ≠ 0 ∧
+      padicValRat p (coeffMat fun a => (Ez a).map (Int.castRingHom ℚ)).det = 0 := by
+  set Mz : Matrix (Fin h) (Fin h) ℤ := Matrix.of fun a k => (Ez a).coeff k with hMz
+  have hQ : coeffMat (fun a => (Ez a).map (Int.castRingHom ℚ)) = Mz.map (Int.castRingHom ℚ) := by
+    ext a k; simp [coeffMat, hMz, coeff_map]
+  have hdetQ : (coeffMat fun a => (Ez a).map (Int.castRingHom ℚ)).det = (Mz.det : ℚ) := by
+    rw [hQ, ← RingHom.mapMatrix_apply, ← RingHom.map_det]; simp
+  have hdetP : ((Mz.det : ℤ) : ZMod p) ≠ 0 := by
+    intro h0
+    have h1 : (Mz.map (Int.castRingHom (ZMod p))).transpose.det = 0 := by
+      rw [Matrix.det_transpose, ← RingHom.mapMatrix_apply, ← RingHom.map_det]; simpa using h0
+    obtain ⟨v, hv0, hv⟩ := Matrix.exists_mulVec_eq_zero_iff.mpr h1
+    apply hv0
+    apply classBasis_independent γ hγ L cls idx hidx hinj v
+    rw [← Finset.sum_congr rfl fun a _ => by rw [← hred a]]
+    ext k
+    rw [finsetSum_coeff, coeff_zero]
+    simp only [coeff_C_mul, coeff_map]
+    by_cases hk : k < h
+    · have := congrFun hv ⟨k, hk⟩
+      simp only [Matrix.mulVec, dotProduct, Matrix.transpose_apply, Matrix.map_apply, hMz,
+        Matrix.of_apply, Pi.zero_apply] at this
+      rw [← this]
+      refine Finset.sum_congr rfl fun a _ => ?_
+      simp [mul_comm]
+    · refine Finset.sum_eq_zero fun a _ => ?_
+      rw [coeff_eq_zero_of_natDegree_lt (by have := hdeg a; omega)]; simp
+  have hndvd : ¬(p : ℤ) ∣ Mz.det := fun h => hdetP ((ZMod.intCast_zmod_eq_zero_iff_dvd _ _).mpr h)
+  rw [hdetQ]
+  refine
+    ⟨fun h => hndvd (by exact_mod_cast (show Mz.det = 0 by exact_mod_cast h) ▸ dvd_zero _), ?_⟩
+  exact padicValRat_int_eq_zero hndvd
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Arith/Val.lean b/LeanPool/Zeta5Irrational/Arith/Val.lean
new file mode 100644
index 0000000000..746cdad444
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Arith/Val.lean
@@ -0,0 +1,224 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import Mathlib.Algebra.Polynomial.Basic
+public import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
+public import Mathlib.NumberTheory.Padics.PadicVal.Basic
+import Mathlib.Data.Rat.Star
+import Mathlib.Tactic.ENatToNat
+import Mathlib.Tactic.ReduceModChar
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.Ring.Basic
+
+/-! # A small `p`-adic valuation toolkit on `ℚ` and `ℚ[X]`
+
+* `VG p q r` : `q = 0 ∨ r ≤ v_p(q)` (with a rational lower bound `r`);
+* `GV p f r` : every coefficient of `f ∈ ℚ[X]` satisfies `VG p · r` (Gauss valuation bound);
+* `det_GV` : if `M i j` has Gauss valuation `≥ ρ i + κ j`, then `det M` has `≥ ∑ ρ + ∑ κ`.
+-/
+
+@[expose] public section
+
+open Finset Polynomial
+
+namespace Zeta5Irrational
+
+variable {p : ℕ}
+
+/-- `v_p(q) ≥ r` (vacuous for `q = 0`). -/
+def VG (p : ℕ) (q : ℚ) (r : ℚ) : Prop :=
+  q = 0 ∨ r ≤ (padicValRat p q : ℚ)
+
+namespace VG
+
+lemma zero (r : ℚ) : VG p 0 r :=
+  Or.inl rfl
+
+lemma mono {q r s : ℚ} (h : VG p q r) (hs : s ≤ r) : VG p q s :=
+  h.imp id fun h => hs.trans h
+
+lemma neg {q r : ℚ} (h : VG p q r) : VG p (-q) r := by
+  rcases h with h | h
+  · left; simp [h]
+  · right; rwa [padicValRat.neg]
+
+lemma add [Fact p.Prime] {q q' r : ℚ} (h : VG p q r) (h' : VG p q' r) : VG p (q + q') r := by
+  by_cases hq : q = 0
+  · simpa [hq] using h'
+  by_cases hq' : q' = 0
+  · simpa [hq'] using h
+  by_cases hs : q + q' = 0
+  · exact Or.inl hs
+  right
+  have h1 := h.resolve_left hq
+  have h2 := h'.resolve_left hq'
+  have := padicValRat.min_le_padicValRat_add (p := p) hs
+  have h3 : (min (padicValRat p q) (padicValRat p q') : ℚ) ≤ padicValRat p (q + q') := by
+    exact_mod_cast this
+  exact (le_min h1 h2).trans h3
+
+lemma sub [Fact p.Prime] {q q' r : ℚ} (h : VG p q r) (h' : VG p q' r) : VG p (q - q') r := by
+  rw [sub_eq_add_neg]; exact h.add h'.neg
+
+lemma mul [Fact p.Prime] {q q' r r' : ℚ} (h : VG p q r) (h' : VG p q' r') :
+    VG p (q * q') (r + r') := by
+  by_cases hq : q = 0
+  · left; simp [hq]
+  by_cases hq' : q' = 0
+  · left; simp [hq']
+  right
+  rw [padicValRat.mul hq hq']
+  push_cast
+  exact add_le_add (h.resolve_left hq) (h'.resolve_left hq')
+
+lemma sum [Fact p.Prime] {ι : Type*} (s : Finset ι) {f : ι → ℚ} {r : ℚ}
+    (h : ∀ i ∈ s, VG p (f i) r) : VG p (∑ i ∈ s, f i) r := by
+  classical
+    induction s using Finset.induction_on with
+  | empty => exact zero r
+  | insert a s ha ih =>
+    rw [Finset.sum_insert ha]
+    exact
+      (h a (Finset.mem_insert_self _ _)).add (ih fun i hi => h i (Finset.mem_insert_of_mem hi))
+
+lemma prod [Fact p.Prime] {ι : Type*} (s : Finset ι) {f : ι → ℚ} {r : ι → ℚ}
+    (h : ∀ i ∈ s, VG p (f i) (r i)) : VG p (∏ i ∈ s, f i) (∑ i ∈ s, r i) := by
+  classical
+    induction s using Finset.induction_on with
+  | empty => right; simp
+  | insert a s ha ih =>
+    rw [Finset.prod_insert ha, Finset.sum_insert ha]
+    exact
+      (h a (Finset.mem_insert_self _ _)).mul (ih fun i hi => h i (Finset.mem_insert_of_mem hi))
+
+lemma intCast (z : ℤ) : VG p (z : ℚ) 0 := by
+  by_cases hz : z = 0
+  · left; simp [hz]
+  right
+  rw [padicValRat.of_int]
+  exact_mod_cast Nat.zero_le _
+
+lemma natCast (n : ℕ) : VG p (n : ℚ) 0 := by exact_mod_cast intCast (p := p) (n : ℤ)
+
+lemma one : VG p 1 0 := by exact_mod_cast natCast (p := p) 1
+
+lemma inv [Fact p.Prime] {q r : ℚ} (_h : q ≠ 0) (hv : (padicValRat p q : ℚ) ≤ r) :
+    VG p q⁻¹ (-r) := by
+  right
+  rw [padicValRat.inv]
+  push_cast
+  linarith
+
+lemma of_eq {q : ℚ} (r : ℚ) (h : q ≠ 0 → r ≤ (padicValRat p q : ℚ)) : VG p q r := by
+  by_cases hq : q = 0
+  · exact Or.inl hq
+  · exact Or.inr (h hq)
+
+/-- `v_p(p^k) = k`. -/
+lemma primePow [hp : Fact p.Prime] (k : ℤ) : VG p ((p : ℚ) ^ k) k := by
+  right
+  have : padicValRat p ((p : ℚ) ^ k) = k := by
+    rw [padicValRat.zpow _, padicValRat.self hp.out.one_lt, mul_one]
+  rw [this]
+
+lemma zpow_mul [Fact p.Prime] {q r : ℚ} (k : ℤ) (h : VG p q r) : VG p ((p : ℚ) ^ k * q) (k + r) :=
+  (primePow k).mul h
+
+lemma pow [Fact p.Prime] {q r : ℚ} (h : VG p q r) (n : ℕ) : VG p (q ^ n) (n * r) := by
+  induction n with
+  | zero => right; simp
+  | succ n ih => rw [pow_succ]; convert ih.mul h using 1; push_cast; ring
+
+/-- A lower bound on the valuation of `1 / j` for `1 ≤ j ≤ n`. -/
+lemma inv_nat [hp : Fact p.Prime] {j n : ℕ} (hj : 1 ≤ j) (hjn : j ≤ n) :
+    VG p ((j : ℚ)⁻¹) (-(Nat.log p n : ℚ)) := by
+  apply inv (by exact_mod_cast (by omega : j ≠ 0))
+  rw [padicValRat.of_nat]
+  have h1 : padicValNat p j ≤ Nat.log p j := by
+    rcases Nat.eq_zero_or_pos (padicValNat p j) with h | h
+    · rw [h]; exact Nat.zero_le _
+    · have hdvd : p ^ padicValNat p j ∣ j := pow_padicValNat_dvd
+      exact Nat.le_log_of_pow_le hp.out.one_lt (Nat.le_of_dvd (by omega) hdvd)
+  have h2 : Nat.log p j ≤ Nat.log p n := Nat.log_mono_right hjn
+  exact_mod_cast h1.trans h2
+
+end VG
+
+/-- Gauss valuation bound for polynomials in `X`. -/
+def GV (p : ℕ) (f : ℚ[X]) (r : ℚ) : Prop :=
+  ∀ n, VG p (f.coeff n) r
+
+namespace GV
+
+lemma zero (r : ℚ) : GV p 0 r := fun n => by simpa using VG.zero (p := p) r
+
+lemma mono {f : ℚ[X]} {r s : ℚ} (h : GV p f r) (hs : s ≤ r) : GV p f s := fun n => (h n).mono hs
+
+lemma add [Fact p.Prime] {f g : ℚ[X]} {r : ℚ} (hf : GV p f r) (hg : GV p g r) : GV p (f + g) r :=
+  fun n => by rw [coeff_add]; exact (hf n).add (hg n)
+
+lemma neg {f : ℚ[X]} {r : ℚ} (hf : GV p f r) : GV p (-f) r := fun n => by
+  rw [coeff_neg];
+  exact (hf n).neg
+
+lemma sub [Fact p.Prime] {f g : ℚ[X]} {r : ℚ} (hf : GV p f r) (hg : GV p g r) : GV p (f - g) r :=
+  fun n => by rw [coeff_sub]; exact (hf n).sub (hg n)
+
+lemma C {q r : ℚ} (h : VG p q r) : GV p (Polynomial.C q) r := fun n =>
+  by
+  rw [coeff_C]; split_ifs
+  · exact h
+  · exact VG.zero r
+
+lemma X : GV p (X : ℚ[X]) 0 := fun n => by
+  rw [coeff_X]; split_ifs
+  · exact VG.one
+  · exact VG.zero 0
+
+lemma mul [Fact p.Prime] {f g : ℚ[X]} {r s : ℚ} (hf : GV p f r) (hg : GV p g s) :
+    GV p (f * g) (r + s) := fun n => by
+  rw [coeff_mul]
+  exact VG.sum _ fun x _ => (hf x.1).mul (hg x.2)
+
+lemma C_mul [Fact p.Prime] {q r s : ℚ} {f : ℚ[X]} (hq : VG p q r) (hf : GV p f s) :
+    GV p (Polynomial.C q * f) (r + s) :=
+  (C hq).mul hf
+
+lemma sum [Fact p.Prime] {ι : Type*} (s : Finset ι) {f : ι → ℚ[X]} {r : ℚ}
+    (h : ∀ i ∈ s, GV p (f i) r) : GV p (∑ i ∈ s, f i) r := fun n => by
+  rw [finsetSum_coeff];
+  exact VG.sum s fun i hi => h i hi n
+
+lemma prod [Fact p.Prime] {ι : Type*} (s : Finset ι) {f : ι → ℚ[X]} {r : ι → ℚ}
+    (h : ∀ i ∈ s, GV p (f i) (r i)) : GV p (∏ i ∈ s, f i) (∑ i ∈ s, r i) := by
+  classical
+    induction s using Finset.induction_on with
+  | empty =>
+    simp only [Finset.prod_empty, Finset.sum_empty]
+    simpa using C (p := p) (VG.one (p := p))
+  | insert a s ha ih =>
+    rw [Finset.prod_insert ha, Finset.sum_insert ha]
+    exact
+      (h a (Finset.mem_insert_self _ _)).mul (ih fun i hi => h i (Finset.mem_insert_of_mem hi))
+
+end GV
+
+/-- **Determinant bound with row and column weights.** -/
+theorem det_GV [Fact p.Prime] {ι : Type*} [Fintype ι] [DecidableEq ι] (M : Matrix ι ι ℚ[X])
+    (ρ κ : ι → ℚ) (h : ∀ i j, GV p (M i j) (ρ i + κ j)) : GV p M.det (∑ i, ρ i + ∑ j, κ j) := by
+  rw [Matrix.det_apply]
+  refine GV.sum _ fun σ _ => ?_
+  have hprod : GV p (∏ i, M (σ i) i) (∑ i, (ρ (σ i) + κ i)) := GV.prod _ fun i _ => h (σ i) i
+  have e : ∑ i, (ρ (σ i) + κ i) = ∑ i, ρ i + ∑ j, κ j := by
+    rw [Finset.sum_add_distrib, Equiv.sum_comp σ ρ]
+  rw [e] at hprod
+  rcases Int.units_eq_one_or (Equiv.Perm.sign σ) with hs | hs
+  · rw [hs, one_smul]; exact hprod
+  · rw [hs, Units.neg_smul, one_smul]; exact hprod.neg
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Certificates.lean b/LeanPool/Zeta5Irrational/Certificates.lean
new file mode 100644
index 0000000000..9cc7b07cf8
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Certificates.lean
@@ -0,0 +1,165 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan
+import Mathlib.Analysis.SpecialFunctions.Complex.Arctan
+import Mathlib.Data.Int.Star
+import Mathlib.Tactic.ENatToNat
+import Mathlib.Tactic.FieldSimp
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.Positivity.Basic
+import Mathlib.Tactic.Ring.Basic
+
+/-! # Certified rational bounds for `log`, `arctan` and `sqrt`
+
+Tools for the numerical verification of the potential inequality (6.2) (Lemma 6.1):
+
+* `log_le_artanh` : upper bound for `log r`, `r ≥ 1`, by a partial sum of the artanh series plus
+  an explicit geometric tail (complementing `log_ge_artanh`);
+* `arctan_ge_partial` / `arctan_le_partial` : two-sided bounds for `arctan x`, `0 ≤ x < 1`, by
+  partial sums of the alternating series;
+* `arctan_eq_two_mul_arctan` : the half-angle reduction `arctan x = 2 arctan (x / (1 + √(1+x²)))`;
+* `sqrt_le_of_sq_le`, `le_sqrt_of_sq_le` : rational enclosures of square roots.
+-/
+
+@[expose] public section
+
+open Finset Filter Topology
+
+namespace Zeta5Irrational
+
+/--
+Upper bound for `log r`, `r ≥ 1`: partial sum plus the tail `2 z^(2m+1) / ((2m+1)(1 - z²))`. -/
+lemma log_le_artanh {r : ℝ} (hr : 1 ≤ r) (m : ℕ) :
+    Real.log r ≤
+      2 * ∑ k ∈ range m, (1 / (2 * (k : ℝ) + 1)) * ((r - 1) / (r + 1)) ^ (2 * k + 1) +
+        2 * ((r - 1) / (r + 1)) ^ (2 * m + 1) / ((2 * m + 1) * (1 - ((r - 1) / (r + 1)) ^ 2)) := by
+  set x : ℝ := (r - 1) / (r + 1) with hx
+  have hr0 : 0 < r + 1 := by linarith
+  have hx0 : 0 ≤ x := div_nonneg (by linarith) hr0.le
+  have hx1 : x < 1 := by rw [hx, div_lt_one hr0]; linarith
+  have habs : |x| < 1 := by rw [abs_of_nonneg hx0]; exact hx1
+  have hsum := Real.hasSum_log_sub_log_of_abs_lt_one habs
+  have hr' : (1 + x) / (1 - x) = r := by rw [hx]; field_simp; ring
+  have hlog : Real.log (1 + x) - Real.log (1 - x) = Real.log r := by
+    rw [← Real.log_div (by linarith) (by linarith), hr']
+  rw [hlog] at hsum
+        -- split the series at `m`
+  have hsplit := hsum.summable.sum_add_tsum_nat_add m
+  rw [hsum.tsum_eq] at hsplit
+        -- the tail is bounded by a geometric series
+  have hx2 : x ^ 2 < 1 := by nlinarith
+  have hgeom :
+    Summable fun k : ℕ => 2 * (1 / (2 * (m : ℝ) + 1)) * x ^ (2 * m + 1) * (x ^ 2) ^ k :=
+    (summable_geometric_of_lt_one (by positivity) hx2).mul_left _
+  have htail :
+    ∑' k : ℕ, 2 * (1 / (2 * ((k + m : ℕ) : ℝ) + 1)) * x ^ (2 * (k + m) + 1) ≤
+      ∑' k : ℕ, 2 * (1 / (2 * (m : ℝ) + 1)) * x ^ (2 * m + 1) * (x ^ 2) ^ k := by
+    refine Summable.tsum_le_tsum (fun k => ?_) ((summable_nat_add_iff m).mpr hsum.summable) hgeom
+    have hk : (1 / (2 * ((k + m : ℕ) : ℝ) + 1)) ≤ 1 / (2 * (m : ℝ) + 1) := by
+      gcongr;
+      linarith [(Nat.cast_nonneg k : (0 : ℝ) ≤ k)]
+    have hp : x ^ (2 * (k + m) + 1) = x ^ (2 * m + 1) * (x ^ 2) ^ k := by
+      rw [← pow_mul, ← pow_add]; congr 1; ring
+    rw [hp]
+    have : 0 ≤ x ^ (2 * m + 1) * (x ^ 2) ^ k := by positivity
+    calc
+      2 * (1 / (2 * ((k + m : ℕ) : ℝ) + 1)) * (x ^ (2 * m + 1) * (x ^ 2) ^ k) ≤
+          2 * (1 / (2 * (m : ℝ) + 1)) * (x ^ (2 * m + 1) * (x ^ 2) ^ k) :=
+        by gcongr
+      _ = _ := by ring
+  rw [tsum_mul_left,
+    tsum_geometric_of_lt_one (by positivity) hx2] at htail
+        -- assemble
+  have hsum_eq :
+    ∑ k ∈ range m, 2 * (1 / (2 * (k : ℝ) + 1)) * x ^ (2 * k + 1) =
+      2 * ∑ k ∈ range m, (1 / (2 * (k : ℝ) + 1)) * x ^ (2 * k + 1) :=
+    by rw [Finset.mul_sum]; refine Finset.sum_congr rfl fun k _ => by ring
+  rw [hsum_eq] at hsplit
+  have h2 :
+    2 * (1 / (2 * (m : ℝ) + 1)) * x ^ (2 * m + 1) * (1 - x ^ 2)⁻¹ =
+      2 * x ^ (2 * m + 1) / ((2 * m + 1) * (1 - x ^ 2)) := by
+    have : (1 : ℝ) - x ^ 2 ≠ 0 := by linarith
+    field_simp
+  rw [h2] at htail
+  linarith
+
+/-- Lower bound `arctan x ≥ ∑_{i < 2k} (-1)^i x^(2i+1)/(2i+1)` for `0 ≤ x < 1`. -/
+lemma arctan_ge_partial {x : ℝ} (hx0 : 0 ≤ x) (hx1 : x < 1) (k : ℕ) :
+    ∑ i ∈ range (2 * k), (-1 : ℝ) ^ i * (x ^ (2 * i + 1) / (2 * (i : ℝ) + 1)) ≤ Real.arctan x := by
+  have hs := Real.hasSum_arctan (x := x) (by rw [Real.norm_eq_abs, abs_of_nonneg hx0]; exact hx1)
+  have hanti : Antitone fun i : ℕ => x ^ (2 * i + 1) / (2 * (i : ℝ) + 1) := by
+    intro i j hij
+    have h1 : x ^ (2 * j + 1) ≤ x ^ (2 * i + 1) := pow_le_pow_of_le_one hx0 hx1.le (by omega)
+    have h2 : (2 * (i : ℝ) + 1) ≤ 2 * (j : ℝ) + 1 := by
+      have : (i : ℝ) ≤ j := by exact_mod_cast hij
+      linarith
+    exact div_le_div₀ (by positivity) h1 (by positivity) h2
+  have htend := hs.tendsto_sum_nat
+  have :=
+    Antitone.alternating_series_le_tendsto (f := fun i : ℕ => x ^ (2 * i + 1) / (2 * (i : ℝ) + 1))
+      (by
+        refine htend.congr fun n => ?_
+        refine Finset.sum_congr rfl fun i _ => ?_
+        push_cast; ring)
+      hanti k
+  exact this
+
+/-- Upper bound `arctan x ≤ ∑_{i < 2k+1} (-1)^i x^(2i+1)/(2i+1)` for `0 ≤ x < 1`. -/
+lemma arctan_le_partial {x : ℝ} (hx0 : 0 ≤ x) (hx1 : x < 1) (k : ℕ) :
+    Real.arctan x ≤
+      ∑ i ∈ range (2 * k + 1), (-1 : ℝ) ^ i * (x ^ (2 * i + 1) / (2 * (i : ℝ) + 1)) := by
+  have hs := Real.hasSum_arctan (x := x) (by rw [Real.norm_eq_abs, abs_of_nonneg hx0]; exact hx1)
+  have hanti : Antitone fun i : ℕ => x ^ (2 * i + 1) / (2 * (i : ℝ) + 1) := by
+    intro i j hij
+    have h1 : x ^ (2 * j + 1) ≤ x ^ (2 * i + 1) := pow_le_pow_of_le_one hx0 hx1.le (by omega)
+    have h2 : (2 * (i : ℝ) + 1) ≤ 2 * (j : ℝ) + 1 := by
+      have : (i : ℝ) ≤ j := by exact_mod_cast hij
+      linarith
+    exact div_le_div₀ (by positivity) h1 (by positivity) h2
+  have htend := hs.tendsto_sum_nat
+  exact
+    Antitone.tendsto_le_alternating_series (f := fun i : ℕ => x ^ (2 * i + 1) / (2 * (i : ℝ) + 1))
+      (by
+        refine htend.congr fun n => ?_
+        refine Finset.sum_congr rfl fun i _ => ?_
+        push_cast; ring)
+      hanti k
+
+/-- The half-angle reduction `arctan x = 2 arctan (x / (1 + √(1 + x²)))` for `x ≥ 0`. -/
+lemma arctan_eq_two_mul_arctan {x : ℝ} (hx : 0 ≤ x) :
+    Real.arctan x = 2 * Real.arctan (x / (1 + Real.sqrt (1 + x ^ 2))) := by
+  set s := Real.sqrt (1 + x ^ 2) with hs
+  have hs2 : s ^ 2 = 1 + x ^ 2 := Real.sq_sqrt (by positivity)
+  have hs1 : 1 ≤ s := by
+    rw [hs];
+    exact Real.le_sqrt_of_sq_le (by nlinarith : (1 : ℝ) ^ 2 ≤ 1 + x ^ 2)
+  set y := x / (1 + s) with hy
+  have hy0 : 0 ≤ y := by positivity
+  have hy1 : y < 1 := by
+    rw [hy, div_lt_one (by positivity)]
+    nlinarith [Real.sqrt_nonneg (1 + x ^ 2)]
+  have hyy : y * y < 1 := by nlinarith
+  have hadd := Real.arctan_add (x := y) (y := y) (by nlinarith)
+  have hval : (y + y) / (1 - y * y) = x := by
+    rw [div_eq_iff (by nlinarith : (1 : ℝ) - y * y ≠ 0), hy]
+    have h1s : (1 + s) ≠ 0 := by positivity
+    field_simp
+    nlinarith [hs2, hs1]
+  rw [hval] at hadd
+  linarith
+
+/-- Rational enclosure of a square root: `√q ≤ u` if `q ≤ u²`, `0 ≤ u`. -/
+lemma sqrt_le_of_sq_le {q u : ℝ} (hu : 0 ≤ u) (h : q ≤ u ^ 2) : Real.sqrt q ≤ u :=
+  Real.sqrt_le_iff.mpr ⟨hu, h⟩
+
+/-- Rational enclosure of a square root: `l ≤ √q` if `l² ≤ q`, `0 ≤ l`. -/
+lemma le_sqrt_of_sq_le {q l : ℝ} (_hl : 0 ≤ l) (h : l ^ 2 ≤ q) : l ≤ Real.sqrt q :=
+  Real.le_sqrt_of_sq_le h
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/CircleAtoms.lean b/LeanPool/Zeta5Irrational/CircleAtoms.lean
new file mode 100644
index 0000000000..1c5ab92118
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/CircleAtoms.lean
@@ -0,0 +1,140 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import Mathlib.Analysis.SpecialFunctions.Complex.CircleMap
+public import Mathlib.Analysis.SpecialFunctions.Log.PosLog
+public import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
+import Mathlib.Analysis.CStarAlgebra.Classes
+import Mathlib.Analysis.SpecialFunctions.Integrals.PosLogEqCircleAverage
+import Mathlib.RingTheory.SimpleRing.Principal
+import Mathlib.Tactic.ENatToNat
+import Mathlib.Tactic.Polynomial.Basic
+import Mathlib.Tactic.ReduceModChar
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.Positivity.Basic
+import Mathlib.Tactic.Ring.Basic
+import Mathlib.Tactic.Ring.RingNF
+
+/-! # Circle integrals of `log ‖· - a‖`
+
+Consequences of Mathlib's circle-average identities, in the normalisation used here
+(integrals over `θ ∈ [0, 2π]` without the factor `(2π)⁻¹`), and countability of point
+preimages under `circleMap` and `cos`.
+-/
+
+@[expose] public section
+
+open MeasureTheory Set Real intervalIntegral Filter Topology
+
+namespace Zeta5Irrational
+
+lemma circle_log_integrable (c a : ℂ) (R : ℝ) :
+    IntervalIntegrable (fun θ => Real.log ‖circleMap c R θ - a‖) volume 0 (2 * π) :=
+  circleIntegrable_log_norm_sub_const R
+
+/-- `∫₀^{2π} log ‖circleMap c R θ - a‖ = 2π (log R + log⁺ (R⁻¹ ‖c - a‖))`. -/
+lemma circle_log_integral (c a : ℂ) {R : ℝ} (hR : R ≠ 0) :
+    ∫ θ in (0 : ℝ)..2 * π, Real.log ‖circleMap c R θ - a‖ =
+      2 * π * (Real.log R + log⁺ (R⁻¹ * ‖c - a‖)) := by
+  have h := circleAverage_log_norm_sub_const_eq_log_radius_add_posLog (a := a) (c := c) hR
+  rw [circleAverage_def, smul_eq_mul] at h
+  have hpi : (2 * π : ℝ) ≠ 0 := by positivity
+  rw [← h, ← mul_assoc, mul_inv_cancel₀ hpi, one_mul]
+
+lemma circle_log_integral_zero_one (a : ℂ) :
+    ∫ θ in (0 : ℝ)..2 * π, Real.log ‖circleMap 0 1 θ - a‖ = 2 * π * log⁺ ‖a‖ := by
+  rw [circle_log_integral 0 a one_ne_zero]
+  simp
+
+lemma abs_log_eq (x : ℝ) : |Real.log x| = 2 * log⁺ x - Real.log x := by
+  rw [posLog_apply]
+  rcases le_total 0 (Real.log x) with h | h
+  · rw [max_eq_right h, abs_of_nonneg h]; ring
+  · rw [max_eq_left h, abs_of_nonpos h]; ring
+
+lemma posLog_norm_sub_le (w q : ℂ) (hw : ‖w‖ ≤ 1) : log⁺ ‖w - q‖ ≤ Real.log (1 + ‖q‖) := by
+  calc
+    log⁺ ‖w - q‖ ≤ log⁺ (1 + ‖q‖) := by
+      apply posLog_le_posLog (by linarith [norm_nonneg (w - q)])
+      calc
+        ‖w - q‖ ≤ ‖w‖ + ‖q‖ := norm_sub_le _ _
+        _ ≤ 1 + ‖q‖ := by linarith
+    _ = Real.log (1 + ‖q‖) :=
+      posLog_eq_log (by rw [abs_of_pos (by positivity)]; linarith [norm_nonneg q])
+
+/-- Uniform bound `∫₀^{2π} |log ‖e^{iθ} - q‖| ≤ 4π log (1 + ‖q‖)`. -/
+lemma abs_log_circle_integral_le (q : ℂ) :
+    ∫ θ in (0 : ℝ)..2 * π, |Real.log ‖circleMap 0 1 θ - q‖| ≤ 4 * π * Real.log (1 + ‖q‖) := by
+  simp_rw [abs_log_eq]
+  have hc : Continuous fun θ => log⁺ ‖circleMap 0 1 θ - q‖ := by
+    have := continuous_circleMap 0 1
+    fun_prop
+  rw [intervalIntegral.integral_sub (hc.intervalIntegrable _ _ |>.const_mul 2)
+      (circle_log_integrable 0 q 1),
+    intervalIntegral.integral_const_mul, circle_log_integral_zero_one]
+  have h1 :
+    (∫ θ in (0 : ℝ)..2 * π, log⁺ ‖circleMap 0 1 θ - q‖) ≤
+      ∫ θ in (0 : ℝ)..2 * π, Real.log (1 + ‖q‖) := by
+    apply integral_mono_on (by positivity) (hc.intervalIntegrable _ _) intervalIntegrable_const
+    intro θ _
+    apply posLog_norm_sub_le
+    simp [circleMap]
+  rw [intervalIntegral.integral_const, sub_zero, smul_eq_mul] at h1
+  have h2 := posLog_nonneg (x := ‖q‖)
+  nlinarith [Real.pi_pos]
+
+/-- Point preimages under `circleMap` are countable. -/
+lemma circleMap_preimage_countable (c : ℂ) {R : ℝ} (hR : R ≠ 0) (z : ℂ) :
+    (circleMap c R ⁻¹' { z }).Countable := by
+  rcases (circleMap c R ⁻¹' { z }).eq_empty_or_nonempty with h | ⟨θ₀, hθ₀⟩
+  · rw [h]; exact Set.countable_empty
+  refine (Set.countable_range fun n : ℤ => θ₀ + n * (2 * π)).mono ?_
+  intro θ hθ
+  simp only [Set.mem_preimage, Set.mem_singleton_iff, circleMap] at hθ hθ₀
+  have he : Complex.exp (θ * Complex.I) = Complex.exp (θ₀ * Complex.I) := by
+    have hR' : (R : ℂ) ≠ 0 := by exact_mod_cast hR
+    have := hθ.trans hθ₀.symm
+    have h2 : (R : ℂ) * Complex.exp (θ * Complex.I) = R * Complex.exp (θ₀ * Complex.I) := by
+      linear_combination this
+    exact mul_left_cancel₀ hR' h2
+  obtain ⟨n, hn⟩ := Complex.exp_eq_exp_iff_exists_int.mp he
+  refine ⟨n, ?_⟩
+  have : (θ : ℂ) = θ₀ + n * (2 * π) := by
+    have hI : Complex.I ≠ 0 := Complex.I_ne_zero
+    have h3 : (θ : ℂ) * Complex.I = (θ₀ + n * (2 * π)) * Complex.I := by rw [hn]; ring
+    exact mul_right_cancel₀ hI h3
+  exact_mod_cast this.symm
+
+/-- Point preimages under `cos` are countable. -/
+lemma cos_preimage_countable (c : ℝ) : (Real.cos ⁻¹' { c }).Countable := by
+  rcases (Real.cos ⁻¹' { c }).eq_empty_or_nonempty with h | ⟨θ₀, hθ₀⟩
+  · rw [h]; exact Set.countable_empty
+  refine
+    ((Set.countable_range fun k : ℤ => 2 * k * π + θ₀).union
+          (Set.countable_range fun k : ℤ => 2 * k * π - θ₀)).mono
+      ?_
+  intro θ hθ
+  simp only [Set.mem_preimage, Set.mem_singleton_iff] at hθ hθ₀
+  obtain ⟨k, hk | hk⟩ := Real.cos_eq_cos_iff.mp (hθ₀.trans hθ.symm)
+  · exact Or.inl ⟨k, hk.symm⟩
+  · exact Or.inr ⟨k, hk.symm⟩
+
+lemma circleMap_zero_one_re (θ : ℝ) : (circleMap 0 1 θ).re = Real.cos θ := by
+  simp [circleMap, Complex.exp_ofReal_mul_I_re]
+
+/-- `cos θ = (w + w⁻¹)/2` for `w = e^{iθ}`. -/
+lemma cos_eq_half_add_inv (θ : ℝ) :
+    ((Real.cos θ : ℝ) : ℂ) = (circleMap 0 1 θ + (circleMap 0 1 θ)⁻¹) / 2 := by
+  rw [Complex.ofReal_cos, Complex.cos, circleMap_zero]
+  simp only [Complex.ofReal_one, one_mul]
+  rw [← Complex.exp_neg]
+  ring_nf
+
+lemma norm_circleMap_zero_one (θ : ℝ) : ‖circleMap 0 1 θ‖ = 1 := by simp [circleMap]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Constants.lean b/LeanPool/Zeta5Irrational/Constants.lean
new file mode 100644
index 0000000000..fab114cad5
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Constants.lean
@@ -0,0 +1,58 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import Mathlib.Basic.Real.Basic
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # Numerical constants of the paper
+
+`U` of (6.4), `A*` of (5.19), `A_M` of (5.20), the displayed value `A₂₀₀` of Appendix B.3,
+the identity (5.20) at `M = 200` and the margin (7.2), both checked by `norm_num`.
+-/
+
+@[expose] public section
+
+namespace Zeta5Irrational
+
+/-- The constant `U` of Lemma 6.1 / (6.4). -/
+def U : ℚ :=
+  -2733991 / 2000000
+
+/-- The constant `A*` of (5.19). -/
+def Astar : ℚ :=
+  9928298118277006344769 / 7535670527041937280000
+
+/-- `A_M` of (5.20), with `λ = 37/40`. -/
+def A (M : ℚ) : ℚ :=
+  Astar + 7 * (37 / 40) / M - (2923 / 240 - 1 / 4) / M ^ 2 + 32 / M ^ 3
+
+/-- The value `A₂₀₀` displayed in Appendix B.3. -/
+def A200 : ℚ :=
+  127125602969131786927559 / 94195881588024216000000
+
+lemma A_200_eq : A 200 = A200 := by unfold A Astar A200; norm_num
+
+/-- The rational margin (7.2): `-1600 (A₂₀₀ + U) > 139 / 5`. -/
+lemma margin : (139 : ℚ) / 5 < -1600 * (A200 + U) := by unfold A200 U; norm_num
+
+/--
+The normalisation constant proved here (`Zeta5Irrational.Growth`): `limsup K⁻² log m_K ≤ A_eff`.
+It is weaker than the paper's `A₂₀₀`, but still below `-U`. -/
+def Aeff : ℚ :=
+  136 / 100
+
+/-- The margin used for irrationality: `A_eff + U < 0`. -/
+lemma Aeff_margin : Aeff + U < 0 := by unfold Aeff U; norm_num
+
+/-- `K = 40 n` as a real number. -/
+noncomputable abbrev Kr (n : ℕ) : ℝ :=
+  40 * (n : ℝ)
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Construction.lean b/LeanPool/Zeta5Irrational/Construction.lean
new file mode 100644
index 0000000000..a35e56943a
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Construction.lean
@@ -0,0 +1,89 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import Mathlib.NumberTheory.Bernoulli
+
+/-! # The Hankel determinant construction of the paper "ζ(5) is irrational"
+
+This file *defines* (in Lean) the objects of Section 2.1 of the paper
+"ζ(5) is irrational" (A. Fauzan, 17 September 2026):
+
+* the polynomials `D_m(t) = ∏_{j=1}^m (t + j²)`;
+* the `ℚ`-linear functional `μ_X`, given on monomials by (2.2) and on the simple poles
+  `1/(t + j²)` by
+(2.3), extended to `P / D_K` by polynomial division and partial fractions;
+* the Hankel matrix `G_K(X)` of (2.4), its determinant `Δ_K(X)`, the scalar `S_K` of (2.5)
+  and `F_K = S_K Δ_K`.
+
+Throughout, `K = 40 n`, `N = 3 n`, `h = 37 n` as in (2.1).
+
+**Convention.** Two polynomial rings `ℚ[X]` occur: the paper's variable `t` (the argument of the
+rational functions the functional is applied to) and the paper's indeterminate `X` (the
+value of the functional is affine in `X`). Both are represented by `Polynomial ℚ`; the
+docstrings say which is which.
+
+Nothing is proved about these objects here; see `Zeta5Irrational.MainEstimate` for the main
+estimate.
+-/
+
+@[expose] public section
+
+open Polynomial Finset
+
+namespace Zeta5Irrational
+
+/-- `D_m(t) = ∏_{j=1}^m (t + j²)`, a monic polynomial in `t`. -/
+noncomputable def D (m : ℕ) : ℚ[X] :=
+  ∏ j ∈ Icc 1 m, (X + C ((j : ℚ) ^ 2))
+
+/-- The polynomial moments (2.2): `μ(t^e) = (-1)^e B_{2e+2} (2e+3)(2e+4)(2e+5) / 24`.
+Mathlib's `bernoulli` uses the convention `B₁ = -1/2`, as does the paper. -/
+def μmono (e : ℕ) : ℚ :=
+  (-1) ^ e * _root_.bernoulli (2 * e + 2) * ((2 * e + 3) * (2 * e + 4) * (2 * e + 5)) / 24
+
+/-- `μ` on polynomials in `t`, extended `ℚ`-linearly from the monomials. -/
+noncomputable def μpoly (P : ℚ[X]) : ℚ :=
+  P.sum fun e c => c * μmono e
+
+/-- The generalized harmonic number `H_j^{(5)} = ∑_{v=1}^j v⁻⁵`. -/
+def H5 (j : ℕ) : ℚ :=
+  ∑ v ∈ Icc 1 j, 1 / (v : ℚ) ^ 5
+
+/-- The pole values (2.3), as polynomials in the indeterminate `X`:
+`μ_X(1/(t + j²)) = j⁴ (X - H_j^{(5)}) - 1/4 + 1/(2j)`. -/
+noncomputable def poleValue (j : ℕ) : ℚ[X] :=
+  C ((j : ℚ) ^ 4) * (X - C (H5 j)) - C (1 / 4) + C (1 / (2 * (j : ℚ)))
+
+/-- `μ_X(P(t) / D_K(t))` for a polynomial `P` in `t`, as a polynomial in `X`.
+Since `D_K` is monic with the simple roots `t = -j²` (`1 ≤ j ≤ K`),
+`P / D_K = (P /ₘ D_K) + ∑_j (P(-j²) / D_K'(-j²)) / (t + j²)`, and `μ_X` is applied termwise. -/
+noncomputable def μX (K : ℕ) (P : ℚ[X]) : ℚ[X] :=
+  C (μpoly (P /ₘ D K)) +
+    ∑ j ∈ Icc 1 K,
+      C (P.eval (-(j : ℚ) ^ 2) / (derivative (D K)).eval (-(j : ℚ) ^ 2)) * poleValue j
+
+/-- The `h × h` Hankel matrix `G_K(X)` of (2.4), whose `(i, j)` entry is
+`μ_X (D_N(t)^6 t^(i+j) / D_K(t))`, with `K = 40 n`, `N = 3 n`, `h = 37 n`. -/
+noncomputable def G (n : ℕ) : Matrix (Fin (37 * n)) (Fin (37 * n)) ℚ[X] := fun i j =>
+  μX (40 * n) (D (3 * n) ^ 6 * X ^ ((i : ℕ) + (j : ℕ)))
+
+/-- `Δ_K(X) = det G_K(X)`, a polynomial in `X` (of degree `h`, by (2.9)). -/
+noncomputable def Δ (n : ℕ) : ℚ[X] :=
+  (G n).det
+
+/-- The scalar `S_K` of (2.5). -/
+noncomputable def S (n : ℕ) : ℚ :=
+  ((40 * n).factorial : ℚ) ^ (2 * (37 * n)) * 4 ^ (37 * n - 1) /
+    (((3 * n).factorial : ℚ) ^ (12 * (37 * n)) *
+      ∏ i ∈ Icc 1 (37 * n - 1), ((2 * i).factorial : ℚ) ^ 2)
+
+/-- `F_K = S_K Δ_K` of (2.5). -/
+noncomputable def F (n : ℕ) : ℚ[X] :=
+  C (S n) * Δ n
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Criterion.lean b/LeanPool/Zeta5Irrational/Criterion.lean
new file mode 100644
index 0000000000..4f1eb4a5fe
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Criterion.lean
@@ -0,0 +1,143 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import Mathlib.Analysis.Complex.Exponential
+public import Mathlib.NumberTheory.Real.Irrational
+import Mathlib.Tactic.ENatToNat
+import Mathlib.Tactic.FieldSimp
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.Ring.Basic
+
+import Mathlib.Analysis.SpecialFunctions.Log.Basic
+import Mathlib.Analysis.SpecialFunctions.Exp
+
+/-! # An irrationality criterion via integer polynomials
+
+This file proves the elementary reduction used in the paper
+"ζ(5) is irrational" (A. Fauzan, 17 September 2026), Section 1.1 / proof of Theorem 1.1:
+
+If `ξ` is a real number and for all sufficiently large `n` there is an integer polynomial `Q`
+of degree at most `d n` with `0 < Q(ξ) ≤ ε n`, where `b ^ (d n) * ε n → 0` for every positive
+integer `b`, then `ξ` is irrational.
+
+The point is that if `ξ = a / b` then `b ^ (d n) * Q(a / b)` is a positive integer, hence at
+least `1`, while it tends to zero.
+-/
+
+@[expose] public section
+
+open Polynomial Filter Topology
+
+namespace Zeta5Irrational
+
+/-- Clearing denominators: for `p : ℤ[X]` of degree at most `d`, the number `b ^ d * p(a / b)`
+is the integer `∑ k, coeff k * a ^ k * b ^ (d - k)`. -/
+lemma pow_mul_aeval_div_eq_intCast (p : ℤ[X]) {d : ℕ} (hd : p.natDegree ≤ d) (a b : ℤ)
+    (hb : b ≠ 0) :
+    (b : ℝ) ^ d * aeval ((a : ℝ) / b) p =
+      ((∑ k ∈ Finset.range (d + 1), p.coeff k * a ^ k * b ^ (d - k) : ℤ) : ℝ) := by
+  have hbr : (b : ℝ) ≠ 0 := by exact_mod_cast hb
+  have hlt : (p.map (algebraMap ℤ ℝ)).natDegree < d + 1 :=
+    lt_of_le_of_lt natDegree_map_le (by omega)
+  rw [aeval_def, eval₂_eq_eval_map, eval_eq_sum_range' hlt, Finset.mul_sum]
+  push_cast
+  refine Finset.sum_congr rfl fun k hk => ?_
+  rw [Finset.mem_range] at hk
+  have hpow : (b : ℝ) ^ d = (b : ℝ) ^ (d - k) * (b : ℝ) ^ k := by rw [← pow_add]; congr 1; omega
+  rw [coeff_map, hpow, div_pow]
+  simp only [algebraMap_int_eq, eq_intCast]
+  field_simp
+
+/-- **Irrationality criterion.** Let `ξ : ℝ`, `d : ℕ → ℕ` and `ε : ℕ → ℝ` be such that
+`b ^ (d n) * ε n → 0` for every positive integer `b`. If for all sufficiently large `n` there is
+an integer polynomial `Q` with `natDegree Q ≤ d n` and `0 < Q(ξ) ≤ ε n`, then `ξ` is
+irrational. -/
+theorem irrational_of_eventually_exists_int_poly (ξ : ℝ) (d : ℕ → ℕ) (ε : ℕ → ℝ)
+    (hε : ∀ b : ℕ, 0 < b → Tendsto (fun n => (b : ℝ) ^ d n * ε n) atTop (𝓝 0))
+    (h : ∀ᶠ n in atTop, ∃ Q : ℤ[X], Q.natDegree ≤ d n ∧ 0 < aeval ξ Q ∧ aeval ξ Q ≤ ε n) :
+    Irrational ξ := by
+  rintro ⟨q, hq⟩
+  have hξ : ξ = ((q.num : ℤ) : ℝ) / ((q.den : ℤ) : ℝ) := by
+    rw [← hq, Rat.cast_def, Int.cast_natCast]
+  have hb0 : 0 < q.den := q.den_pos
+  have hbR : (0 : ℝ) < (q.den : ℝ) := by exact_mod_cast hb0
+  have hsmall : ∀ᶠ n in atTop, (q.den : ℝ) ^ d n * ε n < 1 :=
+    (hε q.den hb0).eventually (gt_mem_nhds one_pos)
+  obtain ⟨n, ⟨Q, hdeg, hpos, hle⟩, hlt⟩ :=
+    (h.and hsmall).exists
+      -- the integer `m = b ^ d * Q(a / b)`
+  obtain ⟨m, hm⟩ : ∃ m : ℤ, (m : ℝ) = (q.den : ℝ) ^ d n * aeval ξ Q := by
+    refine ⟨∑ k ∈ Finset.range (d n + 1), Q.coeff k * q.num ^ k * (q.den : ℤ) ^ (d n - k), ?_⟩
+    have key := pow_mul_aeval_div_eq_intCast Q hdeg q.num (q.den : ℤ) (by exact_mod_cast hb0.ne')
+    rw [hξ, ← key]
+    simp only [Int.cast_natCast]
+  have hm_pos : (0 : ℝ) < m := by rw [hm]; exact mul_pos (pow_pos hbR _) hpos
+  have hm_ge : (1 : ℝ) ≤ m := by
+    have : (0 : ℤ) < m := by exact_mod_cast hm_pos
+    exact_mod_cast this
+  have hm_lt : (m : ℝ) < 1 := by
+    rw [hm]
+    calc
+      (q.den : ℝ) ^ d n * aeval ξ Q ≤ (q.den : ℝ) ^ d n * ε n :=
+        mul_le_mul_of_nonneg_left hle (pow_nonneg hbR.le _)
+      _ < 1 := hlt
+  linarith
+
+/-- The specific decay used in the paper: `b ^ (37 n) * exp (-(139/5) n²) → 0`. -/
+theorem tendsto_pow_mul_exp_neg_sq (b : ℕ) (hb : 0 < b) :
+    Tendsto (fun n : ℕ => (b : ℝ) ^ (37 * n) * Real.exp (-(139 / 5) * (n : ℝ) ^ 2)) atTop (𝓝 0) :=
+  by
+  have hbR : (0 : ℝ) < b := by
+    exact_mod_cast
+      hb
+        -- rewrite as a single exponential
+  have hrw :
+    ∀ n : ℕ,
+      (b : ℝ) ^ (37 * n) * Real.exp (-(139 / 5) * (n : ℝ) ^ 2) =
+        Real.exp ((37 * n) * Real.log b - (139 / 5) * (n : ℝ) ^ 2) := by
+    intro n
+    have hbpow : (0 : ℝ) < (b : ℝ) ^ (37 * n) := pow_pos hbR _
+    rw [← Real.exp_log hbpow, ← Real.exp_add, Real.log_pow]
+    congr 1; push_cast; ring
+  simp_rw [hrw]
+    -- for large `n` the exponent is at most `-n`
+  have hexp : Tendsto (fun n : ℕ => Real.exp (-(n : ℝ))) atTop (𝓝 0) :=
+    Real.tendsto_exp_neg_atTop_nhds_zero.comp tendsto_natCast_atTop_atTop
+  refine squeeze_zero' (Eventually.of_forall fun n => (Real.exp_pos _).le) ?_ hexp
+  filter_upwards [eventually_ge_atTop (⌈(37 * Real.log b + 1) * 5 / 139⌉₊)] with n hn
+  apply Real.exp_le_exp.mpr
+  have hn' : (37 * Real.log b + 1) * 5 / 139 ≤ (n : ℝ) :=
+    le_trans (Nat.le_ceil _) (by exact_mod_cast hn)
+  have hn0 : (0 : ℝ) ≤ n := Nat.cast_nonneg n
+  nlinarith [mul_le_mul_of_nonneg_left hn' hn0]
+
+/-- The decay `b ^ (37 n) * exp (-c n²) → 0` for every `c > 0`. -/
+theorem tendsto_pow_mul_exp_neg_sq_of_pos {c : ℝ} (hc : 0 < c) (b : ℕ) (hb : 0 < b) :
+    Tendsto (fun n : ℕ => (b : ℝ) ^ (37 * n) * Real.exp (-c * (n : ℝ) ^ 2)) atTop (𝓝 0) := by
+  have hbR : (0 : ℝ) < b := by exact_mod_cast hb
+  have hrw :
+    ∀ n : ℕ,
+      (b : ℝ) ^ (37 * n) * Real.exp (-c * (n : ℝ) ^ 2) =
+        Real.exp ((37 * n) * Real.log b - c * (n : ℝ) ^ 2) := by
+    intro n
+    have hbpow : (0 : ℝ) < (b : ℝ) ^ (37 * n) := pow_pos hbR _
+    rw [← Real.exp_log hbpow, ← Real.exp_add, Real.log_pow]
+    congr 1; push_cast; ring
+  simp_rw [hrw]
+  have hexp : Tendsto (fun n : ℕ => Real.exp (-(n : ℝ))) atTop (𝓝 0) :=
+    Real.tendsto_exp_neg_atTop_nhds_zero.comp tendsto_natCast_atTop_atTop
+  refine squeeze_zero' (Eventually.of_forall fun n => (Real.exp_pos _).le) ?_ hexp
+  filter_upwards [eventually_ge_atTop (⌈(37 * Real.log b + 1) / c⌉₊)] with n hn
+  apply Real.exp_le_exp.mpr
+  have hn' : (37 * Real.log b + 1) / c ≤ (n : ℝ) :=
+    le_trans (Nat.le_ceil _) (by exact_mod_cast hn)
+  have hn'' : 37 * Real.log b + 1 ≤ c * n := by rw [div_le_iff₀ hc] at hn'; linarith
+  have hn0 : (0 : ℝ) ≤ n := Nat.cast_nonneg n
+  nlinarith [mul_le_mul_of_nonneg_left hn'' hn0]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Degree.lean b/LeanPool/Zeta5Irrational/Degree.lean
new file mode 100644
index 0000000000..aa31570384
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Degree.lean
@@ -0,0 +1,213 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Construction
+public import Mathlib.LinearAlgebra.Vandermonde
+import Mathlib.Algebra.Ring.IsFormallyReal
+import Mathlib.Data.Rat.Star
+import Mathlib.RingTheory.PiTensorProduct
+import Mathlib.Tactic.ENatToNat
+import Mathlib.Tactic.Polynomial.Basic
+import Mathlib.Tactic.ReduceModChar
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+import Mathlib.Tactic.Positivity.Basic
+import Mathlib.Tactic.Ring.Basic
+
+/-! # The degree of `Δ_K` and `F_K` — formula (2.9) of the paper
+
+The entries of `G_K(X)` are affine in `X`; the coefficient of `X^h` in `Δ_K = det G_K` is the
+determinant of the matrix of `X`-coefficients, which is `Vᵀ diag(w) V` for the Vandermonde
+matrix `V` of the nodes `-j²`, `N < j ≤ K`, with nonzero weights `w`. Hence `deg Δ_K = h`.
+-/
+
+@[expose] public section
+
+open Polynomial Matrix Finset
+
+namespace Zeta5Irrational
+
+lemma coeff_det_of_natDegree_le_one {R : Type*} [CommRing R] {n : ℕ}
+    (M : Matrix (Fin n) (Fin n) R[X]) (hM : ∀ i j, (M i j).natDegree ≤ 1) :
+    (M.det).coeff n = (Matrix.of fun i j => (M i j).coeff 1).det ∧ M.det.natDegree ≤ n := by
+  constructor
+  · rw [det_apply', det_apply', finsetSum_coeff]
+    refine Finset.sum_congr rfl fun σ _ => ?_
+    have hc :=
+      coeff_prod_of_natDegree_le (s := Finset.univ) (f := fun i => M (σ i) i) (n := 1)
+        (fun i _ => hM _ _)
+    simp only [Finset.card_univ, Fintype.card_fin, mul_one] at hc
+    rw [← C_eq_intCast, coeff_C_mul, hc]
+    simp
+  · rw [det_apply']
+    refine natDegree_sum_le_of_forall_le _ _ fun σ _ => ?_
+    refine (natDegree_mul_le).trans ?_
+    rw [natDegree_intCast, zero_add]
+    refine (natDegree_prod_le _ _).trans ?_
+    calc
+      ∑ i : Fin n, (M (σ i) i).natDegree ≤ ∑ i : Fin n, 1 := Finset.sum_le_sum fun i _ => hM _ _
+      _ = n := by simp
+
+/-- Residue coefficient at the pole `-j²` of `P / D K`, for `1 ≤ j ≤ K`. -/
+noncomputable def res (K : ℕ) (P : ℚ[X]) (j : ℕ) : ℚ :=
+  P.eval (-(j : ℚ) ^ 2) / (derivative (D K)).eval (-(j : ℚ) ^ 2)
+
+lemma poleValue_eq (j : ℕ) :
+    poleValue j = C ((j : ℚ) ^ 4) * X + C (-(j : ℚ) ^ 4 * H5 j - 1 / 4 + 1 / (2 * (j : ℚ))) := by
+  simp only [poleValue, C_add, C_sub, C_mul, C_neg, C_pow]
+  ring
+
+lemma μX_eq (K : ℕ) (P : ℚ[X]) :
+    μX K P =
+      C (∑ j ∈ Icc 1 K, res K P j * (j : ℚ) ^ 4) * X +
+        C
+          (μpoly (P /ₘ D K) +
+            ∑ j ∈ Icc 1 K, res K P j * (-(j : ℚ) ^ 4 * H5 j - 1 / 4 + 1 / (2 * (j : ℚ)))) := by
+  simp only [μX, res, poleValue_eq, mul_add, ← mul_assoc, ← C_mul, Finset.sum_add_distrib,
+    ← Finset.sum_mul, ← map_sum, C_add]
+  ring
+
+lemma natDegree_μX_le (K : ℕ) (P : ℚ[X]) : (μX K P).natDegree ≤ 1 := by
+  rw [μX_eq]
+  compute_degree!
+
+lemma coeff_μX_one (K : ℕ) (P : ℚ[X]) :
+    (μX K P).coeff 1 = ∑ j ∈ Icc 1 K, res K P j * (j : ℚ) ^ 4 := by
+  rw [μX_eq]
+  simp only [coeff_add, coeff_C_mul, coeff_X_one, coeff_C, mul_one, one_ne_zero, ite_false,
+    add_zero]
+
+lemma eval_D (m : ℕ) (x : ℚ) : (D m).eval x = ∏ j ∈ Icc 1 m, (x + (j : ℚ) ^ 2) := by
+  simp [D, eval_prod]
+
+lemma eval_D_eq_zero {m j : ℕ} (hj : j ∈ Icc 1 m) : (D m).eval (-(j : ℚ) ^ 2) = 0 := by
+  rw [eval_D]
+  exact Finset.prod_eq_zero hj (by ring)
+
+lemma eval_D_ne_zero {m j : ℕ} (hj : m < j) : (D m).eval (-(j : ℚ) ^ 2) ≠ 0 := by
+  rw [eval_D]
+  refine Finset.prod_ne_zero_iff.mpr fun i hi => ?_
+  rw [Finset.mem_Icc] at hi
+  have h1 : (i : ℚ) < j := by exact_mod_cast (by omega : i < j)
+  have h2 : (0 : ℚ) ≤ i := by positivity
+  nlinarith
+
+lemma eval_derivative_D {K j : ℕ} (hj : j ∈ Icc 1 K) :
+    (derivative (D K)).eval (-(j : ℚ) ^ 2) =
+      ∏ i ∈ (Icc 1 K).erase j, ((i : ℚ) ^ 2 - (j : ℚ) ^ 2) := by
+  rw [D, derivative_prod_finset, eval_finsetSum, Finset.sum_eq_single j]
+  · simp only [derivative_add, derivative_X, derivative_C, add_zero, mul_one, eval_prod, eval_add,
+      eval_X, eval_C]
+    exact Finset.prod_congr rfl fun i _ => by ring
+  · intro b _ hb
+    rw [eval_mul, eval_prod]
+    exact
+      mul_eq_zero_of_left (Finset.prod_eq_zero (Finset.mem_erase.mpr ⟨hb.symm, hj⟩) (by simp)) _
+  · intro h; exact absurd hj h
+
+lemma eval_derivative_D_ne_zero {K j : ℕ} (hj : j ∈ Icc 1 K) :
+    (derivative (D K)).eval (-(j : ℚ) ^ 2) ≠ 0 := by
+  rw [eval_derivative_D hj]
+  refine Finset.prod_ne_zero_iff.mpr fun i hi => ?_
+  have hij : i ≠ j := (Finset.mem_erase.mp hi).1
+  have : (i : ℚ) ≠ j := by exact_mod_cast hij
+  have h0i : (0 : ℚ) ≤ i := by positivity
+  have h0j : (0 : ℚ) ≤ j := by positivity
+  intro h
+  have hfac : ((i : ℚ) - j) * ((i : ℚ) + j) = 0 := by linear_combination h
+  rcases mul_eq_zero.mp hfac with h' | h'
+  · exact this (by linarith)
+  · have hj1 : (1 : ℚ) ≤ j := by exact_mod_cast (Finset.mem_Icc.mp hj).1
+    linarith
+
+/-- Reindexing: a sum over `Icc 1 K` whose terms vanish on `Icc 1 N` is a sum over
+`k : Fin (K - N)` of the terms at `N + 1 + k`. -/
+lemma sum_Icc_eq_sum_fin (N K : ℕ) (hNK : N ≤ K) (f : ℕ → ℚ) (hf : ∀ j ∈ Icc 1 N, f j = 0) :
+    ∑ j ∈ Icc 1 K, f j = ∑ k : Fin (K - N), f (N + 1 + k) := by
+  rw [Fin.sum_univ_eq_sum_range (fun k => f (N + 1 + k)) (K - N)]
+  have h1 := Finset.sum_Ico_eq_sum_range f (N + 1) (K + 1)
+  rw [show K + 1 - (N + 1) = K - N by omega] at h1
+  rw [← h1]
+  have h2 : Icc 1 K = Ico 1 (K + 1) := by ext x; simp only [Finset.mem_Icc, Finset.mem_Ico]; omega
+  rw [h2, ← Finset.sum_Ico_consecutive f (show 1 ≤ N + 1 by omega) (show N + 1 ≤ K + 1 by omega)]
+  rw [Finset.sum_eq_zero (fun j hj => hf j ?_), zero_add]
+  rw [Finset.mem_Ico] at hj
+  rw [Finset.mem_Icc]; omega
+
+/-- The nodes `x_k = -(N + 1 + k)²`, `k < h`. -/
+noncomputable def node (n : ℕ) (k : Fin (37 * n)) : ℚ :=
+  -((3 * n + 1 + (k : ℕ) : ℕ) : ℚ) ^ 2
+
+/-- The weights `w_k = D_N(x_k)^6 (N + 1 + k)^4 / D_K'(x_k)`. -/
+noncomputable def weight (n : ℕ) (k : Fin (37 * n)) : ℚ :=
+  (D (3 * n)).eval (node n k) ^ 6 * ((3 * n + 1 + (k : ℕ) : ℕ) : ℚ) ^ 4 /
+    (derivative (D (40 * n))).eval (node n k)
+
+lemma natDegree_G_le (n : ℕ) (i j : Fin (37 * n)) : (G n i j).natDegree ≤ 1 :=
+  natDegree_μX_le _ _
+
+/-- The matrix of `X`-coefficients of `G_K(X)` is `Vᵀ diag(w) V` with `V` the Vandermonde
+matrix of the nodes. -/
+lemma coeff_G_one (n : ℕ) :
+    (Matrix.of fun i j : Fin (37 * n) => (G n i j).coeff 1) =
+      (vandermonde (node n))ᵀ * diagonal (weight n) * vandermonde (node n) := by
+  ext i j
+  simp only [Matrix.of_apply, G, coeff_μX_one, res]
+  rw [sum_Icc_eq_sum_fin (3 * n) (40 * n) (by omega)
+      (fun j' =>
+        eval (-(j' : ℚ) ^ 2) (D (3 * n) ^ 6 * X ^ ((i : ℕ) + (j : ℕ))) /
+            eval (-(j' : ℚ) ^ 2) (derivative (D (40 * n))) *
+          (j' : ℚ) ^ 4)
+      (fun j' hj' => by rw [eval_mul, eval_pow, eval_D_eq_zero hj']; simp)]
+  rw [show 40 * n - 3 * n = 37 * n by omega]
+  rw [Matrix.mul_apply]
+  refine Finset.sum_congr rfl fun k _ => ?_
+  simp only [Matrix.mul_diagonal, Matrix.transpose_apply, vandermonde_apply, weight, node,
+    eval_mul, eval_pow, eval_X]
+  rw [pow_add]
+  ring
+
+lemma node_injective (n : ℕ) : Function.Injective (node n) := by
+  intro k₁ k₂ h
+  simp only [node, neg_inj] at h
+  have h' : (3 * n + 1 + (k₁ : ℕ)) ^ 2 = (3 * n + 1 + (k₂ : ℕ)) ^ 2 := by exact_mod_cast h
+  have := Nat.pow_left_injective (by norm_num : 2 ≠ 0) h'
+  exact Fin.ext (by omega)
+
+lemma weight_ne_zero (n : ℕ) (k : Fin (37 * n)) : weight n k ≠ 0 := by
+  have hk := k.isLt
+  have h1 : (D (3 * n)).eval (node n k) ≠ 0 := eval_D_ne_zero (by omega)
+  have h2 : (derivative (D (40 * n))).eval (node n k) ≠ 0 :=
+    eval_derivative_D_ne_zero (by rw [Finset.mem_Icc]; omega)
+  have h3 : ((3 * n + 1 + (k : ℕ) : ℕ) : ℚ) ≠ 0 := by positivity
+  unfold weight
+  exact div_ne_zero (mul_ne_zero (pow_ne_zero _ h1) (pow_ne_zero _ h3)) h2
+
+lemma det_coeff_G_one_ne_zero (n : ℕ) :
+    (Matrix.of fun i j : Fin (37 * n) => (G n i j).coeff 1).det ≠ 0 := by
+  rw [coeff_G_one, det_mul, det_mul, det_transpose, det_diagonal]
+  have hV : (vandermonde (node n)).det ≠ 0 := det_vandermonde_ne_zero_iff.mpr (node_injective n)
+  exact
+    mul_ne_zero (mul_ne_zero hV (Finset.prod_ne_zero_iff.mpr fun k _ => weight_ne_zero n k)) hV
+
+/-- **(2.9)**: `Δ_K` has degree exactly `h = 37 n`. -/
+theorem natDegree_Δ (n : ℕ) : (Δ n).natDegree = 37 * n := by
+  obtain ⟨hc, hd⟩ := coeff_det_of_natDegree_le_one (G n) (natDegree_G_le n)
+  exact natDegree_eq_of_le_of_coeff_ne_zero hd (hc ▸ det_coeff_G_one_ne_zero n)
+
+lemma S_pos (n : ℕ) : 0 < S n := by
+  unfold S
+  positivity
+
+/-- **(2.9)**: `F_K` has degree exactly `h = 37 n`. -/
+theorem natDegree_F (n : ℕ) : (F n).natDegree = 37 * n := by
+  rw [F, natDegree_C_mul (S_pos n).ne', natDegree_Δ]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/DetIntegral.lean b/LeanPool/Zeta5Irrational/DetIntegral.lean
new file mode 100644
index 0000000000..e6cc133638
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/DetIntegral.lean
@@ -0,0 +1,76 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Positivity
+import LeanPool.Zeta5Irrational.Andreief
+import Mathlib.Tactic.Ring.Basic
+
+/-! # The Hankel determinant as a multiple integral, (6.10)
+
+`Δ_K(ζ(5)) = (1/h!) ∫_{(0,∞)^h} det(V(y))² ∏ᵢ ν(yᵢ) dy`, where `V(y)` is the Vandermonde matrix
+of `y₁², …, y_h²` and `ν(y) = D_N(y²)^6 / D_K(y²) · w(y)`.
+-/
+
+@[expose] public section
+
+open MeasureTheory Set Finset Polynomial
+
+namespace Zeta5Irrational
+
+/-- The measure `dy` on `(0, ∞)`. -/
+noncomputable abbrev μpos : Measure ℝ :=
+  volume.restrict (Ioi 0)
+
+/-- **(6.10)**: the Hankel determinant as a multiple integral. -/
+theorem Δ_eq_integral (n : ℕ) :
+    aeval zeta5 (Δ n) =
+      (1 / ((37 * n).factorial : ℝ)) *
+        ∫ x : Fin (37 * n) → ℝ,
+          (Matrix.vandermonde fun j => x j ^ 2).det ^ 2 *
+            ∏ j, ν n (x j) ∂(Measure.pi fun _ : Fin (37 * n) => μpos) := by
+  -- the matrix `G_K(ζ(5))` has entries `∫ (y²)^i · ((y²)^j ν(y)) dy`
+  have hdet : aeval zeta5 (Δ n) = ((G n).map (aeval zeta5)).det := by
+    have := RingHom.map_det (aeval zeta5 : ℚ[X] →ₐ[ℚ] ℝ).toRingHom (G n)
+    simpa [Δ, RingHom.mapMatrix_apply] using this
+  set f : Fin (37 * n) → ℝ → ℝ := fun i y => (y ^ 2) ^ (i : ℕ) with hf
+  set g : Fin (37 * n) → ℝ → ℝ := fun j y => (y ^ 2) ^ (j : ℕ) * ν n y with hg
+  have hentry : (G n).map (aeval zeta5) = Matrix.of fun i j => ∫ y, f i y * g j y ∂μpos := by
+    ext i j
+    simp only [Matrix.map_apply, Matrix.of_apply, G, hf, hg]
+    rw [(moment_rep _ _).2]
+    refine setIntegral_congr_fun measurableSet_Ioi fun y _ => ?_
+    rw [integrand_eq]; ring
+  have hfg : ∀ i j, Integrable (fun y => f i y * g j y) μpos := by
+    intro i j
+    have := (moment_rep (40 * n) (D (3 * n) ^ 6 * X ^ ((i : ℕ) + (j : ℕ)))).1
+    refine this.congr_fun (fun y _ => ?_) measurableSet_Ioi
+    simp only [hf, hg]; rw [integrand_eq]; ring
+  rw [hdet, hentry, andreief f g hfg]
+  congr 1
+  refine
+    integral_congr_ae
+      (Filter.Eventually.of_forall fun x => ?_)
+        -- `det (f i (x j)) = det Vᵀ` and `det (g i (x j)) = (∏ ν) det Vᵀ`
+  have h1 : (Matrix.of fun i j => f i (x j)).det = (Matrix.vandermonde fun j => x j ^ 2).det := by
+    rw [← Matrix.det_transpose (Matrix.vandermonde _)]
+    refine congrArg Matrix.det ?_
+    ext i j; simp [hf, Matrix.vandermonde, Matrix.transpose_apply]
+  have h2 :
+    (Matrix.of fun i j => g i (x j)).det =
+      (∏ j, ν n (x j)) * (Matrix.vandermonde fun j => x j ^ 2).det := by
+    have :
+      (Matrix.of fun i j => g i (x j)) =
+        Matrix.of fun i j => ν n (x j) * (Matrix.of fun i j => f i (x j)) i j :=
+      by ext i j; simp [hf, hg]; ring
+    rw [this, Matrix.det_mul_row, h1]
+  change
+    (Matrix.of fun i j => f i (x j)).det * (Matrix.of fun i j => g i (x j)).det =
+      (Matrix.vandermonde fun j => x j ^ 2).det ^ 2 * ∏ j, ν n (x j)
+  rw [h1, h2]; ring
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Energy.lean b/LeanPool/Zeta5Irrational/Energy.lean
new file mode 100644
index 0000000000..66431c4370
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Energy.lean
@@ -0,0 +1,356 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.DetIntegral
+import LeanPool.Zeta5Irrational.HermiteIBP
+import LeanPool.Zeta5Irrational.Moments
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Analysis.SumIntegralComparisons
+import Mathlib.Tactic.ENatToNat
+import Mathlib.Tactic.Polynomial.Basic
+import Mathlib.Tactic.ReduceModChar
+import Mathlib.Tactic.FieldSimp
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+import Mathlib.Tactic.Positivity.Basic
+import Mathlib.Tactic.Ring.Basic
+import Mathlib.Tactic.Ring.RingNF
+
+/-! # Towards the energy bound (6.14): elementary ingredients
+
+* the external field `V(t) = 2π√t + ∫₀¹ log(t+u²) du - 6 ∫₀^α log(t+u²) du` of (6.1);
+* the bound `w(y) ≤ (1/π) (1 + 2πy)^5 e^{-2πy}` (a form of (6.11));
+* a variant of the sum–integral comparison (6.12) for `u ↦ log (t + u²)`, using
+  `2 log(2K) + 2` in place of `2 log K + 2` in the upper error bound.
+-/
+
+@[expose] public section
+
+open Filter Topology Set MeasureTheory Finset
+
+namespace Zeta5Irrational
+
+/-- The external field of (6.1). -/
+noncomputable def Vfield (t : ℝ) : ℝ :=
+  2 * Real.pi * Real.sqrt t + (∫ u in (0 : ℝ)..1, Real.log (t + u ^ 2)) -
+    6 * ∫ u in (0 : ℝ)..(3 / 40), Real.log (t + u ^ 2)
+
+/-- `w(y) ≤ (1/π) (1 + 2πy)^5 e^{-2πy}` for `y > 0`. -/
+lemma w_le' {y : ℝ} (hy : 0 < y) :
+    w y ≤ 1 / Real.pi * (1 + 2 * Real.pi * y) ^ 5 * Real.exp (-(2 * Real.pi * y)) := by
+  rw [w_eq_F4 hy]
+  have h := abs_F4_le hy
+  have hF : F4 y ≤ 24 * (1 + 2 * Real.pi * y) ^ (4 + 1) / (2 * Real.pi * y ^ (4 + 1) * qe y) :=
+    (le_abs_self _).trans h
+  have hq : qe y = Real.exp (2 * Real.pi * y) := rfl
+  have hqinv : 1 / qe y = Real.exp (-(2 * Real.pi * y)) := by rw [hq, Real.exp_neg, one_div]
+  have hqpos := qe_pos y
+  calc
+    y ^ 5 * F4 y / 12 ≤
+        y ^ 5 * (24 * (1 + 2 * Real.pi * y) ^ (4 + 1) / (2 * Real.pi * y ^ (4 + 1) * qe y)) /
+          12 :=
+      by gcongr
+    _ = 1 / Real.pi * (1 + 2 * Real.pi * y) ^ 5 * (1 / qe y) := by
+      field_simp
+      ring
+    _ = _ := by rw [hqinv]
+
+/-- The monotone comparison (6.12): for `t > 0` and `1 ≤ m`,
+`K ∫₀^{m/K} log(t+u²) du ≤ ∑_{j=1}^m log(t + (j/K)²)`. -/
+lemma integral_le_sum_log {t : ℝ} (ht : 0 < t) {K : ℝ} (hK : 0 < K) (m : ℕ) :
+    K * ∫ u in (0 : ℝ)..((m : ℝ) / K), Real.log (t + u ^ 2) ≤
+      ∑ j ∈ range m, Real.log (t + (((j : ℝ) + 1) / K) ^ 2) := by
+  -- substitution `u = v / K` and the monotone comparison on `[0, m]`
+  have hmono : MonotoneOn (fun v : ℝ => Real.log (t + (v / K) ^ 2)) (Icc 0 (0 + (m : ℝ))) := by
+    intro x hx y hy hxy
+    have hx0 : 0 ≤ x := hx.1
+    refine Real.log_le_log (by positivity) ?_
+    gcongr
+  have hcmp := hmono.integral_le_sum
+  simp only [zero_add] at hcmp
+        -- change of variables in the integral
+  have hsub :
+    ∫ u in (0 : ℝ)..((m : ℝ) / K), Real.log (t + u ^ 2) =
+      (1 / K) * ∫ v in (0 : ℝ)..(m : ℝ), Real.log (t + (v / K) ^ 2) := by
+    have :=
+      intervalIntegral.integral_comp_div (fun u => Real.log (t + u ^ 2)) (a := 0) (b := (m : ℝ))
+        hK.ne'
+    simp only [zero_div] at this
+    rw [this, smul_eq_mul]; field_simp
+  rw [hsub]
+  have h1 :
+    K * (1 / K * ∫ v in (0 : ℝ)..(m : ℝ), Real.log (t + (v / K) ^ 2)) =
+      ∫ v in (0 : ℝ)..(m : ℝ), Real.log (t + (v / K) ^ 2) :=
+    by field_simp
+  rw [h1]
+  refine hcmp.trans (le_of_eq ?_)
+  refine Finset.sum_congr rfl fun j _ => ?_
+  push_cast; ring_nf
+
+/-- A weakening of the reverse comparison (6.12): for `t > 0`, `1 ≤ m ≤ K`,
+`∑_{j=1}^m log(t + (j/K)²) ≤ K ∫₀^{m/K} log(t+u²) du + 2 log(2K) + 2`.
+The paper has the sharper remainder `2 log K + 2`. -/
+lemma sum_log_le_integral {t : ℝ} (ht : 0 < t) {K : ℝ} (hK : 1 ≤ K) {m : ℕ} (hm : 1 ≤ m)
+    (hmK : (m : ℝ) ≤ K) :
+    ∑ j ∈ range m, Real.log (t + (((j : ℝ) + 1) / K) ^ 2) ≤
+      K * (∫ u in (0 : ℝ)..((m : ℝ) / K), Real.log (t + u ^ 2)) + 2 * Real.log (2 * K) + 2 := by
+  have hK0 : 0 < K := by linarith
+  set f : ℝ → ℝ := fun u => Real.log (t + u ^ 2) with hf
+  have hfcont : Continuous f := by
+    rw [hf];
+    exact Continuous.log (by fun_prop) (fun u => by positivity)
+  have hfint : ∀ a b : ℝ, IntervalIntegrable f volume a b := fun a b =>
+    hfcont.intervalIntegrable a b
+  set c : ℝ := (m : ℝ) / K with hc
+  have hc0 : 0 ≤ c := by
+    positivity
+      -- (L1) right-endpoint sums vs the shifted integral
+  have hmono : MonotoneOn (fun v : ℝ => f ((v + 1) / K)) (Icc 0 (0 + (m : ℝ))) := by
+    intro x hx y hy hxy
+    have hx0 : 0 ≤ x := hx.1
+    simp only [hf]
+    refine Real.log_le_log (by positivity) ?_
+    gcongr
+  have hcmp := hmono.sum_le_integral
+  simp only [zero_add] at hcmp
+  have hshift :
+    ∫ v in (0 : ℝ)..(m : ℝ), f ((v + 1) / K) = K * ∫ u in (1 / K : ℝ)..(((m : ℝ) + 1) / K), f u :=
+  by
+    have h1 :=
+      intervalIntegral.integral_comp_add_right (fun v => f (v / K)) (a := 0) (b := (m : ℝ)) 1
+    have h2 := intervalIntegral.integral_comp_div f (a := 1) (b := (m : ℝ) + 1) hK0.ne'
+    simp only [zero_add] at h1
+    rw [h1, h2, smul_eq_mul]
+      -- (L2) splitting
+  have hfint' : IntervalIntegrable (fun u => f (u + c)) volume 0 (1 / K) :=
+    (hfcont.comp (continuous_id.add continuous_const)).intervalIntegrable _ _
+  have hsplit :
+    ∫ u in (1 / K : ℝ)..(((m : ℝ) + 1) / K), f u =
+      (∫ u in (0 : ℝ)..c, f u) + ∫ u in (0 : ℝ)..(1 / K : ℝ), (f (u + c) - f u) := by
+    rw [intervalIntegral.integral_sub hfint' (hfint _ _),
+      intervalIntegral.integral_comp_add_right f c]
+    have e1 :=
+      intervalIntegral.integral_add_adjacent_intervals (hfint 0 (1 / K))
+        (hfint (1 / K) (((m : ℝ) + 1) / K))
+    have e2 :=
+      intervalIntegral.integral_add_adjacent_intervals (hfint 0 c) (hfint c (((m : ℝ) + 1) / K))
+    have e3 : (0 : ℝ) + c = c := by ring
+    have e4 : 1 / K + c = ((m : ℝ) + 1) / K := by rw [hc]; field_simp; ring
+    rw [e3, e4]
+    linarith
+      -- (L3) pointwise bound on `(0, 1/K)`
+  have hpt : ∀ u ∈ Ioo (0 : ℝ) (1 / K), f (u + c) - f u ≤ 2 * (Real.log (u + c) - Real.log u) := by
+    intro u hu
+    have hu0 : 0 < u := hu.1
+    simp only [hf]
+    rw [← Real.log_div (by positivity) (by positivity), ← Real.log_div (by positivity) hu0.ne',
+      show (2 : ℝ) * Real.log ((u + c) / u) = Real.log (((u + c) / u) ^ 2) by
+        rw [Real.log_pow];
+        push_cast; ring]
+    refine Real.log_le_log (by positivity) ?_
+    rw [div_pow, div_le_div_iff₀ (by positivity) (by positivity)]
+    nlinarith [sq_nonneg u,
+      mul_nonneg ht.le (mul_nonneg hc0 (by positivity : (0 : ℝ) ≤ 2 * u + c))]
+      -- (L4) the two logarithmic integrals
+  have hI1 : ∫ u in (0 : ℝ)..(1 / K : ℝ), Real.log u = 1 / K * Real.log (1 / K) - 1 / K := by
+    rw [integral_log]; simp
+  have hm' : (1 : ℝ) ≤ m := by exact_mod_cast hm
+  have hcpos : 0 < c := by rw [hc]; exact div_pos (by linarith) hK0
+  have hcont2 : IntervalIntegrable (fun u => Real.log (u + c)) volume 0 (1 / K) := by
+    refine ContinuousOn.intervalIntegrable (ContinuousOn.log (by fun_prop) (fun u hu => ?_))
+    rw [Set.uIcc_of_le (by positivity)] at hu
+    have := hu.1
+    exact (by linarith : (0 : ℝ) < u + c).ne'
+  have hI2 :
+    ∫ u in (0 : ℝ)..(1 / K : ℝ), Real.log (u + c) ≤ 1 / K * Real.log (((m : ℝ) + 1) / K) := by
+    have hle : ∀ u ∈ Icc (0 : ℝ) (1 / K), Real.log (u + c) ≤ Real.log (((m : ℝ) + 1) / K) := by
+      intro u hu
+      refine Real.log_le_log (by linarith [hu.1]) ?_
+      have e : ((m : ℝ) + 1) / K = 1 / K + c := by rw [hc]; field_simp; ring
+      rw [e]; linarith [hu.2]
+    have := intervalIntegral.integral_mono_on (by positivity) hcont2 intervalIntegrable_const hle
+    simpa [intervalIntegral.integral_const] using this
+  have hlogint : IntervalIntegrable Real.log volume 0 (1 / K) :=
+    intervalIntegral.intervalIntegrable_log'
+  have hdiff :
+    ∫ u in (0 : ℝ)..(1 / K : ℝ), (f (u + c) - f u) ≤
+      2 *
+        ((∫ u in (0 : ℝ)..(1 / K : ℝ), Real.log (u + c)) -
+          ∫ u in (0 : ℝ)..(1 / K : ℝ), Real.log u) := by
+    rw [← intervalIntegral.integral_sub hcont2 hlogint, ← intervalIntegral.integral_const_mul]
+    exact
+      intervalIntegral.integral_mono_on_of_le_Ioo (by positivity) (hfint'.sub (hfint _ _))
+        ((hcont2.sub hlogint).const_mul 2)
+        hpt
+          -- combine
+  have hlog2 : Real.log (((m : ℝ) + 1) / K) - Real.log (1 / K) ≤ Real.log (2 * K) := by
+    rw [← Real.log_div (by positivity) (by positivity)]
+    refine Real.log_le_log (by positivity) ?_
+    have e : ((m : ℝ) + 1) / K / (1 / K) = (m : ℝ) + 1 := by field_simp
+    rw [e]; linarith
+  calc
+    ∑ x ∈ range m, f ((↑x + 1) / K) ≤ ∫ v in (0 : ℝ)..(m : ℝ), f ((v + 1) / K) := hcmp
+    _ = K * ((∫ u in (0 : ℝ)..c, f u) + ∫ u in (0 : ℝ)..(1 / K : ℝ), (f (u + c) - f u)) := by
+      rw [hshift, hsplit]
+    _ ≤
+        K *
+          ((∫ u in (0 : ℝ)..c, f u) +
+            2 * ((1 / K * Real.log (((m : ℝ) + 1) / K)) - (1 / K * Real.log (1 / K) - 1 / K))) := by
+      gcongr
+      rw [← hI1]
+      exact hdiff.trans (by gcongr)
+    _ =
+        K * (∫ u in (0 : ℝ)..c, f u) + 2 * (Real.log (((m : ℝ) + 1) / K) - Real.log (1 / K)) +
+          2 := by
+      field_simp
+      ring
+    _ ≤ K * (∫ u in (0 : ℝ)..c, f u) + 2 * Real.log (2 * K) + 2 := by gcongr
+
+/-- `∫₀^∞ (1 + 2πy)^5 e^{-y/K} dy ≤ (2π)^5 · 32 · 121 · K^6` for `K ≥ 1`. -/
+lemma integral_one_add_pow_mul_exp {K : ℝ} (hK : 1 ≤ K) :
+    ∫ y in Ioi 0, (1 + 2 * Real.pi * y) ^ 5 * Real.exp (-(1 / K * y)) ≤
+      (2 * Real.pi) ^ 5 * 32 * 121 * K ^ 6 := by
+  have hK0 : 0 < K := by linarith
+  have hr : 0 < 1 / K := by positivity
+  have hpi :=
+    Real.pi_gt_three
+      -- the dominating function
+  have hg :
+    IntegrableOn
+      (fun y : ℝ =>
+        (2 * Real.pi) ^ 5 * 32 * (y ^ 0 * Real.exp (-(1 / K * y))) +
+          (2 * Real.pi) ^ 5 * 32 * (y ^ 5 * Real.exp (-(1 / K * y))))
+      (Ioi 0) :=
+    ((integrableOn_pow_mul_exp_neg_mul 0 hr).const_mul _).add
+      ((integrableOn_pow_mul_exp_neg_mul 5 hr).const_mul _)
+  have hle :
+    ∀ y : ℝ,
+      0 < y →
+        (1 + 2 * Real.pi * y) ^ 5 * Real.exp (-(1 / K * y)) ≤
+          (2 * Real.pi) ^ 5 * 32 * (y ^ 0 * Real.exp (-(1 / K * y))) +
+            (2 * Real.pi) ^ 5 * 32 * (y ^ 5 * Real.exp (-(1 / K * y))) := by
+    intro y hy
+    have h1 : (1 + 2 * Real.pi * y) ^ 5 ≤ (2 * Real.pi) ^ 5 * (1 + y) ^ 5 := by
+      rw [← mul_pow]
+      exact pow_le_pow_left₀ (by positivity) (by nlinarith) 5
+    have h2 : (1 + y) ^ 5 ≤ 32 * (1 + y ^ 5) := by
+      rcases le_or_gt y 1 with h | h
+      · have : (1 + y) ^ 5 ≤ 2 ^ 5 := pow_le_pow_left₀ (by linarith) (by linarith) 5
+        have : 0 ≤ y ^ 5 := by positivity
+        nlinarith
+      · have : (1 + y) ^ 5 ≤ (2 * y) ^ 5 := pow_le_pow_left₀ (by linarith) (by linarith) 5
+        have : 0 ≤ y ^ 5 := by positivity
+        nlinarith
+    have he : 0 ≤ Real.exp (-(1 / K * y)) := (Real.exp_pos _).le
+    calc
+      (1 + 2 * Real.pi * y) ^ 5 * Real.exp (-(1 / K * y)) ≤
+          (2 * Real.pi) ^ 5 * (32 * (1 + y ^ 5)) * Real.exp (-(1 / K * y)) := by
+        gcongr
+        exact h1.trans (by gcongr)
+      _ = _ := by simp only [pow_zero, one_mul]; ring
+  have hnn :
+    0 ≤ᵐ[volume.restrict (Ioi 0)] fun y : ℝ =>
+      (1 + 2 * Real.pi * y) ^ 5 * Real.exp (-(1 / K * y)) :=
+    (ae_restrict_iff' measurableSet_Ioi).mpr
+      (Filter.Eventually.of_forall fun y hy =>
+        by
+        have : (0 : ℝ) < y := hy
+        positivity)
+  have hmono :
+    ∀ᵐ y ∂(volume.restrict (Ioi 0)),
+      (1 + 2 * Real.pi * y) ^ 5 * Real.exp (-(1 / K * y)) ≤
+        (2 * Real.pi) ^ 5 * 32 * (y ^ 0 * Real.exp (-(1 / K * y))) +
+          (2 * Real.pi) ^ 5 * 32 * (y ^ 5 * Real.exp (-(1 / K * y))) :=
+    (ae_restrict_iff' measurableSet_Ioi).mpr (Filter.Eventually.of_forall fun y hy => hle y hy)
+  calc
+    ∫ y in Ioi 0, (1 + 2 * Real.pi * y) ^ 5 * Real.exp (-(1 / K * y)) ≤
+        ∫ y in Ioi 0,
+          ((2 * Real.pi) ^ 5 * 32 * (y ^ 0 * Real.exp (-(1 / K * y))) +
+            (2 * Real.pi) ^ 5 * 32 * (y ^ 5 * Real.exp (-(1 / K * y)))) :=
+      integral_mono_of_nonneg hnn hg hmono
+    _ =
+        (2 * Real.pi) ^ 5 * 32 * ((0 : ℕ).factorial / (1 / K) ^ (0 + 1)) +
+          (2 * Real.pi) ^ 5 * 32 * ((5 : ℕ).factorial / (1 / K) ^ (5 + 1)) := by
+      rw [integral_add ((integrableOn_pow_mul_exp_neg_mul 0 hr).const_mul _)
+          ((integrableOn_pow_mul_exp_neg_mul 5 hr).const_mul _),
+        integral_const_mul, integral_const_mul, integral_pow_mul_exp_neg_mul 0 hr,
+        integral_pow_mul_exp_neg_mul 5 hr]
+    _ = (2 * Real.pi) ^ 5 * 32 * (K + 120 * K ^ 6) := by
+      simp only [Nat.factorial]; norm_num
+      all_goals field_simp
+    _ ≤ (2 * Real.pi) ^ 5 * 32 * 121 * K ^ 6 := by
+      have : K ≤ K ^ 6 := by
+        calc
+          K = K ^ 1 := (pow_one K).symm
+          _ ≤ K ^ 6 := pow_le_pow_right₀ hK (by norm_num)
+      nlinarith [pow_pos (by positivity : (0 : ℝ) < 2 * Real.pi) 5]
+
+/-! ### Almost every configuration has positive, pairwise distinct coordinates -/
+
+/-- The hyperplane `{x | x i = x j}` is null for Lebesgue measure on `Fin h → ℝ`. -/
+lemma volume_eq_coord_null {h : ℕ} {i j : Fin h} (hij : i ≠ j) :
+    (volume : Measure (Fin h → ℝ)) {x | x i = x j} = 0 := by
+  set L : (Fin h → ℝ) →ₗ[ℝ] ℝ :=
+    LinearMap.proj (R := ℝ) (φ := fun _ : Fin h => ℝ) i -
+      LinearMap.proj (R := ℝ) (φ := fun _ : Fin h => ℝ) j with
+    hL
+  have hset : {x : Fin h → ℝ | x i = x j} = (LinearMap.ker L : Set (Fin h → ℝ)) := by
+    ext x
+    simp [hL, LinearMap.mem_ker, sub_eq_zero]
+  rw [hset]
+  refine Measure.addHaar_submodule volume _ ?_
+  intro htop
+  have hmem : (Pi.single i (1 : ℝ) : Fin h → ℝ) ∈ LinearMap.ker L := by
+    rw [htop];
+    exact Submodule.mem_top
+  rw [LinearMap.mem_ker] at hmem
+  simp [hL, hij.symm] at hmem
+
+/-- Almost every `x` (for `dy` on `(0,∞)^h`) has positive, pairwise distinct coordinates. -/
+lemma ae_pos_injective (h : ℕ) :
+    ∀ᵐ x ∂(Measure.pi fun _ : Fin h => μpos), (∀ i, 0 < x i) ∧ Function.Injective x := by
+  have hpi :
+    (Measure.pi fun _ : Fin h => μpos) =
+      (volume : Measure (Fin h → ℝ)).restrict (Set.pi univ fun _ => Ioi 0) :=
+    by rw [volume_pi, Measure.restrict_pi_pi]
+  rw [hpi]
+  have h1 :
+    ∀ᵐ x ∂((volume : Measure (Fin h → ℝ)).restrict (Set.pi univ fun _ => Ioi 0)), ∀ i, 0 < x i := by
+    have :=
+      ae_restrict_mem (μ := (volume : Measure (Fin h → ℝ))) (s :=
+        Set.pi univ fun _ : Fin h => Ioi (0 : ℝ))
+        (MeasurableSet.univ_pi fun _ => measurableSet_Ioi)
+    filter_upwards [this] with x hx
+    intro i
+    exact hx i (Set.mem_univ i)
+  have hac : (volume : Measure (Fin h → ℝ)).restrict (Set.pi univ fun _ => Ioi 0) ≪ volume :=
+    Measure.absolutelyContinuous_of_le Measure.restrict_le_self
+  have h2 :
+    ∀ᵐ x ∂((volume : Measure (Fin h → ℝ)).restrict (Set.pi univ fun _ => Ioi 0)),
+      Function.Injective x := by
+    have hall :
+      ∀ᵐ x ∂((volume : Measure (Fin h → ℝ)).restrict (Set.pi univ fun _ => Ioi 0)),
+        ∀ i j, i ≠ j → x i ≠ x j := by
+      rw [ae_all_iff]; intro i
+      rw [ae_all_iff]; intro j
+      by_cases hij : i = j
+      · exact Filter.Eventually.of_forall fun x h => absurd hij h
+      · have h0 : (volume : Measure (Fin h → ℝ)) {x | x i = x j} = 0 := volume_eq_coord_null hij
+        have h1 := hac h0
+        rw [ae_iff]
+        convert h1 using 2
+        ext x; simp [hij]
+    filter_upwards [hall] with x hx
+    intro i j hxy
+    by_contra hij
+    exact hx i j hij hxy
+  exact h1.and h2
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/EnergyAtoms.lean b/LeanPool/Zeta5Irrational/EnergyAtoms.lean
new file mode 100644
index 0000000000..0787016cf4
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/EnergyAtoms.lean
@@ -0,0 +1,314 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Constants
+public import LeanPool.Zeta5Irrational.ArcsineAtoms
+public import LeanPool.Zeta5Irrational.EnergyLimit
+public import LeanPool.Zeta5Irrational.Measure
+import LeanPool.Zeta5Irrational.CircleAtoms
+import LeanPool.Zeta5Irrational.Potential
+import Mathlib.Tactic.ENatToNat
+import Mathlib.Tactic.ReduceModChar
+import Mathlib.Tactic.FieldSimp
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+import Mathlib.Tactic.Positivity.Basic
+import Mathlib.Tactic.Ring.Basic
+
+/-! # The atoms of the configuration energy
+
+For a configuration `t : Fin h → ℝ` and a radius `ε > 0`, the atoms are the circles of radius
+`ε` around the `t i` (weights `1/K`) and the sixteen arcsine curves (weights `-c_j`).
+This file verifies the hypotheses of `energy_log_nonpos` for these atoms.
+-/
+
+@[expose] public section
+
+open MeasureTheory Set Real intervalIntegral Filter Topology
+
+namespace Zeta5Irrational
+
+/-- Reindexing `∑_{j ∈ Icc 1 m} f j = ∑_{j < m} f (j + 1)`. -/
+lemma sum_Icc_eq_sum_range' (m : ℕ) (f : ℕ → ℝ) :
+    ∑ j ∈ Finset.Icc 1 m, f j = ∑ j ∈ Finset.range m, f (j + 1) := by
+  have e1 : Finset.Icc 1 m = Finset.Ico 1 (m + 1) := by
+    ext x;
+    simp only [Finset.mem_Icc, Finset.mem_Ico]; omega
+  rw [e1, Finset.sum_Ico_eq_sum_range, show m + 1 - 1 = m by omega]
+  refine Finset.sum_congr rfl fun j _ => ?_
+  rw [add_comm]
+
+/-- Midpoints and half-lengths of the intervals of Table 1. -/
+noncomputable def mρ (j : ℕ) : ℝ :=
+  (aρ j + bρ j) / 2
+
+/-- Half-width of the support of the `j`th arcsine component. -/
+noncomputable def rρ (j : ℕ) : ℝ :=
+  (bρ j - aρ j) / 2
+
+lemma rρ_pos (j : Fin 16) : 0 < rρ (j + 1) := by
+  have := aρ_lt_bρ (j + 1) (by simp only [Finset.mem_Icc]; omega)
+  unfold rρ; linarith
+
+lemma rρ_ge (j : Fin 16) : 1 / 450 ≤ rρ (j + 1) := by
+  unfold rρ;
+  fin_cases j <;> norm_num [aρ, bρ, table1]
+
+lemma mρ_bounds (j : Fin 16) : 0 < mρ (j + 1) ∧ mρ (j + 1) < 2 := by
+  unfold mρ;
+  fin_cases j <;> norm_num [aρ, bρ, table1]
+
+lemma rρ_le (j : Fin 16) : rρ (j + 1) ≤ 1 := by
+  unfold rρ;
+  fin_cases j <;> norm_num [aρ, bρ, table1]
+
+/-- The index type of the atoms. -/
+abbrev Idx (h : ℕ) :=
+  Fin h ⊕ Fin 16
+
+/-- The weights. -/
+noncomputable def atomS (h : ℕ) (K : ℝ) : Idx h → ℝ
+  | Sum.inl _ => 1 / K
+  | Sum.inr j => -cρ (j + 1)
+
+/-- The curves. -/
+noncomputable def atomγ {h : ℕ} (t : Fin h → ℝ) (ε : ℝ) : Idx h → ℝ → ℂ
+  | Sum.inl i => circleMap (t i) ε
+  | Sum.inr j => arcCurve (mρ (j + 1)) (rρ (j + 1))
+
+lemma continuous_atomγ {h : ℕ} (t : Fin h → ℝ) (ε : ℝ) (k : Idx h) : Continuous (atomγ t ε k) := by
+  cases k with
+  | inl i => exact continuous_circleMap _ _
+  | inr j => exact continuous_arcCurve _ _
+
+lemma norm_atomγ_le {h : ℕ} (t : Fin h → ℝ) (ε : ℝ) (k : Idx h) (θ : ℝ) :
+    ‖atomγ t ε k θ‖ ≤ (∑ i, |t i|) + |ε| + 3 := by
+  cases k with
+  | inl i =>
+    simp only [atomγ, circleMap]
+    calc
+      ‖(t i : ℂ) + ε * Complex.exp (θ * Complex.I)‖ ≤
+          ‖(t i : ℂ)‖ + ‖(ε : ℂ) * Complex.exp (θ * Complex.I)‖ :=
+        norm_add_le _ _
+      _ = |t i| + |ε| := by
+        rw [norm_mul, Complex.norm_exp_ofReal_mul_I, mul_one, Complex.norm_real,
+          Complex.norm_real, Real.norm_eq_abs, Real.norm_eq_abs]
+      _ ≤ (∑ i, |t i|) + |ε| + 3 := by
+        have : |t i| ≤ ∑ i, |t i| :=
+          Finset.single_le_sum (fun i _ => abs_nonneg (t i)) (Finset.mem_univ i)
+        linarith
+  | inr j =>
+    simp only [atomγ]
+    refine (norm_arcCurve_le _ _ _).trans ?_
+    have h1 := mρ_bounds j
+    have h2 := rρ_pos j
+    have h3 := rρ_le j
+    rw [abs_of_pos h1.1, abs_of_pos h2]
+    have : 0 ≤ ∑ i, |t i| := Finset.sum_nonneg fun i _ => abs_nonneg _
+    linarith [abs_nonneg ε]
+
+/-- Zero total mass when `h = 37 n` and `K = 40 n`. -/
+lemma atomS_mass (n : ℕ) (hn : 0 < n) : ∑ k, atomS (37 * n) (Kr n) k = 0 := by
+  rw [Fintype.sum_sum_type]
+  simp only [atomS, Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul,
+    Finset.sum_neg_distrib]
+  have h1 : ∑ j : Fin 16, cρ (j + 1) = lam := by
+    rw [← Sρ_sixteen, Sρ, sum_Icc_eq_sum_range', Fin.sum_univ_eq_sum_range (fun j => cρ (j + 1))]
+  rw [h1]
+  simp only [Kr, lam]
+  push_cast
+  field_simp
+  ring
+
+/-! ### Pointwise integrability and bounds for the atoms -/
+
+/-- `∫₀^{2π} |log ‖z - circleMap c ε φ‖| ≤ 2π |log ε| + 4π log (1 + ‖(z - c)/ε‖)`. -/
+lemma circle_point_abs_log_le (z c : ℂ) {ε : ℝ} (hε : 0 < ε) :
+    ∫ φ in (0 : ℝ)..2 * π, |Real.log ‖z - circleMap c ε φ‖| ≤
+      2 * π * |Real.log ε| + 4 * π * Real.log (1 + ‖(z - c) / ε‖) := by
+  have hε' : (ε : ℂ) ≠ 0 := by exact_mod_cast hε.ne'
+  have hfac : ∀ φ, ‖z - circleMap c ε φ‖ = ε * ‖circleMap 0 1 φ - (z - c) / ε‖ := by
+    intro φ
+    have e : z - circleMap c ε φ = -((ε : ℂ) * (circleMap 0 1 φ - (z - c) / ε)) := by
+      simp only [circleMap, Complex.ofReal_one, one_mul, zero_add]; field_simp; ring
+    rw [e, norm_neg, norm_mul, Complex.norm_real, Real.norm_eq_abs, abs_of_pos hε]
+  have hpt :
+    ∀ φ,
+      |Real.log ‖z - circleMap c ε φ‖| ≤
+        |Real.log ε| + |Real.log ‖circleMap 0 1 φ - (z - c) / ε‖| := by
+    intro φ
+    rw [hfac]
+    by_cases h0 : ‖circleMap 0 1 φ - (z - c) / ε‖ = 0
+    · rw [h0, mul_zero, Real.log_zero, abs_zero]; positivity
+    · rw [Real.log_mul hε.ne' h0]; exact abs_add_le _ _
+  have hi : IntervalIntegrable (fun φ => |Real.log ‖z - circleMap c ε φ‖|) volume 0 (2 * π) := by
+    have := (circle_log_integrable c z ε).abs
+    refine this.congr ?_
+    intro φ _; simp only [norm_sub_rev]
+  calc
+    ∫ φ in (0 : ℝ)..2 * π, |Real.log ‖z - circleMap c ε φ‖| ≤
+        ∫ φ in (0 : ℝ)..2 * π, (|Real.log ε| + |Real.log ‖circleMap 0 1 φ - (z - c) / ε‖|) :=
+      integral_mono_on (by positivity) hi
+        (intervalIntegrable_const.add (circle_log_integrable 0 _ 1).abs) (fun φ _ => hpt φ)
+    _ =
+        2 * π * |Real.log ε| +
+          ∫ φ in (0 : ℝ)..2 * π, |Real.log ‖circleMap 0 1 φ - (z - c) / ε‖| := by
+      rw [intervalIntegral.integral_add intervalIntegrable_const
+          (circle_log_integrable 0 _ 1).abs,
+        intervalIntegral.integral_const]
+      simp
+    _ ≤ _ := by linarith [abs_log_circle_integral_le ((z - c) / ε)]
+
+/-- Integrability of `φ ↦ log ‖z - atomγ k φ‖`. -/
+lemma atom_point_integrable {h : ℕ} (t : Fin h → ℝ) {ε : ℝ} (_hε : 0 < ε) (k : Idx h) (z : ℂ) :
+    IntervalIntegrable (fun φ => Real.log ‖z - atomγ t ε k φ‖) volume 0 (2 * π) := by
+  cases k with
+  | inl i =>
+    have := circle_log_integrable (t i) z ε
+    refine this.congr ?_
+    intro φ _; simp only [atomγ, norm_sub_rev]
+  | inr j => exact arc_log_integrable (rρ_pos j)
+
+lemma norm_root_le {ζ sq : ℂ} (hsq : sq ^ 2 = ζ ^ 2 - 1) : ‖sq‖ ≤ ‖ζ‖ + 1 := by
+  have h1 : ‖sq‖ ^ 2 = ‖ζ ^ 2 - 1‖ := by rw [← norm_pow, hsq]
+  have h2 : ‖ζ ^ 2 - 1‖ ≤ ‖ζ‖ ^ 2 + 1 := by
+    calc
+      ‖ζ ^ 2 - 1‖ ≤ ‖ζ ^ 2‖ + ‖(1 : ℂ)‖ := norm_sub_le _ _
+      _ = ‖ζ‖ ^ 2 + 1 := by rw [norm_pow, norm_one]
+  nlinarith [norm_nonneg sq, norm_nonneg ζ]
+
+/-- A uniform bound for `∫₀^{2π} |log ‖z - atomγ k φ‖| dφ` over `‖z‖ ≤ R`. -/
+lemma atom_point_abs_log_le {h : ℕ} (t : Fin h → ℝ) {ε : ℝ} (hε : 0 < ε) (k : Idx h) {R : ℝ}
+    (_hR : 0 ≤ R) :
+    ∃ C, ∀ z : ℂ, ‖z‖ ≤ R → ∫ φ in (0 : ℝ)..2 * π, |Real.log ‖z - atomγ t ε k φ‖| ≤ C := by
+  cases k with
+  | inl i =>
+    refine ⟨2 * π * |Real.log ε| + 4 * π * ((R + |t i|) / ε), fun z hz => ?_⟩
+    refine (circle_point_abs_log_le z (t i) hε).trans ?_
+    have h1 : Real.log (1 + ‖(z - t i) / ε‖) ≤ ‖(z - t i) / ε‖ := by
+      have := Real.log_le_sub_one_of_pos (show 0 < 1 + ‖(z - t i) / ε‖ by positivity)
+      linarith
+    have h2 : ‖(z - t i) / ε‖ ≤ (R + |t i|) / ε := by
+      rw [norm_div, Complex.norm_real, Real.norm_eq_abs, abs_of_pos hε]
+      apply div_le_div_of_nonneg_right _ hε.le
+      calc
+        ‖z - t i‖ ≤ ‖z‖ + ‖(t i : ℂ)‖ := norm_sub_le _ _
+        _ ≤ R + |t i| := by rw [Complex.norm_real, Real.norm_eq_abs]; linarith
+    nlinarith [Real.pi_pos]
+  | inr j =>
+    set m := mρ (j + 1)
+    set r := rρ (j + 1)
+    have hr := rρ_pos j
+    refine ⟨2 * π * |Real.log (r / 2)| + 8 * π * (2 * ((R + |m|) / r) + 1), fun z hz => ?_⟩
+    set sq : ℂ := (((z - m) / r) ^ 2 - 1) ^ ((2 : ℕ) : ℂ)⁻¹ with hsq_def
+    have hsq : sq ^ 2 = ((z - m) / r) ^ 2 - 1 := Complex.cpow_nat_inv_pow _ two_ne_zero
+    refine (arc_abs_log_integral_le hr hsq).trans ?_
+    have hζ : ‖(z - m) / r‖ ≤ (R + |m|) / r := by
+      rw [norm_div, Complex.norm_real, Real.norm_eq_abs, abs_of_pos hr]
+      apply div_le_div_of_nonneg_right _ hr.le
+      calc
+        ‖z - m‖ ≤ ‖z‖ + ‖(m : ℂ)‖ := norm_sub_le _ _
+        _ ≤ R + |m| := by rw [Complex.norm_real, Real.norm_eq_abs]; linarith
+    have hs := norm_root_le hsq
+    have hq1 : Real.log (1 + ‖(z - m) / r + sq‖) ≤ 2 * ((R + |m|) / r) + 1 := by
+      have := Real.log_le_sub_one_of_pos (show 0 < 1 + ‖(z - m) / r + sq‖ by positivity)
+      have := norm_add_le ((z - m) / r) sq
+      linarith
+    have hq2 : Real.log (1 + ‖(z - m) / r - sq‖) ≤ 2 * ((R + |m|) / r) + 1 := by
+      have := Real.log_le_sub_one_of_pos (show 0 < 1 + ‖(z - m) / r - sq‖ by positivity)
+      have := norm_sub_le ((z - m) / r) sq
+      linarith
+    nlinarith [Real.pi_pos]
+
+/-! ### The hypotheses of `energy_log_nonpos` -/
+
+lemma measurable_atom_kernel {h : ℕ} (t : Fin h → ℝ) (ε : ℝ) (k l : Idx h) :
+    Measurable fun p : ℝ × ℝ => Real.log ‖atomγ t ε k p.1 - atomγ t ε l p.2‖ := by
+  apply Real.measurable_log.comp
+  exact
+    (continuous_norm.comp
+        (((continuous_atomγ t ε k).comp continuous_fst).sub
+          ((continuous_atomγ t ε l).comp continuous_snd))).measurable
+
+/-- Torus integrability of the logarithmic kernel for each pair of atoms. -/
+lemma atom_pair_integrable {h : ℕ} (t : Fin h → ℝ) {ε : ℝ} (hε : 0 < ε) (k l : Idx h) :
+    Integrable (fun p : ℝ × ℝ => Real.log ‖atomγ t ε k p.1 - atomγ t ε l p.2‖)
+      (μcirc.prod μcirc) := by
+  have hR0 : 0 ≤ (∑ i, |t i|) + |ε| + 3 := by
+    have : 0 ≤ ∑ i, |t i| := Finset.sum_nonneg fun i _ => abs_nonneg _
+    linarith [abs_nonneg ε]
+  obtain ⟨C, hC⟩ := atom_point_abs_log_le t hε l hR0
+  have hmeas := (measurable_atom_kernel t ε k l).aestronglyMeasurable (μ := μcirc.prod μcirc)
+  rw [integrable_prod_iff hmeas]
+  constructor
+  · filter_upwards with θ
+    have := atom_point_integrable t hε l (atomγ t ε k θ)
+    exact (intervalIntegrable_iff_integrableOn_Ioc_of_le (by positivity)).mp this
+  · refine Integrable.mono' (integrable_const C) hmeas.norm.integral_prod_right' ?_
+    filter_upwards with θ
+    rw [Real.norm_eq_abs, abs_of_nonneg (integral_nonneg fun _ => norm_nonneg _)]
+    have := hC (atomγ t ε k θ) (norm_atomγ_le t ε k θ)
+    rw [intervalIntegral.integral_of_le (by positivity)] at this
+    simpa [Real.norm_eq_abs] using this
+
+/-- Point preimages under the atoms are countable. -/
+lemma atom_preimage_countable {h : ℕ} (t : Fin h → ℝ) {ε : ℝ} (hε : 0 < ε) (l : Idx h) (c : ℂ) :
+    {φ : ℝ | c = atomγ t ε l φ}.Countable := by
+  cases l with
+  | inl i =>
+    refine (circleMap_preimage_countable (t i) hε.ne' c).mono ?_
+    intro φ hφ
+    simp only [Set.mem_ofPred_eq, atomγ] at hφ
+    simp [hφ]
+  | inr j =>
+    refine (cos_preimage_countable ((c.re - mρ (j + 1)) / rρ (j + 1))).mono ?_
+    intro φ hφ
+    simp only [Set.mem_ofPred_eq, atomγ, arcCurve] at hφ
+    have hr := rρ_pos j
+    have : c.re = mρ (j + 1) + rρ (j + 1) * Real.cos φ := by rw [hφ, Complex.ofReal_re]
+    simp only [Set.mem_preimage, Set.mem_singleton_iff]
+    rw [this]; field_simp; ring
+
+/-- Almost every pair of parameters gives distinct points. -/
+lemma atom_pair_ae_ne {h : ℕ} (t : Fin h → ℝ) {ε : ℝ} (hε : 0 < ε) (k l : Idx h) :
+    ∀ᵐ p ∂(μcirc.prod μcirc), atomγ t ε k p.1 ≠ atomγ t ε l p.2 := by
+  rw [ae_iff]
+  simp only [not_not]
+  have hS : MeasurableSet {p : ℝ × ℝ | atomγ t ε k p.1 = atomγ t ε l p.2} :=
+    (isClosed_eq ((continuous_atomγ t ε k).comp continuous_fst)
+        ((continuous_atomγ t ε l).comp continuous_snd)).measurableSet
+  rw [Measure.prod_apply hS]
+  have : ∀ θ, μcirc (Prod.mk θ ⁻¹' {p : ℝ × ℝ | atomγ t ε k p.1 = atomγ t ε l p.2}) = 0 := by
+    intro θ
+    have hc := (atom_preimage_countable t hε l (atomγ t ε k θ)).measure_zero (volume : Measure ℝ)
+    refine le_antisymm ?_ bot_le
+    calc
+      μcirc (Prod.mk θ ⁻¹' {p : ℝ × ℝ | atomγ t ε k p.1 = atomγ t ε l p.2}) ≤
+          volume (Prod.mk θ ⁻¹' {p : ℝ × ℝ | atomγ t ε k p.1 = atomγ t ε l p.2}) :=
+        (Measure.le_iff'.mp Measure.restrict_le_self) _
+      _ = volume {φ : ℝ | atomγ t ε k θ = atomγ t ε l φ} := rfl
+      _ = 0 := hc
+  simp only [this, lintegral_zero]
+
+/-- Symmetry of the pair integrals of the logarithmic kernel. -/
+lemma pairInt_log_symm {h : ℕ} (t : Fin h → ℝ) {ε : ℝ} (hε : 0 < ε) (k l : Idx h) :
+    pairInt (atomγ t ε) (fun z w => Real.log ‖z - w‖) k l =
+      pairInt (atomγ t ε) (fun z w => Real.log ‖z - w‖) l k := by
+  rw [pairInt_eq_prod _ _ _ _ (atom_pair_integrable t hε k l),
+    pairInt_eq_prod _ _ _ _ (atom_pair_integrable t hε l k), ← integral_prod_swap]
+  simp only [Prod.fst_swap, Prod.snd_swap, norm_sub_rev]
+
+/-- **Lemma 6.2 for the configuration atoms.** -/
+theorem atom_energy_nonpos (n : ℕ) (hn : 0 < n) (t : Fin (37 * n) → ℝ) {ε : ℝ} (hε : 0 < ε) :
+    energy (atomS (37 * n) (Kr n)) (atomγ t ε) (fun z w => Real.log ‖z - w‖) ≤ 0 :=
+  energy_log_nonpos (continuous_atomγ t ε) (norm_atomγ_le t ε) (atomS_mass n hn)
+    (atom_pair_integrable t hε) (atom_pair_ae_ne t hε)
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/EnergyBlocks.lean b/LeanPool/Zeta5Irrational/EnergyBlocks.lean
new file mode 100644
index 0000000000..0541757aa3
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/EnergyBlocks.lean
@@ -0,0 +1,239 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.EnergyAtoms
+public import LeanPool.Zeta5Irrational.Potential
+import LeanPool.Zeta5Irrational.ArcsinePotential
+import LeanPool.Zeta5Irrational.CircleAtoms
+import Mathlib.Tactic.Polynomial.Basic
+import Mathlib.Tactic.ReduceModChar
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.Positivity.Basic
+import Mathlib.Tactic.Ring.Basic
+import Mathlib.Tactic.Ring.RingNF
+
+/-! # The four blocks of the configuration energy
+
+With `P k l = pairInt (atomγ t ε) (log ‖· - ·‖) k l`:
+
+* (A) `P (inl i) (inl i) = (2π)² log ε`;
+* (B) `(2π)² log |tᵢ - tᵢ'| ≤ P (inl i) (inl i')` for `tᵢ ≠ tᵢ'`;
+* (C) `P (inl i) (inr j) ≤ (2π)² (Uω aⱼ bⱼ tᵢ + errⱼ(tᵢ))` (circle regularisation error);
+* (D) `(2π)² log ((bⱼ' - aⱼ')/4) ≤ P (inr j) (inr j')`.
+-/
+
+@[expose] public section
+
+open MeasureTheory Set Real intervalIntegral Filter Topology
+
+namespace Zeta5Irrational
+
+/-- The pair integrals of the logarithmic kernel for the configuration atoms. -/
+noncomputable def Plog {h : ℕ} (t : Fin h → ℝ) (ε : ℝ) (k l : Idx h) : ℝ :=
+  pairInt (atomγ t ε) (fun z w => Real.log ‖z - w‖) k l
+
+lemma log_le_log_add_posLog {ε x : ℝ} (hε : 0 < ε) (hx : 0 < x) :
+    Real.log x ≤ Real.log ε + log⁺ (ε⁻¹ * x) := by
+  have h1 : Real.log (ε⁻¹ * x) ≤ log⁺ (ε⁻¹ * x) := le_max_right _ _
+  rw [Real.log_mul (inv_ne_zero hε.ne') hx.ne', Real.log_inv] at h1
+  linarith
+
+/-- The inner integrals are interval integrable in the outer variable. -/
+lemma inner_intervalIntegrable {h : ℕ} (t : Fin h → ℝ) {ε : ℝ} (hε : 0 < ε) (k l : Idx h) :
+    IntervalIntegrable (fun θ => ∫ φ in (0 : ℝ)..2 * π, Real.log ‖atomγ t ε k θ - atomγ t ε l φ‖)
+      volume 0 (2 * π) := by
+  have := (atom_pair_integrable t hε k l).integral_prod_left
+  rw [intervalIntegrable_iff_integrableOn_Ioc_of_le (by positivity)]
+  simp_rw [intervalIntegral.integral_of_le (by positivity : (0 : ℝ) ≤ 2 * π)]
+  exact this
+
+lemma norm_circleMap_sub_center' (c : ℝ) {ε : ℝ} (hε : 0 < ε) (θ : ℝ) :
+    ‖(c : ℂ) - circleMap c ε θ‖ = ε := by
+  rw [norm_sub_rev, circleMap_sub_center, norm_circleMap_zero, abs_of_pos hε]
+
+lemma circle_log_integral' (c a : ℂ) {R : ℝ} (hR : R ≠ 0) :
+    ∫ θ in (0 : ℝ)..2 * π, Real.log ‖a - circleMap c R θ‖ =
+      2 * π * (Real.log R + log⁺ (R⁻¹ * ‖c - a‖)) := by
+  rw [← circle_log_integral c a hR]
+  apply integral_congr
+  intro θ _
+  simp only
+  rw [norm_sub_rev]
+
+/-- (A): the self-energy of a circle. -/
+lemma block_A {h : ℕ} (t : Fin h → ℝ) {ε : ℝ} (hε : 0 < ε) (i : Fin h) :
+    Plog t ε (Sum.inl i) (Sum.inl i) = (2 * π) ^ 2 * Real.log ε := by
+  unfold Plog pairInt
+  have hin :
+    ∀ θ,
+      (∫ φ in (0 : ℝ)..2 * π, Real.log ‖atomγ t ε (Sum.inl i) θ - atomγ t ε (Sum.inl i) φ‖) =
+        2 * π * Real.log ε := by
+    intro θ
+    simp only [atomγ]
+    rw [circle_log_integral' _ _ hε.ne', norm_circleMap_sub_center' _ hε, inv_mul_cancel₀ hε.ne',
+      (posLog_eq_zero_iff _).mpr (by simp), add_zero]
+  simp_rw [hin]
+  rw [intervalIntegral.integral_const]
+  simp only [sub_zero, smul_eq_mul]
+  ring
+
+/-- (B): the mutual energy of two circles dominates `log |tᵢ - tᵢ'|`. -/
+lemma block_B {h : ℕ} (t : Fin h → ℝ) {ε : ℝ} (hε : 0 < ε) (i i' : Fin h) (hne : t i ≠ t i') :
+    (2 * π) ^ 2 * Real.log |t i - t i'| ≤ Plog t ε (Sum.inl i) (Sum.inl i') := by
+  unfold Plog pairInt
+  have hin :
+    ∀ θ,
+      circleMap (t i) ε θ ≠ (t i' : ℂ) →
+        2 * π * Real.log ‖circleMap (t i) ε θ - t i'‖ ≤
+          ∫ φ in (0 : ℝ)..2 * π, Real.log ‖atomγ t ε (Sum.inl i) θ - atomγ t ε (Sum.inl i') φ‖ := by
+    intro θ hθ
+    simp only [atomγ]
+    rw [circle_log_integral' _ _ hε.ne', norm_sub_rev (t i' : ℂ)]
+    have := log_le_log_add_posLog hε (norm_pos_iff.mpr (sub_ne_zero.mpr hθ))
+    nlinarith [Real.pi_pos]
+  have h1 :
+    (∫ θ in (0 : ℝ)..2 * π, 2 * π * Real.log ‖circleMap (t i) ε θ - t i'‖) ≤
+      ∫ θ in (0 : ℝ)..2 * π,
+        ∫ φ in (0 : ℝ)..2 * π, Real.log ‖atomγ t ε (Sum.inl i) θ - atomγ t ε (Sum.inl i') φ‖ := by
+    refine
+      intervalIntegral.integral_mono_ae (by positivity)
+        ((circle_log_integrable _ _ _).const_mul _) (inner_intervalIntegrable t hε _ _) ?_
+    have hc :=
+      (circleMap_preimage_countable (t i) hε.ne' (t i')).measure_zero (volume : Measure ℝ)
+    filter_upwards [compl_mem_ae_iff.mpr hc] with θ hθ
+    exact hin θ hθ
+  have h2 :
+    (∫ θ in (0 : ℝ)..2 * π, 2 * π * Real.log ‖circleMap (t i) ε θ - t i'‖) =
+      2 * π * (2 * π * (Real.log ε + log⁺ (ε⁻¹ * ‖(t i : ℂ) - t i'‖))) :=
+    by rw [intervalIntegral.integral_const_mul, circle_log_integral _ _ hε.ne']
+  have h3 : Real.log |t i - t i'| ≤ Real.log ε + log⁺ (ε⁻¹ * ‖(t i : ℂ) - t i'‖) := by
+    rw [← Complex.ofReal_sub, Complex.norm_real, Real.norm_eq_abs]
+    exact log_le_log_add_posLog hε (abs_pos.mpr (sub_ne_zero.mpr hne))
+  nlinarith [Real.pi_pos]
+
+/-- The regularisation error of the arcsine potential at a real point `t`. -/
+noncomputable def errj (ε t m r : ℝ) : ℝ :=
+  ε / r + Real.sqrt (ε / r * ((ε + 2 * |t - m|) / r))
+
+lemma errj_nonneg {ε t m r : ℝ} (hε : 0 < ε) (hr : 0 < r) : 0 ≤ errj ε t m r := by
+  unfold errj;
+  positivity
+
+/-- (C): circle–arcsine energy. -/
+lemma block_C {h : ℕ} (t : Fin h → ℝ) {ε : ℝ} (hε : 0 < ε) (i : Fin h) (j : Fin 16) :
+    Plog t ε (Sum.inl i) (Sum.inr j) ≤
+      (2 * π) ^ 2 *
+        (Uω (aρ (j + 1)) (bρ (j + 1)) (t i) + errj ε (t i) (mρ (j + 1)) (rρ (j + 1))) := by
+  unfold Plog pairInt
+  have hr := rρ_pos j
+  have hab : aρ (j + 1) < bρ (j + 1) := aρ_lt_bρ (j + 1) (by simp only [Finset.mem_Icc]; omega)
+  have hm' : mρ (j + 1) = (aρ (j + 1) + bρ (j + 1)) / 2 := rfl
+  have hr' : rρ (j + 1) = (bρ (j + 1) - aρ (j + 1)) / 2 := rfl
+  set sq' : ℂ := ((((t i : ℂ) - mρ (j + 1)) / rρ (j + 1)) ^ 2 - 1) ^ ((2 : ℕ) : ℂ)⁻¹ with hsq'_def
+  have hsq' : sq' ^ 2 = (((t i : ℂ) - mρ (j + 1)) / rρ (j + 1)) ^ 2 - 1 :=
+    Complex.cpow_nat_inv_pow _ two_ne_zero
+  have hU :
+    2 * π *
+        (Real.log (rρ (j + 1) / 2) + log⁺ ‖((t i : ℂ) - mρ (j + 1)) / rρ (j + 1) + sq'‖ +
+          log⁺ ‖((t i : ℂ) - mρ (j + 1)) / rρ (j + 1) - sq'‖) =
+      2 * π * Uω (aρ (j + 1)) (bρ (j + 1)) (t i) :=
+    by rw [← arc_log_integral hr hsq', hm', hr', arc_potential_real hab (t i)]
+  have hin :
+    ∀ θ,
+      (∫ φ in (0 : ℝ)..2 * π, Real.log ‖atomγ t ε (Sum.inl i) θ - atomγ t ε (Sum.inr j) φ‖) ≤
+        2 * π * (Uω (aρ (j + 1)) (bρ (j + 1)) (t i) + errj ε (t i) (mρ (j + 1)) (rρ (j + 1))) := by
+    intro θ
+    simp only [atomγ]
+    set z := circleMap (t i) ε θ with hz
+    set sq : ℂ := (((z - mρ (j + 1)) / rρ (j + 1)) ^ 2 - 1) ^ ((2 : ℕ) : ℂ)⁻¹ with hsq_def
+    have hsq : sq ^ 2 = ((z - mρ (j + 1)) / rρ (j + 1)) ^ 2 - 1 :=
+      Complex.cpow_nat_inv_pow _ two_ne_zero
+    rw [arc_log_integral hr hsq]
+    have hG := Gpot_sub_le hsq hsq'
+    have hzc : ‖z - (t i)‖ = ε := by
+      rw [hz, circleMap_sub_center, norm_circleMap_zero, abs_of_pos hε]
+    have hd :
+      ‖(z - mρ (j + 1)) / rρ (j + 1) - ((t i : ℂ) - mρ (j + 1)) / rρ (j + 1)‖ = ε / rρ (j + 1) := by
+      rw [← sub_div, show z - mρ (j + 1) - ((t i : ℂ) - mρ (j + 1)) = z - (t i) by ring, norm_div,
+        hzc, Complex.norm_real, Real.norm_eq_abs, abs_of_pos hr]
+    have hs :
+      ‖(z - mρ (j + 1)) / rρ (j + 1) + ((t i : ℂ) - mρ (j + 1)) / rρ (j + 1)‖ ≤
+        (ε + 2 * |(t i) - mρ (j + 1)|) / rρ (j + 1) := by
+      rw [← add_div, norm_div, Complex.norm_real, Real.norm_eq_abs, abs_of_pos hr]
+      apply div_le_div_of_nonneg_right _ hr.le
+      calc
+        ‖z - mρ (j + 1) + ((t i : ℂ) - mρ (j + 1))‖ =
+            ‖(z - t i) + 2 * ((t i : ℂ) - mρ (j + 1))‖ :=
+          by ring_nf
+        _ ≤ ‖z - (t i)‖ + ‖(2 : ℂ) * ((t i : ℂ) - mρ (j + 1))‖ := (norm_add_le _ _)
+        _ = ε + 2 * |(t i) - mρ (j + 1)| := by
+          rw [hzc, norm_mul, ← Complex.ofReal_sub, Complex.norm_real, Real.norm_eq_abs]
+          simp
+    have hsqrt :
+      Real.sqrt
+          (‖(z - mρ (j + 1)) / rρ (j + 1) - ((t i : ℂ) - mρ (j + 1)) / rρ (j + 1)‖ *
+            ‖(z - mρ (j + 1)) / rρ (j + 1) + ((t i : ℂ) - mρ (j + 1)) / rρ (j + 1)‖) ≤
+        Real.sqrt (ε / rρ (j + 1) * ((ε + 2 * |(t i) - mρ (j + 1)|) / rρ (j + 1))) := by
+      apply Real.sqrt_le_sqrt
+      rw [hd]
+      exact mul_le_mul_of_nonneg_left hs (by positivity)
+    rw [hd] at hG hsqrt
+    unfold errj
+    nlinarith [Real.pi_pos]
+  calc
+    (∫ θ in (0 : ℝ)..2 * π,
+          ∫ φ in (0 : ℝ)..2 * π, Real.log ‖atomγ t ε (Sum.inl i) θ - atomγ t ε (Sum.inr j) φ‖) ≤
+        ∫ θ in (0 : ℝ)..2 * π,
+          2 * π * (Uω (aρ (j + 1)) (bρ (j + 1)) (t i) + errj ε (t i) (mρ (j + 1)) (rρ (j + 1))) :=
+      integral_mono_on (by positivity) (inner_intervalIntegrable t hε _ _)
+        intervalIntegrable_const (fun θ _ => hin θ)
+    _ = _ := by rw [intervalIntegral.integral_const]; simp only [sub_zero, smul_eq_mul]; ring
+
+/-- `Uω a b t ≥ log ((b - a)/4)` for all `t`. -/
+lemma log_le_Uω {a b : ℝ} (hab : a < b) (t : ℝ) : Real.log ((b - a) / 4) ≤ Uω a b t := by
+  by_cases h : a ≤ t ∧ t ≤ b
+  · rw [Uω_of_mem h.1 h.2]
+  · unfold Uω
+    rw [ite_eq_right h]
+    have hs := Real.sqrt_nonneg ((t - a) * (t - b))
+    have h1 : (b - a) / 2 ≤ |t - (a + b) / 2| := by
+      rw [le_abs]
+      rcases not_and_or.mp h with h | h
+      · right; linarith [not_le.mp h]
+      · left; linarith [not_le.mp h]
+    apply Real.log_le_log (by linarith)
+    linarith
+
+/-- (D): arcsine–arcsine energy is at least `log ((bⱼ' - aⱼ')/4)`. -/
+lemma block_D {h : ℕ} (t : Fin h → ℝ) {ε : ℝ} (hε : 0 < ε) (j j' : Fin 16) :
+    (2 * π) ^ 2 * Real.log ((bρ (j' + 1) - aρ (j' + 1)) / 4) ≤
+      Plog t ε (Sum.inr j) (Sum.inr j') := by
+  unfold Plog pairInt
+  have hab : aρ (j' + 1) < bρ (j' + 1) := aρ_lt_bρ (j' + 1) (by simp only [Finset.mem_Icc]; omega)
+  have hin :
+    ∀ θ,
+      2 * π * Real.log ((bρ (j' + 1) - aρ (j' + 1)) / 4) ≤
+        ∫ φ in (0 : ℝ)..2 * π, Real.log ‖atomγ t ε (Sum.inr j) θ - atomγ t ε (Sum.inr j') φ‖ := by
+    intro θ
+    change
+      2 * π * Real.log ((bρ (j' + 1) - aρ (j' + 1)) / 4) ≤
+        ∫ φ in (0 : ℝ)..2 * π,
+          Real.log
+            ‖((mρ (j + 1) + rρ (j + 1) * Real.cos θ : ℝ) : ℂ) -
+                arcCurve ((aρ (j' + 1) + bρ (j' + 1)) / 2) ((bρ (j' + 1) - aρ (j' + 1)) / 2) φ‖
+    rw [arc_potential_real hab]
+    exact mul_le_mul_of_nonneg_left (log_le_Uω hab _) (by positivity)
+  calc
+    (2 * π) ^ 2 * Real.log ((bρ (j' + 1) - aρ (j' + 1)) / 4) =
+        ∫ θ in (0 : ℝ)..2 * π, 2 * π * Real.log ((bρ (j' + 1) - aρ (j' + 1)) / 4) :=
+      by rw [intervalIntegral.integral_const]; simp only [sub_zero, smul_eq_mul]; ring
+    _ ≤ _ :=
+      integral_mono_on (by positivity) intervalIntegrable_const
+        (inner_intervalIntegrable t hε _ _) (fun θ _ => hin θ)
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/EnergyBound.lean b/LeanPool/Zeta5Irrational/EnergyBound.lean
new file mode 100644
index 0000000000..2d368ea630
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/EnergyBound.lean
@@ -0,0 +1,372 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+import LeanPool.Zeta5Irrational.EnergyFinal
+import LeanPool.Zeta5Irrational.Moments
+public import LeanPool.Zeta5Irrational.Constants
+import LeanPool.Zeta5Irrational.PartialFractions
+import Mathlib.Tactic.ReduceModChar
+import Mathlib.Tactic.Polynomial.Basic
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Algebra.Ring.IsFormallyReal
+import Mathlib.Tactic.ENatToNat
+public import LeanPool.Zeta5Irrational.Energy
+public import LeanPool.Zeta5Irrational.Measure
+import Mathlib.Tactic.FieldSimp
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+import Mathlib.Tactic.Positivity.Basic
+import Mathlib.Tactic.Ring.Basic
+
+/-! # Variants of the energy bound (6.14) and configuration inequality (6.9)
+
+* `energy_ineq`: a variant of (6.9), with `5h + 6h log K` replacing
+  `(120 + √2)h + 2h log K`, for positive pairwise distinct configurations. It follows from
+  `energy_ineq_final` (Lemma 6.1, Lemma 6.2, circle regularisation).
+* `log_Δ_le'`: a variant of (6.14), with `22 h log K + 50 h` replacing
+  `18 h log K + 160 h`.
+* everything else: the pointwise bound for the integrand of (6.10) and its integration.
+-/
+
+@[expose] public section
+
+open Filter Topology Set MeasureTheory Finset Polynomial
+
+namespace Zeta5Irrational
+
+/-- A variant of **(6.9)** with lower-order terms `5h + 6h log K` instead of
+`(120 + √2)h + 2h log K`, as proved in `energy_ineq_final`. -/
+theorem energy_ineq (n : ℕ) (hn : 0 < n) (t : Fin (37 * n) → ℝ) (ht : ∀ i, 0 < t i)
+    (hinj : Function.Injective t) :
+    (∑ i, ∑ j ∈ Finset.Ioi i, 2 * Real.log |t j - t i|) - Kr n * ∑ i, Vfield (t i) +
+        ∑ i, Real.sqrt (t i) ≤
+      (lam * M0 - Irho) * Kr n ^ 2 + 5 * (37 * n) + 6 * (37 * n) * Real.log (Kr n) :=
+  energy_ineq_final n hn t ht hinj
+
+lemma aeval_D_eq (m : ℕ) (s : ℝ) : aeval s (D m) = ∏ j ∈ Icc 1 m, (s + (j : ℝ) ^ 2) := by
+  simp only [D, map_prod, map_add, aeval_X, aeval_C, eq_ratCast, Rat.cast_pow, Rat.cast_natCast]
+
+lemma sum_Ioi_card_mul_two (h : ℕ) : (∑ i : Fin h, (Finset.Ioi i).card) * 2 = h * (h - 1) := by
+  simp only [Fin.card_Ioi]
+  rw [Fin.sum_univ_eq_sum_range (fun i => h - 1 - i) h, Finset.sum_range_reflect (fun i => i) h,
+    Finset.sum_range_id_mul_two]
+
+lemma sum_Ioi_const (h : ℕ) (c : ℝ) :
+    ∑ i : Fin h, ∑ _j ∈ Finset.Ioi i, c = (h : ℝ) * ((h : ℝ) - 1) / 2 * c := by
+  simp only [Finset.sum_const, nsmul_eq_mul, ← Finset.sum_mul]
+  have := sum_Ioi_card_mul_two h
+  rcases Nat.eq_zero_or_pos h with rfl | hh
+  · simp
+  have hcast : ((∑ i : Fin h, (Finset.Ioi i).card : ℕ) : ℝ) = (h : ℝ) * ((h : ℝ) - 1) / 2 := by
+    have h2 : (((∑ i : Fin h, (Finset.Ioi i).card) * 2 : ℕ) : ℝ) = ((h * (h - 1) : ℕ) : ℝ) := by
+      rw [this]
+    push_cast [Nat.cast_sub (by omega : 1 ≤ h)] at h2 ⊢
+    linarith
+  rw [← hcast]
+  push_cast
+  ring
+
+/-- `log D_m(y²) = 2m log K + ∑_{j<m} log(t + ((j+1)/K)²)` with `t = (y/K)²`. -/
+lemma log_aeval_D (m : ℕ) {K y : ℝ} (hK : 0 < K) (hy : 0 < y) :
+    Real.log (aeval (y ^ 2) (D m)) =
+      2 * m * Real.log K + ∑ j ∈ range m, Real.log ((y / K) ^ 2 + (((j : ℝ) + 1) / K) ^ 2) := by
+  rw [aeval_D_eq, Real.log_prod (fun j _ => by positivity), sum_Icc_eq_sum_range']
+  have :
+    ∀ j : ℕ,
+      Real.log (y ^ 2 + ((j + 1 : ℕ) : ℝ) ^ 2) =
+        2 * Real.log K + Real.log ((y / K) ^ 2 + (((j : ℝ) + 1) / K) ^ 2) := by
+    intro j
+    have e : y ^ 2 + ((j + 1 : ℕ) : ℝ) ^ 2 = K ^ 2 * ((y / K) ^ 2 + (((j : ℝ) + 1) / K) ^ 2) := by
+      push_cast; field_simp
+    rw [e, Real.log_mul (by positivity) (by positivity), Real.log_pow]
+    push_cast; ring
+  simp_rw [this]
+  rw [Finset.sum_add_distrib, Finset.sum_const, Finset.card_range, nsmul_eq_mul]
+  ring
+
+/-- The constant `A_n` in the pointwise bound. -/
+noncomputable def Aconst (n : ℕ) : ℝ :=
+  2 * (37 * n) * (37 * n + 6 * (3 * n) - 40 * n : ℝ) * Real.log (Kr n) +
+        (lam * M0 - Irho) * Kr n ^ 2 +
+      16 * (37 * n) * Real.log (Kr n) +
+    (12 * Real.log 2 + 17) * (37 * n)
+
+/-- The one-dimensional factor `(1 + 2πy)^5 e^{-y/K}`. -/
+noncomputable def gK (n : ℕ) (y : ℝ) : ℝ :=
+  (1 + 2 * Real.pi * y) ^ 5 * Real.exp (-(1 / Kr n * y))
+
+/-- **Pointwise bound** for the integrand of (6.10), for positive pairwise distinct `y`. -/
+theorem integrand_le (n : ℕ) (hn : 0 < n) (y : Fin (37 * n) → ℝ) (hy : ∀ i, 0 < y i)
+    (hinj : Function.Injective y) :
+    (Matrix.vandermonde fun j => y j ^ 2).det ^ 2 * ∏ j, ν n (y j) ≤
+      Real.exp (Aconst n) * ∏ i, gK n (y i) := by
+  set K : ℝ := Kr n with hKdef
+  have hK : 0 < K := by simp only [hKdef, Kr]; positivity
+  have h1n : (1 : ℝ) ≤ n := by exact_mod_cast hn
+  have hK1 : 1 ≤ K := by
+    simp only [hKdef, Kr]
+    linarith
+  set t : Fin (37 * n) → ℝ := fun i => (y i / K) ^ 2 with htdef
+  have ht : ∀ i, 0 < t i := fun i => by simp only [htdef]; have := hy i; positivity
+  have htinj : Function.Injective t := by
+    intro i j hij
+    simp only [htdef] at hij
+    have := (pow_left_inj₀ (div_pos (hy i) hK).le (div_pos (hy j) hK).le two_ne_zero).mp hij
+    exact
+      hinj
+        ((div_left_inj' hK.ne').mp this)
+          -- positivity of both sides
+  have hdet : (Matrix.vandermonde fun j => y j ^ 2).det ≠ 0 := by
+    rw [Matrix.det_vandermonde_ne_zero_iff]
+    intro i j hij
+    exact hinj ((pow_left_inj₀ (hy i).le (hy j).le two_ne_zero).mp hij)
+  have hν : ∀ j, 0 < ν n (y j) := fun j => ν_pos n (hy j)
+  have hL : 0 < (Matrix.vandermonde fun j => y j ^ 2).det ^ 2 * ∏ j, ν n (y j) :=
+    mul_pos (by positivity) (Finset.prod_pos fun j _ => hν j)
+  have hg : ∀ i, 0 < gK n (y i) := fun i => by unfold gK; have := hy i; positivity
+  have hR : 0 < Real.exp (Aconst n) * ∏ i, gK n (y i) :=
+    mul_pos (Real.exp_pos _) (Finset.prod_pos fun i _ => hg i)
+  rw [← Real.log_le_log_iff hL hR]
+    -- log of the left-hand side
+  have hne : ∀ i j : Fin (37 * n), j ∈ Finset.Ioi i → y j ^ 2 - y i ^ 2 ≠ 0 := by
+    intro i j hj h0
+    have : y j ^ 2 = y i ^ 2 := by linarith
+    exact
+      (Finset.mem_Ioi.mp hj).ne' (hinj ((pow_left_inj₀ (hy j).le (hy i).le two_ne_zero).mp this))
+  have hdet2 :
+    Real.log ((Matrix.vandermonde fun j => y j ^ 2).det ^ 2) =
+      ∑ i, ∑ j ∈ Finset.Ioi i, (4 * Real.log K + 2 * Real.log |t j - t i|) := by
+    rw [Matrix.det_vandermonde]
+    simp only [← Finset.prod_pow]
+    rw [Real.log_prod
+        (fun i _ =>
+          (Finset.prod_pos fun j hj =>
+              lt_of_le_of_ne (sq_nonneg _) (Ne.symm (pow_ne_zero 2 (hne i j hj)))).ne')]
+    refine Finset.sum_congr rfl fun i _ => ?_
+    rw [Real.log_prod (fun j hj => pow_ne_zero 2 (hne i j hj))]
+    refine Finset.sum_congr rfl fun j hj => ?_
+    have e : y j ^ 2 - y i ^ 2 = K ^ 2 * (t j - t i) := by simp only [htdef]; field_simp
+    have hne' : t j - t i ≠ 0 := by intro h0; apply hne i j hj; rw [e, h0, mul_zero]
+    rw [Real.log_pow, e, ← Real.log_abs, abs_mul,
+      Real.log_mul (by positivity) (abs_pos.mpr hne').ne',
+      abs_of_pos (by positivity : (0 : ℝ) < K ^ 2), Real.log_pow]
+    push_cast;
+    ring
+      -- log of each weight
+  have hν_le :
+    ∀ i,
+      Real.log (ν n (y i)) ≤
+        (12 * (3 * n) - 2 * (40 * n)) * Real.log K - K * Vfield (t i) + 12 * Real.log (2 * K) +
+            12 +
+          5 * Real.log (1 + 2 * Real.pi * y i) := by
+    intro i
+    have hyi := hy i
+    have hDN := aeval_D_pos (3 * n) (t := y i ^ 2) (by positivity)
+    have hDK := aeval_D_pos (40 * n) (t := y i ^ 2) (by positivity)
+    have hw := w_pos hyi
+    unfold ν
+    rw [Real.log_mul (by positivity) hw.ne', Real.log_div (by positivity) hDK.ne', Real.log_pow,
+      log_aeval_D _ hK hyi, log_aeval_D _ hK hyi]
+      -- comparison lemmas
+    have hN : ((3 * n : ℕ) : ℝ) / K = 3 / 40 := by simp only [hKdef, Kr]; push_cast; field_simp
+    have hKK : ((40 * n : ℕ) : ℝ) / K = 1 := by simp only [hKdef, Kr]; push_cast; field_simp
+    have h1 :=
+      sum_log_le_integral (ht i) hK1 (m := 3 * n) (by omega)
+        (by simp only [hKdef, Kr]; push_cast; linarith [(by exact_mod_cast hn : (1 : ℝ) ≤ n)])
+    have h2 := integral_le_sum_log (ht i) hK (40 * n)
+    rw [hN] at h1
+    rw [hKK] at h2
+          -- the weight
+    have hw' : Real.log (w (y i)) ≤ 5 * Real.log (1 + 2 * Real.pi * y i) - 2 * Real.pi * y i := by
+      calc
+        Real.log (w (y i)) ≤
+            Real.log
+              (1 / Real.pi * (1 + 2 * Real.pi * y i) ^ 5 * Real.exp (-(2 * Real.pi * y i))) :=
+          Real.log_le_log hw (w_le' hyi)
+        _ = Real.log (1 / Real.pi) + 5 * Real.log (1 + 2 * Real.pi * y i) - 2 * Real.pi * y i := by
+          rw [Real.log_mul (by positivity) (by positivity),
+            Real.log_mul (by positivity) (by positivity), Real.log_exp, Real.log_pow]
+          push_cast; ring
+        _ ≤ 5 * Real.log (1 + 2 * Real.pi * y i) - 2 * Real.pi * y i := by
+          have : Real.log (1 / Real.pi) ≤ 0 := by
+            rw [one_div, Real.log_inv];
+            linarith [Real.log_nonneg (by linarith [Real.pi_gt_three] : (1 : ℝ) ≤ Real.pi)]
+          linarith
+            -- `√t = y/K`
+    have hsqrt : Real.sqrt (t i) = y i / K := by
+      simp only [htdef];
+      exact Real.sqrt_sq (by positivity)
+    have hV :
+      K * Vfield (t i) =
+        2 * Real.pi * y i + K * (∫ u in (0 : ℝ)..1, Real.log (t i + u ^ 2)) -
+          6 * (K * ∫ u in (0 : ℝ)..(3 / 40), Real.log (t i + u ^ 2)) :=
+      by unfold Vfield; rw [hsqrt]; field_simp
+    push_cast at h1 h2 ⊢
+    rw [hV]
+    nlinarith [h1, h2, hw']
+      -- assemble
+  rw [Real.log_mul (by positivity) (Finset.prod_pos fun j _ => hν j).ne', hdet2,
+    Real.log_prod (fun j _ => (hν j).ne'),
+    Real.log_mul (Real.exp_pos _).ne' (Finset.prod_pos fun i _ => hg i).ne', Real.log_exp,
+    Real.log_prod (fun i _ => (hg i).ne')]
+  have hE := energy_ineq n hn t ht htinj
+  have hsqrt : ∀ i, Real.sqrt (t i) = y i / K := fun i => by
+    simp only [htdef];
+    exact Real.sqrt_sq (div_pos (hy i) hK).le
+  simp_rw [hsqrt] at hE
+  have hgi : ∀ i, Real.log (gK n (y i)) = 5 * Real.log (1 + 2 * Real.pi * y i) - y i / K := by
+    intro i
+    unfold gK
+    rw [Real.log_mul (by have := hy i; positivity) (Real.exp_pos _).ne', Real.log_exp,
+      Real.log_pow]
+    push_cast; ring
+  simp_rw [hgi]
+  simp only [Finset.sum_add_distrib, sum_Ioi_const]
+  have hsum_le := Finset.sum_le_sum fun i (_ : i ∈ Finset.univ) => hν_le i
+  simp only [Finset.sum_add_distrib, Finset.sum_sub_distrib, Finset.sum_const, Finset.card_univ,
+    Fintype.card_fin, nsmul_eq_mul, ← Finset.mul_sum] at hsum_le
+  have hlog2K : Real.log (2 * K) = Real.log 2 + Real.log K := Real.log_mul two_ne_zero hK.ne'
+  rw [hlog2K] at hsum_le
+  unfold Aconst
+  simp only [Finset.sum_sub_distrib, ← Finset.mul_sum]
+  simp only [← Finset.mul_sum] at hE
+  rw [← hKdef] at *
+  push_cast at *
+  linarith [hE, hsum_le]
+
+lemma integrableOn_gK (n : ℕ) (hn : 0 < n) : IntegrableOn (gK n) (Ioi 0) := by
+  have hK : 0 < Kr n := by simp only [Kr]; positivity
+  have hr : 0 < 1 / Kr n := by positivity
+  have hg :
+    IntegrableOn
+      (fun y : ℝ =>
+        (2 * Real.pi) ^ 5 * 32 * (y ^ 0 * Real.exp (-(1 / Kr n * y))) +
+          (2 * Real.pi) ^ 5 * 32 * (y ^ 5 * Real.exp (-(1 / Kr n * y))))
+      (Ioi 0) :=
+    ((integrableOn_pow_mul_exp_neg_mul 0 hr).const_mul _).add
+      ((integrableOn_pow_mul_exp_neg_mul 5 hr).const_mul _)
+  have hcont : Continuous (gK n) := by unfold gK; fun_prop
+  refine hg.mono' hcont.aestronglyMeasurable ?_
+  refine (ae_restrict_iff' measurableSet_Ioi).mpr (Filter.Eventually.of_forall fun y hy => ?_)
+  have hy : 0 < y := hy
+  rw [Real.norm_of_nonneg (by unfold gK; positivity)]
+  unfold gK
+  have h1 : (1 + 2 * Real.pi * y) ^ 5 ≤ (2 * Real.pi) ^ 5 * (1 + y) ^ 5 := by
+    rw [← mul_pow]
+    exact pow_le_pow_left₀ (by positivity) (by nlinarith [Real.pi_gt_three]) 5
+  have h2 : (1 + y) ^ 5 ≤ 32 * (1 + y ^ 5) := by
+    rcases le_or_gt y 1 with h | h
+    · have : (1 + y) ^ 5 ≤ 2 ^ 5 := pow_le_pow_left₀ (by linarith) (by linarith) 5
+      have : 0 ≤ y ^ 5 := by positivity
+      nlinarith
+    · have : (1 + y) ^ 5 ≤ (2 * y) ^ 5 := pow_le_pow_left₀ (by linarith) (by linarith) 5
+      have : 0 ≤ y ^ 5 := by positivity
+      nlinarith
+  have he : 0 ≤ Real.exp (-(1 / Kr n * y)) := (Real.exp_pos _).le
+  calc
+    (1 + 2 * Real.pi * y) ^ 5 * Real.exp (-(1 / Kr n * y)) ≤
+        (2 * Real.pi) ^ 5 * (32 * (1 + y ^ 5)) * Real.exp (-(1 / Kr n * y)) := by
+      gcongr
+      exact h1.trans (by gcongr)
+    _ = _ := by simp only [pow_zero, one_mul]; ring
+
+/-- `log ((2π)^5 · 32 · 121) ≤ 18`. -/
+lemma log_C_le : Real.log ((2 * Real.pi) ^ 5 * 32 * 121) ≤ 18 := by
+  rw [Real.log_le_iff_le_exp (by positivity)]
+  have hpi : Real.pi < 3.15 := Real.pi_lt_d2
+  have he : (2.7182818283 : ℝ) < Real.exp 1 := Real.exp_one_gt_d9
+  have h18 : Real.exp 18 = Real.exp 1 ^ 18 := by rw [← Real.exp_nat_mul]; norm_num
+  rw [h18]
+  have hpow : (2.7182818283 : ℝ) ^ 18 ≤ Real.exp 1 ^ 18 := pow_le_pow_left₀ (by norm_num) he.le 18
+  have h2pi : (2 * Real.pi) ^ 5 ≤ 6.3 ^ 5 := pow_le_pow_left₀ (by positivity) (by linarith) 5
+  calc
+    (2 * Real.pi) ^ 5 * 32 * 121 ≤ 6.3 ^ 5 * 32 * 121 := by gcongr
+    _ ≤ (2.7182818283 : ℝ) ^ 18 := by norm_num
+    _ ≤ Real.exp 1 ^ 18 := hpow
+
+/-- A variant of **(6.14)** with `22 h log K + 50 h` instead of
+`18 h log K + 160 h`: the energy bound for `Δ_K(ζ(5))` from `energy_ineq`. -/
+theorem log_Δ_le' (n : ℕ) (hn : 0 < n) :
+    Real.log (aeval zeta5 (Δ n)) ≤
+      2 * (37 * n) * (37 * n + 6 * (3 * n) - 40 * n : ℝ) * Real.log (Kr n) +
+            (lam * M0 - Irho) * Kr n ^ 2 +
+          22 * (37 * n) * Real.log (Kr n) +
+        50 * (37 * n) := by
+  have hK : 0 < Kr n := by simp only [Kr]; positivity
+  have h1n : (1 : ℝ) ≤ n := by exact_mod_cast hn
+  have hK1 : 1 ≤ Kr n := by simp only [Kr]; linarith
+  have hlogK : 0 ≤ Real.log (Kr n) := Real.log_nonneg hK1
+  have hΔ := Δ_pos n hn
+  rw [Δ_eq_integral] at hΔ ⊢
+  set I :=
+    ∫ x : Fin (37 * n) → ℝ,
+      (Matrix.vandermonde fun j => x j ^ 2).det ^ 2 *
+        ∏ j, ν n (x j) ∂(Measure.pi fun _ : Fin (37 * n) => μpos) with
+    hI
+  have hfac : (0 : ℝ) < ((37 * n).factorial : ℝ) := by positivity
+  have hIpos : 0 < I := by
+    by_contra hneg
+    push Not at hneg
+    have : (1 / ((37 * n).factorial : ℝ)) * I ≤ 0 :=
+      mul_nonpos_of_nonneg_of_nonpos (by positivity) hneg
+    linarith
+  have hgint := integrableOn_gK n hn
+  have hg0 : 0 ≤ ∫ y in Ioi 0, gK n y :=
+    setIntegral_nonneg measurableSet_Ioi fun y hy =>
+      by
+      have : (0 : ℝ) < y := hy
+      unfold gK; positivity
+  have hg1 : ∫ y in Ioi 0, gK n y ≤ (2 * Real.pi) ^ 5 * 32 * 121 * Kr n ^ 6 :=
+    integral_one_add_pow_mul_exp
+      hK1
+        -- the integral bound
+  have hbound : I ≤ Real.exp (Aconst n) * (∫ y in Ioi 0, gK n y) ^ (37 * n) := by
+    calc
+      I ≤
+          ∫ x : Fin (37 * n) → ℝ,
+            Real.exp (Aconst n) * ∏ i, gK n (x i) ∂(Measure.pi fun _ : Fin (37 * n) => μpos) := by
+        refine
+          integral_mono_of_nonneg ?_ ((Integrable.fintype_prod fun _ => hgint).const_mul _) ?_
+        · filter_upwards [ae_pos_injective (37 * n)] with x hx
+          exact mul_nonneg (sq_nonneg _) (Finset.prod_pos fun j _ => ν_pos n (hx.1 j)).le
+        · filter_upwards [ae_pos_injective (37 * n)] with x hx
+          exact integrand_le n hn x hx.1 hx.2
+      _ = Real.exp (Aconst n) * (∫ y in Ioi 0, gK n y) ^ (37 * n) := by
+        rw [integral_const_mul, integral_fintype_prod_eq_prod (fun _ => gK n), Finset.prod_const,
+          Finset.card_univ, Fintype.card_fin]
+  have hlog1 : Real.log (1 / ((37 * n).factorial : ℝ) * I) ≤ Real.log I := by
+    rw [Real.log_mul (by positivity) hIpos.ne']
+    have : Real.log (1 / ((37 * n).factorial : ℝ)) ≤ 0 := by
+      rw [one_div, Real.log_inv]
+      linarith [Real.log_nonneg
+          (by exact_mod_cast Nat.one_le_iff_ne_zero.mpr (Nat.factorial_ne_zero _) :
+            (1 : ℝ) ≤ ((37 * n).factorial : ℝ))]
+    linarith
+  have hlog2 : Real.log 2 ≤ 0.7 := by have := Real.log_two_lt_d9; linarith
+  have hsqrt2 : Real.sqrt 2 ≤ 1.5 := by
+    rw [show (1.5 : ℝ) = Real.sqrt (1.5 ^ 2) by rw [Real.sqrt_sq (by norm_num)]]
+    exact Real.sqrt_le_sqrt (by norm_num)
+  have hC := log_C_le
+  have hCpos : (0 : ℝ) < (2 * Real.pi) ^ 5 * 32 * 121 := by positivity
+  calc
+    Real.log (1 / ((37 * n).factorial : ℝ) * I) ≤ Real.log I := hlog1
+    _ ≤ Real.log (Real.exp (Aconst n) * ((2 * Real.pi) ^ 5 * 32 * 121 * Kr n ^ 6) ^ (37 * n)) := by
+      refine Real.log_le_log hIpos (hbound.trans ?_)
+      gcongr
+    _ = Aconst n + (37 * n) * (Real.log ((2 * Real.pi) ^ 5 * 32 * 121) + 6 * Real.log (Kr n)) := by
+      rw [Real.log_mul (Real.exp_pos _).ne' (by positivity), Real.log_exp, Real.log_pow,
+        Real.log_mul hCpos.ne' (by positivity), Real.log_pow]
+      push_cast; ring
+    _ ≤ _ := by
+      unfold Aconst
+      nlinarith [mul_nonneg (by positivity : (0 : ℝ) ≤ 37 * n) hlogK]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/EnergyConst.lean b/LeanPool/Zeta5Irrational/EnergyConst.lean
new file mode 100644
index 0000000000..0484b2917b
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/EnergyConst.lean
@@ -0,0 +1,352 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Measure
+public import LeanPool.Zeta5Irrational.Constants
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Tactic.ENatToNat
+import Mathlib.Tactic.ReduceModChar
+import Mathlib.Tactic.FieldSimp
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+import Mathlib.Tactic.Positivity.Basic
+import Mathlib.Tactic.Ring.Basic
+
+/-! # The numerical inequality (6.4): `λ M₀ - I(ρ) + C* ≤ U`
+
+Only *lower* bounds for logarithms are needed. They come from the series
+`log ((1+x)/(1-x)) = 2 ∑ x^(2k+1)/(2k+1)`, whose terms are nonnegative for `0 ≤ x < 1`, so that
+partial sums are lower bounds, together with Mathlib's `Real.log_two_lt_d9` for the range
+reduction `log y = log (2^k y) - k log 2`. The certificates (`k`, partial sums with 10 terms,
+rounded down) were generated by exact rational arithmetic; `norm_num` checks each one.
+-/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+/-- Lower bound for `log r`, `r ≥ 1`, by a partial sum of the artanh series. -/
+lemma log_ge_artanh {r : ℝ} (hr : 1 ≤ r) (m : ℕ) :
+    2 * ∑ k ∈ range m, (1 / (2 * (k : ℝ) + 1)) * ((r - 1) / (r + 1)) ^ (2 * k + 1) ≤ Real.log r :=
+  by
+  set x : ℝ := (r - 1) / (r + 1) with hx
+  have hr0 : 0 < r + 1 := by linarith
+  have hx0 : 0 ≤ x := div_nonneg (by linarith) hr0.le
+  have hx1 : x < 1 := by rw [hx, div_lt_one hr0]; linarith
+  have habs : |x| < 1 := by rw [abs_of_nonneg hx0]; exact hx1
+  have hsum := Real.hasSum_log_sub_log_of_abs_lt_one habs
+  have hr' : (1 + x) / (1 - x) = r := by rw [hx]; field_simp; ring
+  have hlog : Real.log (1 + x) - Real.log (1 - x) = Real.log r := by
+    rw [← Real.log_div (by linarith) (by linarith), hr']
+  rw [hlog] at hsum
+  have key :=
+    sum_le_hasSum (range m)
+      (fun k _ => by
+        have : 0 ≤ x ^ (2 * k + 1) := pow_nonneg hx0 _
+        positivity)
+      hsum
+  rw [Finset.mul_sum]
+  refine le_trans (le_of_eq ?_) key
+  exact Finset.sum_congr rfl fun k _ => by ring
+
+/-- Lower bound for `log y`, `y > 0`, after the range reduction by `2^k`. -/
+lemma log_ge_of_pow_two {y : ℝ} (hy : 0 < y) (k m : ℕ) (hk : 1 ≤ 2 ^ k * y) :
+    2 *
+          ∑ j ∈ range m,
+            (1 / (2 * (j : ℝ) + 1)) * ((2 ^ k * y - 1) / (2 ^ k * y + 1)) ^ (2 * j + 1) -
+        k * 0.6931471808 ≤
+      Real.log y := by
+  have h1 := log_ge_artanh hk m
+  have h2 : Real.log (2 ^ k * y) = k * Real.log 2 + Real.log y := by
+    rw [Real.log_mul (by positivity) hy.ne', Real.log_pow]
+  have h3 := Real.log_two_lt_d9
+  have hk0 : (0 : ℝ) ≤ k := Nat.cast_nonneg k
+  nlinarith [mul_le_mul_of_nonneg_left h3.le hk0]
+
+private lemma energy_log_width_1 :
+    (-66675683088239 / 10000000000000 : ℝ) ≤ Real.log ((bρ 1 - aρ 1) / 4) := by
+  have e : ((bρ 1 - aρ 1) / 4) = (1017189489 / 800000000000 : ℝ) := by
+    simp [aρ, bρ, table1];
+    norm_num
+  rw [e]
+  refine le_trans ?_ (log_ge_of_pow_two (by norm_num) 10 10 (by norm_num))
+  simp only [Finset.sum_range_succ, Finset.sum_range_zero]
+  norm_num
+
+private lemma energy_log_width_2 :
+    (-57268912195833 / 10000000000000 : ℝ) ≤ Real.log ((bρ 2 - aρ 2) / 4) := by
+  have e : ((bρ 2 - aρ 2) / 4) = (13028749591 / 4000000000000 : ℝ) := by
+    simp [aρ, bρ, table1];
+    norm_num
+  rw [e]
+  refine le_trans ?_ (log_ge_of_pow_two (by norm_num) 9 10 (by norm_num))
+  simp only [Finset.sum_range_succ, Finset.sum_range_zero]
+  norm_num
+
+private lemma energy_log_width_3 :
+    (-51024260463511 / 10000000000000 : ℝ) ≤ Real.log ((bρ 3 - aρ 3) / 4) := by
+  have e : ((bρ 3 - aρ 3) / 4) = (24327894059 / 4000000000000 : ℝ) := by
+    simp [aρ, bρ, table1];
+    norm_num
+  rw [e]
+  refine le_trans ?_ (log_ge_of_pow_two (by norm_num) 8 10 (by norm_num))
+  simp only [Finset.sum_range_succ, Finset.sum_range_zero]
+  norm_num
+
+private lemma energy_log_width_4 :
+    (-22898992334719 / 5000000000000 : ℝ) ≤ Real.log ((bρ 4 - aρ 4) / 4) := by
+  have e : ((bρ 4 - aρ 4) / 4) = (4102785289 / 400000000000 : ℝ) := by
+    simp [aρ, bρ, table1];
+    norm_num
+  rw [e]
+  refine le_trans ?_ (log_ge_of_pow_two (by norm_num) 7 10 (by norm_num))
+  simp only [Finset.sum_range_succ, Finset.sum_range_zero]
+  norm_num
+
+private lemma energy_log_width_5 :
+    (-41154728268737 / 10000000000000 : ℝ) ≤ Real.log ((bρ 5 - aρ 5) / 4) := by
+  have e : ((bρ 5 - aρ 5) / 4) = (65272891657 / 4000000000000 : ℝ) := by
+    simp [aρ, bρ, table1];
+    norm_num
+  rw [e]
+  refine le_trans ?_ (log_ge_of_pow_two (by norm_num) 6 10 (by norm_num))
+  simp only [Finset.sum_range_succ, Finset.sum_range_zero]
+  norm_num
+
+private lemma energy_log_width_6 :
+    (-18490372338477 / 5000000000000 : ℝ) ≤ Real.log ((bρ 6 - aρ 6) / 4) := by
+  have e : ((bρ 6 - aρ 6) / 4) = (99084713271 / 4000000000000 : ℝ) := by
+    simp [aρ, bρ, table1];
+    norm_num
+  rw [e]
+  refine le_trans ?_ (log_ge_of_pow_two (by norm_num) 6 10 (by norm_num))
+  simp only [Finset.sum_range_succ, Finset.sum_range_zero]
+  norm_num
+
+private lemma energy_log_width_7 :
+    (-6647891986277 / 2000000000000 : ℝ) ≤ Real.log ((bρ 7 - aρ 7) / 4) := by
+  have e : ((bρ 7 - aρ 7) / 4) = (144041816267 / 4000000000000 : ℝ) := by
+    simp [aρ, bρ, table1];
+    norm_num
+  rw [e]
+  refine le_trans ?_ (log_ge_of_pow_two (by norm_num) 5 10 (by norm_num))
+  simp only [Finset.sum_range_succ, Finset.sum_range_zero]
+  norm_num
+
+private lemma energy_log_width_8 :
+    (-239301955437 / 80000000000 : ℝ) ≤ Real.log ((bρ 8 - aρ 8) / 4) := by
+  have e : ((bρ 8 - aρ 8) / 4) = (200893556541 / 4000000000000 : ℝ) := by
+    simp [aρ, bρ, table1];
+    norm_num
+  rw [e]
+  refine le_trans ?_ (log_ge_of_pow_two (by norm_num) 5 10 (by norm_num))
+  simp only [Finset.sum_range_succ, Finset.sum_range_zero]
+  norm_num
+
+private lemma energy_log_width_9 :
+    (-5397331999367 / 2000000000000 : ℝ) ≤ Real.log ((bρ 9 - aρ 9) / 4) := by
+  have e : ((bρ 9 - aρ 9) / 4) = (269180899217 / 4000000000000 : ℝ) := by
+    simp [aρ, bρ, table1];
+    norm_num
+  rw [e]
+  refine le_trans ?_ (log_ge_of_pow_two (by norm_num) 4 10 (by norm_num))
+  simp only [Finset.sum_range_succ, Finset.sum_range_zero]
+  norm_num
+
+private lemma energy_log_width_10 :
+    (-24448510736217 / 10000000000000 : ℝ) ≤ Real.log ((bρ 10 - aρ 10) / 4) := by
+  have e : ((bρ 10 - aρ 10) / 4) = (21684762939 / 250000000000 : ℝ) := by
+    simp [aρ, bρ, table1];
+    norm_num
+  rw [e]
+  refine le_trans ?_ (log_ge_of_pow_two (by norm_num) 4 10 (by norm_num))
+  simp only [Finset.sum_range_succ, Finset.sum_range_zero]
+  norm_num
+
+private lemma energy_log_width_11 :
+    (-22286490348067 / 10000000000000 : ℝ) ≤ Real.log ((bρ 11 - aρ 11) / 4) := by
+  have e : ((bρ 11 - aρ 11) / 4) = (430695182301 / 4000000000000 : ℝ) := by
+    simp [aρ, bρ, table1]; norm_num
+  rw [e]
+  refine le_trans ?_ (log_ge_of_pow_two (by norm_num) 4 10 (by norm_num))
+  simp only [Finset.sum_range_succ, Finset.sum_range_zero]
+  norm_num
+
+private lemma energy_log_width_12 :
+    (-20489791734377 / 10000000000000 : ℝ) ≤ Real.log ((bρ 12 - aρ 12) / 4) := by
+  have e : ((bρ 12 - aρ 12) / 4) = (128866386789 / 1000000000000 : ℝ) := by
+    simp [aρ, bρ, table1]; norm_num
+  rw [e]
+  refine le_trans ?_ (log_ge_of_pow_two (by norm_num) 3 10 (by norm_num))
+  simp only [Finset.sum_range_succ, Finset.sum_range_zero]
+  norm_num
+
+private lemma energy_log_width_13 :
+    (-9524391997859 / 5000000000000 : ℝ) ≤ Real.log ((bρ 13 - aρ 13) / 4) := by
+  have e : ((bρ 13 - aρ 13) / 4) = (595362962907 / 4000000000000 : ℝ) := by
+    simp [aρ, bρ, table1]; norm_num
+  rw [e]
+  refine le_trans ?_ (log_ge_of_pow_two (by norm_num) 3 10 (by norm_num))
+  simp only [Finset.sum_range_succ, Finset.sum_range_zero]
+  norm_num
+
+private lemma energy_log_width_14 :
+    (-17955224674931 / 10000000000000 : ℝ) ≤ Real.log ((bρ 14 - aρ 14) / 4) := by
+  have e : ((bρ 14 - aρ 14) / 4) = (166040678943 / 1000000000000 : ℝ) := by
+    simp [aρ, bρ, table1]; norm_num
+  rw [e]
+  refine le_trans ?_ (log_ge_of_pow_two (by norm_num) 3 10 (by norm_num))
+  simp only [Finset.sum_range_succ, Finset.sum_range_zero]
+  norm_num
+
+private lemma energy_log_width_15 :
+    (-8601243721761 / 5000000000000 : ℝ) ≤ Real.log ((bρ 15 - aρ 15) / 4) := by
+  have e : ((bρ 15 - aρ 15) / 4) = (716086447547 / 4000000000000 : ℝ) := by
+    simp [aρ, bρ, table1]; norm_num
+  rw [e]
+  refine le_trans ?_ (log_ge_of_pow_two (by norm_num) 3 10 (by norm_num))
+  simp only [Finset.sum_range_succ, Finset.sum_range_zero]
+  norm_num
+
+private lemma energy_log_width_16 :
+    (-16785662120803 / 10000000000000 : ℝ) ≤ Real.log ((bρ 16 - aρ 16) / 4) := by
+  have e : ((bρ 16 - aρ 16) / 4) = (746565554359 / 4000000000000 : ℝ) := by
+    simp [aρ, bρ, table1]; norm_num
+  rw [e]
+  refine le_trans ?_ (log_ge_of_pow_two (by norm_num) 3 10 (by norm_num))
+  simp only [Finset.sum_range_succ, Finset.sum_range_zero]
+  norm_num
+
+private lemma energy_log_alpha : (-25902671664061 / 10000000000000 : ℝ) ≤ Real.log alph := by
+  have e : alph = (3 / 40 : ℝ) := by norm_num [alph]
+  rw [e]
+  refine le_trans ?_ (log_ge_of_pow_two (by norm_num) 4 10 (by norm_num))
+  simp only [Finset.sum_range_succ, Finset.sum_range_zero]
+  norm_num
+
+private lemma energy_log_twice_lambda : (1537964097723 / 2500000000000 : ℝ) ≤ Real.log (2 * lam) :=
+  by
+  have e : (2 * lam) = (37 / 20 : ℝ) := by norm_num [lam]
+  rw [e]
+  refine le_trans ?_ (log_ge_of_pow_two (by norm_num) 0 10 (by norm_num))
+  simp only [Finset.sum_range_succ, Finset.sum_range_zero]
+  norm_num
+
+/-- **(6.4)**: `λ M₀ - I(ρ) + C* ≤ U`. -/
+theorem energy_const : lam * M0 - Irho + Cstar ≤ (U : ℝ) := by
+  have h1 := energy_log_width_1
+  have h2 := energy_log_width_2
+  have h3 := energy_log_width_3
+  have h4 := energy_log_width_4
+  have h5 := energy_log_width_5
+  have h6 := energy_log_width_6
+  have h7 := energy_log_width_7
+  have h8 := energy_log_width_8
+  have h9 := energy_log_width_9
+  have h10 := energy_log_width_10
+  have h11 := energy_log_width_11
+  have h12 := energy_log_width_12
+  have h13 := energy_log_width_13
+  have h14 := energy_log_width_14
+  have h15 := energy_log_width_15
+  have h16 := energy_log_width_16
+  have hα := energy_log_alpha
+  have hl := energy_log_twice_lambda
+  have hS0 : Sρ 0 = (0 : ℝ) := by simp [Sρ]
+  have hS1 : Sρ 1 = (525779809 / 50000000000 : ℝ) := by
+    simp [Sρ, cρ, table1]; norm_num
+  have hS2 : Sρ 2 = (39987333973 / 1000000000000 : ℝ) := by
+    simp [Sρ, cρ, table1, Finset.sum_Icc_succ_top]; norm_num
+  have hS3 : Sρ 3 = (323912887 / 3906250000 : ℝ) := by
+    simp [Sρ, cρ, table1, Finset.sum_Icc_succ_top]; norm_num
+  have hS4 : Sρ 4 = (70562965519 / 500000000000 : ℝ) := by
+    simp [Sρ, cρ, table1, Finset.sum_Icc_succ_top]; norm_num
+  have hS5 : Sρ 5 = (52540888087 / 250000000000 : ℝ) := by
+    simp [Sρ, cρ, table1, Finset.sum_Icc_succ_top]; norm_num
+  have hS6 : Sρ 6 = (289036651537 / 1000000000000 : ℝ) := by
+    simp [Sρ, cρ, table1, Finset.sum_Icc_succ_top]; norm_num
+  have hS7 : Sρ 7 = (46736596531 / 125000000000 : ℝ) := by
+    simp [Sρ, cρ, table1, Finset.sum_Icc_succ_top]; norm_num
+  have hS8 : Sρ 8 = (739662167 / 1600000000 : ℝ) := by
+    simp [Sρ, cρ, table1, Finset.sum_Icc_succ_top]; norm_num
+  have hS9 : Sρ 9 = (1101184473 / 2000000000 : ℝ) := by
+    simp [Sρ, cρ, table1, Finset.sum_Icc_succ_top]; norm_num
+  have hS10 : Sρ 10 = (127212911551 / 200000000000 : ℝ) := by
+    simp [Sρ, cρ, table1, Finset.sum_Icc_succ_top]; norm_num
+  have hS11 : Sρ 11 = (714963741993 / 1000000000000 : ℝ) := by
+    simp [Sρ, cρ, table1, Finset.sum_Icc_succ_top]; norm_num
+  have hS12 : Sρ 12 = (785317213911 / 1000000000000 : ℝ) := by
+    simp [Sρ, cρ, table1, Finset.sum_Icc_succ_top]; norm_num
+  have hS13 : Sρ 13 = (422078095263 / 500000000000 : ℝ) := by
+    simp [Sρ, cρ, table1, Finset.sum_Icc_succ_top]; norm_num
+  have hS14 : Sρ 14 = (55536094477 / 62500000000 : ℝ) := by
+    simp [Sρ, cρ, table1, Finset.sum_Icc_succ_top]; norm_num
+  have hS15 : Sρ 15 = (919040377423 / 1000000000000 : ℝ) := by
+    simp [Sρ, cρ, table1, Finset.sum_Icc_succ_top]; norm_num
+  have hS16 : Sρ 16 = (37 / 40 : ℝ) := by simp [Sρ, cρ, table1, Finset.sum_Icc_succ_top]; norm_num
+  have hI :
+    Irho =
+      ((276444407552076481 / 2500000000000000000000 : ℝ)) * Real.log ((bρ 1 - aρ 1) / 4) +
+                                    ((1488409115247409372329 / 1000000000000000000000000 : ℝ)) *
+                                      Real.log ((bρ 2 - aρ 2) / 4) +
+                                  ((1055404259743817139291 / 200000000000000000000000 : ℝ)) *
+                                    Real.log ((bρ 3 - aρ 3) / 4) +
+                                ((652026011717750304813 / 50000000000000000000000 : ℝ)) *
+                                  Real.log ((bρ 4 - aρ 4) / 4) +
+                              ((1212609516209410227783 / 50000000000000000000000 : ℝ)) *
+                                Real.log ((bρ 5 - aρ 5) / 4) +
+                            ((7874693439238125629853 / 200000000000000000000000 : ℝ)) *
+                              Real.log ((bρ 6 - aρ 6) / 4) +
+                          ((11250723841514726902227 / 200000000000000000000000 : ℝ)) *
+                            Real.log ((bρ 7 - aρ 7) / 4) +
+                        ((73915179740055157667121 / 1000000000000000000000000 : ℝ)) *
+                          Real.log ((bρ 8 - aρ 8) / 4) +
+                      ((5724212864942206439 / 64000000000000000000 : ℝ)) *
+                        Real.log ((bρ 9 - aρ 9) / 4) +
+                    ((4057052429519671935601 / 40000000000000000000000 : ℝ)) *
+                      Real.log ((bρ 10 - aρ 10) / 4) +
+                  ((13324378841571167621503 / 125000000000000000000000 : ℝ)) *
+                    Real.log ((bρ 11 - aρ 11) / 4) +
+                ((51538073291163213039 / 488281250000000000000 : ℝ)) *
+                  Real.log ((bρ 12 - aρ 12) / 4) +
+              ((19175309507686616048151 / 200000000000000000000000 : ℝ)) *
+                Real.log ((bρ 13 - aρ 13) / 4) +
+            ((19242580043687170786687 / 250000000000000000000000 : ℝ)) *
+              Real.log ((bρ 14 - aρ 14) / 4) +
+          ((11013044231138638563501 / 200000000000000000000000 : ℝ)) *
+            Real.log ((bρ 15 - aρ 15) / 4) +
+        ((10989784666189711879071 / 1000000000000000000000000 : ℝ)) *
+          Real.log ((bρ 16 - aρ 16) / 4) := by
+    simp only [Irho, Finset.sum_Icc_succ_top (show 1 ≤ 16 by norm_num),
+      Finset.sum_Icc_succ_top (show 1 ≤ 15 by norm_num),
+      Finset.sum_Icc_succ_top (show 1 ≤ 14 by norm_num),
+      Finset.sum_Icc_succ_top (show 1 ≤ 13 by norm_num),
+      Finset.sum_Icc_succ_top (show 1 ≤ 12 by norm_num),
+      Finset.sum_Icc_succ_top (show 1 ≤ 11 by norm_num),
+      Finset.sum_Icc_succ_top (show 1 ≤ 10 by norm_num),
+      Finset.sum_Icc_succ_top (show 1 ≤ 9 by norm_num),
+      Finset.sum_Icc_succ_top (show 1 ≤ 8 by norm_num),
+      Finset.sum_Icc_succ_top (show 1 ≤ 7 by norm_num),
+      Finset.sum_Icc_succ_top (show 1 ≤ 6 by norm_num),
+      Finset.sum_Icc_succ_top (show 1 ≤ 5 by norm_num),
+      Finset.sum_Icc_succ_top (show 1 ≤ 4 by norm_num),
+      Finset.sum_Icc_succ_top (show 1 ≤ 3 by norm_num),
+      Finset.sum_Icc_succ_top (show 1 ≤ 2 by norm_num), Finset.Icc_self, Finset.sum_singleton]
+    simp only [hS0, hS1, hS2, hS3, hS4, hS5, hS6, hS7, hS8, hS9, hS10, hS11, hS12, hS13, hS14,
+      hS15, hS16]
+    norm_num
+  rw [hI]
+  unfold Cstar
+  norm_num [U, M0, lam, alph] at *
+  linarith
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/EnergyFinal.lean b/LeanPool/Zeta5Irrational/EnergyFinal.lean
new file mode 100644
index 0000000000..333219f6d3
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/EnergyFinal.lean
@@ -0,0 +1,253 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+import LeanPool.Zeta5Irrational.Table.Intervals
+import LeanPool.Zeta5Irrational.Table.Tail
+import LeanPool.Zeta5Irrational.Certificates
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Data.Int.Star
+import Mathlib.Tactic.ENatToNat
+import Std.Tactic.BVDecide.Normalize.Prop
+public import LeanPool.Zeta5Irrational.EnergyRaw
+import Mathlib.Tactic.FieldSimp
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+import Mathlib.Tactic.Positivity.Basic
+import Mathlib.Tactic.Ring.Basic
+
+/-! # A variant of the configuration inequality (6.9)
+
+From the raw inequality `energy_raw` with `ε = K⁻⁶`, the potential inequality `potential_ineq`
+(Lemma 6.1), the tail bound (6.8) and the bound for the regularisation error we obtain
+`energy_ineq_final`: for every configuration of `h = 37 n` positive, pairwise distinct points,
+
+`2∑_{i<j} log|tᵢ - tⱼ| - K∑V(tᵢ) + ∑√tᵢ ≤ (λM₀ - I(ρ))K² + 5h + 6h log K`.
+
+The paper uses `(120 + √2)h + 2h log K` for the lower-order terms and `ε = K⁻²`.
+Here the different regularisation estimate and `ε = K⁻⁶` give `5h + 6h log K`.
+-/
+
+@[expose] public section
+
+open MeasureTheory Set Real Filter Topology Finset
+
+namespace Zeta5Irrational
+
+lemma sqrt_add_le' {a b : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) :
+    Real.sqrt (a + b) ≤ Real.sqrt a + Real.sqrt b := by
+  rw [Real.sqrt_le_iff]
+  refine ⟨by positivity, ?_⟩
+  have h1 := Real.sq_sqrt ha
+  have h2 := Real.sq_sqrt hb
+  nlinarith [Real.sqrt_nonneg a, Real.sqrt_nonneg b]
+
+/-- The regularisation error at one interval. -/
+lemma errj_le {ε t m r : ℝ} (hε : 0 < ε) (hε1 : ε ≤ 1) (ht : 0 < t) (hm : 0 < m ∧ m < 2)
+    (hr : 1 / 450 ≤ r) : errj ε t m r ≤ Real.sqrt ε * (637 * Real.sqrt t + 1457) := by
+  unfold errj
+  have hr0 : 0 < r := by linarith
+  have hse : 0 ≤ Real.sqrt ε := Real.sqrt_nonneg ε
+  have hst : 0 ≤ Real.sqrt t := Real.sqrt_nonneg t
+  have hεs : ε ≤ Real.sqrt ε := by rw [Real.le_sqrt hε.le hε.le]; nlinarith
+  have h1 : ε / r ≤ 450 * ε := by rw [div_le_iff₀ hr0]; nlinarith
+  have habs : |t - m| ≤ t + 2 := by rw [abs_le]; constructor <;> linarith
+  have h2 : ε / r * ((ε + 2 * |t - m|) / r) ≤ 450 ^ 2 * ε * (2 * t + 5) := by
+    have : (ε + 2 * |t - m|) / r ≤ 450 * (2 * t + 5) := by rw [div_le_iff₀ hr0]; nlinarith
+    calc
+      ε / r * ((ε + 2 * |t - m|) / r) ≤ (450 * ε) * (450 * (2 * t + 5)) :=
+        mul_le_mul h1 this (by positivity) (by positivity)
+      _ = 450 ^ 2 * ε * (2 * t + 5) := by ring
+  have hs2 : Real.sqrt 2 ≤ 14143 / 10000 := by rw [Real.sqrt_le_iff]; constructor <;> norm_num
+  have hs5 : Real.sqrt 5 ≤ 22361 / 10000 := by rw [Real.sqrt_le_iff]; constructor <;> norm_num
+  have h3 :
+    Real.sqrt (ε / r * ((ε + 2 * |t - m|) / r)) ≤
+      450 * Real.sqrt ε * (14143 / 10000 * Real.sqrt t + 22361 / 10000) := by
+    calc
+      Real.sqrt (ε / r * ((ε + 2 * |t - m|) / r)) ≤ Real.sqrt (450 ^ 2 * ε * (2 * t + 5)) :=
+        Real.sqrt_le_sqrt h2
+      _ = 450 * Real.sqrt ε * Real.sqrt (2 * t + 5) := by
+        rw [Real.sqrt_mul (by positivity), Real.sqrt_mul (by positivity),
+          Real.sqrt_sq (by norm_num)]
+      _ ≤ 450 * Real.sqrt ε * (Real.sqrt (2 * t) + Real.sqrt 5) := by
+        apply mul_le_mul_of_nonneg_left (sqrt_add_le' (by positivity) (by norm_num))
+        positivity
+      _ ≤ 450 * Real.sqrt ε * (14143 / 10000 * Real.sqrt t + 22361 / 10000) := by
+        apply mul_le_mul_of_nonneg_left _ (by positivity)
+        rw [Real.sqrt_mul (by norm_num)]
+        have := mul_le_mul_of_nonneg_right hs2 hst
+        linarith
+  nlinarith
+
+/-- The total mass of the sixteen weights. -/
+lemma sum_cρ_fin : ∑ j : Fin 16, cρ (j + 1) = lam := by
+  rw [← Sρ_sixteen, Sρ, sum_Icc_eq_sum_range', Fin.sum_univ_eq_sum_range (fun j => cρ (j + 1))]
+
+/-- The total regularisation error. -/
+lemma Err_le {ε t : ℝ} (hε : 0 < ε) (hε1 : ε ≤ 1) (ht : 0 < t) :
+    Err ε t ≤ Real.sqrt ε * (590 * Real.sqrt t + 1348) := by
+  unfold Err
+  have hse : 0 ≤ Real.sqrt ε := Real.sqrt_nonneg ε
+  have hst : 0 ≤ Real.sqrt t := Real.sqrt_nonneg t
+  calc
+    ∑ j : Fin 16, cρ (j + 1) * errj ε t (mρ (j + 1)) (rρ (j + 1)) ≤
+        ∑ j : Fin 16, cρ (j + 1) * (Real.sqrt ε * (637 * Real.sqrt t + 1457)) := by
+      refine Finset.sum_le_sum fun j _ => ?_
+      exact
+        mul_le_mul_of_nonneg_left (errj_le hε hε1 ht (mρ_bounds j) (rρ_ge j))
+          (cρ_pos _ (by simp only [Finset.mem_Icc]; omega)).le
+    _ = lam * (Real.sqrt ε * (637 * Real.sqrt t + 1457)) := by rw [← Finset.sum_mul, sum_cρ_fin]
+    _ ≤ _ := by
+      simp only [lam]
+      nlinarith
+
+/-- The tail bound with an extra `c √t`, `0 ≤ c ≤ 1/20`. -/
+lemma potential_ineq_tail' {t c : ℝ} (ht : 2 ≤ t) (_hc0 : 0 ≤ c) (hc : c ≤ 1 / 20) :
+    2 * Uρ t - Vfield t + c * Real.sqrt t ≤ M0 := by
+  have ht0 : 0 < t := by linarith
+  have h1 := Uρ_le_log ht
+  have h2 := Vfield_ge_tail ht
+  have hs : Real.sqrt t ^ 2 = t := Real.sq_sqrt ht0.le
+  have hs0 : 0 < Real.sqrt t := Real.sqrt_pos.mpr ht0
+  have hs2 : (141 / 100 : ℝ) ≤ Real.sqrt t :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num; linarith)
+  have hlog : Real.log t ≤ 2 * (Real.sqrt t - 1) := by
+    have e : Real.log (Real.sqrt t ^ 2) = 2 * Real.log (Real.sqrt t) := by
+      rw [Real.log_pow];
+      norm_num
+    rw [hs] at e
+    rw [e]
+    linarith [Real.log_le_sub_one_of_pos hs0]
+  have hq : (3 / 40 : ℝ) ^ 2 / t ≤ (3 / 40) ^ 2 / 2 :=
+    div_le_div_of_nonneg_left (by norm_num) (by norm_num) ht
+  have hpi := Real.pi_gt_d2
+  have hprod : 314 / 100 * Real.sqrt t ≤ Real.pi * Real.sqrt t :=
+    mul_le_mul_of_nonneg_right (by norm_num at hpi; linarith) hs0.le
+  have hcs : c * Real.sqrt t ≤ 1 / 20 * Real.sqrt t := mul_le_mul_of_nonneg_right hc hs0.le
+  unfold M0
+  linarith
+
+/-- A variant of (6.7): `2Uρ(t) - V(t) + (2/K)√t ≤ M₀ + 3/K` for `K ≥ 40`.
+The paper instead uses `√t/K` and `√2/K`, and assumes `K ≥ 2`. -/
+lemma pointwise_bound {K t : ℝ} (hK : 40 ≤ K) (ht : 0 < t) :
+    2 * Uρ t - Vfield t + 2 / K * Real.sqrt t ≤ M0 + 3 / K := by
+  have hK0 : 0 < K := by linarith
+  rcases le_or_gt t 2 with h | h
+  · have h1 := potential_ineq t ht
+    have hs : Real.sqrt t ≤ 3 / 2 := by
+      rw [Real.sqrt_le_iff];
+      constructor <;> [norm_num; nlinarith]
+    have : 2 / K * Real.sqrt t ≤ 3 / K := by
+      rw [show 3 / K = 2 / K * (3 / 2) by ring]
+      exact mul_le_mul_of_nonneg_left hs (by positivity)
+    exact add_le_add h1 this
+  · have h1 :=
+      potential_ineq_tail' (c := 2 / K) h.le (by positivity) (by rw [div_le_iff₀ hK0]; linarith)
+    have : (0 : ℝ) ≤ 3 / K := by positivity
+    linarith
+
+/-- The double sum over all ordered pairs equals twice the sum over `i < j`. -/
+lemma sum_pairs_eq {h : ℕ} (t : Fin h → ℝ) :
+    ∑ i, ∑ i', Real.log |t i - t i'| = ∑ i, ∑ j ∈ Finset.Ioi i, 2 * Real.log |t j - t i| := by
+  have hsplit :
+    ∀ i : Fin h,
+      ∑ i', Real.log |t i - t i'| =
+        ∑ i' ∈ Finset.Iio i, Real.log |t i - t i'| + ∑ i' ∈ Finset.Ioi i, Real.log |t i - t i'| :=
+  by
+    intro i
+    have e1 : (Finset.univ : Finset (Fin h)).filter (fun x => x < i) = Finset.Iio i := by
+      ext x;
+      simp
+    have e2 : (Finset.univ : Finset (Fin h)).filter (fun x => ¬x < i) = Finset.Ici i := by
+      ext x;
+      simp [not_lt]
+    rw [← Finset.sum_filter_add_sum_filter_not Finset.univ (fun x => x < i), e1, e2, ←
+      Finset.add_sum_Ioi_eq_sum_Ici, sub_self, abs_zero, Real.log_zero, zero_add]
+  simp_rw [hsplit]
+  rw [Finset.sum_add_distrib]
+  have hcomm :
+    ∑ i, ∑ i' ∈ Finset.Iio i, Real.log |t i - t i'| =
+      ∑ i, ∑ i' ∈ Finset.Ioi i, Real.log |t i - t i'| := by
+    rw [Finset.sum_comm' (t' := Finset.univ) (s' := fun i' => Finset.Ioi i')]
+    · refine Finset.sum_congr rfl fun i _ => Finset.sum_congr rfl fun i' _ => ?_
+      rw [abs_sub_comm]
+    · intro x y
+      simp only [Finset.mem_univ, Finset.mem_Iio, Finset.mem_Ioi, true_and, and_true]
+  rw [hcomm, ← Finset.sum_add_distrib]
+  refine Finset.sum_congr rfl fun i _ => ?_
+  rw [← Finset.sum_add_distrib]
+  refine Finset.sum_congr rfl fun j _ => ?_
+  rw [abs_sub_comm]; ring
+
+/-- A variant of **(6.9)** with lower-order terms `5h + 6h log K` instead of
+`(120 + √2)h + 2h log K`, for positive pairwise distinct configurations and `K = 40n`. -/
+theorem energy_ineq_final (n : ℕ) (hn : 0 < n) (t : Fin (37 * n) → ℝ) (ht : ∀ i, 0 < t i)
+    (hinj : Function.Injective t) :
+    (∑ i, ∑ j ∈ Finset.Ioi i, 2 * Real.log |t j - t i|) - Kr n * ∑ i, Vfield (t i) +
+        ∑ i, Real.sqrt (t i) ≤
+      (lam * M0 - Irho) * Kr n ^ 2 + 5 * (37 * n) + 6 * (37 * n) * Real.log (Kr n) := by
+  have hK : 40 ≤ Kr n := by
+    simp only [Kr]; have : (1 : ℝ) ≤ n := by exact_mod_cast hn
+    linarith
+  have hK0 : 0 < Kr n := by linarith
+  set K := Kr n with hKdef
+  set ε : ℝ := K ^ (-6 : ℤ) with hε_def
+  have hε : 0 < ε := by rw [hε_def]; positivity
+  have hε1 : ε ≤ 1 := by
+    rw [hε_def, zpow_neg, inv_le_one_iff₀]
+    right; exact one_le_zpow₀ (by linarith) (by norm_num)
+  have hsqrtε : Real.sqrt ε = K ^ (-3 : ℤ) := by
+    have e : (K ^ (-3 : ℤ)) ^ 2 = ε := by rw [hε_def, ← zpow_natCast, ← zpow_mul]; norm_num
+    rw [← e, Real.sqrt_sq (by positivity)]
+  have hlogε : Real.log ε = -6 * Real.log K := by rw [hε_def, Real.log_zpow]; push_cast; ring
+  have hraw := energy_raw n hn t hinj hε
+  rw [← hKdef, sum_pairs_eq,
+    hlogε] at hraw
+        -- bound the error terms
+  have hErr : ∀ i, Err ε (t i) ≤ 1 / K ^ 3 * (590 * Real.sqrt (t i) + 1348) := fun i =>
+    by
+    have := Err_le hε hε1 (ht i)
+    rw [hsqrtε, zpow_neg, zpow_ofNat] at this
+    rw [one_div]; exact this
+  have hK2 : 1 / K ^ 2 ≤ 1 / 1600 :=
+    one_div_le_one_div_of_le (by norm_num)
+      (by nlinarith)
+        -- pointwise
+  have hpt :
+    ∀ i, 2 * K * (Uρ (t i) + Err ε (t i)) - K * Vfield (t i) + Real.sqrt (t i) ≤ K * M0 + 5 := by
+    intro i
+    have h1 := pointwise_bound hK (ht i)
+    have h2 := hErr i
+    have hs := Real.sqrt_nonneg (t i)
+    have e1 :
+      2 * K * (1 / K ^ 3 * (590 * Real.sqrt (t i) + 1348)) =
+        1 / K ^ 2 * (1180 * Real.sqrt (t i) + 2696) :=
+      by field_simp; ring
+    have e2 :
+      1 / K ^ 2 * (1180 * Real.sqrt (t i) + 2696) ≤ 1 / 1600 * (1180 * Real.sqrt (t i) + 2696) :=
+      mul_le_mul_of_nonneg_right hK2 (by positivity)
+    have e3 : K * (2 / K * Real.sqrt (t i)) = 2 * Real.sqrt (t i) := by field_simp
+    have e4 : K * (3 / K) = 3 := by field_simp
+    have h1' := mul_le_mul_of_nonneg_left h1 hK0.le
+    rw [mul_add, mul_add, mul_sub, e3, e4] at h1'
+    have h2' := mul_le_mul_of_nonneg_left h2 (by positivity : (0 : ℝ) ≤ 2 * K)
+    rw [e1] at h2'
+    nlinarith
+  have hsum := Finset.sum_le_sum fun i (_ : i ∈ Finset.univ) => hpt i
+  simp only [Finset.sum_add_distrib, Finset.sum_sub_distrib, Finset.sum_const, Finset.card_univ,
+    Fintype.card_fin, nsmul_eq_mul, ← Finset.mul_sum] at hsum
+  simp only [Finset.sum_add_distrib] at hraw
+  have hlam : ((37 * n : ℕ) : ℝ) * (K * M0) = lam * M0 * K ^ 2 := by
+    rw [hKdef];
+    simp only [Kr, lam]; push_cast; ring
+  rw [hlam] at hsum
+  push_cast at hraw hsum ⊢
+  linarith [hraw, hsum]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/EnergyLimit.lean b/LeanPool/Zeta5Irrational/EnergyLimit.lean
new file mode 100644
index 0000000000..a6a7a9750d
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/EnergyLimit.lean
@@ -0,0 +1,103 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.ZeroMass
+import LeanPool.Zeta5Irrational.Kernel
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.Positivity.Basic
+
+import Mathlib.MeasureTheory.Integral.Prod
+import Mathlib.MeasureTheory.Integral.DominatedConvergence
+
+/-! # Lemma 6.2 for the logarithmic kernel
+
+Passing to the limit `a → 0`, `b → ∞` in `energy_Ltr_nonpos` by dominated convergence.
+The hypotheses are integrability of `log ‖γ k θ - γ l φ‖` on the torus and the a.e.
+distinctness `γ k θ ≠ γ l φ`.
+-/
+
+@[expose] public section
+
+open MeasureTheory Set Real intervalIntegral Filter Topology
+
+namespace Zeta5Irrational
+
+/-- The measure on `[0, 2π]`. -/
+noncomputable abbrev μcirc : Measure ℝ :=
+  volume.restrict (Ioc 0 (2 * π))
+
+variable {ι : Type*}
+
+/-- `pairInt` as an integral over the product measure. -/
+lemma pairInt_eq_prod (γ : ι → ℝ → ℂ) (f : ℂ → ℂ → ℝ) (k l : ι)
+    (hf : Integrable (fun p : ℝ × ℝ => f (γ k p.1) (γ l p.2)) (μcirc.prod μcirc)) :
+    pairInt γ f k l = ∫ p : ℝ × ℝ, f (γ k p.1) (γ l p.2) ∂(μcirc.prod μcirc) := by
+  unfold pairInt
+  rw [integral_prod _ hf, intervalIntegral.integral_of_le (by positivity)]
+  simp_rw [intervalIntegral.integral_of_le (by positivity : (0 : ℝ) ≤ 2 * π)]
+
+lemma integrable_of_continuous_torus {F : ℝ × ℝ → ℝ} (hF : Continuous F) :
+    Integrable F (μcirc.prod μcirc) := by
+  rw [Measure.prod_restrict]
+  refine IntegrableOn.mono_set ?_ (Set.prod_mono Ioc_subset_Icc_self Ioc_subset_Icc_self)
+  exact ContinuousOn.integrableOn_compact (isCompact_Icc.prod isCompact_Icc) hF.continuousOn
+
+/-- Limit of the truncated pair integrals. -/
+lemma tendsto_pairInt_Ltr {γ : ι → ℝ → ℂ} (hγ : ∀ k, Continuous (γ k)) (k l : ι)
+    (hint : Integrable (fun p : ℝ × ℝ => Real.log ‖γ k p.1 - γ l p.2‖) (μcirc.prod μcirc))
+    (hae : ∀ᵐ p ∂(μcirc.prod μcirc), γ k p.1 ≠ γ l p.2) :
+    Tendsto
+      (fun n : ℕ => pairInt γ (fun z w => Ltr (1 / ((n : ℝ) + 1)) ((n : ℝ) + 1) ‖z - w‖) k l)
+      atTop (𝓝 (pairInt γ (fun z w => Real.log ‖z - w‖) k l)) := by
+  have hn : ∀ n : ℕ, (0 : ℝ) < 1 / ((n : ℝ) + 1) := fun n => by positivity
+  have hn' : ∀ n : ℕ, 1 / ((n : ℝ) + 1) ≤ (n : ℝ) + 1 := fun n => by
+    rw [div_le_iff₀ (by positivity)]; nlinarith
+  have hcont :
+    ∀ n : ℕ,
+      Continuous fun p : ℝ × ℝ => Ltr (1 / ((n : ℝ) + 1)) ((n : ℝ) + 1) ‖γ k p.1 - γ l p.2‖ :=
+    fun n => (continuous_Ltr (hn n) (hn' n)).comp (by fun_prop)
+  rw [pairInt_eq_prod γ _ k l hint]
+  have e :
+    ∀ n : ℕ,
+      pairInt γ (fun z w => Ltr (1 / ((n : ℝ) + 1)) ((n : ℝ) + 1) ‖z - w‖) k l =
+        ∫ p : ℝ × ℝ,
+          Ltr (1 / ((n : ℝ) + 1)) ((n : ℝ) + 1) ‖γ k p.1 - γ l p.2‖ ∂(μcirc.prod μcirc) :=
+    fun n =>
+    pairInt_eq_prod γ (fun z w => Ltr (1 / ((n : ℝ) + 1)) ((n : ℝ) + 1) ‖z - w‖) k l
+      (integrable_of_continuous_torus (hcont n))
+  simp_rw [e]
+  refine
+    tendsto_integral_of_dominated_convergence (fun p => |Real.log ‖γ k p.1 - γ l p.2‖|)
+      (fun n => (hcont n).aestronglyMeasurable) hint.abs ?_ ?_
+  · intro n
+    filter_upwards [hae] with p hp
+    rw [Real.norm_eq_abs]
+    exact abs_Ltr_le (hn n) (hn' n) (norm_pos_iff.mpr (sub_ne_zero.mpr hp))
+  · filter_upwards [hae] with p hp
+    exact tendsto_Ltr (norm_pos_iff.mpr (sub_ne_zero.mpr hp))
+
+/-- **Lemma 6.2** (zero-mass logarithmic energy) for explicit atoms. -/
+theorem energy_log_nonpos [Fintype ι] {s : ι → ℝ} {γ : ι → ℝ → ℂ} (hγ : ∀ k, Continuous (γ k))
+    {R : ℝ} (hR : ∀ k θ, ‖γ k θ‖ ≤ R) (hmass : ∑ k, s k = 0)
+    (hint : ∀ k l, Integrable (fun p : ℝ × ℝ => Real.log ‖γ k p.1 - γ l p.2‖) (μcirc.prod μcirc))
+    (hae : ∀ k l, ∀ᵐ p ∂(μcirc.prod μcirc), γ k p.1 ≠ γ l p.2) :
+    energy s γ (fun z w => Real.log ‖z - w‖) ≤ 0 := by
+  have hn : ∀ n : ℕ, (0 : ℝ) < 1 / ((n : ℝ) + 1) := fun n => by positivity
+  have hn' : ∀ n : ℕ, 1 / ((n : ℝ) + 1) ≤ (n : ℝ) + 1 := fun n => by
+    rw [div_le_iff₀ (by positivity)]; nlinarith
+  have hle : ∀ n : ℕ, energy s γ (fun z w => Ltr (1 / ((n : ℝ) + 1)) ((n : ℝ) + 1) ‖z - w‖) ≤ 0 :=
+    fun n => energy_Ltr_nonpos hγ hR hmass (hn n) (hn' n)
+  have hlim :
+    Tendsto (fun n : ℕ => energy s γ (fun z w => Ltr (1 / ((n : ℝ) + 1)) ((n : ℝ) + 1) ‖z - w‖))
+      atTop (𝓝 (energy s γ (fun z w => Real.log ‖z - w‖))) := by
+    unfold energy
+    refine tendsto_finsetSum _ fun k _ => tendsto_finsetSum _ fun l _ => ?_
+    exact (tendsto_pairInt_Ltr hγ k l (hint k l) (hae k l)).const_mul _
+  exact le_of_tendsto' hlim hle
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/EnergyRaw.lean b/LeanPool/Zeta5Irrational/EnergyRaw.lean
new file mode 100644
index 0000000000..49cb00dce5
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/EnergyRaw.lean
@@ -0,0 +1,255 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+import Mathlib.RingTheory.PiTensorProduct
+import Mathlib.Tactic.ENatToNat
+import Std.Tactic.BVDecide.Normalize.Prop
+public import LeanPool.Zeta5Irrational.EnergyBlocks
+import Mathlib.Tactic.FieldSimp
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.Positivity.Basic
+import Mathlib.Tactic.Ring.Basic
+
+/-! # The raw configuration inequality
+
+Combining `atom_energy_nonpos` with the four block estimates gives, for every configuration
+`t : Fin (37 n) → ℝ` of pairwise distinct points and every `ε > 0`:
+
+`∑ᵢ ∑ᵢ' log |tᵢ - tᵢ'| ≤ -h log ε + 2K ∑ᵢ (Uρ(tᵢ) + Err ε (tᵢ)) - K² I(ρ)`
+
+(`energy_raw`), where the diagonal terms are `log 0 = 0`.
+-/
+
+@[expose] public section
+
+open MeasureTheory Set Real intervalIntegral Filter Topology Finset
+
+namespace Zeta5Irrational
+
+/-- The total regularisation error at a point. -/
+noncomputable def Err (ε t : ℝ) : ℝ :=
+  ∑ j : Fin 16, cρ (j + 1) * errj ε t (mρ (j + 1)) (rρ (j + 1))
+
+/--
+`∑_{j,j' < N} c_j c_j' f (max j j') = ∑_{j < N} (S_{j+1}² - S_j²) f_j` with `S_k = ∑_{i<k} c_i`. -/
+lemma sum_sum_max_eq (c f : ℕ → ℝ) (N : ℕ) :
+    ∑ j ∈ range N, ∑ j' ∈ range N, c j * c j' * f (max j j') =
+      ∑ j ∈ range N, ((∑ i ∈ range (j + 1), c i) ^ 2 - (∑ i ∈ range j, c i) ^ 2) * f j := by
+  induction N with
+  | zero => simp
+  | succ N ih =>
+    simp only [Finset.sum_range_succ] at ih ⊢
+    rw [Finset.sum_add_distrib, ih]
+    have h1 : ∀ j ∈ range N, c j * c N * f (max j N) = c j * c N * f N := by
+      intro j hj
+      rw [max_eq_right (Finset.mem_range.mp hj).le]
+    have h2 : ∀ j' ∈ range N, c N * c j' * f (max N j') = c N * c j' * f N := by
+      intro j' hj'
+      rw [max_eq_left (Finset.mem_range.mp hj').le]
+    rw [Finset.sum_congr rfl h1, Finset.sum_congr rfl h2, max_self]
+    simp only [← Finset.sum_mul, ← Finset.mul_sum]
+    ring
+
+/-- `I(ρ)` as a double sum over the sixteen intervals. -/
+lemma Irho_eq_double_sum :
+    Irho =
+      ∑ j : Fin 16,
+        ∑ j' : Fin 16,
+          cρ (j + 1) * cρ (j' + 1) *
+            Real.log ((bρ (max (j : ℕ) j' + 1) - aρ (max (j : ℕ) j' + 1)) / 4) := by
+  have hS : ∀ k, Sρ k = ∑ i ∈ range k, cρ (i + 1) := fun k => by rw [Sρ, sum_Icc_eq_sum_range']
+  unfold Irho
+  rw [sum_Icc_eq_sum_range']
+  simp only [Nat.add_sub_cancel, hS]
+  rw [←
+    sum_sum_max_eq (fun i => cρ (i + 1)) (fun j => Real.log ((bρ (j + 1) - aρ (j + 1)) / 4)) 16, ←
+    Fin.sum_univ_eq_sum_range
+      (fun j =>
+        ∑ j' ∈ range 16,
+          cρ (j + 1) * cρ (j' + 1) * Real.log ((bρ (max j j' + 1) - aρ (max j j' + 1)) / 4))
+      16]
+  refine Finset.sum_congr rfl fun j _ => ?_
+  rw [←
+    Fin.sum_univ_eq_sum_range
+      (fun j' =>
+        cρ ((j : ℕ) + 1) * cρ (j' + 1) *
+          Real.log ((bρ (max (j : ℕ) j' + 1) - aρ (max (j : ℕ) j' + 1)) / 4))
+      16]
+
+/-- The arcsine–arcsine block dominates `(2π)² I(ρ)`. -/
+lemma block_D_sum {h : ℕ} (t : Fin h → ℝ) {ε : ℝ} (hε : 0 < ε) :
+    (2 * π) ^ 2 * Irho ≤
+      ∑ j : Fin 16, ∑ j' : Fin 16, cρ (j + 1) * cρ (j' + 1) * Plog t ε (Sum.inr j) (Sum.inr j') :=
+  by
+  rw [Irho_eq_double_sum, Finset.mul_sum]
+  refine Finset.sum_le_sum fun j _ => ?_
+  rw [Finset.mul_sum]
+  refine Finset.sum_le_sum fun j' _ => ?_
+  have hc : 0 ≤ cρ (j + 1) * cρ (j' + 1) :=
+    mul_nonneg (cρ_pos _ (by simp only [Finset.mem_Icc]; omega)).le
+      (cρ_pos _ (by simp only [Finset.mem_Icc]; omega)).le
+  have hP :
+    (2 * π) ^ 2 * Real.log ((bρ (max (j : ℕ) j' + 1) - aρ (max (j : ℕ) j' + 1)) / 4) ≤
+      Plog t ε (Sum.inr j) (Sum.inr j') := by
+    rcases le_total (j : ℕ) j' with hjj | hjj
+    · rw [max_eq_right hjj]; exact block_D t hε j j'
+    · rw [max_eq_left hjj, Plog, pairInt_log_symm t hε]; exact block_D t hε j' j
+  calc
+    (2 * π) ^ 2 *
+          (cρ (j + 1) * cρ (j' + 1) *
+            Real.log ((bρ (max (j : ℕ) j' + 1) - aρ (max (j : ℕ) j' + 1)) / 4)) =
+        cρ (j + 1) * cρ (j' + 1) *
+          ((2 * π) ^ 2 * Real.log ((bρ (max (j : ℕ) j' + 1) - aρ (max (j : ℕ) j' + 1)) / 4)) :=
+      by ring
+    _ ≤ _ := mul_le_mul_of_nonneg_left hP hc
+
+/-- The circle–circle block. -/
+lemma block_AB_sum {h : ℕ} (t : Fin h → ℝ) (hinj : Function.Injective t) {ε : ℝ} (hε : 0 < ε) :
+    (2 * π) ^ 2 * ((h : ℝ) * Real.log ε + ∑ i, ∑ i', Real.log |t i - t i'|) ≤
+      ∑ i, ∑ i', Plog t ε (Sum.inl i) (Sum.inl i') := by
+  have hi :
+    ∀ i,
+      (2 * π) ^ 2 * (Real.log ε + ∑ i', Real.log |t i - t i'|) ≤
+        ∑ i', Plog t ε (Sum.inl i) (Sum.inl i') := by
+    intro i
+    rw [←
+      Finset.add_sum_erase (s := Finset.univ) (f := fun i' => Plog t ε (Sum.inl i) (Sum.inl i'))
+        (Finset.mem_univ i),
+      block_A t hε i, ←
+      Finset.sum_erase (s := Finset.univ) (a := i) (f := fun i' => Real.log |t i - t i'|)
+        (by simp),
+      mul_add, Finset.mul_sum]
+    refine add_le_add le_rfl (Finset.sum_le_sum fun i' hi' => ?_)
+    exact block_B t hε i i' (fun h => Finset.ne_of_mem_erase hi' (hinj h).symm)
+  calc
+    (2 * π) ^ 2 * ((h : ℝ) * Real.log ε + ∑ i, ∑ i', Real.log |t i - t i'|) =
+        ∑ i : Fin h, (2 * π) ^ 2 * (Real.log ε + ∑ i', Real.log |t i - t i'|) := by
+      simp only [mul_add, Finset.sum_add_distrib, Finset.sum_const, Finset.card_univ,
+        Fintype.card_fin, nsmul_eq_mul, Finset.mul_sum]
+      ring
+    _ ≤ _ := Finset.sum_le_sum fun i _ => hi i
+
+/-- The circle–arcsine block. -/
+lemma block_C_sum {h : ℕ} (t : Fin h → ℝ) {ε : ℝ} (hε : 0 < ε) (i : Fin h) :
+    ∑ j : Fin 16, cρ (j + 1) * Plog t ε (Sum.inl i) (Sum.inr j) ≤
+      (2 * π) ^ 2 * (Uρ (t i) + Err ε (t i)) := by
+  have hU : Uρ (t i) = ∑ j : Fin 16, cρ (j + 1) * Uω (aρ (j + 1)) (bρ (j + 1)) (t i) := by
+    rw [Uρ, sum_Icc_eq_sum_range',
+      Fin.sum_univ_eq_sum_range (fun j => cρ (j + 1) * Uω (aρ (j + 1)) (bρ (j + 1)) (t i)) 16]
+  rw [hU, Err, ← Finset.sum_add_distrib, Finset.mul_sum]
+  refine Finset.sum_le_sum fun j _ => ?_
+  have hc : 0 ≤ cρ (j + 1) := (cρ_pos _ (by simp only [Finset.mem_Icc]; omega)).le
+  calc
+    cρ (j + 1) * Plog t ε (Sum.inl i) (Sum.inr j) ≤
+        cρ (j + 1) *
+          ((2 * π) ^ 2 *
+            (Uω (aρ (j + 1)) (bρ (j + 1)) (t i) + errj ε (t i) (mρ (j + 1)) (rρ (j + 1)))) :=
+      mul_le_mul_of_nonneg_left (block_C t hε i j) hc
+    _ = _ := by ring
+
+/-- **The raw configuration inequality.** -/
+theorem energy_raw (n : ℕ) (hn : 0 < n) (t : Fin (37 * n) → ℝ) (hinj : Function.Injective t)
+    {ε : ℝ} (hε : 0 < ε) :
+    ∑ i, ∑ i', Real.log |t i - t i'| ≤
+      -(37 * n : ℝ) * Real.log ε + 2 * Kr n * ∑ i, (Uρ (t i) + Err ε (t i)) - Kr n ^ 2 * Irho := by
+  have hK : 0 < Kr n := by simp only [Kr]; positivity
+  set AB := ∑ i, ∑ i', Plog t ε (Sum.inl i) (Sum.inl i') with hAB
+  set C := ∑ i, ∑ j : Fin 16, cρ (j + 1) * Plog t ε (Sum.inl i) (Sum.inr j) with hC
+  set D :=
+    ∑ j : Fin 16, ∑ j' : Fin 16, cρ (j + 1) * cρ (j' + 1) * Plog t ε (Sum.inr j) (Sum.inr j') with
+    hD
+  have hE :
+    ∑ i,
+          (∑ i', 1 / Kr n * (1 / Kr n) * Plog t ε (Sum.inl i) (Sum.inl i') +
+            ∑ j : Fin 16, 1 / Kr n * -cρ (j + 1) * Plog t ε (Sum.inl i) (Sum.inr j)) +
+        ∑ j : Fin 16,
+          (∑ i, -cρ (j + 1) * (1 / Kr n) * Plog t ε (Sum.inr j) (Sum.inl i) +
+            ∑ j' : Fin 16, -cρ (j + 1) * -cρ (j' + 1) * Plog t ε (Sum.inr j) (Sum.inr j')) ≤
+      0 := by
+    have := atom_energy_nonpos n hn t hε
+    unfold energy at this
+    simpa only [Fintype.sum_sum_type, atomS, Plog] using this
+  rw [Finset.sum_add_distrib, Finset.sum_add_distrib] at hE
+  have hL1 :
+    ∑ i, ∑ i', 1 / Kr n * (1 / Kr n) * Plog t ε (Sum.inl i) (Sum.inl i') = (1 / Kr n) ^ 2 * AB := by
+    rw [hAB, Finset.mul_sum]
+    refine Finset.sum_congr rfl fun i _ => ?_
+    rw [Finset.mul_sum]
+    exact Finset.sum_congr rfl fun i' _ => by ring
+  have hL2 :
+    ∑ i, ∑ j : Fin 16, 1 / Kr n * -cρ (j + 1) * Plog t ε (Sum.inl i) (Sum.inr j) =
+      -(1 / Kr n) * C := by
+    rw [hC, Finset.mul_sum]
+    refine Finset.sum_congr rfl fun i _ => ?_
+    rw [Finset.mul_sum]
+    exact Finset.sum_congr rfl fun j _ => by ring
+  have hL3 :
+    ∑ j : Fin 16, ∑ i, -cρ (j + 1) * (1 / Kr n) * Plog t ε (Sum.inr j) (Sum.inl i) =
+      -(1 / Kr n) * C := by
+    rw [hC, Finset.mul_sum, Finset.sum_comm]
+    refine Finset.sum_congr rfl fun j _ => ?_
+    rw [Finset.mul_sum]
+    refine Finset.sum_congr rfl fun i _ => ?_
+    simp only [Plog]
+    rw [pairInt_log_symm t hε]
+    ring
+  have hL4 :
+    ∑ j : Fin 16, ∑ j' : Fin 16, -cρ (j + 1) * -cρ (j' + 1) * Plog t ε (Sum.inr j) (Sum.inr j') =
+      D := by
+    rw [hD]
+    refine Finset.sum_congr rfl fun j _ => Finset.sum_congr rfl fun j' _ => by ring
+  rw [hL1, hL2, hL3, hL4] at hE
+  have hE' : (1 / Kr n) ^ 2 * AB - 2 * (1 / Kr n) * C + D ≤ 0 := by linarith
+  have h1 := block_AB_sum t hinj hε
+  have h2 : C ≤ (2 * π) ^ 2 * ∑ i, (Uρ (t i) + Err ε (t i)) := by
+    rw [hC, Finset.mul_sum]
+    exact Finset.sum_le_sum fun i _ => block_C_sum t hε i
+  have h3 := block_D_sum t hε
+  rw [← hAB] at h1
+  rw [←
+    hD] at h3
+        -- combine: (1/K)² AB ≤ 2 (1/K) C - D
+  have hpi : 0 < (2 * π) ^ 2 := by positivity
+  have hK2 : 0 < (1 / Kr n) ^ 2 := by positivity
+  have key :
+    (1 / Kr n) ^ 2 *
+        ((2 * π) ^ 2 * ((37 * n : ℝ) * Real.log ε + ∑ i, ∑ i', Real.log |t i - t i'|)) ≤
+      2 * (1 / Kr n) * ((2 * π) ^ 2 * ∑ i, (Uρ (t i) + Err ε (t i))) - (2 * π) ^ 2 * Irho := by
+    have := mul_le_mul_of_nonneg_left h1 hK2.le
+    push_cast at this
+    have h2' := mul_le_mul_of_nonneg_left h2 (by positivity : (0 : ℝ) ≤ 2 * (1 / Kr n))
+    linarith
+  have key2 :
+    (1 / Kr n) ^ 2 * ((37 * n : ℝ) * Real.log ε + ∑ i, ∑ i', Real.log |t i - t i'|) ≤
+      2 * (1 / Kr n) * ∑ i, (Uρ (t i) + Err ε (t i)) - Irho := by
+    have := key
+    have e :
+      (1 / Kr n) ^ 2 *
+          ((2 * π) ^ 2 * ((37 * n : ℝ) * Real.log ε + ∑ i, ∑ i', Real.log |t i - t i'|)) =
+        (2 * π) ^ 2 *
+          ((1 / Kr n) ^ 2 * ((37 * n : ℝ) * Real.log ε + ∑ i, ∑ i', Real.log |t i - t i'|)) :=
+      by ring
+    have e2 :
+      2 * (1 / Kr n) * ((2 * π) ^ 2 * ∑ i, (Uρ (t i) + Err ε (t i))) - (2 * π) ^ 2 * Irho =
+        (2 * π) ^ 2 * (2 * (1 / Kr n) * ∑ i, (Uρ (t i) + Err ε (t i)) - Irho) :=
+      by ring
+    rw [e, e2] at this
+    exact le_of_mul_le_mul_left this hpi
+  have this := mul_le_mul_of_nonneg_left key2 (sq_nonneg (Kr n))
+  have e1 :
+    Kr n ^ 2 * ((1 / Kr n) ^ 2 * ((37 * n : ℝ) * Real.log ε + ∑ i, ∑ i', Real.log |t i - t i'|)) =
+      (37 * n : ℝ) * Real.log ε + ∑ i, ∑ i', Real.log |t i - t i'| :=
+    by field_simp
+  have e2 :
+    Kr n ^ 2 * (2 * (1 / Kr n) * ∑ i, (Uρ (t i) + Err ε (t i)) - Irho) =
+      2 * Kr n * ∑ i, (Uρ (t i) + Err ε (t i)) - Kr n ^ 2 * Irho :=
+    by field_simp
+  rw [e1, e2] at this
+  linarith
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Gaussian.lean b/LeanPool/Zeta5Irrational/Gaussian.lean
new file mode 100644
index 0000000000..1046a5770f
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Gaussian.lean
@@ -0,0 +1,105 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import Mathlib.LinearAlgebra.Complex.FiniteDimensional
+public import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
+import Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral
+import Mathlib.Tactic.Positivity.Basic
+import Mathlib.Tactic.Ring.Basic
+
+/-! # Gaussian identity and Fubini helpers for the logarithmic energy
+
+* `integral_gaussian_sub` : `∫ x, exp (-b (x - c)²) = √(π/b)`;
+* `gaussian_conv` : the completing-the-square identity
+  `∫ u : ℂ, exp (-2s‖z-u‖²) exp (-2s‖w-u‖²) = π/(4s) exp (-s‖z-w‖²)`;
+* `intervalIntegral_swap_of_continuous` : Fubini for interval integrals of continuous functions.
+-/
+
+@[expose] public section
+
+open MeasureTheory Set Real
+
+namespace Zeta5Irrational
+
+lemma integral_gaussian_sub {b : ℝ} (c : ℝ) :
+    ∫ x : ℝ, Real.exp (-b * (x - c) ^ 2) = Real.sqrt (π / b) := by
+  rw [integral_sub_right_eq_self (fun x => Real.exp (-b * x ^ 2)) c]
+  exact integral_gaussian b
+
+lemma integrable_gaussian_sub {b : ℝ} (hb : 0 < b) (c : ℝ) :
+    Integrable (fun x : ℝ => Real.exp (-b * (x - c) ^ 2)) := by
+  have := (integrable_exp_neg_mul_sq hb).comp_sub_right c
+  simpa using this
+
+/-- Completing the square in one variable. -/
+lemma sq_add_sq_eq (a b x : ℝ) :
+    (a - x) ^ 2 + (b - x) ^ 2 = 2 * (x - (a + b) / 2) ^ 2 + (a - b) ^ 2 / 2 := by ring
+
+lemma gaussian_conv_one (s a b : ℝ) :
+    ∫ x : ℝ, Real.exp (-(2 * s) * (a - x) ^ 2) * Real.exp (-(2 * s) * (b - x) ^ 2) =
+      Real.sqrt (π / (4 * s)) * Real.exp (-s * (a - b) ^ 2) := by
+  have :
+    ∀ x : ℝ,
+      Real.exp (-(2 * s) * (a - x) ^ 2) * Real.exp (-(2 * s) * (b - x) ^ 2) =
+        Real.exp (-(4 * s) * (x - (a + b) / 2) ^ 2) * Real.exp (-s * (a - b) ^ 2) := by
+    intro x
+    rw [← Real.exp_add, ← Real.exp_add]
+    congr 1
+    ring
+  simp_rw [this]
+  rw [integral_mul_const, integral_gaussian_sub]
+
+/-- The norm squared on `ℂ` in terms of real and imaginary parts. -/
+lemma normSq_eq (z : ℂ) : ‖z‖ ^ 2 = z.re ^ 2 + z.im ^ 2 := by
+  rw [Complex.sq_norm, Complex.normSq_apply]; ring
+
+/-- The completing-the-square identity on `ℂ`. -/
+lemma gaussian_conv {s : ℝ} (hs : 0 < s) (z w : ℂ) :
+    ∫ u : ℂ, Real.exp (-(2 * s) * ‖z - u‖ ^ 2) * Real.exp (-(2 * s) * ‖w - u‖ ^ 2) =
+      π / (4 * s) * Real.exp (-s * ‖z - w‖ ^ 2) := by
+  have h := Complex.volume_preserving_equiv_real_prod
+  have key :
+    ∀ u : ℂ,
+      Real.exp (-(2 * s) * ‖z - u‖ ^ 2) * Real.exp (-(2 * s) * ‖w - u‖ ^ 2) =
+        (Real.exp (-(2 * s) * (z.re - u.re) ^ 2) * Real.exp (-(2 * s) * (w.re - u.re) ^ 2)) *
+          (Real.exp (-(2 * s) * (z.im - u.im) ^ 2) * Real.exp (-(2 * s) * (w.im - u.im) ^ 2)) := by
+    intro u
+    rw [normSq_eq, normSq_eq]
+    simp only [Complex.sub_re, Complex.sub_im]
+    rw [← Real.exp_add, ← Real.exp_add, ← Real.exp_add, ← Real.exp_add]
+    congr 1; ring
+  simp_rw [key]
+  have e :=
+    h.integral_comp'
+      (fun p : ℝ × ℝ =>
+        (Real.exp (-(2 * s) * (z.re - p.1) ^ 2) * Real.exp (-(2 * s) * (w.re - p.1) ^ 2)) *
+          (Real.exp (-(2 * s) * (z.im - p.2) ^ 2) * Real.exp (-(2 * s) * (w.im - p.2) ^ 2)))
+  simp only [Complex.measurableEquivRealProd_apply] at e
+  rw [e, Measure.volume_eq_prod]
+  rw [integral_prod_mul
+      (fun x : ℝ => Real.exp (-(2 * s) * (z.re - x) ^ 2) * Real.exp (-(2 * s) * (w.re - x) ^ 2))
+      (fun y : ℝ => Real.exp (-(2 * s) * (z.im - y) ^ 2) * Real.exp (-(2 * s) * (w.im - y) ^ 2)),
+    gaussian_conv_one s _ _, gaussian_conv_one s _ _, normSq_eq]
+  simp only [Complex.sub_re, Complex.sub_im]
+  rw [mul_mul_mul_comm, ← Real.sqrt_mul (by positivity), ← Real.exp_add, ← sq,
+    Real.sqrt_sq (by positivity)]
+  congr 2
+  ring
+
+/-- Fubini for interval integrals of a continuous function of two variables. -/
+lemma intervalIntegral_swap_of_continuous {f : ℝ → ℝ → ℝ} (hf : Continuous (Function.uncurry f))
+    {a b c d : ℝ} (hab : a ≤ b) (hcd : c ≤ d) :
+    ∫ x in a..b, ∫ y in c..d, f x y = ∫ y in c..d, ∫ x in a..b, f x y := by
+  rw [intervalIntegral.integral_of_le hab, intervalIntegral.integral_of_le hcd]
+  simp_rw [intervalIntegral.integral_of_le hab, intervalIntegral.integral_of_le hcd]
+  apply integral_integral_swap
+  rw [Measure.prod_restrict]
+  refine IntegrableOn.mono_set ?_ (Set.prod_mono Ioc_subset_Icc_self Ioc_subset_Icc_self)
+  exact ContinuousOn.integrableOn_compact (isCompact_Icc.prod isCompact_Icc) hf.continuousOn
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Growth.lean b/LeanPool/Zeta5Irrational/Growth.lean
new file mode 100644
index 0000000000..27e01f8508
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Growth.lean
@@ -0,0 +1,152 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+import LeanPool.Zeta5Irrational.Growth.Assembly1
+import Mathlib.Analysis.Complex.ExponentialBounds
+public import LeanPool.Zeta5Irrational.Growth.Assembly4
+public import LeanPool.Zeta5Irrational.Constants
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+import Mathlib.Tactic.Positivity.Basic
+import Mathlib.Tactic.Ring.Basic
+
+/-! # Growth of the normaliser (Proposition 5.2, in the form needed here)
+
+`log m_K ≤ (A_eff + ε) K²` for all large `K`, with `A_eff = 1.36 < -U`.
+
+The bound assembles the four prime ranges:
+
+* small primes `p ≤ K/400`: `(6λ/400)K²` by
+the prime number theorem;
+* `K/400 < p ≤ K/20`: the tail bound, `≤ -0.05346 K²`;
+* `K/20 < p ≤ K/3`: the exact table, `(5.18) K² = 0.04279 K²`;
+* `K/3 < p ≤ 2h`: the outer table, `129101/96000 K² = 1.34480 K²`;
+
+total `≈ 1.3480 K² + o(K²)`.
+-/
+
+@[expose] public section
+
+open Filter Finset
+
+namespace Zeta5Irrational
+
+/-- The lower-order terms are `o(K²)`. -/
+lemma low_order {δ : ℝ} (hδ : 0 < δ) :
+    ∀ᶠ n : ℕ in atTop,
+      6 * (37 * (n : ℝ)) * ((Real.sqrt (200 * n) + 1) * Real.log (200 * n)) + 3 * 10 ^ 9 * n ≤
+        δ * Kr n ^ 2 := by
+  set A : ℝ := (1776 + 3 * 10 ^ 9) / (8 * δ) + 1 with hA
+  have hy : Tendsto (fun n : ℕ => (200 * (n : ℝ)) ^ ((1 : ℝ) / 4)) atTop atTop :=
+    (tendsto_rpow_atTop (by norm_num)).comp
+      ((tendsto_natCast_atTop_atTop).const_mul_atTop (by norm_num))
+  filter_upwards [hy.eventually_ge_atTop A, eventually_ge_atTop 1] with n hs hn
+  set s := (200 * (n : ℝ)) ^ ((1 : ℝ) / 4) with hsdef
+  have hn1 : (1 : ℝ) ≤ n := by exact_mod_cast hn
+  have hy0 : (0 : ℝ) ≤ 200 * n := by positivity
+  have hA1 : 1 ≤ A := by
+    rw [hA]; have : 0 ≤ (1776 + 3 * 10 ^ 9) / (8 * δ) := by positivity
+    linarith
+  have hs1 : 1 ≤ s := le_trans hA1 hs
+  have hs4 : s ^ 4 = 200 * n := by rw [hsdef, ← Real.rpow_natCast, ← Real.rpow_mul hy0]; norm_num
+  have hsqrt : Real.sqrt (200 * n) = s ^ 2 := by
+    rw [← hs4, show s ^ 4 = (s ^ 2) ^ 2 by ring, Real.sqrt_sq (by positivity)]
+  have hlog : Real.log (200 * n) ≤ 4 * s := by
+    rw [← hs4, Real.log_pow]
+    have := Real.log_le_sub_one_of_pos (show 0 < s by linarith)
+    push_cast; linarith
+  have hlog0 : 0 ≤ Real.log (200 * n) := Real.log_nonneg (by linarith)
+  rw [hsqrt]
+  have h1 : (s ^ 2 + 1) * Real.log (200 * n) ≤ 8 * s ^ 3 := by
+    have := mul_le_mul_of_nonneg_left hlog (show (0 : ℝ) ≤ s ^ 2 + 1 by positivity)
+    nlinarith [pow_le_pow_left₀ (by norm_num : (0 : ℝ) ≤ 1) hs1 2]
+  have hKr : Kr n ^ 2 = 8 * s ^ 4 * n := by simp only [Kr]; rw [hs4]; ring
+  rw [hKr]
+  have hs3 : 1 ≤ s ^ 3 := one_le_pow₀ hs1
+  have hkey : 1776 * s ^ 3 + 3 * 10 ^ 9 ≤ 8 * δ * s ^ 4 := by
+    have hAs : (1776 + 3 * 10 ^ 9) / (8 * δ) ≤ s := by linarith
+    have : 1776 + 3 * 10 ^ 9 ≤ 8 * δ * s := by rw [div_le_iff₀ (by positivity)] at hAs; linarith
+    nlinarith [mul_le_mul_of_nonneg_right this (show (0 : ℝ) ≤ s ^ 3 by positivity)]
+  have hn0 : (0 : ℝ) ≤ n := by positivity
+  nlinarith [mul_le_mul_of_nonneg_left h1 (show (0 : ℝ) ≤ 6 * (37 * n) by positivity),
+    mul_le_mul_of_nonneg_left hkey hn0]
+
+lemma theta_le_two (y : ℝ) (hy : 0 ≤ y) : Chebyshev.theta y ≤ 2 * y := by
+  have h := Chebyshev.theta_le_log4_mul_x hy
+  have : Real.log 4 < 2 := by
+    rw [show (4 : ℝ) = 2 ^ 2 by norm_num, Real.log_pow]
+    have := Real.log_two_lt_d9; push_cast; linarith
+  nlinarith
+
+/-- **Growth of the normaliser.** -/
+theorem growth_mN :
+    ∀ ε : ℝ, 0 < ε → ∀ᶠ n in atTop, Real.log (mN n) ≤ ((Aeff : ℝ) + ε) * Kr n ^ 2 := by
+  intro ε hε
+  set δ : ℝ := min (ε / 20) (1 / 10000) with hδdef
+  have hδ : 0 < δ := lt_min (by positivity) (by norm_num)
+  have hδ1 : δ ≤ ε / 20 := min_le_left _ _
+  have hδ2 : δ ≤ 1 / 10000 := min_le_right _ _
+  have hKr : Tendsto Kr atTop atTop :=
+    (tendsto_natCast_atTop_atTop).const_mul_atTop (by norm_num : (0 : ℝ) < 40)
+  have hy10 : Tendsto (fun n : ℕ => ((n / 10 : ℕ) : ℝ)) atTop atTop :=
+    tendsto_natCast_atTop_atTop.comp (Nat.tendsto_div_const_atTop (by norm_num))
+  have Eθ := hy10.eventually (theta_eventually_close hδ)
+  filter_upwards [hKr.eventually (tail_psum_limit hδ), hKr.eventually (inner_psum_limit hδ),
+    hKr.eventually (outer_psum_limit hδ), Eθ, low_order hδ, eventually_ge_atTop 10400] with n h2
+    h3 h4 hθ hlow hn
+  have hK0 : 0 < Kr n ^ 2 := by
+    simp only [Kr]; have : (1 : ℝ) ≤ n := by exact_mod_cast (by omega : 1 ≤ n)
+    positivity
+  rw [div_le_iff₀ hK0] at h2 h3 h4
+  have hK : Kr n ^ 2 = 1600 * (n : ℝ) ^ 2 := by simp only [Kr]; ring
+  rw [log_mN,
+    sum_split4 _ (a := n / 10) (b := 2 * n) (c := 40 * n / 3) (N := 2 * (37 * n)) (by omega)
+      (by omega) (by omega)]
+  have r1 := small_range_le n (by omega)
+  have r2 := range2_le hn
+  have r3 := range3_le hn
+  have r4 :=
+    range4_le
+      hn
+        -- the `θ` terms
+  have hθ1 : Chebyshev.theta ((n / 10 : ℕ) : ℝ) ≤ (1 + δ) * (n / 10 : ℝ) := by
+    have h10 : ((n / 10 : ℕ) : ℝ) ≤ (n : ℝ) / 10 := by
+      rw [le_div_iff₀ (by norm_num)];
+      exact_mod_cast (Nat.div_mul_le_self n 10)
+    have := (abs_le.mp hθ).2
+    nlinarith
+  have t2 := theta_le_two ((2 * n : ℕ) : ℝ) (by positivity)
+  have t3 := theta_le_two ((40 * n / 3 : ℕ) : ℝ) (by positivity)
+  have t4 := theta_le_two ((2 * (37 * n) : ℕ) : ℝ) (by positivity)
+  have hn3 : ((40 * n / 3 : ℕ) : ℝ) ≤ 40 * n / 3 := by
+    rw [le_div_iff₀ (by norm_num)];
+    exact_mod_cast (Nat.div_mul_le_self (40 * n) 3)
+  push_cast at t2 t4 r1 r2 r4
+  have hA : (Aeff : ℝ) = 136 / 100 := by unfold Aeff; norm_num
+  rw [hA]
+  have hn0 : (0 : ℝ) ≤ n := by
+    positivity
+      -- the main terms
+  have hmain : 6 * (37 * (n : ℝ)) * ((1 + δ) * (n / 10)) = 22.2 * (1 + δ) / 1600 * Kr n ^ 2 := by
+    rw [hK]; ring
+  have hsum :
+    -6843153 / 128000000 + 322437603634266857629 / 7535670527041937280000 + 129101 / 96000 +
+        22.2 / 1600 <
+      (136 / 100 : ℝ) - 1 / 1000 :=
+    by norm_num
+  have i1 := mul_le_mul_of_nonneg_left hθ1 (show (0 : ℝ) ≤ 6 * (37 * n) by positivity)
+  have i2 := mul_le_mul_of_nonneg_left hδ2 hK0.le
+  have i3 := mul_le_mul_of_nonneg_left hδ1 hK0.le
+  have i4 : (10 : ℝ) ^ 8 * Chebyshev.theta ((40 * n / 3 : ℕ) : ℝ) ≤ 10 ^ 8 * (2 * (40 * n / 3)) :=
+    by have := t3.trans (mul_le_mul_of_nonneg_left hn3 (by norm_num)); linarith
+  have hε0 : 0 ≤ ε * Kr n ^ 2 := by positivity
+  linarith
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Growth/Assembly1.lean b/LeanPool/Zeta5Irrational/Growth/Assembly1.lean
new file mode 100644
index 0000000000..b5130e1a38
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Growth/Assembly1.lean
@@ -0,0 +1,199 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.PrimeSum
+public import LeanPool.Zeta5Irrational.Norm
+import Mathlib.Tactic.ReduceModChar
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Tactic.ENatToNat
+import Std.Tactic.BVDecide.Normalize.Prop
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+import Mathlib.Tactic.Positivity.Basic
+import Mathlib.Tactic.Ring.Basic
+
+/-! # Growth: decomposition of `log m_K` and the small primes
+-/
+
+@[expose] public section
+
+open Finset Filter
+
+namespace Zeta5Irrational
+
+/-- `log m_K = ∑_{k ≤ 2h} c(k) (-L_k)`. -/
+lemma log_mN (n : ℕ) :
+    Real.log (mN n : ℝ) = ∑ k ∈ Icc 1 (2 * (37 * n)), cPrime k * (-(Lp n k : ℝ)) := by
+  unfold mN
+  push_cast
+  rw [Real.log_prod
+      (fun p hp => by
+        have := (Finset.mem_filter.mp hp).2.pos
+        exact zpow_ne_zero _ (by exact_mod_cast this.ne'))]
+  simp only [Real.log_zpow]
+  rw [Finset.sum_filter]
+  have e : range (2 * (37 * n) + 1) = insert 0 (Icc 1 (2 * (37 * n))) := by
+    ext k;
+    simp only [Finset.mem_range, Finset.mem_insert, Finset.mem_Icc]; omega
+  rw [e, Finset.sum_insert (by simp)]
+  simp only [Nat.not_prime_zero, ite_false, zero_add]
+  refine Finset.sum_congr rfl fun k _ => ?_
+  unfold cPrime
+  split_ifs <;> push_cast <;> ring
+
+/-- Splitting `(0, N]` at `a ≤ b ≤ c`. -/
+lemma sum_split4 (F : ℕ → ℝ) {a b c N : ℕ} (hab : a ≤ b) (hbc : b ≤ c) (hcN : c ≤ N) :
+    ∑ k ∈ Icc 1 N, F k =
+      ∑ k ∈ Ioc 0 a, F k + ∑ k ∈ Ioc a b, F k + ∑ k ∈ Ioc b c, F k + ∑ k ∈ Ioc c N, F k := by
+  have e : Icc 1 N = Ioc 0 N := by ext k; simp only [Finset.mem_Icc, Finset.mem_Ioc]; omega
+  rw [e, ← Finset.sum_Ioc_consecutive _ (Nat.zero_le a) (hab.trans (hbc.trans hcN)), ←
+    Finset.sum_Ioc_consecutive _ hab (hbc.trans hcN), ← Finset.sum_Ioc_consecutive _ hbc hcN]
+  ring
+
+/-! ### Small primes -/
+
+lemma log_mul_natLog_le {k Z : ℕ} (hk : 2 ≤ k) (hZ : 1 ≤ Z) :
+    (Nat.log k Z : ℝ) * Real.log k ≤ Real.log Z := by
+  have h := Nat.pow_log_le_self k (show Z ≠ 0 by omega)
+  have h' : ((k ^ Nat.log k Z : ℕ) : ℝ) ≤ (Z : ℝ) := by exact_mod_cast h
+  have hk0 : (0 : ℝ) < k := by exact_mod_cast (show 0 < k by omega)
+  rw [← Real.log_pow]
+  push_cast at h'
+  exact Real.log_le_log (pow_pos hk0 _) h'
+
+lemma cPrime_natLog_le {k Z : ℕ} (hZ : 1 ≤ Z) :
+    cPrime k * (Nat.log k Z : ℝ) ≤ cPrime k + (if k * k ≤ Z then Real.log Z else 0) := by
+  unfold cPrime
+  split_ifs with hp hkk
+  · have := log_mul_natLog_le hp.two_le hZ
+    have : 0 ≤ Real.log k := Real.log_nonneg (by exact_mod_cast hp.one_lt.le)
+    nlinarith
+  · have hlt : Nat.log k Z ≤ 1 := by
+      have : Z < k ^ 2 := by rw [sq]; omega
+      exact Nat.lt_succ_iff.mp (Nat.log_lt_of_lt_pow (by omega) this)
+    have : 0 ≤ Real.log k := Real.log_nonneg (by exact_mod_cast hp.one_lt.le)
+    have : (Nat.log k Z : ℝ) ≤ 1 := by exact_mod_cast hlt
+    nlinarith
+  · simp only [zero_add, zero_mul]
+    have : (1 : ℝ) ≤ (Z : ℝ) := by exact_mod_cast hZ
+    exact Real.log_nonneg this
+  · simp
+
+lemma sum_cPrime_le_theta (M : ℕ) : ∑ k ∈ Ioc 0 M, cPrime k ≤ Chebyshev.theta M := by
+  rw [theta_eq_sum_cPrime, Nat.floor_natCast]
+  apply Finset.sum_le_sum_of_subset_of_nonneg
+  · intro k hk; simp only [Finset.mem_Ioc, Finset.mem_Icc] at hk ⊢; omega
+  · intro k _ _; unfold cPrime; split_ifs with h
+    · exact Real.log_nonneg (by exact_mod_cast h.one_lt.le)
+    · exact le_refl _
+
+lemma cPrime_nonneg' (k : ℕ) : 0 ≤ cPrime k := by
+  unfold cPrime; split_ifs with h
+  · exact Real.log_nonneg (by exact_mod_cast h.one_lt.le)
+  · exact le_refl _
+
+/-- The small-prime range. -/
+lemma small_range_le (n : ℕ) (hn : 1 ≤ n) :
+    ∑ k ∈ Ioc 0 (n / 10), cPrime k * (-(Lp n k : ℝ)) ≤
+      6 * (37 * n) *
+          (Chebyshev.theta (n / 10 : ℕ) + (Real.sqrt (200 * n) + 1) * Real.log (200 * n)) +
+        (37 * n) * 100 := by
+  have hZ : 1 ≤ 200 * n := by omega
+  have hterm :
+    ∀ k ∈ Ioc 0 (n / 10),
+      cPrime k * (-(Lp n k : ℝ)) ≤
+        6 * (37 * n) * (cPrime k + (if k * k ≤ 200 * n then Real.log (200 * n : ℕ) else 0)) +
+          (37 * n) * (if k ≤ 24 then 4 else 0) := by
+    intro k hk
+    have hk' := Finset.mem_Ioc.mp hk
+    have hLp : Lp n k = Lsmall n k := by unfold Lp Mcut; rw [ite_eq_left (by omega)]
+    rw [hLp]; unfold Lsmall
+    simp only [Int.cast_neg, Int.cast_add, Int.cast_mul, Int.cast_natCast, Int.cast_ofNat,
+      neg_neg, Nat.cast_mul, Nat.cast_ofNat]
+    have h1 := cPrime_natLog_le (k := k) hZ
+    have hc0 := cPrime_nonneg' k
+    have h24 : cPrime k * (padicValNat k 24 : ℝ) ≤ (if k ≤ 24 then 4 else 0) := by
+      unfold cPrime
+      split_ifs with hp h24
+      · have hv : k ^ padicValNat k 24 ∣ 24 := by have := Fact.mk hp; exact pow_padicValNat_dvd
+        have := Nat.le_of_dvd (by norm_num) hv
+        have h' : ((k ^ padicValNat k 24 : ℕ) : ℝ) ≤ 24 := by exact_mod_cast this
+        have hk0 : (0 : ℝ) < k := by exact_mod_cast hp.pos
+        have : (padicValNat k 24 : ℝ) * Real.log k ≤ Real.log 24 := by
+          rw [← Real.log_pow];
+          push_cast at h'; exact Real.log_le_log (pow_pos hk0 _) h'
+        have : Real.log 24 < 4 := by
+          have := Real.log_le_sub_one_of_pos (show (0 : ℝ) < 24 / 16 by norm_num)
+          have h16 : Real.log 16 = 4 * Real.log 2 := by
+            rw [show (16 : ℝ) = 2 ^ 4 by norm_num, Real.log_pow]; norm_num
+          have := Real.log_two_lt_d9
+          have e : Real.log 24 = Real.log (24 / 16) + Real.log 16 := by
+            rw [← Real.log_mul (by norm_num) (by norm_num)]; norm_num
+          linarith
+        linarith
+      · have : padicValNat k 24 = 0 :=
+          padicValNat.eq_zero_of_not_dvd
+            (fun h => by have := Nat.le_of_dvd (by norm_num) h; omega)
+        rw [this]; simp
+      · simp
+      · simp
+    have hn0 : (0 : ℝ) ≤ 37 * n := by positivity
+    have e1 := mul_le_mul_of_nonneg_left h1 (show (0 : ℝ) ≤ 6 * (37 * n) by positivity)
+    have e2 := mul_le_mul_of_nonneg_left h24 hn0
+    push_cast at e1
+    nlinarith
+  refine (Finset.sum_le_sum hterm).trans ?_
+  rw [Finset.sum_add_distrib, ← Finset.mul_sum, ← Finset.mul_sum, Finset.sum_add_distrib]
+  have hθ :=
+    sum_cPrime_le_theta
+      (n / 10)
+        -- the square-root part
+  have hsq :
+    ∑ k ∈ Ioc 0 (n / 10), (if k * k ≤ 200 * n then Real.log (200 * n : ℕ) else 0) ≤
+      (Real.sqrt (200 * n) + 1) * Real.log (200 * n) := by
+    rw [Finset.sum_ite, Finset.sum_const_zero, add_zero, Finset.sum_const, nsmul_eq_mul]
+    have hlog : 0 ≤ Real.log (200 * n : ℕ) := Real.log_nonneg (by exact_mod_cast hZ)
+    have hcard :
+      (((Ioc 0 (n / 10)).filter fun k => k * k ≤ 200 * n).card : ℝ) ≤ Real.sqrt (200 * n) + 1 := by
+      have hsub :
+        (Ioc 0 (n / 10)).filter (fun k => k * k ≤ 200 * n) ⊆ Icc 1 (Nat.sqrt (200 * n)) := by
+        intro k hk
+        simp only [Finset.mem_filter, Finset.mem_Ioc, Finset.mem_Icc] at hk ⊢
+        exact ⟨hk.1.1, Nat.le_sqrt.mpr hk.2⟩
+      have := Finset.card_le_card hsub
+      rw [Nat.card_Icc] at this
+      have h2 : ((Nat.sqrt (200 * n) : ℕ) : ℝ) ≤ Real.sqrt (200 * n) := by
+        have := Real.nat_sqrt_le_real_sqrt (a := 200 * n)
+        push_cast at this; exact this
+      have :
+        (((Ioc 0 (n / 10)).filter fun k => k * k ≤ 200 * n).card : ℝ) ≤
+          (Nat.sqrt (200 * n) : ℝ) :=
+        by exact_mod_cast (show _ ≤ Nat.sqrt (200 * n) by omega)
+      linarith
+    push_cast at hlog ⊢
+    exact mul_le_mul_of_nonneg_right hcard hlog
+  have h24 : ∑ k ∈ Ioc 0 (n / 10), (if k ≤ 24 then (4 : ℝ) else 0) ≤ 100 := by
+    rw [Finset.sum_ite, Finset.sum_const_zero, add_zero, Finset.sum_const, nsmul_eq_mul]
+    have : ((Ioc 0 (n / 10)).filter fun k => k ≤ 24).card ≤ 24 := by
+      calc
+        _ ≤ (Icc 1 24).card :=
+          Finset.card_le_card
+            (fun k hk => by
+              simp only [Finset.mem_filter, Finset.mem_Ioc, Finset.mem_Icc] at hk ⊢;
+              omega)
+        _ = 24 := by simp
+    have : (((Ioc 0 (n / 10)).filter fun k => k ≤ 24).card : ℝ) ≤ 24 := by exact_mod_cast this
+    linarith
+  have hn0 : (0 : ℝ) ≤ 37 * n := by positivity
+  push_cast at hsq ⊢
+  nlinarith [mul_le_mul_of_nonneg_left h24 hn0,
+    mul_le_mul_of_nonneg_left (add_le_add hθ hsq) (show (0 : ℝ) ≤ 6 * (37 * n) by positivity)]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Growth/Assembly2.lean b/LeanPool/Zeta5Irrational/Growth/Assembly2.lean
new file mode 100644
index 0000000000..6685482501
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Growth/Assembly2.lean
@@ -0,0 +1,97 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+import LeanPool.Zeta5Irrational.Growth.Constants
+public import LeanPool.Zeta5Irrational.Growth.OuterPrime
+public import LeanPool.Zeta5Irrational.Growth.TablePrime
+public import LeanPool.Zeta5Irrational.Constants
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+import Mathlib.Tactic.Positivity.Basic
+import Mathlib.Tactic.Ring.Basic
+
+import Mathlib.Algebra.Order.Floor.Semifield
+
+/-! # Growth: the three prime ranges `K/400 < p ≤ K/20`, `K/20 < p ≤ K/3`, `K/3 < p ≤ 2h`
+-/
+
+@[expose] public section
+
+open Finset Filter
+
+namespace Zeta5Irrational
+
+lemma floor_Kr_div (n d : ℕ) (_hd : 0 < d) : ⌊Kr n / d⌋₊ = 40 * n / d := by
+  rw [show Kr n / d = ((40 * n : ℕ) : ℝ) / (d : ℕ) by simp [Kr]]
+  exact Nat.floor_div_eq_div _ _
+
+lemma floor_Kr_outer (n : ℕ) : ⌊Kr n / (20 / 37)⌋₊ = 74 * n := by
+  rw [show Kr n / (20 / 37) = ((74 * n : ℕ) : ℝ) by simp [Kr]; ring]
+  exact Nat.floor_natCast _
+
+/-! ### Bounds on the additive constants -/
+
+lemma Bmax_le {n p Q : ℕ} (hq : 40 * n / p ≤ Q) : Bmax n p ≤ (5 * (Q : ℝ) + 10) * (3 * Q + 6) := by
+  unfold Bmax ktopI
+  simp only [Int.cast_add, Int.cast_mul, Int.cast_ofNat, Int.cast_natCast]
+  have hQ : ((40 * n / p : ℕ) : ℝ) ≤ Q := by exact_mod_cast hq
+  have h0' : (0 : ℝ) ≤ ((3 * n / p : ℕ) : ℝ) := by positivity
+  generalize ((40 * n / p : ℕ) : ℝ) = qR at hQ ⊢
+  generalize ((3 * n / p : ℕ) : ℝ) = qR' at h0' ⊢
+  set D := max 0 ((8 * qR + 20) / 2 - ((5 + 12 * qR' - 2 * qR) - 4) / 2) with hD
+  have hD0 : 0 ≤ D := le_max_left _ _
+  have hD1 : D ≤ 5 * (Q : ℝ) + 10 := max_le (by positivity) (by linarith)
+  nlinarith
+
+lemma ktop_sub_klo (n p : ℕ) :
+    ((ktopI n p - kloI n p : ℤ) : ℝ) = 10 * ((40 * n / p : ℕ) : ℝ) + 26 := by
+  unfold ktopI kloI
+  simp only [Int.cast_sub, Int.cast_add, Int.cast_mul, Int.cast_ofNat, Int.cast_natCast,
+    Int.cast_neg]
+  ring
+
+lemma Ctail_le {n p : ℕ} (hp : 0 < p) (hq : 40 * n / p < 400) (hn : (n : ℝ) ≤ 10 * p) :
+    Ctail n p ≤ 10 ^ 7 := by
+  unfold Ctail
+  rw [ktop_sub_klo]
+  unfold L0I
+  have hB := Bmax_le (n := n) (p := p) (Q := 399) (by omega)
+  have hQ : ((40 * n / p : ℕ) : ℝ) ≤ 399 := by exact_mod_cast (show 40 * n / p ≤ 399 by omega)
+  have hpR : (0 : ℝ) < p := by exact_mod_cast hp
+  have h1 : 4 * (37 * (n : ℝ)) / p ≤ 1480 := by rw [div_le_iff₀ hpR]; linarith
+  have h0 : (0 : ℝ) ≤ ((40 * n / p : ℕ) : ℝ) := by positivity
+  simp only [Nat.cast_add, Nat.cast_mul, Nat.cast_ofNat]
+  generalize ((40 * n / p : ℕ) : ℝ) = qR at hQ h0 ⊢
+  norm_num at hB
+  nlinarith
+
+lemma Ctab_le {n p : ℕ} (hq : 40 * n / p ≤ 20) {k : ℤ} (hk1 : kloI n p ≤ k)
+    (hk2 : k ≤ ktopI n p) : Ctab n p k ≤ 10 ^ 8 := by
+  unfold Ctab
+  have hq' : 3 * n / p ≤ 40 * n / p := Nat.div_le_div_right (by omega)
+  have hk1' : -2 * ((40 * n / p : ℕ) : ℤ) - 6 ≤ k := hk1
+  have hk2' : k ≤ 8 * ((40 * n / p : ℕ) : ℤ) + 20 := hk2
+  have hs := sum_abs_bIn_le hq' hk1' hk2'
+  have hB := Bmax_le (n := n) (p := p) (Q := 20) hq
+  rw [ktop_sub_klo]
+  unfold L0I
+  have hQ : ((40 * n / p : ℕ) : ℝ) ≤ 20 := by exact_mod_cast hq
+  have h0 : (0 : ℝ) ≤ ((40 * n / p : ℕ) : ℝ) := by positivity
+  simp only [Nat.cast_add, Nat.cast_mul, Nat.cast_ofNat]
+  generalize ((40 * n / p : ℕ) : ℝ) = qR at hQ h0 hs ⊢
+  norm_num at hB
+  nlinarith
+
+lemma Cout_le {n p : ℕ} (hq : 40 * n / p ≤ 2) : Cout n p ≤ 10 ^ 5 := by
+  unfold Cout
+  have := sum_abs_bGO_le hq
+  linarith
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Growth/Assembly3.lean b/LeanPool/Zeta5Irrational/Growth/Assembly3.lean
new file mode 100644
index 0000000000..bacfb775b8
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Growth/Assembly3.lean
@@ -0,0 +1,181 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+import LeanPool.Zeta5Irrational.Growth.Assembly2
+import LeanPool.Zeta5Irrational.Growth.Assembly1
+public import LeanPool.Zeta5Irrational.PrimeSumStep
+public import LeanPool.Zeta5Irrational.Growth.PieceIdx
+public import LeanPool.Zeta5Irrational.Growth.InnerSum
+public import LeanPool.Zeta5Irrational.Growth.OuterPrime
+public import LeanPool.Zeta5Irrational.Constants
+import Mathlib.Tactic.ReduceModChar
+import Mathlib.Data.Int.Star
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+import Mathlib.Tactic.Ring.Basic
+
+/-! # Growth: the per-range sums as prime sums
+-/
+
+@[expose] public section
+
+open Finset Filter
+
+namespace Zeta5Irrational
+
+lemma range_sum_le {a b : ℕ} (F : ℕ → ℝ) (f : ℝ → ℝ) (K C : ℝ) (hC : 0 ≤ C)
+    (h : ∀ k ∈ Ioc a b, k.Prime → F k ≤ k * f (K / k) + C) :
+    ∑ k ∈ Ioc a b, cPrime k * F k ≤
+      ∑ k ∈ Ioc a b, (k : ℝ) * cPrime k * f (K / k) + C * Chebyshev.theta b := by
+  have hθ : ∑ k ∈ Ioc a b, cPrime k ≤ Chebyshev.theta b := by
+    refine le_trans ?_ (sum_cPrime_le_theta b)
+    apply Finset.sum_le_sum_of_subset_of_nonneg
+    · intro k hk; simp only [Finset.mem_Ioc] at hk ⊢; omega
+    · intro k _ _; exact cPrime_nonneg' k
+  have hterm : ∀ k ∈ Ioc a b, cPrime k * F k ≤ (k : ℝ) * cPrime k * f (K / k) + C * cPrime k := by
+    intro k hk
+    by_cases hp : k.Prime
+    · have := h k hk hp
+      have hc := cPrime_nonneg' k
+      nlinarith [mul_le_mul_of_nonneg_left this hc]
+    · simp [cPrime, hp]
+  refine (Finset.sum_le_sum hterm).trans ?_
+  rw [Finset.sum_add_distrib, ← Finset.mul_sum]
+  nlinarith [mul_le_mul_of_nonneg_left hθ hC]
+
+/-- Facts about primes above `n/10` for large `n`. -/
+lemma big_prime_facts {n k : ℕ} (hn : 10400 ≤ n) (hk : n / 10 < k) (hp : k.Prime) :
+    7 ≤ k ∧
+      2 * (k / 2) + 1 = k ∧
+        ¬k * Mcut ≤ 40 * n ∧
+          2 * (40 * n) < k ^ 2 ∧
+            5 + 2 * (18 * n + 2 * (37 * n)) - 2 * (40 * n) + 1 < k ^ 2 ∧ (n : ℝ) ≤ 10 * k := by
+  have h10 : n + 1 ≤ 10 * k := by omega
+  have hsq : 10400 * n + 700 ≤ k ^ 2 * 100 := by nlinarith
+  have hodd' : 2 * (k / 2) + 1 = k := by
+    rcases Nat.even_or_odd k with ⟨j, hj⟩ | ⟨j, hj⟩
+    · exfalso
+      have : 2 ∣ k := ⟨j, by omega⟩
+      rcases hp.eq_one_or_self_of_dvd 2 this with h | h
+      · omega
+      · omega
+    · omega
+  refine ⟨by omega, hodd', by unfold Mcut; omega, by omega, by omega, ?_⟩
+  exact_mod_cast (show n ≤ 10 * k by omega)
+
+/-- **Range 2** (inner tail). -/
+lemma range2_le {n : ℕ} (hn : 10400 ≤ n) :
+    ∑ k ∈ Ioc (n / 10) (2 * n), cPrime k * (-(Lp n k : ℝ)) ≤
+      psum (fun x => x * Ftail x + 27 / 16) 20 400 (Kr n) +
+        10 ^ 7 * Chebyshev.theta (2 * n : ℕ) := by
+  have hps :
+    psum (fun x => x * Ftail x + 27 / 16) 20 400 (Kr n) =
+      ∑ k ∈ Ioc (n / 10) (2 * n),
+        (k : ℝ) * cPrime k * ((fun x => x * Ftail x + 27 / 16) (Kr n / k)) := by
+    unfold psum
+    rw [show (400 : ℝ) = ((400 : ℕ) : ℝ) by norm_num, show (20 : ℝ) = ((20 : ℕ) : ℝ) by norm_num,
+      floor_Kr_div n 400 (by norm_num), floor_Kr_div n 20 (by norm_num),
+      show 40 * n / 400 = n / 10 by omega, show 40 * n / 20 = 2 * n by omega]
+  rw [hps]
+  refine
+    range_sum_le (fun k => -(Lp n k : ℝ)) (fun x => x * Ftail x + 27 / 16) (Kr n) (10 ^ 7)
+      (by norm_num) fun k hk hp => ?_
+  have hk' := Finset.mem_Ioc.mp hk
+  obtain ⟨hp7, hodd, hsmall, hsq, hdeg, hn10⟩ := big_prime_facts hn hk'.1 hp
+  have := Fact.mk hp
+  have h := tail_Lp_le (p := k) (n := n) (by omega) hodd hp7 hsmall (by omega) hsq hdeg
+  have hC :=
+    Ctail_le (n := n) (p := k) hp.pos
+      (by
+        unfold Mcut at hsmall; rw [not_le] at hsmall
+        exact (Nat.div_lt_iff_lt_mul hp.pos).mpr (by linarith))
+      hn10
+  have e : Kr n / k = 40 * n / k := by simp [Kr]
+  simp only [e]
+  linarith
+
+/-- The inner table as a function. -/
+noncomputable def fIn (x : ℝ) : ℝ :=
+  gIn (pieceIdx tIn 125 x) x
+
+lemma tIn_zero : tIn 0 = 3 := by simp [tIn, tInL]
+
+lemma tIn_last : tIn 125 = 20 := by simp [tIn, tInL]
+
+/-- **Range 3** (inner table). -/
+lemma range3_le {n : ℕ} (hn : 10400 ≤ n) :
+    ∑ k ∈ Ioc (2 * n) (40 * n / 3), cPrime k * (-(Lp n k : ℝ)) ≤
+      psum fIn 3 20 (Kr n) + 10 ^ 8 * Chebyshev.theta (40 * n / 3 : ℕ) := by
+  have hps :
+    psum fIn 3 20 (Kr n) = ∑ k ∈ Ioc (2 * n) (40 * n / 3), (k : ℝ) * cPrime k * fIn (Kr n / k) := by
+    unfold psum
+    rw [show (3 : ℝ) = ((3 : ℕ) : ℝ) by norm_num, show (20 : ℝ) = ((20 : ℕ) : ℝ) by norm_num,
+      floor_Kr_div n 3 (by norm_num), floor_Kr_div n 20 (by norm_num),
+      show 40 * n / 20 = 2 * n by omega]
+  rw [hps]
+  refine range_sum_le (fun k => -(Lp n k : ℝ)) fIn (Kr n) (10 ^ 8) (by norm_num) fun k hk hp => ?_
+  have hk' := Finset.mem_Ioc.mp hk
+  obtain ⟨hp7, hodd, hsmall, hsq, hdeg, hn10⟩ := big_prime_facts hn (by omega) hp
+  have := Fact.mk hp
+  have hin : 3 * k ≤ 40 * n := by have := Nat.div_mul_le_self (40 * n) 3; omega
+  have hkR : (0 : ℝ) < k := by exact_mod_cast hp.pos
+  set x : ℝ := 40 * n / k with hx
+  have hx3 : 3 ≤ x := by rw [hx, le_div_iff₀ hkR]; exact_mod_cast (show 3 * k ≤ 40 * n by omega)
+  have hx20 : x < 20 := by
+    rw [hx, div_lt_iff₀ hkR];
+    exact_mod_cast (show 40 * n < 20 * k by omega)
+  obtain ⟨hi, hx1, hx2⟩ :=
+    pieceIdx_spec tIn_mono (x := x) (by rw [tIn_zero]; exact hx3) (by rw [tIn_last]; exact hx20)
+  set i := pieceIdx tIn 125 x with hidef
+  have hmem : x ∈ Set.Ico (tIn i) (tIn (i + 1)) := ⟨hx1, hx2⟩
+  have hkr := inner_table_k i hi x hmem
+  have hfl : ⌊x⌋₊ = 40 * n / k := by
+    rw [hx, show (40 * n : ℝ) / k = ((40 * n : ℕ) : ℝ) / (k : ℕ) by push_cast; ring,
+      Nat.floor_div_eq_div]
+  rw [hfl] at hkr
+  have hk1 : kloI n k ≤ kIn i := hkr.1
+  have hk2 : kIn i ≤ ktopI n k := hkr.2
+  have h := table_Lp_le (p := k) (n := n) (by omega) hodd hp7 hsmall hin hsq hdeg (kIn i) hk1 hk2
+  have htab := inner_table i hi x hmem
+  have hq20 : 40 * n / k ≤ 20 := by
+    have : 40 * n < 20 * k := by omega
+    exact Nat.le_of_lt_succ ((Nat.div_lt_iff_lt_mul hp.pos).mpr (by omega))
+  have hC := Ctab_le (n := n) (p := k) hq20 hk1 hk2
+  unfold fIn
+  rw [← hidef, ← htab]
+  rw [← hx] at h
+  linarith
+
+/-- **Range 4** (outer range). -/
+lemma range4_le {n : ℕ} (hn : 10400 ≤ n) :
+    ∑ k ∈ Ioc (40 * n / 3) (2 * (37 * n)), cPrime k * (-(Lp n k : ℝ)) ≤
+      psum Eout (20 / 37) 3 (Kr n) + 10 ^ 5 * Chebyshev.theta (2 * (37 * n) : ℕ) := by
+  have hps :
+    psum Eout (20 / 37) 3 (Kr n) =
+      ∑ k ∈ Ioc (40 * n / 3) (2 * (37 * n)), (k : ℝ) * cPrime k * Eout (Kr n / k) := by
+    unfold psum
+    rw [show (3 : ℝ) = ((3 : ℕ) : ℝ) by norm_num, floor_Kr_div n 3 (by norm_num), floor_Kr_outer,
+      show 74 * n = 2 * (37 * n) by ring]
+  rw [hps]
+  refine
+    range_sum_le (fun k => -(Lp n k : ℝ)) Eout (Kr n) (10 ^ 5) (by norm_num) fun k hk hp => ?_
+  have hk' := Finset.mem_Ioc.mp hk
+  obtain ⟨hp7, hodd, _, hsq, _, _⟩ := big_prime_facts hn (by omega) hp
+  have := Fact.mk hp
+  have hout : 40 * n < 3 * k := by
+    have := Nat.lt_div_mul_add (a := 40 * n) (show 0 < 3 by norm_num); omega
+  have h := outer_Lp_le (p := k) (n := n) (by omega) hodd hp7 hout hsq
+  have hq2 : 40 * n / k ≤ 2 := Nat.le_of_lt_succ ((Nat.div_lt_iff_lt_mul hp.pos).mpr (by omega))
+  have hC := Cout_le (n := n) (p := k) hq2
+  have e : Kr n / k = 40 * n / k := by simp [Kr]
+  rw [e]
+  linarith
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Growth/Assembly4.lean b/LeanPool/Zeta5Irrational/Growth/Assembly4.lean
new file mode 100644
index 0000000000..2bdd813b1c
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Growth/Assembly4.lean
@@ -0,0 +1,130 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+import LeanPool.Zeta5Irrational.PrimeSumIntegral
+public import LeanPool.Zeta5Irrational.Growth.TailIntegral
+import LeanPool.Zeta5Irrational.Growth.TableIntegrals
+public import LeanPool.Zeta5Irrational.Growth.OuterTable
+import LeanPool.Zeta5Irrational.Growth.TableGen
+import Mathlib.Tactic.ReduceModChar
+public import LeanPool.Zeta5Irrational.Growth.Assembly3
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+import Mathlib.Tactic.Positivity.Basic
+import Mathlib.Tactic.Ring.Basic
+
+/-! # Growth: the limits of the three prime sums
+-/
+
+@[expose] public section
+
+open Finset Filter
+
+namespace Zeta5Irrational
+
+lemma aIn_le : ∀ i < 125, |((aInL.getD i 0 : ℚ) : ℝ)| ≤ 100 := by
+  intro i hi
+  interval_cases i <;> simp only [aInL, List.getD_cons_succ, List.getD_cons_zero] <;>
+    norm_num [abs_le]
+
+lemma aOut_le : ∀ i < 13, |((aOutL.getD i 0 : ℚ) : ℝ)| ≤ 100 := by
+  intro i hi
+  interval_cases i <;> simp only [aOutL, List.getD_cons_succ, List.getD_cons_zero] <;>
+    norm_num [abs_le]
+
+lemma lin_lip (a b : ℝ) (L : ℝ) (ha : |a| ≤ L) (x y : ℝ) :
+    |(a * x + b) - (a * y + b)| ≤ L * |x - y| := by
+  rw [show (a * x + b) - (a * y + b) = a * (x - y) by ring, abs_mul]
+  exact mul_le_mul_of_nonneg_right ha (abs_nonneg _)
+
+/-- **The inner table sum.** -/
+lemma inner_psum_limit {δ : ℝ} (hδ : 0 < δ) :
+    ∀ᶠ K : ℝ in atTop,
+      psum fIn 3 20 K / K ^ 2 ≤ 322437603634266857629 / 7535670527041937280000 + δ := by
+  have h :=
+    psum_piecewise_le (f := fIn) (g := gIn) (t := tIn) (L := 100) 125 (by rw [tIn_zero]; norm_num)
+      tIn_mono (by norm_num) (fun i _ => by unfold gIn; fun_prop)
+      (fun i hi x _ y _ => by unfold gIn; exact lin_lip _ _ _ (aIn_le i hi) x y)
+      (fun i hi x hx => by unfold fIn; rw [pieceIdx_eq tIn_mono hi hx]) hδ
+  rw [tIn_zero, tIn_last, inner_integrals] at h
+  exact h
+
+lemma tOut_zero : tOut 0 = 20 / 37 := by simp [tOut, tOutL]
+
+lemma tOut_last : tOut 13 = 3 := by simp [tOut, tOutL]
+
+/-- **The outer table sum.** -/
+lemma outer_psum_limit {δ : ℝ} (hδ : 0 < δ) :
+    ∀ᶠ K : ℝ in atTop, psum Eout (20 / 37) 3 K / K ^ 2 ≤ 129101 / 96000 + δ := by
+  have h :=
+    psum_piecewise_le (f := Eout) (g := gOut) (t := tOut) (L := 100) 13
+      (by rw [tOut_zero]; norm_num) tOut_mono (by norm_num) (fun i _ => by unfold gOut; fun_prop)
+      (fun i hi x _ y _ => by unfold gOut; exact lin_lip _ _ _ (aOut_le i hi) x y)
+      (fun i hi x hx => outer_table i hi x hx) hδ
+  rw [tOut_zero, tOut_last, outer_integrals] at h
+  exact h
+
+lemma tT_zero : tT 0 = 20 := by unfold tT; norm_num
+
+lemma tT_last : tT 1140 = 400 := by unfold tT; norm_num
+
+lemma qT_le_400 {i : ℕ} (hi : i < 1140) : (qT i : ℝ) ≤ 400 := by
+  unfold qT;
+  exact_mod_cast (show (60 + i) / 3 ≤ 400 by omega)
+
+lemma qT'_le_30 {i : ℕ} (hi : i < 1140) : (qT' i : ℝ) ≤ 30 := by
+  unfold qT';
+  exact_mod_cast (show (60 + i) / 40 ≤ 30 by omega)
+
+lemma tT_le_400 {i : ℕ} (hi : i ≤ 1140) : tT i ≤ 400 := by
+  unfold tT; have : (i : ℝ) ≤ 1140 := by exact_mod_cast hi
+  linarith
+
+/-- **The tail sum.** -/
+lemma tail_psum_limit {δ : ℝ} (hδ : 0 < δ) :
+    ∀ᶠ K : ℝ in atTop,
+      psum (fun x => x * Ftail x + 27 / 16) 20 400 K / K ^ 2 ≤ -6843153 / 128000000 + δ := by
+  have h :=
+    psum_piecewise_le (f := fun x => x * Ftail x + 27 / 16) (g := gT) (t := tT) (L := 3000) 1140
+      (by rw [tT_zero]; norm_num) (fun i _ => by rw [tT_succ]; linarith) (by norm_num)
+      (fun i _ => (continuous_gT i).continuousOn)
+      (fun i hi x hx y hy =>
+        by
+        have hx0 : 0 ≤ x := by linarith [hx.1, tT_pos i]
+        have hy0 : 0 ≤ y := by linarith [hy.1, tT_pos i]
+        have hx4 : x ≤ 400 := hx.2.trans (tT_le_400 (by omega))
+        have hy4 : y ≤ 400 := hy.2.trans (tT_le_400 (by omega))
+        have hq := qT_le_400 hi
+        have hq' := qT'_le_30 hi
+        have hq0 : (0 : ℝ) ≤ qT i := by positivity
+        have hq0' : (0 : ℝ) ≤ qT' i := by positivity
+        have e :
+          gT i x - gT i y =
+            (x - y) *
+              (37 / 10 - 37 / 20 * (qT i : ℝ) + 111 / 10 * (qT' i : ℝ) + 407 / 400 * (x + y)) :=
+          by unfold gT FT; ring
+        rw [e, abs_mul, mul_comm]
+        apply mul_le_mul_of_nonneg_right _ (abs_nonneg _)
+        rw [abs_le]; constructor <;> nlinarith)
+      (fun i hi x hx => by
+        obtain ⟨hx1, hx2⟩ := hx
+        have h1 := qT_le i; have h2 := tT_succ_le i; have h3 := qT'_le i; have h4 := tT_succ_le' i
+        have hf : Int.fract x = x - qT i := fract_of_bounds (by linarith) (by linarith)
+        have hg : Int.fract (3 / 40 * x) = 3 / 40 * x - qT' i :=
+          fract_of_bounds (by linarith) (by linarith)
+        unfold gT FT Ftail
+        rw [hf, hg]; ring)
+      hδ
+  have hsum := tail_sum_le
+  rw [tT_zero, tT_last] at h
+  filter_upwards [h] with K hK
+  linarith
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Growth/ClassData.lean b/LeanPool/Zeta5Irrational/Growth/ClassData.lean
new file mode 100644
index 0000000000..44fc2a75e8
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Growth/ClassData.lean
@@ -0,0 +1,169 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Growth.ClassSum
+public import LeanPool.Zeta5Irrational.Arith.ClassCount
+import Mathlib.Data.Int.Star
+import Mathlib.Tactic.ENatToNat
+import Mathlib.Tactic.ReduceModChar
+import Std.Tactic.BVDecide.Normalize.Prop
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.Positivity.Basic
+import Mathlib.Tactic.Ring.Basic
+
+/-! # Class data and `v_p(S_K)` in the inner and outer ranges
+
+* `SX p X c = 2 ⌊X/p⌋ + a_c(X mod p)` for `1 ≤ c ≤ m`;
+* `-v_p(S_K) ≤ -2h⌊K/p⌋ + 12h⌊N/p⌋ + 2 ∑_{j ≥ 1} (h - jp/2)⁺` when `p² > max(2h, N)`, `p` odd.
+-/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+variable {p : ℕ} [hp : Fact p.Prime]
+
+lemma succ_mod_cases (X q : ℕ) (hq : 0 < q) :
+    ((X + 1) % q = X % q + 1 ∧ (X + 1) / q = X / q) ∨
+      (X % q = q - 1 ∧ (X + 1) % q = 0 ∧ (X + 1) / q = X / q + 1) := by
+  have h := Nat.div_add_mod X q
+  have hl := Nat.mod_lt X hq
+  rcases Nat.lt_or_ge (X % q + 1) q with h1 | h1
+  · left
+    have : (X + 1) / q = X / q ∧ (X + 1) % q = X % q + 1 :=
+      (Nat.div_mod_unique hq).mpr ⟨by linarith, h1⟩
+    exact ⟨this.2, this.1⟩
+  · right
+    have hb : X % q = q - 1 := by omega
+    have : (X + 1) / q = X / q + 1 ∧ (X + 1) % q = 0 :=
+      (Nat.div_mod_unique hq).mpr ⟨by rw [Nat.mul_add, mul_one]; omega, hq⟩
+    exact ⟨hb, this.2, this.1⟩
+
+lemma count_res (r : ℕ) (hr1 : 1 ≤ r) (hrp : r < p) (X : ℕ) :
+    ((Icc 1 X).filter fun j : ℕ => ((j : ℤ) : ZMod p) = ((r : ℤ) : ZMod p)).card =
+      X / p + (if r ≤ X % p then 1 else 0) := by
+  have hp0 : 0 < p := by omega
+  induction X with
+  | zero => simp; omega
+  | succ X ih =>
+    rw [← insert_Icc_right_eq_Icc_add_one (by omega), Finset.filter_insert]
+    have hiff : ((((X + 1 : ℕ) : ℤ) : ZMod p) = ((r : ℤ) : ZMod p)) ↔ (X + 1) % p = r := by
+      rw [Int.cast_natCast, Int.cast_natCast, ZMod.natCast_eq_natCast_iff', Nat.mod_eq_of_lt hrp]
+    have hnot : X + 1 ∉ (Icc 1 X).filter fun j : ℕ => ((j : ℤ) : ZMod p) = ((r : ℤ) : ZMod p) :=
+      by simp
+    rcases succ_mod_cases X p hp0 with ⟨h1, h2⟩ | ⟨h1, h2, h3⟩
+    · by_cases h : (X + 1) % p = r
+      · rw [ite_eq_left (hiff.mpr h), Finset.card_insert_of_notMem hnot, ih, h2]
+        split_ifs <;> omega
+      · rw [ite_eq_right (fun h' => h (hiff.mp h')), ih, h2]
+        split_ifs <;> omega
+    · rw [ite_eq_right (fun h' => by have := hiff.mp h'; omega), ih, h3, h2, h1]
+      split_ifs <;> omega
+
+/-- **`SX` on a class**: `SX p X c = 2 ⌊X/p⌋ + a_c(X mod p)`. -/
+lemma SX_class {m : ℕ} (hm : 2 * m + 1 = p) {c : ℕ} (hc1 : 1 ≤ c) (hcm : c ≤ m) (X : ℕ) :
+    SX p X (((c : ℕ) : ℤ) : ZMod p) = 2 * (X / p) + aCnt p (X % p) c := by
+  unfold SX aCnt
+  have hneg : -(((c : ℕ) : ℤ) : ZMod p) = (((p - c : ℕ) : ℤ) : ZMod p) := by
+    rw [Nat.cast_sub (by omega), Int.cast_sub]; simp
+  rw [hneg, count_res c hc1 (by omega) X, count_res (p - c) (by omega) (by omega) X]
+  have := Nat.mod_lt X (show 0 < p by omega)
+  split_ifs <;> omega
+
+lemma SX_zero (X : ℕ) : SX p X 0 = 2 * (X / p) := by
+  unfold SX
+  have h : ((Icc 1 X).filter fun j : ℕ => ((j : ℤ) : ZMod p) = 0).card = X / p := by
+    rw [← Nat.Ioc_filter_dvd_card_eq_div X p]
+    congr 1
+    ext j
+    simp only [Finset.mem_filter, Finset.mem_Icc, Finset.mem_Ioc, Int.cast_natCast,
+      ZMod.natCast_eq_zero_iff]
+    omega
+  rw [neg_zero, h]; ring
+
+/-! ### `v_p(S_K)` -/
+
+lemma padicValNat_factorial_ge {k : ℕ} : k / p ≤ padicValNat p k.factorial := by
+  rcases Nat.lt_or_ge k p with h | h
+  · rw [Nat.div_eq_of_lt h]; exact Nat.zero_le _
+  · rw [padicValNat_factorial (b := Nat.log p k + 1) (by omega)]
+    have h1 : 1 ∈ Ico 1 (Nat.log p k + 1) := by
+      simp only [Finset.mem_Ico]
+      exact ⟨le_rfl, by have := Nat.log_pos hp.out.one_lt h; omega⟩
+    have := Finset.single_le_sum (f := fun i => k / p ^ i) (fun i _ => Nat.zero_le _) h1
+    simpa using this
+
+lemma padicValNat_factorial_small {k : ℕ} (hk : k < p ^ 2) : padicValNat p k.factorial = k / p := by
+  rw [padicValNat_factorial (b := 2)]
+  · simp
+  · rcases Nat.eq_zero_or_pos k with h0 | h0
+    · subst h0; simp
+    · exact Nat.log_lt_of_lt_pow h0.ne' hk
+
+/-- The layer-cake function `T(t) = ∑_{j ≤ J} (t - j p/2)⁺`. -/
+noncomputable def layer (p J : ℕ) (t : ℝ) : ℝ :=
+  ∑ j ∈ Icc 1 J, max 0 (t - j * p / 2)
+
+lemma layer_step (J : ℕ) (i : ℕ) (hJ : 2 * i / p ≤ J) :
+    ((2 * i / p : ℕ) : ℝ) ≤ layer p J (i + 1) - layer p J i := by
+  unfold layer
+  rw [← Finset.sum_sub_distrib]
+  have hp0 : (0 : ℝ) < p := by exact_mod_cast hp.out.pos
+  calc
+    ((2 * i / p : ℕ) : ℝ) = ∑ j ∈ Icc 1 J, (if j * p ≤ 2 * i then (1 : ℝ) else 0) := by
+      rw [Finset.sum_ite, Finset.sum_const_zero, add_zero, Finset.sum_const, nsmul_eq_mul,
+        mul_one]
+      congr 1
+      have : (Icc 1 J).filter (fun j => j * p ≤ 2 * i) = Icc 1 (2 * i / p) := by
+        ext j
+        simp only [Finset.mem_filter, Finset.mem_Icc]
+        constructor
+        · rintro ⟨⟨h1, _⟩, h3⟩; exact ⟨h1, (Nat.le_div_iff_mul_le hp.out.pos).mpr h3⟩
+        · rintro ⟨h1, h2⟩; exact ⟨⟨h1, h2.trans hJ⟩, (Nat.le_div_iff_mul_le hp.out.pos).mp h2⟩
+      rw [this, Nat.card_Icc]; simp
+    _ ≤ _ := by
+      refine Finset.sum_le_sum fun j _ => ?_
+      split_ifs with h
+      · have h' : (j : ℝ) * p ≤ 2 * i := by exact_mod_cast h
+        rw [max_eq_right (by linarith), max_eq_right (by linarith)]
+        linarith
+      · have := le_max_left (0 : ℝ) ((i : ℝ) + 1 - j * p / 2)
+        have :=
+          max_le_max (le_refl (0 : ℝ))
+            (show (i : ℝ) - j * p / 2 ≤ (i : ℝ) + 1 - j * p / 2 by linarith)
+        linarith
+
+/-- **Layer cake**: `∑_{i < H} ⌊2i/p⌋ ≤ ∑_{j ≤ J} (H - jp/2)⁺` if `2H ≤ Jp`. -/
+lemma sum_floor_le_layer (J H : ℕ) (hJ : 2 * H ≤ J * p) :
+    (∑ i ∈ range H, ((2 * i / p : ℕ) : ℝ)) ≤ layer p J H := by
+  have h0 : layer p J 0 = 0 := by
+    unfold layer
+    refine Finset.sum_eq_zero fun j hj => ?_
+    have := (Finset.mem_Icc.mp hj).1
+    have : (0 : ℝ) ≤ j * p / 2 := by positivity
+    exact max_eq_left (by linarith)
+  have : ∀ H' ≤ H, (∑ i ∈ range H', ((2 * i / p : ℕ) : ℝ)) ≤ layer p J H' := by
+    intro H'
+    induction H' with
+    | zero => intro _; simp [h0]
+    | succ H' ih =>
+      intro hH
+      rw [Finset.sum_range_succ]
+      have := ih (by omega)
+      have hstep :=
+        layer_step J H'
+          (by
+            rw [Nat.div_le_iff_le_mul_add_pred hp.out.pos]
+            nlinarith [hp.out.pos])
+      push_cast at hstep ⊢
+      linarith
+  exact this H le_rfl
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Growth/ClassSum.lean b/LeanPool/Zeta5Irrational/Growth/ClassSum.lean
new file mode 100644
index 0000000000..454b5c8437
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Growth/ClassSum.lean
@@ -0,0 +1,225 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import Mathlib.Algebra.BigOperators.Group.Finset.Defs
+public import Mathlib.Basic.Real.Basic
+public import Mathlib.Data.Fin.VecNotation
+public import Mathlib.Order.Interval.Finset.Nat
+import Mathlib.Tactic.Polynomial.Basic
+import Mathlib.Tactic.ReduceModChar
+import Std.Tactic.BVDecide.Normalize.Prop
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+import Mathlib.Tactic.Ring.Basic
+
+/-! # Class sums and their continuous limits
+
+For `p = 2m + 1` and `0 ≤ v, v' < p` put, for `1 ≤ c ≤ m`,
+`a_c = [c ≤ v] + [p - v ≤ c]` and `a'_c = [c ≤ v'] + [p - v' ≤ c]`.
+Every function `h` of `(a_c, a'_c)` is a combination of the products of the interval indicators
+`1 = [1 ≤ c ≤ m]`, `A = [c ≤ v]`, `E = [p - v ≤ c]`, so `∑_c h(a_c, a'_c)` is a combination of
+nine interval counts. Each count is within `5/2` of `p` times its continuous analogue, where
+`v/p = f`, `v'/p = g`.
+
+We also prove `SX p X c = 2 ⌊X/p⌋ + a_c(X mod p)`.
+-/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+/-- The interval `[lo i, hi i]` of the indicator `i ∈ {1, A, E}`. -/
+def ivLo (p v : ℕ) : Fin 3 → ℕ :=
+  ![1, 1, p - v]
+
+/-- Upper endpoints of the three discrete intervals used in the class-count expansion. -/
+def ivHi (m v : ℕ) : Fin 3 → ℕ :=
+  ![m, v, m]
+
+/-- The continuous intervals. -/
+noncomputable def cvLo (f : ℝ) : Fin 3 → ℝ :=
+  ![0, 0, 1 - f]
+
+/-- Upper endpoints of the continuous intervals corresponding to `ivHi`. -/
+noncomputable def cvHi (f : ℝ) : Fin 3 → ℝ :=
+  ![1 / 2, f, 1 / 2]
+
+/-- The coefficients of `h` in the basis `1, A, E`. -/
+def bcoef (h : ℕ → ℝ) : Fin 3 → ℝ :=
+  ![h 0, h 1 - h 0, h 2 - h 1]
+
+/-- `a(c) = [c ≤ v] + [p - v ≤ c]`. -/
+def aCnt (p v c : ℕ) : ℕ :=
+  (if c ≤ v then 1 else 0) + (if p - v ≤ c then 1 else 0)
+
+/-- The interval indicator. -/
+def ivInd (p m v : ℕ) (i : Fin 3) (c : ℕ) : ℝ :=
+  if ivLo p v i ≤ c ∧ c ≤ ivHi m v i then 1 else 0
+
+lemma expand_one {p m v c : ℕ} (hm : 2 * m + 1 = p) (hv : v < p) (hc1 : 1 ≤ c) (hcm : c ≤ m)
+    (h : ℕ → ℝ) : h (aCnt p v c) = ∑ i, bcoef h i * ivInd p m v i c := by
+  simp only [Fin.sum_univ_three, bcoef, ivInd, ivLo, ivHi, aCnt, Matrix.cons_val_zero,
+    Matrix.cons_val_one, Matrix.cons_val_two, Matrix.head_cons, Matrix.tail_cons]
+  by_cases h1 : c ≤ v <;> by_cases h2 : p - v ≤ c
+  · simp [h1, h2, hc1, hcm]
+  · simp [h1, h2, hc1, hcm]
+  · exfalso; omega
+  · simp [h1, h2, hc1, hcm]
+
+lemma expand_two {p m v v' c : ℕ} (hm : 2 * m + 1 = p) (hv : v < p) (hv' : v' < p) (hc1 : 1 ≤ c)
+    (hcm : c ≤ m) (h : ℕ → ℕ → ℝ) :
+    h (aCnt p v c) (aCnt p v' c) =
+      ∑ i,
+        ∑ j, bcoef (fun a => bcoef (fun b => h a b) j) i * (ivInd p m v i c * ivInd p m v' j c) :=
+  by
+  rw [expand_one hm hv hc1 hcm (fun a => h a (aCnt p v' c))]
+  refine Finset.sum_congr rfl fun i _ => ?_
+  have e : ∀ a, h a (aCnt p v' c) = ∑ j, bcoef (fun b => h a b) j * ivInd p m v' j c := fun a =>
+    expand_one hm hv' hc1 hcm (fun b => h a b)
+  simp only [e]
+  simp only [bcoef, Fin.sum_univ_three, Matrix.cons_val_zero, Matrix.cons_val_one,
+    Matrix.cons_val_two, Matrix.head_cons, Matrix.tail_cons]
+  fin_cases i <;> simp <;> ring
+
+/-- The discrete count of an intersection of two intervals within `[1, m]`. -/
+lemma card_two_intervals (m lo hi lo' hi' : ℕ) :
+    ((Icc 1 m).filter fun c => (lo ≤ c ∧ c ≤ hi) ∧ (lo' ≤ c ∧ c ≤ hi')).card =
+      min (min hi hi') m + 1 - max (max lo lo') 1 := by
+  have :
+    (Icc 1 m).filter (fun c => (lo ≤ c ∧ c ≤ hi) ∧ (lo' ≤ c ∧ c ≤ hi')) =
+      Icc (max (max lo lo') 1) (min (min hi hi') m) :=
+    by ext c; simp only [Finset.mem_filter, Finset.mem_Icc]; omega
+  rw [this, Nat.card_Icc]
+
+/-- **Class sums as interval counts.** -/
+theorem class_sum {p m v v' : ℕ} (hm : 2 * m + 1 = p) (hv : v < p) (hv' : v' < p)
+    (h : ℕ → ℕ → ℝ) :
+    ∑ c ∈ Icc 1 m, h (aCnt p v c) (aCnt p v' c) =
+      ∑ i,
+        ∑ j,
+          bcoef (fun a => bcoef (fun b => h a b) j) i *
+            ((min (min (ivHi m v i) (ivHi m v' j)) m + 1 -
+                  max (max (ivLo p v i) (ivLo p v' j)) 1 :
+                ℕ) :
+              ℝ) := by
+  rw [Finset.sum_congr rfl fun c hc =>
+      expand_two hm hv hv' (Finset.mem_Icc.mp hc).1 (Finset.mem_Icc.mp hc).2 h]
+  rw [Finset.sum_comm]
+  refine Finset.sum_congr rfl fun i _ => ?_
+  rw [Finset.sum_comm]
+  refine Finset.sum_congr rfl fun j _ => ?_
+  rw [← Finset.mul_sum, ← card_two_intervals]
+  congr 1
+  rw [Finset.card_filter, Nat.cast_sum]
+  refine Finset.sum_congr rfl fun c _ => ?_
+  simp only [ivInd]
+  split_ifs <;> simp_all
+
+/-- The continuous analogue of an interval count. -/
+noncomputable def ccount (f g : ℝ) (i j : Fin 3) : ℝ :=
+  max 0 (min (min (cvHi f i) (cvHi g j)) (1 / 2) - max (max (cvLo f i) (cvLo g j)) 0)
+
+lemma cast_trunc_sub (a b : ℕ) : ((a - b : ℕ) : ℝ) = max 0 ((a : ℝ) - b) := by
+  rcases le_or_gt b a with h | h
+  · have h' : (b : ℝ) ≤ a := by exact_mod_cast h
+    rw [Nat.cast_sub h, max_eq_right (by linarith)]
+  · have h' : (a : ℝ) < b := by exact_mod_cast h
+    rw [Nat.sub_eq_zero_of_le h.le, max_eq_left (by linarith)]
+    simp
+
+lemma abs_max0_sub (u w : ℝ) : |max 0 u - max 0 w| ≤ |u - w| := by
+  have := abs_max_sub_max_le_max (0 : ℝ) u 0 w
+  simpa using this
+
+lemma hi_approx {p m v : ℕ} (hm : 2 * m + 1 = p) (_hv : v < p) (i : Fin 3) :
+    |(ivHi m v i : ℝ) - p * cvHi ((v : ℝ) / p) i| ≤ 1 / 2 := by
+  have hp : (0 : ℝ) < p := by
+    have : 0 < p := by omega
+    exact_mod_cast this
+  have hmR : (m : ℝ) = (p - 1) / 2 := by
+    have : ((2 * m + 1 : ℕ) : ℝ) = p := by exact_mod_cast hm
+    push_cast at this; linarith
+  fin_cases i <;> simp only [ivHi, Fin.zero_eta, Fin.isValue, Matrix.cons_val_zero, hmR, cvHi,
+      one_div, ge_iff_le, Fin.mk_one, Matrix.cons_val_one, Fin.reduceFinMk, Matrix.cons_val]
+  · rw [abs_le]; constructor <;> linarith
+  · rw [mul_div_cancel₀ _ hp.ne']; simp
+  · rw [abs_le]; constructor <;> linarith
+
+lemma lo_approx {p v : ℕ} (hv : v < p) (i : Fin 3) :
+    |(ivLo p v i : ℝ) - p * cvLo ((v : ℝ) / p) i| ≤ 1 := by
+  have hp : (0 : ℝ) < p := by
+    have : 0 < p := by omega
+    exact_mod_cast this
+  fin_cases i <;> simp only [ivLo, Fin.zero_eta, Fin.isValue, Matrix.cons_val_zero, Nat.cast_one,
+      cvLo, mul_zero, sub_zero, abs_one, Std.le_refl, Fin.mk_one, Matrix.cons_val_one,
+      Fin.reduceFinMk, Matrix.cons_val, ge_iff_le]
+  rw [Nat.cast_sub hv.le, mul_sub, mul_one, mul_div_cancel₀ _ hp.ne']
+  simp
+
+/-- **Each interval count is within `5/2` of `p` times its continuous analogue.** -/
+lemma count_approx {p m v v' : ℕ} (hm : 2 * m + 1 = p) (hv : v < p) (hv' : v' < p) (i j : Fin 3) :
+    |((min (min (ivHi m v i) (ivHi m v' j)) m + 1 - max (max (ivLo p v i) (ivLo p v' j)) 1 : ℕ) :
+            ℝ) -
+          (p : ℝ) * ccount ((v : ℝ) / p) ((v' : ℝ) / p) i j| ≤
+      5 / 2 := by
+  have hp : (0 : ℝ) < p := by
+    have : 0 < p := by omega
+    exact_mod_cast this
+  have hmR : (m : ℝ) = (p - 1) / 2 := by
+    have : ((2 * m + 1 : ℕ) : ℝ) = p := by exact_mod_cast hm
+    push_cast at this; linarith
+  rw [cast_trunc_sub]
+  unfold ccount
+  rw [mul_max_of_nonneg _ _ hp.le, mul_zero, mul_sub, mul_min_of_nonneg _ _ hp.le,
+    mul_min_of_nonneg _ _ hp.le, mul_max_of_nonneg _ _ hp.le, mul_max_of_nonneg _ _ hp.le,
+    mul_zero]
+  refine (abs_max0_sub _ _).trans ?_
+  push_cast
+  have h1 :
+    |min (min (ivHi m v i : ℝ) (ivHi m v' j)) m -
+          min (min (p * cvHi ((v : ℝ) / p) i) (p * cvHi ((v' : ℝ) / p) j)) (p * (1 / 2))| ≤
+      1 / 2 := by
+    refine (abs_min_sub_min_le_max _ _ _ _).trans (max_le ?_ ?_)
+    · exact
+        (abs_min_sub_min_le_max _ _ _ _).trans (max_le (hi_approx hm hv i) (hi_approx hm hv' j))
+    · rw [hmR, abs_le]; constructor <;> linarith
+  have h2 :
+    |max (max (ivLo p v i : ℝ) (ivLo p v' j)) 1 -
+          max (max (p * cvLo ((v : ℝ) / p) i) (p * cvLo ((v' : ℝ) / p) j)) 0| ≤
+      1 := by
+    refine (abs_max_sub_max_le_max _ _ _ _).trans (max_le ?_ (by norm_num))
+    exact (abs_max_sub_max_le_max _ _ _ _).trans (max_le (lo_approx hv i) (lo_approx hv' j))
+  rw [abs_le] at h1 h2 ⊢
+  constructor <;> linarith [h1.1, h1.2, h2.1, h2.2]
+
+/-- **Class sums versus their continuous limit.** -/
+theorem class_sum_approx {p m v v' : ℕ} (hm : 2 * m + 1 = p) (hv : v < p) (hv' : v' < p)
+    (h : ℕ → ℕ → ℝ) :
+    |∑ c ∈ Icc 1 m, h (aCnt p v c) (aCnt p v' c) -
+          (p : ℝ) *
+            ∑ i,
+              ∑ j,
+                bcoef (fun a => bcoef (fun b => h a b) j) i *
+                  ccount ((v : ℝ) / p) ((v' : ℝ) / p) i j| ≤
+      5 / 2 * ∑ i, ∑ j, |bcoef (fun a => bcoef (fun b => h a b) j) i| := by
+  rw [class_sum hm hv hv', Finset.mul_sum, ← Finset.sum_sub_distrib]
+  refine (Finset.abs_sum_le_sum_abs _ _).trans ?_
+  rw [Finset.mul_sum]
+  refine Finset.sum_le_sum fun i _ => ?_
+  rw [Finset.mul_sum, ← Finset.sum_sub_distrib]
+  refine (Finset.abs_sum_le_sum_abs _ _).trans ?_
+  rw [Finset.mul_sum]
+  refine Finset.sum_le_sum fun j _ => ?_
+  rw [mul_left_comm, ← mul_sub, abs_mul, mul_comm (5 / 2 : ℝ)]
+  exact mul_le_mul_of_nonneg_left (count_approx hm hv hv' i j) (abs_nonneg _)
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Growth/Constants.lean b/LeanPool/Zeta5Irrational/Growth/Constants.lean
new file mode 100644
index 0000000000..588a3173ae
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Growth/Constants.lean
@@ -0,0 +1,137 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Growth.TablePrime
+import Mathlib.Data.Rat.Star
+import Mathlib.Data.Int.Star
+public import LeanPool.Zeta5Irrational.Growth.OuterPrime
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+import Mathlib.Tactic.Positivity.Basic
+import Mathlib.Tactic.Ring.Basic
+
+/-! # Uniform bounds for the additive constants of the per-prime bounds
+-/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma abs_bcoef_le (h : ℕ → ℝ) {M : ℝ} (hM : ∀ a ≤ 2, |h a| ≤ M) (i : Fin 3) :
+    |bcoef h i| ≤ 2 * M := by
+  have h0 := hM 0 (by norm_num); have h1 := hM 1 (by norm_num); have h2 := hM 2 (by norm_num)
+  have hM0 : 0 ≤ M := (abs_nonneg _).trans h0
+  fin_cases i <;>
+    simp only [bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+      Matrix.head_cons, Matrix.tail_cons, Fin.zero_eta, Fin.mk_one, Fin.reduceFinMk]
+  · linarith
+  · exact (abs_sub _ _).trans (by linarith)
+  · exact (abs_sub _ _).trans (by linarith)
+
+lemma abs_bcoef2_le (h : ℕ → ℕ → ℝ) {M : ℝ} (hM : ∀ a ≤ 2, ∀ b ≤ 2, |h a b| ≤ M) (i j : Fin 3) :
+    |bcoef (fun a => bcoef (fun b => h a b) j) i| ≤ 4 * M := by
+  have :=
+    abs_bcoef_le (fun a => bcoef (fun b => h a b) j) (M := 2 * M)
+      (fun a ha => abs_bcoef_le (fun b => h a b) (fun b hb => hM a ha b hb) j) i
+  linarith
+
+lemma psiR_nonneg (k β : ℤ) : 0 ≤ psiR k β := by
+  unfold psiR
+  refine Finset.sum_nonneg fun i hi => ?_
+  have := lrow_lt (Finset.mem_range.mp hi)
+  linarith
+
+lemma abs_psiR_le (k β : ℤ) : |(psiR k β : ℝ)| ≤ ((k - β + 1 : ℤ) : ℝ) ^ 2 := by
+  have h1 := psiR_nonneg k β
+  have h2 := psiR_le k β
+  rw [abs_of_nonneg (by exact_mod_cast h1)]
+  have : (psiR k β : ℝ) ≤ ((k - β + 1 : ℤ) : ℝ) ^ 2 / 8 := by exact_mod_cast h2
+  nlinarith [sq_nonneg ((k - β + 1 : ℤ) : ℝ)]
+
+/-- The inner coefficients. -/
+lemma sum_abs_bIn_le {q q' : ℕ} (hq' : q' ≤ q) {k : ℤ} (hk1 : -2 * (q : ℤ) - 6 ≤ k)
+    (hk2 : k ≤ 8 * (q : ℤ) + 20) : ∑ i, ∑ j, |bIn q q' k i j| ≤ 36 * (22 * (q : ℝ) + 30) ^ 2 := by
+  have hM :
+    ∀ a : ℕ,
+      a ≤ 2 →
+        ∀ b : ℕ,
+          b ≤ 2 →
+            |(psiR k (12 * (q' : ℤ) - 2 * (q : ℤ) - 4 + 6 * (b : ℤ) - a) : ℝ)| ≤
+              (22 * (q : ℝ) + 30) ^ 2 := by
+    intro a ha b hb
+    refine (abs_psiR_le _ _).trans ?_
+    have hb' : (b : ℤ) ≤ 2 := by exact_mod_cast hb
+    have ha' : (a : ℤ) ≤ 2 := by exact_mod_cast ha
+    have hq'' : (q' : ℤ) ≤ q := by exact_mod_cast hq'
+    have hb0 : (0 : ℤ) ≤ b := by positivity
+    have ha0 : (0 : ℤ) ≤ a := by positivity
+    have hq0 : (0 : ℤ) ≤ q' := by positivity
+    have hbd :
+      |k - (12 * (q' : ℤ) - 2 * (q : ℤ) - 4 + 6 * (b : ℤ) - a) + 1| ≤ 22 * (q : ℤ) + 30 := by
+      rw [abs_le]; constructor <;> linarith
+    have hbdR :
+      |((k - (12 * (q' : ℤ) - 2 * (q : ℤ) - 4 + 6 * (b : ℤ) - a) + 1 : ℤ) : ℝ)| ≤
+        22 * (q : ℝ) + 30 :=
+      by exact_mod_cast hbd
+    rw [← sq_abs]
+    exact pow_le_pow_left₀ (abs_nonneg _) hbdR 2
+  have hb := fun i j => abs_bcoef2_le _ hM i j
+  calc
+    ∑ i, ∑ j, |bIn q q' k i j| ≤ ∑ _i : Fin 3, ∑ _j : Fin 3, 4 * (22 * (q : ℝ) + 30) ^ 2 :=
+      Finset.sum_le_sum fun i _ => Finset.sum_le_sum fun j _ => hb i j
+    _ = 36 * (22 * (q : ℝ) + 30) ^ 2 := by simp; ring
+
+/-- The outer coefficients. -/
+lemma abs_GO_le (q a d : ℕ) : |(GO q a d : ℝ)| ≤ (2 * (q : ℝ) + a) * ((q : ℝ) + a + 3) := by
+  unfold GO
+  push_cast
+  refine (Finset.abs_sum_le_sum_abs _ _).trans ?_
+  calc
+    ∑ i ∈ range (2 * q + a - 2), |min 0 ((i : ℝ) + 3 * d - (2 * (q : ℝ) + a + 4) / 2)| ≤
+        ∑ _i ∈ range (2 * q + a - 2), ((q : ℝ) + a + 3) := by
+      refine Finset.sum_le_sum fun i _ => ?_
+      rw [abs_le]
+      have : (0 : ℝ) ≤ i := by positivity
+      have : (0 : ℝ) ≤ d := by positivity
+      have ha : (0 : ℝ) ≤ a := by positivity
+      constructor
+      · have := min_le_right (0 : ℝ) ((i : ℝ) + 3 * d - ((2 * q + a : ℝ) + 4) / 2)
+        have hq0 : (0 : ℝ) ≤ q := by positivity
+        have :=
+          le_min (show -((q : ℝ) + a + 3) ≤ 0 by linarith)
+            (show -((q : ℝ) + a + 3) ≤ (i : ℝ) + 3 * d - ((2 * q + a : ℝ) + 4) / 2 by linarith)
+        exact this
+      · have := min_le_left (0 : ℝ) ((i : ℝ) + 3 * d - ((2 * q + a : ℝ) + 4) / 2)
+        linarith [show (0 : ℝ) ≤ (q : ℝ) + a + 3 by positivity]
+    _ = ((2 * q + a - 2 : ℕ) : ℝ) * ((q : ℝ) + a + 3) := by
+      rw [Finset.sum_const, Finset.card_range, nsmul_eq_mul]
+    _ ≤ (2 * (q : ℝ) + a) * ((q : ℝ) + a + 3) := by
+      apply mul_le_mul_of_nonneg_right _ (by positivity)
+      have : 2 * q + a - 2 ≤ 2 * q + a := Nat.sub_le _ _
+      exact_mod_cast this
+
+lemma sum_abs_bGO_le {q : ℕ} (hq : q ≤ 2) : ∑ i, ∑ j, |bGO q i j| ≤ 36 * 70 := by
+  have hM : ∀ a : ℕ, a ≤ 2 → ∀ b : ℕ, b ≤ 2 → |(GO q a b : ℝ)| ≤ 70 := by
+    intro a ha b _
+    refine (abs_GO_le q a b).trans ?_
+    have hq' : (q : ℝ) ≤ 2 := by exact_mod_cast hq
+    have ha' : (a : ℝ) ≤ 2 := by exact_mod_cast ha
+    have : (0 : ℝ) ≤ q := by positivity
+    have : (0 : ℝ) ≤ a := by positivity
+    nlinarith
+  have hb := fun i j => abs_bcoef2_le _ hM i j
+  calc
+    ∑ i, ∑ j, |bGO q i j| ≤ ∑ _i : Fin 3, ∑ _j : Fin 3, 4 * (70 : ℝ) :=
+      Finset.sum_le_sum fun i _ => Finset.sum_le_sum fun j _ => hb i j
+    _ ≤ 36 * 70 := by simp; norm_num
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Growth/InnerPrime.lean b/LeanPool/Zeta5Irrational/Growth/InnerPrime.lean
new file mode 100644
index 0000000000..8fc01213cf
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Growth/InnerPrime.lean
@@ -0,0 +1,324 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+import LeanPool.Zeta5Irrational.Legendre
+import Mathlib.Tactic.ReduceModChar
+import Mathlib.Data.Rat.Star
+import Mathlib.Data.Int.Star
+public import LeanPool.Zeta5Irrational.Growth.ClassData
+public import LeanPool.Zeta5Irrational.Norm
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+import Mathlib.Tactic.Positivity.Basic
+import Mathlib.Tactic.Ring.Basic
+
+/-! # The inner range: a per-prime upper bound for `-L_p`
+
+For `K/400 < p ≤ K/3` (and `n` large enough for the side conditions), every level `k` in
+`[klo, ktop]` gives
+`-L_p ≤ -2h q + 12h q' + 2 T(h) - h k + 2 ∑_{c=1}^m ψ(k, B + 6a'_c - a_c) + (k - klo) L₀ + 2B₀ + 1`,
+with `q = ⌊K/p⌋`, `q' = ⌊N/p⌋`, `B = 12q' - 2q - 4`, the class counts `a_c, a'_c` of
+`ClassSum`, and the layer-cake function `T`.
+-/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+variable {p : ℕ} [hp : Fact p.Prime]
+
+lemma sum_erase_zero_eq_Icc {m : ℕ} (F : ℕ → ℚ) :
+    ∑ c ∈ (univ : Finset (Fin (m + 1))).erase 0, F c = ∑ c ∈ Icc 1 m, F c := by
+  have h1 := Finset.add_sum_erase (univ : Finset (Fin (m + 1))) (fun c => F c) (Finset.mem_univ 0)
+  rw [Fin.sum_univ_eq_sum_range (fun c => F c), Finset.sum_range_eq_add_Ico _ (by omega)] at h1
+  simp only [Fin.val_zero] at h1
+  have h2 : Ico 1 (m + 1) = Icc 1 m := by ext c; simp only [Finset.mem_Ico, Finset.mem_Icc]; omega
+  rw [h2] at h1
+  linarith
+
+/-- `-v_p(S_K)` in the inner and outer ranges. -/
+lemma neg_vSK_le {n : ℕ} (hn : 1 ≤ n) (_hp3 : 3 ≤ p) (hN : 3 * n < p ^ 2)
+    (hH : 2 * (37 * n) < p ^ 2) (J : ℕ) (hJ : 2 * (37 * n) ≤ J * p) :
+    -(vSK n p : ℝ) ≤
+      -2 * (37 * n : ℕ) * ((40 * n / p : ℕ) : ℝ) + 12 * (37 * n : ℕ) * ((3 * n / p : ℕ) : ℝ) +
+        2 * layer p J (37 * n : ℕ) := by
+  have h := vS_eq (by omega : 0 < n) hp.out
+  rw [h]
+  have h1 := padicValNat_factorial_ge (p := p) (k := 40 * n)
+  have h2 := padicValNat_factorial_small (p := p) (k := 3 * n) hN
+  have h3 : ∀ i ∈ Icc 1 (37 * n - 1), padicValNat p (2 * i).factorial = 2 * i / p := fun i hi =>
+    padicValNat_factorial_small (by have := (Finset.mem_Icc.mp hi).2; omega)
+  have hsum : (((∑ i ∈ Icc 1 (37 * n - 1), 2 * i / p : ℕ)) : ℝ) ≤ layer p J (37 * n : ℕ) := by
+    have := sum_floor_le_layer (p := p) J (37 * n) hJ
+    rw [Finset.sum_range_eq_add_Ico _ (by omega)] at this
+    have e2 : Ico 1 (37 * n) = Icc 1 (37 * n - 1) := by
+      ext c;
+      simp only [Finset.mem_Ico, Finset.mem_Icc]; omega
+    rw [e2] at this
+    rw [Nat.cast_sum]
+    simpa using this
+  have hs :
+    ∑ i ∈ Icc 1 (37 * n - 1), (padicValNat p (2 * i).factorial : ℤ) =
+      ((∑ i ∈ Icc 1 (37 * n - 1), 2 * i / p : ℕ) : ℤ) :=
+    by rw [Nat.cast_sum]; exact Finset.sum_congr rfl (fun i hi => by rw [h3 i hi])
+  rw [hs, h2]
+  generalize padicValNat p (40 * n).factorial = A at h1 ⊢
+  generalize padicValNat p 4 = B
+  generalize (∑ i ∈ Icc 1 (37 * n - 1), 2 * i / p) = Sg at hsum ⊢
+  generalize 40 * n / p = q at h1 ⊢
+  generalize 3 * n / p = q'
+  have h1R : (q : ℝ) ≤ (A : ℝ) := by exact_mod_cast h1
+  have hB : (0 : ℝ) ≤ (B : ℝ) := by positivity
+  have hnR : (1 : ℝ) ≤ (n : ℝ) := by exact_mod_cast hn
+  push_cast at hsum ⊢
+  nlinarith [mul_le_mul_of_nonneg_left h1R (show (0 : ℝ) ≤ 74 * n by positivity),
+    mul_nonneg (show (0 : ℝ) ≤ 37 * n - 1 by linarith) hB]
+
+/-- The zero-class deficit bound. -/
+noncomputable def Bmax (n p : ℕ) : ℝ :=
+  max 0
+      ((ktopI n p : ℝ) / 2 -
+        ((5 + 12 * ((3 * n / p : ℕ) : ℝ) - 2 * ((40 * n / p : ℕ) : ℝ)) - 4) / 2) *
+    (max 0
+          ((ktopI n p : ℝ) / 2 -
+            ((5 + 12 * ((3 * n / p : ℕ) : ℝ) - 2 * ((40 * n / p : ℕ) : ℝ)) - 4) / 2) /
+        2 +
+      1)
+
+/-- The class function of the inner range. -/
+noncomputable def hIn (n p : ℕ) (k : ℤ) (a a' : ℕ) : ℝ :=
+  (psiR k (12 * ((3 * n / p : ℕ) : ℤ) - 2 * ((40 * n / p : ℕ) : ℤ) - 4 + 6 * (a' : ℤ) - a) : ℝ)
+
+section
+
+variable {n : ℕ} (hn : 33 ≤ n) (hodd : 2 * (p / 2) + 1 = p) (hp7 : 7 ≤ p)
+  (hsmall : ¬p * Mcut ≤ 40 * n) (hin : 3 * p ≤ 40 * n) (hsq : 2 * (40 * n) < p ^ 2)
+  (hdeg : 5 + 2 * (18 * n + 2 * (37 * n)) - 2 * (40 * n) + 1 < p ^ 2)
+
+include hodd in
+lemma betaI_eq {c : ℕ} (hc1 : 1 ≤ c) (hcm : c ≤ p / 2) :
+    betaI n p (m := p / 2) (⟨c, by omega⟩ : Fin (p / 2 + 1)) =
+      12 * ((3 * n / p : ℕ) : ℤ) - 2 * ((40 * n / p : ℕ) : ℤ) - 4 +
+          6 * (aCnt p (3 * n % p) c : ℤ) -
+        aCnt p (40 * n % p) c := by
+  unfold betaI
+  simp only
+  rw [SX_class hodd hc1 hcm, SX_class hodd hc1 hcm]
+  push_cast; ring
+
+include hodd hsmall in
+omit hp in
+lemma q_lt : 40 * n / p < 400 := by
+  unfold Mcut at hsmall
+  rw [not_le] at hsmall
+  exact (Nat.div_lt_iff_lt_mul (by omega)).mpr (by linarith)
+
+include hodd in
+lemma klo_le_beta (c : Fin (p / 2 + 1)) (hc : c ≠ 0) : kloI n p ≤ betaI n p c := by
+  have hc1 : 1 ≤ (c : ℕ) := by
+    rcases Nat.eq_zero_or_pos (c : ℕ) with h | h
+    · exact absurd (Fin.ext h) hc
+    · exact h
+  have hcm := Nat.lt_succ_iff.mp c.isLt
+  have e := betaI_eq (n := n) hodd hc1 hcm
+  simp only [Fin.eta] at e
+  rw [e]
+  unfold kloI aCnt
+  generalize 3 * n / p = q' at *
+  generalize 40 * n / p = q at *
+  split_ifs <;> omega
+
+include hodd in
+lemma rowsK_klo : rowsK (betaI n p (m := p / 2)) (kloI n p) = 0 := by
+  unfold rowsK
+  refine Finset.sum_eq_zero fun c hc => ?_
+  have := klo_le_beta (n := n) hodd c (Finset.mem_erase.mp hc).1
+  unfold lrow; omega
+
+lemma Bse_eq_beta (c : Fin (p / 2 + 1)) (hc : c ≠ 0) :
+    Bse p (40 * n) (3 * n) (c : ℕ) = (betaI n p c : ℚ) + 4 := by
+  have hc0 : (c : ℕ) ≠ 0 := fun h => hc (Fin.ext h)
+  unfold Bse betaI
+  rw [ite_eq_right hc0]
+  push_cast; ring
+
+lemma Bse_zero_eq :
+    Bse p (40 * n) (3 * n) 0 = 5 + 12 * ((3 * n / p : ℕ) : ℚ) - 2 * ((40 * n / p : ℕ) : ℚ) := by
+  unfold Bse
+  simp only [ite_eq_left, Nat.cast_zero, Int.cast_zero]
+  rw [SX_zero, SX_zero]
+  push_cast; ring
+
+include hn hodd hp7 hsmall hin hsq hdeg in
+/-- The inner side conditions hold. -/
+lemma innerOK_of : InnerOK n p := by
+  have hm1 : 1 ≤ p / 2 := by omega
+  set β := betaI n p (m := p / 2) with hβdef
+  have hq := q_lt hodd hsmall
+  have hlo : rowsK β (kloI n p) + L0I n p ≤ 37 * n := by rw [rowsK_klo hodd]; unfold L0I; omega
+  set kf := kfSel β (37 * n) (L0I n p) (kloI n p) (ktopI n p) with hkf
+  have hfeas : rowsK β kf + L0I n p ≤ 37 * n :=
+    kfSel_feasible β (37 * n) (L0I n p) (kloI n p) (ktopI n p) hlo
+  have hkfle : kf ≤ ktopI n p :=
+    kfSel_le β (37 * n) (L0I n p) (kloI n p) (ktopI n p)
+      (by unfold kloI ktopI; generalize 40 * n / p = q; omega)
+  have hβ : ∀ c : Fin (p / 2 + 1), c ≠ 0 → Bse p (40 * n) (3 * n) (c : ℕ) = (β c : ℚ) + 4 :=
+    fun c hc => Bse_eq_beta c hc
+  refine ⟨hodd, hm1, by omega, Sep_PlK hsq, by nlinarith, hdeg, ?_, ?_⟩
+  · exact sum_allocK β (37 * n) kf (by omega)
+  · -- the zero-class condition
+    have h1 := tmin_allocK_le hm1 (Bse p (40 * n) (3 * n)) β hβ (37 * n) kf
+    have hA0 : allocI n p (p / 2) 0 = 37 * n - rowsK β kf := by
+      unfold allocI allocK;
+      rw [ite_eq_left rfl]
+    have hL0 : L0I n p ≤ allocI n p (p / 2) 0 := by rw [hA0]; omega
+    have hL0R : ((L0I n p : ℕ) : ℚ) ≤ (allocI n p (p / 2) 0 : ℚ) := by exact_mod_cast hL0
+    rw [Bse_zero_eq]
+    have hb1 : (β ⟨1, by omega⟩ : ℚ) ≤ 12 * ((3 * n / p : ℕ) : ℚ) + 8 := by
+      have e := betaI_eq (n := n) hodd (c := 1) le_rfl (by omega)
+      have : β ⟨1, by omega⟩ ≤ 12 * ((3 * n / p : ℕ) : ℤ) + 8 := by
+        rw [hβdef, e]; unfold aCnt; generalize 3 * n / p = q' at *; generalize 40 * n / p = q at *
+        split_ifs <;> omega
+      have h' := (Int.cast_le (R := ℚ)).mpr this
+      rw [Int.cast_add, Int.cast_mul, Int.cast_natCast] at h'
+      norm_num at h'
+      exact h'
+    have hkfR : (kf : ℚ) ≤ ktopI n p := by exact_mod_cast hkfle
+    unfold ktopI at hkfR
+    unfold L0I at hL0R
+    rw [Int.cast_add, Int.cast_mul, Int.cast_natCast] at hkfR
+    norm_num at hkfR
+    push_cast at hL0R
+    refine h1.trans ?_
+    rw [div_le_iff₀ (by norm_num : (0 : ℚ) < 2)]
+    have hq0 : (0 : ℚ) ≤ ((40 * n / p : ℕ) : ℚ) := by positivity
+    have hq0' : (0 : ℚ) ≤ ((3 * n / p : ℕ) : ℚ) := by positivity
+    refine max_le ?_ ?_ <;> linarith
+
+include hn hodd hp7 hsmall hin hsq hdeg in
+/-- **The inner per-prime bound** (discrete form). -/
+theorem inner_Lp_le (k : ℤ) (hk1 : kloI n p ≤ k) (hk2 : k ≤ ktopI n p) :
+    -(Lp n p : ℝ) ≤
+      -2 * (37 * n : ℕ) * ((40 * n / p : ℕ) : ℝ) + 12 * (37 * n : ℕ) * ((3 * n / p : ℕ) : ℝ) +
+                  2 * layer p 1000 (37 * n : ℕ) -
+                (37 * n : ℕ) * (k : ℝ) +
+              2 * ∑ c ∈ Icc 1 (p / 2), hIn n p k (aCnt p (40 * n % p) c) (aCnt p (3 * n % p) c) +
+            ((k - kloI n p : ℤ) : ℝ) * (L0I n p : ℝ) +
+          2 * Bmax n p +
+        1 := by
+  have hm1 : 1 ≤ p / 2 := by omega
+  have hOK := innerOK_of hn hodd hp7 hsmall hin hsq hdeg
+  have hLp : Lp n p = vSK n p + ⌊2 * WI n p⌋ := by
+    unfold Lp;
+    rw [ite_eq_right hsmall, ite_eq_left hin, ite_eq_left hOK]
+  have hq :=
+    q_lt hodd
+      hsmall
+        -- `v_p(S_K)`
+  have hv :=
+    neg_vSK_le (p := p) (n := n) (by omega) (by omega) (by nlinarith) (by nlinarith) 1000
+      (by
+        unfold Mcut at hsmall; rw [not_le] at hsmall
+        nlinarith)
+        -- the allocation
+  set β := betaI n p (m := p / 2) with hβdef
+  have hlo : rowsK β (kloI n p) + L0I n p ≤ 37 * n := by rw [rowsK_klo hodd]; unfold L0I; omega
+  set kf := kfSel β (37 * n) (L0I n p) (kloI n p) (ktopI n p) with hkf
+  have hfeas : rowsK β kf + L0I n p ≤ 37 * n :=
+    kfSel_feasible β (37 * n) (L0I n p) (kloI n p) (ktopI n p) hlo
+  have hkfle : kf ≤ ktopI n p :=
+    kfSel_le β (37 * n) (L0I n p) (kloI n p) (ktopI n p)
+      (by unfold kloI ktopI; generalize 40 * n / p = q; omega)
+  have hβ : ∀ c : Fin (p / 2 + 1), c ≠ 0 → Bse p (40 * n) (3 * n) (c : ℕ) = (β c : ℚ) + 4 :=
+    fun c hc => Bse_eq_beta c hc
+  have hW := sum_wcap_allocK hm1 (Bse p (40 * n) (3 * n)) β hβ (37 * n) kf (by omega)
+  have hWI : WI n p = ∑ s, wcap hm1 (allocK β (37 * n) kf) (Bse p (40 * n) (3 * n)) s := by
+    unfold WI; rw [dite_eq_left hm1]; rfl
+  rw [← hWI] at hW
+  have hPhi :=
+    PhiK_le_kf β (37 * n) (L0I n p) (kloI n p) (ktopI n p) hlo hk1
+      hk2
+        -- `Φ(k)` as a class sum
+  have hsumQ :
+    ∑ c ∈ (univ : Finset (Fin (p / 2 + 1))).erase 0, psiR k (β c) =
+      ∑ c ∈ Icc 1 (p / 2),
+        psiR k
+          (12 * ((3 * n / p : ℕ) : ℤ) - 2 * ((40 * n / p : ℕ) : ℤ) - 4 +
+              6 * (aCnt p (3 * n % p) c : ℤ) -
+            aCnt p (40 * n % p) c) := by
+    have e :=
+      sum_erase_zero_eq_Icc (m := p / 2)
+        (fun c =>
+          psiR k
+            (6 * (SX p (3 * n) (((c : ℕ) : ℤ) : ZMod p) : ℤ) -
+                (SX p (40 * n) (((c : ℕ) : ℤ) : ZMod p) : ℤ) -
+              4))
+    refine e.trans (Finset.sum_congr rfl fun c hc => ?_)
+    have hc' := Finset.mem_Icc.mp hc
+    have := betaI_eq (n := n) hodd hc'.1 hc'.2
+    unfold betaI at this
+    simp only at this ⊢
+    rw [this]
+  have hPhik :
+    (PhiK β (37 * n) k : ℝ) =
+      ((37 * n : ℕ) : ℝ) * k / 2 -
+        ∑ c ∈ Icc 1 (p / 2), hIn n p k (aCnt p (40 * n % p) c) (aCnt p (3 * n % p) c) := by
+    unfold PhiK
+    rw [hsumQ]
+    unfold hIn
+    push_cast
+    ring
+      -- the zero-class deficit
+  have hB0 :
+    max 0 ((kf : ℚ) / 2 - (Bse p (40 * n) (3 * n) 0 - 4) / 2) *
+        (max 0 ((kf : ℚ) / 2 - (Bse p (40 * n) (3 * n) 0 - 4) / 2) / 2 + 1) ≤
+      (Bmax n p : ℝ) := by
+    rw [Bse_zero_eq]
+    unfold Bmax
+    have hkR : ((kf : ℤ) : ℝ) ≤ ((ktopI n p : ℤ) : ℝ) := by exact_mod_cast hkfle
+    push_cast
+    generalize ((40 * n / p : ℕ) : ℝ) = qR
+    generalize ((3 * n / p : ℕ) : ℝ) = qR'
+    generalize ((ktopI n p : ℤ) : ℝ) = tR at hkR ⊢
+    generalize ((kf : ℤ) : ℝ) = kR at hkR ⊢
+    set D1 := max 0 (kR / 2 - (5 + 12 * qR' - 2 * qR - 4) / 2) with hD1
+    set D2 := max 0 (tR / 2 - (5 + 12 * qR' - 2 * qR - 4) / 2) with hD2
+    have h0 : 0 ≤ D1 := le_max_left _ _
+    have h12 : D1 ≤ D2 := max_le_max (le_refl 0) (by linarith)
+    nlinarith
+  have hfl : ((⌊2 * WI n p⌋ : ℤ) : ℝ) ≥ 2 * (WI n p : ℝ) - 1 := by
+    have := Int.lt_floor_add_one (2 * WI n p)
+    have : ((2 * WI n p : ℚ) : ℝ) < ((⌊2 * WI n p⌋ : ℤ) : ℝ) + 1 := by exact_mod_cast this
+    push_cast at this; linarith
+  have hWR :
+    ((PhiK β (37 * n) kf : ℚ) : ℝ) -
+        ((max 0 ((kf : ℚ) / 2 - (Bse p (40 * n) (3 * n) 0 - 4) / 2) *
+              (max 0 ((kf : ℚ) / 2 - (Bse p (40 * n) (3 * n) 0 - 4) / 2) / 2 + 1) :
+            ℚ) :
+          ℝ) ≤
+      (WI n p : ℝ) :=
+    by exact_mod_cast hW
+  have hPR :
+    ((PhiK β (37 * n) k : ℚ) : ℝ) ≤
+      ((PhiK β (37 * n) kf : ℚ) : ℝ) + ((k - kloI n p : ℤ) : ℝ) * (L0I n p : ℝ) / 2 := by
+    have := (Rat.cast_le (K := ℝ)).mpr hPhi
+    push_cast at this ⊢
+    linarith
+  rw [hLp]
+  push_cast
+  rw [hPhik] at hPR
+  push_cast at hWR hB0 hv hPR
+  linarith
+
+end
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Growth/InnerSum.lean b/LeanPool/Zeta5Irrational/Growth/InnerSum.lean
new file mode 100644
index 0000000000..de90a84b39
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Growth/InnerSum.lean
@@ -0,0 +1,2242 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+import LeanPool.Zeta5Irrational.Growth.InnerTable
+import LeanPool.Zeta5Irrational.Growth.TableGen
+public import LeanPool.Zeta5Irrational.Growth.TablePrime
+import Mathlib.Data.Int.Star
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-!Generated by `gen_inner_sum.py`: the table of `Ê_in` on `[3, 20]`. -/
+
+
+@[expose] public section
+
+namespace Zeta5Irrational
+
+/-- The 126 rational endpoints of the 125 inner-range linear pieces. -/
+def tInL : List ℚ :=
+  [(3), (70 / 23), (120 / 37), (10 / 3), (80 / 23), (7 / 2), (160 / 43), (140 / 37), (90 / 23),
+    (4), (160 / 37), (100 / 23), (9 / 2), (200 / 43), (110 / 23), (180 / 37), (5), (120 / 23),
+    (200 / 37), (11 / 2), (240 / 43), (130 / 23), (220 / 37), (6), (140 / 23), (240 / 37),
+    (13 / 2), (280 / 43), (150 / 23), (20 / 3), (160 / 23), (7), (260 / 37), (170 / 23),
+    (320 / 43), (15 / 2), (280 / 37), (180 / 23), (8), (300 / 37), (190 / 23), (360 / 43),
+    (17 / 2), (320 / 37), (200 / 23), (9), (210 / 23), (340 / 37), (400 / 43), (19 / 2),
+    (220 / 23), (360 / 37), (10), (440 / 43), (380 / 37), (240 / 23), (21 / 2), (400 / 37),
+    (250 / 23), (11), (480 / 43), (260 / 23), (420 / 37), (23 / 2), (270 / 23), (440 / 37), (12),
+    (520 / 43), (280 / 23), (460 / 37), (25 / 2), (290 / 23), (480 / 37), (13), (560 / 43),
+    (300 / 23), (40 / 3), (310 / 23), (27 / 2), (500 / 37), (320 / 23), (600 / 43), (14),
+    (520 / 37), (330 / 23), (29 / 2), (540 / 37), (340 / 23), (640 / 43), (15), (560 / 37),
+    (350 / 23), (31 / 2), (360 / 23), (580 / 37), (680 / 43), (16), (370 / 23), (600 / 37),
+    (33 / 2), (380 / 23), (50 / 3), (720 / 43), (620 / 37), (390 / 23), (17), (640 / 37),
+    (400 / 23), (35 / 2), (760 / 43), (410 / 23), (660 / 37), (18), (420 / 23), (680 / 37),
+    (37 / 2), (800 / 43), (430 / 23), (700 / 37), (19), (440 / 23), (720 / 37), (39 / 2),
+    (840 / 43), (450 / 23), (20)]
+
+/-- Integer allocation levels chosen for the 125 inner-range pieces. -/
+def kInL : List ℤ :=
+  [2, 3, 3, 4, 5, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 11, 11, 11, 11, 12,
+    13, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 17, 17, 17, 17, 18, 18, 19, 19, 19, 19, 20, 20,
+    22, 22, 22, 23, 23, 23, 24, 24, 24, 25, 25, 25, 26, 26, 26, 26, 27, 27, 27, 28, 28, 28, 28,
+    29, 30, 31, 31, 31, 32, 32, 32, 32, 33, 33, 33, 34, 34, 34, 34, 35, 35, 36, 36, 36, 36, 37,
+    37, 37, 38, 39, 39, 39, 40, 40, 40, 41, 41, 41, 42, 42, 42, 43, 43, 43, 43, 44, 44, 44, 45,
+    45, 45, 45, 46]
+
+/-- Rational slopes of the 125 inner-range linear pieces. -/
+def aInL : List ℚ :=
+  [(18 / 5), (49 / 20), (283 / 40), (277 / 40), (231 / 40), (271 / 40), (71 / 20), (27 / 5),
+    (17 / 4), (17 / 5), (321 / 40), (55 / 8), (63 / 8), (93 / 20), (7 / 2), (107 / 20), (9 / 2),
+    (67 / 20), (319 / 40), (359 / 40), (23 / 4), (23 / 5), (129 / 20), (28 / 5), (89 / 20),
+    (363 / 40), (403 / 40), (137 / 20), (57 / 10), (69 / 10), (23 / 4), (49 / 10), (27 / 4),
+    (28 / 5), (19 / 8), (27 / 8), (8), (137 / 20), (6), (157 / 20), (67 / 10), (139 / 40),
+    (179 / 40), (91 / 10), (159 / 20), (71 / 10), (119 / 20), (39 / 5), (183 / 40), (223 / 40),
+    (177 / 40), (181 / 20), (69 / 10), (147 / 40), (221 / 40), (35 / 8), (43 / 8), (10),
+    (177 / 20), (8), (191 / 40), (29 / 8), (219 / 40), (259 / 40), (213 / 40), (199 / 20),
+    (91 / 10), (47 / 8), (189 / 40), (263 / 40), (303 / 40), (257 / 40), (221 / 20), (51 / 5),
+    (279 / 40), (233 / 40), (145 / 8), (679 / 40), (719 / 40), (793 / 40), (747 / 40), (309 / 20),
+    (73 / 5), (769 / 40), (723 / 40), (763 / 40), (837 / 40), (791 / 40), (331 / 20), (157 / 10),
+    (813 / 40), (767 / 40), (807 / 40), (761 / 40), (167 / 8), (353 / 20), (84 / 5), (313 / 20),
+    (811 / 40), (851 / 40), (161 / 8), (799 / 40), (67 / 4), (93 / 5), (349 / 20), (83 / 5),
+    (849 / 40), (803 / 40), (843 / 40), (357 / 20), (167 / 10), (371 / 20), (177 / 10),
+    (331 / 20), (847 / 40), (887 / 40), (379 / 20), (89 / 5), (393 / 20), (94 / 5), (353 / 20),
+    (891 / 40), (931 / 40), (401 / 20), (189 / 10)]
+
+/-- Rational intercepts of the 125 inner-range linear pieces. -/
+def bInL : List ℚ :=
+  [(-6), (-5 / 2), (-35 / 2), (-17), (-13), (-33 / 2), (-9 / 2), (-23 / 2), (-7), (-11), (-31),
+    (-26), (-61 / 2), (-31 / 2), (-10), (-19), (-24), (-18), (-43), (-97 / 2), (-61 / 2), (-24),
+    (-35), (-41), (-34), (-64), (-141 / 2), (-99 / 2), (-42), (-50), (-42), (-49), (-62),
+    (-107 / 2), (-59 / 2), (-37), (-72), (-63), (-71), (-86), (-153 / 2), (-99 / 2), (-58), (-98),
+    (-88), (-97), (-173 / 2), (-207 / 2), (-147 / 2), (-83), (-72), (-117), (-114), (-81), (-100),
+    (-88), (-197 / 2), (-297 / 2), (-136), (-147), (-111), (-98), (-119), (-261 / 2), (-117),
+    (-172), (-184), (-145), (-131), (-154), (-333 / 2), (-152), (-212), (-225), (-183), (-168),
+    (-184), (-337 / 2), (-182), (-207), (-191), (-146), (-160), (-225), (-417 / 2), (-223),
+    (-250), (-233), (-185), (-200), (-270), (-505 / 2), (-268), (-250), (-279), (-228), (-244),
+    (-451 / 2), (-601 / 2), (-317), (-298), (-591 / 2), (-483 / 2), (-545 / 2), (-253), (-270),
+    (-350), (-330), (-695 / 2), (-581 / 2), (-270), (-303), (-321), (-300), (-385), (-807 / 2),
+    (-687 / 2), (-322), (-357), (-376), (-354), (-444), (-927 / 2), (-801 / 2), (-378)]
+
+/-- The `i`th inner-range endpoint as a real number; zero outside the table. -/
+noncomputable def tIn (i : ℕ) : ℝ :=
+  ((tInL.getD i 0 : ℚ) : ℝ)
+
+/-- The allocation level for inner piece `i`; zero outside the table. -/
+def kIn (i : ℕ) : ℤ :=
+  kInL.getD i 0
+
+/-- The affine bound on inner piece `i`, with coefficients from `aInL` and `bInL`. -/
+noncomputable def gIn (i : ℕ) (x : ℝ) : ℝ :=
+  ((aInL.getD i 0 : ℚ) : ℝ) * x + ((bInL.getD i 0 : ℚ) : ℝ)
+
+theorem tIn_mono : ∀ i < 125, tIn i < tIn (i + 1) := by
+  have certificate : ∀ i : Fin 125,
+      tInL.getD i.val 0 < tInL.getD (i.val + 1) 0 := by decide +kernel
+  intro i hi
+  simp only [tIn]
+  exact_mod_cast certificate ⟨i, hi⟩
+
+private lemma inner_table_0 (x : ℝ) (hx : x ∈ Set.Ico (tIn 0) (tIn (0 + 1))) :
+    Ein x (kIn 0) = gIn 0 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_0 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_1 (x : ℝ) (hx : x ∈ Set.Ico (tIn 1) (tIn (1 + 1))) :
+    Ein x (kIn 1) = gIn 1 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_1 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_2 (x : ℝ) (hx : x ∈ Set.Ico (tIn 2) (tIn (2 + 1))) :
+    Ein x (kIn 2) = gIn 2 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_2 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_3 (x : ℝ) (hx : x ∈ Set.Ico (tIn 3) (tIn (3 + 1))) :
+    Ein x (kIn 3) = gIn 3 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_3 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_4 (x : ℝ) (hx : x ∈ Set.Ico (tIn 4) (tIn (4 + 1))) :
+    Ein x (kIn 4) = gIn 4 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_4 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_5 (x : ℝ) (hx : x ∈ Set.Ico (tIn 5) (tIn (5 + 1))) :
+    Ein x (kIn 5) = gIn 5 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_5 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_6 (x : ℝ) (hx : x ∈ Set.Ico (tIn 6) (tIn (6 + 1))) :
+    Ein x (kIn 6) = gIn 6 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_6 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_7 (x : ℝ) (hx : x ∈ Set.Ico (tIn 7) (tIn (7 + 1))) :
+    Ein x (kIn 7) = gIn 7 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_7 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_8 (x : ℝ) (hx : x ∈ Set.Ico (tIn 8) (tIn (8 + 1))) :
+    Ein x (kIn 8) = gIn 8 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_8 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_9 (x : ℝ) (hx : x ∈ Set.Ico (tIn 9) (tIn (9 + 1))) :
+    Ein x (kIn 9) = gIn 9 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_9 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_10 (x : ℝ) (hx : x ∈ Set.Ico (tIn 10) (tIn (10 + 1))) :
+    Ein x (kIn 10) = gIn 10 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_10 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_11 (x : ℝ) (hx : x ∈ Set.Ico (tIn 11) (tIn (11 + 1))) :
+    Ein x (kIn 11) = gIn 11 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_11 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_12 (x : ℝ) (hx : x ∈ Set.Ico (tIn 12) (tIn (12 + 1))) :
+    Ein x (kIn 12) = gIn 12 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_12 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_13 (x : ℝ) (hx : x ∈ Set.Ico (tIn 13) (tIn (13 + 1))) :
+    Ein x (kIn 13) = gIn 13 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_13 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_14 (x : ℝ) (hx : x ∈ Set.Ico (tIn 14) (tIn (14 + 1))) :
+    Ein x (kIn 14) = gIn 14 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_14 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_15 (x : ℝ) (hx : x ∈ Set.Ico (tIn 15) (tIn (15 + 1))) :
+    Ein x (kIn 15) = gIn 15 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_15 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_16 (x : ℝ) (hx : x ∈ Set.Ico (tIn 16) (tIn (16 + 1))) :
+    Ein x (kIn 16) = gIn 16 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_16 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_17 (x : ℝ) (hx : x ∈ Set.Ico (tIn 17) (tIn (17 + 1))) :
+    Ein x (kIn 17) = gIn 17 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_17 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_18 (x : ℝ) (hx : x ∈ Set.Ico (tIn 18) (tIn (18 + 1))) :
+    Ein x (kIn 18) = gIn 18 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_18 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_19 (x : ℝ) (hx : x ∈ Set.Ico (tIn 19) (tIn (19 + 1))) :
+    Ein x (kIn 19) = gIn 19 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_19 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_20 (x : ℝ) (hx : x ∈ Set.Ico (tIn 20) (tIn (20 + 1))) :
+    Ein x (kIn 20) = gIn 20 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_20 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_21 (x : ℝ) (hx : x ∈ Set.Ico (tIn 21) (tIn (21 + 1))) :
+    Ein x (kIn 21) = gIn 21 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_21 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_22 (x : ℝ) (hx : x ∈ Set.Ico (tIn 22) (tIn (22 + 1))) :
+    Ein x (kIn 22) = gIn 22 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_22 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_23 (x : ℝ) (hx : x ∈ Set.Ico (tIn 23) (tIn (23 + 1))) :
+    Ein x (kIn 23) = gIn 23 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_23 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_24 (x : ℝ) (hx : x ∈ Set.Ico (tIn 24) (tIn (24 + 1))) :
+    Ein x (kIn 24) = gIn 24 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_24 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_25 (x : ℝ) (hx : x ∈ Set.Ico (tIn 25) (tIn (25 + 1))) :
+    Ein x (kIn 25) = gIn 25 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_25 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_26 (x : ℝ) (hx : x ∈ Set.Ico (tIn 26) (tIn (26 + 1))) :
+    Ein x (kIn 26) = gIn 26 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_26 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_27 (x : ℝ) (hx : x ∈ Set.Ico (tIn 27) (tIn (27 + 1))) :
+    Ein x (kIn 27) = gIn 27 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_27 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_28 (x : ℝ) (hx : x ∈ Set.Ico (tIn 28) (tIn (28 + 1))) :
+    Ein x (kIn 28) = gIn 28 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_28 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_29 (x : ℝ) (hx : x ∈ Set.Ico (tIn 29) (tIn (29 + 1))) :
+    Ein x (kIn 29) = gIn 29 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_29 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_30 (x : ℝ) (hx : x ∈ Set.Ico (tIn 30) (tIn (30 + 1))) :
+    Ein x (kIn 30) = gIn 30 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_30 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_31 (x : ℝ) (hx : x ∈ Set.Ico (tIn 31) (tIn (31 + 1))) :
+    Ein x (kIn 31) = gIn 31 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_31 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_32 (x : ℝ) (hx : x ∈ Set.Ico (tIn 32) (tIn (32 + 1))) :
+    Ein x (kIn 32) = gIn 32 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_32 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_33 (x : ℝ) (hx : x ∈ Set.Ico (tIn 33) (tIn (33 + 1))) :
+    Ein x (kIn 33) = gIn 33 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_33 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_34 (x : ℝ) (hx : x ∈ Set.Ico (tIn 34) (tIn (34 + 1))) :
+    Ein x (kIn 34) = gIn 34 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_34 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_35 (x : ℝ) (hx : x ∈ Set.Ico (tIn 35) (tIn (35 + 1))) :
+    Ein x (kIn 35) = gIn 35 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_35 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_36 (x : ℝ) (hx : x ∈ Set.Ico (tIn 36) (tIn (36 + 1))) :
+    Ein x (kIn 36) = gIn 36 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_36 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_37 (x : ℝ) (hx : x ∈ Set.Ico (tIn 37) (tIn (37 + 1))) :
+    Ein x (kIn 37) = gIn 37 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_37 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_38 (x : ℝ) (hx : x ∈ Set.Ico (tIn 38) (tIn (38 + 1))) :
+    Ein x (kIn 38) = gIn 38 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_38 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_39 (x : ℝ) (hx : x ∈ Set.Ico (tIn 39) (tIn (39 + 1))) :
+    Ein x (kIn 39) = gIn 39 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_39 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_40 (x : ℝ) (hx : x ∈ Set.Ico (tIn 40) (tIn (40 + 1))) :
+    Ein x (kIn 40) = gIn 40 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_40 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_41 (x : ℝ) (hx : x ∈ Set.Ico (tIn 41) (tIn (41 + 1))) :
+    Ein x (kIn 41) = gIn 41 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_41 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_42 (x : ℝ) (hx : x ∈ Set.Ico (tIn 42) (tIn (42 + 1))) :
+    Ein x (kIn 42) = gIn 42 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_42 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_43 (x : ℝ) (hx : x ∈ Set.Ico (tIn 43) (tIn (43 + 1))) :
+    Ein x (kIn 43) = gIn 43 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_43 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_44 (x : ℝ) (hx : x ∈ Set.Ico (tIn 44) (tIn (44 + 1))) :
+    Ein x (kIn 44) = gIn 44 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_44 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_45 (x : ℝ) (hx : x ∈ Set.Ico (tIn 45) (tIn (45 + 1))) :
+    Ein x (kIn 45) = gIn 45 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_45 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_46 (x : ℝ) (hx : x ∈ Set.Ico (tIn 46) (tIn (46 + 1))) :
+    Ein x (kIn 46) = gIn 46 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_46 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_47 (x : ℝ) (hx : x ∈ Set.Ico (tIn 47) (tIn (47 + 1))) :
+    Ein x (kIn 47) = gIn 47 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_47 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_48 (x : ℝ) (hx : x ∈ Set.Ico (tIn 48) (tIn (48 + 1))) :
+    Ein x (kIn 48) = gIn 48 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_48 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_49 (x : ℝ) (hx : x ∈ Set.Ico (tIn 49) (tIn (49 + 1))) :
+    Ein x (kIn 49) = gIn 49 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_49 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_50 (x : ℝ) (hx : x ∈ Set.Ico (tIn 50) (tIn (50 + 1))) :
+    Ein x (kIn 50) = gIn 50 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_50 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_51 (x : ℝ) (hx : x ∈ Set.Ico (tIn 51) (tIn (51 + 1))) :
+    Ein x (kIn 51) = gIn 51 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_51 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_52 (x : ℝ) (hx : x ∈ Set.Ico (tIn 52) (tIn (52 + 1))) :
+    Ein x (kIn 52) = gIn 52 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_52 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_53 (x : ℝ) (hx : x ∈ Set.Ico (tIn 53) (tIn (53 + 1))) :
+    Ein x (kIn 53) = gIn 53 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_53 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_54 (x : ℝ) (hx : x ∈ Set.Ico (tIn 54) (tIn (54 + 1))) :
+    Ein x (kIn 54) = gIn 54 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_54 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_55 (x : ℝ) (hx : x ∈ Set.Ico (tIn 55) (tIn (55 + 1))) :
+    Ein x (kIn 55) = gIn 55 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_55 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_56 (x : ℝ) (hx : x ∈ Set.Ico (tIn 56) (tIn (56 + 1))) :
+    Ein x (kIn 56) = gIn 56 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_56 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_57 (x : ℝ) (hx : x ∈ Set.Ico (tIn 57) (tIn (57 + 1))) :
+    Ein x (kIn 57) = gIn 57 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_57 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_58 (x : ℝ) (hx : x ∈ Set.Ico (tIn 58) (tIn (58 + 1))) :
+    Ein x (kIn 58) = gIn 58 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_58 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_59 (x : ℝ) (hx : x ∈ Set.Ico (tIn 59) (tIn (59 + 1))) :
+    Ein x (kIn 59) = gIn 59 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_59 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_60 (x : ℝ) (hx : x ∈ Set.Ico (tIn 60) (tIn (60 + 1))) :
+    Ein x (kIn 60) = gIn 60 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_60 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_61 (x : ℝ) (hx : x ∈ Set.Ico (tIn 61) (tIn (61 + 1))) :
+    Ein x (kIn 61) = gIn 61 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_61 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_62 (x : ℝ) (hx : x ∈ Set.Ico (tIn 62) (tIn (62 + 1))) :
+    Ein x (kIn 62) = gIn 62 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_62 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_63 (x : ℝ) (hx : x ∈ Set.Ico (tIn 63) (tIn (63 + 1))) :
+    Ein x (kIn 63) = gIn 63 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_63 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_64 (x : ℝ) (hx : x ∈ Set.Ico (tIn 64) (tIn (64 + 1))) :
+    Ein x (kIn 64) = gIn 64 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_64 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_65 (x : ℝ) (hx : x ∈ Set.Ico (tIn 65) (tIn (65 + 1))) :
+    Ein x (kIn 65) = gIn 65 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_65 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_66 (x : ℝ) (hx : x ∈ Set.Ico (tIn 66) (tIn (66 + 1))) :
+    Ein x (kIn 66) = gIn 66 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_66 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_67 (x : ℝ) (hx : x ∈ Set.Ico (tIn 67) (tIn (67 + 1))) :
+    Ein x (kIn 67) = gIn 67 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_67 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_68 (x : ℝ) (hx : x ∈ Set.Ico (tIn 68) (tIn (68 + 1))) :
+    Ein x (kIn 68) = gIn 68 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_68 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_69 (x : ℝ) (hx : x ∈ Set.Ico (tIn 69) (tIn (69 + 1))) :
+    Ein x (kIn 69) = gIn 69 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_69 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_70 (x : ℝ) (hx : x ∈ Set.Ico (tIn 70) (tIn (70 + 1))) :
+    Ein x (kIn 70) = gIn 70 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_70 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_71 (x : ℝ) (hx : x ∈ Set.Ico (tIn 71) (tIn (71 + 1))) :
+    Ein x (kIn 71) = gIn 71 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_71 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_72 (x : ℝ) (hx : x ∈ Set.Ico (tIn 72) (tIn (72 + 1))) :
+    Ein x (kIn 72) = gIn 72 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_72 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_73 (x : ℝ) (hx : x ∈ Set.Ico (tIn 73) (tIn (73 + 1))) :
+    Ein x (kIn 73) = gIn 73 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_73 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_74 (x : ℝ) (hx : x ∈ Set.Ico (tIn 74) (tIn (74 + 1))) :
+    Ein x (kIn 74) = gIn 74 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_74 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_75 (x : ℝ) (hx : x ∈ Set.Ico (tIn 75) (tIn (75 + 1))) :
+    Ein x (kIn 75) = gIn 75 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_75 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_76 (x : ℝ) (hx : x ∈ Set.Ico (tIn 76) (tIn (76 + 1))) :
+    Ein x (kIn 76) = gIn 76 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_76 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_77 (x : ℝ) (hx : x ∈ Set.Ico (tIn 77) (tIn (77 + 1))) :
+    Ein x (kIn 77) = gIn 77 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_77 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_78 (x : ℝ) (hx : x ∈ Set.Ico (tIn 78) (tIn (78 + 1))) :
+    Ein x (kIn 78) = gIn 78 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_78 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_79 (x : ℝ) (hx : x ∈ Set.Ico (tIn 79) (tIn (79 + 1))) :
+    Ein x (kIn 79) = gIn 79 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_79 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_80 (x : ℝ) (hx : x ∈ Set.Ico (tIn 80) (tIn (80 + 1))) :
+    Ein x (kIn 80) = gIn 80 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_80 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_81 (x : ℝ) (hx : x ∈ Set.Ico (tIn 81) (tIn (81 + 1))) :
+    Ein x (kIn 81) = gIn 81 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_81 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_82 (x : ℝ) (hx : x ∈ Set.Ico (tIn 82) (tIn (82 + 1))) :
+    Ein x (kIn 82) = gIn 82 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_82 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_83 (x : ℝ) (hx : x ∈ Set.Ico (tIn 83) (tIn (83 + 1))) :
+    Ein x (kIn 83) = gIn 83 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_83 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_84 (x : ℝ) (hx : x ∈ Set.Ico (tIn 84) (tIn (84 + 1))) :
+    Ein x (kIn 84) = gIn 84 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_84 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_85 (x : ℝ) (hx : x ∈ Set.Ico (tIn 85) (tIn (85 + 1))) :
+    Ein x (kIn 85) = gIn 85 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_85 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_86 (x : ℝ) (hx : x ∈ Set.Ico (tIn 86) (tIn (86 + 1))) :
+    Ein x (kIn 86) = gIn 86 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_86 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_87 (x : ℝ) (hx : x ∈ Set.Ico (tIn 87) (tIn (87 + 1))) :
+    Ein x (kIn 87) = gIn 87 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_87 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_88 (x : ℝ) (hx : x ∈ Set.Ico (tIn 88) (tIn (88 + 1))) :
+    Ein x (kIn 88) = gIn 88 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_88 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_89 (x : ℝ) (hx : x ∈ Set.Ico (tIn 89) (tIn (89 + 1))) :
+    Ein x (kIn 89) = gIn 89 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_89 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_90 (x : ℝ) (hx : x ∈ Set.Ico (tIn 90) (tIn (90 + 1))) :
+    Ein x (kIn 90) = gIn 90 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_90 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_91 (x : ℝ) (hx : x ∈ Set.Ico (tIn 91) (tIn (91 + 1))) :
+    Ein x (kIn 91) = gIn 91 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_91 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_92 (x : ℝ) (hx : x ∈ Set.Ico (tIn 92) (tIn (92 + 1))) :
+    Ein x (kIn 92) = gIn 92 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_92 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_93 (x : ℝ) (hx : x ∈ Set.Ico (tIn 93) (tIn (93 + 1))) :
+    Ein x (kIn 93) = gIn 93 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_93 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_94 (x : ℝ) (hx : x ∈ Set.Ico (tIn 94) (tIn (94 + 1))) :
+    Ein x (kIn 94) = gIn 94 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_94 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_95 (x : ℝ) (hx : x ∈ Set.Ico (tIn 95) (tIn (95 + 1))) :
+    Ein x (kIn 95) = gIn 95 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_95 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_96 (x : ℝ) (hx : x ∈ Set.Ico (tIn 96) (tIn (96 + 1))) :
+    Ein x (kIn 96) = gIn 96 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_96 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_97 (x : ℝ) (hx : x ∈ Set.Ico (tIn 97) (tIn (97 + 1))) :
+    Ein x (kIn 97) = gIn 97 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_97 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_98 (x : ℝ) (hx : x ∈ Set.Ico (tIn 98) (tIn (98 + 1))) :
+    Ein x (kIn 98) = gIn 98 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_98 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_99 (x : ℝ) (hx : x ∈ Set.Ico (tIn 99) (tIn (99 + 1))) :
+    Ein x (kIn 99) = gIn 99 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_99 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_100 (x : ℝ) (hx : x ∈ Set.Ico (tIn 100) (tIn (100 + 1))) :
+    Ein x (kIn 100) = gIn 100 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_100 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_101 (x : ℝ) (hx : x ∈ Set.Ico (tIn 101) (tIn (101 + 1))) :
+    Ein x (kIn 101) = gIn 101 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_101 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_102 (x : ℝ) (hx : x ∈ Set.Ico (tIn 102) (tIn (102 + 1))) :
+    Ein x (kIn 102) = gIn 102 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_102 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_103 (x : ℝ) (hx : x ∈ Set.Ico (tIn 103) (tIn (103 + 1))) :
+    Ein x (kIn 103) = gIn 103 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_103 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_104 (x : ℝ) (hx : x ∈ Set.Ico (tIn 104) (tIn (104 + 1))) :
+    Ein x (kIn 104) = gIn 104 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_104 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_105 (x : ℝ) (hx : x ∈ Set.Ico (tIn 105) (tIn (105 + 1))) :
+    Ein x (kIn 105) = gIn 105 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_105 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_106 (x : ℝ) (hx : x ∈ Set.Ico (tIn 106) (tIn (106 + 1))) :
+    Ein x (kIn 106) = gIn 106 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_106 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_107 (x : ℝ) (hx : x ∈ Set.Ico (tIn 107) (tIn (107 + 1))) :
+    Ein x (kIn 107) = gIn 107 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_107 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_108 (x : ℝ) (hx : x ∈ Set.Ico (tIn 108) (tIn (108 + 1))) :
+    Ein x (kIn 108) = gIn 108 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_108 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_109 (x : ℝ) (hx : x ∈ Set.Ico (tIn 109) (tIn (109 + 1))) :
+    Ein x (kIn 109) = gIn 109 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_109 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_110 (x : ℝ) (hx : x ∈ Set.Ico (tIn 110) (tIn (110 + 1))) :
+    Ein x (kIn 110) = gIn 110 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_110 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_111 (x : ℝ) (hx : x ∈ Set.Ico (tIn 111) (tIn (111 + 1))) :
+    Ein x (kIn 111) = gIn 111 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_111 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_112 (x : ℝ) (hx : x ∈ Set.Ico (tIn 112) (tIn (112 + 1))) :
+    Ein x (kIn 112) = gIn 112 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_112 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_113 (x : ℝ) (hx : x ∈ Set.Ico (tIn 113) (tIn (113 + 1))) :
+    Ein x (kIn 113) = gIn 113 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_113 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_114 (x : ℝ) (hx : x ∈ Set.Ico (tIn 114) (tIn (114 + 1))) :
+    Ein x (kIn 114) = gIn 114 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_114 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_115 (x : ℝ) (hx : x ∈ Set.Ico (tIn 115) (tIn (115 + 1))) :
+    Ein x (kIn 115) = gIn 115 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_115 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_116 (x : ℝ) (hx : x ∈ Set.Ico (tIn 116) (tIn (116 + 1))) :
+    Ein x (kIn 116) = gIn 116 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_116 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_117 (x : ℝ) (hx : x ∈ Set.Ico (tIn 117) (tIn (117 + 1))) :
+    Ein x (kIn 117) = gIn 117 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_117 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_118 (x : ℝ) (hx : x ∈ Set.Ico (tIn 118) (tIn (118 + 1))) :
+    Ein x (kIn 118) = gIn 118 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_118 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_119 (x : ℝ) (hx : x ∈ Set.Ico (tIn 119) (tIn (119 + 1))) :
+    Ein x (kIn 119) = gIn 119 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_119 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_120 (x : ℝ) (hx : x ∈ Set.Ico (tIn 120) (tIn (120 + 1))) :
+    Ein x (kIn 120) = gIn 120 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_120 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_121 (x : ℝ) (hx : x ∈ Set.Ico (tIn 121) (tIn (121 + 1))) :
+    Ein x (kIn 121) = gIn 121 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_121 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_122 (x : ℝ) (hx : x ∈ Set.Ico (tIn 122) (tIn (122 + 1))) :
+    Ein x (kIn 122) = gIn 122 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_122 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_123 (x : ℝ) (hx : x ∈ Set.Ico (tIn 123) (tIn (123 + 1))) :
+    Ein x (kIn 123) = gIn 123 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_123 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+private lemma inner_table_124 (x : ℝ) (hx : x ∈ Set.Ico (tIn 124) (tIn (124 + 1))) :
+    Ein x (kIn 124) = gIn 124 x := by
+  simp only [tIn, tInL, kIn, kInL, gIn, aInL, bInL, List.getD_cons_succ, List.getD_cons_zero,
+    Set.mem_Ico] at hx ⊢
+  push_cast at hx ⊢
+  have := inPiece_124 x (by linarith [hx.1]) (by linarith [hx.2])
+  linarith
+
+theorem inner_table : ∀ i < 125, ∀ x ∈ Set.Ico (tIn i) (tIn (i + 1)), Ein x (kIn i) = gIn i x := by
+  intro i hi x hx
+  interval_cases i
+  · exact inner_table_0 x hx
+  · exact inner_table_1 x hx
+  · exact inner_table_2 x hx
+  · exact inner_table_3 x hx
+  · exact inner_table_4 x hx
+  · exact inner_table_5 x hx
+  · exact inner_table_6 x hx
+  · exact inner_table_7 x hx
+  · exact inner_table_8 x hx
+  · exact inner_table_9 x hx
+  · exact inner_table_10 x hx
+  · exact inner_table_11 x hx
+  · exact inner_table_12 x hx
+  · exact inner_table_13 x hx
+  · exact inner_table_14 x hx
+  · exact inner_table_15 x hx
+  · exact inner_table_16 x hx
+  · exact inner_table_17 x hx
+  · exact inner_table_18 x hx
+  · exact inner_table_19 x hx
+  · exact inner_table_20 x hx
+  · exact inner_table_21 x hx
+  · exact inner_table_22 x hx
+  · exact inner_table_23 x hx
+  · exact inner_table_24 x hx
+  · exact inner_table_25 x hx
+  · exact inner_table_26 x hx
+  · exact inner_table_27 x hx
+  · exact inner_table_28 x hx
+  · exact inner_table_29 x hx
+  · exact inner_table_30 x hx
+  · exact inner_table_31 x hx
+  · exact inner_table_32 x hx
+  · exact inner_table_33 x hx
+  · exact inner_table_34 x hx
+  · exact inner_table_35 x hx
+  · exact inner_table_36 x hx
+  · exact inner_table_37 x hx
+  · exact inner_table_38 x hx
+  · exact inner_table_39 x hx
+  · exact inner_table_40 x hx
+  · exact inner_table_41 x hx
+  · exact inner_table_42 x hx
+  · exact inner_table_43 x hx
+  · exact inner_table_44 x hx
+  · exact inner_table_45 x hx
+  · exact inner_table_46 x hx
+  · exact inner_table_47 x hx
+  · exact inner_table_48 x hx
+  · exact inner_table_49 x hx
+  · exact inner_table_50 x hx
+  · exact inner_table_51 x hx
+  · exact inner_table_52 x hx
+  · exact inner_table_53 x hx
+  · exact inner_table_54 x hx
+  · exact inner_table_55 x hx
+  · exact inner_table_56 x hx
+  · exact inner_table_57 x hx
+  · exact inner_table_58 x hx
+  · exact inner_table_59 x hx
+  · exact inner_table_60 x hx
+  · exact inner_table_61 x hx
+  · exact inner_table_62 x hx
+  · exact inner_table_63 x hx
+  · exact inner_table_64 x hx
+  · exact inner_table_65 x hx
+  · exact inner_table_66 x hx
+  · exact inner_table_67 x hx
+  · exact inner_table_68 x hx
+  · exact inner_table_69 x hx
+  · exact inner_table_70 x hx
+  · exact inner_table_71 x hx
+  · exact inner_table_72 x hx
+  · exact inner_table_73 x hx
+  · exact inner_table_74 x hx
+  · exact inner_table_75 x hx
+  · exact inner_table_76 x hx
+  · exact inner_table_77 x hx
+  · exact inner_table_78 x hx
+  · exact inner_table_79 x hx
+  · exact inner_table_80 x hx
+  · exact inner_table_81 x hx
+  · exact inner_table_82 x hx
+  · exact inner_table_83 x hx
+  · exact inner_table_84 x hx
+  · exact inner_table_85 x hx
+  · exact inner_table_86 x hx
+  · exact inner_table_87 x hx
+  · exact inner_table_88 x hx
+  · exact inner_table_89 x hx
+  · exact inner_table_90 x hx
+  · exact inner_table_91 x hx
+  · exact inner_table_92 x hx
+  · exact inner_table_93 x hx
+  · exact inner_table_94 x hx
+  · exact inner_table_95 x hx
+  · exact inner_table_96 x hx
+  · exact inner_table_97 x hx
+  · exact inner_table_98 x hx
+  · exact inner_table_99 x hx
+  · exact inner_table_100 x hx
+  · exact inner_table_101 x hx
+  · exact inner_table_102 x hx
+  · exact inner_table_103 x hx
+  · exact inner_table_104 x hx
+  · exact inner_table_105 x hx
+  · exact inner_table_106 x hx
+  · exact inner_table_107 x hx
+  · exact inner_table_108 x hx
+  · exact inner_table_109 x hx
+  · exact inner_table_110 x hx
+  · exact inner_table_111 x hx
+  · exact inner_table_112 x hx
+  · exact inner_table_113 x hx
+  · exact inner_table_114 x hx
+  · exact inner_table_115 x hx
+  · exact inner_table_116 x hx
+  · exact inner_table_117 x hx
+  · exact inner_table_118 x hx
+  · exact inner_table_119 x hx
+  · exact inner_table_120 x hx
+  · exact inner_table_121 x hx
+  · exact inner_table_122 x hx
+  · exact inner_table_123 x hx
+  · exact inner_table_124 x hx
+
+private lemma inner_table_k_0 (x : ℝ) (hx : x ∈ Set.Ico (tIn 0) (tIn (0 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 0 ∧ kIn 0 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 3) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_1 (x : ℝ) (hx : x ∈ Set.Ico (tIn 1) (tIn (1 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 1 ∧ kIn 1 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 3) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_2 (x : ℝ) (hx : x ∈ Set.Ico (tIn 2) (tIn (2 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 2 ∧ kIn 2 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 3) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_3 (x : ℝ) (hx : x ∈ Set.Ico (tIn 3) (tIn (3 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 3 ∧ kIn 3 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 3) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_4 (x : ℝ) (hx : x ∈ Set.Ico (tIn 4) (tIn (4 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 4 ∧ kIn 4 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 3) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_5 (x : ℝ) (hx : x ∈ Set.Ico (tIn 5) (tIn (5 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 5 ∧ kIn 5 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 3) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_6 (x : ℝ) (hx : x ∈ Set.Ico (tIn 6) (tIn (6 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 6 ∧ kIn 6 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 3) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_7 (x : ℝ) (hx : x ∈ Set.Ico (tIn 7) (tIn (7 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 7 ∧ kIn 7 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 3) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_8 (x : ℝ) (hx : x ∈ Set.Ico (tIn 8) (tIn (8 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 8 ∧ kIn 8 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 3) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_9 (x : ℝ) (hx : x ∈ Set.Ico (tIn 9) (tIn (9 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 9 ∧ kIn 9 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 4) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_10 (x : ℝ) (hx : x ∈ Set.Ico (tIn 10) (tIn (10 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 10 ∧ kIn 10 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 4) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_11 (x : ℝ) (hx : x ∈ Set.Ico (tIn 11) (tIn (11 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 11 ∧ kIn 11 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 4) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_12 (x : ℝ) (hx : x ∈ Set.Ico (tIn 12) (tIn (12 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 12 ∧ kIn 12 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 4) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_13 (x : ℝ) (hx : x ∈ Set.Ico (tIn 13) (tIn (13 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 13 ∧ kIn 13 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 4) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_14 (x : ℝ) (hx : x ∈ Set.Ico (tIn 14) (tIn (14 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 14 ∧ kIn 14 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 4) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_15 (x : ℝ) (hx : x ∈ Set.Ico (tIn 15) (tIn (15 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 15 ∧ kIn 15 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 4) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_16 (x : ℝ) (hx : x ∈ Set.Ico (tIn 16) (tIn (16 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 16 ∧ kIn 16 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 5) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_17 (x : ℝ) (hx : x ∈ Set.Ico (tIn 17) (tIn (17 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 17 ∧ kIn 17 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 5) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_18 (x : ℝ) (hx : x ∈ Set.Ico (tIn 18) (tIn (18 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 18 ∧ kIn 18 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 5) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_19 (x : ℝ) (hx : x ∈ Set.Ico (tIn 19) (tIn (19 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 19 ∧ kIn 19 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 5) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_20 (x : ℝ) (hx : x ∈ Set.Ico (tIn 20) (tIn (20 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 20 ∧ kIn 20 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 5) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_21 (x : ℝ) (hx : x ∈ Set.Ico (tIn 21) (tIn (21 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 21 ∧ kIn 21 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 5) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_22 (x : ℝ) (hx : x ∈ Set.Ico (tIn 22) (tIn (22 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 22 ∧ kIn 22 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 5) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_23 (x : ℝ) (hx : x ∈ Set.Ico (tIn 23) (tIn (23 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 23 ∧ kIn 23 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 6) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_24 (x : ℝ) (hx : x ∈ Set.Ico (tIn 24) (tIn (24 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 24 ∧ kIn 24 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 6) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_25 (x : ℝ) (hx : x ∈ Set.Ico (tIn 25) (tIn (25 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 25 ∧ kIn 25 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 6) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_26 (x : ℝ) (hx : x ∈ Set.Ico (tIn 26) (tIn (26 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 26 ∧ kIn 26 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 6) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_27 (x : ℝ) (hx : x ∈ Set.Ico (tIn 27) (tIn (27 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 27 ∧ kIn 27 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 6) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_28 (x : ℝ) (hx : x ∈ Set.Ico (tIn 28) (tIn (28 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 28 ∧ kIn 28 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 6) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_29 (x : ℝ) (hx : x ∈ Set.Ico (tIn 29) (tIn (29 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 29 ∧ kIn 29 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 6) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_30 (x : ℝ) (hx : x ∈ Set.Ico (tIn 30) (tIn (30 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 30 ∧ kIn 30 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 6) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_31 (x : ℝ) (hx : x ∈ Set.Ico (tIn 31) (tIn (31 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 31 ∧ kIn 31 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 7) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_32 (x : ℝ) (hx : x ∈ Set.Ico (tIn 32) (tIn (32 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 32 ∧ kIn 32 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 7) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_33 (x : ℝ) (hx : x ∈ Set.Ico (tIn 33) (tIn (33 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 33 ∧ kIn 33 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 7) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_34 (x : ℝ) (hx : x ∈ Set.Ico (tIn 34) (tIn (34 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 34 ∧ kIn 34 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 7) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_35 (x : ℝ) (hx : x ∈ Set.Ico (tIn 35) (tIn (35 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 35 ∧ kIn 35 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 7) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_36 (x : ℝ) (hx : x ∈ Set.Ico (tIn 36) (tIn (36 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 36 ∧ kIn 36 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 7) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_37 (x : ℝ) (hx : x ∈ Set.Ico (tIn 37) (tIn (37 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 37 ∧ kIn 37 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 7) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_38 (x : ℝ) (hx : x ∈ Set.Ico (tIn 38) (tIn (38 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 38 ∧ kIn 38 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 8) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_39 (x : ℝ) (hx : x ∈ Set.Ico (tIn 39) (tIn (39 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 39 ∧ kIn 39 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 8) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_40 (x : ℝ) (hx : x ∈ Set.Ico (tIn 40) (tIn (40 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 40 ∧ kIn 40 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 8) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_41 (x : ℝ) (hx : x ∈ Set.Ico (tIn 41) (tIn (41 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 41 ∧ kIn 41 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 8) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_42 (x : ℝ) (hx : x ∈ Set.Ico (tIn 42) (tIn (42 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 42 ∧ kIn 42 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 8) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_43 (x : ℝ) (hx : x ∈ Set.Ico (tIn 43) (tIn (43 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 43 ∧ kIn 43 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 8) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_44 (x : ℝ) (hx : x ∈ Set.Ico (tIn 44) (tIn (44 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 44 ∧ kIn 44 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 8) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_45 (x : ℝ) (hx : x ∈ Set.Ico (tIn 45) (tIn (45 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 45 ∧ kIn 45 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 9) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_46 (x : ℝ) (hx : x ∈ Set.Ico (tIn 46) (tIn (46 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 46 ∧ kIn 46 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 9) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_47 (x : ℝ) (hx : x ∈ Set.Ico (tIn 47) (tIn (47 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 47 ∧ kIn 47 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 9) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_48 (x : ℝ) (hx : x ∈ Set.Ico (tIn 48) (tIn (48 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 48 ∧ kIn 48 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 9) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_49 (x : ℝ) (hx : x ∈ Set.Ico (tIn 49) (tIn (49 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 49 ∧ kIn 49 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 9) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_50 (x : ℝ) (hx : x ∈ Set.Ico (tIn 50) (tIn (50 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 50 ∧ kIn 50 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 9) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_51 (x : ℝ) (hx : x ∈ Set.Ico (tIn 51) (tIn (51 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 51 ∧ kIn 51 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 9) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_52 (x : ℝ) (hx : x ∈ Set.Ico (tIn 52) (tIn (52 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 52 ∧ kIn 52 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 10) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_53 (x : ℝ) (hx : x ∈ Set.Ico (tIn 53) (tIn (53 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 53 ∧ kIn 53 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 10) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_54 (x : ℝ) (hx : x ∈ Set.Ico (tIn 54) (tIn (54 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 54 ∧ kIn 54 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 10) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_55 (x : ℝ) (hx : x ∈ Set.Ico (tIn 55) (tIn (55 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 55 ∧ kIn 55 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 10) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_56 (x : ℝ) (hx : x ∈ Set.Ico (tIn 56) (tIn (56 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 56 ∧ kIn 56 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 10) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_57 (x : ℝ) (hx : x ∈ Set.Ico (tIn 57) (tIn (57 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 57 ∧ kIn 57 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 10) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_58 (x : ℝ) (hx : x ∈ Set.Ico (tIn 58) (tIn (58 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 58 ∧ kIn 58 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 10) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_59 (x : ℝ) (hx : x ∈ Set.Ico (tIn 59) (tIn (59 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 59 ∧ kIn 59 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 11) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_60 (x : ℝ) (hx : x ∈ Set.Ico (tIn 60) (tIn (60 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 60 ∧ kIn 60 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 11) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_61 (x : ℝ) (hx : x ∈ Set.Ico (tIn 61) (tIn (61 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 61 ∧ kIn 61 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 11) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_62 (x : ℝ) (hx : x ∈ Set.Ico (tIn 62) (tIn (62 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 62 ∧ kIn 62 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 11) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_63 (x : ℝ) (hx : x ∈ Set.Ico (tIn 63) (tIn (63 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 63 ∧ kIn 63 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 11) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_64 (x : ℝ) (hx : x ∈ Set.Ico (tIn 64) (tIn (64 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 64 ∧ kIn 64 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 11) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_65 (x : ℝ) (hx : x ∈ Set.Ico (tIn 65) (tIn (65 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 65 ∧ kIn 65 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 11) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_66 (x : ℝ) (hx : x ∈ Set.Ico (tIn 66) (tIn (66 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 66 ∧ kIn 66 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 12) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_67 (x : ℝ) (hx : x ∈ Set.Ico (tIn 67) (tIn (67 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 67 ∧ kIn 67 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 12) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_68 (x : ℝ) (hx : x ∈ Set.Ico (tIn 68) (tIn (68 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 68 ∧ kIn 68 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 12) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_69 (x : ℝ) (hx : x ∈ Set.Ico (tIn 69) (tIn (69 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 69 ∧ kIn 69 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 12) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_70 (x : ℝ) (hx : x ∈ Set.Ico (tIn 70) (tIn (70 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 70 ∧ kIn 70 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 12) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_71 (x : ℝ) (hx : x ∈ Set.Ico (tIn 71) (tIn (71 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 71 ∧ kIn 71 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 12) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_72 (x : ℝ) (hx : x ∈ Set.Ico (tIn 72) (tIn (72 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 72 ∧ kIn 72 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 12) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_73 (x : ℝ) (hx : x ∈ Set.Ico (tIn 73) (tIn (73 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 73 ∧ kIn 73 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 13) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_74 (x : ℝ) (hx : x ∈ Set.Ico (tIn 74) (tIn (74 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 74 ∧ kIn 74 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 13) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_75 (x : ℝ) (hx : x ∈ Set.Ico (tIn 75) (tIn (75 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 75 ∧ kIn 75 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 13) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_76 (x : ℝ) (hx : x ∈ Set.Ico (tIn 76) (tIn (76 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 76 ∧ kIn 76 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 13) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_77 (x : ℝ) (hx : x ∈ Set.Ico (tIn 77) (tIn (77 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 77 ∧ kIn 77 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 13) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_78 (x : ℝ) (hx : x ∈ Set.Ico (tIn 78) (tIn (78 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 78 ∧ kIn 78 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 13) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_79 (x : ℝ) (hx : x ∈ Set.Ico (tIn 79) (tIn (79 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 79 ∧ kIn 79 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 13) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_80 (x : ℝ) (hx : x ∈ Set.Ico (tIn 80) (tIn (80 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 80 ∧ kIn 80 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 13) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_81 (x : ℝ) (hx : x ∈ Set.Ico (tIn 81) (tIn (81 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 81 ∧ kIn 81 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 13) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_82 (x : ℝ) (hx : x ∈ Set.Ico (tIn 82) (tIn (82 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 82 ∧ kIn 82 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 14) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_83 (x : ℝ) (hx : x ∈ Set.Ico (tIn 83) (tIn (83 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 83 ∧ kIn 83 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 14) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_84 (x : ℝ) (hx : x ∈ Set.Ico (tIn 84) (tIn (84 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 84 ∧ kIn 84 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 14) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_85 (x : ℝ) (hx : x ∈ Set.Ico (tIn 85) (tIn (85 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 85 ∧ kIn 85 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 14) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_86 (x : ℝ) (hx : x ∈ Set.Ico (tIn 86) (tIn (86 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 86 ∧ kIn 86 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 14) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_87 (x : ℝ) (hx : x ∈ Set.Ico (tIn 87) (tIn (87 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 87 ∧ kIn 87 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 14) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_88 (x : ℝ) (hx : x ∈ Set.Ico (tIn 88) (tIn (88 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 88 ∧ kIn 88 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 14) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_89 (x : ℝ) (hx : x ∈ Set.Ico (tIn 89) (tIn (89 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 89 ∧ kIn 89 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 15) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_90 (x : ℝ) (hx : x ∈ Set.Ico (tIn 90) (tIn (90 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 90 ∧ kIn 90 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 15) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_91 (x : ℝ) (hx : x ∈ Set.Ico (tIn 91) (tIn (91 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 91 ∧ kIn 91 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 15) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_92 (x : ℝ) (hx : x ∈ Set.Ico (tIn 92) (tIn (92 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 92 ∧ kIn 92 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 15) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_93 (x : ℝ) (hx : x ∈ Set.Ico (tIn 93) (tIn (93 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 93 ∧ kIn 93 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 15) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_94 (x : ℝ) (hx : x ∈ Set.Ico (tIn 94) (tIn (94 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 94 ∧ kIn 94 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 15) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_95 (x : ℝ) (hx : x ∈ Set.Ico (tIn 95) (tIn (95 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 95 ∧ kIn 95 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 15) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_96 (x : ℝ) (hx : x ∈ Set.Ico (tIn 96) (tIn (96 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 96 ∧ kIn 96 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 16) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_97 (x : ℝ) (hx : x ∈ Set.Ico (tIn 97) (tIn (97 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 97 ∧ kIn 97 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 16) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_98 (x : ℝ) (hx : x ∈ Set.Ico (tIn 98) (tIn (98 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 98 ∧ kIn 98 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 16) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_99 (x : ℝ) (hx : x ∈ Set.Ico (tIn 99) (tIn (99 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 99 ∧ kIn 99 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 16) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_100 (x : ℝ) (hx : x ∈ Set.Ico (tIn 100) (tIn (100 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 100 ∧ kIn 100 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 16) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_101 (x : ℝ) (hx : x ∈ Set.Ico (tIn 101) (tIn (101 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 101 ∧ kIn 101 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 16) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_102 (x : ℝ) (hx : x ∈ Set.Ico (tIn 102) (tIn (102 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 102 ∧ kIn 102 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 16) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_103 (x : ℝ) (hx : x ∈ Set.Ico (tIn 103) (tIn (103 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 103 ∧ kIn 103 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 16) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_104 (x : ℝ) (hx : x ∈ Set.Ico (tIn 104) (tIn (104 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 104 ∧ kIn 104 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 16) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_105 (x : ℝ) (hx : x ∈ Set.Ico (tIn 105) (tIn (105 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 105 ∧ kIn 105 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 17) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_106 (x : ℝ) (hx : x ∈ Set.Ico (tIn 106) (tIn (106 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 106 ∧ kIn 106 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 17) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_107 (x : ℝ) (hx : x ∈ Set.Ico (tIn 107) (tIn (107 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 107 ∧ kIn 107 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 17) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_108 (x : ℝ) (hx : x ∈ Set.Ico (tIn 108) (tIn (108 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 108 ∧ kIn 108 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 17) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_109 (x : ℝ) (hx : x ∈ Set.Ico (tIn 109) (tIn (109 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 109 ∧ kIn 109 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 17) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_110 (x : ℝ) (hx : x ∈ Set.Ico (tIn 110) (tIn (110 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 110 ∧ kIn 110 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 17) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_111 (x : ℝ) (hx : x ∈ Set.Ico (tIn 111) (tIn (111 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 111 ∧ kIn 111 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 17) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_112 (x : ℝ) (hx : x ∈ Set.Ico (tIn 112) (tIn (112 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 112 ∧ kIn 112 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 18) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_113 (x : ℝ) (hx : x ∈ Set.Ico (tIn 113) (tIn (113 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 113 ∧ kIn 113 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 18) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_114 (x : ℝ) (hx : x ∈ Set.Ico (tIn 114) (tIn (114 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 114 ∧ kIn 114 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 18) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_115 (x : ℝ) (hx : x ∈ Set.Ico (tIn 115) (tIn (115 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 115 ∧ kIn 115 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 18) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_116 (x : ℝ) (hx : x ∈ Set.Ico (tIn 116) (tIn (116 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 116 ∧ kIn 116 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 18) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_117 (x : ℝ) (hx : x ∈ Set.Ico (tIn 117) (tIn (117 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 117 ∧ kIn 117 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 18) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_118 (x : ℝ) (hx : x ∈ Set.Ico (tIn 118) (tIn (118 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 118 ∧ kIn 118 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 18) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_119 (x : ℝ) (hx : x ∈ Set.Ico (tIn 119) (tIn (119 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 119 ∧ kIn 119 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 19) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_120 (x : ℝ) (hx : x ∈ Set.Ico (tIn 120) (tIn (120 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 120 ∧ kIn 120 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 19) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_121 (x : ℝ) (hx : x ∈ Set.Ico (tIn 121) (tIn (121 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 121 ∧ kIn 121 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 19) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_122 (x : ℝ) (hx : x ∈ Set.Ico (tIn 122) (tIn (122 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 122 ∧ kIn 122 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 19) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_123 (x : ℝ) (hx : x ∈ Set.Ico (tIn 123) (tIn (123 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 123 ∧ kIn 123 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 19) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+private lemma inner_table_k_124 (x : ℝ) (hx : x ∈ Set.Ico (tIn 124) (tIn (124 + 1))) :
+    -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn 124 ∧ kIn 124 ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  simp only [tIn, tInL, kIn, kInL, List.getD_cons_succ, List.getD_cons_zero, Set.mem_Ico] at hx ⊢
+  push_cast at hx
+  rw [nat_floor_of_bounds (q := 19) (by norm_num; linarith [hx.1]) (by norm_num; linarith [hx.2])]
+  norm_num
+
+theorem inner_table_k :
+    ∀ i < 125,
+      ∀ x ∈ Set.Ico (tIn i) (tIn (i + 1)),
+        -2 * (⌊x⌋₊ : ℤ) - 6 ≤ kIn i ∧ kIn i ≤ 8 * (⌊x⌋₊ : ℤ) + 20 := by
+  intro i hi x hx
+  interval_cases i
+  · exact inner_table_k_0 x hx
+  · exact inner_table_k_1 x hx
+  · exact inner_table_k_2 x hx
+  · exact inner_table_k_3 x hx
+  · exact inner_table_k_4 x hx
+  · exact inner_table_k_5 x hx
+  · exact inner_table_k_6 x hx
+  · exact inner_table_k_7 x hx
+  · exact inner_table_k_8 x hx
+  · exact inner_table_k_9 x hx
+  · exact inner_table_k_10 x hx
+  · exact inner_table_k_11 x hx
+  · exact inner_table_k_12 x hx
+  · exact inner_table_k_13 x hx
+  · exact inner_table_k_14 x hx
+  · exact inner_table_k_15 x hx
+  · exact inner_table_k_16 x hx
+  · exact inner_table_k_17 x hx
+  · exact inner_table_k_18 x hx
+  · exact inner_table_k_19 x hx
+  · exact inner_table_k_20 x hx
+  · exact inner_table_k_21 x hx
+  · exact inner_table_k_22 x hx
+  · exact inner_table_k_23 x hx
+  · exact inner_table_k_24 x hx
+  · exact inner_table_k_25 x hx
+  · exact inner_table_k_26 x hx
+  · exact inner_table_k_27 x hx
+  · exact inner_table_k_28 x hx
+  · exact inner_table_k_29 x hx
+  · exact inner_table_k_30 x hx
+  · exact inner_table_k_31 x hx
+  · exact inner_table_k_32 x hx
+  · exact inner_table_k_33 x hx
+  · exact inner_table_k_34 x hx
+  · exact inner_table_k_35 x hx
+  · exact inner_table_k_36 x hx
+  · exact inner_table_k_37 x hx
+  · exact inner_table_k_38 x hx
+  · exact inner_table_k_39 x hx
+  · exact inner_table_k_40 x hx
+  · exact inner_table_k_41 x hx
+  · exact inner_table_k_42 x hx
+  · exact inner_table_k_43 x hx
+  · exact inner_table_k_44 x hx
+  · exact inner_table_k_45 x hx
+  · exact inner_table_k_46 x hx
+  · exact inner_table_k_47 x hx
+  · exact inner_table_k_48 x hx
+  · exact inner_table_k_49 x hx
+  · exact inner_table_k_50 x hx
+  · exact inner_table_k_51 x hx
+  · exact inner_table_k_52 x hx
+  · exact inner_table_k_53 x hx
+  · exact inner_table_k_54 x hx
+  · exact inner_table_k_55 x hx
+  · exact inner_table_k_56 x hx
+  · exact inner_table_k_57 x hx
+  · exact inner_table_k_58 x hx
+  · exact inner_table_k_59 x hx
+  · exact inner_table_k_60 x hx
+  · exact inner_table_k_61 x hx
+  · exact inner_table_k_62 x hx
+  · exact inner_table_k_63 x hx
+  · exact inner_table_k_64 x hx
+  · exact inner_table_k_65 x hx
+  · exact inner_table_k_66 x hx
+  · exact inner_table_k_67 x hx
+  · exact inner_table_k_68 x hx
+  · exact inner_table_k_69 x hx
+  · exact inner_table_k_70 x hx
+  · exact inner_table_k_71 x hx
+  · exact inner_table_k_72 x hx
+  · exact inner_table_k_73 x hx
+  · exact inner_table_k_74 x hx
+  · exact inner_table_k_75 x hx
+  · exact inner_table_k_76 x hx
+  · exact inner_table_k_77 x hx
+  · exact inner_table_k_78 x hx
+  · exact inner_table_k_79 x hx
+  · exact inner_table_k_80 x hx
+  · exact inner_table_k_81 x hx
+  · exact inner_table_k_82 x hx
+  · exact inner_table_k_83 x hx
+  · exact inner_table_k_84 x hx
+  · exact inner_table_k_85 x hx
+  · exact inner_table_k_86 x hx
+  · exact inner_table_k_87 x hx
+  · exact inner_table_k_88 x hx
+  · exact inner_table_k_89 x hx
+  · exact inner_table_k_90 x hx
+  · exact inner_table_k_91 x hx
+  · exact inner_table_k_92 x hx
+  · exact inner_table_k_93 x hx
+  · exact inner_table_k_94 x hx
+  · exact inner_table_k_95 x hx
+  · exact inner_table_k_96 x hx
+  · exact inner_table_k_97 x hx
+  · exact inner_table_k_98 x hx
+  · exact inner_table_k_99 x hx
+  · exact inner_table_k_100 x hx
+  · exact inner_table_k_101 x hx
+  · exact inner_table_k_102 x hx
+  · exact inner_table_k_103 x hx
+  · exact inner_table_k_104 x hx
+  · exact inner_table_k_105 x hx
+  · exact inner_table_k_106 x hx
+  · exact inner_table_k_107 x hx
+  · exact inner_table_k_108 x hx
+  · exact inner_table_k_109 x hx
+  · exact inner_table_k_110 x hx
+  · exact inner_table_k_111 x hx
+  · exact inner_table_k_112 x hx
+  · exact inner_table_k_113 x hx
+  · exact inner_table_k_114 x hx
+  · exact inner_table_k_115 x hx
+  · exact inner_table_k_116 x hx
+  · exact inner_table_k_117 x hx
+  · exact inner_table_k_118 x hx
+  · exact inner_table_k_119 x hx
+  · exact inner_table_k_120 x hx
+  · exact inner_table_k_121 x hx
+  · exact inner_table_k_122 x hx
+  · exact inner_table_k_123 x hx
+  · exact inner_table_k_124 x hx
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Growth/InnerTable.lean b/LeanPool/Zeta5Irrational/Growth/InnerTable.lean
new file mode 100644
index 0000000000..973705cd6e
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Growth/InnerTable.lean
@@ -0,0 +1,8269 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+import LeanPool.Zeta5Irrational.Growth.TableGen
+public import LeanPool.Zeta5Irrational.Growth.TablePrime
+import Mathlib.Tactic.ReduceModChar
+import Mathlib.Tactic.Polynomial.Basic
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+import Mathlib.Tactic.Ring.Basic
+
+/-!Generated by `gen_lean_inner.py`: the pieces of `Ê_in` on `[3, 20]`. -/
+
+@[expose] public section
+
+namespace Zeta5Irrational
+
+theorem inPiece_0 (x : ℝ) (h1 : (3 : ℝ) ≤ x) (h2 : x < ((70) / 23 : ℝ)) :
+    Ein x (2) = ((18) / 5 : ℝ) * x + ((-6) : ℝ) := by
+  rw [Ein_expand (2) (q := 3) (q' := 0) (μ := 5) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (3 : ℝ) := by linarith
+  have hf1 : x - (3 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-3) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 = ((1 : ℝ) * x + ((-3) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_1 (x : ℝ) (h1 : ((70) / 23 : ℝ) ≤ x) (h2 : x < ((120) / 37 : ℝ)) :
+    Ein x (3) = ((49) / 20 : ℝ) * x + ((-5) / 2 : ℝ) := by
+  rw [Ein_expand (3) (q := 3) (q' := 0) (μ := 5) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (3 : ℝ) := by linarith
+  have hf1 : x - (3 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-3) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 = ((1 : ℝ) * x + ((-3) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_2 (x : ℝ) (h1 : ((120) / 37 : ℝ) ≤ x) (h2 : x < ((10) / 3 : ℝ)) :
+    Ein x (3) = ((283) / 40 : ℝ) * x + ((-35) / 2 : ℝ) := by
+  rw [Ein_expand (3) (q := 3) (q' := 0) (μ := 6) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (3 : ℝ) := by linarith
+  have hf1 : x - (3 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-3) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_3 (x : ℝ) (h1 : ((10) / 3 : ℝ) ≤ x) (h2 : x < ((80) / 23 : ℝ)) :
+    Ein x (4) = ((277) / 40 : ℝ) * x + ((-17) : ℝ) := by
+  rw [Ein_expand (4) (q := 3) (q' := 0) (μ := 6) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (3 : ℝ) := by linarith
+  have hf1 : x - (3 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-3) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_4 (x : ℝ) (h1 : ((80) / 23 : ℝ) ≤ x) (h2 : x < ((7) / 2 : ℝ)) :
+    Ein x (5) = ((231) / 40 : ℝ) * x + ((-13) : ℝ) := by
+  rw [Ein_expand (5) (q := 3) (q' := 0) (μ := 6) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (3 : ℝ) := by linarith
+  have hf1 : x - (3 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-3) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_5 (x : ℝ) (h1 : ((7) / 2 : ℝ) ≤ x) (h2 : x < ((160) / 43 : ℝ)) :
+    Ein x (5) = ((271) / 40 : ℝ) * x + ((-33) / 2 : ℝ) := by
+  rw [Ein_expand (5) (q := 3) (q' := 0) (μ := 6) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (3 : ℝ) := by linarith
+  have hf1 : x - (3 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-7) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_6 (x : ℝ) (h1 : ((160) / 43 : ℝ) ≤ x) (h2 : x < ((140) / 37 : ℝ)) :
+    Ein x (5) = ((71) / 20 : ℝ) * x + ((-9) / 2 : ℝ) := by
+  rw [Ein_expand (5) (q := 3) (q' := 0) (μ := 6) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (3 : ℝ) := by linarith
+  have hf1 : x - (3 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-7) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 =
+      (((43) / 40 : ℝ) * x + ((-4) : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_7 (x : ℝ) (h1 : ((140) / 37 : ℝ) ≤ x) (h2 : x < ((90) / 23 : ℝ)) :
+    Ein x (5) = ((27) / 5 : ℝ) * x + ((-23) / 2 : ℝ) := by
+  rw [Ein_expand (5) (q := 3) (q' := 0) (μ := 7) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (3 : ℝ) := by linarith
+  have hf1 : x - (3 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-7) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 =
+      (((43) / 40 : ℝ) * x + ((-4) : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_8 (x : ℝ) (h1 : ((90) / 23 : ℝ) ≤ x) (h2 : x < (4 : ℝ)) :
+    Ein x (6) = ((17) / 4 : ℝ) * x + ((-7) : ℝ) := by
+  rw [Ein_expand (6) (q := 3) (q' := 0) (μ := 7) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (3 : ℝ) := by linarith
+  have hf1 : x - (3 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-7) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 =
+      (((43) / 40 : ℝ) * x + ((-4) : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (3 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_9 (x : ℝ) (h1 : (4 : ℝ) ≤ x) (h2 : x < ((160) / 37 : ℝ)) :
+    Ein x (6) = ((17) / 5 : ℝ) * x + ((-11) : ℝ) := by
+  rw [Ein_expand (6) (q := 4) (q' := 0) (μ := 7) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (4 : ℝ) := by linarith
+  have hf1 : x - (4 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-4) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 = ((1 : ℝ) * x + ((-4) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_10 (x : ℝ) (h1 : ((160) / 37 : ℝ) ≤ x) (h2 : x < ((100) / 23 : ℝ)) :
+    Ein x (6) = ((321) / 40 : ℝ) * x + ((-31) : ℝ) := by
+  rw [Ein_expand (6) (q := 4) (q' := 0) (μ := 8) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (4 : ℝ) := by linarith
+  have hf1 : x - (4 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-4) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_11 (x : ℝ) (h1 : ((100) / 23 : ℝ) ≤ x) (h2 : x < ((9) / 2 : ℝ)) :
+    Ein x (7) = ((55) / 8 : ℝ) * x + ((-26) : ℝ) := by
+  rw [Ein_expand (7) (q := 4) (q' := 0) (μ := 8) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (4 : ℝ) := by linarith
+  have hf1 : x - (4 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-4) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_12 (x : ℝ) (h1 : ((9) / 2 : ℝ) ≤ x) (h2 : x < ((200) / 43 : ℝ)) :
+    Ein x (7) = ((63) / 8 : ℝ) * x + ((-61) / 2 : ℝ) := by
+  rw [Ein_expand (7) (q := 4) (q' := 0) (μ := 8) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (4 : ℝ) := by linarith
+  have hf1 : x - (4 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-9) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_13 (x : ℝ) (h1 : ((200) / 43 : ℝ) ≤ x) (h2 : x < ((110) / 23 : ℝ)) :
+    Ein x (7) = ((93) / 20 : ℝ) * x + ((-31) / 2 : ℝ) := by
+  rw [Ein_expand (7) (q := 4) (q' := 0) (μ := 8) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (4 : ℝ) := by linarith
+  have hf1 : x - (4 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-9) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 =
+      (((43) / 40 : ℝ) * x + ((-5) : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_14 (x : ℝ) (h1 : ((110) / 23 : ℝ) ≤ x) (h2 : x < ((180) / 37 : ℝ)) :
+    Ein x (8) = ((7) / 2 : ℝ) * x + ((-10) : ℝ) := by
+  rw [Ein_expand (8) (q := 4) (q' := 0) (μ := 8) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (4 : ℝ) := by linarith
+  have hf1 : x - (4 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-9) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 =
+      (((43) / 40 : ℝ) * x + ((-5) : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_15 (x : ℝ) (h1 : ((180) / 37 : ℝ) ≤ x) (h2 : x < (5 : ℝ)) :
+    Ein x (8) = ((107) / 20 : ℝ) * x + ((-19) : ℝ) := by
+  rw [Ein_expand (8) (q := 4) (q' := 0) (μ := 9) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (4 : ℝ) := by linarith
+  have hf1 : x - (4 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-9) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 =
+      (((43) / 40 : ℝ) * x + ((-5) : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (4 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_16 (x : ℝ) (h1 : (5 : ℝ) ≤ x) (h2 : x < ((120) / 23 : ℝ)) :
+    Ein x (8) = ((9) / 2 : ℝ) * x + ((-24) : ℝ) := by
+  rw [Ein_expand (8) (q := 5) (q' := 0) (μ := 9) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (5 : ℝ) := by linarith
+  have hf1 : x - (5 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-5) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 = ((1 : ℝ) * x + ((-5) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_17 (x : ℝ) (h1 : ((120) / 23 : ℝ) ≤ x) (h2 : x < ((200) / 37 : ℝ)) :
+    Ein x (9) = ((67) / 20 : ℝ) * x + ((-18) : ℝ) := by
+  rw [Ein_expand (9) (q := 5) (q' := 0) (μ := 9) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (5 : ℝ) := by linarith
+  have hf1 : x - (5 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-5) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 = ((1 : ℝ) * x + ((-5) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_18 (x : ℝ) (h1 : ((200) / 37 : ℝ) ≤ x) (h2 : x < ((11) / 2 : ℝ)) :
+    Ein x (9) = ((319) / 40 : ℝ) * x + ((-43) : ℝ) := by
+  rw [Ein_expand (9) (q := 5) (q' := 0) (μ := 10) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (5 : ℝ) := by linarith
+  have hf1 : x - (5 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-5) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_19 (x : ℝ) (h1 : ((11) / 2 : ℝ) ≤ x) (h2 : x < ((240) / 43 : ℝ)) :
+    Ein x (9) = ((359) / 40 : ℝ) * x + ((-97) / 2 : ℝ) := by
+  rw [Ein_expand (9) (q := 5) (q' := 0) (μ := 10) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (5 : ℝ) := by linarith
+  have hf1 : x - (5 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-11) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_20 (x : ℝ) (h1 : ((240) / 43 : ℝ) ≤ x) (h2 : x < ((130) / 23 : ℝ)) :
+    Ein x (9) = ((23) / 4 : ℝ) * x + ((-61) / 2 : ℝ) := by
+  rw [Ein_expand (9) (q := 5) (q' := 0) (μ := 10) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (5 : ℝ) := by linarith
+  have hf1 : x - (5 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-11) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 =
+      (((43) / 40 : ℝ) * x + ((-6) : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_21 (x : ℝ) (h1 : ((130) / 23 : ℝ) ≤ x) (h2 : x < ((220) / 37 : ℝ)) :
+    Ein x (10) = ((23) / 5 : ℝ) * x + ((-24) : ℝ) := by
+  rw [Ein_expand (10) (q := 5) (q' := 0) (μ := 10) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (5 : ℝ) := by linarith
+  have hf1 : x - (5 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-11) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 =
+      (((43) / 40 : ℝ) * x + ((-6) : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_22 (x : ℝ) (h1 : ((220) / 37 : ℝ) ≤ x) (h2 : x < (6 : ℝ)) :
+    Ein x (10) = ((129) / 20 : ℝ) * x + ((-35) : ℝ) := by
+  rw [Ein_expand (10) (q := 5) (q' := 0) (μ := 11) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (5 : ℝ) := by linarith
+  have hf1 : x - (5 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-11) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 =
+      (((43) / 40 : ℝ) * x + ((-6) : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (5 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_23 (x : ℝ) (h1 : (6 : ℝ) ≤ x) (h2 : x < ((140) / 23 : ℝ)) :
+    Ein x (10) = ((28) / 5 : ℝ) * x + ((-41) : ℝ) := by
+  rw [Ein_expand (10) (q := 6) (q' := 0) (μ := 11) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (6 : ℝ) := by linarith
+  have hf1 : x - (6 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-6) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 = ((1 : ℝ) * x + ((-6) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_24 (x : ℝ) (h1 : ((140) / 23 : ℝ) ≤ x) (h2 : x < ((240) / 37 : ℝ)) :
+    Ein x (11) = ((89) / 20 : ℝ) * x + ((-34) : ℝ) := by
+  rw [Ein_expand (11) (q := 6) (q' := 0) (μ := 11) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (6 : ℝ) := by linarith
+  have hf1 : x - (6 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-6) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 = ((1 : ℝ) * x + ((-6) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_25 (x : ℝ) (h1 : ((240) / 37 : ℝ) ≤ x) (h2 : x < ((13) / 2 : ℝ)) :
+    Ein x (11) = ((363) / 40 : ℝ) * x + ((-64) : ℝ) := by
+  rw [Ein_expand (11) (q := 6) (q' := 0) (μ := 12) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (6 : ℝ) := by linarith
+  have hf1 : x - (6 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-6) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_26 (x : ℝ) (h1 : ((13) / 2 : ℝ) ≤ x) (h2 : x < ((280) / 43 : ℝ)) :
+    Ein x (11) = ((403) / 40 : ℝ) * x + ((-141) / 2 : ℝ) := by
+  rw [Ein_expand (11) (q := 6) (q' := 0) (μ := 12) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (6 : ℝ) := by linarith
+  have hf1 : x - (6 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-13) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_27 (x : ℝ) (h1 : ((280) / 43 : ℝ) ≤ x) (h2 : x < ((150) / 23 : ℝ)) :
+    Ein x (11) = ((137) / 20 : ℝ) * x + ((-99) / 2 : ℝ) := by
+  rw [Ein_expand (11) (q := 6) (q' := 0) (μ := 12) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (6 : ℝ) := by linarith
+  have hf1 : x - (6 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-13) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 =
+      (((43) / 40 : ℝ) * x + ((-7) : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_28 (x : ℝ) (h1 : ((150) / 23 : ℝ) ≤ x) (h2 : x < ((20) / 3 : ℝ)) :
+    Ein x (12) = ((57) / 10 : ℝ) * x + ((-42) : ℝ) := by
+  rw [Ein_expand (12) (q := 6) (q' := 0) (μ := 12) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (6 : ℝ) := by linarith
+  have hf1 : x - (6 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-13) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 =
+      (((43) / 40 : ℝ) * x + ((-7) : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_29 (x : ℝ) (h1 : ((20) / 3 : ℝ) ≤ x) (h2 : x < ((160) / 23 : ℝ)) :
+    Ein x (13) = ((69) / 10 : ℝ) * x + ((-50) : ℝ) := by
+  rw [Ein_expand (13) (q := 6) (q' := 0) (μ := 12) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (6 : ℝ) := by linarith
+  have hf1 : x - (6 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-13) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 =
+      ((1 : ℝ) * x + ((-13) / 2 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_30 (x : ℝ) (h1 : ((160) / 23 : ℝ) ≤ x) (h2 : x < (7 : ℝ)) :
+    Ein x (14) = ((23) / 4 : ℝ) * x + ((-42) : ℝ) := by
+  rw [Ein_expand (14) (q := 6) (q' := 0) (μ := 12) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (6 : ℝ) := by linarith
+  have hf1 : x - (6 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-13) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 =
+      ((1 : ℝ) * x + ((-13) / 2 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (6 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_31 (x : ℝ) (h1 : (7 : ℝ) ≤ x) (h2 : x < ((260) / 37 : ℝ)) :
+    Ein x (14) = ((49) / 10 : ℝ) * x + ((-49) : ℝ) := by
+  rw [Ein_expand (14) (q := 7) (q' := 0) (μ := 12) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (7 : ℝ) := by linarith
+  have hf1 : x - (7 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-7) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 = ((1 : ℝ) * x + ((-7) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_32 (x : ℝ) (h1 : ((260) / 37 : ℝ) ≤ x) (h2 : x < ((170) / 23 : ℝ)) :
+    Ein x (14) = ((27) / 4 : ℝ) * x + ((-62) : ℝ) := by
+  rw [Ein_expand (14) (q := 7) (q' := 0) (μ := 13) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (7 : ℝ) := by linarith
+  have hf1 : x - (7 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-7) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 = ((1 : ℝ) * x + ((-7) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_33 (x : ℝ) (h1 : ((170) / 23 : ℝ) ≤ x) (h2 : x < ((320) / 43 : ℝ)) :
+    Ein x (15) = ((28) / 5 : ℝ) * x + ((-107) / 2 : ℝ) := by
+  rw [Ein_expand (15) (q := 7) (q' := 0) (μ := 13) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (7 : ℝ) := by linarith
+  have hf1 : x - (7 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-7) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 = ((1 : ℝ) * x + ((-7) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_34 (x : ℝ) (h1 : ((320) / 43 : ℝ) ≤ x) (h2 : x < ((15) / 2 : ℝ)) :
+    Ein x (15) = ((19) / 8 : ℝ) * x + ((-59) / 2 : ℝ) := by
+  rw [Ein_expand (15) (q := 7) (q' := 0) (μ := 13) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (7 : ℝ) := by linarith
+  have hf1 : x - (7 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-7) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 = ((1 : ℝ) * x + ((-7) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 =
+      (((43) / 40 : ℝ) * x + ((-8) : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_35 (x : ℝ) (h1 : ((15) / 2 : ℝ) ≤ x) (h2 : x < ((280) / 37 : ℝ)) :
+    Ein x (15) = ((27) / 8 : ℝ) * x + ((-37) : ℝ) := by
+  rw [Ein_expand (15) (q := 7) (q' := 0) (μ := 13) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (7 : ℝ) := by linarith
+  have hf1 : x - (7 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-15) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 =
+      ((1 : ℝ) * x + ((-15) / 2 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 =
+      ((1 : ℝ) * x + ((-15) / 2 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_36 (x : ℝ) (h1 : ((280) / 37 : ℝ) ≤ x) (h2 : x < ((180) / 23 : ℝ)) :
+    Ein x (15) = (8 : ℝ) * x + ((-72) : ℝ) := by
+  rw [Ein_expand (15) (q := 7) (q' := 0) (μ := 14) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (7 : ℝ) := by linarith
+  have hf1 : x - (7 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-15) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 =
+      ((1 : ℝ) * x + ((-15) / 2 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_37 (x : ℝ) (h1 : ((180) / 23 : ℝ) ≤ x) (h2 : x < (8 : ℝ)) :
+    Ein x (16) = ((137) / 20 : ℝ) * x + ((-63) : ℝ) := by
+  rw [Ein_expand (16) (q := 7) (q' := 0) (μ := 14) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (7 : ℝ) := by linarith
+  have hf1 : x - (7 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-15) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 =
+      ((1 : ℝ) * x + ((-15) / 2 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (7 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_38 (x : ℝ) (h1 : (8 : ℝ) ≤ x) (h2 : x < ((300) / 37 : ℝ)) :
+    Ein x (16) = (6 : ℝ) * x + ((-71) : ℝ) := by
+  rw [Ein_expand (16) (q := 8) (q' := 0) (μ := 14) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (8 : ℝ) := by linarith
+  have hf1 : x - (8 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-8) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 = ((1 : ℝ) * x + ((-8) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_39 (x : ℝ) (h1 : ((300) / 37 : ℝ) ≤ x) (h2 : x < ((190) / 23 : ℝ)) :
+    Ein x (16) = ((157) / 20 : ℝ) * x + ((-86) : ℝ) := by
+  rw [Ein_expand (16) (q := 8) (q' := 0) (μ := 15) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (8 : ℝ) := by linarith
+  have hf1 : x - (8 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-8) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 = ((1 : ℝ) * x + ((-8) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_40 (x : ℝ) (h1 : ((190) / 23 : ℝ) ≤ x) (h2 : x < ((360) / 43 : ℝ)) :
+    Ein x (17) = ((67) / 10 : ℝ) * x + ((-153) / 2 : ℝ) := by
+  rw [Ein_expand (17) (q := 8) (q' := 0) (μ := 15) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (8 : ℝ) := by linarith
+  have hf1 : x - (8 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-8) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 = ((1 : ℝ) * x + ((-8) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_41 (x : ℝ) (h1 : ((360) / 43 : ℝ) ≤ x) (h2 : x < ((17) / 2 : ℝ)) :
+    Ein x (17) = ((139) / 40 : ℝ) * x + ((-99) / 2 : ℝ) := by
+  rw [Ein_expand (17) (q := 8) (q' := 0) (μ := 15) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (8 : ℝ) := by linarith
+  have hf1 : x - (8 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-8) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 = ((1 : ℝ) * x + ((-8) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 =
+      (((43) / 40 : ℝ) * x + ((-9) : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_42 (x : ℝ) (h1 : ((17) / 2 : ℝ) ≤ x) (h2 : x < ((320) / 37 : ℝ)) :
+    Ein x (17) = ((179) / 40 : ℝ) * x + ((-58) : ℝ) := by
+  rw [Ein_expand (17) (q := 8) (q' := 0) (μ := 15) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (8 : ℝ) := by linarith
+  have hf1 : x - (8 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-17) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 =
+      ((1 : ℝ) * x + ((-17) / 2 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 =
+      ((1 : ℝ) * x + ((-17) / 2 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_43 (x : ℝ) (h1 : ((320) / 37 : ℝ) ≤ x) (h2 : x < ((200) / 23 : ℝ)) :
+    Ein x (17) = ((91) / 10 : ℝ) * x + ((-98) : ℝ) := by
+  rw [Ein_expand (17) (q := 8) (q' := 0) (μ := 16) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (8 : ℝ) := by linarith
+  have hf1 : x - (8 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-17) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 =
+      ((1 : ℝ) * x + ((-17) / 2 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_44 (x : ℝ) (h1 : ((200) / 23 : ℝ) ≤ x) (h2 : x < (9 : ℝ)) :
+    Ein x (18) = ((159) / 20 : ℝ) * x + ((-88) : ℝ) := by
+  rw [Ein_expand (18) (q := 8) (q' := 0) (μ := 16) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (8 : ℝ) := by linarith
+  have hf1 : x - (8 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-17) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 =
+      ((1 : ℝ) * x + ((-17) / 2 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (8 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_45 (x : ℝ) (h1 : (9 : ℝ) ≤ x) (h2 : x < ((210) / 23 : ℝ)) :
+    Ein x (18) = ((71) / 10 : ℝ) * x + ((-97) : ℝ) := by
+  rw [Ein_expand (18) (q := 9) (q' := 0) (μ := 16) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (9 : ℝ) := by linarith
+  have hf1 : x - (9 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-9) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 = ((1 : ℝ) * x + ((-9) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_46 (x : ℝ) (h1 : ((210) / 23 : ℝ) ≤ x) (h2 : x < ((340) / 37 : ℝ)) :
+    Ein x (19) = ((119) / 20 : ℝ) * x + ((-173) / 2 : ℝ) := by
+  rw [Ein_expand (19) (q := 9) (q' := 0) (μ := 16) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (9 : ℝ) := by linarith
+  have hf1 : x - (9 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-9) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 = ((1 : ℝ) * x + ((-9) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_47 (x : ℝ) (h1 : ((340) / 37 : ℝ) ≤ x) (h2 : x < ((400) / 43 : ℝ)) :
+    Ein x (19) = ((39) / 5 : ℝ) * x + ((-207) / 2 : ℝ) := by
+  rw [Ein_expand (19) (q := 9) (q' := 0) (μ := 17) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (9 : ℝ) := by linarith
+  have hf1 : x - (9 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-9) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 = ((1 : ℝ) * x + ((-9) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_48 (x : ℝ) (h1 : ((400) / 43 : ℝ) ≤ x) (h2 : x < ((19) / 2 : ℝ)) :
+    Ein x (19) = ((183) / 40 : ℝ) * x + ((-147) / 2 : ℝ) := by
+  rw [Ein_expand (19) (q := 9) (q' := 0) (μ := 17) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (9 : ℝ) := by linarith
+  have hf1 : x - (9 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-9) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 = ((1 : ℝ) * x + ((-9) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 =
+      (((43) / 40 : ℝ) * x + ((-10) : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_49 (x : ℝ) (h1 : ((19) / 2 : ℝ) ≤ x) (h2 : x < ((220) / 23 : ℝ)) :
+    Ein x (19) = ((223) / 40 : ℝ) * x + ((-83) : ℝ) := by
+  rw [Ein_expand (19) (q := 9) (q' := 0) (μ := 17) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (9 : ℝ) := by linarith
+  have hf1 : x - (9 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-19) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 =
+      ((1 : ℝ) * x + ((-19) / 2 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 =
+      ((1 : ℝ) * x + ((-19) / 2 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_50 (x : ℝ) (h1 : ((220) / 23 : ℝ) ≤ x) (h2 : x < ((360) / 37 : ℝ)) :
+    Ein x (20) = ((177) / 40 : ℝ) * x + ((-72) : ℝ) := by
+  rw [Ein_expand (20) (q := 9) (q' := 0) (μ := 17) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (9 : ℝ) := by linarith
+  have hf1 : x - (9 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-19) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 =
+      ((1 : ℝ) * x + ((-19) / 2 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 =
+      ((1 : ℝ) * x + ((-19) / 2 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_51 (x : ℝ) (h1 : ((360) / 37 : ℝ) ≤ x) (h2 : x < (10 : ℝ)) :
+    Ein x (20) = ((181) / 20 : ℝ) * x + ((-117) : ℝ) := by
+  rw [Ein_expand (20) (q := 9) (q' := 0) (μ := 18) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (9 : ℝ) := by linarith
+  have hf1 : x - (9 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-19) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 =
+      ((1 : ℝ) * x + ((-19) / 2 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (9 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_52 (x : ℝ) (h1 : (10 : ℝ) ≤ x) (h2 : x < ((440) / 43 : ℝ)) :
+    Ein x (22) = ((69) / 10 : ℝ) * x + ((-114) : ℝ) := by
+  rw [Ein_expand (22) (q := 10) (q' := 0) (μ := 18) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (10 : ℝ) := by linarith
+  have hf1 : x - (10 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-10) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 = ((1 : ℝ) * x + ((-10) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_53 (x : ℝ) (h1 : ((440) / 43 : ℝ) ≤ x) (h2 : x < ((380) / 37 : ℝ)) :
+    Ein x (22) = ((147) / 40 : ℝ) * x + ((-81) : ℝ) := by
+  rw [Ein_expand (22) (q := 10) (q' := 0) (μ := 18) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (10 : ℝ) := by linarith
+  have hf1 : x - (10 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-10) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 = ((1 : ℝ) * x + ((-10) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 =
+      (((43) / 40 : ℝ) * x + ((-11) : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_54 (x : ℝ) (h1 : ((380) / 37 : ℝ) ≤ x) (h2 : x < ((240) / 23 : ℝ)) :
+    Ein x (22) = ((221) / 40 : ℝ) * x + ((-100) : ℝ) := by
+  rw [Ein_expand (22) (q := 10) (q' := 0) (μ := 19) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (10 : ℝ) := by linarith
+  have hf1 : x - (10 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-10) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 = ((1 : ℝ) * x + ((-10) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 =
+      (((43) / 40 : ℝ) * x + ((-11) : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_55 (x : ℝ) (h1 : ((240) / 23 : ℝ) ≤ x) (h2 : x < ((21) / 2 : ℝ)) :
+    Ein x (23) = ((35) / 8 : ℝ) * x + ((-88) : ℝ) := by
+  rw [Ein_expand (23) (q := 10) (q' := 0) (μ := 19) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (10 : ℝ) := by linarith
+  have hf1 : x - (10 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-10) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 = ((1 : ℝ) * x + ((-10) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 =
+      (((43) / 40 : ℝ) * x + ((-11) : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_56 (x : ℝ) (h1 : ((21) / 2 : ℝ) ≤ x) (h2 : x < ((400) / 37 : ℝ)) :
+    Ein x (23) = ((43) / 8 : ℝ) * x + ((-197) / 2 : ℝ) := by
+  rw [Ein_expand (23) (q := 10) (q' := 0) (μ := 19) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (10 : ℝ) := by linarith
+  have hf1 : x - (10 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-21) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 =
+      ((1 : ℝ) * x + ((-21) / 2 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 =
+      ((1 : ℝ) * x + ((-21) / 2 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_57 (x : ℝ) (h1 : ((400) / 37 : ℝ) ≤ x) (h2 : x < ((250) / 23 : ℝ)) :
+    Ein x (23) = (10 : ℝ) * x + ((-297) / 2 : ℝ) := by
+  rw [Ein_expand (23) (q := 10) (q' := 0) (μ := 20) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (10 : ℝ) := by linarith
+  have hf1 : x - (10 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-21) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 =
+      ((1 : ℝ) * x + ((-21) / 2 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_58 (x : ℝ) (h1 : ((250) / 23 : ℝ) ≤ x) (h2 : x < (11 : ℝ)) :
+    Ein x (24) = ((177) / 20 : ℝ) * x + ((-136) : ℝ) := by
+  rw [Ein_expand (24) (q := 10) (q' := 0) (μ := 20) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (10 : ℝ) := by linarith
+  have hf1 : x - (10 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-21) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 =
+      ((1 : ℝ) * x + ((-21) / 2 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (10 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_59 (x : ℝ) (h1 : (11 : ℝ) ≤ x) (h2 : x < ((480) / 43 : ℝ)) :
+    Ein x (24) = (8 : ℝ) * x + ((-147) : ℝ) := by
+  rw [Ein_expand (24) (q := 11) (q' := 0) (μ := 20) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (11 : ℝ) := by linarith
+  have hf1 : x - (11 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-11) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 = ((1 : ℝ) * x + ((-11) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_60 (x : ℝ) (h1 : ((480) / 43 : ℝ) ≤ x) (h2 : x < ((260) / 23 : ℝ)) :
+    Ein x (24) = ((191) / 40 : ℝ) * x + ((-111) : ℝ) := by
+  rw [Ein_expand (24) (q := 11) (q' := 0) (μ := 20) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (11 : ℝ) := by linarith
+  have hf1 : x - (11 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-11) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 = ((1 : ℝ) * x + ((-11) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 =
+      (((43) / 40 : ℝ) * x + ((-12) : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_61 (x : ℝ) (h1 : ((260) / 23 : ℝ) ≤ x) (h2 : x < ((420) / 37 : ℝ)) :
+    Ein x (25) = ((29) / 8 : ℝ) * x + ((-98) : ℝ) := by
+  rw [Ein_expand (25) (q := 11) (q' := 0) (μ := 20) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (11 : ℝ) := by linarith
+  have hf1 : x - (11 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-11) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 = ((1 : ℝ) * x + ((-11) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 =
+      (((43) / 40 : ℝ) * x + ((-12) : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_62 (x : ℝ) (h1 : ((420) / 37 : ℝ) ≤ x) (h2 : x < ((23) / 2 : ℝ)) :
+    Ein x (25) = ((219) / 40 : ℝ) * x + ((-119) : ℝ) := by
+  rw [Ein_expand (25) (q := 11) (q' := 0) (μ := 21) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (11 : ℝ) := by linarith
+  have hf1 : x - (11 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-11) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 = ((1 : ℝ) * x + ((-11) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 =
+      (((43) / 40 : ℝ) * x + ((-12) : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_63 (x : ℝ) (h1 : ((23) / 2 : ℝ) ≤ x) (h2 : x < ((270) / 23 : ℝ)) :
+    Ein x (25) = ((259) / 40 : ℝ) * x + ((-261) / 2 : ℝ) := by
+  rw [Ein_expand (25) (q := 11) (q' := 0) (μ := 21) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (11 : ℝ) := by linarith
+  have hf1 : x - (11 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-23) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 =
+      ((1 : ℝ) * x + ((-23) / 2 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 =
+      ((1 : ℝ) * x + ((-23) / 2 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_64 (x : ℝ) (h1 : ((270) / 23 : ℝ) ≤ x) (h2 : x < ((440) / 37 : ℝ)) :
+    Ein x (26) = ((213) / 40 : ℝ) * x + ((-117) : ℝ) := by
+  rw [Ein_expand (26) (q := 11) (q' := 0) (μ := 21) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (11 : ℝ) := by linarith
+  have hf1 : x - (11 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-23) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 =
+      ((1 : ℝ) * x + ((-23) / 2 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 =
+      ((1 : ℝ) * x + ((-23) / 2 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_65 (x : ℝ) (h1 : ((440) / 37 : ℝ) ≤ x) (h2 : x < (12 : ℝ)) :
+    Ein x (26) = ((199) / 20 : ℝ) * x + ((-172) : ℝ) := by
+  rw [Ein_expand (26) (q := 11) (q' := 0) (μ := 22) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (11 : ℝ) := by linarith
+  have hf1 : x - (11 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-23) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 =
+      ((1 : ℝ) * x + ((-23) / 2 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (11 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_66 (x : ℝ) (h1 : (12 : ℝ) ≤ x) (h2 : x < ((520) / 43 : ℝ)) :
+    Ein x (26) = ((91) / 10 : ℝ) * x + ((-184) : ℝ) := by
+  rw [Ein_expand (26) (q := 12) (q' := 0) (μ := 22) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (12 : ℝ) := by linarith
+  have hf1 : x - (12 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-12) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 = ((1 : ℝ) * x + ((-12) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_67 (x : ℝ) (h1 : ((520) / 43 : ℝ) ≤ x) (h2 : x < ((280) / 23 : ℝ)) :
+    Ein x (26) = ((47) / 8 : ℝ) * x + ((-145) : ℝ) := by
+  rw [Ein_expand (26) (q := 12) (q' := 0) (μ := 22) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (12 : ℝ) := by linarith
+  have hf1 : x - (12 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-12) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 = ((1 : ℝ) * x + ((-12) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 =
+      (((43) / 40 : ℝ) * x + ((-13) : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_68 (x : ℝ) (h1 : ((280) / 23 : ℝ) ≤ x) (h2 : x < ((460) / 37 : ℝ)) :
+    Ein x (27) = ((189) / 40 : ℝ) * x + ((-131) : ℝ) := by
+  rw [Ein_expand (27) (q := 12) (q' := 0) (μ := 22) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (12 : ℝ) := by linarith
+  have hf1 : x - (12 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-12) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 = ((1 : ℝ) * x + ((-12) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 =
+      (((43) / 40 : ℝ) * x + ((-13) : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_69 (x : ℝ) (h1 : ((460) / 37 : ℝ) ≤ x) (h2 : x < ((25) / 2 : ℝ)) :
+    Ein x (27) = ((263) / 40 : ℝ) * x + ((-154) : ℝ) := by
+  rw [Ein_expand (27) (q := 12) (q' := 0) (μ := 23) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (12 : ℝ) := by linarith
+  have hf1 : x - (12 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-12) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 = ((1 : ℝ) * x + ((-12) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 =
+      (((43) / 40 : ℝ) * x + ((-13) : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_70 (x : ℝ) (h1 : ((25) / 2 : ℝ) ≤ x) (h2 : x < ((290) / 23 : ℝ)) :
+    Ein x (27) = ((303) / 40 : ℝ) * x + ((-333) / 2 : ℝ) := by
+  rw [Ein_expand (27) (q := 12) (q' := 0) (μ := 23) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (12 : ℝ) := by linarith
+  have hf1 : x - (12 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-25) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 =
+      ((1 : ℝ) * x + ((-25) / 2 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 =
+      ((1 : ℝ) * x + ((-25) / 2 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_71 (x : ℝ) (h1 : ((290) / 23 : ℝ) ≤ x) (h2 : x < ((480) / 37 : ℝ)) :
+    Ein x (28) = ((257) / 40 : ℝ) * x + ((-152) : ℝ) := by
+  rw [Ein_expand (28) (q := 12) (q' := 0) (μ := 23) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (12 : ℝ) := by linarith
+  have hf1 : x - (12 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-25) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 =
+      ((1 : ℝ) * x + ((-25) / 2 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 =
+      ((1 : ℝ) * x + ((-25) / 2 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_72 (x : ℝ) (h1 : ((480) / 37 : ℝ) ≤ x) (h2 : x < (13 : ℝ)) :
+    Ein x (28) = ((221) / 20 : ℝ) * x + ((-212) : ℝ) := by
+  rw [Ein_expand (28) (q := 12) (q' := 0) (μ := 24) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (12 : ℝ) := by linarith
+  have hf1 : x - (12 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-25) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 =
+      ((1 : ℝ) * x + ((-25) / 2 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (12 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_73 (x : ℝ) (h1 : (13 : ℝ) ≤ x) (h2 : x < ((560) / 43 : ℝ)) :
+    Ein x (28) = ((51) / 5 : ℝ) * x + ((-225) : ℝ) := by
+  rw [Ein_expand (28) (q := 13) (q' := 0) (μ := 24) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (13 : ℝ) := by linarith
+  have hf1 : x - (13 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-13) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 = ((1 : ℝ) * x + ((-13) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_74 (x : ℝ) (h1 : ((560) / 43 : ℝ) ≤ x) (h2 : x < ((300) / 23 : ℝ)) :
+    Ein x (28) = ((279) / 40 : ℝ) * x + ((-183) : ℝ) := by
+  rw [Ein_expand (28) (q := 13) (q' := 0) (μ := 24) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (13 : ℝ) := by linarith
+  have hf1 : x - (13 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-13) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 = ((1 : ℝ) * x + ((-13) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 =
+      (((43) / 40 : ℝ) * x + ((-14) : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_75 (x : ℝ) (h1 : ((300) / 23 : ℝ) ≤ x) (h2 : x < ((40) / 3 : ℝ)) :
+    Ein x (29) = ((233) / 40 : ℝ) * x + ((-168) : ℝ) := by
+  rw [Ein_expand (29) (q := 13) (q' := 0) (μ := 24) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (13 : ℝ) := by linarith
+  have hf1 : x - (13 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 =
+      (((3) / 40 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-13) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 = ((1 : ℝ) * x + ((-13) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 =
+      (((43) / 40 : ℝ) * x + ((-14) : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_76 (x : ℝ) (h1 : ((40) / 3 : ℝ) ≤ x) (h2 : x < ((310) / 23 : ℝ)) :
+    Ein x (30) = ((145) / 8 : ℝ) * x + ((-184) : ℝ) := by
+  rw [Ein_expand (30) (q := 13) (q' := 1) (μ := 24) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (13 : ℝ) := by linarith
+  have hf1 : x - (13 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (1 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (1 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-13) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_77 (x : ℝ) (h1 : ((310) / 23 : ℝ) ≤ x) (h2 : x < ((27) / 2 : ℝ)) :
+    Ein x (31) = ((679) / 40 : ℝ) * x + ((-337) / 2 : ℝ) := by
+  rw [Ein_expand (31) (q := 13) (q' := 1) (μ := 24) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (13 : ℝ) := by linarith
+  have hf1 : x - (13 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (1 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (1 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-13) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_78 (x : ℝ) (h1 : ((27) / 2 : ℝ) ≤ x) (h2 : x < ((500) / 37 : ℝ)) :
+    Ein x (31) = ((719) / 40 : ℝ) * x + ((-182) : ℝ) := by
+  rw [Ein_expand (31) (q := 13) (q' := 1) (μ := 24) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (13 : ℝ) := by linarith
+  have hf1 : x - (13 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (1 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (1 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-27) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_79 (x : ℝ) (h1 : ((500) / 37 : ℝ) ≤ x) (h2 : x < ((320) / 23 : ℝ)) :
+    Ein x (31) = ((793) / 40 : ℝ) * x + ((-207) : ℝ) := by
+  rw [Ein_expand (31) (q := 13) (q' := 1) (μ := 25) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (13 : ℝ) := by linarith
+  have hf1 : x - (13 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (1 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (1 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-27) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_80 (x : ℝ) (h1 : ((320) / 23 : ℝ) ≤ x) (h2 : x < ((600) / 43 : ℝ)) :
+    Ein x (32) = ((747) / 40 : ℝ) * x + ((-191) : ℝ) := by
+  rw [Ein_expand (32) (q := 13) (q' := 1) (μ := 25) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (13 : ℝ) := by linarith
+  have hf1 : x - (13 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (1 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (1 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-27) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_81 (x : ℝ) (h1 : ((600) / 43 : ℝ) ≤ x) (h2 : x < (14 : ℝ)) :
+    Ein x (32) = ((309) / 20 : ℝ) * x + ((-146) : ℝ) := by
+  rw [Ein_expand (32) (q := 13) (q' := 1) (μ := 25) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (13 : ℝ) := by linarith
+  have hf1 : x - (13 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (1 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (1 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-27) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 1 =
+      (((43) / 40 : ℝ) * x + ((-15) : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (13 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_82 (x : ℝ) (h1 : (14 : ℝ) ≤ x) (h2 : x < ((520) / 37 : ℝ)) :
+    Ein x (32) = ((73) / 5 : ℝ) * x + ((-160) : ℝ) := by
+  rw [Ein_expand (32) (q := 14) (q' := 1) (μ := 25) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (14 : ℝ) := by linarith
+  have hf1 : x - (14 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (1 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (1 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-14) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 1 = ((1 : ℝ) * x + ((-14) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_83 (x : ℝ) (h1 : ((520) / 37 : ℝ) ≤ x) (h2 : x < ((330) / 23 : ℝ)) :
+    Ein x (32) = ((769) / 40 : ℝ) * x + ((-225) : ℝ) := by
+  rw [Ein_expand (32) (q := 14) (q' := 1) (μ := 26) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (14 : ℝ) := by linarith
+  have hf1 : x - (14 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (1 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (1 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-14) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_84 (x : ℝ) (h1 : ((330) / 23 : ℝ) ≤ x) (h2 : x < ((29) / 2 : ℝ)) :
+    Ein x (33) = ((723) / 40 : ℝ) * x + ((-417) / 2 : ℝ) := by
+  rw [Ein_expand (33) (q := 14) (q' := 1) (μ := 26) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (14 : ℝ) := by linarith
+  have hf1 : x - (14 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (1 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (1 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-14) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_85 (x : ℝ) (h1 : ((29) / 2 : ℝ) ≤ x) (h2 : x < ((540) / 37 : ℝ)) :
+    Ein x (33) = ((763) / 40 : ℝ) * x + ((-223) : ℝ) := by
+  rw [Ein_expand (33) (q := 14) (q' := 1) (μ := 26) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (14 : ℝ) := by linarith
+  have hf1 : x - (14 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (1 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (1 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-29) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_86 (x : ℝ) (h1 : ((540) / 37 : ℝ) ≤ x) (h2 : x < ((340) / 23 : ℝ)) :
+    Ein x (33) = ((837) / 40 : ℝ) * x + ((-250) : ℝ) := by
+  rw [Ein_expand (33) (q := 14) (q' := 1) (μ := 27) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (14 : ℝ) := by linarith
+  have hf1 : x - (14 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (1 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (1 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-29) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_87 (x : ℝ) (h1 : ((340) / 23 : ℝ) ≤ x) (h2 : x < ((640) / 43 : ℝ)) :
+    Ein x (34) = ((791) / 40 : ℝ) * x + ((-233) : ℝ) := by
+  rw [Ein_expand (34) (q := 14) (q' := 1) (μ := 27) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (14 : ℝ) := by linarith
+  have hf1 : x - (14 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (1 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (1 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-29) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_88 (x : ℝ) (h1 : ((640) / 43 : ℝ) ≤ x) (h2 : x < (15 : ℝ)) :
+    Ein x (34) = ((331) / 20 : ℝ) * x + ((-185) : ℝ) := by
+  rw [Ein_expand (34) (q := 14) (q' := 1) (μ := 27) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (14 : ℝ) := by linarith
+  have hf1 : x - (14 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (1 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (1 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-29) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 1 =
+      (((43) / 40 : ℝ) * x + ((-16) : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (14 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_89 (x : ℝ) (h1 : (15 : ℝ) ≤ x) (h2 : x < ((560) / 37 : ℝ)) :
+    Ein x (34) = ((157) / 10 : ℝ) * x + ((-200) : ℝ) := by
+  rw [Ein_expand (34) (q := 15) (q' := 1) (μ := 27) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (15 : ℝ) := by linarith
+  have hf1 : x - (15 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (1 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (1 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-15) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 1 = ((1 : ℝ) * x + ((-15) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_90 (x : ℝ) (h1 : ((560) / 37 : ℝ) ≤ x) (h2 : x < ((350) / 23 : ℝ)) :
+    Ein x (34) = ((813) / 40 : ℝ) * x + ((-270) : ℝ) := by
+  rw [Ein_expand (34) (q := 15) (q' := 1) (μ := 28) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (15 : ℝ) := by linarith
+  have hf1 : x - (15 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (1 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (1 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-15) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_91 (x : ℝ) (h1 : ((350) / 23 : ℝ) ≤ x) (h2 : x < ((31) / 2 : ℝ)) :
+    Ein x (35) = ((767) / 40 : ℝ) * x + ((-505) / 2 : ℝ) := by
+  rw [Ein_expand (35) (q := 15) (q' := 1) (μ := 28) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (15 : ℝ) := by linarith
+  have hf1 : x - (15 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (1 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (1 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-15) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_92 (x : ℝ) (h1 : ((31) / 2 : ℝ) ≤ x) (h2 : x < ((360) / 23 : ℝ)) :
+    Ein x (35) = ((807) / 40 : ℝ) * x + ((-268) : ℝ) := by
+  rw [Ein_expand (35) (q := 15) (q' := 1) (μ := 28) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (15 : ℝ) := by linarith
+  have hf1 : x - (15 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (1 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (1 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-31) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_93 (x : ℝ) (h1 : ((360) / 23 : ℝ) ≤ x) (h2 : x < ((580) / 37 : ℝ)) :
+    Ein x (36) = ((761) / 40 : ℝ) * x + ((-250) : ℝ) := by
+  rw [Ein_expand (36) (q := 15) (q' := 1) (μ := 28) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (15 : ℝ) := by linarith
+  have hf1 : x - (15 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (1 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (1 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-31) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_94 (x : ℝ) (h1 : ((580) / 37 : ℝ) ≤ x) (h2 : x < ((680) / 43 : ℝ)) :
+    Ein x (36) = ((167) / 8 : ℝ) * x + ((-279) : ℝ) := by
+  rw [Ein_expand (36) (q := 15) (q' := 1) (μ := 29) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (15 : ℝ) := by linarith
+  have hf1 : x - (15 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (1 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (1 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-31) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_95 (x : ℝ) (h1 : ((680) / 43 : ℝ) ≤ x) (h2 : x < (16 : ℝ)) :
+    Ein x (36) = ((353) / 20 : ℝ) * x + ((-228) : ℝ) := by
+  rw [Ein_expand (36) (q := 15) (q' := 1) (μ := 29) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (15 : ℝ) := by linarith
+  have hf1 : x - (15 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (1 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (1 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-31) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 1 =
+      (((43) / 40 : ℝ) * x + ((-17) : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (15 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_96 (x : ℝ) (h1 : (16 : ℝ) ≤ x) (h2 : x < ((370) / 23 : ℝ)) :
+    Ein x (36) = ((84) / 5 : ℝ) * x + ((-244) : ℝ) := by
+  rw [Ein_expand (36) (q := 16) (q' := 1) (μ := 29) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (16 : ℝ) := by linarith
+  have hf1 : x - (16 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (1 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (1 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-16) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 1 = ((1 : ℝ) * x + ((-16) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_97 (x : ℝ) (h1 : ((370) / 23 : ℝ) ≤ x) (h2 : x < ((600) / 37 : ℝ)) :
+    Ein x (37) = ((313) / 20 : ℝ) * x + ((-451) / 2 : ℝ) := by
+  rw [Ein_expand (37) (q := 16) (q' := 1) (μ := 29) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (16 : ℝ) := by linarith
+  have hf1 : x - (16 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (1 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (1 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-16) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 1 = ((1 : ℝ) * x + ((-16) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_98 (x : ℝ) (h1 : ((600) / 37 : ℝ) ≤ x) (h2 : x < ((33) / 2 : ℝ)) :
+    Ein x (37) = ((811) / 40 : ℝ) * x + ((-601) / 2 : ℝ) := by
+  rw [Ein_expand (37) (q := 16) (q' := 1) (μ := 30) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (16 : ℝ) := by linarith
+  have hf1 : x - (16 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (1 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (1 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-16) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_99 (x : ℝ) (h1 : ((33) / 2 : ℝ) ≤ x) (h2 : x < ((380) / 23 : ℝ)) :
+    Ein x (37) = ((851) / 40 : ℝ) * x + ((-317) : ℝ) := by
+  rw [Ein_expand (37) (q := 16) (q' := 1) (μ := 30) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (16 : ℝ) := by linarith
+  have hf1 : x - (16 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (1 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (1 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-33) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_100 (x : ℝ) (h1 : ((380) / 23 : ℝ) ≤ x) (h2 : x < ((50) / 3 : ℝ)) :
+    Ein x (38) = ((161) / 8 : ℝ) * x + ((-298) : ℝ) := by
+  rw [Ein_expand (38) (q := 16) (q' := 1) (μ := 30) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (16 : ℝ) := by linarith
+  have hf1 : x - (16 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (1 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (1 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-33) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_101 (x : ℝ) (h1 : ((50) / 3 : ℝ) ≤ x) (h2 : x < ((720) / 43 : ℝ)) :
+    Ein x (39) = ((799) / 40 : ℝ) * x + ((-591) / 2 : ℝ) := by
+  rw [Ein_expand (39) (q := 16) (q' := 1) (μ := 30) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (16 : ℝ) := by linarith
+  have hf1 : x - (16 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (1 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (1 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-33) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_102 (x : ℝ) (h1 : ((720) / 43 : ℝ) ≤ x) (h2 : x < ((620) / 37 : ℝ)) :
+    Ein x (39) = ((67) / 4 : ℝ) * x + ((-483) / 2 : ℝ) := by
+  rw [Ein_expand (39) (q := 16) (q' := 1) (μ := 30) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (16 : ℝ) := by linarith
+  have hf1 : x - (16 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (1 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (1 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-33) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 1 =
+      (((43) / 40 : ℝ) * x + ((-18) : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_103 (x : ℝ) (h1 : ((620) / 37 : ℝ) ≤ x) (h2 : x < ((390) / 23 : ℝ)) :
+    Ein x (39) = ((93) / 5 : ℝ) * x + ((-545) / 2 : ℝ) := by
+  rw [Ein_expand (39) (q := 16) (q' := 1) (μ := 31) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (16 : ℝ) := by linarith
+  have hf1 : x - (16 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (1 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (1 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-33) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 1 =
+      (((43) / 40 : ℝ) * x + ((-18) : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_104 (x : ℝ) (h1 : ((390) / 23 : ℝ) ≤ x) (h2 : x < (17 : ℝ)) :
+    Ein x (40) = ((349) / 20 : ℝ) * x + ((-253) : ℝ) := by
+  rw [Ein_expand (40) (q := 16) (q' := 1) (μ := 31) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (16 : ℝ) := by linarith
+  have hf1 : x - (16 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (1 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (1 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-33) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 1 =
+      (((43) / 40 : ℝ) * x + ((-18) : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (16 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_105 (x : ℝ) (h1 : (17 : ℝ) ≤ x) (h2 : x < ((640) / 37 : ℝ)) :
+    Ein x (40) = ((83) / 5 : ℝ) * x + ((-270) : ℝ) := by
+  rw [Ein_expand (40) (q := 17) (q' := 1) (μ := 31) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (17 : ℝ) := by linarith
+  have hf1 : x - (17 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (1 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (1 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-17) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 1 = ((1 : ℝ) * x + ((-17) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_106 (x : ℝ) (h1 : ((640) / 37 : ℝ) ≤ x) (h2 : x < ((400) / 23 : ℝ)) :
+    Ein x (40) = ((849) / 40 : ℝ) * x + ((-350) : ℝ) := by
+  rw [Ein_expand (40) (q := 17) (q' := 1) (μ := 32) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (17 : ℝ) := by linarith
+  have hf1 : x - (17 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (1 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (1 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-17) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_107 (x : ℝ) (h1 : ((400) / 23 : ℝ) ≤ x) (h2 : x < ((35) / 2 : ℝ)) :
+    Ein x (41) = ((803) / 40 : ℝ) * x + ((-330) : ℝ) := by
+  rw [Ein_expand (41) (q := 17) (q' := 1) (μ := 32) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (17 : ℝ) := by linarith
+  have hf1 : x - (17 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (1 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (1 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-17) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_108 (x : ℝ) (h1 : ((35) / 2 : ℝ) ≤ x) (h2 : x < ((760) / 43 : ℝ)) :
+    Ein x (41) = ((843) / 40 : ℝ) * x + ((-695) / 2 : ℝ) := by
+  rw [Ein_expand (41) (q := 17) (q' := 1) (μ := 32) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (17 : ℝ) := by linarith
+  have hf1 : x - (17 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (1 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (1 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-35) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_109 (x : ℝ) (h1 : ((760) / 43 : ℝ) ≤ x) (h2 : x < ((410) / 23 : ℝ)) :
+    Ein x (41) = ((357) / 20 : ℝ) * x + ((-581) / 2 : ℝ) := by
+  rw [Ein_expand (41) (q := 17) (q' := 1) (μ := 32) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (17 : ℝ) := by linarith
+  have hf1 : x - (17 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (1 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (1 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-35) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 1 =
+      (((43) / 40 : ℝ) * x + ((-19) : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_110 (x : ℝ) (h1 : ((410) / 23 : ℝ) ≤ x) (h2 : x < ((660) / 37 : ℝ)) :
+    Ein x (42) = ((167) / 10 : ℝ) * x + ((-270) : ℝ) := by
+  rw [Ein_expand (42) (q := 17) (q' := 1) (μ := 32) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (17 : ℝ) := by linarith
+  have hf1 : x - (17 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (1 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (1 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-35) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 1 =
+      (((43) / 40 : ℝ) * x + ((-19) : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_111 (x : ℝ) (h1 : ((660) / 37 : ℝ) ≤ x) (h2 : x < (18 : ℝ)) :
+    Ein x (42) = ((371) / 20 : ℝ) * x + ((-303) : ℝ) := by
+  rw [Ein_expand (42) (q := 17) (q' := 1) (μ := 33) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (17 : ℝ) := by linarith
+  have hf1 : x - (17 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (1 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (1 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-35) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 1 =
+      (((43) / 40 : ℝ) * x + ((-19) : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (17 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_112 (x : ℝ) (h1 : (18 : ℝ) ≤ x) (h2 : x < ((420) / 23 : ℝ)) :
+    Ein x (42) = ((177) / 10 : ℝ) * x + ((-321) : ℝ) := by
+  rw [Ein_expand (42) (q := 18) (q' := 1) (μ := 33) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (18 : ℝ) := by linarith
+  have hf1 : x - (18 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (1 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (1 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-18) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 1 = ((1 : ℝ) * x + ((-18) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_113 (x : ℝ) (h1 : ((420) / 23 : ℝ) ≤ x) (h2 : x < ((680) / 37 : ℝ)) :
+    Ein x (43) = ((331) / 20 : ℝ) * x + ((-300) : ℝ) := by
+  rw [Ein_expand (43) (q := 18) (q' := 1) (μ := 33) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (18 : ℝ) := by linarith
+  have hf1 : x - (18 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (1 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (1 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-18) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 1 = ((1 : ℝ) * x + ((-18) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_114 (x : ℝ) (h1 : ((680) / 37 : ℝ) ≤ x) (h2 : x < ((37) / 2 : ℝ)) :
+    Ein x (43) = ((847) / 40 : ℝ) * x + ((-385) : ℝ) := by
+  rw [Ein_expand (43) (q := 18) (q' := 1) (μ := 34) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (18 : ℝ) := by linarith
+  have hf1 : x - (18 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (1 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (1 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-18) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_115 (x : ℝ) (h1 : ((37) / 2 : ℝ) ≤ x) (h2 : x < ((800) / 43 : ℝ)) :
+    Ein x (43) = ((887) / 40 : ℝ) * x + ((-807) / 2 : ℝ) := by
+  rw [Ein_expand (43) (q := 18) (q' := 1) (μ := 34) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (18 : ℝ) := by linarith
+  have hf1 : x - (18 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (1 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (1 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-37) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_116 (x : ℝ) (h1 : ((800) / 43 : ℝ) ≤ x) (h2 : x < ((430) / 23 : ℝ)) :
+    Ein x (43) = ((379) / 20 : ℝ) * x + ((-687) / 2 : ℝ) := by
+  rw [Ein_expand (43) (q := 18) (q' := 1) (μ := 34) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (18 : ℝ) := by linarith
+  have hf1 : x - (18 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (1 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (1 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-37) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 1 =
+      (((43) / 40 : ℝ) * x + ((-20) : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_117 (x : ℝ) (h1 : ((430) / 23 : ℝ) ≤ x) (h2 : x < ((700) / 37 : ℝ)) :
+    Ein x (44) = ((89) / 5 : ℝ) * x + ((-322) : ℝ) := by
+  rw [Ein_expand (44) (q := 18) (q' := 1) (μ := 34) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (18 : ℝ) := by linarith
+  have hf1 : x - (18 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (1 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (1 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-37) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 1 =
+      (((43) / 40 : ℝ) * x + ((-20) : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_118 (x : ℝ) (h1 : ((700) / 37 : ℝ) ≤ x) (h2 : x < (19 : ℝ)) :
+    Ein x (44) = ((393) / 20 : ℝ) * x + ((-357) : ℝ) := by
+  rw [Ein_expand (44) (q := 18) (q' := 1) (μ := 35) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (18 : ℝ) := by linarith
+  have hf1 : x - (18 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (1 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (1 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-37) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 1 =
+      (((43) / 40 : ℝ) * x + ((-20) : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (18 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_119 (x : ℝ) (h1 : (19 : ℝ) ≤ x) (h2 : x < ((440) / 23 : ℝ)) :
+    Ein x (44) = ((94) / 5 : ℝ) * x + ((-376) : ℝ) := by
+  rw [Ein_expand (44) (q := 19) (q' := 1) (μ := 35) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (19 : ℝ) := by linarith
+  have hf1 : x - (19 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (1 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (1 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (19 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (19 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (19 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (19 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-19) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (19 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (19 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 1 = ((1 : ℝ) * x + ((-19) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (19 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (19 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (19 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_120 (x : ℝ) (h1 : ((440) / 23 : ℝ) ≤ x) (h2 : x < ((720) / 37 : ℝ)) :
+    Ein x (45) = ((353) / 20 : ℝ) * x + ((-354) : ℝ) := by
+  rw [Ein_expand (45) (q := 19) (q' := 1) (μ := 35) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (19 : ℝ) := by linarith
+  have hf1 : x - (19 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (1 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (1 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (19 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (19 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (19 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (19 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-19) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (19 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (19 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 1 = ((1 : ℝ) * x + ((-19) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (19 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (19 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (19 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_121 (x : ℝ) (h1 : ((720) / 37 : ℝ) ≤ x) (h2 : x < ((39) / 2 : ℝ)) :
+    Ein x (45) = ((891) / 40 : ℝ) * x + ((-444) : ℝ) := by
+  rw [Ein_expand (45) (q := 19) (q' := 1) (μ := 36) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (19 : ℝ) := by linarith
+  have hf1 : x - (19 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (1 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (1 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (19 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (19 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (19 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (19 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-19) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (19 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (19 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (19 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (19 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (19 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_122 (x : ℝ) (h1 : ((39) / 2 : ℝ) ≤ x) (h2 : x < ((840) / 43 : ℝ)) :
+    Ein x (45) = ((931) / 40 : ℝ) * x + ((-927) / 2 : ℝ) := by
+  rw [Ein_expand (45) (q := 19) (q' := 1) (μ := 36) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (19 : ℝ) := by linarith
+  have hf1 : x - (19 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (1 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (1 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (19 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (19 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (19 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (19 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (19 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-39) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (19 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (19 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (19 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (19 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_123 (x : ℝ) (h1 : ((840) / 43 : ℝ) ≤ x) (h2 : x < ((450) / 23 : ℝ)) :
+    Ein x (45) = ((401) / 20 : ℝ) * x + ((-801) / 2 : ℝ) := by
+  rw [Ein_expand (45) (q := 19) (q' := 1) (μ := 36) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (19 : ℝ) := by linarith
+  have hf1 : x - (19 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (1 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (1 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (19 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (19 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (19 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (19 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (19 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-39) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (19 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (19 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (19 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 1 =
+      (((43) / 40 : ℝ) * x + ((-21) : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (19 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+theorem inPiece_124 (x : ℝ) (h1 : ((450) / 23 : ℝ) ≤ x) (h2 : x < (20 : ℝ)) :
+    Ein x (46) = ((189) / 10 : ℝ) * x + ((-378) : ℝ) := by
+  rw [Ein_expand (46) (q := 19) (q' := 1) (μ := 36) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (19 : ℝ) := by linarith
+  have hf1 : x - (19 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (1 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (1 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (19 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (19 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (19 : ℝ)) (3 / 40 * x - (1 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (19 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 0 =
+      ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (19 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-39) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (19 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (19 : ℝ)) (3 / 40 * x - (1 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (19 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 1 =
+      (((43) / 40 : ℝ) * x + ((-21) : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (19 : ℝ)) (3 / 40 * x - (1 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22]
+  simp only [bIn, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, psiR_eq, lrow, toNat_cast_q]
+  push_cast
+  norm_num
+  ring
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Growth/OuterPrime.lean b/LeanPool/Zeta5Irrational/Growth/OuterPrime.lean
new file mode 100644
index 0000000000..c170d300b5
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Growth/OuterPrime.lean
@@ -0,0 +1,330 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Growth.TailPrime
+import Mathlib.Tactic.ReduceModChar
+import Mathlib.Tactic.Polynomial.Basic
+import Mathlib.Data.Rat.Star
+import Mathlib.Data.Int.Star
+import Std.Tactic.BVDecide.Normalize.Prop
+import Mathlib.Tactic.FieldSimp
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+import Mathlib.Tactic.Ring.Basic
+
+import Mathlib.Algebra.Order.Floor.Semifield
+
+/-! # The outer range: class counts
+
+For a nonzero class `c`, the number of class members in `(X, Y]` is `SX p Y c - SX p X c`.
+-/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+variable {p : ℕ} [hp : Fact p.Prime] {m : ℕ}
+
+lemma jc_eq_iff (hm : 2 * m + 1 = p) (j : ℕ) (c : Fin (m + 1)) :
+    jc hm j = c ↔
+      ((j : ℤ) : ZMod p) = (((c : ℕ) : ℤ) : ZMod p) ∨
+        ((j : ℤ) : ZMod p) = -(((c : ℕ) : ℤ) : ZMod p) := by
+  unfold jc
+  constructor
+  · intro h
+    rcases ccls_spec hm ((j : ℤ) : ZMod p) with h1 | h1 <;> rw [h] at h1
+    · left; exact h1.symm
+    · right; rw [h1]; ring
+  · intro h
+    refine (ccls_unique hm _ c ?_).symm
+    rcases h with h | h
+    · left; exact h.symm
+    · right; rw [h]; ring
+
+lemma neg_ne_self_zmod (hm : 2 * m + 1 = p) {x : ZMod p} (hx : x ≠ 0) : x ≠ -x := by
+  intro h
+  have h2 : (2 : ZMod p) * x = 0 := by linear_combination h
+  rcases mul_eq_zero.mp h2 with h3 | h3
+  · exact two_ne_zero_zmod hm h3
+  · exact hx h3
+
+lemma card_class (hm : 2 * m + 1 = p) {c : Fin (m + 1)} (hc : c ≠ 0) (X : ℕ) :
+    ((Icc 1 X).filter fun j => jc hm j = c).card = SX p X (((c : ℕ) : ℤ) : ZMod p) := by
+  have hc0 : (((c : ℕ) : ℤ) : ZMod p) ≠ 0 := by
+    intro h
+    apply hc
+    have := (ccls_eq_zero_iff hm _).mpr h
+    rw [ccls_self hm c] at this
+    exact this
+  unfold SX
+  rw [← Finset.card_union_of_disjoint]
+  · congr 1
+    ext j
+    simp only [Finset.mem_filter, Finset.mem_union, jc_eq_iff hm]
+    tauto
+  · rw [Finset.disjoint_left]
+    intro j h1 h2
+    simp only [Finset.mem_filter] at h1 h2
+    exact neg_ne_self_zmod hm hc0 (h1.2.symm.trans h2.2)
+
+lemma card_class_Ioc (hm : 2 * m + 1 = p) {c : Fin (m + 1)} (hc : c ≠ 0) {X Y : ℕ} (hXY : X ≤ Y) :
+    ((Icc (X + 1) Y).filter fun j => jc hm j = c).card =
+      SX p Y (((c : ℕ) : ℤ) : ZMod p) - SX p X (((c : ℕ) : ℤ) : ZMod p) := by
+  rw [← card_class hm hc, ← card_class hm hc]
+  have hsplit :
+    (Icc 1 Y).filter (fun j => jc hm j = c) =
+      (Icc 1 X).filter (fun j => jc hm j = c) ∪ (Icc (X + 1) Y).filter (fun j => jc hm j = c) :=
+    by ext j; simp only [Finset.mem_filter, Finset.mem_union, Finset.mem_Icc]; omega
+  have hdisj :
+    Disjoint ((Icc 1 X).filter (fun j => jc hm j = c))
+      ((Icc (X + 1) Y).filter (fun j => jc hm j = c)) := by
+    rw [Finset.disjoint_left]; intro j h1 h2
+    simp only [Finset.mem_filter, Finset.mem_Icc] at h1 h2; omega
+  rw [hsplit, Finset.card_union_of_disjoint hdisj]
+  omega
+
+/-- The per-class outer weight sum `G_q(a, δ)`. -/
+noncomputable def GO (q a d : ℕ) : ℚ :=
+  ∑ i ∈ range (2 * q + a - 2), min 0 ((i : ℚ) + 3 * d - ((2 * q + a : ℕ) + 4 : ℚ) / 2)
+
+section
+
+variable (hm : 2 * m + 1 = p) {n : ℕ} (hn : 1 ≤ n) (hout : 40 * n < 3 * p)
+
+include hm hn hout in
+omit hp in
+lemma three_n_le_m : 3 * n ≤ m := by omega
+
+include hm in
+lemma SX_small {X : ℕ} (hX : X < p) {c : ℕ} (hc1 : 1 ≤ c) (hcm : c ≤ m) :
+    SX p X (((c : ℕ) : ℤ) : ZMod p) = aCnt p X c := by
+  rw [SX_class hm hc1 hcm, Nat.div_eq_of_lt hX, Nat.mod_eq_of_lt hX]; ring
+
+include hn hout in
+lemma class_data {c : Fin (m + 1)} (hc : c ≠ 0) :
+    LoC hm n c = 2 * (40 * n / p) + aCnt p (40 * n % p) c - aCnt p (3 * n) c ∧
+      nbC hm n c = 2 * (40 * n / p) + aCnt p (40 * n % p) c - 2 ∧
+        ellC hm n c = 2 * (40 * n / p) + aCnt p (40 * n % p) c ∧ delC n c = aCnt p (3 * n) c := by
+  have hc1 : 1 ≤ (c : ℕ) := by
+    rcases Nat.eq_zero_or_pos (c : ℕ) with h | h
+    · exact absurd (Fin.ext h) hc
+    · exact h
+  have hcm := Nat.lt_succ_iff.mp c.isLt
+  have h3m := three_n_le_m hm hn hout
+  have hN : 3 * n < p := by omega
+  have hK := SX_class hm hc1 hcm (40 * n)
+  have hNs := SX_small hm hN hc1 hcm
+  refine ⟨?_, ?_, ?_, ?_⟩
+  · unfold LoC tailC tailO
+    rw [card_class_Ioc hm hc (by omega), hK, hNs]
+  · unfold nbC bigC tailC tailO
+    rw [Finset.filter_filter]
+    rcases le_or_gt p (40 * n) with hpK | hpK
+    · have e :
+        (Icc (3 * n + 1) (40 * n)).filter (fun j => jc hm j = c ∧ p < j) =
+          (Icc (p + 1) (40 * n)).filter (fun j => jc hm j = c) :=
+        by ext j; simp only [Finset.mem_filter, Finset.mem_Icc]; omega
+      rw [e, card_class_Ioc hm hc hpK, hK, SX_class hm hc1 hcm p, Nat.div_self (by omega),
+        Nat.mod_self]
+      unfold aCnt; split_ifs <;> omega
+    · have e : (Icc (3 * n + 1) (40 * n)).filter (fun j => jc hm j = c ∧ p < j) = ∅ := by
+        ext j;
+        simp only [Finset.mem_filter, Finset.mem_Icc, Finset.notMem_empty, iff_false]; omega
+      rw [e, Finset.card_empty, Nat.div_eq_of_lt hpK]
+      unfold aCnt; split_ifs <;> omega
+  · unfold ellC; rw [card_class hm hc, hK]
+  · unfold delC aCnt; split_ifs <;> omega
+
+include hn hout in
+lemma sum_wO_class {c : Fin (m + 1)} (hc : c ≠ 0) :
+    ∑ i ∈ range (LoC hm n c), wO hm n c i =
+      GO (40 * n / p) (aCnt p (40 * n % p) c) (aCnt p (3 * n) c) := by
+  obtain ⟨hL, hnb, hl, hd⟩ := class_data hm hn hout hc
+  have hδ : aCnt p (3 * n) c ≤ 1 := by
+    have := three_n_le_m hm hn hout
+    have hcm := Nat.lt_succ_iff.mp c.isLt
+    unfold aCnt; split_ifs <;> omega
+  have hsub : nbC hm n c ≤ LoC hm n c := by rw [hL, hnb]; omega
+  rw [← Finset.sum_range_add_sum_Ico _ hsub]
+  have h0 : ∑ i ∈ Ico (nbC hm n c) (LoC hm n c), wO hm n c i = 0 := by
+    refine Finset.sum_eq_zero fun i hi => ?_
+    unfold wO; rw [ite_eq_right hc, ite_eq_right (by have := (Finset.mem_Ico.mp hi).1; omega)]
+  rw [h0, add_zero, hnb]
+  unfold GO
+  refine Finset.sum_congr rfl fun i hi => ?_
+  unfold wO
+  rw [ite_eq_right hc, ite_eq_left (by rw [hnb]; exact Finset.mem_range.mp hi), hl, hd]
+
+include hn hout in
+lemma sum_wO_zero : -4 ≤ ∑ i ∈ range (LoC hm n 0), wO hm n 0 i := by
+  have hL : LoC hm n 0 ≤ 2 := by
+    unfold LoC tailC tailO
+    have hsub : (Icc (3 * n + 1) (40 * n)).filter (fun j => jc hm j = 0) ⊆ {p, 2 * p} := by
+      intro j hj
+      obtain ⟨hj1, hj2⟩ := Finset.mem_filter.mp hj
+      have hdvd : p ∣ j := by exact_mod_cast (jc_eq_zero_iff hm j).mp hj2
+      obtain ⟨t, rfl⟩ := hdvd
+      have := Finset.mem_Icc.mp hj1
+      have ht1 : 1 ≤ t := by
+        rcases Nat.eq_zero_or_pos t with h | h
+        · subst h; omega
+        · exact h
+      have ht3 : t < 3 := by
+        by_contra h; rw [not_lt] at h
+        have : p * 3 ≤ p * t := Nat.mul_le_mul_left _ h
+        omega
+      interval_cases t <;> simp [mul_comm]
+    exact (Finset.card_le_card hsub).trans (Finset.card_le_two)
+  have : ∀ i, -2 ≤ wO hm n 0 i := by
+    intro i; unfold wO; rw [ite_eq_left rfl];
+    split_ifs <;> norm_num
+  calc
+    (-4 : ℚ) ≤ ∑ _i ∈ range (LoC hm n 0), (-2 : ℚ) := by
+      rw [Finset.sum_const, Finset.card_range, nsmul_eq_mul]
+      have : (LoC hm n 0 : ℚ) ≤ 2 := by exact_mod_cast hL
+      linarith
+    _ ≤ _ := Finset.sum_le_sum fun i _ => this i
+
+end
+
+/-- The class-function coefficients of the outer range. -/
+noncomputable def bGO (q : ℕ) (i j : Fin 3) : ℝ :=
+  bcoef (fun a => bcoef (fun b => (GO q a b : ℝ)) j) i
+
+/-- **The continuous outer function** (in the variable `x = K/p`). -/
+noncomputable def Eout (x : ℝ) : ℝ :=
+  -2 * (37 / 40 * x) * (⌊x⌋₊ : ℝ) + 2 * Jc (37 / 40 * x) -
+      2 * ∑ i, ∑ j, bGO ⌊x⌋₊ i j * ccount (Int.fract x) (Int.fract (3 / 40 * x)) i j +
+    max 0 (13 / 10 * x - 2)
+
+/-- The additive error of the outer bound. -/
+noncomputable def Cout (n p : ℕ) : ℝ :=
+  5 * ∑ i, ∑ j, |bGO (40 * n / p) i j| + 11
+
+section
+
+variable {n : ℕ} (hn : 33 ≤ n) (hodd : 2 * (p / 2) + 1 = p) (hp7 : 7 ≤ p) (hout : 40 * n < 3 * p)
+  (hle : p ≤ 2 * (37 * n)) (hsq : 2 * (40 * n) < p ^ 2)
+
+include hn hodd hp7 hout hle hsq in
+omit hle in
+/-- **The outer per-prime bound.** -/
+theorem outer_Lp_le : -(Lp n p : ℝ) ≤ (p : ℝ) * Eout (40 * n / p) + Cout n p := by
+  have hp0 : 0 < p := by omega
+  have hpR0 : (0 : ℝ) < p := by exact_mod_cast hp0
+  have hN : 3 * n < p := by omega
+  have hOK : OuterOK n p := ⟨hodd, hp7, hout, hsq, hN⟩
+  have hLp : Lp n p = vSK n p + ⌊WO n p⌋ := by
+    unfold Lp Mcut;
+    rw [ite_eq_right (by omega), ite_eq_right (by omega), ite_eq_left hOK]
+  set m := p / 2 with hmdef
+  have hWO :
+    WO n p =
+      2 * ∑ c, ∑ i ∈ range (LoC hodd n c), wO hodd n c i -
+        ((37 * n - ((2 * p + 22 * n - 3) - (37 * n - 1)) : ℕ) : ℚ) := by
+    unfold WO;
+    rw [dite_eq_left hp.out, dite_eq_left hodd]
+      -- the class sum
+  have hsplit :
+    ∑ c, ∑ i ∈ range (LoC hodd n c), wO hodd n c i =
+      ∑ i ∈ range (LoC hodd n 0), wO hodd n 0 i +
+        ∑ c ∈ Icc 1 m, GO (40 * n / p) (aCnt p (40 * n % p) c) (aCnt p (3 * n) c) := by
+    rw [← Finset.add_sum_erase _ _ (Finset.mem_univ (0 : Fin (m + 1)))]
+    congr 1
+    rw [Finset.sum_congr rfl fun c hc =>
+        sum_wO_class hodd (by omega) hout (Finset.mem_erase.mp hc).1]
+    exact
+      sum_erase_zero_eq_Icc (fun c => GO (40 * n / p) (aCnt p (40 * n % p) c) (aCnt p (3 * n) c))
+  have hZ :=
+    sum_wO_zero hodd (n := n) (by omega)
+      hout
+        -- the continuous class sum
+  have hcs :=
+    class_sum_approx hodd (Nat.mod_lt (40 * n) hp0) (Nat.mod_lt (3 * n) hp0)
+      (fun a b => (GO (40 * n / p) a b : ℝ))
+  rw [Nat.mod_eq_of_lt hN] at hcs
+  rw [abs_le] at hcs
+  have hcsQ :
+    ((∑ c ∈ Icc 1 m, GO (40 * n / p) (aCnt p (40 * n % p) c) (aCnt p (3 * n) c) : ℚ) : ℝ) =
+      ∑ c ∈ Icc 1 m, (GO (40 * n / p) (aCnt p (40 * n % p) c) (aCnt p (3 * n) c) : ℝ) := by
+    push_cast;
+    rfl
+      -- the rank loss
+  have hr :
+    ((37 * n - ((2 * p + 22 * n - 3) - (37 * n - 1)) : ℕ) : ℝ) ≤
+      (p : ℝ) * max (0 : ℝ) (13 / 10 * (40 * (n : ℝ) / (p : ℝ)) - 2) + 2 := by
+    have e :
+      (p : ℝ) * max (0 : ℝ) (13 / 10 * (40 * (n : ℝ) / (p : ℝ)) - 2) =
+        max 0 (52 * (n : ℝ) - 2 * p) :=
+      by rw [mul_max_of_nonneg _ _ hpR0.le, mul_zero]; congr 1; field_simp; ring
+    rw [e]
+    have hm0 := le_max_left (0 : ℝ) (52 * n - 2 * p)
+    have hm1 := le_max_right (0 : ℝ) (52 * n - 2 * p)
+    rcases Nat.lt_or_ge (37 * n) (((2 * p + 22 * n - 3) - (37 * n - 1))) with h | h
+    · rw [Nat.sub_eq_zero_of_le h.le]; push_cast; linarith
+    · have hR : 37 * n - ((2 * p + 22 * n - 3) - (37 * n - 1)) ≤ 52 * n + 2 - 2 * p := by omega
+      rcases le_or_gt (52 * n + 2) (2 * p) with h5 | h5
+      · have : 37 * n - ((2 * p + 22 * n - 3) - (37 * n - 1)) = 0 := by omega
+        rw [this]; push_cast; linarith
+      · have h2 := (Nat.cast_le (α := ℝ)).mpr hR
+        have h6 : ((52 * n + 2 - 2 * p : ℕ) : ℝ) = 52 * (n : ℝ) + 2 - 2 * p := by
+          rw [Nat.cast_sub (by omega)]; push_cast; ring
+        rw [h6] at h2
+        linarith
+          -- `v_p(S_K)`
+  have hv :=
+    neg_vSK_le (p := p) (n := n) (by omega) (by omega) (by nlinarith) (by nlinarith) 1000
+      (by omega)
+  rw [Nat.div_eq_of_lt hN] at hv
+  rw [layer_eq hp0] at hv
+        -- the target
+  have hx : (40 * n / p : ℝ) = ((40 * n : ℕ) : ℝ) / (p : ℕ) := by push_cast; ring
+  have hfl : ⌊(40 * n / p : ℝ)⌋₊ = 40 * n / p := by rw [hx, Nat.floor_div_eq_div]
+  have hfK : Int.fract ((40 * n / p : ℝ)) = ((40 * n % p : ℕ) : ℝ) / p := by
+    rw [hx];
+    exact Int.fract_div_natCast_eq_div_natCast_mod
+  have hfN : Int.fract ((3 : ℝ) / 40 * (40 * n / p)) = ((3 * n : ℕ) : ℝ) / p := by
+    rw [show (3 : ℝ) / 40 * (40 * n / p) = ((3 * n : ℕ) : ℝ) / (p : ℕ) by push_cast; ring,
+      Int.fract_div_natCast_eq_div_natCast_mod, Nat.mod_eq_of_lt hN]
+  have hu : ((37 * n : ℕ) : ℝ) / p = 37 / 40 * (40 * n / p) := by push_cast; ring
+  rw [hu] at hv
+  have e1 : ((37 * n : ℕ) : ℝ) = (p : ℝ) * (37 / 40 * (40 * n / p)) := by push_cast; field_simp
+  rw [e1] at hv
+  have hfl2 : ((40 * n / p : ℕ) : ℝ) = (⌊(40 * n / p : ℝ)⌋₊ : ℝ) := by rw [hfl]
+  unfold Eout Cout
+  rw [hfl, hfK, hfN]
+  have hflo := Int.lt_floor_add_one (WO n p)
+  have hfloR : ((WO n p : ℚ) : ℝ) < ((⌊WO n p⌋ : ℤ) : ℝ) + 1 := by exact_mod_cast hflo
+  have hWOR :
+    ((WO n p : ℚ) : ℝ) =
+      2 *
+          (((∑ i ∈ range (LoC hodd n 0), wO hodd n 0 i : ℚ) : ℝ) +
+            ∑ c ∈ Icc 1 m, (GO (40 * n / p) (aCnt p (40 * n % p) c) (aCnt p (3 * n) c) : ℝ)) -
+        ((37 * n - ((2 * p + 22 * n - 3) - (37 * n - 1)) : ℕ) : ℝ) :=
+    by rw [hWO, hsplit]; push_cast; ring
+  push_cast at hWOR
+  rw [hLp]
+  push_cast
+  have hZR : (-4 : ℝ) ≤ ((∑ i ∈ range (LoC hodd n 0), wO hodd n 0 i : ℚ) : ℝ) := by
+    exact_mod_cast hZ
+  push_cast at hZR
+  have hb :
+    ∀ i j,
+      bcoef (fun a => bcoef (fun b => (GO (40 * n / p) a b : ℝ)) j) i = bGO (40 * n / p) i j :=
+    fun i j => rfl
+  simp only [hb] at hcs
+  push_cast at hcs
+  linarith [hcs.1]
+
+end
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Growth/OuterTable.lean b/LeanPool/Zeta5Irrational/Growth/OuterTable.lean
new file mode 100644
index 0000000000..4dfb67f95a
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Growth/OuterTable.lean
@@ -0,0 +1,983 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+import LeanPool.Zeta5Irrational.Growth.TableGen
+public import LeanPool.Zeta5Irrational.Growth.OuterPrime
+import Mathlib.Tactic.ReduceModChar
+import Mathlib.Tactic.Polynomial.Basic
+import Mathlib.Data.Rat.Star
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+import Mathlib.Tactic.Ring.Basic
+import Mathlib.Tactic.Ring.RingNF
+
+/-!Generated by `gen_lean_outer.py`: the table of `Ê_out` on `[20/37, 3]`. -/
+
+@[expose] public section
+
+namespace Zeta5Irrational
+
+theorem outPiece_0 (x : ℝ) (h1 : ((20) / 37 : ℝ) ≤ x) (h2 : x < ((40) / 43 : ℝ)) :
+    Eout x = ((37) / 20 : ℝ) * x + ((-1) : ℝ) := by
+  rw [Eout_expand (q := 0) (q' := 0) (μ := 1) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (0 : ℝ) := by linarith
+  have hf1 : x - (0 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (0 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (0 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (0 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (0 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (0 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (0 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (0 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (0 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (0 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have em : max 0 (13 / 10 * x - 2) = ((0 : ℝ) * x + (0 : ℝ)) := by
+    simp only [max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22, em]
+  simp only [bGO, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, GO]
+  norm_num
+  ring_nf
+
+theorem outPiece_1 (x : ℝ) (h1 : ((40) / 43 : ℝ) ≤ x) (h2 : x < (1 : ℝ)) :
+    Eout x = ((37) / 20 : ℝ) * x + ((-1) : ℝ) := by
+  rw [Eout_expand (q := 0) (q' := 0) (μ := 1) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (0 : ℝ) := by linarith
+  have hf1 : x - (0 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (0 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (0 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (0 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (0 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (0 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (0 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (0 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-1) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (0 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 =
+      (((43) / 40 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (0 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have em : max 0 (13 / 10 * x - 2) = ((0 : ℝ) * x + (0 : ℝ)) := by
+    simp only [max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22, em]
+  simp only [bGO, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, GO]
+  norm_num
+  ring_nf
+
+theorem outPiece_2 (x : ℝ) (h1 : (1 : ℝ) ≤ x) (h2 : x < ((40) / 37 : ℝ)) :
+    Eout x = (1 : ℝ) * x + ((-2) : ℝ) := by
+  rw [Eout_expand (q := 1) (q' := 0) (μ := 1) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (1 : ℝ) := by linarith
+  have hf1 : x - (1 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (1 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (1 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (1 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (1 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (1 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 = ((1 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (1 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (1 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (1 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (1 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have em : max 0 (13 / 10 * x - 2) = ((0 : ℝ) * x + (0 : ℝ)) := by
+    simp only [max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22, em]
+  simp only [bGO, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, GO, Finset.sum_range_succ, Finset.sum_range_zero]
+  norm_num
+  ring_nf
+
+theorem outPiece_3 (x : ℝ) (h1 : ((40) / 37 : ℝ) ≤ x) (h2 : x < ((3) / 2 : ℝ)) :
+    Eout x = ((42) / 5 : ℝ) * x + ((-10) : ℝ) := by
+  rw [Eout_expand (q := 1) (q' := 0) (μ := 2) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (1 : ℝ) := by linarith
+  have hf1 : x - (1 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (1 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (1 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (1 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (1 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-1) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (1 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (1 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (1 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (1 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (1 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have em : max 0 (13 / 10 * x - 2) = ((0 : ℝ) * x + (0 : ℝ)) := by
+    simp only [max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22, em]
+  simp only [bGO, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, GO, Finset.sum_range_succ, Finset.sum_range_zero]
+  norm_num
+  ring_nf
+
+theorem outPiece_4 (x : ℝ) (h1 : ((3) / 2 : ℝ) ≤ x) (h2 : x < ((20) / 13 : ℝ)) :
+    Eout x = ((42) / 5 : ℝ) * x + ((-10) : ℝ) := by
+  rw [Eout_expand (q := 1) (q' := 0) (μ := 2) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (1 : ℝ) := by linarith
+  have hf1 : x - (1 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (1 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (1 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (1 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (1 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (1 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (1 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (1 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-3) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (1 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (1 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have em : max 0 (13 / 10 * x - 2) = ((0 : ℝ) * x + (0 : ℝ)) := by
+    simp only [max_def]
+    split_ifs <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22, em]
+  simp only [bGO, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, GO, Finset.sum_range_succ, Finset.sum_range_zero]
+  norm_num
+  ring_nf
+
+theorem outPiece_5 (x : ℝ) (h1 : ((20) / 13 : ℝ) ≤ x) (h2 : x < ((60) / 37 : ℝ)) :
+    Eout x = ((97) / 10 : ℝ) * x + ((-12) : ℝ) := by
+  rw [Eout_expand (q := 1) (q' := 0) (μ := 2) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (1 : ℝ) := by linarith
+  have hf1 : x - (1 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (1 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (1 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (1 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (1 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (1 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (1 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (1 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-3) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (1 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (1 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have em : max 0 (13 / 10 * x - 2) = (((13) / 10 : ℝ) * x + ((-2) : ℝ)) := by
+    (try simp only [max_def]); (try split_ifs) <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22, em]
+  simp only [bGO, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, GO, Finset.sum_range_succ, Finset.sum_range_zero]
+  norm_num
+  ring_nf
+
+theorem outPiece_6 (x : ℝ) (h1 : ((60) / 37 : ℝ) ≤ x) (h2 : x < ((80) / 43 : ℝ)) :
+    Eout x = ((231) / 20 : ℝ) * x + ((-15) : ℝ) := by
+  rw [Eout_expand (q := 1) (q' := 0) (μ := 3) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (1 : ℝ) := by linarith
+  have hf1 : x - (1 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (1 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (1 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (1 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (1 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (1 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (1 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (1 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-3) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (1 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (1 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have em : max 0 (13 / 10 * x - 2) = (((13) / 10 : ℝ) * x + ((-2) : ℝ)) := by
+    (try simp only [max_def]); (try split_ifs) <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22, em]
+  simp only [bGO, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, GO, Finset.sum_range_succ, Finset.sum_range_zero]
+  norm_num
+  ring_nf
+
+theorem outPiece_7 (x : ℝ) (h1 : ((80) / 43 : ℝ) ≤ x) (h2 : x < (2 : ℝ)) :
+    Eout x = ((51) / 10 : ℝ) * x + ((-3) : ℝ) := by
+  rw [Eout_expand (q := 1) (q' := 0) (μ := 3) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (1 : ℝ) := by linarith
+  have hf1 : x - (1 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (1 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (1 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (1 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (1 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (1 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (1 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (1 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-3) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (1 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 =
+      (((43) / 40 : ℝ) * x + ((-2) : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (1 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have em : max 0 (13 / 10 * x - 2) = (((13) / 10 : ℝ) * x + ((-2) : ℝ)) := by
+    (try simp only [max_def]); (try split_ifs) <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22, em]
+  simp only [bGO, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, GO, Finset.sum_range_succ, Finset.sum_range_zero]
+  norm_num
+  ring_nf
+
+theorem outPiece_8 (x : ℝ) (h1 : (2 : ℝ) ≤ x) (h2 : x < ((80) / 37 : ℝ)) :
+    Eout x = ((17) / 4 : ℝ) * x + ((-5) : ℝ) := by
+  rw [Eout_expand (q := 2) (q' := 0) (μ := 3) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (2 : ℝ) := by linarith
+  have hf1 : x - (2 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (2 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (2 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (2 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (2 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-2) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (2 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 = ((1 : ℝ) * x + ((-2) : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (2 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (2 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (2 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (2 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have em : max 0 (13 / 10 * x - 2) = (((13) / 10 : ℝ) * x + ((-2) : ℝ)) := by
+    (try simp only [max_def]); (try split_ifs) <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22, em]
+  simp only [bGO, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, GO, Finset.sum_range_succ, Finset.sum_range_zero]
+  norm_num
+  ring_nf
+
+theorem outPiece_9 (x : ℝ) (h1 : ((80) / 37 : ℝ) ≤ x) (h2 : x < ((5) / 2 : ℝ)) :
+    Eout x = ((429) / 40 : ℝ) * x + ((-19) : ℝ) := by
+  rw [Eout_expand (q := 2) (q' := 0) (μ := 4) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (2 : ℝ) := by linarith
+  have hf1 : x - (2 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (2 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (2 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (2 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (2 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((1 : ℝ) * x + ((-2) : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (2 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (2 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (2 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (2 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (2 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have em : max 0 (13 / 10 * x - 2) = (((13) / 10 : ℝ) * x + ((-2) : ℝ)) := by
+    (try simp only [max_def]); (try split_ifs) <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22, em]
+  simp only [bGO, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, GO, Finset.sum_range_succ, Finset.sum_range_zero]
+  norm_num
+  ring_nf
+
+theorem outPiece_10 (x : ℝ) (h1 : ((5) / 2 : ℝ) ≤ x) (h2 : x < ((100) / 37 : ℝ)) :
+    Eout x = ((429) / 40 : ℝ) * x + ((-19) : ℝ) := by
+  rw [Eout_expand (q := 2) (q' := 0) (μ := 4) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (2 : ℝ) := by linarith
+  have hf1 : x - (2 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (2 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (2 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (2 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (2 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (2 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (2 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (2 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-5) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (2 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (2 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have em : max 0 (13 / 10 * x - 2) = (((13) / 10 : ℝ) * x + ((-2) : ℝ)) := by
+    (try simp only [max_def]); (try split_ifs) <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22, em]
+  simp only [bGO, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, GO, Finset.sum_range_succ, Finset.sum_range_zero]
+  norm_num
+  ring_nf
+
+theorem outPiece_11 (x : ℝ) (h1 : ((100) / 37 : ℝ) ≤ x) (h2 : x < ((120) / 43 : ℝ)) :
+    Eout x = ((503) / 40 : ℝ) * x + ((-24) : ℝ) := by
+  rw [Eout_expand (q := 2) (q' := 0) (μ := 5) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (2 : ℝ) := by linarith
+  have hf1 : x - (2 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (2 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (2 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (2 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (2 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (2 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (2 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (2 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-5) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (2 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (2 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have em : max 0 (13 / 10 * x - 2) = (((13) / 10 : ℝ) * x + ((-2) : ℝ)) := by
+    (try simp only [max_def]); (try split_ifs) <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22, em]
+  simp only [bGO, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, GO, Finset.sum_range_succ, Finset.sum_range_zero]
+  norm_num
+  ring_nf
+
+theorem outPiece_12 (x : ℝ) (h1 : ((120) / 43 : ℝ) ≤ x) (h2 : x < (3 : ℝ)) :
+    Eout x = ((36) / 5 : ℝ) * x + ((-9) : ℝ) := by
+  rw [Eout_expand (q := 2) (q' := 0) (μ := 5) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by norm_num; linarith) (by norm_num; linarith)
+      (by norm_num; linarith) (by linarith)]
+  have hf0 : (0 : ℝ) ≤ x - (2 : ℝ) := by linarith
+  have hf1 : x - (2 : ℝ) < 1 := by linarith
+  have hg0 : (0 : ℝ) ≤ 3 / 40 * x - (0 : ℝ) := by linarith
+  have hg1 : 3 / 40 * x - (0 : ℝ) < 1 := by linarith
+  have e00 :
+    ccount (x - (2 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc00]
+    ring
+  have e01 :
+    ccount (x - (2 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc01 hg0 hg1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e02 :
+    ccount (x - (2 : ℝ)) (3 / 40 * x - (0 : ℝ)) 0 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc02 hg1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e10 :
+    ccount (x - (2 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 0 = ((0 : ℝ) * x + ((1) / 2 : ℝ)) := by
+    rw [cc10 hf0 hf1]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e11 :
+    ccount (x - (2 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 1 =
+      (((3) / 40 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc11 hf0 hf1 hg0]
+    simp only [min_def]
+    split_ifs <;> linarith
+  have e12 :
+    ccount (x - (2 : ℝ)) (3 / 40 * x - (0 : ℝ)) 1 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc12 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e20 :
+    ccount (x - (2 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 0 =
+      ((1 : ℝ) * x + ((-5) / 2 : ℝ)) := by
+    rw [cc20 hf1]
+    simp only [max_def]
+    split_ifs <;> linarith
+  have e21 :
+    ccount (x - (2 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 1 =
+      (((43) / 40 : ℝ) * x + ((-3) : ℝ)) := by
+    rw [cc21 hf1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have e22 :
+    ccount (x - (2 : ℝ)) (3 / 40 * x - (0 : ℝ)) 2 2 = ((0 : ℝ) * x + (0 : ℝ)) := by
+    rw [cc22 hg1]
+    simp only [min_def, max_def]
+    split_ifs <;> linarith
+  have em : max 0 (13 / 10 * x - 2) = (((13) / 10 : ℝ) * x + ((-2) : ℝ)) := by
+    (try simp only [max_def]); (try split_ifs) <;> linarith
+  push_cast
+  simp only [Fin.sum_univ_three]
+  rw [e00, e01, e02, e10, e11, e12, e20, e21, e22, em]
+  simp only [bGO, bcoef, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons, GO, Finset.sum_range_succ, Finset.sum_range_zero]
+  norm_num
+  ring_nf
+
+/-- The 14 rational endpoints of the 13 outer-range linear pieces. -/
+def tOutL : List ℚ :=
+  [(20 / 37), (40 / 43), (1), (40 / 37), (3 / 2), (20 / 13), (60 / 37), (80 / 43), (2), (80 / 37),
+    (5 / 2), (100 / 37), (120 / 43), (3)]
+
+/-- Rational slopes of the 13 outer-range linear pieces. -/
+def aOutL : List ℚ :=
+  [(37 / 20), (37 / 20), (1), (42 / 5), (42 / 5), (97 / 10), (231 / 20), (51 / 10), (17 / 4),
+    (429 / 40), (429 / 40), (503 / 40), (36 / 5)]
+
+/-- Rational intercepts of the 13 outer-range linear pieces. -/
+def bOutL : List ℚ :=
+  [(-1), (-1), (-2), (-10), (-10), (-12), (-15), (-3), (-5), (-19), (-19), (-24), (-9)]
+
+/-- The `i`th outer-range endpoint as a real number; zero outside the table. -/
+noncomputable def tOut (i : ℕ) : ℝ :=
+  ((tOutL.getD i 0 : ℚ) : ℝ)
+
+/-- The affine bound on outer piece `i`, with coefficients from `aOutL` and `bOutL`. -/
+noncomputable def gOut (i : ℕ) (x : ℝ) : ℝ :=
+  ((aOutL.getD i 0 : ℚ) : ℝ) * x + ((bOutL.getD i 0 : ℚ) : ℝ)
+
+theorem tOut_mono : ∀ i < 13, tOut i < tOut (i + 1) := by
+  intro i hi
+  interval_cases i <;> simp only [tOut, tOutL, List.getD_cons_succ, List.getD_cons_zero] <;>
+    norm_num
+
+theorem outer_table : ∀ i < 13, ∀ x ∈ Set.Ico (tOut i) (tOut (i + 1)), Eout x = gOut i x := by
+  intro i hi x hx
+  interval_cases i
+  · simp only [tOut, tOutL, gOut, aOutL, bOutL, List.getD_cons_succ, List.getD_cons_zero,
+      Set.mem_Ico] at hx ⊢
+    push_cast at hx ⊢
+    have := outPiece_0 x (by linarith [hx.1]) (by linarith [hx.2])
+    linarith
+  · simp only [tOut, tOutL, gOut, aOutL, bOutL, List.getD_cons_succ, List.getD_cons_zero,
+      Set.mem_Ico] at hx ⊢
+    push_cast at hx ⊢
+    have := outPiece_1 x (by linarith [hx.1]) (by linarith [hx.2])
+    linarith
+  · simp only [tOut, tOutL, gOut, aOutL, bOutL, List.getD_cons_succ, List.getD_cons_zero,
+      Set.mem_Ico] at hx ⊢
+    push_cast at hx ⊢
+    have := outPiece_2 x (by linarith [hx.1]) (by linarith [hx.2])
+    linarith
+  · simp only [tOut, tOutL, gOut, aOutL, bOutL, List.getD_cons_succ, List.getD_cons_zero,
+      Set.mem_Ico] at hx ⊢
+    push_cast at hx ⊢
+    have := outPiece_3 x (by linarith [hx.1]) (by linarith [hx.2])
+    linarith
+  · simp only [tOut, tOutL, gOut, aOutL, bOutL, List.getD_cons_succ, List.getD_cons_zero,
+      Set.mem_Ico] at hx ⊢
+    push_cast at hx ⊢
+    have := outPiece_4 x (by linarith [hx.1]) (by linarith [hx.2])
+    linarith
+  · simp only [tOut, tOutL, gOut, aOutL, bOutL, List.getD_cons_succ, List.getD_cons_zero,
+      Set.mem_Ico] at hx ⊢
+    push_cast at hx ⊢
+    have := outPiece_5 x (by linarith [hx.1]) (by linarith [hx.2])
+    linarith
+  · simp only [tOut, tOutL, gOut, aOutL, bOutL, List.getD_cons_succ, List.getD_cons_zero,
+      Set.mem_Ico] at hx ⊢
+    push_cast at hx ⊢
+    have := outPiece_6 x (by linarith [hx.1]) (by linarith [hx.2])
+    linarith
+  · simp only [tOut, tOutL, gOut, aOutL, bOutL, List.getD_cons_succ, List.getD_cons_zero,
+      Set.mem_Ico] at hx ⊢
+    push_cast at hx ⊢
+    have := outPiece_7 x (by linarith [hx.1]) (by linarith [hx.2])
+    linarith
+  · simp only [tOut, tOutL, gOut, aOutL, bOutL, List.getD_cons_succ, List.getD_cons_zero,
+      Set.mem_Ico] at hx ⊢
+    push_cast at hx ⊢
+    have := outPiece_8 x (by linarith [hx.1]) (by linarith [hx.2])
+    linarith
+  · simp only [tOut, tOutL, gOut, aOutL, bOutL, List.getD_cons_succ, List.getD_cons_zero,
+      Set.mem_Ico] at hx ⊢
+    push_cast at hx ⊢
+    have := outPiece_9 x (by linarith [hx.1]) (by linarith [hx.2])
+    linarith
+  · simp only [tOut, tOutL, gOut, aOutL, bOutL, List.getD_cons_succ, List.getD_cons_zero,
+      Set.mem_Ico] at hx ⊢
+    push_cast at hx ⊢
+    have := outPiece_10 x (by linarith [hx.1]) (by linarith [hx.2])
+    linarith
+  · simp only [tOut, tOutL, gOut, aOutL, bOutL, List.getD_cons_succ, List.getD_cons_zero,
+      Set.mem_Ico] at hx ⊢
+    push_cast at hx ⊢
+    have := outPiece_11 x (by linarith [hx.1]) (by linarith [hx.2])
+    linarith
+  · simp only [tOut, tOutL, gOut, aOutL, bOutL, List.getD_cons_succ, List.getD_cons_zero,
+      Set.mem_Ico] at hx ⊢
+    push_cast at hx ⊢
+    have := outPiece_12 x (by linarith [hx.1]) (by linarith [hx.2])
+    linarith
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Growth/PieceIdx.lean b/LeanPool/Zeta5Irrational/Growth/PieceIdx.lean
new file mode 100644
index 0000000000..7356f741b5
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Growth/PieceIdx.lean
@@ -0,0 +1,70 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import Mathlib.Basic.Real.Basic
+import Mathlib.Tactic.Linarith
+
+/-! # Locating a point in a partition
+
+For a strictly increasing grid `t 0 < t 1 < … < t m`, `pieceIdx t m x` is the index of the piece
+`[t i, t (i+1))` containing `x ∈ [t 0, t m)`.
+-/
+
+@[expose] public section
+
+namespace Zeta5Irrational
+
+open Classical in
+/-- The piece containing `x`. -/
+noncomputable def pieceIdx (t : ℕ → ℝ) (m : ℕ) (x : ℝ) : ℕ :=
+  Nat.findGreatest (fun i => t i ≤ x) m
+
+lemma grid_mono {t : ℕ → ℝ} {m : ℕ} (h : ∀ i < m, t i < t (i + 1)) {i j : ℕ} (hij : i ≤ j)
+    (hj : j ≤ m) : t i ≤ t j := by
+  induction j with
+  | zero =>
+    have : i = 0 := by omega
+    rw [this]
+  | succ j ih =>
+    rcases Nat.lt_or_ge i (j + 1) with h1 | h1
+    · exact (ih (by omega) (by omega)).trans (h j (by omega)).le
+    · have : i = j + 1 := by omega
+      rw [this]
+
+/-- `x ∈ [t 0, t m)` lies in the piece `pieceIdx t m x`. -/
+lemma pieceIdx_spec {t : ℕ → ℝ} {m : ℕ} (_h : ∀ i < m, t i < t (i + 1)) {x : ℝ} (h0 : t 0 ≤ x)
+    (hm : x < t m) : pieceIdx t m x < m ∧ t (pieceIdx t m x) ≤ x ∧ x < t (pieceIdx t m x + 1) := by
+  classical
+  unfold pieceIdx
+  set i := Nat.findGreatest (fun i => t i ≤ x) m with hi
+  have hspec : t i ≤ x := Nat.findGreatest_spec (P := fun i => t i ≤ x) (Nat.zero_le m) h0
+  have hle : i ≤ m := Nat.findGreatest_le m
+  have hlt : i < m := by
+    rcases Nat.lt_or_ge i m with h1 | h1
+    · exact h1
+    · have : i = m := by omega
+      rw [this] at hspec; linarith
+  refine ⟨hlt, hspec, ?_⟩
+  by_contra hc
+  rw [not_lt] at hc
+  have :=
+    Nat.findGreatest_is_greatest (P := fun i => t i ≤ x) (n := m) (k := i + 1) (by omega)
+      (by omega)
+  exact this hc
+
+/-- Inside the piece `i` the index is `i`. -/
+lemma pieceIdx_eq {t : ℕ → ℝ} {m : ℕ} (h : ∀ i < m, t i < t (i + 1)) {i : ℕ} (hi : i < m) {x : ℝ}
+    (hx : x ∈ Set.Ico (t i) (t (i + 1))) : pieceIdx t m x = i := by
+  classical
+  unfold pieceIdx
+  rw [Nat.findGreatest_eq_iff]
+  refine ⟨hi.le, fun _ => hx.1, fun j hj1 hj2 => ?_⟩
+  have := grid_mono h (show i + 1 ≤ j by omega) hj2
+  exact not_le.mpr (lt_of_lt_of_le hx.2 this)
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Growth/TableGen.lean b/LeanPool/Zeta5Irrational/Growth/TableGen.lean
new file mode 100644
index 0000000000..27f2753ce0
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Growth/TableGen.lean
@@ -0,0 +1,146 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Growth.TablePrime
+public import LeanPool.Zeta5Irrational.Growth.OuterPrime
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.Positivity.Basic
+import Mathlib.Tactic.Ring.Basic
+
+/-! # Helpers for the piece tables
+
+On a piece where `⌊x⌋`, `⌊αx⌋`, `⌊2λx⌋` are constant, `Ê_in(x, k)` and `Ê_out(x)` are explicit
+combinations of the interval counts `c_{ij}({x}, {αx})`; the counts have closed forms.
+-/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma nat_floor_of_bounds {x : ℝ} {q : ℕ} (h1 : (q : ℝ) ≤ x) (h2 : x < q + 1) : ⌊x⌋₊ = q := by
+  rw [Nat.floor_eq_iff (by linarith)]; exact ⟨h1, h2⟩
+
+lemma fract_of_bounds {x : ℝ} {q : ℕ} (h1 : (q : ℝ) ≤ x) (h2 : x < q + 1) : Int.fract x = x - q :=
+  by
+  rw [Int.fract, show ⌊x⌋ = (q : ℤ) by rw [Int.floor_eq_iff]; push_cast; exact ⟨h1, h2⟩]
+  push_cast; ring
+
+lemma two_Jc_lin {u : ℝ} (μ : ℕ) (h1 : (μ : ℝ) ≤ 2 * u) (h2 : 2 * u < μ + 1) (hu : u ≤ 500) :
+    2 * Jc u = 2 * μ * u - (μ : ℝ) * (μ + 1) / 2 := by
+  have hu0 : 0 ≤ u := by
+    have : (0 : ℝ) ≤ μ := by positivity
+    linarith
+  have e := two_Jc_eq hu0 hu
+  have hfr : Int.fract (2 * u) = 2 * u - μ := fract_of_bounds h1 h2
+  rw [e, hfr]; ring
+
+/-- `ψ` in closed form. -/
+lemma psiR_eq (k β : ℤ) : psiR k β = (lrow k β : ℚ) * (((k - β : ℤ) : ℚ) - lrow k β + 1) / 2 := by
+  unfold psiR
+  rw [← sum_range_tri]
+  refine Finset.sum_congr rfl fun i _ => ?_; push_cast; ring
+
+/-! ### Closed forms of the interval counts -/
+
+section
+
+variable {f g : ℝ} (hf0 : 0 ≤ f) (hf1 : f < 1) (hg0 : 0 ≤ g) (hg1 : g < 1)
+
+lemma cc00 : ccount f g 0 0 = 1 / 2 := by
+  simp only [ccount, cvLo, cvHi, Matrix.cons_val_zero]
+  simp only [max_def, min_def]
+  split_ifs <;> linarith
+
+include hg0 hg1 in
+lemma cc01 : ccount f g 0 1 = min g (1 / 2) := by
+  simp only [ccount, cvLo, cvHi, Matrix.cons_val_zero, Matrix.cons_val_one]
+  simp only [max_def, min_def]
+  split_ifs <;> linarith
+
+include hg1 in
+lemma cc02 : ccount f g 0 2 = max 0 (g - 1 / 2) := by
+  simp only [ccount, cvLo, cvHi, Matrix.cons_val_zero, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons]
+  simp only [max_def, min_def]
+  split_ifs <;> linarith
+
+include hf0 hf1 in
+lemma cc10 : ccount f g 1 0 = min f (1 / 2) := by
+  simp only [ccount, cvLo, cvHi, Matrix.cons_val_zero, Matrix.cons_val_one]
+  simp only [max_def, min_def]
+  split_ifs <;> linarith
+
+include hf1 in
+lemma cc20 : ccount f g 2 0 = max 0 (f - 1 / 2) := by
+  simp only [ccount, cvLo, cvHi, Matrix.cons_val_zero, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons]
+  simp only [max_def, min_def]
+  split_ifs <;> linarith
+
+include hf0 hf1 hg0 in
+lemma cc11 : ccount f g 1 1 = min (min f g) (1 / 2) := by
+  simp only [ccount, cvLo, cvHi, Matrix.cons_val_zero, Matrix.cons_val_one]
+  simp only [max_def, min_def]
+  split_ifs <;> linarith
+
+include hg1 in
+lemma cc12 : ccount f g 1 2 = max 0 (min f (1 / 2) + g - 1) := by
+  simp only [ccount, cvLo, cvHi, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons]
+  simp only [max_def, min_def]
+  split_ifs <;> linarith
+
+include hf1 in
+lemma cc21 : ccount f g 2 1 = max 0 (min g (1 / 2) + f - 1) := by
+  simp only [ccount, cvLo, cvHi, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons]
+  simp only [max_def, min_def]
+  split_ifs <;> linarith
+
+include hg1 in
+lemma cc22 : ccount f g 2 2 = max 0 (min f g - 1 / 2) := by
+  simp only [ccount, cvLo, cvHi, Matrix.cons_val_two,
+    Matrix.head_cons, Matrix.tail_cons]
+  simp only [max_def, min_def]
+  split_ifs <;> linarith
+
+end
+
+/-- `Ê_in` on a piece. -/
+lemma Ein_expand {x : ℝ} (k : ℤ) {q q' μ : ℕ} (hq1 : (q : ℝ) ≤ x) (hq2 : x < q + 1)
+    (hr1 : (q' : ℝ) ≤ 3 / 40 * x) (hr2 : 3 / 40 * x < q' + 1) (hm1 : (μ : ℝ) ≤ 2 * (37 / 40 * x))
+    (hm2 : 2 * (37 / 40 * x) < μ + 1) (hx : x ≤ 400) :
+    Ein x k =
+      -2 * (37 / 40 * x) * q + 12 * (37 / 40 * x) * q' +
+            (2 * μ * (37 / 40 * x) - (μ : ℝ) * (μ + 1) / 2) -
+          37 / 40 * x * k +
+        2 * ∑ i, ∑ j, bIn q q' k i j * ccount (x - q) (3 / 40 * x - q') i j := by
+  unfold Ein
+  rw [nat_floor_of_bounds hq1 hq2, nat_floor_of_bounds hr1 hr2, fract_of_bounds hq1 hq2,
+    fract_of_bounds hr1 hr2, two_Jc_lin μ hm1 hm2 (by linarith)]
+
+/-- `Ê_out` on a piece. -/
+lemma Eout_expand {x : ℝ} {q q' μ : ℕ} (hq1 : (q : ℝ) ≤ x) (hq2 : x < q + 1)
+    (hr1 : (q' : ℝ) ≤ 3 / 40 * x) (hr2 : 3 / 40 * x < q' + 1) (hm1 : (μ : ℝ) ≤ 2 * (37 / 40 * x))
+    (hm2 : 2 * (37 / 40 * x) < μ + 1) (hx : x ≤ 400) :
+    Eout x =
+      -2 * (37 / 40 * x) * q + (2 * μ * (37 / 40 * x) - (μ : ℝ) * (μ + 1) / 2) -
+          2 * ∑ i, ∑ j, bGO q i j * ccount (x - q) (3 / 40 * x - q') i j +
+        max 0 (13 / 10 * x - 2) := by
+  unfold Eout
+  rw [nat_floor_of_bounds hq1 hq2, fract_of_bounds hq1 hq2, fract_of_bounds hr1 hr2,
+    two_Jc_lin μ hm1 hm2 (by linarith)]
+
+theorem toNat_cast_q (a : ℤ) : ((a.toNat : ℕ) : ℚ) = max (a : ℚ) 0 := by
+  have := Int.toNat_eq_max a
+  have h2 : (((a.toNat : ℕ) : ℤ) : ℚ) = ((max a 0 : ℤ) : ℚ) := by rw [this]
+  push_cast at h2; exact h2
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Growth/TableIntegrals.lean b/LeanPool/Zeta5Irrational/Growth/TableIntegrals.lean
new file mode 100644
index 0000000000..7418d13270
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Growth/TableIntegrals.lean
@@ -0,0 +1,101 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+import LeanPool.Zeta5Irrational.Growth.PieceIdx
+import Mathlib.Tactic.ReduceModChar
+public import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
+import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus
+import Mathlib.Analysis.Calculus.Deriv.Inv
+import Mathlib.Analysis.Calculus.Deriv.Pow
+import Mathlib.Algebra.Ring.IsFormallyReal
+public import LeanPool.Zeta5Irrational.Growth.InnerSum
+public import LeanPool.Zeta5Irrational.Growth.OuterTable
+import Mathlib.Tactic.FieldSimp
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.Ring.Basic
+
+/-! # The integrals of the piece tables
+
+`∑_i ∫_{t_i}^{t_{i+1}} (a_i x + b_i)/x³ dx` equals `(5.18)` for the inner table and
+`127751/96000 + 9/640` for the outer table.
+-/
+
+@[expose] public section
+
+open Finset intervalIntegral
+
+namespace Zeta5Irrational
+
+lemma integral_lin_div_cube (A B : ℝ) {a b : ℝ} (ha : 0 < a) (hab : a ≤ b) :
+    ∫ x in a..b, (A * x + B) / x ^ 3 = A * (1 / a - 1 / b) + B / 2 * (1 / a ^ 2 - 1 / b ^ 2) := by
+  have hpos : ∀ x ∈ Set.uIcc a b, 0 < x := by
+    intro x hx;
+    rcases Set.mem_uIcc.mp hx with h | h <;> linarith [h.1]
+  have hb : 0 < b := by linarith
+  rw [integral_eq_sub_of_hasDerivAt (f := fun x => -A / x - B / 2 / x ^ 2)]
+  · field_simp; ring
+  · intro x hx
+    have hx0 := (hpos x hx).ne'
+    have d1 := (hasDerivAt_const x (-A)).div (hasDerivAt_id x) hx0
+    have d2 := (hasDerivAt_const x (B / 2)).div (hasDerivAt_pow 2 x) (pow_ne_zero 2 hx0)
+    have := d1.sub d2
+    refine this.congr_deriv ?_
+    simp only [id]
+    field_simp
+    ring
+  · apply ContinuousOn.intervalIntegrable
+    apply ContinuousOn.div (by fun_prop) (by fun_prop)
+    intro x hx; exact pow_ne_zero 3 (hpos x hx).ne'
+
+lemma tIn_pos (i : ℕ) (hi : i ≤ 125) : 0 < tIn i := by
+  have h0 : tIn 0 = 3 := by simp [tIn, tInL]
+  have := grid_mono tIn_mono (Nat.zero_le i) hi
+  linarith
+
+lemma tOut_pos (i : ℕ) (hi : i ≤ 13) : 0 < tOut i := by
+  have h0 : tOut 0 = 20 / 37 := by simp [tOut, tOutL]
+  have := grid_mono tOut_mono (Nat.zero_le i) hi
+  linarith
+
+private lemma inner_integrals_rational :
+    (range 125).sum (fun i =>
+      aInL.getD i 0 * (1 / tInL.getD i 0 - 1 / tInL.getD (i + 1) 0) +
+        bInL.getD i 0 / 2 * (1 / (tInL.getD i 0) ^ 2 -
+          1 / (tInL.getD (i + 1) 0) ^ 2)) =
+      (322437603634266857629 / 7535670527041937280000 : ℚ) := by decide +kernel
+
+private lemma outer_integrals_rational :
+    (range 13).sum (fun i =>
+      aOutL.getD i 0 * (1 / tOutL.getD i 0 - 1 / tOutL.getD (i + 1) 0) +
+        bOutL.getD i 0 / 2 * (1 / (tOutL.getD i 0) ^ 2 -
+          1 / (tOutL.getD (i + 1) 0) ^ 2)) = (129101 / 96000 : ℚ) := by decide +kernel
+
+theorem inner_integrals :
+    ∑ i ∈ range 125, ∫ x in tIn i..tIn (i + 1), gIn i x / x ^ 3 =
+      322437603634266857629 / 7535670527041937280000 := by
+  rw [Finset.sum_congr rfl fun i hi =>
+      by
+      have hi' := Finset.mem_range.mp hi
+      exact integral_lin_div_cube _ _ (tIn_pos i hi'.le) (tIn_mono i hi').le]
+  simp only [tIn]
+  have certificate := congrArg (fun q : ℚ => (q : ℝ)) inner_integrals_rational
+  push_cast at certificate
+  exact certificate
+
+theorem outer_integrals :
+    ∑ i ∈ range 13, ∫ x in tOut i..tOut (i + 1), gOut i x / x ^ 3 = 129101 / 96000 := by
+  rw [Finset.sum_congr rfl fun i hi =>
+      by
+      have hi' := Finset.mem_range.mp hi
+      exact integral_lin_div_cube _ _ (tOut_pos i hi'.le) (tOut_mono i hi').le]
+  simp only [tOut]
+  have certificate := congrArg (fun q : ℚ => (q : ℝ)) outer_integrals_rational
+  push_cast at certificate
+  exact certificate
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Growth/TablePrime.lean b/LeanPool/Zeta5Irrational/Growth/TablePrime.lean
new file mode 100644
index 0000000000..4d37eb850c
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Growth/TablePrime.lean
@@ -0,0 +1,107 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Growth.TailPrime
+import Mathlib.Tactic.FieldSimp
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.Positivity.Basic
+import Mathlib.Tactic.Ring.Basic
+
+import Mathlib.Algebra.Order.Floor.Semifield
+
+/-! # The inner range in continuous form: `-L_p ≤ p Ê(K/p, k) + C`
+
+`Ê(x, k) = -2u⌊x⌋ + 12u⌊αx⌋ + 2J(u) - uk + 2 ∑_{i,j} b_{ij}(k) c_{ij}({x}, {αx})`,
+with `u = λx`, the coefficients `b_{ij}` of the class function `ψ(k, B + 6a' - a)` in the basis
+`1, A, E`, and the continuous interval counts `c_{ij}`.
+-/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+/-- The class-function coefficients. -/
+noncomputable def bIn (q q' : ℕ) (k : ℤ) (i j : Fin 3) : ℝ :=
+  bcoef
+    (fun a => bcoef (fun b => (psiR k (12 * (q' : ℤ) - 2 * (q : ℤ) - 4 + 6 * (b : ℤ) - a) : ℝ)) j)
+    i
+
+/-- **The continuous inner function.** -/
+noncomputable def Ein (x : ℝ) (k : ℤ) : ℝ :=
+  -2 * (37 / 40 * x) * (⌊x⌋₊ : ℝ) + 12 * (37 / 40 * x) * (⌊3 / 40 * x⌋₊ : ℝ) +
+        2 * Jc (37 / 40 * x) -
+      37 / 40 * x * k +
+    2 * ∑ i, ∑ j, bIn ⌊x⌋₊ ⌊3 / 40 * x⌋₊ k i j * ccount (Int.fract x) (Int.fract (3 / 40 * x)) i j
+
+/-- The additive error of the table bound. -/
+noncomputable def Ctab (n p : ℕ) (k : ℤ) : ℝ :=
+  5 * ∑ i, ∑ j, |bIn (40 * n / p) (3 * n / p) k i j| +
+        ((ktopI n p - kloI n p : ℤ) : ℝ) * (L0I n p : ℝ) +
+      2 * Bmax n p +
+    1
+
+variable {p : ℕ} [hp : Fact p.Prime]
+
+section
+
+variable {n : ℕ} (hn : 33 ≤ n) (hodd : 2 * (p / 2) + 1 = p) (hp7 : 7 ≤ p)
+  (hsmall : ¬p * Mcut ≤ 40 * n) (hin : 3 * p ≤ 40 * n) (hsq : 2 * (40 * n) < p ^ 2)
+  (hdeg : 5 + 2 * (18 * n + 2 * (37 * n)) - 2 * (40 * n) + 1 < p ^ 2)
+
+include hn hodd hp7 hsmall hin hsq hdeg in
+/-- **The inner bound in continuous form.** -/
+theorem table_Lp_le (k : ℤ) (hk1 : kloI n p ≤ k) (hk2 : k ≤ ktopI n p) :
+    -(Lp n p : ℝ) ≤ (p : ℝ) * Ein (40 * n / p) k + Ctab n p k := by
+  have hp0 : 0 < p := by omega
+  have hpR0 : (0 : ℝ) < p := by exact_mod_cast hp0
+  have main :=
+    inner_Lp_le hn hodd hp7 hsmall hin hsq hdeg k hk1
+      hk2
+        -- the floors and fractional parts
+  have hx : (40 * n / p : ℝ) = ((40 * n : ℕ) : ℝ) / (p : ℕ) := by push_cast; ring
+  have hfl : ⌊(40 * n / p : ℝ)⌋₊ = 40 * n / p := by rw [hx, Nat.floor_div_eq_div]
+  have hfl' : ⌊(3 : ℝ) / 40 * (40 * n / p)⌋₊ = 3 * n / p := by
+    rw [show (3 : ℝ) / 40 * (40 * n / p) = ((3 * n : ℕ) : ℝ) / (p : ℕ) by push_cast; ring,
+      Nat.floor_div_eq_div]
+  have hfK : Int.fract ((40 * n / p : ℝ)) = ((40 * n % p : ℕ) : ℝ) / p := by
+    rw [hx];
+    exact Int.fract_div_natCast_eq_div_natCast_mod
+  have hfN : Int.fract ((3 : ℝ) / 40 * (40 * n / p)) = ((3 * n % p : ℕ) : ℝ) / p := by
+    rw [show (3 : ℝ) / 40 * (40 * n / p) = ((3 * n : ℕ) : ℝ) / (p : ℕ) by push_cast; ring]
+    exact
+      Int.fract_div_natCast_eq_div_natCast_mod
+        -- the class sum
+  have hcs := class_sum_approx hodd (Nat.mod_lt (40 * n) hp0) (Nat.mod_lt (3 * n) hp0) (hIn n p k)
+  have hb :
+    ∀ i j,
+      bcoef (fun a => bcoef (fun b => hIn n p k a b) j) i = bIn (40 * n / p) (3 * n / p) k i j :=
+    fun i j => rfl
+  simp only [hb] at hcs
+  rw [abs_le] at hcs
+        -- the layer term
+  have hlay : 2 * layer p 1000 (37 * n : ℕ) = (p : ℝ) * (2 * Jc (37 / 40 * (40 * n / p))) := by
+    rw [layer_eq hp0]
+    have : ((37 * n : ℕ) : ℝ) / p = 37 / 40 * (40 * n / p) := by push_cast; ring
+    rw [this]; ring
+  unfold Ein Ctab
+  rw [hfl, hfl', hfK, hfN]
+  have e1 : ((37 * n : ℕ) : ℝ) = (p : ℝ) * (37 / 40 * (40 * n / p)) := by push_cast; field_simp
+  rw [hlay] at main
+  rw [e1] at main
+  have hE :
+    ((k - kloI n p : ℤ) : ℝ) * (L0I n p : ℝ) ≤ ((ktopI n p - kloI n p : ℤ) : ℝ) * (L0I n p : ℝ) :=
+  by
+    apply mul_le_mul_of_nonneg_right _ (by positivity)
+    exact_mod_cast (show k - kloI n p ≤ ktopI n p - kloI n p by omega)
+  nlinarith [hcs.2, hE]
+
+end
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Growth/TailIntegral.lean b/LeanPool/Zeta5Irrational/Growth/TailIntegral.lean
new file mode 100644
index 0000000000..ee13f5e198
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Growth/TailIntegral.lean
@@ -0,0 +1,389 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+import Mathlib.Tactic.ReduceModChar
+import Mathlib.Tactic.Polynomial.Basic
+public import Mathlib.Analysis.LocallyConvex.AbsConvexOpen
+public import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
+public import Mathlib.Analysis.Calculus.Deriv.Basic
+import Mathlib.Algebra.Ring.IsFormallyReal
+import Mathlib.Tactic.FieldSimp
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+import Mathlib.Tactic.Positivity.Basic
+import Mathlib.Tactic.Ring.Basic
+
+import Mathlib.Analysis.Calculus.Deriv.Add
+import Mathlib.Analysis.Calculus.Deriv.Pow
+import Mathlib.Analysis.Calculus.Deriv.Comp
+import Mathlib.Analysis.Calculus.Deriv.Inv
+import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus
+
+/-! # The tail integral `∫_{20}^{400} (x F(x) + 27/16)/x³ dx`
+
+On the grid `t_i = 20 + i/3` (`i ≤ 1140`) the fractional parts `{x}` and `{3x/40}` are affine on
+each piece. With `P = 74 g(1-g) - λ f(1-f)` and `C = (74/α) G₀(g) - λ G₀(f)`,
+`G₀(v) = v(1-v)(2v-1)/6`, we have `P' = F + λ` and `C' = P - Pbar` on each piece, and `P`, `C` are
+continuous across the breakpoints. Two integrations by parts on each piece and telescoping give
+`∑_i ∫_{t_i}^{t_{i+1}} (x F + 27/16)/x³ ≤ tailBound`.
+-/
+
+@[expose] public section
+
+open Finset intervalIntegral
+
+namespace Zeta5Irrational
+
+/-- The grid. -/
+noncomputable def tT (i : ℕ) : ℝ :=
+  20 + i / 3
+
+/-- Integer part of the left endpoint `tT i` of a tail interval. -/
+def qT (i : ℕ) : ℕ :=
+  (60 + i) / 3
+
+/-- Integer part of `3 / 40 * tT i`, used to fix the second fractional part. -/
+def qT' (i : ℕ) : ℕ :=
+  (60 + i) / 40
+
+/-- `F` on the piece `i`. -/
+noncomputable def FT (i : ℕ) (x : ℝ) : ℝ :=
+  37 / 10 + 37 / 20 * (x - qT i) - 111 / 10 * (3 / 40 * x - qT' i)
+
+/-- The tail majorant on piece `i`, including the additive error `27 / 16`. -/
+noncomputable def gT (i : ℕ) (x : ℝ) : ℝ :=
+  x * FT i x + 27 / 16
+
+/-- The quadratic oscillation contributed by the two fractional parts on tail piece `i`. -/
+noncomputable def PT (i : ℕ) (x : ℝ) : ℝ :=
+  74 * ((3 / 40 * x - qT' i) * (1 - (3 / 40 * x - qT' i))) -
+    37 / 40 * ((x - qT i) * (1 - (x - qT i)))
+
+/-- Polynomial primitive of the centered quadratic `v * (1 - v) - 1 / 6`. -/
+noncomputable def G0 (v : ℝ) : ℝ :=
+  v * (1 - v) * (2 * v - 1) / 6
+
+/-- Primitive for the centered tail oscillation `PT i - Pbar` on piece `i`. -/
+noncomputable def CT (i : ℕ) (x : ℝ) : ℝ :=
+  74 * 40 / 3 * G0 (3 / 40 * x - qT' i) - 37 / 40 * G0 (x - qT i)
+
+/-- `Pbar = (74 - λ)/6`. -/
+noncomputable def Pbar : ℝ :=
+  2923 / 240
+
+/-! ### The pieces -/
+
+lemma tT_succ (i : ℕ) : tT (i + 1) = tT i + 1 / 3 := by unfold tT; push_cast; ring
+
+lemma qT_le (i : ℕ) : (qT i : ℝ) ≤ tT i := by
+  unfold qT tT
+  have h := Nat.div_mul_le_self (60 + i) 3
+  have : ((60 + i) / 3 : ℕ) * (3 : ℝ) ≤ 60 + i := by exact_mod_cast h
+  linarith
+
+lemma tT_succ_le (i : ℕ) : tT (i + 1) ≤ qT i + 1 := by
+  unfold qT tT
+  have h := Nat.lt_div_mul_add (a := 60 + i) (show 0 < 3 by norm_num)
+  have : ((60 + i : ℕ) : ℝ) + 1 ≤ ((60 + i) / 3 : ℕ) * 3 + 3 := by
+    exact_mod_cast (by omega : 60 + i + 1 ≤ (60 + i) / 3 * 3 + 3)
+  push_cast at this ⊢
+  linarith
+
+lemma qT'_le (i : ℕ) : (qT' i : ℝ) ≤ 3 / 40 * tT i := by
+  unfold qT' tT
+  have h := Nat.div_mul_le_self (60 + i) 40
+  have : ((60 + i) / 40 : ℕ) * (40 : ℝ) ≤ 60 + i := by exact_mod_cast h
+  linarith
+
+lemma tT_succ_le' (i : ℕ) : 3 / 40 * tT (i + 1) ≤ qT' i + 1 := by
+  unfold qT' tT
+  have : ((60 + i : ℕ) : ℝ) + 1 ≤ ((60 + i) / 40 : ℕ) * 40 + 40 := by
+    exact_mod_cast (by omega : 60 + i + 1 ≤ (60 + i) / 40 * 40 + 40)
+  push_cast at this ⊢
+  linarith
+
+lemma tT_pos (i : ℕ) : 20 ≤ tT i := by
+  unfold tT; have : (0 : ℝ) ≤ (i : ℝ) / 3 := by positivity
+  linarith
+
+/-- On the closed piece the local coordinates lie in `[0, 1]`. -/
+lemma piece_coords {i : ℕ} {x : ℝ} (hx : x ∈ Set.Icc (tT i) (tT (i + 1))) :
+    0 ≤ x - qT i ∧ x - qT i ≤ 1 ∧ 0 ≤ 3 / 40 * x - qT' i ∧ 3 / 40 * x - qT' i ≤ 1 := by
+  have h1 := qT_le i; have h2 := tT_succ_le i; have h3 := qT'_le i; have h4 := tT_succ_le' i
+  obtain ⟨hx1, hx2⟩ := hx
+  refine ⟨by linarith, by linarith, by linarith, by linarith⟩
+
+/-! ### Derivatives -/
+
+lemma hasDerivAt_PT (i : ℕ) (x : ℝ) : HasDerivAt (PT i) (FT i x + 37 / 40) x := by
+  have :
+    HasDerivAt (PT i)
+      (74 * ((3 / 40) * (1 - (3 / 40 * x - qT' i)) + (3 / 40 * x - qT' i) * (-(3 / 40))) -
+        37 / 40 * (1 * (1 - (x - qT i)) + (x - qT i) * (-1)))
+      x := by
+    unfold PT
+    apply HasDerivAt.sub
+    · apply HasDerivAt.const_mul
+      apply HasDerivAt.mul
+      · simpa using ((hasDerivAt_id x).const_mul (3 / 40)).sub_const (qT' i : ℝ)
+      · simpa using (((hasDerivAt_id x).const_mul (3 / 40)).sub_const (qT' i : ℝ)).const_sub 1
+    · apply HasDerivAt.const_mul
+      apply HasDerivAt.mul
+      · simpa using (hasDerivAt_id x).sub_const (qT i : ℝ)
+      · simpa using ((hasDerivAt_id x).sub_const (qT i : ℝ)).const_sub 1
+  convert this using 1
+  unfold FT; ring
+
+lemma hasDerivAt_G0 (v : ℝ) : HasDerivAt G0 (v * (1 - v) - 1 / 6) v := by
+  have : HasDerivAt G0 ((-6 * v ^ 2 + 6 * v - 1) / 6) v := by
+    have e : G0 = fun v => (-2 * v ^ 3 + 3 * v ^ 2 - v) / 6 := by funext v; unfold G0; ring
+    rw [e]
+    have :=
+      ((((hasDerivAt_pow 3 v).const_mul (-2)).add ((hasDerivAt_pow 2 v).const_mul 3)).sub
+            (hasDerivAt_id v)).div_const
+        6
+    exact this.congr_deriv (by norm_num; ring)
+  exact this.congr_deriv (by ring)
+
+lemma hasDerivAt_CT (i : ℕ) (x : ℝ) : HasDerivAt (CT i) (PT i x - Pbar) x := by
+  have h1 :=
+    (hasDerivAt_G0 (3 / 40 * x - qT' i)).comp x
+      (((hasDerivAt_id x).const_mul (3 / 40)).sub_const (qT' i : ℝ))
+  have h2 := (hasDerivAt_G0 (x - qT i)).comp x ((hasDerivAt_id x).sub_const (qT i : ℝ))
+  have := (h1.const_mul (74 * 40 / 3)).sub (h2.const_mul (37 / 40))
+  have e :
+    CT i =
+      (fun y => 74 * 40 / 3 * (G0 ∘ fun x => 3 / 40 * id x - (qT' i : ℝ)) y) - fun y =>
+        37 / 40 * (G0 ∘ fun x => id x - (qT i : ℝ)) y :=
+    by funext y; simp [CT, Function.comp]
+  rw [e]
+  exact this.congr_deriv (by unfold PT Pbar; ring)
+
+/-! ### Continuity across breakpoints -/
+
+lemma step3 (i : ℕ) :
+    (qT (i + 1) : ℝ) = qT i ∨ ((qT (i + 1) : ℝ) = qT i + 1 ∧ tT (i + 1) = qT (i + 1)) := by
+  unfold qT tT
+  rcases
+    (show
+      (60 + (i + 1)) / 3 = (60 + i) / 3 ∨
+        ((60 + (i + 1)) / 3 = (60 + i) / 3 + 1 ∧ 60 + (i + 1) = 3 * ((60 + (i + 1)) / 3))
+      by omega) with
+    h | ⟨h1, h2⟩
+  · left; rw [h]
+  · right
+    refine ⟨by rw [h1]; push_cast; ring, ?_⟩
+    have : ((60 + (i + 1) : ℕ) : ℝ) = 3 * (((60 + (i + 1)) / 3 : ℕ) : ℝ) := by exact_mod_cast h2
+    push_cast at this ⊢; linarith
+
+lemma step40 (i : ℕ) :
+    (qT' (i + 1) : ℝ) = qT' i ∨
+      ((qT' (i + 1) : ℝ) = qT' i + 1 ∧ 3 / 40 * tT (i + 1) = qT' (i + 1)) := by
+  unfold qT' tT
+  rcases
+    (show
+      (60 + (i + 1)) / 40 = (60 + i) / 40 ∨
+        ((60 + (i + 1)) / 40 = (60 + i) / 40 + 1 ∧ 60 + (i + 1) = 40 * ((60 + (i + 1)) / 40))
+      by omega) with
+    h | ⟨h1, h2⟩
+  · left; rw [h]
+  · right
+    refine ⟨by rw [h1]; push_cast; ring, ?_⟩
+    have : ((60 + (i + 1) : ℕ) : ℝ) = 40 * (((60 + (i + 1)) / 40 : ℕ) : ℝ) := by exact_mod_cast h2
+    push_cast at this ⊢; linarith
+
+lemma PT_cont (i : ℕ) : PT i (tT (i + 1)) = PT (i + 1) (tT (i + 1)) := by
+  unfold PT
+  rcases step3 i with h3 | ⟨h3, e3⟩ <;> rcases step40 i with h4 | ⟨h4, e4⟩
+  · rw [h3, h4]
+  · rw [h3]; rw [h4] at e4 ⊢; rw [e4]; ring
+  · rw [h4]; rw [h3] at e3 ⊢; rw [e3]; ring
+  · rw [h3] at e3 ⊢; rw [h4] at e4 ⊢; rw [e3] at e4 ⊢; rw [e4]; ring
+
+lemma CT_cont (i : ℕ) : CT i (tT (i + 1)) = CT (i + 1) (tT (i + 1)) := by
+  unfold CT G0
+  rcases step3 i with h3 | ⟨h3, e3⟩ <;> rcases step40 i with h4 | ⟨h4, e4⟩
+  · rw [h3, h4]
+  · rw [h3]; rw [h4] at e4 ⊢; rw [e4]; ring
+  · rw [h4]; rw [h3] at e3 ⊢; rw [e3]; ring
+  · rw [h3] at e3 ⊢; rw [h4] at e4 ⊢; rw [e3] at e4 ⊢; rw [e4]; ring
+
+/-! ### The antiderivative on a piece -/
+
+/-- `Φ_i(x) = P/x² + 2C/x³ - Pbar/x² + λ/x - (27/32)/x²`. -/
+noncomputable def PhiT (i : ℕ) (x : ℝ) : ℝ :=
+  PT i x / x ^ 2 + 2 * CT i x / x ^ 3 - Pbar / x ^ 2 + 37 / 40 / x - 27 / 32 / x ^ 2
+
+lemma hasDerivAt_PhiT (i : ℕ) {x : ℝ} (hx : 0 < x) :
+    HasDerivAt (PhiT i) (gT i x / x ^ 3 - 6 * CT i x / x ^ 4) x := by
+  have hx0 : x ≠ 0 := hx.ne'
+  have h2 : HasDerivAt (fun x : ℝ => x ^ 2) (2 * x) x := by simpa using hasDerivAt_pow 2 x
+  have h3 : HasDerivAt (fun x : ℝ => x ^ 3) (3 * x ^ 2) x := by simpa using hasDerivAt_pow 3 x
+  have d1 := (hasDerivAt_PT i x).div h2 (pow_ne_zero 2 hx0)
+  have d2 := ((hasDerivAt_CT i x).const_mul 2).div h3 (pow_ne_zero 3 hx0)
+  have d3 := (hasDerivAt_const x Pbar).div h2 (pow_ne_zero 2 hx0)
+  have d4 := (hasDerivAt_const x (37 / 40 : ℝ)).div (hasDerivAt_id x) hx0
+  have d5 := (hasDerivAt_const x (27 / 32 : ℝ)).div h2 (pow_ne_zero 2 hx0)
+  have := (((d1.add d2).sub d3).add d4).sub d5
+  have e :
+    PhiT i =
+      (((fun y => PT i y / y ^ 2) + fun y => 2 * CT i y / y ^ 3) - fun y => Pbar / y ^ 2) +
+          (fun y => 37 / 40 / id y) -
+        fun y => 27 / 32 / y ^ 2 :=
+    by funext y; simp [PhiT]
+  rw [e]
+  refine this.congr_deriv ?_
+  simp only [id]
+  unfold gT FT PT
+  field_simp
+  ring
+
+lemma continuous_gT (i : ℕ) : Continuous (gT i) := by unfold gT FT; fun_prop
+
+lemma continuous_CT (i : ℕ) : Continuous (CT i) := by unfold CT G0; fun_prop
+
+lemma continuousOn_div_pow {f : ℝ → ℝ} (hf : Continuous f) (k : ℕ) {a b : ℝ} (ha : 0 < a)
+    (hb : 0 < b) : ContinuousOn (fun x => f x / x ^ k) (Set.uIcc a b) := by
+  apply ContinuousOn.div hf.continuousOn (continuous_pow k).continuousOn
+  intro x hx
+  have : 0 < x := by rcases Set.mem_uIcc.mp hx with h | h <;> linarith [h.1]
+  positivity
+
+lemma abs_G0_le {v : ℝ} (h0 : 0 ≤ v) (h1 : v ≤ 1) : |G0 v| ≤ 1 / 60 := by
+  unfold G0
+  set w := v * (1 - v) with hw
+  have hw0 : 0 ≤ w := mul_nonneg h0 (sub_nonneg.mpr h1)
+  have e : (v * (1 - v) * (2 * v - 1)) ^ 2 = w ^ 2 * (1 - 4 * w) := by rw [hw]; ring
+  have hfac : w ^ 2 * (1 - 4 * w) - 1 / 108 = -((w - 1 / 6) ^ 2 * (4 * w + 1 / 3)) := by ring
+  have h2 : (v * (1 - v) * (2 * v - 1)) ^ 2 ≤ 1 / 100 := by
+    rw [e]
+    have : 0 ≤ (w - 1 / 6) ^ 2 * (4 * w + 1 / 3) := mul_nonneg (sq_nonneg _) (by linarith)
+    linarith
+  rw [abs_le]
+  constructor <;> nlinarith [h2]
+
+lemma abs_CT_le {i : ℕ} {x : ℝ} (hx : x ∈ Set.Icc (tT i) (tT (i + 1))) : |CT i x| ≤ 17 := by
+  obtain ⟨h1, h2, h3, h4⟩ := piece_coords hx
+  have a1 := abs_G0_le h3 h4
+  have a2 := abs_G0_le h1 h2
+  unfold CT
+  rw [abs_le] at a1 a2 ⊢
+  constructor <;> nlinarith [a1.1, a1.2, a2.1, a2.2]
+
+/-- **One piece.** -/
+lemma piece_bound (i : ℕ) :
+    ∫ x in tT i..tT (i + 1), gT i x / x ^ 3 ≤
+      PhiT i (tT (i + 1)) - PhiT i (tT i) + 34 * (1 / tT i ^ 3 - 1 / tT (i + 1) ^ 3) := by
+  have ha := tT_pos i
+  have hb := tT_pos (i + 1)
+  have hab : tT i ≤ tT (i + 1) := by rw [tT_succ]; linarith
+  have ha0 : (0 : ℝ) < tT i := by linarith
+  have hb0 : (0 : ℝ) < tT (i + 1) := by linarith
+  have hpos : ∀ x ∈ Set.uIcc (tT i) (tT (i + 1)), 0 < x := by
+    intro x hx;
+    rcases Set.mem_uIcc.mp hx with h | h <;> linarith [h.1]
+  have hint1 :
+    IntervalIntegrable (fun x => gT i x / x ^ 3) MeasureTheory.volume (tT i) (tT (i + 1)) :=
+    (continuousOn_div_pow (continuous_gT i) 3 ha0 hb0).intervalIntegrable
+  have hint2 :
+    IntervalIntegrable (fun x => CT i x / x ^ 4) MeasureTheory.volume (tT i) (tT (i + 1)) :=
+    (continuousOn_div_pow (continuous_CT i) 4 ha0 hb0).intervalIntegrable
+  have hftc :
+    ∫ x in tT i..tT (i + 1), (gT i x / x ^ 3 - 6 * CT i x / x ^ 4) =
+      PhiT i (tT (i + 1)) - PhiT i (tT i) := by
+    apply integral_eq_sub_of_hasDerivAt
+    · intro x hx; exact hasDerivAt_PhiT i (hpos x hx)
+    · have :
+        (fun x => gT i x / x ^ 3 - 6 * CT i x / x ^ 4) = fun x =>
+          gT i x / x ^ 3 - 6 * (CT i x / x ^ 4) :=
+        by funext x; ring
+      rw [this]; exact hint1.sub (hint2.const_mul 6)
+  have hA :
+    IntervalIntegrable (fun x => gT i x / x ^ 3 - 6 * CT i x / x ^ 4) MeasureTheory.volume (tT i)
+      (tT (i + 1)) := by
+    have :
+      (fun x => gT i x / x ^ 3 - 6 * CT i x / x ^ 4) = fun x =>
+        gT i x / x ^ 3 - 6 * (CT i x / x ^ 4) :=
+      by funext x; ring
+    rw [this]; exact hint1.sub (hint2.const_mul 6)
+  have hsplit :
+    ∫ x in tT i..tT (i + 1), gT i x / x ^ 3 =
+      (∫ x in tT i..tT (i + 1), (gT i x / x ^ 3 - 6 * CT i x / x ^ 4)) +
+        6 * ∫ x in tT i..tT (i + 1), CT i x / x ^ 4 := by
+    have e1 :
+      ∫ x in tT i..tT (i + 1), gT i x / x ^ 3 =
+        ∫ x in tT i..tT (i + 1), ((gT i x / x ^ 3 - 6 * CT i x / x ^ 4) + 6 * (CT i x / x ^ 4)) :=
+      by congr 1; funext x; ring
+    rw [e1, intervalIntegral.integral_add hA (hint2.const_mul 6),
+      intervalIntegral.integral_const_mul]
+      -- the bound on `∫ C/x⁴`
+  have hC :
+    ∫ x in tT i..tT (i + 1), CT i x / x ^ 4 ≤ 17 / 3 * (1 / tT i ^ 3 - 1 / tT (i + 1) ^ 3) := by
+    have hmono : ∫ x in tT i..tT (i + 1), CT i x / x ^ 4 ≤ ∫ x in tT i..tT (i + 1), 17 / x ^ 4 := by
+      apply intervalIntegral.integral_mono_on hab hint2
+      · exact (continuousOn_div_pow continuous_const 4 ha0 hb0).intervalIntegrable
+      · intro x hx
+        have hx0 : 0 < x := by linarith [hx.1]
+        have := abs_CT_le hx
+        rw [abs_le] at this
+        exact div_le_div_of_nonneg_right this.2 (by positivity)
+    have hval :
+      ∫ x in tT i..tT (i + 1), 17 / x ^ 4 = 17 / 3 * (1 / tT i ^ 3 - 1 / tT (i + 1) ^ 3) := by
+      rw [integral_eq_sub_of_hasDerivAt (f := fun x => -(17 / 3) / x ^ 3)]
+      · field_simp; ring
+      · intro x hx
+        have hx0 := hpos x hx
+        have :=
+          ((hasDerivAt_const x (-(17 / 3) : ℝ)).div (hasDerivAt_pow 3 x) (pow_ne_zero 3 hx0.ne'))
+        refine this.congr_deriv ?_
+        norm_num
+        field_simp
+      · exact (continuousOn_div_pow continuous_const 4 ha0 hb0).intervalIntegrable
+    linarith
+  rw [hsplit, hftc]
+  linarith
+
+lemma PhiT_cont (i : ℕ) : PhiT i (tT (i + 1)) = PhiT (i + 1) (tT (i + 1)) := by
+  unfold PhiT;
+  rw [PT_cont, CT_cont]
+
+lemma tele_phi (N : ℕ) :
+    ∑ i ∈ range N, (PhiT i (tT (i + 1)) - PhiT i (tT i)) = PhiT N (tT N) - PhiT 0 (tT 0) := by
+  induction N with
+  | zero => simp
+  | succ N ih => rw [Finset.sum_range_succ, ih, PhiT_cont]; ring
+
+lemma tele_cube (N : ℕ) :
+    ∑ i ∈ range N, 34 * (1 / tT i ^ 3 - 1 / tT (i + 1) ^ 3) =
+      34 * (1 / tT 0 ^ 3 - 1 / tT N ^ 3) := by
+  induction N with
+  | zero => simp
+  | succ N ih => rw [Finset.sum_range_succ, ih]; ring
+
+/-- **The tail sum**: `∑_i ∫_{t_i}^{t_{i+1}} g_i/x³ ≤ -6843153/128000000 ≈ -0.05346`. -/
+theorem tail_sum_le :
+    ∑ i ∈ range 1140, ∫ x in tT i..tT (i + 1), gT i x / x ^ 3 ≤ -6843153 / 128000000 := by
+  have h := Finset.sum_le_sum fun i (_ : i ∈ range 1140) => piece_bound i
+  have tele1 := tele_phi 1140
+  have tele2 := tele_cube 1140
+  rw [Finset.sum_add_distrib, tele1, tele2] at h
+  refine h.trans (le_of_eq ?_)
+  have e0 : tT 0 = 20 := by unfold tT; norm_num
+  have e1 : tT 1140 = 400 := by unfold tT; norm_num
+  have q0 : qT 0 = 20 := by decide
+  have q0' : qT' 0 = 1 := by decide
+  have q1 : qT 1140 = 400 := by decide
+  have q1' : qT' 1140 = 30 := by decide
+  rw [e0, e1]
+  unfold PhiT PT CT G0 Pbar
+  rw [q0, q0', q1, q1']
+  norm_num
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Growth/TailPrime.lean b/LeanPool/Zeta5Irrational/Growth/TailPrime.lean
new file mode 100644
index 0000000000..634fa654b4
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Growth/TailPrime.lean
@@ -0,0 +1,573 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+import Mathlib.Tactic.ReduceModChar
+import Mathlib.Tactic.Polynomial.Basic
+import Mathlib.Data.Rat.Star
+import Mathlib.Data.Int.Star
+import Mathlib.Tactic.ENatToNat
+public import LeanPool.Zeta5Irrational.Growth.InnerPrime
+import Mathlib.Tactic.FieldSimp
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.Positivity.Basic
+import Mathlib.Tactic.Ring.Basic
+import Mathlib.Tactic.Ring.RingNF
+
+/-! # The inner tail: `-L_p ≤ p (x F(x) + 27/16) + O(x²)`
+
+For an explicit level `k` (the rounded minimiser of the quadratic majorant) the discrete bound of
+`inner_Lp_le` is at most `p (x F(x) + 27/16) + C(x)`, where
+`F(x) = 4λ + 2λ{x} - 12λ{αx}`, `λ = 37/40`, `α = 3/40`, `x = K/p`.
+-/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+/-! ### Elementary facts -/
+
+lemma sum_range_tri (t : ℚ) (L : ℕ) :
+    ∑ i ∈ range L, ((t - 2 * i) / 2) = (L : ℚ) * (t - L + 1) / 2 := by
+  induction L with
+  | zero => simp
+  | succ L ih => rw [Finset.sum_range_succ, ih]; push_cast; ring
+
+lemma psiR_le (k β : ℤ) : psiR k β ≤ ((k - β + 1 : ℤ) : ℚ) ^ 2 / 8 := by
+  unfold psiR
+  set t := k - β with ht
+  have hsum :
+    ∑ i ∈ range (lrow k β), (((k : ℚ) - 2 * i - β) / 2) =
+      (lrow k β : ℚ) * ((t : ℚ) - lrow k β + 1) / 2 := by
+    rw [← sum_range_tri]
+    refine Finset.sum_congr rfl fun i _ => ?_
+    rw [ht]; push_cast; ring
+  rw [hsum]
+  unfold lrow
+  rcases le_or_gt (t + 1) 0 with h | h
+  · have : ((k - β + 1) / 2).toNat = 0 := by omega
+    rw [this]; simp only [Nat.cast_zero, zero_mul, zero_div]; positivity
+  · rcases Int.emod_two_eq_zero_or_one t with h2 | h2
+    · have e : (((k - β + 1) / 2).toNat : ℚ) = (t : ℚ) / 2 := by
+        have : (((k - β + 1) / 2).toNat : ℤ) * 2 = t := by omega
+        have : ((((k - β + 1) / 2).toNat : ℤ) : ℚ) * 2 = (t : ℚ) := by exact_mod_cast this
+        push_cast at this; linarith
+      rw [e]; push_cast; nlinarith [sq_nonneg ((t : ℚ) + 1)]
+    · have e : (((k - β + 1) / 2).toNat : ℚ) = ((t : ℚ) + 1) / 2 := by
+        have : (((k - β + 1) / 2).toNat : ℤ) * 2 = t + 1 := by omega
+        have : ((((k - β + 1) / 2).toNat : ℤ) : ℚ) * 2 = (t : ℚ) + 1 := by exact_mod_cast this
+        push_cast at this; linarith
+      rw [e]; push_cast; nlinarith [sq_nonneg ((t : ℚ) + 1)]
+
+lemma sum_aCnt {p m v : ℕ} (hm : 2 * m + 1 = p) (hv : v < p) : ∑ c ∈ Icc 1 m, aCnt p v c = v := by
+  unfold aCnt
+  rw [Finset.sum_add_distrib, Finset.sum_ite, Finset.sum_ite, Finset.sum_const_zero,
+    Finset.sum_const_zero, add_zero, add_zero, Finset.sum_const, Finset.sum_const, smul_eq_mul,
+    smul_eq_mul, mul_one, mul_one]
+  have h1 : (Icc 1 m).filter (fun c => c ≤ v) = Icc 1 (min m v) := by
+    ext c;
+    simp only [Finset.mem_filter, Finset.mem_Icc]; omega
+  have h2 : (Icc 1 m).filter (fun c => p - v ≤ c) = Icc (max 1 (p - v)) m := by
+    ext c;
+    simp only [Finset.mem_filter, Finset.mem_Icc]; omega
+  rw [h1, h2, Nat.card_Icc, Nat.card_Icc]
+  omega
+
+lemma aCnt_range {p m v c : ℕ} (hm : 2 * m + 1 = p) (hv : v < p) (_hc1 : 1 ≤ c) (hcm : c ≤ m) :
+    (if m < v then 1 else 0) ≤ aCnt p v c ∧ aCnt p v c ≤ (if m < v then 1 else 0) + 1 := by
+  unfold aCnt; split_ifs <;> omega
+
+/-- **Popoviciu**: values in an interval of length `7` have `∑ φ² ≤ (∑ φ)²/m + 49m/4`. -/
+lemma popoviciu {ι : Type*} (s : Finset ι) (φ : ι → ℝ) (L : ℝ)
+    (hφ : ∀ c ∈ s, L ≤ φ c ∧ φ c ≤ L + 7) (hs : 0 < s.card) :
+    ∑ c ∈ s, φ c ^ 2 ≤ (∑ c ∈ s, φ c) ^ 2 / s.card + 49 * s.card / 4 := by
+  set m : ℝ := (s.card : ℝ) with hmdef
+  have hm0 : 0 < m := by rw [hmdef]; exact_mod_cast hs
+  have h1 : ∑ c ∈ s, (φ c - L) ^ 2 ≤ 7 * ∑ c ∈ s, (φ c - L) := by
+    rw [Finset.mul_sum]
+    refine Finset.sum_le_sum fun c hc => ?_
+    have := hφ c hc
+    nlinarith
+  set U := ∑ c ∈ s, (φ c - L) with hU
+  have hsumφ : ∑ c ∈ s, φ c = U + m * L := by
+    rw [hU, Finset.sum_sub_distrib, Finset.sum_const, nsmul_eq_mul]; ring
+  have hsq : ∑ c ∈ s, φ c ^ 2 = ∑ c ∈ s, (φ c - L) ^ 2 + 2 * L * U + m * L ^ 2 := by
+    rw [hU, Finset.mul_sum, ← Finset.sum_add_distrib]
+    rw [show m * L ^ 2 = ∑ _c ∈ s, L ^ 2 by rw [Finset.sum_const, nsmul_eq_mul]]
+    rw [← Finset.sum_add_distrib]
+    refine Finset.sum_congr rfl fun c _ => ?_; ring
+  rw [hsq, hsumφ]
+  have h2 : 7 * U ≤ U ^ 2 / m + 49 * m / 4 := by
+    rw [div_add' _ _ _ hm0.ne', le_div_iff₀ hm0]
+    nlinarith [sq_nonneg (U - 7 * m / 2)]
+  have e : (U + m * L) ^ 2 / m = U ^ 2 / m + 2 * L * U + m * L ^ 2 := by field_simp; ring
+  rw [e]
+  linarith
+
+/-- The continuous layer function `J(u) = ∑_{j ≤ 1000} (u - j/2)⁺`. -/
+noncomputable def Jc (u : ℝ) : ℝ :=
+  ∑ j ∈ (Icc 1 1000 : Finset ℕ), max 0 (u - (j : ℝ) / 2)
+
+lemma layer_eq {p : ℕ} (hp : 0 < p) (h : ℕ) : layer p 1000 h = (p : ℝ) * Jc ((h : ℝ) / p) := by
+  unfold layer Jc
+  rw [Finset.mul_sum]
+  refine Finset.sum_congr rfl fun j _ => ?_
+  have hpR : (0 : ℝ) < p := by exact_mod_cast hp
+  rw [mul_max_of_nonneg _ _ hpR.le, mul_zero, mul_sub, mul_div_cancel₀ _ hpR.ne']
+  congr 1; ring
+
+/-- `2 J(u) = 2u² - u + η(1 - η)/2` with `η = {2u}`, for `0 ≤ u ≤ 500`. -/
+lemma two_Jc_eq {u : ℝ} (hu0 : 0 ≤ u) (hu : u ≤ 500) :
+    2 * Jc u = 2 * u ^ 2 - u + Int.fract (2 * u) * (1 - Int.fract (2 * u)) / 2 := by
+  set μ := ⌊2 * u⌋₊ with hμ
+  have hμ1 : (μ : ℝ) ≤ 2 * u := Nat.floor_le (by linarith)
+  have hμ2 : 2 * u < μ + 1 := Nat.lt_floor_add_one _
+  have hμ1000 : μ ≤ 1000 := by
+    have : (μ : ℝ) ≤ 1000 := by linarith
+    exact_mod_cast this
+  have hfr : Int.fract (2 * u) = 2 * u - μ := by
+    rw [Int.fract,
+      show ⌊2 * u⌋ = (μ : ℤ) by
+        rw [hμ, ← Int.floor_toNat, Int.toNat_of_nonneg (Int.floor_nonneg.mpr (by linarith))]]
+    push_cast; ring
+  have hsplit : Jc u = ∑ j ∈ (Icc 1 μ : Finset ℕ), (u - (j : ℝ) / 2) := by
+    unfold Jc
+    rw [←
+      Finset.sum_subset (Finset.Icc_subset_Icc_right hμ1000)
+        (fun j hj hjn => by
+          simp only [Finset.mem_Icc, not_and, not_le] at hj hjn
+          have := hjn hj.1
+          have : (μ : ℝ) + 1 ≤ j := by exact_mod_cast this
+          exact max_eq_left (by linarith))]
+    refine Finset.sum_congr rfl fun j hj => ?_
+    have := (Finset.mem_Icc.mp hj).2
+    have : (j : ℝ) ≤ μ := by exact_mod_cast this
+    exact max_eq_right (by linarith)
+  have hclosed : ∑ j ∈ (Icc 1 μ : Finset ℕ), (u - (j : ℝ) / 2) = μ * u - (μ : ℝ) * (μ + 1) / 4 := by
+    induction μ with
+    | zero => simp
+    | succ k ih => rw [Finset.sum_Icc_succ_top (by omega), ih]; push_cast; ring
+  rw [hsplit, hclosed, hfr]
+  ring
+
+lemma phi_bounds {p m v v' c : ℕ} (hm : 2 * m + 1 = p) (hv : v < p) (hv' : v' < p) (hc1 : 1 ≤ c)
+    (hcm : c ≤ m) :
+    6 * (if m < v' then (1 : ℝ) else 0) - (if m < v then (1 : ℝ) else 0) - 1 ≤
+        6 * (aCnt p v' c : ℝ) - aCnt p v c ∧
+      6 * (aCnt p v' c : ℝ) - aCnt p v c ≤
+        6 * (if m < v' then (1 : ℝ) else 0) - (if m < v then (1 : ℝ) else 0) - 1 + 7 := by
+  have r1 := aCnt_range hm hv hc1 hcm
+  have r2 := aCnt_range hm hv' hc1 hcm
+  split_ifs at r1 r2 ⊢ <;>
+    · have e1 : ((aCnt p v c : ℕ) : ℝ) ≤ 2 := by exact_mod_cast (show aCnt p v c ≤ 2 by omega)
+      have e2 : ((aCnt p v' c : ℕ) : ℝ) ≤ 2 := by exact_mod_cast (show aCnt p v' c ≤ 2 by omega)
+      first
+      | ( have f1 : (1 : ℝ) ≤ aCnt p v c := by exact_mod_cast (show 1 ≤ aCnt p v c by omega)
+          have f2 : (1 : ℝ) ≤ aCnt p v' c := by exact_mod_cast (show 1 ≤ aCnt p v' c by omega)
+          constructor <;> linarith)
+      | ( have f1 : (1 : ℝ) ≤ aCnt p v c := by exact_mod_cast (show 1 ≤ aCnt p v c by omega)
+          have f2 : (aCnt p v' c : ℝ) ≤ 1 := by exact_mod_cast (show aCnt p v' c ≤ 1 by omega)
+          have f3 : (0 : ℝ) ≤ aCnt p v' c := by positivity
+          constructor <;> linarith)
+      | ( have f1 : (aCnt p v c : ℝ) ≤ 1 := by exact_mod_cast (show aCnt p v c ≤ 1 by omega)
+          have f2 : (1 : ℝ) ≤ aCnt p v' c := by exact_mod_cast (show 1 ≤ aCnt p v' c by omega)
+          have f3 : (0 : ℝ) ≤ aCnt p v c := by positivity
+          constructor <;> linarith)
+      | ( have f1 : (aCnt p v c : ℝ) ≤ 1 := by exact_mod_cast (show aCnt p v c ≤ 1 by omega)
+          have f2 : (aCnt p v' c : ℝ) ≤ 1 := by exact_mod_cast (show aCnt p v' c ≤ 1 by omega)
+          have f3 : (0 : ℝ) ≤ aCnt p v c := by positivity
+          have f4 : (0 : ℝ) ≤ aCnt p v' c := by positivity
+          constructor <;> linarith)
+
+lemma hIn_le_sq (n p : ℕ) (k τ : ℤ)
+    (hk : k = τ + (12 * ((3 * n / p : ℕ) : ℤ) - 2 * ((40 * n / p : ℕ) : ℤ) - 4) - 1) (a a' : ℕ) :
+    hIn n p k a a' ≤ ((τ : ℝ) - (6 * (a' : ℝ) - a)) ^ 2 / 8 := by
+  unfold hIn
+  have h1 :=
+    psiR_le k (12 * ((3 * n / p : ℕ) : ℤ) - 2 * ((40 * n / p : ℕ) : ℤ) - 4 + 6 * (a' : ℤ) - a)
+  have hz :
+    k - (12 * ((3 * n / p : ℕ) : ℤ) - 2 * ((40 * n / p : ℕ) : ℤ) - 4 + 6 * (a' : ℤ) - a) + 1 =
+      τ - (6 * (a' : ℤ) - a) :=
+    by rw [hk]; ring
+  rw [hz] at h1
+  have h2 := (Rat.cast_le (K := ℝ)).mpr h1
+  rw [Rat.cast_div, Rat.cast_pow, Rat.cast_intCast] at h2
+  refine h2.trans (le_of_eq ?_)
+  push_cast; ring
+
+lemma layer_le {p : ℕ} (hp0 : 0 < p) (H : ℕ) (hu500 : (H : ℝ) / p ≤ 500) :
+    2 * layer p 1000 H ≤ 2 * (H : ℝ) ^ 2 / p - H + p / 8 := by
+  have hpR0 : (0 : ℝ) < p := by exact_mod_cast hp0
+  have hu0 : (0 : ℝ) ≤ (H : ℝ) / p := by positivity
+  rw [layer_eq hp0]
+  have e := two_Jc_eq hu0 hu500
+  set η := Int.fract (2 * ((H : ℝ) / p)) with hη
+  have hη0 : 0 ≤ η := Int.fract_nonneg _
+  have hη1 : η < 1 := Int.fract_lt_one _
+  have hηb : η * (1 - η) ≤ 1 / 4 := by nlinarith [sq_nonneg (η - 1 / 2)]
+  have : 2 * ((p : ℝ) * Jc ((H : ℝ) / p)) = p * (2 * Jc ((H : ℝ) / p)) := by ring
+  rw [this, e]
+  have e2 : (p : ℝ) * (2 * ((H : ℝ) / p) ^ 2 - (H : ℝ) / p) = 2 * (H : ℝ) ^ 2 / p - H := by
+    field_simp
+  have e3 :
+    (p : ℝ) * (2 * ((H : ℝ) / p) ^ 2 - (H : ℝ) / p + η * (1 - η) / 2) =
+      2 * (H : ℝ) ^ 2 / p - H + p * (η * (1 - η)) / 2 :=
+    by rw [← e2]; ring
+  rw [e3]
+  have := mul_le_mul_of_nonneg_left hηb hpR0.le
+  linarith
+
+lemma quad_le {hR SR m τ : ℝ} (hm0 : 0 < m) (h1 : τ ≤ (SR + 2 * hR) / m + 1 / 2)
+    (h2 : (SR + 2 * hR) / m + 1 / 2 < τ + 1) :
+    -hR * τ + (m * τ ^ 2 - 2 * τ * SR + SR ^ 2 / m) / 4 ≤ -hR * SR / m - hR ^ 2 / m + m / 16 := by
+  set τs := (SR + 2 * hR) / m with hτs
+  have hd : (τ - τs) ^ 2 ≤ 1 / 4 := by nlinarith
+  have e :
+    -hR * τ + (m * τ ^ 2 - 2 * τ * SR + SR ^ 2 / m) / 4 =
+      -hR * SR / m - hR ^ 2 / m + m * (τ - τs) ^ 2 / 4 :=
+    by rw [hτs]; field_simp; ring
+  rw [e]
+  have := mul_le_mul_of_nonneg_left hd hm0.le
+  linarith
+
+lemma diff_le {hR SR p : ℝ} (hp : 1 < p) (hh0 : 0 ≤ hR) (hhS : 0 ≤ hR + SR) :
+    2 * hR ^ 2 / p - hR ^ 2 / ((p - 1) / 2) - hR * SR / ((p - 1) / 2) + 2 * hR * SR / p ≤ 0 := by
+  have hp0 : 0 < p := by linarith
+  have hp1 : 0 < p - 1 := by linarith
+  have e :
+    2 * hR ^ 2 / p - hR ^ 2 / ((p - 1) / 2) - hR * SR / ((p - 1) / 2) + 2 * hR * SR / p =
+      -(2 * (hR * (hR + SR))) / (p * (p - 1)) :=
+    by field_simp; ring
+  rw [e]
+  apply div_nonpos_of_nonpos_of_nonneg
+  · have := mul_nonneg hh0 hhS; linarith
+  · positivity
+
+/-- `F(x) = 4λ + 2λ{x} - 12λ{αx}`. -/
+noncomputable def Ftail (x : ℝ) : ℝ :=
+  4 * (37 / 40) + 2 * (37 / 40) * Int.fract x - 12 * (37 / 40) * Int.fract (3 / 40 * x)
+
+/-- The additive error of the tail bound. -/
+noncomputable def Ctail (n p : ℕ) : ℝ :=
+  4 * (37 * n : ℝ) / p + ((ktopI n p - kloI n p : ℤ) : ℝ) * (L0I n p : ℝ) + 2 * Bmax n p + 1
+
+/-- Sum the quadratic class bound using the seven-unit range of the counts. -/
+lemma tail_class_sum_le (n p v v' : ℕ) (k τ : ℤ)
+    (hodd : 2 * (p / 2) + 1 = p) (hvp : v < p) (hv'p : v' < p)
+    (hm : 0 < p / 2)
+    (hk : k = τ + (12 * ((3 * n / p : ℕ) : ℤ) - 2 * ((40 * n / p : ℕ) : ℤ) - 4) - 1) :
+    2 * ∑ c ∈ Icc 1 (p / 2), hIn n p k (aCnt p v c) (aCnt p v' c) ≤
+      ((p / 2 : ℕ) * (τ : ℝ) ^ 2 - 2 * τ * (6 * (v' : ℝ) - v) +
+        (6 * (v' : ℝ) - v) ^ 2 / (p / 2 : ℕ) + 49 * (p / 2 : ℕ) / 4) / 4 := by
+  let m := p / 2
+  let SR : ℝ := 6 * (v' : ℝ) - v
+  set φ : ℕ → ℝ := fun c => 6 * (aCnt p v' c : ℝ) - aCnt p v c with hφ
+  have hIn_le : ∀ c, hIn n p k (aCnt p v c) (aCnt p v' c) ≤ ((τ : ℝ) - φ c) ^ 2 / 8 := fun c =>
+    hIn_le_sq n p k τ hk _ _
+  have hcard : ((Icc 1 m).card : ℝ) = m := by rw [Nat.card_Icc]; simp
+  have hSphi : ∑ c ∈ Icc 1 m, φ c = SR := by
+    rw [hφ, Finset.sum_sub_distrib, ← Finset.mul_sum]
+    have e1 := sum_aCnt hodd hvp
+    have e2 := sum_aCnt hodd hv'p
+    have e1' : ∑ c ∈ Icc 1 m, (aCnt p v c : ℝ) = v := by exact_mod_cast e1
+    have e2' : ∑ c ∈ Icc 1 m, (aCnt p v' c : ℝ) = v' := by exact_mod_cast e2
+    rw [e1', e2']
+  have hpop : ∑ c ∈ Icc 1 m, φ c ^ 2 ≤ SR ^ 2 / m + 49 * m / 4 := by
+    have hm1 : 0 < (Icc 1 m).card := by rw [Nat.card_Icc]; omega
+    have :=
+      popoviciu (Icc 1 m) φ
+        (6 * (if m < v' then (1 : ℝ) else 0) - (if m < v then (1 : ℝ) else 0) - 1)
+        (fun c hc => phi_bounds hodd hvp hv'p (Finset.mem_Icc.mp hc).1 (Finset.mem_Icc.mp hc).2)
+        hm1
+    rw [hcard, hSphi] at this
+    exact this
+  have hsq :
+    ∑ c ∈ Icc 1 m, ((τ : ℝ) - φ c) ^ 2 =
+      (m : ℝ) * (τ : ℝ) ^ 2 - 2 * τ * SR + ∑ c ∈ Icc 1 m, φ c ^ 2 := by
+    have :
+      ∑ c ∈ Icc 1 m, ((τ : ℝ) - φ c) ^ 2 = ∑ c ∈ Icc 1 m, ((τ : ℝ) ^ 2 - 2 * τ * φ c + φ c ^ 2) :=
+      by refine Finset.sum_congr rfl fun c _ => ?_; ring
+    rw [this, Finset.sum_add_distrib, Finset.sum_sub_distrib, Finset.sum_const, nsmul_eq_mul, ←
+      Finset.mul_sum, hSphi, hcard]
+  have hclass :
+    2 * ∑ c ∈ Icc 1 m, hIn n p k (aCnt p v c) (aCnt p v' c) ≤
+      ((m : ℝ) * (τ : ℝ) ^ 2 - 2 * τ * SR + SR ^ 2 / m + 49 * m / 4) / 4 := by
+    have := Finset.sum_le_sum fun c (_ : c ∈ Icc 1 m) => hIn_le c
+    rw [← Finset.sum_div, hsq] at this
+    linarith
+  exact hclass
+
+/-- The integral level obtained by rounding the quadratic minimizer. -/
+noncomputable def tailLevel (n p : ℕ) : ℤ :=
+  ⌊(6 * ((3 * n % p : ℕ) : ℝ) - ((40 * n % p : ℕ) : ℝ) +
+      2 * (37 * n : ℝ)) / ((p / 2 : ℕ) : ℝ) + 1 / 2⌋ +
+    (12 * ((3 * n / p : ℕ) : ℤ) - 2 * ((40 * n / p : ℕ) : ℤ) - 4) - 1
+
+/-- The rounded minimizer lies in the admissible interval of integral levels. -/
+lemma tailLevel_mem (n p : ℕ) (hodd : 2 * (p / 2) + 1 = p)
+    (hp7 : 7 ≤ p) (hin : 3 * p ≤ 40 * n) :
+    kloI n p ≤ tailLevel n p ∧ tailLevel n p ≤ ktopI n p := by
+  -- notation
+  have hp0 : 0 < p := by omega
+  have hpR : (7 : ℝ) ≤ p := by exact_mod_cast hp7
+  set m := p / 2 with hmdef
+  have hmR : (m : ℝ) = ((p : ℝ) - 1) / 2 := by
+    have : ((2 * m + 1 : ℕ) : ℝ) = p := by exact_mod_cast hodd
+    push_cast at this; linarith
+  have hm0 : (0 : ℝ) < m := by rw [hmR]; linarith
+  set v := 40 * n % p with hv
+  set v' := 3 * n % p with hv'
+  have hvp : v < p := Nat.mod_lt _ hp0
+  have hv'p : v' < p := Nat.mod_lt _ hp0
+  set q := 40 * n / p with hq
+  set q' := 3 * n / p with hq'
+  have hdivK : (40 * n : ℝ) = q * p + v := by
+    have := Nat.div_add_mod (40 * n) p
+    have h' : ((p * q + v : ℕ) : ℝ) = ((40 * n : ℕ) : ℝ) := by rw [this]
+    push_cast at h'; linarith
+  have hdivN : (3 * n : ℝ) = q' * p + v' := by
+    have := Nat.div_add_mod (3 * n) p
+    have h' : ((p * q' + v' : ℕ) : ℝ) = ((3 * n : ℕ) : ℝ) := by rw [this]
+    push_cast at h'; linarith
+  set hR : ℝ := (37 * n : ℝ) with hhR
+  set SR : ℝ := 6 * (v' : ℝ) - v with hSR
+  set τs : ℝ := (SR + 2 * hR) / m with hτs
+  set τ : ℤ := ⌊τs + 1 / 2⌋ with hτ
+  have hδ1 : (τ : ℝ) ≤ τs + 1 / 2 := Int.floor_le _
+  have hδ2 : τs + 1 / 2 < τ + 1 := Int.lt_floor_add_one _
+  set B : ℤ := 12 * (q' : ℤ) - 2 * (q : ℤ) - 4 with hB
+  set k : ℤ := τ + B - 1 with hk
+    -- basic size facts
+  have hqlt : (40 * n : ℝ) < ((q : ℝ) + 1) * p := by
+    have := Nat.lt_div_mul_add (a := 40 * n) hp0
+    have : ((40 * n : ℕ) : ℝ) < ((q * p + p : ℕ) : ℝ) := by exact_mod_cast this
+    push_cast at this; linarith
+  have hq'le : (q' : ℝ) * p ≤ 3 * n := by
+    rw [hdivN]; have : (0 : ℝ) ≤ v' := by positivity
+    linarith
+  have hnin : (3 : ℝ) * p ≤ 40 * n := by exact_mod_cast hin
+  have hvR : (v : ℝ) < p := by exact_mod_cast hvp
+  have hv'R : (v' : ℝ) < p := by exact_mod_cast hv'p
+  have hv0 : (0 : ℝ) ≤ v := by positivity
+  have hv'0 : (0 : ℝ) ≤ v' := by positivity
+  have hq0 : (0 : ℝ) ≤ q := by positivity
+  have hq'0 : (0 : ℝ) ≤ q' := by positivity
+  have hτsm : τs * m = SR + 2 * hR := by
+    rw [hτs];
+    field_simp
+      -- `τs` bounds
+  have hτs_lo : 8 ≤ τs := by
+    have : 8 * (m : ℝ) ≤ SR + 2 * hR := by rw [hmR, hSR, hhR]; linarith
+    rw [hτs, le_div_iff₀ hm0]; linarith
+  have h40 : 40 * (q' : ℝ) ≤ 3 * q + 3 := by
+    have e1 : 40 * ((q' : ℝ) * p) ≤ 120 * n := by linarith
+    have e2 : 120 * (n : ℝ) < 3 * ((q : ℝ) * p) + 3 * p := by linarith
+    have : (40 * (q' : ℝ) - 3 * q - 3) * p < 0 := by ring_nf; ring_nf at e1 e2; linarith
+    by_contra hc
+    rw [not_le] at hc
+    have :=
+      mul_nonneg (show (0 : ℝ) ≤ 40 * q' - 3 * q - 3 by linarith) (show (0 : ℝ) ≤ p by linarith)
+    linarith
+  have hτs_hi : τs ≤ 10 * (q : ℝ) + 24 - 12 * q' := by
+    have hp1 : (0 : ℝ) ≤ (p : ℝ) - 7 := by linarith
+    have f1 := mul_nonneg hq0 hp1
+    have f2 := mul_le_mul_of_nonneg_right h40 (show (0 : ℝ) ≤ (p : ℝ) - 1 by linarith)
+    have : (SR + 2 * hR) ≤ (10 * (q : ℝ) + 24 - 12 * q') * m := by
+      rw [hmR, hSR, hhR]; ring_nf;
+      ring_nf at f1 f2 hqlt; nlinarith
+    rw [hτs, div_le_iff₀ hm0];
+    linarith
+      -- `k` is admissible
+  have hk1 : kloI n p ≤ k := by
+    have : (1 : ℝ) ≤ τ := by
+      have : (8 : ℝ) + 1 / 2 - 1 < τ := by linarith
+      have h1 : (8 : ℤ) ≤ τ := by
+        have : (7 : ℝ) < τ := by linarith
+        exact_mod_cast (show (7 : ℤ) < τ by exact_mod_cast this)
+      exact_mod_cast (show (1 : ℤ) ≤ τ by omega)
+    have hτ1 : (1 : ℤ) ≤ τ := by exact_mod_cast this
+    have hklo : kloI n p = -2 * (q : ℤ) - 6 := rfl
+    rw [hklo, hk, hB]
+    have hq0Z : (0 : ℤ) ≤ q := by positivity
+    have hq'0Z : (0 : ℤ) ≤ q' := by positivity
+    generalize (q : ℤ) = Q at *
+    generalize (q' : ℤ) = Q' at *
+    omega
+  have hk2 : k ≤ ktopI n p := by
+    have hτle : (τ : ℝ) ≤ 10 * (q : ℝ) + 24 - 12 * q' + 1 / 2 := by linarith
+    have : τ ≤ 10 * (q : ℤ) + 24 - 12 * (q' : ℤ) := by
+      have : (τ : ℝ) < 10 * (q : ℝ) + 24 - 12 * q' + 1 := by linarith
+      have : (τ : ℝ) < ((10 * (q : ℤ) + 24 - 12 * (q' : ℤ) + 1 : ℤ) : ℝ) := by push_cast; linarith
+      have := (Int.cast_lt (R := ℝ)).mp this
+      omega
+    have hktop : ktopI n p = 8 * (q : ℤ) + 20 := rfl
+    rw [hktop, hk, hB]
+    have hq0Z : (0 : ℤ) ≤ q := by positivity
+    generalize (q : ℤ) = Q at *
+    generalize (q' : ℤ) = Q' at *
+    omega
+  change kloI n p ≤ k ∧ k ≤ ktopI n p
+  exact ⟨hk1, hk2⟩
+
+variable {p : ℕ} [hp : Fact p.Prime]
+
+section
+
+variable {n : ℕ} (hn : 33 ≤ n) (hodd : 2 * (p / 2) + 1 = p) (hp7 : 7 ≤ p)
+  (hsmall : ¬p * Mcut ≤ 40 * n) (hin : 3 * p ≤ 40 * n) (hsq : 2 * (40 * n) < p ^ 2)
+  (hdeg : 5 + 2 * (18 * n + 2 * (37 * n)) - 2 * (40 * n) + 1 < p ^ 2)
+
+include hn hodd hp7 hsmall hin hsq hdeg in
+/-- **The tail bound**: `-L_p ≤ p (x F(x) + 27/16) + C`. -/
+theorem tail_Lp_le :
+    -(Lp n p : ℝ) ≤ (p : ℝ) * ((40 * n / p : ℝ) * Ftail (40 * n / p) + 27 / 16) + Ctail n p := by
+  -- notation
+  have hp0 : 0 < p := by omega
+  have hpR : (7 : ℝ) ≤ p := by exact_mod_cast hp7
+  set m := p / 2 with hmdef
+  have hmR : (m : ℝ) = ((p : ℝ) - 1) / 2 := by
+    have : ((2 * m + 1 : ℕ) : ℝ) = p := by exact_mod_cast hodd
+    push_cast at this; linarith
+  have hm0 : (0 : ℝ) < m := by rw [hmR]; linarith
+  set v := 40 * n % p with hv
+  set v' := 3 * n % p with hv'
+  have hvp : v < p := Nat.mod_lt _ hp0
+  have hv'p : v' < p := Nat.mod_lt _ hp0
+  set q := 40 * n / p with hq
+  set q' := 3 * n / p with hq'
+  have hdivK : (40 * n : ℝ) = q * p + v := by
+    have := Nat.div_add_mod (40 * n) p
+    have h' : ((p * q + v : ℕ) : ℝ) = ((40 * n : ℕ) : ℝ) := by rw [this]
+    push_cast at h'; linarith
+  have hdivN : (3 * n : ℝ) = q' * p + v' := by
+    have := Nat.div_add_mod (3 * n) p
+    have h' : ((p * q' + v' : ℕ) : ℝ) = ((3 * n : ℕ) : ℝ) := by rw [this]
+    push_cast at h'; linarith
+  set hR : ℝ := (37 * n : ℝ) with hhR
+  set SR : ℝ := 6 * (v' : ℝ) - v with hSR
+  set τs : ℝ := (SR + 2 * hR) / m with hτs
+  set τ : ℤ := ⌊τs + 1 / 2⌋ with hτ
+  have hδ1 : (τ : ℝ) ≤ τs + 1 / 2 := Int.floor_le _
+  have hδ2 : τs + 1 / 2 < τ + 1 := Int.lt_floor_add_one _
+  set B : ℤ := 12 * (q' : ℤ) - 2 * (q : ℤ) - 4 with hB
+  set k : ℤ := τ + B - 1 with hk
+    -- basic size facts
+  have hqlt : (40 * n : ℝ) < ((q : ℝ) + 1) * p := by
+    have := Nat.lt_div_mul_add (a := 40 * n) hp0
+    have : ((40 * n : ℕ) : ℝ) < ((q * p + p : ℕ) : ℝ) := by exact_mod_cast this
+    push_cast at this; linarith
+  have hq'le : (q' : ℝ) * p ≤ 3 * n := by
+    rw [hdivN]; have : (0 : ℝ) ≤ v' := by positivity
+    linarith
+  have hnin : (3 : ℝ) * p ≤ 40 * n := by exact_mod_cast hin
+  have hvR : (v : ℝ) < p := by exact_mod_cast hvp
+  have hv'R : (v' : ℝ) < p := by exact_mod_cast hv'p
+  have hv0 : (0 : ℝ) ≤ v := by positivity
+  have hv'0 : (0 : ℝ) ≤ v' := by positivity
+  have hq0 : (0 : ℝ) ≤ q := by positivity
+  have hq'0 : (0 : ℝ) ≤ q' := by positivity
+  have hτsm : τs * m = SR + 2 * hR := by
+    rw [hτs];
+    field_simp
+      -- `τs` bounds
+  have hkLevel : k = tailLevel n p := rfl
+  obtain ⟨hk1, hk2⟩ := tailLevel_mem n p hodd hp7 hin
+  rw [← hkLevel] at hk1 hk2
+  have main :=
+    inner_Lp_le hn hodd hp7 hsmall hin hsq hdeg k hk1
+      hk2
+        -- the class sum
+  have hclass := tail_class_sum_le n p v v' k τ hodd hvp hv'p
+    (by exact_mod_cast hm0) (by rw [hk, hB])
+  have hpR0 : (0 : ℝ) < p := by linarith
+  have hu500 : ((37 * n : ℕ) : ℝ) / p ≤ 500 := by
+    rw [div_le_iff₀ hpR0]
+    have : (40 * n : ℝ) < 400 * p := by
+      unfold Mcut at hsmall; rw [not_le] at hsmall
+      exact_mod_cast (show 40 * n < 400 * p by linarith)
+    push_cast; linarith
+  have hlay := layer_le hp0 (37 * n) hu500
+  have e37 : ((37 * n : ℕ) : ℝ) = hR := by rw [hhR]; push_cast; ring
+  rw [e37] at hlay
+  have hquad := quad_le (hR := hR) (SR := SR) (τ := (τ : ℝ)) hm0 hδ1 hδ2
+  have hdiff :=
+    diff_le (hR := hR) (SR := SR) (p := (p : ℝ)) (by linarith) (by rw [hhR]; positivity)
+      (by rw [hhR, hSR]; linarith)
+  rw [←
+    hmR] at hdiff
+        -- the target
+  have hfK : Int.fract ((40 * n / p : ℝ)) = (v : ℝ) / p := by
+    have := Int.fract_div_natCast_eq_div_natCast_mod (k := ℝ) (m := 40 * n) (n := p)
+    push_cast at this; exact this
+  have hfN : Int.fract ((3 : ℝ) / 40 * (40 * n / p)) = (v' : ℝ) / p := by
+    have := Int.fract_div_natCast_eq_div_natCast_mod (k := ℝ) (m := 3 * n) (n := p)
+    have e : (3 : ℝ) / 40 * (40 * n / p) = ((3 * n : ℕ) : ℝ) / (p : ℕ) := by push_cast; ring
+    rw [e]; exact this
+  have hT :
+    (p : ℝ) * ((40 * n / p : ℝ) * Ftail (40 * n / p) + 27 / 16) =
+      4 * hR - 2 * hR * SR / p + 27 * p / 16 :=
+    by unfold Ftail; rw [hfK, hfN, hSR, hhR]; field_simp; ring
+  have hE :
+    ((k - kloI n p : ℤ) : ℝ) * (L0I n p : ℝ) ≤ ((ktopI n p - kloI n p : ℤ) : ℝ) * (L0I n p : ℝ) :=
+  by
+    apply mul_le_mul_of_nonneg_right _ (by positivity)
+    exact_mod_cast (show k - kloI n p ≤ ktopI n p - kloI n p by omega)
+  have hkR : (k : ℝ) = τ + 12 * (q' : ℝ) - 2 * q - 5 := by rw [hk, hB]; push_cast; ring
+  have hCt :
+    Ctail n p =
+      4 * hR / p + ((ktopI n p - kloI n p : ℤ) : ℝ) * (L0I n p : ℝ) + 2 * Bmax n p + 1 :=
+    by unfold Ctail; rw [hhR]
+  have h4 : 0 ≤ 4 * hR / p := by positivity
+  rw [hT, hCt]
+  have hmain := main
+  rw [hkR, e37] at hmain
+  simp only [← hq, ← hq', ← hmdef, ← hv, ← hv'] at hmain
+  have hmR16 : (m : ℝ) / 16 + 49 * m / 16 = 25 * (p : ℝ) / 16 - 25 / 16 := by rw [hmR]; ring
+  have s1 :
+    -(Lp n p : ℝ) ≤
+      2 * layer p 1000 hR - hR * τ + 5 * hR +
+              2 * ∑ x ∈ Icc 1 m, hIn n p k (aCnt p v x) (aCnt p v' x) +
+            ((k - kloI n p : ℤ) : ℝ) * (L0I n p : ℝ) +
+          2 * Bmax n p +
+        1 := by
+    have :
+      -2 * hR * (q : ℝ) + 12 * hR * q' - hR * ((τ : ℝ) + 12 * q' - 2 * q - 5) =
+        -hR * τ + 5 * hR :=
+      by ring
+    linarith
+  have s2 :
+    -(Lp n p : ℝ) ≤
+      2 * layer p 1000 hR + 5 * hR + (-hR * SR / m - hR ^ 2 / m + m / 16) + 49 * m / 16 +
+            ((k - kloI n p : ℤ) : ℝ) * (L0I n p : ℝ) +
+          2 * Bmax n p +
+        1 := by
+    have :
+      ((m : ℝ) * (τ : ℝ) ^ 2 - 2 * τ * SR + SR ^ 2 / m + 49 * m / 4) / 4 =
+        ((m : ℝ) * (τ : ℝ) ^ 2 - 2 * τ * SR + SR ^ 2 / m) / 4 + 49 * m / 16 :=
+      by ring
+    linarith
+  have s3 :
+    -(Lp n p : ℝ) ≤
+      2 * hR ^ 2 / p + 4 * hR + p / 8 - hR * SR / m - hR ^ 2 / m + (25 * (p : ℝ) / 16 - 25 / 16) +
+            ((k - kloI n p : ℤ) : ℝ) * (L0I n p : ℝ) +
+          2 * Bmax n p +
+        1 := by
+    have : -hR * SR / m = -(hR * SR / m) := by ring
+    linarith
+  have s4 :
+    -(Lp n p : ℝ) ≤
+      4 * hR - 2 * hR * SR / p + 27 * p / 16 + ((k - kloI n p : ℤ) : ℝ) * (L0I n p : ℝ) +
+          2 * Bmax n p +
+        1 :=
+    by linarith
+  linarith
+
+end
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Hermite.lean b/LeanPool/Zeta5Irrational/Hermite.lean
new file mode 100644
index 0000000000..90b02f0088
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Hermite.lean
@@ -0,0 +1,528 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+import Mathlib.Analysis.SpecialFunctions.Trigonometric.DerivHyp
+
+public import LeanPool.Zeta5Irrational.Weight
+public import Mathlib.Analysis.Calculus.Deriv.Basic
+public import Mathlib.Analysis.LocallyConvex.AbsConvexOpen
+public import Mathlib.MeasureTheory.Integral.Bochner.Basic
+public import Mathlib.MeasureTheory.Integral.IntegrableOn
+import Mathlib.Algebra.Ring.IsFormallyReal
+import Mathlib.Analysis.CStarAlgebra.Classes
+import Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent
+import Mathlib.RingTheory.SimpleRing.Principal
+import Mathlib.Tactic.ENatToNat
+import Mathlib.Tactic.Polynomial.Basic
+import Mathlib.Tactic.ReduceModChar
+import Mathlib.Tactic.FieldSimp
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+import Mathlib.Tactic.Positivity.Basic
+import Mathlib.Tactic.Ring.Basic
+import Mathlib.Tactic.Ring.RingNF
+
+import Mathlib.Analysis.SpecialFunctions.ImproperIntegrals
+
+/-! # Towards Hermite's formula at `s = 5`: the calculus ingredients
+
+* the closed form `∑_{ℓ ≥ 1} ℓ⁴ x^ℓ = x (1 + 11x + 11x² + x³) / (1 - x)⁵` for `0 ≤ x < 1`;
+* `F(y) = 1 / (e^{2πy} - 1)` and its derivatives `F₁, …, F₄` (explicit rational functions of
+  `q = e^{2πy}`), with `w(y) = y⁵ F₄(y) / 12`;
+* `g(y) = y⁵ / (12 (y² + a²))` and its derivatives `g₁, …, g₄`.
+-/
+
+@[expose] public section
+
+open Filter Topology Finset
+
+namespace Zeta5Irrational
+
+/-! ### The series `∑ ℓ⁴ x^ℓ` -/
+
+/--
+The telescoping polynomial: `(1 - x)⁵ ∑_{ℓ < N} (ℓ+1)⁴ x^{ℓ+1} = x(1+11x+11x²+x³) - x^{N+1} P(N,
+x)`. -/
+noncomputable def Ptel (N x : ℝ) : ℝ :=
+  N ^ 4 * x ^ 4 - 4 * N ^ 4 * x ^ 3 + 6 * N ^ 4 * x ^ 2 - 4 * N ^ 4 * x + N ^ 4 -
+                                  4 * N ^ 3 * x ^ 3 +
+                                12 * N ^ 3 * x ^ 2 -
+                              12 * N ^ 3 * x +
+                            4 * N ^ 3 +
+                          6 * N ^ 2 * x ^ 3 -
+                        6 * N ^ 2 * x ^ 2 -
+                      6 * N ^ 2 * x +
+                    6 * N ^ 2 -
+                  4 * N * x ^ 3 -
+                12 * N * x ^ 2 +
+              12 * N * x +
+            4 * N +
+          x ^ 3 +
+        11 * x ^ 2 +
+      11 * x +
+    1
+
+lemma telescope (x : ℝ) (N : ℕ) :
+    (1 - x) ^ 5 * ∑ ℓ ∈ range N, ((ℓ : ℝ) + 1) ^ 4 * x ^ (ℓ + 1) =
+      x * (1 + 11 * x + 11 * x ^ 2 + x ^ 3) - x ^ (N + 1) * Ptel N x := by
+  induction N with
+  | zero => simp [Ptel]; ring
+  | succ N ih =>
+    rw [Finset.sum_range_succ, mul_add, ih]
+    push_cast
+    unfold Ptel
+    ring
+
+lemma tendsto_pow_mul_Ptel {x : ℝ} (hx0 : 0 ≤ x) (hx1 : x < 1) :
+    Tendsto (fun N : ℕ => x ^ (N + 1) * Ptel N x) atTop (𝓝 0) := by
+  have habs : |x| < 1 := by rw [abs_of_nonneg hx0]; exact hx1
+  have h : ∀ k : ℕ, Tendsto (fun N : ℕ => (N : ℝ) ^ k * x ^ N) atTop (𝓝 0) := fun k =>
+    tendsto_pow_const_mul_const_pow_of_abs_lt_one k habs
+  have h0 := h 0; have h1 := h 1; have h2 := h 2; have h3 := h 3; have h4 := h 4
+  have :
+    (fun N : ℕ => x ^ (N + 1) * Ptel N x) = fun N : ℕ =>
+      ((N : ℝ) ^ 4 * x ^ N) * (x ^ 5 - 4 * x ^ 4 + 6 * x ^ 3 - 4 * x ^ 2 + x) +
+              ((N : ℝ) ^ 3 * x ^ N) * (-4 * x ^ 4 + 12 * x ^ 3 - 12 * x ^ 2 + 4 * x) +
+            ((N : ℝ) ^ 2 * x ^ N) * (6 * x ^ 4 - 6 * x ^ 3 - 6 * x ^ 2 + 6 * x) +
+          ((N : ℝ) ^ 1 * x ^ N) * (-4 * x ^ 4 - 12 * x ^ 3 + 12 * x ^ 2 + 4 * x) +
+        ((N : ℝ) ^ 0 * x ^ N) * (x ^ 4 + 11 * x ^ 3 + 11 * x ^ 2 + x) :=
+    by funext N; unfold Ptel; ring
+  rw [this]
+  simpa using
+    ((((h4.mul_const _).add (h3.mul_const _)).add (h2.mul_const _)).add (h1.mul_const _)).add
+      (h0.mul_const _)
+
+/-- `∑_{ℓ ≥ 1} ℓ⁴ x^ℓ = x (1 + 11x + 11x² + x³) / (1 - x)⁵`. -/
+lemma hasSum_pow_four_mul_geometric {x : ℝ} (hx0 : 0 ≤ x) (hx1 : x < 1) :
+    HasSum (fun ℓ : ℕ => ((ℓ : ℝ) + 1) ^ 4 * x ^ (ℓ + 1))
+      (x * (1 + 11 * x + 11 * x ^ 2 + x ^ 3) / (1 - x) ^ 5) := by
+  have h1x : (0 : ℝ) < (1 - x) ^ 5 := pow_pos (by linarith) 5
+  refine (hasSum_iff_tendsto_nat_of_nonneg (fun ℓ => by positivity) _).mpr ?_
+  have hS :
+    ∀ N : ℕ,
+      ∑ ℓ ∈ range N, ((ℓ : ℝ) + 1) ^ 4 * x ^ (ℓ + 1) =
+        (x * (1 + 11 * x + 11 * x ^ 2 + x ^ 3) - x ^ (N + 1) * Ptel N x) / (1 - x) ^ 5 :=
+    by intro N; rw [← telescope, mul_div_cancel_left₀ _ h1x.ne']
+  simp_rw [hS]
+  have :=
+    ((tendsto_const_nhds (x := x * (1 + 11 * x + 11 * x ^ 2 + x ^ 3))).sub
+          (tendsto_pow_mul_Ptel hx0 hx1)).div_const
+      ((1 - x) ^ 5)
+  rw [sub_zero] at this
+  exact this
+
+/-! ### `F(y) = 1 / (e^{2πy} - 1)` and its derivatives -/
+
+/-- `q(y) = e^{2πy}`. -/
+noncomputable def qe (y : ℝ) : ℝ :=
+  Real.exp (2 * Real.pi * y)
+
+lemma qe_pos (y : ℝ) : 0 < qe y :=
+  Real.exp_pos _
+
+lemma qe_gt_one {y : ℝ} (hy : 0 < y) : 1 < qe y := by
+  unfold qe; rw [← Real.exp_zero];
+  exact Real.exp_lt_exp.mpr (by positivity)
+
+lemma hasDerivAt_qe (y : ℝ) : HasDerivAt qe (2 * Real.pi * qe y) y := by
+  have h := ((hasDerivAt_id (𝕜 := ℝ) y).const_mul (2 * Real.pi)).exp
+  simp only [id, mul_one] at h
+  unfold qe
+  refine
+    HasDerivAt.congr_deriv
+      (HasDerivAt.congr_of_eventuallyEq h (Filter.Eventually.of_forall fun _ => rfl)) ?_
+  ring
+
+/-- `F(y) = 1 / (e^{2πy} - 1)` and its first four derivatives. -/
+noncomputable def F0 (y : ℝ) : ℝ :=
+  1 / (qe y - 1)
+
+/-- Expression for the first derivative of `F0` on the positive half-line. -/
+noncomputable def F1 (y : ℝ) : ℝ :=
+  -(2 * Real.pi) * qe y / (qe y - 1) ^ 2
+
+/-- Expression for the second derivative of `F0` on the positive half-line. -/
+noncomputable def F2 (y : ℝ) : ℝ :=
+  (2 * Real.pi) ^ 2 * qe y * (qe y + 1) / (qe y - 1) ^ 3
+
+/-- Expression for the third derivative of `F0` on the positive half-line. -/
+noncomputable def F3 (y : ℝ) : ℝ :=
+  -(2 * Real.pi) ^ 3 * qe y * (qe y ^ 2 + 4 * qe y + 1) / (qe y - 1) ^ 4
+
+/-- Expression for the fourth derivative of `F0` on the positive half-line. -/
+noncomputable def F4 (y : ℝ) : ℝ :=
+  (2 * Real.pi) ^ 4 * qe y * (qe y ^ 3 + 11 * qe y ^ 2 + 11 * qe y + 1) / (qe y - 1) ^ 5
+
+lemma hasDerivAt_F0 {y : ℝ} (hy : 0 < y) : HasDerivAt F0 (F1 y) y := by
+  have hq : qe y - 1 ≠ 0 := by have := qe_gt_one hy; linarith
+  have := ((hasDerivAt_qe y).sub_const 1).inv hq
+  unfold F0 F1
+  simp only [one_div]
+  refine
+    HasDerivAt.congr_deriv
+      (HasDerivAt.congr_of_eventuallyEq this (Filter.Eventually.of_forall fun _ => rfl)) ?_
+  try simp only [Pi.pow_apply, Pi.mul_apply, Pi.div_apply, Pi.sub_apply, Pi.add_apply]
+  field_simp
+
+lemma hasDerivAt_F1 {y : ℝ} (hy : 0 < y) : HasDerivAt F1 (F2 y) y := by
+  have hq : qe y - 1 ≠ 0 := by have := qe_gt_one hy; linarith
+  have hd :=
+    ((hasDerivAt_qe y).const_mul (-(2 * Real.pi))).div (((hasDerivAt_qe y).sub_const 1).pow 2)
+      (pow_ne_zero 2 hq)
+  unfold F1 F2
+  refine
+    HasDerivAt.congr_deriv
+      (HasDerivAt.congr_of_eventuallyEq hd (Filter.Eventually.of_forall fun _ => rfl)) ?_
+  try simp only [Pi.pow_apply]
+  field_simp
+  ring
+
+lemma hasDerivAt_F2 {y : ℝ} (hy : 0 < y) : HasDerivAt F2 (F3 y) y := by
+  have hq : qe y - 1 ≠ 0 := by have := qe_gt_one hy; linarith
+  have hd :=
+    (((hasDerivAt_qe y).const_mul ((2 * Real.pi) ^ 2)).mul ((hasDerivAt_qe y).add_const 1)).div
+      (((hasDerivAt_qe y).sub_const 1).pow 3) (pow_ne_zero 3 hq)
+  unfold F2 F3
+  refine
+    HasDerivAt.congr_deriv
+      (HasDerivAt.congr_of_eventuallyEq hd (Filter.Eventually.of_forall fun _ => rfl)) ?_
+  try simp only [Pi.pow_apply, Pi.mul_apply]
+  field_simp
+  ring
+
+lemma hasDerivAt_F3 {y : ℝ} (hy : 0 < y) : HasDerivAt F3 (F4 y) y := by
+  have hq : qe y - 1 ≠ 0 := by have := qe_gt_one hy; linarith
+  have hd :=
+    (((hasDerivAt_qe y).const_mul (-(2 * Real.pi) ^ 3)).mul
+          ((((hasDerivAt_qe y).pow 2).add ((hasDerivAt_qe y).const_mul 4)).add_const 1)).div
+      (((hasDerivAt_qe y).sub_const 1).pow 4) (pow_ne_zero 4 hq)
+  unfold F3 F4
+  refine
+    HasDerivAt.congr_deriv
+      (HasDerivAt.congr_of_eventuallyEq hd (Filter.Eventually.of_forall fun _ => rfl)) ?_
+  try simp only [Pi.pow_apply, Pi.mul_apply, Pi.add_apply]
+  field_simp
+  ring
+
+/-- The series `wS` in closed form: `wS(y) = q (q³ + 11q² + 11q + 1) / (q - 1)⁵`. -/
+lemma wS_eq {y : ℝ} (hy : 0 < y) :
+    wS y = qe y * (qe y ^ 3 + 11 * qe y ^ 2 + 11 * qe y + 1) / (qe y - 1) ^ 5 := by
+  set x : ℝ := Real.exp (-(2 * Real.pi * y)) with hx
+  have hx0 : 0 ≤ x := (Real.exp_pos _).le
+  have hx1 : x < 1 := by
+    rw [hx, ← Real.exp_zero];
+    exact Real.exp_lt_exp.mpr (by nlinarith [Real.pi_pos])
+  have hterm :
+    ∀ ℓ : ℕ,
+      ((ℓ : ℝ) + 1) ^ 4 * Real.exp (-(2 * Real.pi * ((ℓ : ℝ) + 1) * y)) =
+        ((ℓ : ℝ) + 1) ^ 4 * x ^ (ℓ + 1) := by
+    intro ℓ
+    rw [hx, ← Real.exp_nat_mul]
+    congr 2
+    push_cast; ring
+  unfold wS
+  simp_rw [hterm]
+  rw [(hasSum_pow_four_mul_geometric hx0 hx1).tsum_eq]
+  have hq : qe y = 1 / x := by rw [hx, qe, one_div, ← Real.exp_neg, neg_neg]
+  rw [hq]
+  have hxne : x ≠ 0 := (Real.exp_pos _).ne'
+  have h1x : 1 - x ≠ 0 := by linarith
+  field_simp
+  try ring
+
+/-- `w(y) = y⁵ F₄(y) / 12` for `y > 0`. -/
+lemma w_eq_F4 {y : ℝ} (hy : 0 < y) : w y = y ^ 5 * F4 y / 12 := by
+  unfold w F4
+  rw [wS_eq hy]
+  ring
+
+/-! ### `g(y) = y⁵ / (12 (y² + a²))` and its derivatives -/
+
+/-- Rational factor paired with `F4` in the Hermite integral formula. -/
+noncomputable def g0 (a y : ℝ) : ℝ :=
+  y ^ 5 / (12 * (y ^ 2 + a ^ 2))
+
+/-- First derivative of the rational factor `g0 a`, for positive `a`. -/
+noncomputable def g1 (a y : ℝ) : ℝ :=
+  y ^ 4 * (5 * a ^ 2 + 3 * y ^ 2) / (12 * (y ^ 2 + a ^ 2) ^ 2)
+
+/-- Second derivative of the rational factor `g0 a`, for positive `a`. -/
+noncomputable def g2 (a y : ℝ) : ℝ :=
+  y ^ 3 * (10 * a ^ 4 + 9 * a ^ 2 * y ^ 2 + 3 * y ^ 4) / (6 * (y ^ 2 + a ^ 2) ^ 3)
+
+/-- Third derivative of the rational factor `g0 a`, for positive `a`. -/
+noncomputable def g3 (a y : ℝ) : ℝ :=
+  y ^ 2 * (10 * a ^ 6 + 5 * a ^ 4 * y ^ 2 + 4 * a ^ 2 * y ^ 4 + y ^ 6) / (2 * (y ^ 2 + a ^ 2) ^ 4)
+
+/-- Fourth derivative of the rational factor `g0 a`, for positive `a`. -/
+noncomputable def g4 (a y : ℝ) : ℝ :=
+  2 * a ^ 4 * y * (5 * a ^ 4 - 10 * a ^ 2 * y ^ 2 + y ^ 4) / (y ^ 2 + a ^ 2) ^ 5
+
+lemma hasDerivAt_g0 {a : ℝ} (ha : 0 < a) (y : ℝ) : HasDerivAt (g0 a) (g1 a y) y := by
+  have hne : 12 * (y ^ 2 + a ^ 2) ≠ 0 := by positivity
+  have hd := (hasDerivAt_pow 5 y).div (((hasDerivAt_pow 2 y).add_const (a ^ 2)).const_mul 12) hne
+  unfold g0 g1
+  refine
+    HasDerivAt.congr_deriv
+      (HasDerivAt.congr_of_eventuallyEq hd (Filter.Eventually.of_forall fun _ => rfl)) ?_
+  try simp only [Pi.pow_apply, Pi.mul_apply, Pi.div_apply, Pi.sub_apply, Pi.add_apply]
+  field_simp
+  ring
+
+lemma hasDerivAt_g1 {a : ℝ} (ha : 0 < a) (y : ℝ) : HasDerivAt (g1 a) (g2 a y) y := by
+  have hne : 12 * (y ^ 2 + a ^ 2) ^ 2 ≠ 0 := by positivity
+  have hd :=
+    ((hasDerivAt_pow 4 y).mul (((hasDerivAt_pow 2 y).const_mul 3).const_add (5 * a ^ 2))).div
+      ((((hasDerivAt_pow 2 y).add_const (a ^ 2)).pow 2).const_mul 12) hne
+  unfold g1 g2
+  refine
+    HasDerivAt.congr_deriv
+      (HasDerivAt.congr_of_eventuallyEq hd (Filter.Eventually.of_forall fun _ => rfl)) ?_
+  try simp only [Pi.pow_apply, Pi.mul_apply]
+  field_simp
+  ring
+
+lemma hasDerivAt_g2 {a : ℝ} (ha : 0 < a) (y : ℝ) : HasDerivAt (g2 a) (g3 a y) y := by
+  have hne : 6 * (y ^ 2 + a ^ 2) ^ 3 ≠ 0 := by positivity
+  have hd :=
+    ((hasDerivAt_pow 3 y).mul
+          ((((hasDerivAt_pow 2 y).const_mul (9 * a ^ 2)).const_add (10 * a ^ 4)).add
+            ((hasDerivAt_pow 4 y).const_mul 3))).div
+      ((((hasDerivAt_pow 2 y).add_const (a ^ 2)).pow 3).const_mul 6) hne
+  unfold g2 g3
+  refine
+    HasDerivAt.congr_deriv
+      (HasDerivAt.congr_of_eventuallyEq hd (Filter.Eventually.of_forall fun _ => rfl)) ?_
+  try simp only [Pi.pow_apply, Pi.mul_apply, Pi.add_apply]
+  field_simp
+  ring
+
+lemma hasDerivAt_g3 {a : ℝ} (ha : 0 < a) (y : ℝ) : HasDerivAt (g3 a) (g4 a y) y := by
+  have hne : 2 * (y ^ 2 + a ^ 2) ^ 4 ≠ 0 := by positivity
+  have hd :=
+    ((hasDerivAt_pow 2 y).mul
+          (((((hasDerivAt_pow 2 y).const_mul (5 * a ^ 4)).const_add (10 * a ^ 6)).add
+                ((hasDerivAt_pow 4 y).const_mul (4 * a ^ 2))).add
+            (hasDerivAt_pow 6 y))).div
+      ((((hasDerivAt_pow 2 y).add_const (a ^ 2)).pow 4).const_mul 2) hne
+  unfold g3 g4
+  refine
+    HasDerivAt.congr_deriv
+      (HasDerivAt.congr_of_eventuallyEq hd (Filter.Eventually.of_forall fun _ => rfl)) ?_
+  try simp only [Pi.pow_apply, Pi.mul_apply, Pi.add_apply]
+  field_simp
+  ring
+
+end Zeta5Irrational
+
+namespace Zeta5Irrational
+
+open MeasureTheory Set
+
+/-! ### Bounds, integrability and the first integral `∫₀^∞ g₄ = 1/2` -/
+
+lemma abs_g4_le {a : ℝ} (ha : 0 < a) {y : ℝ} (hy : 0 ≤ y) :
+    |g4 a y| ≤ 10 * a ^ 3 / (y ^ 2 + a ^ 2) ^ 2 := by
+  unfold g4
+  have hden : 0 < y ^ 2 + a ^ 2 := by positivity
+  rw [abs_div, abs_of_pos (by positivity : (0 : ℝ) < (y ^ 2 + a ^ 2) ^ 5),
+    div_le_div_iff₀ (by positivity) (by positivity)]
+  have h1 :
+    |2 * a ^ 4 * y * (5 * a ^ 4 - 10 * a ^ 2 * y ^ 2 + y ^ 4)| ≤
+      2 * a ^ 4 * y * (10 * (y ^ 2 + a ^ 2) ^ 2) := by
+    rw [abs_mul, abs_of_nonneg (by positivity : (0 : ℝ) ≤ 2 * a ^ 4 * y)]
+    refine mul_le_mul_of_nonneg_left ?_ (by positivity)
+    rw [abs_le];
+    constructor <;>
+      nlinarith [sq_nonneg (a ^ 2 - y ^ 2), sq_nonneg y, sq_nonneg a,
+        mul_nonneg (sq_nonneg a) (sq_nonneg y)]
+  have h2 : y ≤ (y ^ 2 + a ^ 2) / (2 * a) := by
+    rw [le_div_iff₀ (by positivity)];
+    nlinarith [sq_nonneg (y - a)]
+  calc
+    |2 * a ^ 4 * y * (5 * a ^ 4 - 10 * a ^ 2 * y ^ 2 + y ^ 4)| * (y ^ 2 + a ^ 2) ^ 2 ≤
+        2 * a ^ 4 * y * (10 * (y ^ 2 + a ^ 2) ^ 2) * (y ^ 2 + a ^ 2) ^ 2 :=
+      by gcongr
+    _ ≤
+        2 * a ^ 4 * ((y ^ 2 + a ^ 2) / (2 * a)) * (10 * (y ^ 2 + a ^ 2) ^ 2) *
+          (y ^ 2 + a ^ 2) ^ 2 :=
+      by gcongr
+    _ = 10 * a ^ 3 * (y ^ 2 + a ^ 2) ^ 5 := by field_simp
+
+lemma inv_sq_add_le {a : ℝ} (ha : 0 < a) (y : ℝ) :
+    1 / (y ^ 2 + a ^ 2) ≤ (1 + 1 / a ^ 2) * (1 / (1 + y ^ 2)) := by
+  rw [div_le_iff₀ (by positivity)]
+  have :
+    (1 + 1 / a ^ 2) * (1 / (1 + y ^ 2)) * (y ^ 2 + a ^ 2) =
+      (y ^ 2 + a ^ 2 + y ^ 2 / a ^ 2 + 1) / (1 + y ^ 2) :=
+    by field_simp; ring
+  rw [this, le_div_iff₀ (by positivity)]
+  have : 0 ≤ y ^ 2 / a ^ 2 := by positivity
+  nlinarith
+
+/-- `g₄ · h` is integrable on `(0, ∞)` whenever `h` is measurable and bounded by `1/(y²+a²)`
+-type control: we only need the three cases below. -/
+lemma integrableOn_g4 {a : ℝ} (ha : 0 < a) : IntegrableOn (g4 a) (Ioi 0) := by
+  have hg :
+    IntegrableOn (fun y : ℝ => (10 * a ^ 3 / a ^ 2 * (1 + 1 / a ^ 2)) * (1 + y ^ 2)⁻¹) (Ioi 0) :=
+    (integrable_inv_one_add_sq.const_mul _).integrableOn
+  have hcont : Continuous (g4 a) := by
+    unfold g4
+    exact
+      (by fun_prop :
+            Continuous fun y : ℝ => 2 * a ^ 4 * y * (5 * a ^ 4 - 10 * a ^ 2 * y ^ 2 + y ^ 4)).div
+        (by fun_prop) (fun y => by positivity)
+  refine hg.mono' hcont.aestronglyMeasurable ?_
+  refine (ae_restrict_iff' measurableSet_Ioi).mpr (Filter.Eventually.of_forall fun y hy => ?_)
+  have hy : 0 < y := hy
+  rw [Real.norm_eq_abs]
+  calc
+    |g4 a y| ≤ 10 * a ^ 3 / (y ^ 2 + a ^ 2) ^ 2 := abs_g4_le ha hy.le
+    _ ≤ 10 * a ^ 3 / (a ^ 2 * (y ^ 2 + a ^ 2)) := by
+      gcongr
+      nlinarith [sq_nonneg y]
+    _ = 10 * a ^ 3 / a ^ 2 * (1 / (y ^ 2 + a ^ 2)) := by field_simp
+    _ ≤ 10 * a ^ 3 / a ^ 2 * ((1 + 1 / a ^ 2) * (1 / (1 + y ^ 2))) := by
+      gcongr;
+      exact inv_sq_add_le ha y
+    _ = _ := by ring
+
+lemma tendsto_g3_atTop {a : ℝ} (ha : 0 < a) : Tendsto (g3 a) atTop (𝓝 (1 / 2)) := by
+  have hφ :
+    Tendsto
+      (fun u : ℝ =>
+        (10 * a ^ 6 * u ^ 6 + 5 * a ^ 4 * u ^ 4 + 4 * a ^ 2 * u ^ 2 + 1) /
+          (2 * (1 + a ^ 2 * u ^ 2) ^ 4))
+      (𝓝 0) (𝓝 (1 / 2)) := by
+    have :
+      ContinuousAt
+        (fun u : ℝ =>
+          (10 * a ^ 6 * u ^ 6 + 5 * a ^ 4 * u ^ 4 + 4 * a ^ 2 * u ^ 2 + 1) /
+            (2 * (1 + a ^ 2 * u ^ 2) ^ 4))
+        0 :=
+      by refine ContinuousAt.div (by fun_prop) (by fun_prop) (by norm_num)
+    have h0 := this.tendsto
+    simpa using h0
+  have := hφ.comp tendsto_inv_atTop_zero
+  refine this.congr' ?_
+  filter_upwards [eventually_gt_atTop 0] with y hy
+  simp only [Function.comp, g3]
+  field_simp
+
+/-- `∫₀^∞ g₄(y) dy = 1/2`. -/
+lemma integral_g4 {a : ℝ} (ha : 0 < a) : ∫ y in Ioi 0, g4 a y = 1 / 2 := by
+  have h :=
+    integral_Ioi_of_hasDerivAt_of_tendsto' (a := 0) (f := g3 a) (f' := g4 a)
+      (fun y _ => hasDerivAt_g3 ha y) (integrableOn_g4 ha) (tendsto_g3_atTop ha)
+  rw [h]; simp [g3]
+
+end Zeta5Irrational
+
+namespace Zeta5Irrational
+
+open Complex in
+/-- The Mittag-Leffler expansion of `F(y) = 1/(e^{2πy} - 1)`:
+`F(y) = -1/2 + 1/(2πy) + (1/π) ∑_{n ≥ 1} y/(y² + n²)`, from Mathlib's `cot_series_rep`. -/
+lemma F0_series {y : ℝ} (hy : 0 < y) :
+    F0 y =
+      -1 / 2 + 1 / (2 * Real.pi * y) +
+        (1 / Real.pi) * ∑' n : ℕ, y / (y ^ 2 + ((n : ℝ) + 1) ^ 2) := by
+  have hx : (I * (y : ℂ)) ∈ integerComplement := by
+    rw [integerComplement_eq]
+    rintro ⟨n, hn⟩
+    have h := congrArg Complex.im hn
+    simp at h
+    linarith
+  have h :=
+    cot_series_rep
+      hx
+        -- the left-hand side
+  have hs : Real.sinh (Real.pi * y) ≠ 0 := (Real.sinh_pos_iff.mpr (by positivity)).ne'
+  have hL :
+    (Real.pi : ℂ) * cot (Real.pi * (I * y)) =
+      -I * ((Real.pi * (Real.cosh (Real.pi * y) / Real.sinh (Real.pi * y)) : ℝ) : ℂ) := by
+    rw [cot_eq_cos_div_sin,
+      show (Real.pi : ℂ) * (I * y) = ((Real.pi * y : ℝ) : ℂ) * I by push_cast; ring, cos_mul_I,
+      sin_mul_I, ← ofReal_cosh, ← ofReal_sinh]
+    have hs' : (Real.sinh (Real.pi * y) : ℂ) ≠ 0 := by exact_mod_cast hs
+    push_cast
+    field_simp
+    ring_nf
+    simp [I_sq]
+      -- the right-hand side
+  have hR :
+    (1 / (I * (y : ℂ)) + ∑' n : ℕ+, (1 / (I * y - n) + 1 / (I * y + n))) =
+      -I * ((1 / y + ∑' n : ℕ, 2 * y / (y ^ 2 + ((n : ℝ) + 1) ^ 2) : ℝ) : ℂ) := by
+    have hterm :
+      ∀ n : ℕ+,
+        (1 / (I * (y : ℂ) - n) + 1 / (I * y + n)) =
+          -I * ((2 * y / (y ^ 2 + ((n : ℕ) : ℝ) ^ 2) : ℝ) : ℂ) := by
+      intro n
+      have h1 : I * (y : ℂ) - n ≠ 0 := by
+        intro h; have := congrArg Complex.im h; simp at this;
+        linarith
+      have h2 : I * (y : ℂ) + n ≠ 0 := by
+        intro h; have := congrArg Complex.im h; simp at this;
+        linarith
+      have h3 : ((y : ℂ) ^ 2 + ((n : ℕ) : ℂ) ^ 2) ≠ 0 := by
+        have : (0 : ℝ) < y ^ 2 + ((n : ℕ) : ℝ) ^ 2 := by positivity
+        exact_mod_cast this.ne'
+      push_cast
+      rw [div_add_div _ _ h1 h2, div_eq_iff (mul_ne_zero h1 h2)]
+      have hIy : (I * (y : ℂ) - n) * (I * y + n) = -((y : ℂ) ^ 2 + ((n : ℕ) : ℂ) ^ 2) := by
+        ring_nf; rw [I_sq]; ring
+      rw [hIy]
+      field_simp
+      ring
+    simp_rw [hterm]
+    rw [tsum_mul_left,
+      tsum_pnat_eq_tsum_succ (f := fun m : ℕ => ((2 * y / (y ^ 2 + (m : ℝ) ^ 2) : ℝ) : ℂ))]
+    push_cast
+    rw [one_div, mul_inv, Complex.inv_I]
+    ring
+  rw [hL, hR] at h
+  have hI : (-I : ℂ) ≠ 0 := by simp
+  have h' := mul_left_cancel₀ hI h
+  have hreal :
+    Real.pi * (Real.cosh (Real.pi * y) / Real.sinh (Real.pi * y)) =
+      1 / y + ∑' n : ℕ, 2 * y / (y ^ 2 + ((n : ℝ) + 1) ^ 2) := by
+    exact_mod_cast
+      h'
+        -- `cosh/sinh = 1 + 2 F₀`
+  have hcoth : Real.cosh (Real.pi * y) / Real.sinh (Real.pi * y) = 1 + 2 * F0 y := by
+    rw [Real.cosh_eq, Real.sinh_eq, F0, qe]
+    have hq : Real.exp (2 * Real.pi * y) = Real.exp (Real.pi * y) ^ 2 := by
+      rw [← Real.exp_nat_mul]; push_cast; ring_nf
+    have hq1 : Real.exp (2 * Real.pi * y) - 1 ≠ 0 := by
+      have := qe_gt_one hy; unfold qe at this;
+      linarith
+    have he : Real.exp (-(Real.pi * y)) = 1 / Real.exp (Real.pi * y) := by
+      rw [Real.exp_neg, one_div]
+    rw [he]
+    have hpos : Real.exp (Real.pi * y) ≠ 0 := (Real.exp_pos _).ne'
+    rw [hq] at hq1 ⊢
+    field_simp
+    ring
+  rw [hcoth] at hreal
+  have hsum :
+    ∑' n : ℕ, 2 * y / (y ^ 2 + ((n : ℝ) + 1) ^ 2) =
+      2 * ∑' n : ℕ, y / (y ^ 2 + ((n : ℝ) + 1) ^ 2) :=
+    by rw [← tsum_mul_left]; congr 1; funext n; ring
+  rw [hsum] at hreal
+  have hpi : Real.pi ≠ 0 := Real.pi_pos.ne'
+  set S := ∑' n : ℕ, y / (y ^ 2 + ((n : ℝ) + 1) ^ 2) with hS
+  have h2 : F0 y = (1 / y + 2 * S) / (2 * Real.pi) - 1 / 2 := by
+    field_simp at hreal ⊢
+    linear_combination hreal
+  rw [h2]
+  field_simp
+  ring
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/HermiteFormula.lean b/LeanPool/Zeta5Irrational/HermiteFormula.lean
new file mode 100644
index 0000000000..a69fef40d5
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/HermiteFormula.lean
@@ -0,0 +1,216 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.HermiteIntegrals
+public import LeanPool.Zeta5Irrational.Zeta5
+public import LeanPool.Zeta5Irrational.Construction
+import LeanPool.Zeta5Irrational.HermiteIBP
+import Mathlib.Tactic.ENatToNat
+import Mathlib.Tactic.Polynomial.Basic
+import Mathlib.Tactic.ReduceModChar
+import Mathlib.Tactic.FieldSimp
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+import Mathlib.Tactic.Positivity.Basic
+import Mathlib.Tactic.Ring.Basic
+import Mathlib.Tactic.Ring.RingNF
+
+/-! # Hermite's formula at `s = 5` for integer arguments
+
+`∫₀^∞ w(y) / (y² + j²) dy = j⁴ (ζ(5) - H_j^{(5)}) - 1/4 + 1/(2j)` for `j ≥ 1`,
+i.e. the pole moments (2.3) of the paper (Proposition 2.2).
+-/
+
+@[expose] public section
+
+open Filter Topology Set MeasureTheory Finset
+
+namespace Zeta5Irrational
+
+/-- `∫₀^∞ y/(y² + a²)² dy = 1/(2a²)`. -/
+lemma integral_y_div_sq_sq {a : ℝ} (ha : 0 < a) :
+    ∫ y in Ioi 0, y / (y ^ 2 + a ^ 2) ^ 2 = 1 / (2 * a ^ 2) := by
+  have hd :
+    ∀ y : ℝ, HasDerivAt (fun y : ℝ => -1 / (2 * (y ^ 2 + a ^ 2))) (y / (y ^ 2 + a ^ 2) ^ 2) y := by
+    intro y
+    have hne : 2 * (y ^ 2 + a ^ 2) ≠ 0 := by positivity
+    have :=
+      (hasDerivAt_const y (-1 : ℝ)).div (((hasDerivAt_pow 2 y).add_const (a ^ 2)).const_mul 2) hne
+    refine this.congr_deriv ?_
+    field_simp
+    ring
+  have hint : IntegrableOn (fun y : ℝ => y / (y ^ 2 + a ^ 2) ^ 2) (Ioi 0) := by
+    refine
+      integrableOn_of_le_inv_sq ha (C := 1 / (2 * a))
+        ((by fun_prop : Continuous fun y : ℝ => y).div (by fun_prop)
+            (fun y => by positivity)).aestronglyMeasurable
+        (fun y hy => ?_)
+    rw [abs_of_nonneg (by positivity)]
+    have h2 : y / (y ^ 2 + a ^ 2) ≤ 1 / (2 * a) := by
+      rw [div_le_div_iff₀ (by positivity) (by positivity)]; nlinarith [sq_nonneg (y - a)]
+    calc
+      y / (y ^ 2 + a ^ 2) ^ 2 = (y / (y ^ 2 + a ^ 2)) * (1 / (y ^ 2 + a ^ 2)) := by field_simp
+      _ ≤ (1 / (2 * a)) * (1 / (y ^ 2 + a ^ 2)) := by gcongr
+      _ = 1 / (2 * a) / (y ^ 2 + a ^ 2) := by ring
+  have hlim : Tendsto (fun y : ℝ => -1 / (2 * (y ^ 2 + a ^ 2))) atTop (𝓝 0) := by
+    refine tendsto_const_nhds.div_atTop ?_
+    exact ((tendsto_pow_atTop two_ne_zero).atTop_add tendsto_const_nhds).const_mul_atTop two_pos
+  have := integral_Ioi_of_hasDerivAt_of_tendsto' (a := 0) (fun y _ => hd y) hint hlim
+  rw [this]
+  field_simp
+  ring
+
+/-- `∫₀^∞ |k4 a n| ≤ 5a / n²`. -/
+lemma integral_abs_k4_le {a n : ℝ} (ha : 0 < a) (hn : 0 < n) :
+    ∫ y in Ioi 0, |k4 a n y| ≤ 5 * a / n ^ 2 := by
+  have hint1 : IntegrableOn (fun y => |k4 a n y|) (Ioi 0) := (integrableOn_k4 ha hn).abs
+  have hbase : IntegrableOn (fun y : ℝ => y / (y ^ 2 + a ^ 2) ^ 2) (Ioi 0) := by
+    refine
+      integrableOn_of_le_inv_sq ha (C := 1 / (2 * a))
+        ((by fun_prop : Continuous fun y : ℝ => y).div (by fun_prop)
+            (fun y => by positivity)).aestronglyMeasurable
+        (fun y hy => ?_)
+    rw [abs_of_nonneg (by positivity)]
+    have h2 : y / (y ^ 2 + a ^ 2) ≤ 1 / (2 * a) := by
+      rw [div_le_div_iff₀ (by positivity) (by positivity)]; nlinarith [sq_nonneg (y - a)]
+    calc
+      y / (y ^ 2 + a ^ 2) ^ 2 = (y / (y ^ 2 + a ^ 2)) * (1 / (y ^ 2 + a ^ 2)) := by field_simp
+      _ ≤ (1 / (2 * a)) * (1 / (y ^ 2 + a ^ 2)) := by gcongr
+      _ = 1 / (2 * a) / (y ^ 2 + a ^ 2) := by ring
+  have hint2 :
+    IntegrableOn (fun y : ℝ => 10 * a ^ 3 / n ^ 2 * (y / (y ^ 2 + a ^ 2) ^ 2)) (Ioi 0) :=
+    hbase.const_mul _
+  calc
+    ∫ y in Ioi 0, |k4 a n y| ≤ ∫ y in Ioi 0, 10 * a ^ 3 / n ^ 2 * (y / (y ^ 2 + a ^ 2) ^ 2) := by
+      refine setIntegral_mono_on hint1 hint2 measurableSet_Ioi fun y hy => ?_
+      have hy : 0 < y := hy
+      unfold k4
+      rw [abs_mul, abs_of_nonneg (by positivity : (0 : ℝ) ≤ y / (y ^ 2 + n ^ 2))]
+      have h1 := abs_g4_le ha hy.le
+      have h2 : y / (y ^ 2 + n ^ 2) ≤ y / n ^ 2 := by gcongr; nlinarith [sq_nonneg y]
+      calc
+        |g4 a y| * (y / (y ^ 2 + n ^ 2)) ≤ 10 * a ^ 3 / (y ^ 2 + a ^ 2) ^ 2 * (y / n ^ 2) := by
+          gcongr
+        _ = 10 * a ^ 3 / n ^ 2 * (y / (y ^ 2 + a ^ 2) ^ 2) := by ring
+    _ = 10 * a ^ 3 / n ^ 2 * (1 / (2 * a ^ 2)) := by
+      rw [integral_const_mul, integral_y_div_sq_sq ha]
+    _ = 5 * a / n ^ 2 := by field_simp; ring
+
+/-- `∑_{n ≥ 0} 1/(n + j + 1)⁵ = ζ(5) - H_j^{(5)}`. -/
+lemma tsum_tail_eq (j : ℕ) : ∑' n : ℕ, 1 / ((j : ℝ) + ((n : ℝ) + 1)) ^ 5 = zeta5 - (H5 j : ℝ) := by
+  have hz : Summable (fun m : ℕ => 1 / (m : ℝ) ^ 5) :=
+    Real.summable_one_div_nat_pow.mpr (by norm_num)
+  have h := hz.sum_add_tsum_nat_add (j + 1)
+  have hH : ∑ i ∈ range (j + 1), 1 / ((i : ℝ)) ^ 5 = (H5 j : ℝ) := by
+    rw [Finset.sum_range_succ']
+    simp only [Nat.cast_zero, ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true, zero_pow, div_zero,
+      add_zero]
+    unfold H5
+    push_cast
+    have e1 : Finset.Icc 1 j = Finset.Ico 1 (j + 1) := by
+      ext x;
+      simp only [Finset.mem_Icc, Finset.mem_Ico]; omega
+    rw [e1, Finset.sum_Ico_eq_sum_range, show j + 1 - 1 = j by omega]
+    refine Finset.sum_congr rfl fun i _ => ?_
+    push_cast; ring_nf
+  have htail :
+    ∑' n : ℕ, 1 / ((j : ℝ) + ((n : ℝ) + 1)) ^ 5 = ∑' n : ℕ, 1 / ((n + (j + 1) : ℕ) : ℝ) ^ 5 := by
+    refine tsum_congr fun n => ?_
+    push_cast; ring_nf
+  rw [htail, ← hH]
+  unfold zeta5
+  linarith
+
+/-- **Hermite's formula at `s = 5`** (the pole moments (2.3) of the paper). -/
+theorem hermite_pole (j : ℕ) (hj : 1 ≤ j) :
+    ∫ y in Ioi 0, w y / (y ^ 2 + (j : ℝ) ^ 2) =
+      (j : ℝ) ^ 4 * (zeta5 - (H5 j : ℝ)) - 1 / 4 + 1 / (2 * (j : ℝ)) := by
+  set a : ℝ := (j : ℝ) with ha_def
+  have ha : 0 < a := by
+    rw [ha_def];
+    exact_mod_cast
+      hj
+        -- Step 1: `w / (y² + a²) = g₀ · F₄`
+  have h1 : ∫ y in Ioi 0, w y / (y ^ 2 + a ^ 2) = ∫ y in Ioi 0, g0 a y * F4 y := by
+    refine setIntegral_congr_fun measurableSet_Ioi fun y hy => ?_
+    have hy : 0 < y := hy
+    have hne : y ^ 2 + a ^ 2 ≠ 0 := by positivity
+    rw [w_eq_F4 hy]; unfold g0; field_simp;
+    try
+      ring
+        -- Step 2: integrate by parts four times
+  rw [h1, integral_g0F4_eq ha]
+    -- Step 3: the pointwise decomposition of `g₄ F₀`
+  have hdec :
+    ∀ y,
+      0 < y →
+        g4 a y * F0 y =
+          -(1 / 2) * g4 a y + 1 / (2 * Real.pi) * h4 a y +
+            1 / Real.pi * ∑' n : ℕ, k4 a ((n : ℝ) + 1) y := by
+    intro y hy
+    rw [F0_series hy, ← g4_div_eq_h4 a hy.ne']
+    simp only [k4]
+    rw [tsum_mul_left]
+    have hy' : y ≠ 0 := hy.ne'
+    field_simp
+    try
+      ring
+        -- Step 4: integrability of the pieces
+  have hA : IntegrableOn (fun y => -(1 / 2) * g4 a y) (Ioi 0) := (integrableOn_g4 ha).const_mul _
+  have hB : IntegrableOn (fun y => 1 / (2 * Real.pi) * h4 a y) (Ioi 0) :=
+    (integrableOn_h4 ha).const_mul _
+  have hprod : IntegrableOn (fun y => g4 a y * F0 y) (Ioi 0) :=
+    integrableOn_of_prod_bound ha ((continuous_g4 a ha).continuousOn.mul continuousOn_F0)
+      (fun y hy => abs_g4F0_le ha hy)
+  have hC : IntegrableOn (fun y => 1 / Real.pi * ∑' n : ℕ, k4 a ((n : ℝ) + 1) y) (Ioi 0) := by
+    have := (hprod.sub hA).sub hB
+    refine this.congr_fun (fun y hy => ?_) measurableSet_Ioi
+    have hy : 0 < y := hy
+    simp only [Pi.sub_apply]
+    rw [hdec y hy];
+    ring
+      -- Step 5: the series of integrals
+  have hnorm : Summable fun n : ℕ => ∫ y in Ioi 0, ‖k4 a ((n : ℝ) + 1) y‖ := by
+    have hs : Summable fun n : ℕ => 5 * a / ((n : ℝ) + 1) ^ 2 := by
+      have :=
+        (summable_nat_add_iff 1).mpr (Real.summable_one_div_nat_pow.mpr (by norm_num : 1 < 2))
+      refine (this.mul_left (5 * a)).congr fun n => ?_
+      push_cast; ring
+    refine
+      Summable.of_nonneg_of_le (fun n => integral_nonneg fun y => norm_nonneg _) (fun n => ?_) hs
+    simp only [Real.norm_eq_abs]
+    exact integral_abs_k4_le ha (by positivity)
+  have hsum :
+    ∫ y in Ioi 0, ∑' n : ℕ, k4 a ((n : ℝ) + 1) y = ∑' n : ℕ, ∫ y in Ioi 0, k4 a ((n : ℝ) + 1) y :=
+    (integral_tsum_of_summable_integral_norm (fun n => integrableOn_k4 ha (by positivity))
+        hnorm).symm
+  have hval :
+    ∀ n : ℕ, ∫ y in Ioi 0, k4 a ((n : ℝ) + 1) y = Real.pi * a ^ 4 / (a + ((n : ℝ) + 1)) ^ 5 :=
+    fun n =>
+    integral_k4 ha
+      (by positivity)
+        -- Step 6: assemble
+  have hAB : IntegrableOn (fun y => -(1 / 2) * g4 a y + 1 / (2 * Real.pi) * h4 a y) (Ioi 0) :=
+    hA.add hB
+  rw [setIntegral_congr_fun measurableSet_Ioi (fun y hy => hdec y hy), integral_add hAB hC,
+    integral_add hA hB, integral_const_mul, integral_const_mul, integral_const_mul,
+    integral_g4 ha, integral_h4 ha, hsum, tsum_congr hval]
+  have htail :
+    ∑' n : ℕ, Real.pi * a ^ 4 / (a + ((n : ℝ) + 1)) ^ 5 =
+      Real.pi * a ^ 4 * (zeta5 - (H5 j : ℝ)) := by
+    rw [← tsum_tail_eq j, ← tsum_mul_left]
+    refine tsum_congr fun n => ?_
+    rw [ha_def]; ring
+  rw [htail]
+  have hpi : Real.pi ≠ 0 := Real.pi_pos.ne'
+  field_simp
+  ring
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/HermiteIBP.lean b/LeanPool/Zeta5Irrational/HermiteIBP.lean
new file mode 100644
index 0000000000..952a616390
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/HermiteIBP.lean
@@ -0,0 +1,468 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.HermiteIntegrals
+import LeanPool.Zeta5Irrational.Moments
+import Mathlib.Algebra.Ring.IsFormallyReal
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.ENatToNat
+import Mathlib.Tactic.Polynomial.Basic
+import Mathlib.Tactic.ReduceModChar
+import Mathlib.Tactic.FieldSimp
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+import Mathlib.Tactic.Positivity.Basic
+import Mathlib.Tactic.Ring.Basic
+import Mathlib.Tactic.Ring.RingNF
+
+import Mathlib.MeasureTheory.Integral.IntegralEqImproper
+
+/-! # Four integrations by parts: `∫₀^∞ g F₄ = ∫₀^∞ g₄ F₀`
+
+Uniform bounds `|F_m(y)| ≤ c_m (1+2πy)^{m+1} / (2π y^{m+1} e^{2πy})` and `|g_j(y)| ≤ c_j y^{5-j}/a²`
+give integrability of the products and vanishing boundary terms at `0⁺` and `∞`.
+-/
+
+@[expose] public section
+
+open Filter Topology Set MeasureTheory
+
+namespace Zeta5Irrational
+
+/-! ### Bounds for `F_m` -/
+
+lemma qe_sub_one_ge {y : ℝ} : 2 * Real.pi * y ≤ qe y - 1 := by
+  unfold qe;
+  linarith [Real.add_one_le_exp (2 * Real.pi * y)]
+
+lemma one_div_qe_sub_one_le {y : ℝ} (hy : 0 < y) :
+    1 / (qe y - 1) ≤ (1 + 2 * Real.pi * y) / (2 * Real.pi * y * qe y) := by
+  have hq := qe_gt_one hy
+  have h := qe_sub_one_ge (y := y)
+  rw [div_le_div_iff₀ (by linarith) (by positivity)]
+  nlinarith [qe_pos y]
+
+/-- Generic estimate `N / (q-1)^(m+1) ≤ c (1+2πy)^(m+1) / (2π y^(m+1) q)` when
+`N ≤ (2π)^m c q^m`. -/
+lemma F_bound_aux {y : ℝ} (hy : 0 < y) (m : ℕ) {N c : ℝ} (hN : 0 ≤ N)
+    (hNc : N ≤ (2 * Real.pi) ^ m * c * qe y ^ m) :
+    N / (qe y - 1) ^ (m + 1) ≤
+      c * (1 + 2 * Real.pi * y) ^ (m + 1) / (2 * Real.pi * y ^ (m + 1) * qe y) := by
+  have hq := qe_gt_one hy
+  have hq0 := qe_pos y
+  have h1 :
+    1 / (qe y - 1) ^ (m + 1) ≤ ((1 + 2 * Real.pi * y) / (2 * Real.pi * y * qe y)) ^ (m + 1) := by
+    rw [one_div, ← inv_pow, ← one_div]
+    exact pow_le_pow_left₀ (by positivity) (one_div_qe_sub_one_le hy) _
+  calc
+    N / (qe y - 1) ^ (m + 1) = N * (1 / (qe y - 1) ^ (m + 1)) := by ring
+    _ ≤
+        ((2 * Real.pi) ^ m * c * qe y ^ m) *
+          ((1 + 2 * Real.pi * y) / (2 * Real.pi * y * qe y)) ^ (m + 1) :=
+      (mul_le_mul hNc h1 (by positivity)
+        (by
+          have : 0 ≤ N := hN
+          exact le_trans this hNc))
+    _ = c * (1 + 2 * Real.pi * y) ^ (m + 1) / (2 * Real.pi * y ^ (m + 1) * qe y) := by
+      have h2pi : (2 * Real.pi) ≠ 0 := by positivity
+      have hqne : qe y ≠ 0 := hq0.ne'
+      rw [div_pow, mul_pow (2 * Real.pi * y) (qe y), mul_pow (2 * Real.pi) y,
+        pow_succ (2 * Real.pi), pow_succ (qe y), pow_succ y]
+      field_simp
+      try ring
+
+lemma abs_F0_le {y : ℝ} (hy : 0 < y) :
+    |F0 y| ≤ 1 * (1 + 2 * Real.pi * y) ^ (0 + 1) / (2 * Real.pi * y ^ (0 + 1) * qe y) := by
+  have hq := qe_gt_one hy
+  rw [F0, abs_of_pos (by positivity)]
+  have := F_bound_aux hy 0 (N := 1) (c := 1) (by norm_num) (by simp)
+  simpa using this
+
+lemma abs_F1_le {y : ℝ} (hy : 0 < y) :
+    |F1 y| ≤ 1 * (1 + 2 * Real.pi * y) ^ (1 + 1) / (2 * Real.pi * y ^ (1 + 1) * qe y) := by
+  have hq := qe_gt_one hy
+  have hq0 := qe_pos y
+  have : F1 y = -((2 * Real.pi) * qe y / (qe y - 1) ^ 2) := by unfold F1; ring
+  rw [this, abs_neg, abs_of_pos (by positivity)]
+  exact F_bound_aux hy 1 (by positivity) (by ring_nf; rfl)
+
+lemma abs_F2_le {y : ℝ} (hy : 0 < y) :
+    |F2 y| ≤ 2 * (1 + 2 * Real.pi * y) ^ (2 + 1) / (2 * Real.pi * y ^ (2 + 1) * qe y) := by
+  have hq := qe_gt_one hy
+  have hq0 := qe_pos y
+  rw [F2, abs_of_pos (by positivity)]
+  refine F_bound_aux hy 2 (by positivity) ?_
+  have h1 : qe y + 1 ≤ 2 * qe y := by linarith
+  calc
+    (2 * Real.pi) ^ 2 * qe y * (qe y + 1) ≤ (2 * Real.pi) ^ 2 * qe y * (2 * qe y) := by gcongr
+    _ = (2 * Real.pi) ^ 2 * 2 * qe y ^ 2 := by ring
+
+lemma abs_F3_le {y : ℝ} (hy : 0 < y) :
+    |F3 y| ≤ 6 * (1 + 2 * Real.pi * y) ^ (3 + 1) / (2 * Real.pi * y ^ (3 + 1) * qe y) := by
+  have hq := qe_gt_one hy
+  have hq0 := qe_pos y
+  have : F3 y = -((2 * Real.pi) ^ 3 * qe y * (qe y ^ 2 + 4 * qe y + 1) / (qe y - 1) ^ 4) := by
+    unfold F3; ring
+  rw [this, abs_neg, abs_of_pos (by positivity)]
+  refine F_bound_aux hy 3 (by positivity) ?_
+  have h1 : qe y ^ 2 + 4 * qe y + 1 ≤ 6 * qe y ^ 2 := by nlinarith
+  calc
+    (2 * Real.pi) ^ 3 * qe y * (qe y ^ 2 + 4 * qe y + 1) ≤
+        (2 * Real.pi) ^ 3 * qe y * (6 * qe y ^ 2) :=
+      by gcongr
+    _ = (2 * Real.pi) ^ 3 * 6 * qe y ^ 3 := by ring
+
+lemma abs_F4_le {y : ℝ} (hy : 0 < y) :
+    |F4 y| ≤ 24 * (1 + 2 * Real.pi * y) ^ (4 + 1) / (2 * Real.pi * y ^ (4 + 1) * qe y) := by
+  have hq := qe_gt_one hy
+  have hq0 := qe_pos y
+  rw [F4, abs_of_pos (by positivity)]
+  refine F_bound_aux hy 4 (by positivity) ?_
+  have h1 : qe y ^ 3 + 11 * qe y ^ 2 + 11 * qe y + 1 ≤ 24 * qe y ^ 3 := by
+    nlinarith [sq_nonneg (qe y), mul_pos hq0 hq0]
+  calc
+    (2 * Real.pi) ^ 4 * qe y * (qe y ^ 3 + 11 * qe y ^ 2 + 11 * qe y + 1) ≤
+        (2 * Real.pi) ^ 4 * qe y * (24 * qe y ^ 3) :=
+      by gcongr
+    _ = (2 * Real.pi) ^ 4 * 24 * qe y ^ 4 := by ring
+
+/-! ### Bounds for `g_j` -/
+
+lemma abs_g0_le {a : ℝ} (ha : 0 < a) {y : ℝ} (hy : 0 ≤ y) : |g0 a y| ≤ 1 / 12 * y ^ 5 / a ^ 2 := by
+  unfold g0
+  rw [abs_of_nonneg (by positivity), div_le_div_iff₀ (by positivity) (by positivity)]
+  nlinarith [pow_nonneg hy 5, sq_nonneg y, mul_nonneg (pow_nonneg hy 5) (sq_nonneg y)]
+
+lemma abs_g1_le {a : ℝ} (ha : 0 < a) {y : ℝ} (_hy : 0 ≤ y) : |g1 a y| ≤ 5 / 12 * y ^ 4 / a ^ 2 := by
+  unfold g1
+  rw [abs_of_nonneg (by positivity), div_le_div_iff₀ (by positivity) (by positivity)]
+  have h4 : 0 ≤ y ^ 4 := by positivity
+  nlinarith [mul_nonneg h4 (sq_nonneg y), mul_nonneg (mul_nonneg h4 (sq_nonneg y)) (sq_nonneg y),
+    mul_nonneg (mul_nonneg h4 (sq_nonneg y)) (sq_nonneg a), mul_nonneg h4 (sq_nonneg a),
+    mul_nonneg (mul_nonneg h4 (sq_nonneg a)) (sq_nonneg a)]
+
+lemma abs_g2_le {a : ℝ} (ha : 0 < a) {y : ℝ} (hy : 0 ≤ y) : |g2 a y| ≤ 10 / 6 * y ^ 3 / a ^ 2 := by
+  unfold g2
+  rw [abs_of_nonneg (by positivity), div_le_div_iff₀ (by positivity) (by positivity)]
+  have h3 : 0 ≤ y ^ 3 := by positivity
+  have hs : 0 ≤ (y ^ 2 + a ^ 2) := by positivity
+  have key : 10 * a ^ 4 + 9 * a ^ 2 * y ^ 2 + 3 * y ^ 4 ≤ 10 * (y ^ 2 + a ^ 2) ^ 2 := by
+    nlinarith [sq_nonneg y, sq_nonneg a, mul_nonneg (sq_nonneg y) (sq_nonneg a)]
+  calc
+    y ^ 3 * (10 * a ^ 4 + 9 * a ^ 2 * y ^ 2 + 3 * y ^ 4) * a ^ 2 ≤
+        y ^ 3 * (10 * (y ^ 2 + a ^ 2) ^ 2) * a ^ 2 :=
+      by gcongr
+    _ ≤ y ^ 3 * (10 * (y ^ 2 + a ^ 2) ^ 2) * (y ^ 2 + a ^ 2) := by gcongr; nlinarith [sq_nonneg y]
+    _ = 10 / 6 * y ^ 3 * (6 * (y ^ 2 + a ^ 2) ^ 3) := by ring
+
+lemma abs_g3_le {a : ℝ} (ha : 0 < a) {y : ℝ} (_hy : 0 ≤ y) : |g3 a y| ≤ 5 * y ^ 2 / a ^ 2 := by
+  unfold g3
+  rw [abs_of_nonneg (by positivity), div_le_div_iff₀ (by positivity) (by positivity)]
+  have hs : 0 ≤ (y ^ 2 + a ^ 2) := by positivity
+  have key :
+    10 * a ^ 6 + 5 * a ^ 4 * y ^ 2 + 4 * a ^ 2 * y ^ 4 + y ^ 6 ≤ 10 * (y ^ 2 + a ^ 2) ^ 3 := by
+    nlinarith [sq_nonneg y, sq_nonneg a, mul_nonneg (sq_nonneg y) (sq_nonneg a),
+      mul_nonneg (mul_nonneg (sq_nonneg y) (sq_nonneg a)) (sq_nonneg y),
+      mul_nonneg (mul_nonneg (sq_nonneg y) (sq_nonneg a)) (sq_nonneg a),
+      pow_nonneg (sq_nonneg y) 3, pow_nonneg (sq_nonneg a) 3]
+  calc
+    y ^ 2 * (10 * a ^ 6 + 5 * a ^ 4 * y ^ 2 + 4 * a ^ 2 * y ^ 4 + y ^ 6) * a ^ 2 ≤
+        y ^ 2 * (10 * (y ^ 2 + a ^ 2) ^ 3) * a ^ 2 :=
+      by gcongr
+    _ ≤ y ^ 2 * (10 * (y ^ 2 + a ^ 2) ^ 3) * (y ^ 2 + a ^ 2) := by gcongr; nlinarith [sq_nonneg y]
+    _ = 5 * y ^ 2 * (2 * (y ^ 2 + a ^ 2) ^ 4) := by ring
+
+lemma g4_eq_mul_h4 (a y : ℝ) : g4 a y = y * h4 a y := by unfold g4 h4; ring
+
+lemma abs_g4_le' {a : ℝ} (ha : 0 < a) {y : ℝ} (hy : 0 ≤ y) : |g4 a y| ≤ 20 * y / a ^ 2 := by
+  rw [g4_eq_mul_h4, abs_mul, abs_of_nonneg hy]
+  have h := abs_h4_le ha y
+  have h2 : 20 / (y ^ 2 + a ^ 2) ≤ 20 / a ^ 2 := by gcongr; nlinarith [sq_nonneg y]
+  calc
+    y * |h4 a y| ≤ y * (20 / a ^ 2) := by gcongr; exact h.trans h2
+    _ = 20 * y / a ^ 2 := by ring
+
+/-! ### Product bounds -/
+
+/--
+`|G F| ≤ 4 y^e (1+2πy)^5 / (a² q)` from `|G| ≤ c y^(m+1+e)/a²`, `|F| ≤ d (1+2πy)^(m+1)/(2π y^(m+1)
+q)`,
+`c d ≤ 8π`, `m + 1 ≤ 5`. -/
+lemma prod_bound_aux {a y : ℝ} (ha : 0 < a) (hy : 0 < y) {G Fv c d : ℝ} {m e : ℕ}
+    (hG : |G| ≤ c * y ^ (m + 1 + e) / a ^ 2)
+    (hF : |Fv| ≤ d * (1 + 2 * Real.pi * y) ^ (m + 1) / (2 * Real.pi * y ^ (m + 1) * qe y))
+    (hc : 0 ≤ c) (_hd : 0 ≤ d) (hcd : c * d ≤ 8 * Real.pi) (hm : m + 1 ≤ 5) :
+    |G * Fv| ≤ 4 * y ^ e * (1 + 2 * Real.pi * y) ^ 5 / (a ^ 2 * qe y) := by
+  have hq := qe_pos y
+  have hpi := Real.pi_pos
+  have hb : 1 ≤ 1 + 2 * Real.pi * y := by nlinarith
+  have hpow : (1 + 2 * Real.pi * y) ^ (m + 1) ≤ (1 + 2 * Real.pi * y) ^ 5 :=
+    pow_le_pow_right₀ hb hm
+  rw [abs_mul]
+  calc
+    |G| * |Fv| ≤
+        (c * y ^ (m + 1 + e) / a ^ 2) *
+          (d * (1 + 2 * Real.pi * y) ^ (m + 1) / (2 * Real.pi * y ^ (m + 1) * qe y)) :=
+      mul_le_mul hG hF (abs_nonneg _) (by positivity)
+    _ = (c * d / (2 * Real.pi)) * (y ^ e * (1 + 2 * Real.pi * y) ^ (m + 1) / (a ^ 2 * qe y)) := by
+      rw [pow_add, pow_add]
+      field_simp
+      try ring
+    _ ≤ (8 * Real.pi / (2 * Real.pi)) * (y ^ e * (1 + 2 * Real.pi * y) ^ 5 / (a ^ 2 * qe y)) := by
+      gcongr
+    _ = 4 * y ^ e * (1 + 2 * Real.pi * y) ^ 5 / (a ^ 2 * qe y) := by
+      field_simp
+      try ring
+
+lemma abs_g0F4_le {a : ℝ} (ha : 0 < a) {y : ℝ} (hy : 0 < y) :
+    |g0 a y * F4 y| ≤ 4 * y ^ 0 * (1 + 2 * Real.pi * y) ^ 5 / (a ^ 2 * qe y) :=
+  prod_bound_aux ha hy (m := 4) (e := 0) (abs_g0_le ha hy.le) (abs_F4_le hy) (by norm_num)
+    (by norm_num) (by nlinarith [Real.pi_gt_three]) (by norm_num)
+
+lemma abs_g1F3_le {a : ℝ} (ha : 0 < a) {y : ℝ} (hy : 0 < y) :
+    |g1 a y * F3 y| ≤ 4 * y ^ 0 * (1 + 2 * Real.pi * y) ^ 5 / (a ^ 2 * qe y) :=
+  prod_bound_aux ha hy (m := 3) (e := 0) (abs_g1_le ha hy.le) (abs_F3_le hy) (by norm_num)
+    (by norm_num) (by nlinarith [Real.pi_gt_three]) (by norm_num)
+
+lemma abs_g2F2_le {a : ℝ} (ha : 0 < a) {y : ℝ} (hy : 0 < y) :
+    |g2 a y * F2 y| ≤ 4 * y ^ 0 * (1 + 2 * Real.pi * y) ^ 5 / (a ^ 2 * qe y) :=
+  prod_bound_aux ha hy (m := 2) (e := 0) (abs_g2_le ha hy.le) (abs_F2_le hy) (by norm_num)
+    (by norm_num) (by nlinarith [Real.pi_gt_three]) (by norm_num)
+
+lemma abs_g3F1_le {a : ℝ} (ha : 0 < a) {y : ℝ} (hy : 0 < y) :
+    |g3 a y * F1 y| ≤ 4 * y ^ 0 * (1 + 2 * Real.pi * y) ^ 5 / (a ^ 2 * qe y) :=
+  prod_bound_aux ha hy (m := 1) (e := 0) (abs_g3_le ha hy.le) (abs_F1_le hy) (by norm_num)
+    (by norm_num) (by nlinarith [Real.pi_gt_three]) (by norm_num)
+
+lemma abs_g4F0_le {a : ℝ} (ha : 0 < a) {y : ℝ} (hy : 0 < y) :
+    |g4 a y * F0 y| ≤ 4 * y ^ 0 * (1 + 2 * Real.pi * y) ^ 5 / (a ^ 2 * qe y) :=
+  prod_bound_aux ha hy (m := 0) (e := 0) (by simpa using abs_g4_le' ha hy.le) (abs_F0_le hy)
+    (by norm_num) (by norm_num) (by nlinarith [Real.pi_gt_three]) (by norm_num)
+
+lemma abs_g0F3_le {a : ℝ} (ha : 0 < a) {y : ℝ} (hy : 0 < y) :
+    |g0 a y * F3 y| ≤ 4 * y ^ 1 * (1 + 2 * Real.pi * y) ^ 5 / (a ^ 2 * qe y) :=
+  prod_bound_aux ha hy (m := 3) (e := 1) (abs_g0_le ha hy.le) (abs_F3_le hy) (by norm_num)
+    (by norm_num) (by nlinarith [Real.pi_gt_three]) (by norm_num)
+
+lemma abs_g1F2_le {a : ℝ} (ha : 0 < a) {y : ℝ} (hy : 0 < y) :
+    |g1 a y * F2 y| ≤ 4 * y ^ 1 * (1 + 2 * Real.pi * y) ^ 5 / (a ^ 2 * qe y) :=
+  prod_bound_aux ha hy (m := 2) (e := 1) (abs_g1_le ha hy.le) (abs_F2_le hy) (by norm_num)
+    (by norm_num) (by nlinarith [Real.pi_gt_three]) (by norm_num)
+
+lemma abs_g2F1_le {a : ℝ} (ha : 0 < a) {y : ℝ} (hy : 0 < y) :
+    |g2 a y * F1 y| ≤ 4 * y ^ 1 * (1 + 2 * Real.pi * y) ^ 5 / (a ^ 2 * qe y) :=
+  prod_bound_aux ha hy (m := 1) (e := 1) (abs_g2_le ha hy.le) (abs_F1_le hy) (by norm_num)
+    (by norm_num) (by nlinarith [Real.pi_gt_three]) (by norm_num)
+
+lemma abs_g3F0_le {a : ℝ} (ha : 0 < a) {y : ℝ} (hy : 0 < y) :
+    |g3 a y * F0 y| ≤ 4 * y ^ 1 * (1 + 2 * Real.pi * y) ^ 5 / (a ^ 2 * qe y) :=
+  prod_bound_aux ha hy (m := 0) (e := 1) (abs_g3_le ha hy.le) (abs_F0_le hy) (by norm_num)
+    (by norm_num) (by nlinarith [Real.pi_gt_three]) (by norm_num)
+
+/-! ### `(1 + x)^5 e^{-x} ≤ C e^{-x/2}` -/
+
+lemma poly_exp_bound {x : ℝ} (hx : 0 ≤ x) :
+    (1 + x) ^ 5 * Real.exp (-x) ≤ 122912 * Real.exp (-(x / 2)) := by
+  have h1 : (1 + x) ^ 5 ≤ 32 * (1 + x ^ 5) := by
+    rcases le_or_gt x 1 with h | h
+    · have : (1 + x) ^ 5 ≤ 2 ^ 5 := pow_le_pow_left₀ (by linarith) (by linarith) 5
+      have : 0 ≤ x ^ 5 := by positivity
+      nlinarith
+    · have : (1 + x) ^ 5 ≤ (2 * x) ^ 5 := pow_le_pow_left₀ (by linarith) (by linarith) 5
+      have : 0 ≤ x ^ 5 := by positivity
+      nlinarith
+  have h2 := Real.pow_div_factorial_le_exp (x := x / 2) (by positivity) 5
+  have h3 : x ^ 5 ≤ 3840 * Real.exp (x / 2) := by
+    rw [Nat.factorial] at h2; norm_num at h2
+    have : (x / 2) ^ 5 = x ^ 5 / 32 := by ring
+    rw [this] at h2
+    linarith
+  have h4 : (1 : ℝ) ≤ Real.exp (x / 2) := Real.one_le_exp (by positivity)
+  have h5 : (1 + x) ^ 5 ≤ 122912 * Real.exp (x / 2) := by nlinarith
+  have he : Real.exp (-x) = Real.exp (-(x / 2)) * Real.exp (-(x / 2)) := by
+    rw [← Real.exp_add];
+    ring_nf
+  have he2 : Real.exp (x / 2) * Real.exp (-(x / 2)) = 1 := by rw [← Real.exp_add]; simp
+  calc
+    (1 + x) ^ 5 * Real.exp (-x) ≤ 122912 * Real.exp (x / 2) * Real.exp (-x) := by gcongr
+    _ = 122912 * Real.exp (-(x / 2)) * (Real.exp (x / 2) * Real.exp (-(x / 2))) := by
+      rw [he];
+      ring
+    _ = 122912 * Real.exp (-(x / 2)) := by rw [he2, mul_one]
+
+lemma bound_exp {a y : ℝ} (ha : 0 < a) (hy : 0 < y) (e : ℕ) :
+    4 * y ^ e * (1 + 2 * Real.pi * y) ^ 5 / (a ^ 2 * qe y) ≤
+      4 * 122912 / a ^ 2 * (y ^ e * Real.exp (-(Real.pi * y))) := by
+  have h := poly_exp_bound (x := 2 * Real.pi * y) (by positivity)
+  have hq : qe y = Real.exp (2 * Real.pi * y) := rfl
+  have hinv : 1 / qe y = Real.exp (-(2 * Real.pi * y)) := by rw [hq, Real.exp_neg, one_div]
+  have hx2 : -(2 * Real.pi * y / 2) = -(Real.pi * y) := by ring
+  rw [hx2] at h
+  calc
+    4 * y ^ e * (1 + 2 * Real.pi * y) ^ 5 / (a ^ 2 * qe y) =
+        4 / a ^ 2 * y ^ e * ((1 + 2 * Real.pi * y) ^ 5 * (1 / qe y)) :=
+      by field_simp
+    _ = 4 / a ^ 2 * y ^ e * ((1 + 2 * Real.pi * y) ^ 5 * Real.exp (-(2 * Real.pi * y))) := by
+      rw [hinv]
+    _ ≤ 4 / a ^ 2 * y ^ e * (122912 * Real.exp (-(Real.pi * y))) := by gcongr
+    _ = _ := by ring
+
+/-! ### Continuity, integrability, limits -/
+
+@[fun_prop]
+lemma continuous_qe : Continuous qe := by unfold qe; fun_prop
+
+lemma qe_sub_one_ne {y : ℝ} (hy : y ∈ Ioi (0 : ℝ)) : qe y - 1 ≠ 0 := by
+  have := qe_gt_one (show (0 : ℝ) < y from hy); linarith
+
+lemma continuousOn_F0 : ContinuousOn F0 (Ioi 0) := by
+  unfold F0;
+  exact ContinuousOn.div continuousOn_const (by fun_prop) fun y hy => qe_sub_one_ne hy
+
+lemma continuousOn_F1 : ContinuousOn F1 (Ioi 0) := by
+  unfold F1;
+  exact ContinuousOn.div (by fun_prop) (by fun_prop) fun y hy => pow_ne_zero _ (qe_sub_one_ne hy)
+
+lemma continuousOn_F2 : ContinuousOn F2 (Ioi 0) := by
+  unfold F2;
+  exact ContinuousOn.div (by fun_prop) (by fun_prop) fun y hy => pow_ne_zero _ (qe_sub_one_ne hy)
+
+lemma continuousOn_F3 : ContinuousOn F3 (Ioi 0) := by
+  unfold F3;
+  exact ContinuousOn.div (by fun_prop) (by fun_prop) fun y hy => pow_ne_zero _ (qe_sub_one_ne hy)
+
+lemma continuousOn_F4 : ContinuousOn F4 (Ioi 0) := by
+  unfold F4;
+  exact ContinuousOn.div (by fun_prop) (by fun_prop) fun y hy => pow_ne_zero _ (qe_sub_one_ne hy)
+
+lemma continuous_g0 {a : ℝ} (ha : 0 < a) : Continuous (g0 a) := by
+  unfold g0;
+  exact (by fun_prop : Continuous fun y : ℝ => y ^ 5).div (by fun_prop) (fun y => by positivity)
+
+lemma continuous_g1 {a : ℝ} (ha : 0 < a) : Continuous (g1 a) := by
+  unfold g1;
+  exact
+    (by fun_prop : Continuous fun y : ℝ => y ^ 4 * (5 * a ^ 2 + 3 * y ^ 2)).div (by fun_prop)
+      (fun y => by positivity)
+
+lemma continuous_g2 {a : ℝ} (ha : 0 < a) : Continuous (g2 a) := by
+  unfold g2;
+  exact
+    (by fun_prop :
+          Continuous fun y : ℝ => y ^ 3 * (10 * a ^ 4 + 9 * a ^ 2 * y ^ 2 + 3 * y ^ 4)).div
+      (by fun_prop) (fun y => by positivity)
+
+lemma continuous_g3 {a : ℝ} (ha : 0 < a) : Continuous (g3 a) := by
+  unfold g3;
+  exact
+    (by fun_prop :
+          Continuous fun y : ℝ =>
+            y ^ 2 * (10 * a ^ 6 + 5 * a ^ 4 * y ^ 2 + 4 * a ^ 2 * y ^ 4 + y ^ 6)).div
+      (by fun_prop) (fun y => by positivity)
+
+/-- Integrability on `(0, ∞)` of a product bounded by `4 y^0 (1+2πy)^5/(a² q)`. -/
+lemma integrableOn_of_prod_bound {a : ℝ} (ha : 0 < a) {f : ℝ → ℝ} (hf : ContinuousOn f (Ioi 0))
+    (hb : ∀ y, 0 < y → |f y| ≤ 4 * y ^ 0 * (1 + 2 * Real.pi * y) ^ 5 / (a ^ 2 * qe y)) :
+    IntegrableOn f (Ioi 0) := by
+  have hg :
+    IntegrableOn (fun y : ℝ => 4 * 122912 / a ^ 2 * (y ^ 0 * Real.exp (-(Real.pi * y))))
+      (Ioi 0) :=
+    (integrableOn_pow_mul_exp_neg_mul 0 Real.pi_pos).const_mul _
+  refine hg.mono' (hf.aestronglyMeasurable measurableSet_Ioi) ?_
+  refine (ae_restrict_iff' measurableSet_Ioi).mpr (Filter.Eventually.of_forall fun y hy => ?_)
+  have hy : 0 < y := hy
+  rw [Real.norm_eq_abs]
+  exact (hb y hy).trans (bound_exp ha hy 0)
+
+/-- Vanishing at `0⁺` and `∞` of a product bounded by `4 y (1+2πy)^5/(a² q)`. -/
+lemma tendsto_of_prod_bound {a : ℝ} (ha : 0 < a) {f : ℝ → ℝ}
+    (hb : ∀ y, 0 < y → |f y| ≤ 4 * y ^ 1 * (1 + 2 * Real.pi * y) ^ 5 / (a ^ 2 * qe y)) :
+    Tendsto f (𝓝[>] 0) (𝓝 0) ∧ Tendsto f atTop (𝓝 0) := by
+  have hb' : ∀ y, 0 < y → |f y| ≤ 4 * 122912 / a ^ 2 * (y ^ 1 * Real.exp (-(Real.pi * y))) :=
+    fun y hy => (hb y hy).trans (bound_exp ha hy 1)
+  constructor
+  · rw [tendsto_zero_iff_abs_tendsto_zero]
+    refine
+      squeeze_zero' (Filter.Eventually.of_forall fun y => abs_nonneg _)
+        (eventually_nhdsWithin_of_forall fun y hy => hb' y hy) ?_
+    have hc : Continuous fun y : ℝ => 4 * 122912 / a ^ 2 * (y ^ 1 * Real.exp (-(Real.pi * y))) :=
+      by fun_prop
+    have := (hc.tendsto 0).mono_left (nhdsWithin_le_nhds (s := Ioi 0))
+    convert this using 2
+    simp
+  · rw [tendsto_zero_iff_abs_tendsto_zero]
+    refine
+      squeeze_zero' (Filter.Eventually.of_forall fun y => abs_nonneg _)
+        ((eventually_gt_atTop 0).mono fun y hy => hb' y hy) ?_
+    have h1 :=
+      (Real.tendsto_pow_mul_exp_neg_atTop_nhds_zero 1).comp
+        (tendsto_id.const_mul_atTop Real.pi_pos)
+    have h2 : Tendsto (fun y : ℝ => y ^ 1 * Real.exp (-(Real.pi * y))) atTop (𝓝 0) := by
+      have := h1.const_mul (1 / Real.pi)
+      rw [mul_zero] at this
+      refine this.congr' (Filter.Eventually.of_forall fun y => ?_)
+      simp only [Function.comp, id, pow_one]
+      have hpi : Real.pi ≠ 0 := Real.pi_pos.ne'
+      field_simp
+    simpa using h2.const_mul (4 * 122912 / a ^ 2)
+
+/-! ### The integrations by parts -/
+
+lemma ibp_step {u u' v v' : ℝ → ℝ} (hu : ∀ y ∈ Ioi (0 : ℝ), HasDerivAt u (u' y) y)
+    (hv : ∀ y ∈ Ioi (0 : ℝ), HasDerivAt v (v' y) y)
+    (h1 : IntegrableOn (fun y => u y * v' y) (Ioi 0))
+    (h2 : IntegrableOn (fun y => u' y * v y) (Ioi 0))
+    (h0 : Tendsto (fun y => u y * v y) (𝓝[>] 0) (𝓝 0))
+    (hinf : Tendsto (fun y => u y * v y) atTop (𝓝 0)) :
+    ∫ y in Ioi 0, u y * v' y = -∫ y in Ioi 0, u' y * v y := by
+  have := integral_Ioi_mul_deriv_eq_deriv_mul hu hv h1 h2 h0 hinf
+  rw [this]; ring
+
+/-- **Four integrations by parts**: `∫₀^∞ g F₄ = ∫₀^∞ g₄ F₀`. -/
+theorem integral_g0F4_eq {a : ℝ} (ha : 0 < a) :
+    ∫ y in Ioi 0, g0 a y * F4 y = ∫ y in Ioi 0, g4 a y * F0 y := by
+  have hg0 : ∀ y ∈ Ioi (0 : ℝ), HasDerivAt (g0 a) (g1 a y) y := fun y _ => hasDerivAt_g0 ha y
+  have hg1 : ∀ y ∈ Ioi (0 : ℝ), HasDerivAt (g1 a) (g2 a y) y := fun y _ => hasDerivAt_g1 ha y
+  have hg2 : ∀ y ∈ Ioi (0 : ℝ), HasDerivAt (g2 a) (g3 a y) y := fun y _ => hasDerivAt_g2 ha y
+  have hg3 : ∀ y ∈ Ioi (0 : ℝ), HasDerivAt (g3 a) (g4 a y) y := fun y _ => hasDerivAt_g3 ha y
+  have hF0 : ∀ y ∈ Ioi (0 : ℝ), HasDerivAt F0 (F1 y) y := fun y hy => hasDerivAt_F0 hy
+  have hF1 : ∀ y ∈ Ioi (0 : ℝ), HasDerivAt F1 (F2 y) y := fun y hy => hasDerivAt_F1 hy
+  have hF2 : ∀ y ∈ Ioi (0 : ℝ), HasDerivAt F2 (F3 y) y := fun y hy => hasDerivAt_F2 hy
+  have hF3 : ∀ y ∈ Ioi (0 : ℝ), HasDerivAt F3 (F4 y) y := fun y hy => hasDerivAt_F3 hy
+  have i04 :=
+    integrableOn_of_prod_bound ha ((continuous_g0 ha).continuousOn.mul continuousOn_F4)
+      (fun y hy => abs_g0F4_le ha hy)
+  have i13 :=
+    integrableOn_of_prod_bound ha ((continuous_g1 ha).continuousOn.mul continuousOn_F3)
+      (fun y hy => abs_g1F3_le ha hy)
+  have i22 :=
+    integrableOn_of_prod_bound ha ((continuous_g2 ha).continuousOn.mul continuousOn_F2)
+      (fun y hy => abs_g2F2_le ha hy)
+  have i31 :=
+    integrableOn_of_prod_bound ha ((continuous_g3 ha).continuousOn.mul continuousOn_F1)
+      (fun y hy => abs_g3F1_le ha hy)
+  have i40 :=
+    integrableOn_of_prod_bound ha ((continuous_g4 a ha).continuousOn.mul continuousOn_F0)
+      (fun y hy => abs_g4F0_le ha hy)
+  obtain ⟨z03, t03⟩ :=
+    tendsto_of_prod_bound ha (f := fun y => g0 a y * F3 y) (fun y hy => abs_g0F3_le ha hy)
+  obtain ⟨z12, t12⟩ :=
+    tendsto_of_prod_bound ha (f := fun y => g1 a y * F2 y) (fun y hy => abs_g1F2_le ha hy)
+  obtain ⟨z21, t21⟩ :=
+    tendsto_of_prod_bound ha (f := fun y => g2 a y * F1 y) (fun y hy => abs_g2F1_le ha hy)
+  obtain ⟨z30, t30⟩ :=
+    tendsto_of_prod_bound ha (f := fun y => g3 a y * F0 y) (fun y hy => abs_g3F0_le ha hy)
+  have s0 := ibp_step hg0 hF3 i04 i13 z03 t03
+  have s1 := ibp_step hg1 hF2 i13 i22 z12 t12
+  have s2 := ibp_step hg2 hF1 i22 i31 z21 t21
+  have s3 := ibp_step hg3 hF0 i31 i40 z30 t30
+  rw [s0, s1, s2, s3]; ring
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/HermiteIntegrals.lean b/LeanPool/Zeta5Irrational/HermiteIntegrals.lean
new file mode 100644
index 0000000000..132f630665
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/HermiteIntegrals.lean
@@ -0,0 +1,418 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Hermite
+public import Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan
+import Mathlib.Algebra.Ring.IsFormallyReal
+import Mathlib.Tactic.ENatToNat
+import Mathlib.Tactic.Polynomial.Basic
+import Mathlib.Tactic.ReduceModChar
+import Std.Tactic.BVDecide.Normalize.Prop
+import Mathlib.Tactic.FieldSimp
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+import Mathlib.Tactic.Positivity.Basic
+import Mathlib.Tactic.Ring.Basic
+import Mathlib.Tactic.Ring.RingNF
+
+import Mathlib.Analysis.SpecialFunctions.ImproperIntegrals
+import Mathlib.Analysis.Calculus.Deriv.Mul
+import Mathlib.MeasureTheory.Integral.IntegralEqImproper
+
+/-! # The rational integrals in Hermite's formula
+
+* `∫₀^∞ g₄(y)/y dy = π/a`;
+* `∫₀^∞ g₄(y) y/(y² + n²) dy = π a⁴/(a + n)⁵` for `n > 0` (separate closed forms for `n ≠ a` and
+  `n = a`).
+The antiderivatives (a rational function plus arctan terms) are verified by differentiation.
+-/
+
+@[expose] public section
+
+open Filter Topology Set MeasureTheory
+
+namespace Zeta5Irrational
+
+/-- `g₄(y) / y` as a rational function (continuous at `0`). -/
+noncomputable def h4 (a y : ℝ) : ℝ :=
+  2 * a ^ 4 * (5 * a ^ 4 - 10 * a ^ 2 * y ^ 2 + y ^ 4) / (y ^ 2 + a ^ 2) ^ 5
+
+lemma g4_div_eq_h4 (a : ℝ) {y : ℝ} (hy : y ≠ 0) : g4 a y / y = h4 a y := by
+  unfold g4 h4;
+  field_simp
+
+lemma abs_h4_le {a : ℝ} (ha : 0 < a) (y : ℝ) : |h4 a y| ≤ 20 / (y ^ 2 + a ^ 2) := by
+  unfold h4
+  have hden : 0 < y ^ 2 + a ^ 2 := by positivity
+  rw [abs_div, abs_of_pos (by positivity : (0 : ℝ) < (y ^ 2 + a ^ 2) ^ 5),
+    div_le_div_iff₀ (by positivity) hden]
+  have h1 :
+    |2 * a ^ 4 * (5 * a ^ 4 - 10 * a ^ 2 * y ^ 2 + y ^ 4)| ≤
+      2 * a ^ 4 * (10 * (y ^ 2 + a ^ 2) ^ 2) := by
+    rw [abs_mul, abs_of_nonneg (by positivity : (0 : ℝ) ≤ 2 * a ^ 4)]
+    refine mul_le_mul_of_nonneg_left ?_ (by positivity)
+    rw [abs_le];
+    constructor <;>
+      nlinarith [sq_nonneg (a ^ 2 - y ^ 2), sq_nonneg y, sq_nonneg a,
+        mul_nonneg (sq_nonneg a) (sq_nonneg y)]
+  have h2 : a ^ 4 ≤ (y ^ 2 + a ^ 2) ^ 2 := by
+    nlinarith [sq_nonneg y, sq_nonneg a, mul_nonneg (sq_nonneg y) (sq_nonneg a)]
+  calc
+    |2 * a ^ 4 * (5 * a ^ 4 - 10 * a ^ 2 * y ^ 2 + y ^ 4)| * (y ^ 2 + a ^ 2) ≤
+        2 * a ^ 4 * (10 * (y ^ 2 + a ^ 2) ^ 2) * (y ^ 2 + a ^ 2) :=
+      by gcongr
+    _ ≤ 2 * (y ^ 2 + a ^ 2) ^ 2 * (10 * (y ^ 2 + a ^ 2) ^ 2) * (y ^ 2 + a ^ 2) := by gcongr
+    _ = 20 * (y ^ 2 + a ^ 2) ^ 5 := by ring
+
+lemma integrableOn_of_le_inv_sq {a : ℝ} (ha : 0 < a) {f : ℝ → ℝ}
+    (hf : AEStronglyMeasurable f (volume.restrict (Ioi 0))) {C : ℝ}
+    (hC : ∀ y, 0 < y → |f y| ≤ C / (y ^ 2 + a ^ 2)) : IntegrableOn f (Ioi 0) := by
+  have hg : IntegrableOn (fun y : ℝ => (C * (1 + 1 / a ^ 2)) * (1 + y ^ 2)⁻¹) (Ioi 0) :=
+    (integrable_inv_one_add_sq.const_mul _).integrableOn
+  refine hg.mono' hf ?_
+  refine (ae_restrict_iff' measurableSet_Ioi).mpr (Filter.Eventually.of_forall fun y hy => ?_)
+  have hy : 0 < y := hy
+  rw [Real.norm_eq_abs]
+  have hC0 : 0 ≤ C := by
+    by_contra hneg
+    push Not at hneg
+    have h1 := hC y hy
+    have h2 : C / (y ^ 2 + a ^ 2) < 0 := div_neg_of_neg_of_pos hneg (by positivity)
+    linarith [abs_nonneg (f y)]
+  calc
+    |f y| ≤ C / (y ^ 2 + a ^ 2) := hC y hy
+    _ = C * (1 / (y ^ 2 + a ^ 2)) := by ring
+    _ ≤ C * ((1 + 1 / a ^ 2) * (1 / (1 + y ^ 2))) := by gcongr; exact inv_sq_add_le ha y
+    _ = _ := by ring
+
+/-- The limit `arctan (y / c) → π/2` as `y → ∞`, for `c > 0`. -/
+lemma tendsto_arctan_div_atTop {c : ℝ} (hc : 0 < c) :
+    Tendsto (fun y : ℝ => Real.arctan (y / c)) atTop (𝓝 (Real.pi / 2)) :=
+  (Real.tendsto_arctan_atTop.mono_right nhdsWithin_le_nhds).comp (tendsto_id.atTop_div_const hc)
+
+/-! ### `∫₀^∞ g₄(y)/y dy = π / a` -/
+
+/-- Antiderivative of `h₄`. -/
+noncomputable def A1 (a y : ℝ) : ℝ :=
+  2 / a * Real.arctan (y / a) +
+    2 * y * (12 * a ^ 6 + 14 * a ^ 4 * y ^ 2 + 11 * a ^ 2 * y ^ 4 + 3 * y ^ 6) /
+      (3 * (a ^ 2 + y ^ 2) ^ 4)
+
+lemma hasDerivAt_A1 {a : ℝ} (ha : 0 < a) (y : ℝ) : HasDerivAt (A1 a) (h4 a y) y := by
+  have h1 := ((hasDerivAt_id y).div_const a).arctan.const_mul (2 / a)
+  have hne : 3 * (a ^ 2 + y ^ 2) ^ 4 ≠ 0 := by positivity
+  have hnum :=
+    ((hasDerivAt_id y).const_mul 2).mul
+      (((((hasDerivAt_pow 2 y).const_mul (14 * a ^ 4)).const_add (12 * a ^ 6)).add
+            ((hasDerivAt_pow 4 y).const_mul (11 * a ^ 2))).add
+        ((hasDerivAt_pow 6 y).const_mul 3))
+  have h2 := hnum.div ((((hasDerivAt_pow 2 y).const_add (a ^ 2)).pow 4).const_mul 3) hne
+  have h := h1.add h2
+  refine (h.congr_of_eventuallyEq (Filter.Eventually.of_forall fun t => ?_)).congr_deriv ?_
+  · simp only [A1, id, Pi.add_apply, Pi.mul_apply, Pi.div_apply, Pi.pow_apply]
+  · simp only [h4, id, Pi.add_apply, Pi.mul_apply, Pi.pow_apply]
+    field_simp
+    ring
+
+lemma tendsto_A1_atTop {a : ℝ} (ha : 0 < a) : Tendsto (A1 a) atTop (𝓝 (Real.pi / a)) := by
+  have h1 :
+    Tendsto (fun y : ℝ => 2 / a * Real.arctan (y / a)) atTop (𝓝 (2 / a * (Real.pi / 2))) :=
+    (tendsto_arctan_div_atTop ha).const_mul _
+  have hφ :
+    Tendsto
+      (fun u : ℝ =>
+        2 * u * (12 * a ^ 6 * u ^ 6 + 14 * a ^ 4 * u ^ 4 + 11 * a ^ 2 * u ^ 2 + 3) /
+          (3 * (a ^ 2 * u ^ 2 + 1) ^ 4))
+      (𝓝 0) (𝓝 0) := by
+    have :
+      ContinuousAt
+        (fun u : ℝ =>
+          2 * u * (12 * a ^ 6 * u ^ 6 + 14 * a ^ 4 * u ^ 4 + 11 * a ^ 2 * u ^ 2 + 3) /
+            (3 * (a ^ 2 * u ^ 2 + 1) ^ 4))
+        0 :=
+      ContinuousAt.div (by fun_prop) (by fun_prop) (by norm_num)
+    simpa using this.tendsto
+  have h2 :
+    Tendsto
+      (fun y : ℝ =>
+        2 * y * (12 * a ^ 6 + 14 * a ^ 4 * y ^ 2 + 11 * a ^ 2 * y ^ 4 + 3 * y ^ 6) /
+          (3 * (a ^ 2 + y ^ 2) ^ 4))
+      atTop (𝓝 0) := by
+    refine (hφ.comp tendsto_inv_atTop_zero).congr' ?_
+    filter_upwards [eventually_gt_atTop 0] with y hy
+    simp only [Function.comp]
+    field_simp
+  have := h1.add h2
+  unfold A1
+  convert this using 2
+  field_simp
+  ring
+
+lemma continuous_g4 (a : ℝ) (ha : 0 < a) : Continuous (g4 a) := by
+  unfold g4
+  exact
+    (by fun_prop :
+          Continuous fun y : ℝ => 2 * a ^ 4 * y * (5 * a ^ 4 - 10 * a ^ 2 * y ^ 2 + y ^ 4)).div
+      (by fun_prop) (fun y => by positivity)
+
+lemma integrableOn_h4 {a : ℝ} (ha : 0 < a) : IntegrableOn (h4 a) (Ioi 0) := by
+  have hcont : Continuous (h4 a) := by
+    unfold h4
+    exact
+      (by fun_prop :
+            Continuous fun y : ℝ => 2 * a ^ 4 * (5 * a ^ 4 - 10 * a ^ 2 * y ^ 2 + y ^ 4)).div
+        (by fun_prop) (fun y => by positivity)
+  exact integrableOn_of_le_inv_sq ha hcont.aestronglyMeasurable (fun y _ => abs_h4_le ha y)
+
+/-- `∫₀^∞ g₄(y)/y dy = π/a`. -/
+lemma integral_h4 {a : ℝ} (ha : 0 < a) : ∫ y in Ioi 0, h4 a y = Real.pi / a := by
+  have h :=
+    integral_Ioi_of_hasDerivAt_of_tendsto' (a := 0) (f := A1 a) (f' := h4 a)
+      (fun y _ => hasDerivAt_A1 ha y) (integrableOn_h4 ha) (tendsto_A1_atTop ha)
+  rw [h]; simp [A1]
+
+/-! ### `∫₀^∞ g₄(y) y/(y² + n²) dy = π a⁴/(a + n)⁵` -/
+
+/-- The integrand `g₄(y) · y/(y² + n²)`. -/
+noncomputable def k4 (a n y : ℝ) : ℝ :=
+  g4 a y * (y / (y ^ 2 + n ^ 2))
+
+lemma abs_k4_le {a n : ℝ} (ha : 0 < a) (hn : 0 < n) {y : ℝ} (hy : 0 < y) :
+    |k4 a n y| ≤ 5 * a / n / (y ^ 2 + a ^ 2) := by
+  unfold k4
+  have h1 := abs_g4_le ha hy.le
+  have h2 : y / (y ^ 2 + n ^ 2) ≤ 1 / (2 * n) := by
+    rw [div_le_div_iff₀ (by positivity) (by positivity)]; nlinarith [sq_nonneg (y - n)]
+  have h3 : 0 ≤ y / (y ^ 2 + n ^ 2) := by positivity
+  rw [abs_mul, abs_of_nonneg h3]
+  calc
+    |g4 a y| * (y / (y ^ 2 + n ^ 2)) ≤ 10 * a ^ 3 / (y ^ 2 + a ^ 2) ^ 2 * (1 / (2 * n)) := by
+      gcongr
+    _ ≤ 10 * a ^ 3 / (a ^ 2 * (y ^ 2 + a ^ 2)) * (1 / (2 * n)) := by
+      gcongr;
+      nlinarith [sq_nonneg y]
+    _ = 5 * a / n / (y ^ 2 + a ^ 2) := by field_simp; ring
+
+lemma integrableOn_k4 {a n : ℝ} (ha : 0 < a) (hn : 0 < n) : IntegrableOn (k4 a n) (Ioi 0) := by
+  have hcont : Continuous (k4 a n) := by
+    unfold k4
+    exact
+      (continuous_g4 a ha).mul
+        ((by fun_prop : Continuous fun y : ℝ => y).div (by fun_prop) (fun y => by positivity))
+  exact integrableOn_of_le_inv_sq ha hcont.aestronglyMeasurable (fun y hy => abs_k4_le ha hn hy)
+
+/-- Antiderivative of `k4 a n` for `n ≠ a`; `D = a² - n²`. -/
+noncomputable def A2 (a n y : ℝ) : ℝ :=
+  (2 * a ^ 9 + 20 * a ^ 7 * n ^ 2 + 10 * a ^ 5 * n ^ 4) / (a ^ 2 - n ^ 2) ^ 5 *
+        Real.arctan (y / a) -
+      (10 * a ^ 8 * n + 20 * a ^ 6 * n ^ 3 + 2 * a ^ 4 * n ^ 5) / (a ^ 2 - n ^ 2) ^ 5 *
+        Real.arctan (y / n) +
+    (y ^ 7 * (3 * (2 * a ^ 8 + 12 * a ^ 6 * n ^ 2 + 2 * a ^ 4 * n ^ 4)) +
+            y ^ 5 * (22 * a ^ 10 + 122 * a ^ 8 * n ^ 2 + 2 * a ^ 6 * n ^ 4 - 2 * a ^ 4 * n ^ 6) +
+          y ^ 3 * (28 * a ^ 12 + 146 * a ^ 10 * n ^ 2 - 40 * a ^ 8 * n ^ 4 + 10 * a ^ 6 * n ^ 6) +
+        y * (3 * (8 * a ^ 14 + 8 * a ^ 12 * n ^ 2))) /
+      (3 * (a ^ 2 - n ^ 2) ^ 4 * (a ^ 2 + y ^ 2) ^ 4)
+
+lemma hasDerivAt_A2 {a n : ℝ} (ha : 0 < a) (hn : 0 < n) (hne : n ≠ a) (y : ℝ) :
+    HasDerivAt (A2 a n) (k4 a n y) y := by
+  have hD : a ^ 2 - n ^ 2 ≠ 0 := by
+    intro h
+    have : (a - n) * (a + n) = 0 := by ring_nf; linarith
+    rcases mul_eq_zero.mp this with h' | h'
+    · exact hne (by linarith)
+    · linarith
+  have h1 :=
+    ((hasDerivAt_id y).div_const a).arctan.const_mul
+      ((2 * a ^ 9 + 20 * a ^ 7 * n ^ 2 + 10 * a ^ 5 * n ^ 4) / (a ^ 2 - n ^ 2) ^ 5)
+  have h2 :=
+    ((hasDerivAt_id y).div_const n).arctan.const_mul
+      ((10 * a ^ 8 * n + 20 * a ^ 6 * n ^ 3 + 2 * a ^ 4 * n ^ 5) / (a ^ 2 - n ^ 2) ^ 5)
+  have hne3 : 3 * (a ^ 2 - n ^ 2) ^ 4 * (a ^ 2 + y ^ 2) ^ 4 ≠ 0 := by positivity
+  have hnum :=
+    ((((hasDerivAt_pow 7 y).mul_const
+                  (3 * (2 * a ^ 8 + 12 * a ^ 6 * n ^ 2 + 2 * a ^ 4 * n ^ 4))).add
+              ((hasDerivAt_pow 5 y).mul_const
+                (22 * a ^ 10 + 122 * a ^ 8 * n ^ 2 + 2 * a ^ 6 * n ^ 4 - 2 * a ^ 4 * n ^ 6))).add
+          ((hasDerivAt_pow 3 y).mul_const
+            (28 * a ^ 12 + 146 * a ^ 10 * n ^ 2 - 40 * a ^ 8 * n ^ 4 + 10 * a ^ 6 * n ^ 6))).add
+      ((hasDerivAt_id y).mul_const (3 * (8 * a ^ 14 + 8 * a ^ 12 * n ^ 2)))
+  have h3 :=
+    hnum.div
+      ((((hasDerivAt_pow 2 y).const_add (a ^ 2)).pow 4).const_mul (3 * (a ^ 2 - n ^ 2) ^ 4)) hne3
+  have h := (h1.sub h2).add h3
+  refine (h.congr_of_eventuallyEq (Filter.Eventually.of_forall fun t => ?_)).congr_deriv ?_
+  · simp only [A2, id, Pi.add_apply, Pi.sub_apply, Pi.div_apply, Pi.pow_apply]
+  · simp only [k4, g4, id, Pi.add_apply, Pi.pow_apply]
+    have hyn : y ^ 2 + n ^ 2 ≠ 0 := by positivity
+    have hya : a ^ 2 + y ^ 2 ≠ 0 := by positivity
+    field_simp
+    ring
+
+lemma tendsto_A2_atTop {a n : ℝ} (ha : 0 < a) (hn : 0 < n) (hne : n ≠ a) :
+    Tendsto (A2 a n) atTop (𝓝 (Real.pi * a ^ 4 / (a + n) ^ 5)) := by
+  have hD : a ^ 2 - n ^ 2 ≠ 0 := by
+    intro h
+    have : (a - n) * (a + n) = 0 := by ring_nf; linarith
+    rcases mul_eq_zero.mp this with h' | h'
+    · exact hne (by linarith)
+    · linarith
+  have h1 :=
+    (tendsto_arctan_div_atTop ha).const_mul
+      ((2 * a ^ 9 + 20 * a ^ 7 * n ^ 2 + 10 * a ^ 5 * n ^ 4) / (a ^ 2 - n ^ 2) ^ 5)
+  have h2 :=
+    (tendsto_arctan_div_atTop hn).const_mul
+      ((10 * a ^ 8 * n + 20 * a ^ 6 * n ^ 3 + 2 * a ^ 4 * n ^ 5) / (a ^ 2 - n ^ 2) ^ 5)
+  have hφ :
+    Tendsto
+      (fun u : ℝ =>
+        u *
+            (3 * (2 * a ^ 8 + 12 * a ^ 6 * n ^ 2 + 2 * a ^ 4 * n ^ 4) +
+                  u ^ 2 *
+                    (22 * a ^ 10 + 122 * a ^ 8 * n ^ 2 + 2 * a ^ 6 * n ^ 4 - 2 * a ^ 4 * n ^ 6) +
+                u ^ 4 *
+                  (28 * a ^ 12 + 146 * a ^ 10 * n ^ 2 - 40 * a ^ 8 * n ^ 4 + 10 * a ^ 6 * n ^ 6) +
+              u ^ 6 * (3 * (8 * a ^ 14 + 8 * a ^ 12 * n ^ 2))) /
+          (3 * (a ^ 2 - n ^ 2) ^ 4 * (a ^ 2 * u ^ 2 + 1) ^ 4))
+      (𝓝 0) (𝓝 0) := by
+    have :
+      ContinuousAt
+        (fun u : ℝ =>
+          u *
+              (3 * (2 * a ^ 8 + 12 * a ^ 6 * n ^ 2 + 2 * a ^ 4 * n ^ 4) +
+                    u ^ 2 *
+                      (22 * a ^ 10 + 122 * a ^ 8 * n ^ 2 + 2 * a ^ 6 * n ^ 4 -
+                        2 * a ^ 4 * n ^ 6) +
+                  u ^ 4 *
+                    (28 * a ^ 12 + 146 * a ^ 10 * n ^ 2 - 40 * a ^ 8 * n ^ 4 +
+                      10 * a ^ 6 * n ^ 6) +
+                u ^ 6 * (3 * (8 * a ^ 14 + 8 * a ^ 12 * n ^ 2))) /
+            (3 * (a ^ 2 - n ^ 2) ^ 4 * (a ^ 2 * u ^ 2 + 1) ^ 4))
+        0 :=
+      ContinuousAt.div (by fun_prop) (by fun_prop) (by
+        simp only [ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true, zero_pow, mul_zero, zero_add,
+          one_pow, mul_one, mul_eq_zero, pow_eq_zero_iff, false_or]
+        positivity)
+    simpa using this.tendsto
+  have h3 :
+    Tendsto
+      (fun y : ℝ =>
+        (y ^ 7 * (3 * (2 * a ^ 8 + 12 * a ^ 6 * n ^ 2 + 2 * a ^ 4 * n ^ 4)) +
+                y ^ 5 *
+                  (22 * a ^ 10 + 122 * a ^ 8 * n ^ 2 + 2 * a ^ 6 * n ^ 4 - 2 * a ^ 4 * n ^ 6) +
+              y ^ 3 *
+                (28 * a ^ 12 + 146 * a ^ 10 * n ^ 2 - 40 * a ^ 8 * n ^ 4 + 10 * a ^ 6 * n ^ 6) +
+            y * (3 * (8 * a ^ 14 + 8 * a ^ 12 * n ^ 2))) /
+          (3 * (a ^ 2 - n ^ 2) ^ 4 * (a ^ 2 + y ^ 2) ^ 4))
+      atTop (𝓝 0) := by
+    refine (hφ.comp tendsto_inv_atTop_zero).congr' ?_
+    filter_upwards [eventually_gt_atTop 0] with y hy
+    simp only [Function.comp]
+    field_simp
+  have := (h1.sub h2).add h3
+  unfold A2
+  convert this using 2
+  have hsum : a + n ≠ 0 := by positivity
+  have hdiff : a - n ≠ 0 := sub_ne_zero.mpr (Ne.symm hne)
+  have hDf : a ^ 2 - n ^ 2 = (a - n) * (a + n) := by ring
+  rw [hDf]
+  field_simp
+  ring
+
+/-- `∫₀^∞ g₄(y) y/(y² + n²) dy = π a⁴/(a + n)⁵` for `n ≠ a`. -/
+lemma integral_k4_ne {a n : ℝ} (ha : 0 < a) (hn : 0 < n) (hne : n ≠ a) :
+    ∫ y in Ioi 0, k4 a n y = Real.pi * a ^ 4 / (a + n) ^ 5 := by
+  have h :=
+    integral_Ioi_of_hasDerivAt_of_tendsto' (a := 0) (f := A2 a n) (f' := k4 a n)
+      (fun y _ => hasDerivAt_A2 ha hn hne y) (integrableOn_k4 ha hn) (tendsto_A2_atTop ha hn hne)
+  rw [h]; simp [A2]
+
+/-- Antiderivative of `k4 a a`. -/
+noncomputable def A2e (a y : ℝ) : ℝ :=
+  Real.arctan (y / a) / (16 * a) +
+    y *
+        (-15 * a ^ 8 + 730 * a ^ 6 * y ^ 2 + 32 * a ^ 4 * y ^ 4 + 70 * a ^ 2 * y ^ 6 +
+          15 * y ^ 8) /
+      (240 * (a ^ 2 + y ^ 2) ^ 5)
+
+lemma hasDerivAt_A2e {a : ℝ} (ha : 0 < a) (y : ℝ) : HasDerivAt (A2e a) (k4 a a y) y := by
+  have h1 := ((hasDerivAt_id y).div_const a).arctan.div_const (16 * a)
+  have hne : 240 * (a ^ 2 + y ^ 2) ^ 5 ≠ 0 := by positivity
+  have hnum :=
+    (hasDerivAt_id y).mul
+      (((((((hasDerivAt_pow 2 y).const_mul (730 * a ^ 6)).const_add (-15 * a ^ 8)).add
+                ((hasDerivAt_pow 4 y).const_mul (32 * a ^ 4))).add
+            ((hasDerivAt_pow 6 y).const_mul (70 * a ^ 2))).add
+        ((hasDerivAt_pow 8 y).const_mul 15)))
+  have h2 := hnum.div ((((hasDerivAt_pow 2 y).const_add (a ^ 2)).pow 5).const_mul 240) hne
+  have h := h1.add h2
+  refine (h.congr_of_eventuallyEq (Filter.Eventually.of_forall fun t => ?_)).congr_deriv ?_
+  · simp only [A2e, id, Pi.add_apply, Pi.mul_apply, Pi.div_apply, Pi.pow_apply]
+  · simp only [k4, g4, id, Pi.add_apply, Pi.mul_apply, Pi.pow_apply]
+    have hya : a ^ 2 + y ^ 2 ≠ 0 := by positivity
+    have hya' : y ^ 2 + a ^ 2 ≠ 0 := by positivity
+    field_simp
+    ring
+
+lemma tendsto_A2e_atTop {a : ℝ} (ha : 0 < a) :
+    Tendsto (A2e a) atTop (𝓝 (Real.pi * a ^ 4 / (a + a) ^ 5)) := by
+  have h1 := (tendsto_arctan_div_atTop ha).div_const (16 * a)
+  have hφ :
+    Tendsto
+      (fun u : ℝ =>
+        u *
+            (-15 * a ^ 8 * u ^ 8 + 730 * a ^ 6 * u ^ 6 + 32 * a ^ 4 * u ^ 4 + 70 * a ^ 2 * u ^ 2 +
+              15) /
+          (240 * (a ^ 2 * u ^ 2 + 1) ^ 5))
+      (𝓝 0) (𝓝 0) := by
+    have :
+      ContinuousAt
+        (fun u : ℝ =>
+          u *
+              (-15 * a ^ 8 * u ^ 8 + 730 * a ^ 6 * u ^ 6 + 32 * a ^ 4 * u ^ 4 +
+                  70 * a ^ 2 * u ^ 2 +
+                15) /
+            (240 * (a ^ 2 * u ^ 2 + 1) ^ 5))
+        0 :=
+      ContinuousAt.div (by fun_prop) (by fun_prop) (by norm_num)
+    simpa using this.tendsto
+  have h2 :
+    Tendsto
+      (fun y : ℝ =>
+        y *
+            (-15 * a ^ 8 + 730 * a ^ 6 * y ^ 2 + 32 * a ^ 4 * y ^ 4 + 70 * a ^ 2 * y ^ 6 +
+              15 * y ^ 8) /
+          (240 * (a ^ 2 + y ^ 2) ^ 5))
+      atTop (𝓝 0) := by
+    refine (hφ.comp tendsto_inv_atTop_zero).congr' ?_
+    filter_upwards [eventually_gt_atTop 0] with y hy
+    simp only [Function.comp]
+    field_simp
+  have := h1.add h2
+  unfold A2e
+  convert this using 2
+  field_simp
+  ring
+
+/-- `∫₀^∞ g₄(y) y/(y² + a²) dy = π a⁴/(2a)⁵`. -/
+lemma integral_k4_eq {a : ℝ} (ha : 0 < a) :
+    ∫ y in Ioi 0, k4 a a y = Real.pi * a ^ 4 / (a + a) ^ 5 := by
+  have h :=
+    integral_Ioi_of_hasDerivAt_of_tendsto' (a := 0) (f := A2e a) (f' := k4 a a)
+      (fun y _ => hasDerivAt_A2e ha y) (integrableOn_k4 ha ha) (tendsto_A2e_atTop ha)
+  rw [h]; simp [A2e]
+
+/-- `∫₀^∞ g₄(y) y/(y² + n²) dy = π a⁴/(a + n)⁵` for all `n > 0`. -/
+lemma integral_k4 {a n : ℝ} (ha : 0 < a) (hn : 0 < n) :
+    ∫ y in Ioi 0, k4 a n y = Real.pi * a ^ 4 / (a + n) ^ 5 := by
+  rcases eq_or_ne n a with rfl | hne
+  · exact integral_k4_eq ha
+  · exact integral_k4_ne ha hn hne
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Kernel.lean b/LeanPool/Zeta5Irrational/Kernel.lean
new file mode 100644
index 0000000000..56e1d42dc6
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Kernel.lean
@@ -0,0 +1,256 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.ZeroMass
+import LeanPool.Zeta5Irrational.Gaussian
+import Mathlib.Analysis.LocallyConvex.AbsConvexOpen
+import Mathlib.Tactic.Polynomial.Basic
+import Mathlib.Tactic.ReduceModChar
+import Mathlib.Tactic.FieldSimp
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+import Mathlib.Tactic.Positivity.Basic
+import Mathlib.Tactic.Ring.Basic
+import Mathlib.Tactic.Ring.RingNF
+
+import Mathlib.Analysis.SpecialFunctions.Integrals.Basic
+import Mathlib.MeasureTheory.Integral.DominatedConvergence
+
+/-! # The truncated logarithmic kernel `L_{a,b}` and its limit `log r`
+
+* `Ltr_eq_v` : `L_{a,b}(r) = (1/2) ∫_1^{r²} (e^{-av} - e^{-bv})/v dv` for `r > 0`;
+* `abs_Ltr_le` : `|L_{a,b}(r)| ≤ |log r|`;
+* `continuous_Ltr` : continuity in `r`;
+* `tendsto_Ltr` : `L_{1/(n+1), n+1}(r) → log r`.
+-/
+
+@[expose] public section
+
+open MeasureTheory Set Real intervalIntegral Filter Topology
+
+namespace Zeta5Irrational
+
+lemma integral_exp_neg_mul' {k : ℝ} (hk : k ≠ 0) (c d : ℝ) :
+    ∫ x in c..d, Real.exp (-k * x) = (Real.exp (-k * c) - Real.exp (-k * d)) / k := by
+  have h : ∀ x ∈ uIcc c d, HasDerivAt (fun x => -Real.exp (-k * x) / k) (Real.exp (-k * x)) x := by
+    intro x _
+    have h1 : HasDerivAt (fun x : ℝ => -k * x) (-k) x := by
+      simpa using (hasDerivAt_id x).const_mul (-k)
+    have h2 := (h1.exp.neg).div_const k
+    refine
+      HasDerivAt.congr_deriv
+        (HasDerivAt.congr_of_eventuallyEq h2 (Filter.Eventually.of_forall fun _ => rfl)) ?_
+    field_simp
+  rw [integral_eq_sub_of_hasDerivAt h (by apply Continuous.intervalIntegrable; fun_prop)]
+  ring
+
+/-- Fubini for interval integrals of continuous functions, arbitrary orientation. -/
+lemma intervalIntegral_swap_of_continuous' {f : ℝ → ℝ → ℝ} (hf : Continuous (Function.uncurry f))
+    (a b c d : ℝ) : ∫ x in a..b, ∫ y in c..d, f x y = ∫ y in c..d, ∫ x in a..b, f x y := by
+  rcases le_total a b with hab | hab <;> rcases le_total c d with hcd | hcd
+  · exact intervalIntegral_swap_of_continuous hf hab hcd
+  · rw [integral_symm d c]
+    simp_rw [integral_symm d c, intervalIntegral.integral_neg]
+    rw [intervalIntegral_swap_of_continuous hf hab hcd]
+  · rw [integral_symm b a]
+    simp_rw [integral_symm b a]
+    rw [intervalIntegral.integral_neg, intervalIntegral_swap_of_continuous hf hab hcd]
+  · rw [integral_symm b a, integral_symm d c]
+    simp_rw [integral_symm b a, integral_symm d c, intervalIntegral.integral_neg]
+    rw [intervalIntegral_swap_of_continuous hf hab hcd]
+
+/-- The substituted form of the truncated kernel. -/
+lemma Ltr_eq_v {a b r : ℝ} (ha : 0 < a) (hab : a ≤ b) (hr : 0 < r) :
+    Ltr a b r = (1 / 2) * ∫ v in (1 : ℝ)..r ^ 2, (Real.exp (-a * v) - Real.exp (-b * v)) / v := by
+  unfold Ltr
+  congr 1
+  have e1 :
+    ∀ s ∈ uIcc a b,
+      (Real.exp (-s) - Real.exp (-s * r ^ 2)) / s = ∫ v in (1 : ℝ)..r ^ 2, Real.exp (-s * v) := by
+    intro s hs
+    rw [uIcc_of_le hab] at hs
+    rw [integral_exp_neg_mul' (ha.trans_le hs.1).ne', mul_one]
+  rw [integral_congr e1]
+  rw [intervalIntegral_swap_of_continuous' (f := fun s v => Real.exp (-s * v)) (by fun_prop)]
+  apply integral_congr
+  intro v hv
+  have hv0 : 0 < v := by
+    rcases le_total 1 (r ^ 2) with h | h
+    · rw [uIcc_of_le h] at hv; linarith [hv.1]
+    · rw [uIcc_of_ge h] at hv; exact lt_of_lt_of_le (by positivity) hv.1
+  simp only
+  simp_rw [show ∀ s : ℝ, -s * v = -v * s from fun s => by ring]
+  rw [integral_exp_neg_mul' hv0.ne']
+
+lemma Ltr_integrand_bounds {a b v : ℝ} (ha : 0 < a) (hab : a ≤ b) (hv : 0 < v) :
+    0 ≤ (Real.exp (-a * v) - Real.exp (-b * v)) / v ∧
+      (Real.exp (-a * v) - Real.exp (-b * v)) / v ≤ 1 / v := by
+  constructor
+  · apply div_nonneg _ hv.le
+    rw [sub_nonneg]
+    apply Real.exp_le_exp.mpr
+    nlinarith
+  · apply div_le_div_of_nonneg_right _ hv.le
+    have := Real.exp_pos (-b * v)
+    have := Real.exp_le_one_iff.mpr (show -a * v ≤ 0 by nlinarith)
+    linarith
+
+lemma continuous_Ltr_integrand {a b : ℝ} (_ha : 0 < a) :
+    Continuous fun v => (Real.exp (-a * v) - Real.exp (-b * v)) * (max v 1)⁻¹ := by
+  have : Continuous fun v : ℝ => (max v 1)⁻¹ :=
+    (continuous_id.max continuous_const).inv₀ fun v => (lt_max_of_lt_right one_pos).ne'
+  fun_prop
+
+lemma intervalIntegrable_of_pos {F : ℝ → ℝ} (hF : Continuous F) {c d : ℝ} (hc : 0 < c)
+    (hd : 0 < d) : IntervalIntegrable (fun v => F v / v) volume c d := by
+  apply ContinuousOn.intervalIntegrable
+  refine hF.continuousOn.div continuousOn_id fun v hv => ?_
+  rcases le_total c d with h | h
+  · rw [uIcc_of_le h] at hv; exact (hc.trans_le hv.1).ne'
+  · rw [uIcc_of_ge h] at hv; exact (hd.trans_le hv.1).ne'
+
+lemma log_eq_half_integral {r : ℝ} (hr : 0 < r) :
+    Real.log r = (1 / 2) * ∫ v in (1 : ℝ)..r ^ 2, 1 / v := by
+  rw [integral_one_div_of_pos one_pos (by positivity), div_one, Real.log_pow]
+  push_cast; ring
+
+/-- `|L_{a,b}(r)| ≤ |log r|` for `r > 0`. -/
+lemma abs_Ltr_le {a b r : ℝ} (ha : 0 < a) (hab : a ≤ b) (hr : 0 < r) :
+    |Ltr a b r| ≤ |Real.log r| := by
+  rw [Ltr_eq_v ha hab hr]
+  have hlog := log_eq_half_integral hr
+  have hF :
+    ∀ c d : ℝ,
+      0 < c →
+        0 < d →
+          IntervalIntegrable (fun v => (Real.exp (-a * v) - Real.exp (-b * v)) / v) volume c d :=
+    fun c d hc hd => intervalIntegrable_of_pos (by fun_prop) hc hd
+  have hG : ∀ c d : ℝ, 0 < c → 0 < d → IntervalIntegrable (fun v : ℝ => 1 / v) volume c d :=
+    fun c d hc hd => intervalIntegrable_of_pos continuous_const hc hd
+  have hr2 : 0 < r ^ 2 := by positivity
+  rcases le_total 1 (r ^ 2) with h | h
+  · have h1 : 0 ≤ ∫ v in (1 : ℝ)..r ^ 2, (Real.exp (-a * v) - Real.exp (-b * v)) / v :=
+      integral_nonneg h fun v hv => (Ltr_integrand_bounds ha hab (by linarith [hv.1])).1
+    have h2 :
+      (∫ v in (1 : ℝ)..r ^ 2, (Real.exp (-a * v) - Real.exp (-b * v)) / v) ≤
+        ∫ v in (1 : ℝ)..r ^ 2, 1 / v :=
+      integral_mono_on h (hF _ _ one_pos hr2) (hG _ _ one_pos hr2) fun v hv =>
+        (Ltr_integrand_bounds ha hab (by linarith [hv.1])).2
+    have hr1 : 0 ≤ Real.log r := Real.log_nonneg (by nlinarith)
+    rw [abs_of_nonneg (by linarith), abs_of_nonneg hr1, hlog]
+    linarith
+  · rw [integral_symm (r ^ 2) 1]
+    rw [integral_symm (r ^ 2) 1] at hlog
+    have h1 : 0 ≤ ∫ v in r ^ 2..(1 : ℝ), (Real.exp (-a * v) - Real.exp (-b * v)) / v :=
+      integral_nonneg h fun v hv => (Ltr_integrand_bounds ha hab (by linarith [hv.1])).1
+    have h2 :
+      (∫ v in r ^ 2..(1 : ℝ), (Real.exp (-a * v) - Real.exp (-b * v)) / v) ≤
+        ∫ v in r ^ 2..(1 : ℝ), 1 / v :=
+      integral_mono_on h (hF _ _ hr2 one_pos) (hG _ _ hr2 one_pos) fun v hv =>
+        (Ltr_integrand_bounds ha hab (by linarith [hv.1])).2
+    have hr1 : Real.log r ≤ 0 := Real.log_nonpos hr.le (by nlinarith)
+    rw [abs_of_nonpos (by linarith), abs_of_nonpos hr1, hlog]
+    linarith
+
+/-- Continuity of the truncated kernel in `r`. -/
+lemma continuous_Ltr {a b : ℝ} (ha : 0 < a) (hab : a ≤ b) : Continuous fun r => Ltr a b r := by
+  simp_rw [Ltr_eq_max ha hab]
+  have hinv := continuous_inv_max ha
+  refine
+    continuous_const.mul
+      (continuous_parametric_intervalIntegral_of_continuous' (f := fun r s =>
+        (Real.exp (-s) - Real.exp (-s * r ^ 2)) * (max s a)⁻¹) ?_ a b)
+  have h1 : Continuous fun p : ℝ × ℝ => Real.exp (-p.2) - Real.exp (-p.2 * p.1 ^ 2) := by fun_prop
+  exact h1.mul (hinv.comp continuous_snd)
+
+/-- The difference `log r - L_{a,b}(r)` as an integral. -/
+lemma log_sub_Ltr {a b r : ℝ} (ha : 0 < a) (hab : a ≤ b) (hr : 0 < r) :
+    Real.log r - Ltr a b r =
+      (1 / 2) * ∫ v in (1 : ℝ)..r ^ 2, (1 - Real.exp (-a * v) + Real.exp (-b * v)) / v := by
+  rw [Ltr_eq_v ha hab hr, log_eq_half_integral hr, ← mul_sub, ←
+    integral_sub (intervalIntegrable_of_pos continuous_const one_pos (by positivity))
+      (intervalIntegrable_of_pos (by fun_prop) one_pos (by positivity))]
+  congr 1
+  apply integral_congr
+  intro v _
+  simp only
+  ring
+
+lemma diff_integrand_bounds {a b m v : ℝ} (ha : 0 < a) (hab : a ≤ b) (hm : 0 < m) (hv : m ≤ v) :
+    0 ≤ (1 - Real.exp (-a * v) + Real.exp (-b * v)) / v ∧
+      (1 - Real.exp (-a * v) + Real.exp (-b * v)) / v ≤ a + Real.exp (-b * m) / m := by
+  have hv0 : 0 < v := hm.trans_le hv
+  have h1 : Real.exp (-a * v) ≤ 1 := Real.exp_le_one_iff.mpr (by nlinarith)
+  have h2 : 0 < Real.exp (-b * v) := Real.exp_pos _
+  have h3 : 1 - a * v ≤ Real.exp (-a * v) := by have := Real.add_one_le_exp (-a * v); linarith
+  have h4 : Real.exp (-b * v) ≤ Real.exp (-b * m) := Real.exp_le_exp.mpr (by nlinarith)
+  constructor
+  · apply div_nonneg _ hv0.le; linarith
+  · rw [div_le_iff₀ hv0]
+    have h5 : Real.exp (-b * m) / m * v ≥ Real.exp (-b * m) := by
+      rw [div_mul_eq_mul_div, ge_iff_le, le_div_iff₀ hm]
+      exact mul_le_mul_of_nonneg_left hv (Real.exp_pos _).le
+    nlinarith
+
+/-- `|log r - L_{a,b}(r)| ≤ (1/2) |r² - 1| (a + e^{-b m}/m)` with `m = min 1 r²`. -/
+lemma abs_log_sub_Ltr_le {a b r : ℝ} (ha : 0 < a) (hab : a ≤ b) (hr : 0 < r) :
+    |Real.log r - Ltr a b r| ≤
+      (1 / 2) * ((a + Real.exp (-b * min 1 (r ^ 2)) / min 1 (r ^ 2)) * |r ^ 2 - 1|) := by
+  rw [log_sub_Ltr ha hab hr, abs_mul, abs_of_pos (by norm_num : (0 : ℝ) < 1 / 2)]
+  apply mul_le_mul_of_nonneg_left _ (by norm_num)
+  have hm : 0 < min 1 (r ^ 2) := lt_min one_pos (by positivity)
+  have :=
+    norm_integral_le_of_norm_le_const (a := 1) (b := r ^ 2) (C :=
+      a + Real.exp (-b * min 1 (r ^ 2)) / min 1 (r ^ 2)) (f := fun v =>
+      (1 - Real.exp (-a * v) + Real.exp (-b * v)) / v)
+      (fun v hv =>
+        by
+        have hv' : min 1 (r ^ 2) ≤ v := by
+          rcases le_total 1 (r ^ 2) with h | h
+          · rw [uIoc_of_le h] at hv; exact (min_le_left _ _).trans hv.1.le
+          · rw [uIoc_of_ge h] at hv; exact (min_le_right _ _).trans hv.1.le
+        rw [Real.norm_eq_abs, abs_of_nonneg (diff_integrand_bounds ha hab hm hv').1]
+        exact (diff_integrand_bounds ha hab hm hv').2)
+  rwa [Real.norm_eq_abs] at this
+
+/-- Convergence of the truncated kernel to the logarithm. -/
+lemma tendsto_Ltr {r : ℝ} (hr : 0 < r) :
+    Tendsto (fun n : ℕ => Ltr (1 / ((n : ℝ) + 1)) ((n : ℝ) + 1) r) atTop (𝓝 (Real.log r)) := by
+  have hm : 0 < min 1 (r ^ 2) := lt_min one_pos (by positivity)
+  rw [tendsto_iff_norm_sub_tendsto_zero]
+  have hbound :
+    ∀ n : ℕ,
+      ‖Ltr (1 / ((n : ℝ) + 1)) ((n : ℝ) + 1) r - Real.log r‖ ≤
+        (1 / 2) *
+          ((1 / ((n : ℝ) + 1) + Real.exp (-((n : ℝ) + 1) * min 1 (r ^ 2)) / min 1 (r ^ 2)) *
+            |r ^ 2 - 1|) := by
+    intro n
+    rw [Real.norm_eq_abs, abs_sub_comm]
+    refine abs_log_sub_Ltr_le (by positivity) ?_ hr
+    rw [div_le_iff₀ (by positivity)]
+    nlinarith
+  refine squeeze_zero (fun n => norm_nonneg _) hbound ?_
+  have h1 : Tendsto (fun n : ℕ => 1 / ((n : ℝ) + 1)) atTop (𝓝 0) :=
+    tendsto_one_div_add_atTop_nhds_zero_nat
+  have h2 :
+    Tendsto (fun n : ℕ => Real.exp (-((n : ℝ) + 1) * min 1 (r ^ 2)) / min 1 (r ^ 2)) atTop
+      (𝓝 0) := by
+    have h3 : Tendsto (fun n : ℕ => ((n : ℝ) + 1) * min 1 (r ^ 2)) atTop atTop := by
+      apply Tendsto.atTop_mul_const hm
+      exact tendsto_natCast_atTop_atTop.atTop_add tendsto_const_nhds
+    have := (tendsto_exp_neg_atTop_nhds_zero.comp h3).div_const (min 1 (r ^ 2))
+    rw [zero_div] at this
+    refine this.congr fun n => ?_
+    simp only [Function.comp_apply]
+    ring_nf
+  have := ((h1.add h2).mul_const |r ^ 2 - 1|).const_mul (1 / 2)
+  simpa using this
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Legendre.lean b/LeanPool/Zeta5Irrational/Legendre.lean
new file mode 100644
index 0000000000..dddf1f9b18
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Legendre.lean
@@ -0,0 +1,84 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Construction
+public import Mathlib.NumberTheory.Padics.PadicVal.Basic
+import Mathlib.Tactic.ENatToNat
+import Mathlib.Tactic.Positivity.Basic
+import Mathlib.Tactic.Ring.Basic
+
+/-! # The `p`-adic valuation of `S_K` (5.3)
+
+`v_p(S_K) = 2h v_p(K!) - 12h v_p(N!) - 2 ∑_{i=1}^{h-1} v_p((2i)!) + (h-1) v_p(4)`,
+where each factorial valuation is given by Legendre's formula (`padicValNat_factorial`).
+-/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+/-- `v_p(S_K)`. -/
+noncomputable def vSK (n p : ℕ) : ℤ :=
+  padicValRat p (S n)
+
+lemma padicValNat_finset_prod (p : ℕ) [hp : Fact p.Prime] {ι : Type*} (s : Finset ι) (f : ι → ℕ)
+    (hf : ∀ i ∈ s, f i ≠ 0) : padicValNat p (∏ i ∈ s, f i) = ∑ i ∈ s, padicValNat p (f i) := by
+  classical
+    induction s using Finset.induction_on with
+  | empty => simp
+  | insert a s ha ih =>
+    rw [Finset.prod_insert ha, Finset.sum_insert ha,
+      padicValNat.mul (hf a (Finset.mem_insert_self _ _))
+        (Finset.prod_ne_zero_iff.mpr fun i hi => hf i (Finset.mem_insert_of_mem hi)),
+      ih fun i hi => hf i (Finset.mem_insert_of_mem hi)]
+
+/-- `S_K` as a quotient of natural numbers. -/
+lemma S_eq_div (n : ℕ) :
+    S n =
+      (((40 * n).factorial ^ (2 * (37 * n)) * 4 ^ (37 * n - 1) : ℕ) : ℚ) /
+        (((3 * n).factorial ^ (12 * (37 * n)) * ∏ i ∈ Icc 1 (37 * n - 1), (2 * i).factorial ^ 2 :
+            ℕ) :
+          ℚ) := by
+  unfold S
+  push_cast
+  ring
+
+/-- **(5.3)**: the valuation of `S_K` in terms of factorial valuations. -/
+theorem vS_eq {n : ℕ} (hn : 0 < n) {p : ℕ} (hp : p.Prime) :
+    vSK n p =
+      2 * (37 * n : ℤ) * padicValNat p (40 * n).factorial -
+            12 * (37 * n : ℤ) * padicValNat p (3 * n).factorial -
+          2 * ∑ i ∈ Icc 1 (37 * n - 1), (padicValNat p (2 * i).factorial : ℤ) +
+        ((37 * n : ℤ) - 1) * padicValNat p 4 := by
+  have := Fact.mk hp
+  unfold vSK
+  rw [S_eq_div]
+  have hA : (((40 * n).factorial ^ (2 * (37 * n)) * 4 ^ (37 * n - 1) : ℕ) : ℚ) ≠ 0 := by
+    norm_cast; positivity
+  have hB :
+    (((3 * n).factorial ^ (12 * (37 * n)) * ∏ i ∈ Icc 1 (37 * n - 1), (2 * i).factorial ^ 2 : ℕ) :
+        ℚ) ≠
+      0 := by
+    norm_cast
+    apply mul_ne_zero (by positivity)
+    exact Finset.prod_ne_zero_iff.mpr fun i _ => by positivity
+  rw [padicValRat.div hA hB, ← padicValRat_of_nat, ← padicValRat_of_nat,
+    padicValNat.mul (by positivity) (by positivity),
+    padicValNat.mul (by positivity) (Finset.prod_ne_zero_iff.mpr fun i _ => by positivity),
+    padicValNat.pow, padicValNat.pow, padicValNat.pow,
+    padicValNat_finset_prod p _ _ (fun i _ => by positivity)]
+  simp only [padicValNat.pow]
+  push_cast
+  rw [Nat.cast_sub (by omega : 1 ≤ 37 * n)]
+  push_cast
+  simp only [Finset.mul_sum]
+  ring
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Main.lean b/LeanPool/Zeta5Irrational/Main.lean
new file mode 100644
index 0000000000..79c04d338f
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Main.lean
@@ -0,0 +1,40 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+import LeanPool.Zeta5Irrational.MainEstimate
+public import LeanPool.Zeta5Irrational.Zeta5
+import LeanPool.Zeta5Irrational.Criterion
+public import Mathlib.NumberTheory.Real.Irrational
+
+/-! # Irrationality of ζ(5)
+
+`Zeta5Irrational.irrational_five` is the target statement. It follows from
+`Zeta5Irrational.main_estimate`
+(Theorem 2.1 of the paper, with a smaller decay rate) and the criterion
+`Zeta5Irrational.irrational_of_eventually_exists_int_poly`. It depends only on the axioms `propext`,
+`Classical.choice` and `Quot.sound`.
+-/
+
+@[expose] public section
+
+open Polynomial Filter Topology
+
+namespace Zeta5Irrational
+
+/-- Theorem 1.1 of the paper, from Theorem 2.1. -/
+theorem irrational_zeta5 : Irrational zeta5 := by
+  obtain ⟨c, hc, h⟩ := main_estimate
+  exact
+    irrational_of_eventually_exists_int_poly zeta5 (fun n => 37 * n)
+      (fun n => Real.exp (-c * (n : ℝ) ^ 2)) (tendsto_pow_mul_exp_neg_sq_of_pos hc)
+      (h.mono fun _ ⟨Q, _, hdeg, hpos, hlt⟩ => ⟨Q, hdeg.le, hpos, hlt.le⟩)
+
+theorem irrational_five : ∃ x, Irrational x ∧ riemannZeta 5 = x :=
+  ⟨zeta5, irrational_zeta5, riemannZeta_five⟩
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/MainEstimate.lean b/LeanPool/Zeta5Irrational/MainEstimate.lean
new file mode 100644
index 0000000000..3afdc0b140
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/MainEstimate.lean
@@ -0,0 +1,122 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+import LeanPool.Zeta5Irrational.RealBound
+import LeanPool.Zeta5Irrational.Growth
+import Mathlib.Tactic.ENatToNat
+public import LeanPool.Zeta5Irrational.Construction
+public import LeanPool.Zeta5Irrational.Zeta5
+public import LeanPool.Zeta5Irrational.Constants
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+import Mathlib.Tactic.Positivity.Basic
+import Mathlib.Tactic.Ring.Basic
+
+/-! # The main estimate (Theorem 2.1 of the paper)
+
+The three inputs of Section 7 of "ζ(5) is irrational" (A. Fauzan, 2026), all proved:
+
+* `degree_F`: `F_K` has degree `h = 37 n`, (2.9) (`Zeta5Irrational.Degree`).
+* `real_bound`: a weakening of Proposition 6.3, `0 < F_K(ζ(5))` and
+  `log F_K(ζ(5)) ≤ U K² + 27 K log K + 200 K` (`Zeta5Irrational.RealBound`),
+  using `27` in place of the paper's `24`.
+* `normalization`: positive rationals `m_K` with `m_K F_K ∈ ℤ[X]` (`integral_mN`) and
+  `log m_K ≤ (A_eff + ε) K²` eventually (`growth_mN`), with `A_eff = 1.36 < -U`.
+
+From these, `main_estimate` proves Theorem 2.1 with decay rate `c = -800 (A_eff + U) > 0` in
+place of the paper's `139/5`. This suffices for the irrationality criterion.
+-/
+
+@[expose] public section
+
+open Polynomial Filter Topology
+
+namespace Zeta5Irrational
+
+/-- **(2.9)**: `F_K` has degree `h = 37 n` (proved in `Zeta5Irrational.Degree`). -/
+theorem degree_F (n : ℕ) : (F n).natDegree = 37 * n :=
+  natDegree_F n
+
+/-- A weakening of **Proposition 6.3**, with `27 K log K` in place of `24 K log K`
+(see `Zeta5Irrational.RealBound` for its decomposition). -/
+theorem real_bound (n : ℕ) (hn : 0 < n) :
+    0 < aeval zeta5 (F n) ∧
+      Real.log (aeval zeta5 (F n)) ≤
+        (U : ℝ) * Kr n ^ 2 + 27 * Kr n * Real.log (Kr n) + 200 * Kr n :=
+  real_bound' n hn
+
+/-- **Propositions 5.1 and 5.2** in the form proved here: a positive rational normalisation
+`m_K` with `m_K F_K ∈ ℤ[X]` for all `n ≥ 1` (`integral_mN`), and for every `ε > 0`,
+eventually `log m_K ≤ (A_eff + ε) K²` (`growth_mN`). -/
+theorem normalization :
+    ∃ m : ℕ → ℚ,
+      (∀ n, 0 < m n) ∧
+        (∀ᶠ n in atTop, ∃ Q : ℤ[X], Q.map (Int.castRingHom ℚ) = C (m n) * F n) ∧
+          ∀ ε : ℝ, 0 < ε → ∀ᶠ n in atTop, Real.log (m n) ≤ ((Aeff : ℝ) + ε) * Kr n ^ 2 :=
+  ⟨mN, mN_pos, (eventually_ge_atTop 1).mono fun _ hn => integral_mN hn, growth_mN⟩
+
+/-- The lower-order terms are eventually dominated: for every `δ > 0`,
+`27 K log K + 200 K < δ K²` for all large `n` (with `K = 40 n`). -/
+lemma eventually_lower_order (δ : ℝ) (hδ : 0 < δ) :
+    ∀ᶠ n : ℕ in atTop, 27 * Kr n * Real.log (Kr n) + 200 * Kr n < δ * Kr n ^ 2 := by
+  have hK : Tendsto Kr atTop atTop :=
+    (tendsto_natCast_atTop_atTop).const_mul_atTop (by norm_num : (0 : ℝ) < 40)
+  have h1 : ∀ᶠ x : ℝ in atTop, ‖Real.log x‖ ≤ δ / 108 * ‖x‖ :=
+    Real.isLittleO_log_id_atTop.def (by positivity)
+  have h2 : ∀ᶠ x : ℝ in atTop, 200 < δ / 4 * x := by
+    have := (tendsto_id.const_mul_atTop (by positivity : (0 : ℝ) < δ / 4)).eventually_gt_atTop 200
+    simpa using this
+  have h3 : ∀ᶠ x : ℝ in atTop, (0 : ℝ) < x := eventually_gt_atTop 0
+  filter_upwards [hK.eventually h1, hK.eventually h2, hK.eventually h3] with n h1 h2 h3
+  rw [Real.norm_eq_abs, Real.norm_eq_abs, abs_of_pos h3] at h1
+  have h4 : Real.log (Kr n) ≤ δ / 108 * Kr n := le_trans (le_abs_self _) h1
+  nlinarith [mul_le_mul_of_nonneg_left h4 (by norm_num : (0 : ℝ) ≤ 24 * Kr n)]
+
+/-- **Theorem 2.1 of the paper**, with the decay rate `c = -800 (A_eff + U) > 0` instead of
+`139/5`: for all large `n`, some positive rational multiple `Q_n` of `F_{40n}` is an integer
+polynomial of degree `37 n` with `0 < Q_n(ζ(5)) < exp(-c n²)`. -/
+theorem main_estimate :
+    ∃ c : ℝ,
+      0 < c ∧
+        ∀ᶠ n in atTop,
+          ∃ Q : ℤ[X],
+            (∃ c' : ℚ, 0 < c' ∧ Q.map (Int.castRingHom ℚ) = C c' * F n) ∧
+              Q.natDegree = 37 * n ∧
+                0 < aeval zeta5 Q ∧ aeval zeta5 Q < Real.exp (-c * (n : ℝ) ^ 2) := by
+  obtain ⟨m, hm_pos, hm_int, hm_log⟩ := normalization
+  set δ : ℝ := -((Aeff : ℝ) + U) with hδ_def
+  have hδ : 0 < δ := by
+    have h : ((Aeff + U : ℚ) : ℝ) < 0 := by exact_mod_cast Aeff_margin
+    push_cast at h
+    rw [hδ_def]; linarith
+  refine ⟨800 * δ, by positivity, ?_⟩
+  filter_upwards [hm_int, hm_log (δ / 4) (by positivity),
+    eventually_lower_order (δ / 4) (by positivity), eventually_gt_atTop 0] with n ⟨Q, hQ⟩ hlog
+    hlow hn
+  obtain ⟨hFpos, hFlog⟩ := real_bound n hn
+  have hc : (0 : ℝ) < m n := by exact_mod_cast hm_pos n
+  have hQeval : aeval zeta5 Q = (m n : ℝ) * aeval zeta5 (F n) := by
+    rw [← aeval_map_algebraMap ℚ zeta5 Q]
+    simp only [algebraMap_int_eq, hQ, map_mul, aeval_C]
+    rfl
+  refine ⟨Q, ⟨m n, hm_pos n, hQ⟩, ?_, ?_, ?_⟩
+  · have h1 : (Q.map (Int.castRingHom ℚ)).natDegree = Q.natDegree :=
+      natDegree_map_eq_of_injective (Int.cast_injective) Q
+    rw [← h1, hQ, natDegree_C_mul (by exact_mod_cast (hm_pos n).ne'), degree_F]
+  · rw [hQeval]; exact mul_pos hc hFpos
+  · rw [hQeval, ← Real.log_lt_iff_lt_exp (mul_pos hc hFpos), Real.log_mul hc.ne' hFpos.ne']
+    have hK2 : Kr n ^ 2 = 1600 * (n : ℝ) ^ 2 := by simp only [Kr]; ring
+    have hn2 : (0 : ℝ) < (n : ℝ) ^ 2 := by positivity
+    rw [hK2] at hlog hFlog hlow
+    have : (U : ℝ) = -((Aeff : ℝ)) - δ := by rw [hδ_def]; ring
+    rw [this] at hFlog
+    nlinarith
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Measure.lean b/LeanPool/Zeta5Irrational/Measure.lean
new file mode 100644
index 0000000000..a8e367177a
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Measure.lean
@@ -0,0 +1,80 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import Mathlib.Analysis.SpecialFunctions.Log.Basic
+import Mathlib.Tactic.ReduceModChar
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # The comparison measure of Appendix A and the constants of Section 6
+
+`λ`, `α`, `C*` (6.3), `M₀` (6.2), Table 1 (the sixteen arcsine components of `ρ`), the partial
+sums `S_j`, and the energy `I(ρ)` by formula (A.2).
+-/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+/-- `λ = 37/40`. -/
+noncomputable abbrev lam : ℝ :=
+  37 / 40
+
+/-- `α = 3/40`. -/
+noncomputable abbrev alph : ℝ :=
+  3 / 40
+
+/-- The constant `C*` of (6.3). -/
+noncomputable def Cstar : ℝ :=
+  -2 * lam + 12 * alph * lam * (1 - Real.log alph) + 3 * lam ^ 2 -
+    2 * lam ^ 2 * Real.log (2 * lam)
+
+/-- `M₀ = -1329/200` of (6.2). -/
+noncomputable abbrev M0 : ℝ :=
+  -1329 / 200
+
+/-- Table 1: `10¹² a_j`, `10¹² b_j`, `10¹² c_j`. -/
+def table1 : List (ℕ × ℕ × ℕ) :=
+  [(3906748086, 8992695531, 10515596180), (2312248264, 15340997855, 29471737793),
+    (1402286665, 25730180724, 42934365099), (881725356, 41909578246, 58204231966),
+    (578197906, 65851089563, 69037621310), (396324613, 99481037884, 78873099189),
+    (283911191, 144325727458, 84856120711), (212206188, 201105762729, 88396082127),
+    (165097686, 269345996903, 88303382125), (133347132, 347089554156, 85472321255),
+    (111522114, 430806704415, 78899184238), (96349355, 515561896511, 70353471918),
+    (85815639, 595448778546, 58838976615), (78667711, 664241383483, 44421321106),
+    (74129565, 716160577112, 30462865791), (71741310, 746637295669, 5959622577)]
+
+/-- `a_j`, `b_j`, `c_j` (1-indexed; `0` outside the table). -/
+noncomputable def aρ (j : ℕ) : ℝ :=
+  ((table1.getD (j - 1) (0, 0, 0)).1 : ℝ) / 10 ^ 12
+
+/-- Right endpoint of the `j`th arcsine component, read from the exact rational table. -/
+noncomputable def bρ (j : ℕ) : ℝ :=
+  ((table1.getD (j - 1) (0, 0, 0)).2.1 : ℝ) / 10 ^ 12
+
+/-- Mass of the `j`th arcsine component, read from the exact rational table. -/
+noncomputable def cρ (j : ℕ) : ℝ :=
+  ((table1.getD (j - 1) (0, 0, 0)).2.2 : ℝ) / 10 ^ 12
+
+/-- Partial sums `S_j = ∑_{i ≤ j} c_i`. -/
+noncomputable def Sρ (j : ℕ) : ℝ :=
+  ∑ i ∈ Icc 1 j, cρ i
+
+/-- The logarithmic energy `I(ρ)` of the measure `ρ`, formula (A.2). -/
+noncomputable def Irho : ℝ :=
+  ∑ j ∈ Icc 1 16, (Sρ j ^ 2 - Sρ (j - 1) ^ 2) * Real.log ((bρ j - aρ j) / 4)
+
+/-- The total mass of `ρ` is `λ` (Appendix A.1). -/
+lemma Sρ_sixteen : Sρ 16 = lam := by
+  simp [Sρ, cρ, table1, Finset.sum_Icc_succ_top]
+  norm_num
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Moments.lean b/LeanPool/Zeta5Irrational/Moments.lean
new file mode 100644
index 0000000000..684400726c
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Moments.lean
@@ -0,0 +1,181 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Weight
+public import LeanPool.Zeta5Irrational.Construction
+public import Mathlib.MeasureTheory.Integral.Bochner.Basic
+public import Mathlib.MeasureTheory.Integral.IntegrableOn
+import Mathlib.Analysis.SpecialFunctions.Gamma.Basic
+import Mathlib.NumberTheory.ZetaValues
+import Mathlib.Tactic.ENatToNat
+import Mathlib.Tactic.Polynomial.Basic
+import Mathlib.Tactic.ReduceModChar
+import Std.Tactic.BVDecide.Normalize.Prop
+import Mathlib.Tactic.FieldSimp
+import Mathlib.Tactic.Positivity.Basic
+import Mathlib.Tactic.Ring.Basic
+import Mathlib.Tactic.Ring.RingNF
+
+/-! # The polynomial moments of the weight `w` (first half of Proposition 2.2)
+
+`∫₀^∞ y^{2e} w(y) dy = μ(t^e) = (-1)^e B_{2e+2} (2e+3)(2e+4)(2e+5) / 24`, from the Gamma
+integral, term-wise integration of the series defining `w`, and Euler's formula
+`ζ(2k) = (-1)^{k+1} 2^{2k-1} π^{2k} B_{2k} / (2k)!` (`hasSum_zeta_nat` in Mathlib).
+-/
+
+@[expose] public section
+
+open MeasureTheory Set Filter Topology Finset
+
+namespace Zeta5Irrational
+
+/-- `∫₀^∞ y^m e^{-ry} dy = m! / r^(m+1)`. -/
+lemma integral_pow_mul_exp_neg_mul (m : ℕ) {r : ℝ} (hr : 0 < r) :
+    ∫ y in Ioi 0, y ^ m * Real.exp (-(r * y)) = (m.factorial : ℝ) / r ^ (m + 1) := by
+  have h := Real.integral_rpow_mul_exp_neg_mul_Ioi (a := (m : ℝ) + 1) (by positivity) hr
+  rw [Real.Gamma_nat_eq_factorial, add_sub_cancel_right] at h
+  have hL :
+    ∫ y in Ioi 0, y ^ m * Real.exp (-(r * y)) = ∫ t in Ioi 0, t ^ (m : ℝ) * Real.exp (-(r * t)) :=
+  by
+    refine setIntegral_congr_fun measurableSet_Ioi fun t _ => ?_
+    simp only [Real.rpow_natCast]
+  rw [hL, h]
+  have : (1 / r) ^ ((m : ℝ) + 1) = (1 / r) ^ (m + 1) := by
+    rw [show ((m : ℝ) + 1) = ((m + 1 : ℕ) : ℝ) by push_cast; ring, Real.rpow_natCast]
+  rw [this, one_div_pow, div_mul_eq_mul_div, one_mul]
+
+lemma integrableOn_pow_mul_exp_neg_mul (m : ℕ) {r : ℝ} (hr : 0 < r) :
+    IntegrableOn (fun y : ℝ => y ^ m * Real.exp (-(r * y))) (Ioi 0) := by
+  have h := Real.GammaIntegral_convergent (s := (m : ℝ) + 1) (by positivity)
+  rw [add_sub_cancel_right] at h
+  have h2 : IntegrableOn (fun x : ℝ => Real.exp (-(r * x)) * (r * x) ^ (m : ℝ)) (Ioi 0) := by
+    have :=
+      (integrableOn_Ioi_comp_mul_left_iff (fun x : ℝ => Real.exp (-x) * x ^ (m : ℝ)) 0 hr).mpr
+        (by simpa using h)
+    simpa using this
+  have h3 :
+    IntegrableOn (fun x : ℝ => (1 / r ^ m) * (Real.exp (-(r * x)) * (r * x) ^ (m : ℝ))) (Ioi 0) :=
+    h2.const_mul _
+  refine h3.congr_fun (fun x hx => ?_) measurableSet_Ioi
+  have hx : 0 < x := hx
+  simp only
+  rw [Real.rpow_natCast, mul_pow]
+  field_simp
+
+/-- The summand `y^{2e} · (2π)⁴ y⁵ / 12 · (ℓ+1)⁴ e^{-2π(ℓ+1)y}`. -/
+noncomputable def momTerm (e ℓ : ℕ) (y : ℝ) : ℝ :=
+  (2 * Real.pi) ^ 4 / 12 * ((ℓ : ℝ) + 1) ^ 4 *
+    (y ^ (2 * e + 5) * Real.exp (-(2 * Real.pi * ((ℓ : ℝ) + 1) * y)))
+
+lemma momTerm_eq (e ℓ : ℕ) (y : ℝ) :
+    momTerm e ℓ y =
+      y ^ (2 * e) * ((2 * Real.pi) ^ 4 * y ^ 5 / 12) *
+        (((ℓ : ℝ) + 1) ^ 4 * Real.exp (-(2 * Real.pi * ((ℓ : ℝ) + 1) * y))) :=
+  by unfold momTerm; ring
+
+lemma integrableOn_momTerm (e ℓ : ℕ) : IntegrableOn (momTerm e ℓ) (Ioi 0) := by
+  unfold momTerm
+  exact
+    (integrableOn_pow_mul_exp_neg_mul (2 * e + 5) (r := 2 * Real.pi * ((ℓ : ℝ) + 1))
+          (by positivity)).const_mul
+      _
+
+lemma integral_momTerm (e ℓ : ℕ) :
+    ∫ y in Ioi 0, momTerm e ℓ y =
+      (2 * Real.pi) ^ 4 / 12 * ((2 * e + 5).factorial : ℝ) / (2 * Real.pi) ^ (2 * e + 6) *
+        (1 / ((ℓ : ℝ) + 1) ^ (2 * e + 2)) := by
+  unfold momTerm
+  rw [integral_const_mul, integral_pow_mul_exp_neg_mul _ (by positivity)]
+  have hl : (0 : ℝ) < (ℓ : ℝ) + 1 := by positivity
+  have hpi : (0 : ℝ) < 2 * Real.pi := by positivity
+  rw [mul_pow (2 * Real.pi) ((ℓ : ℝ) + 1)]
+  have hsplit :
+    ((ℓ : ℝ) + 1) ^ (2 * e + 5 + 1) = ((ℓ : ℝ) + 1) ^ 4 * ((ℓ : ℝ) + 1) ^ (2 * e + 2) := by ring
+  rw [hsplit]
+  field_simp
+
+lemma momTerm_nonneg (e ℓ : ℕ) {y : ℝ} (hy : 0 < y) : 0 ≤ momTerm e ℓ y := by
+  unfold momTerm;
+  positivity
+
+/-- **The polynomial moments**: `∫₀^∞ y^{2e} w(y) dy = μ(t^e)`, with integrability. -/
+theorem mono_moment (e : ℕ) :
+    IntegrableOn (fun y => y ^ (2 * e) * w y) (Ioi 0) ∧
+      ∫ y in Ioi 0, y ^ (2 * e) * w y = ((μmono e : ℚ) : ℝ) := by
+  have hpt : ∀ y, y ^ (2 * e) * w y = ∑' ℓ, momTerm e ℓ y := by
+    intro y
+    simp_rw [momTerm_eq]
+    rw [tsum_mul_left]
+    unfold w wS; ring
+  have hnorm : ∀ ℓ, ∫ y in Ioi 0, ‖momTerm e ℓ y‖ = ∫ y in Ioi 0, momTerm e ℓ y := by
+    intro ℓ
+    refine setIntegral_congr_fun measurableSet_Ioi fun y hy => ?_
+    exact Real.norm_of_nonneg (momTerm_nonneg e ℓ hy)
+  have hsum : Summable fun ℓ => ∫ y in Ioi 0, ‖momTerm e ℓ y‖ := by
+    simp_rw [hnorm, integral_momTerm]
+    refine Summable.mul_left _ ?_
+    have :=
+      (summable_nat_add_iff 1).mpr (Real.summable_one_div_nat_pow.mpr (by omega : 1 < 2 * e + 2))
+    refine this.congr fun ℓ => ?_
+    push_cast; ring
+  have hint := integral_tsum_of_summable_integral_norm (fun ℓ => integrableOn_momTerm e ℓ) hsum
+  constructor
+  · -- integrability by domination
+    have hg :
+      IntegrableOn
+        (fun y : ℝ =>
+          32 * (y ^ (2 * e + 1) * Real.exp (-(Real.pi * y))) +
+            11 * (y ^ (2 * e) * Real.exp (-(Real.pi * y))))
+        (Ioi 0) :=
+      ((integrableOn_pow_mul_exp_neg_mul (2 * e + 1) Real.pi_pos).const_mul 32).add
+        ((integrableOn_pow_mul_exp_neg_mul (2 * e) Real.pi_pos).const_mul 11)
+    refine
+      hg.mono' ((measurable_id.pow_const _).aemeasurable.mul aemeasurable_w).aestronglyMeasurable
+        ?_
+    refine (ae_restrict_iff' measurableSet_Ioi).mpr (Filter.Eventually.of_forall fun y hy => ?_)
+    have hy : 0 < y := hy
+    rw [Real.norm_of_nonneg (mul_nonneg (by positivity) (w_pos hy).le)]
+    have := mul_le_mul_of_nonneg_left (w_le hy) (by positivity : (0 : ℝ) ≤ y ^ (2 * e))
+    calc
+      y ^ (2 * e) * w y ≤ y ^ (2 * e) * ((32 * y + 11) * Real.exp (-(Real.pi * y))) := this
+      _ = _ := by ring
+  · rw [setIntegral_congr_fun measurableSet_Ioi (fun y _ => hpt y), ← hint]
+    simp_rw [integral_momTerm]
+    rw [tsum_mul_left]
+      -- Euler's formula
+    have hz := hasSum_zeta_nat (k := e + 1) (by omega)
+    have hz1 := (hasSum_nat_add_iff' 1).mpr hz
+    simp only [Finset.range_one, Finset.sum_singleton, Nat.cast_zero, ne_eq, Nat.mul_eq_zero,
+      OfNat.ofNat_ne_zero, Nat.add_eq_zero_iff, one_ne_zero, and_false, or_self,
+      not_false_eq_true, zero_pow, div_zero, sub_zero] at hz1
+    have hS :
+      ∑' ℓ : ℕ, 1 / ((ℓ : ℝ) + 1) ^ (2 * e + 2) =
+        (-1 : ℝ) ^ (e + 1 + 1) * (2 : ℝ) ^ (2 * (e + 1) - 1) * Real.pi ^ (2 * (e + 1)) *
+            (bernoulli (2 * (e + 1)) : ℝ) /
+          (2 * (e + 1)).factorial := by
+      rw [← hz1.tsum_eq]
+      refine tsum_congr fun ℓ => ?_
+      push_cast
+      ring_nf
+    rw [hS]
+    have hfac :
+      ((2 * e + 5).factorial : ℝ) =
+        ((2 * e + 5) * (2 * e + 4) * (2 * e + 3) : ℕ) * ((2 * e + 2).factorial : ℝ) := by
+      rw [show 2 * e + 5 = (2 * e + 4) + 1 from rfl, Nat.factorial_succ,
+        show 2 * e + 4 = (2 * e + 3) + 1 from rfl, Nat.factorial_succ,
+        show 2 * e + 3 = (2 * e + 2) + 1 from rfl, Nat.factorial_succ]
+      push_cast; ring
+    rw [show 2 * (e + 1) = 2 * e + 2 by ring, show 2 * e + 2 - 1 = 2 * e + 1 by omega, hfac]
+    unfold μmono
+    push_cast
+    have hfac2 : (0 : ℝ) < ((2 * e + 2).factorial : ℝ) := by positivity
+    have hpi : 0 < Real.pi := Real.pi_pos
+    field_simp
+    ring
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Norm.lean b/LeanPool/Zeta5Irrational/Norm.lean
new file mode 100644
index 0000000000..9fe1c945aa
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Norm.lean
@@ -0,0 +1,330 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Arith.OuterEntryMain
+public import LeanPool.Zeta5Irrational.Arith.InnerAlloc
+import LeanPool.Zeta5Irrational.Arith.SmallPrimeF
+import LeanPool.Zeta5Irrational.Arith.SmallPrimeTau
+public import LeanPool.Zeta5Irrational.Legendre
+import Mathlib.Data.Int.Star
+import Mathlib.Data.Nat.Prime.Factorial
+import Mathlib.Data.Rat.Star
+import Mathlib.Tactic.ENatToNat
+import Mathlib.Tactic.Polynomial.Basic
+import Mathlib.Tactic.ReduceModChar
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # The normalisation actually used
+
+For every prime `p ≤ 2h` the local exponent `Lp n p` is one of the proved lower bounds for
+`v_p^G(F_K)`:
+
+* small primes `p M ≤ K`: the bound (3.12), `small_prime_bound`;
+* inner range `K < pM`, `3p ≤ K`: `v_p(S_K) + ⌊2 ∑ wcap⌋` for the water-filling allocation
+  (`inner_bound_cap`), when its side conditions hold;
+* outer range `K < 3p`: `v_p(S_K) + ⌊2 ∑ w - r⌋` (`outer_bound`), when its side conditions hold;
+
+falling back to (3.12) otherwise. The normaliser is `mN n = ∏_{p ≤ 2h} p^{-L_p}`, and
+`integral_mN` proves `mN n · F_K ∈ ℤ[X]` for every `n ≥ 1`.
+-/
+
+@[expose] public section
+
+open Finset Polynomial
+
+namespace Zeta5Irrational
+
+/-- The cutoff `M` separating small primes from the inner range. -/
+def Mcut : ℕ :=
+  400
+
+/-- The small-prime exponent (3.12). -/
+noncomputable def Lsmall (n p : ℕ) : ℤ :=
+  -(6 * (37 * n : ℕ) * (Nat.log p (200 * n) : ℤ) + (37 * n : ℕ) * (padicValNat p 24 : ℤ))
+
+/-- The class offsets `β_c = 6 ℓ_N(c) - ℓ_K(c) - 4`. -/
+noncomputable def betaI (n p : ℕ) {m : ℕ} (c : Fin (m + 1)) : ℤ :=
+  6 * (SX p (3 * n) (((c : ℕ) : ℤ) : ZMod p) : ℤ) - (SX p (40 * n) (((c : ℕ) : ℤ) : ZMod p) : ℤ) -
+    4
+
+/-- The water-filling parameters. -/
+def kloI (n p : ℕ) : ℤ :=
+  -2 * ((40 * n / p : ℕ) : ℤ) - 6
+
+/-- Upper admissible allocation level in the inner-prime range. -/
+def ktopI (n p : ℕ) : ℤ :=
+  8 * ((40 * n / p : ℕ) : ℤ) + 20
+
+/-- Rows reserved for the zero residue class in the inner allocation. -/
+def L0I (n p : ℕ) : ℕ :=
+  3 * (40 * n / p) + 6
+
+/-- The inner allocation. -/
+noncomputable def allocI (n p m : ℕ) : Fin (m + 1) → ℕ :=
+  allocK (betaI n p (m := m)) (37 * n)
+    (kfSel (betaI n p (m := m)) (37 * n) (L0I n p) (kloI n p) (ktopI n p))
+
+/-- The side conditions of the inner bound. -/
+def InnerOK (n p : ℕ) : Prop :=
+  ∃ _hm : 2 * (p / 2) + 1 = p,
+    ∃ hm1 : 1 ≤ p / 2,
+      5 ≤ p ∧
+        Sep p (PlK (40 * n)) ∧
+          40 * n < p ^ 2 ∧
+            5 + 2 * (18 * n + 2 * (37 * n)) - 2 * (40 * n) + 1 < p ^ 2 ∧
+              ∑ c, allocI n p (p / 2) c = 37 * n ∧
+                tmin hm1 (allocI n p (p / 2)) (Bse p (40 * n) (3 * n)) ≤
+                  2 * (allocI n p (p / 2) 0 : ℚ) + (Bse p (40 * n) (3 * n) 0 - 4) / 2
+
+/-- The inner weight sum (for `m = p / 2`). -/
+noncomputable def WI (n p : ℕ) : ℚ :=
+  if hm1 : 1 ≤ p / 2 then ∑ s, wcap hm1 (allocI n p (p / 2)) (Bse p (40 * n) (3 * n)) s else 0
+
+/-- The side conditions of the outer bound. -/
+def OuterOK (n p : ℕ) : Prop :=
+  2 * (p / 2) + 1 = p ∧ 7 ≤ p ∧ 40 * n < 3 * p ∧ 2 * (40 * n) < p ^ 2 ∧ 3 * n < p
+
+/-- The outer weight sum minus the rank loss (for `m = p / 2`). -/
+noncomputable def WO (n p : ℕ) : ℚ :=
+  if hpp : p.Prime then
+    if hm : 2 * (p / 2) + 1 = p then
+      haveI : Fact p.Prime := ⟨hpp⟩
+      2 * ∑ c, ∑ i ∈ range (LoC hm n c), wO hm n c i -
+        ((37 * n - ((2 * p + 22 * n - 3) - (37 * n - 1)) : ℕ) : ℚ)
+    else 0
+  else 0
+
+open Classical in
+/-- **The local exponent.** -/
+noncomputable def Lp (n p : ℕ) : ℤ :=
+  if p * Mcut ≤ 40 * n then Lsmall n p
+  else
+    if 3 * p ≤ 40 * n then (if InnerOK n p then vSK n p + ⌊2 * WI n p⌋ else Lsmall n p)
+    else (if OuterOK n p then vSK n p + ⌊WO n p⌋ else Lsmall n p)
+
+/-- **The normaliser** `m_K = ∏_{p ≤ 2h} p^{-L_p}`. -/
+noncomputable def mN (n : ℕ) : ℚ :=
+  ∏ p ∈ (range (2 * (37 * n) + 1)).filter Nat.Prime, (p : ℚ) ^ (-Lp n p)
+
+lemma mN_pos (n : ℕ) : 0 < mN n := by
+  unfold mN
+  apply Finset.prod_pos
+  intro p hp
+  have : 0 < p := (Finset.mem_filter.mp hp).2.pos
+  exact zpow_pos (by exact_mod_cast this) _
+
+/-! ### Validity of the local exponents -/
+
+variable {p : ℕ} [hp : Fact p.Prime]
+
+omit hp in
+lemma Sep_PlK {K : ℕ} (hK : 2 * K < p ^ 2) : Sep p (PlK K) := by
+  intro r hr s hs hrs _ hdvd
+  obtain ⟨j, hj, ε, hε, rfl⟩ := mem_PlK_iff hr
+  obtain ⟨j', hj', ε', hε', rfl⟩ := mem_PlK_iff hs
+  have h1 := Finset.mem_Icc.mp hj
+  have h2 := Finset.mem_Icc.mp hj'
+  have hlt : |ε * (j : ℤ) - ε' * j'| < (p : ℤ) ^ 2 := by
+    rcases hε with rfl | rfl <;> rcases hε' with rfl | rfl <;> rw [abs_lt] <;> constructor <;>
+      nlinarith
+  have := Int.eq_zero_of_abs_lt_dvd hdvd hlt
+  exact hrs (by linarith)
+
+lemma GV_F_of_Δ {n : ℕ} {r : ℚ} (h : GV p (Δ n) r) : GV p (F n) ((vSK n p : ℚ) + r) := by
+  unfold F
+  have hS : VG p (S n) (vSK n p) := by unfold vSK; right; exact le_refl _
+  exact GV.C_mul hS h
+
+lemma GV_F_Lsmall {n : ℕ} (hn : 1 ≤ n) : GV p (F n) (Lsmall n p) := by
+  have := small_prime_bound (p := p) hn
+  have e :
+    ((Lsmall n p : ℤ) : ℚ) =
+      -(6 * ((37 * n : ℕ) : ℚ) * (Nat.log p (200 * n) : ℚ) +
+          ((37 * n : ℕ) : ℚ) * (padicValNat p 24 : ℚ)) := by
+    simp only [Lsmall, Int.cast_neg, Int.cast_add, Int.cast_mul, Int.cast_natCast, Int.cast_ofNat]
+  rw [e]; exact this
+
+lemma GV_F_inner {n : ℕ} (h : InnerOK n p) : GV p (F n) (vSK n p + ⌊2 * WI n p⌋) := by
+  obtain ⟨hm, hm1, hp5, hsep, hK, hdeg, hsum, hZ⟩ := h
+  have hΔ := inner_bound_cap hp5 hm hm1 n (allocI n p (p / 2)) hsum hsep hK hdeg hZ
+  have hW : WI n p = ∑ s, wcap hm1 (allocI n p (p / 2)) (Bse p (40 * n) (3 * n)) s := by
+    unfold WI; rw [dite_eq_left hm1]
+  rw [← hW] at hΔ
+  have := GV_F_of_Δ hΔ
+  exact this.mono (by have := Int.floor_le (2 * WI n p); linarith)
+
+lemma GV_F_outer {n : ℕ} (h : OuterOK n p) : GV p (F n) (vSK n p + ⌊WO n p⌋) := by
+  obtain ⟨hm, hp7, hK3, hK, hpN⟩ := h
+  have hΔ := outer_bound hm n hp7 hK3 hK hpN
+  have hW :
+    WO n p =
+      2 * ∑ c, ∑ i ∈ range (LoC hm n c), wO hm n c i -
+        ((37 * n - ((2 * p + 22 * n - 3) - (37 * n - 1)) : ℕ) : ℚ) :=
+    by unfold WO; rw [dite_eq_left hp.out, dite_eq_left hm]
+  rw [← hW] at hΔ
+  have := GV_F_of_Δ hΔ
+  exact this.mono (by have := Int.floor_le (WO n p); linarith)
+
+/-- **Every local exponent is valid.** -/
+theorem GV_F_Lp {n : ℕ} (hn : 1 ≤ n) : GV p (F n) (Lp n p) := by
+  unfold Lp
+  split_ifs with h1 h2 h3 h4
+  · exact GV_F_Lsmall hn
+  · exact_mod_cast GV_F_inner h3
+  · exact GV_F_Lsmall hn
+  · exact_mod_cast GV_F_outer h4
+  · exact GV_F_Lsmall hn
+
+/-! ### Integrality -/
+
+lemma padicValNat_factorial_lt {k : ℕ} (hk : k < p) : padicValNat p k.factorial = 0 :=
+  padicValNat.eq_zero_of_not_dvd
+    (fun h => by have := (Nat.Prime.dvd_factorial hp.out).mp h; omega)
+
+lemma vSK_big {n : ℕ} (hn : 1 ≤ n) (hbig : 2 * (37 * n) < p) : vSK n p = 0 := by
+  have h := vS_eq (by omega : 0 < n) hp.out
+  rw [h, padicValNat_factorial_lt (by omega), padicValNat_factorial_lt (by omega)]
+  have h4 : padicValNat p 4 = 0 :=
+    padicValNat.eq_zero_of_not_dvd (fun hd => by have := Nat.le_of_dvd (by norm_num) hd; omega)
+  rw [h4,
+    Finset.sum_eq_zero
+      (fun i hi => by
+        rw [padicValNat_factorial_lt (by have := Finset.mem_Icc.mp hi; omega)];
+        rfl)]
+  simp
+
+lemma OuterOK_big {n : ℕ} (hn : 1 ≤ n) (hbig : 2 * (37 * n) < p) : OuterOK n p := by
+  have hodd : 2 * (p / 2) + 1 = p := by
+    have := hp.out.eq_one_or_self_of_dvd 2
+    rcases Nat.even_or_odd p with ⟨k, hk⟩ | ⟨k, hk⟩
+    · have := this ⟨k, by omega⟩; omega
+    · omega
+  refine ⟨hodd, by omega, by omega, by nlinarith, by omega⟩
+
+lemma WO_big {n : ℕ} (hbig : 2 * (37 * n) < p) : 0 ≤ WO n p := by
+  unfold WO
+  rw [dite_eq_left hp.out]
+  split_ifs with hm
+  · have hsum : ∑ c, ∑ i ∈ range (LoC hm n c), wO hm n c i = 0 := by
+      refine Finset.sum_eq_zero fun c _ => Finset.sum_eq_zero fun i hi => ?_
+      by_cases hc : c = 0
+      · exfalso
+        subst hc
+        have hL : LoC hm n 0 = 0 := by
+          unfold LoC tailC
+          rw [Finset.card_eq_zero, Finset.filter_eq_empty_iff]
+          intro j hj hj0
+          have hj' := Finset.mem_Icc.mp hj
+          have hd : p ∣ j := by exact_mod_cast (jc_eq_zero_iff hm j).mp hj0
+          have := Nat.le_of_dvd (by omega) hd
+          omega
+        rw [hL] at hi; simp at hi
+      · have hnb : nbC hm n c = 0 := by
+          unfold nbC bigC
+          rw [Finset.card_eq_zero, Finset.filter_eq_empty_iff]
+          intro j hj
+          have := Finset.mem_Icc.mp (Finset.mem_filter.mp hj).1
+          omega
+        unfold wO
+        rw [ite_eq_right hc, hnb, ite_eq_right (by omega)]
+    rw [hsum, show 37 * n - ((2 * p + 22 * n - 3) - (37 * n - 1)) = 0 by omega]
+    simp
+  · exact le_refl _
+
+lemma GV_F_big {n : ℕ} (hn : 1 ≤ n) (hbig : 2 * (37 * n) < p) : GV p (F n) 0 := by
+  have h := GV_F_outer (OuterOK_big hn hbig)
+  rw [vSK_big hn hbig] at h
+  refine h.mono ?_
+  have := WO_big (n := n) hbig
+  have h0 : (0 : ℤ) ≤ ⌊WO n p⌋ := Int.floor_nonneg.mpr this
+  have h0' : (0 : ℚ) ≤ (⌊WO n p⌋ : ℚ) := by exact_mod_cast h0
+  push_cast
+  linarith
+
+omit hp in
+lemma padicValRat_mN (n : ℕ) (p : ℕ) [Fact p.Prime] :
+    padicValRat p (mN n) = if p ≤ 2 * (37 * n) then -Lp n p else 0 := by
+  unfold mN
+  rw [padicValRat_finset_prod _ _
+      (fun q hq => by
+        have := (Finset.mem_filter.mp hq).2.pos
+        exact zpow_ne_zero _ (by exact_mod_cast this.ne'))]
+  simp only [padicValRat.zpow, padicValRat.of_nat]
+  have hval :
+    ∀ q ∈ (range (2 * (37 * n) + 1)).filter Nat.Prime,
+      (padicValNat p q : ℤ) = if q = p then 1 else 0 := by
+    intro q hq
+    have hq' : q.Prime := (Finset.mem_filter.mp hq).2
+    have := Fact.mk hq'
+    split_ifs with h
+    · subst h; simp []
+    · rw [padicValNat_primes (Ne.symm h)]; rfl
+  rw [Finset.sum_congr rfl (fun q hq => by rw [hval q hq])]
+  simp only [mul_ite, mul_one, mul_zero]
+  rw [Finset.sum_ite_eq']
+  have hpp : (Fact.out : p.Prime) = (Fact.out : p.Prime) := rfl
+  by_cases h : p ≤ 2 * (37 * n)
+  · rw [ite_eq_left (Finset.mem_filter.mpr ⟨Finset.mem_range.mpr (by omega), Fact.out⟩),
+      ite_eq_left h]
+  · rw [ite_eq_right
+        (fun hm => h (by have := Finset.mem_range.mp (Finset.mem_filter.mp hm).1; omega)),
+      ite_eq_right h]
+
+/-- A rational number which is `p`-integral at every prime is an integer. -/
+lemma rat_int_of_VG {q : ℚ} (h : ∀ p : ℕ, ∀ _ : Fact p.Prime, VG p q 0) : ∃ z : ℤ, q = z := by
+  by_contra hne
+  have hden : q.den ≠ 1 := fun h1 => hne ⟨q.num, (Rat.coe_int_num_of_den_eq_one h1).symm⟩
+  set p := q.den.minFac with hpdef
+  have hpp : p.Prime := Nat.minFac_prime hden
+  have := Fact.mk hpp
+  have hdvd : p ∣ q.den := Nat.minFac_dvd _
+  rcases h p ‹_› with h0 | hv
+  · exact hne ⟨0, by simp [h0]⟩
+  · have hnum : ¬(p : ℤ) ∣ q.num := by
+      intro hd
+      have hc := q.reduced
+      have : p ∣ Nat.gcd q.num.natAbs q.den := Nat.dvd_gcd (Int.natCast_dvd.mp hd) hdvd
+      rw [hc] at this
+      exact hpp.one_lt.ne' (Nat.dvd_one.mp this)
+    have h1 : padicValInt p q.num = 0 := padicValInt.eq_zero_of_not_dvd hnum
+    have h2 : 1 ≤ padicValNat p q.den := one_le_padicValNat_of_dvd q.den_ne_zero hdvd
+    have e : padicValRat p q = padicValInt p q.num - padicValNat p q.den := rfl
+    have hlt : padicValRat p q < 0 := by rw [e, h1]; omega
+    have : (padicValRat p q : ℚ) < 0 := by exact_mod_cast hlt
+    linarith
+
+/-- **Integrality** (Proposition 5.1 for the normaliser `mN`). -/
+theorem integral_mN {n : ℕ} (hn : 1 ≤ n) :
+    ∃ Q : ℤ[X], Q.map (Int.castRingHom ℚ) = C (mN n) * F n := by
+  have hall : ∀ p : ℕ, ∀ _ : Fact p.Prime, GV p (C (mN n) * F n) 0 := by
+    intro p _
+    have hm : VG p (mN n) (if p ≤ 2 * (37 * n) then -Lp n p else 0 : ℤ) := by
+      right;
+      rw [padicValRat_mN]
+    by_cases hbig : p ≤ 2 * (37 * n)
+    · rw [ite_eq_left hbig] at hm
+      have := GV.C_mul hm (GV_F_Lp (p := p) hn)
+      refine this.mono ?_
+      push_cast; linarith
+    · rw [ite_eq_right hbig] at hm
+      have := GV.C_mul hm (GV_F_big (p := p) hn (by omega))
+      simpa using this
+  have hcoef : ∀ k, ∃ z : ℤ, (C (mN n) * F n).coeff k = z := fun k =>
+    rat_int_of_VG fun p hp => hall p hp k
+  have hlift : C (mN n) * F n ∈ lifts (Int.castRingHom ℚ) := by
+    rw [lifts_iff_coeff_lifts]
+    intro k
+    obtain ⟨z, hz⟩ := hcoef k
+    exact ⟨z, by simp [hz]⟩
+  obtain ⟨Q, hQ⟩ := hlift
+  exact ⟨Q, hQ⟩
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/PNT.lean b/LeanPool/Zeta5Irrational/PNT.lean
new file mode 100644
index 0000000000..67f48e53cb
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/PNT.lean
@@ -0,0 +1,29 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import Mathlib.NumberTheory.Chebyshev
+import LeanPool.MooreBound.PrimeNumberTheoremAnd.Consequences
+
+/-! # The prime number theorem, imported from `PrimeNumberTheoremAnd`
+
+The normalisation step (Proposition 5.2 of the paper) needs the prime number theorem in the
+form `θ(x) ~ x` for Chebyshev's function `θ`. Mathlib does not contain it; the
+`PrimeNumberTheoremAnd` development already preserved in `LeanPool.MooreBound` does.
+-/
+
+@[expose] public section
+
+open Filter Asymptotics
+
+namespace Zeta5Irrational
+
+/-- Chebyshev's function `θ(x) = ∑_{p ≤ x} log p` is asymptotic to `x`. -/
+theorem theta_asymptotic : Chebyshev.theta ~[atTop] id :=
+  MooreBound.chebyshev_asymptotic
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/PartialFractions.lean b/LeanPool/Zeta5Irrational/PartialFractions.lean
new file mode 100644
index 0000000000..77fbebf47b
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/PartialFractions.lean
@@ -0,0 +1,139 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Degree
+import Mathlib.Data.Rat.Star
+import Mathlib.LinearAlgebra.Lagrange
+import Mathlib.Tactic.ENatToNat
+import Mathlib.Tactic.Polynomial.Basic
+import Mathlib.Tactic.ReduceModChar
+import Mathlib.Tactic.FieldSimp
+import Mathlib.Tactic.Positivity.Basic
+import Mathlib.Tactic.Ring.Basic
+
+/-! # Partial fractions for `P / D_K`
+
+For a polynomial `P` and `t ≥ 0`, `P(t) / D_K(t) = (P /ₘ D_K)(t) + ∑_j res_j / (t + j²)` with
+`res_j = P(-j²) / D_K'(-j²)`, the residues used in the definition of `μ_X` (`Zeta5Irrational.res`).
+-/
+
+@[expose] public section
+
+open Polynomial Finset
+
+namespace Zeta5Irrational
+
+lemma D_monic (K : ℕ) : (D K).Monic := by
+  unfold D;
+  exact monic_prod_of_monic _ _ fun j _ => monic_X_add_C _
+
+lemma natDegree_D (K : ℕ) : (D K).natDegree = K := by
+  unfold D; rw [natDegree_prod_of_monic _ _ fun j _ => monic_X_add_C _]
+  simp only [natDegree_X_add_C, Finset.sum_const, smul_eq_mul, mul_one, Nat.card_Icc]
+  omega
+
+/-- The polynomial identity behind the partial fractions: the remainder `P %ₘ D_K` equals
+`∑_j res_j ∏_{i ≠ j} (X + i²)`. -/
+lemma modByMonic_eq_sum_res (K : ℕ) (P : ℚ[X]) :
+    P %ₘ D K = ∑ j ∈ Icc 1 K, C (res K P j) * ∏ i ∈ (Icc 1 K).erase j, (X + C ((i : ℚ) ^ 2)) := by
+  have hnodes : Set.InjOn (fun j : ℕ => -(j : ℚ) ^ 2) (Icc 1 K : Set ℕ) := by
+    intro a _ b _ hab
+    have h1 : (a : ℚ) ^ 2 = (b : ℚ) ^ 2 := by simpa using hab
+    have h2 : a ^ 2 = b ^ 2 := by exact_mod_cast h1
+    exact Nat.pow_left_injective two_ne_zero h2
+  set s : Finset ℚ := (Icc 1 K).image (fun j : ℕ => -(j : ℚ) ^ 2) with hs
+  have hcard : s.card = K := by
+    rw [hs, Finset.card_image_of_injOn hnodes];
+    simp
+      -- degree of each summand
+  have hterm :
+    ∀ j ∈ Icc 1 K, (C (res K P j) * ∏ i ∈ (Icc 1 K).erase j, (X + C ((i : ℚ) ^ 2))).degree < K := by
+    intro j hj
+    have hK : 1 ≤ K := (Finset.mem_Icc.mp hj).2.trans' (Finset.mem_Icc.mp hj).1
+    have hprod : (∏ i ∈ (Icc 1 K).erase j, (X + C ((i : ℚ) ^ 2))).natDegree = K - 1 := by
+      rw [natDegree_prod_of_monic _ _ fun i _ => monic_X_add_C _]
+      simp only [natDegree_X_add_C, Finset.sum_const, smul_eq_mul, mul_one,
+        Finset.card_erase_of_mem hj, Nat.card_Icc]
+      omega
+    calc
+      (C (res K P j) * ∏ i ∈ (Icc 1 K).erase j, (X + C ((i : ℚ) ^ 2))).degree ≤
+          (C (res K P j)).degree + (∏ i ∈ (Icc 1 K).erase j, (X + C ((i : ℚ) ^ 2))).degree :=
+        degree_mul_le _ _
+      _ ≤ 0 + ((K - 1 : ℕ) : WithBot ℕ) := by
+        gcongr
+        · exact degree_C_le
+        · rw [← hprod]; exact degree_le_natDegree
+      _ < K := by rw [zero_add]; exact_mod_cast (by omega : K - 1 < K)
+  refine eq_of_degree_sub_lt_of_eval_finset_eq s ?_ ?_
+  · rw [hcard]
+    refine lt_of_le_of_lt (degree_sub_le _ _) (max_lt ?_ ?_)
+    · rcases eq_or_ne (P %ₘ D K) 0 with h | h
+      · rw [h, degree_zero]; exact WithBot.bot_lt_coe _
+      · have := degree_modByMonic_lt P (D_monic K)
+        rwa [degree_eq_natDegree (D_monic K).ne_zero, natDegree_D] at this
+    · refine lt_of_le_of_lt (degree_sum_le _ _) ?_
+      exact (Finset.sup_lt_iff (WithBot.bot_lt_coe _)).mpr hterm
+  · intro x hx
+    rw [hs, Finset.mem_image] at hx
+    obtain ⟨j₀, hj₀, rfl⟩ :=
+      hx
+        -- LHS: `(P %ₘ D_K)(-j₀²) = P(-j₀²)`
+    have hL : (P %ₘ D K).eval (-(j₀ : ℚ) ^ 2) = P.eval (-(j₀ : ℚ) ^ 2) := by
+      conv_rhs => rw [← modByMonic_add_div P (D K)]
+      rw [eval_add, eval_mul, eval_D_eq_zero hj₀, zero_mul, add_zero]
+    rw [hL, eval_finsetSum, Finset.sum_eq_single j₀]
+    · rw [eval_mul, eval_C, eval_prod, res, eval_derivative_D hj₀]
+      have hne := eval_derivative_D_ne_zero hj₀
+      rw [eval_derivative_D hj₀] at hne
+      have :
+        ∏ i ∈ (Icc 1 K).erase j₀, eval (-(j₀ : ℚ) ^ 2) (X + C ((i : ℚ) ^ 2)) =
+          ∏ i ∈ (Icc 1 K).erase j₀, ((i : ℚ) ^ 2 - (j₀ : ℚ) ^ 2) :=
+        Finset.prod_congr rfl fun i _ => by simp; ring
+      rw [this, div_mul_cancel₀ _ hne]
+    · intro j hj hjne
+      rw [eval_mul, eval_prod]
+      refine mul_eq_zero_of_right _ (Finset.prod_eq_zero (i := j₀) ?_ ?_)
+      · exact Finset.mem_erase.mpr ⟨fun h => hjne h.symm, hj₀⟩
+      · simp
+    · intro h; exact absurd hj₀ h
+
+lemma aeval_D_pos (m : ℕ) {t : ℝ} (ht : 0 ≤ t) : 0 < aeval t (D m) := by
+  simp only [D, map_prod, map_add, aeval_X, aeval_C, eq_ratCast, Rat.cast_pow, Rat.cast_natCast]
+  refine Finset.prod_pos fun j hj => ?_
+  have h1 : (1 : ℝ) ≤ j := by exact_mod_cast (Finset.mem_Icc.mp hj).1
+  positivity
+
+/-- **Partial fractions** as real functions, for `t ≥ 0`. -/
+theorem partial_fractions (K : ℕ) (P : ℚ[X]) {t : ℝ} (ht : 0 ≤ t) :
+    aeval t P / aeval t (D K) =
+      aeval t (P /ₘ D K) + ∑ j ∈ Icc 1 K, (res K P j : ℝ) / (t + (j : ℝ) ^ 2) := by
+  have hD : 0 < aeval t (D K) := aeval_D_pos K ht
+  have hsplit : aeval t P = aeval t (P %ₘ D K) + aeval t (D K) * aeval t (P /ₘ D K) := by
+    conv_lhs => rw [← modByMonic_add_div P (D K)]
+    rw [map_add, map_mul]
+  rw [hsplit, add_div, mul_div_cancel_left₀ _ hD.ne', add_comm]
+  congr 1
+  rw [modByMonic_eq_sum_res, map_sum, Finset.sum_div]
+  refine Finset.sum_congr rfl fun j hj => ?_
+  rw [map_mul, aeval_C, map_prod]
+  simp only [map_add, aeval_X, aeval_C, eq_ratCast, Rat.cast_pow, Rat.cast_natCast]
+  have hj2 : 0 < t + (j : ℝ) ^ 2 := by
+    have : (1 : ℝ) ≤ j := by exact_mod_cast (Finset.mem_Icc.mp hj).1
+    positivity
+  have hDt : aeval t (D K) = (t + (j : ℝ) ^ 2) * ∏ i ∈ (Icc 1 K).erase j, (t + (i : ℝ) ^ 2) := by
+    simp only [D, map_prod, map_add, aeval_X, aeval_C, eq_ratCast, Rat.cast_pow, Rat.cast_natCast]
+    rw [← Finset.mul_prod_erase _ _ hj]
+  rw [hDt]
+  have hprod : 0 < ∏ i ∈ (Icc 1 K).erase j, (t + (i : ℝ) ^ 2) :=
+    Finset.prod_pos fun i hi =>
+      by
+      have : (1 : ℝ) ≤ i := by exact_mod_cast (Finset.mem_Icc.mp (Finset.mem_of_mem_erase hi)).1
+      positivity
+  field_simp
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Positivity.lean b/LeanPool/Zeta5Irrational/Positivity.lean
new file mode 100644
index 0000000000..2ff1a93a95
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Positivity.lean
@@ -0,0 +1,261 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Weight
+public import LeanPool.Zeta5Irrational.Zeta5
+public import LeanPool.Zeta5Irrational.Degree
+import LeanPool.Zeta5Irrational.HermiteFormula
+import LeanPool.Zeta5Irrational.Moments
+import LeanPool.Zeta5Irrational.PartialFractions
+import Mathlib.Algebra.Ring.IsFormallyReal
+import Mathlib.Tactic.ReduceModChar
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.Positivity.Basic
+import Mathlib.Tactic.Ring.Basic
+
+/-! # Proposition 2.2: the moment representation, and positivity of `Δ_K(ζ(5))`
+
+* `pole_moment` : Hermite's formula at `s = 5`,
+  `∫₀^∞ w(y) / (y² + j²) dy = j⁴ (ζ(5) - H_j^{(5)}) - 1/4 + 1/(2j)` — proved in
+  `Zeta5Irrational.HermiteFormula` (four integrations by
+parts and the Mittag-Leffler expansion);
+* `moment_rep` : `μ_{ζ(5)}(P / D_K) = ∫₀^∞ (P / D_K)(y²) w(y) dy` with integrability, proved from
+  the polynomial moments (`mono_moment`), the pole moments and the partial fractions
+  (**Proposition 2.2 is now fully proved**);
+* `Δ_pos` : `Δ_K(ζ(5)) > 0`, since `G_K(ζ(5))` is the Gram matrix of a positive weight.
+-/
+
+@[expose] public section
+
+open Polynomial MeasureTheory Set Finset
+
+open scoped Matrix
+
+namespace Zeta5Irrational
+
+/-- **Hermite's formula at `s = 5`** (the pole moments of Proposition 2.2), proved in
+`Zeta5Irrational.HermiteFormula`. -/
+theorem pole_moment (j : ℕ) (hj : 1 ≤ j) :
+    ∫ y in Ioi 0, w y / (y ^ 2 + (j : ℝ) ^ 2) =
+      (j : ℝ) ^ 4 * (zeta5 - (H5 j : ℝ)) - 1 / 4 + 1 / (2 * (j : ℝ)) :=
+  hermite_pole j hj
+
+lemma integrableOn_pole (j : ℕ) (hj : 1 ≤ j) :
+    IntegrableOn (fun y => w y / (y ^ 2 + (j : ℝ) ^ 2)) (Ioi 0) := by
+  have hg :
+    IntegrableOn
+      (fun y : ℝ =>
+        32 * (y ^ 1 * Real.exp (-(Real.pi * y))) + 11 * (y ^ 0 * Real.exp (-(Real.pi * y))))
+      (Ioi 0) :=
+    ((integrableOn_pow_mul_exp_neg_mul 1 Real.pi_pos).const_mul 32).add
+      ((integrableOn_pow_mul_exp_neg_mul 0 Real.pi_pos).const_mul 11)
+  have hj' : (1 : ℝ) ≤ j := by exact_mod_cast hj
+  refine
+    hg.mono'
+      (aemeasurable_w.div
+          (by fun_prop :
+              Measurable fun y : ℝ => y ^ 2 + (j : ℝ) ^ 2).aemeasurable).aestronglyMeasurable
+      ?_
+  refine (ae_restrict_iff' measurableSet_Ioi).mpr (Filter.Eventually.of_forall fun y hy => ?_)
+  have hy : 0 < y := hy
+  have hw := w_pos hy
+  have hden : (1 : ℝ) ≤ y ^ 2 + (j : ℝ) ^ 2 := by nlinarith
+  rw [Real.norm_of_nonneg (div_nonneg hw.le (by positivity))]
+  calc
+    w y / (y ^ 2 + (j : ℝ) ^ 2) ≤ w y := div_le_self hw.le hden
+    _ ≤ (32 * y + 11) * Real.exp (-(Real.pi * y)) := (w_le hy)
+    _ = _ := by ring
+
+/-- `P(y²) / D_K(y²) · w(y)`. -/
+noncomputable def integrand (K : ℕ) (P : ℚ[X]) (y : ℝ) : ℝ :=
+  aeval (y ^ 2) P / aeval (y ^ 2) (D K) * w y
+
+/-- The pointwise decomposition of the integrand, for `y > 0`. -/
+lemma integrand_eq_sum (K : ℕ) (P : ℚ[X]) {y : ℝ} (hy : 0 < y) :
+    integrand K P y =
+      (∑ e ∈ range ((P /ₘ D K).natDegree + 1), ((P /ₘ D K).coeff e : ℝ) * (y ^ (2 * e) * w y)) +
+        ∑ j ∈ Icc 1 K, (res K P j : ℝ) * (w y / (y ^ 2 + (j : ℝ) ^ 2)) := by
+  unfold integrand
+  rw [partial_fractions K P (t := y ^ 2) (by positivity), aeval_eq_sum_range, add_mul,
+    Finset.sum_mul, Finset.sum_mul]
+  congr 1
+  · refine Finset.sum_congr rfl fun e _ => ?_
+    try simp only [Rat.smul_def, ← pow_mul]
+    ring
+  · refine Finset.sum_congr rfl fun j _ => ?_
+    ring
+
+/-- The value `μ_{ζ(5)}(P / D_K)` in terms of the coefficients of the quotient and residues. -/
+lemma aeval_μX (K : ℕ) (P : ℚ[X]) :
+    aeval zeta5 (μX K P) =
+      (∑ e ∈ range ((P /ₘ D K).natDegree + 1), ((P /ₘ D K).coeff e : ℝ) * (μmono e : ℝ)) +
+        ∑ j ∈ Icc 1 K,
+          (res K P j : ℝ) * ((j : ℝ) ^ 4 * (zeta5 - (H5 j : ℝ)) - 1 / 4 + 1 / (2 * (j : ℝ))) := by
+  simp only [μX, map_add, aeval_C, map_sum, map_mul, poleValue, map_sub, aeval_X, eq_ratCast]
+  congr 1
+  · unfold μpoly
+    rw [Polynomial.sum_over_range' _ (fun n => by simp) _ (Nat.lt_succ_self _)]
+    push_cast
+    rfl
+  · refine Finset.sum_congr rfl fun j _ => ?_
+    simp only [res]
+    push_cast
+    ring
+
+/-- **Proposition 2.2** (the moment representation): for every polynomial `P`, the function
+`(P / D_K)(y²) w(y)` is integrable on `(0, ∞)` and its integral is `μ_{ζ(5)}(P / D_K)`. -/
+theorem moment_rep (K : ℕ) (P : ℚ[X]) :
+    IntegrableOn (integrand K P) (Ioi 0) ∧ aeval zeta5 (μX K P) = ∫ y in Ioi 0, integrand K P y :=
+  by
+  set Q := P /ₘ D K with hQ
+  have hint1 :
+    ∀ e ∈ range (Q.natDegree + 1),
+      IntegrableOn (fun y => (Q.coeff e : ℝ) * (y ^ (2 * e) * w y)) (Ioi 0) :=
+    fun e _ => (mono_moment e).1.const_mul _
+  have hint2 :
+    ∀ j ∈ Finset.Icc 1 K,
+      IntegrableOn (fun y => (res K P j : ℝ) * (w y / (y ^ 2 + (j : ℝ) ^ 2))) (Ioi 0) :=
+    fun j hj => (integrableOn_pole j (Finset.mem_Icc.mp hj).1).const_mul _
+  have hsumint :
+    IntegrableOn
+      (fun y =>
+        (∑ e ∈ range (Q.natDegree + 1), (Q.coeff e : ℝ) * (y ^ (2 * e) * w y)) +
+          ∑ j ∈ Icc 1 K, (res K P j : ℝ) * (w y / (y ^ 2 + (j : ℝ) ^ 2)))
+      (Ioi 0) :=
+    (integrable_finsetSum _ hint1).add (integrable_finsetSum _ hint2)
+  have hEq :
+    EqOn
+      (fun y =>
+        (∑ e ∈ range (Q.natDegree + 1), (Q.coeff e : ℝ) * (y ^ (2 * e) * w y)) +
+          ∑ j ∈ Icc 1 K, (res K P j : ℝ) * (w y / (y ^ 2 + (j : ℝ) ^ 2)))
+      (integrand K P) (Ioi 0) :=
+    fun y hy => (integrand_eq_sum K P hy).symm
+  refine ⟨hsumint.congr_fun hEq measurableSet_Ioi, ?_⟩
+  rw [← setIntegral_congr_fun measurableSet_Ioi hEq,
+    integral_add (integrable_finsetSum _ hint1) (integrable_finsetSum _ hint2),
+    integral_finsetSum _ hint1, integral_finsetSum _ hint2, aeval_μX]
+  congr 1
+  · refine Finset.sum_congr rfl fun e _ => ?_
+    rw [integral_const_mul, (mono_moment e).2]
+  · refine Finset.sum_congr rfl fun j hj => ?_
+    rw [integral_const_mul, pole_moment j (Finset.mem_Icc.mp hj).1]
+
+/-- The positive weight `ν(y) = D_N(y²)^6 / D_K(y²) · w(y)`. -/
+noncomputable def ν (n : ℕ) (y : ℝ) : ℝ :=
+  aeval (y ^ 2) (D (3 * n)) ^ 6 / aeval (y ^ 2) (D (40 * n)) * w y
+
+lemma ν_pos (n : ℕ) {y : ℝ} (hy : 0 < y) : 0 < ν n y := by
+  unfold ν
+  have h1 := aeval_D_pos (3 * n) (t := y ^ 2) (by positivity)
+  have h2 := aeval_D_pos (40 * n) (t := y ^ 2) (by positivity)
+  have h3 := w_pos hy
+  positivity
+
+lemma integrand_eq (n : ℕ) (i j : ℕ) (y : ℝ) :
+    integrand (40 * n) (D (3 * n) ^ 6 * X ^ (i + j)) y = (y ^ 2) ^ i * (y ^ 2) ^ j * ν n y := by
+  simp only [integrand, ν, map_mul, map_pow, aeval_X]
+  rw [pow_add]; ring
+
+/-- **Positivity**: `Δ_K(ζ(5)) > 0`, from the moment representation. -/
+theorem Δ_pos (n : ℕ) (_hn : 0 < n) : 0 < aeval zeta5 (Δ n) := by
+  have hdet : aeval zeta5 (Δ n) = ((G n).map (aeval zeta5)).det := by
+    have := RingHom.map_det (aeval zeta5 : ℚ[X] →ₐ[ℚ] ℝ).toRingHom (G n)
+    simpa [Δ, RingHom.mapMatrix_apply] using this
+  rw [hdet]
+  set M : Matrix (Fin (37 * n)) (Fin (37 * n)) ℝ := (G n).map (aeval zeta5) with hM
+  have hentry :
+    ∀ i j : Fin (37 * n),
+      M i j = ∫ y in Ioi 0, integrand (40 * n) (D (3 * n) ^ 6 * X ^ ((i : ℕ) + (j : ℕ))) y :=
+    fun i j => (moment_rep _ _).2
+  have hint :
+    ∀ i j : Fin (37 * n),
+      IntegrableOn (integrand (40 * n) (D (3 * n) ^ 6 * X ^ ((i : ℕ) + (j : ℕ)))) (Ioi 0) :=
+    fun i j => (moment_rep _ _).1
+  apply Matrix.PosDef.det_pos
+  refine Matrix.PosDef.of_dotProduct_mulVec_pos ?_ ?_
+  · refine Matrix.IsHermitian.ext fun i j => ?_
+    simp only [star_trivial, hM, Matrix.map_apply, G]
+    rw [Nat.add_comm (j : ℕ) i]
+  · intro x hx
+    set f : Fin (37 * n) → Fin (37 * n) → ℝ → ℝ := fun i j y =>
+      x i * x j * integrand (40 * n) (D (3 * n) ^ 6 * X ^ ((i : ℕ) + (j : ℕ))) y with hf
+    have hfint : ∀ i j, IntegrableOn (f i j) (Ioi 0) := fun i j => (hint i j).const_mul _
+    have hpt : ∀ y, ∑ i, ∑ j, f i j y = (∑ i, x i * (y ^ 2) ^ (i : ℕ)) ^ 2 * ν n y := by
+      intro y
+      simp only [hf, integrand_eq]
+      rw [sq (∑ i, x i * (y ^ 2) ^ (i : ℕ)), Finset.sum_mul_sum, Finset.sum_mul]
+      refine Finset.sum_congr rfl fun i _ => ?_
+      rw [Finset.sum_mul]
+      refine Finset.sum_congr rfl fun j _ => ?_
+      ring
+    have hq : star x ⬝ᵥ (M *ᵥ x) = ∫ y in Ioi 0, (∑ i, x i * (y ^ 2) ^ (i : ℕ)) ^ 2 * ν n y := by
+      have h1 : star x ⬝ᵥ (M *ᵥ x) = ∑ i, ∑ j, ∫ y in Set.Ioi 0, f i j y := by
+        simp only [dotProduct, Matrix.mulVec, star_trivial, hentry, hf, Finset.mul_sum]
+        refine Finset.sum_congr rfl fun i _ => Finset.sum_congr rfl fun j _ => ?_
+        rw [integral_const_mul]; ring
+      rw [h1]
+      have h2 : ∀ i, ∑ j, ∫ y in Set.Ioi 0, f i j y = ∫ y in Set.Ioi 0, ∑ j, f i j y := fun i =>
+        (integral_finsetSum _ (fun j _ => hfint i j)).symm
+      simp_rw [h2]
+      rw [← integral_finsetSum _ (fun i _ => integrable_finsetSum _ (fun j _ => hfint i j))]
+      exact integral_congr_ae (Filter.Eventually.of_forall fun y => hpt y)
+    rw [hq]
+    have hsq : IntegrableOn (fun y => (∑ i, x i * (y ^ 2) ^ (i : ℕ)) ^ 2 * ν n y) (Ioi 0) := by
+      have : (fun y => (∑ i, x i * (y ^ 2) ^ (i : ℕ)) ^ 2 * ν n y) = fun y => ∑ i, ∑ j, f i j y :=
+        funext fun y => (hpt y).symm
+      rw [this]
+      exact integrable_finsetSum _ fun i _ => integrable_finsetSum _ fun j _ => hfint i j
+    have hnn :
+      0 ≤ᵐ[volume.restrict (Ioi 0)] fun y => (∑ i, x i * (y ^ 2) ^ (i : ℕ)) ^ 2 * ν n y := by
+      refine (ae_restrict_iff' measurableSet_Ioi).mpr (Filter.Eventually.of_forall fun y hy => ?_)
+      exact mul_nonneg (sq_nonneg _) (ν_pos n hy).le
+    rw [setIntegral_pos_iff_support_of_nonneg_ae hnn hsq]
+    set q : ℝ[X] := ∑ i : Fin (37 * n), C (x i) * X ^ (i : ℕ) with hqdef
+    have hqeval : ∀ y : ℝ, q.eval (y ^ 2) = ∑ i, x i * (y ^ 2) ^ (i : ℕ) := by
+      intro y;
+      simp [hqdef, eval_finsetSum]
+    have hq0 : q ≠ 0 := by
+      intro h0
+      apply hx
+      funext i
+      have hc : q.coeff (i : ℕ) = x i := by
+        simp only [hqdef, finsetSum_coeff, coeff_C_mul_X_pow]
+        rw [Finset.sum_eq_single i]
+        · simp
+        · intro b _ hb; rw [ite_eq_right]; exact fun h => hb (Fin.ext h.symm)
+        · intro h; exact absurd (Finset.mem_univ i) h
+      rw [← hc, h0, coeff_zero]; rfl
+    have hroots : Set.Finite {t : ℝ | IsRoot q t} := Polynomial.finite_setOfPred_isRoot hq0
+    have hB : Set.Finite {y : ℝ | IsRoot q (y ^ 2)} := by
+      refine (hroots.image Real.sqrt |>.union (hroots.image fun t => -Real.sqrt t)).subset ?_
+      intro y hy
+      simp only [Set.mem_ofPred_eq] at hy
+      rcases le_or_gt 0 y with h | h
+      · exact Or.inl ⟨y ^ 2, hy, Real.sqrt_sq h⟩
+      · exact
+          Or.inr
+            ⟨y ^ 2, hy, by
+              change -Real.sqrt (y ^ 2) = y
+              rw [Real.sqrt_sq_eq_abs, abs_of_neg h, neg_neg]⟩
+    have hsub :
+      Set.Ioi 0 \ {y : ℝ | IsRoot q (y ^ 2)} ⊆
+        Function.support (fun y => (∑ i, x i * (y ^ 2) ^ (i : ℕ)) ^ 2 * ν n y) ∩ Set.Ioi 0 := by
+      intro y hy
+      obtain ⟨hy0, hyB⟩ := hy
+      refine ⟨?_, hy0⟩
+      simp only [Function.mem_support]
+      have hqy : q.eval (y ^ 2) ≠ 0 := fun h => hyB h
+      rw [hqeval] at hqy
+      exact mul_ne_zero (pow_ne_zero _ hqy) (ν_pos n hy0).ne'
+    calc
+      (0 : ENNReal) < ⊤ := ENNReal.zero_lt_top
+      _ = volume (Set.Ioi (0 : ℝ) \ {y : ℝ | IsRoot q (y ^ 2)}) := by
+        rw [measure_sdiff_null (hB.measure_zero volume), Real.volume_Ioi]
+      _ ≤ _ := measure_mono hsub
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Potential.lean b/LeanPool/Zeta5Irrational/Potential.lean
new file mode 100644
index 0000000000..7ab8939ef5
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Potential.lean
@@ -0,0 +1,531 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Measure
+public import LeanPool.Zeta5Irrational.Energy
+public import Mathlib.Analysis.LocallyConvex.AbsConvexOpen
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.RingTheory.Etale.Weakly
+import Mathlib.RingTheory.TotallySplit
+import Mathlib.Tactic.ReduceModChar
+import Mathlib.Tactic.FieldSimp
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+import Mathlib.Tactic.Positivity.Basic
+import Mathlib.Tactic.Ring.Basic
+
+/-! # The potential of the comparison measure `ρ` (closed form) and the field `V`
+
+* `Uω a b t` : the closed form (A.1) of the logarithmic potential of the arcsine measure on
+  `[a, b]`;
+* `Uρ t = ∑ⱼ cⱼ Uω aⱼ bⱼ t` for the sixteen intervals of Table 1;
+* monotonicity of `Uω` (nonincreasing left of `[a,b]`, constant on it, nondecreasing right of it);
+* the derivative of `V`: `V'(t) = P(√t)/√t` with `P(y) = π + arctan(1/y) - 6 arctan(α/y)`.
+-/
+
+@[expose] public section
+
+open Filter Topology Set Finset
+
+namespace Zeta5Irrational
+
+/-- The closed form (A.1) of the potential of the arcsine measure on `[a, b]`. -/
+noncomputable def Uω (a b t : ℝ) : ℝ :=
+  if a ≤ t ∧ t ≤ b then Real.log ((b - a) / 4)
+  else Real.log ((|t - (a + b) / 2| + Real.sqrt ((t - a) * (t - b))) / 2)
+
+/-- The potential of `ρ`. -/
+noncomputable def Uρ (t : ℝ) : ℝ :=
+  ∑ j ∈ Icc 1 16, cρ j * Uω (aρ j) (bρ j) t
+
+lemma Uω_of_mem {a b t : ℝ} (h1 : a ≤ t) (h2 : t ≤ b) : Uω a b t = Real.log ((b - a) / 4) := by
+  simp [Uω, h1, h2]
+
+lemma Uω_of_ge {a b t : ℝ} (hab : a < b) (h : b ≤ t) :
+    Uω a b t = Real.log ((t - (a + b) / 2 + Real.sqrt ((t - a) * (t - b))) / 2) := by
+  rcases eq_or_lt_of_le h with rfl | h'
+  · rw [Uω_of_mem hab.le le_rfl]
+    simp only [sub_self, mul_zero, Real.sqrt_zero, add_zero]
+    congr 1; ring
+  · have : ¬(a ≤ t ∧ t ≤ b) := fun hh => absurd hh.2 (not_le.mpr h')
+    simp only [Uω, this, ite_false]
+    rw [abs_of_nonneg (by linarith)]
+
+lemma Uω_of_le {a b t : ℝ} (hab : a < b) (h : t ≤ a) :
+    Uω a b t = Real.log (((a + b) / 2 - t + Real.sqrt ((t - a) * (t - b))) / 2) := by
+  rcases eq_or_lt_of_le h with rfl | h'
+  · rw [Uω_of_mem le_rfl hab.le]
+    simp only [sub_self, zero_mul, Real.sqrt_zero, add_zero]
+    congr 1; ring
+  · have : ¬(a ≤ t ∧ t ≤ b) := fun hh => absurd hh.1 (not_le.mpr h')
+    simp only [Uω, this, ite_false]
+    rw [abs_of_nonpos (by linarith), neg_sub]
+
+/-- `Uω a b` is nondecreasing on `[b, ∞)`. -/
+lemma Uω_monotoneOn_Ici {a b : ℝ} (hab : a < b) : MonotoneOn (Uω a b) (Ici b) := by
+  intro x hx y hy hxy
+  have hx' : b ≤ x := hx
+  have hy' : b ≤ y := hy
+  rw [Uω_of_ge hab hx', Uω_of_ge hab hy']
+  have h1 : Real.sqrt ((x - a) * (x - b)) ≤ Real.sqrt ((y - a) * (y - b)) :=
+    Real.sqrt_le_sqrt (mul_le_mul (by linarith) (by linarith) (by linarith) (by linarith))
+  have hpos : 0 < (x - (a + b) / 2 + Real.sqrt ((x - a) * (x - b))) / 2 := by
+    have := Real.sqrt_nonneg ((x - a) * (x - b)); linarith
+  refine Real.log_le_log hpos ?_
+  linarith
+
+/-- `Uω a b` is nonincreasing on `(-∞, a]`. -/
+lemma Uω_antitoneOn_Iic {a b : ℝ} (hab : a < b) : AntitoneOn (Uω a b) (Iic a) := by
+  intro x hx y hy hxy
+  have hx' : x ≤ a := hx
+  have hy' : y ≤ a := hy
+  rw [Uω_of_le hab hx', Uω_of_le hab hy']
+  have h1 : Real.sqrt ((y - a) * (y - b)) ≤ Real.sqrt ((x - a) * (x - b)) := by
+    refine Real.sqrt_le_sqrt ?_
+    have e1 : (y - a) * (y - b) = (a - y) * (b - y) := by ring
+    have e2 : (x - a) * (x - b) = (a - x) * (b - x) := by ring
+    rw [e1, e2]
+    exact mul_le_mul (by linarith) (by linarith) (by linarith) (by linarith)
+  have hpos : 0 < ((a + b) / 2 - y + Real.sqrt ((y - a) * (y - b))) / 2 := by
+    have := Real.sqrt_nonneg ((y - a) * (y - b)); linarith
+  refine Real.log_le_log hpos ?_
+  linarith
+
+/-! ### The derivative of `V` -/
+
+/-- `P(y) = π + arctan (1/y) - 6 arctan (α/y)`. -/
+noncomputable def Pfun (y : ℝ) : ℝ :=
+  Real.pi + Real.arctan (1 / y) - 6 * Real.arctan ((3 / 40) / y)
+
+/-- The closed form of `V` as a function of `y = √t`: `Φ(y) = V(y²)`. -/
+noncomputable def Φ (y : ℝ) : ℝ :=
+  Real.log (1 + y ^ 2) - 6 * (3 / 40) * Real.log (y ^ 2 + (3 / 40) ^ 2) - 2 + 12 * (3 / 40) +
+    2 * y * Pfun y
+
+lemma Vfield_eq_Φ {t : ℝ} (ht : 0 < t) : Vfield t = Φ (Real.sqrt t) := by
+  rw [Vfield_eq ht]
+  unfold Φ Pfun
+  rw [Real.sq_sqrt ht.le]
+
+lemma hasDerivAt_Φ {y : ℝ} (hy : 0 < y) : HasDerivAt Φ (2 * Pfun y) y := by
+  have h1 := ((hasDerivAt_pow 2 y).const_add (1 : ℝ)).log (by positivity : (1 : ℝ) + y ^ 2 ≠ 0)
+  have h2 :=
+    (((hasDerivAt_pow 2 y).add_const ((3 / 40 : ℝ) ^ 2)).log (by positivity)).const_mul
+      (6 * (3 / 40 : ℝ))
+  have h3 := (hasDerivAt_inv hy.ne').arctan
+  have h4 := ((hasDerivAt_inv hy.ne').const_mul (3 / 40 : ℝ)).arctan
+  have hP0 := ((hasDerivAt_const y Real.pi).add h3).sub (h4.const_mul 6)
+  have hP : HasDerivAt Pfun _ y :=
+    hP0.congr_of_eventuallyEq
+      (Filter.Eventually.of_forall fun x => by
+        simp only [Pfun, Pi.add_apply, Pi.sub_apply, div_eq_mul_inv, one_mul])
+  have h5 := ((hasDerivAt_id y).const_mul (2 : ℝ)).mul hP
+  have h := (((h1.sub h2).sub_const (2 : ℝ)).add_const (12 * (3 / 40 : ℝ))).add h5
+  refine (h.congr_of_eventuallyEq (Filter.Eventually.of_forall fun x => ?_)).congr_deriv ?_
+  · simp only [Φ, Pi.add_apply, Pi.sub_apply, Pi.mul_apply, id]
+    try ring
+  · unfold Pfun
+    have hne1 : (1 : ℝ) + y ^ 2 ≠ 0 := by positivity
+    have hne2 : y ^ 2 + (3 / 40 : ℝ) ^ 2 ≠ 0 := by positivity
+    simp only [Nat.cast_ofNat, show (2 : ℕ) - 1 = 1 from rfl, pow_one, id]
+    field_simp
+    ring
+
+/-- `Uω a b` is nonincreasing on `(-∞, b]` (constant on `[a, b]`). -/
+lemma Uω_antitoneOn_Iic' {a b : ℝ} (hab : a < b) : AntitoneOn (Uω a b) (Iic b) := by
+  intro x hx y hy hxy
+  have hx' : x ≤ b := hx
+  have hy' : y ≤ b := hy
+  by_cases hya : y ≤ a
+  · exact
+      Uω_antitoneOn_Iic hab (show x ∈ Iic a from (hxy.trans hya)) (show y ∈ Iic a from hya) hxy
+  · push Not at hya
+    rw [Uω_of_mem hya.le hy']
+    by_cases hxa : x ≤ a
+    · have := Uω_antitoneOn_Iic hab (show x ∈ Iic a from hxa) (Set.mem_Iic.mpr le_rfl) hxa
+      rwa [Uω_of_mem le_rfl hab.le] at this
+    · push Not at hxa
+      rw [Uω_of_mem hxa.le hx']
+
+/-- `Uω a b` is nondecreasing on `[a, ∞)` (constant on `[a, b]`). -/
+lemma Uω_monotoneOn_Ici' {a b : ℝ} (hab : a < b) : MonotoneOn (Uω a b) (Ici a) := by
+  intro x hx y hy hxy
+  have hx' : a ≤ x := hx
+  have hy' : a ≤ y := hy
+  by_cases hxb : b ≤ x
+  · exact Uω_monotoneOn_Ici hab (show x ∈ Ici b from hxb) (show y ∈ Ici b from hxb.trans hxy) hxy
+  · push Not at hxb
+    rw [Uω_of_mem hx' hxb.le]
+    by_cases hyb : b ≤ y
+    · have := Uω_monotoneOn_Ici hab (Set.mem_Ici.mpr le_rfl) (show y ∈ Ici b from hyb) hyb
+      rwa [Uω_of_mem hab.le le_rfl] at this
+    · push Not at hyb
+      rw [Uω_of_mem hy' hyb.le]
+
+/-! ### Facts about Table 1 -/
+
+lemma aρ_lt_bρ : ∀ j ∈ Finset.Icc 1 16, aρ j < bρ j := by
+  intro j hj
+  rw [Finset.mem_Icc] at hj
+  obtain ⟨hj1, hj2⟩ := hj
+  interval_cases j <;> norm_num [aρ, bρ, table1]
+
+lemma cρ_pos : ∀ j ∈ Finset.Icc 1 16, 0 < cρ j := by
+  intro j hj
+  rw [Finset.mem_Icc] at hj
+  obtain ⟨hj1, hj2⟩ := hj
+  interval_cases j <;> norm_num [cρ, table1]
+
+lemma aρ_one_le_bρ : ∀ j ∈ Finset.Icc 1 16, aρ 1 ≤ bρ j := by
+  intro j hj
+  rw [Finset.mem_Icc] at hj
+  obtain ⟨hj1, hj2⟩ := hj
+  interval_cases j <;> norm_num [aρ, bρ, table1]
+
+lemma aρ_le_bρ_one : ∀ j ∈ Finset.Icc 1 16, aρ j ≤ bρ 1 := by
+  intro j hj
+  rw [Finset.mem_Icc] at hj
+  obtain ⟨hj1, hj2⟩ := hj
+  interval_cases j <;> norm_num [aρ, bρ, table1]
+
+lemma aρ_le_aρ_one : ∀ j ∈ Finset.Icc 1 16, aρ j ≤ aρ 1 := by
+  intro j hj
+  rw [Finset.mem_Icc] at hj
+  obtain ⟨hj1, hj2⟩ := hj
+  interval_cases j <;> norm_num [aρ, bρ, table1]
+
+lemma bρ_one_le_bρ : ∀ j ∈ Finset.Icc 1 16, bρ 1 ≤ bρ j := by
+  intro j hj
+  rw [Finset.mem_Icc] at hj
+  obtain ⟨hj1, hj2⟩ := hj
+  interval_cases j <;> norm_num [aρ, bρ, table1]
+
+/-- `Uρ` is nonincreasing on `(-∞, a₁]`. -/
+lemma Uρ_antitoneOn : AntitoneOn Uρ (Iic (aρ 1)) := by
+  intro x hx y hy hxy
+  unfold Uρ
+  refine Finset.sum_le_sum fun j hj => ?_
+  refine mul_le_mul_of_nonneg_left ?_ (cρ_pos j hj).le
+  exact
+    Uω_antitoneOn_Iic' (aρ_lt_bρ j hj)
+      (show x ∈ Iic (bρ j) from (show x ≤ aρ 1 from hx).trans (aρ_one_le_bρ j hj))
+      (show y ∈ Iic (bρ j) from (show y ≤ aρ 1 from hy).trans (aρ_one_le_bρ j hj)) hxy
+
+/-- `Uρ` is nondecreasing on `[b₁, ∞)`. -/
+lemma Uρ_monotoneOn : MonotoneOn Uρ (Ici (bρ 1)) := by
+  intro x hx y hy hxy
+  unfold Uρ
+  refine Finset.sum_le_sum fun j hj => ?_
+  refine mul_le_mul_of_nonneg_left ?_ (cρ_pos j hj).le
+  exact
+    Uω_monotoneOn_Ici' (aρ_lt_bρ j hj) (show x ∈ Ici (aρ j) from (aρ_le_bρ_one j hj).trans hx)
+      (show y ∈ Ici (aρ j) from (aρ_le_bρ_one j hj).trans hy) hxy
+
+/-- `Uρ` is constant on `[a₁, b₁]`. -/
+lemma Uρ_of_mem {t : ℝ} (h1 : aρ 1 ≤ t) (h2 : t ≤ bρ 1) :
+    Uρ t = ∑ j ∈ Icc 1 16, cρ j * Real.log ((bρ j - aρ j) / 4) := by
+  unfold Uρ
+  refine Finset.sum_congr rfl fun j hj => ?_
+  rw [Uω_of_mem ((aρ_le_aρ_one j hj).trans h1) (h2.trans (bρ_one_le_bρ j hj))]
+
+/-! ### Monotonicity of `P` and unimodality of `Φ` -/
+
+lemma hasDerivAt_Pfun {y : ℝ} (hy : 0 < y) :
+    HasDerivAt Pfun (-1 / (y ^ 2 + 1) + 6 * (3 / 40) / (y ^ 2 + (3 / 40) ^ 2)) y := by
+  have h3 := (hasDerivAt_inv hy.ne').arctan
+  have h4 := ((hasDerivAt_inv hy.ne').const_mul (3 / 40 : ℝ)).arctan
+  have hP0 := ((hasDerivAt_const y Real.pi).add h3).sub (h4.const_mul 6)
+  refine
+    (hP0.congr_of_eventuallyEq
+          (Filter.Eventually.of_forall fun x => by
+            simp only [Pfun, Pi.add_apply, Pi.sub_apply, div_eq_mul_inv, one_mul])).congr_deriv
+      ?_
+  have hne1 : y ^ 2 + (1 : ℝ) ≠ 0 := by positivity
+  have hne2 : y ^ 2 + (3 / 40 : ℝ) ^ 2 ≠ 0 := by positivity
+  field_simp
+  ring
+
+/-- The turning point `y₀² = 711/880` of `P`. -/
+noncomputable abbrev y0 : ℝ :=
+  Real.sqrt (711 / 880)
+
+lemma deriv_Pfun_nonneg {y : ℝ} (hy : 0 < y) (hy0 : y ≤ y0) :
+    0 ≤ -1 / (y ^ 2 + 1) + 6 * (3 / 40) / (y ^ 2 + (3 / 40) ^ 2) := by
+  have hy2 : y ^ 2 ≤ 711 / 880 := by
+    have := Real.sqrt_le_sqrt (le_of_eq (Real.sq_sqrt (by norm_num : (0 : ℝ) ≤ 711 / 880)).symm)
+    calc
+      y ^ 2 ≤ y0 ^ 2 := pow_le_pow_left₀ hy.le hy0 2
+      _ = 711 / 880 := Real.sq_sqrt (by norm_num)
+  rw [div_add_div _ _ (by positivity) (by positivity)]
+  refine div_nonneg ?_ (by positivity)
+  nlinarith
+
+lemma deriv_Pfun_nonpos {y : ℝ} (hy0 : y0 ≤ y) :
+    -1 / (y ^ 2 + 1) + 6 * (3 / 40) / (y ^ 2 + (3 / 40) ^ 2) ≤ 0 := by
+  have hy2 : 711 / 880 ≤ y ^ 2 := by
+    calc
+      (711 / 880 : ℝ) = y0 ^ 2 := (Real.sq_sqrt (by norm_num)).symm
+      _ ≤ y ^ 2 := pow_le_pow_left₀ (Real.sqrt_nonneg _) hy0 2
+  rw [div_add_div _ _ (by positivity) (by positivity)]
+  refine div_nonpos_of_nonpos_of_nonneg ?_ (by positivity)
+  nlinarith
+
+lemma Pfun_monotoneOn : MonotoneOn Pfun (Ioc 0 y0) := by
+  refine monotoneOn_of_deriv_nonneg (convex_Ioc 0 y0) ?_ ?_ ?_
+  · intro y hy; exact (hasDerivAt_Pfun hy.1).continuousAt.continuousWithinAt
+  · intro y hy
+    rw [interior_Ioc] at hy
+    exact (hasDerivAt_Pfun hy.1).differentiableAt.differentiableWithinAt
+  · intro y hy
+    rw [interior_Ioc] at hy
+    rw [(hasDerivAt_Pfun hy.1).deriv]
+    exact deriv_Pfun_nonneg hy.1 hy.2.le
+
+lemma Pfun_antitoneOn : AntitoneOn Pfun (Ici y0) := by
+  have hy0 : 0 < y0 := Real.sqrt_pos.mpr (by norm_num)
+  refine antitoneOn_of_deriv_nonpos (convex_Ici y0) ?_ ?_ ?_
+  · intro y hy; exact (hasDerivAt_Pfun (hy0.trans_le hy)).continuousAt.continuousWithinAt
+  · intro y hy
+    rw [interior_Ici] at hy
+    exact (hasDerivAt_Pfun (hy0.trans hy)).differentiableAt.differentiableWithinAt
+  · intro y hy
+    rw [interior_Ici] at hy
+    rw [(hasDerivAt_Pfun (hy0.trans hy)).deriv]
+    exact deriv_Pfun_nonpos (le_of_lt hy)
+
+lemma arctan_le_self' {x : ℝ} (hx : 0 ≤ x) : Real.arctan x ≤ x := by
+  rcases hx.lt_or_eq with h | h
+  · have ht : 0 < Real.arctan x := Real.arctan_pos.mpr h
+    have := Real.lt_tan ht (Real.arctan_lt_pi_div_two x)
+    rw [Real.tan_arctan] at this
+    exact this.le
+  · rw [← h, Real.arctan_zero]
+
+lemma Pfun_pos_of_one_le {y : ℝ} (hy : 1 ≤ y) : 0 < Pfun y := by
+  unfold Pfun
+  have h1 : 0 ≤ Real.arctan (1 / y) := Real.arctan_nonneg.mpr (by positivity)
+  have h2 : Real.arctan ((3 / 40) / y) ≤ (3 / 40) / y := arctan_le_self' (by positivity)
+  have h3 : (3 / 40 : ℝ) / y ≤ 3 / 40 := by rw [div_le_iff₀ (by linarith)]; nlinarith
+  linarith [Real.pi_gt_three]
+
+lemma Pfun_one_pos : 0 < Pfun 1 :=
+  Pfun_pos_of_one_le le_rfl
+
+/-- The bracket `[q₋, q₊]` of the paper (Appendix A.3). -/
+noncomputable abbrev qm : ℝ :=
+  59205077 / 10 ^ 10
+
+/-- Upper rational endpoint of the bracket around the minimum of the external field. -/
+noncomputable abbrev qp : ℝ :=
+  59205079 / 10 ^ 10
+
+lemma sqrt_qm_le_y0 : Real.sqrt qm ≤ y0 :=
+  Real.sqrt_le_sqrt (by norm_num)
+
+lemma sqrt_qp_le_y0 : Real.sqrt qp ≤ y0 :=
+  Real.sqrt_le_sqrt (by norm_num)
+
+lemma sqrt_qm_pos : 0 < Real.sqrt qm :=
+  Real.sqrt_pos.mpr (by norm_num)
+
+lemma sqrt_qp_pos : 0 < Real.sqrt qp :=
+  Real.sqrt_pos.mpr (by norm_num)
+
+/-- Numerical fact: `P(√q₋) < 0` (certified). -/
+lemma Pfun_sqrt_qm_neg : Pfun (Real.sqrt qm) < 0 := by
+  set s := Real.sqrt qm with hs
+  have hs_pos : 0 < s := sqrt_qm_pos
+  have hs_lo : (7694483543421 / 100000000000000 : ℝ) ≤ s :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi : s ≤ (60113152683 / 781250000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have h1 : Real.arctan (1 / s) = Real.pi / 2 - Real.arctan s := by
+    rw [one_div, Real.arctan_inv_of_pos hs_pos]
+  have hS2 :
+    (76793521753675120919939711999 / 1000000000000000000000000000000 : ℝ) ≤
+      ∑ i ∈ Finset.range (2 * 8),
+        (-1 : ℝ) ^ i *
+          ((7694483543421 / 100000000000000 : ℝ) ^ (2 * i + 1) / (2 * (i : ℝ) + 1)) :=
+    by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num
+  have h2 :
+    (76793521753675120919939711999 / 1000000000000000000000000000000 : ℝ) ≤ Real.arctan s :=
+    hS2.trans
+      ((arctan_ge_partial (by norm_num) (by norm_num) 8).trans
+        (Real.arctan_strictMono.monotone hs_lo))
+  have h3 :
+    Real.arctan ((3 / 40) / (60113152683 / 781250000000 : ℝ)) ≤ Real.arctan ((3 / 40) / s) :=
+    Real.arctan_strictMono.monotone (div_le_div_of_nonneg_left (by norm_num) hs_pos hs_hi)
+  have h4 :
+    Real.arctan ((3 / 40) / (60113152683 / 781250000000 : ℝ)) =
+      2 *
+        Real.arctan
+          (((3 / 40) / (60113152683 / 781250000000 : ℝ)) /
+            (1 + Real.sqrt (1 + ((3 / 40) / (60113152683 / 781250000000 : ℝ)) ^ 2))) :=
+    arctan_eq_two_mul_arctan (by norm_num)
+  have h5 :
+    (1953125000000000000000000 / 4801949469571619658211011 : ℝ) ≤
+      ((3 / 40) / (60113152683 / 781250000000 : ℝ)) /
+        (1 + Real.sqrt (1 + ((3 / 40) / (60113152683 / 781250000000 : ℝ)) ^ 2)) := by
+    have hsq :
+      Real.sqrt (1 + ((3 / 40) / (60113152683 / 781250000000 : ℝ)) ^ 2) ≤
+        (139645531231451 / 100000000000000 : ℝ) :=
+      sqrt_le_of_sq_le (by norm_num) (by norm_num)
+    rw [show
+        (1953125000000000000000000 / 4801949469571619658211011 : ℝ) =
+          ((3 / 40) / (60113152683 / 781250000000 : ℝ)) /
+            (1 + (139645531231451 / 100000000000000 : ℝ))
+        by norm_num]
+    exact div_le_div_of_nonneg_left (by norm_num) (by positivity) (by linarith)
+  have hS6 :
+    (77259924705678505221464494407 / 200000000000000000000000000000 : ℝ) ≤
+      ∑ i ∈ Finset.range (2 * 14),
+        (-1 : ℝ) ^ i *
+          ((1953125000000000000000000 / 4801949469571619658211011 : ℝ) ^ (2 * i + 1) /
+            (2 * (i : ℝ) + 1)) :=
+    by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num
+  have h6 :
+    (77259924705678505221464494407 / 200000000000000000000000000000 : ℝ) ≤
+      Real.arctan
+        (((3 / 40) / (60113152683 / 781250000000 : ℝ)) /
+          (1 + Real.sqrt (1 + ((3 / 40) / (60113152683 / 781250000000 : ℝ)) ^ 2))) :=
+    hS6.trans
+      ((arctan_ge_partial (by norm_num) (by norm_num) 14).trans
+        (Real.arctan_strictMono.monotone h5))
+  have hpi := Real.pi_lt_d20
+  unfold Pfun
+  rw [h1]
+  linarith
+
+/-- Numerical fact: `0 < P(√q₊)` (certified). -/
+lemma Pfun_sqrt_qp_pos : 0 < Pfun (Real.sqrt qp) := by
+  set s := Real.sqrt qp with hs
+  have hs_pos : 0 < s := sqrt_qp_pos
+  have hs_lo : (961810459173 / 12500000000000 : ℝ) ≤ s :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi : s ≤ (7694483673387 / 100000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have h1 : Real.arctan (1 / s) = Real.pi / 2 - Real.arctan s := by
+    rw [one_div, Real.arctan_inv_of_pos hs_pos]
+  have hS2 :
+    ∑ i ∈ Finset.range (2 * 8 + 1),
+        (-1 : ℝ) ^ i * ((7694483673387 / 100000000000000 : ℝ) ^ (2 * i + 1) / (2 * (i : ℝ) + 1)) ≤
+      (76793523045685761842673943627 / 1000000000000000000000000000000 : ℝ) :=
+    by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num
+  have h2 :
+    Real.arctan s ≤ (76793523045685761842673943627 / 1000000000000000000000000000000 : ℝ) :=
+    ((Real.arctan_strictMono.monotone hs_hi).trans
+          (arctan_le_partial (by norm_num) (by norm_num) 8)).trans
+      hS2
+  have h3 :
+    Real.arctan ((3 / 40) / s) ≤ Real.arctan ((3 / 40) / (961810459173 / 12500000000000 : ℝ)) :=
+    Real.arctan_strictMono.monotone (div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo)
+  have h4 :
+    Real.arctan ((3 / 40) / (961810459173 / 12500000000000 : ℝ)) =
+      2 *
+        Real.arctan
+          (((3 / 40) / (961810459173 / 12500000000000 : ℝ)) /
+            (1 + Real.sqrt (1 + ((3 / 40) / (961810459173 / 12500000000000 : ℝ)) ^ 2))) :=
+    arctan_eq_two_mul_arctan (by norm_num)
+  have h5 :
+    ((3 / 40) / (961810459173 / 12500000000000 : ℝ)) /
+        (1 + Real.sqrt (1 + ((3 / 40) / (961810459173 / 12500000000000 : ℝ)) ^ 2)) ≤
+      (7812500000000000000000000 / 19207798110603085910913171 : ℝ) := by
+    have hsq :
+      (34911382520581 / 25000000000000 : ℝ) ≤
+        Real.sqrt (1 + ((3 / 40) / (961810459173 / 12500000000000 : ℝ)) ^ 2) :=
+      le_sqrt_of_sq_le (by norm_num) (by norm_num)
+    rw [show
+        (7812500000000000000000000 / 19207798110603085910913171 : ℝ) =
+          ((3 / 40) / (961810459173 / 12500000000000 : ℝ)) /
+            (1 + (34911382520581 / 25000000000000 : ℝ))
+        by norm_num]
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) (by linarith)
+  have hS6 :
+    ∑ i ∈ Finset.range (2 * 14 + 1),
+        (-1 : ℝ) ^ i *
+          ((7812500000000000000000000 / 19207798110603085910913171 : ℝ) ^ (2 * i + 1) /
+            (2 * (i : ℝ) + 1)) ≤
+      (193149809653637591795814749669 / 500000000000000000000000000000 : ℝ) :=
+    by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num
+  have h6 :
+    Real.arctan
+        (((3 / 40) / (961810459173 / 12500000000000 : ℝ)) /
+          (1 + Real.sqrt (1 + ((3 / 40) / (961810459173 / 12500000000000 : ℝ)) ^ 2))) ≤
+      (193149809653637591795814749669 / 500000000000000000000000000000 : ℝ) :=
+    ((Real.arctan_strictMono.monotone h5).trans
+          (arctan_le_partial (by norm_num) (by norm_num) 14)).trans
+      hS6
+  have hpi := Real.pi_gt_d20
+  unfold Pfun
+  rw [h1]
+  linarith
+
+/-- `P ≤ 0` on `(0, √q₋]`. -/
+lemma Pfun_nonpos {y : ℝ} (hy : 0 < y) (hy' : y ≤ Real.sqrt qm) : Pfun y ≤ 0 :=
+  (Pfun_monotoneOn ⟨hy, hy'.trans sqrt_qm_le_y0⟩ ⟨sqrt_qm_pos, sqrt_qm_le_y0⟩ hy').trans
+    Pfun_sqrt_qm_neg.le
+
+/-- `0 ≤ P` on `[√q₊, ∞)`. -/
+lemma Pfun_nonneg {y : ℝ} (hy : Real.sqrt qp ≤ y) : 0 ≤ Pfun y := by
+  rcases le_or_gt y y0 with h | h
+  · exact
+      Pfun_sqrt_qp_pos.le.trans
+        (Pfun_monotoneOn ⟨sqrt_qp_pos, sqrt_qp_le_y0⟩ ⟨sqrt_qp_pos.trans_le hy, h⟩ hy)
+  · rcases le_or_gt y 1 with h1 | h1
+    · have hy0 : y0 ≤ 1 := by
+        rw [show (1 : ℝ) = Real.sqrt 1 by simp];
+        exact Real.sqrt_le_sqrt (by norm_num)
+      exact
+        Pfun_one_pos.le.trans
+          (Pfun_antitoneOn (show y ∈ Ici y0 from h.le) (show (1 : ℝ) ∈ Ici y0 from hy0) h1)
+    · exact (Pfun_pos_of_one_le h1.le).le
+
+/-- `Φ` is nonincreasing on `(0, √q₋]`. -/
+lemma Φ_antitoneOn : AntitoneOn Φ (Ioc 0 (Real.sqrt qm)) := by
+  refine antitoneOn_of_deriv_nonpos (convex_Ioc _ _) ?_ ?_ ?_
+  · intro y hy; exact (hasDerivAt_Φ hy.1).continuousAt.continuousWithinAt
+  · intro y hy
+    rw [interior_Ioc] at hy
+    exact (hasDerivAt_Φ hy.1).differentiableAt.differentiableWithinAt
+  · intro y hy
+    rw [interior_Ioc] at hy
+    rw [(hasDerivAt_Φ hy.1).deriv]
+    linarith [Pfun_nonpos hy.1 hy.2.le]
+
+/-- `Φ` is nondecreasing on `[√q₊, ∞)`. -/
+lemma Φ_monotoneOn : MonotoneOn Φ (Ici (Real.sqrt qp)) := by
+  refine monotoneOn_of_deriv_nonneg (convex_Ici _) ?_ ?_ ?_
+  · intro y hy; exact (hasDerivAt_Φ (sqrt_qp_pos.trans_le hy)).continuousAt.continuousWithinAt
+  · intro y hy
+    rw [interior_Ici] at hy
+    exact (hasDerivAt_Φ (sqrt_qp_pos.trans hy)).differentiableAt.differentiableWithinAt
+  · intro y hy
+    rw [interior_Ici] at hy
+    rw [(hasDerivAt_Φ (sqrt_qp_pos.trans hy)).deriv]
+    linarith [Pfun_nonneg hy.le]
+
+/-- `V` is nonincreasing on `(0, q₋]`. -/
+lemma Vfield_antitoneOn : AntitoneOn Vfield (Ioc 0 qm) := by
+  intro x hx y hy hxy
+  rw [Vfield_eq_Φ hx.1, Vfield_eq_Φ hy.1]
+  exact
+    Φ_antitoneOn ⟨Real.sqrt_pos.mpr hx.1, Real.sqrt_le_sqrt hx.2⟩
+      ⟨Real.sqrt_pos.mpr hy.1, Real.sqrt_le_sqrt hy.2⟩ (Real.sqrt_le_sqrt hxy)
+
+/-- `V` is nondecreasing on `[q₊, ∞)`. -/
+lemma Vfield_monotoneOn : MonotoneOn Vfield (Ici qp) := by
+  intro x hx y hy hxy
+  have hqp : (0 : ℝ) < qp := by norm_num
+  rw [Vfield_eq_Φ (hqp.trans_le hx), Vfield_eq_Φ (hqp.trans_le hy)]
+  exact Φ_monotoneOn (Real.sqrt_le_sqrt hx) (Real.sqrt_le_sqrt hy) (Real.sqrt_le_sqrt hxy)
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/PotentialV.lean b/LeanPool/Zeta5Irrational/PotentialV.lean
new file mode 100644
index 0000000000..337e2663ab
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/PotentialV.lean
@@ -0,0 +1,80 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Energy
+public import Mathlib.Analysis.LocallyConvex.AbsConvexOpen
+import Mathlib.Tactic.ENatToNat
+import Mathlib.Tactic.Polynomial.Basic
+import Mathlib.Tactic.ReduceModChar
+import Mathlib.Tactic.FieldSimp
+import Mathlib.Tactic.Positivity.Basic
+import Mathlib.Tactic.Ring.Basic
+import Mathlib.Tactic.Ring.RingNF
+
+/-! # The closed form (A.5) of the external field `V`
+
+`∫₀^c log(t + u²) du = c log(t + c²) - 2c + 2√t arctan(c/√t)` for `t > 0`, hence
+`V(t) = log(1+t) - 6α log(t+α²) - 2 + 12α + 2√t (π + arctan(1/√t) - 6 arctan(α/√t))`.
+-/
+
+@[expose] public section
+
+open Filter Topology Set MeasureTheory
+
+namespace Zeta5Irrational
+
+/-- Antiderivative of `u ↦ log (t + u²)`. -/
+noncomputable def GVpot (t u : ℝ) : ℝ :=
+  u * Real.log (t + u ^ 2) - 2 * u + 2 * Real.sqrt t * Real.arctan (u / Real.sqrt t)
+
+lemma hasDerivAt_GV {t : ℝ} (ht : 0 < t) (u : ℝ) :
+    HasDerivAt (GVpot t) (Real.log (t + u ^ 2)) u := by
+  have hs : 0 < Real.sqrt t := Real.sqrt_pos.mpr ht
+  have hs2 : Real.sqrt t ^ 2 = t := Real.sq_sqrt ht.le
+  have h1 :
+    HasDerivAt (fun u : ℝ => u * Real.log (t + u ^ 2))
+      (1 * Real.log (t + u ^ 2) + u * ((2 * u) / (t + u ^ 2))) u := by
+    have hlog : HasDerivAt (fun u : ℝ => Real.log (t + u ^ 2)) ((2 * u) / (t + u ^ 2)) u := by
+      have := ((hasDerivAt_pow 2 u).const_add t).log (by positivity)
+      simpa using this
+    exact (hasDerivAt_id u).mul hlog
+  have h2 : HasDerivAt (fun u : ℝ => 2 * u) 2 u := by simpa using (hasDerivAt_id u).const_mul 2
+  have h3 :
+    HasDerivAt (fun u : ℝ => 2 * Real.sqrt t * Real.arctan (u / Real.sqrt t))
+      (2 * Real.sqrt t * ((1 / (1 + (u / Real.sqrt t) ^ 2)) * (1 / Real.sqrt t))) u := by
+    have := ((hasDerivAt_id u).div_const (Real.sqrt t)).arctan
+    simpa using this.const_mul (2 * Real.sqrt t)
+  have h := (h1.sub h2).add h3
+  refine h.congr_deriv ?_
+  have hne : t + u ^ 2 ≠ 0 := by positivity
+  field_simp
+  rw [hs2]
+  ring
+
+/-- `∫₀^c log(t + u²) du = c log(t + c²) - 2c + 2√t arctan(c/√t)` for `t > 0`. -/
+lemma integral_log_add_sq {t : ℝ} (ht : 0 < t) (c : ℝ) :
+    ∫ u in (0 : ℝ)..c, Real.log (t + u ^ 2) =
+      c * Real.log (t + c ^ 2) - 2 * c + 2 * Real.sqrt t * Real.arctan (c / Real.sqrt t) := by
+  have :=
+    intervalIntegral.integral_eq_sub_of_hasDerivAt (a := 0) (b := c) (f := GVpot t) (f' :=
+      fun u => Real.log (t + u ^ 2)) (fun u _ => hasDerivAt_GV ht u)
+      ((Continuous.log (by fun_prop) (fun u => by positivity)).intervalIntegrable _ _)
+  rw [this]
+  simp [GVpot]
+
+/-- **(A.5)**: the closed form of `V`. -/
+theorem Vfield_eq {t : ℝ} (ht : 0 < t) :
+    Vfield t =
+      Real.log (1 + t) - 6 * (3 / 40) * Real.log (t + (3 / 40) ^ 2) - 2 + 12 * (3 / 40) +
+        2 * Real.sqrt t *
+          (Real.pi + Real.arctan (1 / Real.sqrt t) - 6 * Real.arctan ((3 / 40) / Real.sqrt t)) := by
+  unfold Vfield
+  rw [integral_log_add_sq ht, integral_log_add_sq ht]
+  ring_nf
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/PrimeSum.lean b/LeanPool/Zeta5Irrational/PrimeSum.lean
new file mode 100644
index 0000000000..10f07f08f3
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/PrimeSum.lean
@@ -0,0 +1,151 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import Mathlib.NumberTheory.Chebyshev
+import LeanPool.Zeta5Irrational.PNT
+import Mathlib.Tactic.ENatToNat
+import Mathlib.Tactic.ReduceModChar
+import Mathlib.Tactic.FieldSimp
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.Positivity.Basic
+import Mathlib.Tactic.Ring.Basic
+
+import Mathlib.Analysis.SpecialFunctions.Integrals.Basic
+
+/-! # Weighted prime sums via the prime number theorem
+
+`∑_{a < p ≤ b} p log p = b θ(b) - a θ(a) - ∫_a^b θ` (Abel summation), and with `θ(x) ~ x`
+this gives `∑_{K/d < p ≤ K/c} p log p = K² (1/c² - 1/d²)/2 + o(K²)`.
+-/
+
+@[expose] public section
+
+open Finset Filter Topology MeasureTheory Set Real Asymptotics
+
+namespace Zeta5Irrational
+
+/-- `c p = log p` for primes, `0` otherwise. -/
+noncomputable def cPrime (k : ℕ) : ℝ :=
+  if k.Prime then Real.log k else 0
+
+lemma theta_eq_sum_cPrime (t : ℝ) : Chebyshev.theta t = ∑ k ∈ Icc 0 ⌊t⌋₊, cPrime k := by
+  rw [Chebyshev.theta_eq_sum_Icc, Finset.sum_filter]
+  rfl
+
+/-- The weighted prime sum `∑_{a < p ≤ b} p log p` (as a sum over integers `k` with
+`⌊a⌋₊ < k ≤ ⌊b⌋₊`). -/
+noncomputable def wsum (a b : ℝ) : ℝ :=
+  ∑ k ∈ Ioc ⌊a⌋₊ ⌊b⌋₊, (k : ℝ) * cPrime k
+
+/-- Abel summation for the weighted prime sum. -/
+lemma wsum_eq {a b : ℝ} (ha : 0 ≤ a) (hab : a ≤ b) :
+    wsum a b =
+      b * Chebyshev.theta b - a * Chebyshev.theta a - ∫ t in Ioc a b, Chebyshev.theta t := by
+  unfold wsum
+  have hd : deriv (fun t : ℝ => t) = fun _ => 1 := by funext t; simp
+  have h :=
+    sum_mul_eq_sub_sub_integral_mul cPrime (f := fun t : ℝ => t) ha hab
+      (fun t _ => differentiableAt_id)
+      (by rw [hd]; exact integrableOn_const measure_Icc_lt_top.ne)
+  simp only [hd, one_mul] at h
+  rw [h, ← theta_eq_sum_cPrime, ← theta_eq_sum_cPrime]
+  congr 1
+  refine setIntegral_congr_fun measurableSet_Ioc fun t _ => ?_
+  exact (theta_eq_sum_cPrime t).symm
+
+/-- The prime number theorem in `ε`-form: `|θ(y) - y| ≤ ε y` for all large `y`. -/
+lemma theta_eventually_close {ε : ℝ} (hε : 0 < ε) :
+    ∀ᶠ y : ℝ in atTop, |Chebyshev.theta y - y| ≤ ε * y := by
+  have h := theta_asymptotic
+  rw [IsEquivalent, IsLittleO] at h
+  have h2 := h hε
+  rw [IsBigOWith] at h2
+  filter_upwards [h2, eventually_ge_atTop 0] with y hy hy0
+  simp only [Pi.sub_apply, id, Real.norm_eq_abs, abs_of_nonneg hy0] at hy
+  exact hy
+
+/-- `∫_a^b θ` is close to `(b² - a²)/2` when `θ` is close to the identity on `[a, b]`. -/
+lemma integral_theta_bounds {a b ε : ℝ} (_ha : 0 ≤ a) (hab : a ≤ b)
+    (h : ∀ y ∈ Icc a b, |Chebyshev.theta y - y| ≤ ε * y) :
+    |(∫ t in Ioc a b, Chebyshev.theta t) - (b ^ 2 - a ^ 2) / 2| ≤ ε * (b ^ 2 - a ^ 2) / 2 := by
+  have hint : IntegrableOn Chebyshev.theta (Ioc a b) :=
+    (Chebyshev.theta_mono.intervalIntegrable (μ := volume) (a := a) (b := b)).1
+  have hsub : IntegrableOn (fun t : ℝ => t) (Ioc a b) :=
+    (continuous_id.integrableOn_Icc).mono_set Ioc_subset_Icc_self
+  have hid : ∫ t in Ioc a b, t = (b ^ 2 - a ^ 2) / 2 := by
+    rw [← intervalIntegral.integral_of_le hab, integral_id]
+  have hε : IntegrableOn (fun t : ℝ => ε * t) (Ioc a b) := hsub.const_mul ε
+  calc
+    |(∫ t in Ioc a b, Chebyshev.theta t) - (b ^ 2 - a ^ 2) / 2| =
+        ‖∫ t in Ioc a b, (Chebyshev.theta t - t)‖ :=
+      by rw [← hid, ← integral_sub hint hsub, Real.norm_eq_abs]
+    _ ≤ ∫ t in Ioc a b, ‖Chebyshev.theta t - t‖ := (norm_integral_le_integral_norm _)
+    _ ≤ ∫ t in Ioc a b, ε * t := by
+      refine setIntegral_mono_on (hint.sub hsub).norm hε measurableSet_Ioc fun t ht => ?_
+      rw [Real.norm_eq_abs]
+      exact h t (Ioc_subset_Icc_self ht)
+    _ = ε * (b ^ 2 - a ^ 2) / 2 := by rw [integral_const_mul, hid]; ring
+
+/-- The bound `|wsum a b - (b² - a²)/2| ≤ 2 η b²` when `|θ(y) - y| ≤ η y` on `[a, b]`. -/
+lemma wsum_close {a b η : ℝ} (ha : 0 ≤ a) (hab : a ≤ b) (_hη : 0 ≤ η)
+    (h : ∀ y ∈ Icc a b, |Chebyshev.theta y - y| ≤ η * y) :
+    |wsum a b - (b ^ 2 - a ^ 2) / 2| ≤ 2 * η * b ^ 2 := by
+  rw [wsum_eq ha hab]
+  have h1 := h b ⟨hab, le_rfl⟩
+  have h2 := h a ⟨le_rfl, hab⟩
+  have h3 := integral_theta_bounds ha hab h
+  have hb : 0 ≤ b := ha.trans hab
+  have e1 : |b * Chebyshev.theta b - b ^ 2| ≤ η * b ^ 2 := by
+    rw [show b * Chebyshev.theta b - b ^ 2 = b * (Chebyshev.theta b - b) by ring, abs_mul,
+      abs_of_nonneg hb]
+    nlinarith
+  have e2 : |a * Chebyshev.theta a - a ^ 2| ≤ η * a ^ 2 := by
+    rw [show a * Chebyshev.theta a - a ^ 2 = a * (Chebyshev.theta a - a) by ring, abs_mul,
+      abs_of_nonneg ha]
+    nlinarith
+  have ha2 : a ^ 2 ≤ b ^ 2 := by nlinarith
+  rw [abs_le] at e1 e2 h3 ⊢
+  constructor <;> nlinarith
+
+/-- `∑_{K/d < p ≤ K/c} p log p / K² → (1/c² - 1/d²)/2`. -/
+theorem wsum_tendsto {c d : ℝ} (hc : 0 < c) (hcd : c < d) :
+    Tendsto (fun K : ℝ => wsum (K / d) (K / c) / K ^ 2) atTop (𝓝 ((1 / c ^ 2 - 1 / d ^ 2) / 2)) :=
+  by
+  rw [Metric.tendsto_atTop]
+  intro ε hε
+  have hd : 0 < d := hc.trans hcd
+  set η := ε * c ^ 2 / 8 with hη_def
+  have hη : 0 < η := by positivity
+  obtain ⟨Y₀, hY₀⟩ := eventually_atTop.mp (theta_eventually_close hη)
+  refine ⟨max (d * Y₀) 1, fun K hK => ?_⟩
+  have hK1 : 1 ≤ K := le_trans (le_max_right _ _) hK
+  have hK0 : 0 < K := by linarith
+  have hKY : d * Y₀ ≤ K := le_trans (le_max_left _ _) hK
+  have ha : 0 ≤ K / d := by positivity
+  have hab : K / d ≤ K / c := div_le_div_of_nonneg_left hK0.le hc hcd.le
+  have hbound :=
+    wsum_close ha hab hη.le
+      (fun y hy =>
+        hY₀ y
+          (by
+            have : Y₀ ≤ K / d := by rw [le_div_iff₀ hd]; linarith
+            linarith [hy.1]))
+  rw [Real.dist_eq]
+  have e :
+    wsum (K / d) (K / c) / K ^ 2 - (1 / c ^ 2 - 1 / d ^ 2) / 2 =
+      (wsum (K / d) (K / c) - ((K / c) ^ 2 - (K / d) ^ 2) / 2) / K ^ 2 :=
+    by field_simp
+  rw [e, abs_div, abs_of_pos (by positivity : (0 : ℝ) < K ^ 2), div_lt_iff₀ (by positivity)]
+  calc
+    |wsum (K / d) (K / c) - ((K / c) ^ 2 - (K / d) ^ 2) / 2| ≤ 2 * η * (K / c) ^ 2 := hbound
+    _ = ε * K ^ 2 / 4 := by rw [hη_def]; field_simp; ring
+    _ < ε * K ^ 2 := by
+      have : 0 < ε * K ^ 2 := by positivity
+      linarith
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/PrimeSumIntegral.lean b/LeanPool/Zeta5Irrational/PrimeSumIntegral.lean
new file mode 100644
index 0000000000..237f0ee5da
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/PrimeSumIntegral.lean
@@ -0,0 +1,306 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.PrimeSumStep
+import Mathlib.Tactic.ENatToNat
+import Mathlib.Tactic.ReduceModChar
+import Mathlib.Tactic.FieldSimp
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+import Mathlib.Tactic.Positivity.Basic
+import Mathlib.Tactic.Ring.Basic
+
+import Mathlib.Analysis.SpecialFunctions.Integrals.Basic
+
+/-! # Prime sums and integrals
+
+For `f` Lipschitz on `[a, b]` (`0 < a < b`), `psum f a b K / K² ≤ ∫_a^b f(x)/x³ dx + ε` for all
+large `K`; and the piecewise version for `f` given by Lipschitz pieces on `[t_i, t_{i+1})`.
+-/
+
+@[expose] public section
+
+open Finset Filter Topology Set intervalIntegral MeasureTheory
+
+namespace Zeta5Irrational
+
+lemma integral_inv_cube {a b : ℝ} (ha : 0 < a) (hb : 0 < b) :
+    ∫ x in a..b, x ^ (-3 : ℤ) = (1 / a ^ 2 - 1 / b ^ 2) / 2 := by
+  rw [integral_zpow (Or.inr ⟨by norm_num, fun h => ?_⟩)]
+  · norm_num
+    field_simp
+    ring
+  · rcases le_total a b with hab | hab
+    · rw [uIcc_of_le hab] at h; linarith [h.1]
+    · rw [uIcc_of_ge hab] at h; linarith [h.1]
+
+/-- Upper bound for `psum` by the integral, for `f` Lipschitz on `[a, b]`. -/
+theorem psum_le_integral {f : ℝ → ℝ} {a b L : ℝ} (ha : 0 < a) (hab : a < b) (hL : 0 ≤ L)
+    (hcont : ContinuousOn f (Icc a b))
+    (hLip : ∀ x ∈ Icc a b, ∀ y ∈ Icc a b, |f x - f y| ≤ L * |x - y|) {ε : ℝ} (hε : 0 < ε) :
+    ∀ᶠ K : ℝ in atTop, psum f a b K / K ^ 2 ≤ (∫ x in a..b, f x / x ^ 3) + ε := by
+  -- choose the mesh
+  obtain ⟨r, hr⟩ := exists_nat_gt (2 * L * (b - a) * (1 / a ^ 2 - 1 / b ^ 2) / ε)
+  have hr0 : 0 < r := by
+    have : (0 : ℝ) ≤ 2 * L * (b - a) * (1 / a ^ 2 - 1 / b ^ 2) / ε := by
+      have : 1 / b ^ 2 ≤ 1 / a ^ 2 := one_div_le_one_div_of_le (by positivity) (by nlinarith)
+      have : 0 ≤ 1 / a ^ 2 - 1 / b ^ 2 := by linarith
+      positivity
+    have : (0 : ℝ) < r := lt_of_le_of_lt this hr
+    exact_mod_cast this
+  set δ := (b - a) / r with hδ
+  have hδ0 : 0 < δ := by positivity
+  set c : ℕ → ℝ := fun j => a + j * δ with hc
+  have hc0 : c 0 = a := by simp [hc]
+  have hcr : c r = b := by simp only [hc]; rw [hδ]; field_simp; ring
+  have hmono : ∀ j < r, c j < c (j + 1) := by intro j _; simp only [hc]; push_cast; linarith
+  have hcle : ∀ j, j ≤ r → a ≤ c j ∧ c j ≤ b := by
+    intro j hj
+    simp only [hc]
+    constructor
+    · have : (0 : ℝ) ≤ j * δ := by positivity
+      linarith
+    · have : (j : ℝ) * δ ≤ r * δ := mul_le_mul_of_nonneg_right (by exact_mod_cast hj) hδ0.le
+      have : (r : ℝ) * δ = b - a := by rw [hδ]; field_simp
+      linarith
+  set v : ℕ → ℝ := fun j => f (c j) + L * δ with hv
+  have hf : ∀ j < r, ∀ x ∈ Ico (c j) (c (j + 1)), f x ≤ v j := by
+    intro j hj x hx
+    have hcj := hcle j hj.le
+    have hcj1 := hcle (j + 1) hj
+    have hxI : x ∈ Icc a b := ⟨hcj.1.trans hx.1, hx.2.le.trans hcj1.2⟩
+    have := hLip x hxI (c j) ⟨hcj.1, hcj.2⟩
+    have hd : |x - c j| ≤ δ := by
+      rw [abs_of_nonneg (by linarith [hx.1])]
+      have : c (j + 1) = c j + δ := by simp only [hc]; push_cast; ring
+      linarith [hx.2]
+    simp only [hv]
+    have := abs_le.mp this
+    nlinarith
+      -- the Darboux sum is at most the integral plus the error
+  have hD :
+    (∑ j ∈ range r, v j * ((1 / c j ^ 2 - 1 / c (j + 1) ^ 2) / 2)) ≤
+      (∫ x in a..b, f x / x ^ 3) + 2 * L * δ * ((1 / a ^ 2 - 1 / b ^ 2) / 2) := by
+    have hint :
+      ∀ j < r,
+        IntervalIntegrable (fun x => (f x + 2 * L * δ) * x ^ (-3 : ℤ)) volume (c j) (c (j + 1)) :=
+  by
+      intro j hj
+      apply ContinuousOn.intervalIntegrable
+      rw [uIcc_of_le (hmono j hj).le]
+      have hsub : Icc (c j) (c (j + 1)) ⊆ Icc a b :=
+        Icc_subset_Icc (hcle j hj.le).1 (hcle (j + 1) hj).2
+      refine ((hcont.mono hsub).add continuousOn_const).mul ?_
+      exact
+        continuousOn_id.zpow₀ (-3) fun x hx =>
+          Or.inl (ha.trans_le ((hcle j hj.le).1.trans hx.1)).ne'
+    have h1 :
+      ∀ j < r,
+        v j * ((1 / c j ^ 2 - 1 / c (j + 1) ^ 2) / 2) ≤
+          ∫ x in c j..c (j + 1), (f x + 2 * L * δ) * x ^ (-3 : ℤ) := by
+      intro j hj
+      have hcj := hcle j hj.le
+      have hcj1 := hcle (j + 1) hj
+      rw [← integral_inv_cube (ha.trans_le hcj.1) (ha.trans_le hcj1.1), ←
+        intervalIntegral.integral_const_mul]
+      apply integral_mono_on (hmono j hj).le
+      · apply ContinuousOn.intervalIntegrable
+        rw [uIcc_of_le (hmono j hj).le]
+        exact
+          continuousOn_const.mul
+            (continuousOn_id.zpow₀ (-3) fun x hx => Or.inl (ha.trans_le (hcj.1.trans hx.1)).ne')
+      · exact hint j hj
+      intro x hx
+      have hxI : x ∈ Icc a b := ⟨hcj.1.trans hx.1, hx.2.trans hcj1.2⟩
+      have hx0 : 0 < x := ha.trans_le hxI.1
+      have hLx := hLip (c j) ⟨hcj.1, hcj.2⟩ x hxI
+      have hd : |c j - x| ≤ δ := by
+        rw [abs_sub_comm, abs_of_nonneg (by linarith [hx.1])]
+        have : c (j + 1) = c j + δ := by simp only [hc]; push_cast; ring
+        linarith [hx.2]
+      have hvle : v j ≤ f x + 2 * L * δ := by
+        simp only [hv]
+        have := abs_le.mp hLx
+        nlinarith
+      exact mul_le_mul_of_nonneg_right hvle (by positivity)
+    calc
+      (∑ j ∈ range r, v j * ((1 / c j ^ 2 - 1 / c (j + 1) ^ 2) / 2)) ≤
+          ∑ j ∈ range r, ∫ x in c j..c (j + 1), (f x + 2 * L * δ) * x ^ (-3 : ℤ) :=
+        Finset.sum_le_sum fun j hj => h1 j (Finset.mem_range.mp hj)
+      _ = ∫ x in a..b, (f x + 2 * L * δ) * x ^ (-3 : ℤ) := by
+        rw [sum_integral_adjacent_intervals hint, hc0, hcr]
+      _ = (∫ x in a..b, f x / x ^ 3) + 2 * L * δ * ((1 / a ^ 2 - 1 / b ^ 2) / 2) := by
+        rw [← integral_inv_cube ha (ha.trans hab), ← intervalIntegral.integral_const_mul, ←
+          intervalIntegral.integral_add]
+        · apply integral_congr
+          intro x hx
+          rw [uIcc_of_le hab.le] at hx
+          have hx0 : 0 < x := ha.trans_le hx.1
+          simp only
+          rw [zpow_neg, zpow_ofNat]
+          field_simp
+        · apply ContinuousOn.intervalIntegrable
+          rw [uIcc_of_le hab.le]
+          exact hcont.div (continuousOn_pow 3) fun x hx => by have := ha.trans_le hx.1; positivity
+        · apply ContinuousOn.intervalIntegrable
+          rw [uIcc_of_le hab.le]
+          exact
+            continuousOn_const.mul
+              (continuousOn_id.zpow₀ (-3) fun x hx => Or.inl (ha.trans_le hx.1).ne')
+                -- the error term is at most ε/2
+  have herr : 2 * L * δ * ((1 / a ^ 2 - 1 / b ^ 2) / 2) ≤ ε / 2 := by
+    rw [hδ]
+    have hr' : 2 * L * (b - a) * (1 / a ^ 2 - 1 / b ^ 2) < r * ε := by rwa [div_lt_iff₀ hε] at hr
+    rw [show
+        2 * L * ((b - a) / r) * ((1 / a ^ 2 - 1 / b ^ 2) / 2) =
+          2 * L * (b - a) * (1 / a ^ 2 - 1 / b ^ 2) / r / 2
+        by field_simp]
+    have : 2 * L * (b - a) * (1 / a ^ 2 - 1 / b ^ 2) / r ≤ ε := by
+      rw [div_le_iff₀ (by exact_mod_cast hr0)]; linarith
+    linarith
+  have hev := psum_eventually_le (f := f) r (by rw [hc0]; exact ha) hmono hf (half_pos hε)
+  rw [hc0, hcr] at hev
+  filter_upwards [hev] with K hK
+  linarith
+
+/-- `K/k ∈ [a, b)` for `k ∈ Ioc ⌊K/b⌋₊ ⌊K/a⌋₊`. -/
+lemma div_mem_Ico {a b K : ℝ} (ha : 0 < a) (hab : a ≤ b) (hK : 0 < K) {k : ℕ}
+    (hk : k ∈ Finset.Ioc ⌊K / b⌋₊ ⌊K / a⌋₊) : K / k ∈ Ico a b := by
+  rw [Finset.mem_Ioc] at hk
+  have hb : 0 < b := ha.trans_le hab
+  have hk0 : 0 < k := by have := hk.1; omega
+  have hk0' : (0 : ℝ) < k := by exact_mod_cast hk0
+  constructor
+  · rw [le_div_iff₀ hk0']
+    have h1 : (k : ℝ) ≤ K / a :=
+      (Nat.floor_le (div_nonneg hK.le ha.le)).trans' (Nat.cast_le.mpr hk.2)
+    rw [le_div_iff₀ ha] at h1
+    linarith
+  · rw [div_lt_iff₀ hk0']
+    have h1 : K / b < k := by
+      have := Nat.lt_floor_add_one (K / b)
+      have h2 : (⌊K / b⌋₊ : ℝ) + 1 ≤ k := by exact_mod_cast hk.1
+      linarith
+    rw [div_lt_iff₀ hb] at h1
+    linarith
+
+lemma psum_congr {f g : ℝ → ℝ} {a b K : ℝ} (ha : 0 < a) (hab : a ≤ b) (hK : 0 < K)
+    (h : ∀ x ∈ Ico a b, f x = g x) : psum f a b K = psum g a b K := by
+  unfold psum
+  refine Finset.sum_congr rfl fun k hk => ?_
+  rw [h _ (div_mem_Ico ha hab hK hk)]
+
+lemma psum_sum_split (f : ℝ → ℝ) {t : ℕ → ℝ} (m : ℕ) (ht0 : 0 < t 0)
+    (hmono : ∀ i < m, t i ≤ t (i + 1)) {K : ℝ} (hK : 0 ≤ K) :
+    psum f (t 0) (t m) K = ∑ i ∈ range m, psum f (t i) (t (i + 1)) K := by
+  induction m with
+  | zero => simp [psum]
+  | succ m ih =>
+    rw [Finset.sum_range_succ, ← ih fun i hi => hmono i (by omega)]
+    exact psum_split f ht0 (c_zero_le fun i hi => hmono i (by omega)) (hmono m (by omega)) hK
+
+/-- The piecewise version: `f = g i` on `[t i, t (i+1))`, each `g i` Lipschitz on the closed
+piece. -/
+theorem psum_piecewise_le {f : ℝ → ℝ} {g : ℕ → ℝ → ℝ} {t : ℕ → ℝ} {L : ℝ} (m : ℕ) (ht0 : 0 < t 0)
+    (hmono : ∀ i < m, t i < t (i + 1)) (hL : 0 ≤ L)
+    (hcont : ∀ i < m, ContinuousOn (g i) (Icc (t i) (t (i + 1))))
+    (hLip :
+      ∀ i < m,
+        ∀ x ∈ Icc (t i) (t (i + 1)), ∀ y ∈ Icc (t i) (t (i + 1)), |g i x - g i y| ≤ L * |x - y|)
+    (hfg : ∀ i < m, ∀ x ∈ Ico (t i) (t (i + 1)), f x = g i x) {ε : ℝ} (hε : 0 < ε) :
+    ∀ᶠ K : ℝ in atTop,
+      psum f (t 0) (t m) K / K ^ 2 ≤ (∑ i ∈ range m, ∫ x in t i..t (i + 1), g i x / x ^ 3) + ε := by
+  rcases Nat.eq_zero_or_pos m with rfl | hm
+  · filter_upwards [eventually_gt_atTop 0] with K hK
+    simp only [psum, Std.le_refl, Finset.Ioc_eq_empty_of_le, sum_empty, zero_div,
+        Finset.range_zero, zero_add]; positivity
+  have hpos : ∀ i ≤ m, 0 < t i := by
+    intro i hi
+    induction i with
+    | zero => exact ht0
+    | succ i ih => exact (ih (by omega)).trans (hmono i (by omega))
+  have hε' : 0 < ε / m := by positivity
+  have hev :
+    ∀ i ∈ range m,
+      ∀ᶠ K : ℝ in atTop,
+        psum (g i) (t i) (t (i + 1)) K / K ^ 2 ≤ (∫ x in t i..t (i + 1), g i x / x ^ 3) + ε / m :=
+  by
+    intro i hi
+    have hi := Finset.mem_range.mp hi
+    exact psum_le_integral (hpos i hi.le) (hmono i hi) hL (hcont i hi) (hLip i hi) hε'
+  filter_upwards [(Finset.eventually_all (range m)).mpr hev, eventually_gt_atTop 0] with K hK hK0
+  rw [psum_sum_split f m ht0 (fun i hi => (hmono i hi).le) hK0.le, Finset.sum_div]
+  have hcongr :
+    ∀ i ∈ range m, psum f (t i) (t (i + 1)) K / K ^ 2 = psum (g i) (t i) (t (i + 1)) K / K ^ 2 := by
+    intro i hi
+    have hi := Finset.mem_range.mp hi
+    rw [psum_congr (hpos i hi.le) (hmono i hi).le hK0 (hfg i hi)]
+  rw [Finset.sum_congr rfl hcongr]
+  calc
+    ∑ i ∈ range m, psum (g i) (t i) (t (i + 1)) K / K ^ 2 ≤
+        ∑ i ∈ range m, ((∫ x in t i..t (i + 1), g i x / x ^ 3) + ε / m) :=
+      Finset.sum_le_sum fun i hi => hK i hi
+    _ = _ := by
+      rw [Finset.sum_add_distrib, Finset.sum_const, Finset.card_range, nsmul_eq_mul]
+      field_simp
+
+lemma psum_mono {f g : ℝ → ℝ} {a b K : ℝ} (ha : 0 < a) (hab : a ≤ b) (hK : 0 < K)
+    (h : ∀ x ∈ Ico a b, f x ≤ g x) : psum f a b K ≤ psum g a b K := by
+  unfold psum
+  refine Finset.sum_le_sum fun k hk => ?_
+  have hk0 : (0 : ℝ) ≤ (k : ℝ) * cPrime k := mul_nonneg (Nat.cast_nonneg _) (cPrime_nonneg k)
+  exact mul_le_mul_of_nonneg_left (h _ (div_mem_Ico ha hab hK hk)) hk0
+
+/-- The piecewise version with `f ≤ g i` on the pieces. -/
+theorem psum_piecewise_le' {f : ℝ → ℝ} {g : ℕ → ℝ → ℝ} {t : ℕ → ℝ} {L : ℝ} (m : ℕ) (ht0 : 0 < t 0)
+    (hmono : ∀ i < m, t i < t (i + 1)) (hL : 0 ≤ L)
+    (hcont : ∀ i < m, ContinuousOn (g i) (Icc (t i) (t (i + 1))))
+    (hLip :
+      ∀ i < m,
+        ∀ x ∈ Icc (t i) (t (i + 1)), ∀ y ∈ Icc (t i) (t (i + 1)), |g i x - g i y| ≤ L * |x - y|)
+    (hfg : ∀ i < m, ∀ x ∈ Ico (t i) (t (i + 1)), f x ≤ g i x) {ε : ℝ} (hε : 0 < ε) :
+    ∀ᶠ K : ℝ in atTop,
+      psum f (t 0) (t m) K / K ^ 2 ≤ (∑ i ∈ range m, ∫ x in t i..t (i + 1), g i x / x ^ 3) + ε := by
+  rcases Nat.eq_zero_or_pos m with rfl | hm
+  · filter_upwards [eventually_gt_atTop 0] with K hK
+    simp only [psum, Std.le_refl, Finset.Ioc_eq_empty_of_le, sum_empty, zero_div,
+        Finset.range_zero, zero_add]; positivity
+  have hpos : ∀ i ≤ m, 0 < t i := by
+    intro i hi
+    induction i with
+    | zero => exact ht0
+    | succ i ih => exact (ih (by omega)).trans (hmono i (by omega))
+  have hε' : 0 < ε / m := by positivity
+  have hev :
+    ∀ i ∈ range m,
+      ∀ᶠ K : ℝ in atTop,
+        psum (g i) (t i) (t (i + 1)) K / K ^ 2 ≤ (∫ x in t i..t (i + 1), g i x / x ^ 3) + ε / m :=
+  by
+    intro i hi
+    have hi := Finset.mem_range.mp hi
+    exact psum_le_integral (hpos i hi.le) (hmono i hi) hL (hcont i hi) (hLip i hi) hε'
+  filter_upwards [(Finset.eventually_all (range m)).mpr hev, eventually_gt_atTop 0] with K hK hK0
+  rw [psum_sum_split f m ht0 (fun i hi => (hmono i hi).le) hK0.le, Finset.sum_div]
+  have hK2 : 0 < K ^ 2 := by positivity
+  calc
+    ∑ i ∈ range m, psum f (t i) (t (i + 1)) K / K ^ 2 ≤
+        ∑ i ∈ range m, psum (g i) (t i) (t (i + 1)) K / K ^ 2 := by
+      refine Finset.sum_le_sum fun i hi => ?_
+      have hi := Finset.mem_range.mp hi
+      exact
+        div_le_div_of_nonneg_right (psum_mono (hpos i hi.le) (hmono i hi).le hK0 (hfg i hi))
+          hK2.le
+    _ ≤ ∑ i ∈ range m, ((∫ x in t i..t (i + 1), g i x / x ^ 3) + ε / m) :=
+      (Finset.sum_le_sum fun i hi => hK i hi)
+    _ = _ := by
+      rw [Finset.sum_add_distrib, Finset.sum_const, Finset.card_range, nsmul_eq_mul]
+      field_simp
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/PrimeSumStep.lean b/LeanPool/Zeta5Irrational/PrimeSumStep.lean
new file mode 100644
index 0000000000..137a0812ba
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/PrimeSumStep.lean
@@ -0,0 +1,143 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.PrimeSum
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.Positivity.Basic
+import Mathlib.Tactic.Ring.Basic
+
+/-! # Prime sums with a weight function
+
+`psum f a b K = ∑_{K/b < p ≤ K/a} p f(K/p) log p`. If `f ≤ v_j` on each cell `[c_j, c_{j+1}]`
+of a partition `a = c_0 < ⋯ < c_r = b`, then
+`psum f a b K ≤ ∑_j v_j wsum (K/c_{j+1}) (K/c_j)`, and the right-hand side is
+`K² ∑_j v_j (1/c_j² - 1/c_{j+1}²)/2 + o(K²)`.
+-/
+
+@[expose] public section
+
+open Finset Filter Topology Set
+
+namespace Zeta5Irrational
+
+/-- `∑_{K/b < p ≤ K/a} p f(K/p) log p`. -/
+noncomputable def psum (f : ℝ → ℝ) (a b K : ℝ) : ℝ :=
+  ∑ k ∈ Ioc ⌊K / b⌋₊ ⌊K / a⌋₊, (k : ℝ) * cPrime k * f (K / k)
+
+lemma cPrime_nonneg (k : ℕ) : 0 ≤ cPrime k := by
+  unfold cPrime
+  split_ifs with h
+  · exact Real.log_nonneg (by exact_mod_cast h.one_lt.le)
+  · exact le_rfl
+
+/-- Splitting at an intermediate point. -/
+lemma psum_split (f : ℝ → ℝ) {a c b K : ℝ} (ha : 0 < a) (hac : a ≤ c) (hcb : c ≤ b) (hK : 0 ≤ K) :
+    psum f a b K = psum f a c K + psum f c b K := by
+  unfold psum
+  rw [add_comm,
+    Finset.sum_Ioc_consecutive _
+      (Nat.floor_le_floor (div_le_div_of_nonneg_left hK (by linarith) hcb))
+      (Nat.floor_le_floor (div_le_div_of_nonneg_left hK ha hac))]
+
+/-- If `f ≤ v` on `[a, b)`, then `psum f a b K ≤ v * wsum (K/b) (K/a)`. -/
+lemma psum_le_of_le {f : ℝ → ℝ} {a b v K : ℝ} (ha : 0 < a) (hab : a ≤ b) (hK : 0 < K)
+    (hf : ∀ x ∈ Ico a b, f x ≤ v) : psum f a b K ≤ v * wsum (K / b) (K / a) := by
+  unfold psum wsum
+  rw [Finset.mul_sum]
+  refine Finset.sum_le_sum fun k hk => ?_
+  rw [Finset.mem_Ioc] at hk
+  have hb : 0 < b := ha.trans_le hab
+  have hk0 : 0 < k := by have := hk.1; omega
+  have hk0' : (0 : ℝ) < k := by exact_mod_cast hk0
+  have hx1 : a ≤ K / k := by
+    rw [le_div_iff₀ hk0']
+    have h1 : (k : ℝ) ≤ K / a := by
+      have := Nat.floor_le (div_nonneg hK.le ha.le) |>.trans' (Nat.cast_le.mpr hk.2)
+      exact this
+    rw [le_div_iff₀ ha] at h1
+    linarith
+  have hx2 : K / k < b := by
+    rw [div_lt_iff₀ hk0']
+    have h1 : K / b < k := by
+      have := Nat.lt_floor_add_one (K / b)
+      have h2 : (⌊K / b⌋₊ : ℝ) + 1 ≤ k := by exact_mod_cast hk.1
+      linarith
+    rw [div_lt_iff₀ hb] at h1
+    linarith
+  have hc := cPrime_nonneg k
+  have := hf (K / k) ⟨hx1, hx2⟩
+  calc
+    (k : ℝ) * cPrime k * f (K / k) ≤ (k : ℝ) * cPrime k * v :=
+      mul_le_mul_of_nonneg_left this (by positivity)
+    _ = v * ((k : ℝ) * cPrime k) := by ring
+
+/-- The step sum over a partition `c 0 < c 1 < ⋯ < c r`, with values `v j` on `[c j, c (j+1)]`. -/
+noncomputable def stepSum (c v : ℕ → ℝ) (r : ℕ) (K : ℝ) : ℝ :=
+  ∑ j ∈ range r, v j * wsum (K / c (j + 1)) (K / c j)
+
+lemma c_zero_le {c : ℕ → ℝ} {r : ℕ} (h : ∀ j < r, c j ≤ c (j + 1)) : c 0 ≤ c r := by
+  induction r with
+  | zero => exact le_rfl
+  | succ r ih => exact (ih fun j hj => h j (by omega)).trans (h r (by omega))
+
+/-- Upper bound of `psum` over a partition by the step sum. -/
+lemma psum_le_stepSum {f : ℝ → ℝ} {c v : ℕ → ℝ} (r : ℕ) (hc0 : 0 < c 0)
+    (hmono : ∀ j < r, c j ≤ c (j + 1)) (hf : ∀ j < r, ∀ x ∈ Ico (c j) (c (j + 1)), f x ≤ v j)
+    {K : ℝ} (hK : 0 < K) : psum f (c 0) (c r) K ≤ stepSum c v r K := by
+  induction r with
+  | zero => simp [psum, stepSum]
+  | succ r ih =>
+    have hcr : c 0 ≤ c r := c_zero_le fun j hj => hmono j (by omega)
+    rw [psum_split f hc0 hcr (hmono r (by omega)) hK.le, stepSum, Finset.sum_range_succ]
+    exact
+      add_le_add (ih (fun j hj => hmono j (by omega)) (fun j hj => hf j (by omega)))
+        (psum_le_of_le (hc0.trans_le hcr) (hmono r (by omega)) hK (hf r (by omega)))
+
+/-- The limit of the normalised step sum. -/
+lemma stepSum_tendsto {c v : ℕ → ℝ} (r : ℕ) (hc0 : 0 < c 0) (hmono : ∀ j < r, c j < c (j + 1)) :
+    Tendsto (fun K : ℝ => stepSum c v r K / K ^ 2) atTop
+      (𝓝 (∑ j ∈ range r, v j * ((1 / c j ^ 2 - 1 / c (j + 1) ^ 2) / 2))) := by
+  have hpos : ∀ j ≤ r, 0 < c j := by
+    intro j hj
+    induction j with
+    | zero => exact hc0
+    | succ j ih => exact (ih (by omega)).trans (hmono j (by omega))
+  have e :
+    (fun K : ℝ => stepSum c v r K / K ^ 2) = fun K =>
+      ∑ j ∈ range r, v j * (wsum (K / c (j + 1)) (K / c j) / K ^ 2) := by
+    funext K
+    unfold stepSum
+    rw [Finset.sum_div]
+    refine Finset.sum_congr rfl fun j _ => ?_
+    ring
+  rw [e]
+  refine tendsto_finsetSum _ fun j hj => ?_
+  have hj := Finset.mem_range.mp hj
+  exact (wsum_tendsto (hpos j hj.le) (hmono j hj)).const_mul _
+
+/-- Eventually, `psum f a b K / K² ≤ (upper Darboux sum) + ε`. -/
+theorem psum_eventually_le {f : ℝ → ℝ} {c v : ℕ → ℝ} (r : ℕ) (hc0 : 0 < c 0)
+    (hmono : ∀ j < r, c j < c (j + 1)) (hf : ∀ j < r, ∀ x ∈ Ico (c j) (c (j + 1)), f x ≤ v j)
+    {ε : ℝ} (hε : 0 < ε) :
+    ∀ᶠ K : ℝ in atTop,
+      psum f (c 0) (c r) K / K ^ 2 ≤
+        (∑ j ∈ range r, v j * ((1 / c j ^ 2 - 1 / c (j + 1) ^ 2) / 2)) + ε := by
+  have h :=
+    (stepSum_tendsto (v := v) r hc0 hmono).eventually
+      (eventually_le_nhds
+        (by linarith :
+          (∑ j ∈ range r, v j * ((1 / c j ^ 2 - 1 / c (j + 1) ^ 2) / 2)) <
+            (∑ j ∈ range r, v j * ((1 / c j ^ 2 - 1 / c (j + 1) ^ 2) / 2)) + ε))
+  filter_upwards [h, eventually_gt_atTop 0] with K hK hK0
+  have := psum_le_stepSum (f := f) r hc0 (fun j hj => (hmono j hj).le) hf hK0
+  calc
+    psum f (c 0) (c r) K / K ^ 2 ≤ stepSum c v r K / K ^ 2 :=
+      div_le_div_of_nonneg_right this (by positivity)
+    _ ≤ _ := hK
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/RealBound.lean b/LeanPool/Zeta5Irrational/RealBound.lean
new file mode 100644
index 0000000000..ab4b46933d
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/RealBound.lean
@@ -0,0 +1,136 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+import Mathlib.Analysis.Complex.ExponentialBounds
+
+public import LeanPool.Zeta5Irrational.Constants
+public import LeanPool.Zeta5Irrational.Degree
+public import LeanPool.Zeta5Irrational.Stirling
+public import LeanPool.Zeta5Irrational.Zeta5
+public import LeanPool.Zeta5Irrational.Measure
+public import LeanPool.Zeta5Irrational.EnergyConst
+public import LeanPool.Zeta5Irrational.Positivity
+public import LeanPool.Zeta5Irrational.EnergyBound
+
+/-! # Section 6 of the paper: the real determinant, decomposed
+
+A variant of Proposition 6.3 (`real_bound`) with `27 K log K` in place of the paper's
+`24 K log K` is derived from
+* `Δ_pos` : `Δ_K(ζ(5)) > 0` — proved in `Zeta5Irrational.Positivity` from the moment representation
+  `moment_rep` (Proposition 2.2);
+* `log_Δ_le`: a variant of (6.14) with `22 h log K + 50 h` in place of
+  `18 h log K + 160 h`, proved in `Zeta5Irrational.EnergyBound` from Andréief's identity
+  (6.10), variants of (6.11)–(6.12), and the variant of (6.9) `energy_ineq`
+  (proved in `Zeta5Irrational.EnergyFinal` from Lemmas 6.1 and 6.2);
+* `energy_const` : the numerical inequality (6.4), `λ M₀ - I(ρ) + C* ≤ U` — Appendix A.4 —
+  proved in `Zeta5Irrational.EnergyConst`;
+* `log_S_le` : the Stirling bound (6.15) — proved.
+-/
+
+@[expose] public section
+
+open Polynomial Filter Topology Finset
+
+namespace Zeta5Irrational
+
+/--
+A variant of **(6.14)** with `22 h log K + 50 h` instead of `18 h log K + 160 h`,
+proved in `Zeta5Irrational.EnergyBound` from the configuration inequality `energy_ineq`. -/
+theorem log_Δ_le (n : ℕ) (hn : 0 < n) :
+    Real.log (aeval zeta5 (Δ n)) ≤
+      2 * (37 * n) * (37 * n + 6 * (3 * n) - 40 * n : ℝ) * Real.log (Kr n) +
+            (lam * M0 - Irho) * Kr n ^ 2 +
+          22 * (37 * n) * Real.log (Kr n) +
+        50 * (37 * n) :=
+  log_Δ_le' n hn
+
+/-- **(6.15)**: the Stirling bound for `log S_K`. -/
+theorem log_S_le (n : ℕ) (hn : 0 < n) :
+    Real.log (S n) ≤
+      (2 * lam - 12 * alph * lam - 2 * lam ^ 2) * Kr n ^ 2 * Real.log (Kr n) + Cstar * Kr n ^ 2 +
+          6 * (37 * n) * Real.log (Kr n) +
+        6 * (37 * n) := by
+  have hS :
+    ((S n : ℚ) : ℝ) =
+      ((40 * n).factorial : ℝ) ^ (2 * (37 * n)) * 4 ^ (37 * n - 1) /
+        (((3 * n).factorial : ℝ) ^ (12 * (37 * n)) *
+          ∏ i ∈ Icc 1 (37 * n - 1), ((2 * i).factorial : ℝ) ^ 2) :=
+    by simp only [S]; push_cast; rfl
+  have hprod : (0 : ℝ) < ∏ i ∈ Icc 1 (37 * n - 1), ((2 * i).factorial : ℝ) ^ 2 :=
+    Finset.prod_pos fun i _ => by positivity
+  rw [hS, Real.log_div (by positivity) (by positivity),
+    Real.log_mul (by positivity) (by positivity), Real.log_mul (by positivity) (by positivity),
+    Real.log_pow, Real.log_pow, Real.log_pow, Real.log_prod (fun i _ => by positivity)]
+  simp only [Real.log_pow, ← Finset.mul_sum]
+  have h1 := log_factorial_le (m := 40 * n) (by omega)
+  have h2 := log_factorial_ge (3 * n)
+  have h3 := sum_log_factorial_two_mul_ge (h := 37 * n) (by omega)
+  have hn1 : (1 : ℝ) ≤ n := by exact_mod_cast hn
+  have hN : Real.log (3 * (n : ℝ)) = Real.log (3 / 40) + Real.log (40 * n) := by
+    rw [← Real.log_mul (by norm_num) (by positivity)]; ring_nf
+  have h2h : Real.log (2 * (37 * (n : ℝ))) = Real.log (37 / 20) + Real.log (40 * n) := by
+    rw [← Real.log_mul (by norm_num) (by positivity)]; ring_nf
+  have hlog4 : Real.log 4 ≤ 1.4 := by
+    rw [show (4 : ℝ) = 2 ^ 2 by norm_num, Real.log_pow]
+    have := Real.log_two_lt_d9; push_cast; linarith
+  have hlog37 : Real.log (37 / 20) ≤ 0.85 := by
+    have := Real.log_le_sub_one_of_pos (by norm_num : (0 : ℝ) < 37 / 20); linarith
+  have hlog40 : 3 ≤ Real.log (40 * (n : ℝ)) := by
+    have hexp : Real.exp 3 < 40 := by
+      have h := Real.exp_one_lt_d9
+      have h3 : Real.exp 3 = Real.exp 1 ^ 3 := by rw [← Real.exp_nat_mul]; norm_num
+      rw [h3]
+      calc
+        Real.exp 1 ^ 3 < 2.7182818286 ^ 3 := by gcongr
+        _ < 40 := by norm_num
+    calc
+      (3 : ℝ) = Real.log (Real.exp 3) := (Real.log_exp 3).symm
+      _ ≤ Real.log (40 * n) := Real.log_le_log (Real.exp_pos 3) (by linarith)
+  have hh1 : ((37 * n - 1 : ℕ) : ℝ) = 37 * (n : ℝ) - 1 := by
+    rw [Nat.cast_sub (by omega)];
+    push_cast; ring
+  have hC :
+    Cstar =
+      -2 * (37 / 40) + 12 * (3 / 40) * (37 / 40) * (1 - Real.log (3 / 40)) + 3 * (37 / 40) ^ 2 -
+        2 * (37 / 40) ^ 2 * Real.log (37 / 20) :=
+    by unfold Cstar; norm_num
+  rw [hC, hh1]
+  push_cast at h1 h2 h3 ⊢
+  rw [hN] at h2
+  rw [h2h] at h3
+  simp only [Kr, lam, alph]
+  have hA : 0 ≤ 37 * (n : ℝ) - 1 := by linarith
+  have hB : 0 ≤ 4 * (37 * (n : ℝ)) := by positivity
+  nlinarith [mul_le_mul_of_nonneg_left hlog4 hA, mul_le_mul_of_nonneg_left hlog37 hB,
+    mul_le_mul_of_nonneg_left hlog40 (by positivity : (0 : ℝ) ≤ 37 * n)]
+
+/-- A weakening of **Proposition 6.3**, from the four statements above:
+we use `27 K log K` in place of the paper's `24 K log K`. -/
+theorem real_bound' (n : ℕ) (hn : 0 < n) :
+    0 < aeval zeta5 (F n) ∧
+      Real.log (aeval zeta5 (F n)) ≤
+        (U : ℝ) * Kr n ^ 2 + 27 * Kr n * Real.log (Kr n) + 200 * Kr n := by
+  have hS : (0 : ℝ) < S n := by exact_mod_cast S_pos n
+  have hΔ := Δ_pos n hn
+  have hF : aeval zeta5 (F n) = (S n : ℝ) * aeval zeta5 (Δ n) := by simp [F, aeval_C]
+  refine ⟨by rw [hF]; positivity, ?_⟩
+  rw [hF, Real.log_mul hS.ne' hΔ.ne']
+  have h1 := log_Δ_le n hn
+  have h2 := log_S_le n hn
+  have h3 := energy_const
+  have hK : 0 < Kr n := by simp only [Kr]; positivity
+  have hlogK : 0 ≤ Real.log (Kr n) := Real.log_nonneg (by simp only [Kr]; norm_cast; omega)
+  have hn' : (0 : ℝ) ≤ n := Nat.cast_nonneg n
+  have h3' : (lam * M0 - Irho + Cstar) * Kr n ^ 2 ≤ (U : ℝ) * Kr n ^ 2 :=
+    mul_le_mul_of_nonneg_right h3 (sq_nonneg _)
+  have hlow : 0 ≤ (n : ℝ) * Real.log (Kr n) := mul_nonneg hn' hlogK
+  simp only [Kr] at *
+  norm_num [lam, alph] at h1 h2 h3' hlow ⊢
+  nlinarith
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Stirling.lean b/LeanPool/Zeta5Irrational/Stirling.lean
new file mode 100644
index 0000000000..a31864c460
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Stirling.lean
@@ -0,0 +1,197 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import Mathlib.Analysis.Calculus.Deriv.Basic
+public import Mathlib.Analysis.LocallyConvex.AbsConvexOpen
+public import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Analysis.SpecialFunctions.Stirling
+import Mathlib.Analysis.SumIntegralComparisons
+import Mathlib.RingTheory.PiTensorProduct
+import Mathlib.Tactic.ENatToNat
+import Mathlib.Tactic.Polynomial.Basic
+import Mathlib.Tactic.ReduceModChar
+import Mathlib.Tactic.FieldSimp
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+import Mathlib.Tactic.Positivity.Basic
+import Mathlib.Tactic.Ring.Basic
+import Mathlib.Tactic.Ring.RingNF
+
+/-! # Stirling-type bounds used in Section 6 of the paper
+
+* `log_factorial_le` : `log m! ≤ m log m - m + (log m)/2 + 1` for `m ≥ 1`;
+* `log_factorial_ge` : `m log m - m ≤ log m!`;
+* `sum_log_factorial_two_mul_ge` : `∑_{i=1}^{h-1} log (2i)! ≥ h² log 2h - (3/2) h² - 2h log 2h`,
+  the bound displayed before (6.15) in the paper.
+-/
+
+@[expose] public section
+
+open Real Finset
+
+namespace Zeta5Irrational
+
+lemma log_factorial_le {m : ℕ} (hm : 1 ≤ m) :
+    Real.log (m.factorial) ≤ m * Real.log m - m + Real.log m / 2 + 1 := by
+  obtain ⟨k, rfl⟩ : ∃ k, m = k + 1 := ⟨m - 1, by omega⟩
+  have h1 := Stirling.log_stirlingSeq'_antitone (Nat.zero_le k)
+  simp only [Function.comp_apply, Nat.succ_eq_add_one, zero_add, Stirling.stirlingSeq_one] at h1
+  have hm0 : (0 : ℝ) < ((k + 1 : ℕ) : ℝ) := by positivity
+  have hfac : (0 : ℝ) < ((k + 1).factorial : ℝ) := by positivity
+  have hs : (0 : ℝ) < √(2 * ((k + 1 : ℕ) : ℝ)) * (((k + 1 : ℕ) : ℝ) / exp 1) ^ (k + 1) := by
+    positivity
+  unfold Stirling.stirlingSeq at h1
+  rw [Real.log_div hfac.ne' hs.ne', Real.log_mul (by positivity) (by positivity),
+    Real.log_sqrt (by positivity), Real.log_pow, Real.log_div hm0.ne' (exp_pos 1).ne',
+    Real.log_exp, Real.log_mul (by norm_num) hm0.ne',
+    Real.log_div (exp_pos 1).ne' (by positivity), Real.log_exp, Real.log_sqrt (by norm_num)] at h1
+  push_cast at h1 ⊢
+  linarith
+
+lemma log_factorial_ge (m : ℕ) : m * Real.log m - m ≤ Real.log (m.factorial) := by
+  rcases Nat.eq_zero_or_pos m with rfl | hm
+  · simp
+  have h := Stirling.le_log_factorial_stirling (n := m) hm.ne'
+  have h1 : 0 ≤ Real.log m := Real.log_nonneg (by exact_mod_cast hm)
+  have h2 : 0 ≤ Real.log (2 * π) := Real.log_nonneg (by nlinarith [Real.pi_gt_three])
+  linarith
+
+/-- An antiderivative of `x log (2x)`. -/
+noncomputable def Gaux (x : ℝ) : ℝ :=
+  x ^ 2 / 2 * Real.log (2 * x) - x ^ 2 / 4
+
+lemma hasDerivAt_Gaux {x : ℝ} (hx : 0 < x) : HasDerivAt Gaux (x * Real.log (2 * x)) x := by
+  have h1 : HasDerivAt (fun x : ℝ => 2 * x) 2 x := by simpa using (hasDerivAt_id x).const_mul 2
+  have h2 : HasDerivAt (fun x : ℝ => Real.log (2 * x)) (2 / (2 * x)) x := h1.log (by positivity)
+  have h3 : HasDerivAt (fun x : ℝ => x ^ 2 / 2) (2 * x / 2) x := by
+    simpa using (hasDerivAt_pow 2 x).div_const 2
+  have h4 : HasDerivAt (fun x : ℝ => x ^ 2 / 4) (2 * x / 4) x := by
+    simpa using (hasDerivAt_pow 2 x).div_const 4
+  have key :
+    HasDerivAt (fun x : ℝ => x ^ 2 / 2 * Real.log (2 * x) - x ^ 2 / 4)
+      (2 * x / 2 * Real.log (2 * x) + x ^ 2 / 2 * (2 / (2 * x)) - 2 * x / 4) x :=
+    (h3.mul h2).sub h4
+  have e :
+    2 * x / 2 * Real.log (2 * x) + x ^ 2 / 2 * (2 / (2 * x)) - 2 * x / 4 = x * Real.log (2 * x) :=
+    by field_simp; ring
+  rw [e] at key
+  exact key
+
+lemma integral_x_log {a b : ℝ} (ha : 0 < a) (hab : a ≤ b) :
+    ∫ x in a..b, x * Real.log (2 * x) = Gaux b - Gaux a := by
+  refine intervalIntegral.integral_eq_sub_of_hasDerivAt (fun x hx => hasDerivAt_Gaux ?_) ?_
+  · rw [Set.uIcc_of_le hab] at hx; exact lt_of_lt_of_le ha hx.1
+  · refine ContinuousOn.intervalIntegrable ?_
+    rw [Set.uIcc_of_le hab]
+    refine continuousOn_id.mul (ContinuousOn.log (continuousOn_const.mul continuousOn_id) ?_)
+    intro x hx
+    have hx0 : 0 < x := lt_of_lt_of_le ha hx.1
+    positivity
+
+lemma monotoneOn_x_log : MonotoneOn (fun x : ℝ => x * Real.log (2 * x)) (Set.Ici 1) := by
+  intro x hx y hy hxy
+  have hx1 : (1 : ℝ) ≤ x := hx
+  have hlog : 0 ≤ Real.log (2 * x) := Real.log_nonneg (by linarith)
+  exact mul_le_mul hxy (Real.log_le_log (by linarith) (by linarith)) hlog (by linarith)
+
+lemma sum_range_cast (m : ℕ) : ∑ k ∈ range m, (k : ℝ) = (m : ℝ) * ((m : ℝ) - 1) / 2 := by
+  induction m with
+  | zero => simp
+  | succ m ih => rw [Finset.sum_range_succ, ih]; push_cast; ring
+
+/-- `∑_{i=1}^{h-1} log (2i)! ≥ h² log (2h) - (3/2) h² - 2 h log (2h)`. -/
+lemma sum_log_factorial_two_mul_ge {h : ℕ} (hh : 2 ≤ h) :
+    (h : ℝ) ^ 2 * Real.log (2 * h) - 3 / 2 * (h : ℝ) ^ 2 - 2 * h * Real.log (2 * h) ≤
+      ∑ i ∈ Icc 1 (h - 1), Real.log ((2 * i).factorial) := by
+  have e1 : Icc 1 (h - 1) = Ico 1 h := by
+    ext x; simp only [Finset.mem_Icc, Finset.mem_Ico];
+    omega
+      -- Step 1: `log (2i)! ≥ 2i log (2i) - 2i`
+  have hstep1 :
+    ∑ i ∈ Icc 1 (h - 1), (2 * (i : ℝ) * Real.log (2 * i) - 2 * i) ≤
+      ∑ i ∈ Icc 1 (h - 1), Real.log ((2 * i).factorial) := by
+    refine Finset.sum_le_sum fun i _ => ?_
+    have := log_factorial_ge (2 * i)
+    push_cast at this
+    linarith
+  have hsplit :
+    ∑ i ∈ Icc 1 (h - 1), (2 * (i : ℝ) * Real.log (2 * i) - 2 * i) =
+      2 * ∑ i ∈ Icc 1 (h - 1), ((i : ℝ) * Real.log (2 * i)) -
+        ∑ i ∈ Icc 1 (h - 1), (2 * (i : ℝ)) := by
+    rw [Finset.sum_sub_distrib, Finset.mul_sum]
+    congr 1
+    exact
+      Finset.sum_congr rfl fun i _ => by
+        ring
+          -- Step 2: sum vs integral for `i log (2i)`
+  have hmono :
+    MonotoneOn (fun x : ℝ => x * Real.log (2 * x)) (Set.Icc 1 (1 + ((h - 2 : ℕ) : ℝ))) :=
+    monotoneOn_x_log.mono fun x hx => hx.1
+  have hcmp := hmono.integral_le_sum
+  have hnn : (0 : ℝ) ≤ ((h - 2 : ℕ) : ℝ) := Nat.cast_nonneg _
+  rw [integral_x_log one_pos (by linarith)] at hcmp
+  have hsum :
+    ∑ i ∈ Icc 1 (h - 1), ((i : ℝ) * Real.log (2 * i)) =
+      (1 : ℝ) * Real.log (2 * 1) +
+        ∑ i ∈ range (h - 2), (1 + ((i + 1 : ℕ) : ℝ)) * Real.log (2 * (1 + ((i + 1 : ℕ) : ℝ))) := by
+    rw [e1, Finset.sum_Ico_eq_sum_range, show h - 1 = (h - 2) + 1 by omega,
+      Finset.sum_range_succ', add_comm]
+    congr 1
+    · push_cast; ring_nf
+    · refine Finset.sum_congr rfl fun i _ => ?_
+      push_cast; ring_nf
+  have hcast : (1 : ℝ) + ((h - 2 : ℕ) : ℝ) = (h : ℝ) - 1 := by
+    rw [Nat.cast_sub (by omega)];
+    push_cast; ring
+  rw [hcast] at hcmp
+        -- Step 3: bounds for `Gaux (h - 1)` and `Gaux 1`
+  have hh1 : (1 : ℝ) ≤ (h : ℝ) - 1 := by
+    have : (2 : ℝ) ≤ h := by exact_mod_cast hh
+    linarith
+  have hhpos : (0 : ℝ) < h := by positivity
+  have hlog2h : 0 ≤ Real.log (2 * h) := Real.log_nonneg (by linarith)
+  have hlogratio : Real.log (2 * ((h : ℝ) - 1)) ≥ Real.log (2 * h) - 1 / ((h : ℝ) - 1) := by
+    have h1 : Real.log (2 * ((h : ℝ) - 1)) = Real.log (2 * h) + Real.log (((h : ℝ) - 1) / h) := by
+      rw [← Real.log_mul (by positivity) (by positivity)]
+      congr 1; field_simp
+    have h2 : Real.log (((h : ℝ) - 1) / h) = -Real.log ((h : ℝ) / (h - 1)) := by
+      rw [← Real.log_inv]; congr 1; rw [inv_div]
+    have h3 : Real.log ((h : ℝ) / (h - 1)) ≤ (h : ℝ) / (h - 1) - 1 :=
+      Real.log_le_sub_one_of_pos (by positivity)
+    have h4 : (h : ℝ) / (h - 1) - 1 = 1 / ((h : ℝ) - 1) := by field_simp; ring
+    linarith
+  have hG1 : Gaux 1 ≤ 1 / 4 := by
+    unfold Gaux
+    have := Real.log_two_lt_d9
+    norm_num; linarith
+  have hGh :
+    Gaux ((h : ℝ) - 1) ≥
+      (h : ℝ) ^ 2 / 2 * Real.log (2 * h) - h * Real.log (2 * h) - ((h : ℝ) - 1) / 2 -
+        ((h : ℝ) - 1) ^ 2 / 4 := by
+    unfold Gaux
+    have hpos : (0 : ℝ) < (h : ℝ) - 1 := by linarith
+    have e2 : ((h : ℝ) - 1) ^ 2 * (1 / ((h : ℝ) - 1)) = (h : ℝ) - 1 := by field_simp
+    have e3 :
+      ((h : ℝ) - 1) ^ 2 * Real.log (2 * h) ≥
+        (h : ℝ) ^ 2 * Real.log (2 * h) - 2 * h * Real.log (2 * h) :=
+      by nlinarith
+    nlinarith [mul_le_mul_of_nonneg_left hlogratio.le (sq_nonneg ((h : ℝ) - 1))]
+      -- Step 4: assemble
+  have hsum_i : ∑ i ∈ Icc 1 (h - 1), (2 * (i : ℝ)) = (h : ℝ) * (h - 1) := by
+    rw [e1, Finset.sum_Ico_eq_sum_range]
+    simp only [Nat.cast_add, Nat.cast_one, mul_add, Finset.sum_add_distrib, ← Finset.mul_sum,
+      sum_range_cast, Finset.sum_const, Finset.card_range, nsmul_eq_mul]
+    rw [Nat.cast_sub (by omega)]; push_cast; ring
+  have hf1 : (0 : ℝ) ≤ 1 * Real.log (2 * 1) := by norm_num; exact Real.log_nonneg (by norm_num)
+  nlinarith
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/Bracket.lean b/LeanPool/Zeta5Irrational/Table/Bracket.lean
new file mode 100644
index 0000000000..36600b8ac0
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/Bracket.lean
@@ -0,0 +1,133 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Potential
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # Bracket: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_br :
+    ((236122014583635193 / 40000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(5576823552816418281887 / 1250000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((769448367338577 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (38396760876845215791 / 500000000000000000000 : ℝ)) -
+              6 * (2 * (772599247056886688553 / 2000000000000000000000 : ℝ))))) ≤
+      Real.log (1 + qm) - 6 * (3 / 40) * Real.log (qp + (3 / 40) ^ 2) - 2 + 12 * (3 / 40) +
+        2 * Real.sqrt qp * Pfun (Real.sqrt qm) := by
+  have hPneg := Pfun_sqrt_qm_neg
+  unfold Pfun at hPneg ⊢
+  have e1 : Real.log (1 + qm) = Real.log (10059205077 / 10000000000 : ℝ) := by norm_num [qm]
+  have e2 : Real.log (qp + (3 / 40) ^ 2) = Real.log (115455079 / 10000000000 : ℝ) := by
+    norm_num [qp]
+  have hL1 :
+    (236122014583635193 / 40000000000000000000 : ℝ) ≤ Real.log (10059205077 / 10000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (115455079 / 10000000000 : ℝ) ≤
+      (-(5576823552816418281887 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show (115455079 / 10000000000 : ℝ) = (115455079 / 78125000 : ℝ) / 2 ^ 7 by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (115455079 / 78125000 : ℝ) ≤
+            (488214274808581718113 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo : (769448354342251 / 10000000000000000 : ℝ) ≤ Real.sqrt qm :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num [qm])
+  have hs_hi : Real.sqrt qm ≤ (384724177171127 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num [qm])
+  have hp_hi : Real.sqrt qp ≤ (769448367338577 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num [qp])
+  have hspos : 0 < Real.sqrt qm := sqrt_qm_pos
+  have hau1 :
+    Real.arctan (384724177171127 / 5000000000000000 : ℝ) ≤
+      (38396760876845215791 / 500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (38396760876845215791 / 500000000000000000000 : ℝ)) ≤
+      Real.arctan (5000000000000000 / 384724177171127 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 384724177171127 : ℝ) =
+              (384724177171127 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1)
+  have hz1 : (5000000000000000 / 384724177171127 : ℝ) ≤ 1 / Real.sqrt qm := by
+    rw [show
+        (5000000000000000 / 384724177171127 : ℝ) = 1 / (384724177171127 / 5000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hau2 :
+    Real.arctan (625000000000000000000000000000 / 1536623830262694597801594657771 : ℝ) ≤
+      (772599247056886688553 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat2 :
+    Real.arctan (750000000000000 / 769448354342251 : ℝ) ≤
+      (2 * (772599247056886688553 / 2000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (3491138280786563 / 2500000000000000 : ℝ) ≤
+          Real.sqrt (1 + (750000000000000 / 769448354342251 : ℝ) ^ 2))
+      (by
+        rw [show
+            (750000000000000 / 769448354342251 : ℝ) /
+                (1 + (3491138280786563 / 2500000000000000 : ℝ)) =
+              (625000000000000000000000000000 / 1536623830262694597801594657771 : ℝ)
+            by
+        norm_num];
+        exact hau2)
+  have hz2 : (3 / 40) / Real.sqrt qm ≤ (750000000000000 / 769448354342251 : ℝ) := by
+    rw [show
+        (750000000000000 / 769448354342251 : ℝ) =
+          (3 / 40) / (769448354342251 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (38396760876845215791 / 500000000000000000000 : ℝ)) -
+        6 * (2 * (772599247056886688553 / 2000000000000000000000 : ℝ))) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt qm) - 6 * Real.arctan (3 / 40 / Real.sqrt qm)) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((769448367338577 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (38396760876845215791 / 500000000000000000000 : ℝ)) -
+          6 * (2 * (772599247056886688553 / 2000000000000000000000 : ℝ)))) ≤
+      Real.sqrt qp *
+        (Real.pi + Real.arctan (1 / Real.sqrt qm) - 6 * Real.arctan (3 / 40 / Real.sqrt qm)) := by
+    have h1 := mul_le_mul_of_nonpos_right hp_hi hPneg.le
+    have h2 :=
+      mul_le_mul_of_nonneg_left hP
+        (by norm_num : (0 : ℝ) ≤ (769448367338577 / 10000000000000000 : ℝ))
+    linarith only [h1, h2]
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/Common.lean b/LeanPool/Zeta5Irrational/Table/Common.lean
new file mode 100644
index 0000000000..dcba05280e
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/Common.lean
@@ -0,0 +1,556 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Tactic.ReduceModChar
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! ### The entries of Table 1 as numerals -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma aρ_1 : aρ 1 = (1953374043 / 500000000000 : ℝ) := by norm_num [aρ, table1]
+
+lemma bρ_1 : bρ 1 = (8992695531 / 1000000000000 : ℝ) := by norm_num [bρ, table1]
+
+lemma cρ_1 : cρ 1 = (525779809 / 50000000000 : ℝ) := by norm_num [cρ, table1]
+
+lemma aρ_2 : aρ 2 = (289031033 / 125000000000 : ℝ) := by norm_num [aρ, table1]
+
+lemma bρ_2 : bρ 2 = (3068199571 / 200000000000 : ℝ) := by norm_num [bρ, table1]
+
+lemma cρ_2 : cρ 2 = (29471737793 / 1000000000000 : ℝ) := by norm_num [cρ, table1]
+
+lemma aρ_3 : aρ 3 = (280457333 / 200000000000 : ℝ) := by norm_num [aρ, table1]
+
+lemma bρ_3 : bρ 3 = (6432545181 / 250000000000 : ℝ) := by norm_num [bρ, table1]
+
+lemma cρ_3 : cρ 3 = (42934365099 / 1000000000000 : ℝ) := by norm_num [cρ, table1]
+
+lemma aρ_4 : aρ 4 = (220431339 / 250000000000 : ℝ) := by norm_num [aρ, table1]
+
+lemma bρ_4 : bρ 4 = (20954789123 / 500000000000 : ℝ) := by norm_num [bρ, table1]
+
+lemma cρ_4 : cρ 4 = (29102115983 / 500000000000 : ℝ) := by norm_num [cρ, table1]
+
+lemma aρ_5 : aρ 5 = (289098953 / 500000000000 : ℝ) := by norm_num [aρ, table1]
+
+lemma bρ_5 : bρ 5 = (65851089563 / 1000000000000 : ℝ) := by norm_num [bρ, table1]
+
+lemma cρ_5 : cρ 5 = (6903762131 / 100000000000 : ℝ) := by norm_num [cρ, table1]
+
+lemma aρ_6 : aρ 6 = (396324613 / 1000000000000 : ℝ) := by norm_num [aρ, table1]
+
+lemma bρ_6 : bρ 6 = (24870259471 / 250000000000 : ℝ) := by norm_num [bρ, table1]
+
+lemma cρ_6 : cρ 6 = (78873099189 / 1000000000000 : ℝ) := by norm_num [cρ, table1]
+
+lemma aρ_7 : aρ 7 = (283911191 / 1000000000000 : ℝ) := by norm_num [aρ, table1]
+
+lemma bρ_7 : bρ 7 = (72162863729 / 500000000000 : ℝ) := by norm_num [bρ, table1]
+
+lemma cρ_7 : cρ 7 = (84856120711 / 1000000000000 : ℝ) := by norm_num [cρ, table1]
+
+lemma aρ_8 : aρ 8 = (53051547 / 250000000000 : ℝ) := by norm_num [aρ, table1]
+
+lemma bρ_8 : bρ 8 = (201105762729 / 1000000000000 : ℝ) := by norm_num [bρ, table1]
+
+lemma cρ_8 : cρ 8 = (88396082127 / 1000000000000 : ℝ) := by norm_num [cρ, table1]
+
+lemma aρ_9 : aρ 9 = (82548843 / 500000000000 : ℝ) := by norm_num [aρ, table1]
+
+lemma bρ_9 : bρ 9 = (269345996903 / 1000000000000 : ℝ) := by norm_num [bρ, table1]
+
+lemma cρ_9 : cρ 9 = (706427057 / 8000000000 : ℝ) := by norm_num [cρ, table1]
+
+lemma aρ_10 : aρ 10 = (33336783 / 250000000000 : ℝ) := by norm_num [aρ, table1]
+
+lemma bρ_10 : bρ 10 = (86772388539 / 250000000000 : ℝ) := by norm_num [bρ, table1]
+
+lemma cρ_10 : cρ 10 = (17094464251 / 200000000000 : ℝ) := by norm_num [cρ, table1]
+
+lemma aρ_11 : aρ 11 = (55761057 / 500000000000 : ℝ) := by norm_num [aρ, table1]
+
+lemma bρ_11 : bρ 11 = (86161340883 / 200000000000 : ℝ) := by norm_num [bρ, table1]
+
+lemma cρ_11 : cρ 11 = (39449592119 / 500000000000 : ℝ) := by norm_num [cρ, table1]
+
+lemma aρ_12 : aρ 12 = (19269871 / 200000000000 : ℝ) := by norm_num [aρ, table1]
+
+lemma bρ_12 : bρ 12 = (515561896511 / 1000000000000 : ℝ) := by norm_num [bρ, table1]
+
+lemma cρ_12 : cρ 12 = (35176735959 / 500000000000 : ℝ) := by norm_num [cρ, table1]
+
+lemma aρ_13 : aρ 13 = (85815639 / 1000000000000 : ℝ) := by norm_num [aρ, table1]
+
+lemma bρ_13 : bρ 13 = (297724389273 / 500000000000 : ℝ) := by norm_num [bρ, table1]
+
+lemma cρ_13 : cρ 13 = (11767795323 / 200000000000 : ℝ) := by norm_num [cρ, table1]
+
+lemma aρ_14 : aρ 14 = (78667711 / 1000000000000 : ℝ) := by norm_num [aρ, table1]
+
+lemma bρ_14 : bρ 14 = (664241383483 / 1000000000000 : ℝ) := by norm_num [bρ, table1]
+
+lemma cρ_14 : cρ 14 = (22210660553 / 500000000000 : ℝ) := by norm_num [cρ, table1]
+
+lemma aρ_15 : aρ 15 = (14825913 / 200000000000 : ℝ) := by norm_num [aρ, table1]
+
+lemma bρ_15 : bρ 15 = (89520072139 / 125000000000 : ℝ) := by norm_num [bρ, table1]
+
+lemma cρ_15 : cρ 15 = (30462865791 / 1000000000000 : ℝ) := by norm_num [cρ, table1]
+
+lemma aρ_16 : aρ 16 = (7174131 / 100000000000 : ℝ) := by norm_num [aρ, table1]
+
+lemma bρ_16 : bρ 16 = (746637295669 / 1000000000000 : ℝ) := by norm_num [bρ, table1]
+
+lemma cρ_16 : cρ 16 = (5959622577 / 1000000000000 : ℝ) := by norm_num [cρ, table1]
+
+/-! ### Upper bounds for the constant values `log ((b_j - a_j)/4)` -/
+
+lemma Lmid_1 :
+    Real.log ((bρ 1 - aρ 1) / 4) ≤ (-(66675683038238292067283 / 10000000000000000000000 : ℝ)) := by
+  rw [aρ_1, bρ_1,
+    show
+      ((8992695531 / 1000000000000 : ℝ) - (1953374043 / 500000000000 : ℝ)) / 4 =
+        (1017189489 / 800000000000 : ℝ)
+      by norm_num]
+  have h :
+    Real.log (1017189489 / 800000000000 : ℝ) ≤
+      (-(66675683038238292067283 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 10
+        (show (1017189489 / 800000000000 : ℝ) = (1017189489 / 781250000 : ℝ) / 2 ^ 10 by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1017189489 / 781250000 : ℝ) ≤
+            (2639034991761707932717 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact h
+
+lemma Lmid_2 :
+    Real.log ((bρ 2 - aρ 2) / 4) ≤ (-(57268912150832389796717 / 10000000000000000000000 : ℝ)) := by
+  rw [aρ_2, bρ_2,
+    show
+      ((3068199571 / 200000000000 : ℝ) - (289031033 / 125000000000 : ℝ)) / 4 =
+        (13028749591 / 4000000000000 : ℝ)
+      by norm_num]
+  have h :
+    Real.log (13028749591 / 4000000000000 : ℝ) ≤
+      (-(57268912150832389796717 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 9
+        (show (13028749591 / 4000000000000 : ℝ) = (13028749591 / 7812500000 : ℝ) / 2 ^ 9 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13028749591 / 7812500000 : ℝ) ≤
+            (5114334076167610203283 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact h
+
+lemma Lmid_3 :
+    Real.log ((bρ 3 - aρ 3) / 4) ≤ (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) := by
+  rw [aρ_3, bρ_3,
+    show
+      ((6432545181 / 250000000000 : ℝ) - (280457333 / 200000000000 : ℝ)) / 4 =
+        (24327894059 / 4000000000000 : ℝ)
+      by norm_num]
+  have h :
+    Real.log (24327894059 / 4000000000000 : ℝ) ≤
+      (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show (24327894059 / 4000000000000 : ℝ) = (24327894059 / 15625000000 : ℝ) / 2 ^ 8 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24327894059 / 15625000000 : ℝ) ≤
+            (2213757000244685362471 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact h
+
+lemma Lmid_4 :
+    Real.log ((bρ 4 - aρ 4) / 4) ≤ (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) := by
+  rw [aρ_4, bρ_4,
+    show
+      ((20954789123 / 500000000000 : ℝ) - (220431339 / 250000000000 : ℝ)) / 4 =
+        (4102785289 / 400000000000 : ℝ)
+      by norm_num]
+  have h :
+    Real.log (4102785289 / 400000000000 : ℝ) ≤
+      (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show (4102785289 / 400000000000 : ℝ) = (4102785289 / 3125000000 : ℝ) / 2 ^ 7 by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4102785289 / 3125000000 : ℝ) ≤
+            (680579496640619414763 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact h
+
+lemma Lmid_5 :
+    Real.log ((bρ 5 - aρ 5) / 4) ≤ (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) := by
+  rw [aρ_5, bρ_5,
+    show
+      ((65851089563 / 1000000000000 : ℝ) - (289098953 / 500000000000 : ℝ)) / 4 =
+        (65272891657 / 4000000000000 : ℝ)
+      by norm_num]
+  have h :
+    Real.log (65272891657 / 4000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show (65272891657 / 4000000000000 : ℝ) = (65272891657 / 62500000000 : ℝ) / 2 ^ 6 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (65272891657 / 62500000000 : ℝ) ≤
+            (217051289631660878473 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact h
+
+lemma Lmid_6 :
+    Real.log ((bρ 6 - aρ 6) / 4) ≤ (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) := by
+  rw [aρ_6, bρ_6,
+    show
+      ((24870259471 / 250000000000 : ℝ) - (396324613 / 1000000000000 : ℝ)) / 4 =
+        (99084713271 / 4000000000000 : ℝ)
+      by norm_num]
+  have h :
+    Real.log (99084713271 / 4000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show (99084713271 / 4000000000000 : ℝ) = (99084713271 / 62500000000 : ℝ) / 2 ^ 6 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (99084713271 / 62500000000 : ℝ) ≤
+            (460808617104656470757 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact h
+
+lemma Lmid_7 :
+    Real.log ((bρ 7 - aρ 7) / 4) ≤ (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) := by
+  rw [aρ_7, bρ_7,
+    show
+      ((72162863729 / 500000000000 : ℝ) - (283911191 / 1000000000000 : ℝ)) / 4 =
+        (144041816267 / 4000000000000 : ℝ)
+      by norm_num]
+  have h :
+    Real.log (144041816267 / 4000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show (144041816267 / 4000000000000 : ℝ) = (144041816267 / 125000000000 : ℝ) / 2 ^ 5 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (144041816267 / 125000000000 : ℝ) ≤
+            (1417899108615240212659 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact h
+
+lemma Lmid_8 :
+    Real.log ((bρ 8 - aρ 8) / 4) ≤ (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) := by
+  rw [aρ_8, bρ_8,
+    show
+      ((201105762729 / 1000000000000 : ℝ) - (53051547 / 250000000000 : ℝ)) / 4 =
+        (200893556541 / 4000000000000 : ℝ)
+      by norm_num]
+  have h :
+    Real.log (200893556541 / 4000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show (200893556541 / 4000000000000 : ℝ) = (200893556541 / 125000000000 : ℝ) / 2 ^ 5 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (200893556541 / 125000000000 : ℝ) ≤
+            (2372307305187944700339 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact h
+
+lemma Lmid_9 :
+    Real.log ((bρ 9 - aρ 9) / 4) ≤ (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) := by
+  rw [aρ_9, bρ_9,
+    show
+      ((269345996903 / 1000000000000 : ℝ) - (82548843 / 500000000000 : ℝ)) / 4 =
+        (269180899217 / 4000000000000 : ℝ)
+      by norm_num]
+  have h :
+    Real.log (269180899217 / 4000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (269180899217 / 4000000000000 : ℝ) = (269180899217 / 250000000000 : ℝ) / 2 ^ 4 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (269180899217 / 250000000000 : ℝ) ≤
+            (739227235165382858631 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact h
+
+lemma Lmid_10 :
+    Real.log ((bρ 10 - aρ 10) / 4) ≤ (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) := by
+  rw [aρ_10, bρ_10,
+    show
+      ((86772388539 / 250000000000 : ℝ) - (33336783 / 250000000000 : ℝ)) / 4 =
+        (21684762939 / 250000000000 : ℝ)
+      by norm_num]
+  have h :
+    Real.log (21684762939 / 250000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (21684762939 / 250000000000 : ℝ) = (21684762939 / 15625000000 : ℝ) / 2 ^ 4 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21684762939 / 15625000000 : ℝ) ≤
+            (1638688247891813620441 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact h
+
+lemma Lmid_11 :
+    Real.log ((bρ 11 - aρ 11) / 4) ≤ (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) := by
+  rw [aρ_11, bρ_11,
+    show
+      ((86161340883 / 200000000000 : ℝ) - (55761057 / 500000000000 : ℝ)) / 4 =
+        (430695182301 / 4000000000000 : ℝ)
+      by norm_num]
+  have h :
+    Real.log (430695182301 / 4000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (430695182301 / 4000000000000 : ℝ) = (430695182301 / 250000000000 : ℝ) / 2 ^ 4 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (430695182301 / 250000000000 : ℝ) ≤
+            (1087879376786883877841 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact h
+
+lemma Lmid_12 :
+    Real.log ((bρ 12 - aρ 12) / 4) ≤ (-(640305991230515290559 / 312500000000000000000 : ℝ)) := by
+  rw [aρ_12, bρ_12,
+    show
+      ((515561896511 / 1000000000000 : ℝ) - (19269871 / 200000000000 : ℝ)) / 4 =
+        (128866386789 / 1000000000000 : ℝ)
+      by norm_num]
+  have h :
+    Real.log (128866386789 / 1000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (128866386789 / 1000000000000 : ℝ) = (128866386789 / 125000000000 : ℝ) / 2 ^ 3 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (128866386789 / 125000000000 : ℝ) ≤
+            (9519490300734709441 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact h
+
+lemma Lmid_13 :
+    Real.log ((bρ 13 - aρ 13) / 4) ≤ (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) := by
+  rw [aρ_13, bρ_13,
+    show
+      ((297724389273 / 500000000000 : ℝ) - (85815639 / 1000000000000 : ℝ)) / 4 =
+        (595362962907 / 4000000000000 : ℝ)
+      by norm_num]
+  have h :
+    Real.log (595362962907 / 4000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (595362962907 / 4000000000000 : ℝ) = (595362962907 / 500000000000 : ℝ) / 2 ^ 3 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (595362962907 / 500000000000 : ℝ) ≤
+            (436407857070690432561 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact h
+
+lemma Lmid_14 :
+    Real.log ((bρ 14 - aρ 14) / 4) ≤ (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) := by
+  rw [aρ_14, bρ_14,
+    show
+      ((664241383483 / 1000000000000 : ℝ) - (78667711 / 1000000000000 : ℝ)) / 4 =
+        (166040678943 / 1000000000000 : ℝ)
+      by norm_num]
+  have h :
+    Real.log (166040678943 / 1000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (166040678943 / 1000000000000 : ℝ) = (166040678943 / 125000000000 : ℝ) / 2 ^ 3 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (166040678943 / 125000000000 : ℝ) ≤
+            (1419595374534700017209 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact h
+
+lemma Lmid_15 :
+    Real.log ((bρ 15 - aρ 15) / 4) ≤ (-(268788866070650094343 / 156250000000000000000 : ℝ)) := by
+  rw [aρ_15, bρ_15,
+    show
+      ((89520072139 / 125000000000 : ℝ) - (14825913 / 200000000000 : ℝ)) / 4 =
+        (716086447547 / 4000000000000 : ℝ)
+      by norm_num]
+  have h :
+    Real.log (716086447547 / 4000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (716086447547 / 4000000000000 : ℝ) = (716086447547 / 500000000000 : ℝ) / 2 ^ 3 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (716086447547 / 500000000000 : ℝ) ≤
+            (56123874694974905657 / 156250000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact h
+
+lemma Lmid_16 :
+    Real.log ((bρ 16 - aρ 16) / 4) ≤ (-(419641552645073499393 / 250000000000000000000 : ℝ)) := by
+  rw [aρ_16, bρ_16,
+    show
+      ((746637295669 / 1000000000000 : ℝ) - (7174131 / 100000000000 : ℝ)) / 4 =
+        (746565554359 / 4000000000000 : ℝ)
+      by norm_num]
+  have h :
+    Real.log (746565554359 / 4000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (746565554359 / 4000000000000 : ℝ) = (746565554359 / 500000000000 : ℝ) / 2 ^ 3 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (746565554359 / 500000000000 : ℝ) ≤
+            (100218832579926500607 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact h
+
+/-- Upper bound for the constant value of `Uρ` on `[a₁, b₁]`. -/
+lemma Umid_le : Uρmid ≤ (-(28683108419825598767969 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρmid_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12,
+    cρ_13, cρ_14, cρ_15, cρ_16]
+  have h1 :
+    (525779809 / 50000000000 : ℝ) * Real.log ((bρ 1 - aρ 1) / 4) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(66675683038238292067283 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left Lmid_1 (by norm_num)
+  have h2 :
+    (29471737793 / 1000000000000 : ℝ) * Real.log ((bρ 2 - aρ 2) / 4) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(57268912150832389796717 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left Lmid_2 (by norm_num)
+  have h3 :
+    (42934365099 / 1000000000000 : ℝ) * Real.log ((bρ 3 - aρ 3) / 4) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left Lmid_3 (by norm_num)
+  have h4 :
+    (29102115983 / 500000000000 : ℝ) * Real.log ((bρ 4 - aρ 4) / 4) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left Lmid_4 (by norm_num)
+  have h5 :
+    (6903762131 / 100000000000 : ℝ) * Real.log ((bρ 5 - aρ 5) / 4) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left Lmid_5 (by norm_num)
+  have h6 :
+    (78873099189 / 1000000000000 : ℝ) * Real.log ((bρ 6 - aρ 6) / 4) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left Lmid_6 (by norm_num)
+  have h7 :
+    (84856120711 / 1000000000000 : ℝ) * Real.log ((bρ 7 - aρ 7) / 4) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left Lmid_7 (by norm_num)
+  have h8 :
+    (88396082127 / 1000000000000 : ℝ) * Real.log ((bρ 8 - aρ 8) / 4) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left Lmid_8 (by norm_num)
+  have h9 :
+    (706427057 / 8000000000 : ℝ) * Real.log ((bρ 9 - aρ 9) / 4) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left Lmid_9 (by norm_num)
+  have h10 :
+    (17094464251 / 200000000000 : ℝ) * Real.log ((bρ 10 - aρ 10) / 4) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left Lmid_10 (by norm_num)
+  have h11 :
+    (39449592119 / 500000000000 : ℝ) * Real.log ((bρ 11 - aρ 11) / 4) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left Lmid_11 (by norm_num)
+  have h12 :
+    (35176735959 / 500000000000 : ℝ) * Real.log ((bρ 12 - aρ 12) / 4) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left Lmid_12 (by norm_num)
+  have h13 :
+    (11767795323 / 200000000000 : ℝ) * Real.log ((bρ 13 - aρ 13) / 4) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left Lmid_13 (by norm_num)
+  have h14 :
+    (22210660553 / 500000000000 : ℝ) * Real.log ((bρ 14 - aρ 14) / 4) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left Lmid_14 (by norm_num)
+  have h15 :
+    (30462865791 / 1000000000000 : ℝ) * Real.log ((bρ 15 - aρ 15) / 4) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left Lmid_15 (by norm_num)
+  have h16 :
+    (5959622577 / 1000000000000 : ℝ) * Real.log ((bρ 16 - aρ 16) / 4) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left Lmid_16 (by norm_num)
+  linarith only [h1, h2, h3, h4, h5, h6, h7, h8, h9, h10, h11, h12, h13, h14, h15, h16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/Intervals.lean b/LeanPool/Zeta5Irrational/Table/Intervals.lean
new file mode 100644
index 0000000000..c34b9d8c37
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/Intervals.lean
@@ -0,0 +1,10382 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+import Mathlib.Analysis.Real.Pi.Bounds
+
+public import LeanPool.Zeta5Irrational.Table.U00
+public import LeanPool.Zeta5Irrational.Table.U01
+public import LeanPool.Zeta5Irrational.Table.U02
+public import LeanPool.Zeta5Irrational.Table.U03
+public import LeanPool.Zeta5Irrational.Table.U04
+public import LeanPool.Zeta5Irrational.Table.U05
+public import LeanPool.Zeta5Irrational.Table.U06
+public import LeanPool.Zeta5Irrational.Table.U07
+public import LeanPool.Zeta5Irrational.Table.U08
+public import LeanPool.Zeta5Irrational.Table.U09
+public import LeanPool.Zeta5Irrational.Table.U10
+public import LeanPool.Zeta5Irrational.Table.U11
+public import LeanPool.Zeta5Irrational.Table.U12
+public import LeanPool.Zeta5Irrational.Table.U13
+public import LeanPool.Zeta5Irrational.Table.U14
+public import LeanPool.Zeta5Irrational.Table.U15
+public import LeanPool.Zeta5Irrational.Table.U16
+public import LeanPool.Zeta5Irrational.Table.U17
+public import LeanPool.Zeta5Irrational.Table.U18
+public import LeanPool.Zeta5Irrational.Table.U19
+public import LeanPool.Zeta5Irrational.Table.U20
+public import LeanPool.Zeta5Irrational.Table.U21
+public import LeanPool.Zeta5Irrational.Table.U22
+public import LeanPool.Zeta5Irrational.Table.U23
+public import LeanPool.Zeta5Irrational.Table.U24
+public import LeanPool.Zeta5Irrational.Table.U25
+public import LeanPool.Zeta5Irrational.Table.U26
+public import LeanPool.Zeta5Irrational.Table.U27
+public import LeanPool.Zeta5Irrational.Table.U28
+public import LeanPool.Zeta5Irrational.Table.U29
+public import LeanPool.Zeta5Irrational.Table.U30
+public import LeanPool.Zeta5Irrational.Table.U31
+public import LeanPool.Zeta5Irrational.Table.U32
+public import LeanPool.Zeta5Irrational.Table.U33
+public import LeanPool.Zeta5Irrational.Table.U34
+public import LeanPool.Zeta5Irrational.Table.U35
+public import LeanPool.Zeta5Irrational.Table.U36
+public import LeanPool.Zeta5Irrational.Table.U37
+public import LeanPool.Zeta5Irrational.Table.U38
+public import LeanPool.Zeta5Irrational.Table.U39
+public import LeanPool.Zeta5Irrational.Table.U40
+public import LeanPool.Zeta5Irrational.Table.U41
+public import LeanPool.Zeta5Irrational.Table.U42
+public import LeanPool.Zeta5Irrational.Table.U43
+public import LeanPool.Zeta5Irrational.Table.U44
+public import LeanPool.Zeta5Irrational.Table.U45
+public import LeanPool.Zeta5Irrational.Table.U46
+public import LeanPool.Zeta5Irrational.Table.U47
+public import LeanPool.Zeta5Irrational.Table.U48
+public import LeanPool.Zeta5Irrational.Table.U49
+public import LeanPool.Zeta5Irrational.Table.U50
+public import LeanPool.Zeta5Irrational.Table.U51
+public import LeanPool.Zeta5Irrational.Table.U52
+public import LeanPool.Zeta5Irrational.Table.U53
+public import LeanPool.Zeta5Irrational.Table.U54
+public import LeanPool.Zeta5Irrational.Table.U55
+public import LeanPool.Zeta5Irrational.Table.U56
+public import LeanPool.Zeta5Irrational.Table.V00
+public import LeanPool.Zeta5Irrational.Table.V01
+public import LeanPool.Zeta5Irrational.Table.V02
+public import LeanPool.Zeta5Irrational.Table.V03
+public import LeanPool.Zeta5Irrational.Table.V04
+public import LeanPool.Zeta5Irrational.Table.V05
+public import LeanPool.Zeta5Irrational.Table.V06
+public import LeanPool.Zeta5Irrational.Table.V07
+public import LeanPool.Zeta5Irrational.Table.V08
+public import LeanPool.Zeta5Irrational.Table.V09
+public import LeanPool.Zeta5Irrational.Table.V10
+public import LeanPool.Zeta5Irrational.Table.V11
+public import LeanPool.Zeta5Irrational.Table.V12
+public import LeanPool.Zeta5Irrational.Table.V13
+public import LeanPool.Zeta5Irrational.Table.V14
+public import LeanPool.Zeta5Irrational.Table.V15
+public import LeanPool.Zeta5Irrational.Table.V16
+public import LeanPool.Zeta5Irrational.Table.V17
+public import LeanPool.Zeta5Irrational.Table.V18
+public import LeanPool.Zeta5Irrational.Table.V19
+public import LeanPool.Zeta5Irrational.Table.V20
+public import LeanPool.Zeta5Irrational.Table.V21
+public import LeanPool.Zeta5Irrational.Table.V22
+public import LeanPool.Zeta5Irrational.Table.V23
+public import LeanPool.Zeta5Irrational.Table.V24
+public import LeanPool.Zeta5Irrational.Table.V25
+public import LeanPool.Zeta5Irrational.Table.V26
+public import LeanPool.Zeta5Irrational.Table.V27
+public import LeanPool.Zeta5Irrational.Table.V28
+public import LeanPool.Zeta5Irrational.Table.V29
+public import LeanPool.Zeta5Irrational.Table.V30
+public import LeanPool.Zeta5Irrational.Table.V31
+public import LeanPool.Zeta5Irrational.Table.V32
+public import LeanPool.Zeta5Irrational.Table.V33
+public import LeanPool.Zeta5Irrational.Table.V34
+public import LeanPool.Zeta5Irrational.Table.V35
+public import LeanPool.Zeta5Irrational.Table.V36
+public import LeanPool.Zeta5Irrational.Table.V37
+public import LeanPool.Zeta5Irrational.Table.V38
+public import LeanPool.Zeta5Irrational.Table.V39
+public import LeanPool.Zeta5Irrational.Table.V40
+public import LeanPool.Zeta5Irrational.Table.V41
+public import LeanPool.Zeta5Irrational.Table.V42
+public import LeanPool.Zeta5Irrational.Table.V43
+public import LeanPool.Zeta5Irrational.Table.V44
+public import LeanPool.Zeta5Irrational.Table.V45
+public import LeanPool.Zeta5Irrational.Table.V46
+public import LeanPool.Zeta5Irrational.Table.V47
+public import LeanPool.Zeta5Irrational.Table.V48
+public import LeanPool.Zeta5Irrational.Table.V49
+public import LeanPool.Zeta5Irrational.Table.V50
+public import LeanPool.Zeta5Irrational.Table.V51
+public import LeanPool.Zeta5Irrational.Table.V52
+public import LeanPool.Zeta5Irrational.Table.V53
+public import LeanPool.Zeta5Irrational.Table.V54
+public import LeanPool.Zeta5Irrational.Table.V55
+public import LeanPool.Zeta5Irrational.Table.V56
+public import LeanPool.Zeta5Irrational.Table.Bracket
+public import LeanPool.Zeta5Irrational.Table.Tail
+
+/-! # Intervals: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma tc_0 :
+    ∀ t, (0 : ℝ) ≤ t → t ≤ (7174131 / 200000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_left (by norm_num) (by norm_num [aρ_1]) U_0 V_1
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_1 :
+    ∀ t,
+      (7174131 / 200000000000 : ℝ) ≤ t →
+        t ≤ (21522393 / 400000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_left (by norm_num) (by norm_num [aρ_1]) U_1 V_2
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_2 :
+    ∀ t,
+      (21522393 / 400000000000 : ℝ) ≤ t →
+        t ≤ (7174131 / 100000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_left (by norm_num) (by norm_num [aρ_1]) U_2 V_3
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_3 :
+    ∀ t,
+      (7174131 / 100000000000 : ℝ) ≤ t →
+        t ≤ (14825913 / 200000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_left (by norm_num) (by norm_num [aρ_1]) U_3 V_4
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_4 :
+    ∀ t,
+      (14825913 / 200000000000 : ℝ) ≤ t →
+        t ≤ (78667711 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_left (by norm_num) (by norm_num [aρ_1]) U_4 V_5
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_5 :
+    ∀ t,
+      (78667711 / 1000000000000 : ℝ) ≤ t →
+        t ≤ (85815639 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_left (by norm_num) (by norm_num [aρ_1]) U_5 V_6
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_6 :
+    ∀ t,
+      (85815639 / 1000000000000 : ℝ) ≤ t →
+        t ≤ (19269871 / 200000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_left (by norm_num) (by norm_num [aρ_1]) U_6 V_7
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_7 :
+    ∀ t,
+      (19269871 / 200000000000 : ℝ) ≤ t →
+        t ≤ (55761057 / 500000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_left (by norm_num) (by norm_num [aρ_1]) U_7 V_8
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_8 :
+    ∀ t,
+      (55761057 / 500000000000 : ℝ) ≤ t →
+        t ≤ (33336783 / 250000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_left (by norm_num) (by norm_num [aρ_1]) U_8 V_9
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_9 :
+    ∀ t,
+      (33336783 / 250000000000 : ℝ) ≤ t →
+        t ≤ (82548843 / 500000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_left (by norm_num) (by norm_num [aρ_1]) U_9 V_10
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_10 :
+    ∀ t,
+      (82548843 / 500000000000 : ℝ) ≤ t →
+        t ≤ (53051547 / 250000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_left (by norm_num) (by norm_num [aρ_1]) U_10 V_11
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_11 :
+    ∀ t,
+      (53051547 / 250000000000 : ℝ) ≤ t →
+        t ≤ (496117379 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_left (by norm_num) (by norm_num [aρ_1]) U_11 V_12
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_12 :
+    ∀ t,
+      (496117379 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (283911191 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_left (by norm_num) (by norm_num [aρ_1]) U_12 V_13
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_13 :
+    ∀ t,
+      (283911191 / 1000000000000 : ℝ) ≤ t →
+        t ≤ (170058951 / 500000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_left (by norm_num) (by norm_num [aρ_1]) U_13 V_14
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_14 :
+    ∀ t,
+      (170058951 / 500000000000 : ℝ) ≤ t →
+        t ≤ (396324613 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_left (by norm_num) (by norm_num [aρ_1]) U_14 V_15
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_15 :
+    ∀ t,
+      (396324613 / 1000000000000 : ℝ) ≤ t →
+        t ≤ (974522519 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_left (by norm_num) (by norm_num [aρ_1]) U_15 V_16
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_16 :
+    ∀ t,
+      (974522519 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (289098953 / 500000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_left (by norm_num) (by norm_num [aρ_1]) U_16 V_17
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_17 :
+    ∀ t,
+      (289098953 / 500000000000 : ℝ) ≤ t →
+        t ≤ (729961631 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_left (by norm_num) (by norm_num [aρ_1]) U_17 V_18
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_18 :
+    ∀ t,
+      (729961631 / 1000000000000 : ℝ) ≤ t →
+        t ≤ (1611686987 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_left (by norm_num) (by norm_num [aρ_1]) U_18 V_19
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_19 :
+    ∀ t,
+      (1611686987 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (220431339 / 250000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_left (by norm_num) (by norm_num [aρ_1]) U_19 V_20
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_20 :
+    ∀ t,
+      (220431339 / 250000000000 : ℝ) ≤ t →
+        t ≤ (4047462733 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_left (by norm_num) (by norm_num [aρ_1]) U_20 V_21
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_21 :
+    ∀ t,
+      (4047462733 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (2284012021 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_left (by norm_num) (by norm_num [aρ_1]) U_21 V_22
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_22 :
+    ∀ t,
+      (2284012021 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (5088585351 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_left (by norm_num) (by norm_num [aρ_1]) U_22 V_23
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_23 :
+    ∀ t,
+      (5088585351 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (280457333 / 200000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_left (by norm_num) (by norm_num [aρ_1]) U_23 V_24
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_24 :
+    ∀ t,
+      (280457333 / 200000000000 : ℝ) ≤ t →
+        t ≤ (6519108259 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_left (by norm_num) (by norm_num [aρ_1]) U_24 V_25
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_25 :
+    ∀ t,
+      (6519108259 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (3714534929 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_left (by norm_num) (by norm_num [aρ_1]) U_25 V_26
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_26 :
+    ∀ t,
+      (3714534929 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (8339031457 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_left (by norm_num) (by norm_num [aρ_1]) U_26 V_27
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_27 :
+    ∀ t,
+      (8339031457 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (289031033 / 125000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_left (by norm_num) (by norm_num [aρ_1]) U_27 V_28
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_28 :
+    ∀ t,
+      (289031033 / 125000000000 : ℝ) ≤ t →
+        t ≤ (124379927 / 40000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_left (by norm_num) (by norm_num [aρ_1]) U_28 V_29
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_29 :
+    ∀ t,
+      (124379927 / 40000000000 : ℝ) ≤ t →
+        t ≤ (7016246261 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_left (by norm_num) (by norm_num [aρ_1]) U_29 V_30
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_30 :
+    ∀ t,
+      (7016246261 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (1953374043 / 500000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_left (by norm_num) (by norm_num [aρ_1]) U_30 V_31
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_31 :
+    ∀ t,
+      (1953374043 / 500000000000 : ℝ) ≤ t →
+        t ≤ (59205077 / 10000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 := by
+  intro t h1 h2 h0
+  have eq1 : qm = (59205077 / 10000000000 : ℝ) := by norm_num [qm]
+  exact
+    check_mid_left Umid_le V_32 (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) t
+      (by rw [aρ_1]; exact h1) (by rw [eq1]; exact h2) h0
+
+lemma tc_32 :
+    ∀ t,
+      (59205077 / 10000000000 : ℝ) ≤ t →
+        t ≤ (59205079 / 10000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 := by
+  intro t h1 h2 h0
+  have eq1 : qm = (59205077 / 10000000000 : ℝ) := by norm_num [qm]
+  have eq2 : qp = (59205079 / 10000000000 : ℝ) := by norm_num [qp]
+  exact
+    check_mid Umid_le V_br (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) t
+      (by rw [eq1]; exact h1) (by rw [eq2]; exact h2) h0
+
+lemma tc_33 :
+    ∀ t,
+      (59205079 / 10000000000 : ℝ) ≤ t →
+        t ≤ (8992695531 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 := by
+  intro t h1 h2 h0
+  have eq2 : qp = (59205079 / 10000000000 : ℝ) := by norm_num [qp]
+  exact
+    check_mid_right Umid_le V_33 (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) t
+      (by rw [eq2]; exact h1) (by rw [bρ_1]; exact h2) h0
+
+lemma tc_34 :
+    ∀ t,
+      (8992695531 / 1000000000000 : ℝ) ≤ t →
+        t ≤ (19572466643 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_35 V_34
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_35 :
+    ∀ t,
+      (19572466643 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (40732008867 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_36 V_35
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_36 :
+    ∀ t,
+      (40732008867 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (1322471389 / 125000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_37 V_36
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_37 :
+    ∀ t,
+      (1322471389 / 125000000000 : ℝ) ≤ t →
+        t ≤ (4549323561 / 400000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_38 V_37
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_38 :
+    ∀ t,
+      (4549323561 / 400000000000 : ℝ) ≤ t →
+        t ≤ (12166846693 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_39 V_38
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_39 :
+    ∀ t,
+      (12166846693 / 1000000000000 : ℝ) ≤ t →
+        t ≤ (3068199571 / 200000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_40 V_39
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_40 :
+    ∀ t,
+      (3068199571 / 200000000000 : ℝ) ≤ t →
+        t ≤ (255845148549 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_41 V_40
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_41 :
+    ∀ t,
+      (255845148549 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (133117165709 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_42 V_41
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_42 :
+    ∀ t,
+      (133117165709 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (108571569141 / 6400000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_43 V_42
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_43 :
+    ∀ t,
+      (108571569141 / 6400000000000 : ℝ) ≤ t →
+        t ≤ (276623514287 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_44 V_43
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_44 :
+    ∀ t,
+      (276623514287 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (563636211443 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_45 V_44
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_45 :
+    ∀ t,
+      (563636211443 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (71753174289 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_46 V_45
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_46 :
+    ∀ t,
+      (71753174289 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (584414577181 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_47 V_46
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_47 :
+    ∀ t,
+      (584414577181 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (11896075201 / 640000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_48 V_47
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_48 :
+    ∀ t,
+      (11896075201 / 640000000000 : ℝ) ≤ t →
+        t ≤ (605192942919 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_49 V_48
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_49 :
+    ∀ t,
+      (605192942919 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (153895531447 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_50 V_49
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_50 :
+    ∀ t,
+      (153895531447 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (318180245763 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_51 V_50
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_51 :
+    ∀ t,
+      (318180245763 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (41071178579 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_52 V_51
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_52 :
+    ∀ t,
+      (41071178579 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (34934779437 / 1600000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_53 V_52
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_53 :
+    ∀ t,
+      (34934779437 / 1600000000000 : ℝ) ≤ t →
+        t ≤ (92531540027 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_54 V_53
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_54 :
+    ∀ t,
+      (92531540027 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (6432545181 / 250000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_55 V_54
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_55 :
+    ∀ t,
+      (6432545181 / 250000000000 : ℝ) ≤ t →
+        t ≤ (213931144553 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_56 V_55
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_56 :
+    ∀ t,
+      (213931144553 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (435951987867 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_57 V_56
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_57 :
+    ∀ t,
+      (435951987867 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (111010421657 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_58 V_57
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_58 :
+    ∀ t,
+      (111010421657 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (452131385389 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_59 V_58
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_59 :
+    ∀ t,
+      (452131385389 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (912352469539 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_60 V_59
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_60 :
+    ∀ t,
+      (912352469539 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (9204421683 / 320000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_61 V_60
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_61 :
+    ∀ t,
+      (9204421683 / 320000000000 : ℝ) ≤ t →
+        t ≤ (928531867061 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_62 V_61
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_62 :
+    ∀ t,
+      (928531867061 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (468310782911 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_63 V_62
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_63 :
+    ∀ t,
+      (468310782911 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (944711264583 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_64 V_63
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_64 :
+    ∀ t,
+      (944711264583 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (59550060209 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_65 V_64
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_65 :
+    ∀ t,
+      (59550060209 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (192178132421 / 6400000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_66 V_65
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_66 :
+    ∀ t,
+      (192178132421 / 6400000000000 : ℝ) ≤ t →
+        t ≤ (484490180433 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_67 V_66
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_67 :
+    ∀ t,
+      (484490180433 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (977070059627 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_68 V_67
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_68 :
+    ∀ t,
+      (977070059627 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (246289939597 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_69 V_68
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_69 :
+    ∀ t,
+      (246289939597 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (100133915591 / 3200000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_70 V_69
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_70 :
+    ∀ t,
+      (100133915591 / 3200000000000 : ℝ) ≤ t →
+        t ≤ (127189819179 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_71 V_70
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_71 :
+    ∀ t,
+      (127189819179 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (516848975477 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_72 V_71
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_72 :
+    ∀ t,
+      (516848975477 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (262469337119 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_73 V_72
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_73 :
+    ∀ t,
+      (262469337119 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (6763975897 / 200000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_74 V_73
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_74 :
+    ∀ t,
+      (6763975897 / 200000000000 : ℝ) ≤ t →
+        t ≤ (278648734641 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_75 V_74
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_75 :
+    ∀ t,
+      (278648734641 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (143369216701 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_76 V_75
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_76 :
+    ∀ t,
+      (143369216701 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (75729457731 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_77 V_76
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_77 :
+    ∀ t,
+      (75729457731 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (20954789123 / 500000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_78 V_77
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_78 :
+    ∀ t,
+      (20954789123 / 500000000000 : ℝ) ≤ t →
+        t ≤ (694494763253 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_79 V_78
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_79 :
+    ∀ t,
+      (694494763253 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (1412931037823 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_80 V_79
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_80 :
+    ∀ t,
+      (1412931037823 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (71843627457 / 1600000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_81 V_80
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_81 :
+    ∀ t,
+      (71843627457 / 1600000000000 : ℝ) ≤ t →
+        t ≤ (2897686609597 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_82 V_81
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_82 :
+    ∀ t,
+      (2897686609597 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (1460814060457 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_83 V_82
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_83 :
+    ∀ t,
+      (1460814060457 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (2945569632231 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_84 V_83
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_84 :
+    ∀ t,
+      (2945569632231 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (742377785887 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_85 V_84
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_85 :
+    ∀ t,
+      (742377785887 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (598690530973 / 12800000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_86 V_85
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_86 :
+    ∀ t,
+      (598690530973 / 12800000000000 : ℝ) ≤ t →
+        t ≤ (1508697083091 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_87 V_86
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_87 :
+    ∀ t,
+      (1508697083091 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (3041335677499 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_88 V_87
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_88 :
+    ∀ t,
+      (3041335677499 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (191579824301 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_89 V_88
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_89 :
+    ∀ t,
+      (191579824301 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (3089218700133 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_90 V_89
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_90 :
+    ∀ t,
+      (3089218700133 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (62263204229 / 1280000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_91 V_90
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_91 :
+    ∀ t,
+      (62263204229 / 1280000000000 : ℝ) ≤ t →
+        t ≤ (3137101722767 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_92 V_91
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_92 :
+    ∀ t,
+      (3137101722767 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (790260808521 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_93 V_92
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_93 :
+    ∀ t,
+      (790260808521 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (3184984745401 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_94 V_93
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_94 :
+    ∀ t,
+      (3184984745401 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (1604463128359 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_95 V_94
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_95 :
+    ∀ t,
+      (1604463128359 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (646573553607 / 12800000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_96 V_95
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_96 :
+    ∀ t,
+      (646573553607 / 12800000000000 : ℝ) ≤ t →
+        t ≤ (407101159919 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_97 V_96
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_97 :
+    ∀ t,
+      (407101159919 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (1652346150993 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_98 V_97
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_98 :
+    ∀ t,
+      (1652346150993 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (167628766231 / 3200000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_99 V_98
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_99 :
+    ∀ t,
+      (167628766231 / 3200000000000 : ℝ) ≤ t →
+        t ≤ (1700229173627 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_100 V_99
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_100 :
+    ∀ t,
+      (1700229173627 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (107760667809 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_101 V_100
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_101 :
+    ∀ t,
+      (107760667809 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (886026853789 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_102 V_101
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_102 :
+    ∀ t,
+      (886026853789 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (454984182553 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_103 V_102
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_103 :
+    ∀ t,
+      (454984182553 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (47892569387 / 800000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_104 V_103
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_104 :
+    ∀ t,
+      (47892569387 / 800000000000 : ℝ) ≤ t →
+        t ≤ (502867205187 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_105 V_104
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_105 :
+    ∀ t,
+      (502867205187 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (65851089563 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_106 V_105
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_106 :
+    ∀ t,
+      (65851089563 / 1000000000000 : ℝ) ≤ t →
+        t ≤ (2140864814337 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_107 V_106
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_107 :
+    ∀ t,
+      (2140864814337 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (1087247381329 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_108 V_107
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_108 :
+    ∀ t,
+      (1087247381329 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (2208124710979 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_109 V_108
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_109 :
+    ∀ t,
+      (2208124710979 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (4449879370279 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_110 V_109
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_110 :
+    ∀ t,
+      (4449879370279 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (22417546593 / 320000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_111 V_110
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_111 :
+    ∀ t,
+      (22417546593 / 320000000000 : ℝ) ≤ t →
+        t ≤ (4517139266921 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_112 V_111
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_112 :
+    ∀ t,
+      (4517139266921 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (2275384607621 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_113 V_112
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_113 :
+    ∀ t,
+      (2275384607621 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (4584399163563 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_114 V_113
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_114 :
+    ∀ t,
+      (4584399163563 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (1154507277971 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_115 V_114
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_115 :
+    ∀ t,
+      (1154507277971 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (930331812041 / 12800000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_116 V_115
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_116 :
+    ∀ t,
+      (930331812041 / 12800000000000 : ℝ) ≤ t →
+        t ≤ (2342644504263 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_117 V_116
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_117 :
+    ∀ t,
+      (2342644504263 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (9404207965373 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_118 V_117
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_118 :
+    ∀ t,
+      (9404207965373 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (4718918956847 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_119 V_118
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_119 :
+    ∀ t,
+      (4718918956847 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (1894293572403 / 25600000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_120 V_119
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_120 :
+    ∀ t,
+      (1894293572403 / 25600000000000 : ℝ) ≤ t →
+        t ≤ (297034306573 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_121 V_120
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_121 :
+    ∀ t,
+      (297034306573 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (9538727758657 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_122 V_121
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_122 :
+    ∀ t,
+      (9538727758657 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (4786178853489 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_123 V_122
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_123 :
+    ∀ t,
+      (4786178853489 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (9605987655299 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_124 V_123
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_124 :
+    ∀ t,
+      (9605987655299 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (481980880181 / 6400000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_125 V_124
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_125 :
+    ∀ t,
+      (481980880181 / 6400000000000 : ℝ) ≤ t →
+        t ≤ (9673247551941 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_126 V_125
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_126 :
+    ∀ t,
+      (9673247551941 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (4853438750131 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_127 V_126
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_127 :
+    ∀ t,
+      (4853438750131 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (1221767174613 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_128 V_127
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_128 :
+    ∀ t,
+      (1221767174613 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (4920698646773 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_129 V_128
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_129 :
+    ∀ t,
+      (4920698646773 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (2477164297547 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_130 V_129
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_130 :
+    ∀ t,
+      (2477164297547 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (997591708683 / 12800000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_131 V_130
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_131 :
+    ∀ t,
+      (997591708683 / 12800000000000 : ℝ) ≤ t →
+        t ≤ (627698561467 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_132 V_131
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_132 :
+    ∀ t,
+      (627698561467 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (5055218440057 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_133 V_132
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_133 :
+    ∀ t,
+      (5055218440057 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (2544424194189 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_134 V_133
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_134 :
+    ∀ t,
+      (2544424194189 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (5122478336699 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_135 V_134
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_135 :
+    ∀ t,
+      (5122478336699 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (257805414251 / 3200000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_136 V_135
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_136 :
+    ∀ t,
+      (257805414251 / 3200000000000 : ℝ) ≤ t →
+        t ≤ (2611684090831 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_137 V_136
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_137 :
+    ∀ t,
+      (2611684090831 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (165332127447 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_138 V_137
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_138 :
+    ∀ t,
+      (165332127447 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (2678943987473 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_139 V_138
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_139 :
+    ∀ t,
+      (2678943987473 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (1356286967897 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_140 V_139
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_140 :
+    ∀ t,
+      (1356286967897 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (694958458109 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_141 V_140
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_141 :
+    ∀ t,
+      (694958458109 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (1423546864539 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_142 V_141
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_142 :
+    ∀ t,
+      (1423546864539 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (72858840643 / 800000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_143 V_142
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_143 :
+    ∀ t,
+      (72858840643 / 800000000000 : ℝ) ≤ t →
+        t ≤ (762218354751 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_144 V_143
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_144 :
+    ∀ t,
+      (762218354751 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (24870259471 / 250000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_145 V_144
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_145 :
+    ∀ t,
+      (24870259471 / 250000000000 : ℝ) ≤ t →
+        t ≤ (1614118950931 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_146 V_145
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_146 :
+    ∀ t,
+      (1614118950931 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (818270647859 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_147 V_146
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_147 :
+    ∀ t,
+      (818270647859 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (331792728101 / 3200000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_148 V_147
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_148 :
+    ∀ t,
+      (331792728101 / 3200000000000 : ℝ) ≤ t →
+        t ≤ (3340349625797 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_149 V_148
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_149 :
+    ∀ t,
+      (3340349625797 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (420346496323 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_150 V_149
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_150 :
+    ∀ t,
+      (420346496323 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (3385194315371 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_151 V_150
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_151 :
+    ∀ t,
+      (3385194315371 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (1703808330079 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_152 V_151
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_152 :
+    ∀ t,
+      (1703808330079 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (686007800989 / 6400000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_153 V_152
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_153 :
+    ∀ t,
+      (686007800989 / 6400000000000 : ℝ) ≤ t →
+        t ≤ (863115337433 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_154 V_153
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_154 :
+    ∀ t,
+      (863115337433 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (6927345044251 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_155 V_154
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_155 :
+    ∀ t,
+      (6927345044251 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (3474883694519 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_156 V_155
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_156 :
+    ∀ t,
+      (3474883694519 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (278887589353 / 2560000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_157 V_156
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_157 :
+    ∀ t,
+      (278887589353 / 2560000000000 : ℝ) ≤ t →
+        t ≤ (1748653019653 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_158 V_157
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_158 :
+    ∀ t,
+      (1748653019653 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (7017034423399 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_159 V_158
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_159 :
+    ∀ t,
+      (7017034423399 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (3519728384093 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_160 V_159
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_160 :
+    ∀ t,
+      (3519728384093 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (7061879112973 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_161 V_160
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_161 :
+    ∀ t,
+      (7061879112973 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (44276884111 / 400000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_162 V_161
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_162 :
+    ∀ t,
+      (44276884111 / 400000000000 : ℝ) ≤ t →
+        t ≤ (7106723802547 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_163 V_162
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_163 :
+    ∀ t,
+      (7106723802547 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (3564573073667 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_164 V_163
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_164 :
+    ∀ t,
+      (3564573073667 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (7151568492121 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_165 V_164
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_165 :
+    ∀ t,
+      (7151568492121 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (1793497709227 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_166 V_165
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_166 :
+    ∀ t,
+      (1793497709227 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (1439282636339 / 12800000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_167 V_166
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_167 :
+    ∀ t,
+      (1439282636339 / 12800000000000 : ℝ) ≤ t →
+        t ≤ (3609417763241 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_168 V_167
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_168 :
+    ∀ t,
+      (3609417763241 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (7241257871269 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_169 V_168
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_169 :
+    ∀ t,
+      (7241257871269 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (907960027007 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_170 V_169
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_170 :
+    ∀ t,
+      (907960027007 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (7286102560843 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_171 V_170
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_171 :
+    ∀ t,
+      (7286102560843 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (730852490563 / 6400000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_172 V_171
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_172 :
+    ∀ t,
+      (730852490563 / 6400000000000 : ℝ) ≤ t →
+        t ≤ (7330947250417 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_173 V_172
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_173 :
+    ∀ t,
+      (7330947250417 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (1838342398801 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_174 V_173
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_174 :
+    ∀ t,
+      (1838342398801 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (3699107142389 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_175 V_174
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_175 :
+    ∀ t,
+      (3699107142389 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (465191185897 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_176 V_175
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_176 :
+    ∀ t,
+      (465191185897 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (3743951831963 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_177 V_176
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_177 :
+    ∀ t,
+      (3743951831963 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (15065496707 / 128000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_178 V_177
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_178 :
+    ∀ t,
+      (15065496707 / 128000000000 : ℝ) ≤ t →
+        t ≤ (3788796521537 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_179 V_178
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_179 :
+    ∀ t,
+      (3788796521537 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (952804716581 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_180 V_179
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_180 :
+    ∀ t,
+      (952804716581 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (3833641211111 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_181 V_180
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_181 :
+    ∀ t,
+      (3833641211111 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (1928031777949 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_182 V_181
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_182 :
+    ∀ t,
+      (1928031777949 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (121903382671 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_183 V_182
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_183 :
+    ∀ t,
+      (121903382671 / 1000000000000 : ℝ) ≤ t →
+        t ≤ (1972876467523 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_184 V_183
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_184 :
+    ∀ t,
+      (1972876467523 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (199529881231 / 1600000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_185 V_184
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_185 :
+    ∀ t,
+      (199529881231 / 1600000000000 : ℝ) ≤ t →
+        t ≤ (2017721157097 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_186 V_185
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_186 :
+    ∀ t,
+      (2017721157097 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (510035875471 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_187 V_186
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_187 :
+    ∀ t,
+      (510035875471 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (1042494095729 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_188 V_187
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_188 :
+    ∀ t,
+      (1042494095729 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (266229110129 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_189 V_188
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_189 :
+    ∀ t,
+      (266229110129 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (110976113009 / 800000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_190 V_189
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_190 :
+    ∀ t,
+      (110976113009 / 800000000000 : ℝ) ≤ t →
+        t ≤ (72162863729 / 500000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_191 V_190
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_191 :
+    ∀ t,
+      (72162863729 / 500000000000 : ℝ) ≤ t →
+        t ≤ (4675203313927 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_192 V_191
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_192 :
+    ∀ t,
+      (4675203313927 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (2365991674599 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_193 V_192
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_193 :
+    ∀ t,
+      (2365991674599 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (4788763384469 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_194 V_193
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_194 :
+    ∀ t,
+      (4788763384469 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (9634306804209 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_195 V_194
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_195 :
+    ∀ t,
+      (9634306804209 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (242277170987 / 1600000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_196 V_195
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_196 :
+    ∀ t,
+      (242277170987 / 1600000000000 : ℝ) ≤ t →
+        t ≤ (9747866874751 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_197 V_196
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_197 :
+    ∀ t,
+      (9747866874751 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (4902323455011 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_198 V_197
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_198 :
+    ∀ t,
+      (4902323455011 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (9861426945293 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_199 V_198
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_199 :
+    ∀ t,
+      (9861426945293 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (2479551745141 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_200 V_199
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_200 :
+    ∀ t,
+      (2479551745141 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (19893193996399 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_201 V_200
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_201 :
+    ∀ t,
+      (19893193996399 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (1994997403167 / 12800000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_202 V_201
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_202 :
+    ∀ t,
+      (1994997403167 / 12800000000000 : ℝ) ≤ t →
+        t ≤ (20006754066941 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_203 V_202
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_203 :
+    ∀ t,
+      (20006754066941 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (5015883525553 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_204 V_203
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_204 :
+    ∀ t,
+      (5015883525553 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (20120314137483 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_205 V_204
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_205 :
+    ∀ t,
+      (20120314137483 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (10088547086377 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_206 V_205
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_206 :
+    ∀ t,
+      (10088547086377 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (809354968321 / 5120000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_207 V_206
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_207 :
+    ∀ t,
+      (809354968321 / 5120000000000 : ℝ) ≤ t →
+        t ≤ (634082945103 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_208 V_207
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_208 :
+    ∀ t,
+      (634082945103 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (20347434278567 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_209 V_208
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_209 :
+    ∀ t,
+      (20347434278567 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (10202107156919 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_210 V_209
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_210 :
+    ∀ t,
+      (10202107156919 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (20460994349109 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_211 V_210
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_211 :
+    ∀ t,
+      (20460994349109 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (1025888719219 / 6400000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_212 V_211
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_212 :
+    ∀ t,
+      (1025888719219 / 6400000000000 : ℝ) ≤ t →
+        t ≤ (20574554419651 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_213 V_212
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_213 :
+    ∀ t,
+      (20574554419651 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (10315667227461 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_214 V_213
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_214 :
+    ∀ t,
+      (10315667227461 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (20688114490193 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_215 V_214
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_215 :
+    ∀ t,
+      (20688114490193 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (2593111815683 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_216 V_215
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_216 :
+    ∀ t,
+      (2593111815683 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (4160334912147 / 25600000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_217 V_216
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_217 :
+    ∀ t,
+      (4160334912147 / 25600000000000 : ℝ) ≤ t →
+        t ≤ (10429227298003 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_218 V_217
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_218 :
+    ∀ t,
+      (10429227298003 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (20915234631277 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_219 V_218
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_219 :
+    ∀ t,
+      (20915234631277 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (5243003666637 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_220 V_219
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_220 :
+    ∀ t,
+      (5243003666637 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (21028794701819 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_221 V_220
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_221 :
+    ∀ t,
+      (21028794701819 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (2108557473709 / 12800000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_222 V_221
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_222 :
+    ∀ t,
+      (2108557473709 / 12800000000000 : ℝ) ≤ t →
+        t ≤ (21142354772361 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_223 V_222
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_223 :
+    ∀ t,
+      (21142354772361 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (1324945925477 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_224 V_223
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_224 :
+    ∀ t,
+      (1324945925477 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (10656347439087 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_225 V_224
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_225 :
+    ∀ t,
+      (10656347439087 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (5356563737179 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_226 V_225
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_226 :
+    ∀ t,
+      (5356563737179 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (10769907509629 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_227 V_226
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_227 :
+    ∀ t,
+      (10769907509629 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (108266875449 / 640000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_228 V_227
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_228 :
+    ∀ t,
+      (108266875449 / 640000000000 : ℝ) ≤ t →
+        t ≤ (10883467580171 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_229 V_228
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_229 :
+    ∀ t,
+      (10883467580171 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (5470123807721 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_230 V_229
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_230 :
+    ∀ t,
+      (5470123807721 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (10997027650713 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_231 V_230
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_231 :
+    ∀ t,
+      (10997027650713 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (345431490187 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_232 V_231
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_232 :
+    ∀ t,
+      (345431490187 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (5583683878263 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_233 V_232
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_233 :
+    ∀ t,
+      (5583683878263 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (2820231956767 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_234 V_233
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_234 :
+    ∀ t,
+      (2820231956767 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (1139448789761 / 6400000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_235 V_234
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_235 :
+    ∀ t,
+      (1139448789761 / 6400000000000 : ℝ) ≤ t →
+        t ≤ (1438505996019 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_236 V_235
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_236 :
+    ∀ t,
+      (1438505996019 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (2933792027309 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_237 V_236
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_237 :
+    ∀ t,
+      (2933792027309 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (149528603129 / 800000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_238 V_237
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_238 :
+    ∀ t,
+      (149528603129 / 800000000000 : ℝ) ≤ t →
+        t ≤ (3047352097851 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_239 V_238
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_239 :
+    ∀ t,
+      (3047352097851 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (1552066066561 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_240 V_239
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_240 :
+    ∀ t,
+      (1552066066561 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (201105762729 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_241 V_240
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_241 :
+    ∀ t,
+      (201105762729 / 1000000000000 : ℝ) ≤ t →
+        t ≤ (3251812320751 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_242 V_241
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_242 :
+    ∀ t,
+      (3251812320751 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (1642966218919 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_243 V_242
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_243 :
+    ∀ t,
+      (1642966218919 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (132802102197 / 640000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_244 V_243
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_244 :
+    ∀ t,
+      (132802102197 / 640000000000 : ℝ) ≤ t →
+        t ≤ (6674225226937 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_245 V_244
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_245 :
+    ∀ t,
+      (6674225226937 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (838543168003 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_246 V_245
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_246 :
+    ∀ t,
+      (838543168003 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (6742465461111 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_247 V_246
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_247 :
+    ∀ t,
+      (6742465461111 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (3388292789099 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_248 V_247
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_248 :
+    ∀ t,
+      (3388292789099 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (1362141139057 / 6400000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_249 V_248
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_249 :
+    ∀ t,
+      (1362141139057 / 6400000000000 : ℝ) ≤ t →
+        t ≤ (1711206453093 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_250 V_249
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_250 :
+    ∀ t,
+      (1711206453093 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (13723771741831 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_251 V_250
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_251 :
+    ∀ t,
+      (13723771741831 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (6878945929459 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_252 V_251
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_252 :
+    ∀ t,
+      (6878945929459 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (2758402395201 / 12800000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_253 V_252
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_253 :
+    ∀ t,
+      (2758402395201 / 12800000000000 : ℝ) ≤ t →
+        t ≤ (3456533023273 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_254 V_253
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_254 :
+    ∀ t,
+      (3456533023273 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (13860252210179 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_255 V_254
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_255 :
+    ∀ t,
+      (13860252210179 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (6947186163633 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_256 V_255
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_256 :
+    ∀ t,
+      (6947186163633 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (13928492444353 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_257 V_256
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_257 :
+    ∀ t,
+      (13928492444353 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (87266328509 / 400000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_258 V_257
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_258 :
+    ∀ t,
+      (87266328509 / 400000000000 : ℝ) ≤ t →
+        t ≤ (13996732678527 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_259 V_258
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_259 :
+    ∀ t,
+      (13996732678527 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (7015426397807 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_260 V_259
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_260 :
+    ∀ t,
+      (7015426397807 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (14064972912701 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_261 V_260
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_261 :
+    ∀ t,
+      (14064972912701 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (3524773257447 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_262 V_261
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_262 :
+    ∀ t,
+      (3524773257447 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (4522628207 / 20480000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_263 V_262
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_263 :
+    ∀ t,
+      (4522628207 / 20480000000 : ℝ) ≤ t →
+        t ≤ (7083666631981 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_264 V_263
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_264 :
+    ∀ t,
+      (7083666631981 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (14201453381049 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_265 V_264
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_265 :
+    ∀ t,
+      (14201453381049 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (1779446687267 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_266 V_265
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_266 :
+    ∀ t,
+      (1779446687267 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (14269693615223 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_267 V_266
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_267 :
+    ∀ t,
+      (14269693615223 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (1430381373231 / 6400000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_268 V_267
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_268 :
+    ∀ t,
+      (1430381373231 / 6400000000000 : ℝ) ≤ t →
+        t ≤ (14337933849397 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_269 V_268
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_269 :
+    ∀ t,
+      (14337933849397 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (3593013491621 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_270 V_269
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_270 :
+    ∀ t,
+      (3593013491621 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (14406174083571 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_271 V_270
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_271 :
+    ∀ t,
+      (14406174083571 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (7220147100329 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_272 V_271
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_272 :
+    ∀ t,
+      (7220147100329 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (2894882863549 / 12800000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_273 V_272
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_273 :
+    ∀ t,
+      (2894882863549 / 12800000000000 : ℝ) ≤ t →
+        t ≤ (906783402177 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_274 V_273
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_274 :
+    ∀ t,
+      (906783402177 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (14542654551919 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_275 V_274
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_275 :
+    ∀ t,
+      (14542654551919 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (7288387334503 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_276 V_275
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_276 :
+    ∀ t,
+      (7288387334503 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (732250745159 / 3200000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_277 V_276
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_277 :
+    ∀ t,
+      (732250745159 / 3200000000000 : ℝ) ≤ t →
+        t ≤ (7356627568677 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_278 V_277
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_278 :
+    ∀ t,
+      (7356627568677 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (1847686921441 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_279 V_278
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_279 :
+    ∀ t,
+      (1847686921441 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (7424867802851 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_280 V_279
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_280 :
+    ∀ t,
+      (7424867802851 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (3729493959969 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_281 V_280
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_281 :
+    ∀ t,
+      (3729493959969 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (299724321481 / 1280000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_282 V_281
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_282 :
+    ∀ t,
+      (299724321481 / 1280000000000 : ℝ) ≤ t →
+        t ≤ (29403234977 / 125000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_283 V_282
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_283 :
+    ∀ t,
+      (29403234977 / 125000000000 : ℝ) ≤ t →
+        t ≤ (7561348271199 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_284 V_283
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_284 :
+    ∀ t,
+      (7561348271199 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (3797734194143 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_285 V_284
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_285 :
+    ∀ t,
+      (3797734194143 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (383185431123 / 1600000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_286 V_285
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_286 :
+    ∀ t,
+      (383185431123 / 1600000000000 : ℝ) ≤ t →
+        t ≤ (3865974428317 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_287 V_286
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_287 :
+    ∀ t,
+      (3865974428317 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (975023636351 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_288 V_287
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_288 :
+    ∀ t,
+      (975023636351 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (3934214662491 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_289 V_288
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_289 :
+    ∀ t,
+      (3934214662491 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (1984167389789 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_290 V_289
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_290 :
+    ∀ t,
+      (1984167389789 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (504571876719 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_291 V_290
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_291 :
+    ∀ t,
+      (504571876719 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (2052407623963 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_292 V_291
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_292 :
+    ∀ t,
+      (2052407623963 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (41730554821 / 160000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_293 V_292
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_293 :
+    ∀ t,
+      (41730554821 / 160000000000 : ℝ) ≤ t →
+        t ≤ (269345996903 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_294 V_293
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_294 :
+    ∀ t,
+      (269345996903 / 1000000000000 : ℝ) ≤ t →
+        t ≤ (8696815458149 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_295 V_294
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_295 :
+    ∀ t,
+      (8696815458149 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (4387279507701 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_296 V_295
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_296 :
+    ∀ t,
+      (4387279507701 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (1770460514531 / 6400000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_297 V_296
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_297 :
+    ∀ t,
+      (1770460514531 / 6400000000000 : ℝ) ≤ t →
+        t ≤ (17782348702563 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_298 V_297
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_298 :
+    ∀ t,
+      (17782348702563 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (2232511532477 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_299 V_298
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_299 :
+    ∀ t,
+      (2232511532477 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (17937835817069 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_300 V_299
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_300 :
+    ∀ t,
+      (17937835817069 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (9007789687161 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_301 V_300
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_301 :
+    ∀ t,
+      (9007789687161 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (723732917263 / 2560000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_302 V_301
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_302 :
+    ∀ t,
+      (723732917263 / 2560000000000 : ℝ) ≤ t →
+        t ≤ (4542766622207 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_303 V_302
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_303 :
+    ∀ t,
+      (4542766622207 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (36419876534909 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_304 V_303
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_304 :
+    ∀ t,
+      (36419876534909 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (18248810046081 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_305 V_304
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_305 :
+    ∀ t,
+      (18248810046081 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (7315072729883 / 25600000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_306 V_305
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_306 :
+    ∀ t,
+      (7315072729883 / 25600000000000 : ℝ) ≤ t →
+        t ≤ (9163276801667 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_307 V_306
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_307 :
+    ∀ t,
+      (9163276801667 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (36730850763921 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_308 V_307
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_308 :
+    ∀ t,
+      (36730850763921 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (18404297160587 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_309 V_308
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_309 :
+    ∀ t,
+      (18404297160587 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (36886337878427 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_310 V_309
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_310 :
+    ∀ t,
+      (36886337878427 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (231025508973 / 800000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_311 V_310
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_311 :
+    ∀ t,
+      (231025508973 / 800000000000 : ℝ) ≤ t →
+        t ≤ (37041824992933 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_312 V_311
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_312 :
+    ∀ t,
+      (37041824992933 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (18559784275093 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_313 V_312
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_313 :
+    ∀ t,
+      (18559784275093 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (37197312107439 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_314 V_313
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_314 :
+    ∀ t,
+      (37197312107439 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (9318763916173 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_315 V_314
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_315 :
+    ∀ t,
+      (9318763916173 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (7470559844389 / 25600000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_316 V_315
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_316 :
+    ∀ t,
+      (7470559844389 / 25600000000000 : ℝ) ≤ t →
+        t ≤ (18715271389599 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_317 V_316
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_317 :
+    ∀ t,
+      (18715271389599 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (37508286336451 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_318 V_317
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_318 :
+    ∀ t,
+      (37508286336451 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (4698253736713 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_319 V_318
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_319 :
+    ∀ t,
+      (4698253736713 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (37663773450957 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_320 V_319
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_320 :
+    ∀ t,
+      (37663773450957 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (3774151700821 / 12800000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_321 V_320
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_321 :
+    ∀ t,
+      (3774151700821 / 12800000000000 : ℝ) ≤ t →
+        t ≤ (37819260565463 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_322 V_321
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_322 :
+    ∀ t,
+      (37819260565463 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (9474251030679 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_323 V_322
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_323 :
+    ∀ t,
+      (9474251030679 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (37974747679969 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_324 V_323
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_324 :
+    ∀ t,
+      (37974747679969 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (19026245618611 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_325 V_324
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_325 :
+    ∀ t,
+      (19026245618611 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (1525209391779 / 5120000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_326 V_325
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_326 :
+    ∀ t,
+      (1525209391779 / 5120000000000 : ℝ) ≤ t →
+        t ≤ (2387998646983 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_327 V_326
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_327 :
+    ∀ t,
+      (2387998646983 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (38285721908981 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_328 V_327
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_328 :
+    ∀ t,
+      (38285721908981 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (19181732733117 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_329 V_328
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_329 :
+    ∀ t,
+      (19181732733117 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (38441209023487 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_330 V_329
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_330 :
+    ∀ t,
+      (38441209023487 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (1925947629037 / 6400000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_331 V_330
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_331 :
+    ∀ t,
+      (1925947629037 / 6400000000000 : ℝ) ≤ t →
+        t ≤ (19337219847623 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_332 V_331
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_332 :
+    ∀ t,
+      (19337219847623 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (4853740851219 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_333 V_332
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_333 :
+    ∀ t,
+      (4853740851219 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (19492706962129 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_334 V_333
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_334 :
+    ∀ t,
+      (19492706962129 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (9785225259691 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_335 V_334
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_335 :
+    ∀ t,
+      (9785225259691 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (3929638815327 / 12800000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_336 V_335
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_336 :
+    ∀ t,
+      (3929638815327 / 12800000000000 : ℝ) ≤ t →
+        t ≤ (616435551059 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_337 V_336
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_337 :
+    ∀ t,
+      (616435551059 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (19803681191141 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_338 V_337
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_338 :
+    ∀ t,
+      (19803681191141 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (9940712374197 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_339 V_338
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_339 :
+    ∀ t,
+      (9940712374197 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (200369118629 / 640000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_340 V_339
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_340 :
+    ∀ t,
+      (200369118629 / 640000000000 : ℝ) ≤ t →
+        t ≤ (10096199488703 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_341 V_340
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_341 :
+    ∀ t,
+      (10096199488703 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (2543485761489 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_342 V_341
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_342 :
+    ∀ t,
+      (2543485761489 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (10251686603209 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_343 V_342
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_343 :
+    ∀ t,
+      (10251686603209 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (5164715080231 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_344 V_343
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_344 :
+    ∀ t,
+      (5164715080231 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (1310614659371 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_345 V_344
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_345 :
+    ∀ t,
+      (1310614659371 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (5320202194737 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_346 V_345
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_346 :
+    ∀ t,
+      (5320202194737 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (539794575199 / 1600000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_347 V_346
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_347 :
+    ∀ t,
+      (539794575199 / 1600000000000 : ℝ) ≤ t →
+        t ≤ (5475689309243 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_348 V_347
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_348 :
+    ∀ t,
+      (5475689309243 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (86772388539 / 250000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_349 V_348
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_349 :
+    ∀ t,
+      (86772388539 / 250000000000 : ℝ) ≤ t →
+        t ≤ (11190582883251 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_350 V_349
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_350 :
+    ∀ t,
+      (11190582883251 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (1127430003351 / 3200000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_351 V_350
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_351 :
+    ∀ t,
+      (1127430003351 / 3200000000000 : ℝ) ≤ t →
+        t ≤ (11358017183769 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_352 V_351
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_352 :
+    ∀ t,
+      (11358017183769 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (22799751517797 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_353 V_352
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_353 :
+    ∀ t,
+      (22799751517797 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (2860433583507 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_354 V_353
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_354 :
+    ∀ t,
+      (2860433583507 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (4593437163663 / 12800000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_355 V_354
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_355 :
+    ∀ t,
+      (4593437163663 / 12800000000000 : ℝ) ≤ t →
+        t ≤ (11525451484287 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_356 V_355
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_356 :
+    ∀ t,
+      (11525451484287 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (23134620118833 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_357 V_356
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_357 :
+    ∀ t,
+      (23134620118833 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (5804584317273 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_358 V_357
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_358 :
+    ∀ t,
+      (5804584317273 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (46520391688443 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_359 V_358
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_359 :
+    ∀ t,
+      (46520391688443 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (23302054419351 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_360 V_359
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_360 :
+    ∀ t,
+      (23302054419351 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (46687825988961 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_361 V_360
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_361 :
+    ∀ t,
+      (46687825988961 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (2338577156961 / 6400000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_362 V_361
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_362 :
+    ∀ t,
+      (2338577156961 / 6400000000000 : ℝ) ≤ t →
+        t ≤ (46855260289479 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_363 V_362
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_363 :
+    ∀ t,
+      (46855260289479 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (23469488719869 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_364 V_363
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_364 :
+    ∀ t,
+      (23469488719869 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (47022694589997 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_365 V_364
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_365 :
+    ∀ t,
+      (47022694589997 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (1472075366883 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_366 V_365
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_366 :
+    ∀ t,
+      (1472075366883 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (9438025778103 / 25600000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_367 V_366
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_367 :
+    ∀ t,
+      (9438025778103 / 25600000000000 : ℝ) ≤ t →
+        t ≤ (23636923020387 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_368 V_367
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_368 :
+    ∀ t,
+      (23636923020387 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (47357563191033 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_369 V_368
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_369 :
+    ∀ t,
+      (47357563191033 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (11860320085323 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_370 V_369
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_370 :
+    ∀ t,
+      (11860320085323 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (47524997491551 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_371 V_370
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_371 :
+    ∀ t,
+      (47524997491551 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (4760871464181 / 12800000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_372 V_371
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_372 :
+    ∀ t,
+      (4760871464181 / 12800000000000 : ℝ) ≤ t →
+        t ≤ (47692431792069 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_373 V_372
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_373 :
+    ∀ t,
+      (47692431792069 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (5972018617791 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_374 V_373
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_374 :
+    ∀ t,
+      (5972018617791 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (47859866092587 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_375 V_374
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_375 :
+    ∀ t,
+      (47859866092587 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (23971791621423 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_376 V_375
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_376 :
+    ∀ t,
+      (23971791621423 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (9605460078621 / 25600000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_377 V_376
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_377 :
+    ∀ t,
+      (9605460078621 / 25600000000000 : ℝ) ≤ t →
+        t ≤ (12027754385841 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_378 V_377
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_378 :
+    ∀ t,
+      (12027754385841 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (48194734693623 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_379 V_378
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_379 :
+    ∀ t,
+      (48194734693623 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (24139225921941 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_380 V_379
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_380 :
+    ∀ t,
+      (24139225921941 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (48362168994141 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_381 V_380
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_381 :
+    ∀ t,
+      (48362168994141 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (121114715361 / 320000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_382 V_381
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_382 :
+    ∀ t,
+      (121114715361 / 320000000000 : ℝ) ≤ t →
+        t ≤ (48529603294659 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_383 V_382
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_383 :
+    ∀ t,
+      (48529603294659 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (24306660222459 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_384 V_383
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_384 :
+    ∀ t,
+      (24306660222459 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (48697037595177 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_385 V_384
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_385 :
+    ∀ t,
+      (48697037595177 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (12195188686359 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_386 V_385
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_386 :
+    ∀ t,
+      (12195188686359 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (9772894379139 / 25600000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_387 V_386
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_387 :
+    ∀ t,
+      (9772894379139 / 25600000000000 : ℝ) ≤ t →
+        t ≤ (24474094522977 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_388 V_387
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_388 :
+    ∀ t,
+      (24474094522977 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (6139452918309 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_389 V_388
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_389 :
+    ∀ t,
+      (6139452918309 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (4928305764699 / 12800000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_390 V_389
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_390 :
+    ∀ t,
+      (4928305764699 / 12800000000000 : ℝ) ≤ t →
+        t ≤ (12362622986877 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_391 V_390
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_391 :
+    ∀ t,
+      (12362622986877 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (24808963124013 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_392 V_391
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_392 :
+    ∀ t,
+      (24808963124013 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (777896258571 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_393 V_392
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_393 :
+    ∀ t,
+      (777896258571 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (24976397424531 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_394 V_393
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_394 :
+    ∀ t,
+      (24976397424531 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (2506011457479 / 6400000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_395 V_394
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_395 :
+    ∀ t,
+      (2506011457479 / 6400000000000 : ℝ) ≤ t →
+        t ≤ (25143831725049 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_396 V_395
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_396 :
+    ∀ t,
+      (25143831725049 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (6306887218827 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_397 V_396
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_397 :
+    ∀ t,
+      (6306887218827 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (12697491587913 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_398 V_397
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_398 :
+    ∀ t,
+      (12697491587913 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (3195302184543 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_399 V_398
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_399 :
+    ∀ t,
+      (3195302184543 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (12864925888431 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_400 V_399
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_400 :
+    ∀ t,
+      (12864925888431 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (1294864303869 / 3200000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_401 V_400
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_401 :
+    ∀ t,
+      (1294864303869 / 3200000000000 : ℝ) ≤ t →
+        t ≤ (13032360188949 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_402 V_401
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_402 :
+    ∀ t,
+      (13032360188949 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (1639509667401 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_403 V_402
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_403 :
+    ∀ t,
+      (1639509667401 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (6641755819863 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_404 V_403
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_404 :
+    ∀ t,
+      (6641755819863 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (3362736485061 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_405 V_404
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_405 :
+    ∀ t,
+      (3362736485061 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (6809190120381 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_406 V_405
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_406 :
+    ∀ t,
+      (6809190120381 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (86161340883 / 200000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_407 V_406
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_407 :
+    ∀ t,
+      (86161340883 / 200000000000 : ℝ) ≤ t →
+        t ≤ (54181913021 / 125000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_408 V_407
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_408 :
+    ∀ t,
+      (54181913021 / 125000000000 : ℝ) ≤ t →
+        t ≤ (436103903921 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_409 V_408
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_409 :
+    ∀ t,
+      (436103903921 / 1000000000000 : ℝ) ≤ t →
+        t ≤ (219376251837 / 500000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_410 V_409
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_410 :
+    ∀ t,
+      (219376251837 / 500000000000 : ℝ) ≤ t →
+        t ≤ (880153607101 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_411 V_410
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_411 :
+    ∀ t,
+      (880153607101 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (441401103427 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_412 V_411
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_412 :
+    ∀ t,
+      (441401103427 / 1000000000000 : ℝ) ≤ t →
+        t ≤ (885450806607 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_413 V_412
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_413 :
+    ∀ t,
+      (885450806607 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (22202485159 / 50000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_414 V_413
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_414 :
+    ∀ t,
+      (22202485159 / 50000000000 : ℝ) ≤ t →
+        t ≤ (890748006113 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_415 V_414
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_415 :
+    ∀ t,
+      (890748006113 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (446698302933 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_416 V_415
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_416 :
+    ∀ t,
+      (446698302933 / 1000000000000 : ℝ) ≤ t →
+        t ≤ (896045205619 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_417 V_416
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_417 :
+    ∀ t,
+      (896045205619 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (1794739010991 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_418 V_417
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_418 :
+    ∀ t,
+      (1794739010991 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (224673451343 / 500000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_419 V_418
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_419 :
+    ∀ t,
+      (224673451343 / 500000000000 : ℝ) ≤ t →
+        t ≤ (1800036210497 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_420 V_419
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_420 :
+    ∀ t,
+      (1800036210497 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (7210739241 / 16000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_421 V_420
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_421 :
+    ∀ t,
+      (7210739241 / 16000000000 : ℝ) ≤ t →
+        t ≤ (1805333410003 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_422 V_421
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_422 :
+    ∀ t,
+      (1805333410003 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (451995502439 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_423 V_422
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_423 :
+    ∀ t,
+      (451995502439 / 1000000000000 : ℝ) ≤ t →
+        t ≤ (1810630609509 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_424 V_423
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_424 :
+    ∀ t,
+      (1810630609509 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (906639604631 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_425 V_424
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_425 :
+    ∀ t,
+      (906639604631 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (363185561803 / 800000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_426 V_425
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_426 :
+    ∀ t,
+      (363185561803 / 800000000000 : ℝ) ≤ t →
+        t ≤ (28415256387 / 62500000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_427 V_426
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_427 :
+    ∀ t,
+      (28415256387 / 62500000000 : ℝ) ≤ t →
+        t ≤ (1821225008521 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_428 V_427
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_428 :
+    ∀ t,
+      (1821225008521 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (911936804137 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_429 V_428
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_429 :
+    ∀ t,
+      (911936804137 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (1826522208027 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_430 V_429
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_430 :
+    ∀ t,
+      (1826522208027 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (91458540389 / 200000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_431 V_430
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_431 :
+    ∀ t,
+      (91458540389 / 200000000000 : ℝ) ≤ t →
+        t ≤ (1831819407533 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_432 V_431
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_432 :
+    ∀ t,
+      (1831819407533 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (917234003643 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_433 V_432
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_433 :
+    ∀ t,
+      (917234003643 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (1837116607039 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_434 V_433
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_434 :
+    ∀ t,
+      (1837116607039 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (229970650849 / 500000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_435 V_434
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_435 :
+    ∀ t,
+      (229970650849 / 500000000000 : ℝ) ≤ t →
+        t ≤ (368482761309 / 800000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_436 V_435
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_436 :
+    ∀ t,
+      (368482761309 / 800000000000 : ℝ) ≤ t →
+        t ≤ (922531203149 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_437 V_436
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_437 :
+    ∀ t,
+      (922531203149 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (1847711006051 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_438 V_437
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_438 :
+    ∀ t,
+      (1847711006051 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (462589901451 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_439 V_438
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_439 :
+    ∀ t,
+      (462589901451 / 1000000000000 : ℝ) ≤ t →
+        t ≤ (1853008205557 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_440 V_439
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_440 :
+    ∀ t,
+      (1853008205557 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (185565680531 / 400000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_441 V_440
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_441 :
+    ∀ t,
+      (185565680531 / 400000000000 : ℝ) ≤ t →
+        t ≤ (1858305405063 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_442 V_441
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_442 :
+    ∀ t,
+      (1858305405063 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (116309625301 / 250000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_443 V_442
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_443 :
+    ∀ t,
+      (116309625301 / 250000000000 : ℝ) ≤ t →
+        t ≤ (1863602604569 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_444 V_443
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_444 :
+    ∀ t,
+      (1863602604569 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (933125602161 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_445 V_444
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_445 :
+    ∀ t,
+      (933125602161 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (467887100957 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_446 V_445
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_446 :
+    ∀ t,
+      (467887100957 / 1000000000000 : ℝ) ≤ t →
+        t ≤ (938422801667 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_447 V_446
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_447 :
+    ∀ t,
+      (938422801667 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (47053570071 / 100000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_448 V_447
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_448 :
+    ∀ t,
+      (47053570071 / 100000000000 : ℝ) ≤ t →
+        t ≤ (943720001173 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_449 V_448
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_449 :
+    ∀ t,
+      (943720001173 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (473184300463 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_450 V_449
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_450 :
+    ∀ t,
+      (473184300463 / 1000000000000 : ℝ) ≤ t →
+        t ≤ (949017200679 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_451 V_450
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_451 :
+    ∀ t,
+      (949017200679 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (59479112527 / 125000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_452 V_451
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_452 :
+    ∀ t,
+      (59479112527 / 125000000000 : ℝ) ≤ t →
+        t ≤ (190862880037 / 400000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_453 V_452
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_453 :
+    ∀ t,
+      (190862880037 / 400000000000 : ℝ) ≤ t →
+        t ≤ (478481499969 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_454 V_453
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_454 :
+    ∀ t,
+      (478481499969 / 1000000000000 : ℝ) ≤ t →
+        t ≤ (959611599691 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_455 V_454
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_455 :
+    ∀ t,
+      (959611599691 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (240565049861 / 500000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_456 V_455
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_456 :
+    ∀ t,
+      (240565049861 / 500000000000 : ℝ) ≤ t →
+        t ≤ (19351147979 / 40000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_457 V_456
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_457 :
+    ∀ t,
+      (19351147979 / 40000000000 : ℝ) ≤ t →
+        t ≤ (121606824807 / 250000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_458 V_457
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_458 :
+    ∀ t,
+      (121606824807 / 250000000000 : ℝ) ≤ t →
+        t ≤ (489075898981 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_459 V_458
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_459 :
+    ∀ t,
+      (489075898981 / 1000000000000 : ℝ) ≤ t →
+        t ≤ (245862249367 / 500000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_460 V_459
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_460 :
+    ∀ t,
+      (245862249367 / 500000000000 : ℝ) ≤ t →
+        t ≤ (494373098487 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_461 V_460
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_461 :
+    ∀ t,
+      (494373098487 / 1000000000000 : ℝ) ≤ t →
+        t ≤ (1553192807 / 3125000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_462 V_461
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_462 :
+    ∀ t,
+      (1553192807 / 3125000000 : ℝ) ≤ t →
+        t ≤ (499670297993 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_463 V_462
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_463 :
+    ∀ t,
+      (499670297993 / 1000000000000 : ℝ) ≤ t →
+        t ≤ (504967497499 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_464 V_463
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_464 :
+    ∀ t,
+      (504967497499 / 1000000000000 : ℝ) ≤ t →
+        t ≤ (102052939401 / 200000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_465 V_464
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_465 :
+    ∀ t,
+      (102052939401 / 200000000000 : ℝ) ≤ t →
+        t ≤ (515561896511 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_466 V_465
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_466 :
+    ∀ t,
+      (515561896511 / 1000000000000 : ℝ) ≤ t →
+        t ≤ (16577867570387 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_467 V_466
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_467 :
+    ∀ t,
+      (16577867570387 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (8328877226211 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_468 V_467
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_468 :
+    ∀ t,
+      (8328877226211 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (16737641334457 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_469 V_468
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_469 :
+    ∀ t,
+      (16737641334457 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (33555169550949 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_470 V_469
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_470 :
+    ∀ t,
+      (33555169550949 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (4204382054123 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_471 V_470
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_471 :
+    ∀ t,
+      (4204382054123 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (33714943315019 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_472 V_471
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_472 :
+    ∀ t,
+      (33714943315019 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (16897415098527 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_473 V_472
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_473 :
+    ∀ t,
+      (16897415098527 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (33874717079089 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_474 V_473
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_474 :
+    ∀ t,
+      (33874717079089 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (8488650990281 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_475 V_474
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_475 :
+    ∀ t,
+      (8488650990281 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (34034490843159 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_476 V_475
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_476 :
+    ∀ t,
+      (34034490843159 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (17057188862597 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_477 V_476
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_477 :
+    ∀ t,
+      (17057188862597 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (34194264607229 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_478 V_477
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_478 :
+    ∀ t,
+      (34194264607229 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (68468416096493 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_479 V_478
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_479 :
+    ∀ t,
+      (68468416096493 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (2142134468079 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_480 V_479
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_480 :
+    ∀ t,
+      (2142134468079 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (68628189860563 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_481 V_480
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_481 :
+    ∀ t,
+      (68628189860563 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (34354038371299 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_482 V_481
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_482 :
+    ∀ t,
+      (34354038371299 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (68787963624633 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_483 V_482
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_483 :
+    ∀ t,
+      (68787963624633 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (17216962626667 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_484 V_483
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_484 :
+    ∀ t,
+      (17216962626667 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (68947737388703 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_485 V_484
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_485 :
+    ∀ t,
+      (68947737388703 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (34513812135369 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_486 V_485
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_486 :
+    ∀ t,
+      (34513812135369 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (69107511152773 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_487 V_486
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_487 :
+    ∀ t,
+      (69107511152773 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (8648424754351 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_488 V_487
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_488 :
+    ∀ t,
+      (8648424754351 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (69267284916843 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_489 V_488
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_489 :
+    ∀ t,
+      (69267284916843 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (34673585899439 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_490 V_489
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_490 :
+    ∀ t,
+      (34673585899439 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (69427058680913 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_491 V_490
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_491 :
+    ∀ t,
+      (69427058680913 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (17376736390737 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_492 V_491
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_492 :
+    ∀ t,
+      (17376736390737 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (69586832444983 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_493 V_492
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_493 :
+    ∀ t,
+      (69586832444983 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (34833359663509 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_494 V_493
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_494 :
+    ∀ t,
+      (34833359663509 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (69746606209053 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_495 V_494
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_495 :
+    ∀ t,
+      (69746606209053 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (4364155818193 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_496 V_495
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_496 :
+    ∀ t,
+      (4364155818193 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (69906379973123 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_497 V_496
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_497 :
+    ∀ t,
+      (69906379973123 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (34993133427579 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_498 V_497
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_498 :
+    ∀ t,
+      (34993133427579 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (70066153737193 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_499 V_498
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_499 :
+    ∀ t,
+      (70066153737193 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (17536510154807 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_500 V_499
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_500 :
+    ∀ t,
+      (17536510154807 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (35152907191649 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_501 V_500
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_501 :
+    ∀ t,
+      (35152907191649 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (8808198518421 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_502 V_501
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_502 :
+    ∀ t,
+      (8808198518421 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (35312680955719 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_503 V_502
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_503 :
+    ∀ t,
+      (35312680955719 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (17696283918877 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_504 V_503
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_504 :
+    ∀ t,
+      (17696283918877 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (35472454719789 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_505 V_504
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_505 :
+    ∀ t,
+      (35472454719789 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (1111010675057 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_506 V_505
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_506 :
+    ∀ t,
+      (1111010675057 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (35632228483859 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_507 V_506
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_507 :
+    ∀ t,
+      (35632228483859 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (17856057682947 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_508 V_507
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_508 :
+    ∀ t,
+      (17856057682947 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (35792002247929 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_509 V_508
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_509 :
+    ∀ t,
+      (35792002247929 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (8967972282491 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_510 V_509
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_510 :
+    ∀ t,
+      (8967972282491 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (35951776011999 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_511 V_510
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_511 :
+    ∀ t,
+      (35951776011999 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (18015831447017 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_512 V_511
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_512 :
+    ∀ t,
+      (18015831447017 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (36111549776069 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_513 V_512
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_513 :
+    ∀ t,
+      (36111549776069 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (4523929582263 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_514 V_513
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_514 :
+    ∀ t,
+      (4523929582263 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (18175605211087 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_515 V_514
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_515 :
+    ∀ t,
+      (18175605211087 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (9127746046561 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_516 V_515
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_516 :
+    ∀ t,
+      (9127746046561 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (18335378975157 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_517 V_516
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_517 :
+    ∀ t,
+      (18335378975157 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (2301908232149 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_518 V_517
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_518 :
+    ∀ t,
+      (2301908232149 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (18495152739227 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_519 V_518
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_519 :
+    ∀ t,
+      (18495152739227 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (9287519810631 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_520 V_519
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_520 :
+    ∀ t,
+      (9287519810631 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (4683703346333 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_521 V_520
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_521 :
+    ∀ t,
+      (4683703346333 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (9447293574701 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_522 V_521
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_522 :
+    ∀ t,
+      (9447293574701 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (297724389273 / 500000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_523 V_522
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_523 :
+    ∀ t,
+      (297724389273 / 500000000000 : ℝ) ≤ t →
+        t ≤ (19123153518409 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_524 V_523
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_524 :
+    ∀ t,
+      (19123153518409 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (9595973061673 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_525 V_524
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_525 :
+    ∀ t,
+      (9595973061673 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (19260738728283 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_526 V_525
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_526 :
+    ∀ t,
+      (19260738728283 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (38590270061503 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_527 V_526
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_527 :
+    ∀ t,
+      (38590270061503 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (966476566661 / 1600000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_528 V_527
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_528 :
+    ∀ t,
+      (966476566661 / 1600000000000 : ℝ) ≤ t →
+        t ≤ (38727855271377 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_529 V_528
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_529 :
+    ∀ t,
+      (38727855271377 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (19398323938157 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_530 V_529
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_530 :
+    ∀ t,
+      (19398323938157 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (38865440481251 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_531 V_530
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_531 :
+    ∀ t,
+      (38865440481251 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (9733558271547 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_532 V_531
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_532 :
+    ∀ t,
+      (9733558271547 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (312024205529 / 512000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_533 V_532
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_533 :
+    ∀ t,
+      (312024205529 / 512000000000 : ℝ) ≤ t →
+        t ≤ (19535909148031 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_534 V_533
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_534 :
+    ∀ t,
+      (19535909148031 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (39140610900999 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_535 V_534
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_535 :
+    ∀ t,
+      (39140610900999 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (2450587719121 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_536 V_535
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_536 :
+    ∀ t,
+      (2450587719121 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (39278196110873 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_537 V_536
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_537 :
+    ∀ t,
+      (39278196110873 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (3934698871581 / 6400000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_538 V_537
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_538 :
+    ∀ t,
+      (3934698871581 / 6400000000000 : ℝ) ≤ t →
+        t ≤ (39415781320747 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_539 V_538
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_539 :
+    ∀ t,
+      (39415781320747 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (9871143481421 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_540 V_539
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_540 :
+    ∀ t,
+      (9871143481421 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (39553366530621 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_541 V_540
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_541 :
+    ∀ t,
+      (39553366530621 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (19811079567779 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_542 V_541
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_542 :
+    ∀ t,
+      (19811079567779 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (7938190348099 / 12800000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_543 V_542
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_543 :
+    ∀ t,
+      (7938190348099 / 12800000000000 : ℝ) ≤ t →
+        t ≤ (4969968043179 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_544 V_543
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_544 :
+    ∀ t,
+      (4969968043179 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (39828536950369 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_545 V_544
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_545 :
+    ∀ t,
+      (39828536950369 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (19948664777653 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_546 V_545
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_546 :
+    ∀ t,
+      (19948664777653 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (39966122160243 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_547 V_546
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_547 :
+    ∀ t,
+      (39966122160243 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (2001745738259 / 3200000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_548 V_547
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_548 :
+    ∀ t,
+      (2001745738259 / 3200000000000 : ℝ) ≤ t →
+        t ≤ (40103707370117 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_549 V_548
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_549 :
+    ∀ t,
+      (40103707370117 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (20086249987527 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_550 V_549
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_550 :
+    ∀ t,
+      (20086249987527 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (40241292579991 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_551 V_550
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_551 :
+    ∀ t,
+      (40241292579991 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (1259690162029 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_552 V_551
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_552 :
+    ∀ t,
+      (1259690162029 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (8075775557973 / 12800000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_553 V_552
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_553 :
+    ∀ t,
+      (8075775557973 / 12800000000000 : ℝ) ≤ t →
+        t ≤ (20223835197401 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_554 V_553
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_554 :
+    ∀ t,
+      (20223835197401 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (40516462999739 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_555 V_554
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_555 :
+    ∀ t,
+      (40516462999739 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (10146313901169 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_556 V_555
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_556 :
+    ∀ t,
+      (10146313901169 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (40654048209613 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_557 V_556
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_557 :
+    ∀ t,
+      (40654048209613 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (814456816291 / 1280000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_558 V_557
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_558 :
+    ∀ t,
+      (814456816291 / 1280000000000 : ℝ) ≤ t →
+        t ≤ (40791633419487 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_559 V_558
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_559 :
+    ∀ t,
+      (40791633419487 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (5107553253053 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_560 V_559
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_560 :
+    ∀ t,
+      (5107553253053 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (20499005617149 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_561 V_560
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_561 :
+    ∀ t,
+      (20499005617149 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (10283899111043 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_562 V_561
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_562 :
+    ∀ t,
+      (10283899111043 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (20636590827023 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_563 V_562
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_563 :
+    ∀ t,
+      (20636590827023 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (517634585799 / 800000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_564 V_563
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_564 :
+    ∀ t,
+      (517634585799 / 800000000000 : ℝ) ≤ t →
+        t ≤ (20774176036897 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_565 V_564
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_565 :
+    ∀ t,
+      (20774176036897 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (10421484320917 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_566 V_565
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_566 :
+    ∀ t,
+      (10421484320917 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (20911761246771 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_567 V_566
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_567 :
+    ∀ t,
+      (20911761246771 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (5245138462927 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_568 V_567
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_568 :
+    ∀ t,
+      (5245138462927 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (10559069530791 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_569 V_568
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_569 :
+    ∀ t,
+      (10559069530791 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (664241383483 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_570 V_569
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_570 :
+    ∀ t,
+      (664241383483 / 1000000000000 : ℝ) ≤ t →
+        t ≤ (4261528693017 / 6400000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_571 V_570
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_571 :
+    ∀ t,
+      (4261528693017 / 6400000000000 : ℝ) ≤ t →
+        t ≤ (10679781329357 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_572 V_571
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_572 :
+    ∀ t,
+      (10679781329357 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (21411481852343 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_573 V_572
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_573 :
+    ∀ t,
+      (21411481852343 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (5365850261493 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_574 V_573
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_574 :
+    ∀ t,
+      (5365850261493 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (21515320239601 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_575 V_574
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_575 :
+    ∀ t,
+      (21515320239601 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (43082559672831 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_576 V_575
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_576 :
+    ∀ t,
+      (43082559672831 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (2156723943323 / 3200000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_577 V_576
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_577 :
+    ∀ t,
+      (2156723943323 / 3200000000000 : ℝ) ≤ t →
+        t ≤ (43186398060089 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_578 V_577
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_578 :
+    ∀ t,
+      (43186398060089 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (21619158626859 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_579 V_578
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_579 :
+    ∀ t,
+      (21619158626859 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (43290236447347 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_580 V_579
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_580 :
+    ∀ t,
+      (43290236447347 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (2708884727561 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_581 V_580
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_581 :
+    ∀ t,
+      (2708884727561 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (8678814966921 / 12800000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_582 V_581
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_582 :
+    ∀ t,
+      (8678814966921 / 12800000000000 : ℝ) ≤ t →
+        t ≤ (21722997014117 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_583 V_582
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_583 :
+    ∀ t,
+      (21722997014117 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (43497913221863 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_584 V_583
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_584 :
+    ∀ t,
+      (43497913221863 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (10887458103873 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_585 V_584
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_585 :
+    ∀ t,
+      (10887458103873 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (43601751609121 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_586 V_585
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_586 :
+    ∀ t,
+      (43601751609121 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (174614683211 / 256000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_587 V_586
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_587 :
+    ∀ t,
+      (174614683211 / 256000000000 : ℝ) ≤ t →
+        t ≤ (43705589996379 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_588 V_587
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_588 :
+    ∀ t,
+      (43705589996379 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (5469688648751 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_589 V_588
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_589 :
+    ∀ t,
+      (5469688648751 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (43809428383637 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_590 V_589
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_590 :
+    ∀ t,
+      (43809428383637 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (21930673788633 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_591 V_590
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_591 :
+    ∀ t,
+      (21930673788633 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (8782653354179 / 12800000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_592 V_591
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_592 :
+    ∀ t,
+      (8782653354179 / 12800000000000 : ℝ) ≤ t →
+        t ≤ (10991296491131 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_593 V_592
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_593 :
+    ∀ t,
+      (10991296491131 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (44017105158153 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_594 V_593
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_594 :
+    ∀ t,
+      (44017105158153 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (22034512175891 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_595 V_594
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_595 :
+    ∀ t,
+      (22034512175891 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (44120943545411 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_596 V_595
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_596 :
+    ∀ t,
+      (44120943545411 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (276080392119 / 400000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_597 V_596
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_597 :
+    ∀ t,
+      (276080392119 / 400000000000 : ℝ) ≤ t →
+        t ≤ (44224781932669 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_598 V_597
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_598 :
+    ∀ t,
+      (44224781932669 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (22138350563149 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_599 V_598
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_599 :
+    ∀ t,
+      (22138350563149 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (44328620319927 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_600 V_599
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_600 :
+    ∀ t,
+      (44328620319927 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (11095134878389 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_601 V_600
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_601 :
+    ∀ t,
+      (11095134878389 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (8886491741437 / 12800000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_602 V_601
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_602 :
+    ∀ t,
+      (8886491741437 / 12800000000000 : ℝ) ≤ t →
+        t ≤ (22242188950407 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_603 V_602
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_603 :
+    ∀ t,
+      (22242188950407 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (5573527036009 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_604 V_603
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_604 :
+    ∀ t,
+      (5573527036009 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (4469205467533 / 6400000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_605 V_604
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_605 :
+    ∀ t,
+      (4469205467533 / 6400000000000 : ℝ) ≤ t →
+        t ≤ (11198973265647 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_606 V_605
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_606 :
+    ∀ t,
+      (11198973265647 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (22449865724923 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_607 V_606
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_607 :
+    ∀ t,
+      (22449865724923 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (2812723114819 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_608 V_607
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_608 :
+    ∀ t,
+      (2812723114819 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (22553704112181 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_609 V_608
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_609 :
+    ∀ t,
+      (22553704112181 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (2260562330581 / 3200000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_610 V_609
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_610 :
+    ∀ t,
+      (2260562330581 / 3200000000000 : ℝ) ≤ t →
+        t ≤ (22657542499439 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_611 V_610
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_611 :
+    ∀ t,
+      (22657542499439 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (5677365423267 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_612 V_611
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_612 :
+    ∀ t,
+      (5677365423267 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (22761380886697 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_613 V_612
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_613 :
+    ∀ t,
+      (22761380886697 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (11406650040163 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_614 V_613
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_614 :
+    ∀ t,
+      (11406650040163 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (89520072139 / 125000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_615 V_614
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_615 :
+    ∀ t,
+      (89520072139 / 125000000000 : ℝ) ≤ t →
+        t ≤ (11489045952349 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_616 V_615
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_616 :
+    ∀ t,
+      (11489045952349 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (4601713724651 / 6400000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_617 V_616
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_617 :
+    ∀ t,
+      (4601713724651 / 6400000000000 : ℝ) ≤ t →
+        t ≤ (5759761335453 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_618 V_617
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_618 :
+    ∀ t,
+      (5759761335453 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (23069522060369 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_619 V_618
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_619 :
+    ∀ t,
+      (23069522060369 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (11549999389463 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_620 V_619
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_620 :
+    ∀ t,
+      (11549999389463 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (23130475497483 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_621 V_620
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_621 :
+    ∀ t,
+      (23130475497483 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (579023805401 / 800000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_622 V_621
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_622 :
+    ∀ t,
+      (579023805401 / 800000000000 : ℝ) ≤ t →
+        t ≤ (23191428934597 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_623 V_622
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_623 :
+    ∀ t,
+      (23191428934597 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (11610952826577 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_624 V_623
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_624 :
+    ∀ t,
+      (11610952826577 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (23252382371711 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_625 V_624
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_625 :
+    ∀ t,
+      (23252382371711 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (5820714772567 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_626 V_625
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_626 :
+    ∀ t,
+      (5820714772567 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (932533432353 / 1280000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_627 V_626
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_627 :
+    ∀ t,
+      (932533432353 / 1280000000000 : ℝ) ≤ t →
+        t ≤ (11671906263691 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_628 V_627
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_628 :
+    ∀ t,
+      (11671906263691 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (23374289245939 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_629 V_628
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_629 :
+    ∀ t,
+      (23374289245939 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (1462797872781 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_630 V_629
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_630 :
+    ∀ t,
+      (1462797872781 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (23435242683053 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_631 V_630
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_631 :
+    ∀ t,
+      (23435242683053 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (2346571940161 / 3200000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_632 V_631
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_632 :
+    ∀ t,
+      (2346571940161 / 3200000000000 : ℝ) ≤ t →
+        t ≤ (23496196120167 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_633 V_632
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_633 :
+    ∀ t,
+      (23496196120167 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (5881668209681 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_634 V_633
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_634 :
+    ∀ t,
+      (5881668209681 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (23557149557281 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_635 V_634
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_635 :
+    ∀ t,
+      (23557149557281 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (11793813137919 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_636 V_635
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_636 :
+    ∀ t,
+      (11793813137919 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (4723620598879 / 6400000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_637 V_636
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_637 :
+    ∀ t,
+      (4723620598879 / 6400000000000 : ℝ) ≤ t →
+        t ≤ (2956072464119 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_638 V_637
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_638 :
+    ∀ t,
+      (2956072464119 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (23679056431509 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_639 V_638
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_639 :
+    ∀ t,
+      (23679056431509 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (11854766575033 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_640 V_639
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_640 :
+    ∀ t,
+      (11854766575033 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (23740009868623 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_641 V_640
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_641 :
+    ∀ t,
+      (23740009868623 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (1188524329359 / 1600000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_642 V_641
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_642 :
+    ∀ t,
+      (1188524329359 / 1600000000000 : ℝ) ≤ t →
+        t ≤ (23800963305737 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_643 V_642
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_643 :
+    ∀ t,
+      (23800963305737 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (11915720012147 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_644 V_643
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_644 :
+    ∀ t,
+      (11915720012147 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (746637295669 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_645 V_644
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_645 :
+    ∀ t,
+      (746637295669 / 1000000000000 : ℝ) ≤ t →
+        t ≤ (765809953469387 / 1024000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_646 V_645
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_646 :
+    ∀ t,
+      (765809953469387 / 1024000000000000 : ℝ) ≤ t →
+        t ≤ (383531658086859 / 512000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_647 V_646
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_647 :
+    ∀ t,
+      (383531658086859 / 512000000000000 : ℝ) ≤ t →
+        t ≤ (768316678878049 / 1024000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_648 V_647
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_648 :
+    ∀ t,
+      (768316678878049 / 1024000000000000 : ℝ) ≤ t →
+        t ≤ (38478502079119 / 51200000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_649 V_648
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_649 :
+    ∀ t,
+      (38478502079119 / 51200000000000 : ℝ) ≤ t →
+        t ≤ (770823404286711 / 1024000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_650 V_649
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_650 :
+    ∀ t,
+      (770823404286711 / 1024000000000000 : ℝ) ≤ t →
+        t ≤ (386038383495521 / 512000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_651 V_650
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_651 :
+    ∀ t,
+      (386038383495521 / 512000000000000 : ℝ) ≤ t →
+        t ≤ (773330129695373 / 1024000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_652 V_651
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_652 :
+    ∀ t,
+      (773330129695373 / 1024000000000000 : ℝ) ≤ t →
+        t ≤ (96822936549963 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_653 V_652
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_653 :
+    ∀ t,
+      (96822936549963 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (155167371020807 / 204800000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_654 V_653
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_654 :
+    ∀ t,
+      (155167371020807 / 204800000000000 : ℝ) ≤ t →
+        t ≤ (388545108904183 / 512000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_655 V_654
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_655 :
+    ∀ t,
+      (388545108904183 / 512000000000000 : ℝ) ≤ t →
+        t ≤ (194899235804257 / 256000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_656 V_655
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_656 :
+    ∀ t,
+      (194899235804257 / 256000000000000 : ℝ) ≤ t →
+        t ≤ (78210366862569 / 102400000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_657 V_656
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_657 :
+    ∀ t,
+      (78210366862569 / 102400000000000 : ℝ) ≤ t →
+        t ≤ (49038149627147 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_658 V_657
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_658 :
+    ∀ t,
+      (49038149627147 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (393558559721507 / 512000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_659 V_658
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_659 :
+    ∀ t,
+      (393558559721507 / 512000000000000 : ℝ) ≤ t →
+        t ≤ (197405961212919 / 256000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_660 V_659
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_660 :
+    ∀ t,
+      (197405961212919 / 256000000000000 : ℝ) ≤ t →
+        t ≤ (396065285130169 / 512000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_661 V_660
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_661 :
+    ∀ t,
+      (396065285130169 / 512000000000000 : ℝ) ≤ t →
+        t ≤ (794637295669 / 1024000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_662 V_661
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_662 :
+    ∀ t,
+      (794637295669 / 1024000000000 : ℝ) ≤ t →
+        t ≤ (398572010538831 / 512000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_663 V_662
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_663 :
+    ∀ t,
+      (398572010538831 / 512000000000000 : ℝ) ≤ t →
+        t ≤ (199912686621581 / 256000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_664 V_663
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_664 :
+    ∀ t,
+      (199912686621581 / 256000000000000 : ℝ) ≤ t →
+        t ≤ (25145756165739 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_665 V_664
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_665 :
+    ∀ t,
+      (25145756165739 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (202419412030243 / 256000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_666 V_665
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_666 :
+    ∀ t,
+      (202419412030243 / 256000000000000 : ℝ) ≤ t →
+        t ≤ (101836387367287 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_667 V_666
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_667 :
+    ∀ t,
+      (101836387367287 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (40985227487781 / 51200000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_668 V_667
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_668 :
+    ∀ t,
+      (40985227487781 / 51200000000000 : ℝ) ≤ t →
+        t ≤ (51544875035809 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_669 V_668
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_669 :
+    ∀ t,
+      (51544875035809 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (104343112775949 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_670 V_669
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_670 :
+    ∀ t,
+      (104343112775949 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (2639911887007 / 3200000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_671 V_670
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_671 :
+    ∀ t,
+      (2639911887007 / 3200000000000 : ℝ) ≤ t →
+        t ≤ (106849838184611 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_672 V_671
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_672 :
+    ∀ t,
+      (106849838184611 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (54051600444471 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_673 V_672
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_673 :
+    ∀ t,
+      (54051600444471 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (27652481574401 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_674 V_673
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_674 :
+    ∀ t,
+      (27652481574401 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (56558325853133 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_675 V_674
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_675 :
+    ∀ t,
+      (56558325853133 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (7226461069683 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_676 V_675
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_676 :
+    ∀ t,
+      (7226461069683 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (30159206983063 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_677 V_676
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_677 :
+    ∀ t,
+      (30159206983063 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (15706284843697 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_678 V_677
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_678 :
+    ∀ t,
+      (15706284843697 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (4239911887007 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_679 V_678
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_679 :
+    ∀ t,
+      (4239911887007 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (18213010252359 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_680 V_679
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_680 :
+    ∀ t,
+      (18213010252359 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (1946637295669 / 1600000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_681 V_680
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_681 :
+    ∀ t,
+      (1946637295669 / 1600000000000 : ℝ) ≤ t →
+        t ≤ (2746637295669 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_682 V_681
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_682 :
+    ∀ t,
+      (2746637295669 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (6746637295669 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_683 V_682
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma tc_683 :
+    ∀ t,
+      (6746637295669 / 4000000000000 : ℝ) ≤ t → t ≤ (2 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  check_right (by norm_num [bρ_1]) (by norm_num) U_684 V_683
+    (by unfold M0; linarith only [Real.pi_gt_d20, Real.pi_lt_d20])
+
+lemma cov_0_2 :
+    ∀ t, (0 : ℝ) ≤ t → t ≤ (21522393 / 400000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (7174131 / 200000000000 : ℝ)).elim (fun h => tc_0 t h1 h h0)
+    (fun h => tc_1 t h.le h2 h0)
+
+lemma cov_3_5 :
+    ∀ t,
+      (7174131 / 100000000000 : ℝ) ≤ t →
+        t ≤ (78667711 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (14825913 / 200000000000 : ℝ)).elim (fun h => tc_3 t h1 h h0)
+    (fun h => tc_4 t h.le h2 h0)
+
+lemma cov_2_5 :
+    ∀ t,
+      (21522393 / 400000000000 : ℝ) ≤ t →
+        t ≤ (78667711 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (7174131 / 100000000000 : ℝ)).elim (fun h => tc_2 t h1 h h0)
+    (fun h => cov_3_5 t h.le h2 h0)
+
+lemma cov_0_5 :
+    ∀ t, (0 : ℝ) ≤ t → t ≤ (78667711 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (21522393 / 400000000000 : ℝ)).elim (fun h => cov_0_2 t h1 h h0)
+    (fun h => cov_2_5 t h.le h2 h0)
+
+lemma cov_5_7 :
+    ∀ t,
+      (78667711 / 1000000000000 : ℝ) ≤ t →
+        t ≤ (19269871 / 200000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (85815639 / 1000000000000 : ℝ)).elim (fun h => tc_5 t h1 h h0)
+    (fun h => tc_6 t h.le h2 h0)
+
+lemma cov_8_10 :
+    ∀ t,
+      (55761057 / 500000000000 : ℝ) ≤ t →
+        t ≤ (82548843 / 500000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (33336783 / 250000000000 : ℝ)).elim (fun h => tc_8 t h1 h h0)
+    (fun h => tc_9 t h.le h2 h0)
+
+lemma cov_7_10 :
+    ∀ t,
+      (19269871 / 200000000000 : ℝ) ≤ t →
+        t ≤ (82548843 / 500000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (55761057 / 500000000000 : ℝ)).elim (fun h => tc_7 t h1 h h0)
+    (fun h => cov_8_10 t h.le h2 h0)
+
+lemma cov_5_10 :
+    ∀ t,
+      (78667711 / 1000000000000 : ℝ) ≤ t →
+        t ≤ (82548843 / 500000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (19269871 / 200000000000 : ℝ)).elim (fun h => cov_5_7 t h1 h h0)
+    (fun h => cov_7_10 t h.le h2 h0)
+
+lemma cov_0_10 :
+    ∀ t, (0 : ℝ) ≤ t → t ≤ (82548843 / 500000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (78667711 / 1000000000000 : ℝ)).elim (fun h => cov_0_5 t h1 h h0)
+    (fun h => cov_5_10 t h.le h2 h0)
+
+lemma cov_10_12 :
+    ∀ t,
+      (82548843 / 500000000000 : ℝ) ≤ t →
+        t ≤ (496117379 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (53051547 / 250000000000 : ℝ)).elim (fun h => tc_10 t h1 h h0)
+    (fun h => tc_11 t h.le h2 h0)
+
+lemma cov_13_15 :
+    ∀ t,
+      (283911191 / 1000000000000 : ℝ) ≤ t →
+        t ≤ (396324613 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (170058951 / 500000000000 : ℝ)).elim (fun h => tc_13 t h1 h h0)
+    (fun h => tc_14 t h.le h2 h0)
+
+lemma cov_12_15 :
+    ∀ t,
+      (496117379 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (396324613 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (283911191 / 1000000000000 : ℝ)).elim (fun h => tc_12 t h1 h h0)
+    (fun h => cov_13_15 t h.le h2 h0)
+
+lemma cov_10_15 :
+    ∀ t,
+      (82548843 / 500000000000 : ℝ) ≤ t →
+        t ≤ (396324613 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (496117379 / 2000000000000 : ℝ)).elim (fun h => cov_10_12 t h1 h h0)
+    (fun h => cov_12_15 t h.le h2 h0)
+
+lemma cov_16_18 :
+    ∀ t,
+      (974522519 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (729961631 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (289098953 / 500000000000 : ℝ)).elim (fun h => tc_16 t h1 h h0)
+    (fun h => tc_17 t h.le h2 h0)
+
+lemma cov_15_18 :
+    ∀ t,
+      (396324613 / 1000000000000 : ℝ) ≤ t →
+        t ≤ (729961631 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (974522519 / 2000000000000 : ℝ)).elim (fun h => tc_15 t h1 h h0)
+    (fun h => cov_16_18 t h.le h2 h0)
+
+lemma cov_19_21 :
+    ∀ t,
+      (1611686987 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (4047462733 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (220431339 / 250000000000 : ℝ)).elim (fun h => tc_19 t h1 h h0)
+    (fun h => tc_20 t h.le h2 h0)
+
+lemma cov_18_21 :
+    ∀ t,
+      (729961631 / 1000000000000 : ℝ) ≤ t →
+        t ≤ (4047462733 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1611686987 / 2000000000000 : ℝ)).elim (fun h => tc_18 t h1 h h0)
+    (fun h => cov_19_21 t h.le h2 h0)
+
+lemma cov_15_21 :
+    ∀ t,
+      (396324613 / 1000000000000 : ℝ) ≤ t →
+        t ≤ (4047462733 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (729961631 / 1000000000000 : ℝ)).elim (fun h => cov_15_18 t h1 h h0)
+    (fun h => cov_18_21 t h.le h2 h0)
+
+lemma cov_10_21 :
+    ∀ t,
+      (82548843 / 500000000000 : ℝ) ≤ t →
+        t ≤ (4047462733 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (396324613 / 1000000000000 : ℝ)).elim (fun h => cov_10_15 t h1 h h0)
+    (fun h => cov_15_21 t h.le h2 h0)
+
+lemma cov_0_21 :
+    ∀ t, (0 : ℝ) ≤ t → t ≤ (4047462733 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (82548843 / 500000000000 : ℝ)).elim (fun h => cov_0_10 t h1 h h0)
+    (fun h => cov_10_21 t h.le h2 h0)
+
+lemma cov_21_23 :
+    ∀ t,
+      (4047462733 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (5088585351 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (2284012021 / 2000000000000 : ℝ)).elim (fun h => tc_21 t h1 h h0)
+    (fun h => tc_22 t h.le h2 h0)
+
+lemma cov_24_26 :
+    ∀ t,
+      (280457333 / 200000000000 : ℝ) ≤ t →
+        t ≤ (3714534929 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (6519108259 / 4000000000000 : ℝ)).elim (fun h => tc_24 t h1 h h0)
+    (fun h => tc_25 t h.le h2 h0)
+
+lemma cov_23_26 :
+    ∀ t,
+      (5088585351 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (3714534929 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (280457333 / 200000000000 : ℝ)).elim (fun h => tc_23 t h1 h h0)
+    (fun h => cov_24_26 t h.le h2 h0)
+
+lemma cov_21_26 :
+    ∀ t,
+      (4047462733 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (3714534929 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (5088585351 / 4000000000000 : ℝ)).elim (fun h => cov_21_23 t h1 h h0)
+    (fun h => cov_23_26 t h.le h2 h0)
+
+lemma cov_26_28 :
+    ∀ t,
+      (3714534929 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (289031033 / 125000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (8339031457 / 4000000000000 : ℝ)).elim (fun h => tc_26 t h1 h h0)
+    (fun h => tc_27 t h.le h2 h0)
+
+lemma cov_29_31 :
+    ∀ t,
+      (124379927 / 40000000000 : ℝ) ≤ t →
+        t ≤ (1953374043 / 500000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (7016246261 / 2000000000000 : ℝ)).elim (fun h => tc_29 t h1 h h0)
+    (fun h => tc_30 t h.le h2 h0)
+
+lemma cov_28_31 :
+    ∀ t,
+      (289031033 / 125000000000 : ℝ) ≤ t →
+        t ≤ (1953374043 / 500000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (124379927 / 40000000000 : ℝ)).elim (fun h => tc_28 t h1 h h0)
+    (fun h => cov_29_31 t h.le h2 h0)
+
+lemma cov_26_31 :
+    ∀ t,
+      (3714534929 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (1953374043 / 500000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (289031033 / 125000000000 : ℝ)).elim (fun h => cov_26_28 t h1 h h0)
+    (fun h => cov_28_31 t h.le h2 h0)
+
+lemma cov_21_31 :
+    ∀ t,
+      (4047462733 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (1953374043 / 500000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (3714534929 / 2000000000000 : ℝ)).elim (fun h => cov_21_26 t h1 h h0)
+    (fun h => cov_26_31 t h.le h2 h0)
+
+lemma cov_31_33 :
+    ∀ t,
+      (1953374043 / 500000000000 : ℝ) ≤ t →
+        t ≤ (59205079 / 10000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (59205077 / 10000000000 : ℝ)).elim (fun h => tc_31 t h1 h h0)
+    (fun h => tc_32 t h.le h2 h0)
+
+lemma cov_34_36 :
+    ∀ t,
+      (8992695531 / 1000000000000 : ℝ) ≤ t →
+        t ≤ (40732008867 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (19572466643 / 2000000000000 : ℝ)).elim (fun h => tc_34 t h1 h h0)
+    (fun h => tc_35 t h.le h2 h0)
+
+lemma cov_33_36 :
+    ∀ t,
+      (59205079 / 10000000000 : ℝ) ≤ t →
+        t ≤ (40732008867 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (8992695531 / 1000000000000 : ℝ)).elim (fun h => tc_33 t h1 h h0)
+    (fun h => cov_34_36 t h.le h2 h0)
+
+lemma cov_31_36 :
+    ∀ t,
+      (1953374043 / 500000000000 : ℝ) ≤ t →
+        t ≤ (40732008867 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (59205079 / 10000000000 : ℝ)).elim (fun h => cov_31_33 t h1 h h0)
+    (fun h => cov_33_36 t h.le h2 h0)
+
+lemma cov_37_39 :
+    ∀ t,
+      (1322471389 / 125000000000 : ℝ) ≤ t →
+        t ≤ (12166846693 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (4549323561 / 400000000000 : ℝ)).elim (fun h => tc_37 t h1 h h0)
+    (fun h => tc_38 t h.le h2 h0)
+
+lemma cov_36_39 :
+    ∀ t,
+      (40732008867 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (12166846693 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1322471389 / 125000000000 : ℝ)).elim (fun h => tc_36 t h1 h h0)
+    (fun h => cov_37_39 t h.le h2 h0)
+
+lemma cov_40_42 :
+    ∀ t,
+      (3068199571 / 200000000000 : ℝ) ≤ t →
+        t ≤ (133117165709 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (255845148549 / 16000000000000 : ℝ)).elim (fun h => tc_40 t h1 h h0)
+    (fun h => tc_41 t h.le h2 h0)
+
+lemma cov_39_42 :
+    ∀ t,
+      (12166846693 / 1000000000000 : ℝ) ≤ t →
+        t ≤ (133117165709 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (3068199571 / 200000000000 : ℝ)).elim (fun h => tc_39 t h1 h h0)
+    (fun h => cov_40_42 t h.le h2 h0)
+
+lemma cov_36_42 :
+    ∀ t,
+      (40732008867 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (133117165709 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (12166846693 / 1000000000000 : ℝ)).elim (fun h => cov_36_39 t h1 h h0)
+    (fun h => cov_39_42 t h.le h2 h0)
+
+lemma cov_31_42 :
+    ∀ t,
+      (1953374043 / 500000000000 : ℝ) ≤ t →
+        t ≤ (133117165709 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (40732008867 / 4000000000000 : ℝ)).elim (fun h => cov_31_36 t h1 h h0)
+    (fun h => cov_36_42 t h.le h2 h0)
+
+lemma cov_21_42 :
+    ∀ t,
+      (4047462733 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (133117165709 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1953374043 / 500000000000 : ℝ)).elim (fun h => cov_21_31 t h1 h h0)
+    (fun h => cov_31_42 t h.le h2 h0)
+
+lemma cov_0_42 :
+    ∀ t,
+      (0 : ℝ) ≤ t → t ≤ (133117165709 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (4047462733 / 4000000000000 : ℝ)).elim (fun h => cov_0_21 t h1 h h0)
+    (fun h => cov_21_42 t h.le h2 h0)
+
+lemma cov_42_44 :
+    ∀ t,
+      (133117165709 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (276623514287 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (108571569141 / 6400000000000 : ℝ)).elim (fun h => tc_42 t h1 h h0)
+    (fun h => tc_43 t h.le h2 h0)
+
+lemma cov_45_47 :
+    ∀ t,
+      (563636211443 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (584414577181 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (71753174289 / 4000000000000 : ℝ)).elim (fun h => tc_45 t h1 h h0)
+    (fun h => tc_46 t h.le h2 h0)
+
+lemma cov_44_47 :
+    ∀ t,
+      (276623514287 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (584414577181 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (563636211443 / 32000000000000 : ℝ)).elim (fun h => tc_44 t h1 h h0)
+    (fun h => cov_45_47 t h.le h2 h0)
+
+lemma cov_42_47 :
+    ∀ t,
+      (133117165709 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (584414577181 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (276623514287 / 16000000000000 : ℝ)).elim (fun h => cov_42_44 t h1 h h0)
+    (fun h => cov_44_47 t h.le h2 h0)
+
+lemma cov_47_49 :
+    ∀ t,
+      (584414577181 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (605192942919 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (11896075201 / 640000000000 : ℝ)).elim (fun h => tc_47 t h1 h h0)
+    (fun h => tc_48 t h.le h2 h0)
+
+lemma cov_50_52 :
+    ∀ t,
+      (153895531447 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (41071178579 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (318180245763 / 16000000000000 : ℝ)).elim (fun h => tc_50 t h1 h h0)
+    (fun h => tc_51 t h.le h2 h0)
+
+lemma cov_49_52 :
+    ∀ t,
+      (605192942919 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (41071178579 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (153895531447 / 8000000000000 : ℝ)).elim (fun h => tc_49 t h1 h h0)
+    (fun h => cov_50_52 t h.le h2 h0)
+
+lemma cov_47_52 :
+    ∀ t,
+      (584414577181 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (41071178579 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (605192942919 / 32000000000000 : ℝ)).elim (fun h => cov_47_49 t h1 h h0)
+    (fun h => cov_49_52 t h.le h2 h0)
+
+lemma cov_42_52 :
+    ∀ t,
+      (133117165709 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (41071178579 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (584414577181 / 32000000000000 : ℝ)).elim (fun h => cov_42_47 t h1 h h0)
+    (fun h => cov_47_52 t h.le h2 h0)
+
+lemma cov_52_54 :
+    ∀ t,
+      (41071178579 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (92531540027 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (34934779437 / 1600000000000 : ℝ)).elim (fun h => tc_52 t h1 h h0)
+    (fun h => tc_53 t h.le h2 h0)
+
+lemma cov_55_57 :
+    ∀ t,
+      (6432545181 / 250000000000 : ℝ) ≤ t →
+        t ≤ (435951987867 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (213931144553 / 8000000000000 : ℝ)).elim (fun h => tc_55 t h1 h h0)
+    (fun h => tc_56 t h.le h2 h0)
+
+lemma cov_54_57 :
+    ∀ t,
+      (92531540027 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (435951987867 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (6432545181 / 250000000000 : ℝ)).elim (fun h => tc_54 t h1 h h0)
+    (fun h => cov_55_57 t h.le h2 h0)
+
+lemma cov_52_57 :
+    ∀ t,
+      (41071178579 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (435951987867 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (92531540027 / 4000000000000 : ℝ)).elim (fun h => cov_52_54 t h1 h h0)
+    (fun h => cov_54_57 t h.le h2 h0)
+
+lemma cov_58_60 :
+    ∀ t,
+      (111010421657 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (912352469539 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (452131385389 / 16000000000000 : ℝ)).elim (fun h => tc_58 t h1 h h0)
+    (fun h => tc_59 t h.le h2 h0)
+
+lemma cov_57_60 :
+    ∀ t,
+      (435951987867 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (912352469539 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (111010421657 / 4000000000000 : ℝ)).elim (fun h => tc_57 t h1 h h0)
+    (fun h => cov_58_60 t h.le h2 h0)
+
+lemma cov_61_63 :
+    ∀ t,
+      (9204421683 / 320000000000 : ℝ) ≤ t →
+        t ≤ (468310782911 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (928531867061 / 32000000000000 : ℝ)).elim (fun h => tc_61 t h1 h h0)
+    (fun h => tc_62 t h.le h2 h0)
+
+lemma cov_60_63 :
+    ∀ t,
+      (912352469539 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (468310782911 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (9204421683 / 320000000000 : ℝ)).elim (fun h => tc_60 t h1 h h0)
+    (fun h => cov_61_63 t h.le h2 h0)
+
+lemma cov_57_63 :
+    ∀ t,
+      (435951987867 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (468310782911 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (912352469539 / 32000000000000 : ℝ)).elim (fun h => cov_57_60 t h1 h h0)
+    (fun h => cov_60_63 t h.le h2 h0)
+
+lemma cov_52_63 :
+    ∀ t,
+      (41071178579 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (468310782911 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (435951987867 / 16000000000000 : ℝ)).elim (fun h => cov_52_57 t h1 h h0)
+    (fun h => cov_57_63 t h.le h2 h0)
+
+lemma cov_42_63 :
+    ∀ t,
+      (133117165709 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (468310782911 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (41071178579 / 2000000000000 : ℝ)).elim (fun h => cov_42_52 t h1 h h0)
+    (fun h => cov_52_63 t h.le h2 h0)
+
+lemma cov_63_65 :
+    ∀ t,
+      (468310782911 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (59550060209 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (944711264583 / 32000000000000 : ℝ)).elim (fun h => tc_63 t h1 h h0)
+    (fun h => tc_64 t h.le h2 h0)
+
+lemma cov_66_68 :
+    ∀ t,
+      (192178132421 / 6400000000000 : ℝ) ≤ t →
+        t ≤ (977070059627 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (484490180433 / 16000000000000 : ℝ)).elim (fun h => tc_66 t h1 h h0)
+    (fun h => tc_67 t h.le h2 h0)
+
+lemma cov_65_68 :
+    ∀ t,
+      (59550060209 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (977070059627 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (192178132421 / 6400000000000 : ℝ)).elim (fun h => tc_65 t h1 h h0)
+    (fun h => cov_66_68 t h.le h2 h0)
+
+lemma cov_63_68 :
+    ∀ t,
+      (468310782911 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (977070059627 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (59550060209 / 2000000000000 : ℝ)).elim (fun h => cov_63_65 t h1 h h0)
+    (fun h => cov_65_68 t h.le h2 h0)
+
+lemma cov_69_71 :
+    ∀ t,
+      (246289939597 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (127189819179 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (100133915591 / 3200000000000 : ℝ)).elim (fun h => tc_69 t h1 h h0)
+    (fun h => tc_70 t h.le h2 h0)
+
+lemma cov_68_71 :
+    ∀ t,
+      (977070059627 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (127189819179 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (246289939597 / 8000000000000 : ℝ)).elim (fun h => tc_68 t h1 h h0)
+    (fun h => cov_69_71 t h.le h2 h0)
+
+lemma cov_72_74 :
+    ∀ t,
+      (516848975477 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (6763975897 / 200000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (262469337119 / 8000000000000 : ℝ)).elim (fun h => tc_72 t h1 h h0)
+    (fun h => tc_73 t h.le h2 h0)
+
+lemma cov_71_74 :
+    ∀ t,
+      (127189819179 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (6763975897 / 200000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (516848975477 / 16000000000000 : ℝ)).elim (fun h => tc_71 t h1 h h0)
+    (fun h => cov_72_74 t h.le h2 h0)
+
+lemma cov_68_74 :
+    ∀ t,
+      (977070059627 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (6763975897 / 200000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (127189819179 / 4000000000000 : ℝ)).elim (fun h => cov_68_71 t h1 h h0)
+    (fun h => cov_71_74 t h.le h2 h0)
+
+lemma cov_63_74 :
+    ∀ t,
+      (468310782911 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (6763975897 / 200000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (977070059627 / 32000000000000 : ℝ)).elim (fun h => cov_63_68 t h1 h h0)
+    (fun h => cov_68_74 t h.le h2 h0)
+
+lemma cov_74_76 :
+    ∀ t,
+      (6763975897 / 200000000000 : ℝ) ≤ t →
+        t ≤ (143369216701 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (278648734641 / 8000000000000 : ℝ)).elim (fun h => tc_74 t h1 h h0)
+    (fun h => tc_75 t h.le h2 h0)
+
+lemma cov_77_79 :
+    ∀ t,
+      (75729457731 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (694494763253 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (20954789123 / 500000000000 : ℝ)).elim (fun h => tc_77 t h1 h h0)
+    (fun h => tc_78 t h.le h2 h0)
+
+lemma cov_76_79 :
+    ∀ t,
+      (143369216701 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (694494763253 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (75729457731 / 2000000000000 : ℝ)).elim (fun h => tc_76 t h1 h h0)
+    (fun h => cov_77_79 t h.le h2 h0)
+
+lemma cov_74_79 :
+    ∀ t,
+      (6763975897 / 200000000000 : ℝ) ≤ t →
+        t ≤ (694494763253 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (143369216701 / 4000000000000 : ℝ)).elim (fun h => cov_74_76 t h1 h h0)
+    (fun h => cov_76_79 t h.le h2 h0)
+
+lemma cov_80_82 :
+    ∀ t,
+      (1412931037823 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (2897686609597 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (71843627457 / 1600000000000 : ℝ)).elim (fun h => tc_80 t h1 h h0)
+    (fun h => tc_81 t h.le h2 h0)
+
+lemma cov_79_82 :
+    ∀ t,
+      (694494763253 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (2897686609597 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1412931037823 / 32000000000000 : ℝ)).elim (fun h => tc_79 t h1 h h0)
+    (fun h => cov_80_82 t h.le h2 h0)
+
+lemma cov_83_85 :
+    ∀ t,
+      (1460814060457 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (742377785887 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (2945569632231 / 64000000000000 : ℝ)).elim (fun h => tc_83 t h1 h h0)
+    (fun h => tc_84 t h.le h2 h0)
+
+lemma cov_82_85 :
+    ∀ t,
+      (2897686609597 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (742377785887 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1460814060457 / 32000000000000 : ℝ)).elim (fun h => tc_82 t h1 h h0)
+    (fun h => cov_83_85 t h.le h2 h0)
+
+lemma cov_79_85 :
+    ∀ t,
+      (694494763253 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (742377785887 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (2897686609597 / 64000000000000 : ℝ)).elim (fun h => cov_79_82 t h1 h h0)
+    (fun h => cov_82_85 t h.le h2 h0)
+
+lemma cov_74_85 :
+    ∀ t,
+      (6763975897 / 200000000000 : ℝ) ≤ t →
+        t ≤ (742377785887 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (694494763253 / 16000000000000 : ℝ)).elim (fun h => cov_74_79 t h1 h h0)
+    (fun h => cov_79_85 t h.le h2 h0)
+
+lemma cov_63_85 :
+    ∀ t,
+      (468310782911 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (742377785887 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (6763975897 / 200000000000 : ℝ)).elim (fun h => cov_63_74 t h1 h h0)
+    (fun h => cov_74_85 t h.le h2 h0)
+
+lemma cov_42_85 :
+    ∀ t,
+      (133117165709 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (742377785887 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (468310782911 / 16000000000000 : ℝ)).elim (fun h => cov_42_63 t h1 h h0)
+    (fun h => cov_63_85 t h.le h2 h0)
+
+lemma cov_0_85 :
+    ∀ t,
+      (0 : ℝ) ≤ t → t ≤ (742377785887 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (133117165709 / 8000000000000 : ℝ)).elim (fun h => cov_0_42 t h1 h h0)
+    (fun h => cov_42_85 t h.le h2 h0)
+
+lemma cov_85_87 :
+    ∀ t,
+      (742377785887 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (1508697083091 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (598690530973 / 12800000000000 : ℝ)).elim (fun h => tc_85 t h1 h h0)
+    (fun h => tc_86 t h.le h2 h0)
+
+lemma cov_88_90 :
+    ∀ t,
+      (3041335677499 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (3089218700133 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (191579824301 / 4000000000000 : ℝ)).elim (fun h => tc_88 t h1 h h0)
+    (fun h => tc_89 t h.le h2 h0)
+
+lemma cov_87_90 :
+    ∀ t,
+      (1508697083091 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (3089218700133 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (3041335677499 / 64000000000000 : ℝ)).elim (fun h => tc_87 t h1 h h0)
+    (fun h => cov_88_90 t h.le h2 h0)
+
+lemma cov_85_90 :
+    ∀ t,
+      (742377785887 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (3089218700133 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1508697083091 / 32000000000000 : ℝ)).elim (fun h => cov_85_87 t h1 h h0)
+    (fun h => cov_87_90 t h.le h2 h0)
+
+lemma cov_90_92 :
+    ∀ t,
+      (3089218700133 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (3137101722767 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (62263204229 / 1280000000000 : ℝ)).elim (fun h => tc_90 t h1 h h0)
+    (fun h => tc_91 t h.le h2 h0)
+
+lemma cov_93_95 :
+    ∀ t,
+      (790260808521 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (1604463128359 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (3184984745401 / 64000000000000 : ℝ)).elim (fun h => tc_93 t h1 h h0)
+    (fun h => tc_94 t h.le h2 h0)
+
+lemma cov_92_95 :
+    ∀ t,
+      (3137101722767 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (1604463128359 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (790260808521 / 16000000000000 : ℝ)).elim (fun h => tc_92 t h1 h h0)
+    (fun h => cov_93_95 t h.le h2 h0)
+
+lemma cov_90_95 :
+    ∀ t,
+      (3089218700133 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (1604463128359 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (3137101722767 / 64000000000000 : ℝ)).elim (fun h => cov_90_92 t h1 h h0)
+    (fun h => cov_92_95 t h.le h2 h0)
+
+lemma cov_85_95 :
+    ∀ t,
+      (742377785887 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (1604463128359 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (3089218700133 / 64000000000000 : ℝ)).elim (fun h => cov_85_90 t h1 h h0)
+    (fun h => cov_90_95 t h.le h2 h0)
+
+lemma cov_95_97 :
+    ∀ t,
+      (1604463128359 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (407101159919 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (646573553607 / 12800000000000 : ℝ)).elim (fun h => tc_95 t h1 h h0)
+    (fun h => tc_96 t h.le h2 h0)
+
+lemma cov_98_100 :
+    ∀ t,
+      (1652346150993 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (1700229173627 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (167628766231 / 3200000000000 : ℝ)).elim (fun h => tc_98 t h1 h h0)
+    (fun h => tc_99 t h.le h2 h0)
+
+lemma cov_97_100 :
+    ∀ t,
+      (407101159919 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (1700229173627 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1652346150993 / 32000000000000 : ℝ)).elim (fun h => tc_97 t h1 h h0)
+    (fun h => cov_98_100 t h.le h2 h0)
+
+lemma cov_95_100 :
+    ∀ t,
+      (1604463128359 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (1700229173627 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (407101159919 / 8000000000000 : ℝ)).elim (fun h => cov_95_97 t h1 h h0)
+    (fun h => cov_97_100 t h.le h2 h0)
+
+lemma cov_101_103 :
+    ∀ t,
+      (107760667809 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (454984182553 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (886026853789 / 16000000000000 : ℝ)).elim (fun h => tc_101 t h1 h h0)
+    (fun h => tc_102 t h.le h2 h0)
+
+lemma cov_100_103 :
+    ∀ t,
+      (1700229173627 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (454984182553 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (107760667809 / 2000000000000 : ℝ)).elim (fun h => tc_100 t h1 h h0)
+    (fun h => cov_101_103 t h.le h2 h0)
+
+lemma cov_104_106 :
+    ∀ t,
+      (47892569387 / 800000000000 : ℝ) ≤ t →
+        t ≤ (65851089563 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (502867205187 / 8000000000000 : ℝ)).elim (fun h => tc_104 t h1 h h0)
+    (fun h => tc_105 t h.le h2 h0)
+
+lemma cov_103_106 :
+    ∀ t,
+      (454984182553 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (65851089563 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (47892569387 / 800000000000 : ℝ)).elim (fun h => tc_103 t h1 h h0)
+    (fun h => cov_104_106 t h.le h2 h0)
+
+lemma cov_100_106 :
+    ∀ t,
+      (1700229173627 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (65851089563 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (454984182553 / 8000000000000 : ℝ)).elim (fun h => cov_100_103 t h1 h h0)
+    (fun h => cov_103_106 t h.le h2 h0)
+
+lemma cov_95_106 :
+    ∀ t,
+      (1604463128359 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (65851089563 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1700229173627 / 32000000000000 : ℝ)).elim (fun h => cov_95_100 t h1 h h0)
+    (fun h => cov_100_106 t h.le h2 h0)
+
+lemma cov_85_106 :
+    ∀ t,
+      (742377785887 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (65851089563 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1604463128359 / 32000000000000 : ℝ)).elim (fun h => cov_85_95 t h1 h h0)
+    (fun h => cov_95_106 t h.le h2 h0)
+
+lemma cov_106_108 :
+    ∀ t,
+      (65851089563 / 1000000000000 : ℝ) ≤ t →
+        t ≤ (1087247381329 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (2140864814337 / 32000000000000 : ℝ)).elim (fun h => tc_106 t h1 h h0)
+    (fun h => tc_107 t h.le h2 h0)
+
+lemma cov_109_111 :
+    ∀ t,
+      (2208124710979 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (22417546593 / 320000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (4449879370279 / 64000000000000 : ℝ)).elim (fun h => tc_109 t h1 h h0)
+    (fun h => tc_110 t h.le h2 h0)
+
+lemma cov_108_111 :
+    ∀ t,
+      (1087247381329 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (22417546593 / 320000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (2208124710979 / 32000000000000 : ℝ)).elim (fun h => tc_108 t h1 h h0)
+    (fun h => cov_109_111 t h.le h2 h0)
+
+lemma cov_106_111 :
+    ∀ t,
+      (65851089563 / 1000000000000 : ℝ) ≤ t →
+        t ≤ (22417546593 / 320000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1087247381329 / 16000000000000 : ℝ)).elim (fun h => cov_106_108 t h1 h h0)
+    (fun h => cov_108_111 t h.le h2 h0)
+
+lemma cov_112_114 :
+    ∀ t,
+      (4517139266921 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (4584399163563 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (2275384607621 / 32000000000000 : ℝ)).elim (fun h => tc_112 t h1 h h0)
+    (fun h => tc_113 t h.le h2 h0)
+
+lemma cov_111_114 :
+    ∀ t,
+      (22417546593 / 320000000000 : ℝ) ≤ t →
+        t ≤ (4584399163563 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (4517139266921 / 64000000000000 : ℝ)).elim (fun h => tc_111 t h1 h h0)
+    (fun h => cov_112_114 t h.le h2 h0)
+
+lemma cov_115_117 :
+    ∀ t,
+      (1154507277971 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (2342644504263 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (930331812041 / 12800000000000 : ℝ)).elim (fun h => tc_115 t h1 h h0)
+    (fun h => tc_116 t h.le h2 h0)
+
+lemma cov_114_117 :
+    ∀ t,
+      (4584399163563 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (2342644504263 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1154507277971 / 16000000000000 : ℝ)).elim (fun h => tc_114 t h1 h h0)
+    (fun h => cov_115_117 t h.le h2 h0)
+
+lemma cov_111_117 :
+    ∀ t,
+      (22417546593 / 320000000000 : ℝ) ≤ t →
+        t ≤ (2342644504263 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (4584399163563 / 64000000000000 : ℝ)).elim (fun h => cov_111_114 t h1 h h0)
+    (fun h => cov_114_117 t h.le h2 h0)
+
+lemma cov_106_117 :
+    ∀ t,
+      (65851089563 / 1000000000000 : ℝ) ≤ t →
+        t ≤ (2342644504263 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (22417546593 / 320000000000 : ℝ)).elim (fun h => cov_106_111 t h1 h h0)
+    (fun h => cov_111_117 t h.le h2 h0)
+
+lemma cov_117_119 :
+    ∀ t,
+      (2342644504263 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (4718918956847 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (9404207965373 / 128000000000000 : ℝ)).elim (fun h => tc_117 t h1 h h0)
+    (fun h => tc_118 t h.le h2 h0)
+
+lemma cov_120_122 :
+    ∀ t,
+      (1894293572403 / 25600000000000 : ℝ) ≤ t →
+        t ≤ (9538727758657 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (297034306573 / 4000000000000 : ℝ)).elim (fun h => tc_120 t h1 h h0)
+    (fun h => tc_121 t h.le h2 h0)
+
+lemma cov_119_122 :
+    ∀ t,
+      (4718918956847 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (9538727758657 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1894293572403 / 25600000000000 : ℝ)).elim (fun h => tc_119 t h1 h h0)
+    (fun h => cov_120_122 t h.le h2 h0)
+
+lemma cov_117_122 :
+    ∀ t,
+      (2342644504263 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (9538727758657 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (4718918956847 / 64000000000000 : ℝ)).elim (fun h => cov_117_119 t h1 h h0)
+    (fun h => cov_119_122 t h.le h2 h0)
+
+lemma cov_123_125 :
+    ∀ t,
+      (4786178853489 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (481980880181 / 6400000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (9605987655299 / 128000000000000 : ℝ)).elim (fun h => tc_123 t h1 h h0)
+    (fun h => tc_124 t h.le h2 h0)
+
+lemma cov_122_125 :
+    ∀ t,
+      (9538727758657 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (481980880181 / 6400000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (4786178853489 / 64000000000000 : ℝ)).elim (fun h => tc_122 t h1 h h0)
+    (fun h => cov_123_125 t h.le h2 h0)
+
+lemma cov_126_128 :
+    ∀ t,
+      (9673247551941 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (1221767174613 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (4853438750131 / 64000000000000 : ℝ)).elim (fun h => tc_126 t h1 h h0)
+    (fun h => tc_127 t h.le h2 h0)
+
+lemma cov_125_128 :
+    ∀ t,
+      (481980880181 / 6400000000000 : ℝ) ≤ t →
+        t ≤ (1221767174613 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (9673247551941 / 128000000000000 : ℝ)).elim (fun h => tc_125 t h1 h h0)
+    (fun h => cov_126_128 t h.le h2 h0)
+
+lemma cov_122_128 :
+    ∀ t,
+      (9538727758657 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (1221767174613 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (481980880181 / 6400000000000 : ℝ)).elim (fun h => cov_122_125 t h1 h h0)
+    (fun h => cov_125_128 t h.le h2 h0)
+
+lemma cov_117_128 :
+    ∀ t,
+      (2342644504263 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (1221767174613 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (9538727758657 / 128000000000000 : ℝ)).elim (fun h => cov_117_122 t h1 h h0)
+    (fun h => cov_122_128 t h.le h2 h0)
+
+lemma cov_106_128 :
+    ∀ t,
+      (65851089563 / 1000000000000 : ℝ) ≤ t →
+        t ≤ (1221767174613 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (2342644504263 / 32000000000000 : ℝ)).elim (fun h => cov_106_117 t h1 h h0)
+    (fun h => cov_117_128 t h.le h2 h0)
+
+lemma cov_85_128 :
+    ∀ t,
+      (742377785887 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (1221767174613 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (65851089563 / 1000000000000 : ℝ)).elim (fun h => cov_85_106 t h1 h h0)
+    (fun h => cov_106_128 t h.le h2 h0)
+
+lemma cov_128_130 :
+    ∀ t,
+      (1221767174613 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (2477164297547 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (4920698646773 / 64000000000000 : ℝ)).elim (fun h => tc_128 t h1 h h0)
+    (fun h => tc_129 t h.le h2 h0)
+
+lemma cov_131_133 :
+    ∀ t,
+      (997591708683 / 12800000000000 : ℝ) ≤ t →
+        t ≤ (5055218440057 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (627698561467 / 8000000000000 : ℝ)).elim (fun h => tc_131 t h1 h h0)
+    (fun h => tc_132 t h.le h2 h0)
+
+lemma cov_130_133 :
+    ∀ t,
+      (2477164297547 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (5055218440057 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (997591708683 / 12800000000000 : ℝ)).elim (fun h => tc_130 t h1 h h0)
+    (fun h => cov_131_133 t h.le h2 h0)
+
+lemma cov_128_133 :
+    ∀ t,
+      (1221767174613 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (5055218440057 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (2477164297547 / 32000000000000 : ℝ)).elim (fun h => cov_128_130 t h1 h h0)
+    (fun h => cov_130_133 t h.le h2 h0)
+
+lemma cov_133_135 :
+    ∀ t,
+      (5055218440057 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (5122478336699 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (2544424194189 / 32000000000000 : ℝ)).elim (fun h => tc_133 t h1 h h0)
+    (fun h => tc_134 t h.le h2 h0)
+
+lemma cov_136_138 :
+    ∀ t,
+      (257805414251 / 3200000000000 : ℝ) ≤ t →
+        t ≤ (165332127447 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (2611684090831 / 32000000000000 : ℝ)).elim (fun h => tc_136 t h1 h h0)
+    (fun h => tc_137 t h.le h2 h0)
+
+lemma cov_135_138 :
+    ∀ t,
+      (5122478336699 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (165332127447 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (257805414251 / 3200000000000 : ℝ)).elim (fun h => tc_135 t h1 h h0)
+    (fun h => cov_136_138 t h.le h2 h0)
+
+lemma cov_133_138 :
+    ∀ t,
+      (5055218440057 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (165332127447 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (5122478336699 / 64000000000000 : ℝ)).elim (fun h => cov_133_135 t h1 h h0)
+    (fun h => cov_135_138 t h.le h2 h0)
+
+lemma cov_128_138 :
+    ∀ t,
+      (1221767174613 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (165332127447 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (5055218440057 / 64000000000000 : ℝ)).elim (fun h => cov_128_133 t h1 h h0)
+    (fun h => cov_133_138 t h.le h2 h0)
+
+lemma cov_138_140 :
+    ∀ t,
+      (165332127447 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (1356286967897 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (2678943987473 / 32000000000000 : ℝ)).elim (fun h => tc_138 t h1 h h0)
+    (fun h => tc_139 t h.le h2 h0)
+
+lemma cov_141_143 :
+    ∀ t,
+      (694958458109 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (72858840643 / 800000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1423546864539 / 16000000000000 : ℝ)).elim (fun h => tc_141 t h1 h h0)
+    (fun h => tc_142 t h.le h2 h0)
+
+lemma cov_140_143 :
+    ∀ t,
+      (1356286967897 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (72858840643 / 800000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (694958458109 / 8000000000000 : ℝ)).elim (fun h => tc_140 t h1 h h0)
+    (fun h => cov_141_143 t h.le h2 h0)
+
+lemma cov_138_143 :
+    ∀ t,
+      (165332127447 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (72858840643 / 800000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1356286967897 / 16000000000000 : ℝ)).elim (fun h => cov_138_140 t h1 h h0)
+    (fun h => cov_140_143 t h.le h2 h0)
+
+lemma cov_144_146 :
+    ∀ t,
+      (762218354751 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (1614118950931 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (24870259471 / 250000000000 : ℝ)).elim (fun h => tc_144 t h1 h h0)
+    (fun h => tc_145 t h.le h2 h0)
+
+lemma cov_143_146 :
+    ∀ t,
+      (72858840643 / 800000000000 : ℝ) ≤ t →
+        t ≤ (1614118950931 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (762218354751 / 8000000000000 : ℝ)).elim (fun h => tc_143 t h1 h h0)
+    (fun h => cov_144_146 t h.le h2 h0)
+
+lemma cov_147_149 :
+    ∀ t,
+      (818270647859 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (3340349625797 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (331792728101 / 3200000000000 : ℝ)).elim (fun h => tc_147 t h1 h h0)
+    (fun h => tc_148 t h.le h2 h0)
+
+lemma cov_146_149 :
+    ∀ t,
+      (1614118950931 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (3340349625797 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (818270647859 / 8000000000000 : ℝ)).elim (fun h => tc_146 t h1 h h0)
+    (fun h => cov_147_149 t h.le h2 h0)
+
+lemma cov_143_149 :
+    ∀ t,
+      (72858840643 / 800000000000 : ℝ) ≤ t →
+        t ≤ (3340349625797 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1614118950931 / 16000000000000 : ℝ)).elim (fun h => cov_143_146 t h1 h h0)
+    (fun h => cov_146_149 t h.le h2 h0)
+
+lemma cov_138_149 :
+    ∀ t,
+      (165332127447 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (3340349625797 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (72858840643 / 800000000000 : ℝ)).elim (fun h => cov_138_143 t h1 h h0)
+    (fun h => cov_143_149 t h.le h2 h0)
+
+lemma cov_128_149 :
+    ∀ t,
+      (1221767174613 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (3340349625797 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (165332127447 / 2000000000000 : ℝ)).elim (fun h => cov_128_138 t h1 h h0)
+    (fun h => cov_138_149 t h.le h2 h0)
+
+lemma cov_149_151 :
+    ∀ t,
+      (3340349625797 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (3385194315371 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (420346496323 / 4000000000000 : ℝ)).elim (fun h => tc_149 t h1 h h0)
+    (fun h => tc_150 t h.le h2 h0)
+
+lemma cov_152_154 :
+    ∀ t,
+      (1703808330079 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (863115337433 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (686007800989 / 6400000000000 : ℝ)).elim (fun h => tc_152 t h1 h h0)
+    (fun h => tc_153 t h.le h2 h0)
+
+lemma cov_151_154 :
+    ∀ t,
+      (3385194315371 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (863115337433 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1703808330079 / 16000000000000 : ℝ)).elim (fun h => tc_151 t h1 h h0)
+    (fun h => cov_152_154 t h.le h2 h0)
+
+lemma cov_149_154 :
+    ∀ t,
+      (3340349625797 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (863115337433 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (3385194315371 / 32000000000000 : ℝ)).elim (fun h => cov_149_151 t h1 h h0)
+    (fun h => cov_151_154 t h.le h2 h0)
+
+lemma cov_155_157 :
+    ∀ t,
+      (6927345044251 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (278887589353 / 2560000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (3474883694519 / 32000000000000 : ℝ)).elim (fun h => tc_155 t h1 h h0)
+    (fun h => tc_156 t h.le h2 h0)
+
+lemma cov_154_157 :
+    ∀ t,
+      (863115337433 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (278887589353 / 2560000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (6927345044251 / 64000000000000 : ℝ)).elim (fun h => tc_154 t h1 h h0)
+    (fun h => cov_155_157 t h.le h2 h0)
+
+lemma cov_158_160 :
+    ∀ t,
+      (1748653019653 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (3519728384093 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (7017034423399 / 64000000000000 : ℝ)).elim (fun h => tc_158 t h1 h h0)
+    (fun h => tc_159 t h.le h2 h0)
+
+lemma cov_157_160 :
+    ∀ t,
+      (278887589353 / 2560000000000 : ℝ) ≤ t →
+        t ≤ (3519728384093 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1748653019653 / 16000000000000 : ℝ)).elim (fun h => tc_157 t h1 h h0)
+    (fun h => cov_158_160 t h.le h2 h0)
+
+lemma cov_154_160 :
+    ∀ t,
+      (863115337433 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (3519728384093 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (278887589353 / 2560000000000 : ℝ)).elim (fun h => cov_154_157 t h1 h h0)
+    (fun h => cov_157_160 t h.le h2 h0)
+
+lemma cov_149_160 :
+    ∀ t,
+      (3340349625797 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (3519728384093 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (863115337433 / 8000000000000 : ℝ)).elim (fun h => cov_149_154 t h1 h h0)
+    (fun h => cov_154_160 t h.le h2 h0)
+
+lemma cov_160_162 :
+    ∀ t,
+      (3519728384093 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (44276884111 / 400000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (7061879112973 / 64000000000000 : ℝ)).elim (fun h => tc_160 t h1 h h0)
+    (fun h => tc_161 t h.le h2 h0)
+
+lemma cov_163_165 :
+    ∀ t,
+      (7106723802547 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (7151568492121 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (3564573073667 / 32000000000000 : ℝ)).elim (fun h => tc_163 t h1 h h0)
+    (fun h => tc_164 t h.le h2 h0)
+
+lemma cov_162_165 :
+    ∀ t,
+      (44276884111 / 400000000000 : ℝ) ≤ t →
+        t ≤ (7151568492121 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (7106723802547 / 64000000000000 : ℝ)).elim (fun h => tc_162 t h1 h h0)
+    (fun h => cov_163_165 t h.le h2 h0)
+
+lemma cov_160_165 :
+    ∀ t,
+      (3519728384093 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (7151568492121 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (44276884111 / 400000000000 : ℝ)).elim (fun h => cov_160_162 t h1 h h0)
+    (fun h => cov_162_165 t h.le h2 h0)
+
+lemma cov_166_168 :
+    ∀ t,
+      (1793497709227 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (3609417763241 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1439282636339 / 12800000000000 : ℝ)).elim (fun h => tc_166 t h1 h h0)
+    (fun h => tc_167 t h.le h2 h0)
+
+lemma cov_165_168 :
+    ∀ t,
+      (7151568492121 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (3609417763241 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1793497709227 / 16000000000000 : ℝ)).elim (fun h => tc_165 t h1 h h0)
+    (fun h => cov_166_168 t h.le h2 h0)
+
+lemma cov_169_171 :
+    ∀ t,
+      (7241257871269 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (7286102560843 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (907960027007 / 8000000000000 : ℝ)).elim (fun h => tc_169 t h1 h h0)
+    (fun h => tc_170 t h.le h2 h0)
+
+lemma cov_168_171 :
+    ∀ t,
+      (3609417763241 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (7286102560843 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (7241257871269 / 64000000000000 : ℝ)).elim (fun h => tc_168 t h1 h h0)
+    (fun h => cov_169_171 t h.le h2 h0)
+
+lemma cov_165_171 :
+    ∀ t,
+      (7151568492121 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (7286102560843 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (3609417763241 / 32000000000000 : ℝ)).elim (fun h => cov_165_168 t h1 h h0)
+    (fun h => cov_168_171 t h.le h2 h0)
+
+lemma cov_160_171 :
+    ∀ t,
+      (3519728384093 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (7286102560843 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (7151568492121 / 64000000000000 : ℝ)).elim (fun h => cov_160_165 t h1 h h0)
+    (fun h => cov_165_171 t h.le h2 h0)
+
+lemma cov_149_171 :
+    ∀ t,
+      (3340349625797 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (7286102560843 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (3519728384093 / 32000000000000 : ℝ)).elim (fun h => cov_149_160 t h1 h h0)
+    (fun h => cov_160_171 t h.le h2 h0)
+
+lemma cov_128_171 :
+    ∀ t,
+      (1221767174613 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (7286102560843 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (3340349625797 / 32000000000000 : ℝ)).elim (fun h => cov_128_149 t h1 h h0)
+    (fun h => cov_149_171 t h.le h2 h0)
+
+lemma cov_85_171 :
+    ∀ t,
+      (742377785887 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (7286102560843 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1221767174613 / 16000000000000 : ℝ)).elim (fun h => cov_85_128 t h1 h h0)
+    (fun h => cov_128_171 t h.le h2 h0)
+
+lemma cov_0_171 :
+    ∀ t,
+      (0 : ℝ) ≤ t → t ≤ (7286102560843 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (742377785887 / 16000000000000 : ℝ)).elim (fun h => cov_0_85 t h1 h h0)
+    (fun h => cov_85_171 t h.le h2 h0)
+
+lemma cov_171_173 :
+    ∀ t,
+      (7286102560843 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (7330947250417 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (730852490563 / 6400000000000 : ℝ)).elim (fun h => tc_171 t h1 h h0)
+    (fun h => tc_172 t h.le h2 h0)
+
+lemma cov_174_176 :
+    ∀ t,
+      (1838342398801 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (465191185897 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (3699107142389 / 32000000000000 : ℝ)).elim (fun h => tc_174 t h1 h h0)
+    (fun h => tc_175 t h.le h2 h0)
+
+lemma cov_173_176 :
+    ∀ t,
+      (7330947250417 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (465191185897 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1838342398801 / 16000000000000 : ℝ)).elim (fun h => tc_173 t h1 h h0)
+    (fun h => cov_174_176 t h.le h2 h0)
+
+lemma cov_171_176 :
+    ∀ t,
+      (7286102560843 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (465191185897 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (7330947250417 / 64000000000000 : ℝ)).elim (fun h => cov_171_173 t h1 h h0)
+    (fun h => cov_173_176 t h.le h2 h0)
+
+lemma cov_176_178 :
+    ∀ t,
+      (465191185897 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (15065496707 / 128000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (3743951831963 / 32000000000000 : ℝ)).elim (fun h => tc_176 t h1 h h0)
+    (fun h => tc_177 t h.le h2 h0)
+
+lemma cov_179_181 :
+    ∀ t,
+      (3788796521537 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (3833641211111 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (952804716581 / 8000000000000 : ℝ)).elim (fun h => tc_179 t h1 h h0)
+    (fun h => tc_180 t h.le h2 h0)
+
+lemma cov_178_181 :
+    ∀ t,
+      (15065496707 / 128000000000 : ℝ) ≤ t →
+        t ≤ (3833641211111 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (3788796521537 / 32000000000000 : ℝ)).elim (fun h => tc_178 t h1 h h0)
+    (fun h => cov_179_181 t h.le h2 h0)
+
+lemma cov_176_181 :
+    ∀ t,
+      (465191185897 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (3833641211111 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (15065496707 / 128000000000 : ℝ)).elim (fun h => cov_176_178 t h1 h h0)
+    (fun h => cov_178_181 t h.le h2 h0)
+
+lemma cov_171_181 :
+    ∀ t,
+      (7286102560843 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (3833641211111 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (465191185897 / 4000000000000 : ℝ)).elim (fun h => cov_171_176 t h1 h h0)
+    (fun h => cov_176_181 t h.le h2 h0)
+
+lemma cov_181_183 :
+    ∀ t,
+      (3833641211111 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (121903382671 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1928031777949 / 16000000000000 : ℝ)).elim (fun h => tc_181 t h1 h h0)
+    (fun h => tc_182 t h.le h2 h0)
+
+lemma cov_184_186 :
+    ∀ t,
+      (1972876467523 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (2017721157097 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (199529881231 / 1600000000000 : ℝ)).elim (fun h => tc_184 t h1 h h0)
+    (fun h => tc_185 t h.le h2 h0)
+
+lemma cov_183_186 :
+    ∀ t,
+      (121903382671 / 1000000000000 : ℝ) ≤ t →
+        t ≤ (2017721157097 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1972876467523 / 16000000000000 : ℝ)).elim (fun h => tc_183 t h1 h h0)
+    (fun h => cov_184_186 t h.le h2 h0)
+
+lemma cov_181_186 :
+    ∀ t,
+      (3833641211111 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (2017721157097 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (121903382671 / 1000000000000 : ℝ)).elim (fun h => cov_181_183 t h1 h h0)
+    (fun h => cov_183_186 t h.le h2 h0)
+
+lemma cov_187_189 :
+    ∀ t,
+      (510035875471 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (266229110129 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1042494095729 / 8000000000000 : ℝ)).elim (fun h => tc_187 t h1 h h0)
+    (fun h => tc_188 t h.le h2 h0)
+
+lemma cov_186_189 :
+    ∀ t,
+      (2017721157097 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (266229110129 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (510035875471 / 4000000000000 : ℝ)).elim (fun h => tc_186 t h1 h h0)
+    (fun h => cov_187_189 t h.le h2 h0)
+
+lemma cov_190_192 :
+    ∀ t,
+      (110976113009 / 800000000000 : ℝ) ≤ t →
+        t ≤ (4675203313927 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (72162863729 / 500000000000 : ℝ)).elim (fun h => tc_190 t h1 h h0)
+    (fun h => tc_191 t h.le h2 h0)
+
+lemma cov_189_192 :
+    ∀ t,
+      (266229110129 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (4675203313927 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (110976113009 / 800000000000 : ℝ)).elim (fun h => tc_189 t h1 h h0)
+    (fun h => cov_190_192 t h.le h2 h0)
+
+lemma cov_186_192 :
+    ∀ t,
+      (2017721157097 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (4675203313927 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (266229110129 / 2000000000000 : ℝ)).elim (fun h => cov_186_189 t h1 h h0)
+    (fun h => cov_189_192 t h.le h2 h0)
+
+lemma cov_181_192 :
+    ∀ t,
+      (3833641211111 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (4675203313927 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (2017721157097 / 16000000000000 : ℝ)).elim (fun h => cov_181_186 t h1 h h0)
+    (fun h => cov_186_192 t h.le h2 h0)
+
+lemma cov_171_192 :
+    ∀ t,
+      (7286102560843 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (4675203313927 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (3833641211111 / 32000000000000 : ℝ)).elim (fun h => cov_171_181 t h1 h h0)
+    (fun h => cov_181_192 t h.le h2 h0)
+
+lemma cov_192_194 :
+    ∀ t,
+      (4675203313927 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (4788763384469 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (2365991674599 / 16000000000000 : ℝ)).elim (fun h => tc_192 t h1 h h0)
+    (fun h => tc_193 t h.le h2 h0)
+
+lemma cov_195_197 :
+    ∀ t,
+      (9634306804209 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (9747866874751 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (242277170987 / 1600000000000 : ℝ)).elim (fun h => tc_195 t h1 h h0)
+    (fun h => tc_196 t h.le h2 h0)
+
+lemma cov_194_197 :
+    ∀ t,
+      (4788763384469 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (9747866874751 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (9634306804209 / 64000000000000 : ℝ)).elim (fun h => tc_194 t h1 h h0)
+    (fun h => cov_195_197 t h.le h2 h0)
+
+lemma cov_192_197 :
+    ∀ t,
+      (4675203313927 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (9747866874751 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (4788763384469 / 32000000000000 : ℝ)).elim (fun h => cov_192_194 t h1 h h0)
+    (fun h => cov_194_197 t h.le h2 h0)
+
+lemma cov_197_199 :
+    ∀ t,
+      (9747866874751 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (9861426945293 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (4902323455011 / 32000000000000 : ℝ)).elim (fun h => tc_197 t h1 h h0)
+    (fun h => tc_198 t h.le h2 h0)
+
+lemma cov_200_202 :
+    ∀ t,
+      (2479551745141 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (1994997403167 / 12800000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (19893193996399 / 128000000000000 : ℝ)).elim (fun h => tc_200 t h1 h h0)
+    (fun h => tc_201 t h.le h2 h0)
+
+lemma cov_199_202 :
+    ∀ t,
+      (9861426945293 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (1994997403167 / 12800000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (2479551745141 / 16000000000000 : ℝ)).elim (fun h => tc_199 t h1 h h0)
+    (fun h => cov_200_202 t h.le h2 h0)
+
+lemma cov_197_202 :
+    ∀ t,
+      (9747866874751 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (1994997403167 / 12800000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (9861426945293 / 64000000000000 : ℝ)).elim (fun h => cov_197_199 t h1 h h0)
+    (fun h => cov_199_202 t h.le h2 h0)
+
+lemma cov_192_202 :
+    ∀ t,
+      (4675203313927 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (1994997403167 / 12800000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (9747866874751 / 64000000000000 : ℝ)).elim (fun h => cov_192_197 t h1 h h0)
+    (fun h => cov_197_202 t h.le h2 h0)
+
+lemma cov_202_204 :
+    ∀ t,
+      (1994997403167 / 12800000000000 : ℝ) ≤ t →
+        t ≤ (5015883525553 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (20006754066941 / 128000000000000 : ℝ)).elim (fun h => tc_202 t h1 h h0)
+    (fun h => tc_203 t h.le h2 h0)
+
+lemma cov_205_207 :
+    ∀ t,
+      (20120314137483 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (809354968321 / 5120000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (10088547086377 / 64000000000000 : ℝ)).elim (fun h => tc_205 t h1 h h0)
+    (fun h => tc_206 t h.le h2 h0)
+
+lemma cov_204_207 :
+    ∀ t,
+      (5015883525553 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (809354968321 / 5120000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (20120314137483 / 128000000000000 : ℝ)).elim (fun h => tc_204 t h1 h h0)
+    (fun h => cov_205_207 t h.le h2 h0)
+
+lemma cov_202_207 :
+    ∀ t,
+      (1994997403167 / 12800000000000 : ℝ) ≤ t →
+        t ≤ (809354968321 / 5120000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (5015883525553 / 32000000000000 : ℝ)).elim (fun h => cov_202_204 t h1 h h0)
+    (fun h => cov_204_207 t h.le h2 h0)
+
+lemma cov_208_210 :
+    ∀ t,
+      (634082945103 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (10202107156919 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (20347434278567 / 128000000000000 : ℝ)).elim (fun h => tc_208 t h1 h h0)
+    (fun h => tc_209 t h.le h2 h0)
+
+lemma cov_207_210 :
+    ∀ t,
+      (809354968321 / 5120000000000 : ℝ) ≤ t →
+        t ≤ (10202107156919 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (634082945103 / 4000000000000 : ℝ)).elim (fun h => tc_207 t h1 h h0)
+    (fun h => cov_208_210 t h.le h2 h0)
+
+lemma cov_211_213 :
+    ∀ t,
+      (20460994349109 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (20574554419651 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1025888719219 / 6400000000000 : ℝ)).elim (fun h => tc_211 t h1 h h0)
+    (fun h => tc_212 t h.le h2 h0)
+
+lemma cov_210_213 :
+    ∀ t,
+      (10202107156919 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (20574554419651 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (20460994349109 / 128000000000000 : ℝ)).elim (fun h => tc_210 t h1 h h0)
+    (fun h => cov_211_213 t h.le h2 h0)
+
+lemma cov_207_213 :
+    ∀ t,
+      (809354968321 / 5120000000000 : ℝ) ≤ t →
+        t ≤ (20574554419651 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (10202107156919 / 64000000000000 : ℝ)).elim (fun h => cov_207_210 t h1 h h0)
+    (fun h => cov_210_213 t h.le h2 h0)
+
+lemma cov_202_213 :
+    ∀ t,
+      (1994997403167 / 12800000000000 : ℝ) ≤ t →
+        t ≤ (20574554419651 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (809354968321 / 5120000000000 : ℝ)).elim (fun h => cov_202_207 t h1 h h0)
+    (fun h => cov_207_213 t h.le h2 h0)
+
+lemma cov_192_213 :
+    ∀ t,
+      (4675203313927 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (20574554419651 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1994997403167 / 12800000000000 : ℝ)).elim (fun h => cov_192_202 t h1 h h0)
+    (fun h => cov_202_213 t h.le h2 h0)
+
+lemma cov_171_213 :
+    ∀ t,
+      (7286102560843 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (20574554419651 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (4675203313927 / 32000000000000 : ℝ)).elim (fun h => cov_171_192 t h1 h h0)
+    (fun h => cov_192_213 t h.le h2 h0)
+
+lemma cov_213_215 :
+    ∀ t,
+      (20574554419651 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (20688114490193 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (10315667227461 / 64000000000000 : ℝ)).elim (fun h => tc_213 t h1 h h0)
+    (fun h => tc_214 t h.le h2 h0)
+
+lemma cov_216_218 :
+    ∀ t,
+      (2593111815683 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (10429227298003 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (4160334912147 / 25600000000000 : ℝ)).elim (fun h => tc_216 t h1 h h0)
+    (fun h => tc_217 t h.le h2 h0)
+
+lemma cov_215_218 :
+    ∀ t,
+      (20688114490193 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (10429227298003 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (2593111815683 / 16000000000000 : ℝ)).elim (fun h => tc_215 t h1 h h0)
+    (fun h => cov_216_218 t h.le h2 h0)
+
+lemma cov_213_218 :
+    ∀ t,
+      (20574554419651 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (10429227298003 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (20688114490193 / 128000000000000 : ℝ)).elim (fun h => cov_213_215 t h1 h h0)
+    (fun h => cov_215_218 t h.le h2 h0)
+
+lemma cov_218_220 :
+    ∀ t,
+      (10429227298003 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (5243003666637 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (20915234631277 / 128000000000000 : ℝ)).elim (fun h => tc_218 t h1 h h0)
+    (fun h => tc_219 t h.le h2 h0)
+
+lemma cov_221_223 :
+    ∀ t,
+      (21028794701819 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (21142354772361 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (2108557473709 / 12800000000000 : ℝ)).elim (fun h => tc_221 t h1 h h0)
+    (fun h => tc_222 t h.le h2 h0)
+
+lemma cov_220_223 :
+    ∀ t,
+      (5243003666637 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (21142354772361 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (21028794701819 / 128000000000000 : ℝ)).elim (fun h => tc_220 t h1 h h0)
+    (fun h => cov_221_223 t h.le h2 h0)
+
+lemma cov_218_223 :
+    ∀ t,
+      (10429227298003 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (21142354772361 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (5243003666637 / 32000000000000 : ℝ)).elim (fun h => cov_218_220 t h1 h h0)
+    (fun h => cov_220_223 t h.le h2 h0)
+
+lemma cov_213_223 :
+    ∀ t,
+      (20574554419651 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (21142354772361 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (10429227298003 / 64000000000000 : ℝ)).elim (fun h => cov_213_218 t h1 h h0)
+    (fun h => cov_218_223 t h.le h2 h0)
+
+lemma cov_223_225 :
+    ∀ t,
+      (21142354772361 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (10656347439087 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1324945925477 / 8000000000000 : ℝ)).elim (fun h => tc_223 t h1 h h0)
+    (fun h => tc_224 t h.le h2 h0)
+
+lemma cov_226_228 :
+    ∀ t,
+      (5356563737179 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (108266875449 / 640000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (10769907509629 / 64000000000000 : ℝ)).elim (fun h => tc_226 t h1 h h0)
+    (fun h => tc_227 t h.le h2 h0)
+
+lemma cov_225_228 :
+    ∀ t,
+      (10656347439087 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (108266875449 / 640000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (5356563737179 / 32000000000000 : ℝ)).elim (fun h => tc_225 t h1 h h0)
+    (fun h => cov_226_228 t h.le h2 h0)
+
+lemma cov_223_228 :
+    ∀ t,
+      (21142354772361 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (108266875449 / 640000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (10656347439087 / 64000000000000 : ℝ)).elim (fun h => cov_223_225 t h1 h h0)
+    (fun h => cov_225_228 t h.le h2 h0)
+
+lemma cov_229_231 :
+    ∀ t,
+      (10883467580171 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (10997027650713 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (5470123807721 / 32000000000000 : ℝ)).elim (fun h => tc_229 t h1 h h0)
+    (fun h => tc_230 t h.le h2 h0)
+
+lemma cov_228_231 :
+    ∀ t,
+      (108266875449 / 640000000000 : ℝ) ≤ t →
+        t ≤ (10997027650713 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (10883467580171 / 64000000000000 : ℝ)).elim (fun h => tc_228 t h1 h h0)
+    (fun h => cov_229_231 t h.le h2 h0)
+
+lemma cov_232_234 :
+    ∀ t,
+      (345431490187 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (2820231956767 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (5583683878263 / 32000000000000 : ℝ)).elim (fun h => tc_232 t h1 h h0)
+    (fun h => tc_233 t h.le h2 h0)
+
+lemma cov_231_234 :
+    ∀ t,
+      (10997027650713 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (2820231956767 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (345431490187 / 2000000000000 : ℝ)).elim (fun h => tc_231 t h1 h h0)
+    (fun h => cov_232_234 t h.le h2 h0)
+
+lemma cov_228_234 :
+    ∀ t,
+      (108266875449 / 640000000000 : ℝ) ≤ t →
+        t ≤ (2820231956767 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (10997027650713 / 64000000000000 : ℝ)).elim (fun h => cov_228_231 t h1 h h0)
+    (fun h => cov_231_234 t h.le h2 h0)
+
+lemma cov_223_234 :
+    ∀ t,
+      (21142354772361 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (2820231956767 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (108266875449 / 640000000000 : ℝ)).elim (fun h => cov_223_228 t h1 h h0)
+    (fun h => cov_228_234 t h.le h2 h0)
+
+lemma cov_213_234 :
+    ∀ t,
+      (20574554419651 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (2820231956767 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (21142354772361 / 128000000000000 : ℝ)).elim (fun h => cov_213_223 t h1 h h0)
+    (fun h => cov_223_234 t h.le h2 h0)
+
+lemma cov_234_236 :
+    ∀ t,
+      (2820231956767 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (1438505996019 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1139448789761 / 6400000000000 : ℝ)).elim (fun h => tc_234 t h1 h h0)
+    (fun h => tc_235 t h.le h2 h0)
+
+lemma cov_237_239 :
+    ∀ t,
+      (2933792027309 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (3047352097851 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (149528603129 / 800000000000 : ℝ)).elim (fun h => tc_237 t h1 h h0)
+    (fun h => tc_238 t h.le h2 h0)
+
+lemma cov_236_239 :
+    ∀ t,
+      (1438505996019 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (3047352097851 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (2933792027309 / 16000000000000 : ℝ)).elim (fun h => tc_236 t h1 h h0)
+    (fun h => cov_237_239 t h.le h2 h0)
+
+lemma cov_234_239 :
+    ∀ t,
+      (2820231956767 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (3047352097851 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1438505996019 / 8000000000000 : ℝ)).elim (fun h => cov_234_236 t h1 h h0)
+    (fun h => cov_236_239 t h.le h2 h0)
+
+lemma cov_240_242 :
+    ∀ t,
+      (1552066066561 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (3251812320751 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (201105762729 / 1000000000000 : ℝ)).elim (fun h => tc_240 t h1 h h0)
+    (fun h => tc_241 t h.le h2 h0)
+
+lemma cov_239_242 :
+    ∀ t,
+      (3047352097851 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (3251812320751 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1552066066561 / 8000000000000 : ℝ)).elim (fun h => tc_239 t h1 h h0)
+    (fun h => cov_240_242 t h.le h2 h0)
+
+lemma cov_243_245 :
+    ∀ t,
+      (1642966218919 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (6674225226937 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (132802102197 / 640000000000 : ℝ)).elim (fun h => tc_243 t h1 h h0)
+    (fun h => tc_244 t h.le h2 h0)
+
+lemma cov_242_245 :
+    ∀ t,
+      (3251812320751 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (6674225226937 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1642966218919 / 8000000000000 : ℝ)).elim (fun h => tc_242 t h1 h h0)
+    (fun h => cov_243_245 t h.le h2 h0)
+
+lemma cov_239_245 :
+    ∀ t,
+      (3047352097851 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (6674225226937 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (3251812320751 / 16000000000000 : ℝ)).elim (fun h => cov_239_242 t h1 h h0)
+    (fun h => cov_242_245 t h.le h2 h0)
+
+lemma cov_234_245 :
+    ∀ t,
+      (2820231956767 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (6674225226937 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (3047352097851 / 16000000000000 : ℝ)).elim (fun h => cov_234_239 t h1 h h0)
+    (fun h => cov_239_245 t h.le h2 h0)
+
+lemma cov_245_247 :
+    ∀ t,
+      (6674225226937 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (6742465461111 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (838543168003 / 4000000000000 : ℝ)).elim (fun h => tc_245 t h1 h h0)
+    (fun h => tc_246 t h.le h2 h0)
+
+lemma cov_248_250 :
+    ∀ t,
+      (3388292789099 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (1711206453093 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1362141139057 / 6400000000000 : ℝ)).elim (fun h => tc_248 t h1 h h0)
+    (fun h => tc_249 t h.le h2 h0)
+
+lemma cov_247_250 :
+    ∀ t,
+      (6742465461111 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (1711206453093 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (3388292789099 / 16000000000000 : ℝ)).elim (fun h => tc_247 t h1 h h0)
+    (fun h => cov_248_250 t h.le h2 h0)
+
+lemma cov_245_250 :
+    ∀ t,
+      (6674225226937 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (1711206453093 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (6742465461111 / 32000000000000 : ℝ)).elim (fun h => cov_245_247 t h1 h h0)
+    (fun h => cov_247_250 t h.le h2 h0)
+
+lemma cov_251_253 :
+    ∀ t,
+      (13723771741831 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (2758402395201 / 12800000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (6878945929459 / 32000000000000 : ℝ)).elim (fun h => tc_251 t h1 h h0)
+    (fun h => tc_252 t h.le h2 h0)
+
+lemma cov_250_253 :
+    ∀ t,
+      (1711206453093 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (2758402395201 / 12800000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (13723771741831 / 64000000000000 : ℝ)).elim (fun h => tc_250 t h1 h h0)
+    (fun h => cov_251_253 t h.le h2 h0)
+
+lemma cov_254_256 :
+    ∀ t,
+      (3456533023273 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (6947186163633 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (13860252210179 / 64000000000000 : ℝ)).elim (fun h => tc_254 t h1 h h0)
+    (fun h => tc_255 t h.le h2 h0)
+
+lemma cov_253_256 :
+    ∀ t,
+      (2758402395201 / 12800000000000 : ℝ) ≤ t →
+        t ≤ (6947186163633 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (3456533023273 / 16000000000000 : ℝ)).elim (fun h => tc_253 t h1 h h0)
+    (fun h => cov_254_256 t h.le h2 h0)
+
+lemma cov_250_256 :
+    ∀ t,
+      (1711206453093 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (6947186163633 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (2758402395201 / 12800000000000 : ℝ)).elim (fun h => cov_250_253 t h1 h h0)
+    (fun h => cov_253_256 t h.le h2 h0)
+
+lemma cov_245_256 :
+    ∀ t,
+      (6674225226937 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (6947186163633 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1711206453093 / 8000000000000 : ℝ)).elim (fun h => cov_245_250 t h1 h h0)
+    (fun h => cov_250_256 t h.le h2 h0)
+
+lemma cov_234_256 :
+    ∀ t,
+      (2820231956767 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (6947186163633 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (6674225226937 / 32000000000000 : ℝ)).elim (fun h => cov_234_245 t h1 h h0)
+    (fun h => cov_245_256 t h.le h2 h0)
+
+lemma cov_213_256 :
+    ∀ t,
+      (20574554419651 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (6947186163633 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (2820231956767 / 16000000000000 : ℝ)).elim (fun h => cov_213_234 t h1 h h0)
+    (fun h => cov_234_256 t h.le h2 h0)
+
+lemma cov_171_256 :
+    ∀ t,
+      (7286102560843 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (6947186163633 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (20574554419651 / 128000000000000 : ℝ)).elim (fun h => cov_171_213 t h1 h h0)
+    (fun h => cov_213_256 t h.le h2 h0)
+
+lemma cov_256_258 :
+    ∀ t,
+      (6947186163633 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (87266328509 / 400000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (13928492444353 / 64000000000000 : ℝ)).elim (fun h => tc_256 t h1 h h0)
+    (fun h => tc_257 t h.le h2 h0)
+
+lemma cov_259_261 :
+    ∀ t,
+      (13996732678527 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (14064972912701 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (7015426397807 / 32000000000000 : ℝ)).elim (fun h => tc_259 t h1 h h0)
+    (fun h => tc_260 t h.le h2 h0)
+
+lemma cov_258_261 :
+    ∀ t,
+      (87266328509 / 400000000000 : ℝ) ≤ t →
+        t ≤ (14064972912701 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (13996732678527 / 64000000000000 : ℝ)).elim (fun h => tc_258 t h1 h h0)
+    (fun h => cov_259_261 t h.le h2 h0)
+
+lemma cov_256_261 :
+    ∀ t,
+      (6947186163633 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (14064972912701 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (87266328509 / 400000000000 : ℝ)).elim (fun h => cov_256_258 t h1 h h0)
+    (fun h => cov_258_261 t h.le h2 h0)
+
+lemma cov_261_263 :
+    ∀ t,
+      (14064972912701 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (4522628207 / 20480000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (3524773257447 / 16000000000000 : ℝ)).elim (fun h => tc_261 t h1 h h0)
+    (fun h => tc_262 t h.le h2 h0)
+
+lemma cov_264_266 :
+    ∀ t,
+      (7083666631981 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (1779446687267 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (14201453381049 / 64000000000000 : ℝ)).elim (fun h => tc_264 t h1 h h0)
+    (fun h => tc_265 t h.le h2 h0)
+
+lemma cov_263_266 :
+    ∀ t,
+      (4522628207 / 20480000000 : ℝ) ≤ t →
+        t ≤ (1779446687267 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (7083666631981 / 32000000000000 : ℝ)).elim (fun h => tc_263 t h1 h h0)
+    (fun h => cov_264_266 t h.le h2 h0)
+
+lemma cov_261_266 :
+    ∀ t,
+      (14064972912701 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (1779446687267 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (4522628207 / 20480000000 : ℝ)).elim (fun h => cov_261_263 t h1 h h0)
+    (fun h => cov_263_266 t h.le h2 h0)
+
+lemma cov_256_266 :
+    ∀ t,
+      (6947186163633 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (1779446687267 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (14064972912701 / 64000000000000 : ℝ)).elim (fun h => cov_256_261 t h1 h h0)
+    (fun h => cov_261_266 t h.le h2 h0)
+
+lemma cov_266_268 :
+    ∀ t,
+      (1779446687267 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (1430381373231 / 6400000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (14269693615223 / 64000000000000 : ℝ)).elim (fun h => tc_266 t h1 h h0)
+    (fun h => tc_267 t h.le h2 h0)
+
+lemma cov_269_271 :
+    ∀ t,
+      (14337933849397 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (14406174083571 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (3593013491621 / 16000000000000 : ℝ)).elim (fun h => tc_269 t h1 h h0)
+    (fun h => tc_270 t h.le h2 h0)
+
+lemma cov_268_271 :
+    ∀ t,
+      (1430381373231 / 6400000000000 : ℝ) ≤ t →
+        t ≤ (14406174083571 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (14337933849397 / 64000000000000 : ℝ)).elim (fun h => tc_268 t h1 h h0)
+    (fun h => cov_269_271 t h.le h2 h0)
+
+lemma cov_266_271 :
+    ∀ t,
+      (1779446687267 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (14406174083571 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1430381373231 / 6400000000000 : ℝ)).elim (fun h => cov_266_268 t h1 h h0)
+    (fun h => cov_268_271 t h.le h2 h0)
+
+lemma cov_272_274 :
+    ∀ t,
+      (7220147100329 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (906783402177 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (2894882863549 / 12800000000000 : ℝ)).elim (fun h => tc_272 t h1 h h0)
+    (fun h => tc_273 t h.le h2 h0)
+
+lemma cov_271_274 :
+    ∀ t,
+      (14406174083571 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (906783402177 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (7220147100329 / 32000000000000 : ℝ)).elim (fun h => tc_271 t h1 h h0)
+    (fun h => cov_272_274 t h.le h2 h0)
+
+lemma cov_275_277 :
+    ∀ t,
+      (14542654551919 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (732250745159 / 3200000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (7288387334503 / 32000000000000 : ℝ)).elim (fun h => tc_275 t h1 h h0)
+    (fun h => tc_276 t h.le h2 h0)
+
+lemma cov_274_277 :
+    ∀ t,
+      (906783402177 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (732250745159 / 3200000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (14542654551919 / 64000000000000 : ℝ)).elim (fun h => tc_274 t h1 h h0)
+    (fun h => cov_275_277 t h.le h2 h0)
+
+lemma cov_271_277 :
+    ∀ t,
+      (14406174083571 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (732250745159 / 3200000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (906783402177 / 4000000000000 : ℝ)).elim (fun h => cov_271_274 t h1 h h0)
+    (fun h => cov_274_277 t h.le h2 h0)
+
+lemma cov_266_277 :
+    ∀ t,
+      (1779446687267 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (732250745159 / 3200000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (14406174083571 / 64000000000000 : ℝ)).elim (fun h => cov_266_271 t h1 h h0)
+    (fun h => cov_271_277 t h.le h2 h0)
+
+lemma cov_256_277 :
+    ∀ t,
+      (6947186163633 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (732250745159 / 3200000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1779446687267 / 8000000000000 : ℝ)).elim (fun h => cov_256_266 t h1 h h0)
+    (fun h => cov_266_277 t h.le h2 h0)
+
+lemma cov_277_279 :
+    ∀ t,
+      (732250745159 / 3200000000000 : ℝ) ≤ t →
+        t ≤ (1847686921441 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (7356627568677 / 32000000000000 : ℝ)).elim (fun h => tc_277 t h1 h h0)
+    (fun h => tc_278 t h.le h2 h0)
+
+lemma cov_280_282 :
+    ∀ t,
+      (7424867802851 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (299724321481 / 1280000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (3729493959969 / 16000000000000 : ℝ)).elim (fun h => tc_280 t h1 h h0)
+    (fun h => tc_281 t h.le h2 h0)
+
+lemma cov_279_282 :
+    ∀ t,
+      (1847686921441 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (299724321481 / 1280000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (7424867802851 / 32000000000000 : ℝ)).elim (fun h => tc_279 t h1 h h0)
+    (fun h => cov_280_282 t h.le h2 h0)
+
+lemma cov_277_282 :
+    ∀ t,
+      (732250745159 / 3200000000000 : ℝ) ≤ t →
+        t ≤ (299724321481 / 1280000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1847686921441 / 8000000000000 : ℝ)).elim (fun h => cov_277_279 t h1 h h0)
+    (fun h => cov_279_282 t h.le h2 h0)
+
+lemma cov_283_285 :
+    ∀ t,
+      (29403234977 / 125000000000 : ℝ) ≤ t →
+        t ≤ (3797734194143 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (7561348271199 / 32000000000000 : ℝ)).elim (fun h => tc_283 t h1 h h0)
+    (fun h => tc_284 t h.le h2 h0)
+
+lemma cov_282_285 :
+    ∀ t,
+      (299724321481 / 1280000000000 : ℝ) ≤ t →
+        t ≤ (3797734194143 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (29403234977 / 125000000000 : ℝ)).elim (fun h => tc_282 t h1 h h0)
+    (fun h => cov_283_285 t h.le h2 h0)
+
+lemma cov_286_288 :
+    ∀ t,
+      (383185431123 / 1600000000000 : ℝ) ≤ t →
+        t ≤ (975023636351 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (3865974428317 / 16000000000000 : ℝ)).elim (fun h => tc_286 t h1 h h0)
+    (fun h => tc_287 t h.le h2 h0)
+
+lemma cov_285_288 :
+    ∀ t,
+      (3797734194143 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (975023636351 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (383185431123 / 1600000000000 : ℝ)).elim (fun h => tc_285 t h1 h h0)
+    (fun h => cov_286_288 t h.le h2 h0)
+
+lemma cov_282_288 :
+    ∀ t,
+      (299724321481 / 1280000000000 : ℝ) ≤ t →
+        t ≤ (975023636351 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (3797734194143 / 16000000000000 : ℝ)).elim (fun h => cov_282_285 t h1 h h0)
+    (fun h => cov_285_288 t h.le h2 h0)
+
+lemma cov_277_288 :
+    ∀ t,
+      (732250745159 / 3200000000000 : ℝ) ≤ t →
+        t ≤ (975023636351 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (299724321481 / 1280000000000 : ℝ)).elim (fun h => cov_277_282 t h1 h h0)
+    (fun h => cov_282_288 t h.le h2 h0)
+
+lemma cov_288_290 :
+    ∀ t,
+      (975023636351 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (1984167389789 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (3934214662491 / 16000000000000 : ℝ)).elim (fun h => tc_288 t h1 h h0)
+    (fun h => tc_289 t h.le h2 h0)
+
+lemma cov_291_293 :
+    ∀ t,
+      (504571876719 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (41730554821 / 160000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (2052407623963 / 8000000000000 : ℝ)).elim (fun h => tc_291 t h1 h h0)
+    (fun h => tc_292 t h.le h2 h0)
+
+lemma cov_290_293 :
+    ∀ t,
+      (1984167389789 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (41730554821 / 160000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (504571876719 / 2000000000000 : ℝ)).elim (fun h => tc_290 t h1 h h0)
+    (fun h => cov_291_293 t h.le h2 h0)
+
+lemma cov_288_293 :
+    ∀ t,
+      (975023636351 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (41730554821 / 160000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1984167389789 / 8000000000000 : ℝ)).elim (fun h => cov_288_290 t h1 h h0)
+    (fun h => cov_290_293 t h.le h2 h0)
+
+lemma cov_294_296 :
+    ∀ t,
+      (269345996903 / 1000000000000 : ℝ) ≤ t →
+        t ≤ (4387279507701 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (8696815458149 / 32000000000000 : ℝ)).elim (fun h => tc_294 t h1 h h0)
+    (fun h => tc_295 t h.le h2 h0)
+
+lemma cov_293_296 :
+    ∀ t,
+      (41730554821 / 160000000000 : ℝ) ≤ t →
+        t ≤ (4387279507701 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (269345996903 / 1000000000000 : ℝ)).elim (fun h => tc_293 t h1 h h0)
+    (fun h => cov_294_296 t h.le h2 h0)
+
+lemma cov_297_299 :
+    ∀ t,
+      (1770460514531 / 6400000000000 : ℝ) ≤ t →
+        t ≤ (2232511532477 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (17782348702563 / 64000000000000 : ℝ)).elim (fun h => tc_297 t h1 h h0)
+    (fun h => tc_298 t h.le h2 h0)
+
+lemma cov_296_299 :
+    ∀ t,
+      (4387279507701 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (2232511532477 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1770460514531 / 6400000000000 : ℝ)).elim (fun h => tc_296 t h1 h h0)
+    (fun h => cov_297_299 t h.le h2 h0)
+
+lemma cov_293_299 :
+    ∀ t,
+      (41730554821 / 160000000000 : ℝ) ≤ t →
+        t ≤ (2232511532477 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (4387279507701 / 16000000000000 : ℝ)).elim (fun h => cov_293_296 t h1 h h0)
+    (fun h => cov_296_299 t h.le h2 h0)
+
+lemma cov_288_299 :
+    ∀ t,
+      (975023636351 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (2232511532477 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (41730554821 / 160000000000 : ℝ)).elim (fun h => cov_288_293 t h1 h h0)
+    (fun h => cov_293_299 t h.le h2 h0)
+
+lemma cov_277_299 :
+    ∀ t,
+      (732250745159 / 3200000000000 : ℝ) ≤ t →
+        t ≤ (2232511532477 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (975023636351 / 4000000000000 : ℝ)).elim (fun h => cov_277_288 t h1 h h0)
+    (fun h => cov_288_299 t h.le h2 h0)
+
+lemma cov_256_299 :
+    ∀ t,
+      (6947186163633 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (2232511532477 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (732250745159 / 3200000000000 : ℝ)).elim (fun h => cov_256_277 t h1 h h0)
+    (fun h => cov_277_299 t h.le h2 h0)
+
+lemma cov_299_301 :
+    ∀ t,
+      (2232511532477 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (9007789687161 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (17937835817069 / 64000000000000 : ℝ)).elim (fun h => tc_299 t h1 h h0)
+    (fun h => tc_300 t h.le h2 h0)
+
+lemma cov_302_304 :
+    ∀ t,
+      (723732917263 / 2560000000000 : ℝ) ≤ t →
+        t ≤ (36419876534909 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (4542766622207 / 16000000000000 : ℝ)).elim (fun h => tc_302 t h1 h h0)
+    (fun h => tc_303 t h.le h2 h0)
+
+lemma cov_301_304 :
+    ∀ t,
+      (9007789687161 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (36419876534909 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (723732917263 / 2560000000000 : ℝ)).elim (fun h => tc_301 t h1 h h0)
+    (fun h => cov_302_304 t h.le h2 h0)
+
+lemma cov_299_304 :
+    ∀ t,
+      (2232511532477 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (36419876534909 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (9007789687161 / 32000000000000 : ℝ)).elim (fun h => cov_299_301 t h1 h h0)
+    (fun h => cov_301_304 t h.le h2 h0)
+
+lemma cov_304_306 :
+    ∀ t,
+      (36419876534909 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (7315072729883 / 25600000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (18248810046081 / 64000000000000 : ℝ)).elim (fun h => tc_304 t h1 h h0)
+    (fun h => tc_305 t h.le h2 h0)
+
+lemma cov_307_309 :
+    ∀ t,
+      (9163276801667 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (18404297160587 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (36730850763921 / 128000000000000 : ℝ)).elim (fun h => tc_307 t h1 h h0)
+    (fun h => tc_308 t h.le h2 h0)
+
+lemma cov_306_309 :
+    ∀ t,
+      (7315072729883 / 25600000000000 : ℝ) ≤ t →
+        t ≤ (18404297160587 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (9163276801667 / 32000000000000 : ℝ)).elim (fun h => tc_306 t h1 h h0)
+    (fun h => cov_307_309 t h.le h2 h0)
+
+lemma cov_304_309 :
+    ∀ t,
+      (36419876534909 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (18404297160587 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (7315072729883 / 25600000000000 : ℝ)).elim (fun h => cov_304_306 t h1 h h0)
+    (fun h => cov_306_309 t h.le h2 h0)
+
+lemma cov_299_309 :
+    ∀ t,
+      (2232511532477 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (18404297160587 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (36419876534909 / 128000000000000 : ℝ)).elim (fun h => cov_299_304 t h1 h h0)
+    (fun h => cov_304_309 t h.le h2 h0)
+
+lemma cov_309_311 :
+    ∀ t,
+      (18404297160587 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (231025508973 / 800000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (36886337878427 / 128000000000000 : ℝ)).elim (fun h => tc_309 t h1 h h0)
+    (fun h => tc_310 t h.le h2 h0)
+
+lemma cov_312_314 :
+    ∀ t,
+      (37041824992933 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (37197312107439 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (18559784275093 / 64000000000000 : ℝ)).elim (fun h => tc_312 t h1 h h0)
+    (fun h => tc_313 t h.le h2 h0)
+
+lemma cov_311_314 :
+    ∀ t,
+      (231025508973 / 800000000000 : ℝ) ≤ t →
+        t ≤ (37197312107439 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (37041824992933 / 128000000000000 : ℝ)).elim (fun h => tc_311 t h1 h h0)
+    (fun h => cov_312_314 t h.le h2 h0)
+
+lemma cov_309_314 :
+    ∀ t,
+      (18404297160587 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (37197312107439 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (231025508973 / 800000000000 : ℝ)).elim (fun h => cov_309_311 t h1 h h0)
+    (fun h => cov_311_314 t h.le h2 h0)
+
+lemma cov_315_317 :
+    ∀ t,
+      (9318763916173 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (18715271389599 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (7470559844389 / 25600000000000 : ℝ)).elim (fun h => tc_315 t h1 h h0)
+    (fun h => tc_316 t h.le h2 h0)
+
+lemma cov_314_317 :
+    ∀ t,
+      (37197312107439 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (18715271389599 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (9318763916173 / 32000000000000 : ℝ)).elim (fun h => tc_314 t h1 h h0)
+    (fun h => cov_315_317 t h.le h2 h0)
+
+lemma cov_318_320 :
+    ∀ t,
+      (37508286336451 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (37663773450957 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (4698253736713 / 16000000000000 : ℝ)).elim (fun h => tc_318 t h1 h h0)
+    (fun h => tc_319 t h.le h2 h0)
+
+lemma cov_317_320 :
+    ∀ t,
+      (18715271389599 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (37663773450957 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (37508286336451 / 128000000000000 : ℝ)).elim (fun h => tc_317 t h1 h h0)
+    (fun h => cov_318_320 t h.le h2 h0)
+
+lemma cov_314_320 :
+    ∀ t,
+      (37197312107439 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (37663773450957 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (18715271389599 / 64000000000000 : ℝ)).elim (fun h => cov_314_317 t h1 h h0)
+    (fun h => cov_317_320 t h.le h2 h0)
+
+lemma cov_309_320 :
+    ∀ t,
+      (18404297160587 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (37663773450957 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (37197312107439 / 128000000000000 : ℝ)).elim (fun h => cov_309_314 t h1 h h0)
+    (fun h => cov_314_320 t h.le h2 h0)
+
+lemma cov_299_320 :
+    ∀ t,
+      (2232511532477 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (37663773450957 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (18404297160587 / 64000000000000 : ℝ)).elim (fun h => cov_299_309 t h1 h h0)
+    (fun h => cov_309_320 t h.le h2 h0)
+
+lemma cov_320_322 :
+    ∀ t,
+      (37663773450957 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (37819260565463 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (3774151700821 / 12800000000000 : ℝ)).elim (fun h => tc_320 t h1 h h0)
+    (fun h => tc_321 t h.le h2 h0)
+
+lemma cov_323_325 :
+    ∀ t,
+      (9474251030679 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (19026245618611 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (37974747679969 / 128000000000000 : ℝ)).elim (fun h => tc_323 t h1 h h0)
+    (fun h => tc_324 t h.le h2 h0)
+
+lemma cov_322_325 :
+    ∀ t,
+      (37819260565463 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (19026245618611 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (9474251030679 / 32000000000000 : ℝ)).elim (fun h => tc_322 t h1 h h0)
+    (fun h => cov_323_325 t h.le h2 h0)
+
+lemma cov_320_325 :
+    ∀ t,
+      (37663773450957 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (19026245618611 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (37819260565463 / 128000000000000 : ℝ)).elim (fun h => cov_320_322 t h1 h h0)
+    (fun h => cov_322_325 t h.le h2 h0)
+
+lemma cov_326_328 :
+    ∀ t,
+      (1525209391779 / 5120000000000 : ℝ) ≤ t →
+        t ≤ (38285721908981 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (2387998646983 / 8000000000000 : ℝ)).elim (fun h => tc_326 t h1 h h0)
+    (fun h => tc_327 t h.le h2 h0)
+
+lemma cov_325_328 :
+    ∀ t,
+      (19026245618611 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (38285721908981 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1525209391779 / 5120000000000 : ℝ)).elim (fun h => tc_325 t h1 h h0)
+    (fun h => cov_326_328 t h.le h2 h0)
+
+lemma cov_329_331 :
+    ∀ t,
+      (19181732733117 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (1925947629037 / 6400000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (38441209023487 / 128000000000000 : ℝ)).elim (fun h => tc_329 t h1 h h0)
+    (fun h => tc_330 t h.le h2 h0)
+
+lemma cov_328_331 :
+    ∀ t,
+      (38285721908981 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (1925947629037 / 6400000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (19181732733117 / 64000000000000 : ℝ)).elim (fun h => tc_328 t h1 h h0)
+    (fun h => cov_329_331 t h.le h2 h0)
+
+lemma cov_325_331 :
+    ∀ t,
+      (19026245618611 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (1925947629037 / 6400000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (38285721908981 / 128000000000000 : ℝ)).elim (fun h => cov_325_328 t h1 h h0)
+    (fun h => cov_328_331 t h.le h2 h0)
+
+lemma cov_320_331 :
+    ∀ t,
+      (37663773450957 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (1925947629037 / 6400000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (19026245618611 / 64000000000000 : ℝ)).elim (fun h => cov_320_325 t h1 h h0)
+    (fun h => cov_325_331 t h.le h2 h0)
+
+lemma cov_331_333 :
+    ∀ t,
+      (1925947629037 / 6400000000000 : ℝ) ≤ t →
+        t ≤ (4853740851219 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (19337219847623 / 64000000000000 : ℝ)).elim (fun h => tc_331 t h1 h h0)
+    (fun h => tc_332 t h.le h2 h0)
+
+lemma cov_334_336 :
+    ∀ t,
+      (19492706962129 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (3929638815327 / 12800000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (9785225259691 / 32000000000000 : ℝ)).elim (fun h => tc_334 t h1 h h0)
+    (fun h => tc_335 t h.le h2 h0)
+
+lemma cov_333_336 :
+    ∀ t,
+      (4853740851219 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (3929638815327 / 12800000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (19492706962129 / 64000000000000 : ℝ)).elim (fun h => tc_333 t h1 h h0)
+    (fun h => cov_334_336 t h.le h2 h0)
+
+lemma cov_331_336 :
+    ∀ t,
+      (1925947629037 / 6400000000000 : ℝ) ≤ t →
+        t ≤ (3929638815327 / 12800000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (4853740851219 / 16000000000000 : ℝ)).elim (fun h => cov_331_333 t h1 h h0)
+    (fun h => cov_333_336 t h.le h2 h0)
+
+lemma cov_337_339 :
+    ∀ t,
+      (616435551059 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (9940712374197 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (19803681191141 / 64000000000000 : ℝ)).elim (fun h => tc_337 t h1 h h0)
+    (fun h => tc_338 t h.le h2 h0)
+
+lemma cov_336_339 :
+    ∀ t,
+      (3929638815327 / 12800000000000 : ℝ) ≤ t →
+        t ≤ (9940712374197 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (616435551059 / 2000000000000 : ℝ)).elim (fun h => tc_336 t h1 h h0)
+    (fun h => cov_337_339 t h.le h2 h0)
+
+lemma cov_340_342 :
+    ∀ t,
+      (200369118629 / 640000000000 : ℝ) ≤ t →
+        t ≤ (2543485761489 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (10096199488703 / 32000000000000 : ℝ)).elim (fun h => tc_340 t h1 h h0)
+    (fun h => tc_341 t h.le h2 h0)
+
+lemma cov_339_342 :
+    ∀ t,
+      (9940712374197 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (2543485761489 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (200369118629 / 640000000000 : ℝ)).elim (fun h => tc_339 t h1 h h0)
+    (fun h => cov_340_342 t h.le h2 h0)
+
+lemma cov_336_342 :
+    ∀ t,
+      (3929638815327 / 12800000000000 : ℝ) ≤ t →
+        t ≤ (2543485761489 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (9940712374197 / 32000000000000 : ℝ)).elim (fun h => cov_336_339 t h1 h h0)
+    (fun h => cov_339_342 t h.le h2 h0)
+
+lemma cov_331_342 :
+    ∀ t,
+      (1925947629037 / 6400000000000 : ℝ) ≤ t →
+        t ≤ (2543485761489 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (3929638815327 / 12800000000000 : ℝ)).elim (fun h => cov_331_336 t h1 h h0)
+    (fun h => cov_336_342 t h.le h2 h0)
+
+lemma cov_320_342 :
+    ∀ t,
+      (37663773450957 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (2543485761489 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1925947629037 / 6400000000000 : ℝ)).elim (fun h => cov_320_331 t h1 h h0)
+    (fun h => cov_331_342 t h.le h2 h0)
+
+lemma cov_299_342 :
+    ∀ t,
+      (2232511532477 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (2543485761489 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (37663773450957 / 128000000000000 : ℝ)).elim (fun h => cov_299_320 t h1 h h0)
+    (fun h => cov_320_342 t h.le h2 h0)
+
+lemma cov_256_342 :
+    ∀ t,
+      (6947186163633 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (2543485761489 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (2232511532477 / 8000000000000 : ℝ)).elim (fun h => cov_256_299 t h1 h h0)
+    (fun h => cov_299_342 t h.le h2 h0)
+
+lemma cov_171_342 :
+    ∀ t,
+      (7286102560843 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (2543485761489 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (6947186163633 / 32000000000000 : ℝ)).elim (fun h => cov_171_256 t h1 h h0)
+    (fun h => cov_256_342 t h.le h2 h0)
+
+lemma cov_0_342 :
+    ∀ t,
+      (0 : ℝ) ≤ t → t ≤ (2543485761489 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (7286102560843 / 64000000000000 : ℝ)).elim (fun h => cov_0_171 t h1 h h0)
+    (fun h => cov_171_342 t h.le h2 h0)
+
+lemma cov_342_344 :
+    ∀ t,
+      (2543485761489 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (5164715080231 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (10251686603209 / 32000000000000 : ℝ)).elim (fun h => tc_342 t h1 h h0)
+    (fun h => tc_343 t h.le h2 h0)
+
+lemma cov_345_347 :
+    ∀ t,
+      (1310614659371 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (539794575199 / 1600000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (5320202194737 / 16000000000000 : ℝ)).elim (fun h => tc_345 t h1 h h0)
+    (fun h => tc_346 t h.le h2 h0)
+
+lemma cov_344_347 :
+    ∀ t,
+      (5164715080231 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (539794575199 / 1600000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1310614659371 / 4000000000000 : ℝ)).elim (fun h => tc_344 t h1 h h0)
+    (fun h => cov_345_347 t h.le h2 h0)
+
+lemma cov_342_347 :
+    ∀ t,
+      (2543485761489 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (539794575199 / 1600000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (5164715080231 / 16000000000000 : ℝ)).elim (fun h => cov_342_344 t h1 h h0)
+    (fun h => cov_344_347 t h.le h2 h0)
+
+lemma cov_347_349 :
+    ∀ t,
+      (539794575199 / 1600000000000 : ℝ) ≤ t →
+        t ≤ (86772388539 / 250000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (5475689309243 / 16000000000000 : ℝ)).elim (fun h => tc_347 t h1 h h0)
+    (fun h => tc_348 t h.le h2 h0)
+
+lemma cov_350_352 :
+    ∀ t,
+      (11190582883251 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (11358017183769 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1127430003351 / 3200000000000 : ℝ)).elim (fun h => tc_350 t h1 h h0)
+    (fun h => tc_351 t h.le h2 h0)
+
+lemma cov_349_352 :
+    ∀ t,
+      (86772388539 / 250000000000 : ℝ) ≤ t →
+        t ≤ (11358017183769 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (11190582883251 / 32000000000000 : ℝ)).elim (fun h => tc_349 t h1 h h0)
+    (fun h => cov_350_352 t h.le h2 h0)
+
+lemma cov_347_352 :
+    ∀ t,
+      (539794575199 / 1600000000000 : ℝ) ≤ t →
+        t ≤ (11358017183769 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (86772388539 / 250000000000 : ℝ)).elim (fun h => cov_347_349 t h1 h h0)
+    (fun h => cov_349_352 t h.le h2 h0)
+
+lemma cov_342_352 :
+    ∀ t,
+      (2543485761489 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (11358017183769 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (539794575199 / 1600000000000 : ℝ)).elim (fun h => cov_342_347 t h1 h h0)
+    (fun h => cov_347_352 t h.le h2 h0)
+
+lemma cov_352_354 :
+    ∀ t,
+      (11358017183769 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (2860433583507 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (22799751517797 / 64000000000000 : ℝ)).elim (fun h => tc_352 t h1 h h0)
+    (fun h => tc_353 t h.le h2 h0)
+
+lemma cov_355_357 :
+    ∀ t,
+      (4593437163663 / 12800000000000 : ℝ) ≤ t →
+        t ≤ (23134620118833 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (11525451484287 / 32000000000000 : ℝ)).elim (fun h => tc_355 t h1 h h0)
+    (fun h => tc_356 t h.le h2 h0)
+
+lemma cov_354_357 :
+    ∀ t,
+      (2860433583507 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (23134620118833 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (4593437163663 / 12800000000000 : ℝ)).elim (fun h => tc_354 t h1 h h0)
+    (fun h => cov_355_357 t h.le h2 h0)
+
+lemma cov_352_357 :
+    ∀ t,
+      (11358017183769 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (23134620118833 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (2860433583507 / 8000000000000 : ℝ)).elim (fun h => cov_352_354 t h1 h h0)
+    (fun h => cov_354_357 t h.le h2 h0)
+
+lemma cov_358_360 :
+    ∀ t,
+      (5804584317273 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (23302054419351 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (46520391688443 / 128000000000000 : ℝ)).elim (fun h => tc_358 t h1 h h0)
+    (fun h => tc_359 t h.le h2 h0)
+
+lemma cov_357_360 :
+    ∀ t,
+      (23134620118833 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (23302054419351 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (5804584317273 / 16000000000000 : ℝ)).elim (fun h => tc_357 t h1 h h0)
+    (fun h => cov_358_360 t h.le h2 h0)
+
+lemma cov_361_363 :
+    ∀ t,
+      (46687825988961 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (46855260289479 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (2338577156961 / 6400000000000 : ℝ)).elim (fun h => tc_361 t h1 h h0)
+    (fun h => tc_362 t h.le h2 h0)
+
+lemma cov_360_363 :
+    ∀ t,
+      (23302054419351 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (46855260289479 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (46687825988961 / 128000000000000 : ℝ)).elim (fun h => tc_360 t h1 h h0)
+    (fun h => cov_361_363 t h.le h2 h0)
+
+lemma cov_357_363 :
+    ∀ t,
+      (23134620118833 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (46855260289479 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (23302054419351 / 64000000000000 : ℝ)).elim (fun h => cov_357_360 t h1 h h0)
+    (fun h => cov_360_363 t h.le h2 h0)
+
+lemma cov_352_363 :
+    ∀ t,
+      (11358017183769 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (46855260289479 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (23134620118833 / 64000000000000 : ℝ)).elim (fun h => cov_352_357 t h1 h h0)
+    (fun h => cov_357_363 t h.le h2 h0)
+
+lemma cov_342_363 :
+    ∀ t,
+      (2543485761489 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (46855260289479 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (11358017183769 / 32000000000000 : ℝ)).elim (fun h => cov_342_352 t h1 h h0)
+    (fun h => cov_352_363 t h.le h2 h0)
+
+lemma cov_363_365 :
+    ∀ t,
+      (46855260289479 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (47022694589997 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (23469488719869 / 64000000000000 : ℝ)).elim (fun h => tc_363 t h1 h h0)
+    (fun h => tc_364 t h.le h2 h0)
+
+lemma cov_366_368 :
+    ∀ t,
+      (1472075366883 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (23636923020387 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (9438025778103 / 25600000000000 : ℝ)).elim (fun h => tc_366 t h1 h h0)
+    (fun h => tc_367 t h.le h2 h0)
+
+lemma cov_365_368 :
+    ∀ t,
+      (47022694589997 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (23636923020387 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1472075366883 / 4000000000000 : ℝ)).elim (fun h => tc_365 t h1 h h0)
+    (fun h => cov_366_368 t h.le h2 h0)
+
+lemma cov_363_368 :
+    ∀ t,
+      (46855260289479 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (23636923020387 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (47022694589997 / 128000000000000 : ℝ)).elim (fun h => cov_363_365 t h1 h h0)
+    (fun h => cov_365_368 t h.le h2 h0)
+
+lemma cov_368_370 :
+    ∀ t,
+      (23636923020387 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (11860320085323 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (47357563191033 / 128000000000000 : ℝ)).elim (fun h => tc_368 t h1 h h0)
+    (fun h => tc_369 t h.le h2 h0)
+
+lemma cov_371_373 :
+    ∀ t,
+      (47524997491551 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (47692431792069 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (4760871464181 / 12800000000000 : ℝ)).elim (fun h => tc_371 t h1 h h0)
+    (fun h => tc_372 t h.le h2 h0)
+
+lemma cov_370_373 :
+    ∀ t,
+      (11860320085323 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (47692431792069 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (47524997491551 / 128000000000000 : ℝ)).elim (fun h => tc_370 t h1 h h0)
+    (fun h => cov_371_373 t h.le h2 h0)
+
+lemma cov_368_373 :
+    ∀ t,
+      (23636923020387 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (47692431792069 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (11860320085323 / 32000000000000 : ℝ)).elim (fun h => cov_368_370 t h1 h h0)
+    (fun h => cov_370_373 t h.le h2 h0)
+
+lemma cov_363_373 :
+    ∀ t,
+      (46855260289479 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (47692431792069 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (23636923020387 / 64000000000000 : ℝ)).elim (fun h => cov_363_368 t h1 h h0)
+    (fun h => cov_368_373 t h.le h2 h0)
+
+lemma cov_373_375 :
+    ∀ t,
+      (47692431792069 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (47859866092587 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (5972018617791 / 16000000000000 : ℝ)).elim (fun h => tc_373 t h1 h h0)
+    (fun h => tc_374 t h.le h2 h0)
+
+lemma cov_376_378 :
+    ∀ t,
+      (23971791621423 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (12027754385841 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (9605460078621 / 25600000000000 : ℝ)).elim (fun h => tc_376 t h1 h h0)
+    (fun h => tc_377 t h.le h2 h0)
+
+lemma cov_375_378 :
+    ∀ t,
+      (47859866092587 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (12027754385841 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (23971791621423 / 64000000000000 : ℝ)).elim (fun h => tc_375 t h1 h h0)
+    (fun h => cov_376_378 t h.le h2 h0)
+
+lemma cov_373_378 :
+    ∀ t,
+      (47692431792069 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (12027754385841 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (47859866092587 / 128000000000000 : ℝ)).elim (fun h => cov_373_375 t h1 h h0)
+    (fun h => cov_375_378 t h.le h2 h0)
+
+lemma cov_379_381 :
+    ∀ t,
+      (48194734693623 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (48362168994141 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (24139225921941 / 64000000000000 : ℝ)).elim (fun h => tc_379 t h1 h h0)
+    (fun h => tc_380 t h.le h2 h0)
+
+lemma cov_378_381 :
+    ∀ t,
+      (12027754385841 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (48362168994141 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (48194734693623 / 128000000000000 : ℝ)).elim (fun h => tc_378 t h1 h h0)
+    (fun h => cov_379_381 t h.le h2 h0)
+
+lemma cov_382_384 :
+    ∀ t,
+      (121114715361 / 320000000000 : ℝ) ≤ t →
+        t ≤ (24306660222459 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (48529603294659 / 128000000000000 : ℝ)).elim (fun h => tc_382 t h1 h h0)
+    (fun h => tc_383 t h.le h2 h0)
+
+lemma cov_381_384 :
+    ∀ t,
+      (48362168994141 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (24306660222459 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (121114715361 / 320000000000 : ℝ)).elim (fun h => tc_381 t h1 h h0)
+    (fun h => cov_382_384 t h.le h2 h0)
+
+lemma cov_378_384 :
+    ∀ t,
+      (12027754385841 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (24306660222459 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (48362168994141 / 128000000000000 : ℝ)).elim (fun h => cov_378_381 t h1 h h0)
+    (fun h => cov_381_384 t h.le h2 h0)
+
+lemma cov_373_384 :
+    ∀ t,
+      (47692431792069 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (24306660222459 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (12027754385841 / 32000000000000 : ℝ)).elim (fun h => cov_373_378 t h1 h h0)
+    (fun h => cov_378_384 t h.le h2 h0)
+
+lemma cov_363_384 :
+    ∀ t,
+      (46855260289479 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (24306660222459 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (47692431792069 / 128000000000000 : ℝ)).elim (fun h => cov_363_373 t h1 h h0)
+    (fun h => cov_373_384 t h.le h2 h0)
+
+lemma cov_342_384 :
+    ∀ t,
+      (2543485761489 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (24306660222459 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (46855260289479 / 128000000000000 : ℝ)).elim (fun h => cov_342_363 t h1 h h0)
+    (fun h => cov_363_384 t h.le h2 h0)
+
+lemma cov_384_386 :
+    ∀ t,
+      (24306660222459 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (12195188686359 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (48697037595177 / 128000000000000 : ℝ)).elim (fun h => tc_384 t h1 h h0)
+    (fun h => tc_385 t h.le h2 h0)
+
+lemma cov_387_389 :
+    ∀ t,
+      (9772894379139 / 25600000000000 : ℝ) ≤ t →
+        t ≤ (6139452918309 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (24474094522977 / 64000000000000 : ℝ)).elim (fun h => tc_387 t h1 h h0)
+    (fun h => tc_388 t h.le h2 h0)
+
+lemma cov_386_389 :
+    ∀ t,
+      (12195188686359 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (6139452918309 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (9772894379139 / 25600000000000 : ℝ)).elim (fun h => tc_386 t h1 h h0)
+    (fun h => cov_387_389 t h.le h2 h0)
+
+lemma cov_384_389 :
+    ∀ t,
+      (24306660222459 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (6139452918309 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (12195188686359 / 32000000000000 : ℝ)).elim (fun h => cov_384_386 t h1 h h0)
+    (fun h => cov_386_389 t h.le h2 h0)
+
+lemma cov_389_391 :
+    ∀ t,
+      (6139452918309 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (12362622986877 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (4928305764699 / 12800000000000 : ℝ)).elim (fun h => tc_389 t h1 h h0)
+    (fun h => tc_390 t h.le h2 h0)
+
+lemma cov_392_394 :
+    ∀ t,
+      (24808963124013 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (24976397424531 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (777896258571 / 2000000000000 : ℝ)).elim (fun h => tc_392 t h1 h h0)
+    (fun h => tc_393 t h.le h2 h0)
+
+lemma cov_391_394 :
+    ∀ t,
+      (12362622986877 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (24976397424531 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (24808963124013 / 64000000000000 : ℝ)).elim (fun h => tc_391 t h1 h h0)
+    (fun h => cov_392_394 t h.le h2 h0)
+
+lemma cov_389_394 :
+    ∀ t,
+      (6139452918309 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (24976397424531 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (12362622986877 / 32000000000000 : ℝ)).elim (fun h => cov_389_391 t h1 h h0)
+    (fun h => cov_391_394 t h.le h2 h0)
+
+lemma cov_384_394 :
+    ∀ t,
+      (24306660222459 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (24976397424531 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (6139452918309 / 16000000000000 : ℝ)).elim (fun h => cov_384_389 t h1 h h0)
+    (fun h => cov_389_394 t h.le h2 h0)
+
+lemma cov_394_396 :
+    ∀ t,
+      (24976397424531 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (25143831725049 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (2506011457479 / 6400000000000 : ℝ)).elim (fun h => tc_394 t h1 h h0)
+    (fun h => tc_395 t h.le h2 h0)
+
+lemma cov_397_399 :
+    ∀ t,
+      (6306887218827 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (3195302184543 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (12697491587913 / 32000000000000 : ℝ)).elim (fun h => tc_397 t h1 h h0)
+    (fun h => tc_398 t h.le h2 h0)
+
+lemma cov_396_399 :
+    ∀ t,
+      (25143831725049 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (3195302184543 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (6306887218827 / 16000000000000 : ℝ)).elim (fun h => tc_396 t h1 h h0)
+    (fun h => cov_397_399 t h.le h2 h0)
+
+lemma cov_394_399 :
+    ∀ t,
+      (24976397424531 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (3195302184543 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (25143831725049 / 64000000000000 : ℝ)).elim (fun h => cov_394_396 t h1 h h0)
+    (fun h => cov_396_399 t h.le h2 h0)
+
+lemma cov_400_402 :
+    ∀ t,
+      (12864925888431 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (13032360188949 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1294864303869 / 3200000000000 : ℝ)).elim (fun h => tc_400 t h1 h h0)
+    (fun h => tc_401 t h.le h2 h0)
+
+lemma cov_399_402 :
+    ∀ t,
+      (3195302184543 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (13032360188949 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (12864925888431 / 32000000000000 : ℝ)).elim (fun h => tc_399 t h1 h h0)
+    (fun h => cov_400_402 t h.le h2 h0)
+
+lemma cov_403_405 :
+    ∀ t,
+      (1639509667401 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (3362736485061 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (6641755819863 / 16000000000000 : ℝ)).elim (fun h => tc_403 t h1 h h0)
+    (fun h => tc_404 t h.le h2 h0)
+
+lemma cov_402_405 :
+    ∀ t,
+      (13032360188949 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (3362736485061 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1639509667401 / 4000000000000 : ℝ)).elim (fun h => tc_402 t h1 h h0)
+    (fun h => cov_403_405 t h.le h2 h0)
+
+lemma cov_399_405 :
+    ∀ t,
+      (3195302184543 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (3362736485061 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (13032360188949 / 32000000000000 : ℝ)).elim (fun h => cov_399_402 t h1 h h0)
+    (fun h => cov_402_405 t h.le h2 h0)
+
+lemma cov_394_405 :
+    ∀ t,
+      (24976397424531 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (3362736485061 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (3195302184543 / 8000000000000 : ℝ)).elim (fun h => cov_394_399 t h1 h h0)
+    (fun h => cov_399_405 t h.le h2 h0)
+
+lemma cov_384_405 :
+    ∀ t,
+      (24306660222459 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (3362736485061 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (24976397424531 / 64000000000000 : ℝ)).elim (fun h => cov_384_394 t h1 h h0)
+    (fun h => cov_394_405 t h.le h2 h0)
+
+lemma cov_405_407 :
+    ∀ t,
+      (3362736485061 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (86161340883 / 200000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (6809190120381 / 16000000000000 : ℝ)).elim (fun h => tc_405 t h1 h h0)
+    (fun h => tc_406 t h.le h2 h0)
+
+lemma cov_408_410 :
+    ∀ t,
+      (54181913021 / 125000000000 : ℝ) ≤ t →
+        t ≤ (219376251837 / 500000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (436103903921 / 1000000000000 : ℝ)).elim (fun h => tc_408 t h1 h h0)
+    (fun h => tc_409 t h.le h2 h0)
+
+lemma cov_407_410 :
+    ∀ t,
+      (86161340883 / 200000000000 : ℝ) ≤ t →
+        t ≤ (219376251837 / 500000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (54181913021 / 125000000000 : ℝ)).elim (fun h => tc_407 t h1 h h0)
+    (fun h => cov_408_410 t h.le h2 h0)
+
+lemma cov_405_410 :
+    ∀ t,
+      (3362736485061 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (219376251837 / 500000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (86161340883 / 200000000000 : ℝ)).elim (fun h => cov_405_407 t h1 h h0)
+    (fun h => cov_407_410 t h.le h2 h0)
+
+lemma cov_411_413 :
+    ∀ t,
+      (880153607101 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (885450806607 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (441401103427 / 1000000000000 : ℝ)).elim (fun h => tc_411 t h1 h h0)
+    (fun h => tc_412 t h.le h2 h0)
+
+lemma cov_410_413 :
+    ∀ t,
+      (219376251837 / 500000000000 : ℝ) ≤ t →
+        t ≤ (885450806607 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (880153607101 / 2000000000000 : ℝ)).elim (fun h => tc_410 t h1 h h0)
+    (fun h => cov_411_413 t h.le h2 h0)
+
+lemma cov_414_416 :
+    ∀ t,
+      (22202485159 / 50000000000 : ℝ) ≤ t →
+        t ≤ (446698302933 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (890748006113 / 2000000000000 : ℝ)).elim (fun h => tc_414 t h1 h h0)
+    (fun h => tc_415 t h.le h2 h0)
+
+lemma cov_413_416 :
+    ∀ t,
+      (885450806607 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (446698302933 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (22202485159 / 50000000000 : ℝ)).elim (fun h => tc_413 t h1 h h0)
+    (fun h => cov_414_416 t h.le h2 h0)
+
+lemma cov_410_416 :
+    ∀ t,
+      (219376251837 / 500000000000 : ℝ) ≤ t →
+        t ≤ (446698302933 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (885450806607 / 2000000000000 : ℝ)).elim (fun h => cov_410_413 t h1 h h0)
+    (fun h => cov_413_416 t h.le h2 h0)
+
+lemma cov_405_416 :
+    ∀ t,
+      (3362736485061 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (446698302933 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (219376251837 / 500000000000 : ℝ)).elim (fun h => cov_405_410 t h1 h h0)
+    (fun h => cov_410_416 t h.le h2 h0)
+
+lemma cov_416_418 :
+    ∀ t,
+      (446698302933 / 1000000000000 : ℝ) ≤ t →
+        t ≤ (1794739010991 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (896045205619 / 2000000000000 : ℝ)).elim (fun h => tc_416 t h1 h h0)
+    (fun h => tc_417 t h.le h2 h0)
+
+lemma cov_419_421 :
+    ∀ t,
+      (224673451343 / 500000000000 : ℝ) ≤ t →
+        t ≤ (7210739241 / 16000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1800036210497 / 4000000000000 : ℝ)).elim (fun h => tc_419 t h1 h h0)
+    (fun h => tc_420 t h.le h2 h0)
+
+lemma cov_418_421 :
+    ∀ t,
+      (1794739010991 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (7210739241 / 16000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (224673451343 / 500000000000 : ℝ)).elim (fun h => tc_418 t h1 h h0)
+    (fun h => cov_419_421 t h.le h2 h0)
+
+lemma cov_416_421 :
+    ∀ t,
+      (446698302933 / 1000000000000 : ℝ) ≤ t →
+        t ≤ (7210739241 / 16000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1794739010991 / 4000000000000 : ℝ)).elim (fun h => cov_416_418 t h1 h h0)
+    (fun h => cov_418_421 t h.le h2 h0)
+
+lemma cov_422_424 :
+    ∀ t,
+      (1805333410003 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (1810630609509 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (451995502439 / 1000000000000 : ℝ)).elim (fun h => tc_422 t h1 h h0)
+    (fun h => tc_423 t h.le h2 h0)
+
+lemma cov_421_424 :
+    ∀ t,
+      (7210739241 / 16000000000 : ℝ) ≤ t →
+        t ≤ (1810630609509 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1805333410003 / 4000000000000 : ℝ)).elim (fun h => tc_421 t h1 h h0)
+    (fun h => cov_422_424 t h.le h2 h0)
+
+lemma cov_425_427 :
+    ∀ t,
+      (906639604631 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (28415256387 / 62500000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (363185561803 / 800000000000 : ℝ)).elim (fun h => tc_425 t h1 h h0)
+    (fun h => tc_426 t h.le h2 h0)
+
+lemma cov_424_427 :
+    ∀ t,
+      (1810630609509 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (28415256387 / 62500000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (906639604631 / 2000000000000 : ℝ)).elim (fun h => tc_424 t h1 h h0)
+    (fun h => cov_425_427 t h.le h2 h0)
+
+lemma cov_421_427 :
+    ∀ t,
+      (7210739241 / 16000000000 : ℝ) ≤ t →
+        t ≤ (28415256387 / 62500000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1810630609509 / 4000000000000 : ℝ)).elim (fun h => cov_421_424 t h1 h h0)
+    (fun h => cov_424_427 t h.le h2 h0)
+
+lemma cov_416_427 :
+    ∀ t,
+      (446698302933 / 1000000000000 : ℝ) ≤ t →
+        t ≤ (28415256387 / 62500000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (7210739241 / 16000000000 : ℝ)).elim (fun h => cov_416_421 t h1 h h0)
+    (fun h => cov_421_427 t h.le h2 h0)
+
+lemma cov_405_427 :
+    ∀ t,
+      (3362736485061 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (28415256387 / 62500000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (446698302933 / 1000000000000 : ℝ)).elim (fun h => cov_405_416 t h1 h h0)
+    (fun h => cov_416_427 t h.le h2 h0)
+
+lemma cov_384_427 :
+    ∀ t,
+      (24306660222459 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (28415256387 / 62500000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (3362736485061 / 8000000000000 : ℝ)).elim (fun h => cov_384_405 t h1 h h0)
+    (fun h => cov_405_427 t h.le h2 h0)
+
+lemma cov_342_427 :
+    ∀ t,
+      (2543485761489 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (28415256387 / 62500000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (24306660222459 / 64000000000000 : ℝ)).elim (fun h => cov_342_384 t h1 h h0)
+    (fun h => cov_384_427 t h.le h2 h0)
+
+lemma cov_427_429 :
+    ∀ t,
+      (28415256387 / 62500000000 : ℝ) ≤ t →
+        t ≤ (911936804137 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1821225008521 / 4000000000000 : ℝ)).elim (fun h => tc_427 t h1 h h0)
+    (fun h => tc_428 t h.le h2 h0)
+
+lemma cov_430_432 :
+    ∀ t,
+      (1826522208027 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (1831819407533 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (91458540389 / 200000000000 : ℝ)).elim (fun h => tc_430 t h1 h h0)
+    (fun h => tc_431 t h.le h2 h0)
+
+lemma cov_429_432 :
+    ∀ t,
+      (911936804137 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (1831819407533 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1826522208027 / 4000000000000 : ℝ)).elim (fun h => tc_429 t h1 h h0)
+    (fun h => cov_430_432 t h.le h2 h0)
+
+lemma cov_427_432 :
+    ∀ t,
+      (28415256387 / 62500000000 : ℝ) ≤ t →
+        t ≤ (1831819407533 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (911936804137 / 2000000000000 : ℝ)).elim (fun h => cov_427_429 t h1 h h0)
+    (fun h => cov_429_432 t h.le h2 h0)
+
+lemma cov_432_434 :
+    ∀ t,
+      (1831819407533 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (1837116607039 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (917234003643 / 2000000000000 : ℝ)).elim (fun h => tc_432 t h1 h h0)
+    (fun h => tc_433 t h.le h2 h0)
+
+lemma cov_435_437 :
+    ∀ t,
+      (229970650849 / 500000000000 : ℝ) ≤ t →
+        t ≤ (922531203149 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (368482761309 / 800000000000 : ℝ)).elim (fun h => tc_435 t h1 h h0)
+    (fun h => tc_436 t h.le h2 h0)
+
+lemma cov_434_437 :
+    ∀ t,
+      (1837116607039 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (922531203149 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (229970650849 / 500000000000 : ℝ)).elim (fun h => tc_434 t h1 h h0)
+    (fun h => cov_435_437 t h.le h2 h0)
+
+lemma cov_432_437 :
+    ∀ t,
+      (1831819407533 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (922531203149 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1837116607039 / 4000000000000 : ℝ)).elim (fun h => cov_432_434 t h1 h h0)
+    (fun h => cov_434_437 t h.le h2 h0)
+
+lemma cov_427_437 :
+    ∀ t,
+      (28415256387 / 62500000000 : ℝ) ≤ t →
+        t ≤ (922531203149 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1831819407533 / 4000000000000 : ℝ)).elim (fun h => cov_427_432 t h1 h h0)
+    (fun h => cov_432_437 t h.le h2 h0)
+
+lemma cov_437_439 :
+    ∀ t,
+      (922531203149 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (462589901451 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1847711006051 / 4000000000000 : ℝ)).elim (fun h => tc_437 t h1 h h0)
+    (fun h => tc_438 t h.le h2 h0)
+
+lemma cov_440_442 :
+    ∀ t,
+      (1853008205557 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (1858305405063 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (185565680531 / 400000000000 : ℝ)).elim (fun h => tc_440 t h1 h h0)
+    (fun h => tc_441 t h.le h2 h0)
+
+lemma cov_439_442 :
+    ∀ t,
+      (462589901451 / 1000000000000 : ℝ) ≤ t →
+        t ≤ (1858305405063 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1853008205557 / 4000000000000 : ℝ)).elim (fun h => tc_439 t h1 h h0)
+    (fun h => cov_440_442 t h.le h2 h0)
+
+lemma cov_437_442 :
+    ∀ t,
+      (922531203149 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (1858305405063 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (462589901451 / 1000000000000 : ℝ)).elim (fun h => cov_437_439 t h1 h h0)
+    (fun h => cov_439_442 t h.le h2 h0)
+
+lemma cov_443_445 :
+    ∀ t,
+      (116309625301 / 250000000000 : ℝ) ≤ t →
+        t ≤ (933125602161 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1863602604569 / 4000000000000 : ℝ)).elim (fun h => tc_443 t h1 h h0)
+    (fun h => tc_444 t h.le h2 h0)
+
+lemma cov_442_445 :
+    ∀ t,
+      (1858305405063 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (933125602161 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (116309625301 / 250000000000 : ℝ)).elim (fun h => tc_442 t h1 h h0)
+    (fun h => cov_443_445 t h.le h2 h0)
+
+lemma cov_446_448 :
+    ∀ t,
+      (467887100957 / 1000000000000 : ℝ) ≤ t →
+        t ≤ (47053570071 / 100000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (938422801667 / 2000000000000 : ℝ)).elim (fun h => tc_446 t h1 h h0)
+    (fun h => tc_447 t h.le h2 h0)
+
+lemma cov_445_448 :
+    ∀ t,
+      (933125602161 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (47053570071 / 100000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (467887100957 / 1000000000000 : ℝ)).elim (fun h => tc_445 t h1 h h0)
+    (fun h => cov_446_448 t h.le h2 h0)
+
+lemma cov_442_448 :
+    ∀ t,
+      (1858305405063 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (47053570071 / 100000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (933125602161 / 2000000000000 : ℝ)).elim (fun h => cov_442_445 t h1 h h0)
+    (fun h => cov_445_448 t h.le h2 h0)
+
+lemma cov_437_448 :
+    ∀ t,
+      (922531203149 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (47053570071 / 100000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1858305405063 / 4000000000000 : ℝ)).elim (fun h => cov_437_442 t h1 h h0)
+    (fun h => cov_442_448 t h.le h2 h0)
+
+lemma cov_427_448 :
+    ∀ t,
+      (28415256387 / 62500000000 : ℝ) ≤ t →
+        t ≤ (47053570071 / 100000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (922531203149 / 2000000000000 : ℝ)).elim (fun h => cov_427_437 t h1 h h0)
+    (fun h => cov_437_448 t h.le h2 h0)
+
+lemma cov_448_450 :
+    ∀ t,
+      (47053570071 / 100000000000 : ℝ) ≤ t →
+        t ≤ (473184300463 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (943720001173 / 2000000000000 : ℝ)).elim (fun h => tc_448 t h1 h h0)
+    (fun h => tc_449 t h.le h2 h0)
+
+lemma cov_451_453 :
+    ∀ t,
+      (949017200679 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (190862880037 / 400000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (59479112527 / 125000000000 : ℝ)).elim (fun h => tc_451 t h1 h h0)
+    (fun h => tc_452 t h.le h2 h0)
+
+lemma cov_450_453 :
+    ∀ t,
+      (473184300463 / 1000000000000 : ℝ) ≤ t →
+        t ≤ (190862880037 / 400000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (949017200679 / 2000000000000 : ℝ)).elim (fun h => tc_450 t h1 h h0)
+    (fun h => cov_451_453 t h.le h2 h0)
+
+lemma cov_448_453 :
+    ∀ t,
+      (47053570071 / 100000000000 : ℝ) ≤ t →
+        t ≤ (190862880037 / 400000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (473184300463 / 1000000000000 : ℝ)).elim (fun h => cov_448_450 t h1 h h0)
+    (fun h => cov_450_453 t h.le h2 h0)
+
+lemma cov_454_456 :
+    ∀ t,
+      (478481499969 / 1000000000000 : ℝ) ≤ t →
+        t ≤ (240565049861 / 500000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (959611599691 / 2000000000000 : ℝ)).elim (fun h => tc_454 t h1 h h0)
+    (fun h => tc_455 t h.le h2 h0)
+
+lemma cov_453_456 :
+    ∀ t,
+      (190862880037 / 400000000000 : ℝ) ≤ t →
+        t ≤ (240565049861 / 500000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (478481499969 / 1000000000000 : ℝ)).elim (fun h => tc_453 t h1 h h0)
+    (fun h => cov_454_456 t h.le h2 h0)
+
+lemma cov_457_459 :
+    ∀ t,
+      (19351147979 / 40000000000 : ℝ) ≤ t →
+        t ≤ (489075898981 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (121606824807 / 250000000000 : ℝ)).elim (fun h => tc_457 t h1 h h0)
+    (fun h => tc_458 t h.le h2 h0)
+
+lemma cov_456_459 :
+    ∀ t,
+      (240565049861 / 500000000000 : ℝ) ≤ t →
+        t ≤ (489075898981 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (19351147979 / 40000000000 : ℝ)).elim (fun h => tc_456 t h1 h h0)
+    (fun h => cov_457_459 t h.le h2 h0)
+
+lemma cov_453_459 :
+    ∀ t,
+      (190862880037 / 400000000000 : ℝ) ≤ t →
+        t ≤ (489075898981 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (240565049861 / 500000000000 : ℝ)).elim (fun h => cov_453_456 t h1 h h0)
+    (fun h => cov_456_459 t h.le h2 h0)
+
+lemma cov_448_459 :
+    ∀ t,
+      (47053570071 / 100000000000 : ℝ) ≤ t →
+        t ≤ (489075898981 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (190862880037 / 400000000000 : ℝ)).elim (fun h => cov_448_453 t h1 h h0)
+    (fun h => cov_453_459 t h.le h2 h0)
+
+lemma cov_459_461 :
+    ∀ t,
+      (489075898981 / 1000000000000 : ℝ) ≤ t →
+        t ≤ (494373098487 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (245862249367 / 500000000000 : ℝ)).elim (fun h => tc_459 t h1 h h0)
+    (fun h => tc_460 t h.le h2 h0)
+
+lemma cov_462_464 :
+    ∀ t,
+      (1553192807 / 3125000000 : ℝ) ≤ t →
+        t ≤ (504967497499 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (499670297993 / 1000000000000 : ℝ)).elim (fun h => tc_462 t h1 h h0)
+    (fun h => tc_463 t h.le h2 h0)
+
+lemma cov_461_464 :
+    ∀ t,
+      (494373098487 / 1000000000000 : ℝ) ≤ t →
+        t ≤ (504967497499 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1553192807 / 3125000000 : ℝ)).elim (fun h => tc_461 t h1 h h0)
+    (fun h => cov_462_464 t h.le h2 h0)
+
+lemma cov_459_464 :
+    ∀ t,
+      (489075898981 / 1000000000000 : ℝ) ≤ t →
+        t ≤ (504967497499 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (494373098487 / 1000000000000 : ℝ)).elim (fun h => cov_459_461 t h1 h h0)
+    (fun h => cov_461_464 t h.le h2 h0)
+
+lemma cov_465_467 :
+    ∀ t,
+      (102052939401 / 200000000000 : ℝ) ≤ t →
+        t ≤ (16577867570387 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (515561896511 / 1000000000000 : ℝ)).elim (fun h => tc_465 t h1 h h0)
+    (fun h => tc_466 t h.le h2 h0)
+
+lemma cov_464_467 :
+    ∀ t,
+      (504967497499 / 1000000000000 : ℝ) ≤ t →
+        t ≤ (16577867570387 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (102052939401 / 200000000000 : ℝ)).elim (fun h => tc_464 t h1 h h0)
+    (fun h => cov_465_467 t h.le h2 h0)
+
+lemma cov_468_470 :
+    ∀ t,
+      (8328877226211 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (33555169550949 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (16737641334457 / 32000000000000 : ℝ)).elim (fun h => tc_468 t h1 h h0)
+    (fun h => tc_469 t h.le h2 h0)
+
+lemma cov_467_470 :
+    ∀ t,
+      (16577867570387 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (33555169550949 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (8328877226211 / 16000000000000 : ℝ)).elim (fun h => tc_467 t h1 h h0)
+    (fun h => cov_468_470 t h.le h2 h0)
+
+lemma cov_464_470 :
+    ∀ t,
+      (504967497499 / 1000000000000 : ℝ) ≤ t →
+        t ≤ (33555169550949 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (16577867570387 / 32000000000000 : ℝ)).elim (fun h => cov_464_467 t h1 h h0)
+    (fun h => cov_467_470 t h.le h2 h0)
+
+lemma cov_459_470 :
+    ∀ t,
+      (489075898981 / 1000000000000 : ℝ) ≤ t →
+        t ≤ (33555169550949 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (504967497499 / 1000000000000 : ℝ)).elim (fun h => cov_459_464 t h1 h h0)
+    (fun h => cov_464_470 t h.le h2 h0)
+
+lemma cov_448_470 :
+    ∀ t,
+      (47053570071 / 100000000000 : ℝ) ≤ t →
+        t ≤ (33555169550949 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (489075898981 / 1000000000000 : ℝ)).elim (fun h => cov_448_459 t h1 h h0)
+    (fun h => cov_459_470 t h.le h2 h0)
+
+lemma cov_427_470 :
+    ∀ t,
+      (28415256387 / 62500000000 : ℝ) ≤ t →
+        t ≤ (33555169550949 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (47053570071 / 100000000000 : ℝ)).elim (fun h => cov_427_448 t h1 h h0)
+    (fun h => cov_448_470 t h.le h2 h0)
+
+lemma cov_470_472 :
+    ∀ t,
+      (33555169550949 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (33714943315019 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (4204382054123 / 8000000000000 : ℝ)).elim (fun h => tc_470 t h1 h h0)
+    (fun h => tc_471 t h.le h2 h0)
+
+lemma cov_473_475 :
+    ∀ t,
+      (16897415098527 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (8488650990281 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (33874717079089 / 64000000000000 : ℝ)).elim (fun h => tc_473 t h1 h h0)
+    (fun h => tc_474 t h.le h2 h0)
+
+lemma cov_472_475 :
+    ∀ t,
+      (33714943315019 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (8488650990281 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (16897415098527 / 32000000000000 : ℝ)).elim (fun h => tc_472 t h1 h h0)
+    (fun h => cov_473_475 t h.le h2 h0)
+
+lemma cov_470_475 :
+    ∀ t,
+      (33555169550949 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (8488650990281 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (33714943315019 / 64000000000000 : ℝ)).elim (fun h => cov_470_472 t h1 h h0)
+    (fun h => cov_472_475 t h.le h2 h0)
+
+lemma cov_475_477 :
+    ∀ t,
+      (8488650990281 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (17057188862597 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (34034490843159 / 64000000000000 : ℝ)).elim (fun h => tc_475 t h1 h h0)
+    (fun h => tc_476 t h.le h2 h0)
+
+lemma cov_478_480 :
+    ∀ t,
+      (34194264607229 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (2142134468079 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (68468416096493 / 128000000000000 : ℝ)).elim (fun h => tc_478 t h1 h h0)
+    (fun h => tc_479 t h.le h2 h0)
+
+lemma cov_477_480 :
+    ∀ t,
+      (17057188862597 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (2142134468079 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (34194264607229 / 64000000000000 : ℝ)).elim (fun h => tc_477 t h1 h h0)
+    (fun h => cov_478_480 t h.le h2 h0)
+
+lemma cov_475_480 :
+    ∀ t,
+      (8488650990281 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (2142134468079 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (17057188862597 / 32000000000000 : ℝ)).elim (fun h => cov_475_477 t h1 h h0)
+    (fun h => cov_477_480 t h.le h2 h0)
+
+lemma cov_470_480 :
+    ∀ t,
+      (33555169550949 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (2142134468079 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (8488650990281 / 16000000000000 : ℝ)).elim (fun h => cov_470_475 t h1 h h0)
+    (fun h => cov_475_480 t h.le h2 h0)
+
+lemma cov_480_482 :
+    ∀ t,
+      (2142134468079 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (34354038371299 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (68628189860563 / 128000000000000 : ℝ)).elim (fun h => tc_480 t h1 h h0)
+    (fun h => tc_481 t h.le h2 h0)
+
+lemma cov_483_485 :
+    ∀ t,
+      (68787963624633 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (68947737388703 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (17216962626667 / 32000000000000 : ℝ)).elim (fun h => tc_483 t h1 h h0)
+    (fun h => tc_484 t h.le h2 h0)
+
+lemma cov_482_485 :
+    ∀ t,
+      (34354038371299 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (68947737388703 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (68787963624633 / 128000000000000 : ℝ)).elim (fun h => tc_482 t h1 h h0)
+    (fun h => cov_483_485 t h.le h2 h0)
+
+lemma cov_480_485 :
+    ∀ t,
+      (2142134468079 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (68947737388703 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (34354038371299 / 64000000000000 : ℝ)).elim (fun h => cov_480_482 t h1 h h0)
+    (fun h => cov_482_485 t h.le h2 h0)
+
+lemma cov_486_488 :
+    ∀ t,
+      (34513812135369 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (8648424754351 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (69107511152773 / 128000000000000 : ℝ)).elim (fun h => tc_486 t h1 h h0)
+    (fun h => tc_487 t h.le h2 h0)
+
+lemma cov_485_488 :
+    ∀ t,
+      (68947737388703 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (8648424754351 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (34513812135369 / 64000000000000 : ℝ)).elim (fun h => tc_485 t h1 h h0)
+    (fun h => cov_486_488 t h.le h2 h0)
+
+lemma cov_489_491 :
+    ∀ t,
+      (69267284916843 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (69427058680913 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (34673585899439 / 64000000000000 : ℝ)).elim (fun h => tc_489 t h1 h h0)
+    (fun h => tc_490 t h.le h2 h0)
+
+lemma cov_488_491 :
+    ∀ t,
+      (8648424754351 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (69427058680913 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (69267284916843 / 128000000000000 : ℝ)).elim (fun h => tc_488 t h1 h h0)
+    (fun h => cov_489_491 t h.le h2 h0)
+
+lemma cov_485_491 :
+    ∀ t,
+      (68947737388703 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (69427058680913 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (8648424754351 / 16000000000000 : ℝ)).elim (fun h => cov_485_488 t h1 h h0)
+    (fun h => cov_488_491 t h.le h2 h0)
+
+lemma cov_480_491 :
+    ∀ t,
+      (2142134468079 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (69427058680913 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (68947737388703 / 128000000000000 : ℝ)).elim (fun h => cov_480_485 t h1 h h0)
+    (fun h => cov_485_491 t h.le h2 h0)
+
+lemma cov_470_491 :
+    ∀ t,
+      (33555169550949 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (69427058680913 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (2142134468079 / 4000000000000 : ℝ)).elim (fun h => cov_470_480 t h1 h h0)
+    (fun h => cov_480_491 t h.le h2 h0)
+
+lemma cov_491_493 :
+    ∀ t,
+      (69427058680913 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (69586832444983 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (17376736390737 / 32000000000000 : ℝ)).elim (fun h => tc_491 t h1 h h0)
+    (fun h => tc_492 t h.le h2 h0)
+
+lemma cov_494_496 :
+    ∀ t,
+      (34833359663509 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (4364155818193 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (69746606209053 / 128000000000000 : ℝ)).elim (fun h => tc_494 t h1 h h0)
+    (fun h => tc_495 t h.le h2 h0)
+
+lemma cov_493_496 :
+    ∀ t,
+      (69586832444983 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (4364155818193 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (34833359663509 / 64000000000000 : ℝ)).elim (fun h => tc_493 t h1 h h0)
+    (fun h => cov_494_496 t h.le h2 h0)
+
+lemma cov_491_496 :
+    ∀ t,
+      (69427058680913 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (4364155818193 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (69586832444983 / 128000000000000 : ℝ)).elim (fun h => cov_491_493 t h1 h h0)
+    (fun h => cov_493_496 t h.le h2 h0)
+
+lemma cov_497_499 :
+    ∀ t,
+      (69906379973123 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (70066153737193 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (34993133427579 / 64000000000000 : ℝ)).elim (fun h => tc_497 t h1 h h0)
+    (fun h => tc_498 t h.le h2 h0)
+
+lemma cov_496_499 :
+    ∀ t,
+      (4364155818193 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (70066153737193 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (69906379973123 / 128000000000000 : ℝ)).elim (fun h => tc_496 t h1 h h0)
+    (fun h => cov_497_499 t h.le h2 h0)
+
+lemma cov_500_502 :
+    ∀ t,
+      (17536510154807 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (8808198518421 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (35152907191649 / 64000000000000 : ℝ)).elim (fun h => tc_500 t h1 h h0)
+    (fun h => tc_501 t h.le h2 h0)
+
+lemma cov_499_502 :
+    ∀ t,
+      (70066153737193 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (8808198518421 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (17536510154807 / 32000000000000 : ℝ)).elim (fun h => tc_499 t h1 h h0)
+    (fun h => cov_500_502 t h.le h2 h0)
+
+lemma cov_496_502 :
+    ∀ t,
+      (4364155818193 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (8808198518421 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (70066153737193 / 128000000000000 : ℝ)).elim (fun h => cov_496_499 t h1 h h0)
+    (fun h => cov_499_502 t h.le h2 h0)
+
+lemma cov_491_502 :
+    ∀ t,
+      (69427058680913 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (8808198518421 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (4364155818193 / 8000000000000 : ℝ)).elim (fun h => cov_491_496 t h1 h h0)
+    (fun h => cov_496_502 t h.le h2 h0)
+
+lemma cov_502_504 :
+    ∀ t,
+      (8808198518421 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (17696283918877 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (35312680955719 / 64000000000000 : ℝ)).elim (fun h => tc_502 t h1 h h0)
+    (fun h => tc_503 t h.le h2 h0)
+
+lemma cov_505_507 :
+    ∀ t,
+      (35472454719789 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (35632228483859 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1111010675057 / 2000000000000 : ℝ)).elim (fun h => tc_505 t h1 h h0)
+    (fun h => tc_506 t h.le h2 h0)
+
+lemma cov_504_507 :
+    ∀ t,
+      (17696283918877 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (35632228483859 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (35472454719789 / 64000000000000 : ℝ)).elim (fun h => tc_504 t h1 h h0)
+    (fun h => cov_505_507 t h.le h2 h0)
+
+lemma cov_502_507 :
+    ∀ t,
+      (8808198518421 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (35632228483859 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (17696283918877 / 32000000000000 : ℝ)).elim (fun h => cov_502_504 t h1 h h0)
+    (fun h => cov_504_507 t h.le h2 h0)
+
+lemma cov_508_510 :
+    ∀ t,
+      (17856057682947 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (8967972282491 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (35792002247929 / 64000000000000 : ℝ)).elim (fun h => tc_508 t h1 h h0)
+    (fun h => tc_509 t h.le h2 h0)
+
+lemma cov_507_510 :
+    ∀ t,
+      (35632228483859 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (8967972282491 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (17856057682947 / 32000000000000 : ℝ)).elim (fun h => tc_507 t h1 h h0)
+    (fun h => cov_508_510 t h.le h2 h0)
+
+lemma cov_511_513 :
+    ∀ t,
+      (35951776011999 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (36111549776069 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (18015831447017 / 32000000000000 : ℝ)).elim (fun h => tc_511 t h1 h h0)
+    (fun h => tc_512 t h.le h2 h0)
+
+lemma cov_510_513 :
+    ∀ t,
+      (8967972282491 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (36111549776069 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (35951776011999 / 64000000000000 : ℝ)).elim (fun h => tc_510 t h1 h h0)
+    (fun h => cov_511_513 t h.le h2 h0)
+
+lemma cov_507_513 :
+    ∀ t,
+      (35632228483859 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (36111549776069 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (8967972282491 / 16000000000000 : ℝ)).elim (fun h => cov_507_510 t h1 h h0)
+    (fun h => cov_510_513 t h.le h2 h0)
+
+lemma cov_502_513 :
+    ∀ t,
+      (8808198518421 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (36111549776069 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (35632228483859 / 64000000000000 : ℝ)).elim (fun h => cov_502_507 t h1 h h0)
+    (fun h => cov_507_513 t h.le h2 h0)
+
+lemma cov_491_513 :
+    ∀ t,
+      (69427058680913 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (36111549776069 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (8808198518421 / 16000000000000 : ℝ)).elim (fun h => cov_491_502 t h1 h h0)
+    (fun h => cov_502_513 t h.le h2 h0)
+
+lemma cov_470_513 :
+    ∀ t,
+      (33555169550949 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (36111549776069 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (69427058680913 / 128000000000000 : ℝ)).elim (fun h => cov_470_491 t h1 h h0)
+    (fun h => cov_491_513 t h.le h2 h0)
+
+lemma cov_427_513 :
+    ∀ t,
+      (28415256387 / 62500000000 : ℝ) ≤ t →
+        t ≤ (36111549776069 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (33555169550949 / 64000000000000 : ℝ)).elim (fun h => cov_427_470 t h1 h h0)
+    (fun h => cov_470_513 t h.le h2 h0)
+
+lemma cov_342_513 :
+    ∀ t,
+      (2543485761489 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (36111549776069 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (28415256387 / 62500000000 : ℝ)).elim (fun h => cov_342_427 t h1 h h0)
+    (fun h => cov_427_513 t h.le h2 h0)
+
+lemma cov_513_515 :
+    ∀ t,
+      (36111549776069 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (18175605211087 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (4523929582263 / 8000000000000 : ℝ)).elim (fun h => tc_513 t h1 h h0)
+    (fun h => tc_514 t h.le h2 h0)
+
+lemma cov_516_518 :
+    ∀ t,
+      (9127746046561 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (2301908232149 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (18335378975157 / 32000000000000 : ℝ)).elim (fun h => tc_516 t h1 h h0)
+    (fun h => tc_517 t h.le h2 h0)
+
+lemma cov_515_518 :
+    ∀ t,
+      (18175605211087 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (2301908232149 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (9127746046561 / 16000000000000 : ℝ)).elim (fun h => tc_515 t h1 h h0)
+    (fun h => cov_516_518 t h.le h2 h0)
+
+lemma cov_513_518 :
+    ∀ t,
+      (36111549776069 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (2301908232149 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (18175605211087 / 32000000000000 : ℝ)).elim (fun h => cov_513_515 t h1 h h0)
+    (fun h => cov_515_518 t h.le h2 h0)
+
+lemma cov_518_520 :
+    ∀ t,
+      (2301908232149 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (9287519810631 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (18495152739227 / 32000000000000 : ℝ)).elim (fun h => tc_518 t h1 h h0)
+    (fun h => tc_519 t h.le h2 h0)
+
+lemma cov_521_523 :
+    ∀ t,
+      (4683703346333 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (297724389273 / 500000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (9447293574701 / 16000000000000 : ℝ)).elim (fun h => tc_521 t h1 h h0)
+    (fun h => tc_522 t h.le h2 h0)
+
+lemma cov_520_523 :
+    ∀ t,
+      (9287519810631 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (297724389273 / 500000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (4683703346333 / 8000000000000 : ℝ)).elim (fun h => tc_520 t h1 h h0)
+    (fun h => cov_521_523 t h.le h2 h0)
+
+lemma cov_518_523 :
+    ∀ t,
+      (2301908232149 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (297724389273 / 500000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (9287519810631 / 16000000000000 : ℝ)).elim (fun h => cov_518_520 t h1 h h0)
+    (fun h => cov_520_523 t h.le h2 h0)
+
+lemma cov_513_523 :
+    ∀ t,
+      (36111549776069 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (297724389273 / 500000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (2301908232149 / 4000000000000 : ℝ)).elim (fun h => cov_513_518 t h1 h h0)
+    (fun h => cov_518_523 t h.le h2 h0)
+
+lemma cov_523_525 :
+    ∀ t,
+      (297724389273 / 500000000000 : ℝ) ≤ t →
+        t ≤ (9595973061673 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (19123153518409 / 32000000000000 : ℝ)).elim (fun h => tc_523 t h1 h h0)
+    (fun h => tc_524 t h.le h2 h0)
+
+lemma cov_526_528 :
+    ∀ t,
+      (19260738728283 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (966476566661 / 1600000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (38590270061503 / 64000000000000 : ℝ)).elim (fun h => tc_526 t h1 h h0)
+    (fun h => tc_527 t h.le h2 h0)
+
+lemma cov_525_528 :
+    ∀ t,
+      (9595973061673 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (966476566661 / 1600000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (19260738728283 / 32000000000000 : ℝ)).elim (fun h => tc_525 t h1 h h0)
+    (fun h => cov_526_528 t h.le h2 h0)
+
+lemma cov_523_528 :
+    ∀ t,
+      (297724389273 / 500000000000 : ℝ) ≤ t →
+        t ≤ (966476566661 / 1600000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (9595973061673 / 16000000000000 : ℝ)).elim (fun h => cov_523_525 t h1 h h0)
+    (fun h => cov_525_528 t h.le h2 h0)
+
+lemma cov_529_531 :
+    ∀ t,
+      (38727855271377 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (38865440481251 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (19398323938157 / 32000000000000 : ℝ)).elim (fun h => tc_529 t h1 h h0)
+    (fun h => tc_530 t h.le h2 h0)
+
+lemma cov_528_531 :
+    ∀ t,
+      (966476566661 / 1600000000000 : ℝ) ≤ t →
+        t ≤ (38865440481251 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (38727855271377 / 64000000000000 : ℝ)).elim (fun h => tc_528 t h1 h h0)
+    (fun h => cov_529_531 t h.le h2 h0)
+
+lemma cov_532_534 :
+    ∀ t,
+      (9733558271547 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (19535909148031 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (312024205529 / 512000000000 : ℝ)).elim (fun h => tc_532 t h1 h h0)
+    (fun h => tc_533 t h.le h2 h0)
+
+lemma cov_531_534 :
+    ∀ t,
+      (38865440481251 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (19535909148031 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (9733558271547 / 16000000000000 : ℝ)).elim (fun h => tc_531 t h1 h h0)
+    (fun h => cov_532_534 t h.le h2 h0)
+
+lemma cov_528_534 :
+    ∀ t,
+      (966476566661 / 1600000000000 : ℝ) ≤ t →
+        t ≤ (19535909148031 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (38865440481251 / 64000000000000 : ℝ)).elim (fun h => cov_528_531 t h1 h h0)
+    (fun h => cov_531_534 t h.le h2 h0)
+
+lemma cov_523_534 :
+    ∀ t,
+      (297724389273 / 500000000000 : ℝ) ≤ t →
+        t ≤ (19535909148031 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (966476566661 / 1600000000000 : ℝ)).elim (fun h => cov_523_528 t h1 h h0)
+    (fun h => cov_528_534 t h.le h2 h0)
+
+lemma cov_513_534 :
+    ∀ t,
+      (36111549776069 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (19535909148031 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (297724389273 / 500000000000 : ℝ)).elim (fun h => cov_513_523 t h1 h h0)
+    (fun h => cov_523_534 t h.le h2 h0)
+
+lemma cov_534_536 :
+    ∀ t,
+      (19535909148031 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (2450587719121 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (39140610900999 / 64000000000000 : ℝ)).elim (fun h => tc_534 t h1 h h0)
+    (fun h => tc_535 t h.le h2 h0)
+
+lemma cov_537_539 :
+    ∀ t,
+      (39278196110873 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (39415781320747 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (3934698871581 / 6400000000000 : ℝ)).elim (fun h => tc_537 t h1 h h0)
+    (fun h => tc_538 t h.le h2 h0)
+
+lemma cov_536_539 :
+    ∀ t,
+      (2450587719121 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (39415781320747 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (39278196110873 / 64000000000000 : ℝ)).elim (fun h => tc_536 t h1 h h0)
+    (fun h => cov_537_539 t h.le h2 h0)
+
+lemma cov_534_539 :
+    ∀ t,
+      (19535909148031 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (39415781320747 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (2450587719121 / 4000000000000 : ℝ)).elim (fun h => cov_534_536 t h1 h h0)
+    (fun h => cov_536_539 t h.le h2 h0)
+
+lemma cov_539_541 :
+    ∀ t,
+      (39415781320747 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (39553366530621 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (9871143481421 / 16000000000000 : ℝ)).elim (fun h => tc_539 t h1 h h0)
+    (fun h => tc_540 t h.le h2 h0)
+
+lemma cov_542_544 :
+    ∀ t,
+      (19811079567779 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (4969968043179 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (7938190348099 / 12800000000000 : ℝ)).elim (fun h => tc_542 t h1 h h0)
+    (fun h => tc_543 t h.le h2 h0)
+
+lemma cov_541_544 :
+    ∀ t,
+      (39553366530621 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (4969968043179 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (19811079567779 / 32000000000000 : ℝ)).elim (fun h => tc_541 t h1 h h0)
+    (fun h => cov_542_544 t h.le h2 h0)
+
+lemma cov_539_544 :
+    ∀ t,
+      (39415781320747 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (4969968043179 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (39553366530621 / 64000000000000 : ℝ)).elim (fun h => cov_539_541 t h1 h h0)
+    (fun h => cov_541_544 t h.le h2 h0)
+
+lemma cov_534_544 :
+    ∀ t,
+      (19535909148031 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (4969968043179 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (39415781320747 / 64000000000000 : ℝ)).elim (fun h => cov_534_539 t h1 h h0)
+    (fun h => cov_539_544 t h.le h2 h0)
+
+lemma cov_544_546 :
+    ∀ t,
+      (4969968043179 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (19948664777653 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (39828536950369 / 64000000000000 : ℝ)).elim (fun h => tc_544 t h1 h h0)
+    (fun h => tc_545 t h.le h2 h0)
+
+lemma cov_547_549 :
+    ∀ t,
+      (39966122160243 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (40103707370117 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (2001745738259 / 3200000000000 : ℝ)).elim (fun h => tc_547 t h1 h h0)
+    (fun h => tc_548 t h.le h2 h0)
+
+lemma cov_546_549 :
+    ∀ t,
+      (19948664777653 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (40103707370117 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (39966122160243 / 64000000000000 : ℝ)).elim (fun h => tc_546 t h1 h h0)
+    (fun h => cov_547_549 t h.le h2 h0)
+
+lemma cov_544_549 :
+    ∀ t,
+      (4969968043179 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (40103707370117 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (19948664777653 / 32000000000000 : ℝ)).elim (fun h => cov_544_546 t h1 h h0)
+    (fun h => cov_546_549 t h.le h2 h0)
+
+lemma cov_550_552 :
+    ∀ t,
+      (20086249987527 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (1259690162029 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (40241292579991 / 64000000000000 : ℝ)).elim (fun h => tc_550 t h1 h h0)
+    (fun h => tc_551 t h.le h2 h0)
+
+lemma cov_549_552 :
+    ∀ t,
+      (40103707370117 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (1259690162029 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (20086249987527 / 32000000000000 : ℝ)).elim (fun h => tc_549 t h1 h h0)
+    (fun h => cov_550_552 t h.le h2 h0)
+
+lemma cov_553_555 :
+    ∀ t,
+      (8075775557973 / 12800000000000 : ℝ) ≤ t →
+        t ≤ (40516462999739 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (20223835197401 / 32000000000000 : ℝ)).elim (fun h => tc_553 t h1 h h0)
+    (fun h => tc_554 t h.le h2 h0)
+
+lemma cov_552_555 :
+    ∀ t,
+      (1259690162029 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (40516462999739 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (8075775557973 / 12800000000000 : ℝ)).elim (fun h => tc_552 t h1 h h0)
+    (fun h => cov_553_555 t h.le h2 h0)
+
+lemma cov_549_555 :
+    ∀ t,
+      (40103707370117 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (40516462999739 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1259690162029 / 2000000000000 : ℝ)).elim (fun h => cov_549_552 t h1 h h0)
+    (fun h => cov_552_555 t h.le h2 h0)
+
+lemma cov_544_555 :
+    ∀ t,
+      (4969968043179 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (40516462999739 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (40103707370117 / 64000000000000 : ℝ)).elim (fun h => cov_544_549 t h1 h h0)
+    (fun h => cov_549_555 t h.le h2 h0)
+
+lemma cov_534_555 :
+    ∀ t,
+      (19535909148031 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (40516462999739 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (4969968043179 / 8000000000000 : ℝ)).elim (fun h => cov_534_544 t h1 h h0)
+    (fun h => cov_544_555 t h.le h2 h0)
+
+lemma cov_513_555 :
+    ∀ t,
+      (36111549776069 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (40516462999739 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (19535909148031 / 32000000000000 : ℝ)).elim (fun h => cov_513_534 t h1 h h0)
+    (fun h => cov_534_555 t h.le h2 h0)
+
+lemma cov_555_557 :
+    ∀ t,
+      (40516462999739 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (40654048209613 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (10146313901169 / 16000000000000 : ℝ)).elim (fun h => tc_555 t h1 h h0)
+    (fun h => tc_556 t h.le h2 h0)
+
+lemma cov_558_560 :
+    ∀ t,
+      (814456816291 / 1280000000000 : ℝ) ≤ t →
+        t ≤ (5107553253053 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (40791633419487 / 64000000000000 : ℝ)).elim (fun h => tc_558 t h1 h h0)
+    (fun h => tc_559 t h.le h2 h0)
+
+lemma cov_557_560 :
+    ∀ t,
+      (40654048209613 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (5107553253053 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (814456816291 / 1280000000000 : ℝ)).elim (fun h => tc_557 t h1 h h0)
+    (fun h => cov_558_560 t h.le h2 h0)
+
+lemma cov_555_560 :
+    ∀ t,
+      (40516462999739 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (5107553253053 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (40654048209613 / 64000000000000 : ℝ)).elim (fun h => cov_555_557 t h1 h h0)
+    (fun h => cov_557_560 t h.le h2 h0)
+
+lemma cov_560_562 :
+    ∀ t,
+      (5107553253053 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (10283899111043 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (20499005617149 / 32000000000000 : ℝ)).elim (fun h => tc_560 t h1 h h0)
+    (fun h => tc_561 t h.le h2 h0)
+
+lemma cov_563_565 :
+    ∀ t,
+      (20636590827023 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (20774176036897 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (517634585799 / 800000000000 : ℝ)).elim (fun h => tc_563 t h1 h h0)
+    (fun h => tc_564 t h.le h2 h0)
+
+lemma cov_562_565 :
+    ∀ t,
+      (10283899111043 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (20774176036897 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (20636590827023 / 32000000000000 : ℝ)).elim (fun h => tc_562 t h1 h h0)
+    (fun h => cov_563_565 t h.le h2 h0)
+
+lemma cov_560_565 :
+    ∀ t,
+      (5107553253053 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (20774176036897 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (10283899111043 / 16000000000000 : ℝ)).elim (fun h => cov_560_562 t h1 h h0)
+    (fun h => cov_562_565 t h.le h2 h0)
+
+lemma cov_555_565 :
+    ∀ t,
+      (40516462999739 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (20774176036897 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (5107553253053 / 8000000000000 : ℝ)).elim (fun h => cov_555_560 t h1 h h0)
+    (fun h => cov_560_565 t h.le h2 h0)
+
+lemma cov_565_567 :
+    ∀ t,
+      (20774176036897 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (20911761246771 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (10421484320917 / 16000000000000 : ℝ)).elim (fun h => tc_565 t h1 h h0)
+    (fun h => tc_566 t h.le h2 h0)
+
+lemma cov_568_570 :
+    ∀ t,
+      (5245138462927 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (664241383483 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (10559069530791 / 16000000000000 : ℝ)).elim (fun h => tc_568 t h1 h h0)
+    (fun h => tc_569 t h.le h2 h0)
+
+lemma cov_567_570 :
+    ∀ t,
+      (20911761246771 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (664241383483 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (5245138462927 / 8000000000000 : ℝ)).elim (fun h => tc_567 t h1 h h0)
+    (fun h => cov_568_570 t h.le h2 h0)
+
+lemma cov_565_570 :
+    ∀ t,
+      (20774176036897 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (664241383483 / 1000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (20911761246771 / 32000000000000 : ℝ)).elim (fun h => cov_565_567 t h1 h h0)
+    (fun h => cov_567_570 t h.le h2 h0)
+
+lemma cov_571_573 :
+    ∀ t,
+      (4261528693017 / 6400000000000 : ℝ) ≤ t →
+        t ≤ (21411481852343 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (10679781329357 / 16000000000000 : ℝ)).elim (fun h => tc_571 t h1 h h0)
+    (fun h => tc_572 t h.le h2 h0)
+
+lemma cov_570_573 :
+    ∀ t,
+      (664241383483 / 1000000000000 : ℝ) ≤ t →
+        t ≤ (21411481852343 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (4261528693017 / 6400000000000 : ℝ)).elim (fun h => tc_570 t h1 h h0)
+    (fun h => cov_571_573 t h.le h2 h0)
+
+lemma cov_574_576 :
+    ∀ t,
+      (5365850261493 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (43082559672831 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (21515320239601 / 32000000000000 : ℝ)).elim (fun h => tc_574 t h1 h h0)
+    (fun h => tc_575 t h.le h2 h0)
+
+lemma cov_573_576 :
+    ∀ t,
+      (21411481852343 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (43082559672831 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (5365850261493 / 8000000000000 : ℝ)).elim (fun h => tc_573 t h1 h h0)
+    (fun h => cov_574_576 t h.le h2 h0)
+
+lemma cov_570_576 :
+    ∀ t,
+      (664241383483 / 1000000000000 : ℝ) ≤ t →
+        t ≤ (43082559672831 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (21411481852343 / 32000000000000 : ℝ)).elim (fun h => cov_570_573 t h1 h h0)
+    (fun h => cov_573_576 t h.le h2 h0)
+
+lemma cov_565_576 :
+    ∀ t,
+      (20774176036897 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (43082559672831 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (664241383483 / 1000000000000 : ℝ)).elim (fun h => cov_565_570 t h1 h h0)
+    (fun h => cov_570_576 t h.le h2 h0)
+
+lemma cov_555_576 :
+    ∀ t,
+      (40516462999739 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (43082559672831 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (20774176036897 / 32000000000000 : ℝ)).elim (fun h => cov_555_565 t h1 h h0)
+    (fun h => cov_565_576 t h.le h2 h0)
+
+lemma cov_576_578 :
+    ∀ t,
+      (43082559672831 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (43186398060089 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (2156723943323 / 3200000000000 : ℝ)).elim (fun h => tc_576 t h1 h h0)
+    (fun h => tc_577 t h.le h2 h0)
+
+lemma cov_579_581 :
+    ∀ t,
+      (21619158626859 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (2708884727561 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (43290236447347 / 64000000000000 : ℝ)).elim (fun h => tc_579 t h1 h h0)
+    (fun h => tc_580 t h.le h2 h0)
+
+lemma cov_578_581 :
+    ∀ t,
+      (43186398060089 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (2708884727561 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (21619158626859 / 32000000000000 : ℝ)).elim (fun h => tc_578 t h1 h h0)
+    (fun h => cov_579_581 t h.le h2 h0)
+
+lemma cov_576_581 :
+    ∀ t,
+      (43082559672831 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (2708884727561 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (43186398060089 / 64000000000000 : ℝ)).elim (fun h => cov_576_578 t h1 h h0)
+    (fun h => cov_578_581 t h.le h2 h0)
+
+lemma cov_582_584 :
+    ∀ t,
+      (8678814966921 / 12800000000000 : ℝ) ≤ t →
+        t ≤ (43497913221863 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (21722997014117 / 32000000000000 : ℝ)).elim (fun h => tc_582 t h1 h h0)
+    (fun h => tc_583 t h.le h2 h0)
+
+lemma cov_581_584 :
+    ∀ t,
+      (2708884727561 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (43497913221863 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (8678814966921 / 12800000000000 : ℝ)).elim (fun h => tc_581 t h1 h h0)
+    (fun h => cov_582_584 t h.le h2 h0)
+
+lemma cov_585_587 :
+    ∀ t,
+      (10887458103873 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (174614683211 / 256000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (43601751609121 / 64000000000000 : ℝ)).elim (fun h => tc_585 t h1 h h0)
+    (fun h => tc_586 t h.le h2 h0)
+
+lemma cov_584_587 :
+    ∀ t,
+      (43497913221863 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (174614683211 / 256000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (10887458103873 / 16000000000000 : ℝ)).elim (fun h => tc_584 t h1 h h0)
+    (fun h => cov_585_587 t h.le h2 h0)
+
+lemma cov_581_587 :
+    ∀ t,
+      (2708884727561 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (174614683211 / 256000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (43497913221863 / 64000000000000 : ℝ)).elim (fun h => cov_581_584 t h1 h h0)
+    (fun h => cov_584_587 t h.le h2 h0)
+
+lemma cov_576_587 :
+    ∀ t,
+      (43082559672831 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (174614683211 / 256000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (2708884727561 / 4000000000000 : ℝ)).elim (fun h => cov_576_581 t h1 h h0)
+    (fun h => cov_581_587 t h.le h2 h0)
+
+lemma cov_587_589 :
+    ∀ t,
+      (174614683211 / 256000000000 : ℝ) ≤ t →
+        t ≤ (5469688648751 / 8000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (43705589996379 / 64000000000000 : ℝ)).elim (fun h => tc_587 t h1 h h0)
+    (fun h => tc_588 t h.le h2 h0)
+
+lemma cov_590_592 :
+    ∀ t,
+      (43809428383637 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (8782653354179 / 12800000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (21930673788633 / 32000000000000 : ℝ)).elim (fun h => tc_590 t h1 h h0)
+    (fun h => tc_591 t h.le h2 h0)
+
+lemma cov_589_592 :
+    ∀ t,
+      (5469688648751 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (8782653354179 / 12800000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (43809428383637 / 64000000000000 : ℝ)).elim (fun h => tc_589 t h1 h h0)
+    (fun h => cov_590_592 t h.le h2 h0)
+
+lemma cov_587_592 :
+    ∀ t,
+      (174614683211 / 256000000000 : ℝ) ≤ t →
+        t ≤ (8782653354179 / 12800000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (5469688648751 / 8000000000000 : ℝ)).elim (fun h => cov_587_589 t h1 h h0)
+    (fun h => cov_589_592 t h.le h2 h0)
+
+lemma cov_593_595 :
+    ∀ t,
+      (10991296491131 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (22034512175891 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (44017105158153 / 64000000000000 : ℝ)).elim (fun h => tc_593 t h1 h h0)
+    (fun h => tc_594 t h.le h2 h0)
+
+lemma cov_592_595 :
+    ∀ t,
+      (8782653354179 / 12800000000000 : ℝ) ≤ t →
+        t ≤ (22034512175891 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (10991296491131 / 16000000000000 : ℝ)).elim (fun h => tc_592 t h1 h h0)
+    (fun h => cov_593_595 t h.le h2 h0)
+
+lemma cov_596_598 :
+    ∀ t,
+      (44120943545411 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (44224781932669 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (276080392119 / 400000000000 : ℝ)).elim (fun h => tc_596 t h1 h h0)
+    (fun h => tc_597 t h.le h2 h0)
+
+lemma cov_595_598 :
+    ∀ t,
+      (22034512175891 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (44224781932669 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (44120943545411 / 64000000000000 : ℝ)).elim (fun h => tc_595 t h1 h h0)
+    (fun h => cov_596_598 t h.le h2 h0)
+
+lemma cov_592_598 :
+    ∀ t,
+      (8782653354179 / 12800000000000 : ℝ) ≤ t →
+        t ≤ (44224781932669 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (22034512175891 / 32000000000000 : ℝ)).elim (fun h => cov_592_595 t h1 h h0)
+    (fun h => cov_595_598 t h.le h2 h0)
+
+lemma cov_587_598 :
+    ∀ t,
+      (174614683211 / 256000000000 : ℝ) ≤ t →
+        t ≤ (44224781932669 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (8782653354179 / 12800000000000 : ℝ)).elim (fun h => cov_587_592 t h1 h h0)
+    (fun h => cov_592_598 t h.le h2 h0)
+
+lemma cov_576_598 :
+    ∀ t,
+      (43082559672831 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (44224781932669 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (174614683211 / 256000000000 : ℝ)).elim (fun h => cov_576_587 t h1 h h0)
+    (fun h => cov_587_598 t h.le h2 h0)
+
+lemma cov_555_598 :
+    ∀ t,
+      (40516462999739 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (44224781932669 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (43082559672831 / 64000000000000 : ℝ)).elim (fun h => cov_555_576 t h1 h h0)
+    (fun h => cov_576_598 t h.le h2 h0)
+
+lemma cov_513_598 :
+    ∀ t,
+      (36111549776069 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (44224781932669 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (40516462999739 / 64000000000000 : ℝ)).elim (fun h => cov_513_555 t h1 h h0)
+    (fun h => cov_555_598 t h.le h2 h0)
+
+lemma cov_598_600 :
+    ∀ t,
+      (44224781932669 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (44328620319927 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (22138350563149 / 32000000000000 : ℝ)).elim (fun h => tc_598 t h1 h h0)
+    (fun h => tc_599 t h.le h2 h0)
+
+lemma cov_601_603 :
+    ∀ t,
+      (11095134878389 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (22242188950407 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (8886491741437 / 12800000000000 : ℝ)).elim (fun h => tc_601 t h1 h h0)
+    (fun h => tc_602 t h.le h2 h0)
+
+lemma cov_600_603 :
+    ∀ t,
+      (44328620319927 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (22242188950407 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (11095134878389 / 16000000000000 : ℝ)).elim (fun h => tc_600 t h1 h h0)
+    (fun h => cov_601_603 t h.le h2 h0)
+
+lemma cov_598_603 :
+    ∀ t,
+      (44224781932669 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (22242188950407 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (44328620319927 / 64000000000000 : ℝ)).elim (fun h => cov_598_600 t h1 h h0)
+    (fun h => cov_600_603 t h.le h2 h0)
+
+lemma cov_603_605 :
+    ∀ t,
+      (22242188950407 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (4469205467533 / 6400000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (5573527036009 / 8000000000000 : ℝ)).elim (fun h => tc_603 t h1 h h0)
+    (fun h => tc_604 t h.le h2 h0)
+
+lemma cov_606_608 :
+    ∀ t,
+      (11198973265647 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (2812723114819 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (22449865724923 / 32000000000000 : ℝ)).elim (fun h => tc_606 t h1 h h0)
+    (fun h => tc_607 t h.le h2 h0)
+
+lemma cov_605_608 :
+    ∀ t,
+      (4469205467533 / 6400000000000 : ℝ) ≤ t →
+        t ≤ (2812723114819 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (11198973265647 / 16000000000000 : ℝ)).elim (fun h => tc_605 t h1 h h0)
+    (fun h => cov_606_608 t h.le h2 h0)
+
+lemma cov_603_608 :
+    ∀ t,
+      (22242188950407 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (2812723114819 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (4469205467533 / 6400000000000 : ℝ)).elim (fun h => cov_603_605 t h1 h h0)
+    (fun h => cov_605_608 t h.le h2 h0)
+
+lemma cov_598_608 :
+    ∀ t,
+      (44224781932669 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (2812723114819 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (22242188950407 / 32000000000000 : ℝ)).elim (fun h => cov_598_603 t h1 h h0)
+    (fun h => cov_603_608 t h.le h2 h0)
+
+lemma cov_608_610 :
+    ∀ t,
+      (2812723114819 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (2260562330581 / 3200000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (22553704112181 / 32000000000000 : ℝ)).elim (fun h => tc_608 t h1 h h0)
+    (fun h => tc_609 t h.le h2 h0)
+
+lemma cov_611_613 :
+    ∀ t,
+      (22657542499439 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (22761380886697 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (5677365423267 / 8000000000000 : ℝ)).elim (fun h => tc_611 t h1 h h0)
+    (fun h => tc_612 t h.le h2 h0)
+
+lemma cov_610_613 :
+    ∀ t,
+      (2260562330581 / 3200000000000 : ℝ) ≤ t →
+        t ≤ (22761380886697 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (22657542499439 / 32000000000000 : ℝ)).elim (fun h => tc_610 t h1 h h0)
+    (fun h => cov_611_613 t h.le h2 h0)
+
+lemma cov_608_613 :
+    ∀ t,
+      (2812723114819 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (22761380886697 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (2260562330581 / 3200000000000 : ℝ)).elim (fun h => cov_608_610 t h1 h h0)
+    (fun h => cov_610_613 t h.le h2 h0)
+
+lemma cov_614_616 :
+    ∀ t,
+      (11406650040163 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (11489045952349 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (89520072139 / 125000000000 : ℝ)).elim (fun h => tc_614 t h1 h h0)
+    (fun h => tc_615 t h.le h2 h0)
+
+lemma cov_613_616 :
+    ∀ t,
+      (22761380886697 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (11489045952349 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (11406650040163 / 16000000000000 : ℝ)).elim (fun h => tc_613 t h1 h h0)
+    (fun h => cov_614_616 t h.le h2 h0)
+
+lemma cov_617_619 :
+    ∀ t,
+      (4601713724651 / 6400000000000 : ℝ) ≤ t →
+        t ≤ (23069522060369 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (5759761335453 / 8000000000000 : ℝ)).elim (fun h => tc_617 t h1 h h0)
+    (fun h => tc_618 t h.le h2 h0)
+
+lemma cov_616_619 :
+    ∀ t,
+      (11489045952349 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (23069522060369 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (4601713724651 / 6400000000000 : ℝ)).elim (fun h => tc_616 t h1 h h0)
+    (fun h => cov_617_619 t h.le h2 h0)
+
+lemma cov_613_619 :
+    ∀ t,
+      (22761380886697 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (23069522060369 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (11489045952349 / 16000000000000 : ℝ)).elim (fun h => cov_613_616 t h1 h h0)
+    (fun h => cov_616_619 t h.le h2 h0)
+
+lemma cov_608_619 :
+    ∀ t,
+      (2812723114819 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (23069522060369 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (22761380886697 / 32000000000000 : ℝ)).elim (fun h => cov_608_613 t h1 h h0)
+    (fun h => cov_613_619 t h.le h2 h0)
+
+lemma cov_598_619 :
+    ∀ t,
+      (44224781932669 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (23069522060369 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (2812723114819 / 4000000000000 : ℝ)).elim (fun h => cov_598_608 t h1 h h0)
+    (fun h => cov_608_619 t h.le h2 h0)
+
+lemma cov_619_621 :
+    ∀ t,
+      (23069522060369 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (23130475497483 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (11549999389463 / 16000000000000 : ℝ)).elim (fun h => tc_619 t h1 h h0)
+    (fun h => tc_620 t h.le h2 h0)
+
+lemma cov_622_624 :
+    ∀ t,
+      (579023805401 / 800000000000 : ℝ) ≤ t →
+        t ≤ (11610952826577 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (23191428934597 / 32000000000000 : ℝ)).elim (fun h => tc_622 t h1 h h0)
+    (fun h => tc_623 t h.le h2 h0)
+
+lemma cov_621_624 :
+    ∀ t,
+      (23130475497483 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (11610952826577 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (579023805401 / 800000000000 : ℝ)).elim (fun h => tc_621 t h1 h h0)
+    (fun h => cov_622_624 t h.le h2 h0)
+
+lemma cov_619_624 :
+    ∀ t,
+      (23069522060369 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (11610952826577 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (23130475497483 / 32000000000000 : ℝ)).elim (fun h => cov_619_621 t h1 h h0)
+    (fun h => cov_621_624 t h.le h2 h0)
+
+lemma cov_625_627 :
+    ∀ t,
+      (23252382371711 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (932533432353 / 1280000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (5820714772567 / 8000000000000 : ℝ)).elim (fun h => tc_625 t h1 h h0)
+    (fun h => tc_626 t h.le h2 h0)
+
+lemma cov_624_627 :
+    ∀ t,
+      (11610952826577 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (932533432353 / 1280000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (23252382371711 / 32000000000000 : ℝ)).elim (fun h => tc_624 t h1 h h0)
+    (fun h => cov_625_627 t h.le h2 h0)
+
+lemma cov_628_630 :
+    ∀ t,
+      (11671906263691 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (1462797872781 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (23374289245939 / 32000000000000 : ℝ)).elim (fun h => tc_628 t h1 h h0)
+    (fun h => tc_629 t h.le h2 h0)
+
+lemma cov_627_630 :
+    ∀ t,
+      (932533432353 / 1280000000000 : ℝ) ≤ t →
+        t ≤ (1462797872781 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (11671906263691 / 16000000000000 : ℝ)).elim (fun h => tc_627 t h1 h h0)
+    (fun h => cov_628_630 t h.le h2 h0)
+
+lemma cov_624_630 :
+    ∀ t,
+      (11610952826577 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (1462797872781 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (932533432353 / 1280000000000 : ℝ)).elim (fun h => cov_624_627 t h1 h h0)
+    (fun h => cov_627_630 t h.le h2 h0)
+
+lemma cov_619_630 :
+    ∀ t,
+      (23069522060369 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (1462797872781 / 2000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (11610952826577 / 16000000000000 : ℝ)).elim (fun h => cov_619_624 t h1 h h0)
+    (fun h => cov_624_630 t h.le h2 h0)
+
+lemma cov_630_632 :
+    ∀ t,
+      (1462797872781 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (2346571940161 / 3200000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (23435242683053 / 32000000000000 : ℝ)).elim (fun h => tc_630 t h1 h h0)
+    (fun h => tc_631 t h.le h2 h0)
+
+lemma cov_633_635 :
+    ∀ t,
+      (23496196120167 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (23557149557281 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (5881668209681 / 8000000000000 : ℝ)).elim (fun h => tc_633 t h1 h h0)
+    (fun h => tc_634 t h.le h2 h0)
+
+lemma cov_632_635 :
+    ∀ t,
+      (2346571940161 / 3200000000000 : ℝ) ≤ t →
+        t ≤ (23557149557281 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (23496196120167 / 32000000000000 : ℝ)).elim (fun h => tc_632 t h1 h h0)
+    (fun h => cov_633_635 t h.le h2 h0)
+
+lemma cov_630_635 :
+    ∀ t,
+      (1462797872781 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (23557149557281 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (2346571940161 / 3200000000000 : ℝ)).elim (fun h => cov_630_632 t h1 h h0)
+    (fun h => cov_632_635 t h.le h2 h0)
+
+lemma cov_636_638 :
+    ∀ t,
+      (11793813137919 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (2956072464119 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (4723620598879 / 6400000000000 : ℝ)).elim (fun h => tc_636 t h1 h h0)
+    (fun h => tc_637 t h.le h2 h0)
+
+lemma cov_635_638 :
+    ∀ t,
+      (23557149557281 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (2956072464119 / 4000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (11793813137919 / 16000000000000 : ℝ)).elim (fun h => tc_635 t h1 h h0)
+    (fun h => cov_636_638 t h.le h2 h0)
+
+lemma cov_639_641 :
+    ∀ t,
+      (23679056431509 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (23740009868623 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (11854766575033 / 16000000000000 : ℝ)).elim (fun h => tc_639 t h1 h h0)
+    (fun h => tc_640 t h.le h2 h0)
+
+lemma cov_638_641 :
+    ∀ t,
+      (2956072464119 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (23740009868623 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (23679056431509 / 32000000000000 : ℝ)).elim (fun h => tc_638 t h1 h h0)
+    (fun h => cov_639_641 t h.le h2 h0)
+
+lemma cov_635_641 :
+    ∀ t,
+      (23557149557281 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (23740009868623 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (2956072464119 / 4000000000000 : ℝ)).elim (fun h => cov_635_638 t h1 h h0)
+    (fun h => cov_638_641 t h.le h2 h0)
+
+lemma cov_630_641 :
+    ∀ t,
+      (1462797872781 / 2000000000000 : ℝ) ≤ t →
+        t ≤ (23740009868623 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (23557149557281 / 32000000000000 : ℝ)).elim (fun h => cov_630_635 t h1 h h0)
+    (fun h => cov_635_641 t h.le h2 h0)
+
+lemma cov_619_641 :
+    ∀ t,
+      (23069522060369 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (23740009868623 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1462797872781 / 2000000000000 : ℝ)).elim (fun h => cov_619_630 t h1 h h0)
+    (fun h => cov_630_641 t h.le h2 h0)
+
+lemma cov_598_641 :
+    ∀ t,
+      (44224781932669 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (23740009868623 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (23069522060369 / 32000000000000 : ℝ)).elim (fun h => cov_598_619 t h1 h h0)
+    (fun h => cov_619_641 t h.le h2 h0)
+
+lemma cov_641_643 :
+    ∀ t,
+      (23740009868623 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (23800963305737 / 32000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1188524329359 / 1600000000000 : ℝ)).elim (fun h => tc_641 t h1 h h0)
+    (fun h => tc_642 t h.le h2 h0)
+
+lemma cov_644_646 :
+    ∀ t,
+      (11915720012147 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (765809953469387 / 1024000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (746637295669 / 1000000000000 : ℝ)).elim (fun h => tc_644 t h1 h h0)
+    (fun h => tc_645 t h.le h2 h0)
+
+lemma cov_643_646 :
+    ∀ t,
+      (23800963305737 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (765809953469387 / 1024000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (11915720012147 / 16000000000000 : ℝ)).elim (fun h => tc_643 t h1 h h0)
+    (fun h => cov_644_646 t h.le h2 h0)
+
+lemma cov_641_646 :
+    ∀ t,
+      (23740009868623 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (765809953469387 / 1024000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (23800963305737 / 32000000000000 : ℝ)).elim (fun h => cov_641_643 t h1 h h0)
+    (fun h => cov_643_646 t h.le h2 h0)
+
+lemma cov_646_648 :
+    ∀ t,
+      (765809953469387 / 1024000000000000 : ℝ) ≤ t →
+        t ≤ (768316678878049 / 1024000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (383531658086859 / 512000000000000 : ℝ)).elim (fun h => tc_646 t h1 h h0)
+    (fun h => tc_647 t h.le h2 h0)
+
+lemma cov_649_651 :
+    ∀ t,
+      (38478502079119 / 51200000000000 : ℝ) ≤ t →
+        t ≤ (386038383495521 / 512000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (770823404286711 / 1024000000000000 : ℝ)).elim (fun h => tc_649 t h1 h h0)
+    (fun h => tc_650 t h.le h2 h0)
+
+lemma cov_648_651 :
+    ∀ t,
+      (768316678878049 / 1024000000000000 : ℝ) ≤ t →
+        t ≤ (386038383495521 / 512000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (38478502079119 / 51200000000000 : ℝ)).elim (fun h => tc_648 t h1 h h0)
+    (fun h => cov_649_651 t h.le h2 h0)
+
+lemma cov_646_651 :
+    ∀ t,
+      (765809953469387 / 1024000000000000 : ℝ) ≤ t →
+        t ≤ (386038383495521 / 512000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (768316678878049 / 1024000000000000 : ℝ)).elim (fun h => cov_646_648 t h1 h h0)
+    (fun h => cov_648_651 t h.le h2 h0)
+
+lemma cov_641_651 :
+    ∀ t,
+      (23740009868623 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (386038383495521 / 512000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (765809953469387 / 1024000000000000 : ℝ)).elim (fun h => cov_641_646 t h1 h h0)
+    (fun h => cov_646_651 t h.le h2 h0)
+
+lemma cov_651_653 :
+    ∀ t,
+      (386038383495521 / 512000000000000 : ℝ) ≤ t →
+        t ≤ (96822936549963 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (773330129695373 / 1024000000000000 : ℝ)).elim (fun h => tc_651 t h1 h h0)
+    (fun h => tc_652 t h.le h2 h0)
+
+lemma cov_654_656 :
+    ∀ t,
+      (155167371020807 / 204800000000000 : ℝ) ≤ t →
+        t ≤ (194899235804257 / 256000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (388545108904183 / 512000000000000 : ℝ)).elim (fun h => tc_654 t h1 h h0)
+    (fun h => tc_655 t h.le h2 h0)
+
+lemma cov_653_656 :
+    ∀ t,
+      (96822936549963 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (194899235804257 / 256000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (155167371020807 / 204800000000000 : ℝ)).elim (fun h => tc_653 t h1 h h0)
+    (fun h => cov_654_656 t h.le h2 h0)
+
+lemma cov_651_656 :
+    ∀ t,
+      (386038383495521 / 512000000000000 : ℝ) ≤ t →
+        t ≤ (194899235804257 / 256000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (96822936549963 / 128000000000000 : ℝ)).elim (fun h => cov_651_653 t h1 h h0)
+    (fun h => cov_653_656 t h.le h2 h0)
+
+lemma cov_657_659 :
+    ∀ t,
+      (78210366862569 / 102400000000000 : ℝ) ≤ t →
+        t ≤ (393558559721507 / 512000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (49038149627147 / 64000000000000 : ℝ)).elim (fun h => tc_657 t h1 h h0)
+    (fun h => tc_658 t h.le h2 h0)
+
+lemma cov_656_659 :
+    ∀ t,
+      (194899235804257 / 256000000000000 : ℝ) ≤ t →
+        t ≤ (393558559721507 / 512000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (78210366862569 / 102400000000000 : ℝ)).elim (fun h => tc_656 t h1 h h0)
+    (fun h => cov_657_659 t h.le h2 h0)
+
+lemma cov_660_662 :
+    ∀ t,
+      (197405961212919 / 256000000000000 : ℝ) ≤ t →
+        t ≤ (794637295669 / 1024000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (396065285130169 / 512000000000000 : ℝ)).elim (fun h => tc_660 t h1 h h0)
+    (fun h => tc_661 t h.le h2 h0)
+
+lemma cov_659_662 :
+    ∀ t,
+      (393558559721507 / 512000000000000 : ℝ) ≤ t →
+        t ≤ (794637295669 / 1024000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (197405961212919 / 256000000000000 : ℝ)).elim (fun h => tc_659 t h1 h h0)
+    (fun h => cov_660_662 t h.le h2 h0)
+
+lemma cov_656_662 :
+    ∀ t,
+      (194899235804257 / 256000000000000 : ℝ) ≤ t →
+        t ≤ (794637295669 / 1024000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (393558559721507 / 512000000000000 : ℝ)).elim (fun h => cov_656_659 t h1 h h0)
+    (fun h => cov_659_662 t h.le h2 h0)
+
+lemma cov_651_662 :
+    ∀ t,
+      (386038383495521 / 512000000000000 : ℝ) ≤ t →
+        t ≤ (794637295669 / 1024000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (194899235804257 / 256000000000000 : ℝ)).elim (fun h => cov_651_656 t h1 h h0)
+    (fun h => cov_656_662 t h.le h2 h0)
+
+lemma cov_641_662 :
+    ∀ t,
+      (23740009868623 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (794637295669 / 1024000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (386038383495521 / 512000000000000 : ℝ)).elim (fun h => cov_641_651 t h1 h h0)
+    (fun h => cov_651_662 t h.le h2 h0)
+
+lemma cov_662_664 :
+    ∀ t,
+      (794637295669 / 1024000000000 : ℝ) ≤ t →
+        t ≤ (199912686621581 / 256000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (398572010538831 / 512000000000000 : ℝ)).elim (fun h => tc_662 t h1 h h0)
+    (fun h => tc_663 t h.le h2 h0)
+
+lemma cov_665_667 :
+    ∀ t,
+      (25145756165739 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (101836387367287 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (202419412030243 / 256000000000000 : ℝ)).elim (fun h => tc_665 t h1 h h0)
+    (fun h => tc_666 t h.le h2 h0)
+
+lemma cov_664_667 :
+    ∀ t,
+      (199912686621581 / 256000000000000 : ℝ) ≤ t →
+        t ≤ (101836387367287 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (25145756165739 / 32000000000000 : ℝ)).elim (fun h => tc_664 t h1 h h0)
+    (fun h => cov_665_667 t h.le h2 h0)
+
+lemma cov_662_667 :
+    ∀ t,
+      (794637295669 / 1024000000000 : ℝ) ≤ t →
+        t ≤ (101836387367287 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (199912686621581 / 256000000000000 : ℝ)).elim (fun h => cov_662_664 t h1 h h0)
+    (fun h => cov_664_667 t h.le h2 h0)
+
+lemma cov_668_670 :
+    ∀ t,
+      (40985227487781 / 51200000000000 : ℝ) ≤ t →
+        t ≤ (104343112775949 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (51544875035809 / 64000000000000 : ℝ)).elim (fun h => tc_668 t h1 h h0)
+    (fun h => tc_669 t h.le h2 h0)
+
+lemma cov_667_670 :
+    ∀ t,
+      (101836387367287 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (104343112775949 / 128000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (40985227487781 / 51200000000000 : ℝ)).elim (fun h => tc_667 t h1 h h0)
+    (fun h => cov_668_670 t h.le h2 h0)
+
+lemma cov_671_673 :
+    ∀ t,
+      (2639911887007 / 3200000000000 : ℝ) ≤ t →
+        t ≤ (54051600444471 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (106849838184611 / 128000000000000 : ℝ)).elim (fun h => tc_671 t h1 h h0)
+    (fun h => tc_672 t h.le h2 h0)
+
+lemma cov_670_673 :
+    ∀ t,
+      (104343112775949 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (54051600444471 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (2639911887007 / 3200000000000 : ℝ)).elim (fun h => tc_670 t h1 h h0)
+    (fun h => cov_671_673 t h.le h2 h0)
+
+lemma cov_667_673 :
+    ∀ t,
+      (101836387367287 / 128000000000000 : ℝ) ≤ t →
+        t ≤ (54051600444471 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (104343112775949 / 128000000000000 : ℝ)).elim (fun h => cov_667_670 t h1 h h0)
+    (fun h => cov_670_673 t h.le h2 h0)
+
+lemma cov_662_673 :
+    ∀ t,
+      (794637295669 / 1024000000000 : ℝ) ≤ t →
+        t ≤ (54051600444471 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (101836387367287 / 128000000000000 : ℝ)).elim (fun h => cov_662_667 t h1 h h0)
+    (fun h => cov_667_673 t h.le h2 h0)
+
+lemma cov_673_675 :
+    ∀ t,
+      (54051600444471 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (56558325853133 / 64000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (27652481574401 / 32000000000000 : ℝ)).elim (fun h => tc_673 t h1 h h0)
+    (fun h => tc_674 t h.le h2 h0)
+
+lemma cov_676_678 :
+    ∀ t,
+      (7226461069683 / 8000000000000 : ℝ) ≤ t →
+        t ≤ (15706284843697 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (30159206983063 / 32000000000000 : ℝ)).elim (fun h => tc_676 t h1 h h0)
+    (fun h => tc_677 t h.le h2 h0)
+
+lemma cov_675_678 :
+    ∀ t,
+      (56558325853133 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (15706284843697 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (7226461069683 / 8000000000000 : ℝ)).elim (fun h => tc_675 t h1 h h0)
+    (fun h => cov_676_678 t h.le h2 h0)
+
+lemma cov_673_678 :
+    ∀ t,
+      (54051600444471 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (15706284843697 / 16000000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (56558325853133 / 64000000000000 : ℝ)).elim (fun h => cov_673_675 t h1 h h0)
+    (fun h => cov_675_678 t h.le h2 h0)
+
+lemma cov_679_681 :
+    ∀ t,
+      (4239911887007 / 4000000000000 : ℝ) ≤ t →
+        t ≤ (1946637295669 / 1600000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (18213010252359 / 16000000000000 : ℝ)).elim (fun h => tc_679 t h1 h h0)
+    (fun h => tc_680 t h.le h2 h0)
+
+lemma cov_678_681 :
+    ∀ t,
+      (15706284843697 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (1946637295669 / 1600000000000 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (4239911887007 / 4000000000000 : ℝ)).elim (fun h => tc_678 t h1 h h0)
+    (fun h => cov_679_681 t h.le h2 h0)
+
+lemma cov_682_684 :
+    ∀ t,
+      (2746637295669 / 2000000000000 : ℝ) ≤ t → t ≤ (2 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (6746637295669 / 4000000000000 : ℝ)).elim (fun h => tc_682 t h1 h h0)
+    (fun h => tc_683 t h.le h2 h0)
+
+lemma cov_681_684 :
+    ∀ t,
+      (1946637295669 / 1600000000000 : ℝ) ≤ t → t ≤ (2 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (2746637295669 / 2000000000000 : ℝ)).elim (fun h => tc_681 t h1 h h0)
+    (fun h => cov_682_684 t h.le h2 h0)
+
+lemma cov_678_684 :
+    ∀ t,
+      (15706284843697 / 16000000000000 : ℝ) ≤ t →
+        t ≤ (2 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (1946637295669 / 1600000000000 : ℝ)).elim (fun h => cov_678_681 t h1 h h0)
+    (fun h => cov_681_684 t h.le h2 h0)
+
+lemma cov_673_684 :
+    ∀ t,
+      (54051600444471 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (2 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (15706284843697 / 16000000000000 : ℝ)).elim (fun h => cov_673_678 t h1 h h0)
+    (fun h => cov_678_684 t h.le h2 h0)
+
+lemma cov_662_684 :
+    ∀ t,
+      (794637295669 / 1024000000000 : ℝ) ≤ t → t ≤ (2 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (54051600444471 / 64000000000000 : ℝ)).elim (fun h => cov_662_673 t h1 h h0)
+    (fun h => cov_673_684 t h.le h2 h0)
+
+lemma cov_641_684 :
+    ∀ t,
+      (23740009868623 / 32000000000000 : ℝ) ≤ t →
+        t ≤ (2 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (794637295669 / 1024000000000 : ℝ)).elim (fun h => cov_641_662 t h1 h h0)
+    (fun h => cov_662_684 t h.le h2 h0)
+
+lemma cov_598_684 :
+    ∀ t,
+      (44224781932669 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (2 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (23740009868623 / 32000000000000 : ℝ)).elim (fun h => cov_598_641 t h1 h h0)
+    (fun h => cov_641_684 t h.le h2 h0)
+
+lemma cov_513_684 :
+    ∀ t,
+      (36111549776069 / 64000000000000 : ℝ) ≤ t →
+        t ≤ (2 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (44224781932669 / 64000000000000 : ℝ)).elim (fun h => cov_513_598 t h1 h h0)
+    (fun h => cov_598_684 t h.le h2 h0)
+
+lemma cov_342_684 :
+    ∀ t,
+      (2543485761489 / 8000000000000 : ℝ) ≤ t → t ≤ (2 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (36111549776069 / 64000000000000 : ℝ)).elim (fun h => cov_342_513 t h1 h h0)
+    (fun h => cov_513_684 t h.le h2 h0)
+
+lemma cov_0_684 : ∀ t, (0 : ℝ) ≤ t → t ≤ (2 : ℝ) → 0 < t → 2 * Uρ t - Vfield t ≤ M0 :=
+  fun t h1 h2 h0 =>
+  (le_or_gt t (2543485761489 / 8000000000000 : ℝ)).elim (fun h => cov_0_342 t h1 h h0)
+    (fun h => cov_342_684 t h.le h2 h0)
+
+/-- Lemma 6.1 of the paper: the potential inequality (6.2) on the whole half-line. -/
+theorem potential_ineq (t : ℝ) (ht : 0 < t) : 2 * Uρ t - Vfield t ≤ M0 := by
+  rcases le_or_gt t 2 with h | h
+  · exact cov_0_684 t ht.le h ht
+  · exact potential_ineq_tail h.le
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/Tail.lean b/LeanPool/Zeta5Irrational/Table/Tail.lean
new file mode 100644
index 0000000000..1901976dd8
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/Tail.lean
@@ -0,0 +1,188 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Potential
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.ReduceModChar
+import Mathlib.Tactic.FieldSimp
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+import Mathlib.Tactic.Positivity.Basic
+
+/-! # The tail `t ≥ 2` of the potential inequality (6.2): the bound (6.8)
+-/
+
+@[expose] public section
+
+open Finset Set
+
+namespace Zeta5Irrational
+
+/-- For `t ≥ b`, `Uω a b t ≤ log t`. -/
+lemma Uω_le_log {a b t : ℝ} (ha : 0 < a) (hab : a < b) (h : b ≤ t) : Uω a b t ≤ Real.log t := by
+  rw [Uω_of_ge hab h]
+  have hm : 0 < t - (a + b) / 2 := by linarith
+  have hsq : Real.sqrt ((t - a) * (t - b)) ≤ t - (a + b) / 2 :=
+    sqrt_le_of_sq_le hm.le (by nlinarith)
+  have hs0 := Real.sqrt_nonneg ((t - a) * (t - b))
+  exact Real.log_le_log (by linarith) (by linarith)
+
+/-- For `t ≥ 2`, `Uρ t ≤ (37/40) log t` (total mass of `ρ` is `37/40`). -/
+lemma Uρ_le_log {t : ℝ} (ht : 2 ≤ t) : Uρ t ≤ 37 / 40 * Real.log t := by
+  rw [Uρ_expand]
+  have h1 :=
+    mul_le_mul_of_nonneg_left
+      (Uω_le_log (a := aρ 1) (b := bρ 1) (by norm_num [aρ, table1]) (by norm_num [aρ, bρ, table1])
+        ((by norm_num [bρ, table1] : bρ 1 ≤ 2).trans ht))
+      (cρ_pos 1 (by norm_num)).le
+  have h2 :=
+    mul_le_mul_of_nonneg_left
+      (Uω_le_log (a := aρ 2) (b := bρ 2) (by norm_num [aρ, table1]) (by norm_num [aρ, bρ, table1])
+        ((by norm_num [bρ, table1] : bρ 2 ≤ 2).trans ht))
+      (cρ_pos 2 (by norm_num)).le
+  have h3 :=
+    mul_le_mul_of_nonneg_left
+      (Uω_le_log (a := aρ 3) (b := bρ 3) (by norm_num [aρ, table1]) (by norm_num [aρ, bρ, table1])
+        ((by norm_num [bρ, table1] : bρ 3 ≤ 2).trans ht))
+      (cρ_pos 3 (by norm_num)).le
+  have h4 :=
+    mul_le_mul_of_nonneg_left
+      (Uω_le_log (a := aρ 4) (b := bρ 4) (by norm_num [aρ, table1]) (by norm_num [aρ, bρ, table1])
+        ((by norm_num [bρ, table1] : bρ 4 ≤ 2).trans ht))
+      (cρ_pos 4 (by norm_num)).le
+  have h5 :=
+    mul_le_mul_of_nonneg_left
+      (Uω_le_log (a := aρ 5) (b := bρ 5) (by norm_num [aρ, table1]) (by norm_num [aρ, bρ, table1])
+        ((by norm_num [bρ, table1] : bρ 5 ≤ 2).trans ht))
+      (cρ_pos 5 (by norm_num)).le
+  have h6 :=
+    mul_le_mul_of_nonneg_left
+      (Uω_le_log (a := aρ 6) (b := bρ 6) (by norm_num [aρ, table1]) (by norm_num [aρ, bρ, table1])
+        ((by norm_num [bρ, table1] : bρ 6 ≤ 2).trans ht))
+      (cρ_pos 6 (by norm_num)).le
+  have h7 :=
+    mul_le_mul_of_nonneg_left
+      (Uω_le_log (a := aρ 7) (b := bρ 7) (by norm_num [aρ, table1]) (by norm_num [aρ, bρ, table1])
+        ((by norm_num [bρ, table1] : bρ 7 ≤ 2).trans ht))
+      (cρ_pos 7 (by norm_num)).le
+  have h8 :=
+    mul_le_mul_of_nonneg_left
+      (Uω_le_log (a := aρ 8) (b := bρ 8) (by norm_num [aρ, table1]) (by norm_num [aρ, bρ, table1])
+        ((by norm_num [bρ, table1] : bρ 8 ≤ 2).trans ht))
+      (cρ_pos 8 (by norm_num)).le
+  have h9 :=
+    mul_le_mul_of_nonneg_left
+      (Uω_le_log (a := aρ 9) (b := bρ 9) (by norm_num [aρ, table1]) (by norm_num [aρ, bρ, table1])
+        ((by norm_num [bρ, table1] : bρ 9 ≤ 2).trans ht))
+      (cρ_pos 9 (by norm_num)).le
+  have h10 :=
+    mul_le_mul_of_nonneg_left
+      (Uω_le_log (a := aρ 10) (b := bρ 10) (by norm_num [aρ, table1])
+        (by norm_num [aρ, bρ, table1]) ((by norm_num [bρ, table1] : bρ 10 ≤ 2).trans ht))
+      (cρ_pos 10 (by norm_num)).le
+  have h11 :=
+    mul_le_mul_of_nonneg_left
+      (Uω_le_log (a := aρ 11) (b := bρ 11) (by norm_num [aρ, table1])
+        (by norm_num [aρ, bρ, table1]) ((by norm_num [bρ, table1] : bρ 11 ≤ 2).trans ht))
+      (cρ_pos 11 (by norm_num)).le
+  have h12 :=
+    mul_le_mul_of_nonneg_left
+      (Uω_le_log (a := aρ 12) (b := bρ 12) (by norm_num [aρ, table1])
+        (by norm_num [aρ, bρ, table1]) ((by norm_num [bρ, table1] : bρ 12 ≤ 2).trans ht))
+      (cρ_pos 12 (by norm_num)).le
+  have h13 :=
+    mul_le_mul_of_nonneg_left
+      (Uω_le_log (a := aρ 13) (b := bρ 13) (by norm_num [aρ, table1])
+        (by norm_num [aρ, bρ, table1]) ((by norm_num [bρ, table1] : bρ 13 ≤ 2).trans ht))
+      (cρ_pos 13 (by norm_num)).le
+  have h14 :=
+    mul_le_mul_of_nonneg_left
+      (Uω_le_log (a := aρ 14) (b := bρ 14) (by norm_num [aρ, table1])
+        (by norm_num [aρ, bρ, table1]) ((by norm_num [bρ, table1] : bρ 14 ≤ 2).trans ht))
+      (cρ_pos 14 (by norm_num)).le
+  have h15 :=
+    mul_le_mul_of_nonneg_left
+      (Uω_le_log (a := aρ 15) (b := bρ 15) (by norm_num [aρ, table1])
+        (by norm_num [aρ, bρ, table1]) ((by norm_num [bρ, table1] : bρ 15 ≤ 2).trans ht))
+      (cρ_pos 15 (by norm_num)).le
+  have h16 :=
+    mul_le_mul_of_nonneg_left
+      (Uω_le_log (a := aρ 16) (b := bρ 16) (by norm_num [aρ, table1])
+        (by norm_num [aρ, bρ, table1]) ((by norm_num [bρ, table1] : bρ 16 ≤ 2).trans ht))
+      (cρ_pos 16 (by norm_num)).le
+  norm_num [cρ, table1] at h1 h2 h3 h4 h5 h6 h7 h8 h9 h10 h11 h12 h13 h14 h15 h16 ⊢
+  linarith
+
+/-- The lower bound for `V` used in (6.8): `V(t) ≥ 2π√t + log t - 6α (log t + α²/t)`. -/
+lemma Vfield_ge_tail {t : ℝ} (ht : 2 ≤ t) :
+    2 * Real.pi * Real.sqrt t + Real.log t - 6 * (3 / 40) * (Real.log t + (3 / 40) ^ 2 / t) ≤
+      Vfield t := by
+  unfold Vfield
+  have ht0 : 0 < t := by linarith
+  have hcont : Continuous fun u : ℝ => Real.log (t + u ^ 2) :=
+    (continuous_const.add (continuous_pow 2)).log
+      (fun u => by simp only [Pi.add_apply]; nlinarith [sq_nonneg u])
+  have h1 : Real.log t ≤ ∫ u in (0 : ℝ)..1, Real.log (t + u ^ 2) := by
+    have e : ∫ u in (0 : ℝ)..1, Real.log t = Real.log t := by simp
+    rw [← e]
+    refine
+      intervalIntegral.integral_mono_on zero_le_one (by simp) (hcont.intervalIntegrable _ _) ?_
+    intro u _
+    exact Real.log_le_log ht0 (by nlinarith)
+  have h2 :
+    ∫ u in (0 : ℝ)..(3 / 40), Real.log (t + u ^ 2) ≤ 3 / 40 * (Real.log t + (3 / 40) ^ 2 / t) := by
+    have e :
+      ∫ u in (0 : ℝ)..(3 / 40), Real.log (t + (3 / 40) ^ 2) =
+        3 / 40 * Real.log (t + (3 / 40) ^ 2) :=
+      by simp
+    have hle :
+      ∫ u in (0 : ℝ)..(3 / 40), Real.log (t + u ^ 2) ≤
+        ∫ u in (0 : ℝ)..(3 / 40), Real.log (t + (3 / 40) ^ 2) := by
+      refine
+        intervalIntegral.integral_mono_on (by norm_num) (hcont.intervalIntegrable _ _) (by simp)
+          ?_
+      intro u hu
+      exact Real.log_le_log (by positivity) (by nlinarith [hu.1, hu.2])
+    have hlog : Real.log (t + (3 / 40) ^ 2) ≤ Real.log t + (3 / 40) ^ 2 / t := by
+      have e2 : Real.log (t + (3 / 40) ^ 2) = Real.log t + Real.log (1 + (3 / 40) ^ 2 / t) := by
+        rw [← Real.log_mul ht0.ne' (by positivity)]
+        congr 1; field_simp
+      have := Real.log_le_sub_one_of_pos (show 0 < 1 + (3 / 40) ^ 2 / t by positivity)
+      linarith
+    linarith
+  linarith
+
+/-- The tail `t ≥ 2` of the potential inequality (6.2), via (6.8). -/
+lemma potential_ineq_tail {t : ℝ} (ht : 2 ≤ t) : 2 * Uρ t - Vfield t ≤ M0 := by
+  have ht0 : 0 < t := by linarith
+  have h1 := Uρ_le_log ht
+  have h2 := Vfield_ge_tail ht
+  have hs : Real.sqrt t ^ 2 = t := Real.sq_sqrt ht0.le
+  have hs0 : 0 < Real.sqrt t := Real.sqrt_pos.mpr ht0
+  have hs2 : (141 / 100 : ℝ) ≤ Real.sqrt t :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num; linarith)
+  have hlog : Real.log t ≤ 2 * (Real.sqrt t - 1) := by
+    have e : Real.log (Real.sqrt t ^ 2) = 2 * Real.log (Real.sqrt t) := by
+      rw [Real.log_pow];
+      norm_num
+    rw [hs] at e
+    rw [e]
+    linarith [Real.log_le_sub_one_of_pos hs0]
+  have hq : (3 / 40 : ℝ) ^ 2 / t ≤ (3 / 40) ^ 2 / 2 :=
+    div_le_div_of_nonneg_left (by norm_num) (by norm_num) ht
+  have hpi := Real.pi_gt_d2
+  have hprod : 314 / 100 * Real.sqrt t ≤ Real.pi * Real.sqrt t :=
+    mul_le_mul_of_nonneg_right (by norm_num at hpi; linarith) hs0.le
+  unfold M0
+  linarith
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/U00.lean b/LeanPool/Zeta5Irrational/Table/U00.lean
new file mode 100644
index 0000000000..0a5c8f9651
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/U00.lean
@@ -0,0 +1,4604 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-! # Certified arcsine potential bounds (U00) -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma U_0_1 :
+    Uω (aρ 1) (bρ 1) (0 : ℝ) ≤ (-(12712663703508542025099 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (123769636673849 / 20000000000000000 : ℝ) ≤
+      (-(12712663703508542025099 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show
+          (123769636673849 / 20000000000000000 : ℝ) =
+            (123769636673849 / 78125000000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (123769636673849 / 78125000000000 : ℝ) ≤
+            (1150279902491457974901 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (59272418588849 / 10000000000000000 : ℝ)) (q :=
+      (123769636673849 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_0_2 :
+    Uω (aρ 2) (bρ 2) (0 : ℝ) ≤ (-(1962983986998948336949 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (147824769578301 / 20000000000000000 : ℝ) ≤
+      (-(1962983986998948336949 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show
+          (147824769578301 / 20000000000000000 : ℝ) =
+            (147824769578301 / 78125000000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (147824769578301 / 78125000000000 : ℝ) ≤
+            (255086989961051663051 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (59558538983301 / 10000000000000000 : ℝ)) (q :=
+      (147824769578301 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_0_3 :
+    Uω (aρ 3) (bρ 3) (0 : ℝ) ≤ (-(46267520391835680481277 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (97864936683177 / 10000000000000000 : ℝ) ≤
+      (-(46267520391835680481277 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (97864936683177 / 10000000000000000 : ℝ) = (97864936683177 / 78125000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (97864936683177 / 78125000000000 : ℝ) ≤
+            (2252782229164319518723 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (30033768210677 / 5000000000000000 : ℝ)) (q :=
+      (97864936683177 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_0_4 :
+    Uω (aρ 4) (bρ 4) (0 : ℝ) ≤ (-(5359553913723040081263 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17171580100553 / 1250000000000000 : ℝ) ≤
+      (-(5359553913723040081263 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (17171580100553 / 1250000000000000 : ℝ) = (17171580100553 / 9765625000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17171580100553 / 9765625000000 : ℝ) ≤
+            (705483913901959918737 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (7420503369 / 1220703125000 : ℝ)) (q :=
+      (17171580100553 / 1250000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_0_5 :
+    Uω (aρ 5) (bρ 5) (0 : ℝ) ≤ (-(39275139978819449528723 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (393851349704671 / 20000000000000000 : ℝ) ≤
+      (-(39275139978819449528723 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (393851349704671 / 20000000000000000 : ℝ) =
+            (393851349704671 / 312500000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (393851349704671 / 312500000000000 : ℝ) ≤
+            (2313690839180550471277 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (61704912359671 / 10000000000000000 : ℝ)) (q :=
+      (393851349704671 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_0_6 :
+    Uω (aρ 6) (bρ 6) (0 : ℝ) ≤ (-(7143339604213661208809 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (281088781634513 / 10000000000000000 : ℝ) ≤
+      (-(7143339604213661208809 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (281088781634513 / 10000000000000000 : ℝ) =
+            (281088781634513 / 156250000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (281088781634513 / 156250000000000 : ℝ) ≤
+            (1174426559386338791191 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (31395375392013 / 5000000000000000 : ℝ)) (q :=
+      (281088781634513 / 10000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_0_7 :
+    Uω (aρ 7) (bρ 7) (0 : ℝ) ≤ (-(32351824960621028314427 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (393530224619551 / 10000000000000000 : ℝ) ≤
+      (-(32351824960621028314427 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (393530224619551 / 10000000000000000 : ℝ) =
+            (393530224619551 / 312500000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (393530224619551 / 312500000000000 : ℝ) ≤
+            (2305534054378971685573 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (32006127997051 / 5000000000000000 : ℝ)) (q :=
+      (393530224619551 / 10000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_0_8 :
+    Uω (aρ 8) (bρ 8) (0 : ℝ) ≤ (-(7315709960795885799737 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (214383325750023 / 4000000000000000 : ℝ) ≤
+      (-(7315709960795885799737 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (214383325750023 / 4000000000000000 : ℝ) =
+            (214383325750023 / 125000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (214383325750023 / 125000000000000 : ℝ) ≤
+            (1348629792954114200263 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (13065356833023 / 2000000000000000 : ℝ)) (q :=
+      (214383325750023 / 4000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_0_9 :
+    Uω (aρ 9) (bρ 9) (0 : ℝ) ≤ (-(13245699571552860366621 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1414240104474349 / 20000000000000000 : ℝ) ≤
+      (-(13245699571552860366621 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1414240104474349 / 20000000000000000 : ℝ) =
+            (1414240104474349 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1414240104474349 / 1250000000000000 : ℝ) ≤
+            (617244034447139633379 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (66684631529349 / 10000000000000000 : ℝ)) (q :=
+      (1414240104474349 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_0_10 :
+    Uω (aρ 10) (bρ 10) (0 : ℝ) ≤ (-(4811289393237367803367 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (902073204108467 / 10000000000000000 : ℝ) ≤
+      (-(4811289393237367803367 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (902073204108467 / 10000000000000000 : ℝ) =
+            (902073204108467 / 625000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (902073204108467 / 625000000000000 : ℝ) ≤
+            (733888049162632196633 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (34015950888467 / 5000000000000000 : ℝ)) (q :=
+      (902073204108467 / 10000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_0_11 :
+    Uω (aρ 11) (bρ 11) (0 : ℝ) ≤ (-(21964674985840736154593 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2223905254151181 / 20000000000000000 : ℝ) ≤
+      (-(21964674985840736154593 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2223905254151181 / 20000000000000000 : ℝ) =
+            (2223905254151181 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2223905254151181 / 1250000000000000 : ℝ) ≤
+            (5761212226159263845407 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (69314121506181 / 10000000000000000 : ℝ)) (q :=
+      (2223905254151181 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_0_12 :
+    Uω (aρ 12) (bρ 12) (0 : ℝ) ≤ (-(4043272987456227879851 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1324385526764569 / 10000000000000000 : ℝ) ≤
+      (-(4043272987456227879851 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1324385526764569 / 10000000000000000 : ℝ) =
+            (1324385526764569 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1324385526764569 / 1250000000000000 : ℝ) ≤
+            (115610094343772120149 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (35239912099569 / 5000000000000000 : ℝ)) (q :=
+      (1324385526764569 / 10000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_0_13 :
+    Uω (aρ 13) (bρ 13) (0 : ℝ) ≤ (-(4702168270076774782451 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3049156407763681 / 20000000000000000 : ℝ) ≤
+      (-(4702168270076774782451 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3049156407763681 / 20000000000000000 : ℝ) =
+            (3049156407763681 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3049156407763681 / 2500000000000000 : ℝ) ≤
+            (496435582173225217549 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (71483436838681 / 10000000000000000 : ℝ)) (q :=
+      (3049156407763681 / 20000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_0_14 :
+    Uω (aρ 14) (bρ 14) (0 : ℝ) ≤ (-(2217195343887504332233 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3393887425779091 / 20000000000000000 : ℝ) ≤
+      (-(2217195343887504332233 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3393887425779091 / 20000000000000000 : ℝ) =
+            (3393887425779091 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3393887425779091 / 2500000000000000 : ℝ) ≤
+            (382106582237495667767 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (72287169809091 / 10000000000000000 : ℝ)) (q :=
+      (3393887425779091 / 20000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_0_15 :
+    Uω (aρ 15) (bρ 15) (0 : ℝ) ≤ (-(16999000975580328413317 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1827017753520839 / 10000000000000000 : ℝ) ≤
+      (-(16999000975580328413317 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1827017753520839 / 10000000000000000 : ℝ) =
+            (1827017753520839 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1827017753520839 / 1250000000000000 : ℝ) ≤
+            (3795414433419671586683 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (36430986828339 / 5000000000000000 : ℝ)) (q :=
+      (1827017753520839 / 10000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_0_16 :
+    Uω (aρ 16) (bρ 16) (0 : ℝ) ≤ (-(1036850562756824007821 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1903366559739307 / 10000000000000000 : ℝ) ≤
+      (-(1036850562756824007821 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1903366559739307 / 10000000000000000 : ℝ) =
+            (1903366559739307 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1903366559739307 / 1250000000000000 : ℝ) ≤
+            (262800400305675992179 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (36593967291807 / 5000000000000000 : ℝ)) (q :=
+      (1903366559739307 / 10000000000000000 : ℝ)) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [aρ_16]) (by norm_num) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [aρ_16, bρ_16]) hlg
+
+lemma U_0 : Uρ (0 : ℝ) ≤ (-(27385656762631359502577 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (0 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(12712663703508542025099 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_0_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (0 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1962983986998948336949 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_0_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (0 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(46267520391835680481277 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_0_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (0 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(5359553913723040081263 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_0_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (0 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(39275139978819449528723 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_0_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (0 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(7143339604213661208809 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_0_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (0 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(32351824960621028314427 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_0_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (0 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(7315709960795885799737 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_0_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (0 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(13245699571552860366621 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_0_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (0 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(4811289393237367803367 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_0_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (0 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(21964674985840736154593 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_0_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (0 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(4043272987456227879851 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_0_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (0 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4702168270076774782451 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_0_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (0 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(2217195343887504332233 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_0_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (0 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(16999000975580328413317 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_0_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (0 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) *
+        (-(1036850562756824007821 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_0_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_1_1 :
+    Uω (aρ 1) (bρ 1) (7174131 / 200000000000 : ℝ) ≤
+      (-(50911373318893302541507 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (61510201432479 / 10000000000000000 : ℝ) ≤
+      (-(50911373318893302541507 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show
+          (61510201432479 / 10000000000000000 : ℝ) = (61510201432479 / 39062500000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (61510201432479 / 39062500000000 : ℝ) ≤
+            (4540401105106697458493 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (29440945664979 / 5000000000000000 : ℝ)) (q :=
+      (61510201432479 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_1_2 :
+    Uω (aρ 2) (bρ 2) (7174131 / 200000000000 : ℝ) ≤
+      (-(49135098078763348601453 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (146933153106433 / 20000000000000000 : ℝ) ≤
+      (-(49135098078763348601453 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show
+          (146933153106433 / 20000000000000000 : ℝ) =
+            (146933153106433 / 78125000000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (146933153106433 / 78125000000000 : ℝ) ≤
+            (6316676345236651398547 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (59025629061433 / 10000000000000000 : ℝ)) (q :=
+      (146933153106433 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_1_3 :
+    Uω (aρ 3) (bρ 3) (7174131 / 200000000000 : ℝ) ≤
+      (-(46327645463733161241931 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (194556576856917 / 20000000000000000 : ℝ) ≤
+      (-(46327645463733161241931 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (194556576856917 / 20000000000000000 : ℝ) =
+            (194556576856917 / 156250000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (194556576856917 / 156250000000000 : ℝ) ≤
+            (2192657157266838758069 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (59252946461917 / 10000000000000000 : ℝ)) (q :=
+      (194556576856917 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_1_4 :
+    Uω (aρ 4) (bρ 4) (7174131 / 200000000000 : ℝ) ≤
+      (-(42936065490516840879561 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (273111736229039 / 20000000000000000 : ℝ) ≤
+      (-(42936065490516840879561 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (273111736229039 / 20000000000000000 : ℝ) =
+            (273111736229039 / 156250000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (273111736229039 / 156250000000000 : ℝ) ≤
+            (5584237130483159120439 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (59513924769039 / 10000000000000000 : ℝ)) (q :=
+      (273111736229039 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_1_5 :
+    Uω (aρ 5) (bρ 5) (7174131 / 200000000000 : ℝ) ≤
+      (-(7866842272424100993973 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (97882918342561 / 5000000000000000 : ℝ) ≤
+      (-(7866842272424100993973 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (97882918342561 / 5000000000000000 : ℝ) = (97882918342561 / 78125000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (97882918342561 / 78125000000000 : ℝ) ≤
+            (450923891175899006027 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (14935985643811 / 2500000000000000 : ℝ)) (q :=
+      (97882918342561 / 5000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_1_6 :
+    Uω (aρ 6) (bρ 6) (7174131 / 200000000000 : ℝ) ≤
+      (-(1431007411209519275791 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (139724784359361 / 5000000000000000 : ℝ) ≤
+      (-(1431007411209519275791 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (139724784359361 / 5000000000000000 : ℝ) =
+            (139724784359361 / 78125000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (139724784359361 / 78125000000000 : ℝ) ≤
+            (232545821510480724209 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (14967757875611 / 2500000000000000 : ℝ)) (q :=
+      (139724784359361 / 5000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_1_7 :
+    Uω (aρ 7) (bρ 7) (7174131 / 200000000000 : ℝ) ≤
+      (-(40512196593284858497 / 12500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (195628501984759 / 5000000000000000 : ℝ) ≤
+      (-(40512196593284858497 / 12500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (195628501984759 / 5000000000000000 : ℝ) =
+            (195628501984759 / 156250000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (195628501984759 / 156250000000000 : ℝ) ≤
+            (2809502175465141503 / 12500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (14956130311009 / 2500000000000000 : ℝ)) (q :=
+      (195628501984759 / 5000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_1_8 :
+    Uω (aρ 8) (bρ 8) (7174131 / 200000000000 : ℝ) ≤
+      (-(14660146048277734861007 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (266443970762523 / 5000000000000000 : ℝ) ≤
+      (-(14660146048277734861007 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (266443970762523 / 5000000000000000 : ℝ) =
+            (266443970762523 / 156250000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (266443970762523 / 156250000000000 : ℝ) ≤
+            (2668533459222265138993 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (14886186253773 / 2500000000000000 : ℝ)) (q :=
+      (266443970762523 / 5000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_1_9 :
+    Uω (aρ 9) (bρ 9) (7174131 / 200000000000 : ℝ) ≤
+      (-(5309696547214735406207 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (70309505590531 / 1000000000000000 : ℝ) ≤
+      (-(5309696547214735406207 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (70309505590531 / 1000000000000000 : ℝ) = (70309505590531 / 62500000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (70309505590531 / 62500000000000 : ℝ) ≤
+            (235480895185264593793 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (2949667270781 / 500000000000000 : ℝ)) (q :=
+      (70309505590531 / 1000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_1_10 :
+    Uω (aρ 10) (bρ 10) (7174131 / 200000000000 : ℝ) ≤
+      (-(24113296642670996497633 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1793918993060211 / 20000000000000000 : ℝ) ≤
+      (-(24113296642670996497633 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1793918993060211 / 20000000000000000 : ℝ) =
+            (1793918993060211 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1793918993060211 / 1250000000000000 : ℝ) ≤
+            (3612590569329003502367 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (58163193170211 / 10000000000000000 : ℝ)) (q :=
+      (1793918993060211 / 20000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_1_11 :
+    Uω (aρ 11) (bρ 11) (7174131 / 200000000000 : ℝ) ≤
+      (-(5505358080027453352527 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2211318713475239 / 20000000000000000 : ℝ) ≤
+      (-(5505358080027453352527 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2211318713475239 / 20000000000000000 : ℝ) =
+            (2211318713475239 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2211318713475239 / 1250000000000000 : ℝ) ≤
+            (1426113722972546647473 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (57086287380239 / 10000000000000000 : ℝ)) (q :=
+      (2211318713475239 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_1_12 :
+    Uω (aρ 12) (bρ 12) (7174131 / 200000000000 : ℝ) ≤
+      (-(10136579785760210337841 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (526754018847003 / 4000000000000000 : ℝ) ≤
+      (-(10136579785760210337841 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (526754018847003 / 4000000000000000 : ℝ) =
+            (526754018847003 / 500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (526754018847003 / 500000000000000 : ℝ) ≤
+            (260627918739789662159 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (11167514291003 / 2000000000000000 : ℝ)) (q :=
+      (526754018847003 / 4000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12]) (by norm_num [aρ_12])
+      (by norm_num) (by norm_num [aρ_12, bρ_12]) (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_1_13 :
+    Uω (aρ 13) (bρ 13) (7174131 / 200000000000 : ℝ) ≤
+      (-(75462414402452552303 / 40000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (606369344182109 / 4000000000000000 : ℝ) ≤
+      (-(75462414402452552303 / 40000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (606369344182109 / 4000000000000000 : ℝ) =
+            (606369344182109 / 500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (606369344182109 / 500000000000000 : ℝ) ≤
+            (7715247233547447697 / 40000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (10906491307109 / 2000000000000000 : ℝ)) (q :=
+      (606369344182109 / 4000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13]) (by norm_num [aρ_13])
+      (by norm_num) (by norm_num [aρ_13, bρ_13]) (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_1_14 :
+    Uω (aρ 14) (bρ 14) (7174131 / 200000000000 : ℝ) ≤
+      (-(8897340153662197454051 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3374557626289937 / 20000000000000000 : ℝ) ≤
+      (-(8897340153662197454051 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3374557626289937 / 20000000000000000 : ℝ) =
+            (3374557626289937 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3374557626289937 / 2500000000000000 : ℝ) ≤
+            (1499867550837802545949 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (53316076869937 / 10000000000000000 : ℝ)) (q :=
+      (3374557626289937 / 20000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_1_15 :
+    Uω (aρ 15) (bρ 15) (7174131 / 200000000000 : ℝ) ≤
+      (-(8528150055396143754779 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3633158069658803 / 20000000000000000 : ℝ) ≤
+      (-(8528150055396143754779 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3633158069658803 / 20000000000000000 : ℝ) =
+            (3633158069658803 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3633158069658803 / 2500000000000000 : ℝ) ≤
+            (1869057649103856245221 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (52343242823803 / 10000000000000000 : ℝ)) (q :=
+      (3633158069658803 / 20000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_1_16 :
+    Uω (aρ 16) (bρ 16) (7174131 / 200000000000 : ℝ) ≤
+      (-(16647030518034101616959 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3784936920023489 / 20000000000000000 : ℝ) ≤
+      (-(16647030518034101616959 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3784936920023489 / 20000000000000000 : ℝ) =
+            (3784936920023489 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3784936920023489 / 2500000000000000 : ℝ) ≤
+            (4147384890965898383041 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (51750441678489 / 10000000000000000 : ℝ)) (q :=
+      (3784936920023489 / 20000000000000000 : ℝ)) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [aρ_16]) (by norm_num) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [aρ_16, bρ_16]) hlg
+
+lemma U_1 :
+    Uρ (7174131 / 200000000000 : ℝ) ≤
+      (-(27439164557690169188743 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (7174131 / 200000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(50911373318893302541507 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_1_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (7174131 / 200000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(49135098078763348601453 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_1_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (7174131 / 200000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(46327645463733161241931 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_1_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (7174131 / 200000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(42936065490516840879561 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_1_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (7174131 / 200000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(7866842272424100993973 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_1_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (7174131 / 200000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1431007411209519275791 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_1_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (7174131 / 200000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) * (-(40512196593284858497 / 12500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_1_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (7174131 / 200000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14660146048277734861007 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_1_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (7174131 / 200000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(5309696547214735406207 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_1_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (7174131 / 200000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(24113296642670996497633 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_1_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (7174131 / 200000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(5505358080027453352527 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_1_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (7174131 / 200000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(10136579785760210337841 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_1_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (7174131 / 200000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) * (-(75462414402452552303 / 40000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_1_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (7174131 / 200000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8897340153662197454051 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_1_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (7174131 / 200000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(8528150055396143754779 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_1_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (7174131 / 200000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) *
+        (-(16647030518034101616959 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_1_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_2_1 :
+    Uω (aρ 1) (bρ 1) (21522393 / 400000000000 : ℝ) ≤
+      (-(25470941903623612969977 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (30661408404607 / 5000000000000000 : ℝ) ≤
+      (-(25470941903623612969977 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show
+          (30661408404607 / 5000000000000000 : ℝ) = (30661408404607 / 19531250000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (30661408404607 / 19531250000000 : ℝ) ≤
+            (2254945308376387030023 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (14671618839607 / 2500000000000000 : ℝ)) (q :=
+      (30661408404607 / 5000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_2_2 :
+    Uω (aρ 2) (bρ 2) (21522393 / 400000000000 : ℝ) ≤
+      (-(49165552758527775040867 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (18310794199513 / 2500000000000000 : ℝ) ≤
+      (-(49165552758527775040867 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show
+          (18310794199513 / 2500000000000000 : ℝ) = (18310794199513 / 9765625000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18310794199513 / 9765625000000 : ℝ) ≤
+            (6286221665472224959133 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (7344772853263 / 1250000000000000 : ℝ)) (q :=
+      (18310794199513 / 2500000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_2_3 :
+    Uω (aρ 3) (bρ 3) (21522393 / 400000000000 : ℝ) ≤
+      (-(46358019844614904236579 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (38793303976897 / 4000000000000000 : ℝ) ≤
+      (-(46358019844614904236579 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (38793303976897 / 4000000000000000 : ℝ) = (38793303976897 / 31250000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (38793303976897 / 31250000000000 : ℝ) ≤
+            (2162282776385095763421 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (11768448552897 / 2000000000000000 : ℝ)) (q :=
+      (38793303976897 / 4000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_2_4 :
+    Uω (aρ 4) (bρ 4) (21522393 / 400000000000 : ℝ) ≤
+      (-(2148318328187890160217 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (272285430885961 / 20000000000000000 : ℝ) ≤
+      (-(2148318328187890160217 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (272285430885961 / 20000000000000000 : ℝ) =
+            (272285430885961 / 156250000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (272285430885961 / 156250000000000 : ℝ) ≤
+            (277696802862109839783 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (58866972700961 / 10000000000000000 : ℝ)) (q :=
+      (272285430885961 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_2_5 :
+    Uω (aρ 5) (bρ 5) (21522393 / 400000000000 : ℝ) ≤
+      (-(19682243075206791475787 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (48793514003197 / 2500000000000000 : ℝ) ≤
+      (-(19682243075206791475787 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (48793514003197 / 2500000000000000 : ℝ) = (48793514003197 / 39062500000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (48793514003197 / 39062500000000 : ℝ) ≤
+            (1112172333793208524213 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (7342466813197 / 1250000000000000 : ℝ)) (q :=
+      (48793514003197 / 2500000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_2_6 :
+    Uω (aρ 6) (bρ 6) (21522393 / 400000000000 : ℝ) ≤
+      (-(35805525469465709732703 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (139301499171973 / 5000000000000000 : ℝ) ≤
+      (-(35805525469465709732703 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (139301499171973 / 5000000000000000 : ℝ) =
+            (139301499171973 / 78125000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (139301499171973 / 78125000000000 : ℝ) ≤
+            (5783305348534290267297 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (14589311006973 / 2500000000000000 : ℝ)) (q :=
+      (139301499171973 / 5000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_2_7 :
+    Uω (aρ 7) (bρ 7) (21522393 / 400000000000 : ℝ) ≤
+      (-(2027518787259263390593 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (390063798273459 / 10000000000000000 : ℝ) ≤
+      (-(2027518787259263390593 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (390063798273459 / 10000000000000000 : ℝ) =
+            (390063798273459 / 312500000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (390063798273459 / 312500000000000 : ℝ) ≤
+            (138566151178236609407 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (28808731563459 / 5000000000000000 : ℝ)) (q :=
+      (390063798273459 / 10000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_2_8 :
+    Uω (aρ 8) (bρ 8) (21522393 / 400000000000 : ℝ) ≤
+      (-(183445131700844988143 / 62500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1062484639786859 / 20000000000000000 : ℝ) ≤
+      (-(183445131700844988143 / 62500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (1062484639786859 / 20000000000000000 : ℝ) =
+            (1062484639786859 / 625000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1062484639786859 / 625000000000000 : ℝ) ≤
+            (33163362142905011857 / 62500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (56432855026859 / 10000000000000000 : ℝ)) (q :=
+      (1062484639786859 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_2_9 :
+    Uω (aρ 9) (bρ 9) (21522393 / 400000000000 : ℝ) ≤
+      (-(13290010338352078837471 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (700881131857547 / 10000000000000000 : ℝ) ≤
+      (-(13290010338352078837471 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (700881131857547 / 10000000000000000 : ℝ) =
+            (700881131857547 / 625000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (700881131857547 / 625000000000000 : ℝ) ≤
+            (572933267647921162529 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (27372425297547 / 5000000000000000 : ℝ)) (q :=
+      (700881131857547 / 10000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_2_10 :
+    Uω (aρ 10) (bρ 10) (21522393 / 400000000000 : ℝ) ≤
+      (-(24145699467680874899253 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1788115596139249 / 20000000000000000 : ℝ) ≤
+      (-(24145699467680874899253 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1788115596139249 / 20000000000000000 : ℝ) =
+            (1788115596139249 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1788115596139249 / 1250000000000000 : ℝ) ≤
+            (3580187744319125100747 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (52539149524249 / 10000000000000000 : ℝ)) (q :=
+      (1788115596139249 / 20000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_2_11 :
+    Uω (aρ 11) (bρ 11) (21522393 / 400000000000 : ℝ) ≤
+      (-(22054972789443236367197 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2203914271090737 / 20000000000000000 : ℝ) ≤
+      (-(22054972789443236367197 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2203914271090737 / 20000000000000000 : ℝ) =
+            (2203914271090737 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2203914271090737 / 1250000000000000 : ℝ) ≤
+            (5670914422556763632803 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (49861198270737 / 10000000000000000 : ℝ)) (q :=
+      (2203914271090737 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_2_12 :
+    Uω (aρ 12) (bρ 12) (21522393 / 400000000000 : ℝ) ≤
+      (-(20308097898670736143723 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (656146049612939 / 5000000000000000 : ℝ) ≤
+      (-(20308097898670736143723 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (656146049612939 / 5000000000000000 : ℝ) =
+            (656146049612939 / 625000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (656146049612939 / 625000000000000 : ℝ) ≤
+            (486317510329263856277 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (11707757236689 / 2500000000000000 : ℝ)) (q :=
+      (656146049612939 / 5000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12]) (by norm_num [aρ_12])
+      (by norm_num) (by norm_num [aρ_12, bρ_12]) (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_2_13 :
+    Uω (aρ 13) (bρ 13) (21522393 / 400000000000 : ℝ) ≤
+      (-(18902136129446577859961 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (755197706355697 / 5000000000000000 : ℝ) ≤
+      (-(18902136129446577859961 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (755197706355697 / 5000000000000000 : ℝ) =
+            (755197706355697 / 625000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (755197706355697 / 625000000000000 : ℝ) ≤
+            (1892279279553422140039 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (10913978580697 / 2500000000000000 : ℝ)) (q :=
+      (755197706355697 / 5000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13]) (by norm_num [aρ_13])
+      (by norm_num) (by norm_num [aρ_13, bρ_13]) (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_2_14 :
+    Uω (aρ 14) (bρ 14) (21522393 / 400000000000 : ℝ) ≤
+      (-(17832860086660498318657 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (840424550963387 / 5000000000000000 : ℝ) ≤
+      (-(17832860086660498318657 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (840424550963387 / 5000000000000000 : ℝ) =
+            (840424550963387 / 625000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (840424550963387 / 625000000000000 : ℝ) ≤
+            (2961555322339501681343 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (10159001927137 / 2500000000000000 : ℝ)) (q :=
+      (840424550963387 / 5000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14]) (by norm_num [aρ_14])
+      (by norm_num) (by norm_num [aρ_14, bρ_14]) (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_2_15 :
+    Uω (aρ 15) (bρ 15) (21522393 / 400000000000 : ℝ) ≤
+      (-(17095939417607986453457 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (723756997736773 / 4000000000000000 : ℝ) ≤
+      (-(17095939417607986453457 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (723756997736773 / 4000000000000000 : ℝ) =
+            (723756997736773 / 500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (723756997736773 / 500000000000000 : ℝ) ≤
+            (3698475991392013546543 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (7629903024773 / 2000000000000000 : ℝ)) (q :=
+      (723756997736773 / 4000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15]) (by norm_num [aρ_15])
+      (by norm_num) (by norm_num [aρ_15, bρ_15]) (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_2_16 :
+    Uω (aρ 16) (bρ 16) (21522393 / 400000000000 : ℝ) ≤
+      (-(4171908594379160841333 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1884799886888283 / 10000000000000000 : ℝ) ≤
+      (-(4171908594379160841333 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1884799886888283 / 10000000000000000 : ℝ) =
+            (1884799886888283 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1884799886888283 / 1250000000000000 : ℝ) ≤
+            (1026695257870839158667 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (18296324353283 / 5000000000000000 : ℝ)) (q :=
+      (1884799886888283 / 10000000000000000 : ℝ)) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [aρ_16]) (by norm_num) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [aρ_16, bρ_16]) hlg
+
+lemma U_2 :
+    Uρ (21522393 / 400000000000 : ℝ) ≤
+      (-(27469213456810556512581 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (21522393 / 400000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(25470941903623612969977 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_2_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (21522393 / 400000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(49165552758527775040867 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_2_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (21522393 / 400000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(46358019844614904236579 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_2_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (21522393 / 400000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(2148318328187890160217 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_2_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (21522393 / 400000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(19682243075206791475787 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_2_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (21522393 / 400000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(35805525469465709732703 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_2_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (21522393 / 400000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(2027518787259263390593 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_2_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (21522393 / 400000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) * (-(183445131700844988143 / 62500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_2_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (21522393 / 400000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(13290010338352078837471 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_2_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (21522393 / 400000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(24145699467680874899253 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_2_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (21522393 / 400000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(22054972789443236367197 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_2_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (21522393 / 400000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(20308097898670736143723 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_2_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (21522393 / 400000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(18902136129446577859961 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_2_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (21522393 / 400000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(17832860086660498318657 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_2_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (21522393 / 400000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(17095939417607986453457 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_2_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (21522393 / 400000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) *
+        (-(4171908594379160841333 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_2_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_3_1 :
+    Uω (aρ 1) (bρ 1) (7174131 / 100000000000 : ℝ) ≤
+      (-(50972496086807483658427 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (61135380726717 / 10000000000000000 : ℝ) ≤
+      (-(50972496086807483658427 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show
+          (61135380726717 / 10000000000000000 : ℝ) = (61135380726717 / 39062500000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (61135380726717 / 39062500000000 : ℝ) ≤
+            (4479278337192516341573 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (29245478234217 / 5000000000000000 : ℝ)) (q :=
+      (61135380726717 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_3_2 :
+    Uω (aρ 2) (bρ 2) (7174131 / 100000000000 : ℝ) ≤
+      (-(49196146549098639705941 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (146038881156689 / 20000000000000000 : ℝ) ≤
+      (-(49196146549098639705941 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show
+          (146038881156689 / 20000000000000000 : ℝ) =
+            (146038881156689 / 78125000000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (146038881156689 / 78125000000000 : ℝ) ≤
+            (6255627874901360294059 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (58490063661689 / 10000000000000000 : ℝ)) (q :=
+      (146038881156689 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_3_3 :
+    Uω (aρ 3) (bρ 3) (7174131 / 100000000000 : ℝ) ≤
+      (-(11597151895750616494561 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (193374126628239 / 20000000000000000 : ℝ) ≤
+      (-(11597151895750616494561 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (193374126628239 / 20000000000000000 : ℝ) =
+            (193374126628239 / 156250000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (193374126628239 / 156250000000000 : ℝ) ≤
+            (532923759499383505439 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (58429202783239 / 10000000000000000 : ℝ)) (q :=
+      (193374126628239 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_3_4 :
+    Uω (aρ 4) (bρ 4) (7174131 / 100000000000 : ℝ) ≤
+      (-(429970042250669996509 / 100000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (271452488626557 / 20000000000000000 : ℝ) ≤
+      (-(429970042250669996509 / 100000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (271452488626557 / 20000000000000000 : ℝ) =
+            (271452488626557 / 156250000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (271452488626557 / 156250000000000 : ℝ) ≤
+            (55232983959330003491 / 100000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (58213383716557 / 10000000000000000 : ℝ)) (q :=
+      (271452488626557 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_3_5 :
+    Uω (aρ 5) (bρ 5) (7174131 / 100000000000 : ℝ) ≤
+      (-(9848821862142350549383 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (194573819461431 / 10000000000000000 : ℝ) ≤
+      (-(9848821862142350549383 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (194573819461431 / 10000000000000000 : ℝ) =
+            (194573819461431 / 156250000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (194573819461431 / 156250000000000 : ℝ) ≤
+            (548385842357649450617 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (28859307338931 / 5000000000000000 : ℝ)) (q :=
+      (194573819461431 / 10000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_3_6 :
+    Uω (aρ 6) (bρ 6) (7174131 / 100000000000 : ℝ) ≤
+      (-(17918336888719840156369 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (277736547269241 / 10000000000000000 : ℝ) ≤
+      (-(17918336888719840156369 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (277736547269241 / 10000000000000000 : ℝ) =
+            (277736547269241 / 156250000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (277736547269241 / 156250000000000 : ℝ) ≤
+            (2876078520280159843631 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (28401847576741 / 5000000000000000 : ℝ)) (q :=
+      (277736547269241 / 10000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_3_7 :
+    Uω (aρ 7) (bρ 7) (7174131 / 100000000000 : ℝ) ≤
+      (-(32472061297251956228297 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (77765378718077 / 2000000000000000 : ℝ) ≤
+      (-(32472061297251956228297 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (77765378718077 / 2000000000000000 : ℝ) = (77765378718077 / 62500000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (77765378718077 / 62500000000000 : ℝ) ≤
+            (2185297717748043771703 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (5532300703577 / 1000000000000000 : ℝ)) (q :=
+      (77765378718077 / 2000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_3_8 :
+    Uω (aρ 8) (bρ 8) (7174131 / 100000000000 : ℝ) ≤
+      (-(14691979007557547771577 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (211802415407127 / 4000000000000000 : ℝ) ≤
+      (-(14691979007557547771577 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (211802415407127 / 4000000000000000 : ℝ) =
+            (211802415407127 / 125000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (211802415407127 / 125000000000000 : ℝ) ≤
+            (2636700499942452228423 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (10627929110127 / 2000000000000000 : ℝ)) (q :=
+      (211802415407127 / 4000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_3_9 :
+    Uω (aρ 9) (bρ 9) (7174131 / 100000000000 : ℝ) ≤
+      (-(26614221306142328686011 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (698488168645371 / 10000000000000000 : ℝ) ≤
+      (-(26614221306142328686011 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (698488168645371 / 10000000000000000 : ℝ) =
+            (698488168645371 / 625000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (698488168645371 / 625000000000000 : ℝ) ≤
+            (1111665905857671313989 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (25069138722871 / 5000000000000000 : ℝ)) (q :=
+      (698488168645371 / 10000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_3_10 :
+    Uω (aρ 10) (bρ 10) (7174131 / 100000000000 : ℝ) ≤
+      (-(24182014718027931878799 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (89081689306903 / 1000000000000000 : ℝ) ≤
+      (-(24182014718027931878799 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (89081689306903 / 1000000000000000 : ℝ) = (89081689306903 / 62500000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (89081689306903 / 62500000000000 : ℝ) ≤
+            (3543872493972068121201 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (2311834639903 / 500000000000000 : ℝ)) (q :=
+      (89081689306903 / 1000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10]) (by norm_num [aρ_10])
+      (by norm_num) (by norm_num [aρ_10, bρ_10]) (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_3_11 :
+    Uω (aρ 11) (bρ 11) (7174131 / 100000000000 : ℝ) ≤
+      (-(22094280725116768354743 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2195268143255481 / 20000000000000000 : ℝ) ≤
+      (-(22094280725116768354743 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2195268143255481 / 20000000000000000 : ℝ) =
+            (2195268143255481 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2195268143255481 / 1250000000000000 : ℝ) ≤
+            (5631606486883231645257 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (41394423710481 / 10000000000000000 : ℝ)) (q :=
+      (2195268143255481 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_3_12 :
+    Uω (aρ 12) (bρ 12) (7174131 / 100000000000 : ℝ) ≤
+      (-(20351605328285829232061 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (653297527892073 / 5000000000000000 : ℝ) ≤
+      (-(20351605328285829232061 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (653297527892073 / 5000000000000000 : ℝ) =
+            (653297527892073 / 625000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (653297527892073 / 625000000000000 : ℝ) ≤
+            (442810080714170767939 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (8904073834573 / 2500000000000000 : ℝ)) (q :=
+      (653297527892073 / 5000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12]) (by norm_num [aρ_12])
+      (by norm_num) (by norm_num [aρ_12, bρ_12]) (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_3_13 :
+    Uω (aρ 13) (bρ 13) (7174131 / 100000000000 : ℝ) ≤
+      (-(18951542575309468805869 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1502951489823239 / 10000000000000000 : ℝ) ≤
+      (-(18951542575309468805869 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1502951489823239 / 10000000000000000 : ℝ) =
+            (1502951489823239 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1502951489823239 / 1250000000000000 : ℝ) ≤
+            (1842872833690531194131 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (14473710910739 / 5000000000000000 : ℝ)) (q :=
+      (1502951489823239 / 10000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_3_14 :
+    Uω (aρ 14) (bρ 14) (7174131 / 100000000000 : ℝ) ≤
+      (-(4472659389032352227177 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3342331165120743 / 20000000000000000 : ℝ) ≤
+      (-(4472659389032352227177 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3342331165120743 / 20000000000000000 : ℝ) =
+            (3342331165120743 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3342331165120743 / 2500000000000000 : ℝ) ≤
+            (725944463217647772823 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (21448322250743 / 10000000000000000 : ℝ)) (q :=
+      (3342331165120743 / 20000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_3_15 :
+    Uω (aρ 15) (bρ 15) (7174131 / 100000000000 : ℝ) ≤
+      (-(17165962668190914916479 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (898383398074539 / 5000000000000000 : ℝ) ≤
+      (-(17165962668190914916479 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (898383398074539 / 5000000000000000 : ℝ) =
+            (898383398074539 / 625000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (898383398074539 / 625000000000000 : ℝ) ≤
+            (3628452740809085083521 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (3269368003289 / 2500000000000000 : ℝ)) (q :=
+      (898383398074539 / 5000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15]) (by norm_num [aρ_15])
+      (by norm_num) (by norm_num [aρ_15, bρ_15]) (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_3_16 :
+    Uω (aρ 16) (bρ 16) (7174131 / 100000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_3 :
+    Uρ (7174131 / 100000000000 : ℝ) ≤
+      (-(27504182777521718667561 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (7174131 / 100000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(50972496086807483658427 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_3_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (7174131 / 100000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(49196146549098639705941 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_3_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (7174131 / 100000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(11597151895750616494561 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_3_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (7174131 / 100000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(429970042250669996509 / 100000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_3_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (7174131 / 100000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(9848821862142350549383 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_3_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (7174131 / 100000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(17918336888719840156369 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_3_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (7174131 / 100000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(32472061297251956228297 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_3_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (7174131 / 100000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14691979007557547771577 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_3_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (7174131 / 100000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26614221306142328686011 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_3_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (7174131 / 100000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(24182014718027931878799 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_3_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (7174131 / 100000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(22094280725116768354743 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_3_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (7174131 / 100000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(20351605328285829232061 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_3_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (7174131 / 100000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(18951542575309468805869 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_3_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (7174131 / 100000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(4472659389032352227177 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_3_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (7174131 / 100000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(17165962668190914916479 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_3_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (7174131 / 100000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_3_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_4_1 :
+    Uω (aρ 1) (bρ 1) (14825913 / 200000000000 : ℝ) ≤
+      (-(50976580114456355348677 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (61110417965973 / 10000000000000000 : ℝ) ≤
+      (-(50976580114456355348677 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show
+          (61110417965973 / 10000000000000000 : ℝ) = (61110417965973 / 39062500000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (61110417965973 / 39062500000000 : ℝ) ≤
+            (4475194309543644651323 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (29232456748473 / 5000000000000000 : ℝ)) (q :=
+      (61110417965973 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_4_2 :
+    Uω (aρ 2) (bρ 2) (14825913 / 200000000000 : ℝ) ≤
+      (-(1968009239129451754533 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (145979244790401 / 20000000000000000 : ℝ) ≤
+      (-(1968009239129451754533 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show
+          (145979244790401 / 20000000000000000 : ℝ) =
+            (145979244790401 / 78125000000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (145979244790401 / 78125000000000 : ℝ) ≤
+            (250061737830548245467 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (58454309845401 / 10000000000000000 : ℝ)) (q :=
+      (145979244790401 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_4_3 :
+    Uω (aρ 3) (bρ 3) (14825913 / 200000000000 : ℝ) ≤
+      (-(46392696947832317425569 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (193295065059677 / 20000000000000000 : ℝ) ≤
+      (-(46392696947832317425569 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (193295065059677 / 20000000000000000 : ℝ) =
+            (193295065059677 / 156250000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (193295065059677 / 156250000000000 : ℝ) ≤
+            (2127605673167682574431 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (58374023764677 / 10000000000000000 : ℝ)) (q :=
+      (193295065059677 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_4_4 :
+    Uω (aρ 4) (bρ 4) (14825913 / 200000000000 : ℝ) ≤
+      (-(21500554949887790778621 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (54268212388099 / 4000000000000000 : ℝ) ≤
+      (-(21500554949887790778621 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (54268212388099 / 4000000000000000 : ℝ) = (54268212388099 / 31250000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (54268212388099 / 31250000000000 : ℝ) ≤
+            (2759596360612209221379 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (11625167916099 / 2000000000000000 : ℝ)) (q :=
+      (54268212388099 / 4000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_4_5 :
+    Uω (aρ 5) (bρ 5) (14825913 / 200000000000 : ℝ) ≤
+      (-(9849857532743252544909 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (194493230401519 / 10000000000000000 : ℝ) ≤
+      (-(9849857532743252544909 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (194493230401519 / 10000000000000000 : ℝ) =
+            (194493230401519 / 156250000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (194493230401519 / 156250000000000 : ℝ) ≤
+            (547350171756747455091 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (28790659554019 / 5000000000000000 : ℝ)) (q :=
+      (194493230401519 / 10000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_4_6 :
+    Uω (aρ 6) (bρ 6) (14825913 / 200000000000 : ℝ) ≤
+      (-(17920442983008148696533 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (277619584033359 / 10000000000000000 : ℝ) ≤
+      (-(17920442983008148696533 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (277619584033359 / 10000000000000000 : ℝ) =
+            (277619584033359 / 156250000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (277619584033359 / 156250000000000 : ℝ) ≤
+            (2873972425991851303467 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (28296825615859 / 5000000000000000 : ℝ)) (q :=
+      (277619584033359 / 10000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_4_7 :
+    Uω (aρ 7) (bρ 7) (14825913 / 200000000000 : ℝ) ≤
+      (-(8119097615340380036387 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (194329300239091 / 5000000000000000 : ℝ) ≤
+      (-(8119097615340380036387 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (194329300239091 / 5000000000000000 : ℝ) =
+            (194329300239091 / 156250000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (194329300239091 / 156250000000000 : ℝ) ≤
+            (545242138409619963613 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (13752575840341 / 2500000000000000 : ℝ)) (q :=
+      (194329300239091 / 5000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_4_8 :
+    Uω (aρ 8) (bρ 8) (14825913 / 200000000000 : ℝ) ≤
+      (-(2938847159611171861641 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (264633547803489 / 5000000000000000 : ℝ) ≤
+      (-(2938847159611171861641 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (264633547803489 / 5000000000000000 : ℝ) =
+            (264633547803489 / 156250000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (264633547803489 / 156250000000000 : ℝ) ≤
+            (526888741888828138359 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (13171410569739 / 2500000000000000 : ℝ)) (q :=
+      (264633547803489 / 5000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_4_9 :
+    Uω (aρ 9) (bρ 9) (14825913 / 200000000000 : ℝ) ≤
+      (-(26619015513642536754779 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1396306758365197 / 20000000000000000 : ℝ) ≤
+      (-(26619015513642536754779 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1396306758365197 / 20000000000000000 : ℝ) =
+            (1396306758365197 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1396306758365197 / 1250000000000000 : ℝ) ≤
+            (1106871698357463245221 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (49492581070197 / 10000000000000000 : ℝ)) (q :=
+      (1396306758365197 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_4_10 :
+    Uω (aρ 10) (bρ 10) (14825913 / 200000000000 : ℝ) ≤
+      (-(24187231062536165434337 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (890352333463911 / 10000000000000000 : ℝ) ≤
+      (-(24187231062536165434337 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (890352333463911 / 10000000000000000 : ℝ) =
+            (890352333463911 / 625000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (890352333463911 / 625000000000000 : ℝ) ≤
+            (3538656149463834565663 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (22665728068911 / 5000000000000000 : ℝ)) (q :=
+      (890352333463911 / 10000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_4_11 :
+    Uω (aρ 11) (bρ 11) (14825913 / 200000000000 : ℝ) ≤
+      (-(22100139537083282243419 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2193982353625503 / 20000000000000000 : ℝ) ≤
+      (-(22100139537083282243419 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2193982353625503 / 20000000000000000 : ℝ) =
+            (2193982353625503 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2193982353625503 / 1250000000000000 : ℝ) ≤
+            (5625747674916717756581 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (40132516630503 / 10000000000000000 : ℝ)) (q :=
+      (2193982353625503 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_4_12 :
+    Uω (aρ 12) (bρ 12) (14825913 / 200000000000 : ℝ) ≤
+      (-(20358481952867747024019 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2611393736553067 / 20000000000000000 : ℝ) ≤
+      (-(20358481952867747024019 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2611393736553067 / 20000000000000000 : ℝ) =
+            (2611393736553067 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2611393736553067 / 2500000000000000 : ℝ) ≤
+            (435933456132252975981 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (33843802873067 / 10000000000000000 : ℝ)) (q :=
+      (2611393736553067 / 20000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_4_13 :
+    Uω (aρ 13) (bρ 13) (14825913 / 200000000000 : ℝ) ≤
+      (-(1896017617456667321 / 1000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (750827230859117 / 5000000000000000 : ℝ) ≤
+      (-(1896017617456667321 / 1000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (750827230859117 / 5000000000000000 : ℝ) =
+            (750827230859117 / 625000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (750827230859117 / 625000000000000 : ℝ) ≤
+            (183423923443332679 / 1000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (6594312040367 / 2500000000000000 : ℝ)) (q :=
+      (750827230859117 / 5000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13]) (by norm_num [aρ_13])
+      (by norm_num) (by norm_num [aρ_13, bρ_13]) (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_4_14 :
+    Uω (aρ 14) (bρ 14) (14825913 / 200000000000 : ℝ) ≤
+      (-(8951472576252204766649 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3338220089219641 / 20000000000000000 : ℝ) ≤
+      (-(8951472576252204766649 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3338220089219641 / 20000000000000000 : ℝ) =
+            (3338220089219641 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3338220089219641 / 2500000000000000 : ℝ) ≤
+            (1445735128247795233351 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (17361128899641 / 10000000000000000 : ℝ)) (q :=
+      (3338220089219641 / 20000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_4_15 :
+    Uω (aρ 15) (bρ 15) (14825913 / 200000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_4_16 :
+    Uω (aρ 16) (bρ 16) (14825913 / 200000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_4 :
+    Uρ (14825913 / 200000000000 : ℝ) ≤
+      (-(13755063908664878567303 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (14825913 / 200000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(50976580114456355348677 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_4_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (14825913 / 200000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1968009239129451754533 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_4_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (14825913 / 200000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(46392696947832317425569 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_4_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (14825913 / 200000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(21500554949887790778621 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_4_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (14825913 / 200000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(9849857532743252544909 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_4_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (14825913 / 200000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(17920442983008148696533 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_4_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (14825913 / 200000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(8119097615340380036387 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_4_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (14825913 / 200000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(2938847159611171861641 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_4_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (14825913 / 200000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26619015513642536754779 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_4_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (14825913 / 200000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(24187231062536165434337 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_4_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (14825913 / 200000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(22100139537083282243419 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_4_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (14825913 / 200000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(20358481952867747024019 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_4_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (14825913 / 200000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) * (-(1896017617456667321 / 1000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_4_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (14825913 / 200000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8951472576252204766649 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_4_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (14825913 / 200000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_4_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (14825913 / 200000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_4_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_5_1 :
+    Uω (aρ 1) (bρ 1) (78667711 / 1000000000000 : ℝ) ≤
+      (-(50984345571698067906027 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24425192541173 / 4000000000000000 : ℝ) ≤
+      (-(50984345571698067906027 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show
+          (24425192541173 / 4000000000000000 : ℝ) = (24425192541173 / 15625000000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24425192541173 / 15625000000000 : ℝ) ≤
+            (4467428852301932093973 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (11683084346173 / 2000000000000000 : ℝ)) (q :=
+      (24425192541173 / 4000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_5_2 :
+    Uω (aρ 2) (bρ 2) (78667711 / 1000000000000 : ℝ) ≤
+      (-(24603999536613700227217 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (145865890759319 / 20000000000000000 : ℝ) ≤
+      (-(24603999536613700227217 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show
+          (145865890759319 / 20000000000000000 : ℝ) =
+            (145865890759319 / 78125000000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (145865890759319 / 78125000000000 : ℝ) ≤
+            (3121887675386299772783 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (58386337274319 / 10000000000000000 : ℝ)) (q :=
+      (145865890759319 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_5_3 :
+    Uω (aρ 3) (bρ 3) (78667711 / 1000000000000 : ℝ) ≤
+      (-(11600119550061847085141 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (193144715793361 / 20000000000000000 : ℝ) ≤
+      (-(11600119550061847085141 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (193144715793361 / 20000000000000000 : ℝ) =
+            (193144715793361 / 156250000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (193144715793361 / 156250000000000 : ℝ) ≤
+            (529956105188152914859 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (58269055958361 / 10000000000000000 : ℝ)) (q :=
+      (193144715793361 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_5_4 :
+    Uω (aρ 4) (bρ 4) (78667711 / 1000000000000 : ℝ) ≤
+      (-(10752232140308635873551 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (271128992465947 / 20000000000000000 : ℝ) ≤
+      (-(10752232140308635873551 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (271128992465947 / 20000000000000000 : ℝ) =
+            (271128992465947 / 156250000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (271128992465947 / 156250000000000 : ℝ) ≤
+            (1377843514941364126449 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (57959151565947 / 10000000000000000 : ℝ)) (q :=
+      (271128992465947 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_5_5 :
+    Uω (aρ 5) (bρ 5) (78667711 / 1000000000000 : ℝ) ≤
+      (-(1231479042758063468317 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (388679312495423 / 20000000000000000 : ℝ) ≤
+      (-(1231479042758063468317 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (388679312495423 / 20000000000000000 : ℝ) =
+            (388679312495423 / 312500000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (388679312495423 / 312500000000000 : ℝ) ≤
+            (68171920304436531683 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (57319552260423 / 10000000000000000 : ℝ)) (q :=
+      (388679312495423 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_5_6 :
+    Uω (aρ 6) (bρ 6) (78667711 / 1000000000000 : ℝ) ≤
+      (-(17924466660160898289283 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22191701086949 / 800000000000000 : ℝ) ≤
+      (-(17924466660160898289283 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (22191701086949 / 800000000000000 : ℝ) = (22191701086949 / 12500000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22191701086949 / 12500000000000 : ℝ) ≤
+            (2869948748839101710717 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (2247695671949 / 400000000000000 : ℝ)) (q :=
+      (22191701086949 / 800000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_5_7 :
+    Uω (aρ 7) (bρ 7) (78667711 / 1000000000000 : ℝ) ≤
+      (-(32484685260499643578059 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (776672699289057 / 20000000000000000 : ℝ) ≤
+      (-(32484685260499643578059 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (776672699289057 / 20000000000000000 : ℝ) =
+            (776672699289057 / 625000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (776672699289057 / 625000000000000 : ℝ) ≤
+            (2172673754500356421941 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (54411183154057 / 10000000000000000 : ℝ)) (q :=
+      (776672699289057 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_5_8 :
+    Uω (aρ 8) (bρ 8) (78667711 / 1000000000000 : ℝ) ≤
+      (-(29397157238015419786121 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1057615185486693 / 20000000000000000 : ℝ) ≤
+      (-(29397157238015419786121 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (1057615185486693 / 20000000000000000 : ℝ) =
+            (1057615185486693 / 625000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1057615185486693 / 625000000000000 : ℝ) ≤
+            (5260201776984580213879 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (51812018011693 / 10000000000000000 : ℝ)) (q :=
+      (1057615185486693 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_5_9 :
+    Uω (aρ 9) (bρ 9) (78667711 / 1000000000000 : ℝ) ≤
+      (-(13314151100942602912333 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (348752663432181 / 5000000000000000 : ℝ) ≤
+      (-(13314151100942602912333 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (348752663432181 / 5000000000000000 : ℝ) =
+            (348752663432181 / 312500000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (348752663432181 / 312500000000000 : ℝ) ≤
+            (548792505057397087667 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (12060464473431 / 2500000000000000 : ℝ)) (q :=
+      (348752663432181 / 5000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_5_10 :
+    Uω (aρ 10) (bρ 10) (78667711 / 1000000000000 : ℝ) ≤
+      (-(12098720823673483752803 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (177888739125561 / 2000000000000000 : ℝ) ≤
+      (-(12098720823673483752803 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (177888739125561 / 2000000000000000 : ℝ) =
+            (177888739125561 / 125000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (177888739125561 / 125000000000000 : ℝ) ≤
+            (1764222782326516247197 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (4355956192561 / 1000000000000000 : ℝ)) (q :=
+      (177888739125561 / 2000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10]) (by norm_num [aρ_10])
+      (by norm_num) (by norm_num [aρ_10, bρ_10]) (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_5_11 :
+    Uω (aρ 11) (bρ 11) (78667711 / 1000000000000 : ℝ) ≤
+      (-(22111813113724319814737 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2191422685824679 / 20000000000000000 : ℝ) ≤
+      (-(22111813113724319814737 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2191422685824679 / 20000000000000000 : ℝ) =
+            (2191422685824679 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2191422685824679 / 1250000000000000 : ℝ) ≤
+            (5614074098275680185263 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (37618230289679 / 10000000000000000 : ℝ)) (q :=
+      (2191422685824679 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_5_12 :
+    Uω (aρ 12) (bρ 12) (78667711 / 1000000000000 : ℝ) ≤
+      (-(20372656090798980590323 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2607694933038421 / 20000000000000000 : ℝ) ≤
+      (-(20372656090798980590323 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2607694933038421 / 20000000000000000 : ℝ) =
+            (2607694933038421 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2607694933038421 / 2500000000000000 : ℝ) ≤
+            (421759318201019409677 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (30190380818421 / 10000000000000000 : ℝ)) (q :=
+      (2607694933038421 / 20000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_5_13 :
+    Uω (aρ 13) (bρ 13) (78667711 / 1000000000000 : ℝ) ≤
+      (-(74138612372821309739 / 39062500000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2997515551401257 / 20000000000000000 : ℝ) ≤
+      (-(74138612372821309739 / 39062500000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2997515551401257 / 20000000000000000 : ℝ) =
+            (2997515551401257 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2997515551401257 / 2500000000000000 : ℝ) ≤
+            (7089572818584940261 / 39062500000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (20629257586257 / 10000000000000000 : ℝ)) (q :=
+      (2997515551401257 / 20000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_5_14 :
+    Uω (aρ 14) (bρ 14) (78667711 / 1000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_5_15 :
+    Uω (aρ 15) (bρ 15) (78667711 / 1000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_5_16 :
+    Uω (aρ 16) (bρ 16) (78667711 / 1000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_5 :
+    Uρ (78667711 / 1000000000000 : ℝ) ≤
+      (-(6880237177330760400461 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (78667711 / 1000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(50984345571698067906027 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_5_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (78667711 / 1000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(24603999536613700227217 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_5_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (78667711 / 1000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(11600119550061847085141 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_5_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (78667711 / 1000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(10752232140308635873551 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_5_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (78667711 / 1000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(1231479042758063468317 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_5_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (78667711 / 1000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(17924466660160898289283 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_5_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (78667711 / 1000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(32484685260499643578059 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_5_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (78667711 / 1000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(29397157238015419786121 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_5_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (78667711 / 1000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(13314151100942602912333 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_5_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (78667711 / 1000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12098720823673483752803 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_5_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (78667711 / 1000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(22111813113724319814737 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_5_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (78667711 / 1000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(20372656090798980590323 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_5_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (78667711 / 1000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) * (-(74138612372821309739 / 39062500000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_5_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (78667711 / 1000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_5_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (78667711 / 1000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_5_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (78667711 / 1000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_5_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_6_1 :
+    Uω (aρ 1) (bρ 1) (85815639 / 1000000000000 : ℝ) ≤
+      (-(10199318023215378615661 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12197651654253 / 2000000000000000 : ℝ) ≤
+      (-(10199318023215378615661 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show
+          (12197651654253 / 2000000000000000 : ℝ) = (12197651654253 / 7812500000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12197651654253 / 7812500000000 : ℝ) ≤
+            (891036861584621384339 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (5833745484753 / 1000000000000000 : ℝ)) (q :=
+      (12197651654253 / 2000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_6_2 :
+    Uω (aρ 2) (bρ 2) (85815639 / 1000000000000 : ℝ) ≤
+      (-(49220252785694328798381 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7284363017877 / 1000000000000000 : ℝ) ≤
+      (-(49220252785694328798381 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show (7284363017877 / 1000000000000000 : ℝ) = (7284363017877 / 3906250000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7284363017877 / 3906250000000 : ℝ) ≤
+            (6231521638305671201619 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (2913959307627 / 500000000000000 : ℝ)) (q :=
+      (7284363017877 / 1000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_6_3 :
+    Uω (aρ 3) (bρ 3) (85815639 / 1000000000000 : ℝ) ≤
+      (-(23206381385399735792023 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (38581518296581 / 4000000000000000 : ℝ) ≤
+      (-(23206381385399735792023 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (38581518296581 / 4000000000000000 : ℝ) = (38581518296581 / 31250000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (38581518296581 / 31250000000000 : ℝ) ≤
+            (1053769925100264207977 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (11620682185581 / 2000000000000000 : ℝ)) (q :=
+      (38581518296581 / 4000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_6_4 :
+    Uω (aρ 4) (bρ 4) (85815639 / 1000000000000 : ℝ) ≤
+      (-(21510644677510011539679 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (135397031276029 / 10000000000000000 : ℝ) ≤
+      (-(21510644677510011539679 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (135397031276029 / 10000000000000000 : ℝ) =
+            (135397031276029 / 78125000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (135397031276029 / 78125000000000 : ℝ) ≤
+            (2749506632989988460321 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (28847850466029 / 5000000000000000 : ℝ)) (q :=
+      (135397031276029 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_6_5 :
+    Uω (aρ 5) (bρ 5) (85815639 / 1000000000000 : ℝ) ≤
+      (-(315358759639002277023 / 80000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (194096580910869 / 10000000000000000 : ℝ) ≤
+      (-(315358759639002277023 / 80000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (194096580910869 / 10000000000000000 : ℝ) =
+            (194096580910869 / 156250000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (194096580910869 / 156250000000000 : ℝ) ≤
+            (17351886904997722977 / 80000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (28452440433369 / 5000000000000000 : ℝ)) (q :=
+      (194096580910869 / 10000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_6_6 :
+    Uω (aρ 6) (bρ 6) (85815639 / 1000000000000 : ℝ) ≤
+      (-(35861726381563634540717 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11081664629919 / 400000000000000 : ℝ) ≤
+      (-(35861726381563634540717 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (11081664629919 / 400000000000000 : ℝ) = (11081664629919 / 6250000000000 : ℝ) / 2 ^ 6 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11081664629919 / 6250000000000 : ℝ) ≤
+            (5727104436436365459283 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (1111091508019 / 200000000000000 : ℝ)) (q :=
+      (11081664629919 / 400000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_6_7 :
+    Uω (aρ 7) (bρ 7) (85815639 / 1000000000000 : ℝ) ≤
+      (-(32497938712340042706679 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (775644021694503 / 20000000000000000 : ℝ) ≤
+      (-(32497938712340042706679 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (775644021694503 / 20000000000000000 : ℝ) =
+            (775644021694503 / 625000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (775644021694503 / 625000000000000 : ℝ) ≤
+            (2159420302659957293321 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (53453984839503 / 10000000000000000 : ℝ)) (q :=
+      (775644021694503 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_6_8 :
+    Uω (aρ 8) (bρ 8) (85815639 / 1000000000000 : ℝ) ≤
+      (-(29411142977102807541847 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1056137066353147 / 20000000000000000 : ℝ) ≤
+      (-(29411142977102807541847 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (1056137066353147 / 20000000000000000 : ℝ) =
+            (1056137066353147 / 625000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1056137066353147 / 625000000000000 : ℝ) ≤
+            (5246216037897192458153 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (50405378158147 / 10000000000000000 : ℝ)) (q :=
+      (1056137066353147 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_6_9 :
+    Uω (aρ 9) (bρ 9) (85815639 / 1000000000000 : ℝ) ≤
+      (-(6660859717657794390429 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1392900669621853 / 20000000000000000 : ℝ) ≤
+      (-(6660859717657794390429 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1392900669621853 / 20000000000000000 : ℝ) =
+            (1392900669621853 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1392900669621853 / 1250000000000000 : ℝ) ≤
+            (270612085342205609571 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (46203353066853 / 10000000000000000 : ℝ)) (q :=
+      (1392900669621853 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_6_10 :
+    Uω (aρ 10) (bρ 10) (85815639 / 1000000000000 : ℝ) ≤
+      (-(4842885154875601165287 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (887934335277677 / 10000000000000000 : ℝ) ≤
+      (-(4842885154875601165287 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (887934335277677 / 10000000000000000 : ℝ) =
+            (887934335277677 / 625000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (887934335277677 / 625000000000000 : ℝ) ≤
+            (702292287524398834713 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (20306160252677 / 5000000000000000 : ℝ)) (q :=
+      (887934335277677 / 10000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_6_11 :
+    Uω (aρ 11) (bρ 11) (85815639 / 1000000000000 : ℝ) ≤
+      (-(4426395685635096752489 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1093504032837657 / 10000000000000000 : ℝ) ≤
+      (-(4426395685635096752489 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1093504032837657 / 10000000000000000 : ℝ) =
+            (1093504032837657 / 625000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1093504032837657 / 625000000000000 : ℝ) ≤
+            (1118781756764903247511 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (16637544710157 / 5000000000000000 : ℝ)) (q :=
+      (1093504032837657 / 10000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_6_12 :
+    Uω (aρ 12) (bρ 12) (85815639 / 1000000000000 : ℝ) ≤
+      (-(5099845272237973435631 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2600735172932703 / 20000000000000000 : ℝ) ≤
+      (-(5099845272237973435631 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2600735172932703 / 20000000000000000 : ℝ) =
+            (2600735172932703 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2600735172932703 / 2500000000000000 : ℝ) ≤
+            (98758580012026564369 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (23302099992703 / 10000000000000000 : ℝ)) (q :=
+      (2600735172932703 / 20000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_6_13 :
+    Uω (aρ 13) (bρ 13) (85815639 / 1000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_6_14 :
+    Uω (aρ 14) (bρ 14) (85815639 / 1000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_6_15 :
+    Uω (aρ 15) (bρ 15) (85815639 / 1000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_6_16 :
+    Uω (aρ 16) (bρ 16) (85815639 / 1000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_6 :
+    Uρ (85815639 / 1000000000000 : ℝ) ≤
+      (-(27537256492584381318183 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (85815639 / 1000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(10199318023215378615661 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_6_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (85815639 / 1000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(49220252785694328798381 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_6_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (85815639 / 1000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(23206381385399735792023 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_6_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (85815639 / 1000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(21510644677510011539679 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_6_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (85815639 / 1000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(315358759639002277023 / 80000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_6_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (85815639 / 1000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(35861726381563634540717 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_6_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (85815639 / 1000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(32497938712340042706679 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_6_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (85815639 / 1000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(29411142977102807541847 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_6_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (85815639 / 1000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(6660859717657794390429 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_6_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (85815639 / 1000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(4842885154875601165287 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_6_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (85815639 / 1000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4426395685635096752489 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_6_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (85815639 / 1000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(5099845272237973435631 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_6_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (85815639 / 1000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_6_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (85815639 / 1000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_6_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (85815639 / 1000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_6_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (85815639 / 1000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_6_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_7_1 :
+    Uω (aρ 1) (bρ 1) (19269871 / 200000000000 : ℝ) ≤
+      (-(25507332223904240133213 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6087812562877 / 1000000000000000 : ℝ) ≤
+      (-(25507332223904240133213 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show (6087812562877 / 1000000000000000 : ℝ) = (6087812562877 / 3906250000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6087812562877 / 3906250000000 : ℝ) ≤
+            (2218554988095759866787 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (2911126336127 / 500000000000000 : ℝ)) (q :=
+      (6087812562877 / 1000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_7_2 :
+    Uω (aρ 2) (bρ 2) (19269871 / 200000000000 : ℝ) ≤
+      (-(30773969948474174723 / 6250000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (145423817540153 / 20000000000000000 : ℝ) ≤
+      (-(30773969948474174723 / 6250000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show
+          (145423817540153 / 20000000000000000 : ℝ) =
+            (145423817540153 / 78125000000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (145423817540153 / 78125000000000 : ℝ) ≤
+            (3883389066525825277 / 6250000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (58121080495153 / 10000000000000000 : ℝ)) (q :=
+      (145423817540153 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_7_3 :
+    Uω (aρ 3) (bρ 3) (19269871 / 200000000000 : ℝ) ≤
+      (-(23215465152356514085683 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (192557444123869 / 20000000000000000 : ℝ) ≤
+      (-(23215465152356514085683 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (192557444123869 / 20000000000000000 : ℝ) =
+            (192557444123869 / 156250000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (192557444123869 / 156250000000000 : ℝ) ≤
+            (1044686158143485914317 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (57858600728869 / 10000000000000000 : ℝ)) (q :=
+      (192557444123869 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_7_4 :
+    Uω (aρ 4) (bρ 4) (19269871 / 200000000000 : ℝ) ≤
+      (-(21519804344189444452201 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13514921999703 / 1000000000000000 : ℝ) ≤
+      (-(21519804344189444452201 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (13514921999703 / 1000000000000000 : ℝ) = (13514921999703 / 7812500000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13514921999703 / 7812500000000 : ℝ) ≤
+            (2740346966310555547799 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (2865270776703 / 500000000000000 : ℝ)) (q :=
+      (13514921999703 / 1000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_7_5 :
+    Uω (aρ 5) (bρ 5) (19269871 / 200000000000 : ℝ) ≤
+      (-(315507654980808174019 / 80000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (77494266368633 / 4000000000000000 : ℝ) ≤
+      (-(315507654980808174019 / 80000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (77494266368633 / 4000000000000000 : ℝ) = (77494266368633 / 62500000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (77494266368633 / 62500000000000 : ℝ) ≤
+            (17202991563191825981 / 80000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (11257677609633 / 2000000000000000 : ℝ)) (q :=
+      (77494266368633 / 4000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_7_6 :
+    Uω (aρ 6) (bρ 6) (19269871 / 200000000000 : ℝ) ≤
+      (-(8970212877133828550347 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (553024552971181 / 20000000000000000 : ℝ) ≤
+      (-(8970212877133828550347 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (553024552971181 / 20000000000000000 : ℝ) =
+            (553024552971181 / 312500000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (553024552971181 / 312500000000000 : ℝ) ≤
+            (1426994827366171449653 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (54601234036181 / 10000000000000000 : ℝ)) (q :=
+      (553024552971181 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_7_7 :
+    Uω (aρ 7) (bρ 7) (19269871 / 200000000000 : ℝ) ≤
+      (-(32517914390293590790913 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (154819233332697 / 4000000000000000 : ℝ) ≤
+      (-(32517914390293590790913 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (154819233332697 / 4000000000000000 : ℝ) =
+            (154819233332697 / 125000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (154819233332697 / 125000000000000 : ℝ) ≤
+            (2139444624706409209087 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (10402293393697 / 2000000000000000 : ℝ)) (q :=
+      (154819233332697 / 4000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_7_8 :
+    Uω (aρ 8) (bρ 8) (19269871 / 200000000000 : ℝ) ≤
+      (-(5886499166201685248543 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (210776863658177 / 4000000000000000 : ℝ) ≤
+      (-(5886499166201685248543 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (210776863658177 / 4000000000000000 : ℝ) =
+            (210776863658177 / 125000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (210776863658177 / 125000000000000 : ℝ) ≤
+            (1044972636798314751457 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (9651593451177 / 2000000000000000 : ℝ)) (q :=
+      (210776863658177 / 4000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_7_9 :
+    Uω (aρ 9) (bρ 9) (19269871 / 200000000000 : ℝ) ≤
+      (-(42667279831309115271 / 16000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (694807884018587 / 10000000000000000 : ℝ) ≤
+      (-(42667279831309115271 / 16000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (694807884018587 / 10000000000000000 : ℝ) =
+            (694807884018587 / 625000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (694807884018587 / 625000000000000 : ℝ) ≤
+            (1694139707890884729 / 16000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (21511894321087 / 5000000000000000 : ℝ)) (q :=
+      (694807884018587 / 10000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_7_10 :
+    Uω (aρ 10) (bρ 10) (19269871 / 200000000000 : ℝ) ≤
+      (-(606049641346655113191 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (885490567993697 / 10000000000000000 : ℝ) ≤
+      (-(606049641346655113191 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (885490567993697 / 10000000000000000 : ℝ) =
+            (885490567993697 / 625000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (885490567993697 / 625000000000000 : ℝ) ≤
+            (87097538953344886809 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (17915061548697 / 5000000000000000 : ℝ)) (q :=
+      (885490567993697 / 10000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_7_11 :
+    Uω (aρ 11) (bρ 11) (19269871 / 200000000000 : ℝ) ≤
+      (-(22167783768353140953257 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2179191411148691 / 20000000000000000 : ℝ) ≤
+      (-(22167783768353140953257 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2179191411148691 / 20000000000000000 : ℝ) =
+            (2179191411148691 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2179191411148691 / 1250000000000000 : ℝ) ≤
+            (5558103443646859046743 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (25563772053691 / 10000000000000000 : ℝ)) (q :=
+      (2179191411148691 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_7_12 :
+    Uω (aρ 12) (bρ 12) (19269871 / 200000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_7_13 :
+    Uω (aρ 13) (bρ 13) (19269871 / 200000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_7_14 :
+    Uω (aρ 14) (bρ 14) (19269871 / 200000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_7_15 :
+    Uω (aρ 15) (bρ 15) (19269871 / 200000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_7_16 :
+    Uω (aρ 16) (bρ 16) (19269871 / 200000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_7 :
+    Uρ (19269871 / 200000000000 : ℝ) ≤ (-(6889957110960769864817 / 2500000000000000000000 : ℝ)) :=
+  by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (19269871 / 200000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(25507332223904240133213 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_7_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (19269871 / 200000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) * (-(30773969948474174723 / 6250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_7_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (19269871 / 200000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(23215465152356514085683 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_7_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (19269871 / 200000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(21519804344189444452201 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_7_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (19269871 / 200000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(315507654980808174019 / 80000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_7_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (19269871 / 200000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(8970212877133828550347 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_7_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (19269871 / 200000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(32517914390293590790913 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_7_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (19269871 / 200000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(5886499166201685248543 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_7_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (19269871 / 200000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(42667279831309115271 / 16000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_7_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (19269871 / 200000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) * (-(606049641346655113191 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_7_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (19269871 / 200000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(22167783768353140953257 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_7_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (19269871 / 200000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_7_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (19269871 / 200000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_7_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (19269871 / 200000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_7_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (19269871 / 200000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_7_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (19269871 / 200000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_7_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_8_1 :
+    Uω (aρ 1) (bρ 1) (55761057 / 500000000000 : ℝ) ≤
+      (-(51040761523417755773563 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (30359729325639 / 5000000000000000 : ℝ) ≤
+      (-(51040761523417755773563 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show
+          (30359729325639 / 5000000000000000 : ℝ) = (30359729325639 / 19531250000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (30359729325639 / 19531250000000 : ℝ) ≤
+            (4411012900582244226437 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (14514230089389 / 2500000000000000 : ℝ)) (q :=
+      (30359729325639 / 5000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_8_2 :
+    Uω (aρ 2) (bρ 2) (55761057 / 500000000000 : ℝ) ≤
+      (-(12316127176026878844577 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9065245788109 / 1250000000000000 : ℝ) ≤
+      (-(12316127176026878844577 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show (9065245788109 / 1250000000000000 : ℝ) = (9065245788109 / 4882812500000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9065245788109 / 4882812500000 : ℝ) ≤
+            (1546816429973121155423 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (7236615394343 / 1250000000000000 : ℝ)) (q :=
+      (9065245788109 / 1250000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_8_3 :
+    Uω (aρ 3) (bρ 3) (55761057 / 500000000000 : ℝ) ≤
+      (-(46457234677264518279157 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (96025799718557 / 10000000000000000 : ℝ) ≤
+      (-(46457234677264518279157 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (96025799718557 / 10000000000000000 : ℝ) = (96025799718557 / 78125000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (96025799718557 / 78125000000000 : ℝ) ≤
+            (2063067943735481720843 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (28752241816057 / 5000000000000000 : ℝ)) (q :=
+      (96025799718557 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_8_4 :
+    Uω (aρ 4) (bρ 4) (55761057 / 500000000000 : ℝ) ≤
+      (-(21533108620975871780131 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (269580170970713 / 20000000000000000 : ℝ) ≤
+      (-(21533108620975871780131 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (269580170970713 / 20000000000000000 : ℝ) =
+            (269580170970713 / 156250000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (269580170970713 / 156250000000000 : ℝ) ≤
+            (2727042689524128219869 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (56738874100713 / 10000000000000000 : ℝ)) (q :=
+      (269580170970713 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_8_5 :
+    Uω (aρ 5) (bρ 5) (55761057 / 500000000000 : ℝ) ≤
+      (-(1578625177908711877893 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (96604975401611 / 5000000000000000 : ℝ) ≤
+      (-(1578625177908711877893 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (96604975401611 / 5000000000000000 : ℝ) = (96604975401611 / 78125000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (96604975401611 / 78125000000000 : ℝ) ≤
+            (84928054811288122107 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (13847171350361 / 2500000000000000 : ℝ)) (q :=
+      (96604975401611 / 5000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_8_6 :
+    Uω (aρ 6) (bρ 6) (55761057 / 500000000000 : ℝ) ≤
+      (-(897725036415218135001 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (275734990404539 / 10000000000000000 : ℝ) ≤
+      (-(897725036415218135001 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (275734990404539 / 10000000000000000 : ℝ) =
+            (275734990404539 / 156250000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (275734990404539 / 156250000000000 : ℝ) ≤
+            (141995734034781864999 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (26599194732039 / 5000000000000000 : ℝ)) (q :=
+      (275734990404539 / 10000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_8_7 :
+    Uω (aρ 7) (bρ 7) (55761057 / 500000000000 : ℝ) ≤
+      (-(16273851117356851483897 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (96474216497051 / 2500000000000000 : ℝ) ≤
+      (-(16273851117356851483897 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (96474216497051 / 2500000000000000 : ℝ) = (96474216497051 / 78125000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (96474216497051 / 78125000000000 : ℝ) ≤
+            (1054828390143148516103 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (3116297491963 / 625000000000000 : ℝ)) (q :=
+      (96474216497051 / 2500000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_8_8 :
+    Uω (aρ 8) (bρ 8) (55761057 / 500000000000 : ℝ) ≤
+      (-(7366260065742286983877 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (525230043323873 / 10000000000000000 : ℝ) ≤
+      (-(7366260065742286983877 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (525230043323873 / 10000000000000000 : ℝ) =
+            (525230043323873 / 312500000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (525230043323873 / 312500000000000 : ℝ) ≤
+            (1298079688007713016123 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (22492731601373 / 5000000000000000 : ℝ)) (q :=
+      (525230043323873 / 10000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_8_9 :
+    Uω (aρ 9) (bρ 9) (55761057 / 500000000000 : ℝ) ≤
+      (-(26704512040091709661141 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (276883941615867 / 4000000000000000 : ℝ) ≤
+      (-(26704512040091709661141 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (276883941615867 / 4000000000000000 : ℝ) =
+            (276883941615867 / 250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (276883941615867 / 250000000000000 : ℝ) ≤
+            (1021375171908290338859 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (7595891254867 / 2000000000000000 : ℝ)) (q :=
+      (276883941615867 / 4000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_8_10 :
+    Uω (aρ 10) (bρ 10) (55761057 / 500000000000 : ℝ) ≤
+      (-(24289887987277840384009 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1762518009461781 / 20000000000000000 : ℝ) ≤
+      (-(24289887987277840384009 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1762518009461781 / 20000000000000000 : ℝ) =
+            (1762518009461781 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1762518009461781 / 1250000000000000 : ℝ) ≤
+            (3435999224722159615991 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (27518724161781 / 10000000000000000 : ℝ)) (q :=
+      (1762518009461781 / 20000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_8_11 :
+    Uω (aρ 11) (bρ 11) (55761057 / 500000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_8_12 :
+    Uω (aρ 12) (bρ 12) (55761057 / 500000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_8_13 :
+    Uω (aρ 13) (bρ 13) (55761057 / 500000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_8_14 :
+    Uω (aρ 14) (bρ 14) (55761057 / 500000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_8_15 :
+    Uω (aρ 15) (bρ 15) (55761057 / 500000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_8_16 :
+    Uω (aρ 16) (bρ 16) (55761057 / 500000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_8 :
+    Uρ (55761057 / 500000000000 : ℝ) ≤ (-(5517964148274383717247 / 2000000000000000000000 : ℝ)) :=
+  by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (55761057 / 500000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(51040761523417755773563 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_8_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (55761057 / 500000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(12316127176026878844577 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_8_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (55761057 / 500000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(46457234677264518279157 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_8_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (55761057 / 500000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(21533108620975871780131 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_8_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (55761057 / 500000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(1578625177908711877893 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_8_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (55761057 / 500000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(897725036415218135001 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_8_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (55761057 / 500000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(16273851117356851483897 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_8_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (55761057 / 500000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(7366260065742286983877 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_8_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (55761057 / 500000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26704512040091709661141 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_8_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (55761057 / 500000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(24289887987277840384009 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_8_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (55761057 / 500000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_8_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (55761057 / 500000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_8_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (55761057 / 500000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_8_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (55761057 / 500000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_8_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (55761057 / 500000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_8_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (55761057 / 500000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_8_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_9_1 :
+    Uω (aρ 1) (bρ 1) (33336783 / 250000000000 : ℝ) ≤
+      (-(10215686268319257150869 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (30245579911641 / 5000000000000000 : ℝ) ≤
+      (-(10215686268319257150869 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show
+          (30245579911641 / 5000000000000000 : ℝ) = (30245579911641 / 19531250000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (30245579911641 / 19531250000000 : ℝ) ≤
+            (874668616480742849131 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (14454643220391 / 2500000000000000 : ℝ)) (q :=
+      (30245579911641 / 5000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_9_2 :
+    Uω (aρ 2) (bρ 2) (33336783 / 250000000000 : ℝ) ≤
+      (-(9860463027542453411111 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (72248304249749 / 10000000000000000 : ℝ) ≤
+      (-(9860463027542453411111 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show
+          (72248304249749 / 10000000000000000 : ℝ) = (72248304249749 / 39062500000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (72248304249749 / 39062500000000 : ℝ) ≤
+            (1229891857257546588889 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (28781924612249 / 5000000000000000 : ℝ)) (q :=
+      (72248304249749 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_9_3 :
+    Uω (aρ 3) (bρ 3) (33336783 / 250000000000 : ℝ) ≤
+      (-(46495358210023539812961 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4783020619089 / 500000000000000 : ℝ) ≤
+      (-(46495358210023539812961 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show (4783020619089 / 500000000000000 : ℝ) = (4783020619089 / 3906250000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4783020619089 / 3906250000000 : ℝ) ≤
+            (2024944410976460187039 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (44524968077 / 7812500000000 : ℝ)) (q :=
+      (4783020619089 / 500000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_9_4 :
+    Uω (aρ 4) (bρ 4) (33336783 / 250000000000 : ℝ) ≤
+      (-(21552482231997735895661 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (67134410938101 / 5000000000000000 : ℝ) ≤
+      (-(21552482231997735895661 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (67134410938101 / 5000000000000000 : ℝ) = (67134410938101 / 39062500000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (67134410938101 / 39062500000000 : ℝ) ≤
+            (2707669078502264104339 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (13978649265601 / 2500000000000000 : ℝ)) (q :=
+      (67134410938101 / 5000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_9_5 :
+    Uω (aρ 5) (bρ 5) (33336783 / 250000000000 : ℝ) ≤
+      (-(19752753975247025066761 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (240551240889 / 12500000000000 : ℝ) ≤
+      (-(19752753975247025066761 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show (240551240889 / 12500000000000 : ℝ) = (240551240889 / 195312500000 : ℝ) / 2 ^ 6 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (240551240889 / 195312500000 : ℝ) ≤
+            (1041661433752974933239 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (270345096987 / 50000000000000 : ℝ)) (q :=
+      (240551240889 / 12500000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_9_6 :
+    Uω (aρ 6) (bρ 6) (33336783 / 250000000000 : ℝ) ≤
+      (-(17975423546595465917579 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13729178527969 / 500000000000000 : ℝ) ≤
+      (-(17975423546595465917579 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (13729178527969 / 500000000000000 : ℝ) = (13729178527969 / 7812500000000 : ℝ) / 2 ^ 6 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13729178527969 / 7812500000000 : ℝ) ≤
+            (2818991862404534082421 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (319461249711 / 62500000000000 : ℝ)) (q :=
+      (13729178527969 / 500000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_9_7 :
+    Uω (aρ 7) (bρ 7) (33336783 / 250000000000 : ℝ) ≤
+      (-(6518591326819602089771 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (768308916905443 / 20000000000000000 : ℝ) ≤
+      (-(6518591326819602089771 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (768308916905443 / 20000000000000000 : ℝ) =
+            (768308916905443 / 625000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (768308916905443 / 625000000000000 : ℝ) ≤
+            (412880476180397910229 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (46594194980443 / 10000000000000000 : ℝ)) (q :=
+      (768308916905443 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_9_8 :
+    Uω (aρ 8) (bρ 8) (33336783 / 250000000000 : ℝ) ≤
+      (-(14758258563778426940663 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1045066541533949 / 20000000000000000 : ℝ) ≤
+      (-(14758258563778426940663 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (1045066541533949 / 20000000000000000 : ℝ) =
+            (1045066541533949 / 625000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1045066541533949 / 625000000000000 : ℝ) ≤
+            (2570420943721573059337 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (39810168268949 / 10000000000000000 : ℝ)) (q :=
+      (1045066541533949 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_9_9 :
+    Uω (aρ 9) (bρ 9) (33336783 / 250000000000 : ℝ) ≤
+      (-(5353890477833411319021 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1375458367291817 / 20000000000000000 : ℝ) ≤
+      (-(5353890477833411319021 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1375458367291817 / 20000000000000000 : ℝ) =
+            (1375458367291817 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1375458367291817 / 1250000000000000 : ℝ) ≤
+            (191286964566588680979 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (29236365666817 / 10000000000000000 : ℝ)) (q :=
+      (1375458367291817 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_9_10 :
+    Uω (aρ 10) (bρ 10) (33336783 / 250000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_9_11 :
+    Uω (aρ 11) (bρ 11) (33336783 / 250000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_9_12 :
+    Uω (aρ 12) (bρ 12) (33336783 / 250000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_9_13 :
+    Uω (aρ 13) (bρ 13) (33336783 / 250000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_9_14 :
+    Uω (aρ 14) (bρ 14) (33336783 / 250000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_9_15 :
+    Uω (aρ 15) (bρ 15) (33336783 / 250000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_9_16 :
+    Uω (aρ 16) (bρ 16) (33336783 / 250000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_9 :
+    Uρ (33336783 / 250000000000 : ℝ) ≤ (-(6907239882122678168379 / 2500000000000000000000 : ℝ)) :=
+  by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (33336783 / 250000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(10215686268319257150869 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_9_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (33336783 / 250000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(9860463027542453411111 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_9_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (33336783 / 250000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(46495358210023539812961 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_9_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (33336783 / 250000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(21552482231997735895661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_9_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (33336783 / 250000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(19752753975247025066761 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_9_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (33336783 / 250000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(17975423546595465917579 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_9_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (33336783 / 250000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(6518591326819602089771 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_9_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (33336783 / 250000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14758258563778426940663 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_9_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (33336783 / 250000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(5353890477833411319021 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_9_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (33336783 / 250000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_9_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (33336783 / 250000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_9_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (33336783 / 250000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_9_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (33336783 / 250000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_9_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (33336783 / 250000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_9_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (33336783 / 250000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_9_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (33336783 / 250000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_9_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_10_1 :
+    Uω (aρ 1) (bρ 1) (82548843 / 500000000000 : ℝ) ≤
+      (-(51133510883215947535941 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24063557272999 / 4000000000000000 : ℝ) ≤
+      (-(51133510883215947535941 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show
+          (24063557272999 / 4000000000000000 : ℝ) = (24063557272999 / 15625000000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24063557272999 / 15625000000000 : ℝ) ≤
+            (4318263540784052464059 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (11494309027999 / 2000000000000000 : ℝ)) (q :=
+      (24063557272999 / 4000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_10_2 :
+    Uω (aρ 2) (bρ 2) (82548843 / 500000000000 : ℝ) ≤
+      (-(24678851808854408309657 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (143698476155901 / 20000000000000000 : ℝ) ≤
+      (-(24678851808854408309657 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show
+          (143698476155901 / 20000000000000000 : ℝ) =
+            (143698476155901 / 78125000000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (143698476155901 / 78125000000000 : ℝ) ≤
+            (3047035403145591690343 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (57083222420901 / 10000000000000000 : ℝ)) (q :=
+      (143698476155901 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_10_3 :
+    Uω (aρ 3) (bρ 3) (82548843 / 500000000000 : ℝ) ≤
+      (-(5818929893213673684367 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23781360331077 / 2500000000000000 : ℝ) ≤
+      (-(5818929893213673684367 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (23781360331077 / 2500000000000000 : ℝ) = (23781360331077 / 19531250000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23781360331077 / 19531250000000 : ℝ) ≤
+            (246107934411326315633 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (1757485080113 / 312500000000000 : ℝ)) (q :=
+      (23781360331077 / 2500000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_10_4 :
+    Uω (aρ 4) (bρ 4) (82548843 / 500000000000 : ℝ) ≤
+      (-(43162374832339657159807 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (133500188112619 / 10000000000000000 : ℝ) ≤
+      (-(43162374832339657159807 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (133500188112619 / 10000000000000000 : ℝ) =
+            (133500188112619 / 78125000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (133500188112619 / 78125000000000 : ℝ) ≤
+            (5357927788660342840193 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (27347417537619 / 5000000000000000 : ℝ)) (q :=
+      (133500188112619 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_10_5 :
+    Uω (aρ 5) (bρ 5) (82548843 / 500000000000 : ℝ) ≤
+      (-(39565324440062555612331 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (95646657087479 / 5000000000000000 : ℝ) ≤
+      (-(39565324440062555612331 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (95646657087479 / 5000000000000000 : ℝ) = (95646657087479 / 78125000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (95646657087479 / 78125000000000 : ℝ) ≤
+            (2023506377937444387669 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (13022791966229 / 2500000000000000 : ℝ)) (q :=
+      (95646657087479 / 5000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_10_6 :
+    Uω (aρ 6) (bρ 6) (82548843 / 500000000000 : ℝ) ≤
+      (-(36014967106290014530057 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (272828572881853 / 10000000000000000 : ℝ) ≤
+      (-(36014967106290014530057 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (272828572881853 / 10000000000000000 : ℝ) =
+            (272828572881853 / 156250000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (272828572881853 / 156250000000000 : ℝ) ≤
+            (5573863711709985469943 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (23960655069353 / 5000000000000000 : ℝ)) (q :=
+      (272828572881853 / 10000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_10_7 :
+    Uω (aρ 7) (bρ 7) (82548843 / 500000000000 : ℝ) ≤
+      (-(8166283330872021732883 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (762783481867911 / 20000000000000000 : ℝ) ≤
+      (-(8166283330872021732883 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (762783481867911 / 20000000000000000 : ℝ) =
+            (762783481867911 / 625000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (762783481867911 / 625000000000000 : ℝ) ≤
+            (498056422877978267117 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (41386265482911 / 10000000000000000 : ℝ)) (q :=
+      (762783481867911 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_10_8 :
+    Uω (aρ 8) (bρ 8) (82548843 / 500000000000 : ℝ) ≤
+      (-(592129837744490728151 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (517852877658559 / 10000000000000000 : ℝ) ≤
+      (-(592129837744490728151 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (517852877658559 / 10000000000000000 : ℝ) =
+            (517852877658559 / 312500000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (517852877658559 / 312500000000000 : ℝ) ≤
+            (101017342555509271849 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (15383443796059 / 5000000000000000 : ℝ)) (q :=
+      (517852877658559 / 10000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_10_9 :
+    Uω (aρ 9) (bρ 9) (82548843 / 500000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_10_10 :
+    Uω (aρ 10) (bρ 10) (82548843 / 500000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_10_11 :
+    Uω (aρ 11) (bρ 11) (82548843 / 500000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_10_12 :
+    Uω (aρ 12) (bρ 12) (82548843 / 500000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_10_13 :
+    Uω (aρ 13) (bρ 13) (82548843 / 500000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_10_14 :
+    Uω (aρ 14) (bρ 14) (82548843 / 500000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_10_15 :
+    Uω (aρ 15) (bρ 15) (82548843 / 500000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_10_16 :
+    Uω (aρ 16) (bρ 16) (82548843 / 500000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_10 :
+    Uρ (82548843 / 500000000000 : ℝ) ≤ (-(1107174623610650247861 / 400000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (82548843 / 500000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(51133510883215947535941 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_10_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (82548843 / 500000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(24678851808854408309657 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_10_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (82548843 / 500000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(5818929893213673684367 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_10_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (82548843 / 500000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(43162374832339657159807 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_10_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (82548843 / 500000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(39565324440062555612331 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_10_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (82548843 / 500000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(36014967106290014530057 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_10_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (82548843 / 500000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(8166283330872021732883 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_10_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (82548843 / 500000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(592129837744490728151 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_10_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (82548843 / 500000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_10_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (82548843 / 500000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_10_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (82548843 / 500000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_10_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (82548843 / 500000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_10_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (82548843 / 500000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_10_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (82548843 / 500000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_10_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (82548843 / 500000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_10_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (82548843 / 500000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_10_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_11_1 :
+    Uω (aρ 1) (bρ 1) (53051547 / 250000000000 : ℝ) ≤
+      (-(10243169805498114481613 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (119331179383113 / 20000000000000000 : ℝ) ≤
+      (-(10243169805498114481613 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show
+          (119331179383113 / 20000000000000000 : ℝ) =
+            (119331179383113 / 78125000000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (119331179383113 / 78125000000000 : ℝ) ≤
+            (847185079301885518387 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (56956023178113 / 10000000000000000 : ℝ)) (q :=
+      (119331179383113 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_11_2 :
+    Uω (aρ 2) (bρ 2) (53051547 / 250000000000 : ℝ) ≤
+      (-(24720375964658024966207 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17813753292691 / 2500000000000000 : ℝ) ≤
+      (-(24720375964658024966207 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show
+          (17813753292691 / 2500000000000000 : ℝ) = (17813753292691 / 9765625000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17813753292691 / 9765625000000 : ℝ) ≤
+            (3005511247341975033793 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (1761433050829 / 312500000000000 : ℝ)) (q :=
+      (17813753292691 / 2500000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_11_3 :
+    Uω (aρ 3) (bρ 3) (53051547 / 250000000000 : ℝ) ≤
+      (-(46636055244366115624033 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (94323922802349 / 10000000000000000 : ℝ) ≤
+      (-(46636055244366115624033 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (94323922802349 / 10000000000000000 : ℝ) = (94323922802349 / 78125000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (94323922802349 / 78125000000000 : ℝ) ≤
+            (1884247376633884375967 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (27553785269849 / 5000000000000000 : ℝ)) (q :=
+      (94323922802349 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_11_4 :
+    Uω (aρ 4) (bρ 4) (53051547 / 250000000000 : ℝ) ≤
+      (-(21624996699245277221649 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (33083897021511 / 2500000000000000 : ℝ) ≤
+      (-(21624996699245277221649 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (33083897021511 / 2500000000000000 : ℝ) = (33083897021511 / 19531250000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (33083897021511 / 19531250000000 : ℝ) ≤
+            (2635154611254722778351 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (6604590005261 / 1250000000000000 : ℝ)) (q :=
+      (33083897021511 / 2500000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_11_5 :
+    Uω (aρ 5) (bρ 5) (53051547 / 250000000000 : ℝ) ≤
+      (-(39658512652531402453443 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (189518966230509 / 10000000000000000 : ℝ) ≤
+      (-(39658512652531402453443 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (189518966230509 / 10000000000000000 : ℝ) =
+            (189518966230509 / 156250000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (189518966230509 / 156250000000000 : ℝ) ≤
+            (1930318165468597546557 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (24506778498009 / 5000000000000000 : ℝ)) (q :=
+      (189518966230509 / 10000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_11_6 :
+    Uω (aρ 6) (bρ 6) (53051547 / 250000000000 : ℝ) ≤
+      (-(18059438196821077208529 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (135004154291197 / 5000000000000000 : ℝ) ≤
+      (-(18059438196821077208529 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (135004154291197 / 5000000000000000 : ℝ) =
+            (135004154291197 / 78125000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (135004154291197 / 78125000000000 : ℝ) ≤
+            (2734977212178922791471 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (10687966639947 / 2500000000000000 : ℝ)) (q :=
+      (135004154291197 / 5000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_11_7 :
+    Uω (aρ 7) (bρ 7) (53051547 / 250000000000 : ℝ) ≤
+      (-(6558653041860765113421 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (150614428548557 / 4000000000000000 : ℝ) ≤
+      (-(6558653041860765113421 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (150614428548557 / 4000000000000000 : ℝ) =
+            (150614428548557 / 125000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (150614428548557 / 125000000000000 : ℝ) ≤
+            (372818761139234886579 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (6429202275557 / 2000000000000000 : ℝ)) (q :=
+      (150614428548557 / 4000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_11_8 :
+    Uω (aρ 8) (bρ 8) (53051547 / 250000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_11_9 :
+    Uω (aρ 9) (bρ 9) (53051547 / 250000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_11_10 :
+    Uω (aρ 10) (bρ 10) (53051547 / 250000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_11_11 :
+    Uω (aρ 11) (bρ 11) (53051547 / 250000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_11_12 :
+    Uω (aρ 12) (bρ 12) (53051547 / 250000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_11_13 :
+    Uω (aρ 13) (bρ 13) (53051547 / 250000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_11_14 :
+    Uω (aρ 14) (bρ 14) (53051547 / 250000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_11_15 :
+    Uω (aρ 15) (bρ 15) (53051547 / 250000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_11_16 :
+    Uω (aρ 16) (bρ 16) (53051547 / 250000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_11 :
+    Uρ (53051547 / 250000000000 : ℝ) ≤
+      (-(13871992575163789919237 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (53051547 / 250000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(10243169805498114481613 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_11_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (53051547 / 250000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(24720375964658024966207 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_11_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (53051547 / 250000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(46636055244366115624033 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_11_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (53051547 / 250000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(21624996699245277221649 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_11_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (53051547 / 250000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(39658512652531402453443 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_11_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (53051547 / 250000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(18059438196821077208529 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_11_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (53051547 / 250000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(6558653041860765113421 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_11_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (53051547 / 250000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_11_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (53051547 / 250000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_11_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (53051547 / 250000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_11_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (53051547 / 250000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_11_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (53051547 / 250000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_11_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (53051547 / 250000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_11_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (53051547 / 250000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_11_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (53051547 / 250000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_11_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (53051547 / 250000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_11_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/U01.lean b/LeanPool/Zeta5Irrational/Table/U01.lean
new file mode 100644
index 0000000000..2ec9d88dbf
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/U01.lean
@@ -0,0 +1,2903 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-! # Certified arcsine potential bounds (U01) -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma U_12_1 :
+    Uω (aρ 1) (bρ 1) (496117379 / 2000000000000 : ℝ) ≤
+      (-(51279014775995128440791 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14822473832053 / 2500000000000000 : ℝ) ≤
+      (-(51279014775995128440791 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show
+          (14822473832053 / 2500000000000000 : ℝ) = (14822473832053 / 9765625000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14822473832053 / 9765625000000 : ℝ) ≤
+            (4172759648004871559209 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (7070394933303 / 1250000000000000 : ℝ)) (q :=
+      (14822473832053 / 2500000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_12_2 :
+    Uω (aρ 2) (bρ 2) (496117379 / 2000000000000 : ℝ) ≤
+      (-(3094041904103170921061 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (141602028099197 / 20000000000000000 : ℝ) ≤
+      (-(3094041904103170921061 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show
+          (141602028099197 / 20000000000000000 : ℝ) =
+            (141602028099197 / 78125000000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (141602028099197 / 78125000000000 : ℝ) ≤
+            (371693997396829078939 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (55816384399197 / 10000000000000000 : ℝ)) (q :=
+      (141602028099197 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_12_3 :
+    Uω (aρ 3) (bρ 3) (496117379 / 2000000000000 : ℝ) ≤
+      (-(23350817492635683984117 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (187414745659063 / 20000000000000000 : ℝ) ≤
+      (-(23350817492635683984117 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (187414745659063 / 20000000000000000 : ℝ) =
+            (187414745659063 / 156250000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (187414745659063 / 156250000000000 : ℝ) ≤
+            (909333817864316015883 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (54232995609063 / 10000000000000000 : ℝ)) (q :=
+      (187414745659063 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_12_4 :
+    Uω (aρ 4) (bρ 4) (496117379 / 2000000000000 : ℝ) ≤
+      (-(8663759421672724895473 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (131428195324229 / 10000000000000000 : ℝ) ≤
+      (-(8663759421672724895473 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (131428195324229 / 10000000000000000 : ℝ) =
+            (131428195324229 / 78125000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (131428195324229 / 78125000000000 : ℝ) ≤
+            (1040301102527275104527 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (25690229766729 / 5000000000000000 : ℝ)) (q :=
+      (131428195324229 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_12_5 :
+    Uω (aρ 5) (bρ 5) (496117379 / 2000000000000 : ℝ) ≤
+      (-(39733556029110319524967 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (376204149959763 / 20000000000000000 : ℝ) ≤
+      (-(39733556029110319524967 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (376204149959763 / 20000000000000000 : ℝ) =
+            (376204149959763 / 312500000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (376204149959763 / 312500000000000 : ℝ) ≤
+            (1855274788889680475033 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (46538299509763 / 10000000000000000 : ℝ)) (q :=
+      (376204149959763 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_12_6 :
+    Uω (aρ 6) (bρ 6) (496117379 / 2000000000000 : ℝ) ≤
+      (-(36207282453746924592843 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (535263583817503 / 20000000000000000 : ℝ) ≤
+      (-(36207282453746924592843 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (535263583817503 / 20000000000000000 : ℝ) =
+            (535263583817503 / 312500000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (535263583817503 / 312500000000000 : ℝ) ≤
+            (5381548364253075407157 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (38357358227503 / 10000000000000000 : ℝ)) (q :=
+      (535263583817503 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_12_7 :
+    Uω (aρ 7) (bρ 7) (496117379 / 2000000000000 : ℝ) ≤
+      (-(32923939525185939772853 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (743295441345089 / 20000000000000000 : ℝ) ≤
+      (-(32923939525185939772853 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (743295441345089 / 20000000000000000 : ℝ) =
+            (743295441345089 / 625000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (743295441345089 / 625000000000000 : ℝ) ≤
+            (1733419489814060227147 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (22727834995089 / 10000000000000000 : ℝ)) (q :=
+      (743295441345089 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_12_8 :
+    Uω (aρ 8) (bρ 8) (496117379 / 2000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_12_9 :
+    Uω (aρ 9) (bρ 9) (496117379 / 2000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_12_10 :
+    Uω (aρ 10) (bρ 10) (496117379 / 2000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_12_11 :
+    Uω (aρ 11) (bρ 11) (496117379 / 2000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_12_12 :
+    Uω (aρ 12) (bρ 12) (496117379 / 2000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_12_13 :
+    Uω (aρ 13) (bρ 13) (496117379 / 2000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_12_14 :
+    Uω (aρ 14) (bρ 14) (496117379 / 2000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_12_15 :
+    Uω (aρ 15) (bρ 15) (496117379 / 2000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_12_16 :
+    Uω (aρ 16) (bρ 16) (496117379 / 2000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_12 :
+    Uρ (496117379 / 2000000000000 : ℝ) ≤
+      (-(27776595649486878085457 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (496117379 / 2000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(51279014775995128440791 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_12_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (496117379 / 2000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3094041904103170921061 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_12_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (496117379 / 2000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(23350817492635683984117 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_12_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (496117379 / 2000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(8663759421672724895473 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_12_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (496117379 / 2000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(39733556029110319524967 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_12_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (496117379 / 2000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(36207282453746924592843 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_12_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (496117379 / 2000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(32923939525185939772853 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_12_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (496117379 / 2000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_12_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (496117379 / 2000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_12_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (496117379 / 2000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_12_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (496117379 / 2000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_12_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (496117379 / 2000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_12_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (496117379 / 2000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_12_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (496117379 / 2000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_12_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (496117379 / 2000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_12_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (496117379 / 2000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_12_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_13_1 :
+    Uω (aρ 1) (bρ 1) (283911191 / 1000000000000 : ℝ) ≤
+      (-(25671310519013681683207 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (117827942579917 / 20000000000000000 : ℝ) ≤
+      (-(25671310519013681683207 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show
+          (117827942579917 / 20000000000000000 : ℝ) =
+            (117827942579917 / 78125000000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (117827942579917 / 78125000000000 : ℝ) ≤
+            (2054576692986318316793 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (56169836404917 / 10000000000000000 : ℝ)) (q :=
+      (117827942579917 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_13_2 :
+    Uω (aρ 2) (bρ 2) (283911191 / 1000000000000 : ℝ) ≤
+      (-(9913844635044356752717 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28138178523381 / 4000000000000000 : ℝ) ≤
+      (-(9913844635044356752717 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show
+          (28138178523381 / 4000000000000000 : ℝ) = (28138178523381 / 15625000000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28138178523381 / 15625000000000 : ℝ) ≤
+            (1176510249755643247283 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (11052754786381 / 2000000000000000 : ℝ)) (q :=
+      (28138178523381 / 4000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_13_3 :
+    Uω (aρ 3) (bρ 3) (283911191 / 1000000000000 : ℝ) ≤
+      (-(46768288274873095600687 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (46542429648523 / 5000000000000000 : ℝ) ≤
+      (-(46768288274873095600687 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (46542429648523 / 5000000000000000 : ℝ) = (46542429648523 / 39062500000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (46542429648523 / 39062500000000 : ℝ) ≤
+            (1752014346126904399313 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (13336623389773 / 2500000000000000 : ℝ)) (q :=
+      (46542429648523 / 5000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_13_4 :
+    Uω (aρ 4) (bρ 4) (283911191 / 1000000000000 : ℝ) ≤
+      (-(5423700856563990487439 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (65250421395167 / 5000000000000000 : ℝ) ≤
+      (-(5423700856563990487439 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (65250421395167 / 5000000000000000 : ℝ) = (65250421395167 / 39062500000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (65250421395167 / 39062500000000 : ℝ) ≤
+            (641336971061009512561 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (12471069870167 / 2500000000000000 : ℝ)) (q :=
+      (65250421395167 / 5000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_13_5 :
+    Uω (aρ 5) (bρ 5) (283911191 / 1000000000000 : ℝ) ≤
+      (-(39812819146312822009509 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (93308506303549 / 5000000000000000 : ℝ) ≤
+      (-(39812819146312822009509 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (93308506303549 / 5000000000000000 : ℝ) = (93308506303549 / 78125000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (93308506303549 / 78125000000000 : ℝ) ≤
+            (1776011671687177990491 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (10981674944799 / 2500000000000000 : ℝ)) (q :=
+      (93308506303549 / 5000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_13_6 :
+    Uω (aρ 6) (bρ 6) (283911191 / 1000000000000 : ℝ) ≤
+      (-(18153609865494868493647 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (529940946095807 / 20000000000000000 : ℝ) ≤
+      (-(18153609865494868493647 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (529940946095807 / 20000000000000000 : ℝ) =
+            (529940946095807 / 312500000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (529940946095807 / 312500000000000 : ℝ) ≤
+            (2640805543505131506353 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (33393245520807 / 10000000000000000 : ℝ)) (q :=
+      (529940946095807 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_13_7 :
+    Uω (aρ 7) (bρ 7) (283911191 / 1000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_13_8 :
+    Uω (aρ 8) (bρ 8) (283911191 / 1000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_13_9 :
+    Uω (aρ 9) (bρ 9) (283911191 / 1000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_13_10 :
+    Uω (aρ 10) (bρ 10) (283911191 / 1000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_13_11 :
+    Uω (aρ 11) (bρ 11) (283911191 / 1000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_13_12 :
+    Uω (aρ 12) (bρ 12) (283911191 / 1000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_13_13 :
+    Uω (aρ 13) (bρ 13) (283911191 / 1000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_13_14 :
+    Uω (aρ 14) (bρ 14) (283911191 / 1000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_13_15 :
+    Uω (aρ 15) (bρ 15) (283911191 / 1000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_13_16 :
+    Uω (aρ 16) (bρ 16) (283911191 / 1000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_13 :
+    Uρ (283911191 / 1000000000000 : ℝ) ≤
+      (-(5565255693326323982243 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (283911191 / 1000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(25671310519013681683207 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_13_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (283911191 / 1000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(9913844635044356752717 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_13_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (283911191 / 1000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(46768288274873095600687 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_13_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (283911191 / 1000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(5423700856563990487439 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_13_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (283911191 / 1000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(39812819146312822009509 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_13_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (283911191 / 1000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(18153609865494868493647 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_13_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (283911191 / 1000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_13_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (283911191 / 1000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_13_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (283911191 / 1000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_13_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (283911191 / 1000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_13_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (283911191 / 1000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_13_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (283911191 / 1000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_13_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (283911191 / 1000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_13_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (283911191 / 1000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_13_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (283911191 / 1000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_13_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (283911191 / 1000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_13_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_14_1 :
+    Uω (aρ 1) (bρ 1) (170058951 / 500000000000 : ℝ) ≤
+      (-(5144324067033277739057 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (29162076708097 / 5000000000000000 : ℝ) ≤
+      (-(5144324067033277739057 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show
+          (29162076708097 / 5000000000000000 : ℝ) = (29162076708097 / 19531250000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (29162076708097 / 19531250000000 : ℝ) ≤
+            (400853375366722260943 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (13888066941847 / 2500000000000000 : ℝ)) (q :=
+      (29162076708097 / 5000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_14_2 :
+    Uω (aρ 2) (bρ 2) (170058951 / 500000000000 : ℝ) ≤
+      (-(496717399318518904317 / 100000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13925594310293 / 2000000000000000 : ℝ) ≤
+      (-(496717399318518904317 / 100000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show
+          (13925594310293 / 2000000000000000 : ℝ) = (13925594310293 / 7812500000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13925594310293 / 7812500000000 : ℝ) ≤
+            (57800344921481095683 / 100000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (5439089152793 / 1000000000000000 : ℝ)) (q :=
+      (13925594310293 / 2000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_14_3 :
+    Uω (aρ 3) (bρ 3) (170058951 / 500000000000 : ℝ) ≤
+      (-(1875002690401424918243 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5756012276917 / 625000000000000 : ℝ) ≤
+      (-(1875002690401424918243 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show (5756012276917 / 625000000000000 : ℝ) = (5756012276917 / 4882812500000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5756012276917 / 4882812500000 : ℝ) ≤
+            (65809414438575081757 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (6491404367043 / 1250000000000000 : ℝ)) (q :=
+      (5756012276917 / 625000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_14_4 :
+    Uω (aρ 4) (bρ 4) (170058951 / 500000000000 : ℝ) ≤
+      (-(5438137640616013885629 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (51600921238611 / 4000000000000000 : ℝ) ≤
+      (-(5438137640616013885629 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (51600921238611 / 4000000000000000 : ℝ) = (51600921238611 / 31250000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (51600921238611 / 31250000000000 : ℝ) ≤
+            (626900187008986114371 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (9489853440611 / 2000000000000000 : ℝ)) (q :=
+      (51600921238611 / 4000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_14_5 :
+    Uω (aρ 5) (bρ 5) (170058951 / 500000000000 : ℝ) ≤
+      (-(39947577872317960521831 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (368238108811001 / 20000000000000000 : ℝ) ≤
+      (-(39947577872317960521831 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (368238108811001 / 20000000000000000 : ℝ) =
+            (368238108811001 / 312500000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (368238108811001 / 312500000000000 : ℝ) ≤
+            (1641252945682039478169 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (39492850486001 / 10000000000000000 : ℝ)) (q :=
+      (368238108811001 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_14_6 :
+    Uω (aρ 6) (bρ 6) (170058951 / 500000000000 : ℝ) ≤
+      (-(36504445610949767396391 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (519591533231167 / 20000000000000000 : ℝ) ≤
+      (-(36504445610949767396391 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (519591533231167 / 20000000000000000 : ℝ) =
+            (519591533231167 / 312500000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (519591533231167 / 312500000000000 : ℝ) ≤
+            (5084385207050232603609 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (23605899766167 / 10000000000000000 : ℝ)) (q :=
+      (519591533231167 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_14_7 :
+    Uω (aρ 7) (bρ 7) (170058951 / 500000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_14_8 :
+    Uω (aρ 8) (bρ 8) (170058951 / 500000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_14_9 :
+    Uω (aρ 9) (bρ 9) (170058951 / 500000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_14_10 :
+    Uω (aρ 10) (bρ 10) (170058951 / 500000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_14_11 :
+    Uω (aρ 11) (bρ 11) (170058951 / 500000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_14_12 :
+    Uω (aρ 12) (bρ 12) (170058951 / 500000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_14_13 :
+    Uω (aρ 13) (bρ 13) (170058951 / 500000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_14_14 :
+    Uω (aρ 14) (bρ 14) (170058951 / 500000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_14_15 :
+    Uω (aρ 15) (bρ 15) (170058951 / 500000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_14_16 :
+    Uω (aρ 16) (bρ 16) (170058951 / 500000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_14 :
+    Uρ (170058951 / 500000000000 : ℝ) ≤
+      (-(13933261935333362960321 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (170058951 / 500000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(5144324067033277739057 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_14_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (170058951 / 500000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(496717399318518904317 / 100000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_14_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (170058951 / 500000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1875002690401424918243 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_14_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (170058951 / 500000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(5438137640616013885629 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_14_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (170058951 / 500000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(39947577872317960521831 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_14_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (170058951 / 500000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(36504445610949767396391 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_14_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (170058951 / 500000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_14_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (170058951 / 500000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_14_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (170058951 / 500000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_14_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (170058951 / 500000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_14_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (170058951 / 500000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_14_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (170058951 / 500000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_14_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (170058951 / 500000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_14_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (170058951 / 500000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_14_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (170058951 / 500000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_14_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (170058951 / 500000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_14_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_15_1 :
+    Uω (aρ 1) (bρ 1) (396324613 / 1000000000000 : ℝ) ≤
+      (-(10308997249903148058101 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2886686981611 / 500000000000000 : ℝ) ≤
+      (-(10308997249903148058101 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show (2886686981611 / 500000000000000 : ℝ) = (2886686981611 / 1953125000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2886686981611 / 1953125000000 : ℝ) ≤
+            (781357634896851941899 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (85833605171 / 15625000000000 : ℝ)) (q :=
+      (2886686981611 / 500000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_15_2 :
+    Uω (aρ 2) (bρ 2) (396324613 / 1000000000000 : ℝ) ≤
+      (-(24887961647494144053291 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (137812659217157 / 20000000000000000 : ℝ) ≤
+      (-(24887961647494144053291 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show
+          (137812659217157 / 20000000000000000 : ℝ) =
+            (137812659217157 / 78125000000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (137812659217157 / 78125000000000 : ℝ) ≤
+            (2837925564505855946709 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (53509674752157 / 10000000000000000 : ℝ)) (q :=
+      (137812659217157 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_15_3 :
+    Uω (aρ 3) (bρ 3) (396324613 / 1000000000000 : ℝ) ≤
+      (-(11746208074433576911541 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (182181659963663 / 20000000000000000 : ℝ) ≤
+      (-(11746208074433576911541 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (182181659963663 / 20000000000000000 : ℝ) =
+            (182181659963663 / 156250000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (182181659963663 / 156250000000000 : ℝ) ≤
+            (383867580816423088459 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (50482569148663 / 10000000000000000 : ℝ)) (q :=
+      (182181659963663 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_15_4 :
+    Uω (aρ 4) (bρ 4) (396324613 / 1000000000000 : ℝ) ≤
+      (-(43626843011011594664449 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5097653032921 / 400000000000000 : ℝ) ≤
+      (-(43626843011011594664449 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show (5097653032921 / 400000000000000 : ℝ) = (5097653032921 / 3125000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5097653032921 / 3125000000000 : ℝ) ≤
+            (4893459609988405335551 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (897787595321 / 200000000000000 : ℝ)) (q :=
+      (5097653032921 / 400000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_15_5 :
+    Uω (aρ 5) (bρ 5) (396324613 / 1000000000000 : ℝ) ≤
+      (-(20049749814838098711663 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (362686051450057 / 20000000000000000 : ℝ) ≤
+      (-(20049749814838098711663 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (362686051450057 / 20000000000000000 : ℝ) =
+            (362686051450057 / 312500000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (362686051450057 / 312500000000000 : ℝ) ≤
+            (744665594161901288337 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (34502860235057 / 10000000000000000 : ℝ)) (q :=
+      (362686051450057 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_15_6 :
+    Uω (aρ 6) (bρ 6) (396324613 / 1000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_15_7 :
+    Uω (aρ 7) (bρ 7) (396324613 / 1000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_15_8 :
+    Uω (aρ 8) (bρ 8) (396324613 / 1000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_15_9 :
+    Uω (aρ 9) (bρ 9) (396324613 / 1000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_15_10 :
+    Uω (aρ 10) (bρ 10) (396324613 / 1000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_15_11 :
+    Uω (aρ 11) (bρ 11) (396324613 / 1000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_15_12 :
+    Uω (aρ 12) (bρ 12) (396324613 / 1000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_15_13 :
+    Uω (aρ 13) (bρ 13) (396324613 / 1000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_15_14 :
+    Uω (aρ 14) (bρ 14) (396324613 / 1000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_15_15 :
+    Uω (aρ 15) (bρ 15) (396324613 / 1000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_15_16 :
+    Uω (aρ 16) (bρ 16) (396324613 / 1000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_15 :
+    Uρ (396324613 / 1000000000000 : ℝ) ≤
+      (-(27930518333896095285283 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (396324613 / 1000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(10308997249903148058101 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_15_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (396324613 / 1000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(24887961647494144053291 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_15_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (396324613 / 1000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(11746208074433576911541 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_15_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (396324613 / 1000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(43626843011011594664449 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_15_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (396324613 / 1000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20049749814838098711663 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_15_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (396324613 / 1000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_15_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (396324613 / 1000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_15_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (396324613 / 1000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_15_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (396324613 / 1000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_15_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (396324613 / 1000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_15_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (396324613 / 1000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_15_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (396324613 / 1000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_15_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (396324613 / 1000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_15_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (396324613 / 1000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_15_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (396324613 / 1000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_15_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (396324613 / 1000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_15_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_16_1 :
+    Uω (aρ 1) (bρ 1) (974522519 / 2000000000000 : ℝ) ≤
+      (-(51712055896552965688933 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1774287399843 / 312500000000000 : ℝ) ≤
+      (-(51712055896552965688933 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show (1774287399843 / 312500000000000 : ℝ) = (1774287399843 / 1220703125000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1774287399843 / 1220703125000 : ℝ) ≤
+            (3739718527447034311067 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (3370611756247 / 625000000000000 : ℝ)) (q :=
+      (1774287399843 / 312500000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_16_2 :
+    Uω (aρ 2) (bρ 2) (974522519 / 2000000000000 : ℝ) ≤
+      (-(49948197000145039205691 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (135458842723471 / 20000000000000000 : ℝ) ≤
+      (-(49948197000145039205691 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show
+          (135458842723471 / 20000000000000000 : ℝ) =
+            (135458842723471 / 78125000000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (135458842723471 / 78125000000000 : ℝ) ≤
+            (5503577423854960794309 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (52065224723471 / 10000000000000000 : ℝ)) (q :=
+      (135458842723471 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_16_3 :
+    Uω (aρ 3) (bρ 3) (974522519 / 2000000000000 : ℝ) ≤
+      (-(11792350042914402417241 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2794531439597 / 312500000000000 : ℝ) ≤
+      (-(11792350042914402417241 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show (2794531439597 / 312500000000000 : ℝ) = (2794531439597 / 2441406250000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2794531439597 / 2441406250000 : ℝ) ≤
+            (337725612335597582759 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (3003767986513 / 625000000000000 : ℝ)) (q :=
+      (2794531439597 / 312500000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_16_4 :
+    Uω (aρ 4) (bρ 4) (974522519 / 2000000000000 : ℝ) ≤
+      (-(219200189839688302473 / 50000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (124753098670197 / 10000000000000000 : ℝ) ≤
+      (-(219200189839688302473 / 50000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (124753098670197 / 10000000000000000 : ℝ) =
+            (124753098670197 / 78125000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (124753098670197 / 78125000000000 : ℝ) ≤
+            (23401323265311697527 / 50000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (20211145962697 / 5000000000000000 : ℝ)) (q :=
+      (124753098670197 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_16_5 :
+    Uω (aρ 5) (bρ 5) (974522519 / 2000000000000 : ℝ) ≤
+      (-(808167918466123975469 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (351654077718997 / 20000000000000000 : ℝ) ≤
+      (-(808167918466123975469 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (351654077718997 / 20000000000000000 : ℝ) =
+            (351654077718997 / 312500000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (351654077718997 / 312500000000000 : ℝ) ≤
+            (23608697893876024531 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (24380252968997 / 10000000000000000 : ℝ)) (q :=
+      (351654077718997 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_16_6 :
+    Uω (aρ 6) (bρ 6) (974522519 / 2000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_16_7 :
+    Uω (aρ 7) (bρ 7) (974522519 / 2000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_16_8 :
+    Uω (aρ 8) (bρ 8) (974522519 / 2000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_16_9 :
+    Uω (aρ 9) (bρ 9) (974522519 / 2000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_16_10 :
+    Uω (aρ 10) (bρ 10) (974522519 / 2000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_16_11 :
+    Uω (aρ 11) (bρ 11) (974522519 / 2000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_16_12 :
+    Uω (aρ 12) (bρ 12) (974522519 / 2000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_16_13 :
+    Uω (aρ 13) (bρ 13) (974522519 / 2000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_16_14 :
+    Uω (aρ 14) (bρ 14) (974522519 / 2000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_16_15 :
+    Uω (aρ 15) (bρ 15) (974522519 / 2000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_16_16 :
+    Uω (aρ 16) (bρ 16) (974522519 / 2000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_16 :
+    Uρ (974522519 / 2000000000000 : ℝ) ≤
+      (-(27979010994860448204283 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (974522519 / 2000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(51712055896552965688933 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_16_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (974522519 / 2000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(49948197000145039205691 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_16_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (974522519 / 2000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(11792350042914402417241 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_16_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (974522519 / 2000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(219200189839688302473 / 50000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_16_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (974522519 / 2000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(808167918466123975469 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_16_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (974522519 / 2000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_16_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (974522519 / 2000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_16_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (974522519 / 2000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_16_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (974522519 / 2000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_16_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (974522519 / 2000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_16_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (974522519 / 2000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_16_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (974522519 / 2000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_16_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (974522519 / 2000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_16_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (974522519 / 2000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_16_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (974522519 / 2000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_16_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (974522519 / 2000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_16_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_17_1 :
+    Uω (aρ 1) (bρ 1) (289098953 / 500000000000 : ℝ) ≤
+      (-(1037645395330397781579 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27909474323909 / 5000000000000000 : ℝ) ≤
+      (-(1037645395330397781579 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show
+          (27909474323909 / 5000000000000000 : ℝ) = (27909474323909 / 19531250000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27909474323909 / 19531250000000 : ℝ) ≤
+            (71390093149602218421 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (13230664567659 / 2500000000000000 : ℝ)) (q :=
+      (27909474323909 / 5000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_17_2 :
+    Uω (aρ 2) (bρ 2) (289098953 / 500000000000 : ℝ) ≤
+      (-(50125360529440552437197 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (133080139249959 / 20000000000000000 : ℝ) ≤
+      (-(50125360529440552437197 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show
+          (133080139249959 / 20000000000000000 : ℝ) =
+            (133080139249959 / 78125000000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (133080139249959 / 78125000000000 : ℝ) ≤
+            (5326413894559447562803 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (50595887714959 / 10000000000000000 : ℝ)) (q :=
+      (133080139249959 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_17_3 :
+    Uω (aρ 3) (bρ 3) (289098953 / 500000000000 : ℝ) ≤
+      (-(47363740868116265483369 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (175407785120543 / 20000000000000000 : ℝ) ≤
+      (-(47363740868116265483369 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (175407785120543 / 20000000000000000 : ℝ) =
+            (175407785120543 / 156250000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (175407785120543 / 156250000000000 : ℝ) ≤
+            (1156561752883734516631 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (45527427235543 / 10000000000000000 : ℝ)) (q :=
+      (175407785120543 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_17_4 :
+    Uω (aρ 4) (bρ 4) (289098953 / 500000000000 : ℝ) ≤
+      (-(11019964131301389298539 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (48718750950201 / 4000000000000000 : ℝ) ≤
+      (-(11019964131301389298539 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (48718750950201 / 4000000000000000 : ℝ) = (48718750950201 / 31250000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (48718750950201 / 31250000000000 : ℝ) ≤
+            (1110111523948610701461 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (7083843160201 / 2000000000000000 : ℝ)) (q :=
+      (48718750950201 / 4000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_17_5 :
+    Uω (aρ 5) (bρ 5) (289098953 / 500000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_17_6 :
+    Uω (aρ 6) (bρ 6) (289098953 / 500000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_17_7 :
+    Uω (aρ 7) (bρ 7) (289098953 / 500000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_17_8 :
+    Uω (aρ 8) (bρ 8) (289098953 / 500000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_17_9 :
+    Uω (aρ 9) (bρ 9) (289098953 / 500000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_17_10 :
+    Uω (aρ 10) (bρ 10) (289098953 / 500000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_17_11 :
+    Uω (aρ 11) (bρ 11) (289098953 / 500000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_17_12 :
+    Uω (aρ 12) (bρ 12) (289098953 / 500000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_17_13 :
+    Uω (aρ 13) (bρ 13) (289098953 / 500000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_17_14 :
+    Uω (aρ 14) (bρ 14) (289098953 / 500000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_17_15 :
+    Uω (aρ 15) (bρ 15) (289098953 / 500000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_17_16 :
+    Uω (aρ 16) (bρ 16) (289098953 / 500000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_17 :
+    Uρ (289098953 / 500000000000 : ℝ) ≤
+      (-(14029924784689781065817 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (289098953 / 500000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1037645395330397781579 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_17_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (289098953 / 500000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(50125360529440552437197 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_17_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (289098953 / 500000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(47363740868116265483369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_17_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (289098953 / 500000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11019964131301389298539 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_17_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (289098953 / 500000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_17_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (289098953 / 500000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_17_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (289098953 / 500000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_17_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (289098953 / 500000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_17_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (289098953 / 500000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_17_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (289098953 / 500000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_17_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (289098953 / 500000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_17_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (289098953 / 500000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_17_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (289098953 / 500000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_17_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (289098953 / 500000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_17_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (289098953 / 500000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_17_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (289098953 / 500000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_17_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_18_1 :
+    Uω (aρ 1) (bρ 1) (729961631 / 1000000000000 : ℝ) ≤
+      (-(52173705525642853674791 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (54215661123871 / 10000000000000000 : ℝ) ≤
+      (-(52173705525642853674791 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show
+          (54215661123871 / 10000000000000000 : ℝ) = (54215661123871 / 39062500000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (54215661123871 / 39062500000000 : ℝ) ≤
+            (3278068898357146325209 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (25616860236371 / 5000000000000000 : ℝ)) (q :=
+      (54215661123871 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_18_2 :
+    Uω (aρ 2) (bρ 2) (729961631 / 1000000000000 : ℝ) ≤
+      (-(157603058690652939389 / 31250000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (129048676752739 / 20000000000000000 : ℝ) ≤
+      (-(157603058690652939389 / 31250000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show
+          (129048676752739 / 20000000000000000 : ℝ) =
+            (129048676752739 / 78125000000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (129048676752739 / 78125000000000 : ℝ) ≤
+            (15683736384347060611 / 31250000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (48082062467739 / 10000000000000000 : ℝ)) (q :=
+      (129048676752739 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_18_3 :
+    Uω (aρ 3) (bρ 3) (729961631 / 1000000000000 : ℝ) ≤
+      (-(5964321618170938271907 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (33872122933673 / 4000000000000000 : ℝ) ≤
+      (-(5964321618170938271907 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (33872122933673 / 4000000000000000 : ℝ) = (33872122933673 / 31250000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (33872122933673 / 31250000000000 : ℝ) ≤
+            (100716209454061728093 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (8199578806673 / 2000000000000000 : ℝ)) (q :=
+      (33872122933673 / 4000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_18_4 :
+    Uω (aρ 4) (bρ 4) (729961631 / 1000000000000 : ℝ) ≤
+      (-(44582338407334756483981 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (57914011427119 / 5000000000000000 : ℝ) ≤
+      (-(44582338407334756483981 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (57914011427119 / 5000000000000000 : ℝ) = (57914011427119 / 39062500000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (57914011427119 / 39062500000000 : ℝ) ≤
+            (3937964213665243516019 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (6249786002119 / 2500000000000000 : ℝ)) (q :=
+      (57914011427119 / 5000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_18_5 :
+    Uω (aρ 5) (bρ 5) (729961631 / 1000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_18_6 :
+    Uω (aρ 6) (bρ 6) (729961631 / 1000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_18_7 :
+    Uω (aρ 7) (bρ 7) (729961631 / 1000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_18_8 :
+    Uω (aρ 8) (bρ 8) (729961631 / 1000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_18_9 :
+    Uω (aρ 9) (bρ 9) (729961631 / 1000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_18_10 :
+    Uω (aρ 10) (bρ 10) (729961631 / 1000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_18_11 :
+    Uω (aρ 11) (bρ 11) (729961631 / 1000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_18_12 :
+    Uω (aρ 12) (bρ 12) (729961631 / 1000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_18_13 :
+    Uω (aρ 13) (bρ 13) (729961631 / 1000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_18_14 :
+    Uω (aρ 14) (bρ 14) (729961631 / 1000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_18_15 :
+    Uω (aρ 15) (bρ 15) (729961631 / 1000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_18_16 :
+    Uω (aρ 16) (bρ 16) (729961631 / 1000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_18 :
+    Uρ (729961631 / 1000000000000 : ℝ) ≤
+      (-(1124651582364191246041 / 400000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (729961631 / 1000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(52173705525642853674791 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_18_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (729961631 / 1000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) * (-(157603058690652939389 / 31250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_18_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (729961631 / 1000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(5964321618170938271907 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_18_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (729961631 / 1000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(44582338407334756483981 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_18_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (729961631 / 1000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_18_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (729961631 / 1000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_18_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (729961631 / 1000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_18_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (729961631 / 1000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_18_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (729961631 / 1000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_18_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (729961631 / 1000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_18_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (729961631 / 1000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_18_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (729961631 / 1000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_18_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (729961631 / 1000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_18_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (729961631 / 1000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_18_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (729961631 / 1000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_18_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (729961631 / 1000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_18_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_19_1 :
+    Uω (aρ 1) (bρ 1) (1611686987 / 2000000000000 : ℝ) ≤
+      (-(26161527144129535042071 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (26705986680359 / 5000000000000000 : ℝ) ≤
+      (-(26161527144129535042071 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show
+          (26705986680359 / 5000000000000000 : ℝ) = (26705986680359 / 19531250000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26705986680359 / 19531250000000 : ℝ) ≤
+            (1564360067870464957929 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (12596290892859 / 2500000000000000 : ℝ)) (q :=
+      (26705986680359 / 5000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_19_2 :
+    Uω (aρ 2) (bρ 2) (1611686987 / 2000000000000 : ℝ) ≤
+      (-(10118588778586567237301 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (127000771489839 / 20000000000000000 : ℝ) ≤
+      (-(10118588778586567237301 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show
+          (127000771489839 / 20000000000000000 : ℝ) =
+            (127000771489839 / 78125000000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (127000771489839 / 78125000000000 : ℝ) ≤
+            (971766106213432762699 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (46792975829839 / 10000000000000000 : ℝ)) (q :=
+      (127000771489839 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_19_3 :
+    Uω (aρ 3) (bρ 3) (1611686987 / 2000000000000 : ℝ) ≤
+      (-(47905346243074077148493 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (33232058051079 / 4000000000000000 : ℝ) ≤
+      (-(47905346243074077148493 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (33232058051079 / 4000000000000000 : ℝ) = (33232058051079 / 31250000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (33232058051079 / 31250000000000 : ℝ) ≤
+            (614956377925922851507 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (7711277649079 / 2000000000000000 : ℝ)) (q :=
+      (33232058051079 / 4000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_19_4 :
+    Uω (aρ 4) (bρ 4) (1611686987 / 2000000000000 : ℝ) ≤
+      (-(8987625693749818873013 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (55889713253743 / 5000000000000000 : ℝ) ≤
+      (-(8987625693749818873013 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (55889713253743 / 5000000000000000 : ℝ) = (55889713253743 / 39062500000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (55889713253743 / 39062500000000 : ℝ) ≤
+            (716434830450181126987 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (4415192484993 / 2500000000000000 : ℝ)) (q :=
+      (55889713253743 / 5000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_19_5 :
+    Uω (aρ 5) (bρ 5) (1611686987 / 2000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_19_6 :
+    Uω (aρ 6) (bρ 6) (1611686987 / 2000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_19_7 :
+    Uω (aρ 7) (bρ 7) (1611686987 / 2000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_19_8 :
+    Uω (aρ 8) (bρ 8) (1611686987 / 2000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_19_9 :
+    Uω (aρ 9) (bρ 9) (1611686987 / 2000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_19_10 :
+    Uω (aρ 10) (bρ 10) (1611686987 / 2000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_19_11 :
+    Uω (aρ 11) (bρ 11) (1611686987 / 2000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_19_12 :
+    Uω (aρ 12) (bρ 12) (1611686987 / 2000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_19_13 :
+    Uω (aρ 13) (bρ 13) (1611686987 / 2000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_19_14 :
+    Uω (aρ 14) (bρ 14) (1611686987 / 2000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_19_15 :
+    Uω (aρ 15) (bρ 15) (1611686987 / 2000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_19_16 :
+    Uω (aρ 16) (bρ 16) (1611686987 / 2000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_19 :
+    Uρ (1611686987 / 2000000000000 : ℝ) ≤
+      (-(1126058948715908662283 / 400000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1611686987 / 2000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(26161527144129535042071 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_19_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1611686987 / 2000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(10118588778586567237301 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_19_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1611686987 / 2000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(47905346243074077148493 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_19_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1611686987 / 2000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(8987625693749818873013 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_19_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1611686987 / 2000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_19_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1611686987 / 2000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_19_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1611686987 / 2000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_19_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1611686987 / 2000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_19_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1611686987 / 2000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_19_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1611686987 / 2000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_19_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1611686987 / 2000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_19_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1611686987 / 2000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_19_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1611686987 / 2000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_19_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1611686987 / 2000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_19_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1611686987 / 2000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_19_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1611686987 / 2000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_19_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_20_1 :
+    Uω (aρ 1) (bρ 1) (220431339 / 250000000000 : ℝ) ≤
+      (-(10494988835400103718169 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (52606829630367 / 10000000000000000 : ℝ) ≤
+      (-(10494988835400103718169 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show
+          (52606829630367 / 10000000000000000 : ℝ) = (52606829630367 / 39062500000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (52606829630367 / 39062500000000 : ℝ) ≤
+            (595366049399896281831 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (24766847367867 / 5000000000000000 : ℝ)) (q :=
+      (52606829630367 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_20_2 :
+    Uω (aρ 2) (bρ 2) (220431339 / 250000000000 : ℝ) ≤
+      (-(50757420127736585768143 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12492899522193 / 2000000000000000 : ℝ) ≤
+      (-(50757420127736585768143 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show
+          (12492899522193 / 2000000000000000 : ℝ) = (12492899522193 / 7812500000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12492899522193 / 7812500000000 : ℝ) ≤
+            (4694354296263414231857 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (4548001818693 / 1000000000000000 : ℝ)) (q :=
+      (12492899522193 / 2000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_20_3 :
+    Uω (aρ 3) (bρ 3) (220431339 / 250000000000 : ℝ) ≤
+      (-(24054501625710163523737 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (162810545224097 / 20000000000000000 : ℝ) ≤
+      (-(24054501625710163523737 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (162810545224097 / 20000000000000000 : ℝ) =
+            (162810545224097 / 156250000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (162810545224097 / 156250000000000 : ℝ) ≤
+            (205649684789836476263 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (35965461839097 / 10000000000000000 : ℝ)) (q :=
+      (162810545224097 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_20_4 :
+    Uω (aρ 4) (bρ 4) (220431339 / 250000000000 : ℝ) ≤
+      (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_4]) (by norm_num [bρ_4])]; exact Lmid_4
+
+lemma U_20_5 :
+    Uω (aρ 5) (bρ 5) (220431339 / 250000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_20_6 :
+    Uω (aρ 6) (bρ 6) (220431339 / 250000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_20_7 :
+    Uω (aρ 7) (bρ 7) (220431339 / 250000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_20_8 :
+    Uω (aρ 8) (bρ 8) (220431339 / 250000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_20_9 :
+    Uω (aρ 9) (bρ 9) (220431339 / 250000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_20_10 :
+    Uω (aρ 10) (bρ 10) (220431339 / 250000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_20_11 :
+    Uω (aρ 11) (bρ 11) (220431339 / 250000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_20_12 :
+    Uω (aρ 12) (bρ 12) (220431339 / 250000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_20_13 :
+    Uω (aρ 13) (bρ 13) (220431339 / 250000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_20_14 :
+    Uω (aρ 14) (bρ 14) (220431339 / 250000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_20_15 :
+    Uω (aρ 15) (bρ 15) (220431339 / 250000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_20_16 :
+    Uω (aρ 16) (bρ 16) (220431339 / 250000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_20 :
+    Uρ (220431339 / 250000000000 : ℝ) ≤
+      (-(7054177370793323835461 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (220431339 / 250000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(10494988835400103718169 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_20_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (220431339 / 250000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(50757420127736585768143 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_20_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (220431339 / 250000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(24054501625710163523737 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_20_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (220431339 / 250000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_20_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (220431339 / 250000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_20_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (220431339 / 250000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_20_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (220431339 / 250000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_20_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (220431339 / 250000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_20_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (220431339 / 250000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_20_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (220431339 / 250000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_20_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (220431339 / 250000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_20_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (220431339 / 250000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_20_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (220431339 / 250000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_20_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (220431339 / 250000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_20_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (220431339 / 250000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_20_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (220431339 / 250000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_20_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_21_1 :
+    Uω (aρ 1) (bρ 1) (4047462733 / 4000000000000 : ℝ) ≤
+      (-(329635258503986697477 / 62500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (102444728028721 / 20000000000000000 : ℝ) ≤
+      (-(329635258503986697477 / 62500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show
+          (102444728028721 / 20000000000000000 : ℝ) =
+            (102444728028721 / 78125000000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (102444728028721 / 78125000000000 : ℝ) ≤
+            (16938331646013302523 / 62500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (48066166776221 / 10000000000000000 : ℝ)) (q :=
+      (102444728028721 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_21_2 :
+    Uω (aρ 2) (bρ 2) (4047462733 / 4000000000000 : ℝ) ≤
+      (-(25525528995361187317657 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12131394246941 / 2000000000000000 : ℝ) ≤
+      (-(25525528995361187317657 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show
+          (12131394246941 / 2000000000000000 : ℝ) = (12131394246941 / 7812500000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12131394246941 / 7812500000000 : ℝ) ≤
+            (2200358216638812682343 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (4316636870691 / 1000000000000000 : ℝ)) (q :=
+      (12131394246941 / 2000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_21_3 :
+    Uω (aρ 3) (bρ 3) (4047462733 / 4000000000000 : ℝ) ≤
+      (-(48497352170934822805119 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (78304506299861 / 10000000000000000 : ℝ) ≤
+      (-(48497352170934822805119 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (78304506299861 / 10000000000000000 : ℝ) = (78304506299861 / 78125000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 2
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (78304506299861 / 78125000000000 : ℝ) ≤
+            (22950450065177194881 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (15532666243611 / 5000000000000000 : ℝ)) (q :=
+      (78304506299861 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_21_4 :
+    Uω (aρ 4) (bρ 4) (4047462733 / 4000000000000 : ℝ) ≤
+      (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_4]) (by norm_num [bρ_4])]; exact Lmid_4
+
+lemma U_21_5 :
+    Uω (aρ 5) (bρ 5) (4047462733 / 4000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_21_6 :
+    Uω (aρ 6) (bρ 6) (4047462733 / 4000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_21_7 :
+    Uω (aρ 7) (bρ 7) (4047462733 / 4000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_21_8 :
+    Uω (aρ 8) (bρ 8) (4047462733 / 4000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_21_9 :
+    Uω (aρ 9) (bρ 9) (4047462733 / 4000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_21_10 :
+    Uω (aρ 10) (bρ 10) (4047462733 / 4000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_21_11 :
+    Uω (aρ 11) (bρ 11) (4047462733 / 4000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_21_12 :
+    Uω (aρ 12) (bρ 12) (4047462733 / 4000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_21_13 :
+    Uω (aρ 13) (bρ 13) (4047462733 / 4000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_21_14 :
+    Uω (aρ 14) (bρ 14) (4047462733 / 4000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_21_15 :
+    Uω (aρ 15) (bρ 15) (4047462733 / 4000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_21_16 :
+    Uω (aρ 16) (bρ 16) (4047462733 / 4000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_21 :
+    Uρ (4047462733 / 4000000000000 : ℝ) ≤
+      (-(28244841495459020605873 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (4047462733 / 4000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(329635258503986697477 / 62500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_21_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (4047462733 / 4000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(25525528995361187317657 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_21_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (4047462733 / 4000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(48497352170934822805119 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_21_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (4047462733 / 4000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_21_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (4047462733 / 4000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_21_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (4047462733 / 4000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_21_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (4047462733 / 4000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_21_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (4047462733 / 4000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_21_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (4047462733 / 4000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_21_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (4047462733 / 4000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_21_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (4047462733 / 4000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_21_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (4047462733 / 4000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_21_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (4047462733 / 4000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_21_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (4047462733 / 4000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_21_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (4047462733 / 4000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_21_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (4047462733 / 4000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_21_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_22_1 :
+    Uω (aρ 1) (bρ 1) (2284012021 / 2000000000000 : ℝ) ≤
+      (-(2650831856230185575557 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (49832962239517 / 10000000000000000 : ℝ) ≤
+      (-(2650831856230185575557 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show
+          (49832962239517 / 10000000000000000 : ℝ) = (49832962239517 / 39062500000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (49832962239517 / 39062500000000 : ℝ) ≤
+            (121756864969814424443 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (23294383249517 / 5000000000000000 : ℝ)) (q :=
+      (49832962239517 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_22_2 :
+    Uω (aρ 2) (bρ 2) (2284012021 / 2000000000000 : ℝ) ≤
+      (-(51361201358950362094901 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (58804608871521 / 10000000000000000 : ℝ) ≤
+      (-(51361201358950362094901 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show
+          (58804608871521 / 10000000000000000 : ℝ) = (58804608871521 / 39062500000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (58804608871521 / 39062500000000 : ℝ) ≤
+            (4090573065049637905099 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (20381523626521 / 5000000000000000 : ℝ)) (q :=
+      (58804608871521 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_22_3 :
+    Uω (aρ 3) (bρ 3) (2284012021 / 2000000000000 : ℝ) ≤
+      (-(979184520167902038687 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (37385038657803 / 5000000000000000 : ℝ) ≤
+      (-(979184520167902038687 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show
+          (37385038657803 / 5000000000000000 : ℝ) = (37385038657803 / 19531250000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (37385038657803 / 19531250000000 : ℝ) ≤
+            (129850968312097961313 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (6324469447803 / 2500000000000000 : ℝ)) (q :=
+      (37385038657803 / 5000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_22_4 :
+    Uω (aρ 4) (bρ 4) (2284012021 / 2000000000000 : ℝ) ≤
+      (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_4]) (by norm_num [bρ_4])]; exact Lmid_4
+
+lemma U_22_5 :
+    Uω (aρ 5) (bρ 5) (2284012021 / 2000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_22_6 :
+    Uω (aρ 6) (bρ 6) (2284012021 / 2000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_22_7 :
+    Uω (aρ 7) (bρ 7) (2284012021 / 2000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_22_8 :
+    Uω (aρ 8) (bρ 8) (2284012021 / 2000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_22_9 :
+    Uω (aρ 9) (bρ 9) (2284012021 / 2000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_22_10 :
+    Uω (aρ 10) (bρ 10) (2284012021 / 2000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_22_11 :
+    Uω (aρ 11) (bρ 11) (2284012021 / 2000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_22_12 :
+    Uω (aρ 12) (bρ 12) (2284012021 / 2000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_22_13 :
+    Uω (aρ 13) (bρ 13) (2284012021 / 2000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_22_14 :
+    Uω (aρ 14) (bρ 14) (2284012021 / 2000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_22_15 :
+    Uω (aρ 15) (bρ 15) (2284012021 / 2000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_22_16 :
+    Uω (aρ 16) (bρ 16) (2284012021 / 2000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_22 :
+    Uρ (2284012021 / 2000000000000 : ℝ) ≤
+      (-(2827670396385794487423 / 1000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (2284012021 / 2000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2650831856230185575557 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_22_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (2284012021 / 2000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(51361201358950362094901 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_22_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (2284012021 / 2000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(979184520167902038687 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_22_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (2284012021 / 2000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_22_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (2284012021 / 2000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_22_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (2284012021 / 2000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_22_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (2284012021 / 2000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_22_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (2284012021 / 2000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_22_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (2284012021 / 2000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_22_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (2284012021 / 2000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_22_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (2284012021 / 2000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_22_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (2284012021 / 2000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_22_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (2284012021 / 2000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_22_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (2284012021 / 2000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_22_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (2284012021 / 2000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_22_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (2284012021 / 2000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_22_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_23_1 :
+    Uω (aρ 1) (bρ 1) (5088585351 / 4000000000000 : ℝ) ≤
+      (-(26650264330778109932447 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (48438139215859 / 10000000000000000 : ℝ) ≤
+      (-(26650264330778109932447 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show
+          (48438139215859 / 10000000000000000 : ℝ) = (48438139215859 / 39062500000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (48438139215859 / 39062500000000 : ℝ) ≤
+            (1075622881221890067553 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (22550261862109 / 5000000000000000 : ℝ)) (q :=
+      (48438139215859 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_23_2 :
+    Uω (aρ 2) (bρ 2) (5088585351 / 4000000000000 : ℝ) ≤
+      (-(206762531076714809157 / 40000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4551916935163 / 800000000000000 : ℝ) ≤
+      (-(206762531076714809157 / 40000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show (4551916935163 / 800000000000000 : ℝ) = (4551916935163 / 3125000000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4551916935163 / 3125000000000 : ℝ) ≤
+            (15044566619285190843 / 40000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (1530126246463 / 400000000000000 : ℝ)) (q :=
+      (4551916935163 / 800000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_23_3 :
+    Uω (aρ 3) (bρ 3) (5088585351 / 4000000000000 : ℝ) ≤
+      (-(49562765747407728412987 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (35195443021111 / 5000000000000000 : ℝ) ≤
+      (-(49562765747407728412987 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show
+          (35195443021111 / 5000000000000000 : ℝ) = (35195443021111 / 19531250000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (35195443021111 / 19531250000000 : ℝ) ≤
+            (5889008676592271587013 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (1115056157309 / 625000000000000 : ℝ)) (q :=
+      (35195443021111 / 5000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_23_4 :
+    Uω (aρ 4) (bρ 4) (5088585351 / 4000000000000 : ℝ) ≤
+      (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_4]) (by norm_num [bρ_4])]; exact Lmid_4
+
+lemma U_23_5 :
+    Uω (aρ 5) (bρ 5) (5088585351 / 4000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_23_6 :
+    Uω (aρ 6) (bρ 6) (5088585351 / 4000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_23_7 :
+    Uω (aρ 7) (bρ 7) (5088585351 / 4000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_23_8 :
+    Uω (aρ 8) (bρ 8) (5088585351 / 4000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_23_9 :
+    Uω (aρ 9) (bρ 9) (5088585351 / 4000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_23_10 :
+    Uω (aρ 10) (bρ 10) (5088585351 / 4000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_23_11 :
+    Uω (aρ 11) (bρ 11) (5088585351 / 4000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_23_12 :
+    Uω (aρ 12) (bρ 12) (5088585351 / 4000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_23_13 :
+    Uω (aρ 13) (bρ 13) (5088585351 / 4000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_23_14 :
+    Uω (aρ 14) (bρ 14) (5088585351 / 4000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_23_15 :
+    Uω (aρ 15) (bρ 15) (5088585351 / 4000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_23_16 :
+    Uω (aρ 16) (bρ 16) (5088585351 / 4000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_23 :
+    Uρ (5088585351 / 4000000000000 : ℝ) ≤
+      (-(14157655382134504069693 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (5088585351 / 4000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(26650264330778109932447 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_23_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (5088585351 / 4000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) * (-(206762531076714809157 / 40000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_23_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (5088585351 / 4000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(49562765747407728412987 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_23_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (5088585351 / 4000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_23_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (5088585351 / 4000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_23_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (5088585351 / 4000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_23_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (5088585351 / 4000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_23_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (5088585351 / 4000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_23_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (5088585351 / 4000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_23_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (5088585351 / 4000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_23_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (5088585351 / 4000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_23_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (5088585351 / 4000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_23_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (5088585351 / 4000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_23_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (5088585351 / 4000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_23_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (5088585351 / 4000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_23_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (5088585351 / 4000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_23_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/U02.lean b/LeanPool/Zeta5Irrational/Table/U02.lean
new file mode 100644
index 0000000000..2175276fa2
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/U02.lean
@@ -0,0 +1,2259 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-! # Certified arcsine potential bounds (U02) -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma U_24_1 :
+    Uω (aρ 1) (bρ 1) (280457333 / 200000000000 : ℝ) ≤
+      (-(53593986196108414589993 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23518669903887 / 5000000000000000 : ℝ) ≤
+      (-(53593986196108414589993 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show
+          (23518669903887 / 5000000000000000 : ℝ) = (23518669903887 / 19531250000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23518669903887 / 19531250000000 : ℝ) ≤
+            (1857788227891585410007 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (10900082045137 / 2500000000000000 : ℝ)) (q :=
+      (23518669903887 / 5000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_24_2 :
+    Uω (aρ 2) (bρ 2) (280457333 / 200000000000 : ℝ) ≤
+      (-(52043031588685060884247 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (109857535172831 / 20000000000000000 : ℝ) ≤
+      (-(52043031588685060884247 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show
+          (109857535172831 / 20000000000000000 : ℝ) =
+            (109857535172831 / 78125000000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (109857535172831 / 78125000000000 : ℝ) ≤
+            (3408742835314939115753 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (35614171227831 / 10000000000000000 : ℝ)) (q :=
+      (109857535172831 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_24_3 :
+    Uω (aρ 3) (bρ 3) (280457333 / 200000000000 : ℝ) ≤
+      (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_3]) (by norm_num [bρ_3])]; exact Lmid_3
+
+lemma U_24_4 :
+    Uω (aρ 4) (bρ 4) (280457333 / 200000000000 : ℝ) ≤
+      (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_4]) (by norm_num [bρ_4])]; exact Lmid_4
+
+lemma U_24_5 :
+    Uω (aρ 5) (bρ 5) (280457333 / 200000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_24_6 :
+    Uω (aρ 6) (bρ 6) (280457333 / 200000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_24_7 :
+    Uω (aρ 7) (bρ 7) (280457333 / 200000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_24_8 :
+    Uω (aρ 8) (bρ 8) (280457333 / 200000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_24_9 :
+    Uω (aρ 9) (bρ 9) (280457333 / 200000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_24_10 :
+    Uω (aρ 10) (bρ 10) (280457333 / 200000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_24_11 :
+    Uω (aρ 11) (bρ 11) (280457333 / 200000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_24_12 :
+    Uω (aρ 12) (bρ 12) (280457333 / 200000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_24_13 :
+    Uω (aρ 13) (bρ 13) (280457333 / 200000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_24_14 :
+    Uω (aρ 14) (bρ 14) (280457333 / 200000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_24_15 :
+    Uω (aρ 15) (bρ 15) (280457333 / 200000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_24_16 :
+    Uω (aρ 16) (bρ 16) (280457333 / 200000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_24 :
+    Uρ (280457333 / 200000000000 : ℝ) ≤
+      (-(28391530796819438246561 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (280457333 / 200000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(53593986196108414589993 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_24_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (280457333 / 200000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(52043031588685060884247 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_24_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (280457333 / 200000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_24_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (280457333 / 200000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_24_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (280457333 / 200000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_24_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (280457333 / 200000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_24_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (280457333 / 200000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_24_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (280457333 / 200000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_24_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (280457333 / 200000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_24_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (280457333 / 200000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_24_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (280457333 / 200000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_24_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (280457333 / 200000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_24_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (280457333 / 200000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_24_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (280457333 / 200000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_24_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (280457333 / 200000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_24_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (280457333 / 200000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_24_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_25_1 :
+    Uω (aρ 1) (bρ 1) (6519108259 / 4000000000000 : ℝ) ≤
+      (-(27066132270492516043287 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8914471820163 / 2000000000000000 : ℝ) ≤
+      (-(27066132270492516043287 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show (8914471820163 / 2000000000000000 : ℝ) = (8914471820163 / 7812500000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8914471820163 / 7812500000000 : ℝ) ≤
+            (659754941507483956713 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (4094527076413 / 1000000000000000 : ℝ)) (q :=
+      (8914471820163 / 2000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_25_2 :
+    Uω (aρ 2) (bρ 2) (6519108259 / 4000000000000 : ℝ) ≤
+      (-(52730540587954460599153 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (102558512735791 / 20000000000000000 : ℝ) ≤
+      (-(52730540587954460599153 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show
+          (102558512735791 / 20000000000000000 : ℝ) =
+            (102558512735791 / 78125000000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (102558512735791 / 78125000000000 : ℝ) ≤
+            (2721233836045539400847 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (30590052788291 / 10000000000000000 : ℝ)) (q :=
+      (102558512735791 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_25_3 :
+    Uω (aρ 3) (bρ 3) (6519108259 / 4000000000000 : ℝ) ≤
+      (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_3]) (by norm_num [bρ_3])]; exact Lmid_3
+
+lemma U_25_4 :
+    Uω (aρ 4) (bρ 4) (6519108259 / 4000000000000 : ℝ) ≤
+      (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_4]) (by norm_num [bρ_4])]; exact Lmid_4
+
+lemma U_25_5 :
+    Uω (aρ 5) (bρ 5) (6519108259 / 4000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_25_6 :
+    Uω (aρ 6) (bρ 6) (6519108259 / 4000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_25_7 :
+    Uω (aρ 7) (bρ 7) (6519108259 / 4000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_25_8 :
+    Uω (aρ 8) (bρ 8) (6519108259 / 4000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_25_9 :
+    Uω (aρ 9) (bρ 9) (6519108259 / 4000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_25_10 :
+    Uω (aρ 10) (bρ 10) (6519108259 / 4000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_25_11 :
+    Uω (aρ 11) (bρ 11) (6519108259 / 4000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_25_12 :
+    Uω (aρ 12) (bρ 12) (6519108259 / 4000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_25_13 :
+    Uω (aρ 13) (bρ 13) (6519108259 / 4000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_25_14 :
+    Uω (aρ 14) (bρ 14) (6519108259 / 4000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_25_15 :
+    Uω (aρ 15) (bρ 15) (6519108259 / 4000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_25_16 :
+    Uω (aρ 16) (bρ 16) (6519108259 / 4000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_25 :
+    Uρ (6519108259 / 4000000000000 : ℝ) ≤
+      (-(14208726599741697552523 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (6519108259 / 4000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(27066132270492516043287 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_25_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (6519108259 / 4000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(52730540587954460599153 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_25_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (6519108259 / 4000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_25_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (6519108259 / 4000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_25_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (6519108259 / 4000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_25_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (6519108259 / 4000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_25_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (6519108259 / 4000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_25_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (6519108259 / 4000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_25_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (6519108259 / 4000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_25_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (6519108259 / 4000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_25_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (6519108259 / 4000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_25_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (6519108259 / 4000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_25_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (6519108259 / 4000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_25_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (6519108259 / 4000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_25_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (6519108259 / 4000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_25_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (6519108259 / 4000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_25_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_26_1 :
+    Uω (aρ 1) (bρ 1) (3714534929 / 2000000000000 : ℝ) ≤
+      (-(13676747820096012817053 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4208288987711 / 1000000000000000 : ℝ) ≤
+      (-(13676747820096012817053 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show (4208288987711 / 1000000000000000 : ℝ) = (4208288987711 / 3906250000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4208288987711 / 3906250000000 : ℝ) ≤
+            (186195785903987182947 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (1912061815711 / 500000000000000 : ℝ)) (q :=
+      (4208288987711 / 1000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_26_2 :
+    Uω (aρ 2) (bρ 2) (3714534929 / 2000000000000 : ℝ) ≤
+      (-(26776440993766361792693 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (94462161952989 / 20000000000000000 : ℝ) ≤
+      (-(26776440993766361792693 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show
+          (94462161952989 / 20000000000000000 : ℝ) = (94462161952989 / 78125000000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (94462161952989 / 78125000000000 : ℝ) ≤
+            (949446218233638207307 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (24768606002989 / 10000000000000000 : ℝ)) (q :=
+      (94462161952989 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_26_3 :
+    Uω (aρ 3) (bρ 3) (3714534929 / 2000000000000 : ℝ) ≤
+      (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_3]) (by norm_num [bρ_3])]; exact Lmid_3
+
+lemma U_26_4 :
+    Uω (aρ 4) (bρ 4) (3714534929 / 2000000000000 : ℝ) ≤
+      (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_4]) (by norm_num [bρ_4])]; exact Lmid_4
+
+lemma U_26_5 :
+    Uω (aρ 5) (bρ 5) (3714534929 / 2000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_26_6 :
+    Uω (aρ 6) (bρ 6) (3714534929 / 2000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_26_7 :
+    Uω (aρ 7) (bρ 7) (3714534929 / 2000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_26_8 :
+    Uω (aρ 8) (bρ 8) (3714534929 / 2000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_26_9 :
+    Uω (aρ 9) (bρ 9) (3714534929 / 2000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_26_10 :
+    Uω (aρ 10) (bρ 10) (3714534929 / 2000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_26_11 :
+    Uω (aρ 11) (bρ 11) (3714534929 / 2000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_26_12 :
+    Uω (aρ 12) (bρ 12) (3714534929 / 2000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_26_13 :
+    Uω (aρ 13) (bρ 13) (3714534929 / 2000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_26_14 :
+    Uω (aρ 14) (bρ 14) (3714534929 / 2000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_26_15 :
+    Uω (aρ 15) (bρ 15) (3714534929 / 2000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_26_16 :
+    Uω (aρ 16) (bρ 16) (3714534929 / 2000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_26 :
+    Uρ (3714534929 / 2000000000000 : ℝ) ≤
+      (-(28447732623893462494161 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (3714534929 / 2000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(13676747820096012817053 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_26_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (3714534929 / 2000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(26776440993766361792693 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_26_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (3714534929 / 2000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_26_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (3714534929 / 2000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_26_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (3714534929 / 2000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_26_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (3714534929 / 2000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_26_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (3714534929 / 2000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_26_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (3714534929 / 2000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_26_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (3714534929 / 2000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_26_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (3714534929 / 2000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_26_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (3714534929 / 2000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_26_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (3714534929 / 2000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_26_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (3714534929 / 2000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_26_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (3714534929 / 2000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_26_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (3714534929 / 2000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_26_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (3714534929 / 2000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_26_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_27_1 :
+    Uω (aρ 1) (bρ 1) (8339031457 / 4000000000000 : ℝ) ≤
+      (-(55324376016046487597617 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2472708345429 / 625000000000000 : ℝ) ≤
+      (-(55324376016046487597617 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show (2472708345429 / 625000000000000 : ℝ) = (2472708345429 / 2441406250000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2472708345429 / 2441406250000 : ℝ) ≤
+            (127398407953512402383 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (8869256902807 / 2500000000000000 : ℝ)) (q :=
+      (2472708345429 / 625000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_27_2 :
+    Uω (aρ 2) (bρ 2) (8339031457 / 4000000000000 : ℝ) ≤
+      (-(10926753652502540317131 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (84784328831557 / 20000000000000000 : ℝ) ≤
+      (-(10926753652502540317131 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show
+          (84784328831557 / 20000000000000000 : ℝ) = (84784328831557 / 78125000000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (84784328831557 / 78125000000000 : ℝ) ≤
+            (163601232297459682869 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (17365676879057 / 10000000000000000 : ℝ)) (q :=
+      (84784328831557 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_27_3 :
+    Uω (aρ 3) (bρ 3) (8339031457 / 4000000000000 : ℝ) ≤
+      (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_3]) (by norm_num [bρ_3])]; exact Lmid_3
+
+lemma U_27_4 :
+    Uω (aρ 4) (bρ 4) (8339031457 / 4000000000000 : ℝ) ≤
+      (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_4]) (by norm_num [bρ_4])]; exact Lmid_4
+
+lemma U_27_5 :
+    Uω (aρ 5) (bρ 5) (8339031457 / 4000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_27_6 :
+    Uω (aρ 6) (bρ 6) (8339031457 / 4000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_27_7 :
+    Uω (aρ 7) (bρ 7) (8339031457 / 4000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_27_8 :
+    Uω (aρ 8) (bρ 8) (8339031457 / 4000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_27_9 :
+    Uω (aρ 9) (bρ 9) (8339031457 / 4000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_27_10 :
+    Uω (aρ 10) (bρ 10) (8339031457 / 4000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_27_11 :
+    Uω (aρ 11) (bρ 11) (8339031457 / 4000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_27_12 :
+    Uω (aρ 12) (bρ 12) (8339031457 / 4000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_27_13 :
+    Uω (aρ 13) (bρ 13) (8339031457 / 4000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_27_14 :
+    Uω (aρ 14) (bρ 14) (8339031457 / 4000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_27_15 :
+    Uω (aρ 15) (bρ 15) (8339031457 / 4000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_27_16 :
+    Uω (aρ 16) (bρ 16) (8339031457 / 4000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_27 :
+    Uρ (8339031457 / 4000000000000 : ℝ) ≤
+      (-(5697216077868329425469 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (8339031457 / 4000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(55324376016046487597617 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_27_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (8339031457 / 4000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(10926753652502540317131 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_27_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (8339031457 / 4000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_27_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (8339031457 / 4000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_27_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (8339031457 / 4000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_27_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (8339031457 / 4000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_27_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (8339031457 / 4000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_27_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (8339031457 / 4000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_27_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (8339031457 / 4000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_27_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (8339031457 / 4000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_27_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (8339031457 / 4000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_27_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (8339031457 / 4000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_27_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (8339031457 / 4000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_27_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (8339031457 / 4000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_27_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (8339031457 / 4000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_27_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (8339031457 / 4000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_27_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_28_1 :
+    Uω (aρ 1) (bρ 1) (289031033 / 125000000000 : ℝ) ≤
+      (-(27996295164352877284557 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (578219487291 / 156250000000000 : ℝ) ≤
+      (-(27996295164352877284557 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 9
+        (show (578219487291 / 156250000000000 : ℝ) = (578219487291 / 305175781250 : ℝ) / 2 ^ 9 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (578219487291 / 305175781250 : ℝ) ≤
+            (3195327949147122715443 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (4079669866031 / 1250000000000000 : ℝ)) (q :=
+      (578219487291 / 156250000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_28_2 :
+    Uω (aρ 2) (bρ 2) (289031033 / 125000000000 : ℝ) ≤
+      (-(57268912150832389796717 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_2]) (by norm_num [bρ_2])]; exact Lmid_2
+
+lemma U_28_3 :
+    Uω (aρ 3) (bρ 3) (289031033 / 125000000000 : ℝ) ≤
+      (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_3]) (by norm_num [bρ_3])]; exact Lmid_3
+
+lemma U_28_4 :
+    Uω (aρ 4) (bρ 4) (289031033 / 125000000000 : ℝ) ≤
+      (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_4]) (by norm_num [bρ_4])]; exact Lmid_4
+
+lemma U_28_5 :
+    Uω (aρ 5) (bρ 5) (289031033 / 125000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_28_6 :
+    Uω (aρ 6) (bρ 6) (289031033 / 125000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_28_7 :
+    Uω (aρ 7) (bρ 7) (289031033 / 125000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_28_8 :
+    Uω (aρ 8) (bρ 8) (289031033 / 125000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_28_9 :
+    Uω (aρ 9) (bρ 9) (289031033 / 125000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_28_10 :
+    Uω (aρ 10) (bρ 10) (289031033 / 125000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_28_11 :
+    Uω (aρ 11) (bρ 11) (289031033 / 125000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_28_12 :
+    Uω (aρ 12) (bρ 12) (289031033 / 125000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_28_13 :
+    Uω (aρ 13) (bρ 13) (289031033 / 125000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_28_14 :
+    Uω (aρ 14) (bρ 14) (289031033 / 125000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_28_15 :
+    Uω (aρ 15) (bρ 15) (289031033 / 125000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_28_16 :
+    Uω (aρ 16) (bρ 16) (289031033 / 125000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_28 :
+    Uρ (289031033 / 125000000000 : ℝ) ≤
+      (-(7142692332734663141767 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (289031033 / 125000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(27996295164352877284557 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_28_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (289031033 / 125000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(57268912150832389796717 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_28_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (289031033 / 125000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_28_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (289031033 / 125000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_28_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (289031033 / 125000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_28_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (289031033 / 125000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_28_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (289031033 / 125000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_28_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (289031033 / 125000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_28_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (289031033 / 125000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_28_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (289031033 / 125000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_28_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (289031033 / 125000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_28_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (289031033 / 125000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_28_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (289031033 / 125000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_28_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (289031033 / 125000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_28_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (289031033 / 125000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_28_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (289031033 / 125000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_28_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_29_1 :
+    Uω (aρ 1) (bρ 1) (124379927 / 40000000000 : ℝ) ≤
+      (-(14737682003414632220873 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13764879544613 / 5000000000000000 : ℝ) ≤
+      (-(14737682003414632220873 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 9
+        (show
+          (13764879544613 / 5000000000000000 : ℝ) = (13764879544613 / 9765625000000 : ℝ) / 2 ^ 9
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13764879544613 / 9765625000000 : ℝ) ≤
+            (858129553335367779127 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (5414320460863 / 2500000000000000 : ℝ)) (q :=
+      (13764879544613 / 5000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_29_2 :
+    Uω (aρ 2) (bρ 2) (124379927 / 40000000000 : ℝ) ≤
+      (-(57268912150832389796717 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_2]) (by norm_num [bρ_2])]; exact Lmid_2
+
+lemma U_29_3 :
+    Uω (aρ 3) (bρ 3) (124379927 / 40000000000 : ℝ) ≤
+      (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_3]) (by norm_num [bρ_3])]; exact Lmid_3
+
+lemma U_29_4 :
+    Uω (aρ 4) (bρ 4) (124379927 / 40000000000 : ℝ) ≤
+      (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_4]) (by norm_num [bρ_4])]; exact Lmid_4
+
+lemma U_29_5 :
+    Uω (aρ 5) (bρ 5) (124379927 / 40000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_29_6 :
+    Uω (aρ 6) (bρ 6) (124379927 / 40000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_29_7 :
+    Uω (aρ 7) (bρ 7) (124379927 / 40000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_29_8 :
+    Uω (aρ 8) (bρ 8) (124379927 / 40000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_29_9 :
+    Uω (aρ 9) (bρ 9) (124379927 / 40000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_29_10 :
+    Uω (aρ 10) (bρ 10) (124379927 / 40000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_29_11 :
+    Uω (aρ 11) (bρ 11) (124379927 / 40000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_29_12 :
+    Uω (aρ 12) (bρ 12) (124379927 / 40000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_29_13 :
+    Uω (aρ 13) (bρ 13) (124379927 / 40000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_29_14 :
+    Uω (aρ 14) (bρ 14) (124379927 / 40000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_29_15 :
+    Uω (aρ 15) (bρ 15) (124379927 / 40000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_29_16 :
+    Uω (aρ 16) (bρ 16) (124379927 / 40000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_29 :
+    Uρ (124379927 / 40000000000 : ℝ) ≤ (-(893808622258701750129 / 312500000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (124379927 / 40000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(14737682003414632220873 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_29_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (124379927 / 40000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(57268912150832389796717 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_29_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (124379927 / 40000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_29_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (124379927 / 40000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_29_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (124379927 / 40000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_29_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (124379927 / 40000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_29_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (124379927 / 40000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_29_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (124379927 / 40000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_29_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (124379927 / 40000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_29_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (124379927 / 40000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_29_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (124379927 / 40000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_29_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (124379927 / 40000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_29_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (124379927 / 40000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_29_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (124379927 / 40000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_29_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (124379927 / 40000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_29_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (124379927 / 40000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_29_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_30_1 :
+    Uω (aρ 1) (bρ 1) (7016246261 / 2000000000000 : ℝ) ≤
+      (-(1910848611840935650757 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22101043215827 / 10000000000000000 : ℝ) ≤
+      (-(1910848611840935650757 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 9
+        (show
+          (22101043215827 / 10000000000000000 : ℝ) = (22101043215827 / 19531250000000 : ℝ) / 2 ^ 9
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22101043215827 / 19531250000000 : ℝ) ≤
+            (38627832752814349243 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_left_certificate (s := (7393049825827 / 5000000000000000 : ℝ)) (q :=
+      (22101043215827 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_30_2 :
+    Uω (aρ 2) (bρ 2) (7016246261 / 2000000000000 : ℝ) ≤
+      (-(57268912150832389796717 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_2]) (by norm_num [bρ_2])]; exact Lmid_2
+
+lemma U_30_3 :
+    Uω (aρ 3) (bρ 3) (7016246261 / 2000000000000 : ℝ) ≤
+      (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_3]) (by norm_num [bρ_3])]; exact Lmid_3
+
+lemma U_30_4 :
+    Uω (aρ 4) (bρ 4) (7016246261 / 2000000000000 : ℝ) ≤
+      (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_4]) (by norm_num [bρ_4])]; exact Lmid_4
+
+lemma U_30_5 :
+    Uω (aρ 5) (bρ 5) (7016246261 / 2000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_30_6 :
+    Uω (aρ 6) (bρ 6) (7016246261 / 2000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_30_7 :
+    Uω (aρ 7) (bρ 7) (7016246261 / 2000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_30_8 :
+    Uω (aρ 8) (bρ 8) (7016246261 / 2000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_30_9 :
+    Uω (aρ 9) (bρ 9) (7016246261 / 2000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_30_10 :
+    Uω (aρ 10) (bρ 10) (7016246261 / 2000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_30_11 :
+    Uω (aρ 11) (bρ 11) (7016246261 / 2000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_30_12 :
+    Uω (aρ 12) (bρ 12) (7016246261 / 2000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_30_13 :
+    Uω (aρ 13) (bρ 13) (7016246261 / 2000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_30_14 :
+    Uω (aρ 14) (bρ 14) (7016246261 / 2000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_30_15 :
+    Uω (aρ 15) (bρ 15) (7016246261 / 2000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_30_16 :
+    Uω (aρ 16) (bρ 16) (7016246261 / 2000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_30 :
+    Uρ (7016246261 / 2000000000000 : ℝ) ≤
+      (-(1789060791099578778267 / 625000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (7016246261 / 2000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1910848611840935650757 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_30_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (7016246261 / 2000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(57268912150832389796717 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_30_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (7016246261 / 2000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_30_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (7016246261 / 2000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_30_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (7016246261 / 2000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_30_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (7016246261 / 2000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_30_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (7016246261 / 2000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_30_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (7016246261 / 2000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_30_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (7016246261 / 2000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_30_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (7016246261 / 2000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_30_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (7016246261 / 2000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_30_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (7016246261 / 2000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_30_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (7016246261 / 2000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_30_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (7016246261 / 2000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_30_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (7016246261 / 2000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_30_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (7016246261 / 2000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_30_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_35_1 :
+    Uω (aρ 1) (bρ 1) (19572466643 / 2000000000000 : ℝ) ≤
+      (-(5896789100911232295581 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27482550300017 / 10000000000000000 : ℝ) ≤
+      (-(5896789100911232295581 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 9
+        (show
+          (27482550300017 / 10000000000000000 : ℝ) = (27482550300017 / 19531250000000 : ℝ) / 2 ^ 9
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27482550300017 / 19531250000000 : ℝ) ≤
+            (341535521788767704419 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (10799992735017 / 5000000000000000 : ℝ)) (q :=
+      (27482550300017 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_35_2 :
+    Uω (aρ 2) (bρ 2) (19572466643 / 2000000000000 : ℝ) ≤
+      (-(57268912150832389796717 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_2]) (by norm_num [bρ_2])]; exact Lmid_2
+
+lemma U_35_3 :
+    Uω (aρ 3) (bρ 3) (19572466643 / 2000000000000 : ℝ) ≤
+      (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_3]) (by norm_num [bρ_3])]; exact Lmid_3
+
+lemma U_35_4 :
+    Uω (aρ 4) (bρ 4) (19572466643 / 2000000000000 : ℝ) ≤
+      (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_4]) (by norm_num [bρ_4])]; exact Lmid_4
+
+lemma U_35_5 :
+    Uω (aρ 5) (bρ 5) (19572466643 / 2000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_35_6 :
+    Uω (aρ 6) (bρ 6) (19572466643 / 2000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_35_7 :
+    Uω (aρ 7) (bρ 7) (19572466643 / 2000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_35_8 :
+    Uω (aρ 8) (bρ 8) (19572466643 / 2000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_35_9 :
+    Uω (aρ 9) (bρ 9) (19572466643 / 2000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_35_10 :
+    Uω (aρ 10) (bρ 10) (19572466643 / 2000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_35_11 :
+    Uω (aρ 11) (bρ 11) (19572466643 / 2000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_35_12 :
+    Uω (aρ 12) (bρ 12) (19572466643 / 2000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_35_13 :
+    Uω (aρ 13) (bρ 13) (19572466643 / 2000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_35_14 :
+    Uω (aρ 14) (bρ 14) (19572466643 / 2000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_35_15 :
+    Uω (aρ 15) (bρ 15) (19572466643 / 2000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_35_16 :
+    Uω (aρ 16) (bρ 16) (19572466643 / 2000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_35 :
+    Uρ (19572466643 / 2000000000000 : ℝ) ≤
+      (-(14301028195703943639221 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (19572466643 / 2000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(5896789100911232295581 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_35_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (19572466643 / 2000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(57268912150832389796717 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_35_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (19572466643 / 2000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_35_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (19572466643 / 2000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_35_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (19572466643 / 2000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_35_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (19572466643 / 2000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_35_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (19572466643 / 2000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_35_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (19572466643 / 2000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_35_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (19572466643 / 2000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_35_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (19572466643 / 2000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_35_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (19572466643 / 2000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_35_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (19572466643 / 2000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_35_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (19572466643 / 2000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_35_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (19572466643 / 2000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_35_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (19572466643 / 2000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_35_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (19572466643 / 2000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_35_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_36_1 :
+    Uω (aρ 1) (bρ 1) (40732008867 / 4000000000000 : ℝ) ≤
+      (-(28671312100952179027439 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (64665325468751 / 20000000000000000 : ℝ) ≤
+      (-(28671312100952179027439 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 9
+        (show
+          (64665325468751 / 20000000000000000 : ℝ) = (64665325468751 / 39062500000000 : ℝ) / 2 ^ 9
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (64665325468751 / 39062500000000 : ℝ) ≤
+            (2520311012547820972561 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (27332521386251 / 10000000000000000 : ℝ)) (q :=
+      (64665325468751 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_36_2 :
+    Uω (aρ 2) (bρ 2) (40732008867 / 4000000000000 : ℝ) ≤
+      (-(57268912150832389796717 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_2]) (by norm_num [bρ_2])]; exact Lmid_2
+
+lemma U_36_3 :
+    Uω (aρ 3) (bρ 3) (40732008867 / 4000000000000 : ℝ) ≤
+      (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_3]) (by norm_num [bρ_3])]; exact Lmid_3
+
+lemma U_36_4 :
+    Uω (aρ 4) (bρ 4) (40732008867 / 4000000000000 : ℝ) ≤
+      (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_4]) (by norm_num [bρ_4])]; exact Lmid_4
+
+lemma U_36_5 :
+    Uω (aρ 5) (bρ 5) (40732008867 / 4000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_36_6 :
+    Uω (aρ 6) (bρ 6) (40732008867 / 4000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_36_7 :
+    Uω (aρ 7) (bρ 7) (40732008867 / 4000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_36_8 :
+    Uω (aρ 8) (bρ 8) (40732008867 / 4000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_36_9 :
+    Uω (aρ 9) (bρ 9) (40732008867 / 4000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_36_10 :
+    Uω (aρ 10) (bρ 10) (40732008867 / 4000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_36_11 :
+    Uω (aρ 11) (bρ 11) (40732008867 / 4000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_36_12 :
+    Uω (aρ 12) (bρ 12) (40732008867 / 4000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_36_13 :
+    Uω (aρ 13) (bρ 13) (40732008867 / 4000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_36_14 :
+    Uω (aρ 14) (bρ 14) (40732008867 / 4000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_36_15 :
+    Uω (aρ 15) (bρ 15) (40732008867 / 4000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_36_16 :
+    Uω (aρ 16) (bρ 16) (40732008867 / 4000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_36 :
+    Uρ (40732008867 / 4000000000000 : ℝ) ≤
+      (-(3573120717747316300783 / 1250000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (40732008867 / 4000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(28671312100952179027439 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_36_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (40732008867 / 4000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(57268912150832389796717 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_36_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (40732008867 / 4000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_36_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (40732008867 / 4000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_36_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (40732008867 / 4000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_36_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (40732008867 / 4000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_36_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (40732008867 / 4000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_36_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (40732008867 / 4000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_36_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (40732008867 / 4000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_36_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (40732008867 / 4000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_36_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (40732008867 / 4000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_36_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (40732008867 / 4000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_36_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (40732008867 / 4000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_36_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (40732008867 / 4000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_36_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (40732008867 / 4000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_36_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (40732008867 / 4000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_36_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_37_1 :
+    Uω (aρ 1) (bρ 1) (1322471389 / 125000000000 : ℝ) ≤
+      (-(437620085001547802181 / 78125000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (18460920659749 / 5000000000000000 : ℝ) ≤
+      (-(437620085001547802181 / 78125000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 9
+        (show
+          (18460920659749 / 5000000000000000 : ℝ) = (18460920659749 / 9765625000000 : ℝ) / 2 ^ 9
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18460920659749 / 9765625000000 : ℝ) ≤
+            (49749026146889697819 / 78125000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (8135797400999 / 2500000000000000 : ℝ)) (q :=
+      (18460920659749 / 5000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_37_2 :
+    Uω (aρ 2) (bρ 2) (1322471389 / 125000000000 : ℝ) ≤
+      (-(57268912150832389796717 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_2]) (by norm_num [bρ_2])]; exact Lmid_2
+
+lemma U_37_3 :
+    Uω (aρ 3) (bρ 3) (1322471389 / 125000000000 : ℝ) ≤
+      (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_3]) (by norm_num [bρ_3])]; exact Lmid_3
+
+lemma U_37_4 :
+    Uω (aρ 4) (bρ 4) (1322471389 / 125000000000 : ℝ) ≤
+      (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_4]) (by norm_num [bρ_4])]; exact Lmid_4
+
+lemma U_37_5 :
+    Uω (aρ 5) (bρ 5) (1322471389 / 125000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_37_6 :
+    Uω (aρ 6) (bρ 6) (1322471389 / 125000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_37_7 :
+    Uω (aρ 7) (bρ 7) (1322471389 / 125000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_37_8 :
+    Uω (aρ 8) (bρ 8) (1322471389 / 125000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_37_9 :
+    Uω (aρ 9) (bρ 9) (1322471389 / 125000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_37_10 :
+    Uω (aρ 10) (bρ 10) (1322471389 / 125000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_37_11 :
+    Uω (aρ 11) (bρ 11) (1322471389 / 125000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_37_12 :
+    Uω (aρ 12) (bρ 12) (1322471389 / 125000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_37_13 :
+    Uω (aρ 13) (bρ 13) (1322471389 / 125000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_37_14 :
+    Uω (aρ 14) (bρ 14) (1322471389 / 125000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_37_15 :
+    Uω (aρ 15) (bρ 15) (1322471389 / 125000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_37_16 :
+    Uω (aρ 16) (bρ 16) (1322471389 / 125000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_37 :
+    Uρ (1322471389 / 125000000000 : ℝ) ≤
+      (-(28571008882018903964403 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1322471389 / 125000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(437620085001547802181 / 78125000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_37_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1322471389 / 125000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(57268912150832389796717 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_37_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1322471389 / 125000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_37_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1322471389 / 125000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_37_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1322471389 / 125000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_37_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1322471389 / 125000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_37_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1322471389 / 125000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_37_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1322471389 / 125000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_37_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1322471389 / 125000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_37_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1322471389 / 125000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_37_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1322471389 / 125000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_37_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1322471389 / 125000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_37_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1322471389 / 125000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_37_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1322471389 / 125000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_37_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1322471389 / 125000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_37_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1322471389 / 125000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_37_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_38_1 :
+    Uω (aρ 1) (bρ 1) (4549323561 / 400000000000 : ℝ) ≤
+      (-(53882828561175423362757 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (91396272407347 / 20000000000000000 : ℝ) ≤
+      (-(53882828561175423362757 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show
+          (91396272407347 / 20000000000000000 : ℝ) = (91396272407347 / 78125000000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (91396272407347 / 78125000000000 : ℝ) ≤
+            (1568945862824576637243 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (42160401467347 / 10000000000000000 : ℝ)) (q :=
+      (91396272407347 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_38_2 :
+    Uω (aρ 2) (bρ 2) (4549323561 / 400000000000 : ℝ) ≤
+      (-(57268912150832389796717 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_2]) (by norm_num [bρ_2])]; exact Lmid_2
+
+lemma U_38_3 :
+    Uω (aρ 3) (bρ 3) (4549323561 / 400000000000 : ℝ) ≤
+      (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_3]) (by norm_num [bρ_3])]; exact Lmid_3
+
+lemma U_38_4 :
+    Uω (aρ 4) (bρ 4) (4549323561 / 400000000000 : ℝ) ≤
+      (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_4]) (by norm_num [bρ_4])]; exact Lmid_4
+
+lemma U_38_5 :
+    Uω (aρ 5) (bρ 5) (4549323561 / 400000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_38_6 :
+    Uω (aρ 6) (bρ 6) (4549323561 / 400000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_38_7 :
+    Uω (aρ 7) (bρ 7) (4549323561 / 400000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_38_8 :
+    Uω (aρ 8) (bρ 8) (4549323561 / 400000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_38_9 :
+    Uω (aρ 9) (bρ 9) (4549323561 / 400000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_38_10 :
+    Uω (aρ 10) (bρ 10) (4549323561 / 400000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_38_11 :
+    Uω (aρ 11) (bρ 11) (4549323561 / 400000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_38_12 :
+    Uω (aρ 12) (bρ 12) (4549323561 / 400000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_38_13 :
+    Uω (aρ 13) (bρ 13) (4549323561 / 400000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_38_14 :
+    Uω (aρ 14) (bρ 14) (4549323561 / 400000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_38_15 :
+    Uω (aρ 15) (bρ 15) (4549323561 / 400000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_38_16 :
+    Uω (aρ 16) (bρ 16) (4549323561 / 400000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_38 :
+    Uρ (4549323561 / 400000000000 : ℝ) ≤ (-(142742919640776502841 / 50000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (4549323561 / 400000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(53882828561175423362757 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_38_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (4549323561 / 400000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(57268912150832389796717 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_38_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (4549323561 / 400000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_38_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (4549323561 / 400000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_38_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (4549323561 / 400000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_38_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (4549323561 / 400000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_38_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (4549323561 / 400000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_38_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (4549323561 / 400000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_38_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (4549323561 / 400000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_38_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (4549323561 / 400000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_38_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (4549323561 / 400000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_38_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (4549323561 / 400000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_38_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (4549323561 / 400000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_38_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (4549323561 / 400000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_38_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (4549323561 / 400000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_38_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (4549323561 / 400000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_38_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_39_1 :
+    Uω (aρ 1) (bρ 1) (12166846693 / 1000000000000 : ℝ) ≤
+      (-(52178850672124585216629 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (54187773546961 / 10000000000000000 : ℝ) ≤
+      (-(52178850672124585216629 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show
+          (54187773546961 / 10000000000000000 : ℝ) = (54187773546961 / 39062500000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (54187773546961 / 39062500000000 : ℝ) ≤
+            (3272923751875414783371 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (25602149124461 / 5000000000000000 : ℝ)) (q :=
+      (54187773546961 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_39_2 :
+    Uω (aρ 2) (bρ 2) (12166846693 / 1000000000000 : ℝ) ≤
+      (-(57268912150832389796717 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_2]) (by norm_num [bρ_2])]; exact Lmid_2
+
+lemma U_39_3 :
+    Uω (aρ 3) (bρ 3) (12166846693 / 1000000000000 : ℝ) ≤
+      (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_3]) (by norm_num [bρ_3])]; exact Lmid_3
+
+lemma U_39_4 :
+    Uω (aρ 4) (bρ 4) (12166846693 / 1000000000000 : ℝ) ≤
+      (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_4]) (by norm_num [bρ_4])]; exact Lmid_4
+
+lemma U_39_5 :
+    Uω (aρ 5) (bρ 5) (12166846693 / 1000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_39_6 :
+    Uω (aρ 6) (bρ 6) (12166846693 / 1000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_39_7 :
+    Uω (aρ 7) (bρ 7) (12166846693 / 1000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_39_8 :
+    Uω (aρ 8) (bρ 8) (12166846693 / 1000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_39_9 :
+    Uω (aρ 9) (bρ 9) (12166846693 / 1000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_39_10 :
+    Uω (aρ 10) (bρ 10) (12166846693 / 1000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_39_11 :
+    Uω (aρ 11) (bρ 11) (12166846693 / 1000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_39_12 :
+    Uω (aρ 12) (bρ 12) (12166846693 / 1000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_39_13 :
+    Uω (aρ 13) (bρ 13) (12166846693 / 1000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_39_14 :
+    Uω (aρ 14) (bρ 14) (12166846693 / 1000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_39_15 :
+    Uω (aρ 15) (bρ 15) (12166846693 / 1000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_39_16 :
+    Uω (aρ 16) (bρ 16) (12166846693 / 1000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_39 :
+    Uρ (12166846693 / 1000000000000 : ℝ) ≤
+      (-(5706133116954878622153 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (12166846693 / 1000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(52178850672124585216629 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_39_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (12166846693 / 1000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(57268912150832389796717 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_39_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (12166846693 / 1000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_39_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (12166846693 / 1000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_39_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (12166846693 / 1000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_39_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (12166846693 / 1000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_39_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (12166846693 / 1000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_39_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (12166846693 / 1000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_39_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (12166846693 / 1000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_39_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (12166846693 / 1000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_39_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (12166846693 / 1000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_39_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (12166846693 / 1000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_39_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (12166846693 / 1000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_39_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (12166846693 / 1000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_39_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (12166846693 / 1000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_39_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (12166846693 / 1000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_39_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/U03.lean b/LeanPool/Zeta5Irrational/Table/U03.lean
new file mode 100644
index 0000000000..563e371024
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/U03.lean
@@ -0,0 +1,2393 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-! # Certified arcsine potential bounds (U03) -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma U_40_1 :
+    Uω (aρ 1) (bρ 1) (3068199571 / 200000000000 : ℝ) ≤
+      (-(47437922514341080202681 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (174111395654621 / 20000000000000000 : ℝ) ≤
+      (-(47437922514341080202681 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (174111395654621 / 20000000000000000 : ℝ) =
+            (174111395654621 / 156250000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (174111395654621 / 156250000000000 : ℝ) ≤
+            (1082380106658919797319 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (85198635189621 / 10000000000000000 : ℝ)) (q :=
+      (174111395654621 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_40_2 :
+    Uω (aρ 2) (bρ 2) (3068199571 / 200000000000 : ℝ) ≤
+      (-(57268912150832389796717 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_2]) (by norm_num [bρ_2])]; exact Lmid_2
+
+lemma U_40_3 :
+    Uω (aρ 3) (bρ 3) (3068199571 / 200000000000 : ℝ) ≤
+      (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_3]) (by norm_num [bρ_3])]; exact Lmid_3
+
+lemma U_40_4 :
+    Uω (aρ 4) (bρ 4) (3068199571 / 200000000000 : ℝ) ≤
+      (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_4]) (by norm_num [bρ_4])]; exact Lmid_4
+
+lemma U_40_5 :
+    Uω (aρ 5) (bρ 5) (3068199571 / 200000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_40_6 :
+    Uω (aρ 6) (bρ 6) (3068199571 / 200000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_40_7 :
+    Uω (aρ 7) (bρ 7) (3068199571 / 200000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_40_8 :
+    Uω (aρ 8) (bρ 8) (3068199571 / 200000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_40_9 :
+    Uω (aρ 9) (bρ 9) (3068199571 / 200000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_40_10 :
+    Uω (aρ 10) (bρ 10) (3068199571 / 200000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_40_11 :
+    Uω (aρ 11) (bρ 11) (3068199571 / 200000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_40_12 :
+    Uω (aρ 12) (bρ 12) (3068199571 / 200000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_40_13 :
+    Uω (aρ 13) (bρ 13) (3068199571 / 200000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_40_14 :
+    Uω (aρ 14) (bρ 14) (3068199571 / 200000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_40_15 :
+    Uω (aρ 15) (bρ 15) (3068199571 / 200000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_40_16 :
+    Uω (aρ 16) (bρ 16) (3068199571 / 200000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_40 :
+    Uρ (3068199571 / 200000000000 : ℝ) ≤
+      (-(28480811898748750448173 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (3068199571 / 200000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(47437922514341080202681 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_40_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (3068199571 / 200000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(57268912150832389796717 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_40_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (3068199571 / 200000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_40_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (3068199571 / 200000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_40_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (3068199571 / 200000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_40_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (3068199571 / 200000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_40_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (3068199571 / 200000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_40_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (3068199571 / 200000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_40_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (3068199571 / 200000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_40_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (3068199571 / 200000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_40_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (3068199571 / 200000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_40_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (3068199571 / 200000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_40_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (3068199571 / 200000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_40_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (3068199571 / 200000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_40_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (3068199571 / 200000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_40_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (3068199571 / 200000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_40_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_41_1 :
+    Uω (aρ 1) (bρ 1) (255845148549 / 16000000000000 : ℝ) ≤
+      (-(23352264455452045399363 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7494420682899 / 800000000000000 : ℝ) ≤
+      (-(23352264455452045399363 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show (7494420682899 / 800000000000000 : ℝ) = (7494420682899 / 6250000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7494420682899 / 6250000000000 : ℝ) ≤
+            (907886855047954600637 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1839090346287 / 200000000000000 : ℝ)) (q :=
+      (7494420682899 / 800000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_41_2 :
+    Uω (aρ 2) (bρ 2) (255845148549 / 16000000000000 : ℝ) ≤
+      (-(2113612623409580587049 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (101438833572867 / 20000000000000000 : ℝ) ≤
+      (-(2113612623409580587049 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show
+          (101438833572867 / 20000000000000000 : ℝ) =
+            (101438833572867 / 78125000000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (101438833572867 / 78125000000000 : ℝ) ≤
+            (104458353550419412951 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (14900923162371 / 5000000000000000 : ℝ)) (q :=
+      (101438833572867 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_41_3 :
+    Uω (aρ 3) (bρ 3) (255845148549 / 16000000000000 : ℝ) ≤
+      (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_3]) (by norm_num [bρ_3])]; exact Lmid_3
+
+lemma U_41_4 :
+    Uω (aρ 4) (bρ 4) (255845148549 / 16000000000000 : ℝ) ≤
+      (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_4]) (by norm_num [bρ_4])]; exact Lmid_4
+
+lemma U_41_5 :
+    Uω (aρ 5) (bρ 5) (255845148549 / 16000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_41_6 :
+    Uω (aρ 6) (bρ 6) (255845148549 / 16000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_41_7 :
+    Uω (aρ 7) (bρ 7) (255845148549 / 16000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_41_8 :
+    Uω (aρ 8) (bρ 8) (255845148549 / 16000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_41_9 :
+    Uω (aρ 9) (bρ 9) (255845148549 / 16000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_41_10 :
+    Uω (aρ 10) (bρ 10) (255845148549 / 16000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_41_11 :
+    Uω (aρ 11) (bρ 11) (255845148549 / 16000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_41_12 :
+    Uω (aρ 12) (bρ 12) (255845148549 / 16000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_41_13 :
+    Uω (aρ 13) (bρ 13) (255845148549 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_41_14 :
+    Uω (aρ 14) (bρ 14) (255845148549 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_41_15 :
+    Uω (aρ 15) (bρ 15) (255845148549 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_41_16 :
+    Uω (aρ 16) (bρ 16) (255845148549 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_41 :
+    Uρ (255845148549 / 16000000000000 : ℝ) ≤
+      (-(14171290695500939233193 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (255845148549 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(23352264455452045399363 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_41_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (255845148549 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(2113612623409580587049 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_41_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (255845148549 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_41_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (255845148549 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_41_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (255845148549 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_41_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (255845148549 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_41_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (255845148549 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_41_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (255845148549 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_41_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (255845148549 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_41_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (255845148549 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_41_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (255845148549 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_41_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (255845148549 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_41_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (255845148549 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_41_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (255845148549 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_41_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (255845148549 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_41_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (255845148549 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_41_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_42_1 :
+    Uω (aρ 1) (bρ 1) (133117165709 / 8000000000000 : ℝ) ≤
+      (-(23011512008219083613497 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12535898731833 / 1250000000000000 : ℝ) ≤
+      (-(23011512008219083613497 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (12535898731833 / 1250000000000000 : ℝ) = (12535898731833 / 9765625000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12535898731833 / 9765625000000 : ℝ) ≤
+            (1248639302280916386503 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (49337570329039 / 5000000000000000 : ℝ)) (q :=
+      (12535898731833 / 1250000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_42_2 :
+    Uω (aρ 2) (bρ 2) (133117165709 / 8000000000000 : ℝ) ≤
+      (-(51055078142581221338421 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (60632591112057 / 10000000000000000 : ℝ) ≤
+      (-(51055078142581221338421 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show
+          (60632591112057 / 10000000000000000 : ℝ) = (60632591112057 / 39062500000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (60632591112057 / 39062500000000 : ℝ) ≤
+            (4396696281418778661579 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2695934730179 / 625000000000000 : ℝ)) (q :=
+      (60632591112057 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_42_3 :
+    Uω (aρ 3) (bρ 3) (133117165709 / 8000000000000 : ℝ) ≤
+      (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_3]) (by norm_num [bρ_3])]; exact Lmid_3
+
+lemma U_42_4 :
+    Uω (aρ 4) (bρ 4) (133117165709 / 8000000000000 : ℝ) ≤
+      (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_4]) (by norm_num [bρ_4])]; exact Lmid_4
+
+lemma U_42_5 :
+    Uω (aρ 5) (bρ 5) (133117165709 / 8000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_42_6 :
+    Uω (aρ 6) (bρ 6) (133117165709 / 8000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_42_7 :
+    Uω (aρ 7) (bρ 7) (133117165709 / 8000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_42_8 :
+    Uω (aρ 8) (bρ 8) (133117165709 / 8000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_42_9 :
+    Uω (aρ 9) (bρ 9) (133117165709 / 8000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_42_10 :
+    Uω (aρ 10) (bρ 10) (133117165709 / 8000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_42_11 :
+    Uω (aρ 11) (bρ 11) (133117165709 / 8000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_42_12 :
+    Uω (aρ 12) (bρ 12) (133117165709 / 8000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_42_13 :
+    Uω (aρ 13) (bρ 13) (133117165709 / 8000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_42_14 :
+    Uω (aρ 14) (bρ 14) (133117165709 / 8000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_42_15 :
+    Uω (aρ 15) (bρ 15) (133117165709 / 8000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_42_16 :
+    Uω (aρ 16) (bρ 16) (133117165709 / 8000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_42 :
+    Uρ (133117165709 / 8000000000000 : ℝ) ≤
+      (-(14141400455464355101593 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (133117165709 / 8000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(23011512008219083613497 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_42_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (133117165709 / 8000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(51055078142581221338421 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_42_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (133117165709 / 8000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_42_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (133117165709 / 8000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_42_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (133117165709 / 8000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_42_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (133117165709 / 8000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_42_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (133117165709 / 8000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_42_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (133117165709 / 8000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_42_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (133117165709 / 8000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_42_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (133117165709 / 8000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_42_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (133117165709 / 8000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_42_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (133117165709 / 8000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_42_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (133117165709 / 8000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_42_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (133117165709 / 8000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_42_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (133117165709 / 8000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_42_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (133117165709 / 8000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_42_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_43_1 :
+    Uω (aρ 1) (bρ 1) (108571569141 / 6400000000000 : ℝ) ≤
+      (-(22849732727431537692483 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (414340535730533 / 40000000000000000 : ℝ) ≤
+      (-(22849732727431537692483 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (414340535730533 / 40000000000000000 : ℝ) =
+            (414340535730533 / 312500000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (414340535730533 / 312500000000000 : ℝ) ≤
+            (1410418583068462307517 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (51012204583727 / 5000000000000000 : ℝ)) (q :=
+      (414340535730533 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_43_2 :
+    Uω (aρ 2) (bρ 2) (108571569141 / 6400000000000 : ℝ) ≤
+      (-(5034826491214142248959 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (260293083348717 / 40000000000000000 : ℝ) ≤
+      (-(5034826491214142248959 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show
+          (260293083348717 / 40000000000000000 : ℝ) =
+            (260293083348717 / 156250000000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (260293083348717 / 156250000000000 : ℝ) ≤
+            (510350951185857751041 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (24384847743273 / 5000000000000000 : ℝ)) (q :=
+      (260293083348717 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_43_3 :
+    Uω (aρ 3) (bρ 3) (108571569141 / 6400000000000 : ℝ) ≤
+      (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_3]) (by norm_num [bρ_3])]; exact Lmid_3
+
+lemma U_43_4 :
+    Uω (aρ 4) (bρ 4) (108571569141 / 6400000000000 : ℝ) ≤
+      (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_4]) (by norm_num [bρ_4])]; exact Lmid_4
+
+lemma U_43_5 :
+    Uω (aρ 5) (bρ 5) (108571569141 / 6400000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_43_6 :
+    Uω (aρ 6) (bρ 6) (108571569141 / 6400000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_43_7 :
+    Uω (aρ 7) (bρ 7) (108571569141 / 6400000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_43_8 :
+    Uω (aρ 8) (bρ 8) (108571569141 / 6400000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_43_9 :
+    Uω (aρ 9) (bρ 9) (108571569141 / 6400000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_43_10 :
+    Uω (aρ 10) (bρ 10) (108571569141 / 6400000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_43_11 :
+    Uω (aρ 11) (bρ 11) (108571569141 / 6400000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_43_12 :
+    Uω (aρ 12) (bρ 12) (108571569141 / 6400000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_43_13 :
+    Uω (aρ 13) (bρ 13) (108571569141 / 6400000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_43_14 :
+    Uω (aρ 14) (bρ 14) (108571569141 / 6400000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_43_15 :
+    Uω (aρ 15) (bρ 15) (108571569141 / 6400000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_43_16 :
+    Uω (aρ 16) (bρ 16) (108571569141 / 6400000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_43 :
+    Uρ (108571569141 / 6400000000000 : ℝ) ≤
+      (-(5651713497111691966977 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (108571569141 / 6400000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(22849732727431537692483 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_43_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (108571569141 / 6400000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(5034826491214142248959 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_43_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (108571569141 / 6400000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_43_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (108571569141 / 6400000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_43_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (108571569141 / 6400000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_43_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (108571569141 / 6400000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_43_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (108571569141 / 6400000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_43_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (108571569141 / 6400000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_43_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (108571569141 / 6400000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_43_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (108571569141 / 6400000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_43_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (108571569141 / 6400000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_43_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (108571569141 / 6400000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_43_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (108571569141 / 6400000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_43_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (108571569141 / 6400000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_43_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (108571569141 / 6400000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_43_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (108571569141 / 6400000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_43_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_44_1 :
+    Uω (aρ 1) (bρ 1) (276623514287 / 16000000000000 : ℝ) ≤
+      (-(45386349211403757950859 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (53439932624379 / 5000000000000000 : ℝ) ≤
+      (-(45386349211403757950859 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (53439932624379 / 5000000000000000 : ℝ) = (53439932624379 / 39062500000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (53439932624379 / 39062500000000 : ℝ) ≤
+            (3133953409596242049141 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (105367252153141 / 10000000000000000 : ℝ)) (q :=
+      (53439932624379 / 5000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_44_2 :
+    Uω (aρ 2) (bρ 2) (276623514287 / 16000000000000 : ℝ) ≤
+      (-(49716311348467284244491 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (138636640821569 / 20000000000000000 : ℝ) ≤
+      (-(49716311348467284244491 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show
+          (138636640821569 / 20000000000000000 : ℝ) =
+            (138636640821569 / 78125000000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (138636640821569 / 78125000000000 : ℝ) ≤
+            (5735463075532715755509 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (27006587493597 / 5000000000000000 : ℝ)) (q :=
+      (138636640821569 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_44_3 :
+    Uω (aρ 3) (bρ 3) (276623514287 / 16000000000000 : ℝ) ≤
+      (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_3]) (by norm_num [bρ_3])]; exact Lmid_3
+
+lemma U_44_4 :
+    Uω (aρ 4) (bρ 4) (276623514287 / 16000000000000 : ℝ) ≤
+      (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_4]) (by norm_num [bρ_4])]; exact Lmid_4
+
+lemma U_44_5 :
+    Uω (aρ 5) (bρ 5) (276623514287 / 16000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_44_6 :
+    Uω (aρ 6) (bρ 6) (276623514287 / 16000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_44_7 :
+    Uω (aρ 7) (bρ 7) (276623514287 / 16000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_44_8 :
+    Uω (aρ 8) (bρ 8) (276623514287 / 16000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_44_9 :
+    Uω (aρ 9) (bρ 9) (276623514287 / 16000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_44_10 :
+    Uω (aρ 10) (bρ 10) (276623514287 / 16000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_44_11 :
+    Uω (aρ 11) (bρ 11) (276623514287 / 16000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_44_12 :
+    Uω (aρ 12) (bρ 12) (276623514287 / 16000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_44_13 :
+    Uω (aρ 13) (bρ 13) (276623514287 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_44_14 :
+    Uω (aρ 14) (bρ 14) (276623514287 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_44_15 :
+    Uω (aρ 15) (bρ 15) (276623514287 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_44_16 :
+    Uω (aρ 16) (bρ 16) (276623514287 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_44 :
+    Uρ (276623514287 / 16000000000000 : ℝ) ≤
+      (-(14118325055929443477217 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (276623514287 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(45386349211403757950859 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_44_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (276623514287 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(49716311348467284244491 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_44_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (276623514287 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_44_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (276623514287 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_44_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (276623514287 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_44_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (276623514287 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_44_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (276623514287 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_44_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (276623514287 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_44_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (276623514287 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_44_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (276623514287 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_44_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (276623514287 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_44_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (276623514287 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_44_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (276623514287 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_44_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (276623514287 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_44_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (276623514287 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_44_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (276623514287 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_44_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_45_1 :
+    Uω (aρ 1) (bρ 1) (563636211443 / 32000000000000 : ℝ) ≤
+      (-(45083004919660055486503 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (440686720790797 / 40000000000000000 : ℝ) ≤
+      (-(45083004919660055486503 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (440686720790797 / 40000000000000000 : ℝ) =
+            (440686720790797 / 312500000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (440686720790797 / 312500000000000 : ℝ) ≤
+            (3437297701339944513497 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (108704262404461 / 10000000000000000 : ℝ)) (q :=
+      (440686720790797 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_45_2 :
+    Uω (aρ 2) (bρ 2) (563636211443 / 32000000000000 : ℝ) ≤
+      (-(9828288768212480750473 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (293679884795729 / 40000000000000000 : ℝ) ≤
+      (-(9828288768212480750473 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show
+          (293679884795729 / 40000000000000000 : ℝ) =
+            (293679884795729 / 156250000000000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (293679884795729 / 156250000000000 : ℝ) ≤
+            (1262066116587519249527 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (58969856916927 / 10000000000000000 : ℝ)) (q :=
+      (293679884795729 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_45_3 :
+    Uω (aρ 3) (bρ 3) (563636211443 / 32000000000000 : ℝ) ≤
+      (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_3]) (by norm_num [bρ_3])]; exact Lmid_3
+
+lemma U_45_4 :
+    Uω (aρ 4) (bρ 4) (563636211443 / 32000000000000 : ℝ) ≤
+      (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_4]) (by norm_num [bρ_4])]; exact Lmid_4
+
+lemma U_45_5 :
+    Uω (aρ 5) (bρ 5) (563636211443 / 32000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_45_6 :
+    Uω (aρ 6) (bρ 6) (563636211443 / 32000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_45_7 :
+    Uω (aρ 7) (bρ 7) (563636211443 / 32000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_45_8 :
+    Uω (aρ 8) (bρ 8) (563636211443 / 32000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_45_9 :
+    Uω (aρ 9) (bρ 9) (563636211443 / 32000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_45_10 :
+    Uω (aρ 10) (bρ 10) (563636211443 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_45_11 :
+    Uω (aρ 11) (bρ 11) (563636211443 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_45_12 :
+    Uω (aρ 12) (bρ 12) (563636211443 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_45_13 :
+    Uω (aρ 13) (bρ 13) (563636211443 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_45_14 :
+    Uω (aρ 14) (bρ 14) (563636211443 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_45_15 :
+    Uω (aρ 15) (bρ 15) (563636211443 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_45_16 :
+    Uω (aρ 16) (bρ 16) (563636211443 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_45 :
+    Uρ (563636211443 / 32000000000000 : ℝ) ≤
+      (-(28216517921339449947509 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (563636211443 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(45083004919660055486503 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_45_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (563636211443 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(9828288768212480750473 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_45_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (563636211443 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_45_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (563636211443 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_45_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (563636211443 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_45_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (563636211443 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_45_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (563636211443 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_45_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (563636211443 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_45_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (563636211443 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_45_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (563636211443 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_45_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (563636211443 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_45_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (563636211443 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_45_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (563636211443 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_45_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (563636211443 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_45_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (563636211443 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_45_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (563636211443 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_45_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_46_1 :
+    Uω (aρ 1) (bρ 1) (71753174289 / 4000000000000 : ℝ) ≤
+      (-(44788826203166347861371 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (226921678747097 / 20000000000000000 : ℝ) ≤
+      (-(44788826203166347861371 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (226921678747097 / 20000000000000000 : ℝ) =
+            (226921678747097 / 156250000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (226921678747097 / 156250000000000 : ℝ) ≤
+            (3731476417833652138629 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (112035961109597 / 10000000000000000 : ℝ)) (q :=
+      (226921678747097 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_46_2 :
+    Uω (aρ 2) (bρ 2) (71753174289 / 4000000000000 : ℝ) ≤
+      (-(24306011407961759067497 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2419116003209 / 312500000000000 : ℝ) ≤
+      (-(24306011407961759067497 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show (2419116003209 / 312500000000000 : ℝ) = (2419116003209 / 1220703125000 : ℝ) / 2 ^ 8
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2419116003209 / 1220703125000 : ℝ) ≤
+            (3419875804038240932503 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (15926679769469 / 2500000000000000 : ℝ)) (q :=
+      (2419116003209 / 312500000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_46_3 :
+    Uω (aρ 3) (bρ 3) (71753174289 / 4000000000000 : ℝ) ≤
+      (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_3]) (by norm_num [bρ_3])]; exact Lmid_3
+
+lemma U_46_4 :
+    Uω (aρ 4) (bρ 4) (71753174289 / 4000000000000 : ℝ) ≤
+      (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_4]) (by norm_num [bρ_4])]; exact Lmid_4
+
+lemma U_46_5 :
+    Uω (aρ 5) (bρ 5) (71753174289 / 4000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_46_6 :
+    Uω (aρ 6) (bρ 6) (71753174289 / 4000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_46_7 :
+    Uω (aρ 7) (bρ 7) (71753174289 / 4000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_46_8 :
+    Uω (aρ 8) (bρ 8) (71753174289 / 4000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_46_9 :
+    Uω (aρ 9) (bρ 9) (71753174289 / 4000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_46_10 :
+    Uω (aρ 10) (bρ 10) (71753174289 / 4000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_46_11 :
+    Uω (aρ 11) (bρ 11) (71753174289 / 4000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_46_12 :
+    Uω (aρ 12) (bρ 12) (71753174289 / 4000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_46_13 :
+    Uω (aρ 13) (bρ 13) (71753174289 / 4000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_46_14 :
+    Uω (aρ 14) (bρ 14) (71753174289 / 4000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_46_15 :
+    Uω (aρ 15) (bρ 15) (71753174289 / 4000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_46_16 :
+    Uω (aρ 16) (bρ 16) (71753174289 / 4000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_46 :
+    Uρ (71753174289 / 4000000000000 : ℝ) ≤
+      (-(1762363843694816057143 / 625000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (71753174289 / 4000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(44788826203166347861371 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_46_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (71753174289 / 4000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(24306011407961759067497 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_46_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (71753174289 / 4000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_46_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (71753174289 / 4000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_46_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (71753174289 / 4000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_46_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (71753174289 / 4000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_46_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (71753174289 / 4000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_46_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (71753174289 / 4000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_46_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (71753174289 / 4000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_46_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (71753174289 / 4000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_46_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (71753174289 / 4000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_46_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (71753174289 / 4000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_46_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (71753174289 / 4000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_46_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (71753174289 / 4000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_46_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (71753174289 / 4000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_46_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (71753174289 / 4000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_46_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_47_1 :
+    Uω (aρ 1) (bρ 1) (584414577181 / 32000000000000 : ℝ) ≤
+      (-(4450326262908179514767 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (466990291497527 / 40000000000000000 : ℝ) ≤
+      (-(4450326262908179514767 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (466990291497527 / 40000000000000000 : ℝ) =
+            (466990291497527 / 312500000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (466990291497527 / 312500000000000 : ℝ) ≤
+            (401703999191820485233 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (115362808464701 / 10000000000000000 : ℝ)) (q :=
+      (466990291497527 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_47_2 :
+    Uω (aρ 2) (bρ 2) (584414577181 / 32000000000000 : ℝ) ≤
+      (-(48119922953915799270603 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (325265715969291 / 40000000000000000 : ℝ) ≤
+      (-(48119922953915799270603 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (325265715969291 / 40000000000000000 : ℝ) =
+            (325265715969291 / 312500000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (325265715969291 / 312500000000000 : ℝ) ≤
+            (400379667084200729397 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (68269533210583 / 10000000000000000 : ℝ)) (q :=
+      (325265715969291 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_47_3 :
+    Uω (aρ 3) (bρ 3) (584414577181 / 32000000000000 : ℝ) ≤
+      (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_3]) (by norm_num [bρ_3])]; exact Lmid_3
+
+lemma U_47_4 :
+    Uω (aρ 4) (bρ 4) (584414577181 / 32000000000000 : ℝ) ≤
+      (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_4]) (by norm_num [bρ_4])]; exact Lmid_4
+
+lemma U_47_5 :
+    Uω (aρ 5) (bρ 5) (584414577181 / 32000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_47_6 :
+    Uω (aρ 6) (bρ 6) (584414577181 / 32000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_47_7 :
+    Uω (aρ 7) (bρ 7) (584414577181 / 32000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_47_8 :
+    Uω (aρ 8) (bρ 8) (584414577181 / 32000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_47_9 :
+    Uω (aρ 9) (bρ 9) (584414577181 / 32000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_47_10 :
+    Uω (aρ 10) (bρ 10) (584414577181 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_47_11 :
+    Uω (aρ 11) (bρ 11) (584414577181 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_47_12 :
+    Uω (aρ 12) (bρ 12) (584414577181 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_47_13 :
+    Uω (aρ 13) (bρ 13) (584414577181 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_47_14 :
+    Uω (aρ 14) (bρ 14) (584414577181 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_47_15 :
+    Uω (aρ 15) (bρ 15) (584414577181 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_47_16 :
+    Uω (aρ 16) (bρ 16) (584414577181 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_47 :
+    Uρ (584414577181 / 32000000000000 : ℝ) ≤
+      (-(14090157794893601635931 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (584414577181 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(4450326262908179514767 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_47_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (584414577181 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(48119922953915799270603 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_47_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (584414577181 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_47_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (584414577181 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_47_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (584414577181 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_47_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (584414577181 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_47_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (584414577181 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_47_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (584414577181 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_47_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (584414577181 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_47_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (584414577181 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_47_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (584414577181 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_47_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (584414577181 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_47_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (584414577181 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_47_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (584414577181 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_47_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (584414577181 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_47_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (584414577181 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_47_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_48_1 :
+    Uω (aρ 1) (bρ 1) (11896075201 / 640000000000 : ℝ) ≤
+      (-(44225812907642607522863 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24006416936349 / 2000000000000000 : ℝ) ≤
+      (-(44225812907642607522863 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (24006416936349 / 2000000000000000 : ℝ) = (24006416936349 / 15625000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24006416936349 / 15625000000000 : ℝ) ≤
+            (4294489713357392477137 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (23737042486573 / 2000000000000000 : ℝ)) (q :=
+      (24006416936349 / 2000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_48_2 :
+    Uω (aρ 2) (bρ 2) (11896075201 / 640000000000 : ℝ) ≤
+      (-(47659199379596186942369 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8515051303781 / 1000000000000000 : ℝ) ≤
+      (-(47659199379596186942369 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show (8515051303781 / 1000000000000000 : ℝ) = (8515051303781 / 7812500000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8515051303781 / 7812500000000 : ℝ) ≤
+            (861103241403813057631 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (14538216330999 / 2000000000000000 : ℝ)) (q :=
+      (8515051303781 / 1000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_48_3 :
+    Uω (aρ 3) (bρ 3) (11896075201 / 640000000000 : ℝ) ≤
+      (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_3]) (by norm_num [bρ_3])]; exact Lmid_3
+
+lemma U_48_4 :
+    Uω (aρ 4) (bρ 4) (11896075201 / 640000000000 : ℝ) ≤
+      (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_4]) (by norm_num [bρ_4])]; exact Lmid_4
+
+lemma U_48_5 :
+    Uω (aρ 5) (bρ 5) (11896075201 / 640000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_48_6 :
+    Uω (aρ 6) (bρ 6) (11896075201 / 640000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_48_7 :
+    Uω (aρ 7) (bρ 7) (11896075201 / 640000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_48_8 :
+    Uω (aρ 8) (bρ 8) (11896075201 / 640000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_48_9 :
+    Uω (aρ 9) (bρ 9) (11896075201 / 640000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_48_10 :
+    Uω (aρ 10) (bρ 10) (11896075201 / 640000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_48_11 :
+    Uω (aρ 11) (bρ 11) (11896075201 / 640000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_48_12 :
+    Uω (aρ 12) (bρ 12) (11896075201 / 640000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_48_13 :
+    Uω (aρ 13) (bρ 13) (11896075201 / 640000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_48_14 :
+    Uω (aρ 14) (bρ 14) (11896075201 / 640000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_48_15 :
+    Uω (aρ 15) (bρ 15) (11896075201 / 640000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_48_16 :
+    Uω (aρ 16) (bρ 16) (11896075201 / 640000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_48 :
+    Uρ (11896075201 / 640000000000 : ℝ) ≤
+      (-(28163819716178893923457 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (11896075201 / 640000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(44225812907642607522863 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_48_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (11896075201 / 640000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(47659199379596186942369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_48_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (11896075201 / 640000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_48_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (11896075201 / 640000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_48_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (11896075201 / 640000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_48_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (11896075201 / 640000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_48_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (11896075201 / 640000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_48_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (11896075201 / 640000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_48_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (11896075201 / 640000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_48_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (11896075201 / 640000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_48_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (11896075201 / 640000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_48_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (11896075201 / 640000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_48_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (11896075201 / 640000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_48_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (11896075201 / 640000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_48_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (11896075201 / 640000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_48_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (11896075201 / 640000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_48_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_49_1 :
+    Uω (aρ 1) (bρ 1) (605192942919 / 32000000000000 : ℝ) ≤
+      (-(21978009554983345102107 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (98651645039663 / 8000000000000000 : ℝ) ≤
+      (-(21978009554983345102107 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (98651645039663 / 8000000000000000 : ℝ) = (98651645039663 / 62500000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (98651645039663 / 62500000000000 : ℝ) ≤
+            (2282141755516654897893 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (12200353602197 / 1000000000000000 : ℝ)) (q :=
+      (98651645039663 / 8000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_49_2 :
+    Uω (aρ 2) (bρ 2) (605192942919 / 32000000000000 : ℝ) ≤
+      (-(47225342930112879030383 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (355704538515791 / 40000000000000000 : ℝ) ≤
+      (-(47225342930112879030383 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (355704538515791 / 40000000000000000 : ℝ) =
+            (355704538515791 / 312500000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (355704538515791 / 312500000000000 : ℝ) ≤
+            (1294959690887120969617 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (19248926297677 / 2500000000000000 : ℝ)) (q :=
+      (355704538515791 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_49_3 :
+    Uω (aρ 3) (bρ 3) (605192942919 / 32000000000000 : ℝ) ≤
+      (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_3]) (by norm_num [bρ_3])]; exact Lmid_3
+
+lemma U_49_4 :
+    Uω (aρ 4) (bρ 4) (605192942919 / 32000000000000 : ℝ) ≤
+      (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_4]) (by norm_num [bρ_4])]; exact Lmid_4
+
+lemma U_49_5 :
+    Uω (aρ 5) (bρ 5) (605192942919 / 32000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_49_6 :
+    Uω (aρ 6) (bρ 6) (605192942919 / 32000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_49_7 :
+    Uω (aρ 7) (bρ 7) (605192942919 / 32000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_49_8 :
+    Uω (aρ 8) (bρ 8) (605192942919 / 32000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_49_9 :
+    Uω (aρ 9) (bρ 9) (605192942919 / 32000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_49_10 :
+    Uω (aρ 10) (bρ 10) (605192942919 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_49_11 :
+    Uω (aρ 11) (bρ 11) (605192942919 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_49_12 :
+    Uω (aρ 12) (bρ 12) (605192942919 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_49_13 :
+    Uω (aρ 13) (bρ 13) (605192942919 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_49_14 :
+    Uω (aρ 14) (bρ 14) (605192942919 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_49_15 :
+    Uω (aρ 15) (bρ 15) (605192942919 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_49_16 :
+    Uω (aρ 16) (bρ 16) (605192942919 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_49 :
+    Uρ (605192942919 / 32000000000000 : ℝ) ≤
+      (-(28148196170031691353311 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (605192942919 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(21978009554983345102107 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_49_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (605192942919 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(47225342930112879030383 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_49_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (605192942919 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_49_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (605192942919 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_49_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (605192942919 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_49_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (605192942919 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_49_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (605192942919 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_49_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (605192942919 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_49_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (605192942919 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_49_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (605192942919 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_49_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (605192942919 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_49_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (605192942919 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_49_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (605192942919 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_49_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (605192942919 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_49_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (605192942919 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_49_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (605192942919 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_49_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_50_1 :
+    Uω (aρ 1) (bρ 1) (153895531447 / 8000000000000 : ℝ) ≤
+      (-(10923365431073482723193 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (253190299591751 / 20000000000000000 : ℝ) ≤
+      (-(10923365431073482723193 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (253190299591751 / 20000000000000000 : ℝ) =
+            (253190299591751 / 156250000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (253190299591751 / 156250000000000 : ℝ) ≤
+            (1206710224176517276807 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (125318103368001 / 10000000000000000 : ℝ)) (q :=
+      (253190299591751 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_50_2 :
+    Uω (aρ 2) (bρ 2) (153895531447 / 8000000000000 : ℝ) ≤
+      (-(182870446791635865959 / 39062500000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (92652592342437 / 10000000000000000 : ℝ) ≤
+      (-(182870446791635865959 / 39062500000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (92652592342437 / 10000000000000000 : ℝ) = (92652592342437 / 78125000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (92652592342437 / 78125000000000 : ℝ) ≤
+            (6661985321645384041 / 39062500000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (20300500242781 / 2500000000000000 : ℝ)) (q :=
+      (92652592342437 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_50_3 :
+    Uω (aρ 3) (bρ 3) (153895531447 / 8000000000000 : ℝ) ≤
+      (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_3]) (by norm_num [bρ_3])]; exact Lmid_3
+
+lemma U_50_4 :
+    Uω (aρ 4) (bρ 4) (153895531447 / 8000000000000 : ℝ) ≤
+      (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_4]) (by norm_num [bρ_4])]; exact Lmid_4
+
+lemma U_50_5 :
+    Uω (aρ 5) (bρ 5) (153895531447 / 8000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_50_6 :
+    Uω (aρ 6) (bρ 6) (153895531447 / 8000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_50_7 :
+    Uω (aρ 7) (bρ 7) (153895531447 / 8000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_50_8 :
+    Uω (aρ 8) (bρ 8) (153895531447 / 8000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_50_9 :
+    Uω (aρ 9) (bρ 9) (153895531447 / 8000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_50_10 :
+    Uω (aρ 10) (bρ 10) (153895531447 / 8000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_50_11 :
+    Uω (aρ 11) (bρ 11) (153895531447 / 8000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_50_12 :
+    Uω (aρ 12) (bρ 12) (153895531447 / 8000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_50_13 :
+    Uω (aρ 13) (bρ 13) (153895531447 / 8000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_50_14 :
+    Uω (aρ 14) (bρ 14) (153895531447 / 8000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_50_15 :
+    Uω (aρ 15) (bρ 15) (153895531447 / 8000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_50_16 :
+    Uω (aρ 16) (bρ 16) (153895531447 / 8000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_50 :
+    Uρ (153895531447 / 8000000000000 : ℝ) ≤
+      (-(28133336822199640692941 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (153895531447 / 8000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(10923365431073482723193 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_50_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (153895531447 / 8000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) * (-(182870446791635865959 / 39062500000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_50_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (153895531447 / 8000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_50_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (153895531447 / 8000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_50_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (153895531447 / 8000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_50_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (153895531447 / 8000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_50_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (153895531447 / 8000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_50_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (153895531447 / 8000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_50_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (153895531447 / 8000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_50_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (153895531447 / 8000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_50_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (153895531447 / 8000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_50_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (153895531447 / 8000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_50_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (153895531447 / 8000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_50_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (153895531447 / 8000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_50_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (153895531447 / 8000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_50_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (153895531447 / 8000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_50_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_51_1 :
+    Uω (aρ 1) (bρ 1) (318180245763 / 16000000000000 : ℝ) ≤
+      (-(21594272656989974022877 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (266302536918669 / 20000000000000000 : ℝ) ≤
+      (-(21594272656989974022877 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (266302536918669 / 20000000000000000 : ℝ) =
+            (266302536918669 / 156250000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (266302536918669 / 156250000000000 : ℝ) ≤
+            (2665878653510025977123 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (65968550700897 / 5000000000000000 : ℝ)) (q :=
+      (266302536918669 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_51_2 :
+    Uω (aρ 2) (bρ 2) (318180245763 / 16000000000000 : ℝ) ≤
+      (-(9210627822742137523341 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (199971256625133 / 20000000000000000 : ℝ) ≤
+      (-(9210627822742137523341 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (199971256625133 / 20000000000000000 : ℝ) =
+            (199971256625133 / 156250000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (199971256625133 / 156250000000000 : ℝ) ≤
+            (493432701457862476659 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (44687416809129 / 5000000000000000 : ℝ)) (q :=
+      (199971256625133 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_51_3 :
+    Uω (aρ 3) (bρ 3) (318180245763 / 16000000000000 : ℝ) ≤
+      (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_3]) (by norm_num [bρ_3])]; exact Lmid_3
+
+lemma U_51_4 :
+    Uω (aρ 4) (bρ 4) (318180245763 / 16000000000000 : ℝ) ≤
+      (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_4]) (by norm_num [bρ_4])]; exact Lmid_4
+
+lemma U_51_5 :
+    Uω (aρ 5) (bρ 5) (318180245763 / 16000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_51_6 :
+    Uω (aρ 6) (bρ 6) (318180245763 / 16000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_51_7 :
+    Uω (aρ 7) (bρ 7) (318180245763 / 16000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_51_8 :
+    Uω (aρ 8) (bρ 8) (318180245763 / 16000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_51_9 :
+    Uω (aρ 9) (bρ 9) (318180245763 / 16000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_51_10 :
+    Uω (aρ 10) (bρ 10) (318180245763 / 16000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_51_11 :
+    Uω (aρ 11) (bρ 11) (318180245763 / 16000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_51_12 :
+    Uω (aρ 12) (bρ 12) (318180245763 / 16000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_51_13 :
+    Uω (aρ 13) (bρ 13) (318180245763 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_51_14 :
+    Uω (aρ 14) (bρ 14) (318180245763 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_51_15 :
+    Uω (aρ 15) (bρ 15) (318180245763 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_51_16 :
+    Uω (aρ 16) (bρ 16) (318180245763 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_51 :
+    Uρ (318180245763 / 16000000000000 : ℝ) ≤
+      (-(7026394710499350942541 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (318180245763 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(21594272656989974022877 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_51_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (318180245763 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(9210627822742137523341 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_51_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (318180245763 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_51_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (318180245763 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_51_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (318180245763 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_51_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (318180245763 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_51_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (318180245763 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_51_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (318180245763 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_51_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (318180245763 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_51_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (318180245763 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_51_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (318180245763 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_51_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (318180245763 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_51_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (318180245763 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_51_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (318180245763 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_51_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (318180245763 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_51_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (318180245763 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_51_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/U04.lean b/LeanPool/Zeta5Irrational/Table/U04.lean
new file mode 100644
index 0000000000..31b2bd5fbc
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/U04.lean
@@ -0,0 +1,2547 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-! # Certified arcsine potential bounds (U04) -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma U_52_1 :
+    Uω (aρ 1) (bρ 1) (41071178579 / 2000000000000 : ℝ) ≤
+      (-(10677082056811715580897 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2235222980587 / 160000000000000 : ℝ) ≤
+      (-(10677082056811715580897 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show (2235222980587 / 160000000000000 : ℝ) = (2235222980587 / 1250000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2235222980587 / 1250000000000 : ℝ) ≤
+            (1452993598438284419103 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1108353582107 / 80000000000000 : ℝ)) (q :=
+      (2235222980587 / 160000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_52_2 :
+    Uω (aρ 2) (bρ 2) (41071178579 / 2000000000000 : ℝ) ≤
+      (-(45357165225401575594487 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1674878732681 / 156250000000000 : ℝ) ≤
+      (-(45357165225401575594487 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show (1674878732681 / 156250000000000 : ℝ) = (1674878732681 / 1220703125000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1674878732681 / 1220703125000 : ℝ) ≤
+            (3163137395598424405513 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3040462983849 / 312500000000000 : ℝ)) (q :=
+      (1674878732681 / 156250000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_52_3 :
+    Uω (aρ 3) (bρ 3) (41071178579 / 2000000000000 : ℝ) ≤
+      (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_3]) (by norm_num [bρ_3])]; exact Lmid_3
+
+lemma U_52_4 :
+    Uω (aρ 4) (bρ 4) (41071178579 / 2000000000000 : ℝ) ≤
+      (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_4]) (by norm_num [bρ_4])]; exact Lmid_4
+
+lemma U_52_5 :
+    Uω (aρ 5) (bρ 5) (41071178579 / 2000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_52_6 :
+    Uω (aρ 6) (bρ 6) (41071178579 / 2000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_52_7 :
+    Uω (aρ 7) (bρ 7) (41071178579 / 2000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_52_8 :
+    Uω (aρ 8) (bρ 8) (41071178579 / 2000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_52_9 :
+    Uω (aρ 9) (bρ 9) (41071178579 / 2000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_52_10 :
+    Uω (aρ 10) (bρ 10) (41071178579 / 2000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_52_11 :
+    Uω (aρ 11) (bρ 11) (41071178579 / 2000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_52_12 :
+    Uω (aρ 12) (bρ 12) (41071178579 / 2000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_52_13 :
+    Uω (aρ 13) (bρ 13) (41071178579 / 2000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_52_14 :
+    Uω (aρ 14) (bρ 14) (41071178579 / 2000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_52_15 :
+    Uω (aρ 15) (bρ 15) (41071178579 / 2000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_52_16 :
+    Uω (aρ 16) (bρ 16) (41071178579 / 2000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_52 :
+    Uρ (41071178579 / 2000000000000 : ℝ) ≤
+      (-(28080017513087561787613 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (41071178579 / 2000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(10677082056811715580897 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_52_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (41071178579 / 2000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(45357165225401575594487 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_52_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (41071178579 / 2000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_52_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (41071178579 / 2000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_52_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (41071178579 / 2000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_52_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (41071178579 / 2000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_52_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (41071178579 / 2000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_52_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (41071178579 / 2000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_52_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (41071178579 / 2000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_52_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (41071178579 / 2000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_52_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (41071178579 / 2000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_52_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (41071178579 / 2000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_52_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (41071178579 / 2000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_52_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (41071178579 / 2000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_52_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (41071178579 / 2000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_52_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (41071178579 / 2000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_52_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_53_1 :
+    Uω (aρ 1) (bρ 1) (34934779437 / 1600000000000 : ℝ) ≤
+      (-(8362590788884110065771 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (152787027675153 / 10000000000000000 : ℝ) ≤
+      (-(8362590788884110065771 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (152787027675153 / 10000000000000000 : ℝ) =
+            (152787027675153 / 78125000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (152787027675153 / 78125000000000 : ℝ) ≤
+            (1341469735315889934229 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (18966112744257 / 1250000000000000 : ℝ)) (q :=
+      (152787027675153 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_53_2 :
+    Uω (aρ 2) (bρ 2) (34934779437 / 1600000000000 : ℝ) ≤
+      (-(1764723410583048654431 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (121332152126361 / 10000000000000000 : ℝ) ≤
+      (-(1764723410583048654431 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (121332152126361 / 10000000000000000 : ℝ) =
+            (121332152126361 / 78125000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (121332152126361 / 78125000000000 : ℝ) ≤
+            (176088694256951345569 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (14073520420809 / 1250000000000000 : ℝ)) (q :=
+      (121332152126361 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_53_3 :
+    Uω (aρ 3) (bρ 3) (34934779437 / 1600000000000 : ℝ) ≤
+      (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_3]) (by norm_num [bρ_3])]; exact Lmid_3
+
+lemma U_53_4 :
+    Uω (aρ 4) (bρ 4) (34934779437 / 1600000000000 : ℝ) ≤
+      (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_4]) (by norm_num [bρ_4])]; exact Lmid_4
+
+lemma U_53_5 :
+    Uω (aρ 5) (bρ 5) (34934779437 / 1600000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_53_6 :
+    Uω (aρ 6) (bρ 6) (34934779437 / 1600000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_53_7 :
+    Uω (aρ 7) (bρ 7) (34934779437 / 1600000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_53_8 :
+    Uω (aρ 8) (bρ 8) (34934779437 / 1600000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_53_9 :
+    Uω (aρ 9) (bρ 9) (34934779437 / 1600000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_53_10 :
+    Uω (aρ 10) (bρ 10) (34934779437 / 1600000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_53_11 :
+    Uω (aρ 11) (bρ 11) (34934779437 / 1600000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_53_12 :
+    Uω (aρ 12) (bρ 12) (34934779437 / 1600000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_53_13 :
+    Uω (aρ 13) (bρ 13) (34934779437 / 1600000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_53_14 :
+    Uω (aρ 14) (bρ 14) (34934779437 / 1600000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_53_15 :
+    Uω (aρ 15) (bρ 15) (34934779437 / 1600000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_53_16 :
+    Uω (aρ 16) (bρ 16) (34934779437 / 1600000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_53 :
+    Uρ (34934779437 / 1600000000000 : ℝ) ≤
+      (-(28034084278989397482397 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (34934779437 / 1600000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(8362590788884110065771 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_53_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (34934779437 / 1600000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1764723410583048654431 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_53_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (34934779437 / 1600000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_53_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (34934779437 / 1600000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_53_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (34934779437 / 1600000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_53_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (34934779437 / 1600000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_53_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (34934779437 / 1600000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_53_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (34934779437 / 1600000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_53_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (34934779437 / 1600000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_53_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (34934779437 / 1600000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_53_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (34934779437 / 1600000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_53_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (34934779437 / 1600000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_53_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (34934779437 / 1600000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_53_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (34934779437 / 1600000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_53_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (34934779437 / 1600000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_53_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (34934779437 / 1600000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_53_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_54_1 :
+    Uω (aρ 1) (bρ 1) (92531540027 / 4000000000000 : ℝ) ≤
+      (-(20496075350021699854459 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8292844436299 / 500000000000000 : ℝ) ≤
+      (-(20496075350021699854459 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show (8292844436299 / 500000000000000 : ℝ) = (8292844436299 / 7812500000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8292844436299 / 7812500000000 : ℝ) ≤
+            (298340058978300145541 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (8244107273473 / 500000000000000 : ℝ)) (q :=
+      (8292844436299 / 500000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_54_2 :
+    Uω (aρ 2) (bρ 2) (92531540027 / 4000000000000 : ℝ) ≤
+      (-(21517316490172672414011 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (135216483035671 / 10000000000000000 : ℝ) ≤
+      (-(21517316490172672414011 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (135216483035671 / 10000000000000000 : ℝ) =
+            (135216483035671 / 78125000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (135216483035671 / 78125000000000 : ℝ) ≤
+            (2742834820327327585989 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (63685173299421 / 5000000000000000 : ℝ)) (q :=
+      (135216483035671 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_54_3 :
+    Uω (aρ 3) (bρ 3) (92531540027 / 4000000000000 : ℝ) ≤
+      (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_3]) (by norm_num [bρ_3])]; exact Lmid_3
+
+lemma U_54_4 :
+    Uω (aρ 4) (bρ 4) (92531540027 / 4000000000000 : ℝ) ≤
+      (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_4]) (by norm_num [bρ_4])]; exact Lmid_4
+
+lemma U_54_5 :
+    Uω (aρ 5) (bρ 5) (92531540027 / 4000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_54_6 :
+    Uω (aρ 6) (bρ 6) (92531540027 / 4000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_54_7 :
+    Uω (aρ 7) (bρ 7) (92531540027 / 4000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_54_8 :
+    Uω (aρ 8) (bρ 8) (92531540027 / 4000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_54_9 :
+    Uω (aρ 9) (bρ 9) (92531540027 / 4000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_54_10 :
+    Uω (aρ 10) (bρ 10) (92531540027 / 4000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_54_11 :
+    Uω (aρ 11) (bρ 11) (92531540027 / 4000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_54_12 :
+    Uω (aρ 12) (bρ 12) (92531540027 / 4000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_54_13 :
+    Uω (aρ 13) (bρ 13) (92531540027 / 4000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_54_14 :
+    Uω (aρ 14) (bρ 14) (92531540027 / 4000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_54_15 :
+    Uω (aρ 15) (bρ 15) (92531540027 / 4000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_54_16 :
+    Uω (aρ 16) (bρ 16) (92531540027 / 4000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_54 :
+    Uρ (92531540027 / 4000000000000 : ℝ) ≤
+      (-(27993521821896214356369 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (92531540027 / 4000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(20496075350021699854459 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_54_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (92531540027 / 4000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(21517316490172672414011 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_54_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (92531540027 / 4000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_54_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (92531540027 / 4000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_54_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (92531540027 / 4000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_54_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (92531540027 / 4000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_54_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (92531540027 / 4000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_54_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (92531540027 / 4000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_54_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (92531540027 / 4000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_54_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (92531540027 / 4000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_54_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (92531540027 / 4000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_54_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (92531540027 / 4000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_54_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (92531540027 / 4000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_54_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (92531540027 / 4000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_54_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (92531540027 / 4000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_54_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (92531540027 / 4000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_54_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_55_1 :
+    Uω (aρ 1) (bρ 1) (6432545181 / 250000000000 : ℝ) ≤
+      (-(19765204149885397770799 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (76784961606559 / 4000000000000000 : ℝ) ≤
+      (-(19765204149885397770799 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (76784961606559 / 4000000000000000 : ℝ) = (76784961606559 / 62500000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (76784961606559 / 62500000000000 : ℝ) ≤
+            (1029211259114602229201 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (38224043775559 / 2000000000000000 : ℝ)) (q :=
+      (76784961606559 / 4000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_55_2 :
+    Uω (aρ 2) (bρ 2) (6432545181 / 250000000000 : ℝ) ≤
+      (-(20598094011261554616851 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (325014159386617 / 20000000000000000 : ℝ) ≤
+      (-(20598094011261554616851 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (325014159386617 / 20000000000000000 : ℝ) =
+            (325014159386617 / 312500000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (325014159386617 / 312500000000000 : ℝ) ≤
+            (196321397738445383149 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (155978582741617 / 10000000000000000 : ℝ)) (q :=
+      (325014159386617 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_55_3 :
+    Uω (aρ 3) (bρ 3) (6432545181 / 250000000000 : ℝ) ≤
+      (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_3]) (by norm_num [bρ_3])]; exact Lmid_3
+
+lemma U_55_4 :
+    Uω (aρ 4) (bρ 4) (6432545181 / 250000000000 : ℝ) ≤
+      (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_4]) (by norm_num [bρ_4])]; exact Lmid_4
+
+lemma U_55_5 :
+    Uω (aρ 5) (bρ 5) (6432545181 / 250000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_55_6 :
+    Uω (aρ 6) (bρ 6) (6432545181 / 250000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_55_7 :
+    Uω (aρ 7) (bρ 7) (6432545181 / 250000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_55_8 :
+    Uω (aρ 8) (bρ 8) (6432545181 / 250000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_55_9 :
+    Uω (aρ 9) (bρ 9) (6432545181 / 250000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_55_10 :
+    Uω (aρ 10) (bρ 10) (6432545181 / 250000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_55_11 :
+    Uω (aρ 11) (bρ 11) (6432545181 / 250000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_55_12 :
+    Uω (aρ 12) (bρ 12) (6432545181 / 250000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_55_13 :
+    Uω (aρ 13) (bρ 13) (6432545181 / 250000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_55_14 :
+    Uω (aρ 14) (bρ 14) (6432545181 / 250000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_55_15 :
+    Uω (aρ 15) (bρ 15) (6432545181 / 250000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_55_16 :
+    Uω (aρ 16) (bρ 16) (6432545181 / 250000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_55 :
+    Uρ (6432545181 / 250000000000 : ℝ) ≤
+      (-(3490496070168995482161 / 1250000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (6432545181 / 250000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(19765204149885397770799 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_55_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (6432545181 / 250000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(20598094011261554616851 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_55_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (6432545181 / 250000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(25512130211755314637529 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_55_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (6432545181 / 250000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_55_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (6432545181 / 250000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_55_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (6432545181 / 250000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_55_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (6432545181 / 250000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_55_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (6432545181 / 250000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_55_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (6432545181 / 250000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_55_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (6432545181 / 250000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_55_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (6432545181 / 250000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_55_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (6432545181 / 250000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_55_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (6432545181 / 250000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_55_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (6432545181 / 250000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_55_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (6432545181 / 250000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_55_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (6432545181 / 250000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_55_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_56_1 :
+    Uω (aρ 1) (bρ 1) (213931144553 / 8000000000000 : ℝ) ≤
+      (-(19507472141650856272829 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (101058419601983 / 5000000000000000 : ℝ) ≤
+      (-(19507472141650856272829 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (101058419601983 / 5000000000000000 : ℝ) =
+            (101058419601983 / 78125000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (101058419601983 / 78125000000000 : ℝ) ≤
+            (1286943267349143727171 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (100658482900841 / 5000000000000000 : ℝ)) (q :=
+      (101058419601983 / 5000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_56_2 :
+    Uω (aρ 2) (bρ 2) (213931144553 / 8000000000000 : ℝ) ≤
+      (-(40569578445247696299549 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (43253932961601 / 2500000000000000 : ℝ) ≤
+      (-(40569578445247696299549 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (43253932961601 / 2500000000000000 : ℝ) = (43253932961601 / 39062500000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (43253932961601 / 39062500000000 : ℝ) ≤
+            (1019252372752303700451 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (83441881798279 / 5000000000000000 : ℝ)) (q :=
+      (43253932961601 / 2500000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_56_3 :
+    Uω (aρ 3) (bρ 3) (213931144553 / 8000000000000 : ℝ) ≤
+      (-(46974446160582055541859 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (729483898519 / 80000000000000 : ℝ) ≤
+      (-(46974446160582055541859 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show (729483898519 / 80000000000000 : ℝ) = (729483898519 / 625000000000 : ℝ) / 2 ^ 7 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (729483898519 / 625000000000 : ℝ) ≤
+            (1545856460417944458141 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (101238761767 / 20000000000000 : ℝ)) (q :=
+      (729483898519 / 80000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_56_4 :
+    Uω (aρ 4) (bρ 4) (213931144553 / 8000000000000 : ℝ) ≤
+      (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_4]) (by norm_num [bρ_4])]; exact Lmid_4
+
+lemma U_56_5 :
+    Uω (aρ 5) (bρ 5) (213931144553 / 8000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_56_6 :
+    Uω (aρ 6) (bρ 6) (213931144553 / 8000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_56_7 :
+    Uω (aρ 7) (bρ 7) (213931144553 / 8000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_56_8 :
+    Uω (aρ 8) (bρ 8) (213931144553 / 8000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_56_9 :
+    Uω (aρ 9) (bρ 9) (213931144553 / 8000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_56_10 :
+    Uω (aρ 10) (bρ 10) (213931144553 / 8000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_56_11 :
+    Uω (aρ 11) (bρ 11) (213931144553 / 8000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_56_12 :
+    Uω (aρ 12) (bρ 12) (213931144553 / 8000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_56_13 :
+    Uω (aρ 13) (bρ 13) (213931144553 / 8000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_56_14 :
+    Uω (aρ 14) (bρ 14) (213931144553 / 8000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_56_15 :
+    Uω (aρ 15) (bρ 15) (213931144553 / 8000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_56_16 :
+    Uω (aρ 16) (bρ 16) (213931144553 / 8000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_56 :
+    Uρ (213931144553 / 8000000000000 : ℝ) ≤
+      (-(13863102336300848802043 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (213931144553 / 8000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(19507472141650856272829 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_56_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (213931144553 / 8000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(40569578445247696299549 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_56_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (213931144553 / 8000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(46974446160582055541859 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_56_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (213931144553 / 8000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_56_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (213931144553 / 8000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_56_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (213931144553 / 8000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_56_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (213931144553 / 8000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_56_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (213931144553 / 8000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_56_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (213931144553 / 8000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_56_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (213931144553 / 8000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_56_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (213931144553 / 8000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_56_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (213931144553 / 8000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_56_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (213931144553 / 8000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_56_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (213931144553 / 8000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_56_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (213931144553 / 8000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_56_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (213931144553 / 8000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_56_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_57_1 :
+    Uω (aρ 1) (bρ 1) (435951987867 / 16000000000000 : ℝ) ≤
+      (-(38766920785090621061557 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (103596247833641 / 5000000000000000 : ℝ) ≤
+      (-(38766920785090621061557 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (103596247833641 / 5000000000000000 : ℝ) =
+            (103596247833641 / 78125000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (103596247833641 / 78125000000000 : ℝ) ≤
+            (2821910032909378938443 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (206412217002689 / 10000000000000000 : ℝ)) (q :=
+      (103596247833641 / 5000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_57_2 :
+    Uω (aρ 2) (bρ 2) (435951987867 / 16000000000000 : ℝ) ≤
+      (-(8054285062790282216923 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11140745093543 / 625000000000000 : ℝ) ≤
+      (-(8054285062790282216923 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (11140745093543 / 625000000000000 : ℝ) = (11140745093543 / 9765625000000 : ℝ) / 2 ^ 6 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11140745093543 / 9765625000000 : ℝ) ≤
+            (263481100809717783077 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (172300081171501 / 10000000000000000 : ℝ)) (q :=
+      (11140745093543 / 625000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_57_3 :
+    Uω (aρ 3) (bρ 3) (435951987867 / 16000000000000 : ℝ) ≤
+      (-(46080796762313820607457 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (99709473640723 / 10000000000000000 : ℝ) ≤
+      (-(46080796762313820607457 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (99709473640723 / 10000000000000000 : ℝ) = (99709473640723 / 78125000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (99709473640723 / 78125000000000 : ℝ) ≤
+            (2439505858686179392543 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (62611291809571 / 10000000000000000 : ℝ)) (q :=
+      (99709473640723 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_57_4 :
+    Uω (aρ 4) (bρ 4) (435951987867 / 16000000000000 : ℝ) ≤
+      (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_4]) (by norm_num [bρ_4])]; exact Lmid_4
+
+lemma U_57_5 :
+    Uω (aρ 5) (bρ 5) (435951987867 / 16000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_57_6 :
+    Uω (aρ 6) (bρ 6) (435951987867 / 16000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_57_7 :
+    Uω (aρ 7) (bρ 7) (435951987867 / 16000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_57_8 :
+    Uω (aρ 8) (bρ 8) (435951987867 / 16000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_57_9 :
+    Uω (aρ 9) (bρ 9) (435951987867 / 16000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_57_10 :
+    Uω (aρ 10) (bρ 10) (435951987867 / 16000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_57_11 :
+    Uω (aρ 11) (bρ 11) (435951987867 / 16000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_57_12 :
+    Uω (aρ 12) (bρ 12) (435951987867 / 16000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_57_13 :
+    Uω (aρ 13) (bρ 13) (435951987867 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_57_14 :
+    Uω (aρ 14) (bρ 14) (435951987867 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_57_15 :
+    Uω (aρ 15) (bρ 15) (435951987867 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_57_16 :
+    Uω (aρ 16) (bρ 16) (435951987867 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_57 :
+    Uρ (435951987867 / 16000000000000 : ℝ) ≤
+      (-(5535288239441576547317 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (435951987867 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(38766920785090621061557 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_57_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (435951987867 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(8054285062790282216923 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_57_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (435951987867 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(46080796762313820607457 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_57_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (435951987867 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_57_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (435951987867 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_57_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (435951987867 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_57_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (435951987867 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_57_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (435951987867 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_57_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (435951987867 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_57_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (435951987867 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_57_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (435951987867 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_57_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (435951987867 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_57_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (435951987867 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_57_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (435951987867 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_57_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (435951987867 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_57_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (435951987867 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_57_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_58_1 :
+    Uω (aρ 1) (bρ 1) (111010421657 / 4000000000000 : ℝ) ≤
+      (-(38524944569892604705309 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21226721167587 / 1000000000000000 : ℝ) ≤
+      (-(38524944569892604705309 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (21226721167587 / 1000000000000000 : ℝ) = (21226721167587 / 15625000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21226721167587 / 15625000000000 : ℝ) ≤
+            (3063886248107395294691 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1321909920589 / 62500000000000 : ℝ)) (q :=
+      (21226721167587 / 1000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_58_2 :
+    Uω (aρ 2) (bρ 2) (111010421657 / 4000000000000 : ℝ) ≤
+      (-(19991241739447185349551 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (91738748014241 / 5000000000000000 : ℝ) ≤
+      (-(19991241739447185349551 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (91738748014241 / 5000000000000000 : ℝ) = (91738748014241 / 78125000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (91738748014241 / 78125000000000 : ℝ) ≤
+            (803173669552814650449 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (22211896063683 / 1250000000000000 : ℝ)) (q :=
+      (91738748014241 / 5000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_58_3 :
+    Uω (aρ 3) (bρ 3) (111010421657 / 4000000000000 : ℝ) ≤
+      (-(45334787031774164684501 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (214864768720367 / 20000000000000000 : ℝ) ≤
+      (-(45334787031774164684501 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (214864768720367 / 20000000000000000 : ℝ) =
+            (214864768720367 / 156250000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (214864768720367 / 156250000000000 : ℝ) ≤
+            (3185515589225835315499 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (73001051522867 / 10000000000000000 : ℝ)) (q :=
+      (214864768720367 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_58_4 :
+    Uω (aρ 4) (bρ 4) (111010421657 / 4000000000000 : ℝ) ≤
+      (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_4]) (by norm_num [bρ_4])]; exact Lmid_4
+
+lemma U_58_5 :
+    Uω (aρ 5) (bρ 5) (111010421657 / 4000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_58_6 :
+    Uω (aρ 6) (bρ 6) (111010421657 / 4000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_58_7 :
+    Uω (aρ 7) (bρ 7) (111010421657 / 4000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_58_8 :
+    Uω (aρ 8) (bρ 8) (111010421657 / 4000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_58_9 :
+    Uω (aρ 9) (bρ 9) (111010421657 / 4000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_58_10 :
+    Uω (aρ 10) (bρ 10) (111010421657 / 4000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_58_11 :
+    Uω (aρ 11) (bρ 11) (111010421657 / 4000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_58_12 :
+    Uω (aρ 12) (bρ 12) (111010421657 / 4000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_58_13 :
+    Uω (aρ 13) (bρ 13) (111010421657 / 4000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_58_14 :
+    Uω (aρ 14) (bρ 14) (111010421657 / 4000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_58_15 :
+    Uω (aρ 15) (bρ 15) (111010421657 / 4000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_58_16 :
+    Uω (aρ 16) (bρ 16) (111010421657 / 4000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_58 :
+    Uρ (111010421657 / 4000000000000 : ℝ) ≤
+      (-(27633351600905070070711 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (111010421657 / 4000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(38524944569892604705309 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_58_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (111010421657 / 4000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(19991241739447185349551 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_58_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (111010421657 / 4000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(45334787031774164684501 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_58_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (111010421657 / 4000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_58_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (111010421657 / 4000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_58_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (111010421657 / 4000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_58_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (111010421657 / 4000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_58_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (111010421657 / 4000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_58_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (111010421657 / 4000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_58_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (111010421657 / 4000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_58_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (111010421657 / 4000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_58_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (111010421657 / 4000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_58_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (111010421657 / 4000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_58_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (111010421657 / 4000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_58_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (111010421657 / 4000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_58_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (111010421657 / 4000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_58_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_59_1 :
+    Uω (aρ 1) (bρ 1) (452131385389 / 16000000000000 : ℝ) ≤
+      (-(19144362876016614852763 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (217341053575261 / 10000000000000000 : ℝ) ≤
+      (-(19144362876016614852763 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (217341053575261 / 10000000000000000 : ℝ) =
+            (217341053575261 / 156250000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (217341053575261 / 156250000000000 : ℝ) ≤
+            (1650052532983385147237 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (216597209367397 / 10000000000000000 : ℝ)) (q :=
+      (217341053575261 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_59_2 :
+    Uω (aρ 2) (bρ 2) (452131385389 / 16000000000000 : ℝ) ≤
+      (-(7940433853980462156227 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (377386787854213 / 20000000000000000 : ℝ) ≤
+      (-(7940433853980462156227 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (377386787854213 / 20000000000000000 : ℝ) =
+            (377386787854213 / 312500000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (377386787854213 / 312500000000000 : ℝ) ≤
+            (377332309619537843773 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5720965705659 / 312500000000000 : ℝ)) (q :=
+      (377386787854213 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_59_3 :
+    Uω (aρ 3) (bρ 3) (452131385389 / 16000000000000 : ℝ) ≤
+      (-(44683833187501300822249 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (114658374582899 / 10000000000000000 : ℝ) ≤
+      (-(44683833187501300822249 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (114658374582899 / 10000000000000000 : ℝ) =
+            (114658374582899 / 78125000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (114658374582899 / 78125000000000 : ℝ) ≤
+            (3836469433498699177751 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (82396970242673 / 10000000000000000 : ℝ)) (q :=
+      (114658374582899 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_59_4 :
+    Uω (aρ 4) (bρ 4) (452131385389 / 16000000000000 : ℝ) ≤
+      (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_4]) (by norm_num [bρ_4])]; exact Lmid_4
+
+lemma U_59_5 :
+    Uω (aρ 5) (bρ 5) (452131385389 / 16000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_59_6 :
+    Uω (aρ 6) (bρ 6) (452131385389 / 16000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_59_7 :
+    Uω (aρ 7) (bρ 7) (452131385389 / 16000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_59_8 :
+    Uω (aρ 8) (bρ 8) (452131385389 / 16000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_59_9 :
+    Uω (aρ 9) (bρ 9) (452131385389 / 16000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_59_10 :
+    Uω (aρ 10) (bρ 10) (452131385389 / 16000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_59_11 :
+    Uω (aρ 11) (bρ 11) (452131385389 / 16000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_59_12 :
+    Uω (aρ 12) (bρ 12) (452131385389 / 16000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_59_13 :
+    Uω (aρ 13) (bρ 13) (452131385389 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_59_14 :
+    Uω (aρ 14) (bρ 14) (452131385389 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_59_15 :
+    Uω (aρ 15) (bρ 15) (452131385389 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_59_16 :
+    Uω (aρ 16) (bρ 16) (452131385389 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_59 :
+    Uρ (452131385389 / 16000000000000 : ℝ) ≤
+      (-(3449332247790833623903 / 1250000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (452131385389 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(19144362876016614852763 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_59_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (452131385389 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(7940433853980462156227 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_59_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (452131385389 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(44683833187501300822249 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_59_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (452131385389 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_59_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (452131385389 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_59_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (452131385389 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_59_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (452131385389 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_59_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (452131385389 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_59_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (452131385389 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_59_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (452131385389 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_59_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (452131385389 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_59_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (452131385389 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_59_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (452131385389 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_59_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (452131385389 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_59_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (452131385389 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_59_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (452131385389 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_59_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_60_1 :
+    Uω (aρ 1) (bρ 1) (912352469539 / 32000000000000 : ℝ) ≤
+      (-(19086345216926816612793 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (879510663099267 / 40000000000000000 : ℝ) ≤
+      (-(19086345216926816612793 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (879510663099267 / 40000000000000000 : ℝ) =
+            (879510663099267 / 625000000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (879510663099267 / 625000000000000 : ℝ) ≤
+            (1708070192073183387207 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (13696400181481 / 625000000000000 : ℝ)) (q :=
+      (879510663099267 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_60_2 :
+    Uω (aρ 2) (bρ 2) (912352469539 / 32000000000000 : ℝ) ≤
+      (-(791301614016874710873 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (765191907200309 / 40000000000000000 : ℝ) ≤
+      (-(791301614016874710873 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (765191907200309 / 40000000000000000 : ℝ) =
+            (765191907200309 / 625000000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (765191907200309 / 625000000000000 : ℝ) ≤
+            (40475002343125289127 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (185752037464217 / 10000000000000000 : ℝ)) (q :=
+      (765191907200309 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_60_3 :
+    Uω (aρ 3) (bρ 3) (912352469539 / 32000000000000 : ℝ) ≤
+      (-(22192513593791331396327 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (472544542588009 / 40000000000000000 : ℝ) ≤
+      (-(22192513593791331396327 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (472544542588009 / 40000000000000000 : ℝ) =
+            (472544542588009 / 312500000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (472544542588009 / 312500000000000 : ℝ) ≤
+            (2067637716708668603673 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (86824461508067 / 10000000000000000 : ℝ)) (q :=
+      (472544542588009 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_60_4 :
+    Uω (aρ 4) (bρ 4) (912352469539 / 32000000000000 : ℝ) ≤
+      (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_4]) (by norm_num [bρ_4])]; exact Lmid_4
+
+lemma U_60_5 :
+    Uω (aρ 5) (bρ 5) (912352469539 / 32000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_60_6 :
+    Uω (aρ 6) (bρ 6) (912352469539 / 32000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_60_7 :
+    Uω (aρ 7) (bρ 7) (912352469539 / 32000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_60_8 :
+    Uω (aρ 8) (bρ 8) (912352469539 / 32000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_60_9 :
+    Uω (aρ 9) (bρ 9) (912352469539 / 32000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_60_10 :
+    Uω (aρ 10) (bρ 10) (912352469539 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_60_11 :
+    Uω (aρ 11) (bρ 11) (912352469539 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_60_12 :
+    Uω (aρ 12) (bρ 12) (912352469539 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_60_13 :
+    Uω (aρ 13) (bρ 13) (912352469539 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_60_14 :
+    Uω (aρ 14) (bρ 14) (912352469539 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_60_15 :
+    Uω (aρ 15) (bρ 15) (912352469539 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_60_16 :
+    Uω (aρ 16) (bρ 16) (912352469539 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_60 :
+    Uρ (912352469539 / 32000000000000 : ℝ) ≤
+      (-(27576568517522084070211 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (912352469539 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(19086345216926816612793 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_60_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (912352469539 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(791301614016874710873 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_60_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (912352469539 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(22192513593791331396327 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_60_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (912352469539 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_60_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (912352469539 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_60_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (912352469539 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_60_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (912352469539 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_60_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (912352469539 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_60_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (912352469539 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_60_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (912352469539 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_60_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (912352469539 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_60_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (912352469539 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_60_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (912352469539 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_60_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (912352469539 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_60_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (912352469539 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_60_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (912352469539 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_60_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_61_1 :
+    Uω (aρ 1) (bρ 1) (9204421683 / 320000000000 : ℝ) ≤
+      (-(19028997467622808750993 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13900880099627 / 625000000000000 : ℝ) ≤
+      (-(19028997467622808750993 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (13900880099627 / 625000000000000 : ℝ) = (13900880099627 / 9765625000000 : ℝ) / 2 ^ 6 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13900880099627 / 9765625000000 : ℝ) ≤
+            (1765417941377191249007 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (110843601839657 / 5000000000000000 : ℝ)) (q :=
+      (13900880099627 / 625000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_61_2 :
+    Uω (aρ 2) (bρ 2) (9204421683 / 320000000000 : ℝ) ≤
+      (-(19714977607198277378769 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (387800886794229 / 20000000000000000 : ℝ) ≤
+      (-(19714977607198277378769 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (387800886794229 / 20000000000000000 : ℝ) =
+            (387800886794229 / 312500000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (387800886794229 / 312500000000000 : ℝ) ≤
+            (1079437801801722621231 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (188428939795479 / 10000000000000000 : ℝ)) (q :=
+      (387800886794229 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_61_3 :
+    Uω (aρ 3) (bρ 3) (9204421683 / 320000000000 : ℝ) ≤
+      (-(22050425163131483974353 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15192680956257 / 1250000000000000 : ℝ) ≤
+      (-(22050425163131483974353 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (15192680956257 / 1250000000000000 : ℝ) = (15192680956257 / 9765625000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15192680956257 / 9765625000000 : ℝ) ≤
+            (2209726147368516025647 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (45553527325681 / 5000000000000000 : ℝ)) (q :=
+      (15192680956257 / 1250000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_61_4 :
+    Uω (aρ 4) (bρ 4) (9204421683 / 320000000000 : ℝ) ≤
+      (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_4]) (by norm_num [bρ_4])]; exact Lmid_4
+
+lemma U_61_5 :
+    Uω (aρ 5) (bρ 5) (9204421683 / 320000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_61_6 :
+    Uω (aρ 6) (bρ 6) (9204421683 / 320000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_61_7 :
+    Uω (aρ 7) (bρ 7) (9204421683 / 320000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_61_8 :
+    Uω (aρ 8) (bρ 8) (9204421683 / 320000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_61_9 :
+    Uω (aρ 9) (bρ 9) (9204421683 / 320000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_61_10 :
+    Uω (aρ 10) (bρ 10) (9204421683 / 320000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_61_11 :
+    Uω (aρ 11) (bρ 11) (9204421683 / 320000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_61_12 :
+    Uω (aρ 12) (bρ 12) (9204421683 / 320000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_61_13 :
+    Uω (aρ 13) (bρ 13) (9204421683 / 320000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_61_14 :
+    Uω (aρ 14) (bρ 14) (9204421683 / 320000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_61_15 :
+    Uω (aρ 15) (bρ 15) (9204421683 / 320000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_61_16 :
+    Uω (aρ 16) (bρ 16) (9204421683 / 320000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_61 :
+    Uρ (9204421683 / 320000000000 : ℝ) ≤
+      (-(27559179089952747868089 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (9204421683 / 320000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(19028997467622808750993 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_61_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (9204421683 / 320000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(19714977607198277378769 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_61_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (9204421683 / 320000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(22050425163131483974353 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_61_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (9204421683 / 320000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_61_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (9204421683 / 320000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_61_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (9204421683 / 320000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_61_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (9204421683 / 320000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_61_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (9204421683 / 320000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_61_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (9204421683 / 320000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_61_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (9204421683 / 320000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_61_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (9204421683 / 320000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_61_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (9204421683 / 320000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_61_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (9204421683 / 320000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_61_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (9204421683 / 320000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_61_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (9204421683 / 320000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_61_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (9204421683 / 320000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_61_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_62_1 :
+    Uω (aρ 1) (bρ 1) (928531867061 / 32000000000000 : ℝ) ≤
+      (-(7588921694372028541901 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (899801230876237 / 40000000000000000 : ℝ) ≤
+      (-(7588921694372028541901 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (899801230876237 / 40000000000000000 : ℝ) =
+            (899801230876237 / 625000000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (899801230876237 / 625000000000000 : ℝ) ≤
+            (728844469227971458099 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (56057906266639 / 2500000000000000 : ℝ)) (q :=
+      (899801230876237 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_62_2 :
+    Uω (aρ 2) (bρ 2) (928531867061 / 32000000000000 : ℝ) ≤
+      (-(39296734472550530549671 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (786003530603713 / 40000000000000000 : ℝ) ≤
+      (-(39296734472550530549671 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (786003530603713 / 40000000000000000 : ℝ) =
+            (786003530603713 / 625000000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (786003530603713 / 625000000000000 : ℝ) ≤
+            (2292096345449469450329 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (95550893720147 / 5000000000000000 : ℝ)) (q :=
+      (786003530603713 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_62_3 :
+    Uω (aρ 3) (bρ 3) (928531867061 / 32000000000000 : ℝ) ≤
+      (-(2739346502402930483743 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (499536329628007 / 40000000000000000 : ℝ) ≤
+      (-(2739346502402930483743 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (499536329628007 / 40000000000000000 : ℝ) =
+            (499536329628007 / 312500000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (499536329628007 / 312500000000000 : ℝ) ≤
+            (293172411409569516257 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (95264293302441 / 10000000000000000 : ℝ)) (q :=
+      (499536329628007 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_62_4 :
+    Uω (aρ 4) (bρ 4) (928531867061 / 32000000000000 : ℝ) ≤
+      (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_4]) (by norm_num [bρ_4])]; exact Lmid_4
+
+lemma U_62_5 :
+    Uω (aρ 5) (bρ 5) (928531867061 / 32000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_62_6 :
+    Uω (aρ 6) (bρ 6) (928531867061 / 32000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_62_7 :
+    Uω (aρ 7) (bρ 7) (928531867061 / 32000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_62_8 :
+    Uω (aρ 8) (bρ 8) (928531867061 / 32000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_62_9 :
+    Uω (aρ 9) (bρ 9) (928531867061 / 32000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_62_10 :
+    Uω (aρ 10) (bρ 10) (928531867061 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_62_11 :
+    Uω (aρ 11) (bρ 11) (928531867061 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_62_12 :
+    Uω (aρ 12) (bρ 12) (928531867061 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_62_13 :
+    Uω (aρ 13) (bρ 13) (928531867061 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_62_14 :
+    Uω (aρ 14) (bρ 14) (928531867061 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_62_15 :
+    Uω (aρ 15) (bρ 15) (928531867061 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_62_16 :
+    Uω (aρ 16) (bρ 16) (928531867061 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_62 :
+    Uρ (928531867061 / 32000000000000 : ℝ) ≤
+      (-(6885603038426120732053 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (928531867061 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(7588921694372028541901 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_62_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (928531867061 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(39296734472550530549671 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_62_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (928531867061 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(2739346502402930483743 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_62_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (928531867061 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_62_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (928531867061 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_62_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (928531867061 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_62_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (928531867061 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_62_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (928531867061 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_62_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (928531867061 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_62_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (928531867061 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_62_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (928531867061 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_62_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (928531867061 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_62_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (928531867061 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_62_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (928531867061 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_62_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (928531867061 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_62_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (928531867061 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_62_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_63_1 :
+    Uω (aρ 1) (bρ 1) (468310782911 / 16000000000000 : ℝ) ≤
+      (-(4729062664342235400103 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (45497270107 / 2000000000000 : ℝ) ≤
+      (-(4729062664342235400103 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show (45497270107 / 2000000000000 : ℝ) = (45497270107 / 31250000000 : ℝ) / 2 ^ 6 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (45497270107 / 31250000000 : ℝ) ≤
+            (469541187907764599897 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (362841087737 / 16000000000000 : ℝ)) (q :=
+      (45497270107 / 2000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_63_2 :
+    Uω (aρ 2) (bρ 2) (468310782911 / 16000000000000 : ℝ) ≤
+      (-(9791340702794776363367 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (398198756912509 / 20000000000000000 : ℝ) ≤
+      (-(9791340702794776363367 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (398198756912509 / 20000000000000000 : ℝ) =
+            (398198756912509 / 312500000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (398198756912509 / 312500000000000 : ℝ) ≤
+            (605867001705223636633 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (96885374094067 / 5000000000000000 : ℝ)) (q :=
+      (398198756912509 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_63_3 :
+    Uω (aρ 3) (bρ 3) (468310782911 / 16000000000000 : ℝ) ≤
+      (-(43569679477317919277667 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (128171911629181 / 10000000000000000 : ℝ) ≤
+      (-(43569679477317919277667 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (128171911629181 / 10000000000000000 : ℝ) =
+            (128171911629181 / 78125000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (128171911629181 / 78125000000000 : ℝ) ≤
+            (4950623143682080722333 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (99311920883987 / 10000000000000000 : ℝ)) (q :=
+      (128171911629181 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_63_4 :
+    Uω (aρ 4) (bρ 4) (468310782911 / 16000000000000 : ℝ) ≤
+      (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_4]) (by norm_num [bρ_4])]; exact Lmid_4
+
+lemma U_63_5 :
+    Uω (aρ 5) (bρ 5) (468310782911 / 16000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_63_6 :
+    Uω (aρ 6) (bρ 6) (468310782911 / 16000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_63_7 :
+    Uω (aρ 7) (bρ 7) (468310782911 / 16000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_63_8 :
+    Uω (aρ 8) (bρ 8) (468310782911 / 16000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_63_9 :
+    Uω (aρ 9) (bρ 9) (468310782911 / 16000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_63_10 :
+    Uω (aρ 10) (bρ 10) (468310782911 / 16000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_63_11 :
+    Uω (aρ 11) (bρ 11) (468310782911 / 16000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_63_12 :
+    Uω (aρ 12) (bρ 12) (468310782911 / 16000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_63_13 :
+    Uω (aρ 13) (bρ 13) (468310782911 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_63_14 :
+    Uω (aρ 14) (bρ 14) (468310782911 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_63_15 :
+    Uω (aρ 15) (bρ 15) (468310782911 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_63_16 :
+    Uω (aρ 16) (bρ 16) (468310782911 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_63 :
+    Uρ (468310782911 / 16000000000000 : ℝ) ≤
+      (-(27526204409010125473529 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (468310782911 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(4729062664342235400103 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_63_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (468310782911 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(9791340702794776363367 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_63_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (468310782911 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(43569679477317919277667 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_63_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (468310782911 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_63_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (468310782911 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_63_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (468310782911 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_63_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (468310782911 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_63_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (468310782911 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_63_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (468310782911 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_63_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (468310782911 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_63_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (468310782911 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_63_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (468310782911 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_63_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (468310782911 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_63_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (468310782911 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_63_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (468310782911 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_63_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (468310782911 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_63_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/U05.lean b/LeanPool/Zeta5Irrational/Table/U05.lean
new file mode 100644
index 0000000000..563112de70
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/U05.lean
@@ -0,0 +1,2631 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-! # Certified arcsine potential bounds (U05) -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma U_64_1 :
+    Uω (aρ 1) (bρ 1) (944711264583 / 32000000000000 : ℝ) ≤
+      (-(18860822371165874400701 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (920088864570853 / 40000000000000000 : ℝ) ≤
+      (-(18860822371165874400701 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (920088864570853 / 40000000000000000 : ℝ) =
+            (920088864570853 / 625000000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (920088864570853 / 625000000000000 : ℝ) ≤
+            (1933593037834125599299 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (229319380188239 / 10000000000000000 : ℝ)) (q :=
+      (920088864570853 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_64_2 :
+    Uω (aρ 2) (bρ 2) (944711264583 / 32000000000000 : ℝ) ≤
+      (-(19517893536797811818607 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (806784040118343 / 40000000000000000 : ℝ) ≤
+      (-(19517893536797811818607 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (806784040118343 / 40000000000000000 : ℝ) =
+            (806784040118343 / 625000000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (806784040118343 / 625000000000000 : ℝ) ≤
+            (1276521872202188181393 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (12277248779499 / 625000000000000 : ℝ)) (q :=
+      (806784040118343 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_64_3 :
+    Uω (aρ 3) (bρ 3) (944711264583 / 32000000000000 : ℝ) ≤
+      (-(21660038294705195113549 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (525645521645909 / 40000000000000000 : ℝ) ≤
+      (-(21660038294705195113549 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (525645521645909 / 40000000000000000 : ℝ) =
+            (525645521645909 / 312500000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (525645521645909 / 312500000000000 : ℝ) ≤
+            (2600113015794804886451 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (103262827585767 / 10000000000000000 : ℝ)) (q :=
+      (525645521645909 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_64_4 :
+    Uω (aρ 4) (bρ 4) (944711264583 / 32000000000000 : ℝ) ≤
+      (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_4]) (by norm_num [bρ_4])]; exact Lmid_4
+
+lemma U_64_5 :
+    Uω (aρ 5) (bρ 5) (944711264583 / 32000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_64_6 :
+    Uω (aρ 6) (bρ 6) (944711264583 / 32000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_64_7 :
+    Uω (aρ 7) (bρ 7) (944711264583 / 32000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_64_8 :
+    Uω (aρ 8) (bρ 8) (944711264583 / 32000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_64_9 :
+    Uω (aρ 9) (bρ 9) (944711264583 / 32000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_64_10 :
+    Uω (aρ 10) (bρ 10) (944711264583 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_64_11 :
+    Uω (aρ 11) (bρ 11) (944711264583 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_64_12 :
+    Uω (aρ 12) (bρ 12) (944711264583 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_64_13 :
+    Uω (aρ 13) (bρ 13) (944711264583 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_64_14 :
+    Uω (aρ 14) (bρ 14) (944711264583 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_64_15 :
+    Uω (aρ 15) (bρ 15) (944711264583 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_64_16 :
+    Uω (aρ 16) (bρ 16) (944711264583 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_64 :
+    Uρ (944711264583 / 32000000000000 : ℝ) ≤
+      (-(5502100664475844411457 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (944711264583 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(18860822371165874400701 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_64_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (944711264583 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(19517893536797811818607 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_64_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (944711264583 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(21660038294705195113549 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_64_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (944711264583 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_64_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (944711264583 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_64_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (944711264583 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_64_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (944711264583 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_64_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (944711264583 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_64_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (944711264583 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_64_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (944711264583 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_64_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (944711264583 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_64_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (944711264583 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_64_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (944711264583 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_64_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (944711264583 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_64_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (944711264583 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_64_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (944711264583 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_64_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_65_1 :
+    Uω (aρ 1) (bρ 1) (59550060209 / 2000000000000 : ℝ) ≤
+      (-(37612010995240153131751 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (93023164149801 / 4000000000000000 : ℝ) ≤
+      (-(37612010995240153131751 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (93023164149801 / 4000000000000000 : ℝ) = (93023164149801 / 62500000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (93023164149801 / 62500000000000 : ℝ) ≤
+            (3976819822759846868249 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (46372547557801 / 2000000000000000 : ℝ)) (q :=
+      (93023164149801 / 4000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_65_2 :
+    Uω (aρ 2) (bρ 2) (59550060209 / 2000000000000 : ℝ) ≤
+      (-(7781591291438441479693 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (408581704475813 / 20000000000000000 : ℝ) ≤
+      (-(7781591291438441479693 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (408581704475813 / 20000000000000000 : ℝ) =
+            (408581704475813 / 312500000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (408581704475813 / 312500000000000 : ℝ) ≤
+            (536174872161558520307 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (199097634025813 / 10000000000000000 : ℝ)) (q :=
+      (408581704475813 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_65_3 :
+    Uω (aρ 3) (bρ 3) (59550060209 / 2000000000000 : ℝ) ≤
+      (-(43079747104388786598999 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (134607839669749 / 10000000000000000 : ℝ) ≤
+      (-(43079747104388786598999 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (134607839669749 / 10000000000000000 : ℝ) =
+            (134607839669749 / 78125000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (134607839669749 / 78125000000000 : ℝ) ≤
+            (5440555516611213401001 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (53563857619749 / 5000000000000000 : ℝ)) (q :=
+      (134607839669749 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_65_4 :
+    Uω (aρ 4) (bρ 4) (59550060209 / 2000000000000 : ℝ) ≤
+      (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_4]) (by norm_num [bρ_4])]; exact Lmid_4
+
+lemma U_65_5 :
+    Uω (aρ 5) (bρ 5) (59550060209 / 2000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_65_6 :
+    Uω (aρ 6) (bρ 6) (59550060209 / 2000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_65_7 :
+    Uω (aρ 7) (bρ 7) (59550060209 / 2000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_65_8 :
+    Uω (aρ 8) (bρ 8) (59550060209 / 2000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_65_9 :
+    Uω (aρ 9) (bρ 9) (59550060209 / 2000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_65_10 :
+    Uω (aρ 10) (bρ 10) (59550060209 / 2000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_65_11 :
+    Uω (aρ 11) (bρ 11) (59550060209 / 2000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_65_12 :
+    Uω (aρ 12) (bρ 12) (59550060209 / 2000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_65_13 :
+    Uω (aρ 13) (bρ 13) (59550060209 / 2000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_65_14 :
+    Uω (aρ 14) (bρ 14) (59550060209 / 2000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_65_15 :
+    Uω (aρ 15) (bρ 15) (59550060209 / 2000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_65_16 :
+    Uω (aρ 16) (bρ 16) (59550060209 / 2000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_65 :
+    Uρ (59550060209 / 2000000000000 : ℝ) ≤
+      (-(13747632336952287678791 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (59550060209 / 2000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(37612010995240153131751 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_65_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (59550060209 / 2000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(7781591291438441479693 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_65_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (59550060209 / 2000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(43079747104388786598999 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_65_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (59550060209 / 2000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_65_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (59550060209 / 2000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_65_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (59550060209 / 2000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_65_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (59550060209 / 2000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_65_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (59550060209 / 2000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_65_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (59550060209 / 2000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_65_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (59550060209 / 2000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_65_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (59550060209 / 2000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_65_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (59550060209 / 2000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_65_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (59550060209 / 2000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_65_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (59550060209 / 2000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_65_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (59550060209 / 2000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_65_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (59550060209 / 2000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_65_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_66_1 :
+    Uω (aρ 1) (bρ 1) (192178132421 / 6400000000000 : ℝ) ≤
+      (-(37503573233468039420779 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (940373755235099 / 40000000000000000 : ℝ) ≤
+      (-(37503573233468039420779 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (940373755235099 / 40000000000000000 : ℝ) =
+            (940373755235099 / 625000000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (940373755235099 / 625000000000000 : ℝ) ≤
+            (4085257584531960579221 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (234405763794737 / 10000000000000000 : ℝ)) (q :=
+      (940373755235099 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_66_2 :
+    Uω (aρ 2) (bρ 2) (192178132421 / 6400000000000 : ℝ) ≤
+      (-(4847727796490385371113 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (33101436143929 / 1600000000000000 : ℝ) ≤
+      (-(4847727796490385371113 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (33101436143929 / 1600000000000000 : ℝ) = (33101436143929 / 25000000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (33101436143929 / 25000000000000 : ℝ) ≤
+            (350876055759614628887 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2017558504863 / 100000000000000 : ℝ)) (q :=
+      (33101436143929 / 1600000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_66_3 :
+    Uω (aρ 3) (bρ 3) (192178132421 / 6400000000000 : ℝ) ≤
+      (-(42847853324868564028227 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (551063142507483 / 40000000000000000 : ℝ) ≤
+      (-(42847853324868564028227 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (551063142507483 / 40000000000000000 : ℝ) =
+            (551063142507483 / 312500000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (551063142507483 / 312500000000000 : ℝ) ≤
+            (5672449296131435971773 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (110915576290929 / 10000000000000000 : ℝ)) (q :=
+      (551063142507483 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_66_4 :
+    Uω (aρ 4) (bρ 4) (192178132421 / 6400000000000 : ℝ) ≤
+      (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_4]) (by norm_num [bρ_4])]; exact Lmid_4
+
+lemma U_66_5 :
+    Uω (aρ 5) (bρ 5) (192178132421 / 6400000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_66_6 :
+    Uω (aρ 6) (bρ 6) (192178132421 / 6400000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_66_7 :
+    Uω (aρ 7) (bρ 7) (192178132421 / 6400000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_66_8 :
+    Uω (aρ 8) (bρ 8) (192178132421 / 6400000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_66_9 :
+    Uω (aρ 9) (bρ 9) (192178132421 / 6400000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_66_10 :
+    Uω (aρ 10) (bρ 10) (192178132421 / 6400000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_66_11 :
+    Uω (aρ 11) (bρ 11) (192178132421 / 6400000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_66_12 :
+    Uω (aρ 12) (bρ 12) (192178132421 / 6400000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_66_13 :
+    Uω (aρ 13) (bρ 13) (192178132421 / 6400000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_66_14 :
+    Uω (aρ 14) (bρ 14) (192178132421 / 6400000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_66_15 :
+    Uω (aρ 15) (bρ 15) (192178132421 / 6400000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_66_16 :
+    Uω (aρ 16) (bρ 16) (192178132421 / 6400000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_66 :
+    Uρ (192178132421 / 6400000000000 : ℝ) ≤
+      (-(13740225391654598492339 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (192178132421 / 6400000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(37503573233468039420779 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_66_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (192178132421 / 6400000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(4847727796490385371113 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_66_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (192178132421 / 6400000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(42847853324868564028227 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_66_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (192178132421 / 6400000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_66_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (192178132421 / 6400000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_66_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (192178132421 / 6400000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_66_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (192178132421 / 6400000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_66_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (192178132421 / 6400000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_66_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (192178132421 / 6400000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_66_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (192178132421 / 6400000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_66_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (192178132421 / 6400000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_66_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (192178132421 / 6400000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_66_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (192178132421 / 6400000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_66_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (192178132421 / 6400000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_66_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (192178132421 / 6400000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_66_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (192178132421 / 6400000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_66_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_67_1 :
+    Uω (aρ 1) (bρ 1) (484490180433 / 16000000000000 : ℝ) ≤
+      (-(37396305496047393699967 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (95051522713499 / 4000000000000000 : ℝ) ≤
+      (-(37396305496047393699967 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (95051522713499 / 4000000000000000 : ℝ) = (95051522713499 / 62500000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (95051522713499 / 62500000000000 : ℝ) ≤
+            (4192525321952606300033 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (23694846888187 / 1000000000000000 : ℝ)) (q :=
+      (95051522713499 / 4000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_67_2 :
+    Uω (aρ 2) (bρ 2) (484490180433 / 16000000000000 : ℝ) ≤
+      (-(19328669154765280711259 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (41895089611997 / 2000000000000000 : ℝ) ≤
+      (-(19328669154765280711259 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (41895089611997 / 2000000000000000 : ℝ) = (41895089611997 / 31250000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (41895089611997 / 31250000000000 : ℝ) ≤
+            (1465746254234719288741 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (40882152788869 / 2000000000000000 : ℝ)) (q :=
+      (41895089611997 / 2000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_67_3 :
+    Uω (aρ 3) (bρ 3) (484490180433 / 16000000000000 : ℝ) ≤
+      (-(1704947106755531654199 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8805564762709 / 625000000000000 : ℝ) ≤
+      (-(1704947106755531654199 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show (8805564762709 / 625000000000000 : ℝ) = (8805564762709 / 4882812500000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8805564762709 / 4882812500000 : ℝ) ≤
+            (235864998084468345801 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (114634046581063 / 10000000000000000 : ℝ)) (q :=
+      (8805564762709 / 625000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_67_4 :
+    Uω (aρ 4) (bρ 4) (484490180433 / 16000000000000 : ℝ) ≤
+      (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_4]) (by norm_num [bρ_4])]; exact Lmid_4
+
+lemma U_67_5 :
+    Uω (aρ 5) (bρ 5) (484490180433 / 16000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_67_6 :
+    Uω (aρ 6) (bρ 6) (484490180433 / 16000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_67_7 :
+    Uω (aρ 7) (bρ 7) (484490180433 / 16000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_67_8 :
+    Uω (aρ 8) (bρ 8) (484490180433 / 16000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_67_9 :
+    Uω (aρ 9) (bρ 9) (484490180433 / 16000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_67_10 :
+    Uω (aρ 10) (bρ 10) (484490180433 / 16000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_67_11 :
+    Uω (aρ 11) (bρ 11) (484490180433 / 16000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_67_12 :
+    Uω (aρ 12) (bρ 12) (484490180433 / 16000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_67_13 :
+    Uω (aρ 13) (bρ 13) (484490180433 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_67_14 :
+    Uω (aρ 14) (bρ 14) (484490180433 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_67_15 :
+    Uω (aρ 15) (bρ 15) (484490180433 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_67_16 :
+    Uω (aρ 16) (bρ 16) (484490180433 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_67 :
+    Uρ (484490180433 / 16000000000000 : ℝ) ≤
+      (-(27466029197992934502529 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (484490180433 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(37396305496047393699967 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_67_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (484490180433 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(19328669154765280711259 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_67_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (484490180433 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1704947106755531654199 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_67_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (484490180433 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_67_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (484490180433 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_67_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (484490180433 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_67_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (484490180433 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_67_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (484490180433 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_67_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (484490180433 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_67_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (484490180433 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_67_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (484490180433 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_67_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (484490180433 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_67_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (484490180433 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_67_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (484490180433 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_67_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (484490180433 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_67_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (484490180433 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_67_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_68_1 :
+    Uω (aρ 1) (bρ 1) (977070059627 / 32000000000000 : ℝ) ≤
+      (-(37290182662862082118669 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (960656077641081 / 40000000000000000 : ℝ) ≤
+      (-(37290182662862082118669 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (960656077641081 / 40000000000000000 : ℝ) =
+            (960656077641081 / 625000000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (960656077641081 / 625000000000000 : ℝ) ≤
+            (4298648155137917881331 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (239490863272103 / 10000000000000000 : ℝ)) (q :=
+      (960656077641081 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_68_2 :
+    Uω (aρ 2) (bρ 2) (977070059627 / 32000000000000 : ℝ) ≤
+      (-(19267229861265618418229 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (848261328979417 / 40000000000000000 : ℝ) ≤
+      (-(19267229861265618418229 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (848261328979417 / 40000000000000000 : ℝ) =
+            (848261328979417 / 625000000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (848261328979417 / 625000000000000 : ℝ) ≤
+            (1527185547734381581771 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (207062501451271 / 10000000000000000 : ℝ)) (q :=
+      (848261328979417 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_68_3 :
+    Uω (aρ 3) (bρ 3) (977070059627 / 32000000000000 : ℝ) ≤
+      (-(42406599719954955895771 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (575923453871657 / 40000000000000000 : ℝ) ≤
+      (-(42406599719954955895771 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (575923453871657 / 40000000000000000 : ℝ) =
+            (575923453871657 / 312500000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (575923453871657 / 312500000000000 : ℝ) ≤
+            (6113702901045044104229 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (118289670247391 / 10000000000000000 : ℝ)) (q :=
+      (575923453871657 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_68_4 :
+    Uω (aρ 4) (bρ 4) (977070059627 / 32000000000000 : ℝ) ≤
+      (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_4]) (by norm_num [bρ_4])]; exact Lmid_4
+
+lemma U_68_5 :
+    Uω (aρ 5) (bρ 5) (977070059627 / 32000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_68_6 :
+    Uω (aρ 6) (bρ 6) (977070059627 / 32000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_68_7 :
+    Uω (aρ 7) (bρ 7) (977070059627 / 32000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_68_8 :
+    Uω (aρ 8) (bρ 8) (977070059627 / 32000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_68_9 :
+    Uω (aρ 9) (bρ 9) (977070059627 / 32000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_68_10 :
+    Uω (aρ 10) (bρ 10) (977070059627 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_68_11 :
+    Uω (aρ 11) (bρ 11) (977070059627 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_68_12 :
+    Uω (aρ 12) (bρ 12) (977070059627 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_68_13 :
+    Uω (aρ 13) (bρ 13) (977070059627 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_68_14 :
+    Uω (aρ 14) (bρ 14) (977070059627 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_68_15 :
+    Uω (aρ 15) (bρ 15) (977070059627 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_68_16 :
+    Uω (aρ 16) (bρ 16) (977070059627 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_68 :
+    Uρ (977070059627 / 32000000000000 : ℝ) ≤
+      (-(27451971703722815927747 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (977070059627 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(37290182662862082118669 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_68_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (977070059627 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(19267229861265618418229 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_68_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (977070059627 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(42406599719954955895771 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_68_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (977070059627 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_68_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (977070059627 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_68_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (977070059627 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_68_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (977070059627 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_68_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (977070059627 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_68_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (977070059627 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_68_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (977070059627 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_68_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (977070059627 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_68_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (977070059627 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_68_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (977070059627 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_68_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (977070059627 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_68_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (977070059627 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_68_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (977070059627 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_68_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_69_1 :
+    Uω (aρ 1) (bρ 1) (246289939597 / 8000000000000 : ℝ) ≤
+      (-(37185180418532968481481 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (485398163167677 / 20000000000000000 : ℝ) ≤
+      (-(37185180418532968481481 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (485398163167677 / 20000000000000000 : ℝ) =
+            (485398163167677 / 312500000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (485398163167677 / 312500000000000 : ℝ) ≤
+            (4403650399467031518519 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (242032956756427 / 10000000000000000 : ℝ)) (q :=
+      (485398163167677 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_69_2 :
+    Uω (aρ 2) (bρ 2) (246289939597 / 8000000000000 : ℝ) ≤
+      (-(4801642989010586665173 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17172295095413 / 800000000000000 : ℝ) ≤
+      (-(4801642989010586665173 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (17172295095413 / 800000000000000 : ℝ) = (17172295095413 / 12500000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17172295095413 / 12500000000000 : ℝ) ≤
+            (396960863239413334827 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (8388447339363 / 400000000000000 : ℝ)) (q :=
+      (17172295095413 / 800000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_69_3 :
+    Uω (aρ 3) (bρ 3) (246289939597 / 8000000000000 : ℝ) ≤
+      (-(10549019659401938900341 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (294088189591647 / 20000000000000000 : ℝ) ≤
+      (-(10549019659401938900341 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (294088189591647 / 20000000000000000 : ℝ) =
+            (294088189591647 / 156250000000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (294088189591647 / 156250000000000 : ℝ) ≤
+            (1581055995848061099659 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (121888102040397 / 10000000000000000 : ℝ)) (q :=
+      (294088189591647 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_69_4 :
+    Uω (aρ 4) (bρ 4) (246289939597 / 8000000000000 : ℝ) ≤
+      (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_4]) (by norm_num [bρ_4])]; exact Lmid_4
+
+lemma U_69_5 :
+    Uω (aρ 5) (bρ 5) (246289939597 / 8000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_69_6 :
+    Uω (aρ 6) (bρ 6) (246289939597 / 8000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_69_7 :
+    Uω (aρ 7) (bρ 7) (246289939597 / 8000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_69_8 :
+    Uω (aρ 8) (bρ 8) (246289939597 / 8000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_69_9 :
+    Uω (aρ 9) (bρ 9) (246289939597 / 8000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_69_10 :
+    Uω (aρ 10) (bρ 10) (246289939597 / 8000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_69_11 :
+    Uω (aρ 11) (bρ 11) (246289939597 / 8000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_69_12 :
+    Uω (aρ 12) (bρ 12) (246289939597 / 8000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_69_13 :
+    Uω (aρ 13) (bρ 13) (246289939597 / 8000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_69_14 :
+    Uω (aρ 14) (bρ 14) (246289939597 / 8000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_69_15 :
+    Uω (aρ 15) (bρ 15) (246289939597 / 8000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_69_16 :
+    Uω (aρ 16) (bρ 16) (246289939597 / 8000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_69 :
+    Uρ (246289939597 / 8000000000000 : ℝ) ≤
+      (-(27438253565757300111681 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (246289939597 / 8000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(37185180418532968481481 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_69_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (246289939597 / 8000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(4801642989010586665173 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_69_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (246289939597 / 8000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(10549019659401938900341 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_69_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (246289939597 / 8000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_69_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (246289939597 / 8000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_69_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (246289939597 / 8000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_69_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (246289939597 / 8000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_69_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (246289939597 / 8000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_69_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (246289939597 / 8000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_69_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (246289939597 / 8000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_69_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (246289939597 / 8000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_69_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (246289939597 / 8000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_69_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (246289939597 / 8000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_69_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (246289939597 / 8000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_69_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (246289939597 / 8000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_69_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (246289939597 / 8000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_69_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_70_1 :
+    Uω (aρ 1) (bρ 1) (100133915591 / 3200000000000 : ℝ) ≤
+      (-(7395688851059352324481 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (247768773144139 / 10000000000000000 : ℝ) ≤
+      (-(7395688851059352324481 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (247768773144139 / 10000000000000000 : ℝ) =
+            (247768773144139 / 156250000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (247768773144139 / 156250000000000 : ℝ) ≤
+            (922077312540647675519 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (247116278151403 / 10000000000000000 : ℝ)) (q :=
+      (247768773144139 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_70_2 :
+    Uω (aρ 2) (bρ 2) (100133915591 / 3200000000000 : ℝ) ≤
+      (-(3817503845189056183093 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (109913022081481 / 5000000000000000 : ℝ) ≤
+      (-(3817503845189056183093 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (109913022081481 / 5000000000000000 : ℝ) =
+            (109913022081481 / 78125000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (109913022081481 / 78125000000000 : ℝ) ≤
+            (341379236610943816907 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (214999832699049 / 10000000000000000 : ℝ)) (q :=
+      (109913022081481 / 5000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_70_3 :
+    Uω (aρ 3) (bρ 3) (100133915591 / 3200000000000 : ℝ) ≤
+      (-(2612053898144625343839 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (76547154724709 / 5000000000000000 : ℝ) ≤
+      (-(2612053898144625343839 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (76547154724709 / 5000000000000000 : ℝ) = (76547154724709 / 39062500000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (76547154724709 / 39062500000000 : ℝ) ≤
+            (420465015667874656161 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (128932469621961 / 10000000000000000 : ℝ)) (q :=
+      (76547154724709 / 5000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_70_4 :
+    Uω (aρ 4) (bρ 4) (100133915591 / 3200000000000 : ℝ) ≤
+      (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_4]) (by norm_num [bρ_4])]; exact Lmid_4
+
+lemma U_70_5 :
+    Uω (aρ 5) (bρ 5) (100133915591 / 3200000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_70_6 :
+    Uω (aρ 6) (bρ 6) (100133915591 / 3200000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_70_7 :
+    Uω (aρ 7) (bρ 7) (100133915591 / 3200000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_70_8 :
+    Uω (aρ 8) (bρ 8) (100133915591 / 3200000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_70_9 :
+    Uω (aρ 9) (bρ 9) (100133915591 / 3200000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_70_10 :
+    Uω (aρ 10) (bρ 10) (100133915591 / 3200000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_70_11 :
+    Uω (aρ 11) (bρ 11) (100133915591 / 3200000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_70_12 :
+    Uω (aρ 12) (bρ 12) (100133915591 / 3200000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_70_13 :
+    Uω (aρ 13) (bρ 13) (100133915591 / 3200000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_70_14 :
+    Uω (aρ 14) (bρ 14) (100133915591 / 3200000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_70_15 :
+    Uω (aρ 15) (bρ 15) (100133915591 / 3200000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_70_16 :
+    Uω (aρ 16) (bρ 16) (100133915591 / 3200000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_70 :
+    Uρ (100133915591 / 3200000000000 : ℝ) ≤
+      (-(685293759890628417919 / 250000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (100133915591 / 3200000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(7395688851059352324481 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_70_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (100133915591 / 3200000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3817503845189056183093 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_70_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (100133915591 / 3200000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(2612053898144625343839 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_70_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (100133915591 / 3200000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_70_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (100133915591 / 3200000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_70_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (100133915591 / 3200000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_70_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (100133915591 / 3200000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_70_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (100133915591 / 3200000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_70_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (100133915591 / 3200000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_70_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (100133915591 / 3200000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_70_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (100133915591 / 3200000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_70_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (100133915591 / 3200000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_70_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (100133915591 / 3200000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_70_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (100133915591 / 3200000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_70_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (100133915591 / 3200000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_70_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (100133915591 / 3200000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_70_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_71_1 :
+    Uω (aρ 1) (bρ 1) (127189819179 / 4000000000000 : ℝ) ≤
+      (-(18387958661588722880311 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (505675833464109 / 20000000000000000 : ℝ) ≤
+      (-(18387958661588722880311 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (505675833464109 / 20000000000000000 : ℝ) =
+            (505675833464109 / 312500000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (505675833464109 / 312500000000000 : ℝ) ≤
+            (2406456747411277119689 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (252198503601609 / 10000000000000000 : ℝ)) (q :=
+      (505675833464109 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_71_2 :
+    Uω (aρ 2) (bρ 2) (127189819179 / 4000000000000 : ℝ) ≤
+      (-(7588542707518441531123 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (224992938362713 / 10000000000000000 : ℝ) ≤
+      (-(7588542707518441531123 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (224992938362713 / 10000000000000000 : ℝ) =
+            (224992938362713 / 156250000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (224992938362713 / 156250000000000 : ℝ) ≤
+            (729223456081558468877 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (110138779686463 / 5000000000000000 : ℝ)) (q :=
+      (224992938362713 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_71_3 :
+    Uω (aρ 3) (bρ 3) (127189819179 / 4000000000000 : ℝ) ≤
+      (-(1656433616878713809077 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (79527998946249 / 5000000000000000 : ℝ) ≤
+      (-(1656433616878713809077 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (79527998946249 / 5000000000000000 : ℝ) = (79527998946249 / 78125000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (79527998946249 / 78125000000000 : ℝ) ≤
+            (7119615841286190923 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4243743274453 / 312500000000000 : ℝ)) (q :=
+      (79527998946249 / 5000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_71_4 :
+    Uω (aρ 4) (bρ 4) (127189819179 / 4000000000000 : ℝ) ≤
+      (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_4]) (by norm_num [bρ_4])]; exact Lmid_4
+
+lemma U_71_5 :
+    Uω (aρ 5) (bρ 5) (127189819179 / 4000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_71_6 :
+    Uω (aρ 6) (bρ 6) (127189819179 / 4000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_71_7 :
+    Uω (aρ 7) (bρ 7) (127189819179 / 4000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_71_8 :
+    Uω (aρ 8) (bρ 8) (127189819179 / 4000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_71_9 :
+    Uω (aρ 9) (bρ 9) (127189819179 / 4000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_71_10 :
+    Uω (aρ 10) (bρ 10) (127189819179 / 4000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_71_11 :
+    Uω (aρ 11) (bρ 11) (127189819179 / 4000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_71_12 :
+    Uω (aρ 12) (bρ 12) (127189819179 / 4000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_71_13 :
+    Uω (aρ 13) (bρ 13) (127189819179 / 4000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_71_14 :
+    Uω (aρ 14) (bρ 14) (127189819179 / 4000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_71_15 :
+    Uω (aρ 15) (bρ 15) (127189819179 / 4000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_71_16 :
+    Uω (aρ 16) (bρ 16) (127189819179 / 4000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_71 :
+    Uρ (127189819179 / 4000000000000 : ℝ) ≤
+      (-(13693185907714144032147 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (127189819179 / 4000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(18387958661588722880311 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_71_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (127189819179 / 4000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(7588542707518441531123 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_71_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (127189819179 / 4000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1656433616878713809077 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_71_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (127189819179 / 4000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_71_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (127189819179 / 4000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_71_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (127189819179 / 4000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_71_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (127189819179 / 4000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_71_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (127189819179 / 4000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_71_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (127189819179 / 4000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_71_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (127189819179 / 4000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_71_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (127189819179 / 4000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_71_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (127189819179 / 4000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_71_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (127189819179 / 4000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_71_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (127189819179 / 4000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_71_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (127189819179 / 4000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_71_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (127189819179 / 4000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_71_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_72_1 :
+    Uω (aρ 1) (bρ 1) (516848975477 / 16000000000000 : ℝ) ≤
+      (-(36577430804916419226009 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (257906544821033 / 10000000000000000 : ℝ) ≤
+      (-(36577430804916419226009 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (257906544821033 / 10000000000000000 : ℝ) =
+            (257906544821033 / 156250000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (257906544821033 / 156250000000000 : ℝ) ≤
+            (5011400013083580773991 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (257279698053941 / 10000000000000000 : ℝ)) (q :=
+      (257906544821033 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_72_2 :
+    Uω (aρ 2) (bρ 2) (516848975477 / 16000000000000 : ℝ) ≤
+      (-(9428971103386477538939 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (460309509343253 / 20000000000000000 : ℝ) ≤
+      (-(9428971103386477538939 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (460309509343253 / 20000000000000000 : ℝ) =
+            (460309509343253 / 312500000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (460309509343253 / 312500000000000 : ℝ) ≤
+            (968236601113522461061 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (28193141283141 / 1250000000000000 : ℝ)) (q :=
+      (460309509343253 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_72_3 :
+    Uω (aρ 3) (bρ 3) (516848975477 / 16000000000000 : ℝ) ≤
+      (-(41047467247559701779401 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (65976783420521 / 4000000000000000 : ℝ) ≤
+      (-(41047467247559701779401 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (65976783420521 / 4000000000000000 : ℝ) = (65976783420521 / 62500000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (65976783420521 / 62500000000000 : ℝ) ≤
+            (541363570440298220599 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1781445554681 / 125000000000000 : ℝ)) (q :=
+      (65976783420521 / 4000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_72_4 :
+    Uω (aρ 4) (bρ 4) (516848975477 / 16000000000000 : ℝ) ≤
+      (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_4]) (by norm_num [bρ_4])]; exact Lmid_4
+
+lemma U_72_5 :
+    Uω (aρ 5) (bρ 5) (516848975477 / 16000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_72_6 :
+    Uω (aρ 6) (bρ 6) (516848975477 / 16000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_72_7 :
+    Uω (aρ 7) (bρ 7) (516848975477 / 16000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_72_8 :
+    Uω (aρ 8) (bρ 8) (516848975477 / 16000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_72_9 :
+    Uω (aρ 9) (bρ 9) (516848975477 / 16000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_72_10 :
+    Uω (aρ 10) (bρ 10) (516848975477 / 16000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_72_11 :
+    Uω (aρ 11) (bρ 11) (516848975477 / 16000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_72_12 :
+    Uω (aρ 12) (bρ 12) (516848975477 / 16000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_72_13 :
+    Uω (aρ 13) (bρ 13) (516848975477 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_72_14 :
+    Uω (aρ 14) (bρ 14) (516848975477 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_72_15 :
+    Uω (aρ 15) (bρ 15) (516848975477 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_72_16 :
+    Uω (aρ 16) (bρ 16) (516848975477 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_72 :
+    Uρ (516848975477 / 16000000000000 : ℝ) ≤
+      (-(273619983663501507283 / 100000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (516848975477 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(36577430804916419226009 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_72_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (516848975477 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(9428971103386477538939 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_72_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (516848975477 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(41047467247559701779401 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_72_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (516848975477 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_72_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (516848975477 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_72_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (516848975477 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_72_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (516848975477 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_72_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (516848975477 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_72_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (516848975477 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_72_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (516848975477 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_72_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (516848975477 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_72_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (516848975477 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_72_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (516848975477 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_72_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (516848975477 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_72_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (516848975477 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_72_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (516848975477 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_72_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_73_1 :
+    Uω (aρ 1) (bρ 1) (262469337119 / 8000000000000 : ℝ) ≤
+      (-(1455313035456262498261 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (262974687362493 / 10000000000000000 : ℝ) ≤
+      (-(1455313035456262498261 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (262974687362493 / 10000000000000000 : ℝ) =
+            (262974687362493 / 156250000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (262974687362493 / 156250000000000 : ℝ) ≤
+            (208240197263737501739 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (65589980352809 / 2500000000000000 : ℝ)) (q :=
+      (262974687362493 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_73_2 :
+    Uω (aρ 2) (bρ 2) (262469337119 / 8000000000000 : ℝ) ≤
+      (-(9373571884890926660439 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (235311840764937 / 10000000000000000 : ℝ) ≤
+      (-(9373571884890926660439 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (235311840764937 / 10000000000000000 : ℝ) =
+            (235311840764937 / 156250000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (235311840764937 / 156250000000000 : ℝ) ≤
+            (1023635819609073339561 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (57700810181531 / 2500000000000000 : ℝ)) (q :=
+      (235311840764937 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_73_3 :
+    Uω (aρ 3) (bρ 3) (262469337119 / 8000000000000 : ℝ) ≤
+      (-(8140134214423032915029 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (341524850067617 / 20000000000000000 : ℝ) ≤
+      (-(8140134214423032915029 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (341524850067617 / 20000000000000000 : ℝ) =
+            (341524850067617 / 312500000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (341524850067617 / 312500000000000 : ℝ) ≤
+            (177631949176967084971 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (149100515613867 / 10000000000000000 : ℝ)) (q :=
+      (341524850067617 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_73_4 :
+    Uω (aρ 4) (bρ 4) (262469337119 / 8000000000000 : ℝ) ≤
+      (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_4]) (by norm_num [bρ_4])]; exact Lmid_4
+
+lemma U_73_5 :
+    Uω (aρ 5) (bρ 5) (262469337119 / 8000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_73_6 :
+    Uω (aρ 6) (bρ 6) (262469337119 / 8000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_73_7 :
+    Uω (aρ 7) (bρ 7) (262469337119 / 8000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_73_8 :
+    Uω (aρ 8) (bρ 8) (262469337119 / 8000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_73_9 :
+    Uω (aρ 9) (bρ 9) (262469337119 / 8000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_73_10 :
+    Uω (aρ 10) (bρ 10) (262469337119 / 8000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_73_11 :
+    Uω (aρ 11) (bρ 11) (262469337119 / 8000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_73_12 :
+    Uω (aρ 12) (bρ 12) (262469337119 / 8000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_73_13 :
+    Uω (aρ 13) (bρ 13) (262469337119 / 8000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_73_14 :
+    Uω (aρ 14) (bρ 14) (262469337119 / 8000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_73_15 :
+    Uω (aρ 15) (bρ 15) (262469337119 / 8000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_73_16 :
+    Uω (aρ 16) (bρ 16) (262469337119 / 8000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_73 :
+    Uρ (262469337119 / 8000000000000 : ℝ) ≤
+      (-(27338531661035234682497 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (262469337119 / 8000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1455313035456262498261 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_73_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (262469337119 / 8000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(9373571884890926660439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_73_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (262469337119 / 8000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(8140134214423032915029 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_73_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (262469337119 / 8000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_73_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (262469337119 / 8000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_73_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (262469337119 / 8000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_73_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (262469337119 / 8000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_73_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (262469337119 / 8000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_73_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (262469337119 / 8000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_73_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (262469337119 / 8000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_73_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (262469337119 / 8000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_73_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (262469337119 / 8000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_73_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (262469337119 / 8000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_73_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (262469337119 / 8000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_73_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (262469337119 / 8000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_73_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (262469337119 / 8000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_73_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_74_1 :
+    Uω (aρ 1) (bρ 1) (6763975897 / 200000000000 : ℝ) ≤
+      (-(36004671011973596596201 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13655481220777 / 500000000000000 : ℝ) ≤
+      (-(36004671011973596596201 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (13655481220777 / 500000000000000 : ℝ) = (13655481220777 / 7812500000000 : ℝ) / 2 ^ 6 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13655481220777 / 7812500000000 : ℝ) ≤
+            (5584159806026403403799 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1703235450413 / 62500000000000 : ℝ)) (q :=
+      (13655481220777 / 500000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_74_2 :
+    Uω (aρ 2) (bρ 2) (6763975897 / 200000000000 : ℝ) ≤
+      (-(18532915025022758161627 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (30701632335961 / 1250000000000000 : ℝ) ≤
+      (-(18532915025022758161627 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (30701632335961 / 1250000000000000 : ℝ) = (30701632335961 / 19531250000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (30701632335961 / 19531250000000 : ℝ) ≤
+            (2261500383977241838373 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (30161694140047 / 1250000000000000 : ℝ)) (q :=
+      (30701632335961 / 1250000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_74_3 :
+    Uω (aρ 3) (bρ 3) (6763975897 / 200000000000 : ℝ) ≤
+      (-(801004622320808228037 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (364477360851279 / 20000000000000000 : ℝ) ≤
+      (-(801004622320808228037 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (364477360851279 / 20000000000000000 : ℝ) =
+            (364477360851279 / 312500000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (364477360851279 / 312500000000000 : ℝ) ≤
+            (30771994039191771963 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (161940902946279 / 10000000000000000 : ℝ)) (q :=
+      (364477360851279 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_74_4 :
+    Uω (aρ 4) (bρ 4) (6763975897 / 200000000000 : ℝ) ≤
+      (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_4]) (by norm_num [bρ_4])]; exact Lmid_4
+
+lemma U_74_5 :
+    Uω (aρ 5) (bρ 5) (6763975897 / 200000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_74_6 :
+    Uω (aρ 6) (bρ 6) (6763975897 / 200000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_74_7 :
+    Uω (aρ 7) (bρ 7) (6763975897 / 200000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_74_8 :
+    Uω (aρ 8) (bρ 8) (6763975897 / 200000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_74_9 :
+    Uω (aρ 9) (bρ 9) (6763975897 / 200000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_74_10 :
+    Uω (aρ 10) (bρ 10) (6763975897 / 200000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_74_11 :
+    Uω (aρ 11) (bρ 11) (6763975897 / 200000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_74_12 :
+    Uω (aρ 12) (bρ 12) (6763975897 / 200000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_74_13 :
+    Uω (aρ 13) (bρ 13) (6763975897 / 200000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_74_14 :
+    Uω (aρ 14) (bρ 14) (6763975897 / 200000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_74_15 :
+    Uω (aρ 15) (bρ 15) (6763975897 / 200000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_74_16 :
+    Uω (aρ 16) (bρ 16) (6763975897 / 200000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_74 :
+    Uρ (6763975897 / 200000000000 : ℝ) ≤
+      (-(27294001523746580880381 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (6763975897 / 200000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(36004671011973596596201 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_74_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (6763975897 / 200000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(18532915025022758161627 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_74_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (6763975897 / 200000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(801004622320808228037 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_74_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (6763975897 / 200000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_74_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (6763975897 / 200000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_74_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (6763975897 / 200000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_74_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (6763975897 / 200000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_74_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (6763975897 / 200000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_74_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (6763975897 / 200000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_74_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (6763975897 / 200000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_74_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (6763975897 / 200000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_74_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (6763975897 / 200000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_74_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (6763975897 / 200000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_74_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (6763975897 / 200000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_74_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (6763975897 / 200000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_74_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (6763975897 / 200000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_74_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_75_1 :
+    Uω (aρ 1) (bρ 1) (278648734641 / 8000000000000 : ℝ) ≤
+      (-(35640354467824444665873 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (113297170223489 / 4000000000000000 : ℝ) ≤
+      (-(35640354467824444665873 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (113297170223489 / 4000000000000000 : ℝ) =
+            (113297170223489 / 62500000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (113297170223489 / 62500000000000 : ℝ) ≤
+            (5948476350175555334127 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (56534430180239 / 2000000000000000 : ℝ)) (q :=
+      (113297170223489 / 4000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_75_2 :
+    Uω (aρ 2) (bρ 2) (278648734641 / 8000000000000 : ℝ) ≤
+      (-(36655583055019159093507 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (511797605676663 / 20000000000000000 : ℝ) ≤
+      (-(36655583055019159093507 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (511797605676663 / 20000000000000000 : ℝ) =
+            (511797605676663 / 312500000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (511797605676663 / 312500000000000 : ℝ) ≤
+            (4933247762980840906493 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (251752917970413 / 10000000000000000 : ℝ)) (q :=
+      (511797605676663 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_75_3 :
+    Uω (aρ 3) (bρ 3) (278648734641 / 8000000000000 : ℝ) ≤
+      (-(19724394162041300263193 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (387071224436723 / 20000000000000000 : ℝ) ≤
+      (-(19724394162041300263193 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (387071224436723 / 20000000000000000 : ℝ) =
+            (387071224436723 / 312500000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (387071224436723 / 312500000000000 : ℝ) ≤
+            (1070021246958699736807 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (174422643080473 / 10000000000000000 : ℝ)) (q :=
+      (387071224436723 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_75_4 :
+    Uω (aρ 4) (bρ 4) (278648734641 / 8000000000000 : ℝ) ≤
+      (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_4]) (by norm_num [bρ_4])]; exact Lmid_4
+
+lemma U_75_5 :
+    Uω (aρ 5) (bρ 5) (278648734641 / 8000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_75_6 :
+    Uω (aρ 6) (bρ 6) (278648734641 / 8000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_75_7 :
+    Uω (aρ 7) (bρ 7) (278648734641 / 8000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_75_8 :
+    Uω (aρ 8) (bρ 8) (278648734641 / 8000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_75_9 :
+    Uω (aρ 9) (bρ 9) (278648734641 / 8000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_75_10 :
+    Uω (aρ 10) (bρ 10) (278648734641 / 8000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_75_11 :
+    Uω (aρ 11) (bρ 11) (278648734641 / 8000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_75_12 :
+    Uω (aρ 12) (bρ 12) (278648734641 / 8000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_75_13 :
+    Uω (aρ 13) (bρ 13) (278648734641 / 8000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_75_14 :
+    Uω (aρ 14) (bρ 14) (278648734641 / 8000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_75_15 :
+    Uω (aρ 15) (bρ 15) (278648734641 / 8000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_75_16 :
+    Uω (aρ 16) (bρ 16) (278648734641 / 8000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_75 :
+    Uρ (278648734641 / 8000000000000 : ℝ) ≤
+      (-(27252257261802753728461 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (278648734641 / 8000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(35640354467824444665873 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_75_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (278648734641 / 8000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(36655583055019159093507 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_75_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (278648734641 / 8000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(19724394162041300263193 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_75_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (278648734641 / 8000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_75_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (278648734641 / 8000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_75_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (278648734641 / 8000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_75_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (278648734641 / 8000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_75_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (278648734641 / 8000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_75_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (278648734641 / 8000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_75_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (278648734641 / 8000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_75_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (278648734641 / 8000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_75_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (278648734641 / 8000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_75_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (278648734641 / 8000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_75_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (278648734641 / 8000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_75_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (278648734641 / 8000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_75_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (278648734641 / 8000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_75_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/U06.lean b/LeanPool/Zeta5Irrational/Table/U06.lean
new file mode 100644
index 0000000000..60816ff4e7
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/U06.lean
@@ -0,0 +1,2788 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-! # Certified arcsine potential bounds (U06) -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma U_76_1 :
+    Uω (aρ 1) (bρ 1) (143369216701 / 4000000000000 : ℝ) ≤
+      (-(17644447654831053876417 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (293374760981357 / 10000000000000000 : ℝ) ≤
+      (-(17644447654831053876417 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (293374760981357 / 10000000000000000 : ℝ) =
+            (293374760981357 / 156250000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (293374760981357 / 156250000000000 : ℝ) ≤
+            (3149967754168946123583 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (146411849147607 / 5000000000000000 : ℝ)) (q :=
+      (293374760981357 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_76_2 :
+    Uω (aρ 2) (bρ 2) (143369216701 / 4000000000000 : ℝ) ≤
+      (-(36262016923645122436277 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16635682819909 / 625000000000000 : ℝ) ≤
+      (-(36262016923645122436277 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (16635682819909 / 625000000000000 : ℝ) = (16635682819909 / 9765625000000 : ℝ) / 2 ^ 6 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16635682819909 / 9765625000000 : ℝ) ≤
+            (5326813894354877563723 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (65546259769897 / 2500000000000000 : ℝ)) (q :=
+      (16635682819909 / 625000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_76_3 :
+    Uω (aρ 3) (bρ 3) (143369216701 / 4000000000000 : ℝ) ≤
+      (-(38888475953383457362861 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (51172302248489 / 2500000000000000 : ℝ) ≤
+      (-(38888475953383457362861 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (51172302248489 / 2500000000000000 : ℝ) = (51172302248489 / 39062500000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (51172302248489 / 39062500000000 : ℝ) ≤
+            (2700354864616542637139 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (46654428295103 / 2500000000000000 : ℝ)) (q :=
+      (51172302248489 / 2500000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_76_4 :
+    Uω (aρ 4) (bρ 4) (143369216701 / 4000000000000 : ℝ) ≤
+      (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_4]) (by norm_num [bρ_4])]; exact Lmid_4
+
+lemma U_76_5 :
+    Uω (aρ 5) (bρ 5) (143369216701 / 4000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_76_6 :
+    Uω (aρ 6) (bρ 6) (143369216701 / 4000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_76_7 :
+    Uω (aρ 7) (bρ 7) (143369216701 / 4000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_76_8 :
+    Uω (aρ 8) (bρ 8) (143369216701 / 4000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_76_9 :
+    Uω (aρ 9) (bρ 9) (143369216701 / 4000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_76_10 :
+    Uω (aρ 10) (bρ 10) (143369216701 / 4000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_76_11 :
+    Uω (aρ 11) (bρ 11) (143369216701 / 4000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_76_12 :
+    Uω (aρ 12) (bρ 12) (143369216701 / 4000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_76_13 :
+    Uω (aρ 13) (bρ 13) (143369216701 / 4000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_76_14 :
+    Uω (aρ 14) (bρ 14) (143369216701 / 4000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_76_15 :
+    Uω (aρ 15) (bρ 15) (143369216701 / 4000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_76_16 :
+    Uω (aρ 16) (bρ 16) (143369216701 / 4000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_76 :
+    Uρ (143369216701 / 4000000000000 : ℝ) ≤
+      (-(27212905725500611600959 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (143369216701 / 4000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(17644447654831053876417 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_76_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (143369216701 / 4000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(36262016923645122436277 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_76_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (143369216701 / 4000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(38888475953383457362861 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_76_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (143369216701 / 4000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_76_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (143369216701 / 4000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_76_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (143369216701 / 4000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_76_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (143369216701 / 4000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_76_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (143369216701 / 4000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_76_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (143369216701 / 4000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_76_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (143369216701 / 4000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_76_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (143369216701 / 4000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_76_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (143369216701 / 4000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_76_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (143369216701 / 4000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_76_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (143369216701 / 4000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_76_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (143369216701 / 4000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_76_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (143369216701 / 4000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_76_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_77_1 :
+    Uω (aρ 1) (bρ 1) (75729457731 / 2000000000000 : ℝ) ≤
+      (-(17310558705766996086987 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (313634604861411 / 10000000000000000 : ℝ) ≤
+      (-(17310558705766996086987 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (313634604861411 / 10000000000000000 : ℝ) =
+            (313634604861411 / 312500000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (313634604861411 / 312500000000000 : ℝ) ≤
+            (18120801733003913013 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (156559569576411 / 5000000000000000 : ℝ)) (q :=
+      (313634604861411 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_77_2 :
+    Uω (aρ 2) (bρ 2) (75729457731 / 2000000000000 : ℝ) ≤
+      (-(17759863104573821333591 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (573360652699651 / 20000000000000000 : ℝ) ≤
+      (-(17759863104573821333591 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (573360652699651 / 20000000000000000 : ℝ) =
+            (573360652699651 / 312500000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (573360652699651 / 312500000000000 : ℝ) ≤
+            (3034552304426178666409 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (282979594639651 / 10000000000000000 : ℝ)) (q :=
+      (573360652699651 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_77_3 :
+    Uω (aρ 3) (bρ 3) (75729457731 / 2000000000000 : ℝ) ≤
+      (-(1893432330977563126623 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (453331156866793 / 20000000000000000 : ℝ) ≤
+      (-(1893432330977563126623 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (453331156866793 / 20000000000000000 : ℝ) =
+            (453331156866793 / 312500000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (453331156866793 / 312500000000000 : ℝ) ≤
+            (186009209922436873377 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (210346205156793 / 10000000000000000 : ℝ)) (q :=
+      (453331156866793 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_77_4 :
+    Uω (aρ 4) (bρ 4) (75729457731 / 2000000000000 : ℝ) ≤
+      (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_4]) (by norm_num [bρ_4])]; exact Lmid_4
+
+lemma U_77_5 :
+    Uω (aρ 5) (bρ 5) (75729457731 / 2000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_77_6 :
+    Uω (aρ 6) (bρ 6) (75729457731 / 2000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_77_7 :
+    Uω (aρ 7) (bρ 7) (75729457731 / 2000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_77_8 :
+    Uω (aρ 8) (bρ 8) (75729457731 / 2000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_77_9 :
+    Uω (aρ 9) (bρ 9) (75729457731 / 2000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_77_10 :
+    Uω (aρ 10) (bρ 10) (75729457731 / 2000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_77_11 :
+    Uω (aρ 11) (bρ 11) (75729457731 / 2000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_77_12 :
+    Uω (aρ 12) (bρ 12) (75729457731 / 2000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_77_13 :
+    Uω (aρ 13) (bρ 13) (75729457731 / 2000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_77_14 :
+    Uω (aρ 14) (bρ 14) (75729457731 / 2000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_77_15 :
+    Uω (aρ 15) (bρ 15) (75729457731 / 2000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_77_16 :
+    Uω (aρ 16) (bρ 16) (75729457731 / 2000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_77 :
+    Uρ (75729457731 / 2000000000000 : ℝ) ≤
+      (-(5428044264104939481293 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (75729457731 / 2000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(17310558705766996086987 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_77_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (75729457731 / 2000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(17759863104573821333591 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_77_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (75729457731 / 2000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1893432330977563126623 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_77_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (75729457731 / 2000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_77_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (75729457731 / 2000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_77_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (75729457731 / 2000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_77_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (75729457731 / 2000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_77_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (75729457731 / 2000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_77_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (75729457731 / 2000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_77_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (75729457731 / 2000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_77_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (75729457731 / 2000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_77_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (75729457731 / 2000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_77_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (75729457731 / 2000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_77_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (75729457731 / 2000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_77_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (75729457731 / 2000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_77_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (75729457731 / 2000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_77_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_78_1 :
+    Uω (aρ 1) (bρ 1) (20954789123 / 500000000000 : ℝ) ≤
+      (-(16703211212357127756093 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (354142058631351 / 10000000000000000 : ℝ) ≤
+      (-(16703211212357127756093 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (354142058631351 / 10000000000000000 : ℝ) =
+            (354142058631351 / 312500000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (354142058631351 / 312500000000000 : ℝ) ≤
+            (625468295142872243907 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (176842776443851 / 5000000000000000 : ℝ)) (q :=
+      (354142058631351 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_78_2 :
+    Uω (aρ 2) (bρ 2) (20954789123 / 500000000000 : ℝ) ≤
+      (-(8546436382655285391233 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (655181959732697 / 20000000000000000 : ℝ) ≤
+      (-(8546436382655285391233 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (655181959732697 / 20000000000000000 : ℝ) =
+            (655181959732697 / 625000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (655181959732697 / 625000000000000 : ℝ) ≤
+            (117903371094714608767 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (324352407867697 / 10000000000000000 : ℝ)) (q :=
+      (655181959732697 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_78_3 :
+    Uω (aρ 3) (bρ 3) (20954789123 / 500000000000 : ℝ) ≤
+      (-(36129596021937323314463 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (539438049580741 / 20000000000000000 : ℝ) ≤
+      (-(36129596021937323314463 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (539438049580741 / 20000000000000000 : ℝ) =
+            (539438049580741 / 312500000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (539438049580741 / 312500000000000 : ℝ) ≤
+            (5459234796062676685537 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (256004604065741 / 10000000000000000 : ℝ)) (q :=
+      (539438049580741 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_78_4 :
+    Uω (aρ 4) (bρ 4) (20954789123 / 500000000000 : ℝ) ≤
+      (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_4]) (by norm_num [bρ_4])]; exact Lmid_4
+
+lemma U_78_5 :
+    Uω (aρ 5) (bρ 5) (20954789123 / 500000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_78_6 :
+    Uω (aρ 6) (bρ 6) (20954789123 / 500000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_78_7 :
+    Uω (aρ 7) (bρ 7) (20954789123 / 500000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_78_8 :
+    Uω (aρ 8) (bρ 8) (20954789123 / 500000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_78_9 :
+    Uω (aρ 9) (bρ 9) (20954789123 / 500000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_78_10 :
+    Uω (aρ 10) (bρ 10) (20954789123 / 500000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_78_11 :
+    Uω (aρ 11) (bρ 11) (20954789123 / 500000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_78_12 :
+    Uω (aρ 12) (bρ 12) (20954789123 / 500000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_78_13 :
+    Uω (aρ 13) (bρ 13) (20954789123 / 500000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_78_14 :
+    Uω (aρ 14) (bρ 14) (20954789123 / 500000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_78_15 :
+    Uω (aρ 15) (bρ 15) (20954789123 / 500000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_78_16 :
+    Uω (aρ 16) (bρ 16) (20954789123 / 500000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_78 :
+    Uρ (20954789123 / 500000000000 : ℝ) ≤
+      (-(27013468316499378362551 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (20954789123 / 500000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(16703211212357127756093 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_78_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (20954789123 / 500000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(8546436382655285391233 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_78_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (20954789123 / 500000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(36129596021937323314463 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_78_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (20954789123 / 500000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11449496158609380585237 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_78_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (20954789123 / 500000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_78_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (20954789123 / 500000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_78_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (20954789123 / 500000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_78_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (20954789123 / 500000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_78_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (20954789123 / 500000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_78_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (20954789123 / 500000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_78_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (20954789123 / 500000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_78_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (20954789123 / 500000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_78_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (20954789123 / 500000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_78_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (20954789123 / 500000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_78_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (20954789123 / 500000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_78_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (20954789123 / 500000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_78_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_79_1 :
+    Uω (aρ 1) (bρ 1) (694494763253 / 16000000000000 : ℝ) ≤
+      (-(8248019133811670168773 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (147649612727507 / 4000000000000000 : ℝ) ≤
+      (-(8248019133811670168773 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (147649612727507 / 4000000000000000 : ℝ) =
+            (147649612727507 / 125000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (147649612727507 / 125000000000000 : ℝ) ≤
+            (416320619938329831227 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (36868605468941 / 1000000000000000 : ℝ)) (q :=
+      (147649612727507 / 4000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_79_2 :
+    Uω (aρ 2) (bρ 2) (694494763253 / 16000000000000 : ℝ) ≤
+      (-(33734931655425398758109 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27415774538689 / 800000000000000 : ℝ) ≤
+      (-(33734931655425398758109 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (27415774538689 / 800000000000000 : ℝ) = (27415774538689 / 25000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27415774538689 / 25000000000000 : ℝ) ≤
+            (922427359574601241891 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3396013670291 / 100000000000000 : ℝ)) (q :=
+      (27415774538689 / 800000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_79_3 :
+    Uω (aρ 3) (bρ 3) (694494763253 / 16000000000000 : ℝ) ≤
+      (-(17781582814941265651163 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (285437704511201 / 10000000000000000 : ℝ) ≤
+      (-(17781582814941265651163 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (285437704511201 / 10000000000000000 : ℝ) =
+            (285437704511201 / 156250000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (285437704511201 / 156250000000000 : ℝ) ≤
+            (3012832594058734348837 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (272478518934277 / 10000000000000000 : ℝ)) (q :=
+      (285437704511201 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_79_4 :
+    Uω (aρ 4) (bρ 4) (694494763253 / 16000000000000 : ℝ) ≤
+      (-(2625083115715732209859 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (37483956842069 / 2500000000000000 : ℝ) ≤
+      (-(2625083115715732209859 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show
+          (37483956842069 / 2500000000000000 : ℝ) = (37483956842069 / 19531250000000 : ℝ) / 2 ^ 7
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (37483956842069 / 19531250000000 : ℝ) ≤
+            (407435798096767790141 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (79768945713427 / 10000000000000000 : ℝ)) (q :=
+      (37483956842069 / 2500000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_79_5 :
+    Uω (aρ 5) (bρ 5) (694494763253 / 16000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_79_6 :
+    Uω (aρ 6) (bρ 6) (694494763253 / 16000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_79_7 :
+    Uω (aρ 7) (bρ 7) (694494763253 / 16000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_79_8 :
+    Uω (aρ 8) (bρ 8) (694494763253 / 16000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_79_9 :
+    Uω (aρ 9) (bρ 9) (694494763253 / 16000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_79_10 :
+    Uω (aρ 10) (bρ 10) (694494763253 / 16000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_79_11 :
+    Uω (aρ 11) (bρ 11) (694494763253 / 16000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_79_12 :
+    Uω (aρ 12) (bρ 12) (694494763253 / 16000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_79_13 :
+    Uω (aρ 13) (bρ 13) (694494763253 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_79_14 :
+    Uω (aρ 14) (bρ 14) (694494763253 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_79_15 :
+    Uω (aρ 15) (bρ 15) (694494763253 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_79_16 :
+    Uω (aρ 16) (bρ 16) (694494763253 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_79 :
+    Uρ (694494763253 / 16000000000000 : ℝ) ≤
+      (-(2675052424918429982301 / 1000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (694494763253 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(8248019133811670168773 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_79_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (694494763253 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(33734931655425398758109 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_79_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (694494763253 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(17781582814941265651163 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_79_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (694494763253 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(2625083115715732209859 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_79_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (694494763253 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_79_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (694494763253 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_79_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (694494763253 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_79_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (694494763253 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_79_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (694494763253 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_79_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (694494763253 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_79_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (694494763253 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_79_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (694494763253 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_79_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (694494763253 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_79_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (694494763253 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_79_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (694494763253 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_79_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (694494763253 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_79_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_80_1 :
+    Uω (aρ 1) (bρ 1) (1412931037823 / 32000000000000 : ℝ) ≤
+      (-(6558236691624471248019 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1506457859888209 / 40000000000000000 : ℝ) ≤
+      (-(6558236691624471248019 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (1506457859888209 / 40000000000000000 : ℝ) =
+            (1506457859888209 / 1250000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1506457859888209 / 1250000000000000 : ℝ) ≤
+            (373235111375528751981 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (376185198709417 / 10000000000000000 : ℝ)) (q :=
+      (1506457859888209 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_80_2 :
+    Uω (aρ 2) (bρ 2) (1412931037823 / 32000000000000 : ℝ) ≤
+      (-(33517056778454642676983 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1400982498276359 / 40000000000000000 : ℝ) ≤
+      (-(33517056778454642676983 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (1400982498276359 / 40000000000000000 : ℝ) =
+            (1400982498276359 / 1250000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1400982498276359 / 1250000000000000 : ℝ) ≤
+            (1140302236545357323017 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (86804132603373 / 2500000000000000 : ℝ)) (q :=
+      (1400982498276359 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_80_3 :
+    Uω (aρ 3) (bρ 3) (1412931037823 / 32000000000000 : ℝ) ≤
+      (-(27572364292843690767 / 7812500000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1173061294838367 / 40000000000000000 : ℝ) ≤
+      (-(27572364292843690767 / 7812500000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (1173061294838367 / 40000000000000000 : ℝ) =
+            (1173061294838367 / 625000000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1173061294838367 / 625000000000000 : ℝ) ≤
+            (4918909783718809233 / 7812500000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (17540752190281 / 625000000000000 : ℝ)) (q :=
+      (1173061294838367 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_80_4 :
+    Uω (aρ 4) (bρ 4) (1412931037823 / 32000000000000 : ℝ) ≤
+      (-(20580854203499261973059 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (652273459780041 / 40000000000000000 : ℝ) ≤
+      (-(20580854203499261973059 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (652273459780041 / 40000000000000000 : ℝ) =
+            (652273459780041 / 625000000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (652273459780041 / 625000000000000 : ℝ) ≤
+            (213561205500738026941 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (98552298580333 / 10000000000000000 : ℝ)) (q :=
+      (652273459780041 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_80_5 :
+    Uω (aρ 5) (bρ 5) (1412931037823 / 32000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_80_6 :
+    Uω (aρ 6) (bρ 6) (1412931037823 / 32000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_80_7 :
+    Uω (aρ 7) (bρ 7) (1412931037823 / 32000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_80_8 :
+    Uω (aρ 8) (bρ 8) (1412931037823 / 32000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_80_9 :
+    Uω (aρ 9) (bρ 9) (1412931037823 / 32000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_80_10 :
+    Uω (aρ 10) (bρ 10) (1412931037823 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_80_11 :
+    Uω (aρ 11) (bρ 11) (1412931037823 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_80_12 :
+    Uω (aρ 12) (bρ 12) (1412931037823 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_80_13 :
+    Uω (aρ 13) (bρ 13) (1412931037823 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_80_14 :
+    Uω (aρ 14) (bρ 14) (1412931037823 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_80_15 :
+    Uω (aρ 15) (bρ 15) (1412931037823 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_80_16 :
+    Uω (aρ 16) (bρ 16) (1412931037823 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_80 :
+    Uρ (1412931037823 / 32000000000000 : ℝ) ≤
+      (-(13340752815781598529229 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1412931037823 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(6558236691624471248019 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_80_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1412931037823 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(33517056778454642676983 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_80_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1412931037823 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) * (-(27572364292843690767 / 7812500000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_80_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1412931037823 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(20580854203499261973059 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_80_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1412931037823 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_80_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1412931037823 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_80_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1412931037823 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_80_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1412931037823 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_80_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1412931037823 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_80_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1412931037823 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_80_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1412931037823 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_80_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1412931037823 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_80_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1412931037823 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_80_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1412931037823 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_80_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1412931037823 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_80_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1412931037823 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_80_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_81_1 :
+    Uω (aρ 1) (bρ 1) (71843627457 / 1600000000000 : ℝ) ≤
+      (-(3259425568664305827969 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (192052279005631 / 5000000000000000 : ℝ) ≤
+      (-(3259425568664305827969 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (192052279005631 / 5000000000000000 : ℝ) =
+            (192052279005631 / 156250000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (192052279005631 / 156250000000000 : ℝ) ≤
+            (206310332835694172031 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (191841831250637 / 5000000000000000 : ℝ)) (q :=
+      (192052279005631 / 5000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_81_2 :
+    Uω (aρ 2) (bρ 2) (71843627457 / 1600000000000 : ℝ) ≤
+      (-(6660781429157999445921 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (715582457297373 / 20000000000000000 : ℝ) ≤
+      (-(6660781429157999445921 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (715582457297373 / 20000000000000000 : ℝ) =
+            (715582457297373 / 625000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (715582457297373 / 625000000000000 : ℝ) ≤
+            (270690373842000554079 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (354826016286123 / 10000000000000000 : ℝ)) (q :=
+      (715582457297373 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_81_3 :
+    Uω (aρ 3) (bρ 3) (71843627457 / 1600000000000 : ℝ) ≤
+      (-(35029836476419809769441 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (120429677203231 / 4000000000000000 : ℝ) ≤
+      (-(35029836476419809769441 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (120429677203231 / 4000000000000000 : ℝ) =
+            (120429677203231 / 62500000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (120429677203231 / 62500000000000 : ℝ) ≤
+            (6558994341580190230559 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (57757610270981 / 2000000000000000 : ℝ)) (q :=
+      (120429677203231 / 4000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_81_4 :
+    Uω (aρ 4) (bρ 4) (71843627457 / 1600000000000 : ℝ) ≤
+      (-(8091999804594550308781 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10932628365517 / 625000000000000 : ℝ) ≤
+      (-(8091999804594550308781 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (10932628365517 / 625000000000000 : ℝ) = (10932628365517 / 9765625000000 : ℝ) / 2 ^ 6 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10932628365517 / 9765625000000 : ℝ) ≤
+            (225766359005449691219 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (57388977050147 / 5000000000000000 : ℝ)) (q :=
+      (10932628365517 / 625000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_81_5 :
+    Uω (aρ 5) (bρ 5) (71843627457 / 1600000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_81_6 :
+    Uω (aρ 6) (bρ 6) (71843627457 / 1600000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_81_7 :
+    Uω (aρ 7) (bρ 7) (71843627457 / 1600000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_81_8 :
+    Uω (aρ 8) (bρ 8) (71843627457 / 1600000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_81_9 :
+    Uω (aρ 9) (bρ 9) (71843627457 / 1600000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_81_10 :
+    Uω (aρ 10) (bρ 10) (71843627457 / 1600000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_81_11 :
+    Uω (aρ 11) (bρ 11) (71843627457 / 1600000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_81_12 :
+    Uω (aρ 12) (bρ 12) (71843627457 / 1600000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_81_13 :
+    Uω (aρ 13) (bρ 13) (71843627457 / 1600000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_81_14 :
+    Uω (aρ 14) (bρ 14) (71843627457 / 1600000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_81_15 :
+    Uω (aρ 15) (bρ 15) (71843627457 / 1600000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_81_16 :
+    Uω (aρ 16) (bρ 16) (71843627457 / 1600000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_81 :
+    Uρ (71843627457 / 1600000000000 : ℝ) ≤
+      (-(5324205551757644520397 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (71843627457 / 1600000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3259425568664305827969 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_81_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (71843627457 / 1600000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(6660781429157999445921 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_81_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (71843627457 / 1600000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(35029836476419809769441 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_81_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (71843627457 / 1600000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(8091999804594550308781 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_81_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (71843627457 / 1600000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_81_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (71843627457 / 1600000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_81_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (71843627457 / 1600000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_81_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (71843627457 / 1600000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_81_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (71843627457 / 1600000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_81_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (71843627457 / 1600000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_81_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (71843627457 / 1600000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_81_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (71843627457 / 1600000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_81_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (71843627457 / 1600000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_81_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (71843627457 / 1600000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_81_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (71843627457 / 1600000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_81_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (71843627457 / 1600000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_81_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_82_1 :
+    Uω (aρ 1) (bρ 1) (2897686609597 / 64000000000000 : ℝ) ≤
+      (-(32497230363190579783231 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3102795865275137 / 80000000000000000 : ℝ) ≤
+      (-(32497230363190579783231 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (3102795865275137 / 80000000000000000 : ℝ) =
+            (3102795865275137 / 2500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3102795865275137 / 2500000000000000 : ℝ) ≤
+            (2160128651809420216769 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (387432651654253 / 10000000000000000 : ℝ)) (q :=
+      (3102795865275137 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_82_2 :
+    Uω (aρ 2) (bρ 2) (2897686609597 / 64000000000000 : ℝ) ≤
+      (-(6639808015630789086519 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (578500836206773 / 16000000000000000 : ℝ) ≤
+      (-(6639808015630789086519 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (578500836206773 / 16000000000000000 : ℝ) =
+            (578500836206773 / 500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (578500836206773 / 500000000000000 : ℝ) ≤
+            (291663787369210913481 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (71725748620787 / 2000000000000000 : ℝ)) (q :=
+      (578500836206773 / 16000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_82_3 :
+    Uω (aρ 3) (bρ 3) (2897686609597 / 64000000000000 : ℝ) ≤
+      (-(87253003056903595321 / 25000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2439776680760601 / 80000000000000000 : ℝ) ≤
+      (-(87253003056903595321 / 25000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (2439776680760601 / 80000000000000000 : ℝ) =
+            (2439776680760601 / 1250000000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2439776680760601 / 1250000000000000 : ℝ) ≤
+            (16719073988096404679 / 25000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (292842974385619 / 10000000000000000 : ℝ)) (q :=
+      (2439776680760601 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_82_4 :
+    Uω (aρ 4) (bρ 4) (2897686609597 / 64000000000000 : ℝ) ≤
+      (-(20072167163996576495227 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1444254397221741 / 80000000000000000 : ℝ) ≤
+      (-(20072167163996576495227 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (1444254397221741 / 80000000000000000 : ℝ) =
+            (1444254397221741 / 1250000000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1444254397221741 / 1250000000000000 : ℝ) ≤
+            (722248245003423504773 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7641036535369 / 625000000000000 : ℝ)) (q :=
+      (1444254397221741 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_82_5 :
+    Uω (aρ 5) (bρ 5) (2897686609597 / 64000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_82_6 :
+    Uω (aρ 6) (bρ 6) (2897686609597 / 64000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_82_7 :
+    Uω (aρ 7) (bρ 7) (2897686609597 / 64000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_82_8 :
+    Uω (aρ 8) (bρ 8) (2897686609597 / 64000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_82_9 :
+    Uω (aρ 9) (bρ 9) (2897686609597 / 64000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_82_10 :
+    Uω (aρ 10) (bρ 10) (2897686609597 / 64000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_82_11 :
+    Uω (aρ 11) (bρ 11) (2897686609597 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_82_12 :
+    Uω (aρ 12) (bρ 12) (2897686609597 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_82_13 :
+    Uω (aρ 13) (bρ 13) (2897686609597 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_82_14 :
+    Uω (aρ 14) (bρ 14) (2897686609597 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_82_15 :
+    Uω (aρ 15) (bρ 15) (2897686609597 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_82_16 :
+    Uω (aρ 16) (bρ 16) (2897686609597 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_82 :
+    Uρ (2897686609597 / 64000000000000 : ℝ) ≤
+      (-(6648255242717957936621 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (2897686609597 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(32497230363190579783231 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_82_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (2897686609597 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(6639808015630789086519 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_82_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (2897686609597 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) * (-(87253003056903595321 / 25000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_82_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (2897686609597 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(20072167163996576495227 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_82_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (2897686609597 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_82_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (2897686609597 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_82_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (2897686609597 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_82_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (2897686609597 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_82_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (2897686609597 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_82_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (2897686609597 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_82_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (2897686609597 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_82_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (2897686609597 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_82_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (2897686609597 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_82_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (2897686609597 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_82_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (2897686609597 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_82_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (2897686609597 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_82_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_83_1 :
+    Uω (aρ 1) (bρ 1) (1460814060457 / 32000000000000 : ℝ) ≤
+      (-(8100284844802228771499 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1566377321980367 / 40000000000000000 : ℝ) ≤
+      (-(8100284844802228771499 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (1566377321980367 / 40000000000000000 : ℝ) =
+            (1566377321980367 / 1250000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1566377321980367 / 1250000000000000 : ℝ) ≤
+            (564054908947771228501 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (391181485182371 / 10000000000000000 : ℝ)) (q :=
+      (1566377321980367 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_83_2 :
+    Uω (aρ 2) (bρ 2) (1460814060457 / 32000000000000 : ℝ) ≤
+      (-(4136909867612453369687 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1461336691110951 / 40000000000000000 : ℝ) ≤
+      (-(4136909867612453369687 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (1461336691110951 / 40000000000000000 : ℝ) =
+            (1461336691110951 / 1250000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1461336691110951 / 1250000000000000 : ℝ) ≤
+            (195260009262546630313 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (362430182257663 / 10000000000000000 : ℝ)) (q :=
+      (1461336691110951 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_83_3 :
+    Uω (aρ 3) (bρ 3) (1460814060457 / 32000000000000 : ℝ) ≤
+      (-(34774333175818615317637 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1235463415565381 / 40000000000000000 : ℝ) ≤
+      (-(34774333175818615317637 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (1235463415565381 / 40000000000000000 : ℝ) =
+            (1235463415565381 / 625000000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1235463415565381 / 625000000000000 : ℝ) ≤
+            (6814497642181384682363 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (148444825417439 / 5000000000000000 : ℝ)) (q :=
+      (1235463415565381 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_83_4 :
+    Uω (aρ 4) (bρ 4) (1460814060457 / 32000000000000 : ℝ) ≤
+      (-(19923513604471120193173 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (743918890423493 / 40000000000000000 : ℝ) ≤
+      (-(19923513604471120193173 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (743918890423493 / 40000000000000000 : ℝ) =
+            (743918890423493 / 625000000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (743918890423493 / 625000000000000 : ℝ) ≤
+            (870901804528879806827 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (64705784664467 / 5000000000000000 : ℝ)) (q :=
+      (743918890423493 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_83_5 :
+    Uω (aρ 5) (bρ 5) (1460814060457 / 32000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_83_6 :
+    Uω (aρ 6) (bρ 6) (1460814060457 / 32000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_83_7 :
+    Uω (aρ 7) (bρ 7) (1460814060457 / 32000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_83_8 :
+    Uω (aρ 8) (bρ 8) (1460814060457 / 32000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_83_9 :
+    Uω (aρ 9) (bρ 9) (1460814060457 / 32000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_83_10 :
+    Uω (aρ 10) (bρ 10) (1460814060457 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_83_11 :
+    Uω (aρ 11) (bρ 11) (1460814060457 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_83_12 :
+    Uω (aρ 12) (bρ 12) (1460814060457 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_83_13 :
+    Uω (aρ 13) (bρ 13) (1460814060457 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_83_14 :
+    Uω (aρ 14) (bρ 14) (1460814060457 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_83_15 :
+    Uω (aρ 15) (bρ 15) (1460814060457 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_83_16 :
+    Uω (aρ 16) (bρ 16) (1460814060457 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_83 :
+    Uρ (1460814060457 / 32000000000000 : ℝ) ≤
+      (-(26566200964288256233641 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1460814060457 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(8100284844802228771499 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_83_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1460814060457 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(4136909867612453369687 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_83_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1460814060457 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(34774333175818615317637 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_83_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1460814060457 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(19923513604471120193173 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_83_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1460814060457 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_83_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1460814060457 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_83_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1460814060457 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_83_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1460814060457 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_83_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1460814060457 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_83_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1460814060457 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_83_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1460814060457 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_83_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1460814060457 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_83_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1460814060457 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_83_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1460814060457 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_83_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1460814060457 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_83_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1460814060457 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_83_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_84_1 :
+    Uω (aρ 1) (bρ 1) (2945569632231 / 64000000000000 : ℝ) ≤
+      (-(16152982436976417664993 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3162712817873931 / 80000000000000000 : ℝ) ≤
+      (-(16152982436976417664993 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (3162712817873931 / 80000000000000000 : ℝ) =
+            (3162712817873931 / 2500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3162712817873931 / 2500000000000000 : ℝ) ≤
+            (1175697070523582335007 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (394930167517389 / 10000000000000000 : ℝ)) (q :=
+      (3162712817873931 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_84_2 :
+    Uω (aρ 2) (bρ 2) (2945569632231 / 64000000000000 : ℝ) ≤
+      (-(16496300146835009040687 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2952837593143699 / 80000000000000000 : ℝ) ≤
+      (-(16496300146835009040687 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (2952837593143699 / 80000000000000000 : ℝ) =
+            (2952837593143699 / 2500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2952837593143699 / 2500000000000000 : ℝ) ≤
+            (832379360664990959313 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (366230373844831 / 10000000000000000 : ℝ)) (q :=
+      (2952837593143699 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_84_3 :
+    Uω (aρ 3) (bρ 3) (2945569632231 / 64000000000000 : ℝ) ≤
+      (-(34649181056232259045777 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2502045325907491 / 80000000000000000 : ℝ) ≤
+      (-(34649181056232259045777 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (2502045325907491 / 80000000000000000 : ℝ) =
+            (2502045325907491 / 2500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 2
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2502045325907491 / 2500000000000000 : ℝ) ≤
+            (8177958767740954223 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (300928413385779 / 10000000000000000 : ℝ)) (q :=
+      (2502045325907491 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_84_4 :
+    Uω (aρ 4) (bρ 4) (2945569632231 / 64000000000000 : ℝ) ≤
+      (-(494567863174717687381 / 125000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1530330503929419 / 80000000000000000 : ℝ) ≤
+      (-(494567863174717687381 / 125000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (1530330503929419 / 80000000000000000 : ℝ) =
+            (1530330503929419 / 1250000000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1530330503929419 / 1250000000000000 : ℝ) ≤
+            (25292522050282312619 / 125000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (136293888956261 / 10000000000000000 : ℝ)) (q :=
+      (1530330503929419 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_84_5 :
+    Uω (aρ 5) (bρ 5) (2945569632231 / 64000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_84_6 :
+    Uω (aρ 6) (bρ 6) (2945569632231 / 64000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_84_7 :
+    Uω (aρ 7) (bρ 7) (2945569632231 / 64000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_84_8 :
+    Uω (aρ 8) (bρ 8) (2945569632231 / 64000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_84_9 :
+    Uω (aρ 9) (bρ 9) (2945569632231 / 64000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_84_10 :
+    Uω (aρ 10) (bρ 10) (2945569632231 / 64000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_84_11 :
+    Uω (aρ 11) (bρ 11) (2945569632231 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_84_12 :
+    Uω (aρ 12) (bρ 12) (2945569632231 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_84_13 :
+    Uω (aρ 13) (bρ 13) (2945569632231 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_84_14 :
+    Uω (aρ 14) (bρ 14) (2945569632231 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_84_15 :
+    Uω (aρ 15) (bρ 15) (2945569632231 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_84_16 :
+    Uω (aρ 16) (bρ 16) (2945569632231 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_84 :
+    Uρ (2945569632231 / 64000000000000 : ℝ) ≤
+      (-(26540410498328256391827 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (2945569632231 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(16152982436976417664993 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_84_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (2945569632231 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(16496300146835009040687 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_84_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (2945569632231 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(34649181056232259045777 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_84_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (2945569632231 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(494567863174717687381 / 125000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_84_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (2945569632231 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_84_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (2945569632231 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_84_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (2945569632231 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_84_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (2945569632231 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_84_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (2945569632231 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_84_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (2945569632231 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_84_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (2945569632231 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_84_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (2945569632231 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_84_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (2945569632231 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_84_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (2945569632231 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_84_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (2945569632231 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_84_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (2945569632231 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_84_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_85_1 :
+    Uω (aρ 1) (bρ 1) (742377785887 / 16000000000000 : ℝ) ≤
+      (-(16105844747499819637607 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (39908380050929 / 1000000000000000 : ℝ) ≤
+      (-(16105844747499819637607 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (39908380050929 / 1000000000000000 : ℝ) = (39908380050929 / 31250000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (39908380050929 / 31250000000000 : ℝ) ≤
+            (1222834760000180362393 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (79735740584841 / 2000000000000000 : ℝ)) (q :=
+      (39908380050929 / 1000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_85_2 :
+    Uω (aρ 2) (bρ 2) (742377785887 / 16000000000000 : ℝ) ≤
+      (-(8222745361049200402393 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (18643731047181 / 500000000000000 : ℝ) ≤
+      (-(8222745361049200402393 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (18643731047181 / 500000000000000 : ℝ) = (18643731047181 / 15625000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18643731047181 / 15625000000000 : ℝ) ≤
+            (441594392700799597607 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (74005871260573 / 2000000000000000 : ℝ)) (q :=
+      (18643731047181 / 500000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_85_3 :
+    Uω (aρ 3) (bρ 3) (742377785887 / 16000000000000 : ℝ) ≤
+      (-(17262847949594570677381 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (158320838924591 / 5000000000000000 : ℝ) ≤
+      (-(17262847949594570677381 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (158320838924591 / 5000000000000000 : ℝ) =
+            (158320838924591 / 156250000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (158320838924591 / 156250000000000 : ℝ) ≤
+            (65831557905429322619 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (304959576463989 / 10000000000000000 : ℝ)) (q :=
+      (158320838924591 / 5000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_85_4 :
+    Uω (aρ 4) (bρ 4) (742377785887 / 16000000000000 : ℝ) ≤
+      (-(7859495672749720896173 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (392972531333811 / 20000000000000000 : ℝ) ≤
+      (-(7859495672749720896173 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (392972531333811 / 20000000000000000 : ℝ) =
+            (392972531333811 / 312500000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (392972531333811 / 312500000000000 : ℝ) ≤
+            (458270490850279103827 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (35735733291109 / 2500000000000000 : ℝ)) (q :=
+      (392972531333811 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_85_5 :
+    Uω (aρ 5) (bρ 5) (742377785887 / 16000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_85_6 :
+    Uω (aρ 6) (bρ 6) (742377785887 / 16000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_85_7 :
+    Uω (aρ 7) (bρ 7) (742377785887 / 16000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_85_8 :
+    Uω (aρ 8) (bρ 8) (742377785887 / 16000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_85_9 :
+    Uω (aρ 9) (bρ 9) (742377785887 / 16000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_85_10 :
+    Uω (aρ 10) (bρ 10) (742377785887 / 16000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_85_11 :
+    Uω (aρ 11) (bρ 11) (742377785887 / 16000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_85_12 :
+    Uω (aρ 12) (bρ 12) (742377785887 / 16000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_85_13 :
+    Uω (aρ 13) (bρ 13) (742377785887 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_85_14 :
+    Uω (aρ 14) (bρ 14) (742377785887 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_85_15 :
+    Uω (aρ 15) (bρ 15) (742377785887 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_85_16 :
+    Uω (aρ 16) (bρ 16) (742377785887 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_85 :
+    Uρ (742377785887 / 16000000000000 : ℝ) ≤
+      (-(6628881657870160769053 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (742377785887 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(16105844747499819637607 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_85_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (742377785887 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(8222745361049200402393 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_85_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (742377785887 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(17262847949594570677381 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_85_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (742377785887 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(7859495672749720896173 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_85_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (742377785887 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_85_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (742377785887 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_85_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (742377785887 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_85_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (742377785887 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_85_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (742377785887 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_85_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (742377785887 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_85_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (742377785887 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_85_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (742377785887 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_85_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (742377785887 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_85_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (742377785887 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_85_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (742377785887 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_85_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (742377785887 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_85_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_86_1 :
+    Uω (aρ 1) (bρ 1) (598690530973 / 12800000000000 : ℝ) ≤
+      (-(16059148189551546355899 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3222627418985217 / 80000000000000000 : ℝ) ≤
+      (-(16059148189551546355899 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (3222627418985217 / 80000000000000000 : ℝ) =
+            (3222627418985217 / 2500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3222627418985217 / 2500000000000000 : ℝ) ≤
+            (1269531317948453644101 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (50303386938581 / 1250000000000000 : ℝ)) (q :=
+      (3222627418985217 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_86_2 :
+    Uω (aρ 2) (bρ 2) (598690530973 / 12800000000000 : ℝ) ≤
+      (-(1024700013012026515809 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3013151652889589 / 80000000000000000 : ℝ) ≤
+      (-(1024700013012026515809 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (3013151652889589 / 80000000000000000 : ℝ) =
+            (3013151652889589 / 2500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3013151652889589 / 2500000000000000 : ℝ) ≤
+            (58342456206723484191 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (373827166494741 / 10000000000000000 : ℝ)) (q :=
+      (3013151652889589 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_86_3 :
+    Uω (aρ 3) (bρ 3) (598690530973 / 12800000000000 : ℝ) ≤
+      (-(6880766185247015737031 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2564192311550617 / 80000000000000000 : ℝ) ≤
+      (-(6880766185247015737031 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (2564192311550617 / 80000000000000000 : ℝ) =
+            (2564192311550617 / 2500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2564192311550617 / 2500000000000000 : ℝ) ≤
+            (50705617752984262969 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (154491718754999 / 5000000000000000 : ℝ)) (q :=
+      (2564192311550617 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_86_4 :
+    Uω (aρ 4) (bρ 4) (598690530973 / 12800000000000 : ℝ) ≤
+      (-(39041532706513693929033 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1612641249535897 / 80000000000000000 : ℝ) ≤
+      (-(39041532706513693929033 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (1612641249535897 / 80000000000000000 : ℝ) =
+            (1612641249535897 / 1250000000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1612641249535897 / 1250000000000000 : ℝ) ≤
+            (2547298111486306070967 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (74694926535659 / 5000000000000000 : ℝ)) (q :=
+      (1612641249535897 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_86_5 :
+    Uω (aρ 5) (bρ 5) (598690530973 / 12800000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_86_6 :
+    Uω (aρ 6) (bρ 6) (598690530973 / 12800000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_86_7 :
+    Uω (aρ 7) (bρ 7) (598690530973 / 12800000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_86_8 :
+    Uω (aρ 8) (bρ 8) (598690530973 / 12800000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_86_9 :
+    Uω (aρ 9) (bρ 9) (598690530973 / 12800000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_86_10 :
+    Uω (aρ 10) (bρ 10) (598690530973 / 12800000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_86_11 :
+    Uω (aρ 11) (bρ 11) (598690530973 / 12800000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_86_12 :
+    Uω (aρ 12) (bρ 12) (598690530973 / 12800000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_86_13 :
+    Uω (aρ 13) (bρ 13) (598690530973 / 12800000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_86_14 :
+    Uω (aρ 14) (bρ 14) (598690530973 / 12800000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_86_15 :
+    Uω (aρ 15) (bρ 15) (598690530973 / 12800000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_86_16 :
+    Uω (aρ 16) (bρ 16) (598690530973 / 12800000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_86 :
+    Uρ (598690530973 / 12800000000000 : ℝ) ≤
+      (-(26491450933863278184387 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (598690530973 / 12800000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(16059148189551546355899 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_86_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (598690530973 / 12800000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1024700013012026515809 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_86_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (598690530973 / 12800000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(6880766185247015737031 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_86_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (598690530973 / 12800000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(39041532706513693929033 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_86_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (598690530973 / 12800000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_86_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (598690530973 / 12800000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_86_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (598690530973 / 12800000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_86_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (598690530973 / 12800000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_86_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (598690530973 / 12800000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_86_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (598690530973 / 12800000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_86_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (598690530973 / 12800000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_86_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (598690530973 / 12800000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_86_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (598690530973 / 12800000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_86_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (598690530973 / 12800000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_86_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (598690530973 / 12800000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_86_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (598690530973 / 12800000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_86_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_87_1 :
+    Uω (aρ 1) (bρ 1) (1508697083091 / 32000000000000 : ℝ) ≤
+      (-(6405153826788440415589 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1626291939211537 / 40000000000000000 : ℝ) ≤
+      (-(6405153826788440415589 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (1626291939211537 / 40000000000000000 : ℝ) =
+            (1626291939211537 / 1250000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1626291939211537 / 1250000000000000 : ℝ) ≤
+            (526317976211559584411 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (406175349224831 / 10000000000000000 : ℝ)) (q :=
+      (1626291939211537 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_87_2 :
+    Uω (aρ 2) (bρ 2) (1508697083091 / 32000000000000 : ℝ) ≤
+      (-(6538167184020175693239 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1521650895320563 / 40000000000000000 : ℝ) ≤
+      (-(6538167184020175693239 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (1521650895320563 / 40000000000000000 : ℝ) =
+            (1521650895320563 / 1250000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1521650895320563 / 1250000000000000 : ℝ) ≤
+            (393304618979824306761 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (11800744993417 / 312500000000000 : ℝ)) (q :=
+      (1521650895320563 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_87_3 :
+    Uω (aρ 3) (bρ 3) (1508697083091 / 32000000000000 : ℝ) ≤
+      (-(34283541402370319659499 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1297611559329427 / 40000000000000000 : ℝ) ≤
+      (-(34283541402370319659499 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (1297611559329427 / 40000000000000000 : ℝ) =
+            (1297611559329427 / 1250000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1297611559329427 / 1250000000000000 : ℝ) ≤
+            (373817612629680340501 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (9781258691993 / 312500000000000 : ℝ)) (q :=
+      (1297611559329427 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_87_4 :
+    Uω (aρ 4) (bρ 4) (1508697083091 / 32000000000000 : ℝ) ≤
+      (-(1551850317616410832219 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (165268434059053 / 8000000000000000 : ℝ) ≤
+      (-(1551850317616410832219 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (165268434059053 / 8000000000000000 : ℝ) =
+            (165268434059053 / 125000000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (165268434059053 / 125000000000000 : ℝ) ≤
+            (111702915103589167781 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (31131952938339 / 2000000000000000 : ℝ)) (q :=
+      (165268434059053 / 8000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_87_5 :
+    Uω (aρ 5) (bρ 5) (1508697083091 / 32000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_87_6 :
+    Uω (aρ 6) (bρ 6) (1508697083091 / 32000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_87_7 :
+    Uω (aρ 7) (bρ 7) (1508697083091 / 32000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_87_8 :
+    Uω (aρ 8) (bρ 8) (1508697083091 / 32000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_87_9 :
+    Uω (aρ 9) (bρ 9) (1508697083091 / 32000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_87_10 :
+    Uω (aρ 10) (bρ 10) (1508697083091 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_87_11 :
+    Uω (aρ 11) (bρ 11) (1508697083091 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_87_12 :
+    Uω (aρ 12) (bρ 12) (1508697083091 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_87_13 :
+    Uω (aρ 13) (bρ 13) (1508697083091 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_87_14 :
+    Uω (aρ 14) (bρ 14) (1508697083091 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_87_15 :
+    Uω (aρ 15) (bρ 15) (1508697083091 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_87_16 :
+    Uω (aρ 16) (bρ 16) (1508697083091 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_87 :
+    Uρ (1508697083091 / 32000000000000 : ℝ) ≤
+      (-(3308512879034082580067 / 1250000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1508697083091 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(6405153826788440415589 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_87_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1508697083091 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(6538167184020175693239 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_87_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1508697083091 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(34283541402370319659499 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_87_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1508697083091 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1551850317616410832219 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_87_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1508697083091 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_87_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1508697083091 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_87_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1508697083091 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_87_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1508697083091 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_87_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1508697083091 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_87_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1508697083091 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_87_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1508697083091 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_87_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1508697083091 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_87_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1508697083091 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_87_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1508697083091 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_87_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1508697083091 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_87_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1508697083091 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_87_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/U07.lean b/LeanPool/Zeta5Irrational/Table/U07.lean
new file mode 100644
index 0000000000..a3851f7f6c
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/U07.lean
@@ -0,0 +1,2848 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-! # Certified arcsine potential bounds (U07) -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma U_88_1 :
+    Uω (aρ 1) (bρ 1) (3041335677499 / 64000000000000 : ℝ) ≤
+      (-(31934091820714900994499 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (131301591905011 / 3200000000000000 : ℝ) ≤
+      (-(31934091820714900994499 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (131301591905011 / 3200000000000000 : ℝ) =
+            (131301591905011 / 100000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (131301591905011 / 100000000000000 : ℝ) ≤
+            (2723267194285099005501 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4099234678821 / 100000000000000 : ℝ)) (q :=
+      (131301591905011 / 3200000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_88_2 :
+    Uω (aρ 2) (bρ 2) (3041335677499 / 64000000000000 : ℝ) ≤
+      (-(254627088457477178169 / 78125000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (614689503320911 / 16000000000000000 : ℝ) ≤
+      (-(254627088457477178169 / 78125000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (614689503320911 / 16000000000000000 : ℝ) =
+            (614689503320911 / 500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (614689503320911 / 500000000000000 : ℝ) ≤
+            (16133528847210321831 / 78125000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (9535485253423 / 250000000000000 : ℝ)) (q :=
+      (614689503320911 / 16000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_88_3 :
+    Uω (aρ 3) (bρ 3) (3041335677499 / 64000000000000 : ℝ) ≤
+      (-(34164784516352722640813 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2626226911586943 / 80000000000000000 : ℝ) ≤
+      (-(34164784516352722640813 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (2626226911586943 / 80000000000000000 : ℝ) =
+            (2626226911586943 / 2500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2626226911586943 / 2500000000000000 : ℝ) ≤
+            (492574498647277359187 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (317010365232517 / 10000000000000000 : ℝ)) (q :=
+      (2626226911586943 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_88_4 :
+    Uω (aρ 4) (bρ 4) (3041335677499 / 64000000000000 : ℝ) ≤
+      (-(1928027625307929110733 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1692101728321819 / 80000000000000000 : ℝ) ≤
+      (-(1928027625307929110733 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (1692101728321819 / 80000000000000000 : ℝ) =
+            (1692101728321819 / 1250000000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1692101728321819 / 1250000000000000 : ℝ) ≤
+            (151413915592070889267 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (40443312620309 / 2500000000000000 : ℝ)) (q :=
+      (1692101728321819 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_88_5 :
+    Uω (aρ 5) (bρ 5) (3041335677499 / 64000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_88_6 :
+    Uω (aρ 6) (bρ 6) (3041335677499 / 64000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_88_7 :
+    Uω (aρ 7) (bρ 7) (3041335677499 / 64000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_88_8 :
+    Uω (aρ 8) (bρ 8) (3041335677499 / 64000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_88_9 :
+    Uω (aρ 9) (bρ 9) (3041335677499 / 64000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_88_10 :
+    Uω (aρ 10) (bρ 10) (3041335677499 / 64000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_88_11 :
+    Uω (aρ 11) (bρ 11) (3041335677499 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_88_12 :
+    Uω (aρ 12) (bρ 12) (3041335677499 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_88_13 :
+    Uω (aρ 13) (bρ 13) (3041335677499 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_88_14 :
+    Uω (aρ 14) (bρ 14) (3041335677499 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_88_15 :
+    Uω (aρ 15) (bρ 15) (3041335677499 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_88_16 :
+    Uω (aρ 16) (bρ 16) (3041335677499 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_88 :
+    Uρ (3041335677499 / 64000000000000 : ℝ) ≤
+      (-(826419256172916514983 / 312500000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (3041335677499 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(31934091820714900994499 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_88_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (3041335677499 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) * (-(254627088457477178169 / 78125000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_88_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (3041335677499 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(34164784516352722640813 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_88_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (3041335677499 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1928027625307929110733 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_88_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (3041335677499 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_88_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (3041335677499 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_88_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (3041335677499 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_88_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (3041335677499 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_88_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (3041335677499 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_88_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (3041335677499 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_88_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (3041335677499 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_88_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (3041335677499 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_88_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (3041335677499 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_88_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (3041335677499 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_88_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (3041335677499 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_88_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (3041335677499 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_88_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_89_1 :
+    Uω (aρ 1) (bρ 1) (191579824301 / 4000000000000 : ℝ) ≤
+      (-(796081223438251080513 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (414061898909553 / 10000000000000000 : ℝ) ≤
+      (-(796081223438251080513 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (414061898909553 / 10000000000000000 : ℝ) =
+            (414061898909553 / 312500000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (414061898909553 / 312500000000000 : ℝ) ≤
+            (70352751936748919487 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (206835727575803 / 5000000000000000 : ℝ)) (q :=
+      (414061898909553 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_89_2 :
+    Uω (aρ 2) (bρ 2) (191579824301 / 4000000000000 : ℝ) ≤
+      (-(6498934924237707349841 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (387948620148739 / 10000000000000000 : ℝ) ≤
+      (-(6498934924237707349841 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (387948620148739 / 10000000000000000 : ℝ) =
+            (387948620148739 / 312500000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (387948620148739 / 312500000000000 : ℝ) ≤
+            (432536878762292650159 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (192606955069989 / 5000000000000000 : ℝ)) (q :=
+      (387948620148739 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_89_3 :
+    Uω (aρ 3) (bρ 3) (191579824301 / 4000000000000 : ℝ) ≤
+      (-(17023759633831421878791 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1297463233747 / 39062500000000 : ℝ) ≤
+      (-(17023759633831421878791 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show (1297463233747 / 39062500000000 : ℝ) = (1297463233747 / 1220703125000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1297463233747 / 1220703125000 : ℝ) ≤
+            (304919873668578121209 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (80253487967741 / 2500000000000000 : ℝ)) (q :=
+      (1297463233747 / 39062500000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_89_4 :
+    Uω (aρ 4) (bρ 4) (191579824301 / 4000000000000 : ℝ) ≤
+      (-(19166747049474907076563 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (216370228340183 / 10000000000000000 : ℝ) ≤
+      (-(19166747049474907076563 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (216370228340183 / 10000000000000000 : ℝ) =
+            (216370228340183 / 156250000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (216370228340183 / 156250000000000 : ℝ) ≤
+            (1627668359525092923437 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (83873706968933 / 5000000000000000 : ℝ)) (q :=
+      (216370228340183 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_89_5 :
+    Uω (aρ 5) (bρ 5) (191579824301 / 4000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_89_6 :
+    Uω (aρ 6) (bρ 6) (191579824301 / 4000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_89_7 :
+    Uω (aρ 7) (bρ 7) (191579824301 / 4000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_89_8 :
+    Uω (aρ 8) (bρ 8) (191579824301 / 4000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_89_9 :
+    Uω (aρ 9) (bρ 9) (191579824301 / 4000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_89_10 :
+    Uω (aρ 10) (bρ 10) (191579824301 / 4000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_89_11 :
+    Uω (aρ 11) (bρ 11) (191579824301 / 4000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_89_12 :
+    Uω (aρ 12) (bρ 12) (191579824301 / 4000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_89_13 :
+    Uω (aρ 13) (bρ 13) (191579824301 / 4000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_89_14 :
+    Uω (aρ 14) (bρ 14) (191579824301 / 4000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_89_15 :
+    Uω (aρ 15) (bρ 15) (191579824301 / 4000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_89_16 :
+    Uω (aρ 16) (bρ 16) (191579824301 / 4000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_89 :
+    Uρ (191579824301 / 4000000000000 : ℝ) ≤
+      (-(26423334234748999464827 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (191579824301 / 4000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(796081223438251080513 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_89_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (191579824301 / 4000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(6498934924237707349841 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_89_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (191579824301 / 4000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(17023759633831421878791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_89_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (191579824301 / 4000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(19166747049474907076563 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_89_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (191579824301 / 4000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_89_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (191579824301 / 4000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_89_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (191579824301 / 4000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_89_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (191579824301 / 4000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_89_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (191579824301 / 4000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_89_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (191579824301 / 4000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_89_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (191579824301 / 4000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_89_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (191579824301 / 4000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_89_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (191579824301 / 4000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_89_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (191579824301 / 4000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_89_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (191579824301 / 4000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_89_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (191579824301 / 4000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_89_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_90_1 :
+    Uω (aρ 1) (bρ 1) (3089218700133 / 64000000000000 : ℝ) ≤
+      (-(31753225403554109637937 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3342450073533213 / 80000000000000000 : ℝ) ≤
+      (-(31753225403554109637937 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (3342450073533213 / 80000000000000000 : ℝ) =
+            (3342450073533213 / 2500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3342450073533213 / 2500000000000000 : ℝ) ≤
+            (2904133611445890362063 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (208709657286261 / 5000000000000000 : ℝ)) (q :=
+      (3342450073533213 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_90_2 :
+    Uω (aρ 2) (bρ 2) (3089218700133 / 64000000000000 : ℝ) ≤
+      (-(16199019208965554918851 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3133726249682913 / 80000000000000000 : ℝ) ≤
+      (-(16199019208965554918851 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (3133726249682913 / 80000000000000000 : ℝ) =
+            (3133726249682913 / 2500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3133726249682913 / 2500000000000000 : ℝ) ≤
+            (1129660298534445081149 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (389007371119947 / 10000000000000000 : ℝ)) (q :=
+      (3133726249682913 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_90_3 :
+    Uω (aρ 3) (bρ 3) (3089218700133 / 64000000000000 : ℝ) ≤
+      (-(6786341272376926062427 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2688157452933953 / 80000000000000000 : ℝ) ≤
+      (-(6786341272376926062427 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (2688157452933953 / 80000000000000000 : ℝ) =
+            (2688157452933953 / 2500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2688157452933953 / 2500000000000000 : ℝ) ≤
+            (145130530623073937573 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (325011278282707 / 10000000000000000 : ℝ)) (q :=
+      (2688157452933953 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_90_4 :
+    Uω (aρ 4) (bρ 4) (3089218700133 / 64000000000000 : ℝ) ≤
+      (-(9528575281149508014001 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1769322173922313 / 80000000000000000 : ℝ) ≤
+      (-(9528575281149508014001 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (1769322173922313 / 80000000000000000 : ℝ) =
+            (1769322173922313 / 1250000000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1769322173922313 / 1250000000000000 : ℝ) ≤
+            (868632423350491985999 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (173596639594797 / 10000000000000000 : ℝ)) (q :=
+      (1769322173922313 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_90_5 :
+    Uω (aρ 5) (bρ 5) (3089218700133 / 64000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_90_6 :
+    Uω (aρ 6) (bρ 6) (3089218700133 / 64000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_90_7 :
+    Uω (aρ 7) (bρ 7) (3089218700133 / 64000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_90_8 :
+    Uω (aρ 8) (bρ 8) (3089218700133 / 64000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_90_9 :
+    Uω (aρ 9) (bρ 9) (3089218700133 / 64000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_90_10 :
+    Uω (aρ 10) (bρ 10) (3089218700133 / 64000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_90_11 :
+    Uω (aρ 11) (bρ 11) (3089218700133 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_90_12 :
+    Uω (aρ 12) (bρ 12) (3089218700133 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_90_13 :
+    Uω (aρ 13) (bρ 13) (3089218700133 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_90_14 :
+    Uω (aρ 14) (bρ 14) (3089218700133 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_90_15 :
+    Uω (aρ 15) (bρ 15) (3089218700133 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_90_16 :
+    Uω (aρ 16) (bρ 16) (3089218700133 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_90 :
+    Uρ (3089218700133 / 64000000000000 : ℝ) ≤
+      (-(13200904617235475791283 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (3089218700133 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(31753225403554109637937 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_90_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (3089218700133 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(16199019208965554918851 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_90_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (3089218700133 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(6786341272376926062427 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_90_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (3089218700133 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(9528575281149508014001 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_90_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (3089218700133 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_90_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (3089218700133 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_90_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (3089218700133 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_90_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (3089218700133 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_90_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (3089218700133 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_90_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (3089218700133 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_90_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (3089218700133 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_90_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (3089218700133 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_90_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (3089218700133 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_90_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (3089218700133 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_90_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (3089218700133 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_90_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (3089218700133 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_90_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_91_1 :
+    Uω (aρ 1) (bρ 1) (62263204229 / 1280000000000 : ℝ) ≤
+      (-(1583200327193589922537 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (67448089160959 / 1600000000000000 : ℝ) ≤
+      (-(1583200327193589922537 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (67448089160959 / 1600000000000000 : ℝ) = (67448089160959 / 50000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (67448089160959 / 50000000000000 : ℝ) ≤
+            (149667623556410077463 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (16846681982317 / 400000000000000 : ℝ)) (q :=
+      (67448089160959 / 1600000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_91_2 :
+    Uω (aρ 2) (bρ 2) (62263204229 / 1280000000000 : ℝ) ≤
+      (-(16151169947406386751941 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1581929751247923 / 40000000000000000 : ℝ) ≤
+      (-(16151169947406386751941 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (1581929751247923 / 40000000000000000 : ℝ) =
+            (1581929751247923 / 1250000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1581929751247923 / 1250000000000000 : ℝ) ≤
+            (1177509560093613248059 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (392799823179899 / 10000000000000000 : ℝ)) (q :=
+      (1581929751247923 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_91_3 :
+    Uω (aρ 3) (bρ 3) (62263204229 / 1280000000000 : ℝ) ≤
+      (-(1056790878554544750231 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (271908607497609 / 8000000000000000 : ℝ) ≤
+      (-(1056790878554544750231 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (271908607497609 / 8000000000000000 : ℝ) =
+            (271908607497609 / 250000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (271908607497609 / 250000000000000 : ℝ) ≤
+            (26251590664205249769 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (8225064316249 / 250000000000000 : ℝ)) (q :=
+      (271908607497609 / 8000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_91_4 :
+    Uω (aρ 4) (bρ 4) (62263204229 / 1280000000000 : ℝ) ≤
+      (-(37902304216173367332429 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (903615836022631 / 40000000000000000 : ℝ) ≤
+      (-(37902304216173367332429 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (903615836022631 / 40000000000000000 : ℝ) =
+            (903615836022631 / 625000000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (903615836022631 / 625000000000000 : ℝ) ≤
+            (3686526601826632667571 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (179333152982253 / 10000000000000000 : ℝ)) (q :=
+      (903615836022631 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_91_5 :
+    Uω (aρ 5) (bρ 5) (62263204229 / 1280000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_91_6 :
+    Uω (aρ 6) (bρ 6) (62263204229 / 1280000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_91_7 :
+    Uω (aρ 7) (bρ 7) (62263204229 / 1280000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_91_8 :
+    Uω (aρ 8) (bρ 8) (62263204229 / 1280000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_91_9 :
+    Uω (aρ 9) (bρ 9) (62263204229 / 1280000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_91_10 :
+    Uω (aρ 10) (bρ 10) (62263204229 / 1280000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_91_11 :
+    Uω (aρ 11) (bρ 11) (62263204229 / 1280000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_91_12 :
+    Uω (aρ 12) (bρ 12) (62263204229 / 1280000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_91_13 :
+    Uω (aρ 13) (bρ 13) (62263204229 / 1280000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_91_14 :
+    Uω (aρ 14) (bρ 14) (62263204229 / 1280000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_91_15 :
+    Uω (aρ 15) (bρ 15) (62263204229 / 1280000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_91_16 :
+    Uω (aρ 16) (bρ 16) (62263204229 / 1280000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_91 :
+    Uρ (62263204229 / 1280000000000 : ℝ) ≤
+      (-(412199998590688327739 / 156250000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (62263204229 / 1280000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1583200327193589922537 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_91_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (62263204229 / 1280000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(16151169947406386751941 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_91_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (62263204229 / 1280000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1056790878554544750231 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_91_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (62263204229 / 1280000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(37902304216173367332429 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_91_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (62263204229 / 1280000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_91_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (62263204229 / 1280000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_91_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (62263204229 / 1280000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_91_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (62263204229 / 1280000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_91_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (62263204229 / 1280000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_91_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (62263204229 / 1280000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_91_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (62263204229 / 1280000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_91_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (62263204229 / 1280000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_91_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (62263204229 / 1280000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_91_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (62263204229 / 1280000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_91_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (62263204229 / 1280000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_91_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (62263204229 / 1280000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_91_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_92_1 :
+    Uω (aρ 1) (bρ 1) (3137101722767 / 64000000000000 : ℝ) ≤
+      (-(31575578075022150256599 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3402358357990847 / 80000000000000000 : ℝ) ≤
+      (-(31575578075022150256599 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (3402358357990847 / 80000000000000000 : ℝ) =
+            (3402358357990847 / 2500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3402358357990847 / 2500000000000000 : ℝ) ≤
+            (3081780939977849743401 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (26557166462523 / 625000000000000 : ℝ)) (q :=
+      (3402358357990847 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_92_2 :
+    Uω (aρ 2) (bρ 2) (3137101722767 / 64000000000000 : ℝ) ≤
+      (-(16103780395389085637527 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3193988835403747 / 80000000000000000 : ℝ) ≤
+      (-(16103780395389085637527 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (3193988835403747 / 80000000000000000 : ℝ) =
+            (3193988835403747 / 2500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3193988835403747 / 2500000000000000 : ℝ) ≤
+            (1224899112110914362473 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (396591295263593 / 10000000000000000 : ℝ)) (q :=
+      (3193988835403747 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_92_3 :
+    Uω (aρ 3) (bρ 3) (3137101722767 / 64000000000000 : ℝ) ≤
+      (-(33704288357139015076803 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2749991436463211 / 80000000000000000 : ℝ) ≤
+      (-(33704288357139015076803 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (2749991436463211 / 80000000000000000 : ℝ) =
+            (2749991436463211 / 2500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2749991436463211 / 2500000000000000 : ℝ) ≤
+            (953070657860984923197 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (332988051878459 / 10000000000000000 : ℝ)) (q :=
+      (2749991436463211 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_92_4 :
+    Uω (aρ 4) (bρ 4) (3137101722767 / 64000000000000 : ℝ) ≤
+      (-(18848462387150699761219 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1844732269690687 / 80000000000000000 : ℝ) ≤
+      (-(18848462387150699761219 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (1844732269690687 / 80000000000000000 : ℝ) =
+            (1844732269690687 / 1250000000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1844732269690687 / 1250000000000000 : ℝ) ≤
+            (1945953021849300238781 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (23120930156291 / 1250000000000000 : ℝ)) (q :=
+      (1844732269690687 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_92_5 :
+    Uω (aρ 5) (bρ 5) (3137101722767 / 64000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_92_6 :
+    Uω (aρ 6) (bρ 6) (3137101722767 / 64000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_92_7 :
+    Uω (aρ 7) (bρ 7) (3137101722767 / 64000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_92_8 :
+    Uω (aρ 8) (bρ 8) (3137101722767 / 64000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_92_9 :
+    Uω (aρ 9) (bρ 9) (3137101722767 / 64000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_92_10 :
+    Uω (aρ 10) (bρ 10) (3137101722767 / 64000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_92_11 :
+    Uω (aρ 11) (bρ 11) (3137101722767 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_92_12 :
+    Uω (aρ 12) (bρ 12) (3137101722767 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_92_13 :
+    Uω (aρ 13) (bρ 13) (3137101722767 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_92_14 :
+    Uω (aρ 14) (bρ 14) (3137101722767 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_92_15 :
+    Uω (aρ 15) (bρ 15) (3137101722767 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_92_16 :
+    Uω (aρ 16) (bρ 16) (3137101722767 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_92 :
+    Uρ (3137101722767 / 64000000000000 : ℝ) ≤
+      (-(13180135171331573725517 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (3137101722767 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(31575578075022150256599 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_92_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (3137101722767 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(16103780395389085637527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_92_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (3137101722767 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(33704288357139015076803 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_92_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (3137101722767 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(18848462387150699761219 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_92_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (3137101722767 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_92_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (3137101722767 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_92_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (3137101722767 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_92_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (3137101722767 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_92_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (3137101722767 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_92_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (3137101722767 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_92_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (3137101722767 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_92_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (3137101722767 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_92_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (3137101722767 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_92_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (3137101722767 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_92_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (3137101722767 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_92_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (3137101722767 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_92_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_93_1 :
+    Uω (aρ 1) (bρ 1) (790260808521 / 16000000000000 : ℝ) ≤
+      (-(31487926091174342905069 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (429038973258891 / 10000000000000000 : ℝ) ≤
+      (-(31487926091174342905069 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (429038973258891 / 10000000000000000 : ℝ) =
+            (429038973258891 / 312500000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (429038973258891 / 312500000000000 : ℝ) ≤
+            (3169432923825657094931 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (428662159277157 / 10000000000000000 : ℝ)) (q :=
+      (429038973258891 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_93_2 :
+    Uω (aρ 2) (bρ 2) (790260808521 / 16000000000000 : ℝ) ≤
+      (-(32113683379671497832899 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (32241143597669 / 800000000000000 : ℝ) ≤
+      (-(32113683379671497832899 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (32241143597669 / 800000000000000 : ℝ) = (32241143597669 / 25000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (32241143597669 / 25000000000000 : ℝ) ≤
+            (2543675635328502167101 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4003818152111 / 100000000000000 : ℝ)) (q :=
+      (32241143597669 / 800000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_93_3 :
+    Uω (aρ 3) (bρ 3) (790260808521 / 16000000000000 : ℝ) ≤
+      (-(1679630618076205600743 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (139043718136819 / 4000000000000000 : ℝ) ≤
+      (-(1679630618076205600743 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (139043718136819 / 4000000000000000 : ℝ) =
+            (139043718136819 / 125000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (139043718136819 / 125000000000000 : ℝ) ≤
+            (53237332673794399257 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (33696792230347 / 1000000000000000 : ℝ)) (q :=
+      (139043718136819 / 4000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_93_4 :
+    Uω (aρ 4) (bρ 4) (790260808521 / 16000000000000 : ℝ) ≤
+      (-(18748829261004226629189 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (117616265462433 / 5000000000000000 : ℝ) ≤
+      (-(18748829261004226629189 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (117616265462433 / 5000000000000000 : ℝ) =
+            (117616265462433 / 78125000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (117616265462433 / 78125000000000 : ℝ) ≤
+            (2045586147995773370811 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (190508574534107 / 10000000000000000 : ℝ)) (q :=
+      (117616265462433 / 5000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_93_5 :
+    Uω (aρ 5) (bρ 5) (790260808521 / 16000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_93_6 :
+    Uω (aρ 6) (bρ 6) (790260808521 / 16000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_93_7 :
+    Uω (aρ 7) (bρ 7) (790260808521 / 16000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_93_8 :
+    Uω (aρ 8) (bρ 8) (790260808521 / 16000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_93_9 :
+    Uω (aρ 9) (bρ 9) (790260808521 / 16000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_93_10 :
+    Uω (aρ 10) (bρ 10) (790260808521 / 16000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_93_11 :
+    Uω (aρ 11) (bρ 11) (790260808521 / 16000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_93_12 :
+    Uω (aρ 12) (bρ 12) (790260808521 / 16000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_93_13 :
+    Uω (aρ 13) (bρ 13) (790260808521 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_93_14 :
+    Uω (aρ 14) (bρ 14) (790260808521 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_93_15 :
+    Uω (aρ 15) (bρ 15) (790260808521 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_93_16 :
+    Uω (aρ 16) (bρ 16) (790260808521 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_93 :
+    Uρ (790260808521 / 16000000000000 : ℝ) ≤
+      (-(26340189022211827785901 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (790260808521 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(31487926091174342905069 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_93_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (790260808521 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(32113683379671497832899 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_93_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (790260808521 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1679630618076205600743 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_93_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (790260808521 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(18748829261004226629189 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_93_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (790260808521 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_93_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (790260808521 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_93_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (790260808521 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_93_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (790260808521 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_93_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (790260808521 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_93_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (790260808521 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_93_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (790260808521 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_93_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (790260808521 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_93_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (790260808521 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_93_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (790260808521 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_93_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (790260808521 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_93_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (790260808521 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_93_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_94_1 :
+    Uω (aρ 1) (bρ 1) (3184984745401 / 64000000000000 : ℝ) ≤
+      (-(15700518525454701202457 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (692452950911409 / 16000000000000000 : ℝ) ≤
+      (-(15700518525454701202457 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (692452950911409 / 16000000000000000 : ℝ) =
+            (692452950911409 / 500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (692452950911409 / 500000000000000 : ℝ) ≤
+            (1628160982045298797543 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (86481908051071 / 2000000000000000 : ℝ)) (q :=
+      (692452950911409 / 16000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_94_2 :
+    Uω (aρ 2) (bρ 2) (3184984745401 / 64000000000000 : ℝ) ≤
+      (-(32020690449308287470393 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3254236182740573 / 80000000000000000 : ℝ) ≤
+      (-(32020690449308287470393 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (3254236182740573 / 80000000000000000 : ℝ) =
+            (3254236182740573 / 2500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3254236182740573 / 2500000000000000 : ℝ) ≤
+            (2636668565691712529607 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (404171409811237 / 10000000000000000 : ℝ)) (q :=
+      (3254236182740573 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_94_3 :
+    Uω (aρ 3) (bρ 3) (3184984745401 / 64000000000000 : ℝ) ≤
+      (-(6696449350830705340347 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2811735639464453 / 80000000000000000 : ℝ) ≤
+      (-(6696449350830705340347 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (2811735639464453 / 80000000000000000 : ℝ) =
+            (2811735639464453 / 2500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2811735639464453 / 2500000000000000 : ℝ) ≤
+            (235022452169294659653 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (340942380342207 / 10000000000000000 : ℝ)) (q :=
+      (2811735639464453 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_94_4 :
+    Uω (aρ 4) (bρ 4) (3184984745401 / 64000000000000 : ℝ) ≤
+      (-(37304062711062346239319 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1918647214652921 / 80000000000000000 : ℝ) ≤
+      (-(37304062711062346239319 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (1918647214652921 / 80000000000000000 : ℝ) =
+            (1918647214652921 / 1250000000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1918647214652921 / 1250000000000000 : ℝ) ≤
+            (4284768106937653760681 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (48991113801081 / 2500000000000000 : ℝ)) (q :=
+      (1918647214652921 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_94_5 :
+    Uω (aρ 5) (bρ 5) (3184984745401 / 64000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_94_6 :
+    Uω (aρ 6) (bρ 6) (3184984745401 / 64000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_94_7 :
+    Uω (aρ 7) (bρ 7) (3184984745401 / 64000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_94_8 :
+    Uω (aρ 8) (bρ 8) (3184984745401 / 64000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_94_9 :
+    Uω (aρ 9) (bρ 9) (3184984745401 / 64000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_94_10 :
+    Uω (aρ 10) (bρ 10) (3184984745401 / 64000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_94_11 :
+    Uω (aρ 11) (bρ 11) (3184984745401 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_94_12 :
+    Uω (aρ 12) (bρ 12) (3184984745401 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_94_13 :
+    Uω (aρ 13) (bρ 13) (3184984745401 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_94_14 :
+    Uω (aρ 14) (bρ 14) (3184984745401 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_94_15 :
+    Uω (aρ 15) (bρ 15) (3184984745401 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_94_16 :
+    Uω (aρ 16) (bρ 16) (3184984745401 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_94 :
+    Uρ (3184984745401 / 64000000000000 : ℝ) ≤
+      (-(6580132024030622994311 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (3184984745401 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(15700518525454701202457 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_94_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (3184984745401 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(32020690449308287470393 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_94_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (3184984745401 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(6696449350830705340347 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_94_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (3184984745401 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(37304062711062346239319 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_94_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (3184984745401 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_94_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (3184984745401 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_94_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (3184984745401 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_94_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (3184984745401 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_94_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (3184984745401 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_94_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (3184984745401 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_94_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (3184984745401 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_94_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (3184984745401 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_94_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (3184984745401 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_94_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (3184984745401 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_94_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (3184984745401 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_94_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (3184984745401 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_94_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_95_1 :
+    Uω (aρ 1) (bρ 1) (1604463128359 / 32000000000000 : ℝ) ≤
+      (-(31314897764576203369223 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1746108637647431 / 40000000000000000 : ℝ) ≤
+      (-(31314897764576203369223 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (1746108637647431 / 40000000000000000 : ℝ) =
+            (1746108637647431 / 1250000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1746108637647431 / 1250000000000000 : ℝ) ≤
+            (3342461250423796630777 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (27259800581033 / 625000000000000 : ℝ)) (q :=
+      (1746108637647431 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_95_2 :
+    Uω (aρ 2) (bρ 2) (1604463128359 / 32000000000000 : ℝ) ≤
+      (-(1995535330098219433187 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1642177203736419 / 40000000000000000 : ℝ) ≤
+      (-(1995535330098219433187 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (1642177203736419 / 40000000000000000 : ℝ) =
+            (1642177203736419 / 1250000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1642177203736419 / 1250000000000000 : ℝ) ≤
+            (170549608339280566813 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (203980052425511 / 5000000000000000 : ℝ)) (q :=
+      (1642177203736419 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_95_3 :
+    Uω (aρ 3) (bρ 3) (1604463128359 / 32000000000000 : ℝ) ≤
+      (-(33373159447642545139733 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1421288007529287 / 40000000000000000 : ℝ) ≤
+      (-(33373159447642545139733 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (1421288007529287 / 40000000000000000 : ℝ) =
+            (1421288007529287 / 1250000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1421288007529287 / 1250000000000000 : ℝ) ≤
+            (1284199567357454860267 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (21556975818591 / 625000000000000 : ℝ)) (q :=
+      (1421288007529287 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_95_4 :
+    Uω (aρ 4) (bρ 4) (1604463128359 / 32000000000000 : ℝ) ≤
+      (-(37115746034298764438831 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (977560446800329 / 40000000000000000 : ℝ) ≤
+      (-(37115746034298764438831 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (977560446800329 / 40000000000000000 : ℝ) =
+            (977560446800329 / 625000000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (977560446800329 / 625000000000000 : ℝ) ≤
+            (4473084783701235561169 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (201342013797977 / 10000000000000000 : ℝ)) (q :=
+      (977560446800329 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_95_5 :
+    Uω (aρ 5) (bρ 5) (1604463128359 / 32000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_95_6 :
+    Uω (aρ 6) (bρ 6) (1604463128359 / 32000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_95_7 :
+    Uω (aρ 7) (bρ 7) (1604463128359 / 32000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_95_8 :
+    Uω (aρ 8) (bρ 8) (1604463128359 / 32000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_95_9 :
+    Uω (aρ 9) (bρ 9) (1604463128359 / 32000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_95_10 :
+    Uω (aρ 10) (bρ 10) (1604463128359 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_95_11 :
+    Uω (aρ 11) (bρ 11) (1604463128359 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_95_12 :
+    Uω (aρ 12) (bρ 12) (1604463128359 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_95_13 :
+    Uω (aρ 13) (bρ 13) (1604463128359 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_95_14 :
+    Uω (aρ 14) (bρ 14) (1604463128359 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_95_15 :
+    Uω (aρ 15) (bρ 15) (1604463128359 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_95_16 :
+    Uω (aρ 16) (bρ 16) (1604463128359 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_95 :
+    Uρ (1604463128359 / 32000000000000 : ℝ) ≤
+      (-(1643828923725105092647 / 625000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1604463128359 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(31314897764576203369223 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_95_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1604463128359 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1995535330098219433187 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_95_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1604463128359 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(33373159447642545139733 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_95_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1604463128359 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(37115746034298764438831 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_95_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1604463128359 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_95_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1604463128359 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_95_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1604463128359 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_95_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1604463128359 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_95_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1604463128359 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_95_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1604463128359 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_95_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1604463128359 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_95_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1604463128359 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_95_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1604463128359 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_95_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1604463128359 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_95_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1604463128359 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_95_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1604463128359 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_95_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_96_1 :
+    Uω (aρ 1) (bρ 1) (646573553607 / 12800000000000 : ℝ) ≤
+      (-(31229495382192123462893 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (704433871945367 / 16000000000000000 : ℝ) ≤
+      (-(31229495382192123462893 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (704433871945367 / 16000000000000000 : ℝ) =
+            (704433871945367 / 500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (704433871945367 / 500000000000000 : ℝ) ≤
+            (3427863632807876537107 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (10997599231531 / 250000000000000 : ℝ)) (q :=
+      (704433871945367 / 16000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_96_2 :
+    Uω (aρ 2) (bρ 2) (646573553607 / 12800000000000 : ℝ) ≤
+      (-(31837291633471834828943 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3314469133291263 / 80000000000000000 : ℝ) ≤
+      (-(31837291633471834828943 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (3314469133291263 / 80000000000000000 : ℝ) =
+            (3314469133291263 / 2500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3314469133291263 / 2500000000000000 : ℝ) ≤
+            (2820067381528165171057 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (411747925162347 / 10000000000000000 : ℝ)) (q :=
+      (3314469133291263 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_96_3 :
+    Uω (aρ 3) (bρ 3) (646573553607 / 12800000000000 : ℝ) ≤
+      (-(33265319572434076398587 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (574679240581723 / 16000000000000000 : ℝ) ≤
+      (-(33265319572434076398587 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (574679240581723 / 16000000000000000 : ℝ) =
+            (574679240581723 / 500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (574679240581723 / 500000000000000 : ℝ) ≤
+            (1392039442565923601413 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (69775159783337 / 2000000000000000 : ℝ)) (q :=
+      (574679240581723 / 16000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_96_4 :
+    Uω (aρ 4) (bρ 4) (646573553607 / 12800000000000 : ℝ) ≤
+      (-(18466180287477100611297 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1991305742591499 / 80000000000000000 : ℝ) ≤
+      (-(18466180287477100611297 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (1991305742591499 / 80000000000000000 : ℝ) =
+            (1991305742591499 / 1250000000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1991305742591499 / 1250000000000000 : ℝ) ≤
+            (2328235121522899388703 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (103323682451203 / 5000000000000000 : ℝ)) (q :=
+      (1991305742591499 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_96_5 :
+    Uω (aρ 5) (bρ 5) (646573553607 / 12800000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_96_6 :
+    Uω (aρ 6) (bρ 6) (646573553607 / 12800000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_96_7 :
+    Uω (aρ 7) (bρ 7) (646573553607 / 12800000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_96_8 :
+    Uω (aρ 8) (bρ 8) (646573553607 / 12800000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_96_9 :
+    Uω (aρ 9) (bρ 9) (646573553607 / 12800000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_96_10 :
+    Uω (aρ 10) (bρ 10) (646573553607 / 12800000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_96_11 :
+    Uω (aρ 11) (bρ 11) (646573553607 / 12800000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_96_12 :
+    Uω (aρ 12) (bρ 12) (646573553607 / 12800000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_96_13 :
+    Uω (aρ 13) (bρ 13) (646573553607 / 12800000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_96_14 :
+    Uω (aρ 14) (bρ 14) (646573553607 / 12800000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_96_15 :
+    Uω (aρ 15) (bρ 15) (646573553607 / 12800000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_96_16 :
+    Uω (aρ 16) (bρ 16) (646573553607 / 12800000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_96 :
+    Uρ (646573553607 / 12800000000000 : ℝ) ≤
+      (-(328529636040277536627 / 125000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (646573553607 / 12800000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(31229495382192123462893 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_96_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (646573553607 / 12800000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(31837291633471834828943 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_96_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (646573553607 / 12800000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(33265319572434076398587 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_96_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (646573553607 / 12800000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(18466180287477100611297 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_96_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (646573553607 / 12800000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_96_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (646573553607 / 12800000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_96_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (646573553607 / 12800000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_96_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (646573553607 / 12800000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_96_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (646573553607 / 12800000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_96_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (646573553607 / 12800000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_96_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (646573553607 / 12800000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_96_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (646573553607 / 12800000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_96_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (646573553607 / 12800000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_96_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (646573553607 / 12800000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_96_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (646573553607 / 12800000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_96_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (646573553607 / 12800000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_96_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_97_1 :
+    Uω (aρ 1) (bρ 1) (407101159919 / 8000000000000 : ℝ) ≤
+      (-(31144817381860143498661 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (222007563681771 / 5000000000000000 : ℝ) ≤
+      (-(31144817381860143498661 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (222007563681771 / 5000000000000000 : ℝ) =
+            (222007563681771 / 156250000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (222007563681771 / 156250000000000 : ℝ) ≤
+            (3512541633139856501339 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (221825511456667 / 5000000000000000 : ℝ)) (q :=
+      (222007563681771 / 5000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_97_2 :
+    Uω (aρ 2) (bρ 2) (407101159919 / 8000000000000 : ℝ) ≤
+      (-(1984178357444751687497 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4180725569849 / 100000000000000 : ℝ) ≤
+      (-(1984178357444751687497 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show (4180725569849 / 100000000000000 : ℝ) = (4180725569849 / 3125000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4180725569849 / 3125000000000 : ℝ) ≤
+            (181906580992748312503 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (8310697893321 / 200000000000000 : ℝ)) (q :=
+      (4180725569849 / 100000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_97_3 :
+    Uω (aρ 3) (bρ 3) (407101159919 / 8000000000000 : ℝ) ≤
+      (-(3315869741376076852277 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9075615260803 / 250000000000000 : ℝ) ≤
+      (-(3315869741376076852277 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show (9075615260803 / 250000000000000 : ℝ) = (9075615260803 / 7812500000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9075615260803 / 7812500000000 : ℝ) ≤
+            (149866160123923147723 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (35283510791049 / 1000000000000000 : ℝ)) (q :=
+      (9075615260803 / 250000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_97_4 :
+    Uω (aρ 4) (bρ 4) (407101159919 / 8000000000000 : ℝ) ≤
+      (-(36753595311212100745499 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (506805864423637 / 20000000000000000 : ℝ) ≤
+      (-(36753595311212100745499 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (506805864423637 / 20000000000000000 : ℝ) =
+            (506805864423637 / 312500000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (506805864423637 / 312500000000000 : ℝ) ≤
+            (4835235506787899254501 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (211885932534887 / 10000000000000000 : ℝ)) (q :=
+      (506805864423637 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_97_5 :
+    Uω (aρ 5) (bρ 5) (407101159919 / 8000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_97_6 :
+    Uω (aρ 6) (bρ 6) (407101159919 / 8000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_97_7 :
+    Uω (aρ 7) (bρ 7) (407101159919 / 8000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_97_8 :
+    Uω (aρ 8) (bρ 8) (407101159919 / 8000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_97_9 :
+    Uω (aρ 9) (bρ 9) (407101159919 / 8000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_97_10 :
+    Uω (aρ 10) (bρ 10) (407101159919 / 8000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_97_11 :
+    Uω (aρ 11) (bρ 11) (407101159919 / 8000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_97_12 :
+    Uω (aρ 12) (bρ 12) (407101159919 / 8000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_97_13 :
+    Uω (aρ 13) (bρ 13) (407101159919 / 8000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_97_14 :
+    Uω (aρ 14) (bρ 14) (407101159919 / 8000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_97_15 :
+    Uω (aρ 15) (bρ 15) (407101159919 / 8000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_97_16 :
+    Uω (aρ 16) (bρ 16) (407101159919 / 8000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_97 :
+    Uρ (407101159919 / 8000000000000 : ℝ) ≤
+      (-(26263832431500510068899 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (407101159919 / 8000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(31144817381860143498661 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_97_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (407101159919 / 8000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1984178357444751687497 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_97_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (407101159919 / 8000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3315869741376076852277 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_97_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (407101159919 / 8000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(36753595311212100745499 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_97_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (407101159919 / 8000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_97_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (407101159919 / 8000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_97_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (407101159919 / 8000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_97_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (407101159919 / 8000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_97_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (407101159919 / 8000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_97_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (407101159919 / 8000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_97_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (407101159919 / 8000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_97_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (407101159919 / 8000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_97_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (407101159919 / 8000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_97_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (407101159919 / 8000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_97_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (407101159919 / 8000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_97_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (407101159919 / 8000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_97_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_98_1 :
+    Uω (aρ 1) (bρ 1) (1652346150993 / 32000000000000 : ℝ) ≤
+      (-(15488793007051782660579 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1806011551951707 / 40000000000000000 : ℝ) ≤
+      (-(15488793007051782660579 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (1806011551951707 / 40000000000000000 : ℝ) =
+            (1806011551951707 / 1250000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1806011551951707 / 1250000000000000 : ℝ) ≤
+            (1839886500448217339421 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (451144821875541 / 10000000000000000 : ℝ)) (q :=
+      (1806011551951707 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_98_2 :
+    Uω (aρ 2) (bρ 2) (1652346150993 / 32000000000000 : ℝ) ≤
+      (-(789210603282412178031 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1702396628433119 / 40000000000000000 : ℝ) ≤
+      (-(789210603282412178031 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (1702396628433119 / 40000000000000000 : ℝ) =
+            (1702396628433119 / 1250000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1702396628433119 / 1250000000000000 : ℝ) ≤
+            (77223372092587821969 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (423106372626247 / 10000000000000000 : ℝ)) (q :=
+      (1702396628433119 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_98_3 :
+    Uω (aρ 3) (bρ 3) (1652346150993 / 32000000000000 : ℝ) ≤
+      (-(32948992811289585472047 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1482871145547441 / 40000000000000000 : ℝ) ≤
+      (-(32948992811289585472047 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (1482871145547441 / 40000000000000000 : ℝ) =
+            (1482871145547441 / 1250000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1482871145547441 / 1250000000000000 : ℝ) ≤
+            (1708366203710414527953 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (11273116797919 / 312500000000000 : ℝ)) (q :=
+      (1482871145547441 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_98_4 :
+    Uω (aρ 4) (bρ 4) (1652346150993 / 32000000000000 : ℝ) ≤
+      (-(9102208735492745651249 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1049166414959979 / 40000000000000000 : ℝ) ≤
+      (-(9102208735492745651249 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (1049166414959979 / 40000000000000000 : ℝ) =
+            (1049166414959979 / 625000000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1049166414959979 / 625000000000000 : ℝ) ≤
+            (1294998969007254348751 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (222181553304677 / 10000000000000000 : ℝ)) (q :=
+      (1049166414959979 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_98_5 :
+    Uω (aρ 5) (bρ 5) (1652346150993 / 32000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_98_6 :
+    Uω (aρ 6) (bρ 6) (1652346150993 / 32000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_98_7 :
+    Uω (aρ 7) (bρ 7) (1652346150993 / 32000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_98_8 :
+    Uω (aρ 8) (bρ 8) (1652346150993 / 32000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_98_9 :
+    Uω (aρ 9) (bρ 9) (1652346150993 / 32000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_98_10 :
+    Uω (aρ 10) (bρ 10) (1652346150993 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_98_11 :
+    Uω (aρ 11) (bρ 11) (1652346150993 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_98_12 :
+    Uω (aρ 12) (bρ 12) (1652346150993 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_98_13 :
+    Uω (aρ 13) (bρ 13) (1652346150993 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_98_14 :
+    Uω (aρ 14) (bρ 14) (1652346150993 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_98_15 :
+    Uω (aρ 15) (bρ 15) (1652346150993 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_98_16 :
+    Uω (aρ 16) (bρ 16) (1652346150993 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_98 :
+    Uρ (1652346150993 / 32000000000000 : ℝ) ≤
+      (-(26227745217472389991977 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1652346150993 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(15488793007051782660579 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_98_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1652346150993 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(789210603282412178031 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_98_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1652346150993 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(32948992811289585472047 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_98_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1652346150993 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(9102208735492745651249 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_98_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1652346150993 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_98_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1652346150993 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_98_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1652346150993 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_98_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1652346150993 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_98_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1652346150993 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_98_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1652346150993 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_98_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1652346150993 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_98_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1652346150993 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_98_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1652346150993 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_98_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1652346150993 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_98_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1652346150993 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_98_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1652346150993 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_98_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_99_1 :
+    Uω (aρ 1) (bρ 1) (167628766231 / 3200000000000 : ℝ) ≤
+      (-(30813109637784336635283 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (183596180579089 / 4000000000000000 : ℝ) ≤
+      (-(30813109637784336635283 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (183596180579089 / 4000000000000000 : ℝ) =
+            (183596180579089 / 125000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (183596180579089 / 125000000000000 : ℝ) ≤
+            (3844249377215663364717 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (45863822650857 / 1000000000000000 : ℝ)) (q :=
+      (183596180579089 / 4000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_99_2 :
+    Uω (aρ 2) (bρ 2) (167628766231 / 3200000000000 : ℝ) ≤
+      (-(31393158740030300939493 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (43312418864037 / 1000000000000000 : ℝ) ≤
+      (-(31393158740030300939493 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (43312418864037 / 1000000000000000 : ℝ) = (43312418864037 / 31250000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (43312418864037 / 31250000000000 : ℝ) ≤
+            (3264200274969699060507 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (86134942680773 / 2000000000000000 : ℝ)) (q :=
+      (43312418864037 / 1000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_99_3 :
+    Uω (aρ 3) (bρ 3) (167628766231 / 3200000000000 : ℝ) ≤
+      (-(8185957218103774495557 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (75680427309953 / 2000000000000000 : ℝ) ≤
+      (-(8185957218103774495557 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (75680427309953 / 2000000000000000 : ℝ) = (75680427309953 / 62500000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (75680427309953 / 62500000000000 : ℝ) ≤
+            (478382535646225504443 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (73725343114531 / 2000000000000000 : ℝ)) (q :=
+      (75680427309953 / 2000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_99_4 :
+    Uω (aρ 4) (bρ 4) (167628766231 / 3200000000000 : ℝ) ≤
+      (-(9019883879209497377887 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (542145274302063 / 20000000000000000 : ℝ) ≤
+      (-(9019883879209497377887 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (542145274302063 / 20000000000000000 : ℝ) =
+            (542145274302063 / 312500000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (542145274302063 / 312500000000000 : ℝ) ≤
+            (1377323825290502622113 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (58065474460047 / 2500000000000000 : ℝ)) (q :=
+      (542145274302063 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_99_5 :
+    Uω (aρ 5) (bρ 5) (167628766231 / 3200000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_99_6 :
+    Uω (aρ 6) (bρ 6) (167628766231 / 3200000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_99_7 :
+    Uω (aρ 7) (bρ 7) (167628766231 / 3200000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_99_8 :
+    Uω (aρ 8) (bρ 8) (167628766231 / 3200000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_99_9 :
+    Uω (aρ 9) (bρ 9) (167628766231 / 3200000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_99_10 :
+    Uω (aρ 10) (bρ 10) (167628766231 / 3200000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_99_11 :
+    Uω (aρ 11) (bρ 11) (167628766231 / 3200000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_99_12 :
+    Uω (aρ 12) (bρ 12) (167628766231 / 3200000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_99_13 :
+    Uω (aρ 13) (bρ 13) (167628766231 / 3200000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_99_14 :
+    Uω (aρ 14) (bρ 14) (167628766231 / 3200000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_99_15 :
+    Uω (aρ 15) (bρ 15) (167628766231 / 3200000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_99_16 :
+    Uω (aρ 16) (bρ 16) (167628766231 / 3200000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_99 :
+    Uρ (167628766231 / 3200000000000 : ℝ) ≤
+      (-(3274109383884848166437 / 1250000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (167628766231 / 3200000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(30813109637784336635283 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_99_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (167628766231 / 3200000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(31393158740030300939493 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_99_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (167628766231 / 3200000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(8185957218103774495557 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_99_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (167628766231 / 3200000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(9019883879209497377887 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_99_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (167628766231 / 3200000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_99_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (167628766231 / 3200000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_99_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (167628766231 / 3200000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_99_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (167628766231 / 3200000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_99_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (167628766231 / 3200000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_99_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (167628766231 / 3200000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_99_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (167628766231 / 3200000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_99_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (167628766231 / 3200000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_99_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (167628766231 / 3200000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_99_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (167628766231 / 3200000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_99_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (167628766231 / 3200000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_99_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (167628766231 / 3200000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_99_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/U08.lean b/LeanPool/Zeta5Irrational/Table/U08.lean
new file mode 100644
index 0000000000..10d12131ec
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/U08.lean
@@ -0,0 +1,2934 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-! # Certified arcsine potential bounds (U08) -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma U_100_1 :
+    Uω (aρ 1) (bρ 1) (1700229173627 / 32000000000000 : ℝ) ≤
+      (-(30651298815258709468093 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (373182261801303 / 8000000000000000 : ℝ) ≤
+      (-(30651298815258709468093 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (373182261801303 / 8000000000000000 : ℝ) =
+            (373182261801303 / 250000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (373182261801303 / 250000000000000 : ℝ) ≤
+            (4006060199741290531907 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (23306562791491 / 500000000000000 : ℝ)) (q :=
+      (373182261801303 / 8000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_100_2 :
+    Uω (aρ 2) (bρ 2) (1700229173627 / 32000000000000 : ℝ) ≤
+      (-(78052365153972182917 / 25000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1762590931396799 / 40000000000000000 : ℝ) ≤
+      (-(78052365153972182917 / 25000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (1762590931396799 / 40000000000000000 : ℝ) =
+            (1762590931396799 / 1250000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1762590931396799 / 1250000000000000 : ℝ) ≤
+            (8591032383527817083 / 25000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (219120039767481 / 5000000000000000 : ℝ)) (q :=
+      (1762590931396799 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_100_3 :
+    Uω (aρ 3) (bρ 3) (1700229173627 / 32000000000000 : ℝ) ≤
+      (-(16271501953539078684187 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1544312862311407 / 40000000000000000 : ℝ) ≤
+      (-(16271501953539078684187 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (1544312862311407 / 40000000000000000 : ℝ) =
+            (1544312862311407 / 1250000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1544312862311407 / 1250000000000000 : ℝ) ≤
+            (1057177553960921315813 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (188248575671133 / 5000000000000000 : ℝ)) (q :=
+      (1544312862311407 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_100_4 :
+    Uω (aρ 4) (bρ 4) (1700229173627 / 32000000000000 : ℝ) ≤
+      (-(35764101527639732941787 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1119037901687173 / 40000000000000000 : ℝ) ≤
+      (-(35764101527639732941787 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (1119037901687173 / 40000000000000000 : ℝ) =
+            (1119037901687173 / 625000000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1119037901687173 / 625000000000000 : ℝ) ≤
+            (5824729290360267058213 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (242153852095149 / 10000000000000000 : ℝ)) (q :=
+      (1119037901687173 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_100_5 :
+    Uω (aρ 5) (bρ 5) (1700229173627 / 32000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_100_6 :
+    Uω (aρ 6) (bρ 6) (1700229173627 / 32000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_100_7 :
+    Uω (aρ 7) (bρ 7) (1700229173627 / 32000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_100_8 :
+    Uω (aρ 8) (bρ 8) (1700229173627 / 32000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_100_9 :
+    Uω (aρ 9) (bρ 9) (1700229173627 / 32000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_100_10 :
+    Uω (aρ 10) (bρ 10) (1700229173627 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_100_11 :
+    Uω (aρ 11) (bρ 11) (1700229173627 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_100_12 :
+    Uω (aρ 12) (bρ 12) (1700229173627 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_100_13 :
+    Uω (aρ 13) (bρ 13) (1700229173627 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_100_14 :
+    Uω (aρ 14) (bρ 14) (1700229173627 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_100_15 :
+    Uω (aρ 15) (bρ 15) (1700229173627 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_100_16 :
+    Uω (aρ 16) (bρ 16) (1700229173627 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_100 :
+    Uρ (1700229173627 / 32000000000000 : ℝ) ≤
+      (-(408736191272622967401 / 156250000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1700229173627 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(30651298815258709468093 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_100_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1700229173627 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) * (-(78052365153972182917 / 25000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_100_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1700229173627 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(16271501953539078684187 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_100_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1700229173627 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(35764101527639732941787 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_100_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1700229173627 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_100_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1700229173627 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_100_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1700229173627 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_100_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1700229173627 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_100_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1700229173627 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_100_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1700229173627 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_100_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1700229173627 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_100_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1700229173627 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_100_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1700229173627 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_100_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1700229173627 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_100_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1700229173627 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_100_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1700229173627 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_100_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_101_1 :
+    Uω (aρ 1) (bρ 1) (107760667809 / 2000000000000 : ℝ) ≤
+      (-(3049206840087526189059 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9479300486123 / 200000000000000 : ℝ) ≤
+      (-(3049206840087526189059 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show (9479300486123 / 200000000000000 : ℝ) = (9479300486123 / 6250000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9479300486123 / 6250000000000 : ℝ) ≤
+            (416529061412473810941 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4736239276523 / 100000000000000 : ℝ)) (q :=
+      (9479300486123 / 200000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_101_2 :
+    Uω (aρ 2) (bρ 2) (107760667809 / 2000000000000 : ℝ) ≤
+      (-(3881460057703866310033 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (89633973091073 / 2000000000000000 : ℝ) ≤
+      (-(3881460057703866310033 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (89633973091073 / 2000000000000000 : ℝ) = (89633973091073 / 62500000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (89633973091073 / 62500000000000 : ℝ) ≤
+            (450709819171133689967 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (44580262246073 / 1000000000000000 : ℝ)) (q :=
+      (89633973091073 / 2000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_101_3 :
+    Uω (aρ 3) (bρ 3) (107760667809 / 2000000000000 : ℝ) ≤
+      (-(126352851143046549433 / 39062500000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (787493063159317 / 20000000000000000 : ℝ) ≤
+      (-(126352851143046549433 / 39062500000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (787493063159317 / 20000000000000000 : ℝ) =
+            (787493063159317 / 625000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (787493063159317 / 625000000000000 : ℝ) ≤
+            (9027457509297200567 / 39062500000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (384352061059317 / 10000000000000000 : ℝ)) (q :=
+      (787493063159317 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_101_4 :
+    Uω (aρ 4) (bρ 4) (107760667809 / 2000000000000 : ℝ) ≤
+      (-(2216324696954452647823 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (288363216890881 / 10000000000000000 : ℝ) ≤
+      (-(2216324696954452647823 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (288363216890881 / 10000000000000000 : ℝ) =
+            (288363216890881 / 156250000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (288363216890881 / 156250000000000 : ℝ) ≤
+            (382977229170547352177 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (125939806373381 / 5000000000000000 : ℝ)) (q :=
+      (288363216890881 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_101_5 :
+    Uω (aρ 5) (bρ 5) (107760667809 / 2000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_101_6 :
+    Uω (aρ 6) (bρ 6) (107760667809 / 2000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_101_7 :
+    Uω (aρ 7) (bρ 7) (107760667809 / 2000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_101_8 :
+    Uω (aρ 8) (bρ 8) (107760667809 / 2000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_101_9 :
+    Uω (aρ 9) (bρ 9) (107760667809 / 2000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_101_10 :
+    Uω (aρ 10) (bρ 10) (107760667809 / 2000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_101_11 :
+    Uω (aρ 11) (bρ 11) (107760667809 / 2000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_101_12 :
+    Uω (aρ 12) (bρ 12) (107760667809 / 2000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_101_13 :
+    Uω (aρ 13) (bρ 13) (107760667809 / 2000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_101_14 :
+    Uω (aρ 14) (bρ 14) (107760667809 / 2000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_101_15 :
+    Uω (aρ 15) (bρ 15) (107760667809 / 2000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_101_16 :
+    Uω (aρ 16) (bρ 16) (107760667809 / 2000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_101 :
+    Uρ (107760667809 / 2000000000000 : ℝ) ≤
+      (-(13063189390197467979439 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (107760667809 / 2000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3049206840087526189059 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_101_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (107760667809 / 2000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3881460057703866310033 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_101_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (107760667809 / 2000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) * (-(126352851143046549433 / 39062500000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_101_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (107760667809 / 2000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(2216324696954452647823 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_101_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (107760667809 / 2000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_101_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (107760667809 / 2000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_101_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (107760667809 / 2000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_101_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (107760667809 / 2000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_101_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (107760667809 / 2000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_101_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (107760667809 / 2000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_101_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (107760667809 / 2000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_101_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (107760667809 / 2000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_101_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (107760667809 / 2000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_101_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (107760667809 / 2000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_101_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (107760667809 / 2000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_101_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (107760667809 / 2000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_101_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_102_1 :
+    Uω (aρ 1) (bρ 1) (886026853789 / 16000000000000 : ℝ) ≤
+      (-(7545257017708016980781 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (488938915059511 / 10000000000000000 : ℝ) ≤
+      (-(7545257017708016980781 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (488938915059511 / 10000000000000000 : ℝ) =
+            (488938915059511 / 312500000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (488938915059511 / 312500000000000 : ℝ) ≤
+            (1119082736041983019219 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (488608264585897 / 10000000000000000 : ℝ)) (q :=
+      (488938915059511 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_102_2 :
+    Uω (aρ 2) (bρ 2) (886026853789 / 16000000000000 : ℝ) ≤
+      (-(30721594805733174632861 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (926420347528863 / 20000000000000000 : ℝ) ≤
+      (-(30721594805733174632861 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (926420347528863 / 20000000000000000 : ℝ) =
+            (926420347528863 / 625000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (926420347528863 / 625000000000000 : ℝ) ≤
+            (3935764209266825367139 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (230459897252869 / 5000000000000000 : ℝ)) (q :=
+      (926420347528863 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_102_3 :
+    Uω (aρ 3) (bρ 3) (886026853789 / 16000000000000 : ℝ) ≤
+      (-(15982371816390557011147 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10226542571457 / 250000000000000 : ℝ) ≤
+      (-(15982371816390557011147 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (10226542571457 / 250000000000000 : ℝ) = (10226542571457 / 7812500000000 : ℝ) / 2 ^ 5 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10226542571457 / 7812500000000 : ℝ) ≤
+            (1346307691109442988853 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (80003791808687 / 2000000000000000 : ℝ)) (q :=
+      (10226542571457 / 250000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_102_4 :
+    Uω (aρ 4) (bρ 4) (886026853789 / 16000000000000 : ℝ) ≤
+      (-(34888585885879677851941 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (610714120865997 / 20000000000000000 : ℝ) ≤
+      (-(34888585885879677851941 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (610714120865997 / 20000000000000000 : ℝ) =
+            (610714120865997 / 312500000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (610714120865997 / 312500000000000 : ℝ) ≤
+            (6700244932120322148059 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (16931490953617 / 625000000000000 : ℝ)) (q :=
+      (610714120865997 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_102_5 :
+    Uω (aρ 5) (bρ 5) (886026853789 / 16000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_102_6 :
+    Uω (aρ 6) (bρ 6) (886026853789 / 16000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_102_7 :
+    Uω (aρ 7) (bρ 7) (886026853789 / 16000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_102_8 :
+    Uω (aρ 8) (bρ 8) (886026853789 / 16000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_102_9 :
+    Uω (aρ 9) (bρ 9) (886026853789 / 16000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_102_10 :
+    Uω (aρ 10) (bρ 10) (886026853789 / 16000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_102_11 :
+    Uω (aρ 11) (bρ 11) (886026853789 / 16000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_102_12 :
+    Uω (aρ 12) (bρ 12) (886026853789 / 16000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_102_13 :
+    Uω (aρ 13) (bρ 13) (886026853789 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_102_14 :
+    Uω (aρ 14) (bρ 14) (886026853789 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_102_15 :
+    Uω (aρ 15) (bρ 15) (886026853789 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_102_16 :
+    Uω (aρ 16) (bρ 16) (886026853789 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_102 :
+    Uρ (886026853789 / 16000000000000 : ℝ) ≤
+      (-(26063668361683246214947 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (886026853789 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(7545257017708016980781 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_102_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (886026853789 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(30721594805733174632861 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_102_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (886026853789 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(15982371816390557011147 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_102_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (886026853789 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(34888585885879677851941 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_102_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (886026853789 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_102_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (886026853789 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_102_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (886026853789 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_102_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (886026853789 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_102_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (886026853789 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_102_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (886026853789 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_102_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (886026853789 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_102_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (886026853789 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_102_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (886026853789 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_102_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (886026853789 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_102_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (886026853789 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_102_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (886026853789 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_102_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_103_1 :
+    Uω (aρ 1) (bρ 1) (454984182553 / 8000000000000 : ℝ) ≤
+      (-(14939691773495747728407 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (31494511537239 / 625000000000000 : ℝ) ≤
+      (-(14939691773495747728407 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (31494511537239 / 625000000000000 : ℝ) = (31494511537239 / 19531250000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (31494511537239 / 19531250000000 : ℝ) ≤
+            (2388987734004252271593 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (251795679542699 / 5000000000000000 : ℝ)) (q :=
+      (31494511537239 / 625000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_103_2 :
+    Uω (aρ 2) (bρ 2) (454984182553 / 8000000000000 : ℝ) ≤
+      (-(7600539775674981056863 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (478245625030189 / 10000000000000000 : ℝ) ≤
+      (-(7600539775674981056863 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (478245625030189 / 10000000000000000 : ℝ) =
+            (478245625030189 / 312500000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (478245625030189 / 312500000000000 : ℝ) ≤
+            (1063799978075018943137 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (929740727469 / 19531250000000 : ℝ)) (q :=
+      (478245625030189 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_103_3 :
+    Uω (aρ 3) (bρ 3) (454984182553 / 8000000000000 : ℝ) ≤
+      (-(1974862208469201710963 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (848701910179767 / 20000000000000000 : ℝ) ≤
+      (-(1974862208469201710963 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (848701910179767 / 20000000000000000 : ℝ) =
+            (848701910179767 / 625000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (848701910179767 / 625000000000000 : ℝ) ≤
+            (191222729968298289037 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (415634018933517 / 10000000000000000 : ℝ)) (q :=
+      (848701910179767 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_103_4 :
+    Uω (aρ 4) (bρ 4) (454984182553 / 8000000000000 : ℝ) ≤
+      (-(17177194607766529470513 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (322112688666289 / 10000000000000000 : ℝ) ≤
+      (-(17177194607766529470513 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (322112688666289 / 10000000000000000 : ℝ) =
+            (322112688666289 / 312500000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (322112688666289 / 312500000000000 : ℝ) ≤
+            (151484899733470529487 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (9045364598479 / 312500000000000 : ℝ)) (q :=
+      (322112688666289 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_103_5 :
+    Uω (aρ 5) (bρ 5) (454984182553 / 8000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_103_6 :
+    Uω (aρ 6) (bρ 6) (454984182553 / 8000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_103_7 :
+    Uω (aρ 7) (bρ 7) (454984182553 / 8000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_103_8 :
+    Uω (aρ 8) (bρ 8) (454984182553 / 8000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_103_9 :
+    Uω (aρ 9) (bρ 9) (454984182553 / 8000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_103_10 :
+    Uω (aρ 10) (bρ 10) (454984182553 / 8000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_103_11 :
+    Uω (aρ 11) (bρ 11) (454984182553 / 8000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_103_12 :
+    Uω (aρ 12) (bρ 12) (454984182553 / 8000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_103_13 :
+    Uω (aρ 13) (bρ 13) (454984182553 / 8000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_103_14 :
+    Uω (aρ 14) (bρ 14) (454984182553 / 8000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_103_15 :
+    Uω (aρ 15) (bρ 15) (454984182553 / 8000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_103_16 :
+    Uω (aρ 16) (bρ 16) (454984182553 / 8000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_103 :
+    Uρ (454984182553 / 8000000000000 : ℝ) ≤
+      (-(26004234865315178925071 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (454984182553 / 8000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(14939691773495747728407 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_103_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (454984182553 / 8000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(7600539775674981056863 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_103_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (454984182553 / 8000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1974862208469201710963 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_103_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (454984182553 / 8000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(17177194607766529470513 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_103_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (454984182553 / 8000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_103_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (454984182553 / 8000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_103_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (454984182553 / 8000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_103_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (454984182553 / 8000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_103_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (454984182553 / 8000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_103_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (454984182553 / 8000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_103_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (454984182553 / 8000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_103_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (454984182553 / 8000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_103_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (454984182553 / 8000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_103_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (454984182553 / 8000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_103_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (454984182553 / 8000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_103_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (454984182553 / 8000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_103_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_104_1 :
+    Uω (aρ 1) (bρ 1) (47892569387 / 800000000000 : ℝ) ≤
+      (-(3662765284919319541649 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (533857069350801 / 10000000000000000 : ℝ) ≤
+      (-(3662765284919319541649 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (533857069350801 / 10000000000000000 : ℝ) =
+            (533857069350801 / 312500000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (533857069350801 / 312500000000000 : ℝ) ≤
+            (669404591955680458351 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (266777119724551 / 5000000000000000 : ℝ)) (q :=
+      (533857069350801 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_104_2 :
+    Uω (aρ 2) (bρ 2) (47892569387 / 800000000000 : ℝ) ≤
+      (-(3724076594691395072657 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (508303695593581 / 10000000000000000 : ℝ) ≤
+      (-(3724076594691395072657 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (508303695593581 / 10000000000000000 : ℝ) =
+            (508303695593581 / 312500000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (508303695593581 / 312500000000000 : ℝ) ≤
+            (608093282183604927343 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (253108252222331 / 5000000000000000 : ℝ)) (q :=
+      (508303695593581 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_104_3 :
+    Uω (aρ 3) (bρ 3) (47892569387 / 800000000000 : ℝ) ≤
+      (-(6180690102931800358943 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (181945026151459 / 4000000000000000 : ℝ) ≤
+      (-(6180690102931800358943 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (181945026151459 / 4000000000000000 : ℝ) =
+            (181945026151459 / 125000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (181945026151459 / 125000000000000 : ℝ) ≤
+            (750781700068199641057 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (89346070072959 / 2000000000000000 : ℝ)) (q :=
+      (181945026151459 / 4000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_104_4 :
+    Uω (aρ 4) (bρ 4) (47892569387 / 800000000000 : ℝ) ≤
+      (-(33380218915095679924057 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (710142503980007 / 20000000000000000 : ℝ) ≤
+      (-(33380218915095679924057 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (710142503980007 / 20000000000000000 : ℝ) =
+            (710142503980007 / 625000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (710142503980007 / 625000000000000 : ℝ) ≤
+            (1277140099904320075943 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (325441904652507 / 10000000000000000 : ℝ)) (q :=
+      (710142503980007 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_104_5 :
+    Uω (aρ 5) (bρ 5) (47892569387 / 800000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_104_6 :
+    Uω (aρ 6) (bρ 6) (47892569387 / 800000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_104_7 :
+    Uω (aρ 7) (bρ 7) (47892569387 / 800000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_104_8 :
+    Uω (aρ 8) (bρ 8) (47892569387 / 800000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_104_9 :
+    Uω (aρ 9) (bρ 9) (47892569387 / 800000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_104_10 :
+    Uω (aρ 10) (bρ 10) (47892569387 / 800000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_104_11 :
+    Uω (aρ 11) (bρ 11) (47892569387 / 800000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_104_12 :
+    Uω (aρ 12) (bρ 12) (47892569387 / 800000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_104_13 :
+    Uω (aρ 13) (bρ 13) (47892569387 / 800000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_104_14 :
+    Uω (aρ 14) (bρ 14) (47892569387 / 800000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_104_15 :
+    Uω (aρ 15) (bρ 15) (47892569387 / 800000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_104_16 :
+    Uω (aρ 16) (bρ 16) (47892569387 / 800000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_104 :
+    Uρ (47892569387 / 800000000000 : ℝ) ≤
+      (-(25893688140692916112373 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (47892569387 / 800000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3662765284919319541649 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_104_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (47892569387 / 800000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3724076594691395072657 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_104_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (47892569387 / 800000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(6180690102931800358943 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_104_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (47892569387 / 800000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(33380218915095679924057 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_104_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (47892569387 / 800000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_104_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (47892569387 / 800000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_104_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (47892569387 / 800000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_104_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (47892569387 / 800000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_104_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (47892569387 / 800000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_104_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (47892569387 / 800000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_104_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (47892569387 / 800000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_104_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (47892569387 / 800000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_104_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (47892569387 / 800000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_104_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (47892569387 / 800000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_104_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (47892569387 / 800000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_104_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (47892569387 / 800000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_104_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_105_1 :
+    Uω (aρ 1) (bρ 1) (502867205187 / 8000000000000 : ℝ) ≤
+      (-(7189101795739181296169 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1127600083104993 / 20000000000000000 : ℝ) ≤
+      (-(7189101795739181296169 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (1127600083104993 / 20000000000000000 : ℝ) =
+            (1127600083104993 / 625000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1127600083104993 / 625000000000000 : ℝ) ≤
+            (1475237958010818703831 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (563513294706243 / 10000000000000000 : ℝ)) (q :=
+      (1127600083104993 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_105_2 :
+    Uω (aρ 2) (bρ 2) (502867205187 / 8000000000000 : ℝ) ≤
+      (-(1826148073864402492947 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1076694127943669 / 20000000000000000 : ℝ) ≤
+      (-(1826148073864402492947 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (1076694127943669 / 20000000000000000 : ℝ) =
+            (1076694127943669 / 625000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1076694127943669 / 625000000000000 : ℝ) ≤
+            (339936864573097507053 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (536376352054919 / 10000000000000000 : ℝ)) (q :=
+      (1076694127943669 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_105_3 :
+    Uω (aρ 3) (bρ 3) (502867205187 / 8000000000000 : ℝ) ≤
+      (-(1210229666652526991207 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (970598978574067 / 20000000000000000 : ℝ) ≤
+      (-(1210229666652526991207 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (970598978574067 / 20000000000000000 : ℝ) =
+            (970598978574067 / 625000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (970598978574067 / 625000000000000 : ℝ) ≤
+            (176064693947473008793 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (477677309035317 / 10000000000000000 : ℝ)) (q :=
+      (970598978574067 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_105_4 :
+    Uω (aρ 4) (bρ 4) (502867205187 / 8000000000000 : ℝ) ≤
+      (-(16253431555359706555137 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (38747605742901 / 1000000000000000 : ℝ) ≤
+      (-(16253431555359706555137 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (38747605742901 / 1000000000000000 : ℝ) = (38747605742901 / 31250000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (38747605742901 / 31250000000000 : ℝ) ≤
+            (1075247952140293444863 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (36032462638427 / 1000000000000000 : ℝ)) (q :=
+      (38747605742901 / 1000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_105_5 :
+    Uω (aρ 5) (bρ 5) (502867205187 / 8000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_105_6 :
+    Uω (aρ 6) (bρ 6) (502867205187 / 8000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_105_7 :
+    Uω (aρ 7) (bρ 7) (502867205187 / 8000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_105_8 :
+    Uω (aρ 8) (bρ 8) (502867205187 / 8000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_105_9 :
+    Uω (aρ 9) (bρ 9) (502867205187 / 8000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_105_10 :
+    Uω (aρ 10) (bρ 10) (502867205187 / 8000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_105_11 :
+    Uω (aρ 11) (bρ 11) (502867205187 / 8000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_105_12 :
+    Uω (aρ 12) (bρ 12) (502867205187 / 8000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_105_13 :
+    Uω (aρ 13) (bρ 13) (502867205187 / 8000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_105_14 :
+    Uω (aρ 14) (bρ 14) (502867205187 / 8000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_105_15 :
+    Uω (aρ 15) (bρ 15) (502867205187 / 8000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_105_16 :
+    Uω (aρ 16) (bρ 16) (502867205187 / 8000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_105 :
+    Uρ (502867205187 / 8000000000000 : ℝ) ≤
+      (-(1289619184650899952501 / 500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (502867205187 / 8000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(7189101795739181296169 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_105_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (502867205187 / 8000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1826148073864402492947 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_105_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (502867205187 / 8000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1210229666652526991207 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_105_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (502867205187 / 8000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(16253431555359706555137 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_105_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (502867205187 / 8000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_105_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (502867205187 / 8000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_105_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (502867205187 / 8000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_105_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (502867205187 / 8000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_105_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (502867205187 / 8000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_105_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (502867205187 / 8000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_105_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (502867205187 / 8000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_105_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (502867205187 / 8000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_105_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (502867205187 / 8000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_105_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (502867205187 / 8000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_105_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (502867205187 / 8000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_105_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (502867205187 / 8000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_105_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_106_1 :
+    Uω (aρ 1) (bρ 1) (65851089563 / 1000000000000 : ℝ) ≤
+      (-(28238965150618197558677 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1187482781690451 / 20000000000000000 : ℝ) ≤
+      (-(28238965150618197558677 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (1187482781690451 / 20000000000000000 : ℝ) =
+            (1187482781690451 / 625000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1187482781690451 / 625000000000000 : ℝ) ≤
+            (6418393864381802441323 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (593469104145451 / 10000000000000000 : ℝ)) (q :=
+      (1187482781690451 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_106_2 :
+    Uω (aρ 2) (bρ 2) (65851089563 / 1000000000000 : ℝ) ≤
+      (-(28675535425575934481119 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (568378078312381 / 10000000000000000 : ℝ) ≤
+      (-(28675535425575934481119 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (568378078312381 / 10000000000000000 : ℝ) =
+            (568378078312381 / 312500000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (568378078312381 / 312500000000000 : ℝ) ≤
+            (5981823589424065518881 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (283255745794881 / 5000000000000000 : ℝ)) (q :=
+      (568378078312381 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_106_3 :
+    Uω (aρ 3) (bρ 3) (65851089563 / 1000000000000 : ℝ) ≤
+      (-(29648629442586589394293 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1031350726414239 / 20000000000000000 : ℝ) ≤
+      (-(29648629442586589394293 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (1031350726414239 / 20000000000000000 : ℝ) =
+            (1031350726414239 / 625000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1031350726414239 / 625000000000000 : ℝ) ≤
+            (5008729572413410605707 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (508502167729239 / 10000000000000000 : ℝ)) (q :=
+      (1031350726414239 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_106_4 :
+    Uω (aρ 4) (bρ 4) (65851089563 / 1000000000000 : ℝ) ≤
+      (-(7928346505977301990569 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (419474093477657 / 10000000000000000 : ℝ) ≤
+      (-(7928346505977301990569 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (419474093477657 / 10000000000000000 : ℝ) =
+            (419474093477657 / 312500000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (419474093477657 / 312500000000000 : ℝ) ≤
+            (735993247772698009431 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (197196904667657 / 5000000000000000 : ℝ)) (q :=
+      (419474093477657 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_106_5 :
+    Uω (aρ 5) (bρ 5) (65851089563 / 1000000000000 : ℝ) ≤
+      (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_5]) (by norm_num [bρ_5])]; exact Lmid_5
+
+lemma U_106_6 :
+    Uω (aρ 6) (bρ 6) (65851089563 / 1000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_106_7 :
+    Uω (aρ 7) (bρ 7) (65851089563 / 1000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_106_8 :
+    Uω (aρ 8) (bρ 8) (65851089563 / 1000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_106_9 :
+    Uω (aρ 9) (bρ 9) (65851089563 / 1000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_106_10 :
+    Uω (aρ 10) (bρ 10) (65851089563 / 1000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_106_11 :
+    Uω (aρ 11) (bρ 11) (65851089563 / 1000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_106_12 :
+    Uω (aρ 12) (bρ 12) (65851089563 / 1000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_106_13 :
+    Uω (aρ 13) (bρ 13) (65851089563 / 1000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_106_14 :
+    Uω (aρ 14) (bρ 14) (65851089563 / 1000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_106_15 :
+    Uω (aρ 15) (bρ 15) (65851089563 / 1000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_106_16 :
+    Uω (aρ 16) (bρ 16) (65851089563 / 1000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_106 :
+    Uρ (65851089563 / 1000000000000 : ℝ) ≤
+      (-(25698694525139896448587 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (65851089563 / 1000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(28238965150618197558677 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_106_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (65851089563 / 1000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(28675535425575934481119 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_106_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (65851089563 / 1000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(29648629442586589394293 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_106_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (65851089563 / 1000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(7928346505977301990569 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_106_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (65851089563 / 1000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20577364119368339121527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_106_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (65851089563 / 1000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_106_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (65851089563 / 1000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_106_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (65851089563 / 1000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_106_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (65851089563 / 1000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_106_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (65851089563 / 1000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_106_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (65851089563 / 1000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_106_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (65851089563 / 1000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_106_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (65851089563 / 1000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_106_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (65851089563 / 1000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_106_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (65851089563 / 1000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_106_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (65851089563 / 1000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_106_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_107_1 :
+    Uω (aρ 1) (bρ 1) (2140864814337 / 32000000000000 : ℝ) ≤
+      (-(7015858174040562655969 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (483404390005741 / 8000000000000000 : ℝ) ≤
+      (-(7015858174040562655969 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (483404390005741 / 8000000000000000 : ℝ) =
+            (483404390005741 / 250000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (483404390005741 / 250000000000000 : ℝ) ≤
+            (1648481579709437344031 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3774924616369 / 62500000000000 : ℝ)) (q :=
+      (483404390005741 / 8000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_107_2 :
+    Uω (aρ 2) (bρ 2) (2140864814337 / 32000000000000 : ℝ) ≤
+      (-(3561466998880705627953 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (463137145245269 / 8000000000000000 : ℝ) ≤
+      (-(3561466998880705627953 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (463137145245269 / 8000000000000000 : ℝ) =
+            (463137145245269 / 250000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (463137145245269 / 250000000000000 : ℝ) ≤
+            (770702877994294372047 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (28854441961393 / 500000000000000 : ℝ)) (q :=
+      (463137145245269 / 8000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_107_3 :
+    Uω (aρ 3) (bρ 3) (2140864814337 / 32000000000000 : ℝ) ≤
+      (-(14722060556063304097267 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (421063944006301 / 8000000000000000 : ℝ) ≤
+      (-(14722060556063304097267 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (421063944006301 / 8000000000000000 : ℝ) =
+            (421063944006301 / 250000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (421063944006301 / 250000000000000 : ℝ) ≤
+            (2606618951436695902733 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (12982548562011 / 250000000000000 : ℝ)) (q :=
+      (421063944006301 / 8000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_107_4 :
+    Uω (aρ 4) (bρ 4) (2140864814337 / 32000000000000 : ℝ) ≤
+      (-(6290166566300702968681 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1722533495644141 / 40000000000000000 : ℝ) ≤
+      (-(6290166566300702968681 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (1722533495644141 / 40000000000000000 : ℝ) =
+            (1722533495644141 / 1250000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1722533495644141 / 1250000000000000 : ℝ) ≤
+            (641305236699297031319 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (203101505675879 / 5000000000000000 : ℝ)) (q :=
+      (1722533495644141 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_107_5 :
+    Uω (aρ 5) (bρ 5) (2140864814337 / 32000000000000 : ℝ) ≤
+      (-(38623722698065706631519 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (840723189842747 / 40000000000000000 : ℝ) ≤
+      (-(38623722698065706631519 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (840723189842747 / 40000000000000000 : ℝ) =
+            (840723189842747 / 625000000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (840723189842747 / 625000000000000 : ℝ) ≤
+            (2965108119934293368481 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (83487777786061 / 10000000000000000 : ℝ)) (q :=
+      (840723189842747 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_107_6 :
+    Uω (aρ 6) (bρ 6) (2140864814337 / 32000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_107_7 :
+    Uω (aρ 7) (bρ 7) (2140864814337 / 32000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_107_8 :
+    Uω (aρ 8) (bρ 8) (2140864814337 / 32000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_107_9 :
+    Uω (aρ 9) (bρ 9) (2140864814337 / 32000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_107_10 :
+    Uω (aρ 10) (bρ 10) (2140864814337 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_107_11 :
+    Uω (aρ 11) (bρ 11) (2140864814337 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_107_12 :
+    Uω (aρ 12) (bρ 12) (2140864814337 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_107_13 :
+    Uω (aρ 13) (bρ 13) (2140864814337 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_107_14 :
+    Uω (aρ 14) (bρ 14) (2140864814337 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_107_15 :
+    Uω (aρ 15) (bρ 15) (2140864814337 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_107_16 :
+    Uω (aρ 16) (bρ 16) (2140864814337 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_107 :
+    Uρ (2140864814337 / 32000000000000 : ℝ) ≤
+      (-(12746317531850535099639 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (2140864814337 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(7015858174040562655969 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_107_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (2140864814337 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3561466998880705627953 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_107_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (2140864814337 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(14722060556063304097267 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_107_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (2140864814337 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(6290166566300702968681 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_107_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (2140864814337 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(38623722698065706631519 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_107_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (2140864814337 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_107_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (2140864814337 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_107_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (2140864814337 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_107_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (2140864814337 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_107_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (2140864814337 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_107_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (2140864814337 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_107_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (2140864814337 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_107_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (2140864814337 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_107_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (2140864814337 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_107_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (2140864814337 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_107_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (2140864814337 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_107_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_108_1 :
+    Uω (aρ 1) (bρ 1) (1087247381329 / 16000000000000 : ℝ) ≤
+      (-(13945465483393786464533 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1229538844085863 / 20000000000000000 : ℝ) ≤
+      (-(13945465483393786464533 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (1229538844085863 / 20000000000000000 : ℝ) =
+            (1229538844085863 / 625000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1229538844085863 / 625000000000000 : ℝ) ≤
+            (3383214024106213535467 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (307253224420119 / 5000000000000000 : ℝ)) (q :=
+      (1229538844085863 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_108_2 :
+    Uω (aρ 2) (bρ 2) (1087247381329 / 16000000000000 : ℝ) ≤
+      (-(14155637296061263611593 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1178927130014927 / 20000000000000000 : ℝ) ≤
+      (-(14155637296061263611593 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (1178927130014927 / 20000000000000000 : ℝ) =
+            (1178927130014927 / 625000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1178927130014927 / 625000000000000 : ℝ) ≤
+            (3173042211438736388407 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (293831873639651 / 5000000000000000 : ℝ)) (q :=
+      (1178927130014927 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_108_3 :
+    Uω (aρ 3) (bρ 3) (1087247381329 / 16000000000000 : ℝ) ≤
+      (-(29243820169079584143899 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1073957319292933 / 20000000000000000 : ℝ) ≤
+      (-(29243820169079584143899 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (1073957319292933 / 20000000000000000 : ℝ) =
+            (1073957319292933 / 625000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1073957319292933 / 625000000000000 : ℝ) ≤
+            (5413538845920415856101 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (132522510726827 / 2500000000000000 : ℝ)) (q :=
+      (1073957319292933 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_108_4 :
+    Uω (aρ 4) (bρ 4) (1087247381329 / 16000000000000 : ℝ) ≤
+      (-(31195782612954482994049 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (883515901536191 / 20000000000000000 : ℝ) ≤
+      (-(31195782612954482994049 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (883515901536191 / 20000000000000000 : ℝ) =
+            (883515901536191 / 625000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (883515901536191 / 625000000000000 : ℝ) ≤
+            (3461576402045517005951 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (208971403107783 / 5000000000000000 : ℝ)) (q :=
+      (883515901536191 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_108_5 :
+    Uω (aρ 5) (bρ 5) (1087247381329 / 16000000000000 : ℝ) ≤
+      (-(1174524119975398790487 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (466384484071707 / 20000000000000000 : ℝ) ≤
+      (-(1174524119975398790487 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (466384484071707 / 20000000000000000 : ℝ) =
+            (466384484071707 / 312500000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (466384484071707 / 312500000000000 : ℝ) ≤
+            (125126843087101209513 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (59500654043041 / 5000000000000000 : ℝ)) (q :=
+      (466384484071707 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_108_6 :
+    Uω (aρ 6) (bρ 6) (1087247381329 / 16000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_108_7 :
+    Uω (aρ 7) (bρ 7) (1087247381329 / 16000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_108_8 :
+    Uω (aρ 8) (bρ 8) (1087247381329 / 16000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_108_9 :
+    Uω (aρ 9) (bρ 9) (1087247381329 / 16000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_108_10 :
+    Uω (aρ 10) (bρ 10) (1087247381329 / 16000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_108_11 :
+    Uω (aρ 11) (bρ 11) (1087247381329 / 16000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_108_12 :
+    Uω (aρ 12) (bρ 12) (1087247381329 / 16000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_108_13 :
+    Uω (aρ 13) (bρ 13) (1087247381329 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_108_14 :
+    Uω (aρ 14) (bρ 14) (1087247381329 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_108_15 :
+    Uω (aρ 15) (bρ 15) (1087247381329 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_108_16 :
+    Uω (aρ 16) (bρ 16) (1087247381329 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_108 :
+    Uρ (1087247381329 / 16000000000000 : ℝ) ≤
+      (-(25390331102288743661441 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1087247381329 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(13945465483393786464533 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_108_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1087247381329 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(14155637296061263611593 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_108_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1087247381329 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(29243820169079584143899 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_108_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1087247381329 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(31195782612954482994049 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_108_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1087247381329 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(1174524119975398790487 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_108_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1087247381329 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_108_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1087247381329 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_108_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1087247381329 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_108_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1087247381329 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_108_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1087247381329 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_108_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1087247381329 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_108_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1087247381329 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_108_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1087247381329 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_108_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1087247381329 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_108_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1087247381329 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_108_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1087247381329 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_108_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_109_1 :
+    Uω (aρ 1) (bρ 1) (2208124710979 / 32000000000000 : ℝ) ≤
+      (-(13860678498047024113487 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2501132810550933 / 40000000000000000 : ℝ) ≤
+      (-(13860678498047024113487 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2501132810550933 / 40000000000000000 : ℝ) =
+            (2501132810550933 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 2
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2501132810550933 / 2500000000000000 : ℝ) ≤
+            (2265107952975886513 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (625024651179529 / 10000000000000000 : ℝ)) (q :=
+      (2501132810550933 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_109_2 :
+    Uω (aρ 2) (bρ 2) (2208124710979 / 32000000000000 : ℝ) ≤
+      (-(7033507858092569809221 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2400018173404683 / 40000000000000000 : ℝ) ≤
+      (-(7033507858092569809221 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (2400018173404683 / 40000000000000000 : ℝ) =
+            (2400018173404683 / 1250000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2400018173404683 / 1250000000000000 : ℝ) ≤
+            (1630831895657430190779 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (149559086279101 / 2500000000000000 : ℝ)) (q :=
+      (2400018173404683 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_109_3 :
+    Uω (aρ 3) (bρ 3) (2208124710979 / 32000000000000 : ℝ) ≤
+      (-(29047552827988840937373 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2190487605624223 / 40000000000000000 : ℝ) ≤
+      (-(29047552827988840937373 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (2190487605624223 / 40000000000000000 : ℝ) =
+            (2190487605624223 / 1250000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2190487605624223 / 1250000000000000 : ℝ) ≤
+            (5609806187011159062627 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (270433583788087 / 5000000000000000 : ℝ)) (q :=
+      (2190487605624223 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_109_4 :
+    Uω (aρ 4) (bρ 4) (2208124710979 / 32000000000000 : ℝ) ≤
+      (-(7736944871340474206587 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1811402676018173 / 40000000000000000 : ℝ) ≤
+      (-(7736944871340474206587 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (1811402676018173 / 40000000000000000 : ℝ) =
+            (1811402676018173 / 1250000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1811402676018173 / 1250000000000000 : ℝ) ≤
+            (927394882409525793413 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (429618883838149 / 10000000000000000 : ℝ)) (q :=
+      (1811402676018173 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_109_5 :
+    Uω (aρ 5) (bρ 5) (2208124710979 / 32000000000000 : ℝ) ≤
+      (-(36793825325521051304251 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1009542158843399 / 40000000000000000 : ℝ) ≤
+      (-(36793825325521051304251 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (1009542158843399 / 40000000000000000 : ℝ) =
+            (1009542158843399 / 625000000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1009542158843399 / 625000000000000 : ℝ) ≤
+            (4795005492478948695749 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (73439272292881 / 5000000000000000 : ℝ)) (q :=
+      (1009542158843399 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_109_6 :
+    Uω (aρ 6) (bρ 6) (2208124710979 / 32000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_109_7 :
+    Uω (aρ 7) (bρ 7) (2208124710979 / 32000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_109_8 :
+    Uω (aρ 8) (bρ 8) (2208124710979 / 32000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_109_9 :
+    Uω (aρ 9) (bρ 9) (2208124710979 / 32000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_109_10 :
+    Uω (aρ 10) (bρ 10) (2208124710979 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_109_11 :
+    Uω (aρ 11) (bρ 11) (2208124710979 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_109_12 :
+    Uω (aρ 12) (bρ 12) (2208124710979 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_109_13 :
+    Uω (aρ 13) (bρ 13) (2208124710979 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_109_14 :
+    Uω (aρ 14) (bρ 14) (2208124710979 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_109_15 :
+    Uω (aρ 15) (bρ 15) (2208124710979 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_109_16 :
+    Uω (aρ 16) (bρ 16) (2208124710979 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_109 :
+    Uρ (2208124710979 / 32000000000000 : ℝ) ≤
+      (-(5061171551165133003801 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (2208124710979 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(13860678498047024113487 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_109_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (2208124710979 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(7033507858092569809221 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_109_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (2208124710979 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(29047552827988840937373 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_109_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (2208124710979 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(7736944871340474206587 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_109_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (2208124710979 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(36793825325521051304251 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_109_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (2208124710979 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_109_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (2208124710979 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_109_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (2208124710979 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_109_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (2208124710979 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_109_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (2208124710979 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_109_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (2208124710979 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_109_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (2208124710979 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_109_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (2208124710979 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_109_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (2208124710979 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_109_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (2208124710979 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_109_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (2208124710979 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_109_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_110_1 :
+    Uω (aρ 1) (bρ 1) (4449879370279 / 64000000000000 : ℝ) ≤
+      (-(3454704645218149241057 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5044320300925587 / 80000000000000000 : ℝ) ≤
+      (-(3454704645218149241057 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (5044320300925587 / 80000000000000000 : ℝ) =
+            (5044320300925587 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5044320300925587 / 5000000000000000 : ℝ) ≤
+            (11031256281850758943 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (630283641710303 / 10000000000000000 : ℝ)) (q :=
+      (5044320300925587 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_110_2 :
+    Uω (aρ 2) (bρ 2) (4449879370279 / 64000000000000 : ℝ) ≤
+      (-(28046581010402783172551 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4842196946632063 / 80000000000000000 : ℝ) ≤
+      (-(28046581010402783172551 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (4842196946632063 / 80000000000000000 : ℝ) =
+            (4842196946632063 / 2500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4842196946632063 / 2500000000000000 : ℝ) ≤
+            (6610778004597216827449 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (301760907823461 / 5000000000000000 : ℝ)) (q :=
+      (4842196946632063 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_110_3 :
+    Uω (aρ 3) (bρ 3) (4449879370279 / 64000000000000 : ℝ) ≤
+      (-(28950880422802094840081 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4423532526790299 / 80000000000000000 : ℝ) ≤
+      (-(28950880422802094840081 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (4423532526790299 / 80000000000000000 : ℝ) =
+            (4423532526790299 / 2500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4423532526790299 / 2500000000000000 : ℝ) ≤
+            (5706478592197905159919 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (546251817036481 / 10000000000000000 : ℝ)) (q :=
+      (4423532526790299 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_110_4 :
+    Uω (aρ 4) (bρ 4) (4449879370279 / 64000000000000 : ℝ) ≤
+      (-(30826290029214530827127 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3667087060613467 / 80000000000000000 : ℝ) ≤
+      (-(30826290029214530827127 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (3667087060613467 / 80000000000000000 : ℝ) =
+            (3667087060613467 / 2500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3667087060613467 / 2500000000000000 : ℝ) ≤
+            (3831068985785469172873 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (435434631557273 / 10000000000000000 : ℝ)) (q :=
+      (3667087060613467 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_110_5 :
+    Uω (aρ 5) (bρ 5) (4449879370279 / 64000000000000 : ℝ) ≤
+      (-(36450498271319145357943 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (417921732650731 / 16000000000000000 : ℝ) ≤
+      (-(36450498271319145357943 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (417921732650731 / 16000000000000000 : ℝ) =
+            (417921732650731 / 250000000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (417921732650731 / 250000000000000 : ℝ) ≤
+            (5138332546680854642057 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (497671723601 / 31250000000000 : ℝ)) (q :=
+      (417921732650731 / 16000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_110_6 :
+    Uω (aρ 6) (bρ 6) (4449879370279 / 64000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_110_7 :
+    Uω (aρ 7) (bρ 7) (4449879370279 / 64000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_110_8 :
+    Uω (aρ 8) (bρ 8) (4449879370279 / 64000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_110_9 :
+    Uω (aρ 9) (bρ 9) (4449879370279 / 64000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_110_10 :
+    Uω (aρ 10) (bρ 10) (4449879370279 / 64000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_110_11 :
+    Uω (aρ 11) (bρ 11) (4449879370279 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_110_12 :
+    Uω (aρ 12) (bρ 12) (4449879370279 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_110_13 :
+    Uω (aρ 13) (bρ 13) (4449879370279 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_110_14 :
+    Uω (aρ 14) (bρ 14) (4449879370279 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_110_15 :
+    Uω (aρ 15) (bρ 15) (4449879370279 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_110_16 :
+    Uω (aρ 16) (bρ 16) (4449879370279 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_110 :
+    Uρ (4449879370279 / 64000000000000 : ℝ) ≤
+      (-(12633737911984751501131 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (4449879370279 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3454704645218149241057 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_110_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (4449879370279 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(28046581010402783172551 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_110_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (4449879370279 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(28950880422802094840081 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_110_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (4449879370279 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(30826290029214530827127 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_110_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (4449879370279 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(36450498271319145357943 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_110_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (4449879370279 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_110_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (4449879370279 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_110_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (4449879370279 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_110_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (4449879370279 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_110_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (4449879370279 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_110_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (4449879370279 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_110_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (4449879370279 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_110_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (4449879370279 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_110_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (4449879370279 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_110_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (4449879370279 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_110_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (4449879370279 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_110_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_111_1 :
+    Uω (aρ 1) (bρ 1) (22417546593 / 320000000000 : ℝ) ≤
+      (-(27554612974095860072559 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (63579683693473 / 1000000000000000 : ℝ) ≤
+      (-(27554612974095860072559 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (63579683693473 / 1000000000000000 : ℝ) = (63579683693473 / 62500000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (63579683693473 / 62500000000000 : ℝ) ≤
+            (171274237904139927441 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (63554256092321 / 1000000000000000 : ℝ)) (q :=
+      (63579683693473 / 1000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_111_2 :
+    Uω (aρ 2) (bρ 2) (22417546593 / 320000000000 : ℝ) ≤
+      (-(2795989308656552345633 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (610544426766827 / 10000000000000000 : ℝ) ≤
+      (-(2795989308656552345633 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (610544426766827 / 10000000000000000 : ℝ) =
+            (610544426766827 / 312500000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (610544426766827 / 312500000000000 : ℝ) ≤
+            (669746592843447654367 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (152201688274351 / 2500000000000000 : ℝ)) (q :=
+      (610544426766827 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_111_3 :
+    Uω (aρ 3) (bρ 3) (22417546593 / 320000000000 : ℝ) ≤
+      (-(14427578041578170897611 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1116519953864983 / 20000000000000000 : ℝ) ≤
+      (-(14427578041578170897611 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (1116519953864983 / 20000000000000000 : ℝ) =
+            (1116519953864983 / 625000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1116519953864983 / 625000000000000 : ℝ) ≤
+            (2901101465921829102389 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (551633959778733 / 10000000000000000 : ℝ)) (q :=
+      (1116519953864983 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_111_4 :
+    Uω (aρ 4) (bρ 4) (22417546593 / 320000000000 : ℝ) ≤
+      (-(3838301319880865820711 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (927828115562441 / 20000000000000000 : ℝ) ≤
+      (-(3838301319880865820711 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (927828115562441 / 20000000000000000 : ℝ) =
+            (927828115562441 / 625000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (927828115562441 / 625000000000000 : ℝ) ≤
+            (493868556994134179289 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (441236302541191 / 10000000000000000 : ℝ)) (q :=
+      (927828115562441 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_111_5 :
+    Uω (aρ 5) (bρ 5) (22417546593 / 320000000000 : ℝ) ≤
+      (-(36132153701588461327473 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (134825024062817 / 5000000000000000 : ℝ) ≤
+      (-(36132153701588461327473 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (134825024062817 / 5000000000000000 : ℝ) =
+            (134825024062817 / 78125000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (134825024062817 / 78125000000000 : ℝ) ≤
+            (5456677116411538672527 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (85449101282509 / 5000000000000000 : ℝ)) (q :=
+      (134825024062817 / 5000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_111_6 :
+    Uω (aρ 6) (bρ 6) (22417546593 / 320000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_111_7 :
+    Uω (aρ 7) (bρ 7) (22417546593 / 320000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_111_8 :
+    Uω (aρ 8) (bρ 8) (22417546593 / 320000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_111_9 :
+    Uω (aρ 9) (bρ 9) (22417546593 / 320000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_111_10 :
+    Uω (aρ 10) (bρ 10) (22417546593 / 320000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_111_11 :
+    Uω (aρ 11) (bρ 11) (22417546593 / 320000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_111_12 :
+    Uω (aρ 12) (bρ 12) (22417546593 / 320000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_111_13 :
+    Uω (aρ 13) (bρ 13) (22417546593 / 320000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_111_14 :
+    Uω (aρ 14) (bρ 14) (22417546593 / 320000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_111_15 :
+    Uω (aρ 15) (bρ 15) (22417546593 / 320000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_111_16 :
+    Uω (aρ 16) (bρ 16) (22417546593 / 320000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_111 :
+    Uρ (22417546593 / 320000000000 : ℝ) ≤
+      (-(25230982823289889081561 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (22417546593 / 320000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(27554612974095860072559 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_111_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (22417546593 / 320000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(2795989308656552345633 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_111_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (22417546593 / 320000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(14427578041578170897611 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_111_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (22417546593 / 320000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3838301319880865820711 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_111_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (22417546593 / 320000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(36132153701588461327473 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_111_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (22417546593 / 320000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_111_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (22417546593 / 320000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_111_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (22417546593 / 320000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_111_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (22417546593 / 320000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_111_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (22417546593 / 320000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_111_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (22417546593 / 320000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_111_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (22417546593 / 320000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_111_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (22417546593 / 320000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_111_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (22417546593 / 320000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_111_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (22417546593 / 320000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_111_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (22417546593 / 320000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_111_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/U09.lean b/LeanPool/Zeta5Irrational/Table/U09.lean
new file mode 100644
index 0000000000..0328805a98
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/U09.lean
@@ -0,0 +1,3066 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-! # Certified arcsine potential bounds (U09) -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma U_112_1 :
+    Uω (aρ 1) (bρ 1) (4517139266921 / 64000000000000 : ℝ) ≤
+      (-(5494454591738893614821 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5128428811782121 / 80000000000000000 : ℝ) ≤
+      (-(5494454591738893614821 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (5128428811782121 / 80000000000000000 : ℝ) =
+            (5128428811782121 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5128428811782121 / 5000000000000000 : ℝ) ≤
+            (50722850661106385179 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (160200352643531 / 2500000000000000 : ℝ)) (q :=
+      (5128428811782121 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_112_2 :
+    Uω (aρ 2) (bρ 2) (4517139266921 / 64000000000000 : ℝ) ≤
+      (-(27873954405000162249769 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (985302360873997 / 16000000000000000 : ℝ) ≤
+      (-(27873954405000162249769 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (985302360873997 / 16000000000000000 : ℝ) =
+            (985302360873997 / 500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (985302360873997 / 500000000000000 : ℝ) ≤
+            (6783404609999837750231 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (61409117123109 / 1000000000000000 : ℝ)) (q :=
+      (985302360873997 / 16000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_112_3 :
+    Uω (aρ 3) (bρ 3) (4517139266921 / 64000000000000 : ℝ) ≤
+      (-(179752256073111659559 / 62500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4508617367911213 / 80000000000000000 : ℝ) ≤
+      (-(179752256073111659559 / 62500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (4508617367911213 / 80000000000000000 : ℝ) =
+            (4508617367911213 / 2500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4508617367911213 / 2500000000000000 : ℝ) ≤
+            (36856237770638340441 / 62500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (557013668466397 / 10000000000000000 : ℝ)) (q :=
+      (4508617367911213 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_112_4 :
+    Uω (aρ 4) (bρ 4) (4517139266921 / 64000000000000 : ℝ) ≤
+      (-(15294047642102670013741 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3755483749271957 / 80000000000000000 : ℝ) ≤
+      (-(15294047642102670013741 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (3755483749271957 / 80000000000000000 : ℝ) =
+            (3755483749271957 / 2500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3755483749271957 / 2500000000000000 : ℝ) ≤
+            (2034631865397329986259 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (447024444871583 / 10000000000000000 : ℝ)) (q :=
+      (3755483749271957 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_112_5 :
+    Uω (aρ 5) (bρ 5) (4517139266921 / 64000000000000 : ℝ) ≤
+      (-(35834287293306577019241 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (444484538504269 / 16000000000000000 : ℝ) ≤
+      (-(35834287293306577019241 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (444484538504269 / 16000000000000000 : ℝ) =
+            (444484538504269 / 250000000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (444484538504269 / 250000000000000 : ℝ) ≤
+            (5754543524693422980759 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (18194910001893 / 1000000000000000 : ℝ)) (q :=
+      (444484538504269 / 16000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_112_6 :
+    Uω (aρ 6) (bρ 6) (4517139266921 / 64000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_112_7 :
+    Uω (aρ 7) (bρ 7) (4517139266921 / 64000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_112_8 :
+    Uω (aρ 8) (bρ 8) (4517139266921 / 64000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_112_9 :
+    Uω (aρ 9) (bρ 9) (4517139266921 / 64000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_112_10 :
+    Uω (aρ 10) (bρ 10) (4517139266921 / 64000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_112_11 :
+    Uω (aρ 11) (bρ 11) (4517139266921 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_112_12 :
+    Uω (aρ 12) (bρ 12) (4517139266921 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_112_13 :
+    Uω (aρ 13) (bρ 13) (4517139266921 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_112_14 :
+    Uω (aρ 14) (bρ 14) (4517139266921 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_112_15 :
+    Uω (aρ 15) (bρ 15) (4517139266921 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_112_16 :
+    Uω (aρ 16) (bρ 16) (4517139266921 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_112 :
+    Uρ (4517139266921 / 64000000000000 : ℝ) ≤
+      (-(25196063800726208672673 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (4517139266921 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(5494454591738893614821 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_112_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (4517139266921 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(27873954405000162249769 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_112_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (4517139266921 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) * (-(179752256073111659559 / 62500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_112_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (4517139266921 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(15294047642102670013741 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_112_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (4517139266921 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(35834287293306577019241 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_112_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (4517139266921 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_112_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (4517139266921 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_112_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (4517139266921 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_112_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (4517139266921 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_112_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (4517139266921 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_112_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (4517139266921 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_112_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (4517139266921 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_112_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (4517139266921 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_112_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (4517139266921 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_112_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (4517139266921 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_112_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (4517139266921 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_112_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_113_1 :
+    Uω (aρ 1) (bρ 1) (2275384607621 / 32000000000000 : ℝ) ≤
+      (-(27390605922888495679481 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (517048265663313 / 8000000000000000 : ℝ) ≤
+      (-(27390605922888495679481 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (517048265663313 / 8000000000000000 : ℝ) =
+            (517048265663313 / 500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (517048265663313 / 500000000000000 : ℝ) ≤
+            (335281289111504320519 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (16151504809043 / 250000000000000 : ℝ)) (q :=
+      (517048265663313 / 8000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_113_2 :
+    Uω (aρ 2) (bρ 2) (2275384607621 / 32000000000000 : ℝ) ≤
+      (-(13894376027202381802373 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2484333085253499 / 40000000000000000 : ℝ) ≤
+      (-(13894376027202381802373 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (2484333085253499 / 40000000000000000 : ℝ) =
+            (2484333085253499 / 1250000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2484333085253499 / 1250000000000000 : ℝ) ≤
+            (3434303480297618197627 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (619375083340187 / 10000000000000000 : ℝ)) (q :=
+      (2484333085253499 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_113_3 :
+    Uω (aρ 3) (bρ 3) (2275384607621 / 32000000000000 : ℝ) ≤
+      (-(14333238409323982707187 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2275572731773867 / 40000000000000000 : ℝ) ≤
+      (-(14333238409323982707187 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (2275572731773867 / 40000000000000000 : ℝ) =
+            (2275572731773867 / 1250000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2275572731773867 / 1250000000000000 : ℝ) ≤
+            (2995441098176017292813 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (562391012950371 / 10000000000000000 : ℝ)) (q :=
+      (2275572731773867 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_113_4 :
+    Uω (aρ 4) (bρ 4) (2275384607621 / 32000000000000 : ℝ) ≤
+      (-(15235650232966056679897 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1899801498462243 / 40000000000000000 : ℝ) ≤
+      (-(15235650232966056679897 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (1899801498462243 / 40000000000000000 : ℝ) =
+            (1899801498462243 / 1250000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1899801498462243 / 1250000000000000 : ℝ) ≤
+            (2093029274533943320103 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (452799577359559 / 10000000000000000 : ℝ)) (q :=
+      (1899801498462243 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_113_5 :
+    Uω (aρ 5) (bρ 5) (2275384607621 / 32000000000000 : ℝ) ≤
+      (-(17776807122822487646779 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (228568373818183 / 8000000000000000 : ℝ) ≤
+      (-(17776807122822487646779 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (228568373818183 / 8000000000000000 : ℝ) =
+            (228568373818183 / 125000000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (228568373818183 / 125000000000000 : ℝ) ≤
+            (3017608286177512353221 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (38501936401779 / 2000000000000000 : ℝ)) (q :=
+      (228568373818183 / 8000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_113_6 :
+    Uω (aρ 6) (bρ 6) (2275384607621 / 32000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_113_7 :
+    Uω (aρ 7) (bρ 7) (2275384607621 / 32000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_113_8 :
+    Uω (aρ 8) (bρ 8) (2275384607621 / 32000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_113_9 :
+    Uω (aρ 9) (bρ 9) (2275384607621 / 32000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_113_10 :
+    Uω (aρ 10) (bρ 10) (2275384607621 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_113_11 :
+    Uω (aρ 11) (bρ 11) (2275384607621 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_113_12 :
+    Uω (aρ 12) (bρ 12) (2275384607621 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_113_13 :
+    Uω (aρ 13) (bρ 13) (2275384607621 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_113_14 :
+    Uω (aρ 14) (bρ 14) (2275384607621 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_113_15 :
+    Uω (aρ 15) (bρ 15) (2275384607621 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_113_16 :
+    Uω (aρ 16) (bρ 16) (2275384607621 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_113 :
+    Uρ (2275384607621 / 32000000000000 : ℝ) ≤
+      (-(5032497630608943774487 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (2275384607621 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(27390605922888495679481 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_113_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (2275384607621 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(13894376027202381802373 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_113_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (2275384607621 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(14333238409323982707187 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_113_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (2275384607621 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(15235650232966056679897 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_113_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (2275384607621 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(17776807122822487646779 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_113_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (2275384607621 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_113_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (2275384607621 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_113_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (2275384607621 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_113_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (2275384607621 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_113_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (2275384607621 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_113_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (2275384607621 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_113_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (2275384607621 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_113_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (2275384607621 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_113_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (2275384607621 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_113_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (2275384607621 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_113_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (2275384607621 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_113_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_114_1 :
+    Uω (aρ 1) (bρ 1) (4584399163563 / 64000000000000 : ℝ) ≤
+      (-(853425029583273381449 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5212536236606059 / 80000000000000000 : ℝ) ≤
+      (-(853425029583273381449 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (5212536236606059 / 80000000000000000 : ℝ) =
+            (5212536236606059 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5212536236606059 / 5000000000000000 : ℝ) ≤
+            (13008945791726618551 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (162829726982449 / 2500000000000000 : ℝ)) (q :=
+      (5212536236606059 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_114_2 :
+    Uω (aρ 2) (bρ 2) (4584399163563 / 64000000000000 : ℝ) ≤
+      (-(13852136729366462766811 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (200432742556419 / 3200000000000000 : ℝ) ≤
+      (-(13852136729366462766811 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (200432742556419 / 3200000000000000 : ℝ) =
+            (200432742556419 / 200000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 2
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (200432742556419 / 200000000000000 : ℝ) ≤
+            (10806876633537233189 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6246585022659 / 100000000000000 : ℝ)) (q :=
+      (200432742556419 / 3200000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_114_3 :
+    Uω (aρ 3) (bρ 3) (4584399163563 / 64000000000000 : ℝ) ≤
+      (-(3571685737350867667193 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (918732874213519 / 16000000000000000 : ℝ) ≤
+      (-(3571685737350867667193 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (918732874213519 / 16000000000000000 : ℝ) =
+            (918732874213519 / 500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (918732874213519 / 500000000000000 : ℝ) ≤
+            (760484139524132332807 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (28388303020259 / 500000000000000 : ℝ)) (q :=
+      (918732874213519 / 16000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_114_4 :
+    Uω (aρ 4) (bρ 4) (4584399163563 / 64000000000000 : ℝ) ≤
+      (-(30355984291079358506601 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3843672171359951 / 80000000000000000 : ℝ) ≤
+      (-(30355984291079358506601 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (3843672171359951 / 80000000000000000 : ℝ) =
+            (3843672171359951 / 2500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3843672171359951 / 2500000000000000 : ℝ) ≤
+            (4301374723920641493399 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (458562191543269 / 10000000000000000 : ℝ)) (q :=
+      (3843672171359951 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_114_5 :
+    Uω (aρ 5) (bρ 5) (4584399163563 / 64000000000000 : ℝ) ≤
+      (-(8821912704169286925593 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2347290188291931 / 80000000000000000 : ℝ) ≤
+      (-(8821912704169286925593 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (2347290188291931 / 80000000000000000 : ℝ) =
+            (2347290188291931 / 1250000000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2347290188291931 / 1250000000000000 : ℝ) ≤
+            (1575295000330713074407 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6333019222227 / 312500000000000 : ℝ)) (q :=
+      (2347290188291931 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_114_6 :
+    Uω (aρ 6) (bρ 6) (4584399163563 / 64000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_114_7 :
+    Uω (aρ 7) (bρ 7) (4584399163563 / 64000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_114_8 :
+    Uω (aρ 8) (bρ 8) (4584399163563 / 64000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_114_9 :
+    Uω (aρ 9) (bρ 9) (4584399163563 / 64000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_114_10 :
+    Uω (aρ 10) (bρ 10) (4584399163563 / 64000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_114_11 :
+    Uω (aρ 11) (bρ 11) (4584399163563 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_114_12 :
+    Uω (aρ 12) (bρ 12) (4584399163563 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_114_13 :
+    Uω (aρ 13) (bρ 13) (4584399163563 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_114_14 :
+    Uω (aρ 14) (bρ 14) (4584399163563 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_114_15 :
+    Uω (aρ 15) (bρ 15) (4584399163563 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_114_16 :
+    Uω (aρ 16) (bρ 16) (4584399163563 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_114 :
+    Uρ (4584399163563 / 64000000000000 : ℝ) ≤
+      (-(25130080728420120270839 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (4584399163563 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(853425029583273381449 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_114_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (4584399163563 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(13852136729366462766811 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_114_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (4584399163563 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3571685737350867667193 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_114_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (4584399163563 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(30355984291079358506601 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_114_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (4584399163563 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(8821912704169286925593 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_114_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (4584399163563 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_114_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (4584399163563 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_114_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (4584399163563 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_114_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (4584399163563 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_114_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (4584399163563 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_114_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (4584399163563 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_114_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (4584399163563 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_114_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (4584399163563 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_114_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (4584399163563 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_114_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (4584399163563 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_114_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (4584399163563 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_114_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_115_1 :
+    Uω (aρ 1) (bρ 1) (1154507277971 / 16000000000000 : ℝ) ≤
+      (-(13614623686929618077151 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1313647389516337 / 20000000000000000 : ℝ) ≤
+      (-(13614623686929618077151 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1313647389516337 / 20000000000000000 : ℝ) =
+            (1313647389516337 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1313647389516337 / 1250000000000000 : ℝ) ≤
+            (248319919070381922849 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (328288779434731 / 5000000000000000 : ℝ)) (q :=
+      (1313647389516337 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_115_2 :
+    Uω (aρ 2) (bρ 2) (1154507277971 / 16000000000000 : ℝ) ≤
+      (-(27620506355407504983373 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (631621129277163 / 10000000000000000 : ℝ) ≤
+      (-(27620506355407504983373 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (631621129277163 / 10000000000000000 : ℝ) =
+            (631621129277163 / 625000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (631621129277163 / 625000000000000 : ℝ) ≤
+            (105380856592495016627 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (629941440417451 / 10000000000000000 : ℝ)) (q :=
+      (631621129277163 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_115_3 :
+    Uω (aρ 3) (bρ 3) (1154507277971 / 16000000000000 : ℝ) ≤
+      (-(28481371009729646648441 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1159043587244137 / 20000000000000000 : ℝ) ≤
+      (-(28481371009729646648441 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (1159043587244137 / 20000000000000000 : ℝ) =
+            (1159043587244137 / 625000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1159043587244137 / 625000000000000 : ℝ) ≤
+            (6175988005270353351559 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (286569437728631 / 5000000000000000 : ℝ)) (q :=
+      (1159043587244137 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_115_4 :
+    Uω (aρ 4) (bρ 4) (1154507277971 / 16000000000000 : ℝ) ≤
+      (-(15121053377842898206551 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (971923284241157 / 20000000000000000 : ℝ) ≤
+      (-(15121053377842898206551 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (971923284241157 / 20000000000000000 : ℝ) =
+            (971923284241157 / 625000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (971923284241157 / 625000000000000 : ℝ) ≤
+            (2207626129657101793449 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (232156376759641 / 5000000000000000 : ℝ)) (q :=
+      (971923284241157 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_115_5 :
+    Uω (aρ 5) (bρ 5) (1154507277971 / 16000000000000 : ℝ) ≤
+      (-(8758616067257419857111 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (150467447175143 / 5000000000000000 : ℝ) ≤
+      (-(8758616067257419857111 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (150467447175143 / 5000000000000000 : ℝ) =
+            (150467447175143 / 78125000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (150467447175143 / 78125000000000 : ℝ) ≤
+            (1638591637242580142889 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (212449177313697 / 10000000000000000 : ℝ)) (q :=
+      (150467447175143 / 5000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_115_6 :
+    Uω (aρ 6) (bρ 6) (1154507277971 / 16000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_115_7 :
+    Uω (aρ 7) (bρ 7) (1154507277971 / 16000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_115_8 :
+    Uω (aρ 8) (bρ 8) (1154507277971 / 16000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_115_9 :
+    Uω (aρ 9) (bρ 9) (1154507277971 / 16000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_115_10 :
+    Uω (aρ 10) (bρ 10) (1154507277971 / 16000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_115_11 :
+    Uω (aρ 11) (bρ 11) (1154507277971 / 16000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_115_12 :
+    Uω (aρ 12) (bρ 12) (1154507277971 / 16000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_115_13 :
+    Uω (aρ 13) (bρ 13) (1154507277971 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_115_14 :
+    Uω (aρ 14) (bρ 14) (1154507277971 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_115_15 :
+    Uω (aρ 15) (bρ 15) (1154507277971 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_115_16 :
+    Uω (aρ 16) (bρ 16) (1154507277971 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_115 :
+    Uρ (1154507277971 / 16000000000000 : ℝ) ≤
+      (-(25098704554830257436931 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1154507277971 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(13614623686929618077151 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_115_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1154507277971 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(27620506355407504983373 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_115_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1154507277971 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(28481371009729646648441 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_115_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1154507277971 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(15121053377842898206551 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_115_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1154507277971 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(8758616067257419857111 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_115_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1154507277971 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_115_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1154507277971 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_115_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1154507277971 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_115_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1154507277971 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_115_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1154507277971 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_115_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1154507277971 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_115_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1154507277971 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_115_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1154507277971 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_115_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1154507277971 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_115_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1154507277971 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_115_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1154507277971 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_115_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_116_1 :
+    Uω (aρ 1) (bρ 1) (930331812041 / 12800000000000 : ℝ) ≤
+      (-(6787383700929643976333 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1059328525434621 / 16000000000000000 : ℝ) ≤
+      (-(6787383700929643976333 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1059328525434621 / 16000000000000000 : ℝ) =
+            (1059328525434621 / 1000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1059328525434621 / 1000000000000000 : ℝ) ≤
+            (144088102070356023667 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (132367229344249 / 2000000000000000 : ℝ)) (q :=
+      (1059328525434621 / 16000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_116_2 :
+    Uω (aρ 2) (bρ 2) (930331812041 / 12800000000000 : ℝ) ≤
+      (-(3442179850001808571531 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5095117629408681 / 80000000000000000 : ℝ) ≤
+      (-(3442179850001808571531 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (5095117629408681 / 80000000000000000 : ℝ) =
+            (5095117629408681 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5095117629408681 / 5000000000000000 : ℝ) ≤
+            (23556051498191428469 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (635223909790139 / 10000000000000000 : ℝ)) (q :=
+      (5095117629408681 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_116_3 :
+    Uω (aρ 3) (bρ 3) (930331812041 / 12800000000000 : ℝ) ≤
+      (-(14195057725500391424769 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4678675646071621 / 80000000000000000 : ℝ) ≤
+      (-(14195057725500391424769 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (4678675646071621 / 80000000000000000 : ℝ) =
+            (4678675646071621 / 2500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4678675646071621 / 2500000000000000 : ℝ) ≤
+            (3133621781999608575231 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (289254760152937 / 5000000000000000 : ℝ)) (q :=
+      (4678675646071621 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_116_4 :
+    Uω (aρ 4) (bρ 4) (930331812041 / 12800000000000 : ℝ) ≤
+      (-(30129629557487540673771 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3931667663084301 / 80000000000000000 : ℝ) ≤
+      (-(30129629557487540673771 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (3931667663084301 / 80000000000000000 : ℝ) =
+            (3931667663084301 / 2500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3931667663084301 / 2500000000000000 : ℝ) ≤
+            (4527729457512459326229 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (117512926406011 / 2500000000000000 : ℝ)) (q :=
+      (3931667663084301 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_116_5 :
+    Uω (aρ 5) (bρ 5) (930331812041 / 12800000000000 : ℝ) ≤
+      (-(34792515104179698616019 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2466438291419753 / 80000000000000000 : ℝ) ≤
+      (-(34792515104179698616019 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show
+          (2466438291419753 / 80000000000000000 : ℝ) =
+            (2466438291419753 / 1250000000000000 : ℝ) / 2 ^ 6
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2466438291419753 / 1250000000000000 : ℝ) ≤
+            (6796315713820301383981 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (221934282042907 / 10000000000000000 : ℝ)) (q :=
+      (2466438291419753 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_116_6 :
+    Uω (aρ 6) (bρ 6) (930331812041 / 12800000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_116_7 :
+    Uω (aρ 7) (bρ 7) (930331812041 / 12800000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_116_8 :
+    Uω (aρ 8) (bρ 8) (930331812041 / 12800000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_116_9 :
+    Uω (aρ 9) (bρ 9) (930331812041 / 12800000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_116_10 :
+    Uω (aρ 10) (bρ 10) (930331812041 / 12800000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_116_11 :
+    Uω (aρ 11) (bρ 11) (930331812041 / 12800000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_116_12 :
+    Uω (aρ 12) (bρ 12) (930331812041 / 12800000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_116_13 :
+    Uω (aρ 13) (bρ 13) (930331812041 / 12800000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_116_14 :
+    Uω (aρ 14) (bρ 14) (930331812041 / 12800000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_116_15 :
+    Uω (aρ 15) (bρ 15) (930331812041 / 12800000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_116_16 :
+    Uω (aρ 16) (bρ 16) (930331812041 / 12800000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_116 :
+    Uρ (930331812041 / 12800000000000 : ℝ) ≤
+      (-(25068249941190760208499 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (930331812041 / 12800000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(6787383700929643976333 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_116_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (930331812041 / 12800000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3442179850001808571531 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_116_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (930331812041 / 12800000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(14195057725500391424769 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_116_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (930331812041 / 12800000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(30129629557487540673771 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_116_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (930331812041 / 12800000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(34792515104179698616019 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_116_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (930331812041 / 12800000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_116_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (930331812041 / 12800000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_116_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (930331812041 / 12800000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_116_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (930331812041 / 12800000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_116_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (930331812041 / 12800000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_116_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (930331812041 / 12800000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_116_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (930331812041 / 12800000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_116_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (930331812041 / 12800000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_116_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (930331812041 / 12800000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_116_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (930331812041 / 12800000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_116_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (930331812041 / 12800000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_116_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_117_1 :
+    Uω (aρ 1) (bρ 1) (2342644504263 / 32000000000000 : ℝ) ≤
+      (-(13535226541398232719239 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (106773908997941 / 1600000000000000 : ℝ) ≤
+      (-(13535226541398232719239 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (106773908997941 / 1600000000000000 : ℝ) =
+            (106773908997941 / 100000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (106773908997941 / 100000000000000 : ℝ) ≤
+            (327717064601767280761 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (26683786919083 / 400000000000000 : ℝ)) (q :=
+      (106773908997941 / 1600000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_117_2 :
+    Uω (aρ 2) (bρ 2) (2342644504263 / 32000000000000 : ℝ) ≤
+      (-(6863764786365759850067 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2568632197939373 / 40000000000000000 : ℝ) ≤
+      (-(6863764786365759850067 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2568632197939373 / 40000000000000000 : ℝ) =
+            (2568632197939373 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2568632197939373 / 2500000000000000 : ℝ) ≤
+            (67707016634240149933 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (640505921982499 / 10000000000000000 : ℝ)) (q :=
+      (2568632197939373 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_117_3 :
+    Uω (aρ 3) (bρ 3) (2342644504263 / 32000000000000 : ℝ) ≤
+      (-(14149851502271592942073 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (472116850189721 / 8000000000000000 : ℝ) ≤
+      (-(14149851502271592942073 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (472116850189721 / 8000000000000000 : ℝ) =
+            (472116850189721 / 250000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (472116850189721 / 250000000000000 : ℝ) ≤
+            (3178828005228407057927 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (116775610967423 / 2000000000000000 : ℝ)) (q :=
+      (472116850189721 / 8000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_117_4 :
+    Uω (aρ 4) (bρ 4) (2342644504263 / 32000000000000 : ℝ) ≤
+      (-(30018515996535756825573 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1987798715104119 / 40000000000000000 : ℝ) ≤
+      (-(30018515996535756825573 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (1987798715104119 / 40000000000000000 : ℝ) =
+            (1987798715104119 / 1250000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1987798715104119 / 1250000000000000 : ℝ) ≤
+            (4638843018464243174427 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (14868108374371 / 312500000000000 : ℝ)) (q :=
+      (1987798715104119 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_117_5 :
+    Uω (aρ 5) (bρ 5) (2342644504263 / 32000000000000 : ℝ) ≤
+      (-(34560553073273099210261 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1262159503365177 / 40000000000000000 : ℝ) ≤
+      (-(34560553073273099210261 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (1262159503365177 / 40000000000000000 : ℝ) =
+            (1262159503365177 / 1250000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1262159503365177 / 1250000000000000 : ℝ) ≤
+            (96805941726900789739 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (231149781445401 / 10000000000000000 : ℝ)) (q :=
+      (1262159503365177 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_117_6 :
+    Uω (aρ 6) (bρ 6) (2342644504263 / 32000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_117_7 :
+    Uω (aρ 7) (bρ 7) (2342644504263 / 32000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_117_8 :
+    Uω (aρ 8) (bρ 8) (2342644504263 / 32000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_117_9 :
+    Uω (aρ 9) (bρ 9) (2342644504263 / 32000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_117_10 :
+    Uω (aρ 10) (bρ 10) (2342644504263 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_117_11 :
+    Uω (aρ 11) (bρ 11) (2342644504263 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_117_12 :
+    Uω (aρ 12) (bρ 12) (2342644504263 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_117_13 :
+    Uω (aρ 13) (bρ 13) (2342644504263 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_117_14 :
+    Uω (aρ 14) (bρ 14) (2342644504263 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_117_15 :
+    Uω (aρ 15) (bρ 15) (2342644504263 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_117_16 :
+    Uω (aρ 16) (bρ 16) (2342644504263 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_117 :
+    Uρ (2342644504263 / 32000000000000 : ℝ) ≤
+      (-(1001545091634398708323 / 400000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (2342644504263 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(13535226541398232719239 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_117_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (2342644504263 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(6863764786365759850067 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_117_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (2342644504263 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(14149851502271592942073 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_117_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (2342644504263 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(30018515996535756825573 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_117_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (2342644504263 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(34560553073273099210261 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_117_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (2342644504263 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_117_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (2342644504263 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_117_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (2342644504263 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_117_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (2342644504263 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_117_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (2342644504263 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_117_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (2342644504263 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_117_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (2342644504263 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_117_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (2342644504263 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_117_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (2342644504263 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_117_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (2342644504263 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_117_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (2342644504263 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_117_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_118_1 :
+    Uω (aρ 1) (bρ 1) (9404207965373 / 128000000000000 : ℝ) ≤
+      (-(6757786420885346854021 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2143888708271497 / 32000000000000000 : ℝ) ≤
+      (-(6757786420885346854021 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2143888708271497 / 32000000000000000 : ℝ) =
+            (2143888708271497 / 2000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2143888708271497 / 2000000000000000 : ℝ) ≤
+            (173685382114653145979 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (8371548918249 / 125000000000000 : ℝ)) (q :=
+      (2143888708271497 / 32000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_118_2 :
+    Uω (aρ 2) (bρ 2) (9404207965373 / 128000000000000 : ℝ) ≤
+      (-(1096564948833060777743 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10316674214847229 / 160000000000000000 : ℝ) ≤
+      (-(1096564948833060777743 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (10316674214847229 / 160000000000000000 : ℝ) =
+            (10316674214847229 / 10000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10316674214847229 / 10000000000000000 : ℝ) ≤
+            (12470539646939222257 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (321573380078069 / 5000000000000000 : ℝ)) (q :=
+      (10316674214847229 / 160000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_118_3 :
+    Uω (aρ 3) (bρ 3) (9404207965373 / 128000000000000 : ℝ) ≤
+      (-(110370343779415330577 / 39062500000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (379392946941437 / 6400000000000000 : ℝ) ≤
+      (-(110370343779415330577 / 39062500000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (379392946941437 / 6400000000000000 : ℝ) =
+            (379392946941437 / 200000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (379392946941437 / 200000000000000 : ℝ) ≤
+            (25009964872928419423 / 39062500000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (23462461953689 / 400000000000000 : ℝ)) (q :=
+      (379392946941437 / 6400000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_118_4 :
+    Uω (aρ 4) (bρ 4) (9404207965373 / 128000000000000 : ℝ) ≤
+      (-(1198538381065681381673 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1599018412565393 / 32000000000000000 : ℝ) ≤
+      (-(1198538381065681381673 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (1599018412565393 / 32000000000000000 : ℝ) =
+            (1599018412565393 / 1000000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1599018412565393 / 1000000000000000 : ℝ) ≤
+            (187755979534318618327 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (95727855713721 / 2000000000000000 : ℝ)) (q :=
+      (1599018412565393 / 32000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_118_5 :
+    Uω (aρ 5) (bρ 5) (9404207965373 / 128000000000000 : ℝ) ≤
+      (-(34447987663370110448543 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5105789273487933 / 160000000000000000 : ℝ) ≤
+      (-(34447987663370110448543 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (5105789273487933 / 160000000000000000 : ℝ) =
+            (5105789273487933 / 5000000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5105789273487933 / 5000000000000000 : ℝ) ≤
+            (209371351629889551457 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (117833174618113 / 5000000000000000 : ℝ)) (q :=
+      (5105789273487933 / 160000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_118_6 :
+    Uω (aρ 6) (bρ 6) (9404207965373 / 128000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_118_7 :
+    Uω (aρ 7) (bρ 7) (9404207965373 / 128000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_118_8 :
+    Uω (aρ 8) (bρ 8) (9404207965373 / 128000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_118_9 :
+    Uω (aρ 9) (bρ 9) (9404207965373 / 128000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_118_10 :
+    Uω (aρ 10) (bρ 10) (9404207965373 / 128000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_118_11 :
+    Uω (aρ 11) (bρ 11) (9404207965373 / 128000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_118_12 :
+    Uω (aρ 12) (bρ 12) (9404207965373 / 128000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_118_13 :
+    Uω (aρ 13) (bρ 13) (9404207965373 / 128000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_118_14 :
+    Uω (aρ 14) (bρ 14) (9404207965373 / 128000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_118_15 :
+    Uω (aρ 15) (bρ 15) (9404207965373 / 128000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_118_16 :
+    Uω (aρ 16) (bρ 16) (9404207965373 / 128000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_118 :
+    Uρ (9404207965373 / 128000000000000 : ℝ) ≤
+      (-(625602605153547579439 / 250000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (9404207965373 / 128000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(6757786420885346854021 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_118_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (9404207965373 / 128000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1096564948833060777743 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_118_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (9404207965373 / 128000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) * (-(110370343779415330577 / 39062500000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_118_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (9404207965373 / 128000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1198538381065681381673 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_118_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (9404207965373 / 128000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(34447987663370110448543 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_118_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (9404207965373 / 128000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_118_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (9404207965373 / 128000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_118_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (9404207965373 / 128000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_118_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (9404207965373 / 128000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_118_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (9404207965373 / 128000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_118_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (9404207965373 / 128000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_118_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (9404207965373 / 128000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_118_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (9404207965373 / 128000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_118_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (9404207965373 / 128000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_118_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (9404207965373 / 128000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_118_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (9404207965373 / 128000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_118_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_119_1 :
+    Uω (aρ 1) (bρ 1) (4718918956847 / 64000000000000 : ℝ) ≤
+      (-(26991992297169125119499 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5380748032018147 / 80000000000000000 : ℝ) ≤
+      (-(26991992297169125119499 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (5380748032018147 / 80000000000000000 : ℝ) =
+            (5380748032018147 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5380748032018147 / 5000000000000000 : ℝ) ≤
+            (733894914830874880501 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (672353139082193 / 10000000000000000 : ℝ)) (q :=
+      (5380748032018147 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_119_2 :
+    Uω (aρ 2) (bρ 2) (4718918956847 / 64000000000000 : ℝ) ≤
+      (-(27373356034580663451179 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5179409378499819 / 80000000000000000 : ℝ) ≤
+      (-(27373356034580663451179 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (5179409378499819 / 80000000000000000 : ℝ) =
+            (5179409378499819 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5179409378499819 / 5000000000000000 : ℝ) ≤
+            (352531177419336548821 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (645787488212611 / 10000000000000000 : ℝ)) (q :=
+      (5179409378499819 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_119_3 :
+    Uω (aρ 3) (bρ 3) (4718918956847 / 64000000000000 : ℝ) ≤
+      (-(28210117915897178442261 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4763653147175431 / 80000000000000000 : ℝ) ≤
+      (-(28210117915897178442261 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (4763653147175431 / 80000000000000000 : ℝ) =
+            (4763653147175431 / 2500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4763653147175431 / 2500000000000000 : ℝ) ≤
+            (6447241099102821557739 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (294622268365757 / 5000000000000000 : ℝ)) (q :=
+      (4763653147175431 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_119_4 :
+    Uω (aρ 4) (bρ 4) (4718918956847 / 64000000000000 : ℝ) ≤
+      (-(29908730883182997431499 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (32155872285303 / 640000000000000 : ℝ) ≤
+      (-(29908730883182997431499 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (32155872285303 / 640000000000000 : ℝ) = (32155872285303 / 20000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (32155872285303 / 20000000000000 : ℝ) ≤
+            (4748628131817002568501 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3851971519347 / 80000000000000 : ℝ)) (q :=
+      (32155872285303 / 640000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_119_5 :
+    Uω (aρ 5) (bρ 5) (4718918956847 / 64000000000000 : ℝ) ≤
+      (-(34337546103835562865811 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2581245470590107 / 80000000000000000 : ℝ) ≤
+      (-(34337546103835562865811 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (2581245470590107 / 80000000000000000 : ℝ) =
+            (2581245470590107 / 2500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2581245470590107 / 2500000000000000 : ℝ) ≤
+            (319812911164437134189 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (240126717985183 / 10000000000000000 : ℝ)) (q :=
+      (2581245470590107 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_119_6 :
+    Uω (aρ 6) (bρ 6) (4718918956847 / 64000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_119_7 :
+    Uω (aρ 7) (bρ 7) (4718918956847 / 64000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_119_8 :
+    Uω (aρ 8) (bρ 8) (4718918956847 / 64000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_119_9 :
+    Uω (aρ 9) (bρ 9) (4718918956847 / 64000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_119_10 :
+    Uω (aρ 10) (bρ 10) (4718918956847 / 64000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_119_11 :
+    Uω (aρ 11) (bρ 11) (4718918956847 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_119_12 :
+    Uω (aρ 12) (bρ 12) (4718918956847 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_119_13 :
+    Uω (aρ 13) (bρ 13) (4718918956847 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_119_14 :
+    Uω (aρ 14) (bρ 14) (4718918956847 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_119_15 :
+    Uω (aρ 15) (bρ 15) (4718918956847 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_119_16 :
+    Uω (aρ 16) (bρ 16) (4718918956847 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_119 :
+    Uρ (4718918956847 / 64000000000000 : ℝ) ≤
+      (-(1563110136778042269237 / 625000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (4718918956847 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(26991992297169125119499 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_119_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (4718918956847 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(27373356034580663451179 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_119_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (4718918956847 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(28210117915897178442261 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_119_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (4718918956847 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(29908730883182997431499 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_119_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (4718918956847 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(34337546103835562865811 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_119_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (4718918956847 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_119_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (4718918956847 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_119_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (4718918956847 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_119_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (4718918956847 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_119_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (4718918956847 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_119_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (4718918956847 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_119_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (4718918956847 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_119_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (4718918956847 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_119_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (4718918956847 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_119_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (4718918956847 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_119_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (4718918956847 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_119_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_120_1 :
+    Uω (aρ 1) (bρ 1) (1894293572403 / 25600000000000 : ℝ) ≤
+      (-(26952991720637376599691 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10803548469219799 / 160000000000000000 : ℝ) ≤
+      (-(26952991720637376599691 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (10803548469219799 / 160000000000000000 : ℝ) =
+            (10803548469219799 / 10000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10803548469219799 / 10000000000000000 : ℝ) ≤
+            (772895491362623400309 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (674982350017553 / 10000000000000000 : ℝ)) (q :=
+      (10803548469219799 / 160000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_120_2 :
+    Uω (aρ 2) (bρ 2) (1894293572403 / 25600000000000 : ℝ) ≤
+      (-(13666377355402748850127 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10400962428977599 / 160000000000000000 : ℝ) ≤
+      (-(13666377355402748850127 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (10400962428977599 / 160000000000000000 : ℝ) =
+            (10400962428977599 / 10000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10400962428977599 / 10000000000000000 : ℝ) ≤
+            (196566250597251149873 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (324214053748639 / 5000000000000000 : ℝ)) (q :=
+      (10400962428977599 / 160000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_120_3 :
+    Uω (aρ 3) (bρ 3) (1894293572403 / 25600000000000 : ℝ) ≤
+      (-(28165630829855142778867 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1913956984259227 / 32000000000000000 : ℝ) ≤
+      (-(28165630829855142778867 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (1913956984259227 / 32000000000000000 : ℝ) =
+            (1913956984259227 / 1000000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1913956984259227 / 1000000000000000 : ℝ) ≤
+            (6491728185144857221133 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (118385405077419 / 2000000000000000 : ℝ)) (q :=
+      (1913956984259227 / 32000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_120_4 :
+    Uω (aρ 4) (bρ 4) (1894293572403 / 25600000000000 : ℝ) ≤
+      (-(5970865177888741761793 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8082823260978207 / 160000000000000000 : ℝ) ≤
+      (-(5970865177888741761793 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (8082823260978207 / 160000000000000000 : ℝ) =
+            (8082823260978207 / 5000000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8082823260978207 / 5000000000000000 : ℝ) ≤
+            (960606625111258238207 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (242175499456177 / 5000000000000000 : ℝ)) (q :=
+      (8082823260978207 / 160000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_120_5 :
+    Uω (aρ 5) (bρ 5) (1894293572403 / 25600000000000 : ℝ) ≤
+      (-(3422912528271636769997 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1043753523753371 / 32000000000000000 : ℝ) ≤
+      (-(3422912528271636769997 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (1043753523753371 / 32000000000000000 : ℝ) =
+            (1043753523753371 / 1000000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1043753523753371 / 1000000000000000 : ℝ) ≤
+            (42823373228363230003 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (48906792594187 / 2000000000000000 : ℝ)) (q :=
+      (1043753523753371 / 32000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_120_6 :
+    Uω (aρ 6) (bρ 6) (1894293572403 / 25600000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_120_7 :
+    Uω (aρ 7) (bρ 7) (1894293572403 / 25600000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_120_8 :
+    Uω (aρ 8) (bρ 8) (1894293572403 / 25600000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_120_9 :
+    Uω (aρ 9) (bρ 9) (1894293572403 / 25600000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_120_10 :
+    Uω (aρ 10) (bρ 10) (1894293572403 / 25600000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_120_11 :
+    Uω (aρ 11) (bρ 11) (1894293572403 / 25600000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_120_12 :
+    Uω (aρ 12) (bρ 12) (1894293572403 / 25600000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_120_13 :
+    Uω (aρ 13) (bρ 13) (1894293572403 / 25600000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_120_14 :
+    Uω (aρ 14) (bρ 14) (1894293572403 / 25600000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_120_15 :
+    Uω (aρ 15) (bρ 15) (1894293572403 / 25600000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_120_16 :
+    Uω (aρ 16) (bρ 16) (1894293572403 / 25600000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_120 :
+    Uρ (1894293572403 / 25600000000000 : ℝ) ≤
+      (-(24995593741306154908533 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1894293572403 / 25600000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(26952991720637376599691 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_120_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1894293572403 / 25600000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(13666377355402748850127 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_120_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1894293572403 / 25600000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(28165630829855142778867 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_120_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1894293572403 / 25600000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(5970865177888741761793 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_120_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1894293572403 / 25600000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(3422912528271636769997 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_120_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1894293572403 / 25600000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_120_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1894293572403 / 25600000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_120_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1894293572403 / 25600000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_120_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1894293572403 / 25600000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_120_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1894293572403 / 25600000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_120_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1894293572403 / 25600000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_120_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1894293572403 / 25600000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_120_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1894293572403 / 25600000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_120_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1894293572403 / 25600000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_120_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1894293572403 / 25600000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_120_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1894293572403 / 25600000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_120_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_121_1 :
+    Uω (aρ 1) (bρ 1) (297034306573 / 4000000000000 : ℝ) ≤
+      (-(6728535691238731355147 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1355700094784461 / 20000000000000000 : ℝ) ≤
+      (-(6728535691238731355147 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1355700094784461 / 20000000000000000 : ℝ) =
+            (1355700094784461 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1355700094784461 / 1250000000000000 : ℝ) ≤
+            (202936111761268644853 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (677611546436961 / 10000000000000000 : ℝ)) (q :=
+      (1355700094784461 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_121_2 :
+    Uω (aρ 2) (bρ 2) (297034306573 / 4000000000000 : ℝ) ≤
+      (-(3411539798815513981353 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (652694077585557 / 10000000000000000 : ℝ) ≤
+      (-(3411539798815513981353 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (652694077585557 / 10000000000000000 : ℝ) =
+            (652694077585557 / 625000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (652694077585557 / 625000000000000 : ℝ) ≤
+            (54196102684486018647 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (325534309666807 / 5000000000000000 : ℝ)) (q :=
+      (652694077585557 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_121_3 :
+    Uω (aρ 3) (bρ 3) (297034306573 / 4000000000000 : ℝ) ≤
+      (-(7030336219103306552713 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (600766225526559 / 10000000000000000 : ℝ) ≤
+      (-(7030336219103306552713 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (600766225526559 / 10000000000000000 : ℝ) =
+            (600766225526559 / 312500000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (600766225526559 / 312500000000000 : ℝ) ≤
+            (1634003534646693447287 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (297304510782809 / 5000000000000000 : ℝ)) (q :=
+      (600766225526559 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_121_4 :
+    Uω (aρ 4) (bρ 4) (297034306573 / 4000000000000 : ℝ) ≤
+      (-(3725030056597356522343 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (507916124857619 / 10000000000000000 : ℝ) ≤
+      (-(3725030056597356522343 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (507916124857619 / 10000000000000000 : ℝ) =
+            (507916124857619 / 312500000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (507916124857619 / 312500000000000 : ℝ) ≤
+            (607139820277643477657 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (243601500646369 / 5000000000000000 : ℝ)) (q :=
+      (507916124857619 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_121_5 :
+    Uω (aρ 5) (bρ 5) (297034306573 / 4000000000000 : ℝ) ≤
+      (-(682452605272557036629 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (82416279421253 / 2500000000000000 : ℝ) ≤
+      (-(682452605272557036629 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (82416279421253 / 2500000000000000 : ℝ) = (82416279421253 / 78125000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (82416279421253 / 78125000000000 : ℝ) ≤
+            (10694575027442963371 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (62222726570631 / 2500000000000000 : ℝ)) (q :=
+      (82416279421253 / 2500000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_121_6 :
+    Uω (aρ 6) (bρ 6) (297034306573 / 4000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_121_7 :
+    Uω (aρ 7) (bρ 7) (297034306573 / 4000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_121_8 :
+    Uω (aρ 8) (bρ 8) (297034306573 / 4000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_121_9 :
+    Uω (aρ 9) (bρ 9) (297034306573 / 4000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_121_10 :
+    Uω (aρ 10) (bρ 10) (297034306573 / 4000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_121_11 :
+    Uω (aρ 11) (bρ 11) (297034306573 / 4000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_121_12 :
+    Uω (aρ 12) (bρ 12) (297034306573 / 4000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_121_13 :
+    Uω (aρ 13) (bρ 13) (297034306573 / 4000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_121_14 :
+    Uω (aρ 14) (bρ 14) (297034306573 / 4000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_121_15 :
+    Uω (aρ 15) (bρ 15) (297034306573 / 4000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_121_16 :
+    Uω (aρ 16) (bρ 16) (297034306573 / 4000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_121 :
+    Uρ (297034306573 / 4000000000000 : ℝ) ≤
+      (-(24981591939352846305661 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (297034306573 / 4000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(6728535691238731355147 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_121_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (297034306573 / 4000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3411539798815513981353 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_121_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (297034306573 / 4000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(7030336219103306552713 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_121_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (297034306573 / 4000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3725030056597356522343 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_121_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (297034306573 / 4000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(682452605272557036629 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_121_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (297034306573 / 4000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_121_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (297034306573 / 4000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_121_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (297034306573 / 4000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_121_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (297034306573 / 4000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_121_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (297034306573 / 4000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_121_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (297034306573 / 4000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_121_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (297034306573 / 4000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_121_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (297034306573 / 4000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_121_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (297034306573 / 4000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_121_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (297034306573 / 4000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_121_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (297034306573 / 4000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_121_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_122_1 :
+    Uω (aρ 1) (bρ 1) (9538727758657 / 128000000000000 : ℝ) ≤
+      (-(26875444254963523057139 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10887652932550481 / 160000000000000000 : ℝ) ≤
+      (-(26875444254963523057139 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (10887652932550481 / 160000000000000000 : ℝ) =
+            (10887652932550481 / 10000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10887652932550481 / 10000000000000000 : ℝ) ≤
+            (850442957036476942861 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (170060182127183 / 2500000000000000 : ℝ)) (q :=
+      (10887652932550481 / 160000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_122_2 :
+    Uω (aρ 2) (bρ 2) (9538727758657 / 128000000000000 : ℝ) ≤
+      (-(27252045731424229280841 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10485247204589921 / 160000000000000000 : ℝ) ≤
+      (-(27252045731424229280841 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (10485247204589921 / 160000000000000000 : ℝ) =
+            (10485247204589921 / 10000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10485247204589921 / 10000000000000000 : ℝ) ≤
+            (473841480575770719159 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (326854512511831 / 5000000000000000 : ℝ)) (q :=
+      (10485247204589921 / 160000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_122_3 :
+    Uω (aρ 3) (bρ 3) (9538727758657 / 128000000000000 : ℝ) ≤
+      (-(28077258208867713837771 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9654730408809841 / 160000000000000000 : ℝ) ≤
+      (-(28077258208867713837771 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (9654730408809841 / 160000000000000000 : ℝ) =
+            (9654730408809841 / 5000000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9654730408809841 / 5000000000000000 : ℝ) ≤
+            (6580100806132286162229 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (18665329121911 / 312500000000000 : ℝ)) (q :=
+      (9654730408809841 / 160000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_122_4 :
+    Uω (aρ 4) (bρ 4) (9538727758657 / 128000000000000 : ℝ) ≤
+      (-(14873235281136487034493 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8170472638651729 / 160000000000000000 : ℝ) ≤
+      (-(14873235281136487034493 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (8170472638651729 / 160000000000000000 : ℝ) =
+            (8170472638651729 / 5000000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8170472638651729 / 5000000000000000 : ℝ) ≤
+            (2455444226363512965507 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (122513122924097 / 2500000000000000 : ℝ)) (q :=
+      (8170472638651729 / 160000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_122_5 :
+    Uω (aρ 5) (bρ 5) (9538727758657 / 128000000000000 : ℝ) ≤
+      (-(6803594680306818344917 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5330134507460841 / 160000000000000000 : ℝ) ≤
+      (-(6803594680306818344917 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (5330134507460841 / 160000000000000000 : ℝ) =
+            (5330134507460841 / 5000000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5330134507460841 / 5000000000000000 : ℝ) ≤
+            (127877122693181655083 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (253200144632527 / 10000000000000000 : ℝ)) (q :=
+      (5330134507460841 / 160000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_122_6 :
+    Uω (aρ 6) (bρ 6) (9538727758657 / 128000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_122_7 :
+    Uω (aρ 7) (bρ 7) (9538727758657 / 128000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_122_8 :
+    Uω (aρ 8) (bρ 8) (9538727758657 / 128000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_122_9 :
+    Uω (aρ 9) (bρ 9) (9538727758657 / 128000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_122_10 :
+    Uω (aρ 10) (bρ 10) (9538727758657 / 128000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_122_11 :
+    Uω (aρ 11) (bρ 11) (9538727758657 / 128000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_122_12 :
+    Uω (aρ 12) (bρ 12) (9538727758657 / 128000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_122_13 :
+    Uω (aρ 13) (bρ 13) (9538727758657 / 128000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_122_14 :
+    Uω (aρ 14) (bρ 14) (9538727758657 / 128000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_122_15 :
+    Uω (aρ 15) (bρ 15) (9538727758657 / 128000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_122_16 :
+    Uω (aρ 16) (bρ 16) (9538727758657 / 128000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_122 :
+    Uρ (9538727758657 / 128000000000000 : ℝ) ≤
+      (-(6241937591781720172913 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (9538727758657 / 128000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(26875444254963523057139 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_122_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (9538727758657 / 128000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(27252045731424229280841 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_122_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (9538727758657 / 128000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(28077258208867713837771 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_122_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (9538727758657 / 128000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(14873235281136487034493 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_122_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (9538727758657 / 128000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(6803594680306818344917 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_122_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (9538727758657 / 128000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_122_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (9538727758657 / 128000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_122_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (9538727758657 / 128000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_122_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (9538727758657 / 128000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_122_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (9538727758657 / 128000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_122_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (9538727758657 / 128000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_122_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (9538727758657 / 128000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_122_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (9538727758657 / 128000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_122_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (9538727758657 / 128000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_122_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (9538727758657 / 128000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_122_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (9538727758657 / 128000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_122_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_123_1 :
+    Uω (aρ 1) (bρ 1) (4786178853489 / 64000000000000 : ℝ) ≤
+      (-(5367379005824790420369 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5464852496684989 / 80000000000000000 : ℝ) ≤
+      (-(5367379005824790420369 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (5464852496684989 / 80000000000000000 : ℝ) =
+            (5464852496684989 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5464852496684989 / 5000000000000000 : ℝ) ≤
+            (177798436575209579631 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (682869896398591 / 10000000000000000 : ℝ)) (q :=
+      (5464852496684989 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_123_2 :
+    Uω (aρ 2) (bρ 2) (4786178853489 / 64000000000000 : ℝ) ≤
+      (-(27211935407585621453431 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5263694164444461 / 80000000000000000 : ℝ) ≤
+      (-(27211935407585621453431 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (5263694164444461 / 80000000000000000 : ℝ) =
+            (5263694164444461 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5263694164444461 / 5000000000000000 : ℝ) ≤
+            (513951804414378546569 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (656349325848459 / 10000000000000000 : ℝ)) (q :=
+      (5263694164444461 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_123_3 :
+    Uω (aρ 3) (bρ 3) (4786178853489 / 64000000000000 : ℝ) ≤
+      (-(14016684503152133467349 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4848598687282497 / 80000000000000000 : ℝ) ≤
+      (-(14016684503152133467349 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (4848598687282497 / 80000000000000000 : ℝ) =
+            (4848598687282497 / 2500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4848598687282497 / 2500000000000000 : ℝ) ≤
+            (3311995004347866532651 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (9374555670437 / 156250000000000 : ℝ)) (q :=
+      (4848598687282497 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_123_4 :
+    Uω (aρ 4) (bρ 4) (4786178853489 / 64000000000000 : ℝ) ≤
+      (-(29693012286484085391417 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (821426753229409 / 16000000000000000 : ℝ) ≤
+      (-(29693012286484085391417 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (821426753229409 / 16000000000000000 : ℝ) =
+            (821426753229409 / 500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (821426753229409 / 500000000000000 : ℝ) ≤
+            (4964346728515914608583 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (98579902737821 / 2000000000000000 : ℝ)) (q :=
+      (821426753229409 / 16000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_123_5 :
+    Uω (aρ 5) (bρ 5) (4786178853489 / 64000000000000 : ℝ) ≤
+      (-(16957536789013437420659 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2692632327572469 / 80000000000000000 : ℝ) ≤
+      (-(16957536789013437420659 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (2692632327572469 / 80000000000000000 : ℝ) =
+            (2692632327572469 / 2500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2692632327572469 / 2500000000000000 : ℝ) ≤
+            (371142718486562579341 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (257464073380461 / 10000000000000000 : ℝ)) (q :=
+      (2692632327572469 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_123_6 :
+    Uω (aρ 6) (bρ 6) (4786178853489 / 64000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_123_7 :
+    Uω (aρ 7) (bρ 7) (4786178853489 / 64000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_123_8 :
+    Uω (aρ 8) (bρ 8) (4786178853489 / 64000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_123_9 :
+    Uω (aρ 9) (bρ 9) (4786178853489 / 64000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_123_10 :
+    Uω (aρ 10) (bρ 10) (4786178853489 / 64000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_123_11 :
+    Uω (aρ 11) (bρ 11) (4786178853489 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_123_12 :
+    Uω (aρ 12) (bρ 12) (4786178853489 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_123_13 :
+    Uω (aρ 13) (bρ 13) (4786178853489 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_123_14 :
+    Uω (aρ 14) (bρ 14) (4786178853489 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_123_15 :
+    Uω (aρ 15) (bρ 15) (4786178853489 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_123_16 :
+    Uω (aρ 16) (bρ 16) (4786178853489 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_123 :
+    Uρ (4786178853489 / 64000000000000 : ℝ) ≤
+      (-(311925788326356217817 / 125000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (4786178853489 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(5367379005824790420369 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_123_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (4786178853489 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(27211935407585621453431 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_123_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (4786178853489 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(14016684503152133467349 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_123_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (4786178853489 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(29693012286484085391417 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_123_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (4786178853489 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(16957536789013437420659 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_123_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (4786178853489 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_123_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (4786178853489 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_123_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (4786178853489 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_123_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (4786178853489 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_123_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (4786178853489 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_123_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (4786178853489 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_123_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (4786178853489 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_123_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (4786178853489 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_123_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (4786178853489 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_123_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (4786178853489 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_123_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (4786178853489 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_123_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/U10.lean b/LeanPool/Zeta5Irrational/Table/U10.lean
new file mode 100644
index 0000000000..29afac606a
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/U10.lean
@@ -0,0 +1,3066 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-! # Certified arcsine potential bounds (U10) -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma U_124_1 :
+    Uω (aρ 1) (bρ 1) (9605987655299 / 128000000000000 : ℝ) ≤
+      (-(13399246969653145294021 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10971756942039611 / 160000000000000000 : ℝ) ≤
+      (-(13399246969653145294021 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (10971756942039611 / 160000000000000000 : ℝ) =
+            (10971756942039611 / 10000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10971756942039611 / 10000000000000000 : ℝ) ≤
+            (463696636346854705979 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (685499050269717 / 10000000000000000 : ℝ)) (q :=
+      (10971756942039611 / 160000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_124_2 :
+    Uω (aρ 2) (bρ 2) (9605987655299 / 128000000000000 : ℝ) ≤
+      (-(27171986109213034869943 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2113905724869911 / 32000000000000000 : ℝ) ≤
+      (-(27171986109213034869943 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2113905724869911 / 32000000000000000 : ℝ) =
+            (2113905724869911 / 2000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2113905724869911 / 2000000000000000 : ℝ) ≤
+            (553901102786965130057 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (32949476153423 / 500000000000000 : ℝ)) (q :=
+      (2113905724869911 / 32000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_124_3 :
+    Uω (aρ 3) (bρ 3) (9605987655299 / 128000000000000 : ℝ) ≤
+      (-(13994837736555882275651 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9739660556868003 / 160000000000000000 : ℝ) ≤
+      (-(13994837736555882275651 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (9739660556868003 / 160000000000000000 : ℝ) =
+            (9739660556868003 / 5000000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9739660556868003 / 5000000000000000 : ℝ) ≤
+            (3333841770944117724349 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (301326060491633 / 5000000000000000 : ℝ)) (q :=
+      (9739660556868003 / 160000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_124_4 :
+    Uω (aρ 4) (bρ 4) (9605987655299 / 128000000000000 : ℝ) ≤
+      (-(14819930885633522672703 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8258043018870651 / 160000000000000000 : ℝ) ≤
+      (-(14819930885633522672703 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (8258043018870651 / 160000000000000000 : ℝ) =
+            (8258043018870651 / 5000000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8258043018870651 / 5000000000000000 : ℝ) ≤
+            (2508748621866477327297 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (495744109798597 / 10000000000000000 : ℝ)) (q :=
+      (8258043018870651 / 160000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_124_5 :
+    Uω (aρ 5) (bρ 5) (9605987655299 / 128000000000000 : ℝ) ≤
+      (-(16906927768461829373233 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1088010008964823 / 32000000000000000 : ℝ) ≤
+      (-(16906927768461829373233 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (1088010008964823 / 32000000000000000 : ℝ) =
+            (1088010008964823 / 1000000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1088010008964823 / 1000000000000000 : ℝ) ≤
+            (421751739038170626767 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (13084245368889 / 500000000000000 : ℝ)) (q :=
+      (1088010008964823 / 32000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_124_6 :
+    Uω (aρ 6) (bρ 6) (9605987655299 / 128000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_124_7 :
+    Uω (aρ 7) (bρ 7) (9605987655299 / 128000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_124_8 :
+    Uω (aρ 8) (bρ 8) (9605987655299 / 128000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_124_9 :
+    Uω (aρ 9) (bρ 9) (9605987655299 / 128000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_124_10 :
+    Uω (aρ 10) (bρ 10) (9605987655299 / 128000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_124_11 :
+    Uω (aρ 11) (bρ 11) (9605987655299 / 128000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_124_12 :
+    Uω (aρ 12) (bρ 12) (9605987655299 / 128000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_124_13 :
+    Uω (aρ 13) (bρ 13) (9605987655299 / 128000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_124_14 :
+    Uω (aρ 14) (bρ 14) (9605987655299 / 128000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_124_15 :
+    Uω (aρ 15) (bρ 15) (9605987655299 / 128000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_124_16 :
+    Uω (aρ 16) (bρ 16) (9605987655299 / 128000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_124 :
+    Uρ (9605987655299 / 128000000000000 : ℝ) ≤
+      (-(24940524488693670762209 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (9605987655299 / 128000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(13399246969653145294021 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_124_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (9605987655299 / 128000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(27171986109213034869943 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_124_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (9605987655299 / 128000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(13994837736555882275651 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_124_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (9605987655299 / 128000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(14819930885633522672703 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_124_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (9605987655299 / 128000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(16906927768461829373233 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_124_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (9605987655299 / 128000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_124_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (9605987655299 / 128000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_124_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (9605987655299 / 128000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_124_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (9605987655299 / 128000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_124_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (9605987655299 / 128000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_124_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (9605987655299 / 128000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_124_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (9605987655299 / 128000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_124_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (9605987655299 / 128000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_124_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (9605987655299 / 128000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_124_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (9605987655299 / 128000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_124_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (9605987655299 / 128000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_124_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_125_1 :
+    Uω (aρ 1) (bρ 1) (481980880181 / 6400000000000 : ℝ) ≤
+      (-(6690059962646030184617 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2753452194961219 / 40000000000000000 : ℝ) ≤
+      (-(6690059962646030184617 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2753452194961219 / 40000000000000000 : ℝ) =
+            (2753452194961219 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2753452194961219 / 2500000000000000 : ℝ) ≤
+            (241411840353969815383 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (688128190282797 / 10000000000000000 : ℝ)) (q :=
+      (2753452194961219 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_125_2 :
+    Uω (aρ 2) (bρ 2) (481980880181 / 6400000000000 : ℝ) ≤
+      (-(1085287861694982018329 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2652917025223531 / 40000000000000000 : ℝ) ≤
+      (-(1085287861694982018329 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2652917025223531 / 40000000000000000 : ℝ) =
+            (2652917025223531 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2652917025223531 / 2500000000000000 : ℝ) ≤
+            (23747626785017981671 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (661629617923953 / 10000000000000000 : ℝ)) (q :=
+      (2652917025223531 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_125_3 :
+    Uω (aρ 3) (bρ 3) (481980880181 / 6400000000000 : ℝ) ≤
+      (-(6986543959626957316373 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2445530001495253 / 40000000000000000 : ℝ) ≤
+      (-(6986543959626957316373 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (2445530001495253 / 40000000000000000 : ℝ) =
+            (2445530001495253 / 1250000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2445530001495253 / 1250000000000000 : ℝ) ≤
+            (1677795794123042683627 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (302666106204907 / 5000000000000000 : ℝ)) (q :=
+      (2445530001495253 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_125_4 :
+    Uω (aρ 4) (bρ 4) (481980880181 / 6400000000000 : ℝ) ≤
+      (-(1849188452354571814583 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2075449857638049 / 40000000000000000 : ℝ) ≤
+      (-(1849188452354571814583 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (2075449857638049 / 40000000000000000 : ℝ) =
+            (2075449857638049 / 1250000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2075449857638049 / 1250000000000000 : ℝ) ≤
+            (316896486082928185417 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (124646580386553 / 2500000000000000 : ℝ)) (q :=
+      (2075449857638049 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_125_5 :
+    Uω (aρ 5) (bρ 5) (481980880181 / 6400000000000 : ℝ) ≤
+      (-(33714249304541786608363 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (274725354828193 / 8000000000000000 : ℝ) ≤
+      (-(33714249304541786608363 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (274725354828193 / 8000000000000000 : ℝ) =
+            (274725354828193 / 250000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (274725354828193 / 250000000000000 : ℝ) ≤
+            (943109710458213391637 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (26586469913267 / 1000000000000000 : ℝ)) (q :=
+      (274725354828193 / 8000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_125_6 :
+    Uω (aρ 6) (bρ 6) (481980880181 / 6400000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_125_7 :
+    Uω (aρ 7) (bρ 7) (481980880181 / 6400000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_125_8 :
+    Uω (aρ 8) (bρ 8) (481980880181 / 6400000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_125_9 :
+    Uω (aρ 9) (bρ 9) (481980880181 / 6400000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_125_10 :
+    Uω (aρ 10) (bρ 10) (481980880181 / 6400000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_125_11 :
+    Uω (aρ 11) (bρ 11) (481980880181 / 6400000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_125_12 :
+    Uω (aρ 12) (bρ 12) (481980880181 / 6400000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_125_13 :
+    Uω (aρ 13) (bρ 13) (481980880181 / 6400000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_125_14 :
+    Uω (aρ 14) (bρ 14) (481980880181 / 6400000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_125_15 :
+    Uω (aρ 15) (bρ 15) (481980880181 / 6400000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_125_16 :
+    Uω (aρ 16) (bρ 16) (481980880181 / 6400000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_125 :
+    Uρ (481980880181 / 6400000000000 : ℝ) ≤
+      (-(12463564729009360759451 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (481980880181 / 6400000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(6690059962646030184617 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_125_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (481980880181 / 6400000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1085287861694982018329 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_125_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (481980880181 / 6400000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(6986543959626957316373 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_125_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (481980880181 / 6400000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1849188452354571814583 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_125_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (481980880181 / 6400000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(33714249304541786608363 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_125_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (481980880181 / 6400000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_125_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (481980880181 / 6400000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_125_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (481980880181 / 6400000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_125_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (481980880181 / 6400000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_125_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (481980880181 / 6400000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_125_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (481980880181 / 6400000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_125_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (481980880181 / 6400000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_125_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (481980880181 / 6400000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_125_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (481980880181 / 6400000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_125_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (481980880181 / 6400000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_125_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (481980880181 / 6400000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_125_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_126_1 :
+    Uω (aρ 1) (bρ 1) (9673247551941 / 128000000000000 : ℝ) ≤
+      (-(26722131641032751290407 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11055860508051669 / 160000000000000000 : ℝ) ≤
+      (-(26722131641032751290407 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (11055860508051669 / 160000000000000000 : ℝ) =
+            (11055860508051669 / 10000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11055860508051669 / 10000000000000000 : ℝ) ≤
+            (1003755570967248709593 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (172689329149017 / 2500000000000000 : ℝ)) (q :=
+      (11055860508051669 / 160000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_126_2 :
+    Uω (aρ 2) (bρ 2) (9673247551941 / 128000000000000 : ℝ) ≤
+      (-(3386570678593159343237 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10653806768286813 / 160000000000000000 : ℝ) ≤
+      (-(3386570678593159343237 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (10653806768286813 / 160000000000000000 : ℝ) =
+            (10653806768286813 / 10000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10653806768286813 / 10000000000000000 : ℝ) ≤
+            (79165222906840656763 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (664269611635461 / 10000000000000000 : ℝ)) (q :=
+      (10653806768286813 / 160000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_126_3 :
+    Uω (aρ 3) (bρ 3) (9673247551941 / 128000000000000 : ℝ) ≤
+      (-(697571708901889533809 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9824575771271253 / 160000000000000000 : ℝ) ≤
+      (-(697571708901889533809 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (9824575771271253 / 160000000000000000 : ℝ) =
+            (9824575771271253 / 5000000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9824575771271253 / 5000000000000000 : ℝ) ≤
+            (168862266473110466191 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (152002960839629 / 2500000000000000 : ℝ)) (q :=
+      (9824575771271253 / 160000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_126_4 :
+    Uω (aρ 4) (bρ 4) (9673247551941 / 128000000000000 : ℝ) ≤
+      (-(2953446897992292134223 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8345537091702853 / 160000000000000000 : ℝ) ≤
+      (-(2953446897992292134223 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (8345537091702853 / 160000000000000000 : ℝ) =
+            (8345537091702853 / 5000000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8345537091702853 / 5000000000000000 : ℝ) ≤
+            (512289003507707865777 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (250713094738733 / 5000000000000000 : ℝ)) (q :=
+      (8345537091702853 / 160000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_126_5 :
+    Uω (aρ 5) (bρ 5) (9673247551941 / 128000000000000 : ℝ) ≤
+      (-(6723237936362052732691 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5548651058829061 / 160000000000000000 : ℝ) ≤
+      (-(6723237936362052732691 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (5548651058829061 / 160000000000000000 : ℝ) =
+            (5548651058829061 / 5000000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5548651058829061 / 5000000000000000 : ℝ) ≤
+            (208233866637947267309 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (135002677351621 / 5000000000000000 : ℝ)) (q :=
+      (5548651058829061 / 160000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_126_6 :
+    Uω (aρ 6) (bρ 6) (9673247551941 / 128000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_126_7 :
+    Uω (aρ 7) (bρ 7) (9673247551941 / 128000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_126_8 :
+    Uω (aρ 8) (bρ 8) (9673247551941 / 128000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_126_9 :
+    Uω (aρ 9) (bρ 9) (9673247551941 / 128000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_126_10 :
+    Uω (aρ 10) (bρ 10) (9673247551941 / 128000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_126_11 :
+    Uω (aρ 11) (bρ 11) (9673247551941 / 128000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_126_12 :
+    Uω (aρ 12) (bρ 12) (9673247551941 / 128000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_126_13 :
+    Uω (aρ 13) (bρ 13) (9673247551941 / 128000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_126_14 :
+    Uω (aρ 14) (bρ 14) (9673247551941 / 128000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_126_15 :
+    Uω (aρ 15) (bρ 15) (9673247551941 / 128000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_126_16 :
+    Uω (aρ 16) (bρ 16) (9673247551941 / 128000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_126 :
+    Uρ (9673247551941 / 128000000000000 : ℝ) ≤
+      (-(6228468283187341352323 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (9673247551941 / 128000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(26722131641032751290407 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_126_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (9673247551941 / 128000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3386570678593159343237 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_126_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (9673247551941 / 128000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(697571708901889533809 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_126_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (9673247551941 / 128000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(2953446897992292134223 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_126_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (9673247551941 / 128000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(6723237936362052732691 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_126_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (9673247551941 / 128000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_126_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (9673247551941 / 128000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_126_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (9673247551941 / 128000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_126_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (9673247551941 / 128000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_126_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (9673247551941 / 128000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_126_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (9673247551941 / 128000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_126_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (9673247551941 / 128000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_126_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (9673247551941 / 128000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_126_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (9673247551941 / 128000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_126_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (9673247551941 / 128000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_126_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (9673247551941 / 128000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_126_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_127_1 :
+    Uω (aρ 1) (bρ 1) (4853438750131 / 64000000000000 : ℝ) ≤
+      (-(2668416820153121014947 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5548956063953347 / 80000000000000000 : ℝ) ≤
+      (-(2668416820153121014947 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (5548956063953347 / 80000000000000000 : ℝ) =
+            (5548956063953347 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5548956063953347 / 5000000000000000 : ℝ) ≤
+            (104171901046878985053 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (86673303670671 / 1250000000000000 : ℝ)) (q :=
+      (5548956063953347 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_127_2 :
+    Uω (aρ 2) (bρ 2) (4853438750131 / 64000000000000 : ℝ) ≤
+      (-(13526545752678102531631 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1069594463613683 / 16000000000000000 : ℝ) ≤
+      (-(13526545752678102531631 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1069594463613683 / 16000000000000000 : ℝ) =
+            (1069594463613683 / 1000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1069594463613683 / 1000000000000000 : ℝ) ≤
+            (336397853321897468369 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (133381901080827 / 2000000000000000 : ℝ)) (q :=
+      (1069594463613683 / 16000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_127_3 :
+    Uω (aρ 3) (bρ 3) (4853438750131 / 64000000000000 : ℝ) ≤
+      (-(1392987565165561452741 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (986702790123111 / 16000000000000000 : ℝ) ≤
+      (-(1392987565165561452741 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (986702790123111 / 16000000000000000 : ℝ) =
+            (986702790123111 / 500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (986702790123111 / 500000000000000 : ℝ) ≤
+            (339880385584438547259 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (15267275497273 / 250000000000000 : ℝ)) (q :=
+      (986702790123111 / 16000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_127_4 :
+    Uω (aρ 4) (bρ 4) (4853438750131 / 64000000000000 : ℝ) ≤
+      (-(14741109681724341615633 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4194628159557691 / 80000000000000000 : ℝ) ≤
+      (-(14741109681724341615633 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (4194628159557691 / 80000000000000000 : ℝ) =
+            (4194628159557691 / 2500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4194628159557691 / 2500000000000000 : ℝ) ≤
+            (2587569825775658384367 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (252131876595727 / 5000000000000000 : ℝ)) (q :=
+      (4194628159557691 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_127_5 :
+    Uω (aρ 5) (bρ 5) (4853438750131 / 64000000000000 : ℝ) ≤
+      (-(33519615797620916311619 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2801248060076399 / 80000000000000000 : ℝ) ≤
+      (-(33519615797620916311619 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (2801248060076399 / 80000000000000000 : ℝ) =
+            (2801248060076399 / 2500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2801248060076399 / 2500000000000000 : ℝ) ≤
+            (1137743217379083688381 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (274108647656131 / 10000000000000000 : ℝ)) (q :=
+      (2801248060076399 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_127_6 :
+    Uω (aρ 6) (bρ 6) (4853438750131 / 64000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_127_7 :
+    Uω (aρ 7) (bρ 7) (4853438750131 / 64000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_127_8 :
+    Uω (aρ 8) (bρ 8) (4853438750131 / 64000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_127_9 :
+    Uω (aρ 9) (bρ 9) (4853438750131 / 64000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_127_10 :
+    Uω (aρ 10) (bρ 10) (4853438750131 / 64000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_127_11 :
+    Uω (aρ 11) (bρ 11) (4853438750131 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_127_12 :
+    Uω (aρ 12) (bρ 12) (4853438750131 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_127_13 :
+    Uω (aρ 13) (bρ 13) (4853438750131 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_127_14 :
+    Uω (aρ 14) (bρ 14) (4853438750131 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_127_15 :
+    Uω (aρ 15) (bρ 15) (4853438750131 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_127_16 :
+    Uω (aρ 16) (bρ 16) (4853438750131 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_127 :
+    Uρ (4853438750131 / 64000000000000 : ℝ) ≤
+      (-(24900750976102301112917 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (4853438750131 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2668416820153121014947 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_127_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (4853438750131 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(13526545752678102531631 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_127_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (4853438750131 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1392987565165561452741 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_127_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (4853438750131 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(14741109681724341615633 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_127_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (4853438750131 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(33519615797620916311619 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_127_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (4853438750131 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_127_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (4853438750131 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_127_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (4853438750131 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_127_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (4853438750131 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_127_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (4853438750131 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_127_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (4853438750131 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_127_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (4853438750131 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_127_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (4853438750131 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_127_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (4853438750131 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_127_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (4853438750131 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_127_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (4853438750131 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_127_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_128_1 :
+    Uω (aρ 1) (bρ 1) (1221767174613 / 16000000000000 : ℝ) ≤
+      (-(5321734251810154824021 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (349437970232949 / 5000000000000000 : ℝ) ≤
+      (-(5321734251810154824021 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (349437970232949 / 5000000000000000 : ℝ) =
+            (349437970232949 / 312500000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (349437970232949 / 312500000000000 : ℝ) ≤
+            (223443190589845175979 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (698644614883671 / 10000000000000000 : ℝ)) (q :=
+      (349437970232949 / 5000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_128_2 :
+    Uω (aρ 2) (bρ 2) (1221767174613 / 16000000000000 : ℝ) ≤
+      (-(26974610252730626450719 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (269505450272227 / 4000000000000000 : ℝ) ≤
+      (-(26974610252730626450719 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (269505450272227 / 4000000000000000 : ℝ) =
+            (269505450272227 / 250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (269505450272227 / 250000000000000 : ℝ) ≤
+            (751276959269373549281 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (67218899782301 / 1000000000000000 : ℝ)) (q :=
+      (269505450272227 / 4000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_128_3 :
+    Uω (aρ 3) (bρ 3) (1221767174613 / 16000000000000 : ℝ) ≤
+      (-(27774081713696468351641 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (621995090304443 / 10000000000000000 : ℝ) ≤
+      (-(27774081713696468351641 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (621995090304443 / 10000000000000000 : ℝ) =
+            (621995090304443 / 312500000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (621995090304443 / 312500000000000 : ℝ) ≤
+            (6883277301303531648359 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (616048033420761 / 10000000000000000 : ℝ)) (q :=
+      (621995090304443 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_128_4 :
+    Uω (aρ 4) (bρ 4) (1221767174613 / 16000000000000 : ℝ) ≤
+      (-(14689297930419472306447 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1059580087923863 / 20000000000000000 : ℝ) ≤
+      (-(14689297930419472306447 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (1059580087923863 / 20000000000000000 : ℝ) =
+            (1059580087923863 / 625000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1059580087923863 / 625000000000000 : ℝ) ≤
+            (2639381577080527693553 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (254966060900369 / 5000000000000000 : ℝ)) (q :=
+      (1059580087923863 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_128_5 :
+    Uω (aρ 5) (bρ 5) (1221767174613 / 16000000000000 : ℝ) ≤
+      (-(33330701080104509943517 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (356833848352817 / 10000000000000000 : ℝ) ≤
+      (-(33330701080104509943517 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (356833848352817 / 10000000000000000 : ℝ) =
+            (356833848352817 / 312500000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (356833848352817 / 312500000000000 : ℝ) ≤
+            (1326657934895490056483 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (282209649917509 / 10000000000000000 : ℝ)) (q :=
+      (356833848352817 / 10000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_128_6 :
+    Uω (aρ 6) (bρ 6) (1221767174613 / 16000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_128_7 :
+    Uω (aρ 7) (bρ 7) (1221767174613 / 16000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_128_8 :
+    Uω (aρ 8) (bρ 8) (1221767174613 / 16000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_128_9 :
+    Uω (aρ 9) (bρ 9) (1221767174613 / 16000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_128_10 :
+    Uω (aρ 10) (bρ 10) (1221767174613 / 16000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_128_11 :
+    Uω (aρ 11) (bρ 11) (1221767174613 / 16000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_128_12 :
+    Uω (aρ 12) (bρ 12) (1221767174613 / 16000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_128_13 :
+    Uω (aρ 13) (bρ 13) (1221767174613 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_128_14 :
+    Uω (aρ 14) (bρ 14) (1221767174613 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_128_15 :
+    Uω (aρ 15) (bρ 15) (1221767174613 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_128_16 :
+    Uω (aρ 16) (bρ 16) (1221767174613 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_128 :
+    Uρ (1221767174613 / 16000000000000 : ℝ) ≤
+      (-(24874892383294093360151 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1221767174613 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(5321734251810154824021 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_128_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1221767174613 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(26974610252730626450719 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_128_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1221767174613 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(27774081713696468351641 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_128_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1221767174613 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(14689297930419472306447 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_128_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1221767174613 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(33330701080104509943517 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_128_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1221767174613 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_128_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1221767174613 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_128_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1221767174613 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_128_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1221767174613 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_128_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1221767174613 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_128_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1221767174613 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_128_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1221767174613 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_128_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1221767174613 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_128_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1221767174613 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_128_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1221767174613 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_128_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1221767174613 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_128_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_129_1 :
+    Uω (aρ 1) (bρ 1) (4920698646773 / 64000000000000 : ℝ) ≤
+      (-(26533740398960545318349 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1126611754809097 / 16000000000000000 : ℝ) ≤
+      (-(26533740398960545318349 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1126611754809097 / 16000000000000000 : ℝ) =
+            (1126611754809097 / 1000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1126611754809097 / 1000000000000000 : ℝ) ≤
+            (1192146813039454681651 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (70390274803809 / 1000000000000000 : ℝ)) (q :=
+      (1126611754809097 / 16000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_129_2 :
+    Uω (aρ 2) (bρ 2) (4920698646773 / 64000000000000 : ℝ) ≤
+      (-(26896742978634930657697 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5432244149519873 / 80000000000000000 : ℝ) ≤
+      (-(26896742978634930657697 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (5432244149519873 / 80000000000000000 : ℝ) =
+            (5432244149519873 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5432244149519873 / 5000000000000000 : ℝ) ≤
+            (829144233365069342303 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (677468104416687 / 10000000000000000 : ℝ)) (q :=
+      (5432244149519873 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_129_3 :
+    Uω (aρ 3) (bρ 3) (4920698646773 / 64000000000000 : ℝ) ≤
+      (-(27689153753963773397029 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1003680100806941 / 16000000000000000 : ℝ) ≤
+      (-(27689153753963773397029 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1003680100806941 / 16000000000000000 : ℝ) =
+            (1003680100806941 / 1000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1003680100806941 / 1000000000000000 : ℝ) ≤
+            (36733458036226602971 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (124280659879079 / 2000000000000000 : ℝ)) (q :=
+      (1003680100806941 / 16000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_129_4 :
+    Uω (aρ 4) (bρ 4) (4920698646773 / 64000000000000 : ℝ) ≤
+      (-(14638058503645561336247 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4281977487348057 / 80000000000000000 : ℝ) ≤
+      (-(14638058503645561336247 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (4281977487348057 / 80000000000000000 : ℝ) =
+            (4281977487348057 / 2500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4281977487348057 / 2500000000000000 : ℝ) ≤
+            (2690621003854438663753 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (515591726288733 / 10000000000000000 : ℝ)) (q :=
+      (4281977487348057 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_129_5 :
+    Uω (aρ 5) (bρ 5) (4920698646773 / 64000000000000 : ℝ) ≤
+      (-(8286772710427584474673 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (581513923481993 / 16000000000000000 : ℝ) ≤
+      (-(8286772710427584474673 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (581513923481993 / 16000000000000000 : ℝ) =
+            (581513923481993 / 500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (581513923481993 / 500000000000000 : ℝ) ≤
+            (377567043322415525327 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (29017967813921 / 1000000000000000 : ℝ)) (q :=
+      (581513923481993 / 16000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_129_6 :
+    Uω (aρ 6) (bρ 6) (4920698646773 / 64000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_129_7 :
+    Uω (aρ 7) (bρ 7) (4920698646773 / 64000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_129_8 :
+    Uω (aρ 8) (bρ 8) (4920698646773 / 64000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_129_9 :
+    Uω (aρ 9) (bρ 9) (4920698646773 / 64000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_129_10 :
+    Uω (aρ 10) (bρ 10) (4920698646773 / 64000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_129_11 :
+    Uω (aρ 11) (bρ 11) (4920698646773 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_129_12 :
+    Uω (aρ 12) (bρ 12) (4920698646773 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_129_13 :
+    Uω (aρ 13) (bρ 13) (4920698646773 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_129_14 :
+    Uω (aρ 14) (bρ 14) (4920698646773 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_129_15 :
+    Uω (aρ 15) (bρ 15) (4920698646773 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_129_16 :
+    Uω (aρ 16) (bρ 16) (4920698646773 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_129 :
+    Uρ (4920698646773 / 64000000000000 : ℝ) ≤
+      (-(12424761255821240445021 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (4920698646773 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(26533740398960545318349 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_129_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (4920698646773 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(26896742978634930657697 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_129_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (4920698646773 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(27689153753963773397029 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_129_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (4920698646773 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(14638058503645561336247 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_129_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (4920698646773 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(8286772710427584474673 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_129_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (4920698646773 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_129_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (4920698646773 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_129_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (4920698646773 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_129_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (4920698646773 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_129_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (4920698646773 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_129_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (4920698646773 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_129_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (4920698646773 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_129_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (4920698646773 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_129_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (4920698646773 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_129_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (4920698646773 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_129_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (4920698646773 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_129_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_130_1 :
+    Uω (aρ 1) (bρ 1) (2477164297547 / 32000000000000 : ℝ) ≤
+      (-(5291873437976526183487 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2837554909783663 / 40000000000000000 : ℝ) ≤
+      (-(5291873437976526183487 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2837554909783663 / 40000000000000000 : ℝ) =
+            (2837554909783663 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2837554909783663 / 2500000000000000 : ℝ) ≤
+            (253304004423473816513 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (354580414996697 / 5000000000000000 : ℝ)) (q :=
+      (2837554909783663 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_130_2 :
+    Uω (aρ 2) (bρ 2) (2477164297547 / 32000000000000 : ℝ) ≤
+      (-(26819480107653810847137 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2737188893046777 / 40000000000000000 : ℝ) ≤
+      (-(26819480107653810847137 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2737188893046777 / 40000000000000000 : ℝ) =
+            (2737188893046777 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2737188893046777 / 2500000000000000 : ℝ) ≤
+            (906407104346189152863 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (682746834134951 / 10000000000000000 : ℝ)) (q :=
+      (2737188893046777 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_130_3 :
+    Uω (aρ 3) (bρ 3) (2477164297547 / 32000000000000 : ℝ) ≤
+      (-(13802477227999504398421 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2530416737297469 / 40000000000000000 : ℝ) ≤
+      (-(13802477227999504398421 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2530416737297469 / 40000000000000000 : ℝ) =
+            (2530416737297469 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2530416737297469 / 2500000000000000 : ℝ) ≤
+            (60466378000495601579 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (626756862610297 / 10000000000000000 : ℝ)) (q :=
+      (2530416737297469 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_130_4 :
+    Uω (aρ 4) (bρ 4) (2477164297547 / 32000000000000 : ℝ) ≤
+      (-(5834951221022183615197 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2162800354217801 / 40000000000000000 : ℝ) ≤
+      (-(5834951221022183615197 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (2162800354217801 / 40000000000000000 : ℝ) =
+            (2162800354217801 / 1250000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2162800354217801 / 1250000000000000 : ℝ) ≤
+            (1096520581977816384803 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (521242852135463 / 10000000000000000 : ℝ)) (q :=
+      (2162800354217801 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_130_5 :
+    Uω (aρ 5) (bρ 5) (2477164297547 / 32000000000000 : ℝ) ≤
+      (-(8242104744242823723913 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1479993291679967 / 40000000000000000 : ℝ) ≤
+      (-(8242104744242823723913 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (1479993291679967 / 40000000000000000 : ℝ) =
+            (1479993291679967 / 1250000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1479993291679967 / 1250000000000000 : ℝ) ≤
+            (422235009507176276087 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (149014620100773 / 5000000000000000 : ℝ)) (q :=
+      (1479993291679967 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_130_6 :
+    Uω (aρ 6) (bρ 6) (2477164297547 / 32000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_130_7 :
+    Uω (aρ 7) (bρ 7) (2477164297547 / 32000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_130_8 :
+    Uω (aρ 8) (bρ 8) (2477164297547 / 32000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_130_9 :
+    Uω (aρ 9) (bρ 9) (2477164297547 / 32000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_130_10 :
+    Uω (aρ 10) (bρ 10) (2477164297547 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_130_11 :
+    Uω (aρ 11) (bρ 11) (2477164297547 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_130_12 :
+    Uω (aρ 12) (bρ 12) (2477164297547 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_130_13 :
+    Uω (aρ 13) (bρ 13) (2477164297547 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_130_14 :
+    Uω (aρ 14) (bρ 14) (2477164297547 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_130_15 :
+    Uω (aρ 15) (bρ 15) (2477164297547 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_130_16 :
+    Uω (aρ 16) (bρ 16) (2477164297547 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_130 :
+    Uρ (2477164297547 / 32000000000000 : ℝ) ≤
+      (-(24824613604535232933069 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (2477164297547 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(5291873437976526183487 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_130_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (2477164297547 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(26819480107653810847137 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_130_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (2477164297547 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(13802477227999504398421 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_130_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (2477164297547 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(5834951221022183615197 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_130_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (2477164297547 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(8242104744242823723913 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_130_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (2477164297547 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_130_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (2477164297547 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_130_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (2477164297547 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_130_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (2477164297547 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_130_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (2477164297547 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_130_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (2477164297547 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_130_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (2477164297547 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_130_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (2477164297547 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_130_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (2477164297547 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_130_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (2477164297547 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_130_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (2477164297547 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_130_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_131_1 :
+    Uω (aρ 1) (bρ 1) (997591708683 / 12800000000000 : ℝ) ≤
+      (-(26385543387535192161299 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (228686426592583 / 3200000000000000 : ℝ) ≤
+      (-(26385543387535192161299 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (228686426592583 / 3200000000000000 : ℝ) =
+            (228686426592583 / 200000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (228686426592583 / 200000000000000 : ℝ) ≤
+            (1340343824464807838701 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (14288377237601 / 200000000000000 : ℝ)) (q :=
+      (228686426592583 / 3200000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_131_2 :
+    Uω (aρ 2) (bρ 2) (997591708683 / 12800000000000 : ℝ) ≤
+      (-(1069712491504947096903 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1103301989972711 / 16000000000000000 : ℝ) ≤
+      (-(1069712491504947096903 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1103301989972711 / 16000000000000000 : ℝ) =
+            (1103301989972711 / 1000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1103301989972711 / 1000000000000000 : ℝ) ≤
+            (39322996975052903097 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (137605039130459 / 2000000000000000 : ℝ)) (q :=
+      (1103301989972711 / 16000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_131_3 :
+    Uω (aρ 3) (bρ 3) (997591708683 / 12800000000000 : ℝ) ≤
+      (-(27521471202141578539429 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (326608627659 / 5120000000000 : ℝ) ≤
+      (-(27521471202141578539429 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (326608627659 / 5120000000000 : ℝ) = (326608627659 / 320000000000 : ℝ) / 2 ^ 4 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (326608627659 / 320000000000 : ℝ) ≤
+            (204416009858421460571 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1011374026127 / 16000000000000 : ℝ)) (q :=
+      (326608627659 / 5120000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_131_4 :
+    Uω (aρ 4) (bρ 4) (997591708683 / 12800000000000 : ℝ) ≤
+      (-(1453724371476601447869 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4369191106212779 / 80000000000000000 : ℝ) ≤
+      (-(1453724371476601447869 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (4369191106212779 / 80000000000000000 : ℝ) =
+            (4369191106212779 / 2500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4369191106212779 / 2500000000000000 : ℝ) ≤
+            (279143579273398552131 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (526885772154601 / 10000000000000000 : ℝ)) (q :=
+      (4369191106212779 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_131_5 :
+    Uω (aρ 5) (bρ 5) (997591708683 / 12800000000000 : ℝ) ≤
+      (-(32794360034669585926951 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3011958796030119 / 80000000000000000 : ℝ) ≤
+      (-(32794360034669585926951 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (3011958796030119 / 80000000000000000 : ℝ) =
+            (3011958796030119 / 2500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3011958796030119 / 2500000000000000 : ℝ) ≤
+            (1862998980330414073049 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2388809483937 / 78125000000000 : ℝ)) (q :=
+      (3011958796030119 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_131_6 :
+    Uω (aρ 6) (bρ 6) (997591708683 / 12800000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_131_7 :
+    Uω (aρ 7) (bρ 7) (997591708683 / 12800000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_131_8 :
+    Uω (aρ 8) (bρ 8) (997591708683 / 12800000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_131_9 :
+    Uω (aρ 9) (bρ 9) (997591708683 / 12800000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_131_10 :
+    Uω (aρ 10) (bρ 10) (997591708683 / 12800000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_131_11 :
+    Uω (aρ 11) (bρ 11) (997591708683 / 12800000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_131_12 :
+    Uω (aρ 12) (bρ 12) (997591708683 / 12800000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_131_13 :
+    Uω (aρ 13) (bρ 13) (997591708683 / 12800000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_131_14 :
+    Uω (aρ 14) (bρ 14) (997591708683 / 12800000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_131_15 :
+    Uω (aρ 15) (bρ 15) (997591708683 / 12800000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_131_16 :
+    Uω (aρ 16) (bρ 16) (997591708683 / 12800000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_131 :
+    Uρ (997591708683 / 12800000000000 : ℝ) ≤
+      (-(49600281584509713859 / 20000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (997591708683 / 12800000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(26385543387535192161299 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_131_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (997591708683 / 12800000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1069712491504947096903 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_131_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (997591708683 / 12800000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(27521471202141578539429 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_131_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (997591708683 / 12800000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1453724371476601447869 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_131_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (997591708683 / 12800000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(32794360034669585926951 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_131_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (997591708683 / 12800000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_131_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (997591708683 / 12800000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_131_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (997591708683 / 12800000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_131_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (997591708683 / 12800000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_131_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (997591708683 / 12800000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_131_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (997591708683 / 12800000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_131_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (997591708683 / 12800000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_131_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (997591708683 / 12800000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_131_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (997591708683 / 12800000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_131_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (997591708683 / 12800000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_131_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (997591708683 / 12800000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_131_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_132_1 :
+    Uω (aρ 1) (bρ 1) (627698561467 / 8000000000000 : ℝ) ≤
+      (-(5262452185846792494703 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (359950707136057 / 5000000000000000 : ℝ) ≤
+      (-(5262452185846792494703 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (359950707136057 / 5000000000000000 : ℝ) =
+            (359950707136057 / 312500000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (359950707136057 / 312500000000000 : ℝ) ≤
+            (282725256553207505297 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (359838422397739 / 5000000000000000 : ℝ)) (q :=
+      (359950707136057 / 5000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_132_2 :
+    Uω (aρ 2) (bρ 2) (627698561467 / 8000000000000 : ℝ) ≤
+      (-(6666682595679150360093 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1389660168617201 / 20000000000000000 : ℝ) ≤
+      (-(6666682595679150360093 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1389660168617201 / 20000000000000000 : ℝ) =
+            (1389660168617201 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1389660168617201 / 1250000000000000 : ℝ) ≤
+            (264789207320849639907 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (693303197378451 / 10000000000000000 : ℝ)) (q :=
+      (1389660168617201 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_132_3 :
+    Uω (aρ 3) (bρ 3) (627698561467 / 8000000000000 : ℝ) ≤
+      (-(3429836462828796430307 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (64320995861959 / 1000000000000000 : ℝ) ≤
+      (-(3429836462828796430307 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (64320995861959 / 1000000000000000 : ℝ) = (64320995861959 / 62500000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (64320995861959 / 62500000000000 : ℝ) ≤
+            (35899438671203569693 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (63745905235043 / 1000000000000000 : ℝ)) (q :=
+      (64320995861959 / 1000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_132_4 :
+    Uω (aρ 4) (bρ 4) (627698561467 / 8000000000000 : ℝ) ≤
+      (-(28975286180128640663519 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1103187431032073 / 20000000000000000 : ℝ) ≤
+      (-(28975286180128640663519 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (1103187431032073 / 20000000000000000 : ℝ) =
+            (1103187431032073 / 625000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1103187431032073 / 625000000000000 : ℝ) ≤
+            (5682072834871359336481 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (532520747208323 / 10000000000000000 : ℝ)) (q :=
+      (1103187431032073 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_132_5 :
+    Uω (aρ 5) (bρ 5) (627698561467 / 8000000000000 : ℝ) ≤
+      (-(32624623126546358520591 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (95734975020517 / 2500000000000000 : ℝ) ≤
+      (-(32624623126546358520591 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (95734975020517 / 2500000000000000 : ℝ) = (95734975020517 / 78125000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (95734975020517 / 78125000000000 : ℝ) ≤
+            (2032735888453641479409 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (156701517837693 / 5000000000000000 : ℝ)) (q :=
+      (95734975020517 / 2500000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_132_6 :
+    Uω (aρ 6) (bρ 6) (627698561467 / 8000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_132_7 :
+    Uω (aρ 7) (bρ 7) (627698561467 / 8000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_132_8 :
+    Uω (aρ 8) (bρ 8) (627698561467 / 8000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_132_9 :
+    Uω (aρ 9) (bρ 9) (627698561467 / 8000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_132_10 :
+    Uω (aρ 10) (bρ 10) (627698561467 / 8000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_132_11 :
+    Uω (aρ 11) (bρ 11) (627698561467 / 8000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_132_12 :
+    Uω (aρ 12) (bρ 12) (627698561467 / 8000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_132_13 :
+    Uω (aρ 13) (bρ 13) (627698561467 / 8000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_132_14 :
+    Uω (aρ 14) (bρ 14) (627698561467 / 8000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_132_15 :
+    Uω (aρ 15) (bρ 15) (627698561467 / 8000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_132_16 :
+    Uω (aρ 16) (bρ 16) (627698561467 / 8000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_132 :
+    Uρ (627698561467 / 8000000000000 : ℝ) ≤
+      (-(24776081667392410080527 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (627698561467 / 8000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(5262452185846792494703 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_132_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (627698561467 / 8000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(6666682595679150360093 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_132_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (627698561467 / 8000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3429836462828796430307 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_132_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (627698561467 / 8000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(28975286180128640663519 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_132_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (627698561467 / 8000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(32624623126546358520591 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_132_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (627698561467 / 8000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_132_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (627698561467 / 8000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_132_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (627698561467 / 8000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_132_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (627698561467 / 8000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_132_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (627698561467 / 8000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_132_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (627698561467 / 8000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_132_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (627698561467 / 8000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_132_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (627698561467 / 8000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_132_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (627698561467 / 8000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_132_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (627698561467 / 8000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_132_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (627698561467 / 8000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_132_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_133_1 :
+    Uω (aρ 1) (bρ 1) (5055218440057 / 64000000000000 : ℝ) ≤
+      (-(5247902385718883216969 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (46410094175333 / 640000000000000 : ℝ) ≤
+      (-(5247902385718883216969 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (46410094175333 / 640000000000000 : ℝ) = (46410094175333 / 40000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (46410094175333 / 40000000000000 : ℝ) ≤
+            (297275056681116783031 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2899739119221 / 40000000000000 : ℝ)) (q :=
+      (46410094175333 / 640000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_133_2 :
+    Uω (aρ 2) (bρ 2) (5055218440057 / 64000000000000 : ℝ) ≤
+      (-(5318245093358236445799 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5600769992529341 / 80000000000000000 : ℝ) ≤
+      (-(5318245093358236445799 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (5600769992529341 / 80000000000000000 : ℝ) =
+            (5600769992529341 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5600769992529341 / 5000000000000000 : ℝ) ≤
+            (226932349041763554201 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (698580847468429 / 10000000000000000 : ℝ)) (q :=
+      (5600769992529341 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_133_3 :
+    Uω (aρ 3) (bρ 3) (5055218440057 / 64000000000000 : ℝ) ≤
+      (-(27356603986588711129289 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5188093221524637 / 80000000000000000 : ℝ) ≤
+      (-(27356603986588711129289 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (5188093221524637 / 80000000000000000 : ℝ) =
+            (5188093221524637 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5188093221524637 / 5000000000000000 : ℝ) ≤
+            (369283225411288870711 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (642807761067253 / 10000000000000000 : ℝ)) (q :=
+      (5188093221524637 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_133_4 :
+    Uω (aρ 4) (bρ 4) (5055218440057 / 64000000000000 : ℝ) ≤
+      (-(14438564217655403680117 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4456277560487961 / 80000000000000000 : ℝ) ≤
+      (-(14438564217655403680117 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (4456277560487961 / 80000000000000000 : ℝ) =
+            (4456277560487961 / 2500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4456277560487961 / 2500000000000000 : ℝ) ≤
+            (2890115289844596319883 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (134537006718271 / 2500000000000000 : ℝ)) (q :=
+      (4456277560487961 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_133_5 :
+    Uω (aρ 5) (bρ 5) (5055218440057 / 64000000000000 : ℝ) ≤
+      (-(4057368370546522140043 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (622939437810389 / 16000000000000000 : ℝ) ≤
+      (-(4057368370546522140043 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (622939437810389 / 16000000000000000 : ℝ) =
+            (622939437810389 / 500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (622939437810389 / 500000000000000 : ℝ) ≤
+            (274801506328477859957 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (8023571333727 / 250000000000000 : ℝ)) (q :=
+      (622939437810389 / 16000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_133_6 :
+    Uω (aρ 6) (bρ 6) (5055218440057 / 64000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_133_7 :
+    Uω (aρ 7) (bρ 7) (5055218440057 / 64000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_133_8 :
+    Uω (aρ 8) (bρ 8) (5055218440057 / 64000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_133_9 :
+    Uω (aρ 9) (bρ 9) (5055218440057 / 64000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_133_10 :
+    Uω (aρ 10) (bρ 10) (5055218440057 / 64000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_133_11 :
+    Uω (aρ 11) (bρ 11) (5055218440057 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_133_12 :
+    Uω (aρ 12) (bρ 12) (5055218440057 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_133_13 :
+    Uω (aρ 13) (bρ 13) (5055218440057 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_133_14 :
+    Uω (aρ 14) (bρ 14) (5055218440057 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_133_15 :
+    Uω (aρ 15) (bρ 15) (5055218440057 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_133_16 :
+    Uω (aρ 16) (bρ 16) (5055218440057 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_133 :
+    Uρ (5055218440057 / 64000000000000 : ℝ) ≤
+      (-(24752415938931362015277 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (5055218440057 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(5247902385718883216969 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_133_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (5055218440057 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(5318245093358236445799 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_133_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (5055218440057 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(27356603986588711129289 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_133_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (5055218440057 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(14438564217655403680117 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_133_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (5055218440057 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(4057368370546522140043 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_133_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (5055218440057 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_133_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (5055218440057 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_133_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (5055218440057 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_133_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (5055218440057 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_133_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (5055218440057 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_133_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (5055218440057 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_133_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (5055218440057 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_133_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (5055218440057 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_133_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (5055218440057 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_133_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (5055218440057 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_133_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (5055218440057 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_133_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_134_1 :
+    Uω (aρ 1) (bρ 1) (2544424194189 / 32000000000000 : ℝ) ≤
+      (-(26167288670425871360889 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2921656021086593 / 40000000000000000 : ℝ) ≤
+      (-(26167288670425871360889 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2921656021086593 / 40000000000000000 : ℝ) =
+            (2921656021086593 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2921656021086593 / 2500000000000000 : ℝ) ≤
+            (1558598541574128639111 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (365096333972117 / 5000000000000000 : ℝ)) (q :=
+      (2921656021086593 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_134_2 :
+    Uω (aρ 2) (bρ 2) (2544424194189 / 32000000000000 : ℝ) ≤
+      (-(26516288816997824638059 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2821448967842361 / 40000000000000000 : ℝ) ≤
+      (-(26516288816997824638059 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2821448967842361 / 40000000000000000 : ℝ) =
+            (2821448967842361 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2821448967842361 / 2500000000000000 : ℝ) ≤
+            (1209598395002175361941 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (351929076916059 / 5000000000000000 : ℝ)) (q :=
+      (2821448967842361 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_134_3 :
+    Uω (aρ 3) (bρ 3) (2544424194189 / 32000000000000 : ℝ) ≤
+      (-(2727519639092246926631 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2615250310535261 / 40000000000000000 : ℝ) ≤
+      (-(2727519639092246926631 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2615250310535261 / 40000000000000000 : ℝ) =
+            (2615250310535261 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2615250310535261 / 2500000000000000 : ℝ) ≤
+            (45069082107753073369 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (81019366441071 / 1250000000000000 : ℝ)) (q :=
+      (2615250310535261 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_134_4 :
+    Uω (aρ 4) (bρ 4) (2544424194189 / 32000000000000 : ℝ) ≤
+      (-(28779991109665995308087 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2249887785468721 / 40000000000000000 : ℝ) ≤
+      (-(28779991109665995308087 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (2249887785468721 / 40000000000000000 : ℝ) =
+            (2249887785468721 / 1250000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2249887785468721 / 1250000000000000 : ℝ) ≤
+            (5877367905334004691913 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (271883925030149 / 5000000000000000 : ℝ)) (q :=
+      (2249887785468721 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_134_5 :
+    Uω (aρ 5) (bρ 5) (2544424194189 / 32000000000000 : ℝ) ≤
+      (-(16148547889847596668857 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1582759550979519 / 40000000000000000 : ℝ) ≤
+      (-(16148547889847596668857 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (1582759550979519 / 40000000000000000 : ℝ) =
+            (1582759550979519 / 1250000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1582759550979519 / 1250000000000000 : ℝ) ≤
+            (1180131617652403331143 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (328393652150697 / 10000000000000000 : ℝ)) (q :=
+      (1582759550979519 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_134_6 :
+    Uω (aρ 6) (bρ 6) (2544424194189 / 32000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_134_7 :
+    Uω (aρ 7) (bρ 7) (2544424194189 / 32000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_134_8 :
+    Uω (aρ 8) (bρ 8) (2544424194189 / 32000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_134_9 :
+    Uω (aρ 9) (bρ 9) (2544424194189 / 32000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_134_10 :
+    Uω (aρ 10) (bρ 10) (2544424194189 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_134_11 :
+    Uω (aρ 11) (bρ 11) (2544424194189 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_134_12 :
+    Uω (aρ 12) (bρ 12) (2544424194189 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_134_13 :
+    Uω (aρ 13) (bρ 13) (2544424194189 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_134_14 :
+    Uω (aρ 14) (bρ 14) (2544424194189 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_134_15 :
+    Uω (aρ 15) (bρ 15) (2544424194189 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_134_16 :
+    Uω (aρ 16) (bρ 16) (2544424194189 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_134 :
+    Uρ (2544424194189 / 32000000000000 : ℝ) ≤
+      (-(6182281286838750952367 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (2544424194189 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(26167288670425871360889 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_134_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (2544424194189 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(26516288816997824638059 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_134_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (2544424194189 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(2727519639092246926631 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_134_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (2544424194189 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(28779991109665995308087 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_134_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (2544424194189 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(16148547889847596668857 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_134_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (2544424194189 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_134_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (2544424194189 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_134_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (2544424194189 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_134_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (2544424194189 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_134_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (2544424194189 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_134_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (2544424194189 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_134_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (2544424194189 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_134_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (2544424194189 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_134_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (2544424194189 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_134_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (2544424194189 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_134_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (2544424194189 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_134_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_135_1 :
+    Uω (aρ 1) (bρ 1) (5122478336699 / 64000000000000 : ℝ) ≤
+      (-(13047791802904571252103 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (235414485158707 / 3200000000000000 : ℝ) ≤
+      (-(13047791802904571252103 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (235414485158707 / 3200000000000000 : ℝ) =
+            (235414485158707 / 200000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (235414485158707 / 200000000000000 : ℝ) ≤
+            (815151803095428747897 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7354505102177 / 100000000000000 : ℝ)) (q :=
+      (235414485158707 / 3200000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_135_2 :
+    Uω (aρ 2) (bρ 2) (5122478336699 / 64000000000000 : ℝ) ≤
+      (-(13220955953813518373719 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5685024534630623 / 80000000000000000 : ℝ) ≤
+      (-(13220955953813518373719 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (5685024534630623 / 80000000000000000 : ℝ) =
+            (5685024534630623 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5685024534630623 / 5000000000000000 : ℝ) ≤
+            (641987652186481626281 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (709135124143437 / 10000000000000000 : ℝ)) (q :=
+      (5685024534630623 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_135_3 :
+    Uω (aρ 3) (bρ 3) (5122478336699 / 64000000000000 : ℝ) ≤
+      (-(13597228774851769235177 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5272902018632919 / 80000000000000000 : ℝ) ≤
+      (-(13597228774851769235177 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (5272902018632919 / 80000000000000000 : ℝ) =
+            (5272902018632919 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5272902018632919 / 5000000000000000 : ℝ) ≤
+            (265714831148230764823 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (653500601494011 / 10000000000000000 : ℝ)) (q :=
+      (5272902018632919 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_135_4 :
+    Uω (aρ 4) (bρ 4) (5122478336699 / 64000000000000 : ℝ) ≤
+      (-(7170962978482870943403 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4543244670778931 / 80000000000000000 : ℝ) ≤
+      (-(7170962978482870943403 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (4543244670778931 / 80000000000000000 : ℝ) =
+            (4543244670778931 / 2500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4543244670778931 / 2500000000000000 : ℝ) ≤
+            (1493376775267129056597 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (274690222797757 / 5000000000000000 : ℝ)) (q :=
+      (4543244670778931 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_135_5 :
+    Uω (aρ 5) (bρ 5) (5122478336699 / 64000000000000 : ℝ) ≤
+      (-(6427771188288008699121 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3216008644341211 / 80000000000000000 : ℝ) ≤
+      (-(6427771188288008699121 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (3216008644341211 / 80000000000000000 : ℝ) =
+            (3216008644341211 / 2500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3216008644341211 / 2500000000000000 : ℝ) ≤
+            (503700614711991300879 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (83940339580271 / 2500000000000000 : ℝ)) (q :=
+      (3216008644341211 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_135_6 :
+    Uω (aρ 6) (bρ 6) (5122478336699 / 64000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_135_7 :
+    Uω (aρ 7) (bρ 7) (5122478336699 / 64000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_135_8 :
+    Uω (aρ 8) (bρ 8) (5122478336699 / 64000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_135_9 :
+    Uω (aρ 9) (bρ 9) (5122478336699 / 64000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_135_10 :
+    Uω (aρ 10) (bρ 10) (5122478336699 / 64000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_135_11 :
+    Uω (aρ 11) (bρ 11) (5122478336699 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_135_12 :
+    Uω (aρ 12) (bρ 12) (5122478336699 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_135_13 :
+    Uω (aρ 13) (bρ 13) (5122478336699 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_135_14 :
+    Uω (aρ 14) (bρ 14) (5122478336699 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_135_15 :
+    Uω (aρ 15) (bρ 15) (5122478336699 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_135_16 :
+    Uω (aρ 16) (bρ 16) (5122478336699 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_135 :
+    Uρ (5122478336699 / 64000000000000 : ℝ) ≤
+      (-(3088274053514358657371 / 1250000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (5122478336699 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(13047791802904571252103 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_135_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (5122478336699 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(13220955953813518373719 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_135_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (5122478336699 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(13597228774851769235177 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_135_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (5122478336699 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(7170962978482870943403 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_135_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (5122478336699 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(6427771188288008699121 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_135_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (5122478336699 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_135_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (5122478336699 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_135_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (5122478336699 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_135_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (5122478336699 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_135_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (5122478336699 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_135_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (5122478336699 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_135_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (5122478336699 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_135_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (5122478336699 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_135_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (5122478336699 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_135_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (5122478336699 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_135_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (5122478336699 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_135_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/U11.lean b/LeanPool/Zeta5Irrational/Table/U11.lean
new file mode 100644
index 0000000000..7a371f4c6a
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/U11.lean
@@ -0,0 +1,3096 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-! # Certified arcsine potential bounds (U11) -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma U_136_1 :
+    Uω (aρ 1) (bρ 1) (257805414251 / 3200000000000 : ℝ) ≤
+      (-(26024389347350076319563 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (296370601810347 / 4000000000000000 : ℝ) ≤
+      (-(26024389347350076319563 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (296370601810347 / 4000000000000000 : ℝ) =
+            (296370601810347 / 250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (296370601810347 / 250000000000000 : ℝ) ≤
+            (1701497864649923680437 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (18517707690059 / 250000000000000 : ℝ)) (q :=
+      (296370601810347 / 4000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_136_2 :
+    Uω (aρ 2) (bρ 2) (257805414251 / 3200000000000 : ℝ) ≤
+      (-(26368086404189464320953 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1431787454788469 / 20000000000000000 : ℝ) ≤
+      (-(26368086404189464320953 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1431787454788469 / 20000000000000000 : ℝ) =
+            (1431787454788469 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1431787454788469 / 1250000000000000 : ℝ) ≤
+            (1357800807810535679047 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (357205882924547 / 5000000000000000 : ℝ)) (q :=
+      (1431787454788469 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_136_3 :
+    Uω (aρ 3) (bρ 3) (257805414251 / 3200000000000 : ℝ) ≤
+      (-(3389297048001651951123 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (664412195038341 / 10000000000000000 : ℝ) ≤
+      (-(3389297048001651951123 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (664412195038341 / 10000000000000000 : ℝ) =
+            (664412195038341 / 625000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (664412195038341 / 625000000000000 : ℝ) ≤
+            (76438853498348048877 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (658844807487307 / 10000000000000000 : ℝ)) (q :=
+      (664412195038341 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_136_4 :
+    Uω (aρ 4) (bρ 4) (257805414251 / 3200000000000 : ℝ) ≤
+      (-(7147172329350315520741 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1146671434283737 / 20000000000000000 : ℝ) ≤
+      (-(7147172329350315520741 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (1146671434283737 / 20000000000000000 : ℝ) =
+            (1146671434283737 / 625000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1146671434283737 / 625000000000000 : ℝ) ≤
+            (1517167424399684479259 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (277493016379681 / 5000000000000000 : ℝ)) (q :=
+      (1146671434283737 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_136_5 :
+    Uω (aρ 5) (bρ 5) (257805414251 / 3200000000000 : ℝ) ≤
+      (-(31984033130623965837531 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2551708774461 / 62500000000000 : ℝ) ≤
+      (-(31984033130623965837531 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show (2551708774461 / 62500000000000 : ℝ) = (2551708774461 / 1953125000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2551708774461 / 1953125000000 : ℝ) ≤
+            (2673325884376034162469 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (68610265127629 / 2000000000000000 : ℝ)) (q :=
+      (2551708774461 / 62500000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_136_6 :
+    Uω (aρ 6) (bρ 6) (257805414251 / 3200000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_136_7 :
+    Uω (aρ 7) (bρ 7) (257805414251 / 3200000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_136_8 :
+    Uω (aρ 8) (bρ 8) (257805414251 / 3200000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_136_9 :
+    Uω (aρ 9) (bρ 9) (257805414251 / 3200000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_136_10 :
+    Uω (aρ 10) (bρ 10) (257805414251 / 3200000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_136_11 :
+    Uω (aρ 11) (bρ 11) (257805414251 / 3200000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_136_12 :
+    Uω (aρ 12) (bρ 12) (257805414251 / 3200000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_136_13 :
+    Uω (aρ 13) (bρ 13) (257805414251 / 3200000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_136_14 :
+    Uω (aρ 14) (bρ 14) (257805414251 / 3200000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_136_15 :
+    Uω (aρ 15) (bρ 15) (257805414251 / 3200000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_136_16 :
+    Uω (aρ 16) (bρ 16) (257805414251 / 3200000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_136 :
+    Uρ (257805414251 / 3200000000000 : ℝ) ≤
+      (-(24683602313731326949681 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (257805414251 / 3200000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(26024389347350076319563 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_136_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (257805414251 / 3200000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(26368086404189464320953 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_136_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (257805414251 / 3200000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3389297048001651951123 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_136_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (257805414251 / 3200000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(7147172329350315520741 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_136_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (257805414251 / 3200000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(31984033130623965837531 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_136_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (257805414251 / 3200000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_136_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (257805414251 / 3200000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_136_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (257805414251 / 3200000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_136_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (257805414251 / 3200000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_136_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (257805414251 / 3200000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_136_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (257805414251 / 3200000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_136_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (257805414251 / 3200000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_136_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (257805414251 / 3200000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_136_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (257805414251 / 3200000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_136_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (257805414251 / 3200000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_136_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (257805414251 / 3200000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_136_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_137_1 :
+    Uω (aρ 1) (bρ 1) (2611684090831 / 32000000000000 : ℝ) ≤
+      (-(25883504479644690843899 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3005755663550111 / 40000000000000000 : ℝ) ≤
+      (-(25883504479644690843899 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (3005755663550111 / 40000000000000000 : ℝ) =
+            (3005755663550111 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3005755663550111 / 2500000000000000 : ℝ) ≤
+            (1842382732355309156101 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (93902971434421 / 1250000000000000 : ℝ)) (q :=
+      (3005755663550111 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_137_2 :
+    Uω (aρ 2) (bρ 2) (2611684090831 / 32000000000000 : ℝ) ≤
+      (-(5244411439855569471477 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2905698279867373 / 40000000000000000 : ℝ) ≤
+      (-(5244411439855569471477 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2905698279867373 / 40000000000000000 : ℝ) =
+            (2905698279867373 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2905698279867373 / 2500000000000000 : ℝ) ≤
+            (300766002544430528523 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (724964092143999 / 10000000000000000 : ℝ)) (q :=
+      (2905698279867373 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_137_3 :
+    Uω (aρ 3) (bρ 3) (2611684090831 / 32000000000000 : ℝ) ≤
+      (-(26956144140679057589663 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2700035818426691 / 40000000000000000 : ℝ) ≤
+      (-(26956144140679057589663 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2700035818426691 / 40000000000000000 : ℝ) =
+            (2700035818426691 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2700035818426691 / 2500000000000000 : ℝ) ≤
+            (769743071320942410337 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (334764483886829 / 5000000000000000 : ℝ)) (q :=
+      (2700035818426691 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_137_4 :
+    Uω (aρ 4) (bρ 4) (2611684090831 / 32000000000000 : ℝ) ≤
+      (-(1136048455279144140469 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2336743549494169 / 40000000000000000 : ℝ) ≤
+      (-(1136048455279144140469 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (2336743549494169 / 40000000000000000 : ℝ) =
+            (2336743549494169 / 1250000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2336743549494169 / 1250000000000000 : ℝ) ≤
+            (250245905320855859531 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (566177014372397 / 10000000000000000 : ℝ)) (q :=
+      (2336743549494169 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_137_5 :
+    Uω (aρ 5) (bρ 5) (2611684090831 / 32000000000000 : ℝ) ≤
+      (-(15841971990601673642843 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (336568744601939 / 8000000000000000 : ℝ) ≤
+      (-(15841971990601673642843 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (336568744601939 / 8000000000000000 : ℝ) =
+            (336568744601939 / 250000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (336568744601939 / 250000000000000 : ℝ) ≤
+            (1486707516898326357157 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (8935425511629 / 250000000000000 : ℝ)) (q :=
+      (336568744601939 / 8000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_137_6 :
+    Uω (aρ 6) (bρ 6) (2611684090831 / 32000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_137_7 :
+    Uω (aρ 7) (bρ 7) (2611684090831 / 32000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_137_8 :
+    Uω (aρ 8) (bρ 8) (2611684090831 / 32000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_137_9 :
+    Uω (aρ 9) (bρ 9) (2611684090831 / 32000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_137_10 :
+    Uω (aρ 10) (bρ 10) (2611684090831 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_137_11 :
+    Uω (aρ 11) (bρ 11) (2611684090831 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_137_12 :
+    Uω (aρ 12) (bρ 12) (2611684090831 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_137_13 :
+    Uω (aρ 13) (bρ 13) (2611684090831 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_137_14 :
+    Uω (aρ 14) (bρ 14) (2611684090831 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_137_15 :
+    Uω (aρ 15) (bρ 15) (2611684090831 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_137_16 :
+    Uω (aρ 16) (bρ 16) (2611684090831 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_137 :
+    Uρ (2611684090831 / 32000000000000 : ℝ) ≤
+      (-(12319697019856429271619 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (2611684090831 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(25883504479644690843899 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_137_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (2611684090831 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(5244411439855569471477 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_137_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (2611684090831 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(26956144140679057589663 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_137_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (2611684090831 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1136048455279144140469 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_137_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (2611684090831 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(15841971990601673642843 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_137_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (2611684090831 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_137_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (2611684090831 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_137_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (2611684090831 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_137_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (2611684090831 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_137_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (2611684090831 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_137_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (2611684090831 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_137_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (2611684090831 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_137_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (2611684090831 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_137_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (2611684090831 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_137_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (2611684090831 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_137_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (2611684090831 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_137_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_138_1 :
+    Uω (aρ 1) (bρ 1) (165332127447 / 2000000000000 : ℝ) ≤
+      (-(25744578028657117086197 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (38097562149829 / 500000000000000 : ℝ) ≤
+      (-(25744578028657117086197 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (38097562149829 / 500000000000000 : ℝ) = (38097562149829 / 31250000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (38097562149829 / 31250000000000 : ℝ) ≤
+            (1981309183342882913803 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (19043476671079 / 250000000000000 : ℝ)) (q :=
+      (38097562149829 / 500000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_138_2 :
+    Uω (aρ 2) (bρ 2) (165332127447 / 2000000000000 : ℝ) ≤
+      (-(3259767267427656465799 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1473909594695023 / 20000000000000000 : ℝ) ≤
+      (-(3259767267427656465799 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1473909594695023 / 20000000000000000 : ℝ) =
+            (1473909594695023 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1473909594695023 / 1250000000000000 : ℝ) ≤
+            (205968634072343534201 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (735515188055023 / 10000000000000000 : ℝ)) (q :=
+      (1473909594695023 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_138_3 :
+    Uω (aρ 3) (bρ 3) (165332127447 / 2000000000000 : ℝ) ≤
+      (-(3350052051316778232639 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (685602991004679 / 10000000000000000 : ℝ) ≤
+      (-(3350052051316778232639 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (685602991004679 / 10000000000000000 : ℝ) =
+            (685602991004679 / 625000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (685602991004679 / 625000000000000 : ℝ) ≤
+            (115683850183221767361 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (340103840859679 / 5000000000000000 : ℝ)) (q :=
+      (685602991004679 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_138_4 :
+    Uω (aρ 4) (bρ 4) (165332127447 / 2000000000000 : ℝ) ≤
+      (-(14108699468577326473109 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (297511623973151 / 5000000000000000 : ℝ) ≤
+      (-(14108699468577326473109 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (297511623973151 / 5000000000000000 : ℝ) =
+            (297511623973151 / 156250000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (297511623973151 / 156250000000000 : ℝ) ≤
+            (3219980038922673526891 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (144335594166901 / 2500000000000000 : ℝ)) (q :=
+      (297511623973151 / 5000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_138_5 :
+    Uω (aρ 5) (bρ 5) (165332127447 / 2000000000000 : ℝ) ≤
+      (-(627911566201869230741 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (34641552310843 / 800000000000000 : ℝ) ≤
+      (-(627911566201869230741 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (34641552310843 / 800000000000000 : ℝ) = (34641552310843 / 25000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (34641552310843 / 25000000000000 : ℝ) ≤
+            (65235614098130769259 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (14860984315243 / 400000000000000 : ℝ)) (q :=
+      (34641552310843 / 800000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_138_6 :
+    Uω (aρ 6) (bρ 6) (165332127447 / 2000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_138_7 :
+    Uω (aρ 7) (bρ 7) (165332127447 / 2000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_138_8 :
+    Uω (aρ 8) (bρ 8) (165332127447 / 2000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_138_9 :
+    Uω (aρ 9) (bρ 9) (165332127447 / 2000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_138_10 :
+    Uω (aρ 10) (bρ 10) (165332127447 / 2000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_138_11 :
+    Uω (aρ 11) (bρ 11) (165332127447 / 2000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_138_12 :
+    Uω (aρ 12) (bρ 12) (165332127447 / 2000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_138_13 :
+    Uω (aρ 13) (bρ 13) (165332127447 / 2000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_138_14 :
+    Uω (aρ 14) (bρ 14) (165332127447 / 2000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_138_15 :
+    Uω (aρ 15) (bρ 15) (165332127447 / 2000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_138_16 :
+    Uω (aρ 16) (bρ 16) (165332127447 / 2000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_138 :
+    Uρ (165332127447 / 2000000000000 : ℝ) ≤
+      (-(4919279757412061629749 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (165332127447 / 2000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(25744578028657117086197 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_138_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (165332127447 / 2000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3259767267427656465799 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_138_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (165332127447 / 2000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3350052051316778232639 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_138_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (165332127447 / 2000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(14108699468577326473109 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_138_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (165332127447 / 2000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(627911566201869230741 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_138_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (165332127447 / 2000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_138_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (165332127447 / 2000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_138_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (165332127447 / 2000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_138_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (165332127447 / 2000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_138_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (165332127447 / 2000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_138_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (165332127447 / 2000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_138_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (165332127447 / 2000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_138_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (165332127447 / 2000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_138_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (165332127447 / 2000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_138_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (165332127447 / 2000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_138_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (165332127447 / 2000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_138_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_139_1 :
+    Uω (aρ 1) (bρ 1) (2678943987473 / 32000000000000 : ℝ) ≤
+      (-(25607556263304084967409 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3089853957178669 / 40000000000000000 : ℝ) ≤
+      (-(25607556263304084967409 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (3089853957178669 / 40000000000000000 : ℝ) =
+            (3089853957178669 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3089853957178669 / 2500000000000000 : ℝ) ≤
+            (2118330948695915032591 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (386127100294511 / 5000000000000000 : ℝ)) (q :=
+      (3089853957178669 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_139_2 :
+    Uω (aρ 2) (bρ 2) (2678943987473 / 32000000000000 : ℝ) ≤
+      (-(3242033609723343368109 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (597987548509579 / 8000000000000000 : ℝ) ≤
+      (-(3242033609723343368109 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (597987548509579 / 8000000000000000 : ℝ) =
+            (597987548509579 / 500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (597987548509579 / 500000000000000 : ℝ) ≤
+            (223702291776656631891 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (149213021156727 / 2000000000000000 : ℝ)) (q :=
+      (597987548509579 / 8000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_139_3 :
+    Uω (aρ 3) (bρ 3) (2678943987473 / 32000000000000 : ℝ) ≤
+      (-(5329422790878437414709 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2784777722022853 / 40000000000000000 : ℝ) ≤
+      (-(5329422790878437414709 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2784777722022853 / 40000000000000000 : ℝ) =
+            (2784777722022853 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2784777722022853 / 2500000000000000 : ℝ) ≤
+            (215754651521562585291 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (345440600935557 / 5000000000000000 : ℝ)) (q :=
+      (2784777722022853 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_139_4 :
+    Uω (aρ 4) (bρ 4) (2678943987473 / 32000000000000 : ℝ) ≤
+      (-(350463796210277297691 / 125000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2423394111912659 / 40000000000000000 : ℝ) ≤
+      (-(350463796210277297691 / 125000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (2423394111912659 / 40000000000000000 : ℝ) =
+            (2423394111912659 / 1250000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2423394111912659 / 1250000000000000 : ℝ) ≤
+            (82753191477222702309 / 125000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (588483577881017 / 10000000000000000 : ℝ)) (q :=
+      (2423394111912659 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_139_5 :
+    Uω (aρ 5) (bρ 5) (2678943987473 / 32000000000000 : ℝ) ≤
+      (-(15558937803900991527783 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (356170396638393 / 8000000000000000 : ℝ) ≤
+      (-(15558937803900991527783 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (356170396638393 / 8000000000000000 : ℝ) =
+            (356170396638393 / 250000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (356170396638393 / 250000000000000 : ℝ) ≤
+            (1769741703599008472217 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (38540243285567 / 1000000000000000 : ℝ)) (q :=
+      (356170396638393 / 8000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_139_6 :
+    Uω (aρ 6) (bρ 6) (2678943987473 / 32000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_139_7 :
+    Uω (aρ 7) (bρ 7) (2678943987473 / 32000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_139_8 :
+    Uω (aρ 8) (bρ 8) (2678943987473 / 32000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_139_9 :
+    Uω (aρ 9) (bρ 9) (2678943987473 / 32000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_139_10 :
+    Uω (aρ 10) (bρ 10) (2678943987473 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_139_11 :
+    Uω (aρ 11) (bρ 11) (2678943987473 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_139_12 :
+    Uω (aρ 12) (bρ 12) (2678943987473 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_139_13 :
+    Uω (aρ 13) (bρ 13) (2678943987473 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_139_14 :
+    Uω (aρ 14) (bρ 14) (2678943987473 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_139_15 :
+    Uω (aρ 15) (bρ 15) (2678943987473 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_139_16 :
+    Uω (aρ 16) (bρ 16) (2678943987473 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_139 :
+    Uρ (2678943987473 / 32000000000000 : ℝ) ≤
+      (-(3069316120527173883389 / 1250000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (2678943987473 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(25607556263304084967409 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_139_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (2678943987473 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3242033609723343368109 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_139_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (2678943987473 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(5329422790878437414709 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_139_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (2678943987473 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(350463796210277297691 / 125000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_139_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (2678943987473 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(15558937803900991527783 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_139_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (2678943987473 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_139_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (2678943987473 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_139_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (2678943987473 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_139_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (2678943987473 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_139_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (2678943987473 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_139_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (2678943987473 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_139_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (2678943987473 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_139_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (2678943987473 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_139_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (2678943987473 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_139_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (2678943987473 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_139_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (2678943987473 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_139_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_140_1 :
+    Uω (aρ 1) (bρ 1) (1356286967897 / 16000000000000 : ℝ) ≤
+      (-(25472387635023400139573 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1565951316076903 / 20000000000000000 : ℝ) ≤
+      (-(25472387635023400139573 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1565951316076903 / 20000000000000000 : ℝ) =
+            (1565951316076903 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1565951316076903 / 1250000000000000 : ℝ) ≤
+            (2253499576976599860427 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (391384589613139 / 5000000000000000 : ℝ)) (q :=
+      (1565951316076903 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_140_2 :
+    Uω (aρ 2) (bρ 2) (1356286967897 / 16000000000000 : ℝ) ≤
+      (-(12898195814616624135193 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1516027018954759 / 20000000000000000 : ℝ) ≤
+      (-(12898195814616624135193 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1516027018954759 / 20000000000000000 : ℝ) =
+            (1516027018954759 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1516027018954759 / 1250000000000000 : ℝ) ≤
+            (964747791383375864807 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (378306947307067 / 5000000000000000 : ℝ)) (q :=
+      (1516027018954759 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_140_3 :
+    Uω (aρ 3) (bρ 3) (1356286967897 / 16000000000000 : ℝ) ≤
+      (-(26496161287310828503139 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (141356678327789 / 2000000000000000 : ℝ) ≤
+      (-(26496161287310828503139 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (141356678327789 / 2000000000000000 : ℝ) =
+            (141356678327789 / 125000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (141356678327789 / 125000000000000 : ℝ) ≤
+            (1229725924689171496861 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (140309953057453 / 2000000000000000 : ℝ)) (q :=
+      (141356678327789 / 2000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_140_4 :
+    Uω (aρ 4) (bρ 4) (1356286967897 / 16000000000000 : ℝ) ≤
+      (-(217657707718408659519 / 78125000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (246664960350121 / 4000000000000000 : ℝ) ≤
+      (-(217657707718408659519 / 78125000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (246664960350121 / 4000000000000000 : ℝ) =
+            (246664960350121 / 125000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (246664960350121 / 125000000000000 : ℝ) ≤
+            (53102909586278840481 / 78125000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (29980098241249 / 500000000000000 : ℝ)) (q :=
+      (246664960350121 / 4000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_140_5 :
+    Uω (aρ 5) (bρ 5) (1356286967897 / 16000000000000 : ℝ) ≤
+      (-(30849926669156202687773 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (457303691842081 / 10000000000000000 : ℝ) ≤
+      (-(30849926669156202687773 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (457303691842081 / 10000000000000000 : ℝ) =
+            (457303691842081 / 312500000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (457303691842081 / 312500000000000 : ℝ) ≤
+            (3807432345843797312227 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (399074466093537 / 10000000000000000 : ℝ)) (q :=
+      (457303691842081 / 10000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_140_6 :
+    Uω (aρ 6) (bρ 6) (1356286967897 / 16000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_140_7 :
+    Uω (aρ 7) (bρ 7) (1356286967897 / 16000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_140_8 :
+    Uω (aρ 8) (bρ 8) (1356286967897 / 16000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_140_9 :
+    Uω (aρ 9) (bρ 9) (1356286967897 / 16000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_140_10 :
+    Uω (aρ 10) (bρ 10) (1356286967897 / 16000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_140_11 :
+    Uω (aρ 11) (bρ 11) (1356286967897 / 16000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_140_12 :
+    Uω (aρ 12) (bρ 12) (1356286967897 / 16000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_140_13 :
+    Uω (aρ 13) (bρ 13) (1356286967897 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_140_14 :
+    Uω (aρ 14) (bρ 14) (1356286967897 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_140_15 :
+    Uω (aρ 15) (bρ 15) (1356286967897 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_140_16 :
+    Uω (aρ 16) (bρ 16) (1356286967897 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_140 :
+    Uρ (1356286967897 / 16000000000000 : ℝ) ≤
+      (-(12256854110596322856331 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1356286967897 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(25472387635023400139573 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_140_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1356286967897 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(12898195814616624135193 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_140_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1356286967897 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(26496161287310828503139 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_140_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1356286967897 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(217657707718408659519 / 78125000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_140_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1356286967897 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(30849926669156202687773 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_140_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1356286967897 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_140_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1356286967897 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_140_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1356286967897 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_140_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1356286967897 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_140_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1356286967897 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_140_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1356286967897 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_140_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1356286967897 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_140_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1356286967897 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_140_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1356286967897 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_140_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1356286967897 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_140_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1356286967897 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_140_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_141_1 :
+    Uω (aρ 1) (bρ 1) (694958458109 / 8000000000000 : ℝ) ≤
+      (-(1260370690780799325851 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (200999943759371 / 2500000000000000 : ℝ) ≤
+      (-(1260370690780799325851 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (200999943759371 / 2500000000000000 : ℝ) =
+            (200999943759371 / 156250000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (200999943759371 / 156250000000000 : ℝ) ≤
+            (125923669819200674149 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (401899347761859 / 5000000000000000 : ℝ)) (q :=
+      (200999943759371 / 2500000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_141_2 :
+    Uω (aρ 2) (bρ 2) (694958458109 / 8000000000000 : ℝ) ≤
+      (-(3190299248545577399671 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1558140111364829 / 20000000000000000 : ℝ) ≤
+      (-(3190299248545577399671 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1558140111364829 / 20000000000000000 : ℝ) =
+            (1558140111364829 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1558140111364829 / 1250000000000000 : ℝ) ≤
+            (275436652954422600329 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (777708269323579 / 10000000000000000 : ℝ)) (q :=
+      (1558140111364829 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_141_3 :
+    Uω (aρ 3) (bρ 3) (694958458109 / 8000000000000 : ℝ) ≤
+      (-(3275127593940399875157 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (727954317676977 / 10000000000000000 : ℝ) ≤
+      (-(3275127593940399875157 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (727954317676977 / 10000000000000000 : ℝ) =
+            (727954317676977 / 625000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (727954317676977 / 625000000000000 : ℝ) ≤
+            (190608307559600124843 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (45179556228919 / 625000000000000 : ℝ)) (q :=
+      (727954317676977 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_141_4 :
+    Uω (aρ 4) (bρ 4) (694958458109 / 8000000000000 : ℝ) ≤
+      (-(13757985990762632792871 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (319129185882303 / 5000000000000000 : ℝ) ≤
+      (-(13757985990762632792871 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (319129185882303 / 5000000000000000 : ℝ) =
+            (319129185882303 / 312500000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (319129185882303 / 312500000000000 : ℝ) ≤
+            (104957615237367207129 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (310887594451481 / 5000000000000000 : ℝ)) (q :=
+      (319129185882303 / 5000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_141_5 :
+    Uω (aρ 5) (bρ 5) (694958458109 / 8000000000000 : ℝ) ≤
+      (-(30340243654537166784837 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (240607945013181 / 5000000000000000 : ℝ) ≤
+      (-(30340243654537166784837 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (240607945013181 / 5000000000000000 : ℝ) =
+            (240607945013181 / 156250000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (240607945013181 / 156250000000000 : ℝ) ≤
+            (4317115360462833215163 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (212940072380737 / 5000000000000000 : ℝ)) (q :=
+      (240607945013181 / 5000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_141_6 :
+    Uω (aρ 6) (bρ 6) (694958458109 / 8000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_141_7 :
+    Uω (aρ 7) (bρ 7) (694958458109 / 8000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_141_8 :
+    Uω (aρ 8) (bρ 8) (694958458109 / 8000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_141_9 :
+    Uω (aρ 9) (bρ 9) (694958458109 / 8000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_141_10 :
+    Uω (aρ 10) (bρ 10) (694958458109 / 8000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_141_11 :
+    Uω (aρ 11) (bρ 11) (694958458109 / 8000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_141_12 :
+    Uω (aρ 12) (bρ 12) (694958458109 / 8000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_141_13 :
+    Uω (aρ 13) (bρ 13) (694958458109 / 8000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_141_14 :
+    Uω (aρ 14) (bρ 14) (694958458109 / 8000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_141_15 :
+    Uω (aρ 15) (bρ 15) (694958458109 / 8000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_141_16 :
+    Uω (aρ 16) (bρ 16) (694958458109 / 8000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_141 :
+    Uρ (694958458109 / 8000000000000 : ℝ) ≤
+      (-(24434952955612677719747 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (694958458109 / 8000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1260370690780799325851 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_141_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (694958458109 / 8000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3190299248545577399671 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_141_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (694958458109 / 8000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3275127593940399875157 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_141_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (694958458109 / 8000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(13757985990762632792871 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_141_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (694958458109 / 8000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(30340243654537166784837 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_141_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (694958458109 / 8000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_141_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (694958458109 / 8000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_141_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (694958458109 / 8000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_141_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (694958458109 / 8000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_141_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (694958458109 / 8000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_141_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (694958458109 / 8000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_141_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (694958458109 / 8000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_141_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (694958458109 / 8000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_141_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (694958458109 / 8000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_141_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (694958458109 / 8000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_141_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (694958458109 / 8000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_141_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_142_1 :
+    Uω (aρ 1) (bρ 1) (1423546864539 / 16000000000000 : ℝ) ≤
+      (-(3118660448249061296987 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1650047233580129 / 20000000000000000 : ℝ) ≤
+      (-(3118660448249061296987 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1650047233580129 / 20000000000000000 : ℝ) =
+            (1650047233580129 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1650047233580129 / 1250000000000000 : ℝ) ≤
+            (347075453250938703013 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (412413830664127 / 5000000000000000 : ℝ)) (q :=
+      (1650047233580129 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_142_2 :
+    Uω (aρ 2) (bρ 2) (1423546864539 / 16000000000000 : ℝ) ≤
+      (-(12627864479779576637511 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (800124607553441 / 10000000000000000 : ℝ) ≤
+      (-(12627864479779576637511 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (800124607553441 / 10000000000000000 : ℝ) =
+            (800124607553441 / 625000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (800124607553441 / 625000000000000 : ℝ) ≤
+            (1235079126220423362489 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (798798655365007 / 10000000000000000 : ℝ)) (q :=
+      (800124607553441 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_142_3 :
+    Uω (aρ 3) (bρ 3) (1423546864539 / 16000000000000 : ℝ) ≤
+      (-(25914457472574753600061 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1498233167106849 / 20000000000000000 : ℝ) ≤
+      (-(25914457472574753600061 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1498233167106849 / 20000000000000000 : ℝ) =
+            (1498233167106849 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1498233167106849 / 1250000000000000 : ℝ) ≤
+            (1811429739425246399939 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (372089356857487 / 5000000000000000 : ℝ)) (q :=
+      (1498233167106849 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_142_4 :
+    Uω (aρ 4) (bρ 4) (1423546864539 / 16000000000000 : ℝ) ≤
+      (-(217470394859432947223 / 80000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (659815621873 / 10000000000000 : ℝ) ≤
+      (-(217470394859432947223 / 80000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (659815621873 / 10000000000000 : ℝ) = (659815621873 / 625000000000 : ℝ) / 2 ^ 4 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (659815621873 / 625000000000 : ℝ) ≤
+            (4336702836567052777 / 80000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5150967771353 / 80000000000000 : ℝ)) (q :=
+      (659815621873 / 10000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_142_5 :
+    Uω (aρ 5) (bρ 5) (1423546864539 / 16000000000000 : ℝ) ≤
+      (-(7465335220405714049317 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (252411098596889 / 5000000000000000 : ℝ) ≤
+      (-(7465335220405714049317 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (252411098596889 / 5000000000000000 : ℝ) =
+            (252411098596889 / 156250000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (252411098596889 / 156250000000000 : ℝ) ≤
+            (1199004533344285950683 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (452074041395681 / 10000000000000000 : ℝ)) (q :=
+      (252411098596889 / 5000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_142_6 :
+    Uω (aρ 6) (bρ 6) (1423546864539 / 16000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_142_7 :
+    Uω (aρ 7) (bρ 7) (1423546864539 / 16000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_142_8 :
+    Uω (aρ 8) (bρ 8) (1423546864539 / 16000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_142_9 :
+    Uω (aρ 9) (bρ 9) (1423546864539 / 16000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_142_10 :
+    Uω (aρ 10) (bρ 10) (1423546864539 / 16000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_142_11 :
+    Uω (aρ 11) (bρ 11) (1423546864539 / 16000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_142_12 :
+    Uω (aρ 12) (bρ 12) (1423546864539 / 16000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_142_13 :
+    Uω (aρ 13) (bρ 13) (1423546864539 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_142_14 :
+    Uω (aρ 14) (bρ 14) (1423546864539 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_142_15 :
+    Uω (aρ 15) (bρ 15) (1423546864539 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_142_16 :
+    Uω (aρ 16) (bρ 16) (1423546864539 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_142 :
+    Uρ (1423546864539 / 16000000000000 : ℝ) ≤
+      (-(12179839953679917785211 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1423546864539 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3118660448249061296987 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_142_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1423546864539 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(12627864479779576637511 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_142_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1423546864539 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(25914457472574753600061 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_142_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1423546864539 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(217470394859432947223 / 80000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_142_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1423546864539 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(7465335220405714049317 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_142_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1423546864539 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_142_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1423546864539 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_142_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1423546864539 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_142_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1423546864539 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_142_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1423546864539 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_142_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1423546864539 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_142_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1423546864539 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_142_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1423546864539 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_142_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1423546864539 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_142_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1423546864539 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_142_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1423546864539 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_142_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_143_1 :
+    Uω (aρ 1) (bρ 1) (72858840643 / 800000000000 : ℝ) ≤
+      (-(4939530432806504756481 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (423023601912517 / 5000000000000000 : ℝ) ≤
+      (-(4939530432806504756481 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (423023601912517 / 5000000000000000 : ℝ) =
+            (423023601912517 / 312500000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (423023601912517 / 312500000000000 : ℝ) ≤
+            (605647009593495243519 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (26433003678049 / 312500000000000 : ℝ)) (q :=
+      (423023601912517 / 5000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_143_2 :
+    Uω (aρ 2) (bρ 2) (72858840643 / 800000000000 : ℝ) ≤
+      (-(12498006527585179078093 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (410588659497357 / 5000000000000000 : ℝ) ≤
+      (-(12498006527585179078093 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (410588659497357 / 5000000000000000 : ℝ) =
+            (410588659497357 / 312500000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (410588659497357 / 312500000000000 : ℝ) ≤
+            (1364937078414820921907 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (51242835034183 / 625000000000000 : ℝ)) (q :=
+      (410588659497357 / 5000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_143_3 :
+    Uω (aρ 3) (bρ 3) (72858840643 / 800000000000 : ℝ) ≤
+      (-(25635980836800211611331 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1540541824803243 / 20000000000000000 : ℝ) ≤
+      (-(25635980836800211611331 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1540541824803243 / 20000000000000000 : ℝ) =
+            (1540541824803243 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1540541824803243 / 1250000000000000 : ℝ) ≤
+            (2089906375199788388669 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (765468653710743 / 10000000000000000 : ℝ)) (q :=
+      (1540541824803243 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_143_4 :
+    Uω (aρ 4) (bρ 4) (72858840643 / 800000000000 : ℝ) ≤
+      (-(26862818505948193076349 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1362676011438963 / 20000000000000000 : ℝ) ≤
+      (-(26862818505948193076349 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1362676011438963 / 20000000000000000 : ℝ) =
+            (1362676011438963 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1362676011438963 / 1250000000000000 : ℝ) ≤
+            (863068706051806923651 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (665897021411463 / 10000000000000000 : ℝ)) (q :=
+      (1362676011438963 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_143_5 :
+    Uω (aρ 5) (bρ 5) (72858840643 / 800000000000 : ℝ) ≤
+      (-(3676145772901062746707 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1056345863520347 / 20000000000000000 : ℝ) ≤
+      (-(3676145772901062746707 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (1056345863520347 / 20000000000000000 : ℝ) =
+            (1056345863520347 / 625000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1056345863520347 / 625000000000000 : ℝ) ≤
+            (656024103973937253293 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (477756792827847 / 10000000000000000 : ℝ)) (q :=
+      (1056345863520347 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_143_6 :
+    Uω (aρ 6) (bρ 6) (72858840643 / 800000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_143_7 :
+    Uω (aρ 7) (bρ 7) (72858840643 / 800000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_143_8 :
+    Uω (aρ 8) (bρ 8) (72858840643 / 800000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_143_9 :
+    Uω (aρ 9) (bρ 9) (72858840643 / 800000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_143_10 :
+    Uω (aρ 10) (bρ 10) (72858840643 / 800000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_143_11 :
+    Uω (aρ 11) (bρ 11) (72858840643 / 800000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_143_12 :
+    Uω (aρ 12) (bρ 12) (72858840643 / 800000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_143_13 :
+    Uω (aρ 13) (bρ 13) (72858840643 / 800000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_143_14 :
+    Uω (aρ 14) (bρ 14) (72858840643 / 800000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_143_15 :
+    Uω (aρ 15) (bρ 15) (72858840643 / 800000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_143_16 :
+    Uω (aρ 16) (bρ 16) (72858840643 / 800000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_143 :
+    Uρ (72858840643 / 800000000000 : ℝ) ≤ (-(189746280053298967991 / 78125000000000000000 : ℝ)) :=
+  by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (72858840643 / 800000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(4939530432806504756481 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_143_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (72858840643 / 800000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(12498006527585179078093 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_143_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (72858840643 / 800000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(25635980836800211611331 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_143_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (72858840643 / 800000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(26862818505948193076349 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_143_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (72858840643 / 800000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(3676145772901062746707 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_143_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (72858840643 / 800000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_143_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (72858840643 / 800000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_143_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (72858840643 / 800000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_143_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (72858840643 / 800000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_143_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (72858840643 / 800000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_143_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (72858840643 / 800000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_143_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (72858840643 / 800000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_143_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (72858840643 / 800000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_143_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (72858840643 / 800000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_143_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (72858840643 / 800000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_143_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (72858840643 / 800000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_143_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_144_1 :
+    Uω (aρ 1) (bρ 1) (762218354751 / 8000000000000 : ℝ) ≤
+      (-(24212631311087841507319 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1776187372263171 / 20000000000000000 : ℝ) ≤
+      (-(24212631311087841507319 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1776187372263171 / 20000000000000000 : ℝ) =
+            (1776187372263171 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1776187372263171 / 1250000000000000 : ℝ) ≤
+            (3513255900912158492681 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (887911646909421 / 10000000000000000 : ℝ)) (q :=
+      (1776187372263171 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_144_2 :
+    Uω (aρ 2) (bρ 2) (762218354751 / 8000000000000 : ℝ) ≤
+      (-(24496038767204679231087 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (431638880396551 / 5000000000000000 : ℝ) ≤
+      (-(24496038767204679231087 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (431638880396551 / 5000000000000000 : ℝ) =
+            (431638880396551 / 312500000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (431638880396551 / 312500000000000 : ℝ) ≤
+            (3229848444795320768913 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (431024404371227 / 5000000000000000 : ℝ)) (q :=
+      (431638880396551 / 5000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_144_3 :
+    Uω (aρ 3) (bρ 3) (762218354751 / 8000000000000 : ℝ) ≤
+      (-(25101527446924252214789 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (406279134060903 / 5000000000000000 : ℝ) ≤
+      (-(25101527446924252214789 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (406279134060903 / 5000000000000000 : ℝ) =
+            (406279134060903 / 312500000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (406279134060903 / 312500000000000 : ℝ) ≤
+            (2624359765075747785211 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (404002964874931 / 5000000000000000 : ℝ)) (q :=
+      (406279134060903 / 5000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_144_4 :
+    Uω (aρ 4) (bρ 4) (762218354751 / 8000000000000 : ℝ) ≤
+      (-(13125734602345873881497 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1448582297152821 / 20000000000000000 : ℝ) ≤
+      (-(13125734602345873881497 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1448582297152821 / 20000000000000000 : ℝ) =
+            (1448582297152821 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1448582297152821 / 1250000000000000 : ℝ) ≤
+            (737209003654126118503 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (709765871724071 / 10000000000000000 : ℝ)) (q :=
+      (1448582297152821 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_144_5 :
+    Uω (aρ 5) (bρ 5) (762218354751 / 8000000000000 : ℝ) ≤
+      (-(28572647461235244444859 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1148512384323577 / 20000000000000000 : ℝ) ≤
+      (-(28572647461235244444859 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (1148512384323577 / 20000000000000000 : ℝ) =
+            (1148512384323577 / 625000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1148512384323577 / 625000000000000 : ℝ) ≤
+            (6084711553764755555141 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (527885878229827 / 10000000000000000 : ℝ)) (q :=
+      (1148512384323577 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_144_6 :
+    Uω (aρ 6) (bρ 6) (762218354751 / 8000000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_144_7 :
+    Uω (aρ 7) (bρ 7) (762218354751 / 8000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_144_8 :
+    Uω (aρ 8) (bρ 8) (762218354751 / 8000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_144_9 :
+    Uω (aρ 9) (bρ 9) (762218354751 / 8000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_144_10 :
+    Uω (aρ 10) (bρ 10) (762218354751 / 8000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_144_11 :
+    Uω (aρ 11) (bρ 11) (762218354751 / 8000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_144_12 :
+    Uω (aρ 12) (bρ 12) (762218354751 / 8000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_144_13 :
+    Uω (aρ 13) (bρ 13) (762218354751 / 8000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_144_14 :
+    Uω (aρ 14) (bρ 14) (762218354751 / 8000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_144_15 :
+    Uω (aρ 15) (bρ 15) (762218354751 / 8000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_144_16 :
+    Uω (aρ 16) (bρ 16) (762218354751 / 8000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_144 :
+    Uρ (762218354751 / 8000000000000 : ℝ) ≤
+      (-(6037851914004356594239 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (762218354751 / 8000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(24212631311087841507319 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_144_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (762218354751 / 8000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(24496038767204679231087 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_144_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (762218354751 / 8000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(25101527446924252214789 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_144_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (762218354751 / 8000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(13125734602345873881497 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_144_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (762218354751 / 8000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(28572647461235244444859 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_144_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (762218354751 / 8000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_144_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (762218354751 / 8000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_144_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (762218354751 / 8000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_144_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (762218354751 / 8000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_144_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (762218354751 / 8000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_144_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (762218354751 / 8000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_144_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (762218354751 / 8000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_144_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (762218354751 / 8000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_144_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (762218354751 / 8000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_144_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (762218354751 / 8000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_144_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (762218354751 / 8000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_144_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_145_1 :
+    Uω (aρ 1) (bρ 1) (24870259471 / 250000000000 : ℝ) ≤
+      (-(5937514892280973947171 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1860278700728729 / 20000000000000000 : ℝ) ≤
+      (-(5937514892280973947171 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1860278700728729 / 20000000000000000 : ℝ) =
+            (1860278700728729 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1860278700728729 / 1250000000000000 : ℝ) ≤
+            (993956910719026052829 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (929965539973729 / 10000000000000000 : ℝ)) (q :=
+      (1860278700728729 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_145_2 :
+    Uω (aρ 2) (bρ 2) (24870259471 / 250000000000 : ℝ) ≤
+      (-(24019940915667638914899 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (181074467070649 / 2000000000000000 : ℝ) ≤
+      (-(24019940915667638914899 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (181074467070649 / 2000000000000000 : ℝ) =
+            (181074467070649 / 125000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (181074467070649 / 125000000000000 : ℝ) ≤
+            (3705946296332361085101 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (90420052246149 / 1000000000000000 : ℝ)) (q :=
+      (181074467070649 / 2000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_145_3 :
+    Uω (aρ 3) (bρ 3) (24870259471 / 250000000000 : ℝ) ≤
+      (-(12297242736766437788551 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1709641544020893 / 20000000000000000 : ℝ) ≤
+      (-(12297242736766437788551 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1709641544020893 / 20000000000000000 : ℝ) =
+            (1709641544020893 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1709641544020893 / 1250000000000000 : ℝ) ≤
+            (1565700869233562211449 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (850493502125893 / 10000000000000000 : ℝ)) (q :=
+      (1709641544020893 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_145_4 :
+    Uω (aρ 4) (bρ 4) (24870259471 / 250000000000 : ℝ) ≤
+      (-(25676711922883085692547 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (95892487016369 / 1250000000000000 : ℝ) ≤
+      (-(25676711922883085692547 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (95892487016369 / 1250000000000000 : ℝ) = (95892487016369 / 78125000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (95892487016369 / 78125000000000 : ℝ) ≤
+            (2049175289116914307453 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (23544560357247 / 312500000000000 : ℝ)) (q :=
+      (95892487016369 / 1250000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_145_5 :
+    Uω (aρ 5) (bρ 5) (24870259471 / 250000000000 : ℝ) ≤
+      (-(13905575961864639085561 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (123938721978887 / 2000000000000000 : ℝ) ≤
+      (-(13905575961864639085561 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (123938721978887 / 2000000000000000 : ℝ) =
+            (123938721978887 / 62500000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (123938721978887 / 62500000000000 : ℝ) ≤
+            (3423103545635360914439 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (57672327829387 / 1000000000000000 : ℝ)) (q :=
+      (123938721978887 / 2000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_145_6 :
+    Uω (aρ 6) (bρ 6) (24870259471 / 250000000000 : ℝ) ≤
+      (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_6]) (by norm_num [bρ_6])]; exact Lmid_6
+
+lemma U_145_7 :
+    Uω (aρ 7) (bρ 7) (24870259471 / 250000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_145_8 :
+    Uω (aρ 8) (bρ 8) (24870259471 / 250000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_145_9 :
+    Uω (aρ 9) (bρ 9) (24870259471 / 250000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_145_10 :
+    Uω (aρ 10) (bρ 10) (24870259471 / 250000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_145_11 :
+    Uω (aρ 11) (bρ 11) (24870259471 / 250000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_145_12 :
+    Uω (aρ 12) (bρ 12) (24870259471 / 250000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_145_13 :
+    Uω (aρ 13) (bρ 13) (24870259471 / 250000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_145_14 :
+    Uω (aρ 14) (bρ 14) (24870259471 / 250000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_145_15 :
+    Uω (aρ 15) (bρ 15) (24870259471 / 250000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_145_16 :
+    Uω (aρ 16) (bρ 16) (24870259471 / 250000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_145 :
+    Uρ (24870259471 / 250000000000 : ℝ) ≤
+      (-(6006179333855527615251 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (24870259471 / 250000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(5937514892280973947171 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_145_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (24870259471 / 250000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(24019940915667638914899 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_145_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (24870259471 / 250000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(12297242736766437788551 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_145_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (24870259471 / 250000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(25676711922883085692547 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_145_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (24870259471 / 250000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(13905575961864639085561 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_145_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (24870259471 / 250000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3698074464695343529243 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_145_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (24870259471 / 250000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_145_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (24870259471 / 250000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_145_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (24870259471 / 250000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_145_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (24870259471 / 250000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_145_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (24870259471 / 250000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_145_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (24870259471 / 250000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_145_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (24870259471 / 250000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_145_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (24870259471 / 250000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_145_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (24870259471 / 250000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_145_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (24870259471 / 250000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_145_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_146_1 :
+    Uω (aρ 1) (bρ 1) (1614118950931 / 16000000000000 : ℝ) ≤
+      (-(23600490741746310132633 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (47207794808651 / 500000000000000 : ℝ) ≤
+      (-(23600490741746310132633 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (47207794808651 / 500000000000000 : ℝ) = (47207794808651 / 31250000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (47207794808651 / 31250000000000 : ℝ) ≤
+            (4125396470253689867367 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (188796933219833 / 2000000000000000 : ℝ)) (q :=
+      (47207794808651 / 500000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_146_2 :
+    Uω (aρ 2) (bρ 2) (1614118950931 / 16000000000000 : ℝ) ≤
+      (-(23866145357694081674823 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (229851046232431 / 2500000000000000 : ℝ) ≤
+      (-(23866145357694081674823 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (229851046232431 / 2500000000000000 : ℝ) =
+            (229851046232431 / 156250000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (229851046232431 / 156250000000000 : ℝ) ≤
+            (3859741854305918325177 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (918250256122573 / 10000000000000000 : ℝ)) (q :=
+      (229851046232431 / 2500000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_146_3 :
+    Uω (aρ 3) (bρ 3) (1614118950931 / 16000000000000 : ℝ) ≤
+      (-(4886213425407009894147 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1737809756710397 / 20000000000000000 : ℝ) ≤
+      (-(4886213425407009894147 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1737809756710397 / 20000000000000000 : ℝ) =
+            (1737809756710397 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1737809756710397 / 1250000000000000 : ℝ) ≤
+            (658964016992990105853 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (432323874661761 / 5000000000000000 : ℝ)) (q :=
+      (1737809756710397 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_146_4 :
+    Uω (aρ 4) (bρ 4) (1614118950931 / 16000000000000 : ℝ) ≤
+      (-(25492478055288563125353 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3125616824483 / 40000000000000 : ℝ) ≤
+      (-(25492478055288563125353 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (3125616824483 / 40000000000000 : ℝ) = (3125616824483 / 2500000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3125616824483 / 2500000000000 : ℝ) ≤
+            (2233409156711436874647 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6143524687357 / 80000000000000 : ℝ)) (q :=
+      (3125616824483 / 40000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_146_5 :
+    Uω (aρ 5) (bρ 5) (1614118950931 / 16000000000000 : ℝ) ≤
+      (-(27571482204858512832921 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1269450401433503 / 20000000000000000 : ℝ) ≤
+      (-(27571482204858512832921 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1269450401433503 / 20000000000000000 : ℝ) =
+            (1269450401433503 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1269450401433503 / 1250000000000000 : ℝ) ≤
+            (154405007141487167079 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (148193123611657 / 2500000000000000 : ℝ)) (q :=
+      (1269450401433503 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_146_6 :
+    Uω (aρ 6) (bρ 6) (1614118950931 / 16000000000000 : ℝ) ≤
+      (-(17303896956176881585291 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (125621101760153 / 4000000000000000 : ℝ) ≤
+      (-(17303896956176881585291 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (125621101760153 / 4000000000000000 : ℝ) =
+            (125621101760153 / 125000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (125621101760153 / 125000000000000 : ℝ) ≤
+            (24782551323118414709 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (11866797695389 / 1000000000000000 : ℝ)) (q :=
+      (125621101760153 / 4000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_146_7 :
+    Uω (aρ 7) (bρ 7) (1614118950931 / 16000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_146_8 :
+    Uω (aρ 8) (bρ 8) (1614118950931 / 16000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_146_9 :
+    Uω (aρ 9) (bρ 9) (1614118950931 / 16000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_146_10 :
+    Uω (aρ 10) (bρ 10) (1614118950931 / 16000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_146_11 :
+    Uω (aρ 11) (bρ 11) (1614118950931 / 16000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_146_12 :
+    Uω (aρ 12) (bρ 12) (1614118950931 / 16000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_146_13 :
+    Uω (aρ 13) (bρ 13) (1614118950931 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_146_14 :
+    Uω (aρ 14) (bρ 14) (1614118950931 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_146_15 :
+    Uω (aρ 15) (bρ 15) (1614118950931 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_146_16 :
+    Uω (aρ 16) (bρ 16) (1614118950931 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_146 :
+    Uρ (1614118950931 / 16000000000000 : ℝ) ≤
+      (-(1189858212400226247237 / 500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1614118950931 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(23600490741746310132633 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_146_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1614118950931 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(23866145357694081674823 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_146_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1614118950931 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(4886213425407009894147 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_146_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1614118950931 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(25492478055288563125353 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_146_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1614118950931 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(27571482204858512832921 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_146_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1614118950931 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(17303896956176881585291 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_146_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1614118950931 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_146_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1614118950931 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_146_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1614118950931 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_146_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1614118950931 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_146_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1614118950931 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_146_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1614118950931 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_146_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1614118950931 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_146_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1614118950931 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_146_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1614118950931 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_146_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1614118950931 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_146_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_147_1 :
+    Uω (aρ 1) (bρ 1) (818270647859 / 8000000000000 : ℝ) ≤
+      (-(586328171474169128263 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (958172366476481 / 10000000000000000 : ℝ) ≤
+      (-(586328171474169128263 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (958172366476481 / 10000000000000000 : ℝ) =
+            (958172366476481 / 625000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (958172366476481 / 625000000000000 : ℝ) ≤
+            (106819008825830871737 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (239500910303553 / 2500000000000000 : ℝ)) (q :=
+      (958172366476481 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_147_2 :
+    Uω (aρ 2) (bρ 2) (818270647859 / 8000000000000 : ℝ) ≤
+      (-(23714685092528748377319 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (466717748083379 / 5000000000000000 : ℝ) ≤
+      (-(23714685092528748377319 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (466717748083379 / 5000000000000000 : ℝ) =
+            (466717748083379 / 312500000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (466717748083379 / 312500000000000 : ℝ) ≤
+            (4011202119471251622681 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (466149456552383 / 5000000000000000 : ℝ)) (q :=
+      (466717748083379 / 5000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_147_3 :
+    Uω (aρ 3) (bρ 3) (818270647859 / 8000000000000 : ℝ) ≤
+      (-(3033787739285829484113 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (353194694588661 / 4000000000000000 : ℝ) ≤
+      (-(3033787739285829484113 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (353194694588661 / 4000000000000000 : ℝ) =
+            (353194694588661 / 250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (353194694588661 / 250000000000000 : ℝ) ≤
+            (431948162214170515887 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (175759500012911 / 2000000000000000 : ℝ)) (q :=
+      (353194694588661 / 4000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_147_4 :
+    Uω (aρ 4) (bρ 4) (818270647859 / 8000000000000 : ℝ) ≤
+      (-(25311691807556838955743 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (795659388347077 / 10000000000000000 : ℝ) ≤
+      (-(25311691807556838955743 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (795659388347077 / 10000000000000000 : ℝ) =
+            (795659388347077 / 625000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (795659388347077 / 625000000000000 : ℝ) ≤
+            (2414195404443161044257 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (195609246220101 / 2500000000000000 : ℝ)) (q :=
+      (795659388347077 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_147_5 :
+    Uω (aρ 5) (bρ 5) (818270647859 / 8000000000000 : ℝ) ≤
+      (-(13669097951678662512941 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (649706537778299 / 10000000000000000 : ℝ) ≤
+      (-(13669097951678662512941 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (649706537778299 / 10000000000000000 : ℝ) =
+            (649706537778299 / 625000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (649706537778299 / 625000000000000 : ℝ) ≤
+            (193845654321337487059 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (76090150384731 / 1250000000000000 : ℝ)) (q :=
+      (649706537778299 / 10000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_147_6 :
+    Uω (aρ 6) (bρ 6) (818270647859 / 8000000000000 : ℝ) ≤
+      (-(2102041636154490140537 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (346219772076763 / 10000000000000000 : ℝ) ≤
+      (-(2102041636154490140537 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (346219772076763 / 10000000000000000 : ℝ) =
+            (346219772076763 / 312500000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (346219772076763 / 312500000000000 : ℝ) ≤
+            (64043302283009859463 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (21123505851847 / 1250000000000000 : ℝ)) (q :=
+      (346219772076763 / 10000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_147_7 :
+    Uω (aρ 7) (bρ 7) (818270647859 / 8000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_147_8 :
+    Uω (aρ 8) (bρ 8) (818270647859 / 8000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_147_9 :
+    Uω (aρ 9) (bρ 9) (818270647859 / 8000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_147_10 :
+    Uω (aρ 10) (bρ 10) (818270647859 / 8000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_147_11 :
+    Uω (aρ 11) (bρ 11) (818270647859 / 8000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_147_12 :
+    Uω (aρ 12) (bρ 12) (818270647859 / 8000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_147_13 :
+    Uω (aρ 13) (bρ 13) (818270647859 / 8000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_147_14 :
+    Uω (aρ 14) (bρ 14) (818270647859 / 8000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_147_15 :
+    Uω (aρ 15) (bρ 15) (818270647859 / 8000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_147_16 :
+    Uω (aρ 16) (bρ 16) (818270647859 / 8000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_147 :
+    Uρ (818270647859 / 8000000000000 : ℝ) ≤
+      (-(23680709076845825592017 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (818270647859 / 8000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(586328171474169128263 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_147_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (818270647859 / 8000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(23714685092528748377319 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_147_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (818270647859 / 8000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3033787739285829484113 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_147_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (818270647859 / 8000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(25311691807556838955743 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_147_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (818270647859 / 8000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(13669097951678662512941 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_147_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (818270647859 / 8000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(2102041636154490140537 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_147_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (818270647859 / 8000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_147_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (818270647859 / 8000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_147_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (818270647859 / 8000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_147_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (818270647859 / 8000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_147_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (818270647859 / 8000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_147_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (818270647859 / 8000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_147_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (818270647859 / 8000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_147_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (818270647859 / 8000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_147_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (818270647859 / 8000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_147_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (818270647859 / 8000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_147_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/U12.lean b/LeanPool/Zeta5Irrational/Table/U12.lean
new file mode 100644
index 0000000000..7d08c89cef
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/U12.lean
@@ -0,0 +1,3281 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-! # Certified arcsine potential bounds (U12) -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma U_148_1 :
+    Uω (aρ 1) (bρ 1) (331792728101 / 3200000000000 : ℝ) ≤
+      (-(23307903831274102064587 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (194437752908333 / 2000000000000000 : ℝ) ≤
+      (-(23307903831274102064587 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (194437752908333 / 2000000000000000 : ℝ) =
+            (194437752908333 / 125000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (194437752908333 / 125000000000000 : ℝ) ≤
+            (4417983380725897935413 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (194404494370541 / 2000000000000000 : ℝ)) (q :=
+      (194437752908333 / 2000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_148_2 :
+    Uω (aρ 2) (bρ 2) (331792728101 / 3200000000000 : ℝ) ≤
+      (-(1472843130775668236677 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (947466293039537 / 10000000000000000 : ℝ) ≤
+      (-(1472843130775668236677 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (947466293039537 / 10000000000000000 : ℝ) =
+            (947466293039537 / 625000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (947466293039537 / 625000000000000 : ℝ) ≤
+            (260024819974331763323 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (946346541358449 / 10000000000000000 : ℝ)) (q :=
+      (947466293039537 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_148_3 :
+    Uω (aρ 3) (bρ 3) (331792728101 / 3200000000000 : ℝ) ≤
+      (-(24112104277449450185939 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1794132906474863 / 20000000000000000 : ℝ) ≤
+      (-(24112104277449450185939 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1794132906474863 / 20000000000000000 : ℝ) =
+            (1794132906474863 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1794132906474863 / 1250000000000000 : ℝ) ≤
+            (3613782934550549814061 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (446471484052119 / 5000000000000000 : ℝ)) (q :=
+      (1794132906474863 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_148_4 :
+    Uω (aρ 4) (bρ 4) (331792728101 / 3200000000000 : ℝ) ≤
+      (-(25134222523221480039377 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (404952970464647 / 5000000000000000 : ℝ) ≤
+      (-(25134222523221480039377 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (404952970464647 / 5000000000000000 : ℝ) =
+            (404952970464647 / 312500000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (404952970464647 / 312500000000000 : ℝ) ≤
+            (2591664688778519960623 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (796916124552963 / 10000000000000000 : ℝ)) (q :=
+      (404952970464647 / 5000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_148_5 :
+    Uω (aρ 5) (bρ 5) (331792728101 / 3200000000000 : ℝ) ≤
+      (-(13555463086437859904049 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (664641470824081 / 10000000000000000 : ℝ) ≤
+      (-(13555463086437859904049 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (664641470824081 / 10000000000000000 : ℝ) =
+            (664641470824081 / 625000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (664641470824081 / 625000000000000 : ℝ) ≤
+            (307480519562140095951 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (624577103677537 / 10000000000000000 : ℝ)) (q :=
+      (664641470824081 / 10000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_148_6 :
+    Uω (aρ 6) (bρ 6) (331792728101 / 3200000000000 : ℝ) ≤
+      (-(1644480720294708417623 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (745851195633597 / 20000000000000000 : ℝ) ≤
+      (-(1644480720294708417623 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (745851195633597 / 20000000000000000 : ℝ) =
+            (745851195633597 / 625000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (745851195633597 / 625000000000000 : ℝ) ≤
+            (88387230455291582377 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (52096433200743 / 2500000000000000 : ℝ)) (q :=
+      (745851195633597 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_148_7 :
+    Uω (aρ 7) (bρ 7) (331792728101 / 3200000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_148_8 :
+    Uω (aρ 8) (bρ 8) (331792728101 / 3200000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_148_9 :
+    Uω (aρ 9) (bρ 9) (331792728101 / 3200000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_148_10 :
+    Uω (aρ 10) (bρ 10) (331792728101 / 3200000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_148_11 :
+    Uω (aρ 11) (bρ 11) (331792728101 / 3200000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_148_12 :
+    Uω (aρ 12) (bρ 12) (331792728101 / 3200000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_148_13 :
+    Uω (aρ 13) (bρ 13) (331792728101 / 3200000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_148_14 :
+    Uω (aρ 14) (bρ 14) (331792728101 / 3200000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_148_15 :
+    Uω (aρ 15) (bρ 15) (331792728101 / 3200000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_148_16 :
+    Uω (aρ 16) (bρ 16) (331792728101 / 3200000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_148 :
+    Uρ (331792728101 / 3200000000000 : ℝ) ≤
+      (-(23583366397967708671417 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (331792728101 / 3200000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(23307903831274102064587 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_148_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (331792728101 / 3200000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1472843130775668236677 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_148_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (331792728101 / 3200000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(24112104277449450185939 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_148_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (331792728101 / 3200000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(25134222523221480039377 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_148_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (331792728101 / 3200000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(13555463086437859904049 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_148_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (331792728101 / 3200000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1644480720294708417623 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_148_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (331792728101 / 3200000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_148_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (331792728101 / 3200000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_148_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (331792728101 / 3200000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_148_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (331792728101 / 3200000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_148_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (331792728101 / 3200000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_148_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (331792728101 / 3200000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_148_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (331792728101 / 3200000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_148_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (331792728101 / 3200000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_148_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (331792728101 / 3200000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_148_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (331792728101 / 3200000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_148_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_149_1 :
+    Uω (aρ 1) (bρ 1) (3340349625797 / 32000000000000 : ℝ) ≤
+      (-(2904509483478323326409 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3916787749818561 / 40000000000000000 : ℝ) ≤
+      (-(2904509483478323326409 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (3916787749818561 / 40000000000000000 : ℝ) =
+            (3916787749818561 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3916787749818561 / 2500000000000000 : ℝ) ≤
+            (561226418021676673591 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (489515917466359 / 5000000000000000 : ℝ)) (q :=
+      (3916787749818561 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_149_2 :
+    Uω (aρ 2) (bρ 2) (3340349625797 / 32000000000000 : ℝ) ≤
+      (-(2936465129834332087859 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3817926022778861 / 40000000000000000 : ℝ) ≤
+      (-(2936465129834332087859 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (3817926022778861 / 40000000000000000 : ℝ) =
+            (3817926022778861 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3817926022778861 / 2500000000000000 : ℝ) ≤
+            (529270771665667912141 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (238342495980717 / 2500000000000000 : ℝ)) (q :=
+      (3817926022778861 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_149_3 :
+    Uω (aρ 3) (bρ 3) (3340349625797 / 32000000000000 : ℝ) ≤
+      (-(6008485673465917970041 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3616422160668097 / 40000000000000000 : ℝ) ≤
+      (-(6008485673465917970041 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (3616422160668097 / 40000000000000000 : ℝ) =
+            (3616422160668097 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3616422160668097 / 2500000000000000 : ℝ) ≤
+            (922986129534082029959 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (450007079608743 / 5000000000000000 : ℝ)) (q :=
+      (3616422160668097 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_149_4 :
+    Uω (aρ 4) (bρ 4) (3340349625797 / 32000000000000 : ℝ) ≤
+      (-(25046692976457419243763 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (653620901436473 / 8000000000000000 : ℝ) ≤
+      (-(25046692976457419243763 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (653620901436473 / 8000000000000000 : ℝ) =
+            (653620901436473 / 500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (653620901436473 / 500000000000000 : ℝ) ≤
+            (2679194235542580756237 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (40207475676981 / 500000000000000 : ℝ)) (q :=
+      (653620901436473 / 8000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_149_5 :
+    Uω (aρ 5) (bρ 5) (3340349625797 / 32000000000000 : ℝ) ≤
+      (-(26999442017161149412787 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2688370509070709 / 40000000000000000 : ℝ) ≤
+      (-(26999442017161149412787 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2688370509070709 / 40000000000000000 : ℝ) =
+            (2688370509070709 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2688370509070709 / 2500000000000000 : ℝ) ≤
+            (726445194838850587213 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (79059054227349 / 1250000000000000 : ℝ)) (q :=
+      (2688370509070709 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_149_6 :
+    Uω (aρ 6) (bρ 6) (3340349625797 / 32000000000000 : ℝ) ≤
+      (-(16283427578860002476577 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1540633872166461 / 40000000000000000 : ℝ) ≤
+      (-(16283427578860002476577 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (1540633872166461 / 40000000000000000 : ℝ) =
+            (1540633872166461 / 1250000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1540633872166461 / 1250000000000000 : ℝ) ≤
+            (1045251928639997523423 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (56461122626667 / 2500000000000000 : ℝ)) (q :=
+      (1540633872166461 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_149_7 :
+    Uω (aρ 7) (bρ 7) (3340349625797 / 32000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_149_8 :
+    Uω (aρ 8) (bρ 8) (3340349625797 / 32000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_149_9 :
+    Uω (aρ 9) (bρ 9) (3340349625797 / 32000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_149_10 :
+    Uω (aρ 10) (bρ 10) (3340349625797 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_149_11 :
+    Uω (aρ 11) (bρ 11) (3340349625797 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_149_12 :
+    Uω (aρ 12) (bρ 12) (3340349625797 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_149_13 :
+    Uω (aρ 13) (bρ 13) (3340349625797 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_149_14 :
+    Uω (aρ 14) (bρ 14) (3340349625797 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_149_15 :
+    Uω (aρ 15) (bρ 15) (3340349625797 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_149_16 :
+    Uω (aρ 16) (bρ 16) (3340349625797 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_149 :
+    Uρ (3340349625797 / 32000000000000 : ℝ) ≤
+      (-(2353883295077089915433 / 1000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (3340349625797 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2904509483478323326409 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_149_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (3340349625797 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(2936465129834332087859 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_149_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (3340349625797 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(6008485673465917970041 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_149_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (3340349625797 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(25046692976457419243763 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_149_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (3340349625797 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(26999442017161149412787 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_149_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (3340349625797 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(16283427578860002476577 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_149_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (3340349625797 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_149_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (3340349625797 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_149_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (3340349625797 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_149_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (3340349625797 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_149_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (3340349625797 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_149_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (3340349625797 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_149_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (3340349625797 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_149_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (3340349625797 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_149_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (3340349625797 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_149_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (3340349625797 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_149_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_150_1 :
+    Uω (aρ 1) (bρ 1) (420346496323 / 4000000000000 : ℝ) ≤
+      (-(5791190081092174150987 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (246551273362417 / 2500000000000000 : ℝ) ≤
+      (-(5791190081092174150987 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (246551273362417 / 2500000000000000 : ℝ) =
+            (246551273362417 / 156250000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (246551273362417 / 156250000000000 : ℝ) ≤
+            (1140281721907825849013 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (246510291044209 / 2500000000000000 : ℝ)) (q :=
+      (246551273362417 / 2500000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_150_2 :
+    Uω (aρ 2) (bρ 2) (420346496323 / 4000000000000 : ℝ) ≤
+      (-(2927311680014733944333 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1922993196237621 / 20000000000000000 : ℝ) ≤
+      (-(2927311680014733944333 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1922993196237621 / 20000000000000000 : ℝ) =
+            (1922993196237621 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1922993196237621 / 1250000000000000 : ℝ) ≤
+            (538424221485266055667 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (960393186025121 / 10000000000000000 : ℝ)) (q :=
+      (1922993196237621 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_150_3 :
+    Uω (aρ 3) (bρ 3) (420346496323 / 4000000000000 : ℝ) ≤
+      (-(93579659246677020249 / 39062500000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (113893016104001 / 1250000000000000 : ℝ) ≤
+      (-(93579659246677020249 / 39062500000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (113893016104001 / 1250000000000000 : ℝ) =
+            (113893016104001 / 78125000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (113893016104001 / 78125000000000 : ℝ) ≤
+            (14724587675197979751 / 39062500000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (226771088450379 / 2500000000000000 : ℝ)) (q :=
+      (113893016104001 / 1250000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_150_4 :
+    Uω (aρ 4) (bρ 4) (420346496323 / 4000000000000 : ℝ) ≤
+      (-(4991989407247450880853 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1648288651716599 / 20000000000000000 : ℝ) ≤
+      (-(4991989407247450880853 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1648288651716599 / 20000000000000000 : ℝ) =
+            (1648288651716599 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1648288651716599 / 1250000000000000 : ℝ) ≤
+            (553188035152549119147 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (811378928919099 / 10000000000000000 : ℝ)) (q :=
+      (1648288651716599 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_150_5 :
+    Uω (aρ 5) (bρ 5) (420346496323 / 4000000000000 : ℝ) ≤
+      (-(1344466957828672674537 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1359066893920101 / 20000000000000000 : ℝ) ≤
+      (-(1344466957828672674537 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1359066893920101 / 20000000000000000 : ℝ) =
+            (1359066893920101 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1359066893920101 / 1250000000000000 : ℝ) ≤
+            (41827402771327325463 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (640347090457601 / 10000000000000000 : ℝ)) (q :=
+      (1359066893920101 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_150_6 :
+    Uω (aρ 6) (bρ 6) (420346496323 / 4000000000000 : ℝ) ≤
+      (-(3226744951560163699253 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (793729402033551 / 20000000000000000 : ℝ) ≤
+      (-(3226744951560163699253 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (793729402033551 / 20000000000000000 : ℝ) =
+            (793729402033551 / 625000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (793729402033551 / 625000000000000 : ℝ) ≤
+            (238990949939836300747 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (242249973711051 / 10000000000000000 : ℝ)) (q :=
+      (793729402033551 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_150_7 :
+    Uω (aρ 7) (bρ 7) (420346496323 / 4000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_150_8 :
+    Uω (aρ 8) (bρ 8) (420346496323 / 4000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_150_9 :
+    Uω (aρ 9) (bρ 9) (420346496323 / 4000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_150_10 :
+    Uω (aρ 10) (bρ 10) (420346496323 / 4000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_150_11 :
+    Uω (aρ 11) (bρ 11) (420346496323 / 4000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_150_12 :
+    Uω (aρ 12) (bρ 12) (420346496323 / 4000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_150_13 :
+    Uω (aρ 13) (bρ 13) (420346496323 / 4000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_150_14 :
+    Uω (aρ 14) (bρ 14) (420346496323 / 4000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_150_15 :
+    Uω (aρ 15) (bρ 15) (420346496323 / 4000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_150_16 :
+    Uω (aρ 16) (bρ 16) (420346496323 / 4000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_150 :
+    Uρ (420346496323 / 4000000000000 : ℝ) ≤
+      (-(23496330052534074716271 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (420346496323 / 4000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(5791190081092174150987 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_150_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (420346496323 / 4000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(2927311680014733944333 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_150_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (420346496323 / 4000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) * (-(93579659246677020249 / 39062500000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_150_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (420346496323 / 4000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(4991989407247450880853 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_150_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (420346496323 / 4000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(1344466957828672674537 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_150_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (420346496323 / 4000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3226744951560163699253 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_150_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (420346496323 / 4000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_150_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (420346496323 / 4000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_150_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (420346496323 / 4000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_150_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (420346496323 / 4000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_150_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (420346496323 / 4000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_150_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (420346496323 / 4000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_150_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (420346496323 / 4000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_150_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (420346496323 / 4000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_150_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (420346496323 / 4000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_150_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (420346496323 / 4000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_150_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_151_1 :
+    Uω (aρ 1) (bρ 1) (3385194315371 / 32000000000000 : ℝ) ≤
+      (-(2886743742383077728029 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (794570586307991 / 8000000000000000 : ℝ) ≤
+      (-(2886743742383077728029 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (794570586307991 / 8000000000000000 : ℝ) =
+            (794570586307991 / 500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (794570586307991 / 500000000000000 : ℝ) ≤
+            (578992159116922271971 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (49652523015077 / 500000000000000 : ℝ)) (q :=
+      (794570586307991 / 8000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_151_2 :
+    Uω (aρ 2) (bρ 2) (3385194315371 / 32000000000000 : ℝ) ≤
+      (-(4669159877652684528577 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3874046291721471 / 40000000000000000 : ℝ) ≤
+      (-(4669159877652684528577 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (3874046291721471 / 40000000000000000 : ℝ) =
+            (3874046291721471 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3874046291721471 / 2500000000000000 : ℝ) ≤
+            (876017564747315471423 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (483708076451149 / 5000000000000000 : ℝ)) (q :=
+      (3874046291721471 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_151_3 :
+    Uω (aρ 3) (bρ 3) (3385194315371 / 32000000000000 : ℝ) ≤
+      (-(23879444915266449414989 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3672728923173331 / 40000000000000000 : ℝ) ≤
+      (-(23879444915266449414989 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (3672728923173331 / 40000000000000000 : ℝ) =
+            (3672728923173331 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3672728923173331 / 2500000000000000 : ℝ) ≤
+            (3846442296733550585011 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (228538393744557 / 2500000000000000 : ℝ)) (q :=
+      (3672728923173331 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_151_4 :
+    Uω (aρ 4) (bρ 4) (3385194315371 / 32000000000000 : ℝ) ≤
+      (-(24873970382882411377119 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3325042363024537 / 40000000000000000 : ℝ) ≤
+      (-(24873970382882411377119 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (3325042363024537 / 40000000000000000 : ℝ) =
+            (3325042363024537 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3325042363024537 / 2500000000000000 : ℝ) ≤
+            (2851916829117588622881 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (818604475968831 / 10000000000000000 : ℝ)) (q :=
+      (3325042363024537 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_151_5 :
+    Uω (aρ 5) (bρ 5) (3385194315371 / 32000000000000 : ℝ) ≤
+      (-(26780580354809959357677 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (549571445302699 / 8000000000000000 : ℝ) ≤
+      (-(26780580354809959357677 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (549571445302699 / 8000000000000000 : ℝ) =
+            (549571445302699 / 500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (549571445302699 / 500000000000000 : ℝ) ≤
+            (945306857190040642323 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (64820182704831 / 1000000000000000 : ℝ)) (q :=
+      (549571445302699 / 8000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_151_6 :
+    Uω (aρ 6) (bρ 6) (3385194315371 / 32000000000000 : ℝ) ≤
+      (-(3198718174266971398463 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1632579498774009 / 40000000000000000 : ℝ) ≤
+      (-(3198718174266971398463 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (1632579498774009 / 40000000000000000 : ℝ) =
+            (1632579498774009 / 1250000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1632579498774009 / 1250000000000000 : ℝ) ≤
+            (267017727233028601537 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (257803338318567 / 10000000000000000 : ℝ)) (q :=
+      (1632579498774009 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_151_7 :
+    Uω (aρ 7) (bρ 7) (3385194315371 / 32000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_151_8 :
+    Uω (aρ 8) (bρ 8) (3385194315371 / 32000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_151_9 :
+    Uω (aρ 9) (bρ 9) (3385194315371 / 32000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_151_10 :
+    Uω (aρ 10) (bρ 10) (3385194315371 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_151_11 :
+    Uω (aρ 11) (bρ 11) (3385194315371 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_151_12 :
+    Uω (aρ 12) (bρ 12) (3385194315371 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_151_13 :
+    Uω (aρ 13) (bρ 13) (3385194315371 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_151_14 :
+    Uω (aρ 14) (bρ 14) (3385194315371 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_151_15 :
+    Uω (aρ 15) (bρ 15) (3385194315371 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_151_16 :
+    Uω (aρ 16) (bρ 16) (3385194315371 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_151 :
+    Uρ (3385194315371 / 32000000000000 : ℝ) ≤
+      (-(4691104214003087231947 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (3385194315371 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2886743742383077728029 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_151_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (3385194315371 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(4669159877652684528577 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_151_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (3385194315371 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(23879444915266449414989 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_151_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (3385194315371 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(24873970382882411377119 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_151_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (3385194315371 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(26780580354809959357677 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_151_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (3385194315371 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3198718174266971398463 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_151_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (3385194315371 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_151_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (3385194315371 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_151_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (3385194315371 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_151_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (3385194315371 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_151_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (3385194315371 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_151_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (3385194315371 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_151_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (3385194315371 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_151_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (3385194315371 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_151_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (3385194315371 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_151_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (3385194315371 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_151_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_152_1 :
+    Uω (aρ 1) (bρ 1) (1703808330079 / 16000000000000 : ℝ) ≤
+      (-(4604727520682233243737 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2000442712217593 / 20000000000000000 : ℝ) ≤
+      (-(4604727520682233243737 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2000442712217593 / 20000000000000000 : ℝ) =
+            (2000442712217593 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2000442712217593 / 1250000000000000 : ℝ) ≤
+            (940449921717766756263 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (500029862001609 / 5000000000000000 : ℝ)) (q :=
+      (2000442712217593 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_152_2 :
+    Uω (aρ 2) (bρ 2) (1703808330079 / 16000000000000 : ℝ) ≤
+      (-(5818407786757043322733 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1951052865344707 / 20000000000000000 : ℝ) ≤
+      (-(5818407786757043322733 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1951052865344707 / 20000000000000000 : ℝ) =
+            (1951052865344707 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1951052865344707 / 1250000000000000 : ℝ) ≤
+            (1113064016242956677267 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (243609722410083 / 2500000000000000 : ℝ)) (q :=
+      (1951052865344707 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_152_3 :
+    Uω (aρ 3) (bρ 3) (1703808330079 / 16000000000000 : ℝ) ≤
+      (-(11901544890733755628069 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (925219857255517 / 10000000000000000 : ℝ) ≤
+      (-(11901544890733755628069 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (925219857255517 / 10000000000000000 : ℝ) =
+            (925219857255517 / 625000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (925219857255517 / 625000000000000 : ℝ) ≤
+            (1961398715266244371931 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (921221845156659 / 10000000000000000 : ℝ)) (q :=
+      (925219857255517 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_152_4 :
+    Uω (aρ 4) (bρ 4) (1703808330079 / 16000000000000 : ℝ) ≤
+      (-(309859363700855048777 / 125000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1676749944515773 / 20000000000000000 : ℝ) ≤
+      (-(309859363700855048777 / 125000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1676749944515773 / 20000000000000000 : ℝ) =
+            (1676749944515773 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1676749944515773 / 1250000000000000 : ℝ) ≤
+            (36714226449144951223 / 125000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (412913128113199 / 5000000000000000 : ℝ)) (q :=
+      (1676749944515773 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_152_5 :
+    Uω (aρ 5) (bρ 5) (1703808330079 / 16000000000000 : ℝ) ≤
+      (-(833535311543117281089 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (138877112805401 / 2000000000000000 : ℝ) ≤
+      (-(833535311543117281089 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (138877112805401 / 2000000000000000 : ℝ) =
+            (138877112805401 / 125000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (138877112805401 / 125000000000000 : ℝ) ≤
+            (32898663831882718911 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (131207471819927 / 2000000000000000 : ℝ)) (q :=
+      (138877112805401 / 2000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_152_6 :
+    Uω (aρ 6) (bρ 6) (1703808330079 / 16000000000000 : ℝ) ≤
+      (-(31722978131427743478867 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (419071922331349 / 10000000000000000 : ℝ) ≤
+      (-(31722978131427743478867 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (419071922331349 / 10000000000000000 : ℝ) =
+            (419071922331349 / 312500000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (419071922331349 / 312500000000000 : ℝ) ≤
+            (2934380883572256521133 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (272650450848323 / 10000000000000000 : ℝ)) (q :=
+      (419071922331349 / 10000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_152_7 :
+    Uω (aρ 7) (bρ 7) (1703808330079 / 16000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_152_8 :
+    Uω (aρ 8) (bρ 8) (1703808330079 / 16000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_152_9 :
+    Uω (aρ 9) (bρ 9) (1703808330079 / 16000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_152_10 :
+    Uω (aρ 10) (bρ 10) (1703808330079 / 16000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_152_11 :
+    Uω (aρ 11) (bρ 11) (1703808330079 / 16000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_152_12 :
+    Uω (aρ 12) (bρ 12) (1703808330079 / 16000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_152_13 :
+    Uω (aρ 13) (bρ 13) (1703808330079 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_152_14 :
+    Uω (aρ 14) (bρ 14) (1703808330079 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_152_15 :
+    Uω (aρ 15) (bρ 15) (1703808330079 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_152_16 :
+    Uω (aρ 16) (bρ 16) (1703808330079 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_152 :
+    Uρ (1703808330079 / 16000000000000 : ℝ) ≤
+      (-(23416159595012170387401 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1703808330079 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(4604727520682233243737 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_152_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1703808330079 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(5818407786757043322733 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_152_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1703808330079 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(11901544890733755628069 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_152_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1703808330079 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(309859363700855048777 / 125000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_152_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1703808330079 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(833535311543117281089 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_152_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1703808330079 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(31722978131427743478867 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_152_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1703808330079 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_152_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1703808330079 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_152_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1703808330079 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_152_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1703808330079 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_152_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1703808330079 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_152_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1703808330079 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_152_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1703808330079 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_152_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1703808330079 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_152_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1703808330079 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_152_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1703808330079 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_152_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_153_1 :
+    Uω (aρ 1) (bρ 1) (686007800989 / 6400000000000 : ℝ) ≤
+      (-(286922704474371271327 / 125000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4028917853838367 / 40000000000000000 : ℝ) ≤
+      (-(286922704474371271327 / 125000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (4028917853838367 / 40000000000000000 : ℝ) =
+            (4028917853838367 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4028917853838367 / 2500000000000000 : ℝ) ≤
+            (59650885675628728673 / 125000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1007068955958871 / 10000000000000000 : ℝ)) (q :=
+      (4028917853838367 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_153_2 :
+    Uω (aρ 2) (bρ 2) (686007800989 / 6400000000000 : ℝ) ≤
+      (-(23201981147737440312697 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (786032943851897 / 8000000000000000 : ℝ) ≤
+      (-(23201981147737440312697 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (786032943851897 / 8000000000000000 : ℝ) =
+            (786032943851897 / 500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (786032943851897 / 500000000000000 : ℝ) ≤
+            (4523906064262559687303 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (98146140117943 / 1000000000000000 : ℝ)) (q :=
+      (786032943851897 / 8000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_153_3 :
+    Uω (aρ 3) (bρ 3) (686007800989 / 6400000000000 : ℝ) ≤
+      (-(949092729089347156587 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3729028076321347 / 40000000000000000 : ℝ) ≤
+      (-(949092729089347156587 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (3729028076321347 / 40000000000000000 : ℝ) =
+            (3729028076321347 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3729028076321347 / 2500000000000000 : ℝ) ≤
+            (159942759390652843413 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (928289186060361 / 10000000000000000 : ℝ)) (q :=
+      (3729028076321347 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_153_4 :
+    Uω (aρ 4) (bρ 4) (686007800989 / 6400000000000 : ℝ) ≤
+      (-(6176067409942879644687 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3381950077383383 / 40000000000000000 : ℝ) ≤
+      (-(6176067409942879644687 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (3381950077383383 / 40000000000000000 : ℝ) =
+            (3381950077383383 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3381950077383383 / 2500000000000000 : ℝ) ≤
+            (755404393057120355313 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (833044367656379 / 10000000000000000 : ℝ)) (q :=
+      (3381950077383383 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_153_5 :
+    Uω (aρ 5) (bρ 5) (686007800989 / 6400000000000 : ℝ) ≤
+      (-(5313390771513018446107 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2807190236661777 / 40000000000000000 : ℝ) ≤
+      (-(5313390771513018446107 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2807190236661777 / 40000000000000000 : ℝ) =
+            (2807190236661777 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2807190236661777 / 2500000000000000 : ℝ) ≤
+            (231786670886981553893 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (41490897914411 / 625000000000000 : ℝ)) (q :=
+      (2807190236661777 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_153_6 :
+    Uω (aρ 6) (bρ 6) (686007800989 / 6400000000000 : ℝ) ≤
+      (-(15736257519806563902167 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1718802708717827 / 40000000000000000 : ℝ) ≤
+      (-(15736257519806563902167 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (1718802708717827 / 40000000000000000 : ℝ) =
+            (1718802708717827 / 1250000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1718802708717827 / 1250000000000000 : ℝ) ≤
+            (1592421987693436097833 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (286900977798601 / 10000000000000000 : ℝ)) (q :=
+      (1718802708717827 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_153_7 :
+    Uω (aρ 7) (bρ 7) (686007800989 / 6400000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_153_8 :
+    Uω (aρ 8) (bρ 8) (686007800989 / 6400000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_153_9 :
+    Uω (aρ 9) (bρ 9) (686007800989 / 6400000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_153_10 :
+    Uω (aρ 10) (bρ 10) (686007800989 / 6400000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_153_11 :
+    Uω (aρ 11) (bρ 11) (686007800989 / 6400000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_153_12 :
+    Uω (aρ 12) (bρ 12) (686007800989 / 6400000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_153_13 :
+    Uω (aρ 13) (bρ 13) (686007800989 / 6400000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_153_14 :
+    Uω (aρ 14) (bρ 14) (686007800989 / 6400000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_153_15 :
+    Uω (aρ 15) (bρ 15) (686007800989 / 6400000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_153_16 :
+    Uω (aρ 16) (bρ 16) (686007800989 / 6400000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_153 :
+    Uρ (686007800989 / 6400000000000 : ℝ) ≤
+      (-(23378058521070381777983 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (686007800989 / 6400000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(286922704474371271327 / 125000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_153_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (686007800989 / 6400000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(23201981147737440312697 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_153_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (686007800989 / 6400000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(949092729089347156587 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_153_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (686007800989 / 6400000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(6176067409942879644687 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_153_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (686007800989 / 6400000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(5313390771513018446107 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_153_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (686007800989 / 6400000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(15736257519806563902167 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_153_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (686007800989 / 6400000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_153_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (686007800989 / 6400000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_153_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (686007800989 / 6400000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_153_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (686007800989 / 6400000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_153_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (686007800989 / 6400000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_153_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (686007800989 / 6400000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_153_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (686007800989 / 6400000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_153_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (686007800989 / 6400000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_153_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (686007800989 / 6400000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_153_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (686007800989 / 6400000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_153_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_154_1 :
+    Uω (aρ 1) (bρ 1) (863115337433 / 8000000000000 : ℝ) ≤
+      (-(22884479388127094268989 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1014237555266513 / 10000000000000000 : ℝ) ≤
+      (-(22884479388127094268989 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1014237555266513 / 10000000000000000 : ℝ) =
+            (1014237555266513 / 625000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1014237555266513 / 625000000000000 : ℝ) ≤
+            (4841407823872905731011 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (126759769603347 / 1250000000000000 : ℝ)) (q :=
+      (1014237555266513 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_154_2 :
+    Uω (aρ 2) (bρ 2) (863115337433 / 8000000000000 : ℝ) ≤
+      (-(2891355248039883648733 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1979111633515501 / 20000000000000000 : ℝ) ≤
+      (-(2891355248039883648733 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1979111633515501 / 20000000000000000 : ℝ) =
+            (1979111633515501 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1979111633515501 / 1250000000000000 : ℝ) ≤
+            (574380653460116351267 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (988483692319251 / 10000000000000000 : ℝ)) (q :=
+      (1979111633515501 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_154_3 :
+    Uω (aρ 3) (bρ 3) (863115337433 / 8000000000000 : ℝ) ≤
+      (-(591303033137199082917 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1878587453599769 / 20000000000000000 : ℝ) ≤
+      (-(591303033137199082917 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1878587453599769 / 20000000000000000 : ℝ) =
+            (1878587453599769 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1878587453599769 / 1250000000000000 : ℝ) ≤
+            (101844147162800917083 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (935355618753519 / 10000000000000000 : ℝ)) (q :=
+      (1878587453599769 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_154_4 :
+    Uω (aρ 4) (bρ 4) (863115337433 / 8000000000000 : ℝ) ≤
+      (-(24620518847931819427187 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (852598279294741 / 10000000000000000 : ℝ) ≤
+      (-(24620518847931819427187 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (852598279294741 / 10000000000000000 : ℝ) =
+            (852598279294741 / 625000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (852598279294741 / 625000000000000 : ℝ) ≤
+            (3105368364068180572813 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (105032363101029 / 1250000000000000 : ℝ)) (q :=
+      (852598279294741 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_154_5 :
+    Uω (aρ 5) (bρ 5) (863115337433 / 8000000000000 : ℝ) ≤
+      (-(1653876205589663543733 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (283680246175593 / 4000000000000000 : ℝ) ≤
+      (-(1653876205589663543733 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (283680246175593 / 4000000000000000 : ℝ) =
+            (283680246175593 / 250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (283680246175593 / 250000000000000 : ℝ) ≤
+            (78991745160336456267 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (134330699286343 / 2000000000000000 : ℝ)) (q :=
+      (283680246175593 / 4000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_154_6 :
+    Uω (aρ 6) (bρ 6) (863115337433 / 8000000000000 : ℝ) ≤
+      (-(15616992344826031747609 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (880147126120617 / 20000000000000000 : ℝ) ≤
+      (-(15616992344826031747609 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (880147126120617 / 20000000000000000 : ℝ) =
+            (880147126120617 / 625000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (880147126120617 / 625000000000000 : ℝ) ≤
+            (1711687162673968252391 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (300639766814367 / 10000000000000000 : ℝ)) (q :=
+      (880147126120617 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_154_7 :
+    Uω (aρ 7) (bρ 7) (863115337433 / 8000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_154_8 :
+    Uω (aρ 8) (bρ 8) (863115337433 / 8000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_154_9 :
+    Uω (aρ 9) (bρ 9) (863115337433 / 8000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_154_10 :
+    Uω (aρ 10) (bρ 10) (863115337433 / 8000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_154_11 :
+    Uω (aρ 11) (bρ 11) (863115337433 / 8000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_154_12 :
+    Uω (aρ 12) (bρ 12) (863115337433 / 8000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_154_13 :
+    Uω (aρ 13) (bρ 13) (863115337433 / 8000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_154_14 :
+    Uω (aρ 14) (bρ 14) (863115337433 / 8000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_154_15 :
+    Uω (aρ 15) (bρ 15) (863115337433 / 8000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_154_16 :
+    Uω (aρ 16) (bρ 16) (863115337433 / 8000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_154 :
+    Uρ (863115337433 / 8000000000000 : ℝ) ≤
+      (-(23341071564045878495493 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (863115337433 / 8000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(22884479388127094268989 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_154_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (863115337433 / 8000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(2891355248039883648733 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_154_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (863115337433 / 8000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(591303033137199082917 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_154_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (863115337433 / 8000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(24620518847931819427187 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_154_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (863115337433 / 8000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(1653876205589663543733 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_154_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (863115337433 / 8000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(15616992344826031747609 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_154_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (863115337433 / 8000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_154_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (863115337433 / 8000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_154_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (863115337433 / 8000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_154_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (863115337433 / 8000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_154_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (863115337433 / 8000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_154_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (863115337433 / 8000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_154_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (863115337433 / 8000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_154_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (863115337433 / 8000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_154_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (863115337433 / 8000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_154_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (863115337433 / 8000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_154_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_155_1 :
+    Uω (aρ 1) (bρ 1) (6927345044251 / 64000000000000 : ℝ) ≤
+      (-(22849990415668781578591 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (325677310541247 / 3200000000000000 : ℝ) ≤
+      (-(22849990415668781578591 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (325677310541247 / 3200000000000000 : ℝ) =
+            (325677310541247 / 200000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (325677310541247 / 200000000000000 : ℝ) ≤
+            (4875896796331218421409 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (40703309832143 / 400000000000000 : ℝ)) (q :=
+      (325677310541247 / 3200000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_155_2 :
+    Uω (aρ 2) (bρ 2) (6927345044251 / 64000000000000 : ℝ) ≤
+      (-(23095461694982195708649 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7944504757085371 / 80000000000000000 : ℝ) ≤
+      (-(23095461694982195708649 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (7944504757085371 / 80000000000000000 : ℝ) =
+            (7944504757085371 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7944504757085371 / 5000000000000000 : ℝ) ≤
+            (4630425517017804291351 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (247998689175531 / 2500000000000000 : ℝ)) (q :=
+      (7944504757085371 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_155_3 :
+    Uω (aρ 3) (bρ 3) (6927345044251 / 64000000000000 : ℝ) ≤
+      (-(11807367817896096737809 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1508499061719027 / 16000000000000000 : ℝ) ≤
+      (-(11807367817896096737809 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1508499061719027 / 16000000000000000 : ℝ) =
+            (1508499061719027 / 1000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1508499061719027 / 1000000000000000 : ℝ) ≤
+            (2055575788103903262191 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (187777700185913 / 2000000000000000 : ℝ)) (q :=
+      (1508499061719027 / 16000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_155_4 :
+    Uω (aρ 4) (bρ 4) (6927345044251 / 64000000000000 : ℝ) ≤
+      (-(12289456342288177086599 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6849224027335643 / 80000000000000000 : ℝ) ≤
+      (-(12289456342288177086599 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (6849224027335643 / 80000000000000000 : ℝ) =
+            (6849224027335643 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6849224027335643 / 5000000000000000 : ℝ) ≤
+            (1573487263711822913401 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (210966215419923 / 2500000000000000 : ℝ)) (q :=
+      (6849224027335643 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_155_5 :
+    Uω (aρ 5) (bρ 5) (6927345044251 / 64000000000000 : ℝ) ≤
+      (-(5282001544501092498839 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5703191106221687 / 80000000000000000 : ℝ) ≤
+      (-(5282001544501092498839 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (5703191106221687 / 80000000000000000 : ℝ) =
+            (5703191106221687 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5703191106221687 / 5000000000000000 : ℝ) ≤
+            (263175897898907501161 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (675546550736203 / 10000000000000000 : ℝ)) (q :=
+      (5703191106221687 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_155_6 :
+    Uω (aρ 6) (bρ 6) (6927345044251 / 64000000000000 : ℝ) ≤
+      (-(31118732925594350230399 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (712279725650911 / 16000000000000000 : ℝ) ≤
+      (-(31118732925594350230399 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (712279725650911 / 16000000000000000 : ℝ) =
+            (712279725650911 / 500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (712279725650911 / 500000000000000 : ℝ) ≤
+            (3538626089405649769601 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (15366940319221 / 500000000000000 : ℝ)) (q :=
+      (712279725650911 / 16000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_155_7 :
+    Uω (aρ 7) (bρ 7) (6927345044251 / 64000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_155_8 :
+    Uω (aρ 8) (bρ 8) (6927345044251 / 64000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_155_9 :
+    Uω (aρ 9) (bρ 9) (6927345044251 / 64000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_155_10 :
+    Uω (aρ 10) (bρ 10) (6927345044251 / 64000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_155_11 :
+    Uω (aρ 11) (bρ 11) (6927345044251 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_155_12 :
+    Uω (aρ 12) (bρ 12) (6927345044251 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_155_13 :
+    Uω (aρ 13) (bρ 13) (6927345044251 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_155_14 :
+    Uω (aρ 14) (bρ 14) (6927345044251 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_155_15 :
+    Uω (aρ 15) (bρ 15) (6927345044251 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_155_16 :
+    Uω (aρ 16) (bρ 16) (6927345044251 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_155 :
+    Uρ (6927345044251 / 64000000000000 : ℝ) ≤
+      (-(11661479184507404062013 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (6927345044251 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(22849990415668781578591 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_155_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (6927345044251 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(23095461694982195708649 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_155_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (6927345044251 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(11807367817896096737809 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_155_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (6927345044251 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(12289456342288177086599 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_155_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (6927345044251 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(5282001544501092498839 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_155_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (6927345044251 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(31118732925594350230399 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_155_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (6927345044251 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_155_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (6927345044251 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_155_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (6927345044251 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_155_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (6927345044251 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_155_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (6927345044251 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_155_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (6927345044251 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_155_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (6927345044251 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_155_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (6927345044251 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_155_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (6927345044251 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_155_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (6927345044251 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_155_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_156_1 :
+    Uω (aρ 1) (bρ 1) (3474883694519 / 32000000000000 : ℝ) ≤
+      (-(5703905005074826194883 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4084982527398643 / 40000000000000000 : ℝ) ≤
+      (-(5703905005074826194883 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (4084982527398643 / 40000000000000000 : ℝ) =
+            (4084982527398643 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4084982527398643 / 2500000000000000 : ℝ) ≤
+            (1227566797925173805117 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (510543663623567 / 5000000000000000 : ℝ)) (q :=
+      (4084982527398643 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_156_2 :
+    Uω (aρ 2) (bρ 2) (3474883694519 / 32000000000000 : ℝ) ≤
+      (-(115301032043345084123 / 50000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (797256276666427 / 8000000000000000 : ℝ) ≤
+      (-(115301032043345084123 / 50000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (797256276666427 / 8000000000000000 : ℝ) =
+            (797256276666427 / 500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (797256276666427 / 500000000000000 : ℝ) ≤
+            (23328404016654915877 / 50000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (24887644193097 / 250000000000000 : ℝ)) (q :=
+      (797256276666427 / 8000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_156_3 :
+    Uω (aρ 3) (bρ 3) (3474883694519 / 32000000000000 : ℝ) ≤
+      (-(23577490354072033648947 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3785319962516137 / 40000000000000000 : ℝ) ≤
+      (-(23577490354072033648947 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (3785319962516137 / 40000000000000000 : ℝ) =
+            (3785319962516137 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3785319962516137 / 2500000000000000 : ℝ) ≤
+            (4148396857927966351053 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (942421163665881 / 10000000000000000 : ℝ)) (q :=
+      (3785319962516137 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_156_4 :
+    Uω (aρ 4) (bρ 4) (3474883694519 / 32000000000000 : ℝ) ≤
+      (-(24537483910791336289901 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (687765838197183 / 8000000000000000 : ℝ) ≤
+      (-(24537483910791336289901 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (687765838197183 / 8000000000000000 : ℝ) =
+            (687765838197183 / 500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (687765838197183 / 500000000000000 : ℝ) ≤
+            (3188403301208663710099 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (84746995896577 / 1000000000000000 : ℝ)) (q :=
+      (687765838197183 / 8000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_156_5 :
+    Uω (aρ 5) (bρ 5) (3474883694519 / 32000000000000 : ℝ) ≤
+      (-(26358294867270100871003 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2866380162684629 / 40000000000000000 : ℝ) ≤
+      (-(26358294867270100871003 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2866380162684629 / 40000000000000000 : ℝ) =
+            (2866380162684629 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2866380162684629 / 2500000000000000 : ℝ) ≤
+            (1367592344729899128997 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (679435364150127 / 10000000000000000 : ℝ)) (q :=
+      (2866380162684629 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_156_6 :
+    Uω (aρ 6) (bρ 6) (3474883694519 / 32000000000000 : ℝ) ≤
+      (-(15502973708874956289793 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (360179341265761 / 8000000000000000 : ℝ) ≤
+      (-(15502973708874956289793 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (360179341265761 / 8000000000000000 : ℝ) =
+            (360179341265761 / 250000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (360179341265761 / 250000000000000 : ℝ) ≤
+            (1825705798625043710207 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (62786802222443 / 2000000000000000 : ℝ)) (q :=
+      (360179341265761 / 8000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_156_7 :
+    Uω (aρ 7) (bρ 7) (3474883694519 / 32000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_156_8 :
+    Uω (aρ 8) (bρ 8) (3474883694519 / 32000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_156_9 :
+    Uω (aρ 9) (bρ 9) (3474883694519 / 32000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_156_10 :
+    Uω (aρ 10) (bρ 10) (3474883694519 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_156_11 :
+    Uω (aρ 11) (bρ 11) (3474883694519 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_156_12 :
+    Uω (aρ 12) (bρ 12) (3474883694519 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_156_13 :
+    Uω (aρ 13) (bρ 13) (3474883694519 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_156_14 :
+    Uω (aρ 14) (bρ 14) (3474883694519 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_156_15 :
+    Uω (aρ 15) (bρ 15) (3474883694519 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_156_16 :
+    Uω (aρ 16) (bρ 16) (3474883694519 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_156 :
+    Uρ (3474883694519 / 32000000000000 : ℝ) ≤
+      (-(23305081601715584225839 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (3474883694519 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(5703905005074826194883 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_156_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (3474883694519 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) * (-(115301032043345084123 / 50000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_156_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (3474883694519 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(23577490354072033648947 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_156_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (3474883694519 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(24537483910791336289901 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_156_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (3474883694519 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(26358294867270100871003 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_156_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (3474883694519 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(15502973708874956289793 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_156_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (3474883694519 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_156_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (3474883694519 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_156_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (3474883694519 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_156_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (3474883694519 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_156_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (3474883694519 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_156_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (3474883694519 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_156_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (3474883694519 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_156_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (3474883694519 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_156_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (3474883694519 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_156_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (3474883694519 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_156_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_157_1 :
+    Uω (aρ 1) (bρ 1) (278887589353 / 2560000000000 : ℝ) ≤
+      (-(22781367389197025760763 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8197997316239641 / 80000000000000000 : ℝ) ≤
+      (-(22781367389197025760763 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (8197997316239641 / 80000000000000000 : ℝ) =
+            (8197997316239641 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8197997316239641 / 5000000000000000 : ℝ) ≤
+            (4944519822802974239237 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (512295950617377 / 5000000000000000 : ℝ)) (q :=
+      (8197997316239641 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_157_2 :
+    Uω (aρ 2) (bρ 2) (278887589353 / 2560000000000 : ℝ) ≤
+      (-(11512537621657838256657 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8000620565049141 / 80000000000000000 : ℝ) ≤
+      (-(11512537621657838256657 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (8000620565049141 / 80000000000000000 : ℝ) =
+            (8000620565049141 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8000620565049141 / 5000000000000000 : ℝ) ≤
+            (2350405984342161743343 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (999016725947129 / 10000000000000000 : ℝ)) (q :=
+      (8000620565049141 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_157_3 :
+    Uω (aρ 3) (bρ 3) (278887589353 / 2560000000000 : ℝ) ≤
+      (-(2942548052630913630127 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1519756734708893 / 16000000000000000 : ℝ) ≤
+      (-(2942548052630913630127 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1519756734708893 / 16000000000000000 : ℝ) =
+            (1519756734708893 / 1000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1519756734708893 / 1000000000000000 : ℝ) ≤
+            (523187848869086369873 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5912210058881 / 62500000000000 : ℝ)) (q :=
+      (1519756734708893 / 16000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_157_4 :
+    Uω (aρ 4) (bρ 4) (278887589353 / 2560000000000 : ℝ) ≤
+      (-(3062028872361546944299 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6906089341960289 / 80000000000000000 : ℝ) ≤
+      (-(3062028872361546944299 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (6906089341960289 / 80000000000000000 : ℝ) =
+            (6906089341960289 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6906089341960289 / 5000000000000000 : ℝ) ≤
+            (403707029138453055701 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (212768551897479 / 2500000000000000 : ℝ)) (q :=
+      (6906089341960289 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_157_5 :
+    Uω (aρ 5) (bρ 5) (278887589353 / 2560000000000 : ℝ) ≤
+      (-(26306876994912978474327 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5762312870576409 / 80000000000000000 : ℝ) ≤
+      (-(26306876994912978474327 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (5762312870576409 / 80000000000000000 : ℝ) =
+            (5762312870576409 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5762312870576409 / 5000000000000000 : ℝ) ≤
+            (1419010217087021525673 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (341660004539473 / 5000000000000000 : ℝ)) (q :=
+      (5762312870576409 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_157_6 :
+    Uω (aρ 6) (bρ 6) (278887589353 / 2560000000000 : ℝ) ≤
+      (-(15447744896504633414501 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3641798503794537 / 80000000000000000 : ℝ) ≤
+      (-(15447744896504633414501 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (3641798503794537 / 80000000000000000 : ℝ) =
+            (3641798503794537 / 2500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3641798503794537 / 2500000000000000 : ℝ) ≤
+            (1880934610995366585499 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (160215896261739 / 5000000000000000 : ℝ)) (q :=
+      (3641798503794537 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_157_7 :
+    Uω (aρ 7) (bρ 7) (278887589353 / 2560000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_157_8 :
+    Uω (aρ 8) (bρ 8) (278887589353 / 2560000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_157_9 :
+    Uω (aρ 9) (bρ 9) (278887589353 / 2560000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_157_10 :
+    Uω (aρ 10) (bρ 10) (278887589353 / 2560000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_157_11 :
+    Uω (aρ 11) (bρ 11) (278887589353 / 2560000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_157_12 :
+    Uω (aρ 12) (bρ 12) (278887589353 / 2560000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_157_13 :
+    Uω (aρ 13) (bρ 13) (278887589353 / 2560000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_157_14 :
+    Uω (aρ 14) (bρ 14) (278887589353 / 2560000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_157_15 :
+    Uω (aρ 15) (bρ 15) (278887589353 / 2560000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_157_16 :
+    Uω (aρ 16) (bρ 16) (278887589353 / 2560000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_157 :
+    Uρ (278887589353 / 2560000000000 : ℝ) ≤
+      (-(931497196827963978981 / 400000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (278887589353 / 2560000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(22781367389197025760763 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_157_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (278887589353 / 2560000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(11512537621657838256657 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_157_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (278887589353 / 2560000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(2942548052630913630127 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_157_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (278887589353 / 2560000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3062028872361546944299 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_157_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (278887589353 / 2560000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(26306876994912978474327 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_157_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (278887589353 / 2560000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(15447744896504633414501 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_157_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (278887589353 / 2560000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_157_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (278887589353 / 2560000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_157_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (278887589353 / 2560000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_157_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (278887589353 / 2560000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_157_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (278887589353 / 2560000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_157_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (278887589353 / 2560000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_157_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (278887589353 / 2560000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_157_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (278887589353 / 2560000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_157_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (278887589353 / 2560000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_157_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (278887589353 / 2560000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_157_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_158_1 :
+    Uω (aρ 1) (bρ 1) (1748653019653 / 16000000000000 : ℝ) ≤
+      (-(11373615858935949105329 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2056507387040809 / 20000000000000000 : ℝ) ≤
+      (-(11373615858935949105329 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2056507387040809 / 20000000000000000 : ℝ) =
+            (2056507387040809 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2056507387040809 / 1250000000000000 : ℝ) ≤
+            (2489327747064050894671 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (257024116960671 / 2500000000000000 : ℝ)) (q :=
+      (2056507387040809 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_158_2 :
+    Uω (aρ 2) (bρ 2) (1748653019653 / 16000000000000 : ℝ) ≤
+      (-(22990067326181240245647 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (250896192326839 / 2500000000000000 : ℝ) ≤
+      (-(22990067326181240245647 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (250896192326839 / 2500000000000000 : ℝ) =
+            (250896192326839 / 156250000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (250896192326839 / 156250000000000 : ℝ) ≤
+            (4735819885818759754353 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1002527631926587 / 10000000000000000 : ℝ)) (q :=
+      (250896192326839 / 2500000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_158_3 :
+    Uω (aρ 3) (bρ 3) (1748653019653 / 16000000000000 : ℝ) ≤
+      (-(1175170839473725169687 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1906731640954683 / 20000000000000000 : ℝ) ≤
+      (-(1175170839473725169687 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1906731640954683 / 20000000000000000 : ℝ) =
+            (1906731640954683 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1906731640954683 / 1250000000000000 : ℝ) ≤
+            (211123521126274830313 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (474742920308279 / 5000000000000000 : ℝ)) (q :=
+      (1906731640954683 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_158_4 :
+    Uω (aρ 4) (bρ 4) (1748653019653 / 16000000000000 : ℝ) ≤
+      (-(12227576180934471999219 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (346725847512379 / 4000000000000000 : ℝ) ≤
+      (-(12227576180934471999219 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (346725847512379 / 4000000000000000 : ℝ) =
+            (346725847512379 / 250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (346725847512379 / 250000000000000 : ℝ) ≤
+            (1635367425065528000781 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (85467761828877 / 1000000000000000 : ℝ)) (q :=
+      (346725847512379 / 4000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_158_5 :
+    Uω (aρ 5) (bρ 5) (1748653019653 / 16000000000000 : ℝ) ≤
+      (-(13127875225417848178601 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1447962256152311 / 20000000000000000 : ℝ) ≤
+      (-(13127875225417848178601 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1447962256152311 / 20000000000000000 : ℝ) =
+            (1447962256152311 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1447962256152311 / 1250000000000000 : ℝ) ≤
+            (735068380582151821399 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (343600278107093 / 5000000000000000 : ℝ)) (q :=
+      (1447962256152311 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_158_6 :
+    Uω (aρ 6) (bρ 6) (1748653019653 / 16000000000000 : ℝ) ≤
+      (-(30787234291560992907967 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (230089821502913 / 5000000000000000 : ℝ) ≤
+      (-(30787234291560992907967 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (230089821502913 / 5000000000000000 : ℝ) =
+            (230089821502913 / 156250000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (230089821502913 / 156250000000000 : ℝ) ≤
+            (3870124723439007092033 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (326837961213527 / 10000000000000000 : ℝ)) (q :=
+      (230089821502913 / 5000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_158_7 :
+    Uω (aρ 7) (bρ 7) (1748653019653 / 16000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_158_8 :
+    Uω (aρ 8) (bρ 8) (1748653019653 / 16000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_158_9 :
+    Uω (aρ 9) (bρ 9) (1748653019653 / 16000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_158_10 :
+    Uω (aρ 10) (bρ 10) (1748653019653 / 16000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_158_11 :
+    Uω (aρ 11) (bρ 11) (1748653019653 / 16000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_158_12 :
+    Uω (aρ 12) (bρ 12) (1748653019653 / 16000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_158_13 :
+    Uω (aρ 13) (bρ 13) (1748653019653 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_158_14 :
+    Uω (aρ 14) (bρ 14) (1748653019653 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_158_15 :
+    Uω (aρ 15) (bρ 15) (1748653019653 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_158_16 :
+    Uω (aρ 16) (bρ 16) (1748653019653 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_158 :
+    Uρ (1748653019653 / 16000000000000 : ℝ) ≤
+      (-(23269992986572398537807 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1748653019653 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(11373615858935949105329 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_158_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1748653019653 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(22990067326181240245647 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_158_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1748653019653 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1175170839473725169687 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_158_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1748653019653 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(12227576180934471999219 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_158_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1748653019653 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(13127875225417848178601 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_158_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1748653019653 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(30787234291560992907967 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_158_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1748653019653 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_158_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1748653019653 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_158_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1748653019653 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_158_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1748653019653 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_158_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1748653019653 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_158_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1748653019653 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_158_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1748653019653 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_158_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1748653019653 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_158_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1748653019653 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_158_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1748653019653 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_158_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_159_1 :
+    Uω (aρ 1) (bρ 1) (7017034423399 / 64000000000000 : ℝ) ≤
+      (-(5678303052512994256621 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8254061750868911 / 80000000000000000 : ℝ) ≤
+      (-(5678303052512994256621 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (8254061750868911 / 80000000000000000 : ℝ) =
+            (8254061750868911 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8254061750868911 / 5000000000000000 : ℝ) ≤
+            (1253168750487005743379 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (515800513573067 / 5000000000000000 : ℝ)) (q :=
+      (8254061750868911 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_159_2 :
+    Uω (aρ 2) (bρ 2) (7017034423399 / 64000000000000 : ℝ) ≤
+      (-(22955181793716472346763 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1611347107416251 / 16000000000000000 : ℝ) ≤
+      (-(22955181793716472346763 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1611347107416251 / 16000000000000000 : ℝ) =
+            (1611347107416251 / 1000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1611347107416251 / 1000000000000000 : ℝ) ≤
+            (4770705418283527653237 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (50301924310461 / 500000000000000 : ℝ)) (q :=
+      (1611347107416251 / 16000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_159_3 :
+    Uω (aρ 3) (bρ 3) (7017034423399 / 64000000000000 : ℝ) ≤
+      (-(11733293211981849575587 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7655068605397903 / 80000000000000000 : ℝ) ≤
+      (-(11733293211981849575587 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (7655068605397903 / 80000000000000000 : ℝ) =
+            (7655068605397903 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7655068605397903 / 5000000000000000 : ℝ) ≤
+            (2129650394018150424413 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (476508929819191 / 5000000000000000 : ℝ)) (q :=
+      (7655068605397903 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_159_4 :
+    Uω (aρ 4) (bρ 4) (7017034423399 / 64000000000000 : ℝ) ≤
+      (-(6103561638252420282721 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6962941249049127 / 80000000000000000 : ℝ) ≤
+      (-(6103561638252420282721 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (6962941249049127 / 80000000000000000 : ℝ) =
+            (6962941249049127 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6962941249049127 / 5000000000000000 : ℝ) ≤
+            (827910164747579717279 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (214570050404047 / 2500000000000000 : ℝ)) (q :=
+      (6962941249049127 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_159_5 :
+    Uω (aρ 5) (bρ 5) (7017034423399 / 64000000000000 : ℝ) ≤
+      (-(6551227913151442296359 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1164273812717547 / 16000000000000000 : ℝ) ≤
+      (-(6551227913151442296359 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1164273812717547 / 16000000000000000 : ℝ) =
+            (1164273812717547 / 1000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1164273812717547 / 1000000000000000 : ℝ) ≤
+            (380243889848557703641 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (8638463432323 / 125000000000000 : ℝ)) (q :=
+      (1164273812717547 / 16000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_159_6 :
+    Uω (aρ 6) (bρ 6) (7017034423399 / 64000000000000 : ℝ) ≤
+      (-(6136213241512007746819 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3720730472628851 / 80000000000000000 : ℝ) ≤
+      (-(6136213241512007746819 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (3720730472628851 / 80000000000000000 : ℝ) =
+            (3720730472628851 / 2500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3720730472628851 / 2500000000000000 : ℝ) ≤
+            (795258561487992253181 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (333157801986119 / 10000000000000000 : ℝ)) (q :=
+      (3720730472628851 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_159_7 :
+    Uω (aρ 7) (bρ 7) (7017034423399 / 64000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_159_8 :
+    Uω (aρ 8) (bρ 8) (7017034423399 / 64000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_159_9 :
+    Uω (aρ 9) (bρ 9) (7017034423399 / 64000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_159_10 :
+    Uω (aρ 10) (bρ 10) (7017034423399 / 64000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_159_11 :
+    Uω (aρ 11) (bρ 11) (7017034423399 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_159_12 :
+    Uω (aρ 12) (bρ 12) (7017034423399 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_159_13 :
+    Uω (aρ 13) (bρ 13) (7017034423399 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_159_14 :
+    Uω (aρ 14) (bρ 14) (7017034423399 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_159_15 :
+    Uω (aρ 15) (bρ 15) (7017034423399 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_159_16 :
+    Uω (aρ 16) (bρ 16) (7017034423399 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_159 :
+    Uρ (7017034423399 / 64000000000000 : ℝ) ≤
+      (-(11626380669416506316129 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (7017034423399 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(5678303052512994256621 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_159_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (7017034423399 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(22955181793716472346763 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_159_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (7017034423399 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(11733293211981849575587 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_159_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (7017034423399 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(6103561638252420282721 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_159_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (7017034423399 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(6551227913151442296359 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_159_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (7017034423399 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(6136213241512007746819 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_159_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (7017034423399 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_159_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (7017034423399 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_159_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (7017034423399 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_159_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (7017034423399 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_159_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (7017034423399 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_159_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (7017034423399 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_159_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (7017034423399 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_159_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (7017034423399 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_159_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (7017034423399 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_159_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (7017034423399 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_159_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/U13.lean b/LeanPool/Zeta5Irrational/Table/U13.lean
new file mode 100644
index 0000000000..b18869a0ad
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/U13.lean
@@ -0,0 +1,3288 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-! # Certified arcsine potential bounds (U13) -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma U_160_1 :
+    Uω (aρ 1) (bρ 1) (3519728384093 / 32000000000000 : ℝ) ≤
+      (-(22679308077571814145547 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4141046962326717 / 40000000000000000 : ℝ) ≤
+      (-(22679308077571814145547 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (4141046962326717 / 40000000000000000 : ℝ) =
+            (4141046962326717 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4141046962326717 / 2500000000000000 : ℝ) ≤
+            (5046579134428185854453 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (32347049350603 / 312500000000000 : ℝ)) (q :=
+      (4141046962326717 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_160_2 :
+    Uω (aρ 2) (bρ 2) (3519728384093 / 32000000000000 : ℝ) ≤
+      (-(22920417791434948469573 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4042396357536883 / 40000000000000000 : ℝ) ≤
+      (-(22920417791434948469573 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (4042396357536883 / 40000000000000000 : ℝ) =
+            (4042396357536883 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4042396357536883 / 2500000000000000 : ℝ) ≤
+            (4805469420565051530427 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1009549289334379 / 10000000000000000 : ℝ)) (q :=
+      (4042396357536883 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_160_3 :
+    Uω (aρ 3) (bρ 3) (3519728384093 / 32000000000000 : ℝ) ≤
+      (-(23429892300799022833847 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (768320980768329 / 8000000000000000 : ℝ) ≤
+      (-(23429892300799022833847 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (768320980768329 / 8000000000000000 : ℝ) =
+            (768320980768329 / 500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (768320980768329 / 500000000000000 : ℝ) ≤
+            (4295994911200977166153 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (23913741720919 / 250000000000000 : ℝ)) (q :=
+      (768320980768329 / 8000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_160_4 :
+    Uω (aρ 4) (bρ 4) (3519728384093 / 32000000000000 : ℝ) ≤
+      (-(24373512065536270100317 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (699136227986493 / 8000000000000000 : ℝ) ≤
+      (-(24373512065536270100317 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (699136227986493 / 8000000000000000 : ℝ) =
+            (699136227986493 / 500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (699136227986493 / 500000000000000 : ℝ) ≤
+            (3352375146463729899683 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (86188196794717 / 1000000000000000 : ℝ)) (q :=
+      (699136227986493 / 8000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_160_5 :
+    Uω (aρ 5) (bρ 5) (3519728384093 / 32000000000000 : ℝ) ≤
+      (-(26154357087978048520417 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2925436628593717 / 40000000000000000 : ℝ) ≤
+      (-(26154357087978048520417 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2925436628593717 / 40000000000000000 : ℝ) =
+            (2925436628593717 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2925436628593717 / 2500000000000000 : ℝ) ≤
+            (1571530124021951479583 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (173737407903199 / 2500000000000000 : ℝ)) (q :=
+      (2925436628593717 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_160_6 :
+    Uω (aρ 6) (bρ 6) (3519728384093 / 32000000000000 : ℝ) ≤
+      (-(30576880569535722827721 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1879848889828579 / 40000000000000000 : ℝ) ≤
+      (-(30576880569535722827721 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (1879848889828579 / 40000000000000000 : ℝ) =
+            (1879848889828579 / 1250000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1879848889828579 / 1250000000000000 : ℝ) ≤
+            (4080478445464277172279 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (339396137370227 / 10000000000000000 : ℝ)) (q :=
+      (1879848889828579 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_160_7 :
+    Uω (aρ 7) (bρ 7) (3519728384093 / 32000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_160_8 :
+    Uω (aρ 8) (bρ 8) (3519728384093 / 32000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_160_9 :
+    Uω (aρ 9) (bρ 9) (3519728384093 / 32000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_160_10 :
+    Uω (aρ 10) (bρ 10) (3519728384093 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_160_11 :
+    Uω (aρ 11) (bρ 11) (3519728384093 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_160_12 :
+    Uω (aρ 12) (bρ 12) (3519728384093 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_160_13 :
+    Uω (aρ 13) (bρ 13) (3519728384093 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_160_14 :
+    Uω (aρ 14) (bρ 14) (3519728384093 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_160_15 :
+    Uω (aρ 15) (bρ 15) (3519728384093 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_160_16 :
+    Uω (aρ 16) (bρ 16) (3519728384093 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_160 :
+    Uρ (3519728384093 / 32000000000000 : ℝ) ≤
+      (-(23235726291618061649999 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (3519728384093 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(22679308077571814145547 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_160_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (3519728384093 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(22920417791434948469573 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_160_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (3519728384093 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(23429892300799022833847 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_160_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (3519728384093 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(24373512065536270100317 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_160_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (3519728384093 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(26154357087978048520417 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_160_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (3519728384093 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(30576880569535722827721 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_160_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (3519728384093 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_160_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (3519728384093 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_160_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (3519728384093 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_160_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (3519728384093 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_160_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (3519728384093 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_160_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (3519728384093 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_160_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (3519728384093 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_160_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (3519728384093 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_160_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (3519728384093 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_160_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (3519728384093 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_160_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_161_1 :
+    Uω (aρ 1) (bρ 1) (7061879112973 / 64000000000000 : ℝ) ≤
+      (-(22645518540262755321593 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8310126069809573 / 80000000000000000 : ℝ) ≤
+      (-(22645518540262755321593 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (8310126069809573 / 80000000000000000 : ℝ) =
+            (8310126069809573 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8310126069809573 / 5000000000000000 : ℝ) ≤
+            (5080368671737244678407 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (519305062067681 / 5000000000000000 : ℝ)) (q :=
+      (8310126069809573 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_161_2 :
+    Uω (aρ 2) (bρ 2) (7061879112973 / 64000000000000 : ℝ) ≤
+      (-(22885774473786439499311 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8112849690563841 / 80000000000000000 : ℝ) ≤
+      (-(22885774473786439499311 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (8112849690563841 / 80000000000000000 : ℝ) =
+            (8112849690563841 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8112849690563841 / 5000000000000000 : ℝ) ≤
+            (4840112738213560500689 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1013060041833929 / 10000000000000000 : ℝ)) (q :=
+      (8112849690563841 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_161_3 :
+    Uω (aρ 3) (bρ 3) (7061879112973 / 64000000000000 : ℝ) ≤
+      (-(4678666681552500330723 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7711350179937309 / 80000000000000000 : ℝ) ≤
+      (-(4678666681552500330723 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (7711350179937309 / 80000000000000000 : ℝ) =
+            (7711350179937309 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7711350179937309 / 5000000000000000 : ℝ) ≤
+            (866510760847499669277 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (15001269851989 / 156250000000000 : ℝ)) (q :=
+      (7711350179937309 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_161_4 :
+    Uω (aρ 4) (bρ 4) (7061879112973 / 64000000000000 : ℝ) ≤
+      (-(12166473716122874792837 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (280791203339793 / 3200000000000000 : ℝ) ≤
+      (-(12166473716122874792837 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (280791203339793 / 3200000000000000 : ℝ) =
+            (280791203339793 / 200000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (280791203339793 / 200000000000000 : ℝ) ≤
+            (1696469889877125207163 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (34619317099267 / 400000000000000 : ℝ)) (q :=
+      (280791203339793 / 3200000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_161_5 :
+    Uω (aρ 5) (bρ 5) (7061879112973 / 64000000000000 : ℝ) ≤
+      (-(26104083312985725933909 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5880361868833621 / 80000000000000000 : ℝ) ≤
+      (-(26104083312985725933909 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (5880361868833621 / 80000000000000000 : ℝ) =
+            (5880361868833621 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5880361868833621 / 5000000000000000 : ℝ) ≤
+            (1621803899014274066091 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (349409146575687 / 5000000000000000 : ℝ)) (q :=
+      (5880361868833621 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_161_6 :
+    Uω (aρ 6) (bρ 6) (7061879112973 / 64000000000000 : ℝ) ≤
+      (-(30474581016948763726831 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3798356722211609 / 80000000000000000 : ℝ) ≤
+      (-(30474581016948763726831 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (3798356722211609 / 80000000000000000 : ℝ) =
+            (3798356722211609 / 2500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3798356722211609 / 2500000000000000 : ℝ) ≤
+            (4182777998051236273169 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (345557381635871 / 10000000000000000 : ℝ)) (q :=
+      (3798356722211609 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_161_7 :
+    Uω (aρ 7) (bρ 7) (7061879112973 / 64000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_161_8 :
+    Uω (aρ 8) (bρ 8) (7061879112973 / 64000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_161_9 :
+    Uω (aρ 9) (bρ 9) (7061879112973 / 64000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_161_10 :
+    Uω (aρ 10) (bρ 10) (7061879112973 / 64000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_161_11 :
+    Uω (aρ 11) (bρ 11) (7061879112973 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_161_12 :
+    Uω (aρ 12) (bρ 12) (7061879112973 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_161_13 :
+    Uω (aρ 13) (bρ 13) (7061879112973 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_161_14 :
+    Uω (aρ 14) (bρ 14) (7061879112973 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_161_15 :
+    Uω (aρ 15) (bρ 15) (7061879112973 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_161_16 :
+    Uω (aρ 16) (bρ 16) (7061879112973 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_161 :
+    Uρ (7061879112973 / 64000000000000 : ℝ) ≤
+      (-(725589995154055185789 / 312500000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (7061879112973 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(22645518540262755321593 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_161_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (7061879112973 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(22885774473786439499311 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_161_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (7061879112973 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(4678666681552500330723 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_161_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (7061879112973 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(12166473716122874792837 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_161_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (7061879112973 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(26104083312985725933909 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_161_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (7061879112973 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(30474581016948763726831 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_161_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (7061879112973 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_161_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (7061879112973 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_161_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (7061879112973 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_161_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (7061879112973 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_161_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (7061879112973 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_161_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (7061879112973 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_161_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (7061879112973 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_161_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (7061879112973 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_161_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (7061879112973 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_161_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (7061879112973 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_161_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_162_1 :
+    Uω (aρ 1) (bρ 1) (44276884111 / 400000000000 : ℝ) ≤
+      (-(22611842825845109650421 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2084539546656537 / 20000000000000000 : ℝ) ≤
+      (-(22611842825845109650421 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2084539546656537 / 20000000000000000 : ℝ) =
+            (2084539546656537 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2084539546656537 / 1250000000000000 : ℝ) ≤
+            (5114044386154890349579 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1042114661966537 / 10000000000000000 : ℝ)) (q :=
+      (2084539546656537 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_162_2 :
+    Uω (aρ 2) (bρ 2) (44276884111 / 400000000000 : ℝ) ≤
+      (-(5712812751008120059167 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (101761330820619 / 1000000000000000 : ℝ) ≤
+      (-(5712812751008120059167 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (101761330820619 / 1000000000000000 : ℝ) =
+            (101761330820619 / 62500000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (101761330820619 / 62500000000000 : ℝ) ≤
+            (1218659051991879940833 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (50828537211619 / 500000000000000 : ℝ)) (q :=
+      (101761330820619 / 1000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_162_3 :
+    Uω (aρ 3) (bρ 3) (44276884111 / 400000000000 : ℝ) ≤
+      (-(364951699124394211791 / 156250000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (386974486564303 / 4000000000000000 : ℝ) ≤
+      (-(364951699124394211791 / 156250000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (386974486564303 / 4000000000000000 : ℝ) =
+            (386974486564303 / 250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (386974486564303 / 250000000000000 : ℝ) ≤
+            (68265288563105788209 / 156250000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (192722533398303 / 2000000000000000 : ℝ)) (q :=
+      (386974486564303 / 4000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_162_4 :
+    Uω (aρ 4) (bρ 4) (44276884111 / 400000000000 : ℝ) ≤
+      (-(12146275602580457198217 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (881024337506673 / 10000000000000000 : ℝ) ≤
+      (-(12146275602580457198217 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (881024337506673 / 10000000000000000 : ℝ) =
+            (881024337506673 / 625000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (881024337506673 / 625000000000000 : ℝ) ≤
+            (1716668003419542801783 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (434541545124173 / 5000000000000000 : ℝ)) (q :=
+      (881024337506673 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_162_5 :
+    Uω (aρ 5) (bρ 5) (44276884111 / 400000000000 : ℝ) ≤
+      (-(26054086950097842529433 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (147745878897207 / 2000000000000000 : ℝ) ≤
+      (-(26054086950097842529433 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (147745878897207 / 2000000000000000 : ℝ) =
+            (147745878897207 / 125000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (147745878897207 / 125000000000000 : ℝ) ≤
+            (1671800261902157470567 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (70268312354207 / 1000000000000000 : ℝ)) (q :=
+      (147745878897207 / 2000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_162_6 :
+    Uω (aρ 6) (bρ 6) (44276884111 / 400000000000 : ℝ) ≤
+      (-(7593519709594403117721 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (959180877277593 / 20000000000000000 : ℝ) ≤
+      (-(7593519709594403117721 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (959180877277593 / 20000000000000000 : ℝ) =
+            (959180877277593 / 625000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (959180877277593 / 625000000000000 : ℝ) ≤
+            (1070820044155596882279 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (351645586987593 / 10000000000000000 : ℝ)) (q :=
+      (959180877277593 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_162_7 :
+    Uω (aρ 7) (bρ 7) (44276884111 / 400000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_162_8 :
+    Uω (aρ 8) (bρ 8) (44276884111 / 400000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_162_9 :
+    Uω (aρ 9) (bρ 9) (44276884111 / 400000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_162_10 :
+    Uω (aρ 10) (bρ 10) (44276884111 / 400000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_162_11 :
+    Uω (aρ 11) (bρ 11) (44276884111 / 400000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_162_12 :
+    Uω (aρ 12) (bρ 12) (44276884111 / 400000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_162_13 :
+    Uω (aρ 13) (bρ 13) (44276884111 / 400000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_162_14 :
+    Uω (aρ 14) (bρ 14) (44276884111 / 400000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_162_15 :
+    Uω (aρ 15) (bρ 15) (44276884111 / 400000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_162_16 :
+    Uω (aρ 16) (bρ 16) (44276884111 / 400000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_162 :
+    Uρ (44276884111 / 400000000000 : ℝ) ≤
+      (-(1450138413038403161583 / 625000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (44276884111 / 400000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(22611842825845109650421 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_162_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (44276884111 / 400000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(5712812751008120059167 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_162_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (44276884111 / 400000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(364951699124394211791 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_162_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (44276884111 / 400000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(12146275602580457198217 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_162_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (44276884111 / 400000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(26054086950097842529433 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_162_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (44276884111 / 400000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(7593519709594403117721 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_162_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (44276884111 / 400000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_162_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (44276884111 / 400000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_162_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (44276884111 / 400000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_162_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (44276884111 / 400000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_162_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (44276884111 / 400000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_162_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (44276884111 / 400000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_162_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (44276884111 / 400000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_162_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (44276884111 / 400000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_162_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (44276884111 / 400000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_162_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (44276884111 / 400000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_162_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_163_1 :
+    Uω (aρ 1) (bρ 1) (7106723802547 / 64000000000000 : ℝ) ≤
+      (-(2822285021227761372561 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1673238055077623 / 16000000000000000 : ℝ) ≤
+      (-(2822285021227761372561 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1673238055077623 / 16000000000000000 : ℝ) =
+            (1673238055077623 / 1000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1673238055077623 / 1000000000000000 : ℝ) ≤
+            (643450880272238627439 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (52280959639203 / 500000000000000 : ℝ)) (q :=
+      (1673238055077623 / 16000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_163_2 :
+    Uω (aρ 2) (bρ 2) (7106723802547 / 64000000000000 : ℝ) ≤
+      (-(22816846554124106986407 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8168963042399931 / 80000000000000000 : ℝ) ≤
+      (-(22816846554124106986407 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (8168963042399931 / 80000000000000000 : ℝ) =
+            (8168963042399931 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8168963042399931 / 5000000000000000 : ℝ) ≤
+            (4909040657875893013593 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (510040698523507 / 5000000000000000 : ℝ)) (q :=
+      (8168963042399931 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_163_3 :
+    Uω (aρ 3) (bρ 3) (7106723802547 / 64000000000000 : ℝ) ≤
+      (-(23320617319657831171039 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7767628470721819 / 80000000000000000 : ℝ) ≤
+      (-(23320617319657831171039 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (7767628470721819 / 80000000000000000 : ℝ) =
+            (7767628470721819 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7767628470721819 / 5000000000000000 : ℝ) ≤
+            (4405269892342168828961 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (483571930238743 / 5000000000000000 : ℝ)) (q :=
+      (7767628470721819 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_163_4 :
+    Uω (aρ 4) (bρ 4) (7106723802547 / 64000000000000 : ℝ) ≤
+      (-(24252321955188649295791 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7076606168984419 / 80000000000000000 : ℝ) ≤
+      (-(24252321955188649295791 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (7076606168984419 / 80000000000000000 : ℝ) =
+            (7076606168984419 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7076606168984419 / 5000000000000000 : ℝ) ≤
+            (3473565256811350704209 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (109085308263517 / 1250000000000000 : ℝ)) (q :=
+      (7076606168984419 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_163_5 :
+    Uω (aρ 5) (bρ 5) (7106723802547 / 64000000000000 : ℝ) ≤
+      (-(26004364686440584843293 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5939293369830159 / 80000000000000000 : ℝ) ≤
+      (-(26004364686440584843293 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (5939293369830159 / 80000000000000000 : ℝ) =
+            (5939293369830159 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5939293369830159 / 5000000000000000 : ℝ) ≤
+            (1721522525559415156707 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (706544185654571 / 10000000000000000 : ℝ)) (q :=
+      (5939293369830159 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_163_6 :
+    Uω (aρ 6) (bρ 6) (7106723802547 / 64000000000000 : ℝ) ≤
+      (-(15137646071887000568089 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3874813059751239 / 80000000000000000 : ℝ) ≤
+      (-(15137646071887000568089 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (3874813059751239 / 80000000000000000 : ℝ) =
+            (3874813059751239 / 2500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3874813059751239 / 2500000000000000 : ℝ) ≤
+            (2191033435612999431911 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (357664483274841 / 10000000000000000 : ℝ)) (q :=
+      (3874813059751239 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_163_7 :
+    Uω (aρ 7) (bρ 7) (7106723802547 / 64000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_163_8 :
+    Uω (aρ 8) (bρ 8) (7106723802547 / 64000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_163_9 :
+    Uω (aρ 9) (bρ 9) (7106723802547 / 64000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_163_10 :
+    Uω (aρ 10) (bρ 10) (7106723802547 / 64000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_163_11 :
+    Uω (aρ 11) (bρ 11) (7106723802547 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_163_12 :
+    Uω (aρ 12) (bρ 12) (7106723802547 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_163_13 :
+    Uω (aρ 13) (bρ 13) (7106723802547 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_163_14 :
+    Uω (aρ 14) (bρ 14) (7106723802547 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_163_15 :
+    Uω (aρ 15) (bρ 15) (7106723802547 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_163_16 :
+    Uω (aρ 16) (bρ 16) (7106723802547 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_163 :
+    Uρ (7106723802547 / 64000000000000 : ℝ) ≤
+      (-(23185723736921093732363 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (7106723802547 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2822285021227761372561 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_163_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (7106723802547 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(22816846554124106986407 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_163_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (7106723802547 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(23320617319657831171039 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_163_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (7106723802547 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(24252321955188649295791 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_163_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (7106723802547 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(26004364686440584843293 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_163_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (7106723802547 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(15137646071887000568089 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_163_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (7106723802547 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_163_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (7106723802547 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_163_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (7106723802547 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_163_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (7106723802547 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_163_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (7106723802547 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_163_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (7106723802547 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_163_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (7106723802547 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_163_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (7106723802547 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_163_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (7106723802547 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_163_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (7106723802547 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_163_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_164_1 :
+    Uω (aρ 1) (bρ 1) (3564573073667 / 32000000000000 : ℝ) ≤
+      (-(4508965963075115482201 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4197111168188307 / 40000000000000000 : ℝ) ≤
+      (-(4508965963075115482201 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (4197111168188307 / 40000000000000000 : ℝ) =
+            (4197111168188307 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4197111168188307 / 2500000000000000 : ℝ) ≤
+            (1036211479324884517799 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (131140464582277 / 1250000000000000 : ℝ)) (q :=
+      (4197111168188307 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_164_2 :
+    Uω (aρ 2) (bρ 2) (3564573073667 / 32000000000000 : ℝ) ≤
+      (-(2278256030458170454529 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4098509711427897 / 40000000000000000 : ℝ) ≤
+      (-(2278256030458170454529 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (4098509711427897 / 40000000000000000 : ℝ) =
+            (4098509711427897 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4098509711427897 / 2500000000000000 : ℝ) ≤
+            (494332690741829545471 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1023592000788011 / 10000000000000000 : ℝ)) (q :=
+      (4098509711427897 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_164_3 :
+    Uω (aρ 3) (bρ 3) (3564573073667 / 32000000000000 : ℝ) ≤
+      (-(23284458156104073445189 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3897883203552751 / 40000000000000000 : ℝ) ≤
+      (-(23284458156104073445189 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (3897883203552751 / 40000000000000000 : ℝ) =
+            (3897883203552751 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3897883203552751 / 2500000000000000 : ℝ) ≤
+            (4441429055895926554811 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (485337426600219 / 5000000000000000 : ℝ)) (q :=
+      (3897883203552751 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_164_4 :
+    Uω (aρ 4) (bρ 4) (3564573073667 / 32000000000000 : ℝ) ≤
+      (-(2421225827178593245939 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3552507264537567 / 40000000000000000 : ℝ) ≤
+      (-(2421225827178593245939 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (3552507264537567 / 40000000000000000 : ℝ) =
+            (3552507264537567 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3552507264537567 / 2500000000000000 : ℝ) ≤
+            (351362894021406754061 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (438140532378923 / 5000000000000000 : ℝ)) (q :=
+      (3552507264537567 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_164_5 :
+    Uω (aρ 5) (bρ 5) (3564573073667 / 32000000000000 : ℝ) ≤
+      (-(5190982654415919228157 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2984368378214133 / 40000000000000000 : ℝ) ≤
+      (-(5190982654415919228157 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2984368378214133 / 40000000000000000 : ℝ) =
+            (2984368378214133 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2984368378214133 / 2500000000000000 : ℝ) ≤
+            (354194787984080771843 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (710401540931129 / 10000000000000000 : ℝ)) (q :=
+      (2984368378214133 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_164_6 :
+    Uω (aρ 6) (bρ 6) (3564573073667 / 32000000000000 : ℝ) ≤
+      (-(30178145148616695910741 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1956319570663707 / 40000000000000000 : ℝ) ≤
+      (-(30178145148616695910741 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (1956319570663707 / 40000000000000000 : ℝ) =
+            (1956319570663707 / 1250000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1956319570663707 / 1250000000000000 : ℝ) ≤
+            (4479213866383304089259 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (90904378073979 / 2500000000000000 : ℝ)) (q :=
+      (1956319570663707 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_164_7 :
+    Uω (aρ 7) (bρ 7) (3564573073667 / 32000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_164_8 :
+    Uω (aρ 8) (bρ 8) (3564573073667 / 32000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_164_9 :
+    Uω (aρ 9) (bρ 9) (3564573073667 / 32000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_164_10 :
+    Uω (aρ 10) (bρ 10) (3564573073667 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_164_11 :
+    Uω (aρ 11) (bρ 11) (3564573073667 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_164_12 :
+    Uω (aρ 12) (bρ 12) (3564573073667 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_164_13 :
+    Uω (aρ 13) (bρ 13) (3564573073667 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_164_14 :
+    Uω (aρ 14) (bρ 14) (3564573073667 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_164_15 :
+    Uω (aρ 15) (bρ 15) (3564573073667 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_164_16 :
+    Uω (aρ 16) (bρ 16) (3564573073667 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_164 :
+    Uρ (3564573073667 / 32000000000000 : ℝ) ≤
+      (-(23169400871890508402437 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (3564573073667 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(4508965963075115482201 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_164_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (3564573073667 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(2278256030458170454529 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_164_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (3564573073667 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(23284458156104073445189 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_164_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (3564573073667 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(2421225827178593245939 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_164_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (3564573073667 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(5190982654415919228157 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_164_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (3564573073667 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(30178145148616695910741 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_164_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (3564573073667 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_164_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (3564573073667 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_164_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (3564573073667 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_164_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (3564573073667 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_164_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (3564573073667 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_164_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (3564573073667 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_164_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (3564573073667 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_164_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (3564573073667 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_164_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (3564573073667 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_164_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (3564573073667 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_164_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_165_1 :
+    Uω (aρ 1) (bρ 1) (7151568492121 / 64000000000000 : ℝ) ≤
+      (-(225114910132635713543 / 100000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8422254369869049 / 80000000000000000 : ℝ) ≤
+      (-(225114910132635713543 / 100000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (8422254369869049 / 80000000000000000 : ℝ) =
+            (8422254369869049 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8422254369869049 / 5000000000000000 : ℝ) ≤
+            (52143961987364286457 / 100000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (263157058414589 / 2500000000000000 : ℝ)) (q :=
+      (8422254369869049 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_165_2 :
+    Uω (aρ 2) (bρ 2) (7151568492121 / 64000000000000 : ℝ) ≤
+      (-(11374195722188479028337 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8225075609029889 / 80000000000000000 : ℝ) ≤
+      (-(11374195722188479028337 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (8225075609029889 / 80000000000000000 : ℝ) =
+            (8225075609029889 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8225075609029889 / 5000000000000000 : ℝ) ≤
+            (2488747883811520971663 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (513551277979283 / 5000000000000000 : ℝ)) (q :=
+      (8225075609029889 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_165_3 :
+    Uω (aρ 3) (bρ 3) (7151568492121 / 64000000000000 : ℝ) ≤
+      (-(23248430285377622815949 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7823903549169049 / 80000000000000000 : ℝ) ≤
+      (-(23248430285377622815949 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (7823903549169049 / 80000000000000000 : ℝ) =
+            (7823903549169049 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7823903549169049 / 5000000000000000 : ℝ) ≤
+            (4477456926622377184051 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (243551411835839 / 2500000000000000 : ℝ)) (q :=
+      (7823903549169049 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_165_4 :
+    Uω (aρ 4) (bρ 4) (7151568492121 / 64000000000000 : ℝ) ≤
+      (-(24172358762633393102107 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7133419818469869 / 80000000000000000 : ℝ) ≤
+      (-(24172358762633393102107 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (7133419818469869 / 80000000000000000 : ℝ) =
+            (7133419818469869 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7133419818469869 / 5000000000000000 : ℝ) ≤
+            (3553528449366606897893 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (879878895733561 / 10000000000000000 : ℝ)) (q :=
+      (7133419818469869 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_165_5 :
+    Uω (aρ 5) (bρ 5) (7151568492121 / 64000000000000 : ℝ) ≤
+      (-(25905729518360809311063 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5998165555909053 / 80000000000000000 : ℝ) ≤
+      (-(25905729518360809311063 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (5998165555909053 / 80000000000000000 : ℝ) =
+            (5998165555909053 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5998165555909053 / 5000000000000000 : ℝ) ≤
+            (1820157693639190688937 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (714255249428357 / 10000000000000000 : ℝ)) (q :=
+      (5998165555909053 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_165_6 :
+    Uω (aρ 6) (bρ 6) (7151568492121 / 64000000000000 : ℝ) ≤
+      (-(30082567552000849825557 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3950214487904321 / 80000000000000000 : ℝ) ≤
+      (-(30082567552000849825557 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (3950214487904321 / 80000000000000000 : ℝ) =
+            (3950214487904321 / 2500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3950214487904321 / 2500000000000000 : ℝ) ≤
+            (4574791462999150174443 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (184753928783587 / 5000000000000000 : ℝ)) (q :=
+      (3950214487904321 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_165_7 :
+    Uω (aρ 7) (bρ 7) (7151568492121 / 64000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_165_8 :
+    Uω (aρ 8) (bρ 8) (7151568492121 / 64000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_165_9 :
+    Uω (aρ 9) (bρ 9) (7151568492121 / 64000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_165_10 :
+    Uω (aρ 10) (bρ 10) (7151568492121 / 64000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_165_11 :
+    Uω (aρ 11) (bρ 11) (7151568492121 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_165_12 :
+    Uω (aρ 12) (bρ 12) (7151568492121 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_165_13 :
+    Uω (aρ 13) (bρ 13) (7151568492121 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_165_14 :
+    Uω (aρ 14) (bρ 14) (7151568492121 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_165_15 :
+    Uω (aρ 15) (bρ 15) (7151568492121 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_165_16 :
+    Uω (aρ 16) (bρ 16) (7151568492121 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_165 :
+    Uρ (7151568492121 / 64000000000000 : ℝ) ≤
+      (-(11576620047079105469959 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (7151568492121 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(225114910132635713543 / 100000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_165_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (7151568492121 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(11374195722188479028337 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_165_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (7151568492121 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(23248430285377622815949 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_165_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (7151568492121 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(24172358762633393102107 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_165_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (7151568492121 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(25905729518360809311063 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_165_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (7151568492121 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(30082567552000849825557 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_165_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (7151568492121 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_165_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (7151568492121 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_165_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (7151568492121 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_165_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (7151568492121 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_165_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (7151568492121 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_165_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (7151568492121 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_165_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (7151568492121 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_165_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (7151568492121 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_165_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (7151568492121 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_165_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (7151568492121 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_165_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_166_1 :
+    Uω (aρ 1) (bρ 1) (1793497709227 / 16000000000000 : ℝ) ≤
+      (-(22478263021719406875639 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2112571594034783 / 20000000000000000 : ℝ) ≤
+      (-(22478263021719406875639 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2112571594034783 / 20000000000000000 : ℝ) =
+            (2112571594034783 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2112571594034783 / 1250000000000000 : ℝ) ≤
+            (5247624190280593124361 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (264033185963227 / 2500000000000000 : ℝ)) (q :=
+      (2112571594034783 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_166_2 :
+    Uω (aρ 2) (bρ 2) (1793497709227 / 16000000000000 : ℝ) ≤
+      (-(181714713366534310893 / 80000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (257910362590861 / 2500000000000000 : ℝ) ≤
+      (-(181714713366534310893 / 80000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (257910362590861 / 2500000000000000 : ℝ) =
+            (257910362590861 / 156250000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (257910362590861 / 156250000000000 : ℝ) ≤
+            (40092384329465689107 / 80000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1030613063055013 / 10000000000000000 : ℝ)) (q :=
+      (257910362590861 / 2500000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_166_3 :
+    Uω (aρ 3) (bρ 3) (1793497709227 / 16000000000000 : ℝ) ≤
+      (-(2901566593777723928731 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1963009976379449 / 20000000000000000 : ℝ) ≤
+      (-(2901566593777723928731 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1963009976379449 / 20000000000000000 : ℝ) =
+            (1963009976379449 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1963009976379449 / 1250000000000000 : ℝ) ≤
+            (564169307722276071269 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (488868122528787 / 5000000000000000 : ℝ)) (q :=
+      (1963009976379449 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_166_4 :
+    Uω (aρ 4) (bρ 4) (1793497709227 / 16000000000000 : ℝ) ≤
+      (-(12066311026658035428937 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (223806939833861 / 2500000000000000 : ℝ) ≤
+      (-(12066311026658035428937 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (223806939833861 / 2500000000000000 : ℝ) =
+            (223806939833861 / 156250000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (223806939833861 / 156250000000000 : ℝ) ≤
+            (1796632579341964571063 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (883475968414013 / 10000000000000000 : ℝ)) (q :=
+      (223806939833861 / 2500000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_166_5 :
+    Uω (aρ 5) (bρ 5) (1793497709227 / 16000000000000 : ℝ) ≤
+      (-(12928405148153663752511 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1506895000779397 / 20000000000000000 : ℝ) ≤
+      (-(12928405148153663752511 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1506895000779397 / 20000000000000000 : ℝ) =
+            (1506895000779397 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1506895000779397 / 1250000000000000 : ℝ) ≤
+            (934538457846336247489 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (359052684928761 / 5000000000000000 : ℝ)) (q :=
+      (1506895000779397 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_166_6 :
+    Uω (aρ 6) (bρ 6) (1793497709227 / 16000000000000 : ℝ) ≤
+      (-(14994246997006309491453 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (996887726049841 / 20000000000000000 : ℝ) ≤
+      (-(14994246997006309491453 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (996887726049841 / 20000000000000000 : ℝ) =
+            (996887726049841 / 625000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (996887726049841 / 625000000000000 : ℝ) ≤
+            (2334432510493690508547 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (187669235133983 / 5000000000000000 : ℝ)) (q :=
+      (996887726049841 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_166_7 :
+    Uω (aρ 7) (bρ 7) (1793497709227 / 16000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_166_8 :
+    Uω (aρ 8) (bρ 8) (1793497709227 / 16000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_166_9 :
+    Uω (aρ 9) (bρ 9) (1793497709227 / 16000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_166_10 :
+    Uω (aρ 10) (bρ 10) (1793497709227 / 16000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_166_11 :
+    Uω (aρ 11) (bρ 11) (1793497709227 / 16000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_166_12 :
+    Uω (aρ 12) (bρ 12) (1793497709227 / 16000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_166_13 :
+    Uω (aρ 13) (bρ 13) (1793497709227 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_166_14 :
+    Uω (aρ 14) (bρ 14) (1793497709227 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_166_15 :
+    Uω (aρ 15) (bρ 15) (1793497709227 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_166_16 :
+    Uω (aρ 16) (bρ 16) (1793497709227 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_166 :
+    Uρ (1793497709227 / 16000000000000 : ℝ) ≤
+      (-(4627447176003051539293 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1793497709227 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(22478263021719406875639 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_166_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1793497709227 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) * (-(181714713366534310893 / 80000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_166_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1793497709227 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(2901566593777723928731 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_166_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1793497709227 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(12066311026658035428937 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_166_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1793497709227 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(12928405148153663752511 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_166_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1793497709227 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(14994246997006309491453 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_166_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1793497709227 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_166_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1793497709227 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_166_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1793497709227 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_166_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1793497709227 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_166_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1793497709227 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_166_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1793497709227 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_166_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1793497709227 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_166_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1793497709227 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_166_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1793497709227 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_166_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1793497709227 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_166_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_167_1 :
+    Uω (aρ 1) (bρ 1) (1439282636339 / 12800000000000 : ℝ) ≤
+      (-(22445145106352584507421 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8478318355456963 / 80000000000000000 : ℝ) ≤
+      (-(22445145106352584507421 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (8478318355456963 / 80000000000000000 : ℝ) =
+            (8478318355456963 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8478318355456963 / 5000000000000000 : ℝ) ≤
+            (5280742105647415492579 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1059637247309397 / 10000000000000000 : ℝ)) (q :=
+      (8478318355456963 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_167_2 :
+    Uω (aρ 2) (bρ 2) (1439282636339 / 12800000000000 : ℝ) ≤
+      (-(2835050336178665111439 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8281187406447127 / 80000000000000000 : ℝ) ≤
+      (-(2835050336178665111439 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (8281187406447127 / 80000000000000000 : ℝ) =
+            (8281187406447127 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8281187406447127 / 5000000000000000 : ℝ) ≤
+            (630685565321334888561 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (517061761283469 / 5000000000000000 : ℝ)) (q :=
+      (8281187406447127 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_167_3 :
+    Uω (aρ 3) (bρ 3) (1439282636339 / 12800000000000 : ℝ) ≤
+      (-(2317676460388831441057 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7880175484632751 / 80000000000000000 : ℝ) ≤
+      (-(2317676460388831441057 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (7880175484632751 / 80000000000000000 : ℝ) =
+            (7880175484632751 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7880175484632751 / 5000000000000000 : ℝ) ≤
+            (454912260811168558943 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (61329165528959 / 625000000000000 : ℝ)) (q :=
+      (7880175484632751 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_167_4 :
+    Uω (aρ 4) (bρ 4) (1439282636339 / 12800000000000 : ℝ) ≤
+      (-(24093046787011313355059 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (287608853384591 / 3200000000000000 : ℝ) ≤
+      (-(24093046787011313355059 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (287608853384591 / 3200000000000000 : ℝ) =
+            (287608853384591 / 200000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (287608853384591 / 200000000000000 : ℝ) ≤
+            (3632840424988686644941 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (17741445840477 / 200000000000000 : ℝ)) (q :=
+      (287608853384591 / 3200000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_167_5 :
+    Uω (aρ 5) (bρ 5) (1439282636339 / 12800000000000 : ℝ) ≤
+      (-(6452038133767527073457 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1211396065534587 / 16000000000000000 : ℝ) ≤
+      (-(6452038133767527073457 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1211396065534587 / 16000000000000000 : ℝ) =
+            (1211396065534587 / 1000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1211396065534587 / 1000000000000000 : ℝ) ≤
+            (479433669232472926543 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (72195195962339 / 1000000000000000 : ℝ)) (q :=
+      (1211396065534587 / 16000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_167_6 :
+    Uω (aρ 6) (bρ 6) (1439282636339 / 12800000000000 : ℝ) ≤
+      (-(29895863580366005101939 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4024659356395267 / 80000000000000000 : ℝ) ≤
+      (-(29895863580366005101939 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (4024659356395267 / 80000000000000000 : ℝ) =
+            (4024659356395267 / 2500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4024659356395267 / 2500000000000000 : ℝ) ≤
+            (4761495434633994898061 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (381112091943973 / 10000000000000000 : ℝ)) (q :=
+      (4024659356395267 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_167_7 :
+    Uω (aρ 7) (bρ 7) (1439282636339 / 12800000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_167_8 :
+    Uω (aρ 8) (bρ 8) (1439282636339 / 12800000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_167_9 :
+    Uω (aρ 9) (bρ 9) (1439282636339 / 12800000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_167_10 :
+    Uω (aρ 10) (bρ 10) (1439282636339 / 12800000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_167_11 :
+    Uω (aρ 11) (bρ 11) (1439282636339 / 12800000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_167_12 :
+    Uω (aρ 12) (bρ 12) (1439282636339 / 12800000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_167_13 :
+    Uω (aρ 13) (bρ 13) (1439282636339 / 12800000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_167_14 :
+    Uω (aρ 14) (bρ 14) (1439282636339 / 12800000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_167_15 :
+    Uω (aρ 15) (bρ 15) (1439282636339 / 12800000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_167_16 :
+    Uω (aρ 16) (bρ 16) (1439282636339 / 12800000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_167 :
+    Uρ (1439282636339 / 12800000000000 : ℝ) ≤
+      (-(11560691531889299496159 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1439282636339 / 12800000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(22445145106352584507421 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_167_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1439282636339 / 12800000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(2835050336178665111439 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_167_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1439282636339 / 12800000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(2317676460388831441057 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_167_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1439282636339 / 12800000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(24093046787011313355059 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_167_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1439282636339 / 12800000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(6452038133767527073457 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_167_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1439282636339 / 12800000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(29895863580366005101939 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_167_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1439282636339 / 12800000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_167_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1439282636339 / 12800000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_167_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1439282636339 / 12800000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_167_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1439282636339 / 12800000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_167_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1439282636339 / 12800000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_167_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1439282636339 / 12800000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_167_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1439282636339 / 12800000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_167_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1439282636339 / 12800000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_167_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1439282636339 / 12800000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_167_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1439282636339 / 12800000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_167_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_168_1 :
+    Uω (aρ 1) (bρ 1) (3609417763241 / 32000000000000 : ℝ) ≤
+      (-(22412136540051290770829 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (850635030808907 / 8000000000000000 : ℝ) ≤
+      (-(22412136540051290770829 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (850635030808907 / 8000000000000000 : ℝ) =
+            (850635030808907 / 500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (850635030808907 / 500000000000000 : ℝ) ≤
+            (5313750671948709229171 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (212628348818891 / 2000000000000000 : ℝ)) (q :=
+      (850635030808907 / 8000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_168_2 :
+    Uω (aρ 2) (bρ 2) (3609417763241 / 32000000000000 : ℝ) ≤
+      (-(22646581213851753936333 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4154621510790221 / 40000000000000000 : ℝ) ≤
+      (-(22646581213851753936333 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (4154621510790221 / 40000000000000000 : ℝ) =
+            (4154621510790221 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4154621510790221 / 2500000000000000 : ℝ) ≤
+            (5079305998148246063667 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (518816967488649 / 5000000000000000 : ℝ)) (q :=
+      (4154621510790221 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_168_3 :
+    Uω (aρ 3) (bρ 3) (3609417763241 / 32000000000000 : ℝ) ≤
+      (-(23141124909983015902357 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3954155147436417 / 40000000000000000 : ℝ) ≤
+      (-(23141124909983015902357 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (3954155147436417 / 40000000000000000 : ℝ) =
+            (3954155147436417 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3954155147436417 / 2500000000000000 : ℝ) ≤
+            (4584762302016984097643 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (246199214912599 / 2500000000000000 : ℝ)) (q :=
+      (3954155147436417 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_168_4 :
+    Uω (aρ 4) (bρ 4) (3609417763241 / 32000000000000 : ℝ) ≤
+      (-(12026815812091775261959 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3609308817279277 / 40000000000000000 : ℝ) ≤
+      (-(12026815812091775261959 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (3609308817279277 / 40000000000000000 : ℝ) =
+            (3609308817279277 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3609308817279277 / 2500000000000000 : ℝ) ≤
+            (1836127793908224738041 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (445333937818413 / 5000000000000000 : ℝ)) (q :=
+      (3609308817279277 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_168_5 :
+    Uω (aρ 5) (bρ 5) (3609417763241 / 32000000000000 : ℝ) ≤
+      (-(12879876610215081971411 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3043183377059021 / 40000000000000000 : ℝ) ≤
+      (-(12879876610215081971411 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (3043183377059021 / 40000000000000000 : ℝ) =
+            (3043183377059021 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3043183377059021 / 2500000000000000 : ℝ) ≤
+            (983066995784918028589 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (362897537430849 / 5000000000000000 : ℝ)) (q :=
+      (3043183377059021 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_168_6 :
+    Uω (aρ 6) (bρ 6) (3609417763241 / 32000000000000 : ℝ) ≤
+      (-(7451154866094100783241 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2030775025954889 / 40000000000000000 : ℝ) ≤
+      (-(7451154866094100783241 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (2030775025954889 / 40000000000000000 : ℝ) =
+            (2030775025954889 / 1250000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2030775025954889 / 1250000000000000 : ℝ) ≤
+            (1213184887655899216759 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (12088477326551 / 312500000000000 : ℝ)) (q :=
+      (2030775025954889 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_168_7 :
+    Uω (aρ 7) (bρ 7) (3609417763241 / 32000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_168_8 :
+    Uω (aρ 8) (bρ 8) (3609417763241 / 32000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_168_9 :
+    Uω (aρ 9) (bρ 9) (3609417763241 / 32000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_168_10 :
+    Uω (aρ 10) (bρ 10) (3609417763241 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_168_11 :
+    Uω (aρ 11) (bρ 11) (3609417763241 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_168_12 :
+    Uω (aρ 12) (bρ 12) (3609417763241 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_168_13 :
+    Uω (aρ 13) (bρ 13) (3609417763241 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_168_14 :
+    Uω (aρ 14) (bρ 14) (3609417763241 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_168_15 :
+    Uω (aρ 15) (bρ 15) (3609417763241 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_168_16 :
+    Uω (aρ 16) (bρ 16) (3609417763241 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_168 :
+    Uρ (3609417763241 / 32000000000000 : ℝ) ≤
+      (-(2310567680468791145093 / 1000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (3609417763241 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(22412136540051290770829 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_168_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (3609417763241 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(22646581213851753936333 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_168_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (3609417763241 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(23141124909983015902357 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_168_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (3609417763241 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(12026815812091775261959 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_168_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (3609417763241 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(12879876610215081971411 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_168_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (3609417763241 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(7451154866094100783241 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_168_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (3609417763241 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_168_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (3609417763241 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_168_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (3609417763241 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_168_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (3609417763241 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_168_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (3609417763241 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_168_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (3609417763241 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_168_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (3609417763241 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_168_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (3609417763241 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_168_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (3609417763241 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_168_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (3609417763241 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_168_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_169_1 :
+    Uω (aρ 1) (bρ 1) (7241257871269 / 64000000000000 : ℝ) ≤
+      (-(22379236602886479923973 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1706876446859697 / 16000000000000000 : ℝ) ≤
+      (-(22379236602886479923973 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1706876446859697 / 16000000000000000 : ℝ) =
+            (1706876446859697 / 1000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1706876446859697 / 1000000000000000 : ℝ) ≤
+            (5346650609113520076027 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (13333077928423 / 125000000000000 : ℝ)) (q :=
+      (1706876446859697 / 16000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_169_2 :
+    Uω (aρ 2) (bρ 2) (7241257871269 / 64000000000000 : ℝ) ≤
+      (-(22612873965720201807563 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8337298450213229 / 80000000000000000 : ℝ) ≤
+      (-(22612873965720201807563 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (8337298450213229 / 80000000000000000 : ℝ) =
+            (8337298450213229 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8337298450213229 / 5000000000000000 : ℝ) ≤
+            (5113013246279798192437 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (520572150381263 / 5000000000000000 : ℝ)) (q :=
+      (8337298450213229 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_169_3 :
+    Uω (aρ 3) (bρ 3) (7241257871269 / 64000000000000 : ℝ) ≤
+      (-(23105612742314276111357 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7936444344477101 / 80000000000000000 : ℝ) ≤
+      (-(23105612742314276111357 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (7936444344477101 / 80000000000000000 : ℝ) =
+            (7936444344477101 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7936444344477101 / 5000000000000000 : ℝ) ≤
+            (4620274469685723888643 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (494163440339247 / 5000000000000000 : ℝ)) (q :=
+      (7936444344477101 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_169_4 :
+    Uω (aρ 4) (bρ 4) (7241257871269 / 64000000000000 : ℝ) ≤
+      (-(24014375242285785827619 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1449402202043753 / 16000000000000000 : ℝ) ≤
+      (-(24014375242285785827619 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1449402202043753 / 16000000000000000 : ℝ) =
+            (1449402202043753 / 1000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1449402202043753 / 1000000000000000 : ℝ) ≤
+            (3711511969714214172381 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (89426272817891 / 1000000000000000 : ℝ)) (q :=
+      (1449402202043753 / 16000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_169_5 :
+    Uω (aρ 5) (bρ 5) (7241257871269 / 64000000000000 : ℝ) ≤
+      (-(25711609393350256228287 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6115739502064329 / 80000000000000000 : ℝ) ≤
+      (-(25711609393350256228287 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (6115739502064329 / 80000000000000000 : ℝ) =
+            (6115739502064329 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6115739502064329 / 5000000000000000 : ℝ) ≤
+            (2014277818649743771713 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (729634770475301 / 10000000000000000 : ℝ)) (q :=
+      (6115739502064329 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_169_6 :
+    Uω (aρ 6) (bρ 6) (7241257871269 / 64000000000000 : ℝ) ≤
+      (-(29714708478032058836987 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (32785860077733 / 640000000000000 : ℝ) ≤
+      (-(29714708478032058836987 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (32785860077733 / 640000000000000 : ℝ) = (32785860077733 / 20000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (32785860077733 / 20000000000000 : ℝ) ≤
+            (4942650536967941163013 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3139987180227 / 80000000000000 : ℝ)) (q :=
+      (32785860077733 / 640000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_169_7 :
+    Uω (aρ 7) (bρ 7) (7241257871269 / 64000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_169_8 :
+    Uω (aρ 8) (bρ 8) (7241257871269 / 64000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_169_9 :
+    Uω (aρ 9) (bρ 9) (7241257871269 / 64000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_169_10 :
+    Uω (aρ 10) (bρ 10) (7241257871269 / 64000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_169_11 :
+    Uω (aρ 11) (bρ 11) (7241257871269 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_169_12 :
+    Uω (aρ 12) (bρ 12) (7241257871269 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_169_13 :
+    Uω (aρ 13) (bρ 13) (7241257871269 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_169_14 :
+    Uω (aρ 14) (bρ 14) (7241257871269 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_169_15 :
+    Uω (aρ 15) (bρ 15) (7241257871269 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_169_16 :
+    Uω (aρ 16) (bρ 16) (7241257871269 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_169 :
+    Uρ (7241257871269 / 64000000000000 : ℝ) ≤
+      (-(23090112557678937190821 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (7241257871269 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(22379236602886479923973 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_169_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (7241257871269 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(22612873965720201807563 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_169_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (7241257871269 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(23105612742314276111357 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_169_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (7241257871269 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(24014375242285785827619 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_169_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (7241257871269 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(25711609393350256228287 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_169_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (7241257871269 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(29714708478032058836987 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_169_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (7241257871269 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_169_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (7241257871269 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_169_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (7241257871269 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_169_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (7241257871269 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_169_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (7241257871269 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_169_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (7241257871269 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_169_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (7241257871269 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_169_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (7241257871269 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_169_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (7241257871269 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_169_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (7241257871269 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_169_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_170_1 :
+    Uω (aρ 1) (bρ 1) (907960027007 / 8000000000000 : ℝ) ≤
+      (-(1396652786376097337059 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2140603533586199 / 20000000000000000 : ℝ) ≤
+      (-(1396652786376097337059 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2140603533586199 / 20000000000000000 : ℝ) =
+            (2140603533586199 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2140603533586199 / 1250000000000000 : ℝ) ≤
+            (336215164373902662941 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1070150717912449 / 10000000000000000 : ℝ)) (q :=
+      (2140603533586199 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_170_2 :
+    Uω (aρ 2) (bρ 2) (907960027007 / 8000000000000 : ℝ) ≤
+      (-(22579280174561325422027 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2091338423556399 / 20000000000000000 : ℝ) ≤
+      (-(22579280174561325422027 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2091338423556399 / 20000000000000000 : ℝ) =
+            (2091338423556399 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2091338423556399 / 1250000000000000 : ℝ) ≤
+            (5146607037438674577973 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1044654620392649 / 10000000000000000 : ℝ)) (q :=
+      (2091338423556399 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_170_3 :
+    Uω (aρ 3) (bρ 3) (907960027007 / 8000000000000 : ℝ) ≤
+      (-(4614045436948869884133 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (497786102597929 / 5000000000000000 : ℝ) ≤
+      (-(4614045436948869884133 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (497786102597929 / 5000000000000000 : ℝ) =
+            (497786102597929 / 312500000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (497786102597929 / 312500000000000 : ℝ) ≤
+            (931132005451130115867 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (495928356788983 / 5000000000000000 : ℝ)) (q :=
+      (497786102597929 / 5000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_170_4 :
+    Uω (aρ 4) (bρ 4) (907960027007 / 8000000000000 : ℝ) ≤
+      (-(1498454770966725311139 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (181885037418007 / 2000000000000000 : ℝ) ≤
+      (-(1498454770966725311139 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (181885037418007 / 2000000000000000 : ℝ) =
+            (181885037418007 / 125000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (181885037418007 / 125000000000000 : ℝ) ≤
+            (234413179783274688861 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (22446421460783 / 250000000000000 : ℝ)) (q :=
+      (181885037418007 / 2000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_170_5 :
+    Uω (aρ 5) (bρ 5) (907960027007 / 8000000000000 : ℝ) ≤
+      (-(12831859074287055078487 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1536274696582797 / 20000000000000000 : ℝ) ≤
+      (-(12831859074287055078487 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1536274696582797 / 20000000000000000 : ℝ) =
+            (1536274696582797 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1536274696582797 / 1250000000000000 : ℝ) ≤
+            (1031084531712944921513 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (733471100169047 / 10000000000000000 : ℝ)) (q :=
+      (1536274696582797 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_170_6 :
+    Uω (aρ 6) (bρ 6) (907960027007 / 8000000000000 : ℝ) ≤
+      (-(29626080805288577062113 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (516839452818463 / 10000000000000000 : ℝ) ≤
+      (-(29626080805288577062113 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (516839452818463 / 10000000000000000 : ℝ) =
+            (516839452818463 / 312500000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (516839452818463 / 312500000000000 : ℝ) ≤
+            (5031278209711422937887 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (49764460545397 / 1250000000000000 : ℝ)) (q :=
+      (516839452818463 / 10000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_170_7 :
+    Uω (aρ 7) (bρ 7) (907960027007 / 8000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_170_8 :
+    Uω (aρ 8) (bρ 8) (907960027007 / 8000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_170_9 :
+    Uω (aρ 9) (bρ 9) (907960027007 / 8000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_170_10 :
+    Uω (aρ 10) (bρ 10) (907960027007 / 8000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_170_11 :
+    Uω (aρ 11) (bρ 11) (907960027007 / 8000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_170_12 :
+    Uω (aρ 12) (bρ 12) (907960027007 / 8000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_170_13 :
+    Uω (aρ 13) (bρ 13) (907960027007 / 8000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_170_14 :
+    Uω (aρ 14) (bρ 14) (907960027007 / 8000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_170_15 :
+    Uω (aρ 15) (bρ 15) (907960027007 / 8000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_170_16 :
+    Uω (aρ 16) (bρ 16) (907960027007 / 8000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_170 :
+    Uρ (907960027007 / 8000000000000 : ℝ) ≤
+      (-(23074686047490937971453 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (907960027007 / 8000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1396652786376097337059 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_170_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (907960027007 / 8000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(22579280174561325422027 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_170_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (907960027007 / 8000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(4614045436948869884133 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_170_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (907960027007 / 8000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1498454770966725311139 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_170_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (907960027007 / 8000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(12831859074287055078487 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_170_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (907960027007 / 8000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(29626080805288577062113 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_170_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (907960027007 / 8000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_170_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (907960027007 / 8000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_170_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (907960027007 / 8000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_170_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (907960027007 / 8000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_170_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (907960027007 / 8000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_170_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (907960027007 / 8000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_170_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (907960027007 / 8000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_170_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (907960027007 / 8000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_170_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (907960027007 / 8000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_170_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (907960027007 / 8000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_170_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_171_1 :
+    Uω (aρ 1) (bρ 1) (7286102560843 / 64000000000000 : ℝ) ≤
+      (-(2789219971449951260491 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1718089201696839 / 16000000000000000 : ℝ) ≤
+      (-(2789219971449951260491 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1718089201696839 / 16000000000000000 : ℝ) =
+            (1718089201696839 / 1000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1718089201696839 / 1000000000000000 : ℝ) ≤
+            (676515930050048739509 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (107365519507433 / 1000000000000000 : ℝ)) (q :=
+      (1718089201696839 / 16000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_171_2 :
+    Uω (aρ 2) (bρ 2) (7286102560843 / 64000000000000 : ℝ) ≤
+      (-(11272899538842934213803 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8393408755472431 / 80000000000000000 : ℝ) ≤
+      (-(11272899538842934213803 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (8393408755472431 / 80000000000000000 : ℝ) =
+            (8393408755472431 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8393408755472431 / 5000000000000000 : ℝ) ≤
+            (2590044067157065786197 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1048164894331389 / 10000000000000000 : ℝ)) (q :=
+      (8393408755472431 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_171_3 :
+    Uω (aρ 3) (bρ 3) (7286102560843 / 64000000000000 : ℝ) ≤
+      (-(23034967331043341839631 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7992710194147811 / 80000000000000000 : ℝ) ≤
+      (-(23034967331043341839631 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (7992710194147811 / 80000000000000000 : ℝ) =
+            (7992710194147811 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7992710194147811 / 5000000000000000 : ℝ) ≤
+            (4690919880956658160369 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (497693180175117 / 5000000000000000 : ℝ)) (q :=
+      (7992710194147811 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_171_4 :
+    Uω (aρ 4) (bρ 4) (7286102560843 / 64000000000000 : ℝ) ≤
+      (-(11968166807144784725189 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7303789128620783 / 80000000000000000 : ℝ) ≤
+      (-(11968166807144784725189 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (7303789128620783 / 80000000000000000 : ℝ) =
+            (7303789128620783 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7303789128620783 / 5000000000000000 : ℝ) ≤
+            (1894776798855215274811 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (901450275033477 / 10000000000000000 : ℝ)) (q :=
+      (7303789128620783 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_171_5 :
+    Uω (aρ 5) (bρ 5) (7286102560843 / 64000000000000 : ℝ) ≤
+      (-(25616076633271137021301 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (246977792683227 / 3200000000000000 : ℝ) ≤
+      (-(25616076633271137021301 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (246977792683227 / 3200000000000000 : ℝ) =
+            (246977792683227 / 200000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (246977792683227 / 200000000000000 : ℝ) ≤
+            (2109810578728862978699 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (14746082329669 / 200000000000000 : ℝ)) (q :=
+      (246977792683227 / 3200000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_171_6 :
+    Uω (aρ 6) (bρ 6) (7286102560843 / 64000000000000 : ℝ) ≤
+      (-(29538689691821717904251 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4171007712090927 / 80000000000000000 : ℝ) ≤
+      (-(29538689691821717904251 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (4171007712090927 / 80000000000000000 : ℝ) =
+            (4171007712090927 / 2500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4171007712090927 / 2500000000000000 : ℝ) ≤
+            (5118669323178282095749 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (403685215376013 / 10000000000000000 : ℝ)) (q :=
+      (4171007712090927 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_171_7 :
+    Uω (aρ 7) (bρ 7) (7286102560843 / 64000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_171_8 :
+    Uω (aρ 8) (bρ 8) (7286102560843 / 64000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_171_9 :
+    Uω (aρ 9) (bρ 9) (7286102560843 / 64000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_171_10 :
+    Uω (aρ 10) (bρ 10) (7286102560843 / 64000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_171_11 :
+    Uω (aρ 11) (bρ 11) (7286102560843 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_171_12 :
+    Uω (aρ 12) (bρ 12) (7286102560843 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_171_13 :
+    Uω (aρ 13) (bρ 13) (7286102560843 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_171_14 :
+    Uω (aρ 14) (bρ 14) (7286102560843 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_171_15 :
+    Uω (aρ 15) (bρ 15) (7286102560843 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_171_16 :
+    Uω (aρ 16) (bρ 16) (7286102560843 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_171 :
+    Uρ (7286102560843 / 64000000000000 : ℝ) ≤
+      (-(4611878649130685497723 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (7286102560843 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2789219971449951260491 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_171_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (7286102560843 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(11272899538842934213803 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_171_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (7286102560843 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(23034967331043341839631 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_171_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (7286102560843 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11968166807144784725189 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_171_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (7286102560843 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(25616076633271137021301 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_171_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (7286102560843 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(29538689691821717904251 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_171_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (7286102560843 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_171_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (7286102560843 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_171_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (7286102560843 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_171_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (7286102560843 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_171_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (7286102560843 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_171_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (7286102560843 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_171_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (7286102560843 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_171_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (7286102560843 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_171_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (7286102560843 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_171_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (7286102560843 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_171_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/U14.lean b/LeanPool/Zeta5Irrational/Table/U14.lean
new file mode 100644
index 0000000000..896e134ad1
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/U14.lean
@@ -0,0 +1,3284 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-! # Certified arcsine potential bounds (U14) -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma U_172_1 :
+    Uω (aρ 1) (bρ 1) (730852490563 / 6400000000000 : ℝ) ≤
+      (-(22281181472692134742863 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4309238928484769 / 40000000000000000 : ℝ) ≤
+      (-(22281181472692134742863 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (4309238928484769 / 40000000000000000 : ℝ) =
+            (4309238928484769 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4309238928484769 / 2500000000000000 : ℝ) ≤
+            (5444705739307865257137 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1077159665822697 / 10000000000000000 : ℝ)) (q :=
+      (4309238928484769 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_172_2 :
+    Uω (aρ 2) (bρ 2) (730852490563 / 6400000000000 : ℝ) ≤
+      (-(11256214960041973685579 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (842146363578383 / 8000000000000000 : ℝ) ≤
+      (-(11256214960041973685579 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (842146363578383 / 8000000000000000 : ℝ) =
+            (842146363578383 / 500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (842146363578383 / 500000000000000 : ℝ) ≤
+            (2606728645958026314421 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (105167512303627 / 1000000000000000 : ℝ)) (q :=
+      (842146363578383 / 8000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_172_3 :
+    Uω (aρ 3) (bρ 3) (730852490563 / 6400000000000 : ℝ) ≤
+      (-(1149991614237294033779 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4010421005056027 / 40000000000000000 : ℝ) ≤
+      (-(1149991614237294033779 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (4010421005056027 / 40000000000000000 : ℝ) =
+            (4010421005056027 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4010421005056027 / 2500000000000000 : ℝ) ≤
+            (236302746362705966221 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (499457911484163 / 5000000000000000 : ℝ)) (q :=
+      (4010421005056027 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_172_4 :
+    Uω (aρ 4) (bρ 4) (730852490563 / 6400000000000 : ℝ) ≤
+      (-(11948772902721880539821 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3666086969961161 / 40000000000000000 : ℝ) ≤
+      (-(11948772902721880539821 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (3666086969961161 / 40000000000000000 : ℝ) =
+            (3666086969961161 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3666086969961161 / 2500000000000000 : ℝ) ≤
+            (1914170703278119460179 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (905042986485893 / 10000000000000000 : ℝ)) (q :=
+      (3666086969961161 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_172_5 :
+    Uω (aρ 5) (bρ 5) (730852490563 / 6400000000000 : ℝ) ≤
+      (-(199755328482226702363 / 78125000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3101888899973749 / 40000000000000000 : ℝ) ≤
+      (-(199755328482226702363 / 78125000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (3101888899973749 / 40000000000000000 : ℝ) =
+            (3101888899973749 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3101888899973749 / 2500000000000000 : ℝ) ≤
+            (16853165361523297637 / 78125000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (741133870827187 / 10000000000000000 : ℝ)) (q :=
+      (3101888899973749 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_172_6 :
+    Uω (aρ 6) (bρ 6) (730852490563 / 6400000000000 : ℝ) ≤
+      (-(1840780699148846481243 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2103558289061037 / 40000000000000000 : ℝ) ≤
+      (-(1840780699148846481243 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (2103558289061037 / 40000000000000000 : ℝ) =
+            (2103558289061037 / 1250000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2103558289061037 / 1250000000000000 : ℝ) ≤
+            (325304239288653518757 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (409208940510831 / 10000000000000000 : ℝ)) (q :=
+      (2103558289061037 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_172_7 :
+    Uω (aρ 7) (bρ 7) (730852490563 / 6400000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_172_8 :
+    Uω (aρ 8) (bρ 8) (730852490563 / 6400000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_172_9 :
+    Uω (aρ 9) (bρ 9) (730852490563 / 6400000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_172_10 :
+    Uω (aρ 10) (bρ 10) (730852490563 / 6400000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_172_11 :
+    Uω (aρ 11) (bρ 11) (730852490563 / 6400000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_172_12 :
+    Uω (aρ 12) (bρ 12) (730852490563 / 6400000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_172_13 :
+    Uω (aρ 13) (bρ 13) (730852490563 / 6400000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_172_14 :
+    Uω (aρ 14) (bρ 14) (730852490563 / 6400000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_172_15 :
+    Uω (aρ 15) (bρ 15) (730852490563 / 6400000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_172_16 :
+    Uω (aρ 16) (bρ 16) (730852490563 / 6400000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_172 :
+    Uρ (730852490563 / 6400000000000 : ℝ) ≤
+      (-(23044230349968969239773 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (730852490563 / 6400000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(22281181472692134742863 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_172_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (730852490563 / 6400000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(11256214960041973685579 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_172_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (730852490563 / 6400000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1149991614237294033779 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_172_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (730852490563 / 6400000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11948772902721880539821 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_172_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (730852490563 / 6400000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(199755328482226702363 / 78125000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_172_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (730852490563 / 6400000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1840780699148846481243 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_172_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (730852490563 / 6400000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_172_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (730852490563 / 6400000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_172_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (730852490563 / 6400000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_172_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (730852490563 / 6400000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_172_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (730852490563 / 6400000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_172_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (730852490563 / 6400000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_172_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (730852490563 / 6400000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_172_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (730852490563 / 6400000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_172_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (730852490563 / 6400000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_172_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (730852490563 / 6400000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_172_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_173_1 :
+    Uω (aρ 1) (bρ 1) (7330947250417 / 64000000000000 : ℝ) ≤
+      (-(22248708993169244456213 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8646509680050409 / 80000000000000000 : ℝ) ≤
+      (-(22248708993169244456213 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (8646509680050409 / 80000000000000000 : ℝ) =
+            (8646509680050409 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8646509680050409 / 5000000000000000 : ℝ) ≤
+            (5477178218830755543787 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (540332065109973 / 5000000000000000 : ℝ)) (q :=
+      (8646509680050409 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_173_2 :
+    Uω (aρ 2) (bρ 2) (7330947250417 / 64000000000000 : ℝ) ≤
+      (-(22479171954322094534041 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8449518336965521 / 80000000000000000 : ℝ) ≤
+      (-(22479171954322094534041 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (8449518336965521 / 80000000000000000 : ℝ) =
+            (8449518336965521 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8449518336965521 / 5000000000000000 : ℝ) ≤
+            (5246715257677905465959 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (263796326739681 / 2500000000000000 : ℝ)) (q :=
+      (8449518336965521 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_173_3 :
+    Uω (aρ 3) (bρ 3) (7330947250417 / 64000000000000 : ℝ) ≤
+      (-(22964821159010354163667 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8048973097240157 / 80000000000000000 : ℝ) ≤
+      (-(22964821159010354163667 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (8048973097240157 / 80000000000000000 : ℝ) =
+            (8048973097240157 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8048973097240157 / 5000000000000000 : ℝ) ≤
+            (4761066052989645836333 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1002445103377383 / 10000000000000000 : ℝ)) (q :=
+      (8048973097240157 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_173_4 :
+    Uω (aρ 4) (bρ 4) (7330947250417 / 64000000000000 : ℝ) ≤
+      (-(23858911651480393246513 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7360555964082549 / 80000000000000000 : ℝ) ≤
+      (-(23858911651480393246513 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (7360555964082549 / 80000000000000000 : ℝ) =
+            (7360555964082549 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7360555964082549 / 5000000000000000 : ℝ) ≤
+            (3866975560519606753487 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (908635001152981 / 10000000000000000 : ℝ)) (q :=
+      (7360555964082549 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_173_5 :
+    Uω (aρ 5) (bρ 5) (7330947250417 / 64000000000000 : ℝ) ≤
+      (-(204172253072514234181 / 80000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6233097936164581 / 80000000000000000 : ℝ) ≤
+      (-(204172253072514234181 / 80000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (6233097936164581 / 80000000000000000 : ℝ) =
+            (6233097936164581 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6233097936164581 / 5000000000000000 : ℝ) ≤
+            (17634844623485765819 / 80000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (744960413508489 / 10000000000000000 : ℝ)) (q :=
+      (6233097936164581 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_173_6 :
+    Uω (aρ 6) (bρ 6) (7330947250417 / 64000000000000 : ℝ) ≤
+      (-(5873488781927654282487 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4243049542363997 / 80000000000000000 : ℝ) ≤
+      (-(5873488781927654282487 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (4243049542363997 / 80000000000000000 : ℝ) =
+            (4243049542363997 / 2500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4243049542363997 / 2500000000000000 : ℝ) ≤
+            (1057983021072345717513 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (414688690198343 / 10000000000000000 : ℝ)) (q :=
+      (4243049542363997 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_173_7 :
+    Uω (aρ 7) (bρ 7) (7330947250417 / 64000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_173_8 :
+    Uω (aρ 8) (bρ 8) (7330947250417 / 64000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_173_9 :
+    Uω (aρ 9) (bρ 9) (7330947250417 / 64000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_173_10 :
+    Uω (aρ 10) (bρ 10) (7330947250417 / 64000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_173_11 :
+    Uω (aρ 11) (bρ 11) (7330947250417 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_173_12 :
+    Uω (aρ 12) (bρ 12) (7330947250417 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_173_13 :
+    Uω (aρ 13) (bρ 13) (7330947250417 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_173_14 :
+    Uω (aρ 14) (bρ 14) (7330947250417 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_173_15 :
+    Uω (aρ 15) (bρ 15) (7330947250417 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_173_16 :
+    Uω (aρ 16) (bρ 16) (7330947250417 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_173 :
+    Uρ (7330947250417 / 64000000000000 : ℝ) ≤
+      (-(23029193766167760068613 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (7330947250417 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(22248708993169244456213 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_173_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (7330947250417 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(22479171954322094534041 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_173_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (7330947250417 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(22964821159010354163667 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_173_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (7330947250417 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(23858911651480393246513 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_173_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (7330947250417 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(204172253072514234181 / 80000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_173_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (7330947250417 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(5873488781927654282487 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_173_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (7330947250417 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_173_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (7330947250417 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_173_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (7330947250417 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_173_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (7330947250417 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_173_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (7330947250417 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_173_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (7330947250417 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_173_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (7330947250417 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_173_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (7330947250417 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_173_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (7330947250417 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_173_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (7330947250417 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_173_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_174_1 :
+    Uω (aρ 1) (bρ 1) (1838342398801 / 16000000000000 : ℝ) ≤
+      (-(888653665905254274529 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (216863536949329 / 2000000000000000 : ℝ) ≤
+      (-(888653665905254274529 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (216863536949329 / 2000000000000000 : ℝ) =
+            (216863536949329 / 125000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (216863536949329 / 125000000000000 : ℝ) ≤
+            (220381822574745725471 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (216833717665533 / 2000000000000000 : ℝ)) (q :=
+      (216863536949329 / 2000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_174_2 :
+    Uω (aρ 2) (bρ 2) (1838342398801 / 16000000000000 : ℝ) ≤
+      (-(22446024440442051456091 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1059696607599907 / 10000000000000000 : ℝ) ≤
+      (-(22446024440442051456091 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1059696607599907 / 10000000000000000 : ℝ) =
+            (1059696607599907 / 625000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1059696607599907 / 625000000000000 : ℝ) ≤
+            (5279862771557948543909 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1058695446544189 / 10000000000000000 : ℝ)) (q :=
+      (1059696607599907 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_174_3 :
+    Uω (aρ 3) (bρ 3) (1838342398801 / 16000000000000 : ℝ) ≤
+      (-(22929933076480730785293 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2019275865800773 / 20000000000000000 : ℝ) ≤
+      (-(22929933076480730785293 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2019275865800773 / 20000000000000000 : ℝ) =
+            (2019275865800773 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2019275865800773 / 1250000000000000 : ℝ) ≤
+            (4795954135519269214707 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (251493550873787 / 2500000000000000 : ℝ)) (q :=
+      (2019275865800773 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_174_4 :
+    Uω (aρ 4) (bρ 4) (1838342398801 / 16000000000000 : ℝ) ≤
+      (-(4764085982108051921477 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1847233808506427 / 20000000000000000 : ℝ) ≤
+      (-(4764085982108051921477 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1847233808506427 / 20000000000000000 : ℝ) =
+            (1847233808506427 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1847233808506427 / 1250000000000000 : ℝ) ≤
+            (781091460291948078523 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (456113163632901 / 5000000000000000 : ℝ)) (q :=
+      (1847233808506427 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_174_5 :
+    Uω (aρ 5) (bρ 5) (1838342398801 / 16000000000000 : ℝ) ≤
+      (-(12737311347516656082539 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (782800677835541 / 10000000000000000 : ℝ) ≤
+      (-(12737311347516656082539 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (782800677835541 / 10000000000000000 : ℝ) =
+            (782800677835541 / 625000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (782800677835541 / 625000000000000 : ℝ) ≤
+            (1125632258483343917461 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (748783793765457 / 10000000000000000 : ℝ)) (q :=
+      (782800677835541 / 10000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_174_6 :
+    Uω (aρ 6) (bρ 6) (1838342398801 / 16000000000000 : ℝ) ≤
+      (-(29283508847027304206383 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (213940674387349 / 4000000000000000 : ℝ) ≤
+      (-(29283508847027304206383 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (213940674387349 / 4000000000000000 : ℝ) =
+            (213940674387349 / 125000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (213940674387349 / 125000000000000 : ℝ) ≤
+            (5373850167972695793617 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5251577314639 / 125000000000000 : ℝ)) (q :=
+      (213940674387349 / 4000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_174_7 :
+    Uω (aρ 7) (bρ 7) (1838342398801 / 16000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_174_8 :
+    Uω (aρ 8) (bρ 8) (1838342398801 / 16000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_174_9 :
+    Uω (aρ 9) (bρ 9) (1838342398801 / 16000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_174_10 :
+    Uω (aρ 10) (bρ 10) (1838342398801 / 16000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_174_11 :
+    Uω (aρ 11) (bρ 11) (1838342398801 / 16000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_174_12 :
+    Uω (aρ 12) (bρ 12) (1838342398801 / 16000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_174_13 :
+    Uω (aρ 13) (bρ 13) (1838342398801 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_174_14 :
+    Uω (aρ 14) (bρ 14) (1838342398801 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_174_15 :
+    Uω (aρ 15) (bρ 15) (1838342398801 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_174_16 :
+    Uω (aρ 16) (bρ 16) (1838342398801 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_174 :
+    Uρ (1838342398801 / 16000000000000 : ℝ) ≤
+      (-(23014280091458413181989 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1838342398801 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(888653665905254274529 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_174_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1838342398801 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(22446024440442051456091 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_174_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1838342398801 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(22929933076480730785293 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_174_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1838342398801 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(4764085982108051921477 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_174_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1838342398801 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(12737311347516656082539 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_174_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1838342398801 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(29283508847027304206383 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_174_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1838342398801 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_174_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1838342398801 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_174_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1838342398801 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_174_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1838342398801 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_174_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1838342398801 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_174_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1838342398801 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_174_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1838342398801 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_174_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1838342398801 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_174_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1838342398801 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_174_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1838342398801 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_174_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_175_1 :
+    Uω (aρ 1) (bρ 1) (3699107142389 / 32000000000000 : ℝ) ≤
+      (-(22151919650023756881351 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4365302499656941 / 40000000000000000 : ℝ) ≤
+      (-(22151919650023756881351 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (4365302499656941 / 40000000000000000 : ℝ) =
+            (4365302499656941 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4365302499656941 / 2500000000000000 : ℝ) ≤
+            (5573967561976243118649 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (272794371479227 / 2500000000000000 : ℝ)) (q :=
+      (4365302499656941 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_175_2 :
+    Uω (aρ 2) (bρ 2) (3699107142389 / 32000000000000 : ℝ) ≤
+      (-(11190028922637405674731 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4266840691716199 / 40000000000000000 : ℝ) ≤
+      (-(11190028922637405674731 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (4266840691716199 / 40000000000000000 : ℝ) =
+            (4266840691716199 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4266840691716199 / 2500000000000000 : ℝ) ≤
+            (2672914683362594325269 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1065715594456537 / 10000000000000000 : ℝ)) (q :=
+      (4266840691716199 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_175_3 :
+    Uω (aρ 3) (bρ 3) (3699107142389 / 32000000000000 : ℝ) ≤
+      (-(11430261289115545831809 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4066681030890477 / 40000000000000000 : ℝ) ≤
+      (-(11430261289115545831809 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (4066681030890477 / 40000000000000000 : ℝ) =
+            (4066681030890477 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4066681030890477 / 2500000000000000 : ℝ) ≤
+            (2432682316884454168191 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (253257967598419 / 2500000000000000 : ℝ)) (q :=
+      (4066681030890477 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_175_4 :
+    Uω (aρ 4) (bρ 4) (3699107142389 / 32000000000000 : ℝ) ≤
+      (-(23743918776680516836079 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3722842820177343 / 40000000000000000 : ℝ) ≤
+      (-(23743918776680516836079 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (3722842820177343 / 40000000000000000 : ℝ) =
+            (3722842820177343 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3722842820177343 / 2500000000000000 : ℝ) ≤
+            (3981968435319483163921 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (919406946102109 / 10000000000000000 : ℝ)) (q :=
+      (3722842820177343 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_175_5 :
+    Uω (aρ 5) (bρ 5) (3699107142389 / 32000000000000 : ℝ) ≤
+      (-(5076303731562824760583 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3160491606868909 / 40000000000000000 : ℝ) ≤
+      (-(5076303731562824760583 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (3160491606868909 / 40000000000000000 : ℝ) =
+            (3160491606868909 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3160491606868909 / 2500000000000000 : ℝ) ≤
+            (468873710837175239417 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (189105314695723 / 2500000000000000 : ℝ)) (q :=
+      (3160491606868909 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_175_6 :
+    Uω (aρ 6) (bρ 6) (3699107142389 / 32000000000000 : ℝ) ≤
+      (-(14559415015712096654269 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2174929933788829 / 40000000000000000 : ℝ) ≤
+      (-(14559415015712096654269 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (2174929933788829 / 40000000000000000 : ℝ) =
+            (2174929933788829 / 1250000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2174929933788829 / 1250000000000000 : ℝ) ≤
+            (2769264491787903345731 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (107720199345713 / 2500000000000000 : ℝ)) (q :=
+      (2174929933788829 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_175_7 :
+    Uω (aρ 7) (bρ 7) (3699107142389 / 32000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_175_8 :
+    Uω (aρ 8) (bρ 8) (3699107142389 / 32000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_175_9 :
+    Uω (aρ 9) (bρ 9) (3699107142389 / 32000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_175_10 :
+    Uω (aρ 10) (bρ 10) (3699107142389 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_175_11 :
+    Uω (aρ 11) (bρ 11) (3699107142389 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_175_12 :
+    Uω (aρ 12) (bρ 12) (3699107142389 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_175_13 :
+    Uω (aρ 13) (bρ 13) (3699107142389 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_175_14 :
+    Uω (aρ 14) (bρ 14) (3699107142389 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_175_15 :
+    Uω (aρ 15) (bρ 15) (3699107142389 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_175_16 :
+    Uω (aρ 16) (bρ 16) (3699107142389 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_175 :
+    Uρ (3699107142389 / 32000000000000 : ℝ) ≤
+      (-(11492404364136263761547 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (3699107142389 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(22151919650023756881351 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_175_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (3699107142389 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(11190028922637405674731 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_175_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (3699107142389 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(11430261289115545831809 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_175_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (3699107142389 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(23743918776680516836079 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_175_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (3699107142389 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(5076303731562824760583 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_175_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (3699107142389 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(14559415015712096654269 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_175_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (3699107142389 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_175_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (3699107142389 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_175_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (3699107142389 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_175_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (3699107142389 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_175_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (3699107142389 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_175_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (3699107142389 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_175_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (3699107142389 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_175_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (3699107142389 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_175_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (3699107142389 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_175_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (3699107142389 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_175_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_176_1 :
+    Uω (aρ 1) (bρ 1) (465191185897 / 4000000000000 : ℝ) ≤
+      (-(22087910127931218969373 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2196667105724981 / 20000000000000000 : ℝ) ≤
+      (-(22087910127931218969373 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2196667105724981 / 20000000000000000 : ℝ) =
+            (2196667105724981 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2196667105724981 / 1250000000000000 : ℝ) ≤
+            (5637977084068781030627 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1098186359067481 / 10000000000000000 : ℝ)) (q :=
+      (2196667105724981 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_176_2 :
+    Uω (aρ 2) (bρ 2) (465191185897 / 4000000000000 : ℝ) ≤
+      (-(5578631090169752076037 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (536861826092047 / 5000000000000000 : ℝ) ≤
+      (-(5578631090169752076037 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (536861826092047 / 5000000000000000 : ℝ) =
+            (536861826092047 / 312500000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (536861826092047 / 312500000000000 : ℝ) ≤
+            (1352840712830247923963 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (67045973138793 / 625000000000000 : ℝ)) (q :=
+      (536861826092047 / 5000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_176_3 :
+    Uω (aρ 3) (bρ 3) (465191185897 / 4000000000000 : ℝ) ≤
+      (-(22791593955763525787333 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1023702233136753 / 10000000000000000 : ℝ) ≤
+      (-(22791593955763525787333 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1023702233136753 / 10000000000000000 : ℝ) =
+            (1023702233136753 / 625000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1023702233136753 / 625000000000000 : ℝ) ≤
+            (4934293256236474212667 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (510044419238003 / 5000000000000000 : ℝ)) (q :=
+      (1023702233136753 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_176_4 :
+    Uω (aρ 4) (bρ 4) (465191185897 / 4000000000000 : ℝ) ≤
+      (-(23668002770991622080457 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (234450794126107 / 2500000000000000 : ℝ) ≤
+      (-(23668002770991622080457 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (234450794126107 / 2500000000000000 : ℝ) =
+            (234450794126107 / 156250000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (234450794126107 / 156250000000000 : ℝ) ≤
+            (4057884441008377919543 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (231646226569089 / 2500000000000000 : ℝ)) (q :=
+      (234450794126107 / 2500000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_176_5 :
+    Uω (aρ 5) (bρ 5) (465191185897 / 4000000000000 : ℝ) ≤
+      (-(6322337309009206611799 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (318975633373579 / 4000000000000000 : ℝ) ≤
+      (-(6322337309009206611799 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (318975633373579 / 4000000000000000 : ℝ) =
+            (318975633373579 / 250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (318975633373579 / 250000000000000 : ℝ) ≤
+            (609134493990793388201 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (152809327894079 / 2000000000000000 : ℝ)) (q :=
+      (318975633373579 / 4000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_176_6 :
+    Uω (aρ 6) (bρ 6) (465191185897 / 4000000000000 : ℝ) ≤
+      (-(2895818328965296759049 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (552537907396753 / 10000000000000000 : ℝ) ≤
+      (-(2895818328965296759049 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (552537907396753 / 10000000000000000 : ℝ) =
+            (552537907396753 / 312500000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (552537907396753 / 312500000000000 : ℝ) ≤
+            (569917572534703240951 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (220742331268003 / 5000000000000000 : ℝ)) (q :=
+      (552537907396753 / 10000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_176_7 :
+    Uω (aρ 7) (bρ 7) (465191185897 / 4000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_176_8 :
+    Uω (aρ 8) (bρ 8) (465191185897 / 4000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_176_9 :
+    Uω (aρ 9) (bρ 9) (465191185897 / 4000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_176_10 :
+    Uω (aρ 10) (bρ 10) (465191185897 / 4000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_176_11 :
+    Uω (aρ 11) (bρ 11) (465191185897 / 4000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_176_12 :
+    Uω (aρ 12) (bρ 12) (465191185897 / 4000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_176_13 :
+    Uω (aρ 13) (bρ 13) (465191185897 / 4000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_176_14 :
+    Uω (aρ 14) (bρ 14) (465191185897 / 4000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_176_15 :
+    Uω (aρ 15) (bρ 15) (465191185897 / 4000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_176_16 :
+    Uω (aρ 16) (bρ 16) (465191185897 / 4000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_176 :
+    Uρ (465191185897 / 4000000000000 : ℝ) ≤
+      (-(22955792340828882002671 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (465191185897 / 4000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(22087910127931218969373 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_176_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (465191185897 / 4000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(5578631090169752076037 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_176_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (465191185897 / 4000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(22791593955763525787333 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_176_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (465191185897 / 4000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(23668002770991622080457 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_176_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (465191185897 / 4000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(6322337309009206611799 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_176_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (465191185897 / 4000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(2895818328965296759049 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_176_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (465191185897 / 4000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_176_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (465191185897 / 4000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_176_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (465191185897 / 4000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_176_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (465191185897 / 4000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_176_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (465191185897 / 4000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_176_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (465191185897 / 4000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_176_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (465191185897 / 4000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_176_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (465191185897 / 4000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_176_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (465191185897 / 4000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_176_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (465191185897 / 4000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_176_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_177_1 :
+    Uω (aρ 1) (bρ 1) (3743951831963 / 32000000000000 : ℝ) ≤
+      (-(11012153915808991134077 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (884273175059109 / 8000000000000000 : ℝ) ≤
+      (-(11012153915808991134077 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (884273175059109 / 8000000000000000 : ℝ) =
+            (884273175059109 / 500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (884273175059109 / 500000000000000 : ℝ) ≤
+            (2850789690191008865923 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (221039041648867 / 2000000000000000 : ℝ)) (q :=
+      (884273175059109 / 8000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_177_2 :
+    Uω (aρ 2) (bρ 2) (3743951831963 / 32000000000000 : ℝ) ≤
+      (-(22249418326153667549627 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4322948188175437 / 40000000000000000 : ℝ) ≤
+      (-(22249418326153667549627 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (4322948188175437 / 40000000000000000 : ℝ) =
+            (4322948188175437 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4322948188175437 / 2500000000000000 : ℝ) ≤
+            (5476468885846332450373 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1079755377194281 / 10000000000000000 : ℝ)) (q :=
+      (4322948188175437 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_177_3 :
+    Uω (aρ 3) (bρ 3) (3743951831963 / 32000000000000 : ℝ) ≤
+      (-(1136157025908839802787 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (824587093075663 / 8000000000000000 : ℝ) ≤
+      (-(1136157025908839802787 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (824587093075663 / 8000000000000000 : ℝ) =
+            (824587093075663 / 500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (824587093075663 / 500000000000000 : ℝ) ≤
+            (250137334691160197213 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (25678628053643 / 250000000000000 : ℝ)) (q :=
+      (824587093075663 / 8000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_177_4 :
+    Uω (aρ 4) (bρ 4) (3743951831963 / 32000000000000 : ℝ) ≤
+      (-(5898168123465075345177 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3779577397159251 / 40000000000000000 : ℝ) ≤
+      (-(5898168123465075345177 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (3779577397159251 / 40000000000000000 : ℝ) =
+            (3779577397159251 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3779577397159251 / 2500000000000000 : ℝ) ≤
+            (1033303679534924654823 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (233440067275297 / 2500000000000000 : ℝ)) (q :=
+      (3779577397159251 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_177_5 :
+    Uω (aρ 5) (bρ 5) (3743951831963 / 32000000000000 : ℝ) ≤
+      (-(25198094437116484192507 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3218997608431433 / 40000000000000000 : ℝ) ≤
+      (-(25198094437116484192507 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (3218997608431433 / 40000000000000000 : ℝ) =
+            (3218997608431433 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3218997608431433 / 2500000000000000 : ℝ) ≤
+            (2527792774883515807493 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (771660294072279 / 10000000000000000 : ℝ)) (q :=
+      (3218997608431433 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_177_6 :
+    Uω (aρ 6) (bρ 6) (3743951831963 / 32000000000000 : ℝ) ≤
+      (-(14400662417436316917723 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2245093052986399 / 40000000000000000 : ℝ) ≤
+      (-(14400662417436316917723 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (2245093052986399 / 40000000000000000 : ℝ) =
+            (2245093052986399 / 1250000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2245093052986399 / 1250000000000000 : ℝ) ≤
+            (2928017090063683082277 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (225974195744881 / 5000000000000000 : ℝ)) (q :=
+      (2245093052986399 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_177_7 :
+    Uω (aρ 7) (bρ 7) (3743951831963 / 32000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_177_8 :
+    Uω (aρ 8) (bρ 8) (3743951831963 / 32000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_177_9 :
+    Uω (aρ 9) (bρ 9) (3743951831963 / 32000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_177_10 :
+    Uω (aρ 10) (bρ 10) (3743951831963 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_177_11 :
+    Uω (aρ 11) (bρ 11) (3743951831963 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_177_12 :
+    Uω (aρ 12) (bρ 12) (3743951831963 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_177_13 :
+    Uω (aρ 13) (bρ 13) (3743951831963 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_177_14 :
+    Uω (aρ 14) (bρ 14) (3743951831963 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_177_15 :
+    Uω (aρ 15) (bρ 15) (3743951831963 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_177_16 :
+    Uω (aρ 16) (bρ 16) (3743951831963 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_177 :
+    Uρ (3743951831963 / 32000000000000 : ℝ) ≤
+      (-(716475289508359637971 / 312500000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (3743951831963 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(11012153915808991134077 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_177_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (3743951831963 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(22249418326153667549627 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_177_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (3743951831963 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1136157025908839802787 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_177_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (3743951831963 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(5898168123465075345177 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_177_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (3743951831963 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(25198094437116484192507 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_177_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (3743951831963 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(14400662417436316917723 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_177_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (3743951831963 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_177_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (3743951831963 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_177_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (3743951831963 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_177_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (3743951831963 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_177_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (3743951831963 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_177_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (3743951831963 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_177_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (3743951831963 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_177_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (3743951831963 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_177_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (3743951831963 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_177_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (3743951831963 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_177_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_178_1 :
+    Uω (aρ 1) (bρ 1) (15065496707 / 128000000000 : ℝ) ≤
+      (-(21961107610964149289349 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (556174686512519 / 5000000000000000 : ℝ) ≤
+      (-(21961107610964149289349 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (556174686512519 / 5000000000000000 : ℝ) =
+            (556174686512519 / 312500000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (556174686512519 / 312500000000000 : ℝ) ≤
+            (5764779601035850710651 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1112204033900701 / 10000000000000000 : ℝ)) (q :=
+      (556174686512519 / 5000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_178_2 :
+    Uω (aρ 2) (bρ 2) (15065496707 / 128000000000 : ℝ) ≤
+      (-(5546183547911107213653 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2175500718287497 / 20000000000000000 : ℝ) ≤
+      (-(5546183547911107213653 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2175500718287497 / 20000000000000000 : ℝ) =
+            (2175500718287497 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2175500718287497 / 1250000000000000 : ℝ) ≤
+            (1385288255088892786347 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (543387509324061 / 5000000000000000 : ℝ)) (q :=
+      (2175500718287497 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_178_3 :
+    Uω (aρ 3) (bρ 3) (15065496707 / 128000000000 : ℝ) ≤
+      (-(5663788928473910630479 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1037765164350651 / 10000000000000000 : ℝ) ≤
+      (-(5663788928473910630479 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1037765164350651 / 10000000000000000 : ℝ) =
+            (1037765164350651 / 625000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1037765164350651 / 625000000000000 : ℝ) ≤
+            (1267682874526089369521 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1034200735411927 / 10000000000000000 : ℝ)) (q :=
+      (1037765164350651 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_178_4 :
+    Uω (aρ 4) (bρ 4) (15065496707 / 128000000000 : ℝ) ≤
+      (-(23517918771318664380059 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (380793701244309 / 4000000000000000 : ℝ) ≤
+      (-(23517918771318664380059 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (380793701244309 / 4000000000000000 : ℝ) =
+            (380793701244309 / 250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (380793701244309 / 250000000000000 : ℝ) ≤
+            (4207968440681335619941 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (94093309399717 / 1000000000000000 : ℝ)) (q :=
+      (380793701244309 / 4000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_178_5 :
+    Uω (aρ 5) (bρ 5) (15065496707 / 128000000000 : ℝ) ≤
+      (-(6276933735209036880089 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1624108059181271 / 20000000000000000 : ℝ) ≤
+      (-(6276933735209036880089 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1624108059181271 / 20000000000000000 : ℝ) =
+            (1624108059181271 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1624108059181271 / 1250000000000000 : ℝ) ≤
+            (654538067790963119911 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (97407820786487 / 1250000000000000 : ℝ)) (q :=
+      (1624108059181271 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_178_6 :
+    Uω (aρ 6) (bρ 6) (15065496707 / 128000000000 : ℝ) ≤
+      (-(572960694532787081867 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1139886618028413 / 20000000000000000 : ℝ) ≤
+      (-(572960694532787081867 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (1139886618028413 / 20000000000000000 : ℝ) =
+            (1139886618028413 / 625000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1139886618028413 / 625000000000000 : ℝ) ≤
+            (120186485767212918133 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (231140750139519 / 5000000000000000 : ℝ)) (q :=
+      (1139886618028413 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_178_7 :
+    Uω (aρ 7) (bρ 7) (15065496707 / 128000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_178_8 :
+    Uω (aρ 8) (bρ 8) (15065496707 / 128000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_178_9 :
+    Uω (aρ 9) (bρ 9) (15065496707 / 128000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_178_10 :
+    Uω (aρ 10) (bρ 10) (15065496707 / 128000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_178_11 :
+    Uω (aρ 11) (bρ 11) (15065496707 / 128000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_178_12 :
+    Uω (aρ 12) (bρ 12) (15065496707 / 128000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_178_13 :
+    Uω (aρ 13) (bρ 13) (15065496707 / 128000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_178_14 :
+    Uω (aρ 14) (bρ 14) (15065496707 / 128000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_178_15 :
+    Uω (aρ 15) (bρ 15) (15065496707 / 128000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_178_16 :
+    Uω (aρ 16) (bρ 16) (15065496707 / 128000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_178 :
+    Uρ (15065496707 / 128000000000 : ℝ) ≤
+      (-(22899039784404335583777 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (15065496707 / 128000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(21961107610964149289349 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_178_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (15065496707 / 128000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(5546183547911107213653 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_178_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (15065496707 / 128000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(5663788928473910630479 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_178_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (15065496707 / 128000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(23517918771318664380059 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_178_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (15065496707 / 128000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(6276933735209036880089 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_178_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (15065496707 / 128000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(572960694532787081867 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_178_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (15065496707 / 128000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_178_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (15065496707 / 128000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_178_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (15065496707 / 128000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_178_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (15065496707 / 128000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_178_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (15065496707 / 128000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_178_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (15065496707 / 128000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_178_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (15065496707 / 128000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_178_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (15065496707 / 128000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_178_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (15065496707 / 128000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_178_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (15065496707 / 128000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_178_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_179_1 :
+    Uω (aρ 1) (bρ 1) (3788796521537 / 32000000000000 : ℝ) ≤
+      (-(5474576103240136115701 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (895485812549507 / 8000000000000000 : ℝ) ≤
+      (-(5474576103240136115701 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (895485812549507 / 8000000000000000 : ℝ) =
+            (895485812549507 / 500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (895485812549507 / 500000000000000 : ℝ) ≤
+            (1456895699759863884299 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (223842567295691 / 2000000000000000 : ℝ)) (q :=
+      (895485812549507 / 8000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_179_2 :
+    Uω (aρ 2) (bρ 2) (3788796521537 / 32000000000000 : ℝ) ≤
+      (-(11060233257341866518873 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4379054360308581 / 40000000000000000 : ℝ) ≤
+      (-(11060233257341866518873 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (4379054360308581 / 40000000000000000 : ℝ) =
+            (4379054360308581 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4379054360308581 / 2500000000000000 : ℝ) ≤
+            (2802710348658133481127 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (546897248884489 / 5000000000000000 : ℝ)) (q :=
+      (4379054360308581 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_179_3 :
+    Uω (aρ 3) (bρ 3) (3788796521537 / 32000000000000 : ℝ) ≤
+      (-(2258763312680448441801 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4179184535876051 / 40000000000000000 : ℝ) ≤
+      (-(2258763312680448441801 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (4179184535876051 / 40000000000000000 : ℝ) =
+            (4179184535876051 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4179184535876051 / 2500000000000000 : ℝ) ≤
+            (513825408519551558199 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1041255691902713 / 10000000000000000 : ℝ)) (q :=
+      (4179184535876051 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_179_4 :
+    Uω (aρ 4) (bρ 4) (3788796521537 / 32000000000000 : ℝ) ≤
+      (-(468874652954865998573 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3836291667072607 / 40000000000000000 : ℝ) ≤
+      (-(468874652954865998573 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (3836291667072607 / 40000000000000000 : ℝ) =
+            (3836291667072607 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3836291667072607 / 2500000000000000 : ℝ) ≤
+            (85643091285134001427 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (948103438565991 / 10000000000000000 : ℝ)) (q :=
+      (3836291667072607 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_179_5 :
+    Uω (aρ 5) (bρ 5) (3788796521537 / 32000000000000 : ℝ) ≤
+      (-(2501825206824387380069 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3277412523361897 / 40000000000000000 : ℝ) ≤
+      (-(2501825206824387380069 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (3277412523361897 / 40000000000000000 : ℝ) =
+            (3277412523361897 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3277412523361897 / 2500000000000000 : ℝ) ≤
+            (270763514375612619931 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (196713446511409 / 2500000000000000 : ℝ)) (q :=
+      (3277412523361897 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_179_6 :
+    Uω (aρ 6) (bρ 6) (3788796521537 / 32000000000000 : ℝ) ≤
+      (-(14249056853658479258529 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2314209318428351 / 40000000000000000 : ℝ) ≤
+      (-(14249056853658479258529 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (2314209318428351 / 40000000000000000 : ℝ) =
+            (2314209318428351 / 1250000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2314209318428351 / 1250000000000000 : ℝ) ≤
+            (3079622653841520741471 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (472492558718863 / 10000000000000000 : ℝ)) (q :=
+      (2314209318428351 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_179_7 :
+    Uω (aρ 7) (bρ 7) (3788796521537 / 32000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_179_8 :
+    Uω (aρ 8) (bρ 8) (3788796521537 / 32000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_179_9 :
+    Uω (aρ 9) (bρ 9) (3788796521537 / 32000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_179_10 :
+    Uω (aρ 10) (bρ 10) (3788796521537 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_179_11 :
+    Uω (aρ 11) (bρ 11) (3788796521537 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_179_12 :
+    Uω (aρ 12) (bρ 12) (3788796521537 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_179_13 :
+    Uω (aρ 13) (bρ 13) (3788796521537 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_179_14 :
+    Uω (aρ 14) (bρ 14) (3788796521537 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_179_15 :
+    Uω (aρ 15) (bρ 15) (3788796521537 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_179_16 :
+    Uω (aρ 16) (bρ 16) (3788796521537 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_179 :
+    Uρ (3788796521537 / 32000000000000 : ℝ) ≤
+      (-(11435632942680119457883 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (3788796521537 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(5474576103240136115701 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_179_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (3788796521537 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(11060233257341866518873 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_179_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (3788796521537 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(2258763312680448441801 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_179_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (3788796521537 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(468874652954865998573 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_179_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (3788796521537 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(2501825206824387380069 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_179_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (3788796521537 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(14249056853658479258529 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_179_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (3788796521537 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_179_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (3788796521537 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_179_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (3788796521537 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_179_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (3788796521537 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_179_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (3788796521537 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_179_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (3788796521537 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_179_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (3788796521537 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_179_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (3788796521537 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_179_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (3788796521537 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_179_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (3788796521537 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_179_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_180_1 :
+    Uω (aρ 1) (bρ 1) (952804716581 / 8000000000000 : ℝ) ≤
+      (-(5458973319820460563247 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2252730294049721 / 20000000000000000 : ℝ) ≤
+      (-(5458973319820460563247 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2252730294049721 / 20000000000000000 : ℝ) =
+            (2252730294049721 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2252730294049721 / 1250000000000000 : ℝ) ≤
+            (1472498483179539436753 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1126221616408471 / 10000000000000000 : ℝ)) (q :=
+      (2252730294049721 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_180_2 :
+    Uω (aρ 2) (bρ 2) (952804716581 / 8000000000000 : ℝ) ≤
+      (-(11028304978811575276189 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (220355348279351 / 2000000000000000 : ℝ) ≤
+      (-(11028304978811575276189 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (220355348279351 / 2000000000000000 : ℝ) =
+            (220355348279351 / 125000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (220355348279351 / 125000000000000 : ℝ) ≤
+            (2834638627188424723811 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (55040690883113 / 500000000000000 : ℝ)) (q :=
+      (220355348279351 / 2000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_180_3 :
+    Uω (aρ 3) (bρ 3) (952804716581 / 8000000000000 : ℝ) ≤
+      (-(11260283236257531228509 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (420730712731861 / 4000000000000000 : ℝ) ≤
+      (-(11260283236257531228509 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (420730712731861 / 4000000000000000 : ℝ) =
+            (420730712731861 / 250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (420730712731861 / 250000000000000 : ℝ) ≤
+            (2602660369742468771491 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (209662000975611 / 2000000000000000 : ℝ)) (q :=
+      (420730712731861 / 4000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_180_4 :
+    Uω (aρ 4) (bρ 4) (952804716581 / 8000000000000 : ℝ) ≤
+      (-(23370105378852098455103 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (966160368188257 / 10000000000000000 : ℝ) ≤
+      (-(23370105378852098455103 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (966160368188257 / 10000000000000000 : ℝ) =
+            (966160368188257 / 625000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (966160368188257 / 625000000000000 : ℝ) ≤
+            (4355781833147901544897 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (119408919832533 / 1250000000000000 : ℝ)) (q :=
+      (966160368188257 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_180_5 :
+    Uω (aρ 5) (bρ 5) (952804716581 / 8000000000000 : ℝ) ≤
+      (-(99718511009526859259 / 40000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1653293728550453 / 20000000000000000 : ℝ) ≤
+      (-(99718511009526859259 / 40000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1653293728550453 / 20000000000000000 : ℝ) =
+            (1653293728550453 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1653293728550453 / 1250000000000000 : ℝ) ≤
+            (11185037838473140741 / 40000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (794434270169203 / 10000000000000000 : ℝ)) (q :=
+      (1653293728550453 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_180_6 :
+    Uω (aρ 6) (bρ 6) (952804716581 / 8000000000000 : ℝ) ≤
+      (-(7087845139958546029759 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (587104198732817 / 10000000000000000 : ℝ) ≤
+      (-(7087845139958546029759 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (587104198732817 / 10000000000000000 : ℝ) =
+            (587104198732817 / 312500000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (587104198732817 / 312500000000000 : ℝ) ≤
+            (1576494613791453970241 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1885114508689 / 39062500000000 : ℝ)) (q :=
+      (587104198732817 / 10000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_180_7 :
+    Uω (aρ 7) (bρ 7) (952804716581 / 8000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_180_8 :
+    Uω (aρ 8) (bρ 8) (952804716581 / 8000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_180_9 :
+    Uω (aρ 9) (bρ 9) (952804716581 / 8000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_180_10 :
+    Uω (aρ 10) (bρ 10) (952804716581 / 8000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_180_11 :
+    Uω (aρ 11) (bρ 11) (952804716581 / 8000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_180_12 :
+    Uω (aρ 12) (bρ 12) (952804716581 / 8000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_180_13 :
+    Uω (aρ 13) (bρ 13) (952804716581 / 8000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_180_14 :
+    Uω (aρ 14) (bρ 14) (952804716581 / 8000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_180_15 :
+    Uω (aρ 15) (bρ 15) (952804716581 / 8000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_180_16 :
+    Uω (aρ 16) (bρ 16) (952804716581 / 8000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_180 :
+    Uρ (952804716581 / 8000000000000 : ℝ) ≤
+      (-(5710967759615853715893 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (952804716581 / 8000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(5458973319820460563247 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_180_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (952804716581 / 8000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(11028304978811575276189 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_180_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (952804716581 / 8000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(11260283236257531228509 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_180_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (952804716581 / 8000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(23370105378852098455103 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_180_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (952804716581 / 8000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(99718511009526859259 / 40000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_180_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (952804716581 / 8000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(7087845139958546029759 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_180_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (952804716581 / 8000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_180_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (952804716581 / 8000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_180_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (952804716581 / 8000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_180_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (952804716581 / 8000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_180_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (952804716581 / 8000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_180_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (952804716581 / 8000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_180_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (952804716581 / 8000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_180_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (952804716581 / 8000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_180_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (952804716581 / 8000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_180_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (952804716581 / 8000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_180_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_181_1 :
+    Uω (aρ 1) (bρ 1) (3833641211111 / 32000000000000 : ℝ) ≤
+      (-(21773869343935046491781 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4533492068996301 / 40000000000000000 : ℝ) ≤
+      (-(21773869343935046491781 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (4533492068996301 / 40000000000000000 : ℝ) =
+            (4533492068996301 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4533492068996301 / 2500000000000000 : ℝ) ≤
+            (5952017868064953508219 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1133230374110963 / 10000000000000000 : ℝ)) (q :=
+      (4533492068996301 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_181_2 :
+    Uω (aρ 2) (bρ 2) (3833641211111 / 32000000000000 : ℝ) ≤
+      (-(21993159284954283541103 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (887031851692717 / 8000000000000000 : ℝ) ≤
+      (-(21993159284954283541103 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (887031851692717 / 8000000000000000 : ℝ) =
+            (887031851692717 / 500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (887031851692717 / 500000000000000 : ℝ) ≤
+            (5732727927045716458897 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (221566596270921 / 2000000000000000 : ℝ)) (q :=
+      (887031851692717 / 8000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_181_3 :
+    Uω (aρ 3) (bρ 3) (3833641211111 / 32000000000000 : ℝ) ≤
+      (-(11226974797381275692259 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (847085691507757 / 8000000000000000 : ℝ) ≤
+      (-(11226974797381275692259 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (847085691507757 / 8000000000000000 : ℝ) =
+            (847085691507757 / 500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (847085691507757 / 500000000000000 : ℝ) ≤
+            (2635968808618724307741 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (211072737448441 / 2000000000000000 : ℝ)) (q :=
+      (847085691507757 / 8000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_181_4 :
+    Uω (aρ 4) (bρ 4) (3833641211111 / 32000000000000 : ℝ) ≤
+      (-(1164851421249206849263 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3892986537824593 / 40000000000000000 : ℝ) ≤
+      (-(1164851421249206849263 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (3892986537824593 / 40000000000000000 : ℝ) =
+            (3892986537824593 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3892986537824593 / 2500000000000000 : ℝ) ≤
+            (221442939350793150737 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (962436908450109 / 10000000000000000 : ℝ)) (q :=
+      (3892986537824593 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_181_5 :
+    Uω (aρ 5) (bρ 5) (3833641211111 / 32000000000000 : ℝ) ≤
+      (-(24841844510715917490201 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3335741528413867 / 40000000000000000 : ℝ) ≤
+      (-(24841844510715917490201 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (3335741528413867 / 40000000000000000 : ℝ) =
+            (3335741528413867 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3335741528413867 / 2500000000000000 : ℝ) ≤
+            (2884042701284082509799 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (401002161539873 / 5000000000000000 : ℝ)) (q :=
+      (3335741528413867 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_181_6 :
+    Uω (aρ 6) (bρ 6) (3833641211111 / 32000000000000 : ℝ) ≤
+      (-(28207669885361066139551 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (476481944676323 / 8000000000000000 : ℝ) ≤
+      (-(28207669885361066139551 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (476481944676323 / 8000000000000000 : ℝ) =
+            (476481944676323 / 250000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (476481944676323 / 250000000000000 : ℝ) ≤
+            (6449689129638933860449 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (24628939785181 / 500000000000000 : ℝ)) (q :=
+      (476481944676323 / 8000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_181_7 :
+    Uω (aρ 7) (bρ 7) (3833641211111 / 32000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_181_8 :
+    Uω (aρ 8) (bρ 8) (3833641211111 / 32000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_181_9 :
+    Uω (aρ 9) (bρ 9) (3833641211111 / 32000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_181_10 :
+    Uω (aρ 10) (bρ 10) (3833641211111 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_181_11 :
+    Uω (aρ 11) (bρ 11) (3833641211111 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_181_12 :
+    Uω (aρ 12) (bρ 12) (3833641211111 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_181_13 :
+    Uω (aρ 13) (bρ 13) (3833641211111 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_181_14 :
+    Uω (aρ 14) (bρ 14) (3833641211111 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_181_15 :
+    Uω (aρ 15) (bρ 15) (3833641211111 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_181_16 :
+    Uω (aρ 16) (bρ 16) (3833641211111 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_181 :
+    Uρ (3833641211111 / 32000000000000 : ℝ) ≤
+      (-(22816840024414922085761 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (3833641211111 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(21773869343935046491781 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_181_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (3833641211111 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(21993159284954283541103 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_181_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (3833641211111 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(11226974797381275692259 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_181_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (3833641211111 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1164851421249206849263 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_181_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (3833641211111 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(24841844510715917490201 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_181_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (3833641211111 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(28207669885361066139551 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_181_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (3833641211111 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_181_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (3833641211111 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_181_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (3833641211111 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_181_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (3833641211111 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_181_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (3833641211111 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_181_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (3833641211111 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_181_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (3833641211111 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_181_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (3833641211111 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_181_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (3833641211111 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_181_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (3833641211111 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_181_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_182_1 :
+    Uω (aρ 1) (bρ 1) (1928031777949 / 16000000000000 : ℝ) ≤
+      (-(542805695774513738577 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2280761753128937 / 20000000000000000 : ℝ) ≤
+      (-(542805695774513738577 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2280761753128937 / 20000000000000000 : ℝ) =
+            (2280761753128937 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2280761753128937 / 1250000000000000 : ℝ) ≤
+            (150341484525486261423 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (285059777498953 / 2500000000000000 : ℝ)) (q :=
+      (2280761753128937 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_182_2 :
+    Uω (aρ 2) (bρ 2) (1928031777949 / 16000000000000 : ℝ) ≤
+      (-(10965054680357433308143 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (557901405604871 / 5000000000000000 : ℝ) ≤
+      (-(10965054680357433308143 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (557901405604871 / 5000000000000000 : ℝ) =
+            (557901405604871 / 312500000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (557901405604871 / 312500000000000 : ℝ) ≤
+            (2897888925642566691857 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1114851991796359 / 10000000000000000 : ℝ)) (q :=
+      (557901405604871 / 5000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_182_3 :
+    Uω (aρ 3) (bρ 3) (1928031777949 / 16000000000000 : ℝ) ≤
+      (-(22387776461924809992277 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2131774275828701 / 20000000000000000 : ℝ) ≤
+      (-(22387776461924809992277 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2131774275828701 / 20000000000000000 : ℝ) =
+            (2131774275828701 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2131774275828701 / 1250000000000000 : ℝ) ≤
+            (5338110750075190007723 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (132802093944447 / 1250000000000000 : ℝ)) (q :=
+      (2131774275828701 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_182_4 :
+    Uω (aρ 4) (bρ 4) (1928031777949 / 16000000000000 : ℝ) ≤
+      (-(11612246722324450860351 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1960663483694801 / 20000000000000000 : ℝ) ≤
+      (-(11612246722324450860351 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1960663483694801 / 20000000000000000 : ℝ) =
+            (1960663483694801 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1960663483694801 / 1250000000000000 : ℝ) ≤
+            (2250696883675549139649 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (242400035121669 / 2500000000000000 : ℝ)) (q :=
+      (1960663483694801 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_182_5 :
+    Uω (aρ 5) (bρ 5) (1928031777949 / 16000000000000 : ℝ) ≤
+      (-(6188721354787161308899 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (210304707658777 / 2500000000000000 : ℝ) ≤
+      (-(6188721354787161308899 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (210304707658777 / 2500000000000000 : ℝ) =
+            (210304707658777 / 156250000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (210304707658777 / 156250000000000 : ℝ) ≤
+            (742750448212838691101 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (809564237397091 / 10000000000000000 : ℝ)) (q :=
+      (210304707658777 / 2500000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_182_6 :
+    Uω (aρ 6) (bρ 6) (1928031777949 / 16000000000000 : ℝ) ≤
+      (-(14033415111008318688351 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (151012556253559 / 2500000000000000 : ℝ) ≤
+      (-(14033415111008318688351 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (151012556253559 / 2500000000000000 : ℝ) =
+            (151012556253559 / 78125000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (151012556253559 / 78125000000000 : ℝ) ≤
+            (3295264396491681311649 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (502467401295347 / 10000000000000000 : ℝ)) (q :=
+      (151012556253559 / 2500000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_182_7 :
+    Uω (aρ 7) (bρ 7) (1928031777949 / 16000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_182_8 :
+    Uω (aρ 8) (bρ 8) (1928031777949 / 16000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_182_9 :
+    Uω (aρ 9) (bρ 9) (1928031777949 / 16000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_182_10 :
+    Uω (aρ 10) (bρ 10) (1928031777949 / 16000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_182_11 :
+    Uω (aρ 11) (bρ 11) (1928031777949 / 16000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_182_12 :
+    Uω (aρ 12) (bρ 12) (1928031777949 / 16000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_182_13 :
+    Uω (aρ 13) (bρ 13) (1928031777949 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_182_14 :
+    Uω (aρ 14) (bρ 14) (1928031777949 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_182_15 :
+    Uω (aρ 15) (bρ 15) (1928031777949 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_182_16 :
+    Uω (aρ 16) (bρ 16) (1928031777949 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_182 :
+    Uρ (1928031777949 / 16000000000000 : ℝ) ≤
+      (-(11395079391242920398171 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1928031777949 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(542805695774513738577 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_182_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1928031777949 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(10965054680357433308143 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_182_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1928031777949 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(22387776461924809992277 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_182_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1928031777949 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11612246722324450860351 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_182_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1928031777949 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(6188721354787161308899 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_182_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1928031777949 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(14033415111008318688351 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_182_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1928031777949 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_182_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1928031777949 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_182_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1928031777949 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_182_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1928031777949 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_182_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1928031777949 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_182_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1928031777949 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_182_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1928031777949 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_182_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1928031777949 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_182_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1928031777949 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_182_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1928031777949 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_182_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_183_1 :
+    Uω (aρ 1) (bρ 1) (121903382671 / 1000000000000 : ℝ) ≤
+      (-(431801468111400782417 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2308793126527319 / 20000000000000000 : ℝ) ≤
+      (-(431801468111400782417 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2308793126527319 / 20000000000000000 : ℝ) =
+            (2308793126527319 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2308793126527319 / 1250000000000000 : ℝ) ≤
+            (122716276128599217583 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1154256517902319 / 10000000000000000 : ℝ)) (q :=
+      (2308793126527319 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_183_2 :
+    Uω (aρ 2) (bρ 2) (121903382671 / 1000000000000 : ℝ) ≤
+      (-(10902595848206565310453 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2259657160477287 / 20000000000000000 : ℝ) ≤
+      (-(10902595848206565310453 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2259657160477287 / 20000000000000000 : ℝ) =
+            (2259657160477287 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2259657160477287 / 1250000000000000 : ℝ) ≤
+            (2960347757793434689547 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1128889564362287 / 10000000000000000 : ℝ)) (q :=
+      (2259657160477287 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_183_3 :
+    Uω (aρ 3) (bρ 3) (121903382671 / 1000000000000 : ℝ) ≤
+      (-(5564184476913954951537 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (107994628219273 / 1000000000000000 : ℝ) ≤
+      (-(5564184476913954951537 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (107994628219273 / 1000000000000000 : ℝ) =
+            (107994628219273 / 62500000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (107994628219273 / 62500000000000 : ℝ) ≤
+            (1367287326086045048463 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (53826053731023 / 500000000000000 : ℝ)) (q :=
+      (107994628219273 / 1000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_183_4 :
+    Uω (aρ 4) (bρ 4) (121903382671 / 1000000000000 : ℝ) ≤
+      (-(23081016992543017206069 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1988997162470901 / 20000000000000000 : ℝ) ≤
+      (-(23081016992543017206069 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1988997162470901 / 20000000000000000 : ℝ) =
+            (1988997162470901 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1988997162470901 / 1250000000000000 : ℝ) ≤
+            (4644870219456982793931 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (983919853770901 / 10000000000000000 : ℝ)) (q :=
+      (1988997162470901 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_183_5 :
+    Uω (aρ 5) (bρ 5) (121903382671 / 1000000000000 : ℝ) ≤
+      (-(196666997090944639807 / 80000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (213942769321833 / 2500000000000000 : ℝ) ≤
+      (-(196666997090944639807 / 80000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (213942769321833 / 2500000000000000 : ℝ) =
+            (213942769321833 / 156250000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (213942769321833 / 156250000000000 : ℝ) ≤
+            (25140100605055360193 / 80000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (12885230706401 / 156250000000000 : ℝ)) (q :=
+      (213942769321833 / 2500000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_183_6 :
+    Uω (aρ 6) (bρ 6) (121903382671 / 1000000000000 : ℝ) ≤
+      (-(27793218382762658487643 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (620805937068547 / 10000000000000000 : ℝ) ≤
+      (-(27793218382762658487643 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (620805937068547 / 10000000000000000 : ℝ) =
+            (620805937068547 / 312500000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (620805937068547 / 312500000000000 : ℝ) ≤
+            (6864140632237341512357 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (260982429956047 / 5000000000000000 : ℝ)) (q :=
+      (620805937068547 / 10000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_183_7 :
+    Uω (aρ 7) (bρ 7) (121903382671 / 1000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_183_8 :
+    Uω (aρ 8) (bρ 8) (121903382671 / 1000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_183_9 :
+    Uω (aρ 9) (bρ 9) (121903382671 / 1000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_183_10 :
+    Uω (aρ 10) (bρ 10) (121903382671 / 1000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_183_11 :
+    Uω (aρ 11) (bρ 11) (121903382671 / 1000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_183_12 :
+    Uω (aρ 12) (bρ 12) (121903382671 / 1000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_183_13 :
+    Uω (aρ 13) (bρ 13) (121903382671 / 1000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_183_14 :
+    Uω (aρ 14) (bρ 14) (121903382671 / 1000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_183_15 :
+    Uω (aρ 15) (bρ 15) (121903382671 / 1000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_183_16 :
+    Uω (aρ 16) (bρ 16) (121903382671 / 1000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_183 :
+    Uρ (121903382671 / 1000000000000 : ℝ) ≤
+      (-(22737794411188599503227 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (121903382671 / 1000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(431801468111400782417 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_183_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (121903382671 / 1000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(10902595848206565310453 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_183_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (121903382671 / 1000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(5564184476913954951537 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_183_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (121903382671 / 1000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(23081016992543017206069 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_183_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (121903382671 / 1000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(196666997090944639807 / 80000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_183_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (121903382671 / 1000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(27793218382762658487643 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_183_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (121903382671 / 1000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_183_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (121903382671 / 1000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_183_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (121903382671 / 1000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_183_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (121903382671 / 1000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_183_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (121903382671 / 1000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_183_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (121903382671 / 1000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_183_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (121903382671 / 1000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_183_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (121903382671 / 1000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_183_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (121903382671 / 1000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_183_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (121903382671 / 1000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_183_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/U15.lean b/LeanPool/Zeta5Irrational/Table/U15.lean
new file mode 100644
index 0000000000..1f01f854ec
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/U15.lean
@@ -0,0 +1,3351 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-! # Certified arcsine potential bounds (U15) -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma U_184_1 :
+    Uω (aρ 1) (bρ 1) (1972876467523 / 16000000000000 : ℝ) ≤
+      (-(10734696757160727232713 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1168412208664483 / 10000000000000000 : ℝ) ≤
+      (-(10734696757160727232713 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1168412208664483 / 10000000000000000 : ℝ) =
+            (1168412208664483 / 625000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1168412208664483 / 625000000000000 : ℝ) ≤
+            (3128246848839272767287 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1168273843212091 / 10000000000000000 : ℝ)) (q :=
+      (1168412208664483 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_184_2 :
+    Uω (aρ 2) (bρ 2) (1972876467523 / 16000000000000 : ℝ) ≤
+      (-(10840908887218247258763 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1143854059566267 / 10000000000000000 : ℝ) ≤
+      (-(10840908887218247258763 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1143854059566267 / 10000000000000000 : ℝ) =
+            (1143854059566267 / 625000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1143854059566267 / 625000000000000 : ℝ) ≤
+            (3022034718781752741237 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1142926557525659 / 10000000000000000 : ℝ)) (q :=
+      (1143854059566267 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_184_3 :
+    Uω (aρ 3) (bρ 3) (1972876467523 / 16000000000000 : ℝ) ≤
+      (-(11063702462299722560483 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1094004261679317 / 10000000000000000 : ℝ) ≤
+      (-(11063702462299722560483 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1094004261679317 / 10000000000000000 : ℝ) =
+            (1094004261679317 / 625000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1094004261679317 / 625000000000000 : ℝ) ≤
+            (2799241143700277439517 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1090623068101759 / 10000000000000000 : ℝ)) (q :=
+      (1094004261679317 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_184_4 :
+    Uω (aρ 4) (bρ 4) (1972876467523 / 16000000000000 : ℝ) ≤
+      (-(22939613024631683193711 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2017322162972089 / 20000000000000000 : ℝ) ≤
+      (-(22939613024631683193711 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2017322162972089 / 20000000000000000 : ℝ) =
+            (2017322162972089 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2017322162972089 / 1250000000000000 : ℝ) ≤
+            (4786274187368316806289 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (499115444390107 / 5000000000000000 : ℝ)) (q :=
+      (2017322162972089 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_184_5 :
+    Uω (aρ 5) (bρ 5) (1972876467523 / 16000000000000 : ℝ) ≤
+      (-(4882994056373813382959 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (174060933478281 / 2000000000000000 : ℝ) ≤
+      (-(4882994056373813382959 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (174060933478281 / 2000000000000000 : ℝ) =
+            (174060933478281 / 125000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (174060933478281 / 125000000000000 : ℝ) ≤
+            (662183386026186617041 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (167941595985187 / 2000000000000000 : ℝ)) (q :=
+      (174060933478281 / 2000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_184_6 :
+    Uω (aρ 6) (bρ 6) (1972876467523 / 16000000000000 : ℝ) ≤
+      (-(13764778325010343518127 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3186959537949 / 50000000000000 : ℝ) ≤
+      (-(13764778325010343518127 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (3186959537949 / 50000000000000 : ℝ) = (3186959537949 / 3125000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3186959537949 / 3125000000000 : ℝ) ≤
+            (98165280989656481873 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (21644913418509 / 400000000000000 : ℝ)) (q :=
+      (3186959537949 / 50000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_184_7 :
+    Uω (aρ 7) (bρ 7) (1972876467523 / 16000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_184_8 :
+    Uω (aρ 8) (bρ 8) (1972876467523 / 16000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_184_9 :
+    Uω (aρ 9) (bρ 9) (1972876467523 / 16000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_184_10 :
+    Uω (aρ 10) (bρ 10) (1972876467523 / 16000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_184_11 :
+    Uω (aρ 11) (bρ 11) (1972876467523 / 16000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_184_12 :
+    Uω (aρ 12) (bρ 12) (1972876467523 / 16000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_184_13 :
+    Uω (aρ 13) (bρ 13) (1972876467523 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_184_14 :
+    Uω (aρ 14) (bρ 14) (1972876467523 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_184_15 :
+    Uω (aρ 15) (bρ 15) (1972876467523 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_184_16 :
+    Uω (aρ 16) (bρ 16) (1972876467523 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_184 :
+    Uρ (1972876467523 / 16000000000000 : ℝ) ≤
+      (-(11343342076695662676383 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1972876467523 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(10734696757160727232713 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_184_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1972876467523 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(10840908887218247258763 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_184_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1972876467523 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(11063702462299722560483 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_184_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1972876467523 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(22939613024631683193711 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_184_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1972876467523 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(4882994056373813382959 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_184_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1972876467523 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(13764778325010343518127 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_184_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1972876467523 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_184_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1972876467523 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_184_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1972876467523 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_184_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1972876467523 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_184_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1972876467523 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_184_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1972876467523 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_184_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1972876467523 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_184_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1972876467523 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_184_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1972876467523 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_184_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1972876467523 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_184_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_185_1 :
+    Uω (aρ 1) (bρ 1) (199529881231 / 1600000000000 : ℝ) ≤
+      (-(4270030594954638318783 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1182427814235847 / 10000000000000000 : ℝ) ≤
+      (-(4270030594954638318783 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1182427814235847 / 10000000000000000 : ℝ) =
+            (1182427814235847 / 625000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1182427814235847 / 625000000000000 : ℝ) ≤
+            (1275146847445361681217 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (36946596526967 / 312500000000000 : ℝ)) (q :=
+      (1182427814235847 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_185_2 :
+    Uω (aρ 2) (bρ 2) (199529881231 / 1600000000000 : ℝ) ≤
+      (-(21559949844423918091557 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7236745373357 / 62500000000000 : ℝ) ≤
+      (-(21559949844423918091557 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (7236745373357 / 62500000000000 : ℝ) = (7236745373357 / 3906250000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7236745373357 / 3906250000000 : ℝ) ≤
+            (6165937367576081908443 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (115696299237549 / 1000000000000000 : ℝ)) (q :=
+      (7236745373357 / 62500000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_185_3 :
+    Uω (aρ 3) (bρ 3) (199529881231 / 1600000000000 : ℝ) ≤
+      (-(21999733418108408918803 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (277015280245117 / 2500000000000000 : ℝ) ≤
+      (-(21999733418108408918803 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (277015280245117 / 2500000000000000 : ℝ) =
+            (277015280245117 / 156250000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (277015280245117 / 156250000000000 : ℝ) ≤
+            (5726153793891591081197 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (552361410606093 / 5000000000000000 : ℝ)) (q :=
+      (277015280245117 / 2500000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_185_4 :
+    Uω (aρ 4) (bρ 4) (199529881231 / 1600000000000 : ℝ) ≤
+      (-(22800221341664611774923 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (409127770634337 / 4000000000000000 : ℝ) ≤
+      (-(22800221341664611774923 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (409127770634337 / 4000000000000000 : ℝ) =
+            (409127770634337 / 250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (409127770634337 / 250000000000000 : ℝ) ≤
+            (4925665870335388225077 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (202506722697587 / 2000000000000000 : ℝ)) (q :=
+      (409127770634337 / 4000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_185_5 :
+    Uω (aρ 5) (bρ 5) (199529881231 / 1600000000000 : ℝ) ≤
+      (-(24249554734050633766169 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (884820586671631 / 10000000000000000 : ℝ) ≤
+      (-(24249554734050633766169 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (884820586671631 / 10000000000000000 : ℝ) =
+            (884820586671631 / 625000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (884820586671631 / 625000000000000 : ℝ) ≤
+            (3476332477949366233831 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (53420365812157 / 625000000000000 : ℝ)) (q :=
+      (884820586671631 / 10000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_185_6 :
+    Uω (aρ 6) (bρ 6) (199529881231 / 1600000000000 : ℝ) ≤
+      (-(5454999569515305561689 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (653825558776359 / 10000000000000000 : ℝ) ≤
+      (-(5454999569515305561689 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (653825558776359 / 10000000000000000 : ℝ) =
+            (653825558776359 / 625000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (653825558776359 / 625000000000000 : ℝ) ≤
+            (90177872884694438311 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (17499255385749 / 312500000000000 : ℝ)) (q :=
+      (653825558776359 / 10000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_185_7 :
+    Uω (aρ 7) (bρ 7) (199529881231 / 1600000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_185_8 :
+    Uω (aρ 8) (bρ 8) (199529881231 / 1600000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_185_9 :
+    Uω (aρ 9) (bρ 9) (199529881231 / 1600000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_185_10 :
+    Uω (aρ 10) (bρ 10) (199529881231 / 1600000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_185_11 :
+    Uω (aρ 11) (bρ 11) (199529881231 / 1600000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_185_12 :
+    Uω (aρ 12) (bρ 12) (199529881231 / 1600000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_185_13 :
+    Uω (aρ 13) (bρ 13) (199529881231 / 1600000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_185_14 :
+    Uω (aρ 14) (bρ 14) (199529881231 / 1600000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_185_15 :
+    Uω (aρ 15) (bρ 15) (199529881231 / 1600000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_185_16 :
+    Uω (aρ 16) (bρ 16) (199529881231 / 1600000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_185 :
+    Uρ (199529881231 / 1600000000000 : ℝ) ≤
+      (-(2829593273725598213717 / 1250000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (199529881231 / 1600000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(4270030594954638318783 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_185_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (199529881231 / 1600000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(21559949844423918091557 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_185_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (199529881231 / 1600000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(21999733418108408918803 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_185_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (199529881231 / 1600000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(22800221341664611774923 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_185_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (199529881231 / 1600000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(24249554734050633766169 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_185_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (199529881231 / 1600000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(5454999569515305561689 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_185_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (199529881231 / 1600000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_185_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (199529881231 / 1600000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_185_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (199529881231 / 1600000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_185_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (199529881231 / 1600000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_185_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (199529881231 / 1600000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_185_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (199529881231 / 1600000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_185_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (199529881231 / 1600000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_185_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (199529881231 / 1600000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_185_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (199529881231 / 1600000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_185_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (199529881231 / 1600000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_185_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_186_1 :
+    Uω (aρ 1) (bρ 1) (2017721157097 / 16000000000000 : ℝ) ≤
+      (-(2654039731137533813037 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (478577352551123 / 4000000000000000 : ℝ) ≤
+      (-(2654039731137533813037 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (478577352551123 / 4000000000000000 : ℝ) =
+            (478577352551123 / 250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (478577352551123 / 250000000000000 : ℝ) ≤
+            (811696170362466186963 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (119630825765499 / 1000000000000000 : ℝ)) (q :=
+      (478577352551123 / 4000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_186_2 :
+    Uω (aρ 2) (bρ 2) (2017721157097 / 16000000000000 : ℝ) ≤
+      (-(4287910305227927671211 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (292976047697431 / 2500000000000000 : ℝ) ≤
+      (-(4287910305227927671211 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (292976047697431 / 2500000000000000 : ℝ) =
+            (292976047697431 / 156250000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (292976047697431 / 156250000000000 : ℝ) ≤
+            (1257267137172072328789 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1170998888988823 / 10000000000000000 : ℝ)) (q :=
+      (292976047697431 / 2500000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_186_3 :
+    Uω (aρ 3) (bρ 3) (2017721157097 / 16000000000000 : ℝ) ≤
+      (-(21873680991519987290401 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (280529225611763 / 2500000000000000 : ℝ) ≤
+      (-(21873680991519987290401 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (280529225611763 / 2500000000000000 : ℝ) =
+            (280529225611763 / 156250000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (280529225611763 / 156250000000000 : ℝ) ≤
+            (5852206220480012709599 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1118820418653479 / 10000000000000000 : ℝ)) (q :=
+      (280529225611763 / 2500000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_186_4 :
+    Uω (aρ 4) (bρ 4) (2017721157097 / 16000000000000 : ℝ) ≤
+      (-(22662784376969185616087 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (129621723770819 / 1250000000000000 : ℝ) ≤
+      (-(22662784376969185616087 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (129621723770819 / 1250000000000000 : ℝ) =
+            (129621723770819 / 78125000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (129621723770819 / 78125000000000 : ℝ) ≤
+            (5063102835030814383913 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1026828375157479 / 10000000000000000 : ℝ)) (q :=
+      (129621723770819 / 1250000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_186_5 :
+    Uω (aρ 5) (bρ 5) (2017721157097 / 16000000000000 : ℝ) ≤
+      (-(4817403452281815741051 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (71945580042487 / 800000000000000 : ℝ) ≤
+      (-(4817403452281815741051 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (71945580042487 / 800000000000000 : ℝ) = (71945580042487 / 50000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (71945580042487 / 50000000000000 : ℝ) ≤
+            (727773990118184258949 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (17394204304431 / 200000000000000 : ℝ)) (q :=
+      (71945580042487 / 800000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_186_6 :
+    Uω (aρ 6) (bρ 6) (2017721157097 / 16000000000000 : ℝ) ≤
+      (-(27028809110634695454701 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (670121781883347 / 10000000000000000 : ℝ) ≤
+      (-(27028809110634695454701 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (670121781883347 / 10000000000000000 : ℝ) =
+            (670121781883347 / 625000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (670121781883347 / 625000000000000 : ℝ) ≤
+            (697078101365304545299 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (578554653066069 / 10000000000000000 : ℝ)) (q :=
+      (670121781883347 / 10000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_186_7 :
+    Uω (aρ 7) (bρ 7) (2017721157097 / 16000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_186_8 :
+    Uω (aρ 8) (bρ 8) (2017721157097 / 16000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_186_9 :
+    Uω (aρ 9) (bρ 9) (2017721157097 / 16000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_186_10 :
+    Uω (aρ 10) (bρ 10) (2017721157097 / 16000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_186_11 :
+    Uω (aρ 11) (bρ 11) (2017721157097 / 16000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_186_12 :
+    Uω (aρ 12) (bρ 12) (2017721157097 / 16000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_186_13 :
+    Uω (aρ 13) (bρ 13) (2017721157097 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_186_14 :
+    Uω (aρ 14) (bρ 14) (2017721157097 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_186_15 :
+    Uω (aρ 15) (bρ 15) (2017721157097 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_186_16 :
+    Uω (aρ 16) (bρ 16) (2017721157097 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_186 :
+    Uρ (2017721157097 / 16000000000000 : ℝ) ≤
+      (-(5646977118127164902371 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (2017721157097 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2654039731137533813037 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_186_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (2017721157097 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(4287910305227927671211 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_186_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (2017721157097 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(21873680991519987290401 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_186_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (2017721157097 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(22662784376969185616087 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_186_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (2017721157097 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(4817403452281815741051 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_186_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (2017721157097 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(27028809110634695454701 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_186_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (2017721157097 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_186_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (2017721157097 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_186_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (2017721157097 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_186_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (2017721157097 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_186_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (2017721157097 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_186_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (2017721157097 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_186_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (2017721157097 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_186_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (2017721157097 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_186_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (2017721157097 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_186_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (2017721157097 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_186_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_187_1 :
+    Uω (aρ 1) (bρ 1) (510035875471 / 4000000000000 : ℝ) ≤
+      (-(10557927693038472750489 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (605229455712777 / 5000000000000000 : ℝ) ≤
+      (-(10557927693038472750489 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (605229455712777 / 5000000000000000 : ℝ) =
+            (605229455712777 / 312500000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (605229455712777 / 312500000000000 : ℝ) ≤
+            (3305015912961527249511 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (75645334516163 / 625000000000000 : ℝ)) (q :=
+      (605229455712777 / 5000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_187_2 :
+    Uω (aρ 2) (bρ 2) (510035875471 / 4000000000000 : ℝ) ≤
+      (-(21320587743862525124249 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (474371544914651 / 4000000000000000 : ℝ) ≤
+      (-(21320587743862525124249 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (474371544914651 / 4000000000000000 : ℝ) =
+            (474371544914651 / 250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (474371544914651 / 250000000000000 : ℝ) ≤
+            (6405299468137474875751 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (237006853298151 / 2000000000000000 : ℝ)) (q :=
+      (474371544914651 / 4000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_187_3 :
+    Uω (aρ 3) (bρ 3) (510035875471 / 4000000000000 : ℝ) ≤
+      (-(4349841371920999492129 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2272343292631247 / 20000000000000000 : ℝ) ≤
+      (-(4349841371920999492129 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2272343292631247 / 20000000000000000 : ℝ) =
+            (2272343292631247 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2272343292631247 / 1250000000000000 : ℝ) ≤
+            (1195336070479000507871 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1132915940898747 / 10000000000000000 : ℝ)) (q :=
+      (2272343292631247 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_187_4 :
+    Uω (aρ 4) (bρ 4) (510035875471 / 4000000000000 : ℝ) ≤
+      (-(28159058803363137901 / 12500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1051124336229791 / 10000000000000000 : ℝ) ≤
+      (-(28159058803363137901 / 12500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1051124336229791 / 10000000000000000 : ℝ) =
+            (1051124336229791 / 625000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1051124336229791 / 625000000000000 : ℝ) ≤
+            (6498300211636862099 / 12500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (520557750896041 / 5000000000000000 : ℝ)) (q :=
+      (1051124336229791 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_187_5 :
+    Uω (aρ 5) (bρ 5) (510035875471 / 4000000000000 : ℝ) ≤
+      (-(11963626737311393197339 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1827606020754751 / 20000000000000000 : ℝ) ≤
+      (-(11963626737311393197339 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1827606020754751 / 20000000000000000 : ℝ) =
+            (1827606020754751 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1827606020754751 / 1250000000000000 : ℝ) ≤
+            (1899316868688606802661 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (884662769422251 / 10000000000000000 : ℝ)) (q :=
+      (1827606020754751 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_187_6 :
+    Uω (aρ 6) (bρ 6) (510035875471 / 4000000000000 : ℝ) ≤
+      (-(13395175617456387926221 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (686293409856847 / 10000000000000000 : ℝ) ≤
+      (-(13395175617456387926221 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (686293409856847 / 10000000000000000 : ℝ) =
+            (686293409856847 / 625000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (686293409856847 / 625000000000000 : ℝ) ≤
+            (467767988543612073779 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (298441971760597 / 5000000000000000 : ℝ)) (q :=
+      (686293409856847 / 10000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_187_7 :
+    Uω (aρ 7) (bρ 7) (510035875471 / 4000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_187_8 :
+    Uω (aρ 8) (bρ 8) (510035875471 / 4000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_187_9 :
+    Uω (aρ 9) (bρ 9) (510035875471 / 4000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_187_10 :
+    Uω (aρ 10) (bρ 10) (510035875471 / 4000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_187_11 :
+    Uω (aρ 11) (bρ 11) (510035875471 / 4000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_187_12 :
+    Uω (aρ 12) (bρ 12) (510035875471 / 4000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_187_13 :
+    Uω (aρ 13) (bρ 13) (510035875471 / 4000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_187_14 :
+    Uω (aρ 14) (bρ 14) (510035875471 / 4000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_187_15 :
+    Uω (aρ 15) (bρ 15) (510035875471 / 4000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_187_16 :
+    Uω (aρ 16) (bρ 16) (510035875471 / 4000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_187 :
+    Uρ (510035875471 / 4000000000000 : ℝ) ≤
+      (-(22540107043114398128433 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (510035875471 / 4000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(10557927693038472750489 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_187_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (510035875471 / 4000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(21320587743862525124249 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_187_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (510035875471 / 4000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(4349841371920999492129 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_187_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (510035875471 / 4000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(28159058803363137901 / 12500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_187_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (510035875471 / 4000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(11963626737311393197339 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_187_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (510035875471 / 4000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(13395175617456387926221 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_187_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (510035875471 / 4000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_187_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (510035875471 / 4000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_187_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (510035875471 / 4000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_187_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (510035875471 / 4000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_187_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (510035875471 / 4000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_187_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (510035875471 / 4000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_187_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (510035875471 / 4000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_187_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (510035875471 / 4000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_187_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (510035875471 / 4000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_187_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (510035875471 / 4000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_187_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_188_1 :
+    Uω (aρ 1) (bρ 1) (1042494095729 / 8000000000000 : ℝ) ≤
+      (-(2088692305103148942643 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2476979730553741 / 20000000000000000 : ℝ) ≤
+      (-(2088692305103148942643 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2476979730553741 / 20000000000000000 : ℝ) =
+            (2476979730553741 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2476979730553741 / 1250000000000000 : ℝ) ≤
+            (683896416096851057357 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1238359328977491 / 10000000000000000 : ℝ)) (q :=
+      (2476979730553741 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_188_2 :
+    Uω (aρ 2) (bρ 2) (1042494095729 / 8000000000000 : ℝ) ≤
+      (-(5271707410071795261109 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2427954925298247 / 20000000000000000 : ℝ) ≤
+      (-(5271707410071795261109 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2427954925298247 / 20000000000000000 : ℝ) =
+            (2427954925298247 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2427954925298247 / 1250000000000000 : ℝ) ≤
+            (1659764392928204738891 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1213103536231997 / 10000000000000000 : ℝ)) (q :=
+      (2427954925298247 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_188_3 :
+    Uω (aρ 3) (bρ 3) (1042494095729 / 8000000000000 : ℝ) ≤
+      (-(4300967582928605563113 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (582139086203921 / 5000000000000000 : ℝ) ≤
+      (-(4300967582928605563113 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (582139086203921 / 5000000000000000 : ℝ) =
+            (582139086203921 / 312500000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (582139086203921 / 312500000000000 : ℝ) ≤
+            (1244209859471394436887 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (580550531049717 / 5000000000000000 : ℝ)) (q :=
+      (582139086203921 / 5000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_188_4 :
+    Uω (aρ 4) (bρ 4) (1042494095729 / 8000000000000 : ℝ) ≤
+      (-(22261662463063795315849 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (539707293795511 / 5000000000000000 : ℝ) ≤
+      (-(22261662463063795315849 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (539707293795511 / 5000000000000000 : ℝ) =
+            (539707293795511 / 312500000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (539707293795511 / 312500000000000 : ℝ) ≤
+            (5464224748936204684151 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (534834036765397 / 5000000000000000 : ℝ)) (q :=
+      (539707293795511 / 5000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_188_5 :
+    Uω (aρ 5) (bρ 5) (1042494095729 / 8000000000000 : ℝ) ≤
+      (-(23615658202251705360119 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1885449873843263 / 20000000000000000 : ℝ) ≤
+      (-(23615658202251705360119 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1885449873843263 / 20000000000000000 : ℝ) =
+            (1885449873843263 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1885449873843263 / 1250000000000000 : ℝ) ≤
+            (4110229009748294639881 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (914478691527013 / 10000000000000000 : ℝ)) (q :=
+      (1885449873843263 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_188_6 :
+    Uω (aρ 6) (bρ 6) (1042494095729 / 8000000000000 : ℝ) ≤
+      (-(329180582000059646687 / 125000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1436612075902289 / 20000000000000000 : ℝ) ≤
+      (-(329180582000059646687 / 125000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1436612075902289 / 20000000000000000 : ℝ) =
+            (1436612075902289 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1436612075902289 / 1250000000000000 : ℝ) ≤
+            (17393008149940353313 / 125000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (632881268726039 / 10000000000000000 : ℝ)) (q :=
+      (1436612075902289 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_188_7 :
+    Uω (aρ 7) (bρ 7) (1042494095729 / 8000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_188_8 :
+    Uω (aρ 8) (bρ 8) (1042494095729 / 8000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_188_9 :
+    Uω (aρ 9) (bρ 9) (1042494095729 / 8000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_188_10 :
+    Uω (aρ 10) (bρ 10) (1042494095729 / 8000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_188_11 :
+    Uω (aρ 11) (bρ 11) (1042494095729 / 8000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_188_12 :
+    Uω (aρ 12) (bρ 12) (1042494095729 / 8000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_188_13 :
+    Uω (aρ 13) (bρ 13) (1042494095729 / 8000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_188_14 :
+    Uω (aρ 14) (bρ 14) (1042494095729 / 8000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_188_15 :
+    Uω (aρ 15) (bρ 15) (1042494095729 / 8000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_188_16 :
+    Uω (aρ 16) (bρ 16) (1042494095729 / 8000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_188 :
+    Uρ (1042494095729 / 8000000000000 : ℝ) ≤
+      (-(22447390042549224808151 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1042494095729 / 8000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2088692305103148942643 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_188_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1042494095729 / 8000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(5271707410071795261109 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_188_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1042494095729 / 8000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(4300967582928605563113 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_188_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1042494095729 / 8000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(22261662463063795315849 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_188_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1042494095729 / 8000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(23615658202251705360119 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_188_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1042494095729 / 8000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(329180582000059646687 / 125000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_188_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1042494095729 / 8000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_188_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1042494095729 / 8000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_188_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1042494095729 / 8000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_188_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1042494095729 / 8000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_188_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1042494095729 / 8000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_188_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1042494095729 / 8000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_188_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1042494095729 / 8000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_188_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1042494095729 / 8000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_188_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1042494095729 / 8000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_188_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1042494095729 / 8000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_188_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_189_1 :
+    Uω (aρ 1) (bρ 1) (266229110129 / 2000000000000 : ℝ) ≤
+      (-(20663115692318653602647 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (633260342654493 / 5000000000000000 : ℝ) ≤
+      (-(20663115692318653602647 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (633260342654493 / 5000000000000000 : ℝ) =
+            (633260342654493 / 625000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (633260342654493 / 625000000000000 : ℝ) ≤
+            (131299716681346397353 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (316598259514493 / 2500000000000000 : ℝ)) (q :=
+      (633260342654493 / 5000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_189_2 :
+    Uω (aρ 2) (bρ 2) (266229110129 / 2000000000000 : ℝ) ≤
+      (-(20858418761046195439279 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1242025128818447 / 10000000000000000 : ℝ) ≤
+      (-(20858418761046195439279 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1242025128818447 / 10000000000000000 : ℝ) =
+            (1242025128818447 / 625000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1242025128818447 / 625000000000000 : ℝ) ≤
+            (6867468450953804560721 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (620585468793447 / 5000000000000000 : ℝ)) (q :=
+      (1242025128818447 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_189_3 :
+    Uω (aρ 3) (bρ 3) (266229110129 / 2000000000000 : ℝ) ≤
+      (-(4253265914535006292877 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1192380983595443 / 10000000000000000 : ℝ) ≤
+      (-(4253265914535006292877 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1192380983595443 / 10000000000000000 : ℝ) =
+            (1192380983595443 / 625000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1192380983595443 / 625000000000000 : ℝ) ≤
+            (1291911527864993707123 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (594639376745443 / 5000000000000000 : ℝ)) (q :=
+      (1192380983595443 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_189_4 :
+    Uω (aρ 4) (bρ 4) (266229110129 / 2000000000000 : ℝ) ≤
+      (-(22003071308913592837197 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2215382647996537 / 20000000000000000 : ℝ) ≤
+      (-(22003071308913592837197 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2215382647996537 / 20000000000000000 : ℝ) =
+            (2215382647996537 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2215382647996537 / 1250000000000000 : ℝ) ≤
+            (5722815903086407162803 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1098193615361537 / 10000000000000000 : ℝ)) (q :=
+      (2215382647996537 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_189_5 :
+    Uω (aρ 5) (bρ 5) (266229110129 / 2000000000000 : ℝ) ≤
+      (-(11657021536339170325913 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1943184195125871 / 20000000000000000 : ℝ) ≤
+      (-(11657021536339170325913 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1943184195125871 / 20000000000000000 : ℝ) =
+            (1943184195125871 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1943184195125871 / 1250000000000000 : ℝ) ≤
+            (2205922069660829674087 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (944185081825871 / 10000000000000000 : ℝ)) (q :=
+      (1943184195125871 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_189_6 :
+    Uω (aρ 6) (bρ 6) (266229110129 / 2000000000000 : ℝ) ≤
+      (-(25903511253721066404459 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1499874063837843 / 20000000000000000 : ℝ) ≤
+      (-(25903511253721066404459 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1499874063837843 / 20000000000000000 : ℝ) =
+            (1499874063837843 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1499874063837843 / 1250000000000000 : ℝ) ≤
+            (1822375958278933595541 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (668115325677843 / 10000000000000000 : ℝ)) (q :=
+      (1499874063837843 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_189_7 :
+    Uω (aρ 7) (bρ 7) (266229110129 / 2000000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_189_8 :
+    Uω (aρ 8) (bρ 8) (266229110129 / 2000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_189_9 :
+    Uω (aρ 9) (bρ 9) (266229110129 / 2000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_189_10 :
+    Uω (aρ 10) (bρ 10) (266229110129 / 2000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_189_11 :
+    Uω (aρ 11) (bρ 11) (266229110129 / 2000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_189_12 :
+    Uω (aρ 12) (bρ 12) (266229110129 / 2000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_189_13 :
+    Uω (aρ 13) (bρ 13) (266229110129 / 2000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_189_14 :
+    Uω (aρ 14) (bρ 14) (266229110129 / 2000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_189_15 :
+    Uω (aρ 15) (bρ 15) (266229110129 / 2000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_189_16 :
+    Uω (aρ 16) (bρ 16) (266229110129 / 2000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_189 :
+    Uρ (266229110129 / 2000000000000 : ℝ) ≤
+      (-(11179100805596847419571 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (266229110129 / 2000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(20663115692318653602647 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_189_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (266229110129 / 2000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(20858418761046195439279 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_189_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (266229110129 / 2000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(4253265914535006292877 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_189_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (266229110129 / 2000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(22003071308913592837197 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_189_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (266229110129 / 2000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(11657021536339170325913 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_189_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (266229110129 / 2000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(25903511253721066404459 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_189_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (266229110129 / 2000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_189_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (266229110129 / 2000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_189_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (266229110129 / 2000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_189_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (266229110129 / 2000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_189_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (266229110129 / 2000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_189_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (266229110129 / 2000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_189_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (266229110129 / 2000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_189_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (266229110129 / 2000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_189_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (266229110129 / 2000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_189_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (266229110129 / 2000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_189_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_190_1 :
+    Uω (aρ 1) (bρ 1) (110976113009 / 800000000000 : ℝ) ≤
+      (-(20229992373945278136929 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2645163915912357 / 20000000000000000 : ℝ) ≤
+      (-(20229992373945278136929 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2645163915912357 / 20000000000000000 : ℝ) =
+            (2645163915912357 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2645163915912357 / 2500000000000000 : ℝ) ≤
+            (564423035054721863071 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1322459721384857 / 10000000000000000 : ℝ)) (q :=
+      (2645163915912357 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_190_2 :
+    Uω (aρ 2) (bρ 2) (110976113009 / 800000000000 : ℝ) ≤
+      (-(20416696451517995856967 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2596235802222861 / 20000000000000000 : ℝ) ≤
+      (-(20416696451517995856967 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2596235802222861 / 20000000000000000 : ℝ) =
+            (2596235802222861 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2596235802222861 / 2500000000000000 : ℝ) ≤
+            (377718957482004143033 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1297300620205361 / 10000000000000000 : ℝ)) (q :=
+      (2596235802222861 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_190_3 :
+    Uω (aρ 3) (bρ 3) (110976113009 / 800000000000 : ℝ) ≤
+      (-(2080581013886745405213 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1248576469633047 / 10000000000000000 : ℝ) ≤
+      (-(2080581013886745405213 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1248576469633047 / 10000000000000000 : ℝ) =
+            (1248576469633047 / 625000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 15
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1248576469633047 / 625000000000000 : ℝ) ≤
+            (692007707313254594787 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (622806931799297 / 5000000000000000 : ℝ)) (q :=
+      (1248576469633047 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_190_4 :
+    Uω (aρ 4) (bρ 4) (110976113009 / 800000000000 : ℝ) ≤
+      (-(21505438446981235916047 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1164208255837901 / 10000000000000000 : ℝ) ≤
+      (-(21505438446981235916047 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1164208255837901 / 10000000000000000 : ℝ) =
+            (1164208255837901 / 625000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1164208255837901 / 625000000000000 : ℝ) ≤
+            (6220448765018764083953 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (577585808536651 / 5000000000000000 : ℝ)) (q :=
+      (1164208255837901 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_190_5 :
+    Uω (aρ 5) (bρ 5) (110976113009 / 800000000000 : ℝ) ≤
+      (-(4547642411201558481751 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2058363123946183 / 20000000000000000 : ℝ) ≤
+      (-(4547642411201558481751 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2058363123946183 / 20000000000000000 : ℝ) =
+            (2058363123946183 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2058363123946183 / 1250000000000000 : ℝ) ≤
+            (997535031198441518249 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1003308148678683 / 10000000000000000 : ℝ)) (q :=
+      (2058363123946183 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_190_6 :
+    Uω (aρ 6) (bρ 6) (110976113009 / 800000000000 : ℝ) ≤
+      (-(25105050925041415667209 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (406136007713679 / 5000000000000000 : ℝ) ≤
+      (-(25105050925041415667209 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (406136007713679 / 5000000000000000 : ℝ) =
+            (406136007713679 / 312500000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (406136007713679 / 312500000000000 : ℝ) ≤
+            (2620836286958584332791 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (46045589420451 / 625000000000000 : ℝ)) (q :=
+      (406136007713679 / 5000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_190_7 :
+    Uω (aρ 7) (bρ 7) (110976113009 / 800000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_190_8 :
+    Uω (aρ 8) (bρ 8) (110976113009 / 800000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_190_9 :
+    Uω (aρ 9) (bρ 9) (110976113009 / 800000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_190_10 :
+    Uω (aρ 10) (bρ 10) (110976113009 / 800000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_190_11 :
+    Uω (aρ 11) (bρ 11) (110976113009 / 800000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_190_12 :
+    Uω (aρ 12) (bρ 12) (110976113009 / 800000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_190_13 :
+    Uω (aρ 13) (bρ 13) (110976113009 / 800000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_190_14 :
+    Uω (aρ 14) (bρ 14) (110976113009 / 800000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_190_15 :
+    Uω (aρ 15) (bρ 15) (110976113009 / 800000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_190_16 :
+    Uω (aρ 16) (bρ 16) (110976113009 / 800000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_190 :
+    Uρ (110976113009 / 800000000000 : ℝ) ≤
+      (-(11094580622395912739221 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (110976113009 / 800000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(20229992373945278136929 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_190_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (110976113009 / 800000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(20416696451517995856967 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_190_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (110976113009 / 800000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(2080581013886745405213 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_190_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (110976113009 / 800000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(21505438446981235916047 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_190_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (110976113009 / 800000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(4547642411201558481751 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_190_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (110976113009 / 800000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(25105050925041415667209 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_190_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (110976113009 / 800000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_190_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (110976113009 / 800000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_190_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (110976113009 / 800000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_190_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (110976113009 / 800000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_190_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (110976113009 / 800000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_190_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (110976113009 / 800000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_190_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (110976113009 / 800000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_190_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (110976113009 / 800000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_190_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (110976113009 / 800000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_190_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (110976113009 / 800000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_190_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_191_1 :
+    Uω (aρ 1) (bρ 1) (72162863729 / 500000000000 : ℝ) ≤
+      (-(19814855621105804326481 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (275728558105041 / 2000000000000000 : ℝ) ≤
+      (-(19814855621105804326481 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (275728558105041 / 2000000000000000 : ℝ) =
+            (275728558105041 / 250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (275728558105041 / 250000000000000 : ℝ) ≤
+            (979559787894195673519 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (137852552455541 / 1000000000000000 : ℝ)) (q :=
+      (275728558105041 / 2000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_191_2 :
+    Uω (aρ 2) (bρ 2) (72162863729 / 500000000000 : ℝ) ≤
+      (-(19993685957203362780097 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2708415227817231 / 20000000000000000 : ℝ) ≤
+      (-(19993685957203362780097 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2708415227817231 / 20000000000000000 : ℝ) =
+            (2708415227817231 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2708415227817231 / 2500000000000000 : ℝ) ≤
+            (800729451796637219903 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1353424183832231 / 10000000000000000 : ℝ)) (q :=
+      (2708415227817231 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_191_3 :
+    Uω (aρ 3) (bρ 3) (72162863729 / 500000000000 : ℝ) ≤
+      (-(10182830255675675289117 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (521903960994751 / 4000000000000000 : ℝ) ≤
+      (-(10182830255675675289117 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (521903960994751 / 4000000000000000 : ℝ) =
+            (521903960994751 / 500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (521903960994751 / 500000000000000 : ℝ) ≤
+            (214377448824324710883 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (260384973467751 / 2000000000000000 : ℝ)) (q :=
+      (521903960994751 / 4000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_191_4 :
+    Uω (aρ 4) (bρ 4) (72162863729 / 500000000000 : ℝ) ≤
+      (-(2628969020630553055031 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2441364382080019 / 20000000000000000 : ℝ) ≤
+      (-(2628969020630553055031 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2441364382080019 / 20000000000000000 : ℝ) =
+            (2441364382080019 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2441364382080019 / 1250000000000000 : ℝ) ≤
+            (836766880869446944969 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1212063625510019 / 10000000000000000 : ℝ)) (q :=
+      (2441364382080019 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_191_5 :
+    Uω (aρ 5) (bρ 5) (72162863729 / 500000000000 : ℝ) ≤
+      (-(22195271732922818939507 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2173209482836907 / 20000000000000000 : ℝ) ≤
+      (-(22195271732922818939507 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2173209482836907 / 20000000000000000 : ℝ) =
+            (2173209482836907 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2173209482836907 / 1250000000000000 : ℝ) ≤
+            (5530615479077181060493 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1062098645601907 / 10000000000000000 : ℝ)) (q :=
+      (2173209482836907 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_191_6 :
+    Uω (aρ 6) (bρ 6) (72162863729 / 500000000000 : ℝ) ≤
+      (-(24376790840821149583193 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (349453517251867 / 4000000000000000 : ℝ) ≤
+      (-(24376790840821149583193 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (349453517251867 / 4000000000000000 : ℝ) =
+            (349453517251867 / 250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (349453517251867 / 250000000000000 : ℝ) ≤
+            (3349096371178850416807 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (160679424832867 / 2000000000000000 : ℝ)) (q :=
+      (349453517251867 / 4000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_191_7 :
+    Uω (aρ 7) (bρ 7) (72162863729 / 500000000000 : ℝ) ≤
+      (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_7]) (by norm_num [bρ_7])]; exact Lmid_7
+
+lemma U_191_8 :
+    Uω (aρ 8) (bρ 8) (72162863729 / 500000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_191_9 :
+    Uω (aρ 9) (bρ 9) (72162863729 / 500000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_191_10 :
+    Uω (aρ 10) (bρ 10) (72162863729 / 500000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_191_11 :
+    Uω (aρ 11) (bρ 11) (72162863729 / 500000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_191_12 :
+    Uω (aρ 12) (bρ 12) (72162863729 / 500000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_191_13 :
+    Uω (aρ 13) (bρ 13) (72162863729 / 500000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_191_14 :
+    Uω (aρ 14) (bρ 14) (72162863729 / 500000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_191_15 :
+    Uω (aρ 15) (bρ 15) (72162863729 / 500000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_191_16 :
+    Uω (aρ 16) (bρ 16) (72162863729 / 500000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_191 :
+    Uρ (72162863729 / 500000000000 : ℝ) ≤
+      (-(22030937450651269436371 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (72162863729 / 500000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(19814855621105804326481 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_191_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (72162863729 / 500000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(19993685957203362780097 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_191_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (72162863729 / 500000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(10182830255675675289117 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_191_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (72162863729 / 500000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(2628969020630553055031 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_191_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (72162863729 / 500000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(22195271732922818939507 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_191_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (72162863729 / 500000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(24376790840821149583193 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_191_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (72162863729 / 500000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(33239459906384759787341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_191_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (72162863729 / 500000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_191_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (72162863729 / 500000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_191_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (72162863729 / 500000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_191_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (72162863729 / 500000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_191_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (72162863729 / 500000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_191_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (72162863729 / 500000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_191_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (72162863729 / 500000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_191_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (72162863729 / 500000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_191_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (72162863729 / 500000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_191_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_192_1 :
+    Uω (aρ 1) (bρ 1) (4675203313927 / 32000000000000 : ℝ) ≤
+      (-(787478463849348170387 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5585552167037221 / 40000000000000000 : ℝ) ≤
+      (-(787478463849348170387 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (5585552167037221 / 40000000000000000 : ℝ) =
+            (5585552167037221 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5585552167037221 / 5000000000000000 : ℝ) ≤
+            (44298152510651829613 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1396272266001423 / 10000000000000000 : ℝ)) (q :=
+      (5585552167037221 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_192_2 :
+    Uω (aρ 2) (bρ 2) (4675203313927 / 32000000000000 : ℝ) ≤
+      (-(19863436151075893595487 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5487846051701767 / 40000000000000000 : ℝ) ≤
+      (-(19863436151075893595487 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (5487846051701767 / 40000000000000000 : ℝ) =
+            (5487846051701767 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5487846051701767 / 5000000000000000 : ℝ) ≤
+            (930979257924106404513 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (85699263802731 / 625000000000000 : ℝ)) (q :=
+      (5487846051701767 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_192_3 :
+    Uω (aρ 3) (bρ 3) (4675203313927 / 32000000000000 : ℝ) ≤
+      (-(4046059230732191981441 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5290167124837241 / 40000000000000000 : ℝ) ≤
+      (-(4046059230732191981441 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (5290167124837241 / 40000000000000000 : ℝ) =
+            (5290167124837241 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5290167124837241 / 5000000000000000 : ℝ) ≤
+            (112823851067808018559 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1319744863761433 / 10000000000000000 : ℝ)) (q :=
+      (5290167124837241 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_192_4 :
+    Uω (aρ 4) (bρ 4) (4675203313927 / 32000000000000 : ℝ) ≤
+      (-(20886435181837476410703 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4954201155424141 / 40000000000000000 : ℝ) ≤
+      (-(20886435181837476410703 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (4954201155424141 / 40000000000000000 : ℝ) =
+            (4954201155424141 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4954201155424141 / 2500000000000000 : ℝ) ≤
+            (6839452030162523589297 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1230056060119883 / 10000000000000000 : ℝ)) (q :=
+      (4954201155424141 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_192_5 :
+    Uω (aρ 5) (bρ 5) (4675203313927 / 32000000000000 : ℝ) ≤
+      (-(22029650929542333847147 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (883800825444237 / 8000000000000000 : ℝ) ≤
+      (-(22029650929542333847147 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (883800825444237 / 8000000000000000 : ℝ) =
+            (883800825444237 / 500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (883800825444237 / 500000000000000 : ℝ) ≤
+            (5696236282457666152853 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (216129493070681 / 2000000000000000 : ℝ)) (q :=
+      (883800825444237 / 8000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_192_6 :
+    Uω (aρ 6) (bρ 6) (4675203313927 / 32000000000000 : ℝ) ≤
+      (-(19326993602285207127 / 8000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3571569920567127 / 40000000000000000 : ℝ) ≤
+      (-(19326993602285207127 / 8000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (3571569920567127 / 40000000000000000 : ℝ) =
+            (3571569920567127 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3571569920567127 / 2500000000000000 : ℝ) ≤
+            (2853716167314792873 / 8000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (103021342145797 / 1250000000000000 : ℝ)) (q :=
+      (3571569920567127 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_192_7 :
+    Uω (aρ 7) (bρ 7) (4675203313927 / 32000000000000 : ℝ) ≤
+      (-(969506853701683875103 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (359521826035421 / 8000000000000000 : ℝ) ≤
+      (-(969506853701683875103 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (359521826035421 / 8000000000000000 : ℝ) =
+            (359521826035421 / 250000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (359521826035421 / 250000000000000 : ℝ) ≤
+            (113535615517066124897 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (32170344546273 / 2000000000000000 : ℝ)) (q :=
+      (359521826035421 / 8000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_192_8 :
+    Uω (aρ 8) (bρ 8) (4675203313927 / 32000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_192_9 :
+    Uω (aρ 9) (bρ 9) (4675203313927 / 32000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_192_10 :
+    Uω (aρ 10) (bρ 10) (4675203313927 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_192_11 :
+    Uω (aρ 11) (bρ 11) (4675203313927 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_192_12 :
+    Uω (aρ 12) (bρ 12) (4675203313927 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_192_13 :
+    Uω (aρ 13) (bρ 13) (4675203313927 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_192_14 :
+    Uω (aρ 14) (bρ 14) (4675203313927 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_192_15 :
+    Uω (aρ 15) (bρ 15) (4675203313927 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_192_16 :
+    Uω (aρ 16) (bρ 16) (4675203313927 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_192 :
+    Uρ (4675203313927 / 32000000000000 : ℝ) ≤
+      (-(21794875057781414112513 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (4675203313927 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(787478463849348170387 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_192_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (4675203313927 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(19863436151075893595487 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_192_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (4675203313927 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(4046059230732191981441 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_192_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (4675203313927 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(20886435181837476410703 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_192_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (4675203313927 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(22029650929542333847147 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_192_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (4675203313927 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) * (-(19326993602285207127 / 8000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_192_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (4675203313927 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(969506853701683875103 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_192_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (4675203313927 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_192_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (4675203313927 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_192_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (4675203313927 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_192_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (4675203313927 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_192_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (4675203313927 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_192_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (4675203313927 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_192_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (4675203313927 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_192_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (4675203313927 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_192_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (4675203313927 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_192_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_193_1 :
+    Uω (aρ 1) (bρ 1) (2365991674599 / 16000000000000 : ℝ) ≤
+      (-(19560682890171164556221 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (282826651118107 / 2000000000000000 : ℝ) ≤
+      (-(19560682890171164556221 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (282826651118107 / 2000000000000000 : ℝ) =
+            (282826651118107 / 250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (282826651118107 / 250000000000000 : ℝ) ≤
+            (1233732518828835443779 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (282803786528339 / 2000000000000000 : ℝ)) (q :=
+      (282826651118107 / 2000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_193_2 :
+    Uω (aρ 2) (bρ 2) (2365991674599 / 16000000000000 : ℝ) ≤
+      (-(394697258427317257959 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1389715152769981 / 10000000000000000 : ℝ) ≤
+      (-(394697258427317257959 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1389715152769981 / 10000000000000000 : ℝ) =
+            (1389715152769981 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1389715152769981 / 1250000000000000 : ℝ) ≤
+            (21191049752682742041 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1388951739510587 / 10000000000000000 : ℝ)) (q :=
+      (1389715152769981 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_193_3 :
+    Uω (aρ 3) (bρ 3) (2365991674599 / 16000000000000 : ℝ) ≤
+      (-(2512093407885480065093 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (335080661610237 / 2500000000000000 : ℝ) ≤
+      (-(2512093407885480065093 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (335080661610237 / 2500000000000000 : ℝ) =
+            (335080661610237 / 312500000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (335080661610237 / 312500000000000 : ℝ) ≤
+            (87208518239519934907 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1337562833202521 / 10000000000000000 : ℝ)) (q :=
+      (335080661610237 / 2500000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_193_4 :
+    Uω (aρ 4) (bρ 4) (2365991674599 / 16000000000000 : ℝ) ≤
+      (-(10371614014328306359131 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1256414826444487 / 10000000000000000 : ℝ) ≤
+      (-(10371614014328306359131 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1256414826444487 / 10000000000000000 : ℝ) =
+            (1256414826444487 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1256414826444487 / 1250000000000000 : ℝ) ≤
+            (25593690171693640869 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1248041374274599 / 10000000000000000 : ℝ)) (q :=
+      (1256414826444487 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_193_5 :
+    Uω (aρ 5) (bρ 5) (2365991674599 / 16000000000000 : ℝ) ≤
+      (-(5466711720418222399723 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1122884031566689 / 10000000000000000 : ℝ) ≤
+      (-(5466711720418222399723 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1122884031566689 / 10000000000000000 : ℝ) =
+            (1122884031566689 / 625000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1122884031566689 / 625000000000000 : ℝ) ≤
+            (1464760082581777600277 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1099169703854003 / 10000000000000000 : ℝ)) (q :=
+      (1122884031566689 / 10000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_193_6 :
+    Uω (aρ 6) (bρ 6) (2365991674599 / 16000000000000 : ℝ) ≤
+      (-(23946103601804326428943 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (456041051278531 / 5000000000000000 : ℝ) ≤
+      (-(23946103601804326428943 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (456041051278531 / 5000000000000000 : ℝ) =
+            (456041051278531 / 312500000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (456041051278531 / 312500000000000 : ℝ) ≤
+            (3779783610195673571057 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (844806220974749 / 10000000000000000 : ℝ)) (q :=
+      (456041051278531 / 5000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_193_7 :
+    Uω (aρ 7) (bρ 7) (2365991674599 / 16000000000000 : ℝ) ≤
+      (-(752824408466949677751 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (39382206101409 / 800000000000000 : ℝ) ≤
+      (-(752824408466949677751 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (39382206101409 / 800000000000000 : ℝ) = (39382206101409 / 25000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (39382206101409 / 25000000000000 : ℝ) ≤
+            (113609566908050322249 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4577170983117 / 200000000000000 : ℝ)) (q :=
+      (39382206101409 / 800000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_193_8 :
+    Uω (aρ 8) (bρ 8) (2365991674599 / 16000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_193_9 :
+    Uω (aρ 9) (bρ 9) (2365991674599 / 16000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_193_10 :
+    Uω (aρ 10) (bρ 10) (2365991674599 / 16000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_193_11 :
+    Uω (aρ 11) (bρ 11) (2365991674599 / 16000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_193_12 :
+    Uω (aρ 12) (bρ 12) (2365991674599 / 16000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_193_13 :
+    Uω (aρ 13) (bρ 13) (2365991674599 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_193_14 :
+    Uω (aρ 14) (bρ 14) (2365991674599 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_193_15 :
+    Uω (aρ 15) (bρ 15) (2365991674599 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_193_16 :
+    Uω (aρ 16) (bρ 16) (2365991674599 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_193 :
+    Uρ (2365991674599 / 16000000000000 : ℝ) ≤
+      (-(21670353188008540621837 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (2365991674599 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(19560682890171164556221 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_193_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (2365991674599 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(394697258427317257959 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_193_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (2365991674599 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(2512093407885480065093 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_193_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (2365991674599 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(10371614014328306359131 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_193_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (2365991674599 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(5466711720418222399723 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_193_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (2365991674599 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(23946103601804326428943 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_193_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (2365991674599 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(752824408466949677751 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_193_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (2365991674599 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_193_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (2365991674599 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_193_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (2365991674599 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_193_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (2365991674599 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_193_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (2365991674599 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_193_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (2365991674599 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_193_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (2365991674599 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_193_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (2365991674599 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_193_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (2365991674599 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_193_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_194_1 :
+    Uω (aρ 1) (bρ 1) (4788763384469 / 32000000000000 : ℝ) ≤
+      (-(3887195840299343045543 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5727513733638853 / 40000000000000000 : ℝ) ≤
+      (-(3887195840299343045543 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (5727513733638853 / 40000000000000000 : ℝ) =
+            (5727513733638853 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5727513733638853 / 5000000000000000 : ℝ) ≤
+            (271687241500656954457 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (178970690907233 / 1250000000000000 : ℝ)) (q :=
+      (5727513733638853 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_194_2 :
+    Uω (aρ 2) (bρ 2) (4788763384469 / 32000000000000 : ℝ) ≤
+      (-(1960792360803155468861 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1125974834608363 / 8000000000000000 : ℝ) ≤
+      (-(1960792360803155468861 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1125974834608363 / 8000000000000000 : ℝ) =
+            (1125974834608363 / 1000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1125974834608363 / 1000000000000000 : ℝ) ≤
+            (118649180096844531139 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (281342951893869 / 2000000000000000 : ℝ)) (q :=
+      (1125974834608363 / 8000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_194_3 :
+    Uω (aρ 3) (bρ 3) (4788763384469 / 32000000000000 : ℝ) ≤
+      (-(19964965591887264809447 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5432410152608797 / 40000000000000000 : ℝ) ≤
+      (-(19964965591887264809447 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (5432410152608797 / 40000000000000000 : ℝ) =
+            (5432410152608797 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5432410152608797 / 5000000000000000 : ℝ) ≤
+            (829449817112735190553 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (338844713900709 / 2500000000000000 : ℝ)) (q :=
+      (5432410152608797 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_194_4 :
+    Uω (aρ 4) (bρ 4) (4788763384469 / 32000000000000 : ℝ) ≤
+      (-(20602069529176842578973 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5097103822148609 / 40000000000000000 : ℝ) ≤
+      (-(20602069529176842578973 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (5097103822148609 / 40000000000000000 : ℝ) =
+            (5097103822148609 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5097103822148609 / 5000000000000000 : ℝ) ≤
+            (192345879823157421027 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (633009935718871 / 5000000000000000 : ℝ)) (q :=
+      (5097103822148609 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_194_5 :
+    Uω (aρ 5) (bρ 5) (4788763384469 / 32000000000000 : ℝ) ≤
+      (-(21706761635276424051639 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4564017605877021 / 40000000000000000 : ℝ) ≤
+      (-(21706761635276424051639 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (4564017605877021 / 40000000000000000 : ℝ) =
+            (4564017605877021 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4564017605877021 / 2500000000000000 : ℝ) ≤
+            (6019125576723575948361 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (279416670659237 / 2500000000000000 : ℝ)) (q :=
+      (4564017605877021 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_194_6 :
+    Uω (aρ 6) (bρ 6) (4788763384469 / 32000000000000 : ℝ) ≤
+      (-(4747716261870926325079 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3724830405755021 / 40000000000000000 : ℝ) ≤
+      (-(4747716261870926325079 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (3724830405755021 / 40000000000000000 : ℝ) =
+            (3724830405755021 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3724830405755021 / 2500000000000000 : ℝ) ≤
+            (797461180529073674921 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (216328364428987 / 2500000000000000 : ℝ)) (q :=
+      (3724830405755021 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_194_7 :
+    Uω (aρ 7) (bρ 7) (4788763384469 / 32000000000000 : ℝ) ≤
+      (-(14708997966031985063453 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2110827096636869 / 40000000000000000 : ℝ) ≤
+      (-(14708997966031985063453 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (2110827096636869 / 40000000000000000 : ℝ) =
+            (2110827096636869 / 1250000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2110827096636869 / 1250000000000000 : ℝ) ≤
+            (2619681541468014936547 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (35246647989609 / 1250000000000000 : ℝ)) (q :=
+      (2110827096636869 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_194_8 :
+    Uω (aρ 8) (bρ 8) (4788763384469 / 32000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_194_9 :
+    Uω (aρ 9) (bρ 9) (4788763384469 / 32000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_194_10 :
+    Uω (aρ 10) (bρ 10) (4788763384469 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_194_11 :
+    Uω (aρ 11) (bρ 11) (4788763384469 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_194_12 :
+    Uω (aρ 12) (bρ 12) (4788763384469 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_194_13 :
+    Uω (aρ 13) (bρ 13) (4788763384469 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_194_14 :
+    Uω (aρ 14) (bρ 14) (4788763384469 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_194_15 :
+    Uω (aρ 15) (bρ 15) (4788763384469 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_194_16 :
+    Uω (aρ 16) (bρ 16) (4788763384469 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_194 :
+    Uρ (4788763384469 / 32000000000000 : ℝ) ≤
+      (-(10782516787768049362653 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (4788763384469 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3887195840299343045543 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_194_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (4788763384469 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1960792360803155468861 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_194_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (4788763384469 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(19964965591887264809447 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_194_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (4788763384469 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(20602069529176842578973 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_194_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (4788763384469 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(21706761635276424051639 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_194_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (4788763384469 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(4747716261870926325079 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_194_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (4788763384469 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(14708997966031985063453 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_194_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (4788763384469 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_194_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (4788763384469 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_194_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (4788763384469 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_194_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (4788763384469 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_194_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (4788763384469 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_194_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (4788763384469 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_194_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (4788763384469 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_194_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (4788763384469 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_194_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (4788763384469 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_194_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_195_1 :
+    Uω (aρ 1) (bρ 1) (9634306804209 / 64000000000000 : ℝ) ≤
+      (-(1210887862743762534847 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11526008073845393 / 80000000000000000 : ℝ) ≤
+      (-(1210887862743762534847 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (11526008073845393 / 80000000000000000 : ℝ) =
+            (11526008073845393 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11526008073845393 / 10000000000000000 : ℝ) ≤
+            (88763100318737465153 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (360159699597173 / 2500000000000000 : ℝ)) (q :=
+      (11526008073845393 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_195_2 :
+    Uω (aρ 2) (bρ 2) (9634306804209 / 64000000000000 : ℝ) ≤
+      (-(977252689052743997793 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11330761183450273 / 80000000000000000 : ℝ) ≤
+      (-(977252689052743997793 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (11330761183450273 / 80000000000000000 : ℝ) =
+            (11330761183450273 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11330761183450273 / 10000000000000000 : ℝ) ≤
+            (62468081397256002207 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (176949511037489 / 1250000000000000 : ℝ)) (q :=
+      (11330761183450273 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_195_3 :
+    Uω (aρ 3) (bρ 3) (9634306804209 / 64000000000000 : ℝ) ≤
+      (-(9949861454974666734381 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10935937046850741 / 80000000000000000 : ℝ) ≤
+      (-(9949861454974666734381 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (10935937046850741 / 80000000000000000 : ℝ) =
+            (10935937046850741 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10935937046850741 / 10000000000000000 : ℝ) ≤
+            (447346249525333265619 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1364286160500029 / 10000000000000000 : ℝ)) (q :=
+      (10935937046850741 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_195_4 :
+    Uω (aρ 4) (bρ 4) (9634306804209 / 64000000000000 : ℝ) ≤
+      (-(20532240139569703071389 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10265642295664161 / 80000000000000000 : ℝ) ≤
+      (-(20532240139569703071389 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (10265642295664161 / 80000000000000000 : ℝ) =
+            (10265642295664161 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10265642295664161 / 10000000000000000 : ℝ) ≤
+            (262175269430296928611 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (19921978965131 / 156250000000000 : ℝ)) (q :=
+      (10265642295664161 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_195_5 :
+    Uω (aρ 5) (bρ 5) (9634306804209 / 64000000000000 : ℝ) ≤
+      (-(21627709382413351683267 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9200480356847417 / 80000000000000000 : ℝ) ≤
+      (-(21627709382413351683267 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (9200480356847417 / 80000000000000000 : ℝ) =
+            (9200480356847417 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9200480356847417 / 5000000000000000 : ℝ) ≤
+            (6098177829586648316733 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (563453044199599 / 5000000000000000 : ℝ)) (q :=
+      (9200480356847417 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_195_6 :
+    Uω (aρ 6) (bρ 6) (9634306804209 / 64000000000000 : ℝ) ≤
+      (-(5909163429288754127091 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1505196341988161 / 16000000000000000 : ℝ) ≤
+      (-(5909163429288754127091 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1505196341988161 / 16000000000000000 : ℝ) =
+            (1505196341988161 / 1000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1505196341988161 / 1000000000000000 : ℝ) ≤
+            (1022308373711245872909 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (175104360362509 / 2000000000000000 : ℝ)) (q :=
+      (1505196341988161 / 16000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_195_7 :
+    Uω (aρ 7) (bρ 7) (9634306804209 / 64000000000000 : ℝ) ≤
+      (-(2911592179000382039081 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4351125095812169 / 80000000000000000 : ℝ) ≤
+      (-(2911592179000382039081 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (4351125095812169 / 80000000000000000 : ℝ) =
+            (4351125095812169 / 2500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4351125095812169 / 2500000000000000 : ℝ) ≤
+            (554143722499617960919 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (152734514520193 / 5000000000000000 : ℝ)) (q :=
+      (4351125095812169 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_195_8 :
+    Uω (aρ 8) (bρ 8) (9634306804209 / 64000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_195_9 :
+    Uω (aρ 9) (bρ 9) (9634306804209 / 64000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_195_10 :
+    Uω (aρ 10) (bρ 10) (9634306804209 / 64000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_195_11 :
+    Uω (aρ 11) (bρ 11) (9634306804209 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_195_12 :
+    Uω (aρ 12) (bρ 12) (9634306804209 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_195_13 :
+    Uω (aρ 13) (bρ 13) (9634306804209 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_195_14 :
+    Uω (aρ 14) (bρ 14) (9634306804209 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_195_15 :
+    Uω (aρ 15) (bρ 15) (9634306804209 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_195_16 :
+    Uω (aρ 16) (bρ 16) (9634306804209 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_195 :
+    Uρ (9634306804209 / 64000000000000 : ℝ) ≤
+      (-(21516535824810612976769 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (9634306804209 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1210887862743762534847 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_195_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (9634306804209 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(977252689052743997793 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_195_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (9634306804209 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(9949861454974666734381 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_195_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (9634306804209 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(20532240139569703071389 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_195_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (9634306804209 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(21627709382413351683267 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_195_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (9634306804209 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(5909163429288754127091 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_195_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (9634306804209 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(2911592179000382039081 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_195_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (9634306804209 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_195_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (9634306804209 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_195_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (9634306804209 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_195_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (9634306804209 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_195_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (9634306804209 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_195_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (9634306804209 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_195_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (9634306804209 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_195_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (9634306804209 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_195_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (9634306804209 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_195_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/U16.lean b/LeanPool/Zeta5Irrational/Table/U16.lean
new file mode 100644
index 0000000000..aa65af783a
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/U16.lean
@@ -0,0 +1,3501 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-! # Certified arcsine potential bounds (U16) -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma U_196_1 :
+    Uω (aρ 1) (bρ 1) (242277170987 / 1600000000000 : ℝ) ≤
+      (-(2414101464838475762107 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (181202947067443 / 1250000000000000 : ℝ) ≤
+      (-(2414101464838475762107 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (181202947067443 / 1250000000000000 : ℝ) =
+            (181202947067443 / 156250000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (181202947067443 / 156250000000000 : ℝ) ≤
+            (185200461286524237893 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (724756026247669 / 5000000000000000 : ℝ)) (q :=
+      (181202947067443 / 1250000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_196_2 :
+    Uω (aρ 2) (bρ 2) (242277170987 / 1600000000000 : ℝ) ≤
+      (-(19482577160501416685469 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (356305423431271 / 2500000000000000 : ℝ) ≤
+      (-(19482577160501416685469 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (356305423431271 / 2500000000000000 : ℝ) =
+            (356305423431271 / 312500000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (356305423431271 / 312500000000000 : ℝ) ≤
+            (1311838248498583314531 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (712238649688209 / 5000000000000000 : ℝ)) (q :=
+      (356305423431271 / 2500000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_196_3 :
+    Uω (aρ 3) (bρ 3) (242277170987 / 1600000000000 : ℝ) ≤
+      (-(4958726199469246675321 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2751762988469771 / 20000000000000000 : ℝ) ≤
+      (-(4958726199469246675321 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2751762988469771 / 20000000000000000 : ℝ) =
+            (2751762988469771 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2751762988469771 / 2500000000000000 : ℝ) ≤
+            (239877652780753324679 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1373193006746021 / 10000000000000000 : ℝ)) (q :=
+      (2751762988469771 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_196_4 :
+    Uω (aρ 4) (bρ 4) (242277170987 / 1600000000000 : ℝ) ≤
+      (-(10231450584493624514443 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1292133819311667 / 10000000000000000 : ℝ) ≤
+      (-(10231450584493624514443 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1292133819311667 / 10000000000000000 : ℝ) =
+            (1292133819311667 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1292133819311667 / 1250000000000000 : ℝ) ≤
+            (165757120006375485557 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (160498979745573 / 1250000000000000 : ℝ)) (q :=
+      (1292133819311667 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_196_5 :
+    Uω (aρ 5) (bρ 5) (242277170987 / 1600000000000 : ℝ) ≤
+      (-(2154930242961830379761 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2318225516752991 / 20000000000000000 : ℝ) ≤
+      (-(2154930242961830379761 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2318225516752991 / 20000000000000000 : ℝ) =
+            (2318225516752991 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2318225516752991 / 1250000000000000 : ℝ) ≤
+            (617658478238169620239 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1136139635429241 / 10000000000000000 : ℝ)) (q :=
+      (2318225516752991 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_196_6 :
+    Uω (aρ 6) (bρ 6) (242277170987 / 1600000000000 : ℝ) ≤
+      (-(5883976514400672601683 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (380109372357853 / 4000000000000000 : ℝ) ≤
+      (-(5883976514400672601683 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (380109372357853 / 4000000000000000 : ℝ) =
+            (380109372357853 / 250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (380109372357853 / 250000000000000 : ℝ) ≤
+            (1047495288599327398317 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (177140271121103 / 2000000000000000 : ℝ)) (q :=
+      (380109372357853 / 4000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_196_7 :
+    Uω (aρ 7) (bρ 7) (242277170987 / 1600000000000 : ℝ) ≤
+      (-(28835587643811316305909 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (559353474134973 / 10000000000000000 : ℝ) ≤
+      (-(28835587643811316305909 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (559353474134973 / 10000000000000000 : ℝ) =
+            (559353474134973 / 312500000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (559353474134973 / 312500000000000 : ℝ) ≤
+            (5821771371188683694091 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (81880705711549 / 2500000000000000 : ℝ)) (q :=
+      (559353474134973 / 10000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_196_8 :
+    Uω (aρ 8) (bρ 8) (242277170987 / 1600000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_196_9 :
+    Uω (aρ 9) (bρ 9) (242277170987 / 1600000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_196_10 :
+    Uω (aρ 10) (bρ 10) (242277170987 / 1600000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_196_11 :
+    Uω (aρ 11) (bρ 11) (242277170987 / 1600000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_196_12 :
+    Uω (aρ 12) (bρ 12) (242277170987 / 1600000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_196_13 :
+    Uω (aρ 13) (bρ 13) (242277170987 / 1600000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_196_14 :
+    Uω (aρ 14) (bρ 14) (242277170987 / 1600000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_196_15 :
+    Uω (aρ 15) (bρ 15) (242277170987 / 1600000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_196_16 :
+    Uω (aρ 16) (bρ 16) (242277170987 / 1600000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_196 :
+    Uρ (242277170987 / 1600000000000 : ℝ) ≤
+      (-(21470082810998145944539 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (242277170987 / 1600000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2414101464838475762107 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_196_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (242277170987 / 1600000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(19482577160501416685469 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_196_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (242277170987 / 1600000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(4958726199469246675321 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_196_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (242277170987 / 1600000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(10231450584493624514443 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_196_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (242277170987 / 1600000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(2154930242961830379761 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_196_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (242277170987 / 1600000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(5883976514400672601683 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_196_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (242277170987 / 1600000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(28835587643811316305909 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_196_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (242277170987 / 1600000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_196_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (242277170987 / 1600000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_196_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (242277170987 / 1600000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_196_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (242277170987 / 1600000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_196_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (242277170987 / 1600000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_196_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (242277170987 / 1600000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_196_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (242277170987 / 1600000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_196_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (242277170987 / 1600000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_196_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (242277170987 / 1600000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_196_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_197_1 :
+    Uω (aρ 1) (bρ 1) (9747866874751 / 64000000000000 : ℝ) ≤
+      (-(9625896157689089085363 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11667969083933543 / 80000000000000000 : ℝ) ≤
+      (-(9625896157689089085363 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (11667969083933543 / 80000000000000000 : ℝ) =
+            (11667969083933543 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11667969083933543 / 10000000000000000 : ℝ) ≤
+            (771311546810910914637 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (729192644944271 / 5000000000000000 : ℝ)) (q :=
+      (11667969083933543 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_197_2 :
+    Uω (aρ 2) (bρ 2) (9747866874751 / 64000000000000 : ℝ) ≤
+      (-(9710244426699519995267 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11472785453890199 / 80000000000000000 : ℝ) ≤
+      (-(9710244426699519995267 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (11472785453890199 / 80000000000000000 : ℝ) =
+            (11472785453890199 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11472785453890199 / 10000000000000000 : ℝ) ≤
+            (686963277800480004733 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (716679197443853 / 5000000000000000 : ℝ)) (q :=
+      (11472785453890199 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_197_3 :
+    Uω (aρ 3) (bρ 3) (9747866874751 / 64000000000000 : ℝ) ≤
+      (-(9885252872164051723893 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (443126602470863 / 3200000000000000 : ℝ) ≤
+      (-(9885252872164051723893 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (443126602470863 / 3200000000000000 : ℝ) =
+            (443126602470863 / 400000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (443126602470863 / 400000000000000 : ℝ) ≤
+            (511954832335948276107 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (27641988064161 / 200000000000000 : ℝ)) (q :=
+      (443126602470863 / 3200000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_197_4 :
+    Uω (aρ 4) (bρ 4) (9747866874751 / 64000000000000 : ℝ) ≤
+      (-(5098511423100886032829 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10408492554054399 / 80000000000000000 : ℝ) ≤
+      (-(5098511423100886032829 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (10408492554054399 / 80000000000000000 : ℝ) =
+            (10408492554054399 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10408492554054399 / 10000000000000000 : ℝ) ≤
+            (100092429149113967171 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (323243864335939 / 2500000000000000 : ℝ)) (q :=
+      (10408492554054399 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_197_5 :
+    Uω (aρ 5) (bρ 5) (9747866874751 / 64000000000000 : ℝ) ≤
+      (-(5367882483107103412791 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1869060181804003 / 16000000000000000 : ℝ) ≤
+      (-(5367882483107103412791 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1869060181804003 / 16000000000000000 : ℝ) =
+            (1869060181804003 / 1000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1869060181804003 / 1000000000000000 : ℝ) ≤
+            (1563589319892896587209 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1789636664719 / 15625000000000 : ℝ)) (q :=
+      (1869060181804003 / 16000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_197_6 :
+    Uω (aρ 6) (bρ 6) (9747866874751 / 64000000000000 : ℝ) ≤
+      (-(5859077040181952131849 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7678281948881963 / 80000000000000000 : ℝ) ≤
+      (-(5859077040181952131849 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (7678281948881963 / 80000000000000000 : ℝ) =
+            (7678281948881963 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7678281948881963 / 5000000000000000 : ℝ) ≤
+            (1072394762818047868151 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (895853100525647 / 10000000000000000 : ℝ)) (q :=
+      (7678281948881963 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_197_7 :
+    Uω (aρ 7) (bρ 7) (9747866874751 / 64000000000000 : ℝ) ≤
+      (-(7143266057875241232323 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4593858074959487 / 80000000000000000 : ℝ) ≤
+      (-(7143266057875241232323 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (4593858074959487 / 80000000000000000 : ℝ) =
+            (4593858074959487 / 2500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4593858074959487 / 2500000000000000 : ℝ) ≤
+            (1521073695874758767677 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (87102128201257 / 2500000000000000 : ℝ)) (q :=
+      (4593858074959487 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_197_8 :
+    Uω (aρ 8) (bρ 8) (9747866874751 / 64000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_197_9 :
+    Uω (aρ 9) (bρ 9) (9747866874751 / 64000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_197_10 :
+    Uω (aρ 10) (bρ 10) (9747866874751 / 64000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_197_11 :
+    Uω (aρ 11) (bρ 11) (9747866874751 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_197_12 :
+    Uω (aρ 12) (bρ 12) (9747866874751 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_197_13 :
+    Uω (aρ 13) (bρ 13) (9747866874751 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_197_14 :
+    Uω (aρ 14) (bρ 14) (9747866874751 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_197_15 :
+    Uω (aρ 15) (bρ 15) (9747866874751 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_197_16 :
+    Uω (aρ 16) (bρ 16) (9747866874751 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_197 :
+    Uρ (9747866874751 / 64000000000000 : ℝ) ≤
+      (-(2678167143913165181433 / 1250000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (9747866874751 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(9625896157689089085363 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_197_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (9747866874751 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(9710244426699519995267 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_197_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (9747866874751 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(9885252872164051723893 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_197_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (9747866874751 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(5098511423100886032829 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_197_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (9747866874751 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(5367882483107103412791 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_197_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (9747866874751 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(5859077040181952131849 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_197_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (9747866874751 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(7143266057875241232323 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_197_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (9747866874751 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_197_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (9747866874751 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_197_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (9747866874751 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_197_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (9747866874751 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_197_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (9747866874751 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_197_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (9747866874751 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_197_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (9747866874751 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_197_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (9747866874751 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_197_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (9747866874751 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_197_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_198_1 :
+    Uω (aρ 1) (bρ 1) (4902323455011 / 32000000000000 : ℝ) ≤
+      (-(19191143047661083763181 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5869474744954929 / 40000000000000000 : ℝ) ≤
+      (-(19191143047661083763181 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (5869474744954929 / 40000000000000000 : ℝ) =
+            (5869474744954929 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5869474744954929 / 5000000000000000 : ℝ) ≤
+            (1603272361338916236819 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1467258510871527 / 10000000000000000 : ℝ)) (q :=
+      (5869474744954929 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_198_2 :
+    Uω (aρ 2) (bρ 2) (4902323455011 / 32000000000000 : ℝ) ≤
+      (-(9679392028817212299531 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5771898452129227 / 40000000000000000 : ℝ) ≤
+      (-(9679392028817212299531 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (5771898452129227 / 40000000000000000 : ℝ) =
+            (5771898452129227 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5771898452129227 / 5000000000000000 : ℝ) ≤
+            (717815675682787700469 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (360559844242169 / 2500000000000000 : ℝ)) (q :=
+      (5771898452129227 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_198_3 :
+    Uω (aρ 3) (bρ 3) (4902323455011 / 32000000000000 : ℝ) ≤
+      (-(492663008624633561373 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5574638202543499 / 40000000000000000 : ℝ) ≤
+      (-(492663008624633561373 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (5574638202543499 / 40000000000000000 : ℝ) =
+            (5574638202543499 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5574638202543499 / 5000000000000000 : ℝ) ≤
+            (27197376600366438627 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (347751339631453 / 2500000000000000 : ℝ)) (q :=
+      (5574638202543499 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_198_4 :
+    Uω (aρ 4) (bρ 4) (4902323455011 / 32000000000000 : ℝ) ≤
+      (-(20325666932152984631853 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5239954211957913 / 40000000000000000 : ℝ) ≤
+      (-(20325666932152984631853 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (5239954211957913 / 40000000000000000 : ℝ) =
+            (5239954211957913 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5239954211957913 / 5000000000000000 : ℝ) ≤
+            (468748476847015368147 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1301957544298019 / 10000000000000000 : ℝ)) (q :=
+      (5239954211957913 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_198_5 :
+    Uω (aρ 5) (bρ 5) (4902323455011 / 32000000000000 : ℝ) ≤
+      (-(42788762652914107141 / 20000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4708838715589633 / 40000000000000000 : ℝ) ≤
+      (-(42788762652914107141 / 20000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (4708838715589633 / 40000000000000000 : ℝ) =
+            (4708838715589633 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4708838715589633 / 2500000000000000 : ℝ) ≤
+            (12663011771085892859 / 20000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1154589715448879 / 10000000000000000 : ℝ)) (q :=
+      (4708838715589633 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_198_6 :
+    Uω (aρ 6) (bρ 6) (4902323455011 / 32000000000000 : ℝ) ≤
+      (-(23337831090964090934793 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3877134477197451 / 40000000000000000 : ℝ) ≤
+      (-(23337831090964090934793 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (3877134477197451 / 40000000000000000 : ℝ) =
+            (3877134477197451 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3877134477197451 / 2500000000000000 : ℝ) ≤
+            (4388056121035909065207 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (226494492848197 / 2500000000000000 : ℝ)) (q :=
+      (3877134477197451 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_198_7 :
+    Uω (aρ 7) (bρ 7) (4902323455011 / 32000000000000 : ℝ) ≤
+      (-(113301949068643981267 / 40000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (470901099739561 / 8000000000000000 : ℝ) ≤
+      (-(113301949068643981267 / 40000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (470901099739561 / 8000000000000000 : ℝ) =
+            (470901099739561 / 250000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (470901099739561 / 250000000000000 : ℝ) ≤
+            (25327486991356018733 / 40000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (73664972580593 / 2000000000000000 : ℝ)) (q :=
+      (470901099739561 / 8000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_198_8 :
+    Uω (aρ 8) (bρ 8) (4902323455011 / 32000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_198_9 :
+    Uω (aρ 9) (bρ 9) (4902323455011 / 32000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_198_10 :
+    Uω (aρ 10) (bρ 10) (4902323455011 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_198_11 :
+    Uω (aρ 11) (bρ 11) (4902323455011 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_198_12 :
+    Uω (aρ 12) (bρ 12) (4902323455011 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_198_13 :
+    Uω (aρ 13) (bρ 13) (4902323455011 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_198_14 :
+    Uω (aρ 14) (bρ 14) (4902323455011 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_198_15 :
+    Uω (aρ 15) (bρ 15) (4902323455011 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_198_16 :
+    Uω (aρ 16) (bρ 16) (4902323455011 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_198 :
+    Uρ (4902323455011 / 32000000000000 : ℝ) ≤
+      (-(10691025983055117798807 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (4902323455011 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(19191143047661083763181 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_198_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (4902323455011 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(9679392028817212299531 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_198_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (4902323455011 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(492663008624633561373 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_198_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (4902323455011 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(20325666932152984631853 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_198_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (4902323455011 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(42788762652914107141 / 20000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_198_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (4902323455011 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(23337831090964090934793 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_198_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (4902323455011 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) * (-(113301949068643981267 / 40000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_198_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (4902323455011 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_198_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (4902323455011 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_198_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (4902323455011 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_198_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (4902323455011 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_198_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (4902323455011 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_198_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (4902323455011 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_198_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (4902323455011 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_198_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (4902323455011 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_198_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (4902323455011 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_198_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_199_1 :
+    Uω (aρ 1) (bρ 1) (9861426945293 / 64000000000000 : ℝ) ≤
+      (-(19130859451563604787399 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (472397193257161 / 3200000000000000 : ℝ) ≤
+      (-(19130859451563604787399 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (472397193257161 / 3200000000000000 : ℝ) =
+            (472397193257161 / 400000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (472397193257161 / 400000000000000 : ℝ) ≤
+            (1663555957436395212601 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (59045268629609 / 400000000000000 : ℝ)) (q :=
+      (472397193257161 / 3200000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_199_2 :
+    Uω (aρ 2) (bρ 2) (9861426945293 / 64000000000000 : ℝ) ≤
+      (-(2412182257464382172739 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11614807909235877 / 80000000000000000 : ℝ) ≤
+      (-(2412182257464382172739 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (11614807909235877 / 80000000000000000 : ℝ) =
+            (11614807909235877 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11614807909235877 / 10000000000000000 : ℝ) ≤
+            (187119668660617827261 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (725560123850969 / 5000000000000000 : ℝ)) (q :=
+      (11614807909235877 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_199_3 :
+    Uω (aρ 3) (bρ 3) (9861426945293 / 64000000000000 : ℝ) ≤
+      (-(19642943299792789663527 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11220386017503029 / 80000000000000000 : ℝ) ≤
+      (-(19642943299792789663527 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (11220386017503029 / 80000000000000000 : ℝ) =
+            (11220386017503029 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11220386017503029 / 10000000000000000 : ℝ) ≤
+            (1151472109207210336473 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (699955440559363 / 5000000000000000 : ℝ)) (q :=
+      (11220386017503029 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_199_4 :
+    Uω (aρ 4) (bρ 4) (9861426945293 / 64000000000000 : ℝ) ≤
+      (-(20257758253733241645161 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10551318290073249 / 80000000000000000 : ℝ) ≤
+      (-(20257758253733241645161 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (10551318290073249 / 80000000000000000 : ℝ) =
+            (10551318290073249 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10551318290073249 / 10000000000000000 : ℝ) ≤
+            (536657155266758354839 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1310938130326281 / 10000000000000000 : ℝ)) (q :=
+      (10551318290073249 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_199_5 :
+    Uω (aρ 5) (bρ 5) (9861426945293 / 64000000000000 : ℝ) ≤
+      (-(5329461579391329115803 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1898006432818281 / 16000000000000000 : ℝ) ≤
+      (-(5329461579391329115803 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1898006432818281 / 16000000000000000 : ℝ) =
+            (1898006432818281 / 1000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1898006432818281 / 1000000000000000 : ℝ) ≤
+            (1602010223608670884197 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (58190325908291 / 500000000000000 : ℝ)) (q :=
+      (1898006432818281 / 16000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_199_6 :
+    Uω (aρ 6) (bρ 6) (9861426945293 / 64000000000000 : ℝ) ≤
+      (-(23240447077305945546461 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7830152027979093 / 80000000000000000 : ℝ) ≤
+      (-(23240447077305945546461 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (7830152027979093 / 80000000000000000 : ℝ) =
+            (7830152027979093 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7830152027979093 / 5000000000000000 : ℝ) ≤
+            (4485440134694054453539 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (458038429638871 / 5000000000000000 : ℝ)) (q :=
+      (7830152027979093 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_199_7 :
+    Uω (aρ 7) (bρ 7) (9861426945293 / 64000000000000 : ℝ) ≤
+      (-(28090691845157120020427 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (964176930281977 / 16000000000000000 : ℝ) ≤
+      (-(28090691845157120020427 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (964176930281977 / 16000000000000000 : ℝ) =
+            (964176930281977 / 500000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (964176930281977 / 500000000000000 : ℝ) ≤
+            (6566667169842879979573 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4842767448693 / 125000000000000 : ℝ)) (q :=
+      (964176930281977 / 16000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_199_8 :
+    Uω (aρ 8) (bρ 8) (9861426945293 / 64000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_199_9 :
+    Uω (aρ 9) (bρ 9) (9861426945293 / 64000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_199_10 :
+    Uω (aρ 10) (bρ 10) (9861426945293 / 64000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_199_11 :
+    Uω (aρ 11) (bρ 11) (9861426945293 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_199_12 :
+    Uω (aρ 12) (bρ 12) (9861426945293 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_199_13 :
+    Uω (aρ 13) (bρ 13) (9861426945293 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_199_14 :
+    Uω (aρ 14) (bρ 14) (9861426945293 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_199_15 :
+    Uω (aρ 15) (bρ 15) (9861426945293 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_199_16 :
+    Uω (aρ 16) (bρ 16) (9861426945293 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_199 :
+    Uρ (9861426945293 / 64000000000000 : ℝ) ≤
+      (-(5335009962320976159597 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (9861426945293 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(19130859451563604787399 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_199_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (9861426945293 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(2412182257464382172739 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_199_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (9861426945293 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(19642943299792789663527 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_199_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (9861426945293 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(20257758253733241645161 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_199_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (9861426945293 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(5329461579391329115803 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_199_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (9861426945293 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(23240447077305945546461 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_199_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (9861426945293 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(28090691845157120020427 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_199_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (9861426945293 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_199_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (9861426945293 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_199_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (9861426945293 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_199_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (9861426945293 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_199_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (9861426945293 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_199_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (9861426945293 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_199_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (9861426945293 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_199_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (9861426945293 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_199_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (9861426945293 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_199_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_200_1 :
+    Uω (aρ 1) (bρ 1) (2479551745141 / 16000000000000 : ℝ) ≤
+      (-(9535468571688656894177 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (148511376370581 / 1000000000000000 : ℝ) ≤
+      (-(9535468571688656894177 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (148511376370581 / 1000000000000000 : ℝ) =
+            (148511376370581 / 125000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (148511376370581 / 125000000000000 : ℝ) ≤
+            (861739132811343105823 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (297000980956699 / 2000000000000000 : ℝ)) (q :=
+      (148511376370581 / 1000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_200_2 :
+    Uω (aρ 2) (bρ 2) (2479551745141 / 16000000000000 : ℝ) ≤
+      (-(19236506232599800661231 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (292145461923753 / 2000000000000000 : ℝ) ≤
+      (-(19236506232599800661231 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (292145461923753 / 2000000000000000 : ℝ) =
+            (292145461923753 / 250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (292145461923753 / 250000000000000 : ℝ) ≤
+            (1557909176400199338769 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (292000201823881 / 2000000000000000 : ℝ)) (q :=
+      (292145461923753 / 2000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_200_3 :
+    Uω (aρ 3) (bρ 3) (2479551745141 / 16000000000000 : ℝ) ≤
+      (-(19579769410281550678357 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1411436741480521 / 10000000000000000 : ℝ) ≤
+      (-(19579769410281550678357 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1411436741480521 / 10000000000000000 : ℝ) =
+            (1411436741480521 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1411436741480521 / 1250000000000000 : ℝ) ≤
+            (1214645998718449321643 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1408815979192917 / 10000000000000000 : ℝ)) (q :=
+      (1411436741480521 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_200_4 :
+    Uω (aρ 4) (bρ 4) (2479551745141 / 16000000000000 : ℝ) ≤
+      (-(2019031316175803607789 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2655680568768123 / 20000000000000000 : ℝ) ≤
+      (-(2019031316175803607789 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2655680568768123 / 20000000000000000 : ℝ) =
+            (2655680568768123 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2655680568768123 / 2500000000000000 : ℝ) ≤
+            (60410224724196392211 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (659958623032499 / 5000000000000000 : ℝ)) (q :=
+      (2655680568768123 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_200_5 :
+    Uω (aρ 5) (bρ 5) (2479551745141 / 16000000000000 : ℝ) ≤
+      (-(21241914872470499005599 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1195295702671639 / 10000000000000000 : ℝ) ≤
+      (-(21241914872470499005599 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1195295702671639 / 10000000000000000 : ℝ) =
+            (1195295702671639 / 625000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1195295702671639 / 625000000000000 : ℝ) ≤
+            (6483972339529500994401 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1173018001975153 / 10000000000000000 : ℝ)) (q :=
+      (1195295702671639 / 10000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_200_6 :
+    Uω (aρ 6) (bρ 6) (2479551745141 / 16000000000000 : ℝ) ≤
+      (-(23144129448493782607159 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (395296728475251 / 4000000000000000 : ℝ) ≤
+      (-(23144129448493782607159 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (395296728475251 / 4000000000000000 : ℝ) =
+            (395296728475251 / 250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (395296728475251 / 250000000000000 : ℝ) ≤
+            (4581757763506217392841 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (92615061414813 / 1000000000000000 : ℝ)) (q :=
+      (395296728475251 / 4000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_200_7 :
+    Uω (aρ 7) (bρ 7) (2479551745141 / 16000000000000 : ℝ) ≤
+      (-(13933497005570068083757 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (77030344441781 / 1250000000000000 : ℝ) ≤
+      (-(13933497005570068083757 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (77030344441781 / 1250000000000000 : ℝ) = (77030344441781 / 39062500000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (77030344441781 / 39062500000000 : ℝ) ≤
+            (3395182501929931916243 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (405813863600371 / 10000000000000000 : ℝ)) (q :=
+      (77030344441781 / 1250000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_200_8 :
+    Uω (aρ 8) (bρ 8) (2479551745141 / 16000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_200_9 :
+    Uω (aρ 9) (bρ 9) (2479551745141 / 16000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_200_10 :
+    Uω (aρ 10) (bρ 10) (2479551745141 / 16000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_200_11 :
+    Uω (aρ 11) (bρ 11) (2479551745141 / 16000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_200_12 :
+    Uω (aρ 12) (bρ 12) (2479551745141 / 16000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_200_13 :
+    Uω (aρ 13) (bρ 13) (2479551745141 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_200_14 :
+    Uω (aρ 14) (bρ 14) (2479551745141 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_200_15 :
+    Uω (aρ 15) (bρ 15) (2479551745141 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_200_16 :
+    Uω (aρ 16) (bρ 16) (2479551745141 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_200 :
+    Uρ (2479551745141 / 16000000000000 : ℝ) ≤
+      (-(21299154326957743988867 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (2479551745141 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(9535468571688656894177 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_200_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (2479551745141 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(19236506232599800661231 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_200_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (2479551745141 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(19579769410281550678357 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_200_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (2479551745141 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(2019031316175803607789 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_200_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (2479551745141 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(21241914872470499005599 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_200_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (2479551745141 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(23144129448493782607159 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_200_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (2479551745141 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(13933497005570068083757 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_200_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (2479551745141 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_200_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (2479551745141 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_200_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (2479551745141 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_200_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (2479551745141 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_200_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (2479551745141 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_200_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (2479551745141 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_200_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (2479551745141 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_200_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (2479551745141 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_200_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (2479551745141 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_200_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_201_1 :
+    Uω (aρ 1) (bρ 1) (19893193996399 / 128000000000000 : ℝ) ≤
+      (-(3808222024764513750947 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23832800450741223 / 160000000000000000 : ℝ) ≤
+      (-(3808222024764513750947 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (23832800450741223 / 160000000000000000 : ℝ) =
+            (23832800450741223 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23832800450741223 / 20000000000000000 : ℝ) ≤
+            (350661057035486249053 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1489441493458981 / 10000000000000000 : ℝ)) (q :=
+      (23832800450741223 / 160000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_201_2 :
+    Uω (aρ 2) (bρ 2) (19893193996399 / 128000000000000 : ℝ) ≤
+      (-(19206169210263464948851 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23442647198638999 / 160000000000000000 : ℝ) ≤
+      (-(19206169210263464948851 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (23442647198638999 / 160000000000000000 : ℝ) =
+            (23442647198638999 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23442647198638999 / 20000000000000000 : ℝ) ≤
+            (1588246198736535051149 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1464441349456203 / 10000000000000000 : ℝ)) (q :=
+      (23442647198638999 / 160000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_201_3 :
+    Uω (aρ 3) (bρ 3) (19893193996399 / 128000000000000 : ℝ) ≤
+      (-(9774166025761886195709 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22654094524535327 / 160000000000000000 : ℝ) ≤
+      (-(9774166025761886195709 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (22654094524535327 / 160000000000000000 : ℝ) =
+            (22654094524535327 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22654094524535327 / 20000000000000000 : ℝ) ≤
+            (623041678738113804291 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (353317092885811 / 2500000000000000 : ℝ)) (q :=
+      (22654094524535327 / 160000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_201_4 :
+    Uω (aρ 4) (bρ 4) (19893193996399 / 128000000000000 : ℝ) ≤
+      (-(20156762467954930955953 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21316844199038367 / 160000000000000000 : ℝ) ≤
+      (-(20156762467954930955953 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (21316844199038367 / 160000000000000000 : ℝ) =
+            (21316844199038367 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21316844199038367 / 20000000000000000 : ℝ) ≤
+            (637652941045069044047 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (331101565480281 / 2500000000000000 : ℝ)) (q :=
+      (21316844199038367 / 160000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_201_5 :
+    Uω (aρ 5) (bρ 5) (19893193996399 / 128000000000000 : ℝ) ≤
+      (-(10602086210028209019321 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19197049055365631 / 160000000000000000 : ℝ) ≤
+      (-(10602086210028209019321 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (19197049055365631 / 160000000000000000 : ℝ) =
+            (19197049055365631 / 10000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19197049055365631 / 10000000000000000 : ℝ) ≤
+            (3260857395971790980679 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (147202723537129 / 1250000000000000 : ℝ)) (q :=
+      (19197049055365631 / 160000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_201_6 :
+    Uω (aρ 6) (bρ 6) (19893193996399 / 128000000000000 : ℝ) ≤
+      (-(23096362479439773573517 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3177515664269971 / 32000000000000000 : ℝ) ≤
+      (-(23096362479439773573517 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (3177515664269971 / 32000000000000000 : ℝ) =
+            (3177515664269971 / 2000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3177515664269971 / 2000000000000000 : ℝ) ≤
+            (4629524732560226426483 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (46558916084253 / 500000000000000 : ℝ)) (q :=
+      (3177515664269971 / 32000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_201_7 :
+    Uω (aρ 7) (bρ 7) (19893193996399 / 128000000000000 : ℝ) ≤
+      (-(27758876997846690296493 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9967064568070847 / 160000000000000000 : ℝ) ≤
+      (-(27758876997846690296493 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (9967064568070847 / 160000000000000000 : ℝ) =
+            (9967064568070847 / 5000000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9967064568070847 / 5000000000000000 : ℝ) ≤
+            (6898482017153309703507 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6480866926331 / 156250000000000 : ℝ)) (q :=
+      (9967064568070847 / 160000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_201_8 :
+    Uω (aρ 8) (bρ 8) (19893193996399 / 128000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_201_9 :
+    Uω (aρ 9) (bρ 9) (19893193996399 / 128000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_201_10 :
+    Uω (aρ 10) (bρ 10) (19893193996399 / 128000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_201_11 :
+    Uω (aρ 11) (bρ 11) (19893193996399 / 128000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_201_12 :
+    Uω (aρ 12) (bρ 12) (19893193996399 / 128000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_201_13 :
+    Uω (aρ 13) (bρ 13) (19893193996399 / 128000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_201_14 :
+    Uω (aρ 14) (bρ 14) (19893193996399 / 128000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_201_15 :
+    Uω (aρ 15) (bρ 15) (19893193996399 / 128000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_201_16 :
+    Uω (aρ 16) (bρ 16) (19893193996399 / 128000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_201 :
+    Uρ (19893193996399 / 128000000000000 : ℝ) ≤
+      (-(1063954824476878769469 / 500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (19893193996399 / 128000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3808222024764513750947 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_201_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (19893193996399 / 128000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(19206169210263464948851 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_201_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (19893193996399 / 128000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(9774166025761886195709 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_201_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (19893193996399 / 128000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(20156762467954930955953 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_201_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (19893193996399 / 128000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(10602086210028209019321 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_201_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (19893193996399 / 128000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(23096362479439773573517 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_201_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (19893193996399 / 128000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(27758876997846690296493 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_201_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (19893193996399 / 128000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_201_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (19893193996399 / 128000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_201_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (19893193996399 / 128000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_201_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (19893193996399 / 128000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_201_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (19893193996399 / 128000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_201_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (19893193996399 / 128000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_201_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (19893193996399 / 128000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_201_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (19893193996399 / 128000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_201_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (19893193996399 / 128000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_201_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_202_1 :
+    Uω (aρ 1) (bρ 1) (1994997403167 / 12800000000000 : ℝ) ≤
+      (-(19011371817764254291277 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11951890325690203 / 80000000000000000 : ℝ) ≤
+      (-(19011371817764254291277 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (11951890325690203 / 80000000000000000 : ℝ) =
+            (11951890325690203 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11951890325690203 / 10000000000000000 : ℝ) ≤
+            (1783043591235745708723 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (373469519570833 / 2500000000000000 : ℝ)) (q :=
+      (11951890325690203 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_202_2 :
+    Uω (aρ 2) (bρ 2) (1994997403167 / 12800000000000 : ℝ) ≤
+      (-(19175924033595514993157 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11756828615332171 / 80000000000000000 : ℝ) ≤
+      (-(19175924033595514993157 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (11756828615332171 / 80000000000000000 : ℝ) =
+            (11756828615332171 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11756828615332171 / 10000000000000000 : ℝ) ≤
+            (1618491375404485006843 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (91805103950239 / 625000000000000 : ℝ)) (q :=
+      (11756828615332171 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_202_3 :
+    Uω (aρ 3) (bρ 3) (1994997403167 / 12800000000000 : ℝ) ≤
+      (-(19516993576980804934593 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11362600180108659 / 80000000000000000 : ℝ) ≤
+      (-(19516993576980804934593 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (11362600180108659 / 80000000000000000 : ℝ) =
+            (11362600180108659 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11362600180108659 / 10000000000000000 : ℝ) ≤
+            (1277421832019195065407 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (708860330373973 / 5000000000000000 : ℝ)) (q :=
+      (11362600180108659 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_202_4 :
+    Uω (aρ 4) (bρ 4) (1994997403167 / 12800000000000 : ℝ) ≤
+      (-(10061662648026687643777 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10694120498127499 / 80000000000000000 : ℝ) ≤
+      (-(10061662648026687643777 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (10694120498127499 / 80000000000000000 : ℝ) =
+            (10694120498127499 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10694120498127499 / 10000000000000000 : ℝ) ≤
+            (335545056473312356223 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (166111865164707 / 1250000000000000 : ℝ)) (q :=
+      (10694120498127499 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_202_5 :
+    Uω (aρ 5) (bρ 5) (1994997403167 / 12800000000000 : ℝ) ≤
+      (-(2645822151221018910863 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1926935660068307 / 16000000000000000 : ℝ) ≤
+      (-(2645822151221018910863 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1926935660068307 / 16000000000000000 : ℝ) =
+            (1926935660068307 / 1000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1926935660068307 / 1000000000000000 : ℝ) ≤
+            (819913750278981089137 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (236444858241233 / 2000000000000000 : ℝ)) (q :=
+      (1926935660068307 / 16000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_202_6 :
+    Uω (aρ 6) (bρ 6) (1994997403167 / 12800000000000 : ℝ) ≤
+      (-(5762213143197609709187 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7981619824227011 / 80000000000000000 : ℝ) ≤
+      (-(5762213143197609709187 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (7981619824227011 / 80000000000000000 : ℝ) =
+            (7981619824227011 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7981619824227011 / 5000000000000000 : ℝ) ≤
+            (1169258659802390290813 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (468100023658767 / 5000000000000000 : ℝ)) (q :=
+      (7981619824227011 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_202_7 :
+    Uω (aρ 7) (bρ 7) (1994997403167 / 12800000000000 : ℝ) ≤
+      (-(13826526486999273842053 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5036550062220967 / 80000000000000000 : ℝ) ≤
+      (-(13826526486999273842053 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (5036550062220967 / 80000000000000000 : ℝ) =
+            (5036550062220967 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5036550062220967 / 5000000000000000 : ℝ) ≤
+            (36417119000726157947 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (423593987576023 / 10000000000000000 : ℝ)) (q :=
+      (5036550062220967 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_202_8 :
+    Uω (aρ 8) (bρ 8) (1994997403167 / 12800000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_202_9 :
+    Uω (aρ 9) (bρ 9) (1994997403167 / 12800000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_202_10 :
+    Uω (aρ 10) (bρ 10) (1994997403167 / 12800000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_202_11 :
+    Uω (aρ 11) (bρ 11) (1994997403167 / 12800000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_202_12 :
+    Uω (aρ 12) (bρ 12) (1994997403167 / 12800000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_202_13 :
+    Uω (aρ 13) (bρ 13) (1994997403167 / 12800000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_202_14 :
+    Uω (aρ 14) (bρ 14) (1994997403167 / 12800000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_202_15 :
+    Uω (aρ 15) (bρ 15) (1994997403167 / 12800000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_202_16 :
+    Uω (aρ 16) (bρ 16) (1994997403167 / 12800000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_202 :
+    Uρ (1994997403167 / 12800000000000 : ℝ) ≤
+      (-(21259278159574063461509 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1994997403167 / 12800000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(19011371817764254291277 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_202_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1994997403167 / 12800000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(19175924033595514993157 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_202_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1994997403167 / 12800000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(19516993576980804934593 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_202_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1994997403167 / 12800000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(10061662648026687643777 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_202_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1994997403167 / 12800000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(2645822151221018910863 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_202_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1994997403167 / 12800000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(5762213143197609709187 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_202_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1994997403167 / 12800000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(13826526486999273842053 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_202_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1994997403167 / 12800000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_202_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1994997403167 / 12800000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_202_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1994997403167 / 12800000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_202_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1994997403167 / 12800000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_202_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1994997403167 / 12800000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_202_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1994997403167 / 12800000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_202_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1994997403167 / 12800000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_202_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1994997403167 / 12800000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_202_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1994997403167 / 12800000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_202_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_203_1 :
+    Uω (aρ 1) (bρ 1) (20006754066941 / 128000000000000 : ℝ) ≤
+      (-(18981721698975795827751 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4794952164296841 / 32000000000000000 : ℝ) ≤
+      (-(18981721698975795827751 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4794952164296841 / 32000000000000000 : ℝ) =
+            (4794952164296841 / 4000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4794952164296841 / 4000000000000000 : ℝ) ≤
+            (1812693710024204172249 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (37457866482269 / 250000000000000 : ℝ)) (q :=
+      (4794952164296841 / 32000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_203_2 :
+    Uω (aρ 2) (bρ 2) (20006754066941 / 128000000000000 : ℝ) ≤
+      (-(3829154029520808723151 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23584667051899501 / 160000000000000000 : ℝ) ≤
+      (-(3829154029520808723151 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (23584667051899501 / 160000000000000000 : ℝ) =
+            (23584667051899501 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23584667051899501 / 20000000000000000 : ℝ) ≤
+            (329729052279191276849 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (92082621912667 / 625000000000000 : ℝ)) (q :=
+      (23584667051899501 / 160000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_203_3 :
+    Uω (aρ 3) (bρ 3) (20006754066941 / 128000000000000 : ℝ) ≤
+      (-(19485753364276007235131 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22796305378484141 / 160000000000000000 : ℝ) ≤
+      (-(19485753364276007235131 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (22796305378484141 / 160000000000000000 : ℝ) =
+            (22796305378484141 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22796305378484141 / 20000000000000000 : ℝ) ≤
+            (1308662044723992764869 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (177771605971969 / 1250000000000000 : ℝ)) (q :=
+      (22796305378484141 / 160000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_203_4 :
+    Uω (aρ 4) (bρ 4) (20006754066941 / 128000000000000 : ℝ) ≤
+      (-(4018000174308122372943 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4291926994118857 / 32000000000000000 : ℝ) ≤
+      (-(4018000174308122372943 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4291926994118857 / 32000000000000000 : ℝ) =
+            (4291926994118857 / 4000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4291926994118857 / 4000000000000000 : ℝ) ≤
+            (140882907491877627057 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (33334580696363 / 250000000000000 : ℝ)) (q :=
+      (4291926994118857 / 32000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_203_5 :
+    Uω (aρ 5) (bρ 5) (20006754066941 / 128000000000000 : ℝ) ≤
+      (-(4225825611375491178981 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19341653998148173 / 160000000000000000 : ℝ) ≤
+      (-(4225825611375491178981 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (19341653998148173 / 160000000000000000 : ℝ) =
+            (19341653998148173 / 10000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19341653998148173 / 10000000000000000 : ℝ) ≤
+            (1319351831024508821019 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (296706381408439 / 2500000000000000 : ℝ)) (q :=
+      (19341653998148173 / 160000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_203_6 :
+    Uω (aρ 6) (bρ 6) (20006754066941 / 128000000000000 : ℝ) ≤
+      (-(1437599792363333804987 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16038853886297317 / 160000000000000000 : ℝ) ≤
+      (-(1437599792363333804987 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (16038853886297317 / 160000000000000000 : ℝ) =
+            (16038853886297317 / 10000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16038853886297317 / 10000000000000000 : ℝ) ≤
+            (295268158386666195013 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (941215886792399 / 10000000000000000 : ℝ)) (q :=
+      (16038853886297317 / 160000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_203_7 :
+    Uω (aρ 7) (bρ 7) (20006754066941 / 128000000000000 : ℝ) ≤
+      (-(27549393096344733901133 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10178060827909021 / 160000000000000000 : ℝ) ≤
+      (-(27549393096344733901133 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (10178060827909021 / 160000000000000000 : ℝ) =
+            (10178060827909021 / 10000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10178060827909021 / 10000000000000000 : ℝ) ≤
+            (176494115655266098867 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (216139067626931 / 5000000000000000 : ℝ)) (q :=
+      (10178060827909021 / 160000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_203_8 :
+    Uω (aρ 8) (bρ 8) (20006754066941 / 128000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_203_9 :
+    Uω (aρ 9) (bρ 9) (20006754066941 / 128000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_203_10 :
+    Uω (aρ 10) (bρ 10) (20006754066941 / 128000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_203_11 :
+    Uω (aρ 11) (bρ 11) (20006754066941 / 128000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_203_12 :
+    Uω (aρ 12) (bρ 12) (20006754066941 / 128000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_203_13 :
+    Uω (aρ 13) (bρ 13) (20006754066941 / 128000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_203_14 :
+    Uω (aρ 14) (bρ 14) (20006754066941 / 128000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_203_15 :
+    Uω (aρ 15) (bρ 15) (20006754066941 / 128000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_203_16 :
+    Uω (aρ 16) (bρ 16) (20006754066941 / 128000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_203 :
+    Uρ (20006754066941 / 128000000000000 : ℝ) ≤
+      (-(21239687987825780551489 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (20006754066941 / 128000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(18981721698975795827751 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_203_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (20006754066941 / 128000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3829154029520808723151 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_203_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (20006754066941 / 128000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(19485753364276007235131 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_203_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (20006754066941 / 128000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(4018000174308122372943 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_203_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (20006754066941 / 128000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(4225825611375491178981 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_203_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (20006754066941 / 128000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1437599792363333804987 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_203_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (20006754066941 / 128000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(27549393096344733901133 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_203_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (20006754066941 / 128000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_203_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (20006754066941 / 128000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_203_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (20006754066941 / 128000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_203_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (20006754066941 / 128000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_203_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (20006754066941 / 128000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_203_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (20006754066941 / 128000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_203_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (20006754066941 / 128000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_203_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (20006754066941 / 128000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_203_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (20006754066941 / 128000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_203_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_204_1 :
+    Uω (aρ 1) (bρ 1) (5015883525553 / 32000000000000 : ℝ) ≤
+      (-(18952159245899628942921 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6011435240330763 / 40000000000000000 : ℝ) ≤
+      (-(18952159245899628942921 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (6011435240330763 / 40000000000000000 : ℝ) =
+            (6011435240330763 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6011435240330763 / 5000000000000000 : ℝ) ≤
+            (1842256163100371057079 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1502751236515069 / 10000000000000000 : ℝ)) (q :=
+      (6011435240330763 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_204_2 :
+    Uω (aρ 2) (bρ 2) (5015883525553 / 32000000000000 : ℝ) ≤
+      (-(2389463375289662657993 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5913919166061143 / 40000000000000000 : ℝ) ≤
+      (-(2389463375289662657993 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (5913919166061143 / 40000000000000000 : ℝ) =
+            (5913919166061143 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5913919166061143 / 5000000000000000 : ℝ) ≤
+            (209838550835337342007 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1477762211890259 / 10000000000000000 : ℝ)) (q :=
+      (5913919166061143 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_204_3 :
+    Uω (aρ 3) (bρ 3) (5015883525553 / 32000000000000 : ℝ) ≤
+      (-(19454610796911121675621 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5716852396747183 / 40000000000000000 : ℝ) ≤
+      (-(19454610796911121675621 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (5716852396747183 / 40000000000000000 : ℝ) =
+            (5716852396747183 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5716852396747183 / 5000000000000000 : ℝ) ≤
+            (1339804612088878324379 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1426624933583279 / 10000000000000000 : ℝ)) (q :=
+      (5716852396747183 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_204_4 :
+    Uω (aρ 4) (bρ 4) (5015883525553 / 32000000000000 : ℝ) ≤
+      (-(20056788427891101095629 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5382756537616719 / 40000000000000000 : ℝ) ≤
+      (-(20056788427891101095629 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (5382756537616719 / 40000000000000000 : ℝ) =
+            (5382756537616719 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5382756537616719 / 5000000000000000 : ℝ) ≤
+            (737626981108898904371 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1337871185083047 / 10000000000000000 : ℝ)) (q :=
+      (5382756537616719 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_204_5 :
+    Uω (aρ 5) (bρ 5) (5015883525553 / 32000000000000 : ℝ) ≤
+      (-(4218364758269121785757 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4853485341333163 / 40000000000000000 : ℝ) ≤
+      (-(4218364758269121785757 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (4853485341333163 / 40000000000000000 : ℝ) =
+            (4853485341333163 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4853485341333163 / 2500000000000000 : ℝ) ≤
+            (1326812684130878214243 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1191425506276269 / 10000000000000000 : ℝ)) (q :=
+      (4853485341333163 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_204_6 :
+    Uω (aρ 6) (bρ 6) (5015883525553 / 32000000000000 : ℝ) ≤
+      (-(22954591801955800902821 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4028605445931183 / 40000000000000000 : ℝ) ≤
+      (-(22954591801955800902821 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (4028605445931183 / 40000000000000000 : ℝ) =
+            (4028605445931183 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4028605445931183 / 2500000000000000 : ℝ) ≤
+            (4771295410044199097179 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (946225933715279 / 10000000000000000 : ℝ)) (q :=
+      (4028605445931183 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_204_7 :
+    Uω (aρ 7) (bρ 7) (5015883525553 / 32000000000000 : ℝ) ≤
+      (-(68619450712148196321 / 25000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2570502550129551 / 40000000000000000 : ℝ) ≤
+      (-(68619450712148196321 / 25000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2570502550129551 / 40000000000000000 : ℝ) =
+            (2570502550129551 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2570502550129551 / 2500000000000000 : ℝ) ≤
+            (695267317851803679 / 25000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (440835866574463 / 10000000000000000 : ℝ)) (q :=
+      (2570502550129551 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_204_8 :
+    Uω (aρ 8) (bρ 8) (5015883525553 / 32000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_204_9 :
+    Uω (aρ 9) (bρ 9) (5015883525553 / 32000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_204_10 :
+    Uω (aρ 10) (bρ 10) (5015883525553 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_204_11 :
+    Uω (aρ 11) (bρ 11) (5015883525553 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_204_12 :
+    Uω (aρ 12) (bρ 12) (5015883525553 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_204_13 :
+    Uω (aρ 13) (bρ 13) (5015883525553 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_204_14 :
+    Uω (aρ 14) (bρ 14) (5015883525553 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_204_15 :
+    Uω (aρ 15) (bρ 15) (5015883525553 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_204_16 :
+    Uω (aρ 16) (bρ 16) (5015883525553 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_204 :
+    Uρ (5015883525553 / 32000000000000 : ℝ) ≤
+      (-(848812625190008480521 / 400000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (5015883525553 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(18952159245899628942921 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_204_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (5015883525553 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(2389463375289662657993 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_204_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (5015883525553 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(19454610796911121675621 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_204_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (5015883525553 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(20056788427891101095629 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_204_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (5015883525553 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(4218364758269121785757 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_204_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (5015883525553 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(22954591801955800902821 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_204_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (5015883525553 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) * (-(68619450712148196321 / 25000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_204_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (5015883525553 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_204_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (5015883525553 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_204_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (5015883525553 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_204_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (5015883525553 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_204_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (5015883525553 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_204_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (5015883525553 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_204_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (5015883525553 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_204_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (5015883525553 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_204_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (5015883525553 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_204_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_205_1 :
+    Uω (aρ 1) (bρ 1) (20120314137483 / 128000000000000 : ℝ) ≤
+      (-(4730670985398022320463 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4823344214232847 / 32000000000000000 : ℝ) ≤
+      (-(4730670985398022320463 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4823344214232847 / 32000000000000000 : ℝ) =
+            (4823344214232847 / 4000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4823344214232847 / 4000000000000000 : ℝ) ≤
+            (467932866851977679537 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (150718780998967 / 1000000000000000 : ℝ)) (q :=
+      (4823344214232847 / 32000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_205_2 :
+    Uω (aρ 2) (bρ 2) (20120314137483 / 128000000000000 : ℝ) ≤
+      (-(9542867026361629120287 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23726686069576899 / 160000000000000000 : ℝ) ≤
+      (-(9542867026361629120287 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (23726686069576899 / 160000000000000000 : ℝ) =
+            (23726686069576899 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23726686069576899 / 20000000000000000 : ℝ) ≤
+            (854340678138370879713 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1482202447301253 / 10000000000000000 : ℝ)) (q :=
+      (23726686069576899 / 160000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_205_3 :
+    Uω (aρ 3) (bρ 3) (20120314137483 / 128000000000000 : ℝ) ≤
+      (-(3884713052838447392357 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22938512993288603 / 160000000000000000 : ℝ) ≤
+      (-(3884713052838447392357 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (22938512993288603 / 160000000000000000 : ℝ) =
+            (22938512993288603 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22938512993288603 / 20000000000000000 : ℝ) ≤
+            (274170028961552607643 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (89442307444701 / 625000000000000 : ℝ)) (q :=
+      (22938512993288603 / 160000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_205_4 :
+    Uω (aρ 4) (bρ 4) (20120314137483 / 128000000000000 : ℝ) ≤
+      (-(2502960900806985594063 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4320482912780263 / 32000000000000000 : ℝ) ≤
+      (-(2502960900806985594063 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4320482912780263 / 32000000000000000 : ℝ) =
+            (4320482912780263 / 4000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4320482912780263 / 4000000000000000 : ℝ) ≤
+            (96341025318014405937 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (268471759301361 / 2000000000000000 : ℝ)) (q :=
+      (4320482912780263 / 32000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_205_5 :
+    Uω (aρ 5) (bρ 5) (20120314137483 / 128000000000000 : ℝ) ≤
+      (-(421093265151500674027 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19486218817967691 / 160000000000000000 : ℝ) ≤
+      (-(421093265151500674027 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (19486218817967691 / 160000000000000000 : ℝ) =
+            (19486218817967691 / 10000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19486218817967691 / 10000000000000000 : ℝ) ≤
+            (133424479088499325973 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (598012123800051 / 5000000000000000 : ℝ)) (q :=
+      (19486218817967691 / 160000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_205_6 :
+    Uω (aρ 6) (bρ 6) (20120314137483 / 128000000000000 : ℝ) ≤
+      (-(22907835009299187804699 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16189944072949099 / 160000000000000000 : ℝ) ≤
+      (-(22907835009299187804699 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (16189944072949099 / 160000000000000000 : ℝ) =
+            (16189944072949099 / 10000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16189944072949099 / 10000000000000000 : ℝ) ≤
+            (4818052202700812195301 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (475615139806389 / 5000000000000000 : ℝ)) (q :=
+      (16189944072949099 / 160000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_205_7 :
+    Uω (aρ 7) (bρ 7) (20120314137483 / 128000000000000 : ℝ) ≤
+      (-(27348107758450982129949 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10385006033903507 / 160000000000000000 : ℝ) ≤
+      (-(27348107758450982129949 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (10385006033903507 / 160000000000000000 : ℝ) =
+            (10385006033903507 / 10000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10385006033903507 / 10000000000000000 : ℝ) ≤
+            (377779453549017870051 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (449274405492079 / 10000000000000000 : ℝ)) (q :=
+      (10385006033903507 / 160000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_205_8 :
+    Uω (aρ 8) (bρ 8) (20120314137483 / 128000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_205_9 :
+    Uω (aρ 9) (bρ 9) (20120314137483 / 128000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_205_10 :
+    Uω (aρ 10) (bρ 10) (20120314137483 / 128000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_205_11 :
+    Uω (aρ 11) (bρ 11) (20120314137483 / 128000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_205_12 :
+    Uω (aρ 12) (bρ 12) (20120314137483 / 128000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_205_13 :
+    Uω (aρ 13) (bρ 13) (20120314137483 / 128000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_205_14 :
+    Uω (aρ 14) (bρ 14) (20120314137483 / 128000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_205_15 :
+    Uω (aρ 15) (bρ 15) (20120314137483 / 128000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_205_16 :
+    Uω (aρ 16) (bρ 16) (20120314137483 / 128000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_205 :
+    Uρ (20120314137483 / 128000000000000 : ℝ) ≤
+      (-(165633997039833799489 / 78125000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (20120314137483 / 128000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(4730670985398022320463 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_205_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (20120314137483 / 128000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(9542867026361629120287 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_205_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (20120314137483 / 128000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3884713052838447392357 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_205_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (20120314137483 / 128000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(2502960900806985594063 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_205_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (20120314137483 / 128000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(421093265151500674027 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_205_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (20120314137483 / 128000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(22907835009299187804699 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_205_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (20120314137483 / 128000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(27348107758450982129949 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_205_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (20120314137483 / 128000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_205_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (20120314137483 / 128000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_205_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (20120314137483 / 128000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_205_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (20120314137483 / 128000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_205_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (20120314137483 / 128000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_205_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (20120314137483 / 128000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_205_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (20120314137483 / 128000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_205_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (20120314137483 / 128000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_205_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (20120314137483 / 128000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_205_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_206_1 :
+    Uω (aρ 1) (bρ 1) (10088547086377 / 64000000000000 : ℝ) ≤
+      (-(151146362189353403557 / 80000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12093850575635933 / 80000000000000000 : ℝ) ≤
+      (-(151146362189353403557 / 80000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (12093850575635933 / 80000000000000000 : ℝ) =
+            (12093850575635933 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12093850575635933 / 10000000000000000 : ℝ) ≤
+            (15208961082646596443 / 80000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1511624379747577 / 10000000000000000 : ℝ)) (q :=
+      (12093850575635933 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_206_2 :
+    Uω (aρ 2) (bρ 2) (10088547086377 / 64000000000000 : ℝ) ≤
+      (-(1905585075871011538629 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11898847634875677 / 80000000000000000 : ℝ) ≤
+      (-(1905585075871011538629 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (11898847634875677 / 80000000000000000 : ℝ) =
+            (11898847634875677 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11898847634875677 / 10000000000000000 : ℝ) ≤
+            (173856465028988461371 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1486642657067513 / 10000000000000000 : ℝ)) (q :=
+      (11898847634875677 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_206_3 :
+    Uω (aρ 3) (bρ 3) (10088547086377 / 64000000000000 : ℝ) ≤
+      (-(775704646446275497539 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11504807802423677 / 80000000000000000 : ℝ) ≤
+      (-(775704646446275497539 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (11504807802423677 / 80000000000000000 : ℝ) =
+            (11504807802423677 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11504807802423677 / 10000000000000000 : ℝ) ≤
+            (56071969913724502461 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1435528805304513 / 10000000000000000 : ℝ)) (q :=
+      (11504807802423677 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_206_4 :
+    Uω (aρ 4) (bρ 4) (10088547086377 / 64000000000000 : ℝ) ≤
+      (-(19990696456353944320339 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10836900119275209 / 80000000000000000 : ℝ) ≤
+      (-(19990696456353944320339 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (10836900119275209 / 80000000000000000 : ℝ) =
+            (10836900119275209 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10836900119275209 / 10000000000000000 : ℝ) ≤
+            (803718952646055679661 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (336711516395599 / 2500000000000000 : ℝ)) (q :=
+      (10836900119275209 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_206_5 :
+    Uω (aρ 5) (bρ 5) (10088547086377 / 64000000000000 : ℝ) ≤
+      (-(21017645314166303358547 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1955848646998037 / 16000000000000000 : ℝ) ≤
+      (-(21017645314166303358547 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1955848646998037 / 16000000000000000 : ℝ) =
+            (1955848646998037 / 1000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1955848646998037 / 1000000000000000 : ℝ) ≤
+            (6708241897833696641453 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (60031088192307 / 500000000000000 : ℝ)) (q :=
+      (1955848646998037 / 16000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_206_6 :
+    Uω (aρ 6) (bρ 6) (10088547086377 / 64000000000000 : ℝ) ≤
+      (-(11430661709532966677083 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8132710735015393 / 80000000000000000 : ℝ) ≤
+      (-(11430661709532966677083 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (8132710735015393 / 80000000000000000 : ℝ) =
+            (8132710735015393 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8132710735015393 / 5000000000000000 : ℝ) ≤
+            (2432281896467033322917 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (478114506996221 / 5000000000000000 : ℝ)) (q :=
+      (8132710735015393 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_206_7 :
+    Uω (aρ 7) (bρ 7) (10088547086377 / 64000000000000 : ℝ) ≤
+      (-(13625138903623822801247 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5243550540331877 / 80000000000000000 : ℝ) ≤
+      (-(13625138903623822801247 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (5243550540331877 / 80000000000000000 : ℝ) =
+            (5243550540331877 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5243550540331877 / 5000000000000000 : ℝ) ≤
+            (237804702376177198753 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (457600346081563 / 10000000000000000 : ℝ)) (q :=
+      (5243550540331877 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_206_8 :
+    Uω (aρ 8) (bρ 8) (10088547086377 / 64000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_206_9 :
+    Uω (aρ 9) (bρ 9) (10088547086377 / 64000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_206_10 :
+    Uω (aρ 10) (bρ 10) (10088547086377 / 64000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_206_11 :
+    Uω (aρ 11) (bρ 11) (10088547086377 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_206_12 :
+    Uω (aρ 12) (bρ 12) (10088547086377 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_206_13 :
+    Uω (aρ 13) (bρ 13) (10088547086377 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_206_14 :
+    Uω (aρ 14) (bρ 14) (10088547086377 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_206_15 :
+    Uω (aρ 15) (bρ 15) (10088547086377 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_206_16 :
+    Uω (aρ 16) (bρ 16) (10088547086377 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_206 :
+    Uρ (10088547086377 / 64000000000000 : ℝ) ≤
+      (-(21182187273591068333859 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (10088547086377 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(151146362189353403557 / 80000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_206_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (10088547086377 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1905585075871011538629 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_206_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (10088547086377 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(775704646446275497539 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_206_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (10088547086377 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(19990696456353944320339 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_206_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (10088547086377 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(21017645314166303358547 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_206_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (10088547086377 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(11430661709532966677083 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_206_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (10088547086377 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(13625138903623822801247 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_206_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (10088547086377 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_206_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (10088547086377 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_206_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (10088547086377 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_206_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (10088547086377 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_206_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (10088547086377 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_206_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (10088547086377 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_206_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (10088547086377 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_206_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (10088547086377 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_206_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (10088547086377 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_206_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_207_1 :
+    Uω (aρ 1) (bρ 1) (809354968321 / 5120000000000 : ℝ) ≤
+      (-(9431996367126526943871 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24258681201906977 / 160000000000000000 : ℝ) ≤
+      (-(9431996367126526943871 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (24258681201906977 / 160000000000000000 : ℝ) =
+            (24258681201906977 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24258681201906977 / 20000000000000000 : ℝ) ≤
+            (965211337373473056129 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1516060945821419 / 10000000000000000 : ℝ)) (q :=
+      (24258681201906977 / 160000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_207_2 :
+    Uω (aρ 2) (bρ 2) (809354968321 / 5120000000000 : ℝ) ≤
+      (-(4756514146251913096419 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23868704266600737 / 160000000000000000 : ℝ) ≤
+      (-(4756514146251913096419 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (23868704266600737 / 160000000000000000 : ℝ) =
+            (23868704266600737 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23868704266600737 / 20000000000000000 : ℝ) ≤
+            (442089705998086903581 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1491082841418139 / 10000000000000000 : ℝ)) (q :=
+      (23868704266600737 / 160000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_207_3 :
+    Uω (aρ 3) (bρ 3) (809354968321 / 5120000000000 : ℝ) ≤
+      (-(19361762888502309310361 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23080717429036129 / 160000000000000000 : ℝ) ≤
+      (-(19361762888502309310361 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (23080717429036129 / 160000000000000000 : ℝ) =
+            (23080717429036129 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23080717429036129 / 20000000000000000 : ℝ) ≤
+            (1432652520497690689639 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1439980593072563 / 10000000000000000 : ℝ)) (q :=
+      (23080717429036129 / 160000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_207_4 :
+    Uω (aρ 4) (bρ 4) (809354968321 / 5120000000000 : ℝ) ≤
+      (-(4989453858591348907371 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (173961465613581 / 1280000000000000 : ℝ) ≤
+      (-(4989453858591348907371 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (173961465613581 / 1280000000000000 : ℝ) =
+            (173961465613581 / 160000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (173961465613581 / 160000000000000 : ℝ) ≤
+            (209149993658651092629 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5405331982881 / 40000000000000 : ℝ)) (q :=
+      (173961465613581 / 1280000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_207_5 :
+    Uω (aρ 5) (bρ 5) (809354968321 / 5120000000000 : ℝ) ≤
+      (-(262259610421003907303 / 125000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19630744433529281 / 160000000000000000 : ℝ) ≤
+      (-(262259610421003907303 / 125000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (19630744433529281 / 160000000000000000 : ℝ) =
+            (19630744433529281 / 10000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19630744433529281 / 10000000000000000 : ℝ) ≤
+            (84313979728996092697 / 125000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1205218069034207 / 10000000000000000 : ℝ)) (q :=
+      (19630744433529281 / 160000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_207_6 :
+    Uω (aρ 6) (bρ 6) (809354968321 / 5120000000000 : ℝ) ≤
+      (-(22815054204177024956727 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16340854675340329 / 160000000000000000 : ℝ) ≤
+      (-(22815054204177024956727 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (16340854675340329 / 160000000000000000 : ℝ) =
+            (16340854675340329 / 10000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16340854675340329 / 10000000000000000 : ℝ) ≤
+            (4910833007822975043273 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (240305556100147 / 2500000000000000 : ℝ)) (q :=
+      (16340854675340329 / 160000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_207_7 :
+    Uω (aρ 7) (bρ 7) (809354968321 / 5120000000000 : ℝ) ≤
+      (-(1086168029808033831629 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10588343642389297 / 160000000000000000 : ℝ) ≤
+      (-(1086168029808033831629 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (10588343642389297 / 160000000000000000 : ℝ) =
+            (10588343642389297 / 10000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10588343642389297 / 10000000000000000 : ℝ) ≤
+            (22867458671966168371 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (465819726041709 / 10000000000000000 : ℝ)) (q :=
+      (10588343642389297 / 160000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_207_8 :
+    Uω (aρ 8) (bρ 8) (809354968321 / 5120000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_207_9 :
+    Uω (aρ 9) (bρ 9) (809354968321 / 5120000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_207_10 :
+    Uω (aρ 10) (bρ 10) (809354968321 / 5120000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_207_11 :
+    Uω (aρ 11) (bρ 11) (809354968321 / 5120000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_207_12 :
+    Uω (aρ 12) (bρ 12) (809354968321 / 5120000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_207_13 :
+    Uω (aρ 13) (bρ 13) (809354968321 / 5120000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_207_14 :
+    Uω (aρ 14) (bρ 14) (809354968321 / 5120000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_207_15 :
+    Uω (aρ 15) (bρ 15) (809354968321 / 5120000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_207_16 :
+    Uω (aρ 16) (bρ 16) (809354968321 / 5120000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_207 :
+    Uρ (809354968321 / 5120000000000 : ℝ) ≤
+      (-(169307316687165452409 / 80000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (809354968321 / 5120000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(9431996367126526943871 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_207_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (809354968321 / 5120000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(4756514146251913096419 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_207_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (809354968321 / 5120000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(19361762888502309310361 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_207_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (809354968321 / 5120000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(4989453858591348907371 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_207_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (809354968321 / 5120000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(262259610421003907303 / 125000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_207_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (809354968321 / 5120000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(22815054204177024956727 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_207_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (809354968321 / 5120000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(1086168029808033831629 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_207_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (809354968321 / 5120000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_207_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (809354968321 / 5120000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_207_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (809354968321 / 5120000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_207_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (809354968321 / 5120000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_207_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (809354968321 / 5120000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_207_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (809354968321 / 5120000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_207_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (809354968321 / 5120000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_207_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (809354968321 / 5120000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_207_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (809354968321 / 5120000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_207_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/U17.lean b/LeanPool/Zeta5Irrational/Table/U17.lean
new file mode 100644
index 0000000000..1fc023c61c
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/U17.lean
@@ -0,0 +1,3504 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-! # Certified arcsine potential bounds (U17) -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma U_208_1 :
+    Uω (aρ 1) (bρ 1) (634082945103 / 4000000000000 : ℝ) ≤
+      (-(18834775819919365320531 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1520603826457973 / 10000000000000000 : ℝ) ≤
+      (-(18834775819919365320531 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1520603826457973 / 10000000000000000 : ℝ) =
+            (1520603826457973 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1520603826457973 / 1250000000000000 : ℝ) ≤
+            (1959639589080634679469 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (760248754121723 / 5000000000000000 : ℝ)) (q :=
+      (1520603826457973 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_208_2 :
+    Uω (aρ 2) (bρ 2) (634082945103 / 4000000000000 : ℝ) ≤
+      (-(18996351001129505770501 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (598492826548401 / 4000000000000000 : ℝ) ≤
+      (-(18996351001129505770501 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (598492826548401 / 4000000000000000 : ℝ) =
+            (598492826548401 / 500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (598492826548401 / 500000000000000 : ℝ) ≤
+            (1798064407870494229499 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (299104600115901 / 2000000000000000 : ℝ)) (q :=
+      (598492826548401 / 4000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_208_3 :
+    Uω (aρ 3) (bρ 3) (634082945103 / 4000000000000 : ℝ) ≤
+      (-(19331004852514832193709 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2893977309141877 / 20000000000000000 : ℝ) ≤
+      (-(19331004852514832193709 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2893977309141877 / 20000000000000000 : ℝ) =
+            (2893977309141877 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2893977309141877 / 2500000000000000 : ℝ) ≤
+            (1463410556485167806291 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1444432283329377 / 10000000000000000 : ℝ)) (q :=
+      (2893977309141877 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_208_4 :
+    Uω (aρ 4) (bρ 4) (634082945103 / 4000000000000 : ℝ) ≤
+      (-(19925043404826570845423 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (681767608758223 / 5000000000000000 : ℝ) ≤
+      (-(19925043404826570845423 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (681767608758223 / 5000000000000000 : ℝ) =
+            (681767608758223 / 625000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (681767608758223 / 625000000000000 : ℝ) ≤
+            (869372004173429154577 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (84738724392837 / 625000000000000 : ℝ)) (q :=
+      (681767608758223 / 5000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_208_5 :
+    Uω (aρ 5) (bρ 5) (634082945103 / 4000000000000 : ℝ) ≤
+      (-(2618004087800691493427 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (615718525594977 / 5000000000000000 : ℝ) ≤
+      (-(2618004087800691493427 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (615718525594977 / 5000000000000000 : ℝ) =
+            (615718525594977 / 312500000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (615718525594977 / 312500000000000 : ℝ) ≤
+            (847731813699308506573 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (75613323560463 / 625000000000000 : ℝ)) (q :=
+      (615718525594977 / 5000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_208_6 :
+    Uω (aρ 6) (bρ 6) (634082945103 / 4000000000000 : ℝ) ≤
+      (-(22769024589857949647633 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (102601527337529 / 1000000000000000 : ℝ) ≤
+      (-(22769024589857949647633 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (102601527337529 / 1000000000000000 : ℝ) =
+            (102601527337529 / 62500000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (102601527337529 / 62500000000000 : ℝ) ≤
+            (4956862622142050352367 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1509703119497 / 15625000000000 : ℝ)) (q :=
+      (102601527337529 / 1000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_208_7 :
+    Uω (aρ 7) (bρ 7) (634082945103 / 4000000000000 : ℝ) ≤
+      (-(27059794013414673404811 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1336097259073869 / 20000000000000000 : ℝ) ≤
+      (-(27059794013414673404811 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1336097259073869 / 20000000000000000 : ℝ) =
+            (1336097259073869 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1336097259073869 / 1250000000000000 : ℝ) ≤
+            (666093198585326595189 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (473938089561369 / 10000000000000000 : ℝ)) (q :=
+      (1336097259073869 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_208_8 :
+    Uω (aρ 8) (bρ 8) (634082945103 / 4000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_208_9 :
+    Uω (aρ 9) (bρ 9) (634082945103 / 4000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_208_10 :
+    Uω (aρ 10) (bρ 10) (634082945103 / 4000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_208_11 :
+    Uω (aρ 11) (bρ 11) (634082945103 / 4000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_208_12 :
+    Uω (aρ 12) (bρ 12) (634082945103 / 4000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_208_13 :
+    Uω (aρ 13) (bρ 13) (634082945103 / 4000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_208_14 :
+    Uω (aρ 14) (bρ 14) (634082945103 / 4000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_208_15 :
+    Uω (aρ 15) (bρ 15) (634082945103 / 4000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_208_16 :
+    Uω (aρ 16) (bρ 16) (634082945103 / 4000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_208 :
+    Uρ (634082945103 / 4000000000000 : ℝ) ≤
+      (-(21144826167404248198697 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (634082945103 / 4000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(18834775819919365320531 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_208_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (634082945103 / 4000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(18996351001129505770501 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_208_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (634082945103 / 4000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(19331004852514832193709 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_208_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (634082945103 / 4000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(19925043404826570845423 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_208_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (634082945103 / 4000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(2618004087800691493427 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_208_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (634082945103 / 4000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(22769024589857949647633 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_208_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (634082945103 / 4000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(27059794013414673404811 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_208_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (634082945103 / 4000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_208_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (634082945103 / 4000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_208_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (634082945103 / 4000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_208_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (634082945103 / 4000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_208_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (634082945103 / 4000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_208_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (634082945103 / 4000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_208_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (634082945103 / 4000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_208_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (634082945103 / 4000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_208_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (634082945103 / 4000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_208_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_209_1 :
+    Uω (aρ 1) (bρ 1) (20347434278567 / 128000000000000 : ℝ) ≤
+      (-(9402822015822657623339 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4880128243157723 / 32000000000000000 : ℝ) ≤
+      (-(9402822015822657623339 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4880128243157723 / 32000000000000000 : ℝ) =
+            (4880128243157723 / 4000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4880128243157723 / 4000000000000000 : ℝ) ≤
+            (994385688677342376661 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (152493406704553 / 1000000000000000 : ℝ)) (q :=
+      (4880128243157723 / 32000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_209_2 :
+    Uω (aρ 2) (bρ 2) (20347434278567 / 128000000000000 : ℝ) ≤
+      (-(18966733481316253290297 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24010721657546831 / 160000000000000000 : ℝ) ≤
+      (-(18966733481316253290297 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (24010721657546831 / 160000000000000000 : ℝ) =
+            (24010721657546831 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24010721657546831 / 20000000000000000 : ℝ) ≤
+            (1827681927683746709703 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1499963134775307 / 10000000000000000 : ℝ)) (q :=
+      (24010721657546831 / 160000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_209_3 :
+    Uω (aρ 3) (bρ 3) (20347434278567 / 128000000000000 : ℝ) ≤
+      (-(19300341465000340110197 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4644583748866899 / 32000000000000000 : ℝ) ≤
+      (-(19300341465000340110197 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4644583748866899 / 32000000000000000 : ℝ) =
+            (4644583748866899 / 4000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4644583748866899 / 4000000000000000 : ℝ) ≤
+            (1494073943999659889803 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (289776775394753 / 2000000000000000 : ℝ)) (q :=
+      (4644583748866899 / 32000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_209_4 :
+    Uω (aρ 4) (bρ 4) (20347434278567 / 128000000000000 : ℝ) ≤
+      (-(3978475927905370941629 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21887941100810039 / 160000000000000000 : ℝ) ≤
+      (-(3978475927905370941629 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (21887941100810039 / 160000000000000000 : ℝ) =
+            (21887941100810039 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21887941100810039 / 20000000000000000 : ℝ) ≤
+            (180407153894629058371 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (21254778946847 / 156250000000000 : ℝ)) (q :=
+      (21887941100810039 / 160000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_209_5 :
+    Uω (aρ 5) (bρ 5) (20347434278567 / 128000000000000 : ℝ) ≤
+      (-(5226858955032555311931 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19775231735235271 / 160000000000000000 : ℝ) ≤
+      (-(5226858955032555311931 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (19775231735235271 / 160000000000000000 : ℝ) =
+            (19775231735235271 / 10000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19775231735235271 / 10000000000000000 : ℝ) ≤
+            (1704612847967444688069 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (607203550618181 / 5000000000000000 : ℝ)) (q :=
+      (19775231735235271 / 160000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_209_6 :
+    Uω (aρ 6) (bρ 6) (20347434278567 / 128000000000000 : ℝ) ≤
+      (-(22723231852291062307257 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16491591236337559 / 160000000000000000 : ℝ) ≤
+      (-(22723231852291062307257 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (16491591236337559 / 160000000000000000 : ℝ) =
+            (16491591236337559 / 10000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16491591236337559 / 10000000000000000 : ℝ) ≤
+            (5002655359708937692743 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (242798103503537 / 2500000000000000 : ℝ)) (q :=
+      (16491591236337559 / 160000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_209_7 :
+    Uω (aρ 7) (bρ 7) (20347434278567 / 128000000000000 : ℝ) ≤
+      (-(26966981408494280424653 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10788445209094639 / 160000000000000000 : ℝ) ≤
+      (-(26966981408494280424653 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (10788445209094639 / 160000000000000000 : ℝ) =
+            (10788445209094639 / 10000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10788445209094639 / 10000000000000000 : ℝ) ≤
+            (758905803505719575347 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (481960541368783 / 10000000000000000 : ℝ)) (q :=
+      (10788445209094639 / 160000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_209_8 :
+    Uω (aρ 8) (bρ 8) (20347434278567 / 128000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_209_9 :
+    Uω (aρ 9) (bρ 9) (20347434278567 / 128000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_209_10 :
+    Uω (aρ 10) (bρ 10) (20347434278567 / 128000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_209_11 :
+    Uω (aρ 11) (bρ 11) (20347434278567 / 128000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_209_12 :
+    Uω (aρ 12) (bρ 12) (20347434278567 / 128000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_209_13 :
+    Uω (aρ 13) (bρ 13) (20347434278567 / 128000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_209_14 :
+    Uω (aρ 14) (bρ 14) (20347434278567 / 128000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_209_15 :
+    Uω (aρ 15) (bρ 15) (20347434278567 / 128000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_209_16 :
+    Uω (aρ 16) (bρ 16) (20347434278567 / 128000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_209 :
+    Uρ (20347434278567 / 128000000000000 : ℝ) ≤
+      (-(5281603793154777291613 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (20347434278567 / 128000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(9402822015822657623339 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_209_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (20347434278567 / 128000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(18966733481316253290297 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_209_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (20347434278567 / 128000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(19300341465000340110197 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_209_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (20347434278567 / 128000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3978475927905370941629 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_209_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (20347434278567 / 128000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(5226858955032555311931 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_209_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (20347434278567 / 128000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(22723231852291062307257 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_209_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (20347434278567 / 128000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(26966981408494280424653 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_209_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (20347434278567 / 128000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_209_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (20347434278567 / 128000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_209_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (20347434278567 / 128000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_209_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (20347434278567 / 128000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_209_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (20347434278567 / 128000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_209_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (20347434278567 / 128000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_209_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (20347434278567 / 128000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_209_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (20347434278567 / 128000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_209_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (20347434278567 / 128000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_209_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_210_1 :
+    Uω (aρ 1) (bρ 1) (10202107156919 / 64000000000000 : ℝ) ≤
+      (-(18776596874758489166257 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (489432423590843 / 3200000000000000 : ℝ) ≤
+      (-(18776596874758489166257 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (489432423590843 / 3200000000000000 : ℝ) =
+            (489432423590843 / 400000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (489432423590843 / 400000000000000 : ℝ) ≤
+            (2017818534241510833743 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (61174824890367 / 400000000000000 : ℝ)) (q :=
+      (489432423590843 / 3200000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_210_2 :
+    Uω (aρ 2) (bρ 2) (10202107156919 / 64000000000000 : ℝ) ≤
+      (-(9468601752239696732103 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2408173005520151 / 16000000000000000 : ℝ) ≤
+      (-(9468601752239696732103 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2408173005520151 / 16000000000000000 : ℝ) =
+            (2408173005520151 / 2000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2408173005520151 / 2000000000000000 : ℝ) ≤
+            (928605952260303267897 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (300880648845319 / 2000000000000000 : ℝ)) (q :=
+      (2408173005520151 / 16000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_210_3 :
+    Uω (aρ 3) (bρ 3) (10202107156919 / 64000000000000 : ℝ) ≤
+      (-(19269772143215850917411 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11647009124868387 / 80000000000000000 : ℝ) ≤
+      (-(19269772143215850917411 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (11647009124868387 / 80000000000000000 : ℝ) =
+            (11647009124868387 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11647009124868387 / 10000000000000000 : ℝ) ≤
+            (1524643265784149082589 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1453335374893503 / 10000000000000000 : ℝ)) (q :=
+      (11647009124868387 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_210_4 :
+    Uω (aρ 4) (bρ 4) (10202107156919 / 64000000000000 : ℝ) ≤
+      (-(124123896360046046919 / 62500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (439186321791003 / 3200000000000000 : ℝ) ≤
+      (-(124123896360046046919 / 62500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (439186321791003 / 3200000000000000 : ℝ) =
+            (439186321791003 / 400000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (439186321791003 / 400000000000000 : ℝ) ≤
+            (5841199946203953081 / 62500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (54591671437407 / 400000000000000 : ℝ)) (q :=
+      (439186321791003 / 3200000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_210_5 :
+    Uω (aρ 5) (bρ 5) (10202107156919 / 64000000000000 : ℝ) ≤
+      (-(20870977099919409056433 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1984746128917551 / 16000000000000000 : ℝ) ≤
+      (-(20870977099919409056433 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1984746128917551 / 16000000000000000 : ℝ) =
+            (1984746128917551 / 1000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1984746128917551 / 1000000000000000 : ℝ) ≤
+            (6854910112080590943567 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (243799971044669 / 2000000000000000 : ℝ)) (q :=
+      (1984746128917551 / 16000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_210_6 :
+    Uω (aρ 6) (bρ 6) (10202107156919 / 64000000000000 : ℝ) ≤
+      (-(11338836658656259650589 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8283447959127683 / 80000000000000000 : ℝ) ≤
+      (-(11338836658656259650589 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (8283447959127683 / 80000000000000000 : ℝ) =
+            (8283447959127683 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8283447959127683 / 5000000000000000 : ℝ) ≤
+            (2524106947343740349411 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (976169558998327 / 10000000000000000 : ℝ)) (q :=
+      (8283447959127683 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_210_7 :
+    Uω (aρ 7) (bρ 7) (10202107156919 / 64000000000000 : ℝ) ≤
+      (-(13437846207618377346453 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5443691373805147 / 80000000000000000 : ℝ) ≤
+      (-(13437846207618377346453 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (5443691373805147 / 80000000000000000 : ℝ) =
+            (5443691373805147 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5443691373805147 / 5000000000000000 : ℝ) ≤
+            (425097398381622653547 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (489891793427693 / 10000000000000000 : ℝ)) (q :=
+      (5443691373805147 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_210_8 :
+    Uω (aρ 8) (bρ 8) (10202107156919 / 64000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_210_9 :
+    Uω (aρ 9) (bρ 9) (10202107156919 / 64000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_210_10 :
+    Uω (aρ 10) (bρ 10) (10202107156919 / 64000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_210_11 :
+    Uω (aρ 11) (bρ 11) (10202107156919 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_210_12 :
+    Uω (aρ 12) (bρ 12) (10202107156919 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_210_13 :
+    Uω (aρ 13) (bρ 13) (10202107156919 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_210_14 :
+    Uω (aρ 14) (bρ 14) (10202107156919 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_210_15 :
+    Uω (aρ 15) (bρ 15) (10202107156919 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_210_16 :
+    Uω (aρ 16) (bρ 16) (10202107156919 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_210 :
+    Uρ (10202107156919 / 64000000000000 : ℝ) ≤
+      (-(5277043811099700677133 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (10202107156919 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(18776596874758489166257 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_210_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (10202107156919 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(9468601752239696732103 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_210_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (10202107156919 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(19269772143215850917411 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_210_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (10202107156919 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(124123896360046046919 / 62500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_210_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (10202107156919 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20870977099919409056433 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_210_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (10202107156919 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(11338836658656259650589 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_210_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (10202107156919 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(13437846207618377346453 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_210_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (10202107156919 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_210_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (10202107156919 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_210_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (10202107156919 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_210_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (10202107156919 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_210_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (10202107156919 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_210_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (10202107156919 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_210_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (10202107156919 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_210_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (10202107156919 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_210_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (10202107156919 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_210_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_211_1 :
+    Uω (aρ 1) (bρ 1) (20460994349109 / 128000000000000 : ℝ) ≤
+      (-(9373816929443217670237 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24542601114837277 / 160000000000000000 : ℝ) ≤
+      (-(9373816929443217670237 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (24542601114837277 / 160000000000000000 : ℝ) =
+            (24542601114837277 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24542601114837277 / 20000000000000000 : ℝ) ≤
+            (1023390775056782329763 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1533807173915519 / 10000000000000000 : ℝ)) (q :=
+      (24542601114837277 / 160000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_211_2 :
+    Uω (aρ 2) (bρ 2) (20460994349109 / 128000000000000 : ℝ) ≤
+      (-(1890776055414610494797 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24152738256647677 / 160000000000000000 : ℝ) ≤
+      (-(1890776055414610494797 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (24152738256647677 / 160000000000000000 : ℝ) =
+            (24152738256647677 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24152738256647677 / 20000000000000000 : ℝ) ≤
+            (188665485485389505203 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1508843329151819 / 10000000000000000 : ℝ)) (q :=
+      (24152738256647677 / 160000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_211_3 :
+    Uω (aρ 3) (bρ 3) (20460994349109 / 128000000000000 : ℝ) ≤
+      (-(19239296309802124964013 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23365116996357173 / 160000000000000000 : ℝ) ≤
+      (-(19239296309802124964013 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (23365116996357173 / 160000000000000000 : ℝ) =
+            (23365116996357173 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23365116996357173 / 20000000000000000 : ℝ) ≤
+            (1555119099197875035987 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (728893388982753 / 5000000000000000 : ℝ)) (q :=
+      (23365116996357173 / 160000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_211_4 :
+    Uω (aρ 4) (bρ 4) (20460994349109 / 128000000000000 : ℝ) ≤
+      (-(4956843506365323570481 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (881227538893997 / 6400000000000000 : ℝ) ≤
+      (-(4956843506365323570481 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (881227538893997 / 6400000000000000 : ℝ) =
+            (881227538893997 / 800000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (881227538893997 / 800000000000000 : ℝ) ≤
+            (241760345884676429519 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (213949592739 / 1562500000000 : ℝ)) (q :=
+      (881227538893997 / 6400000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_211_5 :
+    Uω (aρ 5) (bρ 5) (20460994349109 / 128000000000000 : ℝ) ≤
+      (-(10417327733948295803493 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19919681586283741 / 160000000000000000 : ℝ) ≤
+      (-(10417327733948295803493 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (19919681586283741 / 160000000000000000 : ℝ) =
+            (19919681586283741 / 10000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19919681586283741 / 10000000000000000 : ℝ) ≤
+            (3445615872051704196507 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1223591452106327 / 10000000000000000 : ℝ)) (q :=
+      (19919681586283741 / 160000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_211_6 :
+    Uω (aρ 6) (bρ 6) (20460994349109 / 128000000000000 : ℝ) ≤
+      (-(5658086589788000643379 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3328431812335569 / 32000000000000000 : ℝ) ≤
+      (-(5658086589788000643379 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (3328431812335569 / 32000000000000000 : ℝ) =
+            (3328431812335569 / 2000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3328431812335569 / 2000000000000000 : ℝ) ≤
+            (1273385213211999356621 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (98114151167059 / 1000000000000000 : ℝ)) (q :=
+      (3328431812335569 / 32000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_211_7 :
+    Uω (aρ 7) (bρ 7) (20460994349109 / 128000000000000 : ℝ) ≤
+      (-(6696465406727953716517 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10985625565955669 / 160000000000000000 : ℝ) ≤
+      (-(6696465406727953716517 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (10985625565955669 / 160000000000000000 : ℝ) =
+            (10985625565955669 / 10000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10985625565955669 / 10000000000000000 : ℝ) ≤
+            (235006396272046283483 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (248868102732659 / 5000000000000000 : ℝ)) (q :=
+      (10985625565955669 / 160000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_211_8 :
+    Uω (aρ 8) (bρ 8) (20460994349109 / 128000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_211_9 :
+    Uω (aρ 9) (bρ 9) (20460994349109 / 128000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_211_10 :
+    Uω (aρ 10) (bρ 10) (20460994349109 / 128000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_211_11 :
+    Uω (aρ 11) (bρ 11) (20460994349109 / 128000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_211_12 :
+    Uω (aρ 12) (bρ 12) (20460994349109 / 128000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_211_13 :
+    Uω (aρ 13) (bρ 13) (20460994349109 / 128000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_211_14 :
+    Uω (aρ 14) (bρ 14) (20460994349109 / 128000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_211_15 :
+    Uω (aρ 15) (bρ 15) (20460994349109 / 128000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_211_16 :
+    Uω (aρ 16) (bρ 16) (20460994349109 / 128000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_211 :
+    Uρ (20460994349109 / 128000000000000 : ℝ) ≤
+      (-(4218020092928001477611 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (20460994349109 / 128000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(9373816929443217670237 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_211_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (20460994349109 / 128000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1890776055414610494797 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_211_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (20460994349109 / 128000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(19239296309802124964013 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_211_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (20460994349109 / 128000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(4956843506365323570481 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_211_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (20460994349109 / 128000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(10417327733948295803493 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_211_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (20460994349109 / 128000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(5658086589788000643379 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_211_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (20460994349109 / 128000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(6696465406727953716517 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_211_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (20460994349109 / 128000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_211_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (20460994349109 / 128000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_211_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (20460994349109 / 128000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_211_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (20460994349109 / 128000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_211_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (20460994349109 / 128000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_211_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (20460994349109 / 128000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_211_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (20460994349109 / 128000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_211_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (20460994349109 / 128000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_211_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (20460994349109 / 128000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_211_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_212_1 :
+    Uω (aρ 1) (bρ 1) (1025888719219 / 6400000000000 : ℝ) ≤
+      (-(9359377248953476493289 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6153395255480061 / 40000000000000000 : ℝ) ≤
+      (-(9359377248953476493289 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (6153395255480061 / 40000000000000000 : ℝ) =
+            (6153395255480061 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6153395255480061 / 5000000000000000 : ℝ) ≤
+            (1037830455546523506711 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1538243722045343 / 10000000000000000 : ℝ)) (q :=
+      (6153395255480061 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_212_2 :
+    Uω (aρ 2) (bρ 2) (1025888719219 / 6400000000000 : ℝ) ≤
+      (-(943920205920242897763 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6055936565903101 / 40000000000000000 : ℝ) ≤
+      (-(943920205920242897763 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (6055936565903101 / 40000000000000000 : ℝ) =
+            (6055936565903101 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6055936565903101 / 5000000000000000 : ℝ) ≤
+            (95800564529757102237 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1513283389766863 / 10000000000000000 : ℝ)) (q :=
+      (6055936565903101 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_212_3 :
+    Uω (aρ 3) (bρ 3) (1025888719219 / 6400000000000 : ℝ) ≤
+      (-(19208913392717338753281 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5859053747781357 / 40000000000000000 : ℝ) ≤
+      (-(19208913392717338753281 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (5859053747781357 / 40000000000000000 : ℝ) =
+            (5859053747781357 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5859053747781357 / 5000000000000000 : ℝ) ≤
+            (1585502016282661246719 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1462238087055991 / 10000000000000000 : ℝ)) (q :=
+      (5859053747781357 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_212_4 :
+    Uω (aρ 4) (bρ 4) (1025888719219 / 6400000000000 : ℝ) ≤
+      (-(791801230265439478831 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5525514568684023 / 40000000000000000 : ℝ) ≤
+      (-(791801230265439478831 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (5525514568684023 / 40000000000000000 : ℝ) =
+            (5525514568684023 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5525514568684023 / 5000000000000000 : ℝ) ≤
+            (39975386094560521169 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (343440669643081 / 2500000000000000 : ℝ)) (q :=
+      (5525514568684023 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_212_5 :
+    Uω (aρ 5) (bρ 5) (1025888719219 / 6400000000000 : ℝ) ≤
+      (-(20798469863029981377397 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4997973182595211 / 40000000000000000 : ℝ) ≤
+      (-(20798469863029981377397 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (4997973182595211 / 40000000000000000 : ℝ) =
+            (4997973182595211 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 15
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4997973182595211 / 2500000000000000 : ℝ) ≤
+            (6927417348970018622603 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (614090952431459 / 5000000000000000 : ℝ)) (q :=
+      (4997973182595211 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_212_6 :
+    Uω (aρ 6) (bρ 6) (1025888719219 / 6400000000000 : ℝ) ≤
+      (-(2823406049901683619027 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4179345323728897 / 40000000000000000 : ℝ) ≤
+      (-(2823406049901683619027 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (4179345323728897 / 40000000000000000 : ℝ) =
+            (4179345323728897 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4179345323728897 / 2500000000000000 : ℝ) ≤
+            (642329851598316380973 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (986108350569761 / 10000000000000000 : ℝ)) (q :=
+      (4179345323728897 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_212_7 :
+    Uω (aρ 7) (bρ 7) (1025888719219 / 6400000000000 : ℝ) ≤
+      (-(26697428239469773735417 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2770801501665571 / 40000000000000000 : ℝ) ≤
+      (-(26697428239469773735417 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2770801501665571 / 40000000000000000 : ℝ) =
+            (2770801501665571 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2770801501665571 / 2500000000000000 : ℝ) ≤
+            (1028458972530226264583 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (252748910149049 / 5000000000000000 : ℝ)) (q :=
+      (2770801501665571 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_212_8 :
+    Uω (aρ 8) (bρ 8) (1025888719219 / 6400000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_212_9 :
+    Uω (aρ 9) (bρ 9) (1025888719219 / 6400000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_212_10 :
+    Uω (aρ 10) (bρ 10) (1025888719219 / 6400000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_212_11 :
+    Uω (aρ 11) (bρ 11) (1025888719219 / 6400000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_212_12 :
+    Uω (aρ 12) (bρ 12) (1025888719219 / 6400000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_212_13 :
+    Uω (aρ 13) (bρ 13) (1025888719219 / 6400000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_212_14 :
+    Uω (aρ 14) (bρ 14) (1025888719219 / 6400000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_212_15 :
+    Uω (aρ 15) (bρ 15) (1025888719219 / 6400000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_212_16 :
+    Uω (aρ 16) (bρ 16) (1025888719219 / 6400000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_212 :
+    Uρ (1025888719219 / 6400000000000 : ℝ) ≤
+      (-(5268046327809099390991 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1025888719219 / 6400000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(9359377248953476493289 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_212_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1025888719219 / 6400000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(943920205920242897763 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_212_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1025888719219 / 6400000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(19208913392717338753281 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_212_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1025888719219 / 6400000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(791801230265439478831 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_212_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1025888719219 / 6400000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20798469863029981377397 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_212_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1025888719219 / 6400000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(2823406049901683619027 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_212_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1025888719219 / 6400000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(26697428239469773735417 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_212_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1025888719219 / 6400000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_212_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1025888719219 / 6400000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_212_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1025888719219 / 6400000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_212_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1025888719219 / 6400000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_212_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1025888719219 / 6400000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_212_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1025888719219 / 6400000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_212_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1025888719219 / 6400000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_212_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1025888719219 / 6400000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_212_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1025888719219 / 6400000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_212_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_213_1 :
+    Uω (aρ 1) (bρ 1) (20574554419651 / 128000000000000 : ℝ) ≤
+      (-(18689958309899125630867 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24684560901034443 / 160000000000000000 : ℝ) ≤
+      (-(18689958309899125630867 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (24684560901034443 / 160000000000000000 : ℝ) =
+            (24684560901034443 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24684560901034443 / 20000000000000000 : ℝ) ≤
+            (2104457099100874369133 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1542680266679071 / 10000000000000000 : ℝ)) (q :=
+      (24684560901034443 / 160000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_213_2 :
+    Uω (aρ 2) (bρ 2) (20574554419651 / 128000000000000 : ℝ) ≤
+      (-(294517713903934530317 / 156250000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24294754077802539 / 160000000000000000 : ℝ) ≤
+      (-(294517713903934530317 / 156250000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (24294754077802539 / 160000000000000000 : ℝ) =
+            (24294754077802539 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24294754077802539 / 20000000000000000 : ℝ) ≤
+            (30395026861690469683 / 156250000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1517723426285083 / 10000000000000000 : ℝ)) (q :=
+      (24294754077802539 / 160000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_213_3 :
+    Uω (aρ 3) (bρ 3) (20574554419651 / 128000000000000 : ℝ) ≤
+      (-(19178622825171756382173 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23507312240887027 / 160000000000000000 : ℝ) ≤
+      (-(19178622825171756382173 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (23507312240887027 / 160000000000000000 : ℝ) =
+            (23507312240887027 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23507312240887027 / 20000000000000000 : ℝ) ≤
+            (1615792583828243617827 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (366672325755161 / 2500000000000000 : ℝ)) (q :=
+      (23507312240887027 / 160000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_213_4 :
+    Uω (aρ 4) (bρ 4) (20574554419651 / 128000000000000 : ℝ) ≤
+      (-(988139645586834072963 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (886937020876051 / 6400000000000000 : ℝ) ≤
+      (-(988139645586834072963 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (886937020876051 / 6400000000000000 : ℝ) =
+            (886937020876051 / 800000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (886937020876051 / 800000000000000 : ℝ) ≤
+            (51581124863165927037 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (55129905768497 / 400000000000000 : ℝ)) (q :=
+      (886937020876051 / 6400000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_213_5 :
+    Uω (aρ 5) (bρ 5) (20574554419651 / 128000000000000 : ℝ) ≤
+      (-(5190604809880666183443 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4012818964743127 / 32000000000000000 : ℝ) ≤
+      (-(5190604809880666183443 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4012818964743127 / 32000000000000000 : ℝ) =
+            (4012818964743127 / 4000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4012818964743127 / 4000000000000000 : ℝ) ≤
+            (7999042369333816557 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (61638561313711 / 500000000000000 : ℝ)) (q :=
+      (4012818964743127 / 32000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_213_6 :
+    Uω (aρ 6) (bρ 6) (20574554419651 / 128000000000000 : ℝ) ≤
+      (-(11271188452447671165599 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16792563233044027 / 160000000000000000 : ℝ) ≤
+      (-(11271188452447671165599 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (16792563233044027 / 160000000000000000 : ℝ) =
+            (16792563233044027 / 10000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16792563233044027 / 10000000000000000 : ℝ) ≤
+            (2591755153552328834401 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (991070152580269 / 10000000000000000 : ℝ)) (q :=
+      (16792563233044027 / 160000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_213_7 :
+    Uω (aρ 7) (bρ 7) (20574554419651 / 128000000000000 : ℝ) ≤
+      (-(2661033560846422827913 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2236030824860287 / 32000000000000000 : ℝ) ≤
+      (-(2661033560846422827913 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2236030824860287 / 32000000000000000 : ℝ) =
+            (2236030824860287 / 2000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2236030824860287 / 2000000000000000 : ℝ) ≤
+            (111555160353577172087 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (102636078949489 / 2000000000000000 : ℝ)) (q :=
+      (2236030824860287 / 32000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_213_8 :
+    Uω (aρ 8) (bρ 8) (20574554419651 / 128000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_213_9 :
+    Uω (aρ 9) (bρ 9) (20574554419651 / 128000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_213_10 :
+    Uω (aρ 10) (bρ 10) (20574554419651 / 128000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_213_11 :
+    Uω (aρ 11) (bρ 11) (20574554419651 / 128000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_213_12 :
+    Uω (aρ 12) (bρ 12) (20574554419651 / 128000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_213_13 :
+    Uω (aρ 13) (bρ 13) (20574554419651 / 128000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_213_14 :
+    Uω (aρ 14) (bρ 14) (20574554419651 / 128000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_213_15 :
+    Uω (aρ 15) (bρ 15) (20574554419651 / 128000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_213_16 :
+    Uω (aρ 16) (bρ 16) (20574554419651 / 128000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_213 :
+    Uρ (20574554419651 / 128000000000000 : ℝ) ≤
+      (-(21054424620542273751667 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (20574554419651 / 128000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(18689958309899125630867 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_213_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (20574554419651 / 128000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(294517713903934530317 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_213_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (20574554419651 / 128000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(19178622825171756382173 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_213_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (20574554419651 / 128000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(988139645586834072963 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_213_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (20574554419651 / 128000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(5190604809880666183443 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_213_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (20574554419651 / 128000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(11271188452447671165599 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_213_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (20574554419651 / 128000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(2661033560846422827913 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_213_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (20574554419651 / 128000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_213_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (20574554419651 / 128000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_213_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (20574554419651 / 128000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_213_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (20574554419651 / 128000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_213_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (20574554419651 / 128000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_213_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (20574554419651 / 128000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_213_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (20574554419651 / 128000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_213_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (20574554419651 / 128000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_213_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (20574554419651 / 128000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_213_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_214_1 :
+    Uω (aρ 1) (bρ 1) (10315667227461 / 64000000000000 : ℝ) ≤
+      (-(18661244817095025294763 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2475554075242049 / 16000000000000000 : ℝ) ≤
+      (-(18661244817095025294763 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2475554075242049 / 16000000000000000 : ℝ) =
+            (2475554075242049 / 2000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2475554075242049 / 2000000000000000 : ℝ) ≤
+            (2133170591904974705237 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (77355840392339 / 500000000000000 : ℝ)) (q :=
+      (2475554075242049 / 16000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_214_2 :
+    Uω (aρ 2) (bρ 2) (10315667227461 / 64000000000000 : ℝ) ≤
+      (-(4704987191384496238617 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2436576170090501 / 16000000000000000 : ℝ) ≤
+      (-(4704987191384496238617 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2436576170090501 / 16000000000000000 : ℝ) =
+            (2436576170090501 / 2000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2436576170090501 / 2000000000000000 : ℝ) ≤
+            (493616660865503761383 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (304432687783469 / 2000000000000000 : ℝ)) (q :=
+      (2436576170090501 / 16000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_214_3 :
+    Uω (aρ 3) (bρ 3) (10315667227461 / 64000000000000 : ℝ) ≤
+      (-(19148424045563445068967 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11789204376202213 / 80000000000000000 : ℝ) ≤
+      (-(19148424045563445068967 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (11789204376202213 / 80000000000000000 : ℝ) =
+            (11789204376202213 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11789204376202213 / 10000000000000000 : ℝ) ≤
+            (1645991363436554931033 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (367785106676193 / 2500000000000000 : ℝ)) (q :=
+      (11789204376202213 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_214_4 :
+    Uω (aρ 4) (bρ 4) (10315667227461 / 64000000000000 : ℝ) ≤
+      (-(1973065979833186243353 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11122395119354733 / 80000000000000000 : ℝ) ≤
+      (-(1973065979833186243353 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (11122395119354733 / 80000000000000000 : ℝ) =
+            (11122395119354733 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11122395119354733 / 10000000000000000 : ℝ) ≤
+            (106375561066813756647 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (691366146778951 / 5000000000000000 : ℝ)) (q :=
+      (11122395119354733 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_214_5 :
+    Uω (aρ 5) (bρ 5) (10315667227461 / 64000000000000 : ℝ) ≤
+      (-(10363251278104916277097 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10068143983497481 / 80000000000000000 : ℝ) ≤
+      (-(10363251278104916277097 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (10068143983497481 / 80000000000000000 : ℝ) =
+            (10068143983497481 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10068143983497481 / 10000000000000000 : ℝ) ≤
+            (33956426395083722903 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1237359428928589 / 10000000000000000 : ℝ)) (q :=
+      (10068143983497481 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_214_6 :
+    Uω (aρ 6) (bρ 6) (10315667227461 / 64000000000000 : ℝ) ≤
+      (-(703054043389017050231 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8433852739132409 / 80000000000000000 : ℝ) ≤
+      (-(703054043389017050231 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (8433852739132409 / 80000000000000000 : ℝ) =
+            (8433852739132409 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8433852739132409 / 5000000000000000 : ℝ) ≤
+            (163379931985982949769 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (996026992977321 / 10000000000000000 : ℝ)) (q :=
+      (8433852739132409 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_214_7 :
+    Uω (aρ 7) (bρ 7) (10315667227461 / 64000000000000 : ℝ) ≤
+      (-(13262265429716689337577 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5638248951379653 / 80000000000000000 : ℝ) ≤
+      (-(13262265429716689337577 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (5638248951379653 / 80000000000000000 : ℝ) =
+            (5638248951379653 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5638248951379653 / 5000000000000000 : ℝ) ≤
+            (600678176283310662423 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (130196856699783 / 2500000000000000 : ℝ)) (q :=
+      (5638248951379653 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_214_8 :
+    Uω (aρ 8) (bρ 8) (10315667227461 / 64000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_214_9 :
+    Uω (aρ 9) (bρ 9) (10315667227461 / 64000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_214_10 :
+    Uω (aρ 10) (bρ 10) (10315667227461 / 64000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_214_11 :
+    Uω (aρ 11) (bρ 11) (10315667227461 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_214_12 :
+    Uω (aρ 12) (bρ 12) (10315667227461 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_214_13 :
+    Uω (aρ 13) (bρ 13) (10315667227461 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_214_14 :
+    Uω (aρ 14) (bρ 14) (10315667227461 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_214_15 :
+    Uω (aρ 15) (bρ 15) (10315667227461 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_214_16 :
+    Uω (aρ 16) (bρ 16) (10315667227461 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_214 :
+    Uρ (10315667227461 / 64000000000000 : ℝ) ≤
+      (-(21036813553478639506113 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (10315667227461 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(18661244817095025294763 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_214_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (10315667227461 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(4704987191384496238617 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_214_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (10315667227461 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(19148424045563445068967 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_214_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (10315667227461 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1973065979833186243353 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_214_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (10315667227461 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(10363251278104916277097 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_214_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (10315667227461 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(703054043389017050231 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_214_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (10315667227461 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(13262265429716689337577 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_214_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (10315667227461 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_214_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (10315667227461 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_214_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (10315667227461 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_214_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (10315667227461 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_214_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (10315667227461 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_214_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (10315667227461 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_214_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (10315667227461 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_214_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (10315667227461 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_214_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (10315667227461 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_214_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_215_1 :
+    Uω (aρ 1) (bρ 1) (20688114490193 / 128000000000000 : ℝ) ≤
+      (-(18632613545832130859089 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24826520576316249 / 160000000000000000 : ℝ) ≤
+      (-(18632613545832130859089 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (24826520576316249 / 160000000000000000 : ℝ) =
+            (24826520576316249 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24826520576316249 / 20000000000000000 : ℝ) ≤
+            (2161801863167869140911 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1551553345578203 / 10000000000000000 : ℝ)) (q :=
+      (24826520576316249 / 160000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_215_2 :
+    Uω (aρ 2) (bρ 2) (20688114490193 / 128000000000000 : ℝ) ≤
+      (-(18790848846917232790029 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4887353826917421 / 32000000000000000 : ℝ) ≤
+      (-(18790848846917232790029 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4887353826917421 / 32000000000000000 : ℝ) =
+            (4887353826917421 / 4000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4887353826917421 / 4000000000000000 : ℝ) ≤
+            (2003566562082767209971 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (76330171393603 / 500000000000000 : ℝ)) (q :=
+      (4887353826917421 / 32000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_215_3 :
+    Uω (aρ 3) (bρ 3) (20688114490193 / 128000000000000 : ℝ) ≤
+      (-(19118316497414914090889 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4729900906471677 / 32000000000000000 : ℝ) ≤
+      (-(19118316497414914090889 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4729900906471677 / 32000000000000000 : ℝ) =
+            (4729900906471677 / 4000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4729900906471677 / 4000000000000000 : ℝ) ≤
+            (1676098911585085909111 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (147559145894347 / 1000000000000000 : ℝ)) (q :=
+      (4729900906471677 / 32000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_215_4 :
+    Uω (aρ 4) (bρ 4) (20688114490193 / 128000000000000 : ℝ) ≤
+      (-(19698630730860232821229 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22316152449701369 / 160000000000000000 : ℝ) ≤
+      (-(19698630730860232821229 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (22316152449701369 / 160000000000000000 : ℝ) =
+            (22316152449701369 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22316152449701369 / 20000000000000000 : ℝ) ≤
+            (1095784678139767178771 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1387216629676343 / 10000000000000000 : ℝ)) (q :=
+      (22316152449701369 / 160000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_215_5 :
+    Uω (aρ 5) (bρ 5) (20688114490193 / 128000000000000 : ℝ) ≤
+      (-(20690718791954763536791 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20208472259413113 / 160000000000000000 : ℝ) ≤
+      (-(20690718791954763536791 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (20208472259413113 / 160000000000000000 : ℝ) =
+            (20208472259413113 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20208472259413113 / 20000000000000000 : ℝ) ≤
+            (103696617045236463209 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1241946525225311 / 10000000000000000 : ℝ)) (q :=
+      (20208472259413113 / 160000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_215_6 :
+    Uω (aρ 6) (bρ 6) (20688114490193 / 128000000000000 : ℝ) ≤
+      (-(22453303405870982867681 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16942808620254937 / 160000000000000000 : ℝ) ≤
+      (-(22453303405870982867681 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (16942808620254937 / 160000000000000000 : ℝ) =
+            (16942808620254937 / 10000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16942808620254937 / 10000000000000000 : ℝ) ≤
+            (5272583806129017132319 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1000978945470539 / 10000000000000000 : ℝ)) (q :=
+      (16942808620254937 / 160000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_215_7 :
+    Uω (aρ 7) (bρ 7) (20688114490193 / 128000000000000 : ℝ) ≤
+      (-(6609991136014297365741 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11372263446798641 / 160000000000000000 : ℝ) ≤
+      (-(6609991136014297365741 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (11372263446798641 / 160000000000000000 : ℝ) =
+            (11372263446798641 / 10000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11372263446798641 / 10000000000000000 : ℝ) ≤
+            (321480666985702634259 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (264161089774251 / 5000000000000000 : ℝ)) (q :=
+      (11372263446798641 / 160000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_215_8 :
+    Uω (aρ 8) (bρ 8) (20688114490193 / 128000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_215_9 :
+    Uω (aρ 9) (bρ 9) (20688114490193 / 128000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_215_10 :
+    Uω (aρ 10) (bρ 10) (20688114490193 / 128000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_215_11 :
+    Uω (aρ 11) (bρ 11) (20688114490193 / 128000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_215_12 :
+    Uω (aρ 12) (bρ 12) (20688114490193 / 128000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_215_13 :
+    Uω (aρ 13) (bρ 13) (20688114490193 / 128000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_215_14 :
+    Uω (aρ 14) (bρ 14) (20688114490193 / 128000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_215_15 :
+    Uω (aρ 15) (bρ 15) (20688114490193 / 128000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_215_16 :
+    Uω (aρ 16) (bρ 16) (20688114490193 / 128000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_215 :
+    Uρ (20688114490193 / 128000000000000 : ℝ) ≤
+      (-(21019347567321569209947 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (20688114490193 / 128000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(18632613545832130859089 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_215_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (20688114490193 / 128000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(18790848846917232790029 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_215_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (20688114490193 / 128000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(19118316497414914090889 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_215_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (20688114490193 / 128000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(19698630730860232821229 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_215_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (20688114490193 / 128000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20690718791954763536791 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_215_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (20688114490193 / 128000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(22453303405870982867681 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_215_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (20688114490193 / 128000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(6609991136014297365741 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_215_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (20688114490193 / 128000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_215_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (20688114490193 / 128000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_215_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (20688114490193 / 128000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_215_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (20688114490193 / 128000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_215_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (20688114490193 / 128000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_215_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (20688114490193 / 128000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_215_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (20688114490193 / 128000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_215_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (20688114490193 / 128000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_215_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (20688114490193 / 128000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_215_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_216_1 :
+    Uω (aρ 1) (bρ 1) (2593111815683 / 16000000000000 : ℝ) ≤
+      (-(744162561060257032513 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3112187546619607 / 20000000000000000 : ℝ) ≤
+      (-(744162561060257032513 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3112187546619607 / 20000000000000000 : ℝ) =
+            (3112187546619607 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3112187546619607 / 2500000000000000 : ℝ) ≤
+            (87614055299742967487 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (388997469975683 / 2500000000000000 : ℝ)) (q :=
+      (3112187546619607 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_216_2 :
+    Uω (aρ 2) (bρ 2) (2593111815683 / 16000000000000 : ℝ) ≤
+      (-(18761833439794934696371 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3063472047562097 / 20000000000000000 : ℝ) ≤
+      (-(18761833439794934696371 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3063472047562097 / 20000000000000000 : ℝ) =
+            (3063472047562097 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3063472047562097 / 2500000000000000 : ℝ) ≤
+            (2032581969205065303629 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (765521696677611 / 5000000000000000 : ℝ)) (q :=
+      (3063472047562097 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_216_3 :
+    Uω (aρ 3) (bρ 3) (2593111815683 / 16000000000000 : ℝ) ≤
+      (-(19088299629310782092179 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1482537474209321 / 10000000000000000 : ℝ) ≤
+      (-(19088299629310782092179 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1482537474209321 / 10000000000000000 : ℝ) =
+            (1482537474209321 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1482537474209321 / 1250000000000000 : ℝ) ≤
+            (1706115779689217907821 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1480042400561767 / 10000000000000000 : ℝ)) (q :=
+      (1482537474209321 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_216_4 :
+    Uω (aρ 4) (bρ 4) (2593111815683 / 16000000000000 : ℝ) ≤
+      (-(393334100610784337877 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2798439022387463 / 20000000000000000 : ℝ) ≤
+      (-(393334100610784337877 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2798439022387463 / 20000000000000000 : ℝ) =
+            (2798439022387463 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2798439022387463 / 2500000000000000 : ℝ) ≤
+            (22554207569215662123 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (347925163898897 / 2500000000000000 : ℝ)) (q :=
+      (2798439022387463 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_216_5 :
+    Uω (aρ 5) (bρ 5) (2593111815683 / 16000000000000 : ℝ) ≤
+      (-(20655066935054398764263 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (633770243708767 / 5000000000000000 : ℝ) ≤
+      (-(20655066935054398764263 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (633770243708767 / 5000000000000000 : ℝ) =
+            (633770243708767 / 625000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (633770243708767 / 625000000000000 : ℝ) ≤
+            (139348473945601235737 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1246532527378193 / 10000000000000000 : ℝ)) (q :=
+      (633770243708767 / 5000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_216_6 :
+    Uω (aρ 6) (bρ 6) (2593111815683 / 16000000000000 : ℝ) ≤
+      (-(22409096555836940562381 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1063617077281511 / 10000000000000000 : ℝ) ≤
+      (-(22409096555836940562381 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1063617077281511 / 10000000000000000 : ℝ) =
+            (1063617077281511 / 625000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1063617077281511 / 625000000000000 : ℝ) ≤
+            (5316790656163059437619 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1005926082246147 / 10000000000000000 : ℝ)) (q :=
+      (1063617077281511 / 10000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_216_7 :
+    Uω (aρ 7) (bρ 7) (2593111815683 / 16000000000000 : ℝ) ≤
+      (-(3294573791959917824413 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1433434393940571 / 20000000000000000 : ℝ) ≤
+      (-(3294573791959917824413 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1433434393940571 / 20000000000000000 : ℝ) =
+            (1433434393940571 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1433434393940571 / 1250000000000000 : ℝ) ≤
+            (171162109540082175587 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (33486731398981 / 625000000000000 : ℝ)) (q :=
+      (1433434393940571 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_216_8 :
+    Uω (aρ 8) (bρ 8) (2593111815683 / 16000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_216_9 :
+    Uω (aρ 9) (bρ 9) (2593111815683 / 16000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_216_10 :
+    Uω (aρ 10) (bρ 10) (2593111815683 / 16000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_216_11 :
+    Uω (aρ 11) (bρ 11) (2593111815683 / 16000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_216_12 :
+    Uω (aρ 12) (bρ 12) (2593111815683 / 16000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_216_13 :
+    Uω (aρ 13) (bρ 13) (2593111815683 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_216_14 :
+    Uω (aρ 14) (bρ 14) (2593111815683 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_216_15 :
+    Uω (aρ 15) (bρ 15) (2593111815683 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_216_16 :
+    Uω (aρ 16) (bρ 16) (2593111815683 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_216 :
+    Uρ (2593111815683 / 16000000000000 : ℝ) ≤
+      (-(10501011194519760065581 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (2593111815683 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(744162561060257032513 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_216_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (2593111815683 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(18761833439794934696371 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_216_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (2593111815683 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(19088299629310782092179 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_216_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (2593111815683 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(393334100610784337877 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_216_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (2593111815683 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20655066935054398764263 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_216_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (2593111815683 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(22409096555836940562381 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_216_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (2593111815683 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(3294573791959917824413 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_216_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (2593111815683 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_216_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (2593111815683 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_216_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (2593111815683 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_216_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (2593111815683 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_216_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (2593111815683 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_216_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (2593111815683 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_216_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (2593111815683 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_216_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (2593111815683 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_216_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (2593111815683 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_216_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_217_1 :
+    Uω (aρ 1) (bρ 1) (4160334912147 / 25600000000000 : ℝ) ≤
+      (-(18575595793526144554733 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24968480142574791 / 160000000000000000 : ℝ) ≤
+      (-(18575595793526144554733 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (24968480142574791 / 160000000000000000 : ℝ) =
+            (24968480142574791 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24968480142574791 / 20000000000000000 : ℝ) ≤
+            (2218819615473855445267 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1560426410849427 / 10000000000000000 : ℝ)) (q :=
+      (24968480142574791 / 160000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_217_2 :
+    Uω (aρ 2) (bρ 2) (4160334912147 / 25600000000000 : ℝ) ≤
+      (-(9366451027138736003651 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4915756688052579 / 32000000000000000 : ℝ) ≤
+      (-(9366451027138736003651 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4915756688052579 / 32000000000000000 : ℝ) =
+            (4915756688052579 / 4000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4915756688052579 / 4000000000000000 : ℝ) ≤
+            (1030756677361263996349 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (38387083389261 / 250000000000000 : ℝ)) (q :=
+      (4915756688052579 / 32000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_217_3 :
+    Uω (aρ 3) (bρ 3) (4160334912147 / 25600000000000 : ℝ) ≤
+      (-(19058372894836346185957 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23791693923897639 / 160000000000000000 : ℝ) ≤
+      (-(19058372894836346185957 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (23791693923897639 / 160000000000000000 : ℝ) =
+            (23791693923897639 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23791693923897639 / 20000000000000000 : ℝ) ≤
+            (1736042514163653814043 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1484493252374783 / 10000000000000000 : ℝ)) (q :=
+      (23791693923897639 / 160000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_217_4 :
+    Uω (aρ 4) (bρ 4) (4160334912147 / 25600000000000 : ℝ) ≤
+      (-(490872050631875603417 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22458869450814343 / 160000000000000000 : ℝ) ≤
+      (-(490872050631875603417 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (22458869450814343 / 160000000000000000 : ℝ) =
+            (22458869450814343 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22458869450814343 / 20000000000000000 : ℝ) ≤
+            (28988334593124396583 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1396184374304371 / 10000000000000000 : ℝ)) (q :=
+      (22458869450814343 / 160000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_217_5 :
+    Uω (aρ 5) (bρ 5) (4160334912147 / 25600000000000 : ℝ) ≤
+      (-(20619545985646205896543 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4070562936210387 / 32000000000000000 : ℝ) ≤
+      (-(20619545985646205896543 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4070562936210387 / 32000000000000000 : ℝ) =
+            (4070562936210387 / 4000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4070562936210387 / 4000000000000000 : ℝ) ≤
+            (174869423353794103457 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (125111744741907 / 1000000000000000 : ℝ)) (q :=
+      (4070562936210387 / 32000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_217_6 :
+    Uω (aρ 6) (bρ 6) (4160334912147 / 25600000000000 : ℝ) ≤
+      (-(22365106478660146974889 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17092899892641359 / 160000000000000000 : ℝ) ≤
+      (-(22365106478660146974889 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (17092899892641359 / 160000000000000000 : ℝ) =
+            (17092899892641359 / 10000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17092899892641359 / 10000000000000000 : ℝ) ≤
+            (5360780733339853025111 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (252717118501937 / 2500000000000000 : ℝ)) (q :=
+      (17092899892641359 / 160000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_217_7 :
+    Uω (aρ 7) (bρ 7) (4160334912147 / 25600000000000 : ℝ) ≤
+      (-(26274364758818529627057 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11562155852785783 / 160000000000000000 : ℝ) ≤
+      (-(26274364758818529627057 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (11562155852785783 / 160000000000000000 : ℝ) =
+            (11562155852785783 / 10000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11562155852785783 / 10000000000000000 : ℝ) ≤
+            (1451522453181470372943 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (543186849785801 / 10000000000000000 : ℝ)) (q :=
+      (11562155852785783 / 160000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_217_8 :
+    Uω (aρ 8) (bρ 8) (4160334912147 / 25600000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_217_9 :
+    Uω (aρ 9) (bρ 9) (4160334912147 / 25600000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_217_10 :
+    Uω (aρ 10) (bρ 10) (4160334912147 / 25600000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_217_11 :
+    Uω (aρ 11) (bρ 11) (4160334912147 / 25600000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_217_12 :
+    Uω (aρ 12) (bρ 12) (4160334912147 / 25600000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_217_13 :
+    Uω (aρ 13) (bρ 13) (4160334912147 / 25600000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_217_14 :
+    Uω (aρ 14) (bρ 14) (4160334912147 / 25600000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_217_15 :
+    Uω (aρ 15) (bρ 15) (4160334912147 / 25600000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_217_16 :
+    Uω (aρ 16) (bρ 16) (4160334912147 / 25600000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_217 :
+    Uρ (4160334912147 / 25600000000000 : ℝ) ≤
+      (-(4196966798483047930983 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (4160334912147 / 25600000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(18575595793526144554733 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_217_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (4160334912147 / 25600000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(9366451027138736003651 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_217_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (4160334912147 / 25600000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(19058372894836346185957 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_217_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (4160334912147 / 25600000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(490872050631875603417 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_217_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (4160334912147 / 25600000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20619545985646205896543 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_217_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (4160334912147 / 25600000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(22365106478660146974889 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_217_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (4160334912147 / 25600000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(26274364758818529627057 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_217_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (4160334912147 / 25600000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_217_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (4160334912147 / 25600000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_217_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (4160334912147 / 25600000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_217_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (4160334912147 / 25600000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_217_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (4160334912147 / 25600000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_217_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (4160334912147 / 25600000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_217_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (4160334912147 / 25600000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_217_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (4160334912147 / 25600000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_217_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (4160334912147 / 25600000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_217_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_218_1 :
+    Uω (aρ 1) (bρ 1) (10429227298003 / 64000000000000 : ℝ) ≤
+      (-(18547208385266200982007 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12519729942699943 / 80000000000000000 : ℝ) ≤
+      (-(18547208385266200982007 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (12519729942699943 / 80000000000000000 : ℝ) =
+            (12519729942699943 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12519729942699943 / 10000000000000000 : ℝ) ≤
+            (2247207023733799017993 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1564862938447017 / 10000000000000000 : ℝ)) (q :=
+      (12519729942699943 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_218_2 :
+    Uω (aρ 2) (bρ 2) (10429227298003 / 64000000000000 : ℝ) ≤
+      (-(3740810840944483037073 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12324895157747767 / 80000000000000000 : ℝ) ≤
+      (-(3740810840944483037073 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (12324895157747767 / 80000000000000000 : ℝ) =
+            (12324895157747767 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12324895157747767 / 10000000000000000 : ℝ) ≤
+            (418072240855516962927 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1539923254718973 / 10000000000000000 : ℝ)) (q :=
+      (12324895157747767 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_218_3 :
+    Uω (aρ 3) (bρ 3) (10429227298003 / 64000000000000 : ℝ) ≤
+      (-(19028535752517117294929 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11931393774223367 / 80000000000000000 : ℝ) ≤
+      (-(19028535752517117294929 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (11931393774223367 / 80000000000000000 : ℝ) =
+            (11931393774223367 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11931393774223367 / 10000000000000000 : ℝ) ≤
+            (1765879656482882705071 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1488944015187873 / 10000000000000000 : ℝ)) (q :=
+      (11931393774223367 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_218_4 :
+    Uω (aρ 4) (bρ 4) (10429227298003 / 64000000000000 : ℝ) ≤
+      (-(19603161049574197177753 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11265112144223691 / 80000000000000000 : ℝ) ≤
+      (-(19603161049574197177753 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (11265112144223691 / 80000000000000000 : ℝ) =
+            (11265112144223691 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11265112144223691 / 10000000000000000 : ℝ) ≤
+            (1191254359425802822247 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (700333894376477 / 5000000000000000 : ℝ)) (q :=
+      (11265112144223691 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_218_5 :
+    Uω (aρ 5) (bρ 5) (10429227298003 / 64000000000000 : ℝ) ≤
+      (-(4116830991103902289863 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10212486500676799 / 80000000000000000 : ℝ) ≤
+      (-(4116830991103902289863 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (10212486500676799 / 80000000000000000 : ℝ) =
+            (10212486500676799 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10212486500676799 / 10000000000000000 : ℝ) ≤
+            (42052090696097710137 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1255701297201231 / 10000000000000000 : ℝ)) (q :=
+      (10212486500676799 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_218_6 :
+    Uω (aρ 6) (bρ 6) (10429227298003 / 64000000000000 : ℝ) ≤
+      (-(2790166356911144784423 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8583944571374899 / 80000000000000000 : ℝ) ≤
+      (-(2790166356911144784423 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (8583944571374899 / 80000000000000000 : ℝ) =
+            (8583944571374899 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8583944571374899 / 5000000000000000 : ℝ) ≤
+            (675569544588855215577 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (253951547503939 / 2500000000000000 : ℝ)) (q :=
+      (8583944571374899 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_218_7 :
+    Uω (aρ 7) (bρ 7) (10429227298003 / 64000000000000 : ℝ) ≤
+      (-(2619324694814019901359 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5828163480531791 / 80000000000000000 : ℝ) ≤
+      (-(2619324694814019901359 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (5828163480531791 / 80000000000000000 : ℝ) =
+            (5828163480531791 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5828163480531791 / 5000000000000000 : ℝ) ≤
+            (153264026385980098641 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (550522298064979 / 10000000000000000 : ℝ)) (q :=
+      (5828163480531791 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_218_8 :
+    Uω (aρ 8) (bρ 8) (10429227298003 / 64000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_218_9 :
+    Uω (aρ 9) (bρ 9) (10429227298003 / 64000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_218_10 :
+    Uω (aρ 10) (bρ 10) (10429227298003 / 64000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_218_11 :
+    Uω (aρ 11) (bρ 11) (10429227298003 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_218_12 :
+    Uω (aρ 12) (bρ 12) (10429227298003 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_218_13 :
+    Uω (aρ 13) (bρ 13) (10429227298003 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_218_14 :
+    Uω (aρ 14) (bρ 14) (10429227298003 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_218_15 :
+    Uω (aρ 15) (bρ 15) (10429227298003 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_218_16 :
+    Uω (aρ 16) (bρ 16) (10429227298003 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_218 :
+    Uρ (10429227298003 / 64000000000000 : ℝ) ≤
+      (-(20967778577491089068771 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (10429227298003 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(18547208385266200982007 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_218_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (10429227298003 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3740810840944483037073 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_218_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (10429227298003 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(19028535752517117294929 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_218_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (10429227298003 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(19603161049574197177753 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_218_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (10429227298003 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(4116830991103902289863 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_218_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (10429227298003 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(2790166356911144784423 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_218_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (10429227298003 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(2619324694814019901359 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_218_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (10429227298003 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_218_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (10429227298003 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_218_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (10429227298003 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_218_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (10429227298003 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_218_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (10429227298003 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_218_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (10429227298003 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_218_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (10429227298003 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_218_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (10429227298003 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_218_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (10429227298003 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_218_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_219_1 :
+    Uω (aρ 1) (bρ 1) (20915234631277 / 128000000000000 : ℝ) ≤
+      (-(4629725336005811571373 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25110439601659373 / 160000000000000000 : ℝ) ≤
+      (-(4629725336005811571373 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (25110439601659373 / 160000000000000000 : ℝ) =
+            (25110439601659373 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25110439601659373 / 20000000000000000 : ℝ) ≤
+            (568878516244188428627 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (784649731361953 / 5000000000000000 : ℝ)) (q :=
+      (25110439601659373 / 160000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_219_2 :
+    Uω (aρ 2) (bρ 2) (20915234631277 / 128000000000000 : ℝ) ≤
+      (-(2334411176211178011403 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24720797007786261 / 160000000000000000 : ℝ) ≤
+      (-(2334411176211178011403 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (24720797007786261 / 160000000000000000 : ℝ) =
+            (24720797007786261 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24720797007786261 / 20000000000000000 : ℝ) ≤
+            (264890749913821988597 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1544363150999767 / 10000000000000000 : ℝ)) (q :=
+      (24720797007786261 / 160000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_219_3 :
+    Uω (aρ 3) (bρ 3) (20915234631277 / 128000000000000 : ℝ) ≤
+      (-(18998787665759244029903 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23933880467362333 / 160000000000000000 : ℝ) ≤
+      (-(18998787665759244029903 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (23933880467362333 / 160000000000000000 : ℝ) =
+            (23933880467362333 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23933880467362333 / 20000000000000000 : ℝ) ≤
+            (1795627743240755970097 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (186674336224597 / 1250000000000000 : ℝ)) (q :=
+      (23933880467362333 / 160000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_219_4 :
+    Uω (aρ 4) (bρ 4) (20915234631277 / 128000000000000 : ℝ) ≤
+      (-(19571541444456643283281 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4520315343059617 / 32000000000000000 : ℝ) ≤
+      (-(19571541444456643283281 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4520315343059617 / 32000000000000000 : ℝ) =
+            (4520315343059617 / 4000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4520315343059617 / 4000000000000000 : ℝ) ≤
+            (1222873964543356716719 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (281030180370749 / 2000000000000000 : ℝ)) (q :=
+      (4520315343059617 / 32000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_219_5 :
+    Uω (aρ 5) (bρ 5) (20915234631277 / 128000000000000 : ℝ) ≤
+      (-(821955714717225692007 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4099424570602057 / 32000000000000000 : ℝ) ≤
+      (-(821955714717225692007 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4099424570602057 / 32000000000000000 : ℝ) =
+            (4099424570602057 / 4000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4099424570602057 / 4000000000000000 : ℝ) ≤
+            (9820901642774307993 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (126028408840277 / 1000000000000000 : ℝ)) (q :=
+      (4099424570602057 / 32000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_219_6 :
+    Uω (aρ 6) (bρ 6) (20915234631277 / 128000000000000 : ℝ) ≤
+      (-(44555534812667741173 / 20000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3448568305934473 / 32000000000000000 : ℝ) ≤
+      (-(44555534812667741173 / 20000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (3448568305934473 / 32000000000000000 : ℝ) =
+            (3448568305934473 / 2000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3448568305934473 / 2000000000000000 : ℝ) ≤
+            (10896239611332258827 / 20000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (102073929812553 / 1000000000000000 : ℝ)) (q :=
+      (3448568305934473 / 32000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_219_7 :
+    Uω (aρ 7) (bρ 7) (20915234631277 / 128000000000000 : ℝ) ≤
+      (-(13056599216528297659903 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11750008581007197 / 160000000000000000 : ℝ) ≤
+      (-(13056599216528297659903 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (11750008581007197 / 160000000000000000 : ℝ) =
+            (11750008581007197 / 10000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11750008581007197 / 10000000000000000 : ℝ) ≤
+            (806344389471702340097 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (34862285018899 / 625000000000000 : ℝ)) (q :=
+      (11750008581007197 / 160000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_219_8 :
+    Uω (aρ 8) (bρ 8) (20915234631277 / 128000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_219_9 :
+    Uω (aρ 9) (bρ 9) (20915234631277 / 128000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_219_10 :
+    Uω (aρ 10) (bρ 10) (20915234631277 / 128000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_219_11 :
+    Uω (aρ 11) (bρ 11) (20915234631277 / 128000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_219_12 :
+    Uω (aρ 12) (bρ 12) (20915234631277 / 128000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_219_13 :
+    Uω (aρ 13) (bρ 13) (20915234631277 / 128000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_219_14 :
+    Uω (aρ 14) (bρ 14) (20915234631277 / 128000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_219_15 :
+    Uω (aρ 15) (bρ 15) (20915234631277 / 128000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_219_16 :
+    Uω (aρ 16) (bρ 16) (20915234631277 / 128000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_219 :
+    Uρ (20915234631277 / 128000000000000 : ℝ) ≤
+      (-(20950852552191954841159 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (20915234631277 / 128000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(4629725336005811571373 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_219_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (20915234631277 / 128000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(2334411176211178011403 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_219_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (20915234631277 / 128000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(18998787665759244029903 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_219_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (20915234631277 / 128000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(19571541444456643283281 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_219_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (20915234631277 / 128000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(821955714717225692007 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_219_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (20915234631277 / 128000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) * (-(44555534812667741173 / 20000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_219_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (20915234631277 / 128000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(13056599216528297659903 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_219_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (20915234631277 / 128000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_219_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (20915234631277 / 128000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_219_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (20915234631277 / 128000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_219_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (20915234631277 / 128000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_219_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (20915234631277 / 128000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_219_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (20915234631277 / 128000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_219_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (20915234631277 / 128000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_219_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (20915234631277 / 128000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_219_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (20915234631277 / 128000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_219_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/U18.lean b/LeanPool/Zeta5Irrational/Table/U18.lean
new file mode 100644
index 0000000000..cf6301eec5
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/U18.lean
@@ -0,0 +1,3496 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-! # Certified arcsine potential bounds (U18) -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma U_220_1 :
+    Uω (aρ 1) (bρ 1) (5243003666637 / 32000000000000 : ℝ) ≤
+      (-(18490674215971361238241 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6295354822894483 / 40000000000000000 : ℝ) ≤
+      (-(18490674215971361238241 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (6295354822894483 / 40000000000000000 : ℝ) =
+            (6295354822894483 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6295354822894483 / 5000000000000000 : ℝ) ≤
+            (2303741193028638761759 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1573735983708179 / 10000000000000000 : ℝ)) (q :=
+      (6295354822894483 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_220_2 :
+    Uω (aρ 2) (bρ 2) (5243003666637 / 32000000000000 : ℝ) ≤
+      (-(2330825898986484937893 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6197950879677093 / 40000000000000000 : ℝ) ≤
+      (-(2330825898986484937893 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (6197950879677093 / 40000000000000000 : ℝ) =
+            (6197950879677093 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6197950879677093 / 5000000000000000 : ℝ) ≤
+            (268476027138515062107 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (387200756152371 / 2500000000000000 : ℝ)) (q :=
+      (6197950879677093 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_220_3 :
+    Uω (aρ 3) (bρ 3) (5243003666637 / 32000000000000 : ℝ) ≤
+      (-(9484564051395421285797 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6001243171733643 / 40000000000000000 : ℝ) ≤
+      (-(9484564051395421285797 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (6001243171733643 / 40000000000000000 : ℝ) =
+            (6001243171733643 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6001243171733643 / 5000000000000000 : ℝ) ≤
+            (912643653104578714203 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1497845276987759 / 10000000000000000 : ℝ)) (q :=
+      (6001243171733643 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_220_4 :
+    Uω (aρ 4) (bρ 4) (5243003666637 / 32000000000000 : ℝ) ≤
+      (-(9770011278685046203197 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5668231688591941 / 40000000000000000 : ℝ) ≤
+      (-(9770011278685046203197 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (5668231688591941 / 40000000000000000 : ℝ) =
+            (5668231688591941 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5668231688591941 / 5000000000000000 : ℝ) ≤
+            (627196425814953796803 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (352408429120477 / 2500000000000000 : ℝ)) (q :=
+      (5668231688591941 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_220_5 :
+    Uω (aρ 5) (bρ 5) (5243003666637 / 32000000000000 : ℝ) ≤
+      (-(20513758757421810226879 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5142316082017839 / 40000000000000000 : ℝ) ≤
+      (-(20513758757421810226879 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (5142316082017839 / 40000000000000000 : ℝ) =
+            (5142316082017839 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5142316082017839 / 5000000000000000 : ℝ) ≤
+            (280656651578189773121 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1264865832529857 / 10000000000000000 : ℝ)) (q :=
+      (5142316082017839 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_220_6 :
+    Uω (aρ 6) (bρ 6) (5243003666637 / 32000000000000 : ℝ) ≤
+      (-(22234413891121944816083 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4329439396326663 / 40000000000000000 : ℝ) ≤
+      (-(22234413891121944816083 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (4329439396326663 / 40000000000000000 : ℝ) =
+            (4329439396326663 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4329439396326663 / 2500000000000000 : ℝ) ≤
+            (5491473320878055183917 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1025667864824269 / 10000000000000000 : ℝ)) (q :=
+      (4329439396326663 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_220_7 :
+    Uω (aρ 7) (bρ 7) (5243003666637 / 32000000000000 : ℝ) ≤
+      (-(26034182944698725387439 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (592160980922423 / 8000000000000000 : ℝ) ≤
+      (-(26034182944698725387439 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (592160980922423 / 8000000000000000 : ℝ) =
+            (592160980922423 / 500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (592160980922423 / 500000000000000 : ℝ) ≤
+            (1691704267301274612561 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (113002399945399 / 2000000000000000 : ℝ)) (q :=
+      (592160980922423 / 8000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_220_8 :
+    Uω (aρ 8) (bρ 8) (5243003666637 / 32000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_220_9 :
+    Uω (aρ 9) (bρ 9) (5243003666637 / 32000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_220_10 :
+    Uω (aρ 10) (bρ 10) (5243003666637 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_220_11 :
+    Uω (aρ 11) (bρ 11) (5243003666637 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_220_12 :
+    Uω (aρ 12) (bρ 12) (5243003666637 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_220_13 :
+    Uω (aρ 13) (bρ 13) (5243003666637 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_220_14 :
+    Uω (aρ 14) (bρ 14) (5243003666637 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_220_15 :
+    Uω (aρ 15) (bρ 15) (5243003666637 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_220_16 :
+    Uω (aρ 16) (bρ 16) (5243003666637 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_220 :
+    Uρ (5243003666637 / 32000000000000 : ℝ) ≤
+      (-(10467026257924661355203 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (5243003666637 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(18490674215971361238241 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_220_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (5243003666637 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(2330825898986484937893 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_220_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (5243003666637 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(9484564051395421285797 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_220_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (5243003666637 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(9770011278685046203197 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_220_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (5243003666637 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20513758757421810226879 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_220_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (5243003666637 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(22234413891121944816083 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_220_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (5243003666637 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(26034182944698725387439 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_220_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (5243003666637 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_220_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (5243003666637 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_220_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (5243003666637 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_220_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (5243003666637 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_220_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (5243003666637 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_220_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (5243003666637 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_220_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (5243003666637 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_220_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (5243003666637 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_220_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (5243003666637 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_220_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_221_1 :
+    Uω (aρ 1) (bρ 1) (21028794701819 / 128000000000000 : ℝ) ≤
+      (-(4615631637779595417191 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5050479791075543 / 32000000000000000 : ℝ) ≤
+      (-(4615631637779595417191 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (5050479791075543 / 32000000000000000 : ℝ) =
+            (5050479791075543 / 4000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5050479791075543 / 4000000000000000 : ℝ) ≤
+            (582972214470404582809 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (315634500285521 / 2000000000000000 : ℝ)) (q :=
+      (5050479791075543 / 32000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_221_2 :
+    Uω (aρ 2) (bρ 2) (21028794701819 / 128000000000000 : ℝ) ≤
+      (-(465450176953729785337 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24862809849817163 / 160000000000000000 : ℝ) ≤
+      (-(465450176953729785337 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (24862809849817163 / 160000000000000000 : ℝ) =
+            (24862809849817163 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24862809849817163 / 20000000000000000 : ℝ) ≤
+            (54410208271270214663 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (194155359467817 / 1250000000000000 : ℝ)) (q :=
+      (24862809849817163 / 160000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_221_3 :
+    Uω (aρ 3) (bρ 3) (21028794701819 / 128000000000000 : ℝ) ≤
+      (-(591861141768881967909 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24076064213378907 / 160000000000000000 : ℝ) ≤
+      (-(591861141768881967909 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (24076064213378907 / 160000000000000000 : ℝ) =
+            (24076064213378907 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24076064213378907 / 20000000000000000 : ℝ) ≤
+            (57964339762368032091 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (751147888768877 / 5000000000000000 : ℝ)) (q :=
+      (24076064213378907 / 160000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_221_4 :
+    Uω (aρ 4) (bρ 4) (21028794701819 / 128000000000000 : ℝ) ≤
+      (-(9754301871052985929913 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4548854885672911 / 32000000000000000 : ℝ) ≤
+      (-(9754301871052985929913 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4548854885672911 / 32000000000000000 : ℝ) =
+            (4548854885672911 / 4000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4548854885672911 / 4000000000000000 : ℝ) ≤
+            (642905833447014070087 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (35352905886899 / 250000000000000 : ℝ)) (q :=
+      (4548854885672911 / 32000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_221_5 :
+    Uω (aρ 5) (bρ 5) (21028794701819 / 128000000000000 : ℝ) ≤
+      (-(10239375834821808422919 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20641397517236387 / 160000000000000000 : ℝ) ≤
+      (-(10239375834821808422919 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (20641397517236387 / 160000000000000000 : ℝ) =
+            (20641397517236387 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20641397517236387 / 20000000000000000 : ℝ) ≤
+            (157831869678191577081 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1269446540919939 / 10000000000000000 : ℝ)) (q :=
+      (20641397517236387 / 160000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_221_6 :
+    Uω (aρ 6) (bρ 6) (21028794701819 / 128000000000000 : ℝ) ≤
+      (-(887650724271362820473 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17392637830890523 / 160000000000000000 : ℝ) ≤
+      (-(887650724271362820473 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (17392637830890523 / 160000000000000000 : ℝ) =
+            (17392637830890523 / 10000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17392637830890523 / 10000000000000000 : ℝ) ≤
+            (221384764208637179527 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (515295977633353 / 5000000000000000 : ℝ)) (q :=
+      (17392637830890523 / 160000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_221_7 :
+    Uω (aρ 7) (bρ 7) (21028794701819 / 128000000000000 : ℝ) ≤
+      (-(12978083121240946271847 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2387195575291783 / 32000000000000000 : ℝ) ≤
+      (-(12978083121240946271847 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2387195575291783 / 32000000000000000 : ℝ) =
+            (2387195575291783 / 2000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2387195575291783 / 2000000000000000 : ℝ) ≤
+            (884860484759053728153 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (114434168344551 / 2000000000000000 : ℝ)) (q :=
+      (2387195575291783 / 32000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_221_8 :
+    Uω (aρ 8) (bρ 8) (21028794701819 / 128000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_221_9 :
+    Uω (aρ 9) (bρ 9) (21028794701819 / 128000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_221_10 :
+    Uω (aρ 10) (bρ 10) (21028794701819 / 128000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_221_11 :
+    Uω (aρ 11) (bρ 11) (21028794701819 / 128000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_221_12 :
+    Uω (aρ 12) (bρ 12) (21028794701819 / 128000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_221_13 :
+    Uω (aρ 13) (bρ 13) (21028794701819 / 128000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_221_14 :
+    Uω (aρ 14) (bρ 14) (21028794701819 / 128000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_221_15 :
+    Uω (aρ 15) (bρ 15) (21028794701819 / 128000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_221_16 :
+    Uω (aρ 16) (bρ 16) (21028794701819 / 128000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_221 :
+    Uρ (21028794701819 / 128000000000000 : ℝ) ≤
+      (-(20917375244390658060383 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (21028794701819 / 128000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(4615631637779595417191 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_221_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (21028794701819 / 128000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(465450176953729785337 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_221_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (21028794701819 / 128000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(591861141768881967909 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_221_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (21028794701819 / 128000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(9754301871052985929913 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_221_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (21028794701819 / 128000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(10239375834821808422919 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_221_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (21028794701819 / 128000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(887650724271362820473 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_221_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (21028794701819 / 128000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(12978083121240946271847 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_221_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (21028794701819 / 128000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_221_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (21028794701819 / 128000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_221_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (21028794701819 / 128000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_221_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (21028794701819 / 128000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_221_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (21028794701819 / 128000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_221_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (21028794701819 / 128000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_221_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (21028794701819 / 128000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_221_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (21028794701819 / 128000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_221_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (21028794701819 / 128000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_221_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_222_1 :
+    Uω (aρ 1) (bρ 1) (2108557473709 / 12800000000000 : ℝ) ≤
+      (-(1843445790326284033331 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12661689296639181 / 80000000000000000 : ℝ) ≤
+      (-(1843445790326284033331 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (12661689296639181 / 80000000000000000 : ℝ) =
+            (12661689296639181 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12661689296639181 / 10000000000000000 : ℝ) ≤
+            (235995750573715966669 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1582609015909639 / 10000000000000000 : ℝ)) (q :=
+      (12661689296639181 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_222_2 :
+    Uω (aρ 2) (bρ 2) (2108557473709 / 12800000000000 : ℝ) ≤
+      (-(18589488599339791797713 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12466908001325097 / 80000000000000000 : ℝ) ≤
+      (-(18589488599339791797713 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (12466908001325097 / 80000000000000000 : ℝ) =
+            (12466908001325097 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12466908001325097 / 10000000000000000 : ℝ) ≤
+            (2204926809660208202287 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (778841352295559 / 5000000000000000 : ℝ)) (q :=
+      (12466908001325097 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_222_3 :
+    Uω (aρ 3) (bρ 3) (2108557473709 / 12800000000000 : ℝ) ≤
+      (-(1891007244489895609963 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (96588620211349 / 640000000000000 : ℝ) ≤
+      (-(1891007244489895609963 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (96588620211349 / 640000000000000 : ℝ) = (96588620211349 / 80000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (96588620211349 / 80000000000000 : ℝ) ≤
+            (188434296410104390037 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3013492384429 / 20000000000000 : ℝ)) (q :=
+      (96588620211349 / 640000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_222_4 :
+    Uω (aρ 4) (bρ 4) (2108557473709 / 12800000000000 : ℝ) ≤
+      (-(19477284358716666822331 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11407809879854049 / 80000000000000000 : ℝ) ≤
+      (-(19477284358716666822331 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (11407809879854049 / 80000000000000000 : ℝ) =
+            (11407809879854049 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11407809879854049 / 10000000000000000 : ℝ) ≤
+            (1317131050283333177669 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (354649615409589 / 2500000000000000 : ℝ)) (q :=
+      (11407809879854049 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_222_5 :
+    Uω (aρ 5) (bρ 5) (2108557473709 / 12800000000000 : ℝ) ≤
+      (-(20443870661181125243673 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10356761254940081 / 80000000000000000 : ℝ) ≤
+      (-(20443870661181125243673 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (10356761254940081 / 80000000000000000 : ℝ) =
+            (10356761254940081 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10356761254940081 / 10000000000000000 : ℝ) ≤
+            (350544747818874756327 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (39813319523277 / 312500000000000 : ℝ)) (q :=
+      (10356761254940081 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_222_6 :
+    Uω (aρ 6) (bρ 6) (2108557473709 / 12800000000000 : ℝ) ≤
+      (-(11074163943683477485381 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8733741388639249 / 80000000000000000 : ℝ) ≤
+      (-(11074163943683477485381 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (8733741388639249 / 80000000000000000 : ℝ) =
+            (8733741388639249 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8733741388639249 / 5000000000000000 : ℝ) ≤
+            (2788779662316522514619 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (129438954163707 / 1250000000000000 : ℝ)) (q :=
+      (8733741388639249 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_222_7 :
+    Uω (aρ 7) (bρ 7) (2108557473709 / 12800000000000 : ℝ) ≤
+      (-(12939557978942782377787 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6014150070893521 / 80000000000000000 : ℝ) ≤
+      (-(12939557978942782377787 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (6014150070893521 / 80000000000000000 : ℝ) =
+            (6014150070893521 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6014150070893521 / 5000000000000000 : ℝ) ≤
+            (923385627057217622213 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (72409398079153 / 1250000000000000 : ℝ)) (q :=
+      (6014150070893521 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_222_8 :
+    Uω (aρ 8) (bρ 8) (2108557473709 / 12800000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_222_9 :
+    Uω (aρ 9) (bρ 9) (2108557473709 / 12800000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_222_10 :
+    Uω (aρ 10) (bρ 10) (2108557473709 / 12800000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_222_11 :
+    Uω (aρ 11) (bρ 11) (2108557473709 / 12800000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_222_12 :
+    Uω (aρ 12) (bρ 12) (2108557473709 / 12800000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_222_13 :
+    Uω (aρ 13) (bρ 13) (2108557473709 / 12800000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_222_14 :
+    Uω (aρ 14) (bρ 14) (2108557473709 / 12800000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_222_15 :
+    Uω (aρ 15) (bρ 15) (2108557473709 / 12800000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_222_16 :
+    Uω (aρ 16) (bρ 16) (2108557473709 / 12800000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_222 :
+    Uρ (2108557473709 / 12800000000000 : ℝ) ≤
+      (-(20900817676992104316193 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (2108557473709 / 12800000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1843445790326284033331 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_222_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (2108557473709 / 12800000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(18589488599339791797713 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_222_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (2108557473709 / 12800000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1891007244489895609963 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_222_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (2108557473709 / 12800000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(19477284358716666822331 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_222_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (2108557473709 / 12800000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20443870661181125243673 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_222_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (2108557473709 / 12800000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(11074163943683477485381 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_222_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (2108557473709 / 12800000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(12939557978942782377787 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_222_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (2108557473709 / 12800000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_222_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (2108557473709 / 12800000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_222_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (2108557473709 / 12800000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_222_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (2108557473709 / 12800000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_222_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (2108557473709 / 12800000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_222_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (2108557473709 / 12800000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_222_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (2108557473709 / 12800000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_222_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (2108557473709 / 12800000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_222_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (2108557473709 / 12800000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_222_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_223_1 :
+    Uω (aρ 1) (bρ 1) (21142354772361 / 128000000000000 : ℝ) ≤
+      (-(1840646782995148303141 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25394358205497089 / 160000000000000000 : ℝ) ≤
+      (-(1840646782995148303141 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (25394358205497089 / 160000000000000000 : ℝ) =
+            (25394358205497089 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25394358205497089 / 20000000000000000 : ℝ) ≤
+            (238794757904851696859 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1587045527181433 / 10000000000000000 : ℝ)) (q :=
+      (25394358205497089 / 160000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_223_2 :
+    Uω (aρ 2) (bρ 2) (21142354772361 / 128000000000000 : ℝ) ≤
+      (-(4640262822588700036873 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5000964395745509 / 32000000000000000 : ℝ) ≤
+      (-(4640262822588700036873 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (5000964395745509 / 32000000000000000 : ℝ) =
+            (5000964395745509 / 4000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5000964395745509 / 4000000000000000 : ℝ) ≤
+            (558341029661299963127 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (39053062783631 / 250000000000000 : ℝ)) (q :=
+      (5000964395745509 / 32000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_223_3 :
+    Uω (aρ 3) (bρ 3) (21142354772361 / 128000000000000 : ℝ) ≤
+      (-(18880675310025809271703 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24218245211379041 / 160000000000000000 : ℝ) ≤
+      (-(18880675310025809271703 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (24218245211379041 / 160000000000000000 : ℝ) =
+            (24218245211379041 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24218245211379041 / 20000000000000000 : ℝ) ≤
+            (1913740098974190728297 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1511196521776677 / 10000000000000000 : ℝ)) (q :=
+      (24218245211379041 / 160000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_223_4 :
+    Uω (aρ 4) (bρ 4) (21142354772361 / 128000000000000 : ℝ) ≤
+      (-(19446063773434154064433 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22886962770534089 / 160000000000000000 : ℝ) ≤
+      (-(19446063773434154064433 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (22886962770534089 / 160000000000000000 : ℝ) =
+            (22886962770534089 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22886962770534089 / 20000000000000000 : ℝ) ≤
+            (1348351635565845935567 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (711540198868029 / 5000000000000000 : ℝ)) (q :=
+      (22886962770534089 / 160000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_223_5 :
+    Uω (aρ 5) (bρ 5) (21142354772361 / 128000000000000 : ℝ) ≤
+      (-(20409114799383380511043 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20785639394077129 / 160000000000000000 : ℝ) ≤
+      (-(20409114799383380511043 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (20785639394077129 / 160000000000000000 : ℝ) =
+            (20785639394077129 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20785639394077129 / 20000000000000000 : ℝ) ≤
+            (385300609616619488957 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (639302447506969 / 5000000000000000 : ℝ)) (q :=
+      (20785639394077129 / 160000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_223_6 :
+    Uω (aρ 6) (bρ 6) (21142354772361 / 128000000000000 : ℝ) ≤
+      (-(4421118220594607150783 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17542292925209281 / 160000000000000000 : ℝ) ≤
+      (-(4421118220594607150783 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (17542292925209281 / 160000000000000000 : ℝ) =
+            (17542292925209281 / 10000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17542292925209281 / 10000000000000000 : ℝ) ≤
+            (1124059221805392849217 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1040426961545457 / 10000000000000000 : ℝ)) (q :=
+      (17542292925209281 / 160000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_223_7 :
+    Uω (aρ 7) (bρ 7) (21142354772361 / 128000000000000 : ℝ) ≤
+      (-(25803001453402450152699 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12120202262824801 / 160000000000000000 : ℝ) ≤
+      (-(25803001453402450152699 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (12120202262824801 / 160000000000000000 : ℝ) =
+            (12120202262824801 / 10000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12120202262824801 / 10000000000000000 : ℝ) ≤
+            (1922885758597549847301 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (586327009507397 / 10000000000000000 : ℝ)) (q :=
+      (12120202262824801 / 160000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_223_8 :
+    Uω (aρ 8) (bρ 8) (21142354772361 / 128000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_223_9 :
+    Uω (aρ 9) (bρ 9) (21142354772361 / 128000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_223_10 :
+    Uω (aρ 10) (bρ 10) (21142354772361 / 128000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_223_11 :
+    Uω (aρ 11) (bρ 11) (21142354772361 / 128000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_223_12 :
+    Uω (aρ 12) (bρ 12) (21142354772361 / 128000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_223_13 :
+    Uω (aρ 13) (bρ 13) (21142354772361 / 128000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_223_14 :
+    Uω (aρ 14) (bρ 14) (21142354772361 / 128000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_223_15 :
+    Uω (aρ 15) (bρ 15) (21142354772361 / 128000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_223_16 :
+    Uω (aρ 16) (bρ 16) (21142354772361 / 128000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_223 :
+    Uρ (21142354772361 / 128000000000000 : ℝ) ≤
+      (-(522109422600521759591 / 250000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (21142354772361 / 128000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1840646782995148303141 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_223_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (21142354772361 / 128000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(4640262822588700036873 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_223_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (21142354772361 / 128000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(18880675310025809271703 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_223_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (21142354772361 / 128000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(19446063773434154064433 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_223_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (21142354772361 / 128000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20409114799383380511043 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_223_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (21142354772361 / 128000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(4421118220594607150783 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_223_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (21142354772361 / 128000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(25803001453402450152699 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_223_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (21142354772361 / 128000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_223_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (21142354772361 / 128000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_223_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (21142354772361 / 128000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_223_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (21142354772361 / 128000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_223_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (21142354772361 / 128000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_223_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (21142354772361 / 128000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_223_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (21142354772361 / 128000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_223_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (21142354772361 / 128000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_223_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (21142354772361 / 128000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_223_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_224_1 :
+    Uω (aρ 1) (bρ 1) (1324945925477 / 8000000000000 : ℝ) ≤
+      (-(4594638973109356257451 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1591583612015541 / 10000000000000000 : ℝ) ≤
+      (-(4594638973109356257451 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1591583612015541 / 10000000000000000 : ℝ) =
+            (1591583612015541 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1591583612015541 / 1250000000000000 : ℝ) ≤
+            (603964879140643742549 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (198935254408729 / 1250000000000000 : ℝ)) (q :=
+      (1591583612015541 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_224_2 :
+    Uω (aρ 2) (bρ 2) (1324945925477 / 8000000000000 : ℝ) ≤
+      (-(2316586836256544584097 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1567239236222003 / 10000000000000000 : ℝ) ≤
+      (-(2316586836256544584097 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1567239236222003 / 10000000000000000 : ℝ) =
+            (1567239236222003 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1567239236222003 / 1250000000000000 : ℝ) ≤
+            (282715089868455415903 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (391640574048189 / 2500000000000000 : ℝ)) (q :=
+      (1567239236222003 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_224_3 :
+    Uω (aρ 3) (bρ 3) (1324945925477 / 8000000000000 : ℝ) ≤
+      (-(2356420577366438601497 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (303616683687529 / 2000000000000000 : ℝ) ≤
+      (-(2356420577366438601497 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (303616683687529 / 2000000000000000 : ℝ) =
+            (303616683687529 / 250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (303616683687529 / 250000000000000 : ℝ) ≤
+            (242881348758561398503 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (37891169174351 / 250000000000000 : ℝ)) (q :=
+      (303616683687529 / 2000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_224_4 :
+    Uω (aρ 4) (bρ 4) (1324945925477 / 8000000000000 : ℝ) ≤
+      (-(3882988271717983275987 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2869787935337357 / 20000000000000000 : ℝ) ≤
+      (-(3882988271717983275987 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2869787935337357 / 20000000000000000 : ℝ) =
+            (2869787935337357 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2869787935337357 / 2500000000000000 : ℝ) ≤
+            (275894810082016724013 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1427562046501107 / 10000000000000000 : ℝ)) (q :=
+      (2869787935337357 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_224_5 :
+    Uω (aρ 5) (bρ 5) (1324945925477 / 8000000000000 : ℝ) ≤
+      (-(1018724158109815143011 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (521443706415633 / 4000000000000000 : ℝ) ≤
+      (-(1018724158109815143011 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (521443706415633 / 4000000000000000 : ℝ) =
+            (521443706415633 / 500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (521443706415633 / 500000000000000 : ℝ) ≤
+            (20996612340184856989 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (256636512515383 / 2000000000000000 : ℝ)) (q :=
+      (521443706415633 / 4000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_224_6 :
+    Uω (aρ 6) (bρ 6) (1324945925477 / 8000000000000 : ℝ) ≤
+      (-(882522226357036714549 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5505333989239 / 50000000000000 : ℝ) ≤
+      (-(882522226357036714549 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (5505333989239 / 50000000000000 : ℝ) = (5505333989239 / 3125000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5505333989239 / 3125000000000 : ℝ) ≤
+            (226513262122963285451 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (20906760026687 / 200000000000000 : ℝ)) (q :=
+      (5505333989239 / 50000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_224_7 :
+    Uω (aρ 7) (bρ 7) (1324945925477 / 8000000000000 : ℝ) ≤
+      (-(643194842372322635267 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (305292480502351 / 4000000000000000 : ℝ) ≤
+      (-(643194842372322635267 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (305292480502351 / 4000000000000000 : ℝ) =
+            (305292480502351 / 250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (305292480502351 / 250000000000000 : ℝ) ≤
+            (49952337927677364733 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (118665637782101 / 2000000000000000 : ℝ)) (q :=
+      (305292480502351 / 4000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_224_8 :
+    Uω (aρ 8) (bρ 8) (1324945925477 / 8000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_224_9 :
+    Uω (aρ 9) (bρ 9) (1324945925477 / 8000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_224_10 :
+    Uω (aρ 10) (bρ 10) (1324945925477 / 8000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_224_11 :
+    Uω (aρ 11) (bρ 11) (1324945925477 / 8000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_224_12 :
+    Uω (aρ 12) (bρ 12) (1324945925477 / 8000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_224_13 :
+    Uω (aρ 13) (bρ 13) (1324945925477 / 8000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_224_14 :
+    Uω (aρ 14) (bρ 14) (1324945925477 / 8000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_224_15 :
+    Uω (aρ 15) (bρ 15) (1324945925477 / 8000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_224_16 :
+    Uω (aρ 16) (bρ 16) (1324945925477 / 8000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_224 :
+    Uρ (1324945925477 / 8000000000000 : ℝ) ≤
+      (-(4173610031223508181363 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1324945925477 / 8000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(4594638973109356257451 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_224_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1324945925477 / 8000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(2316586836256544584097 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_224_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1324945925477 / 8000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(2356420577366438601497 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_224_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1324945925477 / 8000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3882988271717983275987 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_224_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1324945925477 / 8000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(1018724158109815143011 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_224_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1324945925477 / 8000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(882522226357036714549 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_224_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1324945925477 / 8000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(643194842372322635267 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_224_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1324945925477 / 8000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_224_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1324945925477 / 8000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_224_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1324945925477 / 8000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_224_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1324945925477 / 8000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_224_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1324945925477 / 8000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_224_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1324945925477 / 8000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_224_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1324945925477 / 8000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_224_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1324945925477 / 8000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_224_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1324945925477 / 8000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_224_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_225_1 :
+    Uω (aρ 1) (bρ 1) (10656347439087 / 64000000000000 : ℝ) ≤
+      (-(4580741172024563874411 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12803648445098779 / 80000000000000000 : ℝ) ≤
+      (-(4580741172024563874411 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (12803648445098779 / 80000000000000000 : ℝ) =
+            (12803648445098779 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12803648445098779 / 10000000000000000 : ℝ) ≤
+            (617862680225436125589 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1600355042002351 / 10000000000000000 : ℝ)) (q :=
+      (12803648445098779 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_225_2 :
+    Uω (aρ 2) (bρ 2) (10656347439087 / 64000000000000 : ℝ) ≤
+      (-(9238110895246623312471 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2521783886136911 / 16000000000000000 : ℝ) ≤
+      (-(9238110895246623312471 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2521783886136911 / 16000000000000000 : ℝ) =
+            (2521783886136911 / 2000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2521783886136911 / 2000000000000000 : ℝ) ≤
+            (1159096809253376687529 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (315088360181759 / 2000000000000000 : ℝ)) (q :=
+      (2521783886136911 / 16000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_225_3 :
+    Uω (aρ 3) (bρ 3) (10656347439087 / 64000000000000 : ℝ) ≤
+      (-(939650026926023325251 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12215755830567379 / 80000000000000000 : ℝ) ≤
+      (-(939650026926023325251 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (12215755830567379 / 80000000000000000 : ℝ) =
+            (12215755830567379 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12215755830567379 / 10000000000000000 : ℝ) ≤
+            (100070743523976674749 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1524547007229501 / 10000000000000000 : ℝ)) (q :=
+      (12215755830567379 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_225_4 :
+    Uω (aρ 4) (bρ 4) (10656347439087 / 64000000000000 : ℝ) ≤
+      (-(9676494279784326839573 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2310097809709927 / 16000000000000000 : ℝ) ≤
+      (-(9676494279784326839573 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2310097809709927 / 16000000000000000 : ℝ) =
+            (2310097809709927 / 2000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2310097809709927 / 2000000000000000 : ℝ) ≤
+            (720713424715673160427 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (287304898558013 / 2000000000000000 : ℝ)) (q :=
+      (2310097809709927 / 16000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_225_5 :
+    Uω (aρ 5) (bρ 5) (10656347439087 / 64000000000000 : ℝ) ≤
+      (-(20305588925442651515513 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10500971128862519 / 80000000000000000 : ℝ) ≤
+      (-(20305588925442651515513 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (10500971128862519 / 80000000000000000 : ℝ) =
+            (10500971128862519 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10500971128862519 / 10000000000000000 : ℝ) ≤
+            (488826483557348484487 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (646167466101643 / 5000000000000000 : ℝ)) (q :=
+      (10500971128862519 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_225_6 :
+    Uω (aρ 6) (bρ 6) (10656347439087 / 64000000000000 : ℝ) ≤
+      (-(5494645146109420751617 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1776651943929107 / 16000000000000000 : ℝ) ≤
+      (-(5494645146109420751617 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1776651943929107 / 16000000000000000 : ℝ) =
+            (1776651943929107 / 1000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1776651943929107 / 1000000000000000 : ℝ) ≤
+            (1436826656890579248383 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3297335796997 / 31250000000000 : ℝ)) (q :=
+      (1776651943929107 / 16000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_225_7 :
+    Uω (aρ 7) (bρ 7) (10656347439087 / 64000000000000 : ℝ) ≤
+      (-(12789994806014067070709 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6196766801680643 / 80000000000000000 : ℝ) ≤
+      (-(12789994806014067070709 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (6196766801680643 / 80000000000000000 : ℝ) =
+            (6196766801680643 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6196766801680643 / 5000000000000000 : ℝ) ≤
+            (1072948799985932929291 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (607185606307817 / 10000000000000000 : ℝ)) (q :=
+      (6196766801680643 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_225_8 :
+    Uω (aρ 8) (bρ 8) (10656347439087 / 64000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_225_9 :
+    Uω (aρ 9) (bρ 9) (10656347439087 / 64000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_225_10 :
+    Uω (aρ 10) (bρ 10) (10656347439087 / 64000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_225_11 :
+    Uω (aρ 11) (bρ 11) (10656347439087 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_225_12 :
+    Uω (aρ 12) (bρ 12) (10656347439087 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_225_13 :
+    Uω (aρ 13) (bρ 13) (10656347439087 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_225_14 :
+    Uω (aρ 14) (bρ 14) (10656347439087 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_225_15 :
+    Uω (aρ 15) (bρ 15) (10656347439087 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_225_16 :
+    Uω (aρ 16) (bρ 16) (10656347439087 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_225 :
+    Uρ (10656347439087 / 64000000000000 : ℝ) ≤
+      (-(1302233018806088858069 / 625000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (10656347439087 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(4580741172024563874411 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_225_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (10656347439087 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(9238110895246623312471 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_225_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (10656347439087 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(939650026926023325251 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_225_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (10656347439087 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(9676494279784326839573 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_225_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (10656347439087 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20305588925442651515513 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_225_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (10656347439087 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(5494645146109420751617 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_225_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (10656347439087 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(12789994806014067070709 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_225_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (10656347439087 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_225_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (10656347439087 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_225_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (10656347439087 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_225_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (10656347439087 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_225_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (10656347439087 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_225_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (10656347439087 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_225_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (10656347439087 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_225_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (10656347439087 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_225_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (10656347439087 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_225_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_226_1 :
+    Uω (aρ 1) (bρ 1) (5356563737179 / 32000000000000 : ℝ) ≤
+      (-(71358128331416751623 / 39062500000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (257492558887961 / 1600000000000000 : ℝ) ≤
+      (-(71358128331416751623 / 39062500000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (257492558887961 / 1600000000000000 : ℝ) =
+            (257492558887961 / 200000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (257492558887961 / 200000000000000 : ℝ) ≤
+            (9870056859989498377 / 39062500000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (64369121452643 / 400000000000000 : ℝ)) (q :=
+      (257492558887961 / 1600000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_226_2 :
+    Uω (aρ 2) (bρ 2) (5356563737179 / 32000000000000 : ℝ) ≤
+      (-(18420066289172837522877 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1267992462989421 / 8000000000000000 : ℝ) ≤
+      (-(18420066289172837522877 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1267992462989421 / 8000000000000000 : ℝ) =
+            (1267992462989421 / 1000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1267992462989421 / 1000000000000000 : ℝ) ≤
+            (2374349119827162477123 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (316864244040023 / 2000000000000000 : ℝ)) (q :=
+      (1267992462989421 / 8000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_226_3 :
+    Uω (aρ 3) (bρ 3) (5356563737179 / 32000000000000 : ℝ) ≤
+      (-(18734976189111055348777 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1228684299891513 / 8000000000000000 : ℝ) ≤
+      (-(18734976189111055348777 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1228684299891513 / 8000000000000000 : ℝ) =
+            (1228684299891513 / 1000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1228684299891513 / 1000000000000000 : ℝ) ≤
+            (2059439219888944651223 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (306689383761069 / 2000000000000000 : ℝ)) (q :=
+      (1228684299891513 / 8000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_226_4 :
+    Uω (aρ 4) (bρ 4) (5356563737179 / 32000000000000 : ℝ) ≤
+      (-(19291421059334564470881 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5810910943084333 / 40000000000000000 : ℝ) ≤
+      (-(19291421059334564470881 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (5810910943084333 / 40000000000000000 : ℝ) =
+            (5810910943084333 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5810910943084333 / 5000000000000000 : ℝ) ≤
+            (1502994349665435529119 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1445485821683729 / 10000000000000000 : ℝ)) (q :=
+      (5810910943084333 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_226_5 :
+    Uω (aρ 5) (bρ 5) (5356563737179 / 32000000000000 : ℝ) ≤
+      (-(10118590390140244503249 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5286526295727557 / 40000000000000000 : ℝ) ≤
+      (-(10118590390140244503249 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (5286526295727557 / 40000000000000000 : ℝ) =
+            (5286526295727557 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5286526295727557 / 5000000000000000 : ℝ) ≤
+            (278617314359755496751 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1301483417340341 / 10000000000000000 : ℝ)) (q :=
+      (5286526295727557 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_226_6 :
+    Uω (aρ 6) (bρ 6) (5356563737179 / 32000000000000 : ℝ) ≤
+      (-(175159092645873246921 / 80000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4478959634428699 / 40000000000000000 : ℝ) ≤
+      (-(175159092645873246921 / 80000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (4478959634428699 / 40000000000000000 : ℝ) =
+            (4478959634428699 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4478959634428699 / 2500000000000000 : ℝ) ≤
+            (46648005050126753079 / 80000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4159923679027 / 39062500000000 : ℝ)) (q :=
+      (4478959634428699 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_226_7 :
+    Uω (aρ 7) (bρ 7) (5356563737179 / 32000000000000 : ℝ) ≤
+      (-(25435499359157548817667 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3143477015809841 / 40000000000000000 : ℝ) ≤
+      (-(25435499359157548817667 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (3143477015809841 / 40000000000000000 : ℝ) =
+            (3143477015809841 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3143477015809841 / 2500000000000000 : ℝ) ≤
+            (2290387852842451182333 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (620860533281483 / 10000000000000000 : ℝ)) (q :=
+      (3143477015809841 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_226_8 :
+    Uω (aρ 8) (bρ 8) (5356563737179 / 32000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_226_9 :
+    Uω (aρ 9) (bρ 9) (5356563737179 / 32000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_226_10 :
+    Uω (aρ 10) (bρ 10) (5356563737179 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_226_11 :
+    Uω (aρ 11) (bρ 11) (5356563737179 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_226_12 :
+    Uω (aρ 12) (bρ 12) (5356563737179 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_226_13 :
+    Uω (aρ 13) (bρ 13) (5356563737179 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_226_14 :
+    Uω (aρ 14) (bρ 14) (5356563737179 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_226_15 :
+    Uω (aρ 15) (bρ 15) (5356563737179 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_226_16 :
+    Uω (aρ 16) (bρ 16) (5356563737179 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_226 :
+    Uρ (5356563737179 / 32000000000000 : ℝ) ≤
+      (-(20803832407117668669149 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (5356563737179 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(71358128331416751623 / 39062500000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_226_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (5356563737179 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(18420066289172837522877 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_226_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (5356563737179 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(18734976189111055348777 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_226_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (5356563737179 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(19291421059334564470881 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_226_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (5356563737179 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(10118590390140244503249 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_226_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (5356563737179 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) * (-(175159092645873246921 / 80000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_226_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (5356563737179 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(25435499359157548817667 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_226_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (5356563737179 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_226_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (5356563737179 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_226_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (5356563737179 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_226_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (5356563737179 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_226_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (5356563737179 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_226_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (5356563737179 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_226_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (5356563737179 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_226_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (5356563737179 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_226_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (5356563737179 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_226_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_227_1 :
+    Uω (aρ 1) (bρ 1) (10769907509629 / 64000000000000 : ℝ) ≤
+      (-(9106350502993110805211 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12945607394839337 / 80000000000000000 : ℝ) ≤
+      (-(9106350502993110805211 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (12945607394839337 / 80000000000000000 : ℝ) =
+            (12945607394839337 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12945607394839337 / 10000000000000000 : ℝ) ≤
+            (1290857201506889194789 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1618101018415303 / 10000000000000000 : ℝ)) (q :=
+      (12945607394839337 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_227_2 :
+    Uω (aρ 2) (bρ 2) (10769907509629 / 64000000000000 : ℝ) ≤
+      (-(18364224635257561809589 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12750929493118181 / 80000000000000000 : ℝ) ≤
+      (-(18364224635257561809589 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (12750929493118181 / 80000000000000000 : ℝ) =
+            (12750929493118181 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12750929493118181 / 10000000000000000 : ℝ) ≤
+            (2430190773742438190411 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (796600277747507 / 5000000000000000 : ℝ)) (q :=
+      (12750929493118181 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_227_3 :
+    Uω (aρ 3) (bρ 3) (10769907509629 / 64000000000000 : ℝ) ≤
+      (-(4669321906570947279869 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2471585775060729 / 16000000000000000 : ℝ) ≤
+      (-(4669321906570947279869 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2471585775060729 / 16000000000000000 : ℝ) =
+            (2471585775060729 / 2000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2471585775060729 / 2000000000000000 : ℝ) ≤
+            (529281945679052720131 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (77117325369569 / 500000000000000 : ℝ)) (q :=
+      (2471585775060729 / 16000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_227_4 :
+    Uω (aρ 4) (bρ 4) (10769907509629 / 64000000000000 : ℝ) ≤
+      (-(9615117024126235910581 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11693150336822277 / 80000000000000000 : ℝ) ≤
+      (-(9615117024126235910581 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (11693150336822277 / 80000000000000000 : ℝ) =
+            (11693150336822277 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11693150336822277 / 10000000000000000 : ℝ) ≤
+            (782090680373764089419 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (727223026918019 / 5000000000000000 : ℝ)) (q :=
+      (11693150336822277 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_227_5 :
+    Uω (aρ 5) (bρ 5) (10769907509629 / 64000000000000 : ℝ) ≤
+      (-(20169251715464789785997 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10645118841465981 / 80000000000000000 : ℝ) ≤
+      (-(20169251715464789785997 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (10645118841465981 / 80000000000000000 : ℝ) =
+            (10645118841465981 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10645118841465981 / 10000000000000000 : ℝ) ≤
+            (625163693535210214003 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (327657024832991 / 2500000000000000 : ℝ)) (q :=
+      (10645118841465981 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_227_6 :
+    Uω (aρ 6) (bρ 6) (10769907509629 / 64000000000000 : ℝ) ≤
+      (-(10905979039386153311897 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1806502965764577 / 16000000000000000 : ℝ) ≤
+      (-(10905979039386153311897 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1806502965764577 / 16000000000000000 : ℝ) =
+            (1806502965764577 / 1000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1806502965764577 / 1000000000000000 : ℝ) ≤
+            (2956964566613846688103 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (107471747131119 / 1000000000000000 : ℝ)) (q :=
+      (1806502965764577 / 16000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_227_7 :
+    Uω (aρ 7) (bρ 7) (10769907509629 / 64000000000000 : ℝ) ≤
+      (-(25294137832826385335031 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6376458507399157 / 80000000000000000 : ℝ) ≤
+      (-(25294137832826385335031 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (6376458507399157 / 80000000000000000 : ℝ) =
+            (6376458507399157 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6376458507399157 / 5000000000000000 : ℝ) ≤
+            (2431749379173614664969 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (317182385857629 / 5000000000000000 : ℝ)) (q :=
+      (6376458507399157 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_227_8 :
+    Uω (aρ 8) (bρ 8) (10769907509629 / 64000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_227_9 :
+    Uω (aρ 9) (bρ 9) (10769907509629 / 64000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_227_10 :
+    Uω (aρ 10) (bρ 10) (10769907509629 / 64000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_227_11 :
+    Uω (aρ 11) (bρ 11) (10769907509629 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_227_12 :
+    Uω (aρ 12) (bρ 12) (10769907509629 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_227_13 :
+    Uω (aρ 13) (bρ 13) (10769907509629 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_227_14 :
+    Uω (aρ 14) (bρ 14) (10769907509629 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_227_15 :
+    Uω (aρ 15) (bρ 15) (10769907509629 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_227_16 :
+    Uω (aρ 16) (bρ 16) (10769907509629 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_227 :
+    Uρ (10769907509629 / 64000000000000 : ℝ) ≤
+      (-(4154468893221481873091 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (10769907509629 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(9106350502993110805211 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_227_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (10769907509629 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(18364224635257561809589 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_227_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (10769907509629 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(4669321906570947279869 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_227_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (10769907509629 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(9615117024126235910581 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_227_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (10769907509629 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(20169251715464789785997 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_227_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (10769907509629 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(10905979039386153311897 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_227_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (10769907509629 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(25294137832826385335031 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_227_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (10769907509629 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_227_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (10769907509629 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_227_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (10769907509629 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_227_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (10769907509629 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_227_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (10769907509629 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_227_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (10769907509629 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_227_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (10769907509629 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_227_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (10769907509629 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_227_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (10769907509629 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_227_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_228_1 :
+    Uω (aρ 1) (bρ 1) (108266875449 / 640000000000 : ℝ) ≤
+      (-(9079010911160674591103 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1627073349652751 / 10000000000000000 : ℝ) ≤
+      (-(9079010911160674591103 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1627073349652751 / 10000000000000000 : ℝ) =
+            (1627073349652751 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1627073349652751 / 1250000000000000 : ℝ) ≤
+            (1318196793339325408897 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1626973988499877 / 10000000000000000 : ℝ)) (q :=
+      (1627073349652751 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_228_2 :
+    Uω (aρ 2) (bρ 2) (108266875449 / 640000000000 : ℝ) ≤
+      (-(18308693337218033383591 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1602741753242869 / 10000000000000000 : ℝ) ≤
+      (-(18308693337218033383591 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1602741753242869 / 10000000000000000 : ℝ) =
+            (1602741753242869 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1602741753242869 / 1250000000000000 : ℝ) ≤
+            (2485722071781966616409 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1602079808190113 / 10000000000000000 : ℝ)) (q :=
+      (1602741753242869 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_228_3 :
+    Uω (aρ 3) (bρ 3) (108266875449 / 640000000000 : ℝ) ≤
+      (-(18619930974125314054273 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (776813342623067 / 5000000000000000 : ℝ) ≤
+      (-(18619930974125314054273 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (776813342623067 / 5000000000000000 : ℝ) =
+            (776813342623067 / 625000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (776813342623067 / 625000000000000 : ℝ) ≤
+            (2174484434874685945727 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1551245778546643 / 10000000000000000 : ℝ)) (q :=
+      (776813342623067 / 5000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_228_4 :
+    Uω (aρ 4) (bρ 4) (108266875449 / 640000000000 : ℝ) ≤
+      (-(766776912281108700213 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1470559310136607 / 10000000000000000 : ℝ) ≤
+      (-(766776912281108700213 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1470559310136607 / 10000000000000000 : ℝ) =
+            (1470559310136607 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1470559310136607 / 1250000000000000 : ℝ) ≤
+            (64999704078891299787 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1463405209392589 / 10000000000000000 : ℝ)) (q :=
+      (1470559310136607 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_228_5 :
+    Uω (aρ 5) (bρ 5) (108266875449 / 640000000000 : ℝ) ≤
+      (-(502544871858661952131 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2679292548779787 / 20000000000000000 : ℝ) ≤
+      (-(502544871858661952131 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2679292548779787 / 20000000000000000 : ℝ) =
+            (2679292548779787 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2679292548779787 / 2500000000000000 : ℝ) ≤
+            (17315513366338047869 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (659884528617081 / 5000000000000000 : ℝ)) (q :=
+      (2679292548779787 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_228_6 :
+    Uω (aρ 6) (bρ 6) (108266875449 / 640000000000 : ℝ) ≤
+      (-(679055625029343955549 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (455352406510399 / 4000000000000000 : ℝ) ≤
+      (-(679055625029343955549 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (455352406510399 / 4000000000000000 : ℝ) =
+            (455352406510399 / 250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (455352406510399 / 250000000000000 : ℝ) ≤
+            (187378350345656044451 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (108447891614637 / 1000000000000000 : ℝ)) (q :=
+      (455352406510399 / 4000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_228_7 :
+    Uω (aρ 7) (bρ 7) (108266875449 / 640000000000 : ℝ) ≤
+      (-(25155736715238488212441 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (808165366760121 / 10000000000000000 : ℝ) ≤
+      (-(25155736715238488212441 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (808165366760121 / 10000000000000000 : ℝ) =
+            (808165366760121 / 625000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (808165366760121 / 625000000000000 : ℝ) ≤
+            (2570150496761511787559 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (647708997874617 / 10000000000000000 : ℝ)) (q :=
+      (808165366760121 / 10000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_228_8 :
+    Uω (aρ 8) (bρ 8) (108266875449 / 640000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_228_9 :
+    Uω (aρ 9) (bρ 9) (108266875449 / 640000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_228_10 :
+    Uω (aρ 10) (bρ 10) (108266875449 / 640000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_228_11 :
+    Uω (aρ 11) (bρ 11) (108266875449 / 640000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_228_12 :
+    Uω (aρ 12) (bρ 12) (108266875449 / 640000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_228_13 :
+    Uω (aρ 13) (bρ 13) (108266875449 / 640000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_228_14 :
+    Uω (aρ 14) (bρ 14) (108266875449 / 640000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_228_15 :
+    Uω (aρ 15) (bρ 15) (108266875449 / 640000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_228_16 :
+    Uω (aρ 16) (bρ 16) (108266875449 / 640000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_228 :
+    Uρ (108266875449 / 640000000000 : ℝ) ≤
+      (-(5185311988381454830147 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (108266875449 / 640000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(9079010911160674591103 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_228_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (108266875449 / 640000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(18308693337218033383591 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_228_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (108266875449 / 640000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(18619930974125314054273 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_228_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (108266875449 / 640000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(766776912281108700213 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_228_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (108266875449 / 640000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(502544871858661952131 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_228_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (108266875449 / 640000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(679055625029343955549 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_228_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (108266875449 / 640000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(25155736715238488212441 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_228_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (108266875449 / 640000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_228_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (108266875449 / 640000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_228_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (108266875449 / 640000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_228_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (108266875449 / 640000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_228_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (108266875449 / 640000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_228_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (108266875449 / 640000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_228_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (108266875449 / 640000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_228_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (108266875449 / 640000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_228_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (108266875449 / 640000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_228_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_229_1 :
+    Uω (aρ 1) (bρ 1) (10883467580171 / 64000000000000 : ℝ) ≤
+      (-(3620728006182036485219 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13087566152328083 / 80000000000000000 : ℝ) ≤
+      (-(3620728006182036485219 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (13087566152328083 / 80000000000000000 : ℝ) =
+            (13087566152328083 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13087566152328083 / 10000000000000000 : ℝ) ≤
+            (538155075617963514781 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (817923473382651 / 5000000000000000 : ℝ)) (q :=
+      (13087566152328083 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_229_2 :
+    Uω (aρ 2) (bρ 2) (10883467580171 / 64000000000000 : ℝ) ≤
+      (-(18253468961514672487477 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12892938233831779 / 80000000000000000 : ℝ) ≤
+      (-(18253468961514672487477 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (12892938233831779 / 80000000000000000 : ℝ) =
+            (12892938233831779 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12892938233831779 / 10000000000000000 : ℝ) ≤
+            (2540946447485327512523 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (805479489825613 / 5000000000000000 : ℝ)) (q :=
+      (12892938233831779 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_229_3 :
+    Uω (aρ 3) (bρ 3) (10883467580171 / 64000000000000 : ℝ) ≤
+      (-(9281451211823238992419 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12500096840639403 / 80000000000000000 : ℝ) ≤
+      (-(9281451211823238992419 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (12500096840639403 / 80000000000000000 : ℝ) =
+            (12500096840639403 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12500096840639403 / 10000000000000000 : ℝ) ≤
+            (1115756492676761007581 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (390036184425783 / 2500000000000000 : ℝ)) (q :=
+      (12500096840639403 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_229_4 :
+    Uω (aρ 4) (bρ 4) (10883467580171 / 64000000000000 : ℝ) ≤
+      (-(19108982704437711202299 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11835794397591891 / 80000000000000000 : ℝ) ≤
+      (-(19108982704437711202299 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (11835794397591891 / 80000000000000000 : ℝ) =
+            (11835794397591891 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11835794397591891 / 10000000000000000 : ℝ) ≤
+            (1685432704562288797701 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (736181654003127 / 5000000000000000 : ℝ)) (q :=
+      (11835794397591891 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_229_5 :
+    Uω (aρ 5) (bρ 5) (10883467580171 / 64000000000000 : ℝ) ≤
+      (-(5008700887564128370101 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10789206959810951 / 80000000000000000 : ℝ) ≤
+      (-(5008700887564128370101 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (10789206959810951 / 80000000000000000 : ℝ) =
+            (10789206959810951 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10789206959810951 / 10000000000000000 : ℝ) ≤
+            (189902964685871629899 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1328906367896019 / 10000000000000000 : ℝ)) (q :=
+      (10789206959810951 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_229_6 :
+    Uω (aρ 6) (bρ 6) (10883467580171 / 64000000000000 : ℝ) ≤
+      (-(21648337739633651589547 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9181520839215787 / 80000000000000000 : ℝ) ≤
+      (-(21648337739633651589547 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (9181520839215787 / 80000000000000000 : ℝ) =
+            (9181520839215787 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9181520839215787 / 5000000000000000 : ℝ) ≤
+            (6077549472366348410453 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (273556303221807 / 2500000000000000 : ℝ)) (q :=
+      (9181520839215787 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_229_7 :
+    Uω (aρ 7) (bρ 7) (10883467580171 / 64000000000000 : ℝ) ≤
+      (-(12510071202273143955923 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1310717216291271 / 16000000000000000 : ℝ) ≤
+      (-(12510071202273143955923 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1310717216291271 / 16000000000000000 : ℝ) =
+            (1310717216291271 / 1000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1310717216291271 / 1000000000000000 : ℝ) ≤
+            (1352872403726856044077 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (66090290420737 / 1000000000000000 : ℝ)) (q :=
+      (1310717216291271 / 16000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_229_8 :
+    Uω (aρ 8) (bρ 8) (10883467580171 / 64000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_229_9 :
+    Uω (aρ 9) (bρ 9) (10883467580171 / 64000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_229_10 :
+    Uω (aρ 10) (bρ 10) (10883467580171 / 64000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_229_11 :
+    Uω (aρ 11) (bρ 11) (10883467580171 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_229_12 :
+    Uω (aρ 12) (bρ 12) (10883467580171 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_229_13 :
+    Uω (aρ 13) (bρ 13) (10883467580171 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_229_14 :
+    Uω (aρ 14) (bρ 14) (10883467580171 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_229_15 :
+    Uω (aρ 15) (bρ 15) (10883467580171 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_229_16 :
+    Uω (aρ 16) (bρ 16) (10883467580171 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_229 :
+    Uρ (10883467580171 / 64000000000000 : ℝ) ≤
+      (-(10355263825737622949503 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (10883467580171 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3620728006182036485219 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_229_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (10883467580171 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(18253468961514672487477 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_229_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (10883467580171 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(9281451211823238992419 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_229_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (10883467580171 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(19108982704437711202299 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_229_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (10883467580171 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(5008700887564128370101 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_229_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (10883467580171 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(21648337739633651589547 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_229_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (10883467580171 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(12510071202273143955923 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_229_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (10883467580171 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_229_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (10883467580171 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_229_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (10883467580171 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_229_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (10883467580171 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_229_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (10883467580171 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_229_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (10883467580171 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_229_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (10883467580171 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_229_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (10883467580171 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_229_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (10883467580171 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_229_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_230_1 :
+    Uω (aρ 1) (bρ 1) (5470123807721 / 32000000000000 : ℝ) ≤
+      (-(18049552413909734351791 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1315854546092271 / 8000000000000000 : ℝ) ≤
+      (-(18049552413909734351791 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1315854546092271 / 8000000000000000 : ℝ) =
+            (1315854546092271 / 1000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1315854546092271 / 1000000000000000 : ℝ) ≤
+            (2744862995090265648209 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (328943978680573 / 2000000000000000 : ℝ)) (q :=
+      (1315854546092271 / 8000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_230_2 :
+    Uω (aρ 2) (bρ 2) (5470123807721 / 32000000000000 : ℝ) ≤
+      (-(727941925252781587649 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6481971061064049 / 40000000000000000 : ℝ) ≤
+      (-(727941925252781587649 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (6481971061064049 / 40000000000000000 : ℝ) =
+            (6481971061064049 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6481971061064049 / 5000000000000000 : ℝ) ≤
+            (103834691107218412351 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (404959517803553 / 2500000000000000 : ℝ)) (q :=
+      (6481971061064049 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_230_3 :
+    Uω (aρ 3) (bρ 3) (5470123807721 / 32000000000000 : ℝ) ≤
+      (-(9253099115622933870931 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1257117897254897 / 8000000000000000 : ℝ) ≤
+      (-(9253099115622933870931 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1257117897254897 / 8000000000000000 : ℝ) =
+            (1257117897254897 / 1000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1257117897254897 / 1000000000000000 : ℝ) ≤
+            (1144108588877066129069 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (156904339016943 / 1000000000000000 : ℝ)) (q :=
+      (1257117897254897 / 8000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_230_4 :
+    Uω (aρ 4) (bρ 4) (5470123807721 / 32000000000000 : ℝ) ≤
+      (-(952445459756741781521 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5953555081510063 / 40000000000000000 : ℝ) ≤
+      (-(952445459756741781521 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (5953555081510063 / 40000000000000000 : ℝ) =
+            (5953555081510063 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5953555081510063 / 5000000000000000 : ℝ) ≤
+            (87275310693258218479 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1481320368852219 / 10000000000000000 : ℝ)) (q :=
+      (5953555081510063 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_230_5 :
+    Uω (aρ 5) (bρ 5) (5470123807721 / 32000000000000 : ℝ) ≤
+      (-(19968271182064297647983 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5430614717210019 / 40000000000000000 : ℝ) ≤
+      (-(19968271182064297647983 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (5430614717210019 / 40000000000000000 : ℝ) =
+            (5430614717210019 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5430614717210019 / 5000000000000000 : ℝ) ≤
+            (826144226935702352017 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1338040106037197 / 10000000000000000 : ℝ)) (q :=
+      (5430614717210019 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_230_6 :
+    Uω (aρ 6) (bρ 6) (5470123807721 / 32000000000000 : ℝ) ≤
+      (-(10783808568520747034887 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4627967280334191 / 40000000000000000 : ℝ) ≤
+      (-(10783808568520747034887 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (4627967280334191 / 40000000000000000 : ℝ) =
+            (4627967280334191 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4627967280334191 / 2500000000000000 : ℝ) ≤
+            (3079135037479252965113 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1103956762739283 / 10000000000000000 : ℝ)) (q :=
+      (4627967280334191 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_230_7 :
+    Uω (aρ 7) (bρ 7) (5470123807721 / 32000000000000 : ℝ) ≤
+      (-(6221803565729163471369 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3320641631589271 / 40000000000000000 : ℝ) ≤
+      (-(6221803565729163471369 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (3320641631589271 / 40000000000000000 : ℝ) =
+            (3320641631589271 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3320641631589271 / 2500000000000000 : ℝ) ≤
+            (709668237270836528631 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (673955319126823 / 10000000000000000 : ℝ)) (q :=
+      (3320641631589271 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_230_8 :
+    Uω (aρ 8) (bρ 8) (5470123807721 / 32000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_230_9 :
+    Uω (aρ 9) (bρ 9) (5470123807721 / 32000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_230_10 :
+    Uω (aρ 10) (bρ 10) (5470123807721 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_230_11 :
+    Uω (aρ 11) (bρ 11) (5470123807721 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_230_12 :
+    Uω (aρ 12) (bρ 12) (5470123807721 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_230_13 :
+    Uω (aρ 13) (bρ 13) (5470123807721 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_230_14 :
+    Uω (aρ 14) (bρ 14) (5470123807721 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_230_15 :
+    Uω (aρ 15) (bρ 15) (5470123807721 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_230_16 :
+    Uω (aρ 16) (bρ 16) (5470123807721 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_230 :
+    Uρ (5470123807721 / 32000000000000 : ℝ) ≤
+      (-(20680169497693025977507 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (5470123807721 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(18049552413909734351791 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_230_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (5470123807721 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(727941925252781587649 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_230_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (5470123807721 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(9253099115622933870931 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_230_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (5470123807721 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(952445459756741781521 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_230_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (5470123807721 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(19968271182064297647983 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_230_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (5470123807721 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(10783808568520747034887 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_230_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (5470123807721 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(6221803565729163471369 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_230_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (5470123807721 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_230_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (5470123807721 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_230_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (5470123807721 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_230_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (5470123807721 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_230_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (5470123807721 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_230_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (5470123807721 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_230_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (5470123807721 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_230_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (5470123807721 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_230_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (5470123807721 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_230_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_231_1 :
+    Uω (aρ 1) (bρ 1) (10997027650713 / 64000000000000 : ℝ) ≤
+      (-(17995755805428837382471 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2645904944750921 / 16000000000000000 : ℝ) ≤
+      (-(17995755805428837382471 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2645904944750921 / 16000000000000000 : ℝ) =
+            (2645904944750921 / 2000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2645904944750921 / 2000000000000000 : ℝ) ≤
+            (2798659603571162617529 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (330718565719949 / 2000000000000000 : ℝ)) (q :=
+      (2645904944750921 / 16000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_231_2 :
+    Uω (aρ 2) (bρ 2) (10997027650713 / 64000000000000 : ℝ) ≤
+      (-(4535981881318316691607 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13034945696058761 / 80000000000000000 : ℝ) ≤
+      (-(4535981881318316691607 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (13034945696058761 / 80000000000000000 : ℝ) =
+            (13034945696058761 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13034945696058761 / 10000000000000000 : ℝ) ≤
+            (662621970931683308393 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (203589635523223 / 1250000000000000 : ℝ)) (q :=
+      (13034945696058761 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_231_3 :
+    Uω (aρ 3) (bρ 3) (10997027650713 / 64000000000000 : ℝ) ≤
+      (-(2306226839652172024137 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12642259898452369 / 80000000000000000 : ℝ) ≤
+      (-(2306226839652172024137 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (12642259898452369 / 80000000000000000 : ℝ) =
+            (12642259898452369 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12642259898452369 / 10000000000000000 : ℝ) ≤
+            (293075086472827975863 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (788970870567093 / 5000000000000000 : ℝ)) (q :=
+      (12642259898452369 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_231_4 :
+    Uω (aρ 4) (bρ 4) (10997027650713 / 64000000000000 : ℝ) ≤
+      (-(18989197817517969399161 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11978421852225553 / 80000000000000000 : ℝ) ≤
+      (-(18989197817517969399161 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (11978421852225553 / 80000000000000000 : ℝ) =
+            (11978421852225553 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11978421852225553 / 10000000000000000 : ℝ) ≤
+            (1805217591482030600839 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (745138205321241 / 5000000000000000 : ℝ)) (q :=
+      (11978421852225553 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_231_5 :
+    Uω (aρ 5) (bρ 5) (10997027650713 / 64000000000000 : ℝ) ≤
+      (-(19902191349902427008359 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10933237909604261 / 80000000000000000 : ℝ) ≤
+      (-(19902191349902427008359 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (10933237909604261 / 80000000000000000 : ℝ) =
+            (10933237909604261 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10933237909604261 / 10000000000000000 : ℝ) ≤
+            (892224059097572991641 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1347170344322159 / 10000000000000000 : ℝ)) (q :=
+      (10933237909604261 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_231_6 :
+    Uω (aρ 6) (bρ 6) (10997027650713 / 64000000000000 : ℝ) ≤
+      (-(10743802233008661327371 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9330290840908217 / 80000000000000000 : ℝ) ≤
+      (-(10743802233008661327371 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (9330290840908217 / 80000000000000000 : ℝ) =
+            (9330290840908217 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9330290840908217 / 5000000000000000 : ℝ) ≤
+            (3119141372991338672629 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (278418488072037 / 2500000000000000 : ℝ)) (q :=
+      (9330290840908217 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_231_7 :
+    Uω (aρ 7) (bρ 7) (10997027650713 / 64000000000000 : ℝ) ≤
+      (-(12378411562319641158947 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1345689348787241 / 16000000000000000 : ℝ) ≤
+      (-(12378411562319641158947 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1345689348787241 / 16000000000000000 : ℝ) =
+            (1345689348787241 / 1000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1345689348787241 / 1000000000000000 : ℝ) ≤
+            (1484532043680358841053 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (137374861761029 / 2000000000000000 : ℝ)) (q :=
+      (1345689348787241 / 16000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_231_8 :
+    Uω (aρ 8) (bρ 8) (10997027650713 / 64000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_231_9 :
+    Uω (aρ 9) (bρ 9) (10997027650713 / 64000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_231_10 :
+    Uω (aρ 10) (bρ 10) (10997027650713 / 64000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_231_11 :
+    Uω (aρ 11) (bρ 11) (10997027650713 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_231_12 :
+    Uω (aρ 12) (bρ 12) (10997027650713 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_231_13 :
+    Uω (aρ 13) (bρ 13) (10997027650713 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_231_14 :
+    Uω (aρ 14) (bρ 14) (10997027650713 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_231_15 :
+    Uω (aρ 15) (bρ 15) (10997027650713 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_231_16 :
+    Uω (aρ 16) (bρ 16) (10997027650713 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_231 :
+    Uρ (10997027650713 / 64000000000000 : ℝ) ≤
+      (-(4130032091383163246483 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (10997027650713 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(17995755805428837382471 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_231_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (10997027650713 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(4535981881318316691607 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_231_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (10997027650713 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(2306226839652172024137 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_231_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (10997027650713 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(18989197817517969399161 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_231_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (10997027650713 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(19902191349902427008359 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_231_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (10997027650713 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(10743802233008661327371 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_231_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (10997027650713 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(12378411562319641158947 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_231_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (10997027650713 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_231_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (10997027650713 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_231_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (10997027650713 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_231_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (10997027650713 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_231_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (10997027650713 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_231_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (10997027650713 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_231_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (10997027650713 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_231_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (10997027650713 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_231_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (10997027650713 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_231_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/U19.lean b/LeanPool/Zeta5Irrational/Table/U19.lean
new file mode 100644
index 0000000000..44902dcf1c
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/U19.lean
@@ -0,0 +1,3536 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-! # Certified arcsine potential bounds (U19) -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma U_232_1 :
+    Uω (aρ 1) (bρ 1) (345431490187 / 2000000000000 : ℝ) ≤
+      (-(1121390443150977258413 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (415640748173641 / 2500000000000000 : ℝ) ≤
+      (-(1121390443150977258413 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (415640748173641 / 2500000000000000 : ℝ) =
+            (415640748173641 / 312500000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (415640748173641 / 312500000000000 : ℝ) ≤
+            (178260519911522741587 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (207808219067391 / 1250000000000000 : ℝ)) (q :=
+      (415640748173641 / 2500000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_232_2 :
+    Uω (aρ 2) (bρ 2) (345431490187 / 2000000000000 : ℝ) ≤
+      (-(113060024226724001509 / 62500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3276487240184307 / 20000000000000000 : ℝ) ≤
+      (-(113060024226724001509 / 62500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3276487240184307 / 20000000000000000 : ℝ) =
+            (3276487240184307 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3276487240184307 / 2500000000000000 : ℝ) ≤
+            (16905072079525998491 / 62500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1637596019844307 / 10000000000000000 : ℝ)) (q :=
+      (3276487240184307 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_232_3 :
+    Uω (aρ 3) (bρ 3) (345431490187 / 2000000000000 : ℝ) ≤
+      (-(3678749652860036091943 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3178334909659493 / 20000000000000000 : ℝ) ≤
+      (-(3678749652860036091943 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3178334909659493 / 20000000000000000 : ℝ) =
+            (3178334909659493 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3178334909659493 / 2500000000000000 : ℝ) ≤
+            (480133428939963908057 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1586839795669493 / 10000000000000000 : ℝ)) (q :=
+      (3178334909659493 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_232_4 :
+    Uω (aρ 4) (bρ 4) (345431490187 / 2000000000000 : ℝ) ≤
+      (-(18929844191670132660891 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3012432384564809 / 20000000000000000 : ℝ) ≤
+      (-(18929844191670132660891 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3012432384564809 / 20000000000000000 : ℝ) =
+            (3012432384564809 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3012432384564809 / 2500000000000000 : ℝ) ≤
+            (1864571217329867339109 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1499231451639809 / 10000000000000000 : ℝ)) (q :=
+      (3012432384564809 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_232_5 :
+    Uω (aρ 5) (bρ 5) (345431490187 / 2000000000000 : ℝ) ≤
+      (-(4959139442762577320171 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2751308167021261 / 20000000000000000 : ℝ) ≤
+      (-(4959139442762577320171 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2751308167021261 / 20000000000000000 : ℝ) =
+            (2751308167021261 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2751308167021261 / 2500000000000000 : ℝ) ≤
+            (239464409487422679829 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1356297153431261 / 10000000000000000 : ℝ)) (q :=
+      (2751308167021261 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_232_6 :
+    Uω (aρ 6) (bρ 6) (345431490187 / 2000000000000 : ℝ) ≤
+      (-(5352071603001420085127 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (587786948158219 / 5000000000000000 : ℝ) ≤
+      (-(5352071603001420085127 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (587786948158219 / 5000000000000000 : ℝ) =
+            (587786948158219 / 312500000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (587786948158219 / 312500000000000 : ℝ) ≤
+            (1579400199998579914873 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (280844288545719 / 2500000000000000 : ℝ)) (q :=
+      (587786948158219 / 5000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_232_7 :
+    Uω (aρ 7) (bρ 7) (345431490187 / 2000000000000 : ℝ) ≤
+      (-(1539303126159486415141 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (212972065231627 / 2500000000000000 : ℝ) ≤
+      (-(1539303126159486415141 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (212972065231627 / 2500000000000000 : ℝ) =
+            (212972065231627 / 156250000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (212972065231627 / 156250000000000 : ℝ) ≤
+            (193564824590513584859 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (87458408020377 / 1250000000000000 : ℝ)) (q :=
+      (212972065231627 / 2500000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_232_8 :
+    Uω (aρ 8) (bρ 8) (345431490187 / 2000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_232_9 :
+    Uω (aρ 9) (bρ 9) (345431490187 / 2000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_232_10 :
+    Uω (aρ 10) (bρ 10) (345431490187 / 2000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_232_11 :
+    Uω (aρ 11) (bρ 11) (345431490187 / 2000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_232_12 :
+    Uω (aρ 12) (bρ 12) (345431490187 / 2000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_232_13 :
+    Uω (aρ 13) (bρ 13) (345431490187 / 2000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_232_14 :
+    Uω (aρ 14) (bρ 14) (345431490187 / 2000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_232_15 :
+    Uω (aρ 15) (bρ 15) (345431490187 / 2000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_232_16 :
+    Uω (aρ 16) (bρ 16) (345431490187 / 2000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_232 :
+    Uρ (345431490187 / 2000000000000 : ℝ) ≤
+      (-(20620488410529718411491 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (345431490187 / 2000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1121390443150977258413 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_232_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (345431490187 / 2000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) * (-(113060024226724001509 / 62500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_232_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (345431490187 / 2000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3678749652860036091943 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_232_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (345431490187 / 2000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(18929844191670132660891 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_232_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (345431490187 / 2000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(4959139442762577320171 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_232_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (345431490187 / 2000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(5352071603001420085127 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_232_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (345431490187 / 2000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(1539303126159486415141 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_232_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (345431490187 / 2000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_232_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (345431490187 / 2000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_232_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (345431490187 / 2000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_232_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (345431490187 / 2000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_232_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (345431490187 / 2000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_232_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (345431490187 / 2000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_232_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (345431490187 / 2000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_232_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (345431490187 / 2000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_232_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (345431490187 / 2000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_232_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_233_1 :
+    Uω (aρ 1) (bρ 1) (5583683878263 / 32000000000000 : ℝ) ≤
+      (-(17836081128313023247777 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6721231122461949 / 40000000000000000 : ℝ) ≤
+      (-(17836081128313023247777 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (6721231122461949 / 40000000000000000 : ℝ) =
+            (6721231122461949 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6721231122461949 / 5000000000000000 : ℝ) ≤
+            (2958334280686976752223 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1680211567358787 / 10000000000000000 : ℝ)) (q :=
+      (6721231122461949 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_233_2 :
+    Uω (aρ 2) (bρ 2) (5583683878263 / 32000000000000 : ℝ) ≤
+      (-(17981834645297691005227 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6623977291121299 / 40000000000000000 : ℝ) ≤
+      (-(17981834645297691005227 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (6623977291121299 / 40000000000000000 : ℝ) =
+            (6623977291121299 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6623977291121299 / 5000000000000000 : ℝ) ≤
+            (2812580763702308994773 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (827676832099231 / 5000000000000000 : ℝ)) (q :=
+      (6623977291121299 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_233_3 :
+    Uω (aρ 3) (bρ 3) (5583683878263 / 32000000000000 : ℝ) ≤
+      (-(3656510475313698243173 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6427747820377973 / 40000000000000000 : ℝ) ≤
+      (-(3656510475313698243173 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (6427747820377973 / 40000000000000000 : ℝ) =
+            (6427747820377973 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6427747820377973 / 5000000000000000 : ℝ) ≤
+            (502372606486301756827 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1604635035176799 / 10000000000000000 : ℝ)) (q :=
+      (6427747820377973 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_233_4 :
+    Uω (aρ 4) (bρ 4) (5583683878263 / 32000000000000 : ℝ) ≤
+      (-(2351524134012850285883 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6096166592176123 / 40000000000000000 : ℝ) ≤
+      (-(2351524134012850285883 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (6096166592176123 / 40000000000000000 : ℝ) =
+            (6096166592176123 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6096166592176123 / 5000000000000000 : ℝ) ≤
+            (247777792112149714117 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (758569301070437 / 5000000000000000 : ℝ)) (q :=
+      (6096166592176123 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_233_5 :
+    Uω (aρ 5) (bρ 5) (5583683878263 / 32000000000000 : ℝ) ≤
+      (-(9853302452241801719929 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5574591063881861 / 40000000000000000 : ℝ) ≤
+      (-(9853302452241801719929 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (5574591063881861 / 40000000000000000 : ℝ) =
+            (5574591063881861 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5574591063881861 / 5000000000000000 : ℝ) ≤
+            (543905252258198280071 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1374540757328743 / 10000000000000000 : ℝ)) (q :=
+      (5574591063881861 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_233_6 :
+    Uω (aρ 6) (bρ 6) (5583683878263 / 32000000000000 : ℝ) ≤
+      (-(425033657159668721931 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4776514838449023 / 40000000000000000 : ℝ) ≤
+      (-(425033657159668721931 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (4776514838449023 / 40000000000000000 : ℝ) =
+            (4776514838449023 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4776514838449023 / 2500000000000000 : ℝ) ≤
+            (129484087080331278069 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (285685754938081 / 2500000000000000 : ℝ)) (q :=
+      (4776514838449023 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_233_7 :
+    Uω (aρ 7) (bρ 7) (5583683878263 / 32000000000000 : ℝ) ≤
+      (-(24379726540844704730177 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (139740377295619 / 1600000000000000 : ℝ) ≤
+      (-(24379726540844704730177 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (139740377295619 / 1600000000000000 : ℝ) =
+            (139740377295619 / 100000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (139740377295619 / 100000000000000 : ℝ) ≤
+            (3346160671155295269823 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (14498033949661 / 200000000000000 : ℝ)) (q :=
+      (139740377295619 / 1600000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_233_8 :
+    Uω (aρ 8) (bρ 8) (5583683878263 / 32000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_233_9 :
+    Uω (aρ 9) (bρ 9) (5583683878263 / 32000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_233_10 :
+    Uω (aρ 10) (bρ 10) (5583683878263 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_233_11 :
+    Uω (aρ 11) (bρ 11) (5583683878263 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_233_12 :
+    Uω (aρ 12) (bρ 12) (5583683878263 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_233_13 :
+    Uω (aρ 13) (bρ 13) (5583683878263 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_233_14 :
+    Uω (aρ 14) (bρ 14) (5583683878263 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_233_15 :
+    Uω (aρ 15) (bρ 15) (5583683878263 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_233_16 :
+    Uω (aρ 16) (bρ 16) (5583683878263 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_233 :
+    Uρ (5583683878263 / 32000000000000 : ℝ) ≤
+      (-(10281055425697362557601 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (5583683878263 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(17836081128313023247777 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_233_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (5583683878263 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(17981834645297691005227 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_233_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (5583683878263 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3656510475313698243173 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_233_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (5583683878263 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(2351524134012850285883 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_233_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (5583683878263 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(9853302452241801719929 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_233_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (5583683878263 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(425033657159668721931 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_233_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (5583683878263 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(24379726540844704730177 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_233_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (5583683878263 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_233_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (5583683878263 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_233_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (5583683878263 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_233_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (5583683878263 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_233_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (5583683878263 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_233_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (5583683878263 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_233_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (5583683878263 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_233_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (5583683878263 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_233_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (5583683878263 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_233_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_234_1 :
+    Uω (aρ 1) (bρ 1) (2820231956767 / 16000000000000 : ℝ) ≤
+      (-(4432757645883601178647 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (212256568384107 / 1250000000000000 : ℝ) ≤
+      (-(4432757645883601178647 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (212256568384107 / 1250000000000000 : ℝ) =
+            (212256568384107 / 156250000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (212256568384107 / 156250000000000 : ℝ) ≤
+            (765846206366398821353 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1697957339251337 / 10000000000000000 : ℝ)) (q :=
+      (212256568384107 / 1250000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_234_2 :
+    Uω (aρ 2) (bρ 2) (2820231956767 / 16000000000000 : ℝ) ≤
+      (-(17875215342854461799939 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (836872439087349 / 5000000000000000 : ℝ) ≤
+      (-(17875215342854461799939 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (836872439087349 / 5000000000000000 : ℝ) =
+            (836872439087349 / 625000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (836872439087349 / 625000000000000 : ℝ) ≤
+            (2919200066145538200061 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1673111013965021 / 10000000000000000 : ℝ)) (q :=
+      (836872439087349 / 5000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_234_3 :
+    Uω (aρ 3) (bρ 3) (2820231956767 / 16000000000000 : ℝ) ≤
+      (-(9086291413154191049523 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3249411783092489 / 20000000000000000 : ℝ) ≤
+      (-(9086291413154191049523 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3249411783092489 / 20000000000000000 : ℝ) =
+            (3249411783092489 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3249411783092489 / 2500000000000000 : ℝ) ≤
+            (1310916291345808950477 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (811214573529057 / 5000000000000000 : ℝ)) (q :=
+      (3249411783092489 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_234_4 :
+    Uω (aρ 4) (bρ 4) (2820231956767 / 16000000000000 : ℝ) ≤
+      (-(4673980589084799832807 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3083730412941973 / 20000000000000000 : ℝ) ≤
+      (-(4673980589084799832807 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3083730412941973 / 20000000000000000 : ℝ) =
+            (3083730412941973 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3083730412941973 / 2500000000000000 : ℝ) ≤
+            (524623263165200167193 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (767520978986299 / 5000000000000000 : ℝ)) (q :=
+      (3083730412941973 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_234_5 :
+    Uω (aρ 5) (bρ 5) (2820231956767 / 16000000000000 : ℝ) ≤
+      (-(19578364916448394077517 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (352908747705531 / 2500000000000000 : ℝ) ≤
+      (-(19578364916448394077517 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (352908747705531 / 2500000000000000 : ℝ) =
+            (352908747705531 / 312500000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (352908747705531 / 312500000000000 : ℝ) ≤
+            (1216050492551605922483 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1392771446009873 / 10000000000000000 : ℝ)) (q :=
+      (352908747705531 / 2500000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_234_6 :
+    Uω (aρ 6) (bρ 6) (2820231956767 / 16000000000000 : ℝ) ≤
+      (-(21097707582265118210857 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2425315245992213 / 20000000000000000 : ℝ) ≤
+      (-(21097707582265118210857 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2425315245992213 / 20000000000000000 : ℝ) =
+            (2425315245992213 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2425315245992213 / 1250000000000000 : ℝ) ≤
+            (6628179629734881789143 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (581028542748919 / 5000000000000000 : ℝ)) (q :=
+      (2425315245992213 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_234_7 :
+    Uω (aρ 7) (bρ 7) (2820231956767 / 16000000000000 : ℝ) ≤
+      (-(24139057684859532006813 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1789303618169693 / 20000000000000000 : ℝ) ≤
+      (-(24139057684859532006813 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1789303618169693 / 20000000000000000 : ℝ) =
+            (1789303618169693 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1789303618169693 / 1250000000000000 : ℝ) ≤
+            (3586829527140467993187 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (374853419217659 / 5000000000000000 : ℝ)) (q :=
+      (1789303618169693 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_234_8 :
+    Uω (aρ 8) (bρ 8) (2820231956767 / 16000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_234_9 :
+    Uω (aρ 9) (bρ 9) (2820231956767 / 16000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_234_10 :
+    Uω (aρ 10) (bρ 10) (2820231956767 / 16000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_234_11 :
+    Uω (aρ 11) (bρ 11) (2820231956767 / 16000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_234_12 :
+    Uω (aρ 12) (bρ 12) (2820231956767 / 16000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_234_13 :
+    Uω (aρ 13) (bρ 13) (2820231956767 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_234_14 :
+    Uω (aρ 14) (bρ 14) (2820231956767 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_234_15 :
+    Uω (aρ 15) (bρ 15) (2820231956767 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_234_16 :
+    Uω (aρ 16) (bρ 16) (2820231956767 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_234 :
+    Uρ (2820231956767 / 16000000000000 : ℝ) ≤
+      (-(20504954889210252469137 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (2820231956767 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(4432757645883601178647 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_234_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (2820231956767 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(17875215342854461799939 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_234_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (2820231956767 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(9086291413154191049523 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_234_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (2820231956767 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(4673980589084799832807 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_234_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (2820231956767 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(19578364916448394077517 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_234_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (2820231956767 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(21097707582265118210857 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_234_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (2820231956767 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(24139057684859532006813 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_234_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (2820231956767 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_234_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (2820231956767 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_234_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (2820231956767 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_234_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (2820231956767 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_234_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (2820231956767 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_234_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (2820231956767 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_234_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (2820231956767 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_234_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (2820231956767 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_234_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (2820231956767 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_234_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_235_1 :
+    Uω (aρ 1) (bρ 1) (1139448789761 / 6400000000000 : ℝ) ≤
+      (-(17627072259042377358077 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6863189170930689 / 40000000000000000 : ℝ) ≤
+      (-(17627072259042377358077 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (6863189170930689 / 40000000000000000 : ℝ) =
+            (6863189170930689 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6863189170930689 / 5000000000000000 : ℝ) ≤
+            (3167343149957622641923 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (857851534774391 / 5000000000000000 : ℝ)) (q :=
+      (6863189170930689 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_235_2 :
+    Uω (aρ 2) (bρ 2) (1139448789761 / 6400000000000 : ℝ) ≤
+      (-(17769721669487918798127 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6765981163663493 / 40000000000000000 : ℝ) ≤
+      (-(17769721669487918798127 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (6765981163663493 / 40000000000000000 : ℝ) =
+            (6765981163663493 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6765981163663493 / 5000000000000000 : ℝ) ≤
+            (3024693739512081201873 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (52839627450787 / 312500000000000 : ℝ)) (q :=
+      (6765981163663493 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_235_3 :
+    Uω (aρ 3) (bρ 3) (1139448789761 / 6400000000000 : ℝ) ≤
+      (-(18063812784732552797297 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6569897130138973 / 40000000000000000 : ℝ) ≤
+      (-(18063812784732552797297 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (6569897130138973 / 40000000000000000 : ℝ) =
+            (6569897130138973 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6569897130138973 / 5000000000000000 : ℝ) ≤
+            (2730602624267447202703 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (410055542003231 / 2500000000000000 : ℝ)) (q :=
+      (6569897130138973 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_235_4 :
+    Uω (aρ 4) (bρ 4) (1139448789761 / 6400000000000 : ℝ) ≤
+      (-(18580999749485371987611 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6238747732738101 / 40000000000000000 : ℝ) ≤
+      (-(18580999749485371987611 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (6238747732738101 / 40000000000000000 : ℝ) =
+            (6238747732738101 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6238747732738101 / 5000000000000000 : ℝ) ≤
+            (2213415659514628012389 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (97058853148593 / 625000000000000 : ℝ)) (q :=
+      (6238747732738101 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_235_5 :
+    Uω (aρ 5) (bρ 5) (1139448789761 / 6400000000000 : ℝ) ≤
+      (-(19451792096157150090147 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5718464033488703 / 40000000000000000 : ℝ) ≤
+      (-(19451792096157150090147 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (5718464033488703 / 40000000000000000 : ℝ) =
+            (5718464033488703 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5718464033488703 / 5000000000000000 : ℝ) ≤
+            (1342623312842849909853 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1410989720087789 / 10000000000000000 : ℝ)) (q :=
+      (5718464033488703 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_235_6 :
+    Uω (aρ 6) (bρ 6) (1139448789761 / 6400000000000 : ℝ) ≤
+      (-(20946267287668847085137 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4924647627296189 / 40000000000000000 : ℝ) ≤
+      (-(20946267287668847085137 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (4924647627296189 / 40000000000000000 : ℝ) =
+            (4924647627296189 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4924647627296189 / 2500000000000000 : ℝ) ≤
+            (6779619924331152914863 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (295330473032883 / 2500000000000000 : ℝ)) (q :=
+      (4924647627296189 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_235_7 :
+    Uω (aρ 7) (bρ 7) (1139448789761 / 6400000000000 : ℝ) ≤
+      (-(23906163545953218465999 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3662928992268993 / 40000000000000000 : ℝ) ≤
+      (-(23906163545953218465999 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (3662928992268993 / 40000000000000000 : ℝ) =
+            (3662928992268993 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3662928992268993 / 2500000000000000 : ℝ) ≤
+            (3819723666046781534001 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (387061977688967 / 5000000000000000 : ℝ)) (q :=
+      (3662928992268993 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_235_8 :
+    Uω (aρ 8) (bρ 8) (1139448789761 / 6400000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_235_9 :
+    Uω (aρ 9) (bρ 9) (1139448789761 / 6400000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_235_10 :
+    Uω (aρ 10) (bρ 10) (1139448789761 / 6400000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_235_11 :
+    Uω (aρ 11) (bρ 11) (1139448789761 / 6400000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_235_12 :
+    Uω (aρ 12) (bρ 12) (1139448789761 / 6400000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_235_13 :
+    Uω (aρ 13) (bρ 13) (1139448789761 / 6400000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_235_14 :
+    Uω (aρ 14) (bρ 14) (1139448789761 / 6400000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_235_15 :
+    Uω (aρ 15) (bρ 15) (1139448789761 / 6400000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_235_16 :
+    Uω (aρ 16) (bρ 16) (1139448789761 / 6400000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_235 :
+    Uρ (1139448789761 / 6400000000000 : ℝ) ≤
+      (-(1278059270240595681709 / 625000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1139448789761 / 6400000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(17627072259042377358077 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_235_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1139448789761 / 6400000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(17769721669487918798127 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_235_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1139448789761 / 6400000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(18063812784732552797297 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_235_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1139448789761 / 6400000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(18580999749485371987611 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_235_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1139448789761 / 6400000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(19451792096157150090147 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_235_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1139448789761 / 6400000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(20946267287668847085137 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_235_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1139448789761 / 6400000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(23906163545953218465999 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_235_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1139448789761 / 6400000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_235_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1139448789761 / 6400000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_235_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1139448789761 / 6400000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_235_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1139448789761 / 6400000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_235_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1139448789761 / 6400000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_235_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1139448789761 / 6400000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_235_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1139448789761 / 6400000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_235_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1139448789761 / 6400000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_235_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1139448789761 / 6400000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_235_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_236_1 :
+    Uω (aρ 1) (bρ 1) (1438505996019 / 8000000000000 : ℝ) ≤
+      (-(3504836734814087840143 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3467084036467329 / 20000000000000000 : ℝ) ≤
+      (-(3504836734814087840143 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3467084036467329 / 20000000000000000 : ℝ) =
+            (3467084036467329 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3467084036467329 / 2500000000000000 : ℝ) ≤
+            (654046346985912159857 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1733448759528579 / 10000000000000000 : ℝ)) (q :=
+      (3467084036467329 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_236_2 :
+    Uω (aρ 2) (bρ 2) (1438505996019 / 8000000000000 : ℝ) ≤
+      (-(8832665044234793364127 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (341849113090293 / 2000000000000000 : ℝ) ≤
+      (-(8832665044234793364127 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (341849113090293 / 2000000000000000 : ℝ) =
+            (341849113090293 / 250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (341849113090293 / 250000000000000 : ℝ) ≤
+            (1564542660265206635873 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (85431243323709 / 500000000000000 : ℝ)) (q :=
+      (341849113090293 / 2000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_236_3 :
+    Uω (aρ 3) (bρ 3) (1438505996019 / 8000000000000 : ℝ) ≤
+      (-(561131759258713630229 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3320484291241897 / 20000000000000000 : ℝ) ≤
+      (-(561131759258713630229 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3320484291241897 / 20000000000000000 : ℝ) =
+            (3320484291241897 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3320484291241897 / 2500000000000000 : ℝ) ≤
+            (88693722272536369771 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1658014133163147 / 10000000000000000 : ℝ)) (q :=
+      (3320484291241897 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_236_4 :
+    Uω (aρ 4) (bρ 4) (1438505996019 / 8000000000000 : ℝ) ≤
+      (-(18467394086119095410203 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3155013781603513 / 20000000000000000 : ℝ) ≤
+      (-(18467394086119095410203 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3155013781603513 / 20000000000000000 : ℝ) =
+            (3155013781603513 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3155013781603513 / 2500000000000000 : ℝ) ≤
+            (2327021322880904589797 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1570837804589763 / 10000000000000000 : ℝ)) (q :=
+      (3155013781603513 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_236_5 :
+    Uω (aρ 5) (bρ 5) (1438505996019 / 8000000000000 : ℝ) ≤
+      (-(19326842580294766154441 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (289518211199629 / 2000000000000000 : ℝ) ≤
+      (-(19326842580294766154441 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (289518211199629 / 2000000000000000 : ℝ) =
+            (289518211199629 / 250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (289518211199629 / 250000000000000 : ℝ) ≤
+            (1467572828705233845559 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (71459802715877 / 500000000000000 : ℝ)) (q :=
+      (289518211199629 / 2000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_236_6 :
+    Uω (aρ 6) (bρ 6) (1438505996019 / 8000000000000 : ℝ) ≤
+      (-(4159454768142052679751 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (624821373386481 / 5000000000000000 : ℝ) ≤
+      (-(4159454768142052679751 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (624821373386481 / 5000000000000000 : ℝ) =
+            (624821373386481 / 312500000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 15
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (624821373386481 / 312500000000000 : ℝ) ≤
+            (1385722674257947320249 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (600269905503587 / 5000000000000000 : ℝ)) (q :=
+      (624821373386481 / 5000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_236_7 :
+    Uω (aρ 7) (bρ 7) (1438505996019 / 8000000000000 : ℝ) ≤
+      (-(23680451252261213551871 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (58539780019197 / 625000000000000 : ℝ) ≤
+      (-(23680451252261213551871 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (58539780019197 / 625000000000000 : ℝ) = (58539780019197 / 39062500000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (58539780019197 / 39062500000000 : ℝ) ≤
+            (4045435959738786448129 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (399094329417777 / 5000000000000000 : ℝ)) (q :=
+      (58539780019197 / 625000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_236_8 :
+    Uω (aρ 8) (bρ 8) (1438505996019 / 8000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_236_9 :
+    Uω (aρ 9) (bρ 9) (1438505996019 / 8000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_236_10 :
+    Uω (aρ 10) (bρ 10) (1438505996019 / 8000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_236_11 :
+    Uω (aρ 11) (bρ 11) (1438505996019 / 8000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_236_12 :
+    Uω (aρ 12) (bρ 12) (1438505996019 / 8000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_236_13 :
+    Uω (aρ 13) (bρ 13) (1438505996019 / 8000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_236_14 :
+    Uω (aρ 14) (bρ 14) (1438505996019 / 8000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_236_15 :
+    Uω (aρ 15) (bρ 15) (1438505996019 / 8000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_236_16 :
+    Uω (aρ 16) (bρ 16) (1438505996019 / 8000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_236 :
+    Uρ (1438505996019 / 8000000000000 : ℝ) ≤
+      (-(815761080340562607109 / 400000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1438505996019 / 8000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3504836734814087840143 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_236_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1438505996019 / 8000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(8832665044234793364127 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_236_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1438505996019 / 8000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(561131759258713630229 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_236_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1438505996019 / 8000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(18467394086119095410203 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_236_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1438505996019 / 8000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(19326842580294766154441 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_236_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1438505996019 / 8000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(4159454768142052679751 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_236_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1438505996019 / 8000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(23680451252261213551871 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_236_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1438505996019 / 8000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_236_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1438505996019 / 8000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_236_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1438505996019 / 8000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_236_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1438505996019 / 8000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_236_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1438505996019 / 8000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_236_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1438505996019 / 8000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_236_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1438505996019 / 8000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_236_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1438505996019 / 8000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_236_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1438505996019 / 8000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_236_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_237_1 :
+    Uω (aρ 1) (bρ 1) (2933792027309 / 16000000000000 : ℝ) ≤
+      (-(216519115091196781593 / 125000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1769031411185933 / 10000000000000000 : ℝ) ≤
+      (-(216519115091196781593 / 125000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1769031411185933 / 10000000000000000 : ℝ) =
+            (1769031411185933 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1769031411185933 / 1250000000000000 : ℝ) ≤
+            (43411077521303218407 / 125000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1768940023388741 / 10000000000000000 : ℝ)) (q :=
+      (1769031411185933 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_237_2 :
+    Uω (aρ 2) (bρ 2) (2933792027309 / 16000000000000 : ℝ) ≤
+      (-(545617583817240004993 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1744745716790423 / 10000000000000000 : ℝ) ≤
+      (-(545617583817240004993 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1744745716790423 / 10000000000000000 : ℝ) =
+            (1744745716790423 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1744745716790423 / 1250000000000000 : ℝ) ≤
+            (104207897714009995007 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1744137647107721 / 10000000000000000 : ℝ)) (q :=
+      (1744745716790423 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_237_3 :
+    Uω (aρ 3) (bρ 3) (2933792027309 / 16000000000000 : ℝ) ≤
+      (-(17744444299101173343339 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (847888177317829 / 5000000000000000 : ℝ) ≤
+      (-(17744444299101173343339 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (847888177317829 / 5000000000000000 : ℝ) =
+            (847888177317829 / 625000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (847888177317829 / 625000000000000 : ℝ) ≤
+            (3049971109898826656661 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1693595029148191 / 10000000000000000 : ℝ)) (q :=
+      (847888177317829 / 5000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_237_4 :
+    Uω (aρ 4) (bρ 4) (2933792027309 / 16000000000000 : ℝ) ≤
+      (-(18244014263751742130657 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3226283470414801 / 20000000000000000 : ℝ) ≤
+      (-(18244014263751742130657 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3226283470414801 / 20000000000000000 : ℝ) =
+            (3226283470414801 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3226283470414801 / 2500000000000000 : ℝ) ≤
+            (2550401145248257869343 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (401654992839169 / 2500000000000000 : ℝ)) (q :=
+      (3226283470414801 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_237_5 :
+    Uω (aρ 5) (bρ 5) (2933792027309 / 16000000000000 : ℝ) ≤
+      (-(9540823325732590924141 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2967048262560857 / 20000000000000000 : ℝ) ≤
+      (-(9540823325732590924141 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2967048262560857 / 20000000000000000 : ℝ) =
+            (2967048262560857 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2967048262560857 / 2500000000000000 : ℝ) ≤
+            (856384378767409075859 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (366393670709433 / 2500000000000000 : ℝ)) (q :=
+      (2967048262560857 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_237_6 :
+    Uω (aρ 6) (bρ 6) (2933792027309 / 16000000000000 : ℝ) ≤
+      (-(20506298475419563252639 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2573076913092093 / 20000000000000000 : ℝ) ≤
+      (-(20506298475419563252639 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2573076913092093 / 20000000000000000 : ℝ) =
+            (2573076913092093 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2573076913092093 / 2500000000000000 : ℝ) ≤
+            (288116933580436747361 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (154855463563621 / 1250000000000000 : ℝ)) (q :=
+      (2573076913092093 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_237_7 :
+    Uω (aρ 7) (bρ 7) (2933792027309 / 16000000000000 : ℝ) ≤
+      (-(2906068668546487972029 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1955952598996313 / 20000000000000000 : ℝ) ≤
+      (-(2906068668546487972029 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1955952598996313 / 20000000000000000 : ℝ) =
+            (1955952598996313 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1955952598996313 / 1250000000000000 : ℝ) ≤
+            (559667232953512027971 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (211345193793297 / 2500000000000000 : ℝ)) (q :=
+      (1955952598996313 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_237_8 :
+    Uω (aρ 8) (bρ 8) (2933792027309 / 16000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_237_9 :
+    Uω (aρ 9) (bρ 9) (2933792027309 / 16000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_237_10 :
+    Uω (aρ 10) (bρ 10) (2933792027309 / 16000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_237_11 :
+    Uω (aρ 11) (bρ 11) (2933792027309 / 16000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_237_12 :
+    Uω (aρ 12) (bρ 12) (2933792027309 / 16000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_237_13 :
+    Uω (aρ 13) (bρ 13) (2933792027309 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_237_14 :
+    Uω (aρ 14) (bρ 14) (2933792027309 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_237_15 :
+    Uω (aρ 15) (bρ 15) (2933792027309 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_237_16 :
+    Uω (aρ 16) (bρ 16) (2933792027309 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_237 :
+    Uρ (2933792027309 / 16000000000000 : ℝ) ≤
+      (-(10143608103708137941839 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (2933792027309 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(216519115091196781593 / 125000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_237_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (2933792027309 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(545617583817240004993 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_237_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (2933792027309 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(17744444299101173343339 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_237_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (2933792027309 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(18244014263751742130657 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_237_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (2933792027309 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(9540823325732590924141 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_237_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (2933792027309 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(20506298475419563252639 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_237_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (2933792027309 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(2906068668546487972029 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_237_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (2933792027309 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_237_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (2933792027309 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_237_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (2933792027309 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_237_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (2933792027309 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_237_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (2933792027309 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_237_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (2933792027309 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_237_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (2933792027309 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_237_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (2933792027309 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_237_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (2933792027309 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_237_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_238_1 :
+    Uω (aρ 1) (bρ 1) (149528603129 / 800000000000 : ℝ) ≤
+      (-(17122900589309473870993 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3609041461089001 / 20000000000000000 : ℝ) ≤
+      (-(17122900589309473870993 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3609041461089001 / 20000000000000000 : ℝ) =
+            (3609041461089001 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3609041461089001 / 2500000000000000 : ℝ) ≤
+            (3671514819690526129007 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1804431140061501 / 10000000000000000 : ℝ)) (q :=
+      (3609041461089001 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_238_2 :
+    Uω (aρ 2) (bρ 2) (149528603129 / 800000000000 : ℝ) ≤
+      (-(8629169461557259480653 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (712098145710151 / 4000000000000000 : ℝ) ≤
+      (-(8629169461557259480653 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (712098145710151 / 4000000000000000 : ℝ) =
+            (712098145710151 / 500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (712098145710151 / 500000000000000 : ℝ) ≤
+            (1768038242942740519347 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (355929884006651 / 2000000000000000 : ℝ)) (q :=
+      (712098145710151 / 4000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_238_3 :
+    Uω (aρ 3) (bρ 3) (149528603129 / 800000000000 : ℝ) ≤
+      (-(3507415055125305177593 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3462617289665989 / 20000000000000000 : ℝ) ≤
+      (-(3507415055125305177593 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3462617289665989 / 20000000000000000 : ℝ) =
+            (3462617289665989 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3462617289665989 / 2500000000000000 : ℝ) ≤
+            (651468026674694822407 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1729172087498489 / 10000000000000000 : ℝ)) (q :=
+      (3462617289665989 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_238_4 :
+    Uω (aρ 4) (bρ 4) (149528603129 / 800000000000 : ℝ) ≤
+      (-(3605110847146402910907 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (103048136671579 / 625000000000000 : ℝ) ≤
+      (-(3605110847146402910907 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (103048136671579 / 625000000000000 : ℝ) = (103048136671579 / 78125000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (103048136671579 / 78125000000000 : ℝ) ≤
+            (553772234653597089093 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (410597338097007 / 2500000000000000 : ℝ)) (q :=
+      (103048136671579 / 625000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_238_5 :
+    Uω (aρ 5) (bρ 5) (149528603129 / 800000000000 : ℝ) ≤
+      (-(9421229765333016833399 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1519435887224519 / 10000000000000000 : ℝ) ≤
+      (-(9421229765333016833399 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1519435887224519 / 10000000000000000 : ℝ) =
+            (1519435887224519 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1519435887224519 / 1250000000000000 : ℝ) ≤
+            (975977939166983166601 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (750955336340769 / 5000000000000000 : ℝ)) (q :=
+      (1519435887224519 / 10000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_238_6 :
+    Uω (aρ 6) (bρ 6) (149528603129 / 800000000000 : ℝ) ≤
+      (-(20224165771428923170819 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1323352798439857 / 10000000000000000 : ℝ) ≤
+      (-(20224165771428923170819 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1323352798439857 / 10000000000000000 : ℝ) =
+            (1323352798439857 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1323352798439857 / 1250000000000000 : ℝ) ≤
+            (570249637571076829181 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (638492435126107 / 5000000000000000 : ℝ)) (q :=
+      (1323352798439857 / 10000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_238_7 :
+    Uω (aρ 7) (bρ 7) (149528603129 / 800000000000 : ℝ) ≤
+      (-(22839854558538710203403 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2037547373500423 / 20000000000000000 : ℝ) ≤
+      (-(22839854558538710203403 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2037547373500423 / 20000000000000000 : ℝ) =
+            (2037547373500423 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2037547373500423 / 1250000000000000 : ℝ) ≤
+            (4886032653461289796597 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (891488027632923 / 10000000000000000 : ℝ)) (q :=
+      (2037547373500423 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_238_8 :
+    Uω (aρ 8) (bρ 8) (149528603129 / 800000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_238_9 :
+    Uω (aρ 9) (bρ 9) (149528603129 / 800000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_238_10 :
+    Uω (aρ 10) (bρ 10) (149528603129 / 800000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_238_11 :
+    Uω (aρ 11) (bρ 11) (149528603129 / 800000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_238_12 :
+    Uω (aρ 12) (bρ 12) (149528603129 / 800000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_238_13 :
+    Uω (aρ 13) (bρ 13) (149528603129 / 800000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_238_14 :
+    Uω (aρ 14) (bρ 14) (149528603129 / 800000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_238_15 :
+    Uω (aρ 15) (bρ 15) (149528603129 / 800000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_238_16 :
+    Uω (aρ 16) (bρ 16) (149528603129 / 800000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_238 :
+    Uρ (149528603129 / 800000000000 : ℝ) ≤
+      (-(10092063400161778436219 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (149528603129 / 800000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(17122900589309473870993 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_238_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (149528603129 / 800000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(8629169461557259480653 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_238_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (149528603129 / 800000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3507415055125305177593 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_238_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (149528603129 / 800000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3605110847146402910907 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_238_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (149528603129 / 800000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(9421229765333016833399 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_238_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (149528603129 / 800000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(20224165771428923170819 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_238_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (149528603129 / 800000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(22839854558538710203403 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_238_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (149528603129 / 800000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_238_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (149528603129 / 800000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_238_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (149528603129 / 800000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_238_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (149528603129 / 800000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_238_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (149528603129 / 800000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_238_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (149528603129 / 800000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_238_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (149528603129 / 800000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_238_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (149528603129 / 800000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_238_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (149528603129 / 800000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_238_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_239_1 :
+    Uω (aρ 1) (bρ 1) (3047352097851 / 16000000000000 : ℝ) ≤
+      (-(423203524096555888311 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (736003992227247 / 4000000000000000 : ℝ) ≤
+      (-(423203524096555888311 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (736003992227247 / 4000000000000000 : ℝ) =
+            (736003992227247 / 500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (736003992227247 / 500000000000000 : ℝ) ≤
+            (96656861128444111689 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (45998052951609 / 250000000000000 : ℝ)) (q :=
+      (736003992227247 / 4000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_239_2 :
+    Uω (aρ 2) (bρ 2) (3047352097851 / 16000000000000 : ℝ) ≤
+      (-(17060894951251626070951 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1815744537478511 / 10000000000000000 : ℝ) ≤
+      (-(17060894951251626070951 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1815744537478511 / 10000000000000000 : ℝ) =
+            (1815744537478511 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1815744537478511 / 1250000000000000 : ℝ) ≤
+            (3733520457748373929049 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1815160244395147 / 10000000000000000 : ℝ)) (q :=
+      (1815744537478511 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_239_3 :
+    Uω (aρ 3) (bρ 3) (3047352097851 / 16000000000000 : ℝ) ≤
+      (-(3466785886107343843369 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3533678264524699 / 20000000000000000 : ℝ) ≤
+      (-(3466785886107343843369 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3533678264524699 / 20000000000000000 : ℝ) =
+            (3533678264524699 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3533678264524699 / 2500000000000000 : ℝ) ≤
+            (692097195692656156631 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (220593192539103 / 1250000000000000 : ℝ)) (q :=
+      (3533678264524699 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_239_4 :
+    Uω (aρ 4) (bρ 4) (3047352097851 / 16000000000000 : ℝ) ≤
+      (-(17811800359939854210869 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1684392654203751 / 10000000000000000 : ℝ) ≤
+      (-(17811800359939854210869 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1684392654203751 / 10000000000000000 : ℝ) =
+            (1684392654203751 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1684392654203751 / 1250000000000000 : ℝ) ≤
+            (2982615049060145789131 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1678146765260627 / 10000000000000000 : ℝ)) (q :=
+      (1684392654203751 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_239_5 :
+    Uω (aρ 5) (bρ 5) (3047352097851 / 16000000000000 : ℝ) ≤
+      (-(9304493713255353838983 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3110655669334747 / 20000000000000000 : ℝ) ≤
+      (-(9304493713255353838983 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3110655669334747 / 20000000000000000 : ℝ) =
+            (3110655669334747 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3110655669334747 / 2500000000000000 : ℝ) ≤
+            (1092713991244646161017 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (192275880690359 / 1250000000000000 : ℝ)) (q :=
+      (3110655669334747 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_239_6 :
+    Uω (aρ 6) (bρ 6) (3047352097851 / 16000000000000 : ℝ) ≤
+      (-(19950321207062330984031 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2720185705526393 / 20000000000000000 : ℝ) ≤
+      (-(19950321207062330984031 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2720185705526393 / 20000000000000000 : ℝ) =
+            (2720185705526393 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2720185705526393 / 2500000000000000 : ℝ) ≤
+            (844094201937669015969 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (657488728427259 / 5000000000000000 : ℝ)) (q :=
+      (2720185705526393 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_239_7 :
+    Uω (aρ 7) (bρ 7) (3047352097851 / 16000000000000 : ℝ) ≤
+      (-(22451573375701112560891 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (84728700154133 / 800000000000000 : ℝ) ≤
+      (-(22451573375701112560891 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (84728700154133 / 800000000000000 : ℝ) = (84728700154133 / 50000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (84728700154133 / 50000000000000 : ℝ) ≤
+            (5274313836298887439109 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (18733412718829 / 200000000000000 : ℝ)) (q :=
+      (84728700154133 / 800000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_239_8 :
+    Uω (aρ 8) (bρ 8) (3047352097851 / 16000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_239_9 :
+    Uω (aρ 9) (bρ 9) (3047352097851 / 16000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_239_10 :
+    Uω (aρ 10) (bρ 10) (3047352097851 / 16000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_239_11 :
+    Uω (aρ 11) (bρ 11) (3047352097851 / 16000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_239_12 :
+    Uω (aρ 12) (bρ 12) (3047352097851 / 16000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_239_13 :
+    Uω (aρ 13) (bρ 13) (3047352097851 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_239_14 :
+    Uω (aρ 14) (bρ 14) (3047352097851 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_239_15 :
+    Uω (aρ 15) (bρ 15) (3047352097851 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_239_16 :
+    Uω (aρ 16) (bρ 16) (3047352097851 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_239 :
+    Uρ (3047352097851 / 16000000000000 : ℝ) ≤
+      (-(20084431088609508863849 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (3047352097851 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(423203524096555888311 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_239_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (3047352097851 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(17060894951251626070951 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_239_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (3047352097851 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3466785886107343843369 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_239_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (3047352097851 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(17811800359939854210869 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_239_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (3047352097851 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(9304493713255353838983 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_239_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (3047352097851 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(19950321207062330984031 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_239_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (3047352097851 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(22451573375701112560891 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_239_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (3047352097851 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_239_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (3047352097851 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_239_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (3047352097851 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_239_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (3047352097851 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_239_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (3047352097851 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_239_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (3047352097851 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_239_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (3047352097851 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_239_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (3047352097851 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_239_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (3047352097851 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_239_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_240_1 :
+    Uω (aρ 1) (bρ 1) (1552066066561 / 8000000000000 : ℝ) ≤
+      (-(3347420493572264953999 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3750998330386281 / 20000000000000000 : ℝ) ≤
+      (-(3347420493572264953999 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3750998330386281 / 20000000000000000 : ℝ) =
+            (3750998330386281 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3750998330386281 / 2500000000000000 : ℝ) ≤
+            (811462588227735046001 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1875412965270031 / 10000000000000000 : ℝ)) (q :=
+      (3750998330386281 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_240_2 :
+    Uω (aρ 2) (bρ 2) (1552066066561 / 8000000000000 : ℝ) ≤
+      (-(4216819111011872001949 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3702486527403017 / 20000000000000000 : ℝ) ≤
+      (-(4216819111011872001949 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3702486527403017 / 20000000000000000 : ℝ) =
+            (3702486527403017 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3702486527403017 / 2500000000000000 : ℝ) ≤
+            (981784741238127998051 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1850670174796767 / 10000000000000000 : ℝ)) (q :=
+      (3702486527403017 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_240_3 :
+    Uω (aρ 3) (bρ 3) (1552066066561 / 8000000000000 : ℝ) ≤
+      (-(3426967557250234145729 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (144189433903157 / 800000000000000 : ℝ) ≤
+      (-(3426967557250234145729 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (144189433903157 / 800000000000000 : ℝ) =
+            (144189433903157 / 100000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (144189433903157 / 100000000000000 : ℝ) ≤
+            (731915524549765854271 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (72012624052907 / 400000000000000 : ℝ)) (q :=
+      (144189433903157 / 800000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_240_4 :
+    Uω (aρ 4) (bρ 4) (1552066066561 / 8000000000000 : ℝ) ≤
+      (-(1100159544436690800209 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (430002378031337 / 2500000000000000 : ℝ) ≤
+      (-(1100159544436690800209 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (430002378031337 / 2500000000000000 : ℝ) =
+            (430002378031337 / 312500000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (430002378031337 / 312500000000000 : ℝ) ≤
+            (199491418625809199791 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (856946479529723 / 5000000000000000 : ℝ)) (q :=
+      (430002378031337 / 2500000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_240_5 :
+    Uω (aρ 5) (bρ 5) (1552066066561 / 8000000000000 : ℝ) ≤
+      (-(18380957992918669968001 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3182402687107411 / 20000000000000000 : ℝ) ≤
+      (-(18380957992918669968001 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3182402687107411 / 20000000000000000 : ℝ) =
+            (3182402687107411 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3182402687107411 / 2500000000000000 : ℝ) ≤
+            (2413457416081330031999 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1574466541251161 / 10000000000000000 : ℝ)) (q :=
+      (3182402687107411 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_240_6 :
+    Uω (aρ 6) (bρ 6) (1552066066561 / 8000000000000 : ℝ) ≤
+      (-(3936852661578481560173 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1396764878355009 / 10000000000000000 : ℝ) ≤
+      (-(3936852661578481560173 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1396764878355009 / 10000000000000000 : ℝ) =
+            (1396764878355009 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1396764878355009 / 1250000000000000 : ℝ) ≤
+            (222030420221518439827 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (169104248249221 / 1250000000000000 : ℝ)) (q :=
+      (1396764878355009 / 10000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_240_7 :
+    Uω (aρ 7) (bρ 7) (1552066066561 / 8000000000000 : ℝ) ≤
+      (-(5520357822357195670947 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2198090751996057 / 20000000000000000 : ℝ) ≤
+      (-(5520357822357195670947 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2198090751996057 / 20000000000000000 : ℝ) =
+            (2198090751996057 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2198090751996057 / 1250000000000000 : ℝ) ≤
+            (1411113980642804329053 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (981056362039807 / 10000000000000000 : ℝ)) (q :=
+      (2198090751996057 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_240_8 :
+    Uω (aρ 8) (bρ 8) (1552066066561 / 8000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_240_9 :
+    Uω (aρ 9) (bρ 9) (1552066066561 / 8000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_240_10 :
+    Uω (aρ 10) (bρ 10) (1552066066561 / 8000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_240_11 :
+    Uω (aρ 11) (bρ 11) (1552066066561 / 8000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_240_12 :
+    Uω (aρ 12) (bρ 12) (1552066066561 / 8000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_240_13 :
+    Uω (aρ 13) (bρ 13) (1552066066561 / 8000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_240_14 :
+    Uω (aρ 14) (bρ 14) (1552066066561 / 8000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_240_15 :
+    Uω (aρ 15) (bρ 15) (1552066066561 / 8000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_240_16 :
+    Uω (aρ 16) (bρ 16) (1552066066561 / 8000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_240 :
+    Uρ (1552066066561 / 8000000000000 : ℝ) ≤
+      (-(4996963179176144483103 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1552066066561 / 8000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3347420493572264953999 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_240_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1552066066561 / 8000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(4216819111011872001949 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_240_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1552066066561 / 8000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3426967557250234145729 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_240_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1552066066561 / 8000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1100159544436690800209 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_240_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1552066066561 / 8000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(18380957992918669968001 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_240_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1552066066561 / 8000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3936852661578481560173 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_240_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1552066066561 / 8000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(5520357822357195670947 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_240_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1552066066561 / 8000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_240_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1552066066561 / 8000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_240_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1552066066561 / 8000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_240_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1552066066561 / 8000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_240_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1552066066561 / 8000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_240_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1552066066561 / 8000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_240_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1552066066561 / 8000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_240_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1552066066561 / 8000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_240_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1552066066561 / 8000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_240_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_241_1 :
+    Uω (aρ 1) (bρ 1) (201105762729 / 1000000000000 : ℝ) ≤
+      (-(8182819195849925653851 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3892954705114493 / 20000000000000000 : ℝ) ≤
+      (-(8182819195849925653851 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3892954705114493 / 20000000000000000 : ℝ) =
+            (3892954705114493 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3892954705114493 / 2500000000000000 : ℝ) ≤
+            (2214388508650074346149 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1946394295909493 / 10000000000000000 : ℝ)) (q :=
+      (3892954705114493 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_241_2 :
+    Uω (aρ 2) (bρ 2) (201105762729 / 1000000000000 : ℝ) ≤
+      (-(16490941930067749877633 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (768895789728329 / 4000000000000000 : ℝ) ≤
+      (-(16490941930067749877633 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (768895789728329 / 4000000000000000 : ℝ) =
+            (768895789728329 / 500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (768895789728329 / 500000000000000 : ℝ) ≤
+            (4303473478932250122367 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (384337510389329 / 2000000000000000 : ℝ)) (q :=
+      (768895789728329 / 4000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_241_3 :
+    Uω (aρ 3) (bρ 3) (201105762729 / 1000000000000 : ℝ) ≤
+      (-(3349638049019418963313 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1873420805930339 / 10000000000000000 : ℝ) ≤
+      (-(3349638049019418963313 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1873420805930339 / 10000000000000000 : ℝ) =
+            (1873420805930339 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1873420805930339 / 1250000000000000 : ℝ) ≤
+            (809245032780581036687 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (935723160757839 / 5000000000000000 : ℝ)) (q :=
+      (1873420805930339 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_241_4 :
+    Uω (aρ 4) (bρ 4) (201105762729 / 1000000000000 : ℝ) ≤
+      (-(8598419076450674071003 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3582455494032877 / 20000000000000000 : ℝ) ≤
+      (-(8598419076450674071003 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3582455494032877 / 20000000000000000 : ℝ) =
+            (3582455494032877 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3582455494032877 / 2500000000000000 : ℝ) ≤
+            (1798788628049325928997 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1785354384752877 / 10000000000000000 : ℝ)) (q :=
+      (3582455494032877 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_241_5 :
+    Uω (aρ 5) (bρ 5) (201105762729 / 1000000000000 : ℝ) ≤
+      (-(17940232793488732042317 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3325795831955751 / 20000000000000000 : ℝ) ≤
+      (-(17940232793488732042317 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3325795831955751 / 20000000000000000 : ℝ) =
+            (3325795831955751 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3325795831955751 / 2500000000000000 : ℝ) ≤
+            (2854182615511267957683 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1646884642010751 / 10000000000000000 : ℝ)) (q :=
+      (3325795831955751 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_241_6 :
+    Uω (aρ 6) (bρ 6) (201105762729 / 1000000000000 : ℝ) ≤
+      (-(19173727179049642941211 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1469926462911759 / 10000000000000000 : ℝ) ≤
+      (-(19173727179049642941211 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1469926462911759 / 10000000000000000 : ℝ) =
+            (1469926462911759 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1469926462911759 / 1250000000000000 : ℝ) ≤
+            (1620688229950357058789 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (714091055509259 / 5000000000000000 : ℝ)) (q :=
+      (1469926462911759 / 10000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_241_7 :
+    Uω (aρ 7) (bρ 7) (201105762729 / 1000000000000 : ℝ) ≤
+      (-(21388339142901952282833 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (588960592762359 / 5000000000000000 : ℝ) ≤
+      (-(21388339142901952282833 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (588960592762359 / 5000000000000000 : ℝ) =
+            (588960592762359 / 312500000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (588960592762359 / 312500000000000 : ℝ) ≤
+            (6337548069098047717167 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (266958234251109 / 2500000000000000 : ℝ)) (q :=
+      (588960592762359 / 5000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_241_8 :
+    Uω (aρ 8) (bρ 8) (201105762729 / 1000000000000 : ℝ) ≤
+      (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_8]) (by norm_num [bρ_8])]; exact Lmid_8
+
+lemma U_241_9 :
+    Uω (aρ 9) (bρ 9) (201105762729 / 1000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_241_10 :
+    Uω (aρ 10) (bρ 10) (201105762729 / 1000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_241_11 :
+    Uω (aρ 11) (bρ 11) (201105762729 / 1000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_241_12 :
+    Uω (aρ 12) (bρ 12) (201105762729 / 1000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_241_13 :
+    Uω (aρ 13) (bρ 13) (201105762729 / 1000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_241_14 :
+    Uω (aρ 14) (bρ 14) (201105762729 / 1000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_241_15 :
+    Uω (aρ 15) (bρ 15) (201105762729 / 1000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_241_16 :
+    Uω (aρ 16) (bρ 16) (201105762729 / 1000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_241 :
+    Uρ (201105762729 / 1000000000000 : ℝ) ≤
+      (-(19803133250419148341029 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (201105762729 / 1000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(8182819195849925653851 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_241_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (201105762729 / 1000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(16490941930067749877633 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_241_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (201105762729 / 1000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3349638049019418963313 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_241_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (201105762729 / 1000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(8598419076450674071003 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_241_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (201105762729 / 1000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(17940232793488732042317 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_241_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (201105762729 / 1000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(19173727179049642941211 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_241_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (201105762729 / 1000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(21388339142901952282833 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_241_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (201105762729 / 1000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14956372202312055299661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_241_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (201105762729 / 1000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_241_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (201105762729 / 1000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_241_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (201105762729 / 1000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_241_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (201105762729 / 1000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_241_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (201105762729 / 1000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_241_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (201105762729 / 1000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_241_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (201105762729 / 1000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_241_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (201105762729 / 1000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_241_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_242_1 :
+    Uω (aρ 1) (bρ 1) (3251812320751 / 16000000000000 : ℝ) ≤
+      (-(16256672346253327359079 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3935606651718043 / 20000000000000000 : ℝ) ≤
+      (-(16256672346253327359079 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3935606651718043 / 20000000000000000 : ℝ) =
+            (3935606651718043 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3935606651718043 / 2500000000000000 : ℝ) ≤
+            (4537743062746672640921 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (491930292333417 / 2500000000000000 : ℝ)) (q :=
+      (3935606651718043 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_242_2 :
+    Uω (aρ 2) (bρ 2) (3251812320751 / 16000000000000 : ℝ) ≤
+      (-(16380582927889856239697 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3887141209948259 / 20000000000000000 : ℝ) ≤
+      (-(16380582927889856239697 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3887141209948259 / 20000000000000000 : ℝ) =
+            (3887141209948259 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3887141209948259 / 2500000000000000 : ℝ) ≤
+            (4413832481110143760303 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (485756185018471 / 2500000000000000 : ℝ)) (q :=
+      (3887141209948259 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_242_3 :
+    Uω (aρ 3) (bρ 3) (3251812320751 / 16000000000000 : ℝ) ≤
+      (-(8317443114412870681471 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (757907249819311 / 4000000000000000 : ℝ) ≤
+      (-(8317443114412870681471 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (757907249819311 / 4000000000000000 : ℝ) =
+            (757907249819311 / 500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (757907249819311 / 500000000000000 : ℝ) ≤
+            (2079764590087129318529 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (94640794278609 / 500000000000000 : ℝ)) (q :=
+      (757907249819311 / 4000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_242_4 :
+    Uω (aρ 4) (bρ 4) (3251812320751 / 16000000000000 : ℝ) ≤
+      (-(17078105971949760117981 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3625244287093789 / 20000000000000000 : ℝ) ≤
+      (-(17078105971949760117981 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3625244287093789 / 20000000000000000 : ℝ) =
+            (3625244287093789 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3625244287093789 / 2500000000000000 : ℝ) ≤
+            (3716309437050239882019 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (903409052317207 / 5000000000000000 : ℝ)) (q :=
+      (3625244287093789 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_242_5 :
+    Uω (aρ 5) (bρ 5) (3251812320751 / 16000000000000 : ℝ) ≤
+      (-(17811592507555697950333 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (673771066028203 / 4000000000000000 : ℝ) ≤
+      (-(17811592507555697950333 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (673771066028203 / 4000000000000000 : ℝ) =
+            (673771066028203 / 500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (673771066028203 / 500000000000000 : ℝ) ≤
+            (2982822901444302049667 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5214434584427 / 31250000000000 : ℝ)) (q :=
+      (673771066028203 / 4000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_242_6 :
+    Uω (aρ 6) (bρ 6) (3251812320751 / 16000000000000 : ℝ) ≤
+      (-(9512788988564780394523 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (596746166723221 / 4000000000000000 : ℝ) ≤
+      (-(9512788988564780394523 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (596746166723221 / 4000000000000000 : ℝ) =
+            (596746166723221 / 500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (596746166723221 / 500000000000000 : ℝ) ≤
+            (884418715935219605477 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (145073494563173 / 1000000000000000 : ℝ)) (q :=
+      (596746166723221 / 4000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_242_7 :
+    Uω (aρ 7) (bρ 7) (3251812320751 / 16000000000000 : ℝ) ≤
+      (-(21190996519392976753163 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (480558989182049 / 4000000000000000 : ℝ) ≤
+      (-(21190996519392976753163 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (480558989182049 / 4000000000000000 : ℝ) =
+            (480558989182049 / 250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (480558989182049 / 250000000000000 : ℝ) ≤
+            (6534890692607023246837 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (109346043868587 / 1000000000000000 : ℝ)) (q :=
+      (480558989182049 / 4000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_242_8 :
+    Uω (aρ 8) (bρ 8) (3251812320751 / 16000000000000 : ℝ) ≤
+      (-(13927889747717311336039 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (616934229595149 / 10000000000000000 : ℝ) ≤
+      (-(13927889747717311336039 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show
+          (616934229595149 / 10000000000000000 : ℝ) =
+            (616934229595149 / 312500000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (616934229595149 / 312500000000000 : ℝ) ≤
+            (3400789759782688663961 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (208075603305923 / 10000000000000000 : ℝ)) (q :=
+      (616934229595149 / 10000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_242_9 :
+    Uω (aρ 9) (bρ 9) (3251812320751 / 16000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_242_10 :
+    Uω (aρ 10) (bρ 10) (3251812320751 / 16000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_242_11 :
+    Uω (aρ 11) (bρ 11) (3251812320751 / 16000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_242_12 :
+    Uω (aρ 12) (bρ 12) (3251812320751 / 16000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_242_13 :
+    Uω (aρ 13) (bρ 13) (3251812320751 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_242_14 :
+    Uω (aρ 14) (bρ 14) (3251812320751 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_242_15 :
+    Uω (aρ 15) (bρ 15) (3251812320751 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_242_16 :
+    Uω (aρ 16) (bρ 16) (3251812320751 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_242 :
+    Uρ (3251812320751 / 16000000000000 : ℝ) ≤
+      (-(2445977526242332705763 / 1250000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (3251812320751 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(16256672346253327359079 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_242_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (3251812320751 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(16380582927889856239697 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_242_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (3251812320751 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(8317443114412870681471 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_242_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (3251812320751 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(17078105971949760117981 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_242_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (3251812320751 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(17811592507555697950333 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_242_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (3251812320751 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(9512788988564780394523 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_242_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (3251812320751 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(21190996519392976753163 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_242_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (3251812320751 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(13927889747717311336039 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_242_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (3251812320751 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_242_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (3251812320751 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_242_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (3251812320751 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_242_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (3251812320751 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_242_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (3251812320751 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_242_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (3251812320751 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_242_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (3251812320751 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_242_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (3251812320751 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_242_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_243_1 :
+    Uω (aρ 1) (bρ 1) (1642966218919 / 8000000000000 : ℝ) ≤
+      (-(16148880970392734533323 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3978258559718229 / 20000000000000000 : ℝ) ≤
+      (-(16148880970392734533323 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3978258559718229 / 20000000000000000 : ℝ) =
+            (3978258559718229 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3978258559718229 / 2500000000000000 : ℝ) ≤
+            (4645534438607265466677 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1989048004154479 / 10000000000000000 : ℝ)) (q :=
+      (3978258559718229 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_243_2 :
+    Uω (aρ 2) (bρ 2) (1642966218919 / 8000000000000 : ℝ) ≤
+      (-(16271429224111498996607 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3929803208140881 / 20000000000000000 : ℝ) ≤
+      (-(16271429224111498996607 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3929803208140881 / 20000000000000000 : ℝ) =
+            (3929803208140881 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3929803208140881 / 2500000000000000 : ℝ) ≤
+            (4522986184888501003393 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1964361665087131 / 10000000000000000 : ℝ)) (q :=
+      (3929803208140881 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_243_3 :
+    Uω (aρ 3) (bρ 3) (1642966218919 / 8000000000000 : ℝ) ≤
+      (-(16522854212519461836571 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3832229893992659 / 20000000000000000 : ℝ) ≤
+      (-(16522854212519461836571 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3832229893992659 / 20000000000000000 : ℝ) =
+            (3832229893992659 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3832229893992659 / 2500000000000000 : ℝ) ≤
+            (4271561196480538163429 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1914184457288909 / 10000000000000000 : ℝ)) (q :=
+      (3832229893992659 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_243_4 :
+    Uω (aρ 4) (bρ 4) (1642966218919 / 8000000000000 : ℝ) ≤
+      (-(16960775844740112262383 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3668029835288273 / 20000000000000000 : ℝ) ≤
+      (-(16960775844740112262383 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3668029835288273 / 20000000000000000 : ℝ) =
+            (3668029835288273 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3668029835288273 / 2500000000000000 : ℝ) ≤
+            (3833639564259887737617 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1828278579649523 / 10000000000000000 : ℝ)) (q :=
+      (3668029835288273 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_243_5 :
+    Uω (aρ 5) (bρ 5) (1642966218919 / 8000000000000 : ℝ) ≤
+      (-(8842308310589361790873 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3411904400602917 / 20000000000000000 : ℝ) ≤
+      (-(8842308310589361790873 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3411904400602917 / 20000000000000000 : ℝ) =
+            (3411904400602917 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3411904400602917 / 2500000000000000 : ℝ) ≤
+            (1554899393910638209127 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1690343064299167 / 10000000000000000 : ℝ)) (q :=
+      (3411904400602917 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_243_6 :
+    Uω (aρ 6) (bρ 6) (1642966218919 / 8000000000000 : ℝ) ≤
+      (-(3775942473366500779873 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3027572175387957 / 20000000000000000 : ℝ) ≤
+      (-(3775942473366500779873 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3027572175387957 / 20000000000000000 : ℝ) =
+            (3027572175387957 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3027572175387957 / 2500000000000000 : ℝ) ≤
+            (382940608433499220127 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1473251214224207 / 10000000000000000 : ℝ)) (q :=
+      (3027572175387957 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_243_7 :
+    Uω (aρ 7) (bρ 7) (1642966218919 / 8000000000000 : ℝ) ≤
+      (-(20998210011066161697331 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (489913398610729 / 4000000000000000 : ℝ) ≤
+      (-(20998210011066161697331 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (489913398610729 / 4000000000000000 : ℝ) =
+            (489913398610729 / 250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (489913398610729 / 250000000000000 : ℝ) ≤
+            (6727677200933838302669 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (223781482529979 / 2000000000000000 : ℝ)) (q :=
+      (489913398610729 / 4000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_243_8 :
+    Uω (aρ 8) (bρ 8) (1642966218919 / 8000000000000 : ℝ) ≤
+      (-(270088395415343064811 / 100000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (671461323264299 / 10000000000000000 : ℝ) ≤
+      (-(270088395415343064811 / 100000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (671461323264299 / 10000000000000000 : ℝ) =
+            (671461323264299 / 625000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (671461323264299 / 625000000000000 : ℝ) ≤
+            (7170476704656935189 / 100000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (18487794841553 / 625000000000000 : ℝ)) (q :=
+      (671461323264299 / 10000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_243_9 :
+    Uω (aρ 9) (bρ 9) (1642966218919 / 8000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_243_10 :
+    Uω (aρ 10) (bρ 10) (1642966218919 / 8000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_243_11 :
+    Uω (aρ 11) (bρ 11) (1642966218919 / 8000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_243_12 :
+    Uω (aρ 12) (bρ 12) (1642966218919 / 8000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_243_13 :
+    Uω (aρ 13) (bρ 13) (1642966218919 / 8000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_243_14 :
+    Uω (aρ 14) (bρ 14) (1642966218919 / 8000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_243_15 :
+    Uω (aρ 15) (bρ 15) (1642966218919 / 8000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_243_16 :
+    Uω (aρ 16) (bρ 16) (1642966218919 / 8000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_243 :
+    Uρ (1642966218919 / 8000000000000 : ℝ) ≤
+      (-(19440334361737891654043 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1642966218919 / 8000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(16148880970392734533323 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_243_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1642966218919 / 8000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(16271429224111498996607 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_243_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1642966218919 / 8000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(16522854212519461836571 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_243_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1642966218919 / 8000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(16960775844740112262383 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_243_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1642966218919 / 8000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(8842308310589361790873 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_243_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1642966218919 / 8000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3775942473366500779873 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_243_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1642966218919 / 8000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(20998210011066161697331 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_243_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1642966218919 / 8000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(270088395415343064811 / 100000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_243_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1642966218919 / 8000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_243_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1642966218919 / 8000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_243_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1642966218919 / 8000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_243_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1642966218919 / 8000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_243_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1642966218919 / 8000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_243_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1642966218919 / 8000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_243_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1642966218919 / 8000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_243_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1642966218919 / 8000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_243_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/U20.lean b/LeanPool/Zeta5Irrational/Table/U20.lean
new file mode 100644
index 0000000000..5b78439083
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/U20.lean
@@ -0,0 +1,3708 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-! # Certified arcsine potential bounds (U20) -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma U_244_1 :
+    Uω (aρ 1) (bρ 1) (132802102197 / 640000000000 : ℝ) ≤
+      (-(80211196028750566739 / 50000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4020910430343609 / 20000000000000000 : ℝ) ≤
+      (-(80211196028750566739 / 50000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4020910430343609 / 20000000000000000 : ℝ) =
+            (4020910430343609 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4020910430343609 / 2500000000000000 : ℝ) ≤
+            (23760881016249433261 / 50000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (502593700400121 / 2500000000000000 : ℝ)) (q :=
+      (4020910430343609 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_244_2 :
+    Uω (aρ 2) (bρ 2) (132802102197 / 640000000000 : ℝ) ≤
+      (-(1010215922572068434639 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (397246495170123 / 2000000000000000 : ℝ) ≤
+      (-(1010215922572068434639 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (397246495170123 / 2000000000000000 : ℝ) =
+            (397246495170123 / 250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (397246495170123 / 250000000000000 : ℝ) ≤
+            (289435040490431565361 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (397139667093621 / 2000000000000000 : ℝ)) (q :=
+      (397246495170123 / 2000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_244_3 :
+    Uω (aρ 3) (bρ 3) (132802102197 / 640000000000 : ℝ) ≤
+      (-(8206032948370699617277 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (968730644853851 / 5000000000000000 : ℝ) ≤
+      (-(8206032948370699617277 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (968730644853851 / 5000000000000000 : ℝ) =
+            (968730644853851 / 625000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (968730644853851 / 625000000000000 : ℝ) ≤
+            (2191174756129300382723 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1935552069532279 / 10000000000000000 : ℝ)) (q :=
+      (968730644853851 / 5000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_244_4 :
+    Uω (aρ 4) (bρ 4) (132802102197 / 640000000000 : ℝ) ≤
+      (-(8422407421941390401861 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1855406125778153 / 10000000000000000 : ℝ) ≤
+      (-(8422407421941390401861 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1855406125778153 / 10000000000000000 : ℝ) =
+            (1855406125778153 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1855406125778153 / 1250000000000000 : ℝ) ≤
+            (1974800282558609598139 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1849735922738181 / 10000000000000000 : ℝ)) (q :=
+      (1855406125778153 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_244_5 :
+    Uω (aρ 5) (bρ 5) (132802102197 / 640000000000 : ℝ) ≤
+      (-(877963093041325964747 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (863735860072067 / 5000000000000000 : ℝ) ≤
+      (-(877963093041325964747 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (863735860072067 / 5000000000000000 : ℝ) =
+            (863735860072067 / 625000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (863735860072067 / 625000000000000 : ℝ) ≤
+            (161757677408674035253 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1712057030805143 / 10000000000000000 : ℝ)) (q :=
+      (863735860072067 / 5000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_244_6 :
+    Uω (aρ 6) (bρ 6) (132802102197 / 640000000000 : ℝ) ≤
+      (-(4684014389455247979343 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1535689301249357 / 10000000000000000 : ℝ) ≤
+      (-(4684014389455247979343 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1535689301249357 / 10000000000000000 : ℝ) =
+            (1535689301249357 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1535689301249357 / 1250000000000000 : ℝ) ≤
+            (514589462794752020657 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1495732568155589 / 10000000000000000 : ℝ)) (q :=
+      (1535689301249357 / 10000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_244_7 :
+    Uω (aρ 7) (bρ 7) (132802102197 / 640000000000 : ℝ) ≤
+      (-(20809744920160066222079 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (124808527874231 / 1000000000000000 : ℝ) ≤
+      (-(20809744920160066222079 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (124808527874231 / 1000000000000000 : ℝ) =
+            (124808527874231 / 62500000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 15
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (124808527874231 / 62500000000000 : ℝ) ≤
+            (6916142291839933777921 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (228837180780299 / 2000000000000000 : ℝ)) (q :=
+      (124808527874231 / 1000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_244_8 :
+    Uω (aρ 8) (bρ 8) (132802102197 / 640000000000 : ℝ) ≤
+      (-(26362368852389073231951 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1432606320755581 / 20000000000000000 : ℝ) ≤
+      (-(26362368852389073231951 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1432606320755581 / 20000000000000000 : ℝ) =
+            (1432606320755581 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1432606320755581 / 1250000000000000 : ℝ) ≤
+            (1363518359610926768049 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (45520414814057 / 1250000000000000 : ℝ)) (q :=
+      (1432606320755581 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_244_9 :
+    Uω (aρ 9) (bρ 9) (132802102197 / 640000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_244_10 :
+    Uω (aρ 10) (bρ 10) (132802102197 / 640000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_244_11 :
+    Uω (aρ 11) (bρ 11) (132802102197 / 640000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_244_12 :
+    Uω (aρ 12) (bρ 12) (132802102197 / 640000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_244_13 :
+    Uω (aρ 13) (bρ 13) (132802102197 / 640000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_244_14 :
+    Uω (aρ 14) (bρ 14) (132802102197 / 640000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_244_15 :
+    Uω (aρ 15) (bρ 15) (132802102197 / 640000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_244_16 :
+    Uω (aρ 16) (bρ 16) (132802102197 / 640000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_244 :
+    Uρ (132802102197 / 640000000000 : ℝ) ≤
+      (-(2416425266355531027053 / 1250000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (132802102197 / 640000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(80211196028750566739 / 50000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_244_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (132802102197 / 640000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1010215922572068434639 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_244_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (132802102197 / 640000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(8206032948370699617277 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_244_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (132802102197 / 640000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(8422407421941390401861 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_244_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (132802102197 / 640000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(877963093041325964747 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_244_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (132802102197 / 640000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(4684014389455247979343 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_244_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (132802102197 / 640000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(20809744920160066222079 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_244_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (132802102197 / 640000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(26362368852389073231951 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_244_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (132802102197 / 640000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_244_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (132802102197 / 640000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_244_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (132802102197 / 640000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_244_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (132802102197 / 640000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_244_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (132802102197 / 640000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_244_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (132802102197 / 640000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_244_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (132802102197 / 640000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_244_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (132802102197 / 640000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_244_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_245_1 :
+    Uω (aρ 1) (bρ 1) (6674225226937 / 32000000000000 : ℝ) ≤
+      (-(15989341814769083441683 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8084472704020983 / 40000000000000000 : ℝ) ≤
+      (-(15989341814769083441683 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (8084472704020983 / 40000000000000000 : ℝ) =
+            (8084472704020983 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8084472704020983 / 5000000000000000 : ℝ) ≤
+            (4805073594230916558317 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2021038186677679 / 10000000000000000 : ℝ)) (q :=
+      (8084472704020983 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_245_2 :
+    Uω (aρ 2) (bρ 2) (6674225226937 / 32000000000000 : ℝ) ≤
+      (-(251717216120021689271 / 156250000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7987591461091323 / 40000000000000000 : ℝ) ≤
+      (-(251717216120021689271 / 156250000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (7987591461091323 / 40000000000000000 : ℝ) =
+            (7987591461091323 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7987591461091323 / 5000000000000000 : ℝ) ≤
+            (73195524645603310729 / 156250000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1996366577722849 / 10000000000000000 : ℝ)) (q :=
+      (7987591461091323 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_245_3 :
+    Uω (aρ 3) (bρ 3) (6674225226937 / 32000000000000 : ℝ) ≤
+      (-(16357129540002021346027 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1558507428879981 / 8000000000000000 : ℝ) ≤
+      (-(16357129540002021346027 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1558507428879981 / 8000000000000000 : ℝ) =
+            (1558507428879981 / 1000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1558507428879981 / 1000000000000000 : ℝ) ≤
+            (4437285868997978653973 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (97311776286357 / 500000000000000 : ℝ)) (q :=
+      (1558507428879981 / 8000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_245_4 :
+    Uω (aρ 4) (bρ 4) (6674225226937 / 32000000000000 : ℝ) ≤
+      (-(839366889642839072517 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1492880927632989 / 8000000000000000 : ℝ) ≤
+      (-(839366889642839072517 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1492880927632989 / 8000000000000000 : ℝ) =
+            (1492880927632989 / 1000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1492880927632989 / 1000000000000000 : ℝ) ≤
+            (200353880807160927483 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (93023172683733 / 500000000000000 : ℝ)) (q :=
+      (1492880927632989 / 8000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_245_5 :
+    Uω (aρ 5) (bρ 5) (6674225226937 / 32000000000000 : ℝ) ≤
+      (-(8748589672698357586477 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6952918634228077 / 40000000000000000 : ℝ) ≤
+      (-(8748589672698357586477 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (6952918634228077 / 40000000000000000 : ℝ) =
+            (6952918634228077 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6952918634228077 / 5000000000000000 : ℝ) ≤
+            (1648618031801642413523 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (861455185520613 / 5000000000000000 : ℝ)) (q :=
+      (6952918634228077 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_245_6 :
+    Uω (aρ 6) (bρ 6) (6674225226937 / 32000000000000 : ℝ) ≤
+      (-(18665037439110731528699 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6186538422931207 / 40000000000000000 : ℝ) ≤
+      (-(18665037439110731528699 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (6186538422931207 / 40000000000000000 : ℝ) =
+            (6186538422931207 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6186538422931207 / 5000000000000000 : ℝ) ≤
+            (2129377969889268471301 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1506960640532791 / 10000000000000000 : ℝ)) (q :=
+      (6186538422931207 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_245_7 :
+    Uω (aρ 7) (bρ 7) (6674225226937 / 32000000000000 : ℝ) ≤
+      (-(129481655524811388393 / 62500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5038825226518599 / 40000000000000000 : ℝ) ≤
+      (-(129481655524811388393 / 62500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (5038825226518599 / 40000000000000000 : ℝ) =
+            (5038825226518599 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5038825226518599 / 5000000000000000 : ℝ) ≤
+            (483440781438611607 / 62500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1156765423086487 / 10000000000000000 : ℝ)) (q :=
+      (5038825226518599 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_245_8 :
+    Uω (aρ 8) (bρ 8) (6674225226937 / 32000000000000 : ℝ) ≤
+      (-(26081207921134859408713 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (117876576485053 / 1600000000000000 : ℝ) ≤
+      (-(26081207921134859408713 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (117876576485053 / 1600000000000000 : ℝ) =
+            (117876576485053 / 100000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (117876576485053 / 100000000000000 : ℝ) ≤
+            (1644679290865140591287 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7887033344607 / 200000000000000 : ℝ)) (q :=
+      (117876576485053 / 1600000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_245_9 :
+    Uω (aρ 9) (bρ 9) (6674225226937 / 32000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_245_10 :
+    Uω (aρ 10) (bρ 10) (6674225226937 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_245_11 :
+    Uω (aρ 11) (bρ 11) (6674225226937 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_245_12 :
+    Uω (aρ 12) (bρ 12) (6674225226937 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_245_13 :
+    Uω (aρ 13) (bρ 13) (6674225226937 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_245_14 :
+    Uω (aρ 14) (bρ 14) (6674225226937 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_245_15 :
+    Uω (aρ 15) (bρ 15) (6674225226937 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_245_16 :
+    Uω (aρ 16) (bρ 16) (6674225226937 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_245 :
+    Uρ (6674225226937 / 32000000000000 : ℝ) ≤
+      (-(19280957920972008264119 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (6674225226937 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(15989341814769083441683 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_245_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (6674225226937 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(251717216120021689271 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_245_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (6674225226937 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(16357129540002021346027 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_245_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (6674225226937 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(839366889642839072517 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_245_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (6674225226937 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(8748589672698357586477 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_245_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (6674225226937 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(18665037439110731528699 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_245_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (6674225226937 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) * (-(129481655524811388393 / 62500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_245_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (6674225226937 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(26081207921134859408713 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_245_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (6674225226937 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_245_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (6674225226937 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_245_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (6674225226937 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_245_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (6674225226937 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_245_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (6674225226937 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_245_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (6674225226937 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_245_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (6674225226937 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_245_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (6674225226937 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_245_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_246_1 :
+    Uω (aρ 1) (bρ 1) (838543168003 / 4000000000000 : ℝ) ≤
+      (-(1992090348424161097649 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (812712452954231 / 4000000000000000 : ℝ) ≤
+      (-(1992090348424161097649 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (812712452954231 / 4000000000000000 : ℝ) =
+            (812712452954231 / 500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (812712452954231 / 500000000000000 : ℝ) ≤
+            (607211577700838902351 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (406340312569731 / 2000000000000000 : ℝ)) (q :=
+      (812712452954231 / 4000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_246_2 :
+    Uω (aρ 2) (bρ 2) (838543168003 / 4000000000000 : ℝ) ≤
+      (-(8028317158933071913149 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4015126448750247 / 20000000000000000 : ℝ) ≤
+      (-(8028317158933071913149 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4015126448750247 / 20000000000000000 : ℝ) =
+            (4015126448750247 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4015126448750247 / 2500000000000000 : ℝ) ≤
+            (2368890545566928086851 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2007034759337747 / 10000000000000000 : ℝ)) (q :=
+      (4015126448750247 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_246_3 :
+    Uω (aρ 3) (bρ 3) (838543168003 / 4000000000000 : ℝ) ≤
+      (-(4075623479472093816281 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3917614336794283 / 20000000000000000 : ℝ) ≤
+      (-(4075623479472093816281 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3917614336794283 / 20000000000000000 : ℝ) =
+            (3917614336794283 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3917614336794283 / 2500000000000000 : ℝ) ≤
+            (1122980372777906183719 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1956918753731783 / 10000000000000000 : ℝ)) (q :=
+      (3917614336794283 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_246_4 :
+    Uω (aρ 4) (bρ 4) (838543168003 / 4000000000000 : ℝ) ≤
+      (-(16730191195247513018179 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (938397910910331 / 5000000000000000 : ℝ) ≤
+      (-(16730191195247513018179 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (938397910910331 / 5000000000000000 : ℝ) =
+            (938397910910331 / 625000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (938397910910331 / 625000000000000 : ℝ) ≤
+            (4064224213752486981821 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (116949390102739 / 625000000000000 : ℝ)) (q :=
+      (938397910910331 / 5000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_246_5 :
+    Uω (aρ 5) (bρ 5) (838543168003 / 4000000000000 : ℝ) ≤
+      (-(8717743323121730710593 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1748986413039669 / 10000000000000000 : ℝ) ≤
+      (-(8717743323121730710593 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1748986413039669 / 10000000000000000 : ℝ) =
+            (1748986413039669 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1748986413039669 / 1250000000000000 : ℝ) ≤
+            (1679464381378269289407 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (866880671708419 / 5000000000000000 : ℝ)) (q :=
+      (1748986413039669 / 10000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_246_6 :
+    Uω (aρ 6) (bρ 6) (838543168003 / 4000000000000 : ℝ) ≤
+      (-(18594544327930489474517 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3115151666007471 / 20000000000000000 : ℝ) ≤
+      (-(18594544327930489474517 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3115151666007471 / 20000000000000000 : ℝ) =
+            (3115151666007471 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3115151666007471 / 2500000000000000 : ℝ) ≤
+            (2199871081069510525483 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1518180558484971 / 10000000000000000 : ℝ)) (q :=
+      (3115151666007471 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_246_7 :
+    Uω (aρ 7) (bρ 7) (838543168003 / 4000000000000 : ℝ) ≤
+      (-(825015431120082058003 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2542616566211877 / 20000000000000000 : ℝ) ≤
+      (-(825015431120082058003 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2542616566211877 / 20000000000000000 : ℝ) =
+            (2542616566211877 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2542616566211877 / 2500000000000000 : ℝ) ≤
+            (6761185239917941997 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1169306839449377 / 10000000000000000 : ℝ)) (q :=
+      (2542616566211877 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_246_8 :
+    Uω (aρ 8) (bρ 8) (838543168003 / 4000000000000 : ℝ) ≤
+      (-(3227522382415953287113 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (756212533019953 / 10000000000000000 : ℝ) ≤
+      (-(3227522382415953287113 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (756212533019953 / 10000000000000000 : ℝ) =
+            (756212533019953 / 625000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (756212533019953 / 625000000000000 : ℝ) ≤
+            (238213519084046712887 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (211328495308703 / 5000000000000000 : ℝ)) (q :=
+      (756212533019953 / 10000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_246_9 :
+    Uω (aρ 9) (bρ 9) (838543168003 / 4000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_246_10 :
+    Uω (aρ 10) (bρ 10) (838543168003 / 4000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_246_11 :
+    Uω (aρ 11) (bρ 11) (838543168003 / 4000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_246_12 :
+    Uω (aρ 12) (bρ 12) (838543168003 / 4000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_246_13 :
+    Uω (aρ 13) (bρ 13) (838543168003 / 4000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_246_14 :
+    Uω (aρ 14) (bρ 14) (838543168003 / 4000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_246_15 :
+    Uω (aρ 15) (bρ 15) (838543168003 / 4000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_246_16 :
+    Uω (aρ 16) (bρ 16) (838543168003 / 4000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_246 :
+    Uρ (838543168003 / 4000000000000 : ℝ) ≤
+      (-(961624510270881730459 / 500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (838543168003 / 4000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1992090348424161097649 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_246_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (838543168003 / 4000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(8028317158933071913149 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_246_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (838543168003 / 4000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(4075623479472093816281 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_246_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (838543168003 / 4000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(16730191195247513018179 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_246_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (838543168003 / 4000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(8717743323121730710593 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_246_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (838543168003 / 4000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(18594544327930489474517 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_246_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (838543168003 / 4000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(825015431120082058003 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_246_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (838543168003 / 4000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(3227522382415953287113 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_246_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (838543168003 / 4000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_246_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (838543168003 / 4000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_246_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (838543168003 / 4000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_246_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (838543168003 / 4000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_246_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (838543168003 / 4000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_246_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (838543168003 / 4000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_246_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (838543168003 / 4000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_246_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (838543168003 / 4000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_246_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_247_1 :
+    Uω (aρ 1) (bρ 1) (6742465461111 / 32000000000000 : ℝ) ≤
+      (-(15884379209042150851217 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8169776337530201 / 40000000000000000 : ℝ) ≤
+      (-(15884379209042150851217 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (8169776337530201 / 40000000000000000 : ℝ) =
+            (8169776337530201 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8169776337530201 / 5000000000000000 : ℝ) ≤
+            (4910036199957849148783 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2042364930252913 / 10000000000000000 : ℝ)) (q :=
+      (8169776337530201 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_247_2 :
+    Uω (aρ 2) (bρ 2) (6742465461111 / 32000000000000 : ℝ) ≤
+      (-(16003649191942420646063 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8072914214553699 / 40000000000000000 : ℝ) ≤
+      (-(16003649191942420646063 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (8072914214553699 / 40000000000000000 : ℝ) =
+            (8072914214553699 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8072914214553699 / 5000000000000000 : ℝ) ≤
+            (4790766217057579353937 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1008851440637331 / 5000000000000000 : ℝ)) (q :=
+      (8072914214553699 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_247_3 :
+    Uω (aρ 3) (bρ 3) (6742465461111 / 32000000000000 : ℝ) ≤
+      (-(8124077874733012623601 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7877919753830639 / 40000000000000000 : ℝ) ≤
+      (-(8124077874733012623601 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (7877919753830639 / 40000000000000000 : ℝ) =
+            (7877919753830639 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7877919753830639 / 5000000000000000 : ℝ) ≤
+            (2273129829766987376399 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (491900439315783 / 2500000000000000 : ℝ)) (q :=
+      (7877919753830639 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_247_4 :
+    Uω (aρ 4) (bρ 4) (6742465461111 / 32000000000000 : ℝ) ≤
+      (-(16673371250703288171891 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7549960475874883 / 40000000000000000 : ℝ) ≤
+      (-(16673371250703288171891 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (7549960475874883 / 40000000000000000 : ℝ) =
+            (7549960475874883 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7549960475874883 / 5000000000000000 : ℝ) ≤
+            (4121044158296711828109 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (940958149675127 / 5000000000000000 : ℝ)) (q :=
+      (7549960475874883 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_247_5 :
+    Uω (aρ 5) (bρ 5) (6742465461111 / 32000000000000 : ℝ) ≤
+      (-(43435447039729323767 / 25000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7038968022732803 / 40000000000000000 : ℝ) ≤
+      (-(43435447039729323767 / 25000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (7038968022732803 / 40000000000000000 : ℝ) =
+            (7038968022732803 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7038968022732803 / 5000000000000000 : ℝ) ≤
+            (8550591482770676233 / 25000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (872304996057107 / 5000000000000000 : ℝ)) (q :=
+      (7038968022732803 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_247_6 :
+    Uω (aρ 6) (bρ 6) (6742465461111 / 32000000000000 : ℝ) ≤
+      (-(4631142522539577805907 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6274052291183249 / 40000000000000000 : ℝ) ≤
+      (-(4631142522539577805907 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (6274052291183249 / 40000000000000000 : ℝ) =
+            (6274052291183249 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6274052291183249 / 5000000000000000 : ℝ) ≤
+            (567461329710422194093 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1529392501479437 / 10000000000000000 : ℝ)) (q :=
+      (6274052291183249 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_247_7 :
+    Uω (aρ 7) (bρ 7) (6742465461111 / 32000000000000 : ℝ) ≤
+      (-(20534683322764266391341 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5131567258775443 / 40000000000000000 : ℝ) ≤
+      (-(20534683322764266391341 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (5131567258775443 / 40000000000000000 : ℝ) =
+            (5131567258775443 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5131567258775443 / 5000000000000000 : ℝ) ≤
+            (259732086235733608659 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (590905683017767 / 5000000000000000 : ℝ)) (q :=
+      (5131567258775443 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_247_8 :
+    Uω (aρ 8) (bρ 8) (6742465461111 / 32000000000000 : ℝ) ≤
+      (-(25575639136592379083153 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3099731638180319 / 40000000000000000 : ℝ) ≤
+      (-(25575639136592379083153 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (3099731638180319 / 40000000000000000 : ℝ) =
+            (3099731638180319 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3099731638180319 / 2500000000000000 : ℝ) ≤
+            (2150248075407620916847 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (112358801769493 / 2500000000000000 : ℝ)) (q :=
+      (3099731638180319 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_247_9 :
+    Uω (aρ 9) (bρ 9) (6742465461111 / 32000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_247_10 :
+    Uω (aρ 10) (bρ 10) (6742465461111 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_247_11 :
+    Uω (aρ 11) (bρ 11) (6742465461111 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_247_12 :
+    Uω (aρ 12) (bρ 12) (6742465461111 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_247_13 :
+    Uω (aρ 13) (bρ 13) (6742465461111 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_247_14 :
+    Uω (aρ 14) (bρ 14) (6742465461111 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_247_15 :
+    Uω (aρ 15) (bρ 15) (6742465461111 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_247_16 :
+    Uω (aρ 16) (bρ 16) (6742465461111 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_247 :
+    Uρ (6742465461111 / 32000000000000 : ℝ) ≤
+      (-(19185673420397004360467 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (6742465461111 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(15884379209042150851217 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_247_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (6742465461111 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(16003649191942420646063 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_247_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (6742465461111 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(8124077874733012623601 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_247_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (6742465461111 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(16673371250703288171891 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_247_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (6742465461111 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(43435447039729323767 / 25000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_247_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (6742465461111 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(4631142522539577805907 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_247_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (6742465461111 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(20534683322764266391341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_247_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (6742465461111 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(25575639136592379083153 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_247_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (6742465461111 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_247_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (6742465461111 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_247_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (6742465461111 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_247_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (6742465461111 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_247_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (6742465461111 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_247_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (6742465461111 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_247_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (6742465461111 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_247_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (6742465461111 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_247_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_248_1 :
+    Uω (aρ 1) (bρ 1) (3388292789099 / 16000000000000 : ℝ) ≤
+      (-(15832308210675581165643 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4106214064128931 / 20000000000000000 : ℝ) ≤
+      (-(15832308210675581165643 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4106214064128931 / 20000000000000000 : ℝ) =
+            (4106214064128931 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4106214064128931 / 2500000000000000 : ℝ) ≤
+            (4962107198324418834357 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (128314268064191 / 625000000000000 : ℝ)) (q :=
+      (4106214064128931 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_248_2 :
+    Uω (aρ 2) (bρ 2) (3388292789099 / 16000000000000 : ℝ) ≤
+      (-(7975471737047489165407 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (253611731691693 / 1250000000000000 : ℝ) ≤
+      (-(7975471737047489165407 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (253611731691693 / 1250000000000000 : ℝ) =
+            (253611731691693 / 156250000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (253611731691693 / 156250000000000 : ℝ) ≤
+            (2421735967452510834593 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2028370944475213 / 10000000000000000 : ℝ)) (q :=
+      (253611731691693 / 1250000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_248_3 :
+    Uω (aρ 3) (bρ 3) (3388292789099 / 16000000000000 : ℝ) ≤
+      (-(3238822361460214488721 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3960305196199621 / 20000000000000000 : ℝ) ≤
+      (-(3238822361460214488721 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3960305196199621 / 20000000000000000 : ℝ) =
+            (3960305196199621 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3960305196199621 / 2500000000000000 : ℝ) ≤
+            (920060720339785511279 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (989142269978873 / 5000000000000000 : ℝ)) (q :=
+      (3960305196199621 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_248_4 :
+    Uω (aρ 4) (bρ 4) (3388292789099 / 16000000000000 : ℝ) ≤
+      (-(3323374844885987262029 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3796368114386551 / 20000000000000000 : ℝ) ≤
+      (-(3323374844885987262029 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3796368114386551 / 20000000000000000 : ℝ) =
+            (3796368114386551 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3796368114386551 / 2500000000000000 : ℝ) ≤
+            (835508236914012737971 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (473160409802419 / 2500000000000000 : ℝ)) (q :=
+      (3796368114386551 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_248_5 :
+    Uω (aρ 5) (bρ 5) (3388292789099 / 16000000000000 : ℝ) ≤
+      (-(3462650200390808338077 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3540992916055939 / 20000000000000000 : ℝ) ≤
+      (-(3462650200390808338077 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3540992916055939 / 20000000000000000 : ℝ) =
+            (3540992916055939 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3540992916055939 / 2500000000000000 : ℝ) ≤
+            (696232881409191661923 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (109716022513379 / 625000000000000 : ℝ)) (q :=
+      (3540992916055939 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_248_6 :
+    Uω (aρ 6) (bρ 6) (3388292789099 / 16000000000000 : ℝ) ≤
+      (-(18455106786157723315289 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (631778564867047 / 4000000000000000 : ℝ) ≤
+      (-(18455106786157723315289 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (631778564867047 / 4000000000000000 : ℝ) =
+            (631778564867047 / 500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (631778564867047 / 500000000000000 : ℝ) ≤
+            (2339308622842276684711 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (19257458045417 / 125000000000000 : ℝ)) (q :=
+      (631778564867047 / 4000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_248_7 :
+    Uω (aρ 7) (bρ 7) (3388292789099 / 16000000000000 : ℝ) ≤
+      (-(20444934186107947075261 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1294457480766791 / 10000000000000000 : ℝ) ≤
+      (-(20444934186107947075261 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1294457480766791 / 10000000000000000 : ℝ) =
+            (1294457480766791 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1294457480766791 / 1250000000000000 : ℝ) ≤
+            (349481222892052924739 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1194280161591707 / 10000000000000000 : ℝ)) (q :=
+      (1294457480766791 / 10000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_248_8 :
+    Uω (aρ 8) (bρ 8) (3388292789099 / 16000000000000 : ℝ) ≤
+      (-(792029153694985536973 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (396509459602709 / 5000000000000000 : ℝ) ≤
+      (-(792029153694985536973 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (396509459602709 / 5000000000000000 : ℝ) =
+            (396509459602709 / 312500000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (396509459602709 / 312500000000000 : ℝ) ≤
+            (74404821680014463027 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (474944689808961 / 10000000000000000 : ℝ)) (q :=
+      (396509459602709 / 5000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_248_9 :
+    Uω (aρ 9) (bρ 9) (3388292789099 / 16000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_248_10 :
+    Uω (aρ 10) (bρ 10) (3388292789099 / 16000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_248_11 :
+    Uω (aρ 11) (bρ 11) (3388292789099 / 16000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_248_12 :
+    Uω (aρ 12) (bρ 12) (3388292789099 / 16000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_248_13 :
+    Uω (aρ 13) (bρ 13) (3388292789099 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_248_14 :
+    Uω (aρ 14) (bρ 14) (3388292789099 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_248_15 :
+    Uω (aρ 15) (bρ 15) (3388292789099 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_248_16 :
+    Uω (aρ 16) (bρ 16) (3388292789099 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_248 :
+    Uρ (3388292789099 / 16000000000000 : ℝ) ≤
+      (-(3828053887706228677047 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (3388292789099 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(15832308210675581165643 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_248_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (3388292789099 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(7975471737047489165407 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_248_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (3388292789099 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3238822361460214488721 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_248_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (3388292789099 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3323374844885987262029 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_248_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (3388292789099 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(3462650200390808338077 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_248_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (3388292789099 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(18455106786157723315289 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_248_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (3388292789099 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(20444934186107947075261 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_248_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (3388292789099 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(792029153694985536973 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_248_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (3388292789099 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_248_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (3388292789099 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_248_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (3388292789099 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_248_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (3388292789099 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_248_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (3388292789099 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_248_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (3388292789099 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_248_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (3388292789099 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_248_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (3388292789099 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_248_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_249_1 :
+    Uω (aρ 1) (bρ 1) (1362141139057 / 6400000000000 : ℝ) ≤
+      (-(7890253483925033164729 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8255079901992847 / 40000000000000000 : ℝ) ≤
+      (-(7890253483925033164729 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (8255079901992847 / 40000000000000000 : ℝ) =
+            (8255079901992847 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8255079901992847 / 5000000000000000 : ℝ) ≤
+            (2506954220574966835271 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2063691639304861 / 10000000000000000 : ℝ)) (q :=
+      (8255079901992847 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_249_2 :
+    Uω (aρ 2) (bρ 2) (1362141139057 / 6400000000000 : ℝ) ≤
+      (-(3179702846290115685933 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8158236498085747 / 40000000000000000 : ℝ) ≤
+      (-(3179702846290115685933 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (8158236498085747 / 40000000000000000 : ℝ) =
+            (8158236498085747 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8158236498085747 / 5000000000000000 : ℝ) ≤
+            (979180235509884314067 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2039038949861311 / 10000000000000000 : ℝ)) (q :=
+      (8158236498085747 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_249_3 :
+    Uω (aρ 3) (bρ 3) (1362141139057 / 6400000000000 : ℝ) ≤
+      (-(3228071783260219938387 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1592660119282217 / 8000000000000000 : ℝ) ≤
+      (-(3228071783260219938387 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1592660119282217 / 8000000000000000 : ℝ) =
+            (1592660119282217 / 1000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1592660119282217 / 1000000000000000 : ℝ) ≤
+            (930811298539780061613 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (99448355268699 / 500000000000000 : ℝ)) (q :=
+      (1592660119282217 / 8000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_249_4 :
+    Uω (aρ 4) (bρ 4) (1362141139057 / 6400000000000 : ℝ) ≤
+      (-(16560696445683286189297 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7635510570247419 / 40000000000000000 : ℝ) ≤
+      (-(16560696445683286189297 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (7635510570247419 / 40000000000000000 : ℝ) =
+            (7635510570247419 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7635510570247419 / 5000000000000000 : ℝ) ≤
+            (4233718963316713810703 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1903366273357147 / 10000000000000000 : ℝ)) (q :=
+      (7635510570247419 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_249_5 :
+    Uω (aρ 5) (bρ 5) (1362141139057 / 6400000000000 : ℝ) ≤
+      (-(1725269844467385308997 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1424999832864611 / 8000000000000000 : ℝ) ≤
+      (-(1725269844467385308997 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1424999832864611 / 8000000000000000 : ℝ) =
+            (1424999832864611 / 1000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1424999832864611 / 1000000000000000 : ℝ) ≤
+            (354171696432614691003 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (353260097945993 / 2000000000000000 : ℝ)) (q :=
+      (1424999832864611 / 8000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_249_6 :
+    Uω (aρ 6) (bρ 6) (1362141139057 / 6400000000000 : ℝ) ≤
+      (-(735445866576952836003 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1272300748482987 / 8000000000000000 : ℝ) ≤
+      (-(735445866576952836003 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1272300748482987 / 8000000000000000 : ℝ) =
+            (1272300748482987 / 1000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1272300748482987 / 1000000000000000 : ℝ) ≤
+            (96330749783047163997 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (310358630783181 / 2000000000000000 : ℝ)) (q :=
+      (1272300748482987 / 8000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_249_7 :
+    Uω (aρ 7) (bρ 7) (1362141139057 / 6400000000000 : ℝ) ≤
+      (-(10178057967148668099869 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5224023340519699 / 40000000000000000 : ℝ) ≤
+      (-(10178057967148668099869 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (5224023340519699 / 40000000000000000 : ℝ) =
+            (5224023340519699 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5224023340519699 / 5000000000000000 : ℝ) ≤
+            (219149737351331900131 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1206714333728287 / 10000000000000000 : ℝ)) (q :=
+      (5224023340519699 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_249_8 :
+    Uω (aρ 8) (bρ 8) (1362141139057 / 6400000000000 : ℝ) ≤
+      (-(201008430917706393269 / 80000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3242271182885199 / 40000000000000000 : ℝ) ≤
+      (-(201008430917706393269 / 80000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (3242271182885199 / 40000000000000000 : ℝ) =
+            (3242271182885199 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3242271182885199 / 2500000000000000 : ℝ) ≤
+            (20798666778293606731 / 80000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (499379906251037 / 10000000000000000 : ℝ)) (q :=
+      (3242271182885199 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_249_9 :
+    Uω (aρ 9) (bρ 9) (1362141139057 / 6400000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_249_10 :
+    Uω (aρ 10) (bρ 10) (1362141139057 / 6400000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_249_11 :
+    Uω (aρ 11) (bρ 11) (1362141139057 / 6400000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_249_12 :
+    Uω (aρ 12) (bρ 12) (1362141139057 / 6400000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_249_13 :
+    Uω (aρ 13) (bρ 13) (1362141139057 / 6400000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_249_14 :
+    Uω (aρ 14) (bρ 14) (1362141139057 / 6400000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_249_15 :
+    Uω (aρ 15) (bρ 15) (1362141139057 / 6400000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_249_16 :
+    Uω (aρ 16) (bρ 16) (1362141139057 / 6400000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_249 :
+    Uρ (1362141139057 / 6400000000000 : ℝ) ≤
+      (-(19096097579846410691343 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1362141139057 / 6400000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(7890253483925033164729 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_249_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1362141139057 / 6400000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3179702846290115685933 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_249_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1362141139057 / 6400000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3228071783260219938387 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_249_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1362141139057 / 6400000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(16560696445683286189297 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_249_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1362141139057 / 6400000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1725269844467385308997 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_249_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1362141139057 / 6400000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(735445866576952836003 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_249_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1362141139057 / 6400000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(10178057967148668099869 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_249_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1362141139057 / 6400000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) * (-(201008430917706393269 / 80000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_249_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1362141139057 / 6400000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_249_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1362141139057 / 6400000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_249_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1362141139057 / 6400000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_249_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1362141139057 / 6400000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_249_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1362141139057 / 6400000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_249_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1362141139057 / 6400000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_249_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1362141139057 / 6400000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_249_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1362141139057 / 6400000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_249_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_250_1 :
+    Uω (aρ 1) (bρ 1) (1711206453093 / 8000000000000 : ℝ) ≤
+      (-(15728972699799062262617 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (259304114343663 / 1250000000000000 : ℝ) ≤
+      (-(15728972699799062262617 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (259304114343663 / 1250000000000000 : ℝ) =
+            (259304114343663 / 156250000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (259304114343663 / 156250000000000 : ℝ) ≤
+            (5065442709200937737383 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1037177490608679 / 5000000000000000 : ℝ)) (q :=
+      (259304114343663 / 1250000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_250_2 :
+    Uω (aρ 2) (bρ 2) (1711206453093 / 8000000000000 : ℝ) ≤
+      (-(7923179288548330911731 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1025112183526729 / 5000000000000000 : ℝ) ≤
+      (-(7923179288548330911731 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1025112183526729 / 5000000000000000 : ℝ) =
+            (1025112183526729 / 625000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1025112183526729 / 625000000000000 : ℝ) ≤
+            (2474028415951669088269 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1024853449167833 / 5000000000000000 : ℝ)) (q :=
+      (1025112183526729 / 5000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_250_3 :
+    Uω (aρ 3) (bρ 3) (1711206453093 / 8000000000000 : ℝ) ≤
+      (-(1608689395258738273783 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (400299518641533 / 2000000000000000 : ℝ) ≤
+      (-(1608689395258738273783 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (400299518641533 / 2000000000000000 : ℝ) =
+            (400299518641533 / 250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (400299518641533 / 250000000000000 : ℝ) ≤
+            (470752145641261726217 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (12497809106213 / 62500000000000 : ℝ)) (q :=
+      (400299518641533 / 2000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_250_4 :
+    Uω (aρ 4) (bρ 4) (1711206453093 / 8000000000000 : ℝ) ≤
+      (-(16504834306304953880329 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (239946360125701 / 1250000000000000 : ℝ) ≤
+      (-(16504834306304953880329 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (239946360125701 / 1250000000000000 : ℝ) =
+            (239946360125701 / 156250000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (239946360125701 / 156250000000000 : ℝ) ≤
+            (4289581102695046119671 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (957045106827483 / 5000000000000000 : ℝ)) (q :=
+      (239946360125701 / 1250000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_250_5 :
+    Uω (aρ 5) (bρ 5) (1711206453093 / 8000000000000 : ℝ) ≤
+      (-(17192516474548806354561 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3584004050662709 / 20000000000000000 : ℝ) ≤
+      (-(17192516474548806354561 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3584004050662709 / 20000000000000000 : ℝ) =
+            (3584004050662709 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3584004050662709 / 2500000000000000 : ℝ) ≤
+            (3601898934451193645439 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1777142421641459 / 10000000000000000 : ℝ)) (q :=
+      (3584004050662709 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_250_6 :
+    Uω (aρ 6) (bρ 6) (1711206453093 / 8000000000000 : ℝ) ≤
+      (-(9158841077746681639 / 5000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3202603450222341 / 20000000000000000 : ℝ) ≤
+      (-(9158841077746681639 / 5000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3202603450222341 / 20000000000000000 : ℝ) =
+            (3202603450222341 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3202603450222341 / 2500000000000000 : ℝ) ≤
+            (1238366626753318361 / 5000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1562982196341091 / 10000000000000000 : ℝ)) (q :=
+      (3202603450222341 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_250_7 :
+    Uω (aρ 7) (bρ 7) (1711206453093 / 8000000000000 : ℝ) ≤
+      (-(4053641396807687271301 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (32938435187193 / 250000000000000 : ℝ) ≤
+      (-(4053641396807687271301 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (32938435187193 / 250000000000000 : ℝ) = (32938435187193 / 31250000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (32938435187193 / 31250000000000 : ℝ) ≤
+            (105241684992312728699 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (121911494185419 / 1000000000000000 : ℝ)) (q :=
+      (32938435187193 / 250000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_250_8 :
+    Uω (aρ 8) (bρ 8) (1711206453093 / 8000000000000 : ℝ) ≤
+      (-(24917441486620581692963 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12932107065137 / 156250000000000 : ℝ) ≤
+      (-(24917441486620581692963 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (12932107065137 / 156250000000000 : ℝ) = (12932107065137 / 9765625000000 : ℝ) / 2 ^ 4 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12932107065137 / 9765625000000 : ℝ) ≤
+            (2808445725379418307037 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (261445741278143 / 5000000000000000 : ℝ)) (q :=
+      (12932107065137 / 156250000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_250_9 :
+    Uω (aρ 9) (bρ 9) (1711206453093 / 8000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_250_10 :
+    Uω (aρ 10) (bρ 10) (1711206453093 / 8000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_250_11 :
+    Uω (aρ 11) (bρ 11) (1711206453093 / 8000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_250_12 :
+    Uω (aρ 12) (bρ 12) (1711206453093 / 8000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_250_13 :
+    Uω (aρ 13) (bρ 13) (1711206453093 / 8000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_250_14 :
+    Uω (aρ 14) (bρ 14) (1711206453093 / 8000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_250_15 :
+    Uω (aρ 15) (bρ 15) (1711206453093 / 8000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_250_16 :
+    Uω (aρ 16) (bρ 16) (1711206453093 / 8000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_250 :
+    Uρ (1711206453093 / 8000000000000 : ℝ) ≤
+      (-(2381627086733213812727 / 1250000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1711206453093 / 8000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(15728972699799062262617 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_250_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1711206453093 / 8000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(7923179288548330911731 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_250_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1711206453093 / 8000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1608689395258738273783 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_250_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1711206453093 / 8000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(16504834306304953880329 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_250_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1711206453093 / 8000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(17192516474548806354561 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_250_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1711206453093 / 8000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) * (-(9158841077746681639 / 5000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_250_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1711206453093 / 8000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(4053641396807687271301 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_250_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1711206453093 / 8000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(24917441486620581692963 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_250_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1711206453093 / 8000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_250_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1711206453093 / 8000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_250_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1711206453093 / 8000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_250_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1711206453093 / 8000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_250_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1711206453093 / 8000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_250_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1711206453093 / 8000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_250_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1711206453093 / 8000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_250_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1711206453093 / 8000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_250_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_251_1 :
+    Uω (aρ 1) (bρ 1) (13723771741831 / 64000000000000 : ℝ) ≤
+      (-(15703304824046411768787 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16638115062611679 / 80000000000000000 : ℝ) ≤
+      (-(15703304824046411768787 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (16638115062611679 / 80000000000000000 : ℝ) =
+            (16638115062611679 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16638115062611679 / 10000000000000000 : ℝ) ≤
+            (5091110584953588231213 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1039843324538413 / 5000000000000000 : ℝ)) (q :=
+      (16638115062611679 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_251_2 :
+    Uω (aρ 2) (bρ 2) (13723771741831 / 64000000000000 : ℝ) ≤
+      (-(15820382455700095831141 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16444455822296023 / 80000000000000000 : ℝ) ≤
+      (-(15820382455700095831141 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (16444455822296023 / 80000000000000000 : ℝ) =
+            (16444455822296023 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16444455822296023 / 10000000000000000 : ℝ) ≤
+            (4974032953299904168859 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (256880106438489 / 1250000000000000 : ℝ)) (q :=
+      (16444455822296023 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_251_3 :
+    Uω (aρ 3) (bρ 3) (13723771741831 / 64000000000000 : ℝ) ≤
+      (-(8030134240491908589773 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3210934041131651 / 16000000000000000 : ℝ) ≤
+      (-(8030134240491908589773 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3210934041131651 / 16000000000000000 : ℝ) =
+            (3210934041131651 / 2000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3210934041131651 / 2000000000000000 : ℝ) ≤
+            (2367073464008091410227 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (200499055369847 / 1000000000000000 : ℝ)) (q :=
+      (3210934041131651 / 16000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_251_4 :
+    Uω (aρ 4) (bρ 4) (13723771741831 / 64000000000000 : ℝ) ≤
+      (-(1029813780661990088497 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15399338975582483 / 80000000000000000 : ℝ) ≤
+      (-(1029813780661990088497 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (15399338975582483 / 80000000000000000 : ℝ) =
+            (15399338975582483 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15399338975582483 / 10000000000000000 : ℝ) ≤
+            (269837182400509911503 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1919451927244527 / 10000000000000000 : ℝ)) (q :=
+      (15399338975582483 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_251_5 :
+    Uω (aρ 5) (bρ 5) (13723771741831 / 64000000000000 : ℝ) ≤
+      (-(17162563024561307747881 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14379021893371507 / 80000000000000000 : ℝ) ≤
+      (-(17162563024561307747881 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (14379021893371507 / 80000000000000000 : ℝ) =
+            (14379021893371507 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14379021893371507 / 10000000000000000 : ℝ) ≤
+            (3631852384438692252119 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1782562576026783 / 10000000000000000 : ℝ)) (q :=
+      (14379021893371507 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_251_6 :
+    Uω (aρ 6) (bρ 6) (13723771741831 / 64000000000000 : ℝ) ≤
+      (-(18283633438707248311039 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12854105956912811 / 80000000000000000 : ℝ) ≤
+      (-(18283633438707248311039 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (12854105956912811 / 80000000000000000 : ℝ) =
+            (12854105956912811 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12854105956912811 / 10000000000000000 : ℝ) ≤
+            (2510781970292751688961 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1568573967052109 / 10000000000000000 : ℝ)) (q :=
+      (12854105956912811 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_251_7 :
+    Uω (aρ 7) (bρ 7) (13723771741831 / 64000000000000 : ℝ) ≤
+      (-(4044917394737842997651 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10586376477557943 / 80000000000000000 : ℝ) ≤
+      (-(4044917394737842997651 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (10586376477557943 / 80000000000000000 : ℝ) =
+            (10586376477557943 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10586376477557943 / 10000000000000000 : ℝ) ≤
+            (113965687062157002349 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (76581436123337 / 625000000000000 : ℝ)) (q :=
+      (10586376477557943 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_251_8 :
+    Uω (aρ 8) (bρ 8) (13723771741831 / 64000000000000 : ℝ) ≤
+      (-(24816587177418204207001 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6688354741662559 / 80000000000000000 : ℝ) ≤
+      (-(24816587177418204207001 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (6688354741662559 / 80000000000000000 : ℝ) =
+            (6688354741662559 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6688354741662559 / 5000000000000000 : ℝ) ≤
+            (2909300034581795792999 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (267169597669773 / 5000000000000000 : ℝ)) (q :=
+      (6688354741662559 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_251_9 :
+    Uω (aρ 9) (bρ 9) (13723771741831 / 64000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_251_10 :
+    Uω (aρ 10) (bρ 10) (13723771741831 / 64000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_251_11 :
+    Uω (aρ 11) (bρ 11) (13723771741831 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_251_12 :
+    Uω (aρ 12) (bρ 12) (13723771741831 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_251_13 :
+    Uω (aρ 13) (bρ 13) (13723771741831 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_251_14 :
+    Uω (aρ 14) (bρ 14) (13723771741831 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_251_15 :
+    Uω (aρ 15) (bρ 15) (13723771741831 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_251_16 :
+    Uω (aρ 16) (bρ 16) (13723771741831 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_251 :
+    Uρ (13723771741831 / 64000000000000 : ℝ) ≤
+      (-(19031849196494214193577 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (13723771741831 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(15703304824046411768787 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_251_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (13723771741831 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(15820382455700095831141 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_251_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (13723771741831 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(8030134240491908589773 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_251_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (13723771741831 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1029813780661990088497 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_251_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (13723771741831 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(17162563024561307747881 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_251_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (13723771741831 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(18283633438707248311039 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_251_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (13723771741831 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(4044917394737842997651 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_251_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (13723771741831 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(24816587177418204207001 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_251_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (13723771741831 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_251_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (13723771741831 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_251_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (13723771741831 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_251_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (13723771741831 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_251_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (13723771741831 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_251_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (13723771741831 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_251_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (13723771741831 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_251_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (13723771741831 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_251_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_252_1 :
+    Uω (aρ 1) (bρ 1) (6878945929459 / 32000000000000 : ℝ) ≤
+      (-(7838851334268503496127 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8340383399527659 / 40000000000000000 : ℝ) ≤
+      (-(7838851334268503496127 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (8340383399527659 / 40000000000000000 : ℝ) =
+            (8340383399527659 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8340383399527659 / 5000000000000000 : ℝ) ≤
+            (2558356370231496503873 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (521254578723223 / 2500000000000000 : ℝ)) (q :=
+      (8340383399527659 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_252_2 :
+    Uω (aρ 2) (bρ 2) (6878945929459 / 32000000000000 : ℝ) ≤
+      (-(3158894733825103070931 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8243558326286459 / 40000000000000000 : ℝ) ≤
+      (-(3158894733825103070931 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (8243558326286459 / 40000000000000000 : ℝ) =
+            (8243558326286459 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8243558326286459 / 5000000000000000 : ℝ) ≤
+            (999988347974896929069 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (515093697695573 / 2500000000000000 : ℝ)) (q :=
+      (8243558326286459 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_252_3 :
+    Uω (aρ 3) (bρ 3) (6878945929459 / 32000000000000 : ℝ) ≤
+      (-(8016856921198687784309 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8048679728474291 / 40000000000000000 : ℝ) ≤
+      (-(8016856921198687784309 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (8048679728474291 / 40000000000000000 : ℝ) =
+            (8048679728474291 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8048679728474291 / 5000000000000000 : ℝ) ≤
+            (2380350783301312215691 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (62822862444569 / 312500000000000 : ℝ)) (q :=
+      (8048679728474291 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_252_4 :
+    Uω (aρ 4) (bρ 4) (6878945929459 / 32000000000000 : ℝ) ≤
+      (-(8224642129649983668167 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7721055113292499 / 40000000000000000 : ℝ) ≤
+      (-(8224642129649983668167 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (7721055113292499 / 40000000000000000 : ℝ) =
+            (7721055113292499 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7721055113292499 / 5000000000000000 : ℝ) ≤
+            (2172565574850016331833 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (240601683962539 / 1250000000000000 : ℝ)) (q :=
+      (7721055113292499 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_252_5 :
+    Uω (aρ 5) (bρ 5) (6878945929459 / 32000000000000 : ℝ) ≤
+      (-(3426540102006141285593 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7211012723073717 / 40000000000000000 : ℝ) ≤
+      (-(3426540102006141285593 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (7211012723073717 / 40000000000000000 : ℝ) =
+            (7211012723073717 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7211012723073717 / 5000000000000000 : ℝ) ≤
+            (732342979793858714407 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1787982195925921 / 10000000000000000 : ℝ)) (q :=
+      (7211012723073717 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_252_6 :
+    Uω (aρ 6) (bρ 6) (6878945929459 / 32000000000000 : ℝ) ≤
+      (-(18249705866105557760711 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6448895441245641 / 40000000000000000 : ℝ) ≤
+      (-(18249705866105557760711 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (6448895441245641 / 40000000000000000 : ℝ) =
+            (6448895441245641 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6448895441245641 / 5000000000000000 : ℝ) ≤
+            (2544709542894442239289 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1574163930151883 / 10000000000000000 : ℝ)) (q :=
+      (6448895441245641 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_252_7 :
+    Uω (aρ 7) (bρ 7) (6878945929459 / 32000000000000 : ℝ) ≤
+      (-(20181186557770656385763 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5316210819229457 / 40000000000000000 : ℝ) ≤
+      (-(20181186557770656385763 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (5316210819229457 / 40000000000000000 : ℝ) =
+            (5316210819229457 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5316210819229457 / 5000000000000000 : ℝ) ≤
+            (613228851229343614237 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1231482999903791 / 10000000000000000 : ℝ)) (q :=
+      (5316210819229457 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_252_8 :
+    Uω (aρ 8) (bρ 8) (6878945929459 / 32000000000000 : ℝ) ≤
+      (-(6179463379973671333507 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (675471839149047 / 8000000000000000 : ℝ) ≤
+      (-(6179463379973671333507 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (675471839149047 / 8000000000000000 : ℝ) =
+            (675471839149047 / 500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (675471839149047 / 500000000000000 : ℝ) ≤
+            (752008423026328666493 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6819985493771 / 125000000000000 : ℝ)) (q :=
+      (675471839149047 / 8000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_252_9 :
+    Uω (aρ 9) (bρ 9) (6878945929459 / 32000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_252_10 :
+    Uω (aρ 10) (bρ 10) (6878945929459 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_252_11 :
+    Uω (aρ 11) (bρ 11) (6878945929459 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_252_12 :
+    Uω (aρ 12) (bρ 12) (6878945929459 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_252_13 :
+    Uω (aρ 13) (bρ 13) (6878945929459 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_252_14 :
+    Uω (aρ 14) (bρ 14) (6878945929459 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_252_15 :
+    Uω (aρ 15) (bρ 15) (6878945929459 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_252_16 :
+    Uω (aρ 16) (bρ 16) (6878945929459 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_252 :
+    Uρ (6878945929459 / 32000000000000 : ℝ) ≤
+      (-(19010913855813137707793 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (6878945929459 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(7838851334268503496127 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_252_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (6878945929459 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3158894733825103070931 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_252_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (6878945929459 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(8016856921198687784309 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_252_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (6878945929459 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(8224642129649983668167 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_252_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (6878945929459 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(3426540102006141285593 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_252_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (6878945929459 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(18249705866105557760711 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_252_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (6878945929459 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(20181186557770656385763 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_252_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (6878945929459 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(6179463379973671333507 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_252_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (6878945929459 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_252_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (6878945929459 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_252_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (6878945929459 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_252_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (6878945929459 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_252_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (6878945929459 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_252_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (6878945929459 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_252_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (6878945929459 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_252_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (6878945929459 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_252_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_253_1 :
+    Uω (aρ 1) (bρ 1) (2758402395201 / 12800000000000 : ℝ) ≤
+      (-(31304331795127864129 / 20000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16723418527387889 / 80000000000000000 : ℝ) ≤
+      (-(31304331795127864129 / 20000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (16723418527387889 / 80000000000000000 : ℝ) =
+            (16723418527387889 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16723418527387889 / 10000000000000000 : ℝ) ≤
+            (10284499022872135871 / 20000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2090349978681191 / 10000000000000000 : ℝ)) (q :=
+      (16723418527387889 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_253_2 :
+    Uω (aρ 2) (bρ 2) (2758402395201 / 12800000000000 : ℝ) ≤
+      (-(3153726373802742696967 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16529777427688981 / 80000000000000000 : ℝ) ≤
+      (-(3153726373802742696967 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (16529777427688981 / 80000000000000000 : ℝ) =
+            (16529777427688981 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16529777427688981 / 10000000000000000 : ℝ) ≤
+            (1005156707997257303033 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (64553397383327 / 312500000000000 : ℝ)) (q :=
+      (16529777427688981 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_253_3 :
+    Uω (aρ 3) (bρ 3) (2758402395201 / 12800000000000 : ℝ) ≤
+      (-(16007229660263138625569 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16140048501191369 / 80000000000000000 : ℝ) ≤
+      (-(16007229660263138625569 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (16140048501191369 / 80000000000000000 : ℝ) =
+            (16140048501191369 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16140048501191369 / 10000000000000000 : ℝ) ≤
+            (4787185748736861374431 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2015672590992061 / 10000000000000000 : ℝ)) (q :=
+      (16140048501191369 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_253_4 :
+    Uω (aρ 4) (bρ 4) (2758402395201 / 12800000000000 : ℝ) ≤
+      (-(4105406294596253327 / 2500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3096976161338029 / 16000000000000000 : ℝ) ≤
+      (-(4105406294596253327 / 2500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3096976161338029 / 16000000000000000 : ℝ) =
+            (3096976161338029 / 2000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3096976161338029 / 2000000000000000 : ℝ) ≤
+            (1093197557653746673 / 2500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (386034969686351 / 2000000000000000 : ℝ)) (q :=
+      (3096976161338029 / 16000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_253_5 :
+    Uω (aρ 5) (bρ 5) (2758402395201 / 12800000000000 : ℝ) ≤
+      (-(4275732092910269981083 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14465026880361081 / 80000000000000000 : ℝ) ≤
+      (-(4275732092910269981083 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (14465026880361081 / 80000000000000000 : ℝ) =
+            (14465026880361081 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14465026880361081 / 10000000000000000 : ℝ) ≤
+            (922871759339730018917 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1793401286184489 / 10000000000000000 : ℝ)) (q :=
+      (14465026880361081 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_253_6 :
+    Uω (aρ 6) (bρ 6) (2758402395201 / 12800000000000 : ℝ) ≤
+      (-(9107949270071717081159 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12941468654379713 / 80000000000000000 : ℝ) ≤
+      (-(9107949270071717081159 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (12941468654379713 / 80000000000000000 : ℝ) =
+            (12941468654379713 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12941468654379713 / 10000000000000000 : ℝ) ≤
+            (1289258434428282918841 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1579752104829147 / 10000000000000000 : ℝ)) (q :=
+      (12941468654379713 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_253_7 :
+    Uω (aρ 7) (bρ 7) (2758402395201 / 12800000000000 : ℝ) ≤
+      (-(2013800325814052412029 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10678435222814689 / 80000000000000000 : ℝ) ≤
+      (-(2013800325814052412029 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (10678435222814689 / 80000000000000000 : ℝ) =
+            (10678435222814689 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10678435222814689 / 10000000000000000 : ℝ) ≤
+            (65641215085947587971 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1237655127697891 / 10000000000000000 : ℝ)) (q :=
+      (10678435222814689 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_253_8 :
+    Uω (aρ 8) (bρ 8) (2758402395201 / 12800000000000 : ℝ) ≤
+      (-(24621121171626664245687 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6820375414613733 / 80000000000000000 : ℝ) ≤
+      (-(24621121171626664245687 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (6820375414613733 / 80000000000000000 : ℝ) =
+            (6820375414613733 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6820375414613733 / 5000000000000000 : ℝ) ≤
+            (3104766040373335754313 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (139170456746913 / 2500000000000000 : ℝ)) (q :=
+      (6820375414613733 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_253_9 :
+    Uω (aρ 9) (bρ 9) (2758402395201 / 12800000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_253_10 :
+    Uω (aρ 10) (bρ 10) (2758402395201 / 12800000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_253_11 :
+    Uω (aρ 11) (bρ 11) (2758402395201 / 12800000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_253_12 :
+    Uω (aρ 12) (bρ 12) (2758402395201 / 12800000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_253_13 :
+    Uω (aρ 13) (bρ 13) (2758402395201 / 12800000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_253_14 :
+    Uω (aρ 14) (bρ 14) (2758402395201 / 12800000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_253_15 :
+    Uω (aρ 15) (bρ 15) (2758402395201 / 12800000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_253_16 :
+    Uω (aρ 16) (bρ 16) (2758402395201 / 12800000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_253 :
+    Uρ (2758402395201 / 12800000000000 : ℝ) ≤
+      (-(4747549936875150188239 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (2758402395201 / 12800000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(31304331795127864129 / 20000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_253_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (2758402395201 / 12800000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3153726373802742696967 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_253_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (2758402395201 / 12800000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(16007229660263138625569 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_253_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (2758402395201 / 12800000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(4105406294596253327 / 2500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_253_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (2758402395201 / 12800000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(4275732092910269981083 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_253_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (2758402395201 / 12800000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(9107949270071717081159 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_253_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (2758402395201 / 12800000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(2013800325814052412029 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_253_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (2758402395201 / 12800000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(24621121171626664245687 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_253_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (2758402395201 / 12800000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_253_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (2758402395201 / 12800000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_253_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (2758402395201 / 12800000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_253_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (2758402395201 / 12800000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_253_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (2758402395201 / 12800000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_253_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (2758402395201 / 12800000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_253_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (2758402395201 / 12800000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_253_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (2758402395201 / 12800000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_253_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_254_1 :
+    Uω (aρ 1) (bρ 1) (3456533023273 / 16000000000000 : ℝ) ≤
+      (-(15626694177986153223101 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (167660702476713 / 800000000000000 : ℝ) ≤
+      (-(15626694177986153223101 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (167660702476713 / 800000000000000 : ℝ) =
+            (167660702476713 / 100000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (167660702476713 / 100000000000000 : ℝ) ≤
+            (5167721231013846776899 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5239204101143 / 25000000000000 : ℝ)) (q :=
+      (167660702476713 / 800000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_254_2 :
+    Uω (aρ 2) (bρ 2) (3456533023273 / 16000000000000 : ℝ) ≤
+      (-(15742856709703212177573 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1035777384254401 / 5000000000000000 : ℝ) ≤
+      (-(15742856709703212177573 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1035777384254401 / 5000000000000000 : ℝ) =
+            (1035777384254401 / 625000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1035777384254401 / 625000000000000 : ℝ) ≤
+            (5051558699296787822427 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2071042628066979 / 10000000000000000 : ℝ)) (q :=
+      (1035777384254401 / 5000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_254_3 :
+    Uω (aρ 3) (bρ 3) (3456533023273 / 16000000000000 : ℝ) ≤
+      (-(1997601945127060592757 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (404568433500703 / 2000000000000000 : ℝ) ≤
+      (-(1997601945127060592757 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (404568433500703 / 2000000000000000 : ℝ) =
+            (404568433500703 / 250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (404568433500703 / 250000000000000 : ℝ) ≤
+            (601699980997939407243 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (404202706481281 / 2000000000000000 : ℝ)) (q :=
+      (404568433500703 / 2000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_254_4 :
+    Uω (aρ 4) (bρ 4) (3456533023273 / 16000000000000 : ℝ) ≤
+      (-(4098510704363691246301 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (30327442815377 / 156250000000000 : ℝ) ≤
+      (-(4098510704363691246301 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (30327442815377 / 156250000000000 : ℝ) = (30327442815377 / 19531250000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (30327442815377 / 19531250000000 : ℝ) ≤
+            (1100093147886308753699 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1935536058832631 / 10000000000000000 : ℝ)) (q :=
+      (30327442815377 / 156250000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_254_5 :
+    Uω (aρ 5) (bρ 5) (3456533023273 / 16000000000000 : ℝ) ≤
+      (-(4268311513824813753719 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1813503276894933 / 10000000000000000 : ℝ) ≤
+      (-(4268311513824813753719 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1813503276894933 / 10000000000000000 : ℝ) =
+            (1813503276894933 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1813503276894933 / 1250000000000000 : ℝ) ≤
+            (930292338425186246281 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1798819851589241 / 10000000000000000 : ℝ)) (q :=
+      (1813503276894933 / 10000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_254_6 :
+    Uω (aρ 6) (bρ 6) (3456533023273 / 16000000000000 : ℝ) ≤
+      (-(1818221057360102248173 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1623142418528303 / 10000000000000000 : ℝ) ≤
+      (-(1818221057360102248173 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1623142418528303 / 10000000000000000 : ℝ) =
+            (1623142418528303 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1623142418528303 / 1250000000000000 : ℝ) ≤
+            (261220483539897751827 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1585338509995981 / 10000000000000000 : ℝ)) (q :=
+      (1623142418528303 / 10000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_254_7 :
+    Uω (aρ 7) (bρ 7) (3456533023273 / 16000000000000 : ℝ) ≤
+      (-(10047517320976467956059 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1340552212587031 / 10000000000000000 : ℝ) ≤
+      (-(10047517320976467956059 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1340552212587031 / 10000000000000000 : ℝ) =
+            (1340552212587031 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1340552212587031 / 1250000000000000 : ℝ) ≤
+            (349690383523532043941 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1243819478873437 / 10000000000000000 : ℝ)) (q :=
+      (1340552212587031 / 10000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_254_8 :
+    Uω (aρ 8) (bρ 8) (3456533023273 / 16000000000000 : ℝ) ≤
+      (-(3065785211127408859203 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1680997852639 / 19531250000000 : ℝ) ≤
+      (-(3065785211127408859203 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (1680997852639 / 19531250000000 : ℝ) = (1680997852639 / 1220703125000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1680997852639 / 1220703125000 : ℝ) ≤
+            (399950690372591140797 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (567598506141711 / 10000000000000000 : ℝ)) (q :=
+      (1680997852639 / 19531250000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_254_9 :
+    Uω (aρ 9) (bρ 9) (3456533023273 / 16000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_254_10 :
+    Uω (aρ 10) (bρ 10) (3456533023273 / 16000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_254_11 :
+    Uω (aρ 11) (bρ 11) (3456533023273 / 16000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_254_12 :
+    Uω (aρ 12) (bρ 12) (3456533023273 / 16000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_254_13 :
+    Uω (aρ 13) (bρ 13) (3456533023273 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_254_14 :
+    Uω (aρ 14) (bρ 14) (3456533023273 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_254_15 :
+    Uω (aρ 15) (bρ 15) (3456533023273 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_254_16 :
+    Uω (aρ 16) (bρ 16) (3456533023273 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_254 :
+    Uρ (3456533023273 / 16000000000000 : ℝ) ≤
+      (-(9484848458057468868523 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (3456533023273 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(15626694177986153223101 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_254_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (3456533023273 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(15742856709703212177573 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_254_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (3456533023273 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1997601945127060592757 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_254_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (3456533023273 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(4098510704363691246301 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_254_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (3456533023273 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(4268311513824813753719 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_254_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (3456533023273 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1818221057360102248173 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_254_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (3456533023273 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(10047517320976467956059 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_254_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (3456533023273 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(3065785211127408859203 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_254_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (3456533023273 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_254_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (3456533023273 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_254_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (3456533023273 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_254_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (3456533023273 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_254_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (3456533023273 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_254_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (3456533023273 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_254_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (3456533023273 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_254_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (3456533023273 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_254_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_255_1 :
+    Uω (aρ 1) (bρ 1) (13860252210179 / 64000000000000 : ℝ) ≤
+      (-(3900321794800609828613 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16808721959966827 / 80000000000000000 : ℝ) ≤
+      (-(3900321794800609828613 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (16808721959966827 / 80000000000000000 : ℝ) =
+            (16808721959966827 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16808721959966827 / 10000000000000000 : ℝ) ≤
+            (1298282057449390171387 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1050506650118119 / 5000000000000000 : ℝ)) (q :=
+      (16808721959966827 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_255_2 :
+    Uω (aρ 2) (bρ 2) (13860252210179 / 64000000000000 : ℝ) ≤
+      (-(15717147848202477416551 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3323019762827807 / 16000000000000000 : ℝ) ≤
+      (-(15717147848202477416551 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3323019762827807 / 16000000000000000 : ℝ) =
+            (3323019762827807 / 2000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3323019762827807 / 2000000000000000 : ℝ) ≤
+            (5077267560797522583449 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (207637652628929 / 1000000000000000 : ℝ)) (q :=
+      (3323019762827807 / 16000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_255_3 :
+    Uω (aρ 3) (bρ 3) (13860252210179 / 64000000000000 : ℝ) ≤
+      (-(15954471174061227251973 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3245085195016607 / 16000000000000000 : ℝ) ≤
+      (-(15954471174061227251973 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3245085195016607 / 16000000000000000 : ℝ) =
+            (3245085195016607 / 2000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3245085195016607 / 2000000000000000 : ℝ) ≤
+            (4839944234938772748027 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (202635442287529 / 1000000000000000 : ℝ)) (q :=
+      (3245085195016607 / 16000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_255_4 :
+    Uω (aρ 4) (bρ 4) (13860252210179 / 64000000000000 : ℝ) ≤
+      (-(255727136714500718647 / 156250000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3114083995289391 / 16000000000000000 : ℝ) ≤
+      (-(255727136714500718647 / 156250000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3114083995289391 / 16000000000000000 : ℝ) =
+            (3114083995289391 / 2000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3114083995289391 / 2000000000000000 : ℝ) ≤
+            (69185604051124281353 / 156250000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (194089710428127 / 1000000000000000 : ℝ)) (q :=
+      (3114083995289391 / 16000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_255_5 :
+    Uω (aρ 5) (bρ 5) (13860252210179 / 64000000000000 : ℝ) ≤
+      (-(17043653012069059339703 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14551023469457759 / 80000000000000000 : ℝ) ≤
+      (-(17043653012069059339703 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (14551023469457759 / 80000000000000000 : ℝ) =
+            (14551023469457759 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14551023469457759 / 10000000000000000 : ℝ) ≤
+            (3750762396930940660297 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1804237896868971 / 10000000000000000 : ℝ)) (q :=
+      (14551023469457759 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_255_6 :
+    Uω (aρ 6) (bρ 6) (13860252210179 / 64000000000000 : ℝ) ≤
+      (-(18148641089420613638953 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2605760607718687 / 16000000000000000 : ℝ) ≤
+      (-(18148641089420613638953 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2605760607718687 / 16000000000000000 : ℝ) =
+            (2605760607718687 / 2000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2605760607718687 / 2000000000000000 : ℝ) ≤
+            (2645774319579386361047 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (159092316429289 / 1000000000000000 : ℝ)) (q :=
+      (2605760607718687 / 16000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_255_7 :
+    Uω (aρ 7) (bρ 7) (13860252210179 / 64000000000000 : ℝ) ≤
+      (-(5013069580125437245247 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10770369532315343 / 80000000000000000 : ℝ) ≤
+      (-(5013069580125437245247 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (10770369532315343 / 80000000000000000 : ℝ) =
+            (10770369532315343 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10770369532315343 / 10000000000000000 : ℝ) ≤
+            (185534272124562754753 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1249976168483367 / 10000000000000000 : ℝ)) (q :=
+      (10770369532315343 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_255_8 :
+    Uω (aρ 8) (bρ 8) (13860252210179 / 64000000000000 : ℝ) ≤
+      (-(6108309044709731099283 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6949731430595183 / 80000000000000000 : ℝ) ≤
+      (-(6108309044709731099283 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (6949731430595183 / 80000000000000000 : ℝ) =
+            (6949731430595183 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6949731430595183 / 5000000000000000 : ℝ) ≤
+            (823162758290268900717 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (578358294393327 / 10000000000000000 : ℝ)) (q :=
+      (6949731430595183 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_255_9 :
+    Uω (aρ 9) (bρ 9) (13860252210179 / 64000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_255_10 :
+    Uω (aρ 10) (bρ 10) (13860252210179 / 64000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_255_11 :
+    Uω (aρ 11) (bρ 11) (13860252210179 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_255_12 :
+    Uω (aρ 12) (bρ 12) (13860252210179 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_255_13 :
+    Uω (aρ 13) (bρ 13) (13860252210179 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_255_14 :
+    Uω (aρ 14) (bρ 14) (13860252210179 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_255_15 :
+    Uω (aρ 15) (bρ 15) (13860252210179 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_255_16 :
+    Uω (aρ 16) (bρ 16) (13860252210179 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_255 :
+    Uρ (13860252210179 / 64000000000000 : ℝ) ≤
+      (-(18949396255775099006671 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (13860252210179 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3900321794800609828613 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_255_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (13860252210179 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(15717147848202477416551 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_255_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (13860252210179 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(15954471174061227251973 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_255_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (13860252210179 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(255727136714500718647 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_255_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (13860252210179 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(17043653012069059339703 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_255_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (13860252210179 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(18148641089420613638953 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_255_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (13860252210179 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(5013069580125437245247 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_255_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (13860252210179 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(6108309044709731099283 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_255_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (13860252210179 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_255_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (13860252210179 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_255_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (13860252210179 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_255_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (13860252210179 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_255_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (13860252210179 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_255_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (13860252210179 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_255_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (13860252210179 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_255_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (13860252210179 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_255_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/U21.lean b/LeanPool/Zeta5Irrational/Table/U21.lean
new file mode 100644
index 0000000000..654ba4ce2b
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/U21.lean
@@ -0,0 +1,3714 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-! # Certified arcsine potential bounds (U21) -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma U_256_1 :
+    Uω (aρ 1) (bρ 1) (6947186163633 / 32000000000000 : ℝ) ≤
+      (-(15575944573125605674177 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8425686832167563 / 40000000000000000 : ℝ) ≤
+      (-(15575944573125605674177 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (8425686832167563 / 40000000000000000 : ℝ) =
+            (8425686832167563 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8425686832167563 / 5000000000000000 : ℝ) ≤
+            (5218470835874394325823 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2106344958033469 / 10000000000000000 : ℝ)) (q :=
+      (8425686832167563 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_256_2 :
+    Uω (aρ 2) (bρ 2) (6947186163633 / 32000000000000 : ℝ) ≤
+      (-(245179764752538628203 / 156250000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8328879713156163 / 40000000000000000 : ℝ) ≤
+      (-(245179764752538628203 / 156250000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (8328879713156163 / 40000000000000000 : ℝ) =
+            (8328879713156163 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8328879713156163 / 5000000000000000 : ℝ) ≤
+            (79732976013086371797 / 156250000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2081710411037769 / 10000000000000000 : ℝ)) (q :=
+      (8328879713156163 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_256_3 :
+    Uω (aρ 3) (bρ 3) (6947186163633 / 32000000000000 : ℝ) ≤
+      (-(995512258233638353213 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8134057203981601 / 40000000000000000 : ℝ) ≤
+      (-(995512258233638353213 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (8134057203981601 / 40000000000000000 : ℝ) =
+            (8134057203981601 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8134057203981601 / 5000000000000000 : ℝ) ≤
+            (304138704828861646787 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (253961907850061 / 1250000000000000 : ℝ)) (q :=
+      (8134057203981601 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_256_4 :
+    Uω (aρ 4) (bρ 4) (6947186163633 / 32000000000000 : ℝ) ≤
+      (-(8169553275997104438517 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7806594288532177 / 40000000000000000 : ℝ) ≤
+      (-(8169553275997104438517 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (7806594288532177 / 40000000000000000 : ℝ) =
+            (7806594288532177 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7806594288532177 / 5000000000000000 : ℝ) ≤
+            (2227654428502895561483 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (243282248267597 / 1250000000000000 : ℝ)) (q :=
+      (7806594288532177 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_256_5 :
+    Uω (aρ 5) (bρ 5) (6947186163633 / 32000000000000 : ℝ) ≤
+      (-(136113189584853233173 / 80000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1459401866194279 / 8000000000000000 : ℝ) ≤
+      (-(136113189584853233173 / 80000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1459401866194279 / 8000000000000000 : ℝ) =
+            (1459401866194279 / 1000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1459401866194279 / 1000000000000000 : ℝ) ≤
+            (30242133687146766827 / 80000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (361931085339077 / 2000000000000000 : ℝ)) (q :=
+      (1459401866194279 / 8000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_256_6 :
+    Uω (aρ 6) (bρ 6) (6947186163633 / 32000000000000 : ℝ) ≤
+      (-(18115189220547125073919 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6536229899488127 / 40000000000000000 : ℝ) ≤
+      (-(18115189220547125073919 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (6536229899488127 / 40000000000000000 : ℝ) =
+            (6536229899488127 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6536229899488127 / 5000000000000000 : ℝ) ≤
+            (2679226188452874926081 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1596506086093751 / 10000000000000000 : ℝ)) (q :=
+      (6536229899488127 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_256_7 :
+    Uω (aρ 7) (bρ 7) (6947186163633 / 32000000000000 : ℝ) ≤
+      (-(10004865974050945407887 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1081629116829469 / 8000000000000000 : ℝ) ≤
+      (-(10004865974050945407887 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1081629116829469 / 8000000000000000 : ℝ) =
+            (1081629116829469 / 1000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1081629116829469 / 1000000000000000 : ℝ) ≤
+            (392341730449054592113 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1962695795599 / 15625000000000 : ℝ)) (q :=
+      (1081629116829469 / 8000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_256_8 :
+    Uω (aρ 8) (bρ 8) (6947186163633 / 32000000000000 : ℝ) ≤
+      (-(24341894157151895026913 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3506751243989351 / 40000000000000000 : ℝ) ≤
+      (-(24341894157151895026913 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (3506751243989351 / 40000000000000000 : ℝ) =
+            (3506751243989351 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3506751243989351 / 2500000000000000 : ℝ) ≤
+            (3383993054848104973087 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (588969790444363 / 10000000000000000 : ℝ)) (q :=
+      (3506751243989351 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_256_9 :
+    Uω (aρ 9) (bρ 9) (6947186163633 / 32000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_256_10 :
+    Uω (aρ 10) (bρ 10) (6947186163633 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_256_11 :
+    Uω (aρ 11) (bρ 11) (6947186163633 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_256_12 :
+    Uω (aρ 12) (bρ 12) (6947186163633 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_256_13 :
+    Uω (aρ 13) (bρ 13) (6947186163633 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_256_14 :
+    Uω (aρ 14) (bρ 14) (6947186163633 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_256_15 :
+    Uω (aρ 15) (bρ 15) (6947186163633 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_256_16 :
+    Uω (aρ 16) (bρ 16) (6947186163633 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_256 :
+    Uρ (6947186163633 / 32000000000000 : ℝ) ≤
+      (-(4732322352294633080727 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (6947186163633 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(15575944573125605674177 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_256_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (6947186163633 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(245179764752538628203 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_256_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (6947186163633 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(995512258233638353213 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_256_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (6947186163633 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(8169553275997104438517 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_256_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (6947186163633 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(136113189584853233173 / 80000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_256_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (6947186163633 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(18115189220547125073919 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_256_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (6947186163633 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(10004865974050945407887 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_256_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (6947186163633 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(24341894157151895026913 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_256_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (6947186163633 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_256_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (6947186163633 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_256_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (6947186163633 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_256_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (6947186163633 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_256_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (6947186163633 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_256_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (6947186163633 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_256_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (6947186163633 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_256_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (6947186163633 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_256_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_257_1 :
+    Uω (aρ 1) (bρ 1) (13928492444353 / 64000000000000 : ℝ) ≤
+      (-(15550666034157081233191 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16894025360836249 / 80000000000000000 : ℝ) ≤
+      (-(15550666034157081233191 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (16894025360836249 / 80000000000000000 : ℝ) =
+            (16894025360836249 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16894025360836249 / 10000000000000000 : ℝ) ≤
+            (5243749374842918766809 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1055838306931953 / 5000000000000000 : ℝ)) (q :=
+      (16894025360836249 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_257_2 :
+    Uω (aρ 2) (bρ 2) (13928492444353 / 64000000000000 : ℝ) ≤
+      (-(7832963829924780234651 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16700419985003521 / 80000000000000000 : ℝ) ≤
+      (-(7832963829924780234651 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (16700419985003521 / 80000000000000000 : ℝ) =
+            (16700419985003521 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16700419985003521 / 10000000000000000 : ℝ) ≤
+            (2564243874575219765349 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (521761070603931 / 2500000000000000 : ℝ)) (q :=
+      (16700419985003521 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_257_3 :
+    Uω (aρ 3) (bρ 3) (13928492444353 / 64000000000000 : ℝ) ≤
+      (-(15901990069294298442933 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16310802640258853 / 80000000000000000 : ℝ) ≤
+      (-(15901990069294298442933 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (16310802640258853 / 80000000000000000 : ℝ) =
+            (16310802640258853 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16310802640258853 / 10000000000000000 : ℝ) ≤
+            (4892425339705701557067 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2037036052579557 / 10000000000000000 : ℝ)) (q :=
+      (16310802640258853 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_257_4 :
+    Uω (aρ 4) (bρ 4) (13928492444353 / 64000000000000 : ℝ) ≤
+      (-(16311751804573218956949 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15655956528717557 / 80000000000000000 : ℝ) ≤
+      (-(16311751804573218956949 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (15655956528717557 / 80000000000000000 : ℝ) =
+            (15655956528717557 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15655956528717557 / 10000000000000000 : ℝ) ≤
+            (4482663604426781043051 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1951618705759233 / 10000000000000000 : ℝ)) (q :=
+      (15655956528717557 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_257_5 :
+    Uω (aρ 5) (bρ 5) (13928492444353 / 64000000000000 : ℝ) ≤
+      (-(16984732574596841281701 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (585480472443061 / 3200000000000000 : ℝ) ≤
+      (-(16984732574596841281701 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (585480472443061 / 3200000000000000 : ℝ) =
+            (585480472443061 / 400000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (585480472443061 / 400000000000000 : ℝ) ≤
+            (3809682834403158718299 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (72602897827359 / 400000000000000 : ℝ)) (q :=
+      (585480472443061 / 3200000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_257_6 :
+    Uω (aρ 6) (bρ 6) (13928492444353 / 64000000000000 : ℝ) ≤
+      (-(18081854109771723636757 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13116109701823297 / 80000000000000000 : ℝ) ≤
+      (-(18081854109771723636757 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (13116109701823297 / 80000000000000000 : ℝ) =
+            (13116109701823297 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13116109701823297 / 10000000000000000 : ℝ) ≤
+            (2712561299228276363243 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (400521823377667 / 2500000000000000 : ℝ)) (q :=
+      (13116109701823297 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_257_7 :
+    Uω (aρ 7) (bρ 7) (13928492444353 / 64000000000000 : ℝ) ≤
+      (-(19967393221007081062849 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10862183049928981 / 80000000000000000 : ℝ) ≤
+      (-(19967393221007081062849 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (10862183049928981 / 80000000000000000 : ℝ) =
+            (10862183049928981 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10862183049928981 / 10000000000000000 : ℝ) ≤
+            (827022187992918937151 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1262267011297089 / 10000000000000000 : ℝ)) (q :=
+      (10862183049928981 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_257_8 :
+    Uω (aρ 8) (bρ 8) (13928492444353 / 64000000000000 : ℝ) ≤
+      (-(4850434515589518544511 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7076711878164501 / 80000000000000000 : ℝ) ≤
+      (-(4850434515589518544511 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (7076711878164501 / 80000000000000000 : ℝ) =
+            (7076711878164501 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7076711878164501 / 5000000000000000 : ℝ) ≤
+            (694742926810481455489 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (599440869695969 / 10000000000000000 : ℝ)) (q :=
+      (7076711878164501 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_257_9 :
+    Uω (aρ 9) (bρ 9) (13928492444353 / 64000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_257_10 :
+    Uω (aρ 10) (bρ 10) (13928492444353 / 64000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_257_11 :
+    Uω (aρ 11) (bρ 11) (13928492444353 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_257_12 :
+    Uω (aρ 12) (bρ 12) (13928492444353 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_257_13 :
+    Uω (aρ 13) (bρ 13) (13928492444353 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_257_14 :
+    Uω (aρ 14) (bρ 14) (13928492444353 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_257_15 :
+    Uω (aρ 15) (bρ 15) (13928492444353 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_257_16 :
+    Uω (aρ 16) (bρ 16) (13928492444353 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_257 :
+    Uρ (13928492444353 / 64000000000000 : ℝ) ≤
+      (-(590917771300687400017 / 312500000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (13928492444353 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(15550666034157081233191 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_257_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (13928492444353 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(7832963829924780234651 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_257_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (13928492444353 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(15901990069294298442933 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_257_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (13928492444353 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(16311751804573218956949 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_257_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (13928492444353 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(16984732574596841281701 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_257_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (13928492444353 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(18081854109771723636757 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_257_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (13928492444353 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(19967393221007081062849 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_257_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (13928492444353 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(4850434515589518544511 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_257_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (13928492444353 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_257_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (13928492444353 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_257_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (13928492444353 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_257_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (13928492444353 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_257_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (13928492444353 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_257_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (13928492444353 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_257_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (13928492444353 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_257_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (13928492444353 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_257_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_258_1 :
+    Uω (aρ 1) (bρ 1) (87266328509 / 400000000000 : ℝ) ≤
+      (-(1552545123916181522009 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4234169262382407 / 20000000000000000 : ℝ) ≤
+      (-(1552545123916181522009 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4234169262382407 / 20000000000000000 : ℝ) =
+            (4234169262382407 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4234169262382407 / 2500000000000000 : ℝ) ≤
+            (526896416983818477991 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2117008267742407 / 10000000000000000 : ℝ)) (q :=
+      (4234169262382407 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_258_2 :
+    Uω (aρ 2) (bρ 2) (87266328509 / 400000000000 : ℝ) ≤
+      (-(15640415660118783464449 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (837154024531081 / 4000000000000000 : ℝ) ≤
+      (-(15640415660118783464449 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (837154024531081 / 4000000000000000 : ℝ) =
+            (837154024531081 / 500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (837154024531081 / 500000000000000 : ℝ) ≤
+            (5153999748881216535551 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (418475628105081 / 2000000000000000 : ℝ)) (q :=
+      (837154024531081 / 4000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_258_3 :
+    Uω (aρ 3) (bρ 3) (87266328509 / 400000000000 : ℝ) ≤
+      (-(15875852624851757203039 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4088372668385891 / 20000000000000000 : ℝ) ≤
+      (-(15875852624851757203039 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4088372668385891 / 20000000000000000 : ℝ) =
+            (4088372668385891 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4088372668385891 / 2500000000000000 : ℝ) ≤
+            (4918562784148242796961 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2042376792605891 / 10000000000000000 : ℝ)) (q :=
+      (4088372668385891 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_258_4 :
+    Uω (aρ 4) (bρ 4) (87266328509 / 400000000000 : ℝ) ≤
+      (-(16284472091276283593661 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3924680959184913 / 20000000000000000 : ℝ) ≤
+      (-(16284472091276283593661 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3924680959184913 / 20000000000000000 : ℝ) =
+            (3924680959184913 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3924680959184913 / 2500000000000000 : ℝ) ≤
+            (4509943317723716406339 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1956979264469913 / 10000000000000000 : ℝ)) (q :=
+      (3924680959184913 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_258_5 :
+    Uω (aρ 5) (bρ 5) (87266328509 / 400000000000 : ℝ) ≤
+      (-(1695540410769040563281 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1835000366887433 / 10000000000000000 : ℝ) ≤
+      (-(1695540410769040563281 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1835000366887433 / 10000000000000000 : ℝ) =
+            (1835000366887433 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1835000366887433 / 1250000000000000 : ℝ) ≤
+            (383901130130959436719 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (910244479197433 / 5000000000000000 : ℝ)) (q :=
+      (1835000366887433 / 10000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_258_6 :
+    Uω (aρ 6) (bρ 6) (87266328509 / 400000000000 : ℝ) ≤
+      (-(18048634909578610900489 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1644969102319353 / 10000000000000000 : ℝ) ≤
+      (-(18048634909578610900489 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1644969102319353 / 10000000000000000 : ℝ) =
+            (1644969102319353 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1644969102319353 / 1250000000000000 : ℝ) ≤
+            (2745780499421389099511 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (803833402199353 / 5000000000000000 : ℝ)) (q :=
+      (1644969102319353 / 10000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_258_7 :
+    Uω (aρ 7) (bρ 7) (87266328509 / 400000000000 : ℝ) ≤
+      (-(19925259876362511462543 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (272701140235927 / 2000000000000000 : ℝ) ≤
+      (-(19925259876362511462543 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (272701140235927 / 2000000000000000 : ℝ) =
+            (272701140235927 / 250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (272701140235927 / 250000000000000 : ℝ) ≤
+            (869155532637488537457 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (126840138287927 / 1000000000000000 : ℝ)) (q :=
+      (272701140235927 / 2000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_258_8 :
+    Uω (aρ 8) (bρ 8) (87266328509 / 400000000000 : ℝ) ≤
+      (-(2416399500157657047719 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1784847134014071 / 20000000000000000 : ℝ) ≤
+      (-(2416399500157657047719 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1784847134014071 / 20000000000000000 : ℝ) =
+            (1784847134014071 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1784847134014071 / 1250000000000000 : ℝ) ≤
+            (356189221042342952281 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (609778765874071 / 10000000000000000 : ℝ)) (q :=
+      (1784847134014071 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_258_9 :
+    Uω (aρ 9) (bρ 9) (87266328509 / 400000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_258_10 :
+    Uω (aρ 10) (bρ 10) (87266328509 / 400000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_258_11 :
+    Uω (aρ 11) (bρ 11) (87266328509 / 400000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_258_12 :
+    Uω (aρ 12) (bρ 12) (87266328509 / 400000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_258_13 :
+    Uω (aρ 13) (bρ 13) (87266328509 / 400000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_258_14 :
+    Uω (aρ 14) (bρ 14) (87266328509 / 400000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_258_15 :
+    Uω (aρ 15) (bρ 15) (87266328509 / 400000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_258_16 :
+    Uω (aρ 16) (bρ 16) (87266328509 / 400000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_258 :
+    Uρ (87266328509 / 400000000000 : ℝ) ≤
+      (-(9444813483692401198133 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (87266328509 / 400000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1552545123916181522009 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_258_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (87266328509 / 400000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(15640415660118783464449 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_258_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (87266328509 / 400000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(15875852624851757203039 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_258_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (87266328509 / 400000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(16284472091276283593661 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_258_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (87266328509 / 400000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1695540410769040563281 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_258_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (87266328509 / 400000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(18048634909578610900489 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_258_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (87266328509 / 400000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(19925259876362511462543 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_258_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (87266328509 / 400000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(2416399500157657047719 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_258_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (87266328509 / 400000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_258_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (87266328509 / 400000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_258_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (87266328509 / 400000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_258_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (87266328509 / 400000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_258_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (87266328509 / 400000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_258_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (87266328509 / 400000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_258_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (87266328509 / 400000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_258_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (87266328509 / 400000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_258_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_259_1 :
+    Uω (aρ 1) (bρ 1) (13996732678527 / 64000000000000 : ℝ) ≤
+      (-(15500299867443468815179 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16979328730474111 / 80000000000000000 : ℝ) ≤
+      (-(15500299867443468815179 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (16979328730474111 / 80000000000000000 : ℝ) =
+            (16979328730474111 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16979328730474111 / 10000000000000000 : ℝ) ≤
+            (5294115541556531184821 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (530584979920921 / 2500000000000000 : ℝ)) (q :=
+      (16979328730474111 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_259_2 :
+    Uω (aρ 2) (bρ 2) (13996732678527 / 64000000000000 : ℝ) ≤
+      (-(15614968612387440695619 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (671429637742859 / 3200000000000000 : ℝ) ≤
+      (-(15614968612387440695619 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (671429637742859 / 3200000000000000 : ℝ) =
+            (671429637742859 / 400000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (671429637742859 / 400000000000000 : ℝ) ≤
+            (5179446796612559304381 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (83908479418721 / 400000000000000 : ℝ)) (q :=
+      (671429637742859 / 3200000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_259_3 :
+    Uω (aρ 3) (bρ 3) (13996732678527 / 64000000000000 : ℝ) ≤
+      (-(15849783439378083627973 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16396178509374483 / 80000000000000000 : ℝ) ≤
+      (-(15849783439378083627973 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (16396178509374483 / 80000000000000000 : ℝ) =
+            (16396178509374483 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16396178509374483 / 10000000000000000 : ℝ) ≤
+            (4944631969621916372027 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2047717483268777 / 10000000000000000 : ℝ)) (q :=
+      (16396178509374483 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_259_4 :
+    Uω (aρ 4) (bρ 4) (13996732678527 / 64000000000000 : ℝ) ≤
+      (-(16257266999367032878831 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3148298101281059 / 16000000000000000 : ℝ) ≤
+      (-(16257266999367032878831 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3148298101281059 / 16000000000000000 : ℝ) =
+            (3148298101281059 / 2000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3148298101281059 / 2000000000000000 : ℝ) ≤
+            (4537148409632967121169 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (49058491589787 / 250000000000000 : ℝ)) (q :=
+      (3148298101281059 / 16000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_259_5 :
+    Uω (aρ 5) (bρ 5) (13996732678527 / 64000000000000 : ℝ) ≤
+      (-(4231540692110611749079 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14722992052033959 / 80000000000000000 : ℝ) ≤
+      (-(4231540692110611749079 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (14722992052033959 / 80000000000000000 : ℝ) =
+            (14722992052033959 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14722992052033959 / 10000000000000000 : ℝ) ≤
+            (967063160139388250921 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (912952484666823 / 5000000000000000 : ℝ)) (q :=
+      (14722992052033959 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_259_6 :
+    Uω (aρ 6) (bρ 6) (13996732678527 / 64000000000000 : ℝ) ≤
+      (-(9007765390997410364347 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13203389219581499 / 80000000000000000 : ℝ) ≤
+      (-(9007765390997410364347 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (13203389219581499 / 80000000000000000 : ℝ) =
+            (13203389219581499 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13203389219581499 / 10000000000000000 : ℝ) ≤
+            (1389442313502589635653 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1613244636360531 / 10000000000000000 : ℝ)) (q :=
+      (13203389219581499 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_259_7 :
+    Uω (aρ 7) (bρ 7) (13996732678527 / 64000000000000 : ℝ) ≤
+      (-(9941664845595572648541 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10953879270205719 / 80000000000000000 : ℝ) ≤
+      (-(9941664845595572648541 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (10953879270205719 / 80000000000000000 : ℝ) =
+            (10953879270205719 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10953879270205719 / 10000000000000000 : ℝ) ≤
+            (455542858904427351459 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (637264264888293 / 5000000000000000 : ℝ)) (q :=
+      (10953879270205719 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_259_8 :
+    Uω (aρ 8) (bρ 8) (13996732678527 / 64000000000000 : ℝ) ≤
+      (-(1203864543957815277543 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7201559110597499 / 80000000000000000 : ℝ) ≤
+      (-(1203864543957815277543 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (7201559110597499 / 80000000000000000 : ℝ) =
+            (7201559110597499 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7201559110597499 / 5000000000000000 : ℝ) ≤
+            (182429816642184722457 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (619990141214531 / 10000000000000000 : ℝ)) (q :=
+      (7201559110597499 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_259_9 :
+    Uω (aρ 9) (bρ 9) (13996732678527 / 64000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_259_10 :
+    Uω (aρ 10) (bρ 10) (13996732678527 / 64000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_259_11 :
+    Uω (aρ 11) (bρ 11) (13996732678527 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_259_12 :
+    Uω (aρ 12) (bρ 12) (13996732678527 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_259_13 :
+    Uω (aρ 13) (bρ 13) (13996732678527 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_259_14 :
+    Uω (aρ 14) (bρ 14) (13996732678527 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_259_15 :
+    Uω (aρ 15) (bρ 15) (13996732678527 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_259_16 :
+    Uω (aρ 16) (bρ 16) (13996732678527 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_259 :
+    Uρ (13996732678527 / 64000000000000 : ℝ) ≤
+      (-(18870057686366365844573 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (13996732678527 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(15500299867443468815179 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_259_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (13996732678527 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(15614968612387440695619 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_259_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (13996732678527 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(15849783439378083627973 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_259_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (13996732678527 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(16257266999367032878831 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_259_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (13996732678527 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(4231540692110611749079 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_259_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (13996732678527 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(9007765390997410364347 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_259_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (13996732678527 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(9941664845595572648541 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_259_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (13996732678527 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(1203864543957815277543 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_259_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (13996732678527 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_259_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (13996732678527 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_259_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (13996732678527 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_259_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (13996732678527 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_259_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (13996732678527 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_259_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (13996732678527 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_259_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (13996732678527 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_259_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (13996732678527 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_259_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_260_1 :
+    Uω (aρ 1) (bρ 1) (7015426397807 / 32000000000000 : ℝ) ≤
+      (-(154752116007199462903 / 100000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (340439608074559 / 1600000000000000 : ℝ) ≤
+      (-(154752116007199462903 / 100000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (340439608074559 / 1600000000000000 : ℝ) =
+            (340439608074559 / 200000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (340439608074559 / 200000000000000 : ℝ) ≤
+            (53192038082800537097 / 100000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (21276715697023 / 100000000000000 : ℝ)) (q :=
+      (340439608074559 / 1600000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_260_2 :
+    Uω (aρ 2) (bρ 2) (7015426397807 / 32000000000000 : ℝ) ≤
+      (-(1948698273326129675261 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8414200672126911 / 40000000000000000 : ℝ) ≤
+      (-(1948698273326129675261 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (8414200672126911 / 40000000000000000 : ℝ) =
+            (8414200672126911 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8414200672126911 / 5000000000000000 : ℝ) ≤
+            (650603652798870324739 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (262880727167971 / 1250000000000000 : ℝ)) (q :=
+      (8414200672126911 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_260_3 :
+    Uω (aρ 3) (bρ 3) (7015426397807 / 32000000000000 : ℝ) ≤
+      (-(3955945539164048809487 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8219433074646267 / 40000000000000000 : ℝ) ≤
+      (-(3955945539164048809487 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (8219433074646267 / 40000000000000000 : ℝ) =
+            (8219433074646267 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8219433074646267 / 5000000000000000 : ℝ) ≤
+            (1242658313085951190513 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1026529062476723 / 5000000000000000 : ℝ)) (q :=
+      (8219433074646267 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_260_4 :
+    Uω (aρ 4) (bρ 4) (7015426397807 / 32000000000000 : ℝ) ≤
+      (-(1623013611952324522931 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7892128271465753 / 40000000000000000 : ℝ) ≤
+      (-(1623013611952324522931 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (7892128271465753 / 40000000000000000 : ℝ) =
+            (7892128271465753 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7892128271465753 / 5000000000000000 : ℝ) ≤
+            (456427928947675477069 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1967699904428189 / 10000000000000000 : ℝ)) (q :=
+      (7892128271465753 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_260_5 :
+    Uω (aρ 5) (bρ 5) (7015426397807 / 32000000000000 : ℝ) ≤
+      (-(16897008032751650925741 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7382989589843757 / 40000000000000000 : ℝ) ≤
+      (-(16897008032751650925741 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (7382989589843757 / 40000000000000000 : ℝ) =
+            (7382989589843757 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7382989589843757 / 5000000000000000 : ℝ) ≤
+            (3897407376248349074259 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1831320482952191 / 10000000000000000 : ℝ)) (q :=
+      (7382989589843757 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_260_6 :
+    Uω (aρ 6) (bρ 6) (7015426397807 / 32000000000000 : ℝ) ≤
+      (-(17982540898443015669829 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6623509487161253 / 40000000000000000 : ℝ) ≤
+      (-(17982540898443015669829 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (6623509487161253 / 40000000000000000 : ℝ) =
+            (6623509487161253 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6623509487161253 / 5000000000000000 : ℝ) ≤
+            (2811874510556984330171 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1618820806750939 / 10000000000000000 : ℝ)) (q :=
+      (6623509487161253 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_260_7 :
+    Uω (aρ 7) (bρ 7) (7015426397807 / 32000000000000 : ℝ) ≤
+      (-(19841600481412250417977 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5499842223512557 / 40000000000000000 : ℝ) ≤
+      (-(19841600481412250417977 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (5499842223512557 / 40000000000000000 : ℝ) =
+            (5499842223512557 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5499842223512557 / 5000000000000000 : ℝ) ≤
+            (952814927587749582023 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1280648555686591 / 10000000000000000 : ℝ)) (q :=
+      (5499842223512557 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_260_8 :
+    Uω (aρ 8) (bρ 8) (7015426397807 / 32000000000000 : ℝ) ≤
+      (-(959679797350474912757 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3631624103671967 / 40000000000000000 : ℝ) ≤
+      (-(959679797350474912757 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (3631624103671967 / 40000000000000000 : ℝ) =
+            (3631624103671967 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3631624103671967 / 2500000000000000 : ℝ) ≤
+            (149355691129525087243 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (78760143388287 / 1250000000000000 : ℝ)) (q :=
+      (3631624103671967 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_260_9 :
+    Uω (aρ 9) (bρ 9) (7015426397807 / 32000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_260_10 :
+    Uω (aρ 10) (bρ 10) (7015426397807 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_260_11 :
+    Uω (aρ 11) (bρ 11) (7015426397807 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_260_12 :
+    Uω (aρ 12) (bρ 12) (7015426397807 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_260_13 :
+    Uω (aρ 13) (bρ 13) (7015426397807 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_260_14 :
+    Uω (aρ 14) (bρ 14) (7015426397807 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_260_15 :
+    Uω (aρ 15) (bρ 15) (7015426397807 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_260_16 :
+    Uω (aρ 16) (bρ 16) (7015426397807 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_260 :
+    Uρ (7015426397807 / 32000000000000 : ℝ) ≤
+      (-(18850654729282272068329 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (7015426397807 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(154752116007199462903 / 100000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_260_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (7015426397807 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1948698273326129675261 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_260_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (7015426397807 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3955945539164048809487 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_260_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (7015426397807 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1623013611952324522931 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_260_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (7015426397807 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(16897008032751650925741 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_260_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (7015426397807 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(17982540898443015669829 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_260_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (7015426397807 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(19841600481412250417977 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_260_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (7015426397807 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(959679797350474912757 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_260_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (7015426397807 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_260_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (7015426397807 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_260_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (7015426397807 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_260_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (7015426397807 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_260_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (7015426397807 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_260_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (7015426397807 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_260_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (7015426397807 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_260_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (7015426397807 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_260_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_261_1 :
+    Uω (aρ 1) (bρ 1) (14064972912701 / 64000000000000 : ℝ) ≤
+      (-(7725093061549607964581 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17064632069348817 / 80000000000000000 : ℝ) ≤
+      (-(7725093061549607964581 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (17064632069348817 / 80000000000000000 : ℝ) =
+            (17064632069348817 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17064632069348817 / 10000000000000000 : ℝ) ≤
+            (2672114642950392035419 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2133003217812673 / 10000000000000000 : ℝ)) (q :=
+      (17064632069348817 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_261_2 :
+    Uω (aρ 2) (bρ 2) (14064972912701 / 64000000000000 : ℝ) ≤
+      (-(622570722209901726493 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3374212338613069 / 16000000000000000 : ℝ) ≤
+      (-(622570722209901726493 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3374212338613069 / 16000000000000000 : ℝ) =
+            (3374212338613069 / 2000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3374212338613069 / 2000000000000000 : ℝ) ≤
+            (209205894150098273507 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (421675927250361 / 2000000000000000 : ℝ)) (q :=
+      (3374212338613069 / 16000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_261_3 :
+    Uω (aρ 3) (bρ 3) (14064972912701 / 64000000000000 : ℝ) ≤
+      (-(157978484232550097711 / 100000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16481553594822649 / 80000000000000000 : ℝ) ≤
+      (-(157978484232550097711 / 100000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (16481553594822649 / 80000000000000000 : ℝ) =
+            (16481553594822649 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16481553594822649 / 10000000000000000 : ℝ) ≤
+            (49965669857449902289 / 100000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2058398718041131 / 10000000000000000 : ℝ)) (q :=
+      (16481553594822649 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_261_4 :
+    Uω (aρ 4) (bρ 4) (14064972912701 / 64000000000000 : ℝ) ≤
+      (-(506346220181221679599 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15827021951478469 / 80000000000000000 : ℝ) ≤
+      (-(506346220181221679599 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (15827021951478469 / 80000000000000000 : ℝ) =
+            (15827021951478469 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15827021951478469 / 10000000000000000 : ℝ) ≤
+            (143479261350028320401 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (986529994135043 / 5000000000000000 : ℝ)) (q :=
+      (15827021951478469 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_261_5 :
+    Uω (aρ 5) (bρ 5) (14064972912701 / 64000000000000 : ℝ) ≤
+      (-(16867939381300538389631 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14808964335655981 / 80000000000000000 : ℝ) ≤
+      (-(16867939381300538389631 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (14808964335655981 / 80000000000000000 : ℝ) =
+            (14808964335655981 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14808964335655981 / 10000000000000000 : ℝ) ≤
+            (3926476027699461610369 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (229591937956183 / 1250000000000000 : ℝ)) (q :=
+      (14808964335655981 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_261_6 :
+    Uω (aρ 6) (bρ 6) (14064972912701 / 64000000000000 : ℝ) ≤
+      (-(17949664439597163688407 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13290642151223293 / 80000000000000000 : ℝ) ≤
+      (-(17949664439597163688407 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (13290642151223293 / 80000000000000000 : ℝ) =
+            (13290642151223293 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13290642151223293 / 10000000000000000 : ℝ) ≤
+            (2844750969402836311593 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (406098833170323 / 2500000000000000 : ℝ)) (q :=
+      (13290642151223293 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_261_7 :
+    Uω (aρ 7) (bρ 7) (14064972912701 / 64000000000000 : ℝ) ≤
+      (-(4950017525222746901699 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2209092309263349 / 16000000000000000 : ℝ) ≤
+      (-(4950017525222746901699 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2209092309263349 / 16000000000000000 : ℝ) =
+            (2209092309263349 / 2000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2209092309263349 / 2000000000000000 : ℝ) ≤
+            (248586327027253098301 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (257352312442931 / 2000000000000000 : ℝ)) (q :=
+      (2209092309263349 / 16000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_261_8 :
+    Uω (aρ 8) (bρ 8) (14064972912701 / 64000000000000 : ℝ) ≤
+      (-(23908046622860256109079 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7324478599017021 / 80000000000000000 : ℝ) ≤
+      (-(23908046622860256109079 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (7324478599017021 / 80000000000000000 : ℝ) =
+            (7324478599017021 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7324478599017021 / 5000000000000000 : ℝ) ≤
+            (3817840589139743890921 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (160014369182431 / 2500000000000000 : ℝ)) (q :=
+      (7324478599017021 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_261_9 :
+    Uω (aρ 9) (bρ 9) (14064972912701 / 64000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_261_10 :
+    Uω (aρ 10) (bρ 10) (14064972912701 / 64000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_261_11 :
+    Uω (aρ 11) (bρ 11) (14064972912701 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_261_12 :
+    Uω (aρ 12) (bρ 12) (14064972912701 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_261_13 :
+    Uω (aρ 13) (bρ 13) (14064972912701 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_261_14 :
+    Uω (aρ 14) (bρ 14) (14064972912701 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_261_15 :
+    Uω (aρ 15) (bρ 15) (14064972912701 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_261_16 :
+    Uω (aρ 16) (bρ 16) (14064972912701 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_261 :
+    Uρ (14064972912701 / 64000000000000 : ℝ) ≤
+      (-(18831412410044943318741 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (14064972912701 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(7725093061549607964581 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_261_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (14064972912701 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(622570722209901726493 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_261_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (14064972912701 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(157978484232550097711 / 100000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_261_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (14064972912701 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(506346220181221679599 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_261_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (14064972912701 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(16867939381300538389631 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_261_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (14064972912701 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(17949664439597163688407 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_261_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (14064972912701 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(4950017525222746901699 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_261_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (14064972912701 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(23908046622860256109079 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_261_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (14064972912701 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_261_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (14064972912701 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_261_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (14064972912701 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_261_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (14064972912701 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_261_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (14064972912701 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_261_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (14064972912701 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_261_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (14064972912701 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_261_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (14064972912701 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_261_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_262_1 :
+    Uω (aρ 1) (bρ 1) (3524773257447 / 16000000000000 : ℝ) ≤
+      (-(15425223121055442148307 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4276820931848451 / 20000000000000000 : ℝ) ≤
+      (-(15425223121055442148307 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4276820931848451 / 20000000000000000 : ℝ) =
+            (4276820931848451 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4276820931848451 / 2500000000000000 : ℝ) ≤
+            (5369192287944557851693 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (534583716007269 / 2500000000000000 : ℝ)) (q :=
+      (4276820931848451 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_262_2 :
+    Uω (aρ 2) (bρ 2) (3524773257447 / 16000000000000 : ℝ) ≤
+      (-(31078027786503970657 / 20000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4228430497599681 / 20000000000000000 : ℝ) ≤
+      (-(31078027786503970657 / 20000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4228430497599681 / 20000000000000000 : ℝ) =
+            (4228430497599681 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4228430497599681 / 2500000000000000 : ℝ) ≤
+            (10510803031496029343 / 20000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1056856721145153 / 5000000000000000 : ℝ)) (q :=
+      (4228430497599681 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_262_3 :
+    Uω (aρ 3) (bρ 3) (3524773257447 / 16000000000000 : ℝ) ≤
+      (-(15771981888500432529827 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (516382526483561 / 2500000000000000 : ℝ) ≤
+      (-(15771981888500432529827 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (516382526483561 / 2500000000000000 : ℝ) =
+            (516382526483561 / 312500000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (516382526483561 / 312500000000000 : ℝ) ≤
+            (5022433520499567470173 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2063739262909113 / 10000000000000000 : ℝ)) (q :=
+      (516382526483561 / 2500000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_262_4 :
+    Uω (aρ 4) (bρ 4) (3524773257447 / 16000000000000 : ℝ) ≤
+      (-(16176095375587935077251 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (991861671071893 / 5000000000000000 : ℝ) ≤
+      (-(16176095375587935077251 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (991861671071893 / 5000000000000000 : ℝ) =
+            (991861671071893 / 625000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (991861671071893 / 625000000000000 : ℝ) ≤
+            (4618320033412064922749 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1978419916393197 / 10000000000000000 : ℝ)) (q :=
+      (991861671071893 / 5000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_262_5 :
+    Uω (aρ 5) (bρ 5) (3524773257447 / 16000000000000 : ℝ) ≤
+      (-(3367791259899323997961 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (928246721082921 / 5000000000000000 : ℝ) ≤
+      (-(3367791259899323997961 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (928246721082921 / 5000000000000000 : ℝ) =
+            (928246721082921 / 625000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (928246721082921 / 625000000000000 : ℝ) ≤
+            (791091821900676002039 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1842150035772309 / 10000000000000000 : ℝ)) (q :=
+      (928246721082921 / 5000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_262_6 :
+    Uω (aρ 6) (bρ 6) (3524773257447 / 16000000000000 : ℝ) ≤
+      (-(17916900595241057764019 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (833391176110807 / 5000000000000000 : ℝ) ≤
+      (-(17916900595241057764019 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (833391176110807 / 5000000000000000 : ℝ) =
+            (833391176110807 / 625000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (833391176110807 / 625000000000000 : ℝ) ≤
+            (2877514813758942235981 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1629968231023853 / 10000000000000000 : ℝ)) (q :=
+      (833391176110807 / 5000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_262_7 :
+    Uω (aρ 7) (bρ 7) (3524773257447 / 16000000000000 : ℝ) ≤
+      (-(9879368220258900866009 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1386401370794209 / 10000000000000000 : ℝ) ≤
+      (-(9879368220258900866009 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1386401370794209 / 10000000000000000 : ℝ) =
+            (1386401370794209 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1386401370794209 / 1250000000000000 : ℝ) ≤
+            (517839484241099133991 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1292867648929043 / 10000000000000000 : ℝ)) (q :=
+      (1386401370794209 / 10000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_262_8 :
+    Uω (aρ 8) (bρ 8) (3524773257447 / 16000000000000 : ℝ) ≤
+      (-(11912694834656871134041 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1846317852262463 / 20000000000000000 : ℝ) ≤
+      (-(11912694834656871134041 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1846317852262463 / 20000000000000000 : ℝ) =
+            (1846317852262463 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1846317852262463 / 1250000000000000 : ℝ) ≤
+            (1950248771343128865959 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (40620275683943 / 625000000000000 : ℝ)) (q :=
+      (1846317852262463 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_262_9 :
+    Uω (aρ 9) (bρ 9) (3524773257447 / 16000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_262_10 :
+    Uω (aρ 10) (bρ 10) (3524773257447 / 16000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_262_11 :
+    Uω (aρ 11) (bρ 11) (3524773257447 / 16000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_262_12 :
+    Uω (aρ 12) (bρ 12) (3524773257447 / 16000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_262_13 :
+    Uω (aρ 13) (bρ 13) (3524773257447 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_262_14 :
+    Uω (aρ 14) (bρ 14) (3524773257447 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_262_15 :
+    Uω (aρ 15) (bρ 15) (3524773257447 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_262_16 :
+    Uω (aρ 16) (bρ 16) (3524773257447 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_262 :
+    Uρ (3524773257447 / 16000000000000 : ℝ) ≤
+      (-(18812325424208424567931 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (3524773257447 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(15425223121055442148307 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_262_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (3524773257447 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) * (-(31078027786503970657 / 20000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_262_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (3524773257447 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(15771981888500432529827 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_262_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (3524773257447 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(16176095375587935077251 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_262_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (3524773257447 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(3367791259899323997961 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_262_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (3524773257447 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(17916900595241057764019 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_262_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (3524773257447 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(9879368220258900866009 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_262_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (3524773257447 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(11912694834656871134041 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_262_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (3524773257447 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_262_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (3524773257447 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_262_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (3524773257447 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_262_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (3524773257447 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_262_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (3524773257447 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_262_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (3524773257447 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_262_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (3524773257447 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_262_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (3524773257447 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_262_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_263_1 :
+    Uω (aρ 1) (bρ 1) (4522628207 / 20480000000 : ℝ) ≤
+      (-(15400322283405405486297 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17149935377919439 / 80000000000000000 : ℝ) ≤
+      (-(15400322283405405486297 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (17149935377919439 / 80000000000000000 : ℝ) =
+            (17149935377919439 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17149935377919439 / 10000000000000000 : ℝ) ≤
+            (5394093125594594513703 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2143666508365641 / 10000000000000000 : ℝ)) (q :=
+      (17149935377919439 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_263_2 :
+    Uω (aρ 2) (bρ 2) (4522628207 / 20480000000 : ℝ) ≤
+      (-(7756911689015681382531 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16956382236642683 / 80000000000000000 : ℝ) ≤
+      (-(7756911689015681382531 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (16956382236642683 / 80000000000000000 : ℝ) =
+            (16956382236642683 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16956382236642683 / 10000000000000000 : ℝ) ≤
+            (2640296015484318617469 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (529761808889113 / 2500000000000000 : ℝ)) (q :=
+      (16956382236642683 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_263_3 :
+    Uω (aρ 3) (bρ 3) (4522628207 / 20480000000 : ℝ) ≤
+      (-(7873091102223352406081 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16566927908739967 / 80000000000000000 : ℝ) ≤
+      (-(7873091102223352406081 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (16566927908739967 / 80000000000000000 : ℝ) =
+            (16566927908739967 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16566927908739967 / 10000000000000000 : ℝ) ≤
+            (2524116602276647593919 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2069079759930773 / 10000000000000000 : ℝ)) (q :=
+      (16566927908739967 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_263_4 :
+    Uω (aρ 4) (bρ 4) (4522628207 / 20480000000 : ℝ) ≤
+      (-(16149184709585599162011 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15912550904995751 / 80000000000000000 : ℝ) ≤
+      (-(16149184709585599162011 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (15912550904995751 / 80000000000000000 : ℝ) =
+            (15912550904995751 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15912550904995751 / 10000000000000000 : ℝ) ≤
+            (4645230699414400837989 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1983779690059719 / 10000000000000000 : ℝ)) (q :=
+      (15912550904995751 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_263_5 :
+    Uω (aρ 5) (bρ 5) (4522628207 / 20480000000 : ℝ) ≤
+      (-(16810058277414486902431 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14894928801881763 / 80000000000000000 : ℝ) ≤
+      (-(16810058277414486902431 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (14894928801881763 / 80000000000000000 : ℝ) =
+            (14894928801881763 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14894928801881763 / 10000000000000000 : ℝ) ≤
+            (3984357131585513097569 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (923782041808111 / 5000000000000000 : ℝ)) (q :=
+      (14894928801881763 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_263_6 :
+    Uω (aρ 6) (bρ 6) (4522628207 / 20480000000 : ℝ) ≤
+      (-(1788424856412957003267 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13377869040521019 / 80000000000000000 : ℝ) ≤
+      (-(1788424856412957003267 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (13377869040521019 / 80000000000000000 : ℝ) =
+            (13377869040521019 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13377869040521019 / 10000000000000000 : ℝ) ≤
+            (291016684487042996733 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (408884879604009 / 2500000000000000 : ℝ)) (q :=
+      (13377869040521019 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_263_7 :
+    Uω (aρ 7) (bρ 7) (4522628207 / 20480000000 : ℝ) ≤
+      (-(19717597427316517697399 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2227386619494727 / 16000000000000000 : ℝ) ≤
+      (-(19717597427316517697399 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2227386619494727 / 16000000000000000 : ℝ) =
+            (2227386619494727 / 2000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2227386619494727 / 2000000000000000 : ℝ) ≤
+            (1076817981683482302601 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (129896691341419 / 1000000000000000 : ℝ)) (q :=
+      (2227386619494727 / 16000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_263_8 :
+    Uω (aρ 8) (bρ 8) (4522628207 / 20480000000 : ℝ) ≤
+      (-(237439716505508975011 / 100000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7445646272237019 / 80000000000000000 : ℝ) ≤
+      (-(237439716505508975011 / 100000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (7445646272237019 / 80000000000000000 : ℝ) =
+            (7445646272237019 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7445646272237019 / 5000000000000000 : ℝ) ≤
+            (39819155614491024989 / 100000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (164921714611259 / 2500000000000000 : ℝ)) (q :=
+      (7445646272237019 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_263_9 :
+    Uω (aρ 9) (bρ 9) (4522628207 / 20480000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_263_10 :
+    Uω (aρ 10) (bρ 10) (4522628207 / 20480000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_263_11 :
+    Uω (aρ 11) (bρ 11) (4522628207 / 20480000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_263_12 :
+    Uω (aρ 12) (bρ 12) (4522628207 / 20480000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_263_13 :
+    Uω (aρ 13) (bρ 13) (4522628207 / 20480000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_263_14 :
+    Uω (aρ 14) (bρ 14) (4522628207 / 20480000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_263_15 :
+    Uω (aρ 15) (bρ 15) (4522628207 / 20480000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_263_16 :
+    Uω (aρ 16) (bρ 16) (4522628207 / 20480000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_263 :
+    Uρ (4522628207 / 20480000000 : ℝ) ≤
+      (-(18793388812557728533729 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (4522628207 / 20480000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(15400322283405405486297 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_263_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (4522628207 / 20480000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(7756911689015681382531 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_263_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (4522628207 / 20480000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(7873091102223352406081 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_263_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (4522628207 / 20480000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(16149184709585599162011 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_263_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (4522628207 / 20480000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(16810058277414486902431 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_263_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (4522628207 / 20480000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1788424856412957003267 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_263_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (4522628207 / 20480000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(19717597427316517697399 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_263_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (4522628207 / 20480000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(237439716505508975011 / 100000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_263_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (4522628207 / 20480000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_263_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (4522628207 / 20480000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_263_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (4522628207 / 20480000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_263_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (4522628207 / 20480000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_263_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (4522628207 / 20480000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_263_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (4522628207 / 20480000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_263_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (4522628207 / 20480000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_263_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (4522628207 / 20480000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_263_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_264_1 :
+    Uω (aρ 1) (bρ 1) (7083666631981 / 32000000000000 : ℝ) ≤
+      (-(7687741650642611239753 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1719258702098169 / 8000000000000000 : ℝ) ≤
+      (-(7687741650642611239753 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1719258702098169 / 8000000000000000 : ℝ) =
+            (1719258702098169 / 1000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1719258702098169 / 1000000000000000 : ℝ) ≤
+            (2709466053857388760247 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (53724953770909 / 250000000000000 : ℝ)) (q :=
+      (1719258702098169 / 8000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_264_2 :
+    Uω (aρ 2) (bρ 2) (7083666631981 / 32000000000000 : ℝ) ≤
+      (-(1936087023678734676367 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8499521216091017 / 40000000000000000 : ℝ) ≤
+      (-(1936087023678734676367 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (8499521216091017 / 40000000000000000 : ℝ) =
+            (8499521216091017 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8499521216091017 / 5000000000000000 : ℝ) ≤
+            (663214902446265323633 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1062190508073223 / 5000000000000000 : ℝ)) (q :=
+      (8499521216091017 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_264_3 :
+    Uω (aρ 3) (bρ 3) (7083666631981 / 32000000000000 : ℝ) ≤
+      (-(15720449025848114804897 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8304807390049419 / 40000000000000000 : ℝ) ≤
+      (-(15720449025848114804897 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (8304807390049419 / 40000000000000000 : ℝ) =
+            (8304807390049419 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8304807390049419 / 5000000000000000 : ℝ) ≤
+            (5073966383151885195103 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2074420209475647 / 10000000000000000 : ℝ)) (q :=
+      (8304807390049419 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_264_4 :
+    Uω (aρ 4) (bρ 4) (7083666631981 / 32000000000000 : ℝ) ≤
+      (-(8061173325877085877879 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7977657230004551 / 40000000000000000 : ℝ) ≤
+      (-(8061173325877085877879 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (7977657230004551 / 40000000000000000 : ℝ) =
+            (7977657230004551 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7977657230004551 / 5000000000000000 : ℝ) ≤
+            (2336034378622914122121 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1989139310518213 / 10000000000000000 : ℝ)) (q :=
+      (7977657230004551 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_264_5 :
+    Uω (aρ 5) (bρ 5) (7083666631981 / 32000000000000 : ℝ) ≤
+      (-(16781244809738776207101 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7468954073150357 / 40000000000000000 : ℝ) ≤
+      (-(16781244809738776207101 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (7468954073150357 / 40000000000000000 : ℝ) =
+            (7468954073150357 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7468954073150357 / 5000000000000000 : ℝ) ≤
+            (4013170599261223792899 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (463244412856529 / 2500000000000000 : ℝ)) (q :=
+      (7468954073150357 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_264_6 :
+    Uω (aρ 6) (bρ 6) (7083666631981 / 32000000000000 : ℝ) ≤
+      (-(446292688846316203507 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6710736442547223 / 40000000000000000 : ℝ) ≤
+      (-(446292688846316203507 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (6710736442547223 / 40000000000000000 : ℝ) =
+            (6710736442547223 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6710736442547223 / 5000000000000000 : ℝ) ≤
+            (73567696378683796493 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1641109211264549 / 10000000000000000 : ℝ)) (q :=
+      (6710736442547223 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_264_7 :
+    Uω (aρ 7) (bρ 7) (7083666631981 / 32000000000000 : ℝ) ≤
+      (-(19676651023580034608527 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5591314161142619 / 40000000000000000 : ℝ) ≤
+      (-(19676651023580034608527 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (5591314161142619 / 40000000000000000 : ℝ) =
+            (5591314161142619 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5591314161142619 / 5000000000000000 : ℝ) ≤
+            (1117764385419965391473 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1305059451322247 / 10000000000000000 : ℝ)) (q :=
+      (5591314161142619 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_264_8 :
+    Uω (aρ 8) (bρ 8) (7083666631981 / 32000000000000 : ℝ) ≤
+      (-(11831871818659477479103 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3752810737940087 / 40000000000000000 : ℝ) ≤
+      (-(11831871818659477479103 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (3752810737940087 / 40000000000000000 : ℝ) =
+            (3752810737940087 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3752810737940087 / 2500000000000000 : ℝ) ≤
+            (2031071787340522520897 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (669349391060981 / 10000000000000000 : ℝ)) (q :=
+      (3752810737940087 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_264_9 :
+    Uω (aρ 9) (bρ 9) (7083666631981 / 32000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_264_10 :
+    Uω (aρ 10) (bρ 10) (7083666631981 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_264_11 :
+    Uω (aρ 11) (bρ 11) (7083666631981 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_264_12 :
+    Uω (aρ 12) (bρ 12) (7083666631981 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_264_13 :
+    Uω (aρ 13) (bρ 13) (7083666631981 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_264_14 :
+    Uω (aρ 14) (bρ 14) (7083666631981 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_264_15 :
+    Uω (aρ 15) (bρ 15) (7083666631981 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_264_16 :
+    Uω (aρ 16) (bρ 16) (7083666631981 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_264 :
+    Uρ (7083666631981 / 32000000000000 : ℝ) ≤
+      (-(18774597929083658498267 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (7083666631981 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(7687741650642611239753 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_264_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (7083666631981 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1936087023678734676367 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_264_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (7083666631981 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(15720449025848114804897 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_264_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (7083666631981 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(8061173325877085877879 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_264_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (7083666631981 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(16781244809738776207101 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_264_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (7083666631981 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(446292688846316203507 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_264_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (7083666631981 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(19676651023580034608527 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_264_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (7083666631981 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(11831871818659477479103 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_264_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (7083666631981 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_264_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (7083666631981 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_264_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (7083666631981 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_264_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (7083666631981 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_264_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (7083666631981 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_264_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (7083666631981 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_264_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (7083666631981 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_264_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (7083666631981 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_264_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_265_1 :
+    Uω (aρ 1) (bρ 1) (14201453381049 / 64000000000000 : ℝ) ≤
+      (-(76753529340636769929 / 50000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17235238656635969 / 80000000000000000 : ℝ) ≤
+      (-(76753529340636769929 / 50000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (17235238656635969 / 80000000000000000 : ℝ) =
+            (17235238656635969 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17235238656635969 / 10000000000000000 : ℝ) ≤
+            (27218547704363230071 / 50000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1077164895727543 / 5000000000000000 : ℝ)) (q :=
+      (17235238656635969 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_265_2 :
+    Uω (aρ 2) (bρ 2) (14201453381049 / 64000000000000 : ℝ) ≤
+      (-(3092726401940492489023 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3408340515479549 / 16000000000000000 : ℝ) ≤
+      (-(3092726401940492489023 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3408340515479549 / 16000000000000000 : ℝ) =
+            (3408340515479549 / 2000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3408340515479549 / 2000000000000000 : ℝ) ≤
+            (1066156679859507510977 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (212971478415553 / 1000000000000000 : ℝ)) (q :=
+      (3408340515479549 / 16000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_265_3 :
+    Uω (aρ 3) (bρ 3) (14201453381049 / 64000000000000 : ℝ) ≤
+      (-(15694782010131099692527 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3330460292602701 / 16000000000000000 : ℝ) ≤
+      (-(15694782010131099692527 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3330460292602701 / 16000000000000000 : ℝ) =
+            (3330460292602701 / 2000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3330460292602701 / 2000000000000000 : ℝ) ≤
+            (5099633398868900307473 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (207976061190947 / 1000000000000000 : ℝ)) (q :=
+      (3330460292602701 / 16000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_265_4 :
+    Uω (aρ 4) (bρ 4) (14201453381049 / 64000000000000 : ℝ) ≤
+      (-(4023895202321576679941 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15998077407130749 / 80000000000000000 : ℝ) ≤
+      (-(4023895202321576679941 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (15998077407130749 / 80000000000000000 : ℝ) =
+            (15998077407130749 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15998077407130749 / 10000000000000000 : ℝ) ≤
+            (1174708649928423320059 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1994498779003781 / 10000000000000000 : ℝ)) (q :=
+      (15998077407130749 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_265_5 :
+    Uω (aρ 5) (bρ 5) (14201453381049 / 64000000000000 : ℝ) ≤
+      (-(16752515395708029767003 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14980885587363893 / 80000000000000000 : ℝ) ≤
+      (-(16752515395708029767003 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (14980885587363893 / 80000000000000000 : ℝ) =
+            (14980885587363893 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14980885587363893 / 10000000000000000 : ℝ) ≤
+            (4041900013291970232997 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1858390743397067 / 10000000000000000 : ℝ)) (q :=
+      (14980885587363893 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_265_6 :
+    Uω (aρ 6) (bρ 6) (14201453381049 / 64000000000000 : ℝ) ≤
+      (-(17819276780701897823461 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13465070416213477 / 80000000000000000 : ℝ) ≤
+      (-(17819276780701897823461 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (13465070416213477 / 80000000000000000 : ℝ) =
+            (13465070416213477 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13465070416213477 / 10000000000000000 : ℝ) ≤
+            (2975138628298102176539 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1646677325749463 / 10000000000000000 : ℝ)) (q :=
+      (13465070416213477 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_265_7 :
+    Uω (aρ 7) (bρ 7) (14201453381049 / 64000000000000 : ℝ) ≤
+      (-(4908973806508137317557 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11228297015867517 / 80000000000000000 : ℝ) ≤
+      (-(4908973806508137317557 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (11228297015867517 / 80000000000000000 : ℝ) =
+            (11228297015867517 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11228297015867517 / 10000000000000000 : ℝ) ≤
+            (289630045741862682443 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1311145356422973 / 10000000000000000 : ℝ)) (q :=
+      (11228297015867517 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_265_8 :
+    Uω (aρ 8) (bρ 8) (14201453381049 / 64000000000000 : ℝ) ≤
+      (-(5896164968722676067157 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7565214084243297 / 80000000000000000 : ℝ) ≤
+      (-(5896164968722676067157 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (7565214084243297 / 80000000000000000 : ℝ) =
+            (7565214084243297 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7565214084243297 / 5000000000000000 : ℝ) ≤
+            (1035306834277323932843 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (339458137428459 / 5000000000000000 : ℝ)) (q :=
+      (7565214084243297 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_265_9 :
+    Uω (aρ 9) (bρ 9) (14201453381049 / 64000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_265_10 :
+    Uω (aρ 10) (bρ 10) (14201453381049 / 64000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_265_11 :
+    Uω (aρ 11) (bρ 11) (14201453381049 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_265_12 :
+    Uω (aρ 12) (bρ 12) (14201453381049 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_265_13 :
+    Uω (aρ 13) (bρ 13) (14201453381049 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_265_14 :
+    Uω (aρ 14) (bρ 14) (14201453381049 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_265_15 :
+    Uω (aρ 15) (bρ 15) (14201453381049 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_265_16 :
+    Uω (aρ 16) (bρ 16) (14201453381049 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_265 :
+    Uρ (14201453381049 / 64000000000000 : ℝ) ≤
+      (-(2344493551589127377827 / 1250000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (14201453381049 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(76753529340636769929 / 50000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_265_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (14201453381049 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3092726401940492489023 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_265_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (14201453381049 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(15694782010131099692527 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_265_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (14201453381049 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(4023895202321576679941 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_265_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (14201453381049 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(16752515395708029767003 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_265_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (14201453381049 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(17819276780701897823461 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_265_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (14201453381049 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(4908973806508137317557 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_265_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (14201453381049 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(5896164968722676067157 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_265_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (14201453381049 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_265_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (14201453381049 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_265_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (14201453381049 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_265_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (14201453381049 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_265_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (14201453381049 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_265_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (14201453381049 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_265_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (14201453381049 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_265_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (14201453381049 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_265_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_266_1 :
+    Uω (aρ 1) (bρ 1) (1779446687267 / 8000000000000 : ℝ) ≤
+      (-(3831497419909473840719 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (863894514246857 / 4000000000000000 : ℝ) ≤
+      (-(3831497419909473840719 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (863894514246857 / 4000000000000000 : ℝ) =
+            (863894514246857 / 500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (863894514246857 / 500000000000000 : ℝ) ≤
+            (1367106432340526159281 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (431932286047107 / 2000000000000000 : ℝ)) (q :=
+      (863894514246857 / 4000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_266_2 :
+    Uω (aρ 2) (bρ 2) (1779446687267 / 8000000000000 : ℝ) ≤
+      (-(15438630523490637756799 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4271090668166743 / 20000000000000000 : ℝ) ≤
+      (-(15438630523490637756799 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4271090668166743 / 20000000000000000 : ℝ) =
+            (4271090668166743 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4271090668166743 / 2500000000000000 : ℝ) ≤
+            (5355784885509362243201 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2135048539677993 / 10000000000000000 : ℝ)) (q :=
+      (4271090668166743 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_266_3 :
+    Uω (aρ 3) (bρ 3) (1779446687267 / 8000000000000 : ℝ) ≤
+      (-(313383616347333776729 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4173746989732977 / 20000000000000000 : ℝ) ≤
+      (-(313383616347333776729 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4173746989732977 / 20000000000000000 : ℝ) =
+            (4173746989732977 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4173746989732977 / 2500000000000000 : ℝ) ≤
+            (102504691832666223271 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2085100967594227 / 10000000000000000 : ℝ)) (q :=
+      (4173746989732977 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_266_4 :
+    Uω (aρ 4) (bρ 4) (1779446687267 / 8000000000000 : ℝ) ≤
+      (-(16068886792570002071263 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1002552484453 / 5000000000000 : ℝ) ≤
+      (-(16068886792570002071263 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (1002552484453 / 5000000000000 : ℝ) = (1002552484453 / 625000000000 : ℝ) / 2 ^ 3 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1002552484453 / 625000000000 : ℝ) ≤
+            (4725528616429997928737 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7999432386953 / 40000000000000 : ℝ)) (q :=
+      (1002552484453 / 5000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_266_5 :
+    Uω (aρ 5) (bρ 5) (1779446687267 / 8000000000000 : ℝ) ≤
+      (-(1045241846191213256377 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1877982642706899 / 10000000000000000 : ℝ) ≤
+      (-(1045241846191213256377 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1877982642706899 / 10000000000000000 : ℝ) =
+            (1877982642706899 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1877982642706899 / 1250000000000000 : ℝ) ≤
+            (254409116871286743623 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (232975420459381 / 1250000000000000 : ℝ)) (q :=
+      (1877982642706899 / 10000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_266_6 :
+    Uω (aρ 6) (bρ 6) (1779446687267 / 8000000000000 : ℝ) ≤
+      (-(4446738867384939903639 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (422145678053367 / 2500000000000000 : ℝ) ≤
+      (-(4446738867384939903639 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (422145678053367 / 2500000000000000 : ℝ) =
+            (422145678053367 / 312500000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (422145678053367 / 312500000000000 : ℝ) ≤
+            (751864984865060096361 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (826121938914093 / 5000000000000000 : ℝ)) (q :=
+      (422145678053367 / 2500000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_266_7 :
+    Uω (aρ 7) (bρ 7) (1779446687267 / 8000000000000 : ℝ) ≤
+      (-(19595328065017379469913 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2818484886490769 / 20000000000000000 : ℝ) ≤
+      (-(19595328065017379469913 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2818484886490769 / 20000000000000000 : ℝ) =
+            (2818484886490769 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2818484886490769 / 2500000000000000 : ℝ) ≤
+            (1199087343982620530087 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1317224720652019 / 10000000000000000 : ℝ)) (q :=
+      (2818484886490769 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_266_8 :
+    Uω (aρ 8) (bρ 8) (1779446687267 / 8000000000000 : ℝ) ≤
+      (-(2938334687597392326683 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (381222002433227 / 4000000000000000 : ℝ) ≤
+      (-(2938334687597392326683 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (381222002433227 / 4000000000000000 : ℝ) =
+            (381222002433227 / 250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (381222002433227 / 250000000000000 : ℝ) ≤
+            (527401213902607673317 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (137678299533477 / 2000000000000000 : ℝ)) (q :=
+      (381222002433227 / 4000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_266_9 :
+    Uω (aρ 9) (bρ 9) (1779446687267 / 8000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_266_10 :
+    Uω (aρ 10) (bρ 10) (1779446687267 / 8000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_266_11 :
+    Uω (aρ 11) (bρ 11) (1779446687267 / 8000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_266_12 :
+    Uω (aρ 12) (bρ 12) (1779446687267 / 8000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_266_13 :
+    Uω (aρ 13) (bρ 13) (1779446687267 / 8000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_266_14 :
+    Uω (aρ 14) (bρ 14) (1779446687267 / 8000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_266_15 :
+    Uω (aρ 15) (bρ 15) (1779446687267 / 8000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_266_16 :
+    Uω (aρ 16) (bρ 16) (1779446687267 / 8000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_266 :
+    Uρ (1779446687267 / 8000000000000 : ℝ) ≤
+      (-(18737436162268459849223 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1779446687267 / 8000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3831497419909473840719 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_266_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1779446687267 / 8000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(15438630523490637756799 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_266_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1779446687267 / 8000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(313383616347333776729 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_266_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1779446687267 / 8000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(16068886792570002071263 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_266_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1779446687267 / 8000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(1045241846191213256377 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_266_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1779446687267 / 8000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(4446738867384939903639 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_266_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1779446687267 / 8000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(19595328065017379469913 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_266_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1779446687267 / 8000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(2938334687597392326683 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_266_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1779446687267 / 8000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_266_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1779446687267 / 8000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_266_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1779446687267 / 8000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_266_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1779446687267 / 8000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_266_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1779446687267 / 8000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_266_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1779446687267 / 8000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_266_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1779446687267 / 8000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_266_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1779446687267 / 8000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_266_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_267_1 :
+    Uω (aρ 1) (bρ 1) (14269693615223 / 64000000000000 : ℝ) ≤
+      (-(3060266886754829329249 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17320541905939527 / 80000000000000000 : ℝ) ≤
+      (-(3060266886754829329249 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (17320541905939527 / 80000000000000000 : ℝ) =
+            (17320541905939527 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17320541905939527 / 10000000000000000 : ℝ) ≤
+            (1098616195045170670751 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (270624133398911 / 1250000000000000 : ℝ)) (q :=
+      (17320541905939527 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_267_2 :
+    Uω (aρ 2) (bρ 2) (14269693615223 / 64000000000000 : ℝ) ≤
+      (-(15413691417798657404249 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17127022718363107 / 80000000000000000 : ℝ) ≤
+      (-(15413691417798657404249 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (17127022718363107 / 80000000000000000 : ℝ) =
+            (17127022718363107 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17127022718363107 / 10000000000000000 : ℝ) ≤
+            (5380723991201342595751 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2140382282807183 / 10000000000000000 : ℝ)) (q :=
+      (17127022718363107 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_267_3 :
+    Uω (aρ 3) (bρ 3) (14269693615223 / 64000000000000 : ℝ) ≤
+      (-(1955455638780413424699 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16737674269287179 / 80000000000000000 : ℝ) ≤
+      (-(1955455638780413424699 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (16737674269287179 / 80000000000000000 : ℝ) =
+            (16737674269287179 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16737674269287179 / 10000000000000000 : ℝ) ≤
+            (643846287344586575301 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2090441276888201 / 10000000000000000 : ℝ)) (q :=
+      (16737674269287179 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_267_4 :
+    Uω (aρ 4) (bρ 4) (14269693615223 / 64000000000000 : ℝ) ≤
+      (-(8021132107576929851351 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16083601497195783 / 80000000000000000 : ℝ) ≤
+      (-(8021132107576929851351 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (16083601497195783 / 80000000000000000 : ℝ) =
+            (16083601497195783 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16083601497195783 / 10000000000000000 : ℝ) ≤
+            (2376075596923070148649 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (125326079058147 / 625000000000000 : ℝ)) (q :=
+      (16083601497195783 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_267_5 :
+    Uω (aρ 5) (bρ 5) (14269693615223 / 64000000000000 : ℝ) ≤
+      (-(16695306747974266726253 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15066834825564987 / 80000000000000000 : ℝ) ≤
+      (-(16695306747974266726253 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (15066834825564987 / 80000000000000000 : ℝ) =
+            (15066834825564987 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15066834825564987 / 10000000000000000 : ℝ) ≤
+            (4099108661025733273747 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1869215516357653 / 10000000000000000 : ℝ)) (q :=
+      (15066834825564987 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_267_6 :
+    Uω (aρ 6) (bρ 6) (14269693615223 / 64000000000000 : ℝ) ≤
+      (-(17754742853671198578557 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13552246792531779 / 80000000000000000 : ℝ) ≤
+      (-(17754742853671198578557 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (13552246792531779 / 80000000000000000 : ℝ) =
+            (13552246792531779 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13552246792531779 / 10000000000000000 : ℝ) ≤
+            (3039672555328801421443 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1657808883239351 / 10000000000000000 : ℝ)) (q :=
+      (13552246792531779 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_267_7 :
+    Uω (aρ 7) (bρ 7) (14269693615223 / 64000000000000 : ℝ) ≤
+      (-(9777473801854687953071 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11319556273165103 / 80000000000000000 : ℝ) ≤
+      (-(9777473801854687953071 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (11319556273165103 / 80000000000000000 : ℝ) =
+            (11319556273165103 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11319556273165103 / 10000000000000000 : ℝ) ≤
+            (619733902645312046929 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (661648817078841 / 5000000000000000 : ℝ)) (q :=
+      (11319556273165103 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_267_8 :
+    Uω (aρ 8) (bρ 8) (14269693615223 / 64000000000000 : ℝ) ≤
+      (-(11714878146986777122497 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7683314305291443 / 80000000000000000 : ℝ) ≤
+      (-(11714878146986777122497 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (7683314305291443 / 80000000000000000 : ℝ) =
+            (7683314305291443 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7683314305291443 / 5000000000000000 : ℝ) ≤
+            (2148065459013222877503 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (697778793529267 / 10000000000000000 : ℝ)) (q :=
+      (7683314305291443 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_267_9 :
+    Uω (aρ 9) (bρ 9) (14269693615223 / 64000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_267_10 :
+    Uω (aρ 10) (bρ 10) (14269693615223 / 64000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_267_11 :
+    Uω (aρ 11) (bρ 11) (14269693615223 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_267_12 :
+    Uω (aρ 12) (bρ 12) (14269693615223 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_267_13 :
+    Uω (aρ 13) (bρ 13) (14269693615223 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_267_14 :
+    Uω (aρ 14) (bρ 14) (14269693615223 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_267_15 :
+    Uω (aρ 15) (bρ 15) (14269693615223 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_267_16 :
+    Uω (aρ 16) (bρ 16) (14269693615223 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_267 :
+    Uρ (14269693615223 / 64000000000000 : ℝ) ≤
+      (-(18719057314217090473663 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (14269693615223 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3060266886754829329249 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_267_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (14269693615223 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(15413691417798657404249 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_267_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (14269693615223 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1955455638780413424699 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_267_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (14269693615223 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(8021132107576929851351 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_267_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (14269693615223 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(16695306747974266726253 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_267_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (14269693615223 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(17754742853671198578557 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_267_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (14269693615223 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(9777473801854687953071 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_267_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (14269693615223 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(11714878146986777122497 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_267_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (14269693615223 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_267_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (14269693615223 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_267_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (14269693615223 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_267_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (14269693615223 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_267_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (14269693615223 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_267_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (14269693615223 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_267_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (14269693615223 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_267_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (14269693615223 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_267_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/U22.lean b/LeanPool/Zeta5Irrational/Table/U22.lean
new file mode 100644
index 0000000000..e8a0c413b6
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/U22.lean
@@ -0,0 +1,3711 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-! # Certified arcsine potential bounds (U22) -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma U_268_1 :
+    Uω (aρ 1) (bρ 1) (1430381373231 / 6400000000000 : ℝ) ≤
+      (-(7638369915361229297179 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8681596759848461 / 40000000000000000 : ℝ) ≤
+      (-(7638369915361229297179 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (8681596759848461 / 40000000000000000 : ℝ) =
+            (8681596759848461 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8681596759848461 / 5000000000000000 : ℝ) ≤
+            (2758837789138770702821 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2170324702335793 / 10000000000000000 : ℝ)) (q :=
+      (8681596759848461 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_268_2 :
+    Uω (aρ 2) (bρ 2) (1430381373231 / 6400000000000 : ℝ) ≤
+      (-(7694407190984975856153 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1716968271485583 / 8000000000000000 : ℝ) ≤
+      (-(7694407190984975856153 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1716968271485583 / 8000000000000000 : ℝ) =
+            (1716968271485583 / 1000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1716968271485583 / 1000000000000000 : ℝ) ≤
+            (2702800513515024143847 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6705362542611 / 31250000000000 : ℝ)) (q :=
+      (1716968271485583 / 8000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_268_3 :
+    Uω (aρ 3) (bρ 3) (1430381373231 / 6400000000000 : ℝ) ≤
+      (-(3123634910807985488307 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8390180197748917 / 40000000000000000 : ℝ) ≤
+      (-(3123634910807985488307 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (8390180197748917 / 40000000000000000 : ℝ) =
+            (8390180197748917 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8390180197748917 / 5000000000000000 : ℝ) ≤
+            (1035248170992014511693 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2095781540146021 / 10000000000000000 : ℝ)) (q :=
+      (8390180197748917 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_268_4 :
+    Uω (aρ 4) (bρ 4) (1430381373231 / 6400000000000 : ℝ) ≤
+      (-(16015712693712828402873 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8063181324878669 / 40000000000000000 : ℝ) ≤
+      (-(16015712693712828402873 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (8063181324878669 / 40000000000000000 : ℝ) =
+            (8063181324878669 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8063181324878669 / 5000000000000000 : ℝ) ≤
+            (4778702715287171597127 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2010576284775897 / 10000000000000000 : ℝ)) (q :=
+      (8063181324878669 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_268_5 :
+    Uω (aρ 5) (bρ 5) (1430381373231 / 6400000000000 : ℝ) ≤
+      (-(16666826535024517328497 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7554903327646481 / 40000000000000000 : ℝ) ≤
+      (-(16666826535024517328497 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (7554903327646481 / 40000000000000000 : ℝ) =
+            (7554903327646481 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7554903327646481 / 5000000000000000 : ℝ) ≤
+            (4127588873975482671503 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1874627205494803 / 10000000000000000 : ℝ)) (q :=
+      (7554903327646481 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_268_6 :
+    Uω (aρ 6) (bρ 6) (1430381373231 / 6400000000000 : ℝ) ≤
+      (-(8861319087358917193727 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (54383303051121 / 320000000000000 : ℝ) ≤
+      (-(8861319087358917193727 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (54383303051121 / 320000000000000 : ℝ) = (54383303051121 / 40000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (54383303051121 / 40000000000000 : ℝ) ≤
+            (1535888617141082806273 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (13306978860053 / 80000000000000 : ℝ)) (q :=
+      (54383303051121 / 320000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_268_7 :
+    Uω (aρ 7) (bρ 7) (1430381373231 / 6400000000000 : ℝ) ≤
+      (-(1951475193735104412929 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5682573775549233 / 40000000000000000 : ℝ) ≤
+      (-(1951475193735104412929 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (5682573775549233 / 40000000000000000 : ℝ) =
+            (5682573775549233 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5682573775549233 / 5000000000000000 : ℝ) ≤
+            (127966347164895587071 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1329364185346179 / 10000000000000000 : ℝ)) (q :=
+      (5682573775549233 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_268_8 :
+    Uω (aρ 8) (bρ 8) (1430381373231 / 6400000000000 : ℝ) ≤
+      (-(23353858451496383736881 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3870925431047819 / 40000000000000000 : ℝ) ≤
+      (-(23353858451496383736881 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (3870925431047819 / 40000000000000000 : ℝ) =
+            (3870925431047819 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3870925431047819 / 2500000000000000 : ℝ) ≤
+            (4372028760503616263119 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (44192604027217 / 625000000000000 : ℝ)) (q :=
+      (3870925431047819 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_268_9 :
+    Uω (aρ 9) (bρ 9) (1430381373231 / 6400000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_268_10 :
+    Uω (aρ 10) (bρ 10) (1430381373231 / 6400000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_268_11 :
+    Uω (aρ 11) (bρ 11) (1430381373231 / 6400000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_268_12 :
+    Uω (aρ 12) (bρ 12) (1430381373231 / 6400000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_268_13 :
+    Uω (aρ 13) (bρ 13) (1430381373231 / 6400000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_268_14 :
+    Uω (aρ 14) (bρ 14) (1430381373231 / 6400000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_268_15 :
+    Uω (aρ 15) (bρ 15) (1430381373231 / 6400000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_268_16 :
+    Uω (aρ 16) (bρ 16) (1430381373231 / 6400000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_268 :
+    Uρ (1430381373231 / 6400000000000 : ℝ) ≤
+      (-(18700808222836457915203 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1430381373231 / 6400000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(7638369915361229297179 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_268_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1430381373231 / 6400000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(7694407190984975856153 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_268_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1430381373231 / 6400000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3123634910807985488307 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_268_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1430381373231 / 6400000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(16015712693712828402873 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_268_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1430381373231 / 6400000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(16666826535024517328497 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_268_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1430381373231 / 6400000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(8861319087358917193727 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_268_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1430381373231 / 6400000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(1951475193735104412929 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_268_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1430381373231 / 6400000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(23353858451496383736881 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_268_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1430381373231 / 6400000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_268_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1430381373231 / 6400000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_268_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1430381373231 / 6400000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_268_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1430381373231 / 6400000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_268_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1430381373231 / 6400000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_268_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1430381373231 / 6400000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_268_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1430381373231 / 6400000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_268_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1430381373231 / 6400000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_268_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_269_1 :
+    Uω (aρ 1) (bρ 1) (14337933849397 / 64000000000000 : ℝ) ≤
+      (-(7626102786438175269679 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17405845126262593 / 80000000000000000 : ℝ) ≤
+      (-(7626102786438175269679 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (17405845126262593 / 80000000000000000 : ℝ) =
+            (17405845126262593 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17405845126262593 / 10000000000000000 : ℝ) ≤
+            (2771104918061824730321 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2175656335682367 / 10000000000000000 : ℝ)) (q :=
+      (17405845126262593 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_269_2 :
+    Uω (aρ 2) (bρ 2) (14337933849397 / 64000000000000 : ℝ) ≤
+      (-(7681999553831932135529 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17212342662511157 / 80000000000000000 : ℝ) ≤
+      (-(7681999553831932135529 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (17212342662511157 / 80000000000000000 : ℝ) =
+            (17212342662511157 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17212342662511157 / 10000000000000000 : ℝ) ≤
+            (2715208150668067864471 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (537762433063627 / 2500000000000000 : ℝ)) (q :=
+      (17212342662511157 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_269_3 :
+    Uω (aρ 3) (bρ 3) (14337933849397 / 64000000000000 : ℝ) ≤
+      (-(7796384408299785382657 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16823046338967949 / 80000000000000000 : ℝ) ≤
+      (-(7796384408299785382657 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (16823046338967949 / 80000000000000000 : ℝ) =
+            (16823046338967949 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16823046338967949 / 10000000000000000 : ℝ) ≤
+            (2600823296200214617343 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1050560878859353 / 5000000000000000 : ℝ)) (q :=
+      (16823046338967949 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_269_4 :
+    Uω (aρ 4) (bρ 4) (14337933849397 / 64000000000000 : ℝ) ≤
+      (-(7994615924007200057329 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16169123213664917 / 80000000000000000 : ℝ) ≤
+      (-(7994615924007200057329 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (16169123213664917 / 80000000000000000 : ℝ) =
+            (16169123213664917 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16169123213664917 / 10000000000000000 : ℝ) ≤
+            (2402591780492799942671 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (503983789364487 / 2500000000000000 : ℝ)) (q :=
+      (16169123213664917 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_269_5 :
+    Uω (aρ 5) (bρ 5) (14337933849397 / 64000000000000 : ℝ) ≤
+      (-(16638428417119872482403 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15152776646850901 / 80000000000000000 : ℝ) ≤
+      (-(16638428417119872482403 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (15152776646850901 / 80000000000000000 : ℝ) =
+            (15152776646850901 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15152776646850901 / 10000000000000000 : ℝ) ≤
+            (4155986991880127517597 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (470009608772361 / 2500000000000000 : ℝ)) (q :=
+      (15152776646850901 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_269_6 :
+    Uω (aρ 6) (bρ 6) (14337933849397 / 64000000000000 : ℝ) ≤
+      (-(3538128136498892697533 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13639398669702877 / 80000000000000000 : ℝ) ≤
+      (-(3538128136498892697533 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (13639398669702877 / 80000000000000000 : ℝ) =
+            (13639398669702877 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13639398669702877 / 10000000000000000 : ℝ) ≤
+            (620754945301107302467 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (834467157971219 / 5000000000000000 : ℝ)) (q :=
+      (13639398669702877 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_269_7 :
+    Uω (aρ 7) (bρ 7) (14337933849397 / 64000000000000 : ℝ) ≤
+      (-(4868684798127873941503 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (91285709812761 / 640000000000000 : ℝ) ≤
+      (-(4868684798127873941503 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (91285709812761 / 640000000000000 : ℝ) = (91285709812761 / 80000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (91285709812761 / 80000000000000 : ℝ) ≤
+            (329919054122126058497 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2670848921851 / 20000000000000 : ℝ)) (q :=
+      (91285709812761 / 640000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_269_8 :
+    Uω (aρ 8) (bρ 8) (14337933849397 / 64000000000000 : ℝ) ≤
+      (-(23278948388726262110201 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1560012575313533 / 16000000000000000 : ℝ) ≤
+      (-(23278948388726262110201 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1560012575313533 / 16000000000000000 : ℝ) =
+            (1560012575313533 / 1000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1560012575313533 / 1000000000000000 : ℝ) ≤
+            (4446938823273737889799 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (143260679951727 / 2000000000000000 : ℝ)) (q :=
+      (1560012575313533 / 16000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_269_9 :
+    Uω (aρ 9) (bρ 9) (14337933849397 / 64000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_269_10 :
+    Uω (aρ 10) (bρ 10) (14337933849397 / 64000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_269_11 :
+    Uω (aρ 11) (bρ 11) (14337933849397 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_269_12 :
+    Uω (aρ 12) (bρ 12) (14337933849397 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_269_13 :
+    Uω (aρ 13) (bρ 13) (14337933849397 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_269_14 :
+    Uω (aρ 14) (bρ 14) (14337933849397 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_269_15 :
+    Uω (aρ 15) (bρ 15) (14337933849397 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_269_16 :
+    Uω (aρ 16) (bρ 16) (14337933849397 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_269 :
+    Uρ (14337933849397 / 64000000000000 : ℝ) ≤
+      (-(3736537088496720148241 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (14337933849397 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(7626102786438175269679 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_269_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (14337933849397 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(7681999553831932135529 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_269_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (14337933849397 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(7796384408299785382657 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_269_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (14337933849397 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(7994615924007200057329 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_269_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (14337933849397 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(16638428417119872482403 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_269_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (14337933849397 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3538128136498892697533 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_269_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (14337933849397 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(4868684798127873941503 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_269_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (14337933849397 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(23278948388726262110201 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_269_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (14337933849397 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_269_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (14337933849397 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_269_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (14337933849397 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_269_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (14337933849397 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_269_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (14337933849397 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_269_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (14337933849397 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_269_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (14337933849397 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_269_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (14337933849397 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_269_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_270_1 :
+    Uω (aρ 1) (bρ 1) (3593013491621 / 16000000000000 : ℝ) ≤
+      (-(15227731364814907513837 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2181062090711159 / 10000000000000000 : ℝ) ≤
+      (-(15227731364814907513837 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2181062090711159 / 10000000000000000 : ℝ) =
+            (2181062090711159 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2181062090711159 / 1250000000000000 : ℝ) ≤
+            (5566684044185092486163 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2180987967244193 / 10000000000000000 : ℝ)) (q :=
+      (2181062090711159 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_270_2 :
+    Uω (aρ 2) (bρ 2) (3593013491621 / 16000000000000 : ℝ) ≤
+      (-(7669622644416340882011 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (431375064042287 / 2000000000000000 : ℝ) ≤
+      (-(7669622644416340882011 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (431375064042287 / 2000000000000000 : ℝ) =
+            (431375064042287 / 250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (431375064042287 / 250000000000000 : ℝ) ≤
+            (2727585060083659117989 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (431276687750949 / 2000000000000000 : ℝ)) (q :=
+      (431375064042287 / 2000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_270_3 :
+    Uω (aρ 3) (bρ 3) (3593013491621 / 16000000000000 : ℝ) ≤
+      (-(15567427568303136711517 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4216433025271839 / 20000000000000000 : ℝ) ≤
+      (-(15567427568303136711517 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4216433025271839 / 20000000000000000 : ℝ) =
+            (4216433025271839 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4216433025271839 / 2500000000000000 : ℝ) ≤
+            (5226987840696863288483 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1053230964976857 / 5000000000000000 : ℝ)) (q :=
+      (4216433025271839 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_270_4 :
+    Uω (aρ 4) (bρ 4) (3593013491621 / 16000000000000 : ℝ) ≤
+      (-(7981410650442638429787 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4052970798400113 / 20000000000000000 : ℝ) ≤
+      (-(7981410650442638429787 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4052970798400113 / 20000000000000000 : ℝ) =
+            (4052970798400113 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4052970798400113 / 2500000000000000 : ℝ) ≤
+            (2415797054057361570213 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (505323471036747 / 2500000000000000 : ℝ)) (q :=
+      (4052970798400113 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_270_5 :
+    Uω (aρ 5) (bρ 5) (3593013491621 / 16000000000000 : ℝ) ≤
+      (-(16610111915455838225121 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (759787240803271 / 4000000000000000 : ℝ) ≤
+      (-(16610111915455838225121 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (759787240803271 / 4000000000000000 : ℝ) =
+            (759787240803271 / 500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (759787240803271 / 500000000000000 : ℝ) ≤
+            (4184303493544161774879 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (188544920909823 / 1000000000000000 : ℝ)) (q :=
+      (759787240803271 / 4000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_270_6 :
+    Uω (aρ 6) (bρ 6) (3593013491621 / 16000000000000 : ℝ) ≤
+      (-(17658749634887927061711 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (855185348357439 / 5000000000000000 : ℝ) ≤
+      (-(17658749634887927061711 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (855185348357439 / 5000000000000000 : ℝ) =
+            (855185348357439 / 625000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (855185348357439 / 625000000000000 : ℝ) ≤
+            (3135665774112072938289 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1674494773651631 / 10000000000000000 : ℝ)) (q :=
+      (855185348357439 / 5000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_270_7 :
+    Uω (aρ 7) (bρ 7) (3593013491621 / 16000000000000 : ℝ) ≤
+      (-(9717453763183694041361 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2864063784966021 / 20000000000000000 : ℝ) ≤
+      (-(9717453763183694041361 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2864063784966021 / 20000000000000000 : ℝ) =
+            (2864063784966021 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2864063784966021 / 2500000000000000 : ℝ) ≤
+            (679753941316305958639 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (167684818243487 / 1250000000000000 : ℝ)) (q :=
+      (2864063784966021 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_270_8 :
+    Uω (aρ 8) (bρ 8) (3593013491621 / 16000000000000 : ℝ) ≤
+      (-(5801248140116723696391 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (491122670330093 / 5000000000000000 : ℝ) ≤
+      (-(5801248140116723696391 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (491122670330093 / 5000000000000000 : ℝ) =
+            (491122670330093 / 312500000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (491122670330093 / 312500000000000 : ℝ) ≤
+            (1130223662883276303609 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (725447093642247 / 10000000000000000 : ℝ)) (q :=
+      (491122670330093 / 5000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_270_9 :
+    Uω (aρ 9) (bρ 9) (3593013491621 / 16000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_270_10 :
+    Uω (aρ 10) (bρ 10) (3593013491621 / 16000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_270_11 :
+    Uω (aρ 11) (bρ 11) (3593013491621 / 16000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_270_12 :
+    Uω (aρ 12) (bρ 12) (3593013491621 / 16000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_270_13 :
+    Uω (aρ 13) (bρ 13) (3593013491621 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_270_14 :
+    Uω (aρ 14) (bρ 14) (3593013491621 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_270_15 :
+    Uω (aρ 15) (bρ 15) (3593013491621 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_270_16 :
+    Uω (aρ 16) (bρ 16) (3593013491621 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_270 :
+    Uρ (3593013491621 / 16000000000000 : ℝ) ≤
+      (-(2333085713962513780869 / 1250000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (3593013491621 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(15227731364814907513837 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_270_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (3593013491621 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(7669622644416340882011 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_270_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (3593013491621 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(15567427568303136711517 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_270_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (3593013491621 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(7981410650442638429787 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_270_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (3593013491621 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(16610111915455838225121 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_270_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (3593013491621 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(17658749634887927061711 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_270_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (3593013491621 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(9717453763183694041361 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_270_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (3593013491621 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(5801248140116723696391 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_270_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (3593013491621 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_270_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (3593013491621 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_270_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (3593013491621 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_270_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (3593013491621 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_270_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (3593013491621 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_270_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (3593013491621 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_270_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (3593013491621 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_270_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (3593013491621 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_270_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_271_1 :
+    Uω (aρ 1) (bρ 1) (14406174083571 / 64000000000000 : ℝ) ≤
+      (-(7601658456640705685281 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3498229663605839 / 16000000000000000 : ℝ) ≤
+      (-(7601658456640705685281 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3498229663605839 / 16000000000000000 : ℝ) =
+            (3498229663605839 / 2000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3498229663605839 / 2000000000000000 : ℝ) ≤
+            (2795549247859294314719 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (218631959703433 / 1000000000000000 : ℝ)) (q :=
+      (3498229663605839 / 16000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_271_2 :
+    Uω (aρ 2) (bρ 2) (14406174083571 / 64000000000000 : ℝ) ≤
+      (-(15314552621698947647151 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17297662412755607 / 80000000000000000 : ℝ) ≤
+      (-(15314552621698947647151 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (17297662412755607 / 80000000000000000 : ℝ) =
+            (17297662412755607 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17297662412755607 / 10000000000000000 : ℝ) ≤
+            (5479862787301052352849 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2161717133225933 / 10000000000000000 : ℝ)) (q :=
+      (17297662412755607 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_271_3 :
+    Uω (aρ 3) (bρ 3) (14406174083571 / 64000000000000 : ℝ) ≤
+      (-(7771075241021788038101 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3381683536646363 / 16000000000000000 : ℝ) ≤
+      (-(7771075241021788038101 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3381683536646363 / 16000000000000000 : ℝ) =
+            (3381683536646363 / 2000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3381683536646363 / 2000000000000000 : ℝ) ≤
+            (2626132463478211961899 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (422360411438997 / 2000000000000000 : ℝ)) (q :=
+      (3381683536646363 / 16000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_271_4 :
+    Uω (aρ 4) (bρ 4) (14406174083571 / 64000000000000 : ℝ) ≤
+      (-(15936480678178473697243 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3250928518839247 / 16000000000000000 : ℝ) ≤
+      (-(15936480678178473697243 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3250928518839247 / 16000000000000000 : ℝ) =
+            (3250928518839247 / 2000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3250928518839247 / 2000000000000000 : ℝ) ≤
+            (4857934730821526302757 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (202665246600109 / 1000000000000000 : ℝ)) (q :=
+      (3250928518839247 / 16000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_271_5 :
+    Uω (aρ 5) (bρ 5) (14406174083571 / 64000000000000 : ℝ) ≤
+      (-(103636728471640727917 / 62500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15238711178580651 / 80000000000000000 : ℝ) ≤
+      (-(103636728471640727917 / 62500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (15238711178580651 / 80000000000000000 : ℝ) =
+            (15238711178580651 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15238711178580651 / 10000000000000000 : ℝ) ≤
+            (26328367834609272083 / 62500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (945429765716097 / 5000000000000000 : ℝ)) (q :=
+      (15238711178580651 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_271_6 :
+    Uω (aρ 6) (bρ 6) (14406174083571 / 64000000000000 : ℝ) ≤
+      (-(17626964297738276232173 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13726526534431967 / 80000000000000000 : ℝ) ≤
+      (-(17626964297738276232173 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (13726526534431967 / 80000000000000000 : ℝ) =
+            (13726526534431967 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13726526534431967 / 10000000000000000 : ℝ) ≤
+            (3167451111261723767827 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1680053745535023 / 10000000000000000 : ℝ)) (q :=
+      (13726526534431967 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_271_7 :
+    Uω (aρ 7) (bρ 7) (14406174083571 / 64000000000000 : ℝ) ≤
+      (-(9697627563002555907191 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11501772124656023 / 80000000000000000 : ℝ) ≤
+      (-(9697627563002555907191 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (11501772124656023 / 80000000000000000 : ℝ) =
+            (11501772124656023 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11501772124656023 / 10000000000000000 : ℝ) ≤
+            (699580141497444092809 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1347526523851037 / 10000000000000000 : ℝ)) (q :=
+      (11501772124656023 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_271_8 :
+    Uω (aρ 8) (bρ 8) (14406174083571 / 64000000000000 : ℝ) ≤
+      (-(23131959300184916443993 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7915562066343051 / 80000000000000000 : ℝ) ≤
+      (-(23131959300184916443993 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (7915562066343051 / 80000000000000000 : ℝ) =
+            (7915562066343051 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7915562066343051 / 5000000000000000 : ℝ) ≤
+            (4593927911815083556007 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (367257830306397 / 5000000000000000 : ℝ)) (q :=
+      (7915562066343051 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_271_9 :
+    Uω (aρ 9) (bρ 9) (14406174083571 / 64000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_271_10 :
+    Uω (aρ 10) (bρ 10) (14406174083571 / 64000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_271_11 :
+    Uω (aρ 11) (bρ 11) (14406174083571 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_271_12 :
+    Uω (aρ 12) (bρ 12) (14406174083571 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_271_13 :
+    Uω (aρ 13) (bρ 13) (14406174083571 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_271_14 :
+    Uω (aρ 14) (bρ 14) (14406174083571 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_271_15 :
+    Uω (aρ 15) (bρ 15) (14406174083571 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_271_16 :
+    Uω (aρ 16) (bρ 16) (14406174083571 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_271 :
+    Uρ (14406174083571 / 64000000000000 : ℝ) ≤
+      (-(18646805938925348800429 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (14406174083571 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(7601658456640705685281 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_271_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (14406174083571 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(15314552621698947647151 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_271_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (14406174083571 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(7771075241021788038101 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_271_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (14406174083571 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(15936480678178473697243 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_271_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (14406174083571 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(103636728471640727917 / 62500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_271_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (14406174083571 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(17626964297738276232173 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_271_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (14406174083571 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(9697627563002555907191 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_271_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (14406174083571 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(23131959300184916443993 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_271_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (14406174083571 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_271_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (14406174083571 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_271_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (14406174083571 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_271_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (14406174083571 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_271_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (14406174083571 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_271_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (14406174083571 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_271_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (14406174083571 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_271_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (14406174083571 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_271_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_272_1 :
+    Uω (aρ 1) (bρ 1) (7220147100329 / 32000000000000 : ℝ) ≤
+      (-(15178961927162266424097 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8766899951667041 / 40000000000000000 : ℝ) ≤
+      (-(15178961927162266424097 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (8766899951667041 / 40000000000000000 : ℝ) =
+            (8766899951667041 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8766899951667041 / 5000000000000000 : ℝ) ≤
+            (5615453481837733575903 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (547912806266427 / 2500000000000000 : ℝ)) (q :=
+      (8766899951667041 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_272_2 :
+    Uω (aρ 2) (bρ 2) (7220147100329 / 32000000000000 : ℝ) ≤
+      (-(1528992080473304163929 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8670161108029413 / 40000000000000000 : ℝ) ≤
+      (-(1528992080473304163929 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (8670161108029413 / 40000000000000000 : ℝ) =
+            (8670161108029413 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8670161108029413 / 5000000000000000 : ℝ) ≤
+            (550449460426695836071 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1083525407878447 / 5000000000000000 : ℝ)) (q :=
+      (8670161108029413 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_272_3 :
+    Uω (aρ 3) (bρ 3) (7220147100329 / 32000000000000 : ℝ) ≤
+      (-(3879234308300097326093 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1695110308676319 / 8000000000000000 : ℝ) ≤
+      (-(3879234308300097326093 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1695110308676319 / 8000000000000000 : ℝ) =
+            (1695110308676319 / 1000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1695110308676319 / 1000000000000000 : ℝ) ≤
+            (1319369543949902673907 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (423428427956597 / 2000000000000000 : ℝ)) (q :=
+      (1695110308676319 / 8000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_272_4 :
+    Uω (aρ 4) (bρ 4) (7220147100329 / 32000000000000 : ℝ) ≤
+      (-(15910209608740880248149 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8148700710017781 / 40000000000000000 : ℝ) ≤
+      (-(15910209608740880248149 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (8148700710017781 / 40000000000000000 : ℝ) =
+            (8148700710017781 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8148700710017781 / 5000000000000000 : ℝ) ≤
+            (4884205800259119751851 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1016005452083039 / 5000000000000000 : ℝ)) (q :=
+      (8148700710017781 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_272_5 :
+    Uω (aρ 5) (bρ 5) (7220147100329 / 32000000000000 : ℝ) ≤
+      (-(16553721866754159805869 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7640837874930451 / 40000000000000000 : ℝ) ≤
+      (-(16553721866754159805869 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (7640837874930451 / 40000000000000000 : ℝ) =
+            (7640837874930451 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7640837874930451 / 5000000000000000 : ℝ) ≤
+            (4240693542245840194131 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1896269405957413 / 10000000000000000 : ℝ)) (q :=
+      (7640837874930451 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_272_6 :
+    Uω (aρ 6) (bρ 6) (7220147100329 / 32000000000000 : ℝ) ≤
+      (-(54985262327256662747 / 31250000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6885040805321457 / 40000000000000000 : ℝ) ≤
+      (-(54985262327256662747 / 31250000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (6885040805321457 / 40000000000000000 : ℝ) =
+            (6885040805321457 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6885040805321457 / 5000000000000000 : ℝ) ≤
+            (9997285825868337253 / 31250000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (421402811573229 / 2500000000000000 : ℝ)) (q :=
+      (6885040805321457 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_272_7 :
+    Uω (aρ 7) (bρ 7) (7220147100329 / 32000000000000 : ℝ) ≤
+      (-(19355780207743348837579 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5773632504211469 / 40000000000000000 : ℝ) ≤
+      (-(19355780207743348837579 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (5773632504211469 / 40000000000000000 : ℝ) =
+            (5773632504211469 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5773632504211469 / 5000000000000000 : ℝ) ≤
+            (1438635201256651162421 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (676784238248961 / 5000000000000000 : ℝ)) (q :=
+      (5773632504211469 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_272_8 :
+    Uω (aρ 8) (bρ 8) (7220147100329 / 32000000000000 : ℝ) ≤
+      (-(23059818675209941582027 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3986435947795567 / 40000000000000000 : ℝ) ≤
+      (-(23059818675209941582027 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (3986435947795567 / 40000000000000000 : ℝ) =
+            (3986435947795567 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3986435947795567 / 2500000000000000 : ℝ) ≤
+            (4666068536790058417973 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (743511849629971 / 10000000000000000 : ℝ)) (q :=
+      (3986435947795567 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_272_9 :
+    Uω (aρ 9) (bρ 9) (7220147100329 / 32000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_272_10 :
+    Uω (aρ 10) (bρ 10) (7220147100329 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_272_11 :
+    Uω (aρ 11) (bρ 11) (7220147100329 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_272_12 :
+    Uω (aρ 12) (bρ 12) (7220147100329 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_272_13 :
+    Uω (aρ 13) (bρ 13) (7220147100329 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_272_14 :
+    Uω (aρ 14) (bρ 14) (7220147100329 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_272_15 :
+    Uω (aρ 15) (bρ 15) (7220147100329 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_272_16 :
+    Uω (aρ 16) (bρ 16) (7220147100329 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_272 :
+    Uρ (7220147100329 / 32000000000000 : ℝ) ≤
+      (-(9314521594811334388811 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (7220147100329 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(15178961927162266424097 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_272_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (7220147100329 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1528992080473304163929 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_272_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (7220147100329 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3879234308300097326093 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_272_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (7220147100329 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(15910209608740880248149 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_272_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (7220147100329 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(16553721866754159805869 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_272_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (7220147100329 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) * (-(54985262327256662747 / 31250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_272_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (7220147100329 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(19355780207743348837579 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_272_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (7220147100329 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(23059818675209941582027 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_272_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (7220147100329 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_272_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (7220147100329 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_272_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (7220147100329 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_272_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (7220147100329 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_272_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (7220147100329 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_272_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (7220147100329 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_272_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (7220147100329 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_272_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (7220147100329 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_272_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_273_1 :
+    Uω (aρ 1) (bρ 1) (2894882863549 / 12800000000000 : ℝ) ≤
+      (-(15154666117466165170289 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3515290296331029 / 16000000000000000 : ℝ) ≤
+      (-(15154666117466165170289 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3515290296331029 / 16000000000000000 : ℝ) =
+            (3515290296331029 / 2000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3515290296331029 / 2000000000000000 : ℝ) ≤
+            (5639749291533834829711 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (219698285135113 / 1000000000000000 : ℝ)) (q :=
+      (3515290296331029 / 16000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_273_2 :
+    Uω (aρ 2) (bρ 2) (2894882863549 / 12800000000000 : ℝ) ≤
+      (-(3816337384657765063509 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (695319278878117 / 3200000000000000 : ℝ) ≤
+      (-(3816337384657765063509 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (695319278878117 / 3200000000000000 : ℝ) =
+            (695319278878117 / 400000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (695319278878117 / 400000000000000 : ℝ) ≤
+            (1382266467592234936491 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (86895379457423 / 400000000000000 : ℝ)) (q :=
+      (695319278878117 / 3200000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_273_3 :
+    Uω (aρ 3) (bρ 3) (2894882863549 / 12800000000000 : ℝ) ≤
+      (-(7745893749807219680829 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16993788313029637 / 80000000000000000 : ℝ) ≤
+      (-(7745893749807219680829 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (16993788313029637 / 80000000000000000 : ℝ) =
+            (16993788313029637 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16993788313029637 / 10000000000000000 : ℝ) ≤
+            (2651313954692780319171 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2122482178054753 / 10000000000000000 : ℝ)) (q :=
+      (16993788313029637 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_273_4 :
+    Uω (aρ 4) (bρ 4) (2894882863549 / 12800000000000 : ℝ) ≤
+      (-(15884007724381241620549 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16340159675653393 / 80000000000000000 : ℝ) ≤
+      (-(15884007724381241620549 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (16340159675653393 / 80000000000000000 : ℝ) =
+            (16340159675653393 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16340159675653393 / 10000000000000000 : ℝ) ≤
+            (4910407684618758379451 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (509342299943923 / 2500000000000000 : ℝ)) (q :=
+      (16340159675653393 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_273_5 :
+    Uω (aρ 5) (bρ 5) (2894882863549 / 12800000000000 : ℝ) ≤
+      (-(4131411845769879803169 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15324638545193217 / 80000000000000000 : ℝ) ≤
+      (-(4131411845769879803169 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (15324638545193217 / 80000000000000000 : ℝ) =
+            (15324638545193217 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15324638545193217 / 10000000000000000 : ℝ) ≤
+            (1067192006480120196831 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (59427463640489 / 312500000000000 : ℝ)) (q :=
+      (15324638545193217 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_273_6 :
+    Uω (aρ 6) (bρ 6) (2894882863549 / 12800000000000 : ℝ) ≤
+      (-(4390926964309566989123 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13813630860364677 / 80000000000000000 : ℝ) ≤
+      (-(4390926964309566989123 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (13813630860364677 / 80000000000000000 : ℝ) =
+            (13813630860364677 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13813630860364677 / 10000000000000000 : ℝ) ≤
+            (807676887940433010877 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1691167290428513 / 10000000000000000 : ℝ)) (q :=
+      (13813630860364677 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_273_7 :
+    Uω (aρ 7) (bρ 7) (2894882863549 / 12800000000000 : ℝ) ≤
+      (-(9658240508237714690711 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11592734112472793 / 80000000000000000 : ℝ) ≤
+      (-(9658240508237714690711 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (11592734112472793 / 80000000000000000 : ℝ) =
+            (11592734112472793 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11592734112472793 / 10000000000000000 : ℝ) ≤
+            (738967196262285309289 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (679802242107771 / 5000000000000000 : ℝ)) (q :=
+      (11592734112472793 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_273_8 :
+    Uω (aρ 8) (bρ 8) (2894882863549 / 12800000000000 : ℝ) ≤
+      (-(22988542356000148652769 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8029902597296329 / 80000000000000000 : ℝ) ≤
+      (-(22988542356000148652769 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (8029902597296329 / 80000000000000000 : ℝ) =
+            (8029902597296329 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8029902597296329 / 5000000000000000 : ℝ) ≤
+            (4737344855999851347231 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (376219128380713 / 5000000000000000 : ℝ)) (q :=
+      (8029902597296329 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_273_9 :
+    Uω (aρ 9) (bρ 9) (2894882863549 / 12800000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_273_10 :
+    Uω (aρ 10) (bρ 10) (2894882863549 / 12800000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_273_11 :
+    Uω (aρ 11) (bρ 11) (2894882863549 / 12800000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_273_12 :
+    Uω (aρ 12) (bρ 12) (2894882863549 / 12800000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_273_13 :
+    Uω (aρ 13) (bρ 13) (2894882863549 / 12800000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_273_14 :
+    Uω (aρ 14) (bρ 14) (2894882863549 / 12800000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_273_15 :
+    Uω (aρ 15) (bρ 15) (2894882863549 / 12800000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_273_16 :
+    Uω (aρ 16) (bρ 16) (2894882863549 / 12800000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_273 :
+    Uρ (2894882863549 / 12800000000000 : ℝ) ≤
+      (-(9305697337325464503709 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (2894882863549 / 12800000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(15154666117466165170289 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_273_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (2894882863549 / 12800000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3816337384657765063509 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_273_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (2894882863549 / 12800000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(7745893749807219680829 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_273_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (2894882863549 / 12800000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(15884007724381241620549 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_273_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (2894882863549 / 12800000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(4131411845769879803169 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_273_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (2894882863549 / 12800000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(4390926964309566989123 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_273_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (2894882863549 / 12800000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(9658240508237714690711 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_273_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (2894882863549 / 12800000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(22988542356000148652769 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_273_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (2894882863549 / 12800000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_273_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (2894882863549 / 12800000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_273_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (2894882863549 / 12800000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_273_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (2894882863549 / 12800000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_273_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (2894882863549 / 12800000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_273_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (2894882863549 / 12800000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_273_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (2894882863549 / 12800000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_273_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (2894882863549 / 12800000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_273_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_274_1 :
+    Uω (aρ 1) (bρ 1) (906783402177 / 4000000000000 : ℝ) ≤
+      (-(15130429197303497441467 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2202387881630389 / 10000000000000000 : ℝ) ≤
+      (-(15130429197303497441467 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2202387881630389 / 10000000000000000 : ℝ) =
+            (2202387881630389 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2202387881630389 / 1250000000000000 : ℝ) ≤
+            (5663986211696502558533 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1101157237951639 / 5000000000000000 : ℝ)) (q :=
+      (2202387881630389 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_274_2 :
+    Uω (aρ 2) (bρ 2) (906783402177 / 4000000000000 : ℝ) ≤
+      (-(15240838526292947970749 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2178205210098281 / 10000000000000000 : ℝ) ≤
+      (-(15240838526292947970749 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2178205210098281 / 10000000000000000 : ℝ) =
+            (2178205210098281 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2178205210098281 / 1250000000000000 : ℝ) ≤
+            (5553576882707052029251 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1088859072674531 / 5000000000000000 : ℝ)) (q :=
+      (2178205210098281 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_274_3 :
+    Uω (aρ 3) (bρ 3) (906783402177 / 4000000000000 : ℝ) ≤
+      (-(3866675240390775093581 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (26619489630259 / 125000000000000 : ℝ) ≤
+      (-(3866675240390775093581 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (26619489630259 / 125000000000000 : ℝ) = (26619489630259 / 15625000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26619489630259 / 15625000000000 : ℝ) ≤
+            (1331928611859224906419 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (106391108617197 / 500000000000000 : ℝ)) (q :=
+      (26619489630259 / 125000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_274_4 :
+    Uω (aρ 4) (bρ 4) (906783402177 / 4000000000000 : ℝ) ≤
+      (-(15857874659838571656279 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4095729341384249 / 20000000000000000 : ℝ) ≤
+      (-(15857874659838571656279 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4095729341384249 / 20000000000000000 : ℝ) =
+            (4095729341384249 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4095729341384249 / 2500000000000000 : ℝ) ≤
+            (4936540749161428343721 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2042727353951749 / 10000000000000000 : ℝ)) (q :=
+      (4095729341384249 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_274_5 :
+    Uω (aρ 5) (bρ 5) (906783402177 / 4000000000000 : ℝ) ≤
+      (-(6444395563387845643 / 3906250000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (960474973730623 / 5000000000000000 : ℝ) ≤
+      (-(6444395563387845643 / 3906250000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (960474973730623 / 5000000000000000 : ℝ) =
+            (960474973730623 / 625000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (960474973730623 / 625000000000000 : ℝ) ≤
+            (1678422955752779357 / 3906250000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (59596494588281 / 312500000000000 : ℝ)) (q :=
+      (960474973730623 / 5000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_274_6 :
+    Uω (aρ 6) (bρ 6) (906783402177 / 4000000000000 : ℝ) ≤
+      (-(17532235324295299403419 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3464293585208773 / 20000000000000000 : ℝ) ≤
+      (-(17532235324295299403419 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3464293585208773 / 20000000000000000 : ℝ) =
+            (3464293585208773 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3464293585208773 / 2500000000000000 : ℝ) ≤
+            (3262180084704700596581 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1696721892251273 / 10000000000000000 : ℝ)) (q :=
+      (3464293585208773 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_274_7 :
+    Uω (aρ 7) (bρ 7) (906783402177 / 4000000000000 : ℝ) ≤
+      (-(4819338956257651832129 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2909544938029893 / 20000000000000000 : ℝ) ≤
+      (-(4819338956257651832129 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2909544938029893 / 20000000000000000 : ℝ) =
+            (2909544938029893 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2909544938029893 / 2500000000000000 : ℝ) ≤
+            (379264895992348167871 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1365634625832393 / 10000000000000000 : ℝ)) (q :=
+      (2909544938029893 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_274_8 :
+    Uω (aρ 8) (bρ 8) (906783402177 / 4000000000000 : ℝ) ≤
+      (-(11459051748918678550413 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2021665997500307 / 20000000000000000 : ℝ) ≤
+      (-(11459051748918678550413 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2021665997500307 / 20000000000000000 : ℝ) =
+            (2021665997500307 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2021665997500307 / 1250000000000000 : ℝ) ≤
+            (2403891857081321449587 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (761297336642807 / 10000000000000000 : ℝ)) (q :=
+      (2021665997500307 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_274_9 :
+    Uω (aρ 9) (bρ 9) (906783402177 / 4000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_274_10 :
+    Uω (aρ 10) (bρ 10) (906783402177 / 4000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_274_11 :
+    Uω (aρ 11) (bρ 11) (906783402177 / 4000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_274_12 :
+    Uω (aρ 12) (bρ 12) (906783402177 / 4000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_274_13 :
+    Uω (aρ 13) (bρ 13) (906783402177 / 4000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_274_14 :
+    Uω (aρ 14) (bρ 14) (906783402177 / 4000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_274_15 :
+    Uω (aρ 15) (bρ 15) (906783402177 / 4000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_274_16 :
+    Uω (aρ 16) (bρ 16) (906783402177 / 4000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_274 :
+    Uρ (906783402177 / 4000000000000 : ℝ) ≤
+      (-(9296928869868299308561 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (906783402177 / 4000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(15130429197303497441467 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_274_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (906783402177 / 4000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(15240838526292947970749 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_274_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (906783402177 / 4000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3866675240390775093581 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_274_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (906783402177 / 4000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(15857874659838571656279 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_274_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (906783402177 / 4000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(6444395563387845643 / 3906250000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_274_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (906783402177 / 4000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(17532235324295299403419 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_274_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (906783402177 / 4000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(4819338956257651832129 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_274_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (906783402177 / 4000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(11459051748918678550413 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_274_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (906783402177 / 4000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_274_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (906783402177 / 4000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_274_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (906783402177 / 4000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_274_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (906783402177 / 4000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_274_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (906783402177 / 4000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_274_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (906783402177 / 4000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_274_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (906783402177 / 4000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_274_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (906783402177 / 4000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_274_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_275_1 :
+    Uω (aρ 1) (bρ 1) (14542654551919 / 64000000000000 : ℝ) ≤
+      (-(15106250881866031193979 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17661754617548211 / 80000000000000000 : ℝ) ≤
+      (-(15106250881866031193979 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (17661754617548211 / 80000000000000000 : ℝ) =
+            (17661754617548211 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17661754617548211 / 10000000000000000 : ℝ) ≤
+            (5688164527133968806021 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2207646098734709 / 10000000000000000 : ℝ)) (q :=
+      (17661754617548211 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_275_2 :
+    Uω (aρ 2) (bρ 2) (14542654551919 / 64000000000000 : ℝ) ≤
+      (-(15216387472800909762541 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17468301342903727 / 80000000000000000 : ℝ) ≤
+      (-(15216387472800909762541 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (17468301342903727 / 80000000000000000 : ℝ) =
+            (17468301342903727 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17468301342903727 / 10000000000000000 : ℝ) ≤
+            (5578027936199090237459 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (545762948145897 / 2500000000000000 : ℝ)) (q :=
+      (17468301342903727 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_275_3 :
+    Uω (aρ 3) (bρ 3) (14542654551919 / 64000000000000 : ℝ) ≤
+      (-(15441677301735706487299 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17079158239092787 / 80000000000000000 : ℝ) ≤
+      (-(15441677301735706487299 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (17079158239092787 / 80000000000000000 : ℝ) =
+            (17079158239092787 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17079158239092787 / 10000000000000000 : ℝ) ≤
+            (5352738107264293512701 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2133162122980853 / 10000000000000000 : ℝ)) (q :=
+      (17079158239092787 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_275_4 :
+    Uω (aρ 4) (bρ 4) (14542654551919 / 64000000000000 : ℝ) ≤
+      (-(633272402110040347591 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16425674494126547 / 80000000000000000 : ℝ) ≤
+      (-(633272402110040347591 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (16425674494126547 / 80000000000000000 : ℝ) =
+            (16425674494126547 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16425674494126547 / 10000000000000000 : ℝ) ≤
+            (198504214249959652409 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2048085367804293 / 10000000000000000 : ℝ)) (q :=
+      (16425674494126547 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_275_5 :
+    Uω (aρ 5) (bρ 5) (14542654551919 / 64000000000000 : ℝ) ≤
+      (-(8234868593102955975117 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15410558868291319 / 80000000000000000 : ℝ) ≤
+      (-(8234868593102955975117 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (15410558868291319 / 80000000000000000 : ℝ) =
+            (15410558868291319 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15410558868291319 / 10000000000000000 : ℝ) ≤
+            (2162339111397044024883 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (956248190340243 / 5000000000000000 : ℝ)) (q :=
+      (15410558868291319 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_275_6 :
+    Uω (aρ 6) (bρ 6) (14542654551919 / 64000000000000 : ℝ) ≤
+      (-(17500865642401456087003 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13900712108530163 / 80000000000000000 : ℝ) ≤
+      (-(17500865642401456087003 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (13900712108530163 / 80000000000000000 : ℝ) =
+            (13900712108530163 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13900712108530163 / 10000000000000000 : ℝ) ≤
+            (3293549766598543912997 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1702275065880197 / 10000000000000000 : ℝ)) (q :=
+      (13900712108530163 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_275_7 :
+    Uω (aρ 7) (bρ 7) (14542654551919 / 64000000000000 : ℝ) ≤
+      (-(384768058671075131277 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11683602236828703 / 80000000000000000 : ℝ) ≤
+      (-(384768058671075131277 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (11683602236828703 / 80000000000000000 : ℝ) =
+            (11683602236828703 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11683602236828703 / 10000000000000000 : ℝ) ≤
+            (31120249508924868723 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (85728686169677 / 625000000000000 : ℝ)) (q :=
+      (11683602236828703 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_275_8 :
+    Uω (aρ 8) (bρ 8) (14542654551919 / 64000000000000 : ℝ) ≤
+      (-(22848476633536171818119 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8143165368057791 / 80000000000000000 : ℝ) ≤
+      (-(22848476633536171818119 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (8143165368057791 / 80000000000000000 : ℝ) =
+            (8143165368057791 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8143165368057791 / 5000000000000000 : ℝ) ≤
+            (4877410578463828181881 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (96261426607763 / 1250000000000000 : ℝ)) (q :=
+      (8143165368057791 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_275_9 :
+    Uω (aρ 9) (bρ 9) (14542654551919 / 64000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_275_10 :
+    Uω (aρ 10) (bρ 10) (14542654551919 / 64000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_275_11 :
+    Uω (aρ 11) (bρ 11) (14542654551919 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_275_12 :
+    Uω (aρ 12) (bρ 12) (14542654551919 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_275_13 :
+    Uω (aρ 13) (bρ 13) (14542654551919 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_275_14 :
+    Uω (aρ 14) (bρ 14) (14542654551919 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_275_15 :
+    Uω (aρ 15) (bρ 15) (14542654551919 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_275_16 :
+    Uω (aρ 16) (bρ 16) (14542654551919 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_275 :
+    Uρ (14542654551919 / 64000000000000 : ℝ) ≤
+      (-(290256716498770197431 / 156250000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (14542654551919 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(15106250881866031193979 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_275_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (14542654551919 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(15216387472800909762541 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_275_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (14542654551919 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(15441677301735706487299 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_275_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (14542654551919 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(633272402110040347591 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_275_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (14542654551919 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(8234868593102955975117 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_275_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (14542654551919 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(17500865642401456087003 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_275_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (14542654551919 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(384768058671075131277 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_275_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (14542654551919 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(22848476633536171818119 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_275_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (14542654551919 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_275_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (14542654551919 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_275_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (14542654551919 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_275_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (14542654551919 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_275_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (14542654551919 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_275_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (14542654551919 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_275_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (14542654551919 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_275_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (14542654551919 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_275_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_276_1 :
+    Uω (aρ 1) (bρ 1) (7288387334503 / 32000000000000 : ℝ) ≤
+      (-(1508213088840681581873 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8852203087610097 / 40000000000000000 : ℝ) ≤
+      (-(1508213088840681581873 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (8852203087610097 / 40000000000000000 : ℝ) =
+            (8852203087610097 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8852203087610097 / 5000000000000000 : ℝ) ≤
+            (571228452059318418127 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2212977719857861 / 10000000000000000 : ℝ)) (q :=
+      (8852203087610097 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_276_2 :
+    Uω (aρ 2) (bρ 2) (7288387334503 / 32000000000000 : ℝ) ≤
+      (-(15191996085398068846649 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8755480479323473 / 40000000000000000 : ℝ) ≤
+      (-(15191996085398068846649 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (8755480479323473 / 40000000000000000 : ℝ) =
+            (8755480479323473 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8755480479323473 / 5000000000000000 : ℝ) ≤
+            (5602419323601931153351 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2188385428224549 / 10000000000000000 : ℝ)) (q :=
+      (8755480479323473 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_276_3 :
+    Uω (aρ 3) (bρ 3) (7288387334503 / 32000000000000 : ℝ) ≤
+      (-(3854179051302327034767 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8560921470759373 / 40000000000000000 : ℝ) ≤
+      (-(3854179051302327034767 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (8560921470759373 / 40000000000000000 : ℝ) =
+            (8560921470759373 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8560921470759373 / 5000000000000000 : ℝ) ≤
+            (1344424800947672965233 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2138502030292499 / 10000000000000000 : ℝ)) (q :=
+      (8560921470759373 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_276_4 :
+    Uω (aρ 4) (bρ 4) (7288387334503 / 32000000000000 : ℝ) ≤
+      (-(1975726692953131972251 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1646843106581577 / 8000000000000000 : ℝ) ≤
+      (-(1975726692953131972251 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1646843106581577 / 8000000000000000 : ℝ) =
+            (1646843106581577 / 1000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1646843106581577 / 1000000000000000 : ℝ) ≤
+            (623575233171868027749 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (410688648486351 / 2000000000000000 : ℝ)) (q :=
+      (1646843106581577 / 8000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_276_5 :
+    Uω (aρ 5) (bρ 5) (7288387334503 / 32000000000000 : ℝ) ≤
+      (-(328838011214802240827 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7726758212883863 / 40000000000000000 : ℝ) ≤
+      (-(328838011214802240827 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (7726758212883863 / 40000000000000000 : ℝ) =
+            (7726758212883863 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7726758212883863 / 5000000000000000 : ℝ) ≤
+            (87050296965197759173 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (239738062719343 / 1250000000000000 : ℝ)) (q :=
+      (7726758212883863 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_276_6 :
+    Uω (aρ 6) (bρ 6) (7288387334503 / 32000000000000 : ℝ) ≤
+      (-(17469598115456413670451 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6972122109588451 / 40000000000000000 : ℝ) ≤
+      (-(17469598115456413670451 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (6972122109588451 / 40000000000000000 : ℝ) =
+            (6972122109588451 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6972122109588451 / 5000000000000000 : ℝ) ≤
+            (3324817293543586329549 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (853913412623519 / 5000000000000000 : ℝ)) (q :=
+      (6972122109588451 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_276_7 :
+    Uω (aρ 7) (bρ 7) (7288387334503 / 32000000000000 : ℝ) ≤
+      (-(3839924133780546951883 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5864500935179073 / 40000000000000000 : ℝ) ≤
+      (-(3839924133780546951883 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (5864500935179073 / 40000000000000000 : ℝ) =
+            (5864500935179073 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5864500935179073 / 5000000000000000 : ℝ) ≤
+            (318958948019453048117 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1377677618802349 / 10000000000000000 : ℝ)) (q :=
+      (5864500935179073 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_276_8 :
+    Uω (aρ 8) (bρ 8) (7288387334503 / 32000000000000 : ℝ) ≤
+      (-(2277963757593818057009 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4099707769672261 / 40000000000000000 : ℝ) ≤
+      (-(2277963757593818057009 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (4099707769672261 / 40000000000000000 : ℝ) =
+            (4099707769672261 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4099707769672261 / 2500000000000000 : ℝ) ≤
+            (494624963606181942991 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (778822687388943 / 10000000000000000 : ℝ)) (q :=
+      (4099707769672261 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_276_9 :
+    Uω (aρ 9) (bρ 9) (7288387334503 / 32000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_276_10 :
+    Uω (aρ 10) (bρ 10) (7288387334503 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_276_11 :
+    Uω (aρ 11) (bρ 11) (7288387334503 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_276_12 :
+    Uω (aρ 12) (bρ 12) (7288387334503 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_276_13 :
+    Uω (aρ 13) (bρ 13) (7288387334503 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_276_14 :
+    Uω (aρ 14) (bρ 14) (7288387334503 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_276_15 :
+    Uω (aρ 15) (bρ 15) (7288387334503 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_276_16 :
+    Uω (aρ 16) (bρ 16) (7288387334503 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_276 :
+    Uρ (7288387334503 / 32000000000000 : ℝ) ≤
+      (-(4639777152719005169681 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (7288387334503 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1508213088840681581873 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_276_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (7288387334503 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(15191996085398068846649 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_276_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (7288387334503 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3854179051302327034767 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_276_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (7288387334503 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1975726692953131972251 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_276_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (7288387334503 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(328838011214802240827 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_276_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (7288387334503 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(17469598115456413670451 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_276_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (7288387334503 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(3839924133780546951883 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_276_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (7288387334503 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(2277963757593818057009 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_276_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (7288387334503 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_276_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (7288387334503 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_276_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (7288387334503 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_276_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (7288387334503 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_276_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (7288387334503 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_276_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (7288387334503 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_276_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (7288387334503 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_276_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (7288387334503 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_276_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_277_1 :
+    Uω (aρ 1) (bρ 1) (732250745159 / 3200000000000 : ℝ) ≤
+      (-(15034064746622953868251 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (88948546351307 / 400000000000000 : ℝ) ≤
+      (-(15034064746622953868251 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (88948546351307 / 400000000000000 : ℝ) = (88948546351307 / 50000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (88948546351307 / 50000000000000 : ℝ) ≤
+            (5760350662377046131749 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (88945638281139 / 400000000000000 : ℝ)) (q :=
+      (88948546351307 / 400000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_277_2 :
+    Uω (aρ 2) (bρ 2) (732250745159 / 3200000000000 : ℝ) ≤
+      (-(3028678229702195023461 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27494187581813 / 125000000000000 : ℝ) ≤
+      (-(3028678229702195023461 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (27494187581813 / 125000000000000 : ℝ) = (27494187581813 / 15625000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27494187581813 / 15625000000000 : ℝ) ≤
+            (1130204852097804976539 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (439810533012641 / 2000000000000000 : ℝ)) (q :=
+      (27494187581813 / 125000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_277_3 :
+    Uω (aρ 3) (bρ 3) (732250745159 / 3200000000000 : ℝ) ≤
+      (-(384174511354075211361 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (34414423663269 / 160000000000000 : ℝ) ≤
+      (-(384174511354075211361 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (34414423663269 / 160000000000000 : ℝ) = (34414423663269 / 20000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (34414423663269 / 20000000000000 : ℝ) ≤
+            (135685873870924788639 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (8596726864927 / 40000000000000 : ℝ)) (q :=
+      (34414423663269 / 160000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_277_4 :
+    Uω (aρ 4) (bρ 4) (732250745159 / 3200000000000 : ℝ) ≤
+      (-(7877011697722102677853 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4138485638959871 / 20000000000000000 : ℝ) ≤
+      (-(7877011697722102677853 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4138485638959871 / 20000000000000000 : ℝ) =
+            (4138485638959871 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4138485638959871 / 2500000000000000 : ℝ) ≤
+            (2520196006777897322147 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (516039644586999 / 2500000000000000 : ℝ)) (q :=
+      (4138485638959871 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_277_5 :
+    Uω (aρ 5) (bρ 5) (732250745159 / 3200000000000 : ℝ) ≤
+      (-(4096615501181864826651 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3884856601398333 / 20000000000000000 : ℝ) ≤
+      (-(4096615501181864826651 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3884856601398333 / 20000000000000000 : ℝ) =
+            (3884856601398333 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3884856601398333 / 2500000000000000 : ℝ) ≤
+            (1101988351068135173349 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (964359730060729 / 5000000000000000 : ℝ)) (q :=
+      (3884856601398333 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_277_6 :
+    Uω (aρ 6) (bρ 6) (732250745159 / 3200000000000 : ℝ) ≤
+      (-(870368338916667326583 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (701564584428197 / 4000000000000000 : ℝ) ≤
+      (-(870368338916667326583 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (701564584428197 / 4000000000000000 : ℝ) =
+            (701564584428197 / 500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (701564584428197 / 500000000000000 : ℝ) ≤
+            (169352431533332673417 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (171892615600411 / 1000000000000000 : ℝ)) (q :=
+      (701564584428197 / 4000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_277_7 :
+    Uω (aρ 7) (bρ 7) (732250745159 / 3200000000000 : ℝ) ≤
+      (-(9561280728788798957347 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2954933442797441 / 20000000000000000 : ℝ) ≤
+      (-(9561280728788798957347 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2954933442797441 / 20000000000000000 : ℝ) =
+            (2954933442797441 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2954933442797441 / 2500000000000000 : ℝ) ≤
+            (835926975711201042653 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (694849028710283 / 5000000000000000 : ℝ)) (q :=
+      (2954933442797441 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_277_8 :
+    Uω (aρ 8) (bρ 8) (732250745159 / 3200000000000 : ℝ) ≤
+      (-(11322116003695119553231 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1038899407952763 / 10000000000000000 : ℝ) ≤
+      (-(11322116003695119553231 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1038899407952763 / 10000000000000000 : ℝ) =
+            (1038899407952763 / 625000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1038899407952763 / 625000000000000 : ℝ) ≤
+            (2540827602304880446769 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (796105081868651 / 10000000000000000 : ℝ)) (q :=
+      (1038899407952763 / 10000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_277_9 :
+    Uω (aρ 9) (bρ 9) (732250745159 / 3200000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_277_10 :
+    Uω (aρ 10) (bρ 10) (732250745159 / 3200000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_277_11 :
+    Uω (aρ 11) (bρ 11) (732250745159 / 3200000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_277_12 :
+    Uω (aρ 12) (bρ 12) (732250745159 / 3200000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_277_13 :
+    Uω (aρ 13) (bρ 13) (732250745159 / 3200000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_277_14 :
+    Uω (aρ 14) (bρ 14) (732250745159 / 3200000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_277_15 :
+    Uω (aρ 15) (bρ 15) (732250745159 / 3200000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_277_16 :
+    Uω (aρ 16) (bρ 16) (732250745159 / 3200000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_277 :
+    Uρ (732250745159 / 3200000000000 : ℝ) ≤
+      (-(4631194231033519735309 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (732250745159 / 3200000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(15034064746622953868251 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_277_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (732250745159 / 3200000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3028678229702195023461 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_277_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (732250745159 / 3200000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(384174511354075211361 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_277_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (732250745159 / 3200000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(7877011697722102677853 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_277_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (732250745159 / 3200000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(4096615501181864826651 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_277_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (732250745159 / 3200000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(870368338916667326583 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_277_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (732250745159 / 3200000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(9561280728788798957347 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_277_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (732250745159 / 3200000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(11322116003695119553231 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_277_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (732250745159 / 3200000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_277_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (732250745159 / 3200000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_277_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (732250745159 / 3200000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_277_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (732250745159 / 3200000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_277_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (732250745159 / 3200000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_277_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (732250745159 / 3200000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_277_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (732250745159 / 3200000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_277_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (732250745159 / 3200000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_277_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_278_1 :
+    Uω (aρ 1) (bρ 1) (7356627568677 / 32000000000000 : ℝ) ≤
+      (-(14986228550453869444753 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8937506169277629 / 40000000000000000 : ℝ) ≤
+      (-(14986228550453869444753 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (8937506169277629 / 40000000000000000 : ℝ) =
+            (8937506169277629 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8937506169277629 / 5000000000000000 : ℝ) ≤
+            (5808186858546130555247 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (558576046878063 / 2500000000000000 : ℝ)) (q :=
+      (8937506169277629 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_278_2 :
+    Uω (aρ 2) (bρ 2) (7356627568677 / 32000000000000 : ℝ) ≤
+      (-(7547510708000491888651 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8840799482296617 / 40000000000000000 : ℝ) ≤
+      (-(7547510708000491888651 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (8840799482296617 / 40000000000000000 : ℝ) =
+            (8840799482296617 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8840799482296617 / 5000000000000000 : ℝ) ≤
+            (2849696996499508111349 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1104859928265873 / 5000000000000000 : ℝ)) (q :=
+      (8840799482296617 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_278_3 :
+    Uω (aρ 3) (bρ 3) (7356627568677 / 32000000000000 : ℝ) ≤
+      (-(7658745617946492467501 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8646290021959631 / 40000000000000000 : ℝ) ≤
+      (-(7658745617946492467501 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (8646290021959631 / 40000000000000000 : ℝ) =
+            (8646290021959631 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8646290021959631 / 5000000000000000 : ℝ) ≤
+            (2738462086553507532499 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2159861232713253 / 10000000000000000 : ℝ)) (q :=
+      (8646290021959631 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_278_4 :
+    Uω (aρ 4) (bρ 4) (7356627568677 / 32000000000000 : ℝ) ≤
+      (-(3140500279114518294383 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1663945186985161 / 8000000000000000 : ℝ) ≤
+      (-(3140500279114518294383 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1663945186985161 / 8000000000000000 : ℝ) =
+            (1663945186985161 / 1000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1663945186985161 / 1000000000000000 : ℝ) ≤
+            (1018382802685481705617 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (103743668513067 / 500000000000000 : ℝ)) (q :=
+      (1663945186985161 / 8000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_278_5 :
+    Uω (aρ 5) (bρ 5) (7356627568677 / 32000000000000 : ℝ) ≤
+      (-(16331333419163968740737 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1562532963402677 / 8000000000000000 : ℝ) ≤
+      (-(16331333419163968740737 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1562532963402677 / 8000000000000000 : ℝ) =
+            (1562532963402677 / 1000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1562532963402677 / 1000000000000000 : ℝ) ≤
+            (4463081989836031259263 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (193953273064013 / 1000000000000000 : ℝ)) (q :=
+      (1562532963402677 / 8000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_278_6 :
+    Uω (aρ 6) (bρ 6) (7356627568677 / 32000000000000 : ℝ) ≤
+      (-(17345535887965030851513 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7059158590152613 / 40000000000000000 : ℝ) ≤
+      (-(17345535887965030851513 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (7059158590152613 / 40000000000000000 : ℝ) =
+            (7059158590152613 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7059158590152613 / 5000000000000000 : ℝ) ≤
+            (3448879521034969148487 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (108126249521859 / 625000000000000 : ℝ)) (q :=
+      (7059158590152613 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_278_7 :
+    Uω (aρ 7) (bρ 7) (7356627568677 / 32000000000000 : ℝ) ≤
+      (-(761846612733371790459 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5955188887814863 / 40000000000000000 : ℝ) ≤
+      (-(761846612733371790459 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (5955188887814863 / 40000000000000000 : ℝ) =
+            (5955188887814863 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5955188887814863 / 5000000000000000 : ℝ) ≤
+            (69930003626628209541 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1401696521940869 / 10000000000000000 : ℝ)) (q :=
+      (5955188887814863 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_278_8 :
+    Uω (aρ 8) (bρ 8) (7356627568677 / 32000000000000 : ℝ) ≤
+      (-(4502343141899525609949 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4211032563555623 / 40000000000000000 : ℝ) ≤
+      (-(4502343141899525609949 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (4211032563555623 / 40000000000000000 : ℝ) =
+            (4211032563555623 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4211032563555623 / 2500000000000000 : ℝ) ≤
+            (1042834300500474390051 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (813160011151249 / 10000000000000000 : ℝ)) (q :=
+      (4211032563555623 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_278_9 :
+    Uω (aρ 9) (bρ 9) (7356627568677 / 32000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_278_10 :
+    Uω (aρ 10) (bρ 10) (7356627568677 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_278_11 :
+    Uω (aρ 11) (bρ 11) (7356627568677 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_278_12 :
+    Uω (aρ 12) (bρ 12) (7356627568677 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_278_13 :
+    Uω (aρ 13) (bρ 13) (7356627568677 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_278_14 :
+    Uω (aρ 14) (bρ 14) (7356627568677 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_278_15 :
+    Uω (aρ 15) (bρ 15) (7356627568677 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_278_16 :
+    Uω (aρ 16) (bρ 16) (7356627568677 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_278 :
+    Uρ (7356627568677 / 32000000000000 : ℝ) ≤
+      (-(2311355678676694845457 / 1250000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (7356627568677 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(14986228550453869444753 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_278_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (7356627568677 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(7547510708000491888651 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_278_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (7356627568677 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(7658745617946492467501 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_278_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (7356627568677 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3140500279114518294383 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_278_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (7356627568677 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(16331333419163968740737 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_278_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (7356627568677 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(17345535887965030851513 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_278_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (7356627568677 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(761846612733371790459 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_278_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (7356627568677 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(4502343141899525609949 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_278_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (7356627568677 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_278_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (7356627568677 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_278_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (7356627568677 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_278_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (7356627568677 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_278_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (7356627568677 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_278_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (7356627568677 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_278_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (7356627568677 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_278_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (7356627568677 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_278_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_279_1 :
+    Uω (aρ 1) (bρ 1) (1847686921441 / 8000000000000 : ℝ) ≤
+      (-(746931005506826067739 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (179603153804829 / 800000000000000 : ℝ) ≤
+      (-(746931005506826067739 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (179603153804829 / 800000000000000 : ℝ) =
+            (179603153804829 / 100000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (179603153804829 / 100000000000000 : ℝ) ≤
+            (292789764943173932261 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (89798696456179 / 400000000000000 : ℝ)) (q :=
+      (179603153804829 / 800000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_279_2 :
+    Uω (aρ 2) (bρ 2) (1847686921441 / 8000000000000 : ℝ) ≤
+      (-(150468846214901381899 / 100000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4441729424490323 / 20000000000000000 : ℝ) ≤
+      (-(150468846214901381899 / 100000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4441729424490323 / 20000000000000000 : ℝ) =
+            (4441729424490323 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4441729424490323 / 2500000000000000 : ℝ) ≤
+            (57475307875098618101 / 100000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2220387003284073 / 10000000000000000 : ℝ)) (q :=
+      (4441729424490323 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_279_3 :
+    Uω (aρ 3) (bρ 3) (1847686921441 / 8000000000000 : ℝ) ≤
+      (-(7634123057285258519997 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4344486897094561 / 20000000000000000 : ℝ) ≤
+      (-(7634123057285258519997 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4344486897094561 / 20000000000000000 : ℝ) =
+            (4344486897094561 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4344486897094561 / 2500000000000000 : ℝ) ≤
+            (2763084647214741480003 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2170540582238311 / 10000000000000000 : ℝ)) (q :=
+      (4344486897094561 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_279_4 :
+    Uω (aρ 4) (bρ 4) (1847686921441 / 8000000000000 : ℝ) ≤
+      (-(3130248953656063937241 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4181239760347549 / 20000000000000000 : ℝ) ≤
+      (-(3130248953656063937241 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4181239760347549 / 20000000000000000 : ℝ) =
+            (4181239760347549 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4181239760347549 / 2500000000000000 : ℝ) ≤
+            (1028634128143936062759 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2085587626556299 / 10000000000000000 : ℝ)) (q :=
+      (4181239760347549 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_279_5 :
+    Uω (aρ 5) (bρ 5) (1847686921441 / 8000000000000 : ℝ) ≤
+      (-(16276511309893494333811 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (785561311168151 / 4000000000000000 : ℝ) ≤
+      (-(16276511309893494333811 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (785561311168151 / 4000000000000000 : ℝ) =
+            (785561311168151 / 500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (785561311168151 / 500000000000000 : ℝ) ≤
+            (4517904099106505666189 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (390068868276901 / 2000000000000000 : ℝ)) (q :=
+      (785561311168151 / 4000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_279_6 :
+    Uω (aρ 6) (bρ 6) (1847686921441 / 8000000000000 : ℝ) ≤
+      (-(17284100131055428922333 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1775665139313283 / 10000000000000000 : ℝ) ≤
+      (-(17284100131055428922333 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1775665139313283 / 10000000000000000 : ℝ) =
+            (1775665139313283 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1775665139313283 / 1250000000000000 : ℝ) ≤
+            (3510315277944571077667 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (435277109827579 / 2500000000000000 : ℝ)) (q :=
+      (1775665139313283 / 10000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_279_7 :
+    Uω (aρ 7) (bρ 7) (1847686921441 / 8000000000000 : ℝ) ≤
+      (-(9485209885025233626713 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (600046806086179 / 4000000000000000 : ℝ) ≤
+      (-(9485209885025233626713 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (600046806086179 / 4000000000000000 : ℝ) =
+            (600046806086179 / 500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (600046806086179 / 500000000000000 : ℝ) ≤
+            (911997819474766373287 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (282734714374929 / 2000000000000000 : ℝ)) (q :=
+      (600046806086179 / 4000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_279_8 :
+    Uω (aρ 8) (bρ 8) (1847686921441 / 8000000000000 : ℝ) ≤
+      (-(11190966569279896228593 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1066510152242077 / 10000000000000000 : ℝ) ≤
+      (-(11190966569279896228593 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1066510152242077 / 10000000000000000 : ℝ) =
+            (1066510152242077 / 625000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1066510152242077 / 625000000000000 : ℝ) ≤
+            (2671977036720103771407 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (12968773394811 / 156250000000000 : ℝ)) (q :=
+      (1066510152242077 / 10000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_279_9 :
+    Uω (aρ 9) (bρ 9) (1847686921441 / 8000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_279_10 :
+    Uω (aρ 10) (bρ 10) (1847686921441 / 8000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_279_11 :
+    Uω (aρ 11) (bρ 11) (1847686921441 / 8000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_279_12 :
+    Uω (aρ 12) (bρ 12) (1847686921441 / 8000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_279_13 :
+    Uω (aρ 13) (bρ 13) (1847686921441 / 8000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_279_14 :
+    Uω (aρ 14) (bρ 14) (1847686921441 / 8000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_279_15 :
+    Uω (aρ 15) (bρ 15) (1847686921441 / 8000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_279_16 :
+    Uω (aρ 16) (bρ 16) (1847686921441 / 8000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_279 :
+    Uρ (1847686921441 / 8000000000000 : ℝ) ≤
+      (-(1153581143868500307221 / 625000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1847686921441 / 8000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(746931005506826067739 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_279_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1847686921441 / 8000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(150468846214901381899 / 100000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_279_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1847686921441 / 8000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(7634123057285258519997 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_279_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1847686921441 / 8000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3130248953656063937241 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_279_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1847686921441 / 8000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(16276511309893494333811 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_279_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1847686921441 / 8000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(17284100131055428922333 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_279_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1847686921441 / 8000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(9485209885025233626713 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_279_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1847686921441 / 8000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(11190966569279896228593 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_279_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1847686921441 / 8000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_279_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1847686921441 / 8000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_279_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1847686921441 / 8000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_279_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1847686921441 / 8000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_279_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1847686921441 / 8000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_279_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1847686921441 / 8000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_279_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1847686921441 / 8000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_279_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1847686921441 / 8000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_279_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/U23.lean b/LeanPool/Zeta5Irrational/Table/U23.lean
new file mode 100644
index 0000000000..14a4f56064
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/U23.lean
@@ -0,0 +1,3707 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-! # Certified arcsine potential bounds (U23) -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma U_280_1 :
+    Uω (aρ 1) (bρ 1) (7424867802851 / 32000000000000 : ℝ) ≤
+      (-(595649490681648873547 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9022809198209127 / 40000000000000000 : ℝ) ≤
+      (-(595649490681648873547 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (9022809198209127 / 40000000000000000 : ℝ) =
+            (9022809198209127 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9022809198209127 / 5000000000000000 : ℝ) ≤
+            (236127125678351126453 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1127815314399313 / 5000000000000000 : ℝ)) (q :=
+      (9022809198209127 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_280_2 :
+    Uω (aρ 2) (bρ 2) (7424867802851 / 32000000000000 : ℝ) ≤
+      (-(14998978531214893129037 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1785223625503007 / 8000000000000000 : ℝ) ≤
+      (-(14998978531214893129037 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1785223625503007 / 8000000000000000 : ℝ) =
+            (1785223625503007 / 1000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1785223625503007 / 1000000000000000 : ℝ) ≤
+            (5795436877785106870963 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (111552705298079 / 500000000000000 : ℝ)) (q :=
+      (1785223625503007 / 8000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_280_3 :
+    Uω (aρ 3) (bρ 3) (7424867802851 / 32000000000000 : ℝ) ≤
+      (-(3804810672586354525097 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8731657237410283 / 40000000000000000 : ℝ) ≤
+      (-(3804810672586354525097 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (8731657237410283 / 40000000000000000 : ℝ) =
+            (8731657237410283 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8731657237410283 / 5000000000000000 : ℝ) ≤
+            (1393793179663645474903 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (545304941814801 / 2500000000000000 : ℝ)) (q :=
+      (8731657237410283 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_280_4 :
+    Uω (aρ 4) (bρ 4) (7424867802851 / 32000000000000 : ℝ) ≤
+      (-(15600250780916396574767 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8405232051652987 / 40000000000000000 : ℝ) ≤
+      (-(15600250780916396574767 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (8405232051652987 / 40000000000000000 : ℝ) =
+            (8405232051652987 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8405232051652987 / 5000000000000000 : ℝ) ≤
+            (5194164628083603425233 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (524075338861389 / 2500000000000000 : ℝ)) (q :=
+      (8405232051652987 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_280_5 :
+    Uω (aρ 5) (bρ 5) (7424867802851 / 32000000000000 : ℝ) ≤
+      (-(16221992242274015104961 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7898558141701877 / 40000000000000000 : ℝ) ≤
+      (-(16221992242274015104961 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (7898558141701877 / 40000000000000000 : ℝ) =
+            (7898558141701877 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7898558141701877 / 5000000000000000 : ℝ) ≤
+            (4572423166725984895039 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1961154319805001 / 10000000000000000 : ℝ)) (q :=
+      (7898558141701877 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_280_6 :
+    Uω (aρ 6) (bρ 6) (7424867802851 / 32000000000000 : ℝ) ≤
+      (-(17223054302843945077151 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7146151950218381 / 40000000000000000 : ℝ) ≤
+      (-(17223054302843945077151 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (7146151950218381 / 40000000000000000 : ℝ) =
+            (7146151950218381 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7146151950218381 / 5000000000000000 : ℝ) ≤
+            (3571361106156054922849 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1752191599203253 / 10000000000000000 : ℝ)) (q :=
+      (7146151950218381 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_280_7 :
+    Uω (aρ 7) (bρ 7) (7424867802851 / 32000000000000 : ℝ) ≤
+      (-(3779062541338344784579 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6045705484189657 / 40000000000000000 : ℝ) ≤
+      (-(3779062541338344784579 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (6045705484189657 / 40000000000000000 : ℝ) =
+            (6045705484189657 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6045705484189657 / 5000000000000000 : ℝ) ≤
+            (379820540461655215421 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1425629746948891 / 10000000000000000 : ℝ)) (q :=
+      (6045705484189657 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_280_8 :
+    Uω (aρ 8) (bρ 8) (7424867802851 / 32000000000000 : ℝ) ≤
+      (-(11127371186098035850237 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4320647243227353 / 40000000000000000 : ℝ) ≤
+      (-(11127371186098035850237 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (4320647243227353 / 40000000000000000 : ℝ) =
+            (4320647243227353 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4320647243227353 / 2500000000000000 : ℝ) ≤
+            (2735572419901964149763 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (846642277807739 / 10000000000000000 : ℝ)) (q :=
+      (4320647243227353 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_280_9 :
+    Uω (aρ 9) (bρ 9) (7424867802851 / 32000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_280_10 :
+    Uω (aρ 10) (bρ 10) (7424867802851 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_280_11 :
+    Uω (aρ 11) (bρ 11) (7424867802851 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_280_12 :
+    Uω (aρ 12) (bρ 12) (7424867802851 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_280_13 :
+    Uω (aρ 13) (bρ 13) (7424867802851 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_280_14 :
+    Uω (aρ 14) (bρ 14) (7424867802851 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_280_15 :
+    Uω (aρ 15) (bρ 15) (7424867802851 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_280_16 :
+    Uω (aρ 16) (bρ 16) (7424867802851 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_280 :
+    Uρ (7424867802851 / 32000000000000 : ℝ) ≤
+      (-(18424120970672703313979 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (7424867802851 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(595649490681648873547 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_280_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (7424867802851 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(14998978531214893129037 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_280_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (7424867802851 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3804810672586354525097 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_280_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (7424867802851 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(15600250780916396574767 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_280_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (7424867802851 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(16221992242274015104961 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_280_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (7424867802851 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(17223054302843945077151 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_280_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (7424867802851 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(3779062541338344784579 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_280_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (7424867802851 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(11127371186098035850237 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_280_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (7424867802851 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_280_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (7424867802851 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_280_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (7424867802851 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_280_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (7424867802851 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_280_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (7424867802851 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_280_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (7424867802851 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_280_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (7424867802851 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_280_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (7424867802851 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_280_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_281_1 :
+    Uω (aρ 1) (bρ 1) (3729493959969 / 16000000000000 : ℝ) ≤
+      (-(14844077893084188706087 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2266365173341027 / 10000000000000000 : ℝ) ≤
+      (-(14844077893084188706087 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2266365173341027 / 10000000000000000 : ℝ) =
+            (2266365173341027 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2266365173341027 / 1250000000000000 : ℝ) ≤
+            (5950337515915811293913 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2266293839786429 / 10000000000000000 : ℝ)) (q :=
+      (2266365173341027 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_281_2 :
+    Uω (aρ 2) (bρ 2) (3729493959969 / 16000000000000 : ℝ) ≤
+      (-(7475650471701528025081 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (179375546383163 / 800000000000000 : ℝ) ≤
+      (-(7475650471701528025081 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (179375546383163 / 800000000000000 : ℝ) =
+            (179375546383163 / 100000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (179375546383163 / 100000000000000 : ℝ) ≤
+            (2921557232798471974919 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (44834423303869 / 200000000000000 : ℝ)) (q :=
+      (179375546383163 / 800000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_281_3 :
+    Uω (aρ 3) (bρ 3) (3729493959969 / 16000000000000 : ℝ) ≤
+      (-(15170478598639876278171 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2193585089108003 / 10000000000000000 : ℝ) ≤
+      (-(15170478598639876278171 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2193585089108003 / 10000000000000000 : ℝ) =
+            (2193585089108003 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2193585089108003 / 1250000000000000 : ℝ) ≤
+            (5623936810360123721829 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2191898790180381 / 10000000000000000 : ℝ)) (q :=
+      (2193585089108003 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_281_4 :
+    Uω (aρ 4) (bρ 4) (3729493959969 / 16000000000000 : ℝ) ≤
+      (-(3109903348603568808439 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4223991771944979 / 20000000000000000 : ℝ) ≤
+      (-(3109903348603568808439 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4223991771944979 / 20000000000000000 : ℝ) =
+            (4223991771944979 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4223991771944979 / 2500000000000000 : ℝ) ≤
+            (1048979733196431191561 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1053507282487177 / 5000000000000000 : ℝ)) (q :=
+      (4223991771944979 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_281_5 :
+    Uω (aρ 5) (bρ 5) (3729493959969 / 16000000000000 : ℝ) ≤
+      (-(16167772839817462190979 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1985374990190821 / 10000000000000000 : ℝ) ≤
+      (-(16167772839817462190979 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1985374990190821 / 10000000000000000 : ℝ) =
+            (1985374990190821 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1985374990190821 / 1250000000000000 : ℝ) ≤
+            (4626642569182537809021 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1971962692746017 / 10000000000000000 : ℝ)) (q :=
+      (1985374990190821 / 10000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_281_6 :
+    Uω (aρ 6) (bρ 6) (3729493959969 / 16000000000000 : ℝ) ≤
+      (-(858119665204360662129 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1797408242110467 / 10000000000000000 : ℝ) ≤
+      (-(858119665204360662129 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1797408242110467 / 10000000000000000 : ℝ) =
+            (1797408242110467 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1797408242110467 / 1250000000000000 : ℝ) ≤
+            (181601105245639337871 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1763269571725309 / 10000000000000000 : ℝ)) (q :=
+      (1797408242110467 / 10000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_281_7 :
+    Uω (aρ 7) (bρ 7) (3729493959969 / 16000000000000 : ℝ) ≤
+      (-(3764166476338037308167 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (304545109974543 / 2000000000000000 : ℝ) ≤
+      (-(3764166476338037308167 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (304545109974543 / 2000000000000000 : ℝ) =
+            (304545109974543 / 250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (304545109974543 / 250000000000000 : ℝ) ≤
+            (394716605461962691833 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (287513113601961 / 2000000000000000 : ℝ)) (q :=
+      (304545109974543 / 2000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_281_8 :
+    Uω (aρ 8) (bρ 8) (3729493959969 / 16000000000000 : ℝ) ≤
+      (-(885200539449577683937 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1093718921142253 / 10000000000000000 : ℝ) ≤
+      (-(885200539449577683937 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1093718921142253 / 10000000000000000 : ℝ) =
+            (1093718921142253 / 625000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1093718921142253 / 625000000000000 : ℝ) ≤
+            (223834949030422316063 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (863093961888881 / 10000000000000000 : ℝ)) (q :=
+      (1093718921142253 / 10000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_281_9 :
+    Uω (aρ 9) (bρ 9) (3729493959969 / 16000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_281_10 :
+    Uω (aρ 10) (bρ 10) (3729493959969 / 16000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_281_11 :
+    Uω (aρ 11) (bρ 11) (3729493959969 / 16000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_281_12 :
+    Uω (aρ 12) (bρ 12) (3729493959969 / 16000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_281_13 :
+    Uω (aρ 13) (bρ 13) (3729493959969 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_281_14 :
+    Uω (aρ 14) (bρ 14) (3729493959969 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_281_15 :
+    Uω (aρ 15) (bρ 15) (3729493959969 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_281_16 :
+    Uω (aρ 16) (bρ 16) (3729493959969 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_281 :
+    Uρ (3729493959969 / 16000000000000 : ℝ) ≤
+      (-(18391299973504200398809 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (3729493959969 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(14844077893084188706087 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_281_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (3729493959969 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(7475650471701528025081 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_281_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (3729493959969 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(15170478598639876278171 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_281_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (3729493959969 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3109903348603568808439 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_281_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (3729493959969 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(16167772839817462190979 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_281_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (3729493959969 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(858119665204360662129 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_281_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (3729493959969 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(3764166476338037308167 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_281_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (3729493959969 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(885200539449577683937 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_281_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (3729493959969 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_281_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (3729493959969 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_281_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (3729493959969 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_281_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (3729493959969 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_281_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (3729493959969 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_281_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (3729493959969 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_281_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (3729493959969 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_281_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (3729493959969 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_281_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_282_1 :
+    Uω (aρ 1) (bρ 1) (299724321481 / 1280000000000 : ℝ) ≤
+      (-(1849642486269237145517 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9108112175886401 / 40000000000000000 : ℝ) ≤
+      (-(1849642486269237145517 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (9108112175886401 / 40000000000000000 : ℝ) =
+            (9108112175886401 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9108112175886401 / 5000000000000000 : ℝ) ≤
+            (749659439855762854483 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (71154907639309 / 312500000000000 : ℝ)) (q :=
+      (9108112175886401 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_282_2 :
+    Uω (aρ 2) (bρ 2) (299724321481 / 1280000000000 : ℝ) ≤
+      (-(14903849687665541604669 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9011436425144511 / 40000000000000000 : ℝ) ≤
+      (-(14903849687665541604669 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (9011436425144511 / 40000000000000000 : ℝ) =
+            (9011436425144511 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9011436425144511 / 5000000000000000 : ℝ) ≤
+            (5890565721334458395331 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2252388181596943 / 10000000000000000 : ℝ)) (q :=
+      (9011436425144511 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_282_3 :
+    Uω (aρ 3) (bρ 3) (299724321481 / 1280000000000 : ℝ) ≤
+      (-(3024390301891908800277 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (70536185247759 / 320000000000000 : ℝ) ≤
+      (-(3024390301891908800277 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (70536185247759 / 320000000000000 : ℝ) = (70536185247759 / 40000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (70536185247759 / 40000000000000 : ℝ) ≤
+            (1134492779908091199723 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (17620621226877 / 80000000000000 : ℝ)) (q :=
+      (70536185247759 / 320000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_282_4 :
+    Uω (aρ 4) (bρ 4) (299724321481 / 1280000000000 : ℝ) ≤
+      (-(193738000068022755659 / 125000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8490734013170131 / 40000000000000000 : ℝ) ≤
+      (-(193738000068022755659 / 125000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (8490734013170131 / 40000000000000000 : ℝ) =
+            (8490734013170131 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8490734013170131 / 5000000000000000 : ℝ) ≤
+            (66192192544477244341 / 125000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2117727263024753 / 10000000000000000 : ℝ)) (q :=
+      (8490734013170131 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_282_5 :
+    Uω (aρ 5) (bρ 5) (299724321481 / 1280000000000 : ℝ) ≤
+      (-(402846244571714183917 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7984438621375987 / 40000000000000000 : ℝ) ≤
+      (-(402846244571714183917 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (7984438621375987 / 40000000000000000 : ℝ) =
+            (7984438621375987 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7984438621375987 / 5000000000000000 : ℝ) ≤
+            (117014140653285816083 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1982769486462681 / 10000000000000000 : ℝ)) (q :=
+      (7984438621375987 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_282_6 :
+    Uω (aρ 6) (bρ 6) (299724321481 / 1280000000000 : ℝ) ≤
+      (-(17102112138147452632809 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7233103806245339 / 40000000000000000 : ℝ) ≤
+      (-(17102112138147452632809 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (7233103806245339 / 40000000000000000 : ℝ) =
+            (7233103806245339 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7233103806245339 / 5000000000000000 : ℝ) ≤
+            (3692303270852547367191 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1774342454037357 / 10000000000000000 : ℝ)) (q :=
+      (7233103806245339 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_282_7 :
+    Uω (aρ 7) (bρ 7) (299724321481 / 1280000000000 : ℝ) ≤
+      (-(1171685462072131543447 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1227211842480363 / 8000000000000000 : ℝ) ≤
+      (-(1171685462072131543447 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1227211842480363 / 8000000000000000 : ℝ) =
+            (1227211842480363 / 1000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1227211842480363 / 1000000000000000 : ℝ) ≤
+            (127965500990368456553 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (289896307575119 / 2000000000000000 : ℝ)) (q :=
+      (1227211842480363 / 8000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_282_8 :
+    Uω (aρ 8) (bρ 8) (299724321481 / 1280000000000 : ℝ) ≤
+      (-(11003813586334001811501 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4428747159444107 / 40000000000000000 : ℝ) ≤
+      (-(11003813586334001811501 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (4428747159444107 / 40000000000000000 : ℝ) =
+            (4428747159444107 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4428747159444107 / 2500000000000000 : ℝ) ≤
+            (2859130019665998188499 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (879367162736741 / 10000000000000000 : ℝ)) (q :=
+      (4428747159444107 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_282_9 :
+    Uω (aρ 9) (bρ 9) (299724321481 / 1280000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_282_10 :
+    Uω (aρ 10) (bρ 10) (299724321481 / 1280000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_282_11 :
+    Uω (aρ 11) (bρ 11) (299724321481 / 1280000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_282_12 :
+    Uω (aρ 12) (bρ 12) (299724321481 / 1280000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_282_13 :
+    Uω (aρ 13) (bρ 13) (299724321481 / 1280000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_282_14 :
+    Uω (aρ 14) (bρ 14) (299724321481 / 1280000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_282_15 :
+    Uω (aρ 15) (bρ 15) (299724321481 / 1280000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_282_16 :
+    Uω (aρ 16) (bρ 16) (299724321481 / 1280000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_282 :
+    Uρ (299724321481 / 1280000000000 : ℝ) ≤
+      (-(18358822832970896564511 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (299724321481 / 1280000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1849642486269237145517 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_282_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (299724321481 / 1280000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(14903849687665541604669 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_282_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (299724321481 / 1280000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3024390301891908800277 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_282_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (299724321481 / 1280000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(193738000068022755659 / 125000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_282_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (299724321481 / 1280000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(402846244571714183917 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_282_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (299724321481 / 1280000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(17102112138147452632809 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_282_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (299724321481 / 1280000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(1171685462072131543447 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_282_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (299724321481 / 1280000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(11003813586334001811501 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_282_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (299724321481 / 1280000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_282_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (299724321481 / 1280000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_282_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (299724321481 / 1280000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_282_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (299724321481 / 1280000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_282_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (299724321481 / 1280000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_282_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (299724321481 / 1280000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_282_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (299724321481 / 1280000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_282_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (299724321481 / 1280000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_282_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_283_1 :
+    Uω (aρ 1) (bρ 1) (29403234977 / 125000000000 : ℝ) ≤
+      (-(14750421189550824741283 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4575381822976329 / 20000000000000000 : ℝ) ≤
+      (-(14750421189550824741283 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4575381822976329 / 20000000000000000 : ℝ) =
+            (4575381822976329 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4575381822976329 / 2500000000000000 : ℝ) ≤
+            (6043994219449175258717 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2287620242901329 / 10000000000000000 : ℝ)) (q :=
+      (4575381822976329 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_283_2 :
+    Uω (aρ 2) (bρ 2) (29403234977 / 125000000000 : ℝ) ≤
+      (-(7428311312201253933377 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4527047723342681 / 20000000000000000 : ℝ) ≤
+      (-(7428311312201253933377 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4527047723342681 / 20000000000000000 : ℝ) =
+            (4527047723342681 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4527047723342681 / 2500000000000000 : ℝ) ≤
+            (2968896392298746066623 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2263055155777681 / 10000000000000000 : ℝ)) (q :=
+      (4527047723342681 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_283_3 :
+    Uω (aρ 3) (bρ 3) (29403234977 / 125000000000 : ℝ) ≤
+      (-(7536829563360700704481 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4429852820324179 / 20000000000000000 : ℝ) ≤
+      (-(7536829563360700704481 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4429852820324179 / 20000000000000000 : ℝ) =
+            (4429852820324179 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4429852820324179 / 2500000000000000 : ℝ) ≤
+            (2860378141139299295519 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2213256359109179 / 10000000000000000 : ℝ)) (q :=
+      (4429852820324179 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_283_4 :
+    Uω (aρ 4) (bρ 4) (29403234977 / 125000000000 : ℝ) ≤
+      (-(1931102244940006175991 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2133370868734877 / 10000000000000000 : ℝ) ≤
+      (-(1931102244940006175991 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2133370868734877 / 10000000000000000 : ℝ) =
+            (2133370868734877 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2133370868734877 / 1250000000000000 : ℝ) ≤
+            (668199681184993824009 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1064219728659877 / 5000000000000000 : ℝ)) (q :=
+      (2133370868734877 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_283_5 :
+    Uω (aρ 5) (bρ 5) (29403234977 / 125000000000 : ℝ) ≤
+      (-(16060219807321465949653 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4013687087452027 / 20000000000000000 : ℝ) ≤
+      (-(16060219807321465949653 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4013687087452027 / 20000000000000000 : ℝ) =
+            (4013687087452027 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4013687087452027 / 2500000000000000 : ℝ) ≤
+            (4734195601678534050347 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1993574726637027 / 10000000000000000 : ℝ)) (q :=
+      (4013687087452027 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_283_6 :
+    Uω (aρ 6) (bρ 6) (29403234977 / 125000000000 : ℝ) ≤
+      (-(8521102954091416726939 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (454785290815107 / 2500000000000000 : ℝ) ≤
+      (-(8521102954091416726939 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (454785290815107 / 2500000000000000 : ℝ) =
+            (454785290815107 / 312500000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (454785290815107 / 312500000000000 : ℝ) ≤
+            (1876104750408583273061 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (55794073151433 / 312500000000000 : ℝ)) (q :=
+      (454785290815107 / 2500000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_283_7 :
+    Uω (aρ 7) (bρ 7) (29403234977 / 125000000000 : ℝ) ≤
+      (-(18673706669854913901521 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12072612293191 / 78125000000000 : ℝ) ≤
+      (-(18673706669854913901521 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (12072612293191 / 78125000000000 : ℝ) = (12072612293191 / 9765625000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12072612293191 / 9765625000000 : ℝ) ≤
+            (2120708739145086098479 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (182672267767737 / 1250000000000000 : ℝ)) (q :=
+      (12072612293191 / 78125000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_283_8 :
+    Uω (aρ 8) (bρ 8) (29403234977 / 125000000000 : ℝ) ≤
+      (-(21887473556177965593987 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2241140564584159 / 20000000000000000 : ℝ) ≤
+      (-(21887473556177965593987 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2241140564584159 / 20000000000000000 : ℝ) =
+            (2241140564584159 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2241140564584159 / 1250000000000000 : ℝ) ≤
+            (5838413655822034406013 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (895471611009159 / 10000000000000000 : ℝ)) (q :=
+      (2241140564584159 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_283_9 :
+    Uω (aρ 9) (bρ 9) (29403234977 / 125000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_283_10 :
+    Uω (aρ 10) (bρ 10) (29403234977 / 125000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_283_11 :
+    Uω (aρ 11) (bρ 11) (29403234977 / 125000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_283_12 :
+    Uω (aρ 12) (bρ 12) (29403234977 / 125000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_283_13 :
+    Uω (aρ 13) (bρ 13) (29403234977 / 125000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_283_14 :
+    Uω (aρ 14) (bρ 14) (29403234977 / 125000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_283_15 :
+    Uω (aρ 15) (bρ 15) (29403234977 / 125000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_283_16 :
+    Uω (aρ 16) (bρ 16) (29403234977 / 125000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_283 :
+    Uρ (29403234977 / 125000000000 : ℝ) ≤
+      (-(3665335590052395257373 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (29403234977 / 125000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(14750421189550824741283 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_283_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (29403234977 / 125000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(7428311312201253933377 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_283_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (29403234977 / 125000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(7536829563360700704481 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_283_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (29403234977 / 125000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1931102244940006175991 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_283_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (29403234977 / 125000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(16060219807321465949653 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_283_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (29403234977 / 125000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(8521102954091416726939 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_283_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (29403234977 / 125000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(18673706669854913901521 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_283_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (29403234977 / 125000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(21887473556177965593987 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_283_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (29403234977 / 125000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_283_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (29403234977 / 125000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_283_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (29403234977 / 125000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_283_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (29403234977 / 125000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_283_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (29403234977 / 125000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_283_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (29403234977 / 125000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_283_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (29403234977 / 125000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_283_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (29403234977 / 125000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_283_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_284_1 :
+    Uω (aρ 1) (bρ 1) (7561348271199 / 32000000000000 : ℝ) ≤
+      (-(2940783950288045734059 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1838683020747251 / 8000000000000000 : ℝ) ≤
+      (-(2940783950288045734059 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1838683020747251 / 8000000000000000 : ℝ) =
+            (1838683020747251 / 1000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1838683020747251 / 1000000000000000 : ℝ) ≤
+            (1218099131511954265941 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (28728542940043 / 125000000000000 : ℝ)) (q :=
+      (1838683020747251 / 8000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_284_2 :
+    Uω (aρ 2) (bρ 2) (7561348271199 / 32000000000000 : ℝ) ≤
+      (-(7404808822111749557067 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9096754384969207 / 40000000000000000 : ℝ) ≤
+      (-(7404808822111749557067 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (9096754384969207 / 40000000000000000 : ℝ) =
+            (9096754384969207 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9096754384969207 / 5000000000000000 : ℝ) ≤
+            (2992398882388250442933 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (568430522082479 / 2500000000000000 : ℝ)) (q :=
+      (9096754384969207 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_284_3 :
+    Uω (aρ 3) (bρ 3) (7561348271199 / 32000000000000 : ℝ) ≤
+      (-(15025599187597887615541 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8902387815003059 / 40000000000000000 : ℝ) ≤
+      (-(15025599187597887615541 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (8902387815003059 / 40000000000000000 : ℝ) =
+            (8902387815003059 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8902387815003059 / 5000000000000000 : ℝ) ≤
+            (5768816221402112384459 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1111967454848421 / 5000000000000000 : ℝ)) (q :=
+      (8902387815003059 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_284_4 :
+    Uω (aρ 4) (bρ 4) (7561348271199 / 32000000000000 : ℝ) ≤
+      (-(15398848036234854074413 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8576231944333977 / 40000000000000000 : ℝ) ≤
+      (-(15398848036234854074413 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (8576231944333977 / 40000000000000000 : ℝ) =
+            (8576231944333977 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8576231944333977 / 5000000000000000 : ℝ) ≤
+            (5395567372765145925587 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2139151155427301 / 10000000000000000 : ℝ)) (q :=
+      (8576231944333977 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_284_5 :
+    Uω (aρ 5) (bρ 5) (7561348271199 / 32000000000000 : ℝ) ≤
+      (-(3201375940674537947543 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8070306671596673 / 40000000000000000 : ℝ) ≤
+      (-(3201375940674537947543 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (8070306671596673 / 40000000000000000 : ℝ) =
+            (8070306671596673 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8070306671596673 / 5000000000000000 : ℝ) ≤
+            (957507141125462052457 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2004378438393649 / 10000000000000000 : ℝ)) (q :=
+      (8070306671596673 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_284_6 :
+    Uω (aρ 6) (bρ 6) (7561348271199 / 32000000000000 : ℝ) ≤
+      (-(16982669814435490042889 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7320015693491687 / 40000000000000000 : ℝ) ≤
+      (-(16982669814435490042889 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (7320015693491687 / 40000000000000000 : ℝ) =
+            (7320015693491687 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7320015693491687 / 5000000000000000 : ℝ) ≤
+            (3811745594564509957111 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (449118331120289 / 2500000000000000 : ℝ)) (q :=
+      (7320015693491687 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_284_7 :
+    Uω (aρ 7) (bρ 7) (7561348271199 / 32000000000000 : ℝ) ≤
+      (-(18601039457942155663267 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6226257982895227 / 40000000000000000 : ℝ) ≤
+      (-(18601039457942155663267 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (6226257982895227 / 40000000000000000 : ℝ) =
+            (6226257982895227 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6226257982895227 / 5000000000000000 : ℝ) ≤
+            (2193375951057844336733 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (736627924971463 / 5000000000000000 : ℝ)) (q :=
+      (6226257982895227 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_284_8 :
+    Uω (aρ 8) (bρ 8) (7561348271199 / 32000000000000 : ℝ) ≤
+      (-(5442362793886692937687 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (36283963877359 / 320000000000000 : ℝ) ≤
+      (-(5442362793886692937687 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (36283963877359 / 320000000000000 : ℝ) = (36283963877359 / 20000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (36283963877359 / 20000000000000 : ℝ) ≤
+            (1489109009113307062313 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3645665008681 / 40000000000000 : ℝ)) (q :=
+      (36283963877359 / 320000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_284_9 :
+    Uω (aρ 9) (bρ 9) (7561348271199 / 32000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_284_10 :
+    Uω (aρ 10) (bρ 10) (7561348271199 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_284_11 :
+    Uω (aρ 11) (bρ 11) (7561348271199 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_284_12 :
+    Uω (aρ 12) (bρ 12) (7561348271199 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_284_13 :
+    Uω (aρ 13) (bρ 13) (7561348271199 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_284_14 :
+    Uω (aρ 14) (bρ 14) (7561348271199 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_284_15 :
+    Uω (aρ 15) (bρ 15) (7561348271199 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_284_16 :
+    Uω (aρ 16) (bρ 16) (7561348271199 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_284 :
+    Uρ (7561348271199 / 32000000000000 : ℝ) ≤
+      (-(9147427256802812168183 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (7561348271199 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2940783950288045734059 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_284_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (7561348271199 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(7404808822111749557067 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_284_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (7561348271199 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(15025599187597887615541 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_284_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (7561348271199 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(15398848036234854074413 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_284_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (7561348271199 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(3201375940674537947543 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_284_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (7561348271199 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(16982669814435490042889 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_284_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (7561348271199 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(18601039457942155663267 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_284_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (7561348271199 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(5442362793886692937687 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_284_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (7561348271199 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_284_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (7561348271199 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_284_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (7561348271199 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_284_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (7561348271199 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_284_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (7561348271199 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_284_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (7561348271199 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_284_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (7561348271199 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_284_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (7561348271199 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_284_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_285_1 :
+    Uω (aρ 1) (bρ 1) (3797734194143 / 16000000000000 : ℝ) ≤
+      (-(3664408391079400055809 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4618033274703681 / 20000000000000000 : ℝ) ≤
+      (-(3664408391079400055809 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4618033274703681 / 20000000000000000 : ℝ) =
+            (4618033274703681 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4618033274703681 / 2500000000000000 : ℝ) ≤
+            (1534195461170599944191 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1154473310724653 / 5000000000000000 : ℝ)) (q :=
+      (4618033274703681 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_285_2 :
+    Uω (aρ 2) (bρ 2) (3797734194143 / 16000000000000 : ℝ) ≤
+      (-(461338520855661178501 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (571213327572647 / 2500000000000000 : ℝ) ≤
+      (-(461338520855661178501 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (571213327572647 / 2500000000000000 : ℝ) =
+            (571213327572647 / 312500000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (571213327572647 / 312500000000000 : ℝ) ≤
+            (188486960675588821499 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2284388979836801 / 10000000000000000 : ℝ)) (q :=
+      (571213327572647 / 2500000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_285_3 :
+    Uω (aρ 3) (bρ 3) (3797734194143 / 16000000000000 : ℝ) ≤
+      (-(14977769461876843290263 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (894506968348281 / 4000000000000000 : ℝ) ≤
+      (-(14977769461876843290263 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (894506968348281 / 4000000000000000 : ℝ) =
+            (894506968348281 / 500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (894506968348281 / 500000000000000 : ℝ) ≤
+            (5816645947123156709737 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (223461330734703 / 1000000000000000 : ℝ)) (q :=
+      (894506968348281 / 4000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_285_4 :
+    Uω (aρ 4) (bρ 4) (3797734194143 / 16000000000000 : ℝ) ≤
+      (-(7674563852708070316761 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2154744859046769 / 10000000000000000 : ℝ) ≤
+      (-(7674563852708070316761 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2154744859046769 / 10000000000000000 : ℝ) =
+            (2154744859046769 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2154744859046769 / 1250000000000000 : ℝ) ≤
+            (2722643851791929683239 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2149862364764163 / 10000000000000000 : ℝ)) (q :=
+      (2154744859046769 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_285_5 :
+    Uω (aρ 5) (bρ 5) (3797734194143 / 16000000000000 : ℝ) ≤
+      (-(3190765262861868759431 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1014154520077303 / 5000000000000000 : ℝ) ≤
+      (-(3190765262861868759431 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1014154520077303 / 5000000000000000 : ℝ) =
+            (1014154520077303 / 625000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1014154520077303 / 625000000000000 : ℝ) ≤
+            (968117818938131240569 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2015180646314837 / 10000000000000000 : ℝ)) (q :=
+      (1014154520077303 / 5000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_285_6 :
+    Uω (aρ 6) (bρ 6) (3797734194143 / 16000000000000 : ℝ) ≤
+      (-(8461749575806555752881 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3681728553827143 / 20000000000000000 : ℝ) ≤
+      (-(8461749575806555752881 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3681728553827143 / 20000000000000000 : ℝ) =
+            (3681728553827143 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3681728553827143 / 2500000000000000 : ℝ) ≤
+            (1935458128693444247119 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (56485359217899 / 312500000000000 : ℝ)) (q :=
+      (3681728553827143 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_285_7 :
+    Uω (aρ 7) (bρ 7) (3797734194143 / 16000000000000 : ℝ) ≤
+      (-(18528955308340390514393 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (627130158553129 / 4000000000000000 : ℝ) ≤
+      (-(18528955308340390514393 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (627130158553129 / 4000000000000000 : ℝ) =
+            (627130158553129 / 500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (627130158553129 / 500000000000000 : ℝ) ≤
+            (2265460100659609485607 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (148511511467127 / 1000000000000000 : ℝ)) (q :=
+      (627130158553129 / 4000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_285_8 :
+    Uω (aρ 8) (bρ 8) (3797734194143 / 16000000000000 : ℝ) ≤
+      (-(21653466102551587295531 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2294203357321019 / 20000000000000000 : ℝ) ≤
+      (-(21653466102551587295531 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2294203357321019 / 20000000000000000 : ℝ) =
+            (2294203357321019 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2294203357321019 / 1250000000000000 : ℝ) ≤
+            (6072421109448412704469 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (231802332641661 / 2500000000000000 : ℝ)) (q :=
+      (2294203357321019 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_285_9 :
+    Uω (aρ 9) (bρ 9) (3797734194143 / 16000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_285_10 :
+    Uω (aρ 10) (bρ 10) (3797734194143 / 16000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_285_11 :
+    Uω (aρ 11) (bρ 11) (3797734194143 / 16000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_285_12 :
+    Uω (aρ 12) (bρ 12) (3797734194143 / 16000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_285_13 :
+    Uω (aρ 13) (bρ 13) (3797734194143 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_285_14 :
+    Uω (aρ 14) (bρ 14) (3797734194143 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_285_15 :
+    Uω (aρ 15) (bρ 15) (3797734194143 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_285_16 :
+    Uω (aρ 16) (bρ 16) (3797734194143 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_285 :
+    Uρ (3797734194143 / 16000000000000 : ℝ) ≤
+      (-(18263342418928709501953 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (3797734194143 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3664408391079400055809 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_285_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (3797734194143 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(461338520855661178501 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_285_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (3797734194143 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(14977769461876843290263 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_285_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (3797734194143 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(7674563852708070316761 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_285_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (3797734194143 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(3190765262861868759431 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_285_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (3797734194143 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(8461749575806555752881 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_285_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (3797734194143 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(18528955308340390514393 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_285_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (3797734194143 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(21653466102551587295531 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_285_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (3797734194143 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_285_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (3797734194143 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_285_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (3797734194143 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_285_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (3797734194143 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_285_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (3797734194143 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_285_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (3797734194143 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_285_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (3797734194143 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_285_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (3797734194143 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_285_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_286_1 :
+    Uω (aρ 1) (bρ 1) (383185431123 / 1600000000000 : ℝ) ≤
+      (-(728284951777396434499 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4660684702538611 / 20000000000000000 : ℝ) ≤
+      (-(728284951777396434499 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4660684702538611 / 20000000000000000 : ℝ) =
+            (4660684702538611 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4660684702538611 / 2500000000000000 : ℝ) ≤
+            (311435818672603565501 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2330272976104861 / 10000000000000000 : ℝ)) (q :=
+      (4660684702538611 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_286_2 :
+    Uω (aρ 2) (bρ 2) (383185431123 / 1600000000000 : ℝ) ≤
+      (-(14669914549627046428131 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4612365355916907 / 20000000000000000 : ℝ) ≤
+      (-(14669914549627046428131 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4612365355916907 / 20000000000000000 : ℝ) =
+            (4612365355916907 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4612365355916907 / 2500000000000000 : ℝ) ≤
+            (6124500859372953571869 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2305722641993157 / 10000000000000000 : ℝ)) (q :=
+      (4612365355916907 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_286_3 :
+    Uω (aρ 3) (bρ 3) (383185431123 / 1600000000000 : ℝ) ≤
+      (-(1488279188913710560159 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2257608130047833 / 10000000000000000 : ℝ) ≤
+      (-(1488279188913710560159 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2257608130047833 / 10000000000000000 : ℝ) =
+            (2257608130047833 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2257608130047833 / 1250000000000000 : ℝ) ≤
+            (591162351986289439841 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (563992413130479 / 2500000000000000 : ℝ)) (q :=
+      (2257608130047833 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_286_4 :
+    Uω (aρ 4) (bρ 4) (383185431123 / 1600000000000 : ℝ) ≤
+      (-(7625212945029603857267 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4352235772568247 / 20000000000000000 : ℝ) ≤
+      (-(7625212945029603857267 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4352235772568247 / 20000000000000000 : ℝ) =
+            (4352235772568247 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4352235772568247 / 2500000000000000 : ℝ) ≤
+            (2771994759470396142733 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2171283346059497 / 10000000000000000 : ℝ)) (q :=
+      (4352235772568247 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_286_5 :
+    Uω (aρ 5) (bρ 5) (383185431123 / 1600000000000 : ℝ) ≤
+      (-(1981070914050747589499 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (51244289419121 / 250000000000000 : ℝ) ≤
+      (-(1981070914050747589499 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (51244289419121 / 250000000000000 : ℝ) = (51244289419121 / 31250000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (51244289419121 / 31250000000000 : ℝ) ≤
+            (618231012074252410501 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (203678064635593 / 1000000000000000 : ℝ)) (q :=
+      (51244289419121 / 250000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_286_6 :
+    Uω (aρ 6) (bρ 6) (383185431123 / 1600000000000 : ℝ) ≤
+      (-(16806235754814884325851 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3725155877604123 / 20000000000000000 : ℝ) ≤
+      (-(16806235754814884325851 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3725155877604123 / 20000000000000000 : ℝ) =
+            (3725155877604123 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3725155877604123 / 2500000000000000 : ℝ) ≤
+            (3988179654185115674149 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1829633745570373 / 10000000000000000 : ℝ)) (q :=
+      (3725155877604123 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_286_7 :
+    Uω (aρ 7) (bρ 7) (383185431123 / 1600000000000 : ℝ) ≤
+      (-(18386495852461762688133 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1590320402548059 / 10000000000000000 : ℝ) ≤
+      (-(18386495852461762688133 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1590320402548059 / 10000000000000000 : ℝ) =
+            (1590320402548059 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1590320402548059 / 1250000000000000 : ℝ) ≤
+            (2407919556538237311867 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (47149376681949 / 312500000000000 : ℝ)) (q :=
+      (1590320402548059 / 10000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_286_8 :
+    Uω (aρ 8) (bρ 8) (383185431123 / 1600000000000 : ℝ) ≤
+      (-(2678408166937960326061 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2346689798958173 / 20000000000000000 : ℝ) ≤
+      (-(2678408166937960326061 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2346689798958173 / 20000000000000000 : ℝ) =
+            (2346689798958173 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2346689798958173 / 1250000000000000 : ℝ) ≤
+            (787327734562039673939 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (958370699024423 / 10000000000000000 : ℝ)) (q :=
+      (2346689798958173 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_286_9 :
+    Uω (aρ 9) (bρ 9) (383185431123 / 1600000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_286_10 :
+    Uω (aρ 10) (bρ 10) (383185431123 / 1600000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_286_11 :
+    Uω (aρ 11) (bρ 11) (383185431123 / 1600000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_286_12 :
+    Uω (aρ 12) (bρ 12) (383185431123 / 1600000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_286_13 :
+    Uω (aρ 13) (bρ 13) (383185431123 / 1600000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_286_14 :
+    Uω (aρ 14) (bρ 14) (383185431123 / 1600000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_286_15 :
+    Uω (aρ 15) (bρ 15) (383185431123 / 1600000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_286_16 :
+    Uω (aρ 16) (bρ 16) (383185431123 / 1600000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_286 :
+    Uρ (383185431123 / 1600000000000 : ℝ) ≤
+      (-(3640242994400076007859 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (383185431123 / 1600000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(728284951777396434499 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_286_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (383185431123 / 1600000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(14669914549627046428131 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_286_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (383185431123 / 1600000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1488279188913710560159 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_286_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (383185431123 / 1600000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(7625212945029603857267 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_286_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (383185431123 / 1600000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1981070914050747589499 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_286_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (383185431123 / 1600000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(16806235754814884325851 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_286_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (383185431123 / 1600000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(18386495852461762688133 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_286_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (383185431123 / 1600000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(2678408166937960326061 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_286_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (383185431123 / 1600000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_286_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (383185431123 / 1600000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_286_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (383185431123 / 1600000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_286_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (383185431123 / 1600000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_286_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (383185431123 / 1600000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_286_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (383185431123 / 1600000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_286_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (383185431123 / 1600000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_286_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (383185431123 / 1600000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_286_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_287_1 :
+    Uω (aρ 1) (bρ 1) (3865974428317 / 16000000000000 : ℝ) ≤
+      (-(7237301029131155242153 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4703336107131151 / 20000000000000000 : ℝ) ≤
+      (-(7237301029131155242153 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4703336107131151 / 20000000000000000 : ℝ) =
+            (4703336107131151 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4703336107131151 / 2500000000000000 : ℝ) ≤
+            (3159906675368844757847 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1175799653759013 / 5000000000000000 : ℝ)) (q :=
+      (4703336107131151 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_287_2 :
+    Uω (aρ 2) (bρ 2) (3865974428317 / 16000000000000 : ℝ) ≤
+      (-(3644463051331309478819 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (931004786760537 / 4000000000000000 : ℝ) ≤
+      (-(3644463051331309478819 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (931004786760537 / 4000000000000000 : ℝ) =
+            (931004786760537 / 500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (931004786760537 / 500000000000000 : ℝ) ≤
+            (1554140800918690521181 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (58176403667489 / 250000000000000 : ℝ)) (q :=
+      (931004786760537 / 4000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_287_3 :
+    Uω (aρ 3) (bρ 3) (3865974428317 / 16000000000000 : ℝ) ≤
+      (-(2957741839148582854709 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4557897092353209 / 20000000000000000 : ℝ) ≤
+      (-(2957741839148582854709 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4557897092353209 / 20000000000000000 : ℝ) =
+            (4557897092353209 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4557897092353209 / 2500000000000000 : ℝ) ≤
+            (1201141242651417145291 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (569331352900021 / 2500000000000000 : ℝ)) (q :=
+      (4557897092353209 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_287_4 :
+    Uω (aρ 4) (bρ 4) (3865974428317 / 16000000000000 : ℝ) ≤
+      (-(378817325494630744883 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1098744989336193 / 5000000000000000 : ℝ) ≤
+      (-(378817325494630744883 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1098744989336193 / 5000000000000000 : ℝ) =
+            (1098744989336193 / 625000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1098744989336193 / 625000000000000 : ℝ) ≤
+            (141043059730369255117 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2192702457656647 / 10000000000000000 : ℝ)) (q :=
+      (1098744989336193 / 5000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_287_5 :
+    Uω (aρ 5) (bρ 5) (3865974428317 / 16000000000000 : ℝ) ≤
+      (-(3936104641443346695053 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (828492498694077 / 4000000000000000 : ℝ) ≤
+      (-(3936104641443346695053 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (828492498694077 / 4000000000000000 : ℝ) =
+            (828492498694077 / 500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (828492498694077 / 500000000000000 : ℝ) ≤
+            (1262499210806653304947 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (102918745655863 / 500000000000000 : ℝ)) (q :=
+      (828492498694077 / 4000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_287_6 :
+    Uω (aρ 6) (bρ 6) (3865974428317 / 16000000000000 : ℝ) ≤
+      (-(8345189935393049647993 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3768564974754009 / 20000000000000000 : ℝ) ≤
+      (-(8345189935393049647993 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3768564974754009 / 20000000000000000 : ℝ) =
+            (3768564974754009 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3768564974754009 / 2500000000000000 : ℝ) ≤
+            (2052017769106950352007 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (462929442385221 / 2500000000000000 : ℝ)) (q :=
+      (3768564974754009 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_287_7 :
+    Uω (aρ 7) (bρ 7) (3865974428317 / 16000000000000 : ℝ) ≤
+      (-(18246250365212857194421 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (129022484854003 / 800000000000000 : ℝ) ≤
+      (-(18246250365212857194421 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (129022484854003 / 800000000000000 : ℝ) =
+            (129022484854003 / 100000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (129022484854003 / 100000000000000 : ℝ) ≤
+            (2548165043787142805579 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (30647525937939 / 200000000000000 : ℝ)) (q :=
+      (129022484854003 / 800000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_287_8 :
+    Uω (aρ 8) (bρ 8) (3865974428317 / 16000000000000 : ℝ) ≤
+      (-(1060412185914542412997 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2398654369369711 / 20000000000000000 : ℝ) ≤
+      (-(1060412185914542412997 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2398654369369711 / 20000000000000000 : ℝ) =
+            (2398654369369711 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2398654369369711 / 1250000000000000 : ℝ) ≤
+            (325882174685457587003 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (494505098128293 / 5000000000000000 : ℝ)) (q :=
+      (2398654369369711 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_287_9 :
+    Uω (aρ 9) (bρ 9) (3865974428317 / 16000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_287_10 :
+    Uω (aρ 10) (bρ 10) (3865974428317 / 16000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_287_11 :
+    Uω (aρ 11) (bρ 11) (3865974428317 / 16000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_287_12 :
+    Uω (aρ 12) (bρ 12) (3865974428317 / 16000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_287_13 :
+    Uω (aρ 13) (bρ 13) (3865974428317 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_287_14 :
+    Uω (aρ 14) (bρ 14) (3865974428317 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_287_15 :
+    Uω (aρ 15) (bρ 15) (3865974428317 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_287_16 :
+    Uω (aρ 16) (bρ 16) (3865974428317 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_287 :
+    Uρ (3865974428317 / 16000000000000 : ℝ) ≤
+      (-(9070113256560938488283 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (3865974428317 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(7237301029131155242153 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_287_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (3865974428317 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3644463051331309478819 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_287_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (3865974428317 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(2957741839148582854709 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_287_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (3865974428317 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(378817325494630744883 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_287_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (3865974428317 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(3936104641443346695053 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_287_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (3865974428317 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(8345189935393049647993 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_287_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (3865974428317 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(18246250365212857194421 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_287_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (3865974428317 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(1060412185914542412997 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_287_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (3865974428317 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_287_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (3865974428317 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_287_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (3865974428317 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_287_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (3865974428317 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_287_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (3865974428317 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_287_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (3865974428317 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_287_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (3865974428317 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_287_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (3865974428317 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_287_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_288_1 :
+    Uω (aρ 1) (bρ 1) (975023636351 / 4000000000000 : ℝ) ≤
+      (-(7192163754279661846917 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4745987489107963 / 20000000000000000 : ℝ) ≤
+      (-(7192163754279661846917 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4745987489107963 / 20000000000000000 : ℝ) =
+            (4745987489107963 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4745987489107963 / 2500000000000000 : ℝ) ≤
+            (3205043950220338153083 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2372925616315463 / 10000000000000000 : ℝ)) (q :=
+      (4745987489107963 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_288_2 :
+    Uω (aρ 2) (bρ 2) (975023636351 / 4000000000000 : ℝ) ≤
+      (-(7243315004429613191293 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2348841179264741 / 10000000000000000 : ℝ) ≤
+      (-(7243315004429613191293 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2348841179264741 / 10000000000000000 : ℝ) =
+            (2348841179264741 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2348841179264741 / 1250000000000000 : ℝ) ≤
+            (3153892700070386808707 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1174194749123491 / 5000000000000000 : ℝ)) (q :=
+      (2348841179264741 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_288_3 :
+    Uω (aρ 3) (bρ 3) (975023636351 / 4000000000000 : ℝ) ≤
+      (-(14695504652313017170091 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2300288677424663 / 10000000000000000 : ℝ) ≤
+      (-(14695504652313017170091 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2300288677424663 / 10000000000000000 : ℝ) =
+            (2300288677424663 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2300288677424663 / 1250000000000000 : ℝ) ≤
+            (6098910756686982829909 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1149340300458413 / 5000000000000000 : ℝ)) (q :=
+      (2300288677424663 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_288_4 :
+    Uω (aρ 4) (bρ 4) (975023636351 / 4000000000000 : ℝ) ≤
+      (-(15055910174395545613739 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2218861163342389 / 10000000000000000 : ℝ) ≤
+      (-(15055910174395545613739 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2218861163342389 / 10000000000000000 : ℝ) =
+            (2218861163342389 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2218861163342389 / 1250000000000000 : ℝ) ≤
+            (5738505234604454386261 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1107059876908639 / 5000000000000000 : ℝ)) (q :=
+      (2218861163342389 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_288_5 :
+    Uω (aρ 5) (bρ 5) (975023636351 / 4000000000000 : ℝ) ≤
+      (-(7820678307979677280879 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1046344069675191 / 5000000000000000 : ℝ) ≤
+      (-(7820678307979677280879 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1046344069675191 / 5000000000000000 : ℝ) =
+            (1046344069675191 / 625000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1046344069675191 / 625000000000000 : ℝ) ≤
+            (2576529396520322719121 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (259995453146033 / 1250000000000000 : ℝ)) (q :=
+      (1046344069675191 / 5000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_288_6 :
+    Uω (aρ 6) (bρ 6) (975023636351 / 4000000000000 : ℝ) ≤
+      (-(8287948505380436413979 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (762391297944483 / 4000000000000000 : ℝ) ≤
+      (-(8287948505380436413979 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (762391297944483 / 4000000000000000 : ℝ) =
+            (762391297944483 / 500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (762391297944483 / 500000000000000 : ℝ) ≤
+            (2109259199119563586021 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (374756842265983 / 2000000000000000 : ℝ)) (q :=
+      (762391297944483 / 4000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_288_7 :
+    Uω (aρ 7) (bρ 7) (975023636351 / 4000000000000 : ℝ) ≤
+      (-(2263518134952506017373 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (408802233372463 / 2500000000000000 : ℝ) ≤
+      (-(2263518134952506017373 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (408802233372463 / 2500000000000000 : ℝ) =
+            (408802233372463 / 312500000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (408802233372463 / 312500000000000 : ℝ) ≤
+            (335783791172493982627 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (388976742336801 / 2500000000000000 : ℝ)) (q :=
+      (408802233372463 / 2500000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_288_8 :
+    Uω (aρ 8) (bρ 8) (975023636351 / 4000000000000 : ℝ) ≤
+      (-(5248963545902056685773 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2450144136751753 / 20000000000000000 : ℝ) ≤
+      (-(5248963545902056685773 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2450144136751753 / 20000000000000000 : ℝ) =
+            (2450144136751753 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2450144136751753 / 1250000000000000 : ℝ) ≤
+            (1682508257097943314227 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1019174890459253 / 10000000000000000 : ℝ)) (q :=
+      (2450144136751753 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_288_9 :
+    Uω (aρ 9) (bρ 9) (975023636351 / 4000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_288_10 :
+    Uω (aρ 10) (bρ 10) (975023636351 / 4000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_288_11 :
+    Uω (aρ 11) (bρ 11) (975023636351 / 4000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_288_12 :
+    Uω (aρ 12) (bρ 12) (975023636351 / 4000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_288_13 :
+    Uω (aρ 13) (bρ 13) (975023636351 / 4000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_288_14 :
+    Uω (aρ 14) (bρ 14) (975023636351 / 4000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_288_15 :
+    Uω (aρ 15) (bρ 15) (975023636351 / 4000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_288_16 :
+    Uω (aρ 16) (bρ 16) (975023636351 / 4000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_288 :
+    Uρ (975023636351 / 4000000000000 : ℝ) ≤
+      (-(1130019727831504586083 / 625000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (975023636351 / 4000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(7192163754279661846917 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_288_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (975023636351 / 4000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(7243315004429613191293 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_288_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (975023636351 / 4000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(14695504652313017170091 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_288_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (975023636351 / 4000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(15055910174395545613739 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_288_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (975023636351 / 4000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(7820678307979677280879 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_288_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (975023636351 / 4000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(8287948505380436413979 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_288_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (975023636351 / 4000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(2263518134952506017373 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_288_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (975023636351 / 4000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(5248963545902056685773 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_288_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (975023636351 / 4000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_288_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (975023636351 / 4000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_288_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (975023636351 / 4000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_288_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (975023636351 / 4000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_288_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (975023636351 / 4000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_288_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (975023636351 / 4000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_288_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (975023636351 / 4000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_288_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (975023636351 / 4000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_288_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_289_1 :
+    Uω (aρ 1) (bρ 1) (3934214662491 / 16000000000000 : ℝ) ≤
+      (-(2858972133708537061617 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (598579856134173 / 2500000000000000 : ℝ) ≤
+      (-(2858972133708537061617 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (598579856134173 / 2500000000000000 : ℝ) =
+            (598579856134173 / 312500000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (598579856134173 / 312500000000000 : ℝ) ≤
+            (1299910948091462938383 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2394251903101509 / 10000000000000000 : ℝ)) (q :=
+      (598579856134173 / 2500000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_289_2 :
+    Uω (aρ 2) (bρ 2) (3934214662491 / 16000000000000 : ℝ) ≤
+      (-(3599058189717142384657 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2370170317116863 / 10000000000000000 : ℝ) ≤
+      (-(3599058189717142384657 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2370170317116863 / 10000000000000000 : ℝ) =
+            (2370170317116863 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2370170317116863 / 1250000000000000 : ℝ) ≤
+            (1599545662532857615343 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2369722700771851 / 10000000000000000 : ℝ)) (q :=
+      (2370170317116863 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_289_3 :
+    Uω (aρ 3) (bρ 3) (3934214662491 / 16000000000000 : ℝ) ≤
+      (-(7301580997444027247323 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4643257063317437 / 20000000000000000 : ℝ) ≤
+      (-(7301580997444027247323 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4643257063317437 / 20000000000000000 : ℝ) =
+            (4643257063317437 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4643257063317437 / 2500000000000000 : ℝ) ≤
+            (3095626707055972752677 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1160017618102781 / 5000000000000000 : ℝ)) (q :=
+      (4643257063317437 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_289_4 :
+    Uω (aρ 4) (bρ 4) (3934214662491 / 16000000000000 : ℝ) ≤
+      (-(1496005898479066858877 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4480462932766027 / 20000000000000000 : ℝ) ≤
+      (-(1496005898479066858877 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4480462932766027 / 20000000000000000 : ℝ) =
+            (4480462932766027 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4480462932766027 / 2500000000000000 : ℝ) ≤
+            (583435642420933141123 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (139720955419947 / 625000000000000 : ℝ)) (q :=
+      (4480462932766027 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_289_5 :
+    Uω (aρ 5) (bρ 5) (3934214662491 / 16000000000000 : ℝ) ≤
+      (-(3884839687161393204871 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4228284680407637 / 20000000000000000 : ℝ) ≤
+      (-(3884839687161393204871 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4228284680407637 / 20000000000000000 : ℝ) =
+            (4228284680407637 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4228284680407637 / 2500000000000000 : ℝ) ≤
+            (1313764165088606795129 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1050773476847881 / 5000000000000000 : ℝ)) (q :=
+      (4228284680407637 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_289_6 :
+    Uω (aρ 6) (bρ 6) (3934214662491 / 16000000000000 : ℝ) ≤
+      (-(16462753974742707140679 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3855331036449663 / 20000000000000000 : ℝ) ≤
+      (-(16462753974742707140679 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3855331036449663 / 20000000000000000 : ℝ) =
+            (3855331036449663 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3855331036449663 / 2500000000000000 : ℝ) ≤
+            (4331661434257292859321 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (473958421219447 / 2500000000000000 : ℝ)) (q :=
+      (3855331036449663 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_289_7 :
+    Uω (aρ 7) (bρ 7) (3934214662491 / 16000000000000 : ℝ) ≤
+      (-(17972110093080970712811 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3315210972743653 / 20000000000000000 : ℝ) ≤
+      (-(17972110093080970712811 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3315210972743653 / 20000000000000000 : ℝ) =
+            (3315210972743653 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3315210972743653 / 2500000000000000 : ℝ) ≤
+            (2822305315919029287189 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (789687500965889 / 5000000000000000 : ℝ)) (q :=
+      (3315210972743653 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_289_8 :
+    Uω (aρ 8) (bρ 8) (3934214662491 / 16000000000000 : ℝ) ≤
+      (-(519740407477164840823 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2501200065417611 / 20000000000000000 : ℝ) ≤
+      (-(519740407477164840823 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2501200065417611 / 20000000000000000 : ℝ) =
+            (2501200065417611 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 2
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2501200065417611 / 2500000000000000 : ℝ) ≤
+            (119977747835159177 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (131113218243217 / 1250000000000000 : ℝ)) (q :=
+      (2501200065417611 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_289_9 :
+    Uω (aρ 9) (bρ 9) (3934214662491 / 16000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_289_10 :
+    Uω (aρ 10) (bρ 10) (3934214662491 / 16000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_289_11 :
+    Uω (aρ 11) (bρ 11) (3934214662491 / 16000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_289_12 :
+    Uω (aρ 12) (bρ 12) (3934214662491 / 16000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_289_13 :
+    Uω (aρ 13) (bρ 13) (3934214662491 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_289_14 :
+    Uω (aρ 14) (bρ 14) (3934214662491 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_289_15 :
+    Uω (aρ 15) (bρ 15) (3934214662491 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_289_16 :
+    Uω (aρ 16) (bρ 16) (3934214662491 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_289 :
+    Uρ (3934214662491 / 16000000000000 : ℝ) ≤
+      (-(1126339213224684761437 / 625000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (3934214662491 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2858972133708537061617 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_289_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (3934214662491 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3599058189717142384657 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_289_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (3934214662491 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(7301580997444027247323 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_289_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (3934214662491 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1496005898479066858877 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_289_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (3934214662491 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(3884839687161393204871 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_289_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (3934214662491 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(16462753974742707140679 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_289_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (3934214662491 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(17972110093080970712811 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_289_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (3934214662491 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(519740407477164840823 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_289_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (3934214662491 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_289_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (3934214662491 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_289_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (3934214662491 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_289_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (3934214662491 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_289_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (3934214662491 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_289_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (3934214662491 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_289_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (3934214662491 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_289_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (3934214662491 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_289_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_290_1 :
+    Uω (aρ 1) (bρ 1) (1984167389789 / 8000000000000 : ℝ) ≤
+      (-(14206187211916833637463 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (966258037522081 / 4000000000000000 : ℝ) ≤
+      (-(14206187211916833637463 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (966258037522081 / 4000000000000000 : ℝ) =
+            (966258037522081 / 500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (966258037522081 / 500000000000000 : ℝ) ≤
+            (6588228197083166362537 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (483115633691831 / 2000000000000000 : ℝ)) (q :=
+      (966258037522081 / 4000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_290_2 :
+    Uω (aρ 2) (bρ 2) (1984167389789 / 8000000000000 : ℝ) ≤
+      (-(14306645663022064867043 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4782998764904191 / 20000000000000000 : ℝ) ≤
+      (-(14306645663022064867043 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4782998764904191 / 20000000000000000 : ℝ) =
+            (4782998764904191 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4782998764904191 / 2500000000000000 : ℝ) ≤
+            (6487769745977935132957 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2391055758262941 / 10000000000000000 : ℝ)) (q :=
+      (4782998764904191 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_290_3 :
+    Uω (aρ 3) (bρ 3) (1984167389789 / 8000000000000 : ℝ) ≤
+      (-(1813958175975205347013 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (146435507278643 / 625000000000000 : ℝ) ≤
+      (-(1813958175975205347013 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (146435507278643 / 625000000000000 : ℝ) = (146435507278643 / 78125000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (146435507278643 / 78125000000000 : ℝ) ≤
+            (785343750149794652987 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1170694666312663 / 5000000000000000 : ℝ)) (q :=
+      (146435507278643 / 625000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_290_4 :
+    Uω (aρ 4) (bρ 4) (1984167389789 / 8000000000000 : ℝ) ≤
+      (-(92907010073051215021 / 62500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2261600912890833 / 10000000000000000 : ℝ) ≤
+      (-(92907010073051215021 / 62500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2261600912890833 / 10000000000000000 : ℝ) =
+            (2261600912890833 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2261600912890833 / 1250000000000000 : ℝ) ≤
+            (37058086233198784979 / 62500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (282118638319427 / 1250000000000000 : ℝ)) (q :=
+      (2261600912890833 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_290_5 :
+    Uω (aρ 5) (bρ 5) (1984167389789 / 8000000000000 : ℝ) ≤
+      (-(15438402962172249138387 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2135593931387483 / 10000000000000000 : ℝ) ≤
+      (-(15438402962172249138387 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2135593931387483 / 10000000000000000 : ℝ) =
+            (2135593931387483 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2135593931387483 / 1250000000000000 : ℝ) ≤
+            (5356012446827750861613 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (530781265720929 / 2500000000000000 : ℝ)) (q :=
+      (2135593931387483 / 10000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_290_6 :
+    Uω (aρ 6) (bρ 6) (1984167389789 / 8000000000000 : ℝ) ≤
+      (-(4087729696628356463639 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (60917018752829 / 312500000000000 : ℝ) ≤
+      (-(4087729696628356463639 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (60917018752829 / 312500000000000 : ℝ) = (60917018752829 / 39062500000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (60917018752829 / 39062500000000 : ℝ) ≤
+            (1110874155621643536361 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (958933387714903 / 5000000000000000 : ℝ)) (q :=
+      (60917018752829 / 312500000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_290_7 :
+    Uω (aρ 7) (bρ 7) (1984167389789 / 8000000000000 : ℝ) ≤
+      (-(1783807909004666494059 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (335994419017261 / 2000000000000000 : ℝ) ≤
+      (-(1783807909004666494059 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (335994419017261 / 2000000000000000 : ℝ) =
+            (335994419017261 / 250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (335994419017261 / 250000000000000 : ℝ) ≤
+            (295633631895333505941 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (20034789327267 / 125000000000000 : ℝ)) (q :=
+      (335994419017261 / 2000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_290_8 :
+    Uω (aρ 8) (bρ 8) (1984167389789 / 8000000000000 : ℝ) ≤
+      (-(1286819084048231859999 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2551858043370813 / 20000000000000000 : ℝ) ≤
+      (-(1286819084048231859999 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2551858043370813 / 20000000000000000 : ℝ) =
+            (2551858043370813 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2551858043370813 / 2500000000000000 : ℝ) ≤
+            (12831879014268140001 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1078238650719563 / 10000000000000000 : ℝ)) (q :=
+      (2551858043370813 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_290_9 :
+    Uω (aρ 9) (bρ 9) (1984167389789 / 8000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_290_10 :
+    Uω (aρ 10) (bρ 10) (1984167389789 / 8000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_290_11 :
+    Uω (aρ 11) (bρ 11) (1984167389789 / 8000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_290_12 :
+    Uω (aρ 12) (bρ 12) (1984167389789 / 8000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_290_13 :
+    Uω (aρ 13) (bρ 13) (1984167389789 / 8000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_290_14 :
+    Uω (aρ 14) (bρ 14) (1984167389789 / 8000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_290_15 :
+    Uω (aρ 15) (bρ 15) (1984167389789 / 8000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_290_16 :
+    Uω (aρ 16) (bρ 16) (1984167389789 / 8000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_290 :
+    Uρ (1984167389789 / 8000000000000 : ℝ) ≤
+      (-(8981756148083042841929 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1984167389789 / 8000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(14206187211916833637463 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_290_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1984167389789 / 8000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(14306645663022064867043 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_290_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1984167389789 / 8000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1813958175975205347013 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_290_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1984167389789 / 8000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(92907010073051215021 / 62500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_290_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1984167389789 / 8000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(15438402962172249138387 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_290_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1984167389789 / 8000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(4087729696628356463639 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_290_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1984167389789 / 8000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(1783807909004666494059 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_290_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1984167389789 / 8000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(1286819084048231859999 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_290_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1984167389789 / 8000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_290_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1984167389789 / 8000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_290_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1984167389789 / 8000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_290_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1984167389789 / 8000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_290_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1984167389789 / 8000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_290_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1984167389789 / 8000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_290_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1984167389789 / 8000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_290_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1984167389789 / 8000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_290_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_291_1 :
+    Uω (aρ 1) (bρ 1) (504571876719 / 2000000000000 : ℝ) ≤
+      (-(3507791254907928832507 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4916592802630059 / 20000000000000000 : ℝ) ≤
+      (-(3507791254907928832507 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4916592802630059 / 20000000000000000 : ℝ) =
+            (4916592802630059 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4916592802630059 / 2500000000000000 : ℝ) ≤
+            (1690812597342071167493 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2458230637120059 / 10000000000000000 : ℝ)) (q :=
+      (4916592802630059 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_291_2 :
+    Uω (aρ 2) (bρ 2) (504571876719 / 2000000000000 : ℝ) ≤
+      (-(7064922361463546649243 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4868314606399477 / 20000000000000000 : ℝ) ≤
+      (-(7064922361463546649243 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4868314606399477 / 20000000000000000 : ℝ) =
+            (4868314606399477 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4868314606399477 / 2500000000000000 : ℝ) ≤
+            (3332285343036453350757 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2433721453399477 / 10000000000000000 : ℝ)) (q :=
+      (4868314606399477 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_291_3 :
+    Uω (aρ 3) (bρ 3) (504571876719 / 2000000000000 : ℝ) ≤
+      (-(14331149326100729740711 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2385646506713341 / 10000000000000000 : ℝ) ≤
+      (-(14331149326100729740711 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2385646506713341 / 10000000000000000 : ℝ) =
+            (2385646506713341 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2385646506713341 / 1250000000000000 : ℝ) ≤
+            (6463266082899270259289 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1192047983388341 / 5000000000000000 : ℝ)) (q :=
+      (2385646506713341 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_291_4 :
+    Uω (aρ 4) (bρ 4) (504571876719 / 2000000000000 : ℝ) ≤
+      (-(14677919486345039845707 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2304337332011151 / 10000000000000000 : ℝ) ≤
+      (-(14677919486345039845707 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2304337332011151 / 10000000000000000 : ℝ) =
+            (2304337332011151 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2304337332011151 / 1250000000000000 : ℝ) ≤
+            (6116495922654960154293 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1149885899218651 / 5000000000000000 : ℝ)) (q :=
+      (2304337332011151 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_291_5 :
+    Uω (aρ 5) (bρ 5) (504571876719 / 2000000000000 : ℝ) ≤
+      (-(3047906602216523433211 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2178489596664727 / 10000000000000000 : ℝ) ≤
+      (-(3047906602216523433211 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2178489596664727 / 10000000000000000 : ℝ) =
+            (2178489596664727 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2178489596664727 / 1250000000000000 : ℝ) ≤
+            (1110976479583476566789 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1083133123539727 / 5000000000000000 : ℝ)) (q :=
+      (2178489596664727 / 10000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_291_6 :
+    Uω (aρ 6) (bρ 6) (504571876719 / 2000000000000 : ℝ) ≤
+      (-(2016381225759584278767 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3985358586130907 / 20000000000000000 : ℝ) ≤
+      (-(2016381225759584278767 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3985358586130907 / 20000000000000000 : ℝ) =
+            (3985358586130907 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3985358586130907 / 2500000000000000 : ℝ) ≤
+            (582920700365415721233 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1961886015020907 / 10000000000000000 : ℝ)) (q :=
+      (3985358586130907 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_291_7 :
+    Uω (aρ 7) (bρ 7) (504571876719 / 2000000000000 : ℝ) ≤
+      (-(8787890108594326096779 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (689848230502033 / 4000000000000000 : ℝ) ≤
+      (-(8787890108594326096779 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (689848230502033 / 4000000000000000 : ℝ) =
+            (689848230502033 / 500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (689848230502033 / 500000000000000 : ℝ) ≤
+            (1609317595905673903221 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (329885992432033 / 2000000000000000 : ℝ)) (q :=
+      (689848230502033 / 4000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_291_8 :
+    Uω (aρ 8) (bρ 8) (504571876719 / 2000000000000 : ℝ) ≤
+      (-(20203793397973935609921 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2652103060454627 / 20000000000000000 : ℝ) ≤
+      (-(20203793397973935609921 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2652103060454627 / 20000000000000000 : ℝ) =
+            (2652103060454627 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2652103060454627 / 2500000000000000 : ℝ) ≤
+            (590622011026064390079 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1135833521444627 / 10000000000000000 : ℝ)) (q :=
+      (2652103060454627 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_291_9 :
+    Uω (aρ 9) (bρ 9) (504571876719 / 2000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_291_10 :
+    Uω (aρ 10) (bρ 10) (504571876719 / 2000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_291_11 :
+    Uω (aρ 11) (bρ 11) (504571876719 / 2000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_291_12 :
+    Uω (aρ 12) (bρ 12) (504571876719 / 2000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_291_13 :
+    Uω (aρ 13) (bρ 13) (504571876719 / 2000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_291_14 :
+    Uω (aρ 14) (bρ 14) (504571876719 / 2000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_291_15 :
+    Uω (aρ 15) (bρ 15) (504571876719 / 2000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_291_16 :
+    Uω (aρ 16) (bρ 16) (504571876719 / 2000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_291 :
+    Uρ (504571876719 / 2000000000000 : ℝ) ≤
+      (-(8925212957824183781139 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (504571876719 / 2000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3507791254907928832507 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_291_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (504571876719 / 2000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(7064922361463546649243 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_291_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (504571876719 / 2000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(14331149326100729740711 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_291_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (504571876719 / 2000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(14677919486345039845707 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_291_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (504571876719 / 2000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(3047906602216523433211 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_291_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (504571876719 / 2000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(2016381225759584278767 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_291_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (504571876719 / 2000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(8787890108594326096779 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_291_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (504571876719 / 2000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(20203793397973935609921 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_291_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (504571876719 / 2000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_291_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (504571876719 / 2000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_291_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (504571876719 / 2000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_291_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (504571876719 / 2000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_291_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (504571876719 / 2000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_291_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (504571876719 / 2000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_291_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (504571876719 / 2000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_291_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (504571876719 / 2000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_291_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/U24.lean b/LeanPool/Zeta5Irrational/Table/U24.lean
new file mode 100644
index 0000000000..c4394f2bf5
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/U24.lean
@@ -0,0 +1,3874 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-! # Certified arcsine potential bounds (U24) -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma U_292_1 :
+    Uω (aρ 1) (bρ 1) (2052407623963 / 8000000000000 : ℝ) ≤
+      (-(3464788411850854761307 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2500947669219157 / 10000000000000000 : ℝ) ≤
+      (-(3464788411850854761307 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2500947669219157 / 10000000000000000 : ℝ) =
+            (2500947669219157 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 2
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2500947669219157 / 2500000000000000 : ℝ) ≤
+            (947489649145238693 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (625220756642391 / 2500000000000000 : ℝ)) (q :=
+      (2500947669219157 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_292_2 :
+    Uω (aρ 2) (bρ 2) (2052407623963 / 8000000000000 : ℝ) ≤
+      (-(3489029123349057418429 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2476814956106143 / 10000000000000000 : ℝ) ≤
+      (-(3489029123349057418429 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2476814956106143 / 10000000000000000 : ℝ) =
+            (2476814956106143 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2476814956106143 / 1250000000000000 : ℝ) ≤
+            (1709574728900942581571 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (77387081651673 / 312500000000000 : ℝ)) (q :=
+      (2476814956106143 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_292_3 :
+    Uω (aρ 3) (bρ 3) (2052407623963 / 8000000000000 : ℝ) ≤
+      (-(14153838244717469109831 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (303540487944847 / 1250000000000000 : ℝ) ≤
+      (-(14153838244717469109831 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (303540487944847 / 1250000000000000 : ℝ) =
+            (303540487944847 / 156250000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (303540487944847 / 156250000000000 : ℝ) ≤
+            (6640577164282530890169 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1213400307054401 / 5000000000000000 : ℝ)) (q :=
+      (303540487944847 / 1250000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_292_4 :
+    Uω (aρ 4) (bρ 4) (2052407623963 / 8000000000000 : ℝ) ≤
+      (-(3623542733545598920361 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (938828241337921 / 4000000000000000 : ℝ) ≤
+      (-(3623542733545598920361 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (938828241337921 / 4000000000000000 : ℝ) =
+            (938828241337921 / 500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (938828241337921 / 500000000000000 : ℝ) ≤
+            (1575061118704401079639 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (468517638949171 / 2000000000000000 : ℝ)) (q :=
+      (938828241337921 / 4000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_292_5 :
+    Uω (aρ 5) (bρ 5) (2052407623963 / 8000000000000 : ℝ) ≤
+      (-(3761145976967960412099 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (138835983024799 / 625000000000000 : ℝ) ≤
+      (-(3761145976967960412099 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (138835983024799 / 625000000000000 : ℝ) = (138835983024799 / 78125000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (138835983024799 / 78125000000000 : ℝ) ≤
+            (1437457875282039587901 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1104694182092409 / 5000000000000000 : ℝ)) (q :=
+      (138835983024799 / 625000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_292_6 :
+    Uω (aρ 6) (bρ 6) (2052407623963 / 8000000000000 : ℝ) ≤
+      (-(1989507064440686596057 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2035984408317511 / 10000000000000000 : ℝ) ≤
+      (-(1989507064440686596057 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2035984408317511 / 10000000000000000 : ℝ) =
+            (2035984408317511 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2035984408317511 / 1250000000000000 : ℝ) ≤
+            (609794861684313403943 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (31341345299473 / 156250000000000 : ℝ)) (q :=
+      (2035984408317511 / 10000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_292_7 :
+    Uω (aρ 7) (bρ 7) (2052407623963 / 8000000000000 : ℝ) ≤
+      (-(8660390319253113093879 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3538327680625391 / 20000000000000000 : ℝ) ≤
+      (-(8660390319253113093879 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3538327680625391 / 20000000000000000 : ℝ) =
+            (3538327680625391 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3538327680625391 / 2500000000000000 : ℝ) ≤
+            (1736817385246886906121 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1695866343916641 / 10000000000000000 : ℝ)) (q :=
+      (3538327680625391 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_292_8 :
+    Uω (aρ 8) (bρ 8) (2052407623963 / 8000000000000 : ℝ) ≤
+      (-(3967468627530778132021 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2751092096951029 / 20000000000000000 : ℝ) ≤
+      (-(3967468627530778132021 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2751092096951029 / 20000000000000000 : ℝ) =
+            (2751092096951029 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2751092096951029 / 2500000000000000 : ℝ) ≤
+            (191414454269221867979 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1192172411582279 / 10000000000000000 : ℝ)) (q :=
+      (2751092096951029 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_292_9 :
+    Uω (aρ 9) (bρ 9) (2052407623963 / 8000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_292_10 :
+    Uω (aρ 10) (bρ 10) (2052407623963 / 8000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_292_11 :
+    Uω (aρ 11) (bρ 11) (2052407623963 / 8000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_292_12 :
+    Uω (aρ 12) (bρ 12) (2052407623963 / 8000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_292_13 :
+    Uω (aρ 13) (bρ 13) (2052407623963 / 8000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_292_14 :
+    Uω (aρ 14) (bρ 14) (2052407623963 / 8000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_292_15 :
+    Uω (aρ 15) (bρ 15) (2052407623963 / 8000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_292_16 :
+    Uω (aρ 16) (bρ 16) (2052407623963 / 8000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_292 :
+    Uρ (2052407623963 / 8000000000000 : ℝ) ≤
+      (-(17740742306804529030479 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (2052407623963 / 8000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3464788411850854761307 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_292_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (2052407623963 / 8000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3489029123349057418429 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_292_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (2052407623963 / 8000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(14153838244717469109831 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_292_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (2052407623963 / 8000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3623542733545598920361 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_292_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (2052407623963 / 8000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(3761145976967960412099 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_292_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (2052407623963 / 8000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1989507064440686596057 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_292_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (2052407623963 / 8000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(8660390319253113093879 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_292_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (2052407623963 / 8000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(3967468627530778132021 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_292_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (2052407623963 / 8000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_292_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (2052407623963 / 8000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_292_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (2052407623963 / 8000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_292_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (2052407623963 / 8000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_292_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (2052407623963 / 8000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_292_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (2052407623963 / 8000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_292_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (2052407623963 / 8000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_292_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (2052407623963 / 8000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_292_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_293_1 :
+    Uω (aρ 1) (bρ 1) (41730554821 / 160000000000 : ℝ) ≤
+      (-(13690051247136983319469 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1271799449755011 / 5000000000000000 : ℝ) ≤
+      (-(13690051247136983319469 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1271799449755011 / 5000000000000000 : ℝ) =
+            (1271799449755011 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1271799449755011 / 1250000000000000 : ℝ) ≤
+            (172892358863016680531 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (79485479399767 / 312500000000000 : ℝ)) (q :=
+      (1271799449755011 / 5000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_293_2 :
+    Uω (aρ 2) (bρ 2) (41730554821 / 160000000000 : ℝ) ≤
+      (-(13785355958980859419847 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5038944709561171 / 20000000000000000 : ℝ) ≤
+      (-(13785355958980859419847 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5038944709561171 / 20000000000000000 : ℝ) =
+            (5038944709561171 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5038944709561171 / 5000000000000000 : ℝ) ≤
+            (77587647019140580153 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2519051263843671 / 10000000000000000 : ℝ)) (q :=
+      (5038944709561171 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_293_3 :
+    Uω (aρ 3) (bρ 3) (41730554821 / 160000000000 : ℝ) ≤
+      (-(13979620201121910036697 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4942000717062239 / 20000000000000000 : ℝ) ≤
+      (-(13979620201121910036697 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4942000717062239 / 20000000000000000 : ℝ) =
+            (4942000717062239 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4942000717062239 / 2500000000000000 : ℝ) ≤
+            (6814795207878089963303 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2469503377694739 / 10000000000000000 : ℝ)) (q :=
+      (4942000717062239 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_293_4 :
+    Uω (aρ 4) (bρ 4) (41730554821 / 160000000000 : ℝ) ≤
+      (-(2862750073378134468569 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (298725112049267 / 1250000000000000 : ℝ) ≤
+      (-(2862750073378134468569 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (298725112049267 / 1250000000000000 : ℝ) =
+            (298725112049267 / 156250000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (298725112049267 / 156250000000000 : ℝ) ≤
+            (1296133008421865531431 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (596349658621443 / 2500000000000000 : ℝ)) (q :=
+      (298725112049267 / 1250000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_293_5 :
+    Uω (aρ 5) (bρ 5) (41730554821 / 160000000000 : ℝ) ≤
+      (-(14853402445948760473001 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4528505748238879 / 20000000000000000 : ℝ) ≤
+      (-(14853402445948760473001 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4528505748238879 / 20000000000000000 : ℝ) =
+            (4528505748238879 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4528505748238879 / 2500000000000000 : ℝ) ≤
+            (5941012963051239526999 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2252492509271379 / 10000000000000000 : ℝ)) (q :=
+      (4528505748238879 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_293_6 :
+    Uω (aρ 6) (bρ 6) (41730554821 / 160000000000 : ℝ) ≤
+      (-(15705721418374445952343 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (831704739547529 / 4000000000000000 : ℝ) ≤
+      (-(15705721418374445952343 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (831704739547529 / 4000000000000000 : ℝ) =
+            (831704739547529 / 500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (831704739547529 / 500000000000000 : ℝ) ≤
+            (5088693990625554047657 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (409950166782029 / 2000000000000000 : ℝ)) (q :=
+      (831704739547529 / 4000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_293_7 :
+    Uω (aρ 7) (bρ 7) (41730554821 / 160000000000 : ℝ) ≤
+      (-(2134081990146632886253 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3627220602139997 / 20000000000000000 : ℝ) ≤
+      (-(2134081990146632886253 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3627220602139997 / 20000000000000000 : ℝ) =
+            (3627220602139997 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3627220602139997 / 2500000000000000 : ℝ) ≤
+            (465219935978367113747 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1742109119072497 / 10000000000000000 : ℝ)) (q :=
+      (3627220602139997 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_293_8 :
+    Uω (aρ 8) (bρ 8) (41730554821 / 160000000000 : ℝ) ≤
+      (-(19487658528202780408503 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1424497670143987 / 10000000000000000 : ℝ) ≤
+      (-(19487658528202780408503 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1424497670143987 / 10000000000000000 : ℝ) =
+            (1424497670143987 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1424497670143987 / 1250000000000000 : ℝ) ≤
+            (1306756880797219591497 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (623712754280237 / 5000000000000000 : ℝ)) (q :=
+      (1424497670143987 / 10000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_293_9 :
+    Uω (aρ 9) (bρ 9) (41730554821 / 160000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_293_10 :
+    Uω (aρ 10) (bρ 10) (41730554821 / 160000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_293_11 :
+    Uω (aρ 11) (bρ 11) (41730554821 / 160000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_293_12 :
+    Uω (aρ 12) (bρ 12) (41730554821 / 160000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_293_13 :
+    Uω (aρ 13) (bρ 13) (41730554821 / 160000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_293_14 :
+    Uω (aρ 14) (bρ 14) (41730554821 / 160000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_293_15 :
+    Uω (aρ 15) (bρ 15) (41730554821 / 160000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_293_16 :
+    Uω (aρ 16) (bρ 16) (41730554821 / 160000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_293 :
+    Uρ (41730554821 / 160000000000 : ℝ) ≤
+      (-(881709807907133594019 / 500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (41730554821 / 160000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(13690051247136983319469 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_293_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (41730554821 / 160000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(13785355958980859419847 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_293_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (41730554821 / 160000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(13979620201121910036697 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_293_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (41730554821 / 160000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(2862750073378134468569 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_293_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (41730554821 / 160000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(14853402445948760473001 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_293_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (41730554821 / 160000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(15705721418374445952343 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_293_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (41730554821 / 160000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(2134081990146632886253 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_293_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (41730554821 / 160000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(19487658528202780408503 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_293_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (41730554821 / 160000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_293_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (41730554821 / 160000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_293_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (41730554821 / 160000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_293_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (41730554821 / 160000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_293_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (41730554821 / 160000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_293_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (41730554821 / 160000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_293_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (41730554821 / 160000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_293_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (41730554821 / 160000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_293_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_294_1 :
+    Uω (aρ 1) (bρ 1) (269345996903 / 1000000000000 : ℝ) ≤
+      (-(13360191075021170630243 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1051560501830063 / 4000000000000000 : ℝ) ≤
+      (-(13360191075021170630243 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1051560501830063 / 4000000000000000 : ℝ) =
+            (1051560501830063 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1051560501830063 / 1000000000000000 : ℝ) ≤
+            (502752530978829369757 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (525767951641063 / 2000000000000000 : ℝ)) (q :=
+      (1051560501830063 / 4000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_294_2 :
+    Uω (aρ 2) (bρ 2) (269345996903 / 1000000000000 : ℝ) ≤
+      (-(26904688033233695969 / 20000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2604786439411037 / 10000000000000000 : ℝ) ≤
+      (-(26904688033233695969 / 20000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2604786439411037 / 10000000000000000 : ℝ) =
+            (2604786439411037 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2604786439411037 / 2500000000000000 : ℝ) ≤
+            (821199178766304031 / 20000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1302189570193537 / 5000000000000000 : ℝ)) (q :=
+      (2604786439411037 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_294_3 :
+    Uω (aρ 3) (bρ 3) (269345996903 / 1000000000000 : ℝ) ≤
+      (-(3410010963726477067487 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (102254025270581 / 400000000000000 : ℝ) ≤
+      (-(3410010963726477067487 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (102254025270581 / 400000000000000 : ℝ) =
+            (102254025270581 / 100000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (102254025270581 / 100000000000000 : ℝ) ≤
+            (55724937773522932513 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (51098072628881 / 200000000000000 : ℝ)) (q :=
+      (102254025270581 / 400000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_294_4 :
+    Uω (aρ 4) (bρ 4) (269345996903 / 1000000000000 : ℝ) ≤
+      (-(13962424118052550522617 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4950506333616661 / 20000000000000000 : ℝ) ≤
+      (-(13962424118052550522617 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4950506333616661 / 20000000000000000 : ℝ) =
+            (4950506333616661 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4950506333616661 / 2500000000000000 : ℝ) ≤
+            (6831991290947449477383 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2471002882596661 / 10000000000000000 : ℝ)) (q :=
+      (4950506333616661 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_294_5 :
+    Uω (aρ 5) (bρ 5) (269345996903 / 1000000000000 : ℝ) ≤
+      (-(14481773407649457578313 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (36718471793851 / 156250000000000 : ℝ) ≤
+      (-(14481773407649457578313 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (36718471793851 / 156250000000000 : ℝ) = (36718471793851 / 19531250000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (36718471793851 / 19531250000000 : ℝ) ≤
+            (6312642001350542421687 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (292331357240991 / 1250000000000000 : ℝ)) (q :=
+      (36718471793851 / 156250000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_294_6 :
+    Uω (aρ 6) (bρ 6) (269345996903 / 1000000000000 : ℝ) ≤
+      (-(1529822745192444888481 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2165740525370507 / 10000000000000000 : ℝ) ≤
+      (-(1529822745192444888481 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2165740525370507 / 10000000000000000 : ℝ) =
+            (2165740525370507 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2165740525370507 / 1250000000000000 : ℝ) ≤
+            (549618795707555111519 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1068703947098007 / 5000000000000000 : ℝ)) (q :=
+      (2165740525370507 / 10000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_294_7 :
+    Uω (aρ 7) (bρ 7) (269345996903 / 1000000000000 : ℝ) ≤
+      (-(16595518643128249441507 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3804484142218451 / 20000000000000000 : ℝ) ≤
+      (-(16595518643128249441507 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3804484142218451 / 20000000000000000 : ℝ) =
+            (3804484142218451 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3804484142218451 / 2500000000000000 : ℝ) ≤
+            (4198896765871750558493 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1834072366433451 / 10000000000000000 : ℝ)) (q :=
+      (3804484142218451 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_294_8 :
+    Uω (aρ 8) (bρ 8) (269345996903 / 1000000000000 : ℝ) ≤
+      (-(18831930265090184175181 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3042073168366019 / 20000000000000000 : ℝ) ≤
+      (-(18831930265090184175181 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3042073168366019 / 20000000000000000 : ℝ) =
+            (3042073168366019 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3042073168366019 / 2500000000000000 : ℝ) ≤
+            (1962485143909815824819 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1355203043921019 / 10000000000000000 : ℝ)) (q :=
+      (3042073168366019 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_294_9 :
+    Uω (aρ 9) (bρ 9) (269345996903 / 1000000000000 : ℝ) ≤
+      (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_9]) (by norm_num [bρ_9])]; exact Lmid_9
+
+lemma U_294_10 :
+    Uω (aρ 10) (bρ 10) (269345996903 / 1000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_294_11 :
+    Uω (aρ 11) (bρ 11) (269345996903 / 1000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_294_12 :
+    Uω (aρ 12) (bρ 12) (269345996903 / 1000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_294_13 :
+    Uω (aρ 13) (bρ 13) (269345996903 / 1000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_294_14 :
+    Uω (aρ 14) (bρ 14) (269345996903 / 1000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_294_15 :
+    Uω (aρ 15) (bρ 15) (269345996903 / 1000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_294_16 :
+    Uω (aρ 16) (bρ 16) (269345996903 / 1000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_294 :
+    Uρ (269345996903 / 1000000000000 : ℝ) ≤
+      (-(4357409086777446195519 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (269345996903 / 1000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(13360191075021170630243 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_294_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (269345996903 / 1000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) * (-(26904688033233695969 / 20000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_294_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (269345996903 / 1000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3410010963726477067487 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_294_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (269345996903 / 1000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(13962424118052550522617 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_294_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (269345996903 / 1000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(14481773407649457578313 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_294_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (269345996903 / 1000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1529822745192444888481 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_294_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (269345996903 / 1000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(16595518643128249441507 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_294_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (269345996903 / 1000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(18831930265090184175181 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_294_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (269345996903 / 1000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(26986659976834617141369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_294_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (269345996903 / 1000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_294_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (269345996903 / 1000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_294_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (269345996903 / 1000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_294_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (269345996903 / 1000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_294_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (269345996903 / 1000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_294_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (269345996903 / 1000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_294_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (269345996903 / 1000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_294_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_295_1 :
+    Uω (aρ 1) (bρ 1) (8696815458149 / 32000000000000 : ℝ) ≤
+      (-(265363977074615904323 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10612786717365443 / 40000000000000000 : ℝ) ≤
+      (-(265363977074615904323 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (10612786717365443 / 40000000000000000 : ℝ) =
+            (10612786717365443 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10612786717365443 / 10000000000000000 : ℝ) ≤
+            (11894895045384095677 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2653135746096159 / 10000000000000000 : ℝ)) (q :=
+      (10612786717365443 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_295_2 :
+    Uω (aρ 2) (bρ 2) (8696815458149 / 32000000000000 : ℝ) ≤
+      (-(6679745953429830672479 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10516340261626047 / 40000000000000000 : ℝ) ≤
+      (-(6679745953429830672479 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (10516340261626047 / 40000000000000000 : ℝ) =
+            (10516340261626047 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10516340261626047 / 10000000000000000 : ℝ) ≤
+            (251725849570169327521 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2628681530736461 / 10000000000000000 : ℝ)) (q :=
+      (10516340261626047 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_295_3 :
+    Uω (aρ 3) (bρ 3) (8696815458149 / 32000000000000 : ℝ) ≤
+      (-(13545402519784924013857 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10322636493641941 / 40000000000000000 : ℝ) ≤
+      (-(13545402519784924013857 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (10322636493641941 / 40000000000000000 : ℝ) =
+            (10322636493641941 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10322636493641941 / 10000000000000000 : ℝ) ≤
+            (317541086215075986143 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (322403219136801 / 1250000000000000 : ℝ)) (q :=
+      (10322636493641941 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_295_4 :
+    Uω (aρ 4) (bρ 4) (8696815458149 / 32000000000000 : ℝ) ≤
+      (-(6932293049890189446321 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9998357638502501 / 40000000000000000 : ℝ) ≤
+      (-(6932293049890189446321 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (9998357638502501 / 40000000000000000 : ℝ) =
+            (9998357638502501 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 15
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9998357638502501 / 5000000000000000 : ℝ) ≤
+            (3464914654609810553679 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (311922563323711 / 1250000000000000 : ℝ)) (q :=
+      (9998357638502501 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_295_5 :
+    Uω (aρ 5) (bρ 5) (8696815458149 / 32000000000000 : ℝ) ≤
+      (-(1437843032608491819939 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9497574219890053 / 40000000000000000 : ℝ) ≤
+      (-(1437843032608491819939 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (9497574219890053 / 40000000000000000 : ℝ) =
+            (9497574219890053 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9497574219890053 / 5000000000000000 : ℝ) ≤
+            (641598508291508180061 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (73849334894327 / 312500000000000 : ℝ)) (q :=
+      (9497574219890053 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_295_6 :
+    Uω (aρ 6) (bρ 6) (8696815458149 / 32000000000000 : ℝ) ≤
+      (-(15185220254473415163101 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8761415055219761 / 40000000000000000 : ℝ) ≤
+      (-(15185220254473415163101 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (8761415055219761 / 40000000000000000 : ℝ) =
+            (8761415055219761 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8761415055219761 / 5000000000000000 : ℝ) ≤
+            (5609195154526584836899 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1081169754711659 / 5000000000000000 : ℝ)) (q :=
+      (8761415055219761 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_295_7 :
+    Uω (aρ 7) (bρ 7) (8696815458149 / 32000000000000 : ℝ) ≤
+      (-(3292797548244679441269 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7709710815941033 / 40000000000000000 : ℝ) ≤
+      (-(3292797548244679441269 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (7709710815941033 / 40000000000000000 : ℝ) =
+            (7709710815941033 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7709710815941033 / 5000000000000000 : ℝ) ≤
+            (866085533555320558731 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (930074385271977 / 5000000000000000 : ℝ)) (q :=
+      (7709710815941033 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_295_8 :
+    Uω (aρ 8) (bρ 8) (8696815458149 / 32000000000000 : ℝ) ≤
+      (-(18654625142945181295753 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1238596676944277 / 8000000000000000 : ℝ) ≤
+      (-(18654625142945181295753 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1238596676944277 / 8000000000000000 : ℝ) =
+            (1238596676944277 / 1000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1238596676944277 / 1000000000000000 : ℝ) ≤
+            (2139790266054818704247 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (138532670627413 / 1000000000000000 : ℝ)) (q :=
+      (1238596676944277 / 8000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_295_9 :
+    Uω (aρ 9) (bρ 9) (8696815458149 / 32000000000000 : ℝ) ≤
+      (-(12544728152889271956379 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3254158843654867 / 40000000000000000 : ℝ) ≤
+      (-(12544728152889271956379 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (3254158843654867 / 40000000000000000 : ℝ) =
+            (3254158843654867 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3254158843654867 / 2500000000000000 : ℝ) ≤
+            (1318215453110728043621 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (256880064100871 / 10000000000000000 : ℝ)) (q :=
+      (3254158843654867 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_295_10 :
+    Uω (aρ 10) (bρ 10) (8696815458149 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_295_11 :
+    Uω (aρ 11) (bρ 11) (8696815458149 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_295_12 :
+    Uω (aρ 12) (bρ 12) (8696815458149 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_295_13 :
+    Uω (aρ 13) (bρ 13) (8696815458149 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_295_14 :
+    Uω (aρ 14) (bρ 14) (8696815458149 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_295_15 :
+    Uω (aρ 15) (bρ 15) (8696815458149 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_295_16 :
+    Uω (aρ 16) (bρ 16) (8696815458149 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_295 :
+    Uρ (8696815458149 / 32000000000000 : ℝ) ≤
+      (-(4301440739823679509847 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (8696815458149 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(265363977074615904323 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_295_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (8696815458149 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(6679745953429830672479 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_295_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (8696815458149 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(13545402519784924013857 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_295_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (8696815458149 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(6932293049890189446321 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_295_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (8696815458149 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1437843032608491819939 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_295_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (8696815458149 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(15185220254473415163101 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_295_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (8696815458149 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(3292797548244679441269 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_295_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (8696815458149 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(18654625142945181295753 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_295_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (8696815458149 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(12544728152889271956379 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_295_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (8696815458149 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_295_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (8696815458149 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_295_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (8696815458149 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_295_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (8696815458149 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_295_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (8696815458149 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_295_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (8696815458149 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_295_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (8696815458149 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_295_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_296_1 :
+    Uω (aρ 1) (bρ 1) (4387279507701 / 16000000000000 : ℝ) ≤
+      (-(6588522609675428053601 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5354984187775611 / 20000000000000000 : ℝ) ≤
+      (-(6588522609675428053601 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5354984187775611 / 20000000000000000 : ℝ) =
+            (5354984187775611 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5354984187775611 / 5000000000000000 : ℝ) ≤
+            (342949193324571946399 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1338715856773743 / 5000000000000000 : ℝ)) (q :=
+      (5354984187775611 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_296_2 :
+    Uω (aρ 2) (bρ 2) (4387279507701 / 16000000000000 : ℝ) ≤
+      (-(1326749428282152019991 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1326691811223351 / 5000000000000000 : ℝ) ≤
+      (-(1326749428282152019991 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1326691811223351 / 5000000000000000 : ℝ) =
+            (1326691811223351 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1326691811223351 / 1250000000000000 : ℝ) ≤
+            (59544932317847980009 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2652983783175279 / 10000000000000000 : ℝ)) (q :=
+      (1326691811223351 / 5000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_296_3 :
+    Uω (aρ 3) (bρ 3) (4387279507701 / 16000000000000 : ℝ) ≤
+      (-(6725824734455487825467 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (130248368026923 / 500000000000000 : ℝ) ≤
+      (-(6725824734455487825467 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (130248368026923 / 500000000000000 : ℝ) =
+            (130248368026923 / 125000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (130248368026923 / 125000000000000 : ℝ) ≤
+            (205647068544512174533 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (520709473141759 / 2000000000000000 : ℝ)) (q :=
+      (130248368026923 / 500000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_296_4 :
+    Uω (aρ 4) (bρ 4) (4387279507701 / 16000000000000 : ℝ) ≤
+      (-(3441924802935448534447 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5047849706220241 / 20000000000000000 : ℝ) ≤
+      (-(3441924802935448534447 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5047849706220241 / 20000000000000000 : ℝ) =
+            (5047849706220241 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5047849706220241 / 5000000000000000 : ℝ) ≤
+            (23811098564551465553 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (629939132979279 / 2500000000000000 : ℝ)) (q :=
+      (5047849706220241 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_296_5 :
+    Uω (aρ 5) (bρ 5) (4387279507701 / 16000000000000 : ℝ) ≤
+      (-(1784519279932689038239 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4797605066056417 / 20000000000000000 : ℝ) ≤
+      (-(1784519279932689038239 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4797605066056417 / 20000000000000000 : ℝ) =
+            (4797605066056417 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4797605066056417 / 2500000000000000 : ℝ) ≤
+            (814782646192310961761 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (596925452772073 / 2500000000000000 : ℝ)) (q :=
+      (4797605066056417 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_296_6 :
+    Uω (aρ 6) (bρ 6) (4387279507701 / 16000000000000 : ℝ) ≤
+      (-(3768377053057300971997 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1107479918431881 / 5000000000000000 : ℝ) ≤
+      (-(3768377053057300971997 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1107479918431881 / 5000000000000000 : ℝ) =
+            (1107479918431881 / 625000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1107479918431881 / 625000000000000 : ℝ) ≤
+            (1430226799192699028003 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2187256793899399 / 10000000000000000 : ℝ)) (q :=
+      (1107479918431881 / 5000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_296_7 :
+    Uω (aρ 7) (bρ 7) (4387279507701 / 16000000000000 : ℝ) ≤
+      (-(16334286269266483063657 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (976294774346881 / 5000000000000000 : ℝ) ≤
+      (-(16334286269266483063657 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (976294774346881 / 5000000000000000 : ℝ) =
+            (976294774346881 / 625000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (976294774346881 / 625000000000000 : ℝ) ≤
+            (4460129139733516936343 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1886177598319399 / 10000000000000000 : ℝ)) (q :=
+      (976294774346881 / 5000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_296_8 :
+    Uω (aρ 8) (bρ 8) (4387279507701 / 16000000000000 : ℝ) ≤
+      (-(4620280082110656607647 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (49229470375789 / 312500000000000 : ℝ) ≤
+      (-(4620280082110656607647 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (49229470375789 / 312500000000000 : ℝ) = (49229470375789 / 39062500000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (49229470375789 / 39062500000000 : ℝ) ≤
+            (578323770139343392353 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1415226256322371 / 10000000000000000 : ℝ)) (q :=
+      (49229470375789 / 312500000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_296_9 :
+    Uω (aρ 9) (bρ 9) (4387279507701 / 16000000000000 : ℝ) ≤
+      (-(24307602199671372227129 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (43984965283121 / 500000000000000 : ℝ) ≤
+      (-(24307602199671372227129 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (43984965283121 / 500000000000000 : ℝ) = (43984965283121 / 31250000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (43984965283121 / 31250000000000 : ℝ) ≤
+            (3418285012328627772871 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (72980878391343 / 2000000000000000 : ℝ)) (q :=
+      (43984965283121 / 500000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_296_10 :
+    Uω (aρ 10) (bρ 10) (4387279507701 / 16000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_296_11 :
+    Uω (aρ 11) (bρ 11) (4387279507701 / 16000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_296_12 :
+    Uω (aρ 12) (bρ 12) (4387279507701 / 16000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_296_13 :
+    Uω (aρ 13) (bρ 13) (4387279507701 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_296_14 :
+    Uω (aρ 14) (bρ 14) (4387279507701 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_296_15 :
+    Uω (aρ 15) (bρ 15) (4387279507701 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_296_16 :
+    Uω (aρ 16) (bρ 16) (4387279507701 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_296 :
+    Uρ (4387279507701 / 16000000000000 : ℝ) ≤
+      (-(17081173195794320474823 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (4387279507701 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(6588522609675428053601 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_296_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (4387279507701 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1326749428282152019991 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_296_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (4387279507701 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(6725824734455487825467 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_296_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (4387279507701 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3441924802935448534447 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_296_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (4387279507701 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1784519279932689038239 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_296_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (4387279507701 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3768377053057300971997 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_296_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (4387279507701 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(16334286269266483063657 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_296_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (4387279507701 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(4620280082110656607647 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_296_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (4387279507701 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(24307602199671372227129 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_296_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (4387279507701 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_296_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (4387279507701 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_296_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (4387279507701 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_296_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (4387279507701 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_296_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (4387279507701 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_296_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (4387279507701 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_296_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (4387279507701 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_296_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_297_1 :
+    Uω (aρ 1) (bρ 1) (1770460514531 / 6400000000000 : ℝ) ≤
+      (-(13086715020354936082899 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2161429998792163 / 8000000000000000 : ℝ) ≤
+      (-(13086715020354936082899 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2161429998792163 / 8000000000000000 : ℝ) =
+            (2161429998792163 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2161429998792163 / 2000000000000000 : ℝ) ≤
+            (776228585645063917101 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (8442898940971 / 31250000000000 : ℝ)) (q :=
+      (2161429998792163 / 8000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_297_2 :
+    Uω (aρ 2) (bρ 2) (1770460514531 / 6400000000000 : ℝ) ≤
+      (-(3294083889020482951309 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10710728449637483 / 40000000000000000 : ℝ) ≤
+      (-(3294083889020482951309 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (10710728449637483 / 40000000000000000 : ℝ) =
+            (10710728449637483 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10710728449637483 / 10000000000000000 : ℝ) ≤
+            (171652012479517048691 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1338642950729527 / 5000000000000000 : ℝ)) (q :=
+      (10710728449637483 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_297_3 :
+    Uω (aρ 3) (bρ 3) (1770460514531 / 6400000000000 : ℝ) ≤
+      (-(13358768164879755402591 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10517101400861367 / 40000000000000000 : ℝ) ≤
+      (-(13358768164879755402591 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (10517101400861367 / 40000000000000000 : ℝ) =
+            (10517101400861367 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10517101400861367 / 10000000000000000 : ℝ) ≤
+            (504175441120244597409 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (656967120855249 / 2500000000000000 : ℝ)) (q :=
+      (10517101400861367 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_297_4 :
+    Uω (aρ 4) (bρ 4) (1770460514531 / 6400000000000 : ℝ) ≤
+      (-(13671745074242828367393 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10193038080950819 / 40000000000000000 : ℝ) ≤
+      (-(13671745074242828367393 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (10193038080950819 / 40000000000000000 : ℝ) =
+            (10193038080950819 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10193038080950819 / 10000000000000000 : ℝ) ≤
+            (191198531757171632607 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1272065502265361 / 5000000000000000 : ℝ)) (q :=
+      (10193038080950819 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_297_5 :
+    Uω (aρ 5) (bρ 5) (1770460514531 / 6400000000000 : ℝ) ≤
+      (-(14174923141875718530619 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9692836806498103 / 40000000000000000 : ℝ) ≤
+      (-(14174923141875718530619 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (9692836806498103 / 40000000000000000 : ℝ) =
+            (9692836806498103 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9692836806498103 / 5000000000000000 : ℝ) ≤
+            (6619492267124281469381 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (603055071659841 / 2500000000000000 : ℝ)) (q :=
+      (9692836806498103 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_297_6 :
+    Uω (aρ 6) (bρ 6) (1770460514531 / 6400000000000 : ℝ) ≤
+      (-(14963061266796872825393 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8958235946698503 / 40000000000000000 : ℝ) ≤
+      (-(14963061266796872825393 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (8958235946698503 / 40000000000000000 : ℝ) =
+            (8958235946698503 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8958235946698503 / 5000000000000000 : ℝ) ≤
+            (5831354142203127174607 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (553040057969891 / 2500000000000000 : ℝ)) (q :=
+      (8958235946698503 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_297_7 :
+    Uω (aρ 7) (bρ 7) (1770460514531 / 6400000000000 : ℝ) ≤
+      (-(1620636089527790470651 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7910914306669263 / 40000000000000000 : ℝ) ≤
+      (-(1620636089527790470651 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (7910914306669263 / 40000000000000000 : ℝ) =
+            (7910914306669263 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7910914306669263 / 5000000000000000 : ℝ) ≤
+            (458805451372209529349 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (119510049539059 / 625000000000000 : ℝ)) (q :=
+      (7910914306669263 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_297_8 :
+    Uω (aρ 8) (bρ 8) (1770460514531 / 6400000000000 : ℝ) ≤
+      (-(4577807632937310211223 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6409340632257907 / 40000000000000000 : ℝ) ≤
+      (-(4577807632937310211223 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (6409340632257907 / 40000000000000000 : ℝ) =
+            (6409340632257907 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6409340632257907 / 5000000000000000 : ℝ) ≤
+            (620796219312689788777 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (722457803379633 / 5000000000000000 : ℝ)) (q :=
+      (6409340632257907 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_297_9 :
+    Uω (aρ 9) (bρ 9) (1770460514531 / 6400000000000 : ℝ) ≤
+      (-(23710349633563607825731 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3735361084134731 / 40000000000000000 : ℝ) ≤
+      (-(23710349633563607825731 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (3735361084134731 / 40000000000000000 : ℝ) =
+            (3735361084134731 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3735361084134731 / 2500000000000000 : ℝ) ≤
+            (4015537578436392174269 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (224445730528839 / 5000000000000000 : ℝ)) (q :=
+      (3735361084134731 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_297_10 :
+    Uω (aρ 10) (bρ 10) (1770460514531 / 6400000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_297_11 :
+    Uω (aρ 11) (bρ 11) (1770460514531 / 6400000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_297_12 :
+    Uω (aρ 12) (bρ 12) (1770460514531 / 6400000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_297_13 :
+    Uω (aρ 13) (bρ 13) (1770460514531 / 6400000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_297_14 :
+    Uω (aρ 14) (bρ 14) (1770460514531 / 6400000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_297_15 :
+    Uω (aρ 15) (bρ 15) (1770460514531 / 6400000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_297_16 :
+    Uω (aρ 16) (bρ 16) (1770460514531 / 6400000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_297 :
+    Uρ (1770460514531 / 6400000000000 : ℝ) ≤
+      (-(16973651665081069339969 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1770460514531 / 6400000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(13086715020354936082899 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_297_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1770460514531 / 6400000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3294083889020482951309 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_297_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1770460514531 / 6400000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(13358768164879755402591 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_297_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1770460514531 / 6400000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(13671745074242828367393 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_297_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1770460514531 / 6400000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(14174923141875718530619 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_297_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1770460514531 / 6400000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(14963061266796872825393 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_297_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1770460514531 / 6400000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(1620636089527790470651 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_297_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1770460514531 / 6400000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(4577807632937310211223 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_297_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1770460514531 / 6400000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(23710349633563607825731 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_297_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1770460514531 / 6400000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_297_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1770460514531 / 6400000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_297_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1770460514531 / 6400000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_297_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1770460514531 / 6400000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_297_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1770460514531 / 6400000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_297_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1770460514531 / 6400000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_297_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1770460514531 / 6400000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_297_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_298_1 :
+    Uω (aρ 1) (bρ 1) (17782348702563 / 64000000000000 : ℝ) ≤
+      (-(6520927042978397703819 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21711481577166763 / 80000000000000000 : ℝ) ≤
+      (-(6520927042978397703819 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (21711481577166763 / 80000000000000000 : ℝ) =
+            (21711481577166763 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21711481577166763 / 20000000000000000 : ℝ) ≤
+            (410544760021602296181 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1356937813800611 / 5000000000000000 : ℝ)) (q :=
+      (21711481577166763 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_298_2 :
+    Uω (aρ 2) (bρ 2) (17782348702563 / 64000000000000 : ℝ) ≤
+      (-(13131066023857589108797 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21518650662438987 / 80000000000000000 : ℝ) ≤
+      (-(13131066023857589108797 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (21518650662438987 / 80000000000000000 : ℝ) =
+            (21518650662438987 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21518650662438987 / 20000000000000000 : ℝ) ≤
+            (731877582142410891203 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1344718455714639 / 5000000000000000 : ℝ)) (q :=
+      (21518650662438987 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_298_3 :
+    Uω (aρ 3) (bρ 3) (17782348702563 / 64000000000000 : ℝ) ≤
+      (-(13312649375703026812221 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21131434035174243 / 80000000000000000 : ℝ) ≤
+      (-(13312649375703026812221 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (21131434035174243 / 80000000000000000 : ℝ) =
+            (21131434035174243 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21131434035174243 / 20000000000000000 : ℝ) ≤
+            (550294230296973187779 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (660007215240773 / 2500000000000000 : ℝ)) (q :=
+      (21131434035174243 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_298_4 :
+    Uω (aρ 4) (bρ 4) (17782348702563 / 64000000000000 : ℝ) ≤
+      (-(13624112191107404353649 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20483412556860871 / 80000000000000000 : ℝ) ≤
+      (-(13624112191107404353649 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (20483412556860871 / 80000000000000000 : ℝ) =
+            (20483412556860871 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20483412556860871 / 20000000000000000 : ℝ) ≤
+            (238831414892595646351 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2556317672449749 / 10000000000000000 : ℝ)) (q :=
+      (20483412556860871 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_298_5 :
+    Uω (aρ 5) (bρ 5) (17782348702563 / 64000000000000 : ℝ) ≤
+      (-(2824938554749409884513 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19483293534172267 / 80000000000000000 : ℝ) ≤
+      (-(2824938554749409884513 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (19483293534172267 / 80000000000000000 : ℝ) =
+            (19483293534172267 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19483293534172267 / 10000000000000000 : ℝ) ≤
+            (1333944527050590115487 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1212238918056299 / 5000000000000000 : ℝ)) (q :=
+      (19483293534172267 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_298_6 :
+    Uω (aρ 6) (bρ 6) (17782348702563 / 64000000000000 : ℝ) ≤
+      (-(14908303102564459612481 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (18014848304206023 / 80000000000000000 : ℝ) ≤
+      (-(14908303102564459612481 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (18014848304206023 / 80000000000000000 : ℝ) =
+            (18014848304206023 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18014848304206023 / 10000000000000000 : ℝ) ≤
+            (5886112306435540387519 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2224606903761037 / 10000000000000000 : ℝ)) (q :=
+      (18014848304206023 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_298_7 :
+    Uω (aρ 7) (bρ 7) (17782348702563 / 64000000000000 : ℝ) ≤
+      (-(16143048243093906018401 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3184463717036787 / 16000000000000000 : ℝ) ≤
+      (-(16143048243093906018401 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3184463717036787 / 16000000000000000 : ℝ) =
+            (3184463717036787 / 2000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3184463717036787 / 2000000000000000 : ℝ) ≤
+            (4651367165906093981599 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (385027170953103 / 2000000000000000 : ℝ)) (q :=
+      (3184463717036787 / 16000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_298_8 :
+    Uω (aρ 8) (bρ 8) (17782348702563 / 64000000000000 : ℝ) ≤
+      (-(18227587364746944480603 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12926350436089839 / 80000000000000000 : ℝ) ≤
+      (-(18227587364746944480603 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (12926350436089839 / 80000000000000000 : ℝ) =
+            (12926350436089839 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12926350436089839 / 10000000000000000 : ℝ) ≤
+            (2566828044253055519397 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1459685468831991 / 10000000000000000 : ℝ)) (q :=
+      (12926350436089839 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_298_9 :
+    Uω (aρ 9) (bρ 9) (17782348702563 / 64000000000000 : ℝ) ≤
+      (-(2345044511846416428901 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7667434863188171 / 80000000000000000 : ℝ) ≤
+      (-(2345044511846416428901 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (7667434863188171 / 80000000000000000 : ℝ) =
+            (7667434863188171 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7667434863188171 / 5000000000000000 : ℝ) ≤
+            (427544209353583571099 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (242961101983287 / 5000000000000000 : ℝ)) (q :=
+      (7667434863188171 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_298_10 :
+    Uω (aρ 10) (bρ 10) (17782348702563 / 64000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_298_11 :
+    Uω (aρ 11) (bρ 11) (17782348702563 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_298_12 :
+    Uω (aρ 12) (bρ 12) (17782348702563 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_298_13 :
+    Uω (aρ 13) (bρ 13) (17782348702563 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_298_14 :
+    Uω (aρ 14) (bρ 14) (17782348702563 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_298_15 :
+    Uω (aρ 15) (bρ 15) (17782348702563 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_298_16 :
+    Uω (aρ 16) (bρ 16) (17782348702563 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_298 :
+    Uρ (17782348702563 / 64000000000000 : ℝ) ≤
+      (-(16923589864234457423761 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (17782348702563 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(6520927042978397703819 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_298_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (17782348702563 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(13131066023857589108797 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_298_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (17782348702563 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(13312649375703026812221 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_298_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (17782348702563 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(13624112191107404353649 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_298_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (17782348702563 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(2824938554749409884513 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_298_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (17782348702563 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(14908303102564459612481 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_298_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (17782348702563 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(16143048243093906018401 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_298_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (17782348702563 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(18227587364746944480603 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_298_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (17782348702563 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(2345044511846416428901 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_298_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (17782348702563 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_298_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (17782348702563 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_298_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (17782348702563 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_298_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (17782348702563 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_298_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (17782348702563 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_298_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (17782348702563 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_298_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (17782348702563 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_298_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_299_1 :
+    Uω (aρ 1) (bρ 1) (2232511532477 / 8000000000000 : ℝ) ≤
+      (-(6498596756097873732649 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2726082893414437 / 10000000000000000 : ℝ) ≤
+      (-(6498596756097873732649 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2726082893414437 / 10000000000000000 : ℝ) =
+            (2726082893414437 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2726082893414437 / 2500000000000000 : ℝ) ≤
+            (432875046902126267351 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (340752948664703 / 1250000000000000 : ℝ)) (q :=
+      (2726082893414437 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_299_2 :
+    Uω (aρ 2) (bρ 2) (2232511532477 / 8000000000000 : ℝ) ≤
+      (-(327150014026541970697 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5403961074209407 / 20000000000000000 : ℝ) ≤
+      (-(327150014026541970697 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5403961074209407 / 20000000000000000 : ℝ) =
+            (5403961074209407 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5403961074209407 / 5000000000000000 : ℝ) ≤
+            (19423576123458029303 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2701587889208157 / 10000000000000000 : ℝ)) (q :=
+      (5403961074209407 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_299_3 :
+    Uω (aρ 3) (bρ 3) (2232511532477 / 8000000000000 : ℝ) ≤
+      (-(1658342816041916903219 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5307166198497323 / 20000000000000000 : ℝ) ≤
+      (-(1658342816041916903219 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5307166198497323 / 20000000000000000 : ℝ) =
+            (5307166198497323 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5307166198497323 / 5000000000000000 : ℝ) ≤
+            (74525134708083096781 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2652189119846073 / 10000000000000000 : ℝ)) (q :=
+      (5307166198497323 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_299_4 :
+    Uω (aρ 4) (bρ 4) (2232511532477 / 8000000000000 : ℝ) ≤
+      (-(2715341168884718786139 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (80393544784709 / 312500000000000 : ℝ) ≤
+      (-(2715341168884718786139 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (80393544784709 / 312500000000000 : ℝ) = (80393544784709 / 78125000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (80393544784709 / 78125000000000 : ℝ) ≤
+            (57247552315281213861 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1284251984317563 / 5000000000000000 : ℝ)) (q :=
+      (80393544784709 / 312500000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_299_5 :
+    Uω (aρ 5) (bρ 5) (2232511532477 / 8000000000000 : ℝ) ≤
+      (-(14074715707090353918203 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4895227260949929 / 20000000000000000 : ℝ) ≤
+      (-(14074715707090353918203 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4895227260949929 / 20000000000000000 : ℝ) =
+            (4895227260949929 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4895227260949929 / 2500000000000000 : ℝ) ≤
+            (6719699701909646081797 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2436734282698679 / 10000000000000000 : ℝ)) (q :=
+      (4895227260949929 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_299_6 :
+    Uω (aρ 6) (bρ 6) (2232511532477 / 8000000000000 : ℝ) ≤
+      (-(7426925208423690697017 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2264151444451861 / 10000000000000000 : ℝ) ≤
+      (-(7426925208423690697017 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2264151444451861 / 10000000000000000 : ℝ) =
+            (2264151444451861 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2264151444451861 / 1250000000000000 : ℝ) ≤
+            (2970282496076309302983 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (279631285724059 / 1250000000000000 : ℝ)) (q :=
+      (2264151444451861 / 10000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_299_7 :
+    Uω (aρ 7) (bρ 7) (2232511532477 / 8000000000000 : ℝ) ≤
+      (-(3216032140314322800777 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2002845705585937 / 10000000000000000 : ℝ) ≤
+      (-(3216032140314322800777 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2002845705585937 / 10000000000000000 : ℝ) =
+            (2002845705585937 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2002845705585937 / 1250000000000000 : ℝ) ≤
+            (942850941485677199223 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (121131261801289 / 625000000000000 : ℝ)) (q :=
+      (2002845705585937 / 10000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_299_8 :
+    Uω (aρ 8) (bρ 8) (2232511532477 / 8000000000000 : ℝ) ≤
+      (-(18144784941387903088781 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (81461425667013 / 500000000000000 : ℝ) ≤
+      (-(18144784941387903088781 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (81461425667013 / 500000000000000 : ℝ) = (81461425667013 / 62500000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (81461425667013 / 62500000000000 : ℝ) ≤
+            (2649630467612096911219 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (147440745566927 / 1000000000000000 : ℝ)) (q :=
+      (81461425667013 / 500000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_299_9 :
+    Uω (aρ 9) (bρ 9) (2232511532477 / 8000000000000 : ℝ) ≤
+      (-(11604530457872435747047 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (98184580963581 / 1000000000000000 : ℝ) ≤
+      (-(11604530457872435747047 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (98184580963581 / 1000000000000000 : ℝ) = (98184580963581 / 62500000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (98184580963581 / 62500000000000 : ℝ) ≤
+            (2258413148127564252953 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (52060767662037 / 1000000000000000 : ℝ)) (q :=
+      (98184580963581 / 1000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_299_10 :
+    Uω (aρ 10) (bρ 10) (2232511532477 / 8000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_299_11 :
+    Uω (aρ 11) (bρ 11) (2232511532477 / 8000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_299_12 :
+    Uω (aρ 12) (bρ 12) (2232511532477 / 8000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_299_13 :
+    Uω (aρ 13) (bρ 13) (2232511532477 / 8000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_299_14 :
+    Uω (aρ 14) (bρ 14) (2232511532477 / 8000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_299_15 :
+    Uω (aρ 15) (bρ 15) (2232511532477 / 8000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_299_16 :
+    Uω (aρ 16) (bρ 16) (2232511532477 / 8000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_299 :
+    Uρ (2232511532477 / 8000000000000 : ℝ) ≤
+      (-(16875345848822549837269 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (2232511532477 / 8000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(6498596756097873732649 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_299_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (2232511532477 / 8000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(327150014026541970697 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_299_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (2232511532477 / 8000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1658342816041916903219 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_299_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (2232511532477 / 8000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(2715341168884718786139 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_299_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (2232511532477 / 8000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(14074715707090353918203 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_299_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (2232511532477 / 8000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(7426925208423690697017 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_299_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (2232511532477 / 8000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(3216032140314322800777 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_299_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (2232511532477 / 8000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(18144784941387903088781 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_299_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (2232511532477 / 8000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(11604530457872435747047 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_299_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (2232511532477 / 8000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_299_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (2232511532477 / 8000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_299_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (2232511532477 / 8000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_299_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (2232511532477 / 8000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_299_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (2232511532477 / 8000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_299_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (2232511532477 / 8000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_299_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (2232511532477 / 8000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_299_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_300_1 :
+    Uω (aρ 1) (bρ 1) (17937835817069 / 64000000000000 : ℝ) ≤
+      (-(12952731517223761911667 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21905844698621997 / 80000000000000000 : ℝ) ≤
+      (-(12952731517223761911667 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (21905844698621997 / 80000000000000000 : ℝ) =
+            (21905844698621997 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21905844698621997 / 20000000000000000 : ℝ) ≤
+            (910212088776238088333 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (684542886580867 / 2500000000000000 : ℝ)) (q :=
+      (21905844698621997 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_300_2 :
+    Uω (aρ 2) (bρ 2) (17937835817069 / 64000000000000 : ℝ) ≤
+      (-(1630142166947326220879 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21713037804200553 / 80000000000000000 : ℝ) ≤
+      (-(1630142166947326220879 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (21713037804200553 / 80000000000000000 : ℝ) =
+            (21713037804200553 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21713037804200553 / 20000000000000000 : ℝ) ≤
+            (102725783802673779121 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2713738835228107 / 10000000000000000 : ℝ)) (q :=
+      (21713037804200553 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_300_3 :
+    Uω (aρ 3) (bρ 3) (17937835817069 / 64000000000000 : ℝ) ≤
+      (-(13221045681674648191651 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21325895084666693 / 80000000000000000 : ℝ) ≤
+      (-(13221045681674648191651 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (21325895084666693 / 80000000000000000 : ℝ) =
+            (21325895084666693 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21325895084666693 / 20000000000000000 : ℝ) ≤
+            (641897924325351808349 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1332174630847321 / 5000000000000000 : ℝ)) (q :=
+      (21325895084666693 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_300_4 :
+    Uω (aρ 4) (bρ 4) (17937835817069 / 64000000000000 : ℝ) ≤
+      (-(3382380970715133003067 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20678080907039941 / 80000000000000000 : ℝ) ≤
+      (-(3382380970715133003067 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (20678080907039941 / 80000000000000000 : ℝ) =
+            (20678080907039941 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20678080907039941 / 20000000000000000 : ℝ) ≤
+            (83354930784866996933 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1290344949176477 / 5000000000000000 : ℝ)) (q :=
+      (20678080907039941 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_300_5 :
+    Uω (aρ 5) (bρ 5) (17937835817069 / 64000000000000 : ℝ) ≤
+      (-(1753123672217734173881 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (787140808324557 / 3200000000000000 : ℝ) ≤
+      (-(1753123672217734173881 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (787140808324557 / 3200000000000000 : ℝ) =
+            (787140808324557 / 400000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (787140808324557 / 400000000000000 : ℝ) ≤
+            (846178253907265826119 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (48979792859129 / 200000000000000 : ℝ)) (q :=
+      (787140808324557 / 3200000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_300_6 :
+    Uω (aρ 6) (bρ 6) (17937835817069 / 64000000000000 : ℝ) ≤
+      (-(14799699741453893134069 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3642312373200597 / 16000000000000000 : ℝ) ≤
+      (-(14799699741453893134069 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3642312373200597 / 16000000000000000 : ℝ) =
+            (3642312373200597 / 2000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3642312373200597 / 2000000000000000 : ℝ) ≤
+            (5994715667546106865931 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (449898086513743 / 2000000000000000 : ℝ)) (q :=
+      (3642312373200597 / 16000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_300_7 :
+    Uω (aρ 7) (bρ 7) (17937835817069 / 64000000000000 : ℝ) ≤
+      (-(320353845175740239713 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3224634129455817 / 16000000000000000 : ℝ) ≤
+      (-(320353845175740239713 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3224634129455817 / 16000000000000000 : ℝ) =
+            (3224634129455817 / 2000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3224634129455817 / 2000000000000000 : ℝ) ≤
+            (95534463004259760287 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (97552700432387 / 500000000000000 : ℝ)) (q :=
+      (3224634129455817 / 16000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_300_8 :
+    Uω (aρ 8) (bρ 8) (17937835817069 / 64000000000000 : ℝ) ≤
+      (-(18062803918743756208487 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13141119956302201 / 80000000000000000 : ℝ) ≤
+      (-(18062803918743756208487 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (13141119956302201 / 80000000000000000 : ℝ) =
+            (13141119956302201 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13141119956302201 / 10000000000000000 : ℝ) ≤
+            (2731611490256243791513 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1489082987243519 / 10000000000000000 : ℝ)) (q :=
+      (13141119956302201 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_300_9 :
+    Uω (aρ 9) (bρ 9) (17937835817069 / 64000000000000 : ℝ) ≤
+      (-(22982841438817210519433 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1606896336678253 / 16000000000000000 : ℝ) ≤
+      (-(22982841438817210519433 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1606896336678253 / 16000000000000000 : ℝ) =
+            (1606896336678253 / 1000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1606896336678253 / 1000000000000000 : ℝ) ≤
+            (4743045773182789480567 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (110677809475157 / 2000000000000000 : ℝ)) (q :=
+      (1606896336678253 / 16000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_300_10 :
+    Uω (aρ 10) (bρ 10) (17937835817069 / 64000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_300_11 :
+    Uω (aρ 11) (bρ 11) (17937835817069 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_300_12 :
+    Uω (aρ 12) (bρ 12) (17937835817069 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_300_13 :
+    Uω (aρ 13) (bρ 13) (17937835817069 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_300_14 :
+    Uω (aρ 14) (bρ 14) (17937835817069 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_300_15 :
+    Uω (aρ 15) (bρ 15) (17937835817069 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_300_16 :
+    Uω (aρ 16) (bρ 16) (17937835817069 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_300 :
+    Uρ (17937835817069 / 64000000000000 : ℝ) ≤
+      (-(8414310178676383152373 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (17937835817069 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(12952731517223761911667 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_300_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (17937835817069 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1630142166947326220879 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_300_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (17937835817069 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(13221045681674648191651 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_300_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (17937835817069 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3382380970715133003067 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_300_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (17937835817069 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1753123672217734173881 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_300_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (17937835817069 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(14799699741453893134069 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_300_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (17937835817069 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(320353845175740239713 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_300_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (17937835817069 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(18062803918743756208487 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_300_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (17937835817069 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(22982841438817210519433 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_300_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (17937835817069 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_300_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (17937835817069 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_300_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (17937835817069 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_300_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (17937835817069 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_300_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (17937835817069 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_300_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (17937835817069 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_300_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (17937835817069 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_300_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_301_1 :
+    Uω (aρ 1) (bρ 1) (9007789687161 / 32000000000000 : ℝ) ≤
+      (-(12908466342858148682791 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11001513115667967 / 40000000000000000 : ℝ) ≤
+      (-(12908466342858148682791 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (11001513115667967 / 40000000000000000 : ℝ) =
+            (11001513115667967 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11001513115667967 / 10000000000000000 : ℝ) ≤
+            (954477263141851317209 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2750319498681171 / 10000000000000000 : ℝ)) (q :=
+      (11001513115667967 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_301_2 :
+    Uω (aρ 2) (bρ 2) (9007789687161 / 32000000000000 : ℝ) ≤
+      (-(12996474539862435628569 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10905115593113297 / 40000000000000000 : ℝ) ≤
+      (-(12996474539862435628569 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (10905115593113297 / 40000000000000000 : ℝ) =
+            (10905115593113297 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10905115593113297 / 10000000000000000 : ℝ) ≤
+            (866469066137564371431 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (681472437478459 / 2500000000000000 : ℝ)) (q :=
+      (10905115593113297 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_301_3 :
+    Uω (aρ 3) (bρ 3) (9007789687161 / 32000000000000 : ℝ) ≤
+      (-(1646944615148955121397 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10711562456793547 / 40000000000000000 : ℝ) ≤
+      (-(1646944615148955121397 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (10711562456793547 / 40000000000000000 : ℝ) =
+            (10711562456793547 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10711562456793547 / 10000000000000000 : ℝ) ≤
+            (85923335601044878603 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2676509288103961 / 10000000000000000 : ℝ)) (q :=
+      (10711562456793547 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_301_4 :
+    Uω (aρ 4) (bρ 4) (9007789687161 / 32000000000000 : ℝ) ≤
+      (-(1348256418568254684709 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (415508258079839 / 1600000000000000 : ℝ) ≤
+      (-(1348256418568254684709 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (415508258079839 / 1600000000000000 : ℝ) =
+            (415508258079839 / 400000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (415508258079839 / 400000000000000 : ℝ) ≤
+            (38037942031745315291 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (103715018670807 / 400000000000000 : ℝ)) (q :=
+      (415508258079839 / 1600000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_301_5 :
+    Uω (aρ 5) (bρ 5) (9007789687161 / 32000000000000 : ℝ) ≤
+      (-(3493877815151177753499 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1977612709202521 / 8000000000000000 : ℝ) ≤
+      (-(3493877815151177753499 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1977612709202521 / 8000000000000000 : ℝ) =
+            (1977612709202521 / 1000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1977612709202521 / 1000000000000000 : ℝ) ≤
+            (1704726037098822246501 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (246124393311349 / 1000000000000000 : ℝ)) (q :=
+      (1977612709202521 / 8000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_301_6 :
+    Uω (aρ 6) (bρ 6) (9007789687161 / 32000000000000 : ℝ) ≤
+      (-(737292383409021471139 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9154949724444683 / 40000000000000000 : ℝ) ≤
+      (-(737292383409021471139 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (9154949724444683 / 40000000000000000 : ℝ) =
+            (9154949724444683 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9154949724444683 / 5000000000000000 : ℝ) ≤
+            (302428387040978528861 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2261927397469529 / 10000000000000000 : ℝ)) (q :=
+      (9154949724444683 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_301_7 :
+    Uω (aρ 7) (bρ 7) (9007789687161 / 32000000000000 : ℝ) ≤
+      (-(7977818517863032778631 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8111767212567727 / 40000000000000000 : ℝ) ≤
+      (-(7977818517863032778631 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (8111767212567727 / 40000000000000000 : ℝ) =
+            (8111767212567727 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8111767212567727 / 5000000000000000 : ℝ) ≤
+            (2419389186636967221369 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1963997522291051 / 10000000000000000 : ℝ)) (q :=
+      (8111767212567727 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_301_8 :
+    Uω (aρ 8) (bρ 8) (9007789687161 / 32000000000000 : ℝ) ≤
+      (-(3596325135396607525091 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6624115712704663 / 40000000000000000 : ℝ) ≤
+      (-(3596325135396607525091 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (6624115712704663 / 40000000000000000 : ℝ) =
+            (6624115712704663 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6624115712704663 / 5000000000000000 : ℝ) ≤
+            (562557946403392474909 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1503713423699519 / 10000000000000000 : ℝ)) (q :=
+      (6624115712704663 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_301_9 :
+    Uω (aρ 9) (bρ 9) (9007789687161 / 32000000000000 : ℝ) ≤
+      (-(2276934142411561312073 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4103931064846051 / 40000000000000000 : ℝ) ≤
+      (-(2276934142411561312073 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (4103931064846051 / 40000000000000000 : ℝ) =
+            (4103931064846051 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4103931064846051 / 2500000000000000 : ℝ) ≤
+            (495654578788438687927 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (584586728130213 / 10000000000000000 : ℝ)) (q :=
+      (4103931064846051 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_301_10 :
+    Uω (aρ 10) (bρ 10) (9007789687161 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_301_11 :
+    Uω (aρ 11) (bρ 11) (9007789687161 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_301_12 :
+    Uω (aρ 12) (bρ 12) (9007789687161 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_301_13 :
+    Uω (aρ 13) (bρ 13) (9007789687161 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_301_14 :
+    Uω (aρ 14) (bρ 14) (9007789687161 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_301_15 :
+    Uω (aρ 15) (bρ 15) (9007789687161 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_301_16 :
+    Uω (aρ 16) (bρ 16) (9007789687161 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_301 :
+    Uρ (9007789687161 / 32000000000000 : ℝ) ≤
+      (-(3356638919890918301819 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (9007789687161 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(12908466342858148682791 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_301_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (9007789687161 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(12996474539862435628569 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_301_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (9007789687161 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1646944615148955121397 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_301_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (9007789687161 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1348256418568254684709 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_301_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (9007789687161 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(3493877815151177753499 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_301_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (9007789687161 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(737292383409021471139 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_301_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (9007789687161 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(7977818517863032778631 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_301_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (9007789687161 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(3596325135396607525091 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_301_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (9007789687161 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(2276934142411561312073 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_301_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (9007789687161 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_301_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (9007789687161 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_301_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (9007789687161 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_301_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (9007789687161 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_301_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (9007789687161 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_301_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (9007789687161 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_301_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (9007789687161 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_301_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_302_1 :
+    Uω (aρ 1) (bρ 1) (723732917263 / 2560000000000 : ℝ) ≤
+      (-(6432198127082150183279 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4420041549140519 / 16000000000000000 : ℝ) ≤
+      (-(6432198127082150183279 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4420041549140519 / 16000000000000000 : ℝ) =
+            (4420041549140519 / 4000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4420041549140519 / 4000000000000000 : ℝ) ≤
+            (499273675917849816721 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (552493489290411 / 2000000000000000 : ℝ)) (q :=
+      (4420041549140519 / 16000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_302_2 :
+    Uω (aρ 2) (bρ 2) (723732917263 / 2560000000000 : ℝ) ≤
+      (-(1619001298812421734721 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21907424444584411 / 80000000000000000 : ℝ) ≤
+      (-(1619001298812421734721 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (21907424444584411 / 80000000000000000 : ℝ) =
+            (21907424444584411 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21907424444584411 / 20000000000000000 : ℝ) ≤
+            (113866651937578265279 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2738040633682509 / 10000000000000000 : ℝ)) (q :=
+      (21907424444584411 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_302_3 :
+    Uω (aρ 3) (bρ 3) (723732917263 / 2560000000000 : ℝ) ≤
+      (-(6565137179223193084959 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21520354287015651 / 80000000000000000 : ℝ) ≤
+      (-(6565137179223193084959 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (21520354287015651 / 80000000000000000 : ℝ) =
+            (21520354287015651 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21520354287015651 / 20000000000000000 : ℝ) ≤
+            (366334623776806915041 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2688669200640319 / 10000000000000000 : ℝ)) (q :=
+      (21520354287015651 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_302_4 :
+    Uω (aρ 4) (bρ 4) (723732917263 / 2560000000000 : ℝ) ≤
+      (-(335895616554249685107 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20872743476022003 / 80000000000000000 : ℝ) ≤
+      (-(335895616554249685107 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (20872743476022003 / 80000000000000000 : ℝ) =
+            (20872743476022003 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20872743476022003 / 20000000000000000 : ℝ) ≤
+            (10677973595750314893 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2605060678956907 / 10000000000000000 : ℝ)) (q :=
+      (20872743476022003 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_302_5 :
+    Uω (aρ 5) (bρ 5) (723732917263 / 2560000000000 : ℝ) ≤
+      (-(6963139434426427686679 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19873729759151967 / 80000000000000000 : ℝ) ≤
+      (-(6963139434426427686679 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (19873729759151967 / 80000000000000000 : ℝ) =
+            (19873729759151967 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19873729759151967 / 10000000000000000 : ℝ) ≤
+            (3434068270073572313321 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1236748584537199 / 5000000000000000 : ℝ)) (q :=
+      (19873729759151967 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_302_6 :
+    Uω (aρ 6) (bρ 6) (723732917263 / 2560000000000 : ℝ) ≤
+      (-(14692290847409693047619 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (18408224513068699 / 80000000000000000 : ℝ) ≤
+      (-(14692290847409693047619 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (18408224513068699 / 80000000000000000 : ℝ) =
+            (18408224513068699 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18408224513068699 / 10000000000000000 : ℝ) ≤
+            (6102124561590306952381 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2274361232693581 / 10000000000000000 : ℝ)) (q :=
+      (18408224513068699 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_302_7 :
+    Uω (aρ 7) (bρ 7) (723732917263 / 2560000000000 : ℝ) ≤
+      (-(7946994641123173545093 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16323857787985871 / 80000000000000000 : ℝ) ≤
+      (-(7946994641123173545093 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (16323857787985871 / 80000000000000000 : ℝ) =
+            (16323857787985871 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16323857787985871 / 10000000000000000 : ℝ) ≤
+            (2450213063376826454907 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (988465466091437 / 5000000000000000 : ℝ)) (q :=
+      (16323857787985871 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_302_8 :
+    Uω (aρ 8) (bρ 8) (723732917263 / 2560000000000 : ℝ) ≤
+      (-(3580246456361628938953 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13355167728518967 / 80000000000000000 : ℝ) ≤
+      (-(3580246456361628938953 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (13355167728518967 / 80000000000000000 : ℝ) =
+            (13355167728518967 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13355167728518967 / 10000000000000000 : ℝ) ≤
+            (578636625438371061047 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (379575017164037 / 2500000000000000 : ℝ)) (q :=
+      (13355167728518967 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_302_9 :
+    Uω (aρ 9) (bρ 9) (723732917263 / 2560000000000 : ℝ) ≤
+      (-(11283356655203721398811 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8375872928878307 / 80000000000000000 : ℝ) ≤
+      (-(11283356655203721398811 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (8375872928878307 / 80000000000000000 : ℝ) =
+            (8375872928878307 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8375872928878307 / 5000000000000000 : ℝ) ≤
+            (2579586950796278601189 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (614441997105983 / 10000000000000000 : ℝ)) (q :=
+      (8375872928878307 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_302_10 :
+    Uω (aρ 10) (bρ 10) (723732917263 / 2560000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_302_11 :
+    Uω (aρ 11) (bρ 11) (723732917263 / 2560000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_302_12 :
+    Uω (aρ 12) (bρ 12) (723732917263 / 2560000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_302_13 :
+    Uω (aρ 13) (bρ 13) (723732917263 / 2560000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_302_14 :
+    Uω (aρ 14) (bρ 14) (723732917263 / 2560000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_302_15 :
+    Uω (aρ 15) (bρ 15) (723732917263 / 2560000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_302_16 :
+    Uω (aρ 16) (bρ 16) (723732917263 / 2560000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_302 :
+    Uρ (723732917263 / 2560000000000 : ℝ) ≤
+      (-(2092362830814368895911 / 1250000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (723732917263 / 2560000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(6432198127082150183279 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_302_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (723732917263 / 2560000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1619001298812421734721 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_302_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (723732917263 / 2560000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(6565137179223193084959 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_302_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (723732917263 / 2560000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(335895616554249685107 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_302_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (723732917263 / 2560000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(6963139434426427686679 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_302_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (723732917263 / 2560000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(14692290847409693047619 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_302_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (723732917263 / 2560000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(7946994641123173545093 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_302_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (723732917263 / 2560000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(3580246456361628938953 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_302_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (723732917263 / 2560000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(11283356655203721398811 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_302_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (723732917263 / 2560000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_302_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (723732917263 / 2560000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_302_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (723732917263 / 2560000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_302_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (723732917263 / 2560000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_302_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (723732917263 / 2560000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_302_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (723732917263 / 2560000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_302_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (723732917263 / 2560000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_302_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_303_1 :
+    Uω (aρ 1) (bρ 1) (4542766622207 / 16000000000000 : ℝ) ≤
+      (-(12820519539047609253381 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5549347310490741 / 20000000000000000 : ℝ) ≤
+      (-(12820519539047609253381 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5549347310490741 / 20000000000000000 : ℝ) =
+            (5549347310490741 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5549347310490741 / 5000000000000000 : ℝ) ≤
+            (1042424066952390746619 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1387307694848183 / 5000000000000000 : ℝ)) (q :=
+      (5549347310490741 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_303_2 :
+    Uω (aρ 2) (bρ 2) (4542766622207 / 16000000000000 : ℝ) ≤
+      (-(12907743127780000518383 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1100230879045659 / 4000000000000000 : ℝ) ≤
+      (-(12907743127780000518383 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1100230879045659 / 4000000000000000 : ℝ) =
+            (1100230879045659 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1100230879045659 / 1000000000000000 : ℝ) ≤
+            (955200478219999481617 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (34377393586799 / 125000000000000 : ℝ)) (q :=
+      (1100230879045659 / 4000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_303_3 :
+    Uω (aρ 3) (bρ 3) (4542766622207 / 16000000000000 : ℝ) ≤
+      (-(6542598065307995406991 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2702197901388079 / 10000000000000000 : ℝ) ≤
+      (-(6542598065307995406991 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2702197901388079 / 10000000000000000 : ℝ) =
+            (2702197901388079 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2702197901388079 / 2500000000000000 : ℝ) ≤
+            (388873737692004593009 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2700829000841783 / 10000000000000000 : ℝ)) (q :=
+      (2702197901388079 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_303_4 :
+    Uω (aρ 4) (bρ 4) (4542766622207 / 16000000000000 : ℝ) ≤
+      (-(13389303251053762690677 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (655314770094759 / 2500000000000000 : ℝ) ≤
+      (-(13389303251053762690677 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (655314770094759 / 2500000000000000 : ℝ) =
+            (655314770094759 / 625000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (655314770094759 / 625000000000000 : ℝ) ≤
+            (473640354946237309323 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2617245539888697 / 10000000000000000 : ℝ)) (q :=
+      (655314770094759 / 2500000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_303_5 :
+    Uω (aρ 5) (bρ 5) (4542766622207 / 16000000000000 : ℝ) ≤
+      (-(13877289753155558398313 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1248208016990797 / 5000000000000000 : ℝ) ≤
+      (-(13877289753155558398313 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1248208016990797 / 5000000000000000 : ℝ) =
+            (1248208016990797 / 625000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 15
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1248208016990797 / 625000000000000 : ℝ) ≤
+            (6917125655844441601687 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2485749366428813 / 10000000000000000 : ℝ)) (q :=
+      (1248208016990797 / 5000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_303_6 :
+    Uω (aρ 6) (bρ 6) (4542766622207 / 16000000000000 : ℝ) ≤
+      (-(7319512993375079674599 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4626634315685649 / 20000000000000000 : ℝ) ≤
+      (-(7319512993375079674599 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4626634315685649 / 20000000000000000 : ℝ) =
+            (4626634315685649 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4626634315685649 / 2500000000000000 : ℝ) ≤
+            (3077694711124920325401 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1143395994645637 / 5000000000000000 : ℝ)) (q :=
+      (4626634315685649 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_303_7 :
+    Uω (aρ 7) (bρ 7) (4542766622207 / 16000000000000 : ℝ) ≤
+      (-(15832743373195214114231 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (205301769048547 / 1000000000000000 : ℝ) ≤
+      (-(15832743373195214114231 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (205301769048547 / 1000000000000000 : ℝ) =
+            (205301769048547 / 125000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (205301769048547 / 125000000000000 : ℝ) ≤
+            (4961672035804785885769 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (397970887067313 / 2000000000000000 : ℝ)) (q :=
+      (205301769048547 / 1000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_303_8 :
+    Uω (aρ 8) (bρ 8) (4542766622207 / 16000000000000 : ℝ) ≤
+      (-(8910803224689173136883 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3365483466576571 / 20000000000000000 : ℝ) ≤
+      (-(8910803224689173136883 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3365483466576571 / 20000000000000000 : ℝ) =
+            (3365483466576571 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3365483466576571 / 2500000000000000 : ℝ) ≤
+            (1486404479810826863117 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (383211043070549 / 2500000000000000 : ℝ)) (q :=
+      (3365483466576571 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_303_9 :
+    Uω (aρ 9) (bρ 9) (4542766622207 / 16000000000000 : ℝ) ≤
+      (-(11186760242463115104121 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (533703873898957 / 5000000000000000 : ℝ) ≤
+      (-(11186760242463115104121 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (533703873898957 / 5000000000000000 : ℝ) =
+            (533703873898957 / 312500000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (533703873898957 / 312500000000000 : ℝ) ≤
+            (2676183363536884895879 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (643141829661453 / 10000000000000000 : ℝ)) (q :=
+      (533703873898957 / 5000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_303_10 :
+    Uω (aρ 10) (bρ 10) (4542766622207 / 16000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_303_11 :
+    Uω (aρ 11) (bρ 11) (4542766622207 / 16000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_303_12 :
+    Uω (aρ 12) (bρ 12) (4542766622207 / 16000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_303_13 :
+    Uω (aρ 13) (bρ 13) (4542766622207 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_303_14 :
+    Uω (aρ 14) (bρ 14) (4542766622207 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_303_15 :
+    Uω (aρ 15) (bρ 15) (4542766622207 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_303_16 :
+    Uω (aρ 16) (bρ 16) (4542766622207 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_303 :
+    Uρ (4542766622207 / 16000000000000 : ℝ) ≤
+      (-(8347807468509570491919 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (4542766622207 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(12820519539047609253381 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_303_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (4542766622207 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(12907743127780000518383 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_303_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (4542766622207 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(6542598065307995406991 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_303_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (4542766622207 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(13389303251053762690677 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_303_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (4542766622207 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(13877289753155558398313 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_303_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (4542766622207 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(7319512993375079674599 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_303_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (4542766622207 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(15832743373195214114231 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_303_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (4542766622207 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(8910803224689173136883 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_303_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (4542766622207 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(11186760242463115104121 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_303_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (4542766622207 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_303_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (4542766622207 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_303_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (4542766622207 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_303_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (4542766622207 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_303_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (4542766622207 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_303_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (4542766622207 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_303_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (4542766622207 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_303_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/U25.lean b/LeanPool/Zeta5Irrational/Table/U25.lean
new file mode 100644
index 0000000000..1b2ee9e135
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/U25.lean
@@ -0,0 +1,3935 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-! # Certified arcsine potential bounds (U25) -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma U_304_1 :
+    Uω (aρ 1) (bρ 1) (36419876534909 / 128000000000000 : ℝ) ≤
+      (-(319966329192450738377 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (44491959966754893 / 160000000000000000 : ℝ) ≤
+      (-(319966329192450738377 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (44491959966754893 / 160000000000000000 : ℝ) =
+            (44491959966754893 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (44491959966754893 / 40000000000000000 : ℝ) ≤
+            (26607260957549261623 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (695172339909899 / 2500000000000000 : ℝ)) (q :=
+      (44491959966754893 / 160000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_304_2 :
+    Uω (aρ 2) (bρ 2) (36419876534909 / 128000000000000 : ℝ) ≤
+      (-(12885682785381020929319 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (44106428207642013 / 160000000000000000 : ℝ) ≤
+      (-(12885682785381020929319 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (44106428207642013 / 160000000000000000 : ℝ) =
+            (44106428207642013 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (44106428207642013 / 40000000000000000 : ℝ) ≤
+            (977260820618979070681 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1378133451130243 / 5000000000000000 : ℝ)) (q :=
+      (44106428207642013 / 160000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_304_3 :
+    Uω (aρ 3) (bρ 3) (36419876534909 / 128000000000000 : ℝ) ≤
+      (-(13062733067052531422153 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8666479002615649 / 32000000000000000 : ℝ) ≤
+      (-(13062733067052531422153 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8666479002615649 / 32000000000000000 : ℝ) =
+            (8666479002615649 / 8000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8666479002615649 / 8000000000000000 : ℝ) ≤
+            (800210538947468577847 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (541381771858003 / 2000000000000000 : ℝ)) (q :=
+      (8666479002615649 / 32000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_304_4 :
+    Uω (aρ 4) (bρ 4) (36419876534909 / 128000000000000 : ℝ) ≤
+      (-(2673224740282882651271 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (42037473411546381 / 160000000000000000 : ℝ) ≤
+      (-(2673224740282882651271 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (42037473411546381 / 160000000000000000 : ℝ) =
+            (42037473411546381 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (42037473411546381 / 40000000000000000 : ℝ) ≤
+            (99363980917117348729 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (655834460040883 / 2500000000000000 : ℝ)) (q :=
+      (42037473411546381 / 160000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_304_5 :
+    Uω (aρ 5) (bρ 5) (36419876534909 / 128000000000000 : ℝ) ≤
+      (-(13852885671077141140659 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (40040251978887333 / 160000000000000000 : ℝ) ≤
+      (-(13852885671077141140659 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (40040251978887333 / 160000000000000000 : ℝ) =
+            (40040251978887333 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 2
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (40040251978887333 / 40000000000000000 : ℝ) ≤
+            (10057934922858859341 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2491875080416151 / 10000000000000000 : ℝ)) (q :=
+      (40040251978887333 / 160000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_304_6 :
+    Uω (aρ 6) (bρ 6) (36419876534909 / 128000000000000 : ℝ) ≤
+      (-(14612502027911626169787 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (37111378164456861 / 160000000000000000 : ℝ) ≤
+      (-(14612502027911626169787 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (37111378164456861 / 160000000000000000 : ℝ) =
+            (37111378164456861 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (37111378164456861 / 20000000000000000 : ℝ) ≤
+            (6181913381088373830213 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1146503114376171 / 5000000000000000 : ℝ)) (q :=
+      (37111378164456861 / 160000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_304_7 :
+    Uω (aρ 7) (bρ 7) (36419876534909 / 128000000000000 : ℝ) ≤
+      (-(7901134692862773503333 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (32948537544495549 / 160000000000000000 : ℝ) ≤
+      (-(7901134692862773503333 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (32948537544495549 / 160000000000000000 : ℝ) =
+            (32948537544495549 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (32948537544495549 / 20000000000000000 : ℝ) ≤
+            (2496073011637226496667 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (998156266008589 / 5000000000000000 : ℝ)) (q :=
+      (32948537544495549 / 160000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_304_8 :
+    Uω (aρ 8) (bρ 8) (36419876534909 / 128000000000000 : ℝ) ≤
+      (-(4445519034434660549931 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27030509260586877 / 160000000000000000 : ℝ) ≤
+      (-(4445519034434660549931 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (27030509260586877 / 160000000000000000 : ℝ) =
+            (27030509260586877 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27030509260586877 / 20000000000000000 : ℝ) ≤
+            (753084817815339450069 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (770050323934297 / 5000000000000000 : ℝ)) (q :=
+      (27030509260586877 / 160000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8])
+      (by norm_num [bρ_8]) (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_304_9 :
+    Uω (aρ 9) (bρ 9) (36419876534909 / 128000000000000 : ℝ) ≤
+      (-(4456019168123497172333 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17238827114719109 / 160000000000000000 : ℝ) ≤
+      (-(4456019168123497172333 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (17238827114719109 / 160000000000000000 : ℝ) =
+            (17238827114719109 / 10000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17238827114719109 / 10000000000000000 : ℝ) ≤
+            (1089158274276502827667 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (657106007995123 / 10000000000000000 : ℝ)) (q :=
+      (17238827114719109 / 160000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9])
+      (by norm_num [bρ_9]) (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_304_10 :
+    Uω (aρ 10) (bρ 10) (36419876534909 / 128000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_304_11 :
+    Uω (aρ 11) (bρ 11) (36419876534909 / 128000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_304_12 :
+    Uω (aρ 12) (bρ 12) (36419876534909 / 128000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_304_13 :
+    Uω (aρ 13) (bρ 13) (36419876534909 / 128000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_304_14 :
+    Uω (aρ 14) (bρ 14) (36419876534909 / 128000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_304_15 :
+    Uω (aρ 15) (bρ 15) (36419876534909 / 128000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_304_16 :
+    Uω (aρ 16) (bρ 16) (36419876534909 / 128000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_304 :
+    Uρ (36419876534909 / 128000000000000 : ℝ) ≤
+      (-(3334862897898375222561 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (36419876534909 / 128000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(319966329192450738377 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_304_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (36419876534909 / 128000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(12885682785381020929319 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_304_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (36419876534909 / 128000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(13062733067052531422153 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_304_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (36419876534909 / 128000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(2673224740282882651271 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_304_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (36419876534909 / 128000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(13852885671077141140659 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_304_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (36419876534909 / 128000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(14612502027911626169787 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_304_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (36419876534909 / 128000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(7901134692862773503333 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_304_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (36419876534909 / 128000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(4445519034434660549931 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_304_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (36419876534909 / 128000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(4456019168123497172333 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_304_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (36419876534909 / 128000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_304_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (36419876534909 / 128000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_304_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (36419876534909 / 128000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_304_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (36419876534909 / 128000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_304_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (36419876534909 / 128000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_304_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (36419876534909 / 128000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_304_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (36419876534909 / 128000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_304_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_305_1 :
+    Uω (aρ 1) (bρ 1) (18248810046081 / 64000000000000 : ℝ) ≤
+      (-(12776834507854284359341 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22294570720353829 / 80000000000000000 : ℝ) ≤
+      (-(12776834507854284359341 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (22294570720353829 / 80000000000000000 : ℝ) =
+            (22294570720353829 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22294570720353829 / 20000000000000000 : ℝ) ≤
+            (1086109098145715640659 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2786763328473301 / 10000000000000000 : ℝ)) (q :=
+      (22294570720353829 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_305_2 :
+    Uω (aρ 2) (bρ 2) (18248810046081 / 64000000000000 : ℝ) ≤
+      (-(200994859613764106703 / 156250000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (884072423872929 / 3200000000000000 : ℝ) ≤
+      (-(200994859613764106703 / 156250000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (884072423872929 / 3200000000000000 : ℝ) =
+            (884072423872929 / 800000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (884072423872929 / 800000000000000 : ℝ) ≤
+            (15613634229985893297 / 156250000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (55246846202013 / 200000000000000 : ℝ)) (q :=
+      (884072423872929 / 3200000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_305_3 :
+    Uω (aρ 3) (bρ 3) (18248810046081 / 64000000000000 : ℝ) ≤
+      (-(13040320400032887200957 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21714811691895949 / 80000000000000000 : ℝ) ≤
+      (-(13040320400032887200957 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (21714811691895949 / 80000000000000000 : ℝ) =
+            (21714811691895949 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21714811691895949 / 20000000000000000 : ℝ) ≤
+            (822623205967112799043 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2712988690218831 / 10000000000000000 : ℝ)) (q :=
+      (21714811691895949 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_305_4 :
+    Uω (aρ 4) (bρ 4) (18248810046081 / 64000000000000 : ℝ) ≤
+      (-(13342997919963146336917 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (842696016982221 / 3200000000000000 : ℝ) ≤
+      (-(13342997919963146336917 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (842696016982221 / 3200000000000000 : ℝ) =
+            (842696016982221 / 800000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (842696016982221 / 800000000000000 : ℝ) ≤
+            (519945686036853663083 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (105177202177949 / 400000000000000 : ℝ)) (q :=
+      (842696016982221 / 3200000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_305_5 :
+    Uω (aρ 5) (bρ 5) (18248810046081 / 64000000000000 : ℝ) ≤
+      (-(3457135375880206226739 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20068922691257481 / 80000000000000000 : ℝ) ≤
+      (-(3457135375880206226739 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (20068922691257481 / 80000000000000000 : ℝ) =
+            (20068922691257481 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20068922691257481 / 20000000000000000 : ℝ) ≤
+            (8600525619793773261 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1249000270229607 / 5000000000000000 : ℝ)) (q :=
+      (20068922691257481 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_305_6 :
+    Uω (aρ 6) (bρ 6) (18248810046081 / 64000000000000 : ℝ) ≤
+      (-(7293024924857735976821 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3720967579529317 / 16000000000000000 : ℝ) ≤
+      (-(7293024924857735976821 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3720967579529317 / 16000000000000000 : ℝ) =
+            (3720967579529317 / 2000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3720967579529317 / 2000000000000000 : ℝ) ≤
+            (3104182779642264023179 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (229921971719649 / 1000000000000000 : ℝ)) (q :=
+      (3720967579529317 / 16000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_305_7 :
+    Uω (aρ 7) (bρ 7) (18248810046081 / 64000000000000 : ℝ) ≤
+      (-(31543787609354435141 / 20000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16524386399945949 / 80000000000000000 : ℝ) ≤
+      (-(31543787609354435141 / 20000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (16524386399945949 / 80000000000000000 : ℝ) =
+            (16524386399945949 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16524386399945949 / 10000000000000000 : ℝ) ≤
+            (10045043208645564859 / 20000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2002768223531331 / 10000000000000000 : ℝ)) (q :=
+      (16524386399945949 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_305_8 :
+    Uω (aρ 8) (bρ 8) (18248810046081 / 64000000000000 : ℝ) ≤
+      (-(17742731513520089426687 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13568534637119313 / 80000000000000000 : ℝ) ≤
+      (-(17742731513520089426687 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (13568534637119313 / 80000000000000000 : ℝ) =
+            (13568534637119313 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13568534637119313 / 10000000000000000 : ℝ) ≤
+            (3051683895479910573313 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (24177295846323 / 156250000000000 : ℝ)) (q :=
+      (13568534637119313 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_305_9 :
+    Uω (aρ 9) (bρ 9) (18248810046081 / 64000000000000 : ℝ) ≤
+      (-(2218861948983381281 / 1000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8698622542271477 / 80000000000000000 : ℝ) ≤
+      (-(2218861948983381281 / 1000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (8698622542271477 / 80000000000000000 : ℝ) =
+            (8698622542271477 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8698622542271477 / 5000000000000000 : ℝ) ≤
+            (553726772216618719 / 1000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (670834538812713 / 10000000000000000 : ℝ)) (q :=
+      (8698622542271477 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_305_10 :
+    Uω (aρ 10) (bρ 10) (18248810046081 / 64000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_305_11 :
+    Uω (aρ 11) (bρ 11) (18248810046081 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_305_12 :
+    Uω (aρ 12) (bρ 12) (18248810046081 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_305_13 :
+    Uω (aρ 13) (bρ 13) (18248810046081 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_305_14 :
+    Uω (aρ 14) (bρ 14) (18248810046081 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_305_15 :
+    Uω (aρ 15) (bρ 15) (18248810046081 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_305_16 :
+    Uω (aρ 16) (bρ 16) (18248810046081 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_305 :
+    Uρ (18248810046081 / 64000000000000 : ℝ) ≤
+      (-(8326613935861223524661 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (18248810046081 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(12776834507854284359341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_305_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (18248810046081 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(200994859613764106703 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_305_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (18248810046081 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(13040320400032887200957 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_305_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (18248810046081 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(13342997919963146336917 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_305_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (18248810046081 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(3457135375880206226739 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_305_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (18248810046081 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(7293024924857735976821 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_305_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (18248810046081 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) * (-(31543787609354435141 / 20000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_305_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (18248810046081 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(17742731513520089426687 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_305_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (18248810046081 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(2218861948983381281 / 1000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_305_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (18248810046081 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_305_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (18248810046081 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_305_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (18248810046081 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_305_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (18248810046081 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_305_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (18248810046081 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_305_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (18248810046081 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_305_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (18248810046081 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_305_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_306_1 :
+    Uω (aρ 1) (bρ 1) (7315072729883 / 25600000000000 : ℝ) ≤
+      (-(6377531675875982707461 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (44686322905842127 / 160000000000000000 : ℝ) ≤
+      (-(6377531675875982707461 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (44686322905842127 / 160000000000000000 : ℝ) =
+            (44686322905842127 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (44686322905842127 / 40000000000000000 : ℝ) ≤
+            (553940127124017292539 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2792837296204719 / 10000000000000000 : ℝ)) (q :=
+      (44686322905842127 / 160000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_306_2 :
+    Uω (aρ 2) (bρ 2) (7315072729883 / 25600000000000 : ℝ) ≤
+      (-(802606725249877005899 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (44300814120233439 / 160000000000000000 : ℝ) ≤
+      (-(802606725249877005899 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (44300814120233439 / 160000000000000000 : ℝ) =
+            (44300814120233439 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (44300814120233439 / 40000000000000000 : ℝ) ≤
+            (63827250125122994101 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2768417710513633 / 10000000000000000 : ℝ)) (q :=
+      (44300814120233439 / 160000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_306_3 :
+    Uω (aρ 3) (bρ 3) (7315072729883 / 25600000000000 : ℝ) ≤
+      (-(2603591580741016814553 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (43526851535827023 / 160000000000000000 : ℝ) ≤
+      (-(2603591580741016814553 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (43526851535827023 / 160000000000000000 : ℝ) =
+            (43526851535827023 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (43526851535827023 / 40000000000000000 : ℝ) ≤
+            (168997140458983185447 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2719068493812831 / 10000000000000000 : ℝ)) (q :=
+      (43526851535827023 / 160000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_306_4 :
+    Uω (aρ 4) (bρ 4) (7315072729883 / 25600000000000 : ℝ) ≤
+      (-(13319925657080517909239 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (42232127603529111 / 160000000000000000 : ℝ) ≤
+      (-(13319925657080517909239 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (42232127603529111 / 160000000000000000 : ℝ) =
+            (42232127603529111 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (42232127603529111 / 40000000000000000 : ℝ) ≤
+            (543017948919482090761 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (164720136458787 / 625000000000000 : ℝ)) (q :=
+      (42232127603529111 / 160000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_306_5 :
+    Uω (aρ 5) (bρ 5) (7315072729883 / 25600000000000 : ℝ) ≤
+      (-(13804256957131762695407 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (40235436769496831 / 160000000000000000 : ℝ) ≤
+      (-(13804256957131762695407 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (40235436769496831 / 160000000000000000 : ℝ) =
+            (40235436769496831 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (40235436769496831 / 40000000000000000 : ℝ) ≤
+            (58686648868237304593 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2504125748421557 / 10000000000000000 : ℝ)) (q :=
+      (40235436769496831 / 160000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_306_6 :
+    Uω (aρ 6) (bρ 6) (7315072729883 / 25600000000000 : ℝ) ≤
+      (-(14559669056125745715263 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7461593493314523 / 32000000000000000 : ℝ) ≤
+      (-(14559669056125745715263 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (7461593493314523 / 32000000000000000 : ℝ) =
+            (7461593493314523 / 4000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7461593493314523 / 4000000000000000 : ℝ) ≤
+            (6234746352874254284737 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (230543246069603 / 1000000000000000 : ℝ)) (q :=
+      (7461593493314523 / 32000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_306_7 :
+    Uω (aρ 7) (bρ 7) (7315072729883 / 25600000000000 : ℝ) ≤
+      (-(3148323192084140152897 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1325959559977023 / 6400000000000000 : ℝ) ≤
+      (-(3148323192084140152897 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1325959559977023 / 6400000000000000 : ℝ) =
+            (1325959559977023 / 800000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1325959559977023 / 800000000000000 : ℝ) ≤
+            (1010559889715859847103 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (40184430661253 / 200000000000000 : ℝ)) (q :=
+      (1325959559977023 / 6400000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_306_8 :
+    Uω (aρ 8) (bρ 8) (7315072729883 / 25600000000000 : ℝ) ≤
+      (-(3540714116985614482459 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27243548913441479 / 160000000000000000 : ℝ) ≤
+      (-(3540714116985614482459 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (27243548913441479 / 160000000000000000 : ℝ) =
+            (27243548913441479 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27243548913441479 / 20000000000000000 : ℝ) ≤
+            (618168964814385517541 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (777291586827319 / 5000000000000000 : ℝ)) (q :=
+      (27243548913441479 / 160000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8])
+      (by norm_num [bρ_8]) (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_306_9 :
+    Uω (aρ 9) (bρ 9) (7315072729883 / 25600000000000 : ℝ) ≤
+      (-(2209898160607914061911 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17553891330828439 / 160000000000000000 : ℝ) ≤
+      (-(2209898160607914061911 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (17553891330828439 / 160000000000000000 : ℝ) =
+            (17553891330828439 / 10000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17553891330828439 / 10000000000000000 : ℝ) ≤
+            (562690560592085938089 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (85542700523501 / 1250000000000000 : ℝ)) (q :=
+      (17553891330828439 / 160000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9])
+      (by norm_num [bρ_9]) (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_306_10 :
+    Uω (aρ 10) (bρ 10) (7315072729883 / 25600000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_306_11 :
+    Uω (aρ 11) (bρ 11) (7315072729883 / 25600000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_306_12 :
+    Uω (aρ 12) (bρ 12) (7315072729883 / 25600000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_306_13 :
+    Uω (aρ 13) (bρ 13) (7315072729883 / 25600000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_306_14 :
+    Uω (aρ 14) (bρ 14) (7315072729883 / 25600000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_306_15 :
+    Uω (aρ 15) (bρ 15) (7315072729883 / 25600000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_306_16 :
+    Uω (aρ 16) (bρ 16) (7315072729883 / 25600000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_306 :
+    Uρ (7315072729883 / 25600000000000 : ℝ) ≤
+      (-(8316172534289768010333 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (7315072729883 / 25600000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(6377531675875982707461 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_306_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (7315072729883 / 25600000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(802606725249877005899 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_306_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (7315072729883 / 25600000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(2603591580741016814553 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_306_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (7315072729883 / 25600000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(13319925657080517909239 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_306_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (7315072729883 / 25600000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(13804256957131762695407 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_306_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (7315072729883 / 25600000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(14559669056125745715263 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_306_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (7315072729883 / 25600000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(3148323192084140152897 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_306_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (7315072729883 / 25600000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(3540714116985614482459 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_306_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (7315072729883 / 25600000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(2209898160607914061911 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_306_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (7315072729883 / 25600000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_306_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (7315072729883 / 25600000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_306_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (7315072729883 / 25600000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_306_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (7315072729883 / 25600000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_306_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (7315072729883 / 25600000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_306_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (7315072729883 / 25600000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_306_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (7315072729883 / 25600000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_306_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_307_1 :
+    Uω (aρ 1) (bρ 1) (9163276801667 / 32000000000000 : ℝ) ≤
+      (-(3183334873245216212523 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11195876090553929 / 40000000000000000 : ℝ) ≤
+      (-(3183334873245216212523 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (11195876090553929 / 40000000000000000 : ℝ) =
+            (11195876090553929 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11195876090553929 / 10000000000000000 : ℝ) ≤
+            (282401028254783787477 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2798911262841027 / 10000000000000000 : ℝ)) (q :=
+      (11195876090553929 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_307_2 :
+    Uω (aρ 2) (bρ 2) (9163276801667 / 32000000000000 : ℝ) ≤
+      (-(12819792339455511980223 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11099501746948329 / 40000000000000000 : ℝ) ≤
+      (-(12819792339455511980223 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (11099501746948329 / 40000000000000000 : ℝ) =
+            (11099501746948329 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11099501746948329 / 10000000000000000 : ℝ) ≤
+            (1043151266544488019777 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2774493103548227 / 10000000000000000 : ℝ)) (q :=
+      (11099501746948329 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_307_3 :
+    Uω (aρ 3) (bρ 3) (9163276801667 / 32000000000000 : ℝ) ≤
+      (-(3248911338433372805021 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10906019867661809 / 40000000000000000 : ℝ) ≤
+      (-(3248911338433372805021 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (10906019867661809 / 40000000000000000 : ℝ) =
+            (10906019867661809 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10906019867661809 / 10000000000000000 : ℝ) ≤
+            (216824563066627194979 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2725148270254967 / 10000000000000000 : ℝ)) (q :=
+      (10906019867661809 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_307_4 :
+    Uω (aρ 4) (bρ 4) (9163276801667 / 32000000000000 : ℝ) ≤
+      (-(6648453332443258315593 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10582363419881751 / 40000000000000000 : ℝ) ≤
+      (-(6648453332443258315593 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (10582363419881751 / 40000000000000000 : ℝ) =
+            (10582363419881751 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10582363419881751 / 10000000000000000 : ℝ) ≤
+            (283018470556741684407 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1320807113714969 / 5000000000000000 : ℝ)) (q :=
+      (10582363419881751 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_307_5 :
+    Uω (aρ 5) (bρ 5) (9163276801667 / 32000000000000 : ℝ) ≤
+      (-(13780031738160772080827 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2016651307729763 / 8000000000000000 : ℝ) ≤
+      (-(13780031738160772080827 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2016651307729763 / 8000000000000000 : ℝ) =
+            (2016651307729763 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2016651307729763 / 2000000000000000 : ℝ) ≤
+            (82911867839227919173 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (251025070614847 / 1000000000000000 : ℝ)) (q :=
+      (2016651307729763 / 8000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_307_6 :
+    Uω (aρ 6) (bρ 6) (9163276801667 / 32000000000000 : ℝ) ≤
+      (-(14533359254442350572721 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1870312661317283 / 8000000000000000 : ℝ) ≤
+      (-(14533359254442350572721 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1870312661317283 / 8000000000000000 : ℝ) =
+            (1870312661317283 / 1000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1870312661317283 / 1000000000000000 : ℝ) ≤
+            (6261056154557649427279 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (231164446525727 / 1000000000000000 : ℝ)) (q :=
+      (1870312661317283 / 8000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_307_7 :
+    Uω (aρ 7) (bρ 7) (9163276801667 / 32000000000000 : ℝ) ≤
+      (-(628457407618548892479 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8312296581530601 / 40000000000000000 : ℝ) ≤
+      (-(628457407618548892479 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (8312296581530601 / 40000000000000000 : ℝ) =
+            (8312296581530601 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8312296581530601 / 5000000000000000 : ℝ) ≤
+            (203319208741451107521 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2015672483489363 / 10000000000000000 : ℝ)) (q :=
+      (8312296581530601 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_307_8 :
+    Uω (aρ 8) (bρ 8) (9163276801667 / 32000000000000 : ℝ) ≤
+      (-(17664591395044278791573 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1367497464768901 / 8000000000000000 : ℝ) ≤
+      (-(17664591395044278791573 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1367497464768901 / 8000000000000000 : ℝ) =
+            (1367497464768901 / 1000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1367497464768901 / 1000000000000000 : ℝ) ≤
+            (3129824013955721208427 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (312361901197263 / 2000000000000000 : ℝ)) (q :=
+      (1367497464768901 / 8000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_307_9 :
+    Uω (aρ 9) (bρ 9) (9163276801667 / 32000000000000 : ℝ) ≤
+      (-(1375692650065395914491 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4427217190492359 / 40000000000000000 : ℝ) ≤
+      (-(1375692650065395914491 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (4427217190492359 / 40000000000000000 : ℝ) =
+            (4427217190492359 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4427217190492359 / 2500000000000000 : ℝ) ≤
+            (357175300684604085509 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (348820033835121 / 5000000000000000 : ℝ)) (q :=
+      (4427217190492359 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_307_10 :
+    Uω (aρ 10) (bρ 10) (9163276801667 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_307_11 :
+    Uω (aρ 11) (bρ 11) (9163276801667 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_307_12 :
+    Uω (aρ 12) (bρ 12) (9163276801667 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_307_13 :
+    Uω (aρ 13) (bρ 13) (9163276801667 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_307_14 :
+    Uω (aρ 14) (bρ 14) (9163276801667 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_307_15 :
+    Uω (aρ 15) (bρ 15) (9163276801667 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_307_16 :
+    Uω (aρ 16) (bρ 16) (9163276801667 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_307 :
+    Uρ (9163276801667 / 32000000000000 : ℝ) ≤
+      (-(8305828477742148551567 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (9163276801667 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3183334873245216212523 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_307_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (9163276801667 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(12819792339455511980223 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_307_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (9163276801667 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3248911338433372805021 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_307_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (9163276801667 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(6648453332443258315593 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_307_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (9163276801667 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(13780031738160772080827 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_307_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (9163276801667 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(14533359254442350572721 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_307_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (9163276801667 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(628457407618548892479 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_307_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (9163276801667 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(17664591395044278791573 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_307_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (9163276801667 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(1375692650065395914491 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_307_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (9163276801667 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_307_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (9163276801667 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_307_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (9163276801667 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_307_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (9163276801667 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_307_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (9163276801667 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_307_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (9163276801667 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_307_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (9163276801667 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_307_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_308_1 :
+    Uω (aρ 1) (bρ 1) (36730850763921 / 128000000000000 : ℝ) ≤
+      (-(1588957840809154527443 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (44880685809885337 / 160000000000000000 : ℝ) ≤
+      (-(1588957840809154527443 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (44880685809885337 / 160000000000000000 : ℝ) =
+            (44880685809885337 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (44880685809885337 / 40000000000000000 : ℝ) ≤
+            (143910109940845472557 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2804985228389339 / 10000000000000000 : ℝ)) (q :=
+      (44880685809885337 / 160000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_308_2 :
+    Uω (aρ 2) (bρ 2) (36730850763921 / 128000000000000 : ℝ) ≤
+      (-(2559585002193774210817 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8899039959342597 / 32000000000000000 : ℝ) ≤
+      (-(2559585002193774210817 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8899039959342597 / 32000000000000000 : ℝ) =
+            (8899039959342597 / 8000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8899039959342597 / 8000000000000000 : ℝ) ≤
+            (213003719006225789183 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (556113697850559 / 2000000000000000 : ℝ)) (q :=
+      (8899039959342597 / 32000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_308_3 :
+    Uω (aρ 3) (bρ 3) (36730850763921 / 128000000000000 : ℝ) ≤
+      (-(12973382527285262823649 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (43721307189703121 / 160000000000000000 : ℝ) ≤
+      (-(12973382527285262823649 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (43721307189703121 / 160000000000000000 : ℝ) =
+            (43721307189703121 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (43721307189703121 / 40000000000000000 : ℝ) ≤
+            (889561078714737176351 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1365614009863281 / 5000000000000000 : ℝ)) (q :=
+      (43721307189703121 / 160000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_308_4 :
+    Uω (aρ 4) (bρ 4) (36730850763921 / 128000000000000 : ℝ) ≤
+      (-(6636970348612193216359 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (42426779081787617 / 160000000000000000 : ℝ) ≤
+      (-(6636970348612193216359 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (42426779081787617 / 160000000000000000 : ℝ) =
+            (42426779081787617 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (42426779081787617 / 40000000000000000 : ℝ) ≤
+            (294501454387806783641 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (661926546825531 / 2500000000000000 : ℝ)) (q :=
+      (42426779081787617 / 160000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_308_5 :
+    Uω (aρ 5) (bρ 5) (36730850763921 / 128000000000000 : ℝ) ≤
+      (-(13755865555041763943411 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (40430613552428249 / 160000000000000000 : ℝ) ≤
+      (-(13755865555041763943411 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (40430613552428249 / 160000000000000000 : ℝ) =
+            (40430613552428249 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (40430613552428249 / 40000000000000000 : ℝ) ≤
+            (107078050958236056589 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2516375415467203 / 10000000000000000 : ℝ)) (q :=
+      (40430613552428249 / 160000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_308_6 :
+    Uω (aρ 6) (bρ 6) (36730850763921 / 128000000000000 : ℝ) ≤
+      (-(7253560027631494245037 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (37504533122141969 / 160000000000000000 : ℝ) ≤
+      (-(7253560027631494245037 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (37504533122141969 / 160000000000000000 : ℝ) =
+            (37504533122141969 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (37504533122141969 / 20000000000000000 : ℝ) ≤
+            (3143647676868505754963 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1158927868410709 / 5000000000000000 : ℝ)) (q :=
+      (37504533122141969 / 160000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_308_7 :
+    Uω (aρ 7) (bρ 7) (36730850763921 / 128000000000000 : ℝ) ≤
+      (-(7840675419673181174101 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (33349364960606209 / 160000000000000000 : ℝ) ≤
+      (-(7840675419673181174101 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (33349364960606209 / 160000000000000000 : ℝ) =
+            (33349364960606209 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (33349364960606209 / 20000000000000000 : ℝ) ≤
+            (2556532284826818825899 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (252765137173681 / 1250000000000000 : ℝ)) (q :=
+      (33349364960606209 / 160000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_308_8 :
+    Uω (aρ 8) (bρ 8) (36730850763921 / 128000000000000 : ℝ) ≤
+      (-(8812896010485875851931 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27456271515134017 / 160000000000000000 : ℝ) ≤
+      (-(8812896010485875851931 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (27456271515134017 / 160000000000000000 : ℝ) =
+            (27456271515134017 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27456271515134017 / 20000000000000000 : ℝ) ≤
+            (1584311694014124148069 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (98064129252839 / 625000000000000 : ℝ)) (q :=
+      (27456271515134017 / 160000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8])
+      (by norm_num [bρ_8]) (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_308_9 :
+    Uω (aρ 9) (bρ 9) (36730850763921 / 128000000000000 : ℝ) ≤
+      (-(2740603860275247694661 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17862271052765049 / 160000000000000000 : ℝ) ≤
+      (-(2740603860275247694661 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (17862271052765049 / 160000000000000000 : ℝ) =
+            (17862271052765049 / 10000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17862271052765049 / 10000000000000000 : ℝ) ≤
+            (725132041224752305339 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (710741638609303 / 10000000000000000 : ℝ)) (q :=
+      (17862271052765049 / 160000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9])
+      (by norm_num [bρ_9]) (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_308_10 :
+    Uω (aρ 10) (bρ 10) (36730850763921 / 128000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_308_11 :
+    Uω (aρ 11) (bρ 11) (36730850763921 / 128000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_308_12 :
+    Uω (aρ 12) (bρ 12) (36730850763921 / 128000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_308_13 :
+    Uω (aρ 13) (bρ 13) (36730850763921 / 128000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_308_14 :
+    Uω (aρ 14) (bρ 14) (36730850763921 / 128000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_308_15 :
+    Uω (aρ 15) (bρ 15) (36730850763921 / 128000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_308_16 :
+    Uω (aρ 16) (bρ 16) (36730850763921 / 128000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_308 :
+    Uρ (36730850763921 / 128000000000000 : ℝ) ≤
+      (-(16591155188789344255453 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (36730850763921 / 128000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1588957840809154527443 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_308_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (36730850763921 / 128000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(2559585002193774210817 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_308_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (36730850763921 / 128000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(12973382527285262823649 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_308_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (36730850763921 / 128000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(6636970348612193216359 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_308_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (36730850763921 / 128000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(13755865555041763943411 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_308_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (36730850763921 / 128000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(7253560027631494245037 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_308_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (36730850763921 / 128000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(7840675419673181174101 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_308_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (36730850763921 / 128000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(8812896010485875851931 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_308_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (36730850763921 / 128000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(2740603860275247694661 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_308_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (36730850763921 / 128000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_308_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (36730850763921 / 128000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_308_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (36730850763921 / 128000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_308_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (36730850763921 / 128000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_308_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (36730850763921 / 128000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_308_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (36730850763921 / 128000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_308_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (36730850763921 / 128000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_308_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_309_1 :
+    Uω (aρ 1) (bρ 1) (18404297160587 / 64000000000000 : ℝ) ≤
+      (-(396563526515380477321 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22488933624453707 / 80000000000000000 : ℝ) ≤
+      (-(396563526515380477321 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (22488933624453707 / 80000000000000000 : ℝ) =
+            (22488933624453707 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22488933624453707 / 20000000000000000 : ℝ) ≤
+            (36653461172119522679 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (702764798214177 / 2500000000000000 : ℝ)) (q :=
+      (22488933624453707 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_309_2 :
+    Uω (aρ 2) (bρ 2) (18404297160587 / 64000000000000 : ℝ) ≤
+      (-(12776105409233890902293 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22296196273687991 / 80000000000000000 : ℝ) ≤
+      (-(12776105409233890902293 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (22296196273687991 / 80000000000000000 : ℝ) =
+            (22296196273687991 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22296196273687991 / 20000000000000000 : ℝ) ≤
+            (1086838196766109097707 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2786643867675279 / 10000000000000000 : ℝ)) (q :=
+      (22296196273687991 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_309_3 :
+    Uω (aρ 3) (bρ 3) (18404297160587 / 64000000000000 : ℝ) ≤
+      (-(12951169203016937992483 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (876370693888647 / 3200000000000000 : ℝ) ≤
+      (-(12951169203016937992483 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (876370693888647 / 3200000000000000 : ℝ) =
+            (876370693888647 / 800000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (876370693888647 / 800000000000000 : ℝ) ≤
+            (911774402983062007517 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (109492309696293 / 400000000000000 : ℝ)) (q :=
+      (876370693888647 / 3200000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_309_4 :
+    Uω (aρ 4) (bρ 4) (18404297160587 / 64000000000000 : ℝ) ≤
+      (-(13251027509644650616501 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21262051907475379 / 80000000000000000 : ℝ) ≤
+      (-(13251027509644650616501 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (21262051907475379 / 80000000000000000 : ℝ) =
+            (21262051907475379 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21262051907475379 / 20000000000000000 : ℝ) ≤
+            (611916096355349383499 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1326899031768563 / 5000000000000000 : ℝ)) (q :=
+      (21262051907475379 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_309_5 :
+    Uω (aρ 5) (bρ 5) (18404297160587 / 64000000000000 : ℝ) ≤
+      (-(13731758118370003654299 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20264099488735607 / 80000000000000000 : ℝ) ≤
+      (-(13731758118370003654299 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (20264099488735607 / 80000000000000000 : ℝ) =
+            (20264099488735607 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20264099488735607 / 20000000000000000 : ℝ) ≤
+            (131185487629996345701 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2522499878187183 / 10000000000000000 : ℝ)) (q :=
+      (20264099488735607 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_309_6 :
+    Uω (aρ 6) (bρ 6) (18404297160587 / 64000000000000 : ℝ) ≤
+      (-(905059442027853315307 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (18801403600488763 / 80000000000000000 : ℝ) ≤
+      (-(905059442027853315307 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (18801403600488763 / 80000000000000000 : ℝ) =
+            (18801403600488763 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18801403600488763 / 10000000000000000 : ℝ) ≤
+            (394591521034646684693 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (36313535644773 / 156250000000000 : ℝ)) (q :=
+      (18801403600488763 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_309_7 :
+    Uω (aρ 7) (bρ 7) (18404297160587 / 64000000000000 : ℝ) ≤
+      (-(2445525352896588967 / 1562500000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3344952508113903 / 16000000000000000 : ℝ) ≤
+      (-(2445525352896588967 / 1562500000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3344952508113903 / 16000000000000000 : ℝ) =
+            (3344952508113903 / 2000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3344952508113903 / 2000000000000000 : ℝ) ≤
+            (803602054759661033 / 1562500000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (101428369852283 / 500000000000000 : ℝ)) (q :=
+      (3344952508113903 / 16000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_309_8 :
+    Uω (aρ 8) (bρ 8) (18404297160587 / 64000000000000 : ℝ) ≤
+      (-(8793585286503424790819 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13781258323136651 / 80000000000000000 : ℝ) ≤
+      (-(8793585286503424790819 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (13781258323136651 / 80000000000000000 : ℝ) =
+            (13781258323136651 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13781258323136651 / 10000000000000000 : ℝ) ≤
+            (1603622417996575209181 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (394058248506861 / 2500000000000000 : ℝ)) (q :=
+      (13781258323136651 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_309_9 :
+    Uω (aρ 9) (bρ 9) (18404297160587 / 64000000000000 : ℝ) ≤
+      (-(873605752059970258149 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9007091878016507 / 80000000000000000 : ℝ) ≤
+      (-(873605752059970258149 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (9007091878016507 / 80000000000000000 : ℝ) =
+            (9007091878016507 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9007091878016507 / 5000000000000000 : ℝ) ≤
+            (235429736420029741851 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (45228563194213 / 625000000000000 : ℝ)) (q :=
+      (9007091878016507 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_309_10 :
+    Uω (aρ 10) (bρ 10) (18404297160587 / 64000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_309_11 :
+    Uω (aρ 11) (bρ 11) (18404297160587 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_309_12 :
+    Uω (aρ 12) (bρ 12) (18404297160587 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_309_13 :
+    Uω (aρ 13) (bρ 13) (18404297160587 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_309_14 :
+    Uω (aρ 14) (bρ 14) (18404297160587 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_309_15 :
+    Uω (aρ 15) (bρ 15) (18404297160587 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_309_16 :
+    Uω (aρ 16) (bρ 16) (18404297160587 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_309 :
+    Uρ (18404297160587 / 64000000000000 : ℝ) ≤
+      (-(16570832112858170651139 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (18404297160587 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(396563526515380477321 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_309_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (18404297160587 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(12776105409233890902293 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_309_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (18404297160587 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(12951169203016937992483 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_309_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (18404297160587 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(13251027509644650616501 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_309_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (18404297160587 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(13731758118370003654299 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_309_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (18404297160587 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(905059442027853315307 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_309_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (18404297160587 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) * (-(2445525352896588967 / 1562500000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_309_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (18404297160587 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(8793585286503424790819 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_309_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (18404297160587 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(873605752059970258149 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_309_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (18404297160587 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_309_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (18404297160587 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_309_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (18404297160587 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_309_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (18404297160587 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_309_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (18404297160587 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_309_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (18404297160587 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_309_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (18404297160587 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_309_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_310_1 :
+    Uω (aρ 1) (bρ 1) (36886337878427 / 128000000000000 : ℝ) ≤
+      (-(791778103538758106661 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (45075048679337883 / 160000000000000000 : ℝ) ≤
+      (-(791778103538758106661 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (45075048679337883 / 160000000000000000 : ℝ) =
+            (45075048679337883 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (45075048679337883 / 40000000000000000 : ℝ) ≤
+            (74655871836241893339 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1408566578125063 / 5000000000000000 : ℝ)) (q :=
+      (45075048679337883 / 160000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_310_2 :
+    Uω (aρ 2) (bρ 2) (36886337878427 / 128000000000000 : ℝ) ≤
+      (-(797145832894660626739 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (44689585240162523 / 160000000000000000 : ℝ) ≤
+      (-(797145832894660626739 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (44689585240162523 / 160000000000000000 : ℝ) =
+            (44689585240162523 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (44689585240162523 / 40000000000000000 : ℝ) ≤
+            (69288142480339373261 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1396359619431603 / 5000000000000000 : ℝ)) (q :=
+      (44689585240162523 / 160000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_310_3 :
+    Uω (aρ 3) (bρ 3) (36886337878427 / 128000000000000 : ℝ) ≤
+      (-(12929005161061176723777 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (43915761986259859 / 160000000000000000 : ℝ) ≤
+      (-(12929005161061176723777 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (43915761986259859 / 160000000000000000 : ℝ) =
+            (43915761986259859 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (43915761986259859 / 40000000000000000 : ℝ) ≤
+            (933938444938823276223 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2743387438475373 / 10000000000000000 : ℝ)) (q :=
+      (43915761986259859 / 160000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_310_4 :
+    Uω (aρ 4) (bρ 4) (36886337878427 / 128000000000000 : ℝ) ≤
+      (-(1322816685938930461697 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (42621427883613691 / 160000000000000000 : ℝ) ≤
+      (-(1322816685938930461697 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (42621427883613691 / 160000000000000000 : ℝ) =
+            (42621427883613691 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (42621427883613691 / 40000000000000000 : ℝ) ≤
+            (63477674661069538303 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1329944928354801 / 5000000000000000 : ℝ)) (q :=
+      (42621427883613691 / 160000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_310_5 :
+    Uω (aρ 5) (bρ 5) (36886337878427 / 128000000000000 : ℝ) ≤
+      (-(13707709140880624061209 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (40625782444058779 / 160000000000000000 : ℝ) ≤
+      (-(13707709140880624061209 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (40625782444058779 / 160000000000000000 : ℝ) =
+            (40625782444058779 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (40625782444058779 / 40000000000000000 : ℝ) ≤
+            (155234465119375938791 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1264312048050119 / 5000000000000000 : ℝ)) (q :=
+      (40625782444058779 / 160000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_310_6 :
+    Uω (aρ 6) (bρ 6) (36886337878427 / 128000000000000 : ℝ) ≤
+      (-(289097038461433344131 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (37701075509361619 / 160000000000000000 : ℝ) ≤
+      (-(289097038461433344131 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (37701075509361619 / 160000000000000000 : ℝ) =
+            (37701075509361619 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (37701075509361619 / 20000000000000000 : ℝ) ≤
+            (126791269718566655869 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2330276104403093 / 10000000000000000 : ℝ)) (q :=
+      (37701075509361619 / 160000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_310_7 :
+    Uω (aρ 7) (bρ 7) (36886337878427 / 128000000000000 : ℝ) ≤
+      (-(3124293761267432852327 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (33549666863660643 / 160000000000000000 : ℝ) ≤
+      (-(3124293761267432852327 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (33549666863660643 / 160000000000000000 : ℝ) =
+            (33549666863660643 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (33549666863660643 / 20000000000000000 : ℝ) ≤
+            (1034589320532567147673 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2035011404450471 / 10000000000000000 : ℝ)) (q :=
+      (33549666863660643 / 160000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_310_8 :
+    Uω (aρ 8) (bρ 8) (36886337878427 / 128000000000000 : ℝ) ≤
+      (-(877436259691853950059 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5533737148278807 / 32000000000000000 : ℝ) ≤
+      (-(877436259691853950059 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (5533737148278807 / 32000000000000000 : ℝ) =
+            (5533737148278807 / 4000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5533737148278807 / 4000000000000000 : ℝ) ≤
+            (162284510758146049941 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (316686083101429 / 2000000000000000 : ℝ)) (q :=
+      (5533737148278807 / 32000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_310_9 :
+    Uω (aρ 9) (bρ 9) (36886337878427 / 128000000000000 : ℝ) ≤
+      (-(1087847238022731517901 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (18164685248242899 / 160000000000000000 : ℝ) ≤
+      (-(1087847238022731517901 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (18164685248242899 / 160000000000000000 : ℝ) =
+            (18164685248242899 / 10000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18164685248242899 / 10000000000000000 : ℝ) ≤
+            (298447122577268482099 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (736395982223253 / 10000000000000000 : ℝ)) (q :=
+      (18164685248242899 / 160000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9])
+      (by norm_num [bρ_9]) (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_310_10 :
+    Uω (aρ 10) (bρ 10) (36886337878427 / 128000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_310_11 :
+    Uω (aρ 11) (bρ 11) (36886337878427 / 128000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_310_12 :
+    Uω (aρ 12) (bρ 12) (36886337878427 / 128000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_310_13 :
+    Uω (aρ 13) (bρ 13) (36886337878427 / 128000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_310_14 :
+    Uω (aρ 14) (bρ 14) (36886337878427 / 128000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_310_15 :
+    Uω (aρ 15) (bρ 15) (36886337878427 / 128000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_310_16 :
+    Uω (aρ 16) (bρ 16) (36886337878427 / 128000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_310 :
+    Uρ (36886337878427 / 128000000000000 : ℝ) ≤
+      (-(4137670170238638207857 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (36886337878427 / 128000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(791778103538758106661 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_310_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (36886337878427 / 128000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(797145832894660626739 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_310_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (36886337878427 / 128000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(12929005161061176723777 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_310_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (36886337878427 / 128000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1322816685938930461697 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_310_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (36886337878427 / 128000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(13707709140880624061209 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_310_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (36886337878427 / 128000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(289097038461433344131 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_310_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (36886337878427 / 128000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(3124293761267432852327 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_310_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (36886337878427 / 128000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(877436259691853950059 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_310_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (36886337878427 / 128000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(1087847238022731517901 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_310_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (36886337878427 / 128000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_310_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (36886337878427 / 128000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_310_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (36886337878427 / 128000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_310_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (36886337878427 / 128000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_310_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (36886337878427 / 128000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_310_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (36886337878427 / 128000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_310_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (36886337878427 / 128000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_310_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_311_1 :
+    Uω (aρ 1) (bρ 1) (231025508973 / 800000000000 : ℝ) ≤
+      (-(632345647487376720253 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5646528762654023 / 20000000000000000 : ℝ) ≤
+      (-(632345647487376720253 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5646528762654023 / 20000000000000000 : ℝ) =
+            (5646528762654023 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5646528762654023 / 5000000000000000 : ℝ) ≤
+            (60801532812623279747 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2823207118576523 / 10000000000000000 : ℝ)) (q :=
+      (5646528762654023 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_311_2 :
+    Uω (aρ 2) (bρ 2) (231025508973 / 800000000000 : ℝ) ≤
+      (-(6366304277815616426409 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2799173617215593 / 10000000000000000 : ℝ) ≤
+      (-(6366304277815616426409 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2799173617215593 / 10000000000000000 : ℝ) =
+            (2799173617215593 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2799173617215593 / 2500000000000000 : ℝ) ≤
+            (565167525184383573591 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1399397301431843 / 5000000000000000 : ℝ)) (q :=
+      (2799173617215593 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_311_3 :
+    Uω (aρ 3) (bρ 3) (231025508973 / 800000000000 : ℝ) ≤
+      (-(12906890183013655558881 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2750811816662373 / 10000000000000000 : ℝ) ≤
+      (-(12906890183013655558881 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2750811816662373 / 10000000000000000 : ℝ) =
+            (2750811816662373 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2750811816662373 / 2500000000000000 : ℝ) ≤
+            (956053422986344441119 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1374733554053623 / 5000000000000000 : ℝ)) (q :=
+      (2750811816662373 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_311_4 :
+    Uω (aρ 4) (bρ 4) (231025508973 / 800000000000 : ℝ) ≤
+      (-(3301339626344053614233 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2669921955770723 / 10000000000000000 : ℝ) ≤
+      (-(3301339626344053614233 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2669921955770723 / 10000000000000000 : ℝ) =
+            (2669921955770723 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2669921955770723 / 2500000000000000 : ℝ) ≤
+            (164396275155946385767 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1332990783694473 / 5000000000000000 : ℝ)) (q :=
+      (2669921955770723 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_311_5 :
+    Uω (aρ 5) (bρ 5) (231025508973 / 800000000000 : ℝ) ≤
+      (-(3420929584356858442139 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (159075640493697 / 625000000000000 : ℝ) ≤
+      (-(3420929584356858442139 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (159075640493697 / 625000000000000 : ℝ) =
+            (159075640493697 / 156250000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (159075640493697 / 156250000000000 : ℝ) ≤
+            (44806317143141557861 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (633687017745201 / 2500000000000000 : ℝ)) (q :=
+      (159075640493697 / 625000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_311_6 :
+    Uω (aρ 6) (bρ 6) (231025508973 / 800000000000 : ℝ) ≤
+      (-(14428822227409249675339 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (944983452332593 / 4000000000000000 : ℝ) ≤
+      (-(14428822227409249675339 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (944983452332593 / 4000000000000000 : ℝ) =
+            (944983452332593 / 500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (944983452332593 / 500000000000000 : ℝ) ≤
+            (6365593181590750324661 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (467297042397093 / 2000000000000000 : ℝ)) (q :=
+      (944983452332593 / 4000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_311_7 :
+    Uω (aρ 7) (bρ 7) (231025508973 / 800000000000 : ℝ) ≤
+      (-(15591669847770762946373 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (841244762045679 / 4000000000000000 : ℝ) ≤
+      (-(15591669847770762946373 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (841244762045679 / 4000000000000000 : ℝ) =
+            (841244762045679 / 500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (841244762045679 / 500000000000000 : ℝ) ≤
+            (5202745561229237053627 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (408290628262179 / 2000000000000000 : ℝ)) (q :=
+      (841244762045679 / 4000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_311_8 :
+    Uω (aρ 8) (bρ 8) (231025508973 / 800000000000 : ℝ) ≤
+      (-(8755227028881924606817 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3471847479083373 / 20000000000000000 : ℝ) ≤
+      (-(8755227028881924606817 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3471847479083373 / 20000000000000000 : ℝ) =
+            (3471847479083373 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3471847479083373 / 2500000000000000 : ℝ) ≤
+            (1641980675618075393183 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1590618461505873 / 10000000000000000 : ℝ)) (q :=
+      (3471847479083373 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_311_9 :
+    Uω (aρ 9) (bρ 9) (231025508973 / 800000000000 : ℝ) ≤
+      (-(1083758172203001746681 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2289230942307729 / 20000000000000000 : ℝ) ≤
+      (-(1083758172203001746681 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2289230942307729 / 20000000000000000 : ℝ) =
+            (2289230942307729 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2289230942307729 / 1250000000000000 : ℝ) ≤
+            (302536188396998253319 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (748967553090229 / 10000000000000000 : ℝ)) (q :=
+      (2289230942307729 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_311_10 :
+    Uω (aρ 10) (bρ 10) (231025508973 / 800000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_311_11 :
+    Uω (aρ 11) (bρ 11) (231025508973 / 800000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_311_12 :
+    Uω (aρ 12) (bρ 12) (231025508973 / 800000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_311_13 :
+    Uω (aρ 13) (bρ 13) (231025508973 / 800000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_311_14 :
+    Uω (aρ 14) (bρ 14) (231025508973 / 800000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_311_15 :
+    Uω (aρ 15) (bρ 15) (231025508973 / 800000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_311_16 :
+    Uω (aρ 16) (bρ 16) (231025508973 / 800000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_311 :
+    Uρ (231025508973 / 800000000000 : ℝ) ≤
+      (-(16530694387511013761449 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (231025508973 / 800000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(632345647487376720253 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_311_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (231025508973 / 800000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(6366304277815616426409 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_311_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (231025508973 / 800000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(12906890183013655558881 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_311_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (231025508973 / 800000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3301339626344053614233 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_311_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (231025508973 / 800000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(3420929584356858442139 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_311_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (231025508973 / 800000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(14428822227409249675339 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_311_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (231025508973 / 800000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(15591669847770762946373 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_311_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (231025508973 / 800000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(8755227028881924606817 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_311_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (231025508973 / 800000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(1083758172203001746681 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_311_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (231025508973 / 800000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_311_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (231025508973 / 800000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_311_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (231025508973 / 800000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_311_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (231025508973 / 800000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_311_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (231025508973 / 800000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_311_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (231025508973 / 800000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_311_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (231025508973 / 800000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_311_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_312_1 :
+    Uω (aρ 1) (bρ 1) (37041824992933 / 128000000000000 : ℝ) ≤
+      (-(12625422528061556459799 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (45269411514645309 / 160000000000000000 : ℝ) ≤
+      (-(12625422528061556459799 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (45269411514645309 / 160000000000000000 : ℝ) =
+            (45269411514645309 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (45269411514645309 / 40000000000000000 : ℝ) ≤
+            (1237521077938443540201 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2829281079842773 / 10000000000000000 : ℝ)) (q :=
+      (45269411514645309 / 160000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_312_2 :
+    Uω (aρ 2) (bρ 2) (37041824992933 / 128000000000000 : ℝ) ≤
+      (-(12710930891948740915369 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1795358818144421 / 6400000000000000 : ℝ) ≤
+      (-(12710930891948740915369 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1795358818144421 / 6400000000000000 : ℝ) =
+            (1795358818144421 / 1600000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1795358818144421 / 1600000000000000 : ℝ) ≤
+            (1152012714051259084631 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (112194798388937 / 400000000000000 : ℝ)) (q :=
+      (1795358818144421 / 6400000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_312_3 :
+    Uω (aρ 3) (bρ 3) (37041824992933 / 128000000000000 : ℝ) ≤
+      (-(12884824051920100839147 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (44110215936846533 / 160000000000000000 : ℝ) ≤
+      (-(12884824051920100839147 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (44110215936846533 / 160000000000000000 : ℝ) =
+            (44110215936846533 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (44110215936846533 / 40000000000000000 : ℝ) ≤
+            (978119554079899160853 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1377773375738963 / 5000000000000000 : ℝ)) (q :=
+      (44110215936846533 / 160000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_312_4 :
+    Uω (aρ 4) (bρ 4) (37041824992933 / 128000000000000 : ℝ) ≤
+      (-(1318260220818366757497 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (42816074045617973 / 160000000000000000 : ℝ) ≤
+      (-(1318260220818366757497 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (42816074045617973 / 160000000000000000 : ℝ) =
+            (42816074045617973 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (42816074045617973 / 40000000000000000 : ℝ) ≤
+            (68034139781633242503 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (668018299034839 / 2500000000000000 : ℝ)) (q :=
+      (42816074045617973 / 160000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_312_5 :
+    Uω (aρ 5) (bρ 5) (37041824992933 / 128000000000000 : ℝ) ≤
+      (-(13659785424961961456927 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (40820943558512269 / 160000000000000000 : ℝ) ≤
+      (-(13659785424961961456927 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (40820943558512269 / 160000000000000000 : ℝ) =
+            (40820943558512269 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (40820943558512269 / 40000000000000000 : ℝ) ≤
+            (203158181038038543073 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2540871804586143 / 10000000000000000 : ℝ)) (q :=
+      (40820943558512269 / 160000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_312_6 :
+    Uω (aρ 6) (bρ 6) (37041824992933 / 128000000000000 : ℝ) ≤
+      (-(14402861608877604147629 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7579518999664049 / 32000000000000000 : ℝ) ≤
+      (-(14402861608877604147629 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (7579518999664049 / 32000000000000000 : ℝ) =
+            (7579518999664049 / 4000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7579518999664049 / 4000000000000000 : ℝ) ≤
+            (6391553800122395852371 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (117134680485107 / 500000000000000 : ℝ)) (q :=
+      (7579518999664049 / 32000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_312_7 :
+    Uω (aρ 7) (bρ 7) (37041824992933 / 128000000000000 : ℝ) ≤
+      (-(7780982377249255875523 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (33749896107048941 / 160000000000000000 : ℝ) ≤
+      (-(7780982377249255875523 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (33749896107048941 / 160000000000000000 : ℝ) =
+            (33749896107048941 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (33749896107048941 / 20000000000000000 : ℝ) ≤
+            (2616225327250744124477 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2047892629053227 / 10000000000000000 : ℝ)) (q :=
+      (33749896107048941 / 160000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_312_8 :
+    Uω (aρ 8) (bρ 8) (37041824992933 / 128000000000000 : ℝ) ≤
+      (-(17472355369949165105051 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27880799932356629 / 160000000000000000 : ℝ) ≤
+      (-(17472355369949165105051 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (27880799932356629 / 160000000000000000 : ℝ) =
+            (27880799932356629 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27880799932356629 / 20000000000000000 : ℝ) ≤
+            (3322060039050834894949 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (99862328659793 / 625000000000000 : ℝ)) (q :=
+      (27880799932356629 / 160000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8])
+      (by norm_num [bρ_8]) (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_312_9 :
+    Uω (aρ 9) (bρ 9) (37041824992933 / 128000000000000 : ℝ) ≤
+      (-(10797367480776861680277 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (29538779142381 / 256000000000000 : ℝ) ≤
+      (-(10797367480776861680277 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (29538779142381 / 256000000000000 : ℝ) = (29538779142381 / 16000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (29538779142381 / 16000000000000 : ℝ) ≤
+            (3065576125223138319723 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1218208025393 / 16000000000000 : ℝ)) (q :=
+      (29538779142381 / 256000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_312_10 :
+    Uω (aρ 10) (bρ 10) (37041824992933 / 128000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_312_11 :
+    Uω (aρ 11) (bρ 11) (37041824992933 / 128000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_312_12 :
+    Uω (aρ 12) (bρ 12) (37041824992933 / 128000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_312_13 :
+    Uω (aρ 13) (bρ 13) (37041824992933 / 128000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_312_14 :
+    Uω (aρ 14) (bρ 14) (37041824992933 / 128000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_312_15 :
+    Uω (aρ 15) (bρ 15) (37041824992933 / 128000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_312_16 :
+    Uω (aρ 16) (bρ 16) (37041824992933 / 128000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_312 :
+    Uρ (37041824992933 / 128000000000000 : ℝ) ≤
+      (-(16510867209835616507937 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (37041824992933 / 128000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(12625422528061556459799 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_312_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (37041824992933 / 128000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(12710930891948740915369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_312_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (37041824992933 / 128000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(12884824051920100839147 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_312_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (37041824992933 / 128000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1318260220818366757497 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_312_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (37041824992933 / 128000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(13659785424961961456927 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_312_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (37041824992933 / 128000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(14402861608877604147629 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_312_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (37041824992933 / 128000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(7780982377249255875523 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_312_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (37041824992933 / 128000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(17472355369949165105051 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_312_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (37041824992933 / 128000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(10797367480776861680277 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_312_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (37041824992933 / 128000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_312_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (37041824992933 / 128000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_312_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (37041824992933 / 128000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_312_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (37041824992933 / 128000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_312_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (37041824992933 / 128000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_312_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (37041824992933 / 128000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_312_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (37041824992933 / 128000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_312_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_313_1 :
+    Uω (aρ 1) (bρ 1) (18559784275093 / 64000000000000 : ℝ) ≤
+      (-(1575497274129372754583 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22683296459815877 / 80000000000000000 : ℝ) ≤
+      (-(1575497274129372754583 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (22683296459815877 / 80000000000000000 : ℝ) =
+            (22683296459815877 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22683296459815877 / 20000000000000000 : ℝ) ≤
+            (157370676620627245417 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (354419380006961 / 1250000000000000 : ℝ)) (q :=
+      (22683296459815877 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_313_2 :
+    Uω (aρ 2) (bρ 2) (18559784275093 / 64000000000000 : ℝ) ≤
+      (-(12689300131364860125261 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22490581487508013 / 80000000000000000 : ℝ) ≤
+      (-(12689300131364860125261 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (22490581487508013 / 80000000000000000 : ℝ) =
+            (22490581487508013 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22490581487508013 / 20000000000000000 : ℝ) ≤
+            (1173643474635139874739 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1405472654744361 / 5000000000000000 : ℝ)) (q :=
+      (22490581487508013 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_313_3 :
+    Uω (aρ 3) (bρ 3) (18559784275093 / 64000000000000 : ℝ) ≤
+      (-(6431403276131736116647 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22103721299196533 / 80000000000000000 : ℝ) ≤
+      (-(6431403276131736116647 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (22103721299196533 / 80000000000000000 : ℝ) =
+            (22103721299196533 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22103721299196533 / 20000000000000000 : ℝ) ≤
+            (500068526868263883353 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (690406592190213 / 2500000000000000 : ℝ)) (q :=
+      (22103721299196533 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_313_4 :
+    Uω (aρ 4) (bρ 4) (18559784275093 / 64000000000000 : ℝ) ≤
+      (-(1644987216254391239947 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21456698073972669 / 80000000000000000 : ℝ) ≤
+      (-(1644987216254391239947 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (21456698073972669 / 80000000000000000 : ℝ) =
+            (21456698073972669 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21456698073972669 / 20000000000000000 : ℝ) ≤
+            (87880734495608760053 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1339082371759943 / 5000000000000000 : ℝ)) (q :=
+      (21456698073972669 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_313_5 :
+    Uω (aρ 5) (bρ 5) (18559784275093 / 64000000000000 : ℝ) ≤
+      (-(13635910122512783687653 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20459260617179309 / 80000000000000000 : ℝ) ≤
+      (-(13635910122512783687653 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (20459260617179309 / 80000000000000000 : ℝ) =
+            (20459260617179309 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20459260617179309 / 20000000000000000 : ℝ) ≤
+            (227033483487216312347 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1273497649328273 / 5000000000000000 : ℝ)) (q :=
+      (20459260617179309 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_313_6 :
+    Uω (aρ 6) (bρ 6) (18559784275093 / 64000000000000 : ℝ) ≤
+      (-(898560605875719717287 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (18997923134720613 / 80000000000000000 : ℝ) ≤
+      (-(898560605875719717287 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (18997923134720613 / 80000000000000000 : ℝ) =
+            (18997923134720613 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18997923134720613 / 10000000000000000 : ℝ) ≤
+            (401090357186780282713 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (146806331448867 / 625000000000000 : ℝ)) (q :=
+      (18997923134720613 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_313_7 :
+    Uω (aρ 7) (bρ 7) (18559784275093 / 64000000000000 : ℝ) ≤
+      (-(155323529047166465161 / 100000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16924991954263829 / 80000000000000000 : ℝ) ≤
+      (-(155323529047166465161 / 100000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (16924991954263829 / 80000000000000000 : ℝ) =
+            (16924991954263829 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16924991954263829 / 10000000000000000 : ℝ) ≤
+            (52620625042833534839 / 100000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (513582472206919 / 2500000000000000 : ℝ)) (q :=
+      (16924991954263829 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_313_8 :
+    Uω (aρ 8) (bρ 8) (18559784275093 / 64000000000000 : ℝ) ≤
+      (-(17434427365685407335693 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13993373516662809 / 80000000000000000 : ℝ) ≤
+      (-(17434427365685407335693 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (13993373516662809 / 80000000000000000 : ℝ) =
+            (13993373516662809 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13993373516662809 / 10000000000000000 : ℝ) ≤
+            (3359988043314592664307 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1604966930767421 / 10000000000000000 : ℝ)) (q :=
+      (13993373516662809 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_313_9 :
+    Uω (aρ 9) (bρ 9) (18559784275093 / 64000000000000 : ℝ) ≤
+      (-(10757799638185708569451 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9304207395734157 / 80000000000000000 : ℝ) ≤
+      (-(10757799638185708569451 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (9304207395734157 / 80000000000000000 : ℝ) =
+            (9304207395734157 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9304207395734157 / 5000000000000000 : ℝ) ≤
+            (3105143967814291430549 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (386820514447629 / 5000000000000000 : ℝ)) (q :=
+      (9304207395734157 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_313_10 :
+    Uω (aρ 10) (bρ 10) (18559784275093 / 64000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_313_11 :
+    Uω (aρ 11) (bρ 11) (18559784275093 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_313_12 :
+    Uω (aρ 12) (bρ 12) (18559784275093 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_313_13 :
+    Uω (aρ 13) (bρ 13) (18559784275093 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_313_14 :
+    Uω (aρ 14) (bρ 14) (18559784275093 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_313_15 :
+    Uω (aρ 15) (bρ 15) (18559784275093 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_313_16 :
+    Uω (aρ 16) (bρ 16) (18559784275093 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_313 :
+    Uρ (18559784275093 / 64000000000000 : ℝ) ≤
+      (-(659647742307838339129 / 400000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (18559784275093 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1575497274129372754583 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_313_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (18559784275093 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(12689300131364860125261 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_313_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (18559784275093 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(6431403276131736116647 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_313_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (18559784275093 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1644987216254391239947 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_313_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (18559784275093 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(13635910122512783687653 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_313_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (18559784275093 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(898560605875719717287 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_313_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (18559784275093 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(155323529047166465161 / 100000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_313_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (18559784275093 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(17434427365685407335693 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_313_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (18559784275093 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(10757799638185708569451 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_313_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (18559784275093 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_313_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (18559784275093 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_313_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (18559784275093 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_313_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (18559784275093 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_313_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (18559784275093 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_313_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (18559784275093 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_313_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (18559784275093 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_313_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_314_1 :
+    Uω (aρ 1) (bρ 1) (37197312107439 / 128000000000000 : ℝ) ≤
+      (-(2516515949483040556447 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (45463774316245559 / 160000000000000000 : ℝ) ≤
+      (-(2516515949483040556447 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (45463774316245559 / 160000000000000000 : ℝ) =
+            (45463774316245559 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (45463774316245559 / 40000000000000000 : ℝ) ≤
+            (256072771716959443553 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2841428999222023 / 10000000000000000 : ℝ)) (q :=
+      (45463774316245559 / 160000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_314_2 :
+    Uω (aρ 2) (bρ 2) (37197312107439 / 128000000000000 : ℝ) ≤
+      (-(6333858035649365264693 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (45078355440033207 / 160000000000000000 : ℝ) ≤
+      (-(6333858035649365264693 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (45078355440033207 / 160000000000000000 : ℝ) =
+            (45078355440033207 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (45078355440033207 / 40000000000000000 : ℝ) ≤
+            (597613767350634735307 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2817020652205479 / 10000000000000000 : ℝ)) (q :=
+      (45078355440033207 / 160000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_314_3 :
+    Uω (aρ 3) (bρ 3) (37197312107439 / 128000000000000 : ℝ) ≤
+      (-(2568167493990255296731 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (44304669052612903 / 160000000000000000 : ℝ) ≤
+      (-(2568167493990255296731 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (44304669052612903 / 160000000000000000 : ℝ) =
+            (44304669052612903 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (44304669052612903 / 40000000000000000 : ℝ) ≤
+            (204421227209744703269 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2767705960127941 / 10000000000000000 : ℝ)) (q :=
+      (44304669052612903 / 160000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_314_4 :
+    Uω (aρ 4) (bρ 4) (37197312107439 / 128000000000000 : ℝ) ≤
+      (-(2627448966956830868993 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8602143520749091 / 32000000000000000 : ℝ) ≤
+      (-(2627448966956830868993 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8602143520749091 / 32000000000000000 : ℝ) =
+            (8602143520749091 / 8000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8602143520749091 / 8000000000000000 : ℝ) ≤
+            (145139754243169131007 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (268425621008451 / 1000000000000000 : ℝ)) (q :=
+      (8602143520749091 / 32000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_314_5 :
+    Uω (aρ 5) (bρ 5) (37197312107439 / 128000000000000 : ℝ) ≤
+      (-(13612092151165089101821 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (41016097007713719 / 160000000000000000 : ℝ) ≤
+      (-(13612092151165089101821 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (41016097007713719 / 160000000000000000 : ℝ) =
+            (41016097007713719 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (41016097007713719 / 40000000000000000 : ℝ) ≤
+            (250851454834910898179 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2553118554915543 / 10000000000000000 : ℝ)) (q :=
+      (41016097007713719 / 160000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_314_6 :
+    Uω (aρ 6) (bρ 6) (37197312107439 / 128000000000000 : ℝ) ≤
+      (-(14351146112426458858217 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (38094091951216839 / 160000000000000000 : ℝ) ≤
+      (-(14351146112426458858217 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (38094091951216839 / 160000000000000000 : ℝ) =
+            (38094091951216839 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (38094091951216839 / 20000000000000000 : ℝ) ≤
+            (6443269296573541141783 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2355108297993433 / 10000000000000000 : ℝ)) (q :=
+      (38094091951216839 / 160000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_314_7 :
+    Uω (aρ 7) (bρ 7) (37197312107439 / 128000000000000 : ℝ) ≤
+      (-(15502833683064404763959 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (33950054053292631 / 160000000000000000 : ℝ) ≤
+      (-(15502833683064404763959 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (33950054053292631 / 160000000000000000 : ℝ) =
+            (33950054053292631 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (33950054053292631 / 20000000000000000 : ℝ) ≤
+            (5291581725935595236041 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2060764941512907 / 10000000000000000 : ℝ)) (q :=
+      (33950054053292631 / 160000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_314_8 :
+    Uω (aρ 8) (bρ 8) (37197312107439 / 128000000000000 : ℝ) ≤
+      (-(695866732387491057881 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28092622109523383 / 160000000000000000 : ℝ) ≤
+      (-(695866732387491057881 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (28092622109523383 / 160000000000000000 : ℝ) =
+            (28092622109523383 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28092622109523383 / 20000000000000000 : ℝ) ≤
+            (135909883972508942119 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1612127599881751 / 10000000000000000 : ℝ)) (q :=
+      (28092622109523383 / 160000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8])
+      (by norm_num [bρ_8]) (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_314_9 :
+    Uω (aρ 9) (bρ 9) (37197312107439 / 128000000000000 : ℝ) ≤
+      (-(21437700710955841835511 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (18753937738718031 / 160000000000000000 : ℝ) ≤
+      (-(21437700710955841835511 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (18753937738718031 / 160000000000000000 : ℝ) =
+            (18753937738718031 / 10000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18753937738718031 / 10000000000000000 : ℝ) ≤
+            (6288186501044158164489 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (392878840945541 / 5000000000000000 : ℝ)) (q :=
+      (18753937738718031 / 160000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9])
+      (by norm_num [bρ_9]) (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_314_10 :
+    Uω (aρ 10) (bρ 10) (37197312107439 / 128000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_314_11 :
+    Uω (aρ 11) (bρ 11) (37197312107439 / 128000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_314_12 :
+    Uω (aρ 12) (bρ 12) (37197312107439 / 128000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_314_13 :
+    Uω (aρ 13) (bρ 13) (37197312107439 / 128000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_314_14 :
+    Uω (aρ 14) (bρ 14) (37197312107439 / 128000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_314_15 :
+    Uω (aρ 15) (bρ 15) (37197312107439 / 128000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_314_16 :
+    Uω (aρ 16) (bρ 16) (37197312107439 / 128000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_314 :
+    Uρ (37197312107439 / 128000000000000 : ℝ) ≤
+      (-(514739632172362430889 / 312500000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (37197312107439 / 128000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2516515949483040556447 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_314_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (37197312107439 / 128000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(6333858035649365264693 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_314_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (37197312107439 / 128000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(2568167493990255296731 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_314_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (37197312107439 / 128000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(2627448966956830868993 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_314_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (37197312107439 / 128000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(13612092151165089101821 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_314_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (37197312107439 / 128000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(14351146112426458858217 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_314_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (37197312107439 / 128000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(15502833683064404763959 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_314_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (37197312107439 / 128000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(695866732387491057881 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_314_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (37197312107439 / 128000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(21437700710955841835511 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_314_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (37197312107439 / 128000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_314_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (37197312107439 / 128000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_314_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (37197312107439 / 128000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_314_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (37197312107439 / 128000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_314_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (37197312107439 / 128000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_314_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (37197312107439 / 128000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_314_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (37197312107439 / 128000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_314_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_315_1 :
+    Uω (aρ 1) (bρ 1) (9318763916173 / 32000000000000 : ℝ) ≤
+      (-(1256122699521332861959 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11390238926135077 / 40000000000000000 : ℝ) ≤
+      (-(1256122699521332861959 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (11390238926135077 / 40000000000000000 : ℝ) =
+            (11390238926135077 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11390238926135077 / 10000000000000000 : ℝ) ≤
+            (130171661078667138041 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (711875739337119 / 2500000000000000 : ℝ)) (q :=
+      (11390238926135077 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_315_2 :
+    Uω (aρ 2) (bρ 2) (9318763916173 / 32000000000000 : ℝ) ≤
+      (-(790386156904967377561 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11293886962256527 / 40000000000000000 : ℝ) ≤
+      (-(790386156904967377561 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (11293886962256527 / 40000000000000000 : ℝ) =
+            (11293886962256527 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11293886962256527 / 10000000000000000 : ℝ) ≤
+            (76047818470032622439 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2823095987919201 / 10000000000000000 : ℝ)) (q :=
+      (11293886962256527 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_315_3 :
+    Uω (aρ 3) (bρ 3) (9318763916173 / 32000000000000 : ℝ) ≤
+      (-(6409458296151520141253 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11100473825217321 / 40000000000000000 : ℝ) ≤
+      (-(6409458296151520141253 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (11100473825217321 / 40000000000000000 : ℝ) =
+            (11100473825217321 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11100473825217321 / 10000000000000000 : ℝ) ≤
+            (522013506848479858747 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1386892762874799 / 5000000000000000 : ℝ)) (q :=
+      (11100473825217321 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_315_4 :
+    Uω (aρ 4) (bρ 4) (9318763916173 / 32000000000000 : ℝ) ≤
+      (-(1311464328789948097699 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (431080384174099 / 1600000000000000 : ℝ) ≤
+      (-(1311464328789948097699 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (431080384174099 / 1600000000000000 : ℝ) =
+            (431080384174099 / 400000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (431080384174099 / 400000000000000 : ℝ) ≤
+            (74830031810051902301 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (107613903855287 / 400000000000000 : ℝ)) (q :=
+      (431080384174099 / 1600000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_315_5 :
+    Uω (aρ 5) (bρ 5) (9318763916173 / 32000000000000 : ℝ) ≤
+      (-(13588331234040514537341 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (411136708922333 / 1600000000000000 : ℝ) ≤
+      (-(13588331234040514537341 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (411136708922333 / 1600000000000000 : ℝ) =
+            (411136708922333 / 400000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (411136708922333 / 400000000000000 : ℝ) ≤
+            (274612371959485462659 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (25592415750701 / 100000000000000 : ℝ)) (q :=
+      (411136708922333 / 1600000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_315_6 :
+    Uω (aρ 6) (bρ 6) (9318763916173 / 32000000000000 : ℝ) ≤
+      (-(7162695248392099559029 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9548083021930961 / 40000000000000000 : ℝ) ≤
+      (-(7162695248392099559029 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (9548083021930961 / 40000000000000000 : ℝ) =
+            (9548083021930961 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9548083021930961 / 5000000000000000 : ℝ) ≤
+            (3234512456107900440971 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1180657299823209 / 5000000000000000 : ℝ)) (q :=
+      (9548083021930961 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_315_7 :
+    Uω (aρ 7) (bρ 7) (9318763916173 / 32000000000000 : ℝ) ≤
+      (-(15473406480532042506811 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8512526676559121 / 40000000000000000 : ℝ) ≤
+      (-(15473406480532042506811 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (8512526676559121 / 40000000000000000 : ℝ) =
+            (8512526676559121 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8512526676559121 / 5000000000000000 : ℝ) ≤
+            (5321008928467957493189 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1033598903860249 / 5000000000000000 : ℝ)) (q :=
+      (8512526676559121 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_315_8 :
+    Uω (aρ 8) (bρ 8) (9318763916173 / 32000000000000 : ℝ) ≤
+      (-(8679538247702567513831 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7049606529114841 / 40000000000000000 : ℝ) ≤
+      (-(8679538247702567513831 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (7049606529114841 / 40000000000000000 : ℝ) =
+            (7049606529114841 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7049606529114841 / 5000000000000000 : ℝ) ≤
+            (1717669456797432486169 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (809639692669179 / 5000000000000000 : ℝ)) (q :=
+      (7049606529114841 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_315_9 :
+    Uω (aρ 9) (bρ 9) (9318763916173 / 32000000000000 : ℝ) ≤
+      (-(21360987514744639833261 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4724589607413553 / 40000000000000000 : ℝ) ≤
+      (-(21360987514744639833261 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (4724589607413553 / 40000000000000000 : ℝ) =
+            (4724589607413553 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4724589607413553 / 2500000000000000 : ℝ) ≤
+            (6364899697255360166739 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (398868276423857 / 5000000000000000 : ℝ)) (q :=
+      (4724589607413553 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_315_10 :
+    Uω (aρ 10) (bρ 10) (9318763916173 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_315_11 :
+    Uω (aρ 11) (bρ 11) (9318763916173 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_315_12 :
+    Uω (aρ 12) (bρ 12) (9318763916173 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_315_13 :
+    Uω (aρ 13) (bρ 13) (9318763916173 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_315_14 :
+    Uω (aρ 14) (bρ 14) (9318763916173 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_315_15 :
+    Uω (aρ 15) (bρ 15) (9318763916173 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_315_16 :
+    Uω (aρ 16) (bρ 16) (9318763916173 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_315 :
+    Uρ (9318763916173 / 32000000000000 : ℝ) ≤
+      (-(4113071593537865275113 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (9318763916173 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1256122699521332861959 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_315_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (9318763916173 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(790386156904967377561 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_315_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (9318763916173 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(6409458296151520141253 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_315_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (9318763916173 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1311464328789948097699 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_315_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (9318763916173 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(13588331234040514537341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_315_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (9318763916173 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(7162695248392099559029 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_315_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (9318763916173 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(15473406480532042506811 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_315_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (9318763916173 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(8679538247702567513831 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_315_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (9318763916173 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(21360987514744639833261 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_315_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (9318763916173 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_315_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (9318763916173 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_315_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (9318763916173 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_315_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (9318763916173 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_315_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (9318763916173 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_315_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (9318763916173 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_315_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (9318763916173 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_315_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/U26.lean b/LeanPool/Zeta5Irrational/Table/U26.lean
new file mode 100644
index 0000000000..c7d7b311dc
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/U26.lean
@@ -0,0 +1,3927 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-! # Certified arcsine potential bounds (U26) -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma U_316_1 :
+    Uω (aρ 1) (bρ 1) (7470559844389 / 25600000000000 : ℝ) ≤
+      (-(313497993542335413487 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (45658137084569129 / 160000000000000000 : ℝ) ≤
+      (-(313497993542335413487 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (45658137084569129 / 160000000000000000 : ℝ) =
+            (45658137084569129 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (45658137084569129 / 40000000000000000 : ℝ) ≤
+            (33075596607664586513 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (356697114305211 / 1250000000000000 : ℝ)) (q :=
+      (45658137084569129 / 160000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_316_2 :
+    Uω (aρ 2) (bρ 2) (7470559844389 / 25600000000000 : ℝ) ≤
+      (-(12624687248934936222979 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (45272740202355649 / 160000000000000000 : ℝ) ≤
+      (-(12624687248934936222979 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (45272740202355649 / 160000000000000000 : ℝ) =
+            (45272740202355649 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (45272740202355649 / 40000000000000000 : ℝ) ≤
+            (1238256357065063777021 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2829171316675003 / 10000000000000000 : ℝ)) (q :=
+      (45272740202355649 / 160000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_316_3 :
+    Uω (aρ 3) (bρ 3) (7470559844389 / 25600000000000 : ℝ) ≤
+      (-(12797043708037897056853 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (44499121344513537 / 160000000000000000 : ℝ) ≤
+      (-(12797043708037897056853 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (44499121344513537 / 160000000000000000 : ℝ) =
+            (44499121344513537 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (44499121344513537 / 40000000000000000 : ℝ) ≤
+            (1065899897962102943147 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2779865065794739 / 10000000000000000 : ℝ)) (q :=
+      (44499121344513537 / 160000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_316_4 :
+    Uω (aρ 4) (bρ 4) (7470559844389 / 25600000000000 : ℝ) ≤
+      (-(13092092856450291711527 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (43205358593290521 / 160000000000000000 : ℝ) ≤
+      (-(13092092856450291711527 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (43205358593290521 / 160000000000000000 : ℝ) =
+            (43205358593290521 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (43205358593290521 / 40000000000000000 : ℝ) ≤
+            (770850749549708288473 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1348219451478431 / 5000000000000000 : ℝ)) (q :=
+      (43205358593290521 / 160000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_316_5 :
+    Uω (aρ 5) (bρ 5) (7470559844389 / 25600000000000 : ℝ) ≤
+      (-(13564627096277214139309 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (41211242901442193 / 160000000000000000 : ℝ) ≤
+      (-(13564627096277214139309 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (41211242901442193 / 160000000000000000 : ℝ) =
+            (41211242901442193 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (41211242901442193 / 40000000000000000 : ℝ) ≤
+            (298316509722785860691 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2565364360810821 / 10000000000000000 : ℝ)) (q :=
+      (41211242901442193 / 160000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_316_6 :
+    Uω (aρ 6) (bρ 6) (7470559844389 / 25600000000000 : ℝ) ≤
+      (-(14299702482758872198407 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (38290566722571937 / 160000000000000000 : ℝ) ≤
+      (-(14299702482758872198407 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (38290566722571937 / 160000000000000000 : ℝ) =
+            (38290566722571937 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (38290566722571937 / 20000000000000000 : ℝ) ≤
+            (6494712926241127801593 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2367520213592039 / 10000000000000000 : ℝ)) (q :=
+      (38290566722571937 / 160000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_316_7 :
+    Uω (aρ 7) (bρ 7) (7470559844389 / 25600000000000 : ℝ) ≤
+      (-(7722035347185278864809 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (34150142030150001 / 160000000000000000 : ℝ) ≤
+      (-(7722035347185278864809 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (34150142030150001 / 160000000000000000 : ℝ) =
+            (34150142030150001 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (34150142030150001 / 20000000000000000 : ℝ) ≤
+            (2675172357314721135191 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2073628507799297 / 10000000000000000 : ℝ)) (q :=
+      (34150142030150001 / 160000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_316_8 :
+    Uω (aρ 8) (bρ 8) (7470559844389 / 25600000000000 : ℝ) ≤
+      (-(17321650244358947368713 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28304159991661457 / 160000000000000000 : ℝ) ≤
+      (-(17321650244358947368713 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (28304159991661457 / 160000000000000000 : ℝ) =
+            (28304159991661457 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28304159991661457 / 20000000000000000 : ℝ) ≤
+            (3472765164641052631287 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1626422404328229 / 10000000000000000 : ℝ)) (q :=
+      (28304159991661457 / 160000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8])
+      (by norm_num [bρ_8]) (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_316_9 :
+    Uω (aρ 9) (bρ 9) (7470559844389 / 25600000000000 : ℝ) ≤
+      (-(2128541148575158751237 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19041725792516417 / 160000000000000000 : ℝ) ≤
+      (-(2128541148575158751237 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (19041725792516417 / 160000000000000000 : ℝ) =
+            (19041725792516417 / 10000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19041725792516417 / 10000000000000000 : ℝ) ≤
+            (644047572624841248763 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (809583757795099 / 10000000000000000 : ℝ)) (q :=
+      (19041725792516417 / 160000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9])
+      (by norm_num [bρ_9]) (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_316_10 :
+    Uω (aρ 10) (bρ 10) (7470559844389 / 25600000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_316_11 :
+    Uω (aρ 11) (bρ 11) (7470559844389 / 25600000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_316_12 :
+    Uω (aρ 12) (bρ 12) (7470559844389 / 25600000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_316_13 :
+    Uω (aρ 13) (bρ 13) (7470559844389 / 25600000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_316_14 :
+    Uω (aρ 14) (bρ 14) (7470559844389 / 25600000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_316_15 :
+    Uω (aρ 15) (bρ 15) (7470559844389 / 25600000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_316_16 :
+    Uω (aρ 16) (bρ 16) (7470559844389 / 25600000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_316 :
+    Uρ (7470559844389 / 25600000000000 : ℝ) ≤
+      (-(16433043457405759501447 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (7470559844389 / 25600000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(313497993542335413487 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_316_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (7470559844389 / 25600000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(12624687248934936222979 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_316_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (7470559844389 / 25600000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(12797043708037897056853 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_316_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (7470559844389 / 25600000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(13092092856450291711527 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_316_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (7470559844389 / 25600000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(13564627096277214139309 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_316_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (7470559844389 / 25600000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(14299702482758872198407 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_316_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (7470559844389 / 25600000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(7722035347185278864809 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_316_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (7470559844389 / 25600000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(17321650244358947368713 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_316_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (7470559844389 / 25600000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(2128541148575158751237 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_316_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (7470559844389 / 25600000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_316_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (7470559844389 / 25600000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_316_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (7470559844389 / 25600000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_316_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (7470559844389 / 25600000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_316_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (7470559844389 / 25600000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_316_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (7470559844389 / 25600000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_316_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (7470559844389 / 25600000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_316_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_317_1 :
+    Uω (aρ 1) (bρ 1) (18715271389599 / 64000000000000 : ℝ) ≤
+      (-(1564832224170227175487 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22877659228192339 / 80000000000000000 : ℝ) ≤
+      (-(1564832224170227175487 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (22877659228192339 / 80000000000000000 : ℝ) =
+            (22877659228192339 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22877659228192339 / 20000000000000000 : ℝ) ≤
+            (168035726579772824513 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2859650870508241 / 10000000000000000 : ℝ)) (q :=
+      (22877659228192339 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_317_2 :
+    Uω (aρ 2) (bρ 2) (18715271389599 / 64000000000000 : ℝ) ≤
+      (-(12603242087980496131311 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22684966250189827 / 80000000000000000 : ℝ) ≤
+      (-(12603242087980496131311 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (22684966250189827 / 80000000000000000 : ℝ) =
+            (22684966250189827 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22684966250189827 / 20000000000000000 : ℝ) ≤
+            (1259701518019503868689 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2835246638517613 / 10000000000000000 : ℝ)) (q :=
+      (22684966250189827 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_317_3 :
+    Uω (aρ 3) (bρ 3) (18715271389599 / 64000000000000 : ℝ) ≤
+      (-(51100874429049288073 / 40000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22298173592442603 / 80000000000000000 : ℝ) ≤
+      (-(51100874429049288073 / 40000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (22298173592442603 / 80000000000000000 : ℝ) =
+            (22298173592442603 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22298173592442603 / 20000000000000000 : ℝ) ≤
+            (4350899994950711927 / 40000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2785944580430807 / 10000000000000000 : ℝ)) (q :=
+      (22298173592442603 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_317_4 :
+    Uω (aρ 4) (bρ 4) (18715271389599 / 64000000000000 : ℝ) ≤
+      (-(653479665454582040087 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21651339067849943 / 80000000000000000 : ℝ) ≤
+      (-(653479665454582040087 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (21651339067849943 / 80000000000000000 : ℝ) =
+            (21651339067849943 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21651339067849943 / 20000000000000000 : ℝ) ≤
+            (39667514845417959913 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1351265065173821 / 5000000000000000 : ℝ)) (q :=
+      (21651339067849943 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_317_5 :
+    Uω (aρ 5) (bρ 5) (18715271389599 / 64000000000000 : ℝ) ≤
+      (-(13540979465010177493251 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4130881304873591 / 16000000000000000 : ℝ) ≤
+      (-(13540979465010177493251 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4130881304873591 / 16000000000000000 : ℝ) =
+            (4130881304873591 / 4000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4130881304873591 / 4000000000000000 : ℝ) ≤
+            (321964140989822506749 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (514297382762429 / 2000000000000000 : ℝ)) (q :=
+      (4130881304873591 / 16000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_317_6 :
+    Uω (aρ 6) (bρ 6) (18715271389599 / 64000000000000 : ℝ) ≤
+      (-(14274081709003551513581 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19194397949454987 / 80000000000000000 : ℝ) ≤
+      (-(14274081709003551513581 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (19194397949454987 / 80000000000000000 : ℝ) =
+            (19194397949454987 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19194397949454987 / 10000000000000000 : ℝ) ≤
+            (6520333699996448486419 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2373725145223903 / 10000000000000000 : ℝ)) (q :=
+      (19194397949454987 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_317_7 :
+    Uω (aρ 7) (bρ 7) (18715271389599 / 64000000000000 : ℝ) ≤
+      (-(3853706432000757266593 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17125080092878191 / 80000000000000000 : ℝ) ≤
+      (-(3853706432000757266593 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (17125080092878191 / 80000000000000000 : ℝ) =
+            (17125080092878191 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17125080092878191 / 10000000000000000 : ℝ) ≤
+            (1344897420249242733407 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (260007132729963 / 1250000000000000 : ℝ)) (q :=
+      (17125080092878191 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_317_8 :
+    Uω (aρ 8) (bρ 8) (18715271389599 / 64000000000000 : ℝ) ≤
+      (-(1728438790549203311371 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14204912327560163 / 80000000000000000 : ℝ) ≤
+      (-(1728438790549203311371 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (14204912327560163 / 80000000000000000 : ℝ) =
+            (14204912327560163 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14204912327560163 / 10000000000000000 : ℝ) ≤
+            (351002750350796688629 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1633556771850197 / 10000000000000000 : ℝ)) (q :=
+      (14204912327560163 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_317_9 :
+    Uω (aρ 9) (bρ 9) (18715271389599 / 64000000000000 : ℝ) ≤
+      (-(21210927637829035889281 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9592042704887499 / 80000000000000000 : ℝ) ≤
+      (-(21210927637829035889281 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (9592042704887499 / 80000000000000000 : ℝ) =
+            (9592042704887499 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9592042704887499 / 5000000000000000 : ℝ) ≤
+            (6514959574170964110719 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (821304994542031 / 10000000000000000 : ℝ)) (q :=
+      (9592042704887499 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_317_10 :
+    Uω (aρ 10) (bρ 10) (18715271389599 / 64000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_317_11 :
+    Uω (aρ 11) (bρ 11) (18715271389599 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_317_12 :
+    Uω (aρ 12) (bρ 12) (18715271389599 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_317_13 :
+    Uω (aρ 13) (bρ 13) (18715271389599 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_317_14 :
+    Uω (aρ 14) (bρ 14) (18715271389599 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_317_15 :
+    Uω (aρ 15) (bρ 15) (18715271389599 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_317_16 :
+    Uω (aρ 16) (bρ 16) (18715271389599 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_317 :
+    Uρ (18715271389599 / 64000000000000 : ℝ) ≤
+      (-(1641393523257367651801 / 1000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (18715271389599 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1564832224170227175487 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_317_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (18715271389599 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(12603242087980496131311 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_317_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (18715271389599 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) * (-(51100874429049288073 / 40000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_317_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (18715271389599 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(653479665454582040087 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_317_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (18715271389599 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(13540979465010177493251 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_317_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (18715271389599 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(14274081709003551513581 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_317_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (18715271389599 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(3853706432000757266593 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_317_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (18715271389599 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(1728438790549203311371 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_317_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (18715271389599 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(21210927637829035889281 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_317_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (18715271389599 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_317_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (18715271389599 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_317_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (18715271389599 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_317_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (18715271389599 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_317_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (18715271389599 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_317_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (18715271389599 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_317_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (18715271389599 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_317_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_318_1 :
+    Uω (aρ 1) (bρ 1) (37508286336451 / 128000000000000 : ℝ) ≤
+      (-(2499488191591324361 / 2000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (45852499820039187 / 160000000000000000 : ℝ) ≤
+      (-(2499488191591324361 / 2000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (45852499820039187 / 160000000000000000 : ℝ) =
+            (45852499820039187 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (45852499820039187 / 40000000000000000 : ℝ) ≤
+            (273100529608675639 / 2000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (358215603194333 / 1250000000000000 : ℝ)) (q :=
+      (45852499820039187 / 160000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_318_2 :
+    Uω (aρ 2) (bρ 2) (37508286336451 / 128000000000000 : ℝ) ≤
+      (-(3145460707552018440057 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (45467124743452883 / 160000000000000000 : ℝ) ≤
+      (-(3145460707552018440057 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (45467124743452883 / 160000000000000000 : ℝ) =
+            (45467124743452883 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (45467124743452883 / 40000000000000000 : ℝ) ≤
+            (320275193947981559943 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (177582622093211 / 625000000000000 : ℝ)) (q :=
+      (45467124743452883 / 160000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_318_3 :
+    Uω (aρ 3) (bρ 3) (37508286336451 / 128000000000000 : ℝ) ≤
+      (-(797090067591125847769 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (44693572823312123 / 160000000000000000 : ℝ) ≤
+      (-(797090067591125847769 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (44693572823312123 / 160000000000000000 : ℝ) =
+            (44693572823312123 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (44693572823312123 / 40000000000000000 : ℝ) ≤
+            (69343907783874152231 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2792024069823781 / 10000000000000000 : ℝ)) (q :=
+      (44693572823312123 / 160000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_318_4 :
+    Uω (aρ 4) (bρ 4) (37508286336451 / 128000000000000 : ℝ) ≤
+      (-(652357220802502956437 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (43399997048911691 / 160000000000000000 : ℝ) ≤
+      (-(652357220802502956437 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (43399997048911691 / 160000000000000000 : ℝ) =
+            (43399997048911691 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (43399997048911691 / 40000000000000000 : ℝ) ≤
+            (40789959497497043563 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2708621279088727 / 10000000000000000 : ℝ)) (q :=
+      (43399997048911691 / 160000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_318_5 :
+    Uω (aρ 5) (bρ 5) (37508286336451 / 128000000000000 : ℝ) ≤
+      (-(13517388069351791248559 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (41406381347382187 / 160000000000000000 : ℝ) ≤
+      (-(13517388069351791248559 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (41406381347382187 / 160000000000000000 : ℝ) =
+            (41406381347382187 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (41406381347382187 / 40000000000000000 : ℝ) ≤
+            (345555536648208751441 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2577609235732539 / 10000000000000000 : ℝ)) (q :=
+      (41406381347382187 / 160000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_318_6 :
+    Uω (aρ 6) (bρ 6) (37508286336451 / 128000000000000 : ℝ) ≤
+      (-(14248527817117285144357 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (38487019659432091 / 160000000000000000 : ℝ) ≤
+      (-(14248527817117285144357 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (38487019659432091 / 160000000000000000 : ℝ) =
+            (38487019659432091 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (38487019659432091 / 20000000000000000 : ℝ) ≤
+            (6545887591882714855643 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2379929399878777 / 10000000000000000 : ℝ)) (q :=
+      (38487019659432091 / 160000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_318_7 :
+    Uω (aρ 7) (bρ 7) (37508286336451 / 128000000000000 : ℝ) ≤
+      (-(7692835495468796043633 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6870032266348943 / 32000000000000000 : ℝ) ≤
+      (-(7692835495468796043633 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (6870032266348943 / 32000000000000000 : ℝ) =
+            (6870032266348943 / 4000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6870032266348943 / 4000000000000000 : ℝ) ≤
+            (2704372209031203956367 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (417296697935571 / 2000000000000000 : ℝ)) (q :=
+      (6870032266348943 / 32000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_318_8 :
+    Uω (aρ 8) (bρ 8) (37508286336451 / 128000000000000 : ℝ) ≤
+      (-(431182196363598939487 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28515421009719987 / 160000000000000000 : ℝ) ≤
+      (-(431182196363598939487 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (28515421009719987 / 160000000000000000 : ℝ) =
+            (28515421009719987 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28515421009719987 / 20000000000000000 : ℝ) ≤
+            (88678188861401060513 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (410170650191191 / 2500000000000000 : ℝ)) (q :=
+      (28515421009719987 / 160000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8])
+      (by norm_num [bρ_8]) (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_318_9 :
+    Uω (aρ 9) (bρ 9) (37508286336451 / 128000000000000 : ℝ) ≤
+      (-(21137493907044255191037 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19325479826693587 / 160000000000000000 : ℝ) ≤
+      (-(21137493907044255191037 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (19325479826693587 / 160000000000000000 : ℝ) =
+            (19325479826693587 / 10000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19325479826693587 / 10000000000000000 : ℝ) ≤
+            (6588393304955744808963 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (406692178343 / 4882812500000 : ℝ)) (q :=
+      (19325479826693587 / 160000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9])
+      (by norm_num [bρ_9]) (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_318_10 :
+    Uω (aρ 10) (bρ 10) (37508286336451 / 128000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_318_11 :
+    Uω (aρ 11) (bρ 11) (37508286336451 / 128000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_318_12 :
+    Uω (aρ 12) (bρ 12) (37508286336451 / 128000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_318_13 :
+    Uω (aρ 13) (bρ 13) (37508286336451 / 128000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_318_14 :
+    Uω (aρ 14) (bρ 14) (37508286336451 / 128000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_318_15 :
+    Uω (aρ 15) (bρ 15) (37508286336451 / 128000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_318_16 :
+    Uω (aρ 16) (bρ 16) (37508286336451 / 128000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_318 :
+    Uρ (37508286336451 / 128000000000000 : ℝ) ≤
+      (-(16394957714446851550457 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (37508286336451 / 128000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2499488191591324361 / 2000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_318_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (37508286336451 / 128000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3145460707552018440057 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_318_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (37508286336451 / 128000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(797090067591125847769 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_318_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (37508286336451 / 128000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(652357220802502956437 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_318_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (37508286336451 / 128000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(13517388069351791248559 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_318_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (37508286336451 / 128000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(14248527817117285144357 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_318_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (37508286336451 / 128000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(7692835495468796043633 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_318_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (37508286336451 / 128000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(431182196363598939487 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_318_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (37508286336451 / 128000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(21137493907044255191037 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_318_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (37508286336451 / 128000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_318_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (37508286336451 / 128000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_318_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (37508286336451 / 128000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_318_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (37508286336451 / 128000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_318_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (37508286336451 / 128000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_318_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (37508286336451 / 128000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_318_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (37508286336451 / 128000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_318_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_319_1 :
+    Uω (aρ 1) (bρ 1) (4698253736713 / 16000000000000 : ℝ) ≤
+      (-(2495253808887446669611 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1148742029389611 / 4000000000000000 : ℝ) ≤
+      (-(2495253808887446669611 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1148742029389611 / 4000000000000000 : ℝ) =
+            (1148742029389611 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1148742029389611 / 1000000000000000 : ℝ) ≤
+            (277334912312553330389 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (287179877958743 / 1000000000000000 : ℝ)) (q :=
+      (1148742029389611 / 4000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_319_2 :
+    Uω (aρ 2) (bρ 2) (4698253736713 / 16000000000000 : ℝ) ≤
+      (-(2512097855895038717213 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1423884904122721 / 5000000000000000 : ℝ) ≤
+      (-(2512097855895038717213 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1423884904122721 / 5000000000000000 : ℝ) =
+            (1423884904122721 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1423884904122721 / 1250000000000000 : ℝ) ≤
+            (260490865304961282787 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2847397261640259 / 10000000000000000 : ℝ)) (q :=
+      (1423884904122721 / 5000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_319_3 :
+    Uω (aρ 3) (bρ 3) (4698253736713 / 16000000000000 : ℝ) ≤
+      (-(12731710923469879778623 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (223953991305553 / 800000000000000 : ℝ) ≤
+      (-(12731710923469879778623 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (223953991305553 / 800000000000000 : ℝ) =
+            (223953991305553 / 200000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (223953991305553 / 200000000000000 : ℝ) ≤
+            (1131232682530120221377 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (13990517670691 / 50000000000000 : ℝ)) (q :=
+      (223953991305553 / 800000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_319_4 :
+    Uω (aρ 4) (bρ 4) (4698253736713 / 16000000000000 : ℝ) ≤
+      (-(3256186487277326469717 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2718582208572577 / 10000000000000000 : ℝ) ≤
+      (-(3256186487277326469717 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2718582208572577 / 10000000000000000 : ℝ) =
+            (2718582208572577 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2718582208572577 / 2500000000000000 : ℝ) ≤
+            (209549414222673530283 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2714712349709529 / 10000000000000000 : ℝ)) (q :=
+      (2718582208572577 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_319_5 :
+    Uω (aρ 5) (bρ 5) (4698253736713 / 16000000000000 : ℝ) ≤
+      (-(421682895011645082857 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5187993476315311 / 20000000000000000 : ℝ) ≤
+      (-(421682895011645082857 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5187993476315311 / 20000000000000000 : ℝ) =
+            (5187993476315311 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5187993476315311 / 5000000000000000 : ℝ) ≤
+            (11534092675854917143 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1291865664107343 / 5000000000000000 : ℝ)) (q :=
+      (5187993476315311 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_319_6 :
+    Uω (aρ 6) (bρ 6) (4698253736713 / 16000000000000 : ℝ) ≤
+      (-(2844608090322516779093 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4823154755797971 / 20000000000000000 : ℝ) ≤
+      (-(2844608090322516779093 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4823154755797971 / 20000000000000000 : ℝ) =
+            (4823154755797971 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4823154755797971 / 2500000000000000 : ℝ) ≤
+            (1314274991477483220907 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1193066491418673 / 5000000000000000 : ℝ)) (q :=
+      (4823154755797971 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_319_7 :
+    Uω (aρ 7) (bρ 7) (4698253736713 / 16000000000000 : ℝ) ≤
+      (-(15356605898681968251623 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1076567050775089 / 5000000000000000 : ℝ) ≤
+      (-(15356605898681968251623 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1076567050775089 / 5000000000000000 : ℝ) =
+            (1076567050775089 / 625000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1076567050775089 / 625000000000000 : ℝ) ≤
+            (5437809510318031748377 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2092907810899731 / 10000000000000000 : ℝ)) (q :=
+      (1076567050775089 / 5000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_319_8 :
+    Uω (aρ 8) (bρ 8) (4698253736713 / 16000000000000 : ℝ) ≤
+      (-(17210348493441849799951 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3577618742706903 / 20000000000000000 : ℝ) ≤
+      (-(17210348493441849799951 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3577618742706903 / 20000000000000000 : ℝ) =
+            (3577618742706903 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3577618742706903 / 2500000000000000 : ℝ) ≤
+            (3584066915558150200049 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (823900000923139 / 5000000000000000 : ℝ)) (q :=
+      (3577618742706903 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_319_9 :
+    Uω (aρ 9) (bρ 9) (4698253736713 / 16000000000000 : ℝ) ≤
+      (-(5266267722922526326391 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9504857824393 / 78125000000000 : ℝ) ≤
+      (-(5266267722922526326391 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (9504857824393 / 78125000000000 : ℝ) = (9504857824393 / 4882812500000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9504857824393 / 4882812500000 : ℝ) ≤
+            (1665204080077473673609 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (844390490543983 / 10000000000000000 : ℝ)) (q :=
+      (9504857824393 / 78125000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_319_10 :
+    Uω (aρ 10) (bρ 10) (4698253736713 / 16000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_319_11 :
+    Uω (aρ 11) (bρ 11) (4698253736713 / 16000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_319_12 :
+    Uω (aρ 12) (bρ 12) (4698253736713 / 16000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_319_13 :
+    Uω (aρ 13) (bρ 13) (4698253736713 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_319_14 :
+    Uω (aρ 14) (bρ 14) (4698253736713 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_319_15 :
+    Uω (aρ 15) (bρ 15) (4698253736713 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_319_16 :
+    Uω (aρ 16) (bρ 16) (4698253736713 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_319 :
+    Uρ (4698253736713 / 16000000000000 : ℝ) ≤
+      (-(2047013394536089855583 / 1250000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (4698253736713 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2495253808887446669611 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_319_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (4698253736713 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(2512097855895038717213 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_319_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (4698253736713 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(12731710923469879778623 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_319_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (4698253736713 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3256186487277326469717 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_319_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (4698253736713 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(421682895011645082857 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_319_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (4698253736713 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(2844608090322516779093 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_319_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (4698253736713 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(15356605898681968251623 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_319_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (4698253736713 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(17210348493441849799951 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_319_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (4698253736713 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(5266267722922526326391 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_319_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (4698253736713 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_319_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (4698253736713 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_319_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (4698253736713 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_319_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (4698253736713 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_319_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (4698253736713 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_319_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (4698253736713 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_319_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (4698253736713 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_319_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_320_1 :
+    Uω (aρ 1) (bρ 1) (37663773450957 / 128000000000000 : ℝ) ≤
+      (-(3113785465743511281037 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (46046862523071781 / 160000000000000000 : ℝ) ≤
+      (-(3113785465743511281037 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (46046862523071781 / 160000000000000000 : ℝ) =
+            (46046862523071781 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (46046862523071781 / 40000000000000000 : ℝ) ≤
+            (351950435756488718963 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2877872732612957 / 10000000000000000 : ℝ)) (q :=
+      (46046862523071781 / 160000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_320_2 :
+    Uω (aρ 2) (bρ 2) (37663773450957 / 128000000000000 : ℝ) ≤
+      (-(12539181240894190232013 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9132301813230193 / 32000000000000000 : ℝ) ≤
+      (-(12539181240894190232013 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9132301813230193 / 32000000000000000 : ℝ) =
+            (9132301813230193 / 8000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9132301813230193 / 8000000000000000 : ℝ) ≤
+            (1323762365105809767987 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (570694512601571 / 2000000000000000 : ℝ)) (q :=
+      (9132301813230193 / 32000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_320_3 :
+    Uω (aρ 3) (bρ 3) (37663773450957 / 128000000000000 : ℝ) ≤
+      (-(198594186367096513121 / 156250000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1795520939983421 / 6400000000000000 : ℝ) ≤
+      (-(198594186367096513121 / 156250000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1795520939983421 / 6400000000000000 : ℝ) =
+            (1795520939983421 / 1600000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1795520939983421 / 1600000000000000 : ℝ) ≤
+            (18014307476653486879 / 156250000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (112167318941487 / 400000000000000 : ℝ)) (q :=
+      (1795520939983421 / 6400000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_320_4 :
+    Uω (aρ 4) (bρ 4) (37663773450957 / 128000000000000 : ℝ) ≤
+      (-(13002397681596293197851 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (43594633004645821 / 160000000000000000 : ℝ) ≤
+      (-(13002397681596293197851 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (43594633004645821 / 160000000000000000 : ℝ) =
+            (43594633004645821 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (43594633004645821 / 40000000000000000 : ℝ) ≤
+            (860545924403706802149 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (340100417841839 / 1250000000000000 : ℝ)) (q :=
+      (43594633004645821 / 160000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_320_5 :
+    Uω (aρ 5) (bρ 5) (37663773450957 / 128000000000000 : ℝ) ≤
+      (-(2694074582216508070629 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (41601512451173557 / 160000000000000000 : ℝ) ≤
+      (-(2694074582216508070629 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (41601512451173557 / 160000000000000000 : ℝ) =
+            (41601512451173557 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (41601512451173557 / 40000000000000000 : ℝ) ≤
+            (78514138983491929371 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2589853192885679 / 10000000000000000 : ℝ)) (q :=
+      (41601512451173557 / 160000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_320_6 :
+    Uω (aρ 6) (bρ 6) (37663773450957 / 128000000000000 : ℝ) ≤
+      (-(7098809629941692031279 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (38683451101567717 / 160000000000000000 : ℝ) ≤
+      (-(7098809629941692031279 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (38683451101567717 / 160000000000000000 : ℝ) =
+            (38683451101567717 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (38683451101567717 / 20000000000000000 : ℝ) ≤
+            (3298398074558307968721 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2392335899324949 / 10000000000000000 : ℝ)) (q :=
+      (38683451101567717 / 160000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_320_7 :
+    Uω (aρ 7) (bρ 7) (37663773450957 / 128000000000000 : ℝ) ≤
+      (-(1532762987265952141441 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6910022643929929 / 32000000000000000 : ℝ) ≤
+      (-(1532762987265952141441 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (6910022643929929 / 32000000000000000 : ℝ) =
+            (6910022643929929 / 4000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6910022643929929 / 4000000000000000 : ℝ) ≤
+            (546678553634047858559 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (209933004484519 / 1000000000000000 : ℝ)) (q :=
+      (6910022643929929 / 32000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_320_8 :
+    Uω (aρ 8) (bρ 8) (37663773450957 / 128000000000000 : ℝ) ≤
+      (-(8586784124854736362831 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5745282464167273 / 32000000000000000 : ℝ) ≤
+      (-(8586784124854736362831 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (5745282464167273 / 32000000000000000 : ℝ) =
+            (5745282464167273 / 4000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5745282464167273 / 4000000000000000 : ℝ) ≤
+            (1810423579645263637169 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (165490908383353 / 1000000000000000 : ℝ)) (q :=
+      (5745282464167273 / 32000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_320_9 :
+    Uω (aρ 9) (bρ 9) (37663773450957 / 128000000000000 : ℝ) ≤
+      (-(1312101351333463023481 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19605529662034021 / 160000000000000000 : ℝ) ≤
+      (-(1312101351333463023481 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (19605529662034021 / 160000000000000000 : ℝ) =
+            (19605529662034021 / 10000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19605529662034021 / 10000000000000000 : ℝ) ≤
+            (420766599416536976519 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (855764379843237 / 10000000000000000 : ℝ)) (q :=
+      (19605529662034021 / 160000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9])
+      (by norm_num [bρ_9]) (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_320_10 :
+    Uω (aρ 10) (bρ 10) (37663773450957 / 128000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_320_11 :
+    Uω (aρ 11) (bρ 11) (37663773450957 / 128000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_320_12 :
+    Uω (aρ 12) (bρ 12) (37663773450957 / 128000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_320_13 :
+    Uω (aρ 13) (bρ 13) (37663773450957 / 128000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_320_14 :
+    Uω (aρ 14) (bρ 14) (37663773450957 / 128000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_320_15 :
+    Uω (aρ 15) (bρ 15) (37663773450957 / 128000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_320_16 :
+    Uω (aρ 16) (bρ 16) (37663773450957 / 128000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_320 :
+    Uρ (37663773450957 / 128000000000000 : ℝ) ≤
+      (-(16357380029375922820787 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (37663773450957 / 128000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3113785465743511281037 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_320_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (37663773450957 / 128000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(12539181240894190232013 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_320_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (37663773450957 / 128000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(198594186367096513121 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_320_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (37663773450957 / 128000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(13002397681596293197851 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_320_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (37663773450957 / 128000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(2694074582216508070629 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_320_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (37663773450957 / 128000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(7098809629941692031279 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_320_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (37663773450957 / 128000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(1532762987265952141441 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_320_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (37663773450957 / 128000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(8586784124854736362831 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_320_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (37663773450957 / 128000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(1312101351333463023481 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_320_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (37663773450957 / 128000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_320_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (37663773450957 / 128000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_320_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (37663773450957 / 128000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_320_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (37663773450957 / 128000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_320_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (37663773450957 / 128000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_320_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (37663773450957 / 128000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_320_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (37663773450957 / 128000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_320_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_321_1 :
+    Uω (aρ 1) (bρ 1) (3774151700821 / 12800000000000 : ℝ) ≤
+      (-(12434059224938226750533 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23072021931276057 / 80000000000000000 : ℝ) ≤
+      (-(12434059224938226750533 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (23072021931276057 / 80000000000000000 : ℝ) =
+            (23072021931276057 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23072021931276057 / 20000000000000000 : ℝ) ≤
+            (1428884381061773249467 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (360493335579701 / 1250000000000000 : ℝ)) (q :=
+      (23072021931276057 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_321_2 :
+    Uω (aρ 2) (bρ 2) (3774151700821 / 12800000000000 : ℝ) ≤
+      (-(12517918520821526742451 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22879350573235169 / 80000000000000000 : ℝ) ≤
+      (-(12517918520821526742451 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (22879350573235169 / 80000000000000000 : ℝ) =
+            (22879350573235169 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22879350573235169 / 20000000000000000 : ℝ) ≤
+            (1345025085178473257549 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1429773928818693 / 5000000000000000 : ℝ)) (q :=
+      (22879350573235169 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_321_3 :
+    Uω (aρ 3) (bρ 3) (3774151700821 / 12800000000000 : ℝ) ≤
+      (-(1268839188906676708531 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22492624270015249 / 80000000000000000 : ℝ) ≤
+      (-(1268839188906676708531 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (22492624270015249 / 80000000000000000 : ℝ) =
+            (22492624270015249 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22492624270015249 / 20000000000000000 : ℝ) ≤
+            (117455171693323291469 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1405131194091203 / 5000000000000000 : ℝ)) (q :=
+      (22492624270015249 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_321_4 :
+    Uω (aρ 4) (bρ 4) (3774151700821 / 12800000000000 : ℝ) ≤
+      (-(3245024847091790789197 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21845975027762613 / 80000000000000000 : ℝ) ≤
+      (-(3245024847091790789197 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (21845975027762613 / 80000000000000000 : ℝ) =
+            (21845975027762613 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21845975027762613 / 20000000000000000 : ℝ) ≤
+            (220711054408209210803 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2726894258684247 / 10000000000000000 : ℝ)) (q :=
+      (21845975027762613 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_321_5 :
+    Uω (aρ 5) (bρ 5) (3774151700821 / 12800000000000 : ℝ) ≤
+      (-(3361737154102947906173 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20849537641114421 / 80000000000000000 : ℝ) ≤
+      (-(3361737154102947906173 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (20849537641114421 / 80000000000000000 : ℝ) =
+            (20849537641114421 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20849537641114421 / 20000000000000000 : ℝ) ≤
+            (103998747397052093827 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2595974831357199 / 10000000000000000 : ℝ)) (q :=
+      (20849537641114421 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_321_6 :
+    Uω (aρ 6) (bρ 6) (3774151700821 / 12800000000000 : ℝ) ≤
+      (-(2834452778434685974403 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19390829433174889 / 80000000000000000 : ℝ) ≤
+      (-(2834452778434685974403 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (19390829433174889 / 80000000000000000 : ℝ) =
+            (19390829433174889 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19390829433174889 / 10000000000000000 : ℝ) ≤
+            (1324430303365314025597 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (599634538628079 / 2500000000000000 : ℝ)) (q :=
+      (19390829433174889 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_321_7 :
+    Uω (aρ 7) (bρ 7) (3774151700821 / 12800000000000 : ℝ) ≤
+      (-(3059748468025370392999 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3465006426906649 / 16000000000000000 : ℝ) ≤
+      (-(3059748468025370392999 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3465006426906649 / 16000000000000000 : ℝ) =
+            (3465006426906649 / 2000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3465006426906649 / 2000000000000000 : ℝ) ≤
+            (1099134613774629607001 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (421150042122381 / 2000000000000000 : ℝ)) (q :=
+      (3465006426906649 / 16000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_321_8 :
+    Uω (aρ 8) (bρ 8) (3774151700821 / 12800000000000 : ℝ) ≤
+      (-(4284236393973365117899 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14415904500640873 / 80000000000000000 : ℝ) ≤
+      (-(4284236393973365117899 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (14415904500640873 / 80000000000000000 : ℝ) =
+            (14415904500640873 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14415904500640873 / 10000000000000000 : ℝ) ≤
+            (914367458276634882101 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (415502488369703 / 2500000000000000 : ℝ)) (q :=
+      (14415904500640873 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_321_9 :
+    Uω (aρ 9) (bρ 9) (3774151700821 / 12800000000000 : ℝ) ≤
+      (-(326923614860068943593 / 156250000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9872128646489561 / 80000000000000000 : ℝ) ≤
+      (-(326923614860068943593 / 156250000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (9872128646489561 / 80000000000000000 : ℝ) =
+            (9872128646489561 / 5000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9872128646489561 / 5000000000000000 : ℝ) ≤
+            (106293372827431056407 / 156250000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (108378952287623 / 1250000000000000 : ℝ)) (q :=
+      (9872128646489561 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_321_10 :
+    Uω (aρ 10) (bρ 10) (3774151700821 / 12800000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_321_11 :
+    Uω (aρ 11) (bρ 11) (3774151700821 / 12800000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_321_12 :
+    Uω (aρ 12) (bρ 12) (3774151700821 / 12800000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_321_13 :
+    Uω (aρ 13) (bρ 13) (3774151700821 / 12800000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_321_14 :
+    Uω (aρ 14) (bρ 14) (3774151700821 / 12800000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_321_15 :
+    Uω (aρ 15) (bρ 15) (3774151700821 / 12800000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_321_16 :
+    Uω (aρ 16) (bρ 16) (3774151700821 / 12800000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_321 :
+    Uρ (3774151700821 / 12800000000000 : ℝ) ≤
+      (-(1021173312797747915237 / 625000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (3774151700821 / 12800000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(12434059224938226750533 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_321_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (3774151700821 / 12800000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(12517918520821526742451 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_321_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (3774151700821 / 12800000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1268839188906676708531 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_321_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (3774151700821 / 12800000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3245024847091790789197 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_321_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (3774151700821 / 12800000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(3361737154102947906173 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_321_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (3774151700821 / 12800000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(2834452778434685974403 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_321_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (3774151700821 / 12800000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(3059748468025370392999 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_321_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (3774151700821 / 12800000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(4284236393973365117899 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_321_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (3774151700821 / 12800000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(326923614860068943593 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_321_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (3774151700821 / 12800000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_321_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (3774151700821 / 12800000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_321_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (3774151700821 / 12800000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_321_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (3774151700821 / 12800000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_321_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (3774151700821 / 12800000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_321_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (3774151700821 / 12800000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_321_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (3774151700821 / 12800000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_321_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_322_1 :
+    Uω (aρ 1) (bρ 1) (37819260565463 / 128000000000000 : ℝ) ≤
+      (-(496520837715664407951 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (46241225194075943 / 160000000000000000 : ℝ) ≤
+      (-(496520837715664407951 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (46241225194075943 / 160000000000000000 : ℝ) =
+            (46241225194075943 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (46241225194075943 / 40000000000000000 : ℝ) ≤
+            (57996906524335592049 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (180626289729231 / 625000000000000 : ℝ)) (q :=
+      (46241225194075943 / 160000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_322_2 :
+    Uω (aρ 2) (bρ 2) (37819260565463 / 128000000000000 : ℝ) ≤
+      (-(124967009268472262333 / 100000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (45855893173228023 / 160000000000000000 : ℝ) ≤
+      (-(124967009268472262333 / 100000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (45855893173228023 / 160000000000000000 : ℝ) =
+            (45855893173228023 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (45855893173228023 / 40000000000000000 : ℝ) ≤
+            (13662426791527737667 / 100000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1432811572785853 / 5000000000000000 : ℝ)) (q :=
+      (45855893173228023 / 160000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_322_3 :
+    Uω (aρ 3) (bρ 3) (37819260565463 / 128000000000000 : ℝ) ≤
+      (-(253336052101030294349 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (45082473383727919 / 160000000000000000 : ℝ) ≤
+      (-(253336052101030294349 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (45082473383727919 / 160000000000000000 : ℝ) =
+            (45082473383727919 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (45082473383727919 / 40000000000000000 : ℝ) ≤
+            (23922820018969705651 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2816341778234193 / 10000000000000000 : ℝ)) (q :=
+      (45082473383727919 / 160000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_322_4 :
+    Uω (aρ 4) (bρ 4) (37819260565463 / 128000000000000 : ℝ) ≤
+      (-(6478925422896764806271 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (43789266493922071 / 160000000000000000 : ℝ) ≤
+      (-(6478925422896764806271 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (43789266493922071 / 160000000000000000 : ℝ) =
+            (43789266493922071 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (43789266493922071 / 40000000000000000 : ℝ) ≤
+            (452546380103235193729 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1366492549036731 / 5000000000000000 : ℝ)) (q :=
+      (43789266493922071 / 160000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_322_5 :
+    Uω (aρ 5) (bρ 5) (37819260565463 / 128000000000000 : ℝ) ≤
+      (-(13423579493192682497527 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (41796636316459999 / 160000000000000000 : ℝ) ≤
+      (-(13423579493192682497527 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (41796636316459999 / 160000000000000000 : ℝ) =
+            (41796636316459999 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (41796636316459999 / 40000000000000000 : ℝ) ≤
+            (439364112807317502473 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2602096245225703 / 10000000000000000 : ℝ)) (q :=
+      (41796636316459999 / 160000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_322_6 :
+    Uω (aρ 6) (bρ 6) (37819260565463 / 128000000000000 : ℝ) ≤
+      (-(2829394800309022849977 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (38879861381664647 / 160000000000000000 : ℝ) ≤
+      (-(2829394800309022849977 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (38879861381664647 / 160000000000000000 : ℝ) =
+            (38879861381664647 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (38879861381664647 / 20000000000000000 : ℝ) ≤
+            (1329488281490977150023 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (601184938379071 / 2500000000000000 : ℝ)) (q :=
+      (38879861381664647 / 160000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_322_7 :
+    Uω (aρ 7) (bρ 7) (37819260565463 / 128000000000000 : ℝ) ≤
+      (-(15269942734092825618979 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6949999784785659 / 32000000000000000 : ℝ) ≤
+      (-(15269942734092825618979 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (6949999784785659 / 32000000000000000 : ℝ) =
+            (6949999784785659 / 4000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6949999784785659 / 4000000000000000 : ℝ) ≤
+            (5524472674907174381021 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (52804208176481 / 250000000000000 : ℝ)) (q :=
+      (6949999784785659 / 32000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_322_8 :
+    Uω (aρ 8) (bρ 8) (37819260565463 / 128000000000000 : ℝ) ≤
+      (-(1710047894900604801367 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28937140821498503 / 160000000000000000 : ℝ) ≤
+      (-(1710047894900604801367 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (28937140821498503 / 160000000000000000 : ℝ) =
+            (28937140821498503 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28937140821498503 / 20000000000000000 : ℝ) ≤
+            (369393645999395198633 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (417275678898879 / 2500000000000000 : ℝ)) (q :=
+      (28937140821498503 / 160000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8])
+      (by norm_num [bρ_8]) (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_322_9 :
+    Uω (aρ 9) (bρ 9) (37819260565463 / 128000000000000 : ℝ) ≤
+      (-(20853507377490201335747 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3976432911427427 / 32000000000000000 : ℝ) ≤
+      (-(20853507377490201335747 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (3976432911427427 / 32000000000000000 : ℝ) =
+            (3976432911427427 / 2000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3976432911427427 / 2000000000000000 : ℝ) ≤
+            (6872379834509798664253 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (175639262182069 / 2000000000000000 : ℝ)) (q :=
+      (3976432911427427 / 32000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_322_10 :
+    Uω (aρ 10) (bρ 10) (37819260565463 / 128000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_322_11 :
+    Uω (aρ 11) (bρ 11) (37819260565463 / 128000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_322_12 :
+    Uω (aρ 12) (bρ 12) (37819260565463 / 128000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_322_13 :
+    Uω (aρ 13) (bρ 13) (37819260565463 / 128000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_322_14 :
+    Uω (aρ 14) (bρ 14) (37819260565463 / 128000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_322_15 :
+    Uω (aρ 15) (bρ 15) (37819260565463 / 128000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_322_16 :
+    Uω (aρ 16) (bρ 16) (37819260565463 / 128000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_322 :
+    Uρ (37819260565463 / 128000000000000 : ℝ) ≤
+      (-(16320282936987819977121 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (37819260565463 / 128000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(496520837715664407951 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_322_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (37819260565463 / 128000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(124967009268472262333 / 100000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_322_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (37819260565463 / 128000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(253336052101030294349 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_322_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (37819260565463 / 128000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(6478925422896764806271 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_322_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (37819260565463 / 128000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(13423579493192682497527 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_322_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (37819260565463 / 128000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(2829394800309022849977 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_322_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (37819260565463 / 128000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(15269942734092825618979 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_322_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (37819260565463 / 128000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(1710047894900604801367 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_322_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (37819260565463 / 128000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(20853507377490201335747 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_322_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (37819260565463 / 128000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_322_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (37819260565463 / 128000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_322_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (37819260565463 / 128000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_322_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (37819260565463 / 128000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_322_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (37819260565463 / 128000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_322_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (37819260565463 / 128000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_322_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (37819260565463 / 128000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_322_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_323_1 :
+    Uω (aρ 1) (bρ 1) (9474251030679 / 32000000000000 : ℝ) ≤
+      (-(3098006707644175633397 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11584601629423329 / 40000000000000000 : ℝ) ≤
+      (-(3098006707644175633397 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (11584601629423329 / 40000000000000000 : ℝ) =
+            (11584601629423329 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11584601629423329 / 10000000000000000 : ℝ) ≤
+            (367729193855824366603 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2896094585709477 / 10000000000000000 : ℝ)) (q :=
+      (11584601629423329 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_323_2 :
+    Uω (aρ 2) (bρ 2) (9474251030679 / 32000000000000 : ℝ) ≤
+      (-(12475528267784406772859 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11488271286690993 / 40000000000000000 : ℝ) ≤
+      (-(12475528267784406772859 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (11488271286690993 / 40000000000000000 : ℝ) =
+            (11488271286690993 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11488271286690993 / 10000000000000000 : ℝ) ≤
+            (1387415338215593227141 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2871698426853309 / 10000000000000000 : ℝ)) (q :=
+      (11488271286690993 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_323_3 :
+    Uω (aρ 3) (bρ 3) (9474251030679 / 32000000000000 : ℝ) ≤
+      (-(2529051974725758156739 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11294924507987287 / 40000000000000000 : ℝ) ≤
+      (-(2529051974725758156739 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (11294924507987287 / 40000000000000000 : ℝ) =
+            (11294924507987287 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11294924507987287 / 10000000000000000 : ℝ) ≤
+            (243536746474241843261 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (44100330372679 / 156250000000000 : ℝ)) (q :=
+      (11294924507987287 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_323_4 :
+    Uω (aρ 4) (bρ 4) (9474251030679 / 32000000000000 : ℝ) ≤
+      (-(6467825915874434987103 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10971645580980563 / 40000000000000000 : ℝ) ≤
+      (-(6467825915874434987103 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (10971645580980563 / 40000000000000000 : ℝ) =
+            (10971645580980563 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10971645580980563 / 10000000000000000 : ℝ) ≤
+            (463645887125565012897 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1369537930706547 / 5000000000000000 : ℝ)) (q :=
+      (10971645580980563 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_323_5 :
+    Uω (aρ 5) (bρ 5) (9474251030679 / 32000000000000 : ℝ) ≤
+      (-(13400265280141196180749 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10473548891629579 / 40000000000000000 : ℝ) ≤
+      (-(13400265280141196180749 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (10473548891629579 / 40000000000000000 : ℝ) =
+            (10473548891629579 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10473548891629579 / 10000000000000000 : ℝ) ≤
+            (462678325858803819251 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1304108718036301 / 5000000000000000 : ℝ)) (q :=
+      (10473548891629579 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_323_6 :
+    Uω (aρ 6) (bρ 6) (9474251030679 / 32000000000000 : ℝ) ≤
+      (-(1412174924384875215969 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9744514672005377 / 40000000000000000 : ℝ) ≤
+      (-(1412174924384875215969 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (9744514672005377 / 40000000000000000 : ℝ) =
+            (9744514672005377 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9744514672005377 / 5000000000000000 : ℝ) ≤
+            (667266616515124784031 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2410940701400501 / 10000000000000000 : ℝ)) (q :=
+      (9744514672005377 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_323_7 :
+    Uω (aρ 7) (bρ 7) (9474251030679 / 32000000000000 : ℝ) ≤
+      (-(15241230493239060186719 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8712479333308459 / 40000000000000000 : ℝ) ≤
+      (-(15241230493239060186719 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (8712479333308459 / 40000000000000000 : ℝ) =
+            (8712479333308459 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8712479333308459 / 5000000000000000 : ℝ) ≤
+            (5553184915760939813281 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1059292206406021 / 5000000000000000 : ℝ)) (q :=
+      (8712479333308459 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_323_8 :
+    Uω (aρ 8) (bρ 8) (9474251030679 / 32000000000000 : ℝ) ≤
+      (-(8532083434991892629273 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7260602151212977 / 40000000000000000 : ℝ) ≤
+      (-(8532083434991892629273 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (7260602151212977 / 40000000000000000 : ℝ) =
+            (7260602151212977 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7260602151212977 / 5000000000000000 : ℝ) ≤
+            (1865124269508107370727 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1676187473104301 / 10000000000000000 : ℝ)) (q :=
+      (7260602151212977 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_323_9 :
+    Uω (aρ 9) (bρ 9) (9474251030679 / 32000000000000 : ℝ) ≤
+      (-(20784778876778594280753 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5004820588425443 / 40000000000000000 : ℝ) ≤
+      (-(20784778876778594280753 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (5004820588425443 / 40000000000000000 : ℝ) =
+            (5004820588425443 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 2
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5004820588425443 / 5000000000000000 : ℝ) ≤
+            (9636532221405719247 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (444631160035267 / 5000000000000000 : ℝ)) (q :=
+      (5004820588425443 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_323_10 :
+    Uω (aρ 10) (bρ 10) (9474251030679 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_323_11 :
+    Uω (aρ 11) (bρ 11) (9474251030679 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_323_12 :
+    Uω (aρ 12) (bρ 12) (9474251030679 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_323_13 :
+    Uω (aρ 13) (bρ 13) (9474251030679 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_323_14 :
+    Uω (aρ 14) (bρ 14) (9474251030679 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_323_15 :
+    Uω (aρ 15) (bρ 15) (9474251030679 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_323_16 :
+    Uω (aρ 16) (bρ 16) (9474251030679 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_323 :
+    Uρ (9474251030679 / 32000000000000 : ℝ) ≤
+      (-(815095342484062068619 / 500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (9474251030679 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3098006707644175633397 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_323_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (9474251030679 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(12475528267784406772859 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_323_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (9474251030679 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(2529051974725758156739 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_323_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (9474251030679 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(6467825915874434987103 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_323_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (9474251030679 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(13400265280141196180749 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_323_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (9474251030679 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1412174924384875215969 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_323_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (9474251030679 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(15241230493239060186719 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_323_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (9474251030679 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(8532083434991892629273 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_323_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (9474251030679 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(20784778876778594280753 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_323_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (9474251030679 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_323_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (9474251030679 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_323_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (9474251030679 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_323_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (9474251030679 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_323_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (9474251030679 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_323_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (9474251030679 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_323_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (9474251030679 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_323_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_324_1 :
+    Uω (aρ 1) (bρ 1) (37974747679969 / 128000000000000 : ℝ) ≤
+      (-(1546384587863347391307 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (46435587833453881 / 160000000000000000 : ℝ) ≤
+      (-(1546384587863347391307 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (46435587833453881 / 160000000000000000 : ℝ) =
+            (46435587833453881 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (46435587833453881 / 40000000000000000 : ℝ) ≤
+            (186483362886652608693 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2902168534769157 / 10000000000000000 : ℝ)) (q :=
+      (46435587833453881 / 160000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_324_2 :
+    Uω (aρ 2) (bρ 2) (37974747679969 / 128000000000000 : ℝ) ≤
+      (-(12454400353658942523317 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (46050277067415249 / 160000000000000000 : ℝ) ≤
+      (-(12454400353658942523317 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (46050277067415249 / 160000000000000000 : ℝ) =
+            (46050277067415249 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (46050277067415249 / 40000000000000000 : ℝ) ≤
+            (1408543252341057476683 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (359721712690541 / 1250000000000000 : ℝ)) (q :=
+      (46050277067415249 / 160000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_324_3 :
+    Uω (aρ 3) (bρ 3) (37974747679969 / 128000000000000 : ℝ) ≤
+      (-(6311881747142055395947 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (362215379887637 / 1280000000000000 : ℝ) ≤
+      (-(6311881747142055395947 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (362215379887637 / 1280000000000000 : ℝ) =
+            (362215379887637 / 320000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (362215379887637 / 320000000000000 : ℝ) ≤
+            (619590055857944604053 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (11314001940767 / 40000000000000 : ℝ)) (q :=
+      (362215379887637 / 1280000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_324_4 :
+    Uω (aρ 4) (bρ 4) (37974747679969 / 128000000000000 : ℝ) ≤
+      (-(6456751062797533349511 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (43983897549575361 / 160000000000000000 : ℝ) ≤
+      (-(6456751062797533349511 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (43983897549575361 / 160000000000000000 : ℝ) =
+            (43983897549575361 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (43983897549575361 / 40000000000000000 : ℝ) ≤
+            (474720740202466650489 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1372583274604671 / 5000000000000000 : ℝ)) (q :=
+      (43983897549575361 / 160000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_324_5 :
+    Uω (aρ 5) (bρ 5) (37974747679969 / 128000000000000 : ℝ) ≤
+      (-(1337700571783894728703 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (41991753044936137 / 160000000000000000 : ℝ) ≤
+      (-(1337700571783894728703 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (41991753044936137 / 160000000000000000 : ℝ) =
+            (41991753044936137 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (41991753044936137 / 40000000000000000 : ℝ) ≤
+            (48593788816105271297 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2614338405464439 / 10000000000000000 : ℝ)) (q :=
+      (41991753044936137 / 160000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_324_6 :
+    Uω (aρ 6) (bρ 6) (37974747679969 / 128000000000000 : ℝ) ≤
+      (-(704829463884613062753 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (39076250825509641 / 160000000000000000 : ℝ) ≤
+      (-(704829463884613062753 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (39076250825509641 / 160000000000000000 : ℝ) =
+            (39076250825509641 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (39076250825509641 / 20000000000000000 : ℝ) ≤
+            (334891306565386937247 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2417141003176127 / 10000000000000000 : ℝ)) (q :=
+      (39076250825509641 / 160000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_324_7 :
+    Uω (aρ 7) (bρ 7) (37974747679969 / 128000000000000 : ℝ) ≤
+      (-(15212605061841808395923 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (34949819644135537 / 160000000000000000 : ℝ) ≤
+      (-(15212605061841808395923 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (34949819644135537 / 160000000000000000 : ℝ) =
+            (34949819644135537 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (34949819644135537 / 20000000000000000 : ℝ) ≤
+            (5581810347158191604077 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (531249621566091 / 2500000000000000 : ℝ)) (q :=
+      (34949819644135537 / 160000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_324_8 :
+    Uω (aρ 8) (bρ 8) (37974747679969 / 128000000000000 : ℝ) ≤
+      (-(17028007863161623062419 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (29147613159923817 / 160000000000000000 : ℝ) ≤
+      (-(17028007863161623062419 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (29147613159923817 / 160000000000000000 : ℝ) =
+            (29147613159923817 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (29147613159923817 / 20000000000000000 : ℝ) ≤
+            (3766407545838376937581 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1683264327077899 / 10000000000000000 : ℝ)) (q :=
+      (29147613159923817 / 160000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8])
+      (by norm_num [bρ_8]) (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_324_9 :
+    Uω (aρ 9) (bρ 9) (37974747679969 / 128000000000000 : ℝ) ≤
+      (-(4143379349202316265587 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20155639796037609 / 160000000000000000 : ℝ) ≤
+      (-(4143379349202316265587 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (20155639796037609 / 160000000000000000 : ℝ) =
+            (20155639796037609 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20155639796037609 / 20000000000000000 : ℝ) ≤
+            (15503732597683734413 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (900233284952123 / 10000000000000000 : ℝ)) (q :=
+      (20155639796037609 / 160000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9])
+      (by norm_num [bρ_9]) (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_324_10 :
+    Uω (aρ 10) (bρ 10) (37974747679969 / 128000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_324_11 :
+    Uω (aρ 11) (bρ 11) (37974747679969 / 128000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_324_12 :
+    Uω (aρ 12) (bρ 12) (37974747679969 / 128000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_324_13 :
+    Uω (aρ 13) (bρ 13) (37974747679969 / 128000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_324_14 :
+    Uω (aρ 14) (bρ 14) (37974747679969 / 128000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_324_15 :
+    Uω (aρ 15) (bρ 15) (37974747679969 / 128000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_324_16 :
+    Uω (aρ 16) (bρ 16) (37974747679969 / 128000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_324 :
+    Uρ (37974747679969 / 128000000000000 : ℝ) ≤
+      (-(8141820960763928849147 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (37974747679969 / 128000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1546384587863347391307 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_324_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (37974747679969 / 128000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(12454400353658942523317 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_324_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (37974747679969 / 128000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(6311881747142055395947 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_324_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (37974747679969 / 128000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(6456751062797533349511 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_324_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (37974747679969 / 128000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1337700571783894728703 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_324_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (37974747679969 / 128000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(704829463884613062753 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_324_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (37974747679969 / 128000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(15212605061841808395923 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_324_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (37974747679969 / 128000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(17028007863161623062419 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_324_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (37974747679969 / 128000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(4143379349202316265587 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_324_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (37974747679969 / 128000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_324_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (37974747679969 / 128000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_324_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (37974747679969 / 128000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_324_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (37974747679969 / 128000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_324_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (37974747679969 / 128000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_324_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (37974747679969 / 128000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_324_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (37974747679969 / 128000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_324_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_325_1 :
+    Uω (aρ 1) (bρ 1) (19026245618611 / 64000000000000 : ℝ) ≤
+      (-(12350170375956099393281 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23266384570703427 / 80000000000000000 : ℝ) ≤
+      (-(12350170375956099393281 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (23266384570703427 / 80000000000000000 : ℝ) =
+            (23266384570703427 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23266384570703427 / 20000000000000000 : ℝ) ≤
+            (1512773230043900606719 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (363530310356611 / 1250000000000000 : ℝ)) (q :=
+      (23266384570703427 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_325_2 :
+    Uω (aρ 2) (bρ 2) (19026245618611 / 64000000000000 : ℝ) ≤
+      (-(248666339913984400827 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23073734467758043 / 80000000000000000 : ℝ) ≤
+      (-(248666339913984400827 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (23073734467758043 / 80000000000000000 : ℝ) =
+            (23073734467758043 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23073734467758043 / 20000000000000000 : ℝ) ≤
+            (28592532206015599173 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1441924484813271 / 5000000000000000 : ℝ)) (q :=
+      (23073734467758043 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_325_3 :
+    Uω (aρ 3) (bρ 3) (19026245618611 / 64000000000000 : ℝ) ≤
+      (-(2520462653559380998639 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22687073373496971 / 80000000000000000 : ℝ) ≤
+      (-(2520462653559380998639 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (22687073373496971 / 80000000000000000 : ℝ) =
+            (22687073373496971 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22687073373496971 / 20000000000000000 : ℝ) ≤
+            (252126067640619001361 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1417289901205637 / 5000000000000000 : ℝ)) (q :=
+      (22687073373496971 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_325_4 :
+    Uω (aρ 4) (bρ 4) (19026245618611 / 64000000000000 : ℝ) ≤
+      (-(12891401508169105599223 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22040606087447539 / 80000000000000000 : ℝ) ≤
+      (-(12891401508169105599223 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (22040606087447539 / 80000000000000000 : ℝ) =
+            (22040606087447539 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22040606087447539 / 20000000000000000 : ℝ) ≤
+            (971542097830894400777 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (687814290490979 / 2500000000000000 : ℝ)) (q :=
+      (22040606087447539 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_325_5 :
+    Uω (aρ 5) (bρ 5) (19026245618611 / 64000000000000 : ℝ) ≤
+      (-(3338450137178833629859 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21044654382064131 / 80000000000000000 : ℝ) ≤
+      (-(3338450137178833629859 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (21044654382064131 / 80000000000000000 : ℝ) =
+            (21044654382064131 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21044654382064131 / 20000000000000000 : ℝ) ≤
+            (127285764321166370141 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (327557394369133 / 1250000000000000 : ℝ)) (q :=
+      (21044654382064131 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_325_6 :
+    Uω (aρ 6) (bρ 6) (19026245618611 / 64000000000000 : ℝ) ≤
+      (-(1758936720551409517731 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19587218916901939 / 80000000000000000 : ℝ) ≤
+      (-(1758936720551409517731 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (19587218916901939 / 80000000000000000 : ℝ) =
+            (19587218916901939 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19587218916901939 / 10000000000000000 : ℝ) ≤
+            (840365205573590482269 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (605835165950629 / 2500000000000000 : ℝ)) (q :=
+      (19587218916901939 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_325_7 :
+    Uω (aρ 7) (bρ 7) (19026245618611 / 64000000000000 : ℝ) ≤
+      (-(15184065889695235582767 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17524853000984307 / 80000000000000000 : ℝ) ≤
+      (-(15184065889695235582767 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (17524853000984307 / 80000000000000000 : ℝ) =
+            (17524853000984307 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17524853000984307 / 10000000000000000 : ℝ) ≤
+            (5610349519304764417233 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (532852641395777 / 2500000000000000 : ℝ)) (q :=
+      (17524853000984307 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_325_8 :
+    Uω (aρ 8) (bρ 8) (19026245618611 / 64000000000000 : ℝ) ≤
+      (-(8496000237880936070873 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14626377640430259 / 80000000000000000 : ℝ) ≤
+      (-(8496000237880936070873 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (14626377640430259 / 80000000000000000 : ℝ) =
+            (14626377640430259 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14626377640430259 / 10000000000000000 : ℝ) ≤
+            (1901207466619063929127 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (422583344196149 / 2500000000000000 : ℝ)) (q :=
+      (14626377640430259 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_325_9 :
+    Uω (aρ 9) (bρ 9) (19026245618611 / 64000000000000 : ℝ) ≤
+      (-(10324916742774442850357 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10145632175558567 / 80000000000000000 : ℝ) ≤
+      (-(10324916742774442850357 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (10145632175558567 / 80000000000000000 : ℝ) =
+            (10145632175558567 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10145632175558567 / 10000000000000000 : ℝ) ≤
+            (72290961725557149643 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (911112638926673 / 10000000000000000 : ℝ)) (q :=
+      (10145632175558567 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_325_10 :
+    Uω (aρ 10) (bρ 10) (19026245618611 / 64000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_325_11 :
+    Uω (aρ 11) (bρ 11) (19026245618611 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_325_12 :
+    Uω (aρ 12) (bρ 12) (19026245618611 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_325_13 :
+    Uω (aρ 13) (bρ 13) (19026245618611 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_325_14 :
+    Uω (aρ 14) (bρ 14) (19026245618611 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_325_15 :
+    Uω (aρ 15) (bρ 15) (19026245618611 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_325_16 :
+    Uω (aρ 16) (bρ 16) (19026245618611 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_325 :
+    Uρ (19026245618611 / 64000000000000 : ℝ) ≤
+      (-(16265485475807527019511 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (19026245618611 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(12350170375956099393281 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_325_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (19026245618611 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(248666339913984400827 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_325_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (19026245618611 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(2520462653559380998639 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_325_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (19026245618611 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(12891401508169105599223 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_325_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (19026245618611 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(3338450137178833629859 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_325_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (19026245618611 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1758936720551409517731 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_325_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (19026245618611 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(15184065889695235582767 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_325_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (19026245618611 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(8496000237880936070873 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_325_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (19026245618611 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(10324916742774442850357 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_325_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (19026245618611 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_325_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (19026245618611 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_325_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (19026245618611 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_325_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (19026245618611 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_325_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (19026245618611 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_325_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (19026245618611 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_325_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (19026245618611 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_325_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_326_1 :
+    Uω (aρ 1) (bρ 1) (1525209391779 / 5120000000000 : ℝ) ≤
+      (-(308232691673755184269 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (46629950441601067 / 160000000000000000 : ℝ) ≤
+      (-(308232691673755184269 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (46629950441601067 / 160000000000000000 : ℝ) =
+            (46629950441601067 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (46629950441601067 / 40000000000000000 : ℝ) ≤
+            (38340898476244815731 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1457158214983387 / 5000000000000000 : ℝ)) (q :=
+      (46629950441601067 / 160000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_326_2 :
+    Uω (aρ 2) (bρ 2) (1525209391779 / 5120000000000 : ℝ) ≤
+      (-(775767375395375690883 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (46244660751397899 / 160000000000000000 : ℝ) ≤
+      (-(775767375395375690883 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (46244660751397899 / 160000000000000000 : ℝ) =
+            (46244660751397899 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (46244660751397899 / 40000000000000000 : ℝ) ≤
+            (90666599979624309117 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1444962115600689 / 5000000000000000 : ℝ)) (q :=
+      (46244660751397899 / 160000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_326_3 :
+    Uω (aρ 3) (bρ 3) (1525209391779 / 5120000000000 : ℝ) ≤
+      (-(2516181799245877235259 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (45471370816306019 / 160000000000000000 : ℝ) ≤
+      (-(2516181799245877235259 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (45471370816306019 / 160000000000000000 : ℝ) =
+            (45471370816306019 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (45471370816306019 / 40000000000000000 : ℝ) ≤
+            (256406921954122764741 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2840659095664893 / 10000000000000000 : ℝ)) (q :=
+      (45471370816306019 / 160000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_326_4 :
+    Uω (aρ 4) (bρ 4) (1525209391779 / 5120000000000 : ℝ) ≤
+      (-(12869349761769917993087 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (44178526203859579 / 160000000000000000 : ℝ) ≤
+      (-(12869349761769917993087 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (44178526203859579 / 160000000000000000 : ℝ) =
+            (44178526203859579 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (44178526203859579 / 40000000000000000 : ℝ) ≤
+            (993593844230082006913 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (344668462521761 / 1250000000000000 : ℝ)) (q :=
+      (44178526203859579 / 160000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_326_5 :
+    Uω (aρ 5) (bρ 5) (1525209391779 / 5120000000000 : ℝ) ≤
+      (-(533225980681196273301 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (42186862736393331 / 160000000000000000 : ℝ) ≤
+      (-(533225980681196273301 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (42186862736393331 / 160000000000000000 : ℝ) =
+            (42186862736393331 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (42186862736393331 / 40000000000000000 : ℝ) ≤
+            (21291763558803726699 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2626579686075807 / 10000000000000000 : ℝ)) (q :=
+      (42186862736393331 / 160000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_326_6 :
+    Uω (aρ 6) (bρ 6) (1525209391779 / 5120000000000 : ℝ) ≤
+      (-(3511615592009915733251 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (39272619752170011 / 160000000000000000 : ℝ) ≤
+      (-(3511615592009915733251 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (39272619752170011 / 160000000000000000 : ℝ) =
+            (39272619752170011 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (39272619752170011 / 20000000000000000 : ℝ) ≤
+            (1686988260240084266749 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (607384922046973 / 2500000000000000 : ℝ)) (q :=
+      (39272619752170011 / 160000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_326_7 :
+    Uω (aρ 7) (bρ 7) (1525209391779 / 5120000000000 : ℝ) ≤
+      (-(473612888501125712317 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (35149576550279731 / 160000000000000000 : ℝ) ≤
+      (-(473612888501125712317 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (35149576550279731 / 160000000000000000 : ℝ) =
+            (35149576550279731 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (35149576550279731 / 20000000000000000 : ℝ) ≤
+            (176212593030124287683 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2137820668711607 / 10000000000000000 : ℝ)) (q :=
+      (35149576550279731 / 160000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_326_8 :
+    Uω (aρ 8) (bρ 8) (1525209391779 / 5120000000000 : ℝ) ≤
+      (-(16956143277397500307727 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (29357835747710419 / 160000000000000000 : ℝ) ≤
+      (-(16956143277397500307727 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (29357835747710419 / 160000000000000000 : ℝ) =
+            (29357835747710419 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (29357835747710419 / 20000000000000000 : ℝ) ≤
+            (3838272131602499692273 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1697394719730443 / 10000000000000000 : ℝ)) (q :=
+      (29357835747710419 / 160000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8])
+      (by norm_num [bρ_8]) (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_326_9 :
+    Uω (aρ 9) (bρ 9) (1525209391779 / 5120000000000 : ℝ) ≤
+      (-(20583563068231126965171 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (817047278612811 / 6400000000000000 : ℝ) ≤
+      (-(20583563068231126965171 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (817047278612811 / 6400000000000000 : ℝ) =
+            (817047278612811 / 800000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (817047278612811 / 800000000000000 : ℝ) ≤
+            (210852340768873034829 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (36876145011667 / 400000000000000 : ℝ)) (q :=
+      (817047278612811 / 6400000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_326_10 :
+    Uω (aρ 10) (bρ 10) (1525209391779 / 5120000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_326_11 :
+    Uω (aρ 11) (bρ 11) (1525209391779 / 5120000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_326_12 :
+    Uω (aρ 12) (bρ 12) (1525209391779 / 5120000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_326_13 :
+    Uω (aρ 13) (bρ 13) (1525209391779 / 5120000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_326_14 :
+    Uω (aρ 14) (bρ 14) (1525209391779 / 5120000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_326_15 :
+    Uω (aρ 15) (bρ 15) (1525209391779 / 5120000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_326_16 :
+    Uω (aρ 16) (bρ 16) (1525209391779 / 5120000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_326 :
+    Uρ (1525209391779 / 5120000000000 : ℝ) ≤
+      (-(162474349687928098737 / 100000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1525209391779 / 5120000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(308232691673755184269 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_326_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1525209391779 / 5120000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(775767375395375690883 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_326_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1525209391779 / 5120000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(2516181799245877235259 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_326_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1525209391779 / 5120000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(12869349761769917993087 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_326_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1525209391779 / 5120000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(533225980681196273301 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_326_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1525209391779 / 5120000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3511615592009915733251 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_326_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1525209391779 / 5120000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(473612888501125712317 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_326_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1525209391779 / 5120000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(16956143277397500307727 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_326_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1525209391779 / 5120000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(20583563068231126965171 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_326_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1525209391779 / 5120000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_326_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1525209391779 / 5120000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_326_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1525209391779 / 5120000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_326_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1525209391779 / 5120000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_326_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1525209391779 / 5120000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_326_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1525209391779 / 5120000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_326_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1525209391779 / 5120000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_326_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_327_1 :
+    Uω (aρ 1) (bρ 1) (2387998646983 / 8000000000000 : ℝ) ≤
+      (-(492339535770254087997 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (730111433345077 / 2500000000000000 : ℝ) ≤
+      (-(492339535770254087997 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (730111433345077 / 2500000000000000 : ℝ) =
+            (730111433345077 / 625000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (730111433345077 / 625000000000000 : ℝ) ≤
+            (62178208469745912003 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1460195188058433 / 5000000000000000 : ℝ)) (q :=
+      (730111433345077 / 2500000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_327_2 :
+    Uω (aρ 2) (bρ 2) (2387998646983 / 8000000000000 : ℝ) ≤
+      (-(12391283199142446414239 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2896365782211833 / 10000000000000000 : ℝ) ≤
+      (-(12391283199142446414239 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2896365782211833 / 10000000000000000 : ℝ) =
+            (2896365782211833 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2896365782211833 / 2500000000000000 : ℝ) ≤
+            (1471660406857553585761 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (723999871572479 / 2500000000000000 : ℝ)) (q :=
+      (2896365782211833 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_327_3 :
+    Uω (aρ 3) (bρ 3) (2387998646983 / 8000000000000 : ℝ) ≤
+      (-(12559550482915562925899 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1139214867377979 / 4000000000000000 : ℝ) ≤
+      (-(12559550482915562925899 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1139214867377979 / 4000000000000000 : ℝ) =
+            (1139214867377979 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1139214867377979 / 1000000000000000 : ℝ) ≤
+            (1303393123084437074101 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (569347673021229 / 2000000000000000 : ℝ)) (q :=
+      (1139214867377979 / 4000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_327_4 :
+    Uω (aρ 4) (bρ 4) (2387998646983 / 8000000000000 : ℝ) ≤
+      (-(2569469334029075128739 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5534479955051489 / 20000000000000000 : ℝ) ≤
+      (-(2569469334029075128739 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5534479955051489 / 20000000000000000 : ℝ) =
+            (5534479955051489 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5534479955051489 / 5000000000000000 : ℝ) ≤
+            (203119387170924871261 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2763438164332739 / 10000000000000000 : ℝ)) (q :=
+      (5534479955051489 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_327_5 :
+    Uω (aρ 5) (bρ 5) (2387998646983 / 8000000000000 : ℝ) ≤
+      (-(532302094754197904583 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1057110374347879 / 4000000000000000 : ℝ) ≤
+      (-(532302094754197904583 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1057110374347879 / 4000000000000000 : ℝ) =
+            (1057110374347879 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1057110374347879 / 1000000000000000 : ℝ) ≤
+            (22215649485802095417 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (526540000071129 / 2000000000000000 : ℝ)) (q :=
+      (1057110374347879 / 4000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_327_6 :
+    Uω (aρ 6) (bρ 6) (2387998646983 / 8000000000000 : ℝ) ≤
+      (-(3505373688820111534097 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (30758434858961 / 125000000000000 : ℝ) ≤
+      (-(3505373688820111534097 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (30758434858961 / 125000000000000 : ℝ) = (30758434858961 / 15625000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (30758434858961 / 15625000000000 : ℝ) ≤
+            (1693230163429888465903 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (243573808119001 / 1000000000000000 : ℝ)) (q :=
+      (30758434858961 / 125000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_327_7 :
+    Uω (aρ 7) (bρ 7) (2387998646983 / 8000000000000 : ℝ) ≤
+      (-(15127244149469302294097 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2203089464428441 / 10000000000000000 : ℝ) ≤
+      (-(15127244149469302294097 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2203089464428441 / 10000000000000000 : ℝ) =
+            (2203089464428441 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2203089464428441 / 1250000000000000 : ℝ) ≤
+            (5667171259530697705903 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (536057203343283 / 2500000000000000 : ℝ)) (q :=
+      (2203089464428441 / 10000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_327_8 :
+    Uω (aρ 8) (bρ 8) (2387998646983 / 8000000000000 : ℝ) ≤
+      (-(52876358936216522127 / 31250000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (920714228960997 / 5000000000000000 : ℝ) ≤
+      (-(52876358936216522127 / 31250000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (920714228960997 / 5000000000000000 : ℝ) =
+            (920714228960997 / 625000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (920714228960997 / 625000000000000 : ℝ) ≤
+            (12106189216908477873 / 31250000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (852224225850119 / 5000000000000000 : ℝ)) (q :=
+      (920714228960997 / 5000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_327_9 :
+    Uω (aρ 9) (bρ 9) (2387998646983 / 8000000000000 : ℝ) ≤
+      (-(4103612166609828334827 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (642513036818419 / 5000000000000000 : ℝ) ≤
+      (-(4103612166609828334827 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (642513036818419 / 5000000000000000 : ℝ) =
+            (642513036818419 / 625000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (642513036818419 / 625000000000000 : ℝ) ≤
+            (55270915190171665173 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (466304655744963 / 5000000000000000 : ℝ)) (q :=
+      (642513036818419 / 5000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_327_10 :
+    Uω (aρ 10) (bρ 10) (2387998646983 / 8000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_327_11 :
+    Uω (aρ 11) (bρ 11) (2387998646983 / 8000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_327_12 :
+    Uω (aρ 12) (bρ 12) (2387998646983 / 8000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_327_13 :
+    Uω (aρ 13) (bρ 13) (2387998646983 / 8000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_327_14 :
+    Uω (aρ 14) (bρ 14) (2387998646983 / 8000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_327_15 :
+    Uω (aρ 15) (bρ 15) (2387998646983 / 8000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_327_16 :
+    Uω (aρ 16) (bρ 16) (2387998646983 / 8000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_327 :
+    Uρ (2387998646983 / 8000000000000 : ℝ) ≤
+      (-(8114743990152761941807 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (2387998646983 / 8000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(492339535770254087997 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_327_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (2387998646983 / 8000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(12391283199142446414239 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_327_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (2387998646983 / 8000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(12559550482915562925899 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_327_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (2387998646983 / 8000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(2569469334029075128739 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_327_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (2387998646983 / 8000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(532302094754197904583 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_327_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (2387998646983 / 8000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3505373688820111534097 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_327_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (2387998646983 / 8000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(15127244149469302294097 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_327_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (2387998646983 / 8000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) * (-(52876358936216522127 / 31250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_327_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (2387998646983 / 8000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(4103612166609828334827 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_327_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (2387998646983 / 8000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_327_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (2387998646983 / 8000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_327_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (2387998646983 / 8000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_327_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (2387998646983 / 8000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_327_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (2387998646983 / 8000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_327_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (2387998646983 / 8000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_327_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (2387998646983 / 8000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_327_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/U27.lean b/LeanPool/Zeta5Irrational/Table/U27.lean
new file mode 100644
index 0000000000..57a0671dec
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/U27.lean
@@ -0,0 +1,3933 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-! # Certified arcsine potential bounds (U27) -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma U_328_1 :
+    Uω (aρ 1) (bρ 1) (38285721908981 / 128000000000000 : ℝ) ≤
+      (-(12287712377373996090711 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (46824313018906437 / 160000000000000000 : ℝ) ≤
+      (-(12287712377373996090711 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (46824313018906437 / 160000000000000000 : ℝ) =
+            (46824313018906437 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (46824313018906437 / 40000000000000000 : ℝ) ≤
+            (1575231228626003909289 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (731616080327291 / 2500000000000000 : ℝ)) (q :=
+      (46824313018906437 / 160000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_328_2 :
+    Uω (aρ 2) (bρ 2) (38285721908981 / 128000000000000 : ℝ) ≤
+      (-(6185166194462050153967 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (46439044227816253 / 160000000000000000 : ℝ) ≤
+      (-(6185166194462050153967 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (46439044227816253 / 160000000000000000 : ℝ) =
+            (46439044227816253 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (46439044227816253 / 40000000000000000 : ℝ) ≤
+            (746305608537949846033 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2902074734932891 / 10000000000000000 : ℝ)) (q :=
+      (46439044227816253 / 160000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_328_3 :
+    Uω (aρ 3) (bρ 3) (38285721908981 / 128000000000000 : ℝ) ≤
+      (-(12538237532450347374363 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (45665818384651213 / 160000000000000000 : ℝ) ≤
+      (-(12538237532450347374363 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (45665818384651213 / 160000000000000000 : ℝ) =
+            (45665818384651213 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (45665818384651213 / 40000000000000000 : ℝ) ≤
+            (1324706073549652625637 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2852817610887261 / 10000000000000000 : ℝ)) (q :=
+      (45665818384651213 / 160000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_328_4 :
+    Uω (aρ 4) (bρ 4) (38285721908981 / 128000000000000 : ℝ) ≤
+      (-(12825392018479425605597 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (44373152488460389 / 160000000000000000 : ℝ) ≤
+      (-(12825392018479425605597 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (44373152488460389 / 160000000000000000 : ℝ) =
+            (44373152488460389 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (44373152488460389 / 40000000000000000 : ℝ) ≤
+            (1037551587520574394403 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (346191069366051 / 1250000000000000 : ℝ)) (q :=
+      (44373152488460389 / 160000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_328_5 :
+    Uω (aρ 5) (bρ 5) (38285721908981 / 128000000000000 : ℝ) ≤
+      (-(6642254426029126533917 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (42381965488764117 / 160000000000000000 : ℝ) ≤
+      (-(6642254426029126533917 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (42381965488764117 / 160000000000000000 : ℝ) =
+            (42381965488764117 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (42381965488764117 / 40000000000000000 : ℝ) ≤
+            (289217376970873466083 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1319410049650687 / 5000000000000000 : ℝ)) (q :=
+      (42381965488764117 / 160000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_328_6 :
+    Uω (aρ 6) (bρ 6) (38285721908981 / 128000000000000 : ℝ) ≤
+      (-(13996590595477121154087 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (39468968474167621 / 160000000000000000 : ℝ) ≤
+      (-(13996590595477121154087 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (39468968474167621 / 160000000000000000 : ℝ) =
+            (39468968474167621 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (39468968474167621 / 20000000000000000 : ℝ) ≤
+            (6797824813522878845913 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (610483961904203 / 2500000000000000 : ℝ)) (q :=
+      (39468968474167621 / 160000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_328_7 :
+    Uω (aρ 7) (bρ 7) (38285721908981 / 128000000000000 : ℝ) ≤
+      (-(7549480253947935622409 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (35349270783747821 / 160000000000000000 : ℝ) ≤
+      (-(7549480253947935622409 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (35349270783747821 / 160000000000000000 : ℝ) =
+            (35349270783747821 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (35349270783747821 / 20000000000000000 : ℝ) ≤
+            (2847727450552064377591 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2150635017074337 / 10000000000000000 : ℝ)) (q :=
+      (35349270783747821 / 160000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_328_8 :
+    Uω (aρ 8) (bρ 8) (38285721908981 / 128000000000000 : ℝ) ≤
+      (-(16884873835296377582209 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (29567814770811661 / 160000000000000000 : ℝ) ≤
+      (-(16884873835296377582209 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (29567814770811661 / 160000000000000000 : ℝ) =
+            (29567814770811661 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (29567814770811661 / 20000000000000000 : ℝ) ≤
+            (3909541573703622417791 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1711494666797317 / 10000000000000000 : ℝ)) (q :=
+      (29567814770811661 / 160000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8])
+      (by norm_num [bρ_8]) (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_328_9 :
+    Uω (aρ 9) (bρ 9) (38285721908981 / 128000000000000 : ℝ) ≤
+      (-(4090660677271729170247 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4138798645103481 / 32000000000000000 : ℝ) ≤
+      (-(4090660677271729170247 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4138798645103481 / 32000000000000000 : ℝ) =
+            (4138798645103481 / 4000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4138798645103481 / 4000000000000000 : ℝ) ≤
+            (68222404528270829753 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (188646520399107 / 2000000000000000 : ℝ)) (q :=
+      (4138798645103481 / 32000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_328_10 :
+    Uω (aρ 10) (bρ 10) (38285721908981 / 128000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_328_11 :
+    Uω (aρ 11) (bρ 11) (38285721908981 / 128000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_328_12 :
+    Uω (aρ 12) (bρ 12) (38285721908981 / 128000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_328_13 :
+    Uω (aρ 13) (bρ 13) (38285721908981 / 128000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_328_14 :
+    Uω (aρ 14) (bρ 14) (38285721908981 / 128000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_328_15 :
+    Uω (aρ 15) (bρ 15) (38285721908981 / 128000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_328_16 :
+    Uω (aρ 16) (bρ 16) (38285721908981 / 128000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_328 :
+    Uρ (38285721908981 / 128000000000000 : ℝ) ≤
+      (-(16211642204940930669019 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (38285721908981 / 128000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(12287712377373996090711 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_328_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (38285721908981 / 128000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(6185166194462050153967 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_328_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (38285721908981 / 128000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(12538237532450347374363 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_328_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (38285721908981 / 128000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(12825392018479425605597 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_328_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (38285721908981 / 128000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(6642254426029126533917 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_328_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (38285721908981 / 128000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(13996590595477121154087 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_328_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (38285721908981 / 128000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(7549480253947935622409 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_328_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (38285721908981 / 128000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(16884873835296377582209 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_328_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (38285721908981 / 128000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(4090660677271729170247 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_328_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (38285721908981 / 128000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_328_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (38285721908981 / 128000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_328_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (38285721908981 / 128000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_328_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (38285721908981 / 128000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_328_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (38285721908981 / 128000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_328_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (38285721908981 / 128000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_328_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (38285721908981 / 128000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_328_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_329_1 :
+    Uω (aρ 1) (bρ 1) (19181732733117 / 64000000000000 : ℝ) ≤
+      (-(6133489718462708656097 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23460747148056609 / 80000000000000000 : ℝ) ≤
+      (-(6133489718462708656097 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (23460747148056609 / 80000000000000000 : ℝ) =
+            (23460747148056609 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23460747148056609 / 20000000000000000 : ℝ) ≤
+            (797982084537291343903 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2932538265549621 / 10000000000000000 : ℝ)) (q :=
+      (23460747148056609 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_329_2 :
+    Uω (aρ 2) (bρ 2) (19181732733117 / 64000000000000 : ℝ) ≤
+      (-(12349425391609171288573 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23268117944500921 / 80000000000000000 : ℝ) ≤
+      (-(12349425391609171288573 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (23268117944500921 / 80000000000000000 : ℝ) =
+            (23268117944500921 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23268117944500921 / 20000000000000000 : ℝ) ≤
+            (1513518214390828711427 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2908149977170699 / 10000000000000000 : ℝ)) (q :=
+      (23268117944500921 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_329_3 :
+    Uω (aρ 3) (bρ 3) (19181732733117 / 64000000000000 : ℝ) ≤
+      (-(6258484975339397372699 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22881520943054829 / 80000000000000000 : ℝ) ≤
+      (-(6258484975339397372699 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (22881520943054829 / 80000000000000000 : ℝ) =
+            (22881520943054829 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22881520943054829 / 20000000000000000 : ℝ) ≤
+            (672986827660602627301 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (357362104144897 / 1250000000000000 : ℝ)) (q :=
+      (22881520943054829 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_329_4 :
+    Uω (aρ 4) (bρ 4) (19181732733117 / 64000000000000 : ℝ) ≤
+      (-(640174279668968429527 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4447046475187897 / 16000000000000000 : ℝ) ≤
+      (-(640174279668968429527 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4447046475187897 / 16000000000000000 : ℝ) =
+            (4447046475187897 / 4000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4447046475187897 / 4000000000000000 : ℝ) ≤
+            (52972900631031570473 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (138780943622267 / 500000000000000 : ℝ)) (q :=
+      (4447046475187897 / 16000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_329_5 :
+    Uω (aρ 5) (bρ 5) (19181732733117 / 64000000000000 : ℝ) ≤
+      (-(13261518716286128424891 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21239757146449213 / 80000000000000000 : ℝ) ≤
+      (-(13261518716286128424891 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (21239757146449213 / 80000000000000000 : ℝ) =
+            (21239757146449213 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21239757146449213 / 20000000000000000 : ℝ) ≤
+            (601424889713871575109 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (661234996101943 / 2500000000000000 : ℝ)) (q :=
+      (21239757146449213 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_329_6 :
+    Uω (aρ 6) (bρ 6) (19181732733117 / 64000000000000 : ℝ) ≤
+      (-(13971749560585366629263 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19783567677166397 / 80000000000000000 : ℝ) ≤
+      (-(13971749560585366629263 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (19783567677166397 / 80000000000000000 : ℝ) =
+            (19783567677166397 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19783567677166397 / 10000000000000000 : ℝ) ≤
+            (6822665848414633370737 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (612033248056767 / 2500000000000000 : ℝ)) (q :=
+      (19783567677166397 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_329_7 :
+    Uω (aρ 7) (bρ 7) (19181732733117 / 64000000000000 : ℝ) ≤
+      (-(15070760978440686554333 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3544909474730529 / 16000000000000000 : ℝ) ≤
+      (-(15070760978440686554333 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3544909474730529 / 16000000000000000 : ℝ) =
+            (3544909474730529 / 2000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3544909474730529 / 2000000000000000 : ℝ) ≤
+            (5723654430559313445667 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (215703929710863 / 1000000000000000 : ℝ)) (q :=
+      (3544909474730529 / 16000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_329_8 :
+    Uω (aρ 8) (bρ 8) (19181732733117 / 64000000000000 : ℝ) ≤
+      (-(2106182354807468528861 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14836357409786957 / 80000000000000000 : ℝ) ≤
+      (-(2106182354807468528861 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (14836357409786957 / 80000000000000000 : ℝ) =
+            (14836357409786957 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14836357409786957 / 10000000000000000 : ℝ) ≤
+            (493119571317531471139 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (53704170546319 / 312500000000000 : ℝ)) (q :=
+      (14836357409786957 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_329_9 :
+    Uω (aρ 9) (bρ 9) (19181732733117 / 64000000000000 : ℝ) ≤
+      (-(20389268512107521157571 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2082693213296429 / 16000000000000000 : ℝ) ≤
+      (-(20389268512107521157571 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2082693213296429 / 16000000000000000 : ℝ) =
+            (2082693213296429 / 2000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2082693213296429 / 2000000000000000 : ℝ) ≤
+            (405146896892478842429 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (190755250003201 / 2000000000000000 : ℝ)) (q :=
+      (2082693213296429 / 16000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_329_10 :
+    Uω (aρ 10) (bρ 10) (19181732733117 / 64000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_329_11 :
+    Uω (aρ 11) (bρ 11) (19181732733117 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_329_12 :
+    Uω (aρ 12) (bρ 12) (19181732733117 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_329_13 :
+    Uω (aρ 13) (bρ 13) (19181732733117 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_329_14 :
+    Uω (aρ 14) (bρ 14) (19181732733117 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_329_15 :
+    Uω (aρ 15) (bρ 15) (19181732733117 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_329_16 :
+    Uω (aρ 16) (bρ 16) (19181732733117 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_329 :
+    Uρ (19181732733117 / 64000000000000 : ℝ) ≤
+      (-(16193895444089082419773 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (19181732733117 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(6133489718462708656097 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_329_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (19181732733117 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(12349425391609171288573 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_329_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (19181732733117 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(6258484975339397372699 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_329_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (19181732733117 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(640174279668968429527 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_329_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (19181732733117 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(13261518716286128424891 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_329_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (19181732733117 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(13971749560585366629263 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_329_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (19181732733117 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(15070760978440686554333 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_329_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (19181732733117 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(2106182354807468528861 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_329_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (19181732733117 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(20389268512107521157571 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_329_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (19181732733117 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_329_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (19181732733117 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_329_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (19181732733117 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_329_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (19181732733117 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_329_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (19181732733117 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_329_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (19181732733117 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_329_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (19181732733117 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_329_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_330_1 :
+    Uω (aρ 1) (bρ 1) (38441209023487 / 128000000000000 : ℝ) ≤
+      (-(61231446973232137127 / 50000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (47018675565752471 / 160000000000000000 : ℝ) ≤
+      (-(61231446973232137127 / 50000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (47018675565752471 / 160000000000000000 : ℝ) =
+            (47018675565752471 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (47018675565752471 / 40000000000000000 : ℝ) ≤
+            (8083271056767862873 / 50000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2938612208844137 / 10000000000000000 : ℝ)) (q :=
+      (47018675565752471 / 160000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_330_2 :
+    Uω (aρ 2) (bρ 2) (38441209023487 / 128000000000000 : ℝ) ≤
+      (-(12328562024288774376437 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (46633427499266559 / 160000000000000000 : ℝ) ≤
+      (-(12328562024288774376437 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (46633427499266559 / 160000000000000000 : ℝ) =
+            (46633427499266559 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (46633427499266559 / 40000000000000000 : ℝ) ≤
+            (1534381581711225623563 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1457112606521699 / 5000000000000000 : ℝ)) (q :=
+      (46633427499266559 / 160000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_330_3 :
+    Uω (aρ 3) (bρ 3) (38441209023487 / 128000000000000 : ℝ) ≤
+      (-(3123936886171363010597 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (45860265200691719 / 160000000000000000 : ℝ) ≤
+      (-(3123936886171363010597 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (45860265200691719 / 160000000000000000 : ℝ) =
+            (45860265200691719 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (45860265200691719 / 40000000000000000 : ℝ) ≤
+            (341799015328636989403 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2864976032071543 / 10000000000000000 : ℝ)) (q :=
+      (45860265200691719 / 160000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_330_4 :
+    Uω (aρ 4) (bρ 4) (38441209023487 / 128000000000000 : ℝ) ≤
+      (-(1278162718286327350039 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (44567776434507663 / 160000000000000000 : ℝ) ≤
+      (-(1278162718286327350039 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (44567776434507663 / 160000000000000000 : ℝ) =
+            (44567776434507663 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (44567776434507663 / 40000000000000000 : ℝ) ≤
+            (108131642313672649961 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (173856819835221 / 625000000000000 : ℝ)) (q :=
+      (44567776434507663 / 160000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_330_5 :
+    Uω (aρ 5) (bρ 5) (38441209023487 / 128000000000000 : ℝ) ≤
+      (-(3309645428236645038509 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (42577061398165479 / 160000000000000000 : ℝ) ≤
+      (-(3309645428236645038509 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (42577061398165479 / 160000000000000000 : ℝ) =
+            (42577061398165479 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (42577061398165479 / 40000000000000000 : ℝ) ≤
+            (156090473263354961491 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2651059657155763 / 10000000000000000 : ℝ)) (q :=
+      (42577061398165479 / 160000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_330_6 :
+    Uω (aρ 6) (bρ 6) (38441209023487 / 128000000000000 : ℝ) ≤
+      (-(2789394265029026856113 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7933059459529491 / 32000000000000000 : ℝ) ≤
+      (-(2789394265029026856113 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (7933059459529491 / 32000000000000000 : ℝ) =
+            (7933059459529491 / 4000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7933059459529491 / 4000000000000000 : ℝ) ≤
+            (1369488816770973143887 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (245432951973101 / 1000000000000000 : ℝ)) (q :=
+      (7933059459529491 / 32000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_330_7 :
+    Uω (aρ 7) (bρ 7) (38441209023487 / 128000000000000 : ℝ) ≤
+      (-(15042645037382581962017 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (35548903458195263 / 160000000000000000 : ℝ) ≤
+      (-(15042645037382581962017 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (35548903458195263 / 160000000000000000 : ℝ) =
+            (35548903458195263 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (35548903458195263 / 20000000000000000 : ℝ) ≤
+            (5751770371617418037983 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1081720835279743 / 5000000000000000 : ℝ)) (q :=
+      (35548903458195263 / 160000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_330_8 :
+    Uω (aρ 8) (bρ 8) (38441209023487 / 128000000000000 : ℝ) ≤
+      (-(67256754094232746029 / 40000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (29777556199880767 / 160000000000000000 : ℝ) ≤
+      (-(67256754094232746029 / 40000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (29777556199880767 / 160000000000000000 : ℝ) =
+            (29777556199880767 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (29777556199880767 / 20000000000000000 : ℝ) ≤
+            (15920907541767253971 / 40000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (862782457305087 / 5000000000000000 : ℝ)) (q :=
+      (29777556199880767 / 160000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8])
+      (by norm_num [bρ_8]) (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_330_9 :
+    Uω (aρ 9) (bρ 9) (38441209023487 / 128000000000000 : ℝ) ≤
+      (-(5081483772522509532501 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4191850960248571 / 32000000000000000 : ℝ) ≤
+      (-(5081483772522509532501 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4191850960248571 / 32000000000000000 : ℝ) =
+            (4191850960248571 / 4000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4191850960248571 / 4000000000000000 : ℝ) ≤
+            (117120079727490467499 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (192848573628087 / 2000000000000000 : ℝ)) (q :=
+      (4191850960248571 / 32000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_330_10 :
+    Uω (aρ 10) (bρ 10) (38441209023487 / 128000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_330_11 :
+    Uω (aρ 11) (bρ 11) (38441209023487 / 128000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_330_12 :
+    Uω (aρ 12) (bρ 12) (38441209023487 / 128000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_330_13 :
+    Uω (aρ 13) (bρ 13) (38441209023487 / 128000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_330_14 :
+    Uω (aρ 14) (bρ 14) (38441209023487 / 128000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_330_15 :
+    Uω (aρ 15) (bρ 15) (38441209023487 / 128000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_330_16 :
+    Uω (aρ 16) (bρ 16) (38441209023487 / 128000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_330 :
+    Uρ (38441209023487 / 128000000000000 : ℝ) ≤
+      (-(8088122799333122230017 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (38441209023487 / 128000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(61231446973232137127 / 50000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_330_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (38441209023487 / 128000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(12328562024288774376437 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_330_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (38441209023487 / 128000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3123936886171363010597 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_330_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (38441209023487 / 128000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1278162718286327350039 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_330_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (38441209023487 / 128000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(3309645428236645038509 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_330_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (38441209023487 / 128000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(2789394265029026856113 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_330_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (38441209023487 / 128000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(15042645037382581962017 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_330_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (38441209023487 / 128000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) * (-(67256754094232746029 / 40000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_330_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (38441209023487 / 128000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(5081483772522509532501 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_330_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (38441209023487 / 128000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_330_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (38441209023487 / 128000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_330_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (38441209023487 / 128000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_330_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (38441209023487 / 128000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_330_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (38441209023487 / 128000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_330_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (38441209023487 / 128000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_330_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (38441209023487 / 128000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_330_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_331_1 :
+    Uω (aρ 1) (bρ 1) (1925947629037 / 6400000000000 : ℝ) ≤
+      (-(6112821036688579436813 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11778964206967761 / 40000000000000000 : ℝ) ≤
+      (-(6112821036688579436813 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (11778964206967761 / 40000000000000000 : ℝ) =
+            (11778964206967761 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11778964206967761 / 10000000000000000 : ℝ) ≤
+            (818650766311420563187 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (368085768899821 / 1250000000000000 : ℝ)) (q :=
+      (11778964206967761 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_331_2 :
+    Uω (aρ 2) (bρ 2) (1925947629037 / 6400000000000 : ℝ) ≤
+      (-(12307742105197329123461 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11682654764732049 / 40000000000000000 : ℝ) ≤
+      (-(12307742105197329123461 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (11682654764732049 / 40000000000000000 : ℝ) =
+            (11682654764732049 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11682654764732049 / 10000000000000000 : ℝ) ≤
+            (1555201500802670876539 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (365037555323839 / 1250000000000000 : ℝ)) (q :=
+      (11682654764732049 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_331_3 :
+    Uω (aρ 3) (bρ 3) (1925947629037 / 6400000000000 : ℝ) ≤
+      (-(3118642530695954496121 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11489372082396121 / 40000000000000000 : ℝ) ≤
+      (-(3118642530695954496121 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (11489372082396121 / 40000000000000000 : ℝ) =
+            (11489372082396121 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11489372082396121 / 10000000000000000 : ℝ) ≤
+            (347093370804045503879 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (717763801943187 / 2500000000000000 : ℝ)) (q :=
+      (11489372082396121 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_331_4 :
+    Uω (aρ 4) (bρ 4) (1925947629037 / 6400000000000 : ℝ) ≤
+      (-(12759816576347544304649 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2233254377007643 / 8000000000000000 : ℝ) ≤
+      (-(12759816576347544304649 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2233254377007643 / 8000000000000000 : ℝ) =
+            (2233254377007643 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2233254377007643 / 2000000000000000 : ℝ) ≤
+            (1103127029652455695351 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (557559858031759 / 2000000000000000 : ℝ)) (q :=
+      (2233254377007643 / 8000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_331_5 :
+    Uω (aρ 5) (bρ 5) (1925947629037 / 6400000000000 : ℝ) ≤
+      (-(2643139519038542533491 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10668651704075777 / 40000000000000000 : ℝ) ≤
+      (-(2643139519038542533491 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (10668651704075777 / 40000000000000000 : ℝ) =
+            (10668651704075777 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10668651704075777 / 10000000000000000 : ℝ) ≤
+            (129449202161457466509 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (83036847469143 / 312500000000000 : ℝ)) (q :=
+      (10668651704075777 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_331_6 :
+    Uω (aρ 6) (bρ 6) (1925947629037 / 6400000000000 : ℝ) ≤
+      (-(2784451113250624090057 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9940863585352533 / 40000000000000000 : ℝ) ≤
+      (-(2784451113250624090057 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (9940863585352533 / 40000000000000000 : ℝ) =
+            (9940863585352533 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9940863585352533 / 5000000000000000 : ℝ) ≤
+            (1374431968549375909943 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1230262717395477 / 5000000000000000 : ℝ)) (q :=
+      (9940863585352533 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_331_7 :
+    Uω (aρ 7) (bρ 7) (1925947629037 / 6400000000000 : ℝ) ≤
+      (-(15014612166085188211901 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (356486970514321 / 1600000000000000 : ℝ) ≤
+      (-(15014612166085188211901 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (356486970514321 / 1600000000000000 : ℝ) =
+            (356486970514321 / 200000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (356486970514321 / 200000000000000 : ℝ) ≤
+            (5779803242914811788099 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (21698421543037 / 100000000000000 : ℝ)) (q :=
+      (356486970514321 / 1600000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_331_8 :
+    Uω (aρ 8) (bρ 8) (1925947629037 / 6400000000000 : ℝ) ≤
+      (-(2097382695647042841857 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7470584906506009 / 40000000000000000 : ℝ) ≤
+      (-(2097382695647042841857 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (7470584906506009 / 40000000000000000 : ℝ) =
+            (7470584906506009 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7470584906506009 / 5000000000000000 : ℝ) ≤
+            (501919230477957158143 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (433147281866923 / 2500000000000000 : ℝ)) (q :=
+      (7470584906506009 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_331_9 :
+    Uω (aρ 9) (bρ 9) (1925947629037 / 6400000000000 : ℝ) ≤
+      (-(20263283021367291826911 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5272745270945327 / 40000000000000000 : ℝ) ≤
+      (-(20263283021367291826911 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (5272745270945327 / 40000000000000000 : ℝ) =
+            (5272745270945327 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5272745270945327 / 5000000000000000 : ℝ) ≤
+            (531132387632708173089 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (974634938047351 / 10000000000000000 : ℝ)) (q :=
+      (5272745270945327 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_331_10 :
+    Uω (aρ 10) (bρ 10) (1925947629037 / 6400000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_331_11 :
+    Uω (aρ 11) (bρ 11) (1925947629037 / 6400000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_331_12 :
+    Uω (aρ 12) (bρ 12) (1925947629037 / 6400000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_331_13 :
+    Uω (aρ 13) (bρ 13) (1925947629037 / 6400000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_331_14 :
+    Uω (aρ 14) (bρ 14) (1925947629037 / 6400000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_331_15 :
+    Uω (aρ 15) (bρ 15) (1925947629037 / 6400000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_331_16 :
+    Uω (aρ 16) (bρ 16) (1925947629037 / 6400000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_331 :
+    Uρ (1925947629037 / 6400000000000 : ℝ) ≤
+      (-(8079345331240174625109 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1925947629037 / 6400000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(6112821036688579436813 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_331_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1925947629037 / 6400000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(12307742105197329123461 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_331_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1925947629037 / 6400000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3118642530695954496121 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_331_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1925947629037 / 6400000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(12759816576347544304649 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_331_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1925947629037 / 6400000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(2643139519038542533491 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_331_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1925947629037 / 6400000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(2784451113250624090057 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_331_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1925947629037 / 6400000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(15014612166085188211901 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_331_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1925947629037 / 6400000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(2097382695647042841857 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_331_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1925947629037 / 6400000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(20263283021367291826911 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_331_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1925947629037 / 6400000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_331_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1925947629037 / 6400000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_331_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1925947629037 / 6400000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_331_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1925947629037 / 6400000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_331_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1925947629037 / 6400000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_331_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1925947629037 / 6400000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_331_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1925947629037 / 6400000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_331_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_332_1 :
+    Uω (aρ 1) (bρ 1) (19337219847623 / 64000000000000 : ℝ) ≤
+      (-(6092237445347728784389 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4731021932973151 / 16000000000000000 : ℝ) ≤
+      (-(6092237445347728784389 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4731021932973151 / 16000000000000000 : ℝ) =
+            (4731021932973151 / 4000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4731021932973151 / 4000000000000000 : ℝ) ≤
+            (839234357652271215611 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (591366806622069 / 2000000000000000 : ℝ)) (q :=
+      (4731021932973151 / 16000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_332_2 :
+    Uω (aρ 2) (bρ 2) (19337219847623 / 64000000000000 : ℝ) ≤
+      (-(12266231890298508999037 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23462501013850143 / 80000000000000000 : ℝ) ≤
+      (-(12266231890298508999037 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (23462501013850143 / 80000000000000000 : ℝ) =
+            (23462501013850143 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23462501013850143 / 20000000000000000 : ℝ) ≤
+            (1596711715701491000963 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1466225441433221 / 5000000000000000 : ℝ)) (q :=
+      (23462501013850143 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_332_3 :
+    Uω (aρ 3) (bρ 3) (19337219847623 / 64000000000000 : ℝ) ≤
+      (-(6216174735296002445103 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (923038680700007 / 3200000000000000 : ℝ) ≤
+      (-(6216174735296002445103 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (923038680700007 / 3200000000000000 : ℝ) =
+            (923038680700007 / 800000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (923038680700007 / 800000000000000 : ℝ) ≤
+            (715297067703997554897 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (57664269802579 / 200000000000000 : ℝ)) (q :=
+      (923038680700007 / 3200000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_332_4 :
+    Uω (aρ 4) (bρ 4) (19337219847623 / 64000000000000 : ℝ) ≤
+      (-(12716337939900776348663 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22429854017776247 / 80000000000000000 : ℝ) ≤
+      (-(12716337939900776348663 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (22429854017776247 / 80000000000000000 : ℝ) =
+            (22429854017776247 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22429854017776247 / 20000000000000000 : ℝ) ≤
+            (1146605666099223651337 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (349997427657871 / 1250000000000000 : ℝ)) (q :=
+      (22429854017776247 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_332_5 :
+    Uω (aρ 5) (bρ 5) (19337219847623 / 64000000000000 : ℝ) ≤
+      (-(3292521759420410891227 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21434846318800643 / 80000000000000000 : ℝ) ≤
+      (-(3292521759420410891227 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (21434846318800643 / 80000000000000000 : ℝ) =
+            (21434846318800643 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21434846318800643 / 20000000000000000 : ℝ) ≤
+            (173214142079589108773 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2669417415854067 / 10000000000000000 : ℝ)) (q :=
+      (21434846318800643 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_332_6 :
+    Uω (aρ 6) (bρ 6) (19337219847623 / 64000000000000 : ℝ) ≤
+      (-(6936505099560823511611 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19979876938793223 / 80000000000000000 : ℝ) ≤
+      (-(6936505099560823511611 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (19979876938793223 / 80000000000000000 : ℝ) =
+            (19979876938793223 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 15
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19979876938793223 / 10000000000000000 : ℝ) ≤
+            (3460702604939176488389 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (618228861498053 / 2500000000000000 : ℝ)) (q :=
+      (19979876938793223 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_332_7 :
+    Uω (aρ 7) (bρ 7) (19337219847623 / 64000000000000 : ℝ) ≤
+      (-(14958793583244939740179 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17924119708444467 / 80000000000000000 : ℝ) ≤
+      (-(14958793583244939740179 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (17924119708444467 / 80000000000000000 : ℝ) =
+            (17924119708444467 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17924119708444467 / 10000000000000000 : ℝ) ≤
+            (5835621825755060259821 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2182637519165023 / 10000000000000000 : ℝ)) (q :=
+      (17924119708444467 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_332_8 :
+    Uω (aρ 8) (bρ 8) (19337219847623 / 64000000000000 : ℝ) ≤
+      (-(8354616257164736756719 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15045867705964499 / 80000000000000000 : ℝ) ≤
+      (-(8354616257164736756719 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (15045867705964499 / 80000000000000000 : ℝ) =
+            (15045867705964499 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15045867705964499 / 10000000000000000 : ℝ) ≤
+            (2042591447335263243281 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1746616169885031 / 10000000000000000 : ℝ)) (q :=
+      (15045867705964499 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_332_9 :
+    Uω (aρ 9) (bρ 9) (19337219847623 / 64000000000000 : ℝ) ≤
+      (-(5034986812783164747793 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10676359544252443 / 80000000000000000 : ℝ) ≤
+      (-(5034986812783164747793 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (10676359544252443 / 80000000000000000 : ℝ) =
+            (10676359544252443 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10676359544252443 / 10000000000000000 : ℝ) ≤
+            (163617039466835252207 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (995204757817017 / 10000000000000000 : ℝ)) (q :=
+      (10676359544252443 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_332_10 :
+    Uω (aρ 10) (bρ 10) (19337219847623 / 64000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_332_11 :
+    Uω (aρ 11) (bρ 11) (19337219847623 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_332_12 :
+    Uω (aρ 12) (bρ 12) (19337219847623 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_332_13 :
+    Uω (aρ 13) (bρ 13) (19337219847623 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_332_14 :
+    Uω (aρ 14) (bρ 14) (19337219847623 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_332_15 :
+    Uω (aρ 15) (bρ 15) (19337219847623 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_332_16 :
+    Uω (aρ 16) (bρ 16) (19337219847623 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_332 :
+    Uρ (19337219847623 / 64000000000000 : ℝ) ≤
+      (-(16123857921615760857801 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (19337219847623 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(6092237445347728784389 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_332_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (19337219847623 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(12266231890298508999037 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_332_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (19337219847623 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(6216174735296002445103 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_332_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (19337219847623 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(12716337939900776348663 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_332_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (19337219847623 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(3292521759420410891227 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_332_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (19337219847623 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(6936505099560823511611 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_332_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (19337219847623 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(14958793583244939740179 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_332_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (19337219847623 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(8354616257164736756719 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_332_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (19337219847623 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(5034986812783164747793 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_332_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (19337219847623 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_332_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (19337219847623 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_332_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (19337219847623 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_332_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (19337219847623 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_332_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (19337219847623 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_332_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (19337219847623 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_332_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (19337219847623 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_332_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_333_1 :
+    Uω (aρ 1) (bρ 1) (4853740851219 / 16000000000000 : ℝ) ≤
+      (-(2428695298668879795403 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2969036362628849 / 10000000000000000 : ℝ) ≤
+      (-(2428695298668879795403 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2969036362628849 / 10000000000000000 : ℝ) =
+            (2969036362628849 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2969036362628849 / 2500000000000000 : ℝ) ≤
+            (343893422531120204597 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2968981911330823 / 10000000000000000 : ℝ)) (q :=
+      (2969036362628849 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_333_2 :
+    Uω (aρ 2) (bρ 2) (4853740851219 / 16000000000000 : ℝ) ≤
+      (-(6112446657647581372819 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (235596923989041 / 800000000000000 : ℝ) ≤
+      (-(6112446657647581372819 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (235596923989041 / 800000000000000 : ℝ) =
+            (235596923989041 / 200000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (235596923989041 / 200000000000000 : ℝ) ≤
+            (819025145352418627181 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (58892025966183 / 200000000000000 : ℝ)) (q :=
+      (235596923989041 / 800000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_333_3 :
+    Uω (aρ 3) (bρ 3) (4853740851219 / 16000000000000 : ℝ) ≤
+      (-(12390306484797001658741 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5793297376456881 / 20000000000000000 : ℝ) ≤
+      (-(12390306484797001658741 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5793297376456881 / 20000000000000000 : ℝ) =
+            (5793297376456881 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5793297376456881 / 5000000000000000 : ℝ) ≤
+            (1472637121202998341259 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1447685840695003 / 5000000000000000 : ℝ)) (q :=
+      (5793297376456881 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_333_4 :
+    Uω (aρ 4) (bρ 4) (4853740851219 / 16000000000000 : ℝ) ≤
+      (-(6336524013436370123083 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (351986923967741 / 1250000000000000 : ℝ) ≤
+      (-(6336524013436370123083 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (351986923967741 / 1250000000000000 : ℝ) =
+            (351986923967741 / 312500000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (351986923967741 / 312500000000000 : ℝ) ≤
+            (594947789563629876917 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2812159269481981 / 10000000000000000 : ℝ)) (q :=
+      (351986923967741 / 1250000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_333_5 :
+    Uω (aρ 5) (bρ 5) (4853740851219 / 16000000000000 : ℝ) ≤
+      (-(13124685103260900577841 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5383096481069049 / 20000000000000000 : ℝ) ≤
+      (-(13124685103260900577841 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5383096481069049 / 20000000000000000 : ℝ) =
+            (5383096481069049 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5383096481069049 / 5000000000000000 : ℝ) ≤
+            (738258502739099422159 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1340827443201087 / 5000000000000000 : ℝ)) (q :=
+      (5383096481069049 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_333_6 :
+    Uω (aρ 6) (bρ 6) (4853740851219 / 16000000000000 : ℝ) ≤
+      (-(1728001366084479726809 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5019504281721021 / 20000000000000000 : ℝ) ≤
+      (-(1728001366084479726809 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5019504281721021 / 20000000000000000 : ℝ) =
+            (5019504281721021 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5019504281721021 / 5000000000000000 : ℝ) ≤
+            (4866584665520273191 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1242651531097073 / 5000000000000000 : ℝ)) (q :=
+      (5019504281721021 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_333_7 :
+    Uω (aρ 7) (bρ 7) (4853740851219 / 16000000000000 : ℝ) ≤
+      (-(7451650589987120200889 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (450596536145979 / 2000000000000000 : ℝ) ≤
+      (-(7451650589987120200889 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (450596536145979 / 2000000000000000 : ℝ) =
+            (450596536145979 / 250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (450596536145979 / 250000000000000 : ℝ) ≤
+            (2945557114512879799111 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (439085104538583 / 2000000000000000 : ℝ)) (q :=
+      (450596536145979 / 2000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_333_8 :
+    Uω (aρ 8) (bρ 8) (4853740851219 / 16000000000000 : ℝ) ≤
+      (-(16639961469023081417581 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1893806728199749 / 10000000000000000 : ℝ) ≤
+      (-(16639961469023081417581 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1893806728199749 / 10000000000000000 : ℝ) =
+            (1893806728199749 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1893806728199749 / 1250000000000000 : ℝ) ≤
+            (4154453939976918582419 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1760615268972623 / 10000000000000000 : ℝ)) (q :=
+      (1893806728199749 / 10000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_333_9 :
+    Uω (aρ 9) (bρ 9) (4853740851219 / 16000000000000 : ℝ) ≤
+      (-(2502389798007060102871 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (168845988894201 / 1250000000000000 : ℝ) ≤
+      (-(2502389798007060102871 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (168845988894201 / 1250000000000000 : ℝ) =
+            (168845988894201 / 156250000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (168845988894201 / 156250000000000 : ℝ) ≤
+            (96912128117939897129 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1015503263240341 / 10000000000000000 : ℝ)) (q :=
+      (168845988894201 / 1250000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_333_10 :
+    Uω (aρ 10) (bρ 10) (4853740851219 / 16000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_333_11 :
+    Uω (aρ 11) (bρ 11) (4853740851219 / 16000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_333_12 :
+    Uω (aρ 12) (bρ 12) (4853740851219 / 16000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_333_13 :
+    Uω (aρ 13) (bρ 13) (4853740851219 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_333_14 :
+    Uω (aρ 14) (bρ 14) (4853740851219 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_333_15 :
+    Uω (aρ 15) (bρ 15) (4853740851219 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_333_16 :
+    Uω (aρ 16) (bρ 16) (4853740851219 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_333 :
+    Uρ (4853740851219 / 16000000000000 : ℝ) ≤
+      (-(2011172851470674920769 / 1250000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (4853740851219 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2428695298668879795403 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_333_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (4853740851219 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(6112446657647581372819 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_333_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (4853740851219 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(12390306484797001658741 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_333_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (4853740851219 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(6336524013436370123083 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_333_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (4853740851219 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(13124685103260900577841 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_333_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (4853740851219 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1728001366084479726809 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_333_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (4853740851219 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(7451650589987120200889 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_333_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (4853740851219 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(16639961469023081417581 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_333_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (4853740851219 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(2502389798007060102871 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_333_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (4853740851219 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_333_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (4853740851219 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_333_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (4853740851219 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_333_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (4853740851219 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_333_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (4853740851219 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_333_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (4853740851219 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_333_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (4853740851219 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_333_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_334_1 :
+    Uω (aρ 1) (bρ 1) (19492706962129 / 64000000000000 : ℝ) ≤
+      (-(12102645502884357136123 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23849472122611141 / 80000000000000000 : ℝ) ≤
+      (-(12102645502884357136123 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (23849472122611141 / 80000000000000000 : ℝ) =
+            (23849472122611141 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23849472122611141 / 20000000000000000 : ℝ) ≤
+            (1760298103115642863877 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2981129785905129 / 10000000000000000 : ℝ)) (q :=
+      (23849472122611141 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_334_2 :
+    Uω (aρ 2) (bρ 2) (19492706962129 / 64000000000000 : ℝ) ≤
+      (-(6091862483132022743739 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23656883685850549 / 80000000000000000 : ℝ) ≤
+      (-(6091862483132022743739 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (23656883685850549 / 80000000000000000 : ℝ) =
+            (23656883685850549 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23656883685850549 / 20000000000000000 : ℝ) ≤
+            (839609319867977256261 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2956751689224981 / 10000000000000000 : ℝ)) (q :=
+      (23656883685850549 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_334_3 :
+    Uω (aρ 3) (bρ 3) (19492706962129 / 64000000000000 : ℝ) ≤
+      (-(12348439675099821250339 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23270411634345401 / 80000000000000000 : ℝ) ≤
+      (-(12348439675099821250339 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (23270411634345401 / 80000000000000000 : ℝ) =
+            (23270411634345401 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23270411634345401 / 20000000000000000 : ℝ) ≤
+            (1514503930900178749661 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1453764891349347 / 5000000000000000 : ℝ)) (q :=
+      (23270411634345401 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_334_4 :
+    Uω (aρ 4) (bρ 4) (19492706962129 / 64000000000000 : ℝ) ≤
+      (-(6314972600848446314769 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22624471133193121 / 80000000000000000 : ℝ) ≤
+      (-(6314972600848446314769 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (22624471133193121 / 80000000000000000 : ℝ) =
+            (22624471133193121 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22624471133193121 / 20000000000000000 : ℝ) ≤
+            (616499202151553685231 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (353042354809453 / 1250000000000000 : ℝ)) (q :=
+      (22624471133193121 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_334_5 :
+    Uω (aρ 5) (bρ 5) (19492706962129 / 64000000000000 : ℝ) ≤
+      (-(13079489878329665778293 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21629922269621181 / 80000000000000000 : ℝ) ≤
+      (-(13079489878329665778293 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (21629922269621181 / 80000000000000000 : ℝ) =
+            (21629922269621181 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21629922269621181 / 20000000000000000 : ℝ) ≤
+            (783453727670334221707 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2693891541917639 / 10000000000000000 : ℝ)) (q :=
+      (21629922269621181 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_334_6 :
+    Uω (aρ 6) (bρ 6) (19492706962129 / 64000000000000 : ℝ) ≤
+      (-(13775255255809352500289 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20176147877517449 / 80000000000000000 : ℝ) ≤
+      (-(13775255255809352500289 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (20176147877517449 / 80000000000000000 : ℝ) =
+            (20176147877517449 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20176147877517449 / 20000000000000000 : ℝ) ≤
+            (87688350190647499711 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1248844159515853 / 5000000000000000 : ℝ)) (q :=
+      (20176147877517449 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_334_7 :
+    Uω (aρ 7) (bρ 7) (19492706962129 / 64000000000000 : ℝ) ≤
+      (-(7424065491729029942457 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3624714849893033 / 16000000000000000 : ℝ) ≤
+      (-(7424065491729029942457 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3624714849893033 / 16000000000000000 : ℝ) =
+            (3624714849893033 / 2000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3624714849893033 / 2000000000000000 : ℝ) ≤
+            (2973142212770970057543 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (441641258555727 / 2000000000000000 : ℝ)) (q :=
+      (3624714849893033 / 16000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_334_8 :
+    Uω (aρ 8) (bρ 8) (19492706962129 / 64000000000000 : ℝ) ≤
+      (-(16571238508300731484711 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15254930817135281 / 80000000000000000 : ℝ) ≤
+      (-(16571238508300731484711 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (15254930817135281 / 80000000000000000 : ℝ) =
+            (15254930817135281 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15254930817135281 / 10000000000000000 : ℝ) ≤
+            (4223176900699268515289 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (443646771509041 / 2500000000000000 : ℝ)) (q :=
+      (15254930817135281 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_334_9 :
+    Uω (aρ 9) (bρ 9) (19492706962129 / 64000000000000 : ℝ) ≤
+      (-(9950333064951701404441 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10934905596091849 / 80000000000000000 : ℝ) ≤
+      (-(9950333064951701404441 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (10934905596091849 / 80000000000000000 : ℝ) =
+            (10934905596091849 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10934905596091849 / 10000000000000000 : ℝ) ≤
+            (446874639548298595559 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (517773204567653 / 5000000000000000 : ℝ)) (q :=
+      (10934905596091849 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_334_10 :
+    Uω (aρ 10) (bρ 10) (19492706962129 / 64000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_334_11 :
+    Uω (aρ 11) (bρ 11) (19492706962129 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_334_12 :
+    Uω (aρ 12) (bρ 12) (19492706962129 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_334_13 :
+    Uω (aρ 13) (bρ 13) (19492706962129 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_334_14 :
+    Uω (aρ 14) (bρ 14) (19492706962129 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_334_15 :
+    Uω (aρ 15) (bρ 15) (19492706962129 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_334_16 :
+    Uω (aρ 16) (bρ 16) (19492706962129 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_334 :
+    Uρ (19492706962129 / 64000000000000 : ℝ) ≤
+      (-(8027626034599468880467 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (19492706962129 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(12102645502884357136123 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_334_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (19492706962129 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(6091862483132022743739 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_334_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (19492706962129 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(12348439675099821250339 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_334_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (19492706962129 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(6314972600848446314769 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_334_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (19492706962129 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(13079489878329665778293 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_334_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (19492706962129 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(13775255255809352500289 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_334_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (19492706962129 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(7424065491729029942457 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_334_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (19492706962129 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(16571238508300731484711 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_334_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (19492706962129 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(9950333064951701404441 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_334_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (19492706962129 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_334_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (19492706962129 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_334_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (19492706962129 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_334_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (19492706962129 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_334_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (19492706962129 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_334_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (19492706962129 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_334_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (19492706962129 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_334_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_335_1 :
+    Uω (aρ 1) (bρ 1) (9785225259691 / 32000000000000 : ℝ) ≤
+      (-(12061980557693874766859 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2394665332978437 / 8000000000000000 : ℝ) ≤
+      (-(12061980557693874766859 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2394665332978437 / 8000000000000000 : ℝ) =
+            (2394665332978437 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2394665332978437 / 2000000000000000 : ℝ) ≤
+            (1800963048306125233141 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (598655531375531 / 2000000000000000 : ℝ)) (q :=
+      (2394665332978437 / 8000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_335_2 :
+    Uω (aρ 2) (bρ 2) (9785225259691 / 32000000000000 : ℝ) ≤
+      (-(1517840680835718813187 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11877037437947009 / 40000000000000000 : ℝ) ≤
+      (-(1517840680835718813187 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (11877037437947009 / 40000000000000000 : ℝ) =
+            (11877037437947009 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11877037437947009 / 10000000000000000 : ℝ) ≤
+            (215027269914281186813 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2968902055915067 / 10000000000000000 : ℝ)) (q :=
+      (11877037437947009 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_335_3 :
+    Uω (aρ 3) (bρ 3) (9785225259691 / 32000000000000 : ℝ) ≤
+      (-(6153373784945376044151 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11683816703774369 / 40000000000000000 : ℝ) ≤
+      (-(6153373784945376044151 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (11683816703774369 / 40000000000000000 : ℝ) =
+            (11683816703774369 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11683816703774369 / 10000000000000000 : ℝ) ≤
+            (778098018054623955849 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2919687795178747 / 10000000000000000 : ℝ)) (q :=
+      (11683816703774369 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_335_4 :
+    Uω (aρ 4) (bρ 4) (9785225259691 / 32000000000000 : ℝ) ≤
+      (-(12587027850031739223471 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (90887112119747 / 320000000000000 : ℝ) ≤
+      (-(12587027850031739223471 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (90887112119747 / 320000000000000 : ℝ) = (90887112119747 / 80000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (90887112119747 / 80000000000000 : ℝ) ≤
+            (1275915755968260776529 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (11346072527363 / 40000000000000 : ℝ)) (q :=
+      (90887112119747 / 320000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_335_5 :
+    Uω (aρ 5) (bρ 5) (9785225259691 / 32000000000000 : ℝ) ≤
+      (-(814656217230021947263 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10863727699530449 / 40000000000000000 : ℝ) ≤
+      (-(814656217230021947263 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (10863727699530449 / 40000000000000000 : ℝ) =
+            (10863727699530449 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10863727699530449 / 10000000000000000 : ℝ) ≤
+            (51777758144978052737 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2706127393456787 / 10000000000000000 : ℝ)) (q :=
+      (10863727699530449 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_335_6 :
+    Uω (aρ 6) (bρ 6) (9785225259691 / 32000000000000 : ℝ) ≤
+      (-(1372674072760321762727 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10137134665196471 / 40000000000000000 : ℝ) ≤
+      (-(1372674072760321762727 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (10137134665196471 / 40000000000000000 : ℝ) =
+            (10137134665196471 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10137134665196471 / 10000000000000000 : ℝ) ≤
+            (13620287839678237273 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1255035625714899 / 5000000000000000 : ℝ)) (q :=
+      (10137134665196471 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_335_7 :
+    Uω (aρ 7) (bρ 7) (9785225259691 / 32000000000000 : ℝ) ≤
+      (-(7396639548000137332547 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9111629309413061 / 40000000000000000 : ℝ) ≤
+      (-(7396639548000137332547 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (9111629309413061 / 40000000000000000 : ℝ) =
+            (9111629309413061 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9111629309413061 / 5000000000000000 : ℝ) ≤
+            (3000568156499862667453 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2220979954298093 / 10000000000000000 : ℝ)) (q :=
+      (9111629309413061 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_335_8 :
+    Uω (aρ 8) (bρ 8) (9785225259691 / 32000000000000 : ℝ) ≤
+      (-(1031440874701677454237 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7679650619033413 / 40000000000000000 : ℝ) ≤
+      (-(1031440874701677454237 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (7679650619033413 / 40000000000000000 : ℝ) =
+            (7679650619033413 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7679650619033413 / 5000000000000000 : ℝ) ≤
+            (268210088360822545763 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1788532260448269 / 10000000000000000 : ℝ)) (q :=
+      (7679650619033413 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_335_9 :
+    Uω (aρ 9) (bρ 9) (9785225259691 / 32000000000000 : ℝ) ≤
+      (-(19784471231578686300367 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5531352331295859 / 40000000000000000 : ℝ) ≤
+      (-(19784471231578686300367 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (5531352331295859 / 40000000000000000 : ℝ) =
+            (5531352331295859 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5531352331295859 / 5000000000000000 : ℝ) ≤
+            (1009944177421313699633 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (263837186234873 / 2500000000000000 : ℝ)) (q :=
+      (5531352331295859 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_335_10 :
+    Uω (aρ 10) (bρ 10) (9785225259691 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_335_11 :
+    Uω (aρ 11) (bρ 11) (9785225259691 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_335_12 :
+    Uω (aρ 12) (bρ 12) (9785225259691 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_335_13 :
+    Uω (aρ 13) (bρ 13) (9785225259691 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_335_14 :
+    Uω (aρ 14) (bρ 14) (9785225259691 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_335_15 :
+    Uω (aρ 15) (bρ 15) (9785225259691 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_335_16 :
+    Uω (aρ 16) (bρ 16) (9785225259691 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_335 :
+    Uρ (9785225259691 / 32000000000000 : ℝ) ≤
+      (-(8010726722061475591077 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (9785225259691 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(12061980557693874766859 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_335_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (9785225259691 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1517840680835718813187 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_335_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (9785225259691 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(6153373784945376044151 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_335_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (9785225259691 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(12587027850031739223471 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_335_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (9785225259691 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(814656217230021947263 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_335_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (9785225259691 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1372674072760321762727 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_335_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (9785225259691 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(7396639548000137332547 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_335_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (9785225259691 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(1031440874701677454237 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_335_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (9785225259691 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(19784471231578686300367 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_335_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (9785225259691 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_335_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (9785225259691 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_335_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (9785225259691 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_335_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (9785225259691 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_335_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (9785225259691 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_335_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (9785225259691 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_335_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (9785225259691 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_335_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_336_1 :
+    Uω (aρ 1) (bρ 1) (3929638815327 / 12800000000000 : ℝ) ≤
+      (-(12021480312697159223583 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24043834522725183 / 80000000000000000 : ℝ) ≤
+      (-(12021480312697159223583 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (24043834522725183 / 80000000000000000 : ℝ) =
+            (24043834522725183 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24043834522725183 / 20000000000000000 : ℝ) ≤
+            (1841463293302840776417 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3005425524292077 / 10000000000000000 : ℝ)) (q :=
+      (24043834522725183 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_336_2 :
+    Uω (aρ 2) (bρ 2) (3929638815327 / 12800000000000 : ℝ) ≤
+      (-(3025473344290043015187 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23851265970219407 / 80000000000000000 : ℝ) ≤
+      (-(3025473344290043015187 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (23851265970219407 / 80000000000000000 : ℝ) =
+            (23851265970219407 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23851265970219407 / 20000000000000000 : ℝ) ≤
+            (440262557209956984813 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2981052398675633 / 10000000000000000 : ℝ)) (q :=
+      (23851265970219407 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_336_3 :
+    Uω (aρ 3) (bρ 3) (3929638815327 / 12800000000000 : ℝ) ≤
+      (-(3066307178984472565569 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23464854829857871 / 80000000000000000 : ℝ) ≤
+      (-(3066307178984472565569 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (23464854829857871 / 80000000000000000 : ℝ) =
+            (23464854829857871 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23464854829857871 / 20000000000000000 : ℝ) ≤
+            (399428722515527434431 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2931845719935249 / 10000000000000000 : ℝ)) (q :=
+      (23464854829857871 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_336_4 :
+    Uω (aρ 4) (bρ 4) (3929638815327 / 12800000000000 : ℝ) ≤
+      (-(12544294378394242768849 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22819083838307347 / 80000000000000000 : ℝ) ≤
+      (-(12544294378394242768849 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (22819083838307347 / 80000000000000000 : ℝ) =
+            (22819083838307347 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22819083838307347 / 20000000000000000 : ℝ) ≤
+            (1318649227605757231151 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1424348576556309 / 5000000000000000 : ℝ)) (q :=
+      (22819083838307347 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_336_5 :
+    Uω (aρ 5) (bρ 5) (3929638815327 / 12800000000000 : ℝ) ≤
+      (-(2597942406802438394041 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21824985356021499 / 80000000000000000 : ℝ) ≤
+      (-(2597942406802438394041 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (21824985356021499 / 80000000000000000 : ℝ) =
+            (21824985356021499 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21824985356021499 / 20000000000000000 : ℝ) ≤
+            (174646314397561605959 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (679590612969039 / 2500000000000000 : ℝ)) (q :=
+      (21824985356021499 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_336_6 :
+    Uω (aρ 6) (bρ 6) (3929638815327 / 12800000000000 : ℝ) ≤
+      (-(6839232463065716702293 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20372381622441031 / 80000000000000000 : ℝ) ≤
+      (-(6839232463065716702293 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (20372381622441031 / 80000000000000000 : ℝ) =
+            (20372381622441031 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20372381622441031 / 20000000000000000 : ℝ) ≤
+            (92239339934283297707 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2522451893621039 / 10000000000000000 : ℝ)) (q :=
+      (20372381622441031 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_336_7 :
+    Uω (aρ 7) (bρ 7) (3929638815327 / 12800000000000 : ℝ) ≤
+      (-(7369370846539389318947 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (18322915041725367 / 80000000000000000 : ℝ) ≤
+      (-(7369370846539389318947 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (18322915041725367 / 80000000000000000 : ℝ) =
+            (18322915041725367 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18322915041725367 / 10000000000000000 : ℝ) ≤
+            (3027836857960610681053 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2233746629202123 / 10000000000000000 : ℝ)) (q :=
+      (18322915041725367 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_336_8 :
+    Uω (aρ 8) (bρ 8) (3929638815327 / 12800000000000 : ℝ) ≤
+      (-(16435398565581580107231 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15463567561933823 / 80000000000000000 : ℝ) ≤
+      (-(16435398565581580107231 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (15463567561933823 / 80000000000000000 : ℝ) =
+            (15463567561933823 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15463567561933823 / 10000000000000000 : ℝ) ≤
+            (4359016843418419892769 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1802451410594237 / 10000000000000000 : ℝ)) (q :=
+      (15463567561933823 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_336_9 :
+    Uω (aρ 9) (bρ 9) (3929638815327 / 12800000000000 : ℝ) ≤
+      (-(19670424193114000292823 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2237918744750327 / 16000000000000000 : ℝ) ≤
+      (-(19670424193114000292823 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2237918744750327 / 16000000000000000 : ℝ) =
+            (2237918744750327 / 2000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2237918744750327 / 2000000000000000 : ℝ) ≤
+            (1123991215885999707177 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (107492357940869 / 1000000000000000 : ℝ)) (q :=
+      (2237918744750327 / 16000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_336_10 :
+    Uω (aρ 10) (bρ 10) (3929638815327 / 12800000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_336_11 :
+    Uω (aρ 11) (bρ 11) (3929638815327 / 12800000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_336_12 :
+    Uω (aρ 12) (bρ 12) (3929638815327 / 12800000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_336_13 :
+    Uω (aρ 13) (bρ 13) (3929638815327 / 12800000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_336_14 :
+    Uω (aρ 14) (bρ 14) (3929638815327 / 12800000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_336_15 :
+    Uω (aρ 15) (bρ 15) (3929638815327 / 12800000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_336_16 :
+    Uω (aρ 16) (bρ 16) (3929638815327 / 12800000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_336 :
+    Uρ (3929638815327 / 12800000000000 : ℝ) ≤
+      (-(15987975586230920657997 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (3929638815327 / 12800000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(12021480312697159223583 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_336_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (3929638815327 / 12800000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3025473344290043015187 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_336_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (3929638815327 / 12800000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3066307178984472565569 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_336_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (3929638815327 / 12800000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(12544294378394242768849 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_336_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (3929638815327 / 12800000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(2597942406802438394041 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_336_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (3929638815327 / 12800000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(6839232463065716702293 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_336_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (3929638815327 / 12800000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(7369370846539389318947 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_336_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (3929638815327 / 12800000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(16435398565581580107231 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_336_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (3929638815327 / 12800000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(19670424193114000292823 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_336_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (3929638815327 / 12800000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_336_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (3929638815327 / 12800000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_336_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (3929638815327 / 12800000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_336_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (3929638815327 / 12800000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_336_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (3929638815327 / 12800000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_336_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (3929638815327 / 12800000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_336_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (3929638815327 / 12800000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_336_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_337_1 :
+    Uω (aρ 1) (bρ 1) (616435551059 / 2000000000000 : ℝ) ≤
+      (-(11981143439097091459783 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1508813481350341 / 5000000000000000 : ℝ) ≤
+      (-(11981143439097091459783 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1508813481350341 / 5000000000000000 : ℝ) =
+            (1508813481350341 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1508813481350341 / 1250000000000000 : ℝ) ≤
+            (1881800166902908540217 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (754393347047841 / 2500000000000000 : ℝ)) (q :=
+      (1508813481350341 / 5000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_337_2 :
+    Uω (aρ 2) (bρ 2) (616435551059 / 2000000000000 : ℝ) ≤
+      (-(12061227395127781664123 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (748389280312261 / 2500000000000000 : ℝ) ≤
+      (-(12061227395127781664123 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (748389280312261 / 2500000000000000 : ℝ) =
+            (748389280312261 / 625000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (748389280312261 / 625000000000000 : ℝ) ≤
+            (1801716210872218335877 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (374150339724761 / 1250000000000000 : ℝ)) (q :=
+      (748389280312261 / 2500000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_337_3 :
+    Uω (aρ 3) (bρ 3) (616435551059 / 2000000000000 : ℝ) ≤
+      (-(12223881678082135699073 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5890518976405023 / 20000000000000000 : ℝ) ≤
+      (-(12223881678082135699073 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5890518976405023 / 20000000000000000 : ℝ) =
+            (5890518976405023 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5890518976405023 / 5000000000000000 : ℝ) ≤
+            (1639061927917864300927 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2944003558055023 / 10000000000000000 : ℝ)) (q :=
+      (5890518976405023 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_337_4 :
+    Uω (aρ 4) (bρ 4) (616435551059 / 2000000000000 : ℝ) ≤
+      (-(3125435803450280365889 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5729097143051207 / 20000000000000000 : ℝ) ≤
+      (-(3125435803450280365889 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5729097143051207 / 20000000000000000 : ℝ) =
+            (5729097143051207 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5729097143051207 / 5000000000000000 : ℝ) ≤
+            (340300098049719634111 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2860875905766207 / 10000000000000000 : ℝ)) (q :=
+      (5729097143051207 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_337_5 :
+    Uω (aρ 5) (bρ 5) (616435551059 / 2000000000000 : ℝ) ≤
+      (-(2589025143491211693361 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1096125609157401 / 4000000000000000 : ℝ) ≤
+      (-(2589025143491211693361 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1096125609157401 / 4000000000000000 : ℝ) =
+            (1096125609157401 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1096125609157401 / 1000000000000000 : ℝ) ≤
+            (183563577708788306639 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (546119345567401 / 2000000000000000 : ℝ)) (q :=
+      (1096125609157401 / 4000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_337_6 :
+    Uω (aρ 6) (bρ 6) (616435551059 / 2000000000000 : ℝ) ≤
+      (-(13630425470284692729279 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (255881061098649 / 1000000000000000 : ℝ) ≤
+      (-(13630425470284692729279 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (255881061098649 / 1000000000000000 : ℝ) =
+            (255881061098649 / 250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (255881061098649 / 250000000000000 : ℝ) ≤
+            (232518135715307270721 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (126741513958149 / 500000000000000 : ℝ)) (q :=
+      (255881061098649 / 1000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_337_7 :
+    Uω (aρ 7) (bρ 7) (616435551059 / 2000000000000 : ℝ) ≤
+      (-(14684515021464761878583 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4605635998653471 / 20000000000000000 : ℝ) ≤
+      (-(14684515021464761878583 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4605635998653471 / 20000000000000000 : ℝ) =
+            (4605635998653471 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4605635998653471 / 2500000000000000 : ℝ) ≤
+            (6109900387535238121417 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2246506436603471 / 10000000000000000 : ℝ)) (q :=
+      (4605635998653471 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_337_8 :
+    Uω (aρ 8) (bρ 8) (616435551059 / 2000000000000 : ℝ) ≤
+      (-(4092065779282799140011 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3891933045477401 / 20000000000000000 : ℝ) ≤
+      (-(4092065779282799140011 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3891933045477401 / 20000000000000000 : ℝ) =
+            (3891933045477401 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3891933045477401 / 2500000000000000 : ℝ) ≤
+            (1106538072967200859989 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1816345134767401 / 10000000000000000 : ℝ)) (q :=
+      (3891933045477401 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_337_9 :
+    Uω (aρ 9) (bρ 9) (616435551059 / 2000000000000 : ℝ) ≤
+      (-(9779212095280143712081 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2828905403778503 / 20000000000000000 : ℝ) ≤
+      (-(9779212095280143712081 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2828905403778503 / 20000000000000000 : ℝ) =
+            (2828905403778503 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2828905403778503 / 2500000000000000 : ℝ) ≤
+            (617995609219856287919 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1094283121428503 / 10000000000000000 : ℝ)) (q :=
+      (2828905403778503 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_337_10 :
+    Uω (aρ 10) (bρ 10) (616435551059 / 2000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_337_11 :
+    Uω (aρ 11) (bρ 11) (616435551059 / 2000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_337_12 :
+    Uω (aρ 12) (bρ 12) (616435551059 / 2000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_337_13 :
+    Uω (aρ 13) (bρ 13) (616435551059 / 2000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_337_14 :
+    Uω (aρ 14) (bρ 14) (616435551059 / 2000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_337_15 :
+    Uω (aρ 15) (bρ 15) (616435551059 / 2000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_337_16 :
+    Uω (aρ 16) (bρ 16) (616435551059 / 2000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_337 :
+    Uρ (616435551059 / 2000000000000 : ℝ) ≤
+      (-(7977403973744688773817 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (616435551059 / 2000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(11981143439097091459783 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_337_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (616435551059 / 2000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(12061227395127781664123 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_337_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (616435551059 / 2000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(12223881678082135699073 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_337_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (616435551059 / 2000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3125435803450280365889 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_337_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (616435551059 / 2000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(2589025143491211693361 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_337_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (616435551059 / 2000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(13630425470284692729279 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_337_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (616435551059 / 2000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(14684515021464761878583 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_337_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (616435551059 / 2000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(4092065779282799140011 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_337_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (616435551059 / 2000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(9779212095280143712081 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_337_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (616435551059 / 2000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_337_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (616435551059 / 2000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_337_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (616435551059 / 2000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_337_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (616435551059 / 2000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_337_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (616435551059 / 2000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_337_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (616435551059 / 2000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_337_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (616435551059 / 2000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_337_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_338_1 :
+    Uω (aρ 1) (bρ 1) (19803681191141 / 64000000000000 : ℝ) ≤
+      (-(46644408687943654039 / 39062500000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24238196866594329 / 80000000000000000 : ℝ) ≤
+      (-(46644408687943654039 / 39062500000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (24238196866594329 / 80000000000000000 : ℝ) =
+            (24238196866594329 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24238196866594329 / 20000000000000000 : ℝ) ≤
+            (7507714772993845961 / 39062500000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3029721248617801 / 10000000000000000 : ℝ)) (q :=
+      (24238196866594329 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_338_2 :
+    Uω (aρ 2) (bρ 2) (19803681191141 / 64000000000000 : ℝ) ≤
+      (-(12020726154596521413683 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24045647876359661 / 80000000000000000 : ℝ) ≤
+      (-(12020726154596521413683 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (24045647876359661 / 80000000000000000 : ℝ) =
+            (24045647876359661 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24045647876359661 / 20000000000000000 : ℝ) ≤
+            (1842217451403478586317 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1502676506784567 / 5000000000000000 : ℝ)) (q :=
+      (24045647876359661 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_338_3 :
+    Uω (aρ 3) (bρ 3) (19803681191141 / 64000000000000 : ℝ) ≤
+      (-(2436541007787700471499 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4731859327822229 / 16000000000000000 : ℝ) ≤
+      (-(2436541007787700471499 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4731859327822229 / 16000000000000000 : ℝ) =
+            (4731859327822229 / 4000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4731859327822229 / 4000000000000000 : ℝ) ≤
+            (336047713412299528501 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (591232262121401 / 2000000000000000 : ℝ)) (q :=
+      (4731859327822229 / 16000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_338_4 :
+    Uω (aρ 4) (bρ 4) (19803681191141 / 64000000000000 : ℝ) ≤
+      (-(6229686401708916302631 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23013692245293137 / 80000000000000000 : ℝ) ≤
+      (-(6229686401708916302631 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (23013692245293137 / 80000000000000000 : ℝ) =
+            (23013692245293137 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23013692245293137 / 20000000000000000 : ℝ) ≤
+            (701785401291083697369 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2873054393217503 / 10000000000000000 : ℝ)) (q :=
+      (23013692245293137 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_338_5 :
+    Uω (aρ 5) (bρ 5) (19803681191141 / 64000000000000 : ℝ) ≤
+      (-(12900738715110178474141 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22020035922321933 / 80000000000000000 : ℝ) ≤
+      (-(12900738715110178474141 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (22020035922321933 / 80000000000000000 : ℝ) =
+            (22020035922321933 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22020035922321933 / 20000000000000000 : ℝ) ≤
+            (962204890889821525859 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1371415115904851 / 5000000000000000 : ℝ)) (q :=
+      (22020035922321933 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_338_6 :
+    Uω (aρ 6) (bρ 6) (19803681191141 / 64000000000000 : ℝ) ≤
+      (-(2716524003002969505043 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20568579258342361 / 80000000000000000 : ℝ) ≤
+      (-(2716524003002969505043 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (20568579258342361 / 80000000000000000 : ℝ) =
+            (20568579258342361 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20568579258342361 / 20000000000000000 : ℝ) ≤
+            (56064718197030494957 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2547206440954809 / 10000000000000000 : ℝ)) (q :=
+      (20568579258342361 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_338_7 :
+    Uω (aρ 7) (bρ 7) (19803681191141 / 64000000000000 : ℝ) ≤
+      (-(14630595397403589959339 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (18522145942925669 / 80000000000000000 : ℝ) ≤
+      (-(14630595397403589959339 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (18522145942925669 / 80000000000000000 : ℝ) =
+            (18522145942925669 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18522145942925669 / 10000000000000000 : ℝ) ≤
+            (6163820011596410040661 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (564814873215159 / 2500000000000000 : ℝ)) (q :=
+      (18522145942925669 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_338_8 :
+    Uω (aρ 8) (bρ 8) (19803681191141 / 64000000000000 : ℝ) ≤
+      (-(16301638799438157530459 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15671797414192209 / 80000000000000000 : ℝ) ≤
+      (-(16301638799438157530459 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (15671797414192209 / 80000000000000000 : ℝ) =
+            (15671797414192209 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15671797414192209 / 10000000000000000 : ℝ) ≤
+            (4492776609561842469541 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1830214012017271 / 10000000000000000 : ℝ)) (q :=
+      (15671797414192209 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_338_9 :
+    Uω (aρ 9) (bρ 9) (19803681191141 / 64000000000000 : ℝ) ≤
+      (-(9724189067304602339073 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11440833256812601 / 80000000000000000 : ℝ) ≤
+      (-(9724189067304602339073 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (11440833256812601 / 80000000000000000 : ℝ) =
+            (11440833256812601 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11440833256812601 / 10000000000000000 : ℝ) ≤
+            (673018637195397660927 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1113438601032369 / 10000000000000000 : ℝ)) (q :=
+      (11440833256812601 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_338_10 :
+    Uω (aρ 10) (bρ 10) (19803681191141 / 64000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_338_11 :
+    Uω (aρ 11) (bρ 11) (19803681191141 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_338_12 :
+    Uω (aρ 12) (bρ 12) (19803681191141 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_338_13 :
+    Uω (aρ 13) (bρ 13) (19803681191141 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_338_14 :
+    Uω (aρ 14) (bρ 14) (19803681191141 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_338_15 :
+    Uω (aρ 15) (bρ 15) (19803681191141 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_338_16 :
+    Uω (aρ 16) (bρ 16) (19803681191141 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_338 :
+    Uρ (19803681191141 / 64000000000000 : ℝ) ≤
+      (-(15921940698173401965557 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (19803681191141 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(46644408687943654039 / 39062500000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_338_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (19803681191141 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(12020726154596521413683 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_338_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (19803681191141 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(2436541007787700471499 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_338_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (19803681191141 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(6229686401708916302631 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_338_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (19803681191141 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(12900738715110178474141 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_338_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (19803681191141 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(2716524003002969505043 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_338_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (19803681191141 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(14630595397403589959339 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_338_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (19803681191141 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(16301638799438157530459 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_338_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (19803681191141 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(9724189067304602339073 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_338_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (19803681191141 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_338_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (19803681191141 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_338_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (19803681191141 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_338_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (19803681191141 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_338_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (19803681191141 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_338_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (19803681191141 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_338_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (19803681191141 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_338_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_339_1 :
+    Uω (aρ 1) (bρ 1) (9940712374197 / 32000000000000 : ℝ) ≤
+      (-(5950477285363575388391 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2433537801785823 / 8000000000000000 : ℝ) ≤
+      (-(5950477285363575388391 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2433537801785823 / 8000000000000000 : ℝ) =
+            (2433537801785823 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2433537801785823 / 2000000000000000 : ℝ) ≤
+            (980994517636424611609 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (608373821122599 / 2000000000000000 : ℝ)) (q :=
+      (2433537801785823 / 8000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_339_2 :
+    Uω (aρ 2) (bρ 2) (9940712374197 / 32000000000000 : ℝ) ≤
+      (-(5990194162937124877343 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (482856773808993 / 1600000000000000 : ℝ) ≤
+      (-(5990194162937124877343 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (482856773808993 / 1600000000000000 : ℝ) =
+            (482856773808993 / 400000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (482856773808993 / 400000000000000 : ℝ) ≤
+            (941277640062875122657 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (60350065725417 / 200000000000000 : ℝ)) (q :=
+      (482856773808993 / 1600000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_339_3 :
+    Uω (aρ 3) (bρ 3) (9940712374197 / 32000000000000 : ℝ) ≤
+      (-(6070848699301788378207 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2375651703453669 / 8000000000000000 : ℝ) ≤
+      (-(6070848699301788378207 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2375651703453669 / 8000000000000000 : ℝ) =
+            (2375651703453669 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2375651703453669 / 2000000000000000 : ℝ) ≤
+            (860623103698211621793 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (296831897864261 / 1000000000000000 : ℝ)) (q :=
+      (2375651703453669 / 8000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_339_4 :
+    Uω (aρ 4) (bρ 4) (9940712374197 / 32000000000000 : ℝ) ≤
+      (-(1552147701776885039179 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11555497435502603 / 40000000000000000 : ℝ) ≤
+      (-(1552147701776885039179 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (11555497435502603 / 40000000000000000 : ℝ) =
+            (11555497435502603 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11555497435502603 / 10000000000000000 : ℝ) ≤
+            (180720248973114960821 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2885232618824739 / 10000000000000000 : ℝ)) (q :=
+      (11555497435502603 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_339_5 :
+    Uω (aρ 5) (bρ 5) (9940712374197 / 32000000000000 : ℝ) ≤
+      (-(6428274620293240149007 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2211755661467821 / 8000000000000000 : ℝ) ≤
+      (-(6428274620293240149007 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2211755661467821 / 8000000000000000 : ℝ) =
+            (2211755661467821 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2211755661467821 / 2000000000000000 : ℝ) ≤
+            (503197182706759850993 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (275506297407799 / 1000000000000000 : ℝ)) (q :=
+      (2211755661467821 / 8000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_339_6 :
+    Uω (aρ 6) (bρ 6) (9940712374197 / 32000000000000 : ℝ) ≤
+      (-(13535046250598920676661 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10333332431410249 / 40000000000000000 : ℝ) ≤
+      (-(13535046250598920676661 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (10333332431410249 / 40000000000000000 : ℝ) =
+            (10333332431410249 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10333332431410249 / 10000000000000000 : ℝ) ≤
+            (327897355401079323339 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1279790205626781 / 5000000000000000 : ℝ)) (q :=
+      (10333332431410249 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_339_7 :
+    Uω (aρ 7) (bρ 7) (9940712374197 / 32000000000000 : ℝ) ≤
+      (-(91106120030342734303 / 62500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1862172134140103 / 8000000000000000 : ℝ) ≤
+      (-(91106120030342734303 / 62500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1862172134140103 / 8000000000000000 : ℝ) =
+            (1862172134140103 / 1000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1862172134140103 / 1000000000000000 : ℝ) ≤
+            (38858976275907265697 / 62500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (454401182331739 / 2000000000000000 : ℝ)) (q :=
+      (1862172134140103 / 8000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_339_8 :
+    Uω (aρ 8) (bρ 8) (9940712374197 / 32000000000000 : ℝ) ≤
+      (-(16235517004178202317679 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7887882750610003 / 40000000000000000 : ℝ) ≤
+      (-(16235517004178202317679 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (7887882750610003 / 40000000000000000 : ℝ) =
+            (7887882750610003 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7887882750610003 / 5000000000000000 : ℝ) ≤
+            (4558898404821797682321 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1844058602953439 / 10000000000000000 : ℝ)) (q :=
+      (7887882750610003 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_339_9 :
+    Uω (aρ 9) (bρ 9) (9940712374197 / 32000000000000 : ℝ) ≤
+      (-(19340199859860171901071 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5782635035815587 / 40000000000000000 : ℝ) ≤
+      (-(19340199859860171901071 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (5782635035815587 / 40000000000000000 : ℝ) =
+            (5782635035815587 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5782635035815587 / 5000000000000000 : ℝ) ≤
+            (1454215549139828098929 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1132400373916231 / 10000000000000000 : ℝ)) (q :=
+      (5782635035815587 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_339_10 :
+    Uω (aρ 10) (bρ 10) (9940712374197 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_339_11 :
+    Uω (aρ 11) (bρ 11) (9940712374197 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_339_12 :
+    Uω (aρ 12) (bρ 12) (9940712374197 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_339_13 :
+    Uω (aρ 13) (bρ 13) (9940712374197 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_339_14 :
+    Uω (aρ 14) (bρ 14) (9940712374197 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_339_15 :
+    Uω (aρ 15) (bρ 15) (9940712374197 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_339_16 :
+    Uω (aρ 16) (bρ 16) (9940712374197 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_339 :
+    Uρ (9940712374197 / 32000000000000 : ℝ) ≤
+      (-(127114917233107029293 / 80000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (9940712374197 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(5950477285363575388391 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_339_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (9940712374197 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(5990194162937124877343 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_339_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (9940712374197 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(6070848699301788378207 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_339_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (9940712374197 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1552147701776885039179 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_339_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (9940712374197 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(6428274620293240149007 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_339_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (9940712374197 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(13535046250598920676661 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_339_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (9940712374197 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) * (-(91106120030342734303 / 62500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_339_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (9940712374197 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(16235517004178202317679 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_339_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (9940712374197 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(19340199859860171901071 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_339_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (9940712374197 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_339_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (9940712374197 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_339_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (9940712374197 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_339_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (9940712374197 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_339_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (9940712374197 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_339_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (9940712374197 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_339_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (9940712374197 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_339_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/U28.lean b/LeanPool/Zeta5Irrational/Table/U28.lean
new file mode 100644
index 0000000000..0cb56450b3
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/U28.lean
@@ -0,0 +1,3960 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-! # Certified arcsine potential bounds (U28) -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma U_340_1 :
+    Uω (aρ 1) (bρ 1) (200369118629 / 640000000000 : ℝ) ≤
+      (-(369418863686504577121 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3066217534982953 / 10000000000000000 : ℝ) ≤
+      (-(369418863686504577121 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3066217534982953 / 10000000000000000 : ℝ) =
+            (3066217534982953 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3066217534982953 / 2500000000000000 : ℝ) ≤
+            (63798124000995422879 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3066164809472781 / 10000000000000000 : ℝ)) (q :=
+      (3066217534982953 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_340_2 :
+    Uω (aρ 2) (bρ 2) (200369118629 / 640000000000 : ℝ) ≤
+      (-(5950098832509869832131 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6084305011555203 / 20000000000000000 : ℝ) ≤
+      (-(5950098832509869832131 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6084305011555203 / 20000000000000000 : ℝ) =
+            (6084305011555203 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6084305011555203 / 5000000000000000 : ℝ) ≤
+            (981372970490130167869 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1520901881786039 / 5000000000000000 : ℝ)) (q :=
+      (6084305011555203 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_340_3 :
+    Uω (aρ 3) (bρ 3) (200369118629 / 640000000000 : ℝ) ≤
+      (-(1507522950055079918221 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5987739206918011 / 20000000000000000 : ℝ) ≤
+      (-(1507522950055079918221 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5987739206918011 / 20000000000000000 : ℝ) =
+            (5987739206918011 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5987739206918011 / 5000000000000000 : ℝ) ≤
+            (225345000694920081779 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1496317032642443 / 5000000000000000 : ℝ)) (q :=
+      (5987739206918011 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_340_4 :
+    Uω (aρ 4) (bρ 4) (200369118629 / 640000000000 : ℝ) ≤
+      (-(616666543212505308213 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2913199629113311 / 10000000000000000 : ℝ) ≤
+      (-(616666543212505308213 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2913199629113311 / 10000000000000000 : ℝ) =
+            (2913199629113311 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2913199629113311 / 2500000000000000 : ℝ) ≤
+            (76480637087494691787 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2909588297658497 / 10000000000000000 : ℝ)) (q :=
+      (2913199629113311 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_340_5 :
+    Uω (aρ 5) (bρ 5) (200369118629 / 640000000000 : ℝ) ≤
+      (-(3192188962342959562939 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (697268406870141 / 2500000000000000 : ℝ) ≤
+      (-(3192188962342959562939 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (697268406870141 / 2500000000000000 : ℝ) =
+            (697268406870141 / 625000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (697268406870141 / 625000000000000 : ℝ) ≤
+            (273546939157040437061 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2779526213728003 / 10000000000000000 : ℝ)) (q :=
+      (697268406870141 / 2500000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_340_6 :
+    Uω (aρ 6) (bρ 6) (200369118629 / 640000000000 : ℝ) ≤
+      (-(13440584727778944987359 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1303925642344057 / 5000000000000000 : ℝ) ≤
+      (-(13440584727778944987359 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1303925642344057 / 5000000000000000 : ℝ) =
+            (1303925642344057 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1303925642344057 / 1250000000000000 : ℝ) ≤
+            (422358878221055012641 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2584321903283103 / 10000000000000000 : ℝ)) (q :=
+      (1303925642344057 / 5000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_340_7 :
+    Uω (aρ 7) (bρ 7) (200369118629 / 640000000000 : ℝ) ≤
+      (-(7235321489269726734981 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4705198564048747 / 20000000000000000 : ℝ) ≤
+      (-(7235321489269726734981 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4705198564048747 / 20000000000000000 : ℝ) =
+            (4705198564048747 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4705198564048747 / 2500000000000000 : ℝ) ≤
+            (3161886215230273265019 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1148739639357811 / 5000000000000000 : ℝ)) (q :=
+      (4705198564048747 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_340_8 :
+    Uω (aρ 8) (bρ 8) (200369118629 / 640000000000 : ℝ) ≤
+      (-(8052373851717295670201 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1997927357326081 / 10000000000000000 : ℝ) ≤
+      (-(8052373851717295670201 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1997927357326081 / 10000000000000000 : ℝ) =
+            (1997927357326081 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1997927357326081 / 1250000000000000 : ℝ) ≤
+            (2344833852782704329799 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1871677080659037 / 10000000000000000 : ℝ)) (q :=
+      (1997927357326081 / 10000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_340_9 :
+    Uω (aρ 9) (bρ 9) (200369118629 / 640000000000 : ℝ) ≤
+      (-(19129132475187687634013 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (738248097158353 / 5000000000000000 : ℝ) ≤
+      (-(19129132475187687634013 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (738248097158353 / 5000000000000000 : ℝ) =
+            (738248097158353 / 625000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (738248097158353 / 625000000000000 : ℝ) ≤
+            (1665282933812312365987 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1169780383000287 / 10000000000000000 : ℝ)) (q :=
+      (738248097158353 / 5000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_340_10 :
+    Uω (aρ 10) (bρ 10) (200369118629 / 640000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_340_11 :
+    Uω (aρ 11) (bρ 11) (200369118629 / 640000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_340_12 :
+    Uω (aρ 12) (bρ 12) (200369118629 / 640000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_340_13 :
+    Uω (aρ 13) (bρ 13) (200369118629 / 640000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_340_14 :
+    Uω (aρ 14) (bρ 14) (200369118629 / 640000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_340_15 :
+    Uω (aρ 15) (bρ 15) (200369118629 / 640000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_340_16 :
+    Uω (aρ 16) (bρ 16) (200369118629 / 640000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_340 :
+    Uρ (200369118629 / 640000000000 : ℝ) ≤
+      (-(3956263075367200025727 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (200369118629 / 640000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(369418863686504577121 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_340_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (200369118629 / 640000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(5950098832509869832131 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_340_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (200369118629 / 640000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1507522950055079918221 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_340_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (200369118629 / 640000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(616666543212505308213 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_340_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (200369118629 / 640000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(3192188962342959562939 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_340_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (200369118629 / 640000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(13440584727778944987359 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_340_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (200369118629 / 640000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(7235321489269726734981 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_340_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (200369118629 / 640000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(8052373851717295670201 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_340_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (200369118629 / 640000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(19129132475187687634013 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_340_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (200369118629 / 640000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_340_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (200369118629 / 640000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_340_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (200369118629 / 640000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_340_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (200369118629 / 640000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_340_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (200369118629 / 640000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_340_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (200369118629 / 640000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_340_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (200369118629 / 640000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_340_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_341_1 :
+    Uω (aρ 1) (bρ 1) (10096199488703 / 32000000000000 : ℝ) ≤
+      (-(11742480570451772639849 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12362051244450521 / 40000000000000000 : ℝ) ≤
+      (-(11742480570451772639849 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (12362051244450521 / 40000000000000000 : ℝ) =
+            (12362051244450521 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12362051244450521 / 10000000000000000 : ℝ) ≤
+            (2120463035548227360151 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3090460500090573 / 10000000000000000 : ℝ)) (q :=
+      (12362051244450521 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_341_2 :
+    Uω (aρ 2) (bρ 2) (10096199488703 / 32000000000000 : ℝ) ≤
+      (-(11820645091192290518187 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12265800522988413 / 40000000000000000 : ℝ) ≤
+      (-(11820645091192290518187 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (12265800522988413 / 40000000000000000 : ℝ) =
+            (12265800522988413 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12265800522988413 / 10000000000000000 : ℝ) ≤
+            (2042298514807709481813 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3066104151869519 / 10000000000000000 : ℝ)) (q :=
+      (12265800522988413 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_341_3 :
+    Uω (aρ 3) (bρ 3) (10096199488703 / 32000000000000 : ℝ) ≤
+      (-(11979329423253933029181 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12072697658661663 / 40000000000000000 : ℝ) ≤
+      (-(11979329423253933029181 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (12072697658661663 / 40000000000000000 : ℝ) =
+            (12072697658661663 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12072697658661663 / 10000000000000000 : ℝ) ≤
+            (1883614182746066970819 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (188559301628509 / 625000000000000 : ℝ)) (q :=
+      (12072697658661663 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_341_4 :
+    Uω (aρ 4) (bρ 4) (10096199488703 / 32000000000000 : ℝ) ≤
+      (-(12250179083177582271047 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11750097580467091 / 40000000000000000 : ℝ) ≤
+      (-(12250179083177582271047 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (11750097580467091 / 40000000000000000 : ℝ) =
+            (11750097580467091 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11750097580467091 / 10000000000000000 : ℝ) ≤
+            (1612764522822417728953 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1466971484011929 / 5000000000000000 : ℝ)) (q :=
+      (11750097580467091 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_341_5 :
+    Uω (aρ 5) (bρ 5) (10096199488703 / 32000000000000 : ℝ) ≤
+      (-(2536346345276691305741 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (450152194268343 / 1600000000000000 : ℝ) ≤
+      (-(2536346345276691305741 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (450152194268343 / 1600000000000000 : ℝ) =
+            (450152194268343 / 400000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (450152194268343 / 400000000000000 : ℝ) ≤
+            (236242375923308694259 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7009966313699 / 25000000000000 : ℝ)) (q :=
+      (450152194268343 / 1600000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_341_6 :
+    Uω (aρ 6) (bρ 6) (10096199488703 / 32000000000000 : ℝ) ≤
+      (-(13347023103751467493021 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10529461057571289 / 40000000000000000 : ℝ) ≤
+      (-(13347023103751467493021 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (10529461057571289 / 40000000000000000 : ℝ) =
+            (10529461057571289 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10529461057571289 / 10000000000000000 : ℝ) ≤
+            (515920502248532506979 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2609055001050957 / 10000000000000000 : ℝ)) (q :=
+      (10529461057571289 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_341_7 :
+    Uω (aρ 7) (bρ 7) (10096199488703 / 32000000000000 : ℝ) ≤
+      (-(1795684838138299536907 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9509883087309847 / 40000000000000000 : ℝ) ≤
+      (-(1795684838138299536907 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (9509883087309847 / 40000000000000000 : ℝ) =
+            (9509883087309847 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9509883087309847 / 5000000000000000 : ℝ) ≤
+            (803617087986700463093 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (580731849170059 / 2500000000000000 : ℝ)) (q :=
+      (9509883087309847 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_341_8 :
+    Uω (aρ 8) (bρ 8) (10096199488703 / 32000000000000 : ℝ) ≤
+      (-(1996986356463040772307 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8095354415229067 / 40000000000000000 : ℝ) ≤
+      (-(1996986356463040772307 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (8095354415229067 / 40000000000000000 : ℝ) =
+            (8095354415229067 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8095354415229067 / 5000000000000000 : ℝ) ≤
+            (602315569661959227693 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (949602355989923 / 5000000000000000 : ℝ)) (q :=
+      (8095354415229067 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_341_9 :
+    Uω (aρ 9) (bρ 9) (10096199488703 / 32000000000000 : ℝ) ≤
+      (-(18924646561087487775289 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6027997085228231 / 40000000000000000 : ℝ) ≤
+      (-(18924646561087487775289 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (6027997085228231 / 40000000000000000 : ℝ) =
+            (6027997085228231 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6027997085228231 / 5000000000000000 : ℝ) ≤
+            (1869768847912512224711 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (301622918834857 / 2500000000000000 : ℝ)) (q :=
+      (6027997085228231 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_341_10 :
+    Uω (aρ 10) (bρ 10) (10096199488703 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_341_11 :
+    Uω (aρ 11) (bρ 11) (10096199488703 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_341_12 :
+    Uω (aρ 12) (bρ 12) (10096199488703 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_341_13 :
+    Uω (aρ 13) (bρ 13) (10096199488703 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_341_14 :
+    Uω (aρ 14) (bρ 14) (10096199488703 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_341_15 :
+    Uω (aρ 15) (bρ 15) (10096199488703 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_341_16 :
+    Uω (aρ 16) (bρ 16) (10096199488703 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_341 :
+    Uρ (10096199488703 / 32000000000000 : ℝ) ≤
+      (-(15761808112806890583937 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (10096199488703 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(11742480570451772639849 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_341_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (10096199488703 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(11820645091192290518187 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_341_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (10096199488703 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(11979329423253933029181 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_341_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (10096199488703 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(12250179083177582271047 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_341_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (10096199488703 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(2536346345276691305741 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_341_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (10096199488703 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(13347023103751467493021 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_341_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (10096199488703 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(1795684838138299536907 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_341_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (10096199488703 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(1996986356463040772307 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_341_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (10096199488703 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(18924646561087487775289 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_341_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (10096199488703 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_341_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (10096199488703 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_341_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (10096199488703 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_341_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (10096199488703 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_341_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (10096199488703 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_341_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (10096199488703 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_341_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (10096199488703 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_341_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_342_1 :
+    Uω (aρ 1) (bρ 1) (2543485761489 / 8000000000000 : ℝ) ≤
+      (-(11664175534405795398607 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6229616161552491 / 20000000000000000 : ℝ) ≤
+      (-(11664175534405795398607 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6229616161552491 / 20000000000000000 : ℝ) =
+            (6229616161552491 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6229616161552491 / 5000000000000000 : ℝ) ≤
+            (2198768071594204601393 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3114756177776241 / 10000000000000000 : ℝ)) (q :=
+      (6229616161552491 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_342_2 :
+    Uω (aρ 2) (bρ 2) (2543485761489 / 8000000000000 : ℝ) ≤
+      (-(5870860263786756796371 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (309074771226449 / 1000000000000000 : ℝ) ≤
+      (-(5870860263786756796371 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (309074771226449 / 1000000000000000 : ℝ) =
+            (309074771226449 / 250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (309074771226449 / 250000000000000 : ℝ) ≤
+            (1060611539213243203629 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (309040445326273 / 1000000000000000 : ℝ)) (q :=
+      (309074771226449 / 1000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_342_3 :
+    Uω (aρ 3) (bρ 3) (2543485761489 / 8000000000000 : ℝ) ≤
+      (-(2974781067166648350813 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1216991626737721 / 4000000000000000 : ℝ) ≤
+      (-(2974781067166648350813 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1216991626737721 / 4000000000000000 : ℝ) =
+            (1216991626737721 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1216991626737721 / 1000000000000000 : ℝ) ≤
+            (490954834333351649187 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (608252653754471 / 2000000000000000 : ℝ)) (q :=
+      (1216991626737721 / 4000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_342_4 :
+    Uω (aρ 4) (bρ 4) (2543485761489 / 8000000000000 : ℝ) ≤
+      (-(6083857343180408154369 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (740462167334903 / 2500000000000000 : ℝ) ≤
+      (-(6083857343180408154369 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (740462167334903 / 2500000000000000 : ℝ) =
+            (740462167334903 / 625000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (740462167334903 / 625000000000000 : ℝ) ≤
+            (847614459819591845631 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1479148327413987 / 5000000000000000 : ℝ)) (q :=
+      (740462167334903 / 2500000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_342_5 :
+    Uω (aρ 5) (bρ 5) (2543485761489 / 8000000000000 : ℝ) ≤
+      (-(12595463419058547117867 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2837827374905103 / 10000000000000000 : ℝ) ≤
+      (-(12595463419058547117867 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2837827374905103 / 10000000000000000 : ℝ) =
+            (2837827374905103 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2837827374905103 / 2500000000000000 : ℝ) ≤
+            (1267480186941452882133 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (707110996323489 / 2500000000000000 : ℝ)) (q :=
+      (2837827374905103 / 10000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_342_6 :
+    Uω (aρ 6) (bρ 6) (2543485761489 / 8000000000000 : ℝ) ≤
+      (-(6627172050807757955697 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (664218791302243 / 2500000000000000 : ℝ) ≤
+      (-(6627172050807757955697 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (664218791302243 / 2500000000000000 : ℝ) =
+            (664218791302243 / 625000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (664218791302243 / 625000000000000 : ℝ) ≤
+            (304299752192242044303 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1316889970520847 / 5000000000000000 : ℝ)) (q :=
+      (664218791302243 / 2500000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_342_7 :
+    Uω (aρ 7) (bρ 7) (2543485761489 / 8000000000000 : ℝ) ≤
+      (-(14261459727060855781449 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1201165023752779 / 5000000000000000 : ℝ) ≤
+      (-(14261459727060855781449 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1201165023752779 / 5000000000000000 : ℝ) =
+            (1201165023752779 / 625000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1201165023752779 / 625000000000000 : ℝ) ≤
+            (6532955681939144218551 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1174175543197433 / 5000000000000000 : ℝ)) (q :=
+      (1201165023752779 / 5000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_342_8 :
+    Uω (aρ 8) (bρ 8) (2543485761489 / 8000000000000 : ℝ) ≤
+      (-(990555338418671925963 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2049706374098079 / 10000000000000000 : ℝ) ≤
+      (-(990555338418671925963 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2049706374098079 / 10000000000000000 : ℝ) =
+            (2049706374098079 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2049706374098079 / 1250000000000000 : ℝ) ≤
+            (309095624643828074037 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (481661347729977 / 2500000000000000 : ℝ)) (q :=
+      (2049706374098079 / 10000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_342_9 :
+    Uω (aρ 9) (bρ 9) (2543485761489 / 8000000000000 : ℝ) ≤
+      (-(4681560142880333083387 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (192149703196409 / 1250000000000000 : ℝ) ≤
+      (-(4681560142880333083387 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (192149703196409 / 1250000000000000 : ℝ) =
+            (192149703196409 / 156250000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (192149703196409 / 156250000000000 : ℝ) ≤
+            (517043709369666916613 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (621296761113147 / 5000000000000000 : ℝ)) (q :=
+      (192149703196409 / 1250000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_342_10 :
+    Uω (aρ 10) (bρ 10) (2543485761489 / 8000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_342_11 :
+    Uω (aρ 11) (bρ 11) (2543485761489 / 8000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_342_12 :
+    Uω (aρ 12) (bρ 12) (2543485761489 / 8000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_342_13 :
+    Uω (aρ 13) (bρ 13) (2543485761489 / 8000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_342_14 :
+    Uω (aρ 14) (bρ 14) (2543485761489 / 8000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_342_15 :
+    Uω (aρ 15) (bρ 15) (2543485761489 / 8000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_342_16 :
+    Uω (aρ 16) (bρ 16) (2543485761489 / 8000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_342 :
+    Uρ (2543485761489 / 8000000000000 : ℝ) ≤
+      (-(15699576321583142466967 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (2543485761489 / 8000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(11664175534405795398607 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_342_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (2543485761489 / 8000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(5870860263786756796371 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_342_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (2543485761489 / 8000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(2974781067166648350813 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_342_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (2543485761489 / 8000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(6083857343180408154369 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_342_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (2543485761489 / 8000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(12595463419058547117867 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_342_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (2543485761489 / 8000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(6627172050807757955697 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_342_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (2543485761489 / 8000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(14261459727060855781449 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_342_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (2543485761489 / 8000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(990555338418671925963 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_342_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (2543485761489 / 8000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(4681560142880333083387 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_342_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (2543485761489 / 8000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_342_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (2543485761489 / 8000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_342_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (2543485761489 / 8000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_342_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (2543485761489 / 8000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_342_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (2543485761489 / 8000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_342_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (2543485761489 / 8000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_342_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (2543485761489 / 8000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_342_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_343_1 :
+    Uω (aρ 1) (bρ 1) (10251686603209 / 32000000000000 : ℝ) ≤
+      (-(5793239462649911037 / 5000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12556413376495749 / 40000000000000000 : ℝ) ≤
+      (-(5793239462649911037 / 5000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (12556413376495749 / 40000000000000000 : ℝ) =
+            (12556413376495749 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12556413376495749 / 10000000000000000 : ℝ) ≤
+            (1138232340350088963 / 5000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1569525921415031 / 5000000000000000 : ℝ)) (q :=
+      (12556413376495749 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_343_2 :
+    Uω (aρ 2) (bρ 2) (10251686603209 / 32000000000000 : ℝ) ≤
+      (-(2332682826846300534697 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12460181005387103 / 40000000000000000 : ℝ) ≤
+      (-(2332682826846300534697 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (12460181005387103 / 40000000000000000 : ℝ) =
+            (12460181005387103 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12460181005387103 / 10000000000000000 : ℝ) ≤
+            (439905894353699465303 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3114704669785739 / 10000000000000000 : ℝ)) (q :=
+      (12460181005387103 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_343_3 :
+    Uω (aρ 3) (bρ 3) (10251686603209 / 32000000000000 : ℝ) ≤
+      (-(2954889447974945029483 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12267134255118471 / 40000000000000000 : ℝ) ≤
+      (-(2954889447974945029483 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (12267134255118471 / 40000000000000000 : ℝ) =
+            (12267134255118471 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12267134255118471 / 10000000000000000 : ℝ) ≤
+            (510846453525054970517 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3065577401001423 / 10000000000000000 : ℝ)) (q :=
+      (12267134255118471 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_343_4 :
+    Uω (aρ 4) (bρ 4) (10251686603209 / 32000000000000 : ℝ) ≤
+      (-(6042963187737180019357 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2388937971062537 / 8000000000000000 : ℝ) ≤
+      (-(6042963187737180019357 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2388937971062537 / 8000000000000000 : ℝ) =
+            (2388937971062537 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2388937971062537 / 2000000000000000 : ℝ) ≤
+            (888508615262819980643 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (298264938216353 / 1000000000000000 : ℝ)) (q :=
+      (2388937971062537 / 8000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_343_5 :
+    Uω (aρ 5) (bρ 5) (10251686603209 / 32000000000000 : ℝ) ≤
+      (-(12509937826887110023689 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (457952343414123 / 1600000000000000 : ℝ) ≤
+      (-(12509937826887110023689 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (457952343414123 / 1600000000000000 : ℝ) =
+            (457952343414123 / 400000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (457952343414123 / 400000000000000 : ℝ) ≤
+            (1353005779112889976311 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (114115946660749 / 400000000000000 : ℝ)) (q :=
+      (457952343414123 / 1600000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_343_6 :
+    Uω (aρ 6) (bρ 6) (10251686603209 / 32000000000000 : ℝ) ≤
+      (-(13162530945191651836923 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10725524403767141 / 40000000000000000 : ℝ) ≤
+      (-(13162530945191651836923 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (10725524403767141 / 40000000000000000 : ℝ) =
+            (10725524403767141 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10725524403767141 / 10000000000000000 : ℝ) ≤
+            (700412660808348163077 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1329248475432879 / 5000000000000000 : ℝ)) (q :=
+      (10725524403767141 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_343_7 :
+    Uω (aρ 7) (bρ 7) (10251686603209 / 32000000000000 : ℝ) ≤
+      (-(14158560319631239190253 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9708710006044581 / 40000000000000000 : ℝ) ≤
+      (-(14158560319631239190253 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (9708710006044581 / 40000000000000000 : ℝ) =
+            (9708710006044581 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9708710006044581 / 5000000000000000 : ℝ) ≤
+            (6635855089368760809747 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1186875566382239 / 5000000000000000 : ℝ)) (q :=
+      (9708710006044581 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_343_8 :
+    Uω (aρ 8) (bρ 8) (10251686603209 / 32000000000000 : ℝ) ≤
+      (-(15723673450207903387083 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1660425999879923 / 8000000000000000 : ℝ) ≤
+      (-(15723673450207903387083 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1660425999879923 / 8000000000000000 : ℝ) =
+            (1660425999879923 / 1000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1660425999879923 / 1000000000000000 : ℝ) ≤
+            (5070741958792096612917 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (390800556156399 / 2000000000000000 : ℝ)) (q :=
+      (1660425999879923 / 8000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_343_9 :
+    Uω (aρ 9) (bρ 9) (10251686603209 / 32000000000000 : ℝ) ≤
+      (-(741338966521064763589 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6268468315680387 / 40000000000000000 : ℝ) ≤
+      (-(741338966521064763589 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (6268468315680387 / 40000000000000000 : ℝ) =
+            (6268468315680387 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6268468315680387 / 5000000000000000 : ℝ) ≤
+            (90437649838935236411 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1278137567282381 / 10000000000000000 : ℝ)) (q :=
+      (6268468315680387 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_343_10 :
+    Uω (aρ 10) (bρ 10) (10251686603209 / 32000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_343_11 :
+    Uω (aρ 11) (bρ 11) (10251686603209 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_343_12 :
+    Uω (aρ 12) (bρ 12) (10251686603209 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_343_13 :
+    Uω (aρ 13) (bρ 13) (10251686603209 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_343_14 :
+    Uω (aρ 14) (bρ 14) (10251686603209 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_343_15 :
+    Uω (aρ 15) (bρ 15) (10251686603209 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_343_16 :
+    Uω (aρ 16) (bρ 16) (10251686603209 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_343 :
+    Uρ (10251686603209 / 32000000000000 : ℝ) ≤
+      (-(7819153509420200613333 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (10251686603209 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(5793239462649911037 / 5000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_343_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (10251686603209 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(2332682826846300534697 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_343_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (10251686603209 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(2954889447974945029483 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_343_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (10251686603209 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(6042963187737180019357 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_343_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (10251686603209 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(12509937826887110023689 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_343_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (10251686603209 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(13162530945191651836923 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_343_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (10251686603209 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(14158560319631239190253 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_343_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (10251686603209 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(15723673450207903387083 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_343_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (10251686603209 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(741338966521064763589 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_343_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (10251686603209 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_343_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (10251686603209 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_343_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (10251686603209 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_343_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (10251686603209 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_343_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (10251686603209 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_343_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (10251686603209 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_343_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (10251686603209 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_343_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_344_1 :
+    Uω (aρ 1) (bρ 1) (5164715080231 / 16000000000000 : ℝ) ≤
+      (-(2301876272154925326387 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3163398601301231 / 10000000000000000 : ℝ) ≤
+      (-(2301876272154925326387 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3163398601301231 / 10000000000000000 : ℝ) =
+            (3163398601301231 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3163398601301231 / 2500000000000000 : ℝ) ≤
+            (470712449045074673613 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3163347495543087 / 10000000000000000 : ℝ)) (q :=
+      (3163398601301231 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_344_2 :
+    Uω (aρ 2) (bρ 2) (5164715080231 / 16000000000000 : ℝ) ≤
+      (-(11585716300751941489211 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6278685497958957 / 20000000000000000 : ℝ) ≤
+      (-(11585716300751941489211 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6278685497958957 / 20000000000000000 : ℝ) =
+            (6278685497958957 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6278685497958957 / 5000000000000000 : ℝ) ≤
+            (2277227305248058510789 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1569502401704791 / 5000000000000000 : ℝ)) (q :=
+      (6278685497958957 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_344_3 :
+    Uω (aρ 3) (bρ 3) (5164715080231 / 16000000000000 : ℝ) ≤
+      (-(11740619893739643235129 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6182175818272327 / 20000000000000000 : ℝ) ≤
+      (-(11740619893739643235129 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6182175818272327 / 20000000000000000 : ℝ) =
+            (6182175818272327 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6182175818272327 / 5000000000000000 : ℝ) ≤
+            (2122323712260356764871 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (386236403759119 / 1250000000000000 : ℝ)) (q :=
+      (6182175818272327 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_344_4 :
+    Uω (aρ 4) (bρ 4) (5164715080231 / 16000000000000 : ℝ) ≤
+      (-(240096062582495861523 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3010495790238087 / 10000000000000000 : ℝ) ≤
+      (-(240096062582495861523 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3010495790238087 / 10000000000000000 : ℝ) =
+            (3010495790238087 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3010495790238087 / 2500000000000000 : ℝ) ≤
+            (37162809537504138477 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3007001173341799 / 10000000000000000 : ℝ)) (q :=
+      (3010495790238087 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_344_5 :
+    Uω (aρ 5) (bρ 5) (5164715080231 / 16000000000000 : ℝ) ≤
+      (-(6212571094579826978127 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5773151127799207 / 20000000000000000 : ℝ) ≤
+      (-(6212571094579826978127 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5773151127799207 / 20000000000000000 : ℝ) =
+            (5773151127799207 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5773151127799207 / 5000000000000000 : ℝ) ≤
+            (718900708420173021873 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (359668829999979 / 1250000000000000 : ℝ)) (q :=
+      (5773151127799207 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_344_6 :
+    Uω (aρ 6) (bρ 6) (5164715080231 / 16000000000000 : ℝ) ≤
+      (-(13071567339577674235373 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (676470795291817 / 2500000000000000 : ℝ) ≤
+      (-(13071567339577674235373 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (676470795291817 / 2500000000000000 : ℝ) =
+            (676470795291817 / 625000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (676470795291817 / 625000000000000 : ℝ) ≤
+            (791376266422325764627 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2683206249675161 / 10000000000000000 : ℝ)) (q :=
+      (676470795291817 / 2500000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_344_7 :
+    Uω (aρ 7) (bρ 7) (5164715080231 / 16000000000000 : ℝ) ≤
+      (-(14056755646450440234437 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2452013509319447 / 10000000000000000 : ℝ) ≤
+      (-(14056755646450440234437 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2452013509319447 / 10000000000000000 : ℝ) =
+            (2452013509319447 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2452013509319447 / 1250000000000000 : ℝ) ≤
+            (6737659762549559765563 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2399128286739519 / 10000000000000000 : ℝ)) (q :=
+      (2452013509319447 / 10000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_344_8 :
+    Uω (aρ 8) (bρ 8) (5164715080231 / 16000000000000 : ℝ) ≤
+      (-(195002498657717841121 / 125000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4202637412278987 / 20000000000000000 : ℝ) ≤
+      (-(195002498657717841121 / 125000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4202637412278987 / 20000000000000000 : ℝ) =
+            (4202637412278987 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4202637412278987 / 2500000000000000 : ℝ) ≤
+            (64927693954782158879 / 125000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (495320082929903 / 2500000000000000 : ℝ)) (q :=
+      (4202637412278987 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_344_9 :
+    Uω (aρ 9) (bρ 9) (5164715080231 / 16000000000000 : ℝ) ≤
+      (-(3669191631964464684769 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (798390139546971 / 5000000000000000 : ℝ) ≤
+      (-(3669191631964464684769 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (798390139546971 / 5000000000000000 : ℝ) =
+            (798390139546971 / 625000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (798390139546971 / 625000000000000 : ℝ) ≤
+            (489691449835535315231 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1313169105988509 / 10000000000000000 : ℝ)) (q :=
+      (798390139546971 / 5000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_344_10 :
+    Uω (aρ 10) (bρ 10) (5164715080231 / 16000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_344_11 :
+    Uω (aρ 11) (bρ 11) (5164715080231 / 16000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_344_12 :
+    Uω (aρ 12) (bρ 12) (5164715080231 / 16000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_344_13 :
+    Uω (aρ 13) (bρ 13) (5164715080231 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_344_14 :
+    Uω (aρ 14) (bρ 14) (5164715080231 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_344_15 :
+    Uω (aρ 15) (bρ 15) (5164715080231 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_344_16 :
+    Uω (aρ 16) (bρ 16) (5164715080231 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_344 :
+    Uρ (5164715080231 / 16000000000000 : ℝ) ≤
+      (-(3115591048149674882067 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (5164715080231 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2301876272154925326387 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_344_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (5164715080231 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(11585716300751941489211 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_344_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (5164715080231 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(11740619893739643235129 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_344_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (5164715080231 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(240096062582495861523 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_344_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (5164715080231 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(6212571094579826978127 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_344_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (5164715080231 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(13071567339577674235373 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_344_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (5164715080231 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(14056755646450440234437 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_344_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (5164715080231 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(195002498657717841121 / 125000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_344_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (5164715080231 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(3669191631964464684769 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_344_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (5164715080231 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_344_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (5164715080231 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_344_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (5164715080231 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_344_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (5164715080231 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_344_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (5164715080231 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_344_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (5164715080231 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_344_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (5164715080231 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_344_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_345_1 :
+    Uω (aρ 1) (bρ 1) (1310614659371 / 4000000000000 : ℝ) ≤
+      (-(11356946906342816493083 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3211989097704713 / 10000000000000000 : ℝ) ≤
+      (-(11356946906342816493083 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3211989097704713 / 10000000000000000 : ℝ) =
+            (3211989097704713 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3211989097704713 / 2500000000000000 : ℝ) ≤
+            (2505996699657183506917 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1605969382533463 / 5000000000000000 : ℝ)) (q :=
+      (3211989097704713 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_345_2 :
+    Uω (aρ 2) (bρ 2) (1310614659371 / 4000000000000 : ℝ) ≤
+      (-(11432108977075101674191 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6375875247375869 / 20000000000000000 : ℝ) ≤
+      (-(11432108977075101674191 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6375875247375869 / 20000000000000000 : ℝ) =
+            (6375875247375869 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6375875247375869 / 5000000000000000 : ℝ) ≤
+            (2430834628924898325809 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3187604829543369 / 10000000000000000 : ℝ)) (q :=
+      (6375875247375869 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_345_3 :
+    Uω (aρ 3) (bρ 3) (1310614659371 / 4000000000000 : ℝ) ≤
+      (-(5792295309103324570207 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3139696159204113 / 10000000000000000 : ℝ) ≤
+      (-(5792295309103324570207 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3139696159204113 / 10000000000000000 : ℝ) =
+            (3139696159204113 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3139696159204113 / 2500000000000000 : ℝ) ≤
+            (1139176493896675429793 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1569259003462863 / 5000000000000000 : ℝ)) (q :=
+      (3139696159204113 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_345_4 :
+    Uω (aρ 4) (bρ 4) (1310614659371 / 4000000000000 : ℝ) ≤
+      (-(11844509075501430543993 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6118282167169283 / 20000000000000000 : ℝ) ≤
+      (-(11844509075501430543993 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6118282167169283 / 20000000000000000 : ℝ) =
+            (6118282167169283 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6118282167169283 / 5000000000000000 : ℝ) ≤
+            (2018434530498569456007 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3055702036751783 / 10000000000000000 : ℝ)) (q :=
+      (6118282167169283 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_345_5 :
+    Uω (aρ 5) (bρ 5) (1310614659371 / 4000000000000 : ℝ) ≤
+      (-(2451538272676043922751 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5870636946319773 / 20000000000000000 : ℝ) ≤
+      (-(2451538272676043922751 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5870636946319773 / 20000000000000000 : ℝ) =
+            (5870636946319773 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5870636946319773 / 5000000000000000 : ℝ) ≤
+            (321050448523956077249 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2926246735237273 / 10000000000000000 : ℝ)) (q :=
+      (5870636946319773 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_345_6 :
+    Uω (aρ 6) (bρ 6) (1310614659371 / 4000000000000 : ℝ) ≤
+      (-(50359866786970237563 / 39062500000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2754876193420369 / 10000000000000000 : ℝ) ≤
+      (-(50359866786970237563 / 39062500000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2754876193420369 / 10000000000000000 : ℝ) =
+            (2754876193420369 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2754876193420369 / 2500000000000000 : ℝ) ≤
+            (3792256673967262437 / 39062500000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1366301275449119 / 5000000000000000 : ℝ)) (q :=
+      (2754876193420369 / 10000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_345_7 :
+    Uω (aρ 7) (bρ 7) (1310614659371 / 4000000000000 : ℝ) ≤
+      (-(3464083838632482367617 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2501652608850121 / 10000000000000000 : ℝ) ≤
+      (-(3464083838632482367617 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2501652608850121 / 10000000000000000 : ℝ) =
+            (2501652608850121 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 2
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2501652608850121 / 2500000000000000 : ℝ) ≤
+            (1652062867517632383 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1224908381258871 / 5000000000000000 : ℝ)) (q :=
+      (2501652608850121 / 10000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_345_8 :
+    Uω (aρ 8) (bρ 8) (1310614659371 / 4000000000000 : ℝ) ≤
+      (-(7679130478477478291329 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1076388887398653 / 5000000000000000 : ℝ) ≤
+      (-(7679130478477478291329 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1076388887398653 / 5000000000000000 : ℝ) =
+            (1076388887398653 / 625000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1076388887398653 / 625000000000000 : ℝ) ≤
+            (2718077226022521708671 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (127225546609507 / 625000000000000 : ℝ)) (q :=
+      (1076388887398653 / 5000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_345_9 :
+    Uω (aρ 9) (bρ 9) (1310614659371 / 4000000000000 : ℝ) ≤
+      (-(449633250021974099083 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25865868672333 / 156250000000000 : ℝ) ≤
+      (-(449633250021974099083 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (25865868672333 / 156250000000000 : ℝ) = (25865868672333 / 19531250000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25865868672333 / 19531250000000 : ℝ) ≤
+            (70227135203025900917 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (345462503644031 / 2500000000000000 : ℝ)) (q :=
+      (25865868672333 / 156250000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_345_10 :
+    Uω (aρ 10) (bρ 10) (1310614659371 / 4000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_345_11 :
+    Uω (aρ 11) (bρ 11) (1310614659371 / 4000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_345_12 :
+    Uω (aρ 12) (bρ 12) (1310614659371 / 4000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_345_13 :
+    Uω (aρ 13) (bρ 13) (1310614659371 / 4000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_345_14 :
+    Uω (aρ 14) (bρ 14) (1310614659371 / 4000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_345_15 :
+    Uω (aρ 15) (bρ 15) (1310614659371 / 4000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_345_16 :
+    Uω (aρ 16) (bρ 16) (1310614659371 / 4000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_345 :
+    Uρ (1310614659371 / 4000000000000 : ℝ) ≤
+      (-(15459844878657915384769 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1310614659371 / 4000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(11356946906342816493083 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_345_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1310614659371 / 4000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(11432108977075101674191 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_345_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1310614659371 / 4000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(5792295309103324570207 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_345_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1310614659371 / 4000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11844509075501430543993 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_345_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1310614659371 / 4000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(2451538272676043922751 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_345_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1310614659371 / 4000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) * (-(50359866786970237563 / 39062500000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_345_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1310614659371 / 4000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(3464083838632482367617 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_345_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1310614659371 / 4000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(7679130478477478291329 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_345_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1310614659371 / 4000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(449633250021974099083 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_345_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1310614659371 / 4000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_345_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1310614659371 / 4000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_345_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1310614659371 / 4000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_345_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1310614659371 / 4000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_345_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1310614659371 / 4000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_345_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1310614659371 / 4000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_345_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1310614659371 / 4000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_345_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_346_1 :
+    Uω (aρ 1) (bρ 1) (5320202194737 / 16000000000000 : ℝ) ≤
+      (-(1400850162952081230453 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (326057957106513 / 1000000000000000 : ℝ) ≤
+      (-(1400850162952081230453 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (326057957106513 / 1000000000000000 : ℝ) =
+            (326057957106513 / 250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (326057957106513 / 250000000000000 : ℝ) ≤
+            (332017787797918769547 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (652105997700927 / 2000000000000000 : ℝ)) (q :=
+      (326057957106513 / 1000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_346_2 :
+    Uω (aρ 2) (bρ 2) (5320202194737 / 16000000000000 : ℝ) ≤
+      (-(11280826001053343395199 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (647306468737741 / 2000000000000000 : ℝ) ≤
+      (-(11280826001053343395199 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (647306468737741 / 2000000000000000 : ℝ) =
+            (647306468737741 / 500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (647306468737741 / 500000000000000 : ℝ) ≤
+            (2582117604946656604801 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (647240909252357 / 2000000000000000 : ℝ)) (q :=
+      (647306468737741 / 2000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_346_3 :
+    Uω (aρ 3) (bρ 3) (5320202194737 / 16000000000000 : ℝ) ≤
+      (-(228619205476329618989 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25506430753241 / 80000000000000 : ℝ) ≤
+      (-(228619205476329618989 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (25506430753241 / 80000000000000 : ℝ) = (25506430753241 / 20000000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25506430753241 / 20000000000000 : ℝ) ≤
+            (48639666643670381011 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (25497149228357 / 80000000000000 : ℝ)) (q :=
+      (25506430753241 / 80000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_346_4 :
+    Uω (aρ 4) (bρ 4) (5320202194737 / 16000000000000 : ℝ) ≤
+      (-(11686749560023923541033 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (621556927077899 / 2000000000000000 : ℝ) ≤
+      (-(11686749560023923541033 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (621556927077899 / 2000000000000000 : ℝ) =
+            (621556927077899 / 500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (621556927077899 / 500000000000000 : ℝ) ≤
+            (2176194045976076458967 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (620879883415673 / 2000000000000000 : ℝ)) (q :=
+      (621556927077899 / 2000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_346_5 :
+    Uω (aρ 5) (bρ 5) (5320202194737 / 16000000000000 : ℝ) ≤
+      (-(12093015219054816054701 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5968112726004279 / 20000000000000000 : ℝ) ≤
+      (-(12093015219054816054701 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5968112726004279 / 20000000000000000 : ℝ) =
+            (5968112726004279 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5968112726004279 / 5000000000000000 : ℝ) ≤
+            (1769928386945183945299 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1487566395819327 / 5000000000000000 : ℝ)) (q :=
+      (5968112726004279 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_346_6 :
+    Uω (aρ 6) (bρ 6) (5320202194737 / 16000000000000 : ℝ) ≤
+      (-(6357949180291555523893 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5607710002359903 / 20000000000000000 : ℝ) ≤
+      (-(6357949180291555523893 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5607710002359903 / 20000000000000000 : ℝ) =
+            (5607710002359903 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5607710002359903 / 5000000000000000 : ℝ) ≤
+            (573522622708444476107 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1390985221567139 / 5000000000000000 : ℝ)) (q :=
+      (5607710002359903 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_346_7 :
+    Uω (aρ 7) (bρ 7) (5320202194737 / 16000000000000 : ℝ) ≤
+      (-(6830008200364275306877 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2551250044033 / 10000000000000 : ℝ) ≤
+      (-(6830008200364275306877 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (2551250044033 / 10000000000000 : ℝ) = (2551250044033 / 2500000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2551250044033 / 2500000000000 : ℝ) ≤
+            (101463602635724693123 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (20003375276803 / 80000000000000 : ℝ)) (q :=
+      (2551250044033 / 10000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_346_8 :
+    Uω (aρ 8) (bρ 8) (5320202194737 / 16000000000000 : ℝ) ≤
+      (-(7561339280388101403409 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4408191068221703 / 20000000000000000 : ℝ) ≤
+      (-(7561339280388101403409 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4408191068221703 / 20000000000000000 : ℝ) =
+            (4408191068221703 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4408191068221703 / 2500000000000000 : ℝ) ≤
+            (2835868424111898596591 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1044827270548039 / 5000000000000000 : ℝ)) (q :=
+      (4408191068221703 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_346_9 :
+    Uω (aρ 9) (bρ 9) (5320202194737 / 16000000000000 : ℝ) ≤
+      (-(2756562137920207449 / 1562500000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (428309575689891 / 2500000000000000 : ℝ) ≤
+      (-(2756562137920207449 / 1562500000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (428309575689891 / 2500000000000000 : ℝ) =
+            (428309575689891 / 312500000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (428309575689891 / 312500000000000 : ℝ) ≤
+            (492565269736042551 / 1562500000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1448905706753503 / 10000000000000000 : ℝ)) (q :=
+      (428309575689891 / 2500000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_346_10 :
+    Uω (aρ 10) (bρ 10) (5320202194737 / 16000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_346_11 :
+    Uω (aρ 11) (bρ 11) (5320202194737 / 16000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_346_12 :
+    Uω (aρ 12) (bρ 12) (5320202194737 / 16000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_346_13 :
+    Uω (aρ 13) (bρ 13) (5320202194737 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_346_14 :
+    Uω (aρ 14) (bρ 14) (5320202194737 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_346_15 :
+    Uω (aρ 15) (bρ 15) (5320202194737 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_346_16 :
+    Uω (aρ 16) (bρ 16) (5320202194737 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_346 :
+    Uρ (5320202194737 / 16000000000000 : ℝ) ≤
+      (-(15344959851341678360121 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (5320202194737 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1400850162952081230453 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_346_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (5320202194737 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(11280826001053343395199 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_346_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (5320202194737 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(228619205476329618989 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_346_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (5320202194737 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11686749560023923541033 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_346_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (5320202194737 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(12093015219054816054701 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_346_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (5320202194737 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(6357949180291555523893 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_346_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (5320202194737 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(6830008200364275306877 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_346_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (5320202194737 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(7561339280388101403409 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_346_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (5320202194737 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(2756562137920207449 / 1562500000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_346_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (5320202194737 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_346_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (5320202194737 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_346_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (5320202194737 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_346_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (5320202194737 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_346_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (5320202194737 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_346_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (5320202194737 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_346_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (5320202194737 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_346_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_347_1 :
+    Uω (aρ 1) (bρ 1) (539794575199 / 1600000000000 : ℝ) ≤
+      (-(5529438415241096726837 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1323668008959031 / 4000000000000000 : ℝ) ≤
+      (-(5529438415241096726837 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1323668008959031 / 4000000000000000 : ℝ) =
+            (1323668008959031 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1323668008959031 / 1000000000000000 : ℝ) ≤
+            (1402033387758903273163 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (661824233577281 / 2000000000000000 : ℝ)) (q :=
+      (1323668008959031 / 4000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_347_2 :
+    Uω (aρ 2) (bρ 2) (539794575199 / 1600000000000 : ℝ) ≤
+      (-(2782949515805895480333 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3285126915848639 / 10000000000000000 : ℝ) ≤
+      (-(2782949515805895480333 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3285126915848639 / 10000000000000000 : ℝ) =
+            (3285126915848639 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3285126915848639 / 2500000000000000 : ℝ) ≤
+            (682786385694104519667 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (102650123978079 / 312500000000000 : ℝ)) (q :=
+      (3285126915848639 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_347_3 :
+    Uω (aρ 3) (bρ 3) (539794575199 / 1600000000000 : ℝ) ≤
+      (-(2819914039390198190657 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1294764395786463 / 4000000000000000 : ℝ) ≤
+      (-(2819914039390198190657 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1294764395786463 / 4000000000000000 : ℝ) =
+            (1294764395786463 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1294764395786463 / 1000000000000000 : ℝ) ≤
+            (645821862109801809343 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (647153644176713 / 2000000000000000 : ℝ)) (q :=
+      (1294764395786463 / 4000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_347_4 :
+    Uω (aρ 4) (bρ 4) (539794575199 / 1600000000000 : ℝ) ≤
+      (-(1153144550858087381687 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6312853052686849 / 20000000000000000 : ℝ) ≤
+      (-(1153144550858087381687 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6312853052686849 / 20000000000000000 : ℝ) =
+            (6312853052686849 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6312853052686849 / 5000000000000000 : ℝ) ≤
+            (233149809741912618313 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3153093475703099 / 10000000000000000 : ℝ)) (q :=
+      (6312853052686849 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_347_5 :
+    Uω (aρ 5) (bρ 5) (539794575199 / 1600000000000 : ℝ) ≤
+      (-(1193102277038038876959 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (303278947685773 / 1000000000000000 : ℝ) ≤
+      (-(1193102277038038876959 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (303278947685773 / 1000000000000000 : ℝ) =
+            (303278947685773 / 250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (303278947685773 / 250000000000000 : ℝ) ≤
+            (193192083561961123041 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (302400929606671 / 1000000000000000 : ℝ)) (q :=
+      (303278947685773 / 1000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_347_6 :
+    Uω (aρ 6) (bρ 6) (539794575199 / 1600000000000 : ℝ) ≤
+      (-(6271384959581393181183 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (228225627797769 / 800000000000000 : ℝ) ≤
+      (-(6271384959581393181183 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (228225627797769 / 800000000000000 : ℝ) =
+            (228225627797769 / 200000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (228225627797769 / 200000000000000 : ℝ) ≤
+            (660086843418606818817 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (113252456497419 / 400000000000000 : ℝ)) (q :=
+      (228225627797769 / 800000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_347_7 :
+    Uω (aρ 7) (bρ 7) (539794575199 / 1600000000000 : ℝ) ≤
+      (-(2693525626556395372243 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1300404147227207 / 5000000000000000 : ℝ) ≤
+      (-(2693525626556395372243 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1300404147227207 / 5000000000000000 : ℝ) =
+            (1300404147227207 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1300404147227207 / 1250000000000000 : ℝ) ≤
+            (79063094643604627757 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1275474343580039 / 5000000000000000 : ℝ)) (q :=
+      (1300404147227207 / 5000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_347_8 :
+    Uω (aρ 8) (bρ 8) (539794575199 / 1600000000000 : ℝ) ≤
+      (-(14893097731127274734401 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4510565346553517 / 20000000000000000 : ℝ) ≤
+      (-(14893097731127274734401 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4510565346553517 / 20000000000000000 : ℝ) =
+            (4510565346553517 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4510565346553517 / 2500000000000000 : ℝ) ≤
+            (5901317677872725265599 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2143439096144767 / 10000000000000000 : ℝ)) (q :=
+      (4510565346553517 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_347_9 :
+    Uω (aρ 9) (bρ 9) (539794575199 / 1600000000000 : ℝ) ≤
+      (-(17314042422240819869213 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3540712685782171 / 20000000000000000 : ℝ) ≤
+      (-(17314042422240819869213 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3540712685782171 / 20000000000000000 : ℝ) =
+            (3540712685782171 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3540712685782171 / 2500000000000000 : ℝ) ≤
+            (3480372986759180130787 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1514552063733421 / 10000000000000000 : ℝ)) (q :=
+      (3540712685782171 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_347_10 :
+    Uω (aρ 10) (bρ 10) (539794575199 / 1600000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_347_11 :
+    Uω (aρ 11) (bρ 11) (539794575199 / 1600000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_347_12 :
+    Uω (aρ 12) (bρ 12) (539794575199 / 1600000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_347_13 :
+    Uω (aρ 13) (bρ 13) (539794575199 / 1600000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_347_14 :
+    Uω (aρ 14) (bρ 14) (539794575199 / 1600000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_347_15 :
+    Uω (aρ 15) (bρ 15) (539794575199 / 1600000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_347_16 :
+    Uω (aρ 16) (bρ 16) (539794575199 / 1600000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_347 :
+    Uρ (539794575199 / 1600000000000 : ℝ) ≤
+      (-(7616529524551265616913 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (539794575199 / 1600000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(5529438415241096726837 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_347_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (539794575199 / 1600000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(2782949515805895480333 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_347_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (539794575199 / 1600000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(2819914039390198190657 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_347_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (539794575199 / 1600000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1153144550858087381687 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_347_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (539794575199 / 1600000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1193102277038038876959 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_347_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (539794575199 / 1600000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(6271384959581393181183 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_347_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (539794575199 / 1600000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(2693525626556395372243 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_347_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (539794575199 / 1600000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14893097731127274734401 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_347_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (539794575199 / 1600000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(17314042422240819869213 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_347_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (539794575199 / 1600000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_347_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (539794575199 / 1600000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_347_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (539794575199 / 1600000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_347_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (539794575199 / 1600000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_347_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (539794575199 / 1600000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_347_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (539794575199 / 1600000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_347_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (539794575199 / 1600000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_347_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_348_1 :
+    Uω (aρ 1) (bρ 1) (5475689309243 / 16000000000000 : ℝ) ≤
+      (-(2728277181692807866107 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6715520905316779 / 20000000000000000 : ℝ) ≤
+      (-(2728277181692807866107 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6715520905316779 / 20000000000000000 : ℝ) =
+            (6715520905316779 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6715520905316779 / 5000000000000000 : ℝ) ≤
+            (737458719807192133893 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (419714038140613 / 1250000000000000 : ℝ)) (q :=
+      (6715520905316779 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_348_2 :
+    Uω (aρ 2) (bρ 2) (5475689309243 / 16000000000000 : ℝ) ≤
+      (-(10984958909324133836937 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6667442693268163 / 20000000000000000 : ℝ) ≤
+      (-(10984958909324133836937 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6667442693268163 / 20000000000000000 : ℝ) =
+            (6667442693268163 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6667442693268163 / 5000000000000000 : ℝ) ≤
+            (2877984696675866163063 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (208337694099143 / 625000000000000 : ℝ)) (q :=
+      (6667442693268163 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_348_3 :
+    Uω (aρ 3) (bρ 3) (5475689309243 / 16000000000000 : ℝ) ≤
+      (-(1113060882446734250669 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6571035238209801 / 20000000000000000 : ℝ) ≤
+      (-(1113060882446734250669 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6571035238209801 / 20000000000000000 : ℝ) =
+            (6571035238209801 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6571035238209801 / 5000000000000000 : ℝ) ≤
+            (273233478153265749331 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1642195878438963 / 5000000000000000 : ℝ)) (q :=
+      (6571035238209801 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_348_4 :
+    Uω (aρ 4) (bρ 4) (5475689309243 / 16000000000000 : ℝ) ≤
+      (-(5689260749156044347883 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (200316677013947 / 625000000000000 : ℝ) ≤
+      (-(5689260749156044347883 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (200316677013947 / 625000000000000 : ℝ) =
+            (200316677013947 / 156250000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (200316677013947 / 156250000000000 : ℝ) ≤
+            (1242211053843955652117 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3201784364179429 / 10000000000000000 : ℝ)) (q :=
+      (200316677013947 / 625000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_348_5 :
+    Uω (aρ 5) (bρ 5) (5475689309243 / 16000000000000 : ℝ) ≤
+      (-(470865098371135128527 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (385189755328009 / 1250000000000000 : ℝ) ≤
+      (-(470865098371135128527 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (385189755328009 / 1250000000000000 : ℝ) =
+            (385189755328009 / 312500000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (385189755328009 / 312500000000000 : ℝ) ≤
+            (83652645868864871473 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3072876704316269 / 10000000000000000 : ℝ)) (q :=
+      (385189755328009 / 1250000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_348_6 :
+    Uω (aρ 6) (bρ 6) (5475689309243 / 16000000000000 : ℝ) ≤
+      (-(12372631891703963693447 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (290177292402339 / 1000000000000000 : ℝ) ≤
+      (-(12372631891703963693447 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (290177292402339 / 1000000000000000 : ℝ) =
+            (290177292402339 / 250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (290177292402339 / 250000000000000 : ℝ) ≤
+            (1490311714296036306553 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (576125368450981 / 2000000000000000 : ℝ)) (q :=
+      (290177292402339 / 1000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_348_7 :
+    Uω (aρ 7) (bρ 7) (5475689309243 / 16000000000000 : ℝ) ≤
+      (-(6639505332453721451779 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1325164821681037 / 5000000000000000 : ℝ) ≤
+      (-(6639505332453721451779 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1325164821681037 / 5000000000000000 : ℝ) =
+            (1325164821681037 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1325164821681037 / 1250000000000000 : ℝ) ≤
+            (291966470546278548221 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2601401661692273 / 10000000000000000 : ℝ)) (q :=
+      (1325164821681037 / 5000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_348_8 :
+    Uω (aρ 8) (bρ 8) (5475689309243 / 16000000000000 : ℝ) ≤
+      (-(14669194304964489016403 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1153174392758377 / 5000000000000000 : ℝ) ≤
+      (-(14669194304964489016403 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1153174392758377 / 5000000000000000 : ℝ) =
+            (1153174392758377 / 625000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1153174392758377 / 625000000000000 : ℝ) ≤
+            (6125221104035510983597 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2196981597341633 / 10000000000000000 : ℝ)) (q :=
+      (1153174392758377 / 5000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_348_9 :
+    Uω (aρ 9) (bρ 9) (5475689309243 / 16000000000000 : ℝ) ≤
+      (-(2124984692148639207169 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (730743044471017 / 4000000000000000 : ℝ) ≤
+      (-(2124984692148639207169 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (730743044471017 / 4000000000000000 : ℝ) =
+            (730743044471017 / 500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (730743044471017 / 500000000000000 : ℝ) ≤
+            (474317233976360792831 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (157896487702321 / 1000000000000000 : ℝ)) (q :=
+      (730743044471017 / 4000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_348_10 :
+    Uω (aρ 10) (bρ 10) (5475689309243 / 16000000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_348_11 :
+    Uω (aρ 11) (bρ 11) (5475689309243 / 16000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_348_12 :
+    Uω (aρ 12) (bρ 12) (5475689309243 / 16000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_348_13 :
+    Uω (aρ 13) (bρ 13) (5475689309243 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_348_14 :
+    Uω (aρ 14) (bρ 14) (5475689309243 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_348_15 :
+    Uω (aρ 15) (bρ 15) (5475689309243 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_348_16 :
+    Uω (aρ 16) (bρ 16) (5475689309243 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_348 :
+    Uρ (5475689309243 / 16000000000000 : ℝ) ≤
+      (-(15123935587556814908631 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (5475689309243 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2728277181692807866107 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_348_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (5475689309243 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(10984958909324133836937 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_348_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (5475689309243 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1113060882446734250669 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_348_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (5475689309243 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(5689260749156044347883 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_348_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (5475689309243 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(470865098371135128527 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_348_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (5475689309243 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(12372631891703963693447 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_348_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (5475689309243 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(6639505332453721451779 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_348_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (5475689309243 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(14669194304964489016403 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_348_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (5475689309243 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(2124984692148639207169 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_348_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (5475689309243 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_348_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (5475689309243 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_348_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (5475689309243 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_348_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (5475689309243 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_348_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (5475689309243 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_348_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (5475689309243 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_348_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (5475689309243 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_348_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_349_1 :
+    Uω (aρ 1) (bρ 1) (86772388539 / 250000000000 : ℝ) ≤
+      (-(2692358756001912750639 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6812701725498661 / 20000000000000000 : ℝ) ≤
+      (-(2692358756001912750639 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6812701725498661 / 20000000000000000 : ℝ) =
+            (6812701725498661 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6812701725498661 / 5000000000000000 : ℝ) ≤
+            (773377145498087249361 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3406303402023661 / 10000000000000000 : ℝ)) (q :=
+      (6812701725498661 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_349_2 :
+    Uω (aρ 2) (bρ 2) (86772388539 / 250000000000 : ℝ) ≤
+      (-(2710061290822971316719 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6764631284279317 / 20000000000000000 : ℝ) ≤
+      (-(2710061290822971316719 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6764631284279317 / 20000000000000000 : ℝ) =
+            (6764631284279317 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6764631284279317 / 5000000000000000 : ℝ) ≤
+            (755674610677028683281 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3382001973314317 / 10000000000000000 : ℝ)) (q :=
+      (6764631284279317 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_349_3 :
+    Uω (aρ 3) (bρ 3) (86772388539 / 250000000000 : ℝ) ≤
+      (-(2745937973915816665167 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1333649502255977 / 4000000000000000 : ℝ) ≤
+      (-(2745937973915816665167 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1333649502255977 / 4000000000000000 : ℝ) =
+            (1333649502255977 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1333649502255977 / 1000000000000000 : ℝ) ≤
+            (719797927584183334833 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (666602861332977 / 2000000000000000 : ℝ)) (q :=
+      (1333649502255977 / 4000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_349_4 :
+    Uω (aρ 4) (bρ 4) (86772388539 / 250000000000 : ℝ) ≤
+      (-(1403488191972210064163 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3253705624260179 / 10000000000000000 : ℝ) ≤
+      (-(1403488191972210064163 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3253705624260179 / 10000000000000000 : ℝ) =
+            (3253705624260179 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3253705624260179 / 2500000000000000 : ℝ) ≤
+            (329379758777789935837 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1625236112485179 / 5000000000000000 : ℝ)) (q :=
+      (3253705624260179 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_349_5 :
+    Uω (aρ 5) (bρ 5) (86772388539 / 250000000000 : ℝ) ≤
+      (-(181480419862451201021 / 156250000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (626048454777501 / 2000000000000000 : ℝ) ≤
+      (-(181480419862451201021 / 156250000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (626048454777501 / 2000000000000000 : ℝ) =
+            (626048454777501 / 500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (626048454777501 / 500000000000000 : ℝ) ≤
+            (35128073981298798979 / 156250000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (312173544356001 / 1000000000000000 : ℝ)) (q :=
+      (626048454777501 / 2000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_349_6 :
+    Uω (aρ 6) (bρ 6) (86772388539 / 250000000000 : ℝ) ≤
+      (-(488215251642030165167 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (590142675128467 / 2000000000000000 : ℝ) ≤
+      (-(488215251642030165167 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (590142675128467 / 2000000000000000 : ℝ) =
+            (590142675128467 / 500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (590142675128467 / 500000000000000 : ℝ) ≤
+            (66302492597969834833 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (292991802220967 / 1000000000000000 : ℝ)) (q :=
+      (590142675128467 / 2000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_349_7 :
+    Uω (aρ 7) (bρ 7) (86772388539 / 250000000000 : ℝ) ≤
+      (-(3273503492390745150533 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1079926478810777 / 4000000000000000 : ℝ) ≤
+      (-(3273503492390745150533 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1079926478810777 / 4000000000000000 : ℝ) =
+            (1079926478810777 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1079926478810777 / 1000000000000000 : ℝ) ≤
+            (192232409109254849467 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (530357009147777 / 2000000000000000 : ℝ)) (q :=
+      (1079926478810777 / 4000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_349_8 :
+    Uω (aρ 8) (bρ 8) (86772388539 / 250000000000 : ℝ) ≤
+      (-(3612667828037066073951 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (294662813728397 / 1250000000000000 : ℝ) ≤
+      (-(3612667828037066073951 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (294662813728397 / 1250000000000000 : ℝ) =
+            (294662813728397 / 156250000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (294662813728397 / 156250000000000 : ℝ) ≤
+            (1585936024212933926049 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (281287415334919 / 1250000000000000 : ℝ)) (q :=
+      (294662813728397 / 1250000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_349_9 :
+    Uω (aρ 9) (bρ 9) (86772388539 / 250000000000 : ℝ) ≤
+      (-(16698172633618705145123 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3765629365421181 / 20000000000000000 : ℝ) ≤
+      (-(16698172633618705145123 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3765629365421181 / 20000000000000000 : ℝ) =
+            (3765629365421181 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3765629365421181 / 2500000000000000 : ℝ) ≤
+            (4096242775381294854877 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1642289296806181 / 10000000000000000 : ℝ)) (q :=
+      (3765629365421181 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_349_10 :
+    Uω (aρ 10) (bρ 10) (86772388539 / 250000000000 : ℝ) ≤
+      (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_10]) (by norm_num [bρ_10])]; exact Lmid_10
+
+lemma U_349_11 :
+    Uω (aρ 11) (bρ 11) (86772388539 / 250000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_349_12 :
+    Uω (aρ 12) (bρ 12) (86772388539 / 250000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_349_13 :
+    Uω (aρ 13) (bρ 13) (86772388539 / 250000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_349_14 :
+    Uω (aρ 14) (bρ 14) (86772388539 / 250000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_349_15 :
+    Uω (aρ 15) (bρ 15) (86772388539 / 250000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_349_16 :
+    Uω (aρ 16) (bρ 16) (86772388539 / 250000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_349 :
+    Uρ (86772388539 / 250000000000 : ℝ) ≤
+      (-(3754352408588390882517 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (86772388539 / 250000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2692358756001912750639 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_349_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (86772388539 / 250000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(2710061290822971316719 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_349_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (86772388539 / 250000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(2745937973915816665167 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_349_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (86772388539 / 250000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1403488191972210064163 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_349_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (86772388539 / 250000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(181480419862451201021 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_349_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (86772388539 / 250000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(488215251642030165167 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_349_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (86772388539 / 250000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(3273503492390745150533 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_349_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (86772388539 / 250000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(3612667828037066073951 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_349_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (86772388539 / 250000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(16698172633618705145123 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_349_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (86772388539 / 250000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12224255358108186379559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_349_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (86772388539 / 250000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_349_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (86772388539 / 250000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_349_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (86772388539 / 250000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_349_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (86772388539 / 250000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_349_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (86772388539 / 250000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_349_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (86772388539 / 250000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_349_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_350_1 :
+    Uω (aρ 1) (bρ 1) (11190582883251 / 32000000000000 : ℝ) ≤
+      (-(10692924948722708581073 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13730051335768191 / 40000000000000000 : ℝ) ≤
+      (-(10692924948722708581073 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (13730051335768191 / 40000000000000000 : ℝ) =
+            (13730051335768191 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13730051335768191 / 10000000000000000 : ℝ) ≤
+            (3170018657277291418927 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1716232867476579 / 5000000000000000 : ℝ)) (q :=
+      (13730051335768191 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_350_2 :
+    Uω (aρ 2) (bρ 2) (11190582883251 / 32000000000000 : ℝ) ≤
+      (-(5381593741458052326591 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13633918637464121 / 40000000000000000 : ℝ) ≤
+      (-(5381593741458052326591 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (13633918637464121 / 40000000000000000 : ℝ) =
+            (13633918637464121 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13633918637464121 / 10000000000000000 : ℝ) ≤
+            (1549878061541947673409 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3408168398311123 / 10000000000000000 : ℝ)) (q :=
+      (13633918637464121 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_350_3 :
+    Uω (aρ 3) (bρ 3) (11190582883251 / 32000000000000 : ℝ) ≤
+      (-(10905566206672627143071 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13441176022308067 / 40000000000000000 : ℝ) ≤
+      (-(10905566206672627143071 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (13441176022308067 / 40000000000000000 : ℝ) =
+            (13441176022308067 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13441176022308067 / 10000000000000000 : ℝ) ≤
+            (2957377399327372856929 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (419899149635387 / 1250000000000000 : ℝ)) (q :=
+      (13441176022308067 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_350_4 :
+    Uω (aρ 4) (bρ 4) (11190582883251 / 32000000000000 : ℝ) ≤
+      (-(11147742773399472978853 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2623914439939867 / 8000000000000000 : ℝ) ≤
+      (-(11147742773399472978853 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2623914439939867 / 8000000000000000 : ℝ) =
+            (2623914439939867 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2623914439939867 / 2000000000000000 : ℝ) ≤
+            (2715200832600527021147 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (327668546684373 / 1000000000000000 : ℝ)) (q :=
+      (2623914439939867 / 8000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_350_5 :
+    Uω (aρ 5) (bρ 5) (11190582883251 / 32000000000000 : ℝ) ≤
+      (-(5765646633728764278609 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12625898322004909 / 40000000000000000 : ℝ) ≤
+      (-(5765646633728764278609 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (12625898322004909 / 40000000000000000 : ℝ) =
+            (12625898322004909 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12625898322004909 / 10000000000000000 : ℝ) ≤
+            (1165825169271235721391 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3148038447331517 / 10000000000000000 : ℝ)) (q :=
+      (12625898322004909 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_350_6 :
+    Uω (aρ 6) (bρ 6) (11190582883251 / 32000000000000 : ℝ) ≤
+      (-(12116491871022092607709 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11908236058652281 / 40000000000000000 : ℝ) ≤
+      (-(12116491871022092607709 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (11908236058652281 / 40000000000000000 : ℝ) =
+            (11908236058652281 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11908236058652281 / 10000000000000000 : ℝ) ≤
+            (1746451734977907392291 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2956447690795203 / 10000000000000000 : ℝ)) (q :=
+      (11908236058652281 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_350_7 :
+    Uω (aρ 7) (bρ 7) (11190582883251 / 32000000000000 : ℝ) ≤
+      (-(12995858129784667405647 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10905787816146419 / 40000000000000000 : ℝ) ≤
+      (-(12995858129784667405647 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (10905787816146419 / 40000000000000000 : ℝ) =
+            (10905787816146419 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10905787816146419 / 10000000000000000 : ℝ) ≤
+            (867085476215332594353 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (41857577348473 / 156250000000000 : ℝ)) (q :=
+      (10905787816146419 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_350_8 :
+    Uω (aρ 8) (bρ 8) (11190582883251 / 32000000000000 : ℝ) ≤
+      (-(7167573149467358629037 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (76310181146337 / 320000000000000 : ℝ) ≤
+      (-(7167573149467358629037 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (76310181146337 / 320000000000000 : ℝ) = (76310181146337 / 40000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (76310181146337 / 40000000000000 : ℝ) ≤
+            (3229634555032641370963 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (18231352121721 / 80000000000000 : ℝ)) (q :=
+      (76310181146337 / 320000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_350_9 :
+    Uω (aρ 9) (bρ 9) (11190582883251 / 32000000000000 : ℝ) ≤
+      (-(8270243481336133218047 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7650957145470101 / 40000000000000000 : ℝ) ≤
+      (-(8270243481336133218047 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (7650957145470101 / 40000000000000000 : ℝ) =
+            (7650957145470101 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7650957145470101 / 5000000000000000 : ℝ) ≤
+            (2126964223163866781953 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1675976894664113 / 10000000000000000 : ℝ)) (q :=
+      (7650957145470101 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_350_10 :
+    Uω (aρ 10) (bρ 10) (11190582883251 / 32000000000000 : ℝ) ≤
+      (-(2271398553081343851221 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4126711735908559 / 40000000000000000 : ℝ) ≤
+      (-(2271398553081343851221 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (4126711735908559 / 40000000000000000 : ℝ) =
+            (4126711735908559 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4126711735908559 / 2500000000000000 : ℝ) ≤
+            (501190168118656148779 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (151206611689171 / 5000000000000000 : ℝ)) (q :=
+      (4126711735908559 / 40000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_350_11 :
+    Uω (aρ 11) (bρ 11) (11190582883251 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_350_12 :
+    Uω (aρ 12) (bρ 12) (11190582883251 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_350_13 :
+    Uω (aρ 13) (bρ 13) (11190582883251 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_350_14 :
+    Uω (aρ 14) (bρ 14) (11190582883251 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_350_15 :
+    Uω (aρ 15) (bρ 15) (11190582883251 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_350_16 :
+    Uω (aρ 16) (bρ 16) (11190582883251 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_350 :
+    Uρ (11190582883251 / 32000000000000 : ℝ) ≤
+      (-(14812819820006037121807 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (11190582883251 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(10692924948722708581073 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_350_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (11190582883251 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(5381593741458052326591 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_350_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (11190582883251 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(10905566206672627143071 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_350_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (11190582883251 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(11147742773399472978853 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_350_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (11190582883251 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(5765646633728764278609 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_350_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (11190582883251 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(12116491871022092607709 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_350_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (11190582883251 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(12995858129784667405647 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_350_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (11190582883251 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(7167573149467358629037 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_350_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (11190582883251 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(8270243481336133218047 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_350_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (11190582883251 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(2271398553081343851221 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_350_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (11190582883251 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_350_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (11190582883251 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_350_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (11190582883251 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_350_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (11190582883251 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_350_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (11190582883251 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_350_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (11190582883251 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_350_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_351_1 :
+    Uω (aρ 1) (bρ 1) (1127430003351 / 3200000000000 : ℝ) ≤
+      (-(2123399165295955899713 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6917349599324441 / 20000000000000000 : ℝ) ≤
+      (-(2123399165295955899713 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6917349599324441 / 20000000000000000 : ℝ) =
+            (6917349599324441 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6917349599324441 / 5000000000000000 : ℝ) ≤
+            (649189555904044100287 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1729314028468783 / 5000000000000000 : ℝ)) (q :=
+      (6917349599324441 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_351_2 :
+    Uω (aρ 2) (bρ 2) (1127430003351 / 3200000000000 : ℝ) ≤
+      (-(2671679790056744699741 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1373857455962341 / 4000000000000000 : ℝ) ≤
+      (-(2671679790056744699741 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1373857455962341 / 4000000000000000 : ℝ) =
+            (1373857455962341 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1373857455962341 / 1000000000000000 : ℝ) ≤
+            (794056111443255300259 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (343433474993483 / 1000000000000000 : ℝ)) (q :=
+      (1373857455962341 / 4000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_351_3 :
+    Uω (aρ 3) (bρ 3) (1127430003351 / 3200000000000 : ℝ) ≤
+      (-(10827987472984737147193 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3386464121925491 / 10000000000000000 : ℝ) ≤
+      (-(10827987472984737147193 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3386464121925491 / 10000000000000000 : ℝ) =
+            (3386464121925491 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3386464121925491 / 2500000000000000 : ℝ) ≤
+            (3034956133015262852807 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3385371820324107 / 10000000000000000 : ℝ)) (q :=
+      (3386464121925491 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_351_4 :
+    Uω (aρ 4) (bρ 4) (1127430003351 / 3200000000000 : ℝ) ≤
+      (-(5534109374562723863797 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (826520016553917 / 2500000000000000 : ℝ) ≤
+      (-(5534109374562723863797 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (826520016553917 / 2500000000000000 : ℝ) =
+            (826520016553917 / 625000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (826520016553917 / 625000000000000 : ℝ) ≤
+            (1397362428437276136203 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3302897889969461 / 10000000000000000 : ℝ)) (q :=
+      (826520016553917 / 2500000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_351_5 :
+    Uω (aρ 5) (bρ 5) (1127430003351 / 3200000000000 : ℝ) ≤
+      (-(2862133506329783265521 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3182705718651991 / 10000000000000000 : ℝ) ≤
+      (-(2862133506329783265521 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3182705718651991 / 10000000000000000 : ℝ) =
+            (3182705718651991 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3182705718651991 / 2500000000000000 : ℝ) ≤
+            (603602395170216734479 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3174339114177107 / 10000000000000000 : ℝ)) (q :=
+      (3182705718651991 / 10000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_351_6 :
+    Uω (aρ 6) (bρ 6) (1127430003351 / 3200000000000 : ℝ) ≤
+      (-(6014198225391302158967 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (300340140298413 / 1000000000000000 : ℝ) ≤
+      (-(6014198225391302158967 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (300340140298413 / 1000000000000000 : ℝ) =
+            (300340140298413 / 250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (300340140298413 / 250000000000000 : ℝ) ≤
+            (917273577608697841033 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (596594171596277 / 2000000000000000 : ℝ)) (q :=
+      (300340140298413 / 1000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_351_7 :
+    Uω (aρ 7) (bρ 7) (1127430003351 / 3200000000000 : ℝ) ≤
+      (-(50385507542380392039 / 39062500000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5506136953700713 / 20000000000000000 : ℝ) ≤
+      (-(50385507542380392039 / 39062500000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5506136953700713 / 20000000000000000 : ℝ) =
+            (5506136953700713 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5506136953700713 / 5000000000000000 : ℝ) ≤
+            (3766615918557107961 / 39062500000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1352983193236919 / 5000000000000000 : ℝ)) (q :=
+      (5506136953700713 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_351_8 :
+    Uω (aρ 8) (bρ 8) (1127430003351 / 3200000000000 : ℝ) ≤
+      (-(2844212325531341812721 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4824109267652769 / 20000000000000000 : ℝ) ≤
+      (-(2844212325531341812721 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4824109267652769 / 20000000000000000 : ℝ) =
+            (4824109267652769 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4824109267652769 / 2500000000000000 : ℝ) ≤
+            (1314670756268658187279 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1153740175882947 / 5000000000000000 : ℝ)) (q :=
+      (4824109267652769 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_351_9 :
+    Uω (aρ 9) (bρ 9) (1127430003351 / 3200000000000 : ℝ) ≤
+      (-(4096481844382657276603 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3885064301949891 / 20000000000000000 : ℝ) ≤
+      (-(4096481844382657276603 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3885064301949891 / 20000000000000000 : ℝ) =
+            (3885064301949891 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3885064301949891 / 2500000000000000 : ℝ) ≤
+            (1102122007867342723397 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (213675126802877 / 1250000000000000 : ℝ)) (q :=
+      (3885064301949891 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_351_10 :
+    Uω (aρ 10) (bρ 10) (1127430003351 / 3200000000000 : ℝ) ≤
+      (-(219985771669146987571 / 100000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (221637849617413 / 2000000000000000 : ℝ) ≤
+      (-(219985771669146987571 / 100000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (221637849617413 / 2000000000000000 : ℝ) =
+            (221637849617413 / 125000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (221637849617413 / 125000000000000 : ℝ) ≤
+            (57273100450853012429 / 100000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (85854848428451 / 2000000000000000 : ℝ)) (q :=
+      (221637849617413 / 2000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10]) (by norm_num [bρ_10])
+      (by norm_num) (by norm_num [aρ_10, bρ_10]) (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_351_11 :
+    Uω (aρ 11) (bρ 11) (1127430003351 / 3200000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_351_12 :
+    Uω (aρ 12) (bρ 12) (1127430003351 / 3200000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_351_13 :
+    Uω (aρ 13) (bρ 13) (1127430003351 / 3200000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_351_14 :
+    Uω (aρ 14) (bρ 14) (1127430003351 / 3200000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_351_15 :
+    Uω (aρ 15) (bρ 15) (1127430003351 / 3200000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_351_16 :
+    Uω (aρ 16) (bρ 16) (1127430003351 / 3200000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_351 :
+    Uρ (1127430003351 / 3200000000000 : ℝ) ≤
+      (-(14696020735188019821327 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1127430003351 / 3200000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2123399165295955899713 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_351_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1127430003351 / 3200000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(2671679790056744699741 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_351_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1127430003351 / 3200000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(10827987472984737147193 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_351_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1127430003351 / 3200000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(5534109374562723863797 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_351_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1127430003351 / 3200000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(2862133506329783265521 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_351_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1127430003351 / 3200000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(6014198225391302158967 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_351_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1127430003351 / 3200000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) * (-(50385507542380392039 / 39062500000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_351_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1127430003351 / 3200000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(2844212325531341812721 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_351_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1127430003351 / 3200000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(4096481844382657276603 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_351_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1127430003351 / 3200000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) * (-(219985771669146987571 / 100000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_351_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1127430003351 / 3200000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_351_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1127430003351 / 3200000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_351_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1127430003351 / 3200000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_351_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1127430003351 / 3200000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_351_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1127430003351 / 3200000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_351_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1127430003351 / 3200000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_351_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/U29.lean b/LeanPool/Zeta5Irrational/Table/U29.lean
new file mode 100644
index 0000000000..2169fe7cbd
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/U29.lean
@@ -0,0 +1,4156 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-! # Certified arcsine potential bounds (U29) -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma U_352_1 :
+    Uω (aρ 1) (bρ 1) (11358017183769 / 32000000000000 : ℝ) ≤
+      (-(10541638900952844384301 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13939347040132423 / 40000000000000000 : ℝ) ≤
+      (-(10541638900952844384301 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (13939347040132423 / 40000000000000000 : ℝ) =
+            (13939347040132423 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13939347040132423 / 10000000000000000 : ℝ) ≤
+            (3321304705047155615699 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3484790368223399 / 10000000000000000 : ℝ)) (q :=
+      (13939347040132423 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_352_2 :
+    Uω (aρ 2) (bρ 2) (11358017183769 / 32000000000000 : ℝ) ≤
+      (-(1061083124688201051313 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13843230338365333 / 40000000000000000 : ℝ) ≤
+      (-(1061083124688201051313 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (13843230338365333 / 40000000000000000 : ℝ) =
+            (13843230338365333 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13843230338365333 / 10000000000000000 : ℝ) ≤
+            (325211235911798948687 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1730250514924927 / 5000000000000000 : ℝ)) (q :=
+      (13843230338365333 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_352_3 :
+    Uω (aρ 3) (bρ 3) (11358017183769 / 32000000000000 : ℝ) ≤
+      (-(2150201267515562619751 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2730107286208521 / 8000000000000000 : ℝ) ≤
+      (-(2150201267515562619751 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2730107286208521 / 8000000000000000 : ℝ) =
+            (2730107286208521 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2730107286208521 / 2000000000000000 : ℝ) ≤
+            (622387453684437380249 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (341155018253849 / 1000000000000000 : ℝ)) (q :=
+      (2730107286208521 / 8000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_352_4 :
+    Uω (aρ 4) (bρ 4) (11358017183769 / 32000000000000 : ℝ) ≤
+      (-(5494661672584152532607 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13329066731210191 / 40000000000000000 : ℝ) ≤
+      (-(5494661672584152532607 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (13329066731210191 / 40000000000000000 : ℝ) =
+            (13329066731210191 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13329066731210191 / 10000000000000000 : ℝ) ≤
+            (1436810130415847467393 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3329109513687283 / 10000000000000000 : ℝ)) (q :=
+      (13329066731210191 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_352_5 :
+    Uω (aρ 5) (bρ 5) (11358017183769 / 32000000000000 : ℝ) ≤
+      (-(5683228812996067015517 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12835742868578809 / 40000000000000000 : ℝ) ≤
+      (-(5683228812996067015517 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (12835742868578809 / 40000000000000000 : ℝ) =
+            (12835742868578809 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12835742868578809 / 10000000000000000 : ℝ) ≤
+            (1248242990003932984483 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (100019921928331 / 312500000000000 : ℝ)) (q :=
+      (12835742868578809 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_352_6 :
+    Uω (aρ 6) (bρ 6) (11358017183769 / 32000000000000 : ℝ) ≤
+      (-(5970540392073785020387 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12118962506210219 / 40000000000000000 : ℝ) ≤
+      (-(5970540392073785020387 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (12118962506210219 / 40000000000000000 : ℝ) =
+            (12118962506210219 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12118962506210219 / 10000000000000000 : ℝ) ≤
+            (960931410926214979613 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3009487695662297 / 10000000000000000 : ℝ)) (q :=
+      (12118962506210219 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_352_7 :
+    Uω (aρ 7) (bρ 7) (11358017183769 / 32000000000000 : ℝ) ≤
+      (-(6401244525886062483541 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11118724159886299 / 40000000000000000 : ℝ) ≤
+      (-(6401244525886062483541 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (11118724159886299 / 40000000000000000 : ℝ) =
+            (11118724159886299 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11118724159886299 / 10000000000000000 : ℝ) ≤
+            (530227277113937516459 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2733029903260337 / 10000000000000000 : ℝ)) (q :=
+      (11118724159886299 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_352_8 :
+    Uω (aρ 8) (bρ 8) (11358017183769 / 32000000000000 : ℝ) ≤
+      (-(2821675808831815775213 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9757551996349087 / 40000000000000000 : ℝ) ≤
+      (-(2821675808831815775213 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (9757551996349087 / 40000000000000000 : ℝ) =
+            (9757551996349087 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9757551996349087 / 5000000000000000 : ℝ) ≤
+            (1337207272968184224787 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2335985472831731 / 10000000000000000 : ℝ)) (q :=
+      (9757551996349087 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_352_9 :
+    Uω (aρ 9) (bρ 9) (11358017183769 / 32000000000000 : ℝ) ≤
+      (-(4058587466679611612501 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7888803428680491 / 40000000000000000 : ℝ) ≤
+      (-(4058587466679611612501 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (7888803428680491 / 40000000000000000 : ℝ) =
+            (7888803428680491 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7888803428680491 / 5000000000000000 : ℝ) ≤
+            (1140016385570388387499 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1742576817357433 / 10000000000000000 : ℝ)) (q :=
+      (7888803428680491 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_352_10 :
+    Uω (aρ 10) (bρ 10) (11358017183769 / 32000000000000 : ℝ) ≤
+      (-(10725842193563147263599 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4681932791743299 / 40000000000000000 : ℝ) ≤
+      (-(10725842193563147263599 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (4681932791743299 / 40000000000000000 : ℝ) =
+            (4681932791743299 / 2500000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4681932791743299 / 2500000000000000 : ℝ) ≤
+            (3137101412436852736401 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (527700532383837 / 10000000000000000 : ℝ)) (q :=
+      (4681932791743299 / 40000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_352_11 :
+    Uω (aρ 11) (bρ 11) (11358017183769 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_352_12 :
+    Uω (aρ 12) (bρ 12) (11358017183769 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_352_13 :
+    Uω (aρ 13) (bρ 13) (11358017183769 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_352_14 :
+    Uω (aρ 14) (bρ 14) (11358017183769 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_352_15 :
+    Uω (aρ 15) (bρ 15) (11358017183769 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_352_16 :
+    Uω (aρ 16) (bρ 16) (11358017183769 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_352 :
+    Uρ (11358017183769 / 32000000000000 : ℝ) ≤
+      (-(14594288429791755881307 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (11358017183769 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(10541638900952844384301 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_352_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (11358017183769 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1061083124688201051313 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_352_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (11358017183769 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(2150201267515562619751 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_352_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (11358017183769 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(5494661672584152532607 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_352_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (11358017183769 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(5683228812996067015517 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_352_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (11358017183769 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(5970540392073785020387 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_352_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (11358017183769 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(6401244525886062483541 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_352_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (11358017183769 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(2821675808831815775213 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_352_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (11358017183769 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(4058587466679611612501 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_352_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (11358017183769 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(10725842193563147263599 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_352_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (11358017183769 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_352_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (11358017183769 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_352_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (11358017183769 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_352_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (11358017183769 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_352_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (11358017183769 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_352_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (11358017183769 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_352_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_353_1 :
+    Uω (aρ 1) (bρ 1) (22799751517797 / 64000000000000 : ℝ) ≤
+      (-(10504172329495590335047 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27983341906000601 / 80000000000000000 : ℝ) ≤
+      (-(10504172329495590335047 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (27983341906000601 / 80000000000000000 : ℝ) =
+            (27983341906000601 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27983341906000601 / 20000000000000000 : ℝ) ≤
+            (3358771276504409664953 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3497871519929369 / 10000000000000000 : ℝ)) (q :=
+      (27983341906000601 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_353_2 :
+    Uω (aρ 2) (bρ 2) (22799751517797 / 64000000000000 : ℝ) ≤
+      (-(10573102202839460368983 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27791116349935233 / 80000000000000000 : ℝ) ≤
+      (-(10573102202839460368983 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (27791116349935233 / 80000000000000000 : ℝ) =
+            (27791116349935233 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27791116349935233 / 20000000000000000 : ℝ) ≤
+            (3289841403160539631017 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3473584143423027 / 10000000000000000 : ℝ)) (q :=
+      (27791116349935233 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_353_3 :
+    Uω (aρ 3) (bρ 3) (22799751517797 / 64000000000000 : ℝ) ≤
+      (-(5356368502045391193103 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27405752421368369 / 80000000000000000 : ℝ) ≤
+      (-(5356368502045391193103 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (27405752421368369 / 80000000000000000 : ℝ) =
+            (27405752421368369 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27405752421368369 / 20000000000000000 : ℝ) ≤
+            (1575103300954608806897 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3424639267631311 / 10000000000000000 : ℝ)) (q :=
+      (27405752421368369 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_353_4 :
+    Uω (aρ 4) (bρ 4) (22799751517797 / 64000000000000 : ℝ) ≤
+      (-(684381767471699544513 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5352575750633297 / 16000000000000000 : ℝ) ≤
+      (-(684381767471699544513 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5352575750633297 / 16000000000000000 : ℝ) =
+            (5352575750633297 / 4000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5352575750633297 / 4000000000000000 : ℝ) ≤
+            (182052207903300455487 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (41777687895573 / 125000000000000 : ℝ)) (q :=
+      (5352575750633297 / 16000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_353_5 :
+    Uω (aρ 5) (bρ 5) (22799751517797 / 64000000000000 : ℝ) ≤
+      (-(11325671971142371422989 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25776402382098673 / 80000000000000000 : ℝ) ≤
+      (-(11325671971142371422989 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (25776402382098673 / 80000000000000000 : ℝ) =
+            (25776402382098673 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25776402382098673 / 20000000000000000 : ℝ) ≤
+            (2537271634857628577011 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3213785858213887 / 10000000000000000 : ℝ)) (q :=
+      (25776402382098673 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_353_6 :
+    Uω (aρ 6) (bρ 6) (22799751517797 / 64000000000000 : ℝ) ≤
+      (-(11897711012095434041723 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24343272620661961 / 80000000000000000 : ℝ) ≤
+      (-(11897711012095434041723 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (24343272620661961 / 80000000000000000 : ℝ) =
+            (24343272620661961 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24343272620661961 / 20000000000000000 : ℝ) ≤
+            (1965232593904565958277 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3022743792994709 / 10000000000000000 : ℝ)) (q :=
+      (24343272620661961 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_353_7 :
+    Uω (aρ 7) (bρ 7) (22799751517797 / 64000000000000 : ℝ) ≤
+      (-(6377372591149652653381 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22343872355375393 / 80000000000000000 : ℝ) ≤
+      (-(6377372591149652653381 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (22343872355375393 / 80000000000000000 : ℝ) =
+            (22343872355375393 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22343872355375393 / 20000000000000000 : ℝ) ≤
+            (554099211850347346619 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2746555107433067 / 10000000000000000 : ℝ)) (q :=
+      (22343872355375393 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_353_8 :
+    Uω (aρ 8) (bρ 8) (22799751517797 / 64000000000000 : ℝ) ≤
+      (-(878284501219037432057 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19624355687976101 / 80000000000000000 : ℝ) ≤
+      (-(878284501219037432057 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (19624355687976101 / 80000000000000000 : ℝ) =
+            (19624355687976101 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19624355687976101 / 10000000000000000 : ℝ) ≤
+            (421366461843462567943 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (587554397980811 / 2500000000000000 : ℝ)) (q :=
+      (19624355687976101 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_353_9 :
+    Uω (aρ 9) (bρ 9) (22799751517797 / 64000000000000 : ℝ) ≤
+      (-(8079818672537926407997 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15895926785369481 / 80000000000000000 : ℝ) ≤
+      (-(8079818672537926407997 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (15895926785369481 / 80000000000000000 : ℝ) =
+            (15895926785369481 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15895926785369481 / 10000000000000000 : ℝ) ≤
+            (2317389031962073592003 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1759075994631589 / 10000000000000000 : ℝ)) (q :=
+      (15895926785369481 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_353_10 :
+    Uω (aρ 10) (bρ 10) (22799751517797 / 64000000000000 : ℝ) ≤
+      (-(10606782922981723231029 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1917902491631817 / 16000000000000000 : ℝ) ≤
+      (-(10606782922981723231029 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (1917902491631817 / 16000000000000000 : ℝ) =
+            (1917902491631817 / 1000000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1917902491631817 / 1000000000000000 : ℝ) ≤
+            (3256160683018276768971 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (57103144632399 / 1000000000000000 : ℝ)) (q :=
+      (1917902491631817 / 16000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_353_11 :
+    Uω (aρ 11) (bρ 11) (22799751517797 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_353_12 :
+    Uω (aρ 12) (bρ 12) (22799751517797 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_353_13 :
+    Uω (aρ 13) (bρ 13) (22799751517797 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_353_14 :
+    Uω (aρ 14) (bρ 14) (22799751517797 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_353_15 :
+    Uω (aρ 15) (bρ 15) (22799751517797 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_353_16 :
+    Uω (aρ 16) (bρ 16) (22799751517797 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_353 :
+    Uρ (22799751517797 / 64000000000000 : ℝ) ≤
+      (-(14546684338120957832659 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (22799751517797 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(10504172329495590335047 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_353_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (22799751517797 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(10573102202839460368983 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_353_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (22799751517797 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(5356368502045391193103 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_353_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (22799751517797 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(684381767471699544513 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_353_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (22799751517797 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(11325671971142371422989 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_353_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (22799751517797 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(11897711012095434041723 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_353_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (22799751517797 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(6377372591149652653381 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_353_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (22799751517797 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(878284501219037432057 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_353_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (22799751517797 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(8079818672537926407997 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_353_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (22799751517797 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(10606782922981723231029 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_353_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (22799751517797 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_353_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (22799751517797 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_353_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (22799751517797 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_353_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (22799751517797 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_353_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (22799751517797 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_353_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (22799751517797 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_353_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_354_1 :
+    Uω (aρ 1) (bρ 1) (2860433583507 / 8000000000000 : ℝ) ≤
+      (-(5233422806154407011667 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7021997430348573 / 20000000000000000 : ℝ) ≤
+      (-(5233422806154407011667 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7021997430348573 / 20000000000000000 : ℝ) =
+            (7021997430348573 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7021997430348573 / 5000000000000000 : ℝ) ≤
+            (1698048996845592988333 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3510952669049823 / 10000000000000000 : ℝ)) (q :=
+      (7021997430348573 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_354_2 :
+    Uω (aρ 2) (bρ 2) (2860433583507 / 8000000000000 : ℝ) ≤
+      (-(2107102999367314674609 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1743485747114849 / 5000000000000000 : ℝ) ≤
+      (-(2107102999367314674609 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1743485747114849 / 5000000000000000 : ℝ) =
+            (1743485747114849 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1743485747114849 / 1250000000000000 : ℝ) ≤
+            (665485721832685325391 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1743333619835323 / 5000000000000000 : ℝ)) (q :=
+      (1743485747114849 / 5000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_354_3 :
+    Uω (aρ 3) (bρ 3) (2860433583507 / 8000000000000 : ℝ) ≤
+      (-(2668653414571848144461 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1719401983032041 / 5000000000000000 : ℝ) ≤
+      (-(2668653414571848144461 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1719401983032041 / 5000000000000000 : ℝ) =
+            (1719401983032041 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1719401983032041 / 1250000000000000 : ℝ) ≤
+            (797082486928151855539 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1718864144844707 / 5000000000000000 : ℝ)) (q :=
+      (1719401983032041 / 5000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_354_4 :
+    Uω (aρ 4) (bρ 4) (2860433583507 / 8000000000000 : ℝ) ≤
+      (-(218220933654013876573 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (671690581810577 / 2000000000000000 : ℝ) ≤
+      (-(218220933654013876573 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (671690581810577 / 2000000000000000 : ℝ) =
+            (671690581810577 / 500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (671690581810577 / 500000000000000 : ℝ) ≤
+            (59037938465986123427 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (167766017836601 / 500000000000000 : ℝ)) (q :=
+      (671690581810577 / 2000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_354_5 :
+    Uω (aρ 5) (bρ 5) (2860433583507 / 8000000000000 : ℝ) ≤
+      (-(5642526418193389259393 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3235164603842813 / 10000000000000000 : ℝ) ≤
+      (-(5642526418193389259393 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3235164603842813 / 10000000000000000 : ℝ) =
+            (3235164603842813 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3235164603842813 / 2500000000000000 : ℝ) ≤
+            (1288945384806610740607 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (806733416411719 / 2500000000000000 : ℝ)) (q :=
+      (3235164603842813 / 10000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_354_6 :
+    Uω (aρ 6) (bρ 6) (2860433583507 / 8000000000000 : ℝ) ≤
+      (-(2963632752427058012139 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1528038384146267 / 5000000000000000 : ℝ) ≤
+      (-(2963632752427058012139 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1528038384146267 / 5000000000000000 : ℝ) =
+            (1528038384146267 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1528038384146267 / 1250000000000000 : ℝ) ≤
+            (502103149072941987861 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1517999184843159 / 5000000000000000 : ℝ)) (q :=
+      (1528038384146267 / 5000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_354_7 :
+    Uω (aρ 7) (bρ 7) (2860433583507 / 8000000000000 : ℝ) ≤
+      (-(12707235810376217255869 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5612569813918199 / 20000000000000000 : ℝ) ≤
+      (-(12707235810376217255869 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5612569813918199 / 20000000000000000 : ℝ) =
+            (5612569813918199 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5612569813918199 / 5000000000000000 : ℝ) ≤
+            (1155707795623782744131 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2760076027779449 / 10000000000000000 : ℝ)) (q :=
+      (5612569813918199 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_354_8 :
+    Uω (aρ 8) (bρ 8) (2860433583507 / 8000000000000 : ℝ) ≤
+      (-(13997061889017450904763 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (123334713659413 / 500000000000000 : ℝ) ≤
+      (-(13997061889017450904763 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (123334713659413 / 500000000000000 : ℝ) = (123334713659413 / 62500000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (123334713659413 / 62500000000000 : ℝ) ≤
+            (6797353519982549095237 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (236443641157777 / 1000000000000000 : ℝ)) (q :=
+      (123334713659413 / 500000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_354_9 :
+    Uω (aρ 9) (bρ 9) (2860433583507 / 8000000000000 : ℝ) ≤
+      (-(4021405282280517378607 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3127738069893 / 15625000000000 : ℝ) ≤
+      (-(4021405282280517378607 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (3127738069893 / 15625000000000 : ℝ) = (3127738069893 / 1953125000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3127738069893 / 1953125000000 : ℝ) ≤
+            (1177198569969482621393 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (177551822302429 / 1000000000000000 : ℝ)) (q :=
+      (3127738069893 / 15625000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_354_10 :
+    Uω (aρ 10) (bρ 10) (2860433583507 / 8000000000000 : ℝ) ≤
+      (-(2099233755464236526453 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2451005913054431 / 20000000000000000 : ℝ) ≤
+      (-(2099233755464236526453 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (2451005913054431 / 20000000000000000 : ℝ) =
+            (2451005913054431 / 1250000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2451005913054431 / 1250000000000000 : ℝ) ≤
+            (673354965735763473547 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (611578440110681 / 10000000000000000 : ℝ)) (q :=
+      (2451005913054431 / 20000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_354_11 :
+    Uω (aρ 11) (bρ 11) (2860433583507 / 8000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_354_12 :
+    Uω (aρ 12) (bρ 12) (2860433583507 / 8000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_354_13 :
+    Uω (aρ 13) (bρ 13) (2860433583507 / 8000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_354_14 :
+    Uω (aρ 14) (bρ 14) (2860433583507 / 8000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_354_15 :
+    Uω (aρ 15) (bρ 15) (2860433583507 / 8000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_354_16 :
+    Uω (aρ 16) (bρ 16) (2860433583507 / 8000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_354 :
+    Uρ (2860433583507 / 8000000000000 : ℝ) ≤
+      (-(7250341187646458758631 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (2860433583507 / 8000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(5233422806154407011667 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_354_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (2860433583507 / 8000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(2107102999367314674609 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_354_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (2860433583507 / 8000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(2668653414571848144461 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_354_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (2860433583507 / 8000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(218220933654013876573 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_354_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (2860433583507 / 8000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(5642526418193389259393 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_354_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (2860433583507 / 8000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(2963632752427058012139 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_354_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (2860433583507 / 8000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(12707235810376217255869 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_354_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (2860433583507 / 8000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(13997061889017450904763 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_354_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (2860433583507 / 8000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(4021405282280517378607 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_354_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (2860433583507 / 8000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(2099233755464236526453 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_354_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (2860433583507 / 8000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_354_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (2860433583507 / 8000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_354_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (2860433583507 / 8000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_354_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (2860433583507 / 8000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_354_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (2860433583507 / 8000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_354_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (2860433583507 / 8000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_354_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_355_1 :
+    Uω (aρ 1) (bρ 1) (4593437163663 / 12800000000000 : ℝ) ≤
+      (-(10429657709158020939677 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28192637526561087 / 80000000000000000 : ℝ) ≤
+      (-(10429657709158020939677 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (28192637526561087 / 80000000000000000 : ℝ) =
+            (28192637526561087 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28192637526561087 / 20000000000000000 : ℝ) ≤
+            (3433285896841979060323 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3524033815613553 / 10000000000000000 : ℝ)) (q :=
+      (28192637526561087 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_355_2 :
+    Uω (aρ 2) (bρ 2) (4593437163663 / 12800000000000 : ℝ) ≤
+      (-(10498068566233714828257 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28000427489214943 / 80000000000000000 : ℝ) ≤
+      (-(10498068566233714828257 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (28000427489214943 / 80000000000000000 : ℝ) =
+            (28000427489214943 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28000427489214943 / 20000000000000000 : ℝ) ≤
+            (3364875039766285171743 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3499750318787017 / 10000000000000000 : ℝ)) (q :=
+      (28000427489214943 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_355_3 :
+    Uω (aρ 3) (bρ 3) (4593437163663 / 12800000000000 : ℝ) ≤
+      (-(10636635189896886432221 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1104604431455487 / 3200000000000000 : ℝ) ≤
+      (-(10636635189896886432221 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1104604431455487 / 3200000000000000 : ℝ) =
+            (1104604431455487 / 800000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1104604431455487 / 800000000000000 : ℝ) ≤
+            (3226308416103113567779 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (138032689977203 / 400000000000000 : ℝ)) (q :=
+      (1104604431455487 / 3200000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_355_4 :
+    Uω (aρ 4) (bρ 4) (4593437163663 / 12800000000000 : ℝ) ≤
+      (-(10872137355885766455087 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (26972367029194927 / 80000000000000000 : ℝ) ≤
+      (-(10872137355885766455087 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (26972367029194927 / 80000000000000000 : ℝ) =
+            (26972367029194927 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26972367029194927 / 20000000000000000 : ℝ) ≤
+            (2990806250114233544913 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3368425491197013 / 10000000000000000 : ℝ)) (q :=
+      (26972367029194927 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_355_5 :
+    Uω (aρ 5) (bρ 5) (4593437163663 / 12800000000000 : ℝ) ≤
+      (-(11244598860684962943507 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25986229109825939 / 80000000000000000 : ℝ) ≤
+      (-(11244598860684962943507 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (25986229109825939 / 80000000000000000 : ℝ) =
+            (25986229109825939 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25986229109825939 / 20000000000000000 : ℝ) ≤
+            (2618344745315037056493 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1620040465344883 / 5000000000000000 : ℝ)) (q :=
+      (25986229109825939 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_355_6 :
+    Uω (aρ 6) (bρ 6) (4593437163663 / 12800000000000 : ℝ) ≤
+      (-(11811539102151936781149 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24553949668774967 / 80000000000000000 : ℝ) ≤
+      (-(11811539102151936781149 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (24553949668774967 / 80000000000000000 : ℝ) =
+            (24553949668774967 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24553949668774967 / 20000000000000000 : ℝ) ≤
+            (2051404503848063218851 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3049251445567023 / 10000000000000000 : ℝ)) (q :=
+      (24553949668774967 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_355_7 :
+    Uω (aρ 7) (bρ 7) (4593437163663 / 12800000000000 : ℝ) ≤
+      (-(12659958571089790164417 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22556669271262279 / 80000000000000000 : ℝ) ≤
+      (-(12659958571089790164417 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (22556669271262279 / 80000000000000000 : ℝ) =
+            (22556669271262279 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22556669271262279 / 20000000000000000 : ℝ) ≤
+            (1202985034910209835583 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2773592726948851 / 10000000000000000 : ℝ)) (q :=
+      (22556669271262279 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_355_8 :
+    Uω (aρ 8) (bρ 8) (4593437163663 / 12800000000000 : ℝ) ≤
+      (-(435684509515087757189 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19842700439289387 / 80000000000000000 : ℝ) ≤
+      (-(435684509515087757189 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (19842700439289387 / 80000000000000000 : ℝ) =
+            (19842700439289387 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19842700439289387 / 10000000000000000 : ℝ) ≤
+            (214140972016162242811 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (594660542573907 / 2500000000000000 : ℝ)) (q :=
+      (19842700439289387 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_355_9 :
+    Uω (aρ 9) (bρ 9) (4593437163663 / 12800000000000 : ℝ) ≤
+      (-(16012286050680742624031 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16131889525455431 / 80000000000000000 : ℝ) ≤
+      (-(16012286050680742624031 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (16131889525455431 / 80000000000000000 : ℝ) =
+            (16131889525455431 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16131889525455431 / 10000000000000000 : ℝ) ≤
+            (4782129358319257375969 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1791905070197139 / 10000000000000000 : ℝ)) (q :=
+      (16131889525455431 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_355_10 :
+    Uω (aρ 10) (bρ 10) (4593437163663 / 12800000000000 : ℝ) ≤
+      (-(20784935530951146153959 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10009484372873479 / 80000000000000000 : ℝ) ≤
+      (-(20784935530951146153959 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (10009484372873479 / 80000000000000000 : ℝ) =
+            (10009484372873479 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 2
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10009484372873479 / 10000000000000000 : ℝ) ≤
+            (9479878048853846041 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (649862815546651 / 10000000000000000 : ℝ)) (q :=
+      (10009484372873479 / 80000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_355_11 :
+    Uω (aρ 11) (bρ 11) (4593437163663 / 12800000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_355_12 :
+    Uω (aρ 12) (bρ 12) (4593437163663 / 12800000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_355_13 :
+    Uω (aρ 13) (bρ 13) (4593437163663 / 12800000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_355_14 :
+    Uω (aρ 14) (bρ 14) (4593437163663 / 12800000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_355_15 :
+    Uω (aρ 15) (bρ 15) (4593437163663 / 12800000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_355_16 :
+    Uω (aρ 16) (bρ 16) (4593437163663 / 12800000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_355 :
+    Uρ (4593437163663 / 12800000000000 : ℝ) ≤
+      (-(722800917271618526219 / 500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (4593437163663 / 12800000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(10429657709158020939677 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_355_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (4593437163663 / 12800000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(10498068566233714828257 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_355_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (4593437163663 / 12800000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(10636635189896886432221 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_355_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (4593437163663 / 12800000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(10872137355885766455087 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_355_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (4593437163663 / 12800000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(11244598860684962943507 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_355_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (4593437163663 / 12800000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(11811539102151936781149 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_355_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (4593437163663 / 12800000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(12659958571089790164417 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_355_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (4593437163663 / 12800000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(435684509515087757189 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_355_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (4593437163663 / 12800000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(16012286050680742624031 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_355_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (4593437163663 / 12800000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(20784935530951146153959 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_355_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (4593437163663 / 12800000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_355_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (4593437163663 / 12800000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_355_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (4593437163663 / 12800000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_355_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (4593437163663 / 12800000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_355_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (4593437163663 / 12800000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_355_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (4593437163663 / 12800000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_355_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_356_1 :
+    Uω (aρ 1) (bρ 1) (11525451484287 / 32000000000000 : ℝ) ≤
+      (-(1299075948921488108637 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14148642660807227 / 40000000000000000 : ℝ) ≤
+      (-(1299075948921488108637 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (14148642660807227 / 40000000000000000 : ℝ) =
+            (14148642660807227 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14148642660807227 / 10000000000000000 : ℝ) ≤
+            (433792001828511891363 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1768557479824463 / 5000000000000000 : ℝ)) (q :=
+      (14148642660807227 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_356_2 :
+    Uω (aρ 2) (bρ 2) (11525451484287 / 32000000000000 : ℝ) ≤
+      (-(1046076186029057426927 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14052541478416471 / 40000000000000000 : ℝ) ≤
+      (-(1046076186029057426927 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (14052541478416471 / 40000000000000000 : ℝ) =
+            (14052541478416471 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14052541478416471 / 10000000000000000 : ℝ) ≤
+            (340218174570942573073 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (878208345240887 / 2500000000000000 : ℝ)) (q :=
+      (14052541478416471 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_356_3 :
+    Uω (aρ 3) (bρ 3) (11525451484287 / 32000000000000 : ℝ) ≤
+      (-(662425031329633293843 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (554395791956353 / 1600000000000000 : ℝ) ≤
+      (-(662425031329633293843 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (554395791956353 / 1600000000000000 : ℝ) =
+            (554395791956353 / 400000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (554395791956353 / 400000000000000 : ℝ) ≤
+            (204008944045366706157 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (138556245902389 / 400000000000000 : ℝ)) (q :=
+      (554395791956353 / 1600000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_356_4 :
+    Uω (aρ 4) (bρ 4) (11525451484287 / 32000000000000 : ℝ) ≤
+      (-(5416689557187702177793 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13538555016173533 / 40000000000000000 : ℝ) ≤
+      (-(5416689557187702177793 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (13538555016173533 / 40000000000000000 : ℝ) =
+            (13538555016173533 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13538555016173533 / 10000000000000000 : ℝ) ≤
+            (1514782245812297822207 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3381530437257079 / 10000000000000000 : ℝ)) (q :=
+      (13538555016173533 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_356_5 :
+    Uω (aρ 5) (bρ 5) (11525451484287 / 32000000000000 : ℝ) ≤
+      (-(2240861739939409583837 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13045569622826253 / 40000000000000000 : ℝ) ≤
+      (-(2240861739939409583837 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (13045569622826253 / 40000000000000000 : ℝ) =
+            (13045569622826253 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13045569622826253 / 10000000000000000 : ℝ) ≤
+            (531726981260590416163 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3253227659918439 / 10000000000000000 : ℝ)) (q :=
+      (13045569622826253 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_356_6 :
+    Uω (aρ 6) (bρ 6) (11525451484287 / 32000000000000 : ℝ) ≤
+      (-(5884366818469516185671 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12329639632951857 / 40000000000000000 : ℝ) ≤
+      (-(5884366818469516185671 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (12329639632951857 / 40000000000000000 : ℝ) =
+            (12329639632951857 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12329639632951857 / 10000000000000000 : ℝ) ≤
+            (1047104984530483814329 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3062503040121241 / 10000000000000000 : ℝ)) (q :=
+      (12329639632951857 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_356_7 :
+    Uω (aρ 7) (bρ 7) (11525451484287 / 32000000000000 : ℝ) ≤
+      (-(6306455568069282526061 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11331521323901253 / 40000000000000000 : ℝ) ≤
+      (-(6306455568069282526061 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (11331521323901253 / 40000000000000000 : ℝ) =
+            (11331521323901253 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11331521323901253 / 10000000000000000 : ℝ) ≤
+            (625016234930717473939 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2787105266355939 / 10000000000000000 : ℝ)) (q :=
+      (11331521323901253 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_356_8 :
+    Uω (aρ 8) (bρ 8) (11525451484287 / 32000000000000 : ℝ) ≤
+      (-(13887075006321262224287 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9975897689903237 / 40000000000000000 : ℝ) ≤
+      (-(13887075006321262224287 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (9975897689903237 / 40000000000000000 : ℝ) =
+            (9975897689903237 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 15
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9975897689903237 / 5000000000000000 : ℝ) ≤
+            (6907340402678737775713 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2392835100696931 / 10000000000000000 : ℝ)) (q :=
+      (9975897689903237 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_356_9 :
+    Uω (aρ 9) (bρ 9) (11525451484287 / 32000000000000 : ℝ) ≤
+      (-(15939617482324508012211 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8124772315387749 / 40000000000000000 : ℝ) ≤
+      (-(15939617482324508012211 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (8124772315387749 / 40000000000000000 : ℝ) =
+            (8124772315387749 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8124772315387749 / 5000000000000000 : ℝ) ≤
+            (4854797926675491987789 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1808238041799187 / 10000000000000000 : ℝ)) (q :=
+      (8124772315387749 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_356_10 :
+    Uω (aρ 10) (bρ 10) (11525451484287 / 32000000000000 : ℝ) ≤
+      (-(102945636677137892163 / 50000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1020740972669511 / 8000000000000000 : ℝ) ≤
+      (-(102945636677137892163 / 50000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1020740972669511 / 8000000000000000 : ℝ) =
+            (1020740972669511 / 1000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1020740972669511 / 1000000000000000 : ℝ) ≤
+            (1026440367862107837 / 50000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (68626334927409 / 1000000000000000 : ℝ)) (q :=
+      (1020740972669511 / 8000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_356_11 :
+    Uω (aρ 11) (bρ 11) (11525451484287 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_356_12 :
+    Uω (aρ 12) (bρ 12) (11525451484287 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_356_13 :
+    Uω (aρ 13) (bρ 13) (11525451484287 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_356_14 :
+    Uω (aρ 14) (bρ 14) (11525451484287 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_356_15 :
+    Uω (aρ 15) (bρ 15) (11525451484287 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_356_16 :
+    Uω (aρ 16) (bρ 16) (11525451484287 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_356 :
+    Uρ (11525451484287 / 32000000000000 : ℝ) ≤
+      (-(3603124797497613790197 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (11525451484287 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1299075948921488108637 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_356_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (11525451484287 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1046076186029057426927 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_356_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (11525451484287 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(662425031329633293843 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_356_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (11525451484287 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(5416689557187702177793 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_356_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (11525451484287 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(2240861739939409583837 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_356_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (11525451484287 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(5884366818469516185671 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_356_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (11525451484287 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(6306455568069282526061 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_356_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (11525451484287 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(13887075006321262224287 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_356_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (11525451484287 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(15939617482324508012211 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_356_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (11525451484287 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) * (-(102945636677137892163 / 50000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_356_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (11525451484287 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_356_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (11525451484287 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_356_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (11525451484287 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_356_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (11525451484287 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_356_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (11525451484287 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_356_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (11525451484287 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_356_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_357_1 :
+    Uω (aρ 1) (bρ 1) (23134620118833 / 64000000000000 : ℝ) ≤
+      (-(5177847120835798780771 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5680386621333237 / 16000000000000000 : ℝ) ≤
+      (-(5177847120835798780771 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5680386621333237 / 16000000000000000 : ℝ) =
+            (5680386621333237 / 4000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5680386621333237 / 4000000000000000 : ℝ) ≤
+            (1753624682164201219229 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (355019610118389 / 1000000000000000 : ℝ)) (q :=
+      (5680386621333237 / 16000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_357_2 :
+    Uω (aρ 2) (bρ 2) (23134620118833 / 64000000000000 : ℝ) ≤
+      (-(10423593839989125093553 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28209738357445857 / 80000000000000000 : ℝ) ≤
+      (-(10423593839989125093553 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (28209738357445857 / 80000000000000000 : ℝ) =
+            (28209738357445857 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28209738357445857 / 20000000000000000 : ℝ) ≤
+            (3439349766010874906447 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (440739553298601 / 1250000000000000 : ℝ)) (q :=
+      (28209738357445857 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_357_3 :
+    Uω (aρ 3) (bρ 3) (23134620118833 / 64000000000000 : ℝ) ≤
+      (-(2640277126802199692219 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5564893633117449 / 16000000000000000 : ℝ) ≤
+      (-(2640277126802199692219 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5564893633117449 / 16000000000000000 : ℝ) =
+            (5564893633117449 / 4000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5564893633117449 / 4000000000000000 : ℝ) ≤
+            (825458774697800307781 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (695398996954831 / 2000000000000000 : ℝ)) (q :=
+      (5564893633117449 / 16000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_357_4 :
+    Uω (aρ 4) (bρ 4) (23134620118833 / 64000000000000 : ℝ) ≤
+      (-(1349346348405073664153 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27181852290607513 / 80000000000000000 : ℝ) ≤
+      (-(1349346348405073664153 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (27181852290607513 / 80000000000000000 : ℝ) =
+            (27181852290607513 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27181852290607513 / 20000000000000000 : ℝ) ≤
+            (383521602344926335847 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1697317598547111 / 5000000000000000 : ℝ)) (q :=
+      (27181852290607513 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_357_5 :
+    Uω (aρ 5) (bρ 5) (23134620118833 / 64000000000000 : ℝ) ≤
+      (-(11164181025510258405207 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (26196047264101189 / 80000000000000000 : ℝ) ≤
+      (-(11164181025510258405207 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (26196047264101189 / 80000000000000000 : ℝ) =
+            (26196047264101189 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26196047264101189 / 20000000000000000 : ℝ) ≤
+            (2698762580489741594793 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3266373859802641 / 10000000000000000 : ℝ)) (q :=
+      (26196047264101189 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_357_6 :
+    Uω (aρ 6) (bρ 6) (23134620118833 / 64000000000000 : ℝ) ≤
+      (-(5863056491764085571929 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24764603014312389 / 80000000000000000 : ℝ) ≤
+      (-(5863056491764085571929 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (24764603014312389 / 80000000000000000 : ℝ) =
+            (24764603014312389 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24764603014312389 / 20000000000000000 : ℝ) ≤
+            (1068415311235914428071 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3075753172495441 / 10000000000000000 : ℝ)) (q :=
+      (24764603014312389 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_357_7 :
+    Uω (aρ 7) (bρ 7) (23134620118833 / 64000000000000 : ℝ) ≤
+      (-(12566091213062274615969 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22769399626136197 / 80000000000000000 : ℝ) ≤
+      (-(12566091213062274615969 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (22769399626136197 / 80000000000000000 : ℝ) =
+            (22769399626136197 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22769399626136197 / 20000000000000000 : ℝ) ≤
+            (1296852392937725384031 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2800613706211393 / 10000000000000000 : ℝ)) (q :=
+      (22769399626136197 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_357_8 :
+    Uω (aρ 8) (bρ 8) (23134620118833 / 64000000000000 : ℝ) ≤
+      (-(13832569823690590154751 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20060839914761097 / 80000000000000000 : ℝ) ≤
+      (-(13832569823690590154751 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (20060839914761097 / 80000000000000000 : ℝ) =
+            (20060839914761097 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20060839914761097 / 20000000000000000 : ℝ) ≤
+            (30373782309409845249 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1203507714853809 / 5000000000000000 : ℝ)) (q :=
+      (20060839914761097 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_357_9 :
+    Uω (aρ 9) (bρ 9) (23134620118833 / 64000000000000 : ℝ) ≤
+      (-(7933800655272450017953 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3273398004260377 / 16000000000000000 : ℝ) ≤
+      (-(7933800655272450017953 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3273398004260377 / 16000000000000000 : ℝ) =
+            (3273398004260377 / 2000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3273398004260377 / 2000000000000000 : ℝ) ≤
+            (2463407049227549982047 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (364903716940563 / 2000000000000000 : ℝ)) (q :=
+      (3273398004260377 / 16000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_357_10 :
+    Uω (aρ 10) (bρ 10) (23134620118833 / 64000000000000 : ℝ) ≤
+      (-(10201613235986337760123 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2079788226324381 / 16000000000000000 : ℝ) ≤
+      (-(10201613235986337760123 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2079788226324381 / 16000000000000000 : ℝ) =
+            (2079788226324381 / 2000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2079788226324381 / 2000000000000000 : ℝ) ≤
+            (195594468513662239877 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (36053269788891 / 500000000000000 : ℝ)) (q :=
+      (2079788226324381 / 16000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_357_11 :
+    Uω (aρ 11) (bρ 11) (23134620118833 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_357_12 :
+    Uω (aρ 12) (bρ 12) (23134620118833 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_357_13 :
+    Uω (aρ 13) (bρ 13) (23134620118833 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_357_14 :
+    Uω (aρ 14) (bρ 14) (23134620118833 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_357_15 :
+    Uω (aρ 15) (bρ 15) (23134620118833 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_357_16 :
+    Uω (aρ 16) (bρ 16) (23134620118833 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_357 :
+    Uρ (23134620118833 / 64000000000000 : ℝ) ≤
+      (-(14369978574398285360333 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (23134620118833 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(5177847120835798780771 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_357_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (23134620118833 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(10423593839989125093553 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_357_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (23134620118833 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(2640277126802199692219 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_357_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (23134620118833 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1349346348405073664153 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_357_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (23134620118833 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(11164181025510258405207 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_357_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (23134620118833 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(5863056491764085571929 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_357_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (23134620118833 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(12566091213062274615969 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_357_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (23134620118833 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(13832569823690590154751 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_357_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (23134620118833 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(7933800655272450017953 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_357_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (23134620118833 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(10201613235986337760123 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_357_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (23134620118833 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_357_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (23134620118833 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_357_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (23134620118833 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_357_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (23134620118833 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_357_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (23134620118833 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_357_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (23134620118833 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_357_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_358_1 :
+    Uω (aρ 1) (bρ 1) (5804584317273 / 16000000000000 : ℝ) ≤
+      (-(5159458327001527128447 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (222707663139269 / 625000000000000 : ℝ) ≤
+      (-(5159458327001527128447 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (222707663139269 / 625000000000000 : ℝ) =
+            (222707663139269 / 156250000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (222707663139269 / 156250000000000 : ℝ) ≤
+            (1772013475998472871553 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3563277240245983 / 10000000000000000 : ℝ)) (q :=
+      (222707663139269 / 625000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_358_2 :
+    Uω (aρ 2) (bρ 2) (5804584317273 / 16000000000000 : ℝ) ≤
+      (-(5193281738929938716601 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7078598422949201 / 20000000000000000 : ℝ) ≤
+      (-(5193281738929938716601 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7078598422949201 / 20000000000000000 : ℝ) =
+            (7078598422949201 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7078598422949201 / 5000000000000000 : ℝ) ≤
+            (1738190064070061283399 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (55296866488259 / 156250000000000 : ℝ)) (q :=
+      (7078598422949201 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_358_3 :
+    Uω (aρ 3) (bρ 3) (5804584317273 / 16000000000000 : ℝ) ≤
+      (-(10523558134738120583187 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6982286623109339 / 20000000000000000 : ℝ) ≤
+      (-(10523558134738120583187 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6982286623109339 / 20000000000000000 : ℝ) =
+            (6982286623109339 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6982286623109339 / 5000000000000000 : ℝ) ≤
+            (3339385471261879416813 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1745041880879357 / 5000000000000000 : ℝ)) (q :=
+      (6982286623109339 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_358_4 :
+    Uω (aρ 4) (bρ 4) (5804584317273 / 16000000000000 : ℝ) ≤
+      (-(10756311217136471123861 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1705412113285619 / 5000000000000000 : ℝ) ≤
+      (-(10756311217136471123861 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1705412113285619 / 5000000000000000 : ℝ) =
+            (1705412113285619 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1705412113285619 / 1250000000000000 : ℝ) ≤
+            (3106632388863528876139 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3407739772856851 / 10000000000000000 : ℝ)) (q :=
+      (1705412113285619 / 5000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_358_5 :
+    Uω (aρ 5) (bρ 5) (5804584317273 / 16000000000000 : ℝ) ≤
+      (-(1390526815796387017917 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6575238297658749 / 20000000000000000 : ℝ) ≤
+      (-(1390526815796387017917 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6575238297658749 / 20000000000000000 : ℝ) =
+            (6575238297658749 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6575238297658749 / 5000000000000000 : ℝ) ≤
+            (342341134953612982083 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (819879884177031 / 2500000000000000 : ℝ)) (q :=
+      (6575238297658749 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_358_6 :
+    Uω (aρ 6) (bρ 6) (5804584317273 / 16000000000000 : ℝ) ≤
+      (-(467347021317313664613 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6217480247316111 / 20000000000000000 : ℝ) ≤
+      (-(467347021317313664613 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6217480247316111 / 20000000000000000 : ℝ) =
+            (6217480247316111 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6217480247316111 / 5000000000000000 : ℝ) ≤
+            (87170722922686335387 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1544500930752743 / 5000000000000000 : ℝ)) (q :=
+      (6217480247316111 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_358_7 :
+    Uω (aρ 7) (bρ 7) (5804584317273 / 16000000000000 : ℝ) ≤
+      (-(6259748272245998521739 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (228757404424093 / 800000000000000 : ℝ) ≤
+      (-(6259748272245998521739 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (228757404424093 / 800000000000000 : ℝ) =
+            (228757404424093 / 200000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (228757404424093 / 200000000000000 : ℝ) ≤
+            (671723530754001478261 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (28141181055517 / 100000000000000 : ℝ)) (q :=
+      (228757404424093 / 800000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_358_8 :
+    Uω (aρ 8) (bρ 8) (5804584317273 / 16000000000000 : ℝ) ≤
+      (-(13778384661680333337793 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (39394208847257 / 156250000000000 : ℝ) ≤
+      (-(13778384661680333337793 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (39394208847257 / 156250000000000 : ℝ) = (39394208847257 / 39062500000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (39394208847257 / 39062500000000 : ℝ) ≤
+            (84558944319666662207 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2421183378738271 / 10000000000000000 : ℝ)) (q :=
+      (39394208847257 / 156250000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_358_9 :
+    Uω (aρ 9) (bρ 9) (5804584317273 / 16000000000000 : ℝ) ≤
+      (-(7898111955309340951673 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (412105781538001 / 2000000000000000 : ℝ) ≤
+      (-(7898111955309340951673 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (412105781538001 / 2000000000000000 : ℝ) =
+            (412105781538001 / 250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (412105781538001 / 250000000000000 : ℝ) ≤
+            (2499095749190659048327 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (368149618005877 / 2000000000000000 : ℝ)) (q :=
+      (412105781538001 / 2000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_358_10 :
+    Uω (aρ 10) (bρ 10) (5804584317273 / 16000000000000 : ℝ) ≤
+      (-(10112960883108076518971 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1323120438779999 / 10000000000000000 : ℝ) ≤
+      (-(10112960883108076518971 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1323120438779999 / 10000000000000000 : ℝ) =
+            (1323120438779999 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1323120438779999 / 1250000000000000 : ℝ) ≤
+            (284246821391923481029 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (754490185704373 / 10000000000000000 : ℝ)) (q :=
+      (1323120438779999 / 10000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_358_11 :
+    Uω (aρ 11) (bρ 11) (5804584317273 / 16000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_358_12 :
+    Uω (aρ 12) (bρ 12) (5804584317273 / 16000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_358_13 :
+    Uω (aρ 13) (bρ 13) (5804584317273 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_358_14 :
+    Uω (aρ 14) (bρ 14) (5804584317273 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_358_15 :
+    Uω (aρ 15) (bρ 15) (5804584317273 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_358_16 :
+    Uω (aρ 16) (bρ 16) (5804584317273 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_358 :
+    Uρ (5804584317273 / 16000000000000 : ℝ) ≤
+      (-(14328342301148102981567 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (5804584317273 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(5159458327001527128447 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_358_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (5804584317273 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(5193281738929938716601 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_358_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (5804584317273 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(10523558134738120583187 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_358_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (5804584317273 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(10756311217136471123861 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_358_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (5804584317273 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1390526815796387017917 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_358_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (5804584317273 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(467347021317313664613 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_358_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (5804584317273 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(6259748272245998521739 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_358_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (5804584317273 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(13778384661680333337793 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_358_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (5804584317273 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(7898111955309340951673 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_358_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (5804584317273 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(10112960883108076518971 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_358_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (5804584317273 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_358_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (5804584317273 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_358_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (5804584317273 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_358_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (5804584317273 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_358_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (5804584317273 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_358_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (5804584317273 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_358_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_359_1 :
+    Uω (aρ 1) (bρ 1) (46520391688443 / 128000000000000 : ℝ) ≤
+      (-(5150289229783770027499 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (57117809531462451 / 160000000000000000 : ℝ) ≤
+      (-(5150289229783770027499 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (57117809531462451 / 160000000000000000 : ℝ) =
+            (57117809531462451 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (57117809531462451 / 40000000000000000 : ℝ) ≤
+            (1781182573216229972501 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3569817808858197 / 10000000000000000 : ℝ)) (q :=
+      (57117809531462451 / 160000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_359_2 :
+    Uω (aρ 2) (bρ 2) (46520391688443 / 128000000000000 : ℝ) ≤
+      (-(5184049800320691189273 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (56733442668706523 / 160000000000000000 : ℝ) ≤
+      (-(5184049800320691189273 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (56733442668706523 / 160000000000000000 : ℝ) =
+            (56733442668706523 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (56733442668706523 / 40000000000000000 : ℝ) ≤
+            (1747422002679308810727 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1772770481761853 / 5000000000000000 : ℝ)) (q :=
+      (56733442668706523 / 160000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_359_3 :
+    Uω (aρ 3) (bρ 3) (46520391688443 / 128000000000000 : ℝ) ≤
+      (-(10504835724603574181621 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (55962971132726131 / 160000000000000000 : ℝ) ≤
+      (-(10504835724603574181621 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (55962971132726131 / 160000000000000000 : ℝ) =
+            (55962971132726131 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (55962971132726131 / 40000000000000000 : ℝ) ≤
+            (3358107881396425818379 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3496628127876157 / 10000000000000000 : ℝ)) (q :=
+      (55962971132726131 / 160000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_359_4 :
+    Uω (aρ 4) (bρ 4) (46520391688443 / 128000000000000 : ℝ) ≤
+      (-(2147427371592678436087 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (54677928600177467 / 160000000000000000 : ℝ) ≤
+      (-(2147427371592678436087 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (54677928600177467 / 160000000000000000 : ℝ) =
+            (54677928600177467 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (54677928600177467 / 40000000000000000 : ℝ) ≤
+            (625161349607321563913 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1707145996186287 / 5000000000000000 : ℝ)) (q :=
+      (54677928600177467 / 160000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_359_5 :
+    Uω (aρ 5) (bρ 5) (46520391688443 / 128000000000000 : ℝ) ≤
+      (-(11104291311875286283503 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (52706810754558643 / 160000000000000000 : ℝ) ≤
+      (-(11104291311875286283503 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (52706810754558643 / 160000000000000000 : ℝ) =
+            (52706810754558643 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (52706810754558643 / 40000000000000000 : ℝ) ≤
+            (2758652294124713716497 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3286092181005221 / 10000000000000000 : ℝ)) (q :=
+      (52706810754558643 / 160000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_359_6 :
+    Uω (aρ 6) (bρ 6) (46520391688443 / 128000000000000 : ℝ) ≤
+      (-(11662525011682297601331 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9969031133943807 / 32000000000000000 : ℝ) ≤
+      (-(11662525011682297601331 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9969031133943807 / 32000000000000000 : ℝ) =
+            (9969031133943807 / 8000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9969031133943807 / 8000000000000000 : ℝ) ≤
+            (2200418594317702398669 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (309562567054027 / 1000000000000000 : ℝ)) (q :=
+      (9969031133943807 / 32000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_359_7 :
+    Uω (aρ 7) (bρ 7) (46520391688443 / 128000000000000 : ℝ) ≤
+      (-(12496282984634089874379 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (45857809724625947 / 160000000000000000 : ℝ) ≤
+      (-(12496282984634089874379 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (45857809724625947 / 160000000000000000 : ℝ) =
+            (45857809724625947 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (45857809724625947 / 40000000000000000 : ℝ) ≤
+            (1366660621365910125621 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1410434404081817 / 5000000000000000 : ℝ)) (q :=
+      (45857809724625947 / 160000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_359_8 :
+    Uω (aρ 8) (bρ 8) (46520391688443 / 128000000000000 : ℝ) ≤
+      (-(13751410833669744054049 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (40448628276486739 / 160000000000000000 : ℝ) ≤
+      (-(13751410833669744054049 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (40448628276486739 / 160000000000000000 : ℝ) =
+            (40448628276486739 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (40448628276486739 / 40000000000000000 : ℝ) ≤
+            (111532772330255945951 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2428262778486233 / 10000000000000000 : ℝ)) (q :=
+      (40448628276486739 / 160000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8])
+      (by norm_num [bρ_8]) (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_359_9 :
+    Uω (aρ 9) (bρ 9) (46520391688443 / 128000000000000 : ℝ) ≤
+      (-(197009632752461729613 / 125000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (33085554011979971 / 160000000000000000 : ℝ) ≤
+      (-(197009632752461729613 / 125000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (33085554011979971 / 160000000000000000 : ℝ) =
+            (33085554011979971 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (33085554011979971 / 20000000000000000 : ℝ) ≤
+            (62920559860038270387 / 125000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1848844123782887 / 10000000000000000 : ℝ)) (q :=
+      (33085554011979971 / 160000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9])
+      (by norm_num [bρ_9]) (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_359_10 :
+    Uω (aρ 10) (bρ 10) (46520391688443 / 128000000000000 : ℝ) ≤
+      (-(1258759909477100777669 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21352267909247611 / 160000000000000000 : ℝ) ≤
+      (-(1258759909477100777669 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (21352267909247611 / 160000000000000000 : ℝ) =
+            (21352267909247611 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21352267909247611 / 20000000000000000 : ℝ) ≤
+            (40891053585399222331 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (385371197218171 / 5000000000000000 : ℝ)) (q :=
+      (21352267909247611 / 160000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_359_11 :
+    Uω (aρ 11) (bρ 11) (46520391688443 / 128000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_359_12 :
+    Uω (aρ 12) (bρ 12) (46520391688443 / 128000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_359_13 :
+    Uω (aρ 13) (bρ 13) (46520391688443 / 128000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_359_14 :
+    Uω (aρ 14) (bρ 14) (46520391688443 / 128000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_359_15 :
+    Uω (aρ 15) (bρ 15) (46520391688443 / 128000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_359_16 :
+    Uω (aρ 16) (bρ 16) (46520391688443 / 128000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_359 :
+    Uρ (46520391688443 / 128000000000000 : ℝ) ≤
+      (-(2861565311915866550951 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (46520391688443 / 128000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(5150289229783770027499 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_359_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (46520391688443 / 128000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(5184049800320691189273 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_359_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (46520391688443 / 128000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(10504835724603574181621 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_359_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (46520391688443 / 128000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(2147427371592678436087 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_359_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (46520391688443 / 128000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(11104291311875286283503 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_359_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (46520391688443 / 128000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(11662525011682297601331 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_359_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (46520391688443 / 128000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(12496282984634089874379 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_359_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (46520391688443 / 128000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(13751410833669744054049 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_359_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (46520391688443 / 128000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(197009632752461729613 / 125000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_359_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (46520391688443 / 128000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(1258759909477100777669 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_359_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (46520391688443 / 128000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_359_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (46520391688443 / 128000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_359_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (46520391688443 / 128000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_359_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (46520391688443 / 128000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_359_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (46520391688443 / 128000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_359_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (46520391688443 / 128000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_359_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_360_1 :
+    Uω (aρ 1) (bρ 1) (23302054419351 / 64000000000000 : ℝ) ≤
+      (-(10282273833372514520303 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28611228647203731 / 80000000000000000 : ℝ) ≤
+      (-(10282273833372514520303 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (28611228647203731 / 80000000000000000 : ℝ) =
+            (28611228647203731 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28611228647203731 / 20000000000000000 : ℝ) ≤
+            (3580669772627485479697 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3576358376862339 / 10000000000000000 : ℝ)) (q :=
+      (28611228647203731 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_360_2 :
+    Uω (aρ 2) (bρ 2) (23302054419351 / 64000000000000 : ℝ) ≤
+      (-(10349669757811332992301 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28419048960617947 / 80000000000000000 : ℝ) ≤
+      (-(10349669757811332992301 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (28419048960617947 / 80000000000000000 : ℝ) =
+            (28419048960617947 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28419048960617947 / 20000000000000000 : ℝ) ≤
+            (3513273848188667007699 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3552082467725893 / 10000000000000000 : ℝ)) (q :=
+      (28419048960617947 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_360_3 :
+    Uω (aρ 3) (bρ 3) (23302054419351 / 64000000000000 : ℝ) ≤
+      (-(1310768540370397377601 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28033824581068403 / 80000000000000000 : ℝ) ≤
+      (-(1310768540370397377601 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (28033824581068403 / 80000000000000000 : ℝ) =
+            (28033824581068403 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28033824581068403 / 20000000000000000 : ℝ) ≤
+            (422099410379602622399 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3503172479188507 / 10000000000000000 : ℝ)) (q :=
+      (28033824581068403 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_360_4 :
+    Uω (aρ 4) (bρ 4) (23302054419351 / 64000000000000 : ℝ) ≤
+      (-(1071799926009268155151 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1095653384267843 / 3200000000000000 : ℝ) ≤
+      (-(1071799926009268155151 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1095653384267843 / 3200000000000000 : ℝ) =
+            (1095653384267843 / 800000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1095653384267843 / 800000000000000 : ℝ) ≤
+            (314494434590731844849 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (136833766666417 / 400000000000000 : ℝ)) (q :=
+      (1095653384267843 / 3200000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_360_5 :
+    Uω (aρ 5) (bρ 5) (23302054419351 / 64000000000000 : ℝ) ≤
+      (-(11084407906422986722507 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (26405857050309291 / 80000000000000000 : ℝ) ≤
+      (-(11084407906422986722507 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (26405857050309291 / 80000000000000000 : ℝ) =
+            (26405857050309291 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26405857050309291 / 20000000000000000 : ℝ) ≤
+            (2778535699577013277493 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3292664696898729 / 10000000000000000 : ℝ)) (q :=
+      (26405857050309291 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_360_6 :
+    Uω (aρ 6) (bρ 6) (23302054419351 / 64000000000000 : ℝ) ≤
+      (-(2328283939467763148137 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24975233264729499 / 80000000000000000 : ℝ) ≤
+      (-(2328283939467763148137 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (24975233264729499 / 80000000000000000 : ℝ) =
+            (24975233264729499 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24975233264729499 / 20000000000000000 : ℝ) ≤
+            (444304781732236851863 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3102249125643781 / 10000000000000000 : ℝ)) (q :=
+      (24975233264729499 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_360_7 :
+    Uω (aρ 7) (bρ 7) (23302054419351 / 64000000000000 : ℝ) ≤
+      (-(3118281226855059681807 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22982065328186551 / 80000000000000000 : ℝ) ≤
+      (-(3118281226855059681807 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (22982065328186551 / 80000000000000000 : ℝ) =
+            (22982065328186551 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22982065328186551 / 20000000000000000 : ℝ) ≤
+            (347454674644940318193 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (706904630567011 / 2500000000000000 : ℝ)) (q :=
+      (22982065328186551 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_360_8 :
+    Uω (aρ 8) (bρ 8) (23302054419351 / 64000000000000 : ℝ) ≤
+      (-(13724515514587036197389 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20278781289178551 / 80000000000000000 : ℝ) ≤
+      (-(13724515514587036197389 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (20278781289178551 / 80000000000000000 : ℝ) =
+            (20278781289178551 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20278781289178551 / 20000000000000000 : ℝ) ≤
+            (138428091412963802611 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (608834790964011 / 2500000000000000 : ℝ)) (q :=
+      (20278781289178551 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_360_9 :
+    Uω (aρ 9) (bρ 9) (23302054419351 / 64000000000000 : ℝ) ≤
+      (-(15725472123040980563639 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16601273704238307 / 80000000000000000 : ℝ) ≤
+      (-(15725472123040980563639 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (16601273704238307 / 80000000000000000 : ℝ) =
+            (16601273704238307 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16601273704238307 / 10000000000000000 : ℝ) ≤
+            (5068943285959019436361 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1856927895980983 / 10000000000000000 : ℝ)) (q :=
+      (16601273704238307 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_360_10 :
+    Uω (aρ 10) (bρ 10) (23302054419351 / 64000000000000 : ℝ) ≤
+      (-(5014042440673356328323 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10766179120663659 / 80000000000000000 : ℝ) ≤
+      (-(5014042440673356328323 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (10766179120663659 / 80000000000000000 : ℝ) =
+            (10766179120663659 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10766179120663659 / 10000000000000000 : ℝ) ≤
+            (184561411576643671677 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (786713283582321 / 10000000000000000 : ℝ)) (q :=
+      (10766179120663659 / 80000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_360_11 :
+    Uω (aρ 11) (bρ 11) (23302054419351 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_360_12 :
+    Uω (aρ 12) (bρ 12) (23302054419351 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_360_13 :
+    Uω (aρ 13) (bρ 13) (23302054419351 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_360_14 :
+    Uω (aρ 14) (bρ 14) (23302054419351 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_360_15 :
+    Uω (aρ 15) (bρ 15) (23302054419351 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_360_16 :
+    Uω (aρ 16) (bρ 16) (23302054419351 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_360 :
+    Uρ (23302054419351 / 64000000000000 : ℝ) ≤
+      (-(14287499111685594879077 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (23302054419351 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(10282273833372514520303 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_360_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (23302054419351 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(10349669757811332992301 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_360_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (23302054419351 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1310768540370397377601 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_360_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (23302054419351 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1071799926009268155151 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_360_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (23302054419351 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(11084407906422986722507 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_360_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (23302054419351 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(2328283939467763148137 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_360_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (23302054419351 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(3118281226855059681807 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_360_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (23302054419351 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(13724515514587036197389 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_360_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (23302054419351 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(15725472123040980563639 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_360_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (23302054419351 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(5014042440673356328323 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_360_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (23302054419351 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_360_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (23302054419351 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_360_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (23302054419351 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_360_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (23302054419351 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_360_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (23302054419351 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_360_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (23302054419351 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_360_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_361_1 :
+    Uω (aρ 1) (bρ 1) (46687825988961 / 128000000000000 : ℝ) ≤
+      (-(2052800530549085513821 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (57327105052514521 / 160000000000000000 : ℝ) ≤
+      (-(2052800530549085513821 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (57327105052514521 / 160000000000000000 : ℝ) =
+            (57327105052514521 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (57327105052514521 / 40000000000000000 : ℝ) ≤
+            (719788190650914486179 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3582898944261737 / 10000000000000000 : ℝ)) (q :=
+      (57327105052514521 / 160000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_361_2 :
+    Uω (aρ 2) (bρ 2) (46687825988961 / 128000000000000 : ℝ) ≤
+      (-(10331273824108637515949 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (56942753141361409 / 160000000000000000 : ℝ) ≤
+      (-(10331273824108637515949 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (56942753141361409 / 160000000000000000 : ℝ) =
+            (56942753141361409 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (56942753141361409 / 40000000000000000 : ℝ) ≤
+            (3531669781891362484051 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1779311983938799 / 5000000000000000 : ℝ)) (q :=
+      (56942753141361409 / 160000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_361_3 :
+    Uω (aρ 3) (bρ 3) (46687825988961 / 128000000000000 : ℝ) ≤
+      (-(5233747899529985342477 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (56172327073769273 / 160000000000000000 : ℝ) ≤
+      (-(5233747899529985342477 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (56172327073769273 / 160000000000000000 : ℝ) =
+            (56172327073769273 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (56172327073769273 / 40000000000000000 : ℝ) ≤
+            (1697723903470014657523 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3509716815778581 / 10000000000000000 : ℝ)) (q :=
+      (56172327073769273 / 160000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_361_4 :
+    Uω (aρ 4) (bρ 4) (46687825988961 / 128000000000000 : ℝ) ≤
+      (-(10698898282584064623033 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (54887409466858969 / 160000000000000000 : ℝ) ≤
+      (-(10698898282584064623033 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (54887409466858969 / 160000000000000000 : ℝ) =
+            (54887409466858969 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (54887409466858969 / 40000000000000000 : ℝ) ≤
+            (3164045323415935376967 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3427396295979793 / 10000000000000000 : ℝ)) (q :=
+      (54887409466858969 / 160000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_361_5 :
+    Uω (aρ 5) (bρ 5) (46687825988961 / 128000000000000 : ℝ) ≤
+      (-(1383070518809231514651 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (52916616425588929 / 160000000000000000 : ℝ) ≤
+      (-(1383070518809231514651 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (52916616425588929 / 160000000000000000 : ℝ) =
+            (52916616425588929 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (52916616425588929 / 40000000000000000 : ℝ) ≤
+            (349797431940768485349 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1649618542578019 / 5000000000000000 : ℝ)) (q :=
+      (52916616425588929 / 160000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_361_6 :
+    Uω (aρ 6) (bρ 6) (46687825988961 / 128000000000000 : ℝ) ≤
+      (-(1452544924334053118787 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (50055774575846001 / 160000000000000000 : ℝ) ≤
+      (-(1452544924334053118787 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (50055774575846001 / 160000000000000000 : ℝ) =
+            (50055774575846001 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (50055774575846001 / 40000000000000000 : ℝ) ≤
+            (280323026415946881213 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (777218057269543 / 2500000000000000 : ℝ)) (q :=
+      (50055774575846001 / 160000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_361_7 :
+    Uω (aρ 7) (bρ 7) (46687825988961 / 128000000000000 : ℝ) ≤
+      (-(778126377514180342509 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (46070443736556433 / 160000000000000000 : ℝ) ≤
+      (-(778126377514180342509 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (46070443736556433 / 160000000000000000 : ℝ) =
+            (46070443736556433 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (46070443736556433 / 40000000000000000 : ℝ) ≤
+            (88307597860819657491 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (22143494179117 / 78125000000000 : ℝ)) (q :=
+      (46070443736556433 / 160000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_361_8 :
+    Uω (aρ 8) (bρ 8) (46687825988961 / 128000000000000 : ℝ) ≤
+      (-(856106138484193971651 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (40666472974807153 / 160000000000000000 : ℝ) ≤
+      (-(856106138484193971651 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (40666472974807153 / 160000000000000000 : ℝ) =
+            (40666472974807153 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (40666472974807153 / 40000000000000000 : ℝ) ≤
+            (10327836890806028349 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (610603140262079 / 2500000000000000 : ℝ)) (q :=
+      (40666472974807153 / 160000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8])
+      (by norm_num [bρ_8]) (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_361_9 :
+    Uω (aρ 9) (bρ 9) (46687825988961 / 128000000000000 : ℝ) ≤
+      (-(15690326843274616969439 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (33319443988064353 / 160000000000000000 : ℝ) ≤
+      (-(15690326843274616969439 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (33319443988064353 / 160000000000000000 : ℝ) =
+            (33319443988064353 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (33319443988064353 / 20000000000000000 : ℝ) ≤
+            (5104088565725383030561 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (932499783032733 / 5000000000000000 : ℝ)) (q :=
+      (33319443988064353 / 160000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9])
+      (by norm_num [bρ_9]) (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_361_10 :
+    Uω (aρ 10) (bρ 10) (46687825988961 / 128000000000000 : ℝ) ≤
+      (-(19973855197920723312371 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21710332399398641 / 160000000000000000 : ℝ) ≤
+      (-(19973855197920723312371 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (21710332399398641 / 160000000000000000 : ℝ) =
+            (21710332399398641 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21710332399398641 / 20000000000000000 : ℝ) ≤
+            (820560211079276687629 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (200604912744313 / 2500000000000000 : ℝ)) (q :=
+      (21710332399398641 / 160000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_361_11 :
+    Uω (aρ 11) (bρ 11) (46687825988961 / 128000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_361_12 :
+    Uω (aρ 12) (bρ 12) (46687825988961 / 128000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_361_13 :
+    Uω (aρ 13) (bρ 13) (46687825988961 / 128000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_361_14 :
+    Uω (aρ 14) (bρ 14) (46687825988961 / 128000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_361_15 :
+    Uω (aρ 15) (bρ 15) (46687825988961 / 128000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_361_16 :
+    Uω (aρ 16) (bρ 16) (46687825988961 / 128000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_361 :
+    Uρ (46687825988961 / 128000000000000 : ℝ) ≤
+      (-(14267351142315811764633 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (46687825988961 / 128000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2052800530549085513821 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_361_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (46687825988961 / 128000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(10331273824108637515949 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_361_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (46687825988961 / 128000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(5233747899529985342477 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_361_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (46687825988961 / 128000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(10698898282584064623033 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_361_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (46687825988961 / 128000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1383070518809231514651 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_361_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (46687825988961 / 128000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1452544924334053118787 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_361_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (46687825988961 / 128000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(778126377514180342509 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_361_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (46687825988961 / 128000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(856106138484193971651 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_361_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (46687825988961 / 128000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(15690326843274616969439 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_361_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (46687825988961 / 128000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(19973855197920723312371 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_361_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (46687825988961 / 128000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_361_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (46687825988961 / 128000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_361_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (46687825988961 / 128000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_361_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (46687825988961 / 128000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_361_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (46687825988961 / 128000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_361_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (46687825988961 / 128000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_361_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_362_1 :
+    Uω (aρ 1) (bρ 1) (2338577156961 / 6400000000000 : ℝ) ≤
+      (-(2049152959136992528997 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14357938201452519 / 40000000000000000 : ℝ) ≤
+      (-(2049152959136992528997 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (14357938201452519 / 40000000000000000 : ℝ) =
+            (14357938201452519 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14357938201452519 / 10000000000000000 : ℝ) ≤
+            (723435762063007471003 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3589439511059697 / 10000000000000000 : ℝ)) (q :=
+      (14357938201452519 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_362_2 :
+    Uω (aρ 2) (bρ 2) (2338577156961 / 6400000000000 : ℝ) ≤
+      (-(5156455837481287013357 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2852370416463071 / 8000000000000000 : ℝ) ≤
+      (-(5156455837481287013357 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2852370416463071 / 8000000000000000 : ℝ) =
+            (2852370416463071 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2852370416463071 / 2000000000000000 : ℝ) ≤
+            (1775015965518712986643 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (713033092800223 / 2000000000000000 : ℝ)) (q :=
+      (2852370416463071 / 8000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_362_3 :
+    Uω (aρ 3) (bρ 3) (2338577156961 / 6400000000000 : ℝ) ≤
+      (-(1306109752858577475561 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14069251217070289 / 40000000000000000 : ℝ) ≤
+      (-(1306109752858577475561 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (14069251217070289 / 40000000000000000 : ℝ) =
+            (14069251217070289 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14069251217070289 / 10000000000000000 : ℝ) ≤
+            (426758197891422524439 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1758130568864291 / 5000000000000000 : ℝ)) (q :=
+      (14069251217070289 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_362_4 :
+    Uω (aρ 4) (bρ 4) (2338577156961 / 6400000000000 : ℝ) ≤
+      (-(10679833785307686060643 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13748037340659291 / 40000000000000000 : ℝ) ≤
+      (-(10679833785307686060643 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (13748037340659291 / 40000000000000000 : ℝ) =
+            (13748037340659291 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13748037340659291 / 10000000000000000 : ℝ) ≤
+            (3183109820692313939357 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3433948380588083 / 10000000000000000 : ℝ)) (q :=
+      (13748037340659291 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_362_5 :
+    Uω (aρ 5) (bρ 5) (2338577156961 / 6400000000000 : ℝ) ≤
+      (-(11044759885449248784207 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13255379433889969 / 40000000000000000 : ℝ) ≤
+      (-(11044759885449248784207 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (13255379433889969 / 40000000000000000 : ℝ) =
+            (13255379433889969 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13255379433889969 / 10000000000000000 : ℝ) ≤
+            (2818183720550751215793 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1652904673269211 / 5000000000000000 : ℝ)) (q :=
+      (13255379433889969 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_362_6 :
+    Uω (aρ 6) (bρ 6) (2338577156961 / 6400000000000 : ℝ) ≤
+      (-(11599343909730129726657 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12540269956705601 / 40000000000000000 : ℝ) ≤
+      (-(11599343909730129726657 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (12540269956705601 / 40000000000000000 : ℝ) =
+            (12540269956705601 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12540269956705601 / 10000000000000000 : ℝ) ≤
+            (2263599696269870273343 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1557747491543119 / 5000000000000000 : ℝ)) (q :=
+      (12540269956705601 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_362_7 :
+    Uω (aρ 7) (bρ 7) (2338577156961 / 6400000000000 : ℝ) ≤
+      (-(3106743528122563498903 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11544187255281831 / 40000000000000000 : ℝ) ≤
+      (-(3106743528122563498903 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (11544187255281831 / 40000000000000000 : ℝ) =
+            (11544187255281831 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11544187255281831 / 10000000000000000 : ℝ) ≤
+            (358992373377436501097 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2841115013134353 / 10000000000000000 : ℝ)) (q :=
+      (11544187255281831 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_362_8 :
+    Uω (aρ 8) (bρ 8) (2338577156961 / 6400000000000 : ℝ) ≤
+      (-(13670958453277325403241 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10193839918233171 / 40000000000000000 : ℝ) ≤
+      (-(13670958453277325403241 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (10193839918233171 / 40000000000000000 : ℝ) =
+            (10193839918233171 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10193839918233171 / 10000000000000000 : ℝ) ≤
+            (191985152722674596759 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2449482995950023 / 10000000000000000 : ℝ)) (q :=
+      (10193839918233171 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_362_9 :
+    Uω (aρ 9) (bρ 9) (2338577156961 / 6400000000000 : ℝ) ≤
+      (-(3913833307851789537159 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1671812250119869 / 8000000000000000 : ℝ) ≤
+      (-(3913833307851789537159 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1671812250119869 / 8000000000000000 : ℝ) =
+            (1671812250119869 / 1000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1671812250119869 / 1000000000000000 : ℝ) ≤
+            (1284770544398210462841 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (187305929049311 / 1000000000000000 : ℝ)) (q :=
+      (1671812250119869 / 8000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_362_10 :
+    Uω (aρ 10) (bρ 10) (2338577156961 / 6400000000000 : ℝ) ≤
+      (-(9946561910692850307419 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8754524920219 / 64000000000000 : ℝ) ≤
+      (-(9946561910692850307419 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (8754524920219 / 64000000000000 : ℝ) = (8754524920219 / 8000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8754524920219 / 8000000000000 : ℝ) ≤
+            (450645793807149692581 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1308602778011 / 16000000000000 : ℝ)) (q :=
+      (8754524920219 / 64000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10]) (by norm_num [bρ_10])
+      (by norm_num) (by norm_num [aρ_10, bρ_10]) (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_362_11 :
+    Uω (aρ 11) (bρ 11) (2338577156961 / 6400000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_362_12 :
+    Uω (aρ 12) (bρ 12) (2338577156961 / 6400000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_362_13 :
+    Uω (aρ 13) (bρ 13) (2338577156961 / 6400000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_362_14 :
+    Uω (aρ 14) (bρ 14) (2338577156961 / 6400000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_362_15 :
+    Uω (aρ 15) (bρ 15) (2338577156961 / 6400000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_362_16 :
+    Uω (aρ 16) (bρ 16) (2338577156961 / 6400000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_362 :
+    Uρ (2338577156961 / 6400000000000 : ℝ) ≤
+      (-(7123687311461268586551 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (2338577156961 / 6400000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2049152959136992528997 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_362_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (2338577156961 / 6400000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(5156455837481287013357 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_362_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (2338577156961 / 6400000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1306109752858577475561 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_362_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (2338577156961 / 6400000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(10679833785307686060643 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_362_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (2338577156961 / 6400000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(11044759885449248784207 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_362_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (2338577156961 / 6400000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(11599343909730129726657 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_362_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (2338577156961 / 6400000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(3106743528122563498903 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_362_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (2338577156961 / 6400000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(13670958453277325403241 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_362_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (2338577156961 / 6400000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(3913833307851789537159 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_362_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (2338577156961 / 6400000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(9946561910692850307419 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_362_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (2338577156961 / 6400000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_362_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (2338577156961 / 6400000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_362_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (2338577156961 / 6400000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_362_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (2338577156961 / 6400000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_362_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (2338577156961 / 6400000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_362_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (2338577156961 / 6400000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_362_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_363_1 :
+    Uω (aρ 1) (bρ 1) (46855260289479 / 128000000000000 : ℝ) ≤
+      (-(2045512028171230018933 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (57536400554320391 / 160000000000000000 : ℝ) ≤
+      (-(2045512028171230018933 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (57536400554320391 / 160000000000000000 : ℝ) =
+            (57536400554320391 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (57536400554320391 / 40000000000000000 : ℝ) ≤
+            (727076693028769981067 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1797990038629751 / 5000000000000000 : ℝ)) (q :=
+      (57536400554320391 / 160000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_363_2 :
+    Uω (aρ 2) (bρ 2) (46855260289479 / 128000000000000 : ℝ) ≤
+      (-(5147291593243858290007 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (57152063485113007 / 160000000000000000 : ℝ) ≤
+      (-(5147291593243858290007 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (57152063485113007 / 160000000000000000 : ℝ) =
+            (57152063485113007 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (57152063485113007 / 40000000000000000 : ℝ) ≤
+            (1784180209756141709993 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3571706956118579 / 10000000000000000 : ℝ)) (q :=
+      (57152063485113007 / 160000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_363_3 :
+    Uω (aρ 3) (bρ 3) (46855260289479 / 128000000000000 : ℝ) ≤
+      (-(325946714534062058909 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (56381682546325183 / 160000000000000000 : ℝ) ≤
+      (-(325946714534062058909 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (56381682546325183 / 160000000000000000 : ℝ) =
+            (56381682546325183 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (56381682546325183 / 40000000000000000 : ℝ) ≤
+            (107270273153437941091 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3522805445120101 / 10000000000000000 : ℝ)) (q :=
+      (56381682546325183 / 160000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_363_4 :
+    Uω (aρ 4) (bρ 4) (46855260289479 / 128000000000000 : ℝ) ≤
+      (-(2665201407234472898471 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (55096888902770287 / 160000000000000000 : ℝ) ≤
+      (-(2665201407234472898471 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (55096888902770287 / 160000000000000000 : ℝ) =
+            (55096888902770287 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (55096888902770287 / 40000000000000000 : ℝ) ≤
+            (800534494265527101529 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3440500420740739 / 10000000000000000 : ℝ)) (q :=
+      (55096888902770287 / 160000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_363_5 :
+    Uω (aρ 5) (bρ 5) (46855260289479 / 128000000000000 : ℝ) ≤
+      (-(5512497476862254615211 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (53126418036573151 / 160000000000000000 : ℝ) ≤
+      (-(5512497476862254615211 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (53126418036573151 / 160000000000000000 : ℝ) =
+            (53126418036573151 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (53126418036573151 / 40000000000000000 : ℝ) ≤
+            (1418974326137745384789 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3312381481801097 / 10000000000000000 : ℝ)) (q :=
+      (53126418036573151 / 160000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_363_6 :
+    Uω (aρ 6) (bρ 6) (46855260289479 / 128000000000000 : ℝ) ≤
+      (-(11578373049824663123831 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10053276460035483 / 32000000000000000 : ℝ) ≤
+      (-(11578373049824663123831 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (10053276460035483 / 32000000000000000 : ℝ) =
+            (10053276460035483 / 8000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10053276460035483 / 8000000000000000 : ℝ) ≤
+            (2284570556175336876169 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (312211738989163 / 1000000000000000 : ℝ)) (q :=
+      (10053276460035483 / 32000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_363_7 :
+    Uω (aρ 7) (bρ 7) (46855260289479 / 128000000000000 : ℝ) ≤
+      (-(12403980855685154507843 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9256609313099763 / 32000000000000000 : ℝ) ≤
+      (-(12403980855685154507843 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9256609313099763 / 32000000000000000 : ℝ) =
+            (9256609313099763 / 8000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9256609313099763 / 8000000000000000 : ℝ) ≤
+            (1458962750314845492157 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (569572360763361 / 2000000000000000 : ℝ)) (q :=
+      (9256609313099763 / 32000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_363_8 :
+    Uω (aρ 8) (bρ 8) (46855260289479 / 128000000000000 : ℝ) ≤
+      (-(13644295748051856774619 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (40884222877358991 / 160000000000000000 : ℝ) ≤
+      (-(13644295748051856774619 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (40884222877358991 / 160000000000000000 : ℝ) =
+            (40884222877358991 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (40884222877358991 / 40000000000000000 : ℝ) ≤
+            (218647857948143225381 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2456550494139327 / 10000000000000000 : ℝ)) (q :=
+      (40884222877358991 / 160000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8])
+      (by norm_num [bρ_8]) (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_363_9 :
+    Uω (aρ 9) (bρ 9) (46855260289479 / 128000000000000 : ℝ) ≤
+      (-(15620489763713965806193 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6710590335968427 / 32000000000000000 : ℝ) ≤
+      (-(15620489763713965806193 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (6710590335968427 / 32000000000000000 : ℝ) =
+            (6710590335968427 / 4000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6710590335968427 / 4000000000000000 : ℝ) ≤
+            (5173925645286034193807 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (47027680570243 / 250000000000000 : ℝ)) (q :=
+      (6710590335968427 / 32000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_363_10 :
+    Uω (aρ 10) (bρ 10) (46855260289479 / 128000000000000 : ℝ) ≤
+      (-(3962778525587527231453 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22060408948427751 / 160000000000000000 : ℝ) ≤
+      (-(3962778525587527231453 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (22060408948427751 / 160000000000000000 : ℝ) =
+            (22060408948427751 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22060408948427751 / 20000000000000000 : ℝ) ≤
+            (196104556212472768547 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (416549207438961 / 5000000000000000 : ℝ)) (q :=
+      (22060408948427751 / 160000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_363_11 :
+    Uω (aρ 11) (bρ 11) (46855260289479 / 128000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_363_12 :
+    Uω (aρ 12) (bρ 12) (46855260289479 / 128000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_363_13 :
+    Uω (aρ 13) (bρ 13) (46855260289479 / 128000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_363_14 :
+    Uω (aρ 14) (bρ 14) (46855260289479 / 128000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_363_15 :
+    Uω (aρ 15) (bρ 15) (46855260289479 / 128000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_363_16 :
+    Uω (aρ 16) (bρ 16) (46855260289479 / 128000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_363 :
+    Uρ (46855260289479 / 128000000000000 : ℝ) ≤
+      (-(14227562214506348845581 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (46855260289479 / 128000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2045512028171230018933 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_363_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (46855260289479 / 128000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(5147291593243858290007 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_363_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (46855260289479 / 128000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(325946714534062058909 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_363_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (46855260289479 / 128000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(2665201407234472898471 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_363_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (46855260289479 / 128000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(5512497476862254615211 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_363_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (46855260289479 / 128000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(11578373049824663123831 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_363_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (46855260289479 / 128000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(12403980855685154507843 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_363_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (46855260289479 / 128000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(13644295748051856774619 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_363_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (46855260289479 / 128000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(15620489763713965806193 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_363_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (46855260289479 / 128000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(3962778525587527231453 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_363_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (46855260289479 / 128000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_363_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (46855260289479 / 128000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_363_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (46855260289479 / 128000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_363_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (46855260289479 / 128000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_363_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (46855260289479 / 128000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_363_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (46855260289479 / 128000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_363_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/U30.lean b/LeanPool/Zeta5Irrational/Table/U30.lean
new file mode 100644
index 0000000000..bb77659786
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/U30.lean
@@ -0,0 +1,4160 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-! # Certified arcsine potential bounds (U30) -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma U_364_1 :
+    Uω (aρ 1) (bρ 1) (23469488719869 / 64000000000000 : ℝ) ≤
+      (-(1020938856758552771767 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28820524149035761 / 80000000000000000 : ℝ) ≤
+      (-(1020938856758552771767 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (28820524149035761 / 80000000000000000 : ℝ) =
+            (28820524149035761 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28820524149035761 / 20000000000000000 : ℝ) ≤
+            (365355503841447228233 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3602520642864409 / 10000000000000000 : ℝ)) (q :=
+      (28820524149035761 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_364_2 :
+    Uω (aρ 2) (bρ 2) (23469488719869 / 64000000000000 : ℝ) ≤
+      (-(1284536029434866094411 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28628359304545961 / 80000000000000000 : ℝ) ≤
+      (-(1284536029434866094411 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (28628359304545961 / 80000000000000000 : ℝ) =
+            (28628359304545961 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28628359304545961 / 20000000000000000 : ℝ) ≤
+            (448331921315133905589 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3578248444251959 / 10000000000000000 : ℝ)) (q :=
+      (28628359304545961 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_364_3 :
+    Uω (aρ 3) (bρ 3) (23469488719869 / 64000000000000 : ℝ) ≤
+      (-(2602936549286429362049 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28243180054274613 / 80000000000000000 : ℝ) ≤
+      (-(2602936549286429362049 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (28243180054274613 / 80000000000000000 : ℝ) =
+            (28243180054274613 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28243180054274613 / 20000000000000000 : ℝ) ≤
+            (862799352213570637951 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1764674869017061 / 5000000000000000 : ℝ)) (q :=
+      (28243180054274613 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_364_4 :
+    Uω (aρ 4) (bρ 4) (23469488719869 / 64000000000000 : ℝ) ≤
+      (-(10641813674947082111827 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27600814044643157 / 80000000000000000 : ℝ) ≤
+      (-(10641813674947082111827 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (27600814044643157 / 80000000000000000 : ℝ) =
+            (27600814044643157 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27600814044643157 / 20000000000000000 : ℝ) ≤
+            (3221129931052917888173 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1723526208345629 / 5000000000000000 : ℝ)) (q :=
+      (27600814044643157 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_364_5 :
+    Uω (aρ 5) (bρ 5) (23469488719869 / 64000000000000 : ℝ) ≤
+      (-(1100526919862125627921 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (26615658667311261 / 80000000000000000 : ℝ) ≤
+      (-(1100526919862125627921 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (26615658667311261 / 80000000000000000 : ℝ) =
+            (26615658667311261 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26615658667311261 / 20000000000000000 : ℝ) ≤
+            (285767440737874372079 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (829738372923321 / 2500000000000000 : ℝ)) (q :=
+      (26615658667311261 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_364_6 :
+    Uω (aρ 6) (bρ 6) (23469488719869 / 64000000000000 : ℝ) ≤
+      (-(11557446623523384046263 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25185841006774341 / 80000000000000000 : ℝ) ≤
+      (-(11557446623523384046263 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (25185841006774341 / 80000000000000000 : ℝ) =
+            (25185841006774341 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25185841006774341 / 20000000000000000 : ℝ) ≤
+            (2305496982476615953737 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1564369725849527 / 5000000000000000 : ℝ)) (q :=
+      (25185841006774341 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_364_7 :
+    Uω (aρ 7) (bρ 7) (23469488719869 / 64000000000000 : ℝ) ≤
+      (-(2476208400660810589321 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23194668212276101 / 80000000000000000 : ℝ) ≤
+      (-(2476208400660810589321 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (23194668212276101 / 80000000000000000 : ℝ) =
+            (23194668212276101 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23194668212276101 / 20000000000000000 : ℝ) ≤
+            (296380320539189410679 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1427303816917247 / 5000000000000000 : ℝ)) (q :=
+      (23194668212276101 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_364_8 :
+    Uω (aρ 8) (bρ 8) (23469488719869 / 64000000000000 : ℝ) ≤
+      (-(1702213703203382586437 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20496531395139397 / 80000000000000000 : ℝ) ≤
+      (-(1702213703203382586437 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (20496531395139397 / 80000000000000000 : ℝ) =
+            (20496531395139397 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20496531395139397 / 20000000000000000 : ℝ) ≤
+            (30654247546617413563 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1231807540445159 / 5000000000000000 : ℝ)) (q :=
+      (20496531395139397 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_364_9 :
+    Uω (aρ 9) (bρ 9) (23469488719869 / 64000000000000 : ℝ) ≤
+      (-(155857949416357088431 / 100000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16834782613025601 / 80000000000000000 : ℝ) ≤
+      (-(155857949416357088431 / 100000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (16834782613025601 / 80000000000000000 : ℝ) =
+            (16834782613025601 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16834782613025601 / 10000000000000000 : ℝ) ≤
+            (52086204673642911569 / 100000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1889143513721869 / 10000000000000000 : ℝ)) (q :=
+      (16834782613025601 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_364_10 :
+    Uω (aρ 10) (bρ 10) (23469488719869 / 64000000000000 : ℝ) ≤
+      (-(19736085683035117132133 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11116361872933557 / 80000000000000000 : ℝ) ≤
+      (-(19736085683035117132133 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (11116361872933557 / 80000000000000000 : ℝ) =
+            (11116361872933557 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11116361872933557 / 10000000000000000 : ℝ) ≤
+            (1058329725964882867867 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (424048681096929 / 5000000000000000 : ℝ)) (q :=
+      (11116361872933557 / 80000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_364_11 :
+    Uω (aρ 11) (bρ 11) (23469488719869 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_364_12 :
+    Uω (aρ 12) (bρ 12) (23469488719869 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_364_13 :
+    Uω (aρ 13) (bρ 13) (23469488719869 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_364_14 :
+    Uω (aρ 14) (bρ 14) (23469488719869 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_364_15 :
+    Uω (aρ 15) (bρ 15) (23469488719869 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_364_16 :
+    Uω (aρ 16) (bρ 16) (23469488719869 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_364 :
+    Uρ (23469488719869 / 64000000000000 : ℝ) ≤
+      (-(142079071854851839327 / 100000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (23469488719869 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1020938856758552771767 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_364_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (23469488719869 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1284536029434866094411 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_364_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (23469488719869 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(2602936549286429362049 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_364_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (23469488719869 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(10641813674947082111827 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_364_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (23469488719869 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1100526919862125627921 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_364_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (23469488719869 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(11557446623523384046263 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_364_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (23469488719869 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(2476208400660810589321 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_364_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (23469488719869 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(1702213703203382586437 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_364_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (23469488719869 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(155857949416357088431 / 100000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_364_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (23469488719869 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(19736085683035117132133 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_364_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (23469488719869 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_364_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (23469488719869 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_364_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (23469488719869 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_364_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (23469488719869 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_364_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (23469488719869 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_364_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (23469488719869 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_364_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_365_1 :
+    Uω (aρ 1) (bρ 1) (47022694589997 / 128000000000000 : ℝ) ≤
+      (-(5095624977928167751467 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (57745696037089349 / 160000000000000000 : ℝ) ≤
+      (-(5095624977928167751467 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (57745696037089349 / 160000000000000000 : ℝ) =
+            (57745696037089349 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (57745696037089349 / 40000000000000000 : ℝ) ≤
+            (1835846825071832248533 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3609061207877653 / 10000000000000000 : ℝ)) (q :=
+      (57745696037089349 / 160000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_365_2 :
+    Uω (aρ 2) (bρ 2) (47022694589997 / 128000000000000 : ℝ) ≤
+      (-(2051605339881276118719 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (57361373701372661 / 160000000000000000 : ℝ) ≤
+      (-(2051605339881276118719 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (57361373701372661 / 160000000000000000 : ℝ) =
+            (57361373701372661 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (57361373701372661 / 40000000000000000 : ℝ) ≤
+            (720983381318723881281 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3584789928423067 / 10000000000000000 : ℝ)) (q :=
+      (57361373701372661 / 160000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_365_3 :
+    Uω (aρ 3) (bρ 3) (47022694589997 / 128000000000000 : ℝ) ≤
+      (-(5196615945586448281083 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (56591037555596373 / 160000000000000000 : ℝ) ≤
+      (-(5196615945586448281083 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (56591037555596373 / 160000000000000000 : ℝ) =
+            (56591037555596373 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (56591037555596373 / 40000000000000000 : ℝ) ≤
+            (1734855857413551718917 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3535894016551031 / 10000000000000000 : ℝ)) (q :=
+      (56591037555596373 / 160000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_365_4 :
+    Uω (aρ 4) (bρ 4) (47022694589997 / 128000000000000 : ℝ) ≤
+      (-(1327857223199951105543 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (55306366924197853 / 160000000000000000 : ℝ) ≤
+      (-(1327857223199951105543 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (55306366924197853 / 160000000000000000 : ℝ) =
+            (55306366924197853 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (55306366924197853 / 40000000000000000 : ℝ) ≤
+            (405010727550048894457 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (215850273043201 / 625000000000000 : ℝ)) (q :=
+      (55306366924197853 / 160000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_365_5 :
+    Uω (aρ 5) (bρ 5) (47022694589997 / 128000000000000 : ℝ) ≤
+      (-(10985582464399812475231 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (53336215635654253 / 160000000000000000 : ℝ) ≤
+      (-(10985582464399812475231 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (53336215635654253 / 160000000000000000 : ℝ) =
+            (53336215635654253 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (53336215635654253 / 40000000000000000 : ℝ) ≤
+            (2877361141600187524769 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1662762688479133 / 5000000000000000 : ℝ)) (q :=
+      (53336215635654253 / 160000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_365_6 :
+    Uω (aρ 6) (bρ 6) (47022694589997 / 128000000000000 : ℝ) ≤
+      (-(11536564440637285168937 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2019079159376957 / 6400000000000000 : ℝ) ≤
+      (-(11536564440637285168937 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2019079159376957 / 6400000000000000 : ℝ) =
+            (2019079159376957 / 1600000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2019079159376957 / 1600000000000000 : ℝ) ≤
+            (2326379165362714831063 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (125414446827779 / 400000000000000 : ℝ)) (q :=
+      (2019079159376957 / 6400000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_365_7 :
+    Uω (aρ 7) (bρ 7) (47022694589997 / 128000000000000 : ℝ) ≤
+      (-(12358157290836437477463 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (46495618652643429 / 160000000000000000 : ℝ) ≤
+      (-(12358157290836437477463 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (46495618652643429 / 160000000000000000 : ℝ) =
+            (46495618652643429 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (46495618652643429 / 40000000000000000 : ℝ) ≤
+            (1504786315163562522537 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2861352509981913 / 10000000000000000 : ℝ)) (q :=
+      (46495618652643429 / 160000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_365_8 :
+    Uω (aρ 8) (bρ 8) (47022694589997 / 128000000000000 : ℝ) ≤
+      (-(6795599808089037888341 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (41101879611489413 / 160000000000000000 : ℝ) ≤
+      (-(6795599808089037888341 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (41101879611489413 / 160000000000000000 : ℝ) =
+            (41101879611489413 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (41101879611489413 / 40000000000000000 : ℝ) ≤
+            (135871994910962111659 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2470676781177661 / 10000000000000000 : ℝ)) (q :=
+      (41101879611489413 / 160000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8])
+      (by norm_num [bρ_8]) (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_365_9 :
+    Uω (aρ 9) (bρ 9) (47022694589997 / 128000000000000 : ℝ) ≤
+      (-(3887811822798896889681 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6757217364903913 / 32000000000000000 : ℝ) ≤
+      (-(3887811822798896889681 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (6757217364903913 / 32000000000000000 : ℝ) =
+            (6757217364903913 / 4000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6757217364903913 / 4000000000000000 : ℝ) ≤
+            (1310792029451103110319 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (189716831116643 / 1000000000000000 : ℝ)) (q :=
+      (6757217364903913 / 32000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_365_10 :
+    Uω (aρ 10) (bρ 10) (47022694589997 / 128000000000000 : ℝ) ≤
+      (-(19659633305412022108387 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22403349611579381 / 160000000000000000 : ℝ) ≤
+      (-(19659633305412022108387 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (22403349611579381 / 160000000000000000 : ℝ) =
+            (22403349611579381 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22403349611579381 / 20000000000000000 : ℝ) ≤
+            (1134782103587977891613 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (862885193043907 / 10000000000000000 : ℝ)) (q :=
+      (22403349611579381 / 160000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_365_11 :
+    Uω (aρ 11) (bρ 11) (47022694589997 / 128000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_365_12 :
+    Uω (aρ 12) (bρ 12) (47022694589997 / 128000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_365_13 :
+    Uω (aρ 13) (bρ 13) (47022694589997 / 128000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_365_14 :
+    Uω (aρ 14) (bρ 14) (47022694589997 / 128000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_365_15 :
+    Uω (aρ 15) (bρ 15) (47022694589997 / 128000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_365_16 :
+    Uω (aρ 16) (bρ 16) (47022694589997 / 128000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_365 :
+    Uρ (47022694589997 / 128000000000000 : ℝ) ≤
+      (-(14188403341773910537759 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (47022694589997 / 128000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(5095624977928167751467 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_365_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (47022694589997 / 128000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(2051605339881276118719 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_365_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (47022694589997 / 128000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(5196615945586448281083 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_365_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (47022694589997 / 128000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1327857223199951105543 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_365_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (47022694589997 / 128000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(10985582464399812475231 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_365_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (47022694589997 / 128000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(11536564440637285168937 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_365_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (47022694589997 / 128000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(12358157290836437477463 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_365_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (47022694589997 / 128000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(6795599808089037888341 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_365_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (47022694589997 / 128000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(3887811822798896889681 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_365_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (47022694589997 / 128000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(19659633305412022108387 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_365_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (47022694589997 / 128000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_365_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (47022694589997 / 128000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_365_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (47022694589997 / 128000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_365_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (47022694589997 / 128000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_365_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (47022694589997 / 128000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_365_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (47022694589997 / 128000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_365_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_366_1 :
+    Uω (aρ 1) (bρ 1) (1472075366883 / 4000000000000 : ℝ) ≤
+      (-(1017314418630376217589 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1446258594284989 / 4000000000000000 : ℝ) ≤
+      (-(1017314418630376217589 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1446258594284989 / 4000000000000000 : ℝ) =
+            (1446258594284989 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1446258594284989 / 1000000000000000 : ℝ) ≤
+            (368979941969623782411 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (723120354460489 / 2000000000000000 : ℝ)) (q :=
+      (1446258594284989 / 4000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_366_2 :
+    Uω (aρ 2) (bρ 2) (1472075366883 / 4000000000000 : ℝ) ≤
+      (-(10239798456410631867589 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1795813398816513 / 5000000000000000 : ℝ) ≤
+      (-(10239798456410631867589 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1795813398816513 / 5000000000000000 : ℝ) =
+            (1795813398816513 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1795813398816513 / 1250000000000000 : ℝ) ≤
+            (3623145149589368132411 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (224458213040847 / 625000000000000 : ℝ)) (q :=
+      (1795813398816513 / 5000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_366_3 :
+    Uω (aρ 3) (bρ 3) (1472075366883 / 4000000000000 : ℝ) ≤
+      (-(64842198875117124881 / 62500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11073381814083 / 31250000000000 : ℝ) ≤
+      (-(64842198875117124881 / 62500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (11073381814083 / 31250000000000 : ℝ) = (11073381814083 / 7812500000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11073381814083 / 7812500000000 : ℝ) ≤
+            (21801198662382875119 / 62500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (177121914037531 / 500000000000000 : ℝ)) (q :=
+      (11073381814083 / 31250000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_366_4 :
+    Uω (aρ 4) (bρ 4) (1472075366883 / 4000000000000 : ℝ) ≤
+      (-(10603937823945748174413 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6926388176187779 / 20000000000000000 : ℝ) ≤
+      (-(10603937823945748174413 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6926388176187779 / 20000000000000000 : ℝ) =
+            (6926388176187779 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6926388176187779 / 5000000000000000 : ℝ) ≤
+            (3259005782054251825587 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3460156276990279 / 10000000000000000 : ℝ)) (q :=
+      (6926388176187779 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_366_5 :
+    Uω (aρ 5) (bρ 5) (1472075366883 / 4000000000000 : ℝ) ≤
+      (-(2741483649062919691693 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6680139118195949 / 20000000000000000 : ℝ) ≤
+      (-(2741483649062919691693 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6680139118195949 / 20000000000000000 : ℝ) =
+            (6680139118195949 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6680139118195949 / 5000000000000000 : ℝ) ≤
+            (724252252437080308307 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3332097138333449 / 10000000000000000 : ℝ)) (q :=
+      (6680139118195949 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_366_6 :
+    Uω (aρ 6) (bρ 6) (1472075366883 / 4000000000000 : ℝ) ≤
+      (-(5757863156105059112447 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6322784153767821 / 20000000000000000 : ℝ) ≤
+      (-(5757863156105059112447 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6322784153767821 / 20000000000000000 : ℝ) =
+            (6322784153767821 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6322784153767821 / 5000000000000000 : ℝ) ≤
+            (1173608646894940887553 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3141982549045321 / 10000000000000000 : ℝ)) (q :=
+      (6322784153767821 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_366_7 :
+    Uω (aρ 7) (bρ 7) (1472075366883 / 4000000000000 : ℝ) ≤
+      (-(3083831613937107456921 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5825236662951187 / 20000000000000000 : ℝ) ≤
+      (-(3083831613937107456921 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5825236662951187 / 20000000000000000 : ℝ) =
+            (5825236662951187 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5825236662951187 / 5000000000000000 : ℝ) ≤
+            (381904287562892543079 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2868096438988687 / 10000000000000000 : ℝ)) (q :=
+      (5825236662951187 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_366_8 :
+    Uω (aρ 8) (bρ 8) (1472075366883 / 4000000000000 : ℝ) ≤
+      (-(1356476525443648126067 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5151334192303659 / 20000000000000000 : ℝ) ≤
+      (-(1356476525443648126067 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5151334192303659 / 20000000000000000 : ℝ) =
+            (5151334192303659 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5151334192303659 / 5000000000000000 : ℝ) ≤
+            (29817835156351873933 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2477735619681159 / 10000000000000000 : ℝ)) (q :=
+      (5151334192303659 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_366_9 :
+    Uω (aρ 9) (bρ 9) (1472075366883 / 4000000000000 : ℝ) ≤
+      (-(969802835152118119703 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4237814704640477 / 20000000000000000 : ℝ) ≤
+      (-(969802835152118119703 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4237814704640477 / 20000000000000000 : ℝ) =
+            (4237814704640477 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4237814704640477 / 2500000000000000 : ℝ) ≤
+            (329848127910381880297 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1905181760377977 / 10000000000000000 : ℝ)) (q :=
+      (4237814704640477 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_366_10 :
+    Uω (aρ 10) (bρ 10) (1472075366883 / 4000000000000 : ℝ) ≤
+      (-(4896117841857481864347 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1410773245971509 / 10000000000000000 : ℝ) ≤
+      (-(4896117841857481864347 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1410773245971509 / 10000000000000000 : ℝ) =
+            (1410773245971509 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1410773245971509 / 1250000000000000 : ℝ) ≤
+            (302486010392518135653 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (438736290587759 / 5000000000000000 : ℝ)) (q :=
+      (1410773245971509 / 10000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_366_11 :
+    Uω (aρ 11) (bρ 11) (1472075366883 / 4000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_366_12 :
+    Uω (aρ 12) (bρ 12) (1472075366883 / 4000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_366_13 :
+    Uω (aρ 13) (bρ 13) (1472075366883 / 4000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_366_14 :
+    Uω (aρ 14) (bρ 14) (1472075366883 / 4000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_366_15 :
+    Uω (aρ 15) (bρ 15) (1472075366883 / 4000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_366_16 :
+    Uω (aρ 16) (bρ 16) (1472075366883 / 4000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_366 :
+    Uρ (1472075366883 / 4000000000000 : ℝ) ≤
+      (-(2833808993384900581253 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1472075366883 / 4000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1017314418630376217589 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_366_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1472075366883 / 4000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(10239798456410631867589 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_366_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1472075366883 / 4000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) * (-(64842198875117124881 / 62500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_366_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1472075366883 / 4000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(10603937823945748174413 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_366_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1472075366883 / 4000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(2741483649062919691693 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_366_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1472075366883 / 4000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(5757863156105059112447 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_366_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1472075366883 / 4000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(3083831613937107456921 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_366_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1472075366883 / 4000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(1356476525443648126067 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_366_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1472075366883 / 4000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(969802835152118119703 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_366_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1472075366883 / 4000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(4896117841857481864347 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_366_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1472075366883 / 4000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_366_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1472075366883 / 4000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_366_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1472075366883 / 4000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_366_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1472075366883 / 4000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_366_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1472075366883 / 4000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_366_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1472075366883 / 4000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_366_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_367_1 :
+    Uω (aρ 1) (bρ 1) (9438025778103 / 25600000000000 : ℝ) ≤
+      (-(10155071140210231970999 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (57954991501027651 / 160000000000000000 : ℝ) ≤
+      (-(10155071140210231970999 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (57954991501027651 / 160000000000000000 : ℝ) =
+            (57954991501027651 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (57954991501027651 / 40000000000000000 : ℝ) ≤
+            (3707872465789768029001 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (905535584035493 / 2500000000000000 : ℝ)) (q :=
+      (57954991501027651 / 160000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_367_2 :
+    Uω (aρ 2) (bρ 2) (9438025778103 / 25600000000000 : ℝ) ≤
+      (-(5110801692648868227249 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (57570683791531163 / 160000000000000000 : ℝ) ≤
+      (-(5110801692648868227249 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (57570683791531163 / 160000000000000000 : ℝ) =
+            (57570683791531163 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (57570683791531163 / 40000000000000000 : ℝ) ≤
+            (1820670110351131772751 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3597872884964911 / 10000000000000000 : ℝ)) (q :=
+      (57570683791531163 / 160000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_367_3 :
+    Uω (aρ 3) (bρ 3) (9438025778103 / 25600000000000 : ℝ) ≤
+      (-(10356305857235350616427 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11360078421341719 / 32000000000000000 : ℝ) ≤
+      (-(10356305857235350616427 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (11360078421341719 / 32000000000000000 : ℝ) =
+            (11360078421341719 / 8000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11360078421341719 / 8000000000000000 : ℝ) ≤
+            (3506637748764649383573 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (354898253071209 / 1000000000000000 : ℝ)) (q :=
+      (11360078421341719 / 32000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_367_4 :
+    Uω (aρ 4) (bρ 4) (9438025778103 / 25600000000000 : ℝ) ≤
+      (-(5292526826907858834617 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2220633741887267 / 6400000000000000 : ℝ) ≤
+      (-(5292526826907858834617 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2220633741887267 / 6400000000000000 : ℝ) =
+            (2220633741887267 / 1600000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2220633741887267 / 1600000000000000 : ℝ) ≤
+            (1638944976092141165383 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (138668325673449 / 400000000000000 : ℝ)) (q :=
+      (2220633741887267 / 6400000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_367_5 :
+    Uω (aρ 5) (bρ 5) (9438025778103 / 25600000000000 : ℝ) ≤
+      (-(10946325440292095100241 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (53546009270215219 / 160000000000000000 : ℝ) ≤
+      (-(10946325440292095100241 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (53546009270215219 / 160000000000000000 : ℝ) =
+            (53546009270215219 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (53546009270215219 / 40000000000000000 : ℝ) ≤
+            (2916618165707904899759 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1669334388275209 / 5000000000000000 : ℝ)) (q :=
+      (53546009270215219 / 160000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_367_6 :
+    Uω (aρ 6) (bρ 6) (9438025778103 / 25600000000000 : ℝ) ≤
+      (-(11494932050507656954393 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (50687564767897259 / 160000000000000000 : ℝ) ≤
+      (-(11494932050507656954393 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (50687564767897259 / 160000000000000000 : ℝ) =
+            (50687564767897259 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (50687564767897259 / 40000000000000000 : ℝ) ≤
+            (2368011555492343045607 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3148603588900673 / 10000000000000000 : ℝ)) (q :=
+      (50687564767897259 / 160000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_367_7 :
+    Uω (aρ 7) (bρ 7) (9438025778103 / 25600000000000 : ℝ) ≤
+      (-(12312549237462678980321 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (46708160430774691 / 160000000000000000 : ℝ) ≤
+      (-(12312549237462678980321 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (46708160430774691 / 160000000000000000 : ℝ) =
+            (46708160430774691 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (46708160430774691 / 40000000000000000 : ℝ) ≤
+            (1550394368537321019679 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (89838732110011 / 312500000000000 : ℝ)) (q :=
+      (46708160430774691 / 160000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_367_8 :
+    Uω (aρ 8) (bρ 8) (9438025778103 / 25600000000000 : ℝ) ≤
+      (-(3384601519907263353979 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1652777790648547 / 6400000000000000 : ℝ) ≤
+      (-(3384601519907263353979 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1652777790648547 / 6400000000000000 : ℝ) =
+            (1652777790648547 / 1600000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1652777790648547 / 1600000000000000 : ℝ) ≤
+            (81134381592736646021 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (99391664831609 / 400000000000000 : ℝ)) (q :=
+      (1652777790648547 / 6400000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_367_9 :
+    Uω (aρ 9) (bρ 9) (9438025778103 / 25600000000000 : ℝ) ≤
+      (-(7741293864429637342667 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (34018858804644867 / 160000000000000000 : ℝ) ≤
+      (-(7741293864429637342667 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (34018858804644867 / 160000000000000000 : ℝ) =
+            (34018858804644867 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (34018858804644867 / 20000000000000000 : ℝ) ≤
+            (2655913840070362657333 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (478296000988531 / 2500000000000000 : ℝ)) (q :=
+      (34018858804644867 / 160000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9])
+      (by norm_num [bρ_9]) (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_367_10 :
+    Uω (aρ 10) (bρ 10) (9438025778103 / 25600000000000 : ℝ) ≤
+      (-(19510540693039739543589 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22739869400635267 / 160000000000000000 : ℝ) ≤
+      (-(19510540693039739543589 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (22739869400635267 / 160000000000000000 : ℝ) =
+            (22739869400635267 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22739869400635267 / 20000000000000000 : ℝ) ≤
+            (1283874715960260456411 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (222967340486981 / 2500000000000000 : ℝ)) (q :=
+      (22739869400635267 / 160000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_367_11 :
+    Uω (aρ 11) (bρ 11) (9438025778103 / 25600000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_367_12 :
+    Uω (aρ 12) (bρ 12) (9438025778103 / 25600000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_367_13 :
+    Uω (aρ 13) (bρ 13) (9438025778103 / 25600000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_367_14 :
+    Uω (aρ 14) (bρ 14) (9438025778103 / 25600000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_367_15 :
+    Uω (aρ 15) (bρ 15) (9438025778103 / 25600000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_367_16 :
+    Uω (aρ 16) (bρ 16) (9438025778103 / 25600000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_367 :
+    Uρ (9438025778103 / 25600000000000 : ℝ) ≤
+      (-(2829965354122144662707 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (9438025778103 / 25600000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(10155071140210231970999 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_367_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (9438025778103 / 25600000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(5110801692648868227249 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_367_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (9438025778103 / 25600000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(10356305857235350616427 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_367_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (9438025778103 / 25600000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(5292526826907858834617 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_367_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (9438025778103 / 25600000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(10946325440292095100241 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_367_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (9438025778103 / 25600000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(11494932050507656954393 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_367_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (9438025778103 / 25600000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(12312549237462678980321 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_367_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (9438025778103 / 25600000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(3384601519907263353979 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_367_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (9438025778103 / 25600000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(7741293864429637342667 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_367_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (9438025778103 / 25600000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(19510540693039739543589 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_367_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (9438025778103 / 25600000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_367_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (9438025778103 / 25600000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_367_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (9438025778103 / 25600000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_367_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (9438025778103 / 25600000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_367_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (9438025778103 / 25600000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_367_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (9438025778103 / 25600000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_367_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_368_1 :
+    Uω (aρ 1) (bρ 1) (23636923020387 / 64000000000000 : ℝ) ≤
+      (-(5068515349750366186051 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (29029819612999467 / 80000000000000000 : ℝ) ≤
+      (-(5068515349750366186051 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (29029819612999467 / 80000000000000000 : ℝ) =
+            (29029819612999467 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (29029819612999467 / 20000000000000000 : ℝ) ≤
+            (1862956453249633813949 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1814341449699699 / 5000000000000000 : ℝ)) (q :=
+      (29029819612999467 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_368_2 :
+    Uω (aρ 2) (bρ 2) (23636923020387 / 64000000000000 : ℝ) ≤
+      (-(10203441365534420333661 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28837669394875791 / 80000000000000000 : ℝ) ≤
+      (-(10203441365534420333661 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (28837669394875791 / 80000000000000000 : ℝ) =
+            (28837669394875791 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28837669394875791 / 20000000000000000 : ℝ) ≤
+            (3659502240465579666339 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3604414357378479 / 10000000000000000 : ℝ)) (q :=
+      (28837669394875791 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_368_3 :
+    Uω (aρ 3) (bρ 3) (23636923020387 / 64000000000000 : ℝ) ≤
+      (-(10337893877074504726461 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28452534606018127 / 80000000000000000 : ℝ) ≤
+      (-(10337893877074504726461 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (28452534606018127 / 80000000000000000 : ℝ) =
+            (28452534606018127 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28452534606018127 / 20000000000000000 : ℝ) ≤
+            (3525049728925495273539 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3555526766514063 / 10000000000000000 : ℝ)) (q :=
+      (28452534606018127 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_368_4 :
+    Uω (aρ 4) (bρ 4) (23636923020387 / 64000000000000 : ℝ) ≤
+      (-(10566205139813757343053 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5562058133920343 / 16000000000000000 : ℝ) ≤
+      (-(10566205139813757343053 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5562058133920343 / 16000000000000000 : ℝ) =
+            (5562058133920343 / 4000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5562058133920343 / 4000000000000000 : ℝ) ≤
+            (3296738466186242656947 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (43415749543437 / 125000000000000 : ℝ)) (q :=
+      (5562058133920343 / 16000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_368_5 :
+    Uω (aρ 5) (bρ 5) (23636923020387 / 64000000000000 : ℝ) ≤
+      (-(2185350968710533704413 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (26825452307701863 / 80000000000000000 : ℝ) ≤
+      (-(2185350968710533704413 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (26825452307701863 / 80000000000000000 : ℝ) =
+            (26825452307701863 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26825452307701863 / 20000000000000000 : ℝ) ≤
+            (587237752489466295587 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3345240292334997 / 10000000000000000 : ℝ)) (q :=
+      (26825452307701863 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_368_6 :
+    Uω (aρ 6) (bρ 6) (23636923020387 / 64000000000000 : ℝ) ≤
+      (-(11474181469007055192813 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25396426807367743 / 80000000000000000 : ℝ) ≤
+      (-(11474181469007055192813 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (25396426807367743 / 80000000000000000 : ℝ) =
+            (25396426807367743 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25396426807367743 / 20000000000000000 : ℝ) ≤
+            (2388762136992944807187 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3155224292391467 / 10000000000000000 : ℝ)) (q :=
+      (25396426807367743 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_368_7 :
+    Uω (aρ 7) (bρ 7) (23636923020387 / 64000000000000 : ℝ) ≤
+      (-(6144912688669251179439 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4681442008695679 / 16000000000000000 : ℝ) ≤
+      (-(6144912688669251179439 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4681442008695679 / 16000000000000000 : ℝ) =
+            (4681442008695679 / 4000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4681442008695679 / 4000000000000000 : ℝ) ≤
+            (786559114330748820561 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (288158148217913 / 1000000000000000 : ℝ)) (q :=
+      (4681442008695679 / 16000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_368_8 :
+    Uω (aρ 8) (bρ 8) (23636923020387 / 64000000000000 : ℝ) ≤
+      (-(13512121635417570816127 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (828563869753399 / 3200000000000000 : ℝ) ≤
+      (-(13512121635417570816127 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (828563869753399 / 3200000000000000 : ℝ) =
+            (828563869753399 / 800000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (828563869753399 / 800000000000000 : ℝ) ≤
+            (350821970582429183873 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (99673792344331 / 400000000000000 : ℝ)) (q :=
+      (828563869753399 / 3200000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_368_9 :
+    Uω (aρ 9) (bρ 9) (23636923020387 / 64000000000000 : ℝ) ≤
+      (-(3089694597383226710451 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17067555723637463 / 80000000000000000 : ℝ) ≤
+      (-(3089694597383226710451 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (17067555723637463 / 80000000000000000 : ℝ) =
+            (17067555723637463 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17067555723637463 / 10000000000000000 : ℝ) ≤
+            (1069188484416773289549 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1921175181918897 / 10000000000000000 : ℝ)) (q :=
+      (17067555723637463 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_368_10 :
+    Uω (aρ 10) (bρ 10) (23636923020387 / 64000000000000 : ℝ) ≤
+      (-(19437786537202591506609 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (458118293853527 / 3200000000000000 : ℝ) ≤
+      (-(19437786537202591506609 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (458118293853527 / 3200000000000000 : ℝ) =
+            (458118293853527 / 400000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (458118293853527 / 400000000000000 : ℝ) ≤
+            (1356628871797408493391 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (36243384843563 / 400000000000000 : ℝ)) (q :=
+      (458118293853527 / 3200000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10]) (by norm_num [bρ_10])
+      (by norm_num) (by norm_num [aρ_10, bρ_10]) (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_368_11 :
+    Uω (aρ 11) (bρ 11) (23636923020387 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_368_12 :
+    Uω (aρ 12) (bρ 12) (23636923020387 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_368_13 :
+    Uω (aρ 13) (bρ 13) (23636923020387 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_368_14 :
+    Uω (aρ 14) (bρ 14) (23636923020387 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_368_15 :
+    Uω (aρ 15) (bρ 15) (23636923020387 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_368_16 :
+    Uω (aρ 16) (bρ 16) (23636923020387 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_368 :
+    Uρ (23636923020387 / 64000000000000 : ℝ) ≤
+      (-(3532685961019217504731 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (23636923020387 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(5068515349750366186051 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_368_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (23636923020387 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(10203441365534420333661 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_368_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (23636923020387 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(10337893877074504726461 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_368_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (23636923020387 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(10566205139813757343053 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_368_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (23636923020387 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(2185350968710533704413 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_368_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (23636923020387 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(11474181469007055192813 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_368_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (23636923020387 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(6144912688669251179439 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_368_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (23636923020387 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(13512121635417570816127 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_368_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (23636923020387 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(3089694597383226710451 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_368_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (23636923020387 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(19437786537202591506609 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_368_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (23636923020387 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_368_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (23636923020387 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_368_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (23636923020387 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_368_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (23636923020387 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_368_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (23636923020387 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_368_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (23636923020387 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_368_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_369_1 :
+    Uω (aρ 1) (bρ 1) (47357563191033 / 128000000000000 : ℝ) ≤
+      (-(5059511373369090753989 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (58164286946338561 / 160000000000000000 : ℝ) ≤
+      (-(5059511373369090753989 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (58164286946338561 / 160000000000000000 : ℝ) =
+            (58164286946338561 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (58164286946338561 / 40000000000000000 : ℝ) ≤
+            (1871960429630909246011 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3635223462077867 / 10000000000000000 : ℝ)) (q :=
+      (58164286946338561 / 160000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_369_2 :
+    Uω (aρ 2) (bρ 2) (47357563191033 / 128000000000000 : ℝ) ≤
+      (-(5092656138621636409351 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (57779993756959153 / 160000000000000000 : ℝ) ≤
+      (-(5092656138621636409351 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (57779993756959153 / 160000000000000000 : ℝ) =
+            (57779993756959153 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (57779993756959153 / 40000000000000000 : ℝ) ≤
+            (1838815664378363590649 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3610955825915441 / 10000000000000000 : ℝ)) (q :=
+      (57779993756959153 / 160000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_369_3 :
+    Uω (aρ 3) (bρ 3) (47357563191033 / 128000000000000 : ℝ) ≤
+      (-(10319515754482508126191 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (57009746204712257 / 160000000000000000 : ℝ) ≤
+      (-(10319515754482508126191 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (57009746204712257 / 160000000000000000 : ℝ) =
+            (57009746204712257 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (57009746204712257 / 40000000000000000 : ℝ) ≤
+            (3543427851517491873809 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3562070988234579 / 10000000000000000 : ℝ)) (q :=
+      (57009746204712257 / 160000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_369_4 :
+    Uω (aρ 4) (bρ 4) (47357563191033 / 128000000000000 : ℝ) ≤
+      (-(10547392147312262476351 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (55725318787519921 / 160000000000000000 : ℝ) ≤
+      (-(10547392147312262476351 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (55725318787519921 / 160000000000000000 : ℝ) =
+            (55725318787519921 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (55725318787519921 / 40000000000000000 : ℝ) ≤
+            (3315551458687737523649 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3479811742150537 / 10000000000000000 : ℝ)) (q :=
+      (55725318787519921 / 160000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_369_5 :
+    Uω (aρ 5) (bρ 5) (47357563191033 / 128000000000000 : ℝ) ≤
+      (-(1090722265397408139433 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (53755798986894057 / 160000000000000000 : ℝ) ≤
+      (-(1090722265397408139433 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (53755798986894057 / 160000000000000000 : ℝ) =
+            (53755798986894057 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (53755798986894057 / 40000000000000000 : ℝ) ≤
+            (295572095202591860567 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (418976460800913 / 1250000000000000 : ℝ)) (q :=
+      (53755798986894057 / 160000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_369_6 :
+    Uω (aρ 6) (bρ 6) (47357563191033 / 128000000000000 : ℝ) ≤
+      (-(1431684297798292659107 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (50898139787561073 / 160000000000000000 : ℝ) ≤
+      (-(1431684297798292659107 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (50898139787561073 / 160000000000000000 : ℝ) =
+            (50898139787561073 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (50898139787561073 / 40000000000000000 : ℝ) ≤
+            (301183652951707340893 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3161844661630681 / 10000000000000000 : ℝ)) (q :=
+      (50898139787561073 / 160000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_369_7 :
+    Uω (aρ 7) (bρ 7) (47357563191033 / 128000000000000 : ℝ) ≤
+      (-(12267154618652294812397 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (46920672324473113 / 160000000000000000 : ℝ) ≤
+      (-(12267154618652294812397 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (46920672324473113 / 160000000000000000 : ℝ) =
+            (46920672324473113 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (46920672324473113 / 40000000000000000 : ℝ) ≤
+            (1595788987347705187603 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1444161304752343 / 5000000000000000 : ℝ)) (q :=
+      (46920672324473113 / 160000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_369_8 :
+    Uω (aρ 8) (bρ 8) (47357563191033 / 128000000000000 : ℝ) ≤
+      (-(6742955734919824914827 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (41536919893371961 / 160000000000000000 : ℝ) ≤
+      (-(6742955734919824914827 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (41536919893371961 / 160000000000000000 : ℝ) =
+            (41536919893371961 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (41536919893371961 / 40000000000000000 : ℝ) ≤
+            (188516068080175085173 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1249447603478521 / 5000000000000000 : ℝ)) (q :=
+      (41536919893371961 / 160000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8])
+      (by norm_num [bρ_8]) (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_369_9 :
+    Uω (aρ 9) (bρ 9) (47357563191033 / 128000000000000 : ℝ) ≤
+      (-(3082899951093501236787 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6850255333021829 / 32000000000000000 : ℝ) ≤
+      (-(3082899951093501236787 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (6850255333021829 / 32000000000000000 : ℝ) =
+            (6850255333021829 / 4000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6850255333021829 / 4000000000000000 : ℝ) ≤
+            (1075983130706498763213 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (192915543178419 / 1000000000000000 : ℝ)) (q :=
+      (6850255333021829 / 32000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_369_10 :
+    Uω (aρ 10) (bρ 10) (47357563191033 / 128000000000000 : ℝ) ≤
+      (-(19366158133686468909589 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4614115023376829 / 32000000000000000 : ℝ) ≤
+      (-(19366158133686468909589 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4614115023376829 / 32000000000000000 : ℝ) =
+            (4614115023376829 / 4000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4614115023376829 / 4000000000000000 : ℝ) ≤
+            (1428257275313531090411 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (184025354350213 / 2000000000000000 : ℝ)) (q :=
+      (4614115023376829 / 32000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_369_11 :
+    Uω (aρ 11) (bρ 11) (47357563191033 / 128000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_369_12 :
+    Uω (aρ 12) (bρ 12) (47357563191033 / 128000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_369_13 :
+    Uω (aρ 13) (bρ 13) (47357563191033 / 128000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_369_14 :
+    Uω (aρ 14) (bρ 14) (47357563191033 / 128000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_369_15 :
+    Uω (aρ 15) (bρ 15) (47357563191033 / 128000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_369_16 :
+    Uω (aρ 16) (bρ 16) (47357563191033 / 128000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_369 :
+    Uρ (47357563191033 / 128000000000000 : ℝ) ≤
+      (-(7055895810716133855411 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (47357563191033 / 128000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(5059511373369090753989 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_369_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (47357563191033 / 128000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(5092656138621636409351 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_369_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (47357563191033 / 128000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(10319515754482508126191 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_369_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (47357563191033 / 128000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(10547392147312262476351 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_369_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (47357563191033 / 128000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1090722265397408139433 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_369_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (47357563191033 / 128000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1431684297798292659107 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_369_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (47357563191033 / 128000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(12267154618652294812397 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_369_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (47357563191033 / 128000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(6742955734919824914827 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_369_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (47357563191033 / 128000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(3082899951093501236787 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_369_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (47357563191033 / 128000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(19366158133686468909589 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_369_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (47357563191033 / 128000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_369_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (47357563191033 / 128000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_369_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (47357563191033 / 128000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_369_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (47357563191033 / 128000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_369_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (47357563191033 / 128000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_369_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (47357563191033 / 128000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_369_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_370_1 :
+    Uω (aρ 1) (bρ 1) (11860320085323 / 32000000000000 : ℝ) ≤
+      (-(10101047165118847851507 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14567233665517871 / 40000000000000000 : ℝ) ≤
+      (-(10101047165118847851507 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (14567233665517871 / 40000000000000000 : ℝ) =
+            (14567233665517871 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14567233665517871 / 10000000000000000 : ℝ) ≤
+            (3761896440881152148493 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1820882012090249 / 5000000000000000 : ℝ)) (q :=
+      (14567233665517871 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_370_2 :
+    Uω (aρ 2) (bρ 2) (11860320085323 / 32000000000000 : ℝ) ≤
+      (-(317725500037437652139 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14471162173330529 / 40000000000000000 : ℝ) ≤
+      (-(317725500037437652139 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (14471162173330529 / 40000000000000000 : ℝ) =
+            (14471162173330529 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14471162173330529 / 10000000000000000 : ℝ) ≤
+            (115491487650062347861 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3617497290596827 / 10000000000000000 : ℝ)) (q :=
+      (14471162173330529 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_370_3 :
+    Uω (aρ 3) (bρ 3) (11860320085323 / 32000000000000 : ℝ) ≤
+      (-(10301171365095080003113 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14278605771339089 / 40000000000000000 : ℝ) ≤
+      (-(10301171365095080003113 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (14278605771339089 / 40000000000000000 : ℝ) =
+            (14278605771339089 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14278605771339089 / 10000000000000000 : ℝ) ≤
+            (3561772240904919996887 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3568615195951107 / 10000000000000000 : ℝ)) (q :=
+      (14278605771339089 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_370_4 :
+    Uω (aρ 4) (bρ 4) (11860320085323 / 32000000000000 : ℝ) ≤
+      (-(2632153635611494003493 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13957513973517219 / 40000000000000000 : ℝ) ≤
+      (-(2632153635611494003493 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (13957513973517219 / 40000000000000000 : ℝ) =
+            (13957513973517219 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13957513973517219 / 10000000000000000 : ℝ) ≤
+            (833582265888505996507 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (871590869526293 / 2500000000000000 : ℝ)) (q :=
+      (13957513973517219 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_370_5 :
+    Uω (aρ 5) (bρ 5) (11860320085323 / 32000000000000 : ℝ) ≤
+      (-(5443864360199435669073 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13465173097600487 / 40000000000000000 : ℝ) ≤
+      (-(5443864360199435669073 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (13465173097600487 / 40000000000000000 : ℝ) =
+            (13465173097600487 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13465173097600487 / 10000000000000000 : ℝ) ≤
+            (1487607442800564330927 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1679191479740903 / 5000000000000000 : ℝ)) (q :=
+      (13465173097600487 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_370_6 :
+    Uω (aρ 6) (bρ 6) (11860320085323 / 32000000000000 : ℝ) ≤
+      (-(11432810606514017181871 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12750855825783927 / 40000000000000000 : ℝ) ≤
+      (-(11432810606514017181871 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (12750855825783927 / 40000000000000000 : ℝ) =
+            (12750855825783927 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12750855825783927 / 10000000000000000 : ℝ) ≤
+            (2430132999485982818129 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1584232349356763 / 5000000000000000 : ℝ)) (q :=
+      (12750855825783927 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_370_7 :
+    Uω (aρ 7) (bρ 7) (11860320085323 / 32000000000000 : ℝ) ≤
+      (-(15305670883222720007 / 12500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11756729298786639 / 40000000000000000 : ℝ) ≤
+      (-(15305670883222720007 / 12500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (11756729298786639 / 40000000000000000 : ℝ) =
+            (11756729298786639 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11756729298786639 / 10000000000000000 : ℝ) ≤
+            (2023008624277279993 / 12500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1447531407987441 / 5000000000000000 : ℝ)) (q :=
+      (11756729298786639 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_370_8 :
+    Uω (aρ 8) (bρ 8) (11860320085323 / 32000000000000 : ℝ) ≤
+      (-(13459775135250613273287 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (416456241716739 / 1600000000000000 : ℝ) ≤
+      (-(13459775135250613273287 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (416456241716739 / 1600000000000000 : ℝ) =
+            (416456241716739 / 400000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (416456241716739 / 400000000000000 : ℝ) ≤
+            (403168470749386726713 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1566214274613 / 6250000000000 : ℝ)) (q :=
+      (416456241716739 / 1600000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_370_9 :
+    Uω (aρ 9) (bρ 9) (11860320085323 / 32000000000000 : ℝ) ≤
+      (-(61522666701171816901 / 40000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8591838884655659 / 40000000000000000 : ℝ) ≤
+      (-(61522666701171816901 / 40000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (8591838884655659 / 40000000000000000 : ℝ) =
+            (8591838884655659 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8591838884655659 / 5000000000000000 : ℝ) ≤
+            (21654994934828183099 / 40000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (121070305538087 / 625000000000000 : ℝ)) (q :=
+      (8591838884655659 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_370_10 :
+    Uω (aρ 10) (bρ 10) (11860320085323 / 32000000000000 : ℝ) ≤
+      (-(3859121660113755047557 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5808478283835511 / 40000000000000000 : ℝ) ≤
+      (-(3859121660113755047557 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (5808478283835511 / 40000000000000000 : ℝ) =
+            (5808478283835511 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5808478283835511 / 5000000000000000 : ℝ) ≤
+            (299761421686244952443 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (467001810847159 / 5000000000000000 : ℝ)) (q :=
+      (5808478283835511 / 40000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_370_11 :
+    Uω (aρ 11) (bρ 11) (11860320085323 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_370_12 :
+    Uω (aρ 12) (bρ 12) (11860320085323 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_370_13 :
+    Uω (aρ 13) (bρ 13) (11860320085323 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_370_14 :
+    Uω (aρ 14) (bρ 14) (11860320085323 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_370_15 :
+    Uω (aρ 15) (bρ 15) (11860320085323 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_370_16 :
+    Uω (aρ 16) (bρ 16) (11860320085323 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_370 :
+    Uρ (11860320085323 / 32000000000000 : ℝ) ≤
+      (-(1761620730734933931937 / 1250000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (11860320085323 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(10101047165118847851507 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_370_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (11860320085323 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(317725500037437652139 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_370_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (11860320085323 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(10301171365095080003113 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_370_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (11860320085323 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(2632153635611494003493 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_370_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (11860320085323 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(5443864360199435669073 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_370_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (11860320085323 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(11432810606514017181871 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_370_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (11860320085323 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) * (-(15305670883222720007 / 12500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_370_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (11860320085323 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(13459775135250613273287 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_370_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (11860320085323 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(61522666701171816901 / 40000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_370_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (11860320085323 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(3859121660113755047557 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_370_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (11860320085323 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_370_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (11860320085323 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_370_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (11860320085323 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_370_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (11860320085323 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_370_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (11860320085323 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_370_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (11860320085323 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_370_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_371_1 :
+    Uω (aρ 1) (bρ 1) (47524997491551 / 128000000000000 : ℝ) ≤
+      (-(10083103838467802937571 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (58373582373222463 / 160000000000000000 : ℝ) ≤
+      (-(10083103838467802937571 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (58373582373222463 / 160000000000000000 : ℝ) =
+            (58373582373222463 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (58373582373222463 / 40000000000000000 : ℝ) ≤
+            (3779839767532197062429 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1824152292855193 / 5000000000000000 : ℝ)) (q :=
+      (58373582373222463 / 160000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_371_2 :
+    Uω (aρ 2) (bρ 2) (47524997491551 / 128000000000000 : ℝ) ≤
+      (-(10149152418818740405747 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (57989303599007487 / 160000000000000000 : ℝ) ≤
+      (-(10149152418818740405747 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (57989303599007487 / 160000000000000000 : ℝ) =
+            (57989303599007487 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (57989303599007487 / 40000000000000000 : ℝ) ≤
+            (3713791187181259594253 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1812019375721757 / 5000000000000000 : ℝ)) (q :=
+      (57989303599007487 / 160000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_371_3 :
+    Uω (aρ 3) (bρ 3) (47524997491551 / 128000000000000 : ℝ) ≤
+      (-(2570715146308073706789 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (57219099854583759 / 160000000000000000 : ℝ) ≤
+      (-(2570715146308073706789 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (57219099854583759 / 160000000000000000 : ℝ) =
+            (57219099854583759 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (57219099854583759 / 40000000000000000 : ℝ) ≤
+            (895020755191926293211 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (893789847435137 / 2500000000000000 : ℝ)) (q :=
+      (57219099854583759 / 160000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_371_4 :
+    Uω (aρ 4) (bρ 4) (47524997491551 / 128000000000000 : ℝ) ≤
+      (-(10509872192106231280577 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (55934792660773479 / 160000000000000000 : ℝ) ≤
+      (-(10509872192106231280577 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (55934792660773479 / 160000000000000000 : ℝ) =
+            (55934792660773479 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (55934792660773479 / 40000000000000000 : ℝ) ≤
+            (3353071413893768719423 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3492915171579263 / 10000000000000000 : ℝ)) (q :=
+      (55934792660773479 / 160000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_371_5 :
+    Uω (aρ 5) (bρ 5) (47524997491551 / 128000000000000 : ℝ) ≤
+      (-(2173654578512854934223 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (53965584831598407 / 160000000000000000 : ℝ) ≤
+      (-(2173654578512854934223 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (53965584831598407 / 160000000000000000 : ℝ) =
+            (53965584831598407 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (53965584831598407 / 40000000000000000 : ℝ) ≤
+            (598934142687145065777 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3364954112267379 / 10000000000000000 : ℝ)) (q :=
+      (53965584831598407 / 160000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_371_6 :
+    Uω (aρ 6) (bρ 6) (47524997491551 / 128000000000000 : ℝ) ≤
+      (-(1426523744804845731913 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (51108704178080479 / 160000000000000000 : ℝ) ≤
+      (-(1426523744804845731913 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (51108704178080479 / 160000000000000000 : ℝ) =
+            (51108704178080479 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (51108704178080479 / 40000000000000000 : ℝ) ≤
+            (306344205945154268087 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1587542202858819 / 5000000000000000 : ℝ)) (q :=
+      (51108704178080479 / 160000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_371_7 :
+    Uω (aρ 7) (bρ 7) (47524997491551 / 128000000000000 : ℝ) ≤
+      (-(381936605880278124109 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (47133154750311471 / 160000000000000000 : ℝ) ≤
+      (-(381936605880278124109 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (47133154750311471 / 160000000000000000 : ℝ) =
+            (47133154750311471 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (47133154750311471 / 40000000000000000 : ℝ) ≤
+            (51280381807221875891 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (181362631750407 / 625000000000000 : ℝ)) (q :=
+      (47133154750311471 / 160000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_371_8 :
+    Uω (aρ 8) (bρ 8) (47524997491551 / 128000000000000 : ℝ) ≤
+      (-(13433712188266326115151 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (41754306508743559 / 160000000000000000 : ℝ) ≤
+      (-(13433712188266326115151 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (41754306508743559 / 160000000000000000 : ℝ) =
+            (41754306508743559 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (41754306508743559 / 40000000000000000 : ℝ) ≤
+            (429231417733673884849 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2512987729150523 / 10000000000000000 : ℝ)) (q :=
+      (41754306508743559 / 160000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8])
+      (by norm_num [bρ_8]) (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_371_9 :
+    Uω (aρ 9) (bρ 9) (47524997491551 / 128000000000000 : ℝ) ≤
+      (-(191837155107511234621 / 125000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (34483349129133271 / 160000000000000000 : ℝ) ≤
+      (-(191837155107511234621 / 125000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (34483349129133271 / 160000000000000000 : ℝ) =
+            (34483349129133271 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (34483349129133271 / 20000000000000000 : ℝ) ≤
+            (68093037504988765379 / 125000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1945083685059237 / 10000000000000000 : ℝ)) (q :=
+      (34483349129133271 / 160000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9])
+      (by norm_num [bρ_9]) (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_371_10 :
+    Uω (aρ 10) (bρ 10) (47524997491551 / 128000000000000 : ℝ) ≤
+      (-(9613046547344408264177 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23395986837515407 / 160000000000000000 : ℝ) ≤
+      (-(9613046547344408264177 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (23395986837515407 / 160000000000000000 : ℝ) =
+            (23395986837515407 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23395986837515407 / 20000000000000000 : ℝ) ≤
+            (784161157155591735823 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (236930608025501 / 2500000000000000 : ℝ)) (q :=
+      (23395986837515407 / 160000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_371_11 :
+    Uω (aρ 11) (bρ 11) (47524997491551 / 128000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_371_12 :
+    Uω (aρ 12) (bρ 12) (47524997491551 / 128000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_371_13 :
+    Uω (aρ 13) (bρ 13) (47524997491551 / 128000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_371_14 :
+    Uω (aρ 14) (bρ 14) (47524997491551 / 128000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_371_15 :
+    Uω (aρ 15) (bρ 15) (47524997491551 / 128000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_371_16 :
+    Uω (aρ 16) (bρ 16) (47524997491551 / 128000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_371 :
+    Uρ (47524997491551 / 128000000000000 : ℝ) ≤
+      (-(112594100321021907347 / 80000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (47524997491551 / 128000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(10083103838467802937571 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_371_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (47524997491551 / 128000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(10149152418818740405747 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_371_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (47524997491551 / 128000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(2570715146308073706789 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_371_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (47524997491551 / 128000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(10509872192106231280577 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_371_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (47524997491551 / 128000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(2173654578512854934223 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_371_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (47524997491551 / 128000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1426523744804845731913 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_371_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (47524997491551 / 128000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(381936605880278124109 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_371_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (47524997491551 / 128000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(13433712188266326115151 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_371_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (47524997491551 / 128000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(191837155107511234621 / 125000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_371_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (47524997491551 / 128000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(9613046547344408264177 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_371_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (47524997491551 / 128000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_371_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (47524997491551 / 128000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_371_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (47524997491551 / 128000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_371_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (47524997491551 / 128000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_371_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (47524997491551 / 128000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_371_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (47524997491551 / 128000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_371_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_372_1 :
+    Uω (aρ 1) (bρ 1) (4760871464181 / 12800000000000 : ℝ) ≤
+      (-(5032596325617203968613 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (29239115039908057 / 80000000000000000 : ℝ) ≤
+      (-(5032596325617203968613 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (29239115039908057 / 80000000000000000 : ℝ) =
+            (29239115039908057 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (29239115039908057 / 20000000000000000 : ℝ) ≤
+            (1898875477382796031387 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (228427821666913 / 625000000000000 : ℝ)) (q :=
+      (29239115039908057 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_372_2 :
+    Uω (aρ 2) (bρ 2) (4760871464181 / 12800000000000 : ℝ) ≤
+      (-(10131121412167346282789 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (29046979237090549 / 80000000000000000 : ℝ) ≤
+      (-(10131121412167346282789 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (29046979237090549 / 80000000000000000 : ℝ) =
+            (29046979237090549 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (29046979237090549 / 20000000000000000 : ℝ) ≤
+            (3731822193832653717211 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3630580208476231 / 10000000000000000 : ℝ)) (q :=
+      (29046979237090549 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_372_3 :
+    Uω (aρ 3) (bρ 3) (4760871464181 / 12800000000000 : ℝ) ≤
+      (-(641536455743347809409 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28661888256502593 / 80000000000000000 : ℝ) ≤
+      (-(641536455743347809409 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (28661888256502593 / 80000000000000000 : ℝ) =
+            (28661888256502593 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28661888256502593 / 20000000000000000 : ℝ) ≤
+            (224897519631652190591 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1790851784839621 / 5000000000000000 : ℝ)) (q :=
+      (28661888256502593 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_372_4 :
+    Uω (aρ 4) (bρ 4) (4760871464181 / 12800000000000 : ℝ) ≤
+      (-(10491164963935251592369 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28019764544771253 / 80000000000000000 : ℝ) ≤
+      (-(10491164963935251592369 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (28019764544771253 / 80000000000000000 : ℝ) =
+            (28019764544771253 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28019764544771253 / 20000000000000000 : ℝ) ≤
+            (3371778642064748407631 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3499466822811407 / 10000000000000000 : ℝ)) (q :=
+      (28019764544771253 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_372_5 :
+    Uω (aρ 5) (bρ 5) (4760871464181 / 12800000000000 : ℝ) ≤
+      (-(10848855021095157620113 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27035238158055049 / 80000000000000000 : ℝ) ≤
+      (-(10848855021095157620113 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (27035238158055049 / 80000000000000000 : ℝ) =
+            (27035238158055049 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27035238158055049 / 20000000000000000 : ℝ) ≤
+            (3014088584904842379887 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (842881286366839 / 2500000000000000 : ℝ)) (q :=
+      (27035238158055049 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_372_6 :
+    Uω (aρ 6) (bρ 6) (4760871464181 / 12800000000000 : ℝ) ≤
+      (-(2847903064094822168529 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25606991214438649 / 80000000000000000 : ℝ) ≤
+      (-(2847903064094822168529 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (25606991214438649 / 80000000000000000 : ℝ) =
+            (25606991214438649 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25606991214438649 / 20000000000000000 : ℝ) ≤
+            (617832837405177831471 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (397712973087907 / 1250000000000000 : ℝ)) (q :=
+      (25606991214438649 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_372_7 :
+    Uω (aρ 7) (bρ 7) (4760871464181 / 12800000000000 : ℝ) ≤
+      (-(12199458412337004357933 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23619692520409741 / 80000000000000000 : ℝ) ≤
+      (-(12199458412337004357933 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (23619692520409741 / 80000000000000000 : ℝ) =
+            (23619692520409741 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23619692520409741 / 20000000000000000 : ℝ) ≤
+            (1663485193662995642067 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2908540491956029 / 10000000000000000 : ℝ)) (q :=
+      (23619692520409741 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_372_8 :
+    Uω (aρ 8) (bρ 8) (4760871464181 / 12800000000000 : ℝ) ≤
+      (-(3351930547426756303613 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20931483544297449 / 80000000000000000 : ℝ) ≤
+      (-(3351930547426756303613 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (20931483544297449 / 80000000000000000 : ℝ) =
+            (20931483544297449 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20931483544297449 / 20000000000000000 : ℝ) ≤
+            (113805354073243696387 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (630007474816989 / 2500000000000000 : ℝ)) (q :=
+      (20931483544297449 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_372_9 :
+    Uω (aρ 9) (bρ 9) (4760871464181 / 12800000000000 : ℝ) ≤
+      (-(7656707819277447128447 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17299629239625421 / 80000000000000000 : ℝ) ≤
+      (-(7656707819277447128447 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (17299629239625421 / 80000000000000000 : ℝ) =
+            (17299629239625421 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17299629239625421 / 10000000000000000 : ℝ) ≤
+            (2740499885222552871553 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1953031951459949 / 10000000000000000 : ℝ)) (q :=
+      (17299629239625421 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_372_10 :
+    Uω (aρ 10) (bρ 10) (4760871464181 / 12800000000000 : ℝ) ≤
+      (-(19157571507822823325749 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2355685035371973 / 16000000000000000 : ℝ) ≤
+      (-(19157571507822823325749 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2355685035371973 / 16000000000000000 : ℝ) =
+            (2355685035371973 / 2000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2355685035371973 / 2000000000000000 : ℝ) ≤
+            (1636843901177176674251 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (24032249231589 / 250000000000000 : ℝ)) (q :=
+      (2355685035371973 / 16000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_372_11 :
+    Uω (aρ 11) (bρ 11) (4760871464181 / 12800000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_372_12 :
+    Uω (aρ 12) (bρ 12) (4760871464181 / 12800000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_372_13 :
+    Uω (aρ 13) (bρ 13) (4760871464181 / 12800000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_372_14 :
+    Uω (aρ 14) (bρ 14) (4760871464181 / 12800000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_372_15 :
+    Uω (aρ 15) (bρ 15) (4760871464181 / 12800000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_372_16 :
+    Uω (aρ 16) (bρ 16) (4760871464181 / 12800000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_372 :
+    Uρ (4760871464181 / 12800000000000 : ℝ) ≤
+      (-(7027838990186991712099 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (4760871464181 / 12800000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(5032596325617203968613 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_372_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (4760871464181 / 12800000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(10131121412167346282789 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_372_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (4760871464181 / 12800000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(641536455743347809409 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_372_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (4760871464181 / 12800000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(10491164963935251592369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_372_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (4760871464181 / 12800000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(10848855021095157620113 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_372_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (4760871464181 / 12800000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(2847903064094822168529 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_372_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (4760871464181 / 12800000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(12199458412337004357933 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_372_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (4760871464181 / 12800000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(3351930547426756303613 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_372_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (4760871464181 / 12800000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(7656707819277447128447 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_372_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (4760871464181 / 12800000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(19157571507822823325749 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_372_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (4760871464181 / 12800000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_372_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (4760871464181 / 12800000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_372_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (4760871464181 / 12800000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_372_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (4760871464181 / 12800000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_372_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (4760871464181 / 12800000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_372_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (4760871464181 / 12800000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_372_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_373_1 :
+    Uω (aρ 1) (bρ 1) (47692431792069 / 128000000000000 : ℝ) ≤
+      (-(2009462697697571396643 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (58582877781876861 / 160000000000000000 : ℝ) ≤
+      (-(2009462697697571396643 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (58582877781876861 / 160000000000000000 : ℝ) =
+            (58582877781876861 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (58582877781876861 / 40000000000000000 : ℝ) ≤
+            (763126023502428603357 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3661385707064217 / 10000000000000000 : ℝ)) (q :=
+      (58582877781876861 / 160000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_373_2 :
+    Uω (aρ 2) (bρ 2) (47692431792069 / 128000000000000 : ℝ) ≤
+      (-(252828071598570374709 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (58198613319007573 / 160000000000000000 : ℝ) ≤
+      (-(252828071598570374709 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (58198613319007573 / 160000000000000000 : ℝ) =
+            (58198613319007573 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (58198613319007573 / 40000000000000000 : ℝ) ≤
+            (93745518551429625291 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (909280415428889 / 2500000000000000 : ℝ)) (q :=
+      (58198613319007573 / 160000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_373_3 :
+    Uω (aρ 3) (bρ 3) (47692431792069 / 128000000000000 : ℝ) ≤
+      (-(512316968137633730297 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (57428453061226893 / 160000000000000000 : ℝ) ≤
+      (-(512316968137633730297 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (57428453061226893 / 160000000000000000 : ℝ) =
+            (57428453061226893 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (57428453061226893 / 40000000000000000 : ℝ) ≤
+            (180830212162366269703 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3588247735842971 / 10000000000000000 : ℝ)) (q :=
+      (57428453061226893 / 160000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_373_4 :
+    Uω (aρ 4) (bρ 4) (47692431792069 / 128000000000000 : ℝ) ≤
+      (-(5236246363160253772561 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (56144265182270437 / 160000000000000000 : ℝ) ≤
+      (-(5236246363160253772561 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (56144265182270437 / 160000000000000000 : ℝ) =
+            (56144265182270437 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (56144265182270437 / 40000000000000000 : ℝ) ≤
+            (1695225439839746227439 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1753009216019207 / 5000000000000000 : ℝ)) (q :=
+      (56144265182270437 / 160000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_373_5 :
+    Uω (aρ 5) (bρ 5) (47692431792069 / 128000000000000 : ℝ) ≤
+      (-(5414737478748498833533 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (54175366849519829 / 160000000000000000 : ℝ) ≤
+      (-(5414737478748498833533 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (54175366849519829 / 160000000000000000 : ℝ) =
+            (54175366849519829 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (54175366849519829 / 40000000000000000 : ℝ) ≤
+            (1516734324251501166467 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (844524014944897 / 2500000000000000 : ℝ)) (q :=
+      (54175366849519829 / 160000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_373_6 :
+    Uω (aρ 6) (bρ 6) (47692431792069 / 128000000000000 : ℝ) ≤
+      (-(2274215463942847126499 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (51319258071870397 / 160000000000000000 : ℝ) ≤
+      (-(2274215463942847126499 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (51319258071870397 / 160000000000000000 : ℝ) =
+            (51319258071870397 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (51319258071870397 / 40000000000000000 : ℝ) ≤
+            (498373257257152873501 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3188322837713409 / 10000000000000000 : ℝ)) (q :=
+      (51319258071870397 / 160000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_373_7 :
+    Uω (aρ 7) (bρ 7) (47692431792069 / 128000000000000 : ℝ) ≤
+      (-(1522124691229523665977 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (47345608117045269 / 160000000000000000 : ℝ) ≤
+      (-(1522124691229523665977 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (47345608117045269 / 160000000000000000 : ℝ) =
+            (47345608117045269 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (47345608117045269 / 40000000000000000 : ℝ) ≤
+            (210743259520476334023 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (728819493530067 / 2500000000000000 : ℝ)) (q :=
+      (47345608117045269 / 160000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_373_8 :
+    Uω (aρ 8) (bρ 8) (47692431792069 / 128000000000000 : ℝ) ≤
+      (-(3345451176135521414619 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (41971606093120077 / 160000000000000000 : ℝ) ≤
+      (-(3345451176135521414619 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (41971606093120077 / 160000000000000000 : ℝ) =
+            (41971606093120077 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (41971606093120077 / 40000000000000000 : ℝ) ≤
+            (120284725364478585381 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2527069372469619 / 10000000000000000 : ℝ)) (q :=
+      (41971606093120077 / 160000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8])
+      (by norm_num [bρ_8]) (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_373_9 :
+    Uω (aρ 9) (bρ 9) (47692431792069 / 128000000000000 : ℝ) ≤
+      (-(15279995068813354745051 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (34715084613313061 / 160000000000000000 : ℝ) ≤
+      (-(15279995068813354745051 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (34715084613313061 / 160000000000000000 : ℝ) =
+            (34715084613313061 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (34715084613313061 / 20000000000000000 : ℝ) ≤
+            (5514420340186645254949 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (980484907926871 / 5000000000000000 : ℝ)) (q :=
+      (34715084613313061 / 160000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9])
+      (by norm_num [bρ_9]) (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_373_10 :
+    Uω (aρ 10) (bρ 10) (47692431792069 / 128000000000000 : ℝ) ≤
+      (-(4772501299645587367841 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23716554219765741 / 160000000000000000 : ℝ) ≤
+      (-(4772501299645587367841 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (23716554219765741 / 160000000000000000 : ℝ) =
+            (23716554219765741 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23716554219765741 / 20000000000000000 : ℝ) ≤
+            (426102552604412632159 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (974712550155327 / 10000000000000000 : ℝ)) (q :=
+      (23716554219765741 / 160000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_373_11 :
+    Uω (aρ 11) (bρ 11) (47692431792069 / 128000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_373_12 :
+    Uω (aρ 12) (bρ 12) (47692431792069 / 128000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_373_13 :
+    Uω (aρ 13) (bρ 13) (47692431792069 / 128000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_373_14 :
+    Uω (aρ 14) (bρ 14) (47692431792069 / 128000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_373_15 :
+    Uω (aρ 15) (bρ 15) (47692431792069 / 128000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_373_16 :
+    Uω (aρ 16) (bρ 16) (47692431792069 / 128000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_373 :
+    Uρ (47692431792069 / 128000000000000 : ℝ) ≤
+      (-(701860433666920155633 / 500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (47692431792069 / 128000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2009462697697571396643 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_373_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (47692431792069 / 128000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(252828071598570374709 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_373_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (47692431792069 / 128000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(512316968137633730297 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_373_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (47692431792069 / 128000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(5236246363160253772561 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_373_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (47692431792069 / 128000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(5414737478748498833533 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_373_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (47692431792069 / 128000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(2274215463942847126499 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_373_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (47692431792069 / 128000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(1522124691229523665977 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_373_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (47692431792069 / 128000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(3345451176135521414619 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_373_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (47692431792069 / 128000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(15279995068813354745051 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_373_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (47692431792069 / 128000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(4772501299645587367841 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_373_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (47692431792069 / 128000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_373_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (47692431792069 / 128000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_373_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (47692431792069 / 128000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_373_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (47692431792069 / 128000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_373_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (47692431792069 / 128000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_373_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (47692431792069 / 128000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_373_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_374_1 :
+    Uω (aρ 1) (bρ 1) (5972018617791 / 16000000000000 : ℝ) ≤
+      (-(10029466235912752073139 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7335940684928617 / 20000000000000000 : ℝ) ≤
+      (-(10029466235912752073139 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7335940684928617 / 20000000000000000 : ℝ) =
+            (7335940684928617 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7335940684928617 / 5000000000000000 : ℝ) ≤
+            (3833477370087247926861 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1833963133447121 / 5000000000000000 : ℝ)) (q :=
+      (7335940684928617 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_374_2 :
+    Uω (aρ 2) (bρ 2) (5972018617791 / 16000000000000 : ℝ) ≤
+      (-(2523789164369170187899 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3643954258353147 / 10000000000000000 : ℝ) ≤
+      (-(2523789164369170187899 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3643954258353147 / 10000000000000000 : ℝ) =
+            (3643954258353147 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3643954258353147 / 2500000000000000 : ℝ) ≤
+            (941946737130829812101 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3643663111181919 / 10000000000000000 : ℝ)) (q :=
+      (3643954258353147 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_374_3 :
+    Uω (aρ 3) (bρ 3) (5972018617791 / 16000000000000 : ℝ) ≤
+      (-(10228128676152854997047 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (359582059374067 / 1000000000000000 : ℝ) ≤
+      (-(10228128676152854997047 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (359582059374067 / 1000000000000000 : ℝ) =
+            (359582059374067 / 250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (359582059374067 / 250000000000000 : ℝ) ≤
+            (3634814929847145002953 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (718958377661393 / 2000000000000000 : ℝ)) (q :=
+      (359582059374067 / 1000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_374_4 :
+    Uω (aρ 4) (bρ 4) (5972018617791 / 16000000000000 : ℝ) ≤
+      (-(10453855348389118523381 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3515562558802351 / 10000000000000000 : ℝ) ≤
+      (-(10453855348389118523381 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3515562558802351 / 10000000000000000 : ℝ) =
+            (3515562558802351 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3515562558802351 / 2500000000000000 : ℝ) ≤
+            (3409088257610881476619 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3512569999495327 / 10000000000000000 : ℝ)) (q :=
+      (3515562558802351 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_374_5 :
+    Uω (aρ 5) (bρ 5) (5972018617791 / 16000000000000 : ℝ) ≤
+      (-(10810132554148952355511 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6785032054670867 / 20000000000000000 : ℝ) ≤
+      (-(10810132554148952355511 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6785032054670867 / 20000000000000000 : ℝ) =
+            (6785032054670867 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6785032054670867 / 5000000000000000 : ℝ) ≤
+            (3052811051851047644489 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (846166713974123 / 2500000000000000 : ℝ)) (q :=
+      (6785032054670867 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_374_6 :
+    Uω (aρ 6) (bρ 6) (5972018617791 / 16000000000000 : ℝ) ≤
+      (-(11350584968972253481687 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (642806639040847 / 2000000000000000 : ℝ) ≤
+      (-(11350584968972253481687 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (642806639040847 / 2000000000000000 : ℝ) =
+            (642806639040847 / 500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (642806639040847 / 500000000000000 : ℝ) ≤
+            (2512358637027746518313 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (638988313354819 / 2000000000000000 : ℝ)) (q :=
+      (642806639040847 / 2000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_374_7 :
+    Uω (aρ 7) (bρ 7) (5972018617791 / 16000000000000 : ℝ) ≤
+      (-(3038647123310739096387 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9267934380643 / 31250000000000 : ℝ) ≤
+      (-(3038647123310739096387 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (9267934380643 / 31250000000000 : ℝ) = (9267934380643 / 7812500000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9267934380643 / 7812500000000 : ℝ) ≤
+            (427088778189260903613 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (584402912147429 / 2000000000000000 : ℝ)) (q :=
+      (9267934380643 / 31250000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_374_8 :
+    Uω (aρ 8) (bρ 8) (5972018617791 / 16000000000000 : ℝ) ≤
+      (-(534238372073429131109 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1315006990691277 / 5000000000000000 : ℝ) ≤
+      (-(534238372073429131109 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1315006990691277 / 5000000000000000 : ℝ) =
+            (1315006990691277 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1315006990691277 / 1250000000000000 : ℝ) ≤
+            (20279372166570868891 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2534106171230733 / 10000000000000000 : ℝ)) (q :=
+      (1315006990691277 / 5000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_374_9 :
+    Uω (aρ 9) (bρ 9) (5972018617791 / 16000000000000 : ℝ) ≤
+      (-(15246709423082283256313 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4353853567226101 / 20000000000000000 : ℝ) ≤
+      (-(15246709423082283256313 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4353853567226101 / 20000000000000000 : ℝ) =
+            (4353853567226101 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4353853567226101 / 2500000000000000 : ℝ) ≤
+            (5547705985917716743687 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (984448702025863 / 5000000000000000 : ℝ)) (q :=
+      (4353853567226101 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_374_10 :
+    Uω (aρ 10) (bρ 10) (5972018617791 / 16000000000000 : ℝ) ≤
+      (-(760934330201234651191 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1492196606227513 / 10000000000000000 : ℝ) ≤
+      (-(760934330201234651191 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1492196606227513 / 10000000000000000 : ℝ) =
+            (1492196606227513 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1492196606227513 / 1250000000000000 : ℝ) ≤
+            (70842286158765348809 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (987996082775651 / 10000000000000000 : ℝ)) (q :=
+      (1492196606227513 / 10000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_374_11 :
+    Uω (aρ 11) (bρ 11) (5972018617791 / 16000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_374_12 :
+    Uω (aρ 12) (bρ 12) (5972018617791 / 16000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_374_13 :
+    Uω (aρ 13) (bρ 13) (5972018617791 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_374_14 :
+    Uω (aρ 14) (bρ 14) (5972018617791 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_374_15 :
+    Uω (aρ 15) (bρ 15) (5972018617791 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_374_16 :
+    Uω (aρ 16) (bρ 16) (5972018617791 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_374 :
+    Uρ (5972018617791 / 16000000000000 : ℝ) ≤
+      (-(14018851335926930024331 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (5972018617791 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(10029466235912752073139 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_374_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (5972018617791 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(2523789164369170187899 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_374_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (5972018617791 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(10228128676152854997047 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_374_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (5972018617791 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(10453855348389118523381 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_374_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (5972018617791 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(10810132554148952355511 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_374_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (5972018617791 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(11350584968972253481687 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_374_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (5972018617791 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(3038647123310739096387 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_374_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (5972018617791 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(534238372073429131109 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_374_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (5972018617791 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(15246709423082283256313 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_374_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (5972018617791 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) * (-(760934330201234651191 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_374_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (5972018617791 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_374_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (5972018617791 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_374_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (5972018617791 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_374_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (5972018617791 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_374_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (5972018617791 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_374_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (5972018617791 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_374_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_375_1 :
+    Uω (aρ 1) (bρ 1) (47859866092587 / 128000000000000 : ℝ) ≤
+      (-(10011650779804707866637 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11758434634499287 / 32000000000000000 : ℝ) ≤
+      (-(10011650779804707866637 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (11758434634499287 / 32000000000000000 : ℝ) =
+            (11758434634499287 / 8000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11758434634499287 / 8000000000000000 : ℝ) ≤
+            (3851292826195292133363 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (734893365232739 / 2000000000000000 : ℝ)) (q :=
+      (11758434634499287 / 32000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_375_2 :
+    Uω (aρ 2) (bρ 2) (47859866092587 / 128000000000000 : ℝ) ≤
+      (-(10077222676728475367993 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (58407922918271723 / 160000000000000000 : ℝ) ≤
+      (-(10077222676728475367993 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (58407922918271723 / 160000000000000000 : ℝ) =
+            (58407922918271723 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (58407922918271723 / 40000000000000000 : ℝ) ≤
+            (3785720929271524632007 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1825102278447803 / 5000000000000000 : ℝ)) (q :=
+      (58407922918271723 / 160000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_375_3 :
+    Uω (aρ 3) (bρ 3) (47859866092587 / 128000000000000 : ℝ) ≤
+      (-(1020995111110190420443 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (57637805829474139 / 160000000000000000 : ℝ) ≤
+      (-(1020995111110190420443 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (57637805829474139 / 160000000000000000 : ℝ) =
+            (57637805829474139 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (57637805829474139 / 40000000000000000 : ℝ) ≤
+            (365299249489809579557 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (900334006786477 / 2500000000000000 : ℝ)) (q :=
+      (57637805829474139 / 160000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_375_4 :
+    Uω (aρ 4) (bρ 4) (47859866092587 / 128000000000000 : ℝ) ≤
+      (-(521762635000115749317 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (56353736367110363 / 160000000000000000 : ℝ) ≤
+      (-(521762635000115749317 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (56353736367110363 / 160000000000000000 : ℝ) =
+            (56353736367110363 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (56353736367110363 / 40000000000000000 : ℝ) ≤
+            (171384545299884250683 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (879780381353859 / 2500000000000000 : ℝ)) (q :=
+      (56353736367110363 / 160000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_375_5 :
+    Uω (aρ 5) (bρ 5) (47859866092587 / 128000000000000 : ℝ) ≤
+      (-(10790827664296983932529 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (54385145085147731 / 160000000000000000 : ℝ) ≤
+      (-(10790827664296983932529 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (54385145085147731 / 160000000000000000 : ℝ) =
+            (54385145085147731 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (54385145085147731 / 40000000000000000 : ℝ) ≤
+            (3072115941703016067471 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3391237534505107 / 10000000000000000 : ℝ)) (q :=
+      (54385145085147731 / 160000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_375_6 :
+    Uω (aρ 6) (bρ 6) (47859866092587 / 128000000000000 : ℝ) ≤
+      (-(1133013502582213637493 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (51529801599142563 / 160000000000000000 : ℝ) ≤
+      (-(1133013502582213637493 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (51529801599142563 / 160000000000000000 : ℝ) =
+            (51529801599142563 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (51529801599142563 / 40000000000000000 : ℝ) ≤
+            (253280858017786362507 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3201559973894461 / 10000000000000000 : ℝ)) (q :=
+      (51529801599142563 / 160000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_375_7 :
+    Uω (aρ 7) (bρ 7) (47859866092587 / 128000000000000 : ℝ) ≤
+      (-(6066115528469227373073 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (47558032825797763 / 160000000000000000 : ℝ) ≤
+      (-(6066115528469227373073 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (47558032825797763 / 160000000000000000 : ℝ) =
+            (47558032825797763 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (47558032825797763 / 40000000000000000 : ℝ) ≤
+            (865356274530772626927 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2928750257986361 / 10000000000000000 : ℝ)) (q :=
+      (47558032825797763 / 160000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_375_8 :
+    Uω (aρ 8) (bρ 8) (47859866092587 / 128000000000000 : ℝ) ≤
+      (-(6665092777346813657789 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (42188820093339523 / 160000000000000000 : ℝ) ≤
+      (-(6665092777346813657789 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (42188820093339523 / 160000000000000000 : ℝ) =
+            (42188820093339523 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (42188820093339523 / 40000000000000000 : ℝ) ≤
+            (266379025653186342211 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2541140317769081 / 10000000000000000 : ℝ)) (q :=
+      (42188820093339523 / 160000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8])
+      (by norm_num [bρ_8]) (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_375_9 :
+    Uω (aρ 9) (bρ 9) (47859866092587 / 128000000000000 : ℝ) ≤
+      (-(15213557444680535887109 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (34946491241789147 / 160000000000000000 : ℝ) ≤
+      (-(15213557444680535887109 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (34946491241789147 / 160000000000000000 : ℝ) =
+            (34946491241789147 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (34946491241789147 / 20000000000000000 : ℝ) ≤
+            (5580857964319464112891 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (494203709921321 / 2500000000000000 : ℝ)) (q :=
+      (34946491241789147 / 160000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9])
+      (by norm_num [bρ_9]) (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_375_10 :
+    Uω (aρ 10) (bρ 10) (47859866092587 / 128000000000000 : ℝ) ≤
+      (-(18957596983821605453081 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4806533814394367 / 32000000000000000 : ℝ) ≤
+      (-(18957596983821605453081 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4806533814394367 / 32000000000000000 : ℝ) =
+            (4806533814394367 / 4000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4806533814394367 / 4000000000000000 : ℝ) ≤
+            (1836818425178394546919 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6257163137207 / 62500000000000 : ℝ)) (q :=
+      (4806533814394367 / 32000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_375_11 :
+    Uω (aρ 11) (bρ 11) (47859866092587 / 128000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_375_12 :
+    Uω (aρ 12) (bρ 12) (47859866092587 / 128000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_375_13 :
+    Uω (aρ 13) (bρ 13) (47859866092587 / 128000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_375_14 :
+    Uω (aρ 14) (bρ 14) (47859866092587 / 128000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_375_15 :
+    Uω (aρ 15) (bρ 15) (47859866092587 / 128000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_375_16 :
+    Uω (aρ 16) (bρ 16) (47859866092587 / 128000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_375 :
+    Uρ (47859866092587 / 128000000000000 : ℝ) ≤
+      (-(14000602877161742308257 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (47859866092587 / 128000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) *
+        (-(10011650779804707866637 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_375_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (47859866092587 / 128000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(10077222676728475367993 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_375_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (47859866092587 / 128000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1020995111110190420443 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_375_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (47859866092587 / 128000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(521762635000115749317 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_375_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (47859866092587 / 128000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(10790827664296983932529 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_375_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (47859866092587 / 128000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1133013502582213637493 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_375_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (47859866092587 / 128000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(6066115528469227373073 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_375_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (47859866092587 / 128000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(6665092777346813657789 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_375_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (47859866092587 / 128000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(15213557444680535887109 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_375_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (47859866092587 / 128000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(18957596983821605453081 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_375_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (47859866092587 / 128000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_375_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (47859866092587 / 128000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_375_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (47859866092587 / 128000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_375_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (47859866092587 / 128000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_375_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (47859866092587 / 128000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_375_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (47859866092587 / 128000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_375_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/U31.lean b/LeanPool/Zeta5Irrational/Table/U31.lean
new file mode 100644
index 0000000000..7ab756e9eb
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/U31.lean
@@ -0,0 +1,4159 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-! # Certified arcsine potential bounds (U31) -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma U_376_1 :
+    Uω (aρ 1) (bρ 1) (23971791621423 / 64000000000000 : ℝ) ≤
+      (-(9993867007066015957191 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (29448410430551627 / 80000000000000000 : ℝ) ≤
+      (-(9993867007066015957191 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (29448410430551627 / 80000000000000000 : ℝ) =
+            (29448410430551627 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (29448410430551627 / 20000000000000000 : ℝ) ≤
+            (3869076598933984042809 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3681007384875563 / 10000000000000000 : ℝ)) (q :=
+      (29448410430551627 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_376_2 :
+    Uω (aρ 2) (bρ 2) (23971791621423 / 64000000000000 : ℝ) ≤
+      (-(10059320806281235657447 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5851257767303279 / 16000000000000000 : ℝ) ≤
+      (-(10059320806281235657447 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5851257767303279 / 16000000000000000 : ℝ) =
+            (5851257767303279 / 4000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5851257767303279 / 4000000000000000 : ℝ) ≤
+            (3803622799718764342553 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (731349199775351 / 2000000000000000 : ℝ)) (q :=
+      (5851257767303279 / 16000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_376_3 :
+    Uω (aρ 3) (bρ 3) (23971791621423 / 64000000000000 : ℝ) ≤
+      (-(2038361309453471391991 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5774248205069027 / 16000000000000000 : ℝ) ≤
+      (-(2038361309453471391991 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5774248205069027 / 16000000000000000 : ℝ) =
+            (5774248205069027 / 4000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5774248205069027 / 4000000000000000 : ℝ) ≤
+            (734227411746528608009 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (180394007621697 / 500000000000000 : ℝ)) (q :=
+      (5774248205069027 / 16000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_376_4 :
+    Uω (aρ 4) (bρ 4) (23971791621423 / 64000000000000 : ℝ) ≤
+      (-(5208342325874971915933 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28229235731470567 / 80000000000000000 : ℝ) ≤
+      (-(5208342325874971915933 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (28229235731470567 / 80000000000000000 : ℝ) =
+            (28229235731470567 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28229235731470567 / 20000000000000000 : ℝ) ≤
+            (1723129477125028084067 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1762836505015149 / 5000000000000000 : ℝ)) (q :=
+      (28229235731470567 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_376_5 :
+    Uω (aρ 5) (bρ 5) (23971791621423 / 64000000000000 : ℝ) ≤
+      (-(2692890035511764538969 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5449003279831591 / 16000000000000000 : ℝ) ≤
+      (-(2692890035511764538969 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5449003279831591 / 16000000000000000 : ℝ) =
+            (5449003279831591 / 4000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5449003279831591 / 4000000000000000 : ℝ) ≤
+            (772845865988235461031 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (679561619257429 / 2000000000000000 : ℝ)) (q :=
+      (5449003279831591 / 16000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_376_6 :
+    Uω (aρ 6) (bρ 6) (23971791621423 / 64000000000000 : ℝ) ≤
+      (-(11309727313063090127271 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25817534757717279 / 80000000000000000 : ℝ) ≤
+      (-(11309727313063090127271 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (25817534757717279 / 80000000000000000 : ℝ) =
+            (25817534757717279 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25817534757717279 / 20000000000000000 : ℝ) ≤
+            (2553216292936909872729 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (100255564408343 / 312500000000000 : ℝ)) (q :=
+      (25817534757717279 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_376_7 :
+    Uω (aρ 7) (bρ 7) (23971791621423 / 64000000000000 : ℝ) ≤
+      (-(12109924977090579279271 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23832117278369707 / 80000000000000000 : ℝ) ≤
+      (-(12109924977090579279271 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (23832117278369707 / 80000000000000000 : ℝ) =
+            (23832117278369707 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23832117278369707 / 20000000000000000 : ℝ) ≤
+            (1753018628909420720729 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2935485071990083 / 10000000000000000 : ℝ)) (q :=
+      (23832117278369707 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_376_8 :
+    Uω (aρ 8) (bρ 8) (23971791621423 / 64000000000000 : ℝ) ≤
+      (-(13304483040210431015309 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21148697721244531 / 80000000000000000 : ℝ) ≤
+      (-(13304483040210431015309 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (21148697721244531 / 80000000000000000 : ℝ) =
+            (21148697721244531 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21148697721244531 / 20000000000000000 : ℝ) ≤
+            (558460565789568984691 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2548171834048789 / 10000000000000000 : ℝ)) (q :=
+      (21148697721244531 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_376_9 :
+    Uω (aρ 9) (bρ 9) (23971791621423 / 64000000000000 : ℝ) ≤
+      (-(15180537896117200035529 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3506207369726651 / 16000000000000000 : ℝ) ≤
+      (-(15180537896117200035529 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3506207369726651 / 16000000000000000 : ℝ) =
+            (3506207369726651 / 2000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3506207369726651 / 2000000000000000 : ℝ) ≤
+            (5613877512882799964471 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (198472224425597 / 1000000000000000 : ℝ)) (q :=
+      (3506207369726651 / 16000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_376_10 :
+    Uω (aρ 10) (bρ 10) (23971791621423 / 64000000000000 : ℝ) ≤
+      (-(18892689722321229897263 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12094582942561791 / 80000000000000000 : ℝ) ≤
+      (-(18892689722321229897263 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (12094582942561791 / 80000000000000000 : ℝ) =
+            (12094582942561791 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12094582942561791 / 10000000000000000 : ℝ) ≤
+            (1901725686678770102737 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (63385487577069 / 625000000000000 : ℝ)) (q :=
+      (12094582942561791 / 80000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_376_11 :
+    Uω (aρ 11) (bρ 11) (23971791621423 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_376_12 :
+    Uω (aρ 12) (bρ 12) (23971791621423 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_376_13 :
+    Uω (aρ 13) (bρ 13) (23971791621423 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_376_14 :
+    Uω (aρ 14) (bρ 14) (23971791621423 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_376_15 :
+    Uω (aρ 15) (bρ 15) (23971791621423 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_376_16 :
+    Uω (aρ 16) (bρ 16) (23971791621423 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_376 :
+    Uρ (23971791621423 / 64000000000000 : ℝ) ≤
+      (-(6991230191038809068327 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (23971791621423 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(9993867007066015957191 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_376_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (23971791621423 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(10059320806281235657447 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_376_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (23971791621423 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(2038361309453471391991 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_376_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (23971791621423 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(5208342325874971915933 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_376_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (23971791621423 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(2692890035511764538969 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_376_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (23971791621423 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(11309727313063090127271 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_376_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (23971791621423 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(12109924977090579279271 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_376_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (23971791621423 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(13304483040210431015309 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_376_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (23971791621423 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(15180537896117200035529 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_376_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (23971791621423 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(18892689722321229897263 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_376_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (23971791621423 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_376_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (23971791621423 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_376_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (23971791621423 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_376_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (23971791621423 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_376_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (23971791621423 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_376_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (23971791621423 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_376_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_377_1 :
+    Uω (aρ 1) (bρ 1) (9605460078621 / 25600000000000 : ℝ) ≤
+      (-(9976114805201331339663 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (59001468545273121 / 160000000000000000 : ℝ) ≤
+      (-(9976114805201331339663 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (59001468545273121 / 160000000000000000 : ℝ) =
+            (59001468545273121 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (59001468545273121 / 40000000000000000 : ℝ) ≤
+            (3886828800798668660337 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (921886985758203 / 2500000000000000 : ℝ)) (q :=
+      (59001468545273121 / 160000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_377_2 :
+    Uω (aρ 2) (bρ 2) (9605460078621 / 25600000000000 : ℝ) ≤
+      (-(2510362732834259353731 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (58617232398093521 / 160000000000000000 : ℝ) ≤
+      (-(2510362732834259353731 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (58617232398093521 / 160000000000000000 : ℝ) =
+            (58617232398093521 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (58617232398093521 / 40000000000000000 : ℝ) ≤
+            (955373168665740646269 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1831643718572681 / 5000000000000000 : ℝ)) (q :=
+      (58617232398093521 / 160000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_377_3 :
+    Uω (aρ 3) (bρ 3) (9605460078621 / 25600000000000 : ℝ) ≤
+      (-(10173694864971688524833 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (57847158164087953 / 160000000000000000 : ℝ) ≤
+      (-(10173694864971688524833 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (57847158164087953 / 160000000000000000 : ℝ) =
+            (57847158164087953 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (57847158164087953 / 40000000000000000 : ℝ) ≤
+            (3689248741028311475167 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1807212132122333 / 5000000000000000 : ℝ)) (q :=
+      (57847158164087953 / 160000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_377_4 :
+    Uω (aρ 4) (bρ 4) (9605460078621 / 25600000000000 : ℝ) ≤
+      (-(5199075537472519501479 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (56563206230168601 / 160000000000000000 : ℝ) ≤
+      (-(5199075537472519501479 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (56563206230168601 / 160000000000000000 : ℝ) =
+            (56563206230168601 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (56563206230168601 / 40000000000000000 : ℝ) ≤
+            (1732396265527480498521 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3532224453569747 / 10000000000000000 : ℝ)) (q :=
+      (56563206230168601 / 160000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_377_5 :
+    Uω (aρ 5) (bρ 5) (9605460078621 / 25600000000000 : ℝ) ≤
+      (-(10752329842358406152703 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (54594919582282977 / 160000000000000000 : ℝ) ≤
+      (-(10752329842358406152703 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (54594919582282977 / 160000000000000000 : ℝ) =
+            (54594919582282977 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (54594919582282977 / 40000000000000000 : ℝ) ≤
+            (3110613763641593847297 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (851094635479761 / 2500000000000000 : ℝ)) (q :=
+      (54594919582282977 / 160000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_377_6 :
+    Uω (aρ 6) (bρ 6) (9605460078621 / 25600000000000 : ℝ) ≤
+      (-(11289361654615094735919 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (51740334887951473 / 160000000000000000 : ℝ) ≤
+      (-(11289361654615094735919 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (51740334887951473 / 160000000000000000 : ℝ) =
+            (51740334887951473 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (51740334887951473 / 40000000000000000 : ℝ) ≤
+            (2573581951384905264081 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1607397915133803 / 5000000000000000 : ℝ)) (q :=
+      (51740334887951473 / 160000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_377_7 :
+    Uω (aρ 7) (bρ 7) (9605460078621 / 25600000000000 : ℝ) ≤
+      (-(12087670011636276249273 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (47770429270239633 / 160000000000000000 : ℝ) ≤
+      (-(12087670011636276249273 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (47770429270239633 / 160000000000000000 : ℝ) =
+            (47770429270239633 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (47770429270239633 / 40000000000000000 : ℝ) ≤
+            (1775273594363723750727 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1471109504406813 / 5000000000000000 : ℝ)) (q :=
+      (47770429270239633 / 160000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_377_8 :
+    Uω (aρ 8) (bρ 8) (9605460078621 / 25600000000000 : ℝ) ≤
+      (-(13278851339418135968547 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1696237996931321 / 6400000000000000 : ℝ) ≤
+      (-(13278851339418135968547 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1696237996931321 / 6400000000000000 : ℝ) =
+            (1696237996931321 / 1600000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1696237996931321 / 1600000000000000 : ℝ) ≤
+            (584092266581864031453 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (51104014835681 / 200000000000000 : ℝ)) (q :=
+      (1696237996931321 / 6400000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_377_9 :
+    Uω (aρ 9) (bρ 9) (9605460078621 / 25600000000000 : ℝ) ≤
+      (-(3029529911736138818707 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (35177576859602393 / 160000000000000000 : ℝ) ≤
+      (-(3029529911736138818707 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (35177576859602393 / 160000000000000000 : ℝ) =
+            (35177576859602393 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (35177576859602393 / 20000000000000000 : ℝ) ≤
+            (1129353170063861181293 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1992619737183971 / 10000000000000000 : ℝ)) (q :=
+      (35177576859602393 / 160000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9])
+      (by norm_num [bρ_9]) (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_377_10 :
+    Uω (aρ 10) (bρ 10) (9605460078621 / 25600000000000 : ℝ) ≤
+      (-(18828606670727822522699 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24344675185006513 / 160000000000000000 : ℝ) ≤
+      (-(18828606670727822522699 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (24344675185006513 / 160000000000000000 : ℝ) =
+            (24344675185006513 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24344675185006513 / 20000000000000000 : ℝ) ≤
+            (1965808738272177477301 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (513533030677243 / 5000000000000000 : ℝ)) (q :=
+      (24344675185006513 / 160000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_377_11 :
+    Uω (aρ 11) (bρ 11) (9605460078621 / 25600000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_377_12 :
+    Uω (aρ 12) (bρ 12) (9605460078621 / 25600000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_377_13 :
+    Uω (aρ 13) (bρ 13) (9605460078621 / 25600000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_377_14 :
+    Uω (aρ 14) (bρ 14) (9605460078621 / 25600000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_377_15 :
+    Uω (aρ 15) (bρ 15) (9605460078621 / 25600000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_377_16 :
+    Uω (aρ 16) (bρ 16) (9605460078621 / 25600000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_377 :
+    Uρ (9605460078621 / 25600000000000 : ℝ) ≤
+      (-(2792884219465679483077 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (9605460078621 / 25600000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(9976114805201331339663 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_377_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (9605460078621 / 25600000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(2510362732834259353731 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_377_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (9605460078621 / 25600000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(10173694864971688524833 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_377_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (9605460078621 / 25600000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(5199075537472519501479 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_377_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (9605460078621 / 25600000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(10752329842358406152703 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_377_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (9605460078621 / 25600000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(11289361654615094735919 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_377_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (9605460078621 / 25600000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(12087670011636276249273 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_377_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (9605460078621 / 25600000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(13278851339418135968547 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_377_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (9605460078621 / 25600000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(3029529911736138818707 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_377_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (9605460078621 / 25600000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(18828606670727822522699 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_377_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (9605460078621 / 25600000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_377_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (9605460078621 / 25600000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_377_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (9605460078621 / 25600000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_377_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (9605460078621 / 25600000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_377_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (9605460078621 / 25600000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_377_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (9605460078621 / 25600000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_377_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_378_1 :
+    Uω (aρ 1) (bρ 1) (12027754385841 / 32000000000000 : ℝ) ≤
+      (-(9958394062313398461891 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2955305811251481 / 8000000000000000 : ℝ) ≤
+      (-(9958394062313398461891 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2955305811251481 / 8000000000000000 : ℝ) =
+            (2955305811251481 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2955305811251481 / 2000000000000000 : ℝ) ≤
+            (3904549543686601538109 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (369408850063839 / 1000000000000000 : ℝ)) (q :=
+      (2955305811251481 / 8000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_378_2 :
+    Uω (aρ 2) (bρ 2) (12027754385841 / 32000000000000 : ℝ) ≤
+      (-(10023612937712580116809 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14680471773403183 / 40000000000000000 : ℝ) ≤
+      (-(10023612937712580116809 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (14680471773403183 / 40000000000000000 : ℝ) =
+            (14680471773403183 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14680471773403183 / 10000000000000000 : ℝ) ≤
+            (3839330668287419883191 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3669828871721279 / 10000000000000000 : ℝ)) (q :=
+      (14680471773403183 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_378_3 :
+    Uω (aρ 3) (bρ 3) (12027754385841 / 32000000000000 : ℝ) ≤
+      (-(10155615945187571679447 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14487958542562941 / 40000000000000000 : ℝ) ≤
+      (-(10155615945187571679447 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (14487958542562941 / 40000000000000000 : ℝ) =
+            (14487958542562941 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14487958542562941 / 10000000000000000 : ℝ) ≤
+            (3707327660812428320553 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1810484181325579 / 5000000000000000 : ℝ)) (q :=
+      (14487958542562941 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_378_4 :
+    Uω (aρ 4) (bρ 4) (12027754385841 / 32000000000000 : ℝ) ≤
+      (-(518982592080921205731 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14166985167654459 / 40000000000000000 : ℝ) ≤
+      (-(518982592080921205731 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (14166985167654459 / 40000000000000000 : ℝ) =
+            (14166985167654459 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14166985167654459 / 10000000000000000 : ℝ) ≤
+            (174164588219078794269 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3538775856261917 / 10000000000000000 : ℝ)) (q :=
+      (14166985167654459 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_378_5 :
+    Uω (aρ 5) (bρ 5) (12027754385841 / 32000000000000 : ℝ) ≤
+      (-(10733136621036858840227 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13674951360604657 / 40000000000000000 : ℝ) ≤
+      (-(10733136621036858840227 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (13674951360604657 / 40000000000000000 : ℝ) =
+            (13674951360604657 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13674951360604657 / 10000000000000000 : ℝ) ≤
+            (3129806984963141159773 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (213184304504501 / 625000000000000 : ℝ)) (q :=
+      (13674951360604657 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_378_6 :
+    Uω (aρ 6) (bρ 6) (12027754385841 / 32000000000000 : ℝ) ≤
+      (-(11269037875509371656561 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12961399433092597 / 40000000000000000 : ℝ) ≤
+      (-(11269037875509371656561 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (12961399433092597 / 40000000000000000 : ℝ) =
+            (12961399433092597 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12961399433092597 / 10000000000000000 : ℝ) ≤
+            (2593905730490628343439 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1610706641727993 / 5000000000000000 : ℝ)) (q :=
+      (12961399433092597 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_378_7 :
+    Uω (aρ 7) (bρ 7) (12027754385841 / 32000000000000 : ℝ) ≤
+      (-(2413093184052817137671 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11969154253592859 / 40000000000000000 : ℝ) ≤
+      (-(2413093184052817137671 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (11969154253592859 / 40000000000000000 : ℝ) =
+            (11969154253592859 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11969154253592859 / 10000000000000000 : ℝ) ≤
+            (359495537147182862329 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2948952074466117 / 10000000000000000 : ℝ)) (q :=
+      (11969154253592859 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_378_8 :
+    Uω (aρ 8) (bρ 8) (12027754385841 / 32000000000000 : ℝ) ≤
+      (-(13253290037235338137851 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10628620926866167 / 40000000000000000 : ℝ) ≤
+      (-(13253290037235338137851 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (10628620926866167 / 40000000000000000 : ℝ) =
+            (10628620926866167 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10628620926866167 / 10000000000000000 : ℝ) ≤
+            (609653568764661862149 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2562227062442771 / 10000000000000000 : ℝ)) (q :=
+      (10628620926866167 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_378_9 :
+    Uω (aρ 9) (bρ 9) (12027754385841 / 32000000000000 : ℝ) ≤
+      (-(7557445616019576398087 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (352930016678841 / 1600000000000000 : ℝ) ≤
+      (-(7557445616019576398087 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (352930016678841 / 1600000000000000 : ℝ) =
+            (352930016678841 / 200000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (352930016678841 / 200000000000000 : ℝ) ≤
+            (2839762088480423601913 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2500634294819 / 12500000000000 : ℝ)) (q :=
+      (352930016678841 / 1600000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_378_10 :
+    Uω (aρ 10) (bρ 10) (12027754385841 / 32000000000000 : ℝ) ≤
+      (-(18765319741641310131049 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6124808429773381 / 40000000000000000 : ℝ) ≤
+      (-(18765319741641310131049 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (6124808429773381 / 40000000000000000 : ℝ) =
+            (6124808429773381 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6124808429773381 / 5000000000000000 : ℝ) ≤
+            (2029095667358689868951 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (519922737875689 / 5000000000000000 : ℝ)) (q :=
+      (6124808429773381 / 40000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_378_11 :
+    Uω (aρ 11) (bρ 11) (12027754385841 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_378_12 :
+    Uω (aρ 12) (bρ 12) (12027754385841 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_378_13 :
+    Uω (aρ 13) (bρ 13) (12027754385841 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_378_14 :
+    Uω (aρ 14) (bρ 14) (12027754385841 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_378_15 :
+    Uω (aρ 15) (bρ 15) (12027754385841 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_378_16 :
+    Uω (aρ 16) (bρ 16) (12027754385841 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_378 :
+    Uρ (12027754385841 / 32000000000000 : ℝ) ≤
+      (-(13946482418286257096903 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (12027754385841 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(9958394062313398461891 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_378_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (12027754385841 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(10023612937712580116809 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_378_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (12027754385841 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(10155615945187571679447 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_378_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (12027754385841 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(518982592080921205731 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_378_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (12027754385841 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(10733136621036858840227 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_378_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (12027754385841 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(11269037875509371656561 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_378_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (12027754385841 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(2413093184052817137671 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_378_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (12027754385841 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(13253290037235338137851 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_378_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (12027754385841 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(7557445616019576398087 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_378_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (12027754385841 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(18765319741641310131049 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_378_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (12027754385841 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_378_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (12027754385841 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_378_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (12027754385841 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_378_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (12027754385841 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_378_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (12027754385841 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_378_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (12027754385841 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_378_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_379_1 :
+    Uω (aρ 1) (bρ 1) (48194734693623 / 128000000000000 : ℝ) ≤
+      (-(9940704667098828597671 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11842152780079227 / 32000000000000000 : ℝ) ≤
+      (-(9940704667098828597671 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (11842152780079227 / 32000000000000000 : ℝ) =
+            (11842152780079227 / 8000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11842152780079227 / 8000000000000000 : ℝ) ≤
+            (3922238938901171402329 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (185031452884761 / 500000000000000 : ℝ)) (q :=
+      (11842152780079227 / 32000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_379_2 :
+    Uω (aρ 2) (bρ 2) (48194734693623 / 128000000000000 : ℝ) ≤
+      (-(2501451677958700695517 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (58826541759748119 / 160000000000000000 : ℝ) ≤
+      (-(2501451677958700695517 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (58826541759748119 / 160000000000000000 : ℝ) =
+            (58826541759748119 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (58826541759748119 / 40000000000000000 : ℝ) ≤
+            (964284223541299304483 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1838185151312109 / 5000000000000000 : ℝ)) (q :=
+      (58826541759748119 / 160000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_379_3 :
+    Uω (aρ 3) (bρ 3) (48194734693623 / 128000000000000 : ℝ) ≤
+      (-(5068784834766586812851 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (58056510069762047 / 160000000000000000 : ℝ) ≤
+      (-(5068784834766586812851 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (58056510069762047 / 160000000000000000 : ℝ) =
+            (58056510069762047 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (58056510069762047 / 40000000000000000 : ℝ) ≤
+            (1862686968233413187149 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3627512447725959 / 10000000000000000 : ℝ)) (q :=
+      (58056510069762047 / 160000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_379_4 :
+    Uω (aρ 4) (bρ 4) (48194734693623 / 128000000000000 : ℝ) ≤
+      (-(414447472980535392993 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (56772674786100383 / 160000000000000000 : ℝ) ≤
+      (-(414447472980535392993 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (56772674786100383 / 160000000000000000 : ℝ) =
+            (56772674786100383 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (56772674786100383 / 40000000000000000 : ℝ) ≤
+            (140070271259464607007 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3545327218333251 / 10000000000000000 : ℝ)) (q :=
+      (56772674786100383 / 160000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_379_5 :
+    Uω (aρ 5) (bρ 5) (48194734693623 / 128000000000000 : ℝ) ≤
+      (-(10713980334728234077113 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (54804690384051199 / 160000000000000000 : ℝ) ≤
+      (-(10713980334728234077113 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (54804690384051199 / 160000000000000000 : ℝ) =
+            (54804690384051199 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (54804690384051199 / 40000000000000000 : ℝ) ≤
+            (3148963271271765922887 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3417519087412103 / 10000000000000000 : ℝ)) (q :=
+      (54804690384051199 / 160000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_379_6 :
+    Uω (aρ 6) (bρ 6) (48194734693623 / 128000000000000 : ℝ) ≤
+      (-(11248755801878956064031 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (51950858064239079 / 160000000000000000 : ℝ) ≤
+      (-(11248755801878956064031 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (51950858064239079 / 160000000000000000 : ℝ) =
+            (51950858064239079 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (51950858064239079 / 40000000000000000 : ℝ) ≤
+            (2614187804121043935969 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (807007605643897 / 2500000000000000 : ℝ)) (q :=
+      (51950858064239079 / 160000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_379_7 :
+    Uω (aρ 7) (bρ 7) (48194734693623 / 128000000000000 : ℝ) ≤
+      (-(602165623219844768649 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (47982797836763607 / 160000000000000000 : ℝ) ≤
+      (-(602165623219844768649 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (47982797836763607 / 160000000000000000 : ℝ) =
+            (47982797836763607 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (47982797836763607 / 40000000000000000 : ℝ) ≤
+            (90981557080155231351 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1477842137450577 / 5000000000000000 : ℝ)) (q :=
+      (47982797836763607 / 160000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_379_8 :
+    Uω (aρ 8) (bρ 8) (48194734693623 / 128000000000000 : ℝ) ≤
+      (-(6613899361208653044927 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (42622996964835687 / 160000000000000000 : ℝ) ≤
+      (-(6613899361208653044927 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (42622996964835687 / 160000000000000000 : ℝ) =
+            (42622996964835687 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (42622996964835687 / 40000000000000000 : ℝ) ≤
+            (317572441791346955073 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (642312704312541 / 2500000000000000 : ℝ)) (q :=
+      (42622996964835687 / 160000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8])
+      (by norm_num [bρ_8]) (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_379_9 :
+    Uω (aρ 9) (bρ 9) (48194734693623 / 128000000000000 : ℝ) ≤
+      (-(3770565433462939589999 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1416333961811631 / 6400000000000000 : ℝ) ≤
+      (-(3770565433462939589999 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1416333961811631 / 6400000000000000 : ℝ) =
+            (1416333961811631 / 800000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1416333961811631 / 800000000000000 : ℝ) ≤
+            (1428038418787060410001 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (40167709113341 / 200000000000000 : ℝ)) (q :=
+      (1416333961811631 / 6400000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_379_10 :
+    Uω (aρ 10) (bρ 10) (48194734693623 / 128000000000000 : ℝ) ≤
+      (-(3740560484983560820433 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24652876119599351 / 160000000000000000 : ℝ) ≤
+      (-(3740560484983560820433 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (24652876119599351 / 160000000000000000 : ℝ) =
+            (24652876119599351 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24652876119599351 / 20000000000000000 : ℝ) ≤
+            (418322596816439179567 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (526255186725311 / 5000000000000000 : ℝ)) (q :=
+      (24652876119599351 / 160000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_379_11 :
+    Uω (aρ 11) (bρ 11) (48194734693623 / 128000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_379_12 :
+    Uω (aρ 12) (bρ 12) (48194734693623 / 128000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_379_13 :
+    Uω (aρ 13) (bρ 13) (48194734693623 / 128000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_379_14 :
+    Uω (aρ 14) (bρ 14) (48194734693623 / 128000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_379_15 :
+    Uω (aρ 15) (bρ 15) (48194734693623 / 128000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_379_16 :
+    Uω (aρ 16) (bρ 16) (48194734693623 / 128000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_379 :
+    Uρ (48194734693623 / 128000000000000 : ℝ) ≤
+      (-(13928641877448414677921 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (48194734693623 / 128000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(9940704667098828597671 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_379_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (48194734693623 / 128000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(2501451677958700695517 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_379_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (48194734693623 / 128000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(5068784834766586812851 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_379_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (48194734693623 / 128000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(414447472980535392993 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_379_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (48194734693623 / 128000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(10713980334728234077113 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_379_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (48194734693623 / 128000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(11248755801878956064031 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_379_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (48194734693623 / 128000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(602165623219844768649 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_379_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (48194734693623 / 128000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(6613899361208653044927 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_379_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (48194734693623 / 128000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(3770565433462939589999 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_379_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (48194734693623 / 128000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(3740560484983560820433 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_379_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (48194734693623 / 128000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_379_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (48194734693623 / 128000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_379_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (48194734693623 / 128000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_379_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (48194734693623 / 128000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_379_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (48194734693623 / 128000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_379_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (48194734693623 / 128000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_379_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_380_1 :
+    Uω (aρ 1) (bρ 1) (24139225921941 / 64000000000000 : ℝ) ≤
+      (-(4961523254421947127759 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (29657705785697957 / 80000000000000000 : ℝ) ≤
+      (-(4961523254421947127759 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (29657705785697957 / 80000000000000000 : ℝ) =
+            (29657705785697957 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (29657705785697957 / 20000000000000000 : ℝ) ≤
+            (1969948548578052872241 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (14481131305493 / 39062500000000 : ℝ)) (q :=
+      (29657705785697957 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_380_2 :
+    Uω (aρ 2) (bρ 2) (24139225921941 / 64000000000000 : ℝ) ≤
+      (-(1248504017592068370069 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (29465598198328121 / 80000000000000000 : ℝ) ≤
+      (-(1248504017592068370069 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (29465598198328121 / 80000000000000000 : ℝ) =
+            (29465598198328121 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (29465598198328121 / 20000000000000000 : ℝ) ≤
+            (484363933157931629931 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3682911729873749 / 10000000000000000 : ℝ)) (q :=
+      (29465598198328121 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_380_3 :
+    Uω (aρ 3) (bρ 3) (24139225921941 / 64000000000000 : ℝ) ≤
+      (-(10119555920267487374541 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (29080592931597497 / 80000000000000000 : ℝ) ≤
+      (-(10119555920267487374541 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (29080592931597497 / 80000000000000000 : ℝ) =
+            (29080592931597497 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (29080592931597497 / 20000000000000000 : ℝ) ≤
+            (3743387685732512625459 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3634056519541093 / 10000000000000000 : ℝ)) (q :=
+      (29080592931597497 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_380_4 :
+    Uω (aρ 4) (bρ 4) (24139225921941 / 64000000000000 : ℝ) ≤
+      (-(10342755897080370757173 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28438704289207221 / 80000000000000000 : ℝ) ≤
+      (-(10342755897080370757173 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (28438704289207221 / 80000000000000000 : ℝ) =
+            (28438704289207221 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28438704289207221 / 20000000000000000 : ℝ) ≤
+            (3520187708919629242827 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (887969635002131 / 2500000000000000 : ℝ)) (q :=
+      (28438704289207221 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_380_5 :
+    Uω (aρ 5) (bρ 5) (24139225921941 / 64000000000000 : ℝ) ≤
+      (-(10694860840911806552079 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5490957441246801 / 16000000000000000 : ℝ) ≤
+      (-(10694860840911806552079 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5490957441246801 / 16000000000000000 : ℝ) =
+            (5490957441246801 / 4000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5490957441246801 / 4000000000000000 : ℝ) ≤
+            (3168082765088193447921 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (171204459430011 / 500000000000000 : ℝ)) (q :=
+      (5490957441246801 / 16000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_380_6 :
+    Uω (aρ 6) (bρ 6) (24139225921941 / 64000000000000 : ℝ) ≤
+      (-(5614257630474681781133 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (26028057949488689 / 80000000000000000 : ℝ) ≤
+      (-(5614257630474681781133 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (26028057949488689 / 80000000000000000 : ℝ) =
+            (26028057949488689 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26028057949488689 / 20000000000000000 : ℝ) ≤
+            (1317214172525318218867 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3234647249553891 / 10000000000000000 : ℝ)) (q :=
+      (26028057949488689 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_380_7 :
+    Uω (aρ 7) (bρ 7) (24139225921941 / 64000000000000 : ℝ) ≤
+      (-(3005302351793728355003 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24044485892302941 / 80000000000000000 : ℝ) ≤
+      (-(3005302351793728355003 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (24044485892302941 / 80000000000000000 : ℝ) =
+            (24044485892302941 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24044485892302941 / 20000000000000000 : ℝ) ≤
+            (460433549706271644997 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1481207808008727 / 5000000000000000 : ℝ)) (q :=
+      (24044485892302941 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_380_8 :
+    Uω (aρ 8) (bρ 8) (24139225921941 / 64000000000000 : ℝ) ≤
+      (-(6601188493753449905703 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21365744931642169 / 80000000000000000 : ℝ) ≤
+      (-(6601188493753449905703 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (21365744931642169 / 80000000000000000 : ℝ) =
+            (21365744931642169 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21365744931642169 / 20000000000000000 : ℝ) ≤
+            (330283309246550094297 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2576272027192261 / 10000000000000000 : ℝ)) (q :=
+      (21365744931642169 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_380_9 :
+    Uω (aρ 9) (bρ 9) (24139225921941 / 64000000000000 : ℝ) ≤
+      (-(601990395975625400427 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17761809949724573 / 80000000000000000 : ℝ) ≤
+      (-(601990395975625400427 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (17761809949724573 / 80000000000000000 : ℝ) =
+            (17761809949724573 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17761809949724573 / 10000000000000000 : ℝ) ≤
+            (229786220384374599573 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1008126955036431 / 5000000000000000 : ℝ)) (q :=
+      (17761809949724573 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_380_10 :
+    Uω (aρ 10) (bρ 10) (24139225921941 / 64000000000000 : ℝ) ≤
+      (-(4660257416487714340737 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12402817534183093 / 80000000000000000 : ℝ) ≤
+      (-(4660257416487714340737 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (12402817534183093 / 80000000000000000 : ℝ) =
+            (12402817534183093 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12402817534183093 / 10000000000000000 : ℝ) ≤
+            (538346435762285659263 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (266266209920623 / 2500000000000000 : ℝ)) (q :=
+      (12402817534183093 / 80000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_380_11 :
+    Uω (aρ 11) (bρ 11) (24139225921941 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_380_12 :
+    Uω (aρ 12) (bρ 12) (24139225921941 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_380_13 :
+    Uω (aρ 13) (bρ 13) (24139225921941 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_380_14 :
+    Uω (aρ 14) (bρ 14) (24139225921941 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_380_15 :
+    Uω (aρ 15) (bρ 15) (24139225921941 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_380_16 :
+    Uω (aρ 16) (bρ 16) (24139225921941 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_380 :
+    Uρ (24139225921941 / 64000000000000 : ℝ) ≤
+      (-(6955448566996252704807 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (24139225921941 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(4961523254421947127759 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_380_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (24139225921941 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1248504017592068370069 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_380_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (24139225921941 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(10119555920267487374541 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_380_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (24139225921941 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(10342755897080370757173 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_380_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (24139225921941 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(10694860840911806552079 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_380_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (24139225921941 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(5614257630474681781133 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_380_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (24139225921941 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(3005302351793728355003 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_380_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (24139225921941 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(6601188493753449905703 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_380_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (24139225921941 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(601990395975625400427 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_380_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (24139225921941 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(4660257416487714340737 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_380_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (24139225921941 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_380_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (24139225921941 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_380_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (24139225921941 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_380_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (24139225921941 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_380_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (24139225921941 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_380_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (24139225921941 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_380_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_381_1 :
+    Uω (aρ 1) (bρ 1) (48362168994141 / 128000000000000 : ℝ) ≤
+      (-(9905419477420379522699 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (59420059238052021 / 160000000000000000 : ℝ) ≤
+      (-(9905419477420379522699 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (59420059238052021 / 160000000000000000 : ℝ) =
+            (59420059238052021 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (59420059238052021 / 40000000000000000 : ℝ) ≤
+            (3957524128579620477301 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3713710170174237 / 10000000000000000 : ℝ)) (q :=
+      (59420059238052021 / 160000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_381_2 :
+    Uω (aρ 2) (bρ 2) (48362168994141 / 128000000000000 : ℝ) ≤
+      (-(997028911205224600551 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (59035851004492581 / 160000000000000000 : ℝ) ≤
+      (-(997028911205224600551 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (59035851004492581 / 160000000000000000 : ℝ) =
+            (59035851004492581 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (59035851004492581 / 40000000000000000 : ℝ) ≤
+            (389265449394775399449 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3689453153489307 / 10000000000000000 : ℝ)) (q :=
+      (59035851004492581 / 160000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_381_3 :
+    Uω (aρ 3) (bρ 3) (48362168994141 / 128000000000000 : ℝ) ≤
+      (-(10101574580285721318899 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (58265861551122629 / 160000000000000000 : ℝ) ≤
+      (-(10101574580285721318899 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (58265861551122629 / 160000000000000000 : ℝ) =
+            (58265861551122629 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (58265861551122629 / 40000000000000000 : ℝ) ≤
+            (3761369025714278681101 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3640600578168063 / 10000000000000000 : ℝ)) (q :=
+      (58265861551122629 / 160000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_381_4 :
+    Uω (aρ 4) (bρ 4) (48362168994141 / 128000000000000 : ℝ) ≤
+      (-(10324358933471767397939 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (56982142049344933 / 160000000000000000 : ℝ) ≤
+      (-(10324358933471767397939 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (56982142049344933 / 160000000000000000 : ℝ) =
+            (56982142049344933 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (56982142049344933 / 40000000000000000 : ℝ) ≤
+            (3538584672528232602061 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3558429821510851 / 10000000000000000 : ℝ)) (q :=
+      (56982142049344933 / 160000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_381_5 :
+    Uω (aρ 5) (bρ 5) (48362168994141 / 128000000000000 : ℝ) ≤
+      (-(2135155599578764903971 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (55014457532915797 / 160000000000000000 : ℝ) ≤
+      (-(2135155599578764903971 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (55014457532915797 / 160000000000000000 : ℝ) =
+            (55014457532915797 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (55014457532915797 / 40000000000000000 : ℝ) ≤
+            (637433121621235096029 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3430659176292209 / 10000000000000000 : ℝ)) (q :=
+      (55014457532915797 / 160000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_381_6 :
+    Uω (aρ 6) (bρ 6) (48362168994141 / 128000000000000 : ℝ) ≤
+      (-(2802079020257341762367 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (52161371251878461 / 160000000000000000 : ℝ) ≤
+      (-(2802079020257341762367 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (52161371251878461 / 160000000000000000 : ℝ) =
+            (52161371251878461 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (52161371251878461 / 40000000000000000 : ℝ) ≤
+            (663656881242658237633 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1620631883151271 / 5000000000000000 : ℝ)) (q :=
+      (52161371251878461 / 160000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_381_7 :
+    Uω (aρ 7) (bρ 7) (48362168994141 / 128000000000000 : ℝ) ≤
+      (-(11999156513438851452341 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (48195138904654069 / 160000000000000000 : ℝ) ≤
+      (-(11999156513438851452341 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (48195138904654069 / 160000000000000000 : ℝ) =
+            (48195138904654069 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (48195138904654069 / 40000000000000000 : ℝ) ≤
+            (1863787092561148547659 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2969146103659493 / 10000000000000000 : ℝ)) (q :=
+      (48195138904654069 / 160000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_381_8 :
+    Uω (aρ 8) (bρ 8) (48362168994141 / 128000000000000 : ℝ) ≤
+      (-(329425610719656756109 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8567992513762633 / 32000000000000000 : ℝ) ≤
+      (-(329425610719656756109 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8567992513762633 / 32000000000000000 : ℝ) =
+            (8567992513762633 / 8000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8567992513762633 / 8000000000000000 : ℝ) ≤
+            (17147979430343243891 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (129164535650969 / 500000000000000 : ℝ)) (q :=
+      (8567992513762633 / 32000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_381_9 :
+    Uω (aρ 9) (bρ 9) (48362168994141 / 128000000000000 : ℝ) ≤
+      (-(15017384581172952189737 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (35638815122779349 / 160000000000000000 : ℝ) ≤
+      (-(15017384581172952189737 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (35638815122779349 / 160000000000000000 : ℝ) =
+            (35638815122779349 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (35638815122779349 / 20000000000000000 : ℝ) ≤
+            (5777030827827047810263 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2024112910625153 / 10000000000000000 : ℝ)) (q :=
+      (35638815122779349 / 160000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9])
+      (by norm_num [bρ_9]) (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_381_10 :
+    Uω (aρ 10) (bρ 10) (48362168994141 / 128000000000000 : ℝ) ≤
+      (-(18579977755783579673383 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24957541445636653 / 160000000000000000 : ℝ) ≤
+      (-(18579977755783579673383 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (24957541445636653 / 160000000000000000 : ℝ) =
+            (24957541445636653 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24957541445636653 / 20000000000000000 : ℝ) ≤
+            (2214437653216420326617 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (269378183619329 / 2500000000000000 : ℝ)) (q :=
+      (24957541445636653 / 160000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_381_11 :
+    Uω (aρ 11) (bρ 11) (48362168994141 / 128000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_381_12 :
+    Uω (aρ 12) (bρ 12) (48362168994141 / 128000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_381_13 :
+    Uω (aρ 13) (bρ 13) (48362168994141 / 128000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_381_14 :
+    Uω (aρ 14) (bρ 14) (48362168994141 / 128000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_381_15 :
+    Uω (aρ 15) (bρ 15) (48362168994141 / 128000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_381_16 :
+    Uω (aρ 16) (bρ 16) (48362168994141 / 128000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_381 :
+    Uρ (48362168994141 / 128000000000000 : ℝ) ≤
+      (-(2778649192868835725683 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (48362168994141 / 128000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(9905419477420379522699 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_381_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (48362168994141 / 128000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(997028911205224600551 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_381_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (48362168994141 / 128000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(10101574580285721318899 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_381_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (48362168994141 / 128000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(10324358933471767397939 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_381_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (48362168994141 / 128000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(2135155599578764903971 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_381_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (48362168994141 / 128000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(2802079020257341762367 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_381_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (48362168994141 / 128000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(11999156513438851452341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_381_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (48362168994141 / 128000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(329425610719656756109 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_381_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (48362168994141 / 128000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(15017384581172952189737 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_381_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (48362168994141 / 128000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(18579977755783579673383 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_381_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (48362168994141 / 128000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_381_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (48362168994141 / 128000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_381_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (48362168994141 / 128000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_381_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (48362168994141 / 128000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_381_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (48362168994141 / 128000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_381_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (48362168994141 / 128000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_381_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_382_1 :
+    Uω (aρ 1) (bρ 1) (121114715361 / 320000000000 : ℝ) ≤
+      (-(395512938531258084853 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3720294181274211 / 10000000000000000 : ℝ) ≤
+      (-(395512938531258084853 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3720294181274211 / 10000000000000000 : ℝ) =
+            (3720294181274211 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3720294181274211 / 2500000000000000 : ℝ) ≤
+            (159004805708741915147 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (930062681400543 / 2500000000000000 : ℝ)) (q :=
+      (3720294181274211 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_382_2 :
+    Uω (aρ 2) (bρ 2) (121114715361 / 320000000000 : ℝ) ≤
+      (-(398103100560546859391 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1478512639585287 / 4000000000000000 : ℝ) ≤
+      (-(398103100560546859391 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1478512639585287 / 4000000000000000 : ℝ) =
+            (1478512639585287 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1478512639585287 / 1000000000000000 : ℝ) ≤
+            (156414643679453140609 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (739198914698037 / 2000000000000000 : ℝ)) (q :=
+      (1478512639585287 / 4000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_382_3 :
+    Uω (aρ 3) (bρ 3) (121114715361 / 320000000000 : ℝ) ≤
+      (-(2520906383278679509249 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1824079285441027 / 5000000000000000 : ℝ) ≤
+      (-(2520906383278679509249 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1824079285441027 / 5000000000000000 : ℝ) =
+            (1824079285441027 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1824079285441027 / 1250000000000000 : ℝ) ≤
+            (944829518221320490751 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1823572311838929 / 5000000000000000 : ℝ)) (q :=
+      (1824079285441027 / 5000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_382_4 :
+    Uω (aρ 4) (bρ 4) (121114715361 / 320000000000 : ℝ) ≤
+      (-(25764989521341751801 / 25000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1783964850020739 / 5000000000000000 : ℝ) ≤
+      (-(25764989521341751801 / 25000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1783964850020739 / 5000000000000000 : ℝ) =
+            (1783964850020739 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1783964850020739 / 1250000000000000 : ℝ) ≤
+            (8892369493658248199 / 25000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1782490531530853 / 5000000000000000 : ℝ)) (q :=
+      (1783964850020739 / 5000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_382_5 :
+    Uω (aρ 5) (bρ 5) (121114715361 / 320000000000 : ℝ) ≤
+      (-(10656731664801098998343 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (430619841801571 / 1250000000000000 : ℝ) ≤
+      (-(10656731664801098998343 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (430619841801571 / 1250000000000000 : ℝ) =
+            (430619841801571 / 312500000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (430619841801571 / 312500000000000 : ℝ) ≤
+            (3206211941198901001657 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1718614525569443 / 5000000000000000 : ℝ)) (q :=
+      (430619841801571 / 1250000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_382_6 :
+    Uω (aρ 6) (bρ 6) (121114715361 / 320000000000 : ℝ) ≤
+      (-(11188158091501860923951 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1633332004315943 / 5000000000000000 : ℝ) ≤
+      (-(11188158091501860923951 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1633332004315943 / 5000000000000000 : ℝ) =
+            (1633332004315943 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1633332004315943 / 1250000000000000 : ℝ) ≤
+            (2674785514498139076049 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1623939987358761 / 5000000000000000 : ℝ)) (q :=
+      (1633332004315943 / 5000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_382_7 :
+    Uω (aρ 7) (bρ 7) (121114715361 / 320000000000 : ℝ) ≤
+      (-(5988576774856659351441 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3018831202702193 / 10000000000000000 : ℝ) ≤
+      (-(5988576774856659351441 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3018831202702193 / 10000000000000000 : ℝ) =
+            (3018831202702193 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3018831202702193 / 2500000000000000 : ℝ) ≤
+            (942895028143340648559 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (371984467952267 / 1250000000000000 : ℝ)) (q :=
+      (3018831202702193 / 10000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_382_8 :
+    Uω (aρ 8) (bρ 8) (121114715361 / 320000000000 : ℝ) ≤
+      (-(13151740646229372841851 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5368551905695767 / 20000000000000000 : ℝ) ≤
+      (-(13151740646229372841851 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5368551905695767 / 20000000000000000 : ℝ) =
+            (5368551905695767 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5368551905695767 / 5000000000000000 : ℝ) ≤
+            (711202959770627158149 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2590306895249517 / 10000000000000000 : ℝ)) (q :=
+      (5368551905695767 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_382_9 :
+    Uω (aρ 9) (bρ 9) (121114715361 / 320000000000 : ℝ) ≤
+      (-(14985134648602909743777 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4469241949103891 / 20000000000000000 : ℝ) ≤
+      (-(14985134648602909743777 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4469241949103891 / 20000000000000000 : ℝ) =
+            (4469241949103891 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4469241949103891 / 2500000000000000 : ℝ) ≤
+            (5809280760397090256223 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2031962567017641 / 10000000000000000 : ℝ)) (q :=
+      (4469241949103891 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_382_10 :
+    Uω (aρ 10) (bρ 10) (121114715361 / 320000000000 : ℝ) ≤
+      (-(72342282154980825373 / 39062500000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3138578058074391 / 20000000000000000 : ℝ) ≤
+      (-(72342282154980825373 / 39062500000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3138578058074391 / 20000000000000000 : ℝ) =
+            (3138578058074391 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3138578058074391 / 2500000000000000 : ℝ) ≤
+            (8885903036425424627 / 39062500000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1089857709483141 / 10000000000000000 : ℝ)) (q :=
+      (3138578058074391 / 20000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_382_11 :
+    Uω (aρ 11) (bρ 11) (121114715361 / 320000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_382_12 :
+    Uω (aρ 12) (bρ 12) (121114715361 / 320000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_382_13 :
+    Uω (aρ 13) (bρ 13) (121114715361 / 320000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_382_14 :
+    Uω (aρ 14) (bρ 14) (121114715361 / 320000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_382_15 :
+    Uω (aρ 15) (bρ 15) (121114715361 / 320000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_382_16 :
+    Uω (aρ 16) (bρ 16) (121114715361 / 320000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_382 :
+    Uρ (121114715361 / 320000000000 : ℝ) ≤
+      (-(2775137250727846017459 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (121114715361 / 320000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(395512938531258084853 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_382_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (121114715361 / 320000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(398103100560546859391 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_382_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (121114715361 / 320000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(2520906383278679509249 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_382_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (121114715361 / 320000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(25764989521341751801 / 25000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_382_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (121114715361 / 320000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(10656731664801098998343 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_382_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (121114715361 / 320000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(11188158091501860923951 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_382_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (121114715361 / 320000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(5988576774856659351441 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_382_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (121114715361 / 320000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(13151740646229372841851 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_382_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (121114715361 / 320000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(14985134648602909743777 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_382_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (121114715361 / 320000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) * (-(72342282154980825373 / 39062500000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_382_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (121114715361 / 320000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_382_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (121114715361 / 320000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_382_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (121114715361 / 320000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_382_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (121114715361 / 320000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_382_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (121114715361 / 320000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_382_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (121114715361 / 320000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_382_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_383_1 :
+    Uω (aρ 1) (bρ 1) (48529603294659 / 128000000000000 : ℝ) ≤
+      (-(38555696708818691983 / 39062500000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11925870911684943 / 32000000000000000 : ℝ) ≤
+      (-(38555696708818691983 / 39062500000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (11925870911684943 / 32000000000000000 : ℝ) =
+            (11925870911684943 / 8000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11925870911684943 / 8000000000000000 : ℝ) ≤
+            (15596426752118808017 / 39062500000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (745358256098571 / 2000000000000000 : ℝ)) (q :=
+      (11925870911684943 / 32000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_383_2 :
+    Uω (aρ 2) (bρ 2) (48529603294659 / 128000000000000 : ℝ) ≤
+      (-(993489723544569558129 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (59245160133566219 / 160000000000000000 : ℝ) ≤
+      (-(993489723544569558129 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (59245160133566219 / 160000000000000000 : ℝ) =
+            (59245160133566219 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (59245160133566219 / 40000000000000000 : ℝ) ≤
+            (392804637055430441871 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3702535989895543 / 10000000000000000 : ℝ)) (q :=
+      (59245160133566219 / 160000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_383_3 :
+    Uω (aρ 3) (bρ 3) (48529603294659 / 128000000000000 : ℝ) ≤
+      (-(2516427165727103197089 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (58475212612729563 / 160000000000000000 : ℝ) ≤
+      (-(2516427165727103197089 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (58475212612729563 / 160000000000000000 : ℝ) =
+            (58475212612729563 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (58475212612729563 / 40000000000000000 : ℝ) ≤
+            (949308735772896802911 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3653688656140961 / 10000000000000000 : ℝ)) (q :=
+      (58475212612729563 / 160000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_383_4 :
+    Uω (aρ 4) (bρ 4) (48529603294659 / 128000000000000 : ℝ) ≤
+      (-(10287666397815893786367 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (57191608034129379 / 160000000000000000 : ℝ) ≤
+      (-(10287666397815893786367 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (57191608034129379 / 160000000000000000 : ℝ) =
+            (57191608034129379 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (57191608034129379 / 40000000000000000 : ℝ) ≤
+            (3575277208184106213633 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1785766132440469 / 5000000000000000 : ℝ)) (q :=
+      (57191608034129379 / 160000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_383_5 :
+    Uω (aρ 5) (bρ 5) (48529603294659 / 128000000000000 : ℝ) ≤
+      (-(5318860850787325044227 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (55224221070690603 / 160000000000000000 : ℝ) ≤
+      (-(5318860850787325044227 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (55224221070690603 / 160000000000000000 : ℝ) =
+            (55224221070690603 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (55224221070690603 / 40000000000000000 : ℝ) ≤
+            (1612610952212674955773 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3443798813786091 / 10000000000000000 : ℝ)) (q :=
+      (55224221070690603 / 160000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_383_6 :
+    Uω (aρ 6) (bρ 6) (48529603294659 / 128000000000000 : ℝ) ≤
+      (-(11168041122814828369441 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (52371874572716323 / 160000000000000000 : ℝ) ≤
+      (-(11168041122814828369441 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (52371874572716323 / 160000000000000000 : ℝ) =
+            (52371874572716323 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (52371874572716323 / 40000000000000000 : ℝ) ≤
+            (2694902483185171630559 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1627247938339653 / 5000000000000000 : ℝ)) (q :=
+      (52371874572716323 / 160000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_383_7 :
+    Uω (aρ 7) (bρ 7) (48529603294659 / 128000000000000 : ℝ) ≤
+      (-(11955200284190424110453 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (48407452846251931 / 160000000000000000 : ℝ) ≤
+      (-(11955200284190424110453 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (48407452846251931 / 160000000000000000 : ℝ) =
+            (48407452846251931 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (48407452846251931 / 40000000000000000 : ℝ) ≤
+            (1907743321809575889547 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2982604541631257 / 10000000000000000 : ℝ)) (q :=
+      (48407452846251931 / 160000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_383_8 :
+    Uω (aρ 8) (bρ 8) (48529603294659 / 128000000000000 : ℝ) ≤
+      (-(3281631310863808718577 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (43056848055855387 / 160000000000000000 : ℝ) ≤
+      (-(3281631310863808718577 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (43056848055855387 / 160000000000000000 : ℝ) =
+            (43056848055855387 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (43056848055855387 / 40000000000000000 : ℝ) ≤
+            (184104590636191281423 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2597320594171689 / 10000000000000000 : ℝ)) (q :=
+      (43056848055855387 / 160000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8])
+      (by norm_num [bρ_8]) (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_383_9 :
+    Uω (aρ 9) (bρ 9) (48529603294659 / 128000000000000 : ℝ) ≤
+      (-(934563061726455595199 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (35868982172610707 / 160000000000000000 : ℝ) ≤
+      (-(934563061726455595199 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (35868982172610707 / 160000000000000000 : ℝ) =
+            (35868982172610707 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (35868982172610707 / 20000000000000000 : ℝ) ≤
+            (365087901336044404801 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (254975373390763 / 1250000000000000 : ℝ)) (q :=
+      (35868982172610707 / 160000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9])
+      (by norm_num [bρ_9]) (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_383_10 :
+    Uω (aρ 10) (bρ 10) (48529603294659 / 128000000000000 : ℝ) ≤
+      (-(9229973893465110443607 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25258911793304411 / 160000000000000000 : ℝ) ≤
+      (-(9229973893465110443607 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (25258911793304411 / 160000000000000000 : ℝ) =
+            (25258911793304411 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25258911793304411 / 20000000000000000 : ℝ) ≤
+            (1167233811034889556393 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1102103223207817 / 10000000000000000 : ℝ)) (q :=
+      (25258911793304411 / 160000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_383_11 :
+    Uω (aρ 11) (bρ 11) (48529603294659 / 128000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_383_12 :
+    Uω (aρ 12) (bρ 12) (48529603294659 / 128000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_383_13 :
+    Uω (aρ 13) (bρ 13) (48529603294659 / 128000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_383_14 :
+    Uω (aρ 14) (bρ 14) (48529603294659 / 128000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_383_15 :
+    Uω (aρ 15) (bρ 15) (48529603294659 / 128000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_383_16 :
+    Uω (aρ 16) (bρ 16) (48529603294659 / 128000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_383 :
+    Uρ (48529603294659 / 128000000000000 : ℝ) ≤
+      (-(13858215987978432094499 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (48529603294659 / 128000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(38555696708818691983 / 39062500000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_383_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (48529603294659 / 128000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(993489723544569558129 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_383_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (48529603294659 / 128000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(2516427165727103197089 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_383_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (48529603294659 / 128000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(10287666397815893786367 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_383_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (48529603294659 / 128000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(5318860850787325044227 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_383_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (48529603294659 / 128000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(11168041122814828369441 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_383_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (48529603294659 / 128000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(11955200284190424110453 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_383_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (48529603294659 / 128000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(3281631310863808718577 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_383_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (48529603294659 / 128000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(934563061726455595199 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_383_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (48529603294659 / 128000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(9229973893465110443607 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_383_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (48529603294659 / 128000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_383_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (48529603294659 / 128000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_383_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (48529603294659 / 128000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_383_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (48529603294659 / 128000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_383_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (48529603294659 / 128000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_383_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (48529603294659 / 128000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_383_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_384_1 :
+    Uω (aρ 1) (bρ 1) (24306660222459 / 64000000000000 : ℝ) ≤
+      (-(9852724051552499823691 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (29867001106093319 / 80000000000000000 : ℝ) ≤
+      (-(9852724051552499823691 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (29867001106093319 / 80000000000000000 : ℝ) =
+            (29867001106093319 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (29867001106093319 / 20000000000000000 : ℝ) ≤
+            (4010219554447500176309 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3733331834849111 / 10000000000000000 : ℝ)) (q :=
+      (29867001106093319 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_384_2 :
+    Uω (aρ 2) (bρ 2) (24306660222459 / 64000000000000 : ℝ) ≤
+      (-(9917248165762109789853 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (29674907327554487 / 80000000000000000 : ℝ) ≤
+      (-(9917248165762109789853 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (29674907327554487 / 80000000000000000 : ℝ) =
+            (29674907327554487 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (29674907327554487 / 20000000000000000 : ℝ) ≤
+            (3945695440237890210147 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3709077402724403 / 10000000000000000 : ℝ)) (q :=
+      (29674907327554487 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_384_3 :
+    Uω (aρ 3) (bρ 3) (24306660222459 / 64000000000000 : ℝ) ≤
+      (-(5023911927221671117419 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1171597759750651 / 3200000000000000 : ℝ) ≤
+      (-(5023911927221671117419 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1171597759750651 / 3200000000000000 : ℝ) =
+            (1171597759750651 / 800000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1171597759750651 / 800000000000000 : ℝ) ≤
+            (1907559875778328882581 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (73204653512547 / 200000000000000 : ℝ)) (q :=
+      (1171597759750651 / 3200000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_384_4 :
+    Uω (aρ 4) (bρ 4) (24306660222459 / 64000000000000 : ℝ) ≤
+      (-(10269370577536574744851 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28648170275744011 / 80000000000000000 : ℝ) ≤
+      (-(10269370577536574744851 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (28648170275744011 / 80000000000000000 : ℝ) =
+            (28648170275744011 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28648170275744011 / 20000000000000000 : ℝ) ≤
+            (3593573028463425255149 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (111815107099587 / 312500000000000 : ℝ)) (q :=
+      (28648170275744011 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_384_5 :
+    Uω (aρ 5) (bρ 5) (24306660222459 / 64000000000000 : ℝ) ≤
+      (-(2654686992240856056823 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27664550749155799 / 80000000000000000 : ℝ) ≤
+      (-(2654686992240856056823 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (27664550749155799 / 80000000000000000 : ℝ) =
+            (27664550749155799 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27664550749155799 / 20000000000000000 : ℝ) ≤
+            (811048909259143943177 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3450368464874731 / 10000000000000000 : ℝ)) (q :=
+      (27664550749155799 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_384_6 :
+    Uω (aρ 6) (bρ 6) (24306660222459 / 64000000000000 : ℝ) ≤
+      (-(11147965006472399763539 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (26238561285308971 / 80000000000000000 : ℝ) ≤
+      (-(11147965006472399763539 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (26238561285308971 / 80000000000000000 : ℝ) =
+            (26238561285308971 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26238561285308971 / 20000000000000000 : ℝ) ≤
+            (2714978599527600236461 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (101909733564157 / 312500000000000 : ℝ)) (q :=
+      (26238561285308971 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_384_7 :
+    Uω (aρ 7) (bρ 7) (24306660222459 / 64000000000000 : ℝ) ≤
+      (-(5966648243356786807413 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4851359975918859 / 16000000000000000 : ℝ) ≤
+      (-(5966648243356786807413 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4851359975918859 / 16000000000000000 : ℝ) =
+            (4851359975918859 / 4000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4851359975918859 / 4000000000000000 : ℝ) ≤
+            (964823559643213192587 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (597866500676871 / 2000000000000000 : ℝ)) (q :=
+      (4851359975918859 / 16000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_384_8 :
+    Uω (aρ 8) (bρ 8) (24306660222459 / 64000000000000 : ℝ) ≤
+      (-(1310137782768200082229 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21582630580093707 / 80000000000000000 : ℝ) ≤
+      (-(1310137782768200082229 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (21582630580093707 / 80000000000000000 : ℝ) =
+            (21582630580093707 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21582630580093707 / 20000000000000000 : ℝ) ≤
+            (76156577831799917771 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (325541478731151 / 1250000000000000 : ℝ)) (q :=
+      (21582630580093707 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_384_9 :
+    Uω (aρ 9) (bρ 9) (24306660222459 / 64000000000000 : ℝ) ≤
+      (-(14921006500376229850049 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17991977855449379 / 80000000000000000 : ℝ) ≤
+      (-(14921006500376229850049 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (17991977855449379 / 80000000000000000 : ℝ) =
+            (17991977855449379 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17991977855449379 / 10000000000000000 : ℝ) ≤
+            (5873408908623770149951 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1023817138524063 / 5000000000000000 : ℝ)) (q :=
+      (17991977855449379 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_384_10 :
+    Uω (aρ 10) (bρ 10) (24306660222459 / 64000000000000 : ℝ) ≤
+      (-(18400928188958520139997 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12704214832724787 / 80000000000000000 : ℝ) ≤
+      (-(18400928188958520139997 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (12704214832724787 / 80000000000000000 : ℝ) =
+            (12704214832724787 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12704214832724787 / 10000000000000000 : ℝ) ≤
+            (2393487220041479860003 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (557126277430989 / 5000000000000000 : ℝ)) (q :=
+      (12704214832724787 / 80000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_384_11 :
+    Uω (aρ 11) (bρ 11) (24306660222459 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_384_12 :
+    Uω (aρ 12) (bρ 12) (24306660222459 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_384_13 :
+    Uω (aρ 13) (bρ 13) (24306660222459 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_384_14 :
+    Uω (aρ 14) (bρ 14) (24306660222459 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_384_15 :
+    Uω (aρ 15) (bρ 15) (24306660222459 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_384_16 :
+    Uω (aρ 16) (bρ 16) (24306660222459 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_384 :
+    Uρ (24306660222459 / 64000000000000 : ℝ) ≤
+      (-(3460208311846665330791 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (24306660222459 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(9852724051552499823691 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_384_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (24306660222459 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(9917248165762109789853 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_384_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (24306660222459 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(5023911927221671117419 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_384_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (24306660222459 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(10269370577536574744851 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_384_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (24306660222459 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(2654686992240856056823 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_384_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (24306660222459 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(11147965006472399763539 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_384_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (24306660222459 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(5966648243356786807413 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_384_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (24306660222459 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(1310137782768200082229 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_384_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (24306660222459 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(14921006500376229850049 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_384_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (24306660222459 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(18400928188958520139997 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_384_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (24306660222459 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_384_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (24306660222459 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_384_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (24306660222459 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_384_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (24306660222459 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_384_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (24306660222459 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_384_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (24306660222459 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_384_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_385_1 :
+    Uω (aρ 1) (bρ 1) (48697037595177 / 128000000000000 : ℝ) ≤
+      (-(9835220437739158478671 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (59838649861695569 / 160000000000000000 : ℝ) ≤
+      (-(9835220437739158478671 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (59838649861695569 / 160000000000000000 : ℝ) =
+            (59838649861695569 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (59838649861695569 / 40000000000000000 : ℝ) ≤
+            (4027723168260841521329 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3739872388673743 / 10000000000000000 : ℝ)) (q :=
+      (59838649861695569 / 160000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_385_2 :
+    Uω (aρ 2) (bρ 2) (48697037595177 / 128000000000000 : ℝ) ≤
+      (-(4949815097480732821211 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11890893829638173 / 32000000000000000 : ℝ) ≤
+      (-(4949815097480732821211 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (11890893829638173 / 32000000000000000 : ℝ) =
+            (11890893829638173 / 8000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11890893829638173 / 8000000000000000 : ℝ) ≤
+            (1981656705519267178789 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (743123762399131 / 2000000000000000 : ℝ)) (q :=
+      (11890893829638173 / 32000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_385_3 :
+    Uω (aρ 3) (bρ 3) (48697037595177 / 128000000000000 : ℝ) ≤
+      (-(5014985496557094599463 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (58684563259077641 / 160000000000000000 : ℝ) ≤
+      (-(5014985496557094599463 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (58684563259077641 / 160000000000000000 : ℝ) =
+            (58684563259077641 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (58684563259077641 / 40000000000000000 : ℝ) ≤
+            (1916486306442905400537 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1833388341103251 / 5000000000000000 : ℝ)) (q :=
+      (58684563259077641 / 160000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_385_4 :
+    Uω (aρ 4) (bρ 4) (48697037595177 / 128000000000000 : ℝ) ≤
+      (-(5125554112303715040283 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (57401072754472689 / 160000000000000000 : ℝ) ≤
+      (-(5125554112303715040283 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (57401072754472689 / 160000000000000000 : ℝ) =
+            (57401072754472689 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (57401072754472689 / 40000000000000000 : ℝ) ≤
+            (1805917690696284959717 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3584634550195883 / 10000000000000000 : ℝ)) (q :=
+      (57401072754472689 / 160000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_385_5 :
+    Uω (aρ 5) (bρ 5) (48697037595177 / 128000000000000 : ℝ) ≤
+      (-(2119962065703611401969 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (55433981038552289 / 160000000000000000 : ℝ) ≤
+      (-(2119962065703611401969 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (55433981038552289 / 160000000000000000 : ℝ) =
+            (55433981038552289 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (55433981038552289 / 40000000000000000 : ℝ) ≤
+            (652626655496388598031 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3456938005040833 / 10000000000000000 : ℝ)) (q :=
+      (55433981038552289 / 160000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_385_6 :
+    Uω (aρ 6) (bρ 6) (48697037595177 / 128000000000000 : ℝ) ≤
+      (-(445117183001040530557 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10516473629322909 / 32000000000000000 : ℝ) ≤
+      (-(445117183001040530557 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (10516473629322909 / 32000000000000000 : ℝ) =
+            (10516473629322909 / 8000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10516473629322909 / 8000000000000000 : ℝ) ≤
+            (109400561238959469443 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (653545353737723 / 2000000000000000 : ℝ)) (q :=
+      (10516473629322909 / 32000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_385_7 :
+    Uω (aρ 7) (bρ 7) (48697037595177 / 128000000000000 : ℝ) ≤
+      (-(1488930241095184756289 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9723948005422973 / 32000000000000000 : ℝ) ≤
+      (-(1488930241095184756289 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9723948005422973 / 32000000000000000 : ℝ) =
+            (9723948005422973 / 8000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9723948005422973 / 8000000000000000 : ℝ) ≤
+            (243937709654815243711 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (599211926902231 / 2000000000000000 : ℝ)) (q :=
+      (9723948005422973 / 32000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_385_8 :
+    Uω (aρ 8) (bρ 8) (48697037595177 / 128000000000000 : ℝ) ≤
+      (-(6538149004840853801523 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8654730943457773 / 32000000000000000 : ℝ) ≤
+      (-(6538149004840853801523 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8654730943457773 / 32000000000000000 : ℝ) =
+            (8654730943457773 / 8000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8654730943457773 / 8000000000000000 : ℝ) ≤
+            (393322798159146198477 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (522268124424581 / 2000000000000000 : ℝ)) (q :=
+      (8654730943457773 / 32000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_385_9 :
+    Uω (aρ 9) (bρ 9) (48697037595177 / 128000000000000 : ℝ) ≤
+      (-(14889126104872971111557 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (36098857042559593 / 160000000000000000 : ℝ) ≤
+      (-(14889126104872971111557 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (36098857042559593 / 160000000000000000 : ℝ) =
+            (36098857042559593 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (36098857042559593 / 20000000000000000 : ℝ) ≤
+            (5905289304127028888443 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1027728270570873 / 5000000000000000 : ℝ)) (q :=
+      (36098857042559593 / 160000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9])
+      (by norm_num [bρ_9]) (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_385_10 :
+    Uω (aρ 10) (bρ 10) (48697037595177 / 128000000000000 : ℝ) ≤
+      (-(18342546204662378687801 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25557202981117793 / 160000000000000000 : ℝ) ≤
+      (-(18342546204662378687801 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (25557202981117793 / 160000000000000000 : ℝ) =
+            (25557202981117793 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25557202981117793 / 20000000000000000 : ℝ) ≤
+            (2451869204337621312199 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1126308816956521 / 10000000000000000 : ℝ)) (q :=
+      (25557202981117793 / 160000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_385_11 :
+    Uω (aρ 11) (bρ 11) (48697037595177 / 128000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_385_12 :
+    Uω (aρ 12) (bρ 12) (48697037595177 / 128000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_385_13 :
+    Uω (aρ 13) (bρ 13) (48697037595177 / 128000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_385_14 :
+    Uω (aρ 14) (bρ 14) (48697037595177 / 128000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_385_15 :
+    Uω (aρ 15) (bρ 15) (48697037595177 / 128000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_385_16 :
+    Uω (aρ 16) (bρ 16) (48697037595177 / 128000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_385 :
+    Uρ (48697037595177 / 128000000000000 : ℝ) ≤
+      (-(13823536199399352962991 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (48697037595177 / 128000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(9835220437739158478671 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_385_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (48697037595177 / 128000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(4949815097480732821211 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_385_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (48697037595177 / 128000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(5014985496557094599463 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_385_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (48697037595177 / 128000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(5125554112303715040283 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_385_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (48697037595177 / 128000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(2119962065703611401969 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_385_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (48697037595177 / 128000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(445117183001040530557 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_385_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (48697037595177 / 128000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(1488930241095184756289 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_385_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (48697037595177 / 128000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(6538149004840853801523 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_385_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (48697037595177 / 128000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(14889126104872971111557 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_385_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (48697037595177 / 128000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(18342546204662378687801 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_385_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (48697037595177 / 128000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_385_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (48697037595177 / 128000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_385_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (48697037595177 / 128000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_385_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (48697037595177 / 128000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_385_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (48697037595177 / 128000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_385_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (48697037595177 / 128000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_385_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_386_1 :
+    Uω (aρ 1) (bρ 1) (12195188686359 / 32000000000000 : ℝ) ≤
+      (-(9817747408755780516049 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14985824376743447 / 40000000000000000 : ℝ) ≤
+      (-(9817747408755780516049 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (14985824376743447 / 40000000000000000 : ℝ) =
+            (14985824376743447 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14985824376743447 / 10000000000000000 : ℝ) ≤
+            (4045196197244219483951 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (29268851109137 / 78125000000000 : ℝ)) (q :=
+      (14985824376743447 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_386_2 :
+    Uω (aρ 2) (bρ 2) (12195188686359 / 32000000000000 : ℝ) ≤
+      (-(4941021606811480716549 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14889780903240487 / 40000000000000000 : ℝ) ≤
+      (-(4941021606811480716549 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (14889780903240487 / 40000000000000000 : ℝ) =
+            (14889780903240487 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14889780903240487 / 10000000000000000 : ℝ) ≤
+            (1990450196188519283451 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (465270027216007 / 1250000000000000 : ℝ)) (q :=
+      (14889780903240487 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_386_3 :
+    Uω (aρ 3) (bρ 3) (12195188686359 / 32000000000000 : ℝ) ≤
+      (-(625759372808085277601 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14697309606979173 / 40000000000000000 : ℝ) ≤
+      (-(625759372808085277601 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (14697309606979173 / 40000000000000000 : ℝ) =
+            (14697309606979173 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14697309606979173 / 10000000000000000 : ℝ) ≤
+            (240674602566914722399 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3673320675947399 / 10000000000000000 : ℝ)) (q :=
+      (14697309606979173 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_386_4 :
+    Uω (aρ 4) (bρ 4) (12195188686359 / 32000000000000 : ℝ) ≤
+      (-(10232879216613597053591 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2875290232240193 / 8000000000000000 : ℝ) ≤
+      (-(10232879216613597053591 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2875290232240193 / 8000000000000000 : ℝ) =
+            (2875290232240193 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2875290232240193 / 2000000000000000 : ℝ) ≤
+            (3630064389386402946409 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (718237126824659 / 2000000000000000 : ℝ)) (q :=
+      (2875290232240193 / 8000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_386_5 :
+    Uω (aρ 5) (bρ 5) (12195188686359 / 32000000000000 : ℝ) ≤
+      (-(10580908642584709892287 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13884714924115549 / 40000000000000000 : ℝ) ≤
+      (-(10580908642584709892287 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (13884714924115549 / 40000000000000000 : ℝ) =
+            (13884714924115549 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13884714924115549 / 10000000000000000 : ℝ) ≤
+            (3282034963415290107713 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3463507434915587 / 10000000000000000 : ℝ)) (q :=
+      (13884714924115549 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_386_6 :
+    Uω (aρ 6) (bρ 6) (12195188686359 / 32000000000000 : ℝ) ≤
+      (-(11107934662065662635813 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13171902828846341 / 40000000000000000 : ℝ) ≤
+      (-(11107934662065662635813 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (13171902828846341 / 40000000000000000 : ℝ) =
+            (13171902828846341 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13171902828846341 / 10000000000000000 : ℝ) ≤
+            (2755008943934337364187 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3274341762420983 / 10000000000000000 : ℝ)) (q :=
+      (13171902828846341 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_386_7 :
+    Uω (aρ 7) (bρ 7) (12195188686359 / 32000000000000 : ℝ) ≤
+      (-(2377927276686470457359 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12181468423672783 / 40000000000000000 : ℝ) ≤
+      (-(2377927276686470457359 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (12181468423672783 / 40000000000000000 : ℝ) =
+            (12181468423672783 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12181468423672783 / 10000000000000000 : ℝ) ≤
+            (394661444513529542641 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (750696485148551 / 2500000000000000 : ℝ)) (q :=
+      (12181468423672783 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_386_8 :
+    Uω (aρ 8) (bρ 8) (12195188686359 / 32000000000000 : ℝ) ≤
+      (-(13051285403735799505811 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10845507221032959 / 40000000000000000 : ℝ) ≤
+      (-(13051285403735799505811 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (10845507221032959 / 40000000000000000 : ℝ) =
+            (10845507221032959 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10845507221032959 / 10000000000000000 : ℝ) ≤
+            (811658202264200494189 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (654586747653573 / 2500000000000000 : ℝ)) (q :=
+      (10845507221032959 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_386_9 :
+    Uω (aρ 9) (bρ 9) (12195188686359 / 32000000000000 : ℝ) ≤
+      (-(7428683367336118564909 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9053421747210509 / 40000000000000000 : ℝ) ≤
+      (-(7428683367336118564909 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (9053421747210509 / 40000000000000000 : ℝ) =
+            (9053421747210509 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9053421747210509 / 5000000000000000 : ℝ) ≤
+            (2968524337163881435091 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2063269882063067 / 10000000000000000 : ℝ)) (q :=
+      (9053421747210509 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_386_10 :
+    Uω (aρ 10) (bρ 10) (12195188686359 / 32000000000000 : ℝ) ≤
+      (-(18284783532368180351299 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6426313849671687 / 40000000000000000 : ℝ) ≤
+      (-(18284783532368180351299 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (6426313849671687 / 40000000000000000 : ℝ) =
+            (6426313849671687 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6426313849671687 / 5000000000000000 : ℝ) ≤
+            (2509631876631819648701 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (71142185424291 / 625000000000000 : ℝ)) (q :=
+      (6426313849671687 / 40000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_386_11 :
+    Uω (aρ 11) (bρ 11) (12195188686359 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_386_12 :
+    Uω (aρ 12) (bρ 12) (12195188686359 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_386_13 :
+    Uω (aρ 13) (bρ 13) (12195188686359 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_386_14 :
+    Uω (aρ 14) (bρ 14) (12195188686359 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_386_15 :
+    Uω (aρ 15) (bρ 15) (12195188686359 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_386_16 :
+    Uω (aρ 16) (bρ 16) (12195188686359 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_386 :
+    Uρ (12195188686359 / 32000000000000 : ℝ) ≤
+      (-(6903161546604532718271 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (12195188686359 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(9817747408755780516049 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_386_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (12195188686359 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(4941021606811480716549 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_386_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (12195188686359 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(625759372808085277601 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_386_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (12195188686359 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(10232879216613597053591 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_386_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (12195188686359 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(10580908642584709892287 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_386_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (12195188686359 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(11107934662065662635813 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_386_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (12195188686359 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(2377927276686470457359 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_386_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (12195188686359 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(13051285403735799505811 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_386_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (12195188686359 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(7428683367336118564909 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_386_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (12195188686359 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(18284783532368180351299 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_386_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (12195188686359 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_386_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (12195188686359 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_386_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (12195188686359 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_386_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (12195188686359 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_386_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (12195188686359 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_386_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (12195188686359 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_386_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_387_1 :
+    Uω (aρ 1) (bρ 1) (9772894379139 / 25600000000000 : ℝ) ≤
+      (-(9800304857901901130603 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12009589029608683 / 32000000000000000 : ℝ) ≤
+      (-(9800304857901901130603 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (12009589029608683 / 32000000000000000 : ℝ) =
+            (12009589029608683 / 8000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12009589029608683 / 8000000000000000 : ℝ) ≤
+            (4062638748098098869397 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (750590698947851 / 2000000000000000 : ℝ)) (q :=
+      (12009589029608683 / 32000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_387_2 :
+    Uω (aρ 2) (bρ 2) (9772894379139 / 25600000000000 : ℝ) ≤
+      (-(4932243556451181186989 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (59663778049571223 / 160000000000000000 : ℝ) ≤
+      (-(4932243556451181186989 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (59663778049571223 / 160000000000000000 : ℝ) =
+            (59663778049571223 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (59663778049571223 / 40000000000000000 : ℝ) ≤
+            (1999228246548818813011 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3728701619940231 / 10000000000000000 : ℝ)) (q :=
+      (59663778049571223 / 160000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_387_3 :
+    Uω (aρ 3) (bρ 3) (9772894379139 / 25600000000000 : ℝ) ≤
+      (-(499718032825331389293 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (58893913494597647 / 160000000000000000 : ℝ) ≤
+      (-(499718032825331389293 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (58893913494597647 / 160000000000000000 : ℝ) =
+            (58893913494597647 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (58893913494597647 / 40000000000000000 : ℝ) ≤
+            (193429147474668610707 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1839932328459267 / 5000000000000000 : ℝ)) (q :=
+      (58893913494597647 / 160000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_387_4 :
+    Uω (aρ 4) (bρ 4) (9772894379139 / 25600000000000 : ℝ) ≤
+      (-(10214683431811717131443 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (57610536224189447 / 160000000000000000 : ℝ) ≤
+      (-(10214683431811717131443 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (57610536224189447 / 160000000000000000 : ℝ) =
+            (57610536224189447 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (57610536224189447 / 40000000000000000 : ℝ) ≤
+            (3648260174188282868557 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3597736679182509 / 10000000000000000 : ℝ)) (q :=
+      (57610536224189447 / 160000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_387_5 :
+    Uω (aρ 5) (bρ 5) (9772894379139 / 25600000000000 : ℝ) ≤
+      (-(10562042774298962654907 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (55643737477052503 / 160000000000000000 : ℝ) ≤
+      (-(10562042774298962654907 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (55643737477052503 / 160000000000000000 : ℝ) =
+            (55643737477052503 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (55643737477052503 / 40000000000000000 : ℝ) ≤
+            (3300900831701037345093 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3470076755125391 / 10000000000000000 : ℝ)) (q :=
+      (55643737477052503 / 160000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_387_6 :
+    Uω (aρ 6) (bρ 6) (9772894379139 / 25600000000000 : ℝ) ≤
+      (-(11087980102211234289417 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (52792852091490599 / 160000000000000000 : ℝ) ≤
+      (-(11087980102211234289417 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (52792852091490599 / 160000000000000000 : ℝ) =
+            (52792852091490599 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (52792852091490599 / 40000000000000000 : ℝ) ≤
+            (2774963503788765710583 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3280956457070153 / 10000000000000000 : ℝ)) (q :=
+      (52792852091490599 / 160000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_387_7 :
+    Uω (aρ 7) (bρ 7) (9772894379139 / 25600000000000 : ℝ) ≤
+      (-(11867879625428304621113 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (48832000806173063 / 160000000000000000 : ℝ) ≤
+      (-(11867879625428304621113 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (48832000806173063 / 160000000000000000 : ℝ) =
+            (48832000806173063 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (48832000806173063 / 40000000000000000 : ℝ) ≤
+            (1995063980571695378887 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3009511427165461 / 10000000000000000 : ℝ)) (q :=
+      (48832000806173063 / 160000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_387_8 :
+    Uω (aρ 8) (bρ 8) (9772894379139 / 25600000000000 : ℝ) ≤
+      (-(3256584906897836630917 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1739615352638351 / 6400000000000000 : ℝ) ≤
+      (-(3256584906897836630917 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1739615352638351 / 6400000000000000 : ℝ) =
+            (1739615352638351 / 1600000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1739615352638351 / 1600000000000000 : ℝ) ≤
+            (209150994602163369083 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (105014038189147 / 400000000000000 : ℝ)) (q :=
+      (1739615352638351 / 6400000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_387_9 :
+    Uω (aρ 9) (bρ 9) (9772894379139 / 25600000000000 : ℝ) ≤
+      (-(593029093542680300733 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7265689271534427 / 32000000000000000 : ℝ) ≤
+      (-(593029093542680300733 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (7265689271534427 / 32000000000000000 : ℝ) =
+            (7265689271534427 / 4000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7265689271534427 / 4000000000000000 : ℝ) ≤
+            (238747522817319699267 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (414214880160569 / 2000000000000000 : ℝ)) (q :=
+      (7265689271534427 / 32000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_387_10 :
+    Uω (aρ 10) (bρ 10) (9772894379139 / 25600000000000 : ℝ) ≤
+      (-(18227622739612179984223 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25852609418760503 / 160000000000000000 : ℝ) ≤
+      (-(18227622739612179984223 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (25852609418760503 / 160000000000000000 : ℝ) =
+            (25852609418760503 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25852609418760503 / 20000000000000000 : ℝ) ≤
+            (2566792669387820015777 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1150153816933891 / 10000000000000000 : ℝ)) (q :=
+      (25852609418760503 / 160000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_387_11 :
+    Uω (aρ 11) (bρ 11) (9772894379139 / 25600000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_387_12 :
+    Uω (aρ 12) (bρ 12) (9772894379139 / 25600000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_387_13 :
+    Uω (aρ 13) (bρ 13) (9772894379139 / 25600000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_387_14 :
+    Uω (aρ 14) (bρ 14) (9772894379139 / 25600000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_387_15 :
+    Uω (aρ 15) (bρ 15) (9772894379139 / 25600000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_387_16 :
+    Uω (aρ 16) (bρ 16) (9772894379139 / 25600000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_387 :
+    Uρ (9772894379139 / 25600000000000 : ℝ) ≤
+      (-(13789192254313298509481 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (9772894379139 / 25600000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(9800304857901901130603 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_387_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (9772894379139 / 25600000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(4932243556451181186989 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_387_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (9772894379139 / 25600000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(499718032825331389293 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_387_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (9772894379139 / 25600000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(10214683431811717131443 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_387_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (9772894379139 / 25600000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(10562042774298962654907 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_387_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (9772894379139 / 25600000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(11087980102211234289417 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_387_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (9772894379139 / 25600000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(11867879625428304621113 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_387_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (9772894379139 / 25600000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(3256584906897836630917 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_387_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (9772894379139 / 25600000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(593029093542680300733 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_387_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (9772894379139 / 25600000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(18227622739612179984223 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_387_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (9772894379139 / 25600000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_387_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (9772894379139 / 25600000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_387_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (9772894379139 / 25600000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_387_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (9772894379139 / 25600000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_387_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (9772894379139 / 25600000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_387_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (9772894379139 / 25600000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_387_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/U32.lean b/LeanPool/Zeta5Irrational/Table/U32.lean
new file mode 100644
index 0000000000..15b5439584
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/U32.lean
@@ -0,0 +1,4150 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-! # Certified arcsine potential bounds (U32) -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma U_388_1 :
+    Uω (aρ 1) (bρ 1) (24474094522977 / 64000000000000 : ℝ) ≤
+      (-(9782892679034460345389 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (30076296392463209 / 80000000000000000 : ℝ) ≤
+      (-(9782892679034460345389 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (30076296392463209 / 80000000000000000 : ℝ) =
+            (30076296392463209 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (30076296392463209 / 20000000000000000 : ℝ) ≤
+            (4080050926965539654611 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1879747023492823 / 5000000000000000 : ℝ)) (q :=
+      (30076296392463209 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_388_2 :
+    Uω (aρ 2) (bρ 2) (24474094522977 / 64000000000000 : ℝ) ≤
+      (-(9846961784527956511941 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (29884216229083321 / 80000000000000000 : ℝ) ≤
+      (-(9846961784527956511941 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (29884216229083321 / 80000000000000000 : ℝ) =
+            (29884216229083321 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (29884216229083321 / 20000000000000000 : ℝ) ≤
+            (4015981821472043488059 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1867621509325337 / 5000000000000000 : ℝ)) (q :=
+      (29884216229083321 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_388_3 :
+    Uω (aρ 3) (bρ 3) (24474094522977 / 64000000000000 : ℝ) ≤
+      (-(9976602955068756301647 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (29499294229832269 / 80000000000000000 : ℝ) ≤
+      (-(9976602955068756301647 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (29499294229832269 / 80000000000000000 : ℝ) =
+            (29499294229832269 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (29499294229832269 / 20000000000000000 : ℝ) ≤
+            (3886340650931243698353 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3686408625187911 / 10000000000000000 : ℝ)) (q :=
+      (29499294229832269 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_388_4 :
+    Uω (aρ 4) (bρ 4) (24474094522977 / 64000000000000 : ℝ) ≤
+      (-(5098260374562509612519 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28857633747162473 / 80000000000000000 : ℝ) ≤
+      (-(5098260374562509612519 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (28857633747162473 / 80000000000000000 : ℝ) =
+            (28857633747162473 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28857633747162473 / 20000000000000000 : ℝ) ≤
+            (1833211428437490387481 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1802143842792731 / 5000000000000000 : ℝ)) (q :=
+      (28857633747162473 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_388_5 :
+    Uω (aρ 5) (bρ 5) (24474094522977 / 64000000000000 : ℝ) ≤
+      (-(10543212587579755254357 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27874307192648233 / 80000000000000000 : ℝ) ≤
+      (-(10543212587579755254357 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (27874307192648233 / 80000000000000000 : ℝ) =
+            (27874307192648233 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27874307192648233 / 20000000000000000 : ℝ) ≤
+            (3319731018420244745643 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1738322983145951 / 5000000000000000 : ℝ)) (q :=
+      (27874307192648233 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_388_6 :
+    Uω (aρ 6) (bρ 6) (24474094522977 / 64000000000000 : ℝ) ≤
+      (-(11068065731103946049403 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (26449045244686297 / 80000000000000000 : ℝ) ≤
+      (-(11068065731103946049403 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (26449045244686297 / 80000000000000000 : ℝ) =
+            (26449045244686297 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26449045244686297 / 20000000000000000 : ℝ) ≤
+            (2794877874896053950597 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1643785427220709 / 5000000000000000 : ℝ)) (q :=
+      (26449045244686297 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_388_7 :
+    Uω (aρ 7) (bρ 7) (24474094522977 / 64000000000000 : ℝ) ≤
+      (-(1480771428879991064171 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24469060702708113 / 80000000000000000 : ℝ) ≤
+      (-(1480771428879991064171 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (24469060702708113 / 80000000000000000 : ℝ) =
+            (24469060702708113 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24469060702708113 / 20000000000000000 : ℝ) ≤
+            (252096521870008935829 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (377029512463359 / 1250000000000000 : ℝ)) (q :=
+      (24469060702708113 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_388_8 :
+    Uω (aρ 8) (bρ 8) (24474094522977 / 64000000000000 : ℝ) ≤
+      (-(13001460302417974526679 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21799359833153453 / 80000000000000000 : ℝ) ≤
+      (-(13001460302417974526679 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (21799359833153453 / 80000000000000000 : ℝ) =
+            (21799359833153453 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21799359833153453 / 20000000000000000 : ℝ) ≤
+            (861483303582025473321 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2632352533658207 / 10000000000000000 : ℝ)) (q :=
+      (21799359833153453 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_388_9 :
+    Uω (aρ 9) (bρ 9) (24474094522977 / 64000000000000 : ℝ) ≤
+      (-(7397103440139706118931 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (18221567971968993 / 80000000000000000 : ℝ) ≤
+      (-(7397103440139706118931 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (18221567971968993 / 80000000000000000 : ℝ) =
+            (18221567971968993 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18221567971968993 / 10000000000000000 : ℝ) ≤
+            (3000104264360293881069 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (519717549180523 / 2500000000000000 : ℝ)) (q :=
+      (18221567971968993 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_388_10 :
+    Uω (aρ 10) (bρ 10) (24474094522977 / 64000000000000 : ℝ) ≤
+      (-(3634209441235360305123 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12999643230488421 / 80000000000000000 : ℝ) ≤
+      (-(3634209441235360305123 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (12999643230488421 / 80000000000000000 : ℝ) =
+            (12999643230488421 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12999643230488421 / 10000000000000000 : ℝ) ≤
+            (524673640564639694877 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1161948044846949 / 10000000000000000 : ℝ)) (q :=
+      (12999643230488421 / 80000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_388_11 :
+    Uω (aρ 11) (bρ 11) (24474094522977 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_388_12 :
+    Uω (aρ 12) (bρ 12) (24474094522977 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_388_13 :
+    Uω (aρ 13) (bρ 13) (24474094522977 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_388_14 :
+    Uω (aρ 14) (bρ 14) (24474094522977 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_388_15 :
+    Uω (aρ 15) (bρ 15) (24474094522977 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_388_16 :
+    Uω (aρ 16) (bρ 16) (24474094522977 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_388 :
+    Uρ (24474094522977 / 64000000000000 : ℝ) ≤
+      (-(6886071039805945481207 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (24474094522977 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(9782892679034460345389 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_388_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (24474094522977 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(9846961784527956511941 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_388_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (24474094522977 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(9976602955068756301647 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_388_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (24474094522977 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(5098260374562509612519 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_388_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (24474094522977 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(10543212587579755254357 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_388_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (24474094522977 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(11068065731103946049403 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_388_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (24474094522977 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(1480771428879991064171 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_388_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (24474094522977 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(13001460302417974526679 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_388_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (24474094522977 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(7397103440139706118931 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_388_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (24474094522977 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(3634209441235360305123 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_388_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (24474094522977 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_388_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (24474094522977 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_388_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (24474094522977 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_388_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (24474094522977 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_388_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (24474094522977 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_388_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (24474094522977 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_388_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_389_1 :
+    Uω (aρ 1) (bρ 1) (6139452918309 / 16000000000000 : ℝ) ≤
+      (-(4874079507725232605329 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7545236005777453 / 20000000000000000 : ℝ) ≤
+      (-(4874079507725232605329 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7545236005777453 / 20000000000000000 : ℝ) =
+            (7545236005777453 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7545236005777453 / 5000000000000000 : ℝ) ≤
+            (2057392295274767394671 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (117892973434979 / 312500000000000 : ℝ)) (q :=
+      (7545236005777453 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_389_2 :
+    Uω (aρ 2) (bρ 2) (6139452918309 / 16000000000000 : ℝ) ≤
+      (-(9812003014569464231753 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7497217648987823 / 20000000000000000 : ℝ) ≤
+      (-(9812003014569464231753 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7497217648987823 / 20000000000000000 : ℝ) =
+            (7497217648987823 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7497217648987823 / 5000000000000000 : ℝ) ≤
+            (4050940591430535768247 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1874162902819849 / 5000000000000000 : ℝ)) (q :=
+      (7497217648987823 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_389_3 :
+    Uω (aρ 3) (bρ 3) (6139452918309 / 16000000000000 : ℝ) ≤
+      (-(310661935157438047953 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3700496130444567 / 10000000000000000 : ℝ) ≤
+      (-(310661935157438047953 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3700496130444567 / 10000000000000000 : ℝ) =
+            (3700496130444567 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3700496130444567 / 2500000000000000 : ℝ) ≤
+            (122555052530061952047 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3699496523891009 / 10000000000000000 : ℝ)) (q :=
+      (3700496130444567 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_389_4 :
+    Uω (aρ 4) (bρ 4) (6139452918309 / 16000000000000 : ℝ) ≤
+      (-(10160294209093882698789 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7240591139195777 / 20000000000000000 : ℝ) ≤
+      (-(10160294209093882698789 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7240591139195777 / 20000000000000000 : ℝ) =
+            (7240591139195777 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7240591139195777 / 5000000000000000 : ℝ) ≤
+            (3702649396906117301211 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (904347395815663 / 2500000000000000 : ℝ)) (q :=
+      (7240591139195777 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_389_5 :
+    Uω (aρ 5) (bρ 5) (6139452918309 / 16000000000000 : ℝ) ≤
+      (-(10505658718397677553489 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3497397850278163 / 10000000000000000 : ℝ) ≤
+      (-(10505658718397677553489 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3497397850278163 / 10000000000000000 : ℝ) =
+            (3497397850278163 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3497397850278163 / 2500000000000000 : ℝ) ≤
+            (3357284887602322446511 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3489784063958201 / 10000000000000000 : ℝ)) (q :=
+      (3497397850278163 / 10000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_389_6 :
+    Uω (aρ 6) (bρ 6) (6139452918309 / 16000000000000 : ℝ) ≤
+      (-(11028356902751511319297 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6638570025956783 / 20000000000000000 : ℝ) ≤
+      (-(11028356902751511319297 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6638570025956783 / 20000000000000000 : ℝ) =
+            (6638570025956783 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6638570025956783 / 5000000000000000 : ℝ) ≤
+            (2834586703248488680703 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1650399382249329 / 5000000000000000 : ℝ)) (q :=
+      (6638570025956783 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_389_7 :
+    Uω (aρ 7) (bρ 7) (6139452918309 / 16000000000000 : ℝ) ≤
+      (-(11802899846123785371143 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6143792905079437 / 20000000000000000 : ℝ) ≤
+      (-(11802899846123785371143 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6143792905079437 / 20000000000000000 : ℝ) =
+            (6143792905079437 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6143792905079437 / 5000000000000000 : ℝ) ≤
+            (2060043759876214628857 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (23669398627979 / 78125000000000 : ℝ)) (q :=
+      (6143792905079437 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_389_8 :
+    Uω (aρ 8) (bρ 8) (6139452918309 / 16000000000000 : ℝ) ≤
+      (-(12951899506522119733701 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2738458420520823 / 10000000000000000 : ℝ) ≤
+      (-(12951899506522119733701 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2738458420520823 / 10000000000000000 : ℝ) =
+            (2738458420520823 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2738458420520823 / 2500000000000000 : ℝ) ≤
+            (911044099477880266299 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2646348611683521 / 10000000000000000 : ℝ)) (q :=
+      (2738458420520823 / 10000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_389_9 :
+    Uω (aρ 9) (bρ 9) (6139452918309 / 16000000000000 : ℝ) ≤
+      (-(14731518704915448957371 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4584038610599473 / 20000000000000000 : ℝ) ≤
+      (-(14731518704915448957371 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4584038610599473 / 20000000000000000 : ℝ) =
+            (4584038610599473 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4584038610599473 / 2500000000000000 : ℝ) ≤
+            (6062896704084551042629 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (523609002400337 / 2500000000000000 : ℝ)) (q :=
+      (4584038610599473 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_389_10 :
+    Uω (aρ 10) (bρ 10) (6139452918309 / 16000000000000 : ℝ) ≤
+      (-(9029794594203179324451 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3286336287777 / 20000000000000 : ℝ) ≤
+      (-(9029794594203179324451 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (3286336287777 / 20000000000000 : ℝ) = (3286336287777 / 2500000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3286336287777 / 2500000000000 : ℝ) ≤
+            (1367413110296820675549 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (9482341762191 / 80000000000000 : ℝ)) (q :=
+      (3286336287777 / 20000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10]) (by norm_num [bρ_10])
+      (by norm_num) (by norm_num [aρ_10, bρ_10]) (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_389_11 :
+    Uω (aρ 11) (bρ 11) (6139452918309 / 16000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_389_12 :
+    Uω (aρ 12) (bρ 12) (6139452918309 / 16000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_389_13 :
+    Uω (aρ 13) (bρ 13) (6139452918309 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_389_14 :
+    Uω (aρ 14) (bρ 14) (6139452918309 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_389_15 :
+    Uω (aρ 15) (bρ 15) (6139452918309 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_389_16 :
+    Uω (aρ 16) (bρ 16) (6139452918309 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_389 :
+    Uρ (6139452918309 / 16000000000000 : ℝ) ≤
+      (-(6869138820388027502371 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (6139452918309 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(4874079507725232605329 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_389_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (6139452918309 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(9812003014569464231753 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_389_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (6139452918309 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(310661935157438047953 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_389_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (6139452918309 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(10160294209093882698789 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_389_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (6139452918309 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(10505658718397677553489 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_389_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (6139452918309 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(11028356902751511319297 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_389_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (6139452918309 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(11802899846123785371143 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_389_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (6139452918309 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(12951899506522119733701 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_389_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (6139452918309 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(14731518704915448957371 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_389_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (6139452918309 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(9029794594203179324451 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_389_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (6139452918309 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_389_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (6139452918309 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_389_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (6139452918309 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_389_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (6139452918309 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_389_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (6139452918309 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_389_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (6139452918309 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_389_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_390_1 :
+    Uω (aρ 1) (bρ 1) (4928305764699 / 12800000000000 : ℝ) ≤
+      (-(9713545579860973935019 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (30285591645513059 / 80000000000000000 : ℝ) ≤
+      (-(9713545579860973935019 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (30285591645513059 / 80000000000000000 : ℝ) =
+            (30285591645513059 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (30285591645513059 / 20000000000000000 : ℝ) ≤
+            (4149398026139026064981 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3785656250792171 / 10000000000000000 : ℝ)) (q :=
+      (30285591645513059 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_390_2 :
+    Uω (aρ 2) (bρ 2) (4928305764699 / 12800000000000 : ℝ) ≤
+      (-(1955433209774552219883 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (30093524907666447 / 80000000000000000 : ℝ) ≤
+      (-(1955433209774552219883 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (30093524907666447 / 80000000000000000 : ℝ) =
+            (30093524907666447 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (30093524907666447 / 20000000000000000 : ℝ) ≤
+            (817155511425447780117 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1880704289420259 / 5000000000000000 : ℝ)) (q :=
+      (30093524907666447 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_390_3 :
+    Uω (aρ 3) (bρ 3) (4928305764699 / 12800000000000 : ℝ) ≤
+      (-(9905885984557209782699 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (29708643657269043 / 80000000000000000 : ℝ) ≤
+      (-(9905885984557209782699 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (29708643657269043 / 80000000000000000 : ℝ) =
+            (29708643657269043 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (29708643657269043 / 20000000000000000 : ℝ) ≤
+            (3957057621442790217301 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3712584372591167 / 10000000000000000 : ℝ)) (q :=
+      (29708643657269043 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_390_4 :
+    Uω (aρ 4) (bρ 4) (4928305764699 / 12800000000000 : ℝ) ≤
+      (-(10124198641054031683193 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1162683790316947 / 3200000000000000 : ℝ) ≤
+      (-(10124198641054031683193 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1162683790316947 / 3200000000000000 : ℝ) =
+            (1162683790316947 / 800000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1162683790316947 / 800000000000000 : ℝ) ≤
+            (3738744964945968316807 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (145219653152793 / 400000000000000 : ℝ)) (q :=
+      (1162683790316947 / 3200000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_390_5 :
+    Uω (aρ 5) (bρ 5) (4928305764699 / 12800000000000 : ℝ) ≤
+      (-(10468245961831071935927 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28084056696482443 / 80000000000000000 : ℝ) ≤
+      (-(10468245961831071935927 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (28084056696482443 / 80000000000000000 : ℝ) =
+            (28084056696482443 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28084056696482443 / 20000000000000000 : ℝ) ≤
+            (3394697644168928064073 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3502921732794517 / 10000000000000000 : ℝ)) (q :=
+      (28084056696482443 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_390_6 :
+    Uω (aρ 6) (bρ 6) (4928305764699 / 12800000000000 : ℝ) ≤
+      (-(1098880688224468762929 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (26659510291728759 / 80000000000000000 : ℝ) ≤
+      (-(1098880688224468762929 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (26659510291728759 / 80000000000000000 : ℝ) =
+            (26659510291728759 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26659510291728759 / 20000000000000000 : ℝ) ≤
+            (287413672375531237071 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (207126594171631 / 625000000000000 : ℝ)) (q :=
+      (26659510291728759 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_390_7 :
+    Uω (aρ 7) (bρ 7) (4928305764699 / 12800000000000 : ℝ) ≤
+      (-(5879909935102141370187 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24681269771681939 / 80000000000000000 : ℝ) ≤
+      (-(5879909935102141370187 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (24681269771681939 / 80000000000000000 : ℝ) =
+            (24681269771681939 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24681269771681939 / 20000000000000000 : ℝ) ≤
+            (1051561867897858629813 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3043126757494391 / 10000000000000000 : ℝ)) (q :=
+      (24681269771681939 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_390_8 :
+    Uω (aρ 8) (bρ 8) (4928305764699 / 12800000000000 : ℝ) ≤
+      (-(12902600052262391086187 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22015937632653179 / 80000000000000000 : ℝ) ≤
+      (-(12902600052262391086187 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (22015937632653179 / 80000000000000000 : ℝ) =
+            (22015937632653179 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22015937632653179 / 20000000000000000 : ℝ) ≤
+            (960343553737608913813 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2660335374077201 / 10000000000000000 : ℝ)) (q :=
+      (22015937632653179 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_390_9 :
+    Uω (aρ 9) (bρ 9) (4928305764699 / 12800000000000 : ℝ) ≤
+      (-(7334647102916508509139 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (18450605960057299 / 80000000000000000 : ℝ) ≤
+      (-(7334647102916508509139 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (18450605960057299 / 80000000000000000 : ℝ) =
+            (18450605960057299 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18450605960057299 / 10000000000000000 : ℝ) ≤
+            (3062560601583491490861 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2109968084288231 / 10000000000000000 : ℝ)) (q :=
+      (18450605960057299 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_390_10 :
+    Uω (aρ 10) (bρ 10) (4928305764699 / 12800000000000 : ℝ) ≤
+      (-(17950290879875761555441 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2657961919588723 / 16000000000000000 : ℝ) ≤
+      (-(17950290879875761555441 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2657961919588723 / 16000000000000000 : ℝ) =
+            (2657961919588723 / 2000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2657961919588723 / 2000000000000000 : ℝ) ≤
+            (2844124529124238444559 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (120832802725481 / 1000000000000000 : ℝ)) (q :=
+      (2657961919588723 / 16000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_390_11 :
+    Uω (aρ 11) (bρ 11) (4928305764699 / 12800000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_390_12 :
+    Uω (aρ 12) (bρ 12) (4928305764699 / 12800000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_390_13 :
+    Uω (aρ 13) (bρ 13) (4928305764699 / 12800000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_390_14 :
+    Uω (aρ 14) (bρ 14) (4928305764699 / 12800000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_390_15 :
+    Uω (aρ 15) (bρ 15) (4928305764699 / 12800000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_390_16 :
+    Uω (aρ 16) (bρ 16) (4928305764699 / 12800000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_390 :
+    Uρ (4928305764699 / 12800000000000 : ℝ) ≤
+      (-(13704718217822075139391 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (4928305764699 / 12800000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(9713545579860973935019 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_390_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (4928305764699 / 12800000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1955433209774552219883 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_390_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (4928305764699 / 12800000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(9905885984557209782699 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_390_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (4928305764699 / 12800000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(10124198641054031683193 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_390_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (4928305764699 / 12800000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(10468245961831071935927 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_390_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (4928305764699 / 12800000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1098880688224468762929 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_390_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (4928305764699 / 12800000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(5879909935102141370187 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_390_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (4928305764699 / 12800000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(12902600052262391086187 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_390_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (4928305764699 / 12800000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(7334647102916508509139 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_390_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (4928305764699 / 12800000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(17950290879875761555441 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_390_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (4928305764699 / 12800000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_390_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (4928305764699 / 12800000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_390_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (4928305764699 / 12800000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_390_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (4928305764699 / 12800000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_390_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (4928305764699 / 12800000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_390_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (4928305764699 / 12800000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_390_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_391_1 :
+    Uω (aρ 1) (bρ 1) (12362622986877 / 32000000000000 : ℝ) ≤
+      (-(9679051542797064085659 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15195119629879051 / 40000000000000000 : ℝ) ≤
+      (-(9679051542797064085659 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (15195119629879051 / 40000000000000000 : ℝ) =
+            (15195119629879051 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15195119629879051 / 10000000000000000 : ℝ) ≤
+            (4183892063202935914341 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3798737349625463 / 10000000000000000 : ℝ)) (q :=
+      (15195119629879051 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_391_2 :
+    Uω (aρ 2) (bρ 2) (12362622986877 / 32000000000000 : ℝ) ≤
+      (-(608903127591907559467 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15099089582401141 / 40000000000000000 : ℝ) ≤
+      (-(608903127591907559467 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (15099089582401141 / 40000000000000000 : ℝ) =
+            (15099089582401141 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15099089582401141 / 10000000000000000 : ℝ) ≤
+            (257530847783092440533 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (943622834599127 / 2500000000000000 : ℝ)) (q :=
+      (15099089582401141 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_391_3 :
+    Uω (aρ 3) (bρ 3) (12362622986877 / 32000000000000 : ℝ) ≤
+      (-(4935357126377696842237 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (596266361461553 / 1600000000000000 : ℝ) ≤
+      (-(4935357126377696842237 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (596266361461553 / 1600000000000000 : ℝ) =
+            (596266361461553 / 400000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (596266361461553 / 400000000000000 : ℝ) ≤
+            (1996114676622303157763 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (74513443436307 / 200000000000000 : ℝ)) (q :=
+      (596266361461553 / 1600000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_391_4 :
+    Uω (aρ 4) (bρ 4) (12362622986877 / 32000000000000 : ℝ) ≤
+      (-(10088233099897932006419 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14585912178574077 / 40000000000000000 : ℝ) ≤
+      (-(10088233099897932006419 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (14585912178574077 / 40000000000000000 : ℝ) =
+            (14585912178574077 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14585912178574077 / 10000000000000000 : ℝ) ≤
+            (3774710506102067993581 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (455449115487247 / 1250000000000000 : ℝ)) (q :=
+      (14585912178574077 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_391_5 :
+    Uω (aρ 5) (bρ 5) (12362622986877 / 32000000000000 : ℝ) ≤
+      (-(10430973256828805279317 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14094464447323703 / 40000000000000000 : ℝ) ≤
+      (-(10430973256828805279317 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (14094464447323703 / 40000000000000000 : ℝ) =
+            (14094464447323703 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14094464447323703 / 10000000000000000 : ℝ) ≤
+            (3431970349171194720683 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3516058977607789 / 10000000000000000 : ℝ)) (q :=
+      (14094464447323703 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_391_6 :
+    Uω (aρ 6) (bρ 6) (12362622986877 / 32000000000000 : ℝ) ≤
+      (-(1094941439062756986717 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13382367932049789 / 40000000000000000 : ℝ) ≤
+      (-(1094941439062756986717 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (13382367932049789 / 40000000000000000 : ℝ) =
+            (13382367932049789 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13382367932049789 / 10000000000000000 : ℝ) ≤
+            (291352921537243013283 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (207953193444427 / 625000000000000 : ℝ)) (q :=
+      (13382367932049789 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_391_7 :
+    Uω (aρ 7) (bρ 7) (12362622986877 / 32000000000000 : ℝ) ≤
+      (-(11716929769262015717329 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12393677662625217 / 40000000000000000 : ℝ) ≤
+      (-(11716929769262015717329 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (12393677662625217 / 40000000000000000 : ℝ) =
+            (12393677662625217 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12393677662625217 / 10000000000000000 : ℝ) ≤
+            (2146013836737984282671 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1528283670579273 / 5000000000000000 : ℝ)) (q :=
+      (12393677662625217 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_391_8 :
+    Uω (aρ 8) (bρ 8) (12362622986877 / 32000000000000 : ℝ) ≤
+      (-(12853559027970004128109 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2212417122326671 / 8000000000000000 : ℝ) ≤
+      (-(12853559027970004128109 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2212417122326671 / 8000000000000000 : ℝ) =
+            (2212417122326671 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2212417122326671 / 2000000000000000 : ℝ) ≤
+            (1009384578029995871891 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (534862593400523 / 2000000000000000 : ℝ)) (q :=
+      (2212417122326671 / 8000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_391_9 :
+    Uω (aρ 9) (bρ 9) (12362622986877 / 32000000000000 : ℝ) ≤
+      (-(14607525600897892519987 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9282462741746189 / 40000000000000000 : ℝ) ≤
+      (-(14607525600897892519987 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (9282462741746189 / 40000000000000000 : ℝ) =
+            (9282462741746189 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9282462741746189 / 5000000000000000 : ℝ) ≤
+            (6186889808102107480013 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (265683395052379 / 1250000000000000 : ℝ)) (q :=
+      (9282462741746189 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_391_10 :
+    Uω (aρ 10) (bρ 10) (12362622986877 / 32000000000000 : ℝ) ≤
+      (-(17843043740896604037687 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1343310603680641 / 8000000000000000 : ℝ) ≤
+      (-(17843043740896604037687 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1343310603680641 / 8000000000000000 : ℝ) =
+            (1343310603680641 / 1000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1343310603680641 / 1000000000000000 : ℝ) ≤
+            (2951371668103395962313 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (61553566612127 / 500000000000000 : ℝ)) (q :=
+      (1343310603680641 / 8000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_391_11 :
+    Uω (aρ 11) (bρ 11) (12362622986877 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_391_12 :
+    Uω (aρ 12) (bρ 12) (12362622986877 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_391_13 :
+    Uω (aρ 13) (bρ 13) (12362622986877 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_391_14 :
+    Uω (aρ 14) (bρ 14) (12362622986877 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_391_15 :
+    Uω (aρ 15) (bρ 15) (12362622986877 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_391_16 :
+    Uω (aρ 16) (bρ 16) (12362622986877 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_391 :
+    Uρ (12362622986877 / 32000000000000 : ℝ) ≤
+      (-(1708931642661093442263 / 1250000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (12362622986877 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(9679051542797064085659 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_391_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (12362622986877 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(608903127591907559467 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_391_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (12362622986877 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(4935357126377696842237 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_391_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (12362622986877 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(10088233099897932006419 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_391_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (12362622986877 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(10430973256828805279317 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_391_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (12362622986877 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1094941439062756986717 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_391_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (12362622986877 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(11716929769262015717329 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_391_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (12362622986877 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(12853559027970004128109 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_391_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (12362622986877 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(14607525600897892519987 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_391_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (12362622986877 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(17843043740896604037687 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_391_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (12362622986877 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_391_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (12362622986877 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_391_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (12362622986877 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_391_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (12362622986877 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_391_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (12362622986877 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_391_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (12362622986877 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_391_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_392_1 :
+    Uω (aρ 1) (bρ 1) (24808963124013 / 64000000000000 : ℝ) ≤
+      (-(9644676083344422677201 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (30494886865928941 / 80000000000000000 : ℝ) ≤
+      (-(9644676083344422677201 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (30494886865928941 / 80000000000000000 : ℝ) =
+            (30494886865928941 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (30494886865928941 / 20000000000000000 : ℝ) ≤
+            (4218267522655577322799 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (952954611610051 / 2500000000000000 : ℝ)) (q :=
+      (30494886865928941 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_392_2 :
+    Uω (aρ 2) (bρ 2) (24808963124013 / 64000000000000 : ℝ) ≤
+      (-(4853927077589738354537 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (30302833367924369 / 80000000000000000 : ℝ) ≤
+      (-(4853927077589738354537 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (30302833367924369 / 80000000000000000 : ℝ) =
+            (30302833367924369 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (30302833367924369 / 20000000000000000 : ℝ) ≤
+            (2077544725410261645463 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3787574084449061 / 10000000000000000 : ℝ)) (q :=
+      (30302833367924369 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_392_3 :
+    Uω (aρ 3) (bρ 3) (24808963124013 / 64000000000000 : ℝ) ≤
+      (-(983566585803844534669 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5983598458612137 / 16000000000000000 : ℝ) ≤
+      (-(983566585803844534669 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5983598458612137 / 16000000000000000 : ℝ) =
+            (5983598458612137 / 4000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5983598458612137 / 4000000000000000 : ℝ) ≤
+            (402727774796155465331 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (186937996104157 / 500000000000000 : ℝ)) (q :=
+      (5983598458612137 / 16000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_392_4 :
+    Uω (aρ 4) (bρ 4) (24808963124013 / 64000000000000 : ℝ) ≤
+      (-(10052396650727247046407 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (29276553360926369 / 80000000000000000 : ℝ) ≤
+      (-(10052396650727247046407 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (29276553360926369 / 80000000000000000 : ℝ) =
+            (29276553360926369 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (29276553360926369 / 20000000000000000 : ℝ) ≤
+            (3810546955272752953593 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3656694370114561 / 10000000000000000 : ℝ)) (q :=
+      (29276553360926369 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_392_5 :
+    Uω (aρ 5) (bρ 5) (24808963124013 / 64000000000000 : ℝ) ≤
+      (-(10393839554312362094071 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28293799415661033 / 80000000000000000 : ℝ) ≤
+      (-(10393839554312362094071 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (28293799415661033 / 80000000000000000 : ℝ) =
+            (28293799415661033 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28293799415661033 / 20000000000000000 : ℝ) ≤
+            (3469104051687637905929 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3529195803133227 / 10000000000000000 : ℝ)) (q :=
+      (28293799415661033 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_392_6 :
+    Uω (aρ 6) (bρ 6) (24808963124013 / 64000000000000 : ℝ) ≤
+      (-(5455089082246591798157 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1074798275030437 / 3200000000000000 : ℝ) ≤
+      (-(5455089082246591798157 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1074798275030437 / 3200000000000000 : ℝ) =
+            (1074798275030437 / 800000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1074798275030437 / 800000000000000 : ℝ) ≤
+            (1476382720753408201843 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (16702377716491 / 50000000000000 : ℝ)) (q :=
+      (1074798275030437 / 3200000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_392_7 :
+    Uω (aρ 7) (bρ 7) (24808963124013 / 64000000000000 : ℝ) ≤
+      (-(11674227833276534326773 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24893428446483941 / 80000000000000000 : ℝ) ≤
+      (-(11674227833276534326773 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (24893428446483941 / 80000000000000000 : ℝ) =
+            (24893428446483941 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24893428446483941 / 20000000000000000 : ℝ) ≤
+            (2188715772723465673227 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1535002408369477 / 5000000000000000 : ℝ)) (q :=
+      (24893428446483941 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_392_8 :
+    Uω (aρ 8) (bρ 8) (24808963124013 / 64000000000000 : ℝ) ≤
+      (-(12804773572830401849953 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22232368708121793 / 80000000000000000 : ℝ) ≤
+      (-(12804773572830401849953 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (22232368708121793 / 80000000000000000 : ℝ) =
+            (22232368708121793 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22232368708121793 / 20000000000000000 : ℝ) ≤
+            (1058170033169598150047 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2688281533488417 / 10000000000000000 : ℝ)) (q :=
+      (22232368708121793 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_392_9 :
+    Uω (aρ 9) (bρ 9) (24808963124013 / 64000000000000 : ℝ) ≤
+      (-(14546205320110649728969 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (18679115879380353 / 80000000000000000 : ℝ) ≤
+      (-(14546205320110649728969 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (18679115879380353 / 80000000000000000 : ℝ) =
+            (18679115879380353 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18679115879380353 / 10000000000000000 : ℝ) ≤
+            (6248210088889350271031 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2140933954663057 / 10000000000000000 : ℝ)) (q :=
+      (18679115879380353 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_392_10 :
+    Uω (aρ 10) (bρ 10) (24808963124013 / 64000000000000 : ℝ) ≤
+      (-(8868874065295747246851 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13575298042598981 / 80000000000000000 : ℝ) ≤
+      (-(8868874065295747246851 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (13575298042598981 / 80000000000000000 : ℝ) =
+            (13575298042598981 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13575298042598981 / 10000000000000000 : ℝ) ≤
+            (1528333639204252753149 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (626769264481357 / 5000000000000000 : ℝ)) (q :=
+      (13575298042598981 / 80000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_392_11 :
+    Uω (aρ 11) (bρ 11) (24808963124013 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_392_12 :
+    Uω (aρ 12) (bρ 12) (24808963124013 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_392_13 :
+    Uω (aρ 13) (bρ 13) (24808963124013 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_392_14 :
+    Uω (aρ 14) (bρ 14) (24808963124013 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_392_15 :
+    Uω (aρ 15) (bρ 15) (24808963124013 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_392_16 :
+    Uω (aρ 16) (bρ 16) (24808963124013 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_392 :
+    Uρ (24808963124013 / 64000000000000 : ℝ) ≤
+      (-(13638472530975199410167 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (24808963124013 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(9644676083344422677201 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_392_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (24808963124013 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(4853927077589738354537 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_392_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (24808963124013 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(983566585803844534669 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_392_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (24808963124013 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(10052396650727247046407 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_392_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (24808963124013 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(10393839554312362094071 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_392_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (24808963124013 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(5455089082246591798157 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_392_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (24808963124013 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(11674227833276534326773 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_392_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (24808963124013 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(12804773572830401849953 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_392_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (24808963124013 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(14546205320110649728969 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_392_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (24808963124013 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(8868874065295747246851 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_392_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (24808963124013 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_392_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (24808963124013 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_392_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (24808963124013 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_392_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (24808963124013 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_392_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (24808963124013 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_392_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (24808963124013 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_392_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_393_1 :
+    Uω (aρ 1) (bρ 1) (777896258571 / 2000000000000 : ℝ) ≤
+      (-(9610418389026122523323 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3824941808013551 / 10000000000000000 : ℝ) ≤
+      (-(9610418389026122523323 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3824941808013551 / 10000000000000000 : ℝ) =
+            (3824941808013551 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3824941808013551 / 2500000000000000 : ℝ) ≤
+            (4252525216973877476677 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1912449770628551 / 5000000000000000 : ℝ)) (q :=
+      (3824941808013551 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_393_2 :
+    Uω (aρ 2) (bρ 2) (777896258571 / 2000000000000 : ℝ) ≤
+      (-(9673377561479211552663 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3800935939698811 / 10000000000000000 : ℝ) ≤
+      (-(9673377561479211552663 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3800935939698811 / 10000000000000000 : ℝ) =
+            (3800935939698811 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3800935939698811 / 2500000000000000 : ℝ) ≤
+            (4189566044520788447337 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1900328408568811 / 5000000000000000 : ℝ)) (q :=
+      (3800935939698811 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_393_3 :
+    Uω (aρ 3) (bρ 3) (777896258571 / 2000000000000 : ℝ) ≤
+      (-(4900369968979588461839 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7505666579816867 / 20000000000000000 : ℝ) ≤
+      (-(4900369968979588461839 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7505666579816867 / 20000000000000000 : ℝ) =
+            (7505666579816867 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7505666579816867 / 5000000000000000 : ℝ) ≤
+            (2031101834020411538161 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3751847623906867 / 10000000000000000 : ℝ)) (q :=
+      (7505666579816867 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_393_4 :
+    Uω (aρ 4) (bρ 4) (777896258571 / 2000000000000 : ℝ) ≤
+      (-(10016688368706092780689 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7345320443908921 / 20000000000000000 : ℝ) ≤
+      (-(10016688368706092780689 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7345320443908921 / 20000000000000000 : ℝ) =
+            (7345320443908921 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7345320443908921 / 5000000000000000 : ℝ) ≤
+            (3846255237293907219311 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3669795669063921 / 10000000000000000 : ℝ)) (q :=
+      (7345320443908921 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_393_5 :
+    Uω (aρ 5) (bρ 5) (777896258571 / 2000000000000 : ℝ) ≤
+      (-(1294605477124456972273 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7099667069545649 / 20000000000000000 : ℝ) ≤
+      (-(1294605477124456972273 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7099667069545649 / 20000000000000000 : ℝ) =
+            (7099667069545649 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7099667069545649 / 5000000000000000 : ℝ) ≤
+            (438262473625543027727 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3542332214035649 / 10000000000000000 : ℝ)) (q :=
+      (7099667069545649 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_393_6 :
+    Uω (aρ 6) (bρ 6) (777896258571 / 2000000000000 : ℝ) ≤
+      (-(10871096955729972325563 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1685948336291547 / 5000000000000000 : ℝ) ≤
+      (-(10871096955729972325563 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1685948336291547 / 5000000000000000 : ℝ) =
+            (1685948336291547 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1685948336291547 / 1250000000000000 : ℝ) ≤
+            (2991846650270027674437 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (838424716199047 / 2500000000000000 : ℝ)) (q :=
+      (1685948336291547 / 5000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_393_7 :
+    Uω (aρ 7) (bρ 7) (777896258571 / 2000000000000 : ℝ) ≤
+      (-(5815856187888038278651 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3124936162240121 / 10000000000000000 : ℝ) ≤
+      (-(5815856187888038278651 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3124936162240121 / 10000000000000000 : ℝ) =
+            (3124936162240121 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3124936162240121 / 2500000000000000 : ℝ) ≤
+            (1115615615111961721349 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1541719612435121 / 5000000000000000 : ℝ)) (q :=
+      (3124936162240121 / 10000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_393_8 :
+    Uω (aρ 8) (bρ 8) (777896258571 / 2000000000000 : ℝ) ≤
+      (-(6378120437829771179201 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5585132661784443 / 20000000000000000 : ℝ) ≤
+      (-(6378120437829771179201 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5585132661784443 / 20000000000000000 : ℝ) =
+            (5585132661784443 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5585132661784443 / 5000000000000000 : ℝ) ≤
+            (553351365170228820799 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2702241213514443 / 10000000000000000 : ℝ)) (q :=
+      (5585132661784443 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_393_9 :
+    Uω (aρ 9) (bρ 9) (777896258571 / 2000000000000 : ℝ) ≤
+      (-(7242662998772565864273 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4698294981566537 / 20000000000000000 : ℝ) ≤
+      (-(7242662998772565864273 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4698294981566537 / 20000000000000000 : ℝ) =
+            (4698294981566537 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4698294981566537 / 2500000000000000 : ℝ) ≤
+            (3154544705727434135727 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2156369161656537 / 10000000000000000 : ℝ)) (q :=
+      (4698294981566537 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_393_10 :
+    Uω (aρ 10) (bρ 10) (777896258571 / 2000000000000 : ℝ) ≤
+      (-(17634312313178388895607 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3429110991584379 / 20000000000000000 : ℝ) ≤
+      (-(17634312313178388895607 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3429110991584379 / 20000000000000000 : ℝ) =
+            (3429110991584379 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3429110991584379 / 2500000000000000 : ℝ) ≤
+            (3160103095821611104393 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1275744205169379 / 10000000000000000 : ℝ)) (q :=
+      (3429110991584379 / 20000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_393_11 :
+    Uω (aρ 11) (bρ 11) (777896258571 / 2000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_393_12 :
+    Uω (aρ 12) (bρ 12) (777896258571 / 2000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_393_13 :
+    Uω (aρ 13) (bρ 13) (777896258571 / 2000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_393_14 :
+    Uω (aρ 14) (bρ 14) (777896258571 / 2000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_393_15 :
+    Uω (aρ 15) (bρ 15) (777896258571 / 2000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_393_16 :
+    Uω (aρ 16) (bρ 16) (777896258571 / 2000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_393 :
+    Uρ (777896258571 / 2000000000000 : ℝ) ≤
+      (-(13605767210096107167543 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (777896258571 / 2000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(9610418389026122523323 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_393_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (777896258571 / 2000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(9673377561479211552663 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_393_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (777896258571 / 2000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(4900369968979588461839 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_393_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (777896258571 / 2000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(10016688368706092780689 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_393_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (777896258571 / 2000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1294605477124456972273 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_393_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (777896258571 / 2000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(10871096955729972325563 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_393_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (777896258571 / 2000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(5815856187888038278651 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_393_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (777896258571 / 2000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(6378120437829771179201 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_393_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (777896258571 / 2000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(7242662998772565864273 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_393_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (777896258571 / 2000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(17634312313178388895607 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_393_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (777896258571 / 2000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_393_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (777896258571 / 2000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_393_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (777896258571 / 2000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_393_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (777896258571 / 2000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_393_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (777896258571 / 2000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_393_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (777896258571 / 2000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_393_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_394_1 :
+    Uω (aρ 1) (bρ 1) (24976397424531 / 64000000000000 : ℝ) ≤
+      (-(478813882784368812437 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (30704182054378223 / 80000000000000000 : ℝ) ≤
+      (-(478813882784368812437 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (30704182054378223 / 80000000000000000 : ℝ) =
+            (30704182054378223 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (30704182054378223 / 20000000000000000 : ℝ) ≤
+            (214333297515631187563 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3837980634096587 / 10000000000000000 : ℝ)) (q :=
+      (30704182054378223 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_394_2 :
+    Uω (aρ 2) (bρ 2) (24976397424531 / 64000000000000 : ℝ) ≤
+      (-(4819509720196513865331 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1220485664574031 / 3200000000000000 : ℝ) ≤
+      (-(4819509720196513865331 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1220485664574031 / 3200000000000000 : ℝ) =
+            (1220485664574031 / 800000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1220485664574031 / 800000000000000 : ℝ) ≤
+            (2111962082803486134669 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (152549581463989 / 400000000000000 : ℝ)) (q :=
+      (1220485664574031 / 3200000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_394_3 :
+    Uω (aρ 3) (bρ 3) (24976397424531 / 64000000000000 : ℝ) ≤
+      (-(152592744360776913309 / 156250000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (30127340153718827 / 80000000000000000 : ℝ) ≤
+      (-(152592744360776913309 / 156250000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (30127340153718827 / 80000000000000000 : ℝ) =
+            (30127340153718827 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (30127340153718827 / 20000000000000000 : ℝ) ≤
+            (64015749482973086691 / 156250000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1882467638895869 / 5000000000000000 : ℝ)) (q :=
+      (30127340153718827 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_394_4 :
+    Uω (aρ 4) (bρ 4) (24976397424531 / 64000000000000 : ℝ) ≤
+      (-(1247638417364616165651 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (29486009607562647 / 80000000000000000 : ℝ) ≤
+      (-(1247638417364616165651 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (29486009607562647 / 80000000000000000 : ℝ) =
+            (29486009607562647 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (29486009607562647 / 20000000000000000 : ℝ) ≤
+            (485229533385383834349 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3682896822317693 / 10000000000000000 : ℝ)) (q :=
+      (29486009607562647 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_394_5 :
+    Uω (aρ 5) (bρ 5) (24976397424531 / 64000000000000 : ℝ) ≤
+      (-(10319985019208236224207 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28503535500595011 / 80000000000000000 : ℝ) ≤
+      (-(10319985019208236224207 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (28503535500595011 / 80000000000000000 : ℝ) =
+            (28503535500595011 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28503535500595011 / 20000000000000000 : ℝ) ≤
+            (3542958586791763775793 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (55554190857981 / 156250000000000 : ℝ)) (q :=
+      (28503535500595011 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_394_6 :
+    Uω (aρ 6) (bρ 6) (24976397424531 / 64000000000000 : ℝ) ≤
+      (-(2708042382818369174677 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27080385431910887 / 80000000000000000 : ℝ) ≤
+      (-(2708042382818369174677 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (27080385431910887 / 80000000000000000 : ℝ) =
+            (27080385431910887 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27080385431910887 / 20000000000000000 : ℝ) ≤
+            (757693518681630825323 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3366921072879753 / 10000000000000000 : ℝ)) (q :=
+      (27080385431910887 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_394_7 :
+    Uω (aρ 7) (bρ 7) (24976397424531 / 64000000000000 : ℝ) ≤
+      (-(11589381733398019179671 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25105538039242823 / 80000000000000000 : ℝ) ≤
+      (-(11589381733398019179671 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (25105538039242823 / 80000000000000000 : ℝ) =
+            (25105538039242823 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25105538039242823 / 20000000000000000 : ℝ) ≤
+            (2273561872601980820329 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3096870605472737 / 10000000000000000 : ℝ)) (q :=
+      (25105538039242823 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_394_8 :
+    Uω (aρ 8) (bρ 8) (24976397424531 / 64000000000000 : ℝ) ≤
+      (-(12707958173717871882943 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22448657588370731 / 80000000000000000 : ℝ) ≤
+      (-(12707958173717871882943 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (22448657588370731 / 80000000000000000 : ℝ) =
+            (22448657588370731 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22448657588370731 / 20000000000000000 : ℝ) ≤
+            (1154985432282128117057 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1358096072047357 / 5000000000000000 : ℝ)) (q :=
+      (22448657588370731 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_394_9 :
+    Uω (aρ 9) (bρ 9) (24976397424531 / 64000000000000 : ℝ) ≤
+      (-(14424880463678895676333 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (18907120318109339 / 80000000000000000 : ℝ) ≤
+      (-(14424880463678895676333 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (18907120318109339 / 80000000000000000 : ℝ) =
+            (18907120318109339 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18907120318109339 / 10000000000000000 : ℝ) ≤
+            (6369534945321104323667 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1085886727444683 / 5000000000000000 : ℝ)) (q :=
+      (18907120318109339 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_394_10 :
+    Uω (aρ 10) (bρ 10) (24976397424531 / 64000000000000 : ℝ) ≤
+      (-(2191581450344132072969 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13856597508524971 / 80000000000000000 : ℝ) ≤
+      (-(2191581450344132072969 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (13856597508524971 / 80000000000000000 : ℝ) =
+            (13856597508524971 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13856597508524971 / 10000000000000000 : ℝ) ≤
+            (407720475780867927031 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (648850892994137 / 5000000000000000 : ℝ)) (q :=
+      (13856597508524971 / 80000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_394_11 :
+    Uω (aρ 11) (bρ 11) (24976397424531 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_394_12 :
+    Uω (aρ 12) (bρ 12) (24976397424531 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_394_13 :
+    Uω (aρ 13) (bρ 13) (24976397424531 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_394_14 :
+    Uω (aρ 14) (bρ 14) (24976397424531 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_394_15 :
+    Uω (aρ 15) (bρ 15) (24976397424531 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_394_16 :
+    Uω (aρ 16) (bρ 16) (24976397424531 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_394 :
+    Uρ (24976397424531 / 64000000000000 : ℝ) ≤
+      (-(3393332157828625400013 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (24976397424531 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(478813882784368812437 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_394_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (24976397424531 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(4819509720196513865331 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_394_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (24976397424531 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(152592744360776913309 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_394_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (24976397424531 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1247638417364616165651 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_394_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (24976397424531 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(10319985019208236224207 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_394_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (24976397424531 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(2708042382818369174677 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_394_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (24976397424531 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(11589381733398019179671 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_394_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (24976397424531 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(12707958173717871882943 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_394_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (24976397424531 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(14424880463678895676333 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_394_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (24976397424531 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(2191581450344132072969 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_394_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (24976397424531 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_394_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (24976397424531 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_394_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (24976397424531 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_394_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (24976397424531 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_394_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (24976397424531 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_394_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (24976397424531 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_394_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_395_1 :
+    Uω (aρ 1) (bρ 1) (2506011457479 / 6400000000000 : ℝ) ≤
+      (-(1908450617476454881111 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15404414818409489 / 40000000000000000 : ℝ) ≤
+      (-(1908450617476454881111 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (15404414818409489 / 40000000000000000 : ℝ) =
+            (15404414818409489 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15404414818409489 / 10000000000000000 : ℝ) ≤
+            (864138103723545118889 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3851061724978807 / 10000000000000000 : ℝ)) (q :=
+      (15404414818409489 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_395_2 :
+    Uω (aρ 2) (bρ 2) (2506011457479 / 6400000000000 : ℝ) ≤
+      (-(9604778980370867307719 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15308397829373917 / 40000000000000000 : ℝ) ≤
+      (-(9604778980370867307719 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (15308397829373917 / 40000000000000000 : ℝ) =
+            (15308397829373917 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15308397829373917 / 10000000000000000 : ℝ) ≤
+            (4258164625629132692281 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3826822242971021 / 10000000000000000 : ℝ)) (q :=
+      (15308397829373917 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_395_3 :
+    Uω (aρ 3) (bρ 3) (2506011457479 / 6400000000000 : ℝ) ≤
+      (-(4865626058448077485459 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15116006899203783 / 40000000000000000 : ℝ) ≤
+      (-(4865626058448077485459 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (15116006899203783 / 40000000000000000 : ℝ) =
+            (15116006899203783 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15116006899203783 / 10000000000000000 : ℝ) ≤
+            (2065845744551922514541 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1889011442117977 / 5000000000000000 : ℝ)) (q :=
+      (15116006899203783 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_395_4 :
+    Uω (aρ 4) (bρ 4) (2506011457479 / 6400000000000 : ℝ) ≤
+      (-(9945652656219027259689 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14795368431452299 / 40000000000000000 : ℝ) ≤
+      (-(9945652656219027259689 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (14795368431452299 / 40000000000000000 : ℝ) =
+            (14795368431452299 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14795368431452299 / 10000000000000000 : ℝ) ≤
+            (3917290949780972740311 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (923999457856303 / 2500000000000000 : ℝ)) (q :=
+      (14795368431452299 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_395_5 :
+    Uω (aρ 5) (bρ 5) (2506011457479 / 6400000000000 : ℝ) ≤
+      (-(2570815536680449644243 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2860840110100997 / 8000000000000000 : ℝ) ≤
+      (-(2570815536680449644243 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2860840110100997 / 8000000000000000 : ℝ) =
+            (2860840110100997 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2860840110100997 / 2000000000000000 : ℝ) ≤
+            (894920364819550355757 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (713720762057311 / 2000000000000000 : ℝ)) (q :=
+      (2860840110100997 / 8000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_395_6 :
+    Uω (aρ 6) (bρ 6) (2506011457479 / 6400000000000 : ℝ) ≤
+      (-(2698348668215771873327 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13592796540881929 / 40000000000000000 : ℝ) ≤
+      (-(2698348668215771873327 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (13592796540881929 / 40000000000000000 : ℝ) =
+            (13592796540881929 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13592796540881929 / 10000000000000000 : ℝ) ≤
+            (767387233284228126673 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3380142180615027 / 10000000000000000 : ℝ)) (q :=
+      (13592796540881929 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_395_7 :
+    Uω (aρ 7) (bρ 7) (2506011457479 / 6400000000000 : ℝ) ≤
+      (-(11547234265459707690501 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12605787413668453 / 40000000000000000 : ℝ) ≤
+      (-(11547234265459707690501 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (12605787413668453 / 40000000000000000 : ℝ) =
+            (12605787413668453 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12605787413668453 / 10000000000000000 : ℝ) ≤
+            (2315709340540292309499 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3110298997768289 / 10000000000000000 : ℝ)) (q :=
+      (12605787413668453 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_395_8 :
+    Uω (aρ 8) (bρ 8) (2506011457479 / 6400000000000 : ℝ) ≤
+      (-(6329961375780254349371 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2255675006833453 / 8000000000000000 : ℝ) ≤
+      (-(6329961375780254349371 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2255675006833453 / 8000000000000000 : ℝ) =
+            (2255675006833453 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2255675006833453 / 2000000000000000 : ℝ) ≤
+            (601510427219745650629 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (546026891871539 / 2000000000000000 : ℝ)) (q :=
+      (2255675006833453 / 8000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_395_9 :
+    Uω (aρ 9) (bρ 9) (2506011457479 / 6400000000000 : ℝ) ≤
+      (-(14364861738092443484433 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1902093966765393 / 8000000000000000 : ℝ) ≤
+      (-(14364861738092443484433 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1902093966765393 / 8000000000000000 : ℝ) =
+            (1902093966765393 / 1000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1902093966765393 / 1000000000000000 : ℝ) ≤
+            (6429553670907556515567 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (437429497509509 / 2000000000000000 : ℝ)) (q :=
+      (1902093966765393 / 8000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_395_10 :
+    Uω (aρ 10) (bρ 10) (2506011457479 / 6400000000000 : ℝ) ≤
+      (-(4358171905916996388157 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1399580821443121 / 8000000000000000 : ℝ) ≤
+      (-(4358171905916996388157 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1399580821443121 / 8000000000000000 : ℝ) =
+            (1399580821443121 / 1000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1399580821443121 / 1000000000000000 : ℝ) ≤
+            (840431946333003611843 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (263884731547373 / 2000000000000000 : ℝ)) (q :=
+      (1399580821443121 / 8000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_395_11 :
+    Uω (aρ 11) (bρ 11) (2506011457479 / 6400000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_395_12 :
+    Uω (aρ 12) (bρ 12) (2506011457479 / 6400000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_395_13 :
+    Uω (aρ 13) (bρ 13) (2506011457479 / 6400000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_395_14 :
+    Uω (aρ 14) (bρ 14) (2506011457479 / 6400000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_395_15 :
+    Uω (aρ 15) (bρ 15) (2506011457479 / 6400000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_395_16 :
+    Uω (aρ 16) (bρ 16) (2506011457479 / 6400000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_395 :
+    Uρ (2506011457479 / 6400000000000 : ℝ) ≤
+      (-(13541148812692470789573 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (2506011457479 / 6400000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1908450617476454881111 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_395_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (2506011457479 / 6400000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(9604778980370867307719 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_395_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (2506011457479 / 6400000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(4865626058448077485459 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_395_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (2506011457479 / 6400000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(9945652656219027259689 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_395_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (2506011457479 / 6400000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(2570815536680449644243 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_395_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (2506011457479 / 6400000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(2698348668215771873327 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_395_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (2506011457479 / 6400000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(11547234265459707690501 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_395_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (2506011457479 / 6400000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(6329961375780254349371 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_395_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (2506011457479 / 6400000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(14364861738092443484433 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_395_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (2506011457479 / 6400000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(4358171905916996388157 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_395_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (2506011457479 / 6400000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_395_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (2506011457479 / 6400000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_395_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (2506011457479 / 6400000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_395_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (2506011457479 / 6400000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_395_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (2506011457479 / 6400000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_395_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (2506011457479 / 6400000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_395_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_396_1 :
+    Uω (aρ 1) (bρ 1) (25143831725049 / 64000000000000 : ℝ) ≤
+      (-(9508343896262421957543 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (30913477211510189 / 80000000000000000 : ℝ) ≤
+      (-(9508343896262421957543 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (30913477211510189 / 80000000000000000 : ℝ) =
+            (30913477211510189 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (30913477211510189 / 20000000000000000 : ℝ) ≤
+            (4354599709737578042457 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3864142813923641 / 10000000000000000 : ℝ)) (q :=
+      (30913477211510189 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_396_2 :
+    Uω (aρ 2) (bρ 2) (25143831725049 / 64000000000000 : ℝ) ≤
+      (-(4785327689087111628957 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (30721449651316873 / 80000000000000000 : ℝ) ≤
+      (-(4785327689087111628957 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (30721449651316873 / 80000000000000000 : ℝ) =
+            (30721449651316873 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (30721449651316873 / 20000000000000000 : ℝ) ≤
+            (2146144113912888371043 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (119997029262041 / 312500000000000 : ℝ)) (q :=
+      (30721449651316873 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_396_3 :
+    Uω (aρ 3) (bρ 3) (25143831725049 / 64000000000000 : ℝ) ≤
+      (-(9696688535615259549621 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (30336687255298973 / 80000000000000000 : ℝ) ≤
+      (-(9696688535615259549621 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (30336687255298973 / 80000000000000000 : ℝ) =
+            (30336687255298973 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (30336687255298973 / 20000000000000000 : ℝ) ≤
+            (4166255070384740450379 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3791110443730837 / 10000000000000000 : ℝ)) (q :=
+      (30336687255298973 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_396_4 :
+    Uω (aρ 4) (bρ 4) (25143831725049 / 64000000000000 : ℝ) ≤
+      (-(9910323425109448560787 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (29695463547771257 / 80000000000000000 : ℝ) ≤
+      (-(9910323425109448560787 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (29695463547771257 / 80000000000000000 : ℝ) =
+            (29695463547771257 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (29695463547771257 / 20000000000000000 : ℝ) ≤
+            (3952620180890551439213 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (927274674478477 / 2500000000000000 : ℝ)) (q :=
+      (29695463547771257 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_396_5 :
+    Uω (aρ 5) (bρ 5) (25143831725049 / 64000000000000 : ℝ) ≤
+      (-(10246674196579949227919 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28713265097272909 / 80000000000000000 : ℝ) ≤
+      (-(10246674196579949227919 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (28713265097272909 / 80000000000000000 : ℝ) =
+            (28713265097272909 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28713265097272909 / 20000000000000000 : ℝ) ≤
+            (3616269409420050772081 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3581739004624321 / 10000000000000000 : ℝ)) (q :=
+      (28713265097272909 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_396_6 :
+    Uω (aρ 6) (bρ 6) (25143831725049 / 64000000000000 : ℝ) ≤
+      (-(5377385588401860750483 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5458159276333873 / 16000000000000000 : ℝ) ≤
+      (-(5377385588401860750483 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5458159276333873 / 16000000000000000 : ℝ) =
+            (5458159276333873 / 4000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5458159276333873 / 4000000000000000 : ℝ) ≤
+            (1554086214598139249517 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (678672440172687 / 2000000000000000 : ℝ)) (q :=
+      (5458159276333873 / 16000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_396_7 :
+    Uω (aρ 7) (bρ 7) (25143831725049 / 64000000000000 : ℝ) ≤
+      (-(2876317088384850507541 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25317599816358261 / 80000000000000000 : ℝ) ≤
+      (-(2876317088384850507541 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (25317599816358261 / 80000000000000000 : ℝ) =
+            (25317599816358261 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25317599816358261 / 20000000000000000 : ℝ) ≤
+            (589418813115149492459 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3123724440295659 / 10000000000000000 : ℝ)) (q :=
+      (25317599816358261 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_396_8 :
+    Uω (aρ 8) (bρ 8) (25143831725049 / 64000000000000 : ℝ) ≤
+      (-(12612131939922347998239 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22664808612311169 / 80000000000000000 : ℝ) ≤
+      (-(12612131939922347998239 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (22664808612311169 / 80000000000000000 : ℝ) =
+            (22664808612311169 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22664808612311169 / 20000000000000000 : ℝ) ≤
+            (1250811666077652001761 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1372034145311943 / 5000000000000000 : ℝ)) (q :=
+      (22664808612311169 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_396_9 :
+    Uω (aρ 9) (bρ 9) (25143831725049 / 64000000000000 : ℝ) ≤
+      (-(7152631511257893339269 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19134640509631401 / 80000000000000000 : ℝ) ≤
+      (-(7152631511257893339269 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (19134640509631401 / 80000000000000000 : ℝ) =
+            (19134640509631401 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19134640509631401 / 10000000000000000 : ℝ) ≤
+            (3244576193242106660731 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (275311486664243 / 1250000000000000 : ℝ)) (q :=
+      (19134640509631401 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_396_10 :
+    Uω (aρ 10) (bρ 10) (25143831725049 / 64000000000000 : ℝ) ≤
+      (-(17334347667952915807461 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14134121903874213 / 80000000000000000 : ℝ) ≤
+      (-(17334347667952915807461 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (14134121903874213 / 80000000000000000 : ℝ) =
+            (14134121903874213 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14134121903874213 / 10000000000000000 : ℝ) ≤
+            (3460067741047084192539 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1340921275369647 / 10000000000000000 : ℝ)) (q :=
+      (14134121903874213 / 80000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_396_11 :
+    Uω (aρ 11) (bρ 11) (25143831725049 / 64000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_396_12 :
+    Uω (aρ 12) (bρ 12) (25143831725049 / 64000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_396_13 :
+    Uω (aρ 13) (bρ 13) (25143831725049 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_396_14 :
+    Uω (aρ 14) (bρ 14) (25143831725049 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_396_15 :
+    Uω (aρ 15) (bρ 15) (25143831725049 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_396_16 :
+    Uω (aρ 16) (bρ 16) (25143831725049 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_396 :
+    Uρ (25143831725049 / 64000000000000 : ℝ) ≤
+      (-(13509220281971782976273 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (25143831725049 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(9508343896262421957543 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_396_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (25143831725049 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(4785327689087111628957 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_396_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (25143831725049 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(9696688535615259549621 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_396_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (25143831725049 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(9910323425109448560787 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_396_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (25143831725049 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(10246674196579949227919 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_396_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (25143831725049 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(5377385588401860750483 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_396_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (25143831725049 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(2876317088384850507541 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_396_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (25143831725049 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(12612131939922347998239 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_396_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (25143831725049 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(7152631511257893339269 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_396_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (25143831725049 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(17334347667952915807461 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_396_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (25143831725049 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_396_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (25143831725049 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_396_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (25143831725049 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_396_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (25143831725049 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_396_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (25143831725049 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_396_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (25143831725049 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_396_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_397_1 :
+    Uω (aρ 1) (bρ 1) (6306887218827 / 16000000000000 : ℝ) ≤
+      (-(9474549302467474731171 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7754531194632573 / 20000000000000000 : ℝ) ≤
+      (-(9474549302467474731171 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7754531194632573 / 20000000000000000 : ℝ) =
+            (7754531194632573 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7754531194632573 / 5000000000000000 : ℝ) ≤
+            (4388394303532525268829 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1938611950475349 / 5000000000000000 : ℝ)) (q :=
+      (7754531194632573 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_397_2 :
+    Uω (aρ 2) (bρ 2) (6306887218827 / 16000000000000 : ℝ) ≤
+      (-(2384161959690752671551 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3853262949073229 / 10000000000000000 : ℝ) ≤
+      (-(2384161959690752671551 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3853262949073229 / 10000000000000000 : ℝ) =
+            (3853262949073229 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3853262949073229 / 2500000000000000 : ℝ) ≤
+            (1081573941809247328449 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3852987616974583 / 10000000000000000 : ℝ)) (q :=
+      (3853262949073229 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_397_3 :
+    Uω (aρ 3) (bρ 3) (6306887218827 / 16000000000000 : ℝ) ≤
+      (-(483112203406672262523 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7610340131582821 / 20000000000000000 : ℝ) ≤
+      (-(483112203406672262523 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7610340131582821 / 20000000000000000 : ℝ) =
+            (7610340131582821 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7610340131582821 / 5000000000000000 : ℝ) ≤
+            (210034976893327737477 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1902098978380473 / 5000000000000000 : ℝ)) (q :=
+      (7610340131582821 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_397_4 :
+    Uω (aρ 4) (bρ 4) (6306887218827 / 16000000000000 : ℝ) ≤
+      (-(395004750383460144211 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3725023708523281 / 10000000000000000 : ℝ) ≤
+      (-(395004750383460144211 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3725023708523281 / 10000000000000000 : ℝ) =
+            (3725023708523281 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3725023708523281 / 2500000000000000 : ℝ) ≤
+            (159512993856539855789 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3722199423289687 / 10000000000000000 : ℝ)) (q :=
+      (3725023708523281 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_397_5 :
+    Uω (aρ 5) (bρ 5) (6306887218827 / 16000000000000 : ℝ) ≤
+      (-(10210220176931123704729 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7204531876741977 / 20000000000000000 : ℝ) ≤
+      (-(10210220176931123704729 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7204531876741977 / 20000000000000000 : ℝ) =
+            (7204531876741977 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7204531876741977 / 5000000000000000 : ℝ) ≤
+            (3652723429068876295271 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1797436901160051 / 5000000000000000 : ℝ)) (q :=
+      (7204531876741977 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_397_6 :
+    Uω (aρ 6) (bρ 6) (6306887218827 / 16000000000000 : ℝ) ≤
+      (-(1071629785373233403567 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6848998845567579 / 20000000000000000 : ℝ) ≤
+      (-(1071629785373233403567 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6848998845567579 / 20000000000000000 : ℝ) =
+            (6848998845567579 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6848998845567579 / 5000000000000000 : ℝ) ≤
+            (314664575226766596433 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (425822643285713 / 1250000000000000 : ℝ)) (q :=
+      (6848998845567579 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_397_7 :
+    Uω (aρ 7) (bρ 7) (6306887218827 / 16000000000000 : ℝ) ≤
+      (-(1146348240106701236677 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (794487911180923 / 2500000000000000 : ℝ) ≤
+      (-(1146348240106701236677 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (794487911180923 / 2500000000000000 : ℝ) =
+            (794487911180923 / 625000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (794487911180923 / 625000000000000 : ℝ) ≤
+            (239946120493298763323 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3137146970925509 / 10000000000000000 : ℝ)) (q :=
+      (794487911180923 / 2500000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_397_8 :
+    Uω (aρ 8) (bρ 8) (6306887218827 / 16000000000000 : ℝ) ≤
+      (-(785286444664797924013 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5693208433662741 / 20000000000000000 : ℝ) ≤
+      (-(785286444664797924013 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5693208433662741 / 20000000000000000 : ℝ) =
+            (5693208433662741 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5693208433662741 / 5000000000000000 : ℝ) ≤
+            (81147530710202075987 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1378996883240433 / 5000000000000000 : ℝ)) (q :=
+      (5693208433662741 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_397_9 :
+    Uω (aρ 9) (bρ 9) (6306887218827 / 16000000000000 : ℝ) ≤
+      (-(14246077694202213291207 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2406028162975879 / 10000000000000000 : ℝ) ≤
+      (-(14246077694202213291207 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2406028162975879 / 10000000000000000 : ℝ) =
+            (2406028162975879 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2406028162975879 / 1250000000000000 : ℝ) ≤
+            (6548337714797786708793 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2217807287129883 / 10000000000000000 : ℝ)) (q :=
+      (2406028162975879 / 10000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_397_10 :
+    Uω (aρ 10) (bρ 10) (6306887218827 / 16000000000000 : ℝ) ≤
+      (-(17237564134643599097387 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (35678952614417 / 200000000000000 : ℝ) ≤
+      (-(17237564134643599097387 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (35678952614417 / 200000000000000 : ℝ) = (35678952614417 / 25000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (35678952614417 / 25000000000000 : ℝ) ≤
+            (3556851274356400902613 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (54488210244593 / 400000000000000 : ℝ)) (q :=
+      (35678952614417 / 200000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10]) (by norm_num [bρ_10])
+      (by norm_num) (by norm_num [aρ_10, bρ_10]) (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_397_11 :
+    Uω (aρ 11) (bρ 11) (6306887218827 / 16000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_397_12 :
+    Uω (aρ 12) (bρ 12) (6306887218827 / 16000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_397_13 :
+    Uω (aρ 13) (bρ 13) (6306887218827 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_397_14 :
+    Uω (aρ 14) (bρ 14) (6306887218827 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_397_15 :
+    Uω (aρ 15) (bρ 15) (6306887218827 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_397_16 :
+    Uω (aρ 16) (bρ 16) (6306887218827 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_397 :
+    Uρ (6306887218827 / 16000000000000 : ℝ) ≤
+      (-(2695507205577800469069 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (6306887218827 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(9474549302467474731171 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_397_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (6306887218827 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(2384161959690752671551 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_397_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (6306887218827 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(483112203406672262523 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_397_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (6306887218827 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(395004750383460144211 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_397_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (6306887218827 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(10210220176931123704729 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_397_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (6306887218827 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1071629785373233403567 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_397_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (6306887218827 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(1146348240106701236677 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_397_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (6306887218827 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(785286444664797924013 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_397_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (6306887218827 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(14246077694202213291207 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_397_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (6306887218827 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(17237564134643599097387 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_397_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (6306887218827 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_397_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (6306887218827 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_397_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (6306887218827 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_397_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (6306887218827 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_397_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (6306887218827 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_397_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (6306887218827 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_397_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_398_1 :
+    Uω (aρ 1) (bρ 1) (12697491587913 / 32000000000000 : ℝ) ≤
+      (-(4703650413353603050421 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15613709944932829 / 40000000000000000 : ℝ) ≤
+      (-(4703650413353603050421 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (15613709944932829 / 40000000000000000 : ℝ) =
+            (15613709944932829 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15613709944932829 / 10000000000000000 : ℝ) ≤
+            (2227821389646396949579 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1951693034664301 / 5000000000000000 : ℝ)) (q :=
+      (15613709944932829 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_398_2 :
+    Uω (aρ 2) (bρ 2) (12697491587913 / 32000000000000 : ℝ) ≤
+      (-(9468977808463272198431 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3103541132329967 / 8000000000000000 : ℝ) ≤
+      (-(9468977808463272198431 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3103541132329967 / 8000000000000000 : ℝ) =
+            (3103541132329967 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3103541132329967 / 2000000000000000 : ℝ) ≤
+            (4393965797536727801569 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (775830588039421 / 2000000000000000 : ℝ)) (q :=
+      (3103541132329967 / 8000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_398_3 :
+    Uω (aρ 3) (bρ 3) (12697491587913 / 32000000000000 : ℝ) ≤
+      (-(1199213651081101815863 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15325353259219533 / 40000000000000000 : ℝ) ≤
+      (-(1199213651081101815863 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (15325353259219533 / 40000000000000000 : ℝ) =
+            (15325353259219533 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15325353259219533 / 10000000000000000 : ℝ) ≤
+            (533654299668898184137 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1915186422665977 / 5000000000000000 : ℝ)) (q :=
+      (15325353259219533 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_398_4 :
+    Uω (aρ 4) (bρ 4) (12697491587913 / 32000000000000 : ℝ) ≤
+      (-(9805079627997499469213 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15004820119667203 / 40000000000000000 : ℝ) ≤
+      (-(9805079627997499469213 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (15004820119667203 / 40000000000000000 : ℝ) =
+            (15004820119667203 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15004820119667203 / 10000000000000000 : ℝ) ≤
+            (4057863978002500530787 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3748400456620789 / 10000000000000000 : ℝ)) (q :=
+      (15004820119667203 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_398_5 :
+    Uω (aρ 5) (bρ 5) (12697491587913 / 32000000000000 : ℝ) ≤
+      (-(10137710016249584730821 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14513923817856079 / 40000000000000000 : ℝ) ≤
+      (-(10137710016249584730821 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (14513923817856079 / 40000000000000000 : ℝ) =
+            (14513923817856079 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14513923817856079 / 10000000000000000 : ℝ) ≤
+            (3725233589750415269179 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3621142225050227 / 10000000000000000 : ℝ)) (q :=
+      (14513923817856079 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_398_6 :
+    Uω (aρ 6) (bρ 6) (12697491587913 / 32000000000000 : ℝ) ≤
+      (-(10639797039393326246521 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13803190342104979 / 40000000000000000 : ℝ) ≤
+      (-(10639797039393326246521 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (13803190342104979 / 40000000000000000 : ℝ) =
+            (13803190342104979 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13803190342104979 / 10000000000000000 : ℝ) ≤
+            (3223146566606673753479 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3433015862314677 / 10000000000000000 : ℝ)) (q :=
+      (13803190342104979 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_398_7 :
+    Uω (aρ 7) (bρ 7) (12697491587913 / 32000000000000 : ℝ) ≤
+      (-(11380444095054808415279 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12817802745399477 / 40000000000000000 : ℝ) ≤
+      (-(11380444095054808415279 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (12817802745399477 / 40000000000000000 : ℝ) =
+            (12817802745399477 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12817802745399477 / 10000000000000000 : ℝ) ≤
+            (2482499510945191584721 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1581991722360963 / 5000000000000000 : ℝ)) (q :=
+      (12817802745399477 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_398_8 :
+    Uω (aρ 8) (bρ 8) (12697491587913 / 32000000000000 : ℝ) ≤
+      (-(12470201145686833426751 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11494392859447927 / 40000000000000000 : ℝ) ≤
+      (-(12470201145686833426751 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (11494392859447927 / 40000000000000000 : ℝ) =
+            (11494392859447927 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11494392859447927 / 10000000000000000 : ℝ) ≤
+            (1392742460313166573249 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2785820153086151 / 10000000000000000 : ℝ)) (q :=
+      (11494392859447927 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_398_9 :
+    Uω (aρ 9) (bρ 9) (12697491587913 / 32000000000000 : ℝ) ≤
+      (-(3532230387094757563699 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9737528115117821 / 40000000000000000 : ℝ) ≤
+      (-(3532230387094757563699 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (9737528115117821 / 40000000000000000 : ℝ) =
+            (9737528115117821 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9737528115117821 / 5000000000000000 : ℝ) ≤
+            (1666373465155242436301 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1124176704640549 / 5000000000000000 : ℝ)) (q :=
+      (9737528115117821 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_398_10 :
+    Uω (aρ 10) (bρ 10) (12697491587913 / 32000000000000 : ℝ) ≤
+      (-(17048418578553716535903 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7272045367266537 / 40000000000000000 : ℝ) ≤
+      (-(17048418578553716535903 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (7272045367266537 / 40000000000000000 : ℝ) =
+            (7272045367266537 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7272045367266537 / 5000000000000000 : ℝ) ≤
+            (3745996830446283464097 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (175521383606307 / 1250000000000000 : ℝ)) (q :=
+      (7272045367266537 / 40000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_398_11 :
+    Uω (aρ 11) (bρ 11) (12697491587913 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_398_12 :
+    Uω (aρ 12) (bρ 12) (12697491587913 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_398_13 :
+    Uω (aρ 13) (bρ 13) (12697491587913 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_398_14 :
+    Uω (aρ 14) (bρ 14) (12697491587913 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_398_15 :
+    Uω (aρ 15) (bρ 15) (12697491587913 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_398_16 :
+    Uω (aρ 16) (bρ 16) (12697491587913 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_398 :
+    Uρ (12697491587913 / 32000000000000 : ℝ) ≤
+      (-(13414874358368062389781 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (12697491587913 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(4703650413353603050421 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_398_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (12697491587913 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(9468977808463272198431 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_398_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (12697491587913 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1199213651081101815863 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_398_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (12697491587913 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(9805079627997499469213 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_398_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (12697491587913 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(10137710016249584730821 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_398_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (12697491587913 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(10639797039393326246521 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_398_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (12697491587913 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(11380444095054808415279 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_398_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (12697491587913 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(12470201145686833426751 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_398_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (12697491587913 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(3532230387094757563699 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_398_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (12697491587913 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(17048418578553716535903 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_398_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (12697491587913 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_398_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (12697491587913 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_398_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (12697491587913 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_398_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (12697491587913 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_398_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (12697491587913 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_398_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (12697491587913 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_398_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_399_1 :
+    Uω (aρ 1) (bρ 1) (3195302184543 / 8000000000000 : ℝ) ≤
+      (-(4670250788467362062751 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7859178742857949 / 20000000000000000 : ℝ) ≤
+      (-(4670250788467362062751 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7859178742857949 / 20000000000000000 : ℝ) =
+            (7859178742857949 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7859178742857949 / 5000000000000000 : ℝ) ≤
+            (2261221014532637937249 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3929548230264199 / 10000000000000000 : ℝ)) (q :=
+      (7859178742857949 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_399_2 :
+    Uω (aρ 2) (bρ 2) (3195302184543 / 8000000000000 : ℝ) ≤
+      (-(376070507557605859629 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (312447188549721 / 800000000000000 : ℝ) ≤
+      (-(376070507557605859629 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (312447188549721 / 800000000000000 : ℝ) =
+            (312447188549721 / 200000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (312447188549721 / 200000000000000 : ℝ) ≤
+            (178447236682394140371 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (156212728546371 / 400000000000000 : ℝ)) (q :=
+      (312447188549721 / 800000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_399_3 :
+    Uω (aρ 3) (bρ 3) (3195302184543 / 8000000000000 : ℝ) ≤
+      (-(1905128218010882884229 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1543002589485657 / 4000000000000000 : ℝ) ≤
+      (-(1905128218010882884229 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1543002589485657 / 4000000000000000 : ℝ) =
+            (1543002589485657 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1543002589485657 / 1000000000000000 : ℝ) ≤
+            (867460503189117115771 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (771309510738907 / 2000000000000000 : ℝ)) (q :=
+      (1543002589485657 / 4000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_399_4 :
+    Uω (aρ 4) (bρ 4) (3195302184543 / 8000000000000 : ℝ) ≤
+      (-(243388208960620545919 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7554772155718287 / 20000000000000000 : ℝ) ≤
+      (-(243388208960620545919 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7554772155718287 / 20000000000000000 : ℝ) =
+            (7554772155718287 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7554772155718287 / 5000000000000000 : ℝ) ≤
+            (103185381189379454081 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3774600943049537 / 10000000000000000 : ℝ)) (q :=
+      (7554772155718287 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_399_5 :
+    Uω (aρ 5) (bρ 5) (3195302184543 / 8000000000000 : ℝ) ≤
+      (-(5032861972902583863707 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (228418450178649 / 625000000000000 : ℝ) ≤
+      (-(5032861972902583863707 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (228418450178649 / 625000000000000 : ℝ) =
+            (228418450178649 / 156250000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (228418450178649 / 156250000000000 : ℝ) ≤
+            (1898609830097416136293 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1823704556191509 / 5000000000000000 : ℝ)) (q :=
+      (228418450178649 / 625000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_399_6 :
+    Uω (aρ 6) (bρ 6) (3195302184543 / 8000000000000 : ℝ) ≤
+      (-(5281941496501066055649 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1390837468862847 / 4000000000000000 : ℝ) ≤
+      (-(5281941496501066055649 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1390837468862847 / 4000000000000000 : ℝ) =
+            (1390837468862847 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1390837468862847 / 1000000000000000 : ℝ) ≤
+            (1649530306498933944351 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (691889285224097 / 2000000000000000 : ℝ)) (q :=
+      (1390837468862847 / 4000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_399_7 :
+    Uω (aρ 7) (bρ 7) (3195302184543 / 8000000000000 : ℝ) ≤
+      (-(1412263374616923864879 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6461888246739339 / 20000000000000000 : ℝ) ≤
+      (-(1412263374616923864879 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6461888246739339 / 20000000000000000 : ℝ) =
+            (6461888246739339 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6461888246739339 / 5000000000000000 : ℝ) ≤
+            (320604576133076135121 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3190808709305589 / 10000000000000000 : ℝ)) (q :=
+      (6461888246739339 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_399_8 :
+    Uω (aρ 8) (bρ 8) (3195302184543 / 8000000000000 : ℝ) ≤
+      (-(3094189178564688703663 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2900576241005023 / 10000000000000000 : ℝ) ≤
+      (-(3094189178564688703663 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2900576241005023 / 10000000000000000 : ℝ) =
+            (2900576241005023 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2900576241005023 / 2500000000000000 : ℝ) ≤
+            (371546722935311296337 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (351701824489537 / 1250000000000000 : ℝ)) (q :=
+      (2900576241005023 / 10000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_399_9 :
+    Uω (aρ 9) (bρ 9) (3195302184543 / 8000000000000 : ℝ) ≤
+      (-(14013343596192001386277 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2462681341559129 / 10000000000000000 : ℝ) ≤
+      (-(14013343596192001386277 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2462681341559129 / 10000000000000000 : ℝ) =
+            (2462681341559129 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2462681341559129 / 1250000000000000 : ℝ) ≤
+            (6781071812807998613723 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (569697606346127 / 2500000000000000 : ℝ)) (q :=
+      (2462681341559129 / 10000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_399_10 :
+    Uω (aρ 10) (bρ 10) (3195302184543 / 8000000000000 : ℝ) ≤
+      (-(843239751853213725649 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (148136215159719 / 800000000000000 : ℝ) ≤
+      (-(843239751853213725649 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (148136215159719 / 800000000000000 : ℝ) =
+            (148136215159719 / 100000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (148136215159719 / 100000000000000 : ℝ) ≤
+            (196481018596786274351 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (57815686190169 / 400000000000000 : ℝ)) (q :=
+      (148136215159719 / 800000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10]) (by norm_num [bρ_10])
+      (by norm_num) (by norm_num [aρ_10, bρ_10]) (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_399_11 :
+    Uω (aρ 11) (bρ 11) (3195302184543 / 8000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_399_12 :
+    Uω (aρ 12) (bρ 12) (3195302184543 / 8000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_399_13 :
+    Uω (aρ 13) (bρ 13) (3195302184543 / 8000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_399_14 :
+    Uω (aρ 14) (bρ 14) (3195302184543 / 8000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_399_15 :
+    Uω (aρ 15) (bρ 15) (3195302184543 / 8000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_399_16 :
+    Uω (aρ 16) (bρ 16) (3195302184543 / 8000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_399 :
+    Uρ (3195302184543 / 8000000000000 : ℝ) ≤
+      (-(13353115432411953119471 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (3195302184543 / 8000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(4670250788467362062751 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_399_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (3195302184543 / 8000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(376070507557605859629 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_399_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (3195302184543 / 8000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1905128218010882884229 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_399_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (3195302184543 / 8000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(243388208960620545919 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_399_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (3195302184543 / 8000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(5032861972902583863707 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_399_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (3195302184543 / 8000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(5281941496501066055649 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_399_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (3195302184543 / 8000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(1412263374616923864879 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_399_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (3195302184543 / 8000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(3094189178564688703663 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_399_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (3195302184543 / 8000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(14013343596192001386277 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_399_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (3195302184543 / 8000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) * (-(843239751853213725649 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_399_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (3195302184543 / 8000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_399_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (3195302184543 / 8000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_399_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (3195302184543 / 8000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_399_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (3195302184543 / 8000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_399_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (3195302184543 / 8000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_399_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (3195302184543 / 8000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_399_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/U33.lean b/LeanPool/Zeta5Irrational/Table/U33.lean
new file mode 100644
index 0000000000..2b3c9df0b7
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/U33.lean
@@ -0,0 +1,4213 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-! # Certified arcsine potential bounds (U33) -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma U_400_1 :
+    Uω (aρ 1) (bρ 1) (12864925888431 / 32000000000000 : ℝ) ≤
+      (-(9274145591088656349597 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15823005011909681 / 40000000000000000 : ℝ) ≤
+      (-(9274145591088656349597 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (15823005011909681 / 40000000000000000 : ℝ) =
+            (15823005011909681 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15823005011909681 / 10000000000000000 : ℝ) ≤
+            (4588798014911343650403 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3955710383905153 / 10000000000000000 : ℝ)) (q :=
+      (15823005011909681 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_400_2 :
+    Uω (aρ 2) (bρ 2) (12864925888431 / 32000000000000 : ℝ) ≤
+      (-(145859318811119573189 / 156250000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15727013095788583 / 40000000000000000 : ℝ) ≤
+      (-(145859318811119573189 / 156250000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (15727013095788583 / 40000000000000000 : ℝ) =
+            (15727013095788583 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15727013095788583 / 10000000000000000 : ℝ) ≤
+            (70749175032630426811 / 156250000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (982870859588651 / 2500000000000000 : ℝ)) (q :=
+      (15727013095788583 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_400_3 :
+    Uω (aρ 3) (bρ 3) (12864925888431 / 32000000000000 : ℝ) ≤
+      (-(9458033394948477753999 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (621387927094633 / 1600000000000000 : ℝ) ≤
+      (-(9458033394948477753999 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (621387927094633 / 1600000000000000 : ℝ) =
+            (621387927094633 / 400000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (621387927094633 / 400000000000000 : ℝ) ≤
+            (4404910211051522246001 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (155308883419729 / 400000000000000 : ℝ)) (q :=
+      (621387927094633 / 1600000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_400_4 :
+    Uω (aρ 4) (bρ 4) (12864925888431 / 32000000000000 : ℝ) ≤
+      (-(4833229095758892645243 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15214267432021373 / 40000000000000000 : ℝ) ≤
+      (-(4833229095758892645243 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (15214267432021373 / 40000000000000000 : ℝ) =
+            (15214267432021373 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15214267432021373 / 10000000000000000 : ℝ) ≤
+            (2098242707241107354757 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3800800893885999 / 10000000000000000 : ℝ)) (q :=
+      (15214267432021373 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_400_5 :
+    Uω (aρ 5) (bρ 5) (12864925888431 / 32000000000000 : ℝ) ≤
+      (-(9994254414217391606101 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14723634800085159 / 40000000000000000 : ℝ) ≤
+      (-(9994254414217391606101 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (14723634800085159 / 40000000000000000 : ℝ) =
+            (14723634800085159 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14723634800085159 / 10000000000000000 : ℝ) ≤
+            (3868689191782608393899 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (918418624313223 / 2500000000000000 : ℝ)) (q :=
+      (14723634800085159 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_400_6 :
+    Uω (aρ 6) (bρ 6) (12864925888431 / 32000000000000 : ℝ) ≤
+      (-(10488546692852194258217 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14013550919603397 / 40000000000000000 : ℝ) ≤
+      (-(10488546692852194258217 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (14013550919603397 / 40000000000000000 : ℝ) =
+            (14013550919603397 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14013550919603397 / 10000000000000000 : ℝ) ≤
+            (3374396913147805741783 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3485872932152011 / 10000000000000000 : ℝ)) (q :=
+      (14013550919603397 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_400_7 :
+    Uω (aρ 7) (bρ 7) (12864925888431 / 32000000000000 : ℝ) ≤
+      (-(2804114770301443257797 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13029728383838411 / 40000000000000000 : ℝ) ≤
+      (-(2804114770301443257797 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (13029728383838411 / 40000000000000000 : ℝ) =
+            (13029728383838411 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13029728383838411 / 10000000000000000 : ℝ) ≤
+            (661621131198556742203 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1608811522514759 / 5000000000000000 : ℝ)) (q :=
+      (13029728383838411 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_400_8 :
+    Uω (aρ 8) (bρ 8) (12864925888431 / 32000000000000 : ℝ) ≤
+      (-(12284230347491191553647 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11710155055891933 / 40000000000000000 : ℝ) ≤
+      (-(12284230347491191553647 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (11710155055891933 / 40000000000000000 : ℝ) =
+            (11710155055891933 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11710155055891933 / 10000000000000000 : ℝ) ≤
+            (1578713258508808446353 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2841378032396279 / 10000000000000000 : ℝ)) (q :=
+      (11710155055891933 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_400_9 :
+    Uω (aρ 9) (bρ 9) (12864925888431 / 32000000000000 : ℝ) ≤
+      (-(6949648283068307006159 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9963713034161937 / 40000000000000000 : ℝ) ≤
+      (-(6949648283068307006159 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (9963713034161937 / 40000000000000000 : ℝ) =
+            (9963713034161937 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9963713034161937 / 5000000000000000 : ℝ) ≤
+            (3447559421431692993841 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2309122649891281 / 10000000000000000 : ℝ)) (q :=
+      (9963713034161937 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_400_10 :
+    Uω (aρ 10) (bρ 10) (12864925888431 / 32000000000000 : ℝ) ≤
+      (-(16686293334959350296841 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7540210654092581 / 40000000000000000 : ℝ) ≤
+      (-(16686293334959350296841 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (7540210654092581 / 40000000000000000 : ℝ) =
+            (7540210654092581 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7540210654092581 / 5000000000000000 : ℝ) ≤
+            (4108122074040649703159 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1485930493351603 / 10000000000000000 : ℝ)) (q :=
+      (7540210654092581 / 40000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_400_11 :
+    Uω (aρ 11) (bρ 11) (12864925888431 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_400_12 :
+    Uω (aρ 12) (bρ 12) (12864925888431 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_400_13 :
+    Uω (aρ 13) (bρ 13) (12864925888431 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_400_14 :
+    Uω (aρ 14) (bρ 14) (12864925888431 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_400_15 :
+    Uω (aρ 15) (bρ 15) (12864925888431 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_400_16 :
+    Uω (aρ 16) (bρ 16) (12864925888431 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_400 :
+    Uρ (12864925888431 / 32000000000000 : ℝ) ≤
+      (-(6646107995395590288459 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (12864925888431 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(9274145591088656349597 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_400_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (12864925888431 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(145859318811119573189 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_400_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (12864925888431 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(9458033394948477753999 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_400_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (12864925888431 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(4833229095758892645243 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_400_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (12864925888431 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(9994254414217391606101 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_400_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (12864925888431 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(10488546692852194258217 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_400_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (12864925888431 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(2804114770301443257797 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_400_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (12864925888431 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(12284230347491191553647 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_400_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (12864925888431 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(6949648283068307006159 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_400_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (12864925888431 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(16686293334959350296841 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_400_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (12864925888431 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_400_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (12864925888431 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_400_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (12864925888431 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_400_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (12864925888431 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_400_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (12864925888431 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_400_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (12864925888431 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_400_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_401_1 :
+    Uω (aρ 1) (bρ 1) (1294864303869 / 3200000000000 : ℝ) ≤
+      (-(2302056756254861910491 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (995478282737609 / 2500000000000000 : ℝ) ≤
+      (-(2302056756254861910491 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (995478282737609 / 2500000000000000 : ℝ) =
+            (995478282737609 / 625000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (995478282737609 / 625000000000000 : ℝ) ≤
+            (1163679145245138089509 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3981872530395247 / 10000000000000000 : ℝ)) (q :=
+      (995478282737609 / 2500000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_401_2 :
+    Uω (aρ 2) (bρ 2) (1294864303869 / 3200000000000 : ℝ) ≤
+      (-(1853734599609437634619 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1978958333561489 / 5000000000000000 : ℝ) ≤
+      (-(1853734599609437634619 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1978958333561489 / 5000000000000000 : ℝ) =
+            (1978958333561489 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1978958333561489 / 1250000000000000 : ℝ) ≤
+            (918854121590562365381 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3957648615250331 / 10000000000000000 : ℝ)) (q :=
+      (1978958333561489 / 5000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_401_3 :
+    Uω (aρ 3) (bρ 3) (1294864303869 / 3200000000000 : ℝ) ≤
+      (-(4695439966694920649199 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3909842528460269 / 10000000000000000 : ℝ) ≤
+      (-(4695439966694920649199 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3909842528460269 / 10000000000000000 : ℝ) =
+            (3909842528460269 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3909842528460269 / 2500000000000000 : ℝ) ≤
+            (2236031836305079350801 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3908896444274913 / 10000000000000000 : ℝ)) (q :=
+      (3909842528460269 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_401_4 :
+    Uω (aρ 4) (bρ 4) (1294864303869 / 3200000000000 : ℝ) ≤
+      (-(9597862507636789230839 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (306379790068437 / 800000000000000 : ℝ) ≤
+      (-(9597862507636789230839 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (306379790068437 / 800000000000000 : ℝ) =
+            (306379790068437 / 200000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (306379790068437 / 200000000000000 : ℝ) ≤
+            (4265081098363210769161 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (38270003201303 / 100000000000000 : ℝ)) (q :=
+      (306379790068437 / 800000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_401_5 :
+    Uω (aρ 5) (bρ 5) (1294864303869 / 3200000000000 : ℝ) ≤
+      (-(1240411754096010472283 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3707121461951423 / 10000000000000000 : ℝ) ≤
+      (-(1240411754096010472283 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3707121461951423 / 10000000000000000 : ℝ) =
+            (3707121461951423 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3707121461951423 / 2500000000000000 : ℝ) ≤
+            (492456196653989527717 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3699938411657221 / 10000000000000000 : ℝ)) (q :=
+      (3707121461951423 / 10000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_401_6 :
+    Uω (aρ 6) (bρ 6) (1294864303869 / 3200000000000 : ℝ) ≤
+      (-(1301722415710541881821 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1411871921821389 / 4000000000000000 : ℝ) ≤
+      (-(1301722415710541881821 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1411871921821389 / 4000000000000000 : ℝ) =
+            (1411871921821389 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1411871921821389 / 1000000000000000 : ℝ) ≤
+            (431145535039458118179 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (87807386800033 / 250000000000000 : ℝ)) (q :=
+      (1411871921821389 / 4000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_401_7 :
+    Uω (aρ 7) (bρ 7) (1294864303869 / 3200000000000 : ℝ) ≤
+      (-(5567744317910864751661 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (410489342450739 / 1250000000000000 : ℝ) ≤
+      (-(5567744317910864751661 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (410489342450739 / 1250000000000000 : ℝ) =
+            (410489342450739 / 312500000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (410489342450739 / 312500000000000 : ℝ) ≤
+            (1363727485089135248339 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3244426722866199 / 10000000000000000 : ℝ)) (q :=
+      (410489342450739 / 1250000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_401_8 :
+    Uω (aρ 8) (bρ 8) (1294864303869 / 3200000000000 : ℝ) ≤
+      (-(12192603200744992393569 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2954486233823141 / 10000000000000000 : ℝ) ≤
+      (-(12192603200744992393569 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2954486233823141 / 10000000000000000 : ℝ) =
+            (2954486233823141 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2954486233823141 / 2500000000000000 : ℝ) ≤
+            (1670340405255007606431 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2869111362640657 / 10000000000000000 : ℝ)) (q :=
+      (2954486233823141 / 10000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_401_9 :
+    Uω (aρ 9) (bρ 9) (1294864303869 / 3200000000000 : ℝ) ≤
+      (-(13786735457843431653397 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (629781204456927 / 2500000000000000 : ℝ) ≤
+      (-(13786735457843431653397 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (629781204456927 / 2500000000000000 : ℝ) =
+            (629781204456927 / 625000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (629781204456927 / 625000000000000 : ℝ) ≤
+            (76208148156568346603 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2339354159009791 / 10000000000000000 : ℝ)) (q :=
+      (629781204456927 / 2500000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_401_10 :
+    Uω (aρ 10) (bρ 10) (1294864303869 / 3200000000000 : ℝ) ≤
+      (-(16512559890281452104139 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (767235389239469 / 4000000000000000 : ℝ) ≤
+      (-(16512559890281452104139 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (767235389239469 / 4000000000000000 : ℝ) =
+            (767235389239469 / 500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (767235389239469 / 500000000000000 : ℝ) ≤
+            (4281855518718547895861 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4768251572021 / 31250000000000 : ℝ)) (q :=
+      (767235389239469 / 4000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10]) (by norm_num [bρ_10])
+      (by norm_num) (by norm_num [aρ_10, bρ_10]) (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_401_11 :
+    Uω (aρ 11) (bρ 11) (1294864303869 / 3200000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_401_12 :
+    Uω (aρ 12) (bρ 12) (1294864303869 / 3200000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_401_13 :
+    Uω (aρ 13) (bρ 13) (1294864303869 / 3200000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_401_14 :
+    Uω (aρ 14) (bρ 14) (1294864303869 / 3200000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_401_15 :
+    Uω (aρ 15) (bρ 15) (1294864303869 / 3200000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_401_16 :
+    Uω (aρ 16) (bρ 16) (1294864303869 / 3200000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_401 :
+    Uρ (1294864303869 / 3200000000000 : ℝ) ≤
+      (-(13232137085469873442083 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1294864303869 / 3200000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2302056756254861910491 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_401_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1294864303869 / 3200000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1853734599609437634619 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_401_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1294864303869 / 3200000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(4695439966694920649199 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_401_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1294864303869 / 3200000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(9597862507636789230839 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_401_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1294864303869 / 3200000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1240411754096010472283 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_401_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1294864303869 / 3200000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1301722415710541881821 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_401_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1294864303869 / 3200000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(5567744317910864751661 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_401_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1294864303869 / 3200000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(12192603200744992393569 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_401_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1294864303869 / 3200000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(13786735457843431653397 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_401_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1294864303869 / 3200000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(16512559890281452104139 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_401_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1294864303869 / 3200000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_401_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1294864303869 / 3200000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_401_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1294864303869 / 3200000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_401_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1294864303869 / 3200000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_401_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1294864303869 / 3200000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_401_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1294864303869 / 3200000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_401_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_402_1 :
+    Uω (aρ 1) (bρ 1) (13032360188949 / 32000000000000 : ℝ) ≤
+      (-(2285685037350096200439 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16032300021672153 / 40000000000000000 : ℝ) ≤
+      (-(2285685037350096200439 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (16032300021672153 / 40000000000000000 : ℝ) =
+            (16032300021672153 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16032300021672153 / 10000000000000000 : ℝ) ≤
+            (1180050864149903799561 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2004017334937257 / 5000000000000000 : ℝ)) (q :=
+      (16032300021672153 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_402_2 :
+    Uω (aρ 2) (bρ 2) (13032360188949 / 32000000000000 : ℝ) ≤
+      (-(4601393316889063213119 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3187264029495937 / 8000000000000000 : ℝ) ≤
+      (-(4601393316889063213119 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3187264029495937 / 8000000000000000 : ℝ) =
+            (3187264029495937 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3187264029495937 / 2000000000000000 : ℝ) ≤
+            (2330078486110936786881 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (99595343632207 / 250000000000000 : ℝ)) (q :=
+      (3187264029495937 / 8000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_402_3 :
+    Uω (aρ 3) (bρ 3) (13032360188949 / 32000000000000 : ℝ) ≤
+      (-(4662087319745159335967 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15744041711187323 / 40000000000000000 : ℝ) ≤
+      (-(4662087319745159335967 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (15744041711187323 / 40000000000000000 : ℝ) =
+            (15744041711187323 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15744041711187323 / 10000000000000000 : ℝ) ≤
+            (2269384483254840664033 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3935070633492099 / 10000000000000000 : ℝ)) (q :=
+      (15744041711187323 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_402_4 :
+    Uω (aρ 4) (bρ 4) (13032360188949 / 32000000000000 : ℝ) ≤
+      (-(1191216852877995912977 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15423710547039463 / 40000000000000000 : ℝ) ≤
+      (-(1191216852877995912977 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (15423710547039463 / 40000000000000000 : ℝ) =
+            (15423710547039463 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15423710547039463 / 10000000000000000 : ℝ) ≤
+            (541651097872004087023 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3853199232483169 / 10000000000000000 : ℝ)) (q :=
+      (15423710547039463 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_402_5 :
+    Uω (aρ 5) (bρ 5) (13032360188949 / 32000000000000 : ℝ) ≤
+      (-(9852835570745593329249 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14933334016782087 / 40000000000000000 : ℝ) ≤
+      (-(9852835570745593329249 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (14933334016782087 / 40000000000000000 : ℝ) =
+            (14933334016782087 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14933334016782087 / 10000000000000000 : ℝ) ≤
+            (4010108035254406670751 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3726200886689481 / 10000000000000000 : ℝ)) (q :=
+      (14933334016782087 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_402_6 :
+    Uω (aρ 6) (bρ 6) (13032360188949 / 32000000000000 : ℝ) ≤
+      (-(10339572280256983547971 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14223879762145861 / 40000000000000000 : ℝ) ≤
+      (-(10339572280256983547971 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (14223879762145861 / 40000000000000000 : ℝ) =
+            (14223879762145861 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14223879762145861 / 10000000000000000 : ℝ) ≤
+            (3523371325743016452029 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (442339266813921 / 1250000000000000 : ℝ)) (q :=
+      (14223879762145861 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_402_7 :
+    Uω (aρ 7) (bρ 7) (13032360188949 / 32000000000000 : ℝ) ≤
+      (-(690949015699183907021 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13241568741205117 / 40000000000000000 : ℝ) ≤
+      (-(690949015699183907021 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (13241568741205117 / 40000000000000000 : ℝ) =
+            (13241568741205117 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13241568741205117 / 10000000000000000 : ℝ) ≤
+            (175484959675816092979 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (817805001200249 / 2500000000000000 : ℝ)) (q :=
+      (13241568741205117 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_402_8 :
+    Uω (aρ 8) (bρ 8) (13032360188949 / 32000000000000 : ℝ) ≤
+      (-(12101857029900682044303 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11925676331596149 / 40000000000000000 : ℝ) ≤
+      (-(12101857029900682044303 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (11925676331596149 / 40000000000000000 : ℝ) =
+            (11925676331596149 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11925676331596149 / 10000000000000000 : ℝ) ≤
+            (1761086576099317955697 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (45262741427133 / 156250000000000 : ℝ)) (q :=
+      (11925676331596149 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_402_9 :
+    Uω (aρ 9) (bρ 9) (13032360188949 / 32000000000000 : ℝ) ≤
+      (-(6837808690690216867231 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10189091787651641 / 40000000000000000 : ℝ) ≤
+      (-(6837808690690216867231 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (10189091787651641 / 40000000000000000 : ℝ) =
+            (10189091787651641 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10189091787651641 / 10000000000000000 : ℝ) ≤
+            (93663112309783132769 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1184744403862129 / 5000000000000000 : ℝ)) (q :=
+      (10189091787651641 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_402_10 :
+    Uω (aρ 10) (bρ 10) (13032360188949 / 32000000000000 : ℝ) ≤
+      (-(2042910051658331651807 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7803336604278519 / 40000000000000000 : ℝ) ≤
+      (-(2042910051658331651807 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (7803336604278519 / 40000000000000000 : ℝ) =
+            (7803336604278519 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7803336604278519 / 5000000000000000 : ℝ) ≤
+            (556391874466668348193 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1565170249532697 / 10000000000000000 : ℝ)) (q :=
+      (7803336604278519 / 40000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_402_11 :
+    Uω (aρ 11) (bρ 11) (13032360188949 / 32000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_402_12 :
+    Uω (aρ 12) (bρ 12) (13032360188949 / 32000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_402_13 :
+    Uω (aρ 13) (bρ 13) (13032360188949 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_402_14 :
+    Uω (aρ 14) (bρ 14) (13032360188949 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_402_15 :
+    Uω (aρ 15) (bρ 15) (13032360188949 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_402_16 :
+    Uω (aρ 16) (bρ 16) (13032360188949 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_402 :
+    Uρ (13032360188949 / 32000000000000 : ℝ) ≤
+      (-(13172843436669313561171 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (13032360188949 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2285685037350096200439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_402_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (13032360188949 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(4601393316889063213119 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_402_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (13032360188949 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(4662087319745159335967 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_402_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (13032360188949 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1191216852877995912977 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_402_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (13032360188949 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(9852835570745593329249 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_402_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (13032360188949 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(10339572280256983547971 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_402_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (13032360188949 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(690949015699183907021 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_402_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (13032360188949 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(12101857029900682044303 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_402_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (13032360188949 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(6837808690690216867231 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_402_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (13032360188949 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(2042910051658331651807 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_402_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (13032360188949 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_402_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (13032360188949 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_402_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (13032360188949 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_402_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (13032360188949 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_402_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (13032360188949 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_402_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (13032360188949 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_402_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_403_1 :
+    Uω (aρ 1) (bρ 1) (1639509667401 / 4000000000000 : ℝ) ≤
+      (-(9077679346739136303621 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8068473752896849 / 20000000000000000 : ℝ) ≤
+      (-(9077679346739136303621 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8068473752896849 / 20000000000000000 : ℝ) =
+            (8068473752896849 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8068473752896849 / 5000000000000000 : ℝ) ≤
+            (4785264259260863696379 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4034196802479349 / 10000000000000000 : ℝ)) (q :=
+      (8068473752896849 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_403_2 :
+    Uω (aρ 2) (bρ 2) (1639509667401 / 4000000000000 : ℝ) ≤
+      (-(2284332897053214077443 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8020486767293193 / 20000000000000000 : ℝ) ≤
+      (-(2284332897053214077443 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8020486767293193 / 20000000000000000 : ℝ) =
+            (8020486767293193 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8020486767293193 / 5000000000000000 : ℝ) ≤
+            (1181403004446785922557 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4009978829385693 / 10000000000000000 : ℝ)) (q :=
+      (8020486767293193 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_403_3 :
+    Uω (aρ 3) (bρ 3) (1639509667401 / 4000000000000 : ℝ) ≤
+      (-(4628955784060132778483 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7924356488063509 / 20000000000000000 : ℝ) ≤
+      (-(4628955784060132778483 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7924356488063509 / 20000000000000000 : ℝ) =
+            (7924356488063509 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7924356488063509 / 5000000000000000 : ℝ) ≤
+            (2302516018939867221517 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3961244656506009 / 10000000000000000 : ℝ)) (q :=
+      (7924356488063509 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_403_4 :
+    Uω (aρ 4) (bρ 4) (1639509667401 / 4000000000000 : ℝ) ≤
+      (-(9462068786104314284357 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1941053822962141 / 5000000000000000 : ℝ) ≤
+      (-(9462068786104314284357 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1941053822962141 / 5000000000000000 : ℝ) =
+            (1941053822962141 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1941053822962141 / 1250000000000000 : ℝ) ≤
+            (4400874819895685715643 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (121231176292377 / 312500000000000 : ℝ)) (q :=
+      (1941053822962141 / 5000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_403_5 :
+    Uω (aρ 5) (bρ 5) (1639509667401 / 4000000000000 : ℝ) ≤
+      (-(4891435975477487785923 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7519089683728511 / 20000000000000000 : ℝ) ≤
+      (-(4891435975477487785923 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7519089683728511 / 20000000000000000 : ℝ) =
+            (7519089683728511 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7519089683728511 / 5000000000000000 : ℝ) ≤
+            (2040035827522512214077 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3752461952571011 / 10000000000000000 : ℝ)) (q :=
+      (7519089683728511 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_403_6 :
+    Uω (aρ 6) (bρ 6) (1639509667401 / 4000000000000 : ℝ) ≤
+      (-(10265917141164475357851 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1432903272379271 / 4000000000000000 : ℝ) ≤
+      (-(10265917141164475357851 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1432903272379271 / 4000000000000000 : ℝ) =
+            (1432903272379271 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1432903272379271 / 1000000000000000 : ℝ) ≤
+            (3597026464835524642149 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (713025801175771 / 2000000000000000 : ℝ)) (q :=
+      (1432903272379271 / 4000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_403_7 :
+    Uω (aρ 7) (bρ 7) (1639509667401 / 4000000000000 : ℝ) ≤
+      (-(1371941851203687294137 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3336864559731609 / 10000000000000000 : ℝ) ≤
+      (-(1371941851203687294137 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3336864559731609 / 10000000000000000 : ℝ) =
+            (3336864559731609 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3336864559731609 / 2500000000000000 : ℝ) ≤
+            (360926099546312705863 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1649001572102859 / 5000000000000000 : ℝ)) (q :=
+      (3336864559731609 / 10000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_403_8 :
+    Uω (aρ 8) (bρ 8) (1639509667401 / 4000000000000 : ℝ) ≤
+      (-(12011974165207489099981 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6016675453428553 / 20000000000000000 : ℝ) ≤
+      (-(12011974165207489099981 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6016675453428553 / 20000000000000000 : ℝ) =
+            (6016675453428553 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6016675453428553 / 5000000000000000 : ℝ) ≤
+            (1850969440792510900019 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2924491129511053 / 10000000000000000 : ℝ)) (q :=
+      (6016675453428553 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_403_9 :
+    Uω (aρ 9) (bρ 9) (1639509667401 / 4000000000000 : ℝ) ≤
+      (-(1695737679398358110773 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (103014978817541 / 400000000000000 : ℝ) ≤
+      (-(1695737679398358110773 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (103014978817541 / 400000000000000 : ℝ) =
+            (103014978817541 / 100000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (103014978817541 / 100000000000000 : ℝ) ≤
+            (37130271351641889227 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (47990604906391 / 200000000000000 : ℝ)) (q :=
+      (103014978817541 / 400000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_403_10 :
+    Uω (aρ 10) (bρ 10) (1639509667401 / 4000000000000 : ℝ) ≤
+      (-(647126960665450513509 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3966622080239831 / 20000000000000000 : ℝ) ≤
+      (-(647126960665450513509 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3966622080239831 / 20000000000000000 : ℝ) =
+            (3966622080239831 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3966622080239831 / 2500000000000000 : ℝ) ≤
+            (184649655694549486491 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1603962418177331 / 10000000000000000 : ℝ)) (q :=
+      (3966622080239831 / 20000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_403_11 :
+    Uω (aρ 11) (bρ 11) (1639509667401 / 4000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_403_12 :
+    Uω (aρ 12) (bρ 12) (1639509667401 / 4000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_403_13 :
+    Uω (aρ 13) (bρ 13) (1639509667401 / 4000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_403_14 :
+    Uω (aρ 14) (bρ 14) (1639509667401 / 4000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_403_15 :
+    Uω (aρ 15) (bρ 15) (1639509667401 / 4000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_403_16 :
+    Uω (aρ 16) (bρ 16) (1639509667401 / 4000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_403 :
+    Uρ (1639509667401 / 4000000000000 : ℝ) ≤
+      (-(3278575728550199324133 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1639509667401 / 4000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(9077679346739136303621 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_403_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1639509667401 / 4000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(2284332897053214077443 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_403_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1639509667401 / 4000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(4628955784060132778483 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_403_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1639509667401 / 4000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(9462068786104314284357 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_403_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1639509667401 / 4000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(4891435975477487785923 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_403_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1639509667401 / 4000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(10265917141164475357851 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_403_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1639509667401 / 4000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(1371941851203687294137 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_403_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1639509667401 / 4000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(12011974165207489099981 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_403_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1639509667401 / 4000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(1695737679398358110773 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_403_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1639509667401 / 4000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) * (-(647126960665450513509 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_403_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1639509667401 / 4000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_403_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1639509667401 / 4000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_403_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1639509667401 / 4000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_403_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1639509667401 / 4000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_403_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1639509667401 / 4000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_403_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1639509667401 / 4000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_403_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_404_1 :
+    Uω (aρ 1) (bρ 1) (6641755819863 / 16000000000000 : ℝ) ≤
+      (-(8948814032234249353119 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8173121216923227 / 20000000000000000 : ℝ) ≤
+      (-(8948814032234249353119 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8173121216923227 / 20000000000000000 : ℝ) =
+            (8173121216923227 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8173121216923227 / 5000000000000000 : ℝ) ≤
+            (4914129573765750646881 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1021630261898463 / 2500000000000000 : ℝ)) (q :=
+      (8173121216923227 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_404_2 :
+    Uω (aρ 2) (bρ 2) (6641755819863 / 16000000000000 : ℝ) ≤
+      (-(1125961639650166118421 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (126955312814517 / 312500000000000 : ℝ) ≤
+      (-(1125961639650166118421 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (126955312814517 / 312500000000000 : ℝ) = (126955312814517 / 78125000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (126955312814517 / 78125000000000 : ℝ) ≤
+            (606906311099833881579 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4062308863309713 / 10000000000000000 : ℝ)) (q :=
+      (126955312814517 / 312500000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_404_3 :
+    Uω (aρ 3) (bρ 3) (6641755819863 / 16000000000000 : ℝ) ≤
+      (-(456334444861630339123 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1003628408424973 / 2500000000000000 : ℝ) ≤
+      (-(456334444861630339123 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1003628408424973 / 2500000000000000 : ℝ) =
+            (1003628408424973 / 625000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1003628408424973 / 625000000000000 : ℝ) ≤
+            (236812735438369660877 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4013592216930409 / 10000000000000000 : ℝ)) (q :=
+      (1003628408424973 / 2500000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_404_4 :
+    Uω (aρ 4) (bρ 4) (6641755819863 / 16000000000000 : ℝ) ≤
+      (-(9328096888636753907783 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (78689338583237 / 200000000000000 : ℝ) ≤
+      (-(9328096888636753907783 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (78689338583237 / 200000000000000 : ℝ) = (78689338583237 / 50000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (78689338583237 / 50000000000000 : ℝ) ≤
+            (4534846717363246092217 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (157271719556773 / 400000000000000 : ℝ)) (q :=
+      (78689338583237 / 200000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_404_5 :
+    Uω (aρ 5) (bρ 5) (6641755819863 / 16000000000000 : ℝ) ≤
+      (-(2411100417759986548877 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1905982730914291 / 5000000000000000 : ℝ) ≤
+      (-(2411100417759986548877 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1905982730914291 / 5000000000000000 : ℝ) =
+            (1905982730914291 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1905982730914291 / 1250000000000000 : ℝ) ≤
+            (1054635483740013451123 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3804979973587789 / 10000000000000000 : ℝ)) (q :=
+      (1905982730914291 / 5000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_404_6 :
+    Uω (aρ 6) (bρ 6) (6641755819863 / 16000000000000 : ℝ) ≤
+      (-(5060114932047977894949 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7269658282256159 / 20000000000000000 : ℝ) ≤
+      (-(5060114932047977894949 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7269658282256159 / 20000000000000000 : ℝ) =
+            (7269658282256159 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7269658282256159 / 5000000000000000 : ℝ) ≤
+            (1871356870952022105051 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (56530432926981 / 156250000000000 : ℝ)) (q :=
+      (7269658282256159 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_404_7 :
+    Uω (aρ 7) (bρ 7) (6641755819863 / 16000000000000 : ℝ) ≤
+      (-(10818157684744796127839 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (423724322632849 / 1250000000000000 : ℝ) ≤
+      (-(10818157684744796127839 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (423724322632849 / 1250000000000000 : ℝ) =
+            (423724322632849 / 312500000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (423724322632849 / 312500000000000 : ℝ) ≤
+            (3044785921255203872161 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3351539967956209 / 10000000000000000 : ℝ)) (q :=
+      (423724322632849 / 1250000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_404_8 :
+    Uω (aρ 8) (bρ 8) (6641755819863 / 16000000000000 : ℝ) ≤
+      (-(5917365200118181528387 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3062133981399583 / 10000000000000000 : ℝ) ≤
+      (-(5917365200118181528387 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3062133981399583 / 10000000000000000 : ℝ) =
+            (3062133981399583 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3062133981399583 / 2500000000000000 : ℝ) ≤
+            (1014106602881818471613 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2979760419969791 / 10000000000000000 : ℝ)) (q :=
+      (3062133981399583 / 10000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_404_9 :
+    Uω (aρ 9) (bρ 9) (6641755819863 / 16000000000000 : ℝ) ≤
+      (-(3337630368056965153349 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13157222635451 / 50000000000000 : ℝ) ≤
+      (-(3337630368056965153349 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (13157222635451 / 50000000000000 : ℝ) = (13157222635451 / 12500000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13157222635451 / 12500000000000 : ℝ) ≤
+            (128105533443034846651 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (98373885588441 / 400000000000000 : ℝ)) (q :=
+      (13157222635451 / 50000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_404_10 :
+    Uω (aρ 10) (bρ 10) (6641755819863 / 16000000000000 : ℝ) ≤
+      (-(15859495990828664169851 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4095065341923027 / 20000000000000000 : ℝ) ≤
+      (-(15859495990828664169851 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4095065341923027 / 20000000000000000 : ℝ) =
+            (4095065341923027 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4095065341923027 / 2500000000000000 : ℝ) ≤
+            (4934919418171335830149 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (420020615237163 / 2500000000000000 : ℝ)) (q :=
+      (4095065341923027 / 20000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_404_11 :
+    Uω (aρ 11) (bρ 11) (6641755819863 / 16000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_404_12 :
+    Uω (aρ 12) (bρ 12) (6641755819863 / 16000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_404_13 :
+    Uω (aρ 13) (bρ 13) (6641755819863 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_404_14 :
+    Uω (aρ 14) (bρ 14) (6641755819863 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_404_15 :
+    Uω (aρ 15) (bρ 15) (6641755819863 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_404_16 :
+    Uω (aρ 16) (bρ 16) (6641755819863 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_404 :
+    Uρ (6641755819863 / 16000000000000 : ℝ) ≤
+      (-(3249841497956135963179 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (6641755819863 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(8948814032234249353119 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_404_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (6641755819863 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1125961639650166118421 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_404_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (6641755819863 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(456334444861630339123 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_404_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (6641755819863 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(9328096888636753907783 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_404_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (6641755819863 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(2411100417759986548877 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_404_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (6641755819863 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(5060114932047977894949 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_404_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (6641755819863 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(10818157684744796127839 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_404_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (6641755819863 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(5917365200118181528387 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_404_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (6641755819863 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(3337630368056965153349 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_404_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (6641755819863 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(15859495990828664169851 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_404_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (6641755819863 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_404_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (6641755819863 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_404_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (6641755819863 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_404_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (6641755819863 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_404_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (6641755819863 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_404_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (6641755819863 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_404_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_405_1 :
+    Uω (aρ 1) (bρ 1) (3362736485061 / 8000000000000 : ℝ) ≤
+      (-(2205397067664007022223 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (66222149240023 / 160000000000000 : ℝ) ≤
+      (-(2205397067664007022223 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (66222149240023 / 160000000000000 : ℝ) = (66222149240023 / 40000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (66222149240023 / 40000000000000 : ℝ) ≤
+            (1260338833835992977777 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (33110762134093 / 80000000000000 : ℝ)) (q :=
+      (66222149240023 / 160000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_405_2 :
+    Uω (aρ 2) (bρ 2) (3362736485061 / 8000000000000 : ℝ) ≤
+      (-(4439856992194217320277 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (822979309962943 / 2000000000000000 : ℝ) ≤
+      (-(4439856992194217320277 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (822979309962943 / 2000000000000000 : ℝ) =
+            (822979309962943 / 500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (822979309962943 / 500000000000000 : ℝ) ≤
+            (2491614810805782679723 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (205731936194909 / 500000000000000 : ℝ)) (q :=
+      (822979309962943 / 2000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_405_3 :
+    Uω (aρ 3) (bρ 3) (3362736485061 / 8000000000000 : ℝ) ≤
+      (-(2249291664094980362187 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4066848710052361 / 10000000000000000 : ℝ) ≤
+      (-(2249291664094980362187 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4066848710052361 / 10000000000000000 : ℝ) =
+            (4066848710052361 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4066848710052361 / 2500000000000000 : ℝ) ≤
+            (1216444237405019637813 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (254121196920217 / 625000000000000 : ℝ)) (q :=
+      (4066848710052361 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_405_4 :
+    Uω (aρ 4) (bρ 4) (3362736485061 / 8000000000000 : ℝ) ≤
+      (-(143685914272745119657 / 156250000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7973650529002219 / 20000000000000000 : ℝ) ≤
+      (-(143685914272745119657 / 156250000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7973650529002219 / 20000000000000000 : ℝ) =
+            (7973650529002219 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7973650529002219 / 5000000000000000 : ℝ) ≤
+            (72922579571004880343 / 156250000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3984186440685969 / 10000000000000000 : ℝ)) (q :=
+      (7973650529002219 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_405_5 :
+    Uω (aρ 5) (bρ 5) (3362736485061 / 8000000000000 : ℝ) ≤
+      (-(4753914742488689626759 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3864383434570763 / 10000000000000000 : ℝ) ≤
+      (-(4753914742488689626759 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3864383434570763 / 10000000000000000 : ℝ) =
+            (3864383434570763 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3864383434570763 / 2500000000000000 : ℝ) ≤
+            (2177557060511310373241 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (964373175040069 / 2500000000000000 : ℝ)) (q :=
+      (3864383434570763 / 10000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_405_6 :
+    Uω (aρ 6) (bρ 6) (3362736485061 / 8000000000000 : ℝ) ≤
+      (-(2494163466899750630607 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1474957202092181 / 4000000000000000 : ℝ) ≤
+      (-(2494163466899750630607 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1474957202092181 / 4000000000000000 : ℝ) =
+            (1474957202092181 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1474957202092181 / 1000000000000000 : ℝ) ≤
+            (971572434600249369393 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (734150443323931 / 2000000000000000 : ℝ)) (q :=
+      (1474957202092181 / 4000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_405_7 :
+    Uω (aρ 7) (bρ 7) (3362736485061 / 8000000000000 : ℝ) ≤
+      (-(2665818446664646446537 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1721352868739247 / 5000000000000000 : ℝ) ≤
+      (-(2665818446664646446537 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1721352868739247 / 5000000000000000 : ℝ) =
+            (1721352868739247 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1721352868739247 / 1250000000000000 : ℝ) ≤
+            (799917454835353553463 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1702519530937869 / 5000000000000000 : ℝ)) (q :=
+      (1721352868739247 / 5000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_405_8 :
+    Uω (aρ 8) (bρ 8) (3362736485061 / 8000000000000 : ℝ) ≤
+      (-(11660741129451976871839 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6231756031451009 / 20000000000000000 : ℝ) ≤
+      (-(11660741129451976871839 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6231756031451009 / 20000000000000000 : ℝ) =
+            (6231756031451009 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6231756031451009 / 5000000000000000 : ℝ) ≤
+            (2202202476548023128161 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3034925269709759 / 10000000000000000 : ℝ)) (q :=
+      (6231756031451009 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_405_9 :
+    Uω (aρ 9) (bρ 9) (3362736485061 / 8000000000000 : ℝ) ≤
+      (-(13140303395378854275257 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5374695565840111 / 20000000000000000 : ℝ) ≤
+      (-(13140303395378854275257 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5374695565840111 / 20000000000000000 : ℝ) =
+            (5374695565840111 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5374695565840111 / 5000000000000000 : ℝ) ≤
+            (722640210621145724743 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2518830432458861 / 10000000000000000 : ℝ)) (q :=
+      (5374695565840111 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_405_10 :
+    Uω (aρ 10) (bρ 10) (3362736485061 / 8000000000000 : ℝ) ≤
+      (-(1555478437684198579321 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (844353462736983 / 4000000000000000 : ℝ) ≤
+      (-(1555478437684198579321 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (844353462736983 / 4000000000000000 : ℝ) =
+            (844353462736983 / 500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (844353462736983 / 500000000000000 : ℝ) ≤
+            (523963103215801420679 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (350892242759733 / 2000000000000000 : ℝ)) (q :=
+      (844353462736983 / 4000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10]) (by norm_num [bρ_10])
+      (by norm_num) (by norm_num [aρ_10, bρ_10]) (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_405_11 :
+    Uω (aρ 11) (bρ 11) (3362736485061 / 8000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_405_12 :
+    Uω (aρ 12) (bρ 12) (3362736485061 / 8000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_405_13 :
+    Uω (aρ 13) (bρ 13) (3362736485061 / 8000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_405_14 :
+    Uω (aρ 14) (bρ 14) (3362736485061 / 8000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_405_15 :
+    Uω (aρ 15) (bρ 15) (3362736485061 / 8000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_405_16 :
+    Uω (aρ 16) (bρ 16) (3362736485061 / 8000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_405 :
+    Uρ (3362736485061 / 8000000000000 : ℝ) ≤
+      (-(12887117814147303746673 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (3362736485061 / 8000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2205397067664007022223 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_405_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (3362736485061 / 8000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(4439856992194217320277 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_405_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (3362736485061 / 8000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(2249291664094980362187 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_405_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (3362736485061 / 8000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(143685914272745119657 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_405_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (3362736485061 / 8000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(4753914742488689626759 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_405_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (3362736485061 / 8000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(2494163466899750630607 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_405_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (3362736485061 / 8000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(2665818446664646446537 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_405_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (3362736485061 / 8000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(11660741129451976871839 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_405_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (3362736485061 / 8000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(13140303395378854275257 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_405_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (3362736485061 / 8000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(1555478437684198579321 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_405_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (3362736485061 / 8000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_405_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (3362736485061 / 8000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_405_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (3362736485061 / 8000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_405_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (3362736485061 / 8000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_405_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (3362736485061 / 8000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_405_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (3362736485061 / 8000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_405_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_406_1 :
+    Uω (aρ 1) (bρ 1) (6809190120381 / 16000000000000 : ℝ) ≤
+      (-(8695960864778360946079 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8382416068107581 / 20000000000000000 : ℝ) ≤
+      (-(8695960864778360946079 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8382416068107581 / 20000000000000000 : ℝ) =
+            (8382416068107581 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8382416068107581 / 5000000000000000 : ℝ) ≤
+            (5166982741221639053921 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (523896182619307 / 1250000000000000 : ℝ)) (q :=
+      (8382416068107581 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_406_2 :
+    Uω (aρ 2) (bρ 2) (6809190120381 / 16000000000000 : ℝ) ≤
+      (-(4376676121341284168073 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4167223006162299 / 10000000000000000 : ℝ) ≤
+      (-(4376676121341284168073 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4167223006162299 / 10000000000000000 : ℝ) =
+            (4167223006162299 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4167223006162299 / 2500000000000000 : ℝ) ≤
+            (2554795681658715831927 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4166968417681473 / 10000000000000000 : ℝ)) (q :=
+      (4167223006162299 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_406_3 :
+    Uω (aρ 3) (bρ 3) (6809190120381 / 16000000000000 : ℝ) ≤
+      (-(2217325329992838613491 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1029795871259173 / 2500000000000000 : ℝ) ≤
+      (-(2217325329992838613491 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1029795871259173 / 2500000000000000 : ℝ) =
+            (1029795871259173 / 625000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1029795871259173 / 625000000000000 : ℝ) ≤
+            (1248410571507161386509 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4118285481780259 / 10000000000000000 : ℝ)) (q :=
+      (1029795871259173 / 2500000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_406_4 :
+    Uω (aρ 4) (bρ 4) (6809190120381 / 16000000000000 : ℝ) ≤
+      (-(9065427257069945571199 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1615673075540913 / 4000000000000000 : ℝ) ≤
+      (-(9065427257069945571199 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1615673075540913 / 4000000000000000 : ℝ) =
+            (1615673075540913 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1615673075540913 / 1000000000000000 : ℝ) ≤
+            (4797516348930054428801 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (100914451761911 / 250000000000000 : ℝ)) (q :=
+      (1615673075540913 / 4000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_406_5 :
+    Uω (aρ 5) (bρ 5) (6809190120381 / 16000000000000 : ℝ) ≤
+      (-(4686551939544443826109 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (391679886675027 / 1000000000000000 : ℝ) ≤
+      (-(4686551939544443826109 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (391679886675027 / 1000000000000000 : ℝ) =
+            (391679886675027 / 250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (391679886675027 / 250000000000000 : ℝ) ≤
+            (2244919863455556173891 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (782000069121483 / 2000000000000000 : ℝ)) (q :=
+      (391679886675027 / 1000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_406_6 :
+    Uω (aρ 6) (bρ 6) (6809190120381 / 16000000000000 : ℝ) ≤
+      (-(9835128300274148067791 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7479900150298289 / 20000000000000000 : ℝ) ≤
+      (-(9835128300274148067791 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7479900150298289 / 20000000000000000 : ℝ) =
+            (7479900150298289 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7479900150298289 / 5000000000000000 : ℝ) ≤
+            (4027815305725851932209 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (930885784386291 / 2500000000000000 : ℝ)) (q :=
+      (7479900150298289 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_406_7 :
+    Uω (aρ 7) (bρ 7) (6809190120381 / 16000000000000 : ℝ) ≤
+      (-(5132228374587151913 / 4882812500000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6991197808870553 / 20000000000000000 : ℝ) ≤
+      (-(5132228374587151913 / 4882812500000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6991197808870553 / 20000000000000000 : ℝ) =
+            (6991197808870553 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6991197808870553 / 5000000000000000 : ℝ) ≤
+            (1636787058030035587 / 4882812500000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (864625544219357 / 2500000000000000 : ℝ)) (q :=
+      (6991197808870553 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_406_8 :
+    Uω (aρ 8) (bρ 8) (6809190120381 / 16000000000000 : ℝ) ≤
+      (-(11489883318043325304749 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6339145253052087 / 20000000000000000 : ℝ) ≤
+      (-(11489883318043325304749 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6339145253052087 / 20000000000000000 : ℝ) =
+            (6339145253052087 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6339145253052087 / 5000000000000000 : ℝ) ≤
+            (2373060287956674695251 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1544995636199481 / 5000000000000000 : ℝ)) (q :=
+      (6339145253052087 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_406_9 :
+    Uω (aρ 9) (bρ 9) (6809190120381 / 16000000000000 : ℝ) ≤
+      (-(404218113034351628443 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (68577394599119 / 250000000000000 : ℝ) ≤
+      (-(404218113034351628443 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (68577394599119 / 250000000000000 : ℝ) = (68577394599119 / 62500000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (68577394599119 / 62500000000000 : ℝ) ≤
+            (28998874653148371557 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (515600643127279 / 2000000000000000 : ℝ)) (q :=
+      (68577394599119 / 250000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_406_10 :
+    Uω (aρ 10) (bρ 10) (6809190120381 / 16000000000000 : ℝ) ≤
+      (-(7631299881357333164523 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2173470319964777 / 10000000000000000 : ℝ) ≤
+      (-(7631299881357333164523 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2173470319964777 / 10000000000000000 : ℝ) =
+            (2173470319964777 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2173470319964777 / 1250000000000000 : ℝ) ≤
+            (2765907823142666835477 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1827311321131429 / 10000000000000000 : ℝ)) (q :=
+      (2173470319964777 / 10000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_406_11 :
+    Uω (aρ 11) (bρ 11) (6809190120381 / 16000000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_406_12 :
+    Uω (aρ 12) (bρ 12) (6809190120381 / 16000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_406_13 :
+    Uω (aρ 13) (bρ 13) (6809190120381 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_406_14 :
+    Uω (aρ 14) (bρ 14) (6809190120381 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_406_15 :
+    Uω (aρ 15) (bρ 15) (6809190120381 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_406_16 :
+    Uω (aρ 16) (bρ 16) (6809190120381 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_406 :
+    Uρ (6809190120381 / 16000000000000 : ℝ) ≤
+      (-(399293109801970771631 / 312500000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (6809190120381 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(8695960864778360946079 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_406_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (6809190120381 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(4376676121341284168073 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_406_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (6809190120381 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(2217325329992838613491 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_406_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (6809190120381 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(9065427257069945571199 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_406_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (6809190120381 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(4686551939544443826109 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_406_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (6809190120381 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(9835128300274148067791 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_406_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (6809190120381 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) * (-(5132228374587151913 / 4882812500000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_406_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (6809190120381 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(11489883318043325304749 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_406_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (6809190120381 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(404218113034351628443 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_406_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (6809190120381 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(7631299881357333164523 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_406_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (6809190120381 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_406_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (6809190120381 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_406_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (6809190120381 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_406_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (6809190120381 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_406_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (6809190120381 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_406_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (6809190120381 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_406_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_407_1 :
+    Uω (aρ 1) (bρ 1) (86161340883 / 200000000000 : ℝ) ≤
+      (-(342875686036299859671 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21217658642903 / 50000000000000 : ℝ) ≤
+      (-(342875686036299859671 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (21217658642903 / 50000000000000 : ℝ) = (21217658642903 / 12500000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21217658642903 / 12500000000000 : ℝ) ≤
+            (211642058203700140329 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (21217468155481 / 50000000000000 : ℝ)) (q :=
+      (21217658642903 / 50000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_407_2 :
+    Uω (aρ 2) (bρ 2) (86161340883 / 200000000000 : ℝ) ≤
+      (-(1725713503188980663307 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (527443672776311 / 1250000000000000 : ℝ) ≤
+      (-(1725713503188980663307 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (527443672776311 / 1250000000000000 : ℝ) =
+            (527443672776311 / 312500000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (527443672776311 / 312500000000000 : ℝ) ≤
+            (1046875218011019336693 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (527412243858247 / 1250000000000000 : ℝ)) (q :=
+      (527443672776311 / 1250000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_407_3 :
+    Uω (aρ 3) (bρ 3) (86161340883 / 200000000000 : ℝ) ≤
+      (-(8743051013203827149901 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8343035940000919 / 20000000000000000 : ℝ) ≤
+      (-(8743051013203827149901 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8343035940000919 / 20000000000000000 : ℝ) =
+            (8343035940000919 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8343035940000919 / 5000000000000000 : ℝ) ≤
+            (5119892592796172850099 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4170631232795919 / 10000000000000000 : ℝ)) (q :=
+      (8343035940000919 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_407_4 :
+    Uω (aρ 4) (bρ 4) (86161340883 / 200000000000 : ℝ) ≤
+      (-(2234159628991301625199 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4091539237232629 / 10000000000000000 : ℝ) ≤
+      (-(2234159628991301625199 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4091539237232629 / 10000000000000000 : ℝ) =
+            (4091539237232629 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4091539237232629 / 2500000000000000 : ℝ) ≤
+            (1231576272508698374801 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2044483974162629 / 5000000000000000 : ℝ)) (q :=
+      (4091539237232629 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_407_5 :
+    Uω (aρ 5) (bρ 5) (86161340883 / 200000000000 : ℝ) ≤
+      (-(1848035083146948254919 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3969211859363087 / 10000000000000000 : ℝ) ≤
+      (-(1848035083146948254919 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3969211859363087 / 10000000000000000 : ℝ) =
+            (3969211859363087 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3969211859363087 / 2500000000000000 : ℝ) ≤
+            (924553638053051745081 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1981251555960587 / 5000000000000000 : ℝ)) (q :=
+      (3969211859363087 / 10000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_407_6 :
+    Uω (aρ 6) (bρ 6) (86161340883 / 200000000000 : ℝ) ≤
+      (-(1939118985162720111611 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7585001271642383 / 20000000000000000 : ℝ) ≤
+      (-(1939118985162720111611 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7585001271642383 / 20000000000000000 : ℝ) =
+            (7585001271642383 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7585001271642383 / 5000000000000000 : ℝ) ≤
+            (833469736037279888389 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3776321039977383 / 10000000000000000 : ℝ)) (q :=
+      (7585001271642383 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_407_7 :
+    Uω (aρ 7) (bρ 7) (86161340883 / 200000000000 : ℝ) ≤
+      (-(10360671859386552173789 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7096949807018291 / 20000000000000000 : ℝ) ≤
+      (-(10360671859386552173789 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7096949807018291 / 20000000000000000 : ℝ) =
+            (7096949807018291 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7096949807018291 / 5000000000000000 : ℝ) ≤
+            (3502271746613447826211 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3511930956113291 / 10000000000000000 : ℝ)) (q :=
+      (7096949807018291 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_407_8 :
+    Uω (aρ 8) (bρ 8) (86161340883 / 200000000000 : ℝ) ≤
+      (-(2264408210820016486429 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (805805102478053 / 2500000000000000 : ℝ) ≤
+      (-(2264408210820016486429 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (805805102478053 / 2500000000000000 : ℝ) =
+            (805805102478053 / 625000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (805805102478053 / 625000000000000 : ℝ) ≤
+            (508180510379983513571 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (98280113133107 / 312500000000000 : ℝ)) (q :=
+      (805805102478053 / 2500000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_407_9 :
+    Uω (aρ 9) (bρ 9) (86161340883 / 200000000000 : ℝ) ≤
+      (-(12734304324888259626497 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (699674745550947 / 2500000000000000 : ℝ) ≤
+      (-(12734304324888259626497 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (699674745550947 / 2500000000000000 : ℝ) =
+            (699674745550947 / 625000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (699674745550947 / 625000000000000 : ℝ) ≤
+            (1128639281111740373503 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (164805399575161 / 625000000000000 : ℝ)) (q :=
+      (699674745550947 / 2500000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_407_10 :
+    Uω (aρ 10) (bρ 10) (86161340883 / 200000000000 : ℝ) ≤
+      (-(3745433928253882754121 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (558845159230569 / 2500000000000000 : ℝ) ≤
+      (-(3745433928253882754121 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (558845159230569 / 2500000000000000 : ℝ) =
+            (558845159230569 / 312500000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (558845159230569 / 312500000000000 : ℝ) ≤
+            (1453169923996117245879 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (237351092016819 / 1250000000000000 : ℝ)) (q :=
+      (558845159230569 / 2500000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10]) (by norm_num [bρ_10])
+      (by norm_num) (by norm_num [aρ_10, bρ_10]) (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_407_11 :
+    Uω (aρ 11) (bρ 11) (86161340883 / 200000000000 : ℝ) ≤
+      (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_11]) (by norm_num [bρ_11])]; exact Lmid_11
+
+lemma U_407_12 :
+    Uω (aρ 12) (bρ 12) (86161340883 / 200000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_407_13 :
+    Uω (aρ 13) (bρ 13) (86161340883 / 200000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_407_14 :
+    Uω (aρ 14) (bρ 14) (86161340883 / 200000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_407_15 :
+    Uω (aρ 15) (bρ 15) (86161340883 / 200000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_407_16 :
+    Uω (aρ 16) (bρ 16) (86161340883 / 200000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_407 :
+    Uρ (86161340883 / 200000000000 : ℝ) ≤
+      (-(791874724932342320321 / 625000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (86161340883 / 200000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(342875686036299859671 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_407_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (86161340883 / 200000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1725713503188980663307 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_407_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (86161340883 / 200000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(8743051013203827149901 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_407_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (86161340883 / 200000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(2234159628991301625199 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_407_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (86161340883 / 200000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1848035083146948254919 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_407_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (86161340883 / 200000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1939118985162720111611 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_407_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (86161340883 / 200000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(10360671859386552173789 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_407_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (86161340883 / 200000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(2264408210820016486429 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_407_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (86161340883 / 200000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(12734304324888259626497 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_407_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (86161340883 / 200000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(3745433928253882754121 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_407_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (86161340883 / 200000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4457298065613116122159 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_407_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (86161340883 / 200000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_407_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (86161340883 / 200000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_407_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (86161340883 / 200000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_407_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (86161340883 / 200000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_407_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (86161340883 / 200000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_407_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_408_1 :
+    Uω (aρ 1) (bρ 1) (54181913021 / 125000000000 : ℝ) ≤
+      (-(1701934117171983693411 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4270017962423161 / 10000000000000000 : ℝ) ≤
+      (-(1701934117171983693411 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4270017962423161 / 10000000000000000 : ℝ) =
+            (4270017962423161 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4270017962423161 / 2500000000000000 : ℝ) ≤
+            (1070654604028016306589 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2134990050625661 / 5000000000000000 : ℝ)) (q :=
+      (4270017962423161 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_408_2 :
+    Uω (aρ 2) (bρ 2) (54181913021 / 125000000000 : ℝ) ≤
+      (-(535374392089433119633 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4246036948215849 / 10000000000000000 : ℝ) ≤
+      (-(535374392089433119633 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4246036948215849 / 10000000000000000 : ℝ) =
+            (4246036948215849 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4246036948215849 / 2500000000000000 : ℝ) ≤
+            (331059583285566880367 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2122893542673349 / 5000000000000000 : ℝ)) (q :=
+      (4246036948215849 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_408_3 :
+    Uω (aρ 3) (bρ 3) (54181913021 / 125000000000 : ℝ) ≤
+      (-(867974593018045612053 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (524751195412123 / 1250000000000000 : ℝ) ≤
+      (-(867974593018045612053 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (524751195412123 / 1250000000000000 : ℝ) =
+            (524751195412123 / 312500000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (524751195412123 / 312500000000000 : ℝ) ≤
+            (518319767581954387947 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (524641052732371 / 1250000000000000 : ℝ)) (q :=
+      (524751195412123 / 1250000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_408_4 :
+    Uω (aρ 4) (bρ 4) (54181913021 / 125000000000 : ℝ) ≤
+      (-(8872073375269568597809 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2059020891423341 / 5000000000000000 : ℝ) ≤
+      (-(8872073375269568597809 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2059020891423341 / 5000000000000000 : ℝ) =
+            (2059020891423341 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2059020891423341 / 1250000000000000 : ℝ) ≤
+            (4990870230730431402191 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1028871760505841 / 2500000000000000 : ℝ)) (q :=
+      (2059020891423341 / 5000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_408_5 :
+    Uω (aρ 5) (bρ 5) (54181913021 / 125000000000 : ℝ) ≤
+      (-(917355697984571850539 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (499467800122701 / 1250000000000000 : ℝ) ≤
+      (-(917355697984571850539 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (499467800122701 / 1250000000000000 : ℝ) =
+            (499467800122701 / 312500000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (499467800122701 / 312500000000000 : ℝ) ≤
+            (468938662615428149461 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (498634774703527 / 1250000000000000 : ℝ)) (q :=
+      (499467800122701 / 1250000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_408_6 :
+    Uω (aρ 6) (bρ 6) (54181913021 / 125000000000 : ℝ) ≤
+      (-(1925140956285161018611 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (238693707408891 / 625000000000000 : ℝ) ≤
+      (-(1925140956285161018611 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (238693707408891 / 625000000000000 : ℝ) =
+            (238693707408891 / 156250000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (238693707408891 / 156250000000000 : ℝ) ≤
+            (847447764914838981389 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (475379050986189 / 1250000000000000 : ℝ)) (q :=
+      (238693707408891 / 625000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_408_7 :
+    Uω (aρ 7) (bρ 7) (54181913021 / 125000000000 : ℝ) ≤
+      (-(10285543461370157781439 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7150468842237723 / 20000000000000000 : ℝ) ≤
+      (-(10285543461370157781439 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7150468842237723 / 20000000000000000 : ℝ) =
+            (7150468842237723 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7150468842237723 / 5000000000000000 : ℝ) ≤
+            (3577400144629842218561 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3538963993802723 / 10000000000000000 : ℝ)) (q :=
+      (7150468842237723 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_408_8 :
+    Uω (aρ 8) (bρ 8) (54181913021 / 125000000000 : ℝ) ≤
+      (-(112381938674848357789 / 100000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (162517991270787 / 500000000000000 : ℝ) ≤
+      (-(112381938674848357789 / 100000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (162517991270787 / 500000000000000 : ℝ) =
+            (162517991270787 / 125000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (162517991270787 / 125000000000000 : ℝ) ≤
+            (26247497385151642211 / 100000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (19829727835853 / 62500000000000 : ℝ)) (q :=
+      (162517991270787 / 500000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_408_9 :
+    Uω (aρ 9) (bρ 9) (54181913021 / 125000000000 : ℝ) ≤
+      (-(12634421825163466454341 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2826793159500669 / 10000000000000000 : ℝ) ≤
+      (-(12634421825163466454341 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2826793159500669 / 10000000000000000 : ℝ) =
+            (2826793159500669 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2826793159500669 / 2500000000000000 : ℝ) ≤
+            (1228521780836533545659 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1333294375133169 / 5000000000000000 : ℝ)) (q :=
+      (2826793159500669 / 10000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_408_10 :
+    Uω (aρ 10) (bρ 10) (54181913021 / 125000000000 : ℝ) ≤
+      (-(7421773000218264526129 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4532971445688459 / 20000000000000000 : ℝ) ≤
+      (-(7421773000218264526129 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4532971445688459 / 20000000000000000 : ℝ) =
+            (4532971445688459 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4532971445688459 / 2500000000000000 : ℝ) ≤
+            (2975434704281735473871 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1934532910448459 / 10000000000000000 : ℝ)) (q :=
+      (4532971445688459 / 20000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_408_11 :
+    Uω (aρ 11) (bρ 11) (54181913021 / 125000000000 : ℝ) ≤
+      (-(10359853335001487342957 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2518747126261537 / 20000000000000000 : ℝ) ≤
+      (-(10359853335001487342957 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2518747126261537 / 20000000000000000 : ℝ) =
+            (2518747126261537 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2518747126261537 / 2500000000000000 : ℝ) ≤
+            (37354369498512657043 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (338785217226537 / 10000000000000000 : ℝ)) (q :=
+      (2518747126261537 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_408_12 :
+    Uω (aρ 12) (bρ 12) (54181913021 / 125000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_408_13 :
+    Uω (aρ 13) (bρ 13) (54181913021 / 125000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_408_14 :
+    Uω (aρ 14) (bρ 14) (54181913021 / 125000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_408_15 :
+    Uω (aρ 15) (bρ 15) (54181913021 / 125000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_408_16 :
+    Uω (aρ 16) (bρ 16) (54181913021 / 125000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_408 :
+    Uρ (54181913021 / 125000000000 : ℝ) ≤
+      (-(12492873305617532369633 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (54181913021 / 125000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1701934117171983693411 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_408_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (54181913021 / 125000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(535374392089433119633 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_408_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (54181913021 / 125000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(867974593018045612053 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_408_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (54181913021 / 125000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(8872073375269568597809 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_408_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (54181913021 / 125000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(917355697984571850539 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_408_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (54181913021 / 125000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1925140956285161018611 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_408_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (54181913021 / 125000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(10285543461370157781439 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_408_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (54181913021 / 125000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(112381938674848357789 / 100000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_408_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (54181913021 / 125000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(12634421825163466454341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_408_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (54181913021 / 125000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(7421773000218264526129 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_408_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (54181913021 / 125000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(10359853335001487342957 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_408_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (54181913021 / 125000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_408_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (54181913021 / 125000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_408_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (54181913021 / 125000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_408_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (54181913021 / 125000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_408_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (54181913021 / 125000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_408_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_409_1 :
+    Uω (aρ 1) (bρ 1) (436103903921 / 1000000000000 : ℝ) ≤
+      (-(8447833787109033449899 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8593008386704303 / 20000000000000000 : ℝ) ≤
+      (-(8447833787109033449899 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8593008386704303 / 20000000000000000 : ℝ) =
+            (8593008386704303 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8593008386704303 / 5000000000000000 : ℝ) ≤
+            (5415109818890966550101 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4296466565579303 / 10000000000000000 : ℝ)) (q :=
+      (8593008386704303 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_409_2 :
+    Uω (aρ 2) (bρ 2) (436103903921 / 1000000000000 : ℝ) ≤
+      (-(531487639599329723427 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1068131123693129 / 2500000000000000 : ℝ) ≤
+      (-(531487639599329723427 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1068131123693129 / 2500000000000000 : ℝ) =
+            (1068131123693129 / 625000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1068131123693129 / 625000000000000 : ℝ) ≤
+            (334946335775670276573 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (267017261308127 / 625000000000000 : ℝ)) (q :=
+      (1068131123693129 / 2500000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_409_3 :
+    Uω (aρ 3) (bρ 3) (436103903921 / 1000000000000 : ℝ) ≤
+      (-(4308419623487413080987 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8449002172795053 / 20000000000000000 : ℝ) ≤
+      (-(4308419623487413080987 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8449002172795053 / 20000000000000000 : ℝ) =
+            (8449002172795053 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8449002172795053 / 5000000000000000 : ℝ) ≤
+            (2623052179512586919013 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4223625470530053 / 10000000000000000 : ℝ)) (q :=
+      (8449002172795053 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_409_4 :
+    Uω (aρ 4) (bρ 4) (436103903921 / 1000000000000 : ℝ) ≤
+      (-(352316917537988224177 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8289088233391073 / 20000000000000000 : ℝ) ≤
+      (-(352316917537988224177 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8289088233391073 / 20000000000000000 : ℝ) =
+            (8289088233391073 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8289088233391073 / 5000000000000000 : ℝ) ≤
+            (202200826702011775823 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4142005712191073 / 10000000000000000 : ℝ)) (q :=
+      (8289088233391073 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_409_5 :
+    Uω (aρ 5) (bρ 5) (436103903921 / 1000000000000 : ℝ) ≤
+      (-(9107380873511893660103 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8044544708027943 / 20000000000000000 : ℝ) ≤
+      (-(9107380873511893660103 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8044544708027943 / 20000000000000000 : ℝ) =
+            (8044544708027943 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8044544708027943 / 5000000000000000 : ℝ) ≤
+            (4755562732488106339897 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4015652106162943 / 10000000000000000 : ℝ)) (q :=
+      (8044544708027943 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_409_6 :
+    Uω (aρ 6) (bρ 6) (436103903921 / 1000000000000 : ℝ) ≤
+      (-(9556303782198983042833 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3845696435999927 / 10000000000000000 : ℝ) ≤
+      (-(9556303782198983042833 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3845696435999927 / 10000000000000000 : ℝ) =
+            (3845696435999927 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3845696435999927 / 2500000000000000 : ℝ) ≤
+            (4306639823801016957167 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1914870322637427 / 5000000000000000 : ℝ)) (q :=
+      (3845696435999927 / 10000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_409_7 :
+    Uω (aρ 7) (bρ 7) (436103903921 / 1000000000000 : ℝ) ≤
+      (-(10210986681307664216749 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7203979667384347 / 20000000000000000 : ℝ) ≤
+      (-(10210986681307664216749 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7203979667384347 / 20000000000000000 : ℝ) =
+            (7203979667384347 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7203979667384347 / 5000000000000000 : ℝ) ≤
+            (3651956924692335783251 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3565988821419347 / 10000000000000000 : ℝ)) (q :=
+      (7203979667384347 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_409_8 :
+    Uω (aρ 8) (bρ 8) (436103903921 / 1000000000000 : ℝ) ≤
+      (-(11155077685116505444933 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6554976318838763 / 20000000000000000 : ℝ) ≤
+      (-(11155077685116505444933 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6554976318838763 / 20000000000000000 : ℝ) =
+            (6554976318838763 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6554976318838763 / 5000000000000000 : ℝ) ≤
+            (2707865920883494555067 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3200527124213763 / 10000000000000000 : ℝ)) (q :=
+      (6554976318838763 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_409_9 :
+    Uω (aρ 9) (bρ 9) (436103903921 / 1000000000000 : ℝ) ≤
+      (-(12535644509160010633761 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5709707646635067 / 20000000000000000 : ℝ) ≤
+      (-(12535644509160010633761 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5709707646635067 / 20000000000000000 : ℝ) =
+            (5709707646635067 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5709707646635067 / 5000000000000000 : ℝ) ≤
+            (1327299096839989366239 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2696224080370067 / 10000000000000000 : ℝ)) (q :=
+      (5709707646635067 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_409_10 :
+    Uω (aρ 10) (bρ 10) (436103903921 / 1000000000000 : ℝ) ≤
+      (-(14707874772608179319249 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (918977980753321 / 4000000000000000 : ℝ) ≤
+      (-(14707874772608179319249 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (918977980753321 / 4000000000000000 : ℝ) =
+            (918977980753321 / 500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (918977980753321 / 500000000000000 : ℝ) ≤
+            (6086540636391820680751 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (393993074199321 / 2000000000000000 : ℝ)) (q :=
+      (918977980753321 / 4000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10]) (by norm_num [bρ_10])
+      (by norm_num) (by norm_num [aρ_10, bρ_10]) (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_409_11 :
+    Uω (aρ 11) (bρ 11) (436103903921 / 1000000000000 : ℝ) ≤
+      (-(10036489124495165724449 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (671756125206833 / 5000000000000000 : ℝ) ≤
+      (-(10036489124495165724449 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (671756125206833 / 5000000000000000 : ℝ) =
+            (671756125206833 / 625000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (671756125206833 / 625000000000000 : ℝ) ≤
+            (360718580004834275551 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (120144148565583 / 2500000000000000 : ℝ)) (q :=
+      (671756125206833 / 5000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11]) (by norm_num [bρ_11])
+      (by norm_num) (by norm_num [aρ_11, bρ_11]) (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_409_12 :
+    Uω (aρ 12) (bρ 12) (436103903921 / 1000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_409_13 :
+    Uω (aρ 13) (bρ 13) (436103903921 / 1000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_409_14 :
+    Uω (aρ 14) (bρ 14) (436103903921 / 1000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_409_15 :
+    Uω (aρ 15) (bρ 15) (436103903921 / 1000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_409_16 :
+    Uω (aρ 16) (bρ 16) (436103903921 / 1000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_409 :
+    Uρ (436103903921 / 1000000000000 : ℝ) ≤
+      (-(6194447236115731929341 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (436103903921 / 1000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(8447833787109033449899 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_409_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (436103903921 / 1000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(531487639599329723427 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_409_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (436103903921 / 1000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(4308419623487413080987 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_409_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (436103903921 / 1000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(352316917537988224177 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_409_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (436103903921 / 1000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(9107380873511893660103 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_409_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (436103903921 / 1000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(9556303782198983042833 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_409_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (436103903921 / 1000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(10210986681307664216749 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_409_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (436103903921 / 1000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(11155077685116505444933 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_409_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (436103903921 / 1000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(12535644509160010633761 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_409_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (436103903921 / 1000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(14707874772608179319249 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_409_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (436103903921 / 1000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(10036489124495165724449 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_409_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (436103903921 / 1000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_409_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (436103903921 / 1000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_409_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (436103903921 / 1000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_409_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (436103903921 / 1000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_409_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (436103903921 / 1000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_409_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_410_1 :
+    Uω (aρ 1) (bρ 1) (219376251837 / 500000000000 : ℝ) ≤
+      (-(2096594256290114153041 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (34583923371369 / 80000000000000 : ℝ) ≤
+      (-(2096594256290114153041 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (34583923371369 / 80000000000000 : ℝ) = (34583923371369 / 20000000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (34583923371369 / 20000000000000 : ℝ) ≤
+            (1369141645209885846959 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (17291812096749 / 40000000000000 : ℝ)) (q :=
+      (34583923371369 / 80000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_410_2 :
+    Uω (aρ 2) (bρ 2) (219376251837 / 500000000000 : ℝ) ≤
+      (-(1055249823054743877783 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8598024044480041 / 20000000000000000 : ℝ) ≤
+      (-(1055249823054743877783 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8598024044480041 / 20000000000000000 : ℝ) =
+            (8598024044480041 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8598024044480041 / 5000000000000000 : ℝ) ≤
+            (677618127695256122217 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4298765238335041 / 10000000000000000 : ℝ)) (q :=
+      (8598024044480041 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_410_3 :
+    Uω (aρ 3) (bρ 3) (219376251837 / 500000000000 : ℝ) ≤
+      (-(4277162989177892952323 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4250992540614981 / 10000000000000000 : ℝ) ≤
+      (-(4277162989177892952323 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4250992540614981 / 10000000000000000 : ℝ) =
+            (4250992540614981 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4250992540614981 / 2500000000000000 : ℝ) ≤
+            (2654308813822107047677 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2125061190717481 / 5000000000000000 : ℝ)) (q :=
+      (4250992540614981 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_410_4 :
+    Uω (aρ 4) (bρ 4) (219376251837 / 500000000000 : ℝ) ≤
+      (-(4372090952870243498743 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2085523121410359 / 5000000000000000 : ℝ) ≤
+      (-(4372090952870243498743 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2085523121410359 / 5000000000000000 : ℝ) =
+            (2085523121410359 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2085523121410359 / 1250000000000000 : ℝ) ≤
+            (2559380850129756501257 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1042130991727859 / 2500000000000000 : ℝ)) (q :=
+      (2085523121410359 / 5000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_410_5 :
+    Uω (aρ 5) (bρ 5) (219376251837 / 500000000000 : ℝ) ≤
+      (-(904164124252520286273 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8097603460136829 / 20000000000000000 : ℝ) ≤
+      (-(904164124252520286273 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8097603460136829 / 20000000000000000 : ℝ) =
+            (8097603460136829 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8097603460136829 / 5000000000000000 : ℝ) ≤
+            (482130236347479713727 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4042224860741829 / 10000000000000000 : ℝ)) (q :=
+      (8097603460136829 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_410_6 :
+    Uω (aρ 6) (bρ 6) (219376251837 / 500000000000 : ℝ) ≤
+      (-(9487385073304664494243 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7744584041430827 / 20000000000000000 : ℝ) ≤
+      (-(9487385073304664494243 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7744584041430827 / 20000000000000000 : ℝ) =
+            (7744584041430827 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7744584041430827 / 5000000000000000 : ℝ) ≤
+            (4375558532695335505757 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3856445817175827 / 10000000000000000 : ℝ)) (q :=
+      (7744584041430827 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_410_7 :
+    Uω (aρ 7) (bρ 7) (219376251837 / 500000000000 : ℝ) ≤
+      (-(5068496359337917405659 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7257482467714601 / 20000000000000000 : ℝ) ≤
+      (-(5068496359337917405659 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7257482467714601 / 20000000000000000 : ℝ) =
+            (7257482467714601 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7257482467714601 / 5000000000000000 : ℝ) ≤
+            (1862975443662082594341 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3593005624219601 / 10000000000000000 : ℝ)) (q :=
+      (7257482467714601 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_410_8 :
+    Uω (aρ 8) (bρ 8) (219376251837 / 500000000000 : ℝ) ≤
+      (-(11072679310070797156689 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6609211395806087 / 20000000000000000 : ℝ) ≤
+      (-(11072679310070797156689 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6609211395806087 / 20000000000000000 : ℝ) =
+            (6609211395806087 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6609211395806087 / 5000000000000000 : ℝ) ≤
+            (2790264295929202843311 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3228276203651087 / 10000000000000000 : ℝ)) (q :=
+      (6609211395806087 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_410_9 :
+    Uω (aρ 9) (bρ 9) (219376251837 / 500000000000 : ℝ) ≤
+      (-(6218972905446657801553 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1441441033374307 / 5000000000000000 : ℝ) ≤
+      (-(6218972905446657801553 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1441441033374307 / 5000000000000000 : ℝ) =
+            (1441441033374307 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1441441033374307 / 1250000000000000 : ℝ) ≤
+            (712498897553342198447 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (681448642425557 / 2500000000000000 : ℝ)) (q :=
+      (1441441033374307 / 5000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_410_10 :
+    Uω (aρ 10) (bρ 10) (219376251837 / 500000000000 : ℝ) ≤
+      (-(14574612834924164906609 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (582066514092719 / 2500000000000000 : ℝ) ≤
+      (-(14574612834924164906609 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (582066514092719 / 2500000000000000 : ℝ) =
+            (582066514092719 / 312500000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (582066514092719 / 312500000000000 : ℝ) ≤
+            (6219802574075835093391 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (250640197805219 / 1250000000000000 : ℝ)) (q :=
+      (582066514092719 / 2500000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10]) (by norm_num [bρ_10])
+      (by norm_num) (by norm_num [aρ_10, bρ_10]) (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_410_11 :
+    Uω (aρ 11) (bρ 11) (219376251837 / 500000000000 : ℝ) ≤
+      (-(611820281831991679449 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1411651351459397 / 10000000000000000 : ℝ) ≤
+      (-(611820281831991679449 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1411651351459397 / 10000000000000000 : ℝ) =
+            (1411651351459397 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1411651351459397 / 1250000000000000 : ℝ) ≤
+            (38005199699258320551 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (295184399411897 / 5000000000000000 : ℝ)) (q :=
+      (1411651351459397 / 10000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_410_12 :
+    Uω (aρ 12) (bρ 12) (219376251837 / 500000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_410_13 :
+    Uω (aρ 13) (bρ 13) (219376251837 / 500000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_410_14 :
+    Uω (aρ 14) (bρ 14) (219376251837 / 500000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_410_15 :
+    Uω (aρ 15) (bρ 15) (219376251837 / 500000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_410_16 :
+    Uω (aρ 16) (bρ 16) (219376251837 / 500000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_410 :
+    Uρ (219376251837 / 500000000000 : ℝ) ≤
+      (-(12297444852051932068801 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (219376251837 / 500000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2096594256290114153041 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_410_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (219376251837 / 500000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1055249823054743877783 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_410_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (219376251837 / 500000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(4277162989177892952323 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_410_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (219376251837 / 500000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(4372090952870243498743 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_410_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (219376251837 / 500000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(904164124252520286273 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_410_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (219376251837 / 500000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(9487385073304664494243 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_410_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (219376251837 / 500000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(5068496359337917405659 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_410_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (219376251837 / 500000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(11072679310070797156689 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_410_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (219376251837 / 500000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(6218972905446657801553 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_410_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (219376251837 / 500000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(14574612834924164906609 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_410_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (219376251837 / 500000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) * (-(611820281831991679449 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_410_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (219376251837 / 500000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_410_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (219376251837 / 500000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_410_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (219376251837 / 500000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_410_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (219376251837 / 500000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_410_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (219376251837 / 500000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_410_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_411_1 :
+    Uω (aρ 1) (bρ 1) (880153607101 / 2000000000000 : ℝ) ≤
+      (-(8355789703777010327123 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4336233534399483 / 10000000000000000 : ℝ) ≤
+      (-(8355789703777010327123 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4336233534399483 / 10000000000000000 : ℝ) =
+            (4336233534399483 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4336233534399483 / 2500000000000000 : ℝ) ≤
+            (5507153902222989672877 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2168098125689483 / 5000000000000000 : ℝ)) (q :=
+      (4336233534399483 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_411_2 :
+    Uω (aρ 2) (bρ 2) (880153607101 / 2000000000000 : ℝ) ≤
+      (-(1682247885388241367499 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8624511557850547 / 20000000000000000 : ℝ) ≤
+      (-(1682247885388241367499 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8624511557850547 / 20000000000000000 : ℝ) =
+            (8624511557850547 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8624511557850547 / 5000000000000000 : ℝ) ≤
+            (1090340835811758632501 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4312009752940547 / 10000000000000000 : ℝ)) (q :=
+      (8624511557850547 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_411_3 :
+    Uω (aρ 3) (bρ 3) (880153607101 / 2000000000000 : ℝ) ≤
+      (-(4261607671066074145483 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4264238242293903 / 10000000000000000 : ℝ) ≤
+      (-(4261607671066074145483 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4264238242293903 / 10000000000000000 : ℝ) =
+            (4264238242293903 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4264238242293903 / 2500000000000000 : ℝ) ≤
+            (2669864131933925854517 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2131685393013903 / 5000000000000000 : ℝ)) (q :=
+      (4264238242293903 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_411_4 :
+    Uω (aρ 4) (bρ 4) (880153607101 / 2000000000000 : ℝ) ≤
+      (-(2178115821864365629041 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4184297229221859 / 10000000000000000 : ℝ) ≤
+      (-(2178115821864365629041 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4184297229221859 / 10000000000000000 : ℝ) =
+            (4184297229221859 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4184297229221859 / 2500000000000000 : ℝ) ≤
+            (1287620079635634370959 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2090891470474359 / 5000000000000000 : ℝ)) (q :=
+      (4184297229221859 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_411_5 :
+    Uω (aρ 5) (bρ 5) (880153607101 / 2000000000000 : ℝ) ≤
+      (-(9008933307601673657931 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2031033102636581 / 5000000000000000 : ℝ) ≤
+      (-(9008933307601673657931 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2031033102636581 / 5000000000000000 : ℝ) =
+            (2031033102636581 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2031033102636581 / 1250000000000000 : ℝ) ≤
+            (4854010298398326342069 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1013877703096581 / 2500000000000000 : ℝ)) (q :=
+      (2031033102636581 / 5000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_411_6 :
+    Uω (aρ 6) (bρ 6) (880153607101 / 2000000000000 : ℝ) ≤
+      (-(4726552237572163998887 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7771178496422463 / 20000000000000000 : ℝ) ≤
+      (-(4726552237572163998887 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7771178496422463 / 20000000000000000 : ℝ) =
+            (7771178496422463 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7771178496422463 / 5000000000000000 : ℝ) ≤
+            (2204919565427836001113 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3869797273402463 / 10000000000000000 : ℝ)) (q :=
+      (7771178496422463 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_411_7 :
+    Uω (aρ 7) (bρ 7) (880153607101 / 2000000000000 : ℝ) ≤
+      (-(404008164002188548969 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7284230914933357 / 20000000000000000 : ℝ) ≤
+      (-(404008164002188548969 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7284230914933357 / 20000000000000000 : ℝ) =
+            (7284230914933357 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7284230914933357 / 5000000000000000 : ℝ) ≤
+            (150509580237811451031 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3606511072673357 / 10000000000000000 : ℝ)) (q :=
+      (7284230914933357 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_411_8 :
+    Uω (aρ 8) (bρ 8) (880153607101 / 2000000000000 : ℝ) ≤
+      (-(5515872638589955285107 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1327264202182369 / 4000000000000000 : ℝ) ≤
+      (-(5515872638589955285107 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1327264202182369 / 4000000000000000 : ℝ) =
+            (1327264202182369 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1327264202182369 / 1000000000000000 : ℝ) ≤
+            (1415599164410044714893 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (648428563998369 / 2000000000000000 : ℝ)) (q :=
+      (1327264202182369 / 4000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_411_9 :
+    Uω (aρ 9) (bρ 9) (880153607101 / 2000000000000 : ℝ) ≤
+      (-(12389492920326356581651 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5793768717721283 / 20000000000000000 : ℝ) ≤
+      (-(12389492920326356581651 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5793768717721283 / 20000000000000000 : ℝ) =
+            (5793768717721283 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5793768717721283 / 5000000000000000 : ℝ) ≤
+            (1473450685673643418349 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2740556155161283 / 10000000000000000 : ℝ)) (q :=
+      (5793768717721283 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_411_10 :
+    Uω (aρ 10) (bρ 10) (880153607101 / 2000000000000 : ℝ) ≤
+      (-(56675208404681107481 / 39062500000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4687254129813093 / 20000000000000000 : ℝ) ≤
+      (-(56675208404681107481 / 39062500000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4687254129813093 / 20000000000000000 : ℝ) =
+            (4687254129813093 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4687254129813093 / 2500000000000000 : ℝ) ≤
+            (24552976786725142519 / 39062500000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2022600600748093 / 10000000000000000 : ℝ)) (q :=
+      (4687254129813093 / 20000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_411_11 :
+    Uω (aρ 11) (bρ 11) (880153607101 / 2000000000000 : ℝ) ≤
+      (-(968136623931926557471 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2884809956741063 / 20000000000000000 : ℝ) ≤
+      (-(968136623931926557471 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2884809956741063 / 20000000000000000 : ℝ) =
+            (2884809956741063 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2884809956741063 / 2500000000000000 : ℝ) ≤
+            (71584146518073442529 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (638633053881063 / 10000000000000000 : ℝ)) (q :=
+      (2884809956741063 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_411_12 :
+    Uω (aρ 12) (bρ 12) (880153607101 / 2000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_411_13 :
+    Uω (aρ 13) (bρ 13) (880153607101 / 2000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_411_14 :
+    Uω (aρ 14) (bρ 14) (880153607101 / 2000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_411_15 :
+    Uω (aρ 15) (bρ 15) (880153607101 / 2000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_411_16 :
+    Uω (aρ 16) (bρ 16) (880153607101 / 2000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_411 :
+    Uρ (880153607101 / 2000000000000 : ℝ) ≤
+      (-(6127214758723858306379 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (880153607101 / 2000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(8355789703777010327123 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_411_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (880153607101 / 2000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1682247885388241367499 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_411_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (880153607101 / 2000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(4261607671066074145483 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_411_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (880153607101 / 2000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(2178115821864365629041 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_411_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (880153607101 / 2000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(9008933307601673657931 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_411_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (880153607101 / 2000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(4726552237572163998887 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_411_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (880153607101 / 2000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(404008164002188548969 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_411_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (880153607101 / 2000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(5515872638589955285107 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_411_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (880153607101 / 2000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(12389492920326356581651 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_411_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (880153607101 / 2000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) * (-(56675208404681107481 / 39062500000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_411_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (880153607101 / 2000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) * (-(968136623931926557471 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_411_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (880153607101 / 2000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_411_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (880153607101 / 2000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_411_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (880153607101 / 2000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_411_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (880153607101 / 2000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_411_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (880153607101 / 2000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_411_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/U34.lean b/LeanPool/Zeta5Irrational/Table/U34.lean
new file mode 100644
index 0000000000..94ec4b3816
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/U34.lean
@@ -0,0 +1,4364 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-! # Certified arcsine potential bounds (U34) -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma U_412_1 :
+    Uω (aρ 1) (bρ 1) (441401103427 / 1000000000000 : ℝ) ≤
+      (-(8325295657190344060219 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8698953293364661 / 20000000000000000 : ℝ) ≤
+      (-(8325295657190344060219 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8698953293364661 / 20000000000000000 : ℝ) =
+            (8698953293364661 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8698953293364661 / 5000000000000000 : ℝ) ≤
+            (5537647948809655939781 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4349439477179661 / 10000000000000000 : ℝ)) (q :=
+      (8698953293364661 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_412_2 :
+    Uω (aρ 2) (bρ 2) (441401103427 / 1000000000000 : ℝ) ≤
+      (-(335222984107915235267 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4325499530969083 / 10000000000000000 : ℝ) ≤
+      (-(335222984107915235267 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4325499530969083 / 10000000000000000 : ℝ) =
+            (4325499530969083 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4325499530969083 / 2500000000000000 : ℝ) ≤
+            (219294760132084764733 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2162627129131583 / 5000000000000000 : ℝ)) (q :=
+      (4325499530969083 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_412_3 :
+    Uω (aρ 3) (bρ 3) (441401103427 / 1000000000000 : ℝ) ≤
+      (-(8492201232119775677881 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8554967854459393 / 20000000000000000 : ℝ) ≤
+      (-(8492201232119775677881 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8554967854459393 / 20000000000000000 : ℝ) =
+            (8554967854459393 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8554967854459393 / 5000000000000000 : ℝ) ≤
+            (5370742373880224322119 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4276619157134393 / 10000000000000000 : ℝ)) (q :=
+      (8554967854459393 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_412_4 :
+    Uω (aρ 4) (bρ 4) (441401103427 / 1000000000000 : ℝ) ≤
+      (-(4340422539223208551097 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2098774082580749 / 5000000000000000 : ℝ) ≤
+      (-(4340422539223208551097 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2098774082580749 / 5000000000000000 : ℝ) =
+            (2098774082580749 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2098774082580749 / 1250000000000000 : ℝ) ≤
+            (2591049263776791448903 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1048760453515749 / 2500000000000000 : ℝ)) (q :=
+      (2098774082580749 / 5000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_412_5 :
+    Uω (aρ 5) (bρ 5) (441401103427 / 1000000000000 : ℝ) ≤
+      (-(4488166174250952639887 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (407533054044943 / 1000000000000000 : ℝ) ≤
+      (-(4488166174250952639887 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (407533054044943 / 1000000000000000 : ℝ) =
+            (407533054044943 / 250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (407533054044943 / 250000000000000 : ℝ) ≤
+            (2443305628749047360113 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (203439824198693 / 500000000000000 : ℝ)) (q :=
+      (407533054044943 / 1000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_412_6 :
+    Uω (aρ 6) (bρ 6) (441401103427 / 1000000000000 : ℝ) ≤
+      (-(470947097206575796623 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3898886104311623 / 10000000000000000 : ℝ) ≤
+      (-(470947097206575796623 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3898886104311623 / 10000000000000000 : ℝ) =
+            (3898886104311623 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3898886104311623 / 2500000000000000 : ℝ) ≤
+            (222200083093424203377 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1941573993419123 / 5000000000000000 : ℝ)) (q :=
+      (3898886104311623 / 10000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_412_7 :
+    Uω (aρ 7) (bρ 7) (441401103427 / 1000000000000 : ℝ) ≤
+      (-(10063552978277690335471 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (45693608893127 / 125000000000000 : ℝ) ≤
+      (-(10063552978277690335471 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (45693608893127 / 125000000000000 : ℝ) = (45693608893127 / 31250000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (45693608893127 / 31250000000000 : ℝ) ≤
+            (3799390627722309664529 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (90500364546883 / 250000000000000 : ℝ)) (q :=
+      (45693608893127 / 125000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_412_8 :
+    Uω (aρ 8) (bρ 8) (441401103427 / 1000000000000 : ℝ) ≤
+      (-(274774647827680332549 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3331712716879051 / 10000000000000000 : ℝ) ≤
+      (-(274774647827680332549 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3331712716879051 / 10000000000000000 : ℝ) =
+            (3331712716879051 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3331712716879051 / 2500000000000000 : ℝ) ≤
+            (71798942322319667451 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1628002122036551 / 5000000000000000 : ℝ)) (q :=
+      (3331712716879051 / 10000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_412_9 :
+    Uω (aρ 9) (bρ 9) (441401103427 / 1000000000000 : ℝ) ≤
+      (-(1234130017850107135807 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5821757867243393 / 20000000000000000 : ℝ) ≤
+      (-(1234130017850107135807 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5821757867243393 / 20000000000000000 : ℝ) =
+            (5821757867243393 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5821757867243393 / 5000000000000000 : ℝ) ≤
+            (152164342749892864193 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2755302305918393 / 10000000000000000 : ℝ)) (q :=
+      (5821757867243393 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_412_10 :
+    Uω (aρ 10) (bρ 10) (441401103427 / 1000000000000 : ℝ) ≤
+      (-(14443658687485022068619 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2358956177358807 / 10000000000000000 : ℝ) ≤
+      (-(14443658687485022068619 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2358956177358807 / 10000000000000000 : ℝ) =
+            (2358956177358807 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2358956177358807 / 1250000000000000 : ℝ) ≤
+            (6350756721514977931381 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1020007913443807 / 5000000000000000 : ℝ)) (q :=
+      (2358956177358807 / 10000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_412_11 :
+    Uω (aρ 11) (bρ 11) (441401103427 / 1000000000000 : ℝ) ≤
+      (-(19162436512716292548947 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (735793522556029 / 5000000000000000 : ℝ) ≤
+      (-(19162436512716292548947 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (735793522556029 / 5000000000000000 : ℝ) =
+            (735793522556029 / 625000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (735793522556029 / 625000000000000 : ℝ) ≤
+            (1631978896283707451053 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (170938547149779 / 2500000000000000 : ℝ)) (q :=
+      (735793522556029 / 5000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11]) (by norm_num [bρ_11])
+      (by norm_num) (by norm_num [aρ_11, bρ_11]) (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_412_12 :
+    Uω (aρ 12) (bρ 12) (441401103427 / 1000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_412_13 :
+    Uω (aρ 13) (bρ 13) (441401103427 / 1000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_412_14 :
+    Uω (aρ 14) (bρ 14) (441401103427 / 1000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_412_15 :
+    Uω (aρ 15) (bρ 15) (441401103427 / 1000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_412_16 :
+    Uω (aρ 16) (bρ 16) (441401103427 / 1000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_412 :
+    Uρ (441401103427 / 1000000000000 : ℝ) ≤
+      (-(6106371938425516455039 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (441401103427 / 1000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(8325295657190344060219 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_412_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (441401103427 / 1000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(335222984107915235267 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_412_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (441401103427 / 1000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(8492201232119775677881 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_412_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (441401103427 / 1000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(4340422539223208551097 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_412_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (441401103427 / 1000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(4488166174250952639887 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_412_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (441401103427 / 1000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(470947097206575796623 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_412_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (441401103427 / 1000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(10063552978277690335471 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_412_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (441401103427 / 1000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(274774647827680332549 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_412_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (441401103427 / 1000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(1234130017850107135807 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_412_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (441401103427 / 1000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(14443658687485022068619 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_412_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (441401103427 / 1000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(19162436512716292548947 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_412_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (441401103427 / 1000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_412_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (441401103427 / 1000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_412_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (441401103427 / 1000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_412_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (441401103427 / 1000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_412_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (441401103427 / 1000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_412_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_413_1 :
+    Uω (aρ 1) (bρ 1) (885450806607 / 2000000000000 : ℝ) ≤
+      (-(829489431824442098373 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8725439516552003 / 20000000000000000 : ℝ) ≤
+      (-(829489431824442098373 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8725439516552003 / 20000000000000000 : ℝ) =
+            (8725439516552003 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8725439516552003 / 5000000000000000 : ℝ) ≤
+            (556804928775557901627 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4362682701602003 / 10000000000000000 : ℝ)) (q :=
+      (8725439516552003 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_413_2 :
+    Uω (aρ 2) (bρ 2) (885450806607 / 2000000000000 : ℝ) ≤
+      (-(417500176740119125927 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1735497311365583 / 4000000000000000 : ℝ) ≤
+      (-(417500176740119125927 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1735497311365583 / 4000000000000000 : ℝ) =
+            (1735497311365583 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1735497311365583 / 1000000000000000 : ℝ) ≤
+            (275647003559880874073 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (867699750877583 / 2000000000000000 : ℝ)) (q :=
+      (1735497311365583 / 4000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_413_3 :
+    Uω (aρ 3) (bρ 3) (885450806607 / 2000000000000 : ℝ) ≤
+      (-(1692256610190248951243 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8581459191154971 / 20000000000000000 : ℝ) ≤
+      (-(1692256610190248951243 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8581459191154971 / 20000000000000000 : ℝ) =
+            (8581459191154971 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8581459191154971 / 5000000000000000 : ℝ) ≤
+            (1080332111009751048757 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4289867495064971 / 10000000000000000 : ℝ)) (q :=
+      (8581459191154971 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_413_4 :
+    Uω (aρ 4) (bρ 4) (885450806607 / 2000000000000 : ℝ) ≤
+      (-(1729865328848618984599 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4210799051116593 / 10000000000000000 : ℝ) ≤
+      (-(1729865328848618984599 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4210799051116593 / 10000000000000000 : ℝ) =
+            (4210799051116593 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4210799051116593 / 2500000000000000 : ℝ) ≤
+            (1042723392351381015401 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2104150293604093 / 5000000000000000 : ℝ)) (q :=
+      (4210799051116593 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_413_5 :
+    Uω (aρ 5) (bρ 5) (885450806607 / 2000000000000 : ℝ) ≤
+      (-(8943837665525954089233 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8177189473928883 / 20000000000000000 : ℝ) ≤
+      (-(8943837665525954089233 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8177189473928883 / 20000000000000000 : ℝ) =
+            (8177189473928883 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8177189473928883 / 5000000000000000 : ℝ) ≤
+            (4919105940474045910767 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4082081878238883 / 10000000000000000 : ℝ)) (q :=
+      (8177189473928883 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_413_6 :
+    Uω (aρ 6) (bρ 6) (885450806607 / 2000000000000 : ℝ) ≤
+      (-(9384896663309727471479 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (782436518566833 / 2000000000000000 : ℝ) ≤
+      (-(9384896663309727471479 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (782436518566833 / 2000000000000000 : ℝ) =
+            (782436518566833 / 500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (782436518566833 / 500000000000000 : ℝ) ≤
+            (4478046942690272528521 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (389649796511833 / 1000000000000000 : ℝ)) (q :=
+      (782436518566833 / 2000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_413_7 :
+    Uω (aρ 7) (bρ 7) (885450806607 / 2000000000000 : ℝ) ≤
+      (-(10027038309970569596151 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1834430503309113 / 5000000000000000 : ℝ) ≤
+      (-(10027038309970569596151 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1834430503309113 / 5000000000000000 : ℝ) =
+            (1834430503309113 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1834430503309113 / 1250000000000000 : ℝ) ≤
+            (3835905296029430403849 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (908379043361613 / 2500000000000000 : ℝ)) (q :=
+      (1834430503309113 / 5000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_413_8 :
+    Uω (aρ 8) (bρ 8) (885450806607 / 2000000000000 : ℝ) ≤
+      (-(10950399669358784600239 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3345262365188567 / 10000000000000000 : ℝ) ≤
+      (-(10950399669358784600239 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3345262365188567 / 10000000000000000 : ℝ) =
+            (3345262365188567 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3345262365188567 / 2500000000000000 : ℝ) ≤
+            (2912543936641215399761 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1634930270963567 / 5000000000000000 : ℝ)) (q :=
+      (3345262365188567 / 10000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_413_9 :
+    Uω (aρ 9) (bρ 9) (885450806607 / 2000000000000 : ℝ) ≤
+      (-(6146682270563129851507 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5849731828561447 / 20000000000000000 : ℝ) ≤
+      (-(6146682270563129851507 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5849731828561447 / 20000000000000000 : ℝ) =
+            (5849731828561447 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5849731828561447 / 5000000000000000 : ℝ) ≤
+            (784789532436870148493 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2770033268471447 / 10000000000000000 : ℝ)) (q :=
+      (5849731828561447 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_413_10 :
+    Uω (aρ 10) (bρ 10) (885450806607 / 2000000000000 : ℝ) ≤
+      (-(3594754308831076809627 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4748508407420689 / 20000000000000000 : ℝ) ≤
+      (-(3594754308831076809627 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4748508407420689 / 20000000000000000 : ℝ) =
+            (4748508407420689 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4748508407420689 / 2500000000000000 : ℝ) ≤
+            (1603849543418923190373 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2057368880825689 / 10000000000000000 : ℝ)) (q :=
+      (4748508407420689 / 20000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_413_11 :
+    Uω (aρ 11) (bρ 11) (885450806607 / 2000000000000 : ℝ) ≤
+      (-(18974596681744203262247 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2998981120853179 / 20000000000000000 : ℝ) ≤
+      (-(18974596681744203262247 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2998981120853179 / 20000000000000000 : ℝ) =
+            (2998981120853179 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2998981120853179 / 2500000000000000 : ℝ) ≤
+            (1819818727255796737753 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (726318220463179 / 10000000000000000 : ℝ)) (q :=
+      (2998981120853179 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_413_12 :
+    Uω (aρ 12) (bρ 12) (885450806607 / 2000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_413_13 :
+    Uω (aρ 13) (bρ 13) (885450806607 / 2000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_413_14 :
+    Uω (aρ 14) (bρ 14) (885450806607 / 2000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_413_15 :
+    Uω (aρ 15) (bρ 15) (885450806607 / 2000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_413_16 :
+    Uω (aρ 16) (bρ 16) (885450806607 / 2000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_413 :
+    Uρ (885450806607 / 2000000000000 : ℝ) ≤
+      (-(1217216813364887052379 / 1000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (885450806607 / 2000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(829489431824442098373 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_413_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (885450806607 / 2000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(417500176740119125927 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_413_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (885450806607 / 2000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1692256610190248951243 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_413_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (885450806607 / 2000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1729865328848618984599 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_413_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (885450806607 / 2000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(8943837665525954089233 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_413_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (885450806607 / 2000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(9384896663309727471479 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_413_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (885450806607 / 2000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(10027038309970569596151 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_413_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (885450806607 / 2000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(10950399669358784600239 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_413_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (885450806607 / 2000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(6146682270563129851507 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_413_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (885450806607 / 2000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(3594754308831076809627 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_413_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (885450806607 / 2000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(18974596681744203262247 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_413_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (885450806607 / 2000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_413_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (885450806607 / 2000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_413_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (885450806607 / 2000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_413_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (885450806607 / 2000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_413_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (885450806607 / 2000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_413_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_414_1 :
+    Uω (aρ 1) (bρ 1) (22202485159 / 50000000000 : ℝ) ≤
+      (-(8264585124940494284273 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1750385147674701 / 4000000000000000 : ℝ) ≤
+      (-(8264585124940494284273 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1750385147674701 / 4000000000000000 : ℝ) =
+            (1750385147674701 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1750385147674701 / 1000000000000000 : ℝ) ≤
+            (5598358481059505715727 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (875185184931701 / 2000000000000000 : ℝ)) (q :=
+      (1750385147674701 / 4000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_414_2 :
+    Uω (aρ 2) (bρ 2) (22202485159 / 50000000000 : ℝ) ≤
+      (-(8319525651625783148213 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4351987021301887 / 10000000000000000 : ℝ) ≤
+      (-(8319525651625783148213 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4351987021301887 / 10000000000000000 : ℝ) =
+            (4351987021301887 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4351987021301887 / 2500000000000000 : ℝ) ≤
+            (5543417954374216851787 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2175871620699387 / 5000000000000000 : ℝ)) (q :=
+      (4351987021301887 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_414_3 :
+    Uω (aρ 3) (bρ 3) (22202485159 / 50000000000 : ℝ) ≤
+      (-(4215230103394798454407 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8607950494980963 / 20000000000000000 : ℝ) ≤
+      (-(4215230103394798454407 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8607950494980963 / 20000000000000000 : ℝ) =
+            (8607950494980963 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8607950494980963 / 5000000000000000 : ℝ) ≤
+            (2716241699605201545593 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4303115800125963 / 10000000000000000 : ℝ)) (q :=
+      (8607950494980963 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_414_4 :
+    Uω (aρ 4) (bρ 4) (22202485159 / 50000000000 : ℝ) ≤
+      (-(8617907356384918787213 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4224049887558109 / 10000000000000000 : ℝ) ≤
+      (-(8617907356384918787213 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4224049887558109 / 10000000000000000 : ℝ) =
+            (4224049887558109 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4224049887558109 / 2500000000000000 : ℝ) ≤
+            (5245036249615081212787 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2110779630663109 / 5000000000000000 : ℝ)) (q :=
+      (4224049887558109 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_414_5 :
+    Uω (aρ 5) (bρ 5) (22202485159 / 50000000000 : ℝ) ≤
+      (-(8911448565837778337919 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (82037175923353 / 200000000000000 : ℝ) ≤
+      (-(8911448565837778337919 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (82037175923353 / 200000000000000 : ℝ) = (82037175923353 / 50000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (82037175923353 / 50000000000000 : ℝ) ≤
+            (4951495040162221662081 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (40953669978803 / 100000000000000 : ℝ)) (q :=
+      (82037175923353 / 200000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_414_6 :
+    Uω (aρ 6) (bρ 6) (22202485159 / 50000000000 : ℝ) ≤
+      (-(9350967824237916076221 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7850957435088173 / 20000000000000000 : ℝ) ≤
+      (-(9350967824237916076221 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7850957435088173 / 20000000000000000 : ℝ) =
+            (7850957435088173 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7850957435088173 / 5000000000000000 : ℝ) ≤
+            (4511975781762083923779 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3909847215773173 / 10000000000000000 : ℝ)) (q :=
+      (7850957435088173 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_414_7 :
+    Uω (aρ 7) (bρ 7) (22202485159 / 50000000000 : ℝ) ≤
+      (-(4995329531900420707149 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1472892941447899 / 4000000000000000 : ℝ) ≤
+      (-(4995329531900420707149 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1472892941447899 / 4000000000000000 : ℝ) =
+            (1472892941447899 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1472892941447899 / 1000000000000000 : ℝ) ≤
+            (1936142271099579292851 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (729403173736899 / 2000000000000000 : ℝ)) (q :=
+      (1472892941447899 / 4000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_414_8 :
+    Uω (aρ 8) (bρ 8) (22202485159 / 50000000000 : ℝ) ≤
+      (-(10909985018713567947419 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3358809482831227 / 10000000000000000 : ℝ) ≤
+      (-(10909985018713567947419 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3358809482831227 / 10000000000000000 : ℝ) =
+            (3358809482831227 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3358809482831227 / 2500000000000000 : ℝ) ≤
+            (2952958587286432052581 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1641855889223727 / 5000000000000000 : ℝ)) (q :=
+      (3358809482831227 / 10000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_414_9 :
+    Uω (aρ 9) (bρ 9) (22202485159 / 50000000000 : ℝ) ≤
+      (-(191338797200200775319 / 156250000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5877690842706347 / 20000000000000000 : ℝ) ≤
+      (-(191338797200200775319 / 156250000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5877690842706347 / 20000000000000000 : ℝ) =
+            (5877690842706347 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5877690842706347 / 5000000000000000 : ℝ) ≤
+            (25269696643549224681 / 156250000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2784749283851347 / 10000000000000000 : ℝ)) (q :=
+      (5877690842706347 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_414_10 :
+    Uω (aρ 10) (bρ 10) (22202485159 / 50000000000 : ℝ) ≤
+      (-(286298355615774617203 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2389521924000261 / 10000000000000000 : ℝ) ≤
+      (-(286298355615774617203 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2389521924000261 / 10000000000000000 : ℝ) =
+            (2389521924000261 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2389521924000261 / 1250000000000000 : ℝ) ≤
+            (129589952564225382797 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1037330661320261 / 5000000000000000 : ℝ)) (q :=
+      (2389521924000261 / 10000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_414_11 :
+    Uω (aρ 11) (bρ 11) (22202485159 / 50000000000 : ℝ) ≤
+      (-(3759439815845592521937 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3052657018071263 / 20000000000000000 : ℝ) ≤
+      (-(3759439815845592521937 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3052657018071263 / 20000000000000000 : ℝ) =
+            (3052657018071263 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3052657018071263 / 2500000000000000 : ℝ) ≤
+            (399443265954407478063 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (766751118916263 / 10000000000000000 : ℝ)) (q :=
+      (3052657018071263 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_414_12 :
+    Uω (aρ 12) (bρ 12) (22202485159 / 50000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_414_13 :
+    Uω (aρ 13) (bρ 13) (22202485159 / 50000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_414_14 :
+    Uω (aρ 14) (bρ 14) (22202485159 / 50000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_414_15 :
+    Uω (aρ 15) (bρ 15) (22202485159 / 50000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_414_16 :
+    Uω (aρ 16) (bρ 16) (22202485159 / 50000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_414 :
+    Uρ (22202485159 / 50000000000 : ℝ) ≤ (-(485301670043732980539 / 400000000000000000000 : ℝ)) :=
+  by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (22202485159 / 50000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(8264585124940494284273 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_414_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (22202485159 / 50000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(8319525651625783148213 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_414_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (22202485159 / 50000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(4215230103394798454407 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_414_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (22202485159 / 50000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(8617907356384918787213 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_414_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (22202485159 / 50000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(8911448565837778337919 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_414_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (22202485159 / 50000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(9350967824237916076221 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_414_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (22202485159 / 50000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(4995329531900420707149 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_414_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (22202485159 / 50000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(10909985018713567947419 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_414_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (22202485159 / 50000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(191338797200200775319 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_414_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (22202485159 / 50000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) * (-(286298355615774617203 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_414_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (22202485159 / 50000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(3759439815845592521937 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_414_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (22202485159 / 50000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_414_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (22202485159 / 50000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_414_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (22202485159 / 50000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_414_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (22202485159 / 50000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_414_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (22202485159 / 50000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_414_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_415_1 :
+    Uω (aρ 1) (bρ 1) (890748006113 / 2000000000000 : ℝ) ≤
+      (-(2058591880093690254427 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2194602989710383 / 5000000000000000 : ℝ) ≤
+      (-(2058591880093690254427 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2194602989710383 / 5000000000000000 : ℝ) =
+            (2194602989710383 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2194602989710383 / 1250000000000000 : ℝ) ≤
+            (1407144021406309745573 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1097292286590383 / 2500000000000000 : ℝ)) (q :=
+      (2194602989710383 / 5000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_415_2 :
+    Uω (aρ 2) (bρ 2) (890748006113 / 2000000000000 : ℝ) ≤
+      (-(4144570193375825698037 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (272826922479647 / 625000000000000 : ℝ) ≤
+      (-(4144570193375825698037 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (272826922479647 / 625000000000000 : ℝ) =
+            (272826922479647 / 156250000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (272826922479647 / 156250000000000 : ℝ) ≤
+            (2786901609624174301963 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (272811732461169 / 625000000000000 : ℝ)) (q :=
+      (272826922479647 / 625000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_415_3 :
+    Uω (aρ 3) (bρ 3) (890748006113 / 2000000000000 : ℝ) ≤
+      (-(4199866056630123266719 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8634441766240031 / 20000000000000000 : ℝ) ≤
+      (-(4199866056630123266719 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8634441766240031 / 20000000000000000 : ℝ) =
+            (8634441766240031 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8634441766240031 / 5000000000000000 : ℝ) ≤
+            (2731605746369876733281 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4316364072620031 / 10000000000000000 : ℝ)) (q :=
+      (8634441766240031 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_415_4 :
+    Uω (aρ 4) (bρ 4) (890748006113 / 2000000000000 : ℝ) ≤
+      (-(8586586592335453971777 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1694920269980443 / 4000000000000000 : ℝ) ≤
+      (-(8586586592335453971777 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1694920269980443 / 4000000000000000 : ℝ) =
+            (1694920269980443 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1694920269980443 / 1000000000000000 : ℝ) ≤
+            (5276357013664546028223 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (846963567469443 / 2000000000000000 : ℝ)) (q :=
+      (1694920269980443 / 4000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_415_5 :
+    Uω (aρ 5) (bρ 5) (890748006113 / 2000000000000 : ℝ) ≤
+      (-(34684235794435448991 / 39062500000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4115122719391029 / 10000000000000000 : ℝ) ≤
+      (-(34684235794435448991 / 39062500000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4115122719391029 / 10000000000000000 : ℝ) =
+            (4115122719391029 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4115122719391029 / 2500000000000000 : ℝ) ≤
+            (19467887666502051009 / 39062500000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2054325922781029 / 5000000000000000 : ℝ)) (q :=
+      (4115122719391029 / 10000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_415_6 :
+    Uω (aρ 6) (bρ 6) (890748006113 / 2000000000000 : ℝ) ≤
+      (-(4658577313435840435391 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (787754896431033 / 2000000000000000 : ℝ) ≤
+      (-(4658577313435840435391 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (787754896431033 / 2000000000000000 : ℝ) =
+            (787754896431033 / 500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (787754896431033 / 500000000000000 : ℝ) ≤
+            (2272894489564159564609 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (392319574623033 / 1000000000000000 : ℝ)) (q :=
+      (787754896431033 / 2000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_415_7 :
+    Uω (aρ 7) (bρ 7) (890748006113 / 2000000000000 : ℝ) ≤
+      (-(9954414220290087389677 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3695602762945023 / 10000000000000000 : ℝ) ≤
+      (-(9954414220290087389677 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3695602762945023 / 10000000000000000 : ℝ) =
+            (3695602762945023 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3695602762945023 / 2500000000000000 : ℝ) ≤
+            (3908529385709912610323 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1830256844285023 / 5000000000000000 : ℝ)) (q :=
+      (3695602762945023 / 10000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_415_8 :
+    Uω (aρ 8) (bρ 8) (890748006113 / 2000000000000 : ℝ) ≤
+      (-(5434870227411690620931 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (421544262712109 / 1250000000000000 : ℝ) ≤
+      (-(5434870227411690620931 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (421544262712109 / 1250000000000000 : ℝ) =
+            (421544262712109 / 312500000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (421544262712109 / 312500000000000 : ℝ) ≤
+            (1496601575588309379069 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (206097376088359 / 625000000000000 : ℝ)) (q :=
+      (421544262712109 / 1250000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_415_9 :
+    Uω (aρ 9) (bρ 9) (890748006113 / 2000000000000 : ℝ) ≤
+      (-(12198252685595503436989 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2952817572699407 / 10000000000000000 : ℝ) ≤
+      (-(12198252685595503436989 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2952817572699407 / 10000000000000000 : ℝ) =
+            (2952817572699407 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2952817572699407 / 2500000000000000 : ℝ) ≤
+            (1664690920404496563011 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1399725293889407 / 5000000000000000 : ℝ)) (q :=
+      (2952817572699407 / 10000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_415_10 :
+    Uω (aρ 10) (bρ 10) (890748006113 / 2000000000000 : ℝ) ≤
+      (-(1781418685494105864799 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (601190022448769 / 2500000000000000 : ℝ) ≤
+      (-(1781418685494105864799 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (601190022448769 / 2500000000000000 : ℝ) =
+            (601190022448769 / 312500000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (601190022448769 / 312500000000000 : ℝ) ≤
+            (817883240630894135201 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (32685853991643 / 156250000000000 : ℝ)) (q :=
+      (601190022448769 / 2500000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10]) (by norm_num [bρ_10])
+      (by norm_num) (by norm_num [aρ_10, bρ_10]) (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_415_11 :
+    Uω (aρ 11) (bρ 11) (890748006113 / 2000000000000 : ℝ) ≤
+      (-(9314361252783403192807 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (31045228193841 / 200000000000000 : ℝ) ≤
+      (-(9314361252783403192807 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (31045228193841 / 200000000000000 : ℝ) = (31045228193841 / 25000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (31045228193841 / 25000000000000 : ℝ) ≤
+            (1082846451716596807193 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (8053739214641 / 100000000000000 : ℝ)) (q :=
+      (31045228193841 / 200000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11]) (by norm_num [bρ_11])
+      (by norm_num) (by norm_num [aρ_11, bρ_11]) (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_415_12 :
+    Uω (aρ 12) (bρ 12) (890748006113 / 2000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_415_13 :
+    Uω (aρ 13) (bρ 13) (890748006113 / 2000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_415_14 :
+    Uω (aρ 14) (bρ 14) (890748006113 / 2000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_415_15 :
+    Uω (aρ 15) (bρ 15) (890748006113 / 2000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_415_16 :
+    Uω (aρ 16) (bρ 16) (890748006113 / 2000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_415 :
+    Uρ (890748006113 / 2000000000000 : ℝ) ≤
+      (-(12093743108271133476063 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (890748006113 / 2000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2058591880093690254427 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_415_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (890748006113 / 2000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(4144570193375825698037 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_415_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (890748006113 / 2000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(4199866056630123266719 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_415_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (890748006113 / 2000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(8586586592335453971777 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_415_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (890748006113 / 2000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(34684235794435448991 / 39062500000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_415_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (890748006113 / 2000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(4658577313435840435391 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_415_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (890748006113 / 2000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(9954414220290087389677 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_415_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (890748006113 / 2000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(5434870227411690620931 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_415_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (890748006113 / 2000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(12198252685595503436989 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_415_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (890748006113 / 2000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(1781418685494105864799 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_415_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (890748006113 / 2000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(9314361252783403192807 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_415_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (890748006113 / 2000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_415_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (890748006113 / 2000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_415_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (890748006113 / 2000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_415_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (890748006113 / 2000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_415_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (890748006113 / 2000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_415_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_416_1 :
+    Uω (aρ 1) (bρ 1) (446698302933 / 1000000000000 : ℝ) ≤
+      (-(8204240952676966510493 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4402449088984149 / 10000000000000000 : ℝ) ≤
+      (-(8204240952676966510493 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4402449088984149 / 10000000000000000 : ℝ) =
+            (4402449088984149 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4402449088984149 / 2500000000000000 : ℝ) ≤
+            (5658702653323033489507 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2201206183361649 / 5000000000000000 : ℝ)) (q :=
+      (4402449088984149 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_416_2 :
+    Uω (aρ 2) (bρ 2) (446698302933 / 1000000000000 : ℝ) ≤
+      (-(1032355897364107694117 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1094618623393083 / 2500000000000000 : ℝ) ≤
+      (-(1032355897364107694117 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1094618623393083 / 2500000000000000 : ℝ) =
+            (1094618623393083 / 625000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1094618623393083 / 625000000000000 : ℝ) ≤
+            (700512053385892305883 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (68409877943901 / 156250000000000 : ℝ)) (q :=
+      (1094618623393083 / 2500000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_416_3 :
+    Uω (aρ 3) (bρ 3) (446698302933 / 1000000000000 : ℝ) ≤
+      (-(8369098189383941263791 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8660933005231133 / 20000000000000000 : ℝ) ≤
+      (-(8369098189383941263791 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8660933005231133 / 20000000000000000 : ℝ) =
+            (8660933005231133 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8660933005231133 / 5000000000000000 : ℝ) ≤
+            (5493845416616058736209 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4329612312846133 / 10000000000000000 : ℝ)) (q :=
+      (8660933005231133 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_416_4 :
+    Uω (aρ 4) (bρ 4) (446698302933 / 1000000000000 : ℝ) ≤
+      (-(8555363735410042263653 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4250551413754841 / 10000000000000000 : ℝ) ≤
+      (-(8555363735410042263653 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4250551413754841 / 10000000000000000 : ℝ) =
+            (4250551413754841 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4250551413754841 / 2500000000000000 : ℝ) ≤
+            (5307579870589957736347 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2124038158094841 / 5000000000000000 : ℝ)) (q :=
+      (4250551413754841 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_416_5 :
+    Uω (aρ 5) (bρ 5) (446698302933 / 1000000000000 : ℝ) ≤
+      (-(884698437876299503827 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4128386507949351 / 10000000000000000 : ℝ) ≤
+      (-(884698437876299503827 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4128386507949351 / 10000000000000000 : ℝ) =
+            (4128386507949351 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4128386507949351 / 2500000000000000 : ℝ) ≤
+            (501595922723700496173 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2060968211956851 / 5000000000000000 : ℝ)) (q :=
+      (4128386507949351 / 10000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_416_6 :
+    Uω (aρ 6) (bρ 6) (446698302933 / 1000000000000 : ℝ) ≤
+      (-(9283456279446531117199 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3952069890330607 / 10000000000000000 : ℝ) ≤
+      (-(9283456279446531117199 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3952069890330607 / 10000000000000000 : ℝ) =
+            (3952069890330607 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3952069890330607 / 2500000000000000 : ℝ) ≤
+            (4579487326553468882801 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1968271781908107 / 5000000000000000 : ℝ)) (q :=
+      (3952069890330607 / 10000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_416_7 :
+    Uω (aρ 7) (bρ 7) (446698302933 / 1000000000000 : ℝ) ≤
+      (-(2479575692907522664859 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3708972244928743 / 10000000000000000 : ℝ) ≤
+      (-(2479575692907522664859 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3708972244928743 / 10000000000000000 : ℝ) =
+            (3708972244928743 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3708972244928743 / 2500000000000000 : ℝ) ≤
+            (986160208592477335141 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1837004826886243 / 5000000000000000 : ℝ)) (q :=
+      (3708972244928743 / 10000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_416_8 :
+    Uω (aρ 8) (bρ 8) (446698302933 / 1000000000000 : ℝ) ≤
+      (-(2165932898364507387119 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (677179250626111 / 2000000000000000 : ℝ) ≤
+      (-(2165932898364507387119 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (677179250626111 / 2000000000000000 : ℝ) =
+            (677179250626111 / 500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (677179250626111 / 500000000000000 : ℝ) ≤
+            (606655822835492612881 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (331139932151611 / 1000000000000000 : ℝ)) (q :=
+      (677179250626111 / 2000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_416_9 :
+    Uω (aρ 9) (bρ 9) (446698302933 / 1000000000000 : ℝ) ≤
+      (-(12151070657503466858303 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5933564967200663 / 20000000000000000 : ℝ) ≤
+      (-(12151070657503466858303 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5933564967200663 / 20000000000000000 : ℝ) =
+            (5933564967200663 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5933564967200663 / 5000000000000000 : ℝ) ≤
+            (1711872948496533141697 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2814137410815663 / 10000000000000000 : ℝ)) (q :=
+      (5933564967200663 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_416_10 :
+    Uω (aρ 10) (bρ 10) (446698302933 / 1000000000000 : ℝ) ≤
+      (-(3547075465469153810511 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (604992356392347 / 2500000000000000 : ℝ) ≤
+      (-(3547075465469153810511 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (604992356392347 / 2500000000000000 : ℝ) =
+            (604992356392347 / 312500000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (604992356392347 / 312500000000000 : ℝ) ≤
+            (1651528386780846189489 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (263633791031097 / 1250000000000000 : ℝ)) (q :=
+      (604992356392347 / 2500000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10]) (by norm_num [bρ_10])
+      (by norm_num) (by norm_num [aρ_10, bρ_10]) (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_416_11 :
+    Uω (aρ 11) (bρ 11) (446698302933 / 1000000000000 : ℝ) ≤
+      (-(738719378596878066367 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1577413760887911 / 10000000000000000 : ℝ) ≤
+      (-(738719378596878066367 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1577413760887911 / 10000000000000000 : ℝ) =
+            (1577413760887911 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1577413760887911 / 1250000000000000 : ℝ) ≤
+            (93057237763121933633 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (421217812545411 / 5000000000000000 : ℝ)) (q :=
+      (1577413760887911 / 10000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_416_12 :
+    Uω (aρ 12) (bρ 12) (446698302933 / 1000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_416_13 :
+    Uω (aρ 13) (bρ 13) (446698302933 / 1000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_416_14 :
+    Uω (aρ 14) (bρ 14) (446698302933 / 1000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_416_15 :
+    Uω (aρ 15) (bρ 15) (446698302933 / 1000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_416_16 :
+    Uω (aρ 16) (bρ 16) (446698302933 / 1000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_416 :
+    Uρ (446698302933 / 1000000000000 : ℝ) ≤
+      (-(1506959668461532668943 / 1250000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (446698302933 / 1000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(8204240952676966510493 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_416_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (446698302933 / 1000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1032355897364107694117 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_416_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (446698302933 / 1000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(8369098189383941263791 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_416_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (446698302933 / 1000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(8555363735410042263653 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_416_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (446698302933 / 1000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(884698437876299503827 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_416_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (446698302933 / 1000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(9283456279446531117199 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_416_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (446698302933 / 1000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(2479575692907522664859 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_416_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (446698302933 / 1000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(2165932898364507387119 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_416_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (446698302933 / 1000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(12151070657503466858303 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_416_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (446698302933 / 1000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(3547075465469153810511 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_416_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (446698302933 / 1000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) * (-(738719378596878066367 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_416_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (446698302933 / 1000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_416_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (446698302933 / 1000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_416_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (446698302933 / 1000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_416_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (446698302933 / 1000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_416_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (446698302933 / 1000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_416_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_417_1 :
+    Uω (aρ 1) (bρ 1) (896045205619 / 2000000000000 : ℝ) ≤
+      (-(2043551218737480737427 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (883138439576587 / 2000000000000000 : ℝ) ≤
+      (-(2043551218737480737427 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (883138439576587 / 2000000000000000 : ℝ) =
+            (883138439576587 / 500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (883138439576587 / 500000000000000 : ℝ) ≤
+            (1422184682762519262573 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (441565558575587 / 1000000000000000 : ℝ)) (q :=
+      (883138439576587 / 2000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_417_2 :
+    Uω (aρ 2) (bρ 2) (896045205619 / 2000000000000 : ℝ) ≤
+      (-(2057161367982340828791 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4391718223036311 / 10000000000000000 : ℝ) ≤
+      (-(2057161367982340828791 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4391718223036311 / 10000000000000000 : ℝ) =
+            (4391718223036311 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4391718223036311 / 2500000000000000 : ℝ) ≤
+            (1408574533517659171209 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2195738324286311 / 5000000000000000 : ℝ)) (q :=
+      (4391718223036311 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_417_3 :
+    Uω (aρ 3) (bρ 3) (896045205619 / 2000000000000 : ℝ) ≤
+      (-(833855785951075132007 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1085928026531197 / 2500000000000000 : ℝ) ≤
+      (-(833855785951075132007 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1085928026531197 / 2500000000000000 : ℝ) =
+            (1085928026531197 / 625000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1085928026531197 / 625000000000000 : ℝ) ≤
+            (552438574648924867993 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (542857565137447 / 1250000000000000 : ℝ)) (q :=
+      (1085928026531197 / 2500000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_417_4 :
+    Uω (aρ 4) (bρ 4) (896045205619 / 2000000000000 : ℝ) ≤
+      (-(1704847634940521090967 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8527604208845687 / 20000000000000000 : ℝ) ≤
+      (-(1704847634940521090967 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8527604208845687 / 20000000000000000 : ℝ) =
+            (8527604208845687 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8527604208845687 / 5000000000000000 : ℝ) ≤
+            (1067741086259478909033 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4261334698760687 / 10000000000000000 : ℝ)) (q :=
+      (8527604208845687 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_417_5 :
+    Uω (aρ 5) (bρ 5) (896045205619 / 2000000000000 : ℝ) ≤
+      (-(1101863492402909633239 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8283300326280939 / 20000000000000000 : ℝ) ≤
+      (-(1101863492402909633239 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8283300326280939 / 20000000000000000 : ℝ) =
+            (8283300326280939 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8283300326280939 / 5000000000000000 : ℝ) ≤
+            (631004458347090366761 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4135220735530939 / 10000000000000000 : ℝ)) (q :=
+      (8283300326280939 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_417_6 :
+    Uω (aρ 6) (bρ 6) (896045205619 / 2000000000000 : ℝ) ≤
+      (-(9249871998363191376847 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7930729891367817 / 20000000000000000 : ℝ) ≤
+      (-(9249871998363191376847 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7930729891367817 / 20000000000000000 : ℝ) =
+            (7930729891367817 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7930729891367817 / 5000000000000000 : ℝ) ≤
+            (4613071607636808623153 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3949890675757817 / 10000000000000000 : ℝ)) (q :=
+      (7930729891367817 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_417_7 :
+    Uω (aρ 7) (bρ 7) (896045205619 / 2000000000000 : ℝ) ≤
+      (-(494116186075126798691 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1488936323901157 / 4000000000000000 : ℝ) ≤
+      (-(494116186075126798691 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1488936323901157 / 4000000000000000 : ℝ) =
+            (1488936323901157 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1488936323901157 / 1000000000000000 : ℝ) ≤
+            (199030994224873201309 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (737500756931157 / 2000000000000000 : ℝ)) (q :=
+      (1488936323901157 / 4000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_417_8 :
+    Uω (aρ 8) (bρ 8) (896045205619 / 2000000000000 : ℝ) ≤
+      (-(10789755663946743181419 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3399435967944357 / 10000000000000000 : ℝ) ≤
+      (-(10789755663946743181419 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3399435967944357 / 10000000000000000 : ℝ) =
+            (3399435967944357 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3399435967944357 / 2500000000000000 : ℝ) ≤
+            (3073187942053256818581 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1662617876189357 / 5000000000000000 : ℝ)) (q :=
+      (3399435967944357 / 10000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_417_9 :
+    Uω (aρ 9) (bρ 9) (896045205619 / 2000000000000 : ℝ) ≤
+      (-(756508381948669364161 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (372592533353797 / 1250000000000000 : ℝ) ≤
+      (-(756508381948669364161 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (372592533353797 / 1250000000000000 : ℝ) =
+            (372592533353797 / 312500000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (372592533353797 / 312500000000000 : ℝ) ≤
+            (109925593426330635839 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (88400311828461 / 312500000000000 : ℝ)) (q :=
+      (372592533353797 / 1250000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_417_10 :
+    Uω (aρ 10) (bρ 10) (896045205619 / 2000000000000 : ℝ) ≤
+      (-(2825152954444311047283 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2435150630002837 / 10000000000000000 : ℝ) ≤
+      (-(2825152954444311047283 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2435150630002837 / 10000000000000000 : ℝ) =
+            (2435150630002837 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2435150630002837 / 1250000000000000 : ℝ) ≤
+            (1333730127355688952717 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1063094869175337 / 5000000000000000 : ℝ)) (q :=
+      (2435150630002837 / 10000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_417_11 :
+    Uω (aρ 11) (bρ 11) (896045205619 / 2000000000000 : ℝ) ≤
+      (-(3662808807969156807193 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3203768806367033 / 20000000000000000 : ℝ) ≤
+      (-(3662808807969156807193 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3203768806367033 / 20000000000000000 : ℝ) =
+            (3203768806367033 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3203768806367033 / 2500000000000000 : ℝ) ≤
+            (496074273830843192807 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (878133910917033 / 10000000000000000 : ℝ)) (q :=
+      (3203768806367033 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_417_12 :
+    Uω (aρ 12) (bρ 12) (896045205619 / 2000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_417_13 :
+    Uω (aρ 13) (bρ 13) (896045205619 / 2000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_417_14 :
+    Uω (aρ 14) (bρ 14) (896045205619 / 2000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_417_15 :
+    Uω (aρ 15) (bρ 15) (896045205619 / 2000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_417_16 :
+    Uω (aρ 16) (bρ 16) (896045205619 / 2000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_417 :
+    Uρ (896045205619 / 2000000000000 : ℝ) ≤
+      (-(2403653742116277630583 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (896045205619 / 2000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2043551218737480737427 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_417_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (896045205619 / 2000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(2057161367982340828791 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_417_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (896045205619 / 2000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(833855785951075132007 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_417_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (896045205619 / 2000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1704847634940521090967 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_417_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (896045205619 / 2000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1101863492402909633239 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_417_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (896045205619 / 2000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(9249871998363191376847 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_417_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (896045205619 / 2000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(494116186075126798691 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_417_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (896045205619 / 2000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(10789755663946743181419 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_417_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (896045205619 / 2000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(756508381948669364161 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_417_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (896045205619 / 2000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(2825152954444311047283 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_417_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (896045205619 / 2000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(3662808807969156807193 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_417_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (896045205619 / 2000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_417_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (896045205619 / 2000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_417_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (896045205619 / 2000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_417_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (896045205619 / 2000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_417_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (896045205619 / 2000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_417_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_418_1 :
+    Uω (aρ 1) (bρ 1) (1794739010991 / 4000000000000 : ℝ) ≤
+      (-(326368824010608195853 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (442231375208497 / 1000000000000000 : ℝ) ≤
+      (-(326368824010608195853 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (442231375208497 / 1000000000000000 : ℝ) =
+            (442231375208497 / 250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (442231375208497 / 250000000000000 : ℝ) ≤
+            (228148920229391804147 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (27639232467359 / 62500000000000 : ℝ)) (q :=
+      (442231375208497 / 1000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_418_2 :
+    Uω (aρ 2) (bρ 2) (1794739010991 / 4000000000000 : ℝ) ≤
+      (-(4106789379413987168301 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8796680172236137 / 20000000000000000 : ℝ) ≤
+      (-(4106789379413987168301 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8796680172236137 / 20000000000000000 : ℝ) =
+            (8796680172236137 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8796680172236137 / 5000000000000000 : ℝ) ≤
+            (2824682423586012831699 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4398098875353637 / 10000000000000000 : ℝ)) (q :=
+      (8796680172236137 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_418_3 :
+    Uω (aρ 3) (bρ 3) (1794739010991 / 4000000000000 : ℝ) ≤
+      (-(8323322613875938862681 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (174013396077207 / 400000000000000 : ℝ) ≤
+      (-(8323322613875938862681 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (174013396077207 / 400000000000000 : ℝ) =
+            (174013396077207 / 100000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (174013396077207 / 100000000000000 : ℝ) ≤
+            (5539620992124061137319 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (86989692266557 / 200000000000000 : ℝ)) (q :=
+      (174013396077207 / 400000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_418_4 :
+    Uω (aρ 4) (bρ 4) (1794739010991 / 4000000000000 : ℝ) ≤
+      (-(8508711691084804030819 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4270427431846133 / 10000000000000000 : ℝ) ≤
+      (-(8508711691084804030819 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4270427431846133 / 10000000000000000 : ℝ) =
+            (4270427431846133 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4270427431846133 / 2500000000000000 : ℝ) ≤
+            (5354231914915195969181 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2133981927112383 / 5000000000000000 : ℝ)) (q :=
+      (4270427431846133 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_418_5 :
+    Uω (aρ 5) (bρ 5) (1794739010991 / 4000000000000 : ℝ) ≤
+      (-(2199727085099057476111 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2074140970562173 / 5000000000000000 : ℝ) ≤
+      (-(2199727085099057476111 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2074140970562173 / 5000000000000000 : ℝ) =
+            (2074140970562173 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2074140970562173 / 1250000000000000 : ℝ) ≤
+            (1266008816400942523889 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (129433212253631 / 312500000000000 : ℝ)) (q :=
+      (2074140970562173 / 5000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_418_6 :
+    Uω (aρ 6) (bρ 6) (1794739010991 / 4000000000000 : ℝ) ≤
+      (-(4616561195014913821701 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3972012342168089 / 10000000000000000 : ℝ) ≤
+      (-(4616561195014913821701 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3972012342168089 / 10000000000000000 : ℝ) =
+            (3972012342168089 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3972012342168089 / 2500000000000000 : ℝ) ≤
+            (2314910607985086178299 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1978281984671839 / 5000000000000000 : ℝ)) (q :=
+      (3972012342168089 / 10000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_418_7 :
+    Uω (aρ 7) (bρ 7) (1794739010991 / 4000000000000 : ℝ) ≤
+      (-(4932191768818572964023 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1491609900548377 / 4000000000000000 : ℝ) ≤
+      (-(4932191768818572964023 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1491609900548377 / 4000000000000000 : ℝ) =
+            (1491609900548377 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1491609900548377 / 1000000000000000 : ℝ) ≤
+            (1999280034181427035977 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (738850033701877 / 2000000000000000 : ℝ)) (q :=
+      (1491609900548377 / 4000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_418_8 :
+    Uω (aρ 8) (bρ 8) (1794739010991 / 4000000000000 : ℝ) ≤
+      (-(2153972694606828066529 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1703102460552339 / 5000000000000000 : ℝ) ≤
+      (-(2153972694606828066529 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1703102460552339 / 5000000000000000 : ℝ) =
+            (1703102460552339 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1703102460552339 / 1250000000000000 : ℝ) ≤
+            (618616026593171933471 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (416519019914607 / 1250000000000000 : ℝ)) (q :=
+      (1703102460552339 / 5000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_418_9 :
+    Uω (aρ 9) (bρ 9) (1794739010991 / 4000000000000 : ℝ) ≤
+      (-(12080757024811163845657 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (597543304028569 / 2000000000000000 : ℝ) ≤
+      (-(12080757024811163845657 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (597543304028569 / 2000000000000000 : ℝ) =
+            (597543304028569 / 500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (597543304028569 / 500000000000000 : ℝ) ≤
+            (1782186581188836154343 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (283614098575319 / 1000000000000000 : ℝ)) (q :=
+      (597543304028569 / 2000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_418_10 :
+    Uω (aρ 10) (bρ 10) (1794739010991 / 4000000000000 : ℝ) ≤
+      (-(14094684596889839880361 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1221365447276197 / 5000000000000000 : ℝ) ≤
+      (-(14094684596889839880361 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1221365447276197 / 5000000000000000 : ℝ) =
+            (1221365447276197 / 625000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1221365447276197 / 625000000000000 : ℝ) ≤
+            (6699730812110160119639 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (266841096008411 / 1250000000000000 : ℝ)) (q :=
+      (1221365447276197 / 5000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_418_11 :
+    Uω (aρ 11) (bρ 11) (1794739010991 / 4000000000000 : ℝ) ≤
+      (-(18239378751248439378963 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (322777936483651 / 2000000000000000 : ℝ) ≤
+      (-(18239378751248439378963 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (322777936483651 / 2000000000000000 : ℝ) =
+            (322777936483651 / 250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (322777936483651 / 250000000000000 : ℝ) ≤
+            (2555036657751560621037 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (89552297000401 / 1000000000000000 : ℝ)) (q :=
+      (322777936483651 / 2000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11]) (by norm_num [bρ_11])
+      (by norm_num) (by norm_num [aρ_11, bρ_11]) (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_418_12 :
+    Uω (aρ 12) (bρ 12) (1794739010991 / 4000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_418_13 :
+    Uω (aρ 13) (bρ 13) (1794739010991 / 4000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_418_14 :
+    Uω (aρ 14) (bρ 14) (1794739010991 / 4000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_418_15 :
+    Uω (aρ 15) (bρ 15) (1794739010991 / 4000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_418_16 :
+    Uω (aρ 16) (bρ 16) (1794739010991 / 4000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_418 :
+    Uρ (1794739010991 / 4000000000000 : ℝ) ≤
+      (-(599989554116664840691 / 500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1794739010991 / 4000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(326368824010608195853 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_418_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1794739010991 / 4000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(4106789379413987168301 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_418_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1794739010991 / 4000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(8323322613875938862681 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_418_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1794739010991 / 4000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(8508711691084804030819 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_418_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1794739010991 / 4000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(2199727085099057476111 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_418_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1794739010991 / 4000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(4616561195014913821701 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_418_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1794739010991 / 4000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(4932191768818572964023 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_418_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1794739010991 / 4000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(2153972694606828066529 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_418_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1794739010991 / 4000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(12080757024811163845657 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_418_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1794739010991 / 4000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(14094684596889839880361 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_418_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1794739010991 / 4000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(18239378751248439378963 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_418_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1794739010991 / 4000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_418_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1794739010991 / 4000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_418_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1794739010991 / 4000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_418_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1794739010991 / 4000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_418_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1794739010991 / 4000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_418_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_419_1 :
+    Uω (aρ 1) (bρ 1) (224673451343 / 500000000000 : ℝ) ≤
+      (-(2036064686302483332461 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2214467653061543 / 5000000000000000 : ℝ) ≤
+      (-(2036064686302483332461 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2214467653061543 / 5000000000000000 : ℝ) =
+            (2214467653061543 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2214467653061543 / 1250000000000000 : ℝ) ≤
+            (1429671215197516667539 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1107224700867793 / 2500000000000000 : ℝ)) (q :=
+      (2214467653061543 / 5000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_419_2 :
+    Uω (aρ 2) (bρ 2) (224673451343 / 500000000000 : ℝ) ≤
+      (-(8198534714646879541903 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4404961948106287 / 10000000000000000 : ℝ) ≤
+      (-(8198534714646879541903 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4404961948106287 / 10000000000000000 : ℝ) =
+            (4404961948106287 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4404961948106287 / 2500000000000000 : ℝ) ≤
+            (5664408891353120458097 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2202360549973787 / 5000000000000000 : ℝ)) (q :=
+      (4404961948106287 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_419_3 :
+    Uω (aρ 3) (bρ 3) (224673451343 / 500000000000 : ℝ) ≤
+      (-(4154055276627529415703 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4356957693793537 / 10000000000000000 : ℝ) ≤
+      (-(4154055276627529415703 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4356957693793537 / 10000000000000000 : ℝ) =
+            (4356957693793537 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4356957693793537 / 2500000000000000 : ℝ) ≤
+            (2777416526372470584297 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2178054348836037 / 5000000000000000 : ℝ)) (q :=
+      (4356957693793537 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_419_4 :
+    Uω (aρ 4) (bρ 4) (224673451343 / 500000000000 : ℝ) ≤
+      (-(53082558156334955823 / 62500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8554105494806037 / 20000000000000000 : ℝ) ≤
+      (-(53082558156334955823 / 62500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8554105494806037 / 20000000000000000 : ℝ) =
+            (8554105494806037 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8554105494806037 / 5000000000000000 : ℝ) ≤
+            (33560839381165044177 / 62500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4274592985956037 / 10000000000000000 : ℝ)) (q :=
+      (8554105494806037 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_419_5 :
+    Uω (aρ 5) (bρ 5) (224673451343 / 500000000000 : ℝ) ≤
+      (-(2195733594623203264179 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2077456843122793 / 5000000000000000 : ℝ) ≤
+      (-(2195733594623203264179 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2077456843122793 / 5000000000000000 : ℝ) =
+            (2077456843122793 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2077456843122793 / 1250000000000000 : ℝ) ≤
+            (1270002306876796735821 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1037126195744043 / 2500000000000000 : ℝ)) (q :=
+      (2077456843122793 / 5000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_419_6 :
+    Uω (aρ 6) (bρ 6) (224673451343 / 500000000000 : ℝ) ≤
+      (-(92164010080748980551 / 100000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7957319303559397 / 20000000000000000 : ℝ) ≤
+      (-(92164010080748980551 / 100000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7957319303559397 / 20000000000000000 : ℝ) =
+            (7957319303559397 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7957319303559397 / 5000000000000000 : ℝ) ≤
+            (46465425979251019449 / 100000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3963237089184397 / 10000000000000000 : ℝ)) (q :=
+      (7957319303559397 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_419_7 :
+    Uω (aρ 7) (bρ 7) (224673451343 / 500000000000 : ℝ) ≤
+      (-(9846476084902234316057 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1867854233724793 / 5000000000000000 : ℝ) ≤
+      (-(9846476084902234316057 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1867854233724793 / 5000000000000000 : ℝ) =
+            (1867854233724793 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1867854233724793 / 1250000000000000 : ℝ) ≤
+            (4016467521097765683943 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (925249025321043 / 2500000000000000 : ℝ)) (q :=
+      (1867854233724793 / 5000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_419_8 :
+    Uω (aρ 8) (bρ 8) (224673451343 / 500000000000 : ℝ) ≤
+      (-(2687503131290268703811 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6825946552857763 / 20000000000000000 : ℝ) ≤
+      (-(2687503131290268703811 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6825946552857763 / 20000000000000000 : ℝ) =
+            (6825946552857763 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6825946552857763 / 5000000000000000 : ℝ) ≤
+            (778232770209731296189 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3339067370582763 / 10000000000000000 : ℝ)) (q :=
+      (6825946552857763 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_419_9 :
+    Uω (aρ 9) (bρ 9) (224673451343 / 500000000000 : ℝ) ≤
+      (-(1507180034067428396347 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5989382065455843 / 20000000000000000 : ℝ) ≤
+      (-(1507180034067428396347 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5989382065455843 / 20000000000000000 : ℝ) =
+            (5989382065455843 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5989382065455843 / 5000000000000000 : ℝ) ≤
+            (225687916682571603653 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2843468511540843 / 10000000000000000 : ℝ)) (q :=
+      (5989382065455843 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_419_10 :
+    Uω (aρ 10) (bρ 10) (224673451343 / 500000000000 : ℝ) ≤
+      (-(7031864198912791906271 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4900608754398927 / 20000000000000000 : ℝ) ≤
+      (-(7031864198912791906271 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4900608754398927 / 20000000000000000 : ℝ) =
+            (4900608754398927 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4900608754398927 / 2500000000000000 : ℝ) ≤
+            (3365343505587208093729 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2143254233978927 / 10000000000000000 : ℝ)) (q :=
+      (4900608754398927 / 20000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_419_11 :
+    Uω (aρ 11) (bρ 11) (224673451343 / 500000000000 : ℝ) ≤
+      (-(9083068948402617052851 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3251506681080341 / 20000000000000000 : ℝ) ≤
+      (-(9083068948402617052851 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3251506681080341 / 20000000000000000 : ℝ) =
+            (3251506681080341 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3251506681080341 / 2500000000000000 : ℝ) ≤
+            (1314138756097382947149 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (912628786865341 / 10000000000000000 : ℝ)) (q :=
+      (3251506681080341 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_419_12 :
+    Uω (aρ 12) (bρ 12) (224673451343 / 500000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_419_13 :
+    Uω (aρ 13) (bρ 13) (224673451343 / 500000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_419_14 :
+    Uω (aρ 14) (bρ 14) (224673451343 / 500000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_419_15 :
+    Uω (aρ 15) (bρ 15) (224673451343 / 500000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_419_16 :
+    Uω (aρ 16) (bρ 16) (224673451343 / 500000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_419 :
+    Uρ (224673451343 / 500000000000 : ℝ) ≤
+      (-(2396291097251474981877 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (224673451343 / 500000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2036064686302483332461 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_419_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (224673451343 / 500000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(8198534714646879541903 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_419_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (224673451343 / 500000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(4154055276627529415703 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_419_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (224673451343 / 500000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(53082558156334955823 / 62500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_419_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (224673451343 / 500000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(2195733594623203264179 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_419_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (224673451343 / 500000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) * (-(92164010080748980551 / 100000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_419_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (224673451343 / 500000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(9846476084902234316057 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_419_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (224673451343 / 500000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(2687503131290268703811 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_419_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (224673451343 / 500000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(1507180034067428396347 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_419_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (224673451343 / 500000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(7031864198912791906271 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_419_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (224673451343 / 500000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(9083068948402617052851 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_419_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (224673451343 / 500000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_419_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (224673451343 / 500000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_419_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (224673451343 / 500000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_419_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (224673451343 / 500000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_419_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (224673451343 / 500000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_419_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_420_1 :
+    Uω (aρ 1) (bρ 1) (1800036210497 / 4000000000000 : ℝ) ≤
+      (-(4064659621397125910963 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4435556859998017 / 10000000000000000 : ℝ) ≤
+      (-(4064659621397125910963 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4435556859998017 / 10000000000000000 : ℝ) =
+            (4435556859998017 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4435556859998017 / 2500000000000000 : ℝ) ≤
+            (2866812181602874089037 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2217760205919267 / 5000000000000000 : ℝ)) (q :=
+      (4435556859998017 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_420_2 :
+    Uω (aρ 2) (bρ 2) (1800036210497 / 4000000000000 : ℝ) ≤
+      (-(4091756635633582914549 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8823167618011783 / 20000000000000000 : ℝ) ≤
+      (-(4091756635633582914549 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8823167618011783 / 20000000000000000 : ℝ) =
+            (8823167618011783 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8823167618011783 / 5000000000000000 : ℝ) ≤
+            (2839715167366417085451 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4411343322364283 / 10000000000000000 : ℝ)) (q :=
+      (8823167618011783 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_420_3 :
+    Uω (aρ 3) (bρ 3) (1800036210497 / 4000000000000 : ℝ) ≤
+      (-(8292921607161953238429 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8727160963465661 / 20000000000000000 : ℝ) ≤
+      (-(8292921607161953238429 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8727160963465661 / 20000000000000000 : ℝ) =
+            (8727160963465661 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8727160963465661 / 5000000000000000 : ℝ) ≤
+            (5570021998838046761571 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4362732774168161 / 10000000000000000 : ℝ)) (q :=
+      (8727160963465661 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_420_4 :
+    Uω (aρ 4) (bρ 4) (1800036210497 / 4000000000000 : ℝ) ≤
+      (-(1059716367714879467071 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1713471220459449 / 4000000000000000 : ℝ) ≤
+      (-(1059716367714879467071 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1713471220459449 / 4000000000000000 : ℝ) =
+            (1713471220459449 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1713471220459449 / 1000000000000000 : ℝ) ≤
+            (673151583035120532929 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (856244418812949 / 2000000000000000 : ℝ)) (q :=
+      (1713471220459449 / 4000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_420_5 :
+    Uω (aρ 5) (bρ 5) (1800036210497 / 4000000000000 : ℝ) ≤
+      (-(1753397194246807411647 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1664618159464713 / 4000000000000000 : ℝ) ≤
+      (-(1753397194246807411647 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1664618159464713 / 4000000000000000 : ℝ) =
+            (1664618159464713 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1664618159464713 / 1000000000000000 : ℝ) ≤
+            (1019191526953192588353 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (831029341685213 / 2000000000000000 : ℝ)) (q :=
+      (1664618159464713 / 4000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_420_6 :
+    Uω (aρ 6) (bρ 6) (1800036210497 / 4000000000000 : ℝ) ≤
+      (-(183994155135901900949 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7970613749913629 / 20000000000000000 : ℝ) ≤
+      (-(183994155135901900949 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7970613749913629 / 20000000000000000 : ℝ) =
+            (7970613749913629 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7970613749913629 / 5000000000000000 : ℝ) ≤
+            (93264716984098099051 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3969910036156129 / 10000000000000000 : ℝ)) (q :=
+      (7970613749913629 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_420_7 :
+    Uω (aρ 7) (bρ 7) (1800036210497 / 4000000000000 : ℝ) ≤
+      (-(9828601241911390730523 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7484783918439749 / 20000000000000000 : ℝ) ≤
+      (-(9828601241911390730523 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7484783918439749 / 20000000000000000 : ℝ) =
+            (7484783918439749 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7484783918439749 / 5000000000000000 : ℝ) ≤
+            (4034342364088609269477 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3707741585442249 / 10000000000000000 : ℝ)) (q :=
+      (7484783918439749 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_420_8 :
+    Uω (aρ 8) (bρ 8) (1800036210497 / 4000000000000 : ℝ) ≤
+      (-(1073020264259629767199 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3419741037623649 / 10000000000000000 : ℝ) ≤
+      (-(1073020264259629767199 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3419741037623649 / 10000000000000000 : ℝ) =
+            (3419741037623649 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3419741037623649 / 2500000000000000 : ℝ) ≤
+            (313274096340370232801 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1672990696794899 / 5000000000000000 : ℝ)) (q :=
+      (3419741037623649 / 10000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_420_9 :
+    Uω (aρ 9) (bρ 9) (1800036210497 / 4000000000000 : ℝ) ≤
+      (-(12034183515403065078679 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (120066552720337 / 400000000000000 : ℝ) ≤
+      (-(12034183515403065078679 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (120066552720337 / 400000000000000 : ℝ) =
+            (120066552720337 / 100000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (120066552720337 / 100000000000000 : ℝ) ≤
+            (1828760090596934921321 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (57015851654387 / 200000000000000 : ℝ)) (q :=
+      (120066552720337 / 400000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_420_10 :
+    Uω (aρ 10) (bρ 10) (1800036210497 / 4000000000000 : ℝ) ≤
+      (-(561315799852319578607 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4915742317110657 / 20000000000000000 : ℝ) ≤
+      (-(561315799852319578607 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4915742317110657 / 20000000000000000 : ℝ) =
+            (4915742317110657 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4915742317110657 / 2500000000000000 : ℝ) ≤
+            (270460816507680421393 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2151766297308157 / 10000000000000000 : ℝ)) (q :=
+      (4915742317110657 / 20000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_420_11 :
+    Uω (aρ 11) (bρ 11) (1800036210497 / 4000000000000 : ℝ) ≤
+      (-(18094246652620039559979 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3274966393514093 / 20000000000000000 : ℝ) ≤
+      (-(18094246652620039559979 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3274966393514093 / 20000000000000000 : ℝ) =
+            (3274966393514093 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3274966393514093 / 2500000000000000 : ℝ) ≤
+            (2700168756379960440021 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (929466999916593 / 10000000000000000 : ℝ)) (q :=
+      (3274966393514093 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_420_12 :
+    Uω (aρ 12) (bρ 12) (1800036210497 / 4000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_420_13 :
+    Uω (aρ 13) (bρ 13) (1800036210497 / 4000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_420_14 :
+    Uω (aρ 14) (bρ 14) (1800036210497 / 4000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_420_15 :
+    Uω (aρ 15) (bρ 15) (1800036210497 / 4000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_420_16 :
+    Uω (aρ 16) (bρ 16) (1800036210497 / 4000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_420 :
+    Uρ (1800036210497 / 4000000000000 : ℝ) ≤
+      (-(5981627919399893279119 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1800036210497 / 4000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(4064659621397125910963 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_420_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1800036210497 / 4000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(4091756635633582914549 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_420_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1800036210497 / 4000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(8292921607161953238429 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_420_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1800036210497 / 4000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1059716367714879467071 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_420_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1800036210497 / 4000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1753397194246807411647 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_420_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1800036210497 / 4000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(183994155135901900949 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_420_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1800036210497 / 4000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(9828601241911390730523 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_420_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1800036210497 / 4000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(1073020264259629767199 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_420_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1800036210497 / 4000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(12034183515403065078679 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_420_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1800036210497 / 4000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) * (-(561315799852319578607 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_420_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1800036210497 / 4000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(18094246652620039559979 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_420_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1800036210497 / 4000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_420_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1800036210497 / 4000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_420_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1800036210497 / 4000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_420_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1800036210497 / 4000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_420_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1800036210497 / 4000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_420_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_421_1 :
+    Uω (aρ 1) (bρ 1) (7210739241 / 16000000000 : ℝ) ≤
+      (-(8114402026328109846991 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1776871365484197 / 4000000000000000 : ℝ) ≤
+      (-(8114402026328109846991 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1776871365484197 / 4000000000000000 : ℝ) =
+            (1776871365484197 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1776871365484197 / 1000000000000000 : ℝ) ≤
+            (5748541579671890153009 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (888428403976197 / 2000000000000000 : ℝ)) (q :=
+      (1776871365484197 / 4000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_421_2 :
+    Uω (aρ 2) (bρ 2) (7210739241 / 16000000000 : ℝ) ≤
+      (-(4084257180438267317039 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4418205668821777 / 10000000000000000 : ℝ) ≤
+      (-(4084257180438267317039 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4418205668821777 / 10000000000000000 : ℝ) =
+            (4418205668821777 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4418205668821777 / 2500000000000000 : ℝ) ≤
+            (2847214622561732682961 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2208982771306777 / 5000000000000000 : ℝ)) (q :=
+      (4418205668821777 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_421_3 :
+    Uω (aρ 3) (bρ 3) (7210739241 / 16000000000 : ℝ) ≤
+      (-(8277755705431535785367 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1748081306306361 / 4000000000000000 : ℝ) ≤
+      (-(8277755705431535785367 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1748081306306361 / 4000000000000000 : ℝ) =
+            (1748081306306361 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1748081306306361 / 1000000000000000 : ℝ) ≤
+            (5585187900568464214633 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (873871368570361 / 2000000000000000 : ℝ)) (q :=
+      (1748081306306361 / 4000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_421_4 :
+    Uω (aρ 4) (bρ 4) (7210739241 / 16000000000 : ℝ) ≤
+      (-(67698212214232369843 / 80000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4290303343137727 / 10000000000000000 : ℝ) ≤
+      (-(67698212214232369843 / 80000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4290303343137727 / 10000000000000000 : ℝ) =
+            (4290303343137727 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4290303343137727 / 2500000000000000 : ℝ) ≤
+            (43205336633767630157 / 80000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2143925589330227 / 5000000000000000 : ℝ)) (q :=
+      (4290303343137727 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_421_5 :
+    Uω (aρ 5) (bρ 5) (7210739241 / 16000000000 : ℝ) ≤
+      (-(8751063036737577972069 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (52102213481619 / 125000000000000 : ℝ) ≤
+      (-(8751063036737577972069 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (52102213481619 / 125000000000000 : ℝ) = (52102213481619 / 31250000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (52102213481619 / 31250000000000 : ℝ) ≤
+            (5111880569262422027931 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (26011178554869 / 62500000000000 : ℝ)) (q :=
+      (52102213481619 / 125000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_421_6 :
+    Uω (aρ 6) (bρ 6) (7210739241 / 16000000000 : ℝ) ≤
+      (-(2295760635244161997167 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (798390802426913 / 2000000000000000 : ℝ) ≤
+      (-(2295760635244161997167 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (798390802426913 / 2000000000000000 : ℝ) =
+            (798390802426913 / 500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (798390802426913 / 500000000000000 : ℝ) ≤
+            (1169975266255838002833 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (397658281112913 / 1000000000000000 : ℝ)) (q :=
+      (798390802426913 / 2000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_421_7 :
+    Uω (aρ 7) (bρ 7) (7210739241 / 16000000000 : ℝ) ≤
+      (-(306586215248868418239 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7498150455807673 / 20000000000000000 : ℝ) ≤
+      (-(306586215248868418239 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7498150455807673 / 20000000000000000 : ℝ) =
+            (7498150455807673 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7498150455807673 / 5000000000000000 : ℝ) ≤
+            (126630772438631581761 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3714486623427673 / 10000000000000000 : ℝ)) (q :=
+      (7498150455807673 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_421_8 :
+    Uω (aρ 8) (bρ 8) (7210739241 / 16000000000 : ℝ) ≤
+      (-(1338804206099587571027 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (685301641672891 / 2000000000000000 : ℝ) ≤
+      (-(1338804206099587571027 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (685301641672891 / 2000000000000000 : ℝ) =
+            (685301641672891 / 500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (685301641672891 / 500000000000000 : ℝ) ≤
+            (394063744650412428973 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (335289423568891 / 1000000000000000 : ℝ)) (q :=
+      (685301641672891 / 2000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_421_9 :
+    Uω (aρ 9) (bρ 9) (7210739241 / 16000000000 : ℝ) ≤
+      (-(12010986417487087268443 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1203453955705317 / 4000000000000000 : ℝ) ≤
+      (-(12010986417487087268443 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1203453955705317 / 4000000000000000 : ℝ) =
+            (1203453955705317 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1203453955705317 / 1000000000000000 : ℝ) ≤
+            (1851957188512912731557 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (571622645169317 / 2000000000000000 : ℝ)) (q :=
+      (1203453955705317 / 4000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_421_10 :
+    Uω (aρ 10) (bρ 10) (7210739241 / 16000000000 : ℝ) ≤
+      (-(350054580804219201741 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2465431317835013 / 10000000000000000 : ℝ) ≤
+      (-(350054580804219201741 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2465431317835013 / 10000000000000000 : ℝ) =
+            (2465431317835013 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2465431317835013 / 1250000000000000 : ℝ) ≤
+            (169805804420780798259 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1080132558242513 / 5000000000000000 : ℝ)) (q :=
+      (2465431317835013 / 10000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_421_11 :
+    Uω (aρ 11) (bρ 11) (7210739241 / 16000000000 : ℝ) ≤
+      (-(4505909143362031622123 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (659634558201427 / 4000000000000000 : ℝ) ≤
+      (-(4505909143362031622123 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (659634558201427 / 4000000000000000 : ℝ) =
+            (659634558201427 / 500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (659634558201427 / 500000000000000 : ℝ) ≤
+            (692694708887968377877 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (189210379605427 / 2000000000000000 : ℝ)) (q :=
+      (659634558201427 / 4000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11]) (by norm_num [bρ_11])
+      (by norm_num) (by norm_num [aρ_11, bρ_11]) (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_421_12 :
+    Uω (aρ 12) (bρ 12) (7210739241 / 16000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_421_13 :
+    Uω (aρ 13) (bρ 13) (7210739241 / 16000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_421_14 :
+    Uω (aρ 14) (bρ 14) (7210739241 / 16000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_421_15 :
+    Uω (aρ 15) (bρ 15) (7210739241 / 16000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_421_16 :
+    Uω (aρ 16) (bρ 16) (7210739241 / 16000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_421 :
+    Uρ (7210739241 / 16000000000 : ℝ) ≤
+      (-(11945186561807156496291 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (7210739241 / 16000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(8114402026328109846991 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_421_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (7210739241 / 16000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(4084257180438267317039 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_421_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (7210739241 / 16000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(8277755705431535785367 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_421_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (7210739241 / 16000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(67698212214232369843 / 80000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_421_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (7210739241 / 16000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(8751063036737577972069 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_421_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (7210739241 / 16000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(2295760635244161997167 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_421_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (7210739241 / 16000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(306586215248868418239 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_421_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (7210739241 / 16000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(1338804206099587571027 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_421_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (7210739241 / 16000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(12010986417487087268443 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_421_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (7210739241 / 16000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) * (-(350054580804219201741 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_421_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (7210739241 / 16000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4505909143362031622123 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_421_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (7210739241 / 16000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_421_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (7210739241 / 16000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_421_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (7210739241 / 16000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_421_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (7210739241 / 16000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_421_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (7210739241 / 16000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_421_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_422_1 :
+    Uω (aρ 1) (bρ 1) (1805333410003 / 4000000000000 : ℝ) ≤
+      (-(2024876757354868561347 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8897599934522477 / 20000000000000000 : ℝ) ≤
+      (-(2024876757354868561347 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8897599934522477 / 20000000000000000 : ℝ) =
+            (8897599934522477 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8897599934522477 / 5000000000000000 : ℝ) ≤
+            (1440859144145131438653 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4448763627599977 / 10000000000000000 : ℝ)) (q :=
+      (8897599934522477 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_422_2 :
+    Uω (aρ 2) (bρ 2) (1805333410003 / 4000000000000 : ℝ) ≤
+      (-(1630707583193492241937 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (442482752755881 / 1000000000000000 : ℝ) ≤
+      (-(1630707583193492241937 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (442482752755881 / 1000000000000000 : ℝ) =
+            (442482752755881 / 250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (442482752755881 / 250000000000000 : ℝ) ≤
+            (1141881138006507758063 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (27653673504407 / 62500000000000 : ℝ)) (q :=
+      (442482752755881 / 1000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_422_3 :
+    Uω (aρ 3) (bρ 3) (1805333410003 / 4000000000000 : ℝ) ≤
+      (-(4131306389108922398569 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4376826045910491 / 10000000000000000 : ℝ) ≤
+      (-(4131306389108922398569 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4376826045910491 / 10000000000000000 : ℝ) =
+            (4376826045910491 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4376826045910491 / 2500000000000000 : ℝ) ≤
+            (2800165413891077601431 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2187990451879241 / 5000000000000000 : ℝ)) (q :=
+      (4376826045910491 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_422_4 :
+    Uω (aρ 4) (bρ 4) (1805333410003 / 4000000000000 : ℝ) ≤
+      (-(8446845986117236546949 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2148464311712387 / 5000000000000000 : ℝ) ≤
+      (-(8446845986117236546949 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2148464311712387 / 5000000000000000 : ℝ) =
+            (2148464311712387 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2148464311712387 / 1250000000000000 : ℝ) ≤
+            (5416097619882763453051 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (268405014990753 / 625000000000000 : ℝ)) (q :=
+      (2148464311712387 / 5000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_422_5 :
+    Uω (aρ 5) (bρ 5) (1805333410003 / 4000000000000 : ℝ) ≤
+      (-(8735165493515219476671 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8349617452008769 / 20000000000000000 : ℝ) ≤
+      (-(8735165493515219476671 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8349617452008769 / 20000000000000000 : ℝ) =
+            (8349617452008769 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8349617452008769 / 5000000000000000 : ℝ) ≤
+            (5127778112484780523329 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4168430364346269 / 10000000000000000 : ℝ)) (q :=
+      (8349617452008769 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_422_6 :
+    Uω (aρ 6) (bρ 6) (1805333410003 / 4000000000000 : ℝ) ≤
+      (-(9166405265891975863527 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7997202127490299 / 20000000000000000 : ℝ) ≤
+      (-(9166405265891975863527 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7997202127490299 / 20000000000000000 : ℝ) =
+            (7997202127490299 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7997202127490299 / 5000000000000000 : ℝ) ≤
+            (4696538340108024136473 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3983255414967799 / 10000000000000000 : ℝ)) (q :=
+      (7997202127490299 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_422_7 :
+    Uω (aρ 7) (bρ 7) (1805333410003 / 4000000000000 : ℝ) ≤
+      (-(4896474451519480225219 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7511516549429117 / 20000000000000000 : ℝ) ≤
+      (-(4896474451519480225219 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7511516549429117 / 20000000000000000 : ℝ) =
+            (7511516549429117 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7511516549429117 / 5000000000000000 : ℝ) ≤
+            (2034997351480519774781 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3721231217666617 / 10000000000000000 : ℝ)) (q :=
+      (7511516549429117 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_422_8 :
+    Uω (aρ 8) (bρ 8) (1805333410003 / 4000000000000 : ℝ) ≤
+      (-(10690705368396436709341 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1373309916918359 / 4000000000000000 : ℝ) ≤
+      (-(10690705368396436709341 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1373309916918359 / 4000000000000000 : ℝ) =
+            (1373309916918359 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1373309916918359 / 1000000000000000 : ℝ) ≤
+            (3172238237603563290659 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (671961180833859 / 2000000000000000 : ℝ)) (q :=
+      (1373309916918359 / 4000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_422_9 :
+    Uω (aρ 9) (bρ 9) (1805333410003 / 4000000000000 : ℝ) ≤
+      (-(5993924322941902594337 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6031208519259153 / 20000000000000000 : ℝ) ≤
+      (-(5993924322941902594337 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6031208519259153 / 20000000000000000 : ℝ) =
+            (6031208519259153 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6031208519259153 / 5000000000000000 : ℝ) ≤
+            (937547480058097405663 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2865430467196653 / 10000000000000000 : ℝ)) (q :=
+      (6031208519259153 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_422_10 :
+    Uω (aρ 10) (bρ 10) (1805333410003 / 4000000000000 : ℝ) ≤
+      (-(2794318392676469736287 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (618246233222387 / 2500000000000000 : ℝ) ≤
+      (-(2794318392676469736287 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (618246233222387 / 2500000000000000 : ℝ) =
+            (618246233222387 / 312500000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (618246233222387 / 312500000000000 : ℝ) ≤
+            (1364564689123530263713 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (542187711802899 / 2500000000000000 : ℝ)) (q :=
+      (618246233222387 / 2500000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10]) (by norm_num [bρ_10])
+      (by norm_num) (by norm_num [aρ_10, bρ_10]) (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_422_11 :
+    Uω (aρ 11) (bρ 11) (1805333410003 / 4000000000000 : ℝ) ≤
+      (-(3590848971023895272907 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3321138969712857 / 20000000000000000 : ℝ) ≤
+      (-(3590848971023895272907 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3321138969712857 / 20000000000000000 : ℝ) =
+            (3321138969712857 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3321138969712857 / 2500000000000000 : ℝ) ≤
+            (568034110776104727093 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (962396577350357 / 10000000000000000 : ℝ)) (q :=
+      (3321138969712857 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_422_12 :
+    Uω (aρ 12) (bρ 12) (1805333410003 / 4000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_422_13 :
+    Uω (aρ 13) (bρ 13) (1805333410003 / 4000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_422_14 :
+    Uω (aρ 14) (bρ 14) (1805333410003 / 4000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_422_15 :
+    Uω (aρ 15) (bρ 15) (1805333410003 / 4000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_422_16 :
+    Uω (aρ 16) (bρ 16) (1805333410003 / 4000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_422 :
+    Uρ (1805333410003 / 4000000000000 : ℝ) ≤
+      (-(11927242524291849257171 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1805333410003 / 4000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2024876757354868561347 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_422_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1805333410003 / 4000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1630707583193492241937 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_422_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1805333410003 / 4000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(4131306389108922398569 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_422_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1805333410003 / 4000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(8446845986117236546949 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_422_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1805333410003 / 4000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(8735165493515219476671 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_422_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1805333410003 / 4000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(9166405265891975863527 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_422_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1805333410003 / 4000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(4896474451519480225219 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_422_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1805333410003 / 4000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(10690705368396436709341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_422_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1805333410003 / 4000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(5993924322941902594337 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_422_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1805333410003 / 4000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(2794318392676469736287 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_422_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1805333410003 / 4000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(3590848971023895272907 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_422_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1805333410003 / 4000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_422_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1805333410003 / 4000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_422_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1805333410003 / 4000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_422_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1805333410003 / 4000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_422_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1805333410003 / 4000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_422_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_423_1 :
+    Uω (aρ 1) (bρ 1) (451995502439 / 1000000000000 : ℝ) ≤
+      (-(4042317092986279800841 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (139231922520343 / 312500000000000 : ℝ) ≤
+      (-(4042317092986279800841 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (139231922520343 / 312500000000000 : ℝ) = (139231922520343 / 78125000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (139231922520343 / 78125000000000 : ℝ) ≤
+            (2889154710013720199159 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (556923154374619 / 1250000000000000 : ℝ)) (q :=
+      (139231922520343 / 312500000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_423_2 :
+    Uω (aρ 2) (bρ 2) (451995502439 / 1000000000000 : ℝ) ≤
+      (-(4069291934667686941101 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1772579754088731 / 4000000000000000 : ℝ) ≤
+      (-(4069291934667686941101 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1772579754088731 / 4000000000000000 : ℝ) =
+            (1772579754088731 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1772579754088731 / 1000000000000000 : ℝ) ≤
+            (2862179868332313058899 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (886241995329731 / 2000000000000000 : ℝ)) (q :=
+      (1772579754088731 / 4000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_423_3 :
+    Uω (aρ 3) (bρ 3) (451995502439 / 1000000000000 : ℝ) ≤
+      (-(4123746377996051450701 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4383448822184229 / 10000000000000000 : ℝ) ≤
+      (-(4123746377996051450701 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4383448822184229 / 10000000000000000 : ℝ) =
+            (4383448822184229 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4383448822184229 / 2500000000000000 : ℝ) ≤
+            (2807725425003948549299 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2191302478461729 / 5000000000000000 : ℝ)) (q :=
+      (4383448822184229 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_423_4 :
+    Uω (aρ 4) (bρ 4) (451995502439 / 1000000000000 : ℝ) ≤
+      (-(4215719623000383988557 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (33621514781749 / 78125000000000 : ℝ) ≤
+      (-(4215719623000383988557 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (33621514781749 / 78125000000000 : ℝ) = (33621514781749 / 19531250000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (33621514781749 / 19531250000000 : ℝ) ≤
+            (2715752179999616011443 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (134409664929617 / 312500000000000 : ℝ)) (q :=
+      (33621514781749 / 78125000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_423_5 :
+    Uω (aρ 5) (bρ 5) (451995502439 / 1000000000000 : ℝ) ≤
+      (-(1743858652094064989949 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8362880682481939 / 20000000000000000 : ℝ) ≤
+      (-(1743858652094064989949 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8362880682481939 / 20000000000000000 : ℝ) =
+            (8362880682481939 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8362880682481939 / 5000000000000000 : ℝ) ≤
+            (1028730069105935010051 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4175072095436939 / 10000000000000000 : ℝ)) (q :=
+      (8362880682481939 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_423_6 :
+    Uω (aρ 6) (bρ 6) (451995502439 / 1000000000000 : ℝ) ≤
+      (-(914979583729635317381 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2002624015108933 / 5000000000000000 : ℝ) ≤
+      (-(914979583729635317381 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2002624015108933 / 5000000000000000 : ℝ) =
+            (2002624015108933 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2002624015108933 / 1250000000000000 : ℝ) ≤
+            (471314776870364682619 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (997481962132683 / 2500000000000000 : ℝ)) (q :=
+      (2002624015108933 / 5000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_423_7 :
+    Uω (aρ 7) (bρ 7) (451995502439 / 1000000000000 : ℝ) ≤
+      (-(9775171167791601484043 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3762441100856269 / 10000000000000000 : ℝ) ≤
+      (-(9775171167791601484043 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3762441100856269 / 10000000000000000 : ℝ) =
+            (3762441100856269 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3762441100856269 / 2500000000000000 : ℝ) ≤
+            (4087772438208398515957 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1863987685283769 / 5000000000000000 : ℝ)) (q :=
+      (3762441100856269 / 10000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_423_8 :
+    Uω (aρ 8) (bρ 8) (451995502439 / 1000000000000 : ℝ) ≤
+      (-(1067101762719614924343 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3440040793032031 / 10000000000000000 : ℝ) ≤
+      (-(1067101762719614924343 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3440040793032031 / 10000000000000000 : ℝ) =
+            (3440040793032031 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3440040793032031 / 2500000000000000 : ℝ) ≤
+            (319192597880385075657 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1683358203129531 / 5000000000000000 : ℝ)) (q :=
+      (3440040793032031 / 10000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_423_9 :
+    Uω (aρ 9) (bρ 9) (451995502439 / 1000000000000 : ℝ) ≤
+      (-(239295397413106100289 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6045143884208817 / 20000000000000000 : ℝ) ≤
+      (-(239295397413106100289 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6045143884208817 / 20000000000000000 : ℝ) =
+            (6045143884208817 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6045143884208817 / 5000000000000000 : ℝ) ≤
+            (37963474706893899711 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2872744332763817 / 10000000000000000 : ℝ)) (q :=
+      (6045143884208817 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_423_10 :
+    Uω (aρ 10) (bρ 10) (451995502439 / 1000000000000 : ℝ) ≤
+      (-(6970560032833848866531 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2480532080237303 / 10000000000000000 : ℝ) ≤
+      (-(6970560032833848866531 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2480532080237303 / 10000000000000000 : ℝ) =
+            (2480532080237303 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2480532080237303 / 1250000000000000 : ℝ) ≤
+            (3426647671666151133469 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1088611821262303 / 5000000000000000 : ℝ)) (q :=
+      (2480532080237303 / 10000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_423_11 :
+    Uω (aρ 11) (bρ 11) (451995502439 / 1000000000000 : ℝ) ≤
+      (-(4471503425795417789409 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (668775393450461 / 4000000000000000 : ℝ) ≤
+      (-(4471503425795417789409 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (668775393450461 / 4000000000000000 : ℝ) =
+            (668775393450461 / 500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (668775393450461 / 500000000000000 : ℝ) ≤
+            (727100426454582210591 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (195702615101461 / 2000000000000000 : ℝ)) (q :=
+      (668775393450461 / 4000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11]) (by norm_num [bρ_11])
+      (by norm_num) (by norm_num [aρ_11, bρ_11]) (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_423_12 :
+    Uω (aρ 12) (bρ 12) (451995502439 / 1000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_423_13 :
+    Uω (aρ 13) (bρ 13) (451995502439 / 1000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_423_14 :
+    Uω (aρ 14) (bρ 14) (451995502439 / 1000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_423_15 :
+    Uω (aρ 15) (bρ 15) (451995502439 / 1000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_423_16 :
+    Uω (aρ 16) (bρ 16) (451995502439 / 1000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_423 :
+    Uρ (451995502439 / 1000000000000 : ℝ) ≤
+      (-(744338687036184402011 / 625000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (451995502439 / 1000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(4042317092986279800841 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_423_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (451995502439 / 1000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(4069291934667686941101 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_423_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (451995502439 / 1000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(4123746377996051450701 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_423_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (451995502439 / 1000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(4215719623000383988557 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_423_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (451995502439 / 1000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1743858652094064989949 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_423_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (451995502439 / 1000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(914979583729635317381 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_423_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (451995502439 / 1000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(9775171167791601484043 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_423_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (451995502439 / 1000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(1067101762719614924343 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_423_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (451995502439 / 1000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(239295397413106100289 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_423_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (451995502439 / 1000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(6970560032833848866531 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_423_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (451995502439 / 1000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4471503425795417789409 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_423_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (451995502439 / 1000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_423_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (451995502439 / 1000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_423_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (451995502439 / 1000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_423_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (451995502439 / 1000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_423_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (451995502439 / 1000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_423_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/U35.lean b/LeanPool/Zeta5Irrational/Table/U35.lean
new file mode 100644
index 0000000000..e6b97745a6
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/U35.lean
@@ -0,0 +1,4371 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-! # Certified arcsine potential bounds (U35) -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma U_424_1 :
+    Uω (aρ 1) (bρ 1) (1810630609509 / 4000000000000 : ℝ) ≤
+      (-(8069783430186067276137 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8924086147760843 / 20000000000000000 : ℝ) ≤
+      (-(8069783430186067276137 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8924086147760843 / 20000000000000000 : ℝ) =
+            (8924086147760843 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8924086147760843 / 5000000000000000 : ℝ) ≤
+            (5793160175813932723863 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4462006842073343 / 10000000000000000 : ℝ)) (q :=
+      (8924086147760843 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_424_2 :
+    Uω (aρ 2) (bρ 2) (1810630609509 / 4000000000000 : ℝ) ≤
+      (-(1624730430815369043297 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8876142483631273 / 20000000000000000 : ℝ) ≤
+      (-(1624730430815369043297 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8876142483631273 / 20000000000000000 : ℝ) =
+            (8876142483631273 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8876142483631273 / 5000000000000000 : ℝ) ≤
+            (1147858290384630956703 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4437832190453773 / 10000000000000000 : ℝ)) (q :=
+      (8876142483631273 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_424_3 :
+    Uω (aρ 3) (bρ 3) (1810630609509 / 4000000000000 : ℝ) ≤
+      (-(4116197784770404230469 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4390071594604641 / 10000000000000000 : ℝ) ≤
+      (-(4116197784770404230469 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4390071594604641 / 10000000000000000 : ℝ) =
+            (4390071594604641 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4390071594604641 / 2500000000000000 : ℝ) ≤
+            (2815274018229595769531 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2194614501190891 / 5000000000000000 : ℝ)) (q :=
+      (4390071594604641 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_424_4 :
+    Uω (aρ 4) (bρ 4) (1810630609509 / 4000000000000 : ℝ) ≤
+      (-(8416056233038239239309 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8620358298217587 / 20000000000000000 : ℝ) ≤
+      (-(8416056233038239239309 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8620358298217587 / 20000000000000000 : ℝ) =
+            (8620358298217587 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8620358298217587 / 5000000000000000 : ℝ) ≤
+            (5446887372961760760691 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4307738292455087 / 10000000000000000 : ℝ)) (q :=
+      (8620358298217587 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_424_5 :
+    Uω (aρ 5) (bρ 5) (1810630609509 / 4000000000000 : ℝ) ≤
+      (-(8703446256895322532487 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (837614384878577 / 2000000000000000 : ℝ) ≤
+      (-(8703446256895322532487 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (837614384878577 / 2000000000000000 : ℝ) =
+            (837614384878577 / 500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (837614384878577 / 500000000000000 : ℝ) ≤
+            (5159497349104677467513 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (418171376235827 / 1000000000000000 : ℝ)) (q :=
+      (837614384878577 / 2000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_424_6 :
+    Uω (aρ 6) (bρ 6) (1810630609509 / 4000000000000 : ℝ) ≤
+      (-(2283303540356153156021 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8023789823958269 / 20000000000000000 : ℝ) ≤
+      (-(2283303540356153156021 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8023789823958269 / 20000000000000000 : ℝ) =
+            (8023789823958269 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8023789823958269 / 5000000000000000 : ℝ) ≤
+            (1182432361143846843979 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3996600112670769 / 10000000000000000 : ℝ)) (q :=
+      (8023789823958269 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_424_7 :
+    Uω (aρ 7) (bρ 7) (1810630609509 / 4000000000000 : ℝ) ≤
+      (-(2439356390886610766343 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (188456185376221 / 500000000000000 : ℝ) ≤
+      (-(2439356390886610766343 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (188456185376221 / 500000000000000 : ℝ) =
+            (188456185376221 / 125000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (188456185376221 / 125000000000000 : ℝ) ≤
+            (1026379510613389233657 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (186735954226067 / 500000000000000 : ℝ)) (q :=
+      (188456185376221 / 500000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_424_8 :
+    Uω (aρ 8) (bρ 8) (1810630609509 / 4000000000000 : ℝ) ≤
+      (-(10651370252152340611 / 10000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (344680621415669 / 1000000000000000 : ℝ) ≤
+      (-(10651370252152340611 / 10000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (344680621415669 / 1000000000000000 : ℝ) =
+            (344680621415669 / 250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (344680621415669 / 250000000000000 : ℝ) ≤
+            (3211573353847659389 / 10000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (84340643728147 / 250000000000000 : ℝ)) (q :=
+      (344680621415669 / 1000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_424_9 :
+    Uω (aρ 9) (bρ 9) (1810630609509 / 4000000000000 : ℝ) ≤
+      (-(1194174976479651820651 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6059075899093861 / 20000000000000000 : ℝ) ≤
+      (-(1194174976479651820651 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6059075899093861 / 20000000000000000 : ℝ) =
+            (6059075899093861 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6059075899093861 / 5000000000000000 : ℝ) ≤
+            (192119384120348179349 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2880054848266361 / 10000000000000000 : ℝ)) (q :=
+      (6059075899093861 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_424_10 :
+    Uω (aρ 10) (bρ 10) (1810630609509 / 4000000000000 : ℝ) ≤
+      (-(13910766432101367482929 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2488072835094411 / 10000000000000000 : ℝ) ≤
+      (-(13910766432101367482929 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2488072835094411 / 10000000000000000 : ℝ) =
+            (2488072835094411 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2488072835094411 / 1250000000000000 : ℝ) ≤
+            (6883648976898632517071 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1092841826428161 / 5000000000000000 : ℝ)) (q :=
+      (2488072835094411 / 10000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_424_11 :
+    Uω (aρ 11) (bρ 11) (1810630609509 / 4000000000000 : ℝ) ≤
+      (-(8909444894819806137933 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3366397878121701 / 20000000000000000 : ℝ) ≤
+      (-(8909444894819806137933 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3366397878121701 / 20000000000000000 : ℝ) =
+            (3366397878121701 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3366397878121701 / 2500000000000000 : ℝ) ≤
+            (1487762809680193862067 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (994412486994201 / 10000000000000000 : ℝ)) (q :=
+      (3366397878121701 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_424_12 :
+    Uω (aρ 12) (bρ 12) (1810630609509 / 4000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_424_13 :
+    Uω (aρ 13) (bρ 13) (1810630609509 / 4000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_424_14 :
+    Uω (aρ 14) (bρ 14) (1810630609509 / 4000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_424_15 :
+    Uω (aρ 15) (bρ 15) (1810630609509 / 4000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_424_16 :
+    Uω (aρ 16) (bρ 16) (1810630609509 / 4000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_424 :
+    Uρ (1810630609509 / 4000000000000 : ℝ) ≤
+      (-(11891711587405053258781 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1810630609509 / 4000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(8069783430186067276137 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_424_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1810630609509 / 4000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1624730430815369043297 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_424_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1810630609509 / 4000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(4116197784770404230469 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_424_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1810630609509 / 4000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(8416056233038239239309 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_424_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1810630609509 / 4000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(8703446256895322532487 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_424_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1810630609509 / 4000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(2283303540356153156021 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_424_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1810630609509 / 4000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(2439356390886610766343 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_424_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1810630609509 / 4000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) * (-(10651370252152340611 / 10000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_424_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1810630609509 / 4000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(1194174976479651820651 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_424_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1810630609509 / 4000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(13910766432101367482929 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_424_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1810630609509 / 4000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(8909444894819806137933 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_424_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1810630609509 / 4000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_424_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1810630609509 / 4000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_424_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1810630609509 / 4000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_424_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1810630609509 / 4000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_424_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1810630609509 / 4000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_424_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_425_1 :
+    Uω (aρ 1) (bρ 1) (906639604631 / 2000000000000 : ℝ) ≤
+      (-(4027477348275718747337 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (357493170156023 / 800000000000000 : ℝ) ≤
+      (-(4027477348275718747337 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (357493170156023 / 800000000000000 : ℝ) =
+            (357493170156023 / 200000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (357493170156023 / 200000000000000 : ℝ) ≤
+            (2903994454724281252663 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (178745137953223 / 400000000000000 : ℝ)) (q :=
+      (357493170156023 / 800000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_425_2 :
+    Uω (aρ 2) (bρ 2) (906639604631 / 2000000000000 : ℝ) ≤
+      (-(4054371351793893970459 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1777877238938007 / 4000000000000000 : ℝ) ≤
+      (-(4054371351793893970459 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1777877238938007 / 4000000000000000 : ℝ) =
+            (1777877238938007 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1777877238938007 / 1000000000000000 : ℝ) ≤
+            (2877100451206106029541 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (888890880426007 / 2000000000000000 : ℝ)) (q :=
+      (1777877238938007 / 4000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_425_3 :
+    Uω (aρ 3) (bρ 3) (906639604631 / 2000000000000 : ℝ) ≤
+      (-(8217321149963825466591 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8793388726378293 / 20000000000000000 : ℝ) ≤
+      (-(8217321149963825466591 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8793388726378293 / 20000000000000000 : ℝ) =
+            (8793388726378293 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8793388726378293 / 5000000000000000 : ℝ) ≤
+            (5645622456036174533409 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4395853040168293 / 10000000000000000 : ℝ)) (q :=
+      (8793388726378293 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_425_4 :
+    Uω (aρ 4) (bρ 4) (906639604631 / 2000000000000 : ℝ) ≤
+      (-(4200348437088783331463 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4316804394612983 / 10000000000000000 : ℝ) ≤
+      (-(4200348437088783331463 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4316804394612983 / 10000000000000000 : ℝ) =
+            (4316804394612983 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4316804394612983 / 2500000000000000 : ℝ) ≤
+            (2731123365911216668537 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2157183642040483 / 5000000000000000 : ℝ)) (q :=
+      (4316804394612983 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_425_5 :
+    Uω (aρ 5) (bρ 5) (906639604631 / 2000000000000 : ℝ) ≤
+      (-(8687624402469212596989 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8389406951225531 / 20000000000000000 : ℝ) ≤
+      (-(8687624402469212596989 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8389406951225531 / 20000000000000000 : ℝ) =
+            (8389406951225531 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8389406951225531 / 5000000000000000 : ℝ) ≤
+            (5175319203530787403011 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4188355365415531 / 10000000000000000 : ℝ)) (q :=
+      (8389406951225531 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_425_6 :
+    Uω (aρ 6) (bρ 6) (906639604631 / 2000000000000 : ℝ) ≤
+      (-(1139582518123484126347 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8037083418905043 / 20000000000000000 : ℝ) ≤
+      (-(1139582518123484126347 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8037083418905043 / 20000000000000000 : ℝ) =
+            (8037083418905043 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8037083418905043 / 5000000000000000 : ℝ) ≤
+            (593285432626515873653 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4003272208235043 / 10000000000000000 : ℝ)) (q :=
+      (8037083418905043 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_425_7 :
+    Uω (aρ 7) (bρ 7) (906639604631 / 2000000000000 : ℝ) ≤
+      (-(1947942394458633086619 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3775806095905767 / 10000000000000000 : ℝ) ≤
+      (-(1947942394458633086619 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3775806095905767 / 10000000000000000 : ℝ) =
+            (3775806095905767 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3775806095905767 / 2500000000000000 : ℝ) ≤
+            (824646326741366913381 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1870731180950767 / 5000000000000000 : ℝ)) (q :=
+      (3775806095905767 / 10000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_425_8 :
+    Uω (aρ 8) (bρ 8) (906639604631 / 2000000000000 : ℝ) ≤
+      (-(10631763071366856419741 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6907142118447623 / 20000000000000000 : ℝ) ≤
+      (-(10631763071366856419741 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6907142118447623 / 20000000000000000 : ℝ) =
+            (6907142118447623 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6907142118447623 / 5000000000000000 : ℝ) ≤
+            (3231180534633143580259 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3380533939877623 / 10000000000000000 : ℝ)) (q :=
+      (6907142118447623 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_425_9 :
+    Uω (aρ 9) (bρ 9) (906639604631 / 2000000000000 : ℝ) ≤
+      (-(11918788004198888387561 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23722674177189 / 78125000000000 : ℝ) ≤
+      (-(11918788004198888387561 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (23722674177189 / 78125000000000 : ℝ) = (23722674177189 / 19531250000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23722674177189 / 19531250000000 : ℝ) ≤
+            (1944155601801111612439 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (180460127446899 / 625000000000000 : ℝ)) (q :=
+      (23722674177189 / 78125000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_425_10 :
+    Uω (aρ 10) (bρ 10) (906639604631 / 2000000000000 : ℝ) ≤
+      (-(3470132493185344546727 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2495607271404321 / 10000000000000000 : ℝ) ≤
+      (-(3470132493185344546727 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2495607271404321 / 10000000000000000 : ℝ) =
+            (2495607271404321 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 15
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2495607271404321 / 1250000000000000 : ℝ) ≤
+            (1728471359064655453273 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1097065513046821 / 5000000000000000 : ℝ)) (q :=
+      (2495607271404321 / 10000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_425_11 :
+    Uω (aρ 11) (bρ 11) (906639604631 / 2000000000000 : ℝ) ≤
+      (-(221910297290179888531 / 125000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3388711953436309 / 20000000000000000 : ℝ) ≤
+      (-(221910297290179888531 / 125000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3388711953436309 / 20000000000000000 : ℝ) =
+            (3388711953436309 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3388711953436309 / 2500000000000000 : ℝ) ≤
+            (38019895322320111469 / 125000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1010105062926309 / 10000000000000000 : ℝ)) (q :=
+      (3388711953436309 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_425_12 :
+    Uω (aρ 12) (bρ 12) (906639604631 / 2000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_425_13 :
+    Uω (aρ 13) (bρ 13) (906639604631 / 2000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_425_14 :
+    Uω (aρ 14) (bρ 14) (906639604631 / 2000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_425_15 :
+    Uω (aρ 15) (bρ 15) (906639604631 / 2000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_425_16 :
+    Uω (aρ 16) (bρ 16) (906639604631 / 2000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_425 :
+    Uρ (906639604631 / 2000000000000 : ℝ) ≤
+      (-(11874116246818642659339 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (906639604631 / 2000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(4027477348275718747337 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_425_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (906639604631 / 2000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(4054371351793893970459 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_425_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (906639604631 / 2000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(8217321149963825466591 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_425_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (906639604631 / 2000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(4200348437088783331463 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_425_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (906639604631 / 2000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(8687624402469212596989 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_425_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (906639604631 / 2000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1139582518123484126347 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_425_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (906639604631 / 2000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(1947942394458633086619 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_425_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (906639604631 / 2000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(10631763071366856419741 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_425_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (906639604631 / 2000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(11918788004198888387561 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_425_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (906639604631 / 2000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(3470132493185344546727 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_425_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (906639604631 / 2000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) * (-(221910297290179888531 / 125000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_425_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (906639604631 / 2000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_425_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (906639604631 / 2000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_425_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (906639604631 / 2000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_425_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (906639604631 / 2000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_425_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (906639604631 / 2000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_425_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_426_1 :
+    Uω (aρ 1) (bρ 1) (363185561803 / 800000000000 : ℝ) ≤
+      (-(1608029583970223099703 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1790114471944513 / 4000000000000000 : ℝ) ≤
+      (-(1608029583970223099703 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1790114471944513 / 4000000000000000 : ℝ) =
+            (1790114471944513 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1790114471944513 / 1000000000000000 : ℝ) ≤
+            (1164559137229776900297 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (895050011054013 / 2000000000000000 : ℝ)) (q :=
+      (1790114471944513 / 4000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_426_2 :
+    Uω (aρ 2) (bρ 2) (363185561803 / 800000000000 : ℝ) ≤
+      (-(252932982861302534687 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8902629903629441 / 20000000000000000 : ℝ) ≤
+      (-(252932982861302534687 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8902629903629441 / 20000000000000000 : ℝ) =
+            (8902629903629441 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8902629903629441 / 5000000000000000 : ℝ) ≤
+            (180284004826197465313 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4451076611686941 / 10000000000000000 : ℝ)) (q :=
+      (8902629903629441 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_426_3 :
+    Uω (aρ 3) (bρ 3) (363185561803 / 800000000000 : ℝ) ≤
+      (-(4101134714336247887123 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (220165856397753 / 500000000000000 : ℝ) ≤
+      (-(4101134714336247887123 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (220165856397753 / 500000000000000 : ℝ) =
+            (220165856397753 / 125000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (220165856397753 / 125000000000000 : ℝ) ≤
+            (2830337088663752112877 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (220123853515881 / 500000000000000 : ℝ)) (q :=
+      (220165856397753 / 500000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_426_4 :
+    Uω (aρ 4) (bρ 4) (363185561803 / 800000000000 : ℝ) ≤
+      (-(8385361096703903811737 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (864685925725911 / 2000000000000000 : ℝ) ≤
+      (-(8385361096703903811737 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (864685925725911 / 2000000000000000 : ℝ) =
+            (864685925725911 / 500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (864685925725911 / 500000000000000 : ℝ) ≤
+            (5477582509296096188263 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (432099625273161 / 1000000000000000 : ℝ)) (q :=
+      (864685925725911 / 2000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_426_5 :
+    Uω (aρ 5) (bρ 5) (363185561803 / 800000000000 : ℝ) ≤
+      (-(541989226078443513091 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1680533998020911 / 4000000000000000 : ℝ) ≤
+      (-(541989226078443513091 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1680533998020911 / 4000000000000000 : ℝ) =
+            (1680533998020911 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1680533998020911 / 1000000000000000 : ℝ) ≤
+            (324444749296556486909 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (838999380982411 / 2000000000000000 : ℝ)) (q :=
+      (1680533998020911 / 4000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_426_6 :
+    Uω (aρ 6) (bρ 6) (363185561803 / 800000000000 : ℝ) ≤
+      (-(2275033423792574258343 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8050376846117527 / 20000000000000000 : ℝ) ≤
+      (-(2275033423792574258343 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8050376846117527 / 20000000000000000 : ℝ) =
+            (8050376846117527 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8050376846117527 / 5000000000000000 : ℝ) ≤
+            (1190702477707425741657 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4009944136065027 / 10000000000000000 : ℝ)) (q :=
+      (8050376846117527 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_426_7 :
+    Uω (aρ 7) (bρ 7) (363185561803 / 800000000000 : ℝ) ≤
+      (-(1215253784585170195443 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3782488267178449 / 10000000000000000 : ℝ) ≤
+      (-(1215253784585170195443 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3782488267178449 / 10000000000000000 : ℝ) =
+            (3782488267178449 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3782488267178449 / 2500000000000000 : ℝ) ≤
+            (517614166164829804557 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1874102602532199 / 5000000000000000 : ℝ)) (q :=
+      (3782488267178449 / 10000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_426_8 :
+    Uω (aρ 8) (bρ 8) (363185561803 / 800000000000 : ℝ) ≤
+      (-(2122439182815299947461 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6920670663515501 / 20000000000000000 : ℝ) ≤
+      (-(2122439182815299947461 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6920670663515501 / 20000000000000000 : ℝ) =
+            (6920670663515501 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6920670663515501 / 5000000000000000 : ℝ) ≤
+            (650149538384700052539 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3387440985563001 / 10000000000000000 : ℝ)) (q :=
+      (6920670663515501 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_426_9 :
+    Uω (aρ 9) (bρ 9) (363185561803 / 800000000000 : ℝ) ≤
+      (-(2973971066903655038759 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1521732495046507 / 5000000000000000 : ℝ) ≤
+      (-(2973971066903655038759 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1521732495046507 / 5000000000000000 : ℝ) =
+            (1521732495046507 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1521732495046507 / 1250000000000000 : ℝ) ≤
+            (491764834596344961241 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (361833241324191 / 1250000000000000 : ℝ)) (q :=
+      (1521732495046507 / 5000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_426_10 :
+    Uω (aρ 10) (bρ 10) (363185561803 / 800000000000 : ℝ) ≤
+      (-(1731301202107627005709 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5006270923733017 / 20000000000000000 : ℝ) ≤
+      (-(1731301202107627005709 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5006270923733017 / 20000000000000000 : ℝ) =
+            (5006270923733017 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 2
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5006270923733017 / 5000000000000000 : ℝ) ≤
+            (1566748642372994291 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2202565907635517 / 10000000000000000 : ℝ)) (q :=
+      (5006270923733017 / 20000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_426_11 :
+    Uω (aρ 11) (bρ 11) (363185561803 / 800000000000 : ℝ) ≤
+      (-(17687769941311851423913 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1705414343765671 / 10000000000000000 : ℝ) ≤
+      (-(17687769941311851423913 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1705414343765671 / 10000000000000000 : ℝ) =
+            (1705414343765671 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1705414343765671 / 1250000000000000 : ℝ) ≤
+            (3106645467688148576087 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (512800148819421 / 5000000000000000 : ℝ)) (q :=
+      (1705414343765671 / 10000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_426_12 :
+    Uω (aρ 12) (bρ 12) (363185561803 / 800000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_426_13 :
+    Uω (aρ 13) (bρ 13) (363185561803 / 800000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_426_14 :
+    Uω (aρ 14) (bρ 14) (363185561803 / 800000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_426_15 :
+    Uω (aρ 15) (bρ 15) (363185561803 / 800000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_426_16 :
+    Uω (aρ 16) (bρ 16) (363185561803 / 800000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_426 :
+    Uρ (363185561803 / 800000000000 : ℝ) ≤
+      (-(11856629194172474159109 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (363185561803 / 800000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1608029583970223099703 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_426_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (363185561803 / 800000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(252932982861302534687 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_426_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (363185561803 / 800000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(4101134714336247887123 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_426_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (363185561803 / 800000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(8385361096703903811737 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_426_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (363185561803 / 800000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(541989226078443513091 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_426_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (363185561803 / 800000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(2275033423792574258343 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_426_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (363185561803 / 800000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(1215253784585170195443 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_426_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (363185561803 / 800000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(2122439182815299947461 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_426_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (363185561803 / 800000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(2973971066903655038759 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_426_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (363185561803 / 800000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(1731301202107627005709 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_426_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (363185561803 / 800000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(17687769941311851423913 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_426_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (363185561803 / 800000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_426_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (363185561803 / 800000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_426_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (363185561803 / 800000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_426_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (363185561803 / 800000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_426_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (363185561803 / 800000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_426_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_427_1 :
+    Uω (aρ 1) (bρ 1) (28415256387 / 62500000000 : ℝ) ≤
+      (-(2006340758789207906781 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8963815465228221 / 20000000000000000 : ℝ) ≤
+      (-(2006340758789207906781 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8963815465228221 / 20000000000000000 : ℝ) =
+            (8963815465228221 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8963815465228221 / 5000000000000000 : ℝ) ≤
+            (1459395142710792093219 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4481871661393221 / 10000000000000000 : ℝ)) (q :=
+      (8963815465228221 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_427_2 :
+    Uω (aρ 2) (bρ 2) (28415256387 / 62500000000 : ℝ) ≤
+      (-(1615798066397558523171 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8915873610458937 / 20000000000000000 : ℝ) ≤
+      (-(1615798066397558523171 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8915873610458937 / 20000000000000000 : ℝ) =
+            (8915873610458937 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8915873610458937 / 5000000000000000 : ℝ) ≤
+            (1156790654802441476829 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4457698819133937 / 10000000000000000 : ℝ)) (q :=
+      (8915873610458937 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_427_3 :
+    Uω (aρ 3) (bρ 3) (28415256387 / 62500000000 : ℝ) ≤
+      (-(818724033738776100043 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4409939888919593 / 10000000000000000 : ℝ) ≤
+      (-(818724033738776100043 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4409939888919593 / 10000000000000000 : ℝ) =
+            (4409939888919593 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4409939888919593 / 2500000000000000 : ℝ) ≤
+            (567570326861223899957 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2204550546432093 / 5000000000000000 : ℝ)) (q :=
+      (4409939888919593 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_427_4 :
+    Uω (aρ 4) (bρ 4) (28415256387 / 62500000000 : ℝ) ≤
+      (-(1046256103529695111109 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4330054851211299 / 10000000000000000 : ℝ) ≤
+      (-(1046256103529695111109 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4330054851211299 / 10000000000000000 : ℝ) =
+            (4330054851211299 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4330054851211299 / 2500000000000000 : ℝ) ≤
+            (686611847220304888891 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2163812599256299 / 5000000000000000 : ℝ)) (q :=
+      (4330054851211299 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_427_5 :
+    Uω (aρ 5) (bρ 5) (28415256387 / 62500000000 : ℝ) ≤
+      (-(2164013955424430799831 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4207966482862127 / 10000000000000000 : ℝ) ≤
+      (-(2164013955424430799831 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4207966482862127 / 10000000000000000 : ℝ) =
+            (4207966482862127 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4207966482862127 / 2500000000000000 : ℝ) ≤
+            (1301721946075569200169 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2100819190574627 / 5000000000000000 : ℝ)) (q :=
+      (4207966482862127 / 10000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_427_6 :
+    Uω (aρ 6) (bρ 6) (28415256387 / 62500000000 : ℝ) ≤
+      (-(9083634719625875003053 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4031835053215791 / 10000000000000000 : ℝ) ≤
+      (-(9083634719625875003053 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4031835053215791 / 10000000000000000 : ℝ) =
+            (4031835053215791 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4031835053215791 / 2500000000000000 : ℝ) ≤
+            (4779308886374124996947 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2008307948498291 / 5000000000000000 : ℝ)) (q :=
+      (4031835053215791 / 10000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_427_7 :
+    Uω (aρ 7) (bρ 7) (28415256387 / 62500000000 : ℝ) ≤
+      (-(4852190180007776394843 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1894585111256033 / 5000000000000000 : ℝ) ≤
+      (-(4852190180007776394843 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1894585111256033 / 5000000000000000 : ℝ) =
+            (1894585111256033 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1894585111256033 / 1250000000000000 : ℝ) ≤
+            (2079281622992223605157 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (938736904087283 / 2500000000000000 : ℝ)) (q :=
+      (1894585111256033 / 5000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_427_8 :
+    Uω (aρ 8) (bρ 8) (28415256387 / 62500000000 : ℝ) ≤
+      (-(10592668610642755392833 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (693419807050719 / 2000000000000000 : ℝ) ≤
+      (-(10592668610642755392833 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (693419807050719 / 2000000000000000 : ℝ) =
+            (693419807050719 / 500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (693419807050719 / 500000000000000 : ℝ) ≤
+            (3270274995357244607167 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (339434689317219 / 1000000000000000 : ℝ)) (q :=
+      (693419807050719 / 2000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_427_9 :
+    Uω (aρ 9) (bρ 9) (28415256387 / 62500000000 : ℝ) ≤
+      (-(5936519118310741301741 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6100852096483651 / 20000000000000000 : ℝ) ≤
+      (-(5936519118310741301741 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6100852096483651 / 20000000000000000 : ℝ) =
+            (6100852096483651 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6100852096483651 / 5000000000000000 : ℝ) ≤
+            (994952684689258698259 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2901966547508651 / 10000000000000000 : ℝ)) (q :=
+      (6100852096483651 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_427_10 :
+    Uω (aρ 10) (bρ 10) (28415256387 / 62500000000 : ℝ) ≤
+      (-(3455101075548872166831 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5021314955928739 / 20000000000000000 : ℝ) ≤
+      (-(3455101075548872166831 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5021314955928739 / 20000000000000000 : ℝ) =
+            (5021314955928739 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5021314955928739 / 5000000000000000 : ℝ) ≤
+            (10634825951127833169 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2210988440448739 / 10000000000000000 : ℝ)) (q :=
+      (5021314955928739 / 20000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_427_11 :
+    Uω (aρ 11) (bρ 11) (28415256387 / 62500000000 : ℝ) ≤
+      (-(8811842877035075998573 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3432756893474963 / 20000000000000000 : ℝ) ≤
+      (-(8811842877035075998573 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3432756893474963 / 20000000000000000 : ℝ) =
+            (3432756893474963 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3432756893474963 / 2500000000000000 : ℝ) ≤
+            (1585364827464924001427 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1040907004199963 / 10000000000000000 : ℝ)) (q :=
+      (3432756893474963 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_427_12 :
+    Uω (aρ 12) (bρ 12) (28415256387 / 62500000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_427_13 :
+    Uω (aρ 13) (bρ 13) (28415256387 / 62500000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_427_14 :
+    Uω (aρ 14) (bρ 14) (28415256387 / 62500000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_427_15 :
+    Uω (aρ 15) (bρ 15) (28415256387 / 62500000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_427_16 :
+    Uω (aρ 16) (bρ 16) (28415256387 / 62500000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_427 :
+    Uρ (28415256387 / 62500000000 : ℝ) ≤
+      (-(11839246909114969195289 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (28415256387 / 62500000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2006340758789207906781 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_427_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (28415256387 / 62500000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1615798066397558523171 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_427_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (28415256387 / 62500000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(818724033738776100043 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_427_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (28415256387 / 62500000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1046256103529695111109 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_427_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (28415256387 / 62500000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(2164013955424430799831 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_427_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (28415256387 / 62500000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(9083634719625875003053 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_427_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (28415256387 / 62500000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(4852190180007776394843 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_427_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (28415256387 / 62500000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(10592668610642755392833 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_427_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (28415256387 / 62500000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(5936519118310741301741 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_427_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (28415256387 / 62500000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(3455101075548872166831 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_427_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (28415256387 / 62500000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(8811842877035075998573 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_427_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (28415256387 / 62500000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_427_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (28415256387 / 62500000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_427_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (28415256387 / 62500000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_427_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (28415256387 / 62500000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_427_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (28415256387 / 62500000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_427_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_428_1 :
+    Uω (aρ 1) (bρ 1) (1821225008521 / 4000000000000 : ℝ) ≤
+      (-(8010599977827891867547 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8977058570418943 / 20000000000000000 : ℝ) ≤
+      (-(8010599977827891867547 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8977058570418943 / 20000000000000000 : ℝ) =
+            (8977058570418943 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8977058570418943 / 5000000000000000 : ℝ) ≤
+            (5852343628172108132453 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4488493267201443 / 10000000000000000 : ℝ)) (q :=
+      (8977058570418943 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_428_2 :
+    Uω (aρ 2) (bρ 2) (1821225008521 / 4000000000000 : ℝ) ≤
+      (-(1612829455829885220953 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1116139664398489 / 2500000000000000 : ℝ) ≤
+      (-(1612829455829885220953 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1116139664398489 / 2500000000000000 : ℝ) =
+            (1116139664398489 / 625000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1116139664398489 / 625000000000000 : ℝ) ≤
+            (1159759265370114779047 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1116080256120103 / 2500000000000000 : ℝ)) (q :=
+      (1116139664398489 / 2500000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_428_3 :
+    Uω (aρ 3) (bρ 3) (1821225008521 / 4000000000000 : ℝ) ≤
+      (-(2043058452034576909719 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1104140661524963 / 2500000000000000 : ℝ) ≤
+      (-(2043058452034576909719 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1104140661524963 / 2500000000000000 : ℝ) =
+            (1104140661524963 / 625000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1104140661524963 / 625000000000000 : ℝ) ≤
+            (1422677449465423090281 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1103931276960551 / 2500000000000000 : ℝ)) (q :=
+      (1104140661524963 / 2500000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_428_4 :
+    Uω (aρ 4) (bρ 4) (1821225008521 / 4000000000000 : ℝ) ≤
+      (-(4177379998365973504417 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8673360124821363 / 20000000000000000 : ℝ) ≤
+      (-(4177379998365973504417 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8673360124821363 / 20000000000000000 : ℝ) =
+            (8673360124821363 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8673360124821363 / 5000000000000000 : ℝ) ≤
+            (2754091804634026495583 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4334254121528863 / 10000000000000000 : ℝ)) (q :=
+      (8673360124821363 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_428_5 :
+    Uω (aρ 5) (bρ 5) (1821225008521 / 4000000000000 : ℝ) ≤
+      (-(8640308936621060834287 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8429195878384133 / 20000000000000000 : ℝ) ≤
+      (-(8640308936621060834287 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8429195878384133 / 20000000000000000 : ℝ) =
+            (8429195878384133 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8429195878384133 / 5000000000000000 : ℝ) ≤
+            (5222634669378939165713 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4208279794426633 / 10000000000000000 : ℝ)) (q :=
+      (8429195878384133 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_428_6 :
+    Uω (aρ 6) (bρ 6) (1821225008521 / 4000000000000 : ℝ) ≤
+      (-(9067163126475235429003 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1615392640135501 / 4000000000000000 : ℝ) ≤
+      (-(9067163126475235429003 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1615392640135501 / 4000000000000000 : ℝ) =
+            (1615392640135501 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1615392640135501 / 1000000000000000 : ℝ) ≤
+            (4795780479524764570997 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (804657498372001 / 2000000000000000 : ℝ)) (q :=
+      (1615392640135501 / 4000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_428_7 :
+    Uω (aρ 7) (bρ 7) (1821225008521 / 4000000000000 : ℝ) ≤
+      (-(9686762106250244520731 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7591703926135517 / 20000000000000000 : ℝ) ≤
+      (-(9686762106250244520731 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7591703926135517 / 20000000000000000 : ℝ) =
+            (7591703926135517 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7591703926135517 / 5000000000000000 : ℝ) ≤
+            (4176181499749755479269 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3761689598078017 / 10000000000000000 : ℝ)) (q :=
+      (7591703926135517 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_428_8 :
+    Uω (aρ 8) (bρ 8) (1821225008521 / 4000000000000 : ℝ) ≤
+      (-(10573180992541647374207 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27139548227949 / 78125000000000 : ℝ) ≤
+      (-(10573180992541647374207 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (27139548227949 / 78125000000000 : ℝ) = (27139548227949 / 19531250000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27139548227949 / 19531250000000 : ℝ) ≤
+            (3289762613458352625793 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (850312917409361 / 2500000000000000 : ℝ)) (q :=
+      (27139548227949 / 78125000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_428_9 :
+    Uω (aρ 9) (bρ 9) (1821225008521 / 4000000000000 : ℝ) ≤
+      (-(11850249595588193161761 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1528692740726231 / 5000000000000000 : ℝ) ≤
+      (-(11850249595588193161761 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1528692740726231 / 5000000000000000 : ℝ) =
+            (1528692740726231 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1528692740726231 / 1250000000000000 : ℝ) ≤
+            (2012694010411806838239 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (90914497329607 / 312500000000000 : ℝ)) (q :=
+      (1528692740726231 / 5000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_428_10 :
+    Uω (aρ 10) (bρ 10) (1821225008521 / 4000000000000 : ℝ) ≤
+      (-(6895256495096437908121 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (251817338999233 / 1000000000000000 : ℝ) ≤
+      (-(6895256495096437908121 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (251817338999233 / 1000000000000000 : ℝ) =
+            (251817338999233 / 250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (251817338999233 / 250000000000000 : ℝ) ≤
+            (36215307903562091879 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (27742484564027 / 125000000000000 : ℝ)) (q :=
+      (251817338999233 / 1000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10]) (by norm_num [bρ_10])
+      (by norm_num) (by norm_num [aρ_10, bρ_10]) (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_428_11 :
+    Uω (aρ 11) (bρ 11) (1821225008521 / 4000000000000 : ℝ) ≤
+      (-(3512106326516641677623 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3454504769179323 / 20000000000000000 : ℝ) ≤
+      (-(3512106326516641677623 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3454504769179323 / 20000000000000000 : ℝ) =
+            (3454504769179323 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3454504769179323 / 2500000000000000 : ℝ) ≤
+            (646776755283358322377 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1056033380521823 / 10000000000000000 : ℝ)) (q :=
+      (3454504769179323 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_428_12 :
+    Uω (aρ 12) (bρ 12) (1821225008521 / 4000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_428_13 :
+    Uω (aρ 13) (bρ 13) (1821225008521 / 4000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_428_14 :
+    Uω (aρ 14) (bρ 14) (1821225008521 / 4000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_428_15 :
+    Uω (aρ 15) (bρ 15) (1821225008521 / 4000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_428_16 :
+    Uω (aρ 16) (bρ 16) (1821225008521 / 4000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_428 :
+    Uρ (1821225008521 / 4000000000000 : ℝ) ≤
+      (-(2364393220857954377449 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1821225008521 / 4000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(8010599977827891867547 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_428_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1821225008521 / 4000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1612829455829885220953 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_428_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1821225008521 / 4000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(2043058452034576909719 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_428_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1821225008521 / 4000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(4177379998365973504417 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_428_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1821225008521 / 4000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(8640308936621060834287 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_428_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1821225008521 / 4000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(9067163126475235429003 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_428_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1821225008521 / 4000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(9686762106250244520731 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_428_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1821225008521 / 4000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(10573180992541647374207 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_428_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1821225008521 / 4000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(11850249595588193161761 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_428_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1821225008521 / 4000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(6895256495096437908121 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_428_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1821225008521 / 4000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(3512106326516641677623 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_428_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1821225008521 / 4000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_428_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1821225008521 / 4000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_428_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1821225008521 / 4000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_428_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1821225008521 / 4000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_428_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1821225008521 / 4000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_428_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_429_1 :
+    Uω (aρ 1) (bρ 1) (911936804137 / 2000000000000 : ℝ) ≤
+      (-(7995858683509481005467 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2247575418824031 / 5000000000000000 : ℝ) ≤
+      (-(7995858683509481005467 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2247575418824031 / 5000000000000000 : ℝ) =
+            (2247575418824031 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2247575418824031 / 1250000000000000 : ℝ) ≤
+            (5867084922490518994533 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1123778718174031 / 2500000000000000 : ℝ)) (q :=
+      (2247575418824031 / 5000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_429_2 :
+    Uω (aρ 2) (bρ 2) (911936804137 / 2000000000000 : ℝ) ≤
+      (-(251541444613192942879 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8942361017825701 / 20000000000000000 : ℝ) ≤
+      (-(251541444613192942879 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8942361017825701 / 20000000000000000 : ℝ) =
+            (8942361017825701 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8942361017825701 / 5000000000000000 : ℝ) ≤
+            (181675543074307057121 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4470943227735701 / 10000000000000000 : ℝ)) (q :=
+      (8942361017825701 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_429_3 :
+    Uω (aρ 3) (bρ 3) (911936804137 / 2000000000000 : ℝ) ≤
+      (-(4078624886629356679157 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2211592699756421 / 5000000000000000 : ℝ) ≤
+      (-(4078624886629356679157 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2211592699756421 / 5000000000000000 : ℝ) =
+            (2211592699756421 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2211592699756421 / 1250000000000000 : ℝ) ≤
+            (2852846916370643320843 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1105587278821421 / 2500000000000000 : ℝ)) (q :=
+      (2211592699756421 / 5000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_429_4 :
+    Uω (aρ 4) (bρ 4) (911936804137 / 2000000000000 : ℝ) ≤
+      (-(8339494530471527057847 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8686610524559697 / 20000000000000000 : ℝ) ≤
+      (-(8339494530471527057847 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8686610524559697 / 20000000000000000 : ℝ) =
+            (8686610524559697 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8686610524559697 / 5000000000000000 : ℝ) ≤
+            (5523449075528472942153 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4340883021884697 / 10000000000000000 : ℝ)) (q :=
+      (8686610524559697 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_429_5 :
+    Uω (aρ 5) (bρ 5) (911936804137 / 2000000000000 : ℝ) ≤
+      (-(8624586883225874612507 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (844245872838181 / 2000000000000000 : ℝ) ≤
+      (-(8624586883225874612507 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (844245872838181 / 2000000000000000 : ℝ) =
+            (844245872838181 / 500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (844245872838181 / 500000000000000 : ℝ) ≤
+            (5238356722774125387493 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (421492114504181 / 1000000000000000 : ℝ)) (q :=
+      (844245872838181 / 2000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_429_6 :
+    Uω (aρ 6) (bρ 6) (911936804137 / 2000000000000 : ℝ) ≤
+      (-(4525359412151244814811 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8090256129680071 / 20000000000000000 : ℝ) ≤
+      (-(4525359412151244814811 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8090256129680071 / 20000000000000000 : ℝ) =
+            (8090256129680071 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8090256129680071 / 5000000000000000 : ℝ) ≤
+            (2406112390848755185189 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4029958921480071 / 10000000000000000 : ℝ)) (q :=
+      (8090256129680071 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_429_7 :
+    Uω (aρ 7) (bρ 7) (911936804137 / 2000000000000 : ℝ) ≤
+      (-(1933835079997006956291 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (950633372499571 / 2500000000000000 : ℝ) ≤
+      (-(1933835079997006956291 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (950633372499571 / 2500000000000000 : ℝ) =
+            (950633372499571 / 625000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (950633372499571 / 625000000000000 : ℝ) ≤
+            (838753641202993043709 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (471053894069571 / 1250000000000000 : ℝ)) (q :=
+      (950633372499571 / 2500000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_429_8 :
+    Uω (aρ 8) (bρ 8) (911936804137 / 2000000000000 : ℝ) ≤
+      (-(5276866446176848011757 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6961249497933711 / 20000000000000000 : ℝ) ≤
+      (-(5276866446176848011757 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6961249497933711 / 20000000000000000 : ℝ) =
+            (6961249497933711 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6961249497933711 / 5000000000000000 : ℝ) ≤
+            (1654605356823151988243 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3408155321833711 / 10000000000000000 : ℝ)) (q :=
+      (6961249497933711 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_429_9 :
+    Uω (aρ 9) (bρ 9) (911936804137 / 2000000000000 : ℝ) ≤
+      (-(5913759015820159389253 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1225737320768777 / 4000000000000000 : ℝ) ≤
+      (-(5913759015820159389253 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1225737320768777 / 4000000000000000 : ℝ) =
+            (1225737320768777 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1225737320768777 / 1000000000000000 : ℝ) ≤
+            (1017712787179840610747 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (583311611220777 / 2000000000000000 : ℝ)) (q :=
+      (1225737320768777 / 4000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_429_10 :
+    Uω (aρ 10) (bρ 10) (911936804137 / 2000000000000 : ℝ) ≤
+      (-(1720091831909998787893 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2525683267082197 / 10000000000000000 : ℝ) ≤
+      (-(1720091831909998787893 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2525683267082197 / 10000000000000000 : ℝ) =
+            (2525683267082197 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2525683267082197 / 2500000000000000 : ℝ) ≤
+            (12776118840001212107 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1113898509959697 / 5000000000000000 : ℝ)) (q :=
+      (2525683267082197 / 10000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_429_11 :
+    Uω (aρ 11) (bρ 11) (911936804137 / 2000000000000 : ℝ) ≤
+      (-(17498270634776701115959 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (434509994437601 / 2500000000000000 : ℝ) ≤
+      (-(17498270634776701115959 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (434509994437601 / 2500000000000000 : ℝ) =
+            (434509994437601 / 312500000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (434509994437601 / 312500000000000 : ℝ) ≤
+            (3296144774223298884041 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (133873383432601 / 1250000000000000 : ℝ)) (q :=
+      (434509994437601 / 2500000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11]) (by norm_num [bρ_11])
+      (by norm_num) (by norm_num [aρ_11, bρ_11]) (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_429_12 :
+    Uω (aρ 12) (bρ 12) (911936804137 / 2000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_429_13 :
+    Uω (aρ 13) (bρ 13) (911936804137 / 2000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_429_14 :
+    Uω (aρ 14) (bρ 14) (911936804137 / 2000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_429_15 :
+    Uω (aρ 15) (bρ 15) (911936804137 / 2000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_429_16 :
+    Uω (aρ 16) (bρ 16) (911936804137 / 2000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_429 :
+    Uρ (911936804137 / 2000000000000 : ℝ) ≤
+      (-(11804783702785120325837 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (911936804137 / 2000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(7995858683509481005467 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_429_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (911936804137 / 2000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(251541444613192942879 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_429_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (911936804137 / 2000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(4078624886629356679157 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_429_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (911936804137 / 2000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(8339494530471527057847 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_429_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (911936804137 / 2000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(8624586883225874612507 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_429_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (911936804137 / 2000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(4525359412151244814811 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_429_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (911936804137 / 2000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(1933835079997006956291 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_429_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (911936804137 / 2000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(5276866446176848011757 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_429_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (911936804137 / 2000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(5913759015820159389253 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_429_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (911936804137 / 2000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(1720091831909998787893 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_429_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (911936804137 / 2000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(17498270634776701115959 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_429_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (911936804137 / 2000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_429_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (911936804137 / 2000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_429_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (911936804137 / 2000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_429_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (911936804137 / 2000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_429_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (911936804137 / 2000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_429_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_430_1 :
+    Uω (aρ 1) (bρ 1) (1826522208027 / 4000000000000 : ℝ) ≤
+      (-(3990569544065491910013 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4501772389930573 / 10000000000000000 : ℝ) ≤
+      (-(3990569544065491910013 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4501772389930573 / 10000000000000000 : ℝ) =
+            (4501772389930573 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4501772389930573 / 2500000000000000 : ℝ) ≤
+            (2940902258934508089987 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2250868238939323 / 5000000000000000 : ℝ)) (q :=
+      (4501772389930573 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_430_2 :
+    Uω (aρ 2) (bρ 2) (1826522208027 / 4000000000000 : ℝ) ≤
+      (-(1004315889034024038327 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8955604718381581 / 20000000000000000 : ℝ) ≤
+      (-(1004315889034024038327 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8955604718381581 / 20000000000000000 : ℝ) =
+            (8955604718381581 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8955604718381581 / 5000000000000000 : ℝ) ≤
+            (728552061715975961673 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4477565428909081 / 10000000000000000 : ℝ)) (q :=
+      (8955604718381581 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_430_3 :
+    Uω (aρ 3) (bρ 3) (1826522208027 / 4000000000000 : ℝ) ≤
+      (-(8142288165387616772923 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2214904074587733 / 5000000000000000 : ℝ) ≤
+      (-(8142288165387616772923 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2214904074587733 / 5000000000000000 : ℝ) =
+            (2214904074587733 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2214904074587733 / 1250000000000000 : ℝ) ≤
+            (5720655440612383227077 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (276810819701777 / 625000000000000 : ℝ)) (q :=
+      (2214904074587733 / 5000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_430_4 :
+    Uω (aρ 4) (bρ 4) (1826522208027 / 4000000000000 : ℝ) ≤
+      (-(4162126179034898229433 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4349930450870627 / 10000000000000000 : ℝ) ≤
+      (-(4162126179034898229433 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4349930450870627 / 10000000000000000 : ℝ) =
+            (4349930450870627 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4349930450870627 / 2500000000000000 : ℝ) ≤
+            (2769345623965101770567 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2173755949841877 / 5000000000000000 : ℝ)) (q :=
+      (4349930450870627 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_430_5 :
+    Uω (aρ 5) (bρ 5) (1826522208027 / 4000000000000 : ℝ) ≤
+      (-(2152222395771836180129 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4227860758006511 / 10000000000000000 : ℝ) ≤
+      (-(2152222395771836180129 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4227860758006511 / 10000000000000000 : ℝ) =
+            (4227860758006511 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4227860758006511 / 2500000000000000 : ℝ) ≤
+            (1313513505728163819871 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2110781216645261 / 5000000000000000 : ℝ)) (q :=
+      (4227860758006511 / 10000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_430_6 :
+    Uω (aρ 6) (bρ 6) (1826522208027 / 4000000000000 : ℝ) ≤
+      (-(9034301722152087805031 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8103548894258587 / 20000000000000000 : ℝ) ≤
+      (-(9034301722152087805031 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8103548894258587 / 20000000000000000 : ℝ) =
+            (8103548894258587 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8103548894258587 / 5000000000000000 : ℝ) ≤
+            (4828641883847912194969 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4036630186676087 / 10000000000000000 : ℝ)) (q :=
+      (8103548894258587 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_430_7 :
+    Uω (aρ 7) (bρ 7) (1826522208027 / 4000000000000 : ℝ) ≤
+      (-(2412905031614942433679 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1523685921779237 / 4000000000000000 : ℝ) ≤
+      (-(2412905031614942433679 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1523685921779237 / 4000000000000000 : ℝ) =
+            (1523685921779237 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1523685921779237 / 1000000000000000 : ℝ) ≤
+            (1052830869885057566321 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (755034456414737 / 2000000000000000 : ℝ)) (q :=
+      (1523685921779237 / 4000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_430_8 :
+    Uω (aρ 8) (bρ 8) (1826522208027 / 4000000000000 : ℝ) ≤
+      (-(2633581035938500616407 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6974773532061733 / 20000000000000000 : ℝ) ≤
+      (-(2633581035938500616407 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6974773532061733 / 20000000000000000 : ℝ) =
+            (6974773532061733 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6974773532061733 / 5000000000000000 : ℝ) ≤
+            (832154865561499383593 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3415057856579233 / 10000000000000000 : ℝ)) (q :=
+      (6974773532061733 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_430_9 :
+    Uω (aρ 9) (bρ 9) (1826522208027 / 4000000000000 : ℝ) ≤
+      (-(11804843234626741708411 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (191956220107513 / 625000000000000 : ℝ) ≤
+      (-(11804843234626741708411 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (191956220107513 / 625000000000000 : ℝ) =
+            (191956220107513 / 156250000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (191956220107513 / 156250000000000 : ℝ) ≤
+            (2058100371373258291589 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (730962249079479 / 2500000000000000 : ℝ)) (q :=
+      (191956220107513 / 625000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_430_10 :
+    Uω (aρ 10) (bρ 10) (1826522208027 / 4000000000000 : ℝ) ≤
+      (-(13731068287134637534901 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (101327487089151 / 400000000000000 : ℝ) ≤
+      (-(13731068287134637534901 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (101327487089151 / 400000000000000 : ℝ) =
+            (101327487089151 / 100000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (101327487089151 / 100000000000000 : ℝ) ≤
+            (131875318865362465099 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (44723666816601 / 200000000000000 : ℝ)) (q :=
+      (101327487089151 / 400000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10]) (by norm_num [bρ_10])
+      (by norm_num) (by norm_num [aρ_10, bρ_10]) (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_430_11 :
+    Uω (aρ 11) (bρ 11) (1826522208027 / 4000000000000 : ℝ) ≤
+      (-(17436868223522066249077 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3497489587483911 / 20000000000000000 : ℝ) ≤
+      (-(17436868223522066249077 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3497489587483911 / 20000000000000000 : ℝ) =
+            (3497489587483911 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3497489587483911 / 2500000000000000 : ℝ) ≤
+            (3357547185477933750923 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1085775200061411 / 10000000000000000 : ℝ)) (q :=
+      (3497489587483911 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_430_12 :
+    Uω (aρ 12) (bρ 12) (1826522208027 / 4000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_430_13 :
+    Uω (aρ 13) (bρ 13) (1826522208027 / 4000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_430_14 :
+    Uω (aρ 14) (bρ 14) (1826522208027 / 4000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_430_15 :
+    Uω (aρ 15) (bρ 15) (1826522208027 / 4000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_430_16 :
+    Uω (aρ 16) (bρ 16) (1826522208027 / 4000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_430 :
+    Uρ (1826522208027 / 4000000000000 : ℝ) ≤
+      (-(5893848409619263390529 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1826522208027 / 4000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3990569544065491910013 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_430_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1826522208027 / 4000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1004315889034024038327 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_430_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1826522208027 / 4000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(8142288165387616772923 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_430_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1826522208027 / 4000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(4162126179034898229433 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_430_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1826522208027 / 4000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(2152222395771836180129 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_430_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1826522208027 / 4000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(9034301722152087805031 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_430_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1826522208027 / 4000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(2412905031614942433679 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_430_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1826522208027 / 4000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(2633581035938500616407 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_430_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1826522208027 / 4000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(11804843234626741708411 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_430_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1826522208027 / 4000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(13731068287134637534901 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_430_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1826522208027 / 4000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(17436868223522066249077 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_430_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1826522208027 / 4000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_430_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1826522208027 / 4000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_430_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1826522208027 / 4000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_430_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1826522208027 / 4000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_430_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1826522208027 / 4000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_430_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_431_1 :
+    Uω (aρ 1) (bρ 1) (91458540389 / 200000000000 : ℝ) ≤
+      (-(7966441127904307519531 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4508393942057693 / 10000000000000000 : ℝ) ≤
+      (-(7966441127904307519531 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4508393942057693 / 10000000000000000 : ℝ) =
+            (4508393942057693 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4508393942057693 / 2500000000000000 : ℝ) ≤
+            (5896502478095692480469 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2254179041375193 / 5000000000000000 : ℝ)) (q :=
+      (4508393942057693 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_431_2 :
+    Uω (aρ 2) (bρ 2) (91458540389 / 200000000000 : ℝ) ≤
+      (-(4009874934127237774067 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1121106052108097 / 2500000000000000 : ℝ) ≤
+      (-(4009874934127237774067 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1121106052108097 / 2500000000000000 : ℝ) =
+            (1121106052108097 / 625000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1121106052108097 / 625000000000000 : ℝ) ≤
+            (2921596868872762225933 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (280261726750611 / 625000000000000 : ℝ)) (q :=
+      (1121106052108097 / 2500000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_431_3 :
+    Uω (aρ 3) (bρ 3) (91458540389 / 200000000000 : ℝ) ≤
+      (-(4063674458732947917093 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8872861790209053 / 20000000000000000 : ℝ) ≤
+      (-(4063674458732947917093 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8872861790209053 / 20000000000000000 : ℝ) =
+            (8872861790209053 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8872861790209053 / 5000000000000000 : ℝ) ≤
+            (2867797344267052082907 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4435597107704053 / 10000000000000000 : ℝ)) (q :=
+      (8872861790209053 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_431_4 :
+    Uω (aρ 4) (bρ 4) (91458540389 / 200000000000 : ℝ) ≤
+      (-(1661806681693453128021 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4356555628234529 / 10000000000000000 : ℝ) ≤
+      (-(1661806681693453128021 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4356555628234529 / 10000000000000000 : ℝ) =
+            (4356555628234529 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4356555628234529 / 2500000000000000 : ℝ) ≤
+            (1110782039506546871979 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2177070377514529 / 5000000000000000 : ℝ)) (q :=
+      (4356555628234529 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_431_5 :
+    Uω (aρ 5) (bρ 5) (91458540389 / 200000000000 : ℝ) ≤
+      (-(8593216958152678828791 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (169379684831433 / 400000000000000 : ℝ) ≤
+      (-(8593216958152678828791 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (169379684831433 / 400000000000000 : ℝ) =
+            (169379684831433 / 100000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (169379684831433 / 100000000000000 : ℝ) ≤
+            (5269726647847321171209 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (84564073189333 / 200000000000000 : ℝ)) (q :=
+      (169379684831433 / 400000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_431_6 :
+    Uω (aρ 6) (bρ 6) (91458540389 / 200000000000 : ℝ) ≤
+      (-(9017911729525724670809 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8116841495226927 / 20000000000000000 : ℝ) ≤
+      (-(9017911729525724670809 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8116841495226927 / 20000000000000000 : ℝ) =
+            (8116841495226927 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8116841495226927 / 5000000000000000 : ℝ) ≤
+            (4845031876474275329191 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4043301288261927 / 10000000000000000 : ℝ)) (q :=
+      (8116841495226927 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_431_7 :
+    Uω (aρ 7) (bρ 7) (91458540389 / 200000000000 : ℝ) ≤
+      (-(38536384686198961727 / 40000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3815895907553403 / 10000000000000000 : ℝ) ≤
+      (-(38536384686198961727 / 40000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3815895907553403 / 10000000000000000 : ℝ) =
+            (3815895907553403 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3815895907553403 / 2500000000000000 : ℝ) ≤
+            (16915389737801038273 / 40000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1890956494450903 / 5000000000000000 : ℝ)) (q :=
+      (3815895907553403 / 10000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_431_8 :
+    Uω (aρ 8) (bρ 8) (91458540389 / 200000000000 : ℝ) ≤
+      (-(10514954581502450073807 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (436768528468821 / 1250000000000000 : ℝ) ≤
+      (-(10514954581502450073807 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (436768528468821 / 1250000000000000 : ℝ) =
+            (436768528468821 / 312500000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (436768528468821 / 312500000000000 : ℝ) ≤
+            (3347989024497549926193 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (427744910079517 / 1250000000000000 : ℝ)) (q :=
+      (436768528468821 / 1250000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_431_9 :
+    Uω (aρ 9) (bρ 9) (91458540389 / 200000000000 : ℝ) ≤
+      (-(5891112448543313736457 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1539127076395919 / 5000000000000000 : ℝ) ≤
+      (-(5891112448543313736457 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1539127076395919 / 5000000000000000 : ℝ) =
+            (1539127076395919 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1539127076395919 / 1250000000000000 : ℝ) ≤
+            (1040359354456686263543 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (732784189769669 / 2500000000000000 : ℝ)) (q :=
+      (1539127076395919 / 5000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_431_10 :
+    Uω (aρ 10) (bρ 10) (91458540389 / 200000000000 : ℝ) ≤
+      (-(13701512890367255710259 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (635171296828693 / 2500000000000000 : ℝ) ≤
+      (-(13701512890367255710259 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (635171296828693 / 2500000000000000 : ℝ) =
+            (635171296828693 / 625000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (635171296828693 / 625000000000000 : ℝ) ≤
+            (161430715632744289741 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (280569732702443 / 1250000000000000 : ℝ)) (q :=
+      (635171296828693 / 2500000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10]) (by norm_num [bρ_10])
+      (by norm_num) (by norm_num [aρ_10, bρ_10]) (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_431_11 :
+    Uω (aρ 11) (bρ 11) (91458540389 / 200000000000 : ℝ) ≤
+      (-(17376292052470984479999 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (109960635615987 / 625000000000000 : ℝ) ≤
+      (-(17376292052470984479999 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (109960635615987 / 625000000000000 : ℝ) = (109960635615987 / 78125000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (109960635615987 / 78125000000000 : ℝ) ≤
+            (3418123356529015520001 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (137550556613323 / 1250000000000000 : ℝ)) (q :=
+      (109960635615987 / 625000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11]) (by norm_num [bρ_11])
+      (by norm_num) (by norm_num [aρ_11, bρ_11]) (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_431_12 :
+    Uω (aρ 12) (bρ 12) (91458540389 / 200000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_431_13 :
+    Uω (aρ 13) (bρ 13) (91458540389 / 200000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_431_14 :
+    Uω (aρ 14) (bρ 14) (91458540389 / 200000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_431_15 :
+    Uω (aρ 15) (bρ 15) (91458540389 / 200000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_431_16 :
+    Uω (aρ 16) (bρ 16) (91458540389 / 200000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_431 :
+    Uρ (91458540389 / 200000000000 : ℝ) ≤
+      (-(2942675685726804811747 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (91458540389 / 200000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(7966441127904307519531 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_431_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (91458540389 / 200000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(4009874934127237774067 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_431_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (91458540389 / 200000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(4063674458732947917093 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_431_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (91458540389 / 200000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1661806681693453128021 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_431_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (91458540389 / 200000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(8593216958152678828791 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_431_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (91458540389 / 200000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(9017911729525724670809 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_431_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (91458540389 / 200000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) * (-(38536384686198961727 / 40000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_431_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (91458540389 / 200000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(10514954581502450073807 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_431_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (91458540389 / 200000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(5891112448543313736457 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_431_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (91458540389 / 200000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(13701512890367255710259 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_431_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (91458540389 / 200000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(17376292052470984479999 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_431_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (91458540389 / 200000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_431_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (91458540389 / 200000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_431_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (91458540389 / 200000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_431_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (91458540389 / 200000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_431_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (91458540389 / 200000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_431_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_432_1 :
+    Uω (aρ 1) (bρ 1) (1831819407533 / 4000000000000 : ℝ) ≤
+      (-(7951764739322229786499 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (903003098806021 / 2000000000000000 : ℝ) ≤
+      (-(7951764739322229786499 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (903003098806021 / 2000000000000000 : ℝ) =
+            (903003098806021 / 500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (903003098806021 / 500000000000000 : ℝ) ≤
+            (5911178866677770213501 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (451497968731271 / 1000000000000000 : ℝ)) (q :=
+      (903003098806021 / 2000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_432_2 :
+    Uω (aρ 2) (bρ 2) (1831819407533 / 4000000000000 : ℝ) ≤
+      (-(1000624303876394457959 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1796418422656891 / 4000000000000000 : ℝ) ≤
+      (-(1000624303876394457959 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1796418422656891 / 4000000000000000 : ℝ) =
+            (1796418422656891 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1796418422656891 / 1000000000000000 : ℝ) ≤
+            (732243646873605542041 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (898161965009391 / 2000000000000000 : ℝ)) (q :=
+      (1796418422656891 / 4000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_432_3 :
+    Uω (aρ 3) (bρ 3) (1831819407533 / 4000000000000 : ℝ) ≤
+      (-(31689187354433060421 / 39062500000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8886107274633451 / 20000000000000000 : ℝ) ≤
+      (-(31689187354433060421 / 39062500000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8886107274633451 / 20000000000000000 : ℝ) =
+            (8886107274633451 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8886107274633451 / 5000000000000000 : ℝ) ≤
+            (22462936106504439579 / 39062500000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4442221092745951 / 10000000000000000 : ℝ)) (q :=
+      (8886107274633451 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_432_4 :
+    Uω (aρ 4) (bρ 4) (1831819407533 / 4000000000000 : ℝ) ≤
+      (-(8293837610929466732469 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1745272317769101 / 4000000000000000 : ℝ) ≤
+      (-(8293837610929466732469 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1745272317769101 / 4000000000000000 : ℝ) =
+            (1745272317769101 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1745272317769101 / 1000000000000000 : ℝ) ≤
+            (5569105995070533267531 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (872153917604601 / 2000000000000000 : ℝ)) (q :=
+      (1745272317769101 / 4000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_432_5 :
+    Uω (aρ 5) (bρ 5) (1831819407533 / 4000000000000 : ℝ) ≤
+      (-(4288784465369381867481 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8482246905349727 / 20000000000000000 : ℝ) ≤
+      (-(4288784465369381867481 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8482246905349727 / 20000000000000000 : ℝ) =
+            (8482246905349727 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8482246905349727 / 5000000000000000 : ℝ) ≤
+            (2642687337630618132519 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4234844823862227 / 10000000000000000 : ℝ)) (q :=
+      (8482246905349727 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_432_6 :
+    Uω (aρ 6) (bρ 6) (1831819407533 / 4000000000000 : ℝ) ≤
+      (-(2250387189094811231179 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2032533483348397 / 5000000000000000 : ℝ) ≤
+      (-(2250387189094811231179 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2032533483348397 / 5000000000000000 : ℝ) =
+            (2032533483348397 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2032533483348397 / 1250000000000000 : ℝ) ≤
+            (1215348712405188768821 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (506246528380761 / 1250000000000000 : ℝ)) (q :=
+      (2032533483348397 / 5000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_432_7 :
+    Uω (aρ 7) (bρ 7) (1831819407533 / 4000000000000 : ℝ) ≤
+      (-(4808301710880464751429 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1529030720176911 / 4000000000000000 : ℝ) ≤
+      (-(4808301710880464751429 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1529030720176911 / 4000000000000000 : ℝ) =
+            (1529030720176911 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1529030720176911 / 1000000000000000 : ℝ) ≤
+            (2123170092119535248571 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (757730655059411 / 2000000000000000 : ℝ)) (q :=
+      (1529030720176911 / 4000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_432_8 :
+    Uω (aρ 8) (bρ 8) (1831819407533 / 4000000000000 : ℝ) ≤
+      (-(10495624041434001507059 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1750454568739631 / 5000000000000000 : ℝ) ≤
+      (-(10495624041434001507059 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1750454568739631 / 5000000000000000 : ℝ) =
+            (1750454568739631 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1750454568739631 / 1250000000000000 : ℝ) ≤
+            (3367319564565998492941 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (214303725044439 / 625000000000000 : ℝ)) (q :=
+      (1750454568739631 / 5000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_432_9 :
+    Uω (aρ 9) (bρ 9) (1831819407533 / 4000000000000 : ℝ) ≤
+      (-(2939915678554231113423 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6170414413915467 / 20000000000000000 : ℝ) ≤
+      (-(2939915678554231113423 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6170414413915467 / 20000000000000000 : ℝ) =
+            (6170414413915467 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6170414413915467 / 5000000000000000 : ℝ) ≤
+            (525820222945768886577 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2938421368027967 / 10000000000000000 : ℝ)) (q :=
+      (6170414413915467 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_432_10 :
+    Uω (aρ 10) (bρ 10) (1831819407533 / 4000000000000 : ℝ) ≤
+      (-(13672067484211088375381 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5096354726270043 / 20000000000000000 : ℝ) ≤
+      (-(13672067484211088375381 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5096354726270043 / 20000000000000000 : ℝ) =
+            (5096354726270043 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5096354726270043 / 5000000000000000 : ℝ) ≤
+            (190876121788911624619 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2252920713877543 / 10000000000000000 : ℝ)) (q :=
+      (5096354726270043 / 20000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_432_11 :
+    Uω (aρ 11) (bρ 11) (1831819407533 / 4000000000000 : ℝ) ≤
+      (-(8658255887911272410519 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3539838466569181 / 20000000000000000 : ℝ) ≤
+      (-(8658255887911272410519 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3539838466569181 / 20000000000000000 : ℝ) =
+            (3539838466569181 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3539838466569181 / 2500000000000000 : ℝ) ≤
+            (1738951816588727589481 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1114881080381681 / 10000000000000000 : ℝ)) (q :=
+      (3539838466569181 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_432_12 :
+    Uω (aρ 12) (bρ 12) (1831819407533 / 4000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_432_13 :
+    Uω (aρ 13) (bρ 13) (1831819407533 / 4000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_432_14 :
+    Uω (aρ 14) (bρ 14) (1831819407533 / 4000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_432_15 :
+    Uω (aρ 15) (bρ 15) (1831819407533 / 4000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_432_16 :
+    Uω (aρ 16) (bρ 16) (1831819407533 / 4000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_432 :
+    Uρ (1831819407533 / 4000000000000 : ℝ) ≤
+      (-(2938449730656911494861 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1831819407533 / 4000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(7951764739322229786499 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_432_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1831819407533 / 4000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1000624303876394457959 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_432_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1831819407533 / 4000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) * (-(31689187354433060421 / 39062500000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_432_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1831819407533 / 4000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(8293837610929466732469 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_432_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1831819407533 / 4000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(4288784465369381867481 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_432_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1831819407533 / 4000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(2250387189094811231179 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_432_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1831819407533 / 4000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(4808301710880464751429 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_432_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1831819407533 / 4000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(10495624041434001507059 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_432_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1831819407533 / 4000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(2939915678554231113423 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_432_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1831819407533 / 4000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(13672067484211088375381 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_432_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1831819407533 / 4000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(8658255887911272410519 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_432_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1831819407533 / 4000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_432_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1831819407533 / 4000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_432_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1831819407533 / 4000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_432_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1831819407533 / 4000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_432_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1831819407533 / 4000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_432_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_433_1 :
+    Uω (aρ 1) (bρ 1) (917234003643 / 2000000000000 : ℝ) ≤
+      (-(1587421971831349571987 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9043274091696977 / 20000000000000000 : ℝ) ≤
+      (-(1587421971831349571987 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9043274091696977 / 20000000000000000 : ℝ) =
+            (9043274091696977 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9043274091696977 / 5000000000000000 : ℝ) ≤
+            (1185166749368650428013 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4521601291566977 / 10000000000000000 : ℝ)) (q :=
+      (9043274091696977 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_433_2 :
+    Uω (aρ 2) (bρ 2) (917234003643 / 2000000000000 : ℝ) ≤
+      (-(998782592033725802447 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4497667903824867 / 10000000000000000 : ℝ) ≤
+      (-(998782592033725802447 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4497667903824867 / 10000000000000000 : ℝ) =
+            (4497667903824867 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4497667903824867 / 2500000000000000 : ℝ) ≤
+            (734085358716274197553 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2248716010014867 / 5000000000000000 : ℝ)) (q :=
+      (4497667903824867 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_433_3 :
+    Uω (aρ 3) (bρ 3) (917234003643 / 2000000000000 : ℝ) ≤
+      (-(8097537234734476602597 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8899352751657329 / 20000000000000000 : ℝ) ≤
+      (-(8097537234734476602597 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8899352751657329 / 20000000000000000 : ℝ) =
+            (8899352751657329 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8899352751657329 / 5000000000000000 : ℝ) ≤
+            (5765406371265523397403 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4448845070387329 / 10000000000000000 : ℝ)) (q :=
+      (8899352751657329 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_433_4 :
+    Uω (aρ 4) (bρ 4) (917234003643 / 2000000000000 : ℝ) ≤
+      (-(8278664895044969696437 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (873961189897237 / 2000000000000000 : ℝ) ≤
+      (-(8278664895044969696437 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (873961189897237 / 2000000000000000 : ℝ) =
+            (873961189897237 / 500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (873961189897237 / 500000000000000 : ℝ) ≤
+            (5584278710955030303563 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (436739839876737 / 1000000000000000 : ℝ)) (q :=
+      (873961189897237 / 2000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_433_5 :
+    Uω (aρ 5) (bρ 5) (917234003643 / 2000000000000 : ℝ) ≤
+      (-(4280972711764913015427 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4247754753818727 / 10000000000000000 : ℝ) ≤
+      (-(4280972711764913015427 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4247754753818727 / 10000000000000000 : ℝ) =
+            (4247754753818727 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4247754753818727 / 2500000000000000 : ℝ) ≤
+            (2650499091235086984573 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2120742963383727 / 5000000000000000 : ℝ)) (q :=
+      (4247754753818727 / 10000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_433_6 :
+    Uω (aρ 6) (bρ 6) (917234003643 / 2000000000000 : ℝ) ≤
+      (-(8985212713119600161899 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (325737048382469 / 800000000000000 : ℝ) ≤
+      (-(8985212713119600161899 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (325737048382469 / 800000000000000 : ℝ) =
+            (325737048382469 / 200000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (325737048382469 / 200000000000000 : ℝ) ≤
+            (4877730892880399838101 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (162265720153269 / 400000000000000 : ℝ)) (q :=
+      (325737048382469 / 800000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_433_7 :
+    Uω (aρ 7) (bρ 7) (917234003643 / 2000000000000 : ℝ) ≤
+      (-(2399785441056326483251 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3829257484234693 / 10000000000000000 : ℝ) ≤
+      (-(2399785441056326483251 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3829257484234693 / 10000000000000000 : ℝ) =
+            (3829257484234693 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3829257484234693 / 2500000000000000 : ℝ) ≤
+            (1065950460443673516749 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1897696571749693 / 5000000000000000 : ℝ)) (q :=
+      (3829257484234693 / 10000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_433_8 :
+    Uω (aρ 8) (bρ 8) (917234003643 / 2000000000000 : ℝ) ≤
+      (-(1309541545056140365681 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3507669498542777 / 10000000000000000 : ℝ) ≤
+      (-(1309541545056140365681 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3507669498542777 / 10000000000000000 : ℝ) =
+            (3507669498542777 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3507669498542777 / 2500000000000000 : ℝ) ≤
+            (423326405693859634319 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1717879411727777 / 5000000000000000 : ℝ)) (q :=
+      (3507669498542777 / 10000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_433_9 :
+    Uω (aρ 9) (bρ 9) (917234003643 / 2000000000000 : ℝ) ≤
+      (-(11737156383840359370543 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (123686347836671 / 400000000000000 : ℝ) ≤
+      (-(11737156383840359370543 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (123686347836671 / 400000000000000 : ℝ) =
+            (123686347836671 / 100000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (123686347836671 / 100000000000000 : ℝ) ≤
+            (2125787222159640629457 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (58914056931271 / 200000000000000 : ℝ)) (q :=
+      (123686347836671 / 400000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_433_10 :
+    Uω (aρ 10) (bρ 10) (917234003643 / 2000000000000 : ℝ) ≤
+      (-(6821365551110305732887 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2555663769420043 / 10000000000000000 : ℝ) ≤
+      (-(6821365551110305732887 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2555663769420043 / 10000000000000000 : ℝ) =
+            (2555663769420043 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2555663769420043 / 2500000000000000 : ℝ) ≤
+            (110106251889694267113 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1130636013532543 / 5000000000000000 : ℝ)) (q :=
+      (2555663769420043 / 10000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_433_11 :
+    Uω (aρ 11) (bρ 11) (917234003643 / 2000000000000 : ℝ) ≤
+      (-(8628749439416123003623 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3560789838101703 / 20000000000000000 : ℝ) ≤
+      (-(8628749439416123003623 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3560789838101703 / 20000000000000000 : ℝ) =
+            (3560789838101703 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3560789838101703 / 2500000000000000 : ℝ) ≤
+            (1768458265083876996377 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1129210952531703 / 10000000000000000 : ℝ)) (q :=
+      (3560789838101703 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_433_12 :
+    Uω (aρ 12) (bρ 12) (917234003643 / 2000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_433_13 :
+    Uω (aρ 13) (bρ 13) (917234003643 / 2000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_433_14 :
+    Uω (aρ 14) (bρ 14) (917234003643 / 2000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_433_15 :
+    Uω (aρ 15) (bρ 15) (917234003643 / 2000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_433_16 :
+    Uω (aρ 16) (bρ 16) (917234003643 / 2000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_433 :
+    Uρ (917234003643 / 2000000000000 : ℝ) ≤
+      (-(2934245738353066200817 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (917234003643 / 2000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1587421971831349571987 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_433_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (917234003643 / 2000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(998782592033725802447 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_433_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (917234003643 / 2000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(8097537234734476602597 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_433_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (917234003643 / 2000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(8278664895044969696437 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_433_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (917234003643 / 2000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(4280972711764913015427 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_433_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (917234003643 / 2000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(8985212713119600161899 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_433_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (917234003643 / 2000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(2399785441056326483251 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_433_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (917234003643 / 2000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(1309541545056140365681 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_433_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (917234003643 / 2000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(11737156383840359370543 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_433_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (917234003643 / 2000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(6821365551110305732887 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_433_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (917234003643 / 2000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(8628749439416123003623 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_433_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (917234003643 / 2000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_433_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (917234003643 / 2000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_433_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (917234003643 / 2000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_433_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (917234003643 / 2000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_433_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (917234003643 / 2000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_433_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_434_1 :
+    Uω (aρ 1) (bρ 1) (1837116607039 / 4000000000000 : ℝ) ≤
+      (-(3961238212228721508109 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (56603232468919 / 125000000000000 : ℝ) ≤
+      (-(3961238212228721508109 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (56603232468919 / 125000000000000 : ℝ) = (56603232468919 / 31250000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (56603232468919 / 31250000000000 : ℝ) ≤
+            (2970233590771278491891 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (226411144775727 / 500000000000000 : ℝ)) (q :=
+      (56603232468919 / 125000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_434_2 :
+    Uω (aρ 2) (bρ 2) (1837116607039 / 4000000000000 : ℝ) ≤
+      (-(7975548720041766875749 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9008579499969673 / 20000000000000000 : ℝ) ≤
+      (-(7975548720041766875749 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9008579499969673 / 20000000000000000 : ℝ) =
+            (9008579499969673 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9008579499969673 / 5000000000000000 : ℝ) ≤
+            (5887394885958233124251 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4504054212967173 / 10000000000000000 : ℝ)) (q :=
+      (9008579499969673 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_434_3 :
+    Uω (aρ 3) (bρ 3) (1837116607039 / 4000000000000 : ℝ) ≤
+      (-(4041332333650775896749 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1782519644262739 / 4000000000000000 : ℝ) ≤
+      (-(4041332333650775896749 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1782519644262739 / 4000000000000000 : ℝ) =
+            (1782519644262739 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1782519644262739 / 1000000000000000 : ℝ) ≤
+            (2890139469349224103251 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (891093808132239 / 2000000000000000 : ℝ)) (q :=
+      (1782519644262739 / 4000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_434_4 :
+    Uω (aρ 4) (bρ 4) (1837116607039 / 4000000000000 : ℝ) ≤
+      (-(4131757595361709041687 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (875286218695081 / 2000000000000000 : ℝ) ≤
+      (-(4131757595361709041687 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (875286218695081 / 2000000000000000 : ℝ) =
+            (875286218695081 / 500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (875286218695081 / 500000000000000 : ℝ) ≤
+            (2799714207638290958313 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (437402718736331 / 1000000000000000 : ℝ)) (q :=
+      (875286218695081 / 2000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_434_5 :
+    Uω (aρ 5) (bρ 5) (1837116607039 / 4000000000000 : ℝ) ≤
+      (-(1709269271915022459983 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1701754409744643 / 4000000000000000 : ℝ) ≤
+      (-(1709269271915022459983 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1701754409744643 / 4000000000000000 : ℝ) =
+            (1701754409744643 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1701754409744643 / 1000000000000000 : ℝ) ≤
+            (1063319449284977540017 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (849625393694143 / 2000000000000000 : ℝ)) (q :=
+      (1701754409744643 / 4000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_434_6 :
+    Uω (aρ 6) (bρ 6) (1837116607039 / 4000000000000 : ℝ) ≤
+      (-(8968903510601811328087 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4078359162264601 / 10000000000000000 : ℝ) ≤
+      (-(8968903510601811328087 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4078359162264601 / 10000000000000000 : ℝ) =
+            (4078359162264601 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4078359162264601 / 2500000000000000 : ℝ) ≤
+            (4894040095398188671913 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2031656809708351 / 5000000000000000 : ℝ)) (q :=
+      (4078359162264601 / 10000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_434_7 :
+    Uω (aρ 7) (bρ 7) (1837116607039 / 4000000000000 : ℝ) ≤
+      (-(4790855543348075876977 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1534375184017047 / 4000000000000000 : ℝ) ≤
+      (-(4790855543348075876977 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1534375184017047 / 4000000000000000 : ℝ) =
+            (1534375184017047 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1534375184017047 / 1000000000000000 : ℝ) ≤
+            (2140616259651924123023 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (760426519146547 / 2000000000000000 : ℝ)) (q :=
+      (1534375184017047 / 4000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_434_8 :
+    Uω (aρ 8) (bρ 8) (1837116607039 / 4000000000000 : ℝ) ≤
+      (-(10457079376504312439157 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7028858628479507 / 20000000000000000 : ℝ) ≤
+      (-(10457079376504312439157 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7028858628479507 / 20000000000000000 : ℝ) =
+            (7028858628479507 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7028858628479507 / 5000000000000000 : ℝ) ≤
+            (3405864229495687560843 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3442656955467007 / 10000000000000000 : ℝ)) (q :=
+      (7028858628479507 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_434_9 :
+    Uω (aρ 9) (bρ 9) (1837116607039 / 4000000000000 : ℝ) ≤
+      (-(2928676401593483871823 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3099108631247451 / 10000000000000000 : ℝ) ≤
+      (-(2928676401593483871823 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3099108631247451 / 10000000000000000 : ℝ) =
+            (3099108631247451 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3099108631247451 / 2500000000000000 : ℝ) ≤
+            (537059499906516128177 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1476490608921201 / 5000000000000000 : ℝ)) (q :=
+      (3099108631247451 / 10000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_434_10 :
+    Uω (aρ 10) (bρ 10) (1837116607039 / 4000000000000 : ℝ) ≤
+      (-(13613502791977963723017 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5126288939717917 / 20000000000000000 : ℝ) ≤
+      (-(13613502791977963723017 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5126288939717917 / 20000000000000000 : ℝ) =
+            (5126288939717917 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5126288939717917 / 5000000000000000 : ℝ) ≤
+            (249440814022036276983 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2269611928560417 / 10000000000000000 : ℝ)) (q :=
+      (5126288939717917 / 20000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_434_11 :
+    Uω (aρ 11) (bρ 11) (1837116607039 / 4000000000000 : ℝ) ≤
+      (-(3439845305272605237621 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1790799986019643 / 10000000000000000 : ℝ) ≤
+      (-(3439845305272605237621 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1790799986019643 / 10000000000000000 : ℝ) =
+            (1790799986019643 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1790799986019643 / 1250000000000000 : ℝ) ≤
+            (719037776527394762379 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (571699793543393 / 5000000000000000 : ℝ)) (q :=
+      (1790799986019643 / 10000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_434_12 :
+    Uω (aρ 12) (bρ 12) (1837116607039 / 4000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_434_13 :
+    Uω (aρ 13) (bρ 13) (1837116607039 / 4000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_434_14 :
+    Uω (aρ 14) (bρ 14) (1837116607039 / 4000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_434_15 :
+    Uω (aρ 15) (bρ 15) (1837116607039 / 4000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_434_16 :
+    Uω (aρ 16) (bρ 16) (1837116607039 / 4000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_434 :
+    Uρ (1837116607039 / 4000000000000 : ℝ) ≤
+      (-(1172025256447057801947 / 1000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1837116607039 / 4000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3961238212228721508109 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_434_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1837116607039 / 4000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(7975548720041766875749 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_434_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1837116607039 / 4000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(4041332333650775896749 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_434_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1837116607039 / 4000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(4131757595361709041687 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_434_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1837116607039 / 4000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1709269271915022459983 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_434_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1837116607039 / 4000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(8968903510601811328087 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_434_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1837116607039 / 4000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(4790855543348075876977 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_434_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1837116607039 / 4000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(10457079376504312439157 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_434_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1837116607039 / 4000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(2928676401593483871823 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_434_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1837116607039 / 4000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(13613502791977963723017 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_434_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1837116607039 / 4000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(3439845305272605237621 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_434_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1837116607039 / 4000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_434_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1837116607039 / 4000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_434_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1837116607039 / 4000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_434_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1837116607039 / 4000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_434_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1837116607039 / 4000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_434_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_435_1 :
+    Uω (aρ 1) (bρ 1) (229970650849 / 500000000000 : ℝ) ≤
+      (-(3953932186274931808203 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9069760298051741 / 20000000000000000 : ℝ) ≤
+      (-(3953932186274931808203 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9069760298051741 / 20000000000000000 : ℝ) =
+            (9069760298051741 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9069760298051741 / 5000000000000000 : ℝ) ≤
+            (2977539616725068191797 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4534844499156741 / 10000000000000000 : ℝ)) (q :=
+      (9069760298051741 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_435_2 :
+    Uω (aρ 2) (bρ 2) (229970650849 / 500000000000 : ℝ) ≤
+      (-(1592171663724093181691 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9021823190253283 / 20000000000000000 : ℝ) ≤
+      (-(1592171663724093181691 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9021823190253283 / 20000000000000000 : ℝ) =
+            (9021823190253283 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9021823190253283 / 5000000000000000 : ℝ) ≤
+            (1180417057475906818309 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4510676403868283 / 10000000000000000 : ℝ)) (q :=
+      (9021823190253283 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_435_3 :
+    Uω (aρ 3) (bρ 3) (229970650849 / 500000000000 : ℝ) ≤
+      (-(1008476774321000244151 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8925843683635361 / 20000000000000000 : ℝ) ≤
+      (-(1008476774321000244151 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8925843683635361 / 20000000000000000 : ℝ) =
+            (8925843683635361 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8925843683635361 / 5000000000000000 : ℝ) ≤
+            (724391176428999755849 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4462093003600361 / 10000000000000000 : ℝ)) (q :=
+      (8925843683635361 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_435_4 :
+    Uω (aρ 4) (bρ 4) (229970650849 / 500000000000 : ℝ) ≤
+      (-(1031048553524197032573 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (876611245288137 / 2000000000000000 : ℝ) ≤
+      (-(1031048553524197032573 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (876611245288137 / 2000000000000000 : ℝ) =
+            (876611245288137 / 500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (876611245288137 / 500000000000000 : ℝ) ≤
+            (701819397225802967427 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (438065595391137 / 1000000000000000 : ℝ)) (q :=
+      (876611245288137 / 2000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_435_5 :
+    Uω (aρ 5) (bρ 5) (229970650849 / 500000000000 : ℝ) ≤
+      (-(8530771662286591006543 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8522034528893589 / 20000000000000000 : ℝ) ≤
+      (-(8530771662286591006543 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8522034528893589 / 20000000000000000 : ℝ) =
+            (8522034528893589 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8522034528893589 / 5000000000000000 : ℝ) ≤
+            (5332171943713408993457 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4254767949258589 / 10000000000000000 : ℝ)) (q :=
+      (8522034528893589 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_435_6 :
+    Uω (aρ 6) (bρ 6) (229970650849 / 500000000000 : ℝ) ≤
+      (-(8952621060125969732773 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1021251284886079 / 2500000000000000 : ℝ) ≤
+      (-(8952621060125969732773 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1021251284886079 / 2500000000000000 : ℝ) =
+            (1021251284886079 / 625000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1021251284886079 / 625000000000000 : ℝ) ≤
+            (4910322545874030267227 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (127187002331051 / 312500000000000 : ℝ)) (q :=
+      (1021251284886079 / 2500000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_435_7 :
+    Uω (aρ 7) (bρ 7) (229970650849 / 500000000000 : ℝ) ≤
+      (-(9564311277543446347187 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24016363931063 / 62500000000000 : ℝ) ≤
+      (-(9564311277543446347187 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (24016363931063 / 62500000000000 : ℝ) = (24016363931063 / 15625000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24016363931063 / 15625000000000 : ℝ) ≤
+            (4298632328456553652813 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (95221790855129 / 250000000000000 : ℝ)) (q :=
+      (24016363931063 / 62500000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_435_8 :
+    Uω (aρ 8) (bρ 8) (229970650849 / 500000000000 : ℝ) ≤
+      (-(10437864928602727238181 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3521188587841929 / 10000000000000000 : ℝ) ≤
+      (-(10437864928602727238181 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3521188587841929 / 10000000000000000 : ℝ) =
+            (3521188587841929 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3521188587841929 / 2500000000000000 : ℝ) ≤
+            (3425078677397272761819 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1724777001644429 / 5000000000000000 : ℝ)) (q :=
+      (3521188587841929 / 10000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_435_9 :
+    Uω (aρ 9) (bρ 9) (229970650849 / 500000000000 : ℝ) ≤
+      (-(11692310084797909684141 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1553028512204731 / 5000000000000000 : ℝ) ≤
+      (-(11692310084797909684141 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1553028512204731 / 5000000000000000 : ℝ) =
+            (1553028512204731 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1553028512204731 / 1250000000000000 : ℝ) ≤
+            (2170633521202090315859 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (740064126195981 / 2500000000000000 : ℝ)) (q :=
+      (1553028512204731 / 5000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_435_10 :
+    Uω (aρ 10) (bρ 10) (229970650849 / 500000000000 : ℝ) ≤
+      (-(13584381614807102870191 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2570619527121791 / 10000000000000000 : ℝ) ≤
+      (-(13584381614807102870191 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2570619527121791 / 10000000000000000 : ℝ) =
+            (2570619527121791 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2570619527121791 / 2500000000000000 : ℝ) ≤
+            (278561991192897129809 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1138970271851791 / 5000000000000000 : ℝ)) (q :=
+      (2570619527121791 / 10000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_435_11 :
+    Uω (aρ 11) (bρ 11) (229970650849 / 500000000000 : ℝ) ≤
+      (-(1714166942718427090193 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (360227406247933 / 2000000000000000 : ℝ) ≤
+      (-(1714166942718427090193 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (360227406247933 / 2000000000000000 : ℝ) =
+            (360227406247933 / 250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (360227406247933 / 250000000000000 : ℝ) ≤
+            (365274598181572909807 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (115745217814433 / 1000000000000000 : ℝ)) (q :=
+      (360227406247933 / 2000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11]) (by norm_num [bρ_11])
+      (by norm_num) (by norm_num [aρ_11, bρ_11]) (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_435_12 :
+    Uω (aρ 12) (bρ 12) (229970650849 / 500000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_435_13 :
+    Uω (aρ 13) (bρ 13) (229970650849 / 500000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_435_14 :
+    Uω (aρ 14) (bρ 14) (229970650849 / 500000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_435_15 :
+    Uω (aρ 15) (bρ 15) (229970650849 / 500000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_435_16 :
+    Uω (aρ 16) (bρ 16) (229970650849 / 500000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_435 :
+    Uρ (229970650849 / 500000000000 : ℝ) ≤
+      (-(1170360560847339253183 / 1000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (229970650849 / 500000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3953932186274931808203 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_435_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (229970650849 / 500000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1592171663724093181691 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_435_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (229970650849 / 500000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1008476774321000244151 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_435_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (229970650849 / 500000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1031048553524197032573 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_435_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (229970650849 / 500000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(8530771662286591006543 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_435_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (229970650849 / 500000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(8952621060125969732773 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_435_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (229970650849 / 500000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(9564311277543446347187 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_435_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (229970650849 / 500000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(10437864928602727238181 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_435_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (229970650849 / 500000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(11692310084797909684141 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_435_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (229970650849 / 500000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(13584381614807102870191 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_435_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (229970650849 / 500000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(1714166942718427090193 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_435_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (229970650849 / 500000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_435_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (229970650849 / 500000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_435_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (229970650849 / 500000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_435_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (229970650849 / 500000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_435_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (229970650849 / 500000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_435_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/U36.lean b/LeanPool/Zeta5Irrational/Table/U36.lean
new file mode 100644
index 0000000000..84aed46b20
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/U36.lean
@@ -0,0 +1,4364 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-! # Certified arcsine potential bounds (U36) -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma U_436_1 :
+    Uω (aρ 1) (bρ 1) (368482761309 / 800000000000 : ℝ) ≤
+      (-(7893273641033907831537 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9083003400772417 / 20000000000000000 : ℝ) ≤
+      (-(7893273641033907831537 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9083003400772417 / 20000000000000000 : ℝ) =
+            (9083003400772417 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9083003400772417 / 5000000000000000 : ℝ) ≤
+            (5969669964966092168463 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4541466102494917 / 10000000000000000 : ℝ)) (q :=
+      (9083003400772417 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_436_2 :
+    Uω (aρ 2) (bρ 2) (368482761309 / 800000000000 : ℝ) ≤
+      (-(3973094734289888371921 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9035066878509517 / 20000000000000000 : ℝ) ≤
+      (-(3973094734289888371921 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9035066878509517 / 20000000000000000 : ℝ) =
+            (9035066878509517 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9035066878509517 / 5000000000000000 : ℝ) ≤
+            (2958377068710111628079 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4517298592742017 / 10000000000000000 : ℝ)) (q :=
+      (9035066878509517 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_436_3 :
+    Uω (aρ 3) (bρ 3) (368482761309 / 800000000000 : ℝ) ≤
+      (-(8052985750959086163193 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4469544569327471 / 10000000000000000 : ℝ) ≤
+      (-(8052985750959086163193 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4469544569327471 / 10000000000000000 : ℝ) =
+            (4469544569327471 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4469544569327471 / 2500000000000000 : ℝ) ≤
+            (5809957855040913836807 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2234358479618721 / 5000000000000000 : ℝ)) (q :=
+      (4469544569327471 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_436_4 :
+    Uω (aρ 4) (bρ 4) (368482761309 / 800000000000 : ℝ) ≤
+      (-(8233284538001393696237 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8779362696863987 / 20000000000000000 : ℝ) ≤
+      (-(8233284538001393696237 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8779362696863987 / 20000000000000000 : ℝ) =
+            (8779362696863987 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8779362696863987 / 5000000000000000 : ℝ) ≤
+            (5629659067998606303763 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4387284698511487 / 10000000000000000 : ℝ)) (q :=
+      (8779362696863987 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_436_5 :
+    Uω (aρ 5) (bρ 5) (368482761309 / 800000000000 : ℝ) ≤
+      (-(2128805313859165745199 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (853529694843337 / 2000000000000000 : ℝ) ≤
+      (-(2128805313859165745199 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (853529694843337 / 2000000000000000 : ℝ) =
+            (853529694843337 / 500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (853529694843337 / 500000000000000 : ℝ) ≤
+            (1336930587640834254801 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (426140886941587 / 1000000000000000 : ℝ)) (q :=
+      (853529694843337 / 2000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_436_6 :
+    Uω (aρ 6) (bρ 6) (368482761309 / 800000000000 : ℝ) ≤
+      (-(4468182636717128945429 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8183302074027421 / 20000000000000000 : ℝ) ≤
+      (-(4468182636717128945429 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8183302074027421 / 20000000000000000 : ℝ) =
+            (8183302074027421 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8183302074027421 / 5000000000000000 : ℝ) ≤
+            (2463289166282871054571 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4076654370149921 / 10000000000000000 : ℝ)) (q :=
+      (8183302074027421 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_436_7 :
+    Uω (aρ 7) (bρ 7) (368482761309 / 800000000000 : ℝ) ≤
+      (-(9546942225749288231573 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1924649146056621 / 5000000000000000 : ℝ) ≤
+      (-(9546942225749288231573 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1924649146056621 / 5000000000000000 : ℝ) =
+            (1924649146056621 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1924649146056621 / 1250000000000000 : ℝ) ≤
+            (4316001380250711768427 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (476951282638623 / 1250000000000000 : ℝ)) (q :=
+      (1924649146056621 / 5000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_436_8 :
+    Uω (aρ 8) (bρ 8) (368482761309 / 800000000000 : ℝ) ≤
+      (-(2083737771356987306073 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (282235785807553 / 800000000000000 : ℝ) ≤
+      (-(2083737771356987306073 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (282235785807553 / 800000000000000 : ℝ) =
+            (282235785807553 / 200000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (282235785807553 / 200000000000000 : ℝ) ≤
+            (688850949843012693927 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (138257998936453 / 400000000000000 : ℝ)) (q :=
+      (282235785807553 / 800000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_436_9 :
+    Uω (aρ 9) (bρ 9) (368482761309 / 800000000000 : ℝ) ≤
+      (-(18234327382226959179 / 15625000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (194562742921581 / 625000000000000 : ℝ) ≤
+      (-(18234327382226959179 / 15625000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (194562742921581 / 625000000000000 : ℝ) =
+            (194562742921581 / 156250000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (194562742921581 / 156250000000000 : ℝ) ≤
+            (3426522002148040821 / 15625000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (741882182518273 / 2500000000000000 : ℝ)) (q :=
+      (194562742921581 / 625000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_436_10 :
+    Uω (aρ 10) (bρ 10) (368482761309 / 800000000000 : ℝ) ≤
+      (-(6777683322747721136581 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5156178005762329 / 20000000000000000 : ℝ) ≤
+      (-(6777683322747721136581 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5156178005762329 / 20000000000000000 : ℝ) =
+            (5156178005762329 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5156178005762329 / 5000000000000000 : ℝ) ≤
+            (153788480252278863419 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2286257995839829 / 10000000000000000 : ℝ)) (q :=
+      (5156178005762329 / 20000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_436_11 :
+    Uω (aρ 11) (bρ 11) (368482761309 / 800000000000 : ℝ) ≤
+      (-(8542401856030899137819 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (724563401128357 / 4000000000000000 : ℝ) ≤
+      (-(8542401856030899137819 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (724563401128357 / 4000000000000000 : ℝ) =
+            (724563401128357 / 500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (724563401128357 / 500000000000000 : ℝ) ≤
+            (1854805848469100862181 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (234274724384857 / 2000000000000000 : ℝ)) (q :=
+      (724563401128357 / 4000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11]) (by norm_num [bρ_11])
+      (by norm_num) (by norm_num [aρ_11, bρ_11]) (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_436_12 :
+    Uω (aρ 12) (bρ 12) (368482761309 / 800000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_436_13 :
+    Uω (aρ 13) (bρ 13) (368482761309 / 800000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_436_14 :
+    Uω (aρ 14) (bρ 14) (368482761309 / 800000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_436_15 :
+    Uω (aρ 15) (bρ 15) (368482761309 / 800000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_436_16 :
+    Uω (aρ 16) (bρ 16) (368482761309 / 800000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_436 :
+    Uρ (368482761309 / 800000000000 : ℝ) ≤
+      (-(11687040051905877864877 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (368482761309 / 800000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(7893273641033907831537 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_436_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (368482761309 / 800000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3973094734289888371921 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_436_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (368482761309 / 800000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(8052985750959086163193 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_436_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (368482761309 / 800000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(8233284538001393696237 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_436_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (368482761309 / 800000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(2128805313859165745199 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_436_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (368482761309 / 800000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(4468182636717128945429 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_436_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (368482761309 / 800000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(9546942225749288231573 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_436_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (368482761309 / 800000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(2083737771356987306073 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_436_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (368482761309 / 800000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(18234327382226959179 / 15625000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_436_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (368482761309 / 800000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(6777683322747721136581 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_436_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (368482761309 / 800000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(8542401856030899137819 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_436_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (368482761309 / 800000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_436_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (368482761309 / 800000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_436_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (368482761309 / 800000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_436_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (368482761309 / 800000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_436_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (368482761309 / 800000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_436_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_437_1 :
+    Uω (aρ 1) (bρ 1) (922531203149 / 2000000000000 : ℝ) ≤
+      (-(7878704167782229152803 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2274061625797599 / 5000000000000000 : ℝ) ≤
+      (-(7878704167782229152803 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2274061625797599 / 5000000000000000 : ℝ) =
+            (2274061625797599 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2274061625797599 / 1250000000000000 : ℝ) ≤
+            (5984239438217770847197 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1137021926382599 / 2500000000000000 : ℝ)) (q :=
+      (2274061625797599 / 5000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_437_2 :
+    Uω (aρ 2) (bρ 2) (922531203149 / 2000000000000 : ℝ) ≤
+      (-(317261684270894545459 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9048310564747279 / 20000000000000000 : ℝ) ≤
+      (-(317261684270894545459 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9048310564747279 / 20000000000000000 : ℝ) =
+            (9048310564747279 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9048310564747279 / 5000000000000000 : ℝ) ≤
+            (237256059969105454541 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4523920779597279 / 10000000000000000 : ℝ)) (q :=
+      (9048310564747279 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_437_3 :
+    Uω (aρ 3) (bρ 3) (922531203149 / 2000000000000 : ℝ) ≤
+      (-(4019089635595831755489 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8952334586404863 / 20000000000000000 : ℝ) ≤
+      (-(4019089635595831755489 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8952334586404863 / 20000000000000000 : ℝ) =
+            (8952334586404863 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8952334586404863 / 5000000000000000 : ℝ) ≤
+            (2912382167404168244511 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4475340907604863 / 10000000000000000 : ℝ)) (q :=
+      (8952334586404863 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_437_4 :
+    Uω (aρ 4) (bρ 4) (922531203149 / 2000000000000 : ℝ) ≤
+      (-(256818857844002620983 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4396306459498997 / 10000000000000000 : ℝ) ≤
+      (-(256818857844002620983 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4396306459498997 / 10000000000000000 : ℝ) =
+            (4396306459498997 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4396306459498997 / 2500000000000000 : ℝ) ≤
+            (176398129843497379017 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2196956710631497 / 5000000000000000 : ℝ)) (q :=
+      (4396306459498997 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_437_5 :
+    Uω (aρ 5) (bρ 5) (922531203149 / 2000000000000 : ℝ) ≤
+      (-(8499695063155904997523 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4274279653812787 / 10000000000000000 : ℝ) ≤
+      (-(8499695063155904997523 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4274279653812787 / 10000000000000000 : ℝ) =
+            (4274279653812787 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4274279653812787 / 2500000000000000 : ℝ) ≤
+            (5363248542844095002477 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2134024864612787 / 5000000000000000 : ℝ)) (q :=
+      (4274279653812787 / 10000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_437_6 :
+    Uω (aρ 6) (bρ 6) (922531203149 / 2000000000000 : ℝ) ≤
+      (-(4460068031353997442141 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2049148427531953 / 5000000000000000 : ℝ) ≤
+      (-(4460068031353997442141 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2049148427531953 / 5000000000000000 : ℝ) =
+            (2049148427531953 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2049148427531953 / 1250000000000000 : ℝ) ≤
+            (2471403771646002557859 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1020831126716953 / 2500000000000000 : ℝ)) (q :=
+      (2049148427531953 / 5000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_437_7 :
+    Uω (aρ 7) (bρ 7) (922531203149 / 2000000000000 : ℝ) ≤
+      (-(4764801910451678914299 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3855978150560469 / 10000000000000000 : ℝ) ≤
+      (-(4764801910451678914299 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3855978150560469 / 10000000000000000 : ℝ) =
+            (3855978150560469 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3855978150560469 / 2500000000000000 : ℝ) ≤
+            (2166669892548321085701 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1911174239310469 / 5000000000000000 : ℝ)) (q :=
+      (3855978150560469 / 10000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_437_8 :
+    Uω (aρ 8) (bρ 8) (922531203149 / 2000000000000 : ℝ) ≤
+      (-(129994387526496953403 / 125000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7069411043431927 / 20000000000000000 : ℝ) ≤
+      (-(129994387526496953403 / 125000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7069411043431927 / 20000000000000000 : ℝ) =
+            (7069411043431927 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7069411043431927 / 5000000000000000 : ℝ) ≤
+            (43292407548503046597 / 125000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3463344872271927 / 10000000000000000 : ℝ)) (q :=
+      (7069411043431927 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_437_9 :
+    Uω (aρ 9) (bρ 9) (922531203149 / 2000000000000 : ℝ) ≤
+      (-(5823841816935791055419 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3119949229481779 / 10000000000000000 : ℝ) ≤
+      (-(5823841816935791055419 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3119949229481779 / 10000000000000000 : ℝ) =
+            (3119949229481779 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3119949229481779 / 2500000000000000 : ℝ) ≤
+            (1107629986064208944581 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1487398958081779 / 5000000000000000 : ℝ)) (q :=
+      (3119949229481779 / 10000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_437_10 :
+    Uω (aρ 10) (bρ 10) (922531203149 / 2000000000000 : ℝ) ≤
+      (-(6763228486011371335239 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5171105915666843 / 20000000000000000 : ℝ) ≤
+      (-(6763228486011371335239 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5171105915666843 / 20000000000000000 : ℝ) =
+            (5171105915666843 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5171105915666843 / 5000000000000000 : ℝ) ≤
+            (168243316988628664761 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2294564406361843 / 10000000000000000 : ℝ)) (q :=
+      (5171105915666843 / 20000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_437_11 :
+    Uω (aρ 11) (bρ 11) (922531203149 / 2000000000000 : ℝ) ≤
+      (-(17028606823962924506891 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3643233423054183 / 20000000000000000 : ℝ) ≤
+      (-(17028606823962924506891 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3643233423054183 / 20000000000000000 : ℝ) =
+            (3643233423054183 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3643233423054183 / 2500000000000000 : ℝ) ≤
+            (3765808585037075493109 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1185168539954183 / 10000000000000000 : ℝ)) (q :=
+      (3643233423054183 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_437_12 :
+    Uω (aρ 12) (bρ 12) (922531203149 / 2000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_437_13 :
+    Uω (aρ 13) (bρ 13) (922531203149 / 2000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_437_14 :
+    Uω (aρ 14) (bρ 14) (922531203149 / 2000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_437_15 :
+    Uω (aρ 15) (bρ 15) (922531203149 / 2000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_437_16 :
+    Uω (aρ 16) (bρ 16) (922531203149 / 2000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_437 :
+    Uρ (922531203149 / 2000000000000 : ℝ) ≤
+      (-(1167055396637717132097 / 1000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (922531203149 / 2000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(7878704167782229152803 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_437_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (922531203149 / 2000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(317261684270894545459 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_437_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (922531203149 / 2000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(4019089635595831755489 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_437_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (922531203149 / 2000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(256818857844002620983 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_437_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (922531203149 / 2000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(8499695063155904997523 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_437_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (922531203149 / 2000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(4460068031353997442141 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_437_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (922531203149 / 2000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(4764801910451678914299 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_437_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (922531203149 / 2000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(129994387526496953403 / 125000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_437_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (922531203149 / 2000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(5823841816935791055419 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_437_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (922531203149 / 2000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(6763228486011371335239 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_437_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (922531203149 / 2000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(17028606823962924506891 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_437_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (922531203149 / 2000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_437_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (922531203149 / 2000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_437_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (922531203149 / 2000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_437_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (922531203149 / 2000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_437_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (922531203149 / 2000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_437_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_438_1 :
+    Uω (aρ 1) (bρ 1) (1847711006051 / 4000000000000 : ℝ) ≤
+      (-(7864155890938650200869 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2277372401326749 / 5000000000000000 : ℝ) ≤
+      (-(7864155890938650200869 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2277372401326749 / 5000000000000000 : ℝ) =
+            (2277372401326749 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2277372401326749 / 1250000000000000 : ℝ) ≤
+            (5998787715061349799131 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (284669331766531 / 625000000000000 : ℝ)) (q :=
+      (2277372401326749 / 5000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_438_2 :
+    Uω (aρ 2) (bρ 2) (1847711006051 / 4000000000000 : ℝ) ≤
+      (-(3958458085164027500651 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (453077712448771 / 1000000000000000 : ℝ) ≤
+      (-(3958458085164027500651 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (453077712448771 / 1000000000000000 : ℝ) =
+            (453077712448771 / 250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (453077712448771 / 250000000000000 : ℝ) ≤
+            (2973013717835972499349 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (113263574111073 / 250000000000000 : ℝ)) (q :=
+      (453077712448771 / 1000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_438_3 :
+    Uω (aρ 3) (bρ 3) (1847711006051 / 4000000000000 : ℝ) ≤
+      (-(250731084071014719003 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2241395006729339 / 5000000000000000 : ℝ) ≤
+      (-(250731084071014719003 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2241395006729339 / 5000000000000000 : ℝ) =
+            (2241395006729339 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2241395006729339 / 1250000000000000 : ℝ) ≤
+            (182485903616485280997 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (560245606091857 / 1250000000000000 : ℝ)) (q :=
+      (2241395006729339 / 5000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_438_4 :
+    Uω (aρ 4) (bρ 4) (1847711006051 / 4000000000000 : ℝ) ≤
+      (-(8203145098388216551507 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (70446904955057 / 160000000000000 : ℝ) ≤
+      (-(8203145098388216551507 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (70446904955057 / 160000000000000 : ℝ) = (70446904955057 / 40000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (70446904955057 / 40000000000000 : ℝ) ≤
+            (5659798507611783448493 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (35204336978117 / 80000000000000 : ℝ)) (q :=
+      (70446904955057 / 160000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_438_5 :
+    Uω (aρ 5) (bρ 5) (1847711006051 / 4000000000000 : ℝ) ≤
+      (-(265131031560338101691 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (267556925210983 / 625000000000000 : ℝ) ≤
+      (-(265131031560338101691 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (267556925210983 / 625000000000000 : ℝ) =
+            (267556925210983 / 156250000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (267556925210983 / 156250000000000 : ℝ) ≤
+            (168085956127161898309 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1068672632242239 / 2500000000000000 : ℝ)) (q :=
+      (267556925210983 / 625000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_438_6 :
+    Uω (aρ 6) (bρ 6) (1847711006051 / 4000000000000 : ℝ) ≤
+      (-(890393334056470965597 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4104942594083463 / 10000000000000000 : ℝ) ≤
+      (-(890393334056470965597 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4104942594083463 / 10000000000000000 : ℝ) =
+            (4104942594083463 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4104942594083463 / 2500000000000000 : ℝ) ≤
+            (495901026543529034403 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2044997242762213 / 5000000000000000 : ℝ)) (q :=
+      (4104942594083463 / 10000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_438_7 :
+    Uω (aρ 7) (bρ 7) (1847711006051 / 4000000000000 : ℝ) ≤
+      (-(4756147976599214266163 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3862657805392399 / 10000000000000000 : ℝ) ≤
+      (-(4756147976599214266163 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3862657805392399 / 10000000000000000 : ℝ) =
+            (3862657805392399 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3862657805392399 / 2500000000000000 : ℝ) ≤
+            (2175323826400785733837 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1914543144451149 / 5000000000000000 : ℝ)) (q :=
+      (3862657805392399 / 10000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_438_8 :
+    Uω (aρ 8) (bρ 8) (1847711006051 / 4000000000000 : ℝ) ≤
+      (-(1297556400836899823467 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7082926376798527 / 20000000000000000 : ℝ) ≤
+      (-(1297556400836899823467 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7082926376798527 / 20000000000000000 : ℝ) =
+            (7082926376798527 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7082926376798527 / 5000000000000000 : ℝ) ≤
+            (435311549913100176533 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3470238706256027 / 10000000000000000 : ℝ)) (q :=
+      (7082926376798527 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_438_9 :
+    Uω (aρ 9) (bρ 9) (1847711006051 / 4000000000000 : ℝ) ≤
+      (-(11625452123025537848547 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7817232659329 / 25000000000000 : ℝ) ≤
+      (-(11625452123025537848547 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (7817232659329 / 25000000000000 : ℝ) = (7817232659329 / 6250000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7817232659329 / 6250000000000 : ℝ) ≤
+            (2237491482974462151453 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (29820640852807 / 100000000000000 : ℝ)) (q :=
+      (7817232659329 / 25000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_438_10 :
+    Uω (aρ 10) (bρ 10) (1847711006051 / 4000000000000 : ℝ) ≤
+      (-(13497651695297014300729 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5186022903438361 / 20000000000000000 : ℝ) ≤
+      (-(13497651695297014300729 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5186022903438361 / 20000000000000000 : ℝ) =
+            (5186022903438361 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5186022903438361 / 5000000000000000 : ℝ) ≤
+            (365291910702985699271 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2302859894750861 / 10000000000000000 : ℝ)) (q :=
+      (5186022903438361 / 20000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_438_11 :
+    Uω (aρ 11) (bρ 11) (1847711006051 / 4000000000000 : ℝ) ≤
+      (-(1060816088683519187111 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (183176384123639 / 1000000000000000 : ℝ) ≤
+      (-(1060816088683519187111 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (183176384123639 / 1000000000000000 : ℝ) =
+            (183176384123639 / 125000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (183176384123639 / 125000000000000 : ℝ) ≤
+            (238834874378980812889 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (29971032499757 / 250000000000000 : ℝ)) (q :=
+      (183176384123639 / 1000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11]) (by norm_num [bρ_11])
+      (by norm_num) (by norm_num [aρ_11, bρ_11]) (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_438_12 :
+    Uω (aρ 12) (bρ 12) (1847711006051 / 4000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_438_13 :
+    Uω (aρ 13) (bρ 13) (1847711006051 / 4000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_438_14 :
+    Uω (aρ 14) (bρ 14) (1847711006051 / 4000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_438_15 :
+    Uω (aρ 15) (bρ 15) (1847711006051 / 4000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_438_16 :
+    Uω (aρ 16) (bρ 16) (1847711006051 / 4000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_438 :
+    Uρ (1847711006051 / 4000000000000 : ℝ) ≤
+      (-(582707276038643686991 / 500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1847711006051 / 4000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(7864155890938650200869 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_438_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1847711006051 / 4000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3958458085164027500651 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_438_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1847711006051 / 4000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(250731084071014719003 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_438_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1847711006051 / 4000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(8203145098388216551507 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_438_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1847711006051 / 4000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(265131031560338101691 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_438_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1847711006051 / 4000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(890393334056470965597 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_438_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1847711006051 / 4000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(4756147976599214266163 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_438_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1847711006051 / 4000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(1297556400836899823467 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_438_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1847711006051 / 4000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(11625452123025537848547 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_438_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1847711006051 / 4000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(13497651695297014300729 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_438_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1847711006051 / 4000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(1060816088683519187111 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_438_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1847711006051 / 4000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_438_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1847711006051 / 4000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_438_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1847711006051 / 4000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_438_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1847711006051 / 4000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_438_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1847711006051 / 4000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_438_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_439_1 :
+    Uω (aρ 1) (bρ 1) (462589901451 / 1000000000000 : ℝ) ≤
+      (-(7665653075113846959 / 9765625000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2280683176780883 / 5000000000000000 : ℝ) ≤
+      (-(7665653075113846959 / 9765625000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2280683176780883 / 5000000000000000 : ℝ) =
+            (2280683176780883 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2280683176780883 / 1250000000000000 : ℝ) ≤
+            (5872377790120528041 / 9765625000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1140332727674633 / 2500000000000000 : ℝ)) (q :=
+      (2280683176780883 / 5000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_439_2 :
+    Uω (aρ 2) (bρ 2) (462589901451 / 1000000000000 : ℝ) ≤
+      (-(7902311596652221809187 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (453739896560137 / 1000000000000000 : ℝ) ≤
+      (-(7902311596652221809187 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (453739896560137 / 1000000000000000 : ℝ) =
+            (453739896560137 / 250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (453739896560137 / 250000000000000 : ℝ) ≤
+            (5960632009347778190813 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (226858257364387 / 500000000000000 : ℝ)) (q :=
+      (453739896560137 / 1000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_439_3 :
+    Uω (aρ 3) (bρ 3) (462589901451 / 1000000000000 : ℝ) ≤
+      (-(1601726388699282176059 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (448941273011223 / 1000000000000000 : ℝ) ≤
+      (-(1601726388699282176059 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (448941273011223 / 1000000000000000 : ℝ) =
+            (448941273011223 / 250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (448941273011223 / 250000000000000 : ℝ) ≤
+            (1170862332500717823941 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (224429439132973 / 500000000000000 : ℝ)) (q :=
+      (448941273011223 / 1000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_439_4 :
+    Uω (aρ 4) (bρ 4) (462589901451 / 1000000000000 : ℝ) ≤
+      (-(8188109411627823297001 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8819113298114521 / 20000000000000000 : ℝ) ≤
+      (-(8188109411627823297001 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8819113298114521 / 20000000000000000 : ℝ) =
+            (8819113298114521 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8819113298114521 / 5000000000000000 : ℝ) ≤
+            (5674834194372176702999 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4407170801614521 / 10000000000000000 : ℝ)) (q :=
+      (8819113298114521 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_439_5 :
+    Uω (aρ 5) (bρ 5) (462589901451 / 1000000000000 : ℝ) ≤
+      (-(8468715020601605035629 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8575083846090523 / 20000000000000000 : ℝ) ≤
+      (-(8468715020601605035629 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8575083846090523 / 20000000000000000 : ℝ) =
+            (8575083846090523 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8575083846090523 / 5000000000000000 : ℝ) ≤
+            (5394228585398394964371 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4281331268925523 / 10000000000000000 : ℝ)) (q :=
+      (8575083846090523 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_439_6 :
+    Uω (aρ 6) (bρ 6) (462589901451 / 1000000000000 : ℝ) ≤
+      (-(2221939255013808664823 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4111588254458403 / 10000000000000000 : ℝ) ≤
+      (-(2221939255013808664823 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4111588254458403 / 10000000000000000 : ℝ) =
+            (4111588254458403 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4111588254458403 / 2500000000000000 : ℝ) ≤
+            (1243796646486191335177 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2048332153445903 / 5000000000000000 : ℝ)) (q :=
+      (4111588254458403 / 10000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_439_7 :
+    Uω (aρ 7) (bρ 7) (462589901451 / 1000000000000 : ℝ) ≤
+      (-(1186877314178239240791 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (309546980614561 / 800000000000000 : ℝ) ≤
+      (-(1186877314178239240791 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (309546980614561 / 800000000000000 : ℝ) =
+            (309546980614561 / 200000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (309546980614561 / 200000000000000 : ℝ) ≤
+            (545990636571760759209 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (153432947763961 / 400000000000000 : ℝ)) (q :=
+      (309546980614561 / 800000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_439_8 :
+    Uω (aρ 8) (bρ 8) (462589901451 / 1000000000000 : ℝ) ≤
+      (-(518069465680476440557 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1774110162905591 / 5000000000000000 : ℝ) ≤
+      (-(518069465680476440557 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1774110162905591 / 5000000000000000 : ℝ) =
+            (1774110162905591 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1774110162905591 / 1250000000000000 : ℝ) ≤
+            (175077714619523559443 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (869282870424341 / 2500000000000000 : ℝ)) (q :=
+      (1774110162905591 / 5000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_439_9 :
+    Uω (aρ 9) (bρ 9) (462589901451 / 1000000000000 : ℝ) ≤
+      (-(11603274705019627962981 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (50141366407917 / 160000000000000 : ℝ) ≤
+      (-(11603274705019627962981 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (50141366407917 / 160000000000000 : ℝ) = (50141366407917 / 40000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (50141366407917 / 40000000000000 : ℝ) ≤
+            (2259668900980372037019 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (23914618075397 / 80000000000000 : ℝ)) (q :=
+      (50141366407917 / 160000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_439_10 :
+    Uω (aρ 10) (bρ 10) (462589901451 / 1000000000000 : ℝ) ≤
+      (-(13468949928897215958101 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5200929086686699 / 20000000000000000 : ℝ) ≤
+      (-(13468949928897215958101 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5200929086686699 / 20000000000000000 : ℝ) =
+            (5200929086686699 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5200929086686699 / 5000000000000000 : ℝ) ≤
+            (393993677102784041899 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2311144578616699 / 10000000000000000 : ℝ)) (q :=
+      (5200929086686699 / 20000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_439_11 :
+    Uω (aρ 11) (bρ 11) (462589901451 / 1000000000000 : ℝ) ≤
+      (-(16918135276424289182913 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (368370391680399 / 2000000000000000 : ℝ) ≤
+      (-(16918135276424289182913 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (368370391680399 / 2000000000000000 : ℝ) =
+            (368370391680399 / 250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (368370391680399 / 250000000000000 : ℝ) ≤
+            (3876280132575710817087 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (121239603493899 / 1000000000000000 : ℝ)) (q :=
+      (368370391680399 / 2000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11]) (by norm_num [bρ_11])
+      (by norm_num) (by norm_num [aρ_11, bρ_11]) (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_439_12 :
+    Uω (aρ 12) (bρ 12) (462589901451 / 1000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_439_13 :
+    Uω (aρ 13) (bρ 13) (462589901451 / 1000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_439_14 :
+    Uω (aρ 14) (bρ 14) (462589901451 / 1000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_439_15 :
+    Uω (aρ 15) (bρ 15) (462589901451 / 1000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_439_16 :
+    Uω (aρ 16) (bρ 16) (462589901451 / 1000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_439 :
+    Uρ (462589901451 / 1000000000000 : ℝ) ≤
+      (-(1454726621769049927881 / 1250000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (462589901451 / 1000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(7665653075113846959 / 9765625000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_439_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (462589901451 / 1000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(7902311596652221809187 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_439_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (462589901451 / 1000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1601726388699282176059 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_439_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (462589901451 / 1000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(8188109411627823297001 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_439_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (462589901451 / 1000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(8468715020601605035629 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_439_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (462589901451 / 1000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(2221939255013808664823 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_439_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (462589901451 / 1000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(1186877314178239240791 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_439_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (462589901451 / 1000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(518069465680476440557 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_439_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (462589901451 / 1000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(11603274705019627962981 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_439_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (462589901451 / 1000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(13468949928897215958101 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_439_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (462589901451 / 1000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(16918135276424289182913 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_439_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (462589901451 / 1000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_439_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (462589901451 / 1000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_439_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (462589901451 / 1000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_439_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (462589901451 / 1000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_439_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (462589901451 / 1000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_439_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_440_1 :
+    Uω (aρ 1) (bρ 1) (1853008205557 / 4000000000000 : ℝ) ≤
+      (-(979390335049681997199 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9135975808641307 / 20000000000000000 : ℝ) ≤
+      (-(979390335049681997199 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9135975808641307 / 20000000000000000 : ℝ) =
+            (9135975808641307 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9135975808641307 / 5000000000000000 : ℝ) ≤
+            (753477615700318002801 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4567952512833807 / 10000000000000000 : ℝ)) (q :=
+      (9135975808641307 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_440_2 :
+    Uω (aρ 2) (bρ 2) (1853008205557 / 4000000000000 : ℝ) ≤
+      (-(3943864161712085556341 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9088041611437987 / 20000000000000000 : ℝ) ≤
+      (-(3943864161712085556341 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9088041611437987 / 20000000000000000 : ℝ) =
+            (9088041611437987 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9088041611437987 / 5000000000000000 : ℝ) ≤
+            (2987607641287914443659 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4543787328140487 / 10000000000000000 : ℝ)) (q :=
+      (9088041611437987 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_440_3 :
+    Uω (aρ 3) (bρ 3) (1853008205557 / 4000000000000 : ℝ) ≤
+      (-(7993890966444840479467 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (899207088635803 / 2000000000000000 : ℝ) ≤
+      (-(7993890966444840479467 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (899207088635803 / 2000000000000000 : ℝ) =
+            (899207088635803 / 500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (899207088635803 / 500000000000000 : ℝ) ≤
+            (5869052639555159520533 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (449521270941053 / 1000000000000000 : ℝ)) (q :=
+      (899207088635803 / 2000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_440_4 :
+    Uω (aρ 4) (bρ 4) (1853008205557 / 4000000000000 : ℝ) ≤
+      (-(4086548161261261473791 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8832363455292731 / 20000000000000000 : ℝ) ≤
+      (-(4086548161261261473791 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8832363455292731 / 20000000000000000 : ℝ) =
+            (8832363455292731 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8832363455292731 / 5000000000000000 : ℝ) ≤
+            (2844923641738738526209 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4413799459410231 / 10000000000000000 : ℝ)) (q :=
+      (8832363455292731 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_440_5 :
+    Uω (aρ 5) (bρ 5) (1853008205557 / 4000000000000 : ℝ) ≤
+      (-(169065220407198884057 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (171766920518411 / 400000000000000 : ℝ) ≤
+      (-(169065220407198884057 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (171766920518411 / 400000000000000 : ℝ) =
+            (171766920518411 / 100000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (171766920518411 / 100000000000000 : ℝ) ≤
+            (108193651712801115943 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (85759438987461 / 200000000000000 : ℝ)) (q :=
+      (171766920518411 / 400000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_440_6 :
+    Uω (aρ 6) (bρ 6) (1853008205557 / 4000000000000 : ℝ) ≤
+      (-(8871607014660831809501 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4118233836572227 / 10000000000000000 : ℝ) ≤
+      (-(8871607014660831809501 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4118233836572227 / 10000000000000000 : ℝ) =
+            (4118233836572227 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4118233836572227 / 2500000000000000 : ℝ) ≤
+            (4991336591339168190499 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2051666985868477 / 5000000000000000 : ℝ)) (q :=
+      (4118233836572227 / 10000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_440_7 :
+    Uω (aρ 7) (bρ 7) (1853008205557 / 4000000000000 : ℝ) ≤
+      (-(9477771392971459838581 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7752033016989401 / 20000000000000000 : ℝ) ≤
+      (-(9477771392971459838581 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7752033016989401 / 20000000000000000 : ℝ) =
+            (7752033016989401 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7752033016989401 / 5000000000000000 : ℝ) ≤
+            (4385172213028540161419 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3842560696341901 / 10000000000000000 : ℝ)) (q :=
+      (7752033016989401 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_440_8 :
+    Uω (aρ 8) (bρ 8) (1853008205557 / 4000000000000 : ℝ) ≤
+      (-(5171182583481206675893 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1421990774837217 / 4000000000000000 : ℝ) ≤
+      (-(5171182583481206675893 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1421990774837217 / 4000000000000000 : ℝ) =
+            (1421990774837217 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1421990774837217 / 1000000000000000 : ℝ) ≤
+            (1760289219518793324107 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (696804640975717 / 2000000000000000 : ℝ)) (q :=
+      (1421990774837217 / 4000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_440_9 :
+    Uω (aρ 9) (bρ 9) (1853008205557 / 4000000000000 : ℝ) ≤
+      (-(11581151095201511045311 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (785194062665077 / 2500000000000000 : ℝ) ≤
+      (-(11581151095201511045311 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (785194062665077 / 2500000000000000 : ℝ) =
+            (785194062665077 / 625000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (785194062665077 / 625000000000000 : ℝ) ≤
+            (2281792510798488954689 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (749146865093279 / 2500000000000000 : ℝ)) (q :=
+      (785194062665077 / 2500000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_440_10 :
+    Uω (aρ 10) (bρ 10) (1853008205557 / 4000000000000 : ℝ) ≤
+      (-(1344035079882253336513 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1303956145297307 / 5000000000000000 : ℝ) ≤
+      (-(1344035079882253336513 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1303956145297307 / 5000000000000000 : ℝ) =
+            (1303956145297307 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1303956145297307 / 1250000000000000 : ℝ) ≤
+            (42259280717746663487 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (289927321717091 / 1250000000000000 : ℝ)) (q :=
+      (1303956145297307 / 5000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_440_11 :
+    Uω (aρ 11) (bρ 11) (1853008205557 / 4000000000000 : ℝ) ≤
+      (-(16863821217932963072499 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3703766041254587 / 20000000000000000 : ℝ) ≤
+      (-(16863821217932963072499 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3703766041254587 / 20000000000000000 : ℝ) =
+            (3703766041254587 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3703766041254587 / 2500000000000000 : ℝ) ≤
+            (3930594191067036927501 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1225836660007087 / 10000000000000000 : ℝ)) (q :=
+      (3703766041254587 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_440_12 :
+    Uω (aρ 12) (bρ 12) (1853008205557 / 4000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_440_13 :
+    Uω (aρ 13) (bρ 13) (1853008205557 / 4000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_440_14 :
+    Uω (aρ 14) (bρ 14) (1853008205557 / 4000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_440_15 :
+    Uω (aρ 15) (bρ 15) (1853008205557 / 4000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_440_16 :
+    Uω (aρ 16) (bρ 16) (1853008205557 / 4000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_440 :
+    Uρ (1853008205557 / 4000000000000 : ℝ) ≤
+      (-(11621554669306362988887 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1853008205557 / 4000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(979390335049681997199 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_440_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1853008205557 / 4000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3943864161712085556341 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_440_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1853008205557 / 4000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(7993890966444840479467 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_440_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1853008205557 / 4000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(4086548161261261473791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_440_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1853008205557 / 4000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(169065220407198884057 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_440_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1853008205557 / 4000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(8871607014660831809501 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_440_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1853008205557 / 4000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(9477771392971459838581 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_440_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1853008205557 / 4000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(5171182583481206675893 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_440_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1853008205557 / 4000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(11581151095201511045311 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_440_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1853008205557 / 4000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(1344035079882253336513 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_440_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1853008205557 / 4000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(16863821217932963072499 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_440_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1853008205557 / 4000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_440_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1853008205557 / 4000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_440_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1853008205557 / 4000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_440_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1853008205557 / 4000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_440_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1853008205557 / 4000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_440_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_441_1 :
+    Uω (aρ 1) (bρ 1) (185565680531 / 400000000000 : ℝ) ≤
+      (-(7820637624329189336659 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (457460945493081 / 1000000000000000 : ℝ) ≤
+      (-(7820637624329189336659 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (457460945493081 / 1000000000000000 : ℝ) =
+            (457460945493081 / 250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (457460945493081 / 250000000000000 : ℝ) ≤
+            (6042305981670810663341 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (228728705733581 / 500000000000000 : ℝ)) (q :=
+      (457460945493081 / 1000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_441_2 :
+    Uω (aρ 2) (bρ 2) (185565680531 / 400000000000 : ℝ) ≤
+      (-(7873166288595550170051 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9101285289689857 / 20000000000000000 : ℝ) ≤
+      (-(7873166288595550170051 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9101285289689857 / 20000000000000000 : ℝ) =
+            (9101285289689857 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9101285289689857 / 5000000000000000 : ℝ) ≤
+            (5989777317404449829949 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4550409507009857 / 10000000000000000 : ℝ)) (q :=
+      (9101285289689857 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_441_3 :
+    Uω (aρ 3) (bρ 3) (185565680531 / 400000000000 : ℝ) ≤
+      (-(3989585847491946285049 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (900531630534973 / 2000000000000000 : ℝ) ≤
+      (-(3989585847491946285049 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (900531630534973 / 2000000000000000 : ℝ) =
+            (900531630534973 / 500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (900531630534973 / 500000000000000 : ℝ) ≤
+            (2941885955508053714951 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (450183662901973 / 1000000000000000 : ℝ)) (q :=
+      (900531630534973 / 2000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_441_4 :
+    Uω (aρ 4) (bρ 4) (185565680531 / 400000000000 : ℝ) ≤
+      (-(1631621152635130144881 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4422806795506861 / 10000000000000000 : ℝ) ≤
+      (-(1631621152635130144881 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4422806795506861 / 10000000000000000 : ℝ) =
+            (4422806795506861 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4422806795506861 / 2500000000000000 : ℝ) ≤
+            (1140967568564869855119 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2210214047874361 / 5000000000000000 : ℝ)) (q :=
+      (4422806795506861 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_441_5 :
+    Uω (aρ 5) (bρ 5) (185565680531 / 400000000000 : ℝ) ≤
+      (-(8437830934746810842847 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (860160814651759 / 2000000000000000 : ℝ) ≤
+      (-(8437830934746810842847 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (860160814651759 / 2000000000000000 : ℝ) =
+            (860160814651759 / 500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (860160814651759 / 500000000000000 : ℝ) ≤
+            (5425112671253189157153 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (429461257058759 / 1000000000000000 : ℝ)) (q :=
+      (860160814651759 / 2000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_441_6 :
+    Uω (aρ 6) (bρ 6) (185565680531 / 400000000000 : ℝ) ≤
+      (-(2213870809572582147229 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8249758681611879 / 20000000000000000 : ℝ) ≤
+      (-(2213870809572582147229 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8249758681611879 / 20000000000000000 : ℝ) =
+            (8249758681611879 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8249758681611879 / 5000000000000000 : ℝ) ≤
+            (1251865091927417852771 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4110003480821879 / 10000000000000000 : ℝ)) (q :=
+      (8249758681611879 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_441_7 :
+    Uω (aρ 7) (bρ 7) (185565680531 / 400000000000 : ℝ) ≤
+      (-(1182569310476322240769 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7765391117776663 / 20000000000000000 : ℝ) ≤
+      (-(1182569310476322240769 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7765391117776663 / 20000000000000000 : ℝ) =
+            (7765391117776663 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7765391117776663 / 5000000000000000 : ℝ) ≤
+            (550298640273677759231 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3849297297746663 / 10000000000000000 : ℝ)) (q :=
+      (7765391117776663 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_441_8 :
+    Uω (aρ 8) (bρ 8) (185565680531 / 400000000000 : ℝ) ≤
+      (-(1032337861184616429291 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1780866512680443 / 5000000000000000 : ℝ) ≤
+      (-(1032337861184616429291 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1780866512680443 / 5000000000000000 : ℝ) =
+            (1780866512680443 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1780866512680443 / 1250000000000000 : ℝ) ≤
+            (353956499415383570709 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (872728470507943 / 2500000000000000 : ℝ)) (q :=
+      (1780866512680443 / 5000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_441_9 :
+    Uω (aρ 9) (bρ 9) (185565680531 / 400000000000 : ℝ) ≤
+      (-(11559081011305702915049 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3147715625007271 / 10000000000000000 : ℝ) ≤
+      (-(11559081011305702915049 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3147715625007271 / 10000000000000000 : ℝ) =
+            (3147715625007271 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3147715625007271 / 2500000000000000 : ℝ) ≤
+            (2303862594694297084951 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1501922354842271 / 5000000000000000 : ℝ)) (q :=
+      (3147715625007271 / 10000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_441_10 :
+    Uω (aρ 10) (bρ 10) (185565680531 / 400000000000 : ℝ) ≤
+      (-(6705926721624019064597 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5230709500928829 / 20000000000000000 : ℝ) ≤
+      (-(6705926721624019064597 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5230709500928829 / 20000000000000000 : ℝ) =
+            (5230709500928829 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5230709500928829 / 5000000000000000 : ℝ) ≤
+            (225545081375980935403 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2327681994093829 / 10000000000000000 : ℝ)) (q :=
+      (5230709500928829 / 20000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_441_11 :
+    Uω (aρ 11) (bρ 11) (185565680531 / 400000000000 : ℝ) ≤
+      (-(4202524258281721346399 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (744743553781797 / 4000000000000000 : ℝ) ≤
+      (-(4202524258281721346399 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (744743553781797 / 4000000000000000 : ℝ) =
+            (744743553781797 / 500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (744743553781797 / 500000000000000 : ℝ) ≤
+            (996079593968278653601 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (247833377655797 / 2000000000000000 : ℝ)) (q :=
+      (744743553781797 / 4000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11]) (by norm_num [bρ_11])
+      (by norm_num) (by norm_num [aρ_11, bρ_11]) (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_441_12 :
+    Uω (aρ 12) (bρ 12) (185565680531 / 400000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_441_13 :
+    Uω (aρ 13) (bρ 13) (185565680531 / 400000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_441_14 :
+    Uω (aρ 14) (bρ 14) (185565680531 / 400000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_441_15 :
+    Uω (aρ 15) (bρ 15) (185565680531 / 400000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_441_16 :
+    Uω (aρ 16) (bρ 16) (185565680531 / 400000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_441 :
+    Uρ (185565680531 / 400000000000 : ℝ) ≤
+      (-(2901342256725090960881 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (185565680531 / 400000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(7820637624329189336659 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_441_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (185565680531 / 400000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(7873166288595550170051 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_441_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (185565680531 / 400000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3989585847491946285049 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_441_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (185565680531 / 400000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1631621152635130144881 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_441_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (185565680531 / 400000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(8437830934746810842847 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_441_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (185565680531 / 400000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(2213870809572582147229 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_441_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (185565680531 / 400000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(1182569310476322240769 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_441_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (185565680531 / 400000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(1032337861184616429291 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_441_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (185565680531 / 400000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(11559081011305702915049 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_441_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (185565680531 / 400000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(6705926721624019064597 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_441_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (185565680531 / 400000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4202524258281721346399 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_441_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (185565680531 / 400000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_441_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (185565680531 / 400000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_441_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (185565680531 / 400000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_441_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (185565680531 / 400000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_441_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (185565680531 / 400000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_441_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_442_1 :
+    Uω (aρ 1) (bρ 1) (1858305405063 / 4000000000000 : ℝ) ≤
+      (-(7806173519924622736079 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (57265387567411 / 125000000000000 : ℝ) ≤
+      (-(7806173519924622736079 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (57265387567411 / 125000000000000 : ℝ) = (57265387567411 / 31250000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (57265387567411 / 31250000000000 : ℝ) ≤
+            (6056770086075377263921 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (229059785810663 / 500000000000000 : ℝ)) (q :=
+      (57265387567411 / 125000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_442_2 :
+    Uω (aρ 2) (bρ 2) (1858305405063 / 4000000000000 : ℝ) ≤
+      (-(3929312715194378892603 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4557264482983499 / 10000000000000000 : ℝ) ≤
+      (-(3929312715194378892603 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4557264482983499 / 10000000000000000 : ℝ) =
+            (4557264482983499 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4557264482983499 / 2500000000000000 : ℝ) ≤
+            (3002159087805621107397 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2278515841952249 / 5000000000000000 : ℝ)) (q :=
+      (4557264482983499 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_442_3 :
+    Uω (aρ 3) (bρ 3) (1858305405063 / 4000000000000 : ℝ) ≤
+      (-(7964474065262793540323 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9018561717231039 / 20000000000000000 : ℝ) ≤
+      (-(7964474065262793540323 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9018561717231039 / 20000000000000000 : ℝ) =
+            (9018561717231039 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9018561717231039 / 5000000000000000 : ℝ) ≤
+            (5898469540737206459677 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4508460541518539 / 10000000000000000 : ℝ)) (q :=
+      (9018561717231039 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_442_4 :
+    Uω (aρ 4) (bρ 4) (1858305405063 / 4000000000000 : ℝ) ≤
+      (-(8143137665996417731169 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (70870909642991 / 160000000000000 : ℝ) ≤
+      (-(8143137665996417731169 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (70870909642991 / 160000000000000 : ℝ) = (70870909642991 / 40000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (70870909642991 / 40000000000000 : ℝ) ≤
+            (5719805940003582268831 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (35416453685811 / 80000000000000 : ℝ)) (q :=
+      (70870909642991 / 160000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_442_5 :
+    Uω (aρ 5) (bρ 5) (1858305405063 / 4000000000000 : ℝ) ≤
+      (-(8422424689650293239287 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2153717552038997 / 5000000000000000 : ℝ) ≤
+      (-(8422424689650293239287 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2153717552038997 / 5000000000000000 : ℝ) =
+            (2153717552038997 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2153717552038997 / 1250000000000000 : ℝ) ≤
+            (5440518916349706760713 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (134414160401359 / 312500000000000 : ℝ)) (q :=
+      (2153717552038997 / 5000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_442_6 :
+    Uω (aρ 6) (bρ 6) (1858305405063 / 4000000000000 : ℝ) ≤
+      (-(8839385605277294481179 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8263049535076131 / 20000000000000000 : ℝ) ≤
+      (-(8839385605277294481179 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8263049535076131 / 20000000000000000 : ℝ) =
+            (8263049535076131 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8263049535076131 / 5000000000000000 : ℝ) ≤
+            (5023558000722705518821 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4116672834903631 / 10000000000000000 : ℝ)) (q :=
+      (8263049535076131 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_442_7 :
+    Uω (aρ 7) (bρ 7) (1858305405063 / 4000000000000 : ℝ) ≤
+      (-(9443367678504315809701 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7778748819826639 / 20000000000000000 : ℝ) ≤
+      (-(9443367678504315809701 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7778748819826639 / 20000000000000000 : ℝ) =
+            (7778748819826639 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7778748819826639 / 5000000000000000 : ℝ) ≤
+            (4419575927495684190299 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3856033500414139 / 10000000000000000 : ℝ)) (q :=
+      (7778748819826639 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_442_8 :
+    Uω (aρ 8) (bρ 8) (1858305405063 / 4000000000000 : ℝ) ≤
+      (-(10304429494337094953457 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7136977187411453 / 20000000000000000 : ℝ) ≤
+      (-(10304429494337094953457 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7136977187411453 / 20000000000000000 : ℝ) =
+            (7136977187411453 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7136977187411453 / 5000000000000000 : ℝ) ≤
+            (3558514111662905046543 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3497803519338953 / 10000000000000000 : ℝ)) (q :=
+      (7136977187411453 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_442_9 :
+    Uω (aρ 9) (bρ 9) (1858305405063 / 4000000000000 : ℝ) ≤
+      (-(180266627709776947529 / 156250000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6309307068413211 / 20000000000000000 : ℝ) ≤
+      (-(180266627709776947529 / 156250000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6309307068413211 / 20000000000000000 : ℝ) =
+            (6309307068413211 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6309307068413211 / 5000000000000000 : ℝ) ≤
+            (36341866133973052471 / 156250000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3011099028700711 / 10000000000000000 : ℝ)) (q :=
+      (6309307068413211 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_442_10 :
+    Uω (aρ 10) (bρ 10) (1858305405063 / 4000000000000 : ℝ) ≤
+      (-(6691728506143263926889 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (262279197906543 / 1000000000000000 : ℝ) ≤
+      (-(6691728506143263926889 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (262279197906543 / 1000000000000000 : ℝ) =
+            (262279197906543 / 250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (262279197906543 / 250000000000000 : ℝ) ≤
+            (239743296856736073111 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (29199186898917 / 125000000000000 : ℝ)) (q :=
+      (262279197906543 / 1000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10]) (by norm_num [bρ_10])
+      (by norm_num) (by norm_num [aρ_10, bρ_10]) (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_442_11 :
+    Uω (aρ 11) (bρ 11) (1858305405063 / 4000000000000 : ℝ) ≤
+      (-(3351389082507226933971 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1871781312453091 / 10000000000000000 : ℝ) ≤
+      (-(3351389082507226933971 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1871781312453091 / 10000000000000000 : ℝ) =
+            (1871781312453091 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1871781312453091 / 1250000000000000 : ℝ) ≤
+            (807493999292773066029 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (626195122446841 / 5000000000000000 : ℝ)) (q :=
+      (1871781312453091 / 10000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_442_12 :
+    Uω (aρ 12) (bρ 12) (1858305405063 / 4000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_442_13 :
+    Uω (aρ 13) (bρ 13) (1858305405063 / 4000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_442_14 :
+    Uω (aρ 14) (bρ 14) (1858305405063 / 4000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_442_15 :
+    Uω (aρ 15) (bρ 15) (1858305405063 / 4000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_442_16 :
+    Uω (aρ 16) (bρ 16) (1858305405063 / 4000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_442 :
+    Uρ (1858305405063 / 4000000000000 : ℝ) ≤
+      (-(5794627270070532732099 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1858305405063 / 4000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(7806173519924622736079 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_442_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1858305405063 / 4000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3929312715194378892603 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_442_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1858305405063 / 4000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(7964474065262793540323 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_442_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1858305405063 / 4000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(8143137665996417731169 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_442_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1858305405063 / 4000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(8422424689650293239287 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_442_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1858305405063 / 4000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(8839385605277294481179 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_442_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1858305405063 / 4000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(9443367678504315809701 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_442_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1858305405063 / 4000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(10304429494337094953457 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_442_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1858305405063 / 4000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(180266627709776947529 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_442_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1858305405063 / 4000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(6691728506143263926889 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_442_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1858305405063 / 4000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(3351389082507226933971 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_442_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1858305405063 / 4000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_442_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1858305405063 / 4000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_442_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1858305405063 / 4000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_442_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1858305405063 / 4000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_442_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1858305405063 / 4000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_442_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_443_1 :
+    Uω (aρ 1) (bρ 1) (116309625301 / 250000000000 : ℝ) ≤
+      (-(7791730306659998859531 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (917570511141501 / 2000000000000000 : ℝ) ≤
+      (-(7791730306659998859531 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (917570511141501 / 2000000000000000 : ℝ) =
+            (917570511141501 / 500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (917570511141501 / 500000000000000 : ℝ) ≤
+            (6071213299340001140469 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (458781731746001 / 1000000000000000 : ℝ)) (q :=
+      (917570511141501 / 2000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_443_2 :
+    Uω (aρ 2) (bρ 2) (116309625301 / 250000000000 : ℝ) ≤
+      (-(7844105687295377049281 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1825554528055601 / 4000000000000000 : ℝ) ≤
+      (-(7844105687295377049281 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1825554528055601 / 4000000000000000 : ℝ) =
+            (1825554528055601 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1825554528055601 / 1000000000000000 : ℝ) ≤
+            (6018837918704622950719 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (912730771766601 / 2000000000000000 : ℝ)) (q :=
+      (1825554528055601 / 4000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_443_3 :
+    Uω (aρ 3) (bρ 3) (116309625301 / 250000000000 : ℝ) ≤
+      (-(3974899006856099753251 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2257951780508313 / 5000000000000000 : ℝ) ≤
+      (-(3974899006856099753251 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2257951780508313 / 5000000000000000 : ℝ) =
+            (2257951780508313 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2257951780508313 / 1250000000000000 : ℝ) ≤
+            (2956572796143900246749 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1128771111734563 / 2500000000000000 : ℝ)) (q :=
+      (2257951780508313 / 5000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_443_4 :
+    Uω (aρ 4) (bρ 4) (116309625301 / 250000000000 : ℝ) ≤
+      (-(254005998865564480957 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8872113798468999 / 20000000000000000 : ℝ) ≤
+      (-(254005998865564480957 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8872113798468999 / 20000000000000000 : ℝ) =
+            (8872113798468999 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8872113798468999 / 5000000000000000 : ℝ) ≤
+            (179210988821935519043 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4433685304438999 / 10000000000000000 : ℝ)) (q :=
+      (8872113798468999 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_443_5 :
+    Uω (aρ 5) (bρ 5) (116309625301 / 250000000000 : ℝ) ≤
+      (-(840704221130343153201 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2157033052777099 / 5000000000000000 : ℝ) ≤
+      (-(840704221130343153201 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2157033052777099 / 5000000000000000 : ℝ) =
+            (2157033052777099 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2157033052777099 / 1250000000000000 : ℝ) ≤
+            (545590139469656846799 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1076973409103349 / 2500000000000000 : ℝ)) (q :=
+      (2157033052777099 / 5000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_443_6 :
+    Uω (aρ 6) (bρ 6) (116309625301 / 250000000000 : ℝ) ≤
+      (-(4411657015188613554371 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4138170117144673 / 10000000000000000 : ℝ) ≤
+      (-(4411657015188613554371 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4138170117144673 / 10000000000000000 : ℝ) =
+            (4138170117144673 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4138170117144673 / 2500000000000000 : ℝ) ≤
+            (2519814787811386445629 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2061671017367173 / 5000000000000000 : ℝ)) (q :=
+      (4138170117144673 / 10000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_443_7 :
+    Uω (aρ 7) (bρ 7) (116309625301 / 250000000000 : ℝ) ≤
+      (-(2356552717548743531489 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7792106125225377 / 20000000000000000 : ℝ) ≤
+      (-(2356552717548743531489 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7792106125225377 / 20000000000000000 : ℝ) =
+            (7792106125225377 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7792106125225377 / 5000000000000000 : ℝ) ≤
+            (1109183183951256468511 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3862769306430377 / 10000000000000000 : ℝ)) (q :=
+      (7792106125225377 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_443_8 :
+    Uω (aρ 8) (bρ 8) (116309625301 / 250000000000 : ℝ) ≤
+      (-(2057103532297391134677 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3575243645193809 / 10000000000000000 : ℝ) ≤
+      (-(2057103532297391134677 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3575243645193809 / 10000000000000000 : ℝ) =
+            (3575243645193809 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3575243645193809 / 2500000000000000 : ℝ) ≤
+            (715485188902608865323 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1752346061466309 / 5000000000000000 : ℝ)) (q :=
+      (3575243645193809 / 10000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_443_9 :
+    Uω (aρ 9) (bρ 9) (116309625301 / 250000000000 : ℝ) ≤
+      (-(2303020060797331947437 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6323179977644683 / 20000000000000000 : ℝ) ≤
+      (-(2303020060797331947437 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6323179977644683 / 20000000000000000 : ℝ) =
+            (6323179977644683 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6323179977644683 / 5000000000000000 : ℝ) ≤
+            (469568660402668052563 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3018350438549683 / 10000000000000000 : ℝ)) (q :=
+      (6323179977644683 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_443_10 :
+    Uω (aρ 10) (bρ 10) (116309625301 / 250000000000 : ℝ) ≤
+      (-(1669395083469542458551 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (210417922531967 / 800000000000000 : ℝ) ≤
+      (-(1669395083469542458551 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (210417922531967 / 800000000000000 : ℝ) =
+            (210417922531967 / 200000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (210417922531967 / 200000000000000 : ℝ) ≤
+            (63472867280457541449 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (93767102307967 / 400000000000000 : ℝ)) (q :=
+      (210417922531967 / 800000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10]) (by norm_num [bρ_10])
+      (by norm_num) (by norm_num [aρ_10, bρ_10]) (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_443_11 :
+    Uω (aρ 11) (bρ 11) (116309625301 / 250000000000 : ℝ) ≤
+      (-(4176087471541592210279 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3763303959367039 / 20000000000000000 : ℝ) ≤
+      (-(4176087471541592210279 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3763303959367039 / 20000000000000000 : ℝ) =
+            (3763303959367039 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3763303959367039 / 2500000000000000 : ℝ) ≤
+            (1022516380708407789721 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1265510079972039 / 10000000000000000 : ℝ)) (q :=
+      (3763303959367039 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_443_12 :
+    Uω (aρ 12) (bρ 12) (116309625301 / 250000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_443_13 :
+    Uω (aρ 13) (bρ 13) (116309625301 / 250000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_443_14 :
+    Uω (aρ 14) (bρ 14) (116309625301 / 250000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_443_15 :
+    Uω (aρ 15) (bρ 15) (116309625301 / 250000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_443_16 :
+    Uω (aρ 16) (bρ 16) (116309625301 / 250000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_443 :
+    Uρ (116309625301 / 250000000000 : ℝ) ≤
+      (-(5786604884954288602901 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (116309625301 / 250000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(7791730306659998859531 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_443_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (116309625301 / 250000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(7844105687295377049281 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_443_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (116309625301 / 250000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3974899006856099753251 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_443_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (116309625301 / 250000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(254005998865564480957 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_443_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (116309625301 / 250000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(840704221130343153201 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_443_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (116309625301 / 250000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(4411657015188613554371 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_443_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (116309625301 / 250000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(2356552717548743531489 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_443_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (116309625301 / 250000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(2057103532297391134677 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_443_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (116309625301 / 250000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(2303020060797331947437 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_443_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (116309625301 / 250000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(1669395083469542458551 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_443_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (116309625301 / 250000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4176087471541592210279 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_443_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (116309625301 / 250000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_443_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (116309625301 / 250000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_443_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (116309625301 / 250000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_443_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (116309625301 / 250000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_443_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (116309625301 / 250000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_443_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_444_1 :
+    Uω (aρ 1) (bρ 1) (1863602604569 / 4000000000000 : ℝ) ≤
+      (-(1944326981068361120593 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9188948211750643 / 20000000000000000 : ℝ) ≤
+      (-(1944326981068361120593 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9188948211750643 / 20000000000000000 : ℝ) =
+            (9188948211750643 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9188948211750643 / 5000000000000000 : ℝ) ≤
+            (1521408920431638879407 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4594438918413143 / 10000000000000000 : ℝ)) (q :=
+      (9188948211750643 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_444_2 :
+    Uω (aρ 2) (bρ 2) (1863602604569 / 4000000000000 : ℝ) ≤
+      (-(1565921399614921480789 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4570508156315713 / 10000000000000000 : ℝ) ≤
+      (-(1565921399614921480789 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4570508156315713 / 10000000000000000 : ℝ) =
+            (4570508156315713 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4570508156315713 / 2500000000000000 : ℝ) ≤
+            (1206667321585078519211 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2285138015901963 / 5000000000000000 : ℝ)) (q :=
+      (4570508156315713 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_444_3 :
+    Uω (aρ 3) (bρ 3) (1863602604569 / 4000000000000 : ℝ) ≤
+      (-(7935143477042547934617 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4522526259893739 / 10000000000000000 : ℝ) ≤
+      (-(7935143477042547934617 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4522526259893739 / 10000000000000000 : ℝ) =
+            (4522526259893739 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4522526259893739 / 2500000000000000 : ℝ) ≤
+            (5927800128957452065383 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2260854172654989 / 5000000000000000 : ℝ)) (q :=
+      (4522526259893739 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_444_4 :
+    Uω (aρ 4) (bρ 4) (1863602604569 / 4000000000000 : ℝ) ≤
+      (-(507079286831002569619 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1110670483799291 / 2500000000000000 : ℝ) ≤
+      (-(507079286831002569619 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1110670483799291 / 2500000000000000 : ℝ) =
+            (1110670483799291 / 625000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1110670483799291 / 625000000000000 : ℝ) ≤
+            (359354688543997430381 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1110078469245457 / 2500000000000000 : ℝ)) (q :=
+      (1110670483799291 / 2500000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_444_5 :
+    Uω (aρ 5) (bρ 5) (1863602604569 / 4000000000000 : ℝ) ≤
+      (-(4195841713141036618307 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8641394155645787 / 20000000000000000 : ℝ) ≤
+      (-(4195841713141036618307 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8641394155645787 / 20000000000000000 : ℝ) =
+            (8641394155645787 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8641394155645787 / 5000000000000000 : ℝ) ≤
+            (2735630089858963381693 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4314534081568287 / 10000000000000000 : ℝ)) (q :=
+      (8641394155645787 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_444_6 :
+    Uω (aρ 6) (bρ 6) (1863602604569 / 4000000000000 : ℝ) ≤
+      (-(880726842876476855731 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1657926155999757 / 4000000000000000 : ℝ) ≤
+      (-(880726842876476855731 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1657926155999757 / 4000000000000000 : ℝ) =
+            (1657926155999757 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1657926155999757 / 1000000000000000 : ℝ) ≤
+            (505567517723523144269 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (826002216212257 / 2000000000000000 : ℝ)) (q :=
+      (1657926155999757 / 4000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_444_7 :
+    Uω (aρ 7) (bρ 7) (1863602604569 / 4000000000000 : ℝ) ≤
+      (-(9409083952601855202521 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7805463036044279 / 20000000000000000 : ℝ) ≤
+      (-(9409083952601855202521 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7805463036044279 / 20000000000000000 : ℝ) =
+            (7805463036044279 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7805463036044279 / 5000000000000000 : ℝ) ≤
+            (4453859653398144797479 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3869504717866779 / 10000000000000000 : ℝ)) (q :=
+      (7805463036044279 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_444_8 :
+    Uω (aρ 8) (bρ 8) (1863602604569 / 4000000000000 : ℝ) ≤
+      (-(10266642961314476776443 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (179099909143343 / 500000000000000 : ℝ) ≤
+      (-(10266642961314476776443 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (179099909143343 / 500000000000000 : ℝ) =
+            (179099909143343 / 125000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (179099909143343 / 125000000000000 : ℝ) ≤
+            (3596300644685523223557 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (175578984944811 / 500000000000000 : ℝ)) (q :=
+      (179099909143343 / 500000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_444_9 :
+    Uω (aρ 9) (bρ 9) (1863602604569 / 4000000000000 : ℝ) ≤
+      (-(5746594563859062553709 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1584262499656509 / 5000000000000000 : ℝ) ≤
+      (-(5746594563859062553709 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1584262499656509 / 5000000000000000 : ℝ) =
+            (1584262499656509 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1584262499656509 / 1250000000000000 : ℝ) ≤
+            (1184877239140937446291 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (378199870018567 / 1250000000000000 : ℝ)) (q :=
+      (1584262499656509 / 5000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_444_10 :
+    Uω (aρ 10) (bρ 10) (1863602604569 / 4000000000000 : ℝ) ≤
+      (-(6663481791477497625633 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5275301925251203 / 20000000000000000 : ℝ) ≤
+      (-(6663481791477497625633 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5275301925251203 / 20000000000000000 : ℝ) =
+            (5275301925251203 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5275301925251203 / 5000000000000000 : ℝ) ≤
+            (267990011522502374367 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2352409920268703 / 10000000000000000 : ℝ)) (q :=
+      (5275301925251203 / 20000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_444_11 :
+    Uω (aρ 11) (bρ 11) (1863602604569 / 4000000000000 : ℝ) ≤
+      (-(4163073691847707252589 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (756588991840753 / 4000000000000000 : ℝ) ≤
+      (-(4163073691847707252589 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (756588991840753 / 4000000000000000 : ℝ) =
+            (756588991840753 / 500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (756588991840753 / 500000000000000 : ℝ) ≤
+            (1035530160402292747411 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (255705916085253 / 2000000000000000 : ℝ)) (q :=
+      (756588991840753 / 4000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11]) (by norm_num [bρ_11])
+      (by norm_num) (by norm_num [aρ_11, bρ_11]) (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_444_12 :
+    Uω (aρ 12) (bρ 12) (1863602604569 / 4000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_444_13 :
+    Uω (aρ 13) (bρ 13) (1863602604569 / 4000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_444_14 :
+    Uω (aρ 14) (bρ 14) (1863602604569 / 4000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_444_15 :
+    Uω (aρ 15) (bρ 15) (1863602604569 / 4000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_444_16 :
+    Uω (aρ 16) (bρ 16) (1863602604569 / 4000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_444 :
+    Uρ (1863602604569 / 4000000000000 : ℝ) ≤
+      (-(5778616670153065444033 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1863602604569 / 4000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1944326981068361120593 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_444_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1863602604569 / 4000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1565921399614921480789 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_444_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1863602604569 / 4000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(7935143477042547934617 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_444_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1863602604569 / 4000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(507079286831002569619 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_444_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1863602604569 / 4000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(4195841713141036618307 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_444_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1863602604569 / 4000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(880726842876476855731 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_444_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1863602604569 / 4000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(9409083952601855202521 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_444_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1863602604569 / 4000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(10266642961314476776443 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_444_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1863602604569 / 4000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(5746594563859062553709 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_444_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1863602604569 / 4000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(6663481791477497625633 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_444_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1863602604569 / 4000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4163073691847707252589 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_444_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1863602604569 / 4000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_444_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1863602604569 / 4000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_444_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1863602604569 / 4000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_444_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1863602604569 / 4000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_444_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1863602604569 / 4000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_444_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_445_1 :
+    Uω (aρ 1) (bρ 1) (933125602161 / 2000000000000 : ℝ) ≤
+      (-(7762906312763454203637 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9202191311793929 / 20000000000000000 : ℝ) ≤
+      (-(7762906312763454203637 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9202191311793929 / 20000000000000000 : ℝ) =
+            (9202191311793929 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9202191311793929 / 5000000000000000 : ℝ) ≤
+            (6100037293236545796363 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4601060519073929 / 10000000000000000 : ℝ)) (q :=
+      (9202191311793929 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_445_2 :
+    Uω (aρ 2) (bρ 2) (933125602161 / 2000000000000 : ℝ) ≤
+      (-(1953782325437929747473 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4577129991517879 / 10000000000000000 : ℝ) ≤
+      (-(1953782325437929747473 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4577129991517879 / 10000000000000000 : ℝ) =
+            (4577129991517879 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4577129991517879 / 2500000000000000 : ℝ) ≤
+            (1511953576062070252527 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2288449101412879 / 5000000000000000 : ℝ)) (q :=
+      (4577129991517879 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_445_3 :
+    Uω (aρ 3) (bρ 3) (933125602161 / 2000000000000 : ℝ) ≤
+      (-(1980127598060602571469 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1132287238815581 / 2500000000000000 : ℝ) ≤
+      (-(1980127598060602571469 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1132287238815581 / 2500000000000000 : ℝ) =
+            (1132287238815581 / 625000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1132287238815581 / 625000000000000 : ℝ) ≤
+            (1485608303439397428531 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (566041529583081 / 1250000000000000 : ℝ)) (q :=
+      (1132287238815581 / 2500000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_445_4 :
+    Uω (aρ 4) (bρ 4) (933125602161 / 2000000000000 : ℝ) ≤
+      (-(202459186902655127247 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8898613921244529 / 20000000000000000 : ℝ) ≤
+      (-(202459186902655127247 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8898613921244529 / 20000000000000000 : ℝ) =
+            (8898613921244529 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8898613921244529 / 5000000000000000 : ℝ) ≤
+            (144114403247344872753 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4446942428449529 / 10000000000000000 : ℝ)) (q :=
+      (8898613921244529 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_445_5 :
+    Uω (aρ 5) (bρ 5) (933125602161 / 2000000000000 : ℝ) ≤
+      (-(1675269652300550677949 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4327328021018731 / 10000000000000000 : ℝ) ≤
+      (-(1675269652300550677949 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4327328021018731 / 10000000000000000 : ℝ) =
+            (4327328021018731 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4327328021018731 / 2500000000000000 : ℝ) ≤
+            (1097319068899449322051 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2160587234288731 / 5000000000000000 : ℝ)) (q :=
+      (4327328021018731 / 10000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_445_6 :
+    Uω (aρ 6) (bρ 6) (933125602161 / 2000000000000 : ℝ) ≤
+      (-(8791248716030945256763 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1037865146618359 / 2500000000000000 : ℝ) ≤
+      (-(8791248716030945256763 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1037865146618359 / 2500000000000000 : ℝ) =
+            (1037865146618359 / 625000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1037865146618359 / 625000000000000 : ℝ) ≤
+            (5071694889969054743237 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (517084996828359 / 1250000000000000 : ℝ)) (q :=
+      (1037865146618359 / 2500000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_445_7 :
+    Uω (aρ 7) (bρ 7) (933125602161 / 2000000000000 : ℝ) ≤
+      (-(4695993410008529072567 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1954704888585057 / 5000000000000000 : ℝ) ≤
+      (-(4695993410008529072567 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1954704888585057 / 5000000000000000 : ℝ) =
+            (1954704888585057 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1954704888585057 / 1250000000000000 : ℝ) ≤
+            (2235478392991470927433 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (969059934195057 / 2500000000000000 : ℝ)) (q :=
+      (1954704888585057 / 5000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_445_8 :
+    Uω (aρ 8) (bρ 8) (933125602161 / 2000000000000 : ℝ) ≤
+      (-(10247805242797003460311 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7177504419484671 / 20000000000000000 : ℝ) ≤
+      (-(10247805242797003460311 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7177504419484671 / 20000000000000000 : ℝ) =
+            (7177504419484671 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7177504419484671 / 5000000000000000 : ℝ) ≤
+            (3615138363202996539689 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3518466253264671 / 10000000000000000 : ℝ)) (q :=
+      (7177504419484671 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_445_9 :
+    Uω (aρ 9) (bρ 9) (933125602161 / 2000000000000 : ℝ) ≤
+      (-(458853214865105613027 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6350917152066091 / 20000000000000000 : ℝ) ≤
+      (-(458853214865105613027 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6350917152066091 / 20000000000000000 : ℝ) =
+            (6350917152066091 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6350917152066091 / 5000000000000000 : ℝ) ≤
+            (95664529374894386973 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3032844614206091 / 10000000000000000 : ℝ)) (q :=
+      (6350917152066091 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_445_10 :
+    Uω (aρ 10) (bρ 10) (933125602161 / 2000000000000 : ℝ) ≤
+      (-(13298864942438442631833 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2645072825576073 / 10000000000000000 : ℝ) ≤
+      (-(13298864942438442631833 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2645072825576073 / 10000000000000000 : ℝ) =
+            (2645072825576073 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2645072825576073 / 2500000000000000 : ℝ) ≤
+            (564078663561557368167 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1180316073393573 / 5000000000000000 : ℝ)) (q :=
+      (2645072825576073 / 10000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_445_11 :
+    Uω (aρ 11) (bρ 11) (933125602161 / 2000000000000 : ℝ) ≤
+      (-(3320153020315792050137 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1901244329463689 / 10000000000000000 : ℝ) ≤
+      (-(3320153020315792050137 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1901244329463689 / 10000000000000000 : ℝ) =
+            (1901244329463689 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1901244329463689 / 1250000000000000 : ℝ) ≤
+            (838730061484207949863 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (645725890383689 / 5000000000000000 : ℝ)) (q :=
+      (1901244329463689 / 10000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_445_12 :
+    Uω (aρ 12) (bρ 12) (933125602161 / 2000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_445_13 :
+    Uω (aρ 13) (bρ 13) (933125602161 / 2000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_445_14 :
+    Uω (aρ 14) (bρ 14) (933125602161 / 2000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_445_15 :
+    Uω (aρ 15) (bρ 15) (933125602161 / 2000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_445_16 :
+    Uω (aρ 16) (bρ 16) (933125602161 / 2000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_445 :
+    Uρ (933125602161 / 2000000000000 : ℝ) ≤
+      (-(5770661967292001257713 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (933125602161 / 2000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(7762906312763454203637 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_445_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (933125602161 / 2000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1953782325437929747473 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_445_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (933125602161 / 2000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1980127598060602571469 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_445_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (933125602161 / 2000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(202459186902655127247 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_445_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (933125602161 / 2000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1675269652300550677949 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_445_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (933125602161 / 2000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(8791248716030945256763 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_445_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (933125602161 / 2000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(4695993410008529072567 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_445_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (933125602161 / 2000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(10247805242797003460311 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_445_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (933125602161 / 2000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(458853214865105613027 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_445_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (933125602161 / 2000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(13298864942438442631833 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_445_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (933125602161 / 2000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(3320153020315792050137 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_445_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (933125602161 / 2000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_445_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (933125602161 / 2000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_445_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (933125602161 / 2000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_445_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (933125602161 / 2000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_445_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (933125602161 / 2000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_445_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_446_1 :
+    Uω (aρ 1) (bρ 1) (467887100957 / 1000000000000 : ℝ) ≤
+      (-(966770645457505427377 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2307169377752123 / 5000000000000000 : ℝ) ≤
+      (-(966770645457505427377 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2307169377752123 / 5000000000000000 : ℝ) =
+            (2307169377752123 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2307169377752123 / 1250000000000000 : ℝ) ≤
+            (766097305292494572623 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1153575929880873 / 2500000000000000 : ℝ)) (q :=
+      (2307169377752123 / 5000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_446_2 :
+    Uω (aρ 2) (bρ 2) (467887100957 / 1000000000000 : ℝ) ≤
+      (-(973279580652723789647 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1836149463606179 / 4000000000000000 : ℝ) ≤
+      (-(973279580652723789647 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1836149463606179 / 4000000000000000 : ℝ) =
+            (1836149463606179 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1836149463606179 / 1000000000000000 : ℝ) ≤
+            (759588370097276210353 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (918028507811179 / 2000000000000000 : ℝ)) (q :=
+      (1836149463606179 / 4000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_446_3 :
+    Uω (aρ 3) (bρ 3) (467887100957 / 1000000000000 : ℝ) ≤
+      (-(7891308327585915285363 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9084788671070627 / 20000000000000000 : ℝ) ≤
+      (-(7891308327585915285363 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9084788671070627 / 20000000000000000 : ℝ) =
+            (9084788671070627 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9084788671070627 / 5000000000000000 : ℝ) ≤
+            (5971635278414084714637 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4541579998445627 / 10000000000000000 : ℝ)) (q :=
+      (9084788671070627 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_446_4 :
+    Uω (aρ 4) (bρ 4) (467887100957 / 1000000000000 : ℝ) ≤
+      (-(4034315884058882351419 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1785022792019079 / 4000000000000000 : ℝ) ≤
+      (-(4034315884058882351419 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1785022792019079 / 4000000000000000 : ℝ) =
+            (1785022792019079 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1785022792019079 / 1000000000000000 : ℝ) ≤
+            (2897155918941117648581 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (892039893707079 / 2000000000000000 : ℝ)) (q :=
+      (1785022792019079 / 4000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_446_5 :
+    Uω (aρ 5) (bρ 5) (467887100957 / 1000000000000 : ℝ) ≤
+      (-(521609281376695497853 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4340589820726307 / 10000000000000000 : ℝ) ≤
+      (-(521609281376695497853 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4340589820726307 / 10000000000000000 : ℝ) =
+            (4340589820726307 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4340589820726307 / 2500000000000000 : ℝ) ≤
+            (344824693998304502147 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2167227534613807 / 5000000000000000 : ℝ)) (q :=
+      (4340589820726307 / 10000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_446_6 :
+    Uω (aρ 6) (bρ 6) (467887100957 / 1000000000000 : ℝ) ≤
+      (-(4379643310814408410447 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (416475075175237 / 1000000000000000 : ℝ) ≤
+      (-(4379643310814408410447 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (416475075175237 / 1000000000000000 : ℝ) =
+            (416475075175237 / 250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (416475075175237 / 250000000000000 : ℝ) ≤
+            (2551828492185591589553 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (207500865320987 / 500000000000000 : ℝ)) (q :=
+      (416475075175237 / 1000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_446_7 :
+    Uω (aρ 7) (bρ 7) (467887100957 / 1000000000000 : ℝ) ≤
+      (-(9357881489879827674907 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (313821256860759 / 800000000000000 : ℝ) ≤
+      (-(9357881489879827674907 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (313821256860759 / 800000000000000 : ℝ) =
+            (313821256860759 / 200000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (313821256860759 / 200000000000000 : ℝ) ≤
+            (4505062116120172325093 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (155588344207759 / 400000000000000 : ℝ)) (q :=
+      (313821256860759 / 800000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_446_8 :
+    Uω (aρ 8) (bρ 8) (467887100957 / 1000000000000 : ℝ) ≤
+      (-(2042048030275987590759 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7204517486101029 / 20000000000000000 : ℝ) ≤
+      (-(2042048030275987590759 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7204517486101029 / 20000000000000000 : ℝ) =
+            (7204517486101029 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7204517486101029 / 5000000000000000 : ℝ) ≤
+            (730540690924012409241 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3532236321116029 / 10000000000000000 : ℝ)) (q :=
+      (7204517486101029 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_446_9 :
+    Uω (aρ 9) (bρ 9) (467887100957 / 1000000000000 : ℝ) ≤
+      (-(5713884519621729279943 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6378642938132311 / 20000000000000000 : ℝ) ≤
+      (-(5713884519621729279943 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6378642938132311 / 20000000000000000 : ℝ) =
+            (6378642938132311 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6378642938132311 / 5000000000000000 : ℝ) ≤
+            (1217587283378270720057 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3047327401507311 / 10000000000000000 : ℝ)) (q :=
+      (6378642938132311 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_446_10 :
+    Uω (aρ 10) (bρ 10) (467887100957 / 1000000000000 : ℝ) ≤
+      (-(13242959787489276770681 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5319803115399497 / 20000000000000000 : ℝ) ≤
+      (-(13242959787489276770681 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5319803115399497 / 20000000000000000 : ℝ) =
+            (5319803115399497 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5319803115399497 / 5000000000000000 : ℝ) ≤
+            (619983818510723229319 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2377046612269497 / 10000000000000000 : ℝ)) (q :=
+      (5319803115399497 / 20000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_446_11 :
+    Uω (aρ 11) (bρ 11) (467887100957 / 1000000000000 : ℝ) ≤
+      (-(16499225691449952125427 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (768259118541357 / 4000000000000000 : ℝ) ≤
+      (-(16499225691449952125427 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (768259118541357 / 4000000000000000 : ℝ) =
+            (768259118541357 / 500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (768259118541357 / 500000000000000 : ℝ) ≤
+            (4295189717550047874573 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (263403143156357 / 2000000000000000 : ℝ)) (q :=
+      (768259118541357 / 4000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11]) (by norm_num [bρ_11])
+      (by norm_num) (by norm_num [aρ_11, bρ_11]) (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_446_12 :
+    Uω (aρ 12) (bρ 12) (467887100957 / 1000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_446_13 :
+    Uω (aρ 13) (bρ 13) (467887100957 / 1000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_446_14 :
+    Uω (aρ 14) (bρ 14) (467887100957 / 1000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_446_15 :
+    Uω (aρ 15) (bρ 15) (467887100957 / 1000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_446_16 :
+    Uω (aρ 16) (bρ 16) (467887100957 / 1000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_446 :
+    Uρ (467887100957 / 1000000000000 : ℝ) ≤
+      (-(11509701201381695884397 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (467887100957 / 1000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(966770645457505427377 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_446_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (467887100957 / 1000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(973279580652723789647 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_446_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (467887100957 / 1000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(7891308327585915285363 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_446_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (467887100957 / 1000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(4034315884058882351419 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_446_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (467887100957 / 1000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(521609281376695497853 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_446_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (467887100957 / 1000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(4379643310814408410447 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_446_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (467887100957 / 1000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(9357881489879827674907 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_446_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (467887100957 / 1000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(2042048030275987590759 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_446_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (467887100957 / 1000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(5713884519621729279943 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_446_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (467887100957 / 1000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(13242959787489276770681 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_446_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (467887100957 / 1000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(16499225691449952125427 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_446_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (467887100957 / 1000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_446_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (467887100957 / 1000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_446_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (467887100957 / 1000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_446_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (467887100957 / 1000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_446_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (467887100957 / 1000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_446_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_447_1 :
+    Uω (aρ 1) (bρ 1) (938422801667 / 2000000000000 : ℝ) ≤
+      (-(7705506384488545139841 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1851032741813739 / 4000000000000000 : ℝ) ≤
+      (-(7705506384488545139841 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1851032741813739 / 4000000000000000 : ℝ) =
+            (1851032741813739 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1851032741813739 / 1000000000000000 : ℝ) ≤
+            (6157437221511454860159 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (925509383763739 / 2000000000000000 : ℝ)) (q :=
+      (1851032741813739 / 4000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_447_2 :
+    Uω (aρ 2) (bρ 2) (938422801667 / 2000000000000 : ℝ) ≤
+      (-(969678404396364492027 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9207234645330419 / 20000000000000000 : ℝ) ≤
+      (-(969678404396364492027 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9207234645330419 / 20000000000000000 : ℝ) =
+            (9207234645330419 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9207234645330419 / 5000000000000000 : ℝ) ≤
+            (763189546353635507973 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4603386867590419 / 10000000000000000 : ℝ)) (q :=
+      (9207234645330419 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_447_3 :
+    Uω (aρ 3) (bρ 3) (938422801667 / 2000000000000 : ℝ) ≤
+      (-(3931095660576254656589 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2277819850978757 / 5000000000000000 : ℝ) ≤
+      (-(3931095660576254656589 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2277819850978757 / 5000000000000000 : ℝ) =
+            (2277819850978757 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2277819850978757 / 1250000000000000 : ℝ) ≤
+            (3000376142423745343411 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1138706933131257 / 2500000000000000 : ℝ)) (q :=
+      (2277819850978757 / 5000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_447_4 :
+    Uω (aρ 4) (bρ 4) (938422801667 / 2000000000000 : ℝ) ≤
+      (-(1004873039024965725031 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (559475869735483 / 1250000000000000 : ℝ) ≤
+      (-(1004873039024965725031 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (559475869735483 / 1250000000000000 : ℝ) =
+            (559475869735483 / 312500000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (559475869735483 / 312500000000000 : ℝ) ≤
+            (727994911725034274969 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (559182053180341 / 1250000000000000 : ℝ)) (q :=
+      (559475869735483 / 1250000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_447_5 :
+    Uω (aρ 5) (bρ 5) (938422801667 / 2000000000000 : ℝ) ≤
+      (-(2078810588735672971517 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4353851505737701 / 10000000000000000 : ℝ) ≤
+      (-(2078810588735672971517 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4353851505737701 / 10000000000000000 : ℝ) =
+            (4353851505737701 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4353851505737701 / 2500000000000000 : ℝ) ≤
+            (1386925312764327028483 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2173867720242701 / 5000000000000000 : ℝ)) (q :=
+      (4353851505737701 / 10000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_447_6 :
+    Uω (aρ 6) (bρ 6) (938422801667 / 2000000000000 : ℝ) ≤
+      (-(8727427080123200181059 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (65281884623493 / 156250000000000 : ℝ) ≤
+      (-(8727427080123200181059 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (65281884623493 / 156250000000000 : ℝ) = (65281884623493 / 39062500000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (65281884623493 / 39062500000000 : ℝ) ≤
+            (5135516525876799818941 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (260209627247319 / 625000000000000 : ℝ)) (q :=
+      (65281884623493 / 156250000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_447_7 :
+    Uω (aρ 7) (bρ 7) (938422801667 / 2000000000000 : ℝ) ≤
+      (-(4661947022709255165381 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3936120871465577 / 10000000000000000 : ℝ) ≤
+      (-(4661947022709255165381 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3936120871465577 / 10000000000000000 : ℝ) =
+            (3936120871465577 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3936120871465577 / 2500000000000000 : ℝ) ≤
+            (2269524780290744834619 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1951587963920577 / 5000000000000000 : ℝ)) (q :=
+      (3936120871465577 / 10000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_447_8 :
+    Uω (aρ 8) (bρ 8) (938422801667 / 2000000000000 : ℝ) ≤
+      (-(2543205299478635574293 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1807881634390941 / 5000000000000000 : ℝ) ≤
+      (-(2543205299478635574293 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1807881634390941 / 5000000000000000 : ℝ) =
+            (1807881634390941 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1807881634390941 / 1250000000000000 : ℝ) ≤
+            (922530602021364425707 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (886500593453441 / 2500000000000000 : ℝ)) (q :=
+      (1807881634390941 / 5000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_447_9 :
+    Uω (aρ 9) (bρ 9) (938422801667 / 2000000000000 : ℝ) ≤
+      (-(11384414167465334094829 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1281271499489889 / 4000000000000000 : ℝ) ≤
+      (-(11384414167465334094829 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1281271499489889 / 4000000000000000 : ℝ) =
+            (1281271499489889 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1281271499489889 / 1000000000000000 : ℝ) ≤
+            (2478529438534665905171 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (612359792411889 / 2000000000000000 : ℝ)) (q :=
+      (1281271499489889 / 4000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_447_10 :
+    Uω (aρ 10) (bρ 10) (938422801667 / 2000000000000 : ℝ) ≤
+      (-(2637487779295076324629 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5349421281563489 / 20000000000000000 : ℝ) ≤
+      (-(2637487779295076324629 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5349421281563489 / 20000000000000000 : ℝ) =
+            (5349421281563489 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5349421281563489 / 5000000000000000 : ℝ) ≤
+            (135100941904923675371 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2393421779668489 / 10000000000000000 : ℝ)) (q :=
+      (5349421281563489 / 20000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_447_11 :
+    Uω (aρ 11) (bρ 11) (938422801667 / 2000000000000 : ℝ) ≤
+      (-(16399624961174716983881 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1939873172807547 / 10000000000000000 : ℝ) ≤
+      (-(16399624961174716983881 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1939873172807547 / 10000000000000000 : ℝ) =
+            (1939873172807547 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1939873172807547 / 1250000000000000 : ℝ) ≤
+            (4394790447825283016119 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (671111734962547 / 5000000000000000 : ℝ)) (q :=
+      (1939873172807547 / 10000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_447_12 :
+    Uω (aρ 12) (bρ 12) (938422801667 / 2000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_447_13 :
+    Uω (aρ 13) (bρ 13) (938422801667 / 2000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_447_14 :
+    Uω (aρ 14) (bρ 14) (938422801667 / 2000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_447_15 :
+    Uω (aρ 15) (bρ 15) (938422801667 / 2000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_447_16 :
+    Uω (aρ 16) (bρ 16) (938422801667 / 2000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_447 :
+    Uρ (938422801667 / 2000000000000 : ℝ) ≤
+      (-(2295666416895180790197 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (938422801667 / 2000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(7705506384488545139841 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_447_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (938422801667 / 2000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(969678404396364492027 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_447_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (938422801667 / 2000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3931095660576254656589 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_447_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (938422801667 / 2000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1004873039024965725031 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_447_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (938422801667 / 2000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(2078810588735672971517 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_447_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (938422801667 / 2000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(8727427080123200181059 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_447_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (938422801667 / 2000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(4661947022709255165381 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_447_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (938422801667 / 2000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(2543205299478635574293 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_447_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (938422801667 / 2000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(11384414167465334094829 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_447_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (938422801667 / 2000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(2637487779295076324629 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_447_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (938422801667 / 2000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(16399624961174716983881 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_447_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (938422801667 / 2000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_447_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (938422801667 / 2000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_447_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (938422801667 / 2000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_447_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (938422801667 / 2000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_447_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (938422801667 / 2000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_447_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/U37.lean b/LeanPool/Zeta5Irrational/Table/U37.lean
new file mode 100644
index 0000000000..fc23ec5bd6
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/U37.lean
@@ -0,0 +1,4361 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-! # Certified arcsine potential bounds (U37) -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma U_448_1 :
+    Uω (aρ 1) (bρ 1) (47053570071 / 100000000000 : ℝ) ≤
+      (-(1919232376114736975373 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9281649905984419 / 20000000000000000 : ℝ) ≤
+      (-(1919232376114736975373 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9281649905984419 / 20000000000000000 : ℝ) =
+            (9281649905984419 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9281649905984419 / 5000000000000000 : ℝ) ≤
+            (1546503525385263024627 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4640790116969419 / 10000000000000000 : ℝ)) (q :=
+      (9281649905984419 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_448_2 :
+    Uω (aρ 2) (bρ 2) (47053570071 / 100000000000 : ℝ) ≤
+      (-(1545740118644833119621 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9233721965000561 / 20000000000000000 : ℝ) ≤
+      (-(1545740118644833119621 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9233721965000561 / 20000000000000000 : ℝ) =
+            (9233721965000561 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9233721965000561 / 5000000000000000 : ℝ) ≤
+            (1226848602555166880379 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4616631188495561 / 10000000000000000 : ℝ)) (q :=
+      (9233721965000561 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_448_3 :
+    Uω (aρ 3) (bρ 3) (47053570071 / 100000000000 : ℝ) ≤
+      (-(391657943934962024627 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9137770109298861 / 20000000000000000 : ℝ) ≤
+      (-(391657943934962024627 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9137770109298861 / 20000000000000000 : ℝ) =
+            (9137770109298861 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9137770109298861 / 5000000000000000 : ℝ) ≤
+            (301489236365037975373 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4568075439143861 / 10000000000000000 : ℝ)) (q :=
+      (9137770109298861 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_448_4 :
+    Uω (aρ 4) (bρ 4) (47053570071 / 100000000000 : ℝ) ≤
+      (-(8009424585511980867437 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4489056894499417 / 10000000000000000 : ℝ) ≤
+      (-(8009424585511980867437 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4489056894499417 / 10000000000000000 : ℝ) =
+            (4489056894499417 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4489056894499417 / 2500000000000000 : ℝ) ≤
+            (5853519020488019132563 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2243356649954417 / 5000000000000000 : ℝ)) (q :=
+      (4489056894499417 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_448_5 :
+    Uω (aρ 5) (bρ 5) (47053570071 / 100000000000 : ℝ) ≤
+      (-(8284829247633143014031 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1746845230840299 / 4000000000000000 : ℝ) ≤
+      (-(8284829247633143014031 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1746845230840299 / 4000000000000000 : ℝ) =
+            (1746845230840299 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1746845230840299 / 1000000000000000 : ℝ) ≤
+            (5578114358366856985969 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (872203116889299 / 2000000000000000 : ℝ)) (q :=
+      (1746845230840299 / 4000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_448_6 :
+    Uω (aρ 6) (bρ 6) (47053570071 / 100000000000 : ℝ) ≤
+      (-(434783471547781231521 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2095665090905803 / 5000000000000000 : ℝ) ≤
+      (-(434783471547781231521 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2095665090905803 / 5000000000000000 : ℝ) =
+            (2095665090905803 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2095665090905803 / 1250000000000000 : ℝ) ≤
+            (258363708752218768479 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1044172542252053 / 2500000000000000 : ℝ)) (q :=
+      (2095665090905803 / 5000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_448_7 :
+    Uω (aρ 7) (bρ 7) (47053570071 / 100000000000 : ℝ) ≤
+      (-(2322505915292548785069 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7898950534522059 / 20000000000000000 : ℝ) ≤
+      (-(2322505915292548785069 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7898950534522059 / 20000000000000000 : ℝ) =
+            (7898950534522059 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7898950534522059 / 5000000000000000 : ℝ) ≤
+            (1143229986207451214931 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3916641720667059 / 10000000000000000 : ℝ)) (q :=
+      (7898950534522059 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_448_8 :
+    Uω (aρ 8) (bρ 8) (47053570071 / 100000000000 : ℝ) ≤
+      (-(5067773603948575405307 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7258531620454173 / 20000000000000000 : ℝ) ≤
+      (-(5067773603948575405307 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7258531620454173 / 20000000000000000 : ℝ) =
+            (7258531620454173 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7258531620454173 / 5000000000000000 : ℝ) ≤
+            (1863698199051424594693 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3559764457939173 / 10000000000000000 : ℝ)) (q :=
+      (7258531620454173 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_448_9 :
+    Uω (aρ 9) (bρ 9) (47053570071 / 100000000000 : ℝ) ≤
+      (-(7088289782332541623 / 6250000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1608515247114633 / 5000000000000000 : ℝ) ≤
+      (-(7088289782332541623 / 6250000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1608515247114633 / 5000000000000000 : ℝ) =
+            (1608515247114633 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1608515247114633 / 1250000000000000 : ℝ) ≤
+            (1576049971417458377 / 6250000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (769064863575883 / 2500000000000000 : ℝ)) (q :=
+      (1608515247114633 / 5000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_448_10 :
+    Uω (aρ 10) (bρ 10) (47053570071 / 100000000000 : ℝ) ≤
+      (-(13132296129856771453691 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (168093779711803 / 625000000000000 : ℝ) ≤
+      (-(13132296129856771453691 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (168093779711803 / 625000000000000 : ℝ) =
+            (168093779711803 / 156250000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (168093779711803 / 156250000000000 : ℝ) ≤
+            (730647476143228546309 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (37652475783089 / 156250000000000 : ℝ)) (q :=
+      (168093779711803 / 625000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10]) (by norm_num [bρ_10])
+      (by norm_num) (by norm_num [aρ_10, bρ_10]) (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_448_11 :
+    Uω (aρ 11) (bρ 11) (47053570071 / 100000000000 : ℝ) ≤
+      (-(16301865280119978924169 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1958930310284389 / 10000000000000000 : ℝ) ≤
+      (-(16301865280119978924169 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1958930310284389 / 10000000000000000 : ℝ) =
+            (1958930310284389 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1958930310284389 / 1250000000000000 : ℝ) ≤
+            (4492550128880021075831 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (683547373056889 / 5000000000000000 : ℝ)) (q :=
+      (1958930310284389 / 10000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_448_12 :
+    Uω (aρ 12) (bρ 12) (47053570071 / 100000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_448_13 :
+    Uω (aρ 13) (bρ 13) (47053570071 / 100000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_448_14 :
+    Uω (aρ 14) (bρ 14) (47053570071 / 100000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_448_15 :
+    Uω (aρ 15) (bρ 15) (47053570071 / 100000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_448_16 :
+    Uω (aρ 16) (bρ 16) (47053570071 / 100000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_448 :
+    Uρ (47053570071 / 100000000000 : ℝ) ≤
+      (-(1430900979261067775189 / 1250000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (47053570071 / 100000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1919232376114736975373 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_448_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (47053570071 / 100000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1545740118644833119621 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_448_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (47053570071 / 100000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(391657943934962024627 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_448_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (47053570071 / 100000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(8009424585511980867437 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_448_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (47053570071 / 100000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(8284829247633143014031 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_448_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (47053570071 / 100000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(434783471547781231521 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_448_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (47053570071 / 100000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(2322505915292548785069 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_448_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (47053570071 / 100000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(5067773603948575405307 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_448_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (47053570071 / 100000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(7088289782332541623 / 6250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_448_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (47053570071 / 100000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(13132296129856771453691 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_448_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (47053570071 / 100000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(16301865280119978924169 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_448_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (47053570071 / 100000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_448_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (47053570071 / 100000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_448_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (47053570071 / 100000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_448_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (47053570071 / 100000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_448_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (47053570071 / 100000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_448_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_449_1 :
+    Uω (aρ 1) (bρ 1) (943720001173 / 2000000000000 : ℝ) ≤
+      (-(7648434056806039205791 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1861627220353087 / 4000000000000000 : ℝ) ≤
+      (-(7648434056806039205791 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1861627220353087 / 4000000000000000 : ℝ) =
+            (1861627220353087 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1861627220353087 / 1000000000000000 : ℝ) ≤
+            (6214509549193960794209 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (930806662797087 / 2000000000000000 : ℝ)) (q :=
+      (1861627220353087 / 4000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_449_2 :
+    Uω (aρ 2) (bρ 2) (943720001173 / 2000000000000 : ℝ) ≤
+      (-(7700056245118668029321 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2315052319276699 / 5000000000000000 : ℝ) ≤
+      (-(7700056245118668029321 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2315052319276699 / 5000000000000000 : ℝ) =
+            (2315052319276699 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2315052319276699 / 1250000000000000 : ℝ) ≤
+            (6162887360881331970679 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1157468875459199 / 2500000000000000 : ℝ)) (q :=
+      (2315052319276699 / 5000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_449_3 :
+    Uω (aρ 3) (bρ 3) (943720001173 / 2000000000000 : ℝ) ≤
+      (-(3902105255140000568633 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9164260787460347 / 20000000000000000 : ℝ) ≤
+      (-(3902105255140000568633 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9164260787460347 / 20000000000000000 : ℝ) =
+            (9164260787460347 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9164260787460347 / 5000000000000000 : ℝ) ≤
+            (3029366547859999431367 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4581323118540347 / 10000000000000000 : ℝ)) (q :=
+      (9164260787460347 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_449_4 :
+    Uω (aρ 4) (bρ 4) (943720001173 / 2000000000000 : ℝ) ≤
+      (-(3989976034925886310423 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4502306790258663 / 10000000000000000 : ℝ) ≤
+      (-(3989976034925886310423 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4502306790258663 / 10000000000000000 : ℝ) =
+            (4502306790258663 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4502306790258663 / 2500000000000000 : ℝ) ≤
+            (2941495768074113689577 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2249985046331163 / 5000000000000000 : ℝ)) (q :=
+      (4502306790258663 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_449_5 :
+    Uω (aρ 5) (bρ 5) (943720001173 / 2000000000000 : ℝ) ≤
+      (-(8254508612734525013499 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8760749071701077 / 20000000000000000 : ℝ) ≤
+      (-(8254508612734525013499 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8760749071701077 / 20000000000000000 : ℝ) =
+            (8760749071701077 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8760749071701077 / 5000000000000000 : ℝ) ≤
+            (5608434993265474986501 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4374295503181077 / 10000000000000000 : ℝ)) (q :=
+      (8760749071701077 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_449_6 :
+    Uω (aρ 6) (bρ 6) (943720001173 / 2000000000000 : ℝ) ≤
+      (-(8664013019972345318039 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8409238904648603 / 20000000000000000 : ℝ) ≤
+      (-(8664013019972345318039 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8409238904648603 / 20000000000000000 : ℝ) =
+            (8409238904648603 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8409238904648603 / 5000000000000000 : ℝ) ≤
+            (5198930586027654681961 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4190025711268603 / 10000000000000000 : ℝ)) (q :=
+      (8409238904648603 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_449_7 :
+    Uω (aρ 7) (bρ 7) (943720001173 / 2000000000000 : ℝ) ≤
+      (-(9256269520445819727293 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3962828906008327 / 10000000000000000 : ℝ) ≤
+      (-(9256269520445819727293 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3962828906008327 / 10000000000000000 : ℝ) =
+            (3962828906008327 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3962828906008327 / 2500000000000000 : ℝ) ≤
+            (4606674085554180272707 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1965052999698327 / 5000000000000000 : ℝ)) (q :=
+      (3962828906008327 / 10000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_449_8 :
+    Uω (aρ 8) (bρ 8) (943720001173 / 2000000000000 : ℝ) ≤
+      (-(2524604255347409636449 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1821383195155667 / 5000000000000000 : ℝ) ≤
+      (-(2524604255347409636449 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1821383195155667 / 5000000000000000 : ℝ) =
+            (1821383195155667 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1821383195155667 / 1250000000000000 : ℝ) ≤
+            (941131646152590363551 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (893380654835667 / 2500000000000000 : ℝ)) (q :=
+      (1821383195155667 / 5000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_449_9 :
+    Uω (aρ 9) (bρ 9) (943720001173 / 2000000000000 : ℝ) ≤
+      (-(11298315421504753008881 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (807719195814421 / 2500000000000000 : ℝ) ≤
+      (-(11298315421504753008881 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (807719195814421 / 2500000000000000 : ℝ) =
+            (807719195814421 / 625000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (807719195814421 / 625000000000000 : ℝ) ≤
+            (2564628184495246991119 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (386338629199421 / 1250000000000000 : ℝ)) (q :=
+      (807719195814421 / 2500000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_449_10 :
+    Uω (aρ 10) (bρ 10) (943720001173 / 2000000000000 : ℝ) ≤
+      (-(6538762754409200956687 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (338033931296161 / 1250000000000000 : ℝ) ≤
+      (-(6538762754409200956687 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (338033931296161 / 1250000000000000 : ℝ) =
+            (338033931296161 / 312500000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (338033931296161 / 312500000000000 : ℝ) ≤
+            (392709048590799043313 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (303257175164197 / 1250000000000000 : ℝ)) (q :=
+      (338033931296161 / 1250000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10]) (by norm_num [bρ_10])
+      (by norm_num) (by norm_num [aρ_10, bρ_10]) (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_449_11 :
+    Uω (aρ 11) (bρ 11) (943720001173 / 2000000000000 : ℝ) ≤
+      (-(8102928517530189352599 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1977828229052923 / 10000000000000000 : ℝ) ≤
+      (-(8102928517530189352599 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1977828229052923 / 10000000000000000 : ℝ) =
+            (1977828229052923 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1977828229052923 / 1250000000000000 : ℝ) ≤
+            (2294279186969810647401 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (695823792442923 / 5000000000000000 : ℝ)) (q :=
+      (1977828229052923 / 10000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_449_12 :
+    Uω (aρ 12) (bρ 12) (943720001173 / 2000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_449_13 :
+    Uω (aρ 13) (bρ 13) (943720001173 / 2000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_449_14 :
+    Uω (aρ 14) (bρ 14) (943720001173 / 2000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_449_15 :
+    Uω (aρ 15) (bρ 15) (943720001173 / 2000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_449_16 :
+    Uω (aρ 16) (bρ 16) (943720001173 / 2000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_449 :
+    Uρ (943720001173 / 2000000000000 : ℝ) ≤
+      (-(11416320353268018089979 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (943720001173 / 2000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(7648434056806039205791 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_449_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (943720001173 / 2000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(7700056245118668029321 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_449_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (943720001173 / 2000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3902105255140000568633 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_449_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (943720001173 / 2000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3989976034925886310423 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_449_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (943720001173 / 2000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(8254508612734525013499 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_449_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (943720001173 / 2000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(8664013019972345318039 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_449_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (943720001173 / 2000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(9256269520445819727293 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_449_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (943720001173 / 2000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(2524604255347409636449 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_449_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (943720001173 / 2000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(11298315421504753008881 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_449_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (943720001173 / 2000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(6538762754409200956687 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_449_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (943720001173 / 2000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(8102928517530189352599 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_449_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (943720001173 / 2000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_449_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (943720001173 / 2000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_449_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (943720001173 / 2000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_449_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (943720001173 / 2000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_449_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (943720001173 / 2000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_449_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_450_1 :
+    Uω (aρ 1) (bρ 1) (473184300463 / 1000000000000 : ℝ) ≤
+      (-(476251223671478843051 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4667311148210701 / 10000000000000000 : ℝ) ≤
+      (-(476251223671478843051 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4667311148210701 / 10000000000000000 : ℝ) =
+            (4667311148210701 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4667311148210701 / 2500000000000000 : ℝ) ≤
+            (390182751703521156949 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2333638254938201 / 5000000000000000 : ℝ)) (q :=
+      (4667311148210701 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_450_2 :
+    Uω (aρ 2) (bρ 2) (473184300463 / 1000000000000 : ℝ) ≤
+      (-(3835746860328307421921 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9286696581713851 / 20000000000000000 : ℝ) ≤
+      (-(3835746860328307421921 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9286696581713851 / 20000000000000000 : ℝ) =
+            (9286696581713851 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9286696581713851 / 5000000000000000 : ℝ) ≤
+            (3095724942671692578079 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4643119807678851 / 10000000000000000 : ℝ)) (q :=
+      (9286696581713851 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_450_3 :
+    Uω (aρ 3) (bρ 3) (473184300463 / 1000000000000 : ℝ) ≤
+      (-(971918216274480697521 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9190751438634959 / 20000000000000000 : ℝ) ≤
+      (-(971918216274480697521 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9190751438634959 / 20000000000000000 : ℝ) =
+            (9190751438634959 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9190751438634959 / 5000000000000000 : ℝ) ≤
+            (760949734475519302479 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4594570770949959 / 10000000000000000 : ℝ)) (q :=
+      (9190751438634959 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_450_4 :
+    Uω (aρ 4) (bρ 4) (473184300463 / 1000000000000 : ℝ) ≤
+      (-(1987641562899738892589 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4515556645521627 / 10000000000000000 : ℝ) ≤
+      (-(1987641562899738892589 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4515556645521627 / 10000000000000000 : ℝ) =
+            (4515556645521627 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4515556645521627 / 2500000000000000 : ℝ) ≤
+            (1478094338600261107411 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2256613402211627 / 5000000000000000 : ℝ)) (q :=
+      (4515556645521627 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_450_5 :
+    Uω (aρ 5) (bρ 5) (473184300463 / 1000000000000 : ℝ) ≤
+      (-(8224279888071012172189 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1757454353203847 / 4000000000000000 : ℝ) ≤
+      (-(8224279888071012172189 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1757454353203847 / 4000000000000000 : ℝ) =
+            (1757454353203847 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1757454353203847 / 1000000000000000 : ℝ) ≤
+            (5638663717928987827811 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (877515039746847 / 2000000000000000 : ℝ)) (q :=
+      (1757454353203847 / 4000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_450_6 :
+    Uω (aρ 6) (bρ 6) (473184300463 / 1000000000000 : ℝ) ≤
+      (-(8632457199341266362179 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8435816860506289 / 20000000000000000 : ℝ) ≤
+      (-(8632457199341266362179 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8435816860506289 / 20000000000000000 : ℝ) =
+            (8435816860506289 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8435816860506289 / 5000000000000000 : ℝ) ≤
+            (5230486406658733637821 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4203360668361289 / 10000000000000000 : ℝ)) (q :=
+      (8435816860506289 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_450_7 :
+    Uω (aρ 7) (bρ 7) (473184300463 / 1000000000000 : ℝ) ≤
+      (-(9222630815204496616637 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (795236359092339 / 2000000000000000 : ℝ) ≤
+      (-(9222630815204496616637 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (795236359092339 / 2000000000000000 : ℝ) =
+            (795236359092339 / 500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (795236359092339 / 500000000000000 : ℝ) ≤
+            (4640312790795503383363 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (394356877953839 / 1000000000000000 : ℝ)) (q :=
+      (795236359092339 / 2000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_450_8 :
+    Uω (aρ 8) (bρ 8) (473184300463 / 1000000000000 : ℝ) ≤
+      (-(10061429492772882776433 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1462506012640659 / 4000000000000000 : ℝ) ≤
+      (-(10061429492772882776433 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1462506012640659 / 4000000000000000 : ℝ) =
+            (1462506012640659 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1462506012640659 / 1000000000000000 : ℝ) ≤
+            (3801514113227117223567 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (717455380631659 / 2000000000000000 : ℝ)) (q :=
+      (1462506012640659 / 4000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_450_9 :
+    Uω (aρ 9) (bρ 9) (473184300463 / 1000000000000 : ℝ) ≤
+      (-(2813891859876060749109 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6489435383967841 / 20000000000000000 : ℝ) ≤
+      (-(2813891859876060749109 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6489435383967841 / 20000000000000000 : ℝ) =
+            (6489435383967841 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6489435383967841 / 5000000000000000 : ℝ) ≤
+            (651844041623939250891 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3105147852282841 / 10000000000000000 : ℝ)) (q :=
+      (6489435383967841 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_450_10 :
+    Uω (aρ 10) (bρ 10) (473184300463 / 1000000000000 : ℝ) ≤
+      (-(6511560604696648093569 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1359511971652677 / 5000000000000000 : ℝ) ≤
+      (-(6511560604696648093569 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1359511971652677 / 5000000000000000 : ℝ) =
+            (1359511971652677 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1359511971652677 / 1250000000000000 : ℝ) ≤
+            (419911198303351906431 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (610579847105177 / 2500000000000000 : ℝ)) (q :=
+      (1359511971652677 / 5000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_450_11 :
+    Uω (aρ 11) (bρ 11) (473184300463 / 1000000000000 : ℝ) ≤
+      (-(16111517734219643503701 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1996575212109423 / 10000000000000000 : ℝ) ≤
+      (-(16111517734219643503701 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1996575212109423 / 10000000000000000 : ℝ) =
+            (1996575212109423 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1996575212109423 / 1250000000000000 : ℝ) ≤
+            (4682897674780356496299 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (707949276116923 / 5000000000000000 : ℝ)) (q :=
+      (1996575212109423 / 10000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_450_12 :
+    Uω (aρ 12) (bρ 12) (473184300463 / 1000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_450_13 :
+    Uω (aρ 13) (bρ 13) (473184300463 / 1000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_450_14 :
+    Uω (aρ 14) (bρ 14) (473184300463 / 1000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_450_15 :
+    Uω (aρ 15) (bρ 15) (473184300463 / 1000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_450_16 :
+    Uω (aρ 16) (bρ 16) (473184300463 / 1000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_450 :
+    Uρ (473184300463 / 1000000000000 : ℝ) ≤
+      (-(11385662126578536729411 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (473184300463 / 1000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(476251223671478843051 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_450_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (473184300463 / 1000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3835746860328307421921 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_450_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (473184300463 / 1000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(971918216274480697521 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_450_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (473184300463 / 1000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1987641562899738892589 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_450_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (473184300463 / 1000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(8224279888071012172189 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_450_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (473184300463 / 1000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(8632457199341266362179 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_450_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (473184300463 / 1000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(9222630815204496616637 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_450_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (473184300463 / 1000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(10061429492772882776433 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_450_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (473184300463 / 1000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(2813891859876060749109 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_450_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (473184300463 / 1000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(6511560604696648093569 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_450_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (473184300463 / 1000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(16111517734219643503701 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_450_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (473184300463 / 1000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_450_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (473184300463 / 1000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_450_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (473184300463 / 1000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_450_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (473184300463 / 1000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_450_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (473184300463 / 1000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_450_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_451_1 :
+    Uω (aρ 1) (bρ 1) (949017200679 / 2000000000000 : ℝ) ≤
+      (-(948960701427451485171 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9361108489961869 / 20000000000000000 : ℝ) ≤
+      (-(948960701427451485171 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9361108489961869 / 20000000000000000 : ℝ) =
+            (9361108489961869 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9361108489961869 / 5000000000000000 : ℝ) ≤
+            (783907249322548514829 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4680519704651869 / 10000000000000000 : ℝ)) (q :=
+      (9361108489961869 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_451_2 :
+    Uω (aρ 2) (bρ 2) (949017200679 / 2000000000000 : ℝ) ≤
+      (-(3821506276829465306197 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2328295969721429 / 5000000000000000 : ℝ) ≤
+      (-(3821506276829465306197 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2328295969721429 / 5000000000000000 : ℝ) =
+            (2328295969721429 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2328295969721429 / 1250000000000000 : ℝ) ≤
+            (3109965526170534693803 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1164091026521429 / 2500000000000000 : ℝ)) (q :=
+      (2328295969721429 / 5000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_451_3 :
+    Uω (aρ 3) (bρ 3) (949017200679 / 2000000000000 : ℝ) ≤
+      (-(242080126779563267641 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4608621031527731 / 10000000000000000 : ℝ) ≤
+      (-(242080126779563267641 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4608621031527731 / 10000000000000000 : ℝ) =
+            (4608621031527731 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4608621031527731 / 2500000000000000 : ℝ) ≤
+            (191136860907936732359 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2303909198302731 / 5000000000000000 : ℝ)) (q :=
+      (4608621031527731 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_451_4 :
+    Uω (aρ 4) (bρ 4) (949017200679 / 2000000000000 : ℝ) ≤
+      (-(7921266621662072203433 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (362304516851529 / 800000000000000 : ℝ) ≤
+      (-(7921266621662072203433 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (362304516851529 / 800000000000000 : ℝ) =
+            (362304516851529 / 200000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (362304516851529 / 200000000000000 : ℝ) ≤
+            (5941676984337927796567 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (181059337436129 / 400000000000000 : ℝ)) (q :=
+      (362304516851529 / 800000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_451_5 :
+    Uω (aρ 5) (bρ 5) (949017200679 / 2000000000000 : ℝ) ≤
+      (-(8194142516591679673101 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (550862139948521 / 1250000000000000 : ℝ) ≤
+      (-(8194142516591679673101 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (550862139948521 / 1250000000000000 : ℝ) =
+            (550862139948521 / 312500000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (550862139948521 / 312500000000000 : ℝ) ≤
+            (5668801089408320326899 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (68763354267599 / 156250000000000 : ℝ)) (q :=
+      (550862139948521 / 1250000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_451_6 :
+    Uω (aρ 6) (bρ 6) (949017200679 / 2000000000000 : ℝ) ≤
+      (-(8601001327470226045849 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1692478847349581 / 4000000000000000 : ℝ) ≤
+      (-(8601001327470226045849 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1692478847349581 / 4000000000000000 : ℝ) =
+            (1692478847349581 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1692478847349581 / 1000000000000000 : ℝ) ≤
+            (5261942278529773954151 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (843339009167581 / 2000000000000000 : ℝ)) (q :=
+      (1692478847349581 / 4000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_451_7 :
+    Uω (aρ 7) (bρ 7) (949017200679 / 2000000000000 : ℝ) ≤
+      (-(9189106745930102249407 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7979067886537949 / 20000000000000000 : ℝ) ≤
+      (-(9189106745930102249407 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7979067886537949 / 20000000000000000 : ℝ) =
+            (7979067886537949 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7979067886537949 / 5000000000000000 : ℝ) ≤
+            (4673836860069897750593 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3957030076387949 / 10000000000000000 : ℝ)) (q :=
+      (7979067886537949 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_451_8 :
+    Uω (aρ 8) (bρ 8) (949017200679 / 2000000000000 : ℝ) ≤
+      (-(2506145872626381178453 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (917440439078489 / 2500000000000000 : ℝ) ≤
+      (-(2506145872626381178453 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (917440439078489 / 2500000000000000 : ℝ) =
+            (917440439078489 / 625000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (917440439078489 / 625000000000000 : ℝ) ≤
+            (959590028873618821547 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (450128419227239 / 1250000000000000 : ℝ)) (q :=
+      (917440439078489 / 2500000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_451_9 :
+    Uω (aρ 9) (bρ 9) (949017200679 / 2000000000000 : ℝ) ≤
+      (-(11213017700970506234207 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (814638323778861 / 2500000000000000 : ℝ) ≤
+      (-(11213017700970506234207 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (814638323778861 / 2500000000000000 : ℝ) =
+            (814638323778861 / 625000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (814638323778861 / 625000000000000 : ℝ) ≤
+            (2649925905029493765793 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (389947007472611 / 1250000000000000 : ℝ)) (q :=
+      (814638323778861 / 2500000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_451_10 :
+    Uω (aρ 10) (bρ 10) (949017200679 / 2000000000000 : ℝ) ≤
+      (-(1296907755686059853951 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2733758320944201 / 10000000000000000 : ℝ) ≤
+      (-(1296907755686059853951 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2733758320944201 / 10000000000000000 : ℝ) =
+            (2733758320944201 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2733758320944201 / 2500000000000000 : ℝ) ≤
+            (89386604913940146049 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1229272572466701 / 5000000000000000 : ℝ)) (q :=
+      (2733758320944201 / 10000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_451_11 :
+    Uω (aρ 11) (bρ 11) (949017200679 / 2000000000000 : ℝ) ≤
+      (-(25630033977972161371 / 16000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2015178885922511 / 10000000000000000 : ℝ) ≤
+      (-(25630033977972161371 / 16000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2015178885922511 / 10000000000000000 : ℝ) =
+            (2015178885922511 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2015178885922511 / 1250000000000000 : ℝ) ≤
+            (7641030676427838629 / 16000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (719931450547511 / 5000000000000000 : ℝ)) (q :=
+      (2015178885922511 / 10000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_451_12 :
+    Uω (aρ 12) (bρ 12) (949017200679 / 2000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_451_13 :
+    Uω (aρ 13) (bρ 13) (949017200679 / 2000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_451_14 :
+    Uω (aρ 14) (bρ 14) (949017200679 / 2000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_451_15 :
+    Uω (aρ 15) (bρ 15) (949017200679 / 2000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_451_16 :
+    Uω (aρ 16) (bρ 16) (949017200679 / 2000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_451 :
+    Uρ (949017200679 / 2000000000000 : ℝ) ≤
+      (-(1135522615867193342733 / 1000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (949017200679 / 2000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(948960701427451485171 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_451_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (949017200679 / 2000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3821506276829465306197 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_451_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (949017200679 / 2000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(242080126779563267641 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_451_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (949017200679 / 2000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(7921266621662072203433 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_451_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (949017200679 / 2000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(8194142516591679673101 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_451_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (949017200679 / 2000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(8601001327470226045849 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_451_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (949017200679 / 2000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(9189106745930102249407 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_451_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (949017200679 / 2000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(2506145872626381178453 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_451_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (949017200679 / 2000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(11213017700970506234207 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_451_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (949017200679 / 2000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(1296907755686059853951 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_451_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (949017200679 / 2000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) * (-(25630033977972161371 / 16000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_451_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (949017200679 / 2000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_451_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (949017200679 / 2000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_451_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (949017200679 / 2000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_451_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (949017200679 / 2000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_451_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (949017200679 / 2000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_451_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_452_1 :
+    Uω (aρ 1) (bρ 1) (59479112527 / 125000000000 : ℝ) ≤
+      (-(1890857924967793981321 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (234689867059907 / 500000000000000 : ℝ) ≤
+      (-(1890857924967793981321 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (234689867059907 / 500000000000000 : ℝ) =
+            (234689867059907 / 125000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (234689867059907 / 125000000000000 : ℝ) ≤
+            (1574877976532206018679 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7334004528627 / 15625000000000 : ℝ)) (q :=
+      (234689867059907 / 500000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_452_2 :
+    Uω (aρ 2) (bρ 2) (59479112527 / 125000000000 : ℝ) ≤
+      (-(1522922456383919548857 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4669835584342829 / 10000000000000000 : ℝ) ≤
+      (-(1522922456383919548857 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4669835584342829 / 10000000000000000 : ℝ) =
+            (4669835584342829 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4669835584342829 / 2500000000000000 : ℝ) ≤
+            (1249666264816080451143 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2334804198560329 / 5000000000000000 : ℝ)) (q :=
+      (4669835584342829 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_452_3 :
+    Uω (aρ 3) (bρ 3) (59479112527 / 125000000000 : ℝ) ≤
+      (-(7717865013179724929687 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9243732660951951 / 20000000000000000 : ℝ) ≤
+      (-(7717865013179724929687 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9243732660951951 / 20000000000000000 : ℝ) =
+            (9243732660951951 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9243732660951951 / 5000000000000000 : ℝ) ≤
+            (6145078592820275070313 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4621065995736951 / 10000000000000000 : ℝ)) (q :=
+      (9243732660951951 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_452_4 :
+    Uω (aρ 4) (bρ 4) (59479112527 / 125000000000 : ℝ) ≤
+      (-(7892052675425208535167 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2271028117988883 / 5000000000000000 : ℝ) ≤
+      (-(7892052675425208535167 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2271028117988883 / 5000000000000000 : ℝ) =
+            (2271028117988883 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2271028117988883 / 1250000000000000 : ℝ) ≤
+            (5970890930574791464833 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1134934996951383 / 2500000000000000 : ℝ)) (q :=
+      (2271028117988883 / 5000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_452_5 :
+    Uω (aρ 5) (bρ 5) (59479112527 / 125000000000 : ℝ) ≤
+      (-(4082047973154115081541 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8840316493168403 / 20000000000000000 : ℝ) ≤
+      (-(4082047973154115081541 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8840316493168403 / 20000000000000000 : ℝ) =
+            (8840316493168403 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8840316493168403 / 5000000000000000 : ℝ) ≤
+            (2849423829845884918459 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4414133928353403 / 10000000000000000 : ℝ)) (q :=
+      (8840316493168403 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_452_6 :
+    Uω (aρ 6) (bρ 6) (59479112527 / 125000000000 : ℝ) ≤
+      (-(8569644768926593276099 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (132640172482107 / 312500000000000 : ℝ) ≤
+      (-(8569644768926593276099 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (132640172482107 / 312500000000000 : ℝ) = (132640172482107 / 78125000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (132640172482107 / 78125000000000 : ℝ) ≤
+            (5293298837073406723901 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (528753606147481 / 1250000000000000 : ℝ)) (q :=
+      (132640172482107 / 312500000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_452_7 :
+    Uω (aρ 7) (bρ 7) (59479112527 / 125000000000 : ℝ) ≤
+      (-(9155696521510087946207 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (800577071394691 / 2000000000000000 : ℝ) ≤
+      (-(9155696521510087946207 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (800577071394691 / 2000000000000000 : ℝ) =
+            (800577071394691 / 500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (800577071394691 / 500000000000000 : ℝ) ≤
+            (4707247084489912053793 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (397048990503191 / 1000000000000000 : ℝ)) (q :=
+      (800577071394691 / 2000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_452_8 :
+    Uω (aρ 8) (bρ 8) (59479112527 / 125000000000 : ℝ) ≤
+      (-(4993938948443957304859 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (368325658632001 / 1000000000000000 : ℝ) ≤
+      (-(4993938948443957304859 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (368325658632001 / 1000000000000000 : ℝ) =
+            (368325658632001 / 250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (368325658632001 / 250000000000000 : ℝ) ≤
+            (1937532854556042695141 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (180738700753251 / 500000000000000 : ℝ)) (q :=
+      (368325658632001 / 1000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_452_9 :
+    Uω (aρ 9) (bρ 9) (59479112527 / 125000000000 : ℝ) ≤
+      (-(2792666058235790186179 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (130895346637183 / 400000000000000 : ℝ) ≤
+      (-(2792666058235790186179 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (130895346637183 / 400000000000000 : ℝ) =
+            (130895346637183 / 100000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (130895346637183 / 100000000000000 : ℝ) ≤
+            (673069843264209813821 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (62679876052883 / 200000000000000 : ℝ)) (q :=
+      (130895346637183 / 400000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_452_10 :
+    Uω (aρ 10) (bρ 10) (59479112527 / 125000000000 : ℝ) ≤
+      (-(12915389020402956627827 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27484749396081 / 100000000000000 : ℝ) ≤
+      (-(12915389020402956627827 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (27484749396081 / 100000000000000 : ℝ) = (27484749396081 / 25000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27484749396081 / 25000000000000 : ℝ) ≤
+            (947554585597043372173 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (12373676917481 / 50000000000000 : ℝ)) (q :=
+      (27484749396081 / 100000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10]) (by norm_num [bρ_10])
+      (by norm_num) (by norm_num [aρ_10, bρ_10]) (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_452_11 :
+    Uω (aρ 11) (bρ 11) (59479112527 / 125000000000 : ℝ) ≤
+      (-(7963773541683381200579 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2033646290193261 / 10000000000000000 : ℝ) ≤
+      (-(7963773541683381200579 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2033646290193261 / 10000000000000000 : ℝ) =
+            (2033646290193261 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2033646290193261 / 1250000000000000 : ℝ) ≤
+            (2433434162816618799421 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (731777355435761 / 5000000000000000 : ℝ)) (q :=
+      (2033646290193261 / 10000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_452_12 :
+    Uω (aρ 12) (bρ 12) (59479112527 / 125000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_452_13 :
+    Uω (aρ 13) (bρ 13) (59479112527 / 125000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_452_14 :
+    Uω (aρ 14) (bρ 14) (59479112527 / 125000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_452_15 :
+    Uω (aρ 15) (bρ 15) (59479112527 / 125000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_452_16 :
+    Uω (aρ 16) (bρ 16) (59479112527 / 125000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_452 :
+    Uρ (59479112527 / 125000000000 : ℝ) ≤
+      (-(2265001184222090608171 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (59479112527 / 125000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1890857924967793981321 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_452_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (59479112527 / 125000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1522922456383919548857 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_452_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (59479112527 / 125000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(7717865013179724929687 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_452_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (59479112527 / 125000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(7892052675425208535167 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_452_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (59479112527 / 125000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(4082047973154115081541 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_452_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (59479112527 / 125000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(8569644768926593276099 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_452_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (59479112527 / 125000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(9155696521510087946207 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_452_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (59479112527 / 125000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(4993938948443957304859 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_452_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (59479112527 / 125000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(2792666058235790186179 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_452_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (59479112527 / 125000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12915389020402956627827 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_452_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (59479112527 / 125000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(7963773541683381200579 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_452_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (59479112527 / 125000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_452_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (59479112527 / 125000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_452_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (59479112527 / 125000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_452_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (59479112527 / 125000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_452_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (59479112527 / 125000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_452_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_453_1 :
+    Uω (aρ 1) (bρ 1) (190862880037 / 400000000000 : ℝ) ≤
+      (-(7535257392981299478961 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (588380054608373 / 1250000000000000 : ℝ) ≤
+      (-(7535257392981299478961 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (588380054608373 / 1250000000000000 : ℝ) =
+            (588380054608373 / 312500000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (588380054608373 / 312500000000000 : ℝ) ≤
+            (6327686213018700521039 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (294187880680873 / 625000000000000 : ℝ)) (q :=
+      (588380054608373 / 1250000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_453_2 :
+    Uω (aρ 2) (bρ 2) (190862880037 / 400000000000 : ℝ) ≤
+      (-(7586292447160640009489 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (292692451599257 / 625000000000000 : ℝ) ≤
+      (-(7586292447160640009489 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (292692451599257 / 625000000000000 : ℝ) =
+            (292692451599257 / 156250000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (292692451599257 / 156250000000000 : ℝ) ≤
+            (6276651158839359990511 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (292678292552889 / 625000000000000 : ℝ)) (q :=
+      (292692451599257 / 625000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_453_3 :
+    Uω (aρ 3) (bρ 3) (190862880037 / 400000000000 : ℝ) ≤
+      (-(3844624062824249615123 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (927022323255189 / 2000000000000000 : ℝ) ≤
+      (-(3844624062824249615123 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (927022323255189 / 2000000000000000 : ℝ) =
+            (927022323255189 / 500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (927022323255189 / 500000000000000 : ℝ) ≤
+            (3086847740175750384877 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (463431356857189 / 1000000000000000 : ℝ)) (q :=
+      (927022323255189 / 2000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_453_4 :
+    Uω (aρ 4) (bρ 4) (190862880037 / 400000000000 : ℝ) ≤
+      (-(1572584782539127471169 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (569413246483767 / 1250000000000000 : ℝ) ≤
+      (-(1572584782539127471169 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (569413246483767 / 1250000000000000 : ℝ) =
+            (569413246483767 / 312500000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (569413246483767 / 312500000000000 : ℝ) ≤
+            (1200003938660872528831 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (569124557603159 / 1250000000000000 : ℝ)) (q :=
+      (569413246483767 / 1250000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_453_5 :
+    Uω (aρ 5) (bρ 5) (190862880037 / 400000000000 : ℝ) ≤
+      (-(1016767453779209346861 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2216709632491869 / 5000000000000000 : ℝ) ≤
+      (-(1016767453779209346861 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2216709632491869 / 5000000000000000 : ℝ) =
+            (2216709632491869 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2216709632491869 / 1250000000000000 : ℝ) ≤
+            (716100496970790653139 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1106853241596869 / 2500000000000000 : ℝ)) (q :=
+      (2216709632491869 / 5000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_453_6 :
+    Uω (aρ 6) (bρ 6) (190862880037 / 400000000000 : ℝ) ≤
+      (-(8538386894358178531511 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8515547272239381 / 20000000000000000 : ℝ) ≤
+      (-(8538386894358178531511 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8515547272239381 / 20000000000000000 : ℝ) =
+            (8515547272239381 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8515547272239381 / 5000000000000000 : ℝ) ≤
+            (5324556711641821468489 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4243362083799381 / 10000000000000000 : ℝ)) (q :=
+      (8515547272239381 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_453_7 :
+    Uω (aρ 7) (bρ 7) (190862880037 / 400000000000 : ℝ) ≤
+      (-(4561199679558233400381 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8032472088031337 / 20000000000000000 : ℝ) ≤
+      (-(4561199679558233400381 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8032472088031337 / 20000000000000000 : ℝ) =
+            (8032472088031337 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8032472088031337 / 5000000000000000 : ℝ) ≤
+            (2370272123441766599619 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3983948280351337 / 10000000000000000 : ℝ)) (q :=
+      (8032472088031337 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_453_8 :
+    Uω (aρ 8) (bρ 8) (190862880037 / 400000000000 : ℝ) ≤
+      (-(995131160783109617529 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7393499086308263 / 20000000000000000 : ℝ) ≤
+      (-(995131160783109617529 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7393499086308263 / 20000000000000000 : ℝ) =
+            (7393499086308263 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7393499086308263 / 5000000000000000 : ℝ) ≤
+            (391163199816890382471 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3628516929968263 / 10000000000000000 : ℝ)) (q :=
+      (7393499086308263 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_453_9 :
+    Uω (aρ 9) (bρ 9) (190862880037 / 400000000000 : ℝ) ≤
+      (-(5564252546781204034449 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (328620887630871 / 1000000000000000 : ℝ) ≤
+      (-(5564252546781204034449 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (328620887630871 / 1000000000000000 : ℝ) =
+            (328620887630871 / 250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (328620887630871 / 250000000000000 : ℝ) ≤
+            (1367219256218795965551 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (157420061231871 / 500000000000000 : ℝ)) (q :=
+      (328620887630871 / 1000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_453_10 :
+    Uω (aρ 10) (bρ 10) (190862880037 / 400000000000 : ℝ) ≤
+      (-(12862050208008657540741 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (86349192049541 / 312500000000000 : ℝ) ≤
+      (-(12862050208008657540741 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (86349192049541 / 312500000000000 : ℝ) = (86349192049541 / 78125000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (86349192049541 / 78125000000000 : ℝ) ≤
+            (1000893397991342459259 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (311361349585703 / 1250000000000000 : ℝ)) (q :=
+      (86349192049541 / 312500000000000 : ℝ)) (by norm_num [aρ_10, bρ_10]) (by norm_num [bρ_10])
+      (by norm_num) (by norm_num [aρ_10, bρ_10]) (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_453_11 :
+    Uω (aρ 11) (bρ 11) (190862880037 / 400000000000 : ℝ) ≤
+      (-(3959444980566102260757 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (64124498075021 / 312500000000000 : ℝ) ≤
+      (-(3959444980566102260757 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (64124498075021 / 312500000000000 : ℝ) = (64124498075021 / 39062500000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (64124498075021 / 39062500000000 : ℝ) ≤
+            (1239158871683897739243 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (11617086004073 / 78125000000000 : ℝ)) (q :=
+      (64124498075021 / 312500000000000 : ℝ)) (by norm_num [aρ_11, bρ_11]) (by norm_num [bρ_11])
+      (by norm_num) (by norm_num [aρ_11, bρ_11]) (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_453_12 :
+    Uω (aρ 12) (bρ 12) (190862880037 / 400000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_453_13 :
+    Uω (aρ 13) (bρ 13) (190862880037 / 400000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_453_14 :
+    Uω (aρ 14) (bρ 14) (190862880037 / 400000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_453_15 :
+    Uω (aρ 15) (bρ 15) (190862880037 / 400000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_453_16 :
+    Uω (aρ 16) (bρ 16) (190862880037 / 400000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_453 :
+    Uρ (190862880037 / 400000000000 : ℝ) ≤
+      (-(705937206632403026211 / 625000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (190862880037 / 400000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(7535257392981299478961 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_453_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (190862880037 / 400000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(7586292447160640009489 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_453_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (190862880037 / 400000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3844624062824249615123 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_453_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (190862880037 / 400000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1572584782539127471169 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_453_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (190862880037 / 400000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1016767453779209346861 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_453_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (190862880037 / 400000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(8538386894358178531511 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_453_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (190862880037 / 400000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(4561199679558233400381 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_453_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (190862880037 / 400000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(995131160783109617529 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_453_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (190862880037 / 400000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(5564252546781204034449 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_453_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (190862880037 / 400000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12862050208008657540741 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_453_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (190862880037 / 400000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(3959444980566102260757 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_453_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (190862880037 / 400000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_453_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (190862880037 / 400000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_453_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (190862880037 / 400000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_453_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (190862880037 / 400000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_453_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (190862880037 / 400000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_453_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_454_1 :
+    Uω (aρ 1) (bρ 1) (478481499969 / 1000000000000 : ℝ) ≤
+      (-(3753581121717678975279 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4720283531992083 / 10000000000000000 : ℝ) ≤
+      (-(3753581121717678975279 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4720283531992083 / 10000000000000000 : ℝ) =
+            (4720283531992083 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4720283531992083 / 2500000000000000 : ℝ) ≤
+            (3177890681282321024721 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2360124641189583 / 5000000000000000 : ℝ)) (q :=
+      (4720283531992083 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_454_2 :
+    Uω (aρ 2) (bρ 2) (478481499969 / 1000000000000 : ℝ) ≤
+      (-(184522768432317577 / 244140625000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9392645726419257 / 20000000000000000 : ℝ) ≤
+      (-(184522768432317577 / 244140625000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9392645726419257 / 20000000000000000 : ℝ) =
+            (9392645726419257 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9392645726419257 / 5000000000000000 : ℝ) ≤
+            (76964001599270899 / 122070312500000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4696096957324257 / 10000000000000000 : ℝ)) (q :=
+      (9392645726419257 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_454_3 :
+    Uω (aρ 3) (bρ 3) (478481499969 / 1000000000000 : ℝ) ≤
+      (-(7660712925159377773271 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2324178444520037 / 5000000000000000 : ℝ) ≤
+      (-(7660712925159377773271 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2324178444520037 / 5000000000000000 : ℝ) =
+            (2324178444520037 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2324178444520037 / 1250000000000000 : ℝ) ≤
+            (6202230680840622226729 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1161890278833787 / 2500000000000000 : ℝ)) (q :=
+      (2324178444520037 / 5000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_454_4 :
+    Uω (aρ 4) (bρ 4) (478481499969 / 1000000000000 : ℝ) ≤
+      (-(156677596753044379399 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2284277834332367 / 5000000000000000 : ℝ) ≤
+      (-(156677596753044379399 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2284277834332367 / 5000000000000000 : ℝ) =
+            (2284277834332367 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2284277834332367 / 1250000000000000 : ℝ) ≤
+            (120581275366955620601 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1141563213912367 / 2500000000000000 : ℝ)) (q :=
+      (2284277834332367 / 5000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_454_5 :
+    Uω (aρ 5) (bρ 5) (478481499969 / 1000000000000 : ℝ) ≤
+      (-(810427302632192858973 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8893360351521979 / 20000000000000000 : ℝ) ≤
+      (-(810427302632192858973 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8893360351521979 / 20000000000000000 : ℝ) =
+            (8893360351521979 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8893360351521979 / 5000000000000000 : ℝ) ≤
+            (575867057967807141027 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4440691789176979 / 10000000000000000 : ℝ)) (q :=
+      (8893360351521979 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_454_6 :
+    Uω (aρ 6) (bρ 6) (478481499969 / 1000000000000 : ℝ) ≤
+      (-(170144541608308100213 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8542122942245727 / 20000000000000000 : ℝ) ≤
+      (-(170144541608308100213 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8542122942245727 / 20000000000000000 : ℝ) =
+            (8542122942245727 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8542122942245727 / 5000000000000000 : ℝ) ≤
+            (107114330511691899787 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4256694755040727 / 10000000000000000 : ℝ)) (q :=
+      (8542122942245727 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_454_7 :
+    Uω (aρ 7) (bρ 7) (478481499969 / 1000000000000 : ℝ) ≤
+      (-(1817842896817785574473 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2014793005867573 / 5000000000000000 : ℝ) ≤
+      (-(1817842896817785574473 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2014793005867573 / 5000000000000000 : ℝ) =
+            (2014793005867573 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2014793005867573 / 1250000000000000 : ℝ) ≤
+            (954745824382214425527 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (999351304256323 / 2500000000000000 : ℝ)) (q :=
+      (2014793005867573 / 5000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_454_8 :
+    Uω (aρ 8) (bρ 8) (478481499969 / 1000000000000 : ℝ) ≤
+      (-(9914883532630687762813 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1484096259207923 / 4000000000000000 : ℝ) ≤
+      (-(9914883532630687762813 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1484096259207923 / 4000000000000000 : ℝ) =
+            (1484096259207923 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1484096259207923 / 1000000000000000 : ℝ) ≤
+            (3948060073369312237187 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (728451228186923 / 2000000000000000 : ℝ)) (q :=
+      (1484096259207923 / 4000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_454_9 :
+    Uω (aρ 9) (bρ 9) (478481499969 / 1000000000000 : ℝ) ≤
+      (-(11086538371389681350551 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1650014498387239 / 5000000000000000 : ℝ) ≤
+      (-(11086538371389681350551 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1650014498387239 / 5000000000000000 : ℝ) =
+            (1650014498387239 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1650014498387239 / 1250000000000000 : ℝ) ≤
+            (2776405234610318649449 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (790699616700989 / 2500000000000000 : ℝ)) (q :=
+      (1650014498387239 / 5000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_454_10 :
+    Uω (aρ 10) (bρ 10) (478481499969 / 1000000000000 : ℝ) ≤
+      (-(1601131982700614138667 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1388928137751341 / 5000000000000000 : ℝ) ≤
+      (-(1601131982700614138667 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1388928137751341 / 5000000000000000 : ℝ) =
+            (1388928137751341 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1388928137751341 / 1250000000000000 : ℝ) ≤
+            (131735968049385861333 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (626753014438841 / 2500000000000000 : ℝ)) (q :=
+      (1388928137751341 / 5000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_454_11 :
+    Uω (aρ 11) (bρ 11) (478481499969 / 1000000000000 : ℝ) ≤
+      (-(15749408998553283782167 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4140395741151013 / 20000000000000000 : ℝ) ≤
+      (-(15749408998553283782167 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4140395741151013 / 20000000000000000 : ℝ) =
+            (4140395741151013 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4140395741151013 / 2500000000000000 : ℝ) ≤
+            (5045006410446716217833 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1510171874106013 / 10000000000000000 : ℝ)) (q :=
+      (4140395741151013 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_454_12 :
+    Uω (aρ 12) (bρ 12) (478481499969 / 1000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_454_13 :
+    Uω (aρ 13) (bρ 13) (478481499969 / 1000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_454_14 :
+    Uω (aρ 14) (bρ 14) (478481499969 / 1000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_454_15 :
+    Uω (aρ 15) (bρ 15) (478481499969 / 1000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_454_16 :
+    Uω (aρ 16) (bρ 16) (478481499969 / 1000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_454 :
+    Uρ (478481499969 / 1000000000000 : ℝ) ≤
+      (-(11265188586184623207311 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (478481499969 / 1000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3753581121717678975279 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_454_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (478481499969 / 1000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) * (-(184522768432317577 / 244140625000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_454_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (478481499969 / 1000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(7660712925159377773271 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_454_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (478481499969 / 1000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(156677596753044379399 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_454_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (478481499969 / 1000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(810427302632192858973 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_454_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (478481499969 / 1000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(170144541608308100213 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_454_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (478481499969 / 1000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(1817842896817785574473 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_454_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (478481499969 / 1000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(9914883532630687762813 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_454_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (478481499969 / 1000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(11086538371389681350551 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_454_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (478481499969 / 1000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(1601131982700614138667 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_454_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (478481499969 / 1000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(15749408998553283782167 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_454_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (478481499969 / 1000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_454_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (478481499969 / 1000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_454_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (478481499969 / 1000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_454_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (478481499969 / 1000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_454_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (478481499969 / 1000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_454_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_455_1 :
+    Uω (aρ 1) (bρ 1) (959611599691 / 2000000000000 : ℝ) ≤
+      (-(1869786451919637710493 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2366763313289 / 5000000000000 : ℝ) ≤
+      (-(1869786451919637710493 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (2366763313289 / 5000000000000 : ℝ) = (2366763313289 / 1250000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2366763313289 / 1250000000000 : ℝ) ≤
+            (1595949449580362289507 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2366746236393 / 5000000000000 : ℝ)) (q :=
+      (2366763313289 / 5000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_455_2 :
+    Uω (aρ 2) (bρ 2) (959611599691 / 2000000000000 : ℝ) ≤
+      (-(3764946137423207459503 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9419132994475907 / 20000000000000000 : ℝ) ≤
+      (-(3764946137423207459503 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9419132994475907 / 20000000000000000 : ℝ) =
+            (9419132994475907 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9419132994475907 / 5000000000000000 : ℝ) ≤
+            (3166525665576792540497 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4709341226615907 / 10000000000000000 : ℝ)) (q :=
+      (9419132994475907 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_455_3 :
+    Uω (aρ 3) (bρ 3) (959611599691 / 2000000000000 : ℝ) ≤
+      (-(3816129473264321412793 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (29135013430497 / 62500000000000 : ℝ) ≤
+      (-(3816129473264321412793 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (29135013430497 / 62500000000000 : ℝ) = (29135013430497 / 15625000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (29135013430497 / 15625000000000 : ℝ) ≤
+            (3115342329735678587207 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (58260107953113 / 125000000000000 : ℝ)) (q :=
+      (29135013430497 / 62500000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_455_4 :
+    Uω (aρ 4) (bρ 4) (959611599691 / 2000000000000 : ℝ) ≤
+      (-(7804919958794561528009 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9163610653402181 / 20000000000000000 : ℝ) ≤
+      (-(7804919958794561528009 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9163610653402181 / 20000000000000000 : ℝ) =
+            (9163610653402181 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9163610653402181 / 5000000000000000 : ℝ) ≤
+            (6058023647205438471991 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4579509172957181 / 10000000000000000 : ℝ)) (q :=
+      (9163610653402181 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_455_5 :
+    Uω (aρ 5) (bρ 5) (959611599691 / 2000000000000 : ℝ) ≤
+      (-(4037247798704166091837 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8919881959757067 / 20000000000000000 : ℝ) ≤
+      (-(4037247798704166091837 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8919881959757067 / 20000000000000000 : ℝ) =
+            (8919881959757067 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8919881959757067 / 5000000000000000 : ℝ) ≤
+            (2894224004295833908163 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4453970398647067 / 10000000000000000 : ℝ)) (q :=
+      (8919881959757067 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_455_6 :
+    Uω (aρ 6) (bρ 6) (959611599691 / 2000000000000 : ℝ) ≤
+      (-(4238082354837365660861 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (428434902707557 / 1000000000000000 : ℝ) ≤
+      (-(4238082354837365660861 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (428434902707557 / 1000000000000000 : ℝ) =
+            (428434902707557 / 250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (428434902707557 / 250000000000000 : ℝ) ≤
+            (2693389448162634339139 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (213501343409057 / 500000000000000 : ℝ)) (q :=
+      (428434902707557 / 1000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_455_7 :
+    Uω (aρ 7) (bρ 7) (959611599691 / 2000000000000 : ℝ) ≤
+      (-(9056141129820026511853 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8085870534744279 / 20000000000000000 : ℝ) ≤
+      (-(9056141129820026511853 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8085870534744279 / 20000000000000000 : ℝ) =
+            (8085870534744279 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8085870534744279 / 5000000000000000 : ℝ) ≤
+            (4806802476179973488147 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4010860729534279 / 10000000000000000 : ℝ)) (q :=
+      (8085870534744279 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_455_8 :
+    Uω (aρ 8) (bρ 8) (959611599691 / 2000000000000 : ℝ) ≤
+      (-(617412037109097621673 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3723729921796121 / 10000000000000000 : ℝ) ≤
+      (-(617412037109097621673 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3723729921796121 / 10000000000000000 : ℝ) =
+            (3723729921796121 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3723729921796121 / 2500000000000000 : ℝ) ≤
+            (249021938265902378327 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1827995844861121 / 5000000000000000 : ℝ)) (q :=
+      (3723729921796121 / 10000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_455_9 :
+    Uω (aρ 9) (bρ 9) (959611599691 / 2000000000000 : ℝ) ≤
+      (-(5522381092373664086313 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (828461024130217 / 2500000000000000 : ℝ) ≤
+      (-(5522381092373664086313 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (828461024130217 / 2500000000000000 : ℝ) =
+            (828461024130217 / 625000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (828461024130217 / 625000000000000 : ℝ) ≤
+            (1409090710626335913687 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (397148208441467 / 1250000000000000 : ℝ)) (q :=
+      (828461024130217 / 2500000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_455_10 :
+    Uω (aρ 10) (bρ 10) (959611599691 / 2000000000000 : ℝ) ≤
+      (-(3189100213102396916873 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1396260828340487 / 5000000000000000 : ℝ) ≤
+      (-(3189100213102396916873 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1396260828340487 / 5000000000000000 : ℝ) =
+            (1396260828340487 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1396260828340487 / 1250000000000000 : ℝ) ≤
+            (276635688397603083127 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (630774955336737 / 2500000000000000 : ℝ)) (q :=
+      (1396260828340487 / 5000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_455_11 :
+    Uω (aρ 11) (bρ 11) (959611599691 / 2000000000000 : ℝ) ≤
+      (-(978898607132638238711 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (104414684974301 / 500000000000000 : ℝ) ≤
+      (-(978898607132638238711 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (104414684974301 / 500000000000000 : ℝ) = (104414684974301 / 62500000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (104414684974301 / 62500000000000 : ℝ) ≤
+            (320752355929861761289 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (38328013329051 / 250000000000000 : ℝ)) (q :=
+      (104414684974301 / 500000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11]) (by norm_num [bρ_11])
+      (by norm_num) (by norm_num [aρ_11, bρ_11]) (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_455_12 :
+    Uω (aρ 12) (bρ 12) (959611599691 / 2000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_455_13 :
+    Uω (aρ 13) (bρ 13) (959611599691 / 2000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_455_14 :
+    Uω (aρ 14) (bρ 14) (959611599691 / 2000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_455_15 :
+    Uω (aρ 15) (bρ 15) (959611599691 / 2000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_455_16 :
+    Uω (aρ 16) (bρ 16) (959611599691 / 2000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_455 :
+    Uρ (959611599691 / 2000000000000 : ℝ) ≤
+      (-(351111886832178083997 / 312500000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (959611599691 / 2000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1869786451919637710493 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_455_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (959611599691 / 2000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3764946137423207459503 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_455_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (959611599691 / 2000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3816129473264321412793 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_455_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (959611599691 / 2000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(7804919958794561528009 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_455_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (959611599691 / 2000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(4037247798704166091837 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_455_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (959611599691 / 2000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(4238082354837365660861 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_455_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (959611599691 / 2000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(9056141129820026511853 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_455_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (959611599691 / 2000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(617412037109097621673 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_455_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (959611599691 / 2000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(5522381092373664086313 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_455_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (959611599691 / 2000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(3189100213102396916873 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_455_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (959611599691 / 2000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) * (-(978898607132638238711 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_455_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (959611599691 / 2000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_455_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (959611599691 / 2000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_455_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (959611599691 / 2000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_455_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (959611599691 / 2000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_455_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (959611599691 / 2000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_455_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_456_1 :
+    Uω (aρ 1) (bρ 1) (240565049861 / 500000000000 : ℝ) ≤
+      (-(7451207645873873946861 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4746769720629249 / 10000000000000000 : ℝ) ≤
+      (-(7451207645873873946861 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4746769720629249 / 10000000000000000 : ℝ) =
+            (4746769720629249 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4746769720629249 / 2500000000000000 : ℝ) ≤
+            (6411735960126126053139 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2373367831061749 / 5000000000000000 : ℝ)) (q :=
+      (4746769720629249 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_456_2 :
+    Uω (aρ 2) (bρ 2) (240565049861 / 500000000000 : ℝ) ≤
+      (-(7501811039978996837427 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9445620255406633 / 20000000000000000 : ℝ) ≤
+      (-(7501811039978996837427 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9445620255406633 / 20000000000000000 : ℝ) =
+            (9445620255406633 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9445620255406633 / 5000000000000000 : ℝ) ≤
+            (6361132566021003162573 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4722585488781633 / 10000000000000000 : ℝ)) (q :=
+      (9445620255406633 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_456_3 :
+    Uω (aρ 3) (bρ 3) (240565049861 / 500000000000 : ℝ) ≤
+      (-(3801942864268135953877 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2337423697952089 / 5000000000000000 : ℝ) ≤
+      (-(3801942864268135953877 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2337423697952089 / 5000000000000000 : ℝ) =
+            (2337423697952089 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2337423697952089 / 1250000000000000 : ℝ) ≤
+            (3129528938731864046123 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1168514032883339 / 2500000000000000 : ℝ)) (q :=
+      (2337423697952089 / 5000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_456_4 :
+    Uω (aρ 4) (bρ 4) (240565049861 / 500000000000 : ℝ) ≤
+      (-(3888021894446453529367 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9190109892629631 / 20000000000000000 : ℝ) ≤
+      (-(3888021894446453529367 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9190109892629631 / 20000000000000000 : ℝ) =
+            (9190109892629631 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9190109892629631 / 5000000000000000 : ℝ) ≤
+            (3043449908553546470633 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4592765413419631 / 10000000000000000 : ℝ)) (q :=
+      (9190109892629631 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_456_5 :
+    Uω (aρ 5) (bρ 5) (240565049861 / 500000000000 : ℝ) ≤
+      (-(1608961362230212211269 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8946403356574973 / 20000000000000000 : ℝ) ≤
+      (-(1608961362230212211269 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8946403356574973 / 20000000000000000 : ℝ) =
+            (8946403356574973 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8946403356574973 / 5000000000000000 : ℝ) ≤
+            (1163627358969787788731 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4467248796699973 / 10000000000000000 : ℝ)) (q :=
+      (8946403356574973 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_456_6 :
+    Uω (aρ 6) (bρ 6) (240565049861 / 500000000000 : ℝ) ≤
+      (-(8445199170563307895169 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2148818153291737 / 5000000000000000 : ℝ) ≤
+      (-(8445199170563307895169 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2148818153291737 / 5000000000000000 : ℝ) =
+            (2148818153291737 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2148818153291737 / 1250000000000000 : ℝ) ≤
+            (5417744435436692104831 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1070839607107987 / 2500000000000000 : ℝ)) (q :=
+      (2148818153291737 / 5000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_456_7 :
+    Uω (aρ 7) (bρ 7) (240565049861 / 500000000000 : ℝ) ≤
+      (-(9023178537642414456629 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4056283818069307 / 10000000000000000 : ℝ) ≤
+      (-(9023178537642414456629 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4056283818069307 / 10000000000000000 : ℝ) =
+            (4056283818069307 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4056283818069307 / 2500000000000000 : ℝ) ≤
+            (4839765068357585543371 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2012157416081807 / 5000000000000000 : ℝ)) (q :=
+      (4056283818069307 / 10000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_456_8 :
+    Uω (aρ 8) (bρ 8) (240565049861 / 500000000000 : ℝ) ≤
+      (-(9842437726581167586699 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (934304346261009 / 2500000000000000 : ℝ) ≤
+      (-(9842437726581167586699 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (934304346261009 / 2500000000000000 : ℝ) =
+            (934304346261009 / 625000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (934304346261009 / 625000000000000 : ℝ) ≤
+            (4020505879418832413301 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (229357726090817 / 625000000000000 : ℝ)) (q :=
+      (934304346261009 / 2500000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_456_9 :
+    Uω (aρ 9) (bρ 9) (240565049861 / 500000000000 : ℝ) ≤
+      (-(27507936702691674071 / 25000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3327654243446367 / 10000000000000000 : ℝ) ≤
+      (-(27507936702691674071 / 25000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3327654243446367 / 10000000000000000 : ℝ) =
+            (3327654243446367 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3327654243446367 / 2500000000000000 : ℝ) ≤
+            (7149422312308325929 / 25000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1595781481308867 / 5000000000000000 : ℝ)) (q :=
+      (3327654243446367 / 10000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_456_10 :
+    Uω (aρ 10) (bρ 10) (240565049861 / 500000000000 : ℝ) ≤
+      (-(12704080176489435428207 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1403585303736957 / 5000000000000000 : ℝ) ≤
+      (-(12704080176489435428207 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1403585303736957 / 5000000000000000 : ℝ) =
+            (1403585303736957 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1403585303736957 / 1250000000000000 : ℝ) ≤
+            (1158863429510564571793 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (634788681041957 / 2500000000000000 : ℝ)) (q :=
+      (1403585303736957 / 5000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_456_11 :
+    Uω (aρ 11) (bρ 11) (240565049861 / 500000000000 : ℝ) ≤
+      (-(15576633237813044959681 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4212553302424011 / 20000000000000000 : ℝ) ≤
+      (-(15576633237813044959681 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4212553302424011 / 20000000000000000 : ℝ) =
+            (4212553302424011 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4212553302424011 / 2500000000000000 : ℝ) ≤
+            (5217782171186955040319 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1555843437849011 / 10000000000000000 : ℝ)) (q :=
+      (4212553302424011 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_456_12 :
+    Uω (aρ 12) (bρ 12) (240565049861 / 500000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_456_13 :
+    Uω (aρ 13) (bρ 13) (240565049861 / 500000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_456_14 :
+    Uω (aρ 14) (bρ 14) (240565049861 / 500000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_456_15 :
+    Uω (aρ 15) (bρ 15) (240565049861 / 500000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_456_16 :
+    Uω (aρ 16) (bρ 16) (240565049861 / 500000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_456 :
+    Uρ (240565049861 / 500000000000 : ℝ) ≤
+      (-(5603082807204348035771 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (240565049861 / 500000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(7451207645873873946861 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_456_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (240565049861 / 500000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(7501811039978996837427 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_456_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (240565049861 / 500000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3801942864268135953877 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_456_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (240565049861 / 500000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3888021894446453529367 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_456_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (240565049861 / 500000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1608961362230212211269 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_456_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (240565049861 / 500000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(8445199170563307895169 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_456_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (240565049861 / 500000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(9023178537642414456629 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_456_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (240565049861 / 500000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(9842437726581167586699 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_456_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (240565049861 / 500000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(27507936702691674071 / 25000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_456_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (240565049861 / 500000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12704080176489435428207 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_456_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (240565049861 / 500000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(15576633237813044959681 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_456_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (240565049861 / 500000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_456_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (240565049861 / 500000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_456_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (240565049861 / 500000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_456_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (240565049861 / 500000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_456_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (240565049861 / 500000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_456_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_457_1 :
+    Uω (aρ 1) (bρ 1) (19351147979 / 40000000000 : ℝ) ≤
+      (-(7395564403113882945229 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4773255907145541 / 10000000000000000 : ℝ) ≤
+      (-(7395564403113882945229 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4773255907145541 / 10000000000000000 : ℝ) =
+            (4773255907145541 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4773255907145541 / 2500000000000000 : ℝ) ≤
+            (6467379202886117054771 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2386611018813041 / 5000000000000000 : ℝ)) (q :=
+      (4773255907145541 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_457_2 :
+    Uω (aρ 2) (bρ 2) (19351147979 / 40000000000 : ℝ) ≤
+      (-(1489176811552806506841 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1187324344516099 / 2500000000000000 : ℝ) ≤
+      (-(1489176811552806506841 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1187324344516099 / 2500000000000000 : ℝ) =
+            (1187324344516099 / 625000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1187324344516099 / 625000000000000 : ℝ) ≤
+            (1283411909647193493159 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (148408562249181 / 312500000000000 : ℝ)) (q :=
+      (1187324344516099 / 2500000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_457_3 :
+    Uω (aρ 3) (bρ 3) (19351147979 / 40000000000 : ℝ) ≤
+      (-(60379037993088768907 / 80000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (117533446298563 / 250000000000000 : ℝ) ≤
+      (-(60379037993088768907 / 80000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (117533446298563 / 250000000000000 : ℝ) = (117533446298563 / 62500000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (117533446298563 / 62500000000000 : ℝ) ≤
+            (50524510854911231093 / 80000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (117513776152001 / 250000000000000 : ℝ)) (q :=
+      (117533446298563 / 250000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_457_4 :
+    Uω (aρ 4) (bρ 4) (19351147979 / 40000000000 : ℝ) ≤
+      (-(482408790506011869709 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2310777035798767 / 5000000000000000 : ℝ) ≤
+      (-(482408790506011869709 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2310777035798767 / 5000000000000000 : ℝ) =
+            (2310777035798767 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2310777035798767 / 1250000000000000 : ℝ) ≤
+            (384025184868988130291 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1154819416613767 / 2500000000000000 : ℝ)) (q :=
+      (2310777035798767 / 5000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_457_5 :
+    Uω (aρ 5) (bρ 5) (19351147979 / 40000000000 : ℝ) ≤
+      (-(798569306100694326047 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8999445523455617 / 20000000000000000 : ℝ) ≤
+      (-(798569306100694326047 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8999445523455617 / 20000000000000000 : ℝ) =
+            (8999445523455617 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8999445523455617 / 5000000000000000 : ℝ) ≤
+            (587725054499305673953 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4493804966050617 / 10000000000000000 : ℝ)) (q :=
+      (8999445523455617 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_457_6 :
+    Uω (aρ 6) (bρ 6) (19351147979 / 40000000000 : ℝ) ≤
+      (-(2095889042436629020461 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8648420093051619 / 20000000000000000 : ℝ) ≤
+      (-(2095889042436629020461 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8648420093051619 / 20000000000000000 : ℝ) =
+            (8648420093051619 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8648420093051619 / 5000000000000000 : ℝ) ≤
+            (1369846859063370979539 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4310019910786619 / 10000000000000000 : ℝ)) (q :=
+      (8648420093051619 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_457_7 :
+    Uω (aρ 7) (bρ 7) (19351147979 / 40000000000 : ℝ) ≤
+      (-(4478791321964708754081 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (40829788327367 / 100000000000000 : ℝ) ≤
+      (-(4478791321964708754081 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (40829788327367 / 100000000000000 : ℝ) = (40829788327367 / 25000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (40829788327367 / 25000000000000 : ℝ) ≤
+            (2452680481035291245919 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (10128047159921 / 25000000000000 : ℝ)) (q :=
+      (40829788327367 / 100000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_457_8 :
+    Uω (aρ 8) (bρ 8) (19351147979 / 40000000000 : ℝ) ≤
+      (-(9770532012500916071231 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (470523370080573 / 1250000000000000 : ℝ) ≤
+      (-(9770532012500916071231 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (470523370080573 / 1250000000000000 : ℝ) =
+            (470523370080573 / 312500000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (470523370080573 / 312500000000000 : ℝ) ≤
+            (4092411593499083928769 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (462147096390521 / 1250000000000000 : ℝ)) (q :=
+      (470523370080573 / 1250000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_457_9 :
+    Uω (aρ 9) (bρ 9) (19351147979 / 40000000000 : ℝ) ≤
+      (-(5460279227141323714019 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6710519888274599 / 20000000000000000 : ℝ) ≤
+      (-(5460279227141323714019 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6710519888274599 / 20000000000000000 : ℝ) =
+            (6710519888274599 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6710519888274599 / 5000000000000000 : ℝ) ≤
+            (1471192575858676285981 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3220288366469599 / 10000000000000000 : ℝ)) (q :=
+      (6710519888274599 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_457_10 :
+    Uω (aρ 10) (bρ 10) (19351147979 / 40000000000 : ℝ) ≤
+      (-(1260042240769293479831 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2836420448410709 / 10000000000000000 : ℝ) ≤
+      (-(1260042240769293479831 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2836420448410709 / 10000000000000000 : ℝ) =
+            (2836420448410709 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2836420448410709 / 2500000000000000 : ℝ) ≤
+            (126252119830706520169 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1285584204255709 / 5000000000000000 : ℝ)) (q :=
+      (2836420448410709 / 10000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_457_11 :
+    Uω (aρ 11) (bρ 11) (19351147979 / 40000000000 : ℝ) ≤
+      (-(15408810197614576507277 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1070961552247659 / 5000000000000000 : ℝ) ≤
+      (-(15408810197614576507277 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1070961552247659 / 5000000000000000 : ℝ) =
+            (1070961552247659 / 625000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1070961552247659 / 625000000000000 : ℝ) ≤
+            (5385605211385423492723 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (400162586721409 / 2500000000000000 : ℝ)) (q :=
+      (1070961552247659 / 5000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_457_12 :
+    Uω (aρ 12) (bρ 12) (19351147979 / 40000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_457_13 :
+    Uω (aρ 13) (bρ 13) (19351147979 / 40000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_457_14 :
+    Uω (aρ 14) (bρ 14) (19351147979 / 40000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_457_15 :
+    Uω (aρ 15) (bρ 15) (19351147979 / 40000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_457_16 :
+    Uω (aρ 16) (bρ 16) (19351147979 / 40000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_457 :
+    Uρ (19351147979 / 40000000000 : ℝ) ≤ (-(278697438641564643689 / 250000000000000000000 : ℝ)) :=
+  by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (19351147979 / 40000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(7395564403113882945229 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_457_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (19351147979 / 40000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1489176811552806506841 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_457_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (19351147979 / 40000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) * (-(60379037993088768907 / 80000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_457_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (19351147979 / 40000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(482408790506011869709 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_457_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (19351147979 / 40000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(798569306100694326047 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_457_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (19351147979 / 40000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(2095889042436629020461 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_457_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (19351147979 / 40000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(4478791321964708754081 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_457_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (19351147979 / 40000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(9770532012500916071231 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_457_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (19351147979 / 40000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(5460279227141323714019 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_457_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (19351147979 / 40000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(1260042240769293479831 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_457_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (19351147979 / 40000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(15408810197614576507277 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_457_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (19351147979 / 40000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_457_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (19351147979 / 40000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_457_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (19351147979 / 40000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_457_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (19351147979 / 40000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_457_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (19351147979 / 40000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_457_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_458_1 :
+    Uω (aρ 1) (bρ 1) (121606824807 / 250000000000 : ℝ) ≤
+      (-(1468045813851038955841 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9599484183152141 / 20000000000000000 : ℝ) ≤
+      (-(1468045813851038955841 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9599484183152141 / 20000000000000000 : ℝ) =
+            (9599484183152141 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9599484183152141 / 5000000000000000 : ℝ) ≤
+            (1304542907348961044159 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4799708408957141 / 10000000000000000 : ℝ)) (q :=
+      (9599484183152141 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_458_2 :
+    Uω (aρ 2) (bρ 2) (121606824807 / 250000000000 : ℝ) ≤
+      (-(7390268148614046540739 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9551569229056067 / 20000000000000000 : ℝ) ≤
+      (-(7390268148614046540739 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9551569229056067 / 20000000000000000 : ℝ) =
+            (9551569229056067 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9551569229056067 / 5000000000000000 : ℝ) ≤
+            (6472675457385953459261 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4775562467371067 / 10000000000000000 : ℝ)) (q :=
+      (9551569229056067 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_458_3 :
+    Uω (aρ 3) (bρ 3) (121606824807 / 250000000000 : ℝ) ≤
+      (-(7491191374781335902451 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4727828258009729 / 10000000000000000 : ℝ) ≤
+      (-(7491191374781335902451 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4727828258009729 / 10000000000000000 : ℝ) =
+            (4727828258009729 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4727828258009729 / 2500000000000000 : ℝ) ≤
+            (6371752231218664097549 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2363522930342229 / 5000000000000000 : ℝ)) (q :=
+      (4727828258009729 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_458_4 :
+    Uω (aρ 4) (bρ 4) (121606824807 / 250000000000 : ℝ) ≤
+      (-(7661366600977939292389 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4648053047121447 / 10000000000000000 : ℝ) ≤
+      (-(7661366600977939292389 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4648053047121447 / 10000000000000000 : ℝ) =
+            (4648053047121447 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4648053047121447 / 2500000000000000 : ℝ) ≤
+            (6201577005022060707611 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2322894809986447 / 5000000000000000 : ℝ)) (q :=
+      (4648053047121447 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_458_5 :
+    Uω (aρ 5) (bρ 5) (121606824807 / 250000000000 : ℝ) ≤
+      (-(7926927611440030024571 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9052486866936173 / 20000000000000000 : ℝ) ≤
+      (-(7926927611440030024571 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9052486866936173 / 20000000000000000 : ℝ) =
+            (9052486866936173 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9052486866936173 / 5000000000000000 : ℝ) ≤
+            (5936015994559969975429 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4520360312001173 / 10000000000000000 : ℝ)) (q :=
+      (9052486866936173 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_458_6 :
+    Uω (aρ 6) (bρ 6) (121606824807 / 250000000000 : ℝ) ≤
+      (-(8322293299607536327821 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8701565422310579 / 20000000000000000 : ℝ) ≤
+      (-(8322293299607536327821 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8701565422310579 / 20000000000000000 : ℝ) =
+            (8701565422310579 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8701565422310579 / 5000000000000000 : ℝ) ≤
+            (5540650306392463672179 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4336679242515579 / 10000000000000000 : ℝ)) (q :=
+      (8701565422310579 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_458_7 :
+    Uω (aρ 7) (bρ 7) (121606824807 / 250000000000 : ℝ) ≤
+      (-(4446210444125925813017 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8219342221977243 / 20000000000000000 : ℝ) ≤
+      (-(4446210444125925813017 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8219342221977243 / 20000000000000000 : ℝ) =
+            (8219342221977243 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8219342221977243 / 5000000000000000 : ℝ) ≤
+            (2485261358874074186983 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4078117422942243 / 10000000000000000 : ℝ)) (q :=
+      (8219342221977243 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_458_8 :
+    Uω (aρ 8) (bρ 8) (121606824807 / 250000000000 : ℝ) ≤
+      (-(4849579062306917657777 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (947787383294077 / 2500000000000000 : ℝ) ≤
+      (-(4849579062306917657777 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (947787383294077 / 2500000000000000 : ℝ) =
+            (947787383294077 / 625000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (947787383294077 / 625000000000000 : ℝ) ≤
+            (2081892740693082342223 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (232788494916101 / 625000000000000 : ℝ)) (q :=
+      (947787383294077 / 2500000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_458_9 :
+    Uω (aρ 9) (bρ 9) (121606824807 / 250000000000 : ℝ) ≤
+      (-(10838675429623092885493 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6765693234574483 / 20000000000000000 : ℝ) ≤
+      (-(10838675429623092885493 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6765693234574483 / 20000000000000000 : ℝ) =
+            (6765693234574483 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6765693234574483 / 5000000000000000 : ℝ) ≤
+            (3024268176376907114507 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3248975715239483 / 10000000000000000 : ℝ)) (q :=
+      (6765693234574483 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_458_10 :
+    Uω (aρ 10) (bρ 10) (121606824807 / 250000000000 : ℝ) ≤
+      (-(2499608972782357176879 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1146243271097001 / 4000000000000000 : ℝ) ≤
+      (-(2499608972782357176879 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1146243271097001 / 4000000000000000 : ℝ) =
+            (1146243271097001 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1146243271097001 / 1000000000000000 : ℝ) ≤
+            (272979748417642823121 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (520611573929001 / 2000000000000000 : ℝ)) (q :=
+      (1146243271097001 / 4000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_458_11 :
+    Uω (aρ 11) (bρ 11) (121606824807 / 250000000000 : ℝ) ≤
+      (-(1905697556669964850993 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2177172565929523 / 10000000000000000 : ℝ) ≤
+      (-(1905697556669964850993 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2177172565929523 / 10000000000000000 : ℝ) =
+            (2177172565929523 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2177172565929523 / 1250000000000000 : ℝ) ≤
+            (693604369455035149007 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (822331636112023 / 5000000000000000 : ℝ)) (q :=
+      (2177172565929523 / 10000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_458_12 :
+    Uω (aρ 12) (bρ 12) (121606824807 / 250000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_458_13 :
+    Uω (aρ 13) (bρ 13) (121606824807 / 250000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_458_14 :
+    Uω (aρ 14) (bρ 14) (121606824807 / 250000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_458_15 :
+    Uω (aρ 15) (bρ 15) (121606824807 / 250000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_458_16 :
+    Uω (aρ 16) (bρ 16) (121606824807 / 250000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_458 :
+    Uρ (121606824807 / 250000000000 : ℝ) ≤
+      (-(2772587281831520040409 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (121606824807 / 250000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1468045813851038955841 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_458_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (121606824807 / 250000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(7390268148614046540739 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_458_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (121606824807 / 250000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(7491191374781335902451 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_458_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (121606824807 / 250000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(7661366600977939292389 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_458_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (121606824807 / 250000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(7926927611440030024571 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_458_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (121606824807 / 250000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(8322293299607536327821 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_458_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (121606824807 / 250000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(4446210444125925813017 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_458_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (121606824807 / 250000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(4849579062306917657777 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_458_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (121606824807 / 250000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(10838675429623092885493 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_458_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (121606824807 / 250000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(2499608972782357176879 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_458_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (121606824807 / 250000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(1905697556669964850993 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_458_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (121606824807 / 250000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_458_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (121606824807 / 250000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_458_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (121606824807 / 250000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_458_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (121606824807 / 250000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_458_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (121606824807 / 250000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_458_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_459_1 :
+    Uω (aρ 1) (bρ 1) (489075898981 / 1000000000000 : ℝ) ≤
+      (-(455324890955604488009 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2413114136977589 / 5000000000000000 : ℝ) ≤
+      (-(455324890955604488009 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2413114136977589 / 5000000000000000 : ℝ) =
+            (2413114136977589 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2413114136977589 / 1250000000000000 : ℝ) ≤
+            (411109084419395511991 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1206548694046339 / 2500000000000000 : ℝ)) (q :=
+      (2413114136977589 / 5000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_459_2 :
+    Uω (aρ 2) (bρ 2) (489075898981 / 1000000000000 : ℝ) ≤
+      (-(7334959870886187479617 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4802271837324207 / 10000000000000000 : ℝ) ≤
+      (-(7334959870886187479617 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4802271837324207 / 10000000000000000 : ℝ) =
+            (4802271837324207 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4802271837324207 / 2500000000000000 : ℝ) ≤
+            (6527983735113812520383 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2401025457716707 / 5000000000000000 : ℝ)) (q :=
+      (4802271837324207 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_459_3 :
+    Uω (aρ 3) (bρ 3) (489075898981 / 1000000000000 : ℝ) ≤
+      (-(3717658526951898766839 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1901727445976551 / 4000000000000000 : ℝ) ≤
+      (-(3717658526951898766839 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1901727445976551 / 4000000000000000 : ℝ) =
+            (1901727445976551 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1901727445976551 / 1000000000000000 : ℝ) ≤
+            (3213813276048101233161 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (950708115403551 / 2000000000000000 : ℝ)) (q :=
+      (1901727445976551 / 4000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_459_4 :
+    Uω (aρ 4) (bρ 4) (489075898981 / 1000000000000 : ℝ) ≤
+      (-(1520903579699353452909 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9349103750871751 / 20000000000000000 : ℝ) ≤
+      (-(1520903579699353452909 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9349103750871751 / 20000000000000000 : ℝ) =
+            (9349103750871751 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9349103750871751 / 5000000000000000 : ℝ) ≤
+            (1251685141500646547091 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4672301279071751 / 10000000000000000 : ℝ)) (q :=
+      (9349103750871751 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_459_5 :
+    Uω (aρ 5) (bρ 5) (489075898981 / 1000000000000 : ℝ) ≤
+      (-(1573701274312190521749 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (182110548028867 / 400000000000000 : ℝ) ≤
+      (-(1573701274312190521749 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (182110548028867 / 400000000000000 : ℝ) =
+            (182110548028867 / 100000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (182110548028867 / 100000000000000 : ℝ) ≤
+            (1198887446887809478251 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (90938296979567 / 200000000000000 : ℝ)) (q :=
+      (182110548028867 / 400000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_459_6 :
+    Uω (aρ 6) (bρ 6) (489075898981 / 1000000000000 : ℝ) ≤
+      (-(4130702935387415911649 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8754708640363831 / 20000000000000000 : ℝ) ≤
+      (-(4130702935387415911649 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8754708640363831 / 20000000000000000 : ℝ) =
+            (8754708640363831 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8754708640363831 / 5000000000000000 : ℝ) ≤
+            (2800768867612584088351 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4363336463038831 / 10000000000000000 : ℝ)) (q :=
+      (8754708640363831 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_459_7 :
+    Uω (aρ 7) (bρ 7) (489075898981 / 1000000000000 : ℝ) ≤
+      (-(4413843738031198771433 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2068180353308557 / 5000000000000000 : ℝ) ≤
+      (-(4413843738031198771433 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2068180353308557 / 5000000000000000 : ℝ) =
+            (2068180353308557 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2068180353308557 / 1250000000000000 : ℝ) ≤
+            (2517628064968801228567 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1026252654167307 / 2500000000000000 : ℝ)) (q :=
+      (2068180353308557 / 5000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_459_8 :
+    Uω (aρ 8) (bρ 8) (489075898981 / 1000000000000 : ℝ) ≤
+      (-(1203538498895373722573 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3818105256282167 / 10000000000000000 : ℝ) ≤
+      (-(1203538498895373722573 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3818105256282167 / 10000000000000000 : ℝ) =
+            (3818105256282167 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3818105256282167 / 2500000000000000 : ℝ) ≤
+            (529329451854626277427 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1876020683669667 / 5000000000000000 : ℝ)) (q :=
+      (3818105256282167 / 10000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_459_9 :
+    Uω (aρ 9) (bρ 9) (489075898981 / 1000000000000 : ℝ) ≤
+      (-(5378755886702058761049 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3410414762510657 / 10000000000000000 : ℝ) ≤
+      (-(5378755886702058761049 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3410414762510657 / 10000000000000000 : ℝ) =
+            (3410414762510657 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3410414762510657 / 2500000000000000 : ℝ) ≤
+            (1552715916297941238951 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1638813004078157 / 5000000000000000 : ℝ)) (q :=
+      (3410414762510657 / 10000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_459_10 :
+    Uω (aρ 10) (bρ 10) (489075898981 / 1000000000000 : ℝ) ≤
+      (-(619845579902810326079 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2894736050691567 / 10000000000000000 : ℝ) ≤
+      (-(619845579902810326079 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2894736050691567 / 10000000000000000 : ℝ) =
+            (2894736050691567 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2894736050691567 / 2500000000000000 : ℝ) ≤
+            (73301600397189673921 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1317413809006567 / 5000000000000000 : ℝ)) (q :=
+      (2894736050691567 / 10000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_459_11 :
+    Uω (aρ 11) (bρ 11) (489075898981 / 1000000000000 : ℝ) ≤
+      (-(1885828314317794521397 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (553014022654023 / 2500000000000000 : ℝ) ≤
+      (-(1885828314317794521397 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (553014022654023 / 2500000000000000 : ℝ) =
+            (553014022654023 / 312500000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (553014022654023 / 312500000000000 : ℝ) ≤
+            (713473611807205478603 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (105496520254199 / 625000000000000 : ℝ)) (q :=
+      (553014022654023 / 2500000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11]) (by norm_num [bρ_11])
+      (by norm_num) (by norm_num [aρ_11, bρ_11]) (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_459_12 :
+    Uω (aρ 12) (bρ 12) (489075898981 / 1000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_459_13 :
+    Uω (aρ 13) (bρ 13) (489075898981 / 1000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_459_14 :
+    Uω (aρ 14) (bρ 14) (489075898981 / 1000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_459_15 :
+    Uω (aρ 15) (bρ 15) (489075898981 / 1000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_459_16 :
+    Uω (aρ 16) (bρ 16) (489075898981 / 1000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_459 :
+    Uρ (489075898981 / 1000000000000 : ℝ) ≤
+      (-(5516744325173004896139 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (489075898981 / 1000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(455324890955604488009 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_459_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (489075898981 / 1000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(7334959870886187479617 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_459_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (489075898981 / 1000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3717658526951898766839 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_459_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (489075898981 / 1000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1520903579699353452909 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_459_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (489075898981 / 1000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1573701274312190521749 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_459_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (489075898981 / 1000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(4130702935387415911649 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_459_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (489075898981 / 1000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(4413843738031198771433 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_459_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (489075898981 / 1000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(1203538498895373722573 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_459_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (489075898981 / 1000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(5378755886702058761049 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_459_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (489075898981 / 1000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) * (-(619845579902810326079 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_459_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (489075898981 / 1000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(1885828314317794521397 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_459_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (489075898981 / 1000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_459_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (489075898981 / 1000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_459_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (489075898981 / 1000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_459_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (489075898981 / 1000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_459_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (489075898981 / 1000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_459_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/U38.lean b/LeanPool/Zeta5Irrational/Table/U38.lean
new file mode 100644
index 0000000000..fa0452ccc2
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/U38.lean
@@ -0,0 +1,4467 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-! # Certified arcsine potential bounds (U38) -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma U_460_1 :
+    Uω (aρ 1) (bρ 1) (245862249367 / 500000000000 : ℝ) ≤
+      (-(3615234313927837667861 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9705428908632907 / 20000000000000000 : ℝ) ≤
+      (-(3615234313927837667861 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9705428908632907 / 20000000000000000 : ℝ) =
+            (9705428908632907 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9705428908632907 / 5000000000000000 : ℝ) ≤
+            (3316237489072162332139 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4852681139377907 / 10000000000000000 : ℝ)) (q :=
+      (9705428908632907 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_460_2 :
+    Uω (aρ 2) (bρ 2) (245862249367 / 500000000000 : ℝ) ≤
+      (-(7279955839745068222781 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4828759046677849 / 10000000000000000 : ℝ) ≤
+      (-(7279955839745068222781 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4828759046677849 / 10000000000000000 : ℝ) =
+            (4828759046677849 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4828759046677849 / 2500000000000000 : ℝ) ≤
+            (6582987766254931777219 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2414269668305349 / 5000000000000000 : ℝ)) (q :=
+      (4828759046677849 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_460_3 :
+    Uω (aρ 3) (bρ 3) (245862249367 / 500000000000 : ℝ) ≤
+      (-(1475950658839148611197 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (478080892355451 / 1000000000000000 : ℝ) ≤
+      (-(1475950658839148611197 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (478080892355451 / 1000000000000000 : ℝ) =
+            (478080892355451 / 250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (478080892355451 / 250000000000000 : ℝ) ≤
+            (1296638062360851388803 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (239001759835701 / 500000000000000 : ℝ)) (q :=
+      (478080892355451 / 1000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_460_4 :
+    Uω (aρ 4) (bρ 4) (245862249367 / 500000000000 : ℝ) ≤
+      (-(3773995427686548841721 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9402101118065153 / 20000000000000000 : ℝ) ≤
+      (-(3773995427686548841721 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9402101118065153 / 20000000000000000 : ℝ) =
+            (9402101118065153 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9402101118065153 / 5000000000000000 : ℝ) ≤
+            (3157476375313451158279 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4698812648735153 / 10000000000000000 : ℝ)) (q :=
+      (9402101118065153 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_460_5 :
+    Uω (aρ 5) (bρ 5) (245862249367 / 500000000000 : ℝ) ≤
+      (-(1249668051570219817 / 1600000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4579283570534183 / 10000000000000000 : ℝ) ≤
+      (-(1249668051570219817 / 1600000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4579283570534183 / 10000000000000000 : ℝ) =
+            (4579283570534183 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4579283570534183 / 2500000000000000 : ℝ) ≤
+            (968402925389780183 / 1600000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2286734295536683 / 5000000000000000 : ℝ)) (q :=
+      (4579283570534183 / 10000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_460_6 :
+    Uω (aρ 6) (bρ 6) (245862249367 / 500000000000 : ℝ) ≤
+      (-(512555580040854648039 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4403924892835409 / 10000000000000000 : ℝ) ≤
+      (-(512555580040854648039 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4403924892835409 / 10000000000000000 : ℝ) =
+            (4403924892835409 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4403924892835409 / 2500000000000000 : ℝ) ≤
+            (353878395334145351961 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2194995805407909 / 5000000000000000 : ℝ)) (q :=
+      (4403924892835409 / 10000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_460_7 :
+    Uω (aρ 7) (bρ 7) (245862249367 / 500000000000 : ℝ) ≤
+      (-(4381688364844639488627 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4163047672003167 / 10000000000000000 : ℝ) ≤
+      (-(4381688364844639488627 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4163047672003167 / 10000000000000000 : ℝ) =
+            (4163047672003167 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4163047672003167 / 2500000000000000 : ℝ) ≤
+            (2549783438155360511373 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2065949274955667 / 5000000000000000 : ℝ)) (q :=
+      (4163047672003167 / 10000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_460_8 :
+    Uω (aρ 8) (bρ 8) (245862249367 / 500000000000 : ℝ) ≤
+      (-(9557973728229765291131 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3845054279069901 / 10000000000000000 : ℝ) ≤
+      (-(9557973728229765291131 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3845054279069901 / 10000000000000000 : ℝ) =
+            (3845054279069901 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3845054279069901 / 2500000000000000 : ℝ) ≤
+            (4304969877770234708869 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1889726707692401 / 5000000000000000 : ℝ)) (q :=
+      (3845054279069901 / 10000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_460_9 :
+    Uω (aρ 9) (bρ 9) (245862249367 / 500000000000 : ℝ) ≤
+      (-(1334631757746133575117 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3437964861470009 / 10000000000000000 : ℝ) ≤
+      (-(1334631757746133575117 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3437964861470009 / 10000000000000000 : ℝ) =
+            (3437964861470009 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3437964861470009 / 2500000000000000 : ℝ) ≤
+            (398236193003866424883 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1653120104272509 / 5000000000000000 : ℝ)) (q :=
+      (3437964861470009 / 10000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_460_10 :
+    Uω (aρ 10) (bρ 10) (245862249367 / 500000000000 : ℝ) ≤
+      (-(12296988295501843449263 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1461903103366413 / 5000000000000000 : ℝ) ≤
+      (-(12296988295501843449263 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1461903103366413 / 5000000000000000 : ℝ) =
+            (1461903103366413 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1461903103366413 / 1250000000000000 : ℝ) ≤
+            (1565955310498156550737 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (666620483141413 / 2500000000000000 : ℝ)) (q :=
+      (1461903103366413 / 5000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_460_11 :
+    Uω (aρ 11) (bρ 11) (245862249367 / 500000000000 : ℝ) ≤
+      (-(14931666246709073635179 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4493202270495279 / 20000000000000000 : ℝ) ≤
+      (-(14931666246709073635179 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4493202270495279 / 20000000000000000 : ℝ) =
+            (4493202270495279 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4493202270495279 / 2500000000000000 : ℝ) ≤
+            (5862749162290926364821 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1730548415800279 / 10000000000000000 : ℝ)) (q :=
+      (4493202270495279 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_460_12 :
+    Uω (aρ 12) (bρ 12) (245862249367 / 500000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_460_13 :
+    Uω (aρ 13) (bρ 13) (245862249367 / 500000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_460_14 :
+    Uω (aρ 14) (bρ 14) (245862249367 / 500000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_460_15 :
+    Uω (aρ 15) (bρ 15) (245862249367 / 500000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_460_16 :
+    Uω (aρ 16) (bρ 16) (245862249367 / 500000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_460 :
+    Uρ (245862249367 / 500000000000 : ℝ) ≤
+      (-(1097728741781144235151 / 1000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (245862249367 / 500000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3615234313927837667861 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_460_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (245862249367 / 500000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(7279955839745068222781 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_460_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (245862249367 / 500000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1475950658839148611197 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_460_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (245862249367 / 500000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3773995427686548841721 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_460_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (245862249367 / 500000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(1249668051570219817 / 1600000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_460_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (245862249367 / 500000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(512555580040854648039 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_460_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (245862249367 / 500000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(4381688364844639488627 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_460_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (245862249367 / 500000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(9557973728229765291131 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_460_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (245862249367 / 500000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(1334631757746133575117 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_460_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (245862249367 / 500000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12296988295501843449263 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_460_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (245862249367 / 500000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(14931666246709073635179 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_460_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (245862249367 / 500000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_460_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (245862249367 / 500000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_460_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (245862249367 / 500000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_460_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (245862249367 / 500000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_460_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (245862249367 / 500000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_460_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_461_1 :
+    Uω (aρ 1) (bρ 1) (494373098487 / 1000000000000 : ℝ) ≤
+      (-(1435207381605281826913 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9758401265385517 / 20000000000000000 : ℝ) ≤
+      (-(1435207381605281826913 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9758401265385517 / 20000000000000000 : ℝ) =
+            (9758401265385517 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9758401265385517 / 5000000000000000 : ℝ) ≤
+            (1337381339594718173087 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4879167498600517 / 10000000000000000 : ℝ)) (q :=
+      (9758401265385517 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_461_2 :
+    Uω (aρ 2) (bρ 2) (494373098487 / 1000000000000 : ℝ) ≤
+      (-(3612626362959643324813 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9710492485617961 / 20000000000000000 : ℝ) ≤
+      (-(3612626362959643324813 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9710492485617961 / 20000000000000000 : ℝ) =
+            (9710492485617961 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9710492485617961 / 5000000000000000 : ℝ) ≤
+            (3318845440040356675187 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4855027731342961 / 10000000000000000 : ℝ)) (q :=
+      (9710492485617961 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_461_3 :
+    Uω (aρ 3) (bρ 3) (494373098487 / 1000000000000 : ℝ) ≤
+      (-(732449666129835809591 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (600912398081019 / 1250000000000000 : ℝ) ≤
+      (-(732449666129835809591 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (600912398081019 / 1250000000000000 : ℝ) =
+            (600912398081019 / 312500000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (600912398081019 / 312500000000000 : ℝ) ≤
+            (653844694470164190409 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (600816215171413 / 1250000000000000 : ℝ)) (q :=
+      (600912398081019 / 1250000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_461_4 :
+    Uω (aρ 4) (bρ 4) (494373098487 / 1000000000000 : ℝ) ≤
+      (-(7491781848650053319527 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9455098200694723 / 20000000000000000 : ℝ) ≤
+      (-(7491781848650053319527 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9455098200694723 / 20000000000000000 : ℝ) =
+            (9455098200694723 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9455098200694723 / 5000000000000000 : ℝ) ≤
+            (6371161757349946680473 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4725323733834723 / 10000000000000000 : ℝ)) (q :=
+      (9455098200694723 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_461_5 :
+    Uω (aρ 5) (bρ 5) (494373098487 / 1000000000000 : ℝ) ≤
+      (-(387634025740016904203 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4605803049788327 / 10000000000000000 : ℝ) ≤
+      (-(387634025740016904203 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4605803049788327 / 10000000000000000 : ℝ) =
+            (4605803049788327 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4605803049788327 / 2500000000000000 : ℝ) ≤
+            (305513154559983095797 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2300010776025827 / 5000000000000000 : ℝ)) (q :=
+      (4605803049788327 / 10000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_461_6 :
+    Uω (aρ 6) (bρ 6) (494373098487 / 1000000000000 : ℝ) ≤
+      (-(814073901128552071001 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8860988895759591 / 20000000000000000 : ℝ) ≤
+      (-(814073901128552071001 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8860988895759591 / 20000000000000000 : ℝ) =
+            (8860988895759591 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8860988895759591 / 5000000000000000 : ℝ) ≤
+            (572220459471447928999 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4416644723374591 / 10000000000000000 : ℝ)) (q :=
+      (8860988895759591 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_461_7 :
+    Uω (aρ 7) (bρ 7) (494373098487 / 1000000000000 : ℝ) ≤
+      (-(8699483085173201573619 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (335178564653041 / 800000000000000 : ℝ) ≤
+      (-(8699483085173201573619 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (335178564653041 / 800000000000000 : ℝ) =
+            (335178564653041 / 200000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (335178564653041 / 200000000000000 : ℝ) ≤
+            (5163460520826798426381 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (166351252988041 / 400000000000000 : ℝ)) (q :=
+      (335178564653041 / 800000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_461_8 :
+    Uω (aρ 8) (bρ 8) (494373098487 / 1000000000000 : ℝ) ≤
+      (-(9488147633802443083583 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7743993492561403 / 20000000000000000 : ℝ) ≤
+      (-(9488147633802443083583 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7743993492561403 / 20000000000000000 : ℝ) =
+            (7743993492561403 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7743993492561403 / 5000000000000000 : ℝ) ≤
+            (4374795972197556916417 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3806852352276403 / 10000000000000000 : ℝ)) (q :=
+      (7743993492561403 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_461_9 :
+    Uω (aρ 9) (bρ 9) (494373098487 / 1000000000000 : ℝ) ≤
+      (-(10597289264973965425953 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6930994757399517 / 20000000000000000 : ℝ) ≤
+      (-(10597289264973965425953 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6930994757399517 / 20000000000000000 : ℝ) =
+            (6930994757399517 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6930994757399517 / 5000000000000000 : ℝ) ≤
+            (3265654341026034574047 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3334819245474517 / 10000000000000000 : ℝ)) (q :=
+      (6930994757399517 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_461_10 :
+    Uω (aρ 10) (bρ 10) (494373098487 / 1000000000000 : ℝ) ≤
+      (-(609912108571579113853 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1476410338670919 / 5000000000000000 : ℝ) ≤
+      (-(609912108571579113853 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1476410338670919 / 5000000000000000 : ℝ) =
+            (1476410338670919 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1476410338670919 / 1250000000000000 : ℝ) ≤
+            (83235071728420886147 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (674506219063419 / 2500000000000000 : ℝ)) (q :=
+      (1476410338670919 / 5000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_461_11 :
+    Uω (aρ 11) (bρ 11) (494373098487 / 1000000000000 : ℝ) ≤
+      (-(1847555945715342408171 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4561664214211997 / 20000000000000000 : ℝ) ≤
+      (-(1847555945715342408171 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4561664214211997 / 20000000000000000 : ℝ) =
+            (4561664214211997 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4561664214211997 / 2500000000000000 : ℝ) ≤
+            (751745980409657591829 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1772524361986997 / 10000000000000000 : ℝ)) (q :=
+      (4561664214211997 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_461_12 :
+    Uω (aρ 12) (bρ 12) (494373098487 / 1000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_461_13 :
+    Uω (aρ 13) (bρ 13) (494373098487 / 1000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_461_14 :
+    Uω (aρ 14) (bρ 14) (494373098487 / 1000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_461_15 :
+    Uω (aρ 15) (bρ 15) (494373098487 / 1000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_461_16 :
+    Uω (aρ 16) (bρ 16) (494373098487 / 1000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_461 :
+    Uρ (494373098487 / 1000000000000 : ℝ) ≤
+      (-(682607457176087644829 / 625000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (494373098487 / 1000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1435207381605281826913 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_461_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (494373098487 / 1000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3612626362959643324813 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_461_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (494373098487 / 1000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(732449666129835809591 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_461_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (494373098487 / 1000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(7491781848650053319527 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_461_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (494373098487 / 1000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(387634025740016904203 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_461_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (494373098487 / 1000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(814073901128552071001 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_461_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (494373098487 / 1000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(8699483085173201573619 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_461_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (494373098487 / 1000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(9488147633802443083583 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_461_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (494373098487 / 1000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(10597289264973965425953 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_461_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (494373098487 / 1000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) * (-(609912108571579113853 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_461_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (494373098487 / 1000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(1847555945715342408171 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_461_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (494373098487 / 1000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_461_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (494373098487 / 1000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_461_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (494373098487 / 1000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_461_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (494373098487 / 1000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_461_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (494373098487 / 1000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_461_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_462_1 :
+    Uω (aρ 1) (bρ 1) (1553192807 / 3125000000 : ℝ) ≤
+      (-(3560949935065535102229 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1226421702279061 / 2500000000000000 : ℝ) ≤
+      (-(3560949935065535102229 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1226421702279061 / 2500000000000000 : ℝ) =
+            (1226421702279061 / 625000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1226421702279061 / 625000000000000 : ℝ) ≤
+            (3370521867934464897771 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (306603365869843 / 625000000000000 : ℝ)) (q :=
+      (1226421702279061 / 2500000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_462_2 :
+    Uω (aρ 2) (bρ 2) (1553192807 / 3125000000 : ℝ) ≤
+      (-(3585423627245885930571 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9763466851865697 / 20000000000000000 : ℝ) ≤
+      (-(3585423627245885930571 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9763466851865697 / 20000000000000000 : ℝ) =
+            (9763466851865697 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9763466851865697 / 5000000000000000 : ℝ) ≤
+            (3346048175754114069429 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4881516100060697 / 10000000000000000 : ℝ)) (q :=
+      (9763466851865697 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_462_3 :
+    Uω (aρ 3) (bρ 3) (1553192807 / 3125000000 : ℝ) ≤
+      (-(1817385944381576922393 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9667578798007611 / 20000000000000000 : ℝ) ≤
+      (-(1817385944381576922393 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9667578798007611 / 20000000000000000 : ℝ) =
+            (9667578798007611 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9667578798007611 / 5000000000000000 : ℝ) ≤
+            (1648349957118423077607 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4833024152552611 / 10000000000000000 : ℝ)) (q :=
+      (9667578798007611 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_462_4 :
+    Uω (aρ 4) (bρ 4) (1553192807 / 3125000000 : ℝ) ≤
+      (-(1487177463258952670079 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9508095003523313 / 20000000000000000 : ℝ) ≤
+      (-(1487177463258952670079 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9508095003523313 / 20000000000000000 : ℝ) =
+            (9508095003523313 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9508095003523313 / 5000000000000000 : ℝ) ≤
+            (1285411257941047329921 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4751834539133313 / 10000000000000000 : ℝ)) (q :=
+      (9508095003523313 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_462_5 :
+    Uω (aρ 5) (bρ 5) (1553192807 / 3125000000 : ℝ) ≤
+      (-(7695268068651840452393 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1852928858083447 / 4000000000000000 : ℝ) ≤
+      (-(7695268068651840452393 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1852928858083447 / 4000000000000000 : ℝ) =
+            (1852928858083447 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1852928858083447 / 1000000000000000 : ℝ) ≤
+            (6167675537348159547607 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (925314749072447 / 2000000000000000 : ℝ)) (q :=
+      (1852928858083447 / 4000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_462_6 :
+    Uω (aρ 6) (bρ 6) (1553192807 / 3125000000 : ℝ) ≤
+      (-(8080950627273355186921 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2228531501813731 / 5000000000000000 : ℝ) ≤
+      (-(8080950627273355186921 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2228531501813731 / 5000000000000000 : ℝ) =
+            (2228531501813731 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2228531501813731 / 1250000000000000 : ℝ) ≤
+            (5781992978726644813079 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1110823959334981 / 2500000000000000 : ℝ)) (q :=
+      (2228531501813731 / 5000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_462_7 :
+    Uω (aρ 7) (bρ 7) (1553192807 / 3125000000 : ℝ) ≤
+      (-(8636001089211850716229 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8432827829585187 / 20000000000000000 : ℝ) ≤
+      (-(8636001089211850716229 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8432827829585187 / 20000000000000000 : ℝ) =
+            (8432827829585187 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8432827829585187 / 5000000000000000 : ℝ) ≤
+            (5226942516788149283771 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4185659040430187 / 10000000000000000 : ℝ)) (q :=
+      (8432827829585187 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_462_8 :
+    Uω (aρ 8) (bρ 8) (1553192807 / 3125000000 : ℝ) ≤
+      (-(9418822182081979184731 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (974733199612577 / 2500000000000000 : ℝ) ≤
+      (-(9418822182081979184731 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (974733199612577 / 2500000000000000 : ℝ) =
+            (974733199612577 / 625000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (974733199612577 / 625000000000000 : ℝ) ≤
+            (4444121423918020815269 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (239639903692851 / 625000000000000 : ℝ)) (q :=
+      (974733199612577 / 2500000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_462_9 :
+    Uω (aρ 9) (bρ 9) (1553192807 / 3125000000 : ℝ) ≤
+      (-(10518204729532456955793 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (54578324412253 / 156250000000000 : ℝ) ≤
+      (-(10518204729532456955793 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (54578324412253 / 156250000000000 : ℝ) = (54578324412253 / 39062500000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (54578324412253 / 39062500000000 : ℝ) ≤
+            (3344738876467543044207 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (420420501914173 / 1250000000000000 : ℝ)) (q :=
+      (54578324412253 / 156250000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_462_10 :
+    Uω (aρ 10) (bρ 10) (1553192807 / 3125000000 : ℝ) ≤
+      (-(2420128375275408177111 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2981781393102121 / 10000000000000000 : ℝ) ≤
+      (-(2420128375275408177111 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2981781393102121 / 10000000000000000 : ℝ) =
+            (2981781393102121 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2981781393102121 / 2500000000000000 : ℝ) ≤
+            (352460345924591822889 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1364730155122121 / 5000000000000000 : ℝ)) (q :=
+      (2981781393102121 / 10000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_462_11 :
+    Uω (aρ 11) (bρ 11) (1553192807 / 3125000000 : ℝ) ≤
+      (-(1829093014460264202961 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1157385405195211 / 5000000000000000 : ℝ) ≤
+      (-(1829093014460264202961 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1157385405195211 / 5000000000000000 : ℝ) =
+            (1157385405195211 / 625000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1157385405195211 / 625000000000000 : ℝ) ≤
+            (770208911664735797039 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (453478942756461 / 2500000000000000 : ℝ)) (q :=
+      (1157385405195211 / 5000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_462_12 :
+    Uω (aρ 12) (bρ 12) (1553192807 / 3125000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_462_13 :
+    Uω (aρ 13) (bρ 13) (1553192807 / 3125000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_462_14 :
+    Uω (aρ 14) (bρ 14) (1553192807 / 3125000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_462_15 :
+    Uω (aρ 15) (bρ 15) (1553192807 / 3125000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_462_16 :
+    Uω (aρ 16) (bρ 16) (1553192807 / 3125000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_462 :
+    Uρ (1553192807 / 3125000000 : ℝ) ≤ (-(5433380228379890830107 / 5000000000000000000000 : ℝ)) :=
+  by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1553192807 / 3125000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3560949935065535102229 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_462_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1553192807 / 3125000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3585423627245885930571 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_462_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1553192807 / 3125000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1817385944381576922393 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_462_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1553192807 / 3125000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1487177463258952670079 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_462_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1553192807 / 3125000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(7695268068651840452393 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_462_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1553192807 / 3125000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(8080950627273355186921 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_462_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1553192807 / 3125000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(8636001089211850716229 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_462_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1553192807 / 3125000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(9418822182081979184731 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_462_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1553192807 / 3125000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(10518204729532456955793 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_462_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1553192807 / 3125000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(2420128375275408177111 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_462_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1553192807 / 3125000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(1829093014460264202961 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_462_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1553192807 / 3125000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_462_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1553192807 / 3125000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_462_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1553192807 / 3125000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_462_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1553192807 / 3125000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_462_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1553192807 / 3125000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_462_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_463_1 :
+    Uω (aρ 1) (bρ 1) (499670297993 / 1000000000000 : ℝ) ≤
+      (-(7068054340607733583047 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4932172983618371 / 10000000000000000 : ℝ) ≤
+      (-(7068054340607733583047 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4932172983618371 / 10000000000000000 : ℝ) =
+            (4932172983618371 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4932172983618371 / 2500000000000000 : ℝ) ≤
+            (6794889265392266416953 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2466070102695871 / 5000000000000000 : ℝ)) (q :=
+      (4932172983618371 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_463_2 :
+    Uω (aρ 2) (bρ 2) (499670297993 / 1000000000000 : ℝ) ≤
+      (-(889592025465360876981 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1963288238504021 / 4000000000000000 : ℝ) ≤
+      (-(889592025465360876981 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1963288238504021 / 4000000000000000 : ℝ) =
+            (1963288238504021 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1963288238504021 / 1000000000000000 : ℝ) ≤
+            (843275925284639123019 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (981600888637021 / 2000000000000000 : ℝ)) (q :=
+      (1963288238504021 / 4000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_463_3 :
+    Uω (aρ 3) (bρ 3) (499670297993 / 1000000000000 : ℝ) ≤
+      (-(3607445660313601547497 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1944111826954371 / 4000000000000000 : ℝ) ≤
+      (-(3607445660313601547497 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1944111826954371 / 4000000000000000 : ℝ) =
+            (1944111826954371 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1944111826954371 / 1000000000000000 : ℝ) ≤
+            (3324026142686398452503 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (971903698357371 / 2000000000000000 : ℝ)) (q :=
+      (1944111826954371 / 4000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_463_4 :
+    Uω (aρ 4) (bρ 4) (499670297993 / 1000000000000 : ℝ) ≤
+      (-(59042430046709734459 / 80000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2390272882802007 / 5000000000000000 : ℝ) ≤
+      (-(59042430046709734459 / 80000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2390272882802007 / 5000000000000000 : ℝ) =
+            (2390272882802007 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2390272882802007 / 1250000000000000 : ℝ) ≤
+            (51861118801290265541 / 80000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1194586267322007 / 2500000000000000 : ℝ)) (q :=
+      (2390272882802007 / 5000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_463_5 :
+    Uω (aρ 5) (bρ 5) (499670297993 / 1000000000000 : ℝ) ≤
+      (-(7638184170449427090801 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9317681726731767 / 20000000000000000 : ℝ) ≤
+      (-(7638184170449427090801 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9317681726731767 / 20000000000000000 : ℝ) =
+            (9317681726731767 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9317681726731767 / 5000000000000000 : ℝ) ≤
+            (6224759435550572909199 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4653125184146767 / 10000000000000000 : ℝ)) (q :=
+      (9317681726731767 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_463_6 :
+    Uω (aρ 6) (bρ 6) (499670297993 / 1000000000000 : ℝ) ≤
+      (-(125336246465165303943 / 156250000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1793452231181081 / 4000000000000000 : ℝ) ≤
+      (-(125336246465165303943 / 156250000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1793452231181081 / 4000000000000000 : ℝ) =
+            (1793452231181081 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1793452231181081 / 1000000000000000 : ℝ) ≤
+            (91272247378584696057 / 156250000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (893988997692081 / 2000000000000000 : ℝ)) (q :=
+      (1793452231181081 / 4000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_463_7 :
+    Uω (aρ 7) (bρ 7) (499670297993 / 1000000000000 : ℝ) ≤
+      (-(2143231349051881482137 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (530386661288787 / 1250000000000000 : ℝ) ≤
+      (-(2143231349051881482137 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (530386661288787 / 1250000000000000 : ℝ) =
+            (530386661288787 / 312500000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (530386661288787 / 312500000000000 : ℝ) ≤
+            (1322504552448118517863 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (526566474241949 / 1250000000000000 : ℝ)) (q :=
+      (530386661288787 / 1250000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_463_8 :
+    Uω (aρ 8) (bρ 8) (499670297993 / 1000000000000 : ℝ) ≤
+      (-(9349990018013522406539 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7851725144125843 / 20000000000000000 : ℝ) ≤
+      (-(9349990018013522406539 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7851725144125843 / 20000000000000000 : ℝ) =
+            (7851725144125843 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7851725144125843 / 5000000000000000 : ℝ) ≤
+            (4512953587986477593461 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3861612008780843 / 10000000000000000 : ℝ)) (q :=
+      (7851725144125843 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_463_9 :
+    Uω (aρ 9) (bρ 9) (499670297993 / 1000000000000 : ℝ) ≤
+      (-(10439788165175504274949 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3520511445092123 / 10000000000000000 : ℝ) ≤
+      (-(10439788165175504274949 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3520511445092123 / 10000000000000000 : ℝ) =
+            (3520511445092123 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3520511445092123 / 2500000000000000 : ℝ) ≤
+            (3423155440824495725051 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1695937691599623 / 5000000000000000 : ℝ)) (q :=
+      (3520511445092123 / 10000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_463_10 :
+    Uω (aρ 10) (bρ 10) (499670297993 / 1000000000000 : ℝ) ≤
+      (-(12004157409170647831141 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (150534509511739 / 500000000000000 : ℝ) ≤
+      (-(12004157409170647831141 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (150534509511739 / 500000000000000 : ℝ) =
+            (150534509511739 / 125000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (150534509511739 / 125000000000000 : ℝ) ≤
+            (1858786196829352168859 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (69019797674489 / 250000000000000 : ℝ)) (q :=
+      (150534509511739 / 500000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10]) (by norm_num [bρ_10])
+      (by norm_num) (by norm_num [aρ_10, bρ_10]) (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_463_11 :
+    Uω (aρ 11) (bρ 11) (499670297993 / 1000000000000 : ℝ) ≤
+      (-(7244175858108899826303 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (939374724944627 / 4000000000000000 : ℝ) ≤
+      (-(7244175858108899826303 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (939374724944627 / 4000000000000000 : ℝ) =
+            (939374724944627 / 500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (939374724944627 / 500000000000000 : ℝ) ≤
+            (3153031846391100173697 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (370952355487627 / 2000000000000000 : ℝ)) (q :=
+      (939374724944627 / 4000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11]) (by norm_num [bρ_11])
+      (by norm_num) (by norm_num [aρ_11, bρ_11]) (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_463_12 :
+    Uω (aρ 12) (bρ 12) (499670297993 / 1000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_463_13 :
+    Uω (aρ 13) (bρ 13) (499670297993 / 1000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_463_14 :
+    Uω (aρ 14) (bρ 14) (499670297993 / 1000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_463_15 :
+    Uω (aρ 15) (bρ 15) (499670297993 / 1000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_463_16 :
+    Uω (aρ 16) (bρ 16) (499670297993 / 1000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_463 :
+    Uρ (499670297993 / 1000000000000 : ℝ) ≤
+      (-(10812388899080794907653 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (499670297993 / 1000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(7068054340607733583047 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_463_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (499670297993 / 1000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(889592025465360876981 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_463_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (499670297993 / 1000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3607445660313601547497 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_463_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (499670297993 / 1000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(59042430046709734459 / 80000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_463_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (499670297993 / 1000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(7638184170449427090801 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_463_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (499670297993 / 1000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(125336246465165303943 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_463_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (499670297993 / 1000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(2143231349051881482137 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_463_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (499670297993 / 1000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(9349990018013522406539 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_463_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (499670297993 / 1000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(10439788165175504274949 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_463_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (499670297993 / 1000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(12004157409170647831141 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_463_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (499670297993 / 1000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(7244175858108899826303 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_463_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (499670297993 / 1000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_463_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (499670297993 / 1000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_463_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (499670297993 / 1000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_463_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (499670297993 / 1000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_463_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (499670297993 / 1000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_463_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_464_1 :
+    Uω (aρ 1) (bρ 1) (504967497499 / 1000000000000 : ℝ) ≤
+      (-(3480612683152575500467 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4985145326981051 / 10000000000000000 : ℝ) ≤
+      (-(3480612683152575500467 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4985145326981051 / 10000000000000000 : ℝ) =
+            (4985145326981051 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 15
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4985145326981051 / 2500000000000000 : ℝ) ≤
+            (3450859119847424499533 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2492556448528551 / 5000000000000000 : ℝ)) (q :=
+      (4985145326981051 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_464_2 :
+    Uω (aρ 2) (bρ 2) (504967497499 / 1000000000000 : ℝ) ≤
+      (-(7009384736248986284349 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9922389798688753 / 20000000000000000 : ℝ) ≤
+      (-(7009384736248986284349 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9922389798688753 / 20000000000000000 : ℝ) =
+            (9922389798688753 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9922389798688753 / 5000000000000000 : ℝ) ≤
+            (6853558869751013715651 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4960981054293753 / 10000000000000000 : ℝ)) (q :=
+      (9922389798688753 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_464_3 :
+    Uω (aρ 3) (bρ 3) (504967497499 / 1000000000000 : ℝ) ≤
+      (-(7106474668395118534257 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4913259769204929 / 10000000000000000 : ℝ) ≤
+      (-(7106474668395118534257 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4913259769204929 / 10000000000000000 : ℝ) =
+            (4913259769204929 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4913259769204929 / 2500000000000000 : ℝ) ≤
+            (6756468937604881465743 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2456253450182429 / 5000000000000000 : ℝ)) (q :=
+      (4913259769204929 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_464_4 :
+    Uω (aρ 4) (bρ 4) (504967497499 / 1000000000000 : ℝ) ≤
+      (-(3635027915338646864123 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4833541889631461 / 10000000000000000 : ℝ) ≤
+      (-(3635027915338646864123 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4833541889631461 / 10000000000000000 : ℝ) =
+            (4833541889631461 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4833541889631461 / 2500000000000000 : ℝ) ≤
+            (3296443887661353135877 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2415682661141461 / 5000000000000000 : ℝ)) (q :=
+      (4833541889631461 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_464_5 :
+    Uω (aρ 5) (bρ 5) (504967497499 / 1000000000000 : ℝ) ≤
+      (-(300999483591545481661 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4711877193432331 / 10000000000000000 : ℝ) ≤
+      (-(300999483591545481661 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4711877193432331 / 10000000000000000 : ℝ) =
+            (4711877193432331 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4711877193432331 / 2500000000000000 : ℝ) ≤
+            (253518260648454518339 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2353112924609831 / 5000000000000000 : ℝ)) (q :=
+      (4711877193432331 / 10000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_464_6 :
+    Uω (aρ 6) (bρ 6) (504967497499 / 1000000000000 : ℝ) ≤
+      (-(987964203752296207611 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9073525703441053 / 20000000000000000 : ℝ) ≤
+      (-(987964203752296207611 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9073525703441053 / 20000000000000000 : ℝ) =
+            (9073525703441053 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9073525703441053 / 5000000000000000 : ℝ) ≤
+            (744903746997703792389 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4523237540936053 / 10000000000000000 : ℝ)) (q :=
+      (9073525703441053 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_464_7 :
+    Uω (aρ 7) (bρ 7) (504967497499 / 1000000000000 : ℝ) ≤
+      (-(168959441163534356641 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4296444785540659 / 10000000000000000 : ℝ) ≤
+      (-(168959441163534356641 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4296444785540659 / 10000000000000000 : ℝ) =
+            (4296444785540659 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4296444785540659 / 2500000000000000 : ℝ) ≤
+            (108299430956465643359 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2133131394668159 / 5000000000000000 : ℝ)) (q :=
+      (4296444785540659 / 10000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_464_8 :
+    Uω (aρ 8) (bρ 8) (504967497499 / 1000000000000 : ℝ) ≤
+      (-(9213776955017311777019 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3979703810172493 / 10000000000000000 : ℝ) ≤
+      (-(9213776955017311777019 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3979703810172493 / 10000000000000000 : ℝ) =
+            (3979703810172493 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3979703810172493 / 2500000000000000 : ℝ) ≤
+            (4649166650982688222981 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1958161244969993 / 5000000000000000 : ℝ)) (q :=
+      (3979703810172493 / 10000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_464_9 :
+    Uω (aρ 9) (bρ 9) (504967497499 / 1000000000000 : ℝ) ≤
+      (-(5142455757492057291689 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3575460363905141 / 10000000000000000 : ℝ) ≤
+      (-(5142455757492057291689 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3575460363905141 / 10000000000000000 : ℝ) =
+            (3575460363905141 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3575460363905141 / 2500000000000000 : ℝ) ≤
+            (1789016045507942708311 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1724400612882641 / 5000000000000000 : ℝ)) (q :=
+      (3575460363905141 / 10000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_464_10 :
+    Uω (aρ 10) (bρ 10) (504967497499 / 1000000000000 : ℝ) ≤
+      (-(11814422219193978853499 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6136717874527443 / 20000000000000000 : ℝ) ≤
+      (-(11814422219193978853499 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6136717874527443 / 20000000000000000 : ℝ) =
+            (6136717874527443 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6136717874527443 / 5000000000000000 : ℝ) ≤
+            (2048521386806021146501 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2823157405977443 / 10000000000000000 : ℝ)) (q :=
+      (6136717874527443 / 20000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_464_11 :
+    Uω (aρ 11) (bρ 11) (504967497499 / 1000000000000 : ℝ) ≤
+      (-(7104388521796205094697 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1207509781695767 / 5000000000000000 : ℝ) ≤
+      (-(7104388521796205094697 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1207509781695767 / 5000000000000000 : ℝ) =
+            (1207509781695767 / 625000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1207509781695767 / 625000000000000 : ℝ) ≤
+            (3292819182703794905303 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (483738821109517 / 2500000000000000 : ℝ)) (q :=
+      (1207509781695767 / 5000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_464_12 :
+    Uω (aρ 12) (bρ 12) (504967497499 / 1000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_464_13 :
+    Uω (aρ 13) (bρ 13) (504967497499 / 1000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_464_14 :
+    Uω (aρ 14) (bρ 14) (504967497499 / 1000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_464_15 :
+    Uω (aρ 15) (bρ 15) (504967497499 / 1000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_464_16 :
+    Uω (aρ 16) (bρ 16) (504967497499 / 1000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_464 :
+    Uρ (504967497499 / 1000000000000 : ℝ) ≤
+      (-(10705328196259241107531 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (504967497499 / 1000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3480612683152575500467 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_464_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (504967497499 / 1000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(7009384736248986284349 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_464_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (504967497499 / 1000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(7106474668395118534257 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_464_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (504967497499 / 1000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3635027915338646864123 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_464_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (504967497499 / 1000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(300999483591545481661 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_464_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (504967497499 / 1000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(987964203752296207611 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_464_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (504967497499 / 1000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(168959441163534356641 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_464_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (504967497499 / 1000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(9213776955017311777019 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_464_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (504967497499 / 1000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(5142455757492057291689 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_464_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (504967497499 / 1000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(11814422219193978853499 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_464_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (504967497499 / 1000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(7104388521796205094697 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_464_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (504967497499 / 1000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_464_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (504967497499 / 1000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_464_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (504967497499 / 1000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_464_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (504967497499 / 1000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_464_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (504967497499 / 1000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_464_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_465_1 :
+    Uω (aρ 1) (bρ 1) (102052939401 / 200000000000 : ℝ) ≤
+      (-(685552559971335199247 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2015247065207743 / 4000000000000000 : ℝ) ≤
+      (-(685552559971335199247 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2015247065207743 / 4000000000000000 : ℝ) =
+            (2015247065207743 / 2000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2015247065207743 / 2000000000000000 : ℝ) ≤
+            (7594620328664800753 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1007617114814743 / 2000000000000000 : ℝ)) (q :=
+      (2015247065207743 / 4000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_465_2 :
+    Uω (aρ 2) (bρ 2) (102052939401 / 200000000000 : ℝ) ≤
+      (-(6903173572969198641041 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1253542288414613 / 2500000000000000 : ℝ) ≤
+      (-(6903173572969198641041 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1253542288414613 / 2500000000000000 : ℝ) =
+            (1253542288414613 / 1250000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1253542288414613 / 1250000000000000 : ℝ) ≤
+            (28298230030801358959 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (313372347991369 / 625000000000000 : ℝ)) (q :=
+      (1253542288414613 / 2500000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_465_3 :
+    Uω (aρ 3) (bρ 3) (102052939401 / 200000000000 : ℝ) ≤
+      (-(3499610595096664125941 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2483119897948543 / 5000000000000000 : ℝ) ≤
+      (-(3499610595096664125941 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2483119897948543 / 5000000000000000 : ℝ) =
+            (2483119897948543 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2483119897948543 / 1250000000000000 : ℝ) ≤
+            (3431861207903335874059 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1241373739672293 / 2500000000000000 : ℝ)) (q :=
+      (2483119897948543 / 5000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_465_4 :
+    Uω (aρ 4) (bρ 4) (102052939401 / 200000000000 : ℝ) ≤
+      (-(7161011195376077060239 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1954614996021967 / 4000000000000000 : ℝ) ≤
+      (-(7161011195376077060239 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1954614996021967 / 4000000000000000 : ℝ) =
+            (1954614996021967 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1954614996021967 / 1000000000000000 : ℝ) ≤
+            (6701932410623922939761 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (976876905613967 / 2000000000000000 : ℝ)) (q :=
+      (1954614996021967 / 4000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_465_5 :
+    Uω (aρ 5) (bρ 5) (102052939401 / 200000000000 : ℝ) ≤
+      (-(3706530023605786136097 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9529824179286207 / 20000000000000000 : ℝ) ≤
+      (-(3706530023605786136097 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9529824179286207 / 20000000000000000 : ℝ) =
+            (9529824179286207 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9529824179286207 / 5000000000000000 : ℝ) ≤
+            (3224941779394213863903 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4759323646581207 / 10000000000000000 : ℝ)) (q :=
+      (9529824179286207 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_465_6 :
+    Uω (aρ 6) (bρ 6) (102052939401 / 200000000000 : ℝ) ≤
+      (-(311491491184998294687 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9179782807801253 / 20000000000000000 : ℝ) ≤
+      (-(311491491184998294687 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9179782807801253 / 20000000000000000 : ℝ) =
+            (9179782807801253 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9179782807801253 / 5000000000000000 : ℝ) ≤
+            (243026253055001705313 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4576522650236253 / 10000000000000000 : ℝ)) (q :=
+      (9179782807801253 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_465_7 :
+    Uω (aρ 7) (bρ 7) (102052939401 / 200000000000 : ℝ) ≤
+      (-(66596658834443514293 / 80000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8699573814343349 / 20000000000000000 : ℝ) ≤
+      (-(66596658834443514293 / 80000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8699573814343349 / 20000000000000000 : ℝ) =
+            (8699573814343349 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8699573814343349 / 5000000000000000 : ℝ) ≤
+            (44306890013556485707 / 80000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4319975037538349 / 10000000000000000 : ℝ)) (q :=
+      (8699573814343349 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_465_8 :
+    Uω (aρ 8) (bρ 8) (102052939401 / 200000000000 : ℝ) ≤
+      (-(4539726412331190736867 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2016760738441907 / 5000000000000000 : ℝ) ≤
+      (-(4539726412331190736867 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2016760738441907 / 5000000000000000 : ℝ) =
+            (2016760738441907 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2016760738441907 / 1250000000000000 : ℝ) ≤
+            (2391745390668809263133 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (992746457075657 / 2500000000000000 : ℝ)) (q :=
+      (2016760738441907 / 5000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_465_9 :
+    Uω (aρ 9) (bρ 9) (102052939401 / 200000000000 : ℝ) ≤
+      (-(5066283857032743146991 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3630347308844513 / 10000000000000000 : ℝ) ≤
+      (-(5066283857032743146991 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3630347308844513 / 10000000000000000 : ℝ) =
+            (3630347308844513 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3630347308844513 / 2500000000000000 : ℝ) ≤
+            (1865187945967256853009 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1752801560292013 / 5000000000000000 : ℝ)) (q :=
+      (3630347308844513 / 10000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_465_10 :
+    Uω (aρ 10) (bρ 10) (102052939401 / 200000000000 : ℝ) ≤
+      (-(11628820653098059451003 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6251679875568011 / 20000000000000000 : ℝ) ≤
+      (-(11628820653098059451003 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6251679875568011 / 20000000000000000 : ℝ) =
+            (6251679875568011 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6251679875568011 / 5000000000000000 : ℝ) ≤
+            (2234122952901940548997 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2885147411958011 / 10000000000000000 : ℝ)) (q :=
+      (6251679875568011 / 20000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_465_11 :
+    Uω (aρ 11) (bρ 11) (102052939401 / 200000000000 : ℝ) ≤
+      (-(871277070376627645763 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2480702483671911 / 10000000000000000 : ℝ) ≤
+      (-(871277070376627645763 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2480702483671911 / 10000000000000000 : ℝ) =
+            (2480702483671911 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2480702483671911 / 1250000000000000 : ℝ) ≤
+            (428373892685872354237 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1006674564969411 / 5000000000000000 : ℝ)) (q :=
+      (2480702483671911 / 10000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_465_12 :
+    Uω (aρ 12) (bρ 12) (102052939401 / 200000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_465_13 :
+    Uω (aρ 13) (bρ 13) (102052939401 / 200000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_465_14 :
+    Uω (aρ 14) (bρ 14) (102052939401 / 200000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_465_15 :
+    Uω (aρ 15) (bρ 15) (102052939401 / 200000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_465_16 :
+    Uω (aρ 16) (bρ 16) (102052939401 / 200000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_465 :
+    Uρ (102052939401 / 200000000000 : ℝ) ≤ (-(80874574167903463 / 76293945312500000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (102052939401 / 200000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(685552559971335199247 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_465_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (102052939401 / 200000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(6903173572969198641041 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_465_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (102052939401 / 200000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3499610595096664125941 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_465_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (102052939401 / 200000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(7161011195376077060239 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_465_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (102052939401 / 200000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(3706530023605786136097 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_465_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (102052939401 / 200000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(311491491184998294687 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_465_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (102052939401 / 200000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) * (-(66596658834443514293 / 80000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_465_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (102052939401 / 200000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(4539726412331190736867 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_465_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (102052939401 / 200000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(5066283857032743146991 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_465_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (102052939401 / 200000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(11628820653098059451003 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_465_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (102052939401 / 200000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) * (-(871277070376627645763 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_465_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (102052939401 / 200000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_465_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (102052939401 / 200000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_465_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (102052939401 / 200000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_465_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (102052939401 / 200000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_465_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (102052939401 / 200000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_465_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_466_1 :
+    Uω (aρ 1) (bρ 1) (515561896511 / 1000000000000 : ℝ) ≤
+      (-(843866426616583228107 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2545544995980961 / 5000000000000000 : ℝ) ≤
+      (-(843866426616583228107 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2545544995980961 / 5000000000000000 : ℝ) =
+            (2545544995980961 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2545544995980961 / 2500000000000000 : ℝ) ≤
+            (22567548758416771893 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1272764559224711 / 2500000000000000 : ℝ)) (q :=
+      (2545544995980961 / 5000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_466_2 :
+    Uω (aρ 2) (bρ 2) (515561896511 / 1000000000000 : ℝ) ≤
+      (-(6798078736206332217349 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2026857344292803 / 4000000000000000 : ℝ) ≤
+      (-(6798078736206332217349 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2026857344292803 / 4000000000000000 : ℝ) =
+            (2026857344292803 / 2000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2026857344292803 / 2000000000000000 : ℝ) ≤
+            (133393066793667782651 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1013386797389803 / 2000000000000000 : ℝ)) (q :=
+      (2026857344292803 / 4000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_466_3 :
+    Uω (aρ 3) (bρ 3) (515561896511 / 1000000000000 : ℝ) ≤
+      (-(430819136700830867653 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5019219653009671 / 10000000000000000 : ℝ) ≤
+      (-(430819136700830867653 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5019219653009671 / 10000000000000000 : ℝ) =
+            (5019219653009671 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5019219653009671 / 5000000000000000 : ℝ) ≤
+            (2397850986669132347 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2509241338927171 / 5000000000000000 : ℝ)) (q :=
+      (5019219653009671 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_466_4 :
+    Uω (aρ 4) (bρ 4) (515561896511 / 1000000000000 : ℝ) ≤
+      (-(3526571922133105569901 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9879065167484401 / 20000000000000000 : ℝ) ≤
+      (-(3526571922133105569901 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9879065167484401 / 20000000000000000 : ℝ) =
+            (9879065167484401 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9879065167484401 / 5000000000000000 : ℝ) ≤
+            (3404899880866894430099 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4937402720384401 / 10000000000000000 : ℝ)) (q :=
+      (9879065167484401 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_466_5 :
+    Uω (aρ 5) (bρ 5) (515561896511 / 1000000000000 : ℝ) ≤
+      (-(3651187397247562346277 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4817945599459521 / 10000000000000000 : ℝ) ≤
+      (-(3651187397247562346277 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4817945599459521 / 10000000000000000 : ℝ) =
+            (4817945599459521 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4817945599459521 / 2500000000000000 : ℝ) ≤
+            (3280284405752437653723 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2406209335577021 / 5000000000000000 : ℝ)) (q :=
+      (4817945599459521 / 10000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_466_6 :
+    Uω (aρ 6) (bρ 6) (515561896511 / 1000000000000 : ℝ) ≤
+      (-(7672208592147029240361 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9286032725980071 / 20000000000000000 : ℝ) ≤
+      (-(7672208592147029240361 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9286032725980071 / 20000000000000000 : ℝ) =
+            (9286032725980071 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9286032725980071 / 5000000000000000 : ℝ) ≤
+            (6190735013852970759639 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4629800573355071 / 10000000000000000 : ℝ)) (q :=
+      (9286032725980071 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_466_7 :
+    Uω (aρ 7) (bρ 7) (515561896511 / 1000000000000 : ℝ) ≤
+      (-(4101358559010733619021 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1100780000137463 / 2500000000000000 : ℝ) ≤
+      (-(4101358559010733619021 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1100780000137463 / 2500000000000000 : ℝ) =
+            (1100780000137463 / 625000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1100780000137463 / 625000000000000 : ℝ) ≤
+            (2830113243989266380979 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (273354326827169 / 625000000000000 : ℝ)) (q :=
+      (1100780000137463 / 2500000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_466_8 :
+    Uω (aρ 8) (bρ 8) (515561896511 / 1000000000000 : ℝ) ≤
+      (-(1789392884586392307439 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (817463306483847 / 2000000000000000 : ℝ) ≤
+      (-(1789392884586392307439 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (817463306483847 / 2000000000000000 : ℝ) =
+            (817463306483847 / 500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (817463306483847 / 500000000000000 : ℝ) ≤
+            (983195836613607692561 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (402560394431347 / 1000000000000000 : ℝ)) (q :=
+      (817463306483847 / 2000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_466_9 :
+    Uω (aρ 9) (bρ 9) (515561896511 / 1000000000000 : ℝ) ≤
+      (-(9982670117907299539191 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1842587622248539 / 5000000000000000 : ℝ) ≤
+      (-(9982670117907299539191 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1842587622248539 / 5000000000000000 : ℝ) =
+            (1842587622248539 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1842587622248539 / 1250000000000000 : ℝ) ≤
+            (3880273488092700460809 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (890571749207289 / 2500000000000000 : ℝ)) (q :=
+      (1842587622248539 / 5000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_466_10 :
+    Uω (aρ 10) (bρ 10) (515561896511 / 1000000000000 : ℝ) ≤
+      (-(715447111252919705633 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (63662900808441 / 200000000000000 : ℝ) ≤
+      (-(715447111252919705633 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (63662900808441 / 200000000000000 : ℝ) = (63662900808441 / 50000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (63662900808441 / 50000000000000 : ℝ) ≤
+            (150986864122080294367 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (29467856221741 / 100000000000000 : ℝ)) (q :=
+      (63662900808441 / 200000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10]) (by norm_num [bρ_10])
+      (by norm_num) (by norm_num [aρ_10, bρ_10]) (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_466_11 :
+    Uω (aρ 11) (bρ 11) (515561896511 / 1000000000000 : ℝ) ≤
+      (-(13682238889552235011001 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2545586826344193 / 10000000000000000 : ℝ) ≤
+      (-(13682238889552235011001 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2545586826344193 / 10000000000000000 : ℝ) =
+            (2545586826344193 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2545586826344193 / 2500000000000000 : ℝ) ≤
+            (180704716447764988999 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1045072910111693 / 5000000000000000 : ℝ)) (q :=
+      (2545586826344193 / 10000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_466_12 :
+    Uω (aρ 12) (bρ 12) (515561896511 / 1000000000000 : ℝ) ≤
+      (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_12]) (by norm_num [bρ_12])]; exact Lmid_12
+
+lemma U_466_13 :
+    Uω (aρ 13) (bρ 13) (515561896511 / 1000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_466_14 :
+    Uω (aρ 14) (bρ 14) (515561896511 / 1000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_466_15 :
+    Uω (aρ 15) (bρ 15) (515561896511 / 1000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_466_16 :
+    Uω (aρ 16) (bρ 16) (515561896511 / 1000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_466 :
+    Uρ (515561896511 / 1000000000000 : ℝ) ≤
+      (-(10497454876210826800467 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (515561896511 / 1000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(843866426616583228107 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_466_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (515561896511 / 1000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(6798078736206332217349 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_466_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (515561896511 / 1000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(430819136700830867653 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_466_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (515561896511 / 1000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3526571922133105569901 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_466_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (515561896511 / 1000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(3651187397247562346277 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_466_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (515561896511 / 1000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(7672208592147029240361 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_466_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (515561896511 / 1000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(4101358559010733619021 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_466_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (515561896511 / 1000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(1789392884586392307439 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_466_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (515561896511 / 1000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(9982670117907299539191 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_466_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (515561896511 / 1000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) * (-(715447111252919705633 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_466_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (515561896511 / 1000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(13682238889552235011001 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_466_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (515561896511 / 1000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(640305991230515290559 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_466_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (515561896511 / 1000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_466_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (515561896511 / 1000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_466_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (515561896511 / 1000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_466_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (515561896511 / 1000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_466_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_467_1 :
+    Uω (aρ 1) (bρ 1) (16577867570387 / 32000000000000 : ℝ) ≤
+      (-(1340402996119632497453 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20464219190211991 / 40000000000000000 : ℝ) ≤
+      (-(1340402996119632497453 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (20464219190211991 / 40000000000000000 : ℝ) =
+            (20464219190211991 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20464219190211991 / 20000000000000000 : ℝ) ≤
+            (45891364480367502547 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2558011598722529 / 5000000000000000 : ℝ)) (q :=
+      (20464219190211991 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_467_2 :
+    Uω (aρ 2) (bρ 2) (16577867570387 / 32000000000000 : ℝ) ≤
+      (-(107982879792002849 / 160000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20368436151556887 / 40000000000000000 : ℝ) ≤
+      (-(107982879792002849 / 160000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (20368436151556887 / 40000000000000000 : ℝ) =
+            (20368436151556887 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20368436151556887 / 20000000000000000 : ℝ) ≤
+            (2920669055997151 / 160000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2545950345313753 / 5000000000000000 : ℝ)) (q :=
+      (20368436151556887 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_467_3 :
+    Uω (aρ 3) (bρ 3) (16577867570387 / 32000000000000 : ℝ) ≤
+      (-(1710871032282440537927 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20176751806438181 / 40000000000000000 : ℝ) ≤
+      (-(1710871032282440537927 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (20176751806438181 / 40000000000000000 : ℝ) =
+            (20176751806438181 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20176751806438181 / 20000000000000000 : ℝ) ≤
+            (21996918467559462073 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5043454624418153 / 10000000000000000 : ℝ)) (q :=
+      (20176751806438181 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_467_4 :
+    Uω (aρ 4) (bρ 4) (16577867570387 / 32000000000000 : ℝ) ≤
+      (-(3501354518816356370809 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19858031797039747 / 40000000000000000 : ℝ) ≤
+      (-(3501354518816356370809 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (19858031797039747 / 40000000000000000 : ℝ) =
+            (19858031797039747 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19858031797039747 / 10000000000000000 : ℝ) ≤
+            (3430117284183643629191 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (77537325012249 / 156250000000000 : ℝ)) (q :=
+      (19858031797039747 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_467_5 :
+    Uω (aρ 5) (bρ 5) (16577867570387 / 32000000000000 : ℝ) ≤
+      (-(7250633714693489879047 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19371755092659821 / 40000000000000000 : ℝ) ≤
+      (-(7250633714693489879047 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (19371755092659821 / 40000000000000000 : ℝ) =
+            (19371755092659821 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19371755092659821 / 10000000000000000 : ℝ) ≤
+            (6612309891306510120953 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4837440367928973 / 10000000000000000 : ℝ)) (q :=
+      (19371755092659821 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_467_6 :
+    Uω (aρ 6) (bρ 6) (16577867570387 / 32000000000000 : ℝ) ≤
+      (-(380921630778540381131 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (18672207568306617 / 40000000000000000 : ℝ) ≤
+      (-(380921630778540381131 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (18672207568306617 / 40000000000000000 : ℝ) =
+            (18672207568306617 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18672207568306617 / 10000000000000000 : ℝ) ≤
+            (312225549521459618869 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4654906980892371 / 10000000000000000 : ℝ)) (q :=
+      (18672207568306617 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_467_7 :
+    Uω (aρ 7) (bρ 7) (16577867570387 / 32000000000000 : ℝ) ≤
+      (-(1629160430370876043877 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17713007176213467 / 40000000000000000 : ℝ) ≤
+      (-(1629160430370876043877 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (17713007176213467 / 40000000000000000 : ℝ) =
+            (17713007176213467 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17713007176213467 / 10000000000000000 : ℝ) ≤
+            (1143428290829123956123 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1099742041401449 / 2500000000000000 : ℝ)) (q :=
+      (17713007176213467 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_467_8 :
+    Uω (aρ 8) (bρ 8) (16577867570387 / 32000000000000 : ℝ) ≤
+      (-(4442573536579649517609 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3290129197766279 / 8000000000000000 : ℝ) ≤
+      (-(4442573536579649517609 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3290129197766279 / 8000000000000000 : ℝ) =
+            (3290129197766279 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3290129197766279 / 2000000000000000 : ℝ) ≤
+            (2488898266420350482391 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (101283230581369 / 250000000000000 : ℝ)) (q :=
+      (3290129197766279 / 8000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_467_9 :
+    Uω (aρ 9) (bρ 9) (16577867570387 / 32000000000000 : ℝ) ≤
+      (-(4956425418371077082301 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14843979615067633 / 40000000000000000 : ℝ) ≤
+      (-(4956425418371077082301 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (14843979615067633 / 40000000000000000 : ℝ) =
+            (14843979615067633 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14843979615067633 / 10000000000000000 : ℝ) ≤
+            (1975046384628922917699 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3588961664732879 / 10000000000000000 : ℝ)) (q :=
+      (14843979615067633 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_467_10 :
+    Uω (aρ 10) (bρ 10) (16577867570387 / 32000000000000 : ℝ) ≤
+      (-(5681424454436483179861 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12840375760971143 / 40000000000000000 : ℝ) ≤
+      (-(5681424454436483179861 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (12840375760971143 / 40000000000000000 : ℝ) =
+            (12840375760971143 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12840375760971143 / 10000000000000000 : ℝ) ≤
+            (1250047348563516820139 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1487859385589817 / 5000000000000000 : ℝ)) (q :=
+      (12840375760971143 / 40000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_467_11 :
+    Uω (aρ 11) (bρ 11) (16577867570387 / 32000000000000 : ℝ) ≤
+      (-(13563808816514087504783 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10303653818006349 / 40000000000000000 : ℝ) ≤
+      (-(13563808816514087504783 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (10303653818006349 / 40000000000000000 : ℝ) =
+            (10303653818006349 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10303653818006349 / 10000000000000000 : ℝ) ≤
+            (299134789485912495217 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2125834425902237 / 10000000000000000 : ℝ)) (q :=
+      (10303653818006349 / 40000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_467_12 :
+    Uω (aρ 12) (bρ 12) (16577867570387 / 32000000000000 : ℝ) ≤
+      (-(2387382650451925792207 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5923771554627799 / 40000000000000000 : ℝ) ≤
+      (-(2387382650451925792207 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (5923771554627799 / 40000000000000000 : ℝ) =
+            (5923771554627799 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5923771554627799 / 5000000000000000 : ℝ) ≤
+            (211919275673074207793 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (179796695448981 / 5000000000000000 : ℝ)) (q :=
+      (5923771554627799 / 40000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_467_13 :
+    Uω (aρ 13) (bρ 13) (16577867570387 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_467_14 :
+    Uω (aρ 14) (bρ 14) (16577867570387 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_467_15 :
+    Uω (aρ 15) (bρ 15) (16577867570387 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_467_16 :
+    Uω (aρ 16) (bρ 16) (16577867570387 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_467 :
+    Uρ (16577867570387 / 32000000000000 : ℝ) ≤
+      (-(10351760659059680238169 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (16577867570387 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1340402996119632497453 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_467_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (16577867570387 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) * (-(107982879792002849 / 160000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_467_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (16577867570387 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1710871032282440537927 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_467_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (16577867570387 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3501354518816356370809 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_467_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (16577867570387 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(7250633714693489879047 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_467_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (16577867570387 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(380921630778540381131 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_467_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (16577867570387 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(1629160430370876043877 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_467_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (16577867570387 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(4442573536579649517609 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_467_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (16577867570387 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(4956425418371077082301 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_467_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (16577867570387 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(5681424454436483179861 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_467_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (16577867570387 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(13563808816514087504783 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_467_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (16577867570387 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(2387382650451925792207 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_467_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (16577867570387 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_467_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (16577867570387 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_467_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (16577867570387 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_467_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (16577867570387 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_467_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_468_1 :
+    Uω (aρ 1) (bρ 1) (8328877226211 / 16000000000000 : ℝ) ≤
+      (-(3326668334306610783819 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10282039203278277 / 20000000000000000 : ℝ) ≤
+      (-(3326668334306610783819 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10282039203278277 / 20000000000000000 : ℝ) =
+            (10282039203278277 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10282039203278277 / 10000000000000000 : ℝ) ≤
+            (139067567193389216181 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2570494077490701 / 5000000000000000 : ℝ)) (q :=
+      (10282039203278277 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_468_2 :
+    Uω (aρ 2) (bρ 2) (8328877226211 / 16000000000000 : ℝ) ≤
+      (-(6700021636373269278519 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10234149410058897 / 20000000000000000 : ℝ) ≤
+      (-(6700021636373269278519 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10234149410058897 / 20000000000000000 : ℝ) =
+            (10234149410058897 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10234149410058897 / 10000000000000000 : ℝ) ≤
+            (231450166626730721481 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2558433687136011 / 5000000000000000 : ℝ)) (q :=
+      (10234149410058897 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_468_3 :
+    Uω (aρ 3) (bρ 3) (8328877226211 / 16000000000000 : ℝ) ≤
+      (-(6794107161476389452203 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10138312428535819 / 20000000000000000 : ℝ) ≤
+      (-(6794107161476389452203 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10138312428535819 / 20000000000000000 : ℝ) =
+            (10138312428535819 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10138312428535819 / 10000000000000000 : ℝ) ≤
+            (137364641523610547797 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (79194164048421 / 156250000000000 : ℝ)) (q :=
+      (10138312428535819 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_468_4 :
+    Uω (aρ 4) (bρ 4) (8328877226211 / 16000000000000 : ℝ) ≤
+      (-(695252753712603288749 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9978966414926927 / 20000000000000000 : ℝ) ≤
+      (-(695252753712603288749 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9978966414926927 / 20000000000000000 : ℝ) =
+            (9978966414926927 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 15
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9978966414926927 / 5000000000000000 : ℝ) ≤
+            (691041606887396711251 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1246843666638763 / 2500000000000000 : ℝ)) (q :=
+      (9978966414926927 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_468_5 :
+    Uω (aρ 5) (bρ 5) (8328877226211 / 16000000000000 : ℝ) ≤
+      (-(1439831914742107004137 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (194717266146073 / 400000000000000 : ℝ) ≤
+      (-(1439831914742107004137 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (194717266146073 / 400000000000000 : ℝ) =
+            (194717266146073 / 100000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (194717266146073 / 100000000000000 : ℝ) ≤
+            (1332756806457892995863 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (194498459130671 / 400000000000000 : ℝ)) (q :=
+      (194717266146073 / 400000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_468_6 :
+    Uω (aρ 6) (bρ 6) (8328877226211 / 16000000000000 : ℝ) ≤
+      (-(1512989178853868331933 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9386173325681481 / 20000000000000000 : ℝ) ≤
+      (-(1512989178853868331933 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9386173325681481 / 20000000000000000 : ℝ) =
+            (9386173325681481 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9386173325681481 / 5000000000000000 : ℝ) ≤
+            (1259599542346131668067 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2340005935892303 / 5000000000000000 : ℝ)) (q :=
+      (9386173325681481 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_468_7 :
+    Uω (aρ 7) (bρ 7) (8328877226211 / 16000000000000 : ℝ) ≤
+      (-(8089213548594952910653 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4453381688665273 / 10000000000000000 : ℝ) ≤
+      (-(8089213548594952910653 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4453381688665273 / 10000000000000000 : ℝ) =
+            (4453381688665273 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4453381688665273 / 2500000000000000 : ℝ) ≤
+            (5773730057405047089347 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4424263304193671 / 10000000000000000 : ℝ)) (q :=
+      (4453381688665273 / 10000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_468_8 :
+    Uω (aρ 8) (bρ 8) (8328877226211 / 16000000000000 : ℝ) ≤
+      (-(8823720942157960686819 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (165520069342137 / 400000000000000 : ℝ) ≤
+      (-(8823720942157960686819 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (165520069342137 / 400000000000000 : ℝ) =
+            (165520069342137 / 100000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (165520069342137 / 100000000000000 : ℝ) ≤
+            (5039222663842039313181 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (163081801812399 / 400000000000000 : ℝ)) (q :=
+      (165520069342137 / 400000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_468_9 :
+    Uω (aρ 9) (bρ 9) (8328877226211 / 16000000000000 : ℝ) ≤
+      (-(9843548329673441358901 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1494720940629161 / 4000000000000000 : ℝ) ≤
+      (-(9843548329673441358901 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1494720940629161 / 4000000000000000 : ℝ) =
+            (1494720940629161 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1494720940629161 / 1000000000000000 : ℝ) ≤
+            (4019395276326558641099 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (361561190970893 / 1000000000000000 : ℝ)) (q :=
+      (1494720940629161 / 4000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_468_10 :
+    Uω (aρ 10) (bρ 10) (8328877226211 / 16000000000000 : ℝ) ≤
+      (-(1409919848820797437973 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (258960579675723 / 800000000000000 : ℝ) ≤
+      (-(1409919848820797437973 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (258960579675723 / 800000000000000 : ℝ) =
+            (258960579675723 / 200000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (258960579675723 / 200000000000000 : ℝ) ≤
+            (322948101929202562027 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3755725914939 / 12500000000000 : ℝ)) (q :=
+      (258960579675723 / 800000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10]) (by norm_num [bρ_10])
+      (by norm_num) (by norm_num [aρ_10, bρ_10]) (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_468_11 :
+    Uω (aρ 11) (bρ 11) (8328877226211 / 16000000000000 : ℝ) ≤
+      (-(13447342272511465768649 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2606089612812773 / 10000000000000000 : ℝ) ≤
+      (-(13447342272511465768649 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2606089612812773 / 10000000000000000 : ℝ) =
+            (2606089612812773 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2606089612812773 / 2500000000000000 : ℝ) ≤
+            (415601333488534231351 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2161222091888671 / 10000000000000000 : ℝ)) (q :=
+      (2606089612812773 / 10000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_468_12 :
+    Uω (aρ 12) (bρ 12) (8328877226211 / 16000000000000 : ℝ) ≤
+      (-(18524580292131774241723 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (784255736301877 / 5000000000000000 : ℝ) ≤
+      (-(18524580292131774241723 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (784255736301877 / 5000000000000000 : ℝ) =
+            (784255736301877 / 625000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (784255736301877 / 625000000000000 : ℝ) ≤
+            (2269835116868225758277 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (509765908155633 / 10000000000000000 : ℝ)) (q :=
+      (784255736301877 / 5000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12]) (by norm_num [bρ_12])
+      (by norm_num) (by norm_num [aρ_12, bρ_12]) (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_468_13 :
+    Uω (aρ 13) (bρ 13) (8328877226211 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_468_14 :
+    Uω (aρ 14) (bρ 14) (8328877226211 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_468_15 :
+    Uω (aρ 15) (bρ 15) (8328877226211 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_468_16 :
+    Uω (aρ 16) (bρ 16) (8328877226211 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_468 :
+    Uρ (8328877226211 / 16000000000000 : ℝ) ≤
+      (-(1026390100268947894541 / 1000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (8328877226211 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3326668334306610783819 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_468_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (8328877226211 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(6700021636373269278519 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_468_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (8328877226211 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(6794107161476389452203 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_468_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (8328877226211 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(695252753712603288749 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_468_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (8328877226211 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1439831914742107004137 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_468_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (8328877226211 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1512989178853868331933 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_468_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (8328877226211 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(8089213548594952910653 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_468_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (8328877226211 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(8823720942157960686819 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_468_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (8328877226211 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(9843548329673441358901 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_468_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (8328877226211 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(1409919848820797437973 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_468_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (8328877226211 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(13447342272511465768649 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_468_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (8328877226211 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(18524580292131774241723 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_468_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (8328877226211 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_468_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (8328877226211 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_468_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (8328877226211 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_468_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (8328877226211 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_468_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_469_1 :
+    Uω (aρ 1) (bρ 1) (16737641334457 / 32000000000000 : ℝ) ≤
+      (-(1651223542470804826287 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20663937616968653 / 40000000000000000 : ℝ) ≤
+      (-(1651223542470804826287 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (20663937616968653 / 40000000000000000 : ℝ) =
+            (20663937616968653 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20663937616968653 / 20000000000000000 : ℝ) ≤
+            (81644408279195173713 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2582976554775757 / 5000000000000000 : ℝ)) (q :=
+      (20663937616968653 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_469_2 :
+    Uω (aρ 2) (bρ 2) (16737641334457 / 32000000000000 : ℝ) ≤
+      (-(1330270268776452498909 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20568161449194411 / 40000000000000000 : ℝ) ≤
+      (-(1330270268776452498909 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (20568161449194411 / 40000000000000000 : ℝ) =
+            (20568161449194411 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20568161449194411 / 20000000000000000 : ℝ) ≤
+            (56024091823547501091 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5141834038174393 / 10000000000000000 : ℝ)) (q :=
+      (20568161449194411 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_469_3 :
+    Uω (aρ 3) (bρ 3) (16737641334457 / 32000000000000 : ℝ) ≤
+      (-(337248643719215027449 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4075299553210723 / 8000000000000000 : ℝ) ≤
+      (-(337248643719215027449 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (4075299553210723 / 8000000000000000 : ℝ) =
+            (4075299553210723 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4075299553210723 / 4000000000000000 : ℝ) ≤
+            (9324946430784972551 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1018679660590799 / 2000000000000000 : ℝ)) (q :=
+      (4075299553210723 / 8000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_469_4 :
+    Uω (aρ 4) (bρ 4) (16737641334457 / 32000000000000 : ℝ) ≤
+      (-(3451298405747701115753 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20057833439830469 / 40000000000000000 : ℝ) ≤
+      (-(3451298405747701115753 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (20057833439830469 / 40000000000000000 : ℝ) =
+            (20057833439830469 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20057833439830469 / 20000000000000000 : ℝ) ≤
+            (14437495752298884247 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2506180160453711 / 5000000000000000 : ℝ)) (q :=
+      (20057833439830469 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_469_5 :
+    Uω (aρ 5) (bρ 5) (16737641334457 / 32000000000000 : ℝ) ≤
+      (-(7147949625221588208791 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3914339396338771 / 8000000000000000 : ℝ) ≤
+      (-(7147949625221588208791 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3914339396338771 / 8000000000000000 : ℝ) =
+            (3914339396338771 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3914339396338771 / 2000000000000000 : ℝ) ≤
+            (6714993980778411791209 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (977496402234823 / 2000000000000000 : ℝ)) (q :=
+      (3914339396338771 / 8000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_469_6 :
+    Uω (aρ 6) (bρ 6) (16737641334457 / 32000000000000 : ℝ) ≤
+      (-(3755872658039831526277 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (18872482749682987 / 40000000000000000 : ℝ) ≤
+      (-(3755872658039831526277 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (18872482749682987 / 40000000000000000 : ℝ) =
+            (18872482749682987 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18872482749682987 / 10000000000000000 : ℝ) ≤
+            (3175599144960168473723 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4705115270308681 / 10000000000000000 : ℝ)) (q :=
+      (18872482749682987 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_469_7 :
+    Uω (aρ 7) (bρ 7) (16737641334457 / 32000000000000 : ℝ) ≤
+      (-(8032947538949810997407 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17914038867081701 / 40000000000000000 : ℝ) ≤
+      (-(8032947538949810997407 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (17914038867081701 / 40000000000000000 : ℝ) =
+            (17914038867081701 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17914038867081701 / 10000000000000000 : ℝ) ≤
+            (5829996067050189002593 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2224777354884019 / 5000000000000000 : ℝ)) (q :=
+      (17914038867081701 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_469_8 :
+    Uω (aρ 8) (bρ 8) (16737641334457 / 32000000000000 : ℝ) ≤
+      (-(8762680969486746709881 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16653349321475323 / 40000000000000000 : ℝ) ≤
+      (-(8762680969486746709881 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (16653349321475323 / 40000000000000000 : ℝ) =
+            (16653349321475323 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16653349321475323 / 10000000000000000 : ℝ) ≤
+            (5100262636513253290119 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4102751588304849 / 10000000000000000 : ℝ)) (q :=
+      (16653349321475323 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_469_9 :
+    Uω (aρ 9) (bρ 9) (16737641334457 / 32000000000000 : ℝ) ≤
+      (-(9774754545064592874979 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15050391423875193 / 40000000000000000 : ℝ) ≤
+      (-(9774754545064592874979 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (15050391423875193 / 40000000000000000 : ℝ) =
+            (15050391423875193 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15050391423875193 / 10000000000000000 : ℝ) ≤
+            (4088189060935407125021 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (227639891741549 / 625000000000000 : ℝ)) (q :=
+      (15050391423875193 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_469_10 :
+    Uω (aρ 10) (bρ 10) (16737641334457 / 32000000000000 : ℝ) ≤
+      (-(2799166476393955032353 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13055543894188003 / 40000000000000000 : ℝ) ≤
+      (-(2799166476393955032353 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (13055543894188003 / 40000000000000000 : ℝ) =
+            (13055543894188003 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13055543894188003 / 10000000000000000 : ℝ) ≤
+            (666569425106044967647 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3033373536516189 / 10000000000000000 : ℝ)) (q :=
+      (13055543894188003 / 40000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_469_11 :
+    Uω (aρ 11) (bρ 11) (16737641334457 / 32000000000000 : ℝ) ≤
+      (-(3333189941798207716189 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2108898059612717 / 8000000000000000 : ℝ) ≤
+      (-(3333189941798207716189 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2108898059612717 / 8000000000000000 : ℝ) =
+            (2108898059612717 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2108898059612717 / 2000000000000000 : ℝ) ≤
+            (132545959701792283811 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (109816168232949 / 500000000000000 : ℝ)) (q :=
+      (2108898059612717 / 8000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_469_12 :
+    Uω (aρ 12) (bρ 12) (16737641334457 / 32000000000000 : ℝ) ≤
+      (-(18084834061631598644507 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6556100873289057 / 40000000000000000 : ℝ) ≤
+      (-(18084834061631598644507 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (6556100873289057 / 40000000000000000 : ℝ) =
+            (6556100873289057 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6556100873289057 / 5000000000000000 : ℝ) ≤
+            (2709581347368401355493 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (156457187239179 / 2500000000000000 : ℝ)) (q :=
+      (6556100873289057 / 40000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_469_13 :
+    Uω (aρ 13) (bρ 13) (16737641334457 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_469_14 :
+    Uω (aρ 14) (bρ 14) (16737641334457 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_469_15 :
+    Uω (aρ 15) (bρ 15) (16737641334457 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_469_16 :
+    Uω (aρ 16) (bρ 16) (16737641334457 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_469 :
+    Uρ (16737641334457 / 32000000000000 : ℝ) ≤
+      (-(5092959469832835324207 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (16737641334457 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1651223542470804826287 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_469_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (16737641334457 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1330270268776452498909 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_469_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (16737641334457 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(337248643719215027449 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_469_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (16737641334457 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3451298405747701115753 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_469_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (16737641334457 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(7147949625221588208791 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_469_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (16737641334457 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3755872658039831526277 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_469_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (16737641334457 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(8032947538949810997407 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_469_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (16737641334457 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(8762680969486746709881 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_469_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (16737641334457 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(9774754545064592874979 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_469_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (16737641334457 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(2799166476393955032353 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_469_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (16737641334457 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(3333189941798207716189 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_469_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (16737641334457 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(18084834061631598644507 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_469_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (16737641334457 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_469_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (16737641334457 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_469_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (16737641334457 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_469_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (16737641334457 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_469_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_470_1 :
+    Uω (aρ 1) (bρ 1) (33555169550949 / 64000000000000 : ℝ) ≤
+      (-(1316152127732697187741 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (41427734439953681 / 80000000000000000 : ℝ) ≤
+      (-(1316152127732697187741 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (41427734439953681 / 80000000000000000 : ℝ) =
+            (41427734439953681 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (41427734439953681 / 40000000000000000 : ℝ) ≤
+            (70142232867302812259 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5178435585737639 / 10000000000000000 : ℝ)) (q :=
+      (41427734439953681 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_470_2 :
+    Uω (aρ 2) (bρ 2) (33555169550949 / 64000000000000 : ℝ) ≤
+      (-(1656776186842113214311 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1649447419928433 / 3200000000000000 : ℝ) ≤
+      (-(1656776186842113214311 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1649447419928433 / 3200000000000000 : ℝ) =
+            (1649447419928433 / 1600000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1649447419928433 / 1600000000000000 : ℝ) ≤
+            (76091763907886785689 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (206172694512477 / 400000000000000 : ℝ)) (q :=
+      (1649447419928433 / 3200000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_470_3 :
+    Uω (aρ 3) (bρ 3) (33555169550949 / 64000000000000 : ℝ) ≤
+      (-(6720495992657626363703 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (40852868336149581 / 80000000000000000 : ℝ) ≤
+      (-(6720495992657626363703 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (40852868336149581 / 80000000000000000 : ℝ) =
+            (40852868336149581 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (40852868336149581 / 40000000000000000 : ℝ) ≤
+            (210975810342373636297 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2552942089323307 / 5000000000000000 : ℝ)) (q :=
+      (40852868336149581 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_470_4 :
+    Uω (aρ 4) (bρ 4) (33555169550949 / 64000000000000 : ℝ) ≤
+      (-(1719431176382352272803 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (40215567176451669 / 80000000000000000 : ℝ) ≤
+      (-(1719431176382352272803 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (40215567176451669 / 80000000000000000 : ℝ) =
+            (40215567176451669 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (40215567176451669 / 40000000000000000 : ℝ) ≤
+            (13436774367647727197 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1226770768991 / 2441406250000 : ℝ)) (q :=
+      (40215567176451669 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_470_5 :
+    Uω (aρ 5) (bρ 5) (33555169550949 / 64000000000000 : ℝ) ≤
+      (-(3561221438257288238383 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (39243363475385473 / 80000000000000000 : ℝ) ≤
+      (-(3561221438257288238383 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (39243363475385473 / 80000000000000000 : ℝ) =
+            (39243363475385473 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (39243363475385473 / 20000000000000000 : ℝ) ≤
+            (3370250364742711761617 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4899992063855587 / 10000000000000000 : ℝ)) (q :=
+      (39243363475385473 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_470_6 :
+    Uω (aρ 6) (bρ 6) (33555169550949 / 64000000000000 : ℝ) ≤
+      (-(7485251371864224695571 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7569019877782629 / 16000000000000000 : ℝ) ≤
+      (-(7485251371864224695571 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7569019877782629 / 16000000000000000 : ℝ) =
+            (7569019877782629 / 4000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7569019877782629 / 4000000000000000 : ℝ) ≤
+            (6377692234135775304429 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (943533283475501 / 2000000000000000 : ℝ)) (q :=
+      (7569019877782629 / 16000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_470_7 :
+    Uω (aρ 7) (bρ 7) (33555169550949 / 64000000000000 : ℝ) ≤
+      (-(8004934346840263548131 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (35928584326409349 / 80000000000000000 : ℝ) ≤
+      (-(8004934346840263548131 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (35928584326409349 / 80000000000000000 : ℝ) =
+            (35928584326409349 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (35928584326409349 / 20000000000000000 : ℝ) ≤
+            (5858009259159736451869 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1115549758127889 / 2500000000000000 : ℝ)) (q :=
+      (35928584326409349 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_470_8 :
+    Uω (aρ 8) (bρ 8) (33555169550949 / 64000000000000 : ℝ) ≤
+      (-(2183076059643294649347 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (33408027328974557 / 80000000000000000 : ℝ) ≤
+      (-(2183076059643294649347 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (33408027328974557 / 80000000000000000 : ℝ) =
+            (33408027328974557 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (33408027328974557 / 20000000000000000 : ℝ) ≤
+            (1282659841856705350653 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2057800717246429 / 5000000000000000 : ℝ)) (q :=
+      (33408027328974557 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_470_9 :
+    Uω (aρ 9) (bρ 9) (33555169550949 / 64000000000000 : ℝ) ≤
+      (-(9740545961313508592881 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (30203929687630289 / 80000000000000000 : ℝ) ≤
+      (-(9740545961313508592881 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (30203929687630289 / 80000000000000000 : ℝ) =
+            (30203929687630289 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (30203929687630289 / 20000000000000000 : ℝ) ≤
+            (4122397644686491407119 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3655542652516791 / 10000000000000000 : ℝ)) (q :=
+      (30203929687630289 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_470_10 :
+    Uω (aρ 10) (bρ 10) (33555169550949 / 64000000000000 : ℝ) ≤
+      (-(139445163964400578137 / 125000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5243700283042729 / 16000000000000000 : ℝ) ≤
+      (-(139445163964400578137 / 125000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5243700283042729 / 16000000000000000 : ℝ) =
+            (5243700283042729 / 4000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5243700283042729 / 4000000000000000 : ℝ) ≤
+            (33841631110599421863 / 125000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (304774461790763 / 1000000000000000 : ℝ)) (q :=
+      (5243700283042729 / 16000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_470_11 :
+    Uω (aρ 11) (bρ 11) (33555169550949 / 64000000000000 : ℝ) ≤
+      (-(13276151640471082417553 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4241739979860997 / 16000000000000000 : ℝ) ≤
+      (-(13276151640471082417553 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4241739979860997 / 16000000000000000 : ℝ) =
+            (4241739979860997 / 4000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4241739979860997 / 4000000000000000 : ℝ) ≤
+            (586791965528917582447 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (442754173027093 / 2000000000000000 : ℝ)) (q :=
+      (4241739979860997 / 16000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_470_12 :
+    Uω (aρ 12) (bρ 12) (33555169550949 / 64000000000000 : ℝ) ≤
+      (-(17893176801322932681187 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13365930292119409 / 80000000000000000 : ℝ) ≤
+      (-(17893176801322932681187 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (13365930292119409 / 80000000000000000 : ℝ) =
+            (13365930292119409 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13365930292119409 / 10000000000000000 : ℝ) ≤
+            (2901238607677067318813 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (676778560024071 / 10000000000000000 : ℝ)) (q :=
+      (13365930292119409 / 80000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_470_13 :
+    Uω (aρ 13) (bρ 13) (33555169550949 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_470_14 :
+    Uω (aρ 14) (bρ 14) (33555169550949 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_470_15 :
+    Uω (aρ 15) (bρ 15) (33555169550949 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_470_16 :
+    Uω (aρ 16) (bρ 16) (33555169550949 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_470 :
+    Uρ (33555169550949 / 64000000000000 : ℝ) ≤
+      (-(1014905945163655209177 / 1000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (33555169550949 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1316152127732697187741 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_470_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (33555169550949 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1656776186842113214311 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_470_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (33555169550949 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(6720495992657626363703 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_470_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (33555169550949 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1719431176382352272803 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_470_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (33555169550949 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(3561221438257288238383 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_470_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (33555169550949 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(7485251371864224695571 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_470_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (33555169550949 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(8004934346840263548131 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_470_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (33555169550949 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(2183076059643294649347 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_470_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (33555169550949 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(9740545961313508592881 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_470_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (33555169550949 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) * (-(139445163964400578137 / 125000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_470_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (33555169550949 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(13276151640471082417553 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_470_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (33555169550949 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(17893176801322932681187 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_470_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (33555169550949 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_470_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (33555169550949 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_470_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (33555169550949 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_470_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (33555169550949 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_470_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_471_1 :
+    Uω (aρ 1) (bρ 1) (4204382054123 / 8000000000000 : ℝ) ≤
+      (-(6556685210681737118977 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10381898410766941 / 20000000000000000 : ℝ) ≤
+      (-(6556685210681737118977 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10381898410766941 / 20000000000000000 : ℝ) =
+            (10381898410766941 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10381898410766941 / 10000000000000000 : ℝ) ≤
+            (374786592318262881023 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5190918061198191 / 10000000000000000 : ℝ)) (q :=
+      (10381898410766941 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_471_2 :
+    Uω (aρ 2) (bρ 2) (4204382054123 / 8000000000000 : ℝ) ≤
+      (-(6602916803099730944047 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (258350300491989 / 500000000000000 : ℝ) ≤
+      (-(6602916803099730944047 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (258350300491989 / 500000000000000 : ℝ) =
+            (258350300491989 / 250000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (258350300491989 / 250000000000000 : ℝ) ≤
+            (328554999900269055953 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (516680068262081 / 1000000000000000 : ℝ)) (q :=
+      (258350300491989 / 500000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_471_3 :
+    Uω (aρ 3) (bρ 3) (4204382054123 / 8000000000000 : ℝ) ≤
+      (-(1339215778672726434163 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2047637053545741 / 4000000000000000 : ℝ) ≤
+      (-(1339215778672726434163 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2047637053545741 / 4000000000000000 : ℝ) =
+            (2047637053545741 / 2000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2047637053545741 / 2000000000000000 : ℝ) ≤
+            (47078581927273565837 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1023674007403991 / 2000000000000000 : ℝ)) (q :=
+      (2047637053545741 / 4000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_471_4 :
+    Uω (aρ 4) (bρ 4) (4204382054123 / 8000000000000 : ℝ) ≤
+      (-(6852914359553545011063 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (251971670415769 / 500000000000000 : ℝ) ≤
+      (-(6852914359553545011063 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (251971670415769 / 500000000000000 : ℝ) =
+            (251971670415769 / 250000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (251971670415769 / 250000000000000 : ℝ) ≤
+            (78557443446454988937 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (503734576698701 / 1000000000000000 : ℝ)) (q :=
+      (251971670415769 / 500000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_471_5 :
+    Uω (aρ 5) (bρ 5) (4204382054123 / 8000000000000 : ℝ) ≤
+      (-(7097001165159826073773 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9835833105782713 / 20000000000000000 : ℝ) ≤
+      (-(7097001165159826073773 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9835833105782713 / 20000000000000000 : ℝ) =
+            (9835833105782713 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9835833105782713 / 5000000000000000 : ℝ) ≤
+            (6765942440840173926227 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4912501975473963 / 10000000000000000 : ℝ)) (q :=
+      (9835833105782713 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_471_6 :
+    Uω (aρ 6) (bρ 6) (4204382054123 / 8000000000000 : ℝ) ≤
+      (-(3729413909532381592127 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9486307955393467 / 20000000000000000 : ℝ) ≤
+      (-(3729413909532381592127 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9486307955393467 / 20000000000000000 : ℝ) =
+            (9486307955393467 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9486307955393467 / 5000000000000000 : ℝ) ≤
+            (3202057893467618407873 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4730217200224717 / 10000000000000000 : ℝ)) (q :=
+      (9486307955393467 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_471_7 :
+    Uω (aρ 7) (bρ 7) (4204382054123 / 8000000000000 : ℝ) ≤
+      (-(15580078944028729029 / 19531250000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9007271820033613 / 20000000000000000 : ℝ) ≤
+      (-(15580078944028729029 / 19531250000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9007271820033613 / 20000000000000000 : ℝ) =
+            (9007271820033613 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9007271820033613 / 5000000000000000 : ℝ) ≤
+            (11495982786440020971 / 19531250000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4474842445624863 / 10000000000000000 : ℝ)) (q :=
+      (9007271820033613 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_471_8 :
+    Uω (aρ 8) (bρ 8) (4204382054123 / 8000000000000 : ℝ) ≤
+      (-(8702022194771938875817 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2094334187160359 / 5000000000000000 : ℝ) ≤
+      (-(8702022194771938875817 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2094334187160359 / 5000000000000000 : ℝ) =
+            (2094334187160359 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2094334187160359 / 1250000000000000 : ℝ) ≤
+            (5160921411228061124183 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2064224512786343 / 5000000000000000 : ℝ)) (q :=
+      (2094334187160359 / 5000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_471_9 :
+    Uω (aρ 9) (bρ 9) (4204382054123 / 8000000000000 : ℝ) ≤
+      (-(9706461627251655129437 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1515352670795999 / 4000000000000000 : ℝ) ≤
+      (-(9706461627251655129437 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1515352670795999 / 4000000000000000 : ℝ) =
+            (1515352670795999 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1515352670795999 / 1000000000000000 : ℝ) ≤
+            (4156481978748344870563 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (733768251854249 / 2000000000000000 : ℝ)) (q :=
+      (1515352670795999 / 4000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_471_10 :
+    Uω (aρ 10) (bρ 10) (4204382054123 / 8000000000000 : ℝ) ≤
+      (-(2778688335021342273317 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1645365549227177 / 5000000000000000 : ℝ) ≤
+      (-(2778688335021342273317 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1645365549227177 / 5000000000000000 : ℝ) =
+            (1645365549227177 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1645365549227177 / 1250000000000000 : ℝ) ≤
+            (687047566478657726683 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1531049567847479 / 5000000000000000 : ℝ)) (q :=
+      (1645365549227177 / 5000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_471_11 :
+    Uω (aρ 11) (bρ 11) (4204382054123 / 8000000000000 : ℝ) ≤
+      (-(6609993535840864133789 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5332038196182559 / 20000000000000000 : ℝ) ≤
+      (-(6609993535840864133789 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5332038196182559 / 20000000000000000 : ℝ) =
+            (5332038196182559 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5332038196182559 / 5000000000000000 : ℝ) ≤
+            (321478267159135866211 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2231151761173809 / 10000000000000000 : ℝ)) (q :=
+      (5332038196182559 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_471_12 :
+    Uω (aρ 12) (bρ 12) (4204382054123 / 8000000000000 : ℝ) ≤
+      (-(3542999804663610854087 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (340155394699111 / 2000000000000000 : ℝ) ≤
+      (-(3542999804663610854087 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (340155394699111 / 2000000000000000 : ℝ) =
+            (340155394699111 / 250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (340155394699111 / 250000000000000 : ℝ) ≤
+            (615883277136389145913 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4527297554171 / 62500000000000 : ℝ)) (q :=
+      (340155394699111 / 2000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12]) (by norm_num [bρ_12])
+      (by norm_num) (by norm_num [aρ_12, bρ_12]) (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_471_13 :
+    Uω (aρ 13) (bρ 13) (4204382054123 / 8000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_471_14 :
+    Uω (aρ 14) (bρ 14) (4204382054123 / 8000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_471_15 :
+    Uω (aρ 15) (bρ 15) (4204382054123 / 8000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_471_16 :
+    Uω (aρ 16) (bρ 16) (4204382054123 / 8000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_471 :
+    Uρ (4204382054123 / 8000000000000 : ℝ) ≤
+      (-(79009721845316621653 / 78125000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (4204382054123 / 8000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(6556685210681737118977 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_471_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (4204382054123 / 8000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(6602916803099730944047 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_471_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (4204382054123 / 8000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1339215778672726434163 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_471_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (4204382054123 / 8000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(6852914359553545011063 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_471_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (4204382054123 / 8000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(7097001165159826073773 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_471_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (4204382054123 / 8000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3729413909532381592127 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_471_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (4204382054123 / 8000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) * (-(15580078944028729029 / 19531250000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_471_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (4204382054123 / 8000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(8702022194771938875817 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_471_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (4204382054123 / 8000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(9706461627251655129437 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_471_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (4204382054123 / 8000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(2778688335021342273317 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_471_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (4204382054123 / 8000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(6609993535840864133789 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_471_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (4204382054123 / 8000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(3542999804663610854087 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_471_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (4204382054123 / 8000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_471_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (4204382054123 / 8000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_471_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (4204382054123 / 8000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_471_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (4204382054123 / 8000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_471_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/U39.lean b/LeanPool/Zeta5Irrational/Table/U39.lean
new file mode 100644
index 0000000000..c2d4b4db6e
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/U39.lean
@@ -0,0 +1,4617 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-! # Certified arcsine potential bounds (U39) -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma U_472_1 :
+    Uω (aρ 1) (bρ 1) (33714943315019 / 64000000000000 : ℝ) ≤
+      (-(3266333803415444261449 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8325490568660087 / 16000000000000000 : ℝ) ≤
+      (-(3266333803415444261449 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (8325490568660087 / 16000000000000000 : ℝ) =
+            (8325490568660087 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8325490568660087 / 8000000000000000 : ℝ) ≤
+            (199402098084555738551 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (520340053593839 / 1000000000000000 : ℝ)) (q :=
+      (8325490568660087 / 16000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_472_2 :
+    Uω (aρ 2) (bρ 2) (33714943315019 / 64000000000000 : ℝ) ≤
+      (-(3289393613990012216949 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (41435910640050719 / 80000000000000000 : ℝ) ≤
+      (-(3289393613990012216949 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (41435910640050719 / 80000000000000000 : ℝ) =
+            (41435910640050719 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (41435910640050719 / 40000000000000000 : ℝ) ≤
+            (176342287509987783051 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5179283997635961 / 10000000000000000 : ℝ)) (q :=
+      (41435910640050719 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_472_3 :
+    Uω (aρ 3) (bρ 3) (33714943315019 / 64000000000000 : ℝ) ≤
+      (-(3335860642552980259753 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (41052613736908707 / 80000000000000000 : ℝ) ≤
+      (-(3335860642552980259753 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (41052613736908707 / 80000000000000000 : ℝ) =
+            (41052613736908707 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (41052613736908707 / 40000000000000000 : ℝ) ≤
+            (129875258947019740247 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2565427939100229 / 5000000000000000 : ℝ)) (q :=
+      (41052613736908707 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_472_4 :
+    Uω (aρ 4) (bρ 4) (33714943315019 / 64000000000000 : ℝ) ≤
+      (-(853520683420242002753 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (40415367151409243 / 80000000000000000 : ℝ) ≤
+      (-(853520683420242002753 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (40415367151409243 / 80000000000000000 : ℝ) =
+            (40415367151409243 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (40415367151409243 / 40000000000000000 : ℝ) ≤
+            (12913291954757997247 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (157807450402831 / 312500000000000 : ℝ)) (q :=
+      (40415367151409243 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_472_5 :
+    Uω (aρ 5) (bρ 5) (33714943315019 / 64000000000000 : ℝ) ≤
+      (-(3535812079804980335119 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (39443300810923579 / 80000000000000000 : ℝ) ≤
+      (-(3535812079804980335119 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (39443300810923579 / 80000000000000000 : ℝ) =
+            (39443300810923579 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (39443300810923579 / 20000000000000000 : ℝ) ≤
+            (3395659723195019664881 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (307813234194011 / 625000000000000 : ℝ)) (q :=
+      (39443300810923579 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_472_6 :
+    Uω (aρ 6) (bρ 6) (33714943315019 / 64000000000000 : ℝ) ≤
+      (-(743247428260968271397 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7609072561782843 / 16000000000000000 : ℝ) ≤
+      (-(743247428260968271397 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7609072561782843 / 16000000000000000 : ℝ) =
+            (7609072561782843 / 4000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7609072561782843 / 4000000000000000 : ℝ) ≤
+            (643046932339031728603 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (948553524348367 / 2000000000000000 : ℝ)) (q :=
+      (7609072561782843 / 16000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_472_7 :
+    Uω (aρ 7) (bρ 7) (33714943315019 / 64000000000000 : ℝ) ≤
+      (-(397457265176319725381 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7225917325218623 / 16000000000000000 : ℝ) ≤
+      (-(397457265176319725381 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7225917325218623 / 16000000000000000 : ℝ) =
+            (7225917325218623 / 4000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7225917325218623 / 4000000000000000 : ℝ) ≤
+            (295689915123680274619 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (56093561959957 / 125000000000000 : ℝ)) (q :=
+      (7225917325218623 / 16000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_472_8 :
+    Uω (aρ 8) (bρ 8) (33714943315019 / 64000000000000 : ℝ) ≤
+      (-(8671834233232153227217 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (33610657723676827 / 80000000000000000 : ℝ) ≤
+      (-(8671834233232153227217 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (33610657723676827 / 80000000000000000 : ℝ) =
+            (33610657723676827 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (33610657723676827 / 20000000000000000 : ℝ) ≤
+            (5191109372767846772783 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (517661797816561 / 1250000000000000 : ℝ)) (q :=
+      (33610657723676827 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_472_9 :
+    Uω (aρ 9) (bρ 9) (33714943315019 / 64000000000000 : ℝ) ≤
+      (-(1209062573808391051687 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (30410154283032899 / 80000000000000000 : ℝ) ≤
+      (-(1209062573808391051687 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (30410154283032899 / 80000000000000000 : ℝ) =
+            (30410154283032899 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (30410154283032899 / 20000000000000000 : ℝ) ≤
+            (523805376941608948313 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1841067075365753 / 5000000000000000 : ℝ)) (q :=
+      (30410154283032899 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_472_10 :
+    Uω (aρ 10) (bρ 10) (33714943315019 / 64000000000000 : ℝ) ≤
+      (-(11074084550028256930173 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5286626166611991 / 16000000000000000 : ℝ) ≤
+      (-(11074084550028256930173 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5286626166611991 / 16000000000000000 : ℝ) =
+            (5286626166611991 / 4000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5286626166611991 / 4000000000000000 : ℝ) ≤
+            (2788859055971743069827 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (307643732173327 / 1000000000000000 : ℝ)) (q :=
+      (5286626166611991 / 16000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_472_11 :
+    Uω (aρ 11) (bρ 11) (33714943315019 / 64000000000000 : ℝ) ≤
+      (-(263285150516408844729 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21447345430718583 / 80000000000000000 : ℝ) ≤
+      (-(263285150516408844729 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (21447345430718583 / 80000000000000000 : ℝ) =
+            (21447345430718583 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21447345430718583 / 20000000000000000 : ℝ) ≤
+            (13973721603591155271 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2248467597352927 / 10000000000000000 : ℝ)) (q :=
+      (21447345430718583 / 80000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_472_12 :
+    Uω (aρ 12) (bρ 12) (33714943315019 / 64000000000000 : ℝ) ≤
+      (-(3509569916371714106323 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13835554275365503 / 80000000000000000 : ℝ) ≤
+      (-(3509569916371714106323 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (13835554275365503 / 80000000000000000 : ℝ) =
+            (13835554275365503 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13835554275365503 / 10000000000000000 : ℝ) ≤
+            (649313165428285893677 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (769219905199657 / 10000000000000000 : ℝ)) (q :=
+      (13835554275365503 / 80000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_472_13 :
+    Uω (aρ 13) (bρ 13) (33714943315019 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_472_14 :
+    Uω (aρ 14) (bρ 14) (33714943315019 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_472_15 :
+    Uω (aρ 15) (bρ 15) (33714943315019 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_472_16 :
+    Uω (aρ 16) (bρ 16) (33714943315019 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_472 :
+    Uρ (33714943315019 / 64000000000000 : ℝ) ≤
+      (-(10078300210227161819509 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (33714943315019 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3266333803415444261449 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_472_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (33714943315019 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3289393613990012216949 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_472_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (33714943315019 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3335860642552980259753 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_472_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (33714943315019 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(853520683420242002753 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_472_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (33714943315019 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(3535812079804980335119 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_472_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (33714943315019 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(743247428260968271397 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_472_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (33714943315019 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(397457265176319725381 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_472_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (33714943315019 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(8671834233232153227217 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_472_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (33714943315019 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(1209062573808391051687 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_472_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (33714943315019 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(11074084550028256930173 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_472_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (33714943315019 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) * (-(263285150516408844729 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_472_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (33714943315019 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(3509569916371714106323 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_472_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (33714943315019 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_472_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (33714943315019 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_472_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (33714943315019 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_472_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (33714943315019 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_472_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_473_1 :
+    Uω (aρ 1) (bρ 1) (16897415098527 / 32000000000000 : ℝ) ≤
+      (-(6508707550010153423257 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4172731204067239 / 8000000000000000 : ℝ) ≤
+      (-(6508707550010153423257 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (4172731204067239 / 8000000000000000 : ℝ) =
+            (4172731204067239 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4172731204067239 / 4000000000000000 : ℝ) ≤
+            (422764252989846576743 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (521588300996341 / 1000000000000000 : ℝ)) (q :=
+      (4172731204067239 / 8000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_473_2 :
+    Uω (aρ 2) (bρ 2) (16897415098527 / 32000000000000 : ℝ) ≤
+      (-(3277357870479055754481 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20767886591173289 / 40000000000000000 : ℝ) ≤
+      (-(3277357870479055754481 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (20767886591173289 / 40000000000000000 : ℝ) =
+            (20767886591173289 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20767886591173289 / 20000000000000000 : ℝ) ≤
+            (188378031020944245519 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5191767307891957 / 10000000000000000 : ℝ)) (q :=
+      (20767886591173289 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_473_3 :
+    Uω (aρ 3) (bρ 3) (16897415098527 / 32000000000000 : ℝ) ≤
+      (-(830927859826774090171 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20576243167316043 / 40000000000000000 : ℝ) ≤
+      (-(830927859826774090171 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (20576243167316043 / 40000000000000000 : ℝ) =
+            (20576243167316043 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20576243167316043 / 20000000000000000 : ℝ) ≤
+            (35506115548225909829 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2571670851156667 / 5000000000000000 : ℝ)) (q :=
+      (20576243167316043 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_473_4 :
+    Uω (aρ 4) (bρ 4) (16897415098527 / 32000000000000 : ℝ) ≤
+      (-(6803477725022190286487 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20257633416314663 / 40000000000000000 : ℝ) ≤
+      (-(6803477725022190286487 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (20257633416314663 / 40000000000000000 : ℝ) =
+            (20257633416314663 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20257633416314663 / 20000000000000000 : ℝ) ≤
+            (127994077977809713513 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1265582751969411 / 2500000000000000 : ℝ)) (q :=
+      (20257633416314663 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_473_5 :
+    Uω (aρ 5) (bρ 5) (16897415098527 / 32000000000000 : ℝ) ≤
+      (-(704631153085188042397 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19771634321509881 / 40000000000000000 : ℝ) ≤
+      (-(704631153085188042397 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (19771634321509881 / 40000000000000000 : ℝ) =
+            (19771634321509881 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19771634321509881 / 10000000000000000 : ℝ) ≤
+            (681663207514811957603 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4937521379810253 / 10000000000000000 : ℝ)) (q :=
+      (19771634321509881 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_473_6 :
+    Uω (aρ 6) (bρ 6) (16897415098527 / 32000000000000 : ℝ) ≤
+      (-(7406190390433035913379 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19072746181188913 / 40000000000000000 : ℝ) ≤
+      (-(7406190390433035913379 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (19072746181188913 / 40000000000000000 : ℝ) =
+            (19072746181188913 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19072746181188913 / 10000000000000000 : ℝ) ≤
+            (6456753215566964086621 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4755317684789769 / 10000000000000000 : ℝ)) (q :=
+      (19072746181188913 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_473_7 :
+    Uω (aρ 7) (bρ 7) (16897415098527 / 32000000000000 : ℝ) ≤
+      (-(7921368550378337604691 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3623008239469197 / 8000000000000000 : ℝ) ≤
+      (-(7921368550378337604691 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3623008239469197 / 8000000000000000 : ℝ) =
+            (3623008239469197 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3623008239469197 / 2000000000000000 : ℝ) ≤
+            (5941575055621662395309 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (900025314725661 / 2000000000000000 : ℝ)) (q :=
+      (3623008239469197 / 8000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_473_8 :
+    Uω (aρ 8) (bρ 8) (16897415098527 / 32000000000000 : ℝ) ≤
+      (-(4320869877513572204633 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16855979799603761 / 40000000000000000 : ℝ) ≤
+      (-(4320869877513572204633 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (16855979799603761 / 40000000000000000 : ℝ) =
+            (16855979799603761 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16855979799603761 / 10000000000000000 : ℝ) ≤
+            (2610601925486427795367 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4154137526097193 / 10000000000000000 : ℝ)) (q :=
+      (16855979799603761 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_473_9 :
+    Uω (aρ 9) (bρ 9) (16897415098527 / 32000000000000 : ℝ) ≤
+      (-(1927732381998579928349 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (610264650713451 / 1600000000000000 : ℝ) ≤
+      (-(1927732381998579928349 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (610264650713451 / 1600000000000000 : ℝ) =
+            (610264650713451 / 400000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (610264650713451 / 400000000000000 : ℝ) ≤
+            (844856339201420071651 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (73908427771469 / 200000000000000 : ℝ)) (q :=
+      (610264650713451 / 1600000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_473_10 :
+    Uω (aρ 10) (bρ 10) (16897415098527 / 32000000000000 : ℝ) ≤
+      (-(11033604756957846854013 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13270174230327283 / 40000000000000000 : ℝ) ≤
+      (-(11033604756957846854013 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (13270174230327283 / 40000000000000000 : ℝ) =
+            (13270174230327283 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13270174230327283 / 10000000000000000 : ℝ) ≤
+            (2829338849042153145987 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1545379701656977 / 5000000000000000 : ℝ)) (q :=
+      (13270174230327283 / 40000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_473_11 :
+    Uω (aρ 11) (bρ 11) (16897415098527 / 32000000000000 : ℝ) ≤
+      (-(262179094799017356733 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10783141901968263 / 40000000000000000 : ℝ) ≤
+      (-(262179094799017356733 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (10783141901968263 / 40000000000000000 : ℝ) =
+            (10783141901968263 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10783141901968263 / 10000000000000000 : ℝ) ≤
+            (15079777320982643267 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (566429966334861 / 2500000000000000 : ℝ)) (q :=
+      (10783141901968263 / 40000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_473_12 :
+    Uω (aρ 12) (bρ 12) (16897415098527 / 32000000000000 : ℝ) ≤
+      (-(17389943031897199826591 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7027880383189607 / 40000000000000000 : ℝ) ≤
+      (-(17389943031897199826591 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (7027880383189607 / 40000000000000000 : ℝ) =
+            (7027880383189607 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7027880383189607 / 5000000000000000 : ℝ) ≤
+            (3404472377102800173409 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (202947300658779 / 2500000000000000 : ℝ)) (q :=
+      (7027880383189607 / 40000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_473_13 :
+    Uω (aρ 13) (bρ 13) (16897415098527 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_473_14 :
+    Uω (aρ 14) (bρ 14) (16897415098527 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_473_15 :
+    Uω (aρ 15) (bρ 15) (16897415098527 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_473_16 :
+    Uω (aρ 16) (bρ 16) (16897415098527 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_473 :
+    Uρ (16897415098527 / 32000000000000 : ℝ) ≤
+      (-(1255512523990723562393 / 1250000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (16897415098527 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(6508707550010153423257 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_473_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (16897415098527 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3277357870479055754481 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_473_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (16897415098527 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(830927859826774090171 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_473_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (16897415098527 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(6803477725022190286487 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_473_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (16897415098527 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(704631153085188042397 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_473_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (16897415098527 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(7406190390433035913379 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_473_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (16897415098527 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(7921368550378337604691 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_473_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (16897415098527 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(4320869877513572204633 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_473_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (16897415098527 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(1927732381998579928349 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_473_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (16897415098527 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(11033604756957846854013 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_473_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (16897415098527 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) * (-(262179094799017356733 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_473_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (16897415098527 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(17389943031897199826591 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_473_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (16897415098527 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_473_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (16897415098527 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_473_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (16897415098527 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_473_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (16897415098527 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_473_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_474_1 :
+    Uω (aρ 1) (bρ 1) (33874717079089 / 64000000000000 : ℝ) ≤
+      (-(25939219060423466283 / 40000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (41827171235204113 / 80000000000000000 : ℝ) ≤
+      (-(25939219060423466283 / 40000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (41827171235204113 / 80000000000000000 : ℝ) =
+            (41827171235204113 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (41827171235204113 / 40000000000000000 : ℝ) ≤
+            (1786668151576533717 / 40000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1307091370819593 / 2500000000000000 : ℝ)) (q :=
+      (41827171235204113 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_474_2 :
+    Uω (aρ 2) (bρ 2) (33874717079089 / 64000000000000 : ℝ) ≤
+      (-(1306140412601594962099 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8327127141148561 / 16000000000000000 : ℝ) ≤
+      (-(1306140412601594962099 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (8327127141148561 / 16000000000000000 : ℝ) =
+            (8327127141148561 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8327127141148561 / 8000000000000000 : ℝ) ≤
+            (80153947998405037901 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1040850122684609 / 2000000000000000 : ℝ)) (q :=
+      (8327127141148561 / 16000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_474_3 :
+    Uω (aρ 3) (bρ 3) (33874717079089 / 64000000000000 : ℝ) ≤
+      (-(3311591693361606929099 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (41252358864580941 / 80000000000000000 : ℝ) ≤
+      (-(3311591693361606929099 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (41252358864580941 / 80000000000000000 : ℝ) =
+            (41252358864580941 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (41252358864580941 / 40000000000000000 : ℝ) ≤
+            (154144208138393070901 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5155827509482579 / 10000000000000000 : ℝ)) (q :=
+      (41252358864580941 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_474_4 :
+    Uω (aρ 4) (bρ 4) (33874717079089 / 64000000000000 : ℝ) ≤
+      (-(677885083085293050957 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (40615166311687373 / 80000000000000000 : ℝ) ≤
+      (-(677885083085293050957 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (40615166311687373 / 80000000000000000 : ℝ) =
+            (40615166311687373 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (40615166311687373 / 40000000000000000 : ℝ) ≤
+            (15262097214706949043 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5074823552324187 / 10000000000000000 : ℝ)) (q :=
+      (40615166311687373 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_474_5 :
+    Uω (aρ 5) (bρ 5) (33874717079089 / 64000000000000 : ℝ) ≤
+      (-(28084251809523764379 / 40000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (39643235923632449 / 80000000000000000 : ℝ) ≤
+      (-(28084251809523764379 / 40000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (39643235923632449 / 80000000000000000 : ℝ) =
+            (39643235923632449 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (39643235923632449 / 20000000000000000 : ℝ) ≤
+            (27367522614476235621 / 40000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (309376929665341 / 625000000000000 : ℝ)) (q :=
+      (39643235923632449 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_474_6 :
+    Uω (aρ 6) (bρ 6) (33874717079089 / 64000000000000 : ℝ) ≤
+      (-(1475995154688466043879 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (38245620493287529 / 80000000000000000 : ℝ) ≤
+      (-(1475995154688466043879 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (38245620493287529 / 80000000000000000 : ℝ) =
+            (38245620493287529 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (38245620493287529 / 20000000000000000 : ℝ) ≤
+            (1296593566511533956121 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2383933696099613 / 5000000000000000 : ℝ)) (q :=
+      (38245620493287529 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_474_7 :
+    Uω (aρ 7) (bρ 7) (33874717079089 / 64000000000000 : ℝ) ≤
+      (-(7893669714757767057957 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (36330574615994797 / 80000000000000000 : ℝ) ≤
+      (-(7893669714757767057957 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (36330574615994797 / 80000000000000000 : ℝ) =
+            (36330574615994797 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (36330574615994797 / 20000000000000000 : ℝ) ≤
+            (5969273891242232942043 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4512767303636043 / 10000000000000000 : ℝ)) (q :=
+      (36330574615994797 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_474_8 :
+    Uω (aρ 8) (bρ 8) (33874717079089 / 64000000000000 : ℝ) ≤
+      (-(8611738167075583192321 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (33813252703021321 / 80000000000000000 : ℝ) ≤
+      (-(8611738167075583192321 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (33813252703021321 / 80000000000000000 : ℝ) =
+            (33813252703021321 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (33813252703021321 / 20000000000000000 : ℝ) ≤
+            (5251205438924416807679 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2083489238366337 / 5000000000000000 : ℝ)) (q :=
+      (33813252703021321 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_474_9 :
+    Uω (aρ 9) (bρ 9) (33874717079089 / 64000000000000 : ℝ) ≤
+      (-(9604944656117145227967 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (30616288416902789 / 80000000000000000 : ℝ) ≤
+      (-(9604944656117145227967 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (30616288416902789 / 80000000000000000 : ℝ) =
+            (30616288416902789 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (30616288416902789 / 20000000000000000 : ℝ) ≤
+            (4257998949882854772033 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3708703033563041 / 10000000000000000 : ℝ)) (q :=
+      (30616288416902789 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_474_10 :
+    Uω (aρ 10) (bρ 10) (33874717079089 / 64000000000000 : ℝ) ≤
+      (-(5496656002181523759141 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5329500512359669 / 16000000000000000 : ℝ) ≤
+      (-(5496656002181523759141 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5329500512359669 / 16000000000000000 : ℝ) =
+            (5329500512359669 / 4000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5329500512359669 / 4000000000000000 : ℝ) ≤
+            (1434815800818476240859 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (310506560328193 / 1000000000000000 : ℝ)) (q :=
+      (5329500512359669 / 16000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_474_11 :
+    Uω (aρ 11) (bρ 11) (33874717079089 / 64000000000000 : ℝ) ≤
+      (-(13054070711118162002301 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4336994733821961 / 16000000000000000 : ℝ) ≤
+      (-(13054070711118162002301 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4336994733821961 / 16000000000000000 : ℝ) =
+            (4336994733821961 / 4000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4336994733821961 / 4000000000000000 : ℝ) ≤
+            (808872894881837997699 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (456582001262959 / 2000000000000000 : ℝ)) (q :=
+      (4336994733821961 / 16000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_474_12 :
+    Uω (aρ 12) (bρ 12) (33874717079089 / 64000000000000 : ℝ) ≤
+      (-(17239930913919872437509 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14268203677230981 / 80000000000000000 : ℝ) ≤
+      (-(17239930913919872437509 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (14268203677230981 / 80000000000000000 : ℝ) =
+            (14268203677230981 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14268203677230981 / 10000000000000000 : ℝ) ≤
+            (3554484495080127562491 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (852417605030089 / 10000000000000000 : ℝ)) (q :=
+      (14268203677230981 / 80000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_474_13 :
+    Uω (aρ 13) (bρ 13) (33874717079089 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_474_14 :
+    Uω (aρ 14) (bρ 14) (33874717079089 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_474_15 :
+    Uω (aρ 15) (bρ 15) (33874717079089 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_474_16 :
+    Uω (aρ 16) (bρ 16) (33874717079089 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_474 :
+    Uρ (33874717079089 / 64000000000000 : ℝ) ≤
+      (-(2002109681776138421131 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (33874717079089 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(25939219060423466283 / 40000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_474_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (33874717079089 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1306140412601594962099 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_474_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (33874717079089 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3311591693361606929099 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_474_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (33874717079089 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(677885083085293050957 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_474_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (33874717079089 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(28084251809523764379 / 40000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_474_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (33874717079089 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1475995154688466043879 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_474_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (33874717079089 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(7893669714757767057957 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_474_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (33874717079089 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(8611738167075583192321 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_474_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (33874717079089 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(9604944656117145227967 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_474_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (33874717079089 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(5496656002181523759141 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_474_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (33874717079089 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(13054070711118162002301 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_474_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (33874717079089 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(17239930913919872437509 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_474_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (33874717079089 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_474_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (33874717079089 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_474_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (33874717079089 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_474_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (33874717079089 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_474_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_475_1 :
+    Uω (aρ 1) (bρ 1) (8488650990281 / 16000000000000 : ℝ) ≤
+      (-(6460958978972525251067 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5240878803364487 / 10000000000000000 : ℝ) ≤
+      (-(6460958978972525251067 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5240878803364487 / 10000000000000000 : ℝ) =
+            (5240878803364487 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5240878803364487 / 5000000000000000 : ℝ) ≤
+            (470512824027474748933 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5240847955888349 / 10000000000000000 : ℝ)) (q :=
+      (5240878803364487 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_475_2 :
+    Uω (aρ 2) (bρ 2) (8488650990281 / 16000000000000 : ℝ) ≤
+      (-(3253372958554693937133 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1043387455259377 / 2000000000000000 : ℝ) ≤
+      (-(3253372958554693937133 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1043387455259377 / 2000000000000000 : ℝ) =
+            (1043387455259377 / 1000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1043387455259377 / 1000000000000000 : ℝ) ≤
+            (212362942945306062867 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1043346782852629 / 2000000000000000 : ℝ)) (q :=
+      (1043387455259377 / 2000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_475_3 :
+    Uω (aρ 3) (bρ 3) (8488650990281 / 16000000000000 : ℝ) ≤
+      (-(1649750631088205402597 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10338057831811619 / 20000000000000000 : ℝ) ≤
+      (-(1649750631088205402597 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10338057831811619 / 20000000000000000 : ℝ) =
+            (10338057831811619 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10338057831811619 / 10000000000000000 : ℝ) ≤
+            (83117319661794597403 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2584156649915497 / 5000000000000000 : ℝ)) (q :=
+      (10338057831811619 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_475_4 :
+    Uω (aρ 4) (bρ 4) (8488650990281 / 16000000000000 : ℝ) ≤
+      (-(3377142242700800895389 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2544691599379543 / 5000000000000000 : ℝ) ≤
+      (-(3377142242700800895389 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2544691599379543 / 5000000000000000 : ℝ) =
+            (2544691599379543 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2544691599379543 / 2500000000000000 : ℝ) ≤
+            (88593658799199104611 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5087316046602547 / 10000000000000000 : ℝ)) (q :=
+      (2544691599379543 / 5000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_475_5 :
+    Uω (aρ 5) (bρ 5) (8488650990281 / 16000000000000 : ℝ) ≤
+      (-(1748969525043867698289 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9935800664233037 / 20000000000000000 : ℝ) ≤
+      (-(1748969525043867698289 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9935800664233037 / 20000000000000000 : ℝ) =
+            (9935800664233037 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9935800664233037 / 5000000000000000 : ℝ) ≤
+            (1716766376456132301711 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1240635058163103 / 2500000000000000 : ℝ)) (q :=
+      (9935800664233037 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_475_6 :
+    Uω (aρ 6) (bρ 6) (8488650990281 / 16000000000000 : ℝ) ≤
+      (-(3676915032743383951457 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2396609200802933 / 5000000000000000 : ℝ) ≤
+      (-(3676915032743383951457 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2396609200802933 / 5000000000000000 : ℝ) =
+            (2396609200802933 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2396609200802933 / 1250000000000000 : ℝ) ≤
+            (3254556770256616048543 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4780416746771107 / 10000000000000000 : ℝ)) (q :=
+      (2396609200802933 / 5000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_475_7 :
+    Uω (aρ 7) (bρ 7) (8488650990281 / 16000000000000 : ℝ) ≤
+      (-(7866048355351224914357 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4553882914965921 / 10000000000000000 : ℝ) ≤
+      (-(7866048355351224914357 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4553882914965921 / 10000000000000000 : ℝ) =
+            (4553882914965921 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4553882914965921 / 2500000000000000 : ℝ) ≤
+            (5996895250648775085643 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4525407154251217 / 10000000000000000 : ℝ)) (q :=
+      (4553882914965921 / 10000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_475_8 :
+    Uω (aρ 8) (bρ 8) (8488650990281 / 16000000000000 : ℝ) ≤
+      (-(858182888206397318407 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4239317139495231 / 10000000000000000 : ℝ) ≤
+      (-(858182888206397318407 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4239317139495231 / 10000000000000000 : ℝ) =
+            (4239317139495231 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4239317139495231 / 2500000000000000 : ℝ) ≤
+            (528111472393602681593 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4179817254649837 / 10000000000000000 : ℝ)) (q :=
+      (4239317139495231 / 10000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_475_9 :
+    Uω (aρ 9) (bρ 9) (8488650990281 / 16000000000000 : ℝ) ≤
+      (-(2392836977549402281269 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1919957635388529 / 5000000000000000 : ℝ) ≤
+      (-(2392836977549402281269 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1919957635388529 / 5000000000000000 : ℝ) =
+            (1919957635388529 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1919957635388529 / 1250000000000000 : ℝ) ≤
+            (1072898923950597718731 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3721979145573491 / 10000000000000000 : ℝ)) (q :=
+      (1919957635388529 / 5000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_475_10 :
+    Uω (aρ 10) (bρ 10) (8488650990281 / 16000000000000 : ℝ) ≤
+      (-(10953204368456377232327 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (267545940105391 / 800000000000000 : ℝ) ≤
+      (-(10953204368456377232327 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (267545940105391 / 800000000000000 : ℝ) =
+            (267545940105391 / 200000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (267545940105391 / 200000000000000 : ℝ) ≤
+            (2909739237543622767673 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (62387122802983 / 200000000000000 : ℝ)) (q :=
+      (267545940105391 / 800000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10]) (by norm_num [bρ_10])
+      (by norm_num) (by norm_num [aρ_10, bρ_10]) (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_475_11 :
+    Uω (aρ 11) (bρ 11) (8488650990281 / 16000000000000 : ℝ) ≤
+      (-(3249899421236699954177 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2725427574771609 / 10000000000000000 : ℝ) ≤
+      (-(3249899421236699954177 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2725427574771609 / 10000000000000000 : ℝ) =
+            (2725427574771609 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2725427574771609 / 2500000000000000 : ℝ) ≤
+            (215836480263300045823 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2300039413262593 / 10000000000000000 : ℝ)) (q :=
+      (2725427574771609 / 10000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_475_12 :
+    Uω (aρ 12) (bρ 12) (8488650990281 / 16000000000000 : ℝ) ≤
+      (-(854838257289714201947 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1809243093871961 / 10000000000000000 : ℝ) ≤
+      (-(854838257289714201947 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1809243093871961 / 10000000000000000 : ℝ) =
+            (1809243093871961 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1809243093871961 / 1250000000000000 : ℝ) ≤
+            (184882513160285798053 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (891370548148297 / 10000000000000000 : ℝ)) (q :=
+      (1809243093871961 / 10000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_475_13 :
+    Uω (aρ 13) (bρ 13) (8488650990281 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_475_14 :
+    Uω (aρ 14) (bρ 14) (8488650990281 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_475_15 :
+    Uω (aρ 15) (bρ 15) (8488650990281 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_475_16 :
+    Uω (aρ 16) (bρ 16) (8488650990281 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_475 :
+    Uρ (8488650990281 / 16000000000000 : ℝ) ≤
+      (-(9977570086192114256609 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (8488650990281 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(6460958978972525251067 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_475_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (8488650990281 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3253372958554693937133 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_475_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (8488650990281 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1649750631088205402597 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_475_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (8488650990281 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3377142242700800895389 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_475_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (8488650990281 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1748969525043867698289 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_475_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (8488650990281 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3676915032743383951457 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_475_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (8488650990281 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(7866048355351224914357 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_475_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (8488650990281 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(858182888206397318407 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_475_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (8488650990281 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(2392836977549402281269 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_475_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (8488650990281 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(10953204368456377232327 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_475_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (8488650990281 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(3249899421236699954177 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_475_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (8488650990281 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(854838257289714201947 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_475_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (8488650990281 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_475_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (8488650990281 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_475_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (8488650990281 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_475_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (8488650990281 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_475_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_476_1 :
+    Uω (aρ 1) (bρ 1) (34034490843159 / 64000000000000 : ℝ) ≤
+      (-(6437169920414059672691 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (42026889615827843 / 80000000000000000 : ℝ) ≤
+      (-(6437169920414059672691 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (42026889615827843 / 80000000000000000 : ℝ) =
+            (42026889615827843 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (42026889615827843 / 40000000000000000 : ℝ) ≤
+            (494301882585940327309 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5253330427798367 / 10000000000000000 : ℝ)) (q :=
+      (42026889615827843 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_476_2 :
+    Uω (aρ 2) (bρ 2) (34034490843159 / 64000000000000 : ℝ) ≤
+      (-(6482847028228737411169 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (41835360696377779 / 80000000000000000 : ℝ) ≤
+      (-(6482847028228737411169 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (41835360696377779 / 80000000000000000 : ℝ) =
+            (41835360696377779 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (41835360696377779 / 40000000000000000 : ℝ) ≤
+            (448624774771262588831 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5229217210445851 / 10000000000000000 : ℝ)) (q :=
+      (41835360696377779 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_476_3 :
+    Uω (aρ 3) (bρ 3) (34034490843159 / 64000000000000 : ℝ) ≤
+      (-(6574880008487592758289 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (41452103723115147 / 80000000000000000 : ℝ) ≤
+      (-(6574880008487592758289 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (41452103723115147 / 80000000000000000 : ℝ) =
+            (41452103723115147 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (41452103723115147 / 40000000000000000 : ℝ) ≤
+            (356591794512407241711 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5180799073480193 / 10000000000000000 : ℝ)) (q :=
+      (41452103723115147 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_476_4 :
+    Uω (aρ 4) (bρ 4) (34034490843159 / 64000000000000 : ℝ) ≤
+      (-(105152787365977387829 / 156250000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1632598586770399 / 3200000000000000 : ℝ) ≤
+      (-(105152787365977387829 / 156250000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1632598586770399 / 3200000000000000 : ℝ) =
+            (1632598586770399 / 1600000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1632598586770399 / 1600000000000000 : ℝ) ≤
+            (3151459555897612171 / 156250000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (25499042455407 / 50000000000000 : ℝ)) (q :=
+      (1632598586770399 / 3200000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_476_5 :
+    Uω (aρ 5) (bρ 5) (34034490843159 / 64000000000000 : ℝ) ≤
+      (-(1742689163167899786689 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (39843168847047383 / 80000000000000000 : ℝ) ≤
+      (-(1742689163167899786689 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (39843168847047383 / 80000000000000000 : ℝ) =
+            (39843168847047383 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (39843168847047383 / 20000000000000000 : ℝ) ≤
+            (1723046738332100213311 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1243762363715813 / 2500000000000000 : ℝ)) (q :=
+      (39843168847047383 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_476_6 :
+    Uω (aρ 6) (bρ 6) (34034490843159 / 64000000000000 : ℝ) ≤
+      (-(7327752903325907666223 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (38445872532141979 / 80000000000000000 : ℝ) ≤
+      (-(7327752903325907666223 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (38445872532141979 / 80000000000000000 : ℝ) =
+            (38445872532141979 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (38445872532141979 / 20000000000000000 : ℝ) ≤
+            (6535190702674092333777 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4792965751276901 / 10000000000000000 : ℝ)) (q :=
+      (38445872532141979 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_476_7 :
+    Uω (aρ 7) (bρ 7) (34034490843159 / 64000000000000 : ℝ) ≤
+      (-(244953251082135055187 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (36531548535282223 / 80000000000000000 : ℝ) ≤
+      (-(244953251082135055187 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (36531548535282223 / 80000000000000000 : ℝ) =
+            (36531548535282223 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (36531548535282223 / 20000000000000000 : ℝ) ≤
+            (188263736605364944813 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2269023066410981 / 5000000000000000 : ℝ)) (q :=
+      (36531548535282223 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_476_8 :
+    Uω (aρ 8) (bρ 8) (34034490843159 / 64000000000000 : ℝ) ≤
+      (-(855201131837044779203 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (34015812917868879 / 80000000000000000 : ℝ) ≤
+      (-(855201131837044779203 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (34015812917868879 / 80000000000000000 : ℝ) =
+            (34015812917868879 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (34015812917868879 / 20000000000000000 : ℝ) ≤
+            (531093228762955220797 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2096326939904313 / 5000000000000000 : ℝ)) (q :=
+      (34015812917868879 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_476_9 :
+    Uω (aρ 9) (bρ 9) (34034490843159 / 64000000000000 : ℝ) ≤
+      (-(190757415289597866371 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (30822334019602451 / 80000000000000000 : ℝ) ≤
+      (-(190757415289597866371 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (30822334019602451 / 80000000000000000 : ℝ) =
+            (30822334019602451 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (30822334019602451 / 20000000000000000 : ℝ) ≤
+            (86501456830402133629 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3735249783602019 / 10000000000000000 : ℝ)) (q :=
+      (30822334019602451 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_476_10 :
+    Uω (aρ 10) (bρ 10) (34034490843159 / 64000000000000 : ℝ) ≤
+      (-(10913279957409600345849 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (26861623664032427 / 80000000000000000 : ℝ) ≤
+      (-(10913279957409600345849 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (26861623664032427 / 80000000000000000 : ℝ) =
+            (26861623664032427 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26861623664032427 / 20000000000000000 : ℝ) ≤
+            (2949663648590399654151 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3133631228204513 / 10000000000000000 : ℝ)) (q :=
+      (26861623664032427 / 80000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_476_11 :
+    Uω (aρ 11) (bρ 11) (34034490843159 / 64000000000000 : ℝ) ≤
+      (-(6472764072437904424893 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21921629978898067 / 80000000000000000 : ℝ) ≤
+      (-(6472764072437904424893 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (21921629978898067 / 80000000000000000 : ℝ) =
+            (21921629978898067 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21921629978898067 / 20000000000000000 : ℝ) ≤
+            (458707730562095575107 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2317109433125923 / 10000000000000000 : ℝ)) (q :=
+      (21921629978898067 / 80000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_476_12 :
+    Uω (aρ 12) (bρ 12) (34034490843159 / 64000000000000 : ℝ) ≤
+      (-(16959611872217714546673 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14673827234924079 / 80000000000000000 : ℝ) ≤
+      (-(16959611872217714546673 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (14673827234924079 / 80000000000000000 : ℝ) =
+            (14673827234924079 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14673827234924079 / 10000000000000000 : ℝ) ≤
+            (3834803536782285453327 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (464429421908713 / 5000000000000000 : ℝ)) (q :=
+      (14673827234924079 / 80000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_476_13 :
+    Uω (aρ 13) (bρ 13) (34034490843159 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_476_14 :
+    Uω (aρ 14) (bρ 14) (34034490843159 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_476_15 :
+    Uω (aρ 15) (bρ 15) (34034490843159 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_476_16 :
+    Uω (aρ 16) (bρ 16) (34034490843159 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_476 :
+    Uρ (34034490843159 / 64000000000000 : ℝ) ≤
+      (-(62156909653359306433 / 62500000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (34034490843159 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(6437169920414059672691 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_476_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (34034490843159 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(6482847028228737411169 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_476_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (34034490843159 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(6574880008487592758289 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_476_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (34034490843159 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(105152787365977387829 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_476_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (34034490843159 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1742689163167899786689 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_476_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (34034490843159 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(7327752903325907666223 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_476_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (34034490843159 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(244953251082135055187 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_476_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (34034490843159 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(855201131837044779203 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_476_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (34034490843159 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(190757415289597866371 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_476_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (34034490843159 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(10913279957409600345849 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_476_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (34034490843159 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(6472764072437904424893 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_476_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (34034490843159 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(16959611872217714546673 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_476_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (34034490843159 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_476_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (34034490843159 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_476_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (34034490843159 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_476_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (34034490843159 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_476_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_477_1 :
+    Uω (aρ 1) (bρ 1) (17057188862597 / 32000000000000 : ℝ) ≤
+      (-(1603359330041331640883 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4212674880195987 / 8000000000000000 : ℝ) ≤
+      (-(1603359330041331640883 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (4212674880195987 / 8000000000000000 : ℝ) =
+            (4212674880195987 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4212674880195987 / 4000000000000000 : ℝ) ≤
+            (129508620708668359117 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1053162579802681 / 2000000000000000 : ℝ)) (q :=
+      (4212674880195987 / 8000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_477_2 :
+    Uω (aρ 2) (bρ 2) (17057188862597 / 32000000000000 : ℝ) ≤
+      (-(6459005123300066603403 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4193522316388401 / 8000000000000000 : ℝ) ≤
+      (-(6459005123300066603403 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (4193522316388401 / 8000000000000000 : ℝ) =
+            (4193522316388401 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4193522316388401 / 4000000000000000 : ℝ) ≤
+            (472466679699933396597 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (131042512550111 / 250000000000000 : ℝ)) (q :=
+      (4193522316388401 / 8000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_477_3 :
+    Uω (aρ 3) (bρ 3) (17057188862597 / 32000000000000 : ℝ) ≤
+      (-(1310163111631396293071 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20775988026334369 / 40000000000000000 : ℝ) ≤
+      (-(1310163111631396293071 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (20775988026334369 / 40000000000000000 : ℝ) =
+            (20775988026334369 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20775988026334369 / 20000000000000000 : ℝ) ≤
+            (76131248968603706929 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2596642415275311 / 5000000000000000 : ℝ)) (q :=
+      (20775988026334369 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_477_4 :
+    Uω (aρ 4) (bρ 4) (17057188862597 / 32000000000000 : ℝ) ≤
+      (-(6705332253855386408567 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20457431775354757 / 40000000000000000 : ℝ) ≤
+      (-(6705332253855386408567 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (20457431775354757 / 40000000000000000 : ℝ) =
+            (20457431775354757 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20457431775354757 / 20000000000000000 : ℝ) ≤
+            (226139549144613591433 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (639037610765727 / 1250000000000000 : ℝ)) (q :=
+      (20457431775354757 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_477_5 :
+    Uω (aρ 5) (bρ 5) (17057188862597 / 32000000000000 : ℝ) ≤
+      (-(3472849145369206145161 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19971567249032607 / 40000000000000000 : ℝ) ≤
+      (-(3472849145369206145161 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (19971567249032607 / 40000000000000000 : ℝ) =
+            (19971567249032607 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 15
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19971567249032607 / 10000000000000000 : ℝ) ≤
+            (3458622657630793854839 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4987558542299741 / 10000000000000000 : ℝ)) (q :=
+      (19971567249032607 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_477_6 :
+    Uω (aρ 6) (bρ 6) (17057188862597 / 32000000000000 : ℝ) ≤
+      (-(7301743926598747382951 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3854599646214253 / 8000000000000000 : ℝ) ≤
+      (-(7301743926598747382951 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3854599646214253 / 8000000000000000 : ℝ) =
+            (3854599646214253 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3854599646214253 / 2000000000000000 : ℝ) ≤
+            (6561199679401252617049 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (480551440845907 / 1000000000000000 : ℝ)) (q :=
+      (3854599646214253 / 8000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_477_7 :
+    Uω (aρ 7) (bρ 7) (17057188862597 / 32000000000000 : ℝ) ≤
+      (-(1562207263759628217761 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (18316015145861673 / 40000000000000000 : ℝ) ≤
+      (-(1562207263759628217761 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (18316015145861673 / 40000000000000000 : ℝ) =
+            (18316015145861673 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18316015145861673 / 10000000000000000 : ℝ) ≤
+            (1210381457440371782239 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2275342123307137 / 5000000000000000 : ℝ)) (q :=
+      (18316015145861673 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_477_8 :
+    Uω (aρ 8) (bρ 8) (17057188862597 / 32000000000000 : ℝ) ≤
+      (-(8522284899989832254591 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17058540093797051 / 40000000000000000 : ℝ) ≤
+      (-(8522284899989832254591 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (17058540093797051 / 40000000000000000 : ℝ) =
+            (17058540093797051 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17058540093797051 / 10000000000000000 : ℝ) ≤
+            (5340658706010167745409 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4205488371921963 / 10000000000000000 : ℝ)) (q :=
+      (17058540093797051 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_477_9 :
+    Uω (aρ 9) (bρ 9) (17057188862597 / 32000000000000 : ℝ) ≤
+      (-(950451232191955561317 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3092532420961119 / 8000000000000000 : ℝ) ≤
+      (-(950451232191955561317 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3092532420961119 / 8000000000000000 : ℝ) =
+            (3092532420961119 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3092532420961119 / 2000000000000000 : ℝ) ≤
+            (435843128408044438683 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (749703001157247 / 2000000000000000 : ℝ)) (q :=
+      (3092532420961119 / 8000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_477_10 :
+    Uω (aρ 10) (bρ 10) (17057188862597 / 32000000000000 : ℝ) ≤
+      (-(10873536910612324331361 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2696859236296747 / 8000000000000000 : ℝ) ≤
+      (-(10873536910612324331361 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2696859236296747 / 8000000000000000 : ℝ) =
+            (2696859236296747 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2696859236296747 / 2000000000000000 : ℝ) ≤
+            (2989406695387675668639 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (629578215524061 / 2000000000000000 : ℝ)) (q :=
+      (2696859236296747 / 8000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_477_11 :
+    Uω (aρ 11) (bρ 11) (17057188862597 / 32000000000000 : ℝ) ≤
+      (-(12891854801988329159687 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11019803511520029 / 40000000000000000 : ℝ) ≤
+      (-(12891854801988329159687 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (11019803511520029 / 40000000000000000 : ℝ) =
+            (11019803511520029 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11019803511520029 / 10000000000000000 : ℝ) ≤
+            (971088804011670840313 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (583530342210863 / 2500000000000000 : ℝ)) (q :=
+      (11019803511520029 / 40000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_477_12 :
+    Uω (aρ 12) (bρ 12) (17057188862597 / 32000000000000 : ℝ) ≤
+      (-(2103474338028338746597 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7434266967437877 / 40000000000000000 : ℝ) ≤
+      (-(2103474338028338746597 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (7434266967437877 / 40000000000000000 : ℝ) =
+            (7434266967437877 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7434266967437877 / 5000000000000000 : ℝ) ≤
+            (495827588096661253403 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (60315824592961 / 625000000000000 : ℝ)) (q :=
+      (7434266967437877 / 40000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_477_13 :
+    Uω (aρ 13) (bρ 13) (17057188862597 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_477_14 :
+    Uω (aρ 14) (bρ 14) (17057188862597 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_477_15 :
+    Uω (aρ 15) (bρ 15) (17057188862597 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_477_16 :
+    Uω (aρ 16) (bρ 16) (17057188862597 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_477 :
+    Uρ (17057188862597 / 32000000000000 : ℝ) ≤
+      (-(2478276551493754470733 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (17057188862597 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1603359330041331640883 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_477_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (17057188862597 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(6459005123300066603403 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_477_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (17057188862597 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1310163111631396293071 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_477_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (17057188862597 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(6705332253855386408567 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_477_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (17057188862597 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(3472849145369206145161 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_477_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (17057188862597 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(7301743926598747382951 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_477_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (17057188862597 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(1562207263759628217761 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_477_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (17057188862597 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(8522284899989832254591 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_477_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (17057188862597 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(950451232191955561317 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_477_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (17057188862597 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(10873536910612324331361 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_477_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (17057188862597 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(12891854801988329159687 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_477_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (17057188862597 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(2103474338028338746597 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_477_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (17057188862597 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_477_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (17057188862597 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_477_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (17057188862597 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_477_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (17057188862597 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_477_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_478_1 :
+    Uω (aρ 1) (bρ 1) (34194264607229 / 64000000000000 : ℝ) ≤
+      (-(6389760910873824690131 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (42226607985331693 / 80000000000000000 : ℝ) ≤
+      (-(6389760910873824690131 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (42226607985331693 / 80000000000000000 : ℝ) =
+            (42226607985331693 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (42226607985331693 / 40000000000000000 : ℝ) ≤
+            (541710892126175309869 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (659786921192299 / 1250000000000000 : ℝ)) (q :=
+      (42226607985331693 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_478_2 :
+    Uω (aρ 2) (bρ 2) (34194264607229 / 64000000000000 : ℝ) ≤
+      (-(6435219931206354359771 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8407017122605121 / 16000000000000000 : ℝ) ≤
+      (-(6435219931206354359771 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (8407017122605121 / 16000000000000000 : ℝ) =
+            (8407017122605121 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8407017122605121 / 8000000000000000 : ℝ) ≤
+            (496251871793645640229 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (525418378897187 / 1000000000000000 : ℝ)) (q :=
+      (8407017122605121 / 16000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_478_3 :
+    Uω (aρ 3) (bρ 3) (34194264607229 / 64000000000000 : ℝ) ≤
+      (-(6526808894415663314091 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (41651848316384409 / 80000000000000000 : ℝ) ≤
+      (-(6526808894415663314091 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (41651848316384409 / 80000000000000000 : ℝ) =
+            (41651848316384409 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (41651848316384409 / 40000000000000000 : ℝ) ≤
+            (404662908584336685909 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5205770571161571 / 10000000000000000 : ℝ)) (q :=
+      (41651848316384409 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_478_4 :
+    Uω (aρ 4) (bρ 4) (34194264607229 / 64000000000000 : ℝ) ≤
+      (-(1670236444950892870231 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (41014762235867349 / 80000000000000000 : ℝ) ≤
+      (-(1670236444950892870231 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (41014762235867349 / 80000000000000000 : ℝ) =
+            (41014762235867349 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (41014762235867349 / 40000000000000000 : ℝ) ≤
+            (62631505799107129769 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2562396616048653 / 5000000000000000 : ℝ)) (q :=
+      (41014762235867349 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_478_5 :
+    Uω (aρ 5) (bρ 5) (34194264607229 / 64000000000000 : ℝ) ≤
+      (-(3460351350126436706987 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (40043099614031753 / 80000000000000000 : ℝ) ≤
+      (-(3460351350126436706987 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (40043099614031753 / 80000000000000000 : ℝ) =
+            (40043099614031753 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 2
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (40043099614031753 / 40000000000000000 : ℝ) ≤
+            (5384551373563293013 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5000067495973407 / 10000000000000000 : ℝ)) (q :=
+      (40043099614031753 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_478_6 :
+    Uω (aρ 6) (bρ 6) (34194264607229 / 64000000000000 : ℝ) ≤
+      (-(7275802777793215124253 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (38646119013703849 / 80000000000000000 : ℝ) ≤
+      (-(7275802777793215124253 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (38646119013703849 / 80000000000000000 : ℝ) =
+            (38646119013703849 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (38646119013703849 / 20000000000000000 : ℝ) ≤
+            (6587140828206784875747 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4818062721031431 / 10000000000000000 : ℝ)) (q :=
+      (38646119013703849 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_478_7 :
+    Uω (aρ 7) (bρ 7) (34194264607229 / 64000000000000 : ℝ) ≤
+      (-(194591119444156857651 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (36732508617790769 / 80000000000000000 : ℝ) ≤
+      (-(194591119444156857651 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (36732508617790769 / 80000000000000000 : ℝ) =
+            (36732508617790769 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (36732508617790769 / 20000000000000000 : ℝ) ≤
+            (151982470705843142349 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4563321502813161 / 10000000000000000 : ℝ)) (q :=
+      (36732508617790769 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_478_8 :
+    Uω (aρ 8) (bρ 8) (34194264607229 / 64000000000000 : ℝ) ≤
+      (-(8492649056459973840639 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1368733560120661 / 3200000000000000 : ℝ) ≤
+      (-(8492649056459973840639 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1368733560120661 / 3200000000000000 : ℝ) =
+            (1368733560120661 / 800000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1368733560120661 / 800000000000000 : ℝ) ≤
+            (5370294549540026159361 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (10545801876149 / 25000000000000 : ℝ)) (q :=
+      (1368733560120661 / 3200000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_478_9 :
+    Uω (aρ 9) (bρ 9) (34194264607229 / 64000000000000 : ℝ) ≤
+      (-(9471271696007942156663 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6205658593083737 / 16000000000000000 : ℝ) ≤
+      (-(9471271696007942156663 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6205658593083737 / 16000000000000000 : ℝ) =
+            (6205658593083737 / 4000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6205658593083737 / 4000000000000000 : ℝ) ≤
+            (4391671909992057843337 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (188088743471007 / 500000000000000 : ℝ)) (q :=
+      (6205658593083737 / 16000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_478_10 :
+    Uω (aρ 10) (bρ 10) (34194264607229 / 64000000000000 : ℝ) ≤
+      (-(5416986698976334785817 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27075500931979433 / 80000000000000000 : ℝ) ≤
+      (-(5416986698976334785817 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (27075500931979433 / 80000000000000000 : ℝ) =
+            (27075500931979433 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27075500931979433 / 20000000000000000 : ℝ) ≤
+            (1514485104023665214183 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3162135894555327 / 10000000000000000 : ℝ)) (q :=
+      (27075500931979433 / 80000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_478_11 :
+    Uω (aρ 11) (bρ 11) (34194264607229 / 64000000000000 : ℝ) ≤
+      (-(12838570585395820165127 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22157356774028233 / 80000000000000000 : ℝ) ≤
+      (-(12838570585395820165127 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (22157356774028233 / 80000000000000000 : ℝ) =
+            (22157356774028233 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22157356774028233 / 20000000000000000 : ℝ) ≤
+            (1024373020604179834873 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2351076481272527 / 10000000000000000 : ℝ)) (q :=
+      (22157356774028233 / 80000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_478_12 :
+    Uω (aρ 12) (bρ 12) (34194264607229 / 64000000000000 : ℝ) ≤
+      (-(4175188987432682907409 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15058626839837021 / 80000000000000000 : ℝ) ≤
+      (-(4175188987432682907409 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (15058626839837021 / 80000000000000000 : ℝ) =
+            (15058626839837021 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15058626839837021 / 10000000000000000 : ℝ) ≤
+            (1023414864817317092591 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (250023523602431 / 2500000000000000 : ℝ)) (q :=
+      (15058626839837021 / 80000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_478_13 :
+    Uω (aρ 13) (bρ 13) (34194264607229 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_478_14 :
+    Uω (aρ 14) (bρ 14) (34194264607229 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_478_15 :
+    Uω (aρ 15) (bρ 15) (34194264607229 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_478_16 :
+    Uω (aρ 16) (bρ 16) (34194264607229 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_478 :
+    Uρ (34194264607229 / 64000000000000 : ℝ) ≤
+      (-(123519148321299201723 / 125000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (34194264607229 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(6389760910873824690131 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_478_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (34194264607229 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(6435219931206354359771 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_478_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (34194264607229 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(6526808894415663314091 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_478_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (34194264607229 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1670236444950892870231 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_478_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (34194264607229 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(3460351350126436706987 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_478_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (34194264607229 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(7275802777793215124253 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_478_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (34194264607229 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(194591119444156857651 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_478_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (34194264607229 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(8492649056459973840639 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_478_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (34194264607229 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(9471271696007942156663 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_478_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (34194264607229 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(5416986698976334785817 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_478_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (34194264607229 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(12838570585395820165127 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_478_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (34194264607229 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(4175188987432682907409 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_478_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (34194264607229 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_478_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (34194264607229 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_478_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (34194264607229 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_478_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (34194264607229 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_478_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_479_1 :
+    Uω (aρ 1) (bρ 1) (68468416096493 / 128000000000000 : ℝ) ≤
+      (-(637794369480791432831 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (84553075151977277 / 160000000000000000 : ℝ) ≤
+      (-(637794369480791432831 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (84553075151977277 / 160000000000000000 : ℝ) =
+            (84553075151977277 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (84553075151977277 / 80000000000000000 : ℝ) ≤
+            (55352810819208567169 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1321134151135911 / 2500000000000000 : ℝ)) (q :=
+      (84553075151977277 / 160000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_479_2 :
+    Uω (aρ 2) (bρ 2) (68468416096493 / 128000000000000 : ℝ) ≤
+      (-(321167425913844597629 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (84170033661501037 / 160000000000000000 : ℝ) ≤
+      (-(321167425913844597629 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (84170033661501037 / 160000000000000000 : ℝ) =
+            (84170033661501037 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (84170033661501037 / 80000000000000000 : ℝ) ≤
+            (25406164236155402371 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2630212715372057 / 5000000000000000 : ℝ)) (q :=
+      (84170033661501037 / 160000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_479_3 :
+    Uω (aρ 3) (bρ 3) (68468416096493 / 128000000000000 : ℝ) ≤
+      (-(1302965429178912640407 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16680713769480353 / 32000000000000000 : ℝ) ≤
+      (-(1302965429178912640407 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (16680713769480353 / 32000000000000000 : ℝ) =
+            (16680713769480353 / 16000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16680713769480353 / 16000000000000000 : ℝ) ≤
+            (83328931421087359593 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1042402687066341 / 2000000000000000 : ℝ)) (q :=
+      (16680713769480353 / 32000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_479_4 :
+    Uω (aρ 4) (bρ 4) (68468416096493 / 128000000000000 : ℝ) ≤
+      (-(3334387412826117338583 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (82129423010569821 / 160000000000000000 : ℝ) ≤
+      (-(3334387412826117338583 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (82129423010569821 / 160000000000000000 : ℝ) =
+            (82129423010569821 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (82129423010569821 / 80000000000000000 : ℝ) ≤
+            (131348488673882661417 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (641379923349089 / 1250000000000000 : ℝ)) (q :=
+      (82129423010569821 / 160000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_479_5 :
+    Uω (aρ 5) (bρ 5) (68468416096493 / 128000000000000 : ℝ) ≤
+      (-(276329133748210301077 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (80186163945248261 / 160000000000000000 : ℝ) ≤
+      (-(276329133748210301077 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (80186163945248261 / 160000000000000000 : ℝ) =
+            (80186163945248261 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 2
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (80186163945248261 / 80000000000000000 : ℝ) ≤
+            (929738371789698923 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5006321922962517 / 10000000000000000 : ℝ)) (q :=
+      (80186163945248261 / 160000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_479_6 :
+    Uω (aρ 6) (bρ 6) (68468416096493 / 128000000000000 : ℝ) ≤
+      (-(7262857527910127196811 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (77392359551862197 / 160000000000000000 : ℝ) ≤
+      (-(7262857527910127196811 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (77392359551862197 / 160000000000000000 : ℝ) =
+            (77392359551862197 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (77392359551862197 / 40000000000000000 : ℝ) ≤
+            (6600086078089872803189 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4824336748929259 / 10000000000000000 : ℝ)) (q :=
+      (77392359551862197 / 160000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_479_7 :
+    Uω (aρ 7) (bρ 7) (68468416096493 / 128000000000000 : ℝ) ≤
+      (-(7769977439395536664309 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (73565493006656277 / 160000000000000000 : ℝ) ≤
+      (-(7769977439395536664309 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (73565493006656277 / 160000000000000000 : ℝ) =
+            (73565493006656277 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (73565493006656277 / 40000000000000000 : ℝ) ≤
+            (6092966166604463335691 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4569639811538519 / 10000000000000000 : ℝ)) (q :=
+      (73565493006656277 / 160000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_479_8 :
+    Uω (aρ 8) (bρ 8) (68468416096493 / 128000000000000 : ℝ) ≤
+      (-(8477864923492468921267 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (68537930528878301 / 160000000000000000 : ℝ) ≤
+      (-(8477864923492468921267 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (68537930528878301 / 160000000000000000 : ℝ) =
+            (68537930528878301 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (68537930528878301 / 40000000000000000 : ℝ) ≤
+            (5385078682507531078733 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (264046009572267 / 625000000000000 : ℝ)) (q :=
+      (68537930528878301 / 160000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8])
+      (by norm_num [bρ_8]) (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_479_9 :
+    Uω (aρ 9) (bρ 9) (68468416096493 / 128000000000000 : ℝ) ≤
+      (-(9454695290100289161531 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (62159538752414389 / 160000000000000000 : ℝ) ≤
+      (-(9454695290100289161531 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (62159538752414389 / 160000000000000000 : ℝ) =
+            (62159538752414389 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (62159538752414389 / 40000000000000000 : ℝ) ≤
+            (4408248315899710838469 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3768402809458283 / 10000000000000000 : ℝ)) (q :=
+      (62159538752414389 / 160000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9])
+      (by norm_num [bρ_9]) (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_479_10 :
+    Uω (aρ 10) (bρ 10) (68468416096493 / 128000000000000 : ℝ) ≤
+      (-(10814258403181725422127 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (54257865842343221 / 160000000000000000 : ℝ) ≤
+      (-(10814258403181725422127 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (54257865842343221 / 160000000000000000 : ℝ) =
+            (54257865842343221 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (54257865842343221 / 40000000000000000 : ℝ) ≤
+            (3048685202818274577873 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3169252729194387 / 10000000000000000 : ℝ)) (q :=
+      (54257865842343221 / 160000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_479_11 :
+    Uω (aρ 11) (bρ 11) (68468416096493 / 128000000000000 : ℝ) ≤
+      (-(12812072247175655142927 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (44432295892864621 / 160000000000000000 : ℝ) ≤
+      (-(12812072247175655142927 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (44432295892864621 / 160000000000000000 : ℝ) =
+            (44432295892864621 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (44432295892864621 / 40000000000000000 : ℝ) ≤
+            (1050871358824344857073 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1179766555857281 / 5000000000000000 : ℝ)) (q :=
+      (44432295892864621 / 160000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_479_12 :
+    Uω (aρ 12) (bρ 12) (68468416096493 / 128000000000000 : ℝ) ≤
+      (-(16638879770636086593247 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (30304185472766581 / 160000000000000000 : ℝ) ≤
+      (-(16638879770636086593247 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (30304185472766581 / 160000000000000000 : ℝ) =
+            (30304185472766581 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (30304185472766581 / 20000000000000000 : ℝ) ≤
+            (4155535638363913406753 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1017219405887307 / 10000000000000000 : ℝ)) (q :=
+      (30304185472766581 / 160000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_479_13 :
+    Uω (aρ 13) (bρ 13) (68468416096493 / 128000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_479_14 :
+    Uω (aρ 14) (bρ 14) (68468416096493 / 128000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_479_15 :
+    Uω (aρ 15) (bρ 15) (68468416096493 / 128000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_479_16 :
+    Uω (aρ 16) (bρ 16) (68468416096493 / 128000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_479 :
+    Uρ (68468416096493 / 128000000000000 : ℝ) ≤
+      (-(2466473327665301070027 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (68468416096493 / 128000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(637794369480791432831 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_479_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (68468416096493 / 128000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(321167425913844597629 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_479_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (68468416096493 / 128000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1302965429178912640407 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_479_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (68468416096493 / 128000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3334387412826117338583 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_479_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (68468416096493 / 128000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(276329133748210301077 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_479_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (68468416096493 / 128000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(7262857527910127196811 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_479_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (68468416096493 / 128000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(7769977439395536664309 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_479_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (68468416096493 / 128000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(8477864923492468921267 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_479_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (68468416096493 / 128000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(9454695290100289161531 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_479_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (68468416096493 / 128000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(10814258403181725422127 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_479_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (68468416096493 / 128000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(12812072247175655142927 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_479_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (68468416096493 / 128000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(16638879770636086593247 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_479_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (68468416096493 / 128000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_479_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (68468416096493 / 128000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_479_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (68468416096493 / 128000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_479_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (68468416096493 / 128000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_479_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_480_1 :
+    Uω (aρ 1) (bρ 1) (2142134468079 / 4000000000000 : ℝ) ≤
+      (-(10185824683330590031 / 16000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1322702098936339 / 2500000000000000 : ℝ) ≤
+      (-(10185824683330590031 / 16000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1322702098936339 / 2500000000000000 : ℝ) =
+            (1322702098936339 / 1250000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1322702098936339 / 1250000000000000 : ℝ) ≤
+            (904530201469409969 / 16000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1322694459844553 / 2500000000000000 : ℝ)) (q :=
+      (1322702098936339 / 2500000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_480_2 :
+    Uω (aρ 2) (bρ 2) (2142134468079 / 4000000000000 : ℝ) ≤
+      (-(1602872795690252040067 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10533737010983287 / 20000000000000000 : ℝ) ≤
+      (-(1602872795690252040067 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10533737010983287 / 20000000000000000 : ℝ) =
+            (10533737010983287 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10533737010983287 / 10000000000000000 : ℝ) ≤
+            (129995155059747959933 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5266667071380787 / 10000000000000000 : ℝ)) (q :=
+      (10533737010983287 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_480_3 :
+    Uω (aρ 3) (bρ 3) (2142134468079 / 4000000000000 : ℝ) ≤
+      (-(6502859740324113881097 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10437930128683687 / 20000000000000000 : ℝ) ≤
+      (-(6502859740324113881097 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10437930128683687 / 20000000000000000 : ℝ) =
+            (10437930128683687 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10437930128683687 / 10000000000000000 : ℝ) ≤
+            (428612062675886118903 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5218256295431187 / 10000000000000000 : ℝ)) (q :=
+      (10437930128683687 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_480_4 :
+    Uω (aρ 4) (bρ 4) (2142134468079 / 4000000000000 : ℝ) ≤
+      (-(6656618678513322478103 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5139332590770681 / 10000000000000000 : ℝ) ≤
+      (-(6656618678513322478103 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5139332590770681 / 10000000000000000 : ℝ) =
+            (5139332590770681 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5139332590770681 / 5000000000000000 : ℝ) ≤
+            (274853124486677521897 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2568642764676931 / 5000000000000000 : ℝ)) (q :=
+      (5139332590770681 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_480_5 :
+    Uω (aρ 5) (bρ 5) (2142134468079 / 4000000000000 : ℝ) ≤
+      (-(21549279880243073947 / 31250000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2508941512434543 / 5000000000000000 : ℝ) ≤
+      (-(21549279880243073947 / 31250000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2508941512434543 / 5000000000000000 : ℝ) =
+            (2508941512434543 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2508941512434543 / 2500000000000000 : ℝ) ≤
+            (111569504131926053 / 31250000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (626572039610709 / 1250000000000000 : ℝ)) (q :=
+      (2508941512434543 / 5000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_480_6 :
+    Uω (aρ 6) (bρ 6) (2142134468079 / 4000000000000 : ℝ) ≤
+      (-(7249929102216071350707 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4843280024696013 / 10000000000000000 : ℝ) ≤
+      (-(7249929102216071350707 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4843280024696013 / 10000000000000000 : ℝ) =
+            (4843280024696013 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4843280024696013 / 2500000000000000 : ℝ) ≤
+            (6613014503783928649293 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2415305345839763 / 5000000000000000 : ℝ)) (q :=
+      (4843280024696013 / 10000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_480_7 :
+    Uω (aρ 7) (bρ 7) (2142134468079 / 4000000000000 : ℝ) ≤
+      (-(7756328985092475948893 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2302061471369069 / 5000000000000000 : ℝ) ≤
+      (-(7756328985092475948893 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2302061471369069 / 5000000000000000 : ℝ) =
+            (2302061471369069 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2302061471369069 / 1250000000000000 : ℝ) ≤
+            (6106614620907524051107 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (17874835580171 / 39062500000000 : ℝ)) (q :=
+      (2302061471369069 / 5000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_480_8 :
+    Uω (aρ 8) (bρ 8) (2142134468079 / 4000000000000 : ℝ) ≤
+      (-(8463103222789283762703 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8579897360264411 / 20000000000000000 : ℝ) ≤
+      (-(8463103222789283762703 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8579897360264411 / 20000000000000000 : ℝ) =
+            (8579897360264411 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8579897360264411 / 5000000000000000 : ℝ) ≤
+            (5399840383210716237297 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4231151034651911 / 10000000000000000 : ℝ)) (q :=
+      (8579897360264411 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_480_9 :
+    Uω (aρ 9) (bρ 9) (2142134468079 / 4000000000000 : ℝ) ≤
+      (-(4719074005300819373633 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1945702532055567 / 5000000000000000 : ℝ) ≤
+      (-(4719074005300819373633 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1945702532055567 / 5000000000000000 : ℝ) =
+            (1945702532055567 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1945702532055567 / 1250000000000000 : ℝ) ≤
+            (2212397797699180626367 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (471878678871221 / 1250000000000000 : ℝ)) (q :=
+      (1945702532055567 / 5000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_480_10 :
+    Uω (aρ 10) (bρ 10) (2142134468079 / 4000000000000 : ℝ) ≤
+      (-(10794587619119258863077 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3397793772506151 / 10000000000000000 : ℝ) ≤
+      (-(10794587619119258863077 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3397793772506151 / 10000000000000000 : ℝ) =
+            (3397793772506151 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3397793772506151 / 2500000000000000 : ℝ) ≤
+            (3068355986880741136923 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1588182940627401 / 5000000000000000 : ℝ)) (q :=
+      (3397793772506151 / 10000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_480_11 :
+    Uω (aρ 11) (bρ 11) (2142134468079 / 4000000000000 : ℝ) ≤
+      (-(12785668633140302686513 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5568721028559307 / 20000000000000000 : ℝ) ≤
+      (-(12785668633140302686513 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5568721028559307 / 20000000000000000 : ℝ) =
+            (5568721028559307 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5568721028559307 / 5000000000000000 : ℝ) ≤
+            (1077274972859697313487 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2367975991006807 / 10000000000000000 : ℝ)) (q :=
+      (5568721028559307 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_480_12 :
+    Uω (aρ 12) (bρ 12) (2142134468079 / 4000000000000 : ℝ) ≤
+      (-(8289014659856279049473 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (476392968685189 / 2500000000000000 : ℝ) ≤
+      (-(8289014659856279049473 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (476392968685189 / 2500000000000000 : ℝ) =
+            (476392968685189 / 312500000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (476392968685189 / 312500000000000 : ℝ) ≤
+            (2108193044643720950527 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (258524702153503 / 2500000000000000 : ℝ)) (q :=
+      (476392968685189 / 2500000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12]) (by norm_num [bρ_12])
+      (by norm_num) (by norm_num [aρ_12, bρ_12]) (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_480_13 :
+    Uω (aρ 13) (bρ 13) (2142134468079 / 4000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_480_14 :
+    Uω (aρ 14) (bρ 14) (2142134468079 / 4000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_480_15 :
+    Uω (aρ 15) (bρ 15) (2142134468079 / 4000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_480_16 :
+    Uω (aρ 16) (bρ 16) (2142134468079 / 4000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_480 :
+    Uρ (2142134468079 / 4000000000000 : ℝ) ≤
+      (-(2462587192580934883643 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (2142134468079 / 4000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(10185824683330590031 / 16000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_480_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (2142134468079 / 4000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1602872795690252040067 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_480_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (2142134468079 / 4000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(6502859740324113881097 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_480_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (2142134468079 / 4000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(6656618678513322478103 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_480_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (2142134468079 / 4000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(21549279880243073947 / 31250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_480_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (2142134468079 / 4000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(7249929102216071350707 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_480_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (2142134468079 / 4000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(7756328985092475948893 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_480_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (2142134468079 / 4000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(8463103222789283762703 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_480_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (2142134468079 / 4000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(4719074005300819373633 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_480_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (2142134468079 / 4000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(10794587619119258863077 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_480_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (2142134468079 / 4000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(12785668633140302686513 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_480_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (2142134468079 / 4000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(8289014659856279049473 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_480_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (2142134468079 / 4000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_480_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (2142134468079 / 4000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_480_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (2142134468079 / 4000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_480_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (2142134468079 / 4000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_480_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_481_1 :
+    Uω (aρ 1) (bρ 1) (68628189860563 / 128000000000000 : ℝ) ≤
+      (-(635435107480579353849 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (84752793510513459 / 160000000000000000 : ℝ) ≤
+      (-(635435107480579353849 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (84752793510513459 / 160000000000000000 : ℝ) =
+            (84752793510513459 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (84752793510513459 / 80000000000000000 : ℝ) ≤
+            (57712072819420646151 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2648509537021349 / 5000000000000000 : ℝ)) (q :=
+      (84752793510513459 / 160000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_481_2 :
+    Uω (aρ 2) (bρ 2) (68628189860563 / 128000000000000 : ℝ) ≤
+      (-(799955986413626136757 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (84369758505179243 / 160000000000000000 : ℝ) ≤
+      (-(799955986413626136757 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (84369758505179243 / 160000000000000000 : ℝ) =
+            (84369758505179243 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (84369758505179243 / 80000000000000000 : ℝ) ≤
+            (66477988961373863243 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5272908710885921 / 10000000000000000 : ℝ)) (q :=
+      (84369758505179243 / 160000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_481_3 :
+    Uω (aρ 3) (bρ 3) (68628189860563 / 128000000000000 : ℝ) ≤
+      (-(6490906643397133693603 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (83603313179088739 / 160000000000000000 : ℝ) ≤
+      (-(6490906643397133693603 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (83603313179088739 / 160000000000000000 : ℝ) =
+            (83603313179088739 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (83603313179088739 / 80000000000000000 : ℝ) ≤
+            (440565159602866306397 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (326531196967163 / 625000000000000 : ℝ)) (q :=
+      (83603313179088739 / 160000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_481_4 :
+    Uω (aρ 4) (bρ 4) (68628189860563 / 128000000000000 : ℝ) ≤
+      (-(6644477302373994372467 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16465843959474311 / 32000000000000000 : ℝ) ≤
+      (-(6644477302373994372467 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (16465843959474311 / 32000000000000000 : ℝ) =
+            (16465843959474311 / 16000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16465843959474311 / 16000000000000000 : ℝ) ≤
+            (286994500626005627533 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (16073536436953 / 31250000000000 : ℝ)) (q :=
+      (16465843959474311 / 32000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_481_5 :
+    Uω (aρ 5) (bρ 5) (68628189860563 / 128000000000000 : ℝ) ≤
+      (-(6883326315246774074947 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (80386092587023787 / 160000000000000000 : ℝ) ≤
+      (-(6883326315246774074947 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (80386092587023787 / 160000000000000000 : ℝ) =
+            (80386092587023787 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (80386092587023787 / 80000000000000000 : ℝ) ≤
+            (48145487753225925053 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5018830677866489 / 10000000000000000 : ℝ)) (q :=
+      (80386092587023787 / 160000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_481_6 :
+    Uω (aρ 6) (bρ 6) (68628189860563 / 128000000000000 : ℝ) ≤
+      (-(7237017456809963206417 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (77592600559880403 / 160000000000000000 : ℝ) ≤
+      (-(7237017456809963206417 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (77592600559880403 / 160000000000000000 : ℝ) =
+            (77592600559880403 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (77592600559880403 / 40000000000000000 : ℝ) ≤
+            (6625926149190036793583 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2418442274806783 / 5000000000000000 : ℝ)) (q :=
+      (77592600559880403 / 160000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_481_7 :
+    Uω (aρ 7) (bρ 7) (68628189860563 / 128000000000000 : ℝ) ≤
+      (-(7742699362116177004641 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (73766439474050131 / 160000000000000000 : ℝ) ≤
+      (-(7742699362116177004641 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (73766439474050131 / 160000000000000000 : ℝ) =
+            (73766439474050131 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (73766439474050131 / 40000000000000000 : ℝ) ≤
+            (6120244243883822995359 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2291137897322391 / 5000000000000000 : ℝ)) (q :=
+      (73766439474050131 / 160000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_481_8 :
+    Uω (aρ 8) (bρ 8) (68628189860563 / 128000000000000 : ℝ) ≤
+      (-(33793455538278231109 / 40000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (68740423084680027 / 160000000000000000 : ℝ) ≤
+      (-(33793455538278231109 / 40000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (68740423084680027 / 160000000000000000 : ℝ) =
+            (68740423084680027 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (68740423084680027 / 40000000000000000 : ℝ) ≤
+            (21658318885721768891 / 40000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4237565397313519 / 10000000000000000 : ℝ)) (q :=
+      (68740423084680027 / 160000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8])
+      (by norm_num [bρ_8]) (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_481_9 :
+    Uω (aρ 9) (bρ 9) (68628189860563 / 128000000000000 : ℝ) ≤
+      (-(9421629749586235676443 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12473082561275951 / 32000000000000000 : ℝ) ≤
+      (-(9421629749586235676443 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (12473082561275951 / 32000000000000000 : ℝ) =
+            (12473082561275951 / 8000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12473082561275951 / 8000000000000000 : ℝ) ≤
+            (4441313856413764323557 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (756330948177197 / 2000000000000000 : ℝ)) (q :=
+      (12473082561275951 / 32000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_481_10 :
+    Uω (aρ 10) (bρ 10) (68628189860563 / 128000000000000 : ℝ) ≤
+      (-(10774960825271525424907 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10894301122940959 / 32000000000000000 : ℝ) ≤
+      (-(10774960825271525424907 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (10894301122940959 / 32000000000000000 : ℝ) =
+            (10894301122940959 / 8000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10894301122940959 / 8000000000000000 : ℝ) ≤
+            (3087982780728474575093 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (636695075084323 / 2000000000000000 : ℝ)) (q :=
+      (10894301122940959 / 32000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_481_11 :
+    Uω (aρ 11) (bρ 11) (68628189860563 / 128000000000000 : ℝ) ≤
+      (-(6379679458388677789959 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (44667131727406203 / 160000000000000000 : ℝ) ≤
+      (-(6379679458388677789959 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (44667131727406203 / 160000000000000000 : ℝ) =
+            (44667131727406203 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (44667131727406203 / 40000000000000000 : ℝ) ≤
+            (551792344611322210041 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2376405265714291 / 10000000000000000 : ℝ)) (q :=
+      (44667131727406203 / 160000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_481_12 :
+    Uω (aρ 12) (bρ 12) (68628189860563 / 128000000000000 : ℝ) ≤
+      (-(3303631472747489113263 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (30672242057537979 / 160000000000000000 : ℝ) ≤
+      (-(3303631472747489113263 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (30672242057537979 / 160000000000000000 : ℝ) =
+            (30672242057537979 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (30672242057537979 / 20000000000000000 : ℝ) ≤
+            (855251609052510886737 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1050744153665763 / 10000000000000000 : ℝ)) (q :=
+      (30672242057537979 / 160000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_481_13 :
+    Uω (aρ 13) (bρ 13) (68628189860563 / 128000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_481_14 :
+    Uω (aρ 14) (bρ 14) (68628189860563 / 128000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_481_15 :
+    Uω (aρ 15) (bρ 15) (68628189860563 / 128000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_481_16 :
+    Uω (aρ 16) (bρ 16) (68628189860563 / 128000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_481 :
+    Uρ (68628189860563 / 128000000000000 : ℝ) ≤
+      (-(4917447403195761472373 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (68628189860563 / 128000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(635435107480579353849 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_481_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (68628189860563 / 128000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(799955986413626136757 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_481_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (68628189860563 / 128000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(6490906643397133693603 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_481_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (68628189860563 / 128000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(6644477302373994372467 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_481_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (68628189860563 / 128000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(6883326315246774074947 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_481_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (68628189860563 / 128000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(7237017456809963206417 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_481_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (68628189860563 / 128000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(7742699362116177004641 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_481_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (68628189860563 / 128000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) * (-(33793455538278231109 / 40000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_481_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (68628189860563 / 128000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(9421629749586235676443 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_481_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (68628189860563 / 128000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(10774960825271525424907 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_481_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (68628189860563 / 128000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(6379679458388677789959 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_481_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (68628189860563 / 128000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(3303631472747489113263 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_481_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (68628189860563 / 128000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_481_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (68628189860563 / 128000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_481_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (68628189860563 / 128000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_481_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (68628189860563 / 128000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_481_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_482_1 :
+    Uω (aρ 1) (bρ 1) (34354038371299 / 64000000000000 : ℝ) ≤
+      (-(3171287802603740539783 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (42426326343872691 / 80000000000000000 : ℝ) ≤
+      (-(3171287802603740539783 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (42426326343872691 / 80000000000000000 : ℝ) =
+            (42426326343872691 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (42426326343872691 / 40000000000000000 : ℝ) ≤
+            (294448098896259460217 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (662907538567213 / 1250000000000000 : ℝ)) (q :=
+      (42426326343872691 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_482_2 :
+    Uω (aρ 2) (bρ 2) (34354038371299 / 64000000000000 : ℝ) ≤
+      (-(159695465267239470827 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (42234810456735991 / 80000000000000000 : ℝ) ≤
+      (-(159695465267239470827 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (42234810456735991 / 80000000000000000 : ℝ) =
+            (42234810456735991 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (42234810456735991 / 40000000000000000 : ℝ) ≤
+            (13591329807760529173 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5279150349263529 / 10000000000000000 : ℝ)) (q :=
+      (42234810456735991 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_482_3 :
+    Uω (aρ 3) (bρ 3) (34354038371299 / 64000000000000 : ℝ) ≤
+      (-(1619741955232353905561 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (41851592648187839 / 80000000000000000 : ℝ) ≤
+      (-(1619741955232353905561 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (41851592648187839 / 80000000000000000 : ℝ) =
+            (41851592648187839 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (41851592648187839 / 40000000000000000 : ℝ) ≤
+            (113125995517646094439 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5230742003476491 / 10000000000000000 : ℝ)) (q :=
+      (41851592648187839 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_482_4 :
+    Uω (aρ 4) (bρ 4) (34354038371299 / 64000000000000 : ℝ) ≤
+      (-(6632350661352732514341 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13188658887367 / 25600000000000 : ℝ) ≤
+      (-(6632350661352732514341 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (13188658887367 / 25600000000000 : ℝ) = (13188658887367 / 12800000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13188658887367 / 12800000000000 : ℝ) ≤
+            (299121141647267485659 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (20599111113 / 40000000000 : ℝ)) (q :=
+      (13188658887367 / 25600000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_482_5 :
+    Uω (aρ 5) (bρ 5) (34354038371299 / 64000000000000 : ℝ) ≤
+      (-(3435449282817292923889 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (40243028256793767 / 80000000000000000 : ℝ) ≤
+      (-(3435449282817292923889 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (40243028256793767 / 80000000000000000 : ℝ) =
+            (40243028256793767 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (40243028256793767 / 40000000000000000 : ℝ) ≤
+            (30286618682707076111 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5025085006027973 / 10000000000000000 : ℝ)) (q :=
+      (40243028256793767 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_482_6 :
+    Uω (aρ 6) (bρ 6) (34354038371299 / 64000000000000 : ℝ) ≤
+      (-(3612061273981600181139 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7769272004873171 / 16000000000000000 : ℝ) ≤
+      (-(3612061273981600181139 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7769272004873171 / 16000000000000000 : ℝ) =
+            (7769272004873171 / 4000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7769272004873171 / 4000000000000000 : ℝ) ≤
+            (3319410529018399818861 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (968631664612199 / 2000000000000000 : ℝ)) (q :=
+      (7769272004873171 / 16000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_482_7 :
+    Uω (aρ 7) (bρ 7) (34354038371299 / 64000000000000 : ℝ) ≤
+      (-(7729088517948928495677 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (36933455092172003 / 80000000000000000 : ℝ) ≤
+      (-(7729088517948928495677 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (36933455092172003 / 80000000000000000 : ℝ) =
+            (36933455092172003 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (36933455092172003 / 20000000000000000 : ℝ) ≤
+            (6133855088051071504323 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1147148367693133 / 2500000000000000 : ℝ)) (q :=
+      (36933455092172003 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_482_8 :
+    Uω (aρ 8) (bρ 8) (34354038371299 / 64000000000000 : ℝ) ≤
+      (-(4216823419692748033009 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1376833263107851 / 3200000000000000 : ℝ) ≤
+      (-(4216823419692748033009 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1376833263107851 / 3200000000000000 : ℝ) =
+            (1376833263107851 / 800000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1376833263107851 / 800000000000000 : ℝ) ≤
+            (2714648383307251966991 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5304974054367 / 12500000000000 : ℝ)) (q :=
+      (1376833263107851 / 3200000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_482_9 :
+    Uω (aρ 9) (bρ 9) (34354038371299 / 64000000000000 : ℝ) ≤
+      (-(2351285099938865213181 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (31234167074635759 / 80000000000000000 : ℝ) ≤
+      (-(2351285099938865213181 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (31234167074635759 / 80000000000000000 : ℝ) =
+            (31234167074635759 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (31234167074635759 / 20000000000000000 : ℝ) ≤
+            (1114450801561134786819 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3788278746088471 / 10000000000000000 : ℝ)) (q :=
+      (31234167074635759 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_482_10 :
+    Uω (aρ 10) (bρ 10) (34354038371299 / 64000000000000 : ℝ) ≤
+      (-(84026389085341504191 / 78125000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27289140900890719 / 80000000000000000 : ℝ) ≤
+      (-(84026389085341504191 / 78125000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (27289140900890719 / 80000000000000000 : ℝ) =
+            (27289140900890719 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27289140900890719 / 20000000000000000 : ℝ) ≤
+            (24277857836533495809 / 78125000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3190581236147211 / 10000000000000000 : ℝ)) (q :=
+      (27289140900890719 / 80000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_482_11 :
+    Uω (aρ 11) (bρ 11) (34354038371299 / 64000000000000 : ℝ) ≤
+      (-(2546628456716178291753 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4478438754372667 / 16000000000000000 : ℝ) ≤
+      (-(2546628456716178291753 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4478438754372667 / 16000000000000000 : ℝ) =
+            (4478438754372667 / 4000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4478438754372667 / 4000000000000000 : ℝ) ≤
+            (225960264483821708247 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (476964216019073 / 2000000000000000 : ℝ)) (q :=
+      (4478438754372667 / 16000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_482_12 :
+    Uω (aρ 12) (bρ 12) (34354038371299 / 64000000000000 : ℝ) ≤
+      (-(16459220212367328245369 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15426774638283691 / 80000000000000000 : ℝ) ≤
+      (-(16459220212367328245369 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (15426774638283691 / 80000000000000000 : ℝ) =
+            (15426774638283691 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15426774638283691 / 10000000000000000 : ℝ) ≤
+            (4335195196632671754631 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (533583196692727 / 5000000000000000 : ℝ)) (q :=
+      (15426774638283691 / 80000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_482_13 :
+    Uω (aρ 13) (bρ 13) (34354038371299 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_482_14 :
+    Uω (aρ 14) (bρ 14) (34354038371299 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_482_15 :
+    Uω (aρ 15) (bρ 15) (34354038371299 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_482_16 :
+    Uω (aρ 16) (bρ 16) (34354038371299 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_482 :
+    Uρ (34354038371299 / 64000000000000 : ℝ) ≤
+      (-(9819528231062824967897 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (34354038371299 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3171287802603740539783 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_482_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (34354038371299 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(159695465267239470827 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_482_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (34354038371299 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1619741955232353905561 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_482_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (34354038371299 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(6632350661352732514341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_482_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (34354038371299 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(3435449282817292923889 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_482_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (34354038371299 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3612061273981600181139 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_482_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (34354038371299 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(7729088517948928495677 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_482_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (34354038371299 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(4216823419692748033009 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_482_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (34354038371299 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(2351285099938865213181 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_482_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (34354038371299 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) * (-(84026389085341504191 / 78125000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_482_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (34354038371299 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(2546628456716178291753 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_482_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (34354038371299 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(16459220212367328245369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_482_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (34354038371299 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_482_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (34354038371299 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_482_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (34354038371299 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_482_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (34354038371299 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_482_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_483_1 :
+    Uω (aρ 1) (bρ 1) (68787963624633 / 128000000000000 : ℝ) ≤
+      (-(6330813985629369780369 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (84952511863626233 / 160000000000000000 : ℝ) ≤
+      (-(6330813985629369780369 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (84952511863626233 / 160000000000000000 : ℝ) =
+            (84952511863626233 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (84952511863626233 / 80000000000000000 : ℝ) ≤
+            (600657817370630219631 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2654750771431913 / 5000000000000000 : ℝ)) (q :=
+      (84952511863626233 / 160000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_483_2 :
+    Uω (aρ 2) (bρ 2) (68787963624633 / 128000000000000 : ℝ) ≤
+      (-(3188001653894608997351 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (84569483312776481 / 160000000000000000 : ℝ) ≤
+      (-(3188001653894608997351 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (84569483312776481 / 160000000000000000 : ℝ) =
+            (84569483312776481 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (84569483312776481 / 80000000000000000 : ℝ) ≤
+            (277734247605391002649 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5285391986517607 / 10000000000000000 : ℝ)) (q :=
+      (84569483312776481 / 160000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_483_3 :
+    Uω (aρ 3) (bρ 3) (68787963624633 / 128000000000000 : ℝ) ≤
+      (-(3233521619429571196959 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (83803057381445937 / 160000000000000000 : ℝ) ≤
+      (-(3233521619429571196959 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (83803057381445937 / 160000000000000000 : ℝ) =
+            (83803057381445937 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (83803057381445937 / 80000000000000000 : ℝ) ≤
+            (232214282070428803041 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5236984851451289 / 10000000000000000 : ℝ)) (q :=
+      (83803057381445937 / 160000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_483_4 :
+    Uω (aρ 4) (bρ 4) (68787963624633 / 128000000000000 : ℝ) ≤
+      (-(6620238719698710557651 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16505803239739541 / 32000000000000000 : ℝ) ≤
+      (-(6620238719698710557651 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (16505803239739541 / 32000000000000000 : ℝ) =
+            (16505803239739541 / 16000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16505803239739541 / 16000000000000000 : ℝ) ≤
+            (311233083301289442349 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (128900597116819 / 250000000000000 : ℝ)) (q :=
+      (16505803239739541 / 32000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_483_5 :
+    Uω (aρ 5) (bρ 5) (68787963624633 / 128000000000000 : ℝ) ≤
+      (-(3429243137104443273827 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (80586020178575737 / 160000000000000000 : ℝ) ≤
+      (-(3429243137104443273827 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (80586020178575737 / 160000000000000000 : ℝ) =
+            (80586020178575737 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (80586020178575737 / 80000000000000000 : ℝ) ≤
+            (36492764395556726173 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2515669650746257 / 5000000000000000 : ℝ)) (q :=
+      (80586020178575737 / 160000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_483_6 :
+    Uω (aρ 6) (bρ 6) (68787963624633 / 128000000000000 : ℝ) ≤
+      (-(7211244332118761894243 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (77792838864313353 / 160000000000000000 : ℝ) ≤
+      (-(7211244332118761894243 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (77792838864313353 / 160000000000000000 : ℝ) =
+            (77792838864313353 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (77792838864313353 / 40000000000000000 : ℝ) ≤
+            (6651699273881238105757 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1212358003087429 / 2500000000000000 : ℝ)) (q :=
+      (77792838864313353 / 160000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_483_7 :
+    Uω (aρ 7) (bρ 7) (68787963624633 / 128000000000000 : ℝ) ≤
+      (-(3857748200147473405413 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (73967379221621257 / 160000000000000000 : ℝ) ≤
+      (-(3857748200147473405413 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (73967379221621257 / 160000000000000000 : ℝ) =
+            (73967379221621257 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (73967379221621257 / 40000000000000000 : ℝ) ≤
+            (3073723602852526594587 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1148727734443301 / 2500000000000000 : ℝ)) (q :=
+      (73967379221621257 / 160000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_483_8 :
+    Uω (aρ 8) (bρ 8) (68787963624633 / 128000000000000 : ℝ) ≤
+      (-(210473800453005356507 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (68942899112957753 / 160000000000000000 : ℝ) ≤
+      (-(210473800453005356507 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (68942899112957753 / 160000000000000000 : ℝ) =
+            (68942899112957753 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (68942899112957753 / 40000000000000000 : ℝ) ≤
+            (136099789696994643493 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2125196287765133 / 5000000000000000 : ℝ)) (q :=
+      (68942899112957753 / 160000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8])
+      (by norm_num [bρ_8]) (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_483_9 :
+    Uω (aρ 9) (bρ 9) (68787963624633 / 128000000000000 : ℝ) ≤
+      (-(2347169963608201734619 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (62571245109110529 / 160000000000000000 : ℝ) ≤
+      (-(2347169963608201734619 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (62571245109110529 / 160000000000000000 : ℝ) =
+            (62571245109110529 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (62571245109110529 / 40000000000000000 : ℝ) ≤
+            (1118565937891798265381 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3794901453409363 / 10000000000000000 : ℝ)) (q :=
+      (62571245109110529 / 160000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9])
+      (by norm_num [bρ_9]) (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_483_10 :
+    Uω (aρ 10) (bρ 10) (68787963624633 / 128000000000000 : ℝ) ≤
+      (-(10735838335119840614069 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (54685029115109073 / 160000000000000000 : ℝ) ≤
+      (-(10735838335119840614069 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (54685029115109073 / 160000000000000000 : ℝ) =
+            (54685029115109073 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (54685029115109073 / 40000000000000000 : ℝ) ≤
+            (3127105270880159385931 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3197683487654181 / 10000000000000000 : ℝ)) (q :=
+      (54685029115109073 / 160000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_483_11 :
+    Uω (aρ 11) (bρ 11) (68787963624633 / 128000000000000 : ℝ) ≤
+      (-(12707017930806615344121 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (44901536813439489 / 160000000000000000 : ℝ) ≤
+      (-(12707017930806615344121 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (44901536813439489 / 160000000000000000 : ℝ) =
+            (44901536813439489 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (44901536813439489 / 40000000000000000 : ℝ) ≤
+            (1155925675193384655879 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2393223576150483 / 10000000000000000 : ℝ)) (q :=
+      (44901536813439489 / 160000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_483_12 :
+    Uω (aρ 12) (bρ 12) (68787963624633 / 128000000000000 : ℝ) ≤
+      (-(3280235471263865683437 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6206630563877357 / 32000000000000000 : ℝ) ≤
+      (-(3280235471263865683437 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (6206630563877357 / 32000000000000000 : ℝ) =
+            (6206630563877357 / 4000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6206630563877357 / 4000000000000000 : ℝ) ≤
+            (878647610536134316563 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (216675134715779 / 2000000000000000 : ℝ)) (q :=
+      (6206630563877357 / 32000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_483_13 :
+    Uω (aρ 13) (bρ 13) (68787963624633 / 128000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_483_14 :
+    Uω (aρ 14) (bρ 14) (68787963624633 / 128000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_483_15 :
+    Uω (aρ 15) (bρ 15) (68787963624633 / 128000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_483_16 :
+    Uω (aρ 16) (bρ 16) (68787963624633 / 128000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_483 :
+    Uρ (68787963624633 / 128000000000000 : ℝ) ≤
+      (-(980424608155701660297 / 1000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (68787963624633 / 128000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(6330813985629369780369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_483_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (68787963624633 / 128000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3188001653894608997351 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_483_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (68787963624633 / 128000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3233521619429571196959 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_483_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (68787963624633 / 128000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(6620238719698710557651 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_483_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (68787963624633 / 128000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(3429243137104443273827 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_483_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (68787963624633 / 128000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(7211244332118761894243 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_483_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (68787963624633 / 128000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(3857748200147473405413 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_483_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (68787963624633 / 128000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(210473800453005356507 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_483_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (68787963624633 / 128000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(2347169963608201734619 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_483_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (68787963624633 / 128000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(10735838335119840614069 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_483_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (68787963624633 / 128000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(12707017930806615344121 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_483_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (68787963624633 / 128000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(3280235471263865683437 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_483_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (68787963624633 / 128000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_483_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (68787963624633 / 128000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_483_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (68787963624633 / 128000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_483_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (68787963624633 / 128000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_483_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/U40.lean b/LeanPool/Zeta5Irrational/Table/U40.lean
new file mode 100644
index 0000000000..6479827caf
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/U40.lean
@@ -0,0 +1,4624 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-! # Certified arcsine potential bounds (U40) -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma U_484_1 :
+    Uω (aρ 1) (bρ 1) (17216962626667 / 32000000000000 : ℝ) ≤
+      (-(49367704558822225637 / 78125000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4252618551908039 / 8000000000000000 : ℝ) ≤
+      (-(49367704558822225637 / 78125000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (4252618551908039 / 8000000000000000 : ℝ) =
+            (4252618551908039 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4252618551908039 / 4000000000000000 : ℝ) ≤
+            (4784418902115274363 / 78125000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (265787138851083 / 500000000000000 : ℝ)) (q :=
+      (4252618551908039 / 8000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_484_2 :
+    Uω (aρ 2) (bρ 2) (17216962626667 / 32000000000000 : ℝ) ≤
+      (-(3182100974805846120097 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21167336425781133 / 40000000000000000 : ℝ) ≤
+      (-(3182100974805846120097 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (21167336425781133 / 40000000000000000 : ℝ) =
+            (21167336425781133 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21167336425781133 / 20000000000000000 : ℝ) ≤
+            (283634926694153879903 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5291633622652129 / 10000000000000000 : ℝ)) (q :=
+      (21167336425781133 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_484_3 :
+    Uω (aρ 3) (bρ 3) (17216962626667 / 32000000000000 : ℝ) ≤
+      (-(129102657264926025039 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20975732358603649 / 40000000000000000 : ℝ) ≤
+      (-(129102657264926025039 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (20975732358603649 / 40000000000000000 : ℝ) =
+            (20975732358603649 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20975732358603649 / 20000000000000000 : ℝ) ≤
+            (9526778795073974961 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5243227695413387 / 10000000000000000 : ℝ)) (q :=
+      (20975732358603649 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_484_4 :
+    Uω (aρ 4) (bρ 4) (17216962626667 / 32000000000000 : ℝ) ≤
+      (-(1652035360447789156013 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20657228563920487 / 40000000000000000 : ℝ) ≤
+      (-(1652035360447789156013 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (20657228563920487 / 40000000000000000 : ℝ) =
+            (20657228563920487 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20657228563920487 / 20000000000000000 : ℝ) ≤
+            (80832590302210843987 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2581134989568403 / 5000000000000000 : ℝ)) (q :=
+      (20657228563920487 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_484_5 :
+    Uω (aρ 5) (bρ 5) (17216962626667 / 32000000000000 : ℝ) ≤
+      (-(1369217880496384891111 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20171495895740663 / 40000000000000000 : ℝ) ≤
+      (-(1369217880496384891111 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (20171495895740663 / 40000000000000000 : ℝ) =
+            (20171495895740663 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20171495895740663 / 20000000000000000 : ℝ) ≤
+            (17076480103615108889 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2518796782190947 / 5000000000000000 : ℝ)) (q :=
+      (20171495895740663 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_484_6 :
+    Uω (aρ 6) (bρ 6) (17216962626667 / 32000000000000 : ℝ) ≤
+      (-(3599191382945151200059 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3894647850461749 / 8000000000000000 : ℝ) ≤
+      (-(3599191382945151200059 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3894647850461749 / 8000000000000000 : ℝ) =
+            (3894647850461749 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3894647850461749 / 2000000000000000 : ℝ) ≤
+            (3332280420054848799941 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (971141123561187 / 2000000000000000 : ℝ)) (q :=
+      (3894647850461749 / 8000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_484_7 :
+    Uω (aρ 7) (bρ 7) (17216962626667 / 32000000000000 : ℝ) ≤
+      (-(3850961478539558197473 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3703392329638651 / 8000000000000000 : ℝ) ≤
+      (-(3850961478539558197473 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3703392329638651 / 8000000000000000 : ℝ) =
+            (3703392329638651 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3703392329638651 / 2000000000000000 : ℝ) ≤
+            (3080510324460441802527 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (460122819650819 / 1000000000000000 : ℝ)) (q :=
+      (3703392329638651 / 8000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_484_8 :
+    Uω (aρ 8) (bρ 8) (17216962626667 / 32000000000000 : ℝ) ≤
+      (-(8404279351985589500927 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17261032743991579 / 40000000000000000 : ℝ) ≤
+      (-(8404279351985589500927 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (17261032743991579 / 40000000000000000 : ℝ) =
+            (17261032743991579 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17261032743991579 / 10000000000000000 : ℝ) ≤
+            (5458664254014410499073 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (532100674468419 / 1250000000000000 : ℝ)) (q :=
+      (17261032743991579 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_484_9 :
+    Uω (aρ 9) (bρ 9) (17216962626667 / 32000000000000 : ℝ) ≤
+      (-(4686124003779445722807 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15668536435040589 / 40000000000000000 : ℝ) ≤
+      (-(4686124003779445722807 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (15668536435040589 / 40000000000000000 : ℝ) =
+            (15668536435040589 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15668536435040589 / 10000000000000000 : ℝ) ≤
+            (2245347799220554277193 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3801522869631857 / 10000000000000000 : ℝ)) (q :=
+      (15668536435040589 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_484_10 :
+    Uω (aρ 10) (bρ 10) (17216962626667 / 32000000000000 : ℝ) ≤
+      (-(10716342206642952717589 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2739587387332633 / 8000000000000000 : ℝ) ≤
+      (-(10716342206642952717589 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2739587387332633 / 8000000000000000 : ℝ) =
+            (2739587387332633 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2739587387332633 / 2000000000000000 : ℝ) ≤
+            (3146601399357047282411 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (640956430787629 / 2000000000000000 : ℝ)) (q :=
+      (2739587387332633 / 8000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_484_11 :
+    Uω (aρ 11) (bρ 11) (17216962626667 / 32000000000000 : ℝ) ≤
+      (-(6340492533616705606711 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11254645163716239 / 40000000000000000 : ℝ) ≤
+      (-(6340492533616705606711 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (11254645163716239 / 40000000000000000 : ℝ) =
+            (11254645163716239 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11254645163716239 / 10000000000000000 : ℝ) ≤
+            (590979269383294393289 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1200806446834841 / 5000000000000000 : ℝ)) (q :=
+      (11254645163716239 / 40000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_484_12 :
+    Uω (aρ 12) (bρ 12) (17216962626667 / 32000000000000 : ℝ) ≤
+      (-(8171995575870492649569 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1560556402166403 / 8000000000000000 : ℝ) ≤
+      (-(8171995575870492649569 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1560556402166403 / 8000000000000000 : ℝ) =
+            (1560556402166403 / 1000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1560556402166403 / 1000000000000000 : ℝ) ≤
+            (2225212128629507350431 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (109938141391257 / 1000000000000000 : ℝ)) (q :=
+      (1560556402166403 / 8000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_484_13 :
+    Uω (aρ 13) (bρ 13) (17216962626667 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_484_14 :
+    Uω (aρ 14) (bρ 14) (17216962626667 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_484_15 :
+    Uω (aρ 15) (bρ 15) (17216962626667 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_484_16 :
+    Uω (aρ 16) (bρ 16) (17216962626667 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_484 :
+    Uρ (17216962626667 / 32000000000000 : ℝ) ≤
+      (-(9789045597887444270013 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (17216962626667 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(49367704558822225637 / 78125000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_484_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (17216962626667 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3182100974805846120097 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_484_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (17216962626667 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(129102657264926025039 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_484_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (17216962626667 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1652035360447789156013 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_484_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (17216962626667 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1369217880496384891111 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_484_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (17216962626667 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3599191382945151200059 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_484_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (17216962626667 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(3850961478539558197473 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_484_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (17216962626667 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(8404279351985589500927 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_484_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (17216962626667 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(4686124003779445722807 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_484_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (17216962626667 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(10716342206642952717589 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_484_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (17216962626667 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(6340492533616705606711 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_484_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (17216962626667 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(8171995575870492649569 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_484_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (17216962626667 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_484_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (17216962626667 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_484_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (17216962626667 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_484_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (17216962626667 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_484_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_485_1 :
+    Uω (aρ 1) (bρ 1) (68947737388703 / 128000000000000 : ℝ) ≤
+      (-(6307332166479452008709 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (85152230211353759 / 160000000000000000 : ℝ) ≤
+      (-(6307332166479452008709 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (85152230211353759 / 160000000000000000 : ℝ) =
+            (85152230211353759 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (85152230211353759 / 80000000000000000 : ℝ) ≤
+            (624139636520547991291 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2660992005505899 / 5000000000000000 : ℝ)) (q :=
+      (85152230211353759 / 160000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_485_2 :
+    Uω (aρ 2) (bρ 2) (68947737388703 / 128000000000000 : ℝ) ≤
+      (-(6352414503277472699569 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (84769208084547807 / 160000000000000000 : ℝ) ≤
+      (-(6352414503277472699569 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (84769208084547807 / 160000000000000000 : ℝ) =
+            (84769208084547807 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (84769208084547807 / 80000000000000000 : ℝ) ≤
+            (579057299722527300431 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2648937628835527 / 5000000000000000 : ℝ)) (q :=
+      (84769208084547807 / 160000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_485_3 :
+    Uω (aρ 3) (bρ 3) (68947737388703 / 128000000000000 : ℝ) ≤
+      (-(644323666027210507003 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (84002801455396183 / 160000000000000000 : ℝ) ≤
+      (-(644323666027210507003 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (84002801455396183 / 160000000000000000 : ℝ) =
+            (84002801455396183 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (84002801455396183 / 80000000000000000 : ℝ) ≤
+            (48823514272789492997 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5249470535377101 / 10000000000000000 : ℝ)) (q :=
+      (84002801455396183 / 160000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_485_4 :
+    Uω (aρ 4) (bρ 4) (68947737388703 / 128000000000000 : ℝ) ≤
+      (-(1319211758427744412517 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (82728812217343327 / 160000000000000000 : ℝ) ≤
+      (-(1319211758427744412517 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (82728812217343327 / 160000000000000000 : ℝ) =
+            (82728812217343327 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (82728812217343327 / 80000000000000000 : ℝ) ≤
+            (67082602172255587483 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2584258030842747 / 5000000000000000 : ℝ)) (q :=
+      (82728812217343327 / 160000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_485_5 :
+    Uω (aρ 5) (bρ 5) (68947737388703 / 128000000000000 : ℝ) ≤
+      (-(3416853956054903156221 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (80785946727717703 / 160000000000000000 : ℝ) ≤
+      (-(3416853956054903156221 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (80785946727717703 / 160000000000000000 : ℝ) =
+            (80785946727717703 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (80785946727717703 / 80000000000000000 : ℝ) ≤
+            (48881945445096843779 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5043847794817291 / 10000000000000000 : ℝ)) (q :=
+      (80785946727717703 / 160000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_485_6 :
+    Uω (aρ 6) (bρ 6) (68947737388703 / 128000000000000 : ℝ) ≤
+      (-(1437107561212252026653 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (77993074486092727 / 160000000000000000 : ℝ) ≤
+      (-(1437107561212252026653 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (77993074486092727 / 160000000000000000 : ℝ) =
+            (77993074486092727 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (77993074486092727 / 40000000000000000 : ℝ) ≤
+            (1335481159987747973347 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4861979139754169 / 10000000000000000 : ℝ)) (q :=
+      (77993074486092727 / 160000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_485_7 :
+    Uω (aρ 7) (bρ 7) (68947737388703 / 128000000000000 : ℝ) ≤
+      (-(7688368136445735574367 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14833662460930483 / 32000000000000000 : ℝ) ≤
+      (-(7688368136445735574367 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (14833662460930483 / 32000000000000000 : ℝ) =
+            (14833662460930483 / 8000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14833662460930483 / 8000000000000000 : ℝ) ≤
+            (6174575469554264425633 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (460754524783413 / 1000000000000000 : ℝ)) (q :=
+      (14833662460930483 / 32000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_485_8 :
+    Uω (aρ 8) (bρ 8) (68947737388703 / 128000000000000 : ℝ) ≤
+      (-(4194814386260065118497 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (69145358762895607 / 160000000000000000 : ℝ) ≤
+      (-(4194814386260065118497 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (69145358762895607 / 160000000000000000 : ℝ) =
+            (69145358762895607 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (69145358762895607 / 40000000000000000 : ℝ) ≤
+            (2736657416739934881503 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4263217706454529 / 10000000000000000 : ℝ)) (q :=
+      (69145358762895607 / 160000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8])
+      (by norm_num [bρ_8]) (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_485_9 :
+    Uω (aρ 9) (bρ 9) (68947737388703 / 128000000000000 : ℝ) ≤
+      (-(9355844753686533661381 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (62777036096304591 / 160000000000000000 : ℝ) ≤
+      (-(9355844753686533661381 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (62777036096304591 / 160000000000000000 : ℝ) =
+            (62777036096304591 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (62777036096304591 / 40000000000000000 : ℝ) ≤
+            (4507098852313466338619 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (952035750372663 / 2500000000000000 : ℝ)) (q :=
+      (62777036096304591 / 160000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9])
+      (by norm_num [bρ_9]) (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_485_10 :
+    Uω (aρ 10) (bρ 10) (68947737388703 / 128000000000000 : ℝ) ≤
+      (-(1337111150499449144981 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (54898437886583623 / 160000000000000000 : ℝ) ≤
+      (-(1337111150499449144981 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (54898437886583623 / 160000000000000000 : ℝ) =
+            (54898437886583623 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (54898437886583623 / 40000000000000000 : ℝ) ≤
+            (395756800250550855019 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3211877258770531 / 10000000000000000 : ℝ)) (q :=
+      (54898437886583623 / 160000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_485_11 :
+    Uω (aρ 11) (bρ 11) (68947737388703 / 128000000000000 : ℝ) ≤
+      (-(6327521456465553556581 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (45135520169009759 / 160000000000000000 : ℝ) ≤
+      (-(6327521456465553556581 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (45135520169009759 / 160000000000000000 : ℝ) =
+            (45135520169009759 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (45135520169009759 / 40000000000000000 : ℝ) ≤
+            (603950346534446443419 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1204994585139399 / 5000000000000000 : ℝ)) (q :=
+      (45135520169009759 / 160000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_485_12 :
+    Uω (aρ 12) (bρ 12) (68947737388703 / 128000000000000 : ℝ) ≤
+      (-(407190663593972207153 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6277509011950307 / 32000000000000000 : ℝ) ≤
+      (-(407190663593972207153 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (6277509011950307 / 32000000000000000 : ℝ) =
+            (6277509011950307 / 4000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6277509011950307 / 4000000000000000 : ℝ) ≤
+            (112669721631027792847 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (27879809457663 / 250000000000000 : ℝ)) (q :=
+      (6277509011950307 / 32000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_485_13 :
+    Uω (aρ 13) (bρ 13) (68947737388703 / 128000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_485_14 :
+    Uω (aρ 14) (bρ 14) (68947737388703 / 128000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_485_15 :
+    Uω (aρ 15) (bρ 15) (68947737388703 / 128000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_485_16 :
+    Uω (aρ 16) (bρ 16) (68947737388703 / 128000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_485 :
+    Uρ (68947737388703 / 128000000000000 : ℝ) ≤
+      (-(4886962101695595637819 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (68947737388703 / 128000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(6307332166479452008709 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_485_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (68947737388703 / 128000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(6352414503277472699569 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_485_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (68947737388703 / 128000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(644323666027210507003 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_485_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (68947737388703 / 128000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1319211758427744412517 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_485_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (68947737388703 / 128000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(3416853956054903156221 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_485_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (68947737388703 / 128000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1437107561212252026653 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_485_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (68947737388703 / 128000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(7688368136445735574367 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_485_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (68947737388703 / 128000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(4194814386260065118497 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_485_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (68947737388703 / 128000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(9355844753686533661381 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_485_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (68947737388703 / 128000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(1337111150499449144981 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_485_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (68947737388703 / 128000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(6327521456465553556581 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_485_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (68947737388703 / 128000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(407190663593972207153 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_485_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (68947737388703 / 128000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_485_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (68947737388703 / 128000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_485_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (68947737388703 / 128000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_485_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (68947737388703 / 128000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_485_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_486_1 :
+    Uω (aρ 1) (bρ 1) (34513812135369 / 64000000000000 : ℝ) ≤
+      (-(786951487770795011317 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (42626044691604937 / 80000000000000000 : ℝ) ≤
+      (-(786951487770795011317 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (42626044691604937 / 80000000000000000 : ℝ) =
+            (42626044691604937 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (42626044691604937 / 40000000000000000 : ℝ) ≤
+            (79482487604204988683 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1332056311208707 / 2500000000000000 : ℝ)) (q :=
+      (42626044691604937 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_486_2 :
+    Uω (aρ 2) (bρ 2) (34513812135369 / 64000000000000 : ℝ) ≤
+      (-(792580117002899229761 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (42434535228538897 / 80000000000000000 : ℝ) ≤
+      (-(792580117002899229761 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (42434535228538897 / 80000000000000000 : ℝ) =
+            (42434535228538897 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (42434535228538897 / 40000000000000000 : ℝ) ≤
+            (73853858372100770239 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2652058445789159 / 5000000000000000 : ℝ)) (q :=
+      (42434535228538897 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_486_3 :
+    Uω (aρ 3) (bρ 3) (34513812135369 / 64000000000000 : ℝ) ≤
+      (-(6431354596238415003153 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (42051336722252341 / 80000000000000000 : ℝ) ≤
+      (-(6431354596238415003153 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (42051336722252341 / 80000000000000000 : ℝ) =
+            (42051336722252341 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (42051336722252341 / 40000000000000000 : ℝ) ≤
+            (500117206761584996847 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5255713371356679 / 10000000000000000 : ℝ)) (q :=
+      (42051336722252341 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_486_4 :
+    Uω (aρ 4) (bρ 4) (34513812135369 / 64000000000000 : ℝ) ≤
+      (-(3291995367689427495473 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (41414355042013509 / 80000000000000000 : ℝ) ≤
+      (-(3291995367689427495473 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (41414355042013509 / 80000000000000000 : ℝ) =
+            (41414355042013509 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (41414355042013509 / 40000000000000000 : ℝ) ≤
+            (173740533810572504527 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5174762132361971 / 10000000000000000 : ℝ)) (q :=
+      (41414355042013509 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_486_5 :
+    Uω (aρ 5) (bρ 5) (34513812135369 / 64000000000000 : ℝ) ≤
+      (-(1705335441222945381761 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (40442954806902749 / 80000000000000000 : ℝ) ≤
+      (-(1705335441222945381761 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (40442954806902749 / 80000000000000000 : ℝ) =
+            (40442954806902749 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (40442954806902749 / 40000000000000000 : ℝ) ≤
+            (27532509527054618239 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5050101992919281 / 10000000000000000 : ℝ)) (q :=
+      (40442954806902749 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_486_6 :
+    Uω (aρ 6) (bρ 6) (34513812135369 / 64000000000000 : ℝ) ≤
+      (-(1793177352395991703339 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (39046595648734657 / 80000000000000000 : ℝ) ≤
+      (-(1793177352395991703339 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (39046595648734657 / 80000000000000000 : ℝ) =
+            (39046595648734657 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (39046595648734657 / 20000000000000000 : ℝ) ≤
+            (1672558549104008296661 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2434126289258629 / 5000000000000000 : ℝ)) (q :=
+      (39046595648734657 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_486_7 :
+    Uω (aρ 7) (bρ 7) (34513812135369 / 64000000000000 : ℝ) ≤
+      (-(7674831886757274297583 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (37134388182037397 / 80000000000000000 : ℝ) ≤
+      (-(7674831886757274297583 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (37134388182037397 / 80000000000000000 : ℝ) =
+            (37134388182037397 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (37134388182037397 / 20000000000000000 : ℝ) ≤
+            (6188111719242725702417 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4613862092602943 / 10000000000000000 : ℝ)) (q :=
+      (37134388182037397 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_486_8 :
+    Uω (aρ 8) (bρ 8) (34513812135369 / 64000000000000 : ℝ) ≤
+      (-(837500021158687506521 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (34623291246055249 / 80000000000000000 : ℝ) ≤
+      (-(837500021158687506521 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (34623291246055249 / 80000000000000000 : ℝ) =
+            (34623291246055249 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (34623291246055249 / 20000000000000000 : ℝ) ≤
+            (548794339441312493479 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2134814754973703 / 5000000000000000 : ℝ)) (q :=
+      (34623291246055249 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_486_9 :
+    Uω (aρ 9) (bρ 9) (34513812135369 / 64000000000000 : ℝ) ≤
+      (-(29185843712424599397 / 31250000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (31439958115515213 / 80000000000000000 : ℝ) ≤
+      (-(29185843712424599397 / 31250000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (31439958115515213 / 80000000000000000 : ℝ) =
+            (31439958115515213 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (31439958115515213 / 20000000000000000 : ℝ) ≤
+            (14135855056325400603 / 31250000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3814761855672397 / 10000000000000000 : ℝ)) (q :=
+      (31439958115515213 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_486_10 :
+    Uω (aρ 10) (bρ 10) (34513812135369 / 64000000000000 : ℝ) ≤
+      (-(10677479115380574000047 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27502549861650857 / 80000000000000000 : ℝ) ≤
+      (-(10677479115380574000047 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (27502549861650857 / 80000000000000000 : ℝ) =
+            (27502549861650857 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27502549861650857 / 20000000000000000 : ℝ) ≤
+            (3185464490619425999953 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (804742206425327 / 2500000000000000 : ℝ)) (q :=
+      (27502549861650857 / 80000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_486_11 :
+    Uω (aρ 11) (bρ 11) (34513812135369 / 64000000000000 : ℝ) ≤
+      (-(2525838139806847648551 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22626178219963501 / 80000000000000000 : ℝ) ≤
+      (-(2525838139806847648551 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (22626178219963501 / 80000000000000000 : ℝ) =
+            (22626178219963501 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22626178219963501 / 20000000000000000 : ℝ) ≤
+            (246750581393152351449 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2418352541484469 / 10000000000000000 : ℝ)) (q :=
+      (22626178219963501 / 80000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_486_12 :
+    Uω (aρ 12) (bρ 12) (34513812135369 / 64000000000000 : ℝ) ≤
+      (-(16232050823419908826121 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15781234614494777 / 80000000000000000 : ℝ) ≤
+      (-(16232050823419908826121 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (15781234614494777 / 80000000000000000 : ℝ) =
+            (15781234614494777 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15781234614494777 / 10000000000000000 : ℝ) ≤
+            (4562364585580091173879 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (70676046050143 / 625000000000000 : ℝ)) (q :=
+      (15781234614494777 / 80000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_486_13 :
+    Uω (aρ 13) (bρ 13) (34513812135369 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_486_14 :
+    Uω (aρ 14) (bρ 14) (34513812135369 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_486_15 :
+    Uω (aρ 15) (bρ 15) (34513812135369 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_486_16 :
+    Uω (aρ 16) (bρ 16) (34513812135369 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_486 :
+    Uρ (34513812135369 / 64000000000000 : ℝ) ≤
+      (-(1951775897521726186223 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (34513812135369 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(786951487770795011317 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_486_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (34513812135369 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(792580117002899229761 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_486_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (34513812135369 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(6431354596238415003153 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_486_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (34513812135369 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3291995367689427495473 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_486_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (34513812135369 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1705335441222945381761 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_486_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (34513812135369 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1793177352395991703339 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_486_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (34513812135369 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(7674831886757274297583 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_486_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (34513812135369 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(837500021158687506521 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_486_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (34513812135369 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(29185843712424599397 / 31250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_486_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (34513812135369 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(10677479115380574000047 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_486_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (34513812135369 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(2525838139806847648551 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_486_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (34513812135369 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(16232050823419908826121 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_486_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (34513812135369 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_486_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (34513812135369 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_486_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (34513812135369 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_486_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (34513812135369 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_486_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_487_1 :
+    Uω (aρ 1) (bρ 1) (69107511152773 / 128000000000000 : ℝ) ≤
+      (-(6283905358389822733931 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (85351948553733837 / 160000000000000000 : ℝ) ≤
+      (-(6283905358389822733931 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (85351948553733837 / 160000000000000000 : ℝ) =
+            (85351948553733837 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (85351948553733837 / 80000000000000000 : ℝ) ≤
+            (647566444610177266069 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5334466478491339 / 10000000000000000 : ℝ)) (q :=
+      (85351948553733837 / 160000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_487_2 :
+    Uω (aρ 2) (bρ 2) (69107511152773 / 128000000000000 : ℝ) ≤
+      (-(6328881215201098496097 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (84968932820745869 / 160000000000000000 : ℝ) ≤
+      (-(6328881215201098496097 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (84968932820745869 / 160000000000000000 : ℝ) =
+            (84968932820745869 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (84968932820745869 / 80000000000000000 : ℝ) ≤
+            (602590587798901503903 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5310358524377843 / 10000000000000000 : ℝ)) (q :=
+      (84968932820745869 / 160000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_487_3 :
+    Uω (aρ 3) (bρ 3) (69107511152773 / 128000000000000 : ℝ) ≤
+      (-(1604871659391791879599 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (84202545401853533 / 160000000000000000 : ℝ) ≤
+      (-(1604871659391791879599 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (84202545401853533 / 160000000000000000 : ℝ) =
+            (84202545401853533 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (84202545401853533 / 80000000000000000 : ℝ) ≤
+            (127996291358208120401 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5261956203366301 / 10000000000000000 : ℝ)) (q :=
+      (84202545401853533 / 160000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_487_4 :
+    Uω (aρ 4) (bρ 4) (69107511152773 / 128000000000000 : ℝ) ≤
+      (-(3285968618138589291277 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3317144314243061 / 6400000000000000 : ℝ) ≤
+      (-(3285968618138589291277 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (3317144314243061 / 6400000000000000 : ℝ) =
+            (3317144314243061 / 3200000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3317144314243061 / 3200000000000000 : ℝ) ≤
+            (179767283361410708723 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (207240327648367 / 400000000000000 : ℝ)) (q :=
+      (3317144314243061 / 6400000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_487_5 :
+    Uω (aρ 5) (bρ 5) (69107511152773 / 128000000000000 : ℝ) ≤
+      (-(6808990922769527856703 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (80985872242185869 / 160000000000000000 : ℝ) ≤
+      (-(6808990922769527856703 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (80985872242185869 / 160000000000000000 : ℝ) =
+            (80985872242185869 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (80985872242185869 / 80000000000000000 : ℝ) ≤
+            (122480880230472143297 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5056356158807843 / 10000000000000000 : ℝ)) (q :=
+      (80985872242185869 / 160000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_487_6 :
+    Uω (aρ 6) (bρ 6) (69107511152773 / 128000000000000 : ℝ) ≤
+      (-(7159897533578763198489 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (78193307445934141 / 160000000000000000 : ℝ) ≤
+      (-(7159897533578763198489 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (78193307445934141 / 160000000000000000 : ℝ) =
+            (78193307445934141 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (78193307445934141 / 40000000000000000 : ℝ) ≤
+            (6703046072421236801511 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4874525934416377 / 10000000000000000 : ℝ)) (q :=
+      (78193307445934141 / 160000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_487_7 :
+    Uω (aρ 7) (bρ 7) (69107511152773 / 128000000000000 : ℝ) ≤
+      (-(7661314156593136158591 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (74369238777818021 / 160000000000000000 : ℝ) ≤
+      (-(7661314156593136158591 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (74369238777818021 / 160000000000000000 : ℝ) =
+            (74369238777818021 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (74369238777818021 / 40000000000000000 : ℝ) ≤
+            (6201629449406863841409 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2310089365830931 / 5000000000000000 : ℝ)) (q :=
+      (74369238777818021 / 160000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_487_8 :
+    Uω (aρ 8) (bρ 8) (69107511152773 / 128000000000000 : ℝ) ≤
+      (-(8360393601371293797627 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (69347802181864277 / 160000000000000000 : ℝ) ≤
+      (-(8360393601371293797627 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (69347802181864277 / 160000000000000000 : ℝ) =
+            (69347802181864277 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (69347802181864277 / 40000000000000000 : ℝ) ≤
+            (5502550004628706202373 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1069010202126911 / 2500000000000000 : ℝ)) (q :=
+      (69347802181864277 / 160000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8])
+      (by norm_num [bρ_8]) (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_487_9 :
+    Uω (aρ 9) (bρ 9) (69107511152773 / 128000000000000 : ℝ) ≤
+      (-(932312360618953778177 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (62982786197452141 / 160000000000000000 : ℝ) ≤
+      (-(932312360618953778177 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (62982786197452141 / 160000000000000000 : ℝ) =
+            (62982786197452141 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (62982786197452141 / 40000000000000000 : ℝ) ≤
+            (453981999981046221823 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3821379438816127 / 10000000000000000 : ℝ)) (q :=
+      (62982786197452141 / 160000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9])
+      (by norm_num [bρ_9]) (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_487_10 :
+    Uω (aρ 10) (bρ 10) (69107511152773 / 128000000000000 : ℝ) ≤
+      (-(10658111730682012403533 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (55111733443456581 / 160000000000000000 : ℝ) ≤
+      (-(10658111730682012403533 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (55111733443456581 / 160000000000000000 : ℝ) =
+            (55111733443456581 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (55111733443456581 / 40000000000000000 : ℝ) ≤
+            (3204831875317987596467 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1613028439030841 / 5000000000000000 : ℝ)) (q :=
+      (55111733443456581 / 160000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_487_11 :
+    Uω (aρ 11) (bρ 11) (69107511152773 / 128000000000000 : ℝ) ≤
+      (-(6301713833760821665297 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1814763621402669 / 6400000000000000 : ℝ) ≤
+      (-(6301713833760821665297 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1814763621402669 / 6400000000000000 : ℝ) =
+            (1814763621402669 / 1600000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1814763621402669 / 1600000000000000 : ℝ) ≤
+            (629757969239178334703 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (48534062814359 / 200000000000000 : ℝ)) (q :=
+      (1814763621402669 / 6400000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_487_12 :
+    Uω (aρ 12) (bρ 12) (69107511152773 / 128000000000000 : ℝ) ≤
+      (-(3235446682644510697477 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6347192319383009 / 32000000000000000 : ℝ) ≤
+      (-(3235446682644510697477 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (6347192319383009 / 32000000000000000 : ℝ) =
+            (6347192319383009 / 4000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6347192319383009 / 4000000000000000 : ℝ) ≤
+            (923436399155489302523 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (114626212013399 / 1000000000000000 : ℝ)) (q :=
+      (6347192319383009 / 32000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_487_13 :
+    Uω (aρ 13) (bρ 13) (69107511152773 / 128000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_487_14 :
+    Uω (aρ 14) (bρ 14) (69107511152773 / 128000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_487_15 :
+    Uω (aρ 15) (bρ 15) (69107511152773 / 128000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_487_16 :
+    Uω (aρ 16) (bρ 16) (69107511152773 / 128000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_487 :
+    Uρ (69107511152773 / 128000000000000 : ℝ) ≤
+      (-(974390919117488362389 / 1000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (69107511152773 / 128000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(6283905358389822733931 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_487_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (69107511152773 / 128000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(6328881215201098496097 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_487_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (69107511152773 / 128000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1604871659391791879599 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_487_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (69107511152773 / 128000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3285968618138589291277 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_487_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (69107511152773 / 128000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(6808990922769527856703 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_487_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (69107511152773 / 128000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(7159897533578763198489 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_487_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (69107511152773 / 128000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(7661314156593136158591 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_487_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (69107511152773 / 128000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(8360393601371293797627 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_487_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (69107511152773 / 128000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(932312360618953778177 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_487_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (69107511152773 / 128000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(10658111730682012403533 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_487_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (69107511152773 / 128000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(6301713833760821665297 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_487_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (69107511152773 / 128000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(3235446682644510697477 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_487_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (69107511152773 / 128000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_487_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (69107511152773 / 128000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_487_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (69107511152773 / 128000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_487_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (69107511152773 / 128000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_487_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_488_1 :
+    Uω (aρ 1) (bρ 1) (8648424754351 / 16000000000000 : ℝ) ≤
+      (-(62722125030626558607 / 100000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10681475965366287 / 20000000000000000 : ℝ) ≤
+      (-(62722125030626558607 / 100000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10681475965366287 / 20000000000000000 : ℝ) =
+            (10681475965366287 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10681475965366287 / 10000000000000000 : ℝ) ≤
+            (6592592999373441393 / 100000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (667588463997739 / 1250000000000000 : ℝ)) (q :=
+      (10681475965366287 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_488_2 :
+    Uω (aρ 2) (bρ 2) (8648424754351 / 16000000000000 : ℝ) ≤
+      (-(1263427061655700437097 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (166149990577311 / 312500000000000 : ℝ) ≤
+      (-(1263427061655700437097 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (166149990577311 / 312500000000000 : ℝ) =
+            (166149990577311 / 156250000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (166149990577311 / 156250000000000 : ℝ) ≤
+            (122867298944299562903 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5316600156073529 / 10000000000000000 : ℝ)) (q :=
+      (166149990577311 / 312500000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_488_3 :
+    Uω (aρ 3) (bρ 3) (8648424754351 / 16000000000000 : ℝ) ≤
+      (-(6407632750799801970241 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10537802165944457 / 20000000000000000 : ℝ) ≤
+      (-(6407632750799801970241 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10537802165944457 / 20000000000000000 : ℝ) =
+            (10537802165944457 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10537802165944457 / 10000000000000000 : ℝ) ≤
+            (523839052200198029759 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2634099515710041 / 5000000000000000 : ℝ)) (q :=
+      (10537802165944457 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_488_4 :
+    Uω (aρ 4) (bρ 4) (8648424754351 / 16000000000000 : ℝ) ≤
+      (-(3279949129863435104927 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10378563191729213 / 20000000000000000 : ℝ) ≤
+      (-(3279949129863435104927 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10378563191729213 / 20000000000000000 : ℝ) =
+            (10378563191729213 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10378563191729213 / 10000000000000000 : ℝ) ≤
+            (185786771636564895073 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2593627119134919 / 5000000000000000 : ℝ)) (q :=
+      (10378563191729213 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_488_5 :
+    Uω (aρ 5) (bρ 5) (8648424754351 / 16000000000000 : ℝ) ≤
+      (-(6796655347826442500329 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5067864663363369 / 10000000000000000 : ℝ) ≤
+      (-(6796655347826442500329 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5067864663363369 / 10000000000000000 : ℝ) =
+            (5067864663363369 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5067864663363369 / 5000000000000000 : ℝ) ≤
+            (134816455173557499671 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5062610292602363 / 10000000000000000 : ℝ)) (q :=
+      (5067864663363369 / 10000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_488_6 :
+    Uω (aρ 6) (bρ 6) (8648424754351 / 16000000000000 : ℝ) ≤
+      (-(3573551067666562741491 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4893338933377709 / 10000000000000000 : ℝ) ≤
+      (-(3573551067666562741491 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4893338933377709 / 10000000000000000 : ℝ) =
+            (4893338933377709 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4893338933377709 / 2500000000000000 : ℝ) ≤
+            (3357920735333437258509 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4880799207771043 / 10000000000000000 : ℝ)) (q :=
+      (4893338933377709 / 10000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_488_7 :
+    Uω (aρ 7) (bρ 7) (8648424754351 / 16000000000000 : ℝ) ≤
+      (-(305912595789937330659 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (116358905550973 / 250000000000000 : ℝ) ≤
+      (-(305912595789937330659 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (116358905550973 / 250000000000000 : ℝ) = (116358905550973 / 62500000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (116358905550973 / 62500000000000 : ℝ) ≤
+            (248605148450062669341 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (925299033170693 / 2000000000000000 : ℝ)) (q :=
+      (116358905550973 / 250000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_488_8 :
+    Uω (aρ 8) (bρ 8) (8648424754351 / 16000000000000 : ℝ) ≤
+      (-(8345808874379213789631 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8681127231287437 / 20000000000000000 : ℝ) ≤
+      (-(8345808874379213789631 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8681127231287437 / 20000000000000000 : ℝ) =
+            (8681127231287437 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8681127231287437 / 5000000000000000 : ℝ) ≤
+            (5517134731620786210369 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2141225802201531 / 5000000000000000 : ℝ)) (q :=
+      (8681127231287437 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_488_9 :
+    Uω (aρ 9) (bρ 9) (8648424754351 / 16000000000000 : ℝ) ≤
+      (-(9306805504687874914537 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3942852878019037 / 10000000000000000 : ℝ) ≤
+      (-(9306805504687874914537 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3942852878019037 / 10000000000000000 : ℝ) =
+            (3942852878019037 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3942852878019037 / 2500000000000000 : ℝ) ≤
+            (4556138101312125085463 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3827995757513699 / 10000000000000000 : ℝ)) (q :=
+      (3942852878019037 / 10000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_488_10 :
+    Uω (aρ 10) (bρ 10) (8648424754351 / 16000000000000 : ℝ) ≤
+      (-(531939342072332120009 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (55218339231969 / 160000000000000 : ℝ) ≤
+      (-(531939342072332120009 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (55218339231969 / 160000000000000 : ℝ) = (55218339231969 / 40000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (55218339231969 / 40000000000000 : ℝ) ≤
+            (161207838227667879991 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (12932565755867 / 40000000000000 : ℝ)) (q :=
+      (55218339231969 / 160000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10]) (by norm_num [bρ_10])
+      (by norm_num) (by norm_num [aρ_10, bρ_10]) (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_488_11 :
+    Uω (aρ 11) (bρ 11) (8648424754351 / 16000000000000 : ℝ) ≤
+      (-(1572219133875210761649 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5685715438202167 / 20000000000000000 : ℝ) ≤
+      (-(1572219133875210761649 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5685715438202167 / 20000000000000000 : ℝ) =
+            (5685715438202167 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5685715438202167 / 5000000000000000 : ℝ) ≤
+            (160648816874789238351 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (19023758588889 / 78125000000000 : ℝ)) (q :=
+      (5685715438202167 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_488_12 :
+    Uω (aρ 12) (bρ 12) (8648424754351 / 16000000000000 : ℝ) ≤
+      (-(16123145677231630302739 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3988509910166249 / 20000000000000000 : ℝ) ≤
+      (-(16123145677231630302739 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3988509910166249 / 20000000000000000 : ℝ) =
+            (3988509910166249 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3988509910166249 / 2500000000000000 : ℝ) ≤
+            (4671269731768369697261 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (580767834013437 / 5000000000000000 : ℝ)) (q :=
+      (3988509910166249 / 20000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_488_13 :
+    Uω (aρ 13) (bρ 13) (8648424754351 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_488_14 :
+    Uω (aρ 14) (bρ 14) (8648424754351 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_488_15 :
+    Uω (aρ 15) (bρ 15) (8648424754351 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_488_16 :
+    Uω (aρ 16) (bρ 16) (8648424754351 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_488 :
+    Uρ (8648424754351 / 16000000000000 : ℝ) ≤
+      (-(194580223848825435147 / 200000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (8648424754351 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(62722125030626558607 / 100000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_488_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (8648424754351 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1263427061655700437097 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_488_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (8648424754351 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(6407632750799801970241 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_488_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (8648424754351 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3279949129863435104927 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_488_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (8648424754351 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(6796655347826442500329 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_488_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (8648424754351 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3573551067666562741491 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_488_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (8648424754351 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(305912595789937330659 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_488_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (8648424754351 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(8345808874379213789631 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_488_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (8648424754351 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(9306805504687874914537 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_488_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (8648424754351 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) * (-(531939342072332120009 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_488_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (8648424754351 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(1572219133875210761649 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_488_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (8648424754351 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(16123145677231630302739 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_488_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (8648424754351 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_488_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (8648424754351 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_488_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (8648424754351 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_488_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (8648424754351 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_488_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_489_1 :
+    Uω (aρ 1) (bρ 1) (69267284916843 / 128000000000000 : ℝ) ≤
+      (-(1252106660842020123547 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17110333378160783 / 32000000000000000 : ℝ) ≤
+      (-(1252106660842020123547 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (17110333378160783 / 32000000000000000 : ℝ) =
+            (17110333378160783 / 16000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17110333378160783 / 16000000000000000 : ℝ) ≤
+            (134187699757979876453 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (534694894530713 / 1000000000000000 : ℝ)) (q :=
+      (17110333378160783 / 32000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_489_2 :
+    Uω (aρ 2) (bρ 2) (69267284916843 / 128000000000000 : ℝ) ≤
+      (-(6305403182837248309091 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (85168657521620947 / 160000000000000000 : ℝ) ≤
+      (-(6305403182837248309091 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (85168657521620947 / 160000000000000000 : ℝ) =
+            (85168657521620947 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (85168657521620947 / 80000000000000000 : ℝ) ≤
+            (626068620162751690909 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5322841786669259 / 10000000000000000 : ℝ)) (q :=
+      (85168657521620947 / 160000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_489_3 :
+    Uω (aρ 3) (bρ 3) (69267284916843 / 128000000000000 : ℝ) ≤
+      (-(3197896451298350055509 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (84402289221723403 / 160000000000000000 : ℝ) ≤
+      (-(3197896451298350055509 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (84402289221723403 / 160000000000000000 : ℝ) =
+            (84402289221723403 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (84402289221723403 / 80000000000000000 : ℝ) ≤
+            (267839450201649944491 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2637220927766033 / 5000000000000000 : ℝ)) (q :=
+      (84402289221723403 / 160000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_489_4 :
+    Uω (aρ 4) (bρ 4) (69267284916843 / 128000000000000 : ℝ) ≤
+      (-(1636968442687012044419 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16625680623527751 / 32000000000000000 : ℝ) ≤
+      (-(1636968442687012044419 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (16625680623527751 / 32000000000000000 : ℝ) =
+            (16625680623527751 / 16000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16625680623527751 / 16000000000000000 : ℝ) ≤
+            (95899508062987955581 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1038700054717297 / 2000000000000000 : ℝ)) (q :=
+      (16625680623527751 / 32000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_489_5 :
+    Uω (aρ 5) (bρ 5) (69267284916843 / 128000000000000 : ℝ) ≤
+      (-(1696083750571734394177 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (81185796729639979 / 160000000000000000 : ℝ) ≤
+      (-(1696083750571734394177 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (81185796729639979 / 160000000000000000 : ℝ) =
+            (81185796729639979 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (81185796729639979 / 80000000000000000 : ℝ) ≤
+            (36784200178265605823 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2534432197210819 / 5000000000000000 : ℝ)) (q :=
+      (81185796729639979 / 160000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_489_6 :
+    Uω (aρ 6) (bρ 6) (69267284916843 / 128000000000000 : ℝ) ≤
+      (-(22294759913439966139 / 31250000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15678707552867983 / 32000000000000000 : ℝ) ≤
+      (-(22294759913439966139 / 31250000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (15678707552867983 / 32000000000000000 : ℝ) =
+            (15678707552867983 / 8000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15678707552867983 / 8000000000000000 : ℝ) ≤
+            (21026938855310033861 / 31250000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (488707239889913 / 1000000000000000 : ℝ)) (q :=
+      (15678707552867983 / 32000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_489_7 :
+    Uω (aρ 7) (bρ 7) (69267284916843 / 128000000000000 : ℝ) ≤
+      (-(7634334050232765267451 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14914031739038511 / 32000000000000000 : ℝ) ≤
+      (-(7634334050232765267451 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (14914031739038511 / 32000000000000000 : ℝ) =
+            (14914031739038511 / 8000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14914031739038511 / 8000000000000000 : ℝ) ≤
+            (6228609555767234732549 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (463281139601571 / 1000000000000000 : ℝ)) (q :=
+      (14914031739038511 / 32000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_489_8 :
+    Uω (aρ 8) (bρ 8) (69267284916843 / 128000000000000 : ℝ) ≤
+      (-(4165622981717381617827 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (69550229515448811 / 160000000000000000 : ℝ) ≤
+      (-(4165622981717381617827 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (69550229515448811 / 160000000000000000 : ℝ) =
+            (69550229515448811 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (69550229515448811 / 40000000000000000 : ℝ) ≤
+            (2765848821282618382173 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2144430949943871 / 5000000000000000 : ℝ)) (q :=
+      (69550229515448811 / 160000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8])
+      (by norm_num [bρ_8]) (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_489_9 :
+    Uω (aρ 9) (bρ 9) (69267284916843 / 128000000000000 : ℝ) ≤
+      (-(9290515580424209657449 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (63188495835948627 / 160000000000000000 : ℝ) ≤
+      (-(9290515580424209657449 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (63188495835948627 / 160000000000000000 : ℝ) =
+            (63188495835948627 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (63188495835948627 / 40000000000000000 : ℝ) ≤
+            (4572428025575790342551 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3834610818310219 / 10000000000000000 : ℝ)) (q :=
+      (63188495835948627 / 160000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9])
+      (by norm_num [bρ_9]) (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_489_10 :
+    Uω (aρ 10) (bρ 10) (69267284916843 / 128000000000000 : ℝ) ≤
+      (-(10619504240865397292307 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (55324917272051771 / 160000000000000000 : ℝ) ≤
+      (-(10619504240865397292307 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (55324917272051771 / 160000000000000000 : ℝ) =
+            (55324917272051771 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (55324917272051771 / 40000000000000000 : ℝ) ≤
+            (3243439365134602707693 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (101256954103691 / 312500000000000 : ℝ)) (q :=
+      (55324917272051771 / 160000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_489_11 :
+    Uω (aρ 11) (bρ 11) (69267284916843 / 128000000000000 : ℝ) ≤
+      (-(12552166172502945745251 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9120451277367631 / 32000000000000000 : ℝ) ≤
+      (-(12552166172502945745251 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9120451277367631 / 32000000000000000 : ℝ) =
+            (9120451277367631 / 8000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9120451277367631 / 8000000000000000 : ℝ) ≤
+            (1310777433497054254749 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (244336654687141 / 1000000000000000 : ℝ)) (q :=
+      (9120451277367631 / 32000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_489_12 :
+    Uω (aρ 12) (bρ 12) (69267284916843 / 128000000000000 : ℝ) ≤
+      (-(16069760752416665406813 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (32078875815043427 / 160000000000000000 : ℝ) ≤
+      (-(16069760752416665406813 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (32078875815043427 / 160000000000000000 : ℝ) =
+            (32078875815043427 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (32078875815043427 / 20000000000000000 : ℝ) ≤
+            (4724654656583334593187 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1176644072082069 / 10000000000000000 : ℝ)) (q :=
+      (32078875815043427 / 160000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_489_13 :
+    Uω (aρ 13) (bρ 13) (69267284916843 / 128000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_489_14 :
+    Uω (aρ 14) (bρ 14) (69267284916843 / 128000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_489_15 :
+    Uω (aρ 15) (bρ 15) (69267284916843 / 128000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_489_16 :
+    Uω (aρ 16) (bρ 16) (69267284916843 / 128000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_489 :
+    Uρ (69267284916843 / 128000000000000 : ℝ) ≤
+      (-(1942836699118080219587 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (69267284916843 / 128000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1252106660842020123547 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_489_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (69267284916843 / 128000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(6305403182837248309091 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_489_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (69267284916843 / 128000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3197896451298350055509 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_489_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (69267284916843 / 128000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1636968442687012044419 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_489_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (69267284916843 / 128000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1696083750571734394177 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_489_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (69267284916843 / 128000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) * (-(22294759913439966139 / 31250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_489_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (69267284916843 / 128000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(7634334050232765267451 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_489_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (69267284916843 / 128000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(4165622981717381617827 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_489_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (69267284916843 / 128000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(9290515580424209657449 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_489_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (69267284916843 / 128000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(10619504240865397292307 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_489_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (69267284916843 / 128000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(12552166172502945745251 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_489_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (69267284916843 / 128000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(16069760752416665406813 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_489_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (69267284916843 / 128000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_489_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (69267284916843 / 128000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_489_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (69267284916843 / 128000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_489_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (69267284916843 / 128000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_489_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_490_1 :
+    Uω (aρ 1) (bρ 1) (34673585899439 / 64000000000000 : ℝ) ≤
+      (-(3124433864984652654599 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (42825763028679659 / 80000000000000000 : ℝ) ≤
+      (-(3124433864984652654599 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (42825763028679659 / 80000000000000000 : ℝ) =
+            (42825763028679659 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (42825763028679659 / 40000000000000000 : ℝ) ≤
+            (341302036515347345401 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5353190178467571 / 10000000000000000 : ℝ)) (q :=
+      (42825763028679659 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_490_2 :
+    Uω (aρ 2) (bρ 2) (34673585899439 / 64000000000000 : ℝ) ≤
+      (-(15734212016432925827 / 25000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1705370397177799 / 3200000000000000 : ℝ) ≤
+      (-(15734212016432925827 / 25000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1705370397177799 / 3200000000000000 : ℝ) =
+            (1705370397177799 / 1600000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1705370397177799 / 1600000000000000 : ℝ) ≤
+            (1594467491067074173 / 25000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (53290834161689 / 100000000000000 : ℝ)) (q :=
+      (1705370397177799 / 3200000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_490_3 :
+    Uω (aρ 3) (bρ 3) (34673585899439 / 64000000000000 : ℝ) ≤
+      (-(797995882467076955683 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (42251080542234311 / 80000000000000000 : ℝ) ≤
+      (-(797995882467076955683 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (42251080542234311 / 80000000000000000 : ℝ) =
+            (42251080542234311 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (42251080542234311 / 40000000000000000 : ℝ) ≤
+            (68438092907923044317 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2640342337858117 / 5000000000000000 : ℝ)) (q :=
+      (42251080542234311 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_490_4 :
+    Uω (aρ 4) (bρ 4) (34673585899439 / 64000000000000 : ℝ) ≤
+      (-(6535863734487156375171 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (41614150303915131 / 80000000000000000 : ℝ) ≤
+      (-(6535863734487156375171 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (41614150303915131 / 80000000000000000 : ℝ) =
+            (41614150303915131 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (41614150303915131 / 40000000000000000 : ℝ) ≤
+            (395608068512843624829 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5199746297201439 / 10000000000000000 : ℝ)) (q :=
+      (41614150303915131 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_490_5 :
+    Uω (aρ 5) (bρ 5) (34673585899439 / 64000000000000 : ℝ) ≤
+      (-(3386014924257870962009 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (40642879295304887 / 80000000000000000 : ℝ) ≤
+      (-(3386014924257870962009 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (40642879295304887 / 80000000000000000 : ℝ) =
+            (40642879295304887 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (40642879295304887 / 40000000000000000 : ℝ) ≤
+            (79720977242129037991 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2537559232191939 / 5000000000000000 : ℝ)) (q :=
+      (40642879295304887 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_490_6 :
+    Uω (aρ 6) (bρ 6) (34673585899439 / 64000000000000 : ℝ) ≤
+      (-(7121560602100884678399 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (39246825969676887 / 80000000000000000 : ℝ) ≤
+      (-(7121560602100884678399 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (39246825969676887 / 80000000000000000 : ℝ) =
+            (39246825969676887 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (39246825969676887 / 20000000000000000 : ℝ) ≤
+            (6741383003899115321601 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2446672754058439 / 5000000000000000 : ℝ)) (q :=
+      (39246825969676887 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_490_7 :
+    Uω (aρ 7) (bρ 7) (34673585899439 / 64000000000000 : ℝ) ≤
+      (-(7620871572269016417851 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (37335308106097219 / 80000000000000000 : ℝ) ≤
+      (-(7620871572269016417851 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (37335308106097219 / 80000000000000000 : ℝ) =
+            (37335308106097219 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (37335308106097219 / 20000000000000000 : ℝ) ≤
+            (6242072033730983582149 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4639127422981961 / 10000000000000000 : ℝ)) (q :=
+      (37335308106097219 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_490_8 :
+    Uω (aρ 8) (bρ 8) (34673585899439 / 64000000000000 : ℝ) ≤
+      (-(2079176200419582819051 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (34825718597617907 / 80000000000000000 : ℝ) ≤
+      (-(2079176200419582819051 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (34825718597617907 / 80000000000000000 : ℝ) =
+            (34825718597617907 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (34825718597617907 / 20000000000000000 : ℝ) ≤
+            (1386559701080417180949 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4295271697202133 / 10000000000000000 : ℝ)) (q :=
+      (34825718597617907 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_490_9 :
+    Uω (aρ 9) (bρ 9) (34673585899439 / 64000000000000 : ℝ) ≤
+      (-(4637126865470083261903 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (31645667806187247 / 80000000000000000 : ℝ) ≤
+      (-(4637126865470083261903 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (31645667806187247 / 80000000000000000 : ℝ) =
+            (31645667806187247 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (31645667806187247 / 20000000000000000 : ℝ) ≤
+            (2294344937529916738097 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (960306156926117 / 2500000000000000 : ℝ)) (q :=
+      (31645667806187247 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_490_10 :
+    Uω (aρ 10) (bρ 10) (34673585899439 / 64000000000000 : ℝ) ≤
+      (-(5300131861877623017487 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1108629354904607 / 3200000000000000 : ℝ) ≤
+      (-(5300131861877623017487 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1108629354904607 / 3200000000000000 : ℝ) =
+            (1108629354904607 / 800000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1108629354904607 / 800000000000000 : ℝ) ≤
+            (1631339941122376982513 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (64946003556129 / 200000000000000 : ℝ)) (q :=
+      (1108629354904607 / 3200000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_490_11 :
+    Uω (aρ 11) (bρ 11) (34673585899439 / 64000000000000 : ℝ) ≤
+      (-(12526666245270194488077 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22859345099191703 / 80000000000000000 : ℝ) ≤
+      (-(12526666245270194488077 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (22859345099191703 / 80000000000000000 : ℝ) =
+            (22859345099191703 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22859345099191703 / 20000000000000000 : ℝ) ≤
+            (1336277360729805511923 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1225839805327791 / 5000000000000000 : ℝ)) (q :=
+      (22859345099191703 / 80000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_490_12 :
+    Uω (aρ 12) (bρ 12) (34673585899439 / 64000000000000 : ℝ) ≤
+      (-(16017053398384051458857 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16124200725712223 / 80000000000000000 : ℝ) ≤
+      (-(16017053398384051458857 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (16124200725712223 / 80000000000000000 : ℝ) =
+            (16124200725712223 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16124200725712223 / 10000000000000000 : ℝ) ≤
+            (4777362010615948541143 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (148949201746339 / 1250000000000000 : ℝ)) (q :=
+      (16124200725712223 / 80000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_490_13 :
+    Uω (aρ 13) (bρ 13) (34673585899439 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_490_14 :
+    Uω (aρ 14) (bρ 14) (34673585899439 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_490_15 :
+    Uω (aρ 15) (bρ 15) (34673585899439 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_490_16 :
+    Uω (aρ 16) (bρ 16) (34673585899439 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_490 :
+    Uρ (34673585899439 / 64000000000000 : ℝ) ≤
+      (-(2424856055011437979509 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (34673585899439 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3124433864984652654599 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_490_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (34673585899439 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) * (-(15734212016432925827 / 25000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_490_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (34673585899439 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(797995882467076955683 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_490_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (34673585899439 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(6535863734487156375171 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_490_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (34673585899439 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(3386014924257870962009 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_490_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (34673585899439 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(7121560602100884678399 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_490_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (34673585899439 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(7620871572269016417851 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_490_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (34673585899439 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(2079176200419582819051 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_490_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (34673585899439 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(4637126865470083261903 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_490_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (34673585899439 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(5300131861877623017487 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_490_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (34673585899439 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(12526666245270194488077 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_490_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (34673585899439 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(16017053398384051458857 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_490_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (34673585899439 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_490_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (34673585899439 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_490_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (34673585899439 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_490_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (34673585899439 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_490_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_491_1 :
+    Uω (aρ 1) (bρ 1) (69427058680913 / 128000000000000 : ℝ) ≤
+      (-(249488629943552077721 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17150277044520221 / 32000000000000000 : ℝ) ≤
+      (-(249488629943552077721 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (17150277044520221 / 32000000000000000 : ℝ) =
+            (17150277044520221 / 16000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17150277044520221 / 16000000000000000 : ℝ) ≤
+            (27770242176447922279 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (535943141146381 / 1000000000000000 : ℝ)) (q :=
+      (17150277044520221 / 32000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_491_2 :
+    Uω (aρ 2) (bρ 2) (69427058680913 / 128000000000000 : ℝ) ≤
+      (-(6281980147295564591557 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (85368382187420993 / 160000000000000000 : ℝ) ≤
+      (-(6281980147295564591557 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (85368382187420993 / 160000000000000000 : ℝ) =
+            (85368382187420993 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (85368382187420993 / 80000000000000000 : ℝ) ≤
+            (649491655704435408443 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (666915630572037 / 1250000000000000 : ℝ)) (q :=
+      (85368382187420993 / 160000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_491_3 :
+    Uω (aρ 3) (bρ 3) (69427058680913 / 128000000000000 : ℝ) ≤
+      (-(6372155189116117571223 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (84602032915902633 / 160000000000000000 : ℝ) ≤
+      (-(6372155189116117571223 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (84602032915902633 / 160000000000000000 : ℝ) =
+            (84602032915902633 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (84602032915902633 / 80000000000000000 : ℝ) ≤
+            (559316613883882428777 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5286927491986501 / 10000000000000000 : ℝ)) (q :=
+      (84602032915902633 / 160000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_491_4 :
+    Uω (aρ 4) (bρ 4) (69427058680913 / 128000000000000 : ℝ) ≤
+      (-(3261934058108181114563 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (83328198004745153 / 160000000000000000 : ℝ) ≤
+      (-(3261934058108181114563 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (83328198004745153 / 160000000000000000 : ℝ) =
+            (83328198004745153 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (83328198004745153 / 80000000000000000 : ℝ) ≤
+            (203801843391818885437 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (325374519322301 / 625000000000000 : ℝ)) (q :=
+      (83328198004745153 / 160000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_491_5 :
+    Uω (aρ 5) (bρ 5) (69427058680913 / 128000000000000 : ℝ) ≤
+      (-(1689934962254301555617 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (81385720197664297 / 160000000000000000 : ℝ) ≤
+      (-(1689934962254301555617 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (81385720197664297 / 160000000000000000 : ℝ) =
+            (81385720197664297 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (81385720197664297 / 80000000000000000 : ℝ) ≤
+            (42932988495698444383 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5081372502606709 / 10000000000000000 : ℝ)) (q :=
+      (81385720197664297 / 160000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_491_6 :
+    Uω (aρ 6) (bρ 6) (69427058680913 / 128000000000000 : ℝ) ≤
+      (-(3554407191258538788113 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (78593765461601849 / 160000000000000000 : ℝ) ≤
+      (-(3554407191258538788113 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (78593765461601849 / 160000000000000000 : ℝ) =
+            (78593765461601849 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (78593765461601849 / 40000000000000000 : ℝ) ≤
+            (3377064611741461211887 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4899618535738903 / 10000000000000000 : ℝ)) (q :=
+      (78593765461601849 / 160000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_491_7 :
+    Uω (aρ 7) (bρ 7) (69427058680913 / 128000000000000 : ℝ) ≤
+      (-(7607427410292149149329 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2990842884410353 / 6400000000000000 : ℝ) ≤
+      (-(7607427410292149149329 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2990842884410353 / 6400000000000000 : ℝ) =
+            (2990842884410353 / 1600000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2990842884410353 / 1600000000000000 : ℝ) ≤
+            (6255516195707850850671 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (185817729903241 / 400000000000000 : ℝ)) (q :=
+      (2990842884410353 / 6400000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_491_8 :
+    Uω (aρ 8) (bρ 8) (69427058680913 / 128000000000000 : ℝ) ≤
+      (-(4151092661282270086939 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (69752640907475889 / 160000000000000000 : ℝ) ≤
+      (-(4151092661282270086939 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (69752640907475889 / 160000000000000000 : ℝ) =
+            (69752640907475889 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (69752640907475889 / 40000000000000000 : ℝ) ≤
+            (2780379141717729913061 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2150840499286579 / 5000000000000000 : ℝ)) (q :=
+      (69752640907475889 / 160000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8])
+      (by norm_num [bρ_8]) (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_491_9 :
+    Uω (aρ 9) (bρ 9) (69427058680913 / 128000000000000 : ℝ) ≤
+      (-(1157252481795129172497 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (63394165429205193 / 160000000000000000 : ℝ) ≤
+      (-(1157252481795129172497 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (63394165429205193 / 160000000000000000 : ℝ) =
+            (63394165429205193 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (63394165429205193 / 40000000000000000 : ℝ) ≤
+            (575615468954870827503 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3847837192149321 / 10000000000000000 : ℝ)) (q :=
+      (63394165429205193 / 160000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9])
+      (by norm_num [bρ_9]) (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_491_10 :
+    Uω (aρ 10) (bρ 10) (69427058680913 / 128000000000000 : ℝ) ≤
+      (-(165329141977207929739 / 156250000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11107598166272613 / 32000000000000000 : ℝ) ≤
+      (-(165329141977207929739 / 156250000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (11107598166272613 / 32000000000000000 : ℝ) =
+            (11107598166272613 / 8000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11107598166272613 / 8000000000000000 : ℝ) ≤
+            (51279351866542070261 / 156250000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (650874880182811 / 2000000000000000 : ℝ)) (q :=
+      (11107598166272613 / 32000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_491_11 :
+    Uω (aρ 11) (bρ 11) (69427058680913 / 128000000000000 : ℝ) ≤
+      (-(6250626286282757761457 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (45835025944617849 / 160000000000000000 : ℝ) ≤
+      (-(6250626286282757761457 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (45835025944617849 / 160000000000000000 : ℝ) =
+            (45835025944617849 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (45835025944617849 / 40000000000000000 : ℝ) ≤
+            (680845516717242238543 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2459980416275903 / 10000000000000000 : ℝ)) (q :=
+      (45835025944617849 / 160000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_491_12 :
+    Uω (aρ 12) (bρ 12) (69427058680913 / 128000000000000 : ℝ) ≤
+      (-(1995624982888555183189 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (32416703437324241 / 160000000000000000 : ℝ) ≤
+      (-(1995624982888555183189 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (32416703437324241 / 160000000000000000 : ℝ) =
+            (32416703437324241 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (32416703437324241 / 20000000000000000 : ℝ) ≤
+            (603676943236444816811 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (603195099774601 / 5000000000000000 : ℝ)) (q :=
+      (32416703437324241 / 160000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_491_13 :
+    Uω (aρ 13) (bρ 13) (69427058680913 / 128000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_491_14 :
+    Uω (aρ 14) (bρ 14) (69427058680913 / 128000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_491_15 :
+    Uω (aρ 15) (bρ 15) (69427058680913 / 128000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_491_16 :
+    Uω (aρ 16) (bρ 16) (69427058680913 / 128000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_491 :
+    Uρ (69427058680913 / 128000000000000 : ℝ) ≤
+      (-(484236579550365468117 / 500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (69427058680913 / 128000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(249488629943552077721 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_491_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (69427058680913 / 128000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(6281980147295564591557 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_491_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (69427058680913 / 128000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(6372155189116117571223 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_491_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (69427058680913 / 128000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3261934058108181114563 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_491_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (69427058680913 / 128000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1689934962254301555617 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_491_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (69427058680913 / 128000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3554407191258538788113 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_491_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (69427058680913 / 128000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(7607427410292149149329 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_491_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (69427058680913 / 128000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(4151092661282270086939 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_491_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (69427058680913 / 128000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(1157252481795129172497 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_491_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (69427058680913 / 128000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) * (-(165329141977207929739 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_491_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (69427058680913 / 128000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(6250626286282757761457 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_491_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (69427058680913 / 128000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(1995624982888555183189 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_491_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (69427058680913 / 128000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_491_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (69427058680913 / 128000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_491_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (69427058680913 / 128000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_491_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (69427058680913 / 128000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_491_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_492_1 :
+    Uω (aρ 1) (bρ 1) (17376736390737 / 32000000000000 : ℝ) ≤
+      (-(6225577328427986757589 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4292562219326693 / 8000000000000000 : ℝ) ≤
+      (-(6225577328427986757589 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (4292562219326693 / 8000000000000000 : ℝ) =
+            (4292562219326693 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4292562219326693 / 4000000000000000 : ℝ) ≤
+            (705894474572013242411 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (268283632214821 / 500000000000000 : ℝ)) (q :=
+      (4292562219326693 / 8000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_492_2 :
+    Uω (aρ 2) (bρ 2) (17376736390737 / 32000000000000 : ℝ) ≤
+      (-(3135144586463325143691 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21367061126811181 / 40000000000000000 : ℝ) ≤
+      (-(3135144586463325143691 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (21367061126811181 / 40000000000000000 : ℝ) =
+            (21367061126811181 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21367061126811181 / 20000000000000000 : ℝ) ≤
+            (330591315036674856309 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2670783335947639 / 5000000000000000 : ℝ)) (q :=
+      (21367061126811181 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_492_3 :
+    Uω (aρ 3) (bρ 3) (17376736390737 / 32000000000000 : ℝ) ≤
+      (-(6360357257749035855581 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21175476179034053 / 40000000000000000 : ℝ) ≤
+      (-(6360357257749035855581 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (21175476179034053 / 40000000000000000 : ℝ) =
+            (21175476179034053 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21175476179034053 / 20000000000000000 : ℝ) ≤
+            (571114545250964144419 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2646585152178357 / 5000000000000000 : ℝ)) (q :=
+      (21175476179034053 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_492_4 :
+    Uω (aρ 4) (bρ 4) (17376736390737 / 32000000000000 : ℝ) ≤
+      (-(3255943440666472898293 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20857023827179691 / 40000000000000000 : ℝ) ≤
+      (-(3255943440666472898293 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (20857023827179691 / 40000000000000000 : ℝ) =
+            (20857023827179691 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20857023827179691 / 20000000000000000 : ℝ) ≤
+            (209792460833527101707 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5212238309494533 / 10000000000000000 : ℝ)) (q :=
+      (20857023827179691 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_492_5 :
+    Uω (aρ 5) (bρ 5) (17376736390737 / 32000000000000 : ℝ) ≤
+      (-(6747464966434608471673 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4074284077586997 / 8000000000000000 : ℝ) ≤
+      (-(6747464966434608471673 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (4074284077586997 / 8000000000000000 : ℝ) =
+            (4074284077586997 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4074284077586997 / 4000000000000000 : ℝ) ≤
+            (184006836565391528327 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (254381325460359 / 500000000000000 : ℝ)) (q :=
+      (4074284077586997 / 8000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_492_6 :
+    Uω (aρ 6) (bρ 6) (17376736390737 / 32000000000000 : ℝ) ≤
+      (-(1419216894299342846093 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19673469583397039 / 40000000000000000 : ℝ) ≤
+      (-(1419216894299342846093 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (19673469583397039 / 40000000000000000 : ℝ) =
+            (19673469583397039 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19673469583397039 / 10000000000000000 : ℝ) ≤
+            (1353371826900657153907 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4905891482078207 / 10000000000000000 : ℝ)) (q :=
+      (19673469583397039 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_492_7 :
+    Uω (aρ 7) (bρ 7) (17376736390737 / 32000000000000 : ℝ) ≤
+      (-(7594001513948014470251 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (18717881598994987 / 40000000000000000 : ℝ) ≤
+      (-(7594001513948014470251 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (18717881598994987 / 40000000000000000 : ℝ) =
+            (18717881598994987 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18717881598994987 / 10000000000000000 : ℝ) ≤
+            (6268942092051985529749 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4651758870637181 / 10000000000000000 : ℝ)) (q :=
+      (18717881598994987 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_492_8 :
+    Uω (aρ 8) (bρ 8) (17376736390737 / 32000000000000 : ℝ) ≤
+      (-(2071921864965059653457 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17463460167469257 / 40000000000000000 : ℝ) ≤
+      (-(2071921864965059653457 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (17463460167469257 / 40000000000000000 : ℝ) =
+            (17463460167469257 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17463460167469257 / 10000000000000000 : ℝ) ≤
+            (1393814036534940346543 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1077022451553579 / 2500000000000000 : ℝ)) (q :=
+      (17463460167469257 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_492_9 :
+    Uω (aρ 9) (bρ 9) (17376736390737 / 32000000000000 : ℝ) ≤
+      (-(9241813849391177808397 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15874246334424941 / 40000000000000000 : ℝ) ≤
+      (-(9241813849391177808397 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (15874246334424941 / 40000000000000000 : ℝ) =
+            (15874246334424941 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15874246334424941 / 10000000000000000 : ℝ) ≤
+            (4621129756608822191603 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1927224259026079 / 5000000000000000 : ℝ)) (q :=
+      (15874246334424941 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_492_10 :
+    Uω (aρ 10) (bρ 10) (17376736390737 / 32000000000000 : ℝ) ≤
+      (-(2640477031809797859469 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2782224335433059 / 8000000000000000 : ℝ) ≤
+      (-(2640477031809797859469 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2782224335433059 / 8000000000000000 : ℝ) =
+            (2782224335433059 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2782224335433059 / 2000000000000000 : ℝ) ≤
+            (825258869690202140531 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (652289044583467 / 2000000000000000 : ℝ)) (q :=
+      (2782224335433059 / 8000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_492_11 :
+    Uω (aρ 11) (bρ 11) (17376736390737 / 32000000000000 : ℝ) ≤
+      (-(2495184889494876055977 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2297563230746221 / 8000000000000000 : ℝ) ≤
+      (-(2495184889494876055977 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2297563230746221 / 8000000000000000 : ℝ) =
+            (2297563230746221 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2297563230746221 / 2000000000000000 : ℝ) ≤
+            (277403831705123944023 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (61706727185131 / 250000000000000 : ℝ)) (q :=
+      (2297563230746221 / 8000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_492_12 :
+    Uω (aρ 12) (bρ 12) (17376736390737 / 32000000000000 : ℝ) ≤
+      (-(1591357776262880250737 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8145956564381167 / 40000000000000000 : ℝ) ≤
+      (-(1591357776262880250737 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (8145956564381167 / 40000000000000000 : ℝ) =
+            (8145956564381167 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8145956564381167 / 5000000000000000 : ℝ) ≤
+            (488083764637119749263 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1221039389415271 / 10000000000000000 : ℝ)) (q :=
+      (8145956564381167 / 40000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_492_13 :
+    Uω (aρ 13) (bρ 13) (17376736390737 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_492_14 :
+    Uω (aρ 14) (bρ 14) (17376736390737 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_492_15 :
+    Uω (aρ 15) (bρ 15) (17376736390737 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_492_16 :
+    Uω (aρ 16) (bρ 16) (17376736390737 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_492 :
+    Uρ (17376736390737 / 32000000000000 : ℝ) ≤
+      (-(9670103930970074289523 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (17376736390737 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(6225577328427986757589 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_492_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (17376736390737 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3135144586463325143691 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_492_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (17376736390737 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(6360357257749035855581 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_492_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (17376736390737 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3255943440666472898293 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_492_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (17376736390737 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(6747464966434608471673 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_492_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (17376736390737 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1419216894299342846093 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_492_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (17376736390737 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(7594001513948014470251 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_492_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (17376736390737 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(2071921864965059653457 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_492_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (17376736390737 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(9241813849391177808397 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_492_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (17376736390737 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(2640477031809797859469 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_492_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (17376736390737 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(2495184889494876055977 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_492_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (17376736390737 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(1591357776262880250737 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_492_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (17376736390737 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_492_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (17376736390737 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_492_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (17376736390737 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_492_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (17376736390737 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_492_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_493_1 :
+    Uω (aρ 1) (bρ 1) (69586832444983 / 128000000000000 : ℝ) ≤
+      (-(1553488109489150942229 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (85951103549162159 / 160000000000000000 : ℝ) ≤
+      (-(1553488109489150942229 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (85951103549162159 / 160000000000000000 : ℝ) =
+            (85951103549162159 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (85951103549162159 / 80000000000000000 : ℝ) ≤
+            (179379841260849057771 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5371913876965973 / 10000000000000000 : ℝ)) (q :=
+      (85951103549162159 / 160000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_493_2 :
+    Uω (aρ 2) (bρ 2) (69586832444983 / 128000000000000 : ℝ) ≤
+      (-(6258611851501049440367 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (85568106818391623 / 160000000000000000 : ℝ) ≤
+      (-(6258611851501049440367 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (85568106818391623 / 160000000000000000 : ℝ) =
+            (85568106818391623 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (85568106818391623 / 80000000000000000 : ℝ) ≤
+            (672859951498950559633 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (668476037266207 / 1250000000000000 : ℝ)) (q :=
+      (85568106818391623 / 160000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_493_3 :
+    Uω (aρ 3) (bρ 3) (69586832444983 / 128000000000000 : ℝ) ≤
+      (-(1587143308191476179277 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (84801776485279623 / 160000000000000000 : ℝ) ≤
+      (-(1587143308191476179277 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (84801776485279623 / 160000000000000000 : ℝ) =
+            (84801776485279623 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (84801776485279623 / 80000000000000000 : ℝ) ≤
+            (145724642558523820723 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (331213319552541 / 625000000000000 : ℝ)) (q :=
+      (84801776485279623 / 160000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_493_4 :
+    Uω (aρ 4) (bρ 4) (69586832444983 / 128000000000000 : ℝ) ≤
+      (-(6499919995358701146053 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (83527992520084831 / 160000000000000000 : ℝ) ≤
+      (-(6499919995358701146053 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (83527992520084831 / 160000000000000000 : ℝ) =
+            (83527992520084831 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (83527992520084831 / 80000000000000000 : ℝ) ≤
+            (431551807641298853947 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5218484298256307 / 10000000000000000 : ℝ)) (q :=
+      (83527992520084831 / 160000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_493_5 :
+    Uω (aρ 5) (bρ 5) (69586832444983 / 128000000000000 : ℝ) ≤
+      (-(1683801290887367162087 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16317128530753699 / 32000000000000000 : ℝ) ≤
+      (-(1683801290887367162087 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (16317128530753699 / 32000000000000000 : ℝ) =
+            (16317128530753699 / 16000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16317128530753699 / 16000000000000000 : ℝ) ≤
+            (49066659862632837913 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1018776096860353 / 2000000000000000 : ℝ)) (q :=
+      (16317128530753699 / 32000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_493_6 :
+    Uω (aρ 6) (bρ 6) (69586832444983 / 128000000000000 : ℝ) ≤
+      (-(7083370827149970993019 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15758798111560779 / 32000000000000000 : ℝ) ≤
+      (-(7083370827149970993019 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (15758798111560779 / 32000000000000000 : ℝ) =
+            (15758798111560779 / 8000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15758798111560779 / 8000000000000000 : ℝ) ≤
+            (6779572778850029006981 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (491216434744619 / 1000000000000000 : ℝ)) (q :=
+      (15758798111560779 / 32000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_493_7 :
+    Uω (aρ 7) (bρ 7) (69586832444983 / 128000000000000 : ℝ) ≤
+      (-(3790296916546084192099 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (74971979075916063 / 160000000000000000 : ℝ) ≤
+      (-(3790296916546084192099 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (74971979075916063 / 160000000000000000 : ℝ) =
+            (74971979075916063 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (74971979075916063 / 40000000000000000 : ℝ) ≤
+            (3141174886453915807901 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4658074292970211 / 10000000000000000 : ℝ)) (q :=
+      (74971979075916063 / 160000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_493_8 :
+    Uω (aρ 8) (bρ 8) (69586832444983 / 128000000000000 : ℝ) ≤
+      (-(8273211147642510212537 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (69955036500040647 / 160000000000000000 : ℝ) ≤
+      (-(8273211147642510212537 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (69955036500040647 / 160000000000000000 : ℝ) =
+            (69955036500040647 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (69955036500040647 / 40000000000000000 : ℝ) ≤
+            (5589732458357489787463 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (539312265290723 / 1250000000000000 : ℝ)) (q :=
+      (69955036500040647 / 160000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8])
+      (by norm_num [bρ_8]) (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_493_9 :
+    Uω (aρ 9) (bρ 9) (69586832444983 / 128000000000000 : ℝ) ≤
+      (-(115320445191368720493 / 125000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12719959077751323 / 32000000000000000 : ℝ) ≤
+      (-(115320445191368720493 / 125000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (12719959077751323 / 32000000000000000 : ℝ) =
+            (12719959077751323 / 8000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12719959077751323 / 8000000000000000 : ℝ) ≤
+            (57966349883631279507 / 125000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (48263232647191 / 125000000000000 : ℝ)) (q :=
+      (12719959077751323 / 32000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_493_10 :
+    Uω (aρ 10) (bρ 10) (69586832444983 / 128000000000000 : ℝ) ≤
+      (-(10542792645437616091663 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (55750955553708511 / 160000000000000000 : ℝ) ≤
+      (-(10542792645437616091663 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (55750955553708511 / 160000000000000000 : ℝ) =
+            (55750955553708511 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (55750955553708511 / 40000000000000000 : ℝ) ≤
+            (3320150960562383908337 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3268512665889267 / 10000000000000000 : ℝ)) (q :=
+      (55750955553708511 / 160000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_493_11 :
+    Uω (aρ 11) (bρ 11) (69586832444983 / 128000000000000 : ℝ) ≤
+      (-(778167573294784877017 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9213481436800803 / 32000000000000000 : ℝ) ≤
+      (-(778167573294784877017 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9213481436800803 / 32000000000000000 : ℝ) =
+            (9213481436800803 / 8000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9213481436800803 / 8000000000000000 : ℝ) ≤
+            (88266402080215122983 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (495309149176241 / 2000000000000000 : ℝ)) (q :=
+      (9213481436800803 / 32000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_493_12 :
+    Uω (aρ 12) (bρ 12) (69586832444983 / 128000000000000 : ℝ) ≤
+      (-(15862765972973506168137 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (32749811854259119 / 160000000000000000 : ℝ) ≤
+      (-(15862765972973506168137 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (32749811854259119 / 160000000000000000 : ℝ) =
+            (32749811854259119 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (32749811854259119 / 20000000000000000 : ℝ) ≤
+            (4931649436026493831863 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1235546426348093 / 10000000000000000 : ℝ)) (q :=
+      (32749811854259119 / 160000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_493_13 :
+    Uω (aρ 13) (bρ 13) (69586832444983 / 128000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_493_14 :
+    Uω (aρ 14) (bρ 14) (69586832444983 / 128000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_493_15 :
+    Uω (aρ 15) (bρ 15) (69586832444983 / 128000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_493_16 :
+    Uω (aρ 16) (bρ 16) (69586832444983 / 128000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_493 :
+    Uρ (69586832444983 / 128000000000000 : ℝ) ≤
+      (-(1931107930420668247501 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (69586832444983 / 128000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1553488109489150942229 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_493_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (69586832444983 / 128000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(6258611851501049440367 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_493_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (69586832444983 / 128000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1587143308191476179277 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_493_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (69586832444983 / 128000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(6499919995358701146053 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_493_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (69586832444983 / 128000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1683801290887367162087 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_493_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (69586832444983 / 128000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(7083370827149970993019 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_493_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (69586832444983 / 128000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(3790296916546084192099 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_493_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (69586832444983 / 128000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(8273211147642510212537 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_493_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (69586832444983 / 128000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(115320445191368720493 / 125000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_493_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (69586832444983 / 128000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(10542792645437616091663 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_493_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (69586832444983 / 128000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) * (-(778167573294784877017 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_493_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (69586832444983 / 128000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(15862765972973506168137 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_493_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (69586832444983 / 128000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_493_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (69586832444983 / 128000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_493_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (69586832444983 / 128000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_493_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (69586832444983 / 128000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_493_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_494_1 :
+    Uω (aρ 1) (bρ 1) (34833359663509 / 64000000000000 : ℝ) ≤
+      (-(1240468209150847565031 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (43025481355245261 / 80000000000000000 : ℝ) ≤
+      (-(1240468209150847565031 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (43025481355245261 / 80000000000000000 : ℝ) =
+            (43025481355245261 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (43025481355245261 / 40000000000000000 : ℝ) ≤
+            (145826151449152434969 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2689077554736517 / 5000000000000000 : ℝ)) (q :=
+      (43025481355245261 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_494_2 :
+    Uω (aρ 2) (bρ 2) (34833359663509 / 64000000000000 : ℝ) ≤
+      (-(1561737037791314671991 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (42833984560446017 / 80000000000000000 : ℝ) ≤
+      (-(1561737037791314671991 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (42833984560446017 / 80000000000000000 : ℝ) =
+            (42833984560446017 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (42833984560446017 / 40000000000000000 : ℝ) ≤
+            (171130912958685328009 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5354049923283223 / 10000000000000000 : ℝ)) (q :=
+      (42833984560446017 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_494_3 :
+    Uω (aρ 3) (bρ 3) (34833359663509 / 64000000000000 : ℝ) ≤
+      (-(792100385176676737587 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8490164822344261 / 16000000000000000 : ℝ) ≤
+      (-(792100385176676737587 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (8490164822344261 / 16000000000000000 : ℝ) =
+            (8490164822344261 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8490164822344261 / 8000000000000000 : ℝ) ≤
+            (74333590198323262413 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1061131183490409 / 2000000000000000 : ℝ)) (q :=
+      (8490164822344261 / 16000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_494_4 :
+    Uω (aρ 4) (bρ 4) (34833359663509 / 64000000000000 : ℝ) ≤
+      (-(3243983711969671616221 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (41813944819587733 / 80000000000000000 : ℝ) ≤
+      (-(3243983711969671616221 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (41813944819587733 / 80000000000000000 : ℝ) =
+            (41813944819587733 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (41813944819587733 / 40000000000000000 : ℝ) ≤
+            (221752189530328383779 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1306182568870913 / 2500000000000000 : ℝ)) (q :=
+      (41813944819587733 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_494_5 :
+    Uω (aρ 5) (bρ 5) (34833359663509 / 64000000000000 : ℝ) ≤
+      (-(1344592080656173711347 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (40842801752338581 / 80000000000000000 : ℝ) ≤
+      (-(1344592080656173711347 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (40842801752338581 / 80000000000000000 : ℝ) =
+            (40842801752338581 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (40842801752338581 / 40000000000000000 : ℝ) ≤
+            (41702279943826288653 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1275033607001591 / 2500000000000000 : ℝ)) (q :=
+      (40842801752338581 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_494_6 :
+    Uω (aρ 6) (bρ 6) (34833359663509 / 64000000000000 : ℝ) ≤
+      (-(1767668351937252963239 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (39447051068363761 / 80000000000000000 : ℝ) ≤
+      (-(1767668351937252963239 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (39447051068363761 / 80000000000000000 : ℝ) =
+            (39447051068363761 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (39447051068363761 / 20000000000000000 : ℝ) ≤
+            (1698067549562747036761 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4918437132152659 / 10000000000000000 : ℝ)) (q :=
+      (39447051068363761 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_494_7 :
+    Uω (aρ 7) (bρ 7) (34833359663509 / 64000000000000 : ℝ) ≤
+      (-(3783602158894347139321 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (37536215078294857 / 80000000000000000 : ℝ) ≤
+      (-(3783602158894347139321 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (37536215078294857 / 80000000000000000 : ℝ) =
+            (37536215078294857 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (37536215078294857 / 20000000000000000 : ℝ) ≤
+            (3147869644105652860679 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4664389515395433 / 10000000000000000 : ℝ)) (q :=
+      (37536215078294857 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_494_8 :
+    Uω (aρ 8) (bρ 8) (34833359663509 / 64000000000000 : ℝ) ≤
+      (-(412937816014834911931 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (35028114207731197 / 80000000000000000 : ℝ) ≤
+      (-(412937816014834911931 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (35028114207731197 / 80000000000000000 : ℝ) =
+            (35028114207731197 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (35028114207731197 / 20000000000000000 : ℝ) ≤
+            (280209364285165088069 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2160452974547259 / 5000000000000000 : ℝ)) (q :=
+      (35028114207731197 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_494_9 :
+    Uω (aρ 9) (bρ 9) (34833359663509 / 64000000000000 : ℝ) ≤
+      (-(920948505196492789883 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (31851297816458373 / 80000000000000000 : ℝ) ≤
+      (-(920948505196492789883 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (31851297816458373 / 80000000000000000 : ℝ) =
+            (31851297816458373 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (31851297816458373 / 20000000000000000 : ℝ) ≤
+            (465345855403507210117 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (483458434954539 / 1250000000000000 : ℝ)) (q :=
+      (31851297816458373 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_494_10 :
+    Uω (aρ 10) (bρ 10) (34833359663509 / 64000000000000 : ℝ) ≤
+      (-(10523718442281250153947 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27928698770740433 / 80000000000000000 : ℝ) ≤
+      (-(10523718442281250153947 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (27928698770740433 / 80000000000000000 : ℝ) =
+            (27928698770740433 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27928698770740433 / 20000000000000000 : ℝ) ≤
+            (3339225163718749846053 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3275576751701827 / 10000000000000000 : ℝ)) (q :=
+      (27928698770740433 / 80000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_494_11 :
+    Uω (aρ 11) (bρ 11) (34833359663509 / 64000000000000 : ℝ) ≤
+      (-(12425522060461687284879 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4618345461216777 / 16000000000000000 : ℝ) ≤
+      (-(12425522060461687284879 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4618345461216777 / 16000000000000000 : ℝ) =
+            (4618345461216777 / 4000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4618345461216777 / 4000000000000000 : ℝ) ≤
+            (1437421545538312715121 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (248481051174269 / 1000000000000000 : ℝ)) (q :=
+      (4618345461216777 / 16000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_494_12 :
+    Uω (aρ 12) (bρ 12) (34833359663509 / 64000000000000 : ℝ) ≤
+      (-(7906272266414078093663 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16457349912420733 / 80000000000000000 : ℝ) ≤
+      (-(7906272266414078093663 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (16457349912420733 / 80000000000000000 : ℝ) =
+            (16457349912420733 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16457349912420733 / 10000000000000000 : ℝ) ≤
+            (2490935438085921906337 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (624958130005951 / 5000000000000000 : ℝ)) (q :=
+      (16457349912420733 / 80000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_494_13 :
+    Uω (aρ 13) (bρ 13) (34833359663509 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_494_14 :
+    Uω (aρ 14) (bρ 14) (34833359663509 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_494_15 :
+    Uω (aρ 15) (bρ 15) (34833359663509 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_494_16 :
+    Uω (aρ 16) (bρ 16) (34833359663509 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_494 :
+    Uρ (34833359663509 / 64000000000000 : ℝ) ≤
+      (-(9641037249386160701971 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (34833359663509 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1240468209150847565031 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_494_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (34833359663509 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1561737037791314671991 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_494_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (34833359663509 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(792100385176676737587 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_494_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (34833359663509 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3243983711969671616221 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_494_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (34833359663509 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1344592080656173711347 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_494_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (34833359663509 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1767668351937252963239 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_494_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (34833359663509 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(3783602158894347139321 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_494_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (34833359663509 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(412937816014834911931 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_494_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (34833359663509 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(920948505196492789883 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_494_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (34833359663509 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(10523718442281250153947 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_494_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (34833359663509 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(12425522060461687284879 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_494_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (34833359663509 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(7906272266414078093663 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_494_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (34833359663509 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_494_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (34833359663509 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_494_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (34833359663509 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_494_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (34833359663509 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_494_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_495_1 :
+    Uω (aρ 1) (bρ 1) (69746606209053 / 128000000000000 : ℝ) ≤
+      (-(123814862410195933307 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17230164374104697 / 32000000000000000 : ℝ) ≤
+      (-(123814862410195933307 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (17230164374104697 / 32000000000000000 : ℝ) =
+            (17230164374104697 / 16000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17230164374104697 / 16000000000000000 : ℝ) ≤
+            (14814573649804066693 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (538439634181817 / 1000000000000000 : ℝ)) (q :=
+      (17230164374104697 / 32000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_495_2 :
+    Uω (aρ 2) (bρ 2) (69746606209053 / 128000000000000 : ℝ) ≤
+      (-(3117649020088563942417 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17153566282955233 / 32000000000000000 : ℝ) ≤
+      (-(3117649020088563942417 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (17153566282955233 / 32000000000000000 : ℝ) =
+            (17153566282955233 / 16000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17153566282955233 / 16000000000000000 : ℝ) ≤
+            (348086881411436057583 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1072058309471951 / 2000000000000000 : ℝ)) (q :=
+      (17153566282955233 / 32000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_495_3 :
+    Uω (aρ 3) (bρ 3) (69746606209053 / 128000000000000 : ℝ) ≤
+      (-(6325046771053860354781 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (85001519930734413 / 160000000000000000 : ℝ) ≤
+      (-(6325046771053860354781 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (85001519930734413 / 160000000000000000 : ℝ) =
+            (85001519930734413 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (85001519930734413 / 80000000000000000 : ℝ) ≤
+            (606425031946139645219 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (663987339775567 / 1250000000000000 : ℝ)) (q :=
+      (85001519930734413 / 160000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_495_4 :
+    Uω (aρ 4) (bρ 4) (69746606209053 / 128000000000000 : ℝ) ≤
+      (-(1619007283210977189607 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (83727786666321213 / 160000000000000000 : ℝ) ≤
+      (-(1619007283210977189607 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (83727786666321213 / 160000000000000000 : ℝ) =
+            (83727786666321213 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (83727786666321213 / 80000000000000000 : ℝ) ≤
+            (113860667539022810393 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2615488120608943 / 5000000000000000 : ℝ)) (q :=
+      (83727786666321213 / 160000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_495_5 :
+    Uω (aρ 5) (bρ 5) (69746606209053 / 128000000000000 : ℝ) ≤
+      (-(6710730648684770959207 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (81785564105388597 / 160000000000000000 : ℝ) ≤
+      (-(6710730648684770959207 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (81785564105388597 / 160000000000000000 : ℝ) =
+            (81785564105388597 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (81785564105388597 / 80000000000000000 : ℝ) ≤
+            (220741154315229040793 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5106388340436309 / 10000000000000000 : ℝ)) (q :=
+      (81785564105388597 / 160000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_495_6 :
+    Uω (aρ 6) (bρ 6) (69746606209053 / 128000000000000 : ℝ) ≤
+      (-(1764498042931787618447 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (78994213072824829 / 160000000000000000 : ℝ) ≤
+      (-(1764498042931787618447 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (78994213072824829 / 160000000000000000 : ℝ) =
+            (78994213072824829 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (78994213072824829 / 40000000000000000 : ℝ) ≤
+            (1701237858568212381553 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2462354918252919 / 5000000000000000 : ℝ)) (q :=
+      (78994213072824829 / 160000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_495_7 :
+    Uω (aρ 7) (bρ 7) (69746606209053 / 128000000000000 : ℝ) ≤
+      (-(1888458229577260486051 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (75172879644487933 / 160000000000000000 : ℝ) ≤
+      (-(1888458229577260486051 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (75172879644487933 / 160000000000000000 : ℝ) =
+            (75172879644487933 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (75172879644487933 / 40000000000000000 : ℝ) ≤
+            (1577277671922739513949 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2335352269361863 / 5000000000000000 : ℝ)) (q :=
+      (75172879644487933 / 160000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_495_8 :
+    Uω (aρ 8) (bρ 8) (69746606209053 / 128000000000000 : ℝ) ≤
+      (-(4122161456257221488253 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (70157416433532973 / 160000000000000000 : ℝ) ≤
+      (-(4122161456257221488253 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (70157416433532973 / 160000000000000000 : ℝ) =
+            (70157416433532973 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (70157416433532973 / 40000000000000000 : ℝ) ≤
+            (2809310346742778511747 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1081828322173589 / 2500000000000000 : ℝ)) (q :=
+      (70157416433532973 / 160000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8])
+      (by norm_num [bρ_8]) (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_495_9 :
+    Uω (aρ 9) (bρ 9) (69746606209053 / 128000000000000 : ℝ) ≤
+      (-(9193362059771989769797 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (63805386120366957 / 160000000000000000 : ℝ) ≤
+      (-(9193362059771989769797 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (63805386120366957 / 160000000000000000 : ℝ) =
+            (63805386120366957 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (63805386120366957 / 40000000000000000 : ℝ) ≤
+            (4669581546228010230203 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (968568781977151 / 2500000000000000 : ℝ)) (q :=
+      (63805386120366957 / 160000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9])
+      (by norm_num [bρ_9]) (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_495_10 :
+    Uω (aρ 10) (bρ 10) (69746606209053 / 128000000000000 : ℝ) ≤
+      (-(5252342660226903689753 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (55963812845365133 / 160000000000000000 : ℝ) ≤
+      (-(5252342660226903689753 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (55963812845365133 / 160000000000000000 : ℝ) =
+            (55963812845365133 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (55963812845365133 / 40000000000000000 : ℝ) ≤
+            (1679129142773096310247 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (410329687753547 / 1250000000000000 : ℝ)) (q :=
+      (55963812845365133 / 160000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_495_11 :
+    Uω (aρ 11) (bρ 11) (69746606209053 / 128000000000000 : ℝ) ≤
+      (-(2480089286429880528157 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (46299407845621909 / 160000000000000000 : ℝ) ≤
+      (-(2480089286429880528157 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (46299407845621909 / 160000000000000000 : ℝ) =
+            (46299407845621909 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (46299407845621909 / 40000000000000000 : ℝ) ≤
+            (292499434770119471843 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2493063503265473 / 10000000000000000 : ℝ)) (q :=
+      (46299407845621909 / 160000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_495_12 :
+    Uω (aρ 12) (bρ 12) (69746606209053 / 128000000000000 : ℝ) ≤
+      (-(3152578911133758865109 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (33078527600011461 / 160000000000000000 : ℝ) ≤
+      (-(3152578911133758865109 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (33078527600011461 / 160000000000000000 : ℝ) =
+            (33078527600011461 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (33078527600011461 / 20000000000000000 : ℝ) ≤
+            (1006304170666241134891 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1264153569249167 / 10000000000000000 : ℝ)) (q :=
+      (33078527600011461 / 160000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_495_13 :
+    Uω (aρ 13) (bρ 13) (69746606209053 / 128000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_495_14 :
+    Uω (aρ 14) (bρ 14) (69746606209053 / 128000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_495_15 :
+    Uω (aρ 15) (bρ 15) (69746606209053 / 128000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_495_16 :
+    Uω (aρ 16) (bρ 16) (69746606209053 / 128000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_495 :
+    Uρ (69746606209053 / 128000000000000 : ℝ) ≤
+      (-(9626595294409110944959 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (69746606209053 / 128000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(123814862410195933307 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_495_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (69746606209053 / 128000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3117649020088563942417 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_495_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (69746606209053 / 128000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(6325046771053860354781 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_495_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (69746606209053 / 128000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1619007283210977189607 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_495_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (69746606209053 / 128000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(6710730648684770959207 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_495_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (69746606209053 / 128000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1764498042931787618447 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_495_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (69746606209053 / 128000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(1888458229577260486051 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_495_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (69746606209053 / 128000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(4122161456257221488253 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_495_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (69746606209053 / 128000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(9193362059771989769797 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_495_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (69746606209053 / 128000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(5252342660226903689753 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_495_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (69746606209053 / 128000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(2480089286429880528157 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_495_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (69746606209053 / 128000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(3152578911133758865109 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_495_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (69746606209053 / 128000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_495_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (69746606209053 / 128000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_495_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (69746606209053 / 128000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_495_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (69746606209053 / 128000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_495_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/U41.lean b/LeanPool/Zeta5Irrational/Table/U41.lean
new file mode 100644
index 0000000000..3284783767
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/U41.lean
@@ -0,0 +1,4612 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-! # Certified arcsine potential bounds (U41) -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma U_496_1 :
+    Uω (aρ 1) (bρ 1) (4364155818193 / 8000000000000 : ℝ) ≤
+      (-(6179158631021010828439 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10781335128658193 / 20000000000000000 : ℝ) ≤
+      (-(6179158631021010828439 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10781335128658193 / 20000000000000000 : ℝ) =
+            (10781335128658193 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10781335128658193 / 10000000000000000 : ℝ) ≤
+            (752313171978989171561 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5390637574001943 / 10000000000000000 : ℝ)) (q :=
+      (10781335128658193 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_496_2 :
+    Uω (aρ 2) (bρ 2) (4364155818193 / 8000000000000 : ℝ) ≤
+      (-(6223661486905342670339 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (536673085625463 / 1000000000000000 : ℝ) ≤
+      (-(6223661486905342670339 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (536673085625463 / 1000000000000000 : ℝ) =
+            (536673085625463 / 500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (536673085625463 / 500000000000000 : ℝ) ≤
+            (707810316094657329661 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (536653317036301 / 1000000000000000 : ℝ)) (q :=
+      (536673085625463 / 1000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_496_3 :
+    Uω (aρ 3) (bρ 3) (4364155818193 / 8000000000000 : ℝ) ≤
+      (-(1578326067291152192927 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10637673950907967 / 20000000000000000 : ℝ) ≤
+      (-(1578326067291152192927 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10637673950907967 / 20000000000000000 : ℝ) =
+            (10637673950907967 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10637673950907967 / 10000000000000000 : ℝ) ≤
+            (154541883458847807073 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5318141515111717 / 10000000000000000 : ℝ)) (q :=
+      (10637673950907967 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_496_4 :
+    Uω (aρ 4) (bρ 4) (4364155818193 / 8000000000000 : ℝ) ≤
+      (-(6464105087964167300301 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10478460450231381 / 20000000000000000 : ℝ) ≤
+      (-(6464105087964167300301 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10478460450231381 / 20000000000000000 : ℝ) =
+            (10478460450231381 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10478460450231381 / 10000000000000000 : ℝ) ≤
+            (467366715035832699699 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5237222195500131 / 10000000000000000 : ℝ)) (q :=
+      (10478460450231381 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_496_5 :
+    Uω (aρ 5) (bρ 5) (4364155818193 / 8000000000000 : ℝ) ≤
+      (-(6698515862953343862829 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10235690557102619 / 20000000000000000 : ℝ) ≤
+      (-(6698515862953343862829 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10235690557102619 / 20000000000000000 : ℝ) =
+            (10235690557102619 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10235690557102619 / 10000000000000000 : ℝ) ≤
+            (232955940046656137171 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5112642221706369 / 10000000000000000 : ℝ)) (q :=
+      (10235690557102619 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_496_6 :
+    Uω (aρ 6) (bρ 6) (4364155818193 / 8000000000000 : ℝ) ≤
+      (-(7045327077678014554961 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9886790421068631 / 20000000000000000 : ℝ) ≤
+      (-(7045327077678014554961 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9886790421068631 / 20000000000000000 : ℝ) =
+            (9886790421068631 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9886790421068631 / 5000000000000000 : ℝ) ≤
+            (6817616528321985445039 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4930982460812381 / 10000000000000000 : ℝ)) (q :=
+      (9886790421068631 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_496_7 :
+    Uω (aρ 7) (bρ 7) (4364155818193 / 8000000000000 : ℝ) ≤
+      (-(1508095917026171862497 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9409165943257819 / 20000000000000000 : ℝ) ≤
+      (-(1508095917026171862497 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9409165943257819 / 20000000000000000 : ℝ) =
+            (9409165943257819 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9409165943257819 / 5000000000000000 : ℝ) ≤
+            (1264492804173828137503 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4677019363761569 / 10000000000000000 : ℝ)) (q :=
+      (9409165943257819 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_496_8 :
+    Uω (aρ 8) (bρ 8) (4364155818193 / 8000000000000 : ℝ) ≤
+      (-(514369428705734080287 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8782325071442361 / 20000000000000000 : ℝ) ≤
+      (-(514369428705734080287 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8782325071442361 / 20000000000000000 : ℝ) =
+            (8782325071442361 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8782325071442361 / 5000000000000000 : ℝ) ≤
+            (352064546669265919713 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4333720143286111 / 10000000000000000 : ℝ)) (q :=
+      (8782325071442361 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_496_9 :
+    Uω (aρ 9) (bρ 9) (4364155818193 / 8000000000000 : ℝ) ≤
+      (-(9177266539706387383039 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (63908166900943 / 160000000000000 : ℝ) ≤
+      (-(9177266539706387383039 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (63908166900943 / 160000000000000 : ℝ) = (63908166900943 / 40000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (63908166900943 / 40000000000000 : ℝ) ≤
+            (4685677066293612616961 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (31047052502573 / 80000000000000 : ℝ)) (q :=
+      (63908166900943 / 160000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_496_10 :
+    Uω (aρ 10) (bρ 10) (4364155818193 / 8000000000000 : ℝ) ≤
+      (-(10485693084161377691493 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7008775204647261 / 20000000000000000 : ℝ) ≤
+      (-(10485693084161377691493 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7008775204647261 / 20000000000000000 : ℝ) =
+            (7008775204647261 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7008775204647261 / 5000000000000000 : ℝ) ≤
+            (3377250521838622308507 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3289694938346011 / 10000000000000000 : ℝ)) (q :=
+      (7008775204647261 / 20000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_496_11 :
+    Uω (aρ 11) (bρ 11) (4364155818193 / 8000000000000 : ℝ) ≤
+      (-(193366462786153684599 / 156250000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5801908477093199 / 20000000000000000 : ℝ) ≤
+      (-(193366462786153684599 / 156250000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5801908477093199 / 20000000000000000 : ℝ) =
+            (5801908477093199 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5801908477093199 / 5000000000000000 : ℝ) ≤
+            (23242031057596315401 / 156250000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2501304836996949 / 10000000000000000 : ℝ)) (q :=
+      (5801908477093199 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_496_12 :
+    Uω (aρ 12) (bρ 12) (4364155818193 / 8000000000000 : ℝ) ≤
+      (-(1964224768782355217967 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (415516632565571 / 2000000000000000 : ℝ) ≤
+      (-(1964224768782355217967 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (415516632565571 / 2000000000000000 : ℝ) =
+            (415516632565571 / 250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (415516632565571 / 250000000000000 : ℝ) ≤
+            (635077157342644782033 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (63913139112223 / 500000000000000 : ℝ)) (q :=
+      (415516632565571 / 2000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12]) (by norm_num [bρ_12])
+      (by norm_num) (by norm_num [aρ_12, bρ_12]) (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_496_13 :
+    Uω (aρ 13) (bρ 13) (4364155818193 / 8000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_496_14 :
+    Uω (aρ 14) (bρ 14) (4364155818193 / 8000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_496_15 :
+    Uω (aρ 15) (bρ 15) (4364155818193 / 8000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_496_16 :
+    Uω (aρ 16) (bρ 16) (4364155818193 / 8000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_496 :
+    Uρ (4364155818193 / 8000000000000 : ℝ) ≤
+      (-(480610621488243035319 / 500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (4364155818193 / 8000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(6179158631021010828439 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_496_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (4364155818193 / 8000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(6223661486905342670339 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_496_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (4364155818193 / 8000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1578326067291152192927 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_496_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (4364155818193 / 8000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(6464105087964167300301 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_496_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (4364155818193 / 8000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(6698515862953343862829 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_496_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (4364155818193 / 8000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(7045327077678014554961 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_496_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (4364155818193 / 8000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(1508095917026171862497 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_496_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (4364155818193 / 8000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(514369428705734080287 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_496_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (4364155818193 / 8000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(9177266539706387383039 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_496_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (4364155818193 / 8000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(10485693084161377691493 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_496_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (4364155818193 / 8000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) * (-(193366462786153684599 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_496_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (4364155818193 / 8000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(1964224768782355217967 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_496_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (4364155818193 / 8000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_496_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (4364155818193 / 8000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_496_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (4364155818193 / 8000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_496_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (4364155818193 / 8000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_496_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_497_1 :
+    Uω (aρ 1) (bρ 1) (69906379973123 / 128000000000000 : ℝ) ≤
+      (-(6167587546193927183517 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (86350540186721179 / 160000000000000000 : ℝ) ≤
+      (-(6167587546193927183517 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (86350540186721179 / 160000000000000000 : ℝ) =
+            (86350540186721179 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (86350540186721179 / 80000000000000000 : ℝ) ≤
+            (763884256806072816483 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5396878806024913 / 10000000000000000 : ℝ)) (q :=
+      (86350540186721179 / 160000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_497_2 :
+    Uω (aρ 2) (bρ 2) (69906379973123 / 128000000000000 : ℝ) ≤
+      (-(6212038459828908181177 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (85967555976815707 / 160000000000000000 : ℝ) ≤
+      (-(6212038459828908181177 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (85967555976815707 / 160000000000000000 : ℝ) =
+            (85967555976815707 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (85967555976815707 / 80000000000000000 : ℝ) ≤
+            (719433343171091818823 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5372774792296729 / 10000000000000000 : ℝ)) (q :=
+      (85967555976815707 / 160000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_497_3 :
+    Uω (aρ 3) (bρ 3) (69906379973123 / 128000000000000 : ℝ) ≤
+      (-(6153882366540572167 / 9765625000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17040252650627759 / 32000000000000000 : ℝ) ≤
+      (-(6153882366540572167 / 9765625000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (17040252650627759 / 32000000000000000 : ℝ) =
+            (17040252650627759 / 16000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17040252650627759 / 16000000000000000 : ℝ) ≤
+            (615133066076615333 / 9765625000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1064876861637423 / 2000000000000000 : ℝ)) (q :=
+      (17040252650627759 / 32000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_497_4 :
+    Uω (aρ 4) (bρ 4) (69906379973123 / 128000000000000 : ℝ) ≤
+      (-(6452195255314039757353 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (83927580446092347 / 160000000000000000 : ℝ) ≤
+      (-(6452195255314039757353 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (83927580446092347 / 160000000000000000 : ℝ) =
+            (83927580446092347 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (83927580446092347 / 80000000000000000 : ℝ) ≤
+            (479276547685960242647 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5243468138371309 / 10000000000000000 : ℝ)) (q :=
+      (83927580446092347 / 160000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_497_5 :
+    Uω (aρ 5) (bρ 5) (69906379973123 / 128000000000000 : ℝ) ≤
+      (-(668631600941429284913 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (81985484559887883 / 160000000000000000 : ℝ) ≤
+      (-(668631600941429284913 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (81985484559887883 / 160000000000000000 : ℝ) =
+            (81985484559887883 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (81985484559887883 / 80000000000000000 : ℝ) ≤
+            (24515579358570715087 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5118896071930751 / 10000000000000000 : ℝ)) (q :=
+      (81985484559887883 / 160000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_497_6 :
+    Uω (aρ 6) (bρ 6) (69906379973123 / 128000000000000 : ℝ) ≤
+      (-(6867849691752660767 / 9765625000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (79194433026340883 / 160000000000000000 : ℝ) ≤
+      (-(6867849691752660767 / 9765625000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (79194433026340883 / 160000000000000000 : ℝ) =
+            (79194433026340883 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (79194433026340883 / 40000000000000000 : ℝ) ≤
+            (6670181173481714233 / 9765625000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (154289218918043 / 312500000000000 : ℝ)) (q :=
+      (79194433026340883 / 160000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_497_7 :
+    Uω (aρ 7) (bρ 7) (69906379973123 / 128000000000000 : ℝ) ≤
+      (-(7527144268936852965187 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (75373773867730363 / 160000000000000000 : ℝ) ≤
+      (-(7527144268936852965187 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (75373773867730363 / 160000000000000000 : ℝ) =
+            (75373773867730363 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (75373773867730363 / 40000000000000000 : ℝ) ≤
+            (6335799337063147034813 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4683333991311061 / 10000000000000000 : ℝ)) (q :=
+      (75373773867730363 / 160000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_497_8 :
+    Uω (aρ 8) (bρ 8) (69906379973123 / 128000000000000 : ℝ) ≤
+      (-(513470005995439371749 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (70359780846663291 / 160000000000000000 : ℝ) ≤
+      (-(513470005995439371749 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (70359780846663291 / 160000000000000000 : ℝ) =
+            (70359780846663291 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (70359780846663291 / 40000000000000000 : ℝ) ≤
+            (352963969379560628251 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4340126515017677 / 10000000000000000 : ℝ)) (q :=
+      (70359780846663291 / 160000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8])
+      (by norm_num [bρ_8]) (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_497_9 :
+    Uω (aρ 9) (bρ 9) (69906379973123 / 128000000000000 : ℝ) ≤
+      (-(366447935732025427061 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (64010938024132779 / 160000000000000000 : ℝ) ≤
+      (-(366447935732025427061 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (64010938024132779 / 160000000000000000 : ℝ) =
+            (64010938024132779 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (64010938024132779 / 40000000000000000 : ℝ) ≤
+            (188069808507974572939 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3887486790561363 / 10000000000000000 : ℝ)) (q :=
+      (64010938024132779 / 160000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9])
+      (by norm_num [bρ_9]) (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_497_10 :
+    Uω (aρ 10) (bρ 10) (69906379973123 / 128000000000000 : ℝ) ≤
+      (-(1308342692389497524921 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (56176564087184987 / 160000000000000000 : ℝ) ≤
+      (-(1308342692389497524921 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (56176564087184987 / 160000000000000000 : ℝ) =
+            (56176564087184987 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (56176564087184987 / 40000000000000000 : ℝ) ≤
+            (424525258360502475079 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3296749081937889 / 10000000000000000 : ℝ)) (q :=
+      (56176564087184987 / 160000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_497_11 :
+    Uω (aρ 11) (bρ 11) (69906379973123 / 128000000000000 : ℝ) ≤
+      (-(3087635739603101573631 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (46531035444361763 / 160000000000000000 : ℝ) ≤
+      (-(3087635739603101573631 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (46531035444361763 / 160000000000000000 : ℝ) =
+            (46531035444361763 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (46531035444361763 / 40000000000000000 : ℝ) ≤
+            (378100161896898426369 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1254767313894993 / 5000000000000000 : ℝ)) (q :=
+      (46531035444361763 / 160000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_497_12 :
+    Uω (aρ 12) (bρ 12) (69906379973123 / 128000000000000 : ℝ) ≤
+      (-(3133047669711312742019 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (33403142406979611 / 160000000000000000 : ℝ) ≤
+      (-(3133047669711312742019 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (33403142406979611 / 160000000000000000 : ℝ) =
+            (33403142406979611 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (33403142406979611 / 20000000000000000 : ℝ) ≤
+            (1025835412088687257981 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1292248094802217 / 10000000000000000 : ℝ)) (q :=
+      (33403142406979611 / 160000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_497_13 :
+    Uω (aρ 13) (bρ 13) (69906379973123 / 128000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_497_14 :
+    Uω (aρ 14) (bρ 14) (69906379973123 / 128000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_497_15 :
+    Uω (aρ 15) (bρ 15) (69906379973123 / 128000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_497_16 :
+    Uω (aρ 16) (bρ 16) (69906379973123 / 128000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_497 :
+    Uρ (69906379973123 / 128000000000000 : ℝ) ≤
+      (-(2399471840990113090949 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (69906379973123 / 128000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(6167587546193927183517 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_497_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (69906379973123 / 128000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(6212038459828908181177 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_497_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (69906379973123 / 128000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) * (-(6153882366540572167 / 9765625000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_497_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (69906379973123 / 128000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(6452195255314039757353 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_497_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (69906379973123 / 128000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(668631600941429284913 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_497_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (69906379973123 / 128000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) * (-(6867849691752660767 / 9765625000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_497_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (69906379973123 / 128000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(7527144268936852965187 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_497_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (69906379973123 / 128000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(513470005995439371749 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_497_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (69906379973123 / 128000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(366447935732025427061 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_497_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (69906379973123 / 128000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(1308342692389497524921 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_497_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (69906379973123 / 128000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(3087635739603101573631 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_497_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (69906379973123 / 128000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(3133047669711312742019 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_497_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (69906379973123 / 128000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_497_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (69906379973123 / 128000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_497_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (69906379973123 / 128000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_497_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (69906379973123 / 128000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_497_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_498_1 :
+    Uω (aρ 1) (bρ 1) (34993133427579 / 64000000000000 : ℝ) ≤
+      (-(6156029835042409091741 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (43225199671447423 / 80000000000000000 : ℝ) ≤
+      (-(6156029835042409091741 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (43225199671447423 / 80000000000000000 : ℝ) =
+            (43225199671447423 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (43225199671447423 / 40000000000000000 : ℝ) ≤
+            (775441967957590908259 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5403120037887637 / 10000000000000000 : ℝ)) (q :=
+      (43225199671447423 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_498_2 :
+    Uω (aρ 2) (bρ 2) (34993133427579 / 64000000000000 : ℝ) ≤
+      (-(1550107231884159405343 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8606741824503083 / 16000000000000000 : ℝ) ≤
+      (-(1550107231884159405343 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (8606741824503083 / 16000000000000000 : ℝ) =
+            (8606741824503083 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8606741824503083 / 8000000000000000 : ℝ) ≤
+            (182760718865840594657 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1075803282632927 / 2000000000000000 : ℝ)) (q :=
+      (8606741824503083 / 16000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_498_3 :
+    Uω (aρ 3) (bρ 3) (34993133427579 / 64000000000000 : ℝ) ≤
+      (-(1257972112255709486683 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (42650567434233643 / 80000000000000000 : ℝ) ≤
+      (-(1257972112255709486683 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (42650567434233643 / 80000000000000000 : ℝ) =
+            (42650567434233643 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (42650567434233643 / 40000000000000000 : ℝ) ≤
+            (128322248344290513317 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (166582096795131 / 312500000000000 : ℝ)) (q :=
+      (42650567434233643 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_498_4 :
+    Uω (aρ 4) (bρ 4) (34993133427579 / 64000000000000 : ℝ) ≤
+      (-(6440299601029007882821 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (42013738599685479 / 80000000000000000 : ℝ) ≤
+      (-(6440299601029007882821 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (42013738599685479 / 80000000000000000 : ℝ) =
+            (42013738599685479 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (42013738599685479 / 40000000000000000 : ℝ) ≤
+            (491172201970992117179 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5249714069872151 / 10000000000000000 : ℝ)) (q :=
+      (42013738599685479 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_498_5 :
+    Uω (aρ 5) (bρ 5) (34993133427579 / 64000000000000 : ℝ) ≤
+      (-(6674131051530196510577 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (41042722207749283 / 80000000000000000 : ℝ) ≤
+      (-(6674131051530196510577 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (41042722207749283 / 80000000000000000 : ℝ) =
+            (41042722207749283 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (41042722207749283 / 40000000000000000 : ℝ) ≤
+            (257340751469803489423 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2562574945611551 / 5000000000000000 : ℝ)) (q :=
+      (41042722207749283 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_498_6 :
+    Uω (aρ 6) (bρ 6) (34993133427579 / 64000000000000 : ℝ) ≤
+      (-(438752821916816559511 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (39647271024314323 / 80000000000000000 : ℝ) ≤
+      (-(438752821916816559511 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (39647271024314323 / 80000000000000000 : ℝ) =
+            (39647271024314323 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (39647271024314323 / 20000000000000000 : ℝ) ≤
+            (427681153458183440489 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2471763735252181 / 5000000000000000 : ℝ)) (q :=
+      (39647271024314323 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_498_7 :
+    Uω (aρ 7) (bρ 7) (34993133427579 / 64000000000000 : ℝ) ≤
+      (-(7513826920613636684123 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (37737109307936711 / 80000000000000000 : ℝ) ≤
+      (-(7513826920613636684123 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (37737109307936711 / 80000000000000000 : ℝ) =
+            (37737109307936711 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (37737109307936711 / 20000000000000000 : ℝ) ≤
+            (6349116685386363315877 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4689648422169959 / 10000000000000000 : ℝ)) (q :=
+      (37737109307936711 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_498_8 :
+    Uω (aρ 8) (bρ 8) (34993133427579 / 64000000000000 : ℝ) ≤
+      (-(2050287639504809022989 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (35230478637993371 / 80000000000000000 : ℝ) ≤
+      (-(2050287639504809022989 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (35230478637993371 / 80000000000000000 : ℝ) =
+            (35230478637993371 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (35230478637993371 / 20000000000000000 : ℝ) ≤
+            (1415448261995190977011 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1086633101506031 / 2500000000000000 : ℝ)) (q :=
+      (35230478637993371 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_498_9 :
+    Uω (aρ 9) (bρ 9) (34993133427579 / 64000000000000 : ℝ) ≤
+      (-(9145157522639751445491 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6411369953907939 / 16000000000000000 : ℝ) ≤
+      (-(9145157522639751445491 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6411369953907939 / 16000000000000000 : ℝ) =
+            (6411369953907939 / 4000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6411369953907939 / 4000000000000000 : ℝ) ≤
+            (4717786083360248554509 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (778818163454141 / 2000000000000000 : ℝ)) (q :=
+      (6411369953907939 / 16000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_498_10 :
+    Uω (aρ 10) (bρ 10) (34993133427579 / 64000000000000 : ℝ) ≤
+      (-(10447830492519363594599 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28141450182058891 / 80000000000000000 : ℝ) ≤
+      (-(10447830492519363594599 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (28141450182058891 / 80000000000000000 : ℝ) =
+            (28141450182058891 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28141450182058891 / 20000000000000000 : ℝ) ≤
+            (3415113113480636405401 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (206487497118469 / 625000000000000 : ℝ)) (q :=
+      (28141450182058891 / 80000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_498_11 :
+    Uω (aρ 11) (bρ 11) (34993133427579 / 64000000000000 : ℝ) ≤
+      (-(12325713800658735169931 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4664671163400139 / 16000000000000000 : ℝ) ≤
+      (-(12325713800658735169931 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4664671163400139 / 16000000000000000 : ℝ) =
+            (4664671163400139 / 4000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4664671163400139 / 4000000000000000 : ℝ) ≤
+            (1537229805341264830069 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (503550597767191 / 2000000000000000 : ℝ)) (q :=
+      (4664671163400139 / 16000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_498_12 :
+    Uω (aρ 12) (bρ 12) (34993133427579 / 64000000000000 : ℝ) ≤
+      (-(7808599520102826369753 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16781997422714663 / 80000000000000000 : ℝ) ≤
+      (-(7808599520102826369753 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (16781997422714663 / 80000000000000000 : ℝ) =
+            (16781997422714663 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16781997422714663 / 10000000000000000 : ℝ) ≤
+            (2588608184397173630247 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1306113486949447 / 10000000000000000 : ℝ)) (q :=
+      (16781997422714663 / 80000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_498_13 :
+    Uω (aρ 13) (bρ 13) (34993133427579 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_498_14 :
+    Uω (aρ 14) (bρ 14) (34993133427579 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_498_15 :
+    Uω (aρ 15) (bρ 15) (34993133427579 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_498_16 :
+    Uω (aρ 16) (bρ 16) (34993133427579 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_498 :
+    Uρ (34993133427579 / 64000000000000 : ℝ) ≤
+      (-(383344754671723342871 / 400000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (34993133427579 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(6156029835042409091741 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_498_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (34993133427579 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1550107231884159405343 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_498_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (34993133427579 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1257972112255709486683 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_498_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (34993133427579 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(6440299601029007882821 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_498_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (34993133427579 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(6674131051530196510577 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_498_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (34993133427579 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(438752821916816559511 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_498_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (34993133427579 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(7513826920613636684123 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_498_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (34993133427579 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(2050287639504809022989 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_498_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (34993133427579 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(9145157522639751445491 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_498_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (34993133427579 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(10447830492519363594599 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_498_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (34993133427579 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(12325713800658735169931 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_498_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (34993133427579 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(7808599520102826369753 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_498_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (34993133427579 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_498_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (34993133427579 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_498_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (34993133427579 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_498_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (34993133427579 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_498_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_499_1 :
+    Uω (aρ 1) (bρ 1) (70066153737193 / 128000000000000 : ℝ) ≤
+      (-(3072242733343819264019 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (86550258497790977 / 160000000000000000 : ℝ) ≤
+      (-(3072242733343819264019 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (86550258497790977 / 160000000000000000 : ℝ) =
+            (86550258497790977 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (86550258497790977 / 80000000000000000 : ℝ) ≤
+            (393493168156180735981 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5409361269590669 / 10000000000000000 : ℝ)) (q :=
+      (86550258497790977 / 160000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_499_2 :
+    Uω (aρ 2) (bρ 2) (70066153737193 / 128000000000000 : ℝ) ≤
+      (-(6188832858726643796807 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (86167280504749081 / 160000000000000000 : ℝ) ≤
+      (-(6188832858726643796807 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (86167280504749081 / 160000000000000000 : ℝ) =
+            (86167280504749081 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (86167280504749081 / 80000000000000000 : ℝ) ≤
+            (742638944273356203193 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (84144656765163 / 156250000000000 : ℝ)) (q :=
+      (86167280504749081 / 160000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_499_3 :
+    Uω (aρ 3) (bρ 3) (70066153737193 / 128000000000000 : ℝ) ≤
+      (-(6278159290806943346951 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (85401006453356393 / 160000000000000000 : ℝ) ≤
+      (-(6278159290806943346951 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (85401006453356393 / 160000000000000000 : ℝ) =
+            (85401006453356393 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (85401006453356393 / 80000000000000000 : ℝ) ≤
+            (653312512193056653049 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2668434941448173 / 5000000000000000 : ℝ)) (q :=
+      (85401006453356393 / 160000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_499_4 :
+    Uω (aρ 4) (bρ 4) (70066153737193 / 128000000000000 : ℝ) ≤
+      (-(6428418091365539699 / 10000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (84127373862011169 / 160000000000000000 : ℝ) ≤
+      (-(6428418091365539699 / 10000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (84127373862011169 / 160000000000000000 : ℝ) =
+            (84127373862011169 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (84127373862011169 / 80000000000000000 : ℝ) ≤
+            (503053711634460301 / 10000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5255959990043193 / 10000000000000000 : ℝ)) (q :=
+      (84127373862011169 / 160000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_499_5 :
+    Uω (aρ 5) (bρ 5) (70066153737193 / 128000000000000 : ℝ) ≤
+      (-(832745119112230134817 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (82185404024557753 / 160000000000000000 : ℝ) ≤
+      (-(832745119112230134817 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (82185404024557753 / 160000000000000000 : ℝ) =
+            (82185404024557753 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (82185404024557753 / 80000000000000000 : ℝ) ≤
+            (33688856262769865183 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1282850919924129 / 2500000000000000 : ℝ)) (q :=
+      (82185404024557753 / 160000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_499_6 :
+    Uω (aρ 6) (bρ 6) (70066153737193 / 128000000000000 : ℝ) ≤
+      (-(3503714117845335411479 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15878930087565661 / 32000000000000000 : ℝ) ≤
+      (-(3503714117845335411479 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (15878930087565661 / 32000000000000000 : ℝ) =
+            (15878930087565661 / 8000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15878930087565661 / 8000000000000000 : ℝ) ≤
+            (3427757685154664588521 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (989959971299067 / 2000000000000000 : ℝ)) (q :=
+      (15878930087565661 / 32000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_499_7 :
+    Uω (aρ 7) (bρ 7) (70066153737193 / 128000000000000 : ℝ) ≤
+      (-(937565936406325238339 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (75574661796839249 / 160000000000000000 : ℝ) ≤
+      (-(937565936406325238339 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (75574661796839249 / 160000000000000000 : ℝ) =
+            (75574661796839249 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (75574661796839249 / 40000000000000000 : ℝ) ≤
+            (795302014343674761661 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4695962657131703 / 10000000000000000 : ℝ)) (q :=
+      (75574661796839249 / 160000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_499_8 :
+    Uω (aρ 8) (bρ 8) (70066153737193 / 128000000000000 : ℝ) ≤
+      (-(1637360436293184612359 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (70562129876487961 / 160000000000000000 : ℝ) ≤
+      (-(1637360436293184612359 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (70562129876487961 / 160000000000000000 : ℝ) =
+            (70562129876487961 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (70562129876487961 / 40000000000000000 : ℝ) ≤
+            (1135228284906815387641 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (272058613651737 / 625000000000000 : ℝ)) (q :=
+      (70562129876487961 / 160000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8])
+      (by norm_num [bρ_8]) (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_499_9 :
+    Uω (aρ 9) (bρ 9) (70066153737193 / 128000000000000 : ℝ) ≤
+      (-(4564571915178486632409 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (64216451494584217 / 160000000000000000 : ℝ) ≤
+      (-(4564571915178486632409 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (64216451494584217 / 160000000000000000 : ℝ) =
+            (64216451494584217 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (64216451494584217 / 40000000000000000 : ℝ) ≤
+            (2366899887821513367591 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (121896676532807 / 312500000000000 : ℝ)) (q :=
+      (64216451494584217 / 160000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9])
+      (by norm_num [bρ_9]) (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_499_10 :
+    Uω (aρ 10) (bρ 10) (70066153737193 / 128000000000000 : ℝ) ≤
+      (-(2607239938261753790789 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (56389210635193161 / 160000000000000000 : ℝ) ≤
+      (-(2607239938261753790789 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (56389210635193161 / 160000000000000000 : ℝ) =
+            (56389210635193161 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (56389210635193161 / 40000000000000000 : ℝ) ≤
+            (858495963238246209211 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1655423787560471 / 5000000000000000 : ℝ)) (q :=
+      (56389210635193161 / 160000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_499_11 :
+    Uω (aρ 11) (bρ 11) (70066153737193 / 128000000000000 : ℝ) ≤
+      (-(6150482750929794706883 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (46762297278161161 / 160000000000000000 : ℝ) ≤
+      (-(6150482750929794706883 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (46762297278161161 / 160000000000000000 : ℝ) =
+            (46762297278161161 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (46762297278161161 / 40000000000000000 : ℝ) ≤
+            (780989052070205293117 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1262980015848471 / 5000000000000000 : ℝ)) (q :=
+      (46762297278161161 / 160000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_499_12 :
+    Uω (aρ 12) (bρ 12) (70066153737193 / 128000000000000 : ℝ) ≤
+      (-(15569664912888544190381 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6744783631096493 / 32000000000000000 : ℝ) ≤
+      (-(15569664912888544190381 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (6744783631096493 / 32000000000000000 : ℝ) =
+            (6744783631096493 / 4000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6744783631096493 / 4000000000000000 : ℝ) ≤
+            (5224750496111455809619 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (263972547609421 / 2000000000000000 : ℝ)) (q :=
+      (6744783631096493 / 32000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_499_13 :
+    Uω (aρ 13) (bρ 13) (70066153737193 / 128000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_499_14 :
+    Uω (aρ 14) (bρ 14) (70066153737193 / 128000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_499_15 :
+    Uω (aρ 15) (bρ 15) (70066153737193 / 128000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_499_16 :
+    Uω (aρ 16) (bρ 16) (70066153737193 / 128000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_499 :
+    Uρ (70066153737193 / 128000000000000 : ℝ) ≤
+      (-(9569405765138696291859 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (70066153737193 / 128000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3072242733343819264019 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_499_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (70066153737193 / 128000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(6188832858726643796807 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_499_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (70066153737193 / 128000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(6278159290806943346951 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_499_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (70066153737193 / 128000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(6428418091365539699 / 10000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_499_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (70066153737193 / 128000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(832745119112230134817 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_499_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (70066153737193 / 128000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3503714117845335411479 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_499_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (70066153737193 / 128000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(937565936406325238339 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_499_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (70066153737193 / 128000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(1637360436293184612359 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_499_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (70066153737193 / 128000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(4564571915178486632409 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_499_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (70066153737193 / 128000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(2607239938261753790789 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_499_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (70066153737193 / 128000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(6150482750929794706883 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_499_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (70066153737193 / 128000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(15569664912888544190381 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_499_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (70066153737193 / 128000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_499_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (70066153737193 / 128000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_499_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (70066153737193 / 128000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_499_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (70066153737193 / 128000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_499_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_500_1 :
+    Uω (aρ 1) (bρ 1) (17536510154807 / 32000000000000 : ℝ) ≤
+      (-(6132954410357619226903 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21662529412853499 / 40000000000000000 : ℝ) ≤
+      (-(6132954410357619226903 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (21662529412853499 / 40000000000000000 : ℝ) =
+            (21662529412853499 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21662529412853499 / 20000000000000000 : ℝ) ≤
+            (798517392642380773097 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2707801250567281 / 5000000000000000 : ℝ)) (q :=
+      (21662529412853499 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_500_2 :
+    Uω (aρ 2) (bρ 2) (17536510154807 / 32000000000000 : ℝ) ≤
+      (-(6177250222205826174879 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21566785688999999 / 40000000000000000 : ℝ) ≤
+      (-(6177250222205826174879 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (21566785688999999 / 40000000000000000 : ℝ) =
+            (21566785688999999 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21566785688999999 / 20000000000000000 : ℝ) ≤
+            (754221580794173825121 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1347874912929453 / 2500000000000000 : ℝ)) (q :=
+      (21566785688999999 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_500_3 :
+    Uω (aρ 3) (bρ 3) (17536510154807 / 32000000000000 : ℝ) ≤
+      (-(195827240620468206163 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21375219501978207 / 40000000000000000 : ℝ) ≤
+      (-(195827240620468206163 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (21375219501978207 / 40000000000000000 : ℝ) =
+            (21375219501978207 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21375219501978207 / 20000000000000000 : ℝ) ≤
+            (20781253223281793837 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1335778166139229 / 2500000000000000 : ℝ)) (q :=
+      (21375219501978207 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_500_4 :
+    Uω (aρ 4) (bρ 4) (17536510154807 / 32000000000000 : ℝ) ≤
+      (-(3208275346350256311827 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21056817608583931 / 40000000000000000 : ℝ) ≤
+      (-(3208275346350256311827 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (21056817608583931 / 40000000000000000 : ℝ) =
+            (21056817608583931 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21056817608583931 / 20000000000000000 : ℝ) ≤
+            (257460555149743688173 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2631102949462389 / 5000000000000000 : ℝ)) (q :=
+      (21056817608583931 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_500_5 :
+    Uω (aρ 5) (bρ 5) (17536510154807 / 32000000000000 : ℝ) ≤
+      (-(831225709655945375553 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20571340846991449 / 40000000000000000 : ℝ) ≤
+      (-(831225709655945375553 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (20571340846991449 / 40000000000000000 : ℝ) =
+            (20571340846991449 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20571340846991449 / 20000000000000000 : ℝ) ≤
+            (35208265719054624447 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5137657437463537 / 10000000000000000 : ℝ)) (q :=
+      (20571340846991449 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_500_6 :
+    Uω (aρ 6) (bρ 6) (17536510154807 / 32000000000000 : ℝ) ≤
+      (-(6994827298646218981741 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19873689549085887 / 40000000000000000 : ℝ) ≤
+      (-(6994827298646218981741 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (19873689549085887 / 40000000000000000 : ℝ) =
+            (19873689549085887 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19873689549085887 / 10000000000000000 : ℝ) ≤
+            (6868116307353781018259 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1239018040912689 / 2500000000000000 : ℝ)) (q :=
+      (19873689549085887 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_500_7 :
+    Uω (aρ 7) (bρ 7) (17536510154807 / 32000000000000 : ℝ) ≤
+      (-(7487245932138787324571 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (18918775854235273 / 40000000000000000 : ℝ) ≤
+      (-(7487245932138787324571 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (18918775854235273 / 40000000000000000 : ℝ) =
+            (18918775854235273 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18918775854235273 / 10000000000000000 : ℝ) ≤
+            (6375697673861212675429 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4702276696985449 / 10000000000000000 : ℝ)) (q :=
+      (18918775854235273 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_500_8 :
+    Uω (aρ 8) (bρ 8) (17536510154807 / 32000000000000 : ℝ) ≤
+      (-(4086237451230692446593 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3533164933252231 / 8000000000000000 : ℝ) ≤
+      (-(4086237451230692446593 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3533164933252231 / 8000000000000000 : ℝ) =
+            (3533164933252231 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3533164933252231 / 2000000000000000 : ℝ) ≤
+            (2845234351769307553407 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (435934275433839 / 1000000000000000 : ℝ)) (q :=
+      (3533164933252231 / 8000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_500_9 :
+    Uω (aρ 9) (bρ 9) (17536510154807 / 32000000000000 : ℝ) ≤
+      (-(2278289304907380917197 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16079798484777499 / 40000000000000000 : ℝ) ≤
+      (-(2278289304907380917197 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (16079798484777499 / 40000000000000000 : ℝ) =
+            (16079798484777499 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16079798484777499 / 10000000000000000 : ℝ) ≤
+            (1187446596592619082803 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1953647645978281 / 5000000000000000 : ℝ)) (q :=
+      (16079798484777499 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_500_10 :
+    Uω (aρ 10) (bρ 10) (17536510154807 / 32000000000000 : ℝ) ≤
+      (-(10410129130832395465343 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (564954950661303 / 1600000000000000 : ℝ) ≤
+      (-(10410129130832395465343 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (564954950661303 / 1600000000000000 : ℝ) =
+            (564954950661303 / 400000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (564954950661303 / 400000000000000 : ℝ) ≤
+            (3452814475167604534657 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (33178919663291 / 100000000000000 : ℝ)) (q :=
+      (564954950661303 / 1600000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10]) (by norm_num [bρ_10])
+      (by norm_num) (by norm_num [aρ_10, bρ_10]) (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_500_11 :
+    Uω (aρ 11) (bρ 11) (17536510154807 / 32000000000000 : ℝ) ≤
+      (-(12276297427255723426533 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (468777932565549 / 1600000000000000 : ℝ) ≤
+      (-(12276297427255723426533 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (468777932565549 / 1600000000000000 : ℝ) =
+            (468777932565549 / 400000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (468777932565549 / 400000000000000 : ℝ) ≤
+            (1586646178744276573467 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (101366234653487 / 400000000000000 : ℝ)) (q :=
+      (468777932565549 / 1600000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11]) (by norm_num [bρ_11])
+      (by norm_num) (by norm_num [aρ_11, bρ_11]) (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_500_12 :
+    Uω (aρ 12) (bρ 12) (17536510154807 / 32000000000000 : ℝ) ≤
+      (-(3880655349478368086291 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8470735269235369 / 40000000000000000 : ℝ) ≤
+      (-(3880655349478368086291 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (8470735269235369 / 40000000000000000 : ℝ) =
+            (8470735269235369 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8470735269235369 / 5000000000000000 : ℝ) ≤
+            (1317948502771631913709 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1333499440570497 / 10000000000000000 : ℝ)) (q :=
+      (8470735269235369 / 40000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_500_13 :
+    Uω (aρ 13) (bρ 13) (17536510154807 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_500_14 :
+    Uω (aρ 14) (bρ 14) (17536510154807 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_500_15 :
+    Uω (aρ 15) (bρ 15) (17536510154807 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_500_16 :
+    Uω (aρ 16) (bρ 16) (17536510154807 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_500 :
+    Uρ (17536510154807 / 32000000000000 : ℝ) ≤
+      (-(382209877564364374023 / 400000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (17536510154807 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(6132954410357619226903 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_500_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (17536510154807 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(6177250222205826174879 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_500_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (17536510154807 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(195827240620468206163 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_500_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (17536510154807 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3208275346350256311827 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_500_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (17536510154807 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(831225709655945375553 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_500_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (17536510154807 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(6994827298646218981741 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_500_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (17536510154807 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(7487245932138787324571 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_500_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (17536510154807 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(4086237451230692446593 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_500_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (17536510154807 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(2278289304907380917197 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_500_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (17536510154807 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(10410129130832395465343 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_500_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (17536510154807 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(12276297427255723426533 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_500_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (17536510154807 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(3880655349478368086291 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_500_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (17536510154807 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_500_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (17536510154807 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_500_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (17536510154807 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_500_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (17536510154807 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_500_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_501_1 :
+    Uω (aρ 1) (bρ 1) (35152907191649 / 64000000000000 : ℝ) ≤
+      (-(1527483027803754788683 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (43424917977429129 / 80000000000000000 : ℝ) ≤
+      (-(1527483027803754788683 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (43424917977429129 / 80000000000000000 : ℝ) =
+            (43424917977429129 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (43424917977429129 / 40000000000000000 : ℝ) ≤
+            (205384922946245211317 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2714042481873563 / 5000000000000000 : ℝ)) (q :=
+      (43424917977429129 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_501_2 :
+    Uω (aρ 2) (bρ 2) (35152907191649 / 64000000000000 : ℝ) ≤
+      (-(3077062560900137648367 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (43233433616608557 / 80000000000000000 : ℝ) ≤
+      (-(3077062560900137648367 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (43233433616608557 / 80000000000000000 : ℝ) =
+            (43233433616608557 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (43233433616608557 / 40000000000000000 : ℝ) ≤
+            (388673340599862351633 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5403982886051983 / 10000000000000000 : ℝ)) (q :=
+      (43233433616608557 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_501_3 :
+    Uω (aρ 3) (bρ 3) (35152907191649 / 64000000000000 : ℝ) ≤
+      (-(6243137428800446165823 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (42850310513225973 / 80000000000000000 : ℝ) ≤
+      (-(6243137428800446165823 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (42850310513225973 / 80000000000000000 : ℝ) =
+            (42850310513225973 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (42850310513225973 / 40000000000000000 : ℝ) ≤
+            (688334374199553834177 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5355598216556337 / 10000000000000000 : ℝ)) (q :=
+      (42850310513225973 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_501_4 :
+    Uω (aρ 4) (bρ 4) (35152907191649 / 64000000000000 : ℝ) ≤
+      (-(3196429047235951039163 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8442706330932121 / 16000000000000000 : ℝ) ≤
+      (-(3196429047235951039163 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (8442706330932121 / 16000000000000000 : ℝ) =
+            (8442706330932121 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8442706330932121 / 8000000000000000 : ℝ) ≤
+            (269306854264048960837 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1054939536595999 / 2000000000000000 : ℝ)) (q :=
+      (8442706330932121 / 16000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_501_5 :
+    Uω (aρ 5) (bρ 5) (35152907191649 / 64000000000000 : ℝ) ≤
+      (-(3312769725239216942567 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (41242640690703941 / 80000000000000000 : ℝ) ≤
+      (-(3312769725239216942567 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (41242640690703941 / 80000000000000000 : ℝ) =
+            (41242640690703941 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (41242640690703941 / 40000000000000000 : ℝ) ≤
+            (152966176260783057433 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5150164861325829 / 10000000000000000 : ℝ)) (q :=
+      (41242640690703941 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_501_6 :
+    Uω (aρ 6) (bρ 6) (35152907191649 / 64000000000000 : ℝ) ≤
+      (-(6969673196046202864671 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (39847485915437389 / 80000000000000000 : ℝ) ≤
+      (-(6969673196046202864671 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (39847485915437389 / 80000000000000000 : ℝ) =
+            (39847485915437389 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (39847485915437389 / 20000000000000000 : ℝ) ≤
+            (6893270409953797135329 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4968616542649191 / 10000000000000000 : ℝ)) (q :=
+      (39847485915437389 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_501_7 :
+    Uω (aρ 7) (bρ 7) (35152907191649 / 64000000000000 : ℝ) ≤
+      (-(746073623083452947819 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (37937990999817837 / 80000000000000000 : ℝ) ≤
+      (-(746073623083452947819 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (37937990999817837 / 80000000000000000 : ℝ) =
+            (37937990999817837 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (37937990999817837 / 20000000000000000 : ℝ) ≤
+            (640220737516547052181 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4714904194504303 / 10000000000000000 : ℝ)) (q :=
+      (37937990999817837 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_501_8 :
+    Uω (aρ 8) (bρ 8) (35152907191649 / 64000000000000 : ℝ) ≤
+      (-(4071941691674149917709 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (35432812436667113 / 80000000000000000 : ℝ) ≤
+      (-(4071941691674149917709 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (35432812436667113 / 80000000000000000 : ℝ) =
+            (35432812436667113 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (35432812436667113 / 20000000000000000 : ℝ) ≤
+            (2859530111325850082291 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2186075602528311 / 5000000000000000 : ℝ)) (q :=
+      (35432812436667113 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_501_9 :
+    Uω (aρ 9) (bρ 9) (35152907191649 / 64000000000000 : ℝ) ≤
+      (-(4540632429122172497341 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6452465048739989 / 16000000000000000 : ℝ) ≤
+      (-(4540632429122172497341 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6452465048739989 / 16000000000000000 : ℝ) =
+            (6452465048739989 / 4000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6452465048739989 / 4000000000000000 : ℝ) ≤
+            (2390839373877827502659 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (392049503517483 / 1000000000000000 : ℝ)) (q :=
+      (6452465048739989 / 16000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_501_10 :
+    Uω (aρ 10) (bρ 10) (35152907191649 / 64000000000000 : ℝ) ≤
+      (-(2074517497181400129451 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28353993531541789 / 80000000000000000 : ℝ) ≤
+      (-(2074517497181400129451 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (28353993531541789 / 80000000000000000 : ℝ) =
+            (28353993531541789 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28353993531541789 / 20000000000000000 : ℝ) ≤
+            (698071224018599870549 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3331971140630291 / 10000000000000000 : ℝ)) (q :=
+      (28353993531541789 / 80000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_501_11 :
+    Uω (aρ 11) (bρ 11) (35152907191649 / 64000000000000 : ℝ) ≤
+      (-(1222719945283823425943 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23554259836225581 / 80000000000000000 : ℝ) ≤
+      (-(1222719945283823425943 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (23554259836225581 / 80000000000000000 : ℝ) =
+            (23554259836225581 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23554259836225581 / 20000000000000000 : ℝ) ≤
+            (163574415316176574057 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2550514343006239 / 10000000000000000 : ℝ)) (q :=
+      (23554259836225581 / 80000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_501_12 :
+    Uω (aρ 12) (bρ 12) (35152907191649 / 64000000000000 : ℝ) ≤
+      (-(15429951354170808899491 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17099196916817357 / 80000000000000000 : ℝ) ≤
+      (-(15429951354170808899491 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (17099196916817357 / 80000000000000000 : ℝ) =
+            (17099196916817357 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17099196916817357 / 10000000000000000 : ℝ) ≤
+            (5364464054829191100509 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1360448709839183 / 10000000000000000 : ℝ)) (q :=
+      (17099196916817357 / 80000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_501_13 :
+    Uω (aρ 13) (bρ 13) (35152907191649 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_501_14 :
+    Uω (aρ 14) (bρ 14) (35152907191649 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_501_15 :
+    Uω (aρ 15) (bρ 15) (35152907191649 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_501_16 :
+    Uω (aρ 16) (bρ 16) (35152907191649 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_501 :
+    Uρ (35152907191649 / 64000000000000 : ℝ) ≤
+      (-(1190885984971035254361 / 1250000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (35152907191649 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1527483027803754788683 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_501_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (35152907191649 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3077062560900137648367 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_501_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (35152907191649 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(6243137428800446165823 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_501_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (35152907191649 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3196429047235951039163 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_501_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (35152907191649 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(3312769725239216942567 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_501_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (35152907191649 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(6969673196046202864671 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_501_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (35152907191649 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(746073623083452947819 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_501_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (35152907191649 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(4071941691674149917709 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_501_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (35152907191649 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(4540632429122172497341 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_501_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (35152907191649 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(2074517497181400129451 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_501_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (35152907191649 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(1222719945283823425943 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_501_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (35152907191649 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(15429951354170808899491 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_501_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (35152907191649 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_501_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (35152907191649 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_501_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (35152907191649 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_501_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (35152907191649 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_501_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_502_1 :
+    Uω (aρ 1) (bρ 1) (8808198518421 / 16000000000000 : ℝ) ≤
+      (-(304348134677823892373 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (544059714082891 / 1000000000000000 : ℝ) ≤
+      (-(304348134677823892373 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (544059714082891 / 1000000000000000 : ℝ) =
+            (544059714082891 / 500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (544059714082891 / 500000000000000 : ℝ) ≤
+            (42225455472176107627 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1088113485145939 / 2000000000000000 : ℝ)) (q :=
+      (544059714082891 / 1000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_502_2 :
+    Uω (aρ 2) (bρ 2) (8808198518421 / 16000000000000 : ℝ) ≤
+      (-(3065526689465984201931 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2166664791922889 / 4000000000000000 : ℝ) ≤
+      (-(3065526689465984201931 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2166664791922889 / 4000000000000000 : ℝ) =
+            (2166664791922889 / 2000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2166664791922889 / 2000000000000000 : ℝ) ≤
+            (400209212034015798069 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (33852913226227 / 62500000000000 : ℝ)) (q :=
+      (2166664791922889 / 4000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_502_3 :
+    Uω (aρ 3) (bρ 3) (8808198518421 / 16000000000000 : ℝ) ≤
+      (-(3109928746905706661603 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2147509098123207 / 4000000000000000 : ℝ) ≤
+      (-(3109928746905706661603 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2147509098123207 / 4000000000000000 : ℝ) =
+            (2147509098123207 / 2000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2147509098123207 / 2000000000000000 : ℝ) ≤
+            (355807154594293338397 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (536808375354791 / 1000000000000000 : ℝ)) (q :=
+      (2147509098123207 / 4000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_502_4 :
+    Uω (aρ 4) (bρ 4) (8808198518421 / 16000000000000 : ℝ) ≤
+      (-(1592305384936174259601 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5289178489179933 / 10000000000000000 : ℝ) ≤
+      (-(1592305384936174259601 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5289178489179933 / 10000000000000000 : ℝ) =
+            (5289178489179933 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5289178489179933 / 5000000000000000 : ℝ) ≤
+            (140562565813825740399 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5287189422356741 / 10000000000000000 : ℝ)) (q :=
+      (5289178489179933 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_502_5 :
+    Uω (aρ 5) (bρ 5) (8808198518421 / 16000000000000 : ℝ) ≤
+      (-(6601332083711793922337 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10335649800367549 / 20000000000000000 : ℝ) ≤
+      (-(6601332083711793922337 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10335649800367549 / 20000000000000000 : ℝ) =
+            (10335649800367549 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10335649800367549 / 10000000000000000 : ℝ) ≤
+            (330139719288206077663 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (161333505115607 / 312500000000000 : ℝ)) (q :=
+      (10335649800367549 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_502_6 :
+    Uω (aρ 6) (bρ 6) (8808198518421 / 16000000000000 : ℝ) ≤
+      (-(6944582519792693022809 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (998689787140121 / 2000000000000000 : ℝ) ≤
+      (-(6944582519792693022809 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (998689787140121 / 2000000000000000 : ℝ) =
+            (998689787140121 / 500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 15
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (998689787140121 / 500000000000000 : ℝ) ≤
+            (6918361086207306977191 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (996232121974617 / 2000000000000000 : ℝ)) (q :=
+      (998689787140121 / 2000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_502_7 :
+    Uω (aρ 7) (bρ 7) (8808198518421 / 16000000000000 : ℝ) ≤
+      (-(7434297431019868095343 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4754803400861627 / 10000000000000000 : ℝ) ≤
+      (-(7434297431019868095343 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4754803400861627 / 10000000000000000 : ℝ) =
+            (4754803400861627 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4754803400861627 / 2500000000000000 : ℝ) ≤
+            (6428646174980131904657 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4727530920955129 / 10000000000000000 : ℝ)) (q :=
+      (4754803400861627 / 10000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_502_8 :
+    Uω (aρ 8) (bρ 8) (8808198518421 / 16000000000000 : ℝ) ≤
+      (-(1014421937012309681831 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2220873001058891 / 5000000000000000 : ℝ) ≤
+      (-(1014421937012309681831 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2220873001058891 / 5000000000000000 : ℝ) =
+            (2220873001058891 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2220873001058891 / 1250000000000000 : ℝ) ≤
+            (718446013737690318169 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4384957774807439 / 10000000000000000 : ℝ)) (q :=
+      (2220873001058891 / 5000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_502_9 :
+    Uω (aρ 9) (bρ 9) (8808198518421 / 16000000000000 : ℝ) ≤
+      (-(4524739837311758777763 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8091258695623841 / 20000000000000000 : ℝ) ≤
+      (-(4524739837311758777763 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8091258695623841 / 20000000000000000 : ℝ) =
+            (8091258695623841 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8091258695623841 / 5000000000000000 : ℝ) ≤
+            (2406731965688241222237 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (983422523638929 / 2500000000000000 : ℝ)) (q :=
+      (8091258695623841 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_502_10 :
+    Uω (aρ 10) (bρ 10) (8808198518421 / 16000000000000 : ℝ) ≤
+      (-(5167602033692465113331 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7115047206429829 / 20000000000000000 : ℝ) ≤
+      (-(5167602033692465113331 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7115047206429829 / 20000000000000000 : ℝ) =
+            (7115047206429829 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7115047206429829 / 5000000000000000 : ℝ) ≤
+            (1763869769307534886669 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (102112965053 / 305175781250 : ℝ)) (q :=
+      (7115047206429829 / 20000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_502_11 :
+    Uω (aρ 11) (bρ 11) (8808198518421 / 16000000000000 : ℝ) ≤
+      (-(1522301870274884627343 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2958681104557373 / 10000000000000000 : ℝ) ≤
+      (-(1522301870274884627343 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2958681104557373 / 10000000000000000 : ℝ) =
+            (2958681104557373 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2958681104557373 / 2500000000000000 : ℝ) ≤
+            (210566080475115372657 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2566829267746621 / 10000000000000000 : ℝ)) (q :=
+      (2958681104557373 / 10000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_502_12 :
+    Uω (aρ 12) (bρ 12) (8808198518421 / 16000000000000 : ℝ) ≤
+      (-(1917385680481320352707 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2156909797056681 / 10000000000000000 : ℝ) ≤
+      (-(1917385680481320352707 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2156909797056681 / 10000000000000000 : ℝ) =
+            (2156909797056681 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2156909797056681 / 1250000000000000 : ℝ) ≤
+            (681916245643679647293 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1386986749430237 / 10000000000000000 : ℝ)) (q :=
+      (2156909797056681 / 10000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_502_13 :
+    Uω (aρ 13) (bρ 13) (8808198518421 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_502_14 :
+    Uω (aρ 14) (bρ 14) (8808198518421 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_502_15 :
+    Uω (aρ 15) (bρ 15) (8808198518421 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_502_16 :
+    Uω (aρ 16) (bρ 16) (8808198518421 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_502 :
+    Uρ (8808198518421 / 16000000000000 : ℝ) ≤
+      (-(189982673360871353711 / 200000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (8808198518421 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(304348134677823892373 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_502_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (8808198518421 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3065526689465984201931 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_502_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (8808198518421 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3109928746905706661603 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_502_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (8808198518421 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1592305384936174259601 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_502_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (8808198518421 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(6601332083711793922337 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_502_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (8808198518421 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(6944582519792693022809 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_502_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (8808198518421 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(7434297431019868095343 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_502_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (8808198518421 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(1014421937012309681831 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_502_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (8808198518421 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(4524739837311758777763 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_502_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (8808198518421 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(5167602033692465113331 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_502_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (8808198518421 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(1522301870274884627343 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_502_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (8808198518421 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(1917385680481320352707 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_502_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (8808198518421 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_502_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (8808198518421 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_502_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (8808198518421 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_502_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (8808198518421 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_502_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_503_1 :
+    Uω (aρ 1) (bρ 1) (35312680955719 / 64000000000000 : ℝ) ≤
+      (-(6064045915001829464321 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8724927254666151 / 16000000000000000 : ℝ) ≤
+      (-(6064045915001829464321 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (8724927254666151 / 16000000000000000 : ℝ) =
+            (8724927254666151 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8724927254666151 / 8000000000000000 : ℝ) ≤
+            (867425887998170535679 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1090609977417319 / 2000000000000000 : ℝ)) (q :=
+      (8724927254666151 / 16000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_503_2 :
+    Uω (aρ 2) (bρ 2) (35312680955719 / 64000000000000 : ℝ) ≤
+      (-(1221606949584324856063 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1737326321746531 / 3200000000000000 : ℝ) ≤
+      (-(1221606949584324856063 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1737326321746531 / 3200000000000000 : ℝ) =
+            (1737326321746531 / 1600000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1737326321746531 / 1600000000000000 : ℝ) ≤
+            (164687411015675143937 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (217157973687189 / 400000000000000 : ℝ)) (q :=
+      (1737326321746531 / 3200000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_503_3 :
+    Uω (aρ 3) (bρ 3) (35312680955719 / 64000000000000 : ℝ) ≤
+      (-(3098315821178827396319 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8610010670417763 / 16000000000000000 : ℝ) ≤
+      (-(3098315821178827396319 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (8610010670417763 / 16000000000000000 : ℝ) =
+            (8610010670417763 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8610010670417763 / 8000000000000000 : ℝ) ≤
+            (367420080321172603681 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (538056927563611 / 1000000000000000 : ℝ)) (q :=
+      (8610010670417763 / 16000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_503_4 :
+    Uω (aρ 4) (bρ 4) (35312680955719 / 64000000000000 : ℝ) ≤
+      (-(634564076381066350629 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (42413323994768167 / 80000000000000000 : ℝ) ≤
+      (-(634564076381066350629 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (42413323994768167 / 80000000000000000 : ℝ) =
+            (42413323994768167 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (42413323994768167 / 40000000000000000 : ℝ) ≤
+            (58583103918933649371 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1324920279342737 / 2500000000000000 : ℝ)) (q :=
+      (42413323994768167 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_503_5 :
+    Uω (aρ 5) (bρ 5) (35312680955719 / 64000000000000 : ℝ) ≤
+      (-(3288591645766013384659 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (41442557229805003 / 80000000000000000 : ℝ) ≤
+      (-(3288591645766013384659 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (41442557229805003 / 80000000000000000 : ℝ) =
+            (41442557229805003 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (41442557229805003 / 40000000000000000 : ℝ) ≤
+            (177144255733986615341 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5175179345465157 / 10000000000000000 : ℝ)) (q :=
+      (41442557229805003 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_503_6 :
+    Uω (aρ 6) (bρ 6) (35312680955719 / 64000000000000 : ℝ) ≤
+      (-(6919554951870387697571 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (40047695818072167 / 80000000000000000 : ℝ) ≤
+      (-(6919554951870387697571 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (40047695818072167 / 80000000000000000 : ℝ) =
+            (40047695818072167 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 2
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (40047695818072167 / 40000000000000000 : ℝ) ≤
+            (11916851129612302429 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1248426091917987 / 2500000000000000 : ℝ)) (q :=
+      (40047695818072167 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_503_7 :
+    Uω (aρ 7) (bρ 7) (35312680955719 / 64000000000000 : ℝ) ≤
+      (-(7407929150169422584079 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (38138860354343527 / 80000000000000000 : ℝ) ≤
+      (-(7407929150169422584079 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (38138860354343527 / 80000000000000000 : ℝ) =
+            (38138860354343527 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (38138860354343527 / 20000000000000000 : ℝ) ≤
+            (6455014455830577415921 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1185039220624947 / 2500000000000000 : ℝ)) (q :=
+      (38138860354343527 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_503_8 :
+    Uω (aρ 8) (bρ 8) (35312680955719 / 64000000000000 : ℝ) ≤
+      (-(8086950740760240682239 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (35635116139077731 / 80000000000000000 : ℝ) ≤
+      (-(8086950740760240682239 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (35635116139077731 / 80000000000000000 : ℝ) =
+            (35635116139077731 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (35635116139077731 / 20000000000000000 : ℝ) ≤
+            (5775992865239759317761 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4397762480023339 / 10000000000000000 : ℝ)) (q :=
+      (35635116139077731 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_503_9 :
+    Uω (aρ 9) (bρ 9) (35312680955719 / 64000000000000 : ℝ) ≤
+      (-(4508900456673071142313 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (32467725773847159 / 80000000000000000 : ℝ) ≤
+      (-(4508900456673071142313 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (32467725773847159 / 80000000000000000 : ℝ) =
+            (32467725773847159 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (32467725773847159 / 20000000000000000 : ℝ) ≤
+            (2422571346326928857687 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (246680032317231 / 625000000000000 : ℝ)) (q :=
+      (32467725773847159 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_503_10 :
+    Uω (aρ 10) (bρ 10) (35312680955719 / 64000000000000 : ℝ) ≤
+      (-(5148988703631236251007 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28566334052394611 / 80000000000000000 : ℝ) ≤
+      (-(5148988703631236251007 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (28566334052394611 / 80000000000000000 : ℝ) =
+            (28566334052394611 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28566334052394611 / 20000000000000000 : ℝ) ≤
+            (1782483099368763748993 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3360091620207559 / 10000000000000000 : ℝ)) (q :=
+      (28566334052394611 / 80000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_503_11 :
+    Uω (aρ 11) (bρ 11) (35312680955719 / 64000000000000 : ℝ) ≤
+      (-(2425987833100888042973 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23784466929877283 / 80000000000000000 : ℝ) ≤
+      (-(2425987833100888042973 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (23784466929877283 / 80000000000000000 : ℝ) =
+            (23784466929877283 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23784466929877283 / 20000000000000000 : ℝ) ≤
+            (346600888099111957027 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2583101465783227 / 10000000000000000 : ℝ)) (q :=
+      (23784466929877283 / 80000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_503_12 :
+    Uω (aρ 12) (bρ 12) (35312680955719 / 64000000000000 : ℝ) ≤
+      (-(15249929318392561796379 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3481961518105123 / 16000000000000000 : ℝ) ≤
+      (-(15249929318392561796379 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3481961518105123 / 16000000000000000 : ℝ) =
+            (3481961518105123 / 2000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3481961518105123 / 2000000000000000 : ℝ) ≤
+            (5544486090607438203621 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (141313672763031 / 1000000000000000 : ℝ)) (q :=
+      (3481961518105123 / 16000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_503_13 :
+    Uω (aρ 13) (bρ 13) (35312680955719 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_503_14 :
+    Uω (aρ 14) (bρ 14) (35312680955719 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_503_15 :
+    Uω (aρ 15) (bρ 15) (35312680955719 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_503_16 :
+    Uω (aρ 16) (bρ 16) (35312680955719 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_503 :
+    Uρ (35312680955719 / 64000000000000 : ℝ) ≤
+      (-(947137693861291331499 / 1000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (35312680955719 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(6064045915001829464321 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_503_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (35312680955719 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1221606949584324856063 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_503_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (35312680955719 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3098315821178827396319 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_503_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (35312680955719 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(634564076381066350629 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_503_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (35312680955719 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(3288591645766013384659 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_503_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (35312680955719 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(6919554951870387697571 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_503_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (35312680955719 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(7407929150169422584079 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_503_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (35312680955719 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(8086950740760240682239 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_503_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (35312680955719 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(4508900456673071142313 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_503_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (35312680955719 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(5148988703631236251007 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_503_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (35312680955719 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(2425987833100888042973 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_503_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (35312680955719 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(15249929318392561796379 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_503_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (35312680955719 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_503_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (35312680955719 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_503_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (35312680955719 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_503_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (35312680955719 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_503_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_504_1 :
+    Uω (aρ 1) (bρ 1) (17696283918877 / 32000000000000 : ℝ) ≤
+      (-(60411815348335259041 / 100000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21862247708772351 / 40000000000000000 : ℝ) ≤
+      (-(60411815348335259041 / 100000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (21862247708772351 / 40000000000000000 : ℝ) =
+            (21862247708772351 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21862247708772351 / 20000000000000000 : ℝ) ≤
+            (8902902681664740959 / 100000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5465532347822113 / 10000000000000000 : ℝ)) (q :=
+      (21862247708772351 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_504_2 :
+    Uω (aρ 2) (bρ 2) (17696283918877 / 32000000000000 : ℝ) ≤
+      (-(3042534492391497808847 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21766510116169791 / 40000000000000000 : ℝ) ≤
+      (-(3042534492391497808847 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (21766510116169791 / 40000000000000000 : ℝ) =
+            (21766510116169791 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21766510116169791 / 20000000000000000 : ℝ) ≤
+            (423201409108502191153 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5441432564030833 / 10000000000000000 : ℝ)) (q :=
+      (21766510116169791 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_504_3 :
+    Uω (aρ 3) (bρ 3) (17696283918877 / 32000000000000 : ℝ) ≤
+      (-(6173459623665779297577 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21574962341257021 / 40000000000000000 : ℝ) ≤
+      (-(6173459623665779297577 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (21574962341257021 / 40000000000000000 : ℝ) =
+            (21574962341257021 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21574962341257021 / 20000000000000000 : ℝ) ≤
+            (758012179334220702423 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (168532961966389 / 312500000000000 : ℝ)) (q :=
+      (21574962341257021 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_504_4 :
+    Uω (aρ 4) (bρ 4) (17696283918877 / 32000000000000 : ℝ) ≤
+      (-(1580528875958493533667 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21256609949949273 / 40000000000000000 : ℝ) ≤
+      (-(1580528875958493533667 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (21256609949949273 / 40000000000000000 : ℝ) =
+            (21256609949949273 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21256609949949273 / 20000000000000000 : ℝ) ≤
+            (152339074791506466333 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2656086384167787 / 5000000000000000 : ℝ)) (q :=
+      (21256609949949273 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_504_5 :
+    Uω (aρ 5) (bρ 5) (17696283918877 / 32000000000000 : ℝ) ≤
+      (-(6553092790598556143351 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20771257389598851 / 40000000000000000 : ℝ) ≤
+      (-(6553092790598556143351 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (20771257389598851 / 40000000000000000 : ℝ) =
+            (20771257389598851 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20771257389598851 / 20000000000000000 : ℝ) ≤
+            (378379012401443856649 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5187686407495363 / 10000000000000000 : ℝ)) (q :=
+      (20771257389598851 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_504_6 :
+    Uω (aρ 6) (bρ 6) (17696283918877 / 32000000000000 : ℝ) ≤
+      (-(3447295084451065829983 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20073899461071497 / 40000000000000000 : ℝ) ≤
+      (-(3447295084451065829983 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (20073899461071497 / 40000000000000000 : ℝ) =
+            (20073899461071497 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20073899461071497 / 20000000000000000 : ℝ) ≤
+            (18440817048934170017 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2503123909185843 / 5000000000000000 : ℝ)) (q :=
+      (20073899461071497 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_504_7 :
+    Uω (aρ 7) (bρ 7) (17696283918877 / 32000000000000 : ℝ) ≤
+      (-(3690815504439367221797 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19119645233276717 / 40000000000000000 : ℝ) ≤
+      (-(3690815504439367221797 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (19119645233276717 / 40000000000000000 : ℝ) =
+            (19119645233276717 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19119645233276717 / 10000000000000000 : ℝ) ≤
+            (3240656298560632778203 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (594097760654287 / 1250000000000000 : ℝ)) (q :=
+      (19119645233276717 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_504_8 :
+    Uω (aρ 8) (bρ 8) (17696283918877 / 32000000000000 : ℝ) ≤
+      (-(1007326077744263590559 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3573625686803071 / 8000000000000000 : ℝ) ≤
+      (-(1007326077744263590559 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3573625686803071 / 8000000000000000 : ℝ) =
+            (3573625686803071 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3573625686803071 / 2000000000000000 : ℝ) ≤
+            (725541873005736409441 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (882113067388723 / 2000000000000000 : ℝ)) (q :=
+      (3573625686803071 / 8000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_504_9 :
+    Uω (aρ 9) (bρ 9) (17696283918877 / 32000000000000 : ℝ) ≤
+      (-(1123278478412851818853 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16285199201546043 / 40000000000000000 : ℝ) ≤
+      (-(1123278478412851818853 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (16285199201546043 / 40000000000000000 : ℝ) =
+            (16285199201546043 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16285199201546043 / 10000000000000000 : ℝ) ≤
+            (609589472337148181147 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3960066349068959 / 10000000000000000 : ℝ)) (q :=
+      (16285199201546043 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_504_10 :
+    Uω (aρ 10) (bρ 10) (17696283918877 / 32000000000000 : ℝ) ≤
+      (-(10260906059422783887687 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14336214918596509 / 40000000000000000 : ℝ) ≤
+      (-(10260906059422783887687 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (14336214918596509 / 40000000000000000 : ℝ) =
+            (14336214918596509 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14336214918596509 / 10000000000000000 : ℝ) ≤
+            (3602037546577216112313 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (421766655136149 / 1250000000000000 : ℝ)) (q :=
+      (14336214918596509 / 40000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_504_11 :
+    Uω (aρ 11) (bρ 11) (17696283918877 / 32000000000000 : ℝ) ≤
+      (-(755110462113986613171 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11949658663093087 / 40000000000000000 : ℝ) ≤
+      (-(755110462113986613171 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (11949658663093087 / 40000000000000000 : ℝ) =
+            (11949658663093087 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11949658663093087 / 10000000000000000 : ℝ) ≤
+            (111323513261013386829 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2599331739542481 / 10000000000000000 : ℝ)) (q :=
+      (11949658663093087 / 40000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_504_12 :
+    Uω (aρ 12) (bρ 12) (17696283918877 / 32000000000000 : ℝ) ≤
+      (-(1516239665633792455279 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8781434594056657 / 40000000000000000 : ℝ) ≤
+      (-(1516239665633792455279 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (8781434594056657 / 40000000000000000 : ℝ) =
+            (8781434594056657 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8781434594056657 / 5000000000000000 : ℝ) ≤
+            (563201875266207544721 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (719459900854633 / 5000000000000000 : ℝ)) (q :=
+      (8781434594056657 / 40000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_504_13 :
+    Uω (aρ 13) (bρ 13) (17696283918877 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_504_14 :
+    Uω (aρ 14) (bρ 14) (17696283918877 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_504_15 :
+    Uω (aρ 15) (bρ 15) (17696283918877 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_504_16 :
+    Uω (aρ 16) (bρ 16) (17696283918877 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_504 :
+    Uρ (17696283918877 / 32000000000000 : ℝ) ≤
+      (-(1180476362959208691143 / 1250000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (17696283918877 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(60411815348335259041 / 100000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_504_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (17696283918877 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3042534492391497808847 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_504_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (17696283918877 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(6173459623665779297577 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_504_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (17696283918877 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1580528875958493533667 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_504_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (17696283918877 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(6553092790598556143351 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_504_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (17696283918877 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3447295084451065829983 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_504_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (17696283918877 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(3690815504439367221797 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_504_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (17696283918877 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(1007326077744263590559 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_504_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (17696283918877 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(1123278478412851818853 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_504_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (17696283918877 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(10260906059422783887687 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_504_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (17696283918877 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) * (-(755110462113986613171 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_504_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (17696283918877 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(1516239665633792455279 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_504_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (17696283918877 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_504_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (17696283918877 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_504_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (17696283918877 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_504_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (17696283918877 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_504_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_505_1 :
+    Uω (aρ 1) (bρ 1) (35472454719789 / 64000000000000 : ℝ) ≤
+      (-(6018369313981465330427 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (43824354559290113 / 80000000000000000 : ℝ) ≤
+      (-(6018369313981465330427 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (43824354559290113 / 80000000000000000 : ℝ) =
+            (43824354559290113 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (43824354559290113 / 40000000000000000 : ℝ) ≤
+            (913102489018534669573 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5478014807940497 / 10000000000000000 : ℝ)) (q :=
+      (43824354559290113 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_505_2 :
+    Uω (aρ 2) (bρ 2) (35472454719789 / 64000000000000 : ℝ) ≤
+      (-(6062155847207341966241 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (43632882404600209 / 80000000000000000 : ℝ) ≤
+      (-(6062155847207341966241 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (43632882404600209 / 80000000000000000 : ℝ) =
+            (43632882404600209 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (43632882404600209 / 40000000000000000 : ℝ) ≤
+            (869315955792658033759 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5453915781778021 / 10000000000000000 : ℝ)) (q :=
+      (43632882404600209 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_505_3 :
+    Uω (aρ 3) (bρ 3) (35472454719789 / 64000000000000 : ℝ) ≤
+      (-(384396324293936033753 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (43249795954150029 / 80000000000000000 : ℝ) ≤
+      (-(384396324293936033753 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (43249795954150029 / 80000000000000000 : ℝ) =
+            (43249795954150029 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (43249795954150029 / 40000000000000000 : ℝ) ≤
+            (48820663393563966247 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1351385068878869 / 2500000000000000 : ℝ)) (q :=
+      (43249795954150029 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_505_4 :
+    Uω (aρ 4) (bρ 4) (35472454719789 / 64000000000000 : ℝ) ≤
+      (-(6298645498833575554591 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8522623126014137 / 16000000000000000 : ℝ) ≤
+      (-(6298645498833575554591 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (8522623126014137 / 16000000000000000 : ℝ) =
+            (8522623126014137 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8522623126014137 / 8000000000000000 : ℝ) ≤
+            (632826304166424445409 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (16639576173627 / 31250000000000 : ℝ)) (q :=
+      (8522623126014137 / 16000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_505_5 :
+    Uω (aρ 5) (bρ 5) (35472454719789 / 64000000000000 : ℝ) ≤
+      (-(6529060299625747032463 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (41642471853104017 / 80000000000000000 : ℝ) ≤
+      (-(6529060299625747032463 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (41642471853104017 / 80000000000000000 : ℝ) =
+            (41642471853104017 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (41642471853104017 / 40000000000000000 : ℝ) ≤
+            (402411503374252967537 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5200193350653973 / 10000000000000000 : ℝ)) (q :=
+      (41642471853104017 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_505_6 :
+    Uω (aρ 6) (bρ 6) (35472454719789 / 64000000000000 : ℝ) ≤
+      (-(858710982215112154227 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (40247900807027701 / 80000000000000000 : ℝ) ≤
+      (-(858710982215112154227 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (40247900807027701 / 80000000000000000 : ℝ) =
+            (40247900807027701 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (40247900807027701 / 40000000000000000 : ℝ) ≤
+            (7722993159887845773 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2509395482137447 / 5000000000000000 : ℝ)) (q :=
+      (40247900807027701 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_505_7 :
+    Uω (aρ 7) (bρ 7) (35472454719789 / 64000000000000 : ℝ) ≤
+      (-(58843221046640534947 / 80000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1533588702705877 / 3200000000000000 : ℝ) ≤
+      (-(58843221046640534947 / 80000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1533588702705877 / 3200000000000000 : ℝ) =
+            (1533588702705877 / 800000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1533588702705877 / 800000000000000 : ℝ) ≤
+            (52060327801359465053 / 80000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (47654065351897 / 100000000000000 : ℝ)) (q :=
+      (1533588702705877 / 3200000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_505_8 :
+    Uω (aρ 8) (bρ 8) (35472454719789 / 64000000000000 : ℝ) ≤
+      (-(2007587162204019625203 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (35837390267996993 / 80000000000000000 : ℝ) ≤
+      (-(2007587162204019625203 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (35837390267996993 / 80000000000000000 : ℝ) =
+            (35837390267996993 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (35837390267996993 / 20000000000000000 : ℝ) ≤
+            (1458148739295980374797 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4423366361617217 / 10000000000000000 : ℝ)) (q :=
+      (35837390267996993 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_505_9 :
+    Uω (aρ 9) (bρ 9) (35472454719789 / 64000000000000 : ℝ) ≤
+      (-(8954759677569814852371 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (32673052853042553 / 80000000000000000 : ℝ) ≤
+      (-(8954759677569814852371 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (32673052853042553 / 80000000000000000 : ℝ) =
+            (32673052853042553 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (32673052853042553 / 20000000000000000 : ℝ) ≤
+            (4908183928430185147629 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3973247636238607 / 10000000000000000 : ℝ)) (q :=
+      (32673052853042553 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_505_10 :
+    Uω (aρ 10) (bρ 10) (35472454719789 / 64000000000000 : ℝ) ≤
+      (-(255599714979519626207 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1151139071792921 / 3200000000000000 : ℝ) ≤
+      (-(255599714979519626207 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1151139071792921 / 3200000000000000 : ℝ) =
+            (1151139071792921 / 800000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1151139071792921 / 800000000000000 : ℝ) ≤
+            (90973875170480373793 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (135526506207149 / 400000000000000 : ℝ)) (q :=
+      (1151139071792921 / 3200000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_505_11 :
+    Uω (aρ 11) (bρ 11) (35472454719789 / 64000000000000 : ℝ) ≤
+      (-(6016947547431752205949 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24014003147253777 / 80000000000000000 : ℝ) ≤
+      (-(6016947547431752205949 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (24014003147253777 / 80000000000000000 : ℝ) =
+            (24014003147253777 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24014003147253777 / 20000000000000000 : ℝ) ≤
+            (914524255568247794051 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2615520869491413 / 10000000000000000 : ℝ)) (q :=
+      (24014003147253777 / 80000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_505_12 :
+    Uω (aρ 12) (bρ 12) (35472454719789 / 64000000000000 : ℝ) ≤
+      (-(1884551029486662392393 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17714540691209181 / 80000000000000000 : ℝ) ≤
+      (-(1884551029486662392393 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (17714540691209181 / 80000000000000000 : ℝ) =
+            (17714540691209181 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17714540691209181 / 10000000000000000 : ℝ) ≤
+            (714750896638337607607 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (91522209510329 / 625000000000000 : ℝ)) (q :=
+      (17714540691209181 / 80000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_505_13 :
+    Uω (aρ 13) (bρ 13) (35472454719789 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_505_14 :
+    Uω (aρ 14) (bρ 14) (35472454719789 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_505_15 :
+    Uω (aρ 15) (bρ 15) (35472454719789 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_505_16 :
+    Uω (aρ 16) (bρ 16) (35472454719789 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_505 :
+    Uρ (35472454719789 / 64000000000000 : ℝ) ≤
+      (-(9416429288524211005069 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (35472454719789 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(6018369313981465330427 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_505_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (35472454719789 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(6062155847207341966241 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_505_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (35472454719789 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(384396324293936033753 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_505_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (35472454719789 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(6298645498833575554591 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_505_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (35472454719789 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(6529060299625747032463 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_505_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (35472454719789 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(858710982215112154227 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_505_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (35472454719789 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) * (-(58843221046640534947 / 80000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_505_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (35472454719789 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(2007587162204019625203 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_505_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (35472454719789 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(8954759677569814852371 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_505_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (35472454719789 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) * (-(255599714979519626207 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_505_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (35472454719789 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(6016947547431752205949 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_505_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (35472454719789 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(1884551029486662392393 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_505_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (35472454719789 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_505_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (35472454719789 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_505_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (35472454719789 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_505_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (35472454719789 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_505_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_506_1 :
+    Uω (aρ 1) (bρ 1) (1111010675057 / 2000000000000 : ℝ) ≤
+      (-(59956090150079933539 / 100000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2745263356161489 / 5000000000000000 : ℝ) ≤
+      (-(59956090150079933539 / 100000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2745263356161489 / 5000000000000000 : ℝ) =
+            (2745263356161489 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2745263356161489 / 2500000000000000 : ℝ) ≤
+            (9358627879920066461 / 100000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1372624316861489 / 2500000000000000 : ℝ)) (q :=
+      (2745263356161489 / 5000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_506_2 :
+    Uω (aρ 2) (bρ 2) (1111010675057 / 2000000000000 : ℝ) ≤
+      (-(6039295094548093923991 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5466593070069701 / 10000000000000000 : ℝ) ≤
+      (-(6039295094548093923991 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5466593070069701 / 10000000000000000 : ℝ) =
+            (5466593070069701 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5466593070069701 / 5000000000000000 : ℝ) ≤
+            (892176708451906076009 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2733199497724701 / 5000000000000000 : ℝ)) (q :=
+      (5466593070069701 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_506_3 :
+    Uω (aρ 3) (bρ 3) (1111010675057 / 2000000000000 : ℝ) ≤
+      (-(6127276090160940364943 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27093541979627 / 50000000000000 : ℝ) ≤
+      (-(6127276090160940364943 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (27093541979627 / 50000000000000 : ℝ) = (27093541979627 / 25000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27093541979627 / 25000000000000 : ℝ) ≤
+            (804195712839059635057 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (13545064383777 / 25000000000000 : ℝ)) (q :=
+      (27093541979627 / 50000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_506_4 :
+    Uω (aρ 4) (bρ 4) (1111010675057 / 2000000000000 : ℝ) ≤
+      (-(3137615244832883098359 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10678252796628263 / 20000000000000000 : ℝ) ≤
+      (-(3137615244832883098359 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10678252796628263 / 20000000000000000 : ℝ) =
+            (10678252796628263 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10678252796628263 / 10000000000000000 : ℝ) ≤
+            (328120656667116901641 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5337155939353263 / 10000000000000000 : ℝ)) (q :=
+      (10678252796628263 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_506_5 :
+    Uω (aρ 5) (bρ 5) (1111010675057 / 2000000000000 : ℝ) ≤
+      (-(6505085539363036308551 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10435607113736623 / 20000000000000000 : ℝ) ≤
+      (-(6505085539363036308551 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10435607113736623 / 20000000000000000 : ℝ) =
+            (10435607113736623 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10435607113736623 / 10000000000000000 : ℝ) ≤
+            (426386263636963691449 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5212700175796623 / 10000000000000000 : ℝ)) (q :=
+      (10435607113736623 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_506_6 :
+    Uω (aρ 6) (bρ 6) (1111010675057 / 2000000000000 : ℝ) ≤
+      (-(6844847704947540843917 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10087000370461153 / 20000000000000000 : ℝ) ≤
+      (-(6844847704947540843917 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10087000370461153 / 20000000000000000 : ℝ) =
+            (10087000370461153 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10087000370461153 / 10000000000000000 : ℝ) ≤
+            (86624098052459156083 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5031333807661153 / 10000000000000000 : ℝ)) (q :=
+      (10087000370461153 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_506_7 :
+    Uω (aρ 7) (bρ 7) (1111010675057 / 2000000000000 : ℝ) ≤
+      (-(3664621821379332739513 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (192200708407459 / 400000000000000 : ℝ) ≤
+      (-(3664621821379332739513 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (192200708407459 / 400000000000000 : ℝ) =
+            (192200708407459 / 100000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (192200708407459 / 100000000000000 : ℝ) ≤
+            (3266849981620667260487 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (95560604766659 / 200000000000000 : ℝ)) (q :=
+      (192200708407459 / 400000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_506_8 :
+    Uω (aρ 8) (bρ 8) (1111010675057 / 2000000000000 : ℝ) ≤
+      (-(4001085167470894044969 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2246157275151393 / 5000000000000000 : ℝ) ≤
+      (-(4001085167470894044969 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2246157275151393 / 5000000000000000 : ℝ) =
+            (2246157275151393 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2246157275151393 / 1250000000000000 : ℝ) ≤
+            (2930386635529105955031 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1109041392476393 / 2500000000000000 : ℝ)) (q :=
+      (2246157275151393 / 5000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_506_9 :
+    Uω (aρ 9) (bρ 9) (1111010675057 / 2000000000000 : ℝ) ≤
+      (-(35693582933142942329 / 40000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8193922326007617 / 20000000000000000 : ℝ) ≤
+      (-(35693582933142942329 / 40000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8193922326007617 / 20000000000000000 : ℝ) =
+            (8193922326007617 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8193922326007617 / 5000000000000000 : ℝ) ≤
+            (19758191490857057671 / 40000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3986424423667617 / 10000000000000000 : ℝ)) (q :=
+      (8193922326007617 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_506_10 :
+    Uω (aρ 10) (bρ 10) (1111010675057 / 2000000000000 : ℝ) ≤
+      (-(10187223622840232274999 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (722111888233093 / 2000000000000000 : ℝ) ≤
+      (-(10187223622840232274999 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (722111888233093 / 2000000000000000 : ℝ) =
+            (722111888233093 / 500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (722111888233093 / 500000000000000 : ℝ) ≤
+            (3675719983159767725001 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (340218001348593 / 1000000000000000 : ℝ)) (q :=
+      (722111888233093 / 2000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10]) (by norm_num [bρ_10])
+      (by norm_num) (by norm_num [aρ_10, bρ_10]) (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_506_11 :
+    Uω (aρ 11) (bρ 11) (1111010675057 / 2000000000000 : ℝ) ≤
+      (-(5993158914446563252359 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (603213185757801 / 2000000000000000 : ℝ) ≤
+      (-(5993158914446563252359 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (603213185757801 / 2000000000000000 : ℝ) =
+            (603213185757801 / 500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (603213185757801 / 500000000000000 : ℝ) ≤
+            (938312888553436747641 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (263166961493801 / 1000000000000000 : ℝ)) (q :=
+      (603213185757801 / 2000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11]) (by norm_num [bρ_11])
+      (by norm_num) (by norm_num [aρ_11, bρ_11]) (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_506_12 :
+    Uω (aρ 12) (bρ 12) (1111010675057 / 2000000000000 : ℝ) ≤
+      (-(14991891141000413337327 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2233111664528861 / 10000000000000000 : ℝ) ≤
+      (-(14991891141000413337327 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2233111664528861 / 10000000000000000 : ℝ) =
+            (2233111664528861 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2233111664528861 / 1250000000000000 : ℝ) ≤
+            (5802524267999586662673 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (744730591551361 / 5000000000000000 : ℝ)) (q :=
+      (2233111664528861 / 10000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_506_13 :
+    Uω (aρ 13) (bρ 13) (1111010675057 / 2000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_506_14 :
+    Uω (aρ 14) (bρ 14) (1111010675057 / 2000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_506_15 :
+    Uω (aρ 15) (bρ 15) (1111010675057 / 2000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_506_16 :
+    Uω (aρ 16) (bρ 16) (1111010675057 / 2000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_506 :
+    Uρ (1111010675057 / 2000000000000 : ℝ) ≤
+      (-(9389226274504482798993 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1111010675057 / 2000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(59956090150079933539 / 100000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_506_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1111010675057 / 2000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(6039295094548093923991 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_506_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1111010675057 / 2000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(6127276090160940364943 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_506_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1111010675057 / 2000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3137615244832883098359 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_506_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1111010675057 / 2000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(6505085539363036308551 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_506_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1111010675057 / 2000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(6844847704947540843917 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_506_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1111010675057 / 2000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(3664621821379332739513 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_506_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1111010675057 / 2000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(4001085167470894044969 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_506_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1111010675057 / 2000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(35693582933142942329 / 40000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_506_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1111010675057 / 2000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(10187223622840232274999 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_506_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1111010675057 / 2000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(5993158914446563252359 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_506_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1111010675057 / 2000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(14991891141000413337327 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_506_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1111010675057 / 2000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_506_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1111010675057 / 2000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_506_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1111010675057 / 2000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_506_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1111010675057 / 2000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_506_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_507_1 :
+    Uω (aρ 1) (bρ 1) (35632228483859 / 64000000000000 : ℝ) ≤
+      (-(373306275130817100621 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (44024072835442519 / 80000000000000000 : ℝ) ≤
+      (-(373306275130817100621 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (44024072835442519 / 80000000000000000 : ℝ) =
+            (44024072835442519 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (44024072835442519 / 40000000000000000 : ℝ) ≤
+            (59910712556682899379 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5502979726342661 / 10000000000000000 : ℝ)) (q :=
+      (44024072835442519 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_507_2 :
+    Uω (aρ 2) (bρ 2) (35632228483859 / 64000000000000 : ℝ) ≤
+      (-(3008243243902837444081 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (43832606700323227 / 80000000000000000 : ℝ) ≤
+      (-(3008243243902837444081 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (43832606700323227 / 80000000000000000 : ℝ) =
+            (43832606700323227 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (43832606700323227 / 40000000000000000 : ℝ) ≤
+            (457492657597162555919 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2739441102536419 / 5000000000000000 : ℝ)) (q :=
+      (43832606700323227 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_507_3 :
+    Uω (aρ 3) (bρ 3) (35632228483859 / 64000000000000 : ℝ) ≤
+      (-(6104264082439977002931 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (43449538322676247 / 80000000000000000 : ℝ) ≤
+      (-(6104264082439977002931 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (43449538322676247 / 80000000000000000 : ℝ) =
+            (43449538322676247 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (43449538322676247 / 40000000000000000 : ℝ) ≤
+            (827207720560022997069 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5430511217011093 / 10000000000000000 : ℝ)) (q :=
+      (43449538322676247 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_507_4 :
+    Uω (aρ 4) (bρ 4) (35632228483859 / 64000000000000 : ℝ) ≤
+      (-(6251870219006976036533 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (42812906570442639 / 80000000000000000 : ℝ) ≤
+      (-(6251870219006976036533 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (42812906570442639 / 80000000000000000 : ℝ) =
+            (42812906570442639 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (42812906570442639 / 40000000000000000 : ℝ) ≤
+            (679601583993023963467 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5349647460017691 / 10000000000000000 : ℝ)) (q :=
+      (42812906570442639 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_507_5 :
+    Uω (aρ 5) (bρ 5) (35632228483859 / 64000000000000 : ℝ) ≤
+      (-(6481168232575320061883 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (41842384588114851 / 80000000000000000 : ℝ) ≤
+      (-(6481168232575320061883 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (41842384588114851 / 80000000000000000 : ℝ) =
+            (41842384588114851 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (41842384588114851 / 40000000000000000 : ℝ) ≤
+            (450303570424679938117 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (653150860471343 / 1250000000000000 : ℝ)) (q :=
+      (41842384588114851 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_507_6 :
+    Uω (aρ 6) (bρ 6) (35632228483859 / 64000000000000 : ℝ) ≤
+      (-(3410034699783073694907 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8089620191124223 / 16000000000000000 : ℝ) ≤
+      (-(3410034699783073694907 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (8089620191124223 / 16000000000000000 : ℝ) =
+            (8089620191124223 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8089620191124223 / 8000000000000000 : ℝ) ≤
+            (55701201716926305093 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (504387635078731 / 1000000000000000 : ℝ)) (q :=
+      (8089620191124223 / 16000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_507_7 :
+    Uω (aρ 7) (bρ 7) (35632228483859 / 64000000000000 : ℝ) ≤
+      (-(1460630734883897880289 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (38540562831702879 / 80000000000000000 : ℝ) ≤
+      (-(1460630734883897880289 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (38540562831702879 / 80000000000000000 : ℝ) =
+            (38540562831702879 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (38540562831702879 / 20000000000000000 : ℝ) ≤
+            (1311957986316102119711 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4790653200567751 / 10000000000000000 : ℝ)) (q :=
+      (38540562831702879 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_507_8 :
+    Uω (aρ 8) (bρ 8) (35632228483859 / 64000000000000 : ℝ) ≤
+      (-(3987036599165787142513 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (36039635334013223 / 80000000000000000 : ℝ) ≤
+      (-(3987036599165787142513 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (36039635334013223 / 80000000000000000 : ℝ) =
+            (36039635334013223 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (36039635334013223 / 20000000000000000 : ℝ) ≤
+            (2944435203834212857487 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4448962977485337 / 10000000000000000 : ℝ)) (q :=
+      (36039635334013223 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_507_9 :
+    Uω (aρ 9) (bρ 9) (35632228483859 / 64000000000000 : ℝ) ≤
+      (-(8892135271530491175893 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (32878307933950143 / 80000000000000000 : ℝ) ≤
+      (-(8892135271530491175893 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (32878307933950143 / 80000000000000000 : ℝ) =
+            (32878307933950143 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (32878307933950143 / 20000000000000000 : ℝ) ≤
+            (4970808334469508824107 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3999596755829567 / 10000000000000000 : ℝ)) (q :=
+      (32878307933950143 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_507_10 :
+    Uω (aρ 10) (bρ 10) (35632228483859 / 64000000000000 : ℝ) ≤
+      (-(5075304873631269541131 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28990426634305347 / 80000000000000000 : ℝ) ≤
+      (-(5075304873631269541131 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (28990426634305347 / 80000000000000000 : ℝ) =
+            (28990426634305347 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28990426634305347 / 20000000000000000 : ℝ) ≤
+            (1856166929368730458869 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (427023183051671 / 1250000000000000 : ℝ)) (q :=
+      (28990426634305347 / 80000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_507_11 :
+    Uω (aρ 11) (bρ 11) (35632228483859 / 64000000000000 : ℝ) ≤
+      (-(11939031264853494913149 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24242893131011923 / 80000000000000000 : ℝ) ≤
+      (-(11939031264853494913149 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (24242893131011923 / 80000000000000000 : ℝ) =
+            (24242893131011923 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24242893131011923 / 20000000000000000 : ℝ) ≤
+            (1923912341146505086851 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (661944678698753 / 2500000000000000 : ℝ)) (q :=
+      (24242893131011923 / 80000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_507_12 :
+    Uω (aρ 12) (bρ 12) (35632228483859 / 64000000000000 : ℝ) ≤
+      (-(2981755615627733624033 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3602798532646971 / 16000000000000000 : ℝ) ≤
+      (-(2981755615627733624033 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3602798532646971 / 16000000000000000 : ℝ) =
+            (3602798532646971 / 2000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3602798532646971 / 2000000000000000 : ℝ) ≤
+            (1177127466172266375967 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (302850738907149 / 2000000000000000 : ℝ)) (q :=
+      (3602798532646971 / 16000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_507_13 :
+    Uω (aρ 13) (bρ 13) (35632228483859 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_507_14 :
+    Uω (aρ 14) (bρ 14) (35632228483859 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_507_15 :
+    Uω (aρ 15) (bρ 15) (35632228483859 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_507_16 :
+    Uω (aρ 16) (bρ 16) (35632228483859 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_507 :
+    Uρ (35632228483859 / 64000000000000 : ℝ) ≤
+      (-(1170274556345986861049 / 1250000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (35632228483859 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(373306275130817100621 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_507_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (35632228483859 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3008243243902837444081 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_507_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (35632228483859 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(6104264082439977002931 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_507_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (35632228483859 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(6251870219006976036533 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_507_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (35632228483859 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(6481168232575320061883 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_507_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (35632228483859 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3410034699783073694907 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_507_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (35632228483859 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(1460630734883897880289 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_507_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (35632228483859 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(3987036599165787142513 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_507_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (35632228483859 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(8892135271530491175893 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_507_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (35632228483859 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(5075304873631269541131 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_507_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (35632228483859 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(11939031264853494913149 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_507_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (35632228483859 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(2981755615627733624033 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_507_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (35632228483859 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_507_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (35632228483859 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_507_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (35632228483859 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_507_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (35632228483859 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_507_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/U42.lean b/LeanPool/Zeta5Irrational/Table/U42.lean
new file mode 100644
index 0000000000..09ae6617cb
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/U42.lean
@@ -0,0 +1,4611 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-! # Certified arcsine potential bounds (U42) -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma U_508_1 :
+    Uω (aρ 1) (bρ 1) (17856057682947 / 32000000000000 : ℝ) ≤
+      (-(5950243241019627244583 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4412393196988273 / 8000000000000000 : ℝ) ≤
+      (-(5950243241019627244583 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (4412393196988273 / 8000000000000000 : ℝ) =
+            (4412393196988273 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4412393196988273 / 4000000000000000 : ℝ) ≤
+            (981228561980372755417 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1103092436926949 / 2000000000000000 : ℝ)) (q :=
+      (4412393196988273 / 8000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_508_2 :
+    Uω (aρ 2) (bρ 2) (17856057682947 / 32000000000000 : ℝ) ≤
+      (-(1498432447403128669001 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21966234412003739 / 40000000000000000 : ℝ) ≤
+      (-(1498432447403128669001 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (21966234412003739 / 40000000000000000 : ℝ) =
+            (21966234412003739 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21966234412003739 / 20000000000000000 : ℝ) ≤
+            (234435503346871330999 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1372841352668983 / 2500000000000000 : ℝ)) (q :=
+      (21966234412003739 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_508_3 :
+    Uω (aρ 3) (bρ 3) (17856057682947 / 32000000000000 : ℝ) ≤
+      (-(3040652460816649757537 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21774704710184083 / 40000000000000000 : ℝ) ≤
+      (-(3040652460816649757537 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (21774704710184083 / 40000000000000000 : ℝ) =
+            (21774704710184083 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21774704710184083 / 20000000000000000 : ℝ) ≤
+            (425083440683350242463 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (680374583264513 / 1250000000000000 : ℝ)) (q :=
+      (21774704710184083 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_508_4 :
+    Uω (aρ 4) (bρ 4) (17856057682947 / 32000000000000 : ℝ) ≤
+      (-(3114282215668375516407 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21456400891532543 / 40000000000000000 : ℝ) ≤
+      (-(3114282215668375516407 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (21456400891532543 / 40000000000000000 : ℝ) =
+            (21456400891532543 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21456400891532543 / 20000000000000000 : ℝ) ≤
+            (351453685831624483593 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2681069468927667 / 5000000000000000 : ℝ)) (q :=
+      (21456400891532543 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_508_5 :
+    Uω (aρ 5) (bρ 5) (17856057682947 / 32000000000000 : ℝ) ≤
+      (-(645730810402355959231 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20971170127983213 / 40000000000000000 : ℝ) ≤
+      (-(645730810402355959231 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (20971170127983213 / 40000000000000000 : ℝ) =
+            (20971170127983213 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20971170127983213 / 20000000000000000 : ℝ) ≤
+            (47416369897644040769 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5237713475415669 / 10000000000000000 : ℝ)) (q :=
+      (20971170127983213 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_508_6 :
+    Uω (aρ 6) (bρ 6) (17856057682947 / 32000000000000 : ℝ) ≤
+      (-(271814105316008456471 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20274099618647413 / 40000000000000000 : ℝ) ≤
+      (-(271814105316008456471 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (20274099618647413 / 40000000000000000 : ℝ) =
+            (20274099618647413 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20274099618647413 / 20000000000000000 : ℝ) ≤
+            (5444766803991543529 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5056418595887769 / 10000000000000000 : ℝ)) (q :=
+      (20274099618647413 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_508_7 :
+    Uω (aρ 7) (bρ 7) (17856057682947 / 32000000000000 : ℝ) ≤
+      (-(7277132358554394793971 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3864098104164537 / 8000000000000000 : ℝ) ≤
+      (-(7277132358554394793971 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3864098104164537 / 8000000000000000 : ℝ) =
+            (3864098104164537 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3864098104164537 / 2000000000000000 : ℝ) ≤
+            (6585811247445605206029 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (960655085547081 / 2000000000000000 : ℝ)) (q :=
+      (3864098104164537 / 8000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_508_8 :
+    Uω (aρ 8) (bρ 8) (17856057682947 / 32000000000000 : ℝ) ≤
+      (-(7946056761336366334497 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (18070373562374121 / 40000000000000000 : ℝ) ≤
+      (-(7946056761336366334497 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (18070373562374121 / 40000000000000000 : ℝ) =
+            (18070373562374121 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18070373562374121 / 10000000000000000 : ℝ) ≤
+            (5916886844663633665503 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4461758599851123 / 10000000000000000 : ℝ)) (q :=
+      (18070373562374121 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_508_9 :
+    Uω (aρ 9) (bρ 9) (17856057682947 / 32000000000000 : ℝ) ≤
+      (-(1772195515441329619083 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16490454459150151 / 40000000000000000 : ℝ) ≤
+      (-(1772195515441329619083 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (16490454459150151 / 40000000000000000 : ℝ) =
+            (16490454459150151 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16490454459150151 / 10000000000000000 : ℝ) ≤
+            (1000393205758670380917 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2006382338299569 / 5000000000000000 : ℝ)) (q :=
+      (16490454459150151 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_508_10 :
+    Uω (aρ 10) (bρ 10) (17856057682947 / 32000000000000 : ℝ) ≤
+      (-(10114145609446987792001 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14548165346591567 / 40000000000000000 : ℝ) ≤
+      (-(10114145609446987792001 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (14548165346591567 / 40000000000000000 : ℝ) =
+            (14548165346591567 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14548165346591567 / 10000000000000000 : ℝ) ≤
+            (3748797996553012207999 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1715089576907423 / 5000000000000000 : ℝ)) (q :=
+      (14548165346591567 / 40000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_508_11 :
+    Uω (aρ 11) (bρ 11) (17856057682947 / 32000000000000 : ℝ) ≤
+      (-(1189203117663709979711 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12178551563170123 / 40000000000000000 : ℝ) ≤
+      (-(1189203117663709979711 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (12178551563170123 / 40000000000000000 : ℝ) =
+            (12178551563170123 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12178551563170123 / 10000000000000000 : ℝ) ≤
+            (197091242936290020289 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (665962222077281 / 2500000000000000 : ℝ)) (q :=
+      (12178551563170123 / 40000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_508_12 :
+    Uω (aρ 12) (bρ 12) (17856057682947 / 32000000000000 : ℝ) ≤
+      (-(2965401357108477598221 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9080949655730931 / 40000000000000000 : ℝ) ≤
+      (-(2965401357108477598221 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (9080949655730931 / 40000000000000000 : ℝ) =
+            (9080949655730931 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9080949655730931 / 5000000000000000 : ℝ) ≤
+            (1193481724691522401779 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (48085875977329 / 312500000000000 : ℝ)) (q :=
+      (9080949655730931 / 40000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_508_13 :
+    Uω (aρ 13) (bρ 13) (17856057682947 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_508_14 :
+    Uω (aρ 14) (bρ 14) (17856057682947 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_508_15 :
+    Uω (aρ 15) (bρ 15) (17856057682947 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_508_16 :
+    Uω (aρ 16) (bρ 16) (17856057682947 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_508 :
+    Uρ (17856057682947 / 32000000000000 : ℝ) ≤
+      (-(9335334772364821229953 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (17856057682947 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(5950243241019627244583 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_508_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (17856057682947 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1498432447403128669001 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_508_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (17856057682947 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3040652460816649757537 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_508_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (17856057682947 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3114282215668375516407 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_508_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (17856057682947 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(645730810402355959231 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_508_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (17856057682947 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(271814105316008456471 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_508_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (17856057682947 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(7277132358554394793971 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_508_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (17856057682947 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(7946056761336366334497 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_508_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (17856057682947 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(1772195515441329619083 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_508_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (17856057682947 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(10114145609446987792001 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_508_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (17856057682947 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(1189203117663709979711 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_508_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (17856057682947 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(2965401357108477598221 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_508_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (17856057682947 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_508_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (17856057682947 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_508_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (17856057682947 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_508_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (17856057682947 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_508_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_509_1 :
+    Uω (aρ 1) (bρ 1) (35792002247929 / 64000000000000 : ℝ) ≤
+      (-(5927637299159037206243 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (44223791101920841 / 80000000000000000 : ℝ) ≤
+      (-(5927637299159037206243 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (44223791101920841 / 80000000000000000 : ℝ) =
+            (44223791101920841 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (44223791101920841 / 40000000000000000 : ℝ) ≤
+            (1003834503840962793757 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (172748270072697 / 312500000000000 : ℝ)) (q :=
+      (44223791101920841 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_509_2 :
+    Uω (aρ 2) (bρ 2) (35792002247929 / 64000000000000 : ℝ) ≤
+      (-(5971024764218213295379 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (44032330931719793 / 80000000000000000 : ℝ) ≤
+      (-(5971024764218213295379 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (44032330931719793 / 80000000000000000 : ℝ) =
+            (44032330931719793 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (44032330931719793 / 40000000000000000 : ℝ) ≤
+            (960447038781786704621 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2751924306143021 / 5000000000000000 : ℝ)) (q :=
+      (44032330931719793 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_509_3 :
+    Uω (aρ 3) (bρ 3) (35792002247929 / 64000000000000 : ℝ) ≤
+      (-(6058398365511501288333 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (43649280460874297 / 80000000000000000 : ℝ) ≤
+      (-(6058398365511501288333 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (43649280460874297 / 80000000000000000 : ℝ) =
+            (43649280460874297 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (43649280460874297 / 40000000000000000 : ℝ) ≤
+            (873073437488498711667 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1363870525231167 / 2500000000000000 : ℝ)) (q :=
+      (43649280460874297 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_509_4 :
+    Uω (aρ 4) (bρ 4) (35792002247929 / 64000000000000 : ℝ) ≤
+      (-(620531287292092823143 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (43012696825574837 / 80000000000000000 : ℝ) ≤
+      (-(620531287292092823143 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (43012696825574837 / 80000000000000000 : ℝ) =
+            (43012696825574837 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (43012696825574837 / 40000000000000000 : ℝ) ≤
+            (72615893007907176857 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5374630373164803 / 10000000000000000 : ℝ)) (q :=
+      (43012696825574837 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_509_5 :
+    Uω (aρ 5) (bρ 5) (35792002247929 / 64000000000000 : ℝ) ≤
+      (-(201047027513926073651 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (42042295461826521 / 80000000000000000 : ℝ) ≤
+      (-(201047027513926073651 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (42042295461826521 / 80000000000000000 : ℝ) =
+            (42042295461826521 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (42042295461826521 / 40000000000000000 : ℝ) ≤
+            (15561466329823926349 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1312554987890681 / 2500000000000000 : ℝ)) (q :=
+      (42042295461826521 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_509_6 :
+    Uω (aρ 6) (bρ 6) (35792002247929 / 64000000000000 : ℝ) ≤
+      (-(1692674274647282850247 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8129659267142937 / 16000000000000000 : ℝ) ≤
+      (-(1692674274647282850247 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (8129659267142937 / 16000000000000000 : ℝ) =
+            (8129659267142937 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8129659267142937 / 8000000000000000 : ℝ) ≤
+            (40193676102717149753 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1013792109034953 / 2000000000000000 : ℝ)) (q :=
+      (8129659267142937 / 16000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_509_7 :
+    Uω (aρ 7) (bρ 7) (35792002247929 / 64000000000000 : ℝ) ≤
+      (-(3625589665429885931267 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7748279266887633 / 16000000000000000 : ℝ) ≤
+      (-(3625589665429885931267 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7748279266887633 / 16000000000000000 : ℝ) =
+            (7748279266887633 / 4000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7748279266887633 / 4000000000000000 : ℝ) ≤
+            (3305882137570114068733 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (963179385123127 / 2000000000000000 : ℝ)) (q :=
+      (7748279266887633 / 16000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_509_8 :
+    Uω (aρ 8) (bρ 8) (35792002247929 / 64000000000000 : ℝ) ≤
+      (-(7918120550604440884439 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (36241851835888241 / 80000000000000000 : ℝ) ≤
+      (-(7918120550604440884439 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (36241851835888241 / 80000000000000000 : ℝ) =
+            (36241851835888241 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (36241851835888241 / 20000000000000000 : ℝ) ≤
+            (5944823055395559115561 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2237276226159077 / 5000000000000000 : ℝ)) (q :=
+      (36241851835888241 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_509_9 :
+    Uω (aρ 9) (bρ 9) (35792002247929 / 64000000000000 : ℝ) ≤
+      (-(220748048573075527627 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (33083492430225213 / 80000000000000000 : ℝ) ≤
+      (-(220748048573075527627 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (33083492430225213 / 80000000000000000 : ℝ) =
+            (33083492430225213 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (33083492430225213 / 20000000000000000 : ℝ) ≤
+            (125825541576924472373 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4025928229262397 / 10000000000000000 : ℝ)) (q :=
+      (33083492430225213 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_509_10 :
+    Uω (aρ 10) (bρ 10) (35792002247929 / 64000000000000 : ℝ) ≤
+      (-(157466091658155803863 / 156250000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (29202188279404597 / 80000000000000000 : ℝ) ≤
+      (-(157466091658155803863 / 156250000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (29202188279404597 / 80000000000000000 : ℝ) =
+            (29202188279404597 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (29202188279404597 / 20000000000000000 : ℝ) ≤
+            (59142402185594196137 / 156250000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3444161225052243 / 10000000000000000 : ℝ)) (q :=
+      (29202188279404597 / 80000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_509_11 :
+    Uω (aρ 11) (bρ 11) (35792002247929 / 64000000000000 : ℝ) ≤
+      (-(11845313439531267758087 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24471160217405897 / 80000000000000000 : ℝ) ≤
+      (-(11845313439531267758087 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (24471160217405897 / 80000000000000000 : ℝ) =
+            (24471160217405897 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24471160217405897 / 20000000000000000 : ℝ) ≤
+            (2017630166468732241913 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5234142257339 / 19531250000000 : ℝ)) (q :=
+      (24471160217405897 / 80000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_509_12 :
+    Uω (aρ 12) (bρ 12) (35792002247929 / 64000000000000 : ℝ) ≤
+      (-(14746519519992178665709 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (18308669336300729 / 80000000000000000 : ℝ) ≤
+      (-(14746519519992178665709 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (18308669336300729 / 80000000000000000 : ℝ) =
+            (18308669336300729 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18308669336300729 / 10000000000000000 : ℝ) ≤
+            (6047895889007821334291 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (390739553041569 / 2500000000000000 : ℝ)) (q :=
+      (18308669336300729 / 80000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_509_13 :
+    Uω (aρ 13) (bρ 13) (35792002247929 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_509_14 :
+    Uω (aρ 14) (bρ 14) (35792002247929 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_509_15 :
+    Uω (aρ 15) (bρ 15) (35792002247929 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_509_16 :
+    Uω (aρ 16) (bρ 16) (35792002247929 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_509 :
+    Uρ (35792002247929 / 64000000000000 : ℝ) ≤
+      (-(1861727304759575972879 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (35792002247929 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(5927637299159037206243 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_509_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (35792002247929 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(5971024764218213295379 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_509_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (35792002247929 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(6058398365511501288333 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_509_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (35792002247929 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(620531287292092823143 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_509_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (35792002247929 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(201047027513926073651 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_509_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (35792002247929 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1692674274647282850247 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_509_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (35792002247929 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(3625589665429885931267 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_509_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (35792002247929 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(7918120550604440884439 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_509_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (35792002247929 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(220748048573075527627 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_509_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (35792002247929 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) * (-(157466091658155803863 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_509_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (35792002247929 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(11845313439531267758087 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_509_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (35792002247929 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(14746519519992178665709 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_509_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (35792002247929 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_509_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (35792002247929 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_509_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (35792002247929 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_509_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (35792002247929 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_509_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_510_1 :
+    Uω (aρ 1) (bρ 1) (8967972282491 / 16000000000000 : ℝ) ≤
+      (-(5905082345456817827057 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11080912557893271 / 20000000000000000 : ℝ) ≤
+      (-(5905082345456817827057 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11080912557893271 / 20000000000000000 : ℝ) =
+            (11080912557893271 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11080912557893271 / 10000000000000000 : ℝ) ≤
+            (1026389457543182172943 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1385106774855349 / 2500000000000000 : ℝ)) (q :=
+      (11080912557893271 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_510_2 :
+    Uω (aρ 2) (bρ 2) (8967972282491 / 16000000000000 : ℝ) ≤
+      (-(5948371177474891842903 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11033048255892149 / 20000000000000000 : ℝ) ≤
+      (-(5948371177474891842903 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11033048255892149 / 20000000000000000 : ℝ) =
+            (11033048255892149 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11033048255892149 / 10000000000000000 : ℝ) ≤
+            (983100625525108157097 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2758165904965137 / 5000000000000000 : ℝ)) (q :=
+      (11033048255892149 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_510_3 :
+    Uω (aρ 3) (bρ 3) (8967972282491 / 16000000000000 : ℝ) ≤
+      (-(150888604337680130333 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10937287861146593 / 20000000000000000 : ℝ) ≤
+      (-(150888604337680130333 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10937287861146593 / 20000000000000000 : ℝ) =
+            (10937287861146593 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10937287861146593 / 10000000000000000 : ℝ) ≤
+            (22398190737319869667 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2733983760767359 / 5000000000000000 : ℝ)) (q :=
+      (10937287861146593 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_510_4 :
+    Uω (aρ 4) (bρ 4) (8967972282491 / 16000000000000 : ℝ) ≤
+      (-(154552882294875529027 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1347268490598601 / 2500000000000000 : ℝ) ≤
+      (-(154552882294875529027 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1347268490598601 / 2500000000000000 : ℝ) =
+            (1347268490598601 / 1250000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1347268490598601 / 1250000000000000 : ℝ) ≤
+            (18733912780124470973 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5387121766241933 / 10000000000000000 : ℝ)) (q :=
+      (1347268490598601 / 2500000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_510_5 :
+    Uω (aρ 5) (bρ 5) (8967972282491 / 16000000000000 : ℝ) ≤
+      (-(100152473289647147279 / 156250000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13169453190309 / 25000000000000 : ℝ) ≤
+      (-(100152473289647147279 / 156250000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (13169453190309 / 25000000000000 : ℝ) = (13169453190309 / 12500000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13169453190309 / 12500000000000 : ℝ) ≤
+            (8151773632227852721 / 156250000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (210509052521413 / 400000000000000 : ℝ)) (q :=
+      (13169453190309 / 25000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_510_6 :
+    Uω (aρ 6) (bρ 6) (8967972282491 / 16000000000000 : ℝ) ≤
+      (-(843262811570624237493 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2037419612982103 / 4000000000000000 : ℝ) ≤
+      (-(843262811570624237493 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2037419612982103 / 4000000000000000 : ℝ) =
+            (2037419612982103 / 2000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2037419612982103 / 2000000000000000 : ℝ) ≤
+            (23171163804375762507 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (63518777510483 / 125000000000000 : ℝ)) (q :=
+      (2037419612982103 / 4000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_510_7 :
+    Uω (aρ 7) (bρ 7) (8967972282491 / 16000000000000 : ℝ) ≤
+      (-(7225294229954618860193 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (388418087329571 / 800000000000000 : ℝ) ≤
+      (-(7225294229954618860193 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (388418087329571 / 800000000000000 : ℝ) =
+            (388418087329571 / 200000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (388418087329571 / 200000000000000 : ℝ) ≤
+            (6637649376045381139807 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (24142588499637 / 50000000000000 : ℝ)) (q :=
+      (388418087329571 / 800000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_510_8 :
+    Uω (aρ 8) (bρ 8) (8967972282491 / 16000000000000 : ℝ) ≤
+      (-(1578052819405799303733 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (227143434549919 / 500000000000000 : ℝ) ≤
+      (-(1578052819405799303733 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (227143434549919 / 500000000000000 : ℝ) =
+            (227143434549919 / 125000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (227143434549919 / 125000000000000 : ℝ) ≤
+            (1194535901794200696267 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (897468910004977 / 2000000000000000 : ℝ)) (q :=
+      (227143434549919 / 500000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_510_9 :
+    Uω (aρ 9) (bρ 9) (8967972282491 / 16000000000000 : ℝ) ≤
+      (-(2199741917220122414509 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (518532166258671 / 1250000000000000 : ℝ) ≤
+      (-(2199741917220122414509 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (518532166258671 / 1250000000000000 : ℝ) =
+            (518532166258671 / 312500000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (518532166258671 / 312500000000000 : ℝ) ≤
+            (1265993984279877585491 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4039087456526861 / 10000000000000000 : ℝ)) (q :=
+      (518532166258671 / 1250000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_510_10 :
+    Uω (aρ 10) (bρ 10) (8967972282491 / 16000000000000 : ℝ) ≤
+      (-(2510415298336719111113 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (366349999458383 / 1000000000000000 : ℝ) ≤
+      (-(2510415298336719111113 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (366349999458383 / 1000000000000000 : ℝ) =
+            (366349999458383 / 250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (366349999458383 / 250000000000000 : ℝ) ≤
+            (955320603163280888887 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (691626363810157 / 2000000000000000 : ℝ)) (q :=
+      (366349999458383 / 1000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10]) (by norm_num [bρ_10])
+      (by norm_num) (by norm_num [aρ_10, bρ_10]) (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_510_11 :
+    Uω (aρ 11) (bρ 11) (8967972282491 / 16000000000000 : ℝ) ≤
+      (-(11798874026815627584347 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1536566695756391 / 5000000000000000 : ℝ) ≤
+      (-(11798874026815627584347 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1536566695756391 / 5000000000000000 : ℝ) =
+            (1536566695756391 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1536566695756391 / 1250000000000000 : ℝ) ≤
+            (2064069579184372415653 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2695875239113689 / 10000000000000000 : ℝ)) (q :=
+      (1536566695756391 / 5000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_510_12 :
+    Uω (aρ 12) (bρ 12) (8967972282491 / 16000000000000 : ℝ) ≤
+      (-(57293994566708123539 / 39062500000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4613588689991969 / 20000000000000000 : ℝ) ≤
+      (-(57293994566708123539 / 39062500000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4613588689991969 / 20000000000000000 : ℝ) =
+            (4613588689991969 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4613588689991969 / 2500000000000000 : ℝ) ≤
+            (23934190624698126461 / 39062500000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (793448621382547 / 5000000000000000 : ℝ)) (q :=
+      (4613588689991969 / 20000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_510_13 :
+    Uω (aρ 13) (bρ 13) (8967972282491 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_510_14 :
+    Uω (aρ 14) (bρ 14) (8967972282491 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_510_15 :
+    Uω (aρ 15) (bρ 15) (8967972282491 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_510_16 :
+    Uω (aρ 16) (bρ 16) (8967972282491 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_510 :
+    Uρ (8967972282491 / 16000000000000 : ℝ) ≤
+      (-(2320524321799194358933 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (8967972282491 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(5905082345456817827057 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_510_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (8967972282491 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(5948371177474891842903 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_510_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (8967972282491 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(150888604337680130333 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_510_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (8967972282491 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(154552882294875529027 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_510_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (8967972282491 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(100152473289647147279 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_510_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (8967972282491 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(843262811570624237493 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_510_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (8967972282491 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(7225294229954618860193 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_510_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (8967972282491 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(1578052819405799303733 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_510_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (8967972282491 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(2199741917220122414509 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_510_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (8967972282491 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(2510415298336719111113 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_510_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (8967972282491 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(11798874026815627584347 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_510_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (8967972282491 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(57293994566708123539 / 39062500000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_510_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (8967972282491 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_510_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (8967972282491 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_510_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (8967972282491 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_510_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (8967972282491 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_510_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_511_1 :
+    Uω (aρ 1) (bρ 1) (35951776011999 / 64000000000000 : ℝ) ≤
+      (-(5882578150418441128189 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8884701871771111 / 16000000000000000 : ℝ) ≤
+      (-(5882578150418441128189 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (8884701871771111 / 16000000000000000 : ℝ) =
+            (8884701871771111 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8884701871771111 / 8000000000000000 : ℝ) ≤
+            (1048893652581558871811 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1110581911184809 / 2000000000000000 : ℝ)) (q :=
+      (8884701871771111 / 16000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_511_2 :
+    Uω (aρ 2) (bρ 2) (35951776011999 / 64000000000000 : ℝ) ≤
+      (-(592576879682269358441 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (44232055099661343 / 80000000000000000 : ℝ) ≤
+      (-(592576879682269358441 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (44232055099661343 / 80000000000000000 : ℝ) =
+            (44232055099661343 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (44232055099661343 / 40000000000000000 : ℝ) ≤
+            (100570300617730641559 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1382203750908873 / 2500000000000000 : ℝ)) (q :=
+      (44232055099661343 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_511_3 :
+    Uω (aρ 3) (bρ 3) (35951776011999 / 64000000000000 : ℝ) ≤
+      (-(300637105334994580849 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (43849022371892547 / 80000000000000000 : ℝ) ≤
+      (-(300637105334994580849 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (43849022371892547 / 80000000000000000 : ℝ) =
+            (43849022371892547 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (43849022371892547 / 40000000000000000 : ℝ) ≤
+            (45936484815005419151 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5480452928043293 / 10000000000000000 : ℝ)) (q :=
+      (43849022371892547 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_511_4 :
+    Uω (aρ 4) (bρ 4) (35951776011999 / 64000000000000 : ℝ) ≤
+      (-(6158971437747780164537 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1728499456199147 / 3200000000000000 : ℝ) ≤
+      (-(6158971437747780164537 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1728499456199147 / 3200000000000000 : ℝ) =
+            (1728499456199147 / 1600000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1728499456199147 / 1600000000000000 : ℝ) ≤
+            (772500365252219835463 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (215984524695193 / 400000000000000 : ℝ)) (q :=
+      (1728499456199147 / 3200000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_511_5 :
+    Uω (aρ 5) (bρ 5) (35951776011999 / 64000000000000 : ℝ) ≤
+      (-(1277213612986815760097 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (42242204500715663 / 80000000000000000 : ℝ) ≤
+      (-(1277213612986815760097 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (42242204500715663 / 80000000000000000 : ℝ) =
+            (42242204500715663 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (42242204500715663 / 40000000000000000 : ℝ) ≤
+            (109080747613184239903 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (329702035040567 / 625000000000000 : ℝ)) (q :=
+      (42242204500715663 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_511_6 :
+    Uω (aρ 6) (bρ 6) (35951776011999 / 64000000000000 : ℝ) ≤
+      (-(6721568513029647007487 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (40848487017751843 / 80000000000000000 : ℝ) ≤
+      (-(6721568513029647007487 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (40848487017751843 / 80000000000000000 : ℝ) =
+            (40848487017751843 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (40848487017751843 / 40000000000000000 : ℝ) ≤
+            (209903289970352992513 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5094043565048117 / 10000000000000000 : ℝ)) (q :=
+      (40848487017751843 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_511_7 :
+    Uω (aρ 7) (bρ 7) (35951776011999 / 64000000000000 : ℝ) ≤
+      (-(359973834867452409587 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (38942218259838171 / 80000000000000000 : ℝ) ≤
+      (-(359973834867452409587 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (38942218259838171 / 80000000000000000 : ℝ) =
+            (38942218259838171 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (38942218259838171 / 20000000000000000 : ℝ) ≤
+            (333173345432547590413 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4841137756329699 / 10000000000000000 : ℝ)) (q :=
+      (38942218259838171 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_511_8 :
+    Uω (aρ 8) (bρ 8) (35951776011999 / 64000000000000 : ℝ) ≤
+      (-(491405433481033414363 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (36444040260901687 / 80000000000000000 : ℝ) ≤
+      (-(491405433481033414363 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (36444040260901687 / 80000000000000000 : ℝ) =
+            (36444040260901687 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (36444040260901687 / 20000000000000000 : ℝ) ≤
+            (375028541893966585637 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2250067453967789 / 5000000000000000 : ℝ)) (q :=
+      (36444040260901687 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_511_9 :
+    Uω (aρ 9) (bρ 9) (35951776011999 / 64000000000000 : ℝ) ≤
+      (-(4384057031379989668599 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1331544308713791 / 3200000000000000 : ℝ) ≤
+      (-(4384057031379989668599 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1331544308713791 / 3200000000000000 : ℝ) =
+            (1331544308713791 / 800000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1331544308713791 / 800000000000000 : ℝ) ≤
+            (2547414771620010331401 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (81044848010627 / 200000000000000 : ℝ)) (q :=
+      (1331544308713791 / 3200000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_511_10 :
+    Uω (aρ 10) (bρ 10) (35951776011999 / 64000000000000 : ℝ) ≤
+      (-(10005638286124311866309 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (29413766279150527 / 80000000000000000 : ℝ) ≤
+      (-(10005638286124311866309 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (29413766279150527 / 80000000000000000 : ℝ) =
+            (29413766279150527 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (29413766279150527 / 20000000000000000 : ℝ) ≤
+            (3857305319875688133691 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (868022768588197 / 2500000000000000 : ℝ)) (q :=
+      (29413766279150527 / 80000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_511_11 :
+    Uω (aρ 11) (bρ 11) (35951776011999 / 64000000000000 : ℝ) ≤
+      (-(11752709006505664412089 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24698826527652611 / 80000000000000000 : ℝ) ≤
+      (-(11752709006505664412089 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (24698826527652611 / 80000000000000000 : ℝ) =
+            (24698826527652611 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24698826527652611 / 20000000000000000 : ℝ) ≤
+            (2110234599494335587911 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2711832762683309 / 10000000000000000 : ℝ)) (q :=
+      (24698826527652611 / 80000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_511_12 :
+    Uω (aρ 12) (bρ 12) (35951776011999 / 64000000000000 : ℝ) ≤
+      (-(3647296514749162681049 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (18599003945990127 / 80000000000000000 : ℝ) ≤
+      (-(3647296514749162681049 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (18599003945990127 / 80000000000000000 : ℝ) =
+            (18599003945990127 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18599003945990127 / 10000000000000000 : ℝ) ≤
+            (1551307337500837318951 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (100661075872043 / 625000000000000 : ℝ)) (q :=
+      (18599003945990127 / 80000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_511_13 :
+    Uω (aρ 13) (bρ 13) (35951776011999 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_511_14 :
+    Uω (aρ 14) (bρ 14) (35951776011999 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_511_15 :
+    Uω (aρ 15) (bρ 15) (35951776011999 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_511_16 :
+    Uω (aρ 16) (bρ 16) (35951776011999 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_511 :
+    Uρ (35951776011999 / 64000000000000 : ℝ) ≤
+      (-(9255712914413268703163 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (35951776011999 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(5882578150418441128189 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_511_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (35951776011999 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(592576879682269358441 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_511_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (35951776011999 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(300637105334994580849 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_511_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (35951776011999 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(6158971437747780164537 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_511_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (35951776011999 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1277213612986815760097 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_511_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (35951776011999 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(6721568513029647007487 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_511_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (35951776011999 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(359973834867452409587 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_511_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (35951776011999 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(491405433481033414363 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_511_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (35951776011999 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(4384057031379989668599 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_511_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (35951776011999 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(10005638286124311866309 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_511_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (35951776011999 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(11752709006505664412089 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_511_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (35951776011999 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(3647296514749162681049 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_511_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (35951776011999 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_511_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (35951776011999 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_511_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (35951776011999 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_511_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (35951776011999 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_511_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_512_1 :
+    Uω (aρ 1) (bρ 1) (18015831447017 / 32000000000000 : ℝ) ≤
+      (-(1465031121523832597231 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22261684241892097 / 40000000000000000 : ℝ) ≤
+      (-(1465031121523832597231 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (22261684241892097 / 40000000000000000 : ℝ) =
+            (22261684241892097 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22261684241892097 / 20000000000000000 : ℝ) ≤
+            (267836829226167402769 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1391348002959559 / 2500000000000000 : ℝ)) (q :=
+      (22261684241892097 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_512_2 :
+    Uω (aρ 2) (bρ 2) (18015831447017 / 32000000000000 : ℝ) ≤
+      (-(368951086954716155551 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22165958580052259 / 40000000000000000 : ℝ) ≤
+      (-(368951086954716155551 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (22165958580052259 / 40000000000000000 : ℝ) =
+            (22165958580052259 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22165958580052259 / 20000000000000000 : ℝ) ≤
+            (64265900732783844449 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5541298193428317 / 10000000000000000 : ℝ)) (q :=
+      (22165958580052259 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_512_3 :
+    Uω (aρ 3) (bρ 3) (18015831447017 / 32000000000000 : ℝ) ≤
+      (-(5989991927800893399251 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21974446621588729 / 40000000000000000 : ℝ) ≤
+      (-(5989991927800893399251 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (21974446621588729 / 40000000000000000 : ℝ) =
+            (21974446621588729 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21974446621588729 / 20000000000000000 : ℝ) ≤
+            (941479875199106600749 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (686617290068319 / 1250000000000000 : ℝ)) (q :=
+      (21974446621588729 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_512_4 :
+    Uω (aρ 4) (bρ 4) (18015831447017 / 32000000000000 : ℝ) ≤
+      (-(6135881062304958525311 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21656190472103367 / 40000000000000000 : ℝ) ≤
+      (-(6135881062304958525311 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (21656190472103367 / 40000000000000000 : ℝ) =
+            (21656190472103367 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21656190472103367 / 20000000000000000 : ℝ) ≤
+            (795590740695041474689 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5412104426868871 / 10000000000000000 : ℝ)) (q :=
+      (21656190472103367 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_512_5 :
+    Uω (aρ 5) (bρ 5) (18015831447017 / 32000000000000 : ℝ) ≤
+      (-(318121696809580573499 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21171079170119309 / 40000000000000000 : ℝ) ≤
+      (-(318121696809580573499 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (21171079170119309 / 40000000000000000 : ℝ) =
+            (21171079170119309 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21171079170119309 / 20000000000000000 : ℝ) ≤
+            (28451893340419426501 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2643869347605921 / 5000000000000000 : ℝ)) (q :=
+      (21171079170119309 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_512_6 :
+    Uω (aρ 6) (bρ 6) (18015831447017 / 32000000000000 : ℝ) ≤
+      (-(1674273715108017435577 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20474290309316747 / 40000000000000000 : ℝ) ≤
+      (-(1674273715108017435577 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (20474290309316747 / 40000000000000000 : ℝ) =
+            (20474290309316747 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20474290309316747 / 20000000000000000 : ℝ) ≤
+            (58594235641982564423 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5106584639950561 / 10000000000000000 : ℝ)) (q :=
+      (20474290309316747 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_512_7 :
+    Uω (aρ 7) (bρ 7) (18015831447017 / 32000000000000 : ℝ) ≤
+      (-(7173726377413230075053 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19521312468740329 / 40000000000000000 : ℝ) ≤
+      (-(7173726377413230075053 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (19521312468740329 / 40000000000000000 : ℝ) =
+            (19521312468740329 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19521312468740329 / 10000000000000000 : ℝ) ≤
+            (6689217228586769924947 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (303359818776397 / 625000000000000 : ℝ)) (q :=
+      (19521312468740329 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_512_8 :
+    Uω (aρ 8) (bρ 8) (18015831447017 / 32000000000000 : ℝ) ≤
+      (-(1958697151459010594993 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (18272562046901279 / 40000000000000000 : ℝ) ≤
+      (-(1958697151459010594993 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (18272562046901279 / 40000000000000000 : ℝ) =
+            (18272562046901279 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18272562046901279 / 10000000000000000 : ℝ) ≤
+            (1507038750040989405007 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4512923540842827 / 10000000000000000 : ℝ)) (q :=
+      (18272562046901279 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_512_9 :
+    Uω (aρ 9) (bρ 9) (18015831447017 / 32000000000000 : ℝ) ≤
+      (-(1747472087922510415983 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3339113982841381 / 8000000000000000 : ℝ) ≤
+      (-(1747472087922510415983 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3339113982841381 / 8000000000000000 : ℝ) =
+            (3339113982841381 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3339113982841381 / 2000000000000000 : ℝ) ≤
+            (1025116633277489584017 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (101634827571391 / 250000000000000 : ℝ)) (q :=
+      (3339113982841381 / 8000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_512_10 :
+    Uω (aρ 10) (bρ 10) (18015831447017 / 32000000000000 : ℝ) ≤
+      (-(9969759858022321014881 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14759743895845517 / 40000000000000000 : ℝ) ≤
+      (-(9969759858022321014881 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (14759743895845517 / 40000000000000000 : ℝ) =
+            (14759743895845517 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14759743895845517 / 10000000000000000 : ℝ) ≤
+            (3893183747977678985119 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1743019563584973 / 5000000000000000 : ℝ)) (q :=
+      (14759743895845517 / 40000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_512_11 :
+    Uω (aρ 11) (bρ 11) (18015831447017 / 32000000000000 : ℝ) ≤
+      (-(2926703634558700865789 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12406220496522309 / 40000000000000000 : ℝ) ≤
+      (-(2926703634558700865789 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (12406220496522309 / 40000000000000000 : ℝ) =
+            (12406220496522309 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12406220496522309 / 10000000000000000 : ℝ) ≤
+            (539032266941299134211 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1363877026856671 / 5000000000000000 : ℝ)) (q :=
+      (12406220496522309 / 40000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_512_12 :
+    Uω (aρ 12) (bρ 12) (18015831447017 / 32000000000000 : ℝ) ≤
+      (-(1814030401212492219359 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9371330972906951 / 40000000000000000 : ℝ) ≤
+      (-(1814030401212492219359 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (9371330972906951 / 40000000000000000 : ℝ) =
+            (9371330972906951 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9371330972906951 / 5000000000000000 : ℝ) ≤
+            (785271524912507780641 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1634009388590663 / 10000000000000000 : ℝ)) (q :=
+      (9371330972906951 / 40000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_512_13 :
+    Uω (aρ 13) (bρ 13) (18015831447017 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_512_14 :
+    Uω (aρ 14) (bρ 14) (18015831447017 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_512_15 :
+    Uω (aρ 15) (bρ 15) (18015831447017 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_512_16 :
+    Uω (aρ 16) (bρ 16) (18015831447017 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_512 :
+    Uρ (18015831447017 / 32000000000000 : ℝ) ≤
+      (-(461473975122939147761 / 500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (18015831447017 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1465031121523832597231 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_512_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (18015831447017 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(368951086954716155551 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_512_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (18015831447017 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(5989991927800893399251 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_512_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (18015831447017 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(6135881062304958525311 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_512_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (18015831447017 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(318121696809580573499 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_512_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (18015831447017 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1674273715108017435577 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_512_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (18015831447017 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(7173726377413230075053 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_512_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (18015831447017 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(1958697151459010594993 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_512_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (18015831447017 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(1747472087922510415983 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_512_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (18015831447017 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(9969759858022321014881 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_512_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (18015831447017 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(2926703634558700865789 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_512_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (18015831447017 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(1814030401212492219359 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_512_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (18015831447017 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_512_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (18015831447017 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_512_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (18015831447017 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_512_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (18015831447017 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_512_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_513_1 :
+    Uω (aρ 1) (bρ 1) (36111549776069 / 64000000000000 : ℝ) ≤
+      (-(2918860563035500800487 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8924645521274961 / 16000000000000000 : ℝ) ≤
+      (-(2918860563035500800487 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (8924645521274961 / 16000000000000000 : ℝ) =
+            (8924645521274961 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8924645521274961 / 8000000000000000 : ℝ) ≤
+            (546875338464499199513 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (69723430839599 / 125000000000000 : ℝ)) (q :=
+      (8924645521274961 / 16000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_513_2 :
+    Uω (aρ 2) (bρ 2) (36111549776069 / 64000000000000 : ℝ) ≤
+      (-(2940358365703278109947 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (44431779205003641 / 80000000000000000 : ℝ) ≤
+      (-(2940358365703278109947 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (44431779205003641 / 80000000000000000 : ℝ) =
+            (44431779205003641 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (44431779205003641 / 40000000000000000 : ℝ) ≤
+            (525377535796721890053 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5553781379335129 / 10000000000000000 : ℝ)) (q :=
+      (44431779205003641 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_513_3 :
+    Uω (aρ 3) (bρ 3) (36111549776069 / 64000000000000 : ℝ) ≤
+      (-(18647791878558030403 / 31250000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (44048764058822241 / 80000000000000000 : ℝ) ≤
+      (-(18647791878558030403 / 31250000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (44048764058822241 / 80000000000000000 : ℝ) =
+            (44048764058822241 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (44048764058822241 / 40000000000000000 : ℝ) ≤
+            (3013057505816969597 / 31250000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5505423699139779 / 10000000000000000 : ℝ)) (q :=
+      (44048764058822241 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_513_4 :
+    Uω (aρ 4) (bρ 4) (36111549776069 / 64000000000000 : ℝ) ≤
+      (-(1222568783742651878327 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (43412275317990269 / 80000000000000000 : ℝ) ≤
+      (-(1222568783742651878327 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (43412275317990269 / 80000000000000000 : ℝ) =
+            (43412275317990269 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (43412275317990269 / 40000000000000000 : ℝ) ≤
+            (163725576857348121673 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2712297847498393 / 5000000000000000 : ℝ)) (q :=
+      (43412275317990269 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_513_5 :
+    Uω (aρ 5) (bρ 5) (36111549776069 / 64000000000000 : ℝ) ≤
+      (-(396178477423036181643 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (42442111730758641 / 80000000000000000 : ℝ) ≤
+      (-(396178477423036181643 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (42442111730758641 / 80000000000000000 : ℝ) =
+            (42442111730758641 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (42442111730758641 / 40000000000000000 : ℝ) ≤
+            (37038510264463818357 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5300244717523879 / 10000000000000000 : ℝ)) (q :=
+      (42442111730758641 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_513_6 :
+    Uω (aρ 6) (bρ 6) (36111549776069 / 64000000000000 : ℝ) ≤
+      (-(133453624748920267399 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (41048673070792113 / 80000000000000000 : ℝ) ≤
+      (-(133453624748920267399 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (41048673070792113 / 80000000000000000 : ℝ) =
+            (41048673070792113 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (41048673070792113 / 40000000000000000 : ℝ) ≤
+            (5175811311079732601 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5119125427672247 / 10000000000000000 : ℝ)) (q :=
+      (41048673070792113 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_513_7 :
+    Uω (aρ 7) (bρ 7) (36111549776069 / 64000000000000 : ℝ) ≤
+      (-(7148042917346742296531 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (39143028788050253 / 80000000000000000 : ℝ) ≤
+      (-(7148042917346742296531 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (39143028788050253 / 80000000000000000 : ℝ) =
+            (39143028788050253 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (39143028788050253 / 20000000000000000 : ℝ) ≤
+            (6714900688653257703469 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2433187868873391 / 5000000000000000 : ℝ)) (q :=
+      (39143028788050253 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_513_8 :
+    Uω (aρ 8) (bρ 8) (36111549776069 / 64000000000000 : ℝ) ≤
+      (-(1561433730153791004607 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7329240217036657 / 16000000000000000 : ℝ) ≤
+      (-(1561433730153791004607 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7329240217036657 / 16000000000000000 : ℝ) =
+            (7329240217036657 / 4000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7329240217036657 / 4000000000000000 : ℝ) ≤
+            (1211154991046208995393 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (113142761584251 / 250000000000000 : ℝ)) (q :=
+      (7329240217036657 / 16000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_513_9 :
+    Uω (aρ 9) (bρ 9) (36111549776069 / 64000000000000 : ℝ) ≤
+      (-(8706706121751562856749 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6698731027276557 / 16000000000000000 : ℝ) ≤
+      (-(8706706121751562856749 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6698731027276557 / 16000000000000000 : ℝ) =
+            (6698731027276557 / 4000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6698731027276557 / 4000000000000000 : ℝ) ≤
+            (5156237484248437143251 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (815707920905983 / 2000000000000000 : ℝ)) (q :=
+      (6698731027276557 / 16000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_513_10 :
+    Uω (aρ 10) (bρ 10) (36111549776069 / 64000000000000 : ℝ) ≤
+      (-(9934024640806293113911 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (29625165030019877 / 80000000000000000 : ℝ) ≤
+      (-(9934024640806293113911 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (29625165030019877 / 80000000000000000 : ℝ) =
+            (29625165030019877 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (29625165030019877 / 20000000000000000 : ℝ) ≤
+            (3928918965193706886089 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (874994027858547 / 2500000000000000 : ℝ)) (q :=
+      (29625165030019877 / 80000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_513_11 :
+    Uω (aρ 11) (bρ 11) (36111549776069 / 64000000000000 : ℝ) ≤
+      (-(5830593435133916158567 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24925913051359713 / 80000000000000000 : ℝ) ≤
+      (-(5830593435133916158567 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (24925913051359713 / 80000000000000000 : ℝ) =
+            (24925913051359713 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24925913051359713 / 20000000000000000 : ℝ) ≤
+            (1100878367866083841433 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2743639742974147 / 10000000000000000 : ℝ)) (q :=
+      (24925913051359713 / 80000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_513_12 :
+    Uω (aρ 12) (bρ 12) (36111549776069 / 64000000000000 : ℝ) ≤
+      (-(288727808611259611257 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (18885370804446089 / 80000000000000000 : ℝ) ≤
+      (-(288727808611259611257 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (18885370804446089 / 80000000000000000 : ℝ) =
+            (18885370804446089 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18885370804446089 / 10000000000000000 : ℝ) ≤
+            (127160499568740388743 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1657204277930741 / 10000000000000000 : ℝ)) (q :=
+      (18885370804446089 / 80000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_513_13 :
+    Uω (aρ 13) (bρ 13) (36111549776069 / 64000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_513_14 :
+    Uω (aρ 14) (bρ 14) (36111549776069 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_513_15 :
+    Uω (aρ 15) (bρ 15) (36111549776069 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_513_16 :
+    Uω (aρ 16) (bρ 16) (36111549776069 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_513 :
+    Uρ (36111549776069 / 64000000000000 : ℝ) ≤
+      (-(4601696685902667401401 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (36111549776069 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2918860563035500800487 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_513_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (36111549776069 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(2940358365703278109947 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_513_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (36111549776069 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) * (-(18647791878558030403 / 31250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_513_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (36111549776069 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1222568783742651878327 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_513_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (36111549776069 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(396178477423036181643 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_513_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (36111549776069 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(133453624748920267399 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_513_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (36111549776069 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(7148042917346742296531 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_513_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (36111549776069 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(1561433730153791004607 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_513_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (36111549776069 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(8706706121751562856749 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_513_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (36111549776069 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(9934024640806293113911 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_513_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (36111549776069 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(5830593435133916158567 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_513_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (36111549776069 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(288727808611259611257 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_513_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (36111549776069 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_513_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (36111549776069 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_513_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (36111549776069 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_513_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (36111549776069 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_513_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_514_1 :
+    Uω (aρ 1) (bρ 1) (4523929582263 / 8000000000000 : ℝ) ≤
+      (-(290768392272368305341 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5590385840830381 / 10000000000000000 : ℝ) ≤
+      (-(290768392272368305341 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5590385840830381 / 10000000000000000 : ℝ) =
+            (5590385840830381 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5590385840830381 / 5000000000000000 : ℝ) ≤
+            (55805197877631694659 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1397589230479253 / 2500000000000000 : ℝ)) (q :=
+      (5590385840830381 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_514_2 :
+    Uω (aρ 2) (bρ 2) (4523929582263 / 8000000000000 : ℝ) ≤
+      (-(234330663573394913113 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (445316412344633 / 800000000000000 : ℝ) ≤
+      (-(234330663573394913113 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (445316412344633 / 800000000000000 : ℝ) =
+            (445316412344633 / 400000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (445316412344633 / 400000000000000 : ℝ) ≤
+            (42928208546605086887 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (222650582455283 / 400000000000000 : ℝ)) (q :=
+      (445316412344633 / 800000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_514_3 :
+    Uω (aρ 3) (bρ 3) (4523929582263 / 8000000000000 : ℝ) ≤
+      (-(2972323146321819319241 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2759289676200287 / 5000000000000000 : ℝ) ≤
+      (-(2972323146321819319241 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2759289676200287 / 5000000000000000 : ℝ) =
+            (2759289676200287 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2759289676200287 / 2500000000000000 : ℝ) ≤
+            (493412755178180680759 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2758954531958699 / 5000000000000000 : ℝ)) (q :=
+      (2759289676200287 / 5000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_514_4 :
+    Uω (aρ 4) (bρ 4) (4523929582263 / 8000000000000 : ℝ) ≤
+      (-(3044929880962231886749 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5439021190933697 / 10000000000000000 : ℝ) ≤
+      (-(3044929880962231886749 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5439021190933697 / 10000000000000000 : ℝ) =
+            (5439021190933697 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5439021190933697 / 5000000000000000 : ℝ) ≤
+            (420806020537768113251 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1359271730512161 / 2500000000000000 : ℝ)) (q :=
+      (5439021190933697 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_514_5 :
+    Uω (aρ 5) (bρ 5) (4523929582263 / 8000000000000 : ℝ) ≤
+      (-(6315332909008076166521 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2658879042215409 / 5000000000000000 : ℝ) ≤
+      (-(6315332909008076166521 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2658879042215409 / 5000000000000000 : ℝ) =
+            (2658879042215409 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2658879042215409 / 2500000000000000 : ℝ) ≤
+            (616138893991923833479 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2656375314188943 / 5000000000000000 : ℝ)) (q :=
+      (2658879042215409 / 5000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_514_6 :
+    Uω (aρ 6) (bρ 6) (4523929582263 / 8000000000000 : ℝ) ≤
+      (-(6648327348947931113759 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2571797773915591 / 5000000000000000 : ℝ) ≤
+      (-(6648327348947931113759 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2571797773915591 / 5000000000000000 : ℝ) =
+            (2571797773915591 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2571797773915591 / 2500000000000000 : ℝ) ≤
+            (283144454052068886241 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2565832965159307 / 5000000000000000 : ℝ)) (q :=
+      (2571797773915591 / 5000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_514_7 :
+    Uω (aρ 7) (bρ 7) (4523929582263 / 8000000000000 : ℝ) ≤
+      (-(7122425967148344153303 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2452714364592631 / 5000000000000000 : ℝ) ≤
+      (-(7122425967148344153303 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2452714364592631 / 5000000000000000 : ℝ) =
+            (2452714364592631 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2452714364592631 / 1250000000000000 : ℝ) ≤
+            (6740517638851655846697 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2439496836893387 / 5000000000000000 : ℝ)) (q :=
+      (2452714364592631 / 5000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_514_8 :
+    Uω (aρ 8) (bρ 8) (4523929582263 / 8000000000000 : ℝ) ≤
+      (-(3889813308929943655439 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (73494542585741 / 160000000000000 : ℝ) ≤
+      (-(3889813308929943655439 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (73494542585741 / 160000000000000 : ℝ) = (73494542585741 / 40000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (73494542585741 / 40000000000000 : ℝ) ≤
+            (3041658494070056344561 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (36307965519791 / 80000000000000 : ℝ)) (q :=
+      (73494542585741 / 160000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_514_9 :
+    Uω (aρ 9) (bρ 9) (4523929582263 / 8000000000000 : ℝ) ≤
+      (-(8676150438646844351813 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4199519225463883 / 10000000000000000 : ℝ) ≤
+      (-(8676150438646844351813 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4199519225463883 / 10000000000000000 : ℝ) =
+            (4199519225463883 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4199519225463883 / 2500000000000000 : ℝ) ≤
+            (5186793167353155648187 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (255730121627751 / 625000000000000 : ℝ)) (q :=
+      (4199519225463883 / 10000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_514_10 :
+    Uω (aρ 10) (bρ 10) (4523929582263 / 8000000000000 : ℝ) ≤
+      (-(395937255363209419699 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7432699630235927 / 20000000000000000 : ℝ) ≤
+      (-(395937255363209419699 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7432699630235927 / 20000000000000000 : ℝ) =
+            (7432699630235927 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7432699630235927 / 5000000000000000 : ℝ) ≤
+            (158580488876790580301 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3513902158847177 / 10000000000000000 : ℝ)) (q :=
+      (7432699630235927 / 20000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_514_11 :
+    Uω (aρ 11) (bρ 11) (4523929582263 / 8000000000000 : ℝ) ≤
+      (-(5807911168319489409469 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6259811290500711 / 20000000000000000 : ℝ) ≤
+      (-(5807911168319489409469 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6259811290500711 / 20000000000000000 : ℝ) =
+            (6259811290500711 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6259811290500711 / 5000000000000000 : ℝ) ≤
+            (1123560634680510590531 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2759490445316961 / 10000000000000000 : ℝ)) (q :=
+      (6259811290500711 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_514_12 :
+    Uω (aρ 12) (bρ 12) (4523929582263 / 8000000000000 : ℝ) ≤
+      (-(179519835636495302763 / 125000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4756792457721397 / 20000000000000000 : ℝ) ≤
+      (-(179519835636495302763 / 125000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4756792457721397 / 20000000000000000 : ℝ) =
+            (4756792457721397 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4756792457721397 / 2500000000000000 : ℝ) ≤
+            (80410356976004697237 / 125000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1680171709222647 / 10000000000000000 : ℝ)) (q :=
+      (4756792457721397 / 20000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_514_13 :
+    Uω (aρ 13) (bρ 13) (4523929582263 / 8000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_514_14 :
+    Uω (aρ 14) (bρ 14) (4523929582263 / 8000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_514_15 :
+    Uω (aρ 15) (bρ 15) (4523929582263 / 8000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_514_16 :
+    Uω (aρ 16) (bρ 16) (4523929582263 / 8000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_514 :
+    Uρ (4523929582263 / 8000000000000 : ℝ) ≤
+      (-(9177451047149676338567 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (4523929582263 / 8000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(290768392272368305341 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_514_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (4523929582263 / 8000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(234330663573394913113 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_514_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (4523929582263 / 8000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(2972323146321819319241 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_514_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (4523929582263 / 8000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3044929880962231886749 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_514_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (4523929582263 / 8000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(6315332909008076166521 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_514_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (4523929582263 / 8000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(6648327348947931113759 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_514_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (4523929582263 / 8000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(7122425967148344153303 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_514_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (4523929582263 / 8000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(3889813308929943655439 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_514_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (4523929582263 / 8000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(8676150438646844351813 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_514_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (4523929582263 / 8000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) * (-(395937255363209419699 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_514_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (4523929582263 / 8000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(5807911168319489409469 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_514_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (4523929582263 / 8000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(179519835636495302763 / 125000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_514_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (4523929582263 / 8000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_514_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (4523929582263 / 8000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_514_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (4523929582263 / 8000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_514_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (4523929582263 / 8000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_514_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_515_1 :
+    Uω (aρ 1) (bρ 1) (18175605211087 / 32000000000000 : ℝ) ≤
+      (-(2885405315160330269137 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4492280496027439 / 8000000000000000 : ℝ) ≤
+      (-(2885405315160330269137 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (4492280496027439 / 8000000000000000 : ℝ) =
+            (4492280496027439 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4492280496027439 / 4000000000000000 : ℝ) ≤
+            (580330586339669730863 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (561532182968891 / 1000000000000000 : ℝ)) (q :=
+      (4492280496027439 / 8000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_515_2 :
+    Uω (aρ 2) (bρ 2) (18175605211087 / 32000000000000 : ℝ) ≤
+      (-(1162703390940672187211 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22365682623738931 / 40000000000000000 : ℝ) ≤
+      (-(1162703390940672187211 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (22365682623738931 / 40000000000000000 : ℝ) =
+            (22365682623738931 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22365682623738931 / 20000000000000000 : ℝ) ≤
+            (223590969659327812789 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2795615456999889 / 5000000000000000 : ℝ)) (q :=
+      (22365682623738931 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_515_3 :
+    Uω (aρ 3) (bρ 3) (18175605211087 / 32000000000000 : ℝ) ≤
+      (-(5899505401803822401993 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4434837617567573 / 8000000000000000 : ℝ) ≤
+      (-(5899505401803822401993 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (4434837617567573 / 8000000000000000 : ℝ) =
+            (4434837617567573 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4434837617567573 / 4000000000000000 : ℝ) ≤
+            (1031966401196177598007 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1108575950479849 / 2000000000000000 : ℝ)) (q :=
+      (4434837617567573 / 8000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_515_4 :
+    Uω (aρ 4) (bρ 4) (18175605211087 / 32000000000000 : ℝ) ≤
+      (-(6044049436994200835527 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21855978729012229 / 40000000000000000 : ℝ) ≤
+      (-(6044049436994200835527 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (21855978729012229 / 40000000000000000 : ℝ) =
+            (21855978729012229 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21855978729012229 / 20000000000000000 : ℝ) ≤
+            (887422366005799164473 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5462069254051427 / 10000000000000000 : ℝ)) (q :=
+      (21855978729012229 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_515_5 :
+    Uω (aρ 5) (bρ 5) (18175605211087 / 32000000000000 : ℝ) ≤
+      (-(6268453107162976316631 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4274196923986051 / 8000000000000000 : ℝ) ≤
+      (-(6268453107162976316631 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (4274196923986051 / 8000000000000000 : ℝ) =
+            (4274196923986051 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4274196923986051 / 4000000000000000 : ℝ) ≤
+            (663018695837023683369 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1042531663837 / 1953125000000 : ℝ)) (q :=
+      (4274196923986051 / 8000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_515_6 :
+    Uω (aρ 6) (bρ 6) (18175605211087 / 32000000000000 : ℝ) ≤
+      (-(6599797605804451610419 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4134894361873683 / 8000000000000000 : ℝ) ≤
+      (-(6599797605804451610419 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (4134894361873683 / 8000000000000000 : ℝ) =
+            (4134894361873683 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4134894361873683 / 4000000000000000 : ℝ) ≤
+            (331674197195548389581 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (128918652217613 / 250000000000000 : ℝ)) (q :=
+      (4134894361873683 / 8000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_515_7 :
+    Uω (aρ 7) (bρ 7) (18175605211087 / 32000000000000 : ℝ) ≤
+      (-(7071390210168205466577 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19722111797769177 / 40000000000000000 : ℝ) ≤
+      (-(7071390210168205466577 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (19722111797769177 / 40000000000000000 : ℝ) =
+            (19722111797769177 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19722111797769177 / 10000000000000000 : ℝ) ≤
+            (6791553395831794533423 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4904227463664901 / 10000000000000000 : ℝ)) (q :=
+      (19722111797769177 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_515_8 :
+    Uω (aρ 8) (bρ 8) (18175605211087 / 32000000000000 : ℝ) ≤
+      (-(1544954905579950569431 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (18474695792596697 / 40000000000000000 : ℝ) ≤
+      (-(1544954905579950569431 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (18474695792596697 / 40000000000000000 : ℝ) =
+            (18474695792596697 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18474695792596697 / 10000000000000000 : ℝ) ≤
+            (1227633815620049430569 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4564061112418661 / 10000000000000000 : ℝ)) (q :=
+      (18474695792596697 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_515_9 :
+    Uω (aρ 9) (bρ 9) (18175605211087 / 32000000000000 : ℝ) ≤
+      (-(1723066465949345500237 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16900550926846531 / 40000000000000000 : ℝ) ≤
+      (-(1723066465949345500237 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (16900550926846531 / 40000000000000000 : ℝ) =
+            (16900550926846531 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16900550926846531 / 10000000000000000 : ℝ) ≤
+            (1049522255250654499763 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2058977153951789 / 5000000000000000 : ℝ)) (q :=
+      (16900550926846531 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_515_10 :
+    Uω (aρ 10) (bρ 10) (18175605211087 / 32000000000000 : ℝ) ≤
+      (-(2456916459404556892093 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14970968162174133 / 40000000000000000 : ℝ) ≤
+      (-(2456916459404556892093 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (14970968162174133 / 40000000000000000 : ℝ) =
+            (14970968162174133 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14970968162174133 / 10000000000000000 : ℝ) ≤
+            (1008819442095443107907 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3541721959062379 / 10000000000000000 : ℝ)) (q :=
+      (14970968162174133 / 40000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_515_11 :
+    Uω (aρ 11) (bρ 11) (18175605211087 / 32000000000000 : ℝ) ≤
+      (-(11525868421030432666923 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12632749536116209 / 40000000000000000 : ℝ) ≤
+      (-(11525868421030432666923 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (12632749536116209 / 40000000000000000 : ℝ) =
+            (12632749536116209 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12632749536116209 / 10000000000000000 : ℝ) ≤
+            (2337075184969567333077 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2791089272238417 / 10000000000000000 : ℝ)) (q :=
+      (12632749536116209 / 40000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_515_12 :
+    Uω (aρ 12) (bρ 12) (18175605211087 / 32000000000000 : ℝ) ≤
+      (-(14214976197084333418893 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9654091678558937 / 40000000000000000 : ℝ) ≤
+      (-(14214976197084333418893 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (9654091678558937 / 40000000000000000 : ℝ) =
+            (9654091678558937 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9654091678558937 / 5000000000000000 : ℝ) ≤
+            (6579439211915666581107 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1725460440144781 / 10000000000000000 : ℝ)) (q :=
+      (9654091678558937 / 40000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_515_13 :
+    Uω (aρ 13) (bρ 13) (18175605211087 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_515_14 :
+    Uω (aρ 14) (bρ 14) (18175605211087 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_515_15 :
+    Uω (aρ 15) (bρ 15) (18175605211087 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_515_16 :
+    Uω (aρ 16) (bρ 16) (18175605211087 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_515 :
+    Uρ (18175605211087 / 32000000000000 : ℝ) ≤
+      (-(9125984834938814482263 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (18175605211087 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2885405315160330269137 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_515_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (18175605211087 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1162703390940672187211 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_515_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (18175605211087 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(5899505401803822401993 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_515_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (18175605211087 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(6044049436994200835527 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_515_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (18175605211087 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(6268453107162976316631 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_515_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (18175605211087 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(6599797605804451610419 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_515_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (18175605211087 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(7071390210168205466577 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_515_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (18175605211087 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(1544954905579950569431 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_515_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (18175605211087 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(1723066465949345500237 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_515_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (18175605211087 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(2456916459404556892093 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_515_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (18175605211087 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(11525868421030432666923 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_515_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (18175605211087 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(14214976197084333418893 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_515_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (18175605211087 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_515_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (18175605211087 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_515_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (18175605211087 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_515_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (18175605211087 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_515_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_516_1 :
+    Uω (aρ 1) (bρ 1) (9127746046561 / 16000000000000 : ℝ) ≤
+      (-(572645107138737305937 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5640315398100093 / 10000000000000000 : ℝ) ≤
+      (-(572645107138737305937 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5640315398100093 / 10000000000000000 : ℝ) =
+            (5640315398100093 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5640315398100093 / 5000000000000000 : ℝ) ≤
+            (120502073161262694063 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5640286735184561 / 10000000000000000 : ℝ)) (q :=
+      (5640315398100093 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_516_2 :
+    Uω (aρ 2) (bρ 2) (9127746046561 / 16000000000000 : ℝ) ≤
+      (-(5768966694719913615483 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11232772299991577 / 20000000000000000 : ℝ) ≤
+      (-(5768966694719913615483 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11232772299991577 / 20000000000000000 : ℝ) =
+            (11232772299991577 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11232772299991577 / 10000000000000000 : ℝ) ≤
+            (1162505108280086384517 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (21938270513617 / 39062500000000 : ℝ)) (q :=
+      (11232772299991577 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_516_3 :
+    Uω (aρ 3) (bρ 3) (9127746046561 / 16000000000000 : ℝ) ≤
+      (-(1170913482845522262301 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5568514664443109 / 10000000000000000 : ℝ) ≤
+      (-(1170913482845522262301 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5568514664443109 / 10000000000000000 : ℝ) =
+            (5568514664443109 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5568514664443109 / 5000000000000000 : ℝ) ≤
+            (215380877754477737699 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5567850386730593 / 10000000000000000 : ℝ)) (q :=
+      (5568514664443109 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_516_4 :
+    Uω (aρ 4) (bρ 4) (9127746046561 / 16000000000000 : ℝ) ≤
+      (-(5998448160638532935769 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10977936186196661 / 20000000000000000 : ℝ) ≤
+      (-(5998448160638532935769 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10977936186196661 / 20000000000000000 : ℝ) =
+            (10977936186196661 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10977936186196661 / 10000000000000000 : ℝ) ≤
+            (933023642361467064231 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1371762856276509 / 2500000000000000 : ℝ)) (q :=
+      (10977936186196661 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_516_5 :
+    Uω (aρ 5) (bρ 5) (9127746046561 / 16000000000000 : ℝ) ≤
+      (-(6221792458419681924697 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10735468014564247 / 20000000000000000 : ℝ) ≤
+      (-(6221792458419681924697 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10735468014564247 / 20000000000000000 : ℝ) =
+            (10735468014564247 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10735468014564247 / 10000000000000000 : ℝ) ≤
+            (709679344580318075303 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2681386586404311 / 5000000000000000 : ℝ)) (q :=
+      (10735468014564247 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_516_6 :
+    Uω (aρ 6) (bρ 6) (9127746046561 / 16000000000000 : ℝ) ≤
+      (-(6551503313245521501489 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5193639799080183 / 10000000000000000 : ℝ) ≤
+      (-(6551503313245521501489 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5193639799080183 / 10000000000000000 : ℝ) =
+            (5193639799080183 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5193639799080183 / 5000000000000000 : ℝ) ≤
+            (379968489754478498511 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5181825131544741 / 10000000000000000 : ℝ)) (q :=
+      (5193639799080183 / 10000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_516_7 :
+    Uω (aρ 7) (bρ 7) (9127746046561 / 16000000000000 : ℝ) ≤
+      (-(1755154090330776418639 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4955625799329087 / 10000000000000000 : ℝ) ≤
+      (-(1755154090330776418639 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4955625799329087 / 10000000000000000 : ℝ) =
+            (4955625799329087 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4955625799329087 / 2500000000000000 : ℝ) ≤
+            (1710581811169223581361 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4929458512802549 / 10000000000000000 : ℝ)) (q :=
+      (4955625799329087 / 10000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_516_8 :
+    Uω (aρ 8) (bρ 8) (9127746046561 / 16000000000000 : ℝ) ≤
+      (-(7670228793910646247569 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9287871355101189 / 20000000000000000 : ℝ) ≤
+      (-(7670228793910646247569 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9287871355101189 / 20000000000000000 : ℝ) =
+            (9287871355101189 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9287871355101189 / 5000000000000000 : ℝ) ≤
+            (6192714812089353752431 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1147404980146391 / 2500000000000000 : ℝ)) (q :=
+      (9287871355101189 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_516_9 :
+    Uω (aρ 9) (bρ 9) (9127746046561 / 16000000000000 : ℝ) ≤
+      (-(34219603551607253439 / 40000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8501496308041777 / 20000000000000000 : ℝ) ≤
+      (-(34219603551607253439 / 40000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8501496308041777 / 20000000000000000 : ℝ) =
+            (8501496308041777 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8501496308041777 / 5000000000000000 : ℝ) ≤
+            (21232170872392746561 / 40000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (518026312735769 / 1250000000000000 : ℝ)) (q :=
+      (8501496308041777 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_516_10 :
+    Uω (aρ 10) (bρ 10) (9127746046561 / 16000000000000 : ℝ) ≤
+      (-(9757453559173009449761 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7538226311282421 / 20000000000000000 : ℝ) ≤
+      (-(9757453559173009449761 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7538226311282421 / 20000000000000000 : ℝ) =
+            (7538226311282421 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7538226311282421 / 5000000000000000 : ℝ) ≤
+            (4105490046826990550239 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (892374884655449 / 2500000000000000 : ℝ)) (q :=
+      (7538226311282421 / 20000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_516_11 :
+    Uω (aρ 11) (bρ 11) (9127746046561 / 16000000000000 : ℝ) ≤
+      (-(5718462538789883425289 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6372805301094043 / 20000000000000000 : ℝ) ≤
+      (-(5718462538789883425289 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6372805301094043 / 20000000000000000 : ℝ) =
+            (6372805301094043 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6372805301094043 / 5000000000000000 : ℝ) ≤
+            (1213009264210116574711 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1411277577319209 / 5000000000000000 : ℝ)) (q :=
+      (6372805301094043 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_516_12 :
+    Uω (aρ 12) (bρ 12) (9127746046561 / 16000000000000 : ℝ) ≤
+      (-(7036065505652730763057 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4896492691807321 / 20000000000000000 : ℝ) ≤
+      (-(7036065505652730763057 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4896492691807321 / 20000000000000000 : ℝ) =
+            (4896492691807321 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4896492691807321 / 2500000000000000 : ℝ) ≤
+            (3361142198847269236943 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (221242830254587 / 1250000000000000 : ℝ)) (q :=
+      (4896492691807321 / 20000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_516_13 :
+    Uω (aρ 13) (bρ 13) (9127746046561 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_516_14 :
+    Uω (aρ 14) (bρ 14) (9127746046561 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_516_15 :
+    Uω (aρ 15) (bρ 15) (9127746046561 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_516_16 :
+    Uω (aρ 16) (bρ 16) (9127746046561 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_516 :
+    Uρ (9127746046561 / 16000000000000 : ℝ) ≤
+      (-(453752827101751907257 / 500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (9127746046561 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(572645107138737305937 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_516_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (9127746046561 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(5768966694719913615483 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_516_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (9127746046561 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1170913482845522262301 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_516_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (9127746046561 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(5998448160638532935769 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_516_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (9127746046561 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(6221792458419681924697 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_516_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (9127746046561 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(6551503313245521501489 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_516_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (9127746046561 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(1755154090330776418639 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_516_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (9127746046561 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(7670228793910646247569 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_516_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (9127746046561 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(34219603551607253439 / 40000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_516_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (9127746046561 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(9757453559173009449761 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_516_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (9127746046561 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(5718462538789883425289 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_516_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (9127746046561 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(7036065505652730763057 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_516_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (9127746046561 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_516_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (9127746046561 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_516_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (9127746046561 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_516_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (9127746046561 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_516_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_517_1 :
+    Uω (aρ 1) (bρ 1) (18335378975157 / 32000000000000 : ℝ) ≤
+      (-(568228742276248743139 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22661120700171239 / 40000000000000000 : ℝ) ≤
+      (-(568228742276248743139 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (22661120700171239 / 40000000000000000 : ℝ) =
+            (22661120700171239 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22661120700171239 / 20000000000000000 : ℝ) ≤
+            (124918438023751256861 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5665251638434057 / 10000000000000000 : ℝ)) (q :=
+      (22661120700171239 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_517_2 :
+    Uω (aρ 2) (bρ 2) (18335378975157 / 32000000000000 : ℝ) ≤
+      (-(2862307020290630033483 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22565406546366123 / 40000000000000000 : ℝ) ≤
+      (-(2862307020290630033483 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (22565406546366123 / 40000000000000000 : ℝ) =
+            (22565406546366123 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22565406546366123 / 20000000000000000 : ℝ) ≤
+            (603428881209369966517 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5641163574041499 / 10000000000000000 : ℝ)) (q :=
+      (22565406546366123 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_517_3 :
+    Uω (aρ 3) (bρ 3) (18335378975157 / 32000000000000 : ℝ) ≤
+      (-(1161966102717595133409 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22373929120856623 / 40000000000000000 : ℝ) ≤
+      (-(1161966102717595133409 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (22373929120856623 / 40000000000000000 : ℝ) =
+            (22373929120856623 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22373929120856623 / 20000000000000000 : ℝ) ≤
+            (224328257882404866591 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5592820967636749 / 10000000000000000 : ℝ)) (q :=
+      (22373929120856623 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_517_4 :
+    Uω (aρ 4) (bρ 4) (18335378975157 / 32000000000000 : ℝ) ≤
+      (-(5953054032277824673143 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4411153139650971 / 8000000000000000 : ℝ) ≤
+      (-(5953054032277824673143 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (4411153139650971 / 8000000000000000 : ℝ) =
+            (4411153139650971 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4411153139650971 / 4000000000000000 : ℝ) ≤
+            (978417770722175326857 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1102406687480173 / 2000000000000000 : ℝ)) (q :=
+      (4411153139650971 / 8000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_517_5 :
+    Uω (aρ 5) (bρ 5) (18335378975157 / 32000000000000 : ℝ) ≤
+      (-(3087674459787515912569 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (862835463099039 / 1600000000000000 : ℝ) ≤
+      (-(3087674459787515912569 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (862835463099039 / 1600000000000000 : ℝ) =
+            (862835463099039 / 800000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (862835463099039 / 800000000000000 : ℝ) ≤
+            (378061441712484087431 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (215511351853857 / 400000000000000 : ℝ)) (q :=
+      (862835463099039 / 1600000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_517_6 :
+    Uω (aρ 6) (bρ 6) (18335378975157 / 32000000000000 : ℝ) ≤
+      (-(1300688437478143619327 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (834985775376793 / 1600000000000000 : ℝ) ≤
+      (-(1300688437478143619327 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (834985775376793 / 1600000000000000 : ℝ) =
+            (834985775376793 / 800000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (834985775376793 / 800000000000000 : ℝ) ≤
+            (85605923121856380673 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (104138061499167 / 200000000000000 : ℝ)) (q :=
+      (834985775376793 / 1600000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_517_7 :
+    Uω (aρ 7) (bρ 7) (18335378975157 / 32000000000000 : ℝ) ≤
+      (-(6970101718382483444987 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19922889199123587 / 40000000000000000 : ℝ) ≤
+      (-(6970101718382483444987 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (19922889199123587 / 40000000000000000 : ℝ) =
+            (19922889199123587 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19922889199123587 / 10000000000000000 : ℝ) ≤
+            (6892841887617516555013 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4954686863070231 / 10000000000000000 : ℝ)) (q :=
+      (19922889199123587 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_517_8 :
+    Uω (aρ 8) (bρ 8) (18335378975157 / 32000000000000 : ℝ) ≤
+      (-(304639437420347823563 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (18676776619031389 / 40000000000000000 : ℝ) ≤
+      (-(304639437420347823563 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (18676776619031389 / 40000000000000000 : ℝ) =
+            (18676776619031389 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18676776619031389 / 10000000000000000 : ℝ) ≤
+            (249878306819652176437 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1153793056091033 / 2500000000000000 : ℝ)) (q :=
+      (18676776619031389 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_517_9 :
+    Uω (aρ 9) (bρ 9) (18335378975157 / 32000000000000 : ℝ) ≤
+      (-(1698970199231094459177 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17105402580301911 / 40000000000000000 : ℝ) ≤
+      (-(1698970199231094459177 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (17105402580301911 / 40000000000000000 : ℝ) =
+            (17105402580301911 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17105402580301911 / 10000000000000000 : ℝ) ≤
+            (1073618521968905540823 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4170450833359393 / 10000000000000000 : ℝ)) (q :=
+      (17105402580301911 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_517_10 :
+    Uω (aρ 10) (bρ 10) (18335378975157 / 32000000000000 : ℝ) ≤
+      (-(4843892576607819671637 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15181854597792241 / 40000000000000000 : ℝ) ≤
+      (-(4843892576607819671637 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (15181854597792241 / 40000000000000000 : ℝ) =
+            (15181854597792241 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15181854597792241 / 10000000000000000 : ℝ) ≤
+            (2087579226392180328363 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1798617937799779 / 5000000000000000 : ℝ)) (q :=
+      (15181854597792241 / 40000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_517_11 :
+    Uω (aρ 11) (bρ 11) (18335378975157 / 32000000000000 : ℝ) ≤
+      (-(2269793160077358042791 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12858214573984021 / 40000000000000000 : ℝ) ≤
+      (-(2269793160077358042791 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (12858214573984021 / 40000000000000000 : ℝ) =
+            (12858214573984021 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12858214573984021 / 10000000000000000 : ℝ) ≤
+            (502795561122641957209 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (89184140309389 / 312500000000000 : ℝ)) (q :=
+      (12858214573984021 / 40000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_517_12 :
+    Uω (aρ 12) (bρ 12) (18335378975157 / 32000000000000 : ℝ) ≤
+      (-(13932802330277890281229 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9930384717987227 / 40000000000000000 : ℝ) ≤
+      (-(13932802330277890281229 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (9930384717987227 / 40000000000000000 : ℝ) =
+            (9930384717987227 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9930384717987227 / 5000000000000000 : ℝ) ≤
+            (6861613078722109718771 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1813677658587051 / 10000000000000000 : ℝ)) (q :=
+      (9930384717987227 / 40000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_517_13 :
+    Uω (aρ 13) (bρ 13) (18335378975157 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_517_14 :
+    Uω (aρ 14) (bρ 14) (18335378975157 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_517_15 :
+    Uω (aρ 15) (bρ 15) (18335378975157 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_517_16 :
+    Uω (aρ 16) (bρ 16) (18335378975157 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_517 :
+    Uρ (18335378975157 / 32000000000000 : ℝ) ≤
+      (-(451232209478411826957 / 500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (18335378975157 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(568228742276248743139 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_517_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (18335378975157 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(2862307020290630033483 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_517_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (18335378975157 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1161966102717595133409 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_517_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (18335378975157 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(5953054032277824673143 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_517_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (18335378975157 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(3087674459787515912569 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_517_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (18335378975157 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1300688437478143619327 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_517_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (18335378975157 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(6970101718382483444987 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_517_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (18335378975157 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(304639437420347823563 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_517_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (18335378975157 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(1698970199231094459177 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_517_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (18335378975157 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(4843892576607819671637 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_517_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (18335378975157 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(2269793160077358042791 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_517_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (18335378975157 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(13932802330277890281229 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_517_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (18335378975157 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_517_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (18335378975157 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_517_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (18335378975157 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_517_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (18335378975157 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_517_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_518_1 :
+    Uω (aρ 1) (bρ 1) (2301908232149 / 4000000000000 : ℝ) ≤
+      (-(704789745198935614057 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11380489901754463 / 20000000000000000 : ℝ) ≤
+      (-(704789745198935614057 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11380489901754463 / 20000000000000000 : ℝ) =
+            (11380489901754463 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11380489901754463 / 10000000000000000 : ℝ) ≤
+            (161644230176064385943 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5690216539466963 / 10000000000000000 : ℝ)) (q :=
+      (11380489901754463 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_518_2 :
+    Uω (aρ 2) (bρ 2) (2301908232149 / 4000000000000 : ℝ) ≤
+      (-(568045724692035134871 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5666317115820643 / 10000000000000000 : ℝ) ≤
+      (-(568045724692035134871 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5666317115820643 / 10000000000000000 : ℝ) =
+            (5666317115820643 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5666317115820643 / 5000000000000000 : ℝ) ≤
+            (125101455607964865129 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2833064940931893 / 5000000000000000 : ℝ)) (q :=
+      (5666317115820643 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_518_3 :
+    Uω (aρ 3) (bρ 3) (2301908232149 / 4000000000000 : ℝ) ≤
+      (-(5765292907843755640569 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11236899739257623 / 20000000000000000 : ℝ) ≤
+      (-(5765292907843755640569 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11236899739257623 / 20000000000000000 : ℝ) =
+            (11236899739257623 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11236899739257623 / 10000000000000000 : ℝ) ≤
+            (1166178895156244359431 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5617791495830123 / 10000000000000000 : ℝ)) (q :=
+      (11236899739257623 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_518_4 :
+    Uω (aρ 4) (bρ 4) (2301908232149 / 4000000000000 : ℝ) ≤
+      (-(184620786785993711109 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11077829355447301 / 20000000000000000 : ℝ) ≤
+      (-(184620786785993711109 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11077829355447301 / 20000000000000000 : ℝ) =
+            (11077829355447301 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11077829355447301 / 10000000000000000 : ℝ) ≤
+            (31987707057756288891 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5537015293084801 / 10000000000000000 : ℝ)) (q :=
+      (11077829355447301 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_518_5 :
+    Uω (aρ 5) (bρ 5) (2301908232149 / 4000000000000 : ℝ) ≤
+      (-(1225824095183702422711 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10835418138452883 / 20000000000000000 : ℝ) ≤
+      (-(1225824095183702422711 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10835418138452883 / 20000000000000000 : ℝ) =
+            (10835418138452883 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10835418138452883 / 10000000000000000 : ℝ) ≤
+            (160470265416297577289 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5412793995425383 / 10000000000000000 : ℝ)) (q :=
+      (10835418138452883 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_518_6 :
+    Uω (aρ 6) (bρ 6) (2301908232149 / 4000000000000 : ℝ) ≤
+      (-(403475748598469510607 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5243681851321087 / 10000000000000000 : ℝ) ≤
+      (-(403475748598469510607 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5243681851321087 / 10000000000000000 : ℝ) =
+            (5243681851321087 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5243681851321087 / 5000000000000000 : ℝ) ≤
+            (29741239089030489393 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2615989967377337 / 5000000000000000 : ℝ)) (q :=
+      (5243681851321087 / 10000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_518_7 :
+    Uω (aρ 7) (bρ 7) (2301908232149 / 4000000000000 : ℝ) ≤
+      (-(6919843623735854358829 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (156431795978329 / 312500000000000 : ℝ) ≤
+      (-(6919843623735854358829 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (156431795978329 / 312500000000000 : ℝ) =
+            (156431795978329 / 156250000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 2
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (156431795978329 / 156250000000000 : ℝ) ≤
+            (11628179264145641171 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1244978138871389 / 2500000000000000 : ℝ)) (q :=
+      (156431795978329 / 312500000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_518_8 :
+    Uω (aρ 8) (bρ 8) (2301908232149 / 4000000000000 : ℝ) ≤
+      (-(7562042532521178980667 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4694449433491747 / 10000000000000000 : ℝ) ≤
+      (-(7562042532521178980667 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4694449433491747 / 10000000000000000 : ℝ) =
+            (4694449433491747 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4694449433491747 / 2500000000000000 : ℝ) ≤
+            (6300901073478821019333 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2320359065597997 / 5000000000000000 : ℝ)) (q :=
+      (4694449433491747 / 10000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_518_9 :
+    Uω (aρ 9) (bρ 9) (2301908232149 / 4000000000000 : ℝ) ≤
+      (-(843517764254758507903 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (860389070729883 / 2000000000000000 : ℝ) ≤
+      (-(843517764254758507903 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (860389070729883 / 2000000000000000 : ℝ) =
+            (860389070729883 / 500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (860389070729883 / 500000000000000 : ℝ) ≤
+            (542776596345241492097 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (419667559987133 / 1000000000000000 : ℝ)) (q :=
+      (860389070729883 / 2000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_518_10 :
+    Uω (aρ 10) (bρ 10) (2301908232149 / 4000000000000 : ℝ) ≤
+      (-(9618651478160744289729 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3821793995318473 / 10000000000000000 : ℝ) ≤
+      (-(9618651478160744289729 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3821793995318473 / 10000000000000000 : ℝ) =
+            (3821793995318473 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3821793995318473 / 2500000000000000 : ℝ) ≤
+            (4244292127839255710271 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1812465958352223 / 5000000000000000 : ℝ)) (q :=
+      (3821793995318473 / 10000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_518_11 :
+    Uω (aρ 11) (bρ 11) (2301908232149 / 4000000000000 : ℝ) ≤
+      (-(2252393043687111150489 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6485284914501151 / 20000000000000000 : ℝ) ≤
+      (-(2252393043687111150489 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6485284914501151 / 20000000000000000 : ℝ) =
+            (6485284914501151 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6485284914501151 / 5000000000000000 : ℝ) ≤
+            (520195677512888849511 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2885105466773651 / 10000000000000000 : ℝ)) (q :=
+      (6485284914501151 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_518_12 :
+    Uω (aρ 12) (bρ 12) (2301908232149 / 4000000000000 : ℝ) ≤
+      (-(6898383884473223033613 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2516598820731029 / 10000000000000000 : ℝ) ≤
+      (-(6898383884473223033613 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2516598820731029 / 10000000000000000 : ℝ) =
+            (2516598820731029 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2516598820731029 / 2500000000000000 : ℝ) ≤
+            (33087918526776966387 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (928359145209779 / 5000000000000000 : ℝ)) (q :=
+      (2516598820731029 / 10000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_518_13 :
+    Uω (aρ 13) (bρ 13) (2301908232149 / 4000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_518_14 :
+    Uω (aρ 14) (bρ 14) (2301908232149 / 4000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_518_15 :
+    Uω (aρ 15) (bρ 15) (2301908232149 / 4000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_518_16 :
+    Uω (aρ 16) (bρ 16) (2301908232149 / 4000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_518 :
+    Uρ (2301908232149 / 4000000000000 : ℝ) ≤
+      (-(8974727806232595687023 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (2301908232149 / 4000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(704789745198935614057 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_518_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (2301908232149 / 4000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(568045724692035134871 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_518_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (2301908232149 / 4000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(5765292907843755640569 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_518_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (2301908232149 / 4000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(184620786785993711109 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_518_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (2301908232149 / 4000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1225824095183702422711 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_518_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (2301908232149 / 4000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(403475748598469510607 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_518_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (2301908232149 / 4000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(6919843623735854358829 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_518_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (2301908232149 / 4000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(7562042532521178980667 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_518_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (2301908232149 / 4000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(843517764254758507903 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_518_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (2301908232149 / 4000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(9618651478160744289729 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_518_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (2301908232149 / 4000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(2252393043687111150489 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_518_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (2301908232149 / 4000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(6898383884473223033613 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_518_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (2301908232149 / 4000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_518_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (2301908232149 / 4000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_518_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (2301908232149 / 4000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_518_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (2301908232149 / 4000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_518_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_519_1 :
+    Uω (aρ 1) (bρ 1) (18495152739227 / 32000000000000 : ℝ) ≤
+      (-(2797270493823523605897 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22860838902471523 / 40000000000000000 : ℝ) ≤
+      (-(2797270493823523605897 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (22860838902471523 / 40000000000000000 : ℝ) =
+            (22860838902471523 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22860838902471523 / 20000000000000000 : ℝ) ≤
+            (668465407676476394103 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1428795359578081 / 2500000000000000 : ℝ)) (q :=
+      (22860838902471523 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_519_2 :
+    Uω (aρ 2) (bρ 2) (18495152739227 / 32000000000000 : ℝ) ≤
+      (-(5636494591394130888957 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22765130351120297 / 40000000000000000 : ℝ) ≤
+      (-(5636494591394130888957 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (22765130351120297 / 40000000000000000 : ℝ) =
+            (22765130351120297 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22765130351120297 / 20000000000000000 : ℝ) ≤
+            (1294977211605869111043 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5691096175146711 / 10000000000000000 : ℝ)) (q :=
+      (22765130351120297 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_519_3 :
+    Uω (aρ 3) (bρ 3) (18495152739227 / 32000000000000 : ℝ) ≤
+      (-(178779775900266116243 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4514733946429581 / 8000000000000000 : ℝ) ≤
+      (-(178779775900266116243 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (4514733946429581 / 8000000000000000 : ℝ) =
+            (4514733946429581 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4514733946429581 / 4000000000000000 : ℝ) ≤
+            (37828717943483883757 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1128552394402103 / 2000000000000000 : ℝ)) (q :=
+      (4514733946429581 / 8000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_519_4 :
+    Uω (aρ 4) (bρ 4) (18495152739227 / 32000000000000 : ℝ) ≤
+      (-(5862879745853656565577 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22255551414533091 / 40000000000000000 : ℝ) ≤
+      (-(5862879745853656565577 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (22255551414533091 / 40000000000000000 : ℝ) =
+            (22255551414533091 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22255551414533091 / 20000000000000000 : ℝ) ≤
+            (1068592057146343434423 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1390499248567027 / 2500000000000000 : ℝ)) (q :=
+      (22255551414533091 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_519_5 :
+    Uω (aρ 5) (bρ 5) (18495152739227 / 32000000000000 : ℝ) ≤
+      (-(6083105140703991195419 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21770785139131201 / 40000000000000000 : ℝ) ≤
+      (-(6083105140703991195419 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (21770785139131201 / 40000000000000000 : ℝ) =
+            (21770785139131201 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21770785139131201 / 20000000000000000 : ℝ) ≤
+            (848366662296008804581 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5437803775902163 / 10000000000000000 : ℝ)) (q :=
+      (21770785139131201 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_519_6 :
+    Uω (aρ 6) (bρ 6) (18495152739227 / 32000000000000 : ℝ) ≤
+      (-(3204005232853766005531 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21074808289928297 / 40000000000000000 : ℝ) ≤
+      (-(3204005232853766005531 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (21074808289928297 / 40000000000000000 : ℝ) =
+            (21074808289928297 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21074808289928297 / 20000000000000000 : ℝ) ≤
+            (261730668646233994469 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5257055726440711 / 10000000000000000 : ℝ)) (q :=
+      (21074808289928297 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_519_7 :
+    Uω (aρ 7) (bρ 7) (18495152739227 / 32000000000000 : ℝ) ≤
+      (-(214682482910002087813 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20123645335996769 / 40000000000000000 : ℝ) ≤
+      (-(214682482910002087813 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (20123645335996769 / 40000000000000000 : ℝ) =
+            (20123645335996769 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20123645335996769 / 20000000000000000 : ℝ) ≤
+            (1926010933747912187 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5005135630234947 / 10000000000000000 : ℝ)) (q :=
+      (20123645335996769 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_519_8 :
+    Uω (aρ 8) (bρ 8) (18495152739227 / 32000000000000 : ℝ) ≤
+      (-(3754197611789808260037 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (18878806265133627 / 40000000000000000 : ℝ) ≤
+      (-(3754197611789808260037 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (18878806265133627 / 40000000000000000 : ℝ) =
+            (18878806265133627 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18878806265133627 / 10000000000000000 : ℝ) ≤
+            (3177274191210191739963 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (291641109133961 / 625000000000000 : ℝ)) (q :=
+      (18878806265133627 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_519_9 :
+    Uω (aρ 9) (bρ 9) (18495152739227 / 32000000000000 : ℝ) ≤
+      (-(1046984489645930925899 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3462025939787131 / 8000000000000000 : ℝ) ≤
+      (-(1046984489645930925899 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3462025939787131 / 8000000000000000 : ℝ) =
+            (3462025939787131 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3462025939787131 / 2000000000000000 : ℝ) ≤
+            (685883461104069074101 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (422288509140439 / 1000000000000000 : ℝ)) (q :=
+      (3462025939787131 / 8000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_519_10 :
+    Uω (aρ 10) (bρ 10) (18495152739227 / 32000000000000 : ℝ) ≤
+      (-(4775021818429599099321 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15392418606289839 / 40000000000000000 : ℝ) ≤
+      (-(4775021818429599099321 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (15392418606289839 / 40000000000000000 : ℝ) =
+            (15392418606289839 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15392418606289839 / 10000000000000000 : ℝ) ≤
+            (2156449984570400900679 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1826294289288241 / 5000000000000000 : ℝ)) (q :=
+      (15392418606289839 / 40000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_519_11 :
+    Uω (aρ 11) (bρ 11) (18495152739227 / 32000000000000 : ℝ) ≤
+      (-(11175899028160388580739 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13082684353527947 / 40000000000000000 : ℝ) ≤
+      (-(11175899028160388580739 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (13082684353527947 / 40000000000000000 : ℝ) =
+            (13082684353527947 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13082684353527947 / 10000000000000000 : ℝ) ≤
+            (2687044577839611419261 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (364524759800067 / 1250000000000000 : ℝ)) (q :=
+      (13082684353527947 / 40000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_519_12 :
+    Uω (aρ 12) (bρ 12) (18495152739227 / 32000000000000 : ℝ) ≤
+      (-(13663827683608571830467 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10201111502974553 / 40000000000000000 : ℝ) ≤
+      (-(13663827683608571830467 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (10201111502974553 / 40000000000000000 : ℝ) =
+            (10201111502974553 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10201111502974553 / 10000000000000000 : ℝ) ≤
+            (199115922391428169533 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1899111749808839 / 10000000000000000 : ℝ)) (q :=
+      (10201111502974553 / 40000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_519_13 :
+    Uω (aρ 13) (bρ 13) (18495152739227 / 32000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_519_14 :
+    Uω (aρ 14) (bρ 14) (18495152739227 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_519_15 :
+    Uω (aρ 15) (bρ 15) (18495152739227 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_519_16 :
+    Uω (aρ 16) (bρ 16) (18495152739227 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_519 :
+    Uρ (18495152739227 / 32000000000000 : ℝ) ≤
+      (-(1785057830144805199833 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (18495152739227 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2797270493823523605897 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_519_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (18495152739227 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(5636494591394130888957 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_519_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (18495152739227 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(178779775900266116243 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_519_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (18495152739227 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(5862879745853656565577 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_519_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (18495152739227 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(6083105140703991195419 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_519_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (18495152739227 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3204005232853766005531 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_519_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (18495152739227 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(214682482910002087813 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_519_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (18495152739227 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(3754197611789808260037 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_519_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (18495152739227 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(1046984489645930925899 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_519_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (18495152739227 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(4775021818429599099321 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_519_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (18495152739227 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(11175899028160388580739 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_519_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (18495152739227 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(13663827683608571830467 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_519_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (18495152739227 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_519_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (18495152739227 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_519_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (18495152739227 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_519_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (18495152739227 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_519_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/U43.lean b/LeanPool/Zeta5Irrational/Table/U43.lean
new file mode 100644
index 0000000000..d16109217a
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/U43.lean
@@ -0,0 +1,4762 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-! # Certified arcsine potential bounds (U43) -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma U_520_1 :
+    Uω (aρ 1) (bρ 1) (9287519810631 / 16000000000000 : ℝ) ≤
+      (-(2775477411467269153561 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11480348998558059 / 20000000000000000 : ℝ) ≤
+      (-(2775477411467269153561 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11480348998558059 / 20000000000000000 : ℝ) =
+            (11480348998558059 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11480348998558059 / 10000000000000000 : ℝ) ≤
+            (690258490032730846439 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1435036583749671 / 2500000000000000 : ℝ)) (q :=
+      (11480348998558059 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_520_2 :
+    Uω (aρ 2) (bρ 2) (9287519810631 / 16000000000000 : ℝ) ≤
+      (-(5592724374280311294667 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11432496105130163 / 20000000000000000 : ℝ) ≤
+      (-(5592724374280311294667 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11432496105130163 / 20000000000000000 : ℝ) =
+            (11432496105130163 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11432496105130163 / 10000000000000000 : ℝ) ≤
+            (1338747428719688705333 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1429015613520197 / 2500000000000000 : ℝ)) (q :=
+      (11432496105130163 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_520_3 :
+    Uω (aρ 3) (bρ 3) (9287519810631 / 16000000000000 : ℝ) ≤
+      (-(5676808531728931494759 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2834192485391191 / 5000000000000000 : ℝ) ≤
+      (-(5676808531728931494759 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2834192485391191 / 5000000000000000 : ℝ) =
+            (2834192485391191 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2834192485391191 / 2500000000000000 : ℝ) ≤
+            (1254663271271068505241 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5667732396865389 / 10000000000000000 : ℝ)) (q :=
+      (2834192485391191 / 5000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_520_4 :
+    Uω (aρ 4) (bρ 4) (9287519810631 / 16000000000000 : ℝ) ≤
+      (-(5818095913874656755391 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11177721906657673 / 20000000000000000 : ℝ) ≤
+      (-(5818095913874656755391 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11177721906657673 / 20000000000000000 : ℝ) =
+            (11177721906657673 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11177721906657673 / 10000000000000000 : ℝ) ≤
+            (1113375889125343244609 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2793489271511649 / 5000000000000000 : ℝ)) (q :=
+      (11177721906657673 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_520_5 :
+    Uω (aρ 5) (bρ 5) (9287519810631 / 16000000000000 : ℝ) ≤
+      (-(3018650477316833315497 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (683460411739091 / 1250000000000000 : ℝ) ≤
+      (-(3018650477316833315497 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (683460411739091 / 1250000000000000 : ℝ) =
+            (683460411739091 / 625000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (683460411739091 / 625000000000000 : ℝ) ≤
+            (447085424183166684503 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5462813143526081 / 10000000000000000 : ℝ)) (q :=
+      (683460411739091 / 1250000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_520_6 :
+    Uω (aρ 6) (bρ 6) (9287519810631 / 16000000000000 : ℝ) ≤
+      (-(795079433204800478269 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10587443534387717 / 20000000000000000 : ℝ) ≤
+      (-(795079433204800478269 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10587443534387717 / 20000000000000000 : ℝ) =
+            (10587443534387717 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10587443534387717 / 10000000000000000 : ℝ) ≤
+            (71354542170199521731 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2641065232614171 / 5000000000000000 : ℝ)) (q :=
+      (10587443534387717 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_520_7 :
+    Uω (aρ 7) (bρ 7) (9287519810631 / 16000000000000 : ℝ) ≤
+      (-(6820086630363946455629 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10112007815094067 / 20000000000000000 : ℝ) ≤
+      (-(6820086630363946455629 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10112007815094067 / 20000000000000000 : ℝ) =
+            (10112007815094067 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10112007815094067 / 10000000000000000 : ℝ) ≤
+            (111385172636053544371 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1257589031673673 / 2500000000000000 : ℝ)) (q :=
+      (10112007815094067 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_520_8 :
+    Uω (aρ 8) (bρ 8) (9287519810631 / 16000000000000 : ℝ) ≤
+      (-(7455040704754400716783 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9489901209014669 / 20000000000000000 : ℝ) ≤
+      (-(7455040704754400716783 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9489901209014669 / 20000000000000000 : ℝ) =
+            (9489901209014669 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9489901209014669 / 5000000000000000 : ℝ) ≤
+            (6407902901245599283217 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2345895585977647 / 5000000000000000 : ℝ)) (q :=
+      (9489901209014669 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_520_9 :
+    Uω (aρ 9) (bρ 9) (9287519810631 / 16000000000000 : ℝ) ≤
+      (-(8316941009312002770807 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8706223999318771 / 20000000000000000 : ℝ) ≤
+      (-(8316941009312002770807 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8706223999318771 / 20000000000000000 : ℝ) =
+            (8706223999318771 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8706223999318771 / 5000000000000000 : ℝ) ≤
+            (5546002596687997229193 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1062269897654849 / 2500000000000000 : ℝ)) (q :=
+      (8706223999318771 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_520_10 :
+    Uω (aρ 10) (bρ 10) (9287519810631 / 16000000000000 : ℝ) ≤
+      (-(4740976483772919301281 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3874396062111003 / 10000000000000000 : ℝ) ≤
+      (-(4740976483772919301281 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3874396062111003 / 10000000000000000 : ℝ) =
+            (3874396062111003 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3874396062111003 / 2500000000000000 : ℝ) ≤
+            (2190495319227080698719 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3680206749017631 / 10000000000000000 : ℝ)) (q :=
+      (3874396062111003 / 10000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_520_11 :
+    Uω (aρ 11) (bρ 11) (9287519810631 / 16000000000000 : ℝ) ≤
+      (-(5545371965551789929277 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3298641441666869 / 10000000000000000 : ℝ) ≤
+      (-(5545371965551789929277 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3298641441666869 / 10000000000000000 : ℝ) =
+            (3298641441666869 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3298641441666869 / 2500000000000000 : ℝ) ≤
+            (1386099837448210070723 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2947174134334363 / 10000000000000000 : ℝ)) (q :=
+      (3298641441666869 / 10000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_520_12 :
+    Uω (aρ 12) (bρ 12) (9287519810631 / 16000000000000 : ℝ) ≤
+      (-(1353380204255246507751 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5167309096466059 / 20000000000000000 : ℝ) ≤
+      (-(1353380204255246507751 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5167309096466059 / 20000000000000000 : ℝ) =
+            (5167309096466059 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5167309096466059 / 5000000000000000 : ℝ) ≤
+            (32914156344753492249 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (485225111037921 / 2500000000000000 : ℝ)) (q :=
+      (5167309096466059 / 20000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_520_13 :
+    Uω (aρ 13) (bρ 13) (9287519810631 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_520_14 :
+    Uω (aρ 14) (bρ 14) (9287519810631 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_520_15 :
+    Uω (aρ 15) (bρ 15) (9287519810631 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_520_16 :
+    Uω (aρ 16) (bρ 16) (9287519810631 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_520 :
+    Uρ (9287519810631 / 16000000000000 : ℝ) ≤
+      (-(8876311486619942590457 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (9287519810631 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2775477411467269153561 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_520_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (9287519810631 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(5592724374280311294667 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_520_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (9287519810631 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(5676808531728931494759 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_520_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (9287519810631 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(5818095913874656755391 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_520_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (9287519810631 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(3018650477316833315497 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_520_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (9287519810631 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(795079433204800478269 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_520_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (9287519810631 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(6820086630363946455629 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_520_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (9287519810631 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(7455040704754400716783 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_520_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (9287519810631 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(8316941009312002770807 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_520_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (9287519810631 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(4740976483772919301281 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_520_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (9287519810631 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(5545371965551789929277 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_520_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (9287519810631 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(1353380204255246507751 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_520_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (9287519810631 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_520_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (9287519810631 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_520_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (9287519810631 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_520_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (9287519810631 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_520_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_521_1 :
+    Uω (aρ 1) (bρ 1) (4683703346333 / 8000000000000 : ℝ) ≤
+      (-(218573932723313454971 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11580208086837357 / 20000000000000000 : ℝ) ≤
+      (-(218573932723313454971 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11580208086837357 / 20000000000000000 : ℝ) =
+            (11580208086837357 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11580208086837357 / 10000000000000000 : ℝ) ≤
+            (58684939396686545029 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5790076122006107 / 10000000000000000 : ℝ)) (q :=
+      (11580208086837357 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_521_2 :
+    Uω (aρ 2) (bρ 2) (4683703346333 / 8000000000000 : ℝ) ≤
+      (-(5505754567146598031943 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5766178960984599 / 10000000000000000 : ℝ) ≤
+      (-(5505754567146598031943 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5766178960984599 / 10000000000000000 : ℝ) =
+            (5766178960984599 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5766178960984599 / 5000000000000000 : ℝ) ≤
+            (1425717235853401968057 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1441498742411987 / 2500000000000000 : ℝ)) (q :=
+      (5766178960984599 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_521_3 :
+    Uω (aρ 3) (bρ 3) (4683703346333 / 8000000000000 : ℝ) ≤
+      (-(698637552390495291897 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (285915998531491 / 500000000000000 : ℝ) ≤
+      (-(698637552390495291897 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (285915998531491 / 500000000000000 : ℝ) =
+            (285915998531491 / 250000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (285915998531491 / 250000000000000 : ℝ) ≤
+            (167796422984504708103 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (571767309528839 / 1000000000000000 : ℝ)) (q :=
+      (285915998531491 / 500000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_521_4 :
+    Uω (aρ 4) (bρ 4) (4683703346333 / 8000000000000 : ℝ) ≤
+      (-(5729125871667981875797 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11277613856261851 / 20000000000000000 : ℝ) ≤
+      (-(5729125871667981875797 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11277613856261851 / 20000000000000000 : ℝ) =
+            (11277613856261851 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11277613856261851 / 10000000000000000 : ℝ) ≤
+            (1202345931332018124203 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5636941191355601 / 10000000000000000 : ℝ)) (q :=
+      (11277613856261851 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_521_5 :
+    Uω (aρ 5) (bρ 5) (4683703346333 / 8000000000000 : ℝ) ≤
+      (-(2973159163481224856561 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2207062681652349 / 4000000000000000 : ℝ) ≤
+      (-(2973159163481224856561 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2207062681652349 / 4000000000000000 : ℝ) =
+            (2207062681652349 / 2000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2207062681652349 / 2000000000000000 : ℝ) ≤
+            (492576738018775143439 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1102566132538099 / 2000000000000000 : ℝ)) (q :=
+      (2207062681652349 / 4000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_521_6 :
+    Uω (aρ 6) (bρ 6) (4683703346333 / 8000000000000 : ℝ) ≤
+      (-(7833195515449707049 / 12500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1068751921395337 / 2000000000000000 : ℝ) ≤
+      (-(7833195515449707049 / 12500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1068751921395337 / 2000000000000000 : ℝ) =
+            (1068751921395337 / 1000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1068751921395337 / 1000000000000000 : ℝ) ≤
+            (831144238300292951 / 12500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (133306921088053 / 250000000000000 : ℝ)) (q :=
+      (1068751921395337 / 2000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_521_7 :
+    Uω (aρ 7) (bρ 7) (4683703346333 / 8000000000000 : ℝ) ≤
+      (-(6721324909858084649313 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5106185263996413 / 10000000000000000 : ℝ) ≤
+      (-(6721324909858084649313 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5106185263996413 / 10000000000000000 : ℝ) =
+            (5106185263996413 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5106185263996413 / 5000000000000000 : ℝ) ≤
+            (210146893141915350687 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (635098692290197 / 1250000000000000 : ℝ)) (q :=
+      (5106185263996413 / 10000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_521_8 :
+    Uω (aρ 8) (bρ 8) (4683703346333 / 8000000000000 : ℝ) ≤
+      (-(1469839420203114228919 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (299714974822367 / 625000000000000 : ℝ) ≤
+      (-(1469839420203114228919 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (299714974822367 / 625000000000000 : ℝ) =
+            (299714974822367 / 156250000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (299714974822367 / 156250000000000 : ℝ) ≤
+            (1302749300996885771081 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2371419927992247 / 5000000000000000 : ℝ)) (q :=
+      (299714974822367 / 625000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_521_9 :
+    Uω (aρ 9) (bρ 9) (4683703346333 / 8000000000000 : ℝ) ≤
+      (-(6406369439884059803 / 7812500000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4404249208749617 / 10000000000000000 : ℝ) ≤
+      (-(6406369439884059803 / 7812500000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4404249208749617 / 10000000000000000 : ℝ) =
+            (4404249208749617 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4404249208749617 / 2500000000000000 : ℝ) ≤
+            (4424055252303440197 / 7812500000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (268839044220499 / 625000000000000 : ℝ)) (q :=
+      (4404249208749617 / 10000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_521_10 :
+    Uω (aρ 10) (bρ 10) (4683703346333 / 8000000000000 : ℝ) ≤
+      (-(9347289623404676103549 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7853845706055591 / 20000000000000000 : ℝ) ≤
+      (-(9347289623404676103549 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7853845706055591 / 20000000000000000 : ℝ) =
+            (7853845706055591 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7853845706055591 / 5000000000000000 : ℝ) ≤
+            (4515653982595323896451 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3735331029579341 / 10000000000000000 : ℝ)) (q :=
+      (7853845706055591 / 20000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_521_11 :
+    Uω (aρ 11) (bρ 11) (4683703346333 / 8000000000000 : ℝ) ≤
+      (-(2730769626571057740701 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6708829015430311 / 20000000000000000 : ℝ) ≤
+      (-(2730769626571057740701 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6708829015430311 / 20000000000000000 : ℝ) =
+            (6708829015430311 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6708829015430311 / 5000000000000000 : ℝ) ≤
+            (734966274928942259299 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3008790965159061 / 10000000000000000 : ℝ)) (q :=
+      (6708829015430311 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_521_12 :
+    Uω (aρ 12) (bρ 12) (4683703346333 / 8000000000000 : ℝ) ≤
+      (-(6640928384740651911177 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (529915087832469 / 2000000000000000 : ℝ) ≤
+      (-(6640928384740651911177 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (529915087832469 / 2000000000000000 : ℝ) =
+            (529915087832469 / 500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (529915087832469 / 500000000000000 : ℝ) ≤
+            (290543418259348088823 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (50570323118461 / 250000000000000 : ℝ)) (q :=
+      (529915087832469 / 2000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12]) (by norm_num [bρ_12])
+      (by norm_num) (by norm_num [aρ_12, bρ_12]) (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_521_13 :
+    Uω (aρ 13) (bρ 13) (4683703346333 / 8000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_521_14 :
+    Uω (aρ 14) (bρ 14) (4683703346333 / 8000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_521_15 :
+    Uω (aρ 15) (bρ 15) (4683703346333 / 8000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_521_16 :
+    Uω (aρ 16) (bρ 16) (4683703346333 / 8000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_521 :
+    Uρ (4683703346333 / 8000000000000 : ℝ) ≤
+      (-(877967859898227552371 / 1000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (4683703346333 / 8000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(218573932723313454971 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_521_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (4683703346333 / 8000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(5505754567146598031943 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_521_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (4683703346333 / 8000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(698637552390495291897 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_521_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (4683703346333 / 8000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(5729125871667981875797 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_521_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (4683703346333 / 8000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(2973159163481224856561 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_521_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (4683703346333 / 8000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) * (-(7833195515449707049 / 12500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_521_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (4683703346333 / 8000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(6721324909858084649313 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_521_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (4683703346333 / 8000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(1469839420203114228919 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_521_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (4683703346333 / 8000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(6406369439884059803 / 7812500000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_521_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (4683703346333 / 8000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(9347289623404676103549 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_521_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (4683703346333 / 8000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(2730769626571057740701 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_521_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (4683703346333 / 8000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(6640928384740651911177 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_521_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (4683703346333 / 8000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_521_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (4683703346333 / 8000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_521_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (4683703346333 / 8000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_521_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (4683703346333 / 8000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_521_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_522_1 :
+    Uω (aρ 1) (bρ 1) (9447293574701 / 16000000000000 : ℝ) ≤
+      (-(2689242726636831311537 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11680067166810993 / 20000000000000000 : ℝ) ≤
+      (-(2689242726636831311537 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11680067166810993 / 20000000000000000 : ℝ) =
+            (11680067166810993 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11680067166810993 / 10000000000000000 : ℝ) ≤
+            (776493174863168688463 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1460001475176967 / 2500000000000000 : ℝ)) (q :=
+      (11680067166810993 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_522_2 :
+    Uω (aρ 2) (bρ 2) (9447293574701 / 16000000000000 : ℝ) ≤
+      (-(2709767332690208033799 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11632219683617487 / 20000000000000000 : ℝ) ≤
+      (-(2709767332690208033799 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11632219683617487 / 20000000000000000 : ℝ) =
+            (11632219683617487 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11632219683617487 / 10000000000000000 : ℝ) ≤
+            (755968568809791966201 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2907963715012181 / 5000000000000000 : ℝ)) (q :=
+      (11632219683617487 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_522_3 :
+    Uω (aρ 3) (bρ 3) (9447293574701 / 16000000000000 : ℝ) ≤
+      (-(275107753260969971343 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11536509743605397 / 20000000000000000 : ℝ) ≤
+      (-(275107753260969971343 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11536509743605397 / 20000000000000000 : ℝ) =
+            (11536509743605397 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11536509743605397 / 10000000000000000 : ℝ) ≤
+            (71465836889030028657 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (180237924886321 / 312500000000000 : ℝ)) (q :=
+      (11536509743605397 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_522_4 :
+    Uω (aρ 4) (bρ 4) (9447293574701 / 16000000000000 : ℝ) ≤
+      (-(5640940935750156606193 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11377505220116183 / 20000000000000000 : ℝ) ≤
+      (-(5640940935750156606193 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11377505220116183 / 20000000000000000 : ℝ) =
+            (11377505220116183 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11377505220116183 / 10000000000000000 : ℝ) ≤
+            (1290530867249843393807 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2843451626969029 / 5000000000000000 : ℝ)) (q :=
+      (11377505220116183 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_522_5 :
+    Uω (aρ 5) (bρ 5) (9447293574701 / 16000000000000 : ℝ) ≤
+      (-(292807872430797669627 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (445410345748031 / 800000000000000 : ℝ) ≤
+      (-(292807872430797669627 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (445410345748031 / 800000000000000 : ℝ) =
+            (445410345748031 / 400000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (445410345748031 / 400000000000000 : ℝ) ≤
+            (53765717719202330373 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (111256931937153 / 200000000000000 : ℝ)) (q :=
+      (445410345748031 / 800000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_522_6 :
+    Uω (aρ 6) (bρ 6) (9447293574701 / 16000000000000 : ℝ) ≤
+      (-(3086678972250842838001 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10787590857392963 / 20000000000000000 : ℝ) ≤
+      (-(3086678972250842838001 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10787590857392963 / 20000000000000000 : ℝ) =
+            (10787590857392963 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10787590857392963 / 10000000000000000 : ℝ) ≤
+            (379056929249157161999 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2691209592844919 / 5000000000000000 : ℝ)) (q :=
+      (10787590857392963 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_522_7 :
+    Uω (aρ 7) (bρ 7) (9447293574701 / 16000000000000 : ℝ) ≤
+      (-(1655884651729656945347 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10312723380877329 / 20000000000000000 : ℝ) ≤
+      (-(1655884651729656945347 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10312723380877329 / 20000000000000000 : ℝ) =
+            (10312723380877329 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10312723380877329 / 10000000000000000 : ℝ) ≤
+            (76983299020343054653 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1282803272483551 / 2500000000000000 : ℝ)) (q :=
+      (10312723380877329 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_522_8 :
+    Uω (aρ 8) (bρ 8) (9447293574701 / 16000000000000 : ℝ) ≤
+      (-(3622243194618693881327 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9691833600993193 / 20000000000000000 : ℝ) ≤
+      (-(3622243194618693881327 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9691833600993193 / 20000000000000000 : ℝ) =
+            (9691833600993193 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9691833600993193 / 5000000000000000 : ℝ) ≤
+            (3309228608381306118673 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1198466240347517 / 2500000000000000 : ℝ)) (q :=
+      (9691833600993193 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_522_9 :
+    Uω (aρ 9) (bρ 9) (9447293574701 / 16000000000000 : ℝ) ≤
+      (-(1616955332102858313771 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4455358042689239 / 10000000000000000 : ℝ) ≤
+      (-(1616955332102858313771 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4455358042689239 / 10000000000000000 : ℝ) =
+            (4455358042689239 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4455358042689239 / 2500000000000000 : ℝ) ≤
+            (1155633389097141686229 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4353713074135353 / 10000000000000000 : ℝ)) (q :=
+      (4455358042689239 / 10000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_522_10 :
+    Uω (aρ 10) (bρ 10) (9447293574701 / 16000000000000 : ℝ) ≤
+      (-(9214596541511311802389 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1591751060955643 / 4000000000000000 : ℝ) ≤
+      (-(9214596541511311802389 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1591751060955643 / 4000000000000000 : ℝ) =
+            (1591751060955643 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1591751060955643 / 1000000000000000 : ℝ) ≤
+            (4648347064488688197611 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (379031132703009 / 1000000000000000 : ℝ)) (q :=
+      (1591751060955643 / 4000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_522_11 :
+    Uω (aρ 11) (bρ 11) (9447293574701 / 16000000000000 : ℝ) ≤
+      (-(10758800567764580269163 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6819950517003007 / 20000000000000000 : ℝ) ≤
+      (-(10758800567764580269163 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6819950517003007 / 20000000000000000 : ℝ) =
+            (6819950517003007 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6819950517003007 / 5000000000000000 : ℝ) ≤
+            (3104143038235419730837 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1534991582729941 / 5000000000000000 : ℝ)) (q :=
+      (6819950517003007 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_522_12 :
+    Uω (aρ 12) (bρ 12) (9447293574701 / 16000000000000 : ℝ) ≤
+      (-(1629974352488310309933 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (271449412615447 / 1000000000000000 : ℝ) ≤
+      (-(1629974352488310309933 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (271449412615447 / 1000000000000000 : ℝ) =
+            (271449412615447 / 250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (271449412615447 / 250000000000000 : ℝ) ≤
+            (102893598261689690067 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (420544199490163 / 2000000000000000 : ℝ)) (q :=
+      (271449412615447 / 1000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12]) (by norm_num [bρ_12])
+      (by norm_num) (by norm_num [aρ_12, bρ_12]) (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_522_13 :
+    Uω (aρ 13) (bρ 13) (9447293574701 / 16000000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_522_14 :
+    Uω (aρ 14) (bρ 14) (9447293574701 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_522_15 :
+    Uω (aρ 15) (bρ 15) (9447293574701 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_522_16 :
+    Uω (aρ 16) (bρ 16) (9447293574701 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_522 :
+    Uρ (9447293574701 / 16000000000000 : ℝ) ≤
+      (-(108558985643967623763 / 125000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (9447293574701 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2689242726636831311537 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_522_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (9447293574701 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(2709767332690208033799 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_522_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (9447293574701 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(275107753260969971343 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_522_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (9447293574701 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(5640940935750156606193 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_522_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (9447293574701 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(292807872430797669627 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_522_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (9447293574701 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3086678972250842838001 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_522_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (9447293574701 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(1655884651729656945347 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_522_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (9447293574701 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(3622243194618693881327 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_522_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (9447293574701 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(1616955332102858313771 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_522_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (9447293574701 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(9214596541511311802389 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_522_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (9447293574701 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(10758800567764580269163 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_522_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (9447293574701 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(1629974352488310309933 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_522_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (9447293574701 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_522_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (9447293574701 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_522_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (9447293574701 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_522_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (9447293574701 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_522_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_523_1 :
+    Uω (aρ 1) (bρ 1) (297724389273 / 500000000000 : ℝ) ≤
+      (-(1323338391653748969901 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5889963119345097 / 10000000000000000 : ℝ) ≤
+      (-(1323338391653748969901 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5889963119345097 / 10000000000000000 : ℝ) =
+            (5889963119345097 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5889963119345097 / 5000000000000000 : ℝ) ≤
+            (409529559096251030099 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2944967835657597 / 5000000000000000 : ℝ)) (q :=
+      (5889963119345097 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_523_2 :
+    Uω (aρ 2) (bρ 2) (297724389273 / 500000000000 : ℝ) ≤
+      (-(2667025923201519139487 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11732081391484447 / 20000000000000000 : ℝ) ≤
+      (-(2667025923201519139487 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11732081391484447 / 20000000000000000 : ℝ) =
+            (11732081391484447 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11732081391484447 / 10000000000000000 : ℝ) ≤
+            (798709978298480860513 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5865859836619447 / 10000000000000000 : ℝ)) (q :=
+      (11732081391484447 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_523_3 :
+    Uω (aρ 3) (bρ 3) (297724389273 / 500000000000 : ℝ) ≤
+      (-(5415959314664550473199 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2909094838421109 / 5000000000000000 : ℝ) ≤
+      (-(5415959314664550473199 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2909094838421109 / 5000000000000000 : ℝ) =
+            (2909094838421109 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2909094838421109 / 2500000000000000 : ℝ) ≤
+            (1515512488335449526801 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1454388476292359 / 2500000000000000 : ℝ)) (q :=
+      (2909094838421109 / 5000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_523_4 :
+    Uω (aρ 4) (bρ 4) (297724389273 / 500000000000 : ℝ) ≤
+      (-(694190920277763514999 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2295479202704893 / 4000000000000000 : ℝ) ≤
+      (-(694190920277763514999 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2295479202704893 / 4000000000000000 : ℝ) =
+            (2295479202704893 / 2000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2295479202704893 / 2000000000000000 : ℝ) ≤
+            (172243055097236485001 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1147372949214893 / 2000000000000000 : ℝ)) (q :=
+      (2295479202704893 / 4000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_523_5 :
+    Uω (aρ 5) (bρ 5) (297724389273 / 500000000000 : ℝ) ≤
+      (-(1441700895845914328219 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11235202336514049 / 20000000000000000 : ℝ) ≤
+      (-(1441700895845914328219 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11235202336514049 / 20000000000000000 : ℝ) =
+            (11235202336514049 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11235202336514049 / 10000000000000000 : ℝ) ≤
+            (291167054904085671781 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5612860988399049 / 10000000000000000 : ℝ)) (q :=
+      (11235202336514049 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_523_6 :
+    Uω (aρ 6) (bρ 6) (297724389273 / 500000000000 : ℝ) ≤
+      (-(1216204732437666014063 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5443829288233231 / 10000000000000000 : ℝ) ≤
+      (-(1216204732437666014063 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5443829288233231 / 10000000000000000 : ℝ) =
+            (5443829288233231 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5443829288233231 / 5000000000000000 : ℝ) ≤
+            (170089628162333985937 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2716278801745731 / 5000000000000000 : ℝ)) (q :=
+      (5443829288233231 / 10000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_523_7 :
+    Uω (aρ 7) (bρ 7) (297724389273 / 500000000000 : ℝ) ≤
+      (-(326335422999999054939 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10413066661597479 / 20000000000000000 : ℝ) ≤
+      (-(326335422999999054939 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10413066661597479 / 20000000000000000 : ℝ) =
+            (10413066661597479 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10413066661597479 / 10000000000000000 : ℝ) ≤
+            (20238167150000945061 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5181627069382479 / 10000000000000000 : ℝ)) (q :=
+      (10413066661597479 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_523_8 :
+    Uω (aρ 8) (bρ 8) (297724389273 / 500000000000 : ℝ) ≤
+      (-(7140884075452489921343 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1224095646752767 / 2500000000000000 : ℝ) ≤
+      (-(7140884075452489921343 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1224095646752767 / 2500000000000000 : ℝ) =
+            (1224095646752767 / 625000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1224095646752767 / 625000000000000 : ℝ) ≤
+            (6722059530547510078657 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (18925262629481 / 39062500000000 : ℝ)) (q :=
+      (1224095646752767 / 2500000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_523_9 :
+    Uω (aρ 9) (bρ 9) (297724389273 / 500000000000 : ℝ) ≤
+      (-(3985388580009479571857 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9012879023440247 / 20000000000000000 : ℝ) ≤
+      (-(3985388580009479571857 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9012879023440247 / 20000000000000000 : ℝ) =
+            (9012879023440247 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9012879023440247 / 5000000000000000 : ℝ) ≤
+            (2946083222990520428143 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4405946710925247 / 10000000000000000 : ℝ)) (q :=
+      (9012879023440247 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_523_10 :
+    Uω (aρ 10) (bρ 10) (297724389273 / 500000000000 : ℝ) ≤
+      (-(9083812071916347774149 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4031763548331721 / 10000000000000000 : ℝ) ≤
+      (-(9083812071916347774149 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4031763548331721 / 10000000000000000 : ℝ) =
+            (4031763548331721 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4031763548331721 / 2500000000000000 : ℝ) ≤
+            (4779131534083652225851 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1922576908821721 / 5000000000000000 : ℝ)) (q :=
+      (4031763548331721 / 10000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_523_11 :
+    Uω (aρ 11) (bρ 11) (297724389273 / 500000000000 : ℝ) ≤
+      (-(10597754534486675358283 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (866334035855753 / 2500000000000000 : ℝ) ≤
+      (-(10597754534486675358283 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (866334035855753 / 2500000000000000 : ℝ) =
+            (866334035855753 / 625000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (866334035855753 / 625000000000000 : ℝ) ≤
+            (3265189071513324641717 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (195673477126939 / 625000000000000 : ℝ)) (q :=
+      (866334035855753 / 2500000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11]) (by norm_num [bρ_11])
+      (by norm_num) (by norm_num [aρ_11, bρ_11]) (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_523_12 :
+    Uω (aρ 12) (bρ 12) (297724389273 / 500000000000 : ℝ) ≤
+      (-(12806664000284943041083 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5557041559334771 / 20000000000000000 : ℝ) ≤
+      (-(12806664000284943041083 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5557041559334771 / 20000000000000000 : ℝ) =
+            (5557041559334771 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5557041559334771 / 5000000000000000 : ℝ) ≤
+            (1056279605715056958917 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2180845003204771 / 10000000000000000 : ℝ)) (q :=
+      (5557041559334771 / 20000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_523_13 :
+    Uω (aρ 13) (bρ 13) (297724389273 / 500000000000 : ℝ) ≤
+      (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_13]) (by norm_num [bρ_13])]; exact Lmid_13
+
+lemma U_523_14 :
+    Uω (aρ 14) (bρ 14) (297724389273 / 500000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_523_15 :
+    Uω (aρ 15) (bρ 15) (297724389273 / 500000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_523_16 :
+    Uω (aρ 16) (bρ 16) (297724389273 / 500000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_523 :
+    Uρ (297724389273 / 500000000000 : ℝ) ≤
+      (-(8591336616545901455247 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (297724389273 / 500000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1323338391653748969901 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_523_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (297724389273 / 500000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(2667025923201519139487 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_523_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (297724389273 / 500000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(5415959314664550473199 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_523_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (297724389273 / 500000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(694190920277763514999 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_523_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (297724389273 / 500000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1441700895845914328219 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_523_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (297724389273 / 500000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1216204732437666014063 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_523_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (297724389273 / 500000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(326335422999999054939 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_523_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (297724389273 / 500000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(7140884075452489921343 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_523_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (297724389273 / 500000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(3985388580009479571857 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_523_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (297724389273 / 500000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(9083812071916347774149 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_523_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (297724389273 / 500000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(10597754534486675358283 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_523_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (297724389273 / 500000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(12806664000284943041083 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_523_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (297724389273 / 500000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4762195995179309567439 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_523_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (297724389273 / 500000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_523_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (297724389273 / 500000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_523_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (297724389273 / 500000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_523_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_524_1 :
+    Uω (aρ 1) (bρ 1) (19123153518409 / 32000000000000 : ℝ) ≤
+      (-(1314230248183163826137 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23645843632824819 / 40000000000000000 : ℝ) ≤
+      (-(1314230248183163826137 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23645843632824819 / 40000000000000000 : ℝ) =
+            (23645843632824819 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23645843632824819 / 20000000000000000 : ℝ) ≤
+            (418637702566836173863 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5911433559994597 / 10000000000000000 : ℝ)) (q :=
+      (23645843632824819 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_524_2 :
+    Uω (aρ 2) (bρ 2) (19123153518409 / 32000000000000 : ℝ) ≤
+      (-(5297470015337856711679 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23550156180768569 / 40000000000000000 : ℝ) ≤
+      (-(5297470015337856711679 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23550156180768569 / 40000000000000000 : ℝ) =
+            (23550156180768569 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23550156180768569 / 20000000000000000 : ℝ) ≤
+            (1634001787662143288321 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (735919855809559 / 1250000000000000 : ℝ)) (q :=
+      (23550156180768569 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_524_3 :
+    Uω (aρ 3) (bρ 3) (19123153518409 / 32000000000000 : ℝ) ≤
+      (-(107581485947053385999 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23358758826441631 / 40000000000000000 : ℝ) ≤
+      (-(107581485947053385999 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23358758826441631 / 40000000000000000 : ℝ) =
+            (23358758826441631 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23358758826441631 / 20000000000000000 : ℝ) ≤
+            (31047950112946614001 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5839056275663003 / 10000000000000000 : ℝ)) (q :=
+      (23358758826441631 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_524_4 :
+    Uω (aρ 4) (bρ 4) (19123153518409 / 32000000000000 : ℝ) ≤
+      (-(5516124554661589611093 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23040810159629221 / 40000000000000000 : ℝ) ≤
+      (-(5516124554661589611093 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23040810159629221 / 40000000000000000 : ℝ) =
+            (23040810159629221 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23040810159629221 / 20000000000000000 : ℝ) ≤
+            (1415347248338410388907 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2879188061660899 / 5000000000000000 : ℝ)) (q :=
+      (23040810159629221 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_524_5 :
+    Uω (aρ 5) (bρ 5) (19123153518409 / 32000000000000 : ℝ) ≤
+      (-(5728576098328493848859 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22556467772897239 / 40000000000000000 : ℝ) ≤
+      (-(5728576098328493848859 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (22556467772897239 / 40000000000000000 : ℝ) =
+            (22556467772897239 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22556467772897239 / 20000000000000000 : ℝ) ≤
+            (1202895704671506151141 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5634394849290807 / 10000000000000000 : ℝ)) (q :=
+      (22556467772897239 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_524_6 :
+    Uω (aρ 6) (bρ 6) (19123153518409 / 32000000000000 : ℝ) ≤
+      (-(6041530126911093500149 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21861485621420399 / 40000000000000000 : ℝ) ≤
+      (-(6041530126911093500149 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (21861485621420399 / 40000000000000000 : ℝ) =
+            (21861485621420399 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21861485621420399 / 20000000000000000 : ℝ) ≤
+            (889941676088906499851 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5454144148692387 / 10000000000000000 : ℝ)) (q :=
+      (21861485621420399 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_524_7 :
+    Uω (aρ 7) (bρ 7) (19123153518409 / 32000000000000 : ℝ) ≤
+      (-(3242653404335615268931 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (836501427633957 / 1600000000000000 : ℝ) ≤
+      (-(3242653404335615268931 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (836501427633957 / 1600000000000000 : ℝ) =
+            (836501427633957 / 800000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (836501427633957 / 800000000000000 : ℝ) ≤
+            (223082497164384731069 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (104066611283333 / 200000000000000 : ℝ)) (q :=
+      (836501427633957 / 1600000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_524_8 :
+    Uω (aρ 8) (bρ 8) (19123153518409 / 32000000000000 : ℝ) ≤
+      (-(7096612213048692962003 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3934486272011177 / 8000000000000000 : ℝ) ≤
+      (-(7096612213048692962003 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3934486272011177 / 8000000000000000 : ℝ) =
+            (3934486272011177 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3934486272011177 / 2000000000000000 : ℝ) ≤
+            (6766331392951307037997 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (486682005011013 / 1000000000000000 : ℝ)) (q :=
+      (3934486272011177 / 8000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_524_9 :
+    Uω (aρ 9) (bρ 9) (19123153518409 / 32000000000000 : ℝ) ≤
+      (-(1584421749117207670283 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (18113700380301757 / 40000000000000000 : ℝ) ≤
+      (-(1584421749117207670283 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (18113700380301757 / 40000000000000000 : ℝ) =
+            (18113700380301757 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18113700380301757 / 10000000000000000 : ℝ) ≤
+            (1188166972082792329717 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2214210094296533 / 5000000000000000 : ℝ)) (q :=
+      (18113700380301757 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_524_10 :
+    Uω (aρ 10) (bρ 10) (19123153518409 / 32000000000000 : ℝ) ≤
+      (-(72224593962241507771 / 80000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16217193864508737 / 40000000000000000 : ℝ) ≤
+      (-(72224593962241507771 / 80000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (16217193864508737 / 40000000000000000 : ℝ) =
+            (16217193864508737 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16217193864508737 / 10000000000000000 : ℝ) ≤
+            (38678954885758492229 / 80000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (967181491047889 / 2500000000000000 : ℝ)) (q :=
+      (16217193864508737 / 40000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_524_11 :
+    Uω (aρ 11) (bρ 11) (19123153518409 / 32000000000000 : ℝ) ≤
+      (-(10529373141229537227421 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13956455198364819 / 40000000000000000 : ℝ) ≤
+      (-(10529373141229537227421 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (13956455198364819 / 40000000000000000 : ℝ) =
+            (13956455198364819 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13956455198364819 / 10000000000000000 : ℝ) ≤
+            (3333570464770462772579 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3156833257324597 / 10000000000000000 : ℝ)) (q :=
+      (13956455198364819 / 40000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_524_12 :
+    Uω (aρ 12) (bρ 12) (19123153518409 / 32000000000000 : ℝ) ≤
+      (-(2541766278107864221893 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2244669741941209 / 8000000000000000 : ℝ) ≤
+      (-(2541766278107864221893 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2244669741941209 / 8000000000000000 : ℝ) =
+            (2244669741941209 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2244669741941209 / 2000000000000000 : ℝ) ≤
+            (230822443092135778107 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (221398010968021 / 1000000000000000 : ℝ)) (q :=
+      (2244669741941209 / 8000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_524_13 :
+    Uω (aρ 13) (bρ 13) (19123153518409 / 32000000000000 : ℝ) ≤
+      (-(3569539676245047689691 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6713427432028769 / 40000000000000000 : ℝ) ≤
+      (-(3569539676245047689691 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (6713427432028769 / 40000000000000000 : ℝ) =
+            (6713427432028769 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6713427432028769 / 5000000000000000 : ℝ) ≤
+            (589343405554952310309 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (89600303109143 / 2500000000000000 : ℝ)) (q :=
+      (6713427432028769 / 40000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_524_14 :
+    Uω (aρ 14) (bρ 14) (19123153518409 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_524_15 :
+    Uω (aρ 15) (bρ 15) (19123153518409 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_524_16 :
+    Uω (aρ 16) (bρ 16) (19123153518409 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_524 :
+    Uρ (19123153518409 / 32000000000000 : ℝ) ≤
+      (-(530057725649425284243 / 625000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (19123153518409 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1314230248183163826137 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_524_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (19123153518409 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(5297470015337856711679 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_524_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (19123153518409 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(107581485947053385999 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_524_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (19123153518409 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(5516124554661589611093 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_524_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (19123153518409 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(5728576098328493848859 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_524_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (19123153518409 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(6041530126911093500149 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_524_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (19123153518409 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(3242653404335615268931 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_524_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (19123153518409 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(7096612213048692962003 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_524_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (19123153518409 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(1584421749117207670283 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_524_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (19123153518409 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) * (-(72224593962241507771 / 80000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_524_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (19123153518409 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(10529373141229537227421 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_524_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (19123153518409 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(2541766278107864221893 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_524_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (19123153518409 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(3569539676245047689691 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_524_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (19123153518409 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_524_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (19123153518409 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_524_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (19123153518409 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_524_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_525_1 :
+    Uω (aρ 1) (bρ 1) (9595973061673 / 16000000000000 : ℝ) ≤
+      (-(522062067163280364799 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (46351239815187 / 78125000000000 : ℝ) ≤
+      (-(522062067163280364799 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (46351239815187 / 78125000000000 : ℝ) = (46351239815187 / 39062500000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (46351239815187 / 39062500000000 : ℝ) ≤
+            (171085113136719635201 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5932931447227247 / 10000000000000000 : ℝ)) (q :=
+      (46351239815187 / 78125000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_525_2 :
+    Uω (aρ 2) (bρ 2) (9595973061673 / 16000000000000 : ℝ) ≤
+      (-(328813845489025310421 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11818074779673017 / 20000000000000000 : ℝ) ≤
+      (-(328813845489025310421 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11818074779673017 / 20000000000000000 : ℝ) =
+            (11818074779673017 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11818074779673017 / 10000000000000000 : ℝ) ≤
+            (104403142198474689579 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (738607230840299 / 1250000000000000 : ℝ)) (q :=
+      (11818074779673017 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_525_3 :
+    Uω (aρ 3) (bρ 3) (9595973061673 / 16000000000000 : ℝ) ≤
+      (-(5342324859967229659017 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11722379438408479 / 20000000000000000 : ℝ) ≤
+      (-(5342324859967229659017 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11722379438408479 / 20000000000000000 : ℝ) =
+            (11722379438408479 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11722379438408479 / 10000000000000000 : ℝ) ≤
+            (1589146943032770340983 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2930279305903927 / 5000000000000000 : ℝ)) (q :=
+      (11722379438408479 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_525_4 :
+    Uω (aρ 4) (bρ 4) (9595973061673 / 16000000000000 : ℝ) ≤
+      (-(2739430605522132083087 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11563414044248639 / 20000000000000000 : ℝ) ≤
+      (-(2739430605522132083087 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11563414044248639 / 20000000000000000 : ℝ) =
+            (11563414044248639 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11563414044248639 / 10000000000000000 : ℝ) ≤
+            (726305295977867916913 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2889943699356507 / 5000000000000000 : ℝ)) (q :=
+      (11563414044248639 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_525_5 :
+    Uω (aρ 5) (bρ 5) (9595973061673 / 16000000000000 : ℝ) ≤
+      (-(5690494434089613539713 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11321265161179799 / 20000000000000000 : ℝ) ≤
+      (-(5690494434089613539713 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11321265161179799 / 20000000000000000 : ℝ) =
+            (11321265161179799 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11321265161179799 / 10000000000000000 : ℝ) ≤
+            (1240977368910386460287 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2827964217489587 / 5000000000000000 : ℝ)) (q :=
+      (11321265161179799 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_525_6 :
+    Uω (aρ 6) (bρ 6) (9595973061673 / 16000000000000 : ℝ) ≤
+      (-(3001096294645691980019 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1371728293226801 / 2500000000000000 : ℝ) ≤
+      (-(3001096294645691980019 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1371728293226801 / 2500000000000000 : ℝ) =
+            (1371728293226801 / 1250000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1371728293226801 / 1250000000000000 : ℝ) ≤
+            (464639606854308019981 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5475729994753783 / 10000000000000000 : ℝ)) (q :=
+      (1371728293226801 / 2500000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_525_7 :
+    Uω (aρ 7) (bρ 7) (9595973061673 / 16000000000000 : ℝ) ≤
+      (-(3222038740598697037627 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10499467327625579 / 20000000000000000 : ℝ) ≤
+      (-(3222038740598697037627 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10499467327625579 / 20000000000000000 : ℝ) =
+            (10499467327625579 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10499467327625579 / 10000000000000000 : ℝ) ≤
+            (243697160901302962373 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2612516178662477 / 5000000000000000 : ℝ)) (q :=
+      (10499467327625579 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_525_8 :
+    Uω (aρ 8) (bρ 8) (9595973061673 / 16000000000000 : ℝ) ≤
+      (-(1763134895300513255911 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4939831070469791 / 10000000000000000 : ℝ) ≤
+      (-(1763134895300513255911 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4939831070469791 / 10000000000000000 : ℝ) =
+            (4939831070469791 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4939831070469791 / 2500000000000000 : ℝ) ≤
+            (1702601006199486744089 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4888768821978957 / 10000000000000000 : ℝ)) (q :=
+      (4939831070469791 / 10000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_525_9 :
+    Uω (aρ 9) (bρ 9) (9595973061673 / 16000000000000 : ℝ) ≤
+      (-(1574737327668269272281 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (568800732303903 / 1250000000000000 : ℝ) ≤
+      (-(1574737327668269272281 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (568800732303903 / 1250000000000000 : ℝ) =
+            (568800732303903 / 312500000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (568800732303903 / 312500000000000 : ℝ) ≤
+            (1197851393531730727719 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4450884026261823 / 10000000000000000 : ℝ)) (q :=
+      (568800732303903 / 1250000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_525_10 :
+    Uω (aρ 10) (bρ 10) (9595973061673 / 16000000000000 : ℝ) ≤
+      (-(8972674828002439749491 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8153642747139649 / 20000000000000000 : ℝ) ≤
+      (-(8972674828002439749491 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8153642747139649 / 20000000000000000 : ℝ) =
+            (8153642747139649 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8153642747139649 / 5000000000000000 : ℝ) ≤
+            (4890268777997560250509 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (486534261254253 / 1250000000000000 : ℝ)) (q :=
+      (8153642747139649 / 20000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_525_11 :
+    Uω (aρ 11) (bρ 11) (9595973061673 / 16000000000000 : ℝ) ≤
+      (-(52307765738945663703 / 50000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1405142956236983 / 4000000000000000 : ℝ) ≤
+      (-(52307765738945663703 / 50000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1405142956236983 / 4000000000000000 : ℝ) =
+            (1405142956236983 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1405142956236983 / 1000000000000000 : ℝ) ≤
+            (17006952291054336297 / 50000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (318282275028429 / 1000000000000000 : ℝ)) (q :=
+      (1405142956236983 / 4000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_525_12 :
+    Uω (aρ 12) (bρ 12) (9595973061673 / 16000000000000 : ℝ) ≤
+      (-(12612446007478394770303 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5666024198846061 / 20000000000000000 : ℝ) ≤
+      (-(12612446007478394770303 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5666024198846061 / 20000000000000000 : ℝ) =
+            (5666024198846061 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5666024198846061 / 5000000000000000 : ℝ) ≤
+            (1250497598521605229697 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (561708066157609 / 2500000000000000 : ℝ)) (q :=
+      (5666024198846061 / 20000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_525_13 :
+    Uω (aρ 13) (bρ 13) (9595973061673 / 16000000000000 : ℝ) ≤
+      (-(17351210317533614871703 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3527577028304457 / 20000000000000000 : ℝ) ≤
+      (-(17351210317533614871703 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3527577028304457 / 20000000000000000 : ℝ) =
+            (3527577028304457 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3527577028304457 / 2500000000000000 : ℝ) ≤
+            (3443205091466385128297 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (63470854460479 / 1250000000000000 : ℝ)) (q :=
+      (3527577028304457 / 20000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_525_14 :
+    Uω (aρ 14) (bρ 14) (9595973061673 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_525_15 :
+    Uω (aρ 15) (bρ 15) (9595973061673 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_525_16 :
+    Uω (aρ 16) (bρ 16) (9595973061673 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_525 :
+    Uρ (9595973061673 / 16000000000000 : ℝ) ≤
+      (-(1682447810411330092499 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (9595973061673 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(522062067163280364799 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_525_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (9595973061673 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(328813845489025310421 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_525_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (9595973061673 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(5342324859967229659017 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_525_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (9595973061673 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(2739430605522132083087 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_525_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (9595973061673 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(5690494434089613539713 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_525_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (9595973061673 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3001096294645691980019 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_525_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (9595973061673 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(3222038740598697037627 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_525_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (9595973061673 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(1763134895300513255911 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_525_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (9595973061673 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(1574737327668269272281 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_525_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (9595973061673 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(8972674828002439749491 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_525_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (9595973061673 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) * (-(52307765738945663703 / 50000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_525_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (9595973061673 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(12612446007478394770303 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_525_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (9595973061673 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(17351210317533614871703 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_525_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (9595973061673 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_525_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (9595973061673 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_525_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (9595973061673 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_525_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_526_1 :
+    Uω (aρ 1) (bρ 1) (19260738728283 / 32000000000000 : ℝ) ≤
+      (-(1296112911651224119971 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23817825935064501 / 40000000000000000 : ℝ) ≤
+      (-(1296112911651224119971 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23817825935064501 / 40000000000000000 : ℝ) =
+            (23817825935064501 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23817825935064501 / 20000000000000000 : ℝ) ≤
+            (436755039098775880029 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5954429333028813 / 10000000000000000 : ℝ)) (q :=
+      (23817825935064501 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_526_2 :
+    Uω (aρ 2) (bρ 2) (19260738728283 / 32000000000000 : ℝ) ≤
+      (-(5224705415238725644339 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4744428583782069 / 8000000000000000 : ℝ) ≤
+      (-(5224705415238725644339 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (4744428583782069 / 8000000000000000 : ℝ) =
+            (4744428583782069 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4744428583782069 / 4000000000000000 : ℝ) ≤
+            (1706766387761274355661 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1186071367492347 / 2000000000000000 : ℝ)) (q :=
+      (4744428583782069 / 8000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_526_3 :
+    Uω (aρ 3) (bρ 3) (19260738728283 / 32000000000000 : ℝ) ≤
+      (-(1326427502307451136827 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23530758859248241 / 40000000000000000 : ℝ) ≤
+      (-(1326427502307451136827 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23530758859248241 / 40000000000000000 : ℝ) =
+            (23530758859248241 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23530758859248241 / 20000000000000000 : ℝ) ≤
+            (406440448442548863173 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5882060913980683 / 10000000000000000 : ℝ)) (q :=
+      (23530758859248241 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_526_4 :
+    Uω (aρ 4) (bρ 4) (19260738728283 / 32000000000000 : ℝ) ≤
+      (-(5441736294544028744673 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23212845815919161 / 40000000000000000 : ℝ) ≤
+      (-(5441736294544028744673 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23212845815919161 / 40000000000000000 : ℝ) =
+            (23212845815919161 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23212845815919161 / 20000000000000000 : ℝ) ≤
+            (1489735508455971255327 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5801398573381143 / 10000000000000000 : ℝ)) (q :=
+      (23212845815919161 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_526_5 :
+    Uω (aρ 5) (bρ 5) (19260738728283 / 32000000000000 : ℝ) ≤
+      (-(113051149611568404339 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4545718465535591 / 8000000000000000 : ℝ) ≤
+      (-(113051149611568404339 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (4545718465535591 / 8000000000000000 : ℝ) =
+            (4545718465535591 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4545718465535591 / 4000000000000000 : ℝ) ≤
+            (25578286448431595661 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (283873087429777 / 500000000000000 : ℝ)) (q :=
+      (4545718465535591 / 8000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_526_6 :
+    Uω (aρ 6) (bρ 6) (19260738728283 / 32000000000000 : ℝ) ≤
+      (-(1192601963369088836747 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22033818380029693 / 40000000000000000 : ℝ) ≤
+      (-(1192601963369088836747 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (22033818380029693 / 40000000000000000 : ℝ) =
+            (22033818380029693 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22033818380029693 / 20000000000000000 : ℝ) ≤
+            (193692397230911163253 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5497315149911409 / 10000000000000000 : ℝ)) (q :=
+      (22033818380029693 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_526_7 :
+    Uω (aρ 7) (bρ 7) (19260738728283 / 32000000000000 : ℝ) ≤
+      (-(6403019035753007704611 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21085330258631749 / 40000000000000000 : ℝ) ≤
+      (-(6403019035753007704611 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (21085330258631749 / 40000000000000000 : ℝ) =
+            (21085330258631749 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21085330258631749 / 20000000000000000 : ℝ) ≤
+            (528452767246992295389 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5246732469972437 / 10000000000000000 : ℝ)) (q :=
+      (21085330258631749 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_526_8 :
+    Uω (aρ 8) (bρ 8) (19260738728283 / 32000000000000 : ℝ) ≤
+      (-(1752166089705982821403 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19846209221993067 / 40000000000000000 : ℝ) ≤
+      (-(1752166089705982821403 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (19846209221993067 / 40000000000000000 : ℝ) =
+            (19846209221993067 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19846209221993067 / 10000000000000000 : ℝ) ≤
+            (1713569811794017178597 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (613839200374137 / 1250000000000000 : ℝ)) (q :=
+      (19846209221993067 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_526_9 :
+    Uω (aρ 9) (bρ 9) (19260738728283 / 32000000000000 : ℝ) ≤
+      (-(1565101650889641952791 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (18289527497605977 / 40000000000000000 : ℝ) ≤
+      (-(1565101650889641952791 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (18289527497605977 / 40000000000000000 : ℝ) =
+            (18289527497605977 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18289527497605977 / 10000000000000000 : ℝ) ≤
+            (1207487070310358047209 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4473338369159551 / 10000000000000000 : ℝ)) (q :=
+      (18289527497605977 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_526_10 :
+    Uω (aρ 10) (bρ 10) (19260738728283 / 32000000000000 : ℝ) ≤
+      (-(2229402349962983556549 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (655893197973881 / 1600000000000000 : ℝ) ≤
+      (-(2229402349962983556549 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (655893197973881 / 1600000000000000 : ℝ) =
+            (655893197973881 / 400000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (655893197973881 / 400000000000000 : ℝ) ≤
+            (1236333551537016443451 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (156631945141003 / 400000000000000 : ℝ)) (q :=
+      (655893197973881 / 1600000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10]) (by norm_num [bρ_10])
+      (by norm_num) (by norm_num [aρ_10, bρ_10]) (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_526_11 :
+    Uω (aρ 11) (bρ 11) (19260738728283 / 32000000000000 : ℝ) ≤
+      (-(5197141997269709507939 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14146270976670461 / 40000000000000000 : ℝ) ≤
+      (-(5197141997269709507939 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (14146270976670461 / 40000000000000000 : ℝ) =
+            (14146270976670461 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14146270976670461 / 10000000000000000 : ℝ) ≤
+            (1734329805730290492061 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3208745768391793 / 10000000000000000 : ℝ)) (q :=
+      (14146270976670461 / 40000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_526_12 :
+    Uω (aρ 12) (bρ 12) (19260738728283 / 32000000000000 : ℝ) ≤
+      (-(12517454039065267076341 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11440206651004759 / 40000000000000000 : ℝ) ≤
+      (-(12517454039065267076341 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (11440206651004759 / 40000000000000000 : ℝ) =
+            (11440206651004759 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11440206651004759 / 10000000000000000 : ℝ) ≤
+            (1345489566934732923659 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1139706851121971 / 5000000000000000 : ℝ)) (q :=
+      (11440206651004759 / 40000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_526_13 :
+    Uω (aρ 13) (bρ 13) (19260738728283 / 32000000000000 : ℝ) ≤
+      (-(16970931968603725312893 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7328612859684809 / 40000000000000000 : ℝ) ≤
+      (-(16970931968603725312893 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (7328612859684809 / 40000000000000000 : ℝ) =
+            (7328612859684809 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7328612859684809 / 5000000000000000 : ℝ) ≤
+            (3823483440396274687107 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (622998548178967 / 10000000000000000 : ℝ)) (q :=
+      (7328612859684809 / 40000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_526_14 :
+    Uω (aρ 14) (bρ 14) (19260738728283 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_526_15 :
+    Uω (aρ 15) (bρ 15) (19260738728283 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_526_16 :
+    Uω (aρ 16) (bρ 16) (19260738728283 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_526 :
+    Uρ (19260738728283 / 32000000000000 : ℝ) ≤
+      (-(1670131396109797597861 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (19260738728283 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1296112911651224119971 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_526_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (19260738728283 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(5224705415238725644339 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_526_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (19260738728283 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1326427502307451136827 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_526_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (19260738728283 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(5441736294544028744673 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_526_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (19260738728283 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(113051149611568404339 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_526_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (19260738728283 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1192601963369088836747 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_526_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (19260738728283 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(6403019035753007704611 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_526_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (19260738728283 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(1752166089705982821403 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_526_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (19260738728283 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(1565101650889641952791 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_526_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (19260738728283 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(2229402349962983556549 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_526_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (19260738728283 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(5197141997269709507939 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_526_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (19260738728283 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(12517454039065267076341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_526_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (19260738728283 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(16970931968603725312893 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_526_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (19260738728283 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_526_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (19260738728283 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_526_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (19260738728283 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_526_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_527_1 :
+    Uω (aρ 1) (bρ 1) (38590270061503 / 64000000000000 : ℝ) ≤
+      (-(645802009247295681979 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1908865720707619 / 3200000000000000 : ℝ) ≤
+      (-(645802009247295681979 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1908865720707619 / 3200000000000000 : ℝ) =
+            (1908865720707619 / 1600000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1908865720707619 / 1600000000000000 : ℝ) ≤
+            (220631966127704318021 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (238607131015911 / 400000000000000 : ℝ)) (q :=
+      (1908865720707619 / 3200000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_527_2 :
+    Uω (aρ 2) (bρ 2) (38590270061503 / 64000000000000 : ℝ) ≤
+      (-(1301649174932163817053 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (47530279183254143 / 80000000000000000 : ℝ) ≤
+      (-(1301649174932163817053 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (47530279183254143 / 80000000000000000 : ℝ) =
+            (47530279183254143 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (47530279183254143 / 40000000000000000 : ℝ) ≤
+            (431218775817836182947 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1485276582324673 / 2500000000000000 : ℝ)) (q :=
+      (47530279183254143 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_527_3 :
+    Uω (aρ 3) (bρ 3) (38590270061503 / 64000000000000 : ℝ) ≤
+      (-(660931593304168506057 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (47147517650434583 / 80000000000000000 : ℝ) ≤
+      (-(660931593304168506057 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (47147517650434583 / 80000000000000000 : ℝ) =
+            (47147517650434583 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (47147517650434583 / 40000000000000000 : ℝ) ≤
+            (205502382070831493943 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2946406026221901 / 5000000000000000 : ℝ)) (q :=
+      (47147517650434583 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_527_4 :
+    Uω (aρ 4) (bρ 4) (38590270061503 / 64000000000000 : ℝ) ≤
+      (-(1355806356402784096183 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (46511709209573511 / 80000000000000000 : ℝ) ≤
+      (-(1355806356402784096183 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (46511709209573511 / 80000000000000000 : ℝ) =
+            (46511709209573511 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (46511709209573511 / 40000000000000000 : ℝ) ≤
+            (377061594347215903817 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2906077061646767 / 5000000000000000 : ℝ)) (q :=
+      (46511709209573511 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_527_5 :
+    Uω (aρ 5) (bρ 5) (38590270061503 / 64000000000000 : ℝ) ≤
+      (-(352102682931682360927 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9108649251285719 / 16000000000000000 : ℝ) ≤
+      (-(352102682931682360927 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (9108649251285719 / 16000000000000000 : ℝ) =
+            (9108649251285719 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9108649251285719 / 8000000000000000 : ℝ) ≤
+            (81114304755817639073 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1137645660868461 / 2000000000000000 : ℝ)) (q :=
+      (9108649251285719 / 16000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_527_6 :
+    Uω (aρ 6) (bρ 6) (38590270061503 / 64000000000000 : ℝ) ≤
+      (-(46433406925091649199 / 78125000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8830760284451563 / 16000000000000000 : ℝ) ≤
+      (-(46433406925091649199 / 78125000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (8830760284451563 / 16000000000000000 : ℝ) =
+            (8830760284451563 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8830760284451563 / 8000000000000000 : ℝ) ≤
+            (7718716535845850801 / 78125000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (550810747093961 / 1000000000000000 : ℝ)) (q :=
+      (8830760284451563 / 16000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_527_7 :
+    Uω (aρ 7) (bρ 7) (38590270061503 / 64000000000000 : ℝ) ≤
+      (-(6382553448201340794523 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8451410725177719 / 16000000000000000 : ℝ) ≤
+      (-(6382553448201340794523 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (8451410725177719 / 16000000000000000 : ℝ) =
+            (8451410725177719 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8451410725177719 / 8000000000000000 : ℝ) ≤
+            (548918354798659205477 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1051516380521461 / 2000000000000000 : ℝ)) (q :=
+      (8451410725177719 / 16000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_527_8 :
+    Uω (aρ 8) (bρ 8) (38590270061503 / 64000000000000 : ℝ) ≤
+      (-(3493400107016278783557 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (39779297464548527 / 80000000000000000 : ℝ) ≤
+      (-(3493400107016278783557 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (39779297464548527 / 80000000000000000 : ℝ) =
+            (39779297464548527 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (39779297464548527 / 20000000000000000 : ℝ) ≤
+            (3438071695983721216443 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (9612665065649 / 19531250000000 : ℝ)) (q :=
+      (39779297464548527 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_527_9 :
+    Uω (aρ 9) (bρ 9) (38590270061503 / 64000000000000 : ℝ) ≤
+      (-(7801509662796789883127 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (36666944995223227 / 80000000000000000 : ℝ) ≤
+      (-(7801509662796789883127 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (36666944995223227 / 80000000000000000 : ℝ) =
+            (36666944995223227 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (36666944995223227 / 20000000000000000 : ℝ) ≤
+            (6061433943203210116873 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4484562024640963 / 10000000000000000 : ℝ)) (q :=
+      (36666944995223227 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_527_10 :
+    Uω (aρ 10) (bρ 10) (38590270061503 / 64000000000000 : ℝ) ≤
+      (-(4445100287735114038913 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (32884669502598183 / 80000000000000000 : ℝ) ≤
+      (-(4445100287735114038913 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (32884669502598183 / 80000000000000000 : ℝ) =
+            (32884669502598183 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (32884669502598183 / 20000000000000000 : ℝ) ≤
+            (2486371515264885961087 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1963776092489851 / 5000000000000000 : ℝ)) (q :=
+      (32884669502598183 / 80000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_527_11 :
+    Uω (aρ 11) (bρ 11) (38590270061503 / 64000000000000 : ℝ) ≤
+      (-(10360852771929530826219 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5677457132364783 / 16000000000000000 : ℝ) ≤
+      (-(10360852771929530826219 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5677457132364783 / 16000000000000000 : ℝ) =
+            (5677457132364783 / 4000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5677457132364783 / 4000000000000000 : ℝ) ≤
+            (3502090834070469173781 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (644336570198227 / 2000000000000000 : ℝ)) (q :=
+      (5677457132364783 / 16000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_527_12 :
+    Uω (aρ 12) (bρ 12) (38590270061503 / 64000000000000 : ℝ) ≤
+      (-(2494092935186941707943 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22988179902838547 / 80000000000000000 : ℝ) ≤
+      (-(2494092935186941707943 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (22988179902838547 / 80000000000000000 : ℝ) =
+            (22988179902838547 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22988179902838547 / 20000000000000000 : ℝ) ≤
+            (278495786013058292057 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2295606507929793 / 10000000000000000 : ℝ)) (q :=
+      (22988179902838547 / 80000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_527_13 :
+    Uω (aρ 13) (bρ 13) (38590270061503 / 64000000000000 : ℝ) ≤
+      (-(16805116772664291941307 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14902290956880979 / 80000000000000000 : ℝ) ≤
+      (-(16805116772664291941307 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (14902290956880979 / 80000000000000000 : ℝ) =
+            (14902290956880979 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14902290956880979 / 10000000000000000 : ℝ) ≤
+            (3989298636335708058693 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (673516013035401 / 10000000000000000 : ℝ)) (q :=
+      (14902290956880979 / 80000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_527_14 :
+    Uω (aρ 14) (bρ 14) (38590270061503 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_527_15 :
+    Uω (aρ 15) (bρ 15) (38590270061503 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_527_16 :
+    Uω (aρ 16) (bρ 16) (38590270061503 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_527 :
+    Uρ (38590270061503 / 64000000000000 : ℝ) ≤
+      (-(832139470866167297877 / 1000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (38590270061503 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(645802009247295681979 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_527_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (38590270061503 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1301649174932163817053 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_527_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (38590270061503 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(660931593304168506057 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_527_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (38590270061503 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1355806356402784096183 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_527_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (38590270061503 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(352102682931682360927 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_527_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (38590270061503 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) * (-(46433406925091649199 / 78125000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_527_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (38590270061503 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(6382553448201340794523 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_527_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (38590270061503 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(3493400107016278783557 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_527_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (38590270061503 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(7801509662796789883127 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_527_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (38590270061503 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(4445100287735114038913 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_527_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (38590270061503 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(10360852771929530826219 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_527_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (38590270061503 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(2494092935186941707943 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_527_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (38590270061503 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(16805116772664291941307 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_527_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (38590270061503 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_527_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (38590270061503 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_527_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (38590270061503 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_527_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_528_1 :
+    Uω (aρ 1) (bρ 1) (966476566661 / 1600000000000 : ℝ) ≤
+      (-(2574206485641140088087 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1195190854096099 / 2000000000000000 : ℝ) ≤
+      (-(2574206485641140088087 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1195190854096099 / 2000000000000000 : ℝ) =
+            (1195190854096099 / 1000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1195190854096099 / 1000000000000000 : ℝ) ≤
+            (891529415858859911913 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (298796360870737 / 500000000000000 : ℝ)) (q :=
+      (1195190854096099 / 2000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_528_2 :
+    Uω (aρ 2) (bρ 2) (966476566661 / 1600000000000 : ℝ) ≤
+      (-(1297130179868403671289 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11904068129833769 / 20000000000000000 : ℝ) ≤
+      (-(1297130179868403671289 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11904068129833769 / 20000000000000000 : ℝ) =
+            (11904068129833769 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11904068129833769 / 10000000000000000 : ℝ) ≤
+            (435737770881596328711 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5951855818797519 / 10000000000000000 : ℝ)) (q :=
+      (11904068129833769 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_528_3 :
+    Uω (aρ 3) (bρ 3) (966476566661 / 1600000000000 : ℝ) ≤
+      (-(5269228762739811181977 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2361675877447789 / 4000000000000000 : ℝ) ≤
+      (-(5269228762739811181977 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2361675877447789 / 4000000000000000 : ℝ) =
+            (2361675877447789 / 2000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2361675877447789 / 2000000000000000 : ℝ) ≤
+            (1662243040260188818023 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1180712636510539 / 2000000000000000 : ℝ)) (q :=
+      (2361675877447789 / 4000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_528_4 :
+    Uω (aρ 4) (bρ 4) (966476566661 / 1600000000000 : ℝ) ≤
+      (-(675593597482667857747 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11649431672063719 / 20000000000000000 : ℝ) ≤
+      (-(675593597482667857747 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11649431672063719 / 20000000000000000 : ℝ) =
+            (11649431672063719 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11649431672063719 / 10000000000000000 : ℝ) ≤
+            (190840377892332142253 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5822909648442469 / 10000000000000000 : ℝ)) (q :=
+      (11649431672063719 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_528_5 :
+    Uω (aρ 5) (bρ 5) (966476566661 / 1600000000000 : ℝ) ≤
+      (-(5614764140355019561613 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5703663448755089 / 10000000000000000 : ℝ) ≤
+      (-(5614764140355019561613 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5703663448755089 / 10000000000000000 : ℝ) =
+            (5703663448755089 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5703663448755089 / 5000000000000000 : ℝ) ≤
+            (1316707662644980438387 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (712374349152991 / 1250000000000000 : ℝ)) (q :=
+      (5703663448755089 / 10000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_528_6 :
+    Uω (aρ 6) (bρ 6) (966476566661 / 1600000000000 : ℝ) ≤
+      (-(5923980591695871839469 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2211998270283633 / 4000000000000000 : ℝ) ≤
+      (-(5923980591695871839469 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2211998270283633 / 4000000000000000 : ℝ) =
+            (2211998270283633 / 2000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2211998270283633 / 2000000000000000 : ℝ) ≤
+            (1007491211304128160531 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1103779924454383 / 2000000000000000 : ℝ)) (q :=
+      (2211998270283633 / 4000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_528_7 :
+    Uω (aρ 7) (bρ 7) (966476566661 / 1600000000000 : ℝ) ≤
+      (-(159053251217352304367 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1058586127126089 / 2000000000000000 : ℝ) ≤
+      (-(159053251217352304367 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1058586127126089 / 2000000000000000 : ℝ) =
+            (1058586127126089 / 1000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1058586127126089 / 1000000000000000 : ℝ) ≤
+            (14233543857647695633 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (65855386535933 / 125000000000000 : ℝ)) (q :=
+      (1058586127126089 / 2000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_528_8 :
+    Uω (aρ 8) (bρ 8) (966476566661 / 1600000000000 : ℝ) ≤
+      (-(6964984750148177349421 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9966543143463281 / 20000000000000000 : ℝ) ≤
+      (-(6964984750148177349421 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9966543143463281 / 20000000000000000 : ℝ) =
+            (9966543143463281 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9966543143463281 / 5000000000000000 : ℝ) ≤
+            (6897958855851822650579 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4932654446417031 / 10000000000000000 : ℝ)) (q :=
+      (9966543143463281 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_528_9 :
+    Uω (aρ 9) (bρ 9) (966476566661 / 1600000000000 : ℝ) ≤
+      (-(7777571051986987253239 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2297176607059593 / 5000000000000000 : ℝ) ≤
+      (-(7777571051986987253239 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2297176607059593 / 5000000000000000 : ℝ) =
+            (2297176607059593 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2297176607059593 / 1250000000000000 : ℝ) ≤
+            (6085372554013012746761 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2247891679776061 / 5000000000000000 : ℝ)) (q :=
+      (2297176607059593 / 5000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_528_10 :
+    Uω (aρ 10) (bρ 10) (966476566661 / 1600000000000 : ℝ) ≤
+      (-(4431436816050858759941 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1648732807485941 / 4000000000000000 : ℝ) ≤
+      (-(4431436816050858759941 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1648732807485941 / 4000000000000000 : ℝ) =
+            (1648732807485941 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1648732807485941 / 1000000000000000 : ℝ) ≤
+            (2500034986949141240059 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (787860000447691 / 2000000000000000 : ℝ)) (q :=
+      (1648732807485941 / 4000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_528_11 :
+    Uω (aρ 11) (bρ 11) (966476566661 / 1600000000000 : ℝ) ≤
+      (-(5163777722653102252189 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7120491318877783 / 20000000000000000 : ℝ) ≤
+      (-(5163777722653102252189 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7120491318877783 / 20000000000000000 : ℝ) =
+            (7120491318877783 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7120491318877783 / 5000000000000000 : ℝ) ≤
+            (1767694080346897747811 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3234603909891533 / 10000000000000000 : ℝ)) (q :=
+      (7120491318877783 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_528_12 :
+    Uω (aρ 12) (bρ 12) (966476566661 / 1600000000000 : ℝ) ≤
+      (-(2484760992263260070481 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5638596856701 / 19531250000000 : ℝ) ≤
+      (-(2484760992263260070481 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (5638596856701 / 19531250000000 : ℝ) = (5638596856701 / 4882812500000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5638596856701 / 4882812500000 : ℝ) ≤
+            (287827728936739929519 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1155867934480287 / 5000000000000000 : ℝ)) (q :=
+      (5638596856701 / 19531250000000 : ℝ)) (by norm_num [aρ_12, bρ_12]) (by norm_num [bρ_12])
+      (by norm_num) (by norm_num [aρ_12, bρ_12]) (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_528_13 :
+    Uω (aρ 13) (bρ 13) (966476566661 / 1600000000000 : ℝ) ≤
+      (-(260170584264644178209 / 156250000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1891733024132149 / 10000000000000000 : ℝ) ≤
+      (-(260170584264644178209 / 156250000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1891733024132149 / 10000000000000000 : ℝ) =
+            (1891733024132149 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1891733024132149 / 1250000000000000 : ℝ) ≤
+            (64742156500980821791 / 156250000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (22520639923689 / 312500000000000 : ℝ)) (q :=
+      (1891733024132149 / 10000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_528_14 :
+    Uω (aρ 14) (bρ 14) (966476566661 / 1600000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_528_15 :
+    Uω (aρ 15) (bρ 15) (966476566661 / 1600000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_528_16 :
+    Uω (aρ 16) (bρ 16) (966476566661 / 1600000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_528 :
+    Uρ (966476566661 / 1600000000000 : ℝ) ≤
+      (-(518305002869011146653 / 625000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (966476566661 / 1600000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2574206485641140088087 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_528_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (966476566661 / 1600000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1297130179868403671289 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_528_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (966476566661 / 1600000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(5269228762739811181977 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_528_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (966476566661 / 1600000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(675593597482667857747 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_528_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (966476566661 / 1600000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(5614764140355019561613 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_528_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (966476566661 / 1600000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(5923980591695871839469 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_528_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (966476566661 / 1600000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(159053251217352304367 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_528_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (966476566661 / 1600000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(6964984750148177349421 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_528_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (966476566661 / 1600000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(7777571051986987253239 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_528_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (966476566661 / 1600000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(4431436816050858759941 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_528_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (966476566661 / 1600000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(5163777722653102252189 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_528_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (966476566661 / 1600000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(2484760992263260070481 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_528_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (966476566661 / 1600000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) * (-(260170584264644178209 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_528_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (966476566661 / 1600000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_528_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (966476566661 / 1600000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_528_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (966476566661 / 1600000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_528_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_529_1 :
+    Uω (aρ 1) (bρ 1) (38727855271377 / 64000000000000 : ℝ) ≤
+      (-(5130442221812664608347 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9578725061719409 / 16000000000000000 : ℝ) ≤
+      (-(5130442221812664608347 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (9578725061719409 / 16000000000000000 : ℝ) =
+            (9578725061719409 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9578725061719409 / 8000000000000000 : ℝ) ≤
+            (1801029581187335391653 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1197335231816321 / 2000000000000000 : ℝ)) (q :=
+      (9578725061719409 / 16000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_529_2 :
+    Uω (aρ 2) (bρ 2) (38727855271377 / 64000000000000 : ℝ) ≤
+      (-(5170477356328555880841 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (47702265846114077 / 80000000000000000 : ℝ) ≤
+      (-(5170477356328555880841 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (47702265846114077 / 80000000000000000 : ℝ) =
+            (47702265846114077 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (47702265846114077 / 40000000000000000 : ℝ) ≤
+            (1760994446671444119159 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5962605305970863 / 10000000000000000 : ℝ)) (q :=
+      (47702265846114077 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_529_3 :
+    Uω (aρ 3) (bρ 3) (38727855271377 / 64000000000000 : ℝ) ≤
+      (-(5251037937022816384007 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (47319517414242301 / 80000000000000000 : ℝ) ≤
+      (-(5251037937022816384007 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (47319517414242301 / 80000000000000000 : ℝ) =
+            (47319517414242301 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (47319517414242301 / 40000000000000000 : ℝ) ≤
+            (1680433865977183615993 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5914314304352919 / 10000000000000000 : ℝ)) (q :=
+      (47319517414242301 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_529_4 :
+    Uω (aρ 4) (bρ 4) (38727855271377 / 64000000000000 : ℝ) ≤
+      (-(5386306230905287627101 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (46683744068430289 / 80000000000000000 : ℝ) ≤
+      (-(5386306230905287627101 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (46683744068430289 / 80000000000000000 : ℝ) =
+            (46683744068430289 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (46683744068430289 / 40000000000000000 : ℝ) ≤
+            (1545165572094712372899 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1458416287241229 / 2500000000000000 : ℝ)) (q :=
+      (46683744068430289 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_529_5 :
+    Uω (aρ 5) (bρ 5) (38727855271377 / 64000000000000 : ℝ) ≤
+      (-(5595920985687641589971 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (45715368657705249 / 80000000000000000 : ℝ) ≤
+      (-(5595920985687641589971 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (45715368657705249 / 80000000000000000 : ℝ) =
+            (45715368657705249 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (45715368657705249 / 40000000000000000 : ℝ) ≤
+            (1335550817312358410029 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (178430037988083 / 312500000000000 : ℝ)) (q :=
+      (45715368657705249 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_529_6 :
+    Uω (aρ 6) (bρ 6) (38727855271377 / 64000000000000 : ℝ) ≤
+      (-(738065397838446361183 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (44326128714278213 / 80000000000000000 : ℝ) ≤
+      (-(738065397838446361183 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (44326128714278213 / 80000000000000000 : ℝ) =
+            (44326128714278213 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (44326128714278213 / 40000000000000000 : ℝ) ≤
+            (128368577536553638817 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5529691604901897 / 10000000000000000 : ℝ)) (q :=
+      (44326128714278213 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_529_7 :
+    Uω (aρ 7) (bρ 7) (38727855271377 / 64000000000000 : ℝ) ≤
+      (-(792718582758281840787 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8485966980979493 / 16000000000000000 : ℝ) ≤
+      (-(792718582758281840787 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (8485966980979493 / 16000000000000000 : ℝ) =
+            (8485966980979493 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8485966980979493 / 8000000000000000 : ℝ) ≤
+            (73715392616718159213 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (527927953331671 / 1000000000000000 : ℝ)) (q :=
+      (8485966980979493 / 16000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_529_8 :
+    Uω (aρ 8) (bρ 8) (38727855271377 / 64000000000000 : ℝ) ≤
+      (-(6943217746578683048433 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7990608759587449 / 16000000000000000 : ℝ) ≤
+      (-(6943217746578683048433 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7990608759587449 / 16000000000000000 : ℝ) =
+            (7990608759587449 / 4000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 15
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7990608759587449 / 4000000000000000 : ℝ) ≤
+            (6919725859421316951567 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (988724681583331 / 2000000000000000 : ℝ)) (q :=
+      (7990608759587449 / 16000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_529_9 :
+    Uω (aρ 9) (bρ 9) (38727855271377 / 64000000000000 : ℝ) ≤
+      (-(7753692110675952395933 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (36842697217734573 / 80000000000000000 : ℝ) ≤
+      (-(7753692110675952395933 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (36842697217734573 / 80000000000000000 : ℝ) =
+            (36842697217734573 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (36842697217734573 / 20000000000000000 : ℝ) ≤
+            (6109251495324047604067 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4507002391225987 / 10000000000000000 : ℝ)) (q :=
+      (36842697217734573 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_529_10 :
+    Uω (aρ 10) (bρ 10) (38727855271377 / 64000000000000 : ℝ) ≤
+      (-(4417814021261383902239 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (33064620044831381 / 80000000000000000 : ℝ) ≤
+      (-(4417814021261383902239 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (33064620044831381 / 80000000000000000 : ℝ) =
+            (33064620044831381 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (33064620044831381 / 20000000000000000 : ℝ) ≤
+            (2513657781738616097761 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3951042131495189 / 10000000000000000 : ℝ)) (q :=
+      (33064620044831381 / 80000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_529_11 :
+    Uω (aρ 11) (bρ 11) (38727855271377 / 64000000000000 : ℝ) ≤
+      (-(10294390783422403344033 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5715316311891197 / 16000000000000000 : ℝ) ≤
+      (-(10294390783422403344033 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5715316311891197 / 16000000000000000 : ℝ) =
+            (5715316311891197 / 4000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5715316311891197 / 4000000000000000 : ℝ) ≤
+            (3568552822577596655967 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (162375456817817 / 500000000000000 : ℝ)) (q :=
+      (5715316311891197 / 16000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_529_12 :
+    Uω (aρ 12) (bρ 12) (38727855271377 / 64000000000000 : ℝ) ≤
+      (-(12377469048134864274801 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4640591408787557 / 16000000000000000 : ℝ) ≤
+      (-(12377469048134864274801 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4640591408787557 / 16000000000000000 : ℝ) =
+            (4640591408787557 / 4000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4640591408787557 / 4000000000000000 : ℝ) ≤
+            (1485474557865135725199 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (232780310416179 / 1000000000000000 : ℝ)) (q :=
+      (4640591408787557 / 16000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_529_13 :
+    Uω (aρ 13) (bρ 13) (38727855271377 / 64000000000000 : ℝ) ≤
+      (-(8253109536821764390211 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3070888133605453 / 16000000000000000 : ℝ) ≤
+      (-(8253109536821764390211 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3070888133605453 / 16000000000000000 : ℝ) =
+            (3070888133605453 / 2000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3070888133605453 / 2000000000000000 : ℝ) ≤
+            (2144098167678235609789 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (19126393794479 / 250000000000000 : ℝ)) (q :=
+      (3070888133605453 / 16000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_529_14 :
+    Uω (aρ 14) (bρ 14) (38727855271377 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_529_15 :
+    Uω (aρ 15) (bρ 15) (38727855271377 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_529_16 :
+    Uω (aρ 16) (bρ 16) (38727855271377 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_529 :
+    Uρ (38727855271377 / 64000000000000 : ℝ) ≤
+      (-(8264987908352755707917 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (38727855271377 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(5130442221812664608347 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_529_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (38727855271377 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(5170477356328555880841 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_529_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (38727855271377 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(5251037937022816384007 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_529_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (38727855271377 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(5386306230905287627101 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_529_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (38727855271377 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(5595920985687641589971 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_529_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (38727855271377 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(738065397838446361183 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_529_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (38727855271377 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(792718582758281840787 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_529_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (38727855271377 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(6943217746578683048433 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_529_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (38727855271377 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(7753692110675952395933 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_529_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (38727855271377 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(4417814021261383902239 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_529_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (38727855271377 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(10294390783422403344033 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_529_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (38727855271377 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(12377469048134864274801 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_529_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (38727855271377 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(8253109536821764390211 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_529_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (38727855271377 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_529_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (38727855271377 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_529_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (38727855271377 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_529_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_530_1 :
+    Uω (aρ 1) (bρ 1) (19398323938157 / 32000000000000 : ℝ) ≤
+      (-(102250074189872219547 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23989808225978627 / 40000000000000000 : ℝ) ≤
+      (-(102250074189872219547 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23989808225978627 / 40000000000000000 : ℝ) =
+            (23989808225978627 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23989808225978627 / 20000000000000000 : ℝ) ≤
+            (36379361870127780453 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5997425100400251 / 10000000000000000 : ℝ)) (q :=
+      (23989808225978627 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_530_2 :
+    Uω (aρ 2) (bρ 2) (19398323938157 / 32000000000000 : ℝ) ≤
+      (-(1288116623196728951863 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23894129581820681 / 40000000000000000 : ℝ) ≤
+      (-(1288116623196728951863 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23894129581820681 / 40000000000000000 : ℝ) =
+            (23894129581820681 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23894129581820681 / 20000000000000000 : ℝ) ≤
+            (444751327553271048137 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2986677395415639 / 5000000000000000 : ℝ)) (q :=
+      (23894129581820681 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_530_3 :
+    Uω (aρ 3) (bρ 3) (19398323938157 / 32000000000000 : ℝ) ≤
+      (-(1308220037204054730923 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23702758623237533 / 40000000000000000 : ℝ) ≤
+      (-(1308220037204054730923 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23702758623237533 / 40000000000000000 : ℝ) =
+            (23702758623237533 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23702758623237533 / 20000000000000000 : ℝ) ≤
+            (424647913545945269077 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (740633177236213 / 1250000000000000 : ℝ)) (q :=
+      (23702758623237533 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_530_4 :
+    Uω (aρ 4) (bρ 4) (19398323938157 / 32000000000000 : ℝ) ≤
+      (-(670987206631612576549 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4676976135064359 / 8000000000000000 : ℝ) ≤
+      (-(670987206631612576549 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (4676976135064359 / 8000000000000000 : ℝ) =
+            (4676976135064359 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4676976135064359 / 4000000000000000 : ℝ) ≤
+            (195446768743387423451 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1168884124999367 / 2000000000000000 : ℝ)) (q :=
+      (4676976135064359 / 8000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_530_5 :
+    Uω (aρ 5) (bρ 5) (19398323938157 / 32000000000000 : ℝ) ≤
+      (-(5577113328436471908801 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22900714730461903 / 40000000000000000 : ℝ) ≤
+      (-(5577113328436471908801 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (22900714730461903 / 40000000000000000 : ℝ) =
+            (22900714730461903 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22900714730461903 / 20000000000000000 : ℝ) ≤
+            (1354358474563528091199 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5720527571901889 / 10000000000000000 : ℝ)) (q :=
+      (22900714730461903 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_530_6 :
+    Uω (aρ 6) (bρ 6) (19398323938157 / 32000000000000 : ℝ) ≤
+      (-(5885103710340832165453 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22206145676008869 / 40000000000000000 : ℝ) ≤
+      (-(5885103710340832165453 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (22206145676008869 / 40000000000000000 : ℝ) =
+            (22206145676008869 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22206145676008869 / 20000000000000000 : ℝ) ≤
+            (1046368092659167834547 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1385120854953843 / 2500000000000000 : ℝ)) (q :=
+      (22206145676008869 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_530_7 :
+    Uω (aρ 7) (bρ 7) (19398323938157 / 32000000000000 : ℝ) ≤
+      (-(1264281822850228723667 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4251622309553559 / 8000000000000000 : ℝ) ≤
+      (-(1264281822850228723667 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (4251622309553559 / 8000000000000000 : ℝ) =
+            (4251622309553559 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4251622309553559 / 4000000000000000 : ℝ) ≤
+            (122012537749771276333 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1058025547290967 / 2000000000000000 : ℝ)) (q :=
+      (4251622309553559 / 8000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_530_8 :
+    Uω (aρ 8) (bρ 8) (19398323938157 / 32000000000000 : ℝ) ≤
+      (-(6921498986855282207523 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20019955581302669 / 40000000000000000 : ℝ) ≤
+      (-(6921498986855282207523 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (20019955581302669 / 40000000000000000 : ℝ) =
+            (20019955581302669 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 2
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20019955581302669 / 20000000000000000 : ℝ) ≤
+            (9972816144717792477 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (154830981392571 / 312500000000000 : ℝ)) (q :=
+      (20019955581302669 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_530_9 :
+    Uω (aρ 9) (bρ 9) (19398323938157 / 32000000000000 : ℝ) ≤
+      (-(1932468132506487573157 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (18465279789097771 / 40000000000000000 : ℝ) ≤
+      (-(1932468132506487573157 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (18465279789097771 / 40000000000000000 : ℝ) =
+            (18465279789097771 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18465279789097771 / 10000000000000000 : ℝ) ≤
+            (1533267768993512426843 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4518219136819823 / 10000000000000000 : ℝ)) (q :=
+      (18465279789097771 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_530_10 :
+    Uω (aρ 10) (bρ 10) (19398323938157 / 32000000000000 : ℝ) ≤
+      (-(8808463284906202056157 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16577280695092817 / 40000000000000000 : ℝ) ≤
+      (-(8808463284906202056157 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (16577280695092817 / 40000000000000000 : ℝ) =
+            (16577280695092817 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16577280695092817 / 10000000000000000 : ℝ) ≤
+            (5054480321093797943843 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1981389311656173 / 5000000000000000 : ℝ)) (q :=
+      (16577280695092817 / 40000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_530_11 :
+    Uω (aρ 11) (bρ 11) (19398323938157 / 32000000000000 : ℝ) ≤
+      (-(513067878678141954839 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14335567632834491 / 40000000000000000 : ℝ) ≤
+      (-(513067878678141954839 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (14335567632834491 / 40000000000000000 : ℝ) =
+            (14335567632834491 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14335567632834491 / 10000000000000000 : ℝ) ≤
+            (180079301621858045161 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3260398718388183 / 10000000000000000 : ℝ)) (q :=
+      (14335567632834491 / 40000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_530_12 :
+    Uω (aρ 12) (bρ 12) (19398323938157 / 32000000000000 : ℝ) ≤
+      (-(1541431407737169729987 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2330997797011611 / 8000000000000000 : ℝ) ≤
+      (-(1541431407737169729987 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2330997797011611 / 8000000000000000 : ℝ) =
+            (2330997797011611 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2330997797011611 / 2000000000000000 : ℝ) ≤
+            (191436543012830270013 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (468761898236993 / 2000000000000000 : ℝ)) (q :=
+      (2330997797011611 / 8000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_530_13 :
+    Uω (aρ 13) (bρ 13) (19398323938157 / 32000000000000 : ℝ) ≤
+      (-(16369481579601011001041 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7782917713863881 / 40000000000000000 : ℝ) ≤
+      (-(16369481579601011001041 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (7782917713863881 / 40000000000000000 : ℝ) =
+            (7782917713863881 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7782917713863881 / 5000000000000000 : ℝ) ≤
+            (4424933829398988998959 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (403577798591439 / 5000000000000000 : ℝ)) (q :=
+      (7782917713863881 / 40000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_530_14 :
+    Uω (aρ 14) (bρ 14) (19398323938157 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_530_15 :
+    Uω (aρ 15) (bρ 15) (19398323938157 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_530_16 :
+    Uω (aρ 16) (bρ 16) (19398323938157 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_530 :
+    Uρ (19398323938157 / 32000000000000 : ℝ) ≤
+      (-(8237627031248805762037 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (19398323938157 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(102250074189872219547 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_530_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (19398323938157 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1288116623196728951863 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_530_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (19398323938157 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1308220037204054730923 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_530_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (19398323938157 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(670987206631612576549 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_530_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (19398323938157 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(5577113328436471908801 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_530_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (19398323938157 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(5885103710340832165453 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_530_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (19398323938157 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(1264281822850228723667 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_530_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (19398323938157 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(6921498986855282207523 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_530_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (19398323938157 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(1932468132506487573157 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_530_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (19398323938157 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(8808463284906202056157 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_530_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (19398323938157 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) * (-(513067878678141954839 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_530_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (19398323938157 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(1541431407737169729987 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_530_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (19398323938157 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(16369481579601011001041 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_530_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (19398323938157 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_530_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (19398323938157 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_530_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (19398323938157 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_530_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_531_1 :
+    Uω (aρ 1) (bρ 1) (38865440481251 / 64000000000000 : ℝ) ≤
+      (-(1273649329718194240281 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (48065607593932067 / 80000000000000000 : ℝ) ≤
+      (-(1273649329718194240281 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (48065607593932067 / 80000000000000000 : ℝ) =
+            (48065607593932067 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (48065607593932067 / 40000000000000000 : ℝ) ≤
+            (459218621031805759719 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1502043510343137 / 2500000000000000 : ℝ)) (q :=
+      (48065607593932067 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_531_2 :
+    Uω (aρ 2) (bρ 2) (38865440481251 / 64000000000000 : ℝ) ≤
+      (-(1026897602395204205191 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (47874252471966791 / 80000000000000000 : ℝ) ≤
+      (-(1026897602395204205191 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (47874252471966791 / 80000000000000000 : ℝ) =
+            (47874252471966791 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (47874252471966791 / 40000000000000000 : ℝ) ≤
+            (359396758204795794809 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5984104273391229 / 10000000000000000 : ℝ)) (q :=
+      (47874252471966791 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_531_3 :
+    Uω (aρ 3) (bρ 3) (38865440481251 / 64000000000000 : ℝ) ≤
+      (-(5214755278309320413741 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (47491517045833679 / 80000000000000000 : ℝ) ≤
+      (-(5214755278309320413741 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (47491517045833679 / 80000000000000000 : ℝ) =
+            (47491517045833679 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (47491517045833679 / 40000000000000000 : ℝ) ≤
+            (1716716524690679586259 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5935816523207951 / 10000000000000000 : ℝ)) (q :=
+      (47491517045833679 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_531_4 :
+    Uω (aρ 4) (bρ 4) (38865440481251 / 64000000000000 : ℝ) ≤
+      (-(5349522921308249214813 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (46855778535434627 / 80000000000000000 : ℝ) ≤
+      (-(5349522921308249214813 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (46855778535434627 / 80000000000000000 : ℝ) =
+            (46855778535434627 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (46855778535434627 / 40000000000000000 : ℝ) ≤
+            (1581948881691750785187 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1463794019168297 / 2500000000000000 : ℝ)) (q :=
+      (46855778535434627 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_531_5 :
+    Uω (aρ 5) (bρ 5) (38865440481251 / 64000000000000 : ℝ) ≤
+      (-(5558341034894179748239 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (45887490001186683 / 80000000000000000 : ℝ) ≤
+      (-(5558341034894179748239 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (45887490001186683 / 80000000000000000 : ℝ) =
+            (45887490001186683 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (45887490001186683 / 40000000000000000 : ℝ) ≤
+            (1373130768105820251761 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2865646931223101 / 5000000000000000 : ℝ)) (q :=
+      (45887490001186683 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_531_6 :
+    Uω (aρ 6) (bρ 6) (38865440481251 / 64000000000000 : ℝ) ≤
+      (-(2932861013183497161807 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1779938132912151 / 3200000000000000 : ℝ) ≤
+      (-(2932861013183497161807 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1779938132912151 / 3200000000000000 : ℝ) =
+            (1779938132912151 / 1600000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1779938132912151 / 1600000000000000 : ℝ) ≤
+            (532874888316502838193 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (222051002719619 / 400000000000000 : ℝ)) (q :=
+      (1779938132912151 / 3200000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_531_7 :
+    Uω (aρ 7) (bρ 7) (38865440481251 / 64000000000000 : ℝ) ≤
+      (-(6301111232271157612451 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (42602609666960279 / 80000000000000000 : ℝ) ≤
+      (-(6301111232271157612451 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (42602609666960279 / 80000000000000000 : ℝ) =
+            (42602609666960279 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (42602609666960279 / 40000000000000000 : ℝ) ≤
+            (630360570728842387549 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5300975534789601 / 10000000000000000 : ℝ)) (q :=
+      (42602609666960279 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_531_8 :
+    Uω (aρ 8) (bρ 8) (38865440481251 / 64000000000000 : ℝ) ≤
+      (-(6899828248220443165641 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (40126774693431603 / 80000000000000000 : ℝ) ≤
+      (-(6899828248220443165641 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (40126774693431603 / 80000000000000000 : ℝ) =
+            (40126774693431603 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (40126774693431603 / 40000000000000000 : ℝ) ≤
+            (31643554779556834359 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (620694805343429 / 1250000000000000 : ℝ)) (q :=
+      (40126774693431603 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_531_9 :
+    Uω (aρ 9) (bρ 9) (38865440481251 / 64000000000000 : ℝ) ≤
+      (-(1926528000919187657291 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (37018412862271711 / 80000000000000000 : ℝ) ≤
+      (-(1926528000919187657291 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (37018412862271711 / 80000000000000000 : ℝ) =
+            (37018412862271711 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (37018412862271711 / 20000000000000000 : ℝ) ≤
+            (1539207900580812342709 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4529433613317459 / 10000000000000000 : ℝ)) (q :=
+      (37018412862271711 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_531_10 :
+    Uω (aρ 10) (bρ 10) (38865440481251 / 64000000000000 : ℝ) ≤
+      (-(8781378842744531549129 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (33244480385545891 / 80000000000000000 : ℝ) ≤
+      (-(8781378842744531549129 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (33244480385545891 / 80000000000000000 : ℝ) =
+            (33244480385545891 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (33244480385545891 / 20000000000000000 : ℝ) ≤
+            (5081564763255468450871 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (993627381907751 / 2500000000000000 : ℝ)) (q :=
+      (33244480385545891 / 80000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_531_11 :
+    Uω (aρ 11) (bρ 11) (38865440481251 / 64000000000000 : ℝ) ≤
+      (-(1278556827644119619991 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28765627133410979 / 80000000000000000 : ℝ) ≤
+      (-(1278556827644119619991 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (28765627133410979 / 80000000000000000 : ℝ) =
+            (28765627133410979 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28765627133410979 / 20000000000000000 : ℝ) ≤
+            (454311123105880380009 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (818318210200569 / 2500000000000000 : ℝ)) (q :=
+      (28765627133410979 / 80000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_531_12 :
+    Uω (aρ 12) (bρ 12) (38865440481251 / 64000000000000 : ℝ) ≤
+      (-(6142873046797036347843 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4683352091286291 / 16000000000000000 : ℝ) ≤
+      (-(6142873046797036347843 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4683352091286291 / 16000000000000000 : ℝ) =
+            (4683352091286291 / 4000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4683352091286291 / 4000000000000000 : ℝ) ≤
+            (788598756202963652157 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (471951253648479 / 2000000000000000 : ℝ)) (q :=
+      (4683352091286291 / 16000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_531_13 :
+    Uω (aρ 13) (bρ 13) (38865440481251 / 64000000000000 : ℝ) ≤
+      (-(16239541782885318070361 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15769417382286619 / 80000000000000000 : ℝ) ≤
+      (-(16239541782885318070361 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (15769417382286619 / 80000000000000000 : ℝ) =
+            (15769417382286619 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15769417382286619 / 10000000000000000 : ℝ) ≤
+            (4554873626114681929639 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (423651120650593 / 5000000000000000 : ℝ)) (q :=
+      (15769417382286619 / 80000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_531_14 :
+    Uω (aρ 14) (bρ 14) (38865440481251 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_531_15 :
+    Uω (aρ 15) (bρ 15) (38865440481251 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_531_16 :
+    Uω (aρ 16) (bρ 16) (38865440481251 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_531 :
+    Uρ (38865440481251 / 64000000000000 : ℝ) ≤
+      (-(1642145670387592022313 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (38865440481251 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1273649329718194240281 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_531_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (38865440481251 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1026897602395204205191 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_531_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (38865440481251 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(5214755278309320413741 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_531_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (38865440481251 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(5349522921308249214813 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_531_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (38865440481251 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(5558341034894179748239 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_531_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (38865440481251 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(2932861013183497161807 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_531_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (38865440481251 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(6301111232271157612451 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_531_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (38865440481251 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(6899828248220443165641 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_531_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (38865440481251 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(1926528000919187657291 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_531_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (38865440481251 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(8781378842744531549129 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_531_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (38865440481251 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(1278556827644119619991 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_531_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (38865440481251 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(6142873046797036347843 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_531_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (38865440481251 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(16239541782885318070361 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_531_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (38865440481251 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_531_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (38865440481251 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_531_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (38865440481251 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_531_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/U44.lean b/LeanPool/Zeta5Irrational/Table/U44.lean
new file mode 100644
index 0000000000..ee2acf913e
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/U44.lean
@@ -0,0 +1,4829 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-! # Certified arcsine potential bounds (U44) -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma U_532_1 :
+    Uω (aρ 1) (bρ 1) (9733558271547 / 16000000000000 : ℝ) ≤
+      (-(1015344587023385956413 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12037899683632227 / 20000000000000000 : ℝ) ≤
+      (-(1015344587023385956413 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (12037899683632227 / 20000000000000000 : ℝ) =
+            (12037899683632227 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12037899683632227 / 10000000000000000 : ℝ) ≤
+            (370949773576614043587 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (188091343187511 / 312500000000000 : ℝ)) (q :=
+      (12037899683632227 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_532_2 :
+    Uω (aρ 2) (bρ 2) (9733558271547 / 16000000000000 : ℝ) ≤
+      (-(2558270898826269298443 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2398012288556993 / 4000000000000000 : ℝ) ≤
+      (-(2558270898826269298443 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2398012288556993 / 4000000000000000 : ℝ) =
+            (2398012288556993 / 2000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2398012288556993 / 2000000000000000 : ℝ) ≤
+            (907465002673730701557 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (599485375366309 / 1000000000000000 : ℝ)) (q :=
+      (2398012288556993 / 4000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_532_3 :
+    Uω (aρ 3) (bρ 3) (9733558271547 / 16000000000000 : ℝ) ≤
+      (-(12991658015855303567 / 25000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1189437920312411 / 2000000000000000 : ℝ) ≤
+      (-(12991658015855303567 / 25000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1189437920312411 / 2000000000000000 : ℝ) =
+            (1189437920312411 / 1000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1189437920312411 / 1000000000000000 : ℝ) ≤
+            (4337021491644696433 / 25000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1189313524070447 / 2000000000000000 : ℝ)) (q :=
+      (1189437920312411 / 2000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_532_4 :
+    Uω (aρ 4) (bρ 4) (9733558271547 / 16000000000000 : ℝ) ≤
+      (-(5331181911364434298457 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5867724452917411 / 10000000000000000 : ℝ) ≤
+      (-(5331181911364434298457 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5867724452917411 / 10000000000000000 : ℝ) =
+            (5867724452917411 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5867724452917411 / 5000000000000000 : ℝ) ≤
+            (1600289891635565701543 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5865931504127947 / 10000000000000000 : ℝ)) (q :=
+      (5867724452917411 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_532_5 :
+    Uω (aρ 5) (bρ 5) (9733558271547 / 16000000000000 : ℝ) ≤
+      (-(2769801986054336931911 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (359168361562289 / 625000000000000 : ℝ) ≤
+      (-(2769801986054336931911 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (359168361562289 / 625000000000000 : ℝ) =
+            (359168361562289 / 312500000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (359168361562289 / 312500000000000 : ℝ) ≤
+            (695933915445663068089 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5742060087621373 / 10000000000000000 : ℝ)) (q :=
+      (359168361562289 / 625000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_532_6 :
+    Uω (aρ 6) (bρ 6) (9733558271547 / 16000000000000 : ℝ) ≤
+      (-(58463779834278348057 / 100000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1114615365762961 / 2000000000000000 : ℝ) ≤
+      (-(58463779834278348057 / 100000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1114615365762961 / 2000000000000000 : ℝ) =
+            (1114615365762961 / 1000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1114615365762961 / 1000000000000000 : ℝ) ≤
+            (10850938195721651943 / 100000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1112413310079547 / 2000000000000000 : ℝ)) (q :=
+      (1114615365762961 / 2000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_532_7 :
+    Uω (aρ 7) (bρ 7) (9733558271547 / 16000000000000 : ℝ) ≤
+      (-(6280854844228632773047 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1334031082159119 / 2500000000000000 : ℝ) ≤
+      (-(6280854844228632773047 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1334031082159119 / 2500000000000000 : ℝ) =
+            (1334031082159119 / 1250000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1334031082159119 / 1250000000000000 : ℝ) ≤
+            (650616958771367226953 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5311822930801077 / 10000000000000000 : ℝ)) (q :=
+      (1334031082159119 / 2500000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_532_8 :
+    Uω (aρ 8) (bρ 8) (9733558271547 / 16000000000000 : ℝ) ≤
+      (-(343910265860526981271 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2010681720788129 / 4000000000000000 : ℝ) ≤
+      (-(343910265860526981271 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2010681720788129 / 4000000000000000 : ℝ) =
+            (2010681720788129 / 2000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2010681720788129 / 2000000000000000 : ℝ) ≤
+            (2663324289473018729 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (497652452880877 / 1000000000000000 : ℝ)) (q :=
+      (2010681720788129 / 4000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_532_9 :
+    Uω (aρ 9) (bρ 9) (9733558271547 / 16000000000000 : ℝ) ≤
+      (-(7682410227718395423797 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2319141071075849 / 5000000000000000 : ℝ) ≤
+      (-(7682410227718395423797 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2319141071075849 / 5000000000000000 : ℝ) =
+            (2319141071075849 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2319141071075849 / 1250000000000000 : ℝ) ≤
+            (6180533378281604576203 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4540645837531521 / 10000000000000000 : ℝ)) (q :=
+      (2319141071075849 / 5000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_532_10 :
+    Uω (aρ 10) (bρ 10) (9733558271547 / 16000000000000 : ℝ) ≤
+      (-(1750874840954918337967 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8333594307057603 / 20000000000000000 : ℝ) ≤
+      (-(1750874840954918337967 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8333594307057603 / 20000000000000000 : ℝ) =
+            (8333594307057603 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8333594307057603 / 5000000000000000 : ℝ) ≤
+            (1021713880245081662033 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (498279361722591 / 1250000000000000 : ℝ)) (q :=
+      (8333594307057603 / 20000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_532_11 :
+    Uω (aρ 11) (bρ 11) (9733558271547 / 16000000000000 : ℝ) ≤
+      (-(2548920187344628660401 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7215014472369557 / 20000000000000000 : ℝ) ≤
+      (-(2548920187344628660401 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7215014472369557 / 20000000000000000 : ℝ) =
+            (7215014472369557 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7215014472369557 / 5000000000000000 : ℝ) ≤
+            (916815714155371339599 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1643065842648841 / 5000000000000000 : ℝ)) (q :=
+      (7215014472369557 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_532_12 :
+    Uω (aρ 12) (bρ 12) (9733558271547 / 16000000000000 : ℝ) ≤
+      (-(3060087048242988541479 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (58808273261367 / 200000000000000 : ℝ) ≤
+      (-(3060087048242988541479 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (58808273261367 / 200000000000000 : ℝ) = (58808273261367 / 50000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (58808273261367 / 50000000000000 : ℝ) ≤
+            (405648853257011458521 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (95025785429993 / 400000000000000 : ℝ)) (q :=
+      (58808273261367 / 200000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12]) (by norm_num [bρ_12])
+      (by norm_num) (by norm_num [aρ_12, bρ_12]) (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_532_13 :
+    Uω (aρ 13) (bρ 13) (9733558271547 / 16000000000000 : ℝ) ≤
+      (-(16115495756291320341457 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3991562256075879 / 20000000000000000 : ℝ) ≤
+      (-(16115495756291320341457 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3991562256075879 / 20000000000000000 : ℝ) =
+            (3991562256075879 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3991562256075879 / 2500000000000000 : ℝ) ≤
+            (4678919652708679658543 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (221440326821001 / 2500000000000000 : ℝ)) (q :=
+      (3991562256075879 / 20000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_532_14 :
+    Uω (aρ 14) (bρ 14) (9733558271547 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_532_15 :
+    Uω (aρ 15) (bρ 15) (9733558271547 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_532_16 :
+    Uω (aρ 16) (bρ 16) (9733558271547 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_532 :
+    Uρ (9733558271547 / 16000000000000 : ℝ) ≤
+      (-(4092119036616476838689 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (9733558271547 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1015344587023385956413 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_532_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (9733558271547 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(2558270898826269298443 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_532_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (9733558271547 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) * (-(12991658015855303567 / 25000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_532_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (9733558271547 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(5331181911364434298457 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_532_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (9733558271547 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(2769801986054336931911 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_532_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (9733558271547 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) * (-(58463779834278348057 / 100000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_532_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (9733558271547 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(6280854844228632773047 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_532_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (9733558271547 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(343910265860526981271 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_532_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (9733558271547 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(7682410227718395423797 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_532_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (9733558271547 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(1750874840954918337967 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_532_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (9733558271547 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(2548920187344628660401 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_532_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (9733558271547 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(3060087048242988541479 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_532_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (9733558271547 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(16115495756291320341457 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_532_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (9733558271547 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_532_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (9733558271547 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_532_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (9733558271547 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_532_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_533_1 :
+    Uω (aρ 1) (bρ 1) (312024205529 / 512000000000 : ℝ) ≤
+      (-(2529440222003763779621 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9647517974751029 / 16000000000000000 : ℝ) ≤
+      (-(2529440222003763779621 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (9647517974751029 / 16000000000000000 : ℝ) =
+            (9647517974751029 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9647517974751029 / 8000000000000000 : ℝ) ≤
+            (936295679496236220379 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1205934384457101 / 2000000000000000 : ℝ)) (q :=
+      (9647517974751029 / 16000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_533_2 :
+    Uω (aρ 2) (bρ 2) (312024205529 / 512000000000 : ℝ) ≤
+      (-(5098627734197961890647 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (48046239061209721 / 80000000000000000 : ℝ) ≤
+      (-(5098627734197961890647 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (48046239061209721 / 80000000000000000 : ℝ) =
+            (48046239061209721 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (48046239061209721 / 40000000000000000 : ℝ) ≤
+            (1832844068802038109353 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6005603231659149 / 10000000000000000 : ℝ)) (q :=
+      (48046239061209721 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_533_3 :
+    Uω (aρ 3) (bρ 3) (312024205529 / 512000000000 : ℝ) ≤
+      (-(5178603814400613640559 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (47663516546640373 / 80000000000000000 : ℝ) ≤
+      (-(5178603814400613640559 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (47663516546640373 / 80000000000000000 : ℝ) =
+            (47663516546640373 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (47663516546640373 / 40000000000000000 : ℝ) ≤
+            (1752867988599386359441 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1489329677341703 / 2500000000000000 : ℝ)) (q :=
+      (47663516546640373 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_533_4 :
+    Uω (aρ 4) (bρ 4) (312024205529 / 512000000000 : ℝ) ≤
+      (-(5312874499598534674163 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1881112504595581 / 3200000000000000 : ℝ) ≤
+      (-(5312874499598534674163 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1881112504595581 / 3200000000000000 : ℝ) =
+            (1881112504595581 / 1600000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1881112504595581 / 1600000000000000 : ℝ) ≤
+            (1618597303401465325837 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (58766869074941 / 100000000000000 : ℝ)) (q :=
+      (1881112504595581 / 3200000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_533_5 :
+    Uω (aρ 5) (bρ 5) (312024205529 / 512000000000 : ℝ) ≤
+      (-(5520902007877414746151 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (46059610298750769 / 80000000000000000 : ℝ) ≤
+      (-(5520902007877414746151 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (46059610298750769 / 80000000000000000 : ℝ) =
+            (46059610298750769 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (46059610298750769 / 40000000000000000 : ℝ) ≤
+            (1410569795122585253849 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5752826247794411 / 10000000000000000 : ℝ)) (q :=
+      (46059610298750769 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_533_6 :
+    Uω (aρ 6) (bρ 6) (312024205529 / 512000000000 : ℝ) ≤
+      (-(2913535717514371392481 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (44670775279013717 / 80000000000000000 : ℝ) ≤
+      (-(2913535717514371392481 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (44670775279013717 / 80000000000000000 : ℝ) =
+            (44670775279013717 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (44670775279013717 / 40000000000000000 : ℝ) ≤
+            (552200183985628607519 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1393214467000037 / 2500000000000000 : ℝ)) (q :=
+      (44670775279013717 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_533_7 :
+    Uω (aρ 7) (bρ 7) (312024205529 / 512000000000 : ℝ) ≤
+      (-(6260639779296802408789 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (42775377991769201 / 80000000000000000 : ℝ) ≤
+      (-(6260639779296802408789 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (42775377991769201 / 80000000000000000 : ℝ) =
+            (42775377991769201 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (42775377991769201 / 40000000000000000 : ℝ) ≤
+            (670832023703197591211 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5322669926949019 / 10000000000000000 : ℝ)) (q :=
+      (42775377991769201 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_533_8 :
+    Uω (aρ 8) (bρ 8) (312024205529 / 512000000000 : ℝ) ≤
+      (-(6856629979244132857753 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (40300490354719693 / 80000000000000000 : ℝ) ≤
+      (-(6856629979244132857753 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (40300490354719693 / 80000000000000000 : ℝ) =
+            (40300490354719693 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (40300490354719693 / 40000000000000000 : ℝ) ≤
+            (74841823755867142247 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2493744834513321 / 5000000000000000 : ℝ)) (q :=
+      (40300490354719693 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_533_9 :
+    Uω (aρ 9) (bρ 9) (312024205529 / 512000000000 : ℝ) ≤
+      (-(3829383450332178808059 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (37194092469595613 / 80000000000000000 : ℝ) ≤
+      (-(3829383450332178808059 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (37194092469595613 / 80000000000000000 : ℝ) =
+            (37194092469595613 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (37194092469595613 / 20000000000000000 : ℝ) ≤
+            (3102088352667821191941 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2275927913052811 / 5000000000000000 : ℝ)) (q :=
+      (37194092469595613 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_533_10 :
+    Uω (aρ 10) (bρ 10) (312024205529 / 512000000000 : ℝ) ≤
+      (-(272732777028236039401 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (33424252113150029 / 80000000000000000 : ℝ) ≤
+      (-(272732777028236039401 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (33424252113150029 / 80000000000000000 : ℝ) =
+            (33424252113150029 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (33424252113150029 / 20000000000000000 : ℝ) ≤
+            (160484210659263960599 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1998977385244613 / 5000000000000000 : ℝ)) (q :=
+      (33424252113150029 / 80000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_533_11 :
+    Uω (aρ 11) (bρ 11) (312024205529 / 512000000000 : ℝ) ≤
+      (-(10163034798815418328541 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28954428248479461 / 80000000000000000 : ℝ) ≤
+      (-(10163034798815418328541 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (28954428248479461 / 80000000000000000 : ℝ) =
+            (28954428248479461 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28954428248479461 / 20000000000000000 : ℝ) ≤
+            (3699908807184581671459 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (412371928815823 / 1250000000000000 : ℝ)) (q :=
+      (28954428248479461 / 80000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_533_12 :
+    Uω (aρ 12) (bρ 12) (312024205529 / 512000000000 : ℝ) ≤
+      (-(12195252362156481322259 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23629629171164773 / 80000000000000000 : ℝ) ≤
+      (-(12195252362156481322259 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (23629629171164773 / 80000000000000000 : ℝ) =
+            (23629629171164773 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23629629171164773 / 20000000000000000 : ℝ) ≤
+            (1667691243843518677741 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (74733617433841 / 312500000000000 : ℝ)) (q :=
+      (23629629171164773 / 80000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_533_13 :
+    Uω (aρ 13) (bρ 13) (312024205529 / 512000000000 : ℝ) ≤
+      (-(3999156105056330198967 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16157174489704889 / 80000000000000000 : ℝ) ≤
+      (-(3999156105056330198967 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (16157174489704889 / 80000000000000000 : ℝ) =
+            (16157174489704889 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16157174489704889 / 10000000000000000 : ℝ) ≤
+            (1199447747193669801033 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (922743829112941 / 10000000000000000 : ℝ)) (q :=
+      (16157174489704889 / 80000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_533_14 :
+    Uω (aρ 14) (bρ 14) (312024205529 / 512000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_533_15 :
+    Uω (aρ 15) (bρ 15) (312024205529 / 512000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_533_16 :
+    Uω (aρ 16) (bρ 16) (312024205529 / 512000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_533 :
+    Uρ (312024205529 / 512000000000 : ℝ) ≤
+      (-(8158113287143502076381 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (312024205529 / 512000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2529440222003763779621 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_533_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (312024205529 / 512000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(5098627734197961890647 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_533_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (312024205529 / 512000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(5178603814400613640559 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_533_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (312024205529 / 512000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(5312874499598534674163 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_533_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (312024205529 / 512000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(5520902007877414746151 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_533_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (312024205529 / 512000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(2913535717514371392481 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_533_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (312024205529 / 512000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(6260639779296802408789 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_533_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (312024205529 / 512000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(6856629979244132857753 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_533_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (312024205529 / 512000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(3829383450332178808059 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_533_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (312024205529 / 512000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) * (-(272732777028236039401 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_533_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (312024205529 / 512000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(10163034798815418328541 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_533_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (312024205529 / 512000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(12195252362156481322259 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_533_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (312024205529 / 512000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(3999156105056330198967 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_533_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (312024205529 / 512000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_533_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (312024205529 / 512000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_533_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (312024205529 / 512000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_533_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_534_1 :
+    Uω (aρ 1) (bρ 1) (19535909148031 / 32000000000000 : ℝ) ≤
+      (-(2520534865968170005413 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4832358101161809 / 8000000000000000 : ℝ) ≤
+      (-(2520534865968170005413 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (4832358101161809 / 8000000000000000 : ℝ) =
+            (4832358101161809 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4832358101161809 / 4000000000000000 : ℝ) ≤
+            (945201035531829994587 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1208084172445967 / 2000000000000000 : ℝ)) (q :=
+      (4832358101161809 / 8000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_534_2 :
+    Uω (aρ 2) (bρ 2) (19535909148031 / 32000000000000 : ℝ) ≤
+      (-(5080745706614145080733 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24066116171112581 / 40000000000000000 : ℝ) ≤
+      (-(5080745706614145080733 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (24066116171112581 / 40000000000000000 : ℝ) =
+            (24066116171112581 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24066116171112581 / 20000000000000000 : ℝ) ≤
+            (1850726096385854919267 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6016352707391603 / 10000000000000000 : ℝ)) (q :=
+      (24066116171112581 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_534_3 :
+    Uω (aρ 3) (bρ 3) (19535909148031 / 32000000000000 : ℝ) ≤
+      (-(258028849230606099 / 500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23874758124220609 / 40000000000000000 : ℝ) ≤
+      (-(258028849230606099 / 500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23874758124220609 / 40000000000000000 : ℝ) =
+            (23874758124220609 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23874758124220609 / 20000000000000000 : ℝ) ≤
+            (88544740919393901 / 500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5968069790295617 / 10000000000000000 : ℝ)) (q :=
+      (23874758124220609 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_534_4 :
+    Uω (aρ 4) (bρ 4) (19535909148031 / 32000000000000 : ℝ) ≤
+      (-(5294600563066589342857 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23556914755306709 / 40000000000000000 : ℝ) ≤
+      (-(5294600563066589342857 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23556914755306709 / 40000000000000000 : ℝ) =
+            (23556914755306709 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23556914755306709 / 20000000000000000 : ℝ) ≤
+            (1636871239933410657143 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5887442286903667 / 10000000000000000 : ℝ)) (q :=
+      (23556914755306709 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_534_5 :
+    Uω (aρ 5) (bρ 5) (19535909148031 / 32000000000000 : ℝ) ≤
+      (-(1100447002148356538251 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23072835029488537 / 40000000000000000 : ℝ) ≤
+      (-(1100447002148356538251 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23072835029488537 / 40000000000000000 : ℝ) =
+            (23072835029488537 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23072835029488537 / 20000000000000000 : ℝ) ≤
+            (285847358451643461749 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5763592343329581 / 10000000000000000 : ℝ)) (q :=
+      (23072835029488537 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_534_6 :
+    Uω (aρ 6) (bρ 6) (19535909148031 / 32000000000000 : ℝ) ≤
+      (-(1161560447106391500073 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22378467636055877 / 40000000000000000 : ℝ) ≤
+      (-(1161560447106391500073 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (22378467636055877 / 40000000000000000 : ℝ) =
+            (22378467636055877 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22378467636055877 / 20000000000000000 : ℝ) ≤
+            (224733913493608499927 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5583649021753251 / 10000000000000000 : ℝ)) (q :=
+      (22378467636055877 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_534_7 :
+    Uω (aρ 7) (bρ 7) (19535909148031 / 32000000000000 : ℝ) ≤
+      (-(1560116466927709879407 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4286175976475109 / 8000000000000000 : ℝ) ≤
+      (-(1560116466927709879407 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (4286175976475109 / 8000000000000000 : ℝ) =
+            (4286175976475109 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4286175976475109 / 4000000000000000 : ℝ) ≤
+            (172751483822290120593 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1066703305134617 / 2000000000000000 : ℝ)) (q :=
+      (4286175976475109 / 8000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_534_8 :
+    Uω (aρ 8) (bρ 8) (19535909148031 / 32000000000000 : ℝ) ≤
+      (-(3417551010603838729737 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20193671267600891 / 40000000000000000 : ℝ) ≤
+      (-(3417551010603838729737 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (20193671267600891 / 40000000000000000 : ℝ) =
+            (20193671267600891 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20193671267600891 / 20000000000000000 : ℝ) ≤
+            (48184890896161270263 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2499226934812879 / 5000000000000000 : ℝ)) (q :=
+      (20193671267600891 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_534_9 :
+    Uω (aρ 9) (bρ 9) (19535909148031 / 32000000000000 : ℝ) ≤
+      (-(7635181723425072818701 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3728191892532483 / 8000000000000000 : ℝ) ≤
+      (-(7635181723425072818701 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3728191892532483 / 8000000000000000 : ℝ) =
+            (3728191892532483 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3728191892532483 / 2000000000000000 : ℝ) ≤
+            (6227761882574927181299 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (114076589887913 / 250000000000000 : ℝ)) (q :=
+      (3728191892532483 / 8000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_534_10 :
+    Uω (aρ 10) (bρ 10) (19535909148031 / 32000000000000 : ℝ) ≤
+      (-(8700602322136267647359 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16757052616423023 / 40000000000000000 : ℝ) ≤
+      (-(8700602322136267647359 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (16757052616423023 / 40000000000000000 : ℝ) =
+            (16757052616423023 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16757052616423023 / 10000000000000000 : ℝ) ≤
+            (5162341283863732352641 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (250604325368239 / 625000000000000 : ℝ)) (q :=
+      (16757052616423023 / 40000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_534_11 :
+    Uω (aρ 11) (bρ 11) (19535909148031 / 32000000000000 : ℝ) ≤
+      (-(10130515627069486294957 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14524369456552627 / 40000000000000000 : ℝ) ≤
+      (-(10130515627069486294957 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (14524369456552627 / 40000000000000000 : ℝ) =
+            (14524369456552627 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14524369456552627 / 10000000000000000 : ℝ) ≤
+            (3732427978930513705043 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1655902126080813 / 5000000000000000 : ℝ)) (q :=
+      (14524369456552627 / 40000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_534_12 :
+    Uω (aρ 12) (bρ 12) (19535909148031 / 32000000000000 : ℝ) ≤
+      (-(6075226774800110837489 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11867862286965257 / 40000000000000000 : ℝ) ≤
+      (-(6075226774800110837489 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (11867862286965257 / 40000000000000000 : ℝ) =
+            (11867862286965257 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11867862286965257 / 10000000000000000 : ℝ) ≤
+            (856245028199889162511 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2407250764052941 / 10000000000000000 : ℝ)) (q :=
+      (11867862286965257 / 40000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_534_13 :
+    Uω (aρ 13) (bρ 13) (19535909148031 / 32000000000000 : ℝ) ≤
+      (-(3970586139317389642427 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (65371510173111 / 320000000000000 : ℝ) ≤
+      (-(3970586139317389642427 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (65371510173111 / 320000000000000 : ℝ) = (65371510173111 / 40000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (65371510173111 / 40000000000000 : ℝ) ≤
+            (1228017712932610357573 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3833682991939 / 40000000000000 : ℝ)) (q :=
+      (65371510173111 / 320000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13]) (by norm_num [bρ_13])
+      (by norm_num) (by norm_num [aρ_13, bρ_13]) (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_534_14 :
+    Uω (aρ 14) (bρ 14) (19535909148031 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_534_15 :
+    Uω (aρ 15) (bρ 15) (19535909148031 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_534_16 :
+    Uω (aρ 16) (bρ 16) (19535909148031 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_534 :
+    Uρ (19535909148031 / 32000000000000 : ℝ) ≤
+      (-(8132319092386912548451 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (19535909148031 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2520534865968170005413 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_534_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (19535909148031 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(5080745706614145080733 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_534_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (19535909148031 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) * (-(258028849230606099 / 500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_534_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (19535909148031 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(5294600563066589342857 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_534_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (19535909148031 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1100447002148356538251 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_534_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (19535909148031 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1161560447106391500073 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_534_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (19535909148031 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(1560116466927709879407 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_534_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (19535909148031 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(3417551010603838729737 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_534_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (19535909148031 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(7635181723425072818701 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_534_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (19535909148031 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(8700602322136267647359 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_534_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (19535909148031 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(10130515627069486294957 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_534_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (19535909148031 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(6075226774800110837489 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_534_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (19535909148031 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(3970586139317389642427 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_534_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (19535909148031 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_534_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (19535909148031 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_534_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (19535909148031 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_534_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_535_1 :
+    Uω (aρ 1) (bρ 1) (39140610900999 / 64000000000000 : ℝ) ≤
+      (-(2511645342950552390927 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9681914429625003 / 16000000000000000 : ℝ) ≤
+      (-(2511645342950552390927 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (9681914429625003 / 16000000000000000 : ℝ) =
+            (9681914429625003 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9681914429625003 / 8000000000000000 : ℝ) ≤
+            (954090558549447609073 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (151279245045879 / 250000000000000 : ℝ)) (q :=
+      (9681914429625003 / 16000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_535_2 :
+    Uω (aρ 2) (bρ 2) (39140610900999 / 64000000000000 : ℝ) ≤
+      (-(5062895600518868362853 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (48218225614234631 / 80000000000000000 : ℝ) ≤
+      (-(5062895600518868362853 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (48218225614234631 / 80000000000000000 : ℝ) =
+            (48218225614234631 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (48218225614234631 / 40000000000000000 : ℝ) ≤
+            (1868576202481131637147 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1506775545218141 / 2500000000000000 : ℝ)) (q :=
+      (48218225614234631 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_535_3 :
+    Uω (aρ 3) (bρ 3) (39140610900999 / 64000000000000 : ℝ) ≤
+      (-(1028516519948136732061 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (47835515918073447 / 80000000000000000 : ℝ) ≤
+      (-(1028516519948136732061 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (47835515918073447 / 80000000000000000 : ℝ) =
+            (47835515918073447 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (47835515918073447 / 40000000000000000 : ℝ) ≤
+            (357777840651863267939 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1494705215795567 / 2500000000000000 : ℝ)) (q :=
+      (47835515918073447 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_535_4 :
+    Uω (aρ 4) (bρ 4) (39140610900999 / 64000000000000 : ℝ) ≤
+      (-(263817998974931911661 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (47199846311035179 / 80000000000000000 : ℝ) ≤
+      (-(263817998974931911661 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (47199846311035179 / 80000000000000000 : ℝ) =
+            (47199846311035179 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (47199846311035179 / 40000000000000000 : ℝ) ≤
+            (82755591175068088339 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5898197642487701 / 10000000000000000 : ℝ)) (q :=
+      (47199846311035179 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_535_5 :
+    Uω (aρ 5) (bρ 5) (39140610900999 / 64000000000000 : ℝ) ≤
+      (-(685450356247687421249 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (46231729562098087 / 80000000000000000 : ℝ) ≤
+      (-(685450356247687421249 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (46231729562098087 / 80000000000000000 : ℝ) =
+            (46231729562098087 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (46231729562098087 / 40000000000000000 : ℝ) ≤
+            (180983619127312578751 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1443589593647107 / 2500000000000000 : ℝ)) (q :=
+      (46231729562098087 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_535_6 :
+    Uω (aρ 6) (bρ 6) (39140610900999 / 64000000000000 : ℝ) ≤
+      (-(5788570240149873008043 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8968618922721031 / 16000000000000000 : ℝ) ≤
+      (-(5788570240149873008043 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (8968618922721031 / 16000000000000000 : ℝ) =
+            (8968618922721031 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8968618922721031 / 8000000000000000 : ℝ) ≤
+            (1142901562850126991957 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1118888002521039 / 2000000000000000 : ℝ)) (q :=
+      (8968618922721031 / 16000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_535_7 :
+    Uω (aρ 7) (bρ 7) (39140610900999 / 64000000000000 : ℝ) ≤
+      (-(3110166470379501080453 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (42948139957716527 / 80000000000000000 : ℝ) ≤
+      (-(3110166470379501080453 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (42948139957716527 / 80000000000000000 : ℝ) =
+            (42948139957716527 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (42948139957716527 / 40000000000000000 : ℝ) ≤
+            (355569431120498919547 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2672181364696519 / 5000000000000000 : ℝ)) (q :=
+      (42948139957716527 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_535_8 :
+    Uω (aρ 8) (bρ 8) (39140610900999 / 64000000000000 : ℝ) ≤
+      (-(6813621231441943836873 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (40474190981887567 / 80000000000000000 : ℝ) ≤
+      (-(6813621231441943836873 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (40474190981887567 / 80000000000000000 : ℝ) =
+            (40474190981887567 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (40474190981887567 / 40000000000000000 : ℝ) ≤
+            (117850571558056163127 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2504708568387899 / 5000000000000000 : ℝ)) (q :=
+      (40474190981887567 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_535_9 :
+    Uω (aρ 9) (bρ 9) (39140610900999 / 64000000000000 : ℝ) ≤
+      (-(190291359982045990177 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (37369736569649347 / 80000000000000000 : ℝ) ≤
+      (-(190291359982045990177 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (37369736569649347 / 80000000000000000 : ℝ) =
+            (37369736569649347 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (37369736569649347 / 20000000000000000 : ℝ) ≤
+            (156282230167954009823 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4574269162076243 / 10000000000000000 : ℝ)) (q :=
+      (37369736569649347 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_535_10 :
+    Uω (aρ 10) (bρ 10) (39140610900999 / 64000000000000 : ℝ) ≤
+      (-(4336917040251970202751 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (33603936777527351 / 80000000000000000 : ℝ) ≤
+      (-(4336917040251970202751 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (33603936777527351 / 80000000000000000 : ℝ) =
+            (33603936777527351 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (33603936777527351 / 20000000000000000 : ℝ) ≤
+            (2594554762748029797249 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (502672280942593 / 1250000000000000 : ℝ)) (q :=
+      (33603936777527351 / 80000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_535_11 :
+    Uω (aρ 11) (bρ 11) (39140610900999 / 64000000000000 : ℝ) ≤
+      (-(5049061054212899413027 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (29142990574389979 / 80000000000000000 : ℝ) ≤
+      (-(5049061054212899413027 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (29142990574389979 / 80000000000000000 : ℝ) =
+            (29142990574389979 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (29142990574389979 / 20000000000000000 : ℝ) ≤
+            (1882410748787100586973 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3324618322961401 / 10000000000000000 : ℝ)) (q :=
+      (29142990574389979 / 80000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_535_12 :
+    Uω (aρ 12) (bρ 12) (39140610900999 / 64000000000000 : ℝ) ≤
+      (-(12105946844337847702467 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23841599897031647 / 80000000000000000 : ℝ) ≤
+      (-(12105946844337847702467 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (23841599897031647 / 80000000000000000 : ℝ) =
+            (23841599897031647 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23841599897031647 / 20000000000000000 : ℝ) ≤
+            (1756996761662152297533 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1211485375153409 / 5000000000000000 : ℝ)) (q :=
+      (23841599897031647 / 80000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_535_13 :
+    Uω (aρ 13) (bρ 13) (39140610900999 / 64000000000000 : ℝ) ≤
+      (-(7886087677459839047893 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16523921161995631 / 80000000000000000 : ℝ) ≤
+      (-(7886087677459839047893 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (16523921161995631 / 80000000000000000 : ℝ) =
+            (16523921161995631 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16523921161995631 / 10000000000000000 : ℝ) ≤
+            (2511120027040160952107 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (496466404071407 / 5000000000000000 : ℝ)) (q :=
+      (16523921161995631 / 80000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_535_14 :
+    Uω (aρ 14) (bρ 14) (39140610900999 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_535_15 :
+    Uω (aρ 15) (bρ 15) (39140610900999 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_535_16 :
+    Uω (aρ 16) (bρ 16) (39140610900999 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_535 :
+    Uρ (39140610900999 / 64000000000000 : ℝ) ≤
+      (-(8106826625353221444803 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (39140610900999 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2511645342950552390927 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_535_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (39140610900999 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(5062895600518868362853 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_535_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (39140610900999 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1028516519948136732061 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_535_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (39140610900999 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(263817998974931911661 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_535_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (39140610900999 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(685450356247687421249 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_535_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (39140610900999 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(5788570240149873008043 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_535_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (39140610900999 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(3110166470379501080453 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_535_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (39140610900999 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(6813621231441943836873 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_535_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (39140610900999 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(190291359982045990177 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_535_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (39140610900999 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(4336917040251970202751 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_535_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (39140610900999 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(5049061054212899413027 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_535_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (39140610900999 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(12105946844337847702467 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_535_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (39140610900999 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(7886087677459839047893 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_535_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (39140610900999 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_535_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (39140610900999 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_535_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (39140610900999 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_535_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_536_1 :
+    Uω (aρ 1) (bρ 1) (2450587719121 / 4000000000000 : ℝ) ≤
+      (-(1251385798375307642803 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6061945410410391 / 10000000000000000 : ℝ) ≤
+      (-(1251385798375307642803 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (6061945410410391 / 10000000000000000 : ℝ) =
+            (6061945410410391 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6061945410410391 / 5000000000000000 : ℝ) ≤
+            (481482152374692357197 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3030959370551641 / 5000000000000000 : ℝ)) (q :=
+      (6061945410410391 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_536_2 :
+    Uω (aρ 2) (bρ 2) (2450587719121 / 4000000000000 : ℝ) ≤
+      (-(5045077302141448493163 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6038027359660779 / 10000000000000000 : ℝ) ≤
+      (-(5045077302141448493163 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (6038027359660779 / 10000000000000000 : ℝ) =
+            (6038027359660779 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6038027359660779 / 5000000000000000 : ℝ) ≤
+            (1886394500858551506837 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3018925826057029 / 5000000000000000 : ℝ)) (q :=
+      (6038027359660779 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_536_3 :
+    Uω (aρ 3) (bρ 3) (2450587719121 / 4000000000000 : ℝ) ≤
+      (-(640577567897808385041 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2995094722231893 / 5000000000000000 : ℝ) ≤
+      (-(640577567897808385041 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2995094722231893 / 5000000000000000 : ℝ) =
+            (2995094722231893 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2995094722231893 / 2500000000000000 : ℝ) ≤
+            (225856407477191614959 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (748696491008759 / 1250000000000000 : ℝ)) (q :=
+      (2995094722231893 / 5000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_536_4 :
+    Uω (aρ 4) (bρ 4) (2450587719121 / 4000000000000 : ℝ) ≤
+      (-(164317269602931110393 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2955366438542201 / 5000000000000000 : ℝ) ≤
+      (-(164317269602931110393 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2955366438542201 / 5000000000000000 : ℝ) =
+            (2955366438542201 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2955366438542201 / 2500000000000000 : ℝ) ≤
+            (52291224240818889607 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (369309560898519 / 625000000000000 : ℝ)) (q :=
+      (2955366438542201 / 5000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_536_5 :
+    Uω (aρ 5) (bρ 5) (2450587719121 / 4000000000000 : ℝ) ≤
+      (-(5465005395609099705269 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11579447202387303 / 20000000000000000 : ℝ) ≤
+      (-(5465005395609099705269 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11579447202387303 / 20000000000000000 : ℝ) =
+            (11579447202387303 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11579447202387303 / 10000000000000000 : ℝ) ≤
+            (1466466407390900294731 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5785124341929803 / 10000000000000000 : ℝ)) (q :=
+      (11579447202387303 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_536_6 :
+    Uω (aρ 6) (bρ 6) (2450587719121 / 4000000000000 : ℝ) ≤
+      (-(1442343826234602812091 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5616156663407157 / 10000000000000000 : ℝ) ≤
+      (-(1442343826234602812091 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5616156663407157 / 10000000000000000 : ℝ) =
+            (5616156663407157 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5616156663407157 / 5000000000000000 : ℝ) ≤
+            (290524124515397187909 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2802615420748407 / 5000000000000000 : ℝ)) (q :=
+      (5616156663407157 / 10000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_536_7 :
+    Uω (aρ 7) (bρ 7) (2450587719121 / 4000000000000 : ℝ) ≤
+      (-(620024083077386747109 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10758629645066451 / 20000000000000000 : ℝ) ≤
+      (-(620024083077386747109 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10758629645066451 / 20000000000000000 : ℝ) =
+            (10758629645066451 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10758629645066451 / 10000000000000000 : ℝ) ≤
+            (73123097222613252891 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5355208540508951 / 10000000000000000 : ℝ)) (q :=
+      (10758629645066451 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_536_8 :
+    Uω (aρ 8) (bρ 8) (2450587719121 / 4000000000000 : ℝ) ≤
+      (-(6792187399728591460427 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1267532366226191 / 2500000000000000 : ℝ) ≤
+      (-(6792187399728591460427 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1267532366226191 / 2500000000000000 : ℝ) =
+            (1267532366226191 / 1250000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1267532366226191 / 1250000000000000 : ℝ) ≤
+            (139284403271408539573 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1255094869148007 / 2500000000000000 : ℝ)) (q :=
+      (1267532366226191 / 2500000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_536_9 :
+    Uω (aρ 9) (bρ 9) (2450587719121 / 4000000000000 : ℝ) ≤
+      (-(7588184633861080601369 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9364386366791683 / 20000000000000000 : ℝ) ≤
+      (-(7588184633861080601369 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9364386366791683 / 20000000000000000 : ℝ) =
+            (9364386366791683 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9364386366791683 / 5000000000000000 : ℝ) ≤
+            (6274758972138919398631 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4585472541934183 / 10000000000000000 : ℝ)) (q :=
+      (9364386366791683 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_536_10 :
+    Uω (aρ 10) (bρ 10) (2450587719121 / 4000000000000 : ℝ) ≤
+      (-(1080892956124258428013 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (105292959172209 / 250000000000000 : ℝ) ≤
+      (-(1080892956124258428013 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (105292959172209 / 250000000000000 : ℝ) = (105292959172209 / 62500000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (105292959172209 / 62500000000000 : ℝ) ≤
+            (651974994625741571987 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (201654097120711 / 500000000000000 : ℝ)) (q :=
+      (105292959172209 / 250000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10]) (by norm_num [bρ_10])
+      (by norm_num) (by norm_num [aρ_10, bρ_10]) (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_536_11 :
+    Uω (aρ 11) (bρ 11) (2450587719121 / 4000000000000 : ℝ) ≤
+      (-(314557910422104071209 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3654647988995821 / 10000000000000000 : ℝ) ≤
+      (-(314557910422104071209 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3654647988995821 / 10000000000000000 : ℝ) =
+            (3654647988995821 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3654647988995821 / 2500000000000000 : ℝ) ≤
+            (118659077265395928791 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1668708906417071 / 5000000000000000 : ℝ)) (q :=
+      (3654647988995821 / 10000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_536_12 :
+    Uω (aρ 12) (bρ 12) (2450587719121 / 4000000000000 : ℝ) ≤
+      (-(6030863735264409501953 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2993407424564263 / 10000000000000000 : ℝ) ≤
+      (-(6030863735264409501953 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2993407424564263 / 10000000000000000 : ℝ) =
+            (2993407424564263 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2993407424564263 / 2500000000000000 : ℝ) ≤
+            (900608067735590498047 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1219318390328013 / 5000000000000000 : ℝ)) (q :=
+      (2993407424564263 / 10000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_536_13 :
+    Uω (aρ 13) (bρ 13) (2450587719121 / 4000000000000 : ℝ) ≤
+      (-(3133142970538085451271 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (521899230802451 / 2500000000000000 : ℝ) ≤
+      (-(3133142970538085451271 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (521899230802451 / 2500000000000000 : ℝ) =
+            (521899230802451 / 312500000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (521899230802451 / 312500000000000 : ℝ) ≤
+            (1025740111261914548729 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (256599379885527 / 2500000000000000 : ℝ)) (q :=
+      (521899230802451 / 2500000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13]) (by norm_num [bρ_13])
+      (by norm_num) (by norm_num [aρ_13, bρ_13]) (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_536_14 :
+    Uω (aρ 14) (bρ 14) (2450587719121 / 4000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_536_15 :
+    Uω (aρ 15) (bρ 15) (2450587719121 / 4000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_536_16 :
+    Uω (aρ 16) (bρ 16) (2450587719121 / 4000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_536 :
+    Uρ (2450587719121 / 4000000000000 : ℝ) ≤
+      (-(252550364807948996239 / 312500000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (2450587719121 / 4000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1251385798375307642803 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_536_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (2450587719121 / 4000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(5045077302141448493163 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_536_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (2450587719121 / 4000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(640577567897808385041 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_536_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (2450587719121 / 4000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(164317269602931110393 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_536_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (2450587719121 / 4000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(5465005395609099705269 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_536_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (2450587719121 / 4000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1442343826234602812091 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_536_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (2450587719121 / 4000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(620024083077386747109 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_536_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (2450587719121 / 4000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(6792187399728591460427 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_536_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (2450587719121 / 4000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(7588184633861080601369 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_536_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (2450587719121 / 4000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(1080892956124258428013 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_536_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (2450587719121 / 4000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) * (-(314557910422104071209 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_536_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (2450587719121 / 4000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(6030863735264409501953 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_536_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (2450587719121 / 4000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(3133142970538085451271 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_536_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (2450587719121 / 4000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_536_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (2450587719121 / 4000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_536_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (2450587719121 / 4000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_536_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_537_1 :
+    Uω (aρ 1) (bρ 1) (39278196110873 / 64000000000000 : ℝ) ≤
+      (-(2493913571466764627649 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (48581554417099593 / 80000000000000000 : ℝ) ≤
+      (-(2493913571466764627649 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (48581554417099593 / 80000000000000000 : ℝ) =
+            (48581554417099593 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (48581554417099593 / 40000000000000000 : ℝ) ≤
+            (971822330033235372351 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (759083460004499 / 1250000000000000 : ℝ)) (q :=
+      (48581554417099593 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_537_2 :
+    Uω (aρ 2) (bρ 2) (39278196110873 / 64000000000000 : ℝ) ≤
+      (-(1005458139663677673063 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (48390212131427721 / 80000000000000000 : ℝ) ≤
+      (-(1005458139663677673063 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (48390212131427721 / 80000000000000000 : ℝ) =
+            (48390212131427721 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (48390212131427721 / 40000000000000000 : ℝ) ≤
+            (380836220936322326937 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (756075140141003 / 1250000000000000 : ℝ)) (q :=
+      (48390212131427721 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_537_3 :
+    Uω (aρ 3) (bρ 3) (39278196110873 / 64000000000000 : ℝ) ≤
+      (-(5106690698961235973153 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (48007515161523729 / 80000000000000000 : ℝ) ≤
+      (-(5106690698961235973153 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (48007515161523729 / 80000000000000000 : ℝ) =
+            (48007515161523729 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (48007515161523729 / 40000000000000000 : ℝ) ≤
+            (1824781104038764026847 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3000161492501013 / 5000000000000000 : ℝ)) (q :=
+      (48007515161523729 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_537_4 :
+    Uω (aρ 4) (bρ 4) (39278196110873 / 64000000000000 : ℝ) ≤
+      (-(5239978385515379585033 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (47371879628050141 / 80000000000000000 : ℝ) ≤
+      (-(5239978385515379585033 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (47371879628050141 / 80000000000000000 : ℝ) =
+            (47371879628050141 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (47371879628050141 / 40000000000000000 : ℝ) ≤
+            (1691493417484620414967 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5919708282698629 / 10000000000000000 : ℝ)) (q :=
+      (47371879628050141 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_537_5 :
+    Uω (aρ 5) (bρ 5) (39278196110873 / 64000000000000 : ℝ) ≤
+      (-(5446442518364453498131 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (46403847802755189 / 80000000000000000 : ℝ) ≤
+      (-(5446442518364453498131 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (46403847802755189 / 80000000000000000 : ℝ) =
+            (46403847802755189 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (46403847802755189 / 40000000000000000 : ℝ) ≤
+            (1485029284635546501869 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5795890245709891 / 10000000000000000 : ℝ)) (q :=
+      (46403847802755189 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_537_6 :
+    Uω (aρ 6) (bρ 6) (39278196110873 / 64000000000000 : ℝ) ≤
+      (-(5750217286790455522099 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (45015411356802417 / 80000000000000000 : ℝ) ≤
+      (-(5750217286790455522099 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (45015411356802417 / 80000000000000000 : ℝ) =
+            (45015411356802417 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (45015411356802417 / 40000000000000000 : ℝ) ≤
+            (1181254516209544477901 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2808010754680849 / 5000000000000000 : ℝ)) (q :=
+      (45015411356802417 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_537_7 :
+    Uω (aρ 7) (bρ 7) (39278196110873 / 64000000000000 : ℝ) ≤
+      (-(6180189371123662487857 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (43120895641921241 / 80000000000000000 : ℝ) ≤
+      (-(6180189371123662487857 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (43120895641921241 / 80000000000000000 : ℝ) =
+            (43120895641921241 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (43120895641921241 / 40000000000000000 : ℝ) ≤
+            (751282431876337512143 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1341513490350351 / 2500000000000000 : ℝ)) (q :=
+      (43120895641921241 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_537_8 :
+    Uω (aρ 8) (bρ 8) (39278196110873 / 64000000000000 : ℝ) ≤
+      (-(1692700079319226165529 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (40647876771499233 / 80000000000000000 : ℝ) ≤
+      (-(1692700079319226165529 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (40647876771499233 / 80000000000000000 : ℝ) =
+            (40647876771499233 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (40647876771499233 / 40000000000000000 : ℝ) ≤
+            (40167871430773834471 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2515670447567951 / 5000000000000000 : ℝ)) (q :=
+      (40647876771499233 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_537_9 :
+    Uω (aρ 9) (bρ 9) (39278196110873 / 64000000000000 : ℝ) ≤
+      (-(3782386067554484225809 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1501813827273289 / 3200000000000000 : ℝ) ≤
+      (-(3782386067554484225809 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1501813827273289 / 3200000000000000 : ℝ) =
+            (1501813827273289 / 800000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1501813827273289 / 800000000000000 : ℝ) ≤
+            (3149085735445515774191 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (91933475021583 / 200000000000000 : ℝ)) (q :=
+      (1501813827273289 / 3200000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_537_10 :
+    Uω (aρ 10) (bρ 10) (39278196110873 / 64000000000000 : ℝ) ≤
+      (-(8620530541481646251921 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (33783535891237313 / 80000000000000000 : ℝ) ≤
+      (-(8620530541481646251921 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (33783535891237313 / 80000000000000000 : ℝ) =
+            (33783535891237313 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (33783535891237313 / 20000000000000000 : ℝ) ≤
+            (5242413064518353748079 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2022390168462711 / 5000000000000000 : ℝ)) (q :=
+      (33783535891237313 / 80000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_537_11 :
+    Uω (aρ 11) (bρ 11) (39278196110873 / 64000000000000 : ℝ) ≤
+      (-(2508426902235644193011 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (29331319594314289 / 80000000000000000 : ℝ) ≤
+      (-(2508426902235644193011 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (29331319594314289 / 80000000000000000 : ℝ) =
+            (29331319594314289 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (29331319594314289 / 20000000000000000 : ℝ) ≤
+            (957308999264355806989 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1675101444449833 / 5000000000000000 : ℝ)) (q :=
+      (29331319594314289 / 80000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_537_12 :
+    Uω (aρ 12) (bρ 12) (39278196110873 / 64000000000000 : ℝ) ≤
+      (-(1502223847783784689383 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4810541441066241 / 16000000000000000 : ℝ) ≤
+      (-(1502223847783784689383 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4810541441066241 / 16000000000000000 : ℝ) =
+            (4810541441066241 / 4000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4810541441066241 / 4000000000000000 : ℝ) ≤
+            (230644102966215310617 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (490849977667779 / 2000000000000000 : ℝ)) (q :=
+      (4810541441066241 / 16000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_537_13 :
+    Uω (aρ 13) (bρ 13) (39278196110873 / 64000000000000 : ℝ) ≤
+      (-(15562622929320063846659 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (674953496939277 / 3200000000000000 : ℝ) ≤
+      (-(15562622929320063846659 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (674953496939277 / 3200000000000000 : ℝ) =
+            (674953496939277 / 400000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (674953496939277 / 400000000000000 : ℝ) ≤
+            (5231792479679936153341 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (42356567378863 / 400000000000000 : ℝ)) (q :=
+      (674953496939277 / 3200000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13]) (by norm_num [bρ_13])
+      (by norm_num) (by norm_num [aρ_13, bρ_13]) (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_537_14 :
+    Uω (aρ 14) (bρ 14) (39278196110873 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_537_15 :
+    Uω (aρ 15) (bρ 15) (39278196110873 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_537_16 :
+    Uω (aρ 16) (bρ 16) (39278196110873 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_537 :
+    Uρ (39278196110873 / 64000000000000 : ℝ) ≤
+      (-(4028326837852990601667 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (39278196110873 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2493913571466764627649 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_537_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (39278196110873 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1005458139663677673063 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_537_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (39278196110873 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(5106690698961235973153 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_537_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (39278196110873 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(5239978385515379585033 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_537_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (39278196110873 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(5446442518364453498131 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_537_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (39278196110873 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(5750217286790455522099 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_537_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (39278196110873 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(6180189371123662487857 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_537_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (39278196110873 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(1692700079319226165529 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_537_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (39278196110873 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(3782386067554484225809 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_537_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (39278196110873 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(8620530541481646251921 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_537_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (39278196110873 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(2508426902235644193011 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_537_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (39278196110873 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(1502223847783784689383 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_537_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (39278196110873 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(15562622929320063846659 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_537_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (39278196110873 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_537_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (39278196110873 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_537_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (39278196110873 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_537_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_538_1 :
+    Uω (aρ 1) (bρ 1) (3934698871581 / 6400000000000 : ℝ) ≤
+      (-(4970142422987990689033 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24333772774790763 / 40000000000000000 : ℝ) ≤
+      (-(4970142422987990689033 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (24333772774790763 / 40000000000000000 : ℝ) =
+            (24333772774790763 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24333772774790763 / 20000000000000000 : ℝ) ≤
+            (1961329380012009310967 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6083416618635069 / 10000000000000000 : ℝ)) (q :=
+      (24333772774790763 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_538_2 :
+    Uω (aρ 2) (bρ 2) (3934698871581 / 6400000000000 : ℝ) ≤
+      (-(626191959561132738727 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24238102688353259 / 40000000000000000 : ℝ) ≤
+      (-(626191959561132738727 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (24238102688353259 / 40000000000000000 : ℝ) =
+            (24238102688353259 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24238102688353259 / 20000000000000000 : ℝ) ≤
+            (240242015813867261273 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6059350587926317 / 10000000000000000 : ℝ)) (q :=
+      (24238102688353259 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_538_3 :
+    Uω (aρ 3) (bρ 3) (3934698871581 / 6400000000000 : ℝ) ≤
+      (-(5088792951723855824927 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24046757367842257 / 40000000000000000 : ℝ) ≤
+      (-(5088792951723855824927 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (24046757367842257 / 40000000000000000 : ℝ) =
+            (24046757367842257 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24046757367842257 / 20000000000000000 : ℝ) ≤
+            (1842678851276144175073 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (375692127126301 / 625000000000000 : ℝ)) (q :=
+      (24046757367842257 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_538_4 :
+    Uω (aρ 4) (bρ 4) (3934698871581 / 6400000000000 : ℝ) ≤
+      (-(1305459283471519444757 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23728948072836411 / 40000000000000000 : ℝ) ≤
+      (-(1305459283471519444757 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23728948072836411 / 40000000000000000 : ℝ) =
+            (23728948072836411 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23728948072836411 / 20000000000000000 : ℝ) ≤
+            (427408667278480555243 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5930463567582893 / 10000000000000000 : ℝ)) (q :=
+      (23728948072836411 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_538_5 :
+    Uω (aρ 5) (bρ 5) (3934698871581 / 6400000000000 : ℝ) ≤
+      (-(339244630606834053439 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23244953271565087 / 40000000000000000 : ℝ) ≤
+      (-(339244630606834053439 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23244953271565087 / 40000000000000000 : ℝ) =
+            (23244953271565087 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23244953271565087 / 20000000000000000 : ℝ) ≤
+            (93972357080665946561 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5806656086282231 / 10000000000000000 : ℝ)) (q :=
+      (23244953271565087 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_538_6 :
+    Uω (aρ 6) (bρ 6) (3934698871581 / 6400000000000 : ℝ) ≤
+      (-(2865548021714676276341 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4510156876594629 / 8000000000000000 : ℝ) ≤
+      (-(2865548021714676276341 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (4510156876594629 / 8000000000000000 : ℝ) =
+            (4510156876594629 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4510156876594629 / 4000000000000000 : ℝ) ≤
+            (600187879785323723659 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (281340600856313 / 500000000000000 : ℝ)) (q :=
+      (4510156876594629 / 8000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_538_7 :
+    Uω (aρ 7) (bρ 7) (3934698871581 / 6400000000000 : ℝ) ≤
+      (-(24640713584814710663 / 40000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21603635576063999 / 40000000000000000 : ℝ) ≤
+      (-(24640713584814710663 / 40000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (21603635576063999 / 40000000000000000 : ℝ) =
+            (21603635576063999 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21603635576063999 / 20000000000000000 : ℝ) ≤
+            (3085173627185289337 / 40000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5376898994431687 / 10000000000000000 : ℝ)) (q :=
+      (21603635576063999 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_538_8 :
+    Uω (aρ 8) (bρ 8) (3934698871581 / 6400000000000 : ℝ) ≤
+      (-(6749459776710677993977 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20367357081351947 / 40000000000000000 : ℝ) ≤
+      (-(6749459776710677993977 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (20367357081351947 / 40000000000000000 : ℝ) =
+            (20367357081351947 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20367357081351947 / 20000000000000000 : ℝ) ≤
+            (182012026289322006023 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5042301398415661 / 10000000000000000 : ℝ)) (q :=
+      (20367357081351947 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_538_9 :
+    Uω (aρ 9) (bρ 9) (3934698871581 / 6400000000000 : ℝ) ≤
+      (-(3770708306633194469347 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (18816568638483439 / 40000000000000000 : ℝ) ≤
+      (-(3770708306633194469347 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (18816568638483439 / 40000000000000000 : ℝ) =
+            (18816568638483439 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18816568638483439 / 10000000000000000 : ℝ) ≤
+            (3160763496366805530653 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4607872805341407 / 10000000000000000 : ℝ)) (q :=
+      (18816568638483439 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_538_10 :
+    Uω (aρ 10) (bρ 10) (3934698871581 / 6400000000000 : ℝ) ≤
+      (-(8593994276661595780249 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (677466076586923 / 1600000000000000 : ℝ) ≤
+      (-(8593994276661595780249 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (677466076586923 / 1600000000000000 : ℝ) =
+            (677466076586923 / 400000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (677466076586923 / 400000000000000 : ℝ) ≤
+            (5268949329338404219751 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (162258939077249 / 400000000000000 : ℝ)) (q :=
+      (677466076586923 / 1600000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10]) (by norm_num [bρ_10])
+      (by norm_num) (by norm_num [aρ_10, bρ_10]) (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_538_11 :
+    Uω (aρ 11) (bρ 11) (3934698871581 / 6400000000000 : ℝ) ≤
+      (-(10001684457041864999147 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14712699139499919 / 40000000000000000 : ℝ) ≤
+      (-(10001684457041864999147 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (14712699139499919 / 40000000000000000 : ℝ) =
+            (14712699139499919 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14712699139499919 / 10000000000000000 : ℝ) ≤
+            (3861259148958135000853 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3362973715549647 / 10000000000000000 : ℝ)) (q :=
+      (14712699139499919 / 40000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_538_12 :
+    Uω (aρ 12) (bρ 12) (3934698871581 / 6400000000000 : ℝ) ≤
+      (-(5987066129331995198681 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12078973669071019 / 40000000000000000 : ℝ) ≤
+      (-(5987066129331995198681 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (12078973669071019 / 40000000000000000 : ℝ) =
+            (12078973669071019 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12078973669071019 / 10000000000000000 : ℝ) ≤
+            (944405673668004801319 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2469811077020197 / 10000000000000000 : ℝ)) (q :=
+      (12078973669071019 / 40000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_538_13 :
+    Uω (aρ 13) (bρ 13) (3934698871581 / 6400000000000 : ℝ) ≤
+      (-(3092521747675457896549 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8521723247589617 / 40000000000000000 : ℝ) ≤
+      (-(3092521747675457896549 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (8521723247589617 / 40000000000000000 : ℝ) =
+            (8521723247589617 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8521723247589617 / 5000000000000000 : ℝ) ≤
+            (1066361334124542103451 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (17040118873039 / 156250000000000 : ℝ)) (q :=
+      (8521723247589617 / 40000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_538_14 :
+    Uω (aρ 14) (bρ 14) (3934698871581 / 6400000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_538_15 :
+    Uω (aρ 15) (bρ 15) (3934698871581 / 6400000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_538_16 :
+    Uω (aρ 16) (bρ 16) (3934698871581 / 6400000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_538 :
+    Uρ (3934698871581 / 6400000000000 : ℝ) ≤
+      (-(1003991872383418357177 / 1250000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (3934698871581 / 6400000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(4970142422987990689033 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_538_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (3934698871581 / 6400000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(626191959561132738727 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_538_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (3934698871581 / 6400000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(5088792951723855824927 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_538_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (3934698871581 / 6400000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1305459283471519444757 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_538_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (3934698871581 / 6400000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(339244630606834053439 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_538_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (3934698871581 / 6400000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(2865548021714676276341 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_538_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (3934698871581 / 6400000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) * (-(24640713584814710663 / 40000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_538_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (3934698871581 / 6400000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(6749459776710677993977 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_538_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (3934698871581 / 6400000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(3770708306633194469347 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_538_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (3934698871581 / 6400000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(8593994276661595780249 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_538_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (3934698871581 / 6400000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(10001684457041864999147 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_538_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (3934698871581 / 6400000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(5987066129331995198681 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_538_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (3934698871581 / 6400000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(3092521747675457896549 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_538_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (3934698871581 / 6400000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_538_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (3934698871581 / 6400000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_538_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (3934698871581 / 6400000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_538_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_539_1 :
+    Uω (aρ 1) (bρ 1) (39415781320747 / 64000000000000 : ℝ) ≤
+      (-(39619911384348771533 / 80000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (48753536680735983 / 80000000000000000 : ℝ) ≤
+      (-(39619911384348771533 / 80000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (48753536680735983 / 80000000000000000 : ℝ) =
+            (48753536680735983 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (48753536680735983 / 40000000000000000 : ℝ) ≤
+            (15831863039651228467 / 80000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6094165556902277 / 10000000000000000 : ℝ)) (q :=
+      (48753536680735983 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_539_2 :
+    Uω (aρ 2) (bρ 2) (39415781320747 / 64000000000000 : ℝ) ≤
+      (-(4991812124691438887987 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (48562198613169711 / 80000000000000000 : ℝ) ≤
+      (-(4991812124691438887987 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (48562198613169711 / 80000000000000000 : ℝ) =
+            (48562198613169711 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (48562198613169711 / 40000000000000000 : ℝ) ≤
+            (1939659678308561112013 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6070100052520709 / 10000000000000000 : ℝ)) (q :=
+      (48562198613169711 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_539_3 :
+    Uω (aρ 3) (bρ 3) (39415781320747 / 64000000000000 : ℝ) ≤
+      (-(2535463593367915656409 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (48179514278362183 / 80000000000000000 : ℝ) ≤
+      (-(2535463593367915656409 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (48179514278362183 / 80000000000000000 : ℝ) =
+            (48179514278362183 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (48179514278362183 / 40000000000000000 : ℝ) ≤
+            (930272308132084343591 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6021825075168827 / 10000000000000000 : ℝ)) (q :=
+      (48179514278362183 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_539_4 :
+    Uω (aρ 4) (bρ 4) (39415781320747 / 64000000000000 : ℝ) ≤
+      (-(162616523524473707163 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (47543912570052419 / 80000000000000000 : ℝ) ≤
+      (-(162616523524473707163 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (47543912570052419 / 80000000000000000 : ℝ) =
+            (47543912570052419 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (47543912570052419 / 40000000000000000 : ℝ) ≤
+            (53991970319276292837 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2970609414578193 / 5000000000000000 : ℝ)) (q :=
+      (47543912570052419 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_539_5 :
+    Uω (aρ 5) (bρ 5) (39415781320747 / 64000000000000 : ℝ) ≤
+      (-(5409419981822095878167 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (46575965032077847 / 80000000000000000 : ℝ) ≤
+      (-(5409419981822095878167 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (46575965032077847 / 80000000000000000 : ℝ) =
+            (46575965032077847 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (46575965032077847 / 40000000000000000 : ℝ) ≤
+            (1522051821177904121833 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5817421863997743 / 10000000000000000 : ℝ)) (q :=
+      (46575965032077847 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_539_6 :
+    Uω (aρ 6) (bρ 6) (39415781320747 / 64000000000000 : ℝ) ≤
+      (-(2856005716701236298161 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (45187725538366107 / 80000000000000000 : ℝ) ≤
+      (-(2856005716701236298161 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (45187725538366107 / 80000000000000000 : ℝ) =
+            (45187725538366107 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (45187725538366107 / 40000000000000000 : ℝ) ≤
+            (609730184798763701839 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (352350147856863 / 625000000000000 : ℝ)) (q :=
+      (45187725538366107 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_539_7 :
+    Uω (aρ 7) (bρ 7) (39415781320747 / 64000000000000 : ℝ) ≤
+      (-(614020774142777776231 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (43293645120254843 / 80000000000000000 : ℝ) ≤
+      (-(614020774142777776231 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (43293645120254843 / 80000000000000000 : ℝ) =
+            (43293645120254843 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (43293645120254843 / 40000000000000000 : ℝ) ≤
+            (79126406157222223769 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (673467955242749 / 1250000000000000 : ℝ)) (q :=
+      (43293645120254843 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_539_8 :
+    Uω (aρ 8) (bρ 8) (39415781320747 / 64000000000000 : ℝ) ≤
+      (-(6728165572055259940237 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (40821547916674547 / 80000000000000000 : ℝ) ≤
+      (-(6728165572055259940237 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (40821547916674547 / 80000000000000000 : ℝ) =
+            (40821547916674547 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (40821547916674547 / 40000000000000000 : ℝ) ≤
+            (203306230944740059763 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2526630496193459 / 5000000000000000 : ℝ)) (q :=
+      (40821547916674547 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_539_9 :
+    Uω (aρ 9) (bρ 9) (39415781320747 / 64000000000000 : ℝ) ≤
+      (-(7518117780844264344539 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (37720920315265511 / 80000000000000000 : ℝ) ≤
+      (-(7518117780844264344539 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (37720920315265511 / 80000000000000000 : ℝ) =
+            (37720920315265511 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (37720920315265511 / 20000000000000000 : ℝ) ≤
+            (6344825825155735655461 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4619069720394659 / 10000000000000000 : ℝ)) (q :=
+      (37720920315265511 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_539_10 :
+    Uω (aρ 10) (bρ 10) (39415781320747 / 64000000000000 : ℝ) ≤
+      (-(8567534377982396619123 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (33963050930670051 / 80000000000000000 : ℝ) ≤
+      (-(8567534377982396619123 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (33963050930670051 / 80000000000000000 : ℝ) =
+            (33963050930670051 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (33963050930670051 / 20000000000000000 : ℝ) ≤
+            (5295409228017603380877 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2034080703870397 / 5000000000000000 : ℝ)) (q :=
+      (33963050930670051 / 80000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_539_11 :
+    Uω (aρ 11) (bρ 11) (39415781320747 / 64000000000000 : ℝ) ≤
+      (-(1993956523096417586819 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (29519420612911899 / 80000000000000000 : ℝ) ≤
+      (-(1993956523096417586819 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (29519420612911899 / 80000000000000000 : ℝ) =
+            (29519420612911899 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (29519420612911899 / 20000000000000000 : ℝ) ≤
+            (778632198103582413181 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (210983153406641 / 625000000000000 : ℝ)) (q :=
+      (29519420612911899 / 80000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_539_12 :
+    Uω (aρ 12) (bρ 12) (39415781320747 / 64000000000000 : ℝ) ≤
+      (-(745671718695141265711 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24262983695874167 / 80000000000000000 : ℝ) ≤
+      (-(745671718695141265711 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (24262983695874167 / 80000000000000000 : ℝ) =
+            (24262983695874167 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24262983695874167 / 20000000000000000 : ℝ) ≤
+            (120762256679858734289 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2485321321931823 / 10000000000000000 : ℝ)) (q :=
+      (24262983695874167 / 80000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_539_13 :
+    Uω (aρ 13) (bρ 13) (39415781320747 / 64000000000000 : ℝ) ≤
+      (-(7682710624029974679527 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17209894999780123 / 80000000000000000 : ℝ) ≤
+      (-(7682710624029974679527 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (17209894999780123 / 80000000000000000 : ℝ) =
+            (17209894999780123 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17209894999780123 / 10000000000000000 : ℝ) ≤
+            (2714497080470025320473 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (70089430593957 / 625000000000000 : ℝ)) (q :=
+      (17209894999780123 / 80000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_539_14 :
+    Uω (aρ 14) (bρ 14) (39415781320747 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_539_15 :
+    Uω (aρ 15) (bρ 15) (39415781320747 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_539_16 :
+    Uω (aρ 16) (bρ 16) (39415781320747 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_539 :
+    Uρ (39415781320747 / 64000000000000 : ℝ) ≤
+      (-(8007440285446487712989 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (39415781320747 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(39619911384348771533 / 80000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_539_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (39415781320747 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(4991812124691438887987 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_539_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (39415781320747 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(2535463593367915656409 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_539_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (39415781320747 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(162616523524473707163 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_539_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (39415781320747 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(5409419981822095878167 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_539_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (39415781320747 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(2856005716701236298161 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_539_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (39415781320747 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(614020774142777776231 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_539_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (39415781320747 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(6728165572055259940237 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_539_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (39415781320747 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(7518117780844264344539 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_539_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (39415781320747 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(8567534377982396619123 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_539_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (39415781320747 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(1993956523096417586819 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_539_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (39415781320747 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(745671718695141265711 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_539_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (39415781320747 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(7682710624029974679527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_539_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (39415781320747 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_539_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (39415781320747 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_539_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (39415781320747 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_539_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_540_1 :
+    Uω (aρ 1) (bρ 1) (9871143481421 / 16000000000000 : ℝ) ≤
+      (-(4934866533064163862467 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2441976390528499 / 4000000000000000 : ℝ) ≤
+      (-(4934866533064163862467 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2441976390528499 / 4000000000000000 : ℝ) =
+            (2441976390528499 / 2000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2441976390528499 / 2000000000000000 : ℝ) ≤
+            (1996605269935836137533 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (610491449483937 / 1000000000000000 : ℝ)) (q :=
+      (2441976390528499 / 4000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_540_2 :
+    Uω (aρ 2) (bρ 2) (9871143481421 / 16000000000000 : ℝ) ≤
+      (-(497411993155784997629 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3040511990054003 / 5000000000000000 : ℝ) ≤
+      (-(497411993155784997629 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (3040511990054003 / 5000000000000000 : ℝ) =
+            (3040511990054003 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3040511990054003 / 2500000000000000 : ℝ) ≤
+            (195735187144215002371 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6080849514922887 / 10000000000000000 : ℝ)) (q :=
+      (3040511990054003 / 5000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_540_3 :
+    Uω (aρ 3) (bρ 3) (9871143481421 / 16000000000000 : ℝ) ≤
+      (-(631636661234611895621 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (377074326482227 / 625000000000000 : ℝ) ≤
+      (-(631636661234611895621 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (377074326482227 / 625000000000000 : ℝ) =
+            (377074326482227 / 312500000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (377074326482227 / 312500000000000 : ℝ) ≤
+            (234797314140388104379 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6032576108488139 / 10000000000000000 : ℝ)) (q :=
+      (377074326482227 / 625000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_540_4 :
+    Uω (aρ 4) (bρ 4) (9871143481421 / 16000000000000 : ℝ) ≤
+      (-(648206640404216422783 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (29768705563559 / 50000000000000 : ℝ) ≤
+      (-(648206640404216422783 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (29768705563559 / 50000000000000 : ℝ) = (29768705563559 / 25000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (29768705563559 / 25000000000000 : ℝ) ≤
+            (218227334970783577217 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (238078962701819 / 400000000000000 : ℝ)) (q :=
+      (29768705563559 / 50000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_540_5 :
+    Uω (aρ 5) (bρ 5) (9871143481421 / 16000000000000 : ℝ) ≤
+      (-(5390960067592240995977 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11665505817747879 / 20000000000000000 : ℝ) ≤
+      (-(5390960067592240995977 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11665505817747879 / 20000000000000000 : ℝ) =
+            (11665505817747879 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11665505817747879 / 10000000000000000 : ℝ) ≤
+            (1540511735407759004023 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2914093789602377 / 5000000000000000 : ℝ)) (q :=
+      (11665505817747879 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_540_6 :
+    Uω (aρ 6) (bρ 6) (9871143481421 / 16000000000000 : ℝ) ≤
+      (-(2846481658037417445579 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5659235209713869 / 10000000000000000 : ℝ) ≤
+      (-(2846481658037417445579 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5659235209713869 / 10000000000000000 : ℝ) =
+            (5659235209713869 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5659235209713869 / 5000000000000000 : ℝ) ≤
+            (619254243462582554421 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5648392556024613 / 10000000000000000 : ℝ)) (q :=
+      (5659235209713869 / 10000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_540_7 :
+    Uω (aρ 7) (bρ 7) (9871143481421 / 16000000000000 : ℝ) ≤
+      (-(6120277243219995072763 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5422502194449367 / 10000000000000000 : ℝ) ≤
+      (-(6120277243219995072763 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5422502194449367 / 10000000000000000 : ℝ) =
+            (5422502194449367 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5422502194449367 / 5000000000000000 : ℝ) ≤
+            (811194559780004927237 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5398587906255609 / 10000000000000000 : ℝ)) (q :=
+      (5422502194449367 / 10000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_540_8 :
+    Uω (aρ 8) (bρ 8) (9871143481421 / 16000000000000 : ℝ) ≤
+      (-(670691749872474504841 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5113547257128181 / 10000000000000000 : ℝ) ≤
+      (-(670691749872474504841 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5113547257128181 / 10000000000000000 : ℝ) =
+            (5113547257128181 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5113547257128181 / 5000000000000000 : ℝ) ≤
+            (22455430427525495159 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5064219682953237 / 10000000000000000 : ℝ)) (q :=
+      (5113547257128181 / 10000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_540_9 :
+    Uω (aρ 9) (bρ 9) (9871143481421 / 16000000000000 : ℝ) ≤
+      (-(7494875352599199835211 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (295380428584411 / 625000000000000 : ℝ) ≤
+      (-(7494875352599199835211 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (295380428584411 / 625000000000000 : ℝ) =
+            (295380428584411 / 156250000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (295380428584411 / 156250000000000 : ℝ) ≤
+            (6368068253400800164789 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4630264511758027 / 10000000000000000 : ℝ)) (q :=
+      (295380428584411 / 625000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_540_10 :
+    Uω (aρ 10) (bρ 10) (9871143481421 / 16000000000000 : ℝ) ≤
+      (-(8541150373580887895981 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4256597171786073 / 10000000000000000 : ℝ) ≤
+      (-(8541150373580887895981 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4256597171786073 / 10000000000000000 : ℝ) =
+            (4256597171786073 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4256597171786073 / 2500000000000000 : ℝ) ≤
+            (5321793232419112104019 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4079844174124021 / 10000000000000000 : ℝ)) (q :=
+      (4256597171786073 / 10000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_540_11 :
+    Uω (aρ 11) (bρ 11) (9871143481421 / 16000000000000 : ℝ) ≤
+      (-(397520041479983402797 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7403346808122357 / 20000000000000000 : ℝ) ≤
+      (-(397520041479983402797 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7403346808122357 / 20000000000000000 : ℝ) =
+            (7403346808122357 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7403346808122357 / 5000000000000000 : ℝ) ≤
+            (156997702760016597203 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (26472447381869 / 78125000000000 : ℝ)) (q :=
+      (7403346808122357 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_540_12 :
+    Uω (aρ 12) (bρ 12) (9871143481421 / 16000000000000 : ℝ) ≤
+      (-(475505288755951282507 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3045977508758077 / 10000000000000000 : ℝ) ≤
+      (-(475505288755951282507 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3045977508758077 / 10000000000000000 : ℝ) =
+            (3045977508758077 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3045977508758077 / 2500000000000000 : ℝ) ≤
+            (79012455484048717493 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2500781570958029 / 10000000000000000 : ℝ)) (q :=
+      (3045977508758077 / 10000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_540_13 :
+    Uω (aρ 13) (bρ 13) (9871143481421 / 16000000000000 : ℝ) ≤
+      (-(7635420998099539770937 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2171679633044277 / 10000000000000000 : ℝ) ≤
+      (-(7635420998099539770937 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2171679633044277 / 10000000000000000 : ℝ) =
+            (2171679633044277 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2171679633044277 / 1250000000000000 : ℝ) ≤
+            (2761786706400460229063 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1151567561125429 / 10000000000000000 : ℝ)) (q :=
+      (2171679633044277 / 10000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_540_14 :
+    Uω (aρ 14) (bρ 14) (9871143481421 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_540_15 :
+    Uω (aρ 15) (bρ 15) (9871143481421 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_540_16 :
+    Uω (aρ 16) (bρ 16) (9871143481421 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_540 :
+    Uρ (9871143481421 / 16000000000000 : ℝ) ≤
+      (-(1596631244612458406567 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (9871143481421 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(4934866533064163862467 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_540_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (9871143481421 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(497411993155784997629 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_540_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (9871143481421 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(631636661234611895621 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_540_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (9871143481421 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(648206640404216422783 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_540_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (9871143481421 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(5390960067592240995977 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_540_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (9871143481421 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(2846481658037417445579 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_540_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (9871143481421 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(6120277243219995072763 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_540_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (9871143481421 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(670691749872474504841 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_540_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (9871143481421 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(7494875352599199835211 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_540_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (9871143481421 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(8541150373580887895981 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_540_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (9871143481421 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) * (-(397520041479983402797 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_540_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (9871143481421 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(475505288755951282507 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_540_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (9871143481421 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(7635420998099539770937 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_540_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (9871143481421 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_540_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (9871143481421 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_540_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (9871143481421 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_540_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_541_1 :
+    Uω (aρ 1) (bρ 1) (39553366530621 / 64000000000000 : ℝ) ≤
+      (-(4917275143594234884783 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (48925518939090473 / 80000000000000000 : ℝ) ≤
+      (-(4917275143594234884783 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (48925518939090473 / 80000000000000000 : ℝ) =
+            (48925518939090473 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (48925518939090473 / 40000000000000000 : ℝ) ≤
+            (2014196659405765115217 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6115663432448087 / 10000000000000000 : ℝ)) (q :=
+      (48925518939090473 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_541_2 :
+    Uω (aρ 2) (bρ 2) (39553366530621 / 64000000000000 : ℝ) ≤
+      (-(77444671661105992217 / 156250000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (48734185059835957 / 80000000000000000 : ℝ) ≤
+      (-(77444671661105992217 / 156250000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (48734185059835957 / 80000000000000000 : ℝ) =
+            (48734185059835957 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (48734185059835957 / 40000000000000000 : ℝ) ≤
+            (30859575260769007783 / 156250000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3045799487572229 / 5000000000000000 : ℝ)) (q :=
+      (48734185059835957 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_541_3 :
+    Uω (aρ 3) (bρ 3) (39553366530621 / 64000000000000 : ℝ) ≤
+      (-(2517645573818314008889 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (48351513269940257 / 80000000000000000 : ℝ) ≤
+      (-(2517645573818314008889 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (48351513269940257 / 80000000000000000 : ℝ) =
+            (48351513269940257 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (48351513269940257 / 40000000000000000 : ℝ) ≤
+            (948090327681685991111 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6043327134020533 / 10000000000000000 : ℝ)) (q :=
+      (48351513269940257 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_541_4 :
+    Uω (aρ 4) (bρ 4) (39553366530621 / 64000000000000 : ℝ) ≤
+      (-(2583805063455020443923 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (47715945141100593 / 80000000000000000 : ℝ) ≤
+      (-(2583805063455020443923 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (47715945141100593 / 80000000000000000 : ℝ) =
+            (47715945141100593 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (47715945141100593 / 40000000000000000 : ℝ) ≤
+            (881930838044979556077 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5962729282875617 / 10000000000000000 : ℝ)) (q :=
+      (47715945141100593 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_541_5 :
+    Uω (aρ 5) (bρ 5) (39553366530621 / 64000000000000 : ℝ) ≤
+      (-(83945847197113369509 / 156250000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (46748081261254197 / 80000000000000000 : ℝ) ≤
+      (-(83945847197113369509 / 156250000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (46748081261254197 / 80000000000000000 : ℝ) =
+            (46748081261254197 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (46748081261254197 / 40000000000000000 : ℝ) ≤
+            (24358399724761630491 / 156250000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2919476616124509 / 5000000000000000 : ℝ)) (q :=
+      (46748081261254197 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_541_6 :
+    Uω (aρ 6) (bρ 6) (39553366530621 / 64000000000000 : ℝ) ≤
+      (-(354621971976424965729 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (45360037187602009 / 80000000000000000 : ℝ) ≤
+      (-(354621971976424965729 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (45360037187602009 / 80000000000000000 : ℝ) =
+            (45360037187602009 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (45360037187602009 / 40000000000000000 : ℝ) ≤
+            (78595015711075034271 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5659182588975971 / 10000000000000000 : ℝ)) (q :=
+      (45360037187602009 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_541_7 :
+    Uω (aρ 7) (bρ 7) (39553366530621 / 64000000000000 : ℝ) ≤
+      (-(3050193369503106150261 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (43466388467366593 / 80000000000000000 : ℝ) ≤
+      (-(3050193369503106150261 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (43466388467366593 / 80000000000000000 : ℝ) =
+            (43466388467366593 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (43466388467366593 / 40000000000000000 : ℝ) ≤
+            (415542531996893849739 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5409431789677117 / 10000000000000000 : ℝ)) (q :=
+      (43466388467366593 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_541_8 :
+    Uω (aρ 8) (bρ 8) (39553366530621 / 64000000000000 : ℝ) ≤
+      (-(3342857676754653328359 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (40995204607164973 / 80000000000000000 : ℝ) ≤
+      (-(3342857676754653328359 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (40995204607164973 / 80000000000000000 : ℝ) =
+            (40995204607164973 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (40995204607164973 / 40000000000000000 : ℝ) ≤
+            (122878224745346671641 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (634397184495839 / 1250000000000000 : ℝ)) (q :=
+      (40995204607164973 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_541_9 :
+    Uω (aρ 9) (bρ 9) (39553366530621 / 64000000000000 : ℝ) ≤
+      (-(116745141336085632999 / 156250000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1515858438762001 / 3200000000000000 : ℝ) ≤
+      (-(116745141336085632999 / 156250000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1515858438762001 / 3200000000000000 : ℝ) =
+            (1515858438762001 / 800000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1515858438762001 / 800000000000000 : ℝ) ≤
+            (99863352507664367001 / 156250000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (185658287791919 / 400000000000000 : ℝ)) (q :=
+      (1515858438762001 / 3200000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_541_10 :
+    Uω (aρ 10) (bρ 10) (39553366530621 / 64000000000000 : ℝ) ≤
+      (-(8514841796218230196619 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (34142483337157277 / 80000000000000000 : ℝ) ≤
+      (-(8514841796218230196619 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (34142483337157277 / 80000000000000000 : ℝ) =
+            (34142483337157277 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (34142483337157277 / 20000000000000000 : ℝ) ≤
+            (5348101809781769803381 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1022880455079947 / 2500000000000000 : ℝ)) (q :=
+      (34142483337157277 / 80000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_541_11 :
+    Uω (aρ 11) (bρ 11) (39553366530621 / 64000000000000 : ℝ) ≤
+      (-(9906338689090923457797 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1188291950557753 / 3200000000000000 : ℝ) ≤
+      (-(9906338689090923457797 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1188291950557753 / 3200000000000000 : ℝ) =
+            (1188291950557753 / 800000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1188291950557753 / 800000000000000 : ℝ) ≤
+            (3956604916909076542203 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (136048092128857 / 400000000000000 : ℝ)) (q :=
+      (1188291950557753 / 3200000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_541_12 :
+    Uω (aρ 12) (bρ 12) (39553366530621 / 64000000000000 : ℝ) ≤
+      (-(2368956448966278179891 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24472460146992109 / 80000000000000000 : ℝ) ≤
+      (-(2368956448966278179891 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (24472460146992109 / 80000000000000000 : ℝ) =
+            (24472460146992109 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24472460146992109 / 20000000000000000 : ℝ) ≤
+            (403632272233721820109 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (157262046604281 / 625000000000000 : ℝ)) (q :=
+      (24472460146992109 / 80000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_541_13 :
+    Uω (aρ 13) (bρ 13) (39553366530621 / 64000000000000 : ℝ) ≤
+      (-(15178679457575972544311 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17534295186849661 / 80000000000000000 : ℝ) ≤
+      (-(15178679457575972544311 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (17534295186849661 / 80000000000000000 : ℝ) =
+            (17534295186849661 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17534295186849661 / 10000000000000000 : ℝ) ≤
+            (5615735951424027455689 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (295258311806971 / 2500000000000000 : ℝ)) (q :=
+      (17534295186849661 / 80000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_541_14 :
+    Uω (aρ 14) (bρ 14) (39553366530621 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_541_15 :
+    Uω (aρ 15) (bρ 15) (39553366530621 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_541_16 :
+    Uω (aρ 16) (bρ 16) (39553366530621 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_541 :
+    Uρ (39553366530621 / 64000000000000 : ℝ) ≤
+      (-(1591814203017836894719 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (39553366530621 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(4917275143594234884783 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_541_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (39553366530621 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) * (-(77444671661105992217 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_541_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (39553366530621 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(2517645573818314008889 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_541_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (39553366530621 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(2583805063455020443923 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_541_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (39553366530621 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(83945847197113369509 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_541_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (39553366530621 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(354621971976424965729 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_541_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (39553366530621 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(3050193369503106150261 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_541_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (39553366530621 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(3342857676754653328359 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_541_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (39553366530621 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(116745141336085632999 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_541_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (39553366530621 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(8514841796218230196619 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_541_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (39553366530621 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(9906338689090923457797 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_541_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (39553366530621 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(2368956448966278179891 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_541_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (39553366530621 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(15178679457575972544311 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_541_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (39553366530621 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_541_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (39553366530621 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_541_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (39553366530621 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_541_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_542_1 :
+    Uω (aρ 1) (bρ 1) (19811079567779 / 32000000000000 : ℝ) ≤
+      (-(195988585830199645053 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24505755033152193 / 40000000000000000 : ℝ) ≤
+      (-(195988585830199645053 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (24505755033152193 / 40000000000000000 : ℝ) =
+            (24505755033152193 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24505755033152193 / 20000000000000000 : ℝ) ≤
+            (81270286289800354947 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6126412369730159 / 10000000000000000 : ℝ)) (q :=
+      (24505755033152193 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_542_2 :
+    Uω (aρ 2) (bρ 2) (19811079567779 / 32000000000000 : ℝ) ≤
+      (-(617353647344841031591 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24410089135065761 / 40000000000000000 : ℝ) ≤
+      (-(617353647344841031591 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (24410089135065761 / 40000000000000000 : ℝ) =
+            (24410089135065761 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24410089135065761 / 20000000000000000 : ℝ) ≤
+            (249080328030158968409 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6102348433196943 / 10000000000000000 : ℝ)) (q :=
+      (24410089135065761 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_542_3 :
+    Uω (aρ 3) (bρ 3) (19811079567779 / 32000000000000 : ℝ) ≤
+      (-(5017520647110118038431 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24218756359586867 / 40000000000000000 : ℝ) ≤
+      (-(5017520647110118038431 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (24218756359586867 / 40000000000000000 : ℝ) =
+            (24218756359586867 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24218756359586867 / 20000000000000000 : ℝ) ≤
+            (1913951155889881961569 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (756759768975937 / 1250000000000000 : ℝ)) (q :=
+      (24218756359586867 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_542_4 :
+    Uω (aρ 4) (bρ 4) (19811079567779 / 32000000000000 : ℝ) ≤
+      (-(514959964612481159213 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23900980644384603 / 40000000000000000 : ℝ) ≤
+      (-(514959964612481159213 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23900980644384603 / 40000000000000000 : ℝ) =
+            (23900980644384603 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23900980644384603 / 20000000000000000 : ℝ) ≤
+            (178187215687518840787 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1493371118817841 / 2500000000000000 : ℝ)) (q :=
+      (23900980644384603 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_542_5 :
+    Uω (aρ 5) (bρ 5) (19811079567779 / 32000000000000 : ℝ) ≤
+      (-(1070828463037465693711 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23417069502119359 / 40000000000000000 : ℝ) ≤
+      (-(1070828463037465693711 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23417069502119359 / 40000000000000000 : ℝ) =
+            (23417069502119359 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23417069502119359 / 20000000000000000 : ℝ) ≤
+            (315465897562534306289 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2924859411736871 / 5000000000000000 : ℝ)) (q :=
+      (23417069502119359 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_542_6 :
+    Uω (aρ 6) (bρ 6) (19811079567779 / 32000000000000 : ℝ) ≤
+      (-(565497600102780771467 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22723096035816431 / 40000000000000000 : ℝ) ≤
+      (-(565497600102780771467 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (22723096035816431 / 40000000000000000 : ℝ) =
+            (22723096035816431 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22723096035816431 / 20000000000000000 : ℝ) ≤
+            (127649580197219228533 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2834986232731139 / 5000000000000000 : ℝ)) (q :=
+      (22723096035816431 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_542_7 :
+    Uω (aρ 7) (bρ 7) (19811079567779 / 32000000000000 : ℝ) ≤
+      (-(6080536067205933051063 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21776378932357017 / 40000000000000000 : ℝ) ≤
+      (-(6080536067205933051063 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (21776378932357017 / 40000000000000000 : ℝ) =
+            (21776378932357017 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21776378932357017 / 20000000000000000 : ℝ) ≤
+            (850935735794066948937 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5420275294492571 / 10000000000000000 : ℝ)) (q :=
+      (21776378932357017 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_542_8 :
+    Uω (aρ 8) (bρ 8) (19811079567779 / 32000000000000 : ℝ) ≤
+      (-(6664558934562687843943 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20541013795148927 / 40000000000000000 : ℝ) ≤
+      (-(6664558934562687843943 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (20541013795148927 / 40000000000000000 : ℝ) =
+            (20541013795148927 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20541013795148927 / 20000000000000000 : ℝ) ≤
+            (266912868437312156057 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2543067188614263 / 5000000000000000 : ℝ)) (q :=
+      (20541013795148927 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_542_9 :
+    Uω (aρ 9) (bρ 9) (19811079567779 / 32000000000000 : ℝ) ≤
+      (-(1862139644687844223167 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (18992109353432327 / 40000000000000000 : ℝ) ≤
+      (-(1862139644687844223167 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (18992109353432327 / 40000000000000000 : ℝ) =
+            (18992109353432327 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18992109353432327 / 10000000000000000 : ℝ) ≤
+            (1603596256812155776833 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2326323892365113 / 5000000000000000 : ℝ)) (q :=
+      (18992109353432327 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_542_10 :
+    Uω (aρ 10) (bρ 10) (19811079567779 / 32000000000000 : ℝ) ≤
+      (-(8488608183216975225863 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17116084497070053 / 40000000000000000 : ℝ) ≤
+      (-(8488608183216975225863 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (17116084497070053 / 40000000000000000 : ℝ) =
+            (17116084497070053 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17116084497070053 / 10000000000000000 : ℝ) ≤
+            (5374335422783024774137 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4103194390044089 / 10000000000000000 : ℝ)) (q :=
+      (17116084497070053 / 40000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_542_11 :
+    Uω (aρ 11) (bρ 11) (19811079567779 / 32000000000000 : ℝ) ≤
+      (-(4937397276860661833359 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14900577911737607 / 40000000000000000 : ℝ) ≤
+      (-(4937397276860661833359 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (14900577911737607 / 40000000000000000 : ℝ) =
+            (14900577911737607 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14900577911737607 / 10000000000000000 : ℝ) ≤
+            (1994074526139338166641 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1706958861791433 / 5000000000000000 : ℝ)) (q :=
+      (14900577911737607 / 40000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_542_12 :
+    Uω (aρ 12) (bρ 12) (19811079567779 / 32000000000000 : ℝ) ≤
+      (-(2950548377821138222619 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12288453755806303 / 40000000000000000 : ℝ) ≤
+      (-(2950548377821138222619 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (12288453755806303 / 40000000000000000 : ℝ) =
+            (12288453755806303 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12288453755806303 / 10000000000000000 : ℝ) ≤
+            (515187523678861777381 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1265777871151107 / 5000000000000000 : ℝ)) (q :=
+      (12288453755806303 / 40000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_542_13 :
+    Uω (aρ 13) (bρ 13) (19811079567779 / 32000000000000 : ℝ) ≤
+      (-(15088764605498135512977 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8846332733846867 / 40000000000000000 : ℝ) ≤
+      (-(15088764605498135512977 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (8846332733846867 / 40000000000000000 : ℝ) =
+            (8846332733846867 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8846332733846867 / 5000000000000000 : ℝ) ≤
+            (5705650803501864487023 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (151234621614687 / 1250000000000000 : ℝ)) (q :=
+      (8846332733846867 / 40000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_542_14 :
+    Uω (aρ 14) (bρ 14) (19811079567779 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_542_15 :
+    Uω (aρ 15) (bρ 15) (19811079567779 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_542_16 :
+    Uω (aρ 16) (bρ 16) (19811079567779 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_542 :
+    Uρ (19811079567779 / 32000000000000 : ℝ) ≤
+      (-(7935174218185580261567 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (19811079567779 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(195988585830199645053 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_542_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (19811079567779 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(617353647344841031591 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_542_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (19811079567779 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(5017520647110118038431 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_542_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (19811079567779 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(514959964612481159213 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_542_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (19811079567779 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1070828463037465693711 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_542_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (19811079567779 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(565497600102780771467 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_542_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (19811079567779 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(6080536067205933051063 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_542_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (19811079567779 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(6664558934562687843943 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_542_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (19811079567779 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(1862139644687844223167 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_542_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (19811079567779 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(8488608183216975225863 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_542_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (19811079567779 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4937397276860661833359 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_542_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (19811079567779 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(2950548377821138222619 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_542_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (19811079567779 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(15088764605498135512977 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_542_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (19811079567779 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_542_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (19811079567779 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_542_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (19811079567779 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_542_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_543_1 :
+    Uω (aρ 1) (bρ 1) (7938190348099 / 12800000000000 : ℝ) ≤
+      (-(195287397249608713577 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1963900047688743 / 3200000000000000 : ℝ) ≤
+      (-(195287397249608713577 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1963900047688743 / 3200000000000000 : ℝ) =
+            (1963900047688743 / 1600000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1963900047688743 / 1600000000000000 : ℝ) ≤
+            (81971474870391286423 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (61371613066873 / 100000000000000 : ℝ)) (q :=
+      (1963900047688743 / 3200000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_543_2 :
+    Uω (aρ 2) (bρ 2) (7938190348099 / 12800000000000 : ℝ) ≤
+      (-(492123039929203977663 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (48906171471796523 / 80000000000000000 : ℝ) ≤
+      (-(492123039929203977663 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (48906171471796523 / 80000000000000000 : ℝ) =
+            (48906171471796523 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (48906171471796523 / 40000000000000000 : ℝ) ≤
+            (201024140370796022337 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6113097889091787 / 10000000000000000 : ℝ)) (q :=
+      (48906171471796523 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_543_3 :
+    Uω (aρ 3) (bρ 3) (7938190348099 / 12800000000000 : ℝ) ≤
+      (-(4999781675993666482479 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (48523512137590243 / 80000000000000000 : ℝ) ≤
+      (-(4999781675993666482479 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (48523512137590243 / 80000000000000000 : ℝ) =
+            (48523512137590243 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (48523512137590243 / 40000000000000000 : ℝ) ≤
+            (1931690127006333517521 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6064829161890217 / 10000000000000000 : ℝ)) (q :=
+      (48523512137590243 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_543_4 :
+    Uω (aρ 4) (bρ 4) (7938190348099 / 12800000000000 : ℝ) ≤
+      (-(2565810781913314626317 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (47887977345194867 / 80000000000000000 : ℝ) ≤
+      (-(2565810781913314626317 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (47887977345194867 / 80000000000000000 : ℝ) =
+            (47887977345194867 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (47887977345194867 / 40000000000000000 : ℝ) ≤
+            (899925119586685373683 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5984239644856373 / 10000000000000000 : ℝ)) (q :=
+      (47887977345194867 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_543_5 :
+    Uω (aρ 5) (bρ 5) (7938190348099 / 12800000000000 : ℝ) ≤
+      (-(5335784226300181600677 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (46920196501307823 / 80000000000000000 : ℝ) ≤
+      (-(5335784226300181600677 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (46920196501307823 / 80000000000000000 : ℝ) =
+            (46920196501307823 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (46920196501307823 / 40000000000000000 : ℝ) ≤
+            (1595687576699818399323 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1465121088304903 / 2500000000000000 : ℝ)) (q :=
+      (46920196501307823 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_543_6 :
+    Uω (aρ 6) (bρ 6) (7938190348099 / 12800000000000 : ℝ) ≤
+      (-(2818018263035098048087 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9106469266673947 / 16000000000000000 : ℝ) ≤
+      (-(2818018263035098048087 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (9106469266673947 / 16000000000000000 : ℝ) =
+            (9106469266673947 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9106469266673947 / 8000000000000000 : ℝ) ≤
+            (647717638464901951913 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (568076218637509 / 1000000000000000 : ℝ)) (q :=
+      (9106469266673947 / 16000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_543_7 :
+    Uω (aρ 7) (bρ 7) (7938190348099 / 12800000000000 : ℝ) ≤
+      (-(1515181266806033804241 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (43639125756708131 / 80000000000000000 : ℝ) ≤
+      (-(1515181266806033804241 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (43639125756708131 / 80000000000000000 : ℝ) =
+            (43639125756708131 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (43639125756708131 / 40000000000000000 : ℝ) ≤
+            (217686683943966195759 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5431118422969689 / 10000000000000000 : ℝ)) (q :=
+      (43639125756708131 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_543_8 :
+    Uω (aρ 8) (bρ 8) (7938190348099 / 12800000000000 : ℝ) ≤
+      (-(830431005173728875531 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (41168847029427407 / 80000000000000000 : ℝ) ≤
+      (-(830431005173728875531 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (41168847029427407 / 80000000000000000 : ℝ) =
+            (41168847029427407 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (41168847029427407 / 40000000000000000 : ℝ) ≤
+            (36002970201271124469 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1274272598122377 / 2500000000000000 : ℝ)) (q :=
+      (41168847029427407 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_543_9 :
+    Uω (aρ 9) (bρ 9) (7938190348099 / 12800000000000 : ℝ) ≤
+      (-(3712741836837896895691 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (38071968132515891 / 80000000000000000 : ℝ) ≤
+      (-(3712741836837896895691 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (38071968132515891 / 80000000000000000 : ℝ) =
+            (38071968132515891 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (38071968132515891 / 20000000000000000 : ℝ) ≤
+            (3218729966162103104309 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4663836296621629 / 10000000000000000 : ℝ)) (q :=
+      (38071968132515891 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_543_10 :
+    Uω (aρ 10) (bρ 10) (7938190348099 / 12800000000000 : ℝ) ≤
+      (-(2115612269099813192683 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6864366903608263 / 16000000000000000 : ℝ) ≤
+      (-(2115612269099813192683 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6864366903608263 / 16000000000000000 : ℝ) =
+            (6864366903608263 / 4000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6864366903608263 / 4000000000000000 : ℝ) ≤
+            (1350123632400186807317 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (822972385299597 / 2000000000000000 : ℝ)) (q :=
+      (6864366903608263 / 16000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_543_11 :
+    Uω (aρ 11) (bρ 11) (7938190348099 / 12800000000000 : ℝ) ≤
+      (-(4921683813520243516333 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5978991803496607 / 16000000000000000 : ℝ) ≤
+      (-(4921683813520243516333 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5978991803496607 / 16000000000000000 : ℝ) =
+            (5978991803496607 / 4000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5978991803496607 / 4000000000000000 : ℝ) ≤
+            (2009787989479756483667 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (685323935512683 / 2000000000000000 : ℝ)) (q :=
+      (5978991803496607 / 16000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_543_12 :
+    Uω (aρ 12) (bρ 12) (7938190348099 / 12800000000000 : ℝ) ≤
+      (-(11759862056280220014139 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24681165651309479 / 80000000000000000 : ℝ) ≤
+      (-(11759862056280220014139 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (24681165651309479 / 80000000000000000 : ℝ) =
+            (24681165651309479 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24681165651309479 / 20000000000000000 : ℝ) ≤
+            (2103081549719779985861 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1273435716352513 / 5000000000000000 : ℝ)) (q :=
+      (24681165651309479 / 80000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_543_13 :
+    Uω (aρ 13) (bρ 13) (7938190348099 / 12800000000000 : ℝ) ≤
+      (-(7500473685979014998829 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17848721780004003 / 80000000000000000 : ℝ) ≤
+      (-(7500473685979014998829 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (17848721780004003 / 80000000000000000 : ℝ) =
+            (17848721780004003 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17848721780004003 / 10000000000000000 : ℝ) ≤
+            (2896734018520985001171 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1238142206473657 / 10000000000000000 : ℝ)) (q :=
+      (17848721780004003 / 80000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_543_14 :
+    Uω (aρ 14) (bρ 14) (7938190348099 / 12800000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_543_15 :
+    Uω (aρ 15) (bρ 15) (7938190348099 / 12800000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_543_16 :
+    Uω (aρ 16) (bρ 16) (7938190348099 / 12800000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_543 :
+    Uρ (7938190348099 / 12800000000000 : ℝ) ≤
+      (-(494466032119251637601 / 625000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (7938190348099 / 12800000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(195287397249608713577 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_543_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (7938190348099 / 12800000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(492123039929203977663 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_543_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (7938190348099 / 12800000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(4999781675993666482479 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_543_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (7938190348099 / 12800000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(2565810781913314626317 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_543_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (7938190348099 / 12800000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(5335784226300181600677 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_543_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (7938190348099 / 12800000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(2818018263035098048087 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_543_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (7938190348099 / 12800000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(1515181266806033804241 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_543_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (7938190348099 / 12800000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(830431005173728875531 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_543_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (7938190348099 / 12800000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(3712741836837896895691 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_543_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (7938190348099 / 12800000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(2115612269099813192683 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_543_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (7938190348099 / 12800000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4921683813520243516333 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_543_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (7938190348099 / 12800000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(11759862056280220014139 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_543_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (7938190348099 / 12800000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(7500473685979014998829 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_543_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (7938190348099 / 12800000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_543_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (7938190348099 / 12800000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_543_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (7938190348099 / 12800000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_543_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/U45.lean b/LeanPool/Zeta5Irrational/Table/U45.lean
new file mode 100644
index 0000000000..840f90bda5
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/U45.lean
@@ -0,0 +1,4829 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-! # Certified arcsine potential bounds (U45) -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma U_544_1 :
+    Uω (aρ 1) (bρ 1) (4969968043179 / 8000000000000 : ℝ) ≤
+      (-(243234294615614466823 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2459174615841993 / 4000000000000000 : ℝ) ≤
+      (-(243234294615614466823 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2459174615841993 / 4000000000000000 : ℝ) =
+            (2459174615841993 / 2000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2459174615841993 / 2000000000000000 : ℝ) ≤
+            (103339295534385533177 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1229582048664243 / 2000000000000000 : ℝ)) (q :=
+      (2459174615841993 / 4000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_544_2 :
+    Uω (aρ 2) (bρ 2) (4969968043179 / 8000000000000 : ℝ) ≤
+      (-(980732507775771540851 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12248041166219101 / 20000000000000000 : ℝ) ≤
+      (-(980732507775771540851 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (12248041166219101 / 20000000000000000 : ℝ) =
+            (12248041166219101 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12248041166219101 / 10000000000000000 : ℝ) ≤
+            (405561852824228459149 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6123847342840351 / 10000000000000000 : ℝ)) (q :=
+      (12248041166219101 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_544_3 :
+    Uω (aρ 3) (bρ 3) (4969968043179 / 8000000000000 : ℝ) ≤
+      (-(2491037061290261113757 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1519047235167293 / 2500000000000000 : ℝ) ≤
+      (-(2491037061290261113757 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1519047235167293 / 2500000000000000 : ℝ) =
+            (1519047235167293 / 1250000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1519047235167293 / 1250000000000000 : ℝ) ≤
+            (974698840209738886243 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3037790082154797 / 5000000000000000 : ℝ)) (q :=
+      (1519047235167293 / 2500000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_544_4 :
+    Uω (aρ 4) (bρ 4) (4969968043179 / 8000000000000 : ℝ) ≤
+      (-(5113675763595366645521 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11993498327717163 / 20000000000000000 : ℝ) ≤
+      (-(5113675763595366645521 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11993498327717163 / 20000000000000000 : ℝ) =
+            (11993498327717163 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11993498327717163 / 10000000000000000 : ℝ) ≤
+            (1817796039404633354479 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5994994791753413 / 10000000000000000 : ℝ)) (q :=
+      (11993498327717163 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_544_5 :
+    Uω (aρ 5) (bρ 5) (4969968043179 / 8000000000000 : ℝ) ≤
+      (-(166170619676123348097 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11751563438453559 / 20000000000000000 : ℝ) ≤
+      (-(166170619676123348097 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11751563438453559 / 20000000000000000 : ℝ) =
+            (11751563438453559 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11751563438453559 / 10000000000000000 : ℝ) ≤
+            (50437874167626651903 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5871249821824809 / 10000000000000000 : ℝ)) (q :=
+      (11751563438453559 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_544_6 :
+    Uω (aρ 6) (bρ 6) (4969968043179 / 8000000000000 : ℝ) ≤
+      (-(2808566494661532032637 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5702312497043969 / 10000000000000000 : ℝ) ≤
+      (-(2808566494661532032637 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5702312497043969 / 10000000000000000 : ℝ) =
+            (5702312497043969 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5702312497043969 / 5000000000000000 : ℝ) ≤
+            (657169406838467967363 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1422887938149797 / 2500000000000000 : ℝ)) (q :=
+      (5702312497043969 / 10000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_544_7 :
+    Uω (aρ 7) (bρ 7) (4969968043179 / 8000000000000 : ℝ) ≤
+      (-(755119197430400781771 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10931373038086789 / 20000000000000000 : ℝ) ≤
+      (-(755119197430400781771 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10931373038086789 / 20000000000000000 : ℝ) =
+            (10931373038086789 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10931373038086789 / 10000000000000000 : ℝ) ≤
+            (111314777944599218229 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5441961177358039 / 10000000000000000 : ℝ)) (q :=
+      (10931373038086789 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_544_8 :
+    Uω (aρ 8) (bρ 8) (4969968043179 / 8000000000000 : ℝ) ≤
+      (-(1655595618708660814971 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10313915736839439 / 20000000000000000 : ℝ) ≤
+      (-(1655595618708660814971 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10313915736839439 / 20000000000000000 : ℝ) =
+            (10313915736839439 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10313915736839439 / 10000000000000000 : ℝ) ≤
+            (77272332041339185029 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5108045527450689 / 10000000000000000 : ℝ)) (q :=
+      (10313915736839439 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_544_9 :
+    Uω (aρ 9) (bρ 9) (4969968043179 / 8000000000000 : ℝ) ≤
+      (-(7402464053785215584387 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9539927326420767 / 20000000000000000 : ℝ) ≤
+      (-(7402464053785215584387 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9539927326420767 / 20000000000000000 : ℝ) =
+            (9539927326420767 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9539927326420767 / 5000000000000000 : ℝ) ≤
+            (6460479552214784415613 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4675022745392017 / 10000000000000000 : ℝ)) (q :=
+      (9539927326420767 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_544_10 :
+    Uω (aρ 10) (bρ 10) (4969968043179 / 8000000000000 : ℝ) ≤
+      (-(4218182011013012442561 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (537679376244323 / 1250000000000000 : ℝ) ≤
+      (-(4218182011013012442561 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (537679376244323 / 1250000000000000 : ℝ) =
+            (537679376244323 / 312500000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (537679376244323 / 312500000000000 : ℝ) ≤
+            (2713289791986987557439 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2063262236187709 / 5000000000000000 : ℝ)) (q :=
+      (537679376244323 / 1250000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10]) (by norm_num [bρ_10])
+      (by norm_num) (by norm_num [aρ_10, bρ_10]) (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_544_11 :
+    Uω (aρ 11) (bρ 11) (4969968043179 / 8000000000000 : ℝ) ≤
+      (-(981205691910561245223 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7497177235692769 / 20000000000000000 : ℝ) ≤
+      (-(981205691910561245223 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7497177235692769 / 20000000000000000 : ℝ) =
+            (7497177235692769 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7497177235692769 / 5000000000000000 : ℝ) ≤
+            (405088668689438754777 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3439308314364019 / 10000000000000000 : ℝ)) (q :=
+      (7497177235692769 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_544_12 :
+    Uω (aρ 12) (bρ 12) (4969968043179 / 8000000000000 : ℝ) ≤
+      (-(292944600445161657491 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1549077372466801 / 5000000000000000 : ℝ) ≤
+      (-(292944600445161657491 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1549077372466801 / 5000000000000000 : ℝ) =
+            (1549077372466801 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1549077372466801 / 1250000000000000 : ℝ) ≤
+            (53628989704838342509 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1281070332611727 / 5000000000000000 : ℝ)) (q :=
+      (1549077372466801 / 5000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_544_13 :
+    Uω (aρ 13) (bρ 13) (4969968043179 / 8000000000000 : ℝ) ≤
+      (-(1864386724188942654493 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (562581847817931 / 2500000000000000 : ℝ) ≤
+      (-(1864386724188942654493 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (562581847817931 / 2500000000000000 : ℝ) =
+            (562581847817931 / 312500000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (562581847817931 / 312500000000000 : ℝ) ≤
+            (734915201936057345507 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (632933849747349 / 5000000000000000 : ℝ)) (q :=
+      (562581847817931 / 2500000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13]) (by norm_num [bρ_13])
+      (by norm_num) (by norm_num [aρ_13, bρ_13]) (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_544_14 :
+    Uω (aρ 14) (bρ 14) (4969968043179 / 8000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_544_15 :
+    Uω (aρ 15) (bρ 15) (4969968043179 / 8000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_544_16 :
+    Uω (aρ 16) (bρ 16) (4969968043179 / 8000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_544 :
+    Uρ (4969968043179 / 8000000000000 : ℝ) ≤
+      (-(7887909540485613658987 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (4969968043179 / 8000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(243234294615614466823 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_544_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (4969968043179 / 8000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(980732507775771540851 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_544_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (4969968043179 / 8000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(2491037061290261113757 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_544_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (4969968043179 / 8000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(5113675763595366645521 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_544_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (4969968043179 / 8000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(166170619676123348097 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_544_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (4969968043179 / 8000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(2808566494661532032637 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_544_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (4969968043179 / 8000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(755119197430400781771 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_544_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (4969968043179 / 8000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(1655595618708660814971 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_544_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (4969968043179 / 8000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(7402464053785215584387 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_544_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (4969968043179 / 8000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(4218182011013012442561 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_544_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (4969968043179 / 8000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(981205691910561245223 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_544_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (4969968043179 / 8000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(292944600445161657491 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_544_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (4969968043179 / 8000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(1864386724188942654493 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_544_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (4969968043179 / 8000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_544_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (4969968043179 / 8000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_544_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (4969968043179 / 8000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_544_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_545_1 :
+    Uω (aρ 1) (bρ 1) (39828536950369 / 64000000000000 : ℝ) ≤
+      (-(4847217421798197610027 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (49269483440175017 / 80000000000000000 : ℝ) ≤
+      (-(4847217421798197610027 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (49269483440175017 / 80000000000000000 : ℝ) =
+            (49269483440175017 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (49269483440175017 / 40000000000000000 : ℝ) ≤
+            (2084254381201802389973 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3079329589816799 / 5000000000000000 : ℝ)) (q :=
+      (49269483440175017 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_545_2 :
+    Uω (aρ 2) (bρ 2) (39828536950369 / 64000000000000 : ℝ) ≤
+      (-(2443062744530523726207 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (49078157849416297 / 80000000000000000 : ℝ) ≤
+      (-(2443062744530523726207 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (49078157849416297 / 80000000000000000 : ℝ) =
+            (49078157849416297 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (49078157849416297 / 40000000000000000 : ℝ) ≤
+            (1022673156969476273793 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3067298397226959 / 5000000000000000 : ℝ)) (q :=
+      (49078157849416297 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_545_3 :
+    Uω (aρ 3) (bρ 3) (39828536950369 / 64000000000000 : ℝ) ≤
+      (-(992879575151331134011 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (48695510882625581 / 80000000000000000 : ℝ) ≤
+      (-(992879575151331134011 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (48695510882625581 / 80000000000000000 : ℝ) =
+            (48695510882625581 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (48695510882625581 / 40000000000000000 : ℝ) ≤
+            (393414785448668865989 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6086331159106239 / 10000000000000000 : ℝ)) (q :=
+      (48695510882625581 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_545_4 :
+    Uω (aρ 4) (bρ 4) (39828536950369 / 64000000000000 : ℝ) ≤
+      (-(1273940532409412364583 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (48060009186278117 / 80000000000000000 : ℝ) ≤
+      (-(1273940532409412364583 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (48060009186278117 / 80000000000000000 : ℝ) =
+            (48060009186278117 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (48060009186278117 / 40000000000000000 : ℝ) ≤
+            (458927418340587635417 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6005749916084373 / 10000000000000000 : ℝ)) (q :=
+      (48060009186278117 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_545_5 :
+    Uω (aρ 5) (bρ 5) (39828536950369 / 64000000000000 : ℝ) ≤
+      (-(2649584500781039761267 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (47092310763100777 / 80000000000000000 : ℝ) ≤
+      (-(2649584500781039761267 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (47092310763100777 / 80000000000000000 : ℝ) =
+            (47092310763100777 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (47092310763100777 / 40000000000000000 : ℝ) ≤
+            (816151400718960238733 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2941007614812519 / 5000000000000000 : ℝ)) (q :=
+      (47092310763100777 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_545_6 :
+    Uω (aρ 6) (bρ 6) (39828536950369 / 64000000000000 : ℝ) ≤
+      (-(559826525414619874691 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9140930600818241 / 16000000000000000 : ℝ) ≤
+      (-(559826525414619874691 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (9140930600818241 / 16000000000000000 : ℝ) =
+            (9140930600818241 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9140930600818241 / 8000000000000000 : ℝ) ≤
+            (133320654885380125309 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1140468233002529 / 2000000000000000 : ℝ)) (q :=
+      (9140930600818241 / 16000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_545_7 :
+    Uω (aρ 7) (bρ 7) (39828536950369 / 64000000000000 : ℝ) ≤
+      (-(240848857808598739739 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (43811857060557489 / 80000000000000000 : ℝ) ≤
+      (-(240848857808598739739 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (43811857060557489 / 80000000000000000 : ℝ) =
+            (43811857060557489 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (43811857060557489 / 40000000000000000 : ℝ) ≤
+            (36410014311401260261 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (85200055623269 / 156250000000000 : ℝ)) (q :=
+      (43811857060557489 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_545_8 :
+    Uω (aρ 8) (bρ 8) (39828536950369 / 64000000000000 : ℝ) ≤
+      (-(3300681018533957643367 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (41342475366695973 / 80000000000000000 : ℝ) ≤
+      (-(3300681018533957643367 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (41342475366695973 / 80000000000000000 : ℝ) =
+            (41342475366695973 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (41342475366695973 / 40000000000000000 : ℝ) ≤
+            (165054882966042356633 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5118999787763837 / 10000000000000000 : ℝ)) (q :=
+      (41342475366695973 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_545_9 :
+    Uω (aρ 9) (bρ 9) (39828536950369 / 64000000000000 : ℝ) ≤
+      (-(1844874861177744237403 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (38247442285464721 / 80000000000000000 : ℝ) ≤
+      (-(1844874861177744237403 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (38247442285464721 / 80000000000000000 : ℝ) =
+            (38247442285464721 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (38247442285464721 / 20000000000000000 : ℝ) ≤
+            (1620861040322255762597 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (585775893227003 / 1250000000000000 : ℝ)) (q :=
+      (38247442285464721 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_545_10 :
+    Uω (aρ 10) (bρ 10) (39828536950369 / 64000000000000 : ℝ) ≤
+      (-(336414102829495814321 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (34501105847704137 / 80000000000000000 : ℝ) ≤
+      (-(336414102829495814321 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (34501105847704137 / 80000000000000000 : ℝ) =
+            (34501105847704137 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (34501105847704137 / 20000000000000000 : ℝ) ≤
+            (218103641410504185679 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2069091034935439 / 5000000000000000 : ℝ)) (q :=
+      (34501105847704137 / 80000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_545_11 :
+    Uω (aρ 11) (bρ 11) (39828536950369 / 64000000000000 : ℝ) ≤
+      (-(1956172290722271357863 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (30082406186747193 / 80000000000000000 : ℝ) ≤
+      (-(1956172290722271357863 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (30082406186747193 / 80000000000000000 : ℝ) =
+            (30082406186747193 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (30082406186747193 / 20000000000000000 : ℝ) ≤
+            (816416430477728642137 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1725991890418321 / 5000000000000000 : ℝ)) (q :=
+      (30082406186747193 / 80000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_545_12 :
+    Uω (aρ 12) (bρ 12) (39828536950369 / 64000000000000 : ℝ) ≤
+      (-(11675955630294015158997 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24889127738889657 / 80000000000000000 : ℝ) ≤
+      (-(11675955630294015158997 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (24889127738889657 / 80000000000000000 : ℝ) =
+            (24889127738889657 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24889127738889657 / 20000000000000000 : ℝ) ≤
+            (2186987975705984841003 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1288682132778629 / 5000000000000000 : ℝ)) (q :=
+      (24889127738889657 / 80000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_545_13 :
+    Uω (aρ 13) (bρ 13) (39828536950369 / 64000000000000 : ℝ) ≤
+      (-(14831083687162373912247 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (18154496392940433 / 80000000000000000 : ℝ) ≤
+      (-(14831083687162373912247 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (18154496392940433 / 80000000000000000 : ℝ) =
+            (18154496392940433 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18154496392940433 / 10000000000000000 : ℝ) ≤
+            (5963331721837626087753 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (161636021333119 / 1250000000000000 : ℝ)) (q :=
+      (18154496392940433 / 80000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_545_14 :
+    Uω (aρ 14) (bρ 14) (39828536950369 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_545_15 :
+    Uω (aρ 15) (bρ 15) (39828536950369 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_545_16 :
+    Uω (aρ 16) (bρ 16) (39828536950369 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_545 :
+    Uρ (39828536950369 / 64000000000000 : ℝ) ≤
+      (-(245766429868497178311 / 312500000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (39828536950369 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(4847217421798197610027 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_545_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (39828536950369 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(2443062744530523726207 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_545_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (39828536950369 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(992879575151331134011 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_545_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (39828536950369 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1273940532409412364583 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_545_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (39828536950369 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(2649584500781039761267 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_545_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (39828536950369 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(559826525414619874691 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_545_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (39828536950369 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(240848857808598739739 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_545_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (39828536950369 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(3300681018533957643367 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_545_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (39828536950369 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(1844874861177744237403 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_545_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (39828536950369 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) * (-(336414102829495814321 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_545_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (39828536950369 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(1956172290722271357863 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_545_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (39828536950369 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(11675955630294015158997 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_545_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (39828536950369 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(14831083687162373912247 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_545_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (39828536950369 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_545_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (39828536950369 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_545_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (39828536950369 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_545_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_546_1 :
+    Uω (aρ 1) (bρ 1) (19948664777653 / 32000000000000 : ℝ) ≤
+      (-(4829779413085618008589 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24677737281115379 / 40000000000000000 : ℝ) ≤
+      (-(4829779413085618008589 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (24677737281115379 / 40000000000000000 : ℝ) =
+            (24677737281115379 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24677737281115379 / 20000000000000000 : ℝ) ≤
+            (2101692389914381991411 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6169408115626127 / 10000000000000000 : ℝ)) (q :=
+      (24677737281115379 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_546_2 :
+    Uω (aρ 2) (bρ 2) (19948664777653 / 32000000000000 : ℝ) ≤
+      (-(2434309570975091985393 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24582075512730507 / 40000000000000000 : ℝ) ≤
+      (-(2434309570975091985393 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (24582075512730507 / 40000000000000000 : ℝ) =
+            (24582075512730507 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24582075512730507 / 20000000000000000 : ℝ) ≤
+            (1031426330524908014607 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6145346243943691 / 10000000000000000 : ℝ)) (q :=
+      (24582075512730507 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_546_3 :
+    Uω (aρ 3) (bρ 3) (19948664777653 / 32000000000000 : ℝ) ≤
+      (-(4946752824996576485697 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (975630204191363 / 1600000000000000 : ℝ) ≤
+      (-(4946752824996576485697 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (975630204191363 / 1600000000000000 : ℝ) =
+            (975630204191363 / 800000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (975630204191363 / 800000000000000 : ℝ) ≤
+            (1984718978003423514303 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (243883285852819 / 400000000000000 : ℝ)) (q :=
+      (975630204191363 / 1600000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_546_4 :
+    Uω (aρ 4) (bρ 4) (19948664777653 / 32000000000000 : ℝ) ≤
+      (-(1269470136695590692193 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4814602497190733 / 8000000000000000 : ℝ) ≤
+      (-(1269470136695590692193 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (4814602497190733 / 8000000000000000 : ℝ) =
+            (4814602497190733 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4814602497190733 / 4000000000000000 : ℝ) ≤
+            (463397814054409307807 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (601650501797027 / 1000000000000000 : ℝ)) (q :=
+      (4814602497190733 / 8000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_546_5 :
+    Uω (aρ 5) (bρ 5) (19948664777653 / 32000000000000 : ℝ) ≤
+      (-(1056182323825263913179 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23589183765250227 / 40000000000000000 : ℝ) ≤
+      (-(1056182323825263913179 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23589183765250227 / 40000000000000000 : ℝ) =
+            (23589183765250227 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23589183765250227 / 20000000000000000 : ℝ) ≤
+            (330112036774736086821 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5892780576953551 / 10000000000000000 : ℝ)) (q :=
+      (23589183765250227 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_546_6 :
+    Uω (aρ 6) (bρ 6) (19948664777653 / 32000000000000 : ℝ) ≤
+      (-(1394858296170015163569 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22895402710036903 / 40000000000000000 : ℝ) ≤
+      (-(1394858296170015163569 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (22895402710036903 / 40000000000000000 : ℝ) =
+            (22895402710036903 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22895402710036903 / 20000000000000000 : ℝ) ≤
+            (338009654579984836431 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5713130424486889 / 10000000000000000 : ℝ)) (q :=
+      (22895402710036903 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_546_7 :
+    Uω (aρ 7) (bρ 7) (19948664777653 / 32000000000000 : ℝ) ≤
+      (-(6001528506852776076673 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21949110245097181 / 40000000000000000 : ℝ) ≤
+      (-(6001528506852776076673 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (21949110245097181 / 40000000000000000 : ℝ) =
+            (21949110245097181 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21949110245097181 / 20000000000000000 : ℝ) ≤
+            (929943296147223923327 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1365911393194257 / 2500000000000000 : ℝ)) (q :=
+      (21949110245097181 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_546_8 :
+    Uω (aρ 8) (bρ 8) (19948664777653 / 32000000000000 : ℝ) ≤
+      (-(1316077306315072704157 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (165717137239417 / 320000000000000 : ℝ) ≤
+      (-(1316077306315072704157 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (165717137239417 / 320000000000000 : ℝ) =
+            (165717137239417 / 160000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (165717137239417 / 160000000000000 : ℝ) ≤
+            (70217054284927295843 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (641244147379 / 1250000000000 : ℝ)) (q :=
+      (165717137239417 / 320000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_546_9 :
+    Uω (aρ 9) (bρ 9) (19948664777653 / 32000000000000 : ℝ) ≤
+      (-(1839147393547709124913 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19167583565192881 / 40000000000000000 : ℝ) ≤
+      (-(1839147393547709124913 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (19167583565192881 / 40000000000000000 : ℝ) =
+            (19167583565192881 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19167583565192881 / 10000000000000000 : ℝ) ≤
+            (1626588507952290875087 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2348694756262439 / 5000000000000000 : ℝ)) (q :=
+      (19167583565192881 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_546_10 :
+    Uω (aρ 10) (bρ 10) (19948664777653 / 32000000000000 : ℝ) ≤
+      (-(8384414277493951212261 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17295355994526979 / 40000000000000000 : ℝ) ≤
+      (-(8384414277493951212261 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (17295355994526979 / 40000000000000000 : ℝ) =
+            (17295355994526979 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17295355994526979 / 10000000000000000 : ℝ) ≤
+            (5478529328506048787739 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4149834760686927 / 10000000000000000 : ℝ)) (q :=
+      (17295355994526979 / 40000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_546_11 :
+    Uω (aρ 11) (bρ 11) (19948664777653 / 32000000000000 : ℝ) ≤
+      (-(9749780267626559126277 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15088025663808893 / 40000000000000000 : ℝ) ≤
+      (-(9749780267626559126277 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (15088025663808893 / 40000000000000000 : ℝ) =
+            (15088025663808893 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15088025663808893 / 10000000000000000 : ℝ) ≤
+            (4113163338373440873723 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (866161555383221 / 2500000000000000 : ℝ)) (q :=
+      (15088025663808893 / 40000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_546_12 :
+    Uω (aρ 12) (bρ 12) (19948664777653 / 32000000000000 : ℝ) ≤
+      (-(2326874644250277370813 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12496419102519193 / 40000000000000000 : ℝ) ≤
+      (-(2326874644250277370813 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (12496419102519193 / 40000000000000000 : ℝ) =
+            (12496419102519193 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12496419102519193 / 10000000000000000 : ℝ) ≤
+            (445714076949722629187 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1296271518786517 / 5000000000000000 : ℝ)) (q :=
+      (12496419102519193 / 40000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_546_13 :
+    Uω (aρ 13) (bρ 13) (19948664777653 / 32000000000000 : ℝ) ≤
+      (-(14748808740681928824749 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9152239278767629 / 40000000000000000 : ℝ) ≤
+      (-(14748808740681928824749 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (9152239278767629 / 40000000000000000 : ℝ) =
+            (9152239278767629 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9152239278767629 / 5000000000000000 : ℝ) ≤
+            (6045606668318071175251 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (329958716823063 / 2500000000000000 : ℝ)) (q :=
+      (9152239278767629 / 40000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_546_14 :
+    Uω (aρ 14) (bρ 14) (19948664777653 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_546_15 :
+    Uω (aρ 15) (bρ 15) (19948664777653 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_546_16 :
+    Uω (aρ 16) (bρ 16) (19948664777653 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_546 :
+    Uρ (19948664777653 / 32000000000000 : ℝ) ≤
+      (-(7841298324713687178439 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (19948664777653 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(4829779413085618008589 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_546_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (19948664777653 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(2434309570975091985393 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_546_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (19948664777653 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(4946752824996576485697 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_546_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (19948664777653 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1269470136695590692193 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_546_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (19948664777653 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1056182323825263913179 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_546_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (19948664777653 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1394858296170015163569 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_546_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (19948664777653 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(6001528506852776076673 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_546_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (19948664777653 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(1316077306315072704157 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_546_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (19948664777653 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(1839147393547709124913 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_546_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (19948664777653 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(8384414277493951212261 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_546_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (19948664777653 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(9749780267626559126277 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_546_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (19948664777653 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(2326874644250277370813 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_546_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (19948664777653 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(14748808740681928824749 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_546_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (19948664777653 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_546_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (19948664777653 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_546_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (19948664777653 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_546_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_547_1 :
+    Uω (aρ 1) (bρ 1) (39966122160243 / 64000000000000 : ℝ) ≤
+      (-(4812371760118998631747 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (49441465683013763 / 80000000000000000 : ℝ) ≤
+      (-(4812371760118998631747 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (49441465683013763 / 80000000000000000 : ℝ) =
+            (49441465683013763 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (49441465683013763 / 40000000000000000 : ℝ) ≤
+            (2119100042881001368253 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (772519631412559 / 1250000000000000 : ℝ)) (q :=
+      (49441465683013763 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_547_2 :
+    Uω (aρ 2) (bρ 2) (39966122160243 / 64000000000000 : ℝ) ≤
+      (-(1212785847555891833407 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9850028838611011 / 16000000000000000 : ℝ) ≤
+      (-(1212785847555891833407 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (9850028838611011 / 16000000000000000 : ℝ) =
+            (9850028838611011 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9850028838611011 / 8000000000000000 : ℝ) ≤
+            (520082103194108166593 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1231219138264159 / 2000000000000000 : ℝ)) (q :=
+      (9850028838611011 / 16000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_547_3 :
+    Uω (aρ 3) (bρ 3) (39966122160243 / 64000000000000 : ℝ) ≤
+      (-(2464569430179590840317 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (48867509506341239 / 80000000000000000 : ℝ) ≤
+      (-(2464569430179590840317 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (48867509506341239 / 80000000000000000 : ℝ) =
+            (48867509506341239 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (48867509506341239 / 40000000000000000 : ℝ) ≤
+            (1001166471320409159683 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6107833125992341 / 10000000000000000 : ℝ)) (q :=
+      (48867509506341239 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_547_4 :
+    Uω (aρ 4) (bρ 4) (39966122160243 / 64000000000000 : ℝ) ≤
+      (-(316251931279762199151 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9646408133647383 / 16000000000000000 : ℝ) ≤
+      (-(316251931279762199151 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (9646408133647383 / 16000000000000000 : ℝ) =
+            (9646408133647383 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9646408133647383 / 8000000000000000 : ℝ) ≤
+            (116965056407737800849 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (301363004876563 / 500000000000000 : ℝ)) (q :=
+      (9646408133647383 / 16000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_547_5 :
+    Uω (aρ 5) (bρ 5) (39966122160243 / 64000000000000 : ℝ) ≤
+      (-(5262687560051708840297 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9452884811467307 / 16000000000000000 : ℝ) ≤
+      (-(5262687560051708840297 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (9452884811467307 / 16000000000000000 : ℝ) =
+            (9452884811467307 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9452884811467307 / 8000000000000000 : ℝ) ≤
+            (1668784242948291159703 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1180709172828233 / 2000000000000000 : ℝ)) (q :=
+      (9452884811467307 / 16000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_547_6 :
+    Uω (aρ 6) (bρ 6) (39966122160243 / 64000000000000 : ℝ) ≤
+      (-(5560636645839827234383 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9175391445551787 / 16000000000000000 : ℝ) ≤
+      (-(5560636645839827234383 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (9175391445551787 / 16000000000000000 : ℝ) =
+            (9175391445551787 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9175391445551787 / 8000000000000000 : ℝ) ≤
+            (1370835157160172765617 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1144783906377353 / 2000000000000000 : ℝ)) (q :=
+      (9175391445551787 / 16000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_547_7 :
+    Uω (aρ 7) (bρ 7) (39966122160243 / 64000000000000 : ℝ) ≤
+      (-(2990937303811851794713 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (43984582450042547 / 80000000000000000 : ℝ) ≤
+      (-(2990937303811851794713 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (43984582450042547 / 80000000000000000 : ℝ) =
+            (43984582450042547 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (43984582450042547 / 40000000000000000 : ℝ) ≤
+            (474798597688148205287 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1368621804554417 / 2500000000000000 : ℝ)) (q :=
+      (43984582450042547 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_547_8 :
+    Uω (aρ 8) (bρ 8) (39966122160243 / 64000000000000 : ℝ) ≤
+      (-(3279727881572910084163 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8303217959810399 / 16000000000000000 : ℝ) ≤
+      (-(3279727881572910084163 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (8303217959810399 / 16000000000000000 : ℝ) =
+            (8303217959810399 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8303217959810399 / 8000000000000000 : ℝ) ≤
+            (186008019927089915837 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (514090570681003 / 1000000000000000 : ℝ)) (q :=
+      (8303217959810399 / 16000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_547_9 :
+    Uω (aρ 9) (bρ 9) (39966122160243 / 64000000000000 : ℝ) ≤
+      (-(7333734172046849876191 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (38422883898404551 / 80000000000000000 : ℝ) ≤
+      (-(7333734172046849876191 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (38422883898404551 / 80000000000000000 : ℝ) =
+            (38422883898404551 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (38422883898404551 / 20000000000000000 : ℝ) ≤
+            (6529209433953150123809 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4708569860008169 / 10000000000000000 : ℝ)) (q :=
+      (38422883898404551 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_547_10 :
+    Uω (aρ 10) (bρ 10) (39966122160243 / 64000000000000 : ℝ) ≤
+      (-(417927435075955961471 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (34680298668558191 / 80000000000000000 : ℝ) ≤
+      (-(417927435075955961471 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (34680298668558191 / 80000000000000000 : ℝ) =
+            (34680298668558191 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (34680298668558191 / 20000000000000000 : ℝ) ≤
+            (275219745224044038529 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4161482586041579 / 10000000000000000 : ℝ)) (q :=
+      (34680298668558191 / 80000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_547_11 :
+    Uω (aρ 11) (bρ 11) (39966122160243 / 64000000000000 : ℝ) ≤
+      (-(4859406205668740621801 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (30269644934577319 / 80000000000000000 : ℝ) ≤
+      (-(4859406205668740621801 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (30269644934577319 / 80000000000000000 : ℝ) =
+            (30269644934577319 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (30269644934577319 / 20000000000000000 : ℝ) ≤
+            (2072065597331259378199 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3477295778751361 / 10000000000000000 : ℝ)) (q :=
+      (30269644934577319 / 80000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_547_12 :
+    Uω (aρ 12) (bρ 12) (39966122160243 / 64000000000000 : ℝ) ≤
+      (-(11593033208050026335313 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25096372489155823 / 80000000000000000 : ℝ) ≤
+      (-(11593033208050026335313 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (25096372489155823 / 80000000000000000 : ℝ) =
+            (25096372489155823 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25096372489155823 / 20000000000000000 : ℝ) ≤
+            (2269910397949973664687 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2607677764080987 / 10000000000000000 : ℝ)) (q :=
+      (25096372489155823 / 80000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_547_13 :
+    Uω (aρ 13) (bρ 13) (39966122160243 / 64000000000000 : ℝ) ≤
+      (-(183352136630615516529 / 125000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (18452678587610763 / 80000000000000000 : ℝ) ≤
+      (-(183352136630615516529 / 125000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (18452678587610763 / 80000000000000000 : ℝ) =
+            (18452678587610763 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18452678587610763 / 10000000000000000 : ℝ) ≤
+            (76578055981884483471 / 125000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (673068015144861 / 5000000000000000 : ℝ)) (q :=
+      (18452678587610763 / 80000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_547_14 :
+    Uω (aρ 14) (bρ 14) (39966122160243 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_547_15 :
+    Uω (aρ 15) (bρ 15) (39966122160243 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_547_16 :
+    Uω (aρ 16) (bρ 16) (39966122160243 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_547 :
+    Uρ (39966122160243 / 64000000000000 : ℝ) ≤
+      (-(1954555256450453128983 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (39966122160243 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(4812371760118998631747 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_547_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (39966122160243 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1212785847555891833407 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_547_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (39966122160243 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(2464569430179590840317 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_547_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (39966122160243 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(316251931279762199151 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_547_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (39966122160243 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(5262687560051708840297 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_547_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (39966122160243 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(5560636645839827234383 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_547_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (39966122160243 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(2990937303811851794713 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_547_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (39966122160243 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(3279727881572910084163 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_547_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (39966122160243 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(7333734172046849876191 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_547_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (39966122160243 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) * (-(417927435075955961471 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_547_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (39966122160243 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4859406205668740621801 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_547_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (39966122160243 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(11593033208050026335313 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_547_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (39966122160243 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) * (-(183352136630615516529 / 125000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_547_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (39966122160243 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_547_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (39966122160243 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_547_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (39966122160243 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_547_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_548_1 :
+    Uω (aρ 1) (bρ 1) (2001745738259 / 3200000000000 : ℝ) ≤
+      (-(2397497178697846703457 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6190932100316333 / 10000000000000000 : ℝ) ≤
+      (-(2397497178697846703457 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (6190932100316333 / 10000000000000000 : ℝ) =
+            (6190932100316333 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6190932100316333 / 5000000000000000 : ℝ) ≤
+            (1068238722802153296543 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6190905986658291 / 10000000000000000 : ℝ)) (q :=
+      (6190932100316333 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_548_2 :
+    Uω (aρ 2) (bρ 2) (2001745738259 / 3200000000000 : ℝ) ≤
+      (-(1208424531780068070263 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2466806867612131 / 4000000000000000 : ℝ) ≤
+      (-(1208424531780068070263 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2466806867612131 / 4000000000000000 : ℝ) =
+            (2466806867612131 / 2000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2466806867612131 / 2000000000000000 : ℝ) ≤
+            (524443418969931929737 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (154171128414907 / 250000000000000 : ℝ)) (q :=
+      (2466806867612131 / 4000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_548_3 :
+    Uω (aρ 3) (bρ 3) (2001745738259 / 3200000000000 : ℝ) ≤
+      (-(613944484060455122537 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12238377193275971 / 20000000000000000 : ℝ) ≤
+      (-(613944484060455122537 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (12238377193275971 / 20000000000000000 : ℝ) =
+            (12238377193275971 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12238377193275971 / 10000000000000000 : ℝ) ≤
+            (252489491314544877463 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1529646024540399 / 2500000000000000 : ℝ)) (q :=
+      (12238377193275971 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_548_4 :
+    Uω (aρ 4) (bρ 4) (2001745738259 / 3200000000000 : ℝ) ≤
+      (-(252110653838961592423 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12079514068936013 / 20000000000000000 : ℝ) ≤
+      (-(252110653838961592423 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (12079514068936013 / 20000000000000000 : ℝ) =
+            (12079514068936013 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12079514068936013 / 10000000000000000 : ℝ) ≤
+            (94462936311038407577 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3019007577443319 / 5000000000000000 : ℝ)) (q :=
+      (12079514068936013 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_548_5 :
+    Uω (aρ 5) (bρ 5) (2001745738259 / 3200000000000 : ℝ) ≤
+      (-(5244496702731643532673 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5918810043115331 / 10000000000000000 : ℝ) ≤
+      (-(5244496702731643532673 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5918810043115331 / 10000000000000000 : ℝ) =
+            (5918810043115331 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5918810043115331 / 5000000000000000 : ℝ) ≤
+            (1686975100268356467327 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5914311091516287 / 10000000000000000 : ℝ)) (q :=
+      (5918810043115331 / 10000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_548_6 :
+    Uω (aρ 6) (bρ 6) (2001745738259 / 3200000000000 : ℝ) ≤
+      (-(55418755033094918291 / 100000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11490777107644973 / 20000000000000000 : ℝ) ≤
+      (-(55418755033094918291 / 100000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11490777107644973 / 20000000000000000 : ℝ) =
+            (11490777107644973 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11490777107644973 / 10000000000000000 : ℝ) ≤
+            (13895962996905081709 / 100000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2867354244035299 / 5000000000000000 : ℝ)) (q :=
+      (11490777107644973 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_548_7 :
+    Uω (aρ 7) (bρ 7) (2001745738259 / 3200000000000 : ℝ) ≤
+      (-(953961534678529139 / 1600000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11017735737204267 / 20000000000000000 : ℝ) ≤
+      (-(953961534678529139 / 1600000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11017735737204267 / 20000000000000000 : ℝ) =
+            (11017735737204267 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11017735737204267 / 10000000000000000 : ℝ) ≤
+            (155073953801470861 / 1600000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1371332124597473 / 2500000000000000 : ℝ)) (q :=
+      (11017735737204267 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_548_8 :
+    Uω (aρ 8) (bρ 8) (2001745738259 / 3200000000000 : ℝ) ≤
+      (-(1634642384464888181893 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (260018074101987 / 500000000000000 : ℝ) ≤
+      (-(1634642384464888181893 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (260018074101987 / 500000000000000 : ℝ) =
+            (260018074101987 / 250000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (260018074101987 / 250000000000000 : ℝ) ≤
+            (98225566285111818107 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1030371475321021 / 2000000000000000 : ℝ)) (q :=
+      (260018074101987 / 500000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_548_9 :
+    Uω (aρ 9) (bρ 9) (2001745738259 / 3200000000000 : ℝ) ≤
+      (-(3655466485081770504419 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2406912040432491 / 5000000000000000 : ℝ) ≤
+      (-(3655466485081770504419 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2406912040432491 / 5000000000000000 : ℝ) =
+            (2406912040432491 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2406912040432491 / 1250000000000000 : ℝ) ≤
+            (3276005317918229495581 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4719748202615589 / 10000000000000000 : ℝ)) (q :=
+      (2406912040432491 / 5000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_548_10 :
+    Uω (aρ 10) (bρ 10) (2001745738259 / 3200000000000 : ℝ) ≤
+      (-(4166377703121249495059 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8692466512294929 / 20000000000000000 : ℝ) ≤
+      (-(4166377703121249495059 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8692466512294929 / 20000000000000000 : ℝ) =
+            (8692466512294929 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8692466512294929 / 5000000000000000 : ℝ) ≤
+            (2765094099878750504941 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2086562793337777 / 5000000000000000 : ℝ)) (q :=
+      (8692466512294929 / 20000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_548_11 :
+    Uω (aρ 11) (bρ 11) (2001745738259 / 3200000000000 : ℝ) ≤
+      (-(121099461847467560769 / 125000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7590796891998231 / 20000000000000000 : ℝ) ≤
+      (-(121099461847467560769 / 125000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7590796891998231 / 20000000000000000 : ℝ) =
+            (7590796891998231 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7590796891998231 / 5000000000000000 : ℝ) ≤
+            (52187333227532439231 / 125000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (218120787036491 / 625000000000000 : ℝ)) (q :=
+      (7590796891998231 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_548_12 :
+    Uω (aρ 12) (bρ 12) (2001745738259 / 3200000000000 : ℝ) ≤
+      (-(5775966047425651859671 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1259986682061253 / 4000000000000000 : ℝ) ≤
+      (-(5775966047425651859671 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1259986682061253 / 4000000000000000 : ℝ) =
+            (1259986682061253 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1259986682061253 / 1000000000000000 : ℝ) ≤
+            (1155505755574348140329 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (262276920757689 / 1000000000000000 : ℝ)) (q :=
+      (1259986682061253 / 4000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_548_13 :
+    Uω (aρ 13) (bρ 13) (2001745738259 / 3200000000000 : ℝ) ≤
+      (-(3647270300163015730903 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2324899871636139 / 10000000000000000 : ℝ) ≤
+      (-(3647270300163015730903 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2324899871636139 / 10000000000000000 : ℝ) =
+            (2324899871636139 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2324899871636139 / 1250000000000000 : ℝ) ≤
+            (1551333552086984269097 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1372017282137903 / 10000000000000000 : ℝ)) (q :=
+      (2324899871636139 / 10000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_548_14 :
+    Uω (aρ 14) (bρ 14) (2001745738259 / 3200000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_548_15 :
+    Uω (aρ 15) (bρ 15) (2001745738259 / 3200000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_548_16 :
+    Uω (aρ 16) (bρ 16) (2001745738259 / 3200000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_548 :
+    Uρ (2001745738259 / 3200000000000 : ℝ) ≤
+      (-(7795288173310978959933 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (2001745738259 / 3200000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2397497178697846703457 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_548_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (2001745738259 / 3200000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1208424531780068070263 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_548_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (2001745738259 / 3200000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(613944484060455122537 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_548_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (2001745738259 / 3200000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(252110653838961592423 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_548_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (2001745738259 / 3200000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(5244496702731643532673 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_548_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (2001745738259 / 3200000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) * (-(55418755033094918291 / 100000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_548_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (2001745738259 / 3200000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) * (-(953961534678529139 / 1600000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_548_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (2001745738259 / 3200000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(1634642384464888181893 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_548_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (2001745738259 / 3200000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(3655466485081770504419 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_548_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (2001745738259 / 3200000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(4166377703121249495059 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_548_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (2001745738259 / 3200000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) * (-(121099461847467560769 / 125000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_548_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (2001745738259 / 3200000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(5775966047425651859671 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_548_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (2001745738259 / 3200000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(3647270300163015730903 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_548_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (2001745738259 / 3200000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_548_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (2001745738259 / 3200000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_548_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (2001745738259 / 3200000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_548_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_549_1 :
+    Uω (aρ 1) (bρ 1) (40103707370117 / 64000000000000 : ℝ) ≤
+      (-(1194411774990531570599 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (49613447920788041 / 80000000000000000 : ℝ) ≤
+      (-(1194411774990531570599 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (49613447920788041 / 80000000000000000 : ℝ) =
+            (49613447920788041 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (49613447920788041 / 40000000000000000 : ℝ) ≤
+            (538456175759468429401 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6201654921701229 / 10000000000000000 : ℝ)) (q :=
+      (49613447920788041 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_549_2 :
+    Uω (aρ 2) (bρ 2) (40103707370117 / 64000000000000 : ℝ) ≤
+      (-(4816283246437239450033 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9884426100613517 / 16000000000000000 : ℝ) ≤
+      (-(4816283246437239450033 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (9884426100613517 / 16000000000000000 : ℝ) =
+            (9884426100613517 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9884426100613517 / 8000000000000000 : ℝ) ≤
+            (2115188556562760549967 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1235518915956223 / 2000000000000000 : ℝ)) (q :=
+      (9884426100613517 / 16000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_549_3 :
+    Uω (aρ 3) (bρ 3) (40103707370117 / 64000000000000 : ℝ) ≤
+      (-(195760150103412896917 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (49039508010013997 / 80000000000000000 : ℝ) ≤
+      (-(195760150103412896917 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (49039508010013997 / 80000000000000000 : ℝ) =
+            (49039508010013997 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (49039508010013997 / 40000000000000000 : ℝ) ≤
+            (81498722016587103083 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3064667531433859 / 5000000000000000 : ℝ)) (q :=
+      (49039508010013997 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_549_4 :
+    Uω (aρ 4) (bρ 4) (40103707370117 / 64000000000000 : ℝ) ≤
+      (-(1256106740590143561431 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (48404071794902537 / 80000000000000000 : ℝ) ≤
+      (-(1256106740590143561431 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (48404071794902537 / 80000000000000000 : ℝ) =
+            (48404071794902537 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (48404071794902537 / 40000000000000000 : ℝ) ≤
+            (476761210159856438569 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6048770190154853 / 10000000000000000 : ℝ)) (q :=
+      (48404071794902537 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_549_5 :
+    Uω (aρ 5) (bρ 5) (40103707370117 / 64000000000000 : ℝ) ≤
+      (-(653292365778121990287 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (47436536394562869 / 80000000000000000 : ℝ) ≤
+      (-(653292365778121990287 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (47436536394562869 / 80000000000000000 : ℝ) =
+            (47436536394562869 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (47436536394562869 / 40000000000000000 : ℝ) ≤
+            (213141609596878009713 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (740634532425617 / 1250000000000000 : ℝ)) (q :=
+      (47436536394562869 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_549_6 :
+    Uω (aρ 6) (bρ 6) (40103707370117 / 64000000000000 : ℝ) ≤
+      (-(690393702942002721149 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (46049259031944133 / 80000000000000000 : ℝ) ≤
+      (-(690393702942002721149 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (46049259031944133 / 80000000000000000 : ℝ) =
+            (46049259031944133 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (46049259031944133 / 40000000000000000 : ℝ) ≤
+            (176040272432997278851 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1436374323472563 / 2500000000000000 : ℝ)) (q :=
+      (46049259031944133 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_549_7 :
+    Uω (aρ 7) (bρ 7) (40103707370117 / 64000000000000 : ℝ) ≤
+      (-(185708853261108282267 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (44157301995163901 / 80000000000000000 : ℝ) ≤
+      (-(185708853261108282267 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (44157301995163901 / 80000000000000000 : ℝ) =
+            (44157301995163901 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (44157301995163901 / 40000000000000000 : ℝ) ≤
+            (30899640582641717733 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2748084707727597 / 5000000000000000 : ℝ)) (q :=
+      (44157301995163901 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_549_8 :
+    Uω (aρ 8) (bρ 8) (40103707370117 / 64000000000000 : ℝ) ≤
+      (-(6517727663076715180429 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8337938100698421 / 16000000000000000 : ℝ) ≤
+      (-(6517727663076715180429 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (8337938100698421 / 16000000000000000 : ℝ) =
+            (8337938100698421 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8337938100698421 / 8000000000000000 : ℝ) ≤
+            (413744139923284819571 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1032561638775449 / 2000000000000000 : ℝ)) (q :=
+      (8337938100698421 / 16000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_549_9 :
+    Uω (aρ 9) (bρ 9) (40103707370117 / 64000000000000 : ℝ) ≤
+      (-(7288185702466368188041 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (38598293432777697 / 80000000000000000 : ℝ) ≤
+      (-(7288185702466368188041 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (38598293432777697 / 80000000000000000 : ℝ) =
+            (38598293432777697 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (38598293432777697 / 20000000000000000 : ℝ) ≤
+            (6574757903533631811959 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4730924554558643 / 10000000000000000 : ℝ)) (q :=
+      (38598293432777697 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_549_10 :
+    Uω (aρ 10) (bρ 10) (40103707370117 / 64000000000000 : ℝ) ≤
+      (-(8307033959244178314683 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (34859414292000669 / 80000000000000000 : ℝ) ≤
+      (-(8307033959244178314683 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (34859414292000669 / 80000000000000000 : ℝ) =
+            (34859414292000669 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (34859414292000669 / 20000000000000000 : ℝ) ≤
+            (5555909646755821685317 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2092381901429693 / 5000000000000000 : ℝ)) (q :=
+      (34859414292000669 / 80000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_549_11 :
+    Uω (aρ 11) (bρ 11) (40103707370117 / 64000000000000 : ℝ) ≤
+      (-(603575809542648455083 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (30456679779584297 / 80000000000000000 : ℝ) ≤
+      (-(603575809542648455083 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (30456679779584297 / 80000000000000000 : ℝ) =
+            (30456679779584297 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (30456679779584297 / 20000000000000000 : ℝ) ≤
+            (262858165832351544917 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3502556800960293 / 10000000000000000 : ℝ)) (q :=
+      (30456679779584297 / 80000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_549_12 :
+    Uω (aρ 12) (bρ 12) (40103707370117 / 64000000000000 : ℝ) ≤
+      (-(5755533234833897135403 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25302924632807569 / 80000000000000000 : ℝ) ≤
+      (-(5755533234833897135403 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (25302924632807569 / 80000000000000000 : ℝ) =
+            (25302924632807569 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25302924632807569 / 20000000000000000 : ℝ) ≤
+            (1175938568166102864597 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2637818110951111 / 10000000000000000 : ℝ)) (q :=
+      (25302924632807569 / 80000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_549_13 :
+    Uω (aρ 13) (bρ 13) (40103707370117 / 64000000000000 : ℝ) ≤
+      (-(14511458352928684789089 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (18744133034056773 / 80000000000000000 : ℝ) ≤
+      (-(14511458352928684789089 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (18744133034056773 / 80000000000000000 : ℝ) =
+            (18744133034056773 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18744133034056773 / 10000000000000000 : ℝ) ≤
+            (6282957056071315210911 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (349375488214603 / 2500000000000000 : ℝ)) (q :=
+      (18744133034056773 / 80000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_549_14 :
+    Uω (aρ 14) (bρ 14) (40103707370117 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_549_15 :
+    Uω (aρ 15) (bρ 15) (40103707370117 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_549_16 :
+    Uω (aρ 16) (bρ 16) (40103707370117 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_549 :
+    Uρ (40103707370117 / 64000000000000 : ℝ) ≤
+      (-(7772494551631630816607 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (40103707370117 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1194411774990531570599 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_549_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (40103707370117 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(4816283246437239450033 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_549_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (40103707370117 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(195760150103412896917 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_549_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (40103707370117 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1256106740590143561431 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_549_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (40103707370117 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(653292365778121990287 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_549_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (40103707370117 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(690393702942002721149 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_549_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (40103707370117 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(185708853261108282267 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_549_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (40103707370117 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(6517727663076715180429 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_549_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (40103707370117 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(7288185702466368188041 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_549_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (40103707370117 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(8307033959244178314683 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_549_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (40103707370117 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) * (-(603575809542648455083 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_549_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (40103707370117 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(5755533234833897135403 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_549_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (40103707370117 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(14511458352928684789089 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_549_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (40103707370117 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_549_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (40103707370117 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_549_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (40103707370117 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_549_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_550_1 :
+    Uω (aρ 1) (bρ 1) (20086249987527 / 32000000000000 : ℝ) ≤
+      (-(4760329883409985412941 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24849719518896217 / 40000000000000000 : ℝ) ≤
+      (-(4760329883409985412941 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (24849719518896217 / 40000000000000000 : ℝ) =
+            (24849719518896217 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24849719518896217 / 20000000000000000 : ℝ) ≤
+            (2171141919590014587059 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6212403856430921 / 10000000000000000 : ℝ)) (q :=
+      (24849719518896217 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_550_2 :
+    Uω (aρ 2) (bρ 2) (20086249987527 / 32000000000000 : ℝ) ≤
+      (-(959779728505078014553 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4950812364557353 / 8000000000000000 : ℝ) ≤
+      (-(959779728505078014553 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (4950812364557353 / 8000000000000000 : ℝ) =
+            (4950812364557353 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4950812364557353 / 4000000000000000 : ℝ) ≤
+            (426514632094921985447 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1237668804177239 / 2000000000000000 : ℝ)) (q :=
+      (4950812364557353 / 8000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_550_3 :
+    Uω (aρ 3) (bρ 3) (20086249987527 / 32000000000000 : ℝ) ≤
+      (-(2438241196225873571523 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24562753608614197 / 40000000000000000 : ℝ) ≤
+      (-(2438241196225873571523 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (24562753608614197 / 40000000000000000 : ℝ) =
+            (24562753608614197 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24562753608614197 / 20000000000000000 : ℝ) ≤
+            (1027494705274126428477 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6140086020149911 / 10000000000000000 : ℝ)) (q :=
+      (24562753608614197 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_550_4 :
+    Uω (aρ 4) (bρ 4) (20086249987527 / 32000000000000 : ℝ) ≤
+      (-(19557314236304715771 / 39062500000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24245043613091401 / 40000000000000000 : ℝ) ≤
+      (-(19557314236304715771 / 39062500000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (24245043613091401 / 40000000000000000 : ℝ) =
+            (24245043613091401 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24245043613091401 / 20000000000000000 : ℝ) ≤
+            (7518747494164034229 / 39062500000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6059525203453513 / 10000000000000000 : ℝ)) (q :=
+      (24245043613091401 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_550_5 :
+    Uω (aρ 5) (bρ 5) (20086249987527 / 32000000000000 : ℝ) ≤
+      (-(520821411025116205967 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23761296103775901 / 40000000000000000 : ℝ) ≤
+      (-(520821411025116205967 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23761296103775901 / 40000000000000000 : ℝ) =
+            (23761296103775901 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23761296103775901 / 20000000000000000 : ℝ) ≤
+            (172325769274883794033 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5935841368130763 / 10000000000000000 : ℝ)) (q :=
+      (23761296103775901 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_550_6 :
+    Uω (aρ 6) (bρ 6) (20086249987527 / 32000000000000 : ℝ) ≤
+      (-(5504458873723566257013 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4613540903523353 / 8000000000000000 : ℝ) ≤
+      (-(5504458873723566257013 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (4613540903523353 / 8000000000000000 : ℝ) =
+            (4613540903523353 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4613540903523353 / 4000000000000000 : ℝ) ≤
+            (1427012929276433742987 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1151257190038239 / 2000000000000000 : ℝ)) (q :=
+      (4613540903523353 / 8000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_550_7 :
+    Uω (aρ 7) (bρ 7) (20086249987527 / 32000000000000 : ℝ) ≤
+      (-(1184629118309960228537 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22121829798830323 / 40000000000000000 : ℝ) ≤
+      (-(1184629118309960228537 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (22121829798830323 / 40000000000000000 : ℝ) =
+            (22121829798830323 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22121829798830323 / 20000000000000000 : ℝ) ≤
+            (201665242290039771463 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2753504985778987 / 5000000000000000 : ℝ)) (q :=
+      (22121829798830323 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_550_8 :
+    Uω (aρ 8) (bρ 8) (20086249987527 / 32000000000000 : ℝ) ≤
+      (-(6496929947425930102249 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20888242881114017 / 40000000000000000 : ℝ) ≤
+      (-(6496929947425930102249 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (20888242881114017 / 40000000000000000 : ℝ) =
+            (20888242881114017 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20888242881114017 / 20000000000000000 : ℝ) ≤
+            (434541855574069897751 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5173758164039821 / 10000000000000000 : ℝ)) (q :=
+      (20888242881114017 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_550_9 :
+    Uω (aρ 9) (bρ 9) (20086249987527 / 32000000000000 : ℝ) ≤
+      (-(3632746052450233074369 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3868598631227813 / 8000000000000000 : ℝ) ≤
+      (-(3632746052450233074369 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3868598631227813 / 8000000000000000 : ℝ) =
+            (3868598631227813 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3868598631227813 / 2000000000000000 : ℝ) ≤
+            (3298725750549766925631 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (948419785982469 / 2000000000000000 : ℝ)) (q :=
+      (3868598631227813 / 8000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_550_10 :
+    Uω (aρ 10) (bρ 10) (20086249987527 / 32000000000000 : ℝ) ≤
+      (-(8281383932199983076381 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17474471778125199 / 40000000000000000 : ℝ) ≤
+      (-(8281383932199983076381 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (17474471778125199 / 40000000000000000 : ℝ) =
+            (17474471778125199 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17474471778125199 / 10000000000000000 : ℝ) ≤
+            (5581559673800016923619 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1049099318600103 / 2500000000000000 : ℝ)) (q :=
+      (17474471778125199 / 40000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_550_11 :
+    Uω (aρ 11) (bρ 11) (20086249987527 / 32000000000000 : ℝ) ≤
+      (-(9626579514052903566141 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15275061056299521 / 40000000000000000 : ℝ) ≤
+      (-(9626579514052903566141 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (15275061056299521 / 40000000000000000 : ℝ) =
+            (15275061056299521 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15275061056299521 / 10000000000000000 : ℝ) ≤
+            (4236364091947096433859 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3515168539692573 / 10000000000000000 : ℝ)) (q :=
+      (15275061056299521 / 40000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_550_12 :
+    Uω (aρ 12) (bρ 12) (20086249987527 / 32000000000000 : ℝ) ≤
+      (-(11470433001551724661083 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12702974179877313 / 40000000000000000 : ℝ) ≤
+      (-(11470433001551724661083 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (12702974179877313 / 40000000000000000 : ℝ) =
+            (12702974179877313 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12702974179877313 / 10000000000000000 : ℝ) ≤
+            (2392510604448275338917 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2652825198166469 / 10000000000000000 : ℝ)) (q :=
+      (12702974179877313 / 40000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_550_13 :
+    Uω (aρ 13) (bρ 13) (20086249987527 / 32000000000000 : ℝ) ≤
+      (-(7217614053428216597823 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9443783011720333 / 40000000000000000 : ℝ) ≤
+      (-(7217614053428216597823 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (9443783011720333 / 40000000000000000 : ℝ) =
+            (9443783011720333 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9443783011720333 / 5000000000000000 : ℝ) ≤
+            (3179593651071783402177 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1422611355682979 / 10000000000000000 : ℝ)) (q :=
+      (9443783011720333 / 40000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_550_14 :
+    Uω (aρ 14) (bρ 14) (20086249987527 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_550_15 :
+    Uω (aρ 15) (bρ 15) (20086249987527 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_550_16 :
+    Uω (aρ 16) (bρ 16) (20086249987527 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_550 :
+    Uρ (20086249987527 / 32000000000000 : ℝ) ≤
+      (-(7749835359783882693713 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (20086249987527 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(4760329883409985412941 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_550_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (20086249987527 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(959779728505078014553 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_550_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (20086249987527 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(2438241196225873571523 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_550_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (20086249987527 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(19557314236304715771 / 39062500000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_550_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (20086249987527 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(520821411025116205967 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_550_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (20086249987527 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(5504458873723566257013 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_550_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (20086249987527 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(1184629118309960228537 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_550_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (20086249987527 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(6496929947425930102249 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_550_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (20086249987527 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(3632746052450233074369 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_550_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (20086249987527 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(8281383932199983076381 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_550_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (20086249987527 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(9626579514052903566141 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_550_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (20086249987527 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(11470433001551724661083 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_550_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (20086249987527 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(7217614053428216597823 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_550_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (20086249987527 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_550_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (20086249987527 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_550_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (20086249987527 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_550_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_551_1 :
+    Uω (aρ 1) (bρ 1) (40241292579991 / 64000000000000 : ℝ) ≤
+      (-(4743042603872454263259 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9957086030710067 / 16000000000000000 : ℝ) ≤
+      (-(4743042603872454263259 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (9957086030710067 / 16000000000000000 : ℝ) =
+            (9957086030710067 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9957086030710067 / 8000000000000000 : ℝ) ≤
+            (2188429199127545736741 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (622315279084899 / 1000000000000000 : ℝ)) (q :=
+      (9957086030710067 / 16000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_551_2 :
+    Uω (aρ 2) (bρ 2) (40241292579991 / 64000000000000 : ℝ) ≤
+      (-(2390772105142890830111 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (49594116779803731 / 80000000000000000 : ℝ) ≤
+      (-(2390772105142890830111 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (49594116779803731 / 80000000000000000 : ℝ) =
+            (49594116779803731 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (49594116779803731 / 40000000000000000 : ℝ) ≤
+            (1074963796357109169889 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6199093459922339 / 10000000000000000 : ℝ)) (q :=
+      (49594116779803731 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_551_3 :
+    Uω (aρ 3) (bρ 3) (40241292579991 / 64000000000000 : ℝ) ≤
+      (-(2429495842219293840209 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (49211506394902783 / 80000000000000000 : ℝ) ≤
+      (-(2429495842219293840209 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (49211506394902783 / 80000000000000000 : ℝ) =
+            (49211506394902783 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (49211506394902783 / 40000000000000000 : ℝ) ≤
+            (1036240059280706159791 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3075418485023551 / 5000000000000000 : ℝ)) (q :=
+      (49211506394902783 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_551_4 :
+    Uω (aρ 4) (bρ 4) (40241292579991 / 64000000000000 : ℝ) ≤
+      (-(2494474705526851839749 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (48576102570051939 / 80000000000000000 : ℝ) ≤
+      (-(2494474705526851839749 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (48576102570051939 / 80000000000000000 : ℝ) =
+            (48576102570051939 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (48576102570051939 / 40000000000000000 : ℝ) ≤
+            (971261195973148160251 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6070280194899391 / 10000000000000000 : ℝ)) (q :=
+      (48576102570051939 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_551_5 :
+    Uω (aρ 5) (bρ 5) (40241292579991 / 64000000000000 : ℝ) ≤
+      (-(259506106759272505429 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1904345911406987 / 3200000000000000 : ℝ) ≤
+      (-(259506106759272505429 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1904345911406987 / 3200000000000000 : ℝ) =
+            (1904345911406987 / 1600000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1904345911406987 / 1600000000000000 : ℝ) ≤
+            (87067483390727494571 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (237864256720603 / 400000000000000 : ℝ)) (q :=
+      (1904345911406987 / 3200000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_551_6 :
+    Uω (aρ 6) (bρ 6) (40241292579991 / 64000000000000 : ℝ) ≤
+      (-(5485803121827726466689 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9244311688760919 / 16000000000000000 : ℝ) ≤
+      (-(5485803121827726466689 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (9244311688760919 / 16000000000000000 : ℝ) =
+            (9244311688760919 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9244311688760919 / 8000000000000000 : ℝ) ≤
+            (1445668681172273533311 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1153414891562511 / 2000000000000000 : ℝ)) (q :=
+      (9244311688760919 / 16000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_551_7 :
+    Uω (aρ 7) (bρ 7) (40241292579991 / 64000000000000 : ℝ) ≤
+      (-(5903646300329178437817 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (44330015764817219 / 80000000000000000 : ℝ) ≤
+      (-(5903646300329178437817 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (44330015764817219 / 80000000000000000 : ℝ) =
+            (44330015764817219 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (44330015764817219 / 40000000000000000 : ℝ) ≤
+            (1027825502670821562183 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5517850168825711 / 10000000000000000 : ℝ)) (q :=
+      (44330015764817219 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_551_8 :
+    Uω (aρ 8) (bρ 8) (40241292579991 / 64000000000000 : ℝ) ≤
+      (-(6476176200793001902877 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8372655530798919 / 16000000000000000 : ℝ) ≤
+      (-(6476176200793001902877 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (8372655530798919 / 16000000000000000 : ℝ) =
+            (8372655530798919 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8372655530798919 / 8000000000000000 : ℝ) ≤
+            (455295602206998097123 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1036941458492011 / 2000000000000000 : ℝ)) (q :=
+      (8372655530798919 / 16000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_551_9 :
+    Uω (aρ 9) (bρ 9) (40241292579991 / 64000000000000 : ℝ) ≤
+      (-(905356489426210870313 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (38773671341181867 / 80000000000000000 : ℝ) ≤
+      (-(905356489426210870313 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (38773671341181867 / 80000000000000000 : ℝ) =
+            (38773671341181867 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (38773671341181867 / 20000000000000000 : ℝ) ≤
+            (827511461323789129687 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4753271342616873 / 10000000000000000 : ℝ)) (q :=
+      (38773671341181867 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_551_10 :
+    Uω (aρ 10) (bρ 10) (40241292579991 / 64000000000000 : ℝ) ≤
+      (-(412790245041383228599 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (280307631994663 / 640000000000000 : ℝ) ≤
+      (-(412790245041383228599 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (280307631994663 / 640000000000000 : ℝ) =
+            (280307631994663 / 160000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (280307631994663 / 160000000000000 : ℝ) ≤
+            (280356935258616771401 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (33664208325197 / 80000000000000 : ℝ)) (q :=
+      (280307631994663 / 640000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10]) (by norm_num [bρ_10])
+      (by norm_num) (by norm_num [aρ_10, bρ_10]) (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_551_11 :
+    Uω (aρ 11) (bρ 11) (40241292579991 / 64000000000000 : ℝ) ≤
+      (-(2399013933030365643021 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (30643515101983563 / 80000000000000000 : ℝ) ≤
+      (-(2399013933030365643021 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (30643515101983563 / 80000000000000000 : ℝ) =
+            (30643515101983563 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (30643515101983563 / 20000000000000000 : ℝ) ≤
+            (1066721968469634356979 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3527767942517297 / 10000000000000000 : ℝ)) (q :=
+      (30643515101983563 / 80000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_551_12 :
+    Uω (aρ 12) (bρ 12) (40241292579991 / 64000000000000 : ℝ) ≤
+      (-(11430028437903927354641 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25508807644800619 / 80000000000000000 : ℝ) ≤
+      (-(11430028437903927354641 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (25508807644800619 / 80000000000000000 : ℝ) =
+            (25508807644800619 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25508807644800619 / 20000000000000000 : ℝ) ≤
+            (2432915168096072645359 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2667791174906561 / 10000000000000000 : ℝ)) (q :=
+      (25508807644800619 / 80000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_551_13 :
+    Uω (aρ 13) (bρ 13) (40241292579991 / 64000000000000 : ℝ) ≤
+      (-(1436032230020782576113 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19029576065137959 / 80000000000000000 : ℝ) ≤
+      (-(1436032230020782576113 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (19029576065137959 / 80000000000000000 : ℝ) =
+            (19029576065137959 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19029576065137959 / 10000000000000000 : ℝ) ≤
+            (643409310879217423887 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (180920627698237 / 1250000000000000 : ℝ)) (q :=
+      (19029576065137959 / 80000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_551_14 :
+    Uω (aρ 14) (bρ 14) (40241292579991 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_551_15 :
+    Uω (aρ 15) (bρ 15) (40241292579991 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_551_16 :
+    Uω (aρ 16) (bρ 16) (40241292579991 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_551 :
+    Uρ (40241292579991 / 64000000000000 : ℝ) ≤
+      (-(7727306164144297624663 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (40241292579991 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(4743042603872454263259 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_551_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (40241292579991 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(2390772105142890830111 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_551_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (40241292579991 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(2429495842219293840209 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_551_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (40241292579991 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(2494474705526851839749 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_551_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (40241292579991 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(259506106759272505429 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_551_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (40241292579991 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(5485803121827726466689 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_551_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (40241292579991 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(5903646300329178437817 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_551_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (40241292579991 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(6476176200793001902877 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_551_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (40241292579991 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(905356489426210870313 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_551_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (40241292579991 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) * (-(412790245041383228599 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_551_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (40241292579991 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(2399013933030365643021 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_551_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (40241292579991 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(11430028437903927354641 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_551_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (40241292579991 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(1436032230020782576113 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_551_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (40241292579991 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_551_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (40241292579991 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_551_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (40241292579991 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_551_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_552_1 :
+    Uω (aρ 1) (bρ 1) (1259690162029 / 2000000000000 : ℝ) ≤
+      (-(4725785158020472823779 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1558481914627131 / 2500000000000000 : ℝ) ≤
+      (-(4725785158020472823779 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1558481914627131 / 2500000000000000 : ℝ) =
+            (1558481914627131 / 1250000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1558481914627131 / 1250000000000000 : ℝ) ≤
+            (2205686644979527176221 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (779237715619631 / 1250000000000000 : ℝ)) (q :=
+      (1558481914627131 / 2500000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_552_2 :
+    Uω (aρ 2) (bρ 2) (1259690162029 / 2000000000000 : ℝ) ≤
+      (-(4764219845165794570447 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12420027476450291 / 20000000000000000 : ℝ) ≤
+      (-(4764219845165794570447 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (12420027476450291 / 20000000000000000 : ℝ) =
+            (12420027476450291 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12420027476450291 / 10000000000000000 : ℝ) ≤
+            (2167251957834205429553 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6209842896900291 / 10000000000000000 : ℝ)) (q :=
+      (12420027476450291 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_552_3 :
+    Uω (aρ 3) (bρ 3) (1259690162029 / 2000000000000 : ℝ) ≤
+      (-(2420765760732845746699 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (246487527715959 / 400000000000000 : ℝ) ≤
+      (-(2420765760732845746699 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (246487527715959 / 400000000000000 : ℝ) =
+            (246487527715959 / 200000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (246487527715959 / 200000000000000 : ℝ) ≤
+            (1044970140767154253301 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (123231758251959 / 200000000000000 : ℝ)) (q :=
+      (246487527715959 / 400000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_552_4 :
+    Uω (aρ 4) (bρ 4) (1259690162029 / 2000000000000 : ℝ) ≤
+      (-(1242814437627491156317 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6082764728371717 / 10000000000000000 : ℝ) ≤
+      (-(1242814437627491156317 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (6082764728371717 / 10000000000000000 : ℝ) =
+            (6082764728371717 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6082764728371717 / 5000000000000000 : ℝ) ≤
+            (490053513122508843683 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3040517582304217 / 5000000000000000 : ℝ)) (q :=
+      (6082764728371717 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_552_5 :
+    Uω (aρ 5) (bρ 5) (1259690162029 / 2000000000000 : ℝ) ≤
+      (-(5172062882054117301997 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11923675782176853 / 20000000000000000 : ℝ) ≤
+      (-(5172062882054117301997 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11923675782176853 / 20000000000000000 : ℝ) =
+            (11923675782176853 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11923675782176853 / 10000000000000000 : ℝ) ≤
+            (1759408920945882698003 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5957371409376853 / 10000000000000000 : ℝ)) (q :=
+      (11923675782176853 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_552_6 :
+    Uω (aρ 6) (bρ 6) (1259690162029 / 2000000000000 : ℝ) ≤
+      (-(136679555913746854773 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5788463407623591 / 10000000000000000 : ℝ) ≤
+      (-(136679555913746854773 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5788463407623591 / 10000000000000000 : ℝ) =
+            (5788463407623591 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5788463407623591 / 5000000000000000 : ℝ) ≤
+            (36607239161253145227 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2888931408793591 / 5000000000000000 : ℝ)) (q :=
+      (5788463407623591 / 10000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_552_7 :
+    Uω (aρ 7) (bρ 7) (1259690162029 / 2000000000000 : ℝ) ≤
+      (-(5884185278614776937653 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (88832741010153 / 160000000000000 : ℝ) ≤
+      (-(5884185278614776937653 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (88832741010153 / 160000000000000 : ℝ) = (88832741010153 / 80000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (88832741010153 / 80000000000000 : ℝ) ≤
+            (1047286524385223062347 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (44229520074953 / 80000000000000 : ℝ)) (q :=
+      (88832741010153 / 160000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_552_8 :
+    Uω (aρ 8) (bρ 8) (1259690162029 / 2000000000000 : ℝ) ≤
+      (-(5164372987447805311 / 8000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5243758275009757 / 10000000000000000 : ℝ) ≤
+      (-(5164372987447805311 / 8000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5243758275009757 / 10000000000000000 : ℝ) =
+            (5243758275009757 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5243758275009757 / 5000000000000000 : ℝ) ≤
+            (380804454952194689 / 8000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2597827792229757 / 5000000000000000 : ℝ)) (q :=
+      (5243758275009757 / 10000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_552_9 :
+    Uω (aρ 9) (bρ 9) (1259690162029 / 2000000000000 : ℝ) ≤
+      (-(3610132436957952754391 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2428834285919803 / 5000000000000000 : ℝ) ≤
+      (-(3610132436957952754391 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2428834285919803 / 5000000000000000 : ℝ) =
+            (2428834285919803 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2428834285919803 / 1250000000000000 : ℝ) ≤
+            (3321339366042047245609 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1191110451619803 / 2500000000000000 : ℝ)) (q :=
+      (2428834285919803 / 5000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_552_10 :
+    Uω (aρ 10) (bρ 10) (1259690162029 / 2000000000000 : ℝ) ≤
+      (-(8230296444833993408677 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2195496611053351 / 5000000000000000 : ℝ) ≤
+      (-(8230296444833993408677 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2195496611053351 / 5000000000000000 : ℝ) =
+            (2195496611053351 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2195496611053351 / 1250000000000000 : ℝ) ≤
+            (5632647161166006591323 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1054912535127101 / 2500000000000000 : ℝ)) (q :=
+      (2195496611053351 / 5000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_552_11 :
+    Uω (aρ 11) (bρ 11) (1259690162029 / 2000000000000 : ℝ) ≤
+      (-(2391410179756399555909 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1536842963727483 / 4000000000000000 : ℝ) ≤
+      (-(2391410179756399555909 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1536842963727483 / 4000000000000000 : ℝ) =
+            (1536842963727483 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1536842963727483 / 1000000000000000 : ℝ) ≤
+            (1074325721743600444091 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (708071028227483 / 2000000000000000 : ℝ)) (q :=
+      (1536842963727483 / 4000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_552_12 :
+    Uω (aρ 12) (bρ 12) (1259690162029 / 2000000000000 : ℝ) ≤
+      (-(11389849601897008077703 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (640287631001113 / 2000000000000000 : ℝ) ≤
+      (-(11389849601897008077703 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (640287631001113 / 2000000000000000 : ℝ) =
+            (640287631001113 / 500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (640287631001113 / 500000000000000 : ℝ) ≤
+            (2473094004102991922297 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (268271672919613 / 1000000000000000 : ℝ)) (q :=
+      (640287631001113 / 2000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12]) (by norm_num [bρ_12])
+      (by norm_num) (by norm_num [aρ_12, bρ_12]) (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_552_13 :
+    Uω (aρ 13) (bρ 13) (1259690162029 / 2000000000000 : ℝ) ≤
+      (-(1428667820436467843137 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4792558739157083 / 20000000000000000 : ℝ) ≤
+      (-(1428667820436467843137 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4792558739157083 / 20000000000000000 : ℝ) =
+            (4792558739157083 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4792558739157083 / 2500000000000000 : ℝ) ≤
+            (650773720463532156863 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1471780899937083 / 10000000000000000 : ℝ)) (q :=
+      (4792558739157083 / 20000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_552_14 :
+    Uω (aρ 14) (bρ 14) (1259690162029 / 2000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_552_15 :
+    Uω (aρ 15) (bρ 15) (1259690162029 / 2000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_552_16 :
+    Uω (aρ 16) (bρ 16) (1259690162029 / 2000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_552 :
+    Uρ (1259690162029 / 2000000000000 : ℝ) ≤
+      (-(61639222863656401809 / 80000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1259690162029 / 2000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(4725785158020472823779 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_552_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1259690162029 / 2000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(4764219845165794570447 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_552_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1259690162029 / 2000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(2420765760732845746699 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_552_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1259690162029 / 2000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1242814437627491156317 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_552_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1259690162029 / 2000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(5172062882054117301997 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_552_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1259690162029 / 2000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(136679555913746854773 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_552_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1259690162029 / 2000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(5884185278614776937653 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_552_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1259690162029 / 2000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) * (-(5164372987447805311 / 8000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_552_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1259690162029 / 2000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(3610132436957952754391 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_552_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1259690162029 / 2000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(8230296444833993408677 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_552_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1259690162029 / 2000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(2391410179756399555909 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_552_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1259690162029 / 2000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(11389849601897008077703 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_552_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1259690162029 / 2000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(1428667820436467843137 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_552_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1259690162029 / 2000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_552_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1259690162029 / 2000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_552_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1259690162029 / 2000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_552_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_553_1 :
+    Uω (aρ 1) (bρ 1) (8075775557973 / 12800000000000 : ℝ) ≤
+      (-(294284840191189489643 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (49957412381352409 / 80000000000000000 : ℝ) ≤
+      (-(294284840191189489643 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (49957412381352409 / 80000000000000000 : ℝ) =
+            (49957412381352409 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (49957412381352409 / 40000000000000000 : ℝ) ≤
+            (138932147496310510357 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (780581332344587 / 1250000000000000 : ℝ)) (q :=
+      (49957412381352409 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_553_2 :
+    Uω (aρ 2) (bρ 2) (8075775557973 / 12800000000000 : ℝ) ≤
+      (-(593365680394418844097 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (49766103023608513 / 80000000000000000 : ℝ) ≤
+      (-(593365680394418844097 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (49766103023608513 / 80000000000000000 : ℝ) =
+            (49766103023608513 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (49766103023608513 / 40000000000000000 : ℝ) ≤
+            (273068294980581155903 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3110296165915361 / 5000000000000000 : ℝ)) (q :=
+      (49766103023608513 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_553_3 :
+    Uω (aρ 3) (bρ 3) (8075775557973 / 12800000000000 : ℝ) ≤
+      (-(4824101797013151299569 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9876700932449797 / 16000000000000000 : ℝ) ≤
+      (-(4824101797013151299569 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (9876700932449797 / 16000000000000000 : ℝ) =
+            (9876700932449797 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9876700932449797 / 8000000000000000 : ℝ) ≤
+            (2107370005986848700431 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (154308471196021 / 250000000000000 : ℝ)) (q :=
+      (9876700932449797 / 16000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_553_4 :
+    Uω (aρ 4) (bρ 4) (8075775557973 / 12800000000000 : ℝ) ≤
+      (-(990719470384999601177 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (48748132997408709 / 80000000000000000 : ℝ) ≤
+      (-(990719470384999601177 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (48748132997408709 / 80000000000000000 : ℝ) =
+            (48748132997408709 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (48748132997408709 / 40000000000000000 : ℝ) ≤
+            (395574890215000398823 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6091790112695771 / 10000000000000000 : ℝ)) (q :=
+      (48748132997408709 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_553_5 :
+    Uω (aρ 5) (bρ 5) (8075775557973 / 12800000000000 : ℝ) ≤
+      (-(16106363226655447673 / 31250000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (47780758239416737 / 80000000000000000 : ℝ) ≤
+      (-(16106363226655447673 / 31250000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (47780758239416737 / 80000000000000000 : ℝ) =
+            (47780758239416737 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (47780758239416737 / 40000000000000000 : ℝ) ≤
+            (5554486157719552327 / 31250000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2984068171266389 / 5000000000000000 : ℝ)) (q :=
+      (47780758239416737 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_553_6 :
+    Uω (aρ 6) (bρ 6) (8075775557973 / 12800000000000 : ℝ) ≤
+      (-(2724298043665763560073 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (46393855490092449 / 80000000000000000 : ℝ) ≤
+      (-(2724298043665763560073 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (46393855490092449 / 80000000000000000 : ℝ) =
+            (46393855490092449 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (46393855490092449 / 40000000000000000 : ℝ) ≤
+            (741437857834236439927 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2894325515170853 / 5000000000000000 : ℝ)) (q :=
+      (46393855490092449 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_553_7 :
+    Uω (aρ 7) (bρ 7) (8075775557973 / 12800000000000 : ℝ) ≤
+      (-(183273824226132342093 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1780108953072601 / 3200000000000000 : ℝ) ≤
+      (-(183273824226132342093 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1780108953072601 / 3200000000000000 : ℝ) =
+            (1780108953072601 / 1600000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1780108953072601 / 1600000000000000 : ℝ) ≤
+            (33334669617617657907 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (110790589905647 / 200000000000000 : ℝ)) (q :=
+      (1780108953072601 / 3200000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_553_8 :
+    Uω (aρ 8) (bρ 8) (8075775557973 / 12800000000000 : ℝ) ≤
+      (-(3217399930171501478579 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (42036851421584033 / 80000000000000000 : ℝ) ≤
+      (-(3217399930171501478579 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (42036851421584033 / 80000000000000000 : ℝ) =
+            (42036851421584033 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (42036851421584033 / 40000000000000000 : ℝ) ≤
+            (248335971328498521421 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2603301522657301 / 5000000000000000 : ℝ)) (q :=
+      (42036851421584033 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_553_9 :
+    Uω (aρ 9) (bρ 9) (8075775557973 / 12800000000000 : ℝ) ≤
+      (-(143954614445972021909 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7789803613516941 / 16000000000000000 : ℝ) ≤
+      (-(143954614445972021909 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7789803613516941 / 16000000000000000 : ℝ) =
+            (7789803613516941 / 4000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7789803613516941 / 4000000000000000 : ℝ) ≤
+            (133304257674027978091 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (477561033517477 / 1000000000000000 : ℝ)) (q :=
+      (7789803613516941 / 16000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_553_10 :
+    Uω (aρ 10) (bρ 10) (8075775557973 / 12800000000000 : ℝ) ≤
+      (-(4102429073931367588247 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (35217419042646141 / 80000000000000000 : ℝ) ≤
+      (-(4102429073931367588247 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (35217419042646141 / 80000000000000000 : ℝ) =
+            (35217419042646141 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (35217419042646141 / 20000000000000000 : ℝ) ≤
+            (2829042729068632411753 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4231269612435129 / 10000000000000000 : ℝ)) (q :=
+      (35217419042646141 / 80000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_553_11 :
+    Uω (aρ 11) (bρ 11) (8075775557973 / 12800000000000 : ℝ) ≤
+      (-(9535333598606493714421 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (30830155149136949 / 80000000000000000 : ℝ) ≤
+      (-(9535333598606493714421 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (30830155149136949 / 80000000000000000 : ℝ) =
+            (30830155149136949 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (30830155149136949 / 20000000000000000 : ℝ) ≤
+            (4327610007393506285579 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3552930265262831 / 10000000000000000 : ℝ)) (q :=
+      (30830155149136949 / 80000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_553_12 :
+    Uω (aρ 12) (bρ 12) (8075775557973 / 12800000000000 : ℝ) ≤
+      (-(11349893390006880405851 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25714043829325249 / 80000000000000000 : ℝ) ≤
+      (-(11349893390006880405851 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (25714043829325249 / 80000000000000000 : ℝ) =
+            (25714043829325249 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25714043829325249 / 20000000000000000 : ℝ) ≤
+            (2513050215993119594149 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1348801265997453 / 5000000000000000 : ℝ)) (q :=
+      (25714043829325249 / 80000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_553_13 :
+    Uω (aρ 13) (bρ 13) (8075775557973 / 12800000000000 : ℝ) ≤
+      (-(7107118967621958687121 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19309608859234949 / 80000000000000000 : ℝ) ≤
+      (-(7107118967621958687121 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (19309608859234949 / 80000000000000000 : ℝ) =
+            (19309608859234949 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19309608859234949 / 10000000000000000 : ℝ) ≤
+            (3290088736878041312879 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1495875531067331 / 10000000000000000 : ℝ)) (q :=
+      (19309608859234949 / 80000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_553_14 :
+    Uω (aρ 14) (bρ 14) (8075775557973 / 12800000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_553_15 :
+    Uω (aρ 15) (bρ 15) (8075775557973 / 12800000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_553_16 :
+    Uω (aρ 16) (bρ 16) (8075775557973 / 12800000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_553 :
+    Uρ (8075775557973 / 12800000000000 : ℝ) ≤
+      (-(3841310813236541712291 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (8075775557973 / 12800000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(294284840191189489643 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_553_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (8075775557973 / 12800000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(593365680394418844097 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_553_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (8075775557973 / 12800000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(4824101797013151299569 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_553_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (8075775557973 / 12800000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(990719470384999601177 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_553_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (8075775557973 / 12800000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(16106363226655447673 / 31250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_553_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (8075775557973 / 12800000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(2724298043665763560073 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_553_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (8075775557973 / 12800000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(183273824226132342093 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_553_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (8075775557973 / 12800000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(3217399930171501478579 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_553_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (8075775557973 / 12800000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(143954614445972021909 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_553_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (8075775557973 / 12800000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(4102429073931367588247 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_553_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (8075775557973 / 12800000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(9535333598606493714421 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_553_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (8075775557973 / 12800000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(11349893390006880405851 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_553_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (8075775557973 / 12800000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(7107118967621958687121 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_553_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (8075775557973 / 12800000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_553_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (8075775557973 / 12800000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_553_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (8075775557973 / 12800000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_553_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_554_1 :
+    Uω (aρ 1) (bρ 1) (20223835197401 / 32000000000000 : ℝ) ≤
+      (-(293209959795218729589 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25021701746704671 / 40000000000000000 : ℝ) ≤
+      (-(293209959795218729589 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (25021701746704671 / 40000000000000000 : ℝ) =
+            (25021701746704671 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25021701746704671 / 20000000000000000 : ℝ) ≤
+            (140007027892281270411 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6255399592249523 / 10000000000000000 : ℝ)) (q :=
+      (25021701746704671 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_554_2 :
+    Uω (aρ 2) (bρ 2) (20223835197401 / 32000000000000 : ℝ) ≤
+      (-(4729660900783168347879 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24926048066634081 / 40000000000000000 : ℝ) ≤
+      (-(4729660900783168347879 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (24926048066634081 / 40000000000000000 : ℝ) =
+            (24926048066634081 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24926048066634081 / 20000000000000000 : ℝ) ≤
+            (2201810902216831652121 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1557835441181057 / 2500000000000000 : ℝ)) (q :=
+      (24926048066634081 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_554_3 :
+    Uω (aρ 3) (bρ 3) (20223835197401 / 32000000000000 : ℝ) ≤
+      (-(2403351202558697929501 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24734751876113413 / 40000000000000000 : ℝ) ≤
+      (-(2403351202558697929501 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (24734751876113413 / 40000000000000000 : ℝ) =
+            (24734751876113413 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24734751876113413 / 20000000000000000 : ℝ) ≤
+            (1062384698941302070499 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3091544887906947 / 5000000000000000 : ℝ)) (q :=
+      (24734751876113413 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_554_4 :
+    Uω (aρ 4) (bρ 4) (20223835197401 / 32000000000000 : ℝ) ≤
+      (-(123399202623718425537 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24417074040907061 / 40000000000000000 : ℝ) ≤
+      (-(123399202623718425537 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (24417074040907061 / 40000000000000000 : ℝ) =
+            (24417074040907061 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24417074040907061 / 20000000000000000 : ℝ) ≤
+            (49887592451281574463 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3051272519637859 / 5000000000000000 : ℝ)) (q :=
+      (24417074040907061 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_554_5 :
+    Uω (aρ 5) (bρ 5) (20223835197401 / 32000000000000 : ℝ) ≤
+      (-(1284010517231633783179 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23933406559280117 / 40000000000000000 : ℝ) ≤
+      (-(1284010517231633783179 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23933406559280117 / 40000000000000000 : ℝ) =
+            (23933406559280117 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23933406559280117 / 20000000000000000 : ℝ) ≤
+            (448857433518366216821 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2989450608898623 / 5000000000000000 : ℝ)) (q :=
+      (23933406559280117 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_554_6 :
+    Uω (aρ 6) (bρ 6) (20223835197401 / 32000000000000 : ℝ) ≤
+      (-(1357511136087193952523 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23240001567198817 / 40000000000000000 : ℝ) ≤
+      (-(1357511136087193952523 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23240001567198817 / 40000000000000000 : ℝ) =
+            (23240001567198817 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23240001567198817 / 20000000000000000 : ℝ) ≤
+            (375356814662806047477 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1449859774224149 / 2500000000000000 : ℝ)) (q :=
+      (23240001567198817 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_554_7 :
+    Uω (aρ 7) (bρ 7) (20223835197401 / 32000000000000 : ℝ) ≤
+      (-(5845377439924378908381 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22294537869171803 / 40000000000000000 : ℝ) ≤
+      (-(5845377439924378908381 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (22294537869171803 / 40000000000000000 : ℝ) =
+            (22294537869171803 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22294537869171803 / 20000000000000000 : ℝ) ≤
+            (1086094363075621091619 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5550368628643089 / 10000000000000000 : ℝ)) (q :=
+      (22294537869171803 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_554_8 :
+    Uω (aρ 8) (bρ 8) (20223835197401 / 32000000000000 : ℝ) ≤
+      (-(3207088446241811533109 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21061816669719703 / 40000000000000000 : ℝ) ≤
+      (-(3207088446241811533109 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (21061816669719703 / 40000000000000000 : ℝ) =
+            (21061816669719703 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21061816669719703 / 20000000000000000 : ℝ) ≤
+            (258647455258188466891 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5217549680257039 / 10000000000000000 : ℝ)) (q :=
+      (21061816669719703 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_554_9 :
+    Uω (aρ 9) (bρ 9) (20223835197401 / 32000000000000 : ℝ) ≤
+      (-(1793812301093678316561 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3903667987396711 / 8000000000000000 : ℝ) ≤
+      (-(1793812301093678316561 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3903667987396711 / 8000000000000000 : ℝ) =
+            (3903667987396711 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3903667987396711 / 2000000000000000 : ℝ) ≤
+            (1671923600406321683439 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (957355388449793 / 2000000000000000 : ℝ)) (q :=
+      (3903667987396711 / 8000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_554_10 :
+    Uω (aρ 10) (bρ 10) (20223835197401 / 32000000000000 : ℝ) ≤
+      (-(4089744798721756566173 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17653436974398951 / 40000000000000000 : ℝ) ≤
+      (-(4089744798721756566173 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (17653436974398951 / 40000000000000000 : ℝ) =
+            (17653436974398951 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17653436974398951 / 10000000000000000 : ℝ) ≤
+            (2841727004278243433827 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4242884494451663 / 10000000000000000 : ℝ)) (q :=
+      (17653436974398951 / 40000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_554_11 :
+    Uω (aρ 11) (bρ 11) (20223835197401 / 32000000000000 : ℝ) ≤
+      (-(190102670123856259617 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15461701618385637 / 40000000000000000 : ℝ) ≤
+      (-(190102670123856259617 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (15461701618385637 / 40000000000000000 : ℝ) =
+            (15461701618385637 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15461701618385637 / 10000000000000000 : ℝ) ≤
+            (87156201996143740383 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1782746721325003 / 5000000000000000 : ℝ)) (q :=
+      (15461701618385637 / 40000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_554_12 :
+    Uω (aρ 12) (bρ 12) (20223835197401 / 32000000000000 : ℝ) ≤
+      (-(113101567696471236689 / 100000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12908213015248219 / 40000000000000000 : ℝ) ≤
+      (-(113101567696471236689 / 100000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (12908213015248219 / 40000000000000000 : ℝ) =
+            (12908213015248219 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12908213015248219 / 10000000000000000 : ℝ) ≤
+            (25527868363528763311 / 100000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2712449237766297 / 10000000000000000 : ℝ)) (q :=
+      (12908213015248219 / 40000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_554_13 :
+    Uω (aρ 13) (bρ 13) (20223835197401 / 32000000000000 : ℝ) ≤
+      (-(7071473971965871018263 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1944775889466269 / 8000000000000000 : ℝ) ≤
+      (-(7071473971965871018263 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1944775889466269 / 8000000000000000 : ℝ) =
+            (1944775889466269 / 1000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1944775889466269 / 1000000000000000 : ℝ) ≤
+            (3325733732534128981737 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (75983209770143 / 500000000000000 : ℝ)) (q :=
+      (1944775889466269 / 8000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_554_14 :
+    Uω (aρ 14) (bρ 14) (20223835197401 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_554_15 :
+    Uω (aρ 15) (bρ 15) (20223835197401 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_554_16 :
+    Uω (aρ 16) (bρ 16) (20223835197401 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_554 :
+    Uρ (20223835197401 / 32000000000000 : ℝ) ≤
+      (-(7660458916787004784987 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (20223835197401 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(293209959795218729589 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_554_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (20223835197401 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(4729660900783168347879 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_554_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (20223835197401 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(2403351202558697929501 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_554_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (20223835197401 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(123399202623718425537 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_554_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (20223835197401 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1284010517231633783179 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_554_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (20223835197401 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1357511136087193952523 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_554_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (20223835197401 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(5845377439924378908381 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_554_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (20223835197401 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(3207088446241811533109 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_554_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (20223835197401 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(1793812301093678316561 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_554_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (20223835197401 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(4089744798721756566173 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_554_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (20223835197401 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) * (-(190102670123856259617 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_554_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (20223835197401 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(113101567696471236689 / 100000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_554_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (20223835197401 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(7071473971965871018263 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_554_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (20223835197401 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_554_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (20223835197401 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_554_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (20223835197401 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_554_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_555_1 :
+    Uω (aρ 1) (bρ 1) (40516462999739 / 64000000000000 : ℝ) ≤
+      (-(1168547699318994396631 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (50129394604245311 / 80000000000000000 : ℝ) ≤
+      (-(1168547699318994396631 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (50129394604245311 / 80000000000000000 : ℝ) =
+            (50129394604245311 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (50129394604245311 / 40000000000000000 : ℝ) ≤
+            (564320251431005603369 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6266148525437109 / 10000000000000000 : ℝ)) (q :=
+      (50129394604245311 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_555_2 :
+    Uω (aρ 2) (bρ 2) (40516462999739 / 64000000000000 : ℝ) ≤
+      (-(1178106528778261540091 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (49938089234822199 / 80000000000000000 : ℝ) ≤
+      (-(1178106528778261540091 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (49938089234822199 / 80000000000000000 : ℝ) =
+            (49938089234822199 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (49938089234822199 / 40000000000000000 : ℝ) ≤
+            (554761421971738459909 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6242091195591331 / 10000000000000000 : ℝ)) (q :=
+      (49938089234822199 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_555_3 :
+    Uω (aρ 3) (bρ 3) (40516462999739 / 64000000000000 : ℝ) ≤
+      (-(4789333240367326840447 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (49555502813276743 / 80000000000000000 : ℝ) ≤
+      (-(4789333240367326840447 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (49555502813276743 / 80000000000000000 : ℝ) =
+            (49555502813276743 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (49555502813276743 / 40000000000000000 : ℝ) ≤
+            (2142138562632673159553 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6193840696554967 / 10000000000000000 : ℝ)) (q :=
+      (49555502813276743 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_555_4 :
+    Uω (aρ 4) (bρ 4) (40516462999739 / 64000000000000 : ℝ) ≤
+      (-(2459184949907346047189 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (48920163080644023 / 80000000000000000 : ℝ) ≤
+      (-(2459184949907346047189 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (48920163080644023 / 80000000000000000 : ℝ) =
+            (48920163080644023 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (48920163080644023 / 40000000000000000 : ℝ) ≤
+            (1006550951592653952811 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6113299944461787 / 10000000000000000 : ℝ)) (q :=
+      (48920163080644023 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_555_5 :
+    Uω (aρ 5) (bρ 5) (40516462999739 / 64000000000000 : ℝ) ≤
+      (-(5118080274195688240163 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (47952867767386439 / 80000000000000000 : ℝ) ≤
+      (-(5118080274195688240163 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (47952867767386439 / 80000000000000000 : ℝ) =
+            (47952867767386439 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (47952867767386439 / 40000000000000000 : ℝ) ≤
+            (1813391528804311759837 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5989666035482391 / 10000000000000000 : ℝ)) (q :=
+      (47952867767386439 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_555_6 :
+    Uω (aρ 6) (bρ 6) (40516462999739 / 64000000000000 : ℝ) ≤
+      (-(1352881869626692158181 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (46566150197161723 / 80000000000000000 : ℝ) ≤
+      (-(1352881869626692158181 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (46566150197161723 / 80000000000000000 : ℝ) =
+            (46566150197161723 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (46566150197161723 / 40000000000000000 : ℝ) ≤
+            (379986081123307841819 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1452556754516553 / 2500000000000000 : ℝ)) (q :=
+      (46566150197161723 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_555_7 :
+    Uω (aρ 7) (bρ 7) (40516462999739 / 64000000000000 : ℝ) ≤
+      (-(1165206064660802796103 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (44675426247907999 / 80000000000000000 : ℝ) ≤
+      (-(1165206064660802796103 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (44675426247907999 / 80000000000000000 : ℝ) =
+            (44675426247907999 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (44675426247907999 / 40000000000000000 : ℝ) ≤
+            (221088295939197203897 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5561207411512781 / 10000000000000000 : ℝ)) (q :=
+      (44675426247907999 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_555_8 :
+    Uω (aρ 8) (bρ 8) (40516462999739 / 64000000000000 : ℝ) ≤
+      (-(6393597145535780670923 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (42210411974394231 / 80000000000000000 : ℝ) ≤
+      (-(6393597145535780670923 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (42210411974394231 / 80000000000000000 : ℝ) =
+            (42210411974394231 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (42210411974394231 / 40000000000000000 : ℝ) ≤
+            (537874657464219329077 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5228495494474339 / 10000000000000000 : ℝ)) (q :=
+      (42210411974394231 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_555_9 :
+    Uω (aρ 9) (bρ 9) (40516462999739 / 64000000000000 : ℝ) ≤
+      (-(7152820065878755550463 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7824866809506411 / 16000000000000000 : ℝ) ≤
+      (-(7152820065878755550463 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7824866809506411 / 16000000000000000 : ℝ) =
+            (7824866809506411 / 4000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7824866809506411 / 4000000000000000 : ℝ) ≤
+            (6710123540121244449537 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (959588328223759 / 2000000000000000 : ℝ)) (q :=
+      (7824866809506411 / 16000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_555_10 :
+    Uω (aρ 10) (bρ 10) (40516462999739 / 64000000000000 : ℝ) ≤
+      (-(1630838076988301541033 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (35396310645675803 / 80000000000000000 : ℝ) ≤
+      (-(1630838076988301541033 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (35396310645675803 / 80000000000000000 : ℝ) =
+            (35396310645675803 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (35396310645675803 / 20000000000000000 : ℝ) ≤
+            (1141750644211698458967 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1063623706037433 / 2500000000000000 : ℝ)) (q :=
+      (35396310645675803 / 80000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_555_11 :
+    Uω (aρ 11) (bρ 11) (40516462999739 / 64000000000000 : ℝ) ≤
+      (-(2368759897097424207653 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (31016604040819147 / 80000000000000000 : ℝ) ≤
+      (-(2368759897097424207653 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (31016604040819147 / 80000000000000000 : ℝ) =
+            (31016604040819147 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (31016604040819147 / 20000000000000000 : ℝ) ≤
+            (1096976004402575792347 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (447255599892571 / 1250000000000000 : ℝ)) (q :=
+      (31016604040819147 / 80000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_555_12 :
+    Uω (aρ 12) (bρ 12) (40516462999739 / 64000000000000 : ℝ) ≤
+      (-(5635318388450483959479 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25918654397605731 / 80000000000000000 : ℝ) ≤
+      (-(5635318388450483959479 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (25918654397605731 / 80000000000000000 : ℝ) =
+            (25918654397605731 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25918654397605731 / 20000000000000000 : ℝ) ≤
+            (1296153414549516040521 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1363628742511107 / 5000000000000000 : ℝ)) (q :=
+      (25918654397605731 / 80000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_555_13 :
+    Uω (aρ 13) (bρ 13) (40516462999739 / 64000000000000 : ℝ) ≤
+      (-(7036379287111782039099 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19584741662899847 / 80000000000000000 : ℝ) ≤
+      (-(7036379287111782039099 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (19584741662899847 / 80000000000000000 : ℝ) =
+            (19584741662899847 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19584741662899847 / 10000000000000000 : ℝ) ≤
+            (3360828417388217960901 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1543161042940743 / 10000000000000000 : ℝ)) (q :=
+      (19584741662899847 / 80000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_555_14 :
+    Uω (aρ 14) (bρ 14) (40516462999739 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_555_15 :
+    Uω (aρ 15) (bρ 15) (40516462999739 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_555_16 :
+    Uω (aρ 16) (bρ 16) (40516462999739 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_555 :
+    Uρ (40516462999739 / 64000000000000 : ℝ) ≤
+      (-(3819205705809023544411 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (40516462999739 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1168547699318994396631 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_555_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (40516462999739 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1178106528778261540091 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_555_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (40516462999739 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(4789333240367326840447 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_555_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (40516462999739 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(2459184949907346047189 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_555_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (40516462999739 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(5118080274195688240163 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_555_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (40516462999739 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1352881869626692158181 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_555_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (40516462999739 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(1165206064660802796103 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_555_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (40516462999739 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(6393597145535780670923 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_555_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (40516462999739 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(7152820065878755550463 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_555_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (40516462999739 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(1630838076988301541033 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_555_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (40516462999739 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(2368759897097424207653 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_555_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (40516462999739 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(5635318388450483959479 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_555_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (40516462999739 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(7036379287111782039099 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_555_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (40516462999739 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_555_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (40516462999739 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_555_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (40516462999739 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_555_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/U46.lean b/LeanPool/Zeta5Irrational/Table/U46.lean
new file mode 100644
index 0000000000..7f3683d39b
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/U46.lean
@@ -0,0 +1,4831 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-! # Certified arcsine potential bounds (U46) -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma U_556_1 :
+    Uω (aρ 1) (bρ 1) (10146313901169 / 16000000000000 : ℝ) ≤
+      (-(291065728968855683219 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12553846428466647 / 20000000000000000 : ℝ) ≤
+      (-(291065728968855683219 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (12553846428466647 / 20000000000000000 : ℝ) =
+            (12553846428466647 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12553846428466647 / 10000000000000000 : ℝ) ≤
+            (142151258718644316781 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3138448729160511 / 5000000000000000 : ℝ)) (q :=
+      (12553846428466647 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_556_2 :
+    Uω (aρ 2) (bρ 2) (10146313901169 / 16000000000000 : ℝ) ≤
+      (-(2347610491870088450017 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12506020582078109 / 20000000000000000 : ℝ) ≤
+      (-(2347610491870088450017 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (12506020582078109 / 20000000000000000 : ℝ) =
+            (12506020582078109 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12506020582078109 / 10000000000000000 : ℝ) ≤
+            (1118125409629911549983 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1563210156110621 / 2500000000000000 : ℝ)) (q :=
+      (12506020582078109 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_556_3 :
+    Uω (aρ 3) (bρ 3) (10146313901169 / 16000000000000 : ℝ) ≤
+      (-(4771994197900481931239 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12410375461387277 / 20000000000000000 : ℝ) ≤
+      (-(4771994197900481931239 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (12410375461387277 / 20000000000000000 : ℝ) =
+            (12410375461387277 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12410375461387277 / 10000000000000000 : ℝ) ≤
+            (2159477605099518068761 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1551147902525413 / 2500000000000000 : ℝ)) (q :=
+      (12410375461387277 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_556_4 :
+    Uω (aρ 4) (bρ 4) (10146313901169 / 16000000000000 : ℝ) ≤
+      (-(2450401313667920874891 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12251544498587317 / 20000000000000000 : ℝ) ≤
+      (-(2450401313667920874891 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (12251544498587317 / 20000000000000000 : ℝ) =
+            (12251544498587317 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12251544498587317 / 10000000000000000 : ℝ) ≤
+            (1015334587832079125109 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1531013707091673 / 2500000000000000 : ℝ)) (q :=
+      (12251544498587317 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_556_5 :
+    Uω (aρ 5) (bρ 5) (10146313901169 / 16000000000000 : ℝ) ≤
+      (-(1275037682980197292503 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3002432636695933 / 5000000000000000 : ℝ) ≤
+      (-(1275037682980197292503 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (3002432636695933 / 5000000000000000 : ℝ) =
+            (3002432636695933 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3002432636695933 / 2500000000000000 : ℝ) ≤
+            (457830267769802707497 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6000430795898107 / 10000000000000000 : ℝ)) (q :=
+      (3002432636695933 / 5000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_556_6 :
+    Uω (aρ 6) (bρ 6) (10146313901169 / 16000000000000 : ℝ) ≤
+      (-(5393044761434235189473 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11663074170404497 / 20000000000000000 : ℝ) ≤
+      (-(5393044761434235189473 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11663074170404497 / 20000000000000000 : ℝ) =
+            (11663074170404497 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11663074170404497 / 10000000000000000 : ℝ) ≤
+            (1538427041565764810527 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (727626849332359 / 1250000000000000 : ℝ)) (q :=
+      (11663074170404497 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_556_7 :
+    Uω (aρ 7) (bρ 7) (10146313901169 / 16000000000000 : ℝ) ≤
+      (-(362920054805424530923 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2238088768184477 / 4000000000000000 : ℝ) ≤
+      (-(362920054805424530923 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2238088768184477 / 4000000000000000 : ℝ) =
+            (2238088768184477 / 2000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2238088768184477 / 2000000000000000 : ℝ) ≤
+            (70296932882075469077 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (139301146148419 / 250000000000000 : ℝ)) (q :=
+      (2238088768184477 / 4000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_556_8 :
+    Uω (aρ 8) (bρ 8) (10146313901169 / 16000000000000 : ℝ) ≤
+      (-(6373060435506251966543 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (165223388074311 / 312500000000000 : ℝ) ≤
+      (-(6373060435506251966543 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (165223388074311 / 312500000000000 : ℝ) =
+            (165223388074311 / 156250000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (165223388074311 / 156250000000000 : ℝ) ≤
+            (558411367493748033457 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5239440493110279 / 10000000000000000 : ℝ)) (q :=
+      (165223388074311 / 312500000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_556_9 :
+    Uω (aρ 9) (bρ 9) (10146313901169 / 16000000000000 : ℝ) ≤
+      (-(1426088610888638727831 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4901497580180003 / 10000000000000000 : ℝ) ≤
+      (-(1426088610888638727831 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4901497580180003 / 10000000000000000 : ℝ) =
+            (4901497580180003 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4901497580180003 / 2500000000000000 : ℝ) ≤
+            (1346500110311361272169 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4809104445074381 / 10000000000000000 : ℝ)) (q :=
+      (4901497580180003 / 10000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_556_10 :
+    Uω (aρ 10) (bρ 10) (10146313901169 / 16000000000000 : ℝ) ≤
+      (-(508060006594250466711 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4435716160243901 / 10000000000000000 : ℝ) ≤
+      (-(508060006594250466711 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4435716160243901 / 10000000000000000 : ℝ) =
+            (4435716160243901 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4435716160243901 / 2500000000000000 : ℝ) ≤
+            (358373968780749533289 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4266100638697177 / 10000000000000000 : ℝ)) (q :=
+      (4435716160243901 / 10000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_556_11 :
+    Uω (aρ 11) (bρ 11) (10146313901169 / 16000000000000 : ℝ) ≤
+      (-(1180631375359092244143 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3888719757142301 / 10000000000000000 : ℝ) ≤
+      (-(1180631375359092244143 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3888719757142301 / 10000000000000000 : ℝ) =
+            (3888719757142301 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3888719757142301 / 2500000000000000 : ℝ) ≤
+            (552236575390907755857 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3590584458698977 / 10000000000000000 : ℝ)) (q :=
+      (3888719757142301 / 10000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_556_12 :
+    Uω (aρ 12) (bρ 12) (10146313901169 / 16000000000000 : ℝ) ≤
+      (-(11231330514346018154391 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3252591427872931 / 10000000000000000 : ℝ) ≤
+      (-(11231330514346018154391 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3252591427872931 / 10000000000000000 : ℝ) =
+            (3252591427872931 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3252591427872931 / 2500000000000000 : ℝ) ≤
+            (2631613091653981845609 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2742027896845237 / 10000000000000000 : ℝ)) (q :=
+      (3252591427872931 / 10000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_556_13 :
+    Uω (aρ 13) (bρ 13) (10146313901169 / 16000000000000 : ℝ) ≤
+      (-(14003623676628956738161 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4930152423445567 / 20000000000000000 : ℝ) ≤
+      (-(14003623676628956738161 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4930152423445567 / 20000000000000000 : ℝ) =
+            (4930152423445567 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4930152423445567 / 2500000000000000 : ℝ) ≤
+            (6790791732371043261839 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (783189603069971 / 5000000000000000 : ℝ)) (q :=
+      (4930152423445567 / 20000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_556_14 :
+    Uω (aρ 14) (bρ 14) (10146313901169 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_556_15 :
+    Uω (aρ 15) (bρ 15) (10146313901169 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_556_16 :
+    Uω (aρ 16) (bρ 16) (10146313901169 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_556 :
+    Uρ (10146313901169 / 16000000000000 : ℝ) ≤
+      (-(7616476006420413902031 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (10146313901169 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(291065728968855683219 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_556_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (10146313901169 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(2347610491870088450017 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_556_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (10146313901169 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(4771994197900481931239 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_556_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (10146313901169 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(2450401313667920874891 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_556_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (10146313901169 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1275037682980197292503 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_556_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (10146313901169 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(5393044761434235189473 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_556_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (10146313901169 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(362920054805424530923 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_556_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (10146313901169 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(6373060435506251966543 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_556_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (10146313901169 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(1426088610888638727831 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_556_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (10146313901169 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) * (-(508060006594250466711 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_556_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (10146313901169 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(1180631375359092244143 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_556_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (10146313901169 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(11231330514346018154391 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_556_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (10146313901169 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(14003623676628956738161 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_556_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (10146313901169 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_556_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (10146313901169 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_556_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (10146313901169 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_556_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_557_1 :
+    Uω (aρ 1) (bρ 1) (40654048209613 / 64000000000000 : ℝ) ≤
+      (-(2319970927352704056403 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (50301376822279401 / 80000000000000000 : ℝ) ≤
+      (-(2319970927352704056403 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (50301376822279401 / 80000000000000000 : ℝ) =
+            (50301376822279401 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (50301376822279401 / 40000000000000000 : ℝ) ≤
+            (1145764974147295943597 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6287646390902819 / 10000000000000000 : ℝ)) (q :=
+      (50301376822279401 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_557_2 :
+    Uω (aρ 2) (bρ 2) (40654048209613 / 64000000000000 : ℝ) ≤
+      (-(1169511351196875418163 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (50110075413780381 / 80000000000000000 : ℝ) ≤
+      (-(1169511351196875418163 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (50110075413780381 / 80000000000000000 : ℝ) =
+            (50110075413780381 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (50110075413780381 / 40000000000000000 : ℝ) ≤
+            (563356599553124581837 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6116787159461 / 9765625000000 : ℝ)) (q :=
+      (50110075413780381 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_557_3 :
+    Uω (aρ 3) (bρ 3) (40654048209613 / 64000000000000 : ℝ) ≤
+      (-(4754685173399231974649 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (49727500849193261 / 80000000000000000 : ℝ) ≤
+      (-(4754685173399231974649 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (49727500849193261 / 80000000000000000 : ℝ) =
+            (49727500849193261 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (49727500849193261 / 40000000000000000 : ℝ) ≤
+            (2176786629600768025351 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1553835629122821 / 2500000000000000 : ℝ)) (q :=
+      (49727500849193261 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_557_4 :
+    Uω (aρ 4) (bρ 4) (40654048209613 / 64000000000000 : ℝ) ≤
+      (-(2441633089450279375213 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (49092192823377549 / 80000000000000000 : ℝ) ≤
+      (-(2441633089450279375213 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (49092192823377549 / 80000000000000000 : ℝ) =
+            (49092192823377549 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (49092192823377549 / 40000000000000000 : ℝ) ≤
+            (1024102812049720624787 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1533702422775589 / 2500000000000000 : ℝ)) (q :=
+      (49092192823377549 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_557_5 :
+    Uω (aρ 5) (bρ 5) (40654048209613 / 64000000000000 : ℝ) ≤
+      (-(5082253326313264360587 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9624995275807277 / 16000000000000000 : ℝ) ≤
+      (-(5082253326313264360587 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (9624995275807277 / 16000000000000000 : ℝ) =
+            (9624995275807277 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9624995275807277 / 8000000000000000 : ℝ) ≤
+            (1849218476686735639413 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1202239099870413 / 2000000000000000 : ℝ)) (q :=
+      (9624995275807277 / 16000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_557_6 :
+    Uω (aρ 6) (bρ 6) (40654048209613 / 64000000000000 : ℝ) ≤
+      (-(5374596265478078590309 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (46738442590975717 / 80000000000000000 : ℝ) ≤
+      (-(5374596265478078590309 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (46738442590975717 / 80000000000000000 : ℝ) =
+            (46738442590975717 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (46738442590975717 / 40000000000000000 : ℝ) ≤
+            (1556875537521921409691 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2915901213738449 / 5000000000000000 : ℝ)) (q :=
+      (46738442590975717 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_557_7 :
+    Uω (aρ 7) (bρ 7) (40654048209613 / 64000000000000 : ℝ) ≤
+      (-(2893724476532077899517 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8969624618761153 / 16000000000000000 : ℝ) ≤
+      (-(2893724476532077899517 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (8969624618761153 / 16000000000000000 : ℝ) =
+            (8969624618761153 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8969624618761153 / 8000000000000000 : ℝ) ≤
+            (572011424967922100483 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (558288393394441 / 1000000000000000 : ℝ)) (q :=
+      (8969624618761153 / 16000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_557_8 :
+    Uω (aρ 8) (bρ 8) (40654048209613 / 64000000000000 : ℝ) ≤
+      (-(6352566579593873370831 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (42383959477729581 / 80000000000000000 : ℝ) ≤
+      (-(6352566579593873370831 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (42383959477729581 / 80000000000000000 : ℝ) =
+            (42383959477729581 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (42383959477729581 / 40000000000000000 : ℝ) ≤
+            (578905223406126629169 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1312596170316341 / 2500000000000000 : ℝ)) (q :=
+      (42383959477729581 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_557_9 :
+    Uω (aρ 9) (bρ 9) (40654048209613 / 64000000000000 : ℝ) ≤
+      (-(177702947989476951683 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (39299619708350081 / 80000000000000000 : ℝ) ≤
+      (-(177702947989476951683 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (39299619708350081 / 80000000000000000 : ℝ) =
+            (39299619708350081 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (39299619708350081 / 20000000000000000 : ℝ) ≤
+            (168870642160523048317 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4820265367280489 / 10000000000000000 : ℝ)) (q :=
+      (39299619708350081 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_557_10 :
+    Uω (aρ 10) (bρ 10) (40654048209613 / 64000000000000 : ℝ) ≤
+      (-(253243698688492737057 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7115026000924909 / 16000000000000000 : ℝ) ≤
+      (-(253243698688492737057 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7115026000924909 / 16000000000000000 : ℝ) =
+            (7115026000924909 / 4000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7115026000924909 / 4000000000000000 : ℝ) ≤
+            (179973288999007262943 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (855540394968821 / 2000000000000000 : ℝ)) (q :=
+      (7115026000924909 / 16000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_557_11 :
+    Uω (aρ 11) (bρ 11) (40654048209613 / 64000000000000 : ℝ) ≤
+      (-(9415166918186793190237 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (31202865774225213 / 80000000000000000 : ℝ) ≤
+      (-(9415166918186793190237 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (31202865774225213 / 80000000000000000 : ℝ) =
+            (31202865774225213 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (31202865774225213 / 20000000000000000 : ℝ) ≤
+            (4447776687813206809763 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (450389067931159 / 1250000000000000 : ℝ)) (q :=
+      (31202865774225213 / 80000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_557_12 :
+    Uω (aρ 12) (bρ 12) (40654048209613 / 64000000000000 : ℝ) ≤
+      (-(11192235148967130137221 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (26122659539245121 / 80000000000000000 : ℝ) ≤
+      (-(11192235148967130137221 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (26122659539245121 / 80000000000000000 : ℝ) =
+            (26122659539245121 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26122659539245121 / 20000000000000000 : ℝ) ≤
+            (2670708457032869862779 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2756761081389249 / 10000000000000000 : ℝ)) (q :=
+      (26122659539245121 / 80000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_557_13 :
+    Uω (aρ 13) (bρ 13) (40654048209613 / 64000000000000 : ℝ) ≤
+      (-(6967750135136901165281 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19855411842309173 / 80000000000000000 : ℝ) ≤
+      (-(6967750135136901165281 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (19855411842309173 / 80000000000000000 : ℝ) =
+            (19855411842309173 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19855411842309173 / 10000000000000000 : ℝ) ≤
+            (3429457569363098834719 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (794665449375131 / 5000000000000000 : ℝ)) (q :=
+      (19855411842309173 / 80000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_557_14 :
+    Uω (aρ 14) (bρ 14) (40654048209613 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_557_15 :
+    Uω (aρ 15) (bρ 15) (40654048209613 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_557_16 :
+    Uω (aρ 16) (bρ 16) (40654048209613 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_557 :
+    Uρ (40654048209613 / 64000000000000 : ℝ) ≤
+      (-(7594649789318552314463 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (40654048209613 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2319970927352704056403 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_557_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (40654048209613 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1169511351196875418163 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_557_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (40654048209613 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(4754685173399231974649 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_557_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (40654048209613 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(2441633089450279375213 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_557_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (40654048209613 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(5082253326313264360587 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_557_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (40654048209613 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(5374596265478078590309 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_557_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (40654048209613 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(2893724476532077899517 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_557_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (40654048209613 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(6352566579593873370831 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_557_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (40654048209613 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(177702947989476951683 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_557_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (40654048209613 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) * (-(253243698688492737057 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_557_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (40654048209613 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(9415166918186793190237 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_557_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (40654048209613 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(11192235148967130137221 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_557_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (40654048209613 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(6967750135136901165281 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_557_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (40654048209613 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_557_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (40654048209613 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_557_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (40654048209613 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_557_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_558_1 :
+    Uω (aρ 1) (bρ 1) (814456816291 / 1280000000000 : ℝ) ≤
+      (-(4622861270707890117367 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25193683964744969 / 40000000000000000 : ℝ) ≤
+      (-(4622861270707890117367 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (25193683964744969 / 40000000000000000 : ℝ) =
+            (25193683964744969 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25193683964744969 / 20000000000000000 : ℝ) ≤
+            (2308610532292109882633 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6298395323184047 / 10000000000000000 : ℝ)) (q :=
+      (25193683964744969 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_558_2 :
+    Uω (aρ 2) (bρ 2) (814456816291 / 1280000000000 : ℝ) ≤
+      (-(4660899276902082966911 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25098034245633637 / 40000000000000000 : ℝ) ≤
+      (-(4660899276902082966911 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (25098034245633637 / 40000000000000000 : ℝ) =
+            (25098034245633637 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25098034245633637 / 20000000000000000 : ℝ) ≤
+            (2270572526097917033089 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6274339476138381 / 10000000000000000 : ℝ)) (q :=
+      (25098034245633637 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_558_3 :
+    Uω (aρ 3) (bρ 3) (814456816291 / 1280000000000 : ℝ) ≤
+      (-(4737406063087016598307 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24906749912178749 / 40000000000000000 : ℝ) ≤
+      (-(4737406063087016598307 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (24906749912178749 / 40000000000000000 : ℝ) =
+            (24906749912178749 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24906749912178749 / 20000000000000000 : ℝ) ≤
+            (2194065739912983401693 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6226093415760937 / 10000000000000000 : ℝ)) (q :=
+      (24906749912178749 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_558_4 :
+    Uω (aρ 4) (bρ 4) (814456816291 / 1280000000000 : ℝ) ≤
+      (-(4865760446468570302513 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4917820756817343 / 8000000000000000 : ℝ) ≤
+      (-(4865760446468570302513 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (4917820756817343 / 8000000000000000 : ℝ) =
+            (4917820756817343 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4917820756817343 / 4000000000000000 : ℝ) ≤
+            (2065711356531429697487 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (76819556659749 / 125000000000000 : ℝ)) (q :=
+      (4917820756817343 / 8000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_558_5 :
+    Uω (aρ 5) (bρ 5) (814456816291 / 1280000000000 : ℝ) ≤
+      (-(2532193971103932224791 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24105515172156347 / 40000000000000000 : ℝ) ≤
+      (-(2532193971103932224791 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (24105515172156347 / 40000000000000000 : ℝ) =
+            (24105515172156347 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24105515172156347 / 20000000000000000 : ℝ) ≤
+            (933541930396067775209 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (752745018268717 / 1250000000000000 : ℝ)) (q :=
+      (24105515172156347 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_558_6 :
+    Uω (aρ 6) (bρ 6) (814456816291 / 1280000000000 : ℝ) ≤
+      (-(1339045465924501048353 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23412293964210231 / 40000000000000000 : ℝ) ≤
+      (-(1339045465924501048353 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23412293964210231 / 40000000000000000 : ℝ) =
+            (23412293964210231 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23412293964210231 / 20000000000000000 : ℝ) ≤
+            (393822484825498951647 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2921294958658339 / 5000000000000000 : ℝ)) (q :=
+      (23412293964210231 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_558_7 :
+    Uω (aρ 7) (bρ 7) (814456816291 / 1280000000000 : ℝ) ≤
+      (-(2884107202550171195271 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22467234723155521 / 40000000000000000 : ℝ) ≤
+      (-(2884107202550171195271 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (22467234723155521 / 40000000000000000 : ℝ) =
+            (22467234723155521 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22467234723155521 / 20000000000000000 : ℝ) ≤
+            (581628698949828804729 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5593721677549323 / 10000000000000000 : ℝ)) (q :=
+      (22467234723155521 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_558_8 :
+    Uω (aρ 8) (bρ 8) (814456816291 / 1280000000000 : ℝ) ≤
+      (-(633211539617910829051 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21235364193371443 / 40000000000000000 : ℝ) ≤
+      (-(633211539617910829051 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (21235364193371443 / 40000000000000000 : ℝ) =
+            (21235364193371443 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21235364193371443 / 20000000000000000 : ℝ) ≤
+            (59935640682089170949 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1315332015999321 / 2500000000000000 : ℝ)) (q :=
+      (21235364193371443 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_558_9 :
+    Uω (aρ 9) (bρ 9) (814456816291 / 1280000000000 : ℝ) ≤
+      (-(3542922206328462858359 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19693625650212941 / 40000000000000000 : ℝ) ≤
+      (-(3542922206328462858359 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (19693625650212941 / 40000000000000000 : ℝ) =
+            (19693625650212941 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19693625650212941 / 10000000000000000 : ℝ) ≤
+            (3388549596671537141641 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4831424420778033 / 10000000000000000 : ℝ)) (q :=
+      (19693625650212941 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_558_10 :
+    Uω (aρ 10) (bρ 10) (814456816291 / 1280000000000 : ℝ) ≤
+      (-(1009838093136397352783 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17832256479524721 / 40000000000000000 : ℝ) ≤
+      (-(1009838093136397352783 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (17832256479524721 / 40000000000000000 : ℝ) =
+            (17832256479524721 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17832256479524721 / 10000000000000000 : ℝ) ≤
+            (723029857613602647217 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4289298868928923 / 10000000000000000 : ℝ)) (q :=
+      (17832256479524721 / 40000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_558_11 :
+    Uω (aρ 11) (bρ 11) (814456816291 / 1280000000000 : ℝ) ≤
+      (-(9385386513549517948139 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15647963836678691 / 40000000000000000 : ℝ) ≤
+      (-(9385386513549517948139 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (15647963836678691 / 40000000000000000 : ℝ) =
+            (15647963836678691 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15647963836678691 / 10000000000000000 : ℝ) ≤
+            (4477557092450482051861 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (903907293427727 / 2500000000000000 : ℝ)) (q :=
+      (15647963836678691 / 40000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_558_12 :
+    Uω (aρ 12) (bρ 12) (814456816291 / 1280000000000 : ℝ) ≤
+      (-(11153347910153076889553 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (104897764484849 / 320000000000000 : ℝ) ≤
+      (-(11153347910153076889553 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (104897764484849 / 320000000000000 : ℝ) = (104897764484849 / 80000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (104897764484849 / 80000000000000 : ℝ) ≤
+            (2709595695846923110447 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (22171661058877 / 80000000000000 : ℝ)) (q :=
+      (104897764484849 / 320000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12]) (by norm_num [bρ_12])
+      (by norm_num) (by norm_num [aρ_12, bρ_12]) (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_558_13 :
+    Uω (aρ 13) (bρ 13) (814456816291 / 1280000000000 : ℝ) ≤
+      (-(13868348245627295984087 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9994596818018049 / 40000000000000000 : ℝ) ≤
+      (-(13868348245627295984087 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (9994596818018049 / 40000000000000000 : ℝ) =
+            (9994596818018049 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 15
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9994596818018049 / 5000000000000000 : ℝ) ≤
+            (6926067163372704015913 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1612027502660587 / 10000000000000000 : ℝ)) (q :=
+      (9994596818018049 / 40000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_558_14 :
+    Uω (aρ 14) (bρ 14) (814456816291 / 1280000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_558_15 :
+    Uω (aρ 15) (bρ 15) (814456816291 / 1280000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_558_16 :
+    Uω (aρ 16) (bρ 16) (814456816291 / 1280000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_558 :
+    Uρ (814456816291 / 1280000000000 : ℝ) ≤
+      (-(1893232505862725806829 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (814456816291 / 1280000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(4622861270707890117367 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_558_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (814456816291 / 1280000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(4660899276902082966911 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_558_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (814456816291 / 1280000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(4737406063087016598307 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_558_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (814456816291 / 1280000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(4865760446468570302513 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_558_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (814456816291 / 1280000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(2532193971103932224791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_558_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (814456816291 / 1280000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1339045465924501048353 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_558_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (814456816291 / 1280000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(2884107202550171195271 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_558_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (814456816291 / 1280000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(633211539617910829051 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_558_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (814456816291 / 1280000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(3542922206328462858359 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_558_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (814456816291 / 1280000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(1009838093136397352783 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_558_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (814456816291 / 1280000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(9385386513549517948139 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_558_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (814456816291 / 1280000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(11153347910153076889553 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_558_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (814456816291 / 1280000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(13868348245627295984087 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_558_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (814456816291 / 1280000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_558_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (814456816291 / 1280000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_558_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (814456816291 / 1280000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_558_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_559_1 :
+    Uω (aρ 1) (bρ 1) (40791633419487 / 64000000000000 : ℝ) ≤
+      (-(2302904905921185912749 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (50473359035504343 / 80000000000000000 : ℝ) ≤
+      (-(2302904905921185912749 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (50473359035504343 / 80000000000000000 : ℝ) =
+            (50473359035504343 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (50473359035504343 / 40000000000000000 : ℝ) ≤
+            (1162830995578814087251 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3154572127583121 / 5000000000000000 : ℝ)) (q :=
+      (50473359035504343 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_559_2 :
+    Uω (aρ 2) (bρ 2) (40791633419487 / 64000000000000 : ℝ) ≤
+      (-(580472812406440178227 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (50282061560814059 / 80000000000000000 : ℝ) ≤
+      (-(580472812406440178227 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (50282061560814059 / 80000000000000000 : ℝ) =
+            (50282061560814059 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (50282061560814059 / 40000000000000000 : ℝ) ≤
+            (285961162968559821773 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6285088899003671 / 10000000000000000 : ℝ)) (q :=
+      (50282061560814059 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_559_3 :
+    Uω (aρ 3) (bρ 3) (40791633419487 / 64000000000000 : ℝ) ≤
+      (-(472015676372460509977 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9979899754237819 / 16000000000000000 : ℝ) ≤
+      (-(472015676372460509977 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (9979899754237819 / 16000000000000000 : ℝ) =
+            (9979899754237819 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9979899754237819 / 8000000000000000 : ℝ) ≤
+            (221131503927539490023 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (623684430794743 / 1000000000000000 : ℝ)) (q :=
+      (9979899754237819 / 16000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_559_4 :
+    Uω (aρ 4) (bρ 4) (40791633419487 / 64000000000000 : ℝ) ≤
+      (-(151508916330217477677 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9852844445835671 / 16000000000000000 : ℝ) ≤
+      (-(151508916330217477677 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (9852844445835671 / 16000000000000000 : ℝ) =
+            (9852844445835671 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9852844445835671 / 8000000000000000 : ℝ) ≤
+            (65099577513532522323 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1231263870701949 / 2000000000000000 : ℝ)) (q :=
+      (9852844445835671 / 16000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_559_5 :
+    Uω (aρ 5) (bρ 5) (40791633419487 / 64000000000000 : ℝ) ≤
+      (-(5046554465058193984063 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (48297084084177007 / 80000000000000000 : ℝ) ≤
+      (-(5046554465058193984063 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (48297084084177007 / 80000000000000000 : ℝ) =
+            (48297084084177007 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (48297084084177007 / 40000000000000000 : ℝ) ≤
+            (1884917337941806015937 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (754090592074301 / 1250000000000000 : ℝ)) (q :=
+      (48297084084177007 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_559_6 :
+    Uω (aρ 6) (bρ 6) (40791633419487 / 64000000000000 : ℝ) ≤
+      (-(533780142986120744697 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (46910732697114243 / 80000000000000000 : ℝ) ≤
+      (-(533780142986120744697 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (46910732697114243 / 80000000000000000 : ℝ) =
+            (46910732697114243 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (46910732697114243 / 40000000000000000 : ℝ) ≤
+            (159367037313879255303 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5853377264968717 / 10000000000000000 : ℝ)) (q :=
+      (46910732697114243 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_559_7 :
+    Uω (aρ 7) (bρ 7) (40791633419487 / 64000000000000 : ℝ) ≤
+      (-(5749017087125469562931 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (45020814429197179 / 80000000000000000 : ℝ) ≤
+      (-(5749017087125469562931 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (45020814429197179 / 80000000000000000 : ℝ) =
+            (45020814429197179 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (45020814429197179 / 40000000000000000 : ℝ) ≤
+            (1182454715874530437069 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5604559078749451 / 10000000000000000 : ℝ)) (q :=
+      (45020814429197179 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_559_8 :
+    Uω (aρ 8) (bρ 8) (40791633419487 / 64000000000000 : ℝ) ≤
+      (-(631170670481372951783 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8511498818824967 / 16000000000000000 : ℝ) ≤
+      (-(631170670481372951783 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (8511498818824967 / 16000000000000000 : ℝ) =
+            (8511498818824967 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8511498818824967 / 8000000000000000 : ℝ) ≤
+            (61976509818627048217 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1054454129264273 / 2000000000000000 : ℝ)) (q :=
+      (8511498818824967 / 16000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_559_9 :
+    Uω (aρ 9) (bρ 9) (40791633419487 / 64000000000000 : ℝ) ≤
+      (-(353181114344393088867 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (39474875469341551 / 80000000000000000 : ℝ) ≤
+      (-(353181114344393088867 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (39474875469341551 / 80000000000000000 : ℝ) =
+            (39474875469341551 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (39474875469341551 / 20000000000000000 : ℝ) ≤
+            (339966065955606911133 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (605322702310693 / 1250000000000000 : ℝ)) (q :=
+      (39474875469341551 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_559_10 :
+    Uω (aρ 10) (bρ 10) (40791633419487 / 64000000000000 : ℝ) ≤
+      (-(402683943645360927051 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (35753878288956451 / 80000000000000000 : ℝ) ≤
+      (-(402683943645360927051 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (35753878288956451 / 80000000000000000 : ℝ) =
+            (35753878288956451 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (35753878288956451 / 20000000000000000 : ℝ) ≤
+            (290463236654639072949 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4300891356884269 / 10000000000000000 : ℝ)) (q :=
+      (35753878288956451 / 80000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_559_11 :
+    Uω (aρ 11) (bρ 11) (40791633419487 / 64000000000000 : ℝ) ≤
+      (-(146182952791553937827 / 156250000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (31388944228734327 / 80000000000000000 : ℝ) ≤
+      (-(146182952791553937827 / 156250000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (31388944228734327 / 80000000000000000 : ℝ) =
+            (31388944228734327 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (31388944228734327 / 20000000000000000 : ℝ) ≤
+            (70425541052196062173 / 156250000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1814067234016869 / 5000000000000000 : ℝ)) (q :=
+      (31388944228734327 / 80000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_559_12 :
+    Uω (aρ 12) (bρ 12) (40791633419487 / 64000000000000 : ℝ) ≤
+      (-(11114666087772948681293 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (26326078487755239 / 80000000000000000 : ℝ) ≤
+      (-(11114666087772948681293 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (26326078487755239 / 80000000000000000 : ℝ) =
+            (26326078487755239 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26326078487755239 / 20000000000000000 : ℝ) ≤
+            (2748277518227051318707 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1393059064736983 / 5000000000000000 : ℝ)) (q :=
+      (26326078487755239 / 80000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_559_13 :
+    Uω (aρ 13) (bρ 13) (40791633419487 / 64000000000000 : ℝ) ≤
+      (-(13802130104783206082597 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20121997581452683 / 80000000000000000 : ℝ) ≤
+      (-(13802130104783206082597 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (20121997581452683 / 80000000000000000 : ℝ) =
+            (20121997581452683 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20121997581452683 / 20000000000000000 : ℝ) ≤
+            (60813501216793917403 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1634479644493327 / 10000000000000000 : ℝ)) (q :=
+      (20121997581452683 / 80000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_559_14 :
+    Uω (aρ 14) (bρ 14) (40791633419487 / 64000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_559_15 :
+    Uω (aρ 15) (bρ 15) (40791633419487 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_559_16 :
+    Uω (aρ 16) (bρ 16) (40791633419487 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_559 :
+    Uρ (40791633419487 / 64000000000000 : ℝ) ≤
+      (-(3775657065764696745661 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (40791633419487 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2302904905921185912749 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_559_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (40791633419487 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(580472812406440178227 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_559_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (40791633419487 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(472015676372460509977 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_559_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (40791633419487 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(151508916330217477677 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_559_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (40791633419487 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(5046554465058193984063 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_559_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (40791633419487 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(533780142986120744697 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_559_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (40791633419487 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(5749017087125469562931 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_559_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (40791633419487 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(631170670481372951783 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_559_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (40791633419487 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(353181114344393088867 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_559_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (40791633419487 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) * (-(402683943645360927051 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_559_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (40791633419487 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) * (-(146182952791553937827 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_559_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (40791633419487 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(11114666087772948681293 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_559_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (40791633419487 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(13802130104783206082597 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_559_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (40791633419487 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_559_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (40791633419487 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_559_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (40791633419487 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_559_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_560_1 :
+    Uω (aρ 1) (bρ 1) (5107553253053 / 8000000000000 : ℝ) ≤
+      (-(2294393689475535923337 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (631991876754109 / 1000000000000000 : ℝ) ≤
+      (-(2294393689475535923337 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (631991876754109 / 1000000000000000 : ℝ) =
+            (631991876754109 / 500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (631991876754109 / 500000000000000 : ℝ) ≤
+            (1171342212024464076663 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (631989318685093 / 1000000000000000 : ℝ)) (q :=
+      (631991876754109 / 1000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_560_2 :
+    Uω (aρ 2) (bρ 2) (5107553253053 / 8000000000000 : ℝ) ≤
+      (-(4626694971520392917771 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1574001706951919 / 2500000000000000 : ℝ) ≤
+      (-(4626694971520392917771 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1574001706951919 / 2500000000000000 : ℝ) =
+            (1574001706951919 / 1250000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1574001706951919 / 1250000000000000 : ℝ) ≤
+            (2304776831479607082229 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3147919159947051 / 5000000000000000 : ℝ)) (q :=
+      (1574001706951919 / 2500000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_560_3 :
+    Uω (aρ 3) (bρ 3) (5107553253053 / 8000000000000 : ℝ) ≤
+      (-(940587434521278228623 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12496374422458581 / 20000000000000000 : ℝ) ≤
+      (-(940587434521278228623 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (12496374422458581 / 20000000000000000 : ℝ) =
+            (12496374422458581 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12496374422458581 / 10000000000000000 : ℝ) ≤
+            (445706926078721771377 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6247595193087331 / 10000000000000000 : ℝ)) (q :=
+      (12496374422458581 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_560_4 :
+    Uω (aρ 4) (bρ 4) (5107553253053 / 8000000000000 : ℝ) ≤
+      (-(1207710175071547666761 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3084389800426919 / 5000000000000000 : ℝ) ≤
+      (-(1207710175071547666761 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (3084389800426919 / 5000000000000000 : ℝ) =
+            (3084389800426919 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3084389800426919 / 2500000000000000 : ℝ) ≤
+            (525157775678452333239 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3083537076700713 / 5000000000000000 : ℝ)) (q :=
+      (3084389800426919 / 5000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_560_5 :
+    Uω (aρ 5) (bρ 5) (5107553253053 / 8000000000000 : ℝ) ≤
+      (-(251437639046613927597 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1511973049994807 / 2500000000000000 : ℝ) ≤
+      (-(251437639046613927597 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1511973049994807 / 2500000000000000 : ℝ) =
+            (1511973049994807 / 1250000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1511973049994807 / 1250000000000000 : ℝ) ≤
+            (95135951103386072403 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3021744635493603 / 5000000000000000 : ℝ)) (q :=
+      (1511973049994807 / 2500000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_560_6 :
+    Uω (aρ 6) (bρ 6) (5107553253053 / 8000000000000 : ℝ) ≤
+      (-(5319454838437104841271 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2349843845009789 / 4000000000000000 : ℝ) ≤
+      (-(5319454838437104841271 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2349843845009789 / 4000000000000000 : ℝ) =
+            (2349843845009789 / 2000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2349843845009789 / 2000000000000000 : ℝ) ≤
+            (1612016964562895158729 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1172832894243539 / 2000000000000000 : ℝ)) (q :=
+      (2349843845009789 / 4000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_560_7 :
+    Uω (aρ 7) (bρ 7) (5107553253053 / 8000000000000 : ℝ) ≤
+      (-(2864928427064339415983 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (704799344537407 / 1250000000000000 : ℝ) ≤
+      (-(2864928427064339415983 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (704799344537407 / 1250000000000000 : ℝ) =
+            (704799344537407 / 625000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (704799344537407 / 625000000000000 : ℝ) ≤
+            (600807474435660584017 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2807698069763631 / 5000000000000000 : ℝ)) (q :=
+      (704799344537407 / 1250000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_560_8 :
+    Uω (aρ 8) (bρ 8) (5107553253053 / 8000000000000 : ℝ) ≤
+      (-(6291340326210612036741 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2665266038735567 / 5000000000000000 : ℝ) ≤
+      (-(6291340326210612036741 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2665266038735567 / 5000000000000000 : ℝ) =
+            (2665266038735567 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2665266038735567 / 2500000000000000 : ℝ) ≤
+            (640131476789387963259 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2641606216605509 / 5000000000000000 : ℝ)) (q :=
+      (2665266038735567 / 5000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_560_9 :
+    Uω (aρ 9) (bρ 9) (5107553253053 / 8000000000000 : ℝ) ≤
+      (-(7041451297304985064883 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (618163941660743 / 1250000000000000 : ℝ) ≤
+      (-(7041451297304985064883 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (618163941660743 / 1250000000000000 : ℝ) =
+            (618163941660743 / 312500000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (618163941660743 / 312500000000000 : ℝ) ≤
+            (6821492308695014935117 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2426868486600319 / 5000000000000000 : ℝ)) (q :=
+      (618163941660743 / 1250000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_560_10 :
+    Uω (aρ 10) (bρ 10) (5107553253053 / 8000000000000 : ℝ) ≤
+      (-(4014360175648586768127 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8960806534119087 / 20000000000000000 : ℝ) ≤
+      (-(4014360175648586768127 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8960806534119087 / 20000000000000000 : ℝ) =
+            (8960806534119087 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8960806534119087 / 5000000000000000 : ℝ) ≤
+            (2917111627351413231873 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4312479474242837 / 10000000000000000 : ℝ)) (q :=
+      (8960806534119087 / 20000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_560_11 :
+    Uω (aρ 11) (bρ 11) (5107553253053 / 8000000000000 : ℝ) ≤
+      (-(932613351350855815937 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7870478976903391 / 20000000000000000 : ℝ) ≤
+      (-(932613351350855815937 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7870478976903391 / 20000000000000000 : ℝ) =
+            (7870478976903391 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7870478976903391 / 5000000000000000 : ℝ) ≤
+            (453681009249144184063 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3640628543232141 / 10000000000000000 : ℝ)) (q :=
+      (7870478976903391 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_560_12 :
+    Uω (aρ 12) (bρ 12) (5107553253053 / 8000000000000 : ℝ) ≤
+      (-(2769046757582107150309 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3303446737945291 / 10000000000000000 : ℝ) ≤
+      (-(2769046757582107150309 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3303446737945291 / 10000000000000000 : ℝ) =
+            (3303446737945291 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3303446737945291 / 2500000000000000 : ℝ) ≤
+            (696689143917892849691 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (700185784726083 / 2500000000000000 : ℝ)) (q :=
+      (3303446737945291 / 10000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_560_13 :
+    Uω (aρ 13) (bρ 13) (5107553253053 / 8000000000000 : ℝ) ≤
+      (-(13736810718800869438119 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5063465858776757 / 20000000000000000 : ℝ) ≤
+      (-(13736810718800869438119 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5063465858776757 / 20000000000000000 : ℝ) =
+            (5063465858776757 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5063465858776757 / 5000000000000000 : ℝ) ≤
+            (126132887199130561881 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1656697263385507 / 10000000000000000 : ℝ)) (q :=
+      (5063465858776757 / 20000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_560_14 :
+    Uω (aρ 14) (bρ 14) (5107553253053 / 8000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_560_15 :
+    Uω (aρ 15) (bρ 15) (5107553253053 / 8000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_560_16 :
+    Uω (aρ 16) (bρ 16) (5107553253053 / 8000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_560 :
+    Uρ (5107553253053 / 8000000000000 : ℝ) ≤
+      (-(1505959936281701094027 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (5107553253053 / 8000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2294393689475535923337 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_560_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (5107553253053 / 8000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(4626694971520392917771 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_560_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (5107553253053 / 8000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(940587434521278228623 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_560_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (5107553253053 / 8000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1207710175071547666761 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_560_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (5107553253053 / 8000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(251437639046613927597 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_560_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (5107553253053 / 8000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(5319454838437104841271 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_560_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (5107553253053 / 8000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(2864928427064339415983 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_560_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (5107553253053 / 8000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(6291340326210612036741 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_560_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (5107553253053 / 8000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(7041451297304985064883 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_560_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (5107553253053 / 8000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(4014360175648586768127 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_560_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (5107553253053 / 8000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(932613351350855815937 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_560_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (5107553253053 / 8000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(2769046757582107150309 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_560_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (5107553253053 / 8000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(13736810718800869438119 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_560_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (5107553253053 / 8000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_560_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (5107553253053 / 8000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_560_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (5107553253053 / 8000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_560_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_561_1 :
+    Uω (aρ 1) (bρ 1) (20499005617149 / 32000000000000 : ℝ) ≤
+      (-(2277414598492095836739 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25365666173215799 / 40000000000000000 : ℝ) ≤
+      (-(2277414598492095836739 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (25365666173215799 / 40000000000000000 : ℝ) =
+            (25365666173215799 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25365666173215799 / 20000000000000000 : ℝ) ≤
+            (1188321303007904163261 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6341391049333837 / 10000000000000000 : ℝ)) (q :=
+      (25365666173215799 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_561_2 :
+    Uω (aρ 2) (bρ 2) (20499005617149 / 32000000000000 : ℝ) ≤
+      (-(4592607267118482431659 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25270020361109551 / 40000000000000000 : ℝ) ≤
+      (-(4592607267118482431659 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (25270020361109551 / 40000000000000000 : ℝ) =
+            (25270020361109551 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25270020361109551 / 20000000000000000 : ℝ) ≤
+            (2338864535881517568341 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6317337155790713 / 10000000000000000 : ℝ)) (q :=
+      (25270020361109551 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_561_3 :
+    Uω (aρ 3) (bρ 3) (20499005617149 / 32000000000000 : ℝ) ≤
+      (-(4668586706926270342089 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25078747721572853 / 40000000000000000 : ℝ) ≤
+      (-(4668586706926270342089 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (25078747721572853 / 40000000000000000 : ℝ) =
+            (25078747721572853 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25078747721572853 / 20000000000000000 : ℝ) ≤
+            (2262885096073729657911 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1567274235593091 / 2500000000000000 : ℝ)) (q :=
+      (25078747721572853 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_561_4 :
+    Uω (aρ 4) (bρ 4) (20499005617149 / 32000000000000 : ℝ) ≤
+      (-(479604253574241296943 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24761132856907953 / 40000000000000000 : ℝ) ≤
+      (-(479604253574241296943 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (24761132856907953 / 40000000000000000 : ℝ) =
+            (24761132856907953 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24761132856907953 / 20000000000000000 : ℝ) ≤
+            (213542926725758703057 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3094291845552457 / 5000000000000000 : ℝ)) (q :=
+      (24761132856907953 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_561_5 :
+    Uω (aρ 5) (bρ 5) (20499005617149 / 32000000000000 : ℝ) ≤
+      (-(4993244339069595102151 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4855524396330021 / 8000000000000000 : ℝ) ≤
+      (-(4993244339069595102151 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (4855524396330021 / 8000000000000000 : ℝ) =
+            (4855524396330021 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4855524396330021 / 4000000000000000 : ℝ) ≤
+            (1938227463930404897849 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (606501817281099 / 1000000000000000 : ℝ)) (q :=
+      (4855524396330021 / 8000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_561_6 :
+    Uω (aρ 6) (bρ 6) (20499005617149 / 32000000000000 : ℝ) ≤
+      (-(5282862684184516705617 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23584581810980857 / 40000000000000000 : ℝ) ≤
+      (-(5282862684184516705617 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23584581810980857 / 40000000000000000 : ℝ) =
+            (23584581810980857 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23584581810980857 / 20000000000000000 : ℝ) ≤
+            (1648609118815483294383 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2942869231308183 / 5000000000000000 : ℝ)) (q :=
+      (23584581810980857 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_561_7 :
+    Uω (aρ 7) (bρ 7) (20499005617149 / 32000000000000 : ℝ) ≤
+      (-(1138329413456083984391 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22639920619566651 / 40000000000000000 : ℝ) ≤
+      (-(1138329413456083984391 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (22639920619566651 / 40000000000000000 : ℝ) =
+            (22639920619566651 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22639920619566651 / 20000000000000000 : ℝ) ≤
+            (247964947143916015609 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5637069247669263 / 10000000000000000 : ℝ)) (q :=
+      (22639920619566651 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_561_8 :
+    Uω (aρ 8) (bρ 8) (20499005617149 / 32000000000000 : ℝ) ≤
+      (-(6250733795887733386939 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21408886102295611 / 40000000000000000 : ℝ) ≤
+      (-(6250733795887733386939 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (21408886102295611 / 40000000000000000 : ℝ) =
+            (21408886102295611 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21408886102295611 / 20000000000000000 : ℝ) ≤
+            (680738007112266613061 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5305093640373743 / 10000000000000000 : ℝ)) (q :=
+      (21408886102295611 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_561_9 :
+    Uω (aρ 9) (bρ 9) (20499005617149 / 32000000000000 : ℝ) ≤
+      (-(3498630877915021785957 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3973770394664829 / 8000000000000000 : ℝ) ≤
+      (-(3498630877915021785957 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3973770394664829 / 8000000000000000 : ℝ) =
+            (3973770394664829 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3973770394664829 / 2000000000000000 : ℝ) ≤
+            (3432840925084978214043 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (487604220424801 / 1000000000000000 : ℝ)) (q :=
+      (3973770394664829 / 8000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_561_10 :
+    Uω (aρ 10) (bρ 10) (20499005617149 / 32000000000000 : ℝ) ≤
+      (-(3989501908388574488219 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (18010934972507951 / 40000000000000000 : ℝ) ≤
+      (-(3989501908388574488219 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (18010934972507951 / 40000000000000000 : ℝ) =
+            (18010934972507951 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18010934972507951 / 10000000000000000 : ℝ) ≤
+            (2941969894611425511781 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4335642737334913 / 10000000000000000 : ℝ)) (q :=
+      (18010934972507951 / 40000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_561_11 :
+    Uω (aρ 11) (bρ 11) (20499005617149 / 32000000000000 : ℝ) ≤
+      (-(4633642821382275522129 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15833863235497879 / 40000000000000000 : ℝ) ≤
+      (-(4633642821382275522129 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (15833863235497879 / 40000000000000000 : ℝ) =
+            (15833863235497879 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15833863235497879 / 10000000000000000 : ℝ) ≤
+            (2297828981617724477871 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3665583495034877 / 10000000000000000 : ℝ)) (q :=
+      (15833863235497879 / 40000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_561_12 :
+    Uω (aρ 12) (bρ 12) (20499005617149 / 32000000000000 : ℝ) ≤
+      (-(10999826886831738863463 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2663014768093521 / 8000000000000000 : ℝ) ≤
+      (-(10999826886831738863463 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2663014768093521 / 8000000000000000 : ℝ) =
+            (2663014768093521 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2663014768093521 / 2000000000000000 : ℝ) ≤
+            (2863116719168261136537 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (141494444710237 / 500000000000000 : ℝ)) (q :=
+      (2663014768093521 / 8000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_561_13 :
+    Uω (aρ 13) (bρ 13) (20499005617149 / 32000000000000 : ℝ) ≤
+      (-(1701092299087291908943 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10257463780728953 / 40000000000000000 : ℝ) ≤
+      (-(1701092299087291908943 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (10257463780728953 / 40000000000000000 : ℝ) =
+            (10257463780728953 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10257463780728953 / 10000000000000000 : ℝ) ≤
+            (31775651662708091057 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (850232802965207 / 5000000000000000 : ℝ)) (q :=
+      (10257463780728953 / 40000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_561_14 :
+    Uω (aρ 14) (bρ 14) (20499005617149 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_561_15 :
+    Uω (aρ 15) (bρ 15) (20499005617149 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_561_16 :
+    Uω (aρ 16) (bρ 16) (20499005617149 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_561 :
+    Uρ (20499005617149 / 32000000000000 : ℝ) ≤
+      (-(1497413207056877704401 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (20499005617149 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2277414598492095836739 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_561_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (20499005617149 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(4592607267118482431659 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_561_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (20499005617149 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(4668586706926270342089 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_561_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (20499005617149 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(479604253574241296943 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_561_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (20499005617149 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(4993244339069595102151 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_561_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (20499005617149 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(5282862684184516705617 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_561_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (20499005617149 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(1138329413456083984391 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_561_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (20499005617149 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(6250733795887733386939 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_561_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (20499005617149 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(3498630877915021785957 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_561_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (20499005617149 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(3989501908388574488219 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_561_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (20499005617149 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4633642821382275522129 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_561_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (20499005617149 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(10999826886831738863463 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_561_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (20499005617149 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(1701092299087291908943 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_561_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (20499005617149 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_561_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (20499005617149 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_561_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (20499005617149 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_561_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_562_1 :
+    Uω (aρ 1) (bρ 1) (10283899111043 / 16000000000000 : ℝ) ≤
+      (-(565123242699242190631 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6362914318480821 / 10000000000000000 : ℝ) ≤
+      (-(565123242699242190631 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (6362914318480821 / 10000000000000000 : ℝ) =
+            (6362914318480821 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6362914318480821 / 5000000000000000 : ℝ) ≤
+            (301310732675757809369 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6362888910644767 / 10000000000000000 : ℝ)) (q :=
+      (6362914318480821 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_562_2 :
+    Uω (aρ 2) (bρ 2) (10283899111043 / 16000000000000 : ℝ) ≤
+      (-(227931768568958541353 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12678006697715053 / 20000000000000000 : ℝ) ≤
+      (-(227931768568958541353 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (12678006697715053 / 20000000000000000 : ℝ) =
+            (12678006697715053 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12678006697715053 / 10000000000000000 : ℝ) ≤
+            (118641821581041458647 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3169417991954089 / 5000000000000000 : ℝ)) (q :=
+      (12678006697715053 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_562_3 :
+    Uω (aρ 3) (bρ 3) (10283899111043 / 16000000000000 : ℝ) ≤
+      (-(4634353854935592034951 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12582373271360453 / 20000000000000000 : ℝ) ≤
+      (-(4634353854935592034951 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (12582373271360453 / 20000000000000000 : ℝ) =
+            (12582373271360453 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12582373271360453 / 10000000000000000 : ℝ) ≤
+            (2297117948064407965049 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3145299331951789 / 5000000000000000 : ℝ)) (q :=
+      (12582373271360453 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_562_4 :
+    Uω (aρ 4) (bρ 4) (10283899111043 / 16000000000000 : ℝ) ≤
+      (-(74396329823126201829 / 156250000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12423573573143853 / 20000000000000000 : ℝ) ≤
+      (-(74396329823126201829 / 156250000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (12423573573143853 / 20000000000000000 : ℝ) =
+            (12423573573143853 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12423573573143853 / 10000000000000000 : ℝ) ≤
+            (33907917098748798171 / 156250000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3105046573375989 / 5000000000000000 : ℝ)) (q :=
+      (12423573573143853 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_562_5 :
+    Uω (aρ 5) (bρ 5) (10283899111043 / 16000000000000 : ℝ) ≤
+      (-(4957861717286478519223 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12181837361044447 / 20000000000000000 : ℝ) ≤
+      (-(4957861717286478519223 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (12181837361044447 / 20000000000000000 : ℝ) =
+            (12181837361044447 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12181837361044447 / 10000000000000000 : ℝ) ≤
+            (1973610085713521480777 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1521636713496893 / 2500000000000000 : ℝ)) (q :=
+      (12181837361044447 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_562_6 :
+    Uω (aρ 6) (bρ 6) (10283899111043 / 16000000000000 : ℝ) ≤
+      (-(5246404410543003654287 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11835362029592483 / 20000000000000000 : ℝ) ≤
+      (-(5246404410543003654287 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11835362029592483 / 20000000000000000 : ℝ) =
+            (11835362029592483 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11835362029592483 / 10000000000000000 : ℝ) ≤
+            (1685067392456996345713 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (738413987209451 / 1250000000000000 : ℝ)) (q :=
+      (11835362029592483 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_562_7 :
+    Uω (aρ 7) (bρ 7) (10283899111043 / 16000000000000 : ℝ) ≤
+      (-(282679195069658106583 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11363129768687869 / 20000000000000000 : ℝ) ≤
+      (-(282679195069658106583 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11363129768687869 / 20000000000000000 : ℝ) =
+            (11363129768687869 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11363129768687869 / 10000000000000000 : ℝ) ≤
+            (63894395080341893417 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2829370508765497 / 5000000000000000 : ℝ)) (q :=
+      (11363129768687869 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_562_8 :
+    Uω (aρ 8) (bρ 8) (10283899111043 / 16000000000000 : ℝ) ≤
+      (-(621029439375479713257 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10747818824267901 / 20000000000000000 : ℝ) ≤
+      (-(621029439375479713257 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10747818824267901 / 20000000000000000 : ℝ) =
+            (10747818824267901 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10747818824267901 / 10000000000000000 : ℝ) ≤
+            (72117740924520286743 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2663485862225513 / 5000000000000000 : ℝ)) (q :=
+      (10747818824267901 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_562_9 :
+    Uω (aρ 9) (bρ 9) (10283899111043 / 16000000000000 : ℝ) ≤
+      (-(6953273864243853381139 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9978221685393537 / 20000000000000000 : ℝ) ≤
+      (-(6953273864243853381139 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9978221685393537 / 20000000000000000 : ℝ) =
+            (9978221685393537 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 15
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9978221685393537 / 5000000000000000 : ℝ) ≤
+            (6909669741756146618861 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2449170106968331 / 5000000000000000 : ℝ)) (q :=
+      (9978221685393537 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_562_10 :
+    Uω (aρ 10) (bρ 10) (10283899111043 / 16000000000000 : ℝ) ≤
+      (-(7929552090284195248573 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4525055686322299 / 10000000000000000 : ℝ) ≤
+      (-(7929552090284195248573 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4525055686322299 / 10000000000000000 : ℝ) =
+            (4525055686322299 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4525055686322299 / 2500000000000000 : ℝ) ≤
+            (5933391515715804751427 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4358788934682723 / 10000000000000000 : ℝ)) (q :=
+      (4525055686322299 / 10000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_562_11 :
+    Uω (aρ 11) (bρ 11) (10283899111043 / 16000000000000 : ℝ) ≤
+      (-(92088367002658787047 / 100000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3981670370963811 / 10000000000000000 : ℝ) ≤
+      (-(92088367002658787047 / 100000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3981670370963811 / 10000000000000000 : ℝ) =
+            (3981670370963811 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3981670370963811 / 2500000000000000 : ℝ) ≤
+            (46541069057341212953 / 100000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3690494930170747 / 10000000000000000 : ℝ)) (q :=
+      (3981670370963811 / 10000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_562_12 :
+    Uω (aρ 12) (bρ 12) (10283899111043 / 16000000000000 : ℝ) ≤
+      (-(5462123687277067907423 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (167701122188047 / 500000000000000 : ℝ) ≤
+      (-(5462123687277067907423 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (167701122188047 / 500000000000000 : ℝ) =
+            (167701122188047 / 125000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (167701122188047 / 125000000000000 : ℝ) ≤
+            (1469348115722932092577 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (571779834490001 / 2000000000000000 : ℝ)) (q :=
+      (167701122188047 / 500000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12]) (by norm_num [bρ_12])
+      (by norm_num) (by norm_num [aρ_12, bρ_12]) (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_562_13 :
+    Uω (aρ 13) (bρ 13) (10283899111043 / 16000000000000 : ℝ) ≤
+      (-(13483890514746875546159 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5193164395830821 / 20000000000000000 : ℝ) ≤
+      (-(13483890514746875546159 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5193164395830821 / 20000000000000000 : ℝ) =
+            (5193164395830821 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5193164395830821 / 5000000000000000 : ℝ) ≤
+            (379053091253124453841 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (871700211176973 / 5000000000000000 : ℝ)) (q :=
+      (5193164395830821 / 20000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_562_14 :
+    Uω (aρ 14) (bρ 14) (10283899111043 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_562_15 :
+    Uω (aρ 15) (bρ 15) (10283899111043 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_562_16 :
+    Uω (aρ 16) (bρ 16) (10283899111043 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_562 :
+    Uρ (10283899111043 / 16000000000000 : ℝ) ≤
+      (-(3722356053754804569431 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (10283899111043 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(565123242699242190631 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_562_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (10283899111043 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(227931768568958541353 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_562_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (10283899111043 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(4634353854935592034951 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_562_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (10283899111043 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(74396329823126201829 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_562_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (10283899111043 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(4957861717286478519223 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_562_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (10283899111043 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(5246404410543003654287 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_562_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (10283899111043 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(282679195069658106583 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_562_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (10283899111043 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(621029439375479713257 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_562_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (10283899111043 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(6953273864243853381139 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_562_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (10283899111043 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(7929552090284195248573 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_562_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (10283899111043 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) * (-(92088367002658787047 / 100000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_562_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (10283899111043 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(5462123687277067907423 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_562_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (10283899111043 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(13483890514746875546159 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_562_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (10283899111043 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_562_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (10283899111043 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_562_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (10283899111043 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_562_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_563_1 :
+    Uω (aρ 1) (bρ 1) (20636590827023 / 32000000000000 : ℝ) ≤
+      (-(1121814209373083173601 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25537648372310497 / 40000000000000000 : ℝ) ≤
+      (-(1121814209373083173601 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (25537648372310497 / 40000000000000000 : ℝ) =
+            (25537648372310497 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25537648372310497 / 20000000000000000 : ℝ) ≤
+            (611053741376916826399 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6384386770795561 / 10000000000000000 : ℝ)) (q :=
+      (25537648372310497 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_563_2 :
+    Uω (aρ 2) (bρ 2) (20636590827023 / 32000000000000 : ℝ) ≤
+      (-(565597312504274044621 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25442006414350139 / 40000000000000000 : ℝ) ≤
+      (-(565597312504274044621 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (25442006414350139 / 40000000000000000 : ℝ) =
+            (25442006414350139 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25442006414350139 / 20000000000000000 : ℝ) ≤
+            (300836662870725955379 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3180167402162691 / 5000000000000000 : ℝ)) (q :=
+      (25442006414350139 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_563_3 :
+    Uω (aρ 3) (bρ 3) (20636590827023 / 32000000000000 : ℝ) ≤
+      (-(184009512553182304903 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25250745308928571 / 40000000000000000 : ℝ) ≤
+      (-(184009512553182304903 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (25250745308928571 / 40000000000000000 : ℝ) =
+            (25250745308928571 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25250745308928571 / 20000000000000000 : ℝ) ≤
+            (93249359566817695097 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3156050178982299 / 5000000000000000 : ℝ)) (q :=
+      (25250745308928571 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_563_4 :
+    Uω (aρ 4) (bρ 4) (20636590827023 / 32000000000000 : ℝ) ≤
+      (-(1181701895927001116443 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4986632254650801 / 8000000000000000 : ℝ) ≤
+      (-(1181701895927001116443 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (4986632254650801 / 8000000000000000 : ℝ) =
+            (4986632254650801 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4986632254650801 / 4000000000000000 : ℝ) ≤
+            (551166054822998883557 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1246320504238463 / 2000000000000000 : ℝ)) (q :=
+      (4986632254650801 / 8000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_563_5 :
+    Uω (aρ 5) (bρ 5) (20636590827023 / 32000000000000 : ℝ) ≤
+      (-(4922604025803230328569 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24449727025899473 / 40000000000000000 : ℝ) ≤
+      (-(4922604025803230328569 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (24449727025899473 / 40000000000000000 : ℝ) =
+            (24449727025899473 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24449727025899473 / 20000000000000000 : ℝ) ≤
+            (2008867777196769671431 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6108075316850049 / 10000000000000000 : ℝ)) (q :=
+      (24449727025899473 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_563_6 :
+    Uω (aρ 6) (bρ 6) (20636590827023 / 32000000000000 : ℝ) ≤
+      (-(2605039519011009743847 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23756865206856329 / 40000000000000000 : ℝ) ≤
+      (-(2605039519011009743847 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23756865206856329 / 40000000000000000 : ℝ) =
+            (23756865206856329 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23756865206856329 / 20000000000000000 : ℝ) ≤
+            (860696382488990256153 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5928884782468477 / 10000000000000000 : ℝ)) (q :=
+      (23756865206856329 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_563_7 :
+    Uω (aρ 7) (bρ 7) (20636590827023 / 32000000000000 : ℝ) ≤
+      (-(1123133245324348551683 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22812595809258933 / 40000000000000000 : ℝ) ≤
+      (-(1123133245324348551683 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (22812595809258933 / 40000000000000000 : ℝ) =
+            (22812595809258933 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22812595809258933 / 20000000000000000 : ℝ) ≤
+            (263161115275651448317 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5680411464429779 / 10000000000000000 : ℝ)) (q :=
+      (22812595809258933 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_563_8 :
+    Uω (aρ 8) (bρ 8) (20636590827023 / 32000000000000 : ℝ) ≤
+      (-(6170020726299281010733 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21582383025250187 / 40000000000000000 : ℝ) ≤
+      (-(6170020726299281010733 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (21582383025250187 / 40000000000000000 : ℝ) =
+            (21582383025250187 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21582383025250187 / 20000000000000000 : ℝ) ≤
+            (761451076700718989267 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2674423361882703 / 5000000000000000 : ℝ)) (q :=
+      (21582383025250187 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_563_9 :
+    Uω (aρ 9) (bρ 9) (20636590827023 / 32000000000000 : ℝ) ≤
+      (-(3454742864578543876601 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20044020521875581 / 40000000000000000 : ℝ) ≤
+      (-(3454742864578543876601 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (20044020521875581 / 40000000000000000 : ℝ) =
+            (20044020521875581 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 2
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20044020521875581 / 20000000000000000 : ℝ) ≤
+            (10993036921456123399 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4920631100438103 / 10000000000000000 : ℝ)) (q :=
+      (20044020521875581 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_563_10 :
+    Uω (aρ 10) (bρ 10) (20636590827023 / 32000000000000 : ℝ) ≤
+      (-(1576072435054941454207 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (18189476927450631 / 40000000000000000 : ℝ) ≤
+      (-(1576072435054941454207 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (18189476927450631 / 40000000000000000 : ℝ) =
+            (18189476927450631 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18189476927450631 / 10000000000000000 : ℝ) ≤
+            (1196516286145058545793 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1095479584180157 / 2500000000000000 : ℝ)) (q :=
+      (18189476927450631 / 40000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_563_11 :
+    Uω (aρ 11) (bρ 11) (20636590827023 / 32000000000000 : ℝ) ≤
+      (-(9150780638360100323413 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16019414449549067 / 40000000000000000 : ℝ) ≤
+      (-(9150780638360100323413 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (16019414449549067 / 40000000000000000 : ℝ) =
+            (16019414449549067 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16019414449549067 / 10000000000000000 : ℝ) ≤
+            (4712162967639899676587 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1857681861987423 / 5000000000000000 : ℝ)) (q :=
+      (16019414449549067 / 40000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_563_12 :
+    Uω (aρ 12) (bρ 12) (20636590827023 / 32000000000000 : ℝ) ≤
+      (-(542471460987771583411 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13516842921434643 / 40000000000000000 : ℝ) ≤
+      (-(542471460987771583411 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (13516842921434643 / 40000000000000000 : ℝ) =
+            (13516842921434643 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13516842921434643 / 10000000000000000 : ℝ) ≤
+            (150675719312228416589 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1443889028301317 / 5000000000000000 : ℝ)) (q :=
+      (13516842921434643 / 40000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_563_13 :
+    Uω (aρ 13) (bρ 13) (20636590827023 / 32000000000000 : ℝ) ≤
+      (-(2672410650574187809643 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2102729401588053 / 8000000000000000 : ℝ) ≤
+      (-(2672410650574187809643 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2102729401588053 / 8000000000000000 : ℝ) =
+            (2102729401588053 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2102729401588053 / 2000000000000000 : ℝ) ≤
+            (100178070625812190357 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (357112368290089 / 2000000000000000 : ℝ)) (q :=
+      (2102729401588053 / 8000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_563_14 :
+    Uω (aρ 14) (bρ 14) (20636590827023 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_563_15 :
+    Uω (aρ 15) (bρ 15) (20636590827023 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_563_16 :
+    Uω (aρ 16) (bρ 16) (20636590827023 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_563 :
+    Uρ (20636590827023 / 32000000000000 : ℝ) ≤
+      (-(740272258788343837463 / 1000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (20636590827023 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1121814209373083173601 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_563_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (20636590827023 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(565597312504274044621 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_563_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (20636590827023 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(184009512553182304903 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_563_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (20636590827023 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1181701895927001116443 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_563_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (20636590827023 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(4922604025803230328569 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_563_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (20636590827023 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(2605039519011009743847 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_563_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (20636590827023 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(1123133245324348551683 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_563_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (20636590827023 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(6170020726299281010733 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_563_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (20636590827023 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(3454742864578543876601 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_563_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (20636590827023 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(1576072435054941454207 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_563_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (20636590827023 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(9150780638360100323413 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_563_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (20636590827023 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(542471460987771583411 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_563_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (20636590827023 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(2672410650574187809643 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_563_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (20636590827023 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_563_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (20636590827023 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_563_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (20636590827023 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_563_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_564_1 :
+    Uω (aρ 1) (bρ 1) (517634585799 / 800000000000 : ℝ) ≤
+      (-(4453641117210170942821 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6405909867100199 / 10000000000000000 : ℝ) ≤
+      (-(4453641117210170942821 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (6405909867100199 / 10000000000000000 : ℝ) =
+            (6405909867100199 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6405909867100199 / 5000000000000000 : ℝ) ≤
+            (2477830685789829057179 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3202942314898949 / 5000000000000000 : ℝ)) (q :=
+      (6405909867100199 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_564_2 :
+    Uω (aρ 2) (bρ 2) (517634585799 / 800000000000 : ℝ) ≤
+      (-(4491035876755540387069 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3190999927253161 / 5000000000000000 : ℝ) ≤
+      (-(4491035876755540387069 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (3190999927253161 / 5000000000000000 : ℝ) =
+            (3190999927253161 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3190999927253161 / 2500000000000000 : ℝ) ≤
+            (2440435926244459612931 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (398864601070009 / 625000000000000 : ℝ)) (q :=
+      (3190999927253161 / 5000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_564_3 :
+    Uω (aρ 3) (bρ 3) (517634585799 / 800000000000 : ℝ) ≤
+      (-(4566237788996584143217 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (395886625324303 / 625000000000000 : ℝ) ≤
+      (-(4566237788996584143217 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (395886625324303 / 625000000000000 : ℝ) =
+            (395886625324303 / 312500000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (395886625324303 / 312500000000000 : ℝ) ≤
+            (2365234014003415856783 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1583400506208799 / 2500000000000000 : ℝ)) (q :=
+      (395886625324303 / 625000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_564_4 :
+    Uω (aρ 4) (bρ 4) (517634585799 / 800000000000 : ℝ) ≤
+      (-(58654614175999811513 / 125000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6254793809870713 / 10000000000000000 : ℝ) ≤
+      (-(58654614175999811513 / 125000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (6254793809870713 / 10000000000000000 : ℝ) =
+            (6254793809870713 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6254793809870713 / 5000000000000000 : ℝ) ≤
+            (27988783361500188487 / 125000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3126555907631963 / 5000000000000000 : ℝ)) (q :=
+      (6254793809870713 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_564_5 :
+    Uω (aρ 5) (bρ 5) (517634585799 / 800000000000 : ℝ) ≤
+      (-(977494076851752214137 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2995090197471 / 4882812500000 : ℝ) ≤
+      (-(977494076851752214137 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (2995090197471 / 4882812500000 : ℝ) = (2995090197471 / 2441406250000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2995090197471 / 2441406250000 : ℝ) ≤
+            (408800283748247785863 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1532400890924679 / 2500000000000000 : ℝ)) (q :=
+      (2995090197471 / 4882812500000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_564_6 :
+    Uω (aρ 6) (bρ 6) (517634585799 / 800000000000 : ℝ) ≤
+      (-(5173885597877607211283 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11921502632982307 / 20000000000000000 : ℝ) ≤
+      (-(5173885597877607211283 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11921502632982307 / 20000000000000000 : ℝ) =
+            (11921502632982307 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11921502632982307 / 10000000000000000 : ℝ) ≤
+            (1757586205122392788717 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5950457122979807 / 10000000000000000 : ℝ)) (q :=
+      (11921502632982307 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_564_7 :
+    Uω (aρ 7) (bρ 7) (517634585799 / 800000000000 : ℝ) ≤
+      (-(1394473231557664473619 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (715591545793229 / 1250000000000000 : ℝ) ≤
+      (-(1394473231557664473619 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (715591545793229 / 1250000000000000 : ℝ) =
+            (715591545793229 / 625000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (715591545793229 / 625000000000000 : ℝ) ≤
+            (338394719192335526381 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1425520150862291 / 2500000000000000 : ℝ)) (q :=
+      (715591545793229 / 1250000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_564_8 :
+    Uω (aρ 8) (bρ 8) (517634585799 / 800000000000 : ℝ) ≤
+      (-(766238927205555868307 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5417280576956103 / 10000000000000000 : ℝ) ≤
+      (-(766238927205555868307 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5417280576956103 / 10000000000000000 : ℝ) =
+            (5417280576956103 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5417280576956103 / 5000000000000000 : ℝ) ≤
+            (100195048169444131693 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2685359338004853 / 5000000000000000 : ℝ)) (q :=
+      (5417280576956103 / 10000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_564_9 :
+    Uω (aρ 9) (bρ 9) (517634585799 / 800000000000 : ℝ) ≤
+      (-(6865895476826353308691 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (503289590483219 / 1000000000000000 : ℝ) ≤
+      (-(6865895476826353308691 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (503289590483219 / 1000000000000000 : ℝ) =
+            (503289590483219 / 500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (503289590483219 / 500000000000000 : ℝ) ≤
+            (65576326173646691309 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (123572874003047 / 250000000000000 : ℝ)) (q :=
+      (503289590483219 / 1000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_564_10 :
+    Uω (aρ 10) (bρ 10) (517634585799 / 800000000000 : ℝ) ≤
+      (-(7831431128554380707361 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9139349024056447 / 20000000000000000 : ℝ) ≤
+      (-(7831431128554380707361 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9139349024056447 / 20000000000000000 : ℝ) =
+            (9139349024056447 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9139349024056447 / 5000000000000000 : ℝ) ≤
+            (6031512477445619292639 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4405031208008947 / 10000000000000000 : ℝ)) (q :=
+      (9139349024056447 / 20000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_564_11 :
+    Uω (aρ 11) (bρ 11) (517634585799 / 800000000000 : ℝ) ≤
+      (-(9093111557506252195301 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8056031916865501 / 20000000000000000 : ℝ) ≤
+      (-(9093111557506252195301 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8056031916865501 / 20000000000000000 : ℝ) =
+            (8056031916865501 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8056031916865501 / 5000000000000000 : ℝ) ≤
+            (4769832048493747804699 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3740190727023001 / 10000000000000000 : ℝ)) (q :=
+      (8056031916865501 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_564_12 :
+    Uω (aρ 12) (bρ 12) (517634585799 / 800000000000 : ℝ) ≤
+      (-(5387676966162376967213 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (136173410858717 / 400000000000000 : ℝ) ≤
+      (-(5387676966162376967213 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (136173410858717 / 400000000000000 : ℝ) =
+            (136173410858717 / 100000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (136173410858717 / 100000000000000 : ℝ) ≤
+            (1543794836837623032787 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (58330588995567 / 200000000000000 : ℝ)) (q :=
+      (136173410858717 / 400000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12]) (by norm_num [bρ_12])
+      (by norm_num) (by norm_num [aρ_12, bρ_12]) (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_564_13 :
+    Uω (aρ 13) (bρ 13) (517634585799 / 800000000000 : ℝ) ≤
+      (-(3310758912492028501331 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5319762758207451 / 20000000000000000 : ℝ) ≤
+      (-(3310758912492028501331 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5319762758207451 / 20000000000000000 : ℝ) =
+            (5319762758207451 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5319762758207451 / 5000000000000000 : ℝ) ≤
+            (154976989007971498669 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1827003406644951 / 10000000000000000 : ℝ)) (q :=
+      (5319762758207451 / 20000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_564_14 :
+    Uω (aρ 14) (bρ 14) (517634585799 / 800000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_564_15 :
+    Uω (aρ 15) (bρ 15) (517634585799 / 800000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_564_16 :
+    Uω (aρ 16) (bρ 16) (517634585799 / 800000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_564 :
+    Uρ (517634585799 / 800000000000 : ℝ) ≤
+      (-(3680541795298583111983 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (517634585799 / 800000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(4453641117210170942821 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_564_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (517634585799 / 800000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(4491035876755540387069 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_564_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (517634585799 / 800000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(4566237788996584143217 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_564_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (517634585799 / 800000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(58654614175999811513 / 125000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_564_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (517634585799 / 800000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(977494076851752214137 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_564_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (517634585799 / 800000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(5173885597877607211283 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_564_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (517634585799 / 800000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(1394473231557664473619 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_564_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (517634585799 / 800000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(766238927205555868307 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_564_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (517634585799 / 800000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(6865895476826353308691 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_564_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (517634585799 / 800000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(7831431128554380707361 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_564_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (517634585799 / 800000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(9093111557506252195301 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_564_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (517634585799 / 800000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(5387676966162376967213 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_564_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (517634585799 / 800000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(3310758912492028501331 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_564_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (517634585799 / 800000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_564_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (517634585799 / 800000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_564_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (517634585799 / 800000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_564_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_565_1 :
+    Uω (aρ 1) (bρ 1) (20774176036897 / 32000000000000 : ℝ) ≤
+      (-(11050345052480570479 / 25000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25709630562217229 / 40000000000000000 : ℝ) ≤
+      (-(11050345052480570479 / 25000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (25709630562217229 / 40000000000000000 : ℝ) =
+            (25709630562217229 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25709630562217229 / 20000000000000000 : ℝ) ≤
+            (6278334455019429521 / 25000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3213691243831651 / 5000000000000000 : ℝ)) (q :=
+      (25709630562217229 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_565_2 :
+    Uω (aρ 2) (bρ 2) (20774176036897 / 32000000000000 : ℝ) ≤
+      (-(4457406733048630161033 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25613992406609107 / 40000000000000000 : ℝ) ≤
+      (-(4457406733048630161033 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (25613992406609107 / 40000000000000000 : ℝ) =
+            (25613992406609107 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25613992406609107 / 20000000000000000 : ℝ) ≤
+            (2474065069951369838967 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6403332422369241 / 10000000000000000 : ℝ)) (q :=
+      (25613992406609107 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_565_3 :
+    Uω (aρ 3) (bρ 3) (20774176036897 / 32000000000000 : ℝ) ≤
+      (-(4532352993907169877393 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25422742678753341 / 40000000000000000 : ℝ) ≤
+      (-(4532352993907169877393 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (25422742678753341 / 40000000000000000 : ℝ) =
+            (25422742678753341 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25422742678753341 / 20000000000000000 : ℝ) ≤
+            (2399118809092830122607 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3177551832395679 / 5000000000000000 : ℝ)) (q :=
+      (25422742678753341 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_565_4 :
+    Uω (aρ 4) (bρ 4) (20774176036897 / 32000000000000 : ℝ) ≤
+      (-(232902447078791866397 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25105189046627263 / 40000000000000000 : ℝ) ≤
+      (-(232902447078791866397 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (25105189046627263 / 40000000000000000 : ℝ) =
+            (25105189046627263 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25105189046627263 / 20000000000000000 : ℝ) ≤
+            (113671143071208133603 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6274621029793319 / 10000000000000000 : ℝ)) (q :=
+      (25105189046627263 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_565_5 :
+    Uω (aρ 5) (bρ 5) (20774176036897 / 32000000000000 : ℝ) ≤
+      (-(4852459921577938326357 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24621830341973913 / 40000000000000000 : ℝ) ≤
+      (-(4852459921577938326357 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (24621830341973913 / 40000000000000000 : ℝ) =
+            (24621830341973913 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24621830341973913 / 20000000000000000 : ℝ) ≤
+            (2079011881422061673643 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1537782899200411 / 2500000000000000 : ℝ)) (q :=
+      (24621830341973913 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_565_6 :
+    Uω (aρ 6) (bρ 6) (20774176036897 / 32000000000000 : ℝ) ≤
+      (-(64222789149440470487 / 125000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4785828849661249 / 8000000000000000 : ℝ) ≤
+      (-(64222789149440470487 / 125000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (4785828849661249 / 8000000000000000 : ℝ) =
+            (4785828849661249 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4785828849661249 / 4000000000000000 : ℝ) ≤
+            (22420608388059529513 / 125000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (597202892510781 / 1000000000000000 : ℝ)) (q :=
+      (4785828849661249 / 8000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_565_7 :
+    Uω (aρ 7) (bρ 7) (20774176036897 / 32000000000000 : ℝ) ≤
+      (-(5540262896390538563607 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22985260535457379 / 40000000000000000 : ℝ) ≤
+      (-(5540262896390538563607 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (22985260535457379 / 40000000000000000 : ℝ) =
+            (22985260535457379 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22985260535457379 / 20000000000000000 : ℝ) ≤
+            (1391208906609461436393 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5723748449443377 / 10000000000000000 : ℝ)) (q :=
+      (22985260535457379 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_565_8 :
+    Uω (aρ 8) (bρ 8) (20774176036897 / 32000000000000 : ℝ) ≤
+      (-(6089965109248601034731 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21755855570410679 / 40000000000000000 : ℝ) ≤
+      (-(6089965109248601034731 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (21755855570410679 / 40000000000000000 : ℝ) =
+            (21755855570410679 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21755855570410679 / 20000000000000000 : ℝ) ≤
+            (841506693751398965269 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5392587618260027 / 10000000000000000 : ℝ)) (q :=
+      (21755855570410679 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_565_9 :
+    Uω (aρ 9) (bρ 9) (20774176036897 / 32000000000000 : ℝ) ≤
+      (-(213203164631554066227 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20219132852365427 / 40000000000000000 : ℝ) ≤
+      (-(213203164631554066227 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (20219132852365427 / 40000000000000000 : ℝ) =
+            (20219132852365427 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20219132852365427 / 20000000000000000 : ℝ) ≤
+            (3405329212195933773 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4965191887597401 / 10000000000000000 : ℝ)) (q :=
+      (20219132852365427 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_565_10 :
+    Uω (aρ 10) (bρ 10) (20774176036897 / 32000000000000 : ℝ) ≤
+      (-(3891378029483553701593 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (18367886624975269 / 40000000000000000 : ℝ) ≤
+      (-(3891378029483553701593 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (18367886624975269 / 40000000000000000 : ℝ) =
+            (18367886624975269 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18367886624975269 / 10000000000000000 : ℝ) ≤
+            (3040093773516446298407 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2214063903698661 / 5000000000000000 : ℝ)) (q :=
+      (18367886624975269 / 40000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_565_11 :
+    Uω (aρ 11) (bρ 11) (20774176036897 / 32000000000000 : ℝ) ≤
+      (-(9035823701220572399023 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16204631289860401 / 40000000000000000 : ℝ) ≤
+      (-(9035823701220572399023 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (16204631289860401 / 40000000000000000 : ℝ) =
+            (16204631289860401 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16204631289860401 / 10000000000000000 : ℝ) ≤
+            (4827119904779427600977 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (470622095755611 / 1250000000000000 : ℝ)) (q :=
+      (16204631289860401 / 40000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_565_12 :
+    Uω (aρ 12) (bρ 12) (20774176036897 / 32000000000000 : ℝ) ≤
+      (-(5351001880761519242997 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13717591735961083 / 40000000000000000 : ℝ) ≤
+      (-(5351001880761519242997 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (13717591735961083 / 40000000000000000 : ℝ) =
+            (13717591735961083 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13717591735961083 / 10000000000000000 : ℝ) ≤
+            (1580469922238480757003 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2945157085780229 / 10000000000000000 : ℝ)) (q :=
+      (13717591735961083 / 40000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_565_13 :
+    Uω (aρ 13) (bρ 13) (20774176036897 / 32000000000000 : ℝ) ≤
+      (-(13126666327036419575017 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10764060149677171 / 40000000000000000 : ℝ) ≤
+      (-(13126666327036419575017 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (10764060149677171 / 40000000000000000 : ℝ) =
+            (10764060149677171 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10764060149677171 / 10000000000000000 : ℝ) ≤
+            (736277278963580424983 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1867773034233273 / 10000000000000000 : ℝ)) (q :=
+      (10764060149677171 / 40000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_565_14 :
+    Uω (aρ 14) (bρ 14) (20774176036897 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_565_15 :
+    Uω (aρ 15) (bρ 15) (20774176036897 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_565_16 :
+    Uω (aρ 16) (bρ 16) (20774176036897 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_565 :
+    Uρ (20774176036897 / 32000000000000 : ℝ) ≤
+      (-(457486403617067305977 / 625000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (20774176036897 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(11050345052480570479 / 25000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_565_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (20774176036897 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(4457406733048630161033 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_565_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (20774176036897 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(4532352993907169877393 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_565_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (20774176036897 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(232902447078791866397 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_565_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (20774176036897 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(4852459921577938326357 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_565_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (20774176036897 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) * (-(64222789149440470487 / 125000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_565_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (20774176036897 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(5540262896390538563607 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_565_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (20774176036897 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(6089965109248601034731 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_565_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (20774176036897 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(213203164631554066227 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_565_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (20774176036897 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(3891378029483553701593 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_565_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (20774176036897 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(9035823701220572399023 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_565_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (20774176036897 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(5351001880761519242997 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_565_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (20774176036897 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(13126666327036419575017 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_565_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (20774176036897 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_565_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (20774176036897 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_565_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (20774176036897 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_565_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_566_1 :
+    Uω (aρ 1) (bρ 1) (10421484320917 / 16000000000000 : ℝ) ≤
+      (-(438674679669422764263 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12897810826891269 / 20000000000000000 : ℝ) ≤
+      (-(438674679669422764263 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (12897810826891269 / 20000000000000000 : ℝ) =
+            (12897810826891269 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12897810826891269 / 10000000000000000 : ℝ) ≤
+            (254472500630577235737 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (806110043050393 / 1250000000000000 : ℝ)) (q :=
+      (12897810826891269 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_566_2 :
+    Uω (aρ 2) (bρ 2) (10421484320917 / 16000000000000 : ℝ) ≤
+      (-(552986288518407495121 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12849992690126547 / 20000000000000000 : ℝ) ≤
+      (-(552986288518407495121 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (12849992690126547 / 20000000000000000 : ℝ) =
+            (12849992690126547 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12849992690126547 / 10000000000000000 : ℝ) ≤
+            (313447686856592504879 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3212415610074211 / 5000000000000000 : ℝ)) (q :=
+      (12849992690126547 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_566_3 :
+    Uω (aρ 3) (bρ 3) (10421484320917 / 16000000000000 : ℝ) ≤
+      (-(1124645662501160291839 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12754370641733471 / 20000000000000000 : ℝ) ≤
+      (-(1124645662501160291839 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (12754370641733471 / 20000000000000000 : ℝ) =
+            (12754370641733471 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12754370641733471 / 10000000000000000 : ℝ) ≤
+            (608222288248839708161 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3188302639052673 / 5000000000000000 : ℝ)) (q :=
+      (12754370641733471 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_566_4 :
+    Uω (aρ 4) (bρ 4) (10421484320917 / 16000000000000 : ℝ) ≤
+      (-(4623846196384515435809 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12595601348158829 / 20000000000000000 : ℝ) ≤
+      (-(4623846196384515435809 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (12595601348158829 / 20000000000000000 : ℝ) =
+            (12595601348158829 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12595601348158829 / 10000000000000000 : ℝ) ≤
+            (2307625606615484564191 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (787016270699463 / 1250000000000000 : ℝ)) (q :=
+      (12595601348158829 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_566_5 :
+    Uω (aρ 5) (bρ 5) (10421484320917 / 16000000000000 : ℝ) ≤
+      (-(963514355168254296311 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12353940681623363 / 20000000000000000 : ℝ) ≤
+      (-(963514355168254296311 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (12353940681623363 / 20000000000000000 : ℝ) =
+            (12353940681623363 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12353940681623363 / 10000000000000000 : ℝ) ≤
+            (422780005431745703689 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3086329709197619 / 5000000000000000 : ℝ)) (q :=
+      (12353940681623363 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_566_6 :
+    Uω (aρ 6) (bρ 6) (10421484320917 / 16000000000000 : ℝ) ≤
+      (-(5101890692535526692447 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6003820541376879 / 10000000000000000 : ℝ) ≤
+      (-(5101890692535526692447 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (6003820541376879 / 10000000000000000 : ℝ) =
+            (6003820541376879 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6003820541376879 / 5000000000000000 : ℝ) ≤
+            (1829581110464473307553 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5993600194665633 / 10000000000000000 : ℝ)) (q :=
+      (6003820541376879 / 10000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_566_7 :
+    Uω (aρ 7) (bρ 7) (10421484320917 / 16000000000000 : ℝ) ≤
+      (-(1375693761494647139359 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11535794524369811 / 20000000000000000 : ℝ) ≤
+      (-(1375693761494647139359 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11535794524369811 / 20000000000000000 : ℝ) =
+            (11535794524369811 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11535794524369811 / 10000000000000000 : ℝ) ≤
+            (357174189255352860641 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2872707508520843 / 5000000000000000 : ℝ)) (q :=
+      (11535794524369811 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_566_8 :
+    Uω (aρ 8) (bρ 8) (10421484320917 / 16000000000000 : ℝ) ≤
+      (-(3025090229805417688551 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1092129144297653 / 2000000000000000 : ℝ) ≤
+      (-(3025090229805417688551 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1092129144297653 / 2000000000000000 : ℝ) =
+            (1092129144297653 / 1000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1092129144297653 / 1000000000000000 : ℝ) ≤
+            (440645671694582311449 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1082890717397681 / 2000000000000000 : ℝ)) (q :=
+      (1092129144297653 / 2000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_566_9 :
+    Uω (aρ 9) (bρ 9) (10421484320917 / 16000000000000 : ℝ) ≤
+      (-(1355860257608270240147 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10153334203382703 / 20000000000000000 : ℝ) ≤
+      (-(1355860257608270240147 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10153334203382703 / 20000000000000000 : ℝ) =
+            (10153334203382703 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10153334203382703 / 10000000000000000 : ℝ) ≤
+            (30434102991729759853 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2493730987877289 / 5000000000000000 : ℝ)) (q :=
+      (10153334203382703 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_566_10 :
+    Uω (aρ 10) (bρ 10) (10421484320917 / 16000000000000 : ℝ) ≤
+      (-(1933583531531276978283 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4614260791157823 / 10000000000000000 : ℝ) ≤
+      (-(1933583531531276978283 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4614260791157823 / 10000000000000000 : ℝ) =
+            (4614260791157823 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4614260791157823 / 2500000000000000 : ℝ) ≤
+            (1532152369968723021717 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4451208388182521 / 10000000000000000 : ℝ)) (q :=
+      (4614260791157823 / 10000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_566_11 :
+    Uω (aρ 11) (bρ 11) (10421484320917 / 16000000000000 : ℝ) ≤
+      (-(140295491425674205323 / 156250000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (407427960636177 / 1000000000000000 : ℝ) ≤
+      (-(140295491425674205323 / 156250000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (407427960636177 / 1000000000000000 : ℝ) =
+            (407427960636177 / 250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (407427960636177 / 250000000000000 : ℝ) ≤
+            (76313002418075794677 / 156250000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (757944528959083 / 2000000000000000 : ℝ)) (q :=
+      (407427960636177 / 1000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11]) (by norm_num [bρ_11])
+      (by norm_num) (by norm_num [aρ_11, bρ_11]) (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_566_12 :
+    Uω (aρ 12) (bρ 12) (10421484320917 / 16000000000000 : ℝ) ≤
+      (-(10629361654901392404073 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (431800063131587 / 1250000000000000 : ℝ) ≤
+      (-(10629361654901392404073 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (431800063131587 / 1250000000000000 : ℝ) =
+            (431800063131587 / 312500000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (431800063131587 / 312500000000000 : ℝ) ≤
+            (3233581951098607595927 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2973664538862267 / 10000000000000000 : ℝ)) (q :=
+      (431800063131587 / 1250000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12]) (by norm_num [bρ_12])
+      (by norm_num) (by norm_num [aρ_12, bρ_12]) (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_566_13 :
+    Uω (aρ 13) (bρ 13) (10421484320917 / 16000000000000 : ℝ) ≤
+      (-(13012790773802567949037 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5443668529638587 / 20000000000000000 : ℝ) ≤
+      (-(13012790773802567949037 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5443668529638587 / 20000000000000000 : ℝ) =
+            (5443668529638587 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5443668529638587 / 5000000000000000 : ℝ) ≤
+            (850152832197432050963 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (953956899995231 / 5000000000000000 : ℝ)) (q :=
+      (5443668529638587 / 20000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_566_14 :
+    Uω (aρ 14) (bρ 14) (10421484320917 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_566_15 :
+    Uω (aρ 15) (bρ 15) (10421484320917 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_566_16 :
+    Uω (aρ 16) (bρ 16) (10421484320917 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_566 :
+    Uρ (10421484320917 / 16000000000000 : ℝ) ≤
+      (-(3639403798061565336769 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (10421484320917 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(438674679669422764263 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_566_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (10421484320917 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(552986288518407495121 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_566_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (10421484320917 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1124645662501160291839 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_566_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (10421484320917 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(4623846196384515435809 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_566_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (10421484320917 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(963514355168254296311 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_566_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (10421484320917 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(5101890692535526692447 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_566_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (10421484320917 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(1375693761494647139359 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_566_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (10421484320917 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(3025090229805417688551 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_566_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (10421484320917 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(1355860257608270240147 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_566_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (10421484320917 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(1933583531531276978283 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_566_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (10421484320917 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) * (-(140295491425674205323 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_566_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (10421484320917 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(10629361654901392404073 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_566_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (10421484320917 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(13012790773802567949037 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_566_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (10421484320917 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_566_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (10421484320917 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_566_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (10421484320917 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_566_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_567_1 :
+    Uω (aρ 1) (bρ 1) (20911761246771 / 32000000000000 : ℝ) ≤
+      (-(4353466699681521755969 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25881612743119157 / 40000000000000000 : ℝ) ≤
+      (-(4353466699681521755969 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (25881612743119157 / 40000000000000000 : ℝ) =
+            (25881612743119157 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25881612743119157 / 20000000000000000 : ℝ) ≤
+            (2578005103318478244031 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6470378200028641 / 10000000000000000 : ℝ)) (q :=
+      (25881612743119157 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_567_2 :
+    Uω (aρ 2) (bρ 2) (20911761246771 / 32000000000000 : ℝ) ≤
+      (-(4390485848910326832271 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25785978339106719 / 40000000000000000 : ℝ) ≤
+      (-(4390485848910326832271 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (25785978339106719 / 40000000000000000 : ℝ) =
+            (25785978339106719 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25785978339106719 / 20000000000000000 : ℝ) ≤
+            (2540985954089673167729 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3223165005266211 / 5000000000000000 : ℝ)) (q :=
+      (25785978339106719 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_567_3 :
+    Uω (aρ 3) (bρ 3) (20911761246771 / 32000000000000 : ℝ) ≤
+      (-(558115748324716916001 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25594739835433403 / 40000000000000000 : ℝ) ≤
+      (-(558115748324716916001 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (25594739835433403 / 40000000000000000 : ℝ) =
+            (25594739835433403 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25594739835433403 / 20000000000000000 : ℝ) ≤
+            (308318227050283083999 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1599526716261441 / 2500000000000000 : ℝ)) (q :=
+      (25594739835433403 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_567_4 :
+    Uω (aρ 4) (bρ 4) (20911761246771 / 32000000000000 : ℝ) ≤
+      (-(1147440024247301742847 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25277216190162251 / 40000000000000000 : ℝ) ≤
+      (-(1147440024247301742847 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (25277216190162251 / 40000000000000000 : ℝ) =
+            (25277216190162251 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25277216190162251 / 20000000000000000 : ℝ) ≤
+            (585427926502698257153 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1579409805868797 / 2500000000000000 : ℝ)) (q :=
+      (25277216190162251 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_567_5 :
+    Uω (aρ 5) (bρ 5) (20911761246771 / 32000000000000 : ℝ) ≤
+      (-(239140254707843448427 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24793931965911463 / 40000000000000000 : ℝ) ≤
+      (-(239140254707843448427 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (24793931965911463 / 40000000000000000 : ℝ) =
+            (24793931965911463 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24793931965911463 / 20000000000000000 : ℝ) ≤
+            (107433335442156551573 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3097093515342397 / 5000000000000000 : ℝ)) (q :=
+      (24793931965911463 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_567_6 :
+    Uω (aρ 6) (bρ 6) (20911761246771 / 32000000000000 : ℝ) ≤
+      (-(5066087342182747444541 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24101419029027671 / 40000000000000000 : ℝ) ≤
+      (-(5066087342182747444541 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (24101419029027671 / 40000000000000000 : ℝ) =
+            (24101419029027671 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24101419029027671 / 20000000000000000 : ℝ) ≤
+            (1865384460817252555459 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3007585468691449 / 5000000000000000 : ℝ)) (q :=
+      (24101419029027671 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_567_7 :
+    Uω (aρ 7) (bρ 7) (20911761246771 / 32000000000000 : ℝ) ≤
+      (-(2732714148191447201139 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23157915034047183 / 40000000000000000 : ℝ) ≤
+      (-(2732714148191447201139 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23157915034047183 / 40000000000000000 : ℝ) =
+            (23157915034047183 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23157915034047183 / 20000000000000000 : ℝ) ≤
+            (733021753308552798861 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2883540160326327 / 5000000000000000 : ℝ)) (q :=
+      (23157915034047183 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_567_8 :
+    Uω (aρ 8) (bρ 8) (20911761246771 / 32000000000000 : ℝ) ≤
+      (-(6010556143983057996553 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21929304326212211 / 40000000000000000 : ℝ) ≤
+      (-(6010556143983057996553 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (21929304326212211 / 40000000000000000 : ℝ) =
+            (21929304326212211 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21929304326212211 / 20000000000000000 : ℝ) ≤
+            (920915659016942003447 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (169884894314849 / 312500000000000 : ℝ)) (q :=
+      (21929304326212211 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_567_9 :
+    Uω (aρ 9) (bρ 9) (20911761246771 / 32000000000000 : ℝ) ≤
+      (-(168407343672909906301 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20394190464948549 / 40000000000000000 : ℝ) ≤
+      (-(168407343672909906301 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (20394190464948549 / 40000000000000000 : ℝ) =
+            (20394190464948549 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20394190464948549 / 20000000000000000 : ℝ) ≤
+            (4879451402090093699 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5009725315803337 / 10000000000000000 : ℝ)) (q :=
+      (20394190464948549 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_567_10 :
+    Uω (aρ 10) (bρ 10) (20911761246771 / 32000000000000 : ℝ) ≤
+      (-(1921540634794681486427 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (18546168162873673 / 40000000000000000 : ℝ) ≤
+      (-(1921540634794681486427 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (18546168162873673 / 40000000000000000 : ℝ) =
+            (18546168162873673 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18546168162873673 / 10000000000000000 : ℝ) ≤
+            (1544195266705318513573 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4474273198260899 / 10000000000000000 : ℝ)) (q :=
+      (18546168162873673 / 40000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_567_11 :
+    Uω (aρ 11) (bρ 11) (20911761246771 / 32000000000000 : ℝ) ≤
+      (-(892236932291373259771 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3277905360743043 / 8000000000000000 : ℝ) ≤
+      (-(892236932291373259771 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3277905360743043 / 8000000000000000 : ℝ) =
+            (3277905360743043 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3277905360743043 / 2000000000000000 : ℝ) ≤
+            (494057428308626740229 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (381442914488667 / 1000000000000000 : ℝ)) (q :=
+      (3277905360743043 / 8000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_567_12 :
+    Uω (aρ 12) (bρ 12) (20911761246771 / 32000000000000 : ℝ) ≤
+      (-(10557411220120595044881 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13917378786151519 / 40000000000000000 : ℝ) ≤
+      (-(10557411220120595044881 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (13917378786151519 / 40000000000000000 : ℝ) =
+            (13917378786151519 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13917378786151519 / 10000000000000000 : ℝ) ≤
+            (3305532385879404955119 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1501027616394911 / 5000000000000000 : ℝ)) (q :=
+      (13917378786151519 / 40000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_567_13 :
+    Uω (aρ 13) (bρ 13) (20911761246771 / 32000000000000 : ℝ) ≤
+      (-(3225317275922843369361 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11009434017866411 / 40000000000000000 : ℝ) ≤
+      (-(3225317275922843369361 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (11009434017866411 / 40000000000000000 : ℝ) =
+            (11009434017866411 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11009434017866411 / 10000000000000000 : ℝ) ≤
+            (240418625577156630639 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (486866147560567 / 2500000000000000 : ℝ)) (q :=
+      (11009434017866411 / 40000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_567_14 :
+    Uω (aρ 14) (bρ 14) (20911761246771 / 32000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_567_15 :
+    Uω (aρ 15) (bρ 15) (20911761246771 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_567_16 :
+    Uω (aρ 16) (bρ 16) (20911761246771 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_567 :
+    Uρ (20911761246771 / 32000000000000 : ℝ) ≤
+      (-(7238148340803391625791 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (20911761246771 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(4353466699681521755969 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_567_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (20911761246771 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(4390485848910326832271 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_567_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (20911761246771 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(558115748324716916001 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_567_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (20911761246771 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1147440024247301742847 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_567_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (20911761246771 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(239140254707843448427 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_567_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (20911761246771 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(5066087342182747444541 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_567_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (20911761246771 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(2732714148191447201139 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_567_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (20911761246771 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(6010556143983057996553 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_567_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (20911761246771 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(168407343672909906301 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_567_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (20911761246771 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(1921540634794681486427 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_567_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (20911761246771 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(892236932291373259771 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_567_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (20911761246771 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(10557411220120595044881 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_567_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (20911761246771 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(3225317275922843369361 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_567_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (20911761246771 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_567_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (20911761246771 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_567_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (20911761246771 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_567_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/U47.lean b/LeanPool/Zeta5Irrational/Table/U47.lean
new file mode 100644
index 0000000000..a027083ec8
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/U47.lean
@@ -0,0 +1,5003 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-! # Certified arcsine potential bounds (U47) -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma U_568_1 :
+    Uω (aρ 1) (bρ 1) (5245138462927 / 8000000000000 : ℝ) ≤
+      (-(4320296992729446895061 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2596760383024923 / 4000000000000000 : ℝ) ≤
+      (-(4320296992729446895061 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2596760383024923 / 4000000000000000 : ℝ) =
+            (2596760383024923 / 2000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2596760383024923 / 2000000000000000 : ℝ) ≤
+            (2611174810270553104939 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1298375210910173 / 2000000000000000 : ℝ)) (q :=
+      (2596760383024923 / 4000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_568_2 :
+    Uω (aρ 2) (bρ 2) (5245138462927 / 8000000000000 : ℝ) ≤
+      (-(2178596304860126658363 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12935985641658737 / 20000000000000000 : ℝ) ≤
+      (-(2178596304860126658363 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (12935985641658737 / 20000000000000000 : ℝ) =
+            (12935985641658737 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12935985641658737 / 10000000000000000 : ℝ) ≤
+            (1287139596639873341637 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6467828793594987 / 10000000000000000 : ℝ)) (q :=
+      (12935985641658737 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_568_3 :
+    Uω (aρ 3) (bρ 3) (5245138462927 / 8000000000000 : ℝ) ≤
+      (-(553922780094374103553 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2568073833518273 / 4000000000000000 : ℝ) ≤
+      (-(553922780094374103553 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2568073833518273 / 4000000000000000 : ℝ) =
+            (2568073833518273 / 2000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2568073833518273 / 2000000000000000 : ℝ) ≤
+            (312511195280625896447 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1283921685175523 / 2000000000000000 : ℝ)) (q :=
+      (2568073833518273 / 4000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_568_4 :
+    Uω (aρ 4) (bρ 4) (5245138462927 / 8000000000000 : ℝ) ≤
+      (-(36446318800435169751 / 80000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3170403691218427 / 5000000000000000 : ℝ) ≤
+      (-(36446318800435169751 / 80000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (3170403691218427 / 5000000000000000 : ℝ) =
+            (3170403691218427 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3170403691218427 / 2500000000000000 : ℝ) ≤
+            (19005455623564830249 / 80000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3169574102112479 / 5000000000000000 : ℝ)) (q :=
+      (3170403691218427 / 5000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_568_5 :
+    Uω (aρ 5) (bρ 5) (5245138462927 / 8000000000000 : ℝ) ≤
+      (-(1187039758133659573379 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12439991077158783 / 20000000000000000 : ℝ) ≤
+      (-(1187039758133659573379 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (12439991077158783 / 20000000000000000 : ℝ) =
+            (12439991077158783 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12439991077158783 / 10000000000000000 : ℝ) ≤
+            (545828192616340426621 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6215714435845033 / 10000000000000000 : ℝ)) (q :=
+      (12439991077158783 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_568_6 :
+    Uω (aρ 6) (bρ 6) (5245138462927 / 8000000000000 : ℝ) ≤
+      (-(5030412153596103091251 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12093777425080943 / 20000000000000000 : ℝ) ≤
+      (-(5030412153596103091251 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (12093777425080943 / 20000000000000000 : ℝ) =
+            (12093777425080943 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12093777425080943 / 10000000000000000 : ℝ) ≤
+            (1901059649403896908749 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6036741158907193 / 10000000000000000 : ℝ)) (q :=
+      (12093777425080943 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_568_7 :
+    Uω (aρ 7) (bρ 7) (5245138462927 / 8000000000000 : ℝ) ≤
+      (-(5428221581310503912809 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11622119259882059 / 20000000000000000 : ℝ) ≤
+      (-(5428221581310503912809 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11622119259882059 / 20000000000000000 : ℝ) =
+            (11622119259882059 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11622119259882059 / 10000000000000000 : ℝ) ≤
+            (1503250221689496087191 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5788744374468309 / 10000000000000000 : ℝ)) (q :=
+      (11622119259882059 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_568_8 :
+    Uω (aρ 8) (bρ 8) (5245138462927 / 8000000000000 : ℝ) ≤
+      (-(597109085408817176559 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5504004990447363 / 10000000000000000 : ℝ) ≤
+      (-(597109085408817176559 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5504004990447363 / 10000000000000000 : ℝ) =
+            (5504004990447363 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5504004990447363 / 5000000000000000 : ℝ) ≤
+            (96038094891182823441 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (341136046676311 / 625000000000000 : ℝ)) (q :=
+      (5504004990447363 / 10000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_568_9 :
+    Uω (aρ 9) (bρ 9) (5245138462927 / 8000000000000 : ℝ) ≤
+      (-(3346738440394695985477 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10240849603025773 / 20000000000000000 : ℝ) ≤
+      (-(3346738440394695985477 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10240849603025773 / 20000000000000000 : ℝ) =
+            (10240849603025773 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10240849603025773 / 10000000000000000 : ℝ) ≤
+            (118997461105304014523 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5031981997312023 / 10000000000000000 : ℝ)) (q :=
+      (10240849603025773 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_568_10 :
+    Uω (aρ 10) (bρ 10) (5245138462927 / 8000000000000 : ℝ) ≤
+      (-(7638238555624362940643 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9317631052494493 / 20000000000000000 : ℝ) ≤
+      (-(7638238555624362940643 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9317631052494493 / 20000000000000000 : ℝ) =
+            (9317631052494493 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9317631052494493 / 5000000000000000 : ℝ) ≤
+            (6224705050375637059357 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4497322480275743 / 10000000000000000 : ℝ)) (q :=
+      (9317631052494493 / 20000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_568_11 :
+    Uω (aρ 11) (bρ 11) (5245138462927 / 8000000000000 : ℝ) ≤
+      (-(2216547990186434725099 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2060232243149073 / 5000000000000000 : ℝ) ≤
+      (-(2216547990186434725099 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2060232243149073 / 5000000000000000 : ℝ) =
+            (2060232243149073 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2060232243149073 / 1250000000000000 : ℝ) ≤
+            (1249187911313565274901 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1919548513291271 / 5000000000000000 : ℝ)) (q :=
+      (2060232243149073 / 5000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_568_12 :
+    Uω (aρ 12) (bρ 12) (5245138462927 / 8000000000000 : ℝ) ≤
+      (-(2621534172356183131653 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1401692859717249 / 4000000000000000 : ℝ) ≤
+      (-(2621534172356183131653 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1401692859717249 / 4000000000000000 : ℝ) =
+            (1401692859717249 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1401692859717249 / 1000000000000000 : ℝ) ≤
+            (844201729143816868347 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (606066489851499 / 2000000000000000 : ℝ)) (q :=
+      (1401692859717249 / 4000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_568_13 :
+    Uω (aρ 13) (bρ 13) (5245138462927 / 8000000000000 : ℝ) ≤
+      (-(12791974179380312102249 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (695651344073957 / 2500000000000000 : ℝ) ≤
+      (-(12791974179380312102249 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (695651344073957 / 2500000000000000 : ℝ) =
+            (695651344073957 / 625000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (695651344073957 / 625000000000000 : ℝ) ≤
+            (1070969426619687897751 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (993230322428953 / 5000000000000000 : ℝ)) (q :=
+      (695651344073957 / 2500000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13]) (by norm_num [bρ_13])
+      (by norm_num) (by norm_num [aρ_13, bρ_13]) (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_568_14 :
+    Uω (aρ 14) (bρ 14) (5245138462927 / 8000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_568_15 :
+    Uω (aρ 15) (bρ 15) (5245138462927 / 8000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_568_16 :
+    Uω (aρ 16) (bρ 16) (5245138462927 / 8000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_568 :
+    Uρ (5245138462927 / 8000000000000 : ℝ) ≤
+      (-(7197794843052013057711 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (5245138462927 / 8000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(4320296992729446895061 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_568_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (5245138462927 / 8000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(2178596304860126658363 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_568_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (5245138462927 / 8000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(553922780094374103553 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_568_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (5245138462927 / 8000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(36446318800435169751 / 80000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_568_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (5245138462927 / 8000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1187039758133659573379 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_568_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (5245138462927 / 8000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(5030412153596103091251 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_568_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (5245138462927 / 8000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(5428221581310503912809 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_568_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (5245138462927 / 8000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(597109085408817176559 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_568_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (5245138462927 / 8000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(3346738440394695985477 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_568_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (5245138462927 / 8000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(7638238555624362940643 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_568_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (5245138462927 / 8000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(2216547990186434725099 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_568_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (5245138462927 / 8000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(2621534172356183131653 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_568_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (5245138462927 / 8000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(12791974179380312102249 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_568_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (5245138462927 / 8000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_568_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (5245138462927 / 8000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_568_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (5245138462927 / 8000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_568_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_569_1 :
+    Uω (aρ 1) (bρ 1) (10559069530791 / 16000000000000 : ℝ) ≤
+      (-(4254285836572129056119 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13069792998988423 / 20000000000000000 : ℝ) ≤
+      (-(4254285836572129056119 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (13069792998988423 / 20000000000000000 : ℝ) =
+            (13069792998988423 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13069792998988423 / 10000000000000000 : ℝ) ≤
+            (2677185966427870943881 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (816858970041131 / 1250000000000000 : ℝ)) (q :=
+      (13069792998988423 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_569_2 :
+    Uω (aρ 2) (bρ 2) (10559069530791 / 16000000000000 : ℝ) ≤
+      (-(4290936846030395396877 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1627747320524411 / 2500000000000000 : ℝ) ≤
+      (-(4290936846030395396877 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1627747320524411 / 2500000000000000 : ℝ) =
+            (1627747320524411 / 1250000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1627747320524411 / 1250000000000000 : ℝ) ≤
+            (2640534956969604603123 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6510826338045913 / 10000000000000000 : ℝ)) (q :=
+      (1627747320524411 / 2500000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_569_3 :
+    Uω (aρ 3) (bρ 3) (10559069530791 / 16000000000000 : ℝ) ≤
+      (-(4364630488213829430073 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12926367590057373 / 20000000000000000 : ℝ) ≤
+      (-(4364630488213829430073 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (12926367590057373 / 20000000000000000 : ℝ) =
+            (12926367590057373 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12926367590057373 / 10000000000000000 : ℝ) ≤
+            (2566841314786170569927 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3231305735128999 / 5000000000000000 : ℝ)) (q :=
+      (12926367590057373 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_569_4 :
+    Uω (aρ 4) (bρ 4) (10559069530791 / 16000000000000 : ℝ) ≤
+      (-(448819378046638877049 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12767627876189017 / 20000000000000000 : ℝ) ≤
+      (-(448819378046638877049 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (12767627876189017 / 20000000000000000 : ℝ) =
+            (12767627876189017 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12767627876189017 / 10000000000000000 : ℝ) ≤
+            (244327802253361122951 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3191082968727321 / 5000000000000000 : ℝ)) (q :=
+      (12767627876189017 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_569_5 :
+    Uω (aρ 5) (bρ 5) (10559069530791 / 16000000000000 : ℝ) ≤
+      (-(935845087457148488507 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3131510163181529 / 5000000000000000 : ℝ) ≤
+      (-(935845087457148488507 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (3131510163181529 / 5000000000000000 : ℝ) =
+            (3131510163181529 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3131510163181529 / 2500000000000000 : ℝ) ≤
+            (450449273142851511493 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6258768633326741 / 10000000000000000 : ℝ)) (q :=
+      (3131510163181529 / 5000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_569_6 :
+    Uω (aρ 6) (bρ 6) (10559069530791 / 16000000000000 : ℝ) ≤
+      (-(1239860650579759773883 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3044977926206299 / 5000000000000000 : ℝ) ≤
+      (-(1239860650579759773883 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (3044977926206299 / 5000000000000000 : ℝ) =
+            (3044977926206299 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3044977926206299 / 2500000000000000 : ℝ) ≤
+            (493007300170240226117 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6079880060565821 / 10000000000000000 : ℝ)) (q :=
+      (3044977926206299 / 5000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_569_7 :
+    Uω (aρ 7) (bρ 7) (10559069530791 / 16000000000000 : ℝ) ≤
+      (-(535422405003292823967 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1170843905192277 / 2000000000000000 : ℝ) ≤
+      (-(535422405003292823967 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1170843905192277 / 2000000000000000 : ℝ) =
+            (1170843905192277 / 1000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1170843905192277 / 1000000000000000 : ℝ) ≤
+            (157724775296707176033 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1166413757684679 / 2000000000000000 : ℝ)) (q :=
+      (1170843905192277 / 2000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_569_8 :
+    Uω (aρ 8) (bρ 8) (10559069530791 / 16000000000000 : ℝ) ≤
+      (-(589263219909437068171 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (221894340956473 / 400000000000000 : ℝ) ≤
+      (-(589263219909437068171 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (221894340956473 / 400000000000000 : ℝ) =
+            (221894340956473 / 200000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (221894340956473 / 200000000000000 : ℝ) ≤
+            (103883960390562931829 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (220075537426571 / 400000000000000 : ℝ)) (q :=
+      (221894340956473 / 400000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_569_9 :
+    Uω (aρ 9) (bρ 9) (10559069530791 / 16000000000000 : ℝ) ≤
+      (-(826051030152510713157 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10328338718797059 / 20000000000000000 : ℝ) ≤
+      (-(826051030152510713157 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10328338718797059 / 20000000000000000 : ℝ) =
+            (10328338718797059 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10328338718797059 / 10000000000000000 : ℝ) ≤
+            (40382945222489286843 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1269118933749421 / 2500000000000000 : ℝ)) (q :=
+      (10328338718797059 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_569_10 :
+    Uω (aρ 10) (bρ 10) (10559069530791 / 16000000000000 : ℝ) ≤
+      (-(7543122663510840268807 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4703339677788387 / 10000000000000000 : ℝ) ≤
+      (-(7543122663510840268807 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4703339677788387 / 10000000000000000 : ℝ) =
+            (4703339677788387 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4703339677788387 / 2500000000000000 : ℝ) ≤
+            (6319820942489159731193 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4543375405272399 / 10000000000000000 : ℝ)) (q :=
+      (4703339677788387 / 10000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_569_11 :
+    Uω (aρ 11) (bρ 11) (10559069530791 / 16000000000000 : ℝ) ≤
+      (-(1750982146713054436347 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8333147197724503 / 20000000000000000 : ℝ) ≤
+      (-(1750982146713054436347 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8333147197724503 / 20000000000000000 : ℝ) =
+            (8333147197724503 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8333147197724503 / 5000000000000000 : ℝ) ≤
+            (1021606574486945563653 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (486039984203141 / 1250000000000000 : ℝ)) (q :=
+      (8333147197724503 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_569_12 :
+    Uω (aρ 12) (bρ 12) (10559069530791 / 16000000000000 : ℝ) ≤
+      (-(5172777597921439708851 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1421537228018487 / 4000000000000000 : ℝ) ≤
+      (-(5172777597921439708851 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1421537228018487 / 4000000000000000 : ℝ) =
+            (1421537228018487 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1421537228018487 / 1000000000000000 : ℝ) ≤
+            (1758694205078560291149 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (154327945633903 / 500000000000000 : ℝ)) (q :=
+      (1421537228018487 / 4000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_569_13 :
+    Uω (aρ 13) (bρ 13) (10559069530791 / 16000000000000 : ℝ) ≤
+      (-(393112829726998233739 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (284232971293049 / 1000000000000000 : ℝ) ≤
+      (-(393112829726998233739 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (284232971293049 / 1000000000000000 : ℝ) =
+            (284232971293049 / 250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (284232971293049 / 250000000000000 : ℝ) ≤
+            (40104157960501766261 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (412582788008321 / 2000000000000000 : ℝ)) (q :=
+      (284232971293049 / 1000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13]) (by norm_num [bρ_13])
+      (by norm_num) (by norm_num [aρ_13, bρ_13]) (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_569_14 :
+    Uω (aρ 14) (bρ 14) (10559069530791 / 16000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_569_15 :
+    Uω (aρ 15) (bρ 15) (10559069530791 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_569_16 :
+    Uω (aρ 16) (bρ 16) (10559069530791 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_569 :
+    Uρ (10559069530791 / 16000000000000 : ℝ) ≤
+      (-(35589846219897266431 / 50000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (10559069530791 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(4254285836572129056119 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_569_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (10559069530791 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(4290936846030395396877 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_569_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (10559069530791 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(4364630488213829430073 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_569_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (10559069530791 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(448819378046638877049 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_569_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (10559069530791 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(935845087457148488507 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_569_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (10559069530791 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1239860650579759773883 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_569_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (10559069530791 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(535422405003292823967 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_569_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (10559069530791 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(589263219909437068171 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_569_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (10559069530791 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(826051030152510713157 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_569_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (10559069530791 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(7543122663510840268807 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_569_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (10559069530791 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(1750982146713054436347 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_569_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (10559069530791 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(5172777597921439708851 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_569_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (10559069530791 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) * (-(393112829726998233739 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_569_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (10559069530791 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_569_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (10559069530791 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_569_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (10559069530791 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_569_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_570_1 :
+    Uω (aρ 1) (bρ 1) (664241383483 / 1000000000000 : ℝ) ≤
+      (-(4188707574941079792709 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6577892039284187 / 10000000000000000 : ℝ) ≤
+      (-(4188707574941079792709 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (6577892039284187 / 10000000000000000 : ℝ) =
+            (6577892039284187 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6577892039284187 / 5000000000000000 : ℝ) ≤
+            (2742764228058920207291 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3288933730911687 / 5000000000000000 : ℝ)) (q :=
+      (6577892039284187 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_570_2 :
+    Uω (aρ 2) (bρ 2) (664241383483 / 1000000000000 : ℝ) ≤
+      (-(4225117198853214904403 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2621594291661379 / 4000000000000000 : ℝ) ≤
+      (-(4225117198853214904403 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2621594291661379 / 4000000000000000 : ℝ) =
+            (2621594291661379 / 2000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2621594291661379 / 2000000000000000 : ℝ) ≤
+            (2706354604146785095597 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1310764770814379 / 2000000000000000 : ℝ)) (q :=
+      (2621594291661379 / 4000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_570_3 :
+    Uω (aρ 3) (bρ 3) (664241383483 / 1000000000000 : ℝ) ≤
+      (-(429832144022243457453 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (65061829555909 / 100000000000000 : ℝ) ≤
+      (-(429832144022243457453 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (65061829555909 / 100000000000000 : ℝ) = (65061829555909 / 50000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (65061829555909 / 50000000000000 : ℝ) ≤
+            (263315036277756542547 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (8132018016621 / 12500000000000 : ℝ)) (q :=
+      (65061829555909 / 100000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_570_4 :
+    Uω (aρ 4) (bρ 4) (664241383483 / 1000000000000 : ℝ) ≤
+      (-(552631475048740095093 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3213410172059717 / 5000000000000000 : ℝ) ≤
+      (-(552631475048740095093 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (3213410172059717 / 5000000000000000 : ℝ) =
+            (3213410172059717 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3213410172059717 / 2500000000000000 : ℝ) ≤
+            (313802500326259904907 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1606295842854717 / 2500000000000000 : ℝ)) (q :=
+      (3213410172059717 / 5000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_570_5 :
+    Uω (aρ 5) (bρ 5) (664241383483 / 1000000000000 : ℝ) ≤
+      (-(4610764411565489378927 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6306044712565743 / 10000000000000000 : ℝ) ≤
+      (-(4610764411565489378927 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (6306044712565743 / 10000000000000000 : ℝ) =
+            (6306044712565743 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6306044712565743 / 5000000000000000 : ℝ) ≤
+            (2320707391434510621073 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3150911013823243 / 5000000000000000 : ℝ)) (q :=
+      (6306044712565743 / 10000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_570_6 :
+    Uω (aρ 6) (bρ 6) (664241383483 / 1000000000000 : ℝ) ≤
+      (-(2444487408756060489599 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6133021982790739 / 10000000000000000 : ℝ) ≤
+      (-(2444487408756060489599 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (6133021982790739 / 10000000000000000 : ℝ) =
+            (6133021982790739 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6133021982790739 / 5000000000000000 : ℝ) ≤
+            (1021248492743939510401 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3061508471618239 / 5000000000000000 : ℝ)) (q :=
+      (6133021982790739 / 10000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_570_7 :
+    Uω (aρ 7) (bρ 7) (664241383483 / 1000000000000 : ℝ) ≤
+      (-(5280774161118552785937 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1474344251231207 / 2500000000000000 : ℝ) ≤
+      (-(5280774161118552785937 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1474344251231207 / 2500000000000000 : ℝ) =
+            (1474344251231207 / 1250000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1474344251231207 / 1250000000000000 : ℝ) ≤
+            (1650697641881447214063 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (367211773016541 / 625000000000000 : ℝ)) (q :=
+      (1474344251231207 / 2500000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_570_8 :
+    Uω (aρ 8) (bρ 8) (664241383483 / 1000000000000 : ℝ) ≤
+      (-(5814794347494712517697 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11181412915014739 / 20000000000000000 : ℝ) ≤
+      (-(5814794347494712517697 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11181412915014739 / 20000000000000000 : ℝ) =
+            (11181412915014739 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11181412915014739 / 10000000000000000 : ℝ) ≤
+            (1116677455505287482303 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5545588924769739 / 10000000000000000 : ℝ)) (q :=
+      (11181412915014739 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_570_9 :
+    Uω (aρ 9) (bρ 9) (664241383483 / 1000000000000 : ℝ) ≤
+      (-(1631020436464782829367 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5207901117902643 / 10000000000000000 : ℝ) ≤
+      (-(1631020436464782829367 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5207901117902643 / 10000000000000000 : ℝ) =
+            (5207901117902643 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5207901117902643 / 5000000000000000 : ℝ) ≤
+            (101847514285217170633 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2560471936960143 / 5000000000000000 : ℝ)) (q :=
+      (5207901117902643 / 10000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_570_10 :
+    Uω (aρ 10) (bρ 10) (664241383483 / 1000000000000 : ℝ) ≤
+      (-(744896543364644492559 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2373917083235651 / 5000000000000000 : ℝ) ≤
+      (-(744896543364644492559 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2373917083235651 / 5000000000000000 : ℝ) =
+            (2373917083235651 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2373917083235651 / 1250000000000000 : ℝ) ≤
+            (641397817235355507441 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1147342251138151 / 2500000000000000 : ℝ)) (q :=
+      (2373917083235651 / 5000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_570_11 :
+    Uω (aρ 11) (bρ 11) (664241383483 / 1000000000000 : ℝ) ≤
+      (-(4322513676307216232947 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (263288111602721 / 625000000000000 : ℝ) ≤
+      (-(4322513676307216232947 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (263288111602721 / 625000000000000 : ℝ) =
+            (263288111602721 / 156250000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (263288111602721 / 156250000000000 : ℝ) ≤
+            (2608958126692783767053 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (492174608637759 / 1250000000000000 : ℝ)) (q :=
+      (263288111602721 / 625000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11]) (by norm_num [bρ_11])
+      (by norm_num) (by norm_num [aρ_11, bρ_11]) (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_570_12 :
+    Uω (aρ 12) (bρ 12) (664241383483 / 1000000000000 : ℝ) ≤
+      (-(10207502320509313781759 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1441298046235793 / 4000000000000000 : ℝ) ≤
+      (-(10207502320509313781759 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1441298046235793 / 4000000000000000 : ℝ) =
+            (1441298046235793 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1441298046235793 / 1000000000000000 : ℝ) ≤
+            (3655441285490686218241 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (628473525135793 / 2000000000000000 : ℝ)) (q :=
+      (1441298046235793 / 4000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_570_13 :
+    Uω (aρ 13) (bρ 13) (664241383483 / 1000000000000 : ℝ) ≤
+      (-(6187441800526216948877 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5802239205317251 / 20000000000000000 : ℝ) ≤
+      (-(6187441800526216948877 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5802239205317251 / 20000000000000000 : ℝ) =
+            (5802239205317251 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5802239205317251 / 5000000000000000 : ℝ) ≤
+            (744030002473783051123 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2137498341412251 / 10000000000000000 : ℝ)) (q :=
+      (5802239205317251 / 20000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_570_14 :
+    Uω (aρ 14) (bρ 14) (664241383483 / 1000000000000 : ℝ) ≤
+      (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_14]) (by norm_num [bρ_14])]; exact Lmid_14
+
+lemma U_570_15 :
+    Uω (aρ 15) (bρ 15) (664241383483 / 1000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_570_16 :
+    Uω (aρ 16) (bρ 16) (664241383483 / 1000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_570 :
+    Uρ (664241383483 / 1000000000000 : ℝ) ≤
+      (-(1407853005784725932979 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (664241383483 / 1000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(4188707574941079792709 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_570_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (664241383483 / 1000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(4225117198853214904403 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_570_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (664241383483 / 1000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(429832144022243457453 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_570_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (664241383483 / 1000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(552631475048740095093 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_570_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (664241383483 / 1000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(4610764411565489378927 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_570_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (664241383483 / 1000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(2444487408756060489599 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_570_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (664241383483 / 1000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(5280774161118552785937 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_570_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (664241383483 / 1000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(5814794347494712517697 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_570_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (664241383483 / 1000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(1631020436464782829367 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_570_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (664241383483 / 1000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(744896543364644492559 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_570_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (664241383483 / 1000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4322513676307216232947 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_570_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (664241383483 / 1000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(10207502320509313781759 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_570_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (664241383483 / 1000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(6187441800526216948877 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_570_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (664241383483 / 1000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8977612329965299982791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_570_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (664241383483 / 1000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_570_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (664241383483 / 1000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_570_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_571_1 :
+    Uω (aρ 1) (bρ 1) (4261528693017 / 6400000000000 : ℝ) ≤
+      (-(4164072279252109276981 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (26376467391064129 / 40000000000000000 : ℝ) ≤
+      (-(4164072279252109276981 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (26376467391064129 / 40000000000000000 : ℝ) =
+            (26376467391064129 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26376467391064129 / 20000000000000000 : ℝ) ≤
+            (2767399523747890723019 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3297046165389001 / 5000000000000000 : ℝ)) (q :=
+      (26376467391064129 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_571_2 :
+    Uω (aρ 2) (bρ 2) (4261528693017 / 6400000000000 : ℝ) ≤
+      (-(1050097909525026300293 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (26280843507653277 / 40000000000000000 : ℝ) ≤
+      (-(1050097909525026300293 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (26280843507653277 / 40000000000000000 : ℝ) =
+            (26280843507653277 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26280843507653277 / 20000000000000000 : ℝ) ≤
+            (682770041224973699707 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (410628087598911 / 625000000000000 : ℝ)) (q :=
+      (26280843507653277 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_571_3 :
+    Uω (aρ 3) (bρ 3) (4261528693017 / 6400000000000 : ℝ) ≤
+      (-(4273412884176880161469 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (26089636471977857 / 40000000000000000 : ℝ) ≤
+      (-(4273412884176880161469 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (26089636471977857 / 40000000000000000 : ℝ) =
+            (26089636471977857 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26089636471977857 / 20000000000000000 : ℝ) ≤
+            (2658058918823119838531 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3260920995047433 / 5000000000000000 : ℝ)) (q :=
+      (26089636471977857 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_571_4 :
+    Uω (aρ 4) (bρ 4) (4261528693017 / 6400000000000 : ℝ) ≤
+      (-(549478980034508885777 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5154439372292431 / 8000000000000000 : ℝ) ≤
+      (-(549478980034508885777 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5154439372292431 / 8000000000000000 : ℝ) =
+            (5154439372292431 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5154439372292431 / 4000000000000000 : ℝ) ≤
+            (316954995340491114223 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1288283273180403 / 2000000000000000 : ℝ)) (q :=
+      (5154439372292431 / 8000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_571_5 :
+    Uω (aρ 5) (bρ 5) (4261528693017 / 6400000000000 : ℝ) ≤
+      (-(4585051419903753167239 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5057824243530933 / 8000000000000000 : ℝ) ≤
+      (-(4585051419903753167239 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5057824243530933 / 8000000000000000 : ℝ) =
+            (5057824243530933 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5057824243530933 / 4000000000000000 : ℝ) ≤
+            (2346420383096246832761 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (631806846333327 / 1000000000000000 : ℝ)) (q :=
+      (5057824243530933 / 8000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_571_6 :
+    Uω (aρ 6) (bρ 6) (4261528693017 / 6400000000000 : ℝ) ≤
+      (-(303907000827735205621 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4919418537551503 / 8000000000000000 : ℝ) ≤
+      (-(303907000827735205621 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (4919418537551503 / 8000000000000000 : ℝ) =
+            (4919418537551503 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4919418537551503 / 4000000000000000 : ℝ) ≤
+            (129309986859764794379 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1227858914704939 / 2000000000000000 : ℝ)) (q :=
+      (4919418537551503 / 8000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_571_7 :
+    Uω (aρ 7) (bρ 7) (4261528693017 / 6400000000000 : ℝ) ≤
+      (-(65664971709746093159 / 125000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23654649224803619 / 40000000000000000 : ℝ) ≤
+      (-(65664971709746093159 / 125000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23654649224803619 / 40000000000000000 : ℝ) =
+            (23654649224803619 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23654649224803619 / 20000000000000000 : ℝ) ≤
+            (20978425827753906841 / 125000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5891734222807747 / 10000000000000000 : ℝ)) (q :=
+      (23654649224803619 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_571_8 :
+    Uω (aρ 8) (bρ 8) (4261528693017 / 6400000000000 : ℝ) ≤
+      (-(1446395180466007862889 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22428251271400941 / 40000000000000000 : ℝ) ≤
+      (-(1446395180466007862889 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (22428251271400941 / 40000000000000000 : ℝ) =
+            (22428251271400941 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22428251271400941 / 20000000000000000 : ℝ) ≤
+            (286472770283992137111 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (695259612180801 / 1250000000000000 : ℝ)) (q :=
+      (22428251271400941 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_571_9 :
+    Uω (aρ 9) (bρ 9) (4261528693017 / 6400000000000 : ℝ) ≤
+      (-(324622520228389607363 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20897601955097881 / 40000000000000000 : ℝ) ≤
+      (-(324622520228389607363 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (20897601955097881 / 40000000000000000 : ℝ) =
+            (20897601955097881 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20897601955097881 / 20000000000000000 : ℝ) ≤
+            (21951069921610392637 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2568858933827439 / 5000000000000000 : ℝ)) (q :=
+      (20897601955097881 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_571_10 :
+    Uω (aρ 10) (bρ 10) (4261528693017 / 6400000000000 : ℝ) ≤
+      (-(1482735833462658137863 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19058468373957353 / 40000000000000000 : ℝ) ≤
+      (-(1482735833462658137863 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (19058468373957353 / 40000000000000000 : ℝ) =
+            (19058468373957353 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19058468373957353 / 10000000000000000 : ℝ) ≤
+            (1289852887737341862137 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2303355055289807 / 5000000000000000 : ℝ)) (q :=
+      (19058468373957353 / 40000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_571_11 :
+    Uω (aρ 11) (bρ 11) (4261528693017 / 6400000000000 : ℝ) ≤
+      (-(4301958498077105265997 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16919854485250991 / 40000000000000000 : ℝ) ≤
+      (-(4301958498077105265997 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (16919854485250991 / 40000000000000000 : ℝ) =
+            (16919854485250991 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16919854485250991 / 10000000000000000 : ℝ) ≤
+            (2629513304922894734003 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3955879792431433 / 10000000000000000 : ℝ)) (q :=
+      (16919854485250991 / 40000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_571_12 :
+    Uω (aρ 12) (bρ 12) (4261528693017 / 6400000000000 : ℝ) ≤
+      (-(10156041558459248697277 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14487341928941459 / 40000000000000000 : ℝ) ≤
+      (-(10156041558459248697277 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (14487341928941459 / 40000000000000000 : ℝ) =
+            (14487341928941459 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14487341928941459 / 10000000000000000 : ℝ) ≤
+            (3706902047540751302723 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3163323610961667 / 10000000000000000 : ℝ)) (q :=
+      (14487341928941459 / 40000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_571_13 :
+    Uω (aρ 13) (bρ 13) (4261528693017 / 6400000000000 : ℝ) ≤
+      (-(614973329293170409787 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11692326769761003 / 40000000000000000 : ℝ) ≤
+      (-(614973329293170409787 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (11692326769761003 / 40000000000000000 : ℝ) =
+            (11692326769761003 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11692326769761003 / 10000000000000000 : ℝ) ≤
+            (78173851006829590213 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2165197772966439 / 10000000000000000 : ℝ)) (q :=
+      (11692326769761003 / 40000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_571_14 :
+    Uω (aρ 14) (bρ 14) (4261528693017 / 6400000000000 : ℝ) ≤
+      (-(16967115179877088640491 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7331410570258827 / 40000000000000000 : ℝ) ≤
+      (-(16967115179877088640491 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (7331410570258827 / 40000000000000000 : ℝ) =
+            (7331410570258827 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7331410570258827 / 5000000000000000 : ℝ) ≤
+            (3827300229122911359509 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (328666958260351 / 10000000000000000 : ℝ)) (q :=
+      (7331410570258827 / 40000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_571_15 :
+    Uω (aρ 15) (bρ 15) (4261528693017 / 6400000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_571_16 :
+    Uω (aρ 16) (bρ 16) (4261528693017 / 6400000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_571 :
+    Uρ (4261528693017 / 6400000000000 : ℝ) ≤
+      (-(435371962683215891223 / 625000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (4261528693017 / 6400000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(4164072279252109276981 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_571_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (4261528693017 / 6400000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1050097909525026300293 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_571_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (4261528693017 / 6400000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(4273412884176880161469 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_571_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (4261528693017 / 6400000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(549478980034508885777 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_571_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (4261528693017 / 6400000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(4585051419903753167239 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_571_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (4261528693017 / 6400000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(303907000827735205621 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_571_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (4261528693017 / 6400000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) * (-(65664971709746093159 / 125000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_571_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (4261528693017 / 6400000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(1446395180466007862889 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_571_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (4261528693017 / 6400000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(324622520228389607363 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_571_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (4261528693017 / 6400000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(1482735833462658137863 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_571_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (4261528693017 / 6400000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4301958498077105265997 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_571_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (4261528693017 / 6400000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(10156041558459248697277 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_571_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (4261528693017 / 6400000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) * (-(614973329293170409787 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_571_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (4261528693017 / 6400000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(16967115179877088640491 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_571_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (4261528693017 / 6400000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_571_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (4261528693017 / 6400000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_571_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_572_1 :
+    Uω (aρ 1) (bρ 1) (10679781329357 / 16000000000000 : ℝ) ≤
+      (-(2069748762339204639219 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (330517082797551 / 500000000000000 : ℝ) ≤
+      (-(2069748762339204639219 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (330517082797551 / 500000000000000 : ℝ) =
+            (330517082797551 / 250000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (330517082797551 / 250000000000000 : ℝ) ≤
+            (1395987139160795360781 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1322063439827783 / 2000000000000000 : ℝ)) (q :=
+      (330517082797551 / 500000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_572_2 :
+    Uω (aρ 2) (bρ 2) (10679781329357 / 16000000000000 : ℝ) ≤
+      (-(4175727064915972273783 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13172872045407273 / 20000000000000000 : ℝ) ≤
+      (-(4175727064915972273783 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (13172872045407273 / 20000000000000000 : ℝ) =
+            (13172872045407273 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13172872045407273 / 10000000000000000 : ℝ) ≤
+            (2755744738084027726217 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1646568736288537 / 2500000000000000 : ℝ)) (q :=
+      (13172872045407273 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_572_3 :
+    Uω (aρ 3) (bρ 3) (10679781329357 / 16000000000000 : ℝ) ≤
+      (-(4248566228358807393553 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13077270546755097 / 20000000000000000 : ℝ) ≤
+      (-(4248566228358807393553 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (13077270546755097 / 20000000000000000 : ℝ) =
+            (13077270546755097 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13077270546755097 / 10000000000000000 : ℝ) ≤
+            (2682905574641192606447 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1634517388212993 / 2500000000000000 : ℝ)) (q :=
+      (13077270546755097 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_572_4 :
+    Uω (aρ 4) (bρ 4) (10679781329357 / 16000000000000 : ℝ) ≤
+      (-(2185337678456050449133 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12918556131774469 / 20000000000000000 : ℝ) ≤
+      (-(2185337678456050449133 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (12918556131774469 / 20000000000000000 : ℝ) =
+            (12918556131774469 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12918556131774469 / 10000000000000000 : ℝ) ≤
+            (1280398223043949550867 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (807206164867043 / 1250000000000000 : ℝ)) (q :=
+      (12918556131774469 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_572_5 :
+    Uω (aρ 5) (bρ 5) (10679781329357 / 16000000000000 : ℝ) ≤
+      (-(2279702231101783945013 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2535406336269411 / 4000000000000000 : ℝ) ≤
+      (-(2279702231101783945013 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2535406336269411 / 4000000000000000 : ℝ) =
+            (2535406336269411 / 2000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2535406336269411 / 2000000000000000 : ℝ) ≤
+            (1186033670398216054987 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (633431478784393 / 1000000000000000 : ℝ)) (q :=
+      (2535406336269411 / 4000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_572_6 :
+    Uω (aρ 6) (bρ 6) (10679781329357 / 16000000000000 : ℝ) ≤
+      (-(4836119278634238343609 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6165524221474687 / 10000000000000000 : ℝ) ≤
+      (-(4836119278634238343609 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (6165524221474687 / 10000000000000000 : ℝ) =
+            (6165524221474687 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6165524221474687 / 5000000000000000 : ℝ) ≤
+            (2095352524365761656391 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6155571924586249 / 10000000000000000 : ℝ)) (q :=
+      (6165524221474687 / 10000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_572_7 :
+    Uω (aρ 7) (bρ 7) (10679781329357 / 16000000000000 : ℝ) ≤
+      (-(2612848856110541349761 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11859894547306783 / 20000000000000000 : ℝ) ≤
+      (-(2612848856110541349761 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11859894547306783 / 20000000000000000 : ℝ) =
+            (11859894547306783 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11859894547306783 / 10000000000000000 : ℝ) ≤
+            (852887045389458650239 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2954039704851829 / 5000000000000000 : ℝ)) (q :=
+      (11859894547306783 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_572_8 :
+    Uω (aρ 8) (bρ 8) (10679781329357 / 16000000000000 : ℝ) ≤
+      (-(5756453563723046984991 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11246836812835769 / 20000000000000000 : ℝ) ≤
+      (-(5756453563723046984991 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11246836812835769 / 20000000000000000 : ℝ) =
+            (11246836812835769 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11246836812835769 / 10000000000000000 : ℝ) ≤
+            (1175018239276953015009 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1394640831643161 / 2500000000000000 : ℝ)) (q :=
+      (11246836812835769 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_572_9 :
+    Uω (aρ 9) (bρ 9) (10679781329357 / 16000000000000 : ℝ) ≤
+      (-(3230461078330779039389 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5240898101526921 / 10000000000000000 : ℝ) ≤
+      (-(3230461078330779039389 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5240898101526921 / 10000000000000000 : ℝ) =
+            (5240898101526921 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5240898101526921 / 5000000000000000 : ℝ) ≤
+            (235274823169220960611 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5154488345150717 / 10000000000000000 : ℝ)) (q :=
+      (5240898101526921 / 10000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_572_10 :
+    Uω (aρ 10) (bρ 10) (10679781329357 / 16000000000000 : ℝ) ≤
+      (-(7378525449464564730863 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1912558387479417 / 4000000000000000 : ℝ) ≤
+      (-(7378525449464564730863 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1912558387479417 / 4000000000000000 : ℝ) =
+            (1912558387479417 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1912558387479417 / 1000000000000000 : ℝ) ≤
+            (6484418156535435269137 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (28900269456181 / 62500000000000 : ℝ)) (q :=
+      (1912558387479417 / 4000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_572_11 :
+    Uω (aρ 11) (bρ 11) (10679781329357 / 16000000000000 : ℝ) ≤
+      (-(4281499084663921275759 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4247307596759467 / 10000000000000000 : ℝ) ≤
+      (-(4281499084663921275759 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4247307596759467 / 10000000000000000 : ℝ) =
+            (4247307596759467 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4247307596759467 / 2500000000000000 : ℝ) ≤
+            (2649972718336078724241 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3974342995315809 / 10000000000000000 : ℝ)) (q :=
+      (4247307596759467 / 10000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_572_12 :
+    Uω (aρ 12) (bρ 12) (10679781329357 / 16000000000000 : ℝ) ≤
+      (-(2020984027645203037053 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (291231858149807 / 800000000000000 : ℝ) ≤
+      (-(2020984027645203037053 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (291231858149807 / 800000000000000 : ℝ) =
+            (291231858149807 / 200000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (291231858149807 / 200000000000000 : ℝ) ≤
+            (751604693554796962947 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (63684487044541 / 200000000000000 : ℝ)) (q :=
+      (291231858149807 / 800000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12]) (by norm_num [bρ_12])
+      (by norm_num) (by norm_num [aρ_12, bρ_12]) (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_572_13 :
+    Uω (aρ 13) (bρ 13) (10679781329357 / 16000000000000 : ℝ) ≤
+      (-(12225004331939791329417 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (736232214017377 / 2500000000000000 : ℝ) ≤
+      (-(12225004331939791329417 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (736232214017377 / 2500000000000000 : ℝ) =
+            (736232214017377 / 625000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (736232214017377 / 625000000000000 : ℝ) ≤
+            (1637939274060208670583 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2192667352215891 / 10000000000000000 : ℝ)) (q :=
+      (736232214017377 / 2500000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13]) (by norm_num [bρ_13])
+      (by norm_num) (by norm_num [aρ_13, bρ_13]) (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_572_14 :
+    Uω (aρ 14) (bρ 14) (10679781329357 / 16000000000000 : ℝ) ≤
+      (-(16558394140482642990089 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (477329293762299 / 2500000000000000 : ℝ) ≤
+      (-(16558394140482642990089 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (477329293762299 / 2500000000000000 : ℝ) =
+            (477329293762299 / 312500000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (477329293762299 / 312500000000000 : ℝ) ≤
+            (4236021268517357009911 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (465371275220267 / 10000000000000000 : ℝ)) (q :=
+      (477329293762299 / 2500000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14]) (by norm_num [bρ_14])
+      (by norm_num) (by norm_num [aρ_14, bρ_14]) (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_572_15 :
+    Uω (aρ 15) (bρ 15) (10679781329357 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_572_16 :
+    Uω (aρ 16) (bρ 16) (10679781329357 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_572 :
+    Uρ (10679781329357 / 16000000000000 : ℝ) ≤
+      (-(13837047140647643897 / 20000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (10679781329357 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2069748762339204639219 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_572_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (10679781329357 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(4175727064915972273783 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_572_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (10679781329357 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(4248566228358807393553 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_572_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (10679781329357 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(2185337678456050449133 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_572_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (10679781329357 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(2279702231101783945013 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_572_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (10679781329357 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(4836119278634238343609 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_572_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (10679781329357 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(2612848856110541349761 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_572_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (10679781329357 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(5756453563723046984991 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_572_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (10679781329357 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(3230461078330779039389 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_572_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (10679781329357 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(7378525449464564730863 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_572_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (10679781329357 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4281499084663921275759 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_572_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (10679781329357 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(2020984027645203037053 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_572_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (10679781329357 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(12225004331939791329417 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_572_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (10679781329357 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(16558394140482642990089 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_572_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (10679781329357 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_572_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (10679781329357 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_572_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_573_1 :
+    Uω (aρ 1) (bρ 1) (21411481852343 / 32000000000000 : ℝ) ≤
+      (-(4114983014388123040979 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (26506265855365327 / 40000000000000000 : ℝ) ≤
+      (-(4114983014388123040979 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (26506265855365327 / 40000000000000000 : ℝ) =
+            (26506265855365327 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26506265855365327 / 20000000000000000 : ℝ) ≤
+            (2816488788611876959021 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1656635516727619 / 2500000000000000 : ℝ)) (q :=
+      (26506265855365327 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_573_2 :
+    Uω (aρ 2) (bρ 2) (21411481852343 / 32000000000000 : ℝ) ≤
+      (-(4151123179164884118219 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (26410644666155673 / 40000000000000000 : ℝ) ≤
+      (-(4151123179164884118219 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (26410644666155673 / 40000000000000000 : ℝ) =
+            (26410644666155673 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26410644666155673 / 20000000000000000 : ℝ) ≤
+            (2780348623835115881781 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6602500484815649 / 10000000000000000 : ℝ)) (q :=
+      (26410644666155673 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_573_3 :
+    Uω (aρ 3) (bρ 3) (21411481852343 / 32000000000000 : ℝ) ≤
+      (-(1055945291455342745149 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (26219445687169191 / 40000000000000000 : ℝ) ≤
+      (-(1055945291455342745149 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (26219445687169191 / 40000000000000000 : ℝ) =
+            (26219445687169191 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26219445687169191 / 20000000000000000 : ℝ) ≤
+            (676922659294657254851 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (819287137709051 / 1250000000000000 : ℝ)) (q :=
+      (26219445687169191 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_573_4 :
+    Uω (aρ 4) (bρ 4) (21411481852343 / 32000000000000 : ℝ) ≤
+      (-(2172791015703680314347 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25902027583361671 / 40000000000000000 : ℝ) ≤
+      (-(2172791015703680314347 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (25902027583361671 / 40000000000000000 : ℝ) =
+            (25902027583361671 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25902027583361671 / 20000000000000000 : ℝ) ≤
+            (1292944885796319685653 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (404617639427103 / 625000000000000 : ℝ)) (q :=
+      (25902027583361671 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_573_5 :
+    Uω (aρ 5) (bρ 5) (21411481852343 / 32000000000000 : ℝ) ≤
+      (-(4533823199721012871099 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25419005287087807 / 40000000000000000 : ℝ) ≤
+      (-(4533823199721012871099 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (25419005287087807 / 40000000000000000 : ℝ) =
+            (25419005287087807 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25419005287087807 / 20000000000000000 : ℝ) ≤
+            (2397648603278987128901 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1587640250507929 / 2500000000000000 : ℝ)) (q :=
+      (25419005287087807 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_573_6 :
+    Uω (aρ 6) (bρ 6) (21411481852343 / 32000000000000 : ℝ) ≤
+      (-(4809796242406969208233 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24727100530005173 / 40000000000000000 : ℝ) ≤
+      (-(4809796242406969208233 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (24727100530005173 / 40000000000000000 : ℝ) =
+            (24727100530005173 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24727100530005173 / 20000000000000000 : ℝ) ≤
+            (2121675560593030791767 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6171848998630399 / 10000000000000000 : ℝ)) (q :=
+      (24727100530005173 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_573_7 :
+    Uω (aρ 7) (bρ 7) (21411481852343 / 32000000000000 : ℝ) ≤
+      (-(324892103887673495129 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23784927640181179 / 40000000000000000 : ℝ) ≤
+      (-(324892103887673495129 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23784927640181179 / 40000000000000000 : ℝ) =
+            (23784927640181179 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23784927640181179 / 20000000000000000 : ℝ) ≤
+            (108324883799826504871 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2962211967239201 / 5000000000000000 : ℝ)) (q :=
+      (23784927640181179 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_573_8 :
+    Uω (aρ 8) (bρ 8) (21411481852343 / 32000000000000 : ℝ) ≤
+      (-(5727412354734878097163 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22559092920130721 / 40000000000000000 : ℝ) ≤
+      (-(5727412354734878097163 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (22559092920130721 / 40000000000000000 : ℝ) =
+            (22559092920130721 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22559092920130721 / 20000000000000000 : ℝ) ≤
+            (1204059448265121902837 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5595048225793173 / 10000000000000000 : ℝ)) (q :=
+      (22559092920130721 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_573_9 :
+    Uω (aρ 9) (bρ 9) (21411481852343 / 32000000000000 : ℝ) ≤
+      (-(803687038877101364811 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21029575893059351 / 40000000000000000 : ℝ) ≤
+      (-(803687038877101364811 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (21029575893059351 / 40000000000000000 : ℝ) =
+            (21029575893059351 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21029575893059351 / 20000000000000000 : ℝ) ≤
+            (62746936497898635189 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (323203458788593 / 625000000000000 : ℝ)) (q :=
+      (21029575893059351 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_573_10 :
+    Uω (aρ 10) (bρ 10) (21411481852343 / 32000000000000 : ℝ) ≤
+      (-(1835875806775894467537 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19192683349971467 / 40000000000000000 : ℝ) ≤
+      (-(1835875806775894467537 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (19192683349971467 / 40000000000000000 : ℝ) =
+            (19192683349971467 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19192683349971467 / 10000000000000000 : ℝ) ≤
+            (1629860094724105532463 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2320684051284273 / 5000000000000000 : ℝ)) (q :=
+      (19192683349971467 / 40000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_573_11 :
+    Uω (aρ 11) (bρ 11) (21411481852343 / 32000000000000 : ℝ) ≤
+      (-(8522268895539115610217 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17058567395075321 / 40000000000000000 : ℝ) ≤
+      (-(8522268895539115610217 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (17058567395075321 / 40000000000000000 : ℝ) =
+            (17058567395075321 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17058567395075321 / 10000000000000000 : ℝ) ≤
+            (5340674710460884389783 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3992786751325473 / 10000000000000000 : ℝ)) (q :=
+      (17058567395075321 / 40000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_573_12 :
+    Uω (aρ 12) (bρ 12) (21411481852343 / 32000000000000 : ℝ) ≤
+      (-(10054132744104097026937 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14635735559531383 / 40000000000000000 : ℝ) ≤
+      (-(10054132744104097026937 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (14635735559531383 / 40000000000000000 : ℝ) =
+            (14635735559531383 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14635735559531383 / 10000000000000000 : ℝ) ≤
+            (3808810861895902973063 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (400633866279813 / 1250000000000000 : ℝ)) (q :=
+      (14635735559531383 / 40000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_573_13 :
+    Uω (aρ 13) (bρ 13) (21411481852343 / 32000000000000 : ℝ) ≤
+      (-(1215146544888570115507 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11866661439586277 / 40000000000000000 : ℝ) ≤
+      (-(1215146544888570115507 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (11866661439586277 / 40000000000000000 : ℝ) =
+            (11866661439586277 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11866661439586277 / 10000000000000000 : ℝ) ≤
+            (171147815711429884493 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2219915611860951 / 10000000000000000 : ℝ)) (q :=
+      (11866661439586277 / 40000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_573_14 :
+    Uω (aρ 14) (bρ 14) (21411481852343 / 32000000000000 : ℝ) ≤
+      (-(8122578466833861866631 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7880282551114177 / 40000000000000000 : ℝ) ≤
+      (-(8122578466833861866631 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (7880282551114177 / 40000000000000000 : ℝ) =
+            (7880282551114177 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7880282551114177 / 5000000000000000 : ℝ) ≤
+            (2274629237666138133369 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (570653452669901 / 10000000000000000 : ℝ)) (q :=
+      (7880282551114177 / 40000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_573_15 :
+    Uω (aρ 15) (bρ 15) (21411481852343 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_573_16 :
+    Uω (aρ 16) (bρ 16) (21411481852343 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_573 :
+    Uρ (21411481852343 / 32000000000000 : ℝ) ≤
+      (-(3437741571450890412419 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (21411481852343 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(4114983014388123040979 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_573_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (21411481852343 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(4151123179164884118219 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_573_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (21411481852343 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1055945291455342745149 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_573_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (21411481852343 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(2172791015703680314347 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_573_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (21411481852343 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(4533823199721012871099 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_573_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (21411481852343 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(4809796242406969208233 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_573_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (21411481852343 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(324892103887673495129 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_573_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (21411481852343 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(5727412354734878097163 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_573_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (21411481852343 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(803687038877101364811 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_573_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (21411481852343 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(1835875806775894467537 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_573_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (21411481852343 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(8522268895539115610217 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_573_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (21411481852343 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(10054132744104097026937 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_573_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (21411481852343 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(1215146544888570115507 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_573_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (21411481852343 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8122578466833861866631 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_573_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (21411481852343 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_573_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (21411481852343 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_573_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_574_1 :
+    Uω (aρ 1) (bρ 1) (5365850261493 / 8000000000000 : ℝ) ≤
+      (-(2045264226863557451219 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13285582542878253 / 20000000000000000 : ℝ) ≤
+      (-(2045264226863557451219 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (13285582542878253 / 20000000000000000 : ℝ) =
+            (13285582542878253 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13285582542878253 / 10000000000000000 : ℝ) ≤
+            (1420471674636442548781 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6642766934097003 / 10000000000000000 : ℝ)) (q :=
+      (13285582542878253 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_574_2 :
+    Uω (aρ 2) (bρ 2) (5365850261493 / 8000000000000 : ℝ) ≤
+      (-(206328984146057335773 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (827360788554193 / 1250000000000000 : ℝ) ≤
+      (-(206328984146057335773 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (827360788554193 / 1250000000000000 : ℝ) =
+            (827360788554193 / 625000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (827360788554193 / 625000000000000 : ℝ) ≤
+            (140244606003942664227 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3309363010297919 / 5000000000000000 : ℝ)) (q :=
+      (827360788554193 / 1250000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_574_3 :
+    Uω (aρ 3) (bρ 3) (5365850261493 / 8000000000000 : ℝ) ≤
+      (-(2099528695947786887917 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13142175126580683 / 20000000000000000 : ℝ) ≤
+      (-(2099528695947786887917 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (13142175126580683 / 20000000000000000 : ℝ) =
+            (13142175126580683 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13142175126580683 / 10000000000000000 : ℝ) ≤
+            (1366207205552213112083 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6570524636659433 / 10000000000000000 : ℝ)) (q :=
+      (13142175126580683 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_574_4 :
+    Uω (aρ 4) (bρ 4) (5365850261493 / 8000000000000 : ℝ) ≤
+      (-(4320551547269413483711 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (259669428215177 / 400000000000000 : ℝ) ≤
+      (-(4320551547269413483711 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (259669428215177 / 400000000000000 : ℝ) =
+            (259669428215177 / 200000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (259669428215177 / 200000000000000 : ℝ) ≤
+            (2610920255730586516289 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (32450575509513 / 50000000000000 : ℝ)) (q :=
+      (259669428215177 / 400000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_574_5 :
+    Uω (aρ 5) (bρ 5) (5365850261493 / 8000000000000 : ℝ) ≤
+      (-(450830729631545839137 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3185493374065603 / 5000000000000000 : ℝ) ≤
+      (-(450830729631545839137 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (3185493374065603 / 5000000000000000 : ℝ) =
+            (3185493374065603 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3185493374065603 / 2500000000000000 : ℝ) ≤
+            (242316450668454160863 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3183403553370581 / 5000000000000000 : ℝ)) (q :=
+      (3185493374065603 / 5000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_574_6 :
+    Uω (aρ 6) (bρ 6) (5365850261493 / 8000000000000 : ℝ) ≤
+      (-(956708507247560145543 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12396051812224373 / 20000000000000000 : ℝ) ≤
+      (-(956708507247560145543 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (12396051812224373 / 20000000000000000 : ℝ) =
+            (12396051812224373 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12396051812224373 / 10000000000000000 : ℝ) ≤
+            (429585853352439854457 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6188125797843123 / 10000000000000000 : ℝ)) (q :=
+      (12396051812224373 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_574_7 :
+    Uω (aρ 7) (bρ 7) (5365850261493 / 8000000000000 : ℝ) ≤
+      (-(5170925165050416482019 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (372657263631819 / 625000000000000 : ℝ) ≤
+      (-(5170925165050416482019 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (372657263631819 / 625000000000000 : ℝ) =
+            (372657263631819 / 312500000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (372657263631819 / 312500000000000 : ℝ) ≤
+            (1760546637949583517981 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2970383901298479 / 5000000000000000 : ℝ)) (q :=
+      (372657263631819 / 625000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_574_8 :
+    Uω (aρ 8) (bρ 8) (5365850261493 / 8000000000000 : ℝ) ≤
+      (-(356153536329907708581 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11312254590872379 / 20000000000000000 : ℝ) ≤
+      (-(356153536329907708581 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11312254590872379 / 20000000000000000 : ℝ) =
+            (11312254590872379 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11312254590872379 / 10000000000000000 : ℝ) ≤
+            (77063451357592291419 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5611531608591129 / 10000000000000000 : ℝ)) (q :=
+      (11312254590872379 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_574_9 :
+    Uω (aρ 9) (bρ 9) (5365850261493 / 8000000000000 : ℝ) ≤
+      (-(6398172183635422242973 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10547776241736821 / 20000000000000000 : ℝ) ≤
+      (-(6398172183635422242973 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10547776241736821 / 20000000000000000 : ℝ) =
+            (10547776241736821 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10547776241736821 / 10000000000000000 : ℝ) ≤
+            (533299619364577757027 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5188018887815571 / 10000000000000000 : ℝ)) (q :=
+      (10547776241736821 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_574_10 :
+    Uω (aρ 10) (bρ 10) (5365850261493 / 8000000000000 : ℝ) ≤
+      (-(7308611460365759401711 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4814941744570177 / 10000000000000000 : ℝ) ≤
+      (-(7308611460365759401711 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4814941744570177 / 10000000000000000 : ℝ) =
+            (4814941744570177 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4814941744570177 / 2500000000000000 : ℝ) ≤
+            (6554332145634240598289 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (582335646089263 / 1250000000000000 : ℝ)) (q :=
+      (4814941744570177 / 10000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_574_11 :
+    Uω (aρ 11) (bρ 11) (5365850261493 / 8000000000000 : ℝ) ≤
+      (-(8481727231061772560251 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2140983255733719 / 5000000000000000 : ℝ) ≤
+      (-(8481727231061772560251 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2140983255733719 / 5000000000000000 : ℝ) =
+            (2140983255733719 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2140983255733719 / 1250000000000000 : ℝ) ≤
+            (5381216374938227439749 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2005605664356813 / 5000000000000000 : ℝ)) (q :=
+      (2140983255733719 / 5000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_574_12 :
+    Uω (aρ 12) (bρ 12) (5365850261493 / 8000000000000 : ℝ) ≤
+      (-(625229637354933601691 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7354885992593791 / 20000000000000000 : ℝ) ≤
+      (-(625229637354933601691 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7354885992593791 / 20000000000000000 : ℝ) =
+            (7354885992593791 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7354885992593791 / 5000000000000000 : ℝ) ≤
+            (241204338020066398309 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3225864395057541 / 10000000000000000 : ℝ)) (q :=
+      (7354885992593791 / 20000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_574_13 :
+    Uω (aρ 13) (bρ 13) (5365850261493 / 8000000000000 : ℝ) ≤
+      (-(12078820249414143656001 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (239063618381401 / 800000000000000 : ℝ) ≤
+      (-(12078820249414143656001 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (239063618381401 / 800000000000000 : ℝ) =
+            (239063618381401 / 200000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (239063618381401 / 200000000000000 : ℝ) ≤
+            (1784123356585856343999 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (89878024143751 / 400000000000000 : ℝ)) (q :=
+      (239063618381401 / 800000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13]) (by norm_num [bρ_13])
+      (by norm_num) (by norm_num [aρ_13, bρ_13]) (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_574_14 :
+    Uω (aρ 14) (bρ 14) (5365850261493 / 8000000000000 : ℝ) ≤
+      (-(1598140732969455004503 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (809088986232721 / 4000000000000000 : ℝ) ≤
+      (-(1598140732969455004503 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (809088986232721 / 4000000000000000 : ℝ) =
+            (809088986232721 / 500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (809088986232721 / 500000000000000 : ℝ) ≤
+            (481300807930544995497 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (131946472053471 / 2000000000000000 : ℝ)) (q :=
+      (809088986232721 / 4000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14]) (by norm_num [bρ_14])
+      (by norm_num) (by norm_num [aρ_14, bρ_14]) (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_574_15 :
+    Uω (aρ 15) (bρ 15) (5365850261493 / 8000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_574_16 :
+    Uω (aρ 16) (bρ 16) (5365850261493 / 8000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_574 :
+    Uρ (5365850261493 / 8000000000000 : ℝ) ≤
+      (-(3417392158413140254523 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (5365850261493 / 8000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2045264226863557451219 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_574_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (5365850261493 / 8000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(206328984146057335773 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_574_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (5365850261493 / 8000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(2099528695947786887917 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_574_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (5365850261493 / 8000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(4320551547269413483711 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_574_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (5365850261493 / 8000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(450830729631545839137 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_574_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (5365850261493 / 8000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(956708507247560145543 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_574_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (5365850261493 / 8000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(5170925165050416482019 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_574_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (5365850261493 / 8000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(356153536329907708581 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_574_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (5365850261493 / 8000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(6398172183635422242973 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_574_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (5365850261493 / 8000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(7308611460365759401711 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_574_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (5365850261493 / 8000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(8481727231061772560251 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_574_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (5365850261493 / 8000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(625229637354933601691 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_574_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (5365850261493 / 8000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(12078820249414143656001 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_574_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (5365850261493 / 8000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(1598140732969455004503 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_574_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (5365850261493 / 8000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_574_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (5365850261493 / 8000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_574_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_575_1 :
+    Uω (aρ 1) (bρ 1) (21515320239601 / 32000000000000 : ℝ) ≤
+      (-(1016533387549429410351 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (26636064314986169 / 40000000000000000 : ℝ) ≤
+      (-(1016533387549429410351 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (26636064314986169 / 40000000000000000 : ℝ) =
+            (26636064314986169 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26636064314986169 / 20000000000000000 : ℝ) ≤
+            (716334563200570589649 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1664747950175693 / 2500000000000000 : ℝ)) (q :=
+      (26636064314986169 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_575_2 :
+    Uω (aρ 2) (bρ 2) (21515320239601 / 32000000000000 : ℝ) ≤
+      (-(4102096280447662030521 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (26540445793607001 / 40000000000000000 : ℝ) ≤
+      (-(4102096280447662030521 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (26540445793607001 / 40000000000000000 : ℝ) =
+            (26540445793607001 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26540445793607001 / 20000000000000000 : ℝ) ≤
+            (2829375522552337969479 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1658737888130797 / 2500000000000000 : ℝ)) (q :=
+      (26540445793607001 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_575_3 :
+    Uω (aρ 3) (bρ 3) (21515320239601 / 32000000000000 : ℝ) ≤
+      (-(4174394604167777532657 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (26349254791691207 / 40000000000000000 : ℝ) ≤
+      (-(4174394604167777532657 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (26349254791691207 / 40000000000000000 : ℝ) =
+            (26349254791691207 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26349254791691207 / 20000000000000000 : ℝ) ≤
+            (2757077198832222467343 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6586752157915291 / 10000000000000000 : ℝ)) (q :=
+      (26349254791691207 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_575_4 :
+    Uω (aρ 4) (bρ 4) (21515320239601 / 32000000000000 : ℝ) ≤
+      (-(4295583590380036756359 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (26031857978628207 / 40000000000000000 : ℝ) ≤
+      (-(4295583590380036756359 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (26031857978628207 / 40000000000000000 : ℝ) =
+            (26031857978628207 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26031857978628207 / 20000000000000000 : ℝ) ≤
+            (2635888212619963243641 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6506347932448791 / 10000000000000000 : ℝ)) (q :=
+      (26031857978628207 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_575_5 :
+    Uω (aρ 5) (bρ 5) (21515320239601 / 32000000000000 : ℝ) ≤
+      (-(448285641842292237797 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25548888480677023 / 40000000000000000 : ℝ) ≤
+      (-(448285641842292237797 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (25548888480677023 / 40000000000000000 : ℝ) =
+            (25548888480677023 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25548888480677023 / 20000000000000000 : ℝ) ≤
+            (244861538457707762203 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6383053102808199 / 10000000000000000 : ℝ)) (q :=
+      (25548888480677023 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_575_6 :
+    Uω (aρ 6) (bρ 6) (21515320239601 / 32000000000000 : ℝ) ≤
+      (-(594669724340460638439 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4971421234711093 / 8000000000000000 : ℝ) ≤
+      (-(594669724340460638439 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (4971421234711093 / 8000000000000000 : ℝ) =
+            (4971421234711093 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4971421234711093 / 4000000000000000 : ℝ) ≤
+            (271764251034539361561 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (310220116219371 / 500000000000000 : ℝ)) (q :=
+      (4971421234711093 / 8000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_575_7 :
+    Uω (aρ 7) (bρ 7) (21515320239601 / 32000000000000 : ℝ) ≤
+      (-(102873036052304889857 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23915200802188847 / 40000000000000000 : ℝ) ≤
+      (-(102873036052304889857 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23915200802188847 / 40000000000000000 : ℝ) =
+            (23915200802188847 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23915200802188847 / 20000000000000000 : ℝ) ≤
+            (35756400007695110143 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5957111019464111 / 10000000000000000 : ℝ)) (q :=
+      (23915200802188847 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_575_8 :
+    Uω (aρ 8) (bρ 8) (21515320239601 / 32000000000000 : ℝ) ≤
+      (-(708698216798880485127 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22689922437161531 / 40000000000000000 : ℝ) ≤
+      (-(708698216798880485127 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (22689922437161531 / 40000000000000000 : ℝ) =
+            (22689922437161531 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22689922437161531 / 20000000000000000 : ℝ) ≤
+            (157735758576119514873 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5628013488290453 / 10000000000000000 : ℝ)) (q :=
+      (22689922437161531 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_575_9 :
+    Uω (aρ 9) (bρ 9) (21515320239601 / 32000000000000 : ℝ) ≤
+      (-(198967159298271784199 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21161522243987643 / 40000000000000000 : ℝ) ≤
+      (-(198967159298271784199 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (21161522243987643 / 40000000000000000 : ℝ) =
+            (21161522243987643 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21161522243987643 / 20000000000000000 : ℝ) ≤
+            (17641334545478215801 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5204779020063509 / 10000000000000000 : ℝ)) (q :=
+      (21161522243987643 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_575_10 :
+    Uω (aρ 10) (bρ 10) (21515320239601 / 32000000000000 : ℝ) ≤
+      (-(290953964891705276277 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19326834935783657 / 40000000000000000 : ℝ) ≤
+      (-(290953964891705276277 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (19326834935783657 / 40000000000000000 : ℝ) =
+            (19326834935783657 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19326834935783657 / 10000000000000000 : ℝ) ≤
+            (263563779348294723723 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1168998599864129 / 2500000000000000 : ℝ)) (q :=
+      (19326834935783657 / 40000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_575_11 :
+    Uω (aρ 11) (bρ 11) (21515320239601 / 32000000000000 : ℝ) ≤
+      (-(4220685632138396378261 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17197126865562489 / 40000000000000000 : ℝ) ≤
+      (-(4220685632138396378261 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (17197126865562489 / 40000000000000000 : ℝ) =
+            (17197126865562489 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17197126865562489 / 10000000000000000 : ℝ) ≤
+            (2710786170861603621739 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1007404247637733 / 2500000000000000 : ℝ)) (q :=
+      (17197126865562489 / 40000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_575_12 :
+    Uω (aρ 12) (bρ 12) (21515320239601 / 32000000000000 : ℝ) ≤
+      (-(9953539452871187747641 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14783704225496667 / 40000000000000000 : ℝ) ≤
+      (-(9953539452871187747641 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (14783704225496667 / 40000000000000000 : ℝ) =
+            (14783704225496667 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14783704225496667 / 10000000000000000 : ℝ) ≤
+            (3909404153128812252359 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3246605767203021 / 10000000000000000 : ℝ)) (q :=
+      (14783704225496667 / 40000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_575_13 :
+    Uω (aρ 13) (bρ 13) (21515320239601 / 32000000000000 : ℝ) ≤
+      (-(6003520311026841393907 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12039289077146017 / 40000000000000000 : ℝ) ≤
+      (-(6003520311026841393907 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (12039289077146017 / 40000000000000000 : ℝ) =
+            (12039289077146017 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12039289077146017 / 10000000000000000 : ℝ) ≤
+            (927951491973158606093 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (284222491827837 / 1250000000000000 : ℝ)) (q :=
+      (12039289077146017 / 40000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_575_14 :
+    Uω (aρ 14) (bρ 14) (21515320239601 / 32000000000000 : ℝ) ≤
+      (-(15749320931358898036187 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8280864409230463 / 40000000000000000 : ℝ) ≤
+      (-(15749320931358898036187 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (8280864409230463 / 40000000000000000 : ℝ) =
+            (8280864409230463 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8280864409230463 / 5000000000000000 : ℝ) ≤
+            (5045094477641101963813 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (738494885709919 / 10000000000000000 : ℝ)) (q :=
+      (8280864409230463 / 40000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_575_15 :
+    Uω (aρ 15) (bρ 15) (21515320239601 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_575_16 :
+    Uω (aρ 16) (bρ 16) (21515320239601 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_575 :
+    Uρ (21515320239601 / 32000000000000 : ℝ) ≤
+      (-(5436506260804101737 / 8000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (21515320239601 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1016533387549429410351 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_575_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (21515320239601 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(4102096280447662030521 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_575_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (21515320239601 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(4174394604167777532657 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_575_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (21515320239601 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(4295583590380036756359 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_575_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (21515320239601 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(448285641842292237797 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_575_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (21515320239601 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(594669724340460638439 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_575_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (21515320239601 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(102873036052304889857 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_575_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (21515320239601 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(708698216798880485127 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_575_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (21515320239601 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(198967159298271784199 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_575_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (21515320239601 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) * (-(290953964891705276277 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_575_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (21515320239601 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4220685632138396378261 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_575_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (21515320239601 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(9953539452871187747641 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_575_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (21515320239601 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(6003520311026841393907 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_575_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (21515320239601 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(15749320931358898036187 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_575_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (21515320239601 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_575_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (21515320239601 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_575_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_576_1 :
+    Uω (aρ 1) (bρ 1) (43082559672831 / 64000000000000 : ℝ) ≤
+      (-(2026979189518170120827 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (53337027858336163 / 80000000000000000 : ℝ) ≤
+      (-(2026979189518170120827 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (53337027858336163 / 80000000000000000 : ℝ) =
+            (53337027858336163 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (53337027858336163 / 40000000000000000 : ℝ) ≤
+            (1438756711981829879173 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6667104233789197 / 10000000000000000 : ℝ)) (q :=
+      (53337027858336163 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_576_2 :
+    Uω (aρ 2) (bρ 2) (43082559672831 / 64000000000000 : ℝ) ≤
+      (-(2044938511275047845963 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (53145792141342859 / 80000000000000000 : ℝ) ≤
+      (-(2044938511275047845963 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (53145792141342859 / 80000000000000000 : ℝ) =
+            (53145792141342859 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (53145792141342859 / 40000000000000000 : ℝ) ≤
+            (1420797390224952154037 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6643064317050871 / 10000000000000000 : ℝ)) (q :=
+      (53145792141342859 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_576_3 :
+    Uω (aρ 3) (bρ 3) (43082559672831 / 64000000000000 : ℝ) ≤
+      (-(4162085986235097085897 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (52763414101442207 / 80000000000000000 : ℝ) ≤
+      (-(4162085986235097085897 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (52763414101442207 / 80000000000000000 : ℝ) =
+            (52763414101442207 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (52763414101442207 / 40000000000000000 : ℝ) ≤
+            (2769385816764902914103 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1648716478356427 / 2500000000000000 : ℝ)) (q :=
+      (52763414101442207 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_576_4 :
+    Uω (aρ 4) (bρ 4) (43082559672831 / 64000000000000 : ℝ) ≤
+      (-(4283122962134978443037 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10425726210792287 / 16000000000000000 : ℝ) ≤
+      (-(4283122962134978443037 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10425726210792287 / 16000000000000000 : ℝ) =
+            (10425726210792287 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10425726210792287 / 8000000000000000 : ℝ) ≤
+            (2648348840865021556963 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1302892866524103 / 2000000000000000 : ℝ)) (q :=
+      (10425726210792287 / 16000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_576_5 :
+    Uω (aρ 5) (bρ 5) (43082559672831 / 64000000000000 : ℝ) ≤
+      (-(4470155260503364583741 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (51162718287578243 / 80000000000000000 : ℝ) ≤
+      (-(4470155260503364583741 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (51162718287578243 / 80000000000000000 : ℝ) =
+            (51162718287578243 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (51162718287578243 / 40000000000000000 : ℝ) ≤
+            (2461316542496635416259 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6391176060359717 / 10000000000000000 : ℝ)) (q :=
+      (51162718287578243 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_576_6 :
+    Uω (aρ 6) (bρ 6) (43082559672831 / 64000000000000 : ℝ) ≤
+      (-(1186072793078736166699 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (49779214489893931 / 80000000000000000 : ℝ) ≤
+      (-(1186072793078736166699 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (49779214489893931 / 80000000000000000 : ℝ) =
+            (49779214489893931 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (49779214489893931 / 40000000000000000 : ℝ) ≤
+            (546795157671263833301 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6212540486078639 / 10000000000000000 : ℝ)) (q :=
+      (49779214489893931 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_576_7 :
+    Uω (aρ 7) (bρ 7) (43082559672831 / 64000000000000 : ℝ) ≤
+      (-(128251079179985541609 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (47895536563943987 / 80000000000000000 : ℝ) ≤
+      (-(128251079179985541609 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (47895536563943987 / 80000000000000000 : ℝ) =
+            (47895536563943987 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (47895536563943987 / 40000000000000000 : ℝ) ≤
+            (45035715895014458391 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5965282385351153 / 10000000000000000 : ℝ)) (q :=
+      (47895536563943987 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_576_8 :
+    Uω (aρ 8) (bρ 8) (43082559672831 / 64000000000000 : ℝ) ≤
+      (-(2827591000309687841803 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (45445255891575023 / 80000000000000000 : ℝ) ≤
+      (-(2827591000309687841803 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (45445255891575023 / 80000000000000000 : ℝ) =
+            (45445255891575023 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (45445255891575023 / 40000000000000000 : ℝ) ≤
+            (638144901190312158197 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (44033233348429 / 78125000000000 : ℝ)) (q :=
+      (45445255891575023 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_576_9 :
+    Uω (aρ 9) (bρ 9) (43082559672831 / 64000000000000 : ℝ) ≤
+      (-(3175687617663083390177 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (42389009167236191 / 80000000000000000 : ℝ) ≤
+      (-(3175687617663083390177 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (42389009167236191 / 80000000000000000 : ℝ) =
+            (42389009167236191 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (42389009167236191 / 40000000000000000 : ℝ) ≤
+            (290048283836916609823 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1303289453968551 / 2500000000000000 : ℝ)) (q :=
+      (42389009167236191 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_576_10 :
+    Uω (aρ 10) (bρ 10) (43082559672831 / 64000000000000 : ℝ) ≤
+      (-(3628258085777660704223 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (38720726184537623 / 80000000000000000 : ℝ) ≤
+      (-(3628258085777660704223 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (38720726184537623 / 80000000000000000 : ℝ) =
+            (38720726184537623 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (38720726184537623 / 20000000000000000 : ℝ) ≤
+            (3303213717222339295777 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2342323051847281 / 5000000000000000 : ℝ)) (q :=
+      (38720726184537623 / 80000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_576_11 :
+    Uω (aρ 11) (bρ 11) (43082559672831 / 64000000000000 : ℝ) ≤
+      (-(4210631164367377930677 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (34463486500790003 / 80000000000000000 : ℝ) ≤
+      (-(4210631164367377930677 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (34463486500790003 / 80000000000000000 : ℝ) =
+            (34463486500790003 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (34463486500790003 / 20000000000000000 : ℝ) ≤
+            (2720840638632622069323 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4038812808962657 / 10000000000000000 : ℝ)) (q :=
+      (34463486500790003 / 80000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_576_12 :
+    Uω (aρ 12) (bρ 12) (43082559672831 / 64000000000000 : ℝ) ≤
+      (-(7756712472024757971 / 7812500000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (29641263796719831 / 80000000000000000 : ℝ) ≤
+      (-(7756712472024757971 / 7812500000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (29641263796719831 / 80000000000000000 : ℝ) =
+            (29641263796719831 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (29641263796719831 / 20000000000000000 : ℝ) ≤
+            (3073712220162742029 / 7812500000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1628478614815057 / 5000000000000000 : ℝ)) (q :=
+      (29641263796719831 / 80000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_576_13 :
+    Uω (aρ 13) (bρ 13) (43082559672831 / 64000000000000 : ℝ) ≤
+      (-(5985733507685321610093 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24164386876963203 / 80000000000000000 : ℝ) ≤
+      (-(5985733507685321610093 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (24164386876963203 / 80000000000000000 : ℝ) =
+            (24164386876963203 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24164386876963203 / 20000000000000000 : ℝ) ≤
+            (945738295314678389907 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2287119741285957 / 10000000000000000 : ℝ)) (q :=
+      (24164386876963203 / 80000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_576_14 :
+    Uω (aρ 14) (bρ 14) (43082559672831 / 64000000000000 : ℝ) ≤
+      (-(15642119863862916581919 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16740227371697279 / 80000000000000000 : ℝ) ≤
+      (-(15642119863862916581919 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (16740227371697279 / 80000000000000000 : ℝ) =
+            (16740227371697279 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16740227371697279 / 10000000000000000 : ℝ) ≤
+            (5152295545137083418081 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (193751787503619 / 2500000000000000 : ℝ)) (q :=
+      (16740227371697279 / 80000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_576_15 :
+    Uω (aρ 15) (bρ 15) (43082559672831 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_576_16 :
+    Uω (aρ 16) (bρ 16) (43082559672831 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_576 :
+    Uρ (43082559672831 / 64000000000000 : ℝ) ≤
+      (-(211765684898012963771 / 312500000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (43082559672831 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2026979189518170120827 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_576_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (43082559672831 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(2044938511275047845963 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_576_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (43082559672831 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(4162085986235097085897 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_576_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (43082559672831 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(4283122962134978443037 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_576_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (43082559672831 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(4470155260503364583741 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_576_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (43082559672831 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1186072793078736166699 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_576_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (43082559672831 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(128251079179985541609 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_576_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (43082559672831 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(2827591000309687841803 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_576_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (43082559672831 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(3175687617663083390177 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_576_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (43082559672831 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(3628258085777660704223 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_576_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (43082559672831 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4210631164367377930677 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_576_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (43082559672831 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(7756712472024757971 / 7812500000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_576_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (43082559672831 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(5985733507685321610093 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_576_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (43082559672831 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(15642119863862916581919 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_576_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (43082559672831 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_576_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (43082559672831 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_576_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_577_1 :
+    Uω (aρ 1) (bρ 1) (2156723943323 / 3200000000000 : ℝ) ≤
+      (-(4041798013437740677517 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (104300638840089 / 156250000000000 : ℝ) ≤
+      (-(4041798013437740677517 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (104300638840089 / 156250000000000 : ℝ) = (104300638840089 / 78125000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (104300638840089 / 78125000000000 : ℝ) ≤
+            (2889673789562259322483 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6675216666732017 / 10000000000000000 : ℝ)) (q :=
+      (104300638840089 / 156250000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_577_2 :
+    Uω (aρ 2) (bρ 2) (2156723943323 / 3200000000000 : ℝ) ≤
+      (-(254854542385908595437 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13302673172915271 / 20000000000000000 : ℝ) ≤
+      (-(254854542385908595437 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (13302673172915271 / 20000000000000000 : ℝ) =
+            (13302673172915271 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13302673172915271 / 10000000000000000 : ℝ) ≤
+            (178362445301591404563 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (831397135078237 / 1250000000000000 : ℝ)) (q :=
+      (13302673172915271 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_577_3 :
+    Uω (aρ 3) (bρ 3) (2156723943323 / 3200000000000 : ℝ) ≤
+      (-(4149792502457496461927 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6603539825740297 / 10000000000000000 : ℝ) ≤
+      (-(4149792502457496461927 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (6603539825740297 / 10000000000000000 : ℝ) =
+            (6603539825740297 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6603539825740297 / 5000000000000000 : ℝ) ≤
+            (2781679300542503538073 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6602979665541219 / 10000000000000000 : ℝ)) (q :=
+      (6603539825740297 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_577_4 :
+    Uω (aρ 4) (bρ 4) (2156723943323 / 3200000000000 : ℝ) ≤
+      (-(4270677848971521522911 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1631048315956143 / 2500000000000000 : ℝ) ≤
+      (-(4270677848971521522911 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1631048315956143 / 2500000000000000 : ℝ) =
+            (1631048315956143 / 1250000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1631048315956143 / 1250000000000000 : ℝ) ≤
+            (2660793954028478477089 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6522580722774769 / 10000000000000000 : ℝ)) (q :=
+      (1631048315956143 / 2500000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_577_5 :
+    Uω (aρ 5) (bρ 5) (2156723943323 / 3200000000000 : ℝ) ≤
+      (-(4457470235029769292473 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12806914876599639 / 20000000000000000 : ℝ) ≤
+      (-(4457470235029769292473 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (12806914876599639 / 20000000000000000 : ℝ) =
+            (12806914876599639 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12806914876599639 / 10000000000000000 : ℝ) ≤
+            (2474001567970230707527 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (799912373882533 / 1250000000000000 : ℝ)) (q :=
+      (12806914876599639 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_577_6 :
+    Uω (aρ 6) (bρ 6) (2156723943323 / 3200000000000 : ℝ) ≤
+      (-(2365620827675887082541 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12461054090802993 / 20000000000000000 : ℝ) ≤
+      (-(2365620827675887082541 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (12461054090802993 / 20000000000000000 : ℝ) =
+            (12461054090802993 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12461054090802993 / 10000000000000000 : ℝ) ≤
+            (1100115073824112917459 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3110339290201809 / 5000000000000000 : ℝ)) (q :=
+      (12461054090802993 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_577_7 :
+    Uω (aρ 7) (bρ 7) (2156723943323 / 3200000000000 : ℝ) ≤
+      (-(1279113290058657057013 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11990167720064659 / 20000000000000000 : ℝ) ≤
+      (-(1279113290058657057013 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11990167720064659 / 20000000000000000 : ℝ) =
+            (11990167720064659 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11990167720064659 / 10000000000000000 : ℝ) ≤
+            (453754660691342942987 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1493363397606321 / 2500000000000000 : ℝ)) (q :=
+      (11990167720064659 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_577_8 :
+    Uω (aρ 8) (bρ 8) (2156723943323 / 3200000000000 : ℝ) ≤
+      (-(1128159861942127147153 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1137766635635763 / 2000000000000000 : ℝ) ≤
+      (-(1128159861942127147153 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1137766635635763 / 2000000000000000 : ℝ) =
+            (1137766635635763 / 1000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1137766635635763 / 1000000000000000 : ℝ) ≤
+            (258134498657872852847 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1128898775611651 / 2000000000000000 : ℝ)) (q :=
+      (1137766635635763 / 2000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_577_9 :
+    Uω (aρ 9) (bρ 9) (2156723943323 / 3200000000000 : ℝ) ≤
+      (-(6335826382699116403269 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (663358913761543 / 1250000000000000 : ℝ) ≤
+      (-(6335826382699116403269 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (663358913761543 / 1250000000000000 : ℝ) =
+            (663358913761543 / 625000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (663358913761543 / 625000000000000 : ℝ) ≤
+            (595645420300883596731 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5221535770245313 / 10000000000000000 : ℝ)) (q :=
+      (663358913761543 / 1250000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_577_10 :
+    Uω (aρ 10) (bρ 10) (2156723943323 / 3200000000000 : ℝ) ≤
+      (-(3619607599305394259193 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9696943697932837 / 20000000000000000 : ℝ) ≤
+      (-(3619607599305394259193 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9696943697932837 / 20000000000000000 : ℝ) =
+            (9696943697932837 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9696943697932837 / 5000000000000000 : ℝ) ≤
+            (3311864203694605740807 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2346647940744231 / 5000000000000000 : ℝ)) (q :=
+      (9696943697932837 / 20000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_577_11 :
+    Uω (aρ 11) (bρ 11) (2156723943323 / 3200000000000 : ℝ) ≤
+      (-(4200599557466197558131 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8633175185095221 / 20000000000000000 : ℝ) ≤
+      (-(4200599557466197558131 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8633175185095221 / 20000000000000000 : ℝ) =
+            (8633175185095221 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8633175185095221 / 5000000000000000 : ℝ) ≤
+            (2730872245533802441869 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2024001997427923 / 5000000000000000 : ℝ)) (q :=
+      (8633175185095221 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_577_12 :
+    Uω (aρ 12) (bρ 12) (2156723943323 / 3200000000000 : ℝ) ≤
+      (-(1980744718241945239729 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3714383566154969 / 10000000000000000 : ℝ) ≤
+      (-(1980744718241945239729 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3714383566154969 / 10000000000000000 : ℝ) =
+            (3714383566154969 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3714383566154969 / 2500000000000000 : ℝ) ≤
+            (791844002958054760271 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3267296038755563 / 10000000000000000 : ℝ)) (q :=
+      (3714383566154969 / 10000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_577_13 :
+    Uω (aρ 13) (bρ 13) (2156723943323 / 3200000000000 : ℝ) ≤
+      (-(596804995792744646391 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3031250076342061 / 10000000000000000 : ℝ) ≤
+      (-(596804995792744646391 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3031250076342061 / 10000000000000000 : ℝ) =
+            (3031250076342061 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3031250076342061 / 2500000000000000 : ℝ) ≤
+            (96342184507255353609 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2300410800724747 / 10000000000000000 : ℝ)) (q :=
+      (3031250076342061 / 10000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_577_14 :
+    Uω (aρ 14) (bρ 14) (2156723943323 / 3200000000000 : ℝ) ≤
+      (-(3884938065980386746899 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4228118294220143 / 20000000000000000 : ℝ) ≤
+      (-(3884938065980386746899 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4228118294220143 / 20000000000000000 : ℝ) =
+            (4228118294220143 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4228118294220143 / 2500000000000000 : ℝ) ≤
+            (1313665786269613253101 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (101244528413221 / 1250000000000000 : ℝ)) (q :=
+      (4228118294220143 / 20000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_577_15 :
+    Uω (aρ 15) (bρ 15) (2156723943323 / 3200000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_577_16 :
+    Uω (aρ 16) (bρ 16) (2156723943323 / 3200000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_577 :
+    Uρ (2156723943323 / 3200000000000 : ℝ) ≤
+      (-(6757620034057452785879 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (2156723943323 / 3200000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(4041798013437740677517 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_577_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (2156723943323 / 3200000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(254854542385908595437 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_577_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (2156723943323 / 3200000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(4149792502457496461927 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_577_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (2156723943323 / 3200000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(4270677848971521522911 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_577_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (2156723943323 / 3200000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(4457470235029769292473 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_577_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (2156723943323 / 3200000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(2365620827675887082541 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_577_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (2156723943323 / 3200000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(1279113290058657057013 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_577_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (2156723943323 / 3200000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(1128159861942127147153 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_577_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (2156723943323 / 3200000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(6335826382699116403269 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_577_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (2156723943323 / 3200000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(3619607599305394259193 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_577_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (2156723943323 / 3200000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4200599557466197558131 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_577_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (2156723943323 / 3200000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(1980744718241945239729 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_577_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (2156723943323 / 3200000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) * (-(596804995792744646391 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_577_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (2156723943323 / 3200000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(3884938065980386746899 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_577_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (2156723943323 / 3200000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_577_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (2156723943323 / 3200000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_577_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_578_1 :
+    Uω (aρ 1) (bρ 1) (43186398060089 / 64000000000000 : ℝ) ≤
+      (-(4029652417436910217643 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (53466826313342649 / 80000000000000000 : ℝ) ≤
+      (-(4029652417436910217643 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (53466826313342649 / 80000000000000000 : ℝ) =
+            (53466826313342649 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (53466826313342649 / 40000000000000000 : ℝ) ≤
+            (2901819385563089782357 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1670832274882939 / 2500000000000000 : ℝ)) (q :=
+      (53466826313342649 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_578_2 :
+    Uω (aρ 2) (bρ 2) (43186398060089 / 64000000000000 : ℝ) ≤
+      (-(4065483210959910322499 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (53275593238182609 / 80000000000000000 : ℝ) ≤
+      (-(4065483210959910322499 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (53275593238182609 / 80000000000000000 : ℝ) =
+            (53275593238182609 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (53275593238182609 / 40000000000000000 : ℝ) ≤
+            (2865988592040089677501 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3329644921625873 / 5000000000000000 : ℝ)) (q :=
+      (53275593238182609 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_578_3 :
+    Uω (aρ 3) (bρ 3) (43186398060089 / 64000000000000 : ℝ) ≤
+      (-(4137514115657745432343 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (52893223096872921 / 80000000000000000 : ℝ) ≤
+      (-(4137514115657745432343 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (52893223096872921 / 80000000000000000 : ℝ) =
+            (52893223096872921 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (52893223096872921 / 40000000000000000 : ℝ) ≤
+            (2793957687342254567657 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1652773353568581 / 2500000000000000 : ℝ)) (q :=
+      (52893223096872921 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_578_4 :
+    Uω (aρ 4) (bρ 4) (43186398060089 / 64000000000000 : ℝ) ≤
+      (-(2129124106141065259827 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (52258461127311233 / 80000000000000000 : ℝ) ≤
+      (-(2129124106141065259827 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (52258461127311233 / 80000000000000000 : ℝ) =
+            (52258461127311233 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (52258461127311233 / 40000000000000000 : ℝ) ≤
+            (1336611795358934740173 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3265348551474451 / 5000000000000000 : ℝ)) (q :=
+      (52258461127311233 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_578_5 :
+    Uω (aρ 5) (bρ 5) (43186398060089 / 64000000000000 : ℝ) ≤
+      (-(1111200325254916125143 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (51292600618223461 / 80000000000000000 : ℝ) ≤
+      (-(1111200325254916125143 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (51292600618223461 / 80000000000000000 : ℝ) =
+            (51292600618223461 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (51292600618223461 / 40000000000000000 : ℝ) ≤
+            (621667625495083874857 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6407421895011959 / 10000000000000000 : ℝ)) (q :=
+      (51292600618223461 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_578_6 :
+    Uω (aρ 6) (bρ 6) (43186398060089 / 64000000000000 : ℝ) ≤
+      (-(1179552299741664472367 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9981843593624249 / 16000000000000000 : ℝ) ≤
+      (-(1179552299741664472367 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (9981843593624249 / 16000000000000000 : ℝ) =
+            (9981843593624249 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9981843593624249 / 8000000000000000 : ℝ) ≤
+            (553315651008335527633 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1245763321525281 / 2000000000000000 : ℝ)) (q :=
+      (9981843593624249 / 16000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_578_7 :
+    Uω (aρ 7) (bρ 7) (43186398060089 / 64000000000000 : ℝ) ≤
+      (-(5102881730426360024451 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9605160911191557 / 16000000000000000 : ℝ) ≤
+      (-(5102881730426360024451 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (9605160911191557 / 16000000000000000 : ℝ) =
+            (9605160911191557 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9605160911191557 / 8000000000000000 : ℝ) ≤
+            (1828590072573639975549 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (299081231767277 / 500000000000000 : ℝ)) (q :=
+      (9605160911191557 / 16000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_578_8 :
+    Uω (aρ 8) (bρ 8) (43186398060089 / 64000000000000 : ℝ) ≤
+      (-(1125287519865866481021 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (45576073482376601 / 80000000000000000 : ℝ) ≤
+      (-(1125287519865866481021 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (45576073482376601 / 80000000000000000 : ℝ) =
+            (45576073482376601 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (45576073482376601 / 40000000000000000 : ℝ) ≤
+            (261006840734133518979 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1413183379572561 / 2500000000000000 : ℝ)) (q :=
+      (45576073482376601 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_578_9 :
+    Uω (aρ 9) (bρ 9) (43186398060089 / 64000000000000 : ℝ) ≤
+      (-(6320302456974391084841 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (42520928444658101 / 80000000000000000 : ℝ) ≤
+      (-(6320302456974391084841 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (42520928444658101 / 80000000000000000 : ℝ) =
+            (42520928444658101 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (42520928444658101 / 40000000000000000 : ℝ) ≤
+            (611169346025608915159 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5229912887220619 / 10000000000000000 : ℝ)) (q :=
+      (42520928444658101 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_578_10 :
+    Uω (aρ 10) (bρ 10) (43186398060089 / 64000000000000 : ℝ) ≤
+      (-(3610973039310716154581 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (38854815735675529 / 80000000000000000 : ℝ) ≤
+      (-(3610973039310716154581 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (38854815735675529 / 80000000000000000 : ℝ) =
+            (38854815735675529 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (38854815735675529 / 20000000000000000 : ℝ) ≤
+            (3320498763689283845419 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (587742967933747 / 1250000000000000 : ℝ)) (q :=
+      (38854815735675529 / 80000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_578_11 :
+    Uω (aρ 11) (bρ 11) (43186398060089 / 64000000000000 : ℝ) ≤
+      (-(8381181392621563129199 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (34601896575832069 / 80000000000000000 : ℝ) ≤
+      (-(8381181392621563129199 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (34601896575832069 / 80000000000000000 : ℝ) =
+            (34601896575832069 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (34601896575832069 / 20000000000000000 : ℝ) ≤
+            (5481762213378436870801 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4057190579714111 / 10000000000000000 : ℝ)) (q :=
+      (34601896575832069 / 80000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_578_12 :
+    Uω (aρ 12) (bρ 12) (43186398060089 / 64000000000000 : ℝ) ≤
+      (-(9878933738315134485679 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (29788823127510321 / 80000000000000000 : ℝ) ≤
+      (-(9878933738315134485679 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (29788823127510321 / 80000000000000000 : ℝ) =
+            (29788823127510321 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (29788823127510321 / 20000000000000000 : ℝ) ≤
+            (3984009867684865514321 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1638811157159337 / 5000000000000000 : ℝ)) (q :=
+      (29788823127510321 / 80000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_578_13 :
+    Uω (aρ 13) (bρ 13) (43186398060089 / 64000000000000 : ℝ) ≤
+      (-(2975234050794647865737 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (973416908640921 / 3200000000000000 : ℝ) ≤
+      (-(2975234050794647865737 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (973416908640921 / 3200000000000000 : ℝ) =
+            (973416908640921 / 800000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (973416908640921 / 800000000000000 : ℝ) ≤
+            (490501850705352134263 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (46273079060837 / 200000000000000 : ℝ)) (q :=
+      (973416908640921 / 3200000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13]) (by norm_num [bρ_13])
+      (by norm_num) (by norm_num [aρ_13, bρ_13]) (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_578_14 :
+    Uω (aρ 14) (bρ 14) (43186398060089 / 64000000000000 : ℝ) ≤
+      (-(15441627376211997410399 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17079243507897601 / 80000000000000000 : ℝ) ≤
+      (-(15441627376211997410399 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (17079243507897601 / 80000000000000000 : ℝ) =
+            (17079243507897601 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17079243507897601 / 10000000000000000 : ℝ) ≤
+            (5352788032788002589601 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (421768218027747 / 5000000000000000 : ℝ)) (q :=
+      (17079243507897601 / 80000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_578_15 :
+    Uω (aρ 15) (bρ 15) (43186398060089 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_578_16 :
+    Uω (aρ 16) (bρ 16) (43186398060089 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_578 :
+    Uρ (43186398060089 / 64000000000000 : ℝ) ≤
+      (-(6738960652673771386763 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (43186398060089 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(4029652417436910217643 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_578_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (43186398060089 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(4065483210959910322499 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_578_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (43186398060089 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(4137514115657745432343 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_578_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (43186398060089 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(2129124106141065259827 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_578_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (43186398060089 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1111200325254916125143 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_578_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (43186398060089 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1179552299741664472367 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_578_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (43186398060089 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(5102881730426360024451 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_578_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (43186398060089 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(1125287519865866481021 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_578_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (43186398060089 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(6320302456974391084841 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_578_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (43186398060089 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(3610973039310716154581 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_578_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (43186398060089 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(8381181392621563129199 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_578_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (43186398060089 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(9878933738315134485679 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_578_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (43186398060089 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(2975234050794647865737 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_578_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (43186398060089 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(15441627376211997410399 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_578_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (43186398060089 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_578_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (43186398060089 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_578_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_579_1 :
+    Uω (aρ 1) (bρ 1) (21619158626859 / 32000000000000 : ℝ) ≤
+      (-(2008760777599865418403 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (26765862769994741 / 40000000000000000 : ℝ) ≤
+      (-(2008760777599865418403 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (26765862769994741 / 40000000000000000 : ℝ) =
+            (26765862769994741 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26765862769994741 / 20000000000000000 : ℝ) ≤
+            (1456975123900134581597 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6691441532188933 / 10000000000000000 : ℝ)) (q :=
+      (26765862769994741 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_579_2 :
+    Uω (aρ 2) (bρ 2) (21619158626859 / 32000000000000 : ℝ) ≤
+      (-(506663573084745193583 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (26670246890460647 / 40000000000000000 : ℝ) ≤
+      (-(506663573084745193583 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (26670246890460647 / 40000000000000000 : ℝ) =
+            (26670246890460647 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26670246890460647 / 20000000000000000 : ℝ) ≤
+            (359770402290254806417 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3333701302465943 / 5000000000000000 : ℝ)) (q :=
+      (26670246890460647 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_579_3 :
+    Uω (aρ 3) (bρ 3) (21619158626859 / 32000000000000 : ℝ) ≤
+      (-(4125250788795458835007 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (26479063787171797 / 40000000000000000 : ℝ) ≤
+      (-(4125250788795458835007 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (26479063787171797 / 40000000000000000 : ℝ) =
+            (26479063787171797 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26479063787171797 / 20000000000000000 : ℝ) ≤
+            (2806221014204541164993 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6619207159637461 / 10000000000000000 : ℝ)) (q :=
+      (26479063787171797 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_579_4 :
+    Uω (aρ 4) (bρ 4) (21619158626859 / 32000000000000 : ℝ) ≤
+      (-(2122917006801633900917 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (26161688052127029 / 40000000000000000 : ℝ) ≤
+      (-(2122917006801633900917 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (26161688052127029 / 40000000000000000 : ℝ) =
+            (26161688052127029 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26161688052127029 / 20000000000000000 : ℝ) ≤
+            (1342818894698366099083 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6538813473180077 / 10000000000000000 : ℝ)) (q :=
+      (26161688052127029 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_579_5 :
+    Uω (aρ 5) (bρ 5) (21619158626859 / 32000000000000 : ℝ) ≤
+      (-(1108037104411686499443 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5135754162345937 / 8000000000000000 : ℝ) ≤
+      (-(1108037104411686499443 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5135754162345937 / 8000000000000000 : ℝ) =
+            (5135754162345937 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5135754162345937 / 4000000000000000 : ℝ) ≤
+            (624830846338313500557 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1283108954463281 / 2000000000000000 : ℝ)) (q :=
+      (5135754162345937 / 8000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_579_6 :
+    Uω (aρ 6) (bρ 6) (21619158626859 / 32000000000000 : ℝ) ≤
+      (-(4705193758468929470479 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24987109652836207 / 40000000000000000 : ℝ) ≤
+      (-(4705193758468929470479 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (24987109652836207 / 40000000000000000 : ℝ) =
+            (24987109652836207 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24987109652836207 / 20000000000000000 : ℝ) ≤
+            (2226278044531070529521 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3118477284004833 / 5000000000000000 : ℝ)) (q :=
+      (24987109652836207 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_579_7 :
+    Uω (aρ 7) (bρ 7) (21619158626859 / 32000000000000 : ℝ) ≤
+      (-(508932882669316172813 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24045468796831573 / 40000000000000000 : ℝ) ≤
+      (-(508932882669316172813 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (24045468796831573 / 40000000000000000 : ℝ) =
+            (24045468796831573 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24045468796831573 / 20000000000000000 : ℝ) ≤
+            (184214297630683827187 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5989795520767349 / 10000000000000000 : ℝ)) (q :=
+      (24045468796831573 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_579_8 :
+    Uω (aρ 8) (bρ 8) (21619158626859 / 32000000000000 : ℝ) ≤
+      (-(5612096807420582264481 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22820740034431137 / 40000000000000000 : ℝ) ≤
+      (-(5612096807420582264481 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (22820740034431137 / 40000000000000000 : ℝ) =
+            (22820740034431137 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22820740034431137 / 20000000000000000 : ℝ) ≤
+            (1319374995579417735519 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5660972790907131 / 10000000000000000 : ℝ)) (q :=
+      (22820740034431137 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_579_9 :
+    Uω (aρ 9) (bρ 9) (21619158626859 / 32000000000000 : ℝ) ≤
+      (-(6304803375883815152733 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21293441537532141 / 40000000000000000 : ℝ) ≤
+      (-(6304803375883815152733 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (21293441537532141 / 40000000000000000 : ℝ) =
+            (21293441537532141 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21293441537532141 / 20000000000000000 : ℝ) ≤
+            (626668427116184847267 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5238289170817633 / 10000000000000000 : ℝ)) (q :=
+      (21293441537532141 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_579_10 :
+    Uω (aρ 10) (bρ 10) (21619158626859 / 32000000000000 : ℝ) ≤
+      (-(1440941737503140562409 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19460924529287769 / 40000000000000000 : ℝ) ≤
+      (-(1440941737503140562409 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (19460924529287769 / 40000000000000000 : ℝ) =
+            (19460924529287769 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19460924529287769 / 10000000000000000 : ℝ) ≤
+            (1331646983696859437591 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4710589700190447 / 10000000000000000 : ℝ)) (q :=
+      (19460924529287769 / 40000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_579_11 :
+    Uω (aρ 11) (bρ 11) (21619158626859 / 32000000000000 : ℝ) ≤
+      (-(4180604466712982170821 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3467107413187723 / 8000000000000000 : ℝ) ≤
+      (-(4180604466712982170821 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3467107413187723 / 8000000000000000 : ℝ) =
+            (3467107413187723 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3467107413187723 / 2000000000000000 : ℝ) ≤
+            (2750867336287017829179 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (406637259472087 / 1000000000000000 : ℝ)) (q :=
+      (3467107413187723 / 8000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_579_12 :
+    Uω (aρ 12) (bρ 12) (21619158626859 / 32000000000000 : ℝ) ≤
+      (-(9854221817320635336431 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14931264031947399 / 40000000000000000 : ℝ) ≤
+      (-(9854221817320635336431 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (14931264031947399 / 40000000000000000 : ℝ) =
+            (14931264031947399 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14931264031947399 / 10000000000000000 : ℝ) ≤
+            (4008721788679364663569 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1643968087205131 / 5000000000000000 : ℝ)) (q :=
+      (14931264031947399 / 40000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_579_13 :
+    Uω (aρ 13) (bρ 13) (21619158626859 / 32000000000000 : ℝ) ≤
+      (-(741623302233828881753 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12210328232377843 / 40000000000000000 : ℝ) ≤
+      (-(741623302233828881753 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (12210328232377843 / 40000000000000000 : ℝ) =
+            (12210328232377843 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12210328232377843 / 10000000000000000 : ℝ) ≤
+            (124810673141171118247 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (581712504055121 / 2500000000000000 : ℝ)) (q :=
+      (12210328232377843 / 40000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_579_14 :
+    Uω (aρ 14) (bρ 14) (21619158626859 / 32000000000000 : ℝ) ≤
+      (-(15347266115033912343549 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8620584086464821 / 40000000000000000 : ℝ) ≤
+      (-(15347266115033912343549 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (8620584086464821 / 40000000000000000 : ℝ) =
+            (8620584086464821 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8620584086464821 / 5000000000000000 : ℝ) ≤
+            (5447149293966087656451 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (875905228308973 / 10000000000000000 : ℝ)) (q :=
+      (8620584086464821 / 40000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_579_15 :
+    Uω (aρ 15) (bρ 15) (21619158626859 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_579_16 :
+    Uω (aρ 16) (bρ 16) (21619158626859 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_579 :
+    Uρ (21619158626859 / 32000000000000 : ℝ) ≤
+      (-(6720502213295018558049 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (21619158626859 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2008760777599865418403 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_579_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (21619158626859 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(506663573084745193583 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_579_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (21619158626859 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(4125250788795458835007 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_579_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (21619158626859 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(2122917006801633900917 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_579_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (21619158626859 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1108037104411686499443 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_579_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (21619158626859 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(4705193758468929470479 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_579_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (21619158626859 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(508932882669316172813 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_579_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (21619158626859 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(5612096807420582264481 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_579_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (21619158626859 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(6304803375883815152733 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_579_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (21619158626859 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(1440941737503140562409 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_579_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (21619158626859 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4180604466712982170821 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_579_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (21619158626859 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(9854221817320635336431 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_579_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (21619158626859 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) * (-(741623302233828881753 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_579_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (21619158626859 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(15347266115033912343549 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_579_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (21619158626859 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_579_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (21619158626859 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_579_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/U48.lean b/LeanPool/Zeta5Irrational/Table/U48.lean
new file mode 100644
index 0000000000..9d659caf5e
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/U48.lean
@@ -0,0 +1,5063 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-! # Certified arcsine potential bounds (U48) -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma U_580_1 :
+    Uω (aρ 1) (bρ 1) (43290236447347 / 64000000000000 : ℝ) ≤
+      (-(4005405391022337117569 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (53596624766068143 / 80000000000000000 : ℝ) ≤
+      (-(4005405391022337117569 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (53596624766068143 / 80000000000000000 : ℝ) =
+            (53596624766068143 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (53596624766068143 / 40000000000000000 : ℝ) ≤
+            (2926066411977662882431 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6699553964704067 / 10000000000000000 : ℝ)) (q :=
+      (53596624766068143 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_580_2 :
+    Uω (aρ 2) (bρ 2) (43290236447347 / 64000000000000 : ℝ) ≤
+      (-(2020574381616309005321 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (53405394319890923 / 80000000000000000 : ℝ) ≤
+      (-(2020574381616309005321 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (53405394319890923 / 80000000000000000 : ℝ) =
+            (53405394319890923 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (53405394319890923 / 40000000000000000 : ℝ) ≤
+            (1445161519883690994679 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3337757682834881 / 5000000000000000 : ℝ)) (q :=
+      (53405394319890923 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_580_3 :
+    Uω (aρ 3) (bρ 3) (43290236447347 / 64000000000000 : ℝ) ≤
+      (-(4113002484966424445947 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (53023032038383907 / 80000000000000000 : ℝ) ≤
+      (-(4113002484966424445947 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (53023032038383907 / 80000000000000000 : ℝ) =
+            (53023032038383907 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (53023032038383907 / 40000000000000000 : ℝ) ≤
+            (2818469318033575554053 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (25887972272043 / 39062500000000 : ℝ)) (q :=
+      (53023032038383907 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_580_4 :
+    Uω (aρ 4) (bρ 4) (43290236447347 / 64000000000000 : ℝ) ≤
+      (-(1058358803653668769017 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (52388291041572979 / 80000000000000000 : ℝ) ≤
+      (-(1058358803653668769017 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (52388291041572979 / 80000000000000000 : ℝ) =
+            (52388291041572979 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (52388291041572979 / 40000000000000000 : ℝ) ≤
+            (674509147096331230983 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1636732458376319 / 2500000000000000 : ℝ)) (q :=
+      (52388291041572979 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_580_5 :
+    Uω (aρ 5) (bρ 5) (43290236447347 / 64000000000000 : ℝ) ≤
+      (-(220975577212004283547 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10284496504502127 / 16000000000000000 : ℝ) ≤
+      (-(220975577212004283547 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10284496504502127 / 16000000000000000 : ℝ) =
+            (10284496504502127 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10284496504502127 / 8000000000000000 : ℝ) ≤
+            (125598012937995716453 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (642366762307469 / 1000000000000000 : ℝ)) (q :=
+      (10284496504502127 / 16000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_580_6 :
+    Uω (aρ 6) (bρ 6) (43290236447347 / 64000000000000 : ℝ) ≤
+      (-(4692195289343839263543 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (50039220376910659 / 80000000000000000 : ℝ) ≤
+      (-(4692195289343839263543 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (50039220376910659 / 80000000000000000 : ℝ) =
+            (50039220376910659 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (50039220376910659 / 40000000000000000 : ℝ) ≤
+            (2239276513656160736457 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (780636557726837 / 1250000000000000 : ℝ)) (q :=
+      (50039220376910659 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_580_7 :
+    Uω (aρ 7) (bρ 7) (43290236447347 / 64000000000000 : ℝ) ≤
+      (-(5075794398165442137843 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (48156069995982091 / 80000000000000000 : ℝ) ≤
+      (-(5075794398165442137843 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (48156069995982091 / 80000000000000000 : ℝ) =
+            (48156069995982091 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (48156069995982091 / 40000000000000000 : ℝ) ≤
+            (1855677404834557862157 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2998983123671277 / 5000000000000000 : ℝ)) (q :=
+      (48156069995982091 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_580_8 :
+    Uω (aρ 8) (bρ 8) (43290236447347 / 64000000000000 : ℝ) ≤
+      (-(1399444218052103159183 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9141377038259751 / 16000000000000000 : ℝ) ≤
+      (-(1399444218052103159183 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (9141377038259751 / 16000000000000000 : ℝ) =
+            (9141377038259751 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9141377038259751 / 8000000000000000 : ℝ) ≤
+            (333423732697896840817 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (141730292437793 / 250000000000000 : ℝ)) (q :=
+      (9141377038259751 / 16000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_580_9 :
+    Uω (aρ 9) (bρ 9) (43290236447347 / 64000000000000 : ℝ) ≤
+      (-(6289329057577266237567 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (42652834387923127 / 80000000000000000 : ℝ) ≤
+      (-(6289329057577266237567 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (42652834387923127 / 80000000000000000 : ℝ) =
+            (42652834387923127 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (42652834387923127 / 40000000000000000 : ℝ) ≤
+            (642142745422733762433 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5246664625027813 / 10000000000000000 : ℝ)) (q :=
+      (42652834387923127 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_580_10 :
+    Uω (aρ 10) (bρ 10) (43290236447347 / 64000000000000 : ℝ) ≤
+      (-(7187502901981604046617 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (38988874802318187 / 80000000000000000 : ℝ) ≤
+      (-(7187502901981604046617 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (38988874802318187 / 80000000000000000 : ℝ) =
+            (38988874802318187 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (38988874802318187 / 20000000000000000 : ℝ) ≤
+            (6675440704018395953383 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2359616881060789 / 5000000000000000 : ℝ)) (q :=
+      (38988874802318187 / 80000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_580_11 :
+    Uω (aρ 11) (bρ 11) (43290236447347 / 64000000000000 : ℝ) ≤
+      (-(8341281510819861628441 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (34740233532064187 / 80000000000000000 : ℝ) ≤
+      (-(8341281510819861628441 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (34740233532064187 / 80000000000000000 : ℝ) =
+            (34740233532064187 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (34740233532064187 / 20000000000000000 : ℝ) ≤
+            (5521662095180138371559 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2037775035381539 / 5000000000000000 : ℝ)) (q :=
+      (34740233532064187 / 80000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_580_12 :
+    Uω (aρ 12) (bρ 12) (43290236447347 / 64000000000000 : ℝ) ≤
+      (-(4914793623665919405137 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (29936183804283567 / 80000000000000000 : ℝ) ≤
+      (-(4914793623665919405137 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (29936183804283567 / 80000000000000000 : ℝ) =
+            (29936183804283567 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (29936183804283567 / 20000000000000000 : ℝ) ≤
+            (2016678179334080594863 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3298237735502923 / 10000000000000000 : ℝ)) (q :=
+      (29936183804283567 / 80000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_580_13 :
+    Uω (aρ 13) (bρ 13) (43290236447347 / 64000000000000 : ℝ) ≤
+      (-(11831206847900431713959 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (980228201742283 / 3200000000000000 : ℝ) ≤
+      (-(11831206847900431713959 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (980228201742283 / 3200000000000000 : ℝ) =
+            (980228201742283 / 800000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (980228201742283 / 800000000000000 : ℝ) ≤
+            (2031736758099568286041 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (23399997869163 / 100000000000000 : ℝ)) (q :=
+      (980228201742283 / 3200000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13]) (by norm_num [bρ_13])
+      (by norm_num) (by norm_num [aρ_13, bρ_13]) (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_580_14 :
+    Uω (aρ 14) (bρ 14) (43290236447347 / 64000000000000 : ℝ) ≤
+      (-(7628136779712544570147 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17398765891887423 / 80000000000000000 : ℝ) ≤
+      (-(7628136779712544570147 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (17398765891887423 / 80000000000000000 : ℝ) =
+            (17398765891887423 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17398765891887423 / 10000000000000000 : ℝ) ≤
+            (2769070924787455429853 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (907192284043887 / 10000000000000000 : ℝ)) (q :=
+      (17398765891887423 / 80000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_580_15 :
+    Uω (aρ 15) (bρ 15) (43290236447347 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_580_16 :
+    Uω (aρ 16) (bρ 16) (43290236447347 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_580 :
+    Uρ (43290236447347 / 64000000000000 : ℝ) ≤
+      (-(837778362582075301389 / 1250000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (43290236447347 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(4005405391022337117569 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_580_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (43290236447347 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(2020574381616309005321 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_580_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (43290236447347 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(4113002484966424445947 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_580_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (43290236447347 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1058358803653668769017 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_580_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (43290236447347 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(220975577212004283547 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_580_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (43290236447347 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(4692195289343839263543 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_580_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (43290236447347 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(5075794398165442137843 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_580_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (43290236447347 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(1399444218052103159183 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_580_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (43290236447347 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(6289329057577266237567 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_580_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (43290236447347 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(7187502901981604046617 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_580_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (43290236447347 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(8341281510819861628441 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_580_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (43290236447347 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(4914793623665919405137 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_580_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (43290236447347 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(11831206847900431713959 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_580_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (43290236447347 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(7628136779712544570147 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_580_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (43290236447347 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_580_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (43290236447347 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_580_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_581_1 :
+    Uω (aρ 1) (bρ 1) (2708884727561 / 4000000000000 : ℝ) ≤
+      (-(3993303889330489471621 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3353845249473793 / 5000000000000000 : ℝ) ≤
+      (-(3993303889330489471621 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (3353845249473793 / 5000000000000000 : ℝ) =
+            (3353845249473793 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3353845249473793 / 2500000000000000 : ℝ) ≤
+            (2938167913669510528379 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (838458299634709 / 1250000000000000 : ℝ)) (q :=
+      (3353845249473793 / 5000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_581_2 :
+    Uω (aρ 2) (bρ 2) (2708884727561 / 4000000000000 : ℝ) ≤
+      (-(4029003710659338433683 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3341893428444077 / 5000000000000000 : ℝ) ≤
+      (-(4029003710659338433683 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (3341893428444077 / 5000000000000000 : ℝ) =
+            (3341893428444077 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3341893428444077 / 2500000000000000 : ℝ) ≤
+            (2902468092340661566317 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (835453515683601 / 1250000000000000 : ℝ)) (q :=
+      (3341893428444077 / 5000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_581_3 :
+    Uω (aρ 3) (bρ 3) (2708884727561 / 4000000000000 : ℝ) ≤
+      (-(4100769167401937979183 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13271984122260781 / 20000000000000000 : ℝ) ≤
+      (-(4100769167401937979183 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (13271984122260781 / 20000000000000000 : ℝ) =
+            (13271984122260781 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13271984122260781 / 10000000000000000 : ℝ) ≤
+            (2830702635598062020817 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6635434640303281 / 10000000000000000 : ℝ)) (q :=
+      (13271984122260781 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_581_4 :
+    Uω (aρ 4) (bρ 4) (2708884727561 / 4000000000000 : ℝ) ≤
+      (-(4221051777138668487973 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13113301484853793 / 20000000000000000 : ℝ) ≤
+      (-(4221051777138668487973 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (13113301484853793 / 20000000000000000 : ℝ) =
+            (13113301484853793 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13113301484853793 / 10000000000000000 : ℝ) ≤
+            (2710420025861331512027 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6555046183961293 / 10000000000000000 : ℝ)) (q :=
+      (13113301484853793 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_581_5 :
+    Uω (aρ 5) (bρ 5) (2708884727561 / 4000000000000 : ℝ) ≤
+      (-(4406890640283341706941 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3217963957236223 / 5000000000000000 : ℝ) ≤
+      (-(4406890640283341706941 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (3217963957236223 / 5000000000000000 : ℝ) =
+            (3217963957236223 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3217963957236223 / 2500000000000000 : ℝ) ≤
+            (2524581162716658293059 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (25124181435107 / 39062500000000 : ℝ)) (q :=
+      (3217963957236223 / 5000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_581_6 :
+    Uω (aρ 6) (bρ 6) (2708884727561 / 4000000000000 : ℝ) ≤
+      (-(292450859203226838757 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (782878455982433 / 1250000000000000 : ℝ) ≤
+      (-(292450859203226838757 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (782878455982433 / 1250000000000000 : ℝ) =
+            (782878455982433 / 625000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (782878455982433 / 625000000000000 : ℝ) ≤
+            (140766128484273161243 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1563307572325357 / 2500000000000000 : ℝ)) (q :=
+      (782878455982433 / 1250000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_581_7 :
+    Uω (aρ 7) (bρ 7) (2708884727561 / 4000000000000 : ℝ) ≤
+      (-(2531139197092148791153 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1506912555172117 / 2500000000000000 : ℝ) ≤
+      (-(2531139197092148791153 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1506912555172117 / 2500000000000000 : ℝ) =
+            (1506912555172117 / 1250000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1506912555172117 / 1250000000000000 : ℝ) ≤
+            (934596704407851208847 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1501534203929859 / 2500000000000000 : ℝ)) (q :=
+      (1506912555172117 / 2500000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_581_8 :
+    Uω (aρ 8) (bρ 8) (2708884727561 / 4000000000000 : ℝ) ≤
+      (-(5583477732194065256547 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5721536107007471 / 10000000000000000 : ℝ) ≤
+      (-(5583477732194065256547 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5721536107007471 / 10000000000000000 : ℝ) =
+            (5721536107007471 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5721536107007471 / 5000000000000000 : ℝ) ≤
+            (1347994070805934743453 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2838725119848721 / 5000000000000000 : ℝ)) (q :=
+      (5721536107007471 / 10000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_581_9 :
+    Uω (aρ 9) (bρ 9) (2708884727561 / 4000000000000 : ℝ) ≤
+      (-(1568469855154944946347 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2669923899943569 / 5000000000000000 : ℝ) ≤
+      (-(1568469855154944946347 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2669923899943569 / 5000000000000000 : ℝ) =
+            (2669923899943569 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2669923899943569 / 2500000000000000 : ℝ) ≤
+            (164398095595055053653 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (656879906727097 / 1250000000000000 : ℝ)) (q :=
+      (2669923899943569 / 5000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_581_10 :
+    Uω (aρ 10) (bρ 10) (2708884727561 / 4000000000000 : ℝ) ≤
+      (-(7170328599460265511283 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4881986626059351 / 10000000000000000 : ℝ) ≤
+      (-(7170328599460265511283 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4881986626059351 / 10000000000000000 : ℝ) =
+            (4881986626059351 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4881986626059351 / 2500000000000000 : ℝ) ≤
+            (6692615006539734488717 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2363937969828101 / 5000000000000000 : ℝ)) (q :=
+      (4881986626059351 / 10000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_581_11 :
+    Uω (aρ 11) (bρ 11) (2708884727561 / 4000000000000 : ℝ) ≤
+      (-(8321398900107086323937 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1087792965586551 / 2500000000000000 : ℝ) ≤
+      (-(8321398900107086323937 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1087792965586551 / 2500000000000000 : ℝ) =
+            (1087792965586551 / 625000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1087792965586551 / 625000000000000 : ℝ) ≤
+            (5541544705892913676063 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1021180759608727 / 2500000000000000 : ℝ)) (q :=
+      (1087792965586551 / 2500000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_581_12 :
+    Uω (aρ 12) (bρ 12) (2708884727561 / 4000000000000 : ℝ) ≤
+      (-(39220117818480693483 / 40000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (937805962756639 / 2500000000000000 : ℝ) ≤
+      (-(39220117818480693483 / 40000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (937805962756639 / 2500000000000000 : ℝ) =
+            (937805962756639 / 625000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (937805962756639 / 625000000000000 : ℝ) ≤
+            (16231656605519306517 / 40000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (827131778120153 / 2500000000000000 : ℝ)) (q :=
+      (937805962756639 / 2500000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12]) (by norm_num [bρ_12])
+      (by norm_num) (by norm_num [aρ_12, bρ_12]) (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_581_13 :
+    Uω (aρ 13) (bρ 13) (2708884727561 / 4000000000000 : ℝ) ≤
+      (-(11796635347326533752359 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1229528577838203 / 4000000000000000 : ℝ) ≤
+      (-(11796635347326533752359 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1229528577838203 / 4000000000000000 : ℝ) =
+            (1229528577838203 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1229528577838203 / 1000000000000000 : ℝ) ≤
+            (2066308258673466247641 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (470620808242703 / 2000000000000000 : ℝ)) (q :=
+      (1229528577838203 / 4000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_581_14 :
+    Uω (aρ 14) (bρ 14) (2708884727561 / 4000000000000 : ℝ) ≤
+      (-(1516831959140549096717 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (219405873684427 / 1000000000000000 : ℝ) ≤
+      (-(1516831959140549096717 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (219405873684427 / 1000000000000000 : ℝ) =
+            (219405873684427 / 125000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (219405873684427 / 125000000000000 : ℝ) ≤
+            (562609581759450903283 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (23437647768901 / 250000000000000 : ℝ)) (q :=
+      (219405873684427 / 1000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14]) (by norm_num [bρ_14])
+      (by norm_num) (by norm_num [aρ_14, bρ_14]) (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_581_15 :
+    Uω (aρ 15) (bρ 15) (2708884727561 / 4000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_581_16 :
+    Uω (aρ 16) (bρ 16) (2708884727561 / 4000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_581 :
+    Uρ (2708884727561 / 4000000000000 : ℝ) ≤
+      (-(41775748645483768217 / 62500000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (2708884727561 / 4000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3993303889330489471621 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_581_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (2708884727561 / 4000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(4029003710659338433683 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_581_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (2708884727561 / 4000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(4100769167401937979183 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_581_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (2708884727561 / 4000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(4221051777138668487973 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_581_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (2708884727561 / 4000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(4406890640283341706941 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_581_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (2708884727561 / 4000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(292450859203226838757 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_581_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (2708884727561 / 4000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(2531139197092148791153 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_581_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (2708884727561 / 4000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(5583477732194065256547 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_581_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (2708884727561 / 4000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(1568469855154944946347 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_581_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (2708884727561 / 4000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(7170328599460265511283 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_581_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (2708884727561 / 4000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(8321398900107086323937 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_581_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (2708884727561 / 4000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(39220117818480693483 / 40000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_581_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (2708884727561 / 4000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(11796635347326533752359 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_581_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (2708884727561 / 4000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(1516831959140549096717 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_581_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (2708884727561 / 4000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_581_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (2708884727561 / 4000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_581_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_582_1 :
+    Uω (aρ 1) (bρ 1) (8678814966921 / 12800000000000 : ℝ) ≤
+      (-(497652126834867869363 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (53726423216529177 / 80000000000000000 : ℝ) ≤
+      (-(497652126834867869363 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (53726423216529177 / 80000000000000000 : ℝ) =
+            (53726423216529177 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (53726423216529177 / 40000000000000000 : ℝ) ≤
+            (368781848540132130637 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6715778829310263 / 10000000000000000 : ℝ)) (q :=
+      (53726423216529177 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_582_2 :
+    Uω (aρ 2) (bρ 2) (8678814966921 / 12800000000000 : ℝ) ≤
+      (-(4016873391124482683507 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (53535195386577869 / 80000000000000000 : ℝ) ≤
+      (-(4016873391124482683507 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (53535195386577869 / 80000000000000000 : ℝ) =
+            (53535195386577869 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (53535195386577869 / 40000000000000000 : ℝ) ≤
+            (2914598411875517316493 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1672935221083109 / 2500000000000000 : ℝ)) (q :=
+      (53535195386577869 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_582_3 :
+    Uω (aρ 3) (bρ 3) (8678814966921 / 12800000000000 : ℝ) ≤
+      (-(4088550799468137633081 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (53152840926370277 / 80000000000000000 : ℝ) ≤
+      (-(4088550799468137633081 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (53152840926370277 / 80000000000000000 : ℝ) =
+            (53152840926370277 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (53152840926370277 / 40000000000000000 : ℝ) ≤
+            (2842921003531862366919 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3321774187815269 / 5000000000000000 : ℝ)) (q :=
+      (53152840926370277 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_582_4 :
+    Uω (aρ 4) (bρ 4) (8678814966921 / 12800000000000 : ℝ) ≤
+      (-(4208683663139425892631 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (52518120797927097 / 80000000000000000 : ℝ) ≤
+      (-(4208683663139425892631 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (52518120797927097 / 80000000000000000 : ℝ) =
+            (52518120797927097 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (52518120797927097 / 40000000000000000 : ℝ) ≤
+            (2722788139860574107369 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6563162524584743 / 10000000000000000 : ℝ)) (q :=
+      (52518120797927097 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_582_5 :
+    Uω (aρ 5) (bρ 5) (8678814966921 / 12800000000000 : ℝ) ≤
+      (-(549285708176747111411 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10310472800733289 / 16000000000000000 : ℝ) ≤
+      (-(549285708176747111411 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10310472800733289 / 16000000000000000 : ℝ) =
+            (10310472800733289 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10310472800733289 / 8000000000000000 : ℝ) ≤
+            (317148267198252888589 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (321995662267729 / 500000000000000 : ℝ)) (q :=
+      (10310472800733289 / 16000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_582_6 :
+    Uω (aρ 6) (bρ 6) (8678814966921 / 12800000000000 : ℝ) ≤
+      (-(466624908802661578893 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (50169221724601889 / 80000000000000000 : ℝ) ≤
+      (-(466624908802661578893 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (50169221724601889 / 80000000000000000 : ℝ) =
+            (50169221724601889 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (50169221724601889 / 40000000000000000 : ℝ) ≤
+            (226522271497338421107 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6261368050728441 / 10000000000000000 : ℝ)) (q :=
+      (50169221724601889 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_582_7 :
+    Uω (aρ 7) (bρ 7) (8678814966921 / 12800000000000 : ℝ) ≤
+      (-(631097595537542197029 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (48286332904819101 / 80000000000000000 : ℝ) ≤
+      (-(631097595537542197029 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (48286332904819101 / 80000000000000000 : ℝ) =
+            (48286332904819101 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (48286332904819101 / 40000000000000000 : ℝ) ≤
+            (235336379837457802971 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (751788403317843 / 1250000000000000 : ℝ)) (q :=
+      (48286332904819101 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_582_8 :
+    Uω (aρ 8) (bρ 8) (8678814966921 / 12800000000000 : ℝ) ≤
+      (-(1392299831538579844629 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (45837691069481813 / 80000000000000000 : ℝ) ≤
+      (-(1392299831538579844629 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (45837691069481813 / 80000000000000000 : ℝ) =
+            (45837691069481813 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (45837691069481813 / 40000000000000000 : ℝ) ≤
+            (340568119211420155371 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2842844209524211 / 5000000000000000 : ℝ)) (q :=
+      (45837691069481813 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_582_9 :
+    Uω (aρ 9) (bρ 9) (8678814966921 / 12800000000000 : ℝ) ≤
+      (-(6258454383988677137099 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8556945424869233 / 16000000000000000 : ℝ) ≤
+      (-(6258454383988677137099 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (8556945424869233 / 16000000000000000 : ℝ) =
+            (8556945424869233 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8556945424869233 / 8000000000000000 : ℝ) ≤
+            (673017419011322862901 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (526341306112451 / 1000000000000000 : ℝ)) (q :=
+      (8556945424869233 / 16000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_582_10 :
+    Uω (aρ 10) (bρ 10) (8678814966921 / 12800000000000 : ℝ) ≤
+      (-(447074103633723834573 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7824580743660861 / 16000000000000000 : ℝ) ≤
+      (-(447074103633723834573 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7824580743660861 / 16000000000000000 : ℝ) =
+            (7824580743660861 / 4000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7824580743660861 / 4000000000000000 : ℝ) ≤
+            (419359871741276165427 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (947303248621809 / 2000000000000000 : ℝ)) (q :=
+      (7824580743660861 / 16000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_582_11 :
+    Uω (aρ 11) (bρ 11) (8678814966921 / 12800000000000 : ℝ) ≤
+      (-(518847554900022932309 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6975699670642717 / 16000000000000000 : ℝ) ≤
+      (-(518847554900022932309 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6975699670642717 / 16000000000000000 : ℝ) =
+            (6975699670642717 / 4000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6975699670642717 / 4000000000000000 : ℝ) ≤
+            (347586420474977067691 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (818778305608273 / 2000000000000000 : ℝ)) (q :=
+      (6975699670642717 / 16000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_582_12 :
+    Uω (aρ 12) (bρ 12) (8678814966921 / 12800000000000 : ℝ) ≤
+      (-(9780547872498677856713 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (30083349528978241 / 80000000000000000 : ℝ) ≤
+      (-(9780547872498677856713 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (30083349528978241 / 80000000000000000 : ℝ) =
+            (30083349528978241 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (30083349528978241 / 20000000000000000 : ℝ) ≤
+            (4082395733501322143287 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3318804418667529 / 10000000000000000 : ℝ)) (q :=
+      (30083349528978241 / 80000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_582_13 :
+    Uω (aρ 13) (bρ 13) (8678814966921 / 12800000000000 : ℝ) ≤
+      (-(5881127756237673751163 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24675259029316617 / 80000000000000000 : ℝ) ≤
+      (-(5881127756237673751163 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (24675259029316617 / 80000000000000000 : ℝ) =
+            (24675259029316617 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24675259029316617 / 20000000000000000 : ℝ) ≤
+            (1050344046762326248837 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2366163535347123 / 10000000000000000 : ℝ)) (q :=
+      (24675259029316617 / 80000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_582_14 :
+    Uω (aρ 14) (bρ 14) (8678814966921 / 12800000000000 : ℝ) ≤
+      (-(3770781227787045933889 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17702646404448193 / 80000000000000000 : ℝ) ≤
+      (-(3770781227787045933889 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (17702646404448193 / 80000000000000000 : ℝ) =
+            (17702646404448193 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17702646404448193 / 10000000000000000 : ℝ) ≤
+            (1427822624462954066111 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (966937664175017 / 10000000000000000 : ℝ)) (q :=
+      (17702646404448193 / 80000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_582_15 :
+    Uω (aρ 15) (bρ 15) (8678814966921 / 12800000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_582_16 :
+    Uω (aρ 16) (bρ 16) (8678814966921 / 12800000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_582 :
+    Uρ (8678814966921 / 12800000000000 : ℝ) ≤
+      (-(1333233638414966829353 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (8678814966921 / 12800000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(497652126834867869363 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_582_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (8678814966921 / 12800000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(4016873391124482683507 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_582_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (8678814966921 / 12800000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(4088550799468137633081 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_582_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (8678814966921 / 12800000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(4208683663139425892631 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_582_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (8678814966921 / 12800000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(549285708176747111411 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_582_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (8678814966921 / 12800000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(466624908802661578893 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_582_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (8678814966921 / 12800000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(631097595537542197029 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_582_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (8678814966921 / 12800000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(1392299831538579844629 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_582_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (8678814966921 / 12800000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(6258454383988677137099 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_582_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (8678814966921 / 12800000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) * (-(447074103633723834573 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_582_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (8678814966921 / 12800000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) * (-(518847554900022932309 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_582_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (8678814966921 / 12800000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(9780547872498677856713 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_582_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (8678814966921 / 12800000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(5881127756237673751163 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_582_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (8678814966921 / 12800000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(3770781227787045933889 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_582_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (8678814966921 / 12800000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_582_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (8678814966921 / 12800000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_582_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_583_1 :
+    Uω (aρ 1) (bρ 1) (21722997014117 / 32000000000000 : ℝ) ≤
+      (-(1984572365875415154351 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (26895661220457821 / 40000000000000000 : ℝ) ≤
+      (-(1984572365875415154351 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (26895661220457821 / 40000000000000000 : ℝ) =
+            (26895661220457821 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26895661220457821 / 20000000000000000 : ℝ) ≤
+            (1481163535624584845649 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1680972815350587 / 2500000000000000 : ℝ)) (q :=
+      (26895661220457821 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_583_2 :
+    Uω (aρ 2) (bρ 2) (21722997014117 / 32000000000000 : ℝ) ≤
+      (-(4004757768924670619983 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5360009591432243 / 8000000000000000 : ℝ) ≤
+      (-(4004757768924670619983 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5360009591432243 / 8000000000000000 : ℝ) =
+            (5360009591432243 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5360009591432243 / 4000000000000000 : ℝ) ≤
+            (2926714034075329380017 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1339970728452809 / 2000000000000000 : ℝ)) (q :=
+      (5360009591432243 / 8000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_583_3 :
+    Uω (aρ 3) (bρ 3) (21722997014117 / 32000000000000 : ℝ) ≤
+      (-(815269468933069742969 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1064354907008283 / 1600000000000000 : ℝ) ≤
+      (-(815269468933069742969 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1064354907008283 / 1600000000000000 : ℝ) =
+            (1064354907008283 / 800000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1064354907008283 / 800000000000000 : ℝ) ≤
+            (571024891666930257031 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (266066484305479 / 400000000000000 : ℝ)) (q :=
+      (1064354907008283 / 1600000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_583_4 :
+    Uω (aρ 4) (bρ 4) (21722997014117 / 32000000000000 : ℝ) ≤
+      (-(2098165417361143833627 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (26291517808627243 / 40000000000000000 : ℝ) ≤
+      (-(2098165417361143833627 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (26291517808627243 / 40000000000000000 : ℝ) =
+            (26291517808627243 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26291517808627243 / 20000000000000000 : ℝ) ≤
+            (1367570484138856166373 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6571278855412059 / 10000000000000000 : ℝ)) (q :=
+      (26291517808627243 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_583_5 :
+    Uω (aρ 5) (bρ 5) (21722997014117 / 32000000000000 : ℝ) ≤
+      (-(4381696579422479512909 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25808652293284761 / 40000000000000000 : ℝ) ≤
+      (-(4381696579422479512909 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (25808652293284761 / 40000000000000000 : ℝ) =
+            (25808652293284761 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25808652293284761 / 20000000000000000 : ℝ) ≤
+            (2549775223577520487091 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3224018008537909 / 5000000000000000 : ℝ)) (q :=
+      (25808652293284761 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_583_6 :
+    Uω (aρ 6) (bρ 6) (21722997014117 / 32000000000000 : ℝ) ≤
+      (-(4653301267676273408001 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25117111001559067 / 40000000000000000 : ℝ) ≤
+      (-(4653301267676273408001 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (25117111001559067 / 40000000000000000 : ℝ) =
+            (25117111001559067 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25117111001559067 / 20000000000000000 : ℝ) ≤
+            (2278170535323726591999 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6269505746352971 / 10000000000000000 : ℝ)) (q :=
+      (25117111001559067 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_583_7 :
+    Uω (aρ 7) (bρ 7) (21722997014117 / 32000000000000 : ℝ) ≤
+      (-(1258825364568131694757 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4835146341648111 / 8000000000000000 : ℝ) ≤
+      (-(1258825364568131694757 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (4835146341648111 / 8000000000000000 : ℝ) =
+            (4835146341648111 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4835146341648111 / 4000000000000000 : ℝ) ≤
+            (474042586181868305243 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1204495496090743 / 2000000000000000 : ℝ)) (q :=
+      (4835146341648111 / 8000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_583_8 :
+    Uω (aρ 8) (bρ 8) (21722997014117 / 32000000000000 : ℝ) ≤
+      (-(1388735398284956359487 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22951545918932221 / 40000000000000000 : ℝ) ≤
+      (-(1388735398284956359487 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (22951545918932221 / 40000000000000000 : ℝ) =
+            (22951545918932221 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22951545918932221 / 20000000000000000 : ℝ) ≤
+            (344132552465043640513 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1423481559284887 / 2500000000000000 : ℝ)) (q :=
+      (22951545918932221 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_583_9 :
+    Uω (aρ 9) (bρ 9) (21722997014117 / 32000000000000 : ℝ) ≤
+      (-(6243053867070716944753 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21425334289664297 / 40000000000000000 : ℝ) ≤
+      (-(6243053867070716944753 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (21425334289664297 / 40000000000000000 : ℝ) =
+            (21425334289664297 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21425334289664297 / 20000000000000000 : ℝ) ≤
+            (688417935929283055247 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2635893025432793 / 5000000000000000 : ℝ)) (q :=
+      (21425334289664297 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_583_10 :
+    Uω (aρ 10) (bρ 10) (21722997014117 / 32000000000000 : ℝ) ≤
+      (-(7136073956947920528813 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19594953486378081 / 40000000000000000 : ℝ) ≤
+      (-(7136073956947920528813 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (19594953486378081 / 40000000000000000 : ℝ) =
+            (19594953486378081 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19594953486378081 / 10000000000000000 : ℝ) ≤
+            (6726869649052079471187 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2372577341358739 / 5000000000000000 : ℝ)) (q :=
+      (19594953486378081 / 40000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_583_11 :
+    Uω (aρ 11) (bρ 11) (21722997014117 / 32000000000000 : ℝ) ≤
+      (-(8281767224600949070467 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17473802007736831 / 40000000000000000 : ℝ) ≤
+      (-(8281767224600949070467 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (17473802007736831 / 40000000000000000 : ℝ) =
+            (17473802007736831 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17473802007736831 / 10000000000000000 : ℝ) ≤
+            (5581176381399050929533 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4103055569601853 / 10000000000000000 : ℝ)) (q :=
+      (17473802007736831 / 40000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_583_12 :
+    Uω (aρ 12) (bρ 12) (21722997014117 / 32000000000000 : ℝ) ≤
+      (-(9756141941200541419451 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3015686041375169 / 8000000000000000 : ℝ) ≤
+      (-(9756141941200541419451 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3015686041375169 / 8000000000000000 : ℝ) =
+            (3015686041375169 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3015686041375169 / 2000000000000000 : ℝ) ≤
+            (4106801664799458580549 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (83226744146409 / 250000000000000 : ℝ)) (q :=
+      (3015686041375169 / 8000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_583_13 :
+    Uω (aρ 13) (bρ 13) (21722997014117 / 32000000000000 : ℝ) ≤
+      (-(586403229508580354881 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2475977040951843 / 8000000000000000 : ℝ) ≤
+      (-(586403229508580354881 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2475977040951843 / 8000000000000000 : ℝ) =
+            (2475977040951843 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2475977040951843 / 2000000000000000 : ℝ) ≤
+            (106743950791419645119 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (475835801278609 / 2000000000000000 : ℝ)) (q :=
+      (2475977040951843 / 8000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_583_14 :
+    Uω (aρ 14) (bρ 14) (21722997014117 / 32000000000000 : ℝ) ≤
+      (-(3000090141854125438813 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8924804140715513 / 40000000000000000 : ℝ) ≤
+      (-(3000090141854125438813 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (8924804140715513 / 40000000000000000 : ℝ) =
+            (8924804140715513 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8924804140715513 / 5000000000000000 : ℝ) ≤
+            (1158792939945874561187 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (497782879708097 / 5000000000000000 : ℝ)) (q :=
+      (8924804140715513 / 40000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_583_15 :
+    Uω (aρ 15) (bρ 15) (21722997014117 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_583_16 :
+    Uω (aρ 16) (bρ 16) (21722997014117 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_583 :
+    Uρ (21722997014117 / 32000000000000 : ℝ) ≤
+      (-(3324180630720396764653 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (21722997014117 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1984572365875415154351 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_583_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (21722997014117 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(4004757768924670619983 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_583_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (21722997014117 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(815269468933069742969 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_583_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (21722997014117 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(2098165417361143833627 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_583_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (21722997014117 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(4381696579422479512909 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_583_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (21722997014117 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(4653301267676273408001 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_583_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (21722997014117 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(1258825364568131694757 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_583_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (21722997014117 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(1388735398284956359487 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_583_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (21722997014117 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(6243053867070716944753 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_583_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (21722997014117 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(7136073956947920528813 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_583_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (21722997014117 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(8281767224600949070467 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_583_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (21722997014117 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(9756141941200541419451 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_583_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (21722997014117 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) * (-(586403229508580354881 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_583_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (21722997014117 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(3000090141854125438813 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_583_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (21722997014117 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_583_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (21722997014117 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_583_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_584_1 :
+    Uω (aρ 1) (bρ 1) (43497913221863 / 64000000000000 : ℝ) ≤
+      (-(3957087005357036086949 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (53856221664742123 / 80000000000000000 : ℝ) ≤
+      (-(3957087005357036086949 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (53856221664742123 / 80000000000000000 : ℝ) =
+            (53856221664742123 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (53856221664742123 / 40000000000000000 : ℝ) ≤
+            (2974384797642963913051 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6732003693354437 / 10000000000000000 : ℝ)) (q :=
+      (53856221664742123 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_584_2 :
+    Uω (aρ 2) (bρ 2) (43497913221863 / 64000000000000 : ℝ) ≤
+      (-(1996328404243077264093 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (53664996438352443 / 80000000000000000 : ℝ) ≤
+      (-(1996328404243077264093 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (53664996438352443 / 80000000000000000 : ℝ) =
+            (53664996438352443 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (53664996438352443 / 40000000000000000 : ℝ) ≤
+            (1469407497256922735907 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6707966399267017 / 10000000000000000 : ℝ)) (q :=
+      (53664996438352443 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_584_3 :
+    Uω (aρ 3) (bρ 3) (43497913221863 / 64000000000000 : ℝ) ≤
+      (-(4064158766627427001413 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (53282649761223291 / 80000000000000000 : ℝ) ≤
+      (-(4064158766627427001413 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (53282649761223291 / 80000000000000000 : ℝ) =
+            (53282649761223291 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (53282649761223291 / 40000000000000000 : ℝ) ≤
+            (2867313036372572998587 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6659775836334729 / 10000000000000000 : ℝ)) (q :=
+      (53282649761223291 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_584_4 :
+    Uω (aρ 4) (bρ 4) (43497913221863 / 64000000000000 : ℝ) ≤
+      (-(167359730165322291151 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10529590079508471 / 16000000000000000 : ℝ) ≤
+      (-(167359730165322291151 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10529590079508471 / 16000000000000000 : ℝ) =
+            (10529590079508471 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10529590079508471 / 8000000000000000 : ℝ) ≤
+            (109899141954677708849 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1315879035295899 / 2000000000000000 : ℝ)) (q :=
+      (10529590079508471 / 16000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_584_5 :
+    Uω (aρ 5) (bρ 5) (43497913221863 / 64000000000000 : ℝ) ≤
+      (-(4369123342251587632137 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (51682245064885059 / 80000000000000000 : ℝ) ≤
+      (-(4369123342251587632137 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (51682245064885059 / 80000000000000000 : ℝ) =
+            (51682245064885059 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (51682245064885059 / 40000000000000000 : ℝ) ≤
+            (2562348460748412367863 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6456158762650171 / 10000000000000000 : ℝ)) (q :=
+      (51682245064885059 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_584_6 :
+    Uω (aρ 6) (bρ 6) (43497913221863 / 64000000000000 : ℝ) ≤
+      (-(1160092560595083378649 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10059844403889607 / 16000000000000000 : ℝ) ≤
+      (-(1160092560595083378649 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10059844403889607 / 16000000000000000 : ℝ) =
+            (10059844403889607 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10059844403889607 / 8000000000000000 : ℝ) ≤
+            (572775390154916621351 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1255528675286183 / 2000000000000000 : ℝ)) (q :=
+      (10059844403889607 / 16000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_584_7 :
+    Uω (aρ 7) (bρ 7) (43497913221863 / 64000000000000 : ℝ) ≤
+      (-(5021840426067032372317 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (48416593303044763 / 80000000000000000 : ℝ) ≤
+      (-(5021840426067032372317 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (48416593303044763 / 80000000000000000 : ℝ) =
+            (48416593303044763 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (48416593303044763 / 40000000000000000 : ℝ) ≤
+            (1909631376932967627683 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6030647578090097 / 10000000000000000 : ℝ)) (q :=
+      (48416593303044763 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_584_8 :
+    Uω (aρ 8) (bρ 8) (43497913221863 / 64000000000000 : ℝ) ≤
+      (-(2770352236236726762283 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (45968491167470523 / 80000000000000000 : ℝ) ≤
+      (-(2770352236236726762283 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (45968491167470523 / 80000000000000000 : ℝ) =
+            (45968491167470523 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (45968491167470523 / 40000000000000000 : ℝ) ≤
+            (695383665263273237717 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5702163695536537 / 10000000000000000 : ℝ)) (q :=
+      (45968491167470523 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_584_9 :
+    Uω (aρ 9) (bρ 9) (43497913221863 / 64000000000000 : ℝ) ≤
+      (-(6227677789659275289543 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (42916606779601831 / 80000000000000000 : ℝ) ≤
+      (-(6227677789659275289543 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (42916606779601831 / 80000000000000000 : ℝ) =
+            (42916606779601831 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (42916606779601831 / 40000000000000000 : ℝ) ≤
+            (703794013340724710457 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1320039556732341 / 2500000000000000 : ℝ)) (q :=
+      (42916606779601831 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_584_10 :
+    Uω (aρ 10) (bρ 10) (43497913221863 / 64000000000000 : ℝ) ≤
+      (-(7118993375547906127047 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (39256902812473411 / 80000000000000000 : ℝ) ≤
+      (-(7118993375547906127047 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (39256902812473411 / 80000000000000000 : ℝ) =
+            (39256902812473411 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (39256902812473411 / 20000000000000000 : ℝ) ≤
+            (6743950230452093872953 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4753791268642259 / 10000000000000000 : ℝ)) (q :=
+      (39256902812473411 / 80000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_584_11 :
+    Uω (aρ 11) (bρ 11) (43497913221863 / 64000000000000 : ℝ) ≤
+      (-(1652403543875702179933 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (35016692004499127 / 80000000000000000 : ℝ) ≤
+      (-(1652403543875702179933 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (35016692004499127 / 80000000000000000 : ℝ) =
+            (35016692004499127 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (35016692004499127 / 20000000000000000 : ℝ) ≤
+            (1120185177324297820067 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (514026899106711 / 1250000000000000 : ℝ)) (q :=
+      (35016692004499127 / 80000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_584_12 :
+    Uω (aρ 12) (bρ 12) (43497913221863 / 64000000000000 : ℝ) ≤
+      (-(9731811107760359174813 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (30230323903687811 / 80000000000000000 : ℝ) ≤
+      (-(9731811107760359174813 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (30230323903687811 / 80000000000000000 : ℝ) =
+            (30230323903687811 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (30230323903687811 / 20000000000000000 : ℝ) ≤
+            (4131132498239640825187 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3339323264335859 / 10000000000000000 : ℝ)) (q :=
+      (30230323903687811 / 80000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_584_13 :
+    Uω (aρ 13) (bρ 13) (43497913221863 / 64000000000000 : ℝ) ≤
+      (-(11694059893297898993841 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4968821714335203 / 16000000000000000 : ℝ) ≤
+      (-(11694059893297898993841 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4968821714335203 / 16000000000000000 : ℝ) =
+            (4968821714335203 / 4000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4968821714335203 / 4000000000000000 : ℝ) ≤
+            (2168883712702101006159 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (239215117292791 / 1000000000000000 : ℝ)) (q :=
+      (4968821714335203 / 16000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_584_14 :
+    Uω (aρ 14) (bρ 14) (43497913221863 / 64000000000000 : ℝ) ≤
+      (-(14920090891057678189429 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3598725058625411 / 16000000000000000 : ℝ) ≤
+      (-(14920090891057678189429 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (3598725058625411 / 16000000000000000 : ℝ) =
+            (3598725058625411 / 2000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3598725058625411 / 2000000000000000 : ℝ) ≤
+            (5874324517942321810571 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (102345763833567 / 1000000000000000 : ℝ)) (q :=
+      (3598725058625411 / 16000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_584_15 :
+    Uω (aρ 15) (bρ 15) (43497913221863 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_584_16 :
+    Uω (aρ 16) (bρ 16) (43497913221863 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_584 :
+    Uρ (43497913221863 / 64000000000000 : ℝ) ≤
+      (-(663068958368421010261 / 1000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (43497913221863 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3957087005357036086949 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_584_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (43497913221863 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1996328404243077264093 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_584_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (43497913221863 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(4064158766627427001413 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_584_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (43497913221863 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(167359730165322291151 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_584_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (43497913221863 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(4369123342251587632137 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_584_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (43497913221863 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1160092560595083378649 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_584_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (43497913221863 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(5021840426067032372317 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_584_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (43497913221863 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(2770352236236726762283 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_584_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (43497913221863 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(6227677789659275289543 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_584_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (43497913221863 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(7118993375547906127047 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_584_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (43497913221863 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(1652403543875702179933 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_584_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (43497913221863 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(9731811107760359174813 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_584_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (43497913221863 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(11694059893297898993841 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_584_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (43497913221863 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(14920090891057678189429 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_584_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (43497913221863 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_584_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (43497913221863 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_584_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_585_1 :
+    Uω (aρ 1) (bρ 1) (10887458103873 / 16000000000000 : ℝ) ≤
+      (-(986260950108896607393 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (674014011100133 / 1000000000000000 : ℝ) ≤
+      (-(986260950108896607393 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (674014011100133 / 1000000000000000 : ℝ) =
+            (674014011100133 / 500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (674014011100133 / 500000000000000 : ℝ) ≤
+            (746607000641103392607 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1348023225033407 / 2000000000000000 : ℝ)) (q :=
+      (674014011100133 / 1000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_585_2 :
+    Uω (aρ 2) (bρ 2) (10887458103873 / 16000000000000 : ℝ) ≤
+      (-(995142618591047904077 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13432474239670343 / 20000000000000000 : ℝ) ≤
+      (-(995142618591047904077 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (13432474239670343 / 20000000000000000 : ℝ) =
+            (13432474239670343 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13432474239670343 / 10000000000000000 : ℝ) ≤
+            (737725332158952095923 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3358039577672359 / 5000000000000000 : ℝ)) (q :=
+      (13432474239670343 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_585_3 :
+    Uω (aρ 3) (bρ 3) (10887458103873 / 16000000000000 : ℝ) ≤
+      (-(4051985029121106850783 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2667377707942301 / 4000000000000000 : ℝ) ≤
+      (-(4051985029121106850783 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2667377707942301 / 4000000000000000 : ℝ) =
+            (2667377707942301 / 2000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2667377707942301 / 2000000000000000 : ℝ) ≤
+            (2879486773878893149217 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (166697239043397 / 250000000000000 : ℝ)) (q :=
+      (2667377707942301 / 4000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_585_4 :
+    Uω (aρ 4) (bρ 4) (10887458103873 / 16000000000000 : ℝ) ≤
+      (-(1042917720939327110799 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13178216284733751 / 20000000000000000 : ℝ) ≤
+      (-(1042917720939327110799 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (13178216284733751 / 20000000000000000 : ℝ) =
+            (13178216284733751 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13178216284733751 / 10000000000000000 : ℝ) ≤
+            (689950229810672889201 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3293755743911563 / 5000000000000000 : ℝ)) (q :=
+      (13178216284733751 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_585_5 :
+    Uω (aρ 5) (bρ 5) (10887458103873 / 16000000000000 : ℝ) ≤
+      (-(4356565913995524418331 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3234199089937957 / 5000000000000000 : ℝ) ≤
+      (-(4356565913995524418331 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (3234199089937957 / 5000000000000000 : ℝ) =
+            (3234199089937957 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3234199089937957 / 2500000000000000 : ℝ) ≤
+            (2574905889004475581669 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6464281482176203 / 10000000000000000 : ℝ)) (q :=
+      (3234199089937957 / 5000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_585_6 :
+    Uω (aρ 6) (bρ 6) (10887458103873 / 16000000000000 : ℝ) ≤
+      (-(4627455968489878491293 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1259105544365247 / 2000000000000000 : ℝ) ≤
+      (-(4627455968489878491293 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1259105544365247 / 2000000000000000 : ℝ) =
+            (1259105544365247 / 1000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1259105544365247 / 1000000000000000 : ℝ) ≤
+            (2304015834510121508707 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1257156188243369 / 2000000000000000 : ℝ)) (q :=
+      (1259105544365247 / 2000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_585_7 :
+    Uω (aρ 7) (bρ 7) (10887458103873 / 16000000000000 : ℝ) ≤
+      (-(500839761785607599 / 1000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12120430641761801 / 20000000000000000 : ℝ) ≤
+      (-(500839761785607599 / 1000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (12120430641761801 / 20000000000000000 : ℝ) =
+            (12120430641761801 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12120430641761801 / 10000000000000000 : ℝ) ≤
+            (192307418514392401 / 1000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (188713047502693 / 312500000000000 : ℝ)) (q :=
+      (12120430641761801 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_585_8 :
+    Uω (aρ 8) (bρ 8) (10887458103873 / 16000000000000 : ℝ) ≤
+      (-(2763243951874325661233 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11508472266131629 / 20000000000000000 : ℝ) ≤
+      (-(2763243951874325661233 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11508472266131629 / 20000000000000000 : ℝ) =
+            (11508472266131629 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11508472266131629 / 10000000000000000 : ℝ) ≤
+            (702491949625674338767 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1427600198949001 / 2500000000000000 : ℝ)) (q :=
+      (11508472266131629 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_585_9 :
+    Uω (aρ 9) (bρ 9) (10887458103873 / 16000000000000 : ℝ) ≤
+      (-(6212326071951551424699 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10745635435155821 / 20000000000000000 : ℝ) ≤
+      (-(6212326071951551424699 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10745635435155821 / 20000000000000000 : ℝ) =
+            (10745635435155821 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10745635435155821 / 10000000000000000 : ℝ) ≤
+            (719145731048448575301 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1322132398295049 / 2500000000000000 : ℝ)) (q :=
+      (10745635435155821 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_585_10 :
+    Uω (aρ 10) (bρ 10) (10887458103873 / 16000000000000 : ℝ) ≤
+      (-(443871487145641250981 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9830972819448893 / 20000000000000000 : ℝ) ≤
+      (-(443871487145641250981 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9830972819448893 / 20000000000000000 : ℝ) =
+            (9830972819448893 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9830972819448893 / 5000000000000000 : ℝ) ≤
+            (422562488229358749019 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1190606502742067 / 2500000000000000 : ℝ)) (q :=
+      (9830972819448893 / 20000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_585_11 :
+    Uω (aρ 11) (bρ 11) (10887458103873 / 16000000000000 : ℝ) ≤
+      (-(257572254535980578843 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4385720304765587 / 10000000000000000 : ℝ) ≤
+      (-(257572254535980578843 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4385720304765587 / 10000000000000000 : ℝ) =
+            (4385720304765587 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4385720304765587 / 2500000000000000 : ℝ) ≤
+            (175644733151519421157 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4121370427255549 / 10000000000000000 : ℝ)) (q :=
+      (4385720304765587 / 10000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_585_12 :
+    Uω (aρ 12) (bρ 12) (10887458103873 / 16000000000000 : ℝ) ≤
+      (-(9707554825897988957459 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1515187021701689 / 4000000000000000 : ℝ) ≤
+      (-(9707554825897988957459 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1515187021701689 / 4000000000000000 : ℝ) =
+            (1515187021701689 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1515187021701689 / 1000000000000000 : ℝ) ≤
+            (4155388780102011042541 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (167478251145891 / 500000000000000 : ℝ)) (q :=
+      (1515187021701689 / 4000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_585_13 :
+    Uω (aρ 13) (bρ 13) (10887458103873 / 16000000000000 : ℝ) ≤
+      (-(11660238798583253649693 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6232069079655727 / 20000000000000000 : ℝ) ≤
+      (-(11660238798583253649693 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6232069079655727 / 20000000000000000 : ℝ) =
+            (6232069079655727 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6232069079655727 / 5000000000000000 : ℝ) ≤
+            (2202704807416746350307 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1202540367830051 / 5000000000000000 : ℝ)) (q :=
+      (6232069079655727 / 20000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_585_14 :
+    Uω (aρ 14) (bρ 14) (10887458103873 / 16000000000000 : ℝ) ≤
+      (-(14841866121726024649593 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4533732993874791 / 20000000000000000 : ℝ) ≤
+      (-(14841866121726024649593 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4533732993874791 / 20000000000000000 : ℝ) =
+            (4533732993874791 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4533732993874791 / 2500000000000000 : ℝ) ≤
+            (5952549287273975350407 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (525335967462083 / 5000000000000000 : ℝ)) (q :=
+      (4533732993874791 / 20000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_585_15 :
+    Uω (aρ 15) (bρ 15) (10887458103873 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_585_16 :
+    Uω (aρ 16) (bρ 16) (10887458103873 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_585 :
+    Uρ (10887458103873 / 16000000000000 : ℝ) ≤
+      (-(6613144944372741958823 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (10887458103873 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(986260950108896607393 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_585_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (10887458103873 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(995142618591047904077 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_585_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (10887458103873 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(4051985029121106850783 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_585_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (10887458103873 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1042917720939327110799 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_585_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (10887458103873 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(4356565913995524418331 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_585_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (10887458103873 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(4627455968489878491293 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_585_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (10887458103873 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) * (-(500839761785607599 / 1000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_585_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (10887458103873 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(2763243951874325661233 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_585_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (10887458103873 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(6212326071951551424699 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_585_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (10887458103873 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) * (-(443871487145641250981 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_585_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (10887458103873 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) * (-(257572254535980578843 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_585_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (10887458103873 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(9707554825897988957459 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_585_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (10887458103873 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(11660238798583253649693 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_585_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (10887458103873 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(14841866121726024649593 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_585_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (10887458103873 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_585_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (10887458103873 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_585_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_586_1 :
+    Uω (aρ 1) (bρ 1) (43601751609121 / 64000000000000 : ℝ) ≤
+      (-(1966507541025517927749 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (53986020110723201 / 80000000000000000 : ℝ) ≤
+      (-(1966507541025517927749 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (53986020110723201 / 80000000000000000 : ℝ) =
+            (53986020110723201 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (53986020110723201 / 40000000000000000 : ℝ) ≤
+            (1499228360474482072251 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1687057139210161 / 2500000000000000 : ℝ)) (q :=
+      (53986020110723201 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_586_2 :
+    Uω (aρ 2) (bρ 2) (43601751609121 / 64000000000000 : ℝ) ≤
+      (-(158739949249696931061 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10758959495064521 / 16000000000000000 : ℝ) ≤
+      (-(158739949249696931061 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10758959495064521 / 16000000000000000 : ℝ) =
+            (10758959495064521 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10758959495064521 / 8000000000000000 : ℝ) ≤
+            (118518922870303068939 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1344838382100099 / 2000000000000000 : ℝ)) (q :=
+      (10758959495064521 / 16000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_586_3 :
+    Uω (aρ 3) (bρ 3) (43601751609121 / 64000000000000 : ℝ) ≤
+      (-(4039826096045358521497 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (53412458543330409 / 80000000000000000 : ℝ) ≤
+      (-(4039826096045358521497 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (53412458543330409 / 80000000000000000 : ℝ) =
+            (53412458543330409 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (53412458543330409 / 40000000000000000 : ℝ) ≤
+            (2891645706954641478503 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3338001641926223 / 5000000000000000 : ℝ)) (q :=
+      (53412458543330409 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_586_4 :
+    Uω (aρ 4) (bρ 4) (43601751609121 / 64000000000000 : ℝ) ≤
+      (-(1039840921529924137879 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (52777779841576017 / 80000000000000000 : ℝ) ≤
+      (-(1039840921529924137879 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (52777779841576017 / 80000000000000000 : ℝ) =
+            (52777779841576017 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (52777779841576017 / 40000000000000000 : ℝ) ≤
+            (693027029220075862121 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (103056684210607 / 156250000000000 : ℝ)) (q :=
+      (52777779841576017 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_586_5 :
+    Uω (aρ 5) (bρ 5) (43601751609121 / 64000000000000 : ℝ) ≤
+      (-(4344024254899230569613 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10362425141865713 / 16000000000000000 : ℝ) ≤
+      (-(4344024254899230569613 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10362425141865713 / 16000000000000000 : ℝ) =
+            (10362425141865713 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10362425141865713 / 8000000000000000 : ℝ) ≤
+            (2587447548100769430387 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1294480835150397 / 2000000000000000 : ℝ)) (q :=
+      (10362425141865713 / 16000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_586_6 :
+    Uω (aρ 6) (bρ 6) (43601751609121 / 64000000000000 : ℝ) ≤
+      (-(4614558402526447368687 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (50429221269616689 / 80000000000000000 : ℝ) ≤
+      (-(4614558402526447368687 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (50429221269616689 / 80000000000000000 : ℝ) =
+            (50429221269616689 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (50429221269616689 / 40000000000000000 : ℝ) ≤
+            (2316913400473552631313 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (393369902560251 / 625000000000000 : ℝ)) (q :=
+      (50429221269616689 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_586_7 :
+    Uω (aρ 7) (bρ 7) (43601751609121 / 64000000000000 : ℝ) ≤
+      (-(998994596803359351861 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (48546851211011821 / 80000000000000000 : ℝ) ≤
+      (-(998994596803359351861 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (48546851211011821 / 80000000000000000 : ℝ) =
+            (48546851211011821 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (48546851211011821 / 40000000000000000 : ℝ) ≤
+            (387299763796640648139 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6046987307072799 / 10000000000000000 : ℝ)) (q :=
+      (48546851211011821 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_586_8 :
+    Uω (aρ 8) (bρ 8) (43601751609121 / 64000000000000 : ℝ) ≤
+      (-(2756145913413908321303 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (46099285535222737 / 80000000000000000 : ℝ) ≤
+      (-(2756145913413908321303 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (46099285535222737 / 80000000000000000 : ℝ) =
+            (46099285535222737 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (46099285535222737 / 40000000000000000 : ℝ) ≤
+            (709589988086091678697 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (714829692433191 / 1250000000000000 : ℝ)) (q :=
+      (46099285535222737 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_586_9 :
+    Uω (aρ 9) (bρ 9) (43601751609121 / 64000000000000 : ℝ) ≤
+      (-(774624829318223816123 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8609694695550233 / 16000000000000000 : ℝ) ≤
+      (-(774624829318223816123 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (8609694695550233 / 16000000000000000 : ℝ) =
+            (8609694695550233 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8609694695550233 / 8000000000000000 : ℝ) ≤
+            (91809146056776183877 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1059380030691527 / 2000000000000000 : ℝ)) (q :=
+      (8609694695550233 / 16000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_586_10 :
+    Uω (aρ 10) (bρ 10) (43601751609121 / 64000000000000 : ℝ) ≤
+      (-(3542462547203858067021 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (39390872408761549 / 80000000000000000 : ℝ) ≤
+      (-(3542462547203858067021 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (39390872408761549 / 80000000000000000 : ℝ) =
+            (39390872408761549 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (39390872408761549 / 20000000000000000 : ℝ) ≤
+            (3389009255796141932979 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4771058919705231 / 10000000000000000 : ℝ)) (q :=
+      (39390872408761549 / 80000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_586_11 :
+    Uω (aρ 11) (bρ 11) (43601751609121 / 64000000000000 : ℝ) ≤
+      (-(8222650286067020655741 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (35154815433084153 / 80000000000000000 : ℝ) ≤
+      (-(8222650286067020655741 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (35154815433084153 / 80000000000000000 : ℝ) =
+            (35154815433084153 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (35154815433084153 / 20000000000000000 : ℝ) ≤
+            (5640293319932979344259 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2065260650995441 / 5000000000000000 : ℝ)) (q :=
+      (35154815433084153 / 80000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_586_12 :
+    Uω (aρ 12) (bρ 12) (43601751609121 / 64000000000000 : ℝ) ≤
+      (-(591026156976621987 / 610351562500000000 : ℝ)) := by
+  have hlg :
+    Real.log (30377110434234341 / 80000000000000000 : ℝ) ≤
+      (-(591026156976621987 / 610351562500000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (30377110434234341 / 80000000000000000 : ℝ) =
+            (30377110434234341 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (30377110434234341 / 20000000000000000 : ℝ) ≤
+            (510201544201052901 / 1220703125000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3359795148963429 / 10000000000000000 : ℝ)) (q :=
+      (30377110434234341 / 80000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_586_13 :
+    Uω (aρ 13) (bρ 13) (43601751609121 / 64000000000000 : ℝ) ≤
+      (-(11626598744486248832609 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25012276384134889 / 80000000000000000 : ℝ) ≤
+      (-(11626598744486248832609 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (25012276384134889 / 80000000000000000 : ℝ) =
+            (25012276384134889 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25012276384134889 / 20000000000000000 : ℝ) ≤
+            (2236344861513751167391 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1208984189016783 / 5000000000000000 : ℝ)) (q :=
+      (25012276384134889 / 80000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_586_14 :
+    Uω (aρ 14) (bρ 14) (43601751609121 / 64000000000000 : ℝ) ≤
+      (-(7382809598534463882491 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (18273733742646113 / 80000000000000000 : ℝ) ≤
+      (-(7382809598534463882491 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (18273733742646113 / 80000000000000000 : ℝ) =
+            (18273733742646113 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18273733742646113 / 10000000000000000 : ℝ) ≤
+            (3014398105965536117509 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (269315000676593 / 2500000000000000 : ℝ)) (q :=
+      (18273733742646113 / 80000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_586_15 :
+    Uω (aρ 15) (bρ 15) (43601751609121 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_586_16 :
+    Uω (aρ 16) (bρ 16) (43601751609121 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_586 :
+    Uρ (43601751609121 / 64000000000000 : ℝ) ≤
+      (-(329786005827769626429 / 500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (43601751609121 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1966507541025517927749 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_586_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (43601751609121 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(158739949249696931061 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_586_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (43601751609121 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(4039826096045358521497 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_586_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (43601751609121 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1039840921529924137879 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_586_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (43601751609121 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(4344024254899230569613 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_586_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (43601751609121 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(4614558402526447368687 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_586_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (43601751609121 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(998994596803359351861 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_586_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (43601751609121 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(2756145913413908321303 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_586_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (43601751609121 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(774624829318223816123 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_586_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (43601751609121 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(3542462547203858067021 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_586_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (43601751609121 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(8222650286067020655741 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_586_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (43601751609121 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(591026156976621987 / 610351562500000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_586_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (43601751609121 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(11626598744486248832609 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_586_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (43601751609121 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(7382809598534463882491 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_586_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (43601751609121 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_586_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (43601751609121 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_586_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_587_1 :
+    Uω (aρ 1) (bρ 1) (174614683211 / 256000000000 : ℝ) ≤
+      (-(1960500407696928441209 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27025459666440907 / 40000000000000000 : ℝ) ≤
+      (-(1960500407696928441209 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (27025459666440907 / 40000000000000000 : ℝ) =
+            (27025459666440907 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27025459666440907 / 20000000000000000 : ℝ) ≤
+            (1505235493803071558791 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3378170494187883 / 5000000000000000 : ℝ)) (q :=
+      (27025459666440907 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_587_2 :
+    Uω (aρ 2) (bρ 2) (174614683211 / 256000000000 : ℝ) ≤
+      (-(3956441543932251401597 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (26929848994144739 / 40000000000000000 : ℝ) ≤
+      (-(3956441543932251401597 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (26929848994144739 / 40000000000000000 : ℝ) =
+            (26929848994144739 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26929848994144739 / 20000000000000000 : ℝ) ≤
+            (2975030259067748598403 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3366152332368841 / 5000000000000000 : ℝ)) (q :=
+      (26929848994144739 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_587_3 :
+    Uω (aρ 3) (bρ 3) (174614683211 / 256000000000 : ℝ) ≤
+      (-(2013840965715370581603 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (26738681457362153 / 40000000000000000 : ℝ) ≤
+      (-(2013840965715370581603 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (26738681457362153 / 40000000000000000 : ℝ) =
+            (26738681457362153 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26738681457362153 / 20000000000000000 : ℝ) ≤
+            (1451894935784629418397 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6684117002696389 / 10000000000000000 : ℝ)) (q :=
+      (26738681457362153 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_587_4 :
+    Uω (aρ 4) (bρ 4) (174614683211 / 256000000000 : ℝ) ≤
+      (-(4147071623883270545931 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (26421347252804141 / 40000000000000000 : ℝ) ≤
+      (-(4147071623883270545931 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (26421347252804141 / 40000000000000000 : ℝ) =
+            (26421347252804141 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26421347252804141 / 20000000000000000 : ℝ) ≤
+            (2784400179116729454069 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6603744081482383 / 10000000000000000 : ℝ)) (q :=
+      (26421347252804141 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_587_5 :
+    Uω (aρ 5) (bρ 5) (174614683211 / 256000000000 : ℝ) ≤
+      (-(866299665071521699251 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25938532938119563 / 40000000000000000 : ℝ) ≤
+      (-(866299665071521699251 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (25938532938119563 / 40000000000000000 : ℝ) =
+            (25938532938119563 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25938532938119563 / 20000000000000000 : ℝ) ≤
+            (519994695528478300749 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3240263421737547 / 5000000000000000 : ℝ)) (q :=
+      (25938532938119563 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_587_6 :
+    Uω (aρ 6) (bρ 6) (174614683211 / 256000000000 : ℝ) ≤
+      (-(4601677501181152033433 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25247110252738111 / 40000000000000000 : ℝ) ≤
+      (-(4601677501181152033433 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (25247110252738111 / 40000000000000000 : ℝ) =
+            (25247110252738111 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25247110252738111 / 20000000000000000 : ℝ) ≤
+            (2329794301818847966567 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (393878492245273 / 625000000000000 : ℝ)) (q :=
+      (25247110252738111 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_587_7 :
+    Uω (aρ 7) (bρ 7) (174614683211 / 256000000000 : ℝ) ≤
+      (-(1245391618782531133827 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24305989618724163 / 40000000000000000 : ℝ) ≤
+      (-(1245391618782531133827 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (24305989618724163 / 40000000000000000 : ℝ) =
+            (24305989618724163 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24305989618724163 / 20000000000000000 : ℝ) ≤
+            (487476331967468866173 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3027578469838697 / 5000000000000000 : ℝ)) (q :=
+      (24305989618724163 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_587_8 :
+    Uω (aρ 8) (bρ 8) (174614683211 / 256000000000 : ℝ) ≤
+      (-(2749058090920340042383 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4616468058571363 / 8000000000000000 : ℝ) ≤
+      (-(2749058090920340042383 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (4616468058571363 / 8000000000000000 : ℝ) =
+            (4616468058571363 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4616468058571363 / 4000000000000000 : ℝ) ≤
+            (716677810579659957617 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (143171848202093 / 250000000000000 : ℝ)) (q :=
+      (4616468058571363 / 8000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_587_9 :
+    Uω (aρ 9) (bρ 9) (174614683211 / 256000000000 : ℝ) ≤
+      (-(6181695398438540011231 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21557201003122639 / 40000000000000000 : ℝ) ≤
+      (-(6181695398438540011231 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (21557201003122639 / 40000000000000000 : ℝ) =
+            (21557201003122639 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21557201003122639 / 20000000000000000 : ℝ) ≤
+            (749776404561459988769 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (663158738947079 / 1250000000000000 : ℝ)) (q :=
+      (21557201003122639 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_587_10 :
+    Uω (aρ 10) (bρ 10) (174614683211 / 256000000000 : ℝ) ≤
+      (-(7067937157608990501517 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19728923122556263 / 40000000000000000 : ℝ) ≤
+      (-(7067937157608990501517 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (19728923122556263 / 40000000000000000 : ℝ) =
+            (19728923122556263 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19728923122556263 / 10000000000000000 : ℝ) ≤
+            (6795006448391009498483 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1194922501197111 / 2500000000000000 : ℝ)) (q :=
+      (19728923122556263 / 40000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_587_11 :
+    Uω (aρ 11) (bρ 11) (174614683211 / 256000000000 : ℝ) ≤
+      (-(1640606385596563904279 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17611925552511887 / 40000000000000000 : ℝ) ≤
+      (-(1640606385596563904279 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (17611925552511887 / 40000000000000000 : ℝ) =
+            (17611925552511887 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17611925552511887 / 10000000000000000 : ℝ) ≤
+            (1131982335603436095721 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (517458480746407 / 1250000000000000 : ℝ)) (q :=
+      (17611925552511887 / 40000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_587_12 :
+    Uω (aρ 12) (bρ 12) (174614683211 / 256000000000 : ℝ) ≤
+      (-(965926376453343920951 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15225217164017443 / 40000000000000000 : ℝ) ≤
+      (-(965926376453343920951 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (15225217164017443 / 40000000000000000 : ℝ) =
+            (15225217164017443 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15225217164017443 / 10000000000000000 : ℝ) ≤
+            (420367984146656079049 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1685006874204517 / 5000000000000000 : ℝ)) (q :=
+      (15225217164017443 / 40000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_587_13 :
+    Uω (aρ 13) (bρ 13) (174614683211 / 256000000000 : ℝ) ≤
+      (-(724571076822981242243 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12548055717620997 / 40000000000000000 : ℝ) ≤
+      (-(724571076822981242243 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (12548055717620997 / 40000000000000000 : ℝ) =
+            (12548055717620997 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12548055717620997 / 10000000000000000 : ℝ) ≤
+            (141862898552018757757 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2430814766805811 / 10000000000000000 : ℝ)) (q :=
+      (12548055717620997 / 40000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_587_14 :
+    Uω (aρ 14) (bρ 14) (174614683211 / 256000000000 : ℝ) ≤
+      (-(7345605697896120243709 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9205105849546417 / 40000000000000000 : ℝ) ≤
+      (-(7345605697896120243709 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (9205105849546417 / 40000000000000000 : ℝ) =
+            (9205105849546417 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9205105849546417 / 5000000000000000 : ℝ) ≤
+            (3051602006603879756291 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1103267117813521 / 10000000000000000 : ℝ)) (q :=
+      (9205105849546417 / 40000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_587_15 :
+    Uω (aρ 15) (bρ 15) (174614683211 / 256000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_587_16 :
+    Uω (aρ 16) (bρ 16) (174614683211 / 256000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_587 :
+    Uρ (174614683211 / 256000000000 : ℝ) ≤
+      (-(1644602174649657013571 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (174614683211 / 256000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1960500407696928441209 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_587_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (174614683211 / 256000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3956441543932251401597 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_587_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (174614683211 / 256000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(2013840965715370581603 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_587_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (174614683211 / 256000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(4147071623883270545931 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_587_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (174614683211 / 256000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(866299665071521699251 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_587_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (174614683211 / 256000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(4601677501181152033433 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_587_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (174614683211 / 256000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(1245391618782531133827 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_587_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (174614683211 / 256000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(2749058090920340042383 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_587_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (174614683211 / 256000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(6181695398438540011231 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_587_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (174614683211 / 256000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(7067937157608990501517 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_587_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (174614683211 / 256000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(1640606385596563904279 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_587_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (174614683211 / 256000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(965926376453343920951 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_587_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (174614683211 / 256000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) * (-(724571076822981242243 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_587_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (174614683211 / 256000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(7345605697896120243709 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_587_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (174614683211 / 256000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_587_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (174614683211 / 256000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_587_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_588_1 :
+    Uω (aρ 1) (bρ 1) (43705589996379 / 64000000000000 : ℝ) ≤
+      (-(39090009657798394581 / 100000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (54115818554488467 / 80000000000000000 : ℝ) ≤
+      (-(39090009657798394581 / 100000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (54115818554488467 / 80000000000000000 : ℝ) =
+            (54115818554488467 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (54115818554488467 / 40000000000000000 : ℝ) ≤
+            (30224708372201605419 / 100000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3382226709886449 / 5000000000000000 : ℝ)) (q :=
+      (54115818554488467 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_588_2 :
+    Uω (aρ 2) (bρ 2) (43705589996379 / 64000000000000 : ℝ) ≤
+      (-(394439887737222588027 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (53924598497595259 / 80000000000000000 : ℝ) ≤
+      (-(394439887737222588027 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (53924598497595259 / 80000000000000000 : ℝ) =
+            (53924598497595259 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (53924598497595259 / 40000000000000000 : ℝ) ≤
+            (298707292562777411973 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1685104354514899 / 2500000000000000 : ℝ)) (q :=
+      (53924598497595259 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_588_3 :
+    Uω (aρ 3) (bρ 3) (43705589996379 / 64000000000000 : ℝ) ≤
+      (-(4015552499438766461231 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (53542267273075323 / 80000000000000000 : ℝ) ≤
+      (-(4015552499438766461231 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (53542267273075323 / 80000000000000000 : ℝ) =
+            (53542267273075323 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (53542267273075323 / 40000000000000000 : ℝ) ≤
+            (2915919303561233538769 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1673057679569903 / 2500000000000000 : ℝ)) (q :=
+      (53542267273075323 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_588_4 :
+    Uω (aρ 4) (bρ 4) (43705589996379 / 64000000000000 : ℝ) ≤
+      (-(413479465984879583207 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2116304365246959 / 3200000000000000 : ℝ) ≤
+      (-(413479465984879583207 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2116304365246959 / 3200000000000000 : ℝ) =
+            (2116304365246959 / 1600000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2116304365246959 / 1600000000000000 : ℝ) ≤
+            (279667714315120416793 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (264474414554771 / 400000000000000 : ℝ)) (q :=
+      (2116304365246959 / 3200000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_588_5 :
+    Uω (aρ 5) (bρ 5) (43705589996379 / 64000000000000 : ℝ) ≤
+      (-(539873510739345538629 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (51942005940127359 / 80000000000000000 : ℝ) ≤
+      (-(539873510739345538629 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (51942005940127359 / 80000000000000000 : ℝ) =
+            (51942005940127359 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (51942005940127359 / 40000000000000000 : ℝ) ≤
+            (326560464635654461371 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6488649485442621 / 10000000000000000 : ℝ)) (q :=
+      (51942005940127359 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_588_6 :
+    Uω (aρ 6) (bρ 6) (43705589996379 / 64000000000000 : ℝ) ≤
+      (-(573601652664223193247 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10111843896638211 / 16000000000000000 : ℝ) ≤
+      (-(573601652664223193247 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10111843896638211 / 16000000000000000 : ℝ) =
+            (10111843896638211 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10111843896638211 / 8000000000000000 : ℝ) ≤
+            (292832322710776806753 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1262038649269709 / 2000000000000000 : ℝ)) (q :=
+      (10111843896638211 / 16000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_588_7 :
+    Uω (aρ 7) (bρ 7) (43705589996379 / 64000000000000 : ℝ) ≤
+      (-(4968178041979655025511 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (48677106648852863 / 80000000000000000 : ℝ) ≤
+      (-(4968178041979655025511 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (48677106648852863 / 80000000000000000 : ℝ) =
+            (48677106648852863 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (48677106648852863 / 40000000000000000 : ℝ) ≤
+            (1963293761020344974489 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6063326418523997 / 10000000000000000 : ℝ)) (q :=
+      (48677106648852863 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_588_8 :
+    Uω (aρ 8) (bρ 8) (43705589996379 / 64000000000000 : ℝ) ≤
+      (-(2741980454591354062683 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (46230074222117999 / 80000000000000000 : ℝ) ≤
+      (-(2741980454591354062683 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (46230074222117999 / 80000000000000000 : ℝ) =
+            (46230074222117999 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (46230074222117999 / 40000000000000000 : ℝ) ≤
+            (723755446908645937317 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5735109963180281 / 10000000000000000 : ℝ)) (q :=
+      (46230074222117999 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_588_9 :
+    Uω (aρ 9) (bρ 9) (43705589996379 / 64000000000000 : ℝ) ≤
+      (-(6166416285021924208357 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (43180327341267823 / 80000000000000000 : ℝ) ≤
+      (-(6166416285021924208357 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (43180327341267823 / 80000000000000000 : ℝ) =
+            (43180327341267823 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (43180327341267823 / 40000000000000000 : ℝ) ≤
+            (765055517978075791643 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5313638871327737 / 10000000000000000 : ℝ)) (q :=
+      (43180327341267823 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_588_10 :
+    Uω (aρ 10) (bρ 10) (43705589996379 / 64000000000000 : ℝ) ≤
+      (-(7050979866472832607387 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (39524812826294763 / 80000000000000000 : ℝ) ≤
+      (-(7050979866472832607387 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (39524812826294763 / 80000000000000000 : ℝ) =
+            (39524812826294763 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (39524812826294763 / 20000000000000000 : ℝ) ≤
+            (6811963739527167392613 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (299269954754967 / 625000000000000 : ℝ)) (q :=
+      (39524812826294763 / 80000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_588_11 :
+    Uω (aρ 11) (bρ 11) (43705589996379 / 64000000000000 : ℝ) ≤
+      (-(8183456858447127867951 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (35292869568515511 / 80000000000000000 : ℝ) ≤
+      (-(8183456858447127867951 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (35292869568515511 / 80000000000000000 : ℝ) =
+            (35292869568515511 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (35292869568515511 / 20000000000000000 : ℝ) ≤
+            (5679486747552872132049 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4148810087839659 / 10000000000000000 : ℝ)) (q :=
+      (35292869568515511 / 80000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_588_12 :
+    Uω (aρ 12) (bρ 12) (43705589996379 / 64000000000000 : ℝ) ≤
+      (-(1204403490610925670401 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (30523712533582199 / 80000000000000000 : ℝ) ≤
+      (-(1204403490610925670401 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (30523712533582199 / 80000000000000000 : ℝ) =
+            (30523712533582199 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (30523712533582199 / 20000000000000000 : ℝ) ≤
+            (528464460139074329599 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3380220925791331 / 10000000000000000 : ℝ)) (q :=
+      (30523712533582199 / 80000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_588_13 :
+    Uω (aρ 13) (bρ 13) (43705589996379 / 64000000000000 : ℝ) ≤
+      (-(11559851808548369618159 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25179784074442571 / 80000000000000000 : ℝ) ≤
+      (-(11559851808548369618159 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (25179784074442571 / 80000000000000000 : ℝ) =
+            (25179784074442571 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25179784074442571 / 20000000000000000 : ℝ) ≤
+            (2303091797451630381841 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (152726284537589 / 625000000000000 : ℝ)) (q :=
+      (25179784074442571 / 80000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_588_14 :
+    Uω (aρ 14) (bρ 14) (43705589996379 / 64000000000000 : ℝ) ≤
+      (-(1461851957046835182271 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (18544526476265923 / 80000000000000000 : ℝ) ≤
+      (-(1461851957046835182271 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (18544526476265923 / 80000000000000000 : ℝ) =
+            (18544526476265923 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18544526476265923 / 10000000000000000 : ℝ) ≤
+            (617589583853164817729 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (564366719051131 / 5000000000000000 : ℝ)) (q :=
+      (18544526476265923 / 80000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_588_15 :
+    Uω (aρ 15) (bρ 15) (43705589996379 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_588_16 :
+    Uω (aρ 16) (bρ 16) (43705589996379 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_588 :
+    Uρ (43705589996379 / 64000000000000 : ℝ) ≤
+      (-(6561204984838086247927 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (43705589996379 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(39090009657798394581 / 100000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_588_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (43705589996379 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(394439887737222588027 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_588_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (43705589996379 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(4015552499438766461231 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_588_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (43705589996379 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(413479465984879583207 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_588_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (43705589996379 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(539873510739345538629 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_588_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (43705589996379 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(573601652664223193247 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_588_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (43705589996379 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(4968178041979655025511 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_588_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (43705589996379 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(2741980454591354062683 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_588_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (43705589996379 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(6166416285021924208357 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_588_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (43705589996379 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(7050979866472832607387 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_588_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (43705589996379 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(8183456858447127867951 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_588_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (43705589996379 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(1204403490610925670401 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_588_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (43705589996379 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(11559851808548369618159 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_588_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (43705589996379 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(1461851957046835182271 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_588_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (43705589996379 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_588_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (43705589996379 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_588_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_589_1 :
+    Uω (aρ 1) (bρ 1) (5469688648751 / 8000000000000 : ℝ) ≤
+      (-(1948507749324743187031 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13545179443886287 / 20000000000000000 : ℝ) ≤
+      (-(1948507749324743187031 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (13545179443886287 / 20000000000000000 : ℝ) =
+            (13545179443886287 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13545179443886287 / 10000000000000000 : ℝ) ≤
+            (1517228152175256812969 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6772565851032537 / 10000000000000000 : ℝ)) (q :=
+      (13545179443886287 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_589_2 :
+    Uω (aρ 2) (bρ 2) (5469688648751 / 8000000000000 : ℝ) ≤
+      (-(245773168539214552107 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13497374750813287 / 20000000000000000 : ℝ) ≤
+      (-(245773168539214552107 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (13497374750813287 / 20000000000000000 : ℝ) =
+            (13497374750813287 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13497374750813287 / 10000000000000000 : ℝ) ≤
+            (187443819148285447893 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6748530170469537 / 10000000000000000 : ℝ)) (q :=
+      (13497374750813287 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_589_3 :
+    Uω (aρ 3) (bρ 3) (5469688648751 / 8000000000000 : ℝ) ≤
+      (-(2001718882180630828503 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (418806028268991 / 625000000000000 : ℝ) ≤
+      (-(2001718882180630828503 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (418806028268991 / 625000000000000 : ℝ) =
+            (418806028268991 / 312500000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (418806028268991 / 312500000000000 : ℝ) ≤
+            (1464017019319369171497 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3350172215306981 / 5000000000000000 : ℝ)) (q :=
+      (418806028268991 / 625000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_589_4 :
+    Uω (aρ 4) (bρ 4) (5469688648751 / 8000000000000 : ℝ) ≤
+      (-(824506551390814758131 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6621565464801823 / 10000000000000000 : ℝ) ≤
+      (-(824506551390814758131 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (6621565464801823 / 10000000000000000 : ℝ) =
+            (6621565464801823 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6621565464801823 / 5000000000000000 : ℝ) ≤
+            (561787809209185241869 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (413748539792181 / 625000000000000 : ℝ)) (q :=
+      (6621565464801823 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_589_5 :
+    Uω (aρ 5) (bρ 5) (5469688648751 / 8000000000000 : ℝ) ≤
+      (-(4306493497263263020103 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6500868237672459 / 10000000000000000 : ℝ) ≤
+      (-(4306493497263263020103 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (6500868237672459 / 10000000000000000 : ℝ) =
+            (6500868237672459 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6500868237672459 / 5000000000000000 : ℝ) ≤
+            (2624978305736736979897 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (50756032044931 / 78125000000000 : ℝ)) (q :=
+      (6500868237672459 / 10000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_589_6 :
+    Uω (aρ 6) (bρ 6) (5469688648751 / 8000000000000 : ℝ) ≤
+      (-(4575965519951952109527 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3164013637734911 / 5000000000000000 : ℝ) ≤
+      (-(4575965519951952109527 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (3164013637734911 / 5000000000000000 : ℝ) =
+            (3164013637734911 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3164013637734911 / 2500000000000000 : ℝ) ≤
+            (2355506283048047890473 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3159165276242947 / 5000000000000000 : ℝ)) (q :=
+      (3164013637734911 / 5000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_589_7 :
+    Uω (aρ 7) (bρ 7) (5469688648751 / 8000000000000 : ℝ) ≤
+      (-(1238701908887633955801 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (487422334477081 / 800000000000000 : ℝ) ≤
+      (-(1238701908887633955801 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (487422334477081 / 800000000000000 : ℝ) =
+            (487422334477081 / 400000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (487422334477081 / 400000000000000 : ℝ) ≤
+            (494166041862366044199 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (242859829769331 / 400000000000000 : ℝ)) (q :=
+      (487422334477081 / 800000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_589_8 :
+    Uω (aρ 8) (bρ 8) (5469688648751 / 8000000000000 : ℝ) ≤
+      (-(5469825949513502250433 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2893466653157457 / 5000000000000000 : ℝ) ≤
+      (-(5469825949513502250433 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2893466653157457 / 5000000000000000 : ℝ) =
+            (2893466653157457 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2893466653157457 / 2500000000000000 : ℝ) ≤
+            (1461645853486497749567 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2871672823138039 / 5000000000000000 : ℝ)) (q :=
+      (2893466653157457 / 5000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_589_9 :
+    Uω (aρ 9) (bρ 9) (5469688648751 / 8000000000000 : ℝ) ≤
+      (-(6151161216080942852501 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1351445296808881 / 2500000000000000 : ℝ) ≤
+      (-(6151161216080942852501 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1351445296808881 / 2500000000000000 : ℝ) =
+            (1351445296808881 / 1250000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1351445296808881 / 1250000000000000 : ℝ) ≤
+            (780310586919057147499 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2661003518238649 / 5000000000000000 : ℝ)) (q :=
+      (1351445296808881 / 2500000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_589_10 :
+    Uω (aρ 10) (bρ 10) (5469688648751 / 8000000000000 : ℝ) ≤
+      (-(7034053104242121893583 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1237242880983201 / 2500000000000000 : ℝ) ≤
+      (-(7034053104242121893583 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1237242880983201 / 2500000000000000 : ℝ) =
+            (1237242880983201 / 625000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1237242880983201 / 625000000000000 : ℝ) ≤
+            (6828890501757878106417 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2398473371683429 / 5000000000000000 : ℝ)) (q :=
+      (1237242880983201 / 2500000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_589_11 :
+    Uω (aρ 11) (bρ 11) (5469688648751 / 8000000000000 : ℝ) ≤
+      (-(4081962433340292229409 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1768093546853501 / 4000000000000000 : ℝ) ≤
+      (-(4081962433340292229409 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1768093546853501 / 4000000000000000 : ℝ) =
+            (1768093546853501 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1768093546853501 / 1000000000000000 : ℝ) ≤
+            (2849509369659707770591 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (831589611194751 / 2000000000000000 : ℝ)) (q :=
+      (1768093546853501 / 4000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_589_12 :
+    Uω (aρ 12) (bρ 12) (5469688648751 / 8000000000000 : ℝ) ≤
+      (-(1201408064539557201607 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1912309091470187 / 5000000000000000 : ℝ) ≤
+      (-(1201408064539557201607 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1912309091470187 / 5000000000000000 : ℝ) =
+            (1912309091470187 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1912309091470187 / 1250000000000000 : ℝ) ≤
+            (531459886210442798393 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1695208392135999 / 5000000000000000 : ℝ)) (q :=
+      (1912309091470187 / 5000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_589_13 :
+    Uω (aρ 13) (bρ 13) (5469688648751 / 8000000000000 : ℝ) ≤
+      (-(11526740094447365904059 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (789478026307013 / 2500000000000000 : ℝ) ≤
+      (-(11526740094447365904059 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (789478026307013 / 2500000000000000 : ℝ) =
+            (789478026307013 / 625000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (789478026307013 / 625000000000000 : ℝ) ≤
+            (2336203511552634095941 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1228193185221177 / 5000000000000000 : ℝ)) (q :=
+      (789478026307013 / 2500000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13]) (by norm_num [bρ_13])
+      (by norm_num) (by norm_num [aρ_13, bρ_13]) (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_589_14 :
+    Uω (aρ 14) (bρ 14) (5469688648751 / 8000000000000 : ℝ) ≤
+      (-(1818429225264942628537 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (583650666392007 / 2500000000000000 : ℝ) ≤
+      (-(1818429225264942628537 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (583650666392007 / 2500000000000000 : ℝ) =
+            (583650666392007 / 312500000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (583650666392007 / 312500000000000 : ℝ) ≤
+            (780872700860057371463 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (576847388083653 / 5000000000000000 : ℝ)) (q :=
+      (583650666392007 / 2500000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14]) (by norm_num [bρ_14])
+      (by norm_num) (by norm_num [aρ_14, bρ_14]) (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_589_15 :
+    Uω (aρ 15) (bρ 15) (5469688648751 / 8000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_589_16 :
+    Uω (aρ 16) (bρ 16) (5469688648751 / 8000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_589 :
+    Uρ (5469688648751 / 8000000000000 : ℝ) ≤
+      (-(818012982659048695281 / 1250000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (5469688648751 / 8000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1948507749324743187031 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_589_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (5469688648751 / 8000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(245773168539214552107 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_589_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (5469688648751 / 8000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(2001718882180630828503 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_589_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (5469688648751 / 8000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(824506551390814758131 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_589_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (5469688648751 / 8000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(4306493497263263020103 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_589_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (5469688648751 / 8000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(4575965519951952109527 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_589_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (5469688648751 / 8000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(1238701908887633955801 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_589_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (5469688648751 / 8000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(5469825949513502250433 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_589_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (5469688648751 / 8000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(6151161216080942852501 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_589_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (5469688648751 / 8000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(7034053104242121893583 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_589_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (5469688648751 / 8000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4081962433340292229409 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_589_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (5469688648751 / 8000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(1201408064539557201607 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_589_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (5469688648751 / 8000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(11526740094447365904059 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_589_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (5469688648751 / 8000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(1818429225264942628537 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_589_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (5469688648751 / 8000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_589_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (5469688648751 / 8000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_589_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_590_1 :
+    Uω (aρ 1) (bρ 1) (43809428383637 / 64000000000000 : ℝ) ≤
+      (-(1942522189783709894671 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2169824679842153 / 3200000000000000 : ℝ) ≤
+      (-(1942522189783709894671 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2169824679842153 / 3200000000000000 : ℝ) =
+            (2169824679842153 / 1600000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2169824679842153 / 1600000000000000 : ℝ) ≤
+            (1523213711716290105329 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (271227131286207 / 400000000000000 : ℝ)) (q :=
+      (2169824679842153 / 3200000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_590_2 :
+    Uω (aρ 2) (bρ 2) (43809428383637 / 64000000000000 : ℝ) ≤
+      (-(980089241722220861977 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (54054399505276293 / 80000000000000000 : ℝ) ≤
+      (-(980089241722220861977 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (54054399505276293 / 80000000000000000 : ℝ) =
+            (54054399505276293 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (54054399505276293 / 40000000000000000 : ℝ) ≤
+            (752778709027779138023 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (844580365246349 / 1250000000000000 : ℝ)) (q :=
+      (54054399505276293 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_590_3 :
+    Uω (aρ 3) (bρ 3) (43809428383637 / 64000000000000 : ℝ) ≤
+      (-(1995668845309871194029 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2146883038033521 / 3200000000000000 : ℝ) ≤
+      (-(1995668845309871194029 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2146883038033521 / 3200000000000000 : ℝ) =
+            (2146883038033521 / 1600000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2146883038033521 / 1600000000000000 : ℝ) ≤
+            (1470067056190128805971 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (268338325588449 / 400000000000000 : ℝ)) (q :=
+      (2146883038033521 / 3200000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_590_4 :
+    Uω (aρ 4) (bρ 4) (43809428383637 / 64000000000000 : ℝ) ≤
+      (-(4110285878273273006483 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (53037438267470893 / 80000000000000000 : ℝ) ≤
+      (-(4110285878273273006483 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (53037438267470893 / 80000000000000000 : ℝ) =
+            (53037438267470893 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (53037438267470893 / 40000000000000000 : ℝ) ≤
+            (2821185924726726993517 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3314046449967221 / 5000000000000000 : ℝ)) (q :=
+      (53037438267470893 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_590_5 :
+    Uω (aρ 5) (bρ 5) (43809428383637 / 64000000000000 : ℝ) ≤
+      (-(4294014520243379947173 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (52071885760380549 / 80000000000000000 : ℝ) ≤
+      (-(4294014520243379947173 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (52071885760380549 / 80000000000000000 : ℝ) =
+            (52071885760380549 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (52071885760380549 / 40000000000000000 : ℝ) ≤
+            (2637457282756620052827 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (813111836562107 / 1250000000000000 : ℝ)) (q :=
+      (52071885760380549 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_590_6 :
+    Uω (aρ 6) (bρ 6) (43809428383637 / 64000000000000 : ℝ) ≤
+      (-(912626870858038316583 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (50689216668171037 / 80000000000000000 : ℝ) ≤
+      (-(912626870858038316583 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (50689216668171037 / 80000000000000000 : ℝ) =
+            (50689216668171037 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (50689216668171037 / 40000000000000000 : ℝ) ≤
+            (473667489741961683417 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3163233897292239 / 5000000000000000 : ℝ)) (q :=
+      (50689216668171037 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_590_7 :
+    Uω (aρ 7) (bρ 7) (43809428383637 / 64000000000000 : ℝ) ≤
+      (-(4941455207028364076573 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (48807359636483301 / 80000000000000000 : ℝ) ≤
+      (-(4941455207028364076573 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (48807359636483301 / 80000000000000000 : ℝ) =
+            (48807359636483301 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (48807359636483301 / 40000000000000000 : ℝ) ≤
+            (1990016595971635923427 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (379979057338909 / 625000000000000 : ℝ)) (q :=
+      (48807359636483301 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_590_8 :
+    Uω (aρ 8) (bρ 8) (43809428383637 / 64000000000000 : ℝ) ≤
+      (-(545571124375522783539 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (46360857276965921 / 80000000000000000 : ℝ) ≤
+      (-(545571124375522783539 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (46360857276965921 / 80000000000000000 : ℝ) =
+            (46360857276965921 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (46360857276965921 / 40000000000000000 : ℝ) ≤
+            (147576055924477216461 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5751580978883199 / 10000000000000000 : ℝ)) (q :=
+      (46360857276965921 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_590_9 :
+    Uω (aρ 9) (bρ 9) (43809428383637 / 64000000000000 : ℝ) ≤
+      (-(766991264223849596283 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (43312168491063741 / 80000000000000000 : ℝ) ≤
+      (-(766991264223849596283 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (43312168491063741 / 80000000000000000 : ℝ) =
+            (43312168491063741 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (43312168491063741 / 40000000000000000 : ℝ) ≤
+            (99442711151150403717 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2665187205383827 / 5000000000000000 : ℝ)) (q :=
+      (43312168491063741 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_590_10 :
+    Uω (aρ 10) (bρ 10) (43809428383637 / 64000000000000 : ℝ) ≤
+      (-(3508578377429020554659 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7931744875896073 / 16000000000000000 : ℝ) ≤
+      (-(3508578377429020554659 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7931744875896073 / 16000000000000000 : ℝ) =
+            (7931744875896073 / 4000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7931744875896073 / 4000000000000000 : ℝ) ≤
+            (3422893425570979445341 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (480557241636681 / 1000000000000000 : ℝ)) (q :=
+      (7931744875896073 / 16000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_590_11 :
+    Uω (aρ 11) (bρ 11) (43809428383637 / 64000000000000 : ℝ) ≤
+      (-(8144435743557710104167 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (35430855323149441 / 80000000000000000 : ℝ) ≤
+      (-(8144435743557710104167 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (35430855323149441 / 80000000000000000 : ℝ) =
+            (35430855323149441 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (35430855323149441 / 20000000000000000 : ℝ) ≤
+            (5718507862442289895833 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4167081778489079 / 10000000000000000 : ℝ)) (q :=
+      (35430855323149441 / 80000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_590_12 :
+    Uω (aρ 12) (bρ 12) (43809428383637 / 64000000000000 : ℝ) ≤
+      (-(2396843256077924959477 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6134026705020049 / 16000000000000000 : ℝ) ≤
+      (-(2396843256077924959477 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6134026705020049 / 16000000000000000 : ℝ) =
+            (6134026705020049 / 4000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6134026705020049 / 4000000000000000 : ℝ) ≤
+            (1068892645422075040523 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (170030071283089 / 500000000000000 : ℝ)) (q :=
+      (6134026705020049 / 16000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_590_13 :
+    Uω (aρ 13) (bρ 13) (43809428383637 / 64000000000000 : ℝ) ≤
+      (-(5746899876398573163821 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25346652217097741 / 80000000000000000 : ℝ) ≤
+      (-(5746899876398573163821 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (25346652217097741 / 80000000000000000 : ℝ) =
+            (25346652217097741 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25346652217097741 / 20000000000000000 : ℝ) ≤
+            (1184571926601426836179 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1234556420128077 / 5000000000000000 : ℝ)) (q :=
+      (25346652217097741 / 80000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_590_14 :
+    Uω (aρ 14) (bρ 14) (43809428383637 / 64000000000000 : ℝ) ≤
+      (-(14477855490426510840571 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (18807224630108041 / 80000000000000000 : ℝ) ≤
+      (-(14477855490426510840571 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (18807224630108041 / 80000000000000000 : ℝ) =
+            (18807224630108041 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18807224630108041 / 10000000000000000 : ℝ) ≤
+            (6316559918573489159429 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1178183228553729 / 10000000000000000 : ℝ)) (q :=
+      (18807224630108041 / 80000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_590_15 :
+    Uω (aρ 15) (bρ 15) (43809428383637 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_590_16 :
+    Uω (aρ 16) (bρ 16) (43809428383637 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_590 :
+    Uρ (43809428383637 / 64000000000000 : ℝ) ≤
+      (-(326355036030635433631 / 500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (43809428383637 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1942522189783709894671 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_590_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (43809428383637 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(980089241722220861977 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_590_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (43809428383637 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1995668845309871194029 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_590_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (43809428383637 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(4110285878273273006483 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_590_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (43809428383637 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(4294014520243379947173 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_590_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (43809428383637 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(912626870858038316583 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_590_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (43809428383637 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(4941455207028364076573 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_590_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (43809428383637 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(545571124375522783539 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_590_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (43809428383637 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(766991264223849596283 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_590_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (43809428383637 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(3508578377429020554659 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_590_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (43809428383637 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(8144435743557710104167 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_590_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (43809428383637 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(2396843256077924959477 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_590_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (43809428383637 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(5746899876398573163821 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_590_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (43809428383637 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(14477855490426510840571 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_590_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (43809428383637 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_590_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (43809428383637 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_590_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_591_1 :
+    Uω (aρ 1) (bρ 1) (21930673788633 / 32000000000000 : ℝ) ≤
+      (-(3873087574221783174133 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5431051621601647 / 8000000000000000 : ℝ) ≤
+      (-(3873087574221783174133 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5431051621601647 / 8000000000000000 : ℝ) =
+            (5431051621601647 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5431051621601647 / 4000000000000000 : ℝ) ≤
+            (3058384228778216825867 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1357758142628261 / 2000000000000000 : ℝ)) (q :=
+      (5431051621601647 / 8000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_591_2 :
+    Uω (aρ 2) (bρ 2) (21930673788633 / 32000000000000 : ℝ) ≤
+      (-(3908357653472914841141 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27059650001838881 / 40000000000000000 : ℝ) ≤
+      (-(3908357653472914841141 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (27059650001838881 / 40000000000000000 : ℝ) =
+            (27059650001838881 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27059650001838881 / 20000000000000000 : ℝ) ≤
+            (3023114149527085158859 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1691188918141657 / 2500000000000000 : ℝ)) (q :=
+      (27059650001838881 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_591_3 :
+    Uω (aρ 3) (bρ 3) (21930673788633 / 32000000000000 : ℝ) ≤
+      (-(497406530345597454679 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (26868490135171893 / 40000000000000000 : ℝ) ≤
+      (-(497406530345597454679 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (26868490135171893 / 40000000000000000 : ℝ) =
+            (26868490135171893 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26868490135171893 / 20000000000000000 : ℝ) ≤
+            (369027445029402545321 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3358285922791567 / 5000000000000000 : ℝ)) (q :=
+      (26868490135171893 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_591_4 :
+    Uω (aρ 4) (bρ 4) (21930673788633 / 32000000000000 : ℝ) ≤
+      (-(512256748377032037619 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5310235277848301 / 8000000000000000 : ℝ) ≤
+      (-(512256748377032037619 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5310235277848301 / 8000000000000000 : ℝ) =
+            (5310235277848301 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5310235277848301 / 4000000000000000 : ℝ) ≤
+            (354177226997967962381 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (331810457684147 / 500000000000000 : ℝ)) (q :=
+      (5310235277848301 / 8000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_591_5 :
+    Uω (aρ 5) (bρ 5) (21930673788633 / 32000000000000 : ℝ) ≤
+      (-(2140775557921181468381 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1042736510350251 / 1600000000000000 : ℝ) ≤
+      (-(2140775557921181468381 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1042736510350251 / 1600000000000000 : ℝ) =
+            (1042736510350251 / 800000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1042736510350251 / 800000000000000 : ℝ) ≤
+            (1324960343578818531619 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (260520690311013 / 400000000000000 : ℝ)) (q :=
+      (1042736510350251 / 1600000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_591_6 :
+    Uω (aρ 6) (bρ 6) (21930673788633 / 32000000000000 : ℝ) ≤
+      (-(4550319681689112793929 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5075421487741559 / 8000000000000000 : ℝ) ≤
+      (-(4550319681689112793929 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5075421487741559 / 8000000000000000 : ℝ) =
+            (5075421487741559 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5075421487741559 / 4000000000000000 : ℝ) ≤
+            (2381152121310887206071 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1266920994578217 / 2000000000000000 : ℝ)) (q :=
+      (5075421487741559 / 8000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_591_7 :
+    Uω (aρ 7) (bρ 7) (21930673788633 / 32000000000000 : ℝ) ≤
+      (-(4928120707798083417323 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24436242608817219 / 40000000000000000 : ℝ) ≤
+      (-(4928120707798083417323 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (24436242608817219 / 40000000000000000 : ℝ) =
+            (24436242608817219 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24436242608817219 / 20000000000000000 : ℝ) ≤
+            (2003351095201916582677 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6087833938705797 / 10000000000000000 : ℝ)) (q :=
+      (24436242608817219 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_591_8 :
+    Uω (aρ 8) (bρ 8) (21930673788633 / 32000000000000 : ℝ) ≤
+      (-(272080836654552202981 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4642624670747133 / 8000000000000000 : ℝ) ≤
+      (-(272080836654552202981 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (4642624670747133 / 8000000000000000 : ℝ) =
+            (4642624670747133 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4642624670747133 / 4000000000000000 : ℝ) ≤
+            (74492753495447797019 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (287990798125251 / 500000000000000 : ℝ)) (q :=
+      (4642624670747133 / 8000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_591_9 :
+    Uω (aρ 9) (bρ 9) (21930673788633 / 32000000000000 : ℝ) ≤
+      (-(6120722900714232247101 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21689042167840287 / 40000000000000000 : ℝ) ≤
+      (-(6120722900714232247101 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (21689042167840287 / 40000000000000000 : ℝ) =
+            (21689042167840287 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21689042167840287 / 20000000000000000 : ℝ) ≤
+            (810748902285767752899 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5338740997917331 / 10000000000000000 : ℝ)) (q :=
+      (21689042167840287 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_591_10 :
+    Uω (aρ 10) (bρ 10) (21930673788633 / 32000000000000 : ℝ) ≤
+      (-(7000290702954301500691 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3972566942892681 / 8000000000000000 : ℝ) ≤
+      (-(7000290702954301500691 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3972566942892681 / 8000000000000000 : ℝ) =
+            (3972566942892681 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3972566942892681 / 2000000000000000 : ℝ) ≤
+            (6862652903045698499309 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (481419630472389 / 1000000000000000 : ℝ)) (q :=
+      (3972566942892681 / 8000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_591_11 :
+    Uω (aρ 11) (bρ 11) (21930673788633 / 32000000000000 : ℝ) ≤
+      (-(2031247320397186581489 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17749911419090027 / 40000000000000000 : ℝ) ≤
+      (-(2031247320397186581489 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (17749911419090027 / 40000000000000000 : ℝ) =
+            (17749911419090027 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17749911419090027 / 10000000000000000 : ℝ) ≤
+            (1434488581102813418511 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4176211283242201 / 10000000000000000 : ℝ)) (q :=
+      (17749911419090027 / 40000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_591_12 :
+    Uω (aρ 12) (bρ 12) (21930673788633 / 32000000000000 : ℝ) ≤
+      (-(9563552940403943445557 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15371638560123711 / 40000000000000000 : ℝ) ≤
+      (-(9563552940403943445557 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (15371638560123711 / 40000000000000000 : ℝ) =
+            (15371638560123711 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15371638560123711 / 10000000000000000 : ℝ) ≤
+            (4299390665596056554443 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3410774950444043 / 10000000000000000 : ℝ)) (q :=
+      (15371638560123711 / 40000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_591_13 :
+    Uω (aρ 13) (bρ 13) (21930673788633 / 32000000000000 : ℝ) ≤
+      (-(2865257125482834296423 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12714926310772163 / 40000000000000000 : ℝ) ≤
+      (-(2865257125482834296423 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (12714926310772163 / 40000000000000000 : ℝ) =
+            (12714926310772163 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12714926310772163 / 10000000000000000 : ℝ) ≤
+            (600478776017165703577 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2481800567363269 / 10000000000000000 : ℝ)) (q :=
+      (12714926310772163 / 40000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_591_14 :
+    Uω (aρ 14) (bρ 14) (21930673788633 / 32000000000000 : ℝ) ≤
+      (-(225151496633754432083 / 156250000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9467925991076191 / 40000000000000000 : ℝ) ≤
+      (-(225151496633754432083 / 156250000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (9467925991076191 / 40000000000000000 : ℝ) =
+            (9467925991076191 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9467925991076191 / 5000000000000000 : ℝ) ≤
+            (99761244131870567917 / 156250000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1202227692560283 / 10000000000000000 : ℝ)) (q :=
+      (9467925991076191 / 40000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_591_15 :
+    Uω (aρ 15) (bρ 15) (21930673788633 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_591_16 :
+    Uω (aρ 16) (bρ 16) (21930673788633 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_591 :
+    Uρ (21930673788633 / 32000000000000 : ℝ) ≤
+      (-(1302038278493546697749 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (21930673788633 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3873087574221783174133 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_591_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (21930673788633 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3908357653472914841141 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_591_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (21930673788633 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(497406530345597454679 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_591_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (21930673788633 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(512256748377032037619 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_591_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (21930673788633 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(2140775557921181468381 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_591_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (21930673788633 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(4550319681689112793929 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_591_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (21930673788633 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(4928120707798083417323 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_591_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (21930673788633 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(272080836654552202981 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_591_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (21930673788633 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(6120722900714232247101 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_591_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (21930673788633 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(7000290702954301500691 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_591_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (21930673788633 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(2031247320397186581489 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_591_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (21930673788633 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(9563552940403943445557 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_591_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (21930673788633 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(2865257125482834296423 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_591_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (21930673788633 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) * (-(225151496633754432083 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_591_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (21930673788633 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_591_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (21930673788633 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_591_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/U49.lean b/LeanPool/Zeta5Irrational/Table/U49.lean
new file mode 100644
index 0000000000..512aa227c9
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/U49.lean
@@ -0,0 +1,5051 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-! # Certified arcsine potential bounds (U49) -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma U_592_1 :
+    Uω (aρ 1) (bρ 1) (8782653354179 / 12800000000000 : ℝ) ≤
+      (-(3861145048423655657401 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10875083087087007 / 16000000000000000 : ℝ) ≤
+      (-(3861145048423655657401 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10875083087087007 / 16000000000000000 : ℝ) =
+            (10875083087087007 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10875083087087007 / 8000000000000000 : ℝ) ≤
+            (3070326754576344342599 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1359380628798283 / 2000000000000000 : ℝ)) (q :=
+      (10875083087087007 / 16000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_592_2 :
+    Uω (aρ 2) (bρ 2) (8782653354179 / 12800000000000 : ℝ) ≤
+      (-(243523295113786450747 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2167368019938823 / 3200000000000000 : ℝ) ≤
+      (-(243523295113786450747 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2167368019938823 / 3200000000000000 : ℝ) =
+            (2167368019938823 / 1600000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2167368019938823 / 1600000000000000 : ℝ) ≤
+            (189693692573713549253 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (67728684222603 / 100000000000000 : ℝ)) (q :=
+      (2167368019938823 / 3200000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_592_3 :
+    Uω (aρ 3) (bρ 3) (8782653354179 / 12800000000000 : ℝ) ≤
+      (-(991795346368845415999 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (53801884576994823 / 80000000000000000 : ℝ) ≤
+      (-(991795346368845415999 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (53801884576994823 / 80000000000000000 : ℝ) =
+            (53801884576994823 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (53801884576994823 / 40000000000000000 : ℝ) ≤
+            (741072604381154584001 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3362342774120681 / 5000000000000000 : ℝ)) (q :=
+      (53801884576994823 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_592_4 :
+    Uω (aρ 4) (bρ 4) (8782653354179 / 12800000000000 : ℝ) ≤
+      (-(51072963081599011411 / 125000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (53167267251590347 / 80000000000000000 : ℝ) ≤
+      (-(51072963081599011411 / 125000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (53167267251590347 / 80000000000000000 : ℝ) =
+            (53167267251590347 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (53167267251590347 / 40000000000000000 : ℝ) ≤
+            (35570434455900988589 / 125000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6644325397955243 / 10000000000000000 : ℝ)) (q :=
+      (53167267251590347 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_592_5 :
+    Uω (aρ 5) (bρ 5) (8782653354179 / 12800000000000 : ℝ) ≤
+      (-(4269103245193693824213 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (52201765173156339 / 80000000000000000 : ℝ) ≤
+      (-(4269103245193693824213 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (52201765173156339 / 80000000000000000 : ℝ) =
+            (52201765173156339 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (52201765173156339 / 40000000000000000 : ℝ) ≤
+            (2662368557806306175787 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6521139797681741 / 10000000000000000 : ℝ)) (q :=
+      (52201765173156339 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_592_6 :
+    Uω (aρ 6) (bρ 6) (8782653354179 / 12800000000000 : ℝ) ≤
+      (-(4537521459674536368377 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (50819212832474303 / 80000000000000000 : ℝ) ≤
+      (-(4537521459674536368377 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (50819212832474303 / 80000000000000000 : ℝ) =
+            (50819212832474303 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (50819212832474303 / 40000000000000000 : ℝ) ≤
+            (2393950343325463631623 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (198210690239101 / 312500000000000 : ℝ)) (q :=
+      (50819212832474303 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_592_7 :
+    Uω (aρ 7) (bρ 7) (8782653354179 / 12800000000000 : ℝ) ≤
+      (-(614350511180361655339 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (48937610193604279 / 80000000000000000 : ℝ) ≤
+      (-(614350511180361655339 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (48937610193604279 / 80000000000000000 : ℝ) =
+            (48937610193604279 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (48937610193604279 / 40000000000000000 : ℝ) ≤
+            (252083464194638344661 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3048001404346863 / 5000000000000000 : ℝ)) (q :=
+      (48937610193604279 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_592_8 :
+    Uω (aρ 8) (bρ 8) (8782653354179 / 12800000000000 : ℝ) ≤
+      (-(1356885589740886941109 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (46491634748014451 / 80000000000000000 : ℝ) ≤
+      (-(1356885589740886941109 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (46491634748014451 / 80000000000000000 : ℝ) =
+            (46491634748014451 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (46491634748014451 / 40000000000000000 : ℝ) ≤
+            (375982361009113058891 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5768050598636269 / 10000000000000000 : ℝ)) (q :=
+      (46491634748014451 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_592_9 :
+    Uω (aρ 9) (bρ 9) (8782653354179 / 12800000000000 : ℝ) ≤
+      (-(6105539499798913087489 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (43443997046514299 / 80000000000000000 : ℝ) ≤
+      (-(6105539499798913087489 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (43443997046514299 / 80000000000000000 : ℝ) =
+            (43443997046514299 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (43443997046514299 / 40000000000000000 : ℝ) ≤
+            (825932303201086912511 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5347106801621231 / 10000000000000000 : ℝ)) (q :=
+      (43443997046514299 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_592_10 :
+    Uω (aρ 10) (bρ 10) (8782653354179 / 12800000000000 : ℝ) ≤
+      (-(43646592711571477181 / 62500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (39792607378096127 / 80000000000000000 : ℝ) ≤
+      (-(43646592711571477181 / 62500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (39792607378096127 / 80000000000000000 : ℝ) =
+            (39792607378096127 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (39792607378096127 / 20000000000000000 : ℝ) ≤
+            (42996804825928522819 / 62500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (602852302251461 / 1250000000000000 : ℝ)) (q :=
+      (39792607378096127 / 80000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_592_11 :
+    Uω (aρ 11) (bρ 11) (8782653354179 / 12800000000000 : ℝ) ≤
+      (-(253299539840680361623 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (35568773592564671 / 80000000000000000 : ℝ) ≤
+      (-(253299539840680361623 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (35568773592564671 / 80000000000000000 : ℝ) =
+            (35568773592564671 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (35568773592564671 / 20000000000000000 : ℝ) ≤
+            (179917447846819638377 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (130791768682307 / 312500000000000 : ℝ)) (q :=
+      (35568773592564671 / 80000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_592_12 :
+    Uω (aρ 12) (bρ 12) (8782653354179 / 12800000000000 : ℝ) ≤
+      (-(9539803762064064091603 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (30816376645680631 / 80000000000000000 : ℝ) ≤
+      (-(9539803762064064091603 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (30816376645680631 / 80000000000000000 : ℝ) =
+            (30816376645680631 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (30816376645680631 / 20000000000000000 : ℝ) ≤
+            (4323139843935935908397 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1710468728898907 / 5000000000000000 : ℝ)) (q :=
+      (30816376645680631 / 80000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_592_13 :
+    Uω (aρ 13) (bρ 13) (8782653354179 / 12800000000000 : ℝ) ≤
+      (-(11428424110941748224189 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25512900419887071 / 80000000000000000 : ℝ) ≤
+      (-(11428424110941748224189 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (25512900419887071 / 80000000000000000 : ℝ) =
+            (25512900419887071 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25512900419887071 / 20000000000000000 : ℝ) ≤
+            (2434519495058251775811 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (311806267868053 / 1250000000000000 : ℝ)) (q :=
+      (25512900419887071 / 80000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_592_14 :
+    Uω (aρ 14) (bρ 14) (8782653354179 / 12800000000000 : ℝ) ≤
+      (-(7171437141596351498281 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19062807884872551 / 80000000000000000 : ℝ) ≤
+      (-(7171437141596351498281 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (19062807884872551 / 80000000000000000 : ℝ) =
+            (19062807884872551 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19062807884872551 / 10000000000000000 : ℝ) ≤
+            (3225770562903648501719 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (612927147117897 / 5000000000000000 : ℝ)) (q :=
+      (19062807884872551 / 80000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_592_15 :
+    Uω (aρ 15) (bρ 15) (8782653354179 / 12800000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_592_16 :
+    Uω (aρ 16) (bρ 16) (8782653354179 / 12800000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_592 :
+    Uρ (8782653354179 / 12800000000000 : ℝ) ≤
+      (-(6493372085490171878343 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (8782653354179 / 12800000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3861145048423655657401 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_592_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (8782653354179 / 12800000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(243523295113786450747 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_592_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (8782653354179 / 12800000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(991795346368845415999 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_592_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (8782653354179 / 12800000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(51072963081599011411 / 125000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_592_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (8782653354179 / 12800000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(4269103245193693824213 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_592_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (8782653354179 / 12800000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(4537521459674536368377 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_592_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (8782653354179 / 12800000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(614350511180361655339 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_592_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (8782653354179 / 12800000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(1356885589740886941109 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_592_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (8782653354179 / 12800000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(6105539499798913087489 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_592_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (8782653354179 / 12800000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) * (-(43646592711571477181 / 62500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_592_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (8782653354179 / 12800000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) * (-(253299539840680361623 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_592_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (8782653354179 / 12800000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(9539803762064064091603 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_592_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (8782653354179 / 12800000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(11428424110941748224189 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_592_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (8782653354179 / 12800000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(7171437141596351498281 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_592_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (8782653354179 / 12800000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_592_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (8782653354179 / 12800000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_592_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_593_1 :
+    Uω (aρ 1) (bρ 1) (10991296491131 / 16000000000000 : ℝ) ≤
+      (-(76984335362129264989 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6805039331788933 / 10000000000000000 : ℝ) ≤
+      (-(76984335362129264989 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (6805039331788933 / 10000000000000000 : ℝ) =
+            (6805039331788933 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6805039331788933 / 5000000000000000 : ℝ) ≤
+            (61645100697870735011 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6805015574705991 / 10000000000000000 : ℝ)) (q :=
+      (6805039331788933 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_593_2 :
+    Uω (aρ 2) (bρ 2) (10991296491131 / 16000000000000 : ℝ) ≤
+      (-(9711005343742677657 / 25000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (339056881185423 / 500000000000000 : ℝ) ≤
+      (-(9711005343742677657 / 25000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (339056881185423 / 500000000000000 : ℝ) =
+            (339056881185423 / 250000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (339056881185423 / 250000000000000 : ℝ) ≤
+            (7617674163757322343 / 25000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1356196234211009 / 2000000000000000 : ℝ)) (q :=
+      (339056881185423 / 500000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_593_3 :
+    Uω (aρ 3) (bρ 3) (10991296491131 / 16000000000000 : ℝ) ≤
+      (-(31641000668466948117 / 80000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2693339443541881 / 4000000000000000 : ℝ) ≤
+      (-(31641000668466948117 / 80000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2693339443541881 / 4000000000000000 : ℝ) =
+            (2693339443541881 / 2000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2693339443541881 / 2000000000000000 : ℝ) ≤
+            (23810773755533051883 / 80000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (673279924769753 / 1000000000000000 : ℝ)) (q :=
+      (2693339443541881 / 4000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_593_4 :
+    Uω (aρ 4) (bρ 4) (10991296491131 / 16000000000000 : ℝ) ≤
+      (-(4073635020287535188957 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1330804542173291 / 2000000000000000 : ℝ) ≤
+      (-(4073635020287535188957 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1330804542173291 / 2000000000000000 : ℝ) =
+            (1330804542173291 / 1000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1330804542173291 / 1000000000000000 : ℝ) ≤
+            (2857836782712464811043 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1330488326557207 / 2000000000000000 : ℝ)) (q :=
+      (1330804542173291 / 2000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_593_5 :
+    Uω (aρ 5) (bρ 5) (10991296491131 / 16000000000000 : ℝ) ≤
+      (-(4256670869576362918179 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13066676181922667 / 20000000000000000 : ℝ) ≤
+      (-(4256670869576362918179 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (13066676181922667 / 20000000000000000 : ℝ) =
+            (13066676181922667 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13066676181922667 / 10000000000000000 : ℝ) ≤
+            (2674800933423637081821 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (816157789038849 / 1250000000000000 : ℝ)) (q :=
+      (13066676181922667 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_593_6 :
+    Uω (aρ 6) (bρ 6) (10991296491131 / 16000000000000 : ℝ) ≤
+      (-(452473964593663599121 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12721052633581053 / 20000000000000000 : ℝ) ≤
+      (-(452473964593663599121 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (12721052633581053 / 20000000000000000 : ℝ) =
+            (12721052633581053 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12721052633581053 / 10000000000000000 : ℝ) ≤
+            (240673215706336400879 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3175439569554589 / 5000000000000000 : ℝ)) (q :=
+      (12721052633581053 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_593_7 :
+    Uω (aρ 7) (bρ 7) (10991296491131 / 16000000000000 : ℝ) ≤
+      (-(4901505303743168612503 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2450136728341123 / 4000000000000000 : ℝ) ≤
+      (-(4901505303743168612503 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2450136728341123 / 4000000000000000 : ℝ) =
+            (2450136728341123 / 2000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2450136728341123 / 2000000000000000 : ℝ) ≤
+            (2029966499256831387497 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (305208576399687 / 500000000000000 : ℝ)) (q :=
+      (2450136728341123 / 4000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_593_8 :
+    Uω (aρ 8) (bρ 8) (10991296491131 / 16000000000000 : ℝ) ≤
+      (-(2706744031536615135693 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11639255351134963 / 20000000000000000 : ℝ) ≤
+      (-(2706744031536615135693 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11639255351134963 / 20000000000000000 : ℝ) =
+            (11639255351134963 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11639255351134963 / 10000000000000000 : ℝ) ≤
+            (758991869963384864307 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (361017805547693 / 625000000000000 : ℝ)) (q :=
+      (11639255351134963 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_593_9 :
+    Uω (aρ 9) (bρ 9) (10991296491131 / 16000000000000 : ℝ) ≤
+      (-(1522594958593701806993 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10877476659562703 / 20000000000000000 : ℝ) ≤
+      (-(1522594958593701806993 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10877476659562703 / 20000000000000000 : ℝ) =
+            (10877476659562703 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10877476659562703 / 10000000000000000 : ℝ) ≤
+            (210272992156298193007 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1338867956387707 / 2500000000000000 : ℝ)) (q :=
+      (10877476659562703 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_593_10 :
+    Uω (aρ 10) (bρ 10) (10991296491131 / 16000000000000 : ℝ) ≤
+      (-(6966649033551154060933 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2491221141562591 / 5000000000000000 : ℝ) ≤
+      (-(6966649033551154060933 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2491221141562591 / 5000000000000000 : ℝ) =
+            (2491221141562591 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2491221141562591 / 1250000000000000 : ℝ) ≤
+            (6896294572448845939067 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4831438765733489 / 10000000000000000 : ℝ)) (q :=
+      (2491221141562591 / 5000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_593_11 :
+    Uω (aρ 11) (bρ 11) (10991296491131 / 16000000000000 : ℝ) ≤
+      (-(8086223519225050504341 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (556839182745233 / 1250000000000000 : ℝ) ≤
+      (-(8086223519225050504341 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (556839182745233 / 1250000000000000 : ℝ) =
+            (556839182745233 / 312500000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (556839182745233 / 312500000000000 : ℝ) ≤
+            (5776720086774949495659 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4194457749611853 / 10000000000000000 : ℝ)) (q :=
+      (556839182745233 / 1250000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11]) (by norm_num [bρ_11])
+      (by norm_num) (by norm_num [aρ_11, bρ_11]) (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_593_12 :
+    Uω (aρ 12) (bρ 12) (10991296491131 / 16000000000000 : ℝ) ≤
+      (-(1903224998521094270931 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1544471624649439 / 4000000000000000 : ℝ) ≤
+      (-(1903224998521094270931 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1544471624649439 / 4000000000000000 : ℝ) =
+            (1544471624649439 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1544471624649439 / 1000000000000000 : ℝ) ≤
+            (869363722678905729069 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (21444306535127 / 62500000000000 : ℝ)) (q :=
+      (1544471624649439 / 4000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_593_13 :
+    Uω (aρ 13) (bρ 13) (10991296491131 / 16000000000000 : ℝ) ≤
+      (-(1424498049762649303159 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6398949480520967 / 20000000000000000 : ℝ) ≤
+      (-(1424498049762649303159 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6398949480520967 / 20000000000000000 : ℝ) =
+            (6398949480520967 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6398949480520967 / 5000000000000000 : ℝ) ≤
+            (308369900987350696841 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (626765536122273 / 2500000000000000 : ℝ)) (q :=
+      (6398949480520967 / 20000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_593_14 :
+    Uω (aρ 14) (bρ 14) (10991296491131 / 16000000000000 : ℝ) ≤
+      (-(713865897465215119503 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1199261699129129 / 5000000000000000 : ℝ) ≤
+      (-(713865897465215119503 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1199261699129129 / 5000000000000000 : ℝ) =
+            (1199261699129129 / 625000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1199261699129129 / 625000000000000 : ℝ) ≤
+            (325854872984784880497 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1249086745529641 / 10000000000000000 : ℝ)) (q :=
+      (1199261699129129 / 5000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_593_15 :
+    Uω (aρ 15) (bρ 15) (10991296491131 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_593_16 :
+    Uω (aρ 16) (bρ 16) (10991296491131 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_593 :
+    Uρ (10991296491131 / 16000000000000 : ℝ) ≤
+      (-(6476639339068879338977 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (10991296491131 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(76984335362129264989 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_593_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (10991296491131 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) * (-(9711005343742677657 / 25000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_593_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (10991296491131 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) * (-(31641000668466948117 / 80000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_593_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (10991296491131 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(4073635020287535188957 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_593_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (10991296491131 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(4256670869576362918179 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_593_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (10991296491131 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(452473964593663599121 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_593_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (10991296491131 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(4901505303743168612503 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_593_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (10991296491131 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(2706744031536615135693 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_593_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (10991296491131 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(1522594958593701806993 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_593_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (10991296491131 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(6966649033551154060933 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_593_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (10991296491131 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(8086223519225050504341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_593_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (10991296491131 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(1903224998521094270931 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_593_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (10991296491131 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(1424498049762649303159 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_593_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (10991296491131 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) * (-(713865897465215119503 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_593_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (10991296491131 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_593_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (10991296491131 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_593_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_594_1 :
+    Uω (aρ 1) (bρ 1) (44017105158153 / 64000000000000 : ℝ) ≤
+      (-(3837302699325395042261 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (54505213872647697 / 80000000000000000 : ℝ) ≤
+      (-(3837302699325395042261 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (54505213872647697 / 80000000000000000 : ℝ) =
+            (54505213872647697 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (54505213872647697 / 40000000000000000 : ℝ) ≤
+            (3094169103674604957739 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3406564002642759 / 5000000000000000 : ℝ)) (q :=
+      (54505213872647697 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_594_2 :
+    Uω (aρ 2) (bρ 2) (44017105158153 / 64000000000000 : ℝ) ≤
+      (-(3872445866191089838999 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (54314001477281973 / 80000000000000000 : ℝ) ≤
+      (-(3872445866191089838999 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (54314001477281973 / 80000000000000000 : ℝ) =
+            (54314001477281973 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (54314001477281973 / 40000000000000000 : ℝ) ≤
+            (3059025936808910161001 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6789093918954087 / 10000000000000000 : ℝ)) (q :=
+      (54314001477281973 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_594_3 :
+    Uω (aρ 3) (bρ 3) (44017105158153 / 64000000000000 : ℝ) ≤
+      (-(1971541650973881459497 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (53931693151918417 / 80000000000000000 : ℝ) ≤
+      (-(1971541650973881459497 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (53931693151918417 / 80000000000000000 : ℝ) =
+            (53931693151918417 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (53931693151918417 / 40000000000000000 : ℝ) ≤
+            (1494194250526118540503 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3370456471981599 / 5000000000000000 : ℝ)) (q :=
+      (53931693151918417 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_594_4 :
+    Uω (aρ 4) (bρ 4) (44017105158153 / 64000000000000 : ℝ) ≤
+      (-(2030723935954041162807 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (53297096084644953 / 80000000000000000 : ℝ) ≤
+      (-(2030723935954041162807 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (53297096084644953 / 80000000000000000 : ℝ) =
+            (53297096084644953 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (53297096084644953 / 40000000000000000 : ℝ) ≤
+            (1435011965545958837193 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (832569732276229 / 1250000000000000 : ℝ)) (q :=
+      (53297096084644953 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_594_5 :
+    Uω (aρ 5) (bρ 5) (44017105158153 / 64000000000000 : ℝ) ≤
+      (-(4244253950414137293169 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (52331644181492417 / 80000000000000000 : ℝ) ≤
+      (-(4244253950414137293169 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (52331644181492417 / 80000000000000000 : ℝ) =
+            (52331644181492417 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (52331644181492417 / 40000000000000000 : ℝ) ≤
+            (2687217852585862706831 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3268692400878349 / 5000000000000000 : ℝ)) (q :=
+      (52331644181492417 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_594_6 :
+    Uω (aρ 6) (bρ 6) (44017105158153 / 64000000000000 : ℝ) ≤
+      (-(4511974198329091467151 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (50949207983937341 / 80000000000000000 : ℝ) ≤
+      (-(4511974198329091467151 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (50949207983937341 / 80000000000000000 : ℝ) =
+            (50949207983937341 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (50949207983937341 / 40000000000000000 : ℝ) ≤
+            (2419497604670908532849 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6359016127507929 / 10000000000000000 : ℝ)) (q :=
+      (50949207983937341 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_594_7 :
+    Uω (aρ 7) (bρ 7) (44017105158153 / 64000000000000 : ℝ) ≤
+      (-(2444112151337690036797 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (49067858339705593 / 80000000000000000 : ℝ) ≤
+      (-(2444112151337690036797 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (49067858339705593 / 80000000000000000 : ℝ) =
+            (49067858339705593 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (49067858339705593 / 40000000000000000 : ℝ) ≤
+            (1021623750162309963203 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (764042512151249 / 1250000000000000 : ℝ)) (q :=
+      (49067858339705593 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_594_8 :
+    Uω (aρ 8) (bρ 8) (44017105158153 / 64000000000000 : ℝ) ≤
+      (-(43195630299015583237 / 80000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (46622406682957993 / 80000000000000000 : ℝ) ≤
+      (-(43195630299015583237 / 80000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (46622406682957993 / 80000000000000000 : ℝ) =
+            (46622406682957993 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (46622406682957993 / 40000000000000000 : ℝ) ≤
+            (12256144124984416763 / 80000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1446129708590773 / 2500000000000000 : ℝ)) (q :=
+      (46622406682957993 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_594_9 :
+    Uω (aρ 9) (bρ 9) (44017105158153 / 64000000000000 : ℝ) ≤
+      (-(1215048765630322143733 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (43575813125483009 / 80000000000000000 : ℝ) ≤
+      (-(1215048765630322143733 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (43575813125483009 / 80000000000000000 : ℝ) =
+            (43575813125483009 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (43575813125483009 / 40000000000000000 : ℝ) ≤
+            (171245594969677856267 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2681918036677173 / 5000000000000000 : ℝ)) (q :=
+      (43575813125483009 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_594_10 :
+    Uω (aρ 10) (bρ 10) (44017105158153 / 64000000000000 : ℝ) ≤
+      (-(1389974637746149243031 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (39926462127377373 / 80000000000000000 : ℝ) ≤
+      (-(1389974637746149243031 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (39926462127377373 / 80000000000000000 : ℝ) =
+            (39926462127377373 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 15
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (39926462127377373 / 20000000000000000 : ℝ) ≤
+            (1382614083453850756969 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4840057357322937 / 10000000000000000 : ℝ)) (q :=
+      (39926462127377373 / 80000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_594_11 :
+    Uω (aρ 11) (bρ 11) (44017105158153 / 64000000000000 : ℝ) ≤
+      (-(8066903811869647594891 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (35706625255963241 / 80000000000000000 : ℝ) ≤
+      (-(8066903811869647594891 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (35706625255963241 / 80000000000000000 : ℝ) =
+            (35706625255963241 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (35706625255963241 / 20000000000000000 : ℝ) ≤
+            (5796039794130352405109 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1050893691418601 / 2500000000000000 : ℝ)) (q :=
+      (35706625255963241 / 80000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_594_12 :
+    Uω (aρ 12) (bρ 12) (44017105158153 / 64000000000000 : ℝ) ≤
+      (-(4746258070544322986259 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (30962445048721753 / 80000000000000000 : ℝ) ≤
+      (-(4746258070544322986259 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (30962445048721753 / 80000000000000000 : ℝ) =
+            (30962445048721753 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (30962445048721753 / 20000000000000000 : ℝ) ≤
+            (2185213732455677013741 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (430153726318629 / 1250000000000000 : ℝ)) (q :=
+      (30962445048721753 / 80000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_594_13 :
+    Uω (aρ 13) (bρ 13) (44017105158153 / 64000000000000 : ℝ) ≤
+      (-(2272741445869784425119 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25678547385049509 / 80000000000000000 : ℝ) ≤
+      (-(2272741445869784425119 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (25678547385049509 / 80000000000000000 : ℝ) =
+            (25678547385049509 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25678547385049509 / 20000000000000000 : ℝ) ≤
+            (499847275330215574881 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2519637136225971 / 10000000000000000 : ℝ)) (q :=
+      (25678547385049509 / 80000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_594_14 :
+    Uω (aρ 14) (bρ 14) (44017105158153 / 64000000000000 : ℝ) ≤
+      (-(14212960197935088470847 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19312076277633053 / 80000000000000000 : ℝ) ≤
+      (-(14212960197935088470847 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (19312076277633053 / 80000000000000000 : ℝ) =
+            (19312076277633053 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19312076277633053 / 10000000000000000 : ℝ) ≤
+            (6581455211064911529153 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1271946644416857 / 10000000000000000 : ℝ)) (q :=
+      (19312076277633053 / 80000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_594_15 :
+    Uω (aρ 15) (bρ 15) (44017105158153 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_594_16 :
+    Uω (aρ 16) (bρ 16) (44017105158153 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_594 :
+    Uρ (44017105158153 / 64000000000000 : ℝ) ≤
+      (-(3229994991353601011403 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (44017105158153 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3837302699325395042261 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_594_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (44017105158153 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3872445866191089838999 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_594_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (44017105158153 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1971541650973881459497 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_594_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (44017105158153 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(2030723935954041162807 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_594_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (44017105158153 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(4244253950414137293169 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_594_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (44017105158153 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(4511974198329091467151 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_594_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (44017105158153 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(2444112151337690036797 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_594_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (44017105158153 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) * (-(43195630299015583237 / 80000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_594_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (44017105158153 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(1215048765630322143733 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_594_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (44017105158153 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(1389974637746149243031 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_594_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (44017105158153 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(8066903811869647594891 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_594_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (44017105158153 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(4746258070544322986259 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_594_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (44017105158153 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(2272741445869784425119 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_594_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (44017105158153 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(14212960197935088470847 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_594_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (44017105158153 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_594_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (44017105158153 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_594_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_595_1 :
+    Uω (aρ 1) (bρ 1) (22034512175891 / 32000000000000 : ℝ) ≤
+      (-(3825402808256825816339 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27285056545222829 / 40000000000000000 : ℝ) ≤
+      (-(3825402808256825816339 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (27285056545222829 / 40000000000000000 : ℝ) =
+            (27285056545222829 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27285056545222829 / 20000000000000000 : ℝ) ≤
+            (3106068994743174183661 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6821240435730477 / 10000000000000000 : ℝ)) (q :=
+      (27285056545222829 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_595_2 :
+    Uω (aρ 2) (bρ 2) (22034512175891 / 32000000000000 : ℝ) ≤
+      (-(965125968428573373733 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27189450980663139 / 40000000000000000 : ℝ) ≤
+      (-(965125968428573373733 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (27189450980663139 / 40000000000000000 : ℝ) =
+            (27189450980663139 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27189450980663139 / 20000000000000000 : ℝ) ≤
+            (767741982321426626267 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (849650833245079 / 1250000000000000 : ℝ)) (q :=
+      (27189450980663139 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_595_3 :
+    Uω (aρ 3) (bρ 3) (22034512175891 / 32000000000000 : ℝ) ≤
+      (-(786211201140834157933 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (215986389681133 / 320000000000000 : ℝ) ≤
+      (-(786211201140834157933 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (215986389681133 / 320000000000000 : ℝ) =
+            (215986389681133 / 160000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (215986389681133 / 160000000000000 : ℝ) ≤
+            (600083159459165842067 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (53992213096399 / 80000000000000 : ℝ)) (q :=
+      (215986389681133 / 320000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_595_4 :
+    Uω (aρ 4) (bρ 4) (22034512175891 / 32000000000000 : ℝ) ≤
+      (-(2024637782567805178767 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5336201044486767 / 8000000000000000 : ℝ) ≤
+      (-(2024637782567805178767 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5336201044486767 / 8000000000000000 : ℝ) =
+            (5336201044486767 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5336201044486767 / 4000000000000000 : ℝ) ≤
+            (1441098118932194821233 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (333433703713049 / 500000000000000 : ℝ)) (q :=
+      (5336201044486767 / 8000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_595_5 :
+    Uω (aρ 5) (bρ 5) (22034512175891 / 32000000000000 : ℝ) ≤
+      (-(528981556159355254323 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5239658353493659 / 8000000000000000 : ℝ) ≤
+      (-(528981556159355254323 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5239658353493659 / 8000000000000000 : ℝ) =
+            (5239658353493659 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5239658353493659 / 4000000000000000 : ℝ) ≤
+            (337452419215644745677 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (654550726611321 / 1000000000000000 : ℝ)) (q :=
+      (5239658353493659 / 8000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_595_6 :
+    Uω (aρ 6) (bρ 6) (22034512175891 / 32000000000000 : ℝ) ≤
+      (-(4499225074868244073119 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25507102591140369 / 40000000000000000 : ℝ) ≤
+      (-(4499225074868244073119 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (25507102591140369 / 40000000000000000 : ℝ) =
+            (25507102591140369 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25507102591140369 / 20000000000000000 : ℝ) ≤
+            (2432246728131755926881 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6367153053089247 / 10000000000000000 : ℝ)) (q :=
+      (25507102591140369 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_595_7 :
+    Uω (aρ 7) (bρ 7) (22034512175891 / 32000000000000 : ℝ) ≤
+      (-(97499220768220577801 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24566490757328667 / 40000000000000000 : ℝ) ≤
+      (-(97499220768220577801 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (24566490757328667 / 40000000000000000 : ℝ) =
+            (24566490757328667 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24566490757328667 / 20000000000000000 : ℝ) ≤
+            (41130215291779422199 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1530127129235849 / 2500000000000000 : ℝ)) (q :=
+      (24566490757328667 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_595_8 :
+    Uω (aρ 8) (bρ 8) (22034512175891 / 32000000000000 : ℝ) ≤
+      (-(5385439474086395814851 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23343895294572749 / 40000000000000000 : ℝ) ≤
+      (-(5385439474086395814851 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23343895294572749 / 40000000000000000 : ℝ) =
+            (23343895294572749 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23343895294572749 / 20000000000000000 : ℝ) ≤
+            (1546032328913604185149 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5792752436905437 / 10000000000000000 : ℝ)) (q :=
+      (23343895294572749 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_595_9 :
+    Uω (aρ 9) (bρ 9) (22034512175891 / 32000000000000 : ℝ) ≤
+      (-(94689553206502731041 / 156250000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21820858261355787 / 40000000000000000 : ℝ) ≤
+      (-(94689553206502731041 / 156250000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (21820858261355787 / 40000000000000000 : ℝ) =
+            (21820858261355787 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21820858261355787 / 20000000000000000 : ℝ) ≤
+            (13614693715372268959 / 156250000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1343049887164239 / 2500000000000000 : ℝ)) (q :=
+      (21820858261355787 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_595_10 :
+    Uω (aρ 10) (bρ 10) (22034512175891 / 32000000000000 : ℝ) ≤
+      (-(3466563593368778755967 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19996689501341251 / 40000000000000000 : ℝ) ≤
+      (-(3466563593368778755967 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (19996689501341251 / 40000000000000000 : ℝ) =
+            (19996689501341251 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 15
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19996689501341251 / 10000000000000000 : ℝ) ≤
+            (3464908209631221244033 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (303042137634043 / 625000000000000 : ℝ)) (q :=
+      (19996689501341251 / 40000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_595_11 :
+    Uω (aρ 11) (bρ 11) (22034512175891 / 32000000000000 : ℝ) ≤
+      (-(502976621982016817177 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17887763190387439 / 40000000000000000 : ℝ) ≤
+      (-(502976621982016817177 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (17887763190387439 / 40000000000000000 : ℝ) =
+            (17887763190387439 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17887763190387439 / 10000000000000000 : ℝ) ≤
+            (363457353392983182823 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2106343836436391 / 5000000000000000 : ℝ)) (q :=
+      (17887763190387439 / 40000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_595_12 :
+    Uω (aρ 12) (bρ 12) (22034512175891 / 32000000000000 : ℝ) ≤
+      (-(9468976722227678351319 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15517707347238349 / 40000000000000000 : ℝ) ≤
+      (-(9468976722227678351319 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (15517707347238349 / 40000000000000000 : ℝ) =
+            (15517707347238349 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15517707347238349 / 10000000000000000 : ℝ) ≤
+            (4393966883772321648681 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3451359847983237 / 10000000000000000 : ℝ)) (q :=
+      (15517707347238349 / 40000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_595_13 :
+    Uω (aρ 13) (bρ 13) (22034512175891 / 32000000000000 : ℝ) ≤
+      (-(708224407302227743551 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2576115101430907 / 8000000000000000 : ℝ) ≤
+      (-(708224407302227743551 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2576115101430907 / 8000000000000000 : ℝ) =
+            (2576115101430907 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2576115101430907 / 2000000000000000 : ℝ) ≤
+            (158209568072772256449 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (253217566953633 / 1000000000000000 : ℝ)) (q :=
+      (2576115101430907 / 8000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_595_14 :
+    Uω (aρ 14) (bρ 14) (22034512175891 / 32000000000000 : ℝ) ≤
+      (-(1768717515541818394809 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9717277055474861 / 40000000000000000 : ℝ) ≤
+      (-(1768717515541818394809 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (9717277055474861 / 40000000000000000 : ℝ) =
+            (9717277055474861 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9717277055474861 / 5000000000000000 : ℝ) ≤
+            (830584410583181605191 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1294453728741493 / 10000000000000000 : ℝ)) (q :=
+      (9717277055474861 / 40000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_595_15 :
+    Uω (aρ 15) (bρ 15) (22034512175891 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_595_16 :
+    Uω (aρ 16) (bρ 16) (22034512175891 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_595 :
+    Uρ (22034512175891 / 32000000000000 : ℝ) ≤
+      (-(3221710550820856703479 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (22034512175891 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3825402808256825816339 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_595_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (22034512175891 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(965125968428573373733 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_595_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (22034512175891 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(786211201140834157933 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_595_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (22034512175891 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(2024637782567805178767 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_595_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (22034512175891 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(528981556159355254323 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_595_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (22034512175891 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(4499225074868244073119 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_595_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (22034512175891 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) * (-(97499220768220577801 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_595_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (22034512175891 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(5385439474086395814851 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_595_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (22034512175891 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(94689553206502731041 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_595_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (22034512175891 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(3466563593368778755967 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_595_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (22034512175891 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) * (-(502976621982016817177 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_595_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (22034512175891 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(9468976722227678351319 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_595_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (22034512175891 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) * (-(708224407302227743551 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_595_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (22034512175891 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(1768717515541818394809 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_595_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (22034512175891 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_595_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (22034512175891 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_595_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_596_1 :
+    Uω (aρ 1) (bρ 1) (44120943545411 / 64000000000000 : ℝ) ≤
+      (-(3813517061197737683561 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (54635012307707267 / 80000000000000000 : ℝ) ≤
+      (-(3813517061197737683561 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (54635012307707267 / 80000000000000000 : ℝ) =
+            (54635012307707267 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (54635012307707267 / 40000000000000000 : ℝ) ≤
+            (3117954741802262316439 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1707338216510337 / 2500000000000000 : ℝ)) (q :=
+      (54635012307707267 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_596_2 :
+    Uω (aρ 2) (bρ 2) (44120943545411 / 64000000000000 : ℝ) ≤
+      (-(962144031500172235307 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (54443802441813363 / 80000000000000000 : ℝ) ≤
+      (-(962144031500172235307 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (54443802441813363 / 80000000000000000 : ℝ) =
+            (54443802441813363 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (54443802441813363 / 40000000000000000 : ℝ) ≤
+            (770723919249827764693 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (425332463254867 / 625000000000000 : ℝ)) (q :=
+      (54443802441813363 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_596_3 :
+    Uω (aρ 3) (bρ 3) (44120943545411 / 64000000000000 : ℝ) ≤
+      (-(1959521580007089325513 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (54061501675977919 / 80000000000000000 : ℝ) ≤
+      (-(1959521580007089325513 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (54061501675977919 / 80000000000000000 : ℝ) =
+            (54061501675977919 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (54061501675977919 / 40000000000000000 : ℝ) ≤
+            (1506214321492910674487 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6757140326969011 / 10000000000000000 : ℝ)) (q :=
+      (54061501675977919 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_596_4 :
+    Uω (aρ 4) (bρ 4) (44120943545411 / 64000000000000 : ℝ) ≤
+      (-(2018559031924290803153 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (53426924767736511 / 80000000000000000 : ℝ) ≤
+      (-(2018559031924290803153 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (53426924767736511 / 80000000000000000 : ℝ) =
+            (53426924767736511 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (53426924767736511 / 40000000000000000 : ℝ) ≤
+            (1447176869575709196847 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6676790280973659 / 10000000000000000 : ℝ)) (q :=
+      (53426924767736511 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_596_5 :
+    Uω (aρ 5) (bρ 5) (44120943545411 / 64000000000000 : ℝ) ≤
+      (-(105486658196741034191 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10492304557679267 / 16000000000000000 : ℝ) ≤
+      (-(105486658196741034191 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10492304557679267 / 16000000000000000 : ℝ) =
+            (10492304557679267 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10492304557679267 / 8000000000000000 : ℝ) ≤
+            (67800136878258965809 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1310725941094723 / 2000000000000000 : ℝ)) (q :=
+      (10492304557679267 / 16000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_596_6 :
+    Uω (aρ 6) (bρ 6) (44120943545411 / 64000000000000 : ℝ) ≤
+      (-(4486492233732258104221 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (51079202130316507 / 80000000000000000 : ℝ) ≤
+      (-(4486492233732258104221 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (51079202130316507 / 80000000000000000 : ℝ) =
+            (51079202130316507 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (51079202130316507 / 40000000000000000 : ℝ) ≤
+            (2444979569267741895779 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3187644958046829 / 5000000000000000 : ℝ)) (q :=
+      (51079202130316507 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_596_7 :
+    Uω (aρ 7) (bρ 7) (44120943545411 / 64000000000000 : ℝ) ≤
+      (-(4861715463315582096821 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (49198104094068487 / 80000000000000000 : ℝ) ≤
+      (-(4861715463315582096821 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (49198104094068487 / 80000000000000000 : ℝ) =
+            (49198104094068487 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (49198104094068487 / 40000000000000000 : ℝ) ≤
+            (2069756339684417903179 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6128676787791653 / 10000000000000000 : ℝ)) (q :=
+      (49198104094068487 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_596_8 :
+    Uω (aρ 8) (bρ 8) (44120943545411 / 64000000000000 : ℝ) ≤
+      (-(42971560525332844949 / 80000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (46753173128945367 / 80000000000000000 : ℝ) ≤
+      (-(42971560525332844949 / 80000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (46753173128945367 / 80000000000000000 : ℝ) =
+            (46753173128945367 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (46753173128945367 / 40000000000000000 : ℝ) ≤
+            (12480213898667155051 / 80000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5800985697850873 / 10000000000000000 : ℝ)) (q :=
+      (46753173128945367 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_596_9 :
+    Uω (aρ 9) (bρ 9) (44120943545411 / 64000000000000 : ℝ) ≤
+      (-(1209008498005994149171 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (43707616844345691 / 80000000000000000 : ℝ) ≤
+      (-(1209008498005994149171 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (43707616844345691 / 80000000000000000 : ℝ) =
+            (43707616844345691 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (43707616844345691 / 40000000000000000 : ℝ) ≤
+            (177285862594005850829 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2690281127530477 / 5000000000000000 : ℝ)) (q :=
+      (43707616844345691 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_596_10 :
+    Uω (aρ 10) (bρ 10) (44120943545411 / 64000000000000 : ℝ) ≤
+      (-(3458205459091484569321 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (40060288928102091 / 80000000000000000 : ℝ) ≤
+      (-(3458205459091484569321 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (40060288928102091 / 80000000000000000 : ℝ) =
+            (40060288928102091 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 2
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (40060288928102091 / 40000000000000000 : ℝ) ≤
+            (7530442408515430679 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2428644654747527 / 5000000000000000 : ℝ)) (q :=
+      (40060288928102091 / 80000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_596_11 :
+    Uω (aρ 11) (bρ 11) (44120943545411 / 64000000000000 : ℝ) ≤
+      (-(8028389739178359166687 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (35844411176559483 / 80000000000000000 : ℝ) ≤
+      (-(8028389739178359166687 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (35844411176559483 / 80000000000000000 : ℝ) =
+            (35844411176559483 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (35844411176559483 / 20000000000000000 : ℝ) ≤
+            (5834553866821640833313 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2110898248907201 / 5000000000000000 : ℝ)) (q :=
+      (35844411176559483 / 80000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_596_12 :
+    Uω (aρ 12) (bρ 12) (44120943545411 / 64000000000000 : ℝ) ≤
+      (-(295172070509336324951 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (31108341806982459 / 80000000000000000 : ℝ) ≤
+      (-(295172070509336324951 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (31108341806982459 / 80000000000000000 : ℝ) =
+            (31108341806982459 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (31108341806982459 / 20000000000000000 : ℝ) ≤
+            (138044917178163675049 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1730739626052573 / 5000000000000000 : ℝ)) (q :=
+      (31108341806982459 / 80000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_596_13 :
+    Uω (aρ 13) (bρ 13) (44120943545411 / 64000000000000 : ℝ) ≤
+      (-(2259926443505063997727 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25843610965586107 / 80000000000000000 : ℝ) ≤
+      (-(2259926443505063997727 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (25843610965586107 / 80000000000000000 : ℝ) =
+            (25843610965586107 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25843610965586107 / 20000000000000000 : ℝ) ≤
+            (512662277694936002273 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1272339141675529 / 5000000000000000 : ℝ)) (q :=
+      (25843610965586107 / 80000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_596_14 :
+    Uω (aρ 14) (bρ 14) (44120943545411 / 64000000000000 : ℝ) ≤
+      (-(7043800923483328933247 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19555693060838851 / 80000000000000000 : ℝ) ≤
+      (-(7043800923483328933247 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (19555693060838851 / 80000000000000000 : ℝ) =
+            (19555693060838851 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19555693060838851 / 10000000000000000 : ℝ) ≤
+            (3353406781016671066753 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (329156523052311 / 2500000000000000 : ℝ)) (q :=
+      (19555693060838851 / 80000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_596_15 :
+    Uω (aρ 15) (bρ 15) (44120943545411 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_596_16 :
+    Uω (aρ 16) (bρ 16) (44120943545411 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_596 :
+    Uρ (44120943545411 / 64000000000000 : ℝ) ≤
+      (-(1606732501947162911213 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (44120943545411 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3813517061197737683561 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_596_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (44120943545411 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(962144031500172235307 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_596_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (44120943545411 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1959521580007089325513 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_596_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (44120943545411 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(2018559031924290803153 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_596_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (44120943545411 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(105486658196741034191 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_596_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (44120943545411 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(4486492233732258104221 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_596_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (44120943545411 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(4861715463315582096821 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_596_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (44120943545411 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) * (-(42971560525332844949 / 80000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_596_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (44120943545411 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(1209008498005994149171 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_596_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (44120943545411 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(3458205459091484569321 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_596_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (44120943545411 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(8028389739178359166687 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_596_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (44120943545411 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(295172070509336324951 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_596_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (44120943545411 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(2259926443505063997727 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_596_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (44120943545411 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(7043800923483328933247 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_596_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (44120943545411 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_596_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (44120943545411 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_596_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_597_1 :
+    Uω (aρ 1) (bρ 1) (276080392119 / 400000000000 : ℝ) ≤
+      (-(475205678070643602751 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (106835764696161 / 156250000000000 : ℝ) ≤
+      (-(475205678070643602751 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (106835764696161 / 156250000000000 : ℝ) = (106835764696161 / 78125000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (106835764696161 / 78125000000000 : ℝ) ≤
+            (391228297304356397249 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (427341581013663 / 625000000000000 : ℝ)) (q :=
+      (106835764696161 / 156250000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_597_2 :
+    Uω (aρ 2) (bρ 2) (276080392119 / 400000000000 : ℝ) ≤
+      (-(959165647276513419161 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1703396966211123 / 2500000000000000 : ℝ) ≤
+      (-(959165647276513419161 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1703396966211123 / 2500000000000000 : ℝ) =
+            (1703396966211123 / 1250000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1703396966211123 / 1250000000000000 : ℝ) ≤
+            (773702303473486580839 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (851679019663623 / 1250000000000000 : ℝ)) (q :=
+      (1703396966211123 / 2500000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_597_3 :
+    Uω (aρ 3) (bρ 3) (276080392119 / 400000000000 : ℝ) ≤
+      (-(195352236509487228353 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13531601479762003 / 20000000000000000 : ℝ) ≤
+      (-(195352236509487228353 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (13531601479762003 / 20000000000000000 : ℝ) =
+            (13531601479762003 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13531601479762003 / 10000000000000000 : ℝ) ≤
+            (151221353640512771647 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6765254013732003 / 10000000000000000 : ℝ)) (q :=
+      (13531601479762003 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_597_4 :
+    Uω (aρ 4) (bρ 4) (276080392119 / 400000000000 : ℝ) ≤
+      (-(4024975332057229824131 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6686479881673441 / 10000000000000000 : ℝ) ≤
+      (-(4024975332057229824131 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (6686479881673441 / 10000000000000000 : ℝ) =
+            (6686479881673441 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6686479881673441 / 5000000000000000 : ℝ) ≤
+            (2906496470942770175869 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3342453239190941 / 5000000000000000 : ℝ)) (q :=
+      (6686479881673441 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_597_5 :
+    Uω (aρ 5) (bρ 5) (276080392119 / 400000000000 : ℝ) ≤
+      (-(4207095548052328053339 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (410362983923773 / 625000000000000 : ℝ) ≤
+      (-(4207095548052328053339 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (410362983923773 / 625000000000000 : ℝ) =
+            (410362983923773 / 312500000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (410362983923773 / 312500000000000 : ℝ) ≤
+            (2724376254947671946661 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (410109507495671 / 625000000000000 : ℝ)) (q :=
+      (410362983923773 / 625000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_597_6 :
+    Uω (aρ 6) (bρ 6) (276080392119 / 400000000000 : ℝ) ≤
+      (-(4473775633260287782671 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (639302485362523 / 1000000000000000 : ℝ) ≤
+      (-(4473775633260287782671 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (639302485362523 / 1000000000000000 : ℝ) =
+            (639302485362523 / 500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (639302485362523 / 500000000000000 : ℝ) ≤
+            (2457696169739712217329 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (319171335838023 / 500000000000000 : ℝ)) (q :=
+      (639302485362523 / 1000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_597_7 :
+    Uω (aρ 7) (bρ 7) (276080392119 / 400000000000 : ℝ) ≤
+      (-(4848487529947418754959 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1539475815009909 / 2500000000000000 : ℝ) ≤
+      (-(4848487529947418754959 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1539475815009909 / 2500000000000000 : ℝ) =
+            (1539475815009909 / 1250000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1539475815009909 / 1250000000000000 : ℝ) ≤
+            (2082984273052581245041 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (767105613793659 / 1250000000000000 : ℝ)) (q :=
+      (1539475815009909 / 2500000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_597_8 :
+    Uω (aρ 8) (bρ 8) (276080392119 / 400000000000 : ℝ) ≤
+      (-(214298820193377640081 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2926159644260453 / 5000000000000000 : ℝ) ≤
+      (-(214298820193377640081 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2926159644260453 / 5000000000000000 : ℝ) =
+            (2926159644260453 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2926159644260453 / 2500000000000000 : ℝ) ≤
+            (62960051926622359919 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1452304654662953 / 2500000000000000 : ℝ)) (q :=
+      (2926159644260453 / 5000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_597_9 :
+    Uω (aρ 9) (bρ 9) (276080392119 / 400000000000 : ℝ) ≤
+      (-(6029977007426569530737 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2188675705235189 / 4000000000000000 : ℝ) ≤
+      (-(6029977007426569530737 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2188675705235189 / 4000000000000000 : ℝ) =
+            (2188675705235189 / 2000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2188675705235189 / 2000000000000000 : ℝ) ≤
+            (901494795573430469263 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1077784839229189 / 2000000000000000 : ℝ)) (q :=
+      (2188675705235189 / 4000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_597_10 :
+    Uω (aρ 10) (bρ 10) (276080392119 / 400000000000 : ℝ) ≤
+      (-(107808191664674528683 / 156250000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10031797985137643 / 20000000000000000 : ℝ) ≤
+      (-(107808191664674528683 / 156250000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10031797985137643 / 20000000000000000 : ℝ) =
+            (10031797985137643 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10031797985137643 / 10000000000000000 : ℝ) ≤
+            (496055257200471317 / 156250000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4865902688602643 / 10000000000000000 : ℝ)) (q :=
+      (10031797985137643 / 20000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_597_11 :
+    Uω (aρ 11) (bρ 11) (276080392119 / 400000000000 : ℝ) ≤
+      (-(2002298744056355109751 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8978319937195677 / 20000000000000000 : ℝ) ≤
+      (-(2002298744056355109751 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8978319937195677 / 20000000000000000 : ℝ) =
+            (8978319937195677 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8978319937195677 / 5000000000000000 : ℝ) ≤
+            (1463437157443644890249 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4230901266865677 / 10000000000000000 : ℝ)) (q :=
+      (8978319937195677 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_597_12 :
+    Uω (aρ 12) (bρ 12) (276080392119 / 400000000000 : ℝ) ≤
+      (-(4711052134525300289301 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3897653344772773 / 10000000000000000 : ℝ) ≤
+      (-(4711052134525300289301 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3897653344772773 / 10000000000000000 : ℝ) =
+            (3897653344772773 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3897653344772773 / 2500000000000000 : ℝ) ≤
+            (2220419668474699710699 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1735794057950273 / 5000000000000000 : ℝ)) (q :=
+      (3897653344772773 / 10000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_597_13 :
+    Uω (aρ 13) (bρ 13) (276080392119 / 400000000000 : ℝ) ≤
+      (-(1408478791481375428463 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6481482336582191 / 20000000000000000 : ℝ) ≤
+      (-(1408478791481375428463 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6481482336582191 / 20000000000000000 : ℝ) =
+            (6481482336582191 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6481482336582191 / 5000000000000000 : ℝ) ≤
+            (324389159268624571537 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2557145504532191 / 10000000000000000 : ℝ)) (q :=
+      (6481482336582191 / 20000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_597_14 :
+    Uω (aρ 14) (bρ 14) (276080392119 / 400000000000 : ℝ) ≤
+      (-(7013246972571734108403 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (491888991620867 / 2000000000000000 : ℝ) ≤
+      (-(7013246972571734108403 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (491888991620867 / 2000000000000000 : ℝ) =
+            (491888991620867 / 250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (491888991620867 / 250000000000000 : ℝ) ≤
+            (3383960731928265891597 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (133848036920367 / 1000000000000000 : ℝ)) (q :=
+      (491888991620867 / 2000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14]) (by norm_num [bρ_14])
+      (by norm_num) (by norm_num [aρ_14, bρ_14]) (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_597_15 :
+    Uω (aρ 15) (bρ 15) (276080392119 / 400000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_597_16 :
+    Uω (aρ 16) (bρ 16) (276080392119 / 400000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_597 :
+    Uρ (276080392119 / 400000000000 : ℝ) ≤
+      (-(6410514213786317389521 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (276080392119 / 400000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(475205678070643602751 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_597_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (276080392119 / 400000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(959165647276513419161 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_597_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (276080392119 / 400000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(195352236509487228353 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_597_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (276080392119 / 400000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(4024975332057229824131 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_597_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (276080392119 / 400000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(4207095548052328053339 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_597_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (276080392119 / 400000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(4473775633260287782671 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_597_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (276080392119 / 400000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(4848487529947418754959 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_597_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (276080392119 / 400000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(214298820193377640081 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_597_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (276080392119 / 400000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(6029977007426569530737 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_597_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (276080392119 / 400000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) * (-(107808191664674528683 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_597_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (276080392119 / 400000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(2002298744056355109751 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_597_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (276080392119 / 400000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(4711052134525300289301 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_597_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (276080392119 / 400000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(1408478791481375428463 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_597_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (276080392119 / 400000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(7013246972571734108403 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_597_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (276080392119 / 400000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_597_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (276080392119 / 400000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_597_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_598_1 :
+    Uω (aρ 1) (bρ 1) (44224781932669 / 64000000000000 : ℝ) ≤
+      (-(3789787864895543404819 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (54764810740629053 / 80000000000000000 : ℝ) ≤
+      (-(3789787864895543404819 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (54764810740629053 / 80000000000000000 : ℝ) =
+            (54764810740629053 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (54764810740629053 / 40000000000000000 : ℝ) ≤
+            (3141683938104456595181 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1711394431565683 / 2500000000000000 : ℝ)) (q :=
+      (54764810740629053 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_598_2 :
+    Uω (aρ 2) (bρ 2) (44224781932669 / 64000000000000 : ℝ) ≤
+      (-(3824763229207356151267 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10914720678433329 / 16000000000000000 : ℝ) ≤
+      (-(3824763229207356151267 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10914720678433329 / 16000000000000000 : ℝ) =
+            (10914720678433329 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10914720678433329 / 8000000000000000 : ℝ) ≤
+            (3106708573792643848733 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (682154490165713 / 1000000000000000 : ℝ)) (q :=
+      (10914720678433329 / 16000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_598_3 :
+    Uω (aρ 3) (bρ 3) (44224781932669 / 64000000000000 : ℝ) ≤
+      (-(3895060681667598397467 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (54191310149538897 / 80000000000000000 : ℝ) ≤
+      (-(3895060681667598397467 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (54191310149538897 / 80000000000000000 : ℝ) =
+            (54191310149538897 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (54191310149538897 / 40000000000000000 : ℝ) ≤
+            (3036411121332401602533 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6773367697350193 / 10000000000000000 : ℝ)) (q :=
+      (54191310149538897 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_598_4 :
+    Uω (aρ 4) (bρ 4) (44224781932669 / 64000000000000 : ℝ) ≤
+      (-(250802958368932343957 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (53556753301956121 / 80000000000000000 : ℝ) ≤
+      (-(250802958368932343957 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (53556753301956121 / 80000000000000000 : ℝ) =
+            (53556753301956121 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (53556753301956121 / 40000000000000000 : ℝ) ≤
+            (182414029318567656043 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6693022666519499 / 10000000000000000 : ℝ)) (q :=
+      (53556753301956121 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_598_5 :
+    Uω (aρ 5) (bρ 5) (44224781932669 / 64000000000000 : ℝ) ≤
+      (-(2097370035909306309519 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (52591400996845877 / 80000000000000000 : ℝ) ≤
+      (-(2097370035909306309519 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (52591400996845877 / 80000000000000000 : ℝ) =
+            (52591400996845877 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (52591400996845877 / 40000000000000000 : ℝ) ≤
+            (1368365865590693690481 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3284937254788469 / 5000000000000000 : ℝ)) (q :=
+      (52591400996845877 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_598_6 :
+    Uω (aρ 6) (bρ 6) (44224781932669 / 64000000000000 : ℝ) ≤
+      (-(4461075231951651050771 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (51209195279289041 / 80000000000000000 : ℝ) ≤
+      (-(4461075231951651050771 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (51209195279289041 / 80000000000000000 : ℝ) =
+            (51209195279289041 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (51209195279289041 / 40000000000000000 : ℝ) ≤
+            (2470396571048348949229 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6391563455327729 / 10000000000000000 : ℝ)) (q :=
+      (51209195279289041 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_598_7 :
+    Uω (aρ 7) (bρ 7) (44224781932669 / 64000000000000 : ℝ) ≤
+      (-(967055438211356976147 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (49328347475768481 / 80000000000000000 : ℝ) ≤
+      (-(967055438211356976147 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (49328347475768481 / 80000000000000000 : ℝ) =
+            (49328347475768481 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (49328347475768481 / 40000000000000000 : ℝ) ≤
+            (419238922388643023853 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6145012885207589 / 10000000000000000 : ℝ)) (q :=
+      (49328347475768481 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_598_8 :
+    Uω (aρ 8) (bρ 8) (44224781932669 / 64000000000000 : ℝ) ≤
+      (-(534351573455711606301 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (46883934132587653 / 80000000000000000 : ℝ) ≤
+      (-(534351573455711606301 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (46883934132587653 / 80000000000000000 : ℝ) =
+            (46883934132587653 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (46883934132587653 / 40000000000000000 : ℝ) ≤
+            (158795606844288393699 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2908725600376191 / 5000000000000000 : ℝ)) (q :=
+      (46883934132587653 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_598_9 :
+    Uω (aρ 9) (bρ 9) (44224781932669 / 64000000000000 : ℝ) ≤
+      (-(187966715081535804873 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (43839408318014213 / 80000000000000000 : ℝ) ≤
+      (-(187966715081535804873 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (43839408318014213 / 80000000000000000 : ℝ) =
+            (43839408318014213 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (43839408318014213 / 40000000000000000 : ℝ) ≤
+            (28641778762214195127 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2698642687734511 / 5000000000000000 : ℝ)) (q :=
+      (43839408318014213 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_598_10 :
+    Uω (aρ 10) (bρ 10) (44224781932669 / 64000000000000 : ℝ) ≤
+      (-(1720766780932629166881 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8038817615334817 / 16000000000000000 : ℝ) ≤
+      (-(1720766780932629166881 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (8038817615334817 / 16000000000000000 : ℝ) =
+            (8038817615334817 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8038817615334817 / 8000000000000000 : ℝ) ≤
+            (12101169817370833119 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (487451434862899 / 1000000000000000 : ℝ)) (q :=
+      (8038817615334817 / 16000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_598_11 :
+    Uω (aρ 11) (bρ 11) (44224781932669 / 64000000000000 : ℝ) ≤
+      (-(7990041466326574167593 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (35982132201957549 / 80000000000000000 : ℝ) ≤
+      (-(7990041466326574167593 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (35982132201957549 / 80000000000000000 : ℝ) =
+            (35982132201957549 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (35982132201957549 / 20000000000000000 : ℝ) ≤
+            (5872902139673425832407 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (530000250769357 / 1250000000000000 : ℝ)) (q :=
+      (35982132201957549 / 80000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_598_12 :
+    Uω (aρ 12) (bρ 12) (44224781932669 / 64000000000000 : ℝ) ≤
+      (-(469938514580832882803 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (31254069915314197 / 80000000000000000 : ℝ) ≤
+      (-(469938514580832882803 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (31254069915314197 / 80000000000000000 : ℝ) =
+            (31254069915314197 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (31254069915314197 / 20000000000000000 : ℝ) ≤
+            (223208665719167117197 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1740843265589509 / 5000000000000000 : ℝ)) (q :=
+      (31254069915314197 / 80000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_598_13 :
+    Uω (aρ 13) (bρ 13) (44224781932669 / 64000000000000 : ℝ) ≤
+      (-(11236182902422223580447 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (26008108217176697 / 80000000000000000 : ℝ) ≤
+      (-(11236182902422223580447 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (26008108217176697 / 80000000000000000 : ℝ) =
+            (26008108217176697 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26008108217176697 / 20000000000000000 : ℝ) ≤
+            (2626760703577776419553 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2569577848239643 / 10000000000000000 : ℝ)) (q :=
+      (26008108217176697 / 80000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_598_14 :
+    Uω (aρ 14) (bρ 14) (44224781932669 / 64000000000000 : ℝ) ≤
+      (-(2793273795028130683401 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19794215259049549 / 80000000000000000 : ℝ) ≤
+      (-(2793273795028130683401 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (19794215259049549 / 80000000000000000 : ℝ) =
+            (19794215259049549 / 10000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19794215259049549 / 10000000000000000 : ℝ) ≤
+            (1365609286771869316599 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (170003986719107 / 1250000000000000 : ℝ)) (q :=
+      (19794215259049549 / 80000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_598_15 :
+    Uω (aρ 15) (bρ 15) (44224781932669 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_598_16 :
+    Uω (aρ 16) (bρ 16) (44224781932669 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_598 :
+    Uρ (44224781932669 / 64000000000000 : ℝ) ≤
+      (-(39963571329679689973 / 62500000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (44224781932669 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3789787864895543404819 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_598_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (44224781932669 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3824763229207356151267 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_598_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (44224781932669 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3895060681667598397467 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_598_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (44224781932669 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(250802958368932343957 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_598_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (44224781932669 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(2097370035909306309519 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_598_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (44224781932669 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(4461075231951651050771 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_598_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (44224781932669 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(967055438211356976147 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_598_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (44224781932669 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(534351573455711606301 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_598_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (44224781932669 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(187966715081535804873 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_598_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (44224781932669 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(1720766780932629166881 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_598_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (44224781932669 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(7990041466326574167593 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_598_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (44224781932669 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(469938514580832882803 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_598_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (44224781932669 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(11236182902422223580447 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_598_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (44224781932669 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(2793273795028130683401 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_598_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (44224781932669 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_598_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (44224781932669 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_598_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_599_1 :
+    Uω (aρ 1) (bρ 1) (22138350563149 / 32000000000000 : ℝ) ≤
+      (-(944486087211076226443 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27414854978146513 / 40000000000000000 : ℝ) ≤
+      (-(944486087211076226443 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (27414854978146513 / 40000000000000000 : ℝ) =
+            (27414854978146513 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27414854978146513 / 20000000000000000 : ℝ) ≤
+            (788381863538923773557 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3426845078087097 / 5000000000000000 : ℝ)) (q :=
+      (27414854978146513 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_599_2 :
+    Uω (aρ 2) (bρ 2) (22138350563149 / 32000000000000 : ℝ) ≤
+      (-(381287801260218030659 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27319251931029037 / 40000000000000000 : ℝ) ≤
+      (-(381287801260218030659 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (27319251931029037 / 40000000000000000 : ℝ) =
+            (27319251931029037 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27319251931029037 / 20000000000000000 : ℝ) ≤
+            (311859379039781969341 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (426853602820341 / 625000000000000 : ℝ)) (q :=
+      (27319251931029037 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_599_3 :
+    Uω (aρ 3) (bρ 3) (22138350563149 / 32000000000000 : ℝ) ≤
+      (-(1941545490004321403837 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5425621436749573 / 8000000000000000 : ℝ) ≤
+      (-(1941545490004321403837 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5425621436749573 / 8000000000000000 : ℝ) =
+            (5425621436749573 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5425621436749573 / 4000000000000000 : ℝ) ≤
+            (1524190411495678596163 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (678148137783487 / 1000000000000000 : ℝ)) (q :=
+      (5425621436749573 / 8000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_599_4 :
+    Uω (aρ 4) (bρ 4) (22138350563149 / 32000000000000 : ℝ) ≤
+      (-(4000734033657501560457 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5362166751357703 / 8000000000000000 : ℝ) ≤
+      (-(4000734033657501560457 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5362166751357703 / 8000000000000000 : ℝ) =
+            (5362166751357703 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5362166751357703 / 4000000000000000 : ℝ) ≤
+            (2930737769342498439543 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1340227769084039 / 2000000000000000 : ℝ)) (q :=
+      (5362166751357703 / 8000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_599_5 :
+    Uω (aρ 5) (bρ 5) (22138350563149 / 32000000000000 : ℝ) ≤
+      (-(418239986130542002681 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5265633995257277 / 8000000000000000 : ℝ) ≤
+      (-(418239986130542002681 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5265633995257277 / 8000000000000000 : ℝ) =
+            (5265633995257277 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5265633995257277 / 4000000000000000 : ℝ) ≤
+            (274907194169457997319 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (657799687450413 / 1000000000000000 : ℝ)) (q :=
+      (5265633995257277 / 8000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_599_6 :
+    Uω (aρ 6) (bρ 6) (22138350563149 / 32000000000000 : ℝ) ≤
+      (-(4448390988465003952377 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25637095741062787 / 40000000000000000 : ℝ) ≤
+      (-(4448390988465003952377 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (25637095741062787 / 40000000000000000 : ℝ) =
+            (25637095741062787 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25637095741062787 / 20000000000000000 : ℝ) ≤
+            (2483080814534996047623 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6399700132032331 / 10000000000000000 : ℝ)) (q :=
+      (25637095741062787 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_599_7 :
+    Uω (aρ 7) (bρ 7) (22138350563149 / 32000000000000 : ℝ) ≤
+      (-(602760549948094116003 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24696734141387703 / 40000000000000000 : ℝ) ≤
+      (-(602760549948094116003 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (24696734141387703 / 40000000000000000 : ℝ) =
+            (24696734141387703 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24696734141387703 / 20000000000000000 : ℝ) ≤
+            (263673425426905883997 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6153180712954789 / 10000000000000000 : ℝ)) (q :=
+      (24696734141387703 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_599_8 :
+    Uω (aρ 8) (bρ 8) (22138350563149 / 32000000000000 : ℝ) ≤
+      (-(1065916139610143317519 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4694931260795021 / 8000000000000000 : ℝ) ≤
+      (-(1065916139610143317519 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (4694931260795021 / 8000000000000000 : ℝ) =
+            (4694931260795021 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4694931260795021 / 4000000000000000 : ℝ) ≤
+            (320378220989856682481 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (582568344558849 / 1000000000000000 : ℝ)) (q :=
+      (4694931260795021 / 8000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_599_9 :
+    Uω (aρ 9) (bρ 9) (22138350563149 / 32000000000000 : ℝ) ≤
+      (-(5999916041147995132601 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21952649749208029 / 40000000000000000 : ℝ) ≤
+      (-(5999916041147995132601 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (21952649749208029 / 40000000000000000 : ℝ) =
+            (21952649749208029 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21952649749208029 / 20000000000000000 : ℝ) ≤
+            (931555761852004867399 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (675705724570619 / 1250000000000000 : ℝ)) (q :=
+      (21952649749208029 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_599_10 :
+    Uω (aρ 10) (bρ 10) (22138350563149 / 32000000000000 : ℝ) ≤
+      (-(6866439379730423357791 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20130488686426481 / 40000000000000000 : ℝ) ≤
+      (-(6866439379730423357791 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (20130488686426481 / 40000000000000000 : ℝ) =
+            (20130488686426481 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20130488686426481 / 20000000000000000 : ℝ) ≤
+            (65032423269576642209 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2441562149334589 / 5000000000000000 : ℝ)) (q :=
+      (20130488686426481 / 40000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_599_11 :
+    Uω (aρ 11) (bρ 11) (22138350563149 / 32000000000000 : ℝ) ≤
+      (-(7970929014454339731257 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (18025484319827781 / 40000000000000000 : ℝ) ≤
+      (-(7970929014454339731257 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (18025484319827781 / 40000000000000000 : ℝ) =
+            (18025484319827781 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18025484319827781 / 10000000000000000 : ℝ) ≤
+            (5892014591545660268743 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1062274685393707 / 2500000000000000 : ℝ)) (q :=
+      (18025484319827781 / 40000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_599_12 :
+    Uω (aρ 12) (bρ 12) (22138350563149 / 32000000000000 : ℝ) ≤
+      (-(1875100772085846270361 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15663435820495567 / 40000000000000000 : ℝ) ≤
+      (-(1875100772085846270361 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (15663435820495567 / 40000000000000000 : ℝ) =
+            (15663435820495567 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15663435820495567 / 10000000000000000 : ℝ) ≤
+            (897487949114153729639 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3491774588593721 / 10000000000000000 : ℝ)) (q :=
+      (15663435820495567 / 40000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_599_13 :
+    Uω (aρ 13) (bρ 13) (22138350563149 / 32000000000000 : ℝ) ≤
+      (-(2240937602556298459121 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13045074796685817 / 40000000000000000 : ℝ) ≤
+      (-(2240937602556298459121 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (13045074796685817 / 40000000000000000 : ℝ) =
+            (13045074796685817 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13045074796685817 / 10000000000000000 : ℝ) ≤
+            (531651118643701540879 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1290987909141923 / 5000000000000000 : ℝ)) (q :=
+      (13045074796685817 / 40000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_599_14 :
+    Uω (aρ 14) (bρ 14) (22138350563149 / 32000000000000 : ℝ) ≤
+      (-(13907183051807184338793 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1991171652777751 / 8000000000000000 : ℝ) ≤
+      (-(13907183051807184338793 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1991171652777751 / 8000000000000000 : ℝ) =
+            (1991171652777751 / 1000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1991171652777751 / 1000000000000000 : ℝ) ≤
+            (6887232357192815661207 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (276258967386063 / 2000000000000000 : ℝ)) (q :=
+      (1991171652777751 / 8000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_599_15 :
+    Uω (aρ 15) (bρ 15) (22138350563149 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_599_16 :
+    Uω (aρ 16) (bρ 16) (22138350563149 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_599 :
+    Uρ (22138350563149 / 32000000000000 : ℝ) ≤
+      (-(6377899458709971549779 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (22138350563149 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(944486087211076226443 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_599_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (22138350563149 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(381287801260218030659 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_599_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (22138350563149 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1941545490004321403837 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_599_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (22138350563149 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(4000734033657501560457 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_599_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (22138350563149 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(418239986130542002681 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_599_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (22138350563149 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(4448390988465003952377 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_599_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (22138350563149 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(602760549948094116003 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_599_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (22138350563149 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(1065916139610143317519 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_599_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (22138350563149 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(5999916041147995132601 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_599_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (22138350563149 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(6866439379730423357791 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_599_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (22138350563149 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(7970929014454339731257 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_599_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (22138350563149 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(1875100772085846270361 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_599_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (22138350563149 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(2240937602556298459121 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_599_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (22138350563149 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(13907183051807184338793 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_599_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (22138350563149 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_599_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (22138350563149 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_599_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_600_1 :
+    Uω (aρ 1) (bρ 1) (44328620319927 / 64000000000000 : ℝ) ≤
+      (-(3766114843185157129627 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (54894609171428223 / 80000000000000000 : ℝ) ≤
+      (-(3766114843185157129627 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (54894609171428223 / 80000000000000000 : ℝ) =
+            (54894609171428223 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (54894609171428223 / 40000000000000000 : ℝ) ≤
+            (3165356959814842870373 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3430901292976731 / 5000000000000000 : ℝ)) (q :=
+      (54894609171428223 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_600_2 :
+    Uω (aρ 2) (bρ 2) (44328620319927 / 64000000000000 : ℝ) ≤
+      (-(19005034528540760341 / 50000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (54703404328442747 / 80000000000000000 : ℝ) ≤
+      (-(19005034528540760341 / 50000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (54703404328442747 / 80000000000000000 : ℝ) =
+            (54703404328442747 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (54703404328442747 / 40000000000000000 : ℝ) ≤
+            (15652324486459239659 / 50000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6837770387717093 / 10000000000000000 : ℝ)) (q :=
+      (54703404328442747 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_600_3 :
+    Uω (aρ 3) (bρ 3) (44328620319927 / 64000000000000 : ℝ) ≤
+      (-(38711355908973622577 / 100000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (54321118572963439 / 80000000000000000 : ℝ) ≤
+      (-(38711355908973622577 / 100000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (54321118572963439 / 80000000000000000 : ℝ) =
+            (54321118572963439 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (54321118572963439 / 40000000000000000 : ℝ) ≤
+            (30603362121026377423 / 100000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3394797527598633 / 5000000000000000 : ℝ)) (q :=
+      (54321118572963439 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_600_4 :
+    Uω (aρ 4) (bρ 4) (44328620319927 / 64000000000000 : ℝ) ≤
+      (-(3988635395722701368637 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10737316337676867 / 16000000000000000 : ℝ) ≤
+      (-(3988635395722701368637 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10737316337676867 / 16000000000000000 : ℝ) =
+            (10737316337676867 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10737316337676867 / 8000000000000000 : ℝ) ≤
+            (2942836407277298631363 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (670925501511749 / 1000000000000000 : ℝ)) (q :=
+      (10737316337676867 / 16000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_600_5 :
+    Uω (aρ 5) (bρ 5) (44328620319927 / 64000000000000 : ℝ) ≤
+      (-(834014975758038440671 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10544255761957887 / 16000000000000000 : ℝ) ≤
+      (-(834014975758038440671 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10544255761957887 / 16000000000000000 : ℝ) =
+            (10544255761957887 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10544255761957887 / 8000000000000000 : ℝ) ≤
+            (552279384841961559329 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1317223842960753 / 2000000000000000 : ℝ)) (q :=
+      (10544255761957887 / 16000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_600_6 :
+    Uω (aρ 6) (bρ 6) (44328620319927 / 64000000000000 : ℝ) ≤
+      (-(887144572323505302381 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (51339187438454079 / 80000000000000000 : ℝ) ≤
+      (-(887144572323505302381 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (51339187438454079 / 80000000000000000 : ℝ) =
+            (51339187438454079 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (51339187438454079 / 40000000000000000 : ℝ) ≤
+            (499149788276494697619 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3203918373554963 / 5000000000000000 : ℝ)) (q :=
+      (51339187438454079 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_600_7 :
+    Uω (aρ 7) (bρ 7) (44328620319927 / 64000000000000 : ℝ) ≤
+      (-(4808909108662188216689 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (49458588503678087 / 80000000000000000 : ℝ) ≤
+      (-(4808909108662188216689 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (49458588503678087 / 80000000000000000 : ℝ) =
+            (49458588503678087 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (49458588503678087 / 40000000000000000 : ℝ) ≤
+            (2122562694337811783311 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (770168549271991 / 1250000000000000 : ℝ)) (q :=
+      (49458588503678087 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_600_8 :
+    Uω (aρ 8) (bρ 8) (44328620319927 / 64000000000000 : ℝ) ≤
+      (-(5315665338778738707151 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (47014689739965883 / 80000000000000000 : ℝ) ≤
+      (-(5315665338778738707151 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (47014689739965883 / 80000000000000000 : ℝ) =
+            (47014689739965883 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (47014689739965883 / 40000000000000000 : ℝ) ≤
+            (1615806464221261292849 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5833915354587877 / 10000000000000000 : ℝ)) (q :=
+      (47014689739965883 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_600_9 :
+    Uω (aρ 9) (bρ 9) (44328620319927 / 64000000000000 : ℝ) ≤
+      (-(598492040897808695309 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8794237531991951 / 16000000000000000 : ℝ) ≤
+      (-(598492040897808695309 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (8794237531991951 / 16000000000000000 : ℝ) =
+            (8794237531991951 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8794237531991951 / 8000000000000000 : ℝ) ≤
+            (94655139402191304691 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1082801092589269 / 2000000000000000 : ℝ)) (q :=
+      (8794237531991951 / 16000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_600_10 :
+    Uω (aρ 10) (bρ 10) (44328620319927 / 64000000000000 : ℝ) ≤
+      (-(6849840925155560255049 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (40327859865204219 / 80000000000000000 : ℝ) ≤
+      (-(6849840925155560255049 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (40327859865204219 / 80000000000000000 : ℝ) =
+            (40327859865204219 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (40327859865204219 / 40000000000000000 : ℝ) ≤
+            (81630877844439744951 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4891732547752461 / 10000000000000000 : ℝ)) (q :=
+      (40327859865204219 / 80000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_600_11 :
+    Uω (aρ 11) (bρ 11) (44328620319927 / 64000000000000 : ℝ) ≤
+      (-(3975928713532320038747 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (36119789164517907 / 80000000000000000 : ℝ) ≤
+      (-(3975928713532320038747 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (36119789164517907 / 80000000000000000 : ℝ) =
+            (36119789164517907 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (36119789164517907 / 20000000000000000 : ℝ) ≤
+            (2955543089467679961253 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4258191498785883 / 10000000000000000 : ℝ)) (q :=
+      (36119789164517907 / 80000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_600_12 :
+    Uω (aρ 12) (bρ 12) (44328620319927 / 64000000000000 : ℝ) ≤
+      (-(9352304517135290160901 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (31399632293277399 / 80000000000000000 : ℝ) ≤
+      (-(9352304517135290160901 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (31399632293277399 / 80000000000000000 : ℝ) =
+            (31399632293277399 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (31399632293277399 / 20000000000000000 : ℝ) ≤
+            (4510639088864709839099 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (875463094415189 / 2500000000000000 : ℝ)) (q :=
+      (31399632293277399 / 80000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_600_13 :
+    Uω (aρ 13) (bρ 13) (44328620319927 / 64000000000000 : ℝ) ≤
+      (-(5586671893103846167657 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (26172055446114123 / 80000000000000000 : ℝ) ≤
+      (-(5586671893103846167657 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (26172055446114123 / 80000000000000000 : ℝ) =
+            (26172055446114123 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26172055446114123 / 20000000000000000 : ℝ) ≤
+            (1344799909896153832343 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2594339907464937 / 10000000000000000 : ℝ)) (q :=
+      (26172055446114123 / 80000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_600_14 :
+    Uω (aρ 14) (bρ 14) (44328620319927 / 64000000000000 : ℝ) ≤
+      (-(2769779097549425911313 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4005623196142587 / 16000000000000000 : ℝ) ≤
+      (-(2769779097549425911313 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4005623196142587 / 16000000000000000 : ℝ) =
+            (4005623196142587 / 4000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 2
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4005623196142587 / 4000000000000000 : ℝ) ≤
+            (2809623650574088687 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (35057058153991 / 250000000000000 : ℝ)) (q :=
+      (4005623196142587 / 16000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_600_15 :
+    Uω (aρ 15) (bρ 15) (44328620319927 / 64000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_600_16 :
+    Uω (aρ 16) (bρ 16) (44328620319927 / 64000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_600 :
+    Uρ (44328620319927 / 64000000000000 : ℝ) ≤
+      (-(3180848175369736682521 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (44328620319927 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3766114843185157129627 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_600_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (44328620319927 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) * (-(19005034528540760341 / 50000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_600_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (44328620319927 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) * (-(38711355908973622577 / 100000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_600_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (44328620319927 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3988635395722701368637 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_600_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (44328620319927 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(834014975758038440671 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_600_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (44328620319927 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(887144572323505302381 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_600_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (44328620319927 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(4808909108662188216689 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_600_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (44328620319927 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(5315665338778738707151 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_600_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (44328620319927 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(598492040897808695309 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_600_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (44328620319927 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(6849840925155560255049 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_600_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (44328620319927 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(3975928713532320038747 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_600_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (44328620319927 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(9352304517135290160901 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_600_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (44328620319927 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(5586671893103846167657 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_600_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (44328620319927 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(2769779097549425911313 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_600_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (44328620319927 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_600_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (44328620319927 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_600_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_601_1 :
+    Uω (aρ 1) (bρ 1) (11095134878389 / 16000000000000 : ℝ) ≤
+      (-(938574828702399587789 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3434969274127283 / 5000000000000000 : ℝ) ≤
+      (-(938574828702399587789 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (3434969274127283 / 5000000000000000 : ℝ) =
+            (3434969274127283 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3434969274127283 / 2500000000000000 : ℝ) ≤
+            (794293122047600412211 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6869915015601007 / 10000000000000000 : ℝ)) (q :=
+      (3434969274127283 / 5000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_601_2 :
+    Uω (aρ 2) (bρ 2) (11095134878389 / 16000000000000 : ℝ) ≤
+      (-(757829975012473701853 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3423019049458321 / 5000000000000000 : ℝ) ≤
+      (-(757829975012473701853 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (3423019049458321 / 5000000000000000 : ℝ) =
+            (3423019049458321 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3423019049458321 / 2500000000000000 : ℝ) ≤
+            (628464385587526298147 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6845883129435159 / 10000000000000000 : ℝ)) (q :=
+      (3423019049458321 / 5000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_601_3 :
+    Uω (aρ 3) (bρ 3) (11095134878389 / 16000000000000 : ℝ) ≤
+      (-(964798620035307040553 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13596505691496687 / 20000000000000000 : ℝ) ≤
+      (-(964798620035307040553 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (13596505691496687 / 20000000000000000 : ℝ) =
+            (13596505691496687 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13596505691496687 / 10000000000000000 : ℝ) ≤
+            (768069330714692959447 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3398854364724281 / 5000000000000000 : ℝ)) (q :=
+      (13596505691496687 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_601_4 :
+    Uω (aρ 4) (bρ 4) (11095134878389 / 16000000000000 : ℝ) ≤
+      (-(994137846157366722441 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13437873956627869 / 20000000000000000 : ℝ) ≤
+      (-(994137846157366722441 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (13437873956627869 / 20000000000000000 : ℝ) =
+            (13437873956627869 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13437873956627869 / 10000000000000000 : ℝ) ≤
+            (738730104592633277559 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (839671396955593 / 1250000000000000 : ℝ)) (q :=
+      (13437873956627869 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_601_5 :
+    Uω (aρ 5) (bρ 5) (11095134878389 / 16000000000000 : ℝ) ≤
+      (-(4157765086690188808493 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13196554392214973 / 20000000000000000 : ℝ) ≤
+      (-(4157765086690188808493 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (13196554392214973 / 20000000000000000 : ℝ) =
+            (13196554392214973 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13196554392214973 / 10000000000000000 : ℝ) ≤
+            (2773706716309811191507 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (103035023915107 / 156250000000000 : ℝ)) (q :=
+      (13196554392214973 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_601_6 :
+    Uω (aρ 6) (bρ 6) (11095134878389 / 16000000000000 : ℝ) ≤
+      (-(2211535405192055338863 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12851045787303101 / 20000000000000000 : ℝ) ≤
+      (-(2211535405192055338863 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (12851045787303101 / 20000000000000000 : ℝ) =
+            (12851045787303101 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12851045787303101 / 10000000000000000 : ℝ) ≤
+            (1254200496307944661137 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (200499165649843 / 312500000000000 : ℝ)) (q :=
+      (12851045787303101 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_601_7 :
+    Uω (aρ 7) (bρ 7) (11095134878389 / 16000000000000 : ℝ) ≤
+      (-(959150254321744450281 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (309523175880027 / 500000000000000 : ℝ) ≤
+      (-(959150254321744450281 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (309523175880027 / 500000000000000 : ℝ) =
+            (309523175880027 / 250000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (309523175880027 / 250000000000000 : ℝ) ≤
+            (427144106278255549719 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1233903185890591 / 2000000000000000 : ℝ)) (q :=
+      (309523175880027 / 500000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_601_8 :
+    Uω (aρ 8) (bρ 8) (11095134878389 / 16000000000000 : ℝ) ≤
+      (-(265088480022531474809 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2942504095894577 / 5000000000000000 : ℝ) ≤
+      (-(265088480022531474809 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2942504095894577 / 5000000000000000 : ℝ) =
+            (2942504095894577 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2942504095894577 / 2500000000000000 : ℝ) ≤
+            (81485110127468525191 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5842146929170183 / 10000000000000000 : ℝ)) (q :=
+      (2942504095894577 / 5000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_601_9 :
+    Uω (aρ 9) (bρ 9) (11095134878389 / 16000000000000 : ℝ) ≤
+      (-(5969947912396625863489 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11009268204151959 / 20000000000000000 : ℝ) ≤
+      (-(5969947912396625863489 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11009268204151959 / 20000000000000000 : ℝ) =
+            (11009268204151959 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11009268204151959 / 10000000000000000 : ℝ) ≤
+            (961523890603374136511 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2711182189051917 / 5000000000000000 : ℝ)) (q :=
+      (11009268204151959 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_601_10 :
+    Uω (aρ 10) (bρ 10) (11095134878389 / 16000000000000 : ℝ) ≤
+      (-(3416635825630334296627 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1262335487174499 / 2500000000000000 : ℝ) ≤
+      (-(3416635825630334296627 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1262335487174499 / 2500000000000000 : ℝ) =
+            (1262335487174499 / 1250000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1262335487174499 / 1250000000000000 : ℝ) ≤
+            (49100075869665703373 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4900339104842867 / 10000000000000000 : ℝ)) (q :=
+      (1262335487174499 / 2500000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_601_11 :
+    Uω (aρ 11) (bρ 11) (11095134878389 / 16000000000000 : ℝ) ≤
+      (-(7932826512081039083167 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9047148469566547 / 20000000000000000 : ℝ) ≤
+      (-(7932826512081039083167 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9047148469566547 / 20000000000000000 : ℝ) =
+            (9047148469566547 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9047148469566547 / 5000000000000000 : ℝ) ≤
+            (5930117093918960916833 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2133640151609211 / 5000000000000000 : ℝ)) (q :=
+      (9047148469566547 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_601_12 :
+    Uω (aρ 12) (bρ 12) (11095134878389 / 16000000000000 : ℝ) ≤
+      (-(9329171808514020960711 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (983511007055157 / 2500000000000000 : ℝ) ≤
+      (-(9329171808514020960711 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (983511007055157 / 2500000000000000 : ℝ) =
+            (983511007055157 / 625000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (983511007055157 / 625000000000000 : ℝ) ≤
+            (4533771797485979039289 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3511919986778131 / 10000000000000000 : ℝ)) (q :=
+      (983511007055157 / 2500000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12]) (by norm_num [bρ_12])
+      (by norm_num) (by norm_num [aρ_12, bρ_12]) (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_601_13 :
+    Uω (aρ 13) (bρ 13) (11095134878389 / 16000000000000 : ℝ) ≤
+      (-(11142148384564178563437 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6563456925967253 / 20000000000000000 : ℝ) ≤
+      (-(11142148384564178563437 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6563456925967253 / 20000000000000000 : ℝ) =
+            (6563456925967253 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6563456925967253 / 5000000000000000 : ℝ) ≤
+            (2720795221435821436563 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (325833824737391 / 1250000000000000 : ℝ)) (q :=
+      (6563456925967253 / 20000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_601_14 :
+    Uω (aρ 14) (bρ 14) (11095134878389 / 16000000000000 : ℝ) ≤
+      (-(551658738635868358737 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5035865592274749 / 20000000000000000 : ℝ) ≤
+      (-(551658738635868358737 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5035865592274749 / 20000000000000000 : ℝ) =
+            (5035865592274749 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5035865592274749 / 5000000000000000 : ℝ) ≤
+            (2859005604131641263 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (177875818656453 / 1250000000000000 : ℝ)) (q :=
+      (5035865592274749 / 20000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_601_15 :
+    Uω (aρ 15) (bρ 15) (11095134878389 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_601_16 :
+    Uω (aρ 16) (bρ 16) (11095134878389 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_601 :
+    Uρ (11095134878389 / 16000000000000 : ℝ) ≤
+      (-(6345560218321767044647 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (11095134878389 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(938574828702399587789 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_601_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (11095134878389 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(757829975012473701853 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_601_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (11095134878389 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(964798620035307040553 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_601_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (11095134878389 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(994137846157366722441 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_601_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (11095134878389 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(4157765086690188808493 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_601_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (11095134878389 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(2211535405192055338863 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_601_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (11095134878389 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(959150254321744450281 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_601_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (11095134878389 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(265088480022531474809 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_601_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (11095134878389 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(5969947912396625863489 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_601_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (11095134878389 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(3416635825630334296627 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_601_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (11095134878389 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(7932826512081039083167 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_601_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (11095134878389 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(9329171808514020960711 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_601_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (11095134878389 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(11142148384564178563437 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_601_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (11095134878389 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) * (-(551658738635868358737 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_601_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (11095134878389 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_601_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (11095134878389 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_601_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_602_1 :
+    Uω (aρ 1) (bρ 1) (8886491741437 / 12800000000000 : ℝ) ≤
+      (-(233906108170397038313 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (55024407600119797 / 80000000000000000 : ℝ) ≤
+      (-(233906108170397038313 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (55024407600119797 / 80000000000000000 : ℝ) =
+            (55024407600119797 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (55024407600119797 / 40000000000000000 : ℝ) ≤
+            (199310879517102961687 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6878027445117293 / 10000000000000000 : ℝ)) (q :=
+      (55024407600119797 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_602_2 :
+    Uω (aρ 2) (bρ 2) (8886491741437 / 12800000000000 : ℝ) ≤
+      (-(944326721830208510903 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (54833205250741649 / 80000000000000000 : ℝ) ≤
+      (-(944326721830208510903 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (54833205250741649 / 80000000000000000 : ℝ) =
+            (54833205250741649 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (54833205250741649 / 40000000000000000 : ℝ) ≤
+            (788541228919791489097 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1713498967570689 / 2500000000000000 : ℝ)) (q :=
+      (54833205250741649 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_602_3 :
+    Uω (aρ 3) (bρ 3) (8886491741437 / 12800000000000 : ℝ) ≤
+      (-(384726761367011421543 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10890185389322033 / 16000000000000000 : ℝ) ≤
+      (-(384726761367011421543 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10890185389322033 / 16000000000000000 : ℝ) =
+            (10890185389322033 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10890185389322033 / 8000000000000000 : ℝ) ≤
+            (308420418932988578457 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1361164480119977 / 2000000000000000 : ℝ)) (q :=
+      (10890185389322033 / 16000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_602_4 :
+    Uω (aρ 4) (bρ 4) (8886491741437 / 12800000000000 : ℝ) ≤
+      (-(792896393007470868719 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (53816409928091253 / 80000000000000000 : ℝ) ≤
+      (-(792896393007470868719 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (53816409928091253 / 80000000000000000 : ℝ) =
+            (53816409928091253 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (53816409928091253 / 40000000000000000 : ℝ) ≤
+            (593397967592529131281 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6725487327035157 / 10000000000000000 : ℝ)) (q :=
+      (53816409928091253 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_602_5 :
+    Uω (aρ 5) (bρ 5) (8886491741437 / 12800000000000 : ℝ) ≤
+      (-(165818817902471995431 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (52851156230146361 / 80000000000000000 : ℝ) ≤
+      (-(165818817902471995431 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (52851156230146361 / 80000000000000000 : ℝ) =
+            (52851156230146361 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (52851156230146361 / 40000000000000000 : ℝ) ≤
+            (111440054217528004569 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3301181910941967 / 5000000000000000 : ℝ)) (q :=
+      (52851156230146361 / 80000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_602_6 :
+    Uω (aρ 6) (bρ 6) (8886491741437 / 12800000000000 : ℝ) ≤
+      (-(88208695877931068533 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (51469178615333637 / 80000000000000000 : ℝ) ≤
+      (-(88208695877931068533 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (51469178615333637 / 80000000000000000 : ℝ) =
+            (51469178615333637 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (51469178615333637 / 40000000000000000 : ℝ) ≤
+            (50420740182068931467 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6424109793320753 / 10000000000000000 : ℝ)) (q :=
+      (51469178615333637 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_602_7 :
+    Uω (aρ 7) (bρ 7) (8886491741437 / 12800000000000 : ℝ) ≤
+      (-(478261084193173771567 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (49588827196469537 / 80000000000000000 : ℝ) ≤
+      (-(478261084193173771567 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (49588827196469537 / 80000000000000000 : ℝ) =
+            (49588827196469537 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (49588827196469537 / 40000000000000000 : ℝ) ≤
+            (214886096106826228433 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (772210414920591 / 1250000000000000 : ℝ)) (q :=
+      (49588827196469537 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_602_8 :
+    Uω (aρ 8) (bρ 8) (8886491741437 / 12800000000000 : ℝ) ≤
+      (-(5287893427020350660491 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (47145439996638613 / 80000000000000000 : ℝ) ≤
+      (-(5287893427020350660491 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (47145439996638613 / 80000000000000000 : ℝ) =
+            (47145439996638613 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (47145439996638613 / 40000000000000000 : ℝ) ≤
+            (1643578375979649339509 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5850378170746997 / 10000000000000000 : ℝ)) (q :=
+      (47145439996638613 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_602_9 :
+    Uω (aρ 9) (bρ 9) (8886491741437 / 12800000000000 : ℝ) ≤
+      (-(5954998478060639171699 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (44102954982235617 / 80000000000000000 : ℝ) ≤
+      (-(5954998478060639171699 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (44102954982235617 / 80000000000000000 : ℝ) =
+            (44102954982235617 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (44102954982235617 / 40000000000000000 : ℝ) ≤
+            (976473324939360828301 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (678840318188281 / 1250000000000000 : ℝ)) (q :=
+      (44102954982235617 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_602_10 :
+    Uω (aρ 10) (bρ 10) (8886491741437 / 12800000000000 : ℝ) ≤
+      (-(6816731449933417676493 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8092320916317969 / 16000000000000000 : ℝ) ≤
+      (-(6816731449933417676493 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (8092320916317969 / 16000000000000000 : ℝ) =
+            (8092320916317969 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8092320916317969 / 8000000000000000 : ℝ) ≤
+            (114740353066582323507 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (981788795767961 / 2000000000000000 : ℝ)) (q :=
+      (8092320916317969 / 16000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_602_11 :
+    Uω (aρ 11) (bρ 11) (8886491741437 / 12800000000000 : ℝ) ≤
+      (-(3956918039439585658247 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (36257382881713221 / 80000000000000000 : ℝ) ≤
+      (-(3956918039439585658247 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (36257382881713221 / 80000000000000000 : ℝ) =
+            (36257382881713221 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (36257382881713221 / 20000000000000000 : ℝ) ≤
+            (2974553763560414341753 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4276365180075649 / 10000000000000000 : ℝ)) (q :=
+      (36257382881713221 / 80000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_602_12 :
+    Uω (aρ 12) (bρ 12) (8886491741437 / 12800000000000 : ℝ) ≤
+      (-(9306105286396064384171 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (31545031787647909 / 80000000000000000 : ℝ) ≤
+      (-(9306105286396064384171 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (31545031787647909 / 80000000000000000 : ℝ) =
+            (31545031787647909 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (31545031787647909 / 20000000000000000 : ℝ) ≤
+            (4556838319603935615829 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3521977503244321 / 10000000000000000 : ℝ)) (q :=
+      (31545031787647909 / 80000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_602_13 :
+    Uω (aρ 13) (bρ 13) (8886491741437 / 12800000000000 : ℝ) ≤
+      (-(5555550003594789216321 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5267093650724393 / 16000000000000000 : ℝ) ≤
+      (-(5555550003594789216321 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5267093650724393 / 16000000000000000 : ℝ) =
+            (5267093650724393 / 4000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5267093650724393 / 4000000000000000 : ℝ) ≤
+            (1375921799405210783679 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (523793672266567 / 2000000000000000 : ℝ)) (q :=
+      (5267093650724393 / 16000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_602_14 :
+    Uω (aρ 14) (bρ 14) (8886491741437 / 12800000000000 : ℝ) ≤
+      (-(6867433388723770022023 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20257801050088877 / 80000000000000000 : ℝ) ≤
+      (-(6867433388723770022023 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (20257801050088877 / 80000000000000000 : ℝ) =
+            (20257801050088877 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20257801050088877 / 20000000000000000 : ℝ) ≤
+            (64038414276229977977 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1443478845494563 / 10000000000000000 : ℝ)) (q :=
+      (20257801050088877 / 80000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_602_15 :
+    Uω (aρ 15) (bρ 15) (8886491741437 / 12800000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_602_16 :
+    Uω (aρ 16) (bρ 16) (8886491741437 / 12800000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_602 :
+    Uρ (8886491741437 / 12800000000000 : ℝ) ≤
+      (-(6329489309641351499701 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (8886491741437 / 12800000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(233906108170397038313 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_602_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (8886491741437 / 12800000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(944326721830208510903 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_602_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (8886491741437 / 12800000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(384726761367011421543 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_602_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (8886491741437 / 12800000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(792896393007470868719 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_602_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (8886491741437 / 12800000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(165818817902471995431 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_602_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (8886491741437 / 12800000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) * (-(88208695877931068533 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_602_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (8886491741437 / 12800000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(478261084193173771567 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_602_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (8886491741437 / 12800000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(5287893427020350660491 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_602_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (8886491741437 / 12800000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(5954998478060639171699 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_602_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (8886491741437 / 12800000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(6816731449933417676493 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_602_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (8886491741437 / 12800000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(3956918039439585658247 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_602_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (8886491741437 / 12800000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(9306105286396064384171 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_602_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (8886491741437 / 12800000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(5555550003594789216321 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_602_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (8886491741437 / 12800000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(6867433388723770022023 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_602_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (8886491741437 / 12800000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_602_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (8886491741437 / 12800000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_602_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_603_1 :
+    Uω (aρ 1) (bρ 1) (22242188950407 / 32000000000000 : ℝ) ≤
+      (-(3730710058060810453223 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27544653406839947 / 40000000000000000 : ℝ) ≤
+      (-(3730710058060810453223 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (27544653406839947 / 40000000000000000 : ℝ) =
+            (27544653406839947 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27544653406839947 / 20000000000000000 : ℝ) ≤
+            (3200761744939189546777 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3443069937251393 / 5000000000000000 : ℝ)) (q :=
+      (27544653406839947 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_603_2 :
+    Uω (aρ 2) (bρ 2) (22242188950407 / 32000000000000 : ℝ) ≤
+      (-(3765477909257897765203 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5489810570668063 / 8000000000000000 : ℝ) ≤
+      (-(3765477909257897765203 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5489810570668063 / 8000000000000000 : ℝ) =
+            (5489810570668063 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5489810570668063 / 4000000000000000 : ℝ) ≤
+            (3165993893742102234797 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (686210861026297 / 1000000000000000 : ℝ)) (q :=
+      (5489810570668063 / 8000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_603_3 :
+    Uω (aρ 3) (bρ 3) (22242188950407 / 32000000000000 : ℝ) ≤
+      (-(1917677478767856130919 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27257915557438993 / 40000000000000000 : ℝ) ≤
+      (-(1917677478767856130919 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (27257915557438993 / 40000000000000000 : ℝ) =
+            (27257915557438993 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27257915557438993 / 20000000000000000 : ℝ) ≤
+            (1548058422732143869081 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6813936068662309 / 10000000000000000 : ℝ)) (q :=
+      (27257915557438993 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_603_4 :
+    Uω (aρ 4) (bρ 4) (22242188950407 / 32000000000000 : ℝ) ≤
+      (-(4940533877167384837 / 12500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (26940661996627907 / 40000000000000000 : ℝ) ≤
+      (-(4940533877167384837 / 12500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (26940661996627907 / 40000000000000000 : ℝ) =
+            (26940661996627907 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26940661996627907 / 20000000000000000 : ℝ) ≤
+            (3723805876582615163 / 12500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3366801734660883 / 5000000000000000 : ℝ)) (q :=
+      (26940661996627907 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_603_5 :
+    Uω (aρ 5) (bρ 5) (22242188950407 / 32000000000000 : ℝ) ≤
+      (-(413319092409985669183 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (26458047397004643 / 40000000000000000 : ℝ) ≤
+      (-(413319092409985669183 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (26458047397004643 / 40000000000000000 : ℝ) =
+            (26458047397004643 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26458047397004643 / 20000000000000000 : ℝ) ≤
+            (279828087890014330817 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3305243044422567 / 5000000000000000 : ℝ)) (q :=
+      (26458047397004643 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_603_6 :
+    Uω (aρ 6) (bρ 6) (22242188950407 / 32000000000000 : ℝ) ≤
+      (-(87956295428855053233 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1030683476754923 / 1600000000000000 : ℝ) ≤
+      (-(87956295428855053233 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1030683476754923 / 1600000000000000 : ℝ) =
+            (1030683476754923 / 800000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1030683476754923 / 800000000000000 : ℝ) ≤
+            (50673140631144946767 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (128644924498387 / 200000000000000 : ℝ)) (q :=
+      (1030683476754923 / 1600000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_603_7 :
+    Uω (aρ 7) (bρ 7) (22242188950407 / 32000000000000 : ℝ) ≤
+      (-(4769487773325352152819 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4965394567297691 / 8000000000000000 : ℝ) ≤
+      (-(4769487773325352152819 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (4965394567297691 / 8000000000000000 : ℝ) =
+            (4965394567297691 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4965394567297691 / 4000000000000000 : ℝ) ≤
+            (2161984029674647847181 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (9665391507011 / 15625000000000 : ℝ)) (q :=
+      (4965394567297691 / 8000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_603_8 :
+    Uω (aρ 8) (bρ 8) (22242188950407 / 32000000000000 : ℝ) ≤
+      (-(5274036762684941512903 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4721081313255643 / 8000000000000000 : ℝ) ≤
+      (-(5274036762684941512903 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (4721081313255643 / 8000000000000000 : ℝ) =
+            (4721081313255643 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4721081313255643 / 4000000000000000 : ℝ) ≤
+            (1657435040315058487097 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2288519172157 / 3906250000000 : ℝ)) (q :=
+      (4721081313255643 / 8000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_603_9 :
+    Uω (aρ 9) (bρ 9) (22242188950407 / 32000000000000 : ℝ) ≤
+      (-(5940072032984582198809 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22084417085315947 / 40000000000000000 : ℝ) ≤
+      (-(5940072032984582198809 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (22084417085315947 / 40000000000000000 : ℝ) =
+            (22084417085315947 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22084417085315947 / 20000000000000000 : ℝ) ≤
+            (991399770015417801191 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2719539984300393 / 5000000000000000 : ℝ)) (q :=
+      (22084417085315947 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_603_10 :
+    Uω (aρ 10) (bρ 10) (22242188950407 / 32000000000000 : ℝ) ≤
+      (-(6800220213689320315221 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4052846687656339 / 8000000000000000 : ℝ) ≤
+      (-(6800220213689320315221 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (4052846687656339 / 8000000000000000 : ℝ) =
+            (4052846687656339 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4052846687656339 / 4000000000000000 : ℝ) ≤
+            (131251589310679684779 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (245877358928933 / 500000000000000 : ℝ)) (q :=
+      (4052846687656339 / 8000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_603_11 :
+    Uω (aρ 11) (bρ 11) (22242188950407 / 32000000000000 : ℝ) ≤
+      (-(3947442969135704496559 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (18163078137386779 / 40000000000000000 : ℝ) ≤
+      (-(3947442969135704496559 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (18163078137386779 / 40000000000000000 : ℝ) =
+            (18163078137386779 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18163078137386779 / 10000000000000000 : ℝ) ≤
+            (2984028833864295503441 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2142723077168101 / 5000000000000000 : ℝ)) (q :=
+      (18163078137386779 / 40000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_603_12 :
+    Uω (aρ 12) (bρ 12) (22242188950407 / 32000000000000 : ℝ) ≤
+      (-(9283104507584379732129 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15808835661897267 / 40000000000000000 : ℝ) ≤
+      (-(9283104507584379732129 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (15808835661897267 / 40000000000000000 : ℝ) =
+            (15808835661897267 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15808835661897267 / 10000000000000000 : ℝ) ≤
+            (4579839098415620267871 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1766012506638223 / 5000000000000000 : ℝ)) (q :=
+      (15808835661897267 / 40000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_603_13 :
+    Uω (aρ 13) (bρ 13) (22242188950407 / 32000000000000 : ℝ) ≤
+      (-(11080196889829367210413 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13208489471044621 / 40000000000000000 : ℝ) ≤
+      (-(11080196889829367210413 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (13208489471044621 / 40000000000000000 : ℝ) =
+            (13208489471044621 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13208489471044621 / 10000000000000000 : ℝ) ≤
+            (2782746716170632789587 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2631233659445123 / 10000000000000000 : ℝ)) (q :=
+      (13208489471044621 / 40000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_603_14 :
+    Uω (aρ 14) (bρ 14) (22242188950407 / 32000000000000 : ℝ) ≤
+      (-(341976439182884107769 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10185587153490103 / 40000000000000000 : ℝ) ≤
+      (-(341976439182884107769 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (10185587153490103 / 40000000000000000 : ℝ) =
+            (10185587153490103 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10185587153490103 / 10000000000000000 : ℝ) ≤
+            (4597150967115892231 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (45740930803527 / 312500000000000 : ℝ)) (q :=
+      (10185587153490103 / 40000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_603_15 :
+    Uω (aρ 15) (bρ 15) (22242188950407 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_603_16 :
+    Uω (aρ 16) (bρ 16) (22242188950407 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_603 :
+    Uρ (22242188950407 / 32000000000000 : ℝ) ≤
+      (-(1578370495051350464269 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (22242188950407 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3730710058060810453223 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_603_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (22242188950407 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3765477909257897765203 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_603_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (22242188950407 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1917677478767856130919 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_603_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (22242188950407 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(4940533877167384837 / 12500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_603_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (22242188950407 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(413319092409985669183 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_603_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (22242188950407 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) * (-(87956295428855053233 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_603_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (22242188950407 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(4769487773325352152819 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_603_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (22242188950407 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(5274036762684941512903 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_603_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (22242188950407 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(5940072032984582198809 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_603_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (22242188950407 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(6800220213689320315221 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_603_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (22242188950407 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(3947442969135704496559 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_603_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (22242188950407 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(9283104507584379732129 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_603_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (22242188950407 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(11080196889829367210413 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_603_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (22242188950407 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) * (-(341976439182884107769 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_603_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (22242188950407 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_603_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (22242188950407 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_603_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/U50.lean b/LeanPool/Zeta5Irrational/Table/U50.lean
new file mode 100644
index 0000000000..b5dcad2c41
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/U50.lean
@@ -0,0 +1,5058 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-! # Certified arcsine potential bounds (U50) -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma U_604_1 :
+    Uω (aρ 1) (bρ 1) (5573527036009 / 8000000000000 : ℝ) ≤
+      (-(185358815803223170541 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6902388154904741 / 10000000000000000 : ℝ) ≤
+      (-(185358815803223170541 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (6902388154904741 / 10000000000000000 : ℝ) =
+            (6902388154904741 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6902388154904741 / 5000000000000000 : ℝ) ≤
+            (161214774346776829459 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (215698897902601 / 312500000000000 : ℝ)) (q :=
+      (6902388154904741 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_604_2 :
+    Uω (aρ 2) (bρ 2) (5573527036009 / 8000000000000 : ℝ) ≤
+      (-(3741861849853575925477 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6878488326024889 / 10000000000000000 : ℝ) ≤
+      (-(3741861849853575925477 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (6878488326024889 / 10000000000000000 : ℝ) =
+            (6878488326024889 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6878488326024889 / 5000000000000000 : ℝ) ≤
+            (3189609953146424074523 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (859791760954191 / 1250000000000000 : ℝ)) (q :=
+      (6878488326024889 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_604_3 :
+    Uω (aρ 3) (bρ 3) (5573527036009 / 8000000000000 : ℝ) ≤
+      (-(762314428217885278549 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13661409853630741 / 20000000000000000 : ℝ) ≤
+      (-(762314428217885278549 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (13661409853630741 / 20000000000000000 : ℝ) =
+            (13661409853630741 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13661409853630741 / 10000000000000000 : ℝ) ≤
+            (623979932382114721451 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6830163395564491 / 10000000000000000 : ℝ)) (q :=
+      (13661409853630741 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_604_4 :
+    Uω (aρ 4) (bρ 4) (5573527036009 / 8000000000000 : ℝ) ≤
+      (-(392836090377941199433 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6751394001858091 / 10000000000000000 : ℝ) ≤
+      (-(392836090377941199433 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (6751394001858091 / 10000000000000000 : ℝ) =
+            (6751394001858091 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6751394001858091 / 5000000000000000 : ℝ) ≤
+            (300311089922058800567 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1687458931678733 / 2500000000000000 : ℝ)) (q :=
+      (6751394001858091 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_604_5 :
+    Uω (aρ 5) (bρ 5) (5573527036009 / 8000000000000 : ℝ) ≤
+      (-(2054338537821379043431 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13261492907724353 / 20000000000000000 : ℝ) ≤
+      (-(2054338537821379043431 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (13261492907724353 / 20000000000000000 : ℝ) =
+            (13261492907724353 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13261492907724353 / 10000000000000000 : ℝ) ≤
+            (1411397363678620956569 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6626730550058103 / 10000000000000000 : ℝ)) (q :=
+      (13261492907724353 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_604_6 :
+    Uω (aρ 6) (bρ 6) (5573527036009 / 8000000000000 : ℝ) ≤
+      (-(1093155636640941648659 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12916040888783747 / 20000000000000000 : ℝ) ≤
+      (-(1093155636640941648659 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (12916040888783747 / 20000000000000000 : ℝ) =
+            (12916040888783747 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12916040888783747 / 10000000000000000 : ℝ) ≤
+            (639712314109058351341 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6448518906257497 / 10000000000000000 : ℝ)) (q :=
+      (12916040888783747 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_604_7 :
+    Uω (aρ 7) (bρ 7) (5573527036009 / 8000000000000 : ℝ) ≤
+      (-(948658707005702538837 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12446045224417483 / 20000000000000000 : ℝ) ≤
+      (-(948658707005702538837 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (12446045224417483 / 20000000000000000 : ℝ) =
+            (12446045224417483 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12446045224417483 / 10000000000000000 : ℝ) ≤
+            (437635653594297461163 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6202184622651233 / 10000000000000000 : ℝ)) (q :=
+      (12446045224417483 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_604_8 :
+    Uω (aρ 8) (bρ 8) (5573527036009 / 8000000000000 : ℝ) ≤
+      (-(5246381739293769484407 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5917694430933567 / 10000000000000000 : ℝ) ≤
+      (-(5246381739293769484407 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5917694430933567 / 10000000000000000 : ℝ) =
+            (5917694430933567 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5917694430933567 / 5000000000000000 : ℝ) ≤
+            (1685090063706230515593 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1468767477860221 / 2500000000000000 : ℝ)) (q :=
+      (5917694430933567 / 10000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_604_9 :
+    Uω (aρ 9) (bρ 9) (5573527036009 / 8000000000000 : ℝ) ≤
+      (-(591028782044301194203 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11075145917614141 / 20000000000000000 : ℝ) ≤
+      (-(591028782044301194203 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11075145917614141 / 20000000000000000 : ℝ) =
+            (11075145917614141 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11075145917614141 / 10000000000000000 : ℝ) ≤
+            (102118398255698805797 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5455792595547891 / 10000000000000000 : ℝ)) (q :=
+      (11075145917614141 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_604_10 :
+    Uω (aρ 10) (bρ 10) (5573527036009 / 8000000000000 : ℝ) ≤
+      (-(6767284209621708981663 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10165542878878319 / 20000000000000000 : ℝ) ≤
+      (-(6767284209621708981663 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10165542878878319 / 20000000000000000 : ℝ) =
+            (10165542878878319 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10165542878878319 / 10000000000000000 : ℝ) ≤
+            (164187593378291018337 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4934748590307069 / 10000000000000000 : ℝ)) (q :=
+      (10165542878878319 / 20000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_604_11 :
+    Uω (aρ 11) (bρ 11) (5573527036009 / 8000000000000 : ℝ) ≤
+      (-(1964276446295185408741 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4557957078120433 / 10000000000000000 : ℝ) ≤
+      (-(1964276446295185408741 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4557957078120433 / 10000000000000000 : ℝ) =
+            (4557957078120433 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4557957078120433 / 2500000000000000 : ℝ) ≤
+            (1501459455204814591259 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (537949561734327 / 1250000000000000 : ℝ)) (q :=
+      (4557957078120433 / 10000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_604_12 :
+    Uω (aρ 12) (bρ 12) (5573527036009 / 8000000000000 : ℝ) ≤
+      (-(9237298431489610316859 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1588141583857573 / 4000000000000000 : ℝ) ≤
+      (-(9237298431489610316859 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1588141583857573 / 4000000000000000 : ℝ) =
+            (1588141583857573 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1588141583857573 / 1000000000000000 : ℝ) ≤
+            (4625645174510389683141 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (710418070721323 / 2000000000000000 : ℝ)) (q :=
+      (1588141583857573 / 4000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_604_13 :
+    Uω (aρ 13) (bρ 13) (5573527036009 / 8000000000000 : ℝ) ≤
+      (-(440752782161147218229 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6644904481908243 / 20000000000000000 : ℝ) ≤
+      (-(440752782161147218229 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6644904481908243 / 20000000000000000 : ℝ) =
+            (6644904481908243 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6644904481908243 / 5000000000000000 : ℝ) ≤
+            (113764962078852781771 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2655668657821993 / 10000000000000000 : ℝ)) (q :=
+      (6644904481908243 / 20000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_604_14 :
+    Uω (aρ 14) (bρ 14) (5573527036009 / 8000000000000 : ℝ) ≤
+      (-(13569695675472187431089 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1029758998734929 / 4000000000000000 : ℝ) ≤
+      (-(13569695675472187431089 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1029758998734929 / 4000000000000000 : ℝ) =
+            (1029758998734929 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1029758998734929 / 1000000000000000 : ℝ) ≤
+            (293247930527812568911 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (300697290926679 / 2000000000000000 : ℝ)) (q :=
+      (1029758998734929 / 4000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_604_15 :
+    Uω (aρ 15) (bρ 15) (5573527036009 / 8000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_604_16 :
+    Uω (aρ 16) (bρ 16) (5573527036009 / 8000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_604 :
+    Uρ (5573527036009 / 8000000000000 : ℝ) ≤
+      (-(392603247597578016043 / 625000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (5573527036009 / 8000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(185358815803223170541 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_604_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (5573527036009 / 8000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3741861849853575925477 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_604_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (5573527036009 / 8000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(762314428217885278549 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_604_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (5573527036009 / 8000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(392836090377941199433 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_604_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (5573527036009 / 8000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(2054338537821379043431 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_604_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (5573527036009 / 8000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1093155636640941648659 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_604_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (5573527036009 / 8000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(948658707005702538837 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_604_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (5573527036009 / 8000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(5246381739293769484407 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_604_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (5573527036009 / 8000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(591028782044301194203 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_604_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (5573527036009 / 8000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(6767284209621708981663 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_604_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (5573527036009 / 8000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(1964276446295185408741 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_604_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (5573527036009 / 8000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(9237298431489610316859 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_604_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (5573527036009 / 8000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) * (-(440752782161147218229 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_604_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (5573527036009 / 8000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(13569695675472187431089 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_604_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (5573527036009 / 8000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_604_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (5573527036009 / 8000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_604_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_605_1 :
+    Uω (aρ 1) (bρ 1) (4469205467533 / 6400000000000 : ℝ) ≤
+      (-(3683697828136284471879 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27674451831362653 / 40000000000000000 : ℝ) ≤
+      (-(3683697828136284471879 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (27674451831362653 / 40000000000000000 : ℝ) =
+            (27674451831362653 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27674451831362653 / 20000000000000000 : ℝ) ≤
+            (3247773974863715528121 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3459294795373007 / 5000000000000000 : ℝ)) (q :=
+      (27674451831362653 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_605_2 :
+    Uω (aρ 2) (bρ 2) (4469205467533 / 6400000000000 : ℝ) ≤
+      (-(743660286677977302009 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27578853747993111 / 40000000000000000 : ℝ) ≤
+      (-(743660286677977302009 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (27578853747993111 / 40000000000000000 : ℝ) =
+            (27578853747993111 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27578853747993111 / 20000000000000000 : ℝ) ≤
+            (642634073922022697991 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6894559561571243 / 10000000000000000 : ℝ)) (q :=
+      (27578853747993111 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_605_3 :
+    Uω (aρ 3) (bρ 3) (4469205467533 / 6400000000000 : ℝ) ≤
+      (-(236740360101894332047 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27387723832635927 / 40000000000000000 : ℝ) ≤
+      (-(236740360101894332047 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (27387723832635927 / 40000000000000000 : ℝ) =
+            (27387723832635927 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27387723832635927 / 20000000000000000 : ℝ) ≤
+            (196476627585605667953 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6846390710242651 / 10000000000000000 : ℝ)) (q :=
+      (27387723832635927 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_605_4 :
+    Uω (aρ 4) (bρ 4) (4469205467533 / 6400000000000 : ℝ) ≤
+      (-(488044063950332874121 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27070489946191399 / 40000000000000000 : ℝ) ≤
+      (-(488044063950332874121 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (27070489946191399 / 40000000000000000 : ℝ) =
+            (27070489946191399 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27070489946191399 / 20000000000000000 : ℝ) ≤
+            (378389911424667125879 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6766067948085387 / 10000000000000000 : ℝ)) (q :=
+      (27070489946191399 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_605_5 :
+    Uω (aρ 5) (bρ 5) (4469205467533 / 6400000000000 : ℝ) ≤
+      (-(1021055811405140322069 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (26587924041186297 / 40000000000000000 : ℝ) ≤
+      (-(1021055811405140322069 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (26587924041186297 / 40000000000000000 : ℝ) =
+            (26587924041186297 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26587924041186297 / 20000000000000000 : ℝ) ≤
+            (711812139344859677931 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1660743728729459 / 2500000000000000 : ℝ)) (q :=
+      (26587924041186297 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_605_6 :
+    Uω (aρ 6) (bρ 6) (4469205467533 / 6400000000000 : ℝ) ≤
+      (-(4347493813142281048483 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25897076154356201 / 40000000000000000 : ℝ) ≤
+      (-(4347493813142281048483 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (25897076154356201 / 40000000000000000 : ℝ) =
+            (25897076154356201 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25897076154356201 / 20000000000000000 : ℝ) ≤
+            (2583977989857718951517 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1616197836660697 / 2500000000000000 : ℝ)) (q :=
+      (25897076154356201 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_605_7 :
+    Uω (aρ 7) (bρ 7) (4469205467533 / 6400000000000 : ℝ) ≤
+      (-(235858409498393209151 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24957206916533209 / 40000000000000000 : ℝ) ≤
+      (-(235858409498393209151 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (24957206916533209 / 40000000000000000 : ℝ) =
+            (24957206916533209 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24957206916533209 / 20000000000000000 : ℝ) ≤
+            (110715180651606790849 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1554629527122823 / 2500000000000000 : ℝ)) (q :=
+      (24957206916533209 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_605_8 :
+    Uω (aρ 8) (bρ 8) (4469205467533 / 6400000000000 : ℝ) ≤
+      (-(5218804088661256319739 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23736146261667077 / 40000000000000000 : ℝ) ≤
+      (-(5218804088661256319739 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23736146261667077 / 40000000000000000 : ℝ) =
+            (23736146261667077 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23736146261667077 / 20000000000000000 : ℝ) ≤
+            (1712667714338743680261 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2945764716199113 / 5000000000000000 : ℝ)) (q :=
+      (23736146261667077 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_605_9 :
+    Uω (aρ 9) (bρ 9) (4469205467533 / 6400000000000 : ℝ) ≤
+      (-(1470148674486619797487 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22216160712343729 / 40000000000000000 : ℝ) ≤
+      (-(1470148674486619797487 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (22216160712343729 / 40000000000000000 : ℝ) =
+            (22216160712343729 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22216160712343729 / 20000000000000000 : ℝ) ≤
+            (262719276263380202513 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (684062785762069 / 1250000000000000 : ℝ)) (q :=
+      (22216160712343729 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_605_10 :
+    Uω (aρ 10) (bρ 10) (4469205467533 / 6400000000000 : ℝ) ≤
+      (-(3367231395774420070013 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20397924892066461 / 40000000000000000 : ℝ) ≤
+      (-(3367231395774420070013 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (20397924892066461 / 40000000000000000 : ℝ) =
+            (20397924892066461 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20397924892066461 / 20000000000000000 : ℝ) ≤
+            (98504505725579929987 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2475971704726459 / 5000000000000000 : ℝ)) (q :=
+      (20397924892066461 / 40000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_605_11 :
+    Uω (aρ 11) (bρ 11) (4469205467533 / 6400000000000 : ℝ) ≤
+      (-(977435570934079413611 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (18300547855289677 / 40000000000000000 : ℝ) ≤
+      (-(977435570934079413611 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (18300547855289677 / 40000000000000000 : ℝ) =
+            (18300547855289677 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18300547855289677 / 10000000000000000 : ℝ) ≤
+            (755432379815920586389 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2160865758634763 / 5000000000000000 : ℝ)) (q :=
+      (18300547855289677 / 40000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_605_12 :
+    Uω (aρ 12) (bρ 12) (4469205467533 / 6400000000000 : ℝ) ≤
+      (-(4595875065596004995123 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15953917980471991 / 40000000000000000 : ℝ) ≤
+      (-(4595875065596004995123 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (15953917980471991 / 40000000000000000 : ℝ) =
+            (15953917980471991 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15953917980471991 / 10000000000000000 : ℝ) ≤
+            (2335596737403995004877 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3572116676545683 / 10000000000000000 : ℝ)) (q :=
+      (15953917980471991 / 40000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_605_13 :
+    Uω (aρ 13) (bρ 13) (4469205467533 / 6400000000000 : ℝ) ≤
+      (-(10958002953060884909373 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2674175867206639 / 8000000000000000 : ℝ) ≤
+      (-(10958002953060884909373 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2674175867206639 / 8000000000000000 : ℝ) =
+            (2674175867206639 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2674175867206639 / 2000000000000000 : ℝ) ≤
+            (2904940652939115090627 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (535995819184257 / 2000000000000000 : ℝ)) (q :=
+      (2674175867206639 / 8000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_605_14 :
+    Uω (aρ 14) (bρ 14) (4469205467533 / 6400000000000 : ℝ) ≤
+      (-(6731579740574347026997 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10407883059168421 / 40000000000000000 : ℝ) ≤
+      (-(6731579740574347026997 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (10407883059168421 / 40000000000000000 : ℝ) =
+            (10407883059168421 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10407883059168421 / 10000000000000000 : ℝ) ≤
+            (199892062425652973003 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (771204121266949 / 5000000000000000 : ℝ)) (q :=
+      (10407883059168421 / 40000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_605_15 :
+    Uω (aρ 15) (bρ 15) (4469205467533 / 6400000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_605_16 :
+    Uω (aρ 16) (bρ 16) (4469205467533 / 6400000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_605 :
+    Uρ (4469205467533 / 6400000000000 : ℝ) ≤
+      (-(781257351911533191943 / 1250000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (4469205467533 / 6400000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3683697828136284471879 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_605_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (4469205467533 / 6400000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(743660286677977302009 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_605_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (4469205467533 / 6400000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(236740360101894332047 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_605_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (4469205467533 / 6400000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(488044063950332874121 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_605_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (4469205467533 / 6400000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1021055811405140322069 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_605_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (4469205467533 / 6400000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(4347493813142281048483 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_605_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (4469205467533 / 6400000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(235858409498393209151 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_605_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (4469205467533 / 6400000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(5218804088661256319739 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_605_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (4469205467533 / 6400000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(1470148674486619797487 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_605_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (4469205467533 / 6400000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(3367231395774420070013 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_605_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (4469205467533 / 6400000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(977435570934079413611 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_605_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (4469205467533 / 6400000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(4595875065596004995123 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_605_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (4469205467533 / 6400000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(10958002953060884909373 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_605_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (4469205467533 / 6400000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(6731579740574347026997 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_605_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (4469205467533 / 6400000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_605_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (4469205467533 / 6400000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_605_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_606_1 :
+    Uω (aρ 1) (bρ 1) (11198973265647 / 16000000000000 : ℝ) ≤
+      (-(36602743354238826769 / 100000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (693483776051957 / 1000000000000000 : ℝ) ≤
+      (-(36602743354238826769 / 100000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (693483776051957 / 1000000000000000 : ℝ) =
+            (693483776051957 / 500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (693483776051957 / 500000000000000 : ℝ) ≤
+            (32711974675761173231 / 100000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1386962889618953 / 2000000000000000 : ℝ)) (q :=
+      (693483776051957 / 1000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_606_2 :
+    Uω (aρ 2) (bρ 2) (11198973265647 / 16000000000000 : ℝ) ≤
+      (-(1847398199132677266083 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3455469273133667 / 5000000000000000 : ℝ) ≤
+      (-(1847398199132677266083 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (3455469273133667 / 5000000000000000 : ℝ) =
+            (3455469273133667 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3455469273133667 / 2500000000000000 : ℝ) ≤
+            (1618337702367322733917 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6910785032100293 / 10000000000000000 : ℝ)) (q :=
+      (3455469273133667 / 5000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_606_3 :
+    Uω (aρ 3) (bρ 3) (11198973265647 / 16000000000000 : ℝ) ≤
+      (-(47052194398732723493 / 125000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6863156983433939 / 10000000000000000 : ℝ) ≤
+      (-(47052194398732723493 / 125000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (6863156983433939 / 10000000000000000 : ℝ) =
+            (6863156983433939 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6863156983433939 / 5000000000000000 : ℝ) ≤
+            (39591203138767276507 / 125000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6862618012783503 / 10000000000000000 : ℝ)) (q :=
+      (6863156983433939 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_606_4 :
+    Uω (aρ 4) (bρ 4) (11198973265647 / 16000000000000 : ℝ) ≤
+      (-(1940200824024491061991 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6783850953355573 / 10000000000000000 : ℝ) ≤
+      (-(1940200824024491061991 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (6783850953355573 / 10000000000000000 : ℝ) =
+            (6783850953355573 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6783850953355573 / 5000000000000000 : ℝ) ≤
+            (1525535077475508938009 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6782300133691771 / 10000000000000000 : ℝ)) (q :=
+      (6783850953355573 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_606_5 :
+    Uω (aρ 5) (bρ 5) (11198973265647 / 16000000000000 : ℝ) ≤
+      (-(4059829140506769287007 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13326431037813833 / 20000000000000000 : ℝ) ≤
+      (-(4059829140506769287007 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (13326431037813833 / 20000000000000000 : ℝ) =
+            (13326431037813833 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13326431037813833 / 10000000000000000 : ℝ) ≤
+            (2871642662493230712993 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3329609592064729 / 5000000000000000 : ℝ)) (q :=
+      (13326431037813833 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_606_6 :
+    Uω (aρ 6) (bρ 6) (11198973265647 / 16000000000000 : ℝ) ≤
+      (-(2161214125510321744941 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (519241401057381 / 800000000000000 : ℝ) ≤
+      (-(2161214125510321744941 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (519241401057381 / 800000000000000 : ℝ) =
+            (519241401057381 / 400000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (519241401057381 / 400000000000000 : ℝ) ≤
+            (1304521775989678255059 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (129621270957803 / 200000000000000 : ℝ)) (q :=
+      (519241401057381 / 800000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_606_7 :
+    Uω (aρ 7) (bρ 7) (11198973265647 / 16000000000000 : ℝ) ≤
+      (-(4691111374317235430867 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12511161124289557 / 20000000000000000 : ℝ) ≤
+      (-(4691111374317235430867 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (12511161124289557 / 20000000000000000 : ℝ) =
+            (12511161124289557 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12511161124289557 / 10000000000000000 : ℝ) ≤
+            (2240360428682764569133 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3117425513252591 / 5000000000000000 : ℝ)) (q :=
+      (12511161124289557 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_606_8 :
+    Uω (aρ 8) (bρ 8) (11198973265647 / 16000000000000 : ℝ) ≤
+      (-(2595651686485454946769 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11900756100985221 / 20000000000000000 : ℝ) ≤
+      (-(2595651686485454946769 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11900756100985221 / 20000000000000000 : ℝ) =
+            (11900756100985221 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11900756100985221 / 10000000000000000 : ℝ) ≤
+            (870084215014545053231 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2953993827270423 / 5000000000000000 : ℝ)) (q :=
+      (11900756100985221 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_606_9 :
+    Uω (aρ 9) (bρ 9) (11198973265647 / 16000000000000 : ℝ) ≤
+      (-(5850992094238623056519 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2785252971286799 / 5000000000000000 : ℝ) ≤
+      (-(5850992094238623056519 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2785252971286799 / 5000000000000000 : ℝ) =
+            (2785252971286799 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2785252971286799 / 2500000000000000 : ℝ) ≤
+            (1080479708761376943481 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5489209067062821 / 10000000000000000 : ℝ)) (q :=
+      (2785252971286799 / 5000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_606_10 :
+    Uω (aρ 10) (bρ 10) (11198973265647 / 16000000000000 : ℝ) ≤
+      (-(837719390211303050617 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (204647509780861 / 400000000000000 : ℝ) ≤
+      (-(837719390211303050617 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (204647509780861 / 400000000000000 : ℝ) =
+            (204647509780861 / 200000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (204647509780861 / 200000000000000 : ℝ) ≤
+            (28714585163696949383 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (198765268178147 / 400000000000000 : ℝ)) (q :=
+      (204647509780861 / 400000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10]) (by norm_num [bρ_10])
+      (by norm_num) (by norm_num [aρ_10, bρ_10]) (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_606_11 :
+    Uω (aρ 11) (bρ 11) (11198973265647 / 16000000000000 : ℝ) ≤
+      (-(7782020821850845249701 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9184618574910929 / 20000000000000000 : ℝ) ≤
+      (-(7782020821850845249701 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9184618574910929 / 20000000000000000 : ℝ) =
+            (9184618574910929 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9184618574910929 / 5000000000000000 : ℝ) ≤
+            (6080922784149154750299 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2169925708263277 / 5000000000000000 : ℝ)) (q :=
+      (9184618574910929 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_606_12 :
+    Uω (aρ 12) (bρ 12) (11198973265647 / 16000000000000 : ℝ) ≤
+      (-(2286614058101796570093 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1602634339274003 / 4000000000000000 : ℝ) ≤
+      (-(2286614058101796570093 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1602634339274003 / 4000000000000000 : ℝ) =
+            (1602634339274003 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1602634339274003 / 1000000000000000 : ℝ) ≤
+            (1179121843398203429907 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (44901307933383 / 125000000000000 : ℝ)) (q :=
+      (1602634339274003 / 4000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_606_13 :
+    Uω (aρ 13) (bρ 13) (11198973265647 / 16000000000000 : ℝ) ≤
+      (-(5448867091589979705297 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6725853653235283 / 20000000000000000 : ℝ) ≤
+      (-(5448867091589979705297 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6725853653235283 / 20000000000000000 : ℝ) =
+            (6725853653235283 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6725853653235283 / 5000000000000000 : ℝ) ≤
+            (1482604711410020294703 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (676042083282727 / 2500000000000000 : ℝ)) (q :=
+      (6725853653235283 / 20000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_606_14 :
+    Uω (aρ 14) (bρ 14) (11198973265647 / 16000000000000 : ℝ) ≤
+      (-(13359251881499432432203 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (657287042718211 / 2500000000000000 : ℝ) ≤
+      (-(13359251881499432432203 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (657287042718211 / 2500000000000000 : ℝ) =
+            (657287042718211 / 625000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (657287042718211 / 625000000000000 : ℝ) ≤
+            (503691724500567567797 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1580538306686313 / 10000000000000000 : ℝ)) (q :=
+      (657287042718211 / 2500000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14]) (by norm_num [bρ_14])
+      (by norm_num) (by norm_num [aρ_14, bρ_14]) (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_606_15 :
+    Uω (aρ 15) (bρ 15) (11198973265647 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_606_16 :
+    Uω (aρ 16) (bρ 16) (11198973265647 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_606 :
+    Uρ (11198973265647 / 16000000000000 : ℝ) ≤
+      (-(1554673099640420038419 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (11198973265647 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(36602743354238826769 / 100000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_606_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (11198973265647 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1847398199132677266083 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_606_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (11198973265647 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) * (-(47052194398732723493 / 125000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_606_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (11198973265647 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1940200824024491061991 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_606_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (11198973265647 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(4059829140506769287007 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_606_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (11198973265647 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(2161214125510321744941 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_606_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (11198973265647 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(4691111374317235430867 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_606_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (11198973265647 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(2595651686485454946769 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_606_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (11198973265647 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(5850992094238623056519 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_606_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (11198973265647 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(837719390211303050617 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_606_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (11198973265647 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(7782020821850845249701 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_606_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (11198973265647 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(2286614058101796570093 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_606_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (11198973265647 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(5448867091589979705297 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_606_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (11198973265647 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(13359251881499432432203 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_606_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (11198973265647 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_606_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (11198973265647 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_606_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_607_1 :
+    Uω (aρ 1) (bρ 1) (22449865724923 / 32000000000000 : ℝ) ≤
+      (-(727381116177924738757 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27804250251773043 / 40000000000000000 : ℝ) ≤
+      (-(727381116177924738757 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (27804250251773043 / 40000000000000000 : ℝ) =
+            (27804250251773043 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27804250251773043 / 20000000000000000 : ℝ) ≤
+            (658913244422075261243 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1737759826233271 / 2500000000000000 : ℝ)) (q :=
+      (27804250251773043 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_607_2 :
+    Uω (aρ 2) (bρ 2) (22449865724923 / 32000000000000 : ℝ) ≤
+      (-(1835673242359555576149 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27708654615376139 / 40000000000000000 : ℝ) ≤
+      (-(1835673242359555576149 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (27708654615376139 / 40000000000000000 : ℝ) =
+            (27708654615376139 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27708654615376139 / 20000000000000000 : ℝ) ≤
+            (1630062659140444423851 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (865876312405579 / 1250000000000000 : ℝ)) (q :=
+      (27708654615376139 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_607_3 :
+    Uω (aρ 3) (bρ 3) (22449865724923 / 32000000000000 : ℝ) ≤
+      (-(748112249305770301769 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27517532010732761 / 40000000000000000 : ℝ) ≤
+      (-(748112249305770301769 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (27517532010732761 / 40000000000000000 : ℝ) =
+            (27517532010732761 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27517532010732761 / 20000000000000000 : ℝ) ≤
+            (638182111294229698231 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6878845303272943 / 10000000000000000 : ℝ)) (q :=
+      (27517532010732761 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_607_4 :
+    Uω (aρ 4) (bρ 4) (22449865724923 / 32000000000000 : ℝ) ≤
+      (-(3856508037953090883837 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5440063521927477 / 8000000000000000 : ℝ) ≤
+      (-(3856508037953090883837 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5440063521927477 / 8000000000000000 : ℝ) =
+            (5440063521927477 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5440063521927477 / 4000000000000000 : ℝ) ≤
+            (3074963765046909116163 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1359706456758051 / 2000000000000000 : ℝ)) (q :=
+      (5440063521927477 / 8000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_607_5 :
+    Uω (aρ 5) (bρ 5) (22449865724923 / 32000000000000 : ℝ) ≤
+      (-(4035494468925474550957 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (26717799920169329 / 40000000000000000 : ℝ) ≤
+      (-(4035494468925474550957 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (26717799920169329 / 40000000000000000 : ℝ) =
+            (26717799920169329 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26717799920169329 / 20000000000000000 : ℝ) ≤
+            (2895977334074525449043 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6675463358391227 / 10000000000000000 : ℝ)) (q :=
+      (26717799920169329 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_607_6 :
+    Uω (aρ 6) (bρ 6) (22449865724923 / 32000000000000 : ℝ) ≤
+      (-(4297425542463654092731 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (26027063476699487 / 40000000000000000 : ℝ) ≤
+      (-(4297425542463654092731 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (26027063476699487 / 40000000000000000 : ℝ) =
+            (26027063476699487 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26027063476699487 / 20000000000000000 : ℝ) ≤
+            (2634046260536345907269 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3248667755898153 / 5000000000000000 : ℝ)) (q :=
+      (26027063476699487 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_607_7 :
+    Uω (aρ 7) (bρ 7) (22449865724923 / 32000000000000 : ℝ) ≤
+      (-(466512272714342381551 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25087436453874287 / 40000000000000000 : ℝ) ≤
+      (-(466512272714342381551 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (25087436453874287 / 40000000000000000 : ℝ) =
+            (25087436453874287 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25087436453874287 / 20000000000000000 : ℝ) ≤
+            (226634907585657618449 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3125591690571853 / 5000000000000000 : ℝ)) (q :=
+      (25087436453874287 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_607_8 :
+    Uω (aρ 8) (bρ 8) (22449865724923 / 32000000000000 : ℝ) ≤
+      (-(1032775831632319691089 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23866875566293157 / 40000000000000000 : ℝ) ≤
+      (-(1032775831632319691089 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23866875566293157 / 40000000000000000 : ℝ) =
+            (23866875566293157 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23866875566293157 / 20000000000000000 : ℝ) ≤
+            (353518528967680308911 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5924444588693141 / 10000000000000000 : ℝ)) (q :=
+      (23866875566293157 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_607_9 :
+    Uω (aρ 9) (bρ 9) (22449865724923 / 32000000000000 : ℝ) ≤
+      (-(29107397217795261913 / 50000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22347881062052147 / 40000000000000000 : ℝ) ≤
+      (-(29107397217795261913 / 50000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (22347881062052147 / 40000000000000000 : ℝ) =
+            (22347881062052147 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22347881062052147 / 20000000000000000 : ℝ) ≤
+            (5549961797204738087 / 50000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1376478241233159 / 2500000000000000 : ℝ)) (q :=
+      (22347881062052147 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_607_10 :
+    Uω (aρ 10) (bρ 10) (22449865724923 / 32000000000000 : ℝ) ≤
+      (-(666916037184109468323 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20531564150751633 / 40000000000000000 : ℝ) ≤
+      (-(666916037184109468323 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (20531564150751633 / 40000000000000000 : ℝ) =
+            (20531564150751633 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20531564150751633 / 20000000000000000 : ℝ) ≤
+            (26231143115890531677 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4986313542777379 / 10000000000000000 : ℝ)) (q :=
+      (20531564150751633 / 40000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_607_11 :
+    Uω (aρ 11) (bρ 11) (22449865724923 / 32000000000000 : ℝ) ≤
+      (-(193617827665090779411 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (18437896573385211 / 40000000000000000 : ℝ) ≤
+      (-(193617827665090779411 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (18437896573385211 / 40000000000000000 : ℝ) =
+            (18437896573385211 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18437896573385211 / 10000000000000000 : ℝ) ≤
+            (152955762484909220589 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (136186136884349 / 312500000000000 : ℝ)) (q :=
+      (18437896573385211 / 40000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_607_12 :
+    Uω (aρ 12) (bρ 12) (22449865724923 / 32000000000000 : ℝ) ≤
+      (-(9101413433336903069591 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16098693349171741 / 40000000000000000 : ℝ) ≤
+      (-(9101413433336903069591 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (16098693349171741 / 40000000000000000 : ℝ) =
+            (16098693349171741 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16098693349171741 / 10000000000000000 : ℝ) ≤
+            (4761530172663096930409 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3612054864877433 / 10000000000000000 : ℝ)) (q :=
+      (16098693349171741 / 40000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_607_13 :
+    Uω (aρ 13) (bρ 13) (22449865724923 / 32000000000000 : ℝ) ≤
+      (-(2709500208247898408319 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13532299322712177 / 40000000000000000 : ℝ) ≤
+      (-(2709500208247898408319 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (13532299322712177 / 40000000000000000 : ℝ) =
+            (13532299322712177 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13532299322712177 / 10000000000000000 : ℝ) ≤
+            (756235693252101591681 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2728239593242651 / 10000000000000000 : ℝ)) (q :=
+      (13532299322712177 / 40000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_607_14 :
+    Uω (aρ 14) (bρ 14) (22449865724923 / 32000000000000 : ℝ) ≤
+      (-(1325779785845717507177 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10623830814298537 / 40000000000000000 : ℝ) ≤
+      (-(1325779785845717507177 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (10623830814298537 / 40000000000000000 : ℝ) =
+            (10623830814298537 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10623830814298537 / 10000000000000000 : ℝ) ≤
+            (60514574754282492823 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1617932624080831 / 10000000000000000 : ℝ)) (q :=
+      (10623830814298537 / 40000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_607_15 :
+    Uω (aρ 15) (bρ 15) (22449865724923 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_607_16 :
+    Uω (aρ 16) (bρ 16) (22449865724923 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_607 :
+    Uρ (22449865724923 / 32000000000000 : ℝ) ≤
+      (-(6187543587831002428899 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (22449865724923 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(727381116177924738757 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_607_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (22449865724923 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1835673242359555576149 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_607_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (22449865724923 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(748112249305770301769 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_607_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (22449865724923 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3856508037953090883837 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_607_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (22449865724923 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(4035494468925474550957 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_607_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (22449865724923 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(4297425542463654092731 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_607_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (22449865724923 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(466512272714342381551 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_607_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (22449865724923 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(1032775831632319691089 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_607_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (22449865724923 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(29107397217795261913 / 50000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_607_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (22449865724923 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(666916037184109468323 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_607_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (22449865724923 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) * (-(193617827665090779411 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_607_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (22449865724923 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(9101413433336903069591 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_607_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (22449865724923 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(2709500208247898408319 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_607_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (22449865724923 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(1325779785845717507177 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_607_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (22449865724923 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_607_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (22449865724923 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_607_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_608_1 :
+    Uω (aρ 1) (bρ 1) (2812723114819 / 4000000000000 : ℝ) ≤
+      (-(3613591309293704288903 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6967287365113519 / 10000000000000000 : ℝ) ≤
+      (-(3613591309293704288903 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (6967287365113519 / 10000000000000000 : ℝ) =
+            (6967287365113519 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6967287365113519 / 5000000000000000 : ℝ) ≤
+            (3317880493706295711097 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3483632080632269 / 5000000000000000 : ℝ)) (q :=
+      (6967287365113519 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_608_2 :
+    Uω (aρ 2) (bρ 2) (2812723114819 / 4000000000000 : ℝ) ≤
+      (-(1823975717406834395501 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13886777519480489 / 20000000000000000 : ℝ) ≤
+      (-(1823975717406834395501 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (13886777519480489 / 20000000000000000 : ℝ) =
+            (13886777519480489 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13886777519480489 / 10000000000000000 : ℝ) ≤
+            (1641760184093165604499 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6943235963027989 / 10000000000000000 : ℝ)) (q :=
+      (13886777519480489 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_608_3 :
+    Uω (aρ 3) (bρ 3) (2812723114819 / 4000000000000 : ℝ) ≤
+      (-(29039082672124137669 / 78125000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (43097556349683 / 62500000000000 : ℝ) ≤
+      (-(29039082672124137669 / 78125000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (43097556349683 / 62500000000000 : ℝ) = (43097556349683 / 31250000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (43097556349683 / 31250000000000 : ℝ) ≤
+            (25113040788813362331 / 78125000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (344753629089803 / 500000000000000 : ℝ)) (q :=
+      (43097556349683 / 62500000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_608_4 :
+    Uω (aρ 4) (bρ 4) (2812723114819 / 4000000000000 : ℝ) ≤
+      (-(479083926015013648739 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13632615667672069 / 20000000000000000 : ℝ) ≤
+      (-(479083926015013648739 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (13632615667672069 / 20000000000000000 : ℝ) =
+            (13632615667672069 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13632615667672069 / 10000000000000000 : ℝ) ≤
+            (387350049359986351261 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6814764398634569 / 10000000000000000 : ℝ)) (q :=
+      (13632615667672069 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_608_5 :
+    Uω (aρ 5) (bρ 5) (2812723114819 / 4000000000000 : ℝ) ≤
+      (-(2005609470815187270507 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (83696054925607 / 125000000000000 : ℝ) ≤
+      (-(2005609470815187270507 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (83696054925607 / 125000000000000 : ℝ) = (83696054925607 / 62500000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (83696054925607 / 62500000000000 : ℝ) ≤
+            (1460126430684812729493 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (334585371919731 / 500000000000000 : ℝ)) (q :=
+      (83696054925607 / 125000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_608_6 :
+    Uω (aρ 6) (bρ 6) (2812723114819 / 4000000000000 : ℝ) ≤
+      (-(4272485372134123980927 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13046028214702497 / 20000000000000000 : ℝ) ≤
+      (-(4272485372134123980927 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (13046028214702497 / 20000000000000000 : ℝ) =
+            (13046028214702497 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13046028214702497 / 10000000000000000 : ℝ) ≤
+            (2658986430865876019073 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6513607240139997 / 10000000000000000 : ℝ)) (q :=
+      (13046028214702497 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_608_7 :
+    Uω (aρ 7) (bρ 7) (2812723114819 / 4000000000000 : ℝ) ≤
+      (-(4639201890375419235181 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6288137385306809 / 10000000000000000 : ℝ) ≤
+      (-(4639201890375419235181 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (6288137385306809 / 10000000000000000 : ℝ) =
+            (6288137385306809 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6288137385306809 / 5000000000000000 : ℝ) ≤
+            (2292269912624580764819 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3133757588405559 / 5000000000000000 : ℝ)) (q :=
+      (6288137385306809 / 10000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_608_8 :
+    Uω (aρ 8) (bρ 8) (2812723114819 / 4000000000000 : ℝ) ≤
+      (-(1027306202776836780947 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11966118188021229 / 20000000000000000 : ℝ) ≤
+      (-(1027306202776836780947 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11966118188021229 / 20000000000000000 : ℝ) =
+            (11966118188021229 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11966118188021229 / 10000000000000000 : ℝ) ≤
+            (358988157823163219053 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5940900245558729 / 10000000000000000 : ℝ)) (q :=
+      (11966118188021229 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_608_9 :
+    Uω (aρ 9) (bρ 9) (2812723114819 / 4000000000000 : ℝ) ≤
+      (-(579205618557146584257 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2801716579992363 / 5000000000000000 : ℝ) ≤
+      (-(579205618557146584257 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2801716579992363 / 5000000000000000 : ℝ) =
+            (2801716579992363 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2801716579992363 / 2500000000000000 : ℝ) ≤
+            (113941561742853415743 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (690326750733369 / 1250000000000000 : ℝ)) (q :=
+      (2801716579992363 / 5000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_608_10 :
+    Uω (aρ 10) (bρ 10) (2812723114819 / 4000000000000 : ℝ) ≤
+      (-(3318338861609870053377 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (643698891971729 / 1250000000000000 : ℝ) ≤
+      (-(3318338861609870053377 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (643698891971729 / 1250000000000000 : ℝ) =
+            (643698891971729 / 625000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (643698891971729 / 625000000000000 : ℝ) ≤
+            (147397039890129946623 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1250872247735041 / 2500000000000000 : ℝ)) (q :=
+      (643698891971729 / 1250000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10]) (by norm_num [bρ_10])
+      (by norm_num) (by norm_num [aρ_10, bρ_10]) (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_608_11 :
+    Uω (aρ 11) (bρ 11) (2812723114819 / 4000000000000 : ℝ) ≤
+      (-(481722500072513733311 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9253263248367061 / 20000000000000000 : ℝ) ≤
+      (-(481722500072513733311 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9253263248367061 / 20000000000000000 : ℝ) =
+            (9253263248367061 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9253263248367061 / 5000000000000000 : ℝ) ≤
+            (384711475302486266689 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4376046593964561 / 10000000000000000 : ℝ)) (q :=
+      (9253263248367061 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_608_12 :
+    Uω (aρ 12) (bρ 12) (2812723114819 / 4000000000000 : ℝ) ≤
+      (-(4528309251245035771429 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8085484546596681 / 20000000000000000 : ℝ) ≤
+      (-(4528309251245035771429 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8085484546596681 / 20000000000000000 : ℝ) =
+            (8085484546596681 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8085484546596681 / 5000000000000000 : ℝ) ≤
+            (2403162551754964228571 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3631967988879181 / 10000000000000000 : ℝ)) (q :=
+      (8085484546596681 / 20000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_608_13 :
+    Uω (aρ 13) (bρ 13) (2812723114819 / 4000000000000 : ℝ) ≤
+      (-(538939547868169931167 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3403165393967557 / 10000000000000000 : ℝ) ≤
+      (-(538939547868169931167 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3403165393967557 / 10000000000000000 : ℝ) =
+            (3403165393967557 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3403165393967557 / 2500000000000000 : ℝ) ≤
+            (154207632431830068833 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1376097985906307 / 5000000000000000 : ℝ)) (q :=
+      (3403165393967557 / 10000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_608_14 :
+    Uω (aρ 14) (bρ 14) (2812723114819 / 4000000000000 : ℝ) ≤
+      (-(13158641155386919367299 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (536484860941541 / 2000000000000000 : ℝ) ≤
+      (-(13158641155386919367299 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (536484860941541 / 2000000000000000 : ℝ) =
+            (536484860941541 / 500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (536484860941541 / 500000000000000 : ℝ) ≤
+            (704302450613080632701 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (165464107833791 / 1000000000000000 : ℝ)) (q :=
+      (536484860941541 / 2000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14]) (by norm_num [bρ_14])
+      (by norm_num) (by norm_num [aρ_14, bρ_14]) (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_608_15 :
+    Uω (aρ 15) (bρ 15) (2812723114819 / 4000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_608_16 :
+    Uω (aρ 16) (bρ 16) (2812723114819 / 4000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_608 :
+    Uρ (2812723114819 / 4000000000000 : ℝ) ≤
+      (-(6156604129477609587247 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (2812723114819 / 4000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3613591309293704288903 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_608_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (2812723114819 / 4000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1823975717406834395501 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_608_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (2812723114819 / 4000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) * (-(29039082672124137669 / 78125000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_608_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (2812723114819 / 4000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(479083926015013648739 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_608_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (2812723114819 / 4000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(2005609470815187270507 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_608_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (2812723114819 / 4000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(4272485372134123980927 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_608_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (2812723114819 / 4000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(4639201890375419235181 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_608_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (2812723114819 / 4000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(1027306202776836780947 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_608_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (2812723114819 / 4000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(579205618557146584257 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_608_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (2812723114819 / 4000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(3318338861609870053377 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_608_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (2812723114819 / 4000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) * (-(481722500072513733311 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_608_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (2812723114819 / 4000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(4528309251245035771429 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_608_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (2812723114819 / 4000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) * (-(538939547868169931167 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_608_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (2812723114819 / 4000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(13158641155386919367299 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_608_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (2812723114819 / 4000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_608_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (2812723114819 / 4000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_608_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_609_1 :
+    Uω (aρ 1) (bρ 1) (22553704112181 / 32000000000000 : ℝ) ≤
+      (-(718066253435485445603 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27934048668128443 / 40000000000000000 : ℝ) ≤
+      (-(718066253435485445603 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (27934048668128443 / 40000000000000000 : ℝ) =
+            (27934048668128443 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27934048668128443 / 20000000000000000 : ℝ) ≤
+            (668228107164514554397 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6983489017092659 / 10000000000000000 : ℝ)) (q :=
+      (27934048668128443 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_609_2 :
+    Uω (aρ 2) (bρ 2) (22553704112181 / 32000000000000 : ℝ) ≤
+      (-(1812305496208947558473 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5567691091174173 / 8000000000000000 : ℝ) ≤
+      (-(1812305496208947558473 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5567691091174173 / 8000000000000000 : ℝ) =
+            (5567691091174173 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5567691091174173 / 4000000000000000 : ℝ) ≤
+            (1653430405291052441527 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (695946142347387 / 1000000000000000 : ℝ)) (q :=
+      (5567691091174173 / 8000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_609_3 :
+    Uω (aρ 3) (bρ 3) (22553704112181 / 32000000000000 : ℝ) ≤
+      (-(1846749648388671706299 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27647340093097407 / 40000000000000000 : ℝ) ≤
+      (-(1846749648388671706299 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (27647340093097407 / 40000000000000000 : ℝ) =
+            (27647340093097407 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27647340093097407 / 20000000000000000 : ℝ) ≤
+            (1618986253111328293701 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6911299848437141 / 10000000000000000 : ℝ)) (q :=
+      (27647340093097407 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_609_4 :
+    Uω (aρ 4) (bρ 4) (22553704112181 / 32000000000000 : ℝ) ≤
+      (-(119027858978336842039 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27330144991045187 / 40000000000000000 : ℝ) ≤
+      (-(119027858978336842039 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (27330144991045187 / 40000000000000000 : ℝ) =
+            (27330144991045187 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27330144991045187 / 20000000000000000 : ℝ) ≤
+            (97580634865413157961 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6830996478476031 / 10000000000000000 : ℝ)) (q :=
+      (27330144991045187 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_609_5 :
+    Uω (aρ 5) (bρ 5) (22553704112181 / 32000000000000 : ℝ) ≤
+      (-(797400454296810172779 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (26847675045071939 / 40000000000000000 : ℝ) ≤
+      (-(797400454296810172779 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (26847675045071939 / 40000000000000000 : ℝ) =
+            (26847675045071939 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26847675045071939 / 20000000000000000 : ℝ) ≤
+            (588893906303189827221 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6707951424824407 / 10000000000000000 : ℝ)) (q :=
+      (26847675045071939 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_609_6 :
+    Uω (aρ 6) (bρ 6) (22553704112181 / 32000000000000 : ℝ) ≤
+      (-(212380371353436732361 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5231409782901509 / 8000000000000000 : ℝ) ≤
+      (-(212380371353436732361 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5231409782901509 / 8000000000000000 : ℝ) =
+            (5231409782901509 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5231409782901509 / 4000000000000000 : ℝ) ≤
+            (134193218796563267639 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (652987873468221 / 1000000000000000 : ℝ)) (q :=
+      (5231409782901509 / 8000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_609_7 :
+    Uω (aρ 7) (bρ 7) (22553704112181 / 32000000000000 : ℝ) ≤
+      (-(2306674254387015104923 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25217661519354599 / 40000000000000000 : ℝ) ≤
+      (-(2306674254387015104923 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (25217661519354599 / 40000000000000000 : ℝ) =
+            (25217661519354599 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25217661519354599 / 20000000000000000 : ℝ) ≤
+            (1159061647112984895077 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6283846417865737 / 10000000000000000 : ℝ)) (q :=
+      (25217661519354599 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_609_8 :
+    Uω (aρ 8) (bρ 8) (22553704112181 / 32000000000000 : ℝ) ≤
+      (-(5109258513458795500777 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23997594652387391 / 40000000000000000 : ℝ) ≤
+      (-(5109258513458795500777 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23997594652387391 / 40000000000000000 : ℝ) =
+            (23997594652387391 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23997594652387391 / 20000000000000000 : ℝ) ≤
+            (1822213289541204499223 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5957354635722133 / 10000000000000000 : ℝ)) (q :=
+      (23997594652387391 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_609_9 :
+    Uω (aρ 9) (bρ 9) (22553704112181 / 32000000000000 : ℝ) ≤
+      (-(720340220661621150801 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22479578555636669 / 40000000000000000 : ℝ) ≤
+      (-(720340220661621150801 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (22479578555636669 / 40000000000000000 : ℝ) =
+            (22479578555636669 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22479578555636669 / 20000000000000000 : ℝ) ≤
+            (146093754713378849199 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1384828053926693 / 2500000000000000 : ℝ)) (q :=
+      (22479578555636669 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_609_10 :
+    Uω (aρ 10) (bρ 10) (22553704112181 / 32000000000000 : ℝ) ≤
+      (-(1651076591580467623333 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (826606091451197 / 1600000000000000 : ℝ) ≤
+      (-(1651076591580467623333 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (826606091451197 / 1600000000000000 : ℝ) =
+            (826606091451197 / 800000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (826606091451197 / 800000000000000 : ℝ) ≤
+            (81791359169532376667 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (25103290572617 / 50000000000000 : ℝ)) (q :=
+      (826606091451197 / 1600000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10]) (by norm_num [bρ_10])
+      (by norm_num) (by norm_num [aρ_10, bρ_10]) (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_609_11 :
+    Uω (aρ 11) (bρ 11) (22553704112181 / 32000000000000 : ℝ) ≤
+      (-(3835280052829488580907 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (18575127284217941 / 40000000000000000 : ℝ) ≤
+      (-(3835280052829488580907 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (18575127284217941 / 40000000000000000 : ℝ) =
+            (18575127284217941 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18575127284217941 / 10000000000000000 : ℝ) ≤
+            (3096191750170511419093 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (8582269999409 / 19531250000000 : ℝ)) (q :=
+      (18575127284217941 / 40000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_609_12 :
+    Uω (aρ 12) (bρ 12) (22553704112181 / 32000000000000 : ℝ) ≤
+      (-(2253017069146923309059 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16243171838822357 / 40000000000000000 : ℝ) ≤
+      (-(2253017069146923309059 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (16243171838822357 / 40000000000000000 : ℝ) =
+            (16243171838822357 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16243171838822357 / 10000000000000000 : ℝ) ≤
+            (1212718832353076690941 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (456480576710577 / 1250000000000000 : ℝ)) (q :=
+      (16243171838822357 / 40000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_609_13 :
+    Uω (aρ 13) (bρ 13) (22553704112181 / 32000000000000 : ℝ) ≤
+      (-(2144018610618247468463 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13692800014302969 / 40000000000000000 : ℝ) ≤
+      (-(2144018610618247468463 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (13692800014302969 / 40000000000000000 : ℝ) =
+            (13692800014302969 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13692800014302969 / 10000000000000000 : ℝ) ≤
+            (628570110581752531537 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1388020221509961 / 5000000000000000 : ℝ)) (q :=
+      (13692800014302969 / 40000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_609_14 :
+    Uω (aρ 14) (bρ 14) (22553704112181 / 32000000000000 : ℝ) ≤
+      (-(6530820785372481273627 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10834281246531611 / 40000000000000000 : ℝ) ≤
+      (-(6530820785372481273627 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (10834281246531611 / 40000000000000000 : ℝ) =
+            (10834281246531611 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10834281246531611 / 10000000000000000 : ℝ) ≤
+            (400651017627518726373 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1690708344179243 / 10000000000000000 : ℝ)) (q :=
+      (10834281246531611 / 40000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_609_15 :
+    Uω (aρ 15) (bρ 15) (22553704112181 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_609_16 :
+    Uω (aρ 16) (bρ 16) (22553704112181 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_609 :
+    Uρ (22553704112181 / 32000000000000 : ℝ) ≤
+      (-(6125866517931195617109 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (22553704112181 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(718066253435485445603 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_609_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (22553704112181 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1812305496208947558473 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_609_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (22553704112181 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1846749648388671706299 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_609_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (22553704112181 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(119027858978336842039 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_609_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (22553704112181 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(797400454296810172779 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_609_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (22553704112181 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(212380371353436732361 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_609_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (22553704112181 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(2306674254387015104923 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_609_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (22553704112181 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(5109258513458795500777 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_609_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (22553704112181 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(720340220661621150801 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_609_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (22553704112181 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(1651076591580467623333 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_609_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (22553704112181 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(3835280052829488580907 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_609_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (22553704112181 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(2253017069146923309059 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_609_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (22553704112181 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(2144018610618247468463 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_609_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (22553704112181 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(6530820785372481273627 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_609_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (22553704112181 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_609_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (22553704112181 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_609_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_610_1 :
+    Uω (aρ 1) (bρ 1) (2260562330581 / 3200000000000 : ℝ) ≤
+      (-(3567125202846665950607 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3499868484350393 / 5000000000000000 : ℝ) ≤
+      (-(3567125202846665950607 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (3499868484350393 / 5000000000000000 : ℝ) =
+            (3499868484350393 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3499868484350393 / 2500000000000000 : ℝ) ≤
+            (3364346600153334049393 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6999713872420947 / 10000000000000000 : ℝ)) (q :=
+      (3499868484350393 / 5000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_610_2 :
+    Uω (aρ 2) (bρ 2) (2260562330581 / 3200000000000 : ℝ) ≤
+      (-(720264980638036197189 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13951677933076191 / 20000000000000000 : ℝ) ≤
+      (-(720264980638036197189 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (13951677933076191 / 20000000000000000 : ℝ) =
+            (13951677933076191 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13951677933076191 / 10000000000000000 : ℝ) ≤
+            (666029379961963802811 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3487843440302783 / 5000000000000000 : ℝ)) (q :=
+      (13951677933076191 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_610_3 :
+    Uω (aρ 3) (bρ 3) (2260562330581 / 3200000000000 : ℝ) ≤
+      (-(146802045239044294797 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13856122049400311 / 20000000000000000 : ℝ) ≤
+      (-(146802045239044294797 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (13856122049400311 / 20000000000000000 : ℝ) =
+            (13856122049400311 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13856122049400311 / 10000000000000000 : ℝ) ≤
+            (130456826880955705203 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3463763551639843 / 5000000000000000 : ℝ)) (q :=
+      (13856122049400311 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_610_4 :
+    Uω (aρ 4) (bρ 4) (2260562330581 / 3200000000000 : ℝ) ≤
+      (-(1892584003101455628531 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8560955805387 / 12500000000000 : ℝ) ≤
+      (-(1892584003101455628531 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (8560955805387 / 12500000000000 : ℝ) = (8560955805387 / 6250000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8560955805387 / 6250000000000 : ℝ) ≤
+            (1573151898398544371469 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (273889140942543 / 400000000000000 : ℝ)) (q :=
+      (8560955805387 / 12500000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_610_5 :
+    Uω (aρ 5) (bρ 5) (2260562330581 / 3200000000000 : ℝ) ≤
+      (-(3962844173437486658913 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13456306164079367 / 20000000000000000 : ℝ) ≤
+      (-(3962844173437486658913 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (13456306164079367 / 20000000000000000 : ℝ) =
+            (13456306164079367 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13456306164079367 / 10000000000000000 : ℝ) ≤
+            (2968627629562513341087 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3362097659179371 / 5000000000000000 : ℝ)) (q :=
+      (13456306164079367 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_610_6 :
+    Uω (aρ 6) (bρ 6) (2260562330581 / 3200000000000 : ℝ) ≤
+      (-(2111395698327103507627 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (655551023387351 / 1000000000000000 : ℝ) ≤
+      (-(2111395698327103507627 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (655551023387351 / 1000000000000000 : ℝ) =
+            (655551023387351 / 500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (655551023387351 / 500000000000000 : ℝ) ≤
+            (1354340203172896492373 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1309229999433279 / 2000000000000000 : ℝ)) (q :=
+      (655551023387351 / 1000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_610_7 :
+    Uω (aρ 7) (bρ 7) (2260562330581 / 3200000000000 : ℝ) ≤
+      (-(4587562229901919172883 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1264138619844117 / 2000000000000000 : ℝ) ≤
+      (-(4587562229901919172883 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1264138619844117 / 2000000000000000 : ℝ) =
+            (1264138619844117 / 1000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1264138619844117 / 1000000000000000 : ℝ) ≤
+            (2343909573098080827117 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1260035421724109 / 2000000000000000 : ℝ)) (q :=
+      (1264138619844117 / 2000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_610_8 :
+    Uω (aρ 8) (bρ 8) (2260562330581 / 3200000000000 : ℝ) ≤
+      (-(2541030616916354549827 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12031475208131079 / 20000000000000000 : ℝ) ≤
+      (-(2541030616916354549827 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (12031475208131079 / 20000000000000000 : ℝ) =
+            (12031475208131079 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12031475208131079 / 10000000000000000 : ℝ) ≤
+            (924705284583645450173 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2986903884825227 / 5000000000000000 : ℝ)) (q :=
+      (12031475208131079 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_610_9 :
+    Uω (aρ 9) (bρ 9) (2260562330581 / 3200000000000 : ℝ) ≤
+      (-(179171113532017318017 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5636354715049359 / 10000000000000000 : ℝ) ≤
+      (-(179171113532017318017 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5636354715049359 / 10000000000000000 : ℝ) =
+            (5636354715049359 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5636354715049359 / 5000000000000000 : ℝ) ≤
+            (37437380311732681983 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5556007619978093 / 10000000000000000 : ℝ)) (q :=
+      (5636354715049359 / 10000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_610_10 :
+    Uω (aρ 10) (bρ 10) (2260562330581 / 3200000000000 : ℝ) ≤
+      (-(821505687596867965743 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10365963754816079 / 20000000000000000 : ℝ) ≤
+      (-(821505687596867965743 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10365963754816079 / 20000000000000000 : ℝ) =
+            (10365963754816079 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10365963754816079 / 10000000000000000 : ℝ) ≤
+            (44928287778132034257 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2518910489095227 / 5000000000000000 : ℝ)) (q :=
+      (10365963754816079 / 20000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_610_11 :
+    Uω (aρ 11) (bρ 11) (2260562330581 / 3200000000000 : ℝ) ≤
+      (-(7633712040526915893759 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2330462411740549 / 5000000000000000 : ℝ) ≤
+      (-(7633712040526915893759 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2330462411740549 / 5000000000000000 : ℝ) =
+            (2330462411740549 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2330462411740549 / 1250000000000000 : ℝ) ≤
+            (6229231565473084106241 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4412183496541571 / 10000000000000000 : ℝ)) (q :=
+      (2330462411740549 / 5000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_610_12 :
+    Uω (aρ 12) (bρ 12) (2260562330581 / 3200000000000 : ℝ) ≤
+      (-(8967759658547838351263 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4078825692896653 / 10000000000000000 : ℝ) ≤
+      (-(8967759658547838351263 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4078825692896653 / 10000000000000000 : ℝ) =
+            (4078825692896653 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4078825692896653 / 2500000000000000 : ℝ) ≤
+            (4895183947452161648737 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3671685332057681 / 10000000000000000 : ℝ)) (q :=
+      (4078825692896653 / 10000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_610_13 :
+    Uω (aρ 13) (bρ 13) (2260562330581 / 3200000000000 : ℝ) ≤
+      (-(10661896036179101416151 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1721590044551353 / 5000000000000000 : ℝ) ≤
+      (-(10661896036179101416151 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1721590044551353 / 5000000000000000 : ℝ) =
+            (1721590044551353 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1721590044551353 / 1250000000000000 : ℝ) ≤
+            (3201047569820898583849 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2799775866064787 / 10000000000000000 : ℝ)) (q :=
+      (1721590044551353 / 5000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_610_14 :
+    Uω (aρ 14) (bρ 14) (2260562330581 / 3200000000000 : ℝ) ≤
+      (-(6483336367110149200723 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5468831640779683 / 20000000000000000 : ℝ) ≤
+      (-(6483336367110149200723 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5468831640779683 / 20000000000000000 : ℝ) =
+            (5468831640779683 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5468831640779683 / 5000000000000000 : ℝ) ≤
+            (448135435889850799277 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (863087306842029 / 5000000000000000 : ℝ)) (q :=
+      (5468831640779683 / 20000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_610_15 :
+    Uω (aρ 15) (bρ 15) (2260562330581 / 3200000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_610_16 :
+    Uω (aρ 16) (bρ 16) (2260562330581 / 3200000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_610 :
+    Uρ (2260562330581 / 3200000000000 : ℝ) ≤
+      (-(3047661947672350956073 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (2260562330581 / 3200000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3567125202846665950607 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_610_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (2260562330581 / 3200000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(720264980638036197189 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_610_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (2260562330581 / 3200000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(146802045239044294797 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_610_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (2260562330581 / 3200000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1892584003101455628531 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_610_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (2260562330581 / 3200000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(3962844173437486658913 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_610_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (2260562330581 / 3200000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(2111395698327103507627 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_610_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (2260562330581 / 3200000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(4587562229901919172883 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_610_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (2260562330581 / 3200000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(2541030616916354549827 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_610_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (2260562330581 / 3200000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(179171113532017318017 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_610_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (2260562330581 / 3200000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(821505687596867965743 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_610_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (2260562330581 / 3200000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(7633712040526915893759 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_610_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (2260562330581 / 3200000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(8967759658547838351263 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_610_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (2260562330581 / 3200000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(10661896036179101416151 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_610_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (2260562330581 / 3200000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(6483336367110149200723 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_610_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (2260562330581 / 3200000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_610_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (2260562330581 / 3200000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_610_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_611_1 :
+    Uω (aρ 1) (bρ 1) (22657542499439 / 32000000000000 : ℝ) ≤
+      (-(1771986433177761046279 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28063847080485117 / 40000000000000000 : ℝ) ≤
+      (-(1771986433177761046279 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (28063847080485117 / 40000000000000000 : ℝ) =
+            (28063847080485117 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28063847080485117 / 20000000000000000 : ℝ) ≤
+            (1693749468322238953721 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7015938727252871 / 10000000000000000 : ℝ)) (q :=
+      (28063847080485117 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_611_2 :
+    Uω (aρ 2) (bρ 2) (22657542499439 / 32000000000000 : ℝ) ≤
+      (-(715618582912362501781 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27968256269851673 / 40000000000000000 : ℝ) ≤
+      (-(715618582912362501781 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (27968256269851673 / 40000000000000000 : ℝ) =
+            (27968256269851673 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27968256269851673 / 20000000000000000 : ℝ) ≤
+            (670675777687637498219 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6991912334446149 / 10000000000000000 : ℝ)) (q :=
+      (27968256269851673 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_611_3 :
+    Uω (aρ 3) (bρ 3) (22657542499439 / 32000000000000 : ℝ) ≤
+      (-(91166445666578071919 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27777148081072199 / 40000000000000000 : ℝ) ≤
+      (-(91166445666578071919 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (27777148081072199 / 40000000000000000 : ℝ) =
+            (27777148081072199 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27777148081072199 / 20000000000000000 : ℝ) ≤
+            (82120349408421928081 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1735938586601603 / 2500000000000000 : ℝ)) (q :=
+      (27777148081072199 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_611_4 :
+    Uω (aρ 4) (bρ 4) (22657542499439 / 32000000000000 : ℝ) ≤
+      (-(3761500697413053400093 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27459972094416933 / 40000000000000000 : ℝ) ≤
+      (-(3761500697413053400093 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (27459972094416933 / 40000000000000000 : ℝ) =
+            (27459972094416933 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27459972094416933 / 20000000000000000 : ℝ) ≤
+            (3169971105586946599907 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6863460534143779 / 10000000000000000 : ℝ)) (q :=
+      (27459972094416933 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_611_5 :
+    Uω (aρ 5) (bρ 5) (22657542499439 / 32000000000000 : ℝ) ≤
+      (-(1969372182254922753893 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5395509885359569 / 8000000000000000 : ℝ) ≤
+      (-(1969372182254922753893 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5395509885359569 / 8000000000000000 : ℝ) =
+            (5395509885359569 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5395509885359569 / 4000000000000000 : ℝ) ≤
+            (1496363719245077246107 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1348087823933847 / 2000000000000000 : ℝ)) (q :=
+      (5395509885359569 / 8000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_611_6 :
+    Uω (aρ 6) (bρ 6) (22657542499439 / 32000000000000 : ℝ) ≤
+      (-(4198036972603761804577 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (26287032495816753 / 40000000000000000 : ℝ) ≤
+      (-(4198036972603761804577 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (26287032495816753 / 40000000000000000 : ℝ) =
+            (26287032495816753 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26287032495816753 / 20000000000000000 : ℝ) ≤
+            (2733434830396238195423 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6562421029318689 / 10000000000000000 : ℝ)) (q :=
+      (26287032495816753 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_611_7 :
+    Uω (aρ 7) (bρ 7) (22657542499439 / 32000000000000 : ℝ) ≤
+      (-(912368540818807209291 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1013915287293877 / 1600000000000000 : ℝ) ≤
+      (-(912368540818807209291 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1013915287293877 / 1600000000000000 : ℝ) =
+            (1013915287293877 / 800000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1013915287293877 / 800000000000000 : ℝ) ≤
+            (473925819781192790709 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (252660290133751 / 400000000000000 : ℝ)) (q :=
+      (1013915287293877 / 1600000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_611_8 :
+    Uω (aρ 8) (bρ 8) (22657542499439 / 32000000000000 : ℝ) ≤
+      (-(5054938755538850791503 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24128303688369371 / 40000000000000000 : ℝ) ≤
+      (-(5054938755538850791503 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (24128303688369371 / 40000000000000000 : ℝ) =
+            (24128303688369371 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24128303688369371 / 20000000000000000 : ℝ) ≤
+            (1876533047461149208497 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2995129828847499 / 5000000000000000 : ℝ)) (q :=
+      (24128303688369371 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_611_9 :
+    Uω (aρ 9) (bρ 9) (22657542499439 / 32000000000000 : ℝ) ≤
+      (-(713039655535341428553 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22611253604052877 / 40000000000000000 : ℝ) ≤
+      (-(713039655535341428553 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (22611253604052877 / 40000000000000000 : ℝ) =
+            (22611253604052877 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22611253604052877 / 20000000000000000 : ℝ) ≤
+            (153394319839658571447 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5572700243896751 / 10000000000000000 : ℝ)) (q :=
+      (22611253604052877 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_611_10 :
+    Uω (aρ 10) (bρ 10) (22657542499439 / 32000000000000 : ℝ) ≤
+      (-(6539894335196580513757 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20798690340675543 / 40000000000000000 : ℝ) ≤
+      (-(6539894335196580513757 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (20798690340675543 / 40000000000000000 : ℝ) =
+            (20798690340675543 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20798690340675543 / 20000000000000000 : ℝ) ≤
+            (391577467803419486243 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1263744411425771 / 2500000000000000 : ℝ)) (q :=
+      (20798690340675543 / 40000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_611_11 :
+    Uω (aρ 11) (bρ 11) (22657542499439 / 32000000000000 : ℝ) ≤
+      (-(1899253611517541986453 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (18712242877819023 / 40000000000000000 : ℝ) ≤
+      (-(1899253611517541986453 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (18712242877819023 / 40000000000000000 : ℝ) =
+            (18712242877819023 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18712242877819023 / 10000000000000000 : ℝ) ≤
+            (1566482289982458013547 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (276889408779989 / 625000000000000 : ℝ)) (q :=
+      (18712242877819023 / 40000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_611_12 :
+    Uω (aρ 12) (bρ 12) (22657542499439 / 32000000000000 : ℝ) ≤
+      (-(8923689615546948989069 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16387363049407717 / 40000000000000000 : ℝ) ≤
+      (-(8923689615546948989069 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (16387363049407717 / 40000000000000000 : ℝ) =
+            (16387363049407717 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16387363049407717 / 10000000000000000 : ℝ) ≤
+            (4939253990453051010931 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3691490722959171 / 10000000000000000 : ℝ)) (q :=
+      (16387363049407717 / 40000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_611_13 :
+    Uω (aρ 13) (bρ 13) (22657542499439 / 32000000000000 : ℝ) ≤
+      (-(1325523652575441464269 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13852428102585417 / 40000000000000000 : ℝ) ≤
+      (-(1325523652575441464269 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (13852428102585417 / 40000000000000000 : ℝ) =
+            (13852428102585417 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13852428102585417 / 10000000000000000 : ℝ) ≤
+            (407344298174558535731 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2823404991143021 / 10000000000000000 : ℝ)) (q :=
+      (13852428102585417 / 40000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_611_14 :
+    Uω (aρ 14) (bρ 14) (22657542499439 / 32000000000000 : ℝ) ≤
+      (-(3218405066046634124583 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11039915945363911 / 40000000000000000 : ℝ) ≤
+      (-(3218405066046634124583 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (11039915945363911 / 40000000000000000 : ℝ) =
+            (11039915945363911 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11039915945363911 / 10000000000000000 : ℝ) ≤
+            (247330835453365875417 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (440269049394317 / 2500000000000000 : ℝ)) (q :=
+      (11039915945363911 / 40000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_611_15 :
+    Uω (aρ 15) (bρ 15) (22657542499439 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_611_16 :
+    Uω (aρ 16) (bρ 16) (22657542499439 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_611 :
+    Uρ (22657542499439 / 32000000000000 : ℝ) ≤
+      (-(1516242492074414445773 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (22657542499439 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1771986433177761046279 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_611_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (22657542499439 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(715618582912362501781 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_611_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (22657542499439 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) * (-(91166445666578071919 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_611_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (22657542499439 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3761500697413053400093 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_611_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (22657542499439 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1969372182254922753893 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_611_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (22657542499439 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(4198036972603761804577 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_611_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (22657542499439 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(912368540818807209291 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_611_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (22657542499439 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(5054938755538850791503 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_611_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (22657542499439 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(713039655535341428553 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_611_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (22657542499439 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(6539894335196580513757 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_611_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (22657542499439 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(1899253611517541986453 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_611_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (22657542499439 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(8923689615546948989069 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_611_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (22657542499439 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(1325523652575441464269 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_611_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (22657542499439 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(3218405066046634124583 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_611_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (22657542499439 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_611_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (22657542499439 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_611_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_612_1 :
+    Uω (aρ 1) (bρ 1) (5677365423267 / 8000000000000 : ℝ) ≤
+      (-(704174801898035062819 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2812874628518123 / 4000000000000000 : ℝ) ≤
+      (-(704174801898035062819 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2812874628518123 / 4000000000000000 : ℝ) =
+            (2812874628518123 / 2000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2812874628518123 / 2000000000000000 : ℝ) ≤
+            (682119558701964937181 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1406432716318373 / 2000000000000000 : ℝ)) (q :=
+      (2812874628518123 / 4000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_612_2 :
+    Uω (aρ 2) (bρ 2) (5677365423267 / 8000000000000 : ℝ) ≤
+      (-(888728693930132351883 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14016578333507227 / 20000000000000000 : ℝ) ≤
+      (-(888728693930132351883 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (14016578333507227 / 20000000000000000 : ℝ) =
+            (14016578333507227 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14016578333507227 / 10000000000000000 : ℝ) ≤
+            (844139256819867648117 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7008137785018477 / 10000000000000000 : ℝ)) (q :=
+      (14016578333507227 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_612_3 :
+    Uω (aρ 3) (bρ 3) (5677365423267 / 8000000000000 : ℝ) ≤
+      (-(3623319127680336539177 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (108758015781547 / 156250000000000 : ℝ) ≤
+      (-(3623319127680336539177 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (108758015781547 / 156250000000000 : ℝ) = (108758015781547 / 78125000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (108758015781547 / 78125000000000 : ℝ) ≤
+            (3308152675319663460823 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3479990788949633 / 5000000000000000 : ℝ)) (q :=
+      (108758015781547 / 156250000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_612_4 :
+    Uω (aρ 4) (bρ 4) (5677365423267 / 8000000000000 : ℝ) ≤
+      (-(116809040482449137779 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13762442771534643 / 20000000000000000 : ℝ) ≤
+      (-(116809040482449137779 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (13762442771534643 / 20000000000000000 : ℝ) =
+            (13762442771534643 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13762442771534643 / 10000000000000000 : ℝ) ≤
+            (99799453361300862221 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6879692510460893 / 10000000000000000 : ℝ)) (q :=
+      (13762442771534643 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_612_5 :
+    Uω (aρ 5) (bρ 5) (5677365423267 / 8000000000000 : ℝ) ≤
+      (-(1957351281884243234707 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1690155396394973 / 2500000000000000 : ℝ) ≤
+      (-(1957351281884243234707 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1690155396394973 / 2500000000000000 : ℝ) =
+            (1690155396394973 / 1250000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1690155396394973 / 1250000000000000 : ℝ) ≤
+            (1508384619615756765293 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3378341414710517 / 5000000000000000 : ℝ)) (q :=
+      (1690155396394973 / 2500000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_612_6 :
+    Uω (aρ 6) (bρ 6) (5677365423267 / 8000000000000 : ℝ) ≤
+      (-(4173343848933882049953 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13176011799446877 / 20000000000000000 : ℝ) ≤
+      (-(4173343848933882049953 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (13176011799446877 / 20000000000000000 : ℝ) =
+            (13176011799446877 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13176011799446877 / 10000000000000000 : ℝ) ≤
+            (2758127954066117950047 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6578691832848127 / 10000000000000000 : ℝ)) (q :=
+      (13176011799446877 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_612_7 :
+    Uω (aρ 7) (bρ 7) (5677365423267 / 8000000000000 : ℝ) ≤
+      (-(2268094792214219806731 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3176623860524561 / 5000000000000000 : ℝ) ≤
+      (-(2268094792214219806731 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (3176623860524561 / 5000000000000000 : ℝ) =
+            (3176623860524561 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3176623860524561 / 2500000000000000 : ℝ) ≤
+            (1197641109285780193269 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3166418428129747 / 5000000000000000 : ℝ)) (q :=
+      (3176623860524561 / 5000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_612_8 :
+    Uω (aρ 8) (bρ 8) (5677365423267 / 8000000000000 : ℝ) ≤
+      (-(2513945331327445238627 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3024206811147909 / 5000000000000000 : ℝ) ≤
+      (-(2513945331327445238627 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (3024206811147909 / 5000000000000000 : ℝ) =
+            (3024206811147909 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3024206811147909 / 2500000000000000 : ℝ) ≤
+            (951790570172554761373 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3003355155046443 / 5000000000000000 : ℝ)) (q :=
+      (3024206811147909 / 5000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_612_9 :
+    Uω (aρ 9) (bρ 9) (5677365423267 / 8000000000000 : ℝ) ≤
+      (-(5675246059732281515241 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2834635354627983 / 5000000000000000 : ℝ) ≤
+      (-(5675246059732281515241 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2834635354627983 / 5000000000000000 : ℝ) =
+            (2834635354627983 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2834635354627983 / 2500000000000000 : ℝ) ≤
+            (1256225743267718484759 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2794695056186591 / 5000000000000000 : ℝ)) (q :=
+      (2834635354627983 / 5000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_612_10 :
+    Uω (aρ 10) (bρ 10) (5677365423267 / 8000000000000 : ℝ) ≤
+      (-(650785208705559288509 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10432720452581229 / 20000000000000000 : ℝ) ≤
+      (-(650785208705559288509 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10432720452581229 / 20000000000000000 : ℝ) =
+            (10432720452581229 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10432720452581229 / 10000000000000000 : ℝ) ≤
+            (42361971594440711491 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5072128179937479 / 10000000000000000 : ℝ)) (q :=
+      (10432720452581229 / 20000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_612_11 :
+    Uω (aρ 11) (bρ 11) (5677365423267 / 8000000000000 : ℝ) ≤
+      (-(7560465982075085006291 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1878075838188081 / 4000000000000000 : ℝ) ≤
+      (-(7560465982075085006291 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1878075838188081 / 4000000000000000 : ℝ) =
+            (1878075838188081 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1878075838188081 / 1000000000000000 : ℝ) ≤
+            (6302477623924914993709 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (889652708900331 / 2000000000000000 : ℝ)) (q :=
+      (1878075838188081 / 4000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_612_12 :
+    Uω (aρ 12) (bρ 12) (5677365423267 / 8000000000000 : ℝ) ≤
+      (-(8879855177153937258143 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4114838450862511 / 10000000000000000 : ℝ) ≤
+      (-(8879855177153937258143 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4114838450862511 / 10000000000000000 : ℝ) =
+            (4114838450862511 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4114838450862511 / 2500000000000000 : ℝ) ≤
+            (4983088428846062741857 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (463907668996409 / 1250000000000000 : ℝ)) (q :=
+      (4114838450862511 / 10000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_612_13 :
+    Uω (aρ 13) (bρ 13) (5677365423267 / 8000000000000 : ℝ) ≤
+      (-(2636740574611385909277 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3482982136594709 / 10000000000000000 : ℝ) ≤
+      (-(2636740574611385909277 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3482982136594709 / 10000000000000000 : ℝ) =
+            (3482982136594709 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3482982136594709 / 2500000000000000 : ℝ) ≤
+            (828995326888614090723 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (711732616257667 / 2500000000000000 : ℝ)) (q :=
+      (3482982136594709 / 10000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_612_14 :
+    Uω (aρ 14) (bρ 14) (5677365423267 / 8000000000000 : ℝ) ≤
+      (-(12782380229451382714503 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5570552549959177 / 20000000000000000 : ℝ) ≤
+      (-(12782380229451382714503 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5570552549959177 / 20000000000000000 : ℝ) =
+            (5570552549959177 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5570552549959177 / 5000000000000000 : ℝ) ≤
+            (1080563376548617285497 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1795446026845427 / 10000000000000000 : ℝ)) (q :=
+      (5570552549959177 / 20000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_612_15 :
+    Uω (aρ 15) (bρ 15) (5677365423267 / 8000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_612_16 :
+    Uω (aρ 16) (bρ 16) (5677365423267 / 8000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_612 :
+    Uρ (5677365423267 / 8000000000000 : ℝ) ≤
+      (-(48278391504829887333 / 80000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (5677365423267 / 8000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(704174801898035062819 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_612_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (5677365423267 / 8000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(888728693930132351883 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_612_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (5677365423267 / 8000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3623319127680336539177 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_612_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (5677365423267 / 8000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(116809040482449137779 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_612_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (5677365423267 / 8000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1957351281884243234707 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_612_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (5677365423267 / 8000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(4173343848933882049953 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_612_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (5677365423267 / 8000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(2268094792214219806731 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_612_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (5677365423267 / 8000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(2513945331327445238627 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_612_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (5677365423267 / 8000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(5675246059732281515241 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_612_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (5677365423267 / 8000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(650785208705559288509 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_612_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (5677365423267 / 8000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(7560465982075085006291 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_612_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (5677365423267 / 8000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(8879855177153937258143 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_612_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (5677365423267 / 8000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(2636740574611385909277 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_612_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (5677365423267 / 8000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(12782380229451382714503 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_612_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (5677365423267 / 8000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_612_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (5677365423267 / 8000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_612_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_613_1 :
+    Uω (aρ 1) (bρ 1) (22761380886697 / 32000000000000 : ℝ) ≤
+      (-(874457096438228208609 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28193645488898291 / 40000000000000000 : ℝ) ≤
+      (-(874457096438228208609 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (28193645488898291 / 40000000000000000 : ℝ) =
+            (28193645488898291 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28193645488898291 / 20000000000000000 : ℝ) ≤
+            (858410854311771791391 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7048388435441333 / 10000000000000000 : ℝ)) (q :=
+      (28193645488898291 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_613_2 :
+    Uω (aρ 2) (bρ 2) (22761380886697 / 32000000000000 : ℝ) ≤
+      (-(3531790237594273493879 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28098057057686023 / 40000000000000000 : ℝ) ≤
+      (-(3531790237594273493879 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (28098057057686023 / 40000000000000000 : ℝ) =
+            (28098057057686023 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28098057057686023 / 20000000000000000 : ℝ) ≤
+            (3399681565405726506121 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7024363232345199 / 10000000000000000 : ℝ)) (q :=
+      (28098057057686023 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_613_3 :
+    Uω (aρ 3) (bρ 3) (22761380886697 / 32000000000000 : ℝ) ≤
+      (-(3600034779659862000939 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27906955975974487 / 40000000000000000 : ℝ) ≤
+      (-(3600034779659862000939 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (27906955975974487 / 40000000000000000 : ℝ) =
+            (27906955975974487 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27906955975974487 / 20000000000000000 : ℝ) ≤
+            (3331437023340137999061 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6976208797839431 / 10000000000000000 : ℝ)) (q :=
+      (27906955975974487 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_613_4 :
+    Uω (aρ 4) (bρ 4) (22761380886697 / 32000000000000 : ℝ) ≤
+      (-(371433353665875078499 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27589798923679363 / 40000000000000000 : ℝ) ≤
+      (-(371433353665875078499 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (27589798923679363 / 40000000000000000 : ℝ) =
+            (27589798923679363 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27589798923679363 / 20000000000000000 : ℝ) ≤
+            (321713826634124921501 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6895924452756869 / 10000000000000000 : ℝ)) (q :=
+      (27589798923679363 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_613_5 :
+    Uω (aρ 5) (bρ 5) (22761380886697 / 32000000000000 : ℝ) ≤
+      (-(486339811538652721501 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27107423076041433 / 40000000000000000 : ℝ) ≤
+      (-(486339811538652721501 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (27107423076041433 / 40000000000000000 : ℝ) =
+            (27107423076041433 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27107423076041433 / 20000000000000000 : ℝ) ≤
+            (380094163836347278499 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (846615806034113 / 1250000000000000 : ℝ)) (q :=
+      (27107423076041433 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_613_6 :
+    Uω (aρ 6) (bρ 6) (22761380886697 / 32000000000000 : ℝ) ≤
+      (-(1037177930485340854641 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (26417014248109331 / 40000000000000000 : ℝ) ≤
+      (-(1037177930485340854641 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (26417014248109331 / 40000000000000000 : ℝ) =
+            (26417014248109331 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26417014248109331 / 20000000000000000 : ℝ) ≤
+            (695690020264659145359 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6594962409446853 / 10000000000000000 : ℝ)) (q :=
+      (26417014248109331 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_613_7 :
+    Uω (aρ 7) (bρ 7) (22761380886697 / 32000000000000 : ℝ) ≤
+      (-(902120505339501589781 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25478098510791969 / 40000000000000000 : ℝ) ≤
+      (-(902120505339501589781 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (25478098510791969 / 40000000000000000 : ℝ) =
+            (25478098510791969 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25478098510791969 / 20000000000000000 : ℝ) ≤
+            (484173855260498410219 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1587291480387043 / 2500000000000000 : ℝ)) (q :=
+      (25478098510791969 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_613_8 :
+    Uω (aρ 8) (bρ 8) (22761380886697 / 32000000000000 : ℝ) ≤
+      (-(1250229135690731567857 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24259002838952893 / 40000000000000000 : ℝ) ≤
+      (-(1250229135690731567857 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (24259002838952893 / 40000000000000000 : ℝ) =
+            (24259002838952893 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24259002838952893 / 20000000000000000 : ℝ) ≤
+            (482638815059268432143 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3011579868484317 / 5000000000000000 : ℝ)) (q :=
+      (24259002838952893 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_613_9 :
+    Uω (aρ 9) (bρ 9) (22761380886697 / 32000000000000 : ℝ) ≤
+      (-(141156538628002730241 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22742906608329809 / 40000000000000000 : ℝ) ≤
+      (-(141156538628002730241 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (22742906608329809 / 40000000000000000 : ℝ) =
+            (22742906608329809 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22742906608329809 / 20000000000000000 : ℝ) ≤
+            (32130256446997269759 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1401519312504273 / 2500000000000000 : ℝ)) (q :=
+      (22742906608329809 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_613_10 :
+    Uω (aρ 10) (bρ 10) (22761380886697 / 32000000000000 : ℝ) ≤
+      (-(6475917982535767971497 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20932179327105513 / 40000000000000000 : ℝ) ≤
+      (-(6475917982535767971497 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (20932179327105513 / 40000000000000000 : ℝ) =
+            (20932179327105513 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20932179327105513 / 20000000000000000 : ℝ) ≤
+            (455553820464232028503 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (636159080362493 / 1250000000000000 : ℝ)) (q :=
+      (20932179327105513 / 40000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_613_11 :
+    Uω (aρ 11) (bρ 11) (22761380886697 / 32000000000000 : ℝ) ≤
+      (-(7524065327419626836843 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3769849229246809 / 8000000000000000 : ℝ) ≤
+      (-(7524065327419626836843 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3769849229246809 / 8000000000000000 : ℝ) =
+            (3769849229246809 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3769849229246809 / 2000000000000000 : ℝ) ≤
+            (6338878278580373163157 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (446628267866921 / 1000000000000000 : ℝ)) (q :=
+      (3769849229246809 / 8000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_613_12 :
+    Uω (aρ 12) (bρ 12) (22761380886697 / 32000000000000 : ℝ) ≤
+      (-(8836253433533761614819 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16531276138937201 / 40000000000000000 : ℝ) ≤
+      (-(8836253433533761614819 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (16531276138937201 / 40000000000000000 : ℝ) =
+            (16531276138937201 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16531276138937201 / 10000000000000000 : ℝ) ≤
+            (5026690172466238385181 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (932749442926447 / 2500000000000000 : ℝ)) (q :=
+      (16531276138937201 / 40000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_613_13 :
+    Uω (aρ 13) (bρ 13) (22761380886697 / 32000000000000 : ℝ) ≤
+      (-(2098041064256764806811 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14011226784954181 / 40000000000000000 : ℝ) ≤
+      (-(2098041064256764806811 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (14011226784954181 / 40000000000000000 : ℝ) =
+            (14011226784954181 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14011226784954181 / 10000000000000000 : ℝ) ≤
+            (674547656943235193189 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1435177418154639 / 5000000000000000 : ℝ)) (q :=
+      (14011226784954181 / 40000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_613_14 :
+    Uω (aρ 14) (bρ 14) (22761380886697 / 32000000000000 : ℝ) ≤
+      (-(1586607232003425158267 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11241290690528949 / 40000000000000000 : ℝ) ≤
+      (-(1586607232003425158267 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (11241290690528949 / 40000000000000000 : ℝ) =
+            (11241290690528949 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11241290690528949 / 10000000000000000 : ℝ) ≤
+            (146260718746574841733 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (914657037070831 / 5000000000000000 : ℝ)) (q :=
+      (11241290690528949 / 40000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_613_15 :
+    Uω (aρ 15) (bρ 15) (22761380886697 / 32000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_613_16 :
+    Uω (aρ 16) (bρ 16) (22761380886697 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_613 :
+    Uρ (22761380886697 / 32000000000000 : ℝ) ≤
+      (-(6004805442082874770883 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (22761380886697 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(874457096438228208609 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_613_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (22761380886697 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3531790237594273493879 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_613_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (22761380886697 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3600034779659862000939 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_613_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (22761380886697 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(371433353665875078499 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_613_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (22761380886697 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(486339811538652721501 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_613_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (22761380886697 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1037177930485340854641 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_613_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (22761380886697 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(902120505339501589781 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_613_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (22761380886697 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(1250229135690731567857 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_613_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (22761380886697 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(141156538628002730241 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_613_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (22761380886697 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(6475917982535767971497 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_613_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (22761380886697 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(7524065327419626836843 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_613_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (22761380886697 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(8836253433533761614819 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_613_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (22761380886697 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(2098041064256764806811 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_613_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (22761380886697 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(1586607232003425158267 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_613_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (22761380886697 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_613_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (22761380886697 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_613_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_614_1 :
+    Uω (aρ 1) (bρ 1) (11406650040163 / 16000000000000 : ℝ) ≤
+      (-(347483575034634540937 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (565170893832861 / 800000000000000 : ℝ) ≤
+      (-(347483575034634540937 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (565170893832861 / 800000000000000 : ℝ) =
+            (565170893832861 / 400000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (565170893832861 / 400000000000000 : ℝ) ≤
+            (345663605265365459063 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (141292265776093 / 200000000000000 : ℝ)) (q :=
+      (565170893832861 / 800000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_614_2 :
+    Uω (aρ 2) (bρ 2) (11406650040163 / 16000000000000 : ℝ) ≤
+      (-(350871905283512890253 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14081478720955629 / 20000000000000000 : ℝ) ≤
+      (-(350871905283512890253 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (14081478720955629 / 20000000000000000 : ℝ) =
+            (14081478720955629 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14081478720955629 / 10000000000000000 : ℝ) ≤
+            (342275275016487109747 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3520294338224377 / 5000000000000000 : ℝ)) (q :=
+      (14081478720955629 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_614_3 :
+    Uω (aρ 3) (bρ 3) (11406650040163 / 16000000000000 : ℝ) ≤
+      (-(894201132501833096559 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6992964972232107 / 10000000000000000 : ℝ) ≤
+      (-(894201132501833096559 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (6992964972232107 / 10000000000000000 : ℝ) =
+            (6992964972232107 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6992964972232107 / 5000000000000000 : ℝ) ≤
+            (838666818248166903441 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6992436006307339 / 10000000000000000 : ℝ)) (q :=
+      (6992964972232107 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_614_4 :
+    Uω (aρ 4) (bρ 4) (11406650040163 / 16000000000000 : ℝ) ≤
+      (-(3690833159315097836869 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13827356118363259 / 20000000000000000 : ℝ) ≤
+      (-(3690833159315097836869 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (13827356118363259 / 20000000000000000 : ℝ) =
+            (13827356118363259 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13827356118363259 / 10000000000000000 : ℝ) ≤
+            (3240638643684902163131 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (864019545158923 / 1250000000000000 : ℝ)) (q :=
+      (13827356118363259 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_614_5 :
+    Uω (aρ 5) (bρ 5) (11406650040163 / 16000000000000 : ℝ) ≤
+      (-(3866791873236816182813 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6793089907317087 / 10000000000000000 : ℝ) ≤
+      (-(3866791873236816182813 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (6793089907317087 / 10000000000000000 : ℝ) =
+            (6793089907317087 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6793089907317087 / 5000000000000000 : ℝ) ≤
+            (3064679929763183817187 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6789169976877299 / 10000000000000000 : ℝ)) (q :=
+      (6793089907317087 / 10000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_614_6 :
+    Uω (aρ 6) (bρ 6) (11406650040163 / 16000000000000 : ℝ) ≤
+      (-(412414029018064730309 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13241002223407207 / 20000000000000000 : ℝ) ≤
+      (-(412414029018064730309 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (13241002223407207 / 20000000000000000 : ℝ) =
+            (13241002223407207 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13241002223407207 / 10000000000000000 : ℝ) ≤
+            (280733151281935269691 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1652808190197583 / 2500000000000000 : ℝ)) (q :=
+      (13241002223407207 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_614_7 :
+    Uω (aρ 7) (bρ 7) (11406650040163 / 16000000000000 : ℝ) ≤
+      (-(897016237875904629603 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12771602535204123 / 20000000000000000 : ℝ) ≤
+      (-(897016237875904629603 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (12771602535204123 / 20000000000000000 : ℝ) =
+            (12771602535204123 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12771602535204123 / 10000000000000000 : ℝ) ≤
+            (489278122724095370397 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (397843403334203 / 625000000000000 : ℝ)) (q :=
+      (12771602535204123 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_614_8 :
+    Uω (aρ 8) (bρ 8) (11406650040163 / 16000000000000 : ℝ) ≤
+      (-(4974015986909744955749 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1520271797356573 / 2500000000000000 : ℝ) ≤
+      (-(4974015986909744955749 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1520271797356573 / 2500000000000000 : ℝ) =
+            (1520271797356573 / 1250000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1520271797356573 / 1250000000000000 : ℝ) ≤
+            (1957455816090255044251 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6039607948335709 / 10000000000000000 : ℝ)) (q :=
+      (1520271797356573 / 2500000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_614_9 :
+    Uω (aρ 9) (bρ 9) (11406650040163 / 16000000000000 : ℝ) ≤
+      (-(1404340792802545254119 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11404362483298911 / 20000000000000000 : ℝ) ≤
+      (-(1404340792802545254119 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11404362483298911 / 20000000000000000 : ℝ) =
+            (11404362483298911 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11404362483298911 / 10000000000000000 : ℝ) ≤
+            (328527157947454745881 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1405690420285509 / 2500000000000000 : ℝ)) (q :=
+      (11404362483298911 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_614_10 :
+    Uω (aρ 10) (bρ 10) (11406650040163 / 16000000000000 : ℝ) ≤
+      (-(6444091256402327224063 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (328107902012629 / 625000000000000 : ℝ) ≤
+      (-(6444091256402327224063 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (328107902012629 / 625000000000000 : ℝ) =
+            (328107902012629 / 312500000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (328107902012629 / 312500000000000 : ℝ) ≤
+            (487380546597672775937 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5106411095742253 / 10000000000000000 : ℝ)) (q :=
+      (328107902012629 / 625000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10]) (by norm_num [bρ_10])
+      (by norm_num) (by norm_num [aρ_10, bρ_10]) (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_614_11 :
+    Uω (aρ 11) (bρ 11) (11406650040163 / 16000000000000 : ℝ) ≤
+      (-(7487811179694703890357 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2364713313159581 / 5000000000000000 : ℝ) ≤
+      (-(7487811179694703890357 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2364713313159581 / 5000000000000000 : ℝ) =
+            (2364713313159581 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2364713313159581 / 1250000000000000 : ℝ) ≤
+            (6375132426305296109643 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4484288110181449 / 10000000000000000 : ℝ)) (q :=
+      (2364713313159581 / 5000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_614_12 :
+    Uω (aρ 12) (bρ 12) (11406650040163 / 16000000000000 : ℝ) ≤
+      (-(8792881533717337725777 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4150782783984341 / 10000000000000000 : ℝ) ≤
+      (-(8792881533717337725777 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4150782783984341 / 10000000000000000 : ℝ) =
+            (4150782783984341 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4150782783984341 / 2500000000000000 : ℝ) ≤
+            (5070062072282662274223 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3750700522196807 / 10000000000000000 : ℝ)) (q :=
+      (4150782783984341 / 10000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_614_13 :
+    Uω (aρ 13) (bρ 13) (11406650040163 / 16000000000000 : ℝ) ≤
+      (-(5216954341376271766899 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3522581932218237 / 10000000000000000 : ℝ) ≤
+      (-(5216954341376271766899 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3522581932218237 / 10000000000000000 : ℝ) =
+            (3522581932218237 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3522581932218237 / 2500000000000000 : ℝ) ≤
+            (1714517461623728233101 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2893680560259599 / 10000000000000000 : ℝ)) (q :=
+      (3522581932218237 / 10000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_614_14 :
+    Uω (aρ 14) (bρ 14) (11406650040163 / 16000000000000 : ℝ) ≤
+      (-(6302483216415656992739 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5670263729237359 / 20000000000000000 : ℝ) ≤
+      (-(6302483216415656992739 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5670263729237359 / 20000000000000000 : ℝ) =
+            (5670263729237359 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5670263729237359 / 5000000000000000 : ℝ) ≤
+            (628988586584343007261 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (465676927526371 / 2500000000000000 : ℝ)) (q :=
+      (5670263729237359 / 20000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_614_15 :
+    Uω (aρ 15) (bρ 15) (11406650040163 / 16000000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_614_16 :
+    Uω (aρ 16) (bρ 16) (11406650040163 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_614 :
+    Uρ (11406650040163 / 16000000000000 : ℝ) ≤
+      (-(5974984503744520764127 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (11406650040163 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(347483575034634540937 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_614_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (11406650040163 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(350871905283512890253 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_614_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (11406650040163 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(894201132501833096559 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_614_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (11406650040163 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3690833159315097836869 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_614_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (11406650040163 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(3866791873236816182813 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_614_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (11406650040163 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(412414029018064730309 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_614_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (11406650040163 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(897016237875904629603 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_614_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (11406650040163 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(4974015986909744955749 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_614_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (11406650040163 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(1404340792802545254119 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_614_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (11406650040163 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(6444091256402327224063 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_614_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (11406650040163 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(7487811179694703890357 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_614_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (11406650040163 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(8792881533717337725777 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_614_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (11406650040163 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(5216954341376271766899 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_614_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (11406650040163 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(6302483216415656992739 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_614_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (11406650040163 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_614_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (11406650040163 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_614_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_615_1 :
+    Uω (aρ 1) (bρ 1) (89520072139 / 125000000000 : ℝ) ≤
+      (-(3429008473743938263037 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7097085773560581 / 10000000000000000 : ℝ) ≤
+      (-(3429008473743938263037 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (7097085773560581 / 10000000000000000 : ℝ) =
+            (7097085773560581 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7097085773560581 / 5000000000000000 : ℝ) ≤
+            (3502463329256061736963 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3548531497043081 / 5000000000000000 : ℝ)) (q :=
+      (7097085773560581 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_615_2 :
+    Uω (aρ 2) (bρ 2) (89520072139 / 125000000000 : ℝ) ≤
+      (-(432841970319017025881 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14146379095600089 / 20000000000000000 : ℝ) ≤
+      (-(432841970319017025881 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (14146379095600089 / 20000000000000000 : ℝ) =
+            (14146379095600089 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14146379095600089 / 10000000000000000 : ℝ) ≤
+            (433592005055982974119 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7073039555075089 / 10000000000000000 : ℝ)) (q :=
+      (14146379095600089 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_615_3 :
+    Uω (aρ 3) (bρ 3) (89520072139 / 125000000000 : ℝ) ≤
+      (-(882626331049433886807 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7025416911659693 / 10000000000000000 : ℝ) ≤
+      (-(882626331049433886807 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (7025416911659693 / 10000000000000000 : ℝ) =
+            (7025416911659693 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7025416911659693 / 5000000000000000 : ℝ) ≤
+            (850241619700566113193 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3512445194572193 / 5000000000000000 : ℝ)) (q :=
+      (7025416911659693 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_615_4 :
+    Uω (aρ 4) (bρ 4) (89520072139 / 125000000000 : ℝ) ≤
+      (-(728799502219952959019 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13892269331013533 / 20000000000000000 : ℝ) ≤
+      (-(728799502219952959019 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (13892269331013533 / 20000000000000000 : ℝ) =
+            (13892269331013533 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13892269331013533 / 10000000000000000 : ℝ) ≤
+            (657494858380047040981 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6944620077903533 / 10000000000000000 : ℝ)) (q :=
+      (13892269331013533 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_615_5 :
+    Uω (aρ 5) (bρ 5) (89520072139 / 125000000000 : ℝ) ≤
+      (-(954777473650250097291 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6825558049848699 / 10000000000000000 : ℝ) ≤
+      (-(954777473650250097291 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (6825558049848699 / 10000000000000000 : ℝ) =
+            (6825558049848699 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6825558049848699 / 5000000000000000 : ℝ) ≤
+            (778090477099749902709 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3410828382961199 / 5000000000000000 : ℝ)) (q :=
+      (6825558049848699 / 10000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_615_6 :
+    Uω (aρ 6) (bρ 6) (89520072139 / 125000000000 : ℝ) ≤
+      (-(1018794579431653135779 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (532239670118649 / 800000000000000 : ℝ) ≤
+      (-(1018794579431653135779 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (532239670118649 / 800000000000000 : ℝ) =
+            (532239670118649 / 400000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (532239670118649 / 400000000000000 : ℝ) ≤
+            (714073371318346864221 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (265750911773249 / 400000000000000 : ℝ)) (q :=
+      (532239670118649 / 800000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_615_7 :
+    Uω (aρ 7) (bρ 7) (89520072139 / 125000000000 : ℝ) ≤
+      (-(4434234323157178674073 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12836707510689501 / 20000000000000000 : ℝ) ≤
+      (-(4434234323157178674073 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (12836707510689501 / 20000000000000000 : ℝ) =
+            (12836707510689501 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12836707510689501 / 10000000000000000 : ℝ) ≤
+            (2497237479842821325927 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6398149932814501 / 10000000000000000 : ℝ)) (q :=
+      (12836707510689501 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_615_8 :
+    Uω (aρ 8) (bρ 8) (89520072139 / 125000000000 : ℝ) ≤
+      (-(307527121787241753561 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12227516690586597 / 20000000000000000 : ℝ) ≤
+      (-(307527121787241753561 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (12227516690586597 / 20000000000000000 : ℝ) =
+            (12227516690586597 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12227516690586597 / 10000000000000000 : ℝ) ≤
+            (125689865900258246439 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6072500764051597 / 10000000000000000 : ℝ)) (q :=
+      (12227516690586597 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_615_9 :
+    Uω (aρ 9) (bρ 9) (89520072139 / 125000000000 : ℝ) ≤
+      (-(5559822753255195245599 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5735086408905213 / 10000000000000000 : ℝ) ≤
+      (-(5559822753255195245599 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5735086408905213 / 10000000000000000 : ℝ) =
+            (5735086408905213 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5735086408905213 / 5000000000000000 : ℝ) ≤
+            (1371649049744804754401 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2828061259817713 / 5000000000000000 : ℝ)) (q :=
+      (5735086408905213 / 10000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_615_10 :
+    Uω (aρ 10) (bρ 10) (89520072139 / 125000000000 : ℝ) ≤
+      (-(6380756920479711852741 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10566161476170617 / 20000000000000000 : ℝ) ≤
+      (-(6380756920479711852741 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10566161476170617 / 20000000000000000 : ℝ) =
+            (10566161476170617 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10566161476170617 / 10000000000000000 : ℝ) ≤
+            (550714882520288147259 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5140670211490617 / 10000000000000000 : ℝ)) (q :=
+      (10566161476170617 / 20000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_615_11 :
+    Uω (aρ 11) (bρ 11) (89520072139 / 125000000000 : ℝ) ≤
+      (-(1853934321690048881789 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9527273158567701 / 20000000000000000 : ℝ) ≤
+      (-(1853934321690048881789 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9527273158567701 / 20000000000000000 : ℝ) =
+            (9527273158567701 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9527273158567701 / 5000000000000000 : ℝ) ≤
+            (1611801579809951118211 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4520258520092701 / 10000000000000000 : ℝ)) (q :=
+      (9527273158567701 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_615_12 :
+    Uω (aρ 12) (bρ 12) (89520072139 / 125000000000 : ℝ) ≤
+      (-(272088004562792016049 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (209333041418471 / 500000000000000 : ℝ) ≤
+      (-(272088004562792016049 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (209333041418471 / 500000000000000 : ℝ) =
+            (209333041418471 / 125000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (209333041418471 / 125000000000000 : ℝ) ≤
+            (161128983124707983951 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (94750177873721 / 250000000000000 : ℝ)) (q :=
+      (209333041418471 / 500000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12]) (by norm_num [bρ_12])
+      (by norm_num) (by norm_num [aρ_12, bρ_12]) (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_615_13 :
+    Uω (aρ 13) (bρ 13) (89520072139 / 125000000000 : ℝ) ≤
+      (-(5161329804599260628999 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3561989124115103 / 10000000000000000 : ℝ) ≤
+      (-(5161329804599260628999 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3561989124115103 / 10000000000000000 : ℝ) =
+            (3561989124115103 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3561989124115103 / 2500000000000000 : ℝ) ≤
+            (1770141998400739371001 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1470022724017603 / 5000000000000000 : ℝ)) (q :=
+      (3561989124115103 / 10000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_615_14 :
+    Uω (aρ 14) (bρ 14) (89520072139 / 125000000000 : ℝ) ≤
+      (-(6216882226043502495963 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2884087755233591 / 10000000000000000 : ℝ) ≤
+      (-(6216882226043502495963 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2884087755233591 / 10000000000000000 : ℝ) =
+            (2884087755233591 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2884087755233591 / 2500000000000000 : ℝ) ≤
+            (714589576956497504037 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (964084997658591 / 5000000000000000 : ℝ)) (q :=
+      (2884087755233591 / 10000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_615_15 :
+    Uω (aρ 15) (bρ 15) (89520072139 / 125000000000 : ℝ) ≤
+      (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_15]) (by norm_num [bρ_15])]; exact Lmid_15
+
+lemma U_615_16 :
+    Uω (aρ 16) (bρ 16) (89520072139 / 125000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_615 :
+    Uρ (89520072139 / 125000000000 : ℝ) ≤
+      (-(5915842076011854104723 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (89520072139 / 125000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3429008473743938263037 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_615_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (89520072139 / 125000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(432841970319017025881 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_615_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (89520072139 / 125000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(882626331049433886807 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_615_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (89520072139 / 125000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(728799502219952959019 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_615_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (89520072139 / 125000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(954777473650250097291 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_615_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (89520072139 / 125000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1018794579431653135779 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_615_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (89520072139 / 125000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(4434234323157178674073 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_615_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (89520072139 / 125000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(307527121787241753561 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_615_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (89520072139 / 125000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(5559822753255195245599 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_615_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (89520072139 / 125000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(6380756920479711852741 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_615_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (89520072139 / 125000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(1853934321690048881789 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_615_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (89520072139 / 125000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(272088004562792016049 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_615_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (89520072139 / 125000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(5161329804599260628999 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_615_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (89520072139 / 125000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(6216882226043502495963 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_615_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (89520072139 / 125000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(268788866070650094343 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_615_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (89520072139 / 125000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_615_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/U51.lean b/LeanPool/Zeta5Irrational/Table/U51.lean
new file mode 100644
index 0000000000..0fb1081be0
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/U51.lean
@@ -0,0 +1,5291 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-! # Certified arcsine potential bounds (U51) -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma U_616_1 :
+    Uω (aρ 1) (bρ 1) (11489045952349 / 16000000000000 : ℝ) ≤
+      (-(106318913346585029177 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7116133783633341 / 10000000000000000 : ℝ) ≤
+      (-(106318913346585029177 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (7116133783633341 / 10000000000000000 : ℝ) =
+            (7116133783633341 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7116133783633341 / 5000000000000000 : ℝ) ≤
+            (110289580497164970823 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7116111065133557 / 10000000000000000 : ℝ)) (q :=
+      (7116133783633341 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_616_2 :
+    Uω (aρ 2) (bρ 2) (11489045952349 / 16000000000000 : ℝ) ≤
+      (-(3435841602808840498403 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14184475799498081 / 20000000000000000 : ℝ) ≤
+      (-(3435841602808840498403 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (14184475799498081 / 20000000000000000 : ℝ) =
+            (14184475799498081 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14184475799498081 / 10000000000000000 : ℝ) ≤
+            (3495630200191159501597 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1773022077468739 / 2500000000000000 : ℝ)) (q :=
+      (14184475799498081 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_616_3 :
+    Uω (aρ 3) (bρ 3) (11489045952349 / 16000000000000 : ℝ) ≤
+      (-(3503427083001394159311 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1761116571139903 / 2500000000000000 : ℝ) ≤
+      (-(3503427083001394159311 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1761116571139903 / 2500000000000000 : ℝ) =
+            (1761116571139903 / 1250000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1761116571139903 / 1250000000000000 : ℝ) ≤
+            (3428044719998605840689 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7043941185846099 / 10000000000000000 : ℝ)) (q :=
+      (1761116571139903 / 2500000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_616_4 :
+    Uω (aρ 4) (bρ 4) (11489045952349 / 16000000000000 : ℝ) ≤
+      (-(904151680374748223911 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13930373515006951 / 20000000000000000 : ℝ) ≤
+      (-(904151680374748223911 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (13930373515006951 / 20000000000000000 : ℝ) =
+            (13930373515006951 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13930373515006951 / 10000000000000000 : ℝ) ≤
+            (828716270375251776089 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3481838156399413 / 5000000000000000 : ℝ)) (q :=
+      (13930373515006951 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_616_5 :
+    Uω (aρ 5) (bρ 5) (11489045952349 / 16000000000000 : ℝ) ≤
+      (-(3791226089629521932043 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1368923372411711 / 2000000000000000 : ℝ) ≤
+      (-(3791226089629521932043 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1368923372411711 / 2000000000000000 : ℝ) =
+            (1368923372411711 / 1000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1368923372411711 / 1000000000000000 : ℝ) ≤
+            (3140245713370478067957 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1368145288248797 / 2000000000000000 : ℝ)) (q :=
+      (1368923372411711 / 2000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_616_6 :
+    Uω (aρ 6) (bρ 6) (11489045952349 / 16000000000000 : ℝ) ≤
+      (-(202327453614113887511 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3336035096403903 / 5000000000000000 : ℝ) ≤
+      (-(202327453614113887511 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (3336035096403903 / 5000000000000000 : ℝ) =
+            (3336035096403903 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3336035096403903 / 2500000000000000 : ℝ) ≤
+            (144246136535886112489 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6662873477882487 / 10000000000000000 : ℝ)) (q :=
+      (3336035096403903 / 5000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_616_7 :
+    Uω (aρ 7) (bρ 7) (11489045952349 / 16000000000000 : ℝ) ≤
+      (-(2202253901422898501509 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1609365418549547 / 2500000000000000 : ℝ) ≤
+      (-(2202253901422898501509 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1609365418549547 / 2500000000000000 : ℝ) =
+            (1609365418549547 / 1250000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1609365418549547 / 1250000000000000 : ℝ) ≤
+            (1263482000077101498491 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6417317821423251 / 10000000000000000 : ℝ)) (q :=
+      (1609365418549547 / 2500000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_616_8 :
+    Uω (aρ 8) (bρ 8) (11489045952349 / 16000000000000 : ℝ) ≤
+      (-(4889116153186919824119 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3066467650961623 / 5000000000000000 : ℝ) ≤
+      (-(4889116153186919824119 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (3066467650961623 / 5000000000000000 : ℝ) =
+            (3066467650961623 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3066467650961623 / 2500000000000000 : ℝ) ≤
+            (2042355649813080175881 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6091806728213367 / 10000000000000000 : ℝ)) (q :=
+      (3066467650961623 / 5000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_616_9 :
+    Uω (aρ 9) (bρ 9) (11489045952349 / 16000000000000 : ℝ) ≤
+      (-(1105240842738283865227 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2877199688669567 / 5000000000000000 : ℝ) ≤
+      (-(1105240842738283865227 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2877199688669567 / 5000000000000000 : ℝ) =
+            (2877199688669567 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2877199688669567 / 2500000000000000 : ℝ) ≤
+            (281053517861716134773 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5675700507405143 / 10000000000000000 : ℝ)) (q :=
+      (2877199688669567 / 5000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_616_10 :
+    Uω (aρ 10) (bρ 10) (11489045952349 / 16000000000000 : ℝ) ≤
+      (-(6343775738609869476143 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10605308731064837 / 20000000000000000 : ℝ) ≤
+      (-(6343775738609869476143 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10605308731064837 / 20000000000000000 : ℝ) =
+            (10605308731064837 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10605308731064837 / 10000000000000000 : ℝ) ≤
+            (587696064390130523857 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (645096189660839 / 1250000000000000 : ℝ)) (q :=
+      (10605308731064837 / 20000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_616_11 :
+    Uω (aρ 11) (bρ 11) (11489045952349 / 16000000000000 : ℝ) ≤
+      (-(3686848114669659335923 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9567411135391271 / 20000000000000000 : ℝ) ≤
+      (-(3686848114669659335923 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9567411135391271 / 20000000000000000 : ℝ) =
+            (9567411135391271 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9567411135391271 / 5000000000000000 : ℝ) ≤
+            (3244623688330340664077 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2270674273909073 / 5000000000000000 : ℝ)) (q :=
+      (9567411135391271 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_616_12 :
+    Uω (aρ 12) (bρ 12) (11489045952349 / 16000000000000 : ℝ) ≤
+      (-(4328354908750573873291 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2103845646309407 / 5000000000000000 : ℝ) ≤
+      (-(4328354908750573873291 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2103845646309407 / 5000000000000000 : ℝ) =
+            (2103845646309407 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2103845646309407 / 1250000000000000 : ℝ) ≤
+            (2603116894249426126709 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3813020094349503 / 10000000000000000 : ℝ)) (q :=
+      (2103845646309407 / 5000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_616_13 :
+    Uω (aρ 13) (bρ 13) (11489045952349 / 16000000000000 : ℝ) ≤
+      (-(10258168058087591677167 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (286802814271449 / 800000000000000 : ℝ) ≤
+      (-(10258168058087591677167 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (286802814271449 / 800000000000000 : ℝ) =
+            (286802814271449 / 200000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (286802814271449 / 200000000000000 : ℝ) ≤
+            (3604775547912408322833 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (29670896074931 / 100000000000000 : ℝ)) (q :=
+      (286802814271449 / 800000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13]) (by norm_num [bρ_13])
+      (by norm_num) (by norm_num [aρ_13, bρ_13]) (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_616_14 :
+    Uω (aρ 14) (bρ 14) (11489045952349 / 16000000000000 : ℝ) ≤
+      (-(6167965695391489451341 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2912442142317699 / 10000000000000000 : ℝ) ≤
+      (-(6167965695391489451341 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2912442142317699 / 10000000000000000 : ℝ) =
+            (2912442142317699 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2912442142317699 / 2500000000000000 : ℝ) ≤
+            (763506107608510548659 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1965830820387273 / 10000000000000000 : ℝ)) (q :=
+      (2912442142317699 / 10000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_616_15 :
+    Uω (aρ 15) (bρ 15) (11489045952349 / 16000000000000 : ℝ) ≤
+      (-(16171439084303750819383 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1984647236594003 / 10000000000000000 : ℝ) ≤
+      (-(16171439084303750819383 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1984647236594003 / 10000000000000000 : ℝ) =
+            (1984647236594003 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1984647236594003 / 1250000000000000 : ℝ) ≤
+            (4622976324696249180617 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (369814286354881 / 10000000000000000 : ℝ)) (q :=
+      (1984647236594003 / 10000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_616_16 :
+    Uω (aρ 16) (bρ 16) (11489045952349 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_616 :
+    Uρ (11489045952349 / 16000000000000 : ℝ) ≤
+      (-(292500798363171186281 / 500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (11489045952349 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(106318913346585029177 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_616_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (11489045952349 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3435841602808840498403 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_616_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (11489045952349 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3503427083001394159311 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_616_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (11489045952349 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(904151680374748223911 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_616_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (11489045952349 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(3791226089629521932043 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_616_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (11489045952349 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(202327453614113887511 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_616_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (11489045952349 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(2202253901422898501509 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_616_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (11489045952349 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(4889116153186919824119 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_616_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (11489045952349 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(1105240842738283865227 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_616_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (11489045952349 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(6343775738609869476143 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_616_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (11489045952349 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(3686848114669659335923 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_616_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (11489045952349 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(4328354908750573873291 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_616_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (11489045952349 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(10258168058087591677167 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_616_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (11489045952349 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(6167965695391489451341 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_616_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (11489045952349 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(16171439084303750819383 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_616_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (11489045952349 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_616_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_617_1 :
+    Uω (aρ 1) (bρ 1) (4601713724651 / 6400000000000 : ℝ) ≤
+      (-(847207624102700801571 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28502631154189897 / 40000000000000000 : ℝ) ≤
+      (-(847207624102700801571 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (28502631154189897 / 40000000000000000 : ℝ) =
+            (28502631154189897 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28502631154189897 / 20000000000000000 : ℝ) ≤
+            (885660326647299198429 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7125635100412761 / 10000000000000000 : ℝ)) (q :=
+      (28502631154189897 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_617_2 :
+    Uω (aρ 2) (bρ 2) (4601713724651 / 6400000000000 : ℝ) ≤
+      (-(3422421599130554510801 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5681409659930501 / 8000000000000000 : ℝ) ≤
+      (-(3422421599130554510801 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5681409659930501 / 8000000000000000 : ℝ) =
+            (5681409659930501 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5681409659930501 / 4000000000000000 : ℝ) ≤
+            (3509050203869445489199 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1420322537130813 / 2000000000000000 : ℝ)) (q :=
+      (5681409659930501 / 8000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_617_3 :
+    Uω (aρ 3) (bρ 3) (4601713724651 / 6400000000000 : ℝ) ≤
+      (-(109059856659853529079 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28215963872502457 / 40000000000000000 : ℝ) ≤
+      (-(109059856659853529079 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (28215963872502457 / 40000000000000000 : ℝ) =
+            (28215963872502457 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28215963872502457 / 20000000000000000 : ℝ) ≤
+            (107548637183896470921 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7053466578429041 / 10000000000000000 : ℝ)) (q :=
+      (28215963872502457 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_617_4 :
+    Uω (aρ 4) (bρ 4) (4601713724651 / 6400000000000 : ℝ) ≤
+      (-(3602939422167406557651 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27898851180049809 / 40000000000000000 : ℝ) ≤
+      (-(3602939422167406557651 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (27898851180049809 / 40000000000000000 : ℝ) =
+            (27898851180049809 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27898851180049809 / 20000000000000000 : ℝ) ≤
+            (3328532380832593442349 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6973204413267717 / 10000000000000000 : ℝ)) (q :=
+      (27898851180049809 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_617_5 :
+    Uω (aρ 5) (bρ 5) (4601713724651 / 6400000000000 : ℝ) ≤
+      (-(1888656661307937322841 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27416584981983811 / 40000000000000000 : ℝ) ≤
+      (-(1888656661307937322841 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (27416584981983811 / 40000000000000000 : ℝ) =
+            (27416584981983811 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27416584981983811 / 20000000000000000 : ℝ) ≤
+            (1577079240192062677159 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3425130616784859 / 5000000000000000 : ℝ)) (q :=
+      (27416584981983811 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_617_6 :
+    Uω (aρ 6) (bρ 6) (4601713724651 / 6400000000000 : ℝ) ≤
+      (-(1008066302970724523623 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (26726429177754317 / 40000000000000000 : ℝ) ≤
+      (-(1008066302970724523623 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (26726429177754317 / 40000000000000000 : ℝ) =
+            (26726429177754317 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26726429177754317 / 20000000000000000 : ℝ) ≤
+            (724801647779275476377 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6672423706594971 / 10000000000000000 : ℝ)) (q :=
+      (26726429177754317 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_617_7 :
+    Uω (aρ 7) (bρ 7) (4601713724651 / 6400000000000 : ℝ) ≤
+      (-(1097419455511665411643 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25788061999608657 / 40000000000000000 : ℝ) ≤
+      (-(1097419455511665411643 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (25788061999608657 / 40000000000000000 : ℝ) =
+            (25788061999608657 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25788061999608657 / 20000000000000000 : ℝ) ≤
+            (635448495238334588357 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6426901498282141 / 10000000000000000 : ℝ)) (q :=
+      (25788061999608657 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_617_8 :
+    Uω (aρ 8) (bρ 8) (4601713724651 / 6400000000000 : ℝ) ≤
+      (-(2436747227048758859281 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24570093904536439 / 40000000000000000 : ℝ) ≤
+      (-(2436747227048758859281 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (24570093904536439 / 40000000000000000 : ℝ) =
+            (24570093904536439 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24570093904536439 / 20000000000000000 : ℝ) ≤
+            (1028988674451241140719 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (381341193880377 / 625000000000000 : ℝ)) (q :=
+      (24570093904536439 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_617_9 :
+    Uω (aρ 9) (bρ 9) (4601713724651 / 6400000000000 : ℝ) ≤
+      (-(1377359600808868290677 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23056220745501421 / 40000000000000000 : ℝ) ≤
+      (-(1377359600808868290677 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23056220745501421 / 40000000000000000 : ℝ) =
+            (23056220745501421 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23056220745501421 / 20000000000000000 : ℝ) ≤
+            (355508349941131709323 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5685488150928523 / 10000000000000000 : ℝ)) (q :=
+      (23056220745501421 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_617_10 :
+    Uω (aρ 10) (bρ 10) (4601713724651 / 6400000000000 : ℝ) ≤
+      (-(158133477903931247259 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21249758747311093 / 40000000000000000 : ℝ) ≤
+      (-(158133477903931247259 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (21249758747311093 / 40000000000000000 : ℝ) =
+            (21249758747311093 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21249758747311093 / 20000000000000000 : ℝ) ≤
+            (15153317171068752741 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5170816185328359 / 10000000000000000 : ℝ)) (q :=
+      (21249758747311093 / 40000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_617_11 :
+    Uω (aρ 11) (bρ 11) (4601713724651 / 6400000000000 : ℝ) ≤
+      (-(3676374418941192739521 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19174946742517903 / 40000000000000000 : ℝ) ≤
+      (-(3676374418941192739521 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (19174946742517903 / 40000000000000000 : ℝ) =
+            (19174946742517903 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19174946742517903 / 10000000000000000 : ℝ) ≤
+            (3255097384058807260479 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1137971702284191 / 2500000000000000 : ℝ)) (q :=
+      (19174946742517903 / 40000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_617_12 :
+    Uω (aρ 12) (bρ 12) (4601713724651 / 6400000000000 : ℝ) ≤
+      (-(431588493602501169943 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16872793394296307 / 40000000000000000 : ℝ) ≤
+      (-(431588493602501169943 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (16872793394296307 / 40000000000000000 : ℝ) =
+            (16872793394296307 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16872793394296307 / 10000000000000000 : ℝ) ≤
+            (261558686697498830057 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1912255115855483 / 5000000000000000 : ℝ)) (q :=
+      (16872793394296307 / 40000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_617_13 :
+    Uω (aρ 13) (bρ 13) (4601713724651 / 6400000000000 : ℝ) ≤
+      (-(10226142048473857112987 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14386140081501321 / 40000000000000000 : ℝ) ≤
+      (-(10226142048473857112987 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (14386140081501321 / 40000000000000000 : ℝ) =
+            (14386140081501321 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14386140081501321 / 10000000000000000 : ℝ) ≤
+            (3636801557526142887013 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2980565316908473 / 10000000000000000 : ℝ)) (q :=
+      (14386140081501321 / 40000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_617_14 :
+    Uω (aρ 14) (bρ 14) (4601713724651 / 6400000000000 : ℝ) ≤
+      (-(12287712213345314582721 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11706078446739869 / 40000000000000000 : ℝ) ≤
+      (-(12287712213345314582721 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (11706078446739869 / 40000000000000000 : ℝ) =
+            (11706078446739869 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11706078446739869 / 10000000000000000 : ℝ) ≤
+            (1575231392654685417279 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1984461784572747 / 10000000000000000 : ℝ)) (q :=
+      (11706078446739869 / 40000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_617_15 :
+    Uω (aρ 15) (bρ 15) (4601713724651 / 6400000000000 : ℝ) ≤
+      (-(15939995460286118426859 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8124465222703667 / 40000000000000000 : ℝ) ≤
+      (-(15939995460286118426859 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (8124465222703667 / 40000000000000000 : ℝ) =
+            (8124465222703667 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8124465222703667 / 5000000000000000 : ℝ) ≤
+            (4854419948713881573141 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (226614224984823 / 5000000000000000 : ℝ)) (q :=
+      (8124465222703667 / 40000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_617_16 :
+    Uω (aρ 16) (bρ 16) (4601713724651 / 6400000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_617 :
+    Uρ (4601713724651 / 6400000000000 : ℝ) ≤
+      (-(728229669929658414691 / 1250000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (4601713724651 / 6400000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(847207624102700801571 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_617_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (4601713724651 / 6400000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3422421599130554510801 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_617_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (4601713724651 / 6400000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(109059856659853529079 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_617_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (4601713724651 / 6400000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3602939422167406557651 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_617_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (4601713724651 / 6400000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1888656661307937322841 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_617_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (4601713724651 / 6400000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1008066302970724523623 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_617_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (4601713724651 / 6400000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(1097419455511665411643 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_617_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (4601713724651 / 6400000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(2436747227048758859281 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_617_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (4601713724651 / 6400000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(1377359600808868290677 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_617_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (4601713724651 / 6400000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) * (-(158133477903931247259 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_617_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (4601713724651 / 6400000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(3676374418941192739521 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_617_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (4601713724651 / 6400000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(431588493602501169943 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_617_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (4601713724651 / 6400000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(10226142048473857112987 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_617_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (4601713724651 / 6400000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(12287712213345314582721 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_617_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (4601713724651 / 6400000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(15939995460286118426859 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_617_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (4601713724651 / 6400000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_617_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_618_1 :
+    Uω (aρ 1) (bρ 1) (5759761335453 / 8000000000000 : ℝ) ≤
+      (-(337547363029600780631 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1427036358676109 / 2000000000000000 : ℝ) ≤
+      (-(337547363029600780631 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1427036358676109 / 2000000000000000 : ℝ) =
+            (1427036358676109 / 1000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1427036358676109 / 1000000000000000 : ℝ) ≤
+            (355599817270399219369 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (89189489194123 / 125000000000000 : ℝ)) (q :=
+      (1427036358676109 / 2000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_618_2 :
+    Uω (aρ 2) (bρ 2) (5759761335453 / 8000000000000 : ℝ) ≤
+      (-(3409019581723940330361 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14222572499079663 / 20000000000000000 : ℝ) ≤
+      (-(3409019581723940330361 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (14222572499079663 / 20000000000000000 : ℝ) =
+            (14222572499079663 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14222572499079663 / 10000000000000000 : ℝ) ≤
+            (3522452221276059669639 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7111137060358413 / 10000000000000000 : ℝ)) (q :=
+      (14222572499079663 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_618_3 :
+    Uω (aρ 3) (bρ 3) (5759761335453 / 8000000000000 : ℝ) ≤
+      (-(173821098891361459987 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14127031299558699 / 20000000000000000 : ℝ) ≤
+      (-(173821098891361459987 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (14127031299558699 / 20000000000000000 : ℝ) =
+            (14127031299558699 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14127031299558699 / 10000000000000000 : ℝ) ≤
+            (172752491258638540013 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7062991967187449 / 10000000000000000 : ℝ)) (q :=
+      (14127031299558699 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_618_4 :
+    Uω (aρ 4) (bρ 4) (5759761335453 / 8000000000000 : ℝ) ≤
+      (-(179464539245497928923 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1396847765378547 / 2000000000000000 : ℝ) ≤
+      (-(179464539245497928923 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1396847765378547 / 2000000000000000 : ℝ) =
+            (1396847765378547 / 1000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1396847765378547 / 1000000000000000 : ℝ) ≤
+            (167109050904502071077 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (349136625123961 / 500000000000000 : ℝ)) (q :=
+      (1396847765378547 / 2000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_618_5 :
+    Uω (aρ 5) (bρ 5) (5759761335453 / 8000000000000 : ℝ) ≤
+      (-(940854976778895516571 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13727351227811363 / 20000000000000000 : ℝ) ≤
+      (-(940854976778895516571 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (13727351227811363 / 20000000000000000 : ℝ) =
+            (13727351227811363 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13727351227811363 / 10000000000000000 : ℝ) ≤
+            (792012973971104483429 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6859795995840113 / 10000000000000000 : ℝ)) (q :=
+      (13727351227811363 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_618_6 :
+    Uω (aρ 6) (bρ 6) (5759761335453 / 8000000000000 : ℝ) ≤
+      (-(803600356250984112259 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3345572179298549 / 5000000000000000 : ℝ) ≤
+      (-(803600356250984112259 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (3345572179298549 / 5000000000000000 : ℝ) =
+            (3345572179298549 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3345572179298549 / 2500000000000000 : ℝ) ≤
+            (582694004349015887741 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3340986930181473 / 5000000000000000 : ℝ)) (q :=
+      (3345572179298549 / 5000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_618_7 :
+    Uω (aρ 7) (bρ 7) (5759761335453 / 8000000000000 : ℝ) ≤
+      (-(2187434969384649431987 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2582627694795441 / 4000000000000000 : ℝ) ≤
+      (-(2187434969384649431987 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2582627694795441 / 4000000000000000 : ℝ) =
+            (2582627694795441 / 2000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2582627694795441 / 2000000000000000 : ℝ) ≤
+            (1278300932115350568013 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1287296999581191 / 2000000000000000 : ℝ)) (q :=
+      (2582627694795441 / 4000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_618_8 :
+    Uω (aρ 8) (bρ 8) (5759761335453 / 8000000000000 : ℝ) ≤
+      (-(1214474362038736148431 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12304222897779567 / 20000000000000000 : ℝ) ≤
+      (-(1214474362038736148431 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (12304222897779567 / 20000000000000000 : ℝ) =
+            (12304222897779567 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12304222897779567 / 10000000000000000 : ℝ) ≤
+            (518393588711263851569 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6111111073048317 / 10000000000000000 : ℝ)) (q :=
+      (12304222897779567 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_618_9 :
+    Uω (aρ 9) (bρ 9) (5759761335453 / 8000000000000 : ℝ) ≤
+      (-(1373175357305435607103 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11547421096770763 / 20000000000000000 : ℝ) ≤
+      (-(1373175357305435607103 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11547421096770763 / 20000000000000000 : ℝ) =
+            (11547421096770763 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11547421096770763 / 10000000000000000 : ℝ) ≤
+            (359692593444564392897 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5695274900399513 / 10000000000000000 : ℝ)) (q :=
+      (11547421096770763 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_618_10 :
+    Uω (aρ 10) (bρ 10) (5759761335453 / 8000000000000 : ℝ) ≤
+      (-(3153469138497098411407 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2128889608355591 / 4000000000000000 : ℝ) ≤
+      (-(3153469138497098411407 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2128889608355591 / 4000000000000000 : ℝ) =
+            (2128889608355591 / 2000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2128889608355591 / 2000000000000000 : ℝ) ≤
+            (312266763002901588593 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1036172175780341 / 2000000000000000 : ℝ)) (q :=
+      (2128889608355591 / 4000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_618_11 :
+    Uω (aρ 11) (bρ 11) (5759761335453 / 8000000000000 : ℝ) ≤
+      (-(7331849876332865554959 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4803765573513503 / 10000000000000000 : ℝ) ≤
+      (-(7331849876332865554959 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4803765573513503 / 10000000000000000 : ℝ) =
+            (4803765573513503 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4803765573513503 / 2500000000000000 : ℝ) ≤
+            (6531093729667134445041 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1140605152588939 / 2500000000000000 : ℝ)) (q :=
+      (4803765573513503 / 10000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_618_12 :
+    Uω (aρ 12) (bρ 12) (5759761335453 / 8000000000000 : ℝ) ≤
+      (-(8606904707458635042521 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1057175004768303 / 2500000000000000 : ℝ) ≤
+      (-(8606904707458635042521 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1057175004768303 / 2500000000000000 : ℝ) =
+            (1057175004768303 / 625000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1057175004768303 / 625000000000000 : ℝ) ≤
+            (5256038898541364957479 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1917994799080087 / 5000000000000000 : ℝ)) (q :=
+      (1057175004768303 / 2500000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_618_13 :
+    Uω (aρ 13) (bρ 13) (5759761335453 / 8000000000000 : ℝ) ≤
+      (-(2038852069256468634701 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7216039368045631 / 20000000000000000 : ℝ) ≤
+      (-(2038852069256468634701 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7216039368045631 / 20000000000000000 : ℝ) =
+            (7216039368045631 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7216039368045631 / 5000000000000000 : ℝ) ≤
+            (733736651943531365299 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2994010669654381 / 10000000000000000 : ℝ)) (q :=
+      (7216039368045631 / 20000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_618_14 :
+    Uω (aρ 14) (bρ 14) (5759761335453 / 8000000000000 : ℝ) ≤
+      (-(3059985522729845712219 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5881066152590859 / 20000000000000000 : ℝ) ≤
+      (-(3059985522729845712219 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5881066152590859 / 20000000000000000 : ℝ) =
+            (5881066152590859 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5881066152590859 / 5000000000000000 : ℝ) ≤
+            (405750378770154287781 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2002964739244609 / 10000000000000000 : ℝ)) (q :=
+      (5881066152590859 / 20000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_618_15 :
+    Uω (aρ 15) (bρ 15) (5759761335453 / 8000000000000 : ℝ) ≤
+      (-(15745009285213127229491 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (517777224673183 / 2500000000000000 : ℝ) ≤
+      (-(15745009285213127229491 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (517777224673183 / 2500000000000000 : ℝ) =
+            (517777224673183 / 312500000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (517777224673183 / 312500000000000 : ℝ) ≤
+            (5049406123786872770509 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (261844830727107 / 5000000000000000 : ℝ)) (q :=
+      (517777224673183 / 2500000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15]) (by norm_num [bρ_15])
+      (by norm_num) (by norm_num [aρ_15, bρ_15]) (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_618_16 :
+    Uω (aρ 16) (bρ 16) (5759761335453 / 8000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_618 :
+    Uρ (5759761335453 / 8000000000000 : ℝ) ≤
+      (-(580282206778786842289 / 1000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (5759761335453 / 8000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(337547363029600780631 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_618_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (5759761335453 / 8000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3409019581723940330361 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_618_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (5759761335453 / 8000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(173821098891361459987 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_618_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (5759761335453 / 8000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(179464539245497928923 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_618_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (5759761335453 / 8000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(940854976778895516571 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_618_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (5759761335453 / 8000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(803600356250984112259 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_618_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (5759761335453 / 8000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(2187434969384649431987 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_618_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (5759761335453 / 8000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(1214474362038736148431 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_618_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (5759761335453 / 8000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(1373175357305435607103 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_618_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (5759761335453 / 8000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(3153469138497098411407 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_618_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (5759761335453 / 8000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(7331849876332865554959 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_618_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (5759761335453 / 8000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(8606904707458635042521 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_618_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (5759761335453 / 8000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(2038852069256468634701 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_618_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (5759761335453 / 8000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(3059985522729845712219 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_618_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (5759761335453 / 8000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(15745009285213127229491 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_618_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (5759761335453 / 8000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_618_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_619_1 :
+    Uω (aρ 1) (bρ 1) (23069522060369 / 32000000000000 : ℝ) ≤
+      (-(840533645271613096197 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28578823192531509 / 40000000000000000 : ℝ) ≤
+      (-(840533645271613096197 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (28578823192531509 / 40000000000000000 : ℝ) =
+            (28578823192531509 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28578823192531509 / 20000000000000000 : ℝ) ≤
+            (892334305478386903803 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3572341585242721 / 5000000000000000 : ℝ)) (q :=
+      (28578823192531509 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_619_2 :
+    Uω (aρ 2) (bρ 2) (23069522060369 / 32000000000000 : ℝ) ≤
+      (-(1697817751219501552147 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28483241694525251 / 40000000000000000 : ℝ) ≤
+      (-(1697817751219501552147 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (28483241694525251 / 40000000000000000 : ℝ) =
+            (28483241694525251 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28483241694525251 / 20000000000000000 : ℝ) ≤
+            (1767918150280498447853 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7120661433992313 / 10000000000000000 : ℝ)) (q :=
+      (28483241694525251 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_619_3 :
+    Uω (aρ 3) (bρ 3) (23069522060369 / 32000000000000 : ℝ) ≤
+      (-(3462946727979391038407 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28292161318114177 / 40000000000000000 : ℝ) ≤
+      (-(3462946727979391038407 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (28292161318114177 / 40000000000000000 : ℝ) =
+            (28292161318114177 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28292161318114177 / 20000000000000000 : ℝ) ≤
+            (3468525075020608961593 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (884064669017097 / 1250000000000000 : ℝ)) (q :=
+      (28292161318114177 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_619_4 :
+    Uω (aρ 4) (bρ 4) (23069522060369 / 32000000000000 : ℝ) ≤
+      (-(3575660758809798843429 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5595011882533867 / 8000000000000000 : ℝ) ≤
+      (-(3575660758809798843429 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5595011882533867 / 8000000000000000 : ℝ) =
+            (5595011882533867 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5595011882533867 / 4000000000000000 : ℝ) ≤
+            (3355811044190201156571 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1398452116095871 / 2000000000000000 : ℝ)) (q :=
+      (5595011882533867 / 8000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_619_5 :
+    Uω (aρ 5) (bρ 5) (23069522060369 / 32000000000000 : ℝ) ≤
+      (-(3749545789310065404481 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27492819869401271 / 40000000000000000 : ℝ) ≤
+      (-(3749545789310065404481 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (27492819869401271 / 40000000000000000 : ℝ) =
+            (27492819869401271 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27492819869401271 / 20000000000000000 : ℝ) ≤
+            (3181926013689934595519 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6869330728180323 / 10000000000000000 : ℝ)) (q :=
+      (27492819869401271 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_619_6 :
+    Uω (aρ 6) (bρ 6) (23069522060369 / 32000000000000 : ℝ) ≤
+      (-(4003758721882345394807 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5360545108355043 / 8000000000000000 : ℝ) ≤
+      (-(4003758721882345394807 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5360545108355043 / 8000000000000000 : ℝ) =
+            (5360545108355043 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5360545108355043 / 4000000000000000 : ℝ) ≤
+            (2927713081117654605193 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1338304787901459 / 2000000000000000 : ℝ)) (q :=
+      (5360545108355043 / 8000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_619_7 :
+    Uω (aρ 7) (bρ 7) (23069522060369 / 32000000000000 : ℝ) ≤
+      (-(872016817370518790959 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25864491543411003 / 40000000000000000 : ℝ) ≤
+      (-(872016817370518790959 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (25864491543411003 / 40000000000000000 : ℝ) =
+            (25864491543411003 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25864491543411003 / 20000000000000000 : ℝ) ≤
+            (514277543229481209041 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6446068321085189 / 10000000000000000 : ℝ)) (q :=
+      (25864491543411003 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_619_8 :
+    Uω (aρ 8) (bρ 8) (23069522060369 / 32000000000000 : ℝ) ≤
+      (-(4842325056394238771433 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24646796884573221 / 40000000000000000 : ℝ) ≤
+      (-(4842325056394238771433 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (24646796884573221 / 40000000000000000 : ℝ) =
+            (24646796884573221 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24646796884573221 / 20000000000000000 : ℝ) ≤
+            (2089146746605761228567 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3060381321503149 / 5000000000000000 : ℝ)) (q :=
+      (24646796884573221 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_619_9 :
+    Uω (aρ 9) (bρ 9) (23069522060369 / 32000000000000 : ℝ) ≤
+      (-(5475993190011855433897 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23133461862679071 / 40000000000000000 : ℝ) ≤
+      (-(5475993190011855433897 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23133461862679071 / 40000000000000000 : ℝ) =
+            (23133461862679071 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23133461862679071 / 20000000000000000 : ℝ) ≤
+            (1455478612988144566103 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5705060760419223 / 10000000000000000 : ℝ)) (q :=
+      (23133461862679071 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_619_10 :
+    Uω (aρ 10) (bρ 10) (23069522060369 / 32000000000000 : ℝ) ≤
+      (-(6288573075515474583353 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (170624235950307 / 320000000000000 : ℝ) ≤
+      (-(6288573075515474583353 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (170624235950307 / 320000000000000 : ℝ) =
+            (170624235950307 / 160000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (170624235950307 / 160000000000000 : ℝ) ≤
+            (642898727484525416647 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (41527228875751 / 80000000000000 : ℝ)) (q :=
+      (170624235950307 / 320000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10]) (by norm_num [bρ_10])
+      (by norm_num) (by norm_num [aρ_10, bρ_10]) (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_619_11 :
+    Uω (aρ 11) (bρ 11) (23069522060369 / 32000000000000 : ℝ) ≤
+      (-(1827749775245038889841 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19255168987033969 / 40000000000000000 : ℝ) ≤
+      (-(1827749775245038889841 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (19255168987033969 / 40000000000000000 : ℝ) =
+            (19255168987033969 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19255168987033969 / 10000000000000000 : ℝ) ≤
+            (1637986126254961110159 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (142904686946771 / 312500000000000 : ℝ)) (q :=
+      (19255168987033969 / 40000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_619_12 :
+    Uω (aρ 12) (bρ 12) (23069522060369 / 32000000000000 : ℝ) ≤
+      (-(343284552296869609383 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16956785409282959 / 40000000000000000 : ℝ) ≤
+      (-(343284552296869609383 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (16956785409282959 / 40000000000000000 : ℝ) =
+            (16956785409282959 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16956785409282959 / 10000000000000000 : ℝ) ≤
+            (211233191943130390617 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3847458290106167 / 10000000000000000 : ℝ)) (q :=
+      (16956785409282959 / 40000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_619_13 :
+    Uω (aρ 13) (bρ 13) (23069522060369 / 32000000000000 : ℝ) ≤
+      (-(1016252133793236407923 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14477957491638911 / 40000000000000000 : ℝ) ≤
+      (-(1016252133793236407923 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (14477957491638911 / 40000000000000000 : ℝ) =
+            (14477957491638911 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14477957491638911 / 10000000000000000 : ℝ) ≤
+            (370042226806763592077 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3007426072879143 / 10000000000000000 : ℝ)) (q :=
+      (14477957491638911 / 40000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_619_14 :
+    Uω (aρ 14) (bρ 14) (23069522060369 / 32000000000000 : ℝ) ≤
+      (-(12192609767079604075557 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (472717487009481 / 1600000000000000 : ℝ) ≤
+      (-(12192609767079604075557 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (472717487009481 / 1600000000000000 : ℝ) =
+            (472717487009481 / 400000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (472717487009481 / 400000000000000 : ℝ) ≤
+            (1670333838920395924443 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1263339499827 / 6250000000000 : ℝ)) (q :=
+      (472717487009481 / 1600000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14]) (by norm_num [bρ_14])
+      (by norm_num) (by norm_num [aρ_14, bρ_14]) (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_619_15 :
+    Uω (aρ 15) (bρ 15) (23069522060369 / 32000000000000 : ℝ) ≤
+      (-(3893334229564846221071 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8427884246986999 / 40000000000000000 : ℝ) ≤
+      (-(3893334229564846221071 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (8427884246986999 / 40000000000000000 : ℝ) =
+            (8427884246986999 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8427884246986999 / 5000000000000000 : ℝ) ≤
+            (1305269622685153778929 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (585890013013187 / 10000000000000000 : ℝ)) (q :=
+      (8427884246986999 / 40000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_619_16 :
+    Uω (aρ 16) (bρ 16) (23069522060369 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_619 :
+    Uρ (23069522060369 / 32000000000000 : ℝ) ≤
+      (-(5780568999489780155329 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (23069522060369 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(840533645271613096197 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_619_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (23069522060369 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1697817751219501552147 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_619_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (23069522060369 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3462946727979391038407 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_619_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (23069522060369 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3575660758809798843429 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_619_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (23069522060369 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(3749545789310065404481 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_619_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (23069522060369 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(4003758721882345394807 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_619_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (23069522060369 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(872016817370518790959 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_619_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (23069522060369 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(4842325056394238771433 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_619_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (23069522060369 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(5475993190011855433897 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_619_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (23069522060369 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(6288573075515474583353 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_619_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (23069522060369 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(1827749775245038889841 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_619_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (23069522060369 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(343284552296869609383 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_619_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (23069522060369 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(1016252133793236407923 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_619_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (23069522060369 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(12192609767079604075557 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_619_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (23069522060369 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(3893334229564846221071 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_619_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (23069522060369 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_619_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_620_1 :
+    Uω (aρ 1) (bρ 1) (11549999389463 / 16000000000000 : ℝ) ≤
+      (-(133952532052508978969 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3577114901402397 / 5000000000000000 : ℝ) ≤
+      (-(133952532052508978969 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (3577114901402397 / 5000000000000000 : ℝ) =
+            (3577114901402397 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3577114901402397 / 2500000000000000 : ℝ) ≤
+            (143306340067491021031 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7154207205280213 / 10000000000000000 : ℝ)) (q :=
+      (3577114901402397 / 5000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_620_2 :
+    Uω (aρ 2) (bρ 2) (11549999389463 / 16000000000000 : ℝ) ≤
+      (-(3382269313318846887479 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1426066919437943 / 2000000000000000 : ℝ) ≤
+      (-(3382269313318846887479 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1426066919437943 / 2000000000000000 : ℝ) =
+            (1426066919437943 / 1000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1426066919437943 / 1000000000000000 : ℝ) ≤
+            (3549202489681153112521 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1426037161312011 / 2000000000000000 : ℝ)) (q :=
+      (1426066919437943 / 2000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_620_3 :
+    Uω (aρ 3) (bρ 3) (11549999389463 / 16000000000000 : ℝ) ≤
+      (-(1724744807306295194459 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1770641251845221 / 2500000000000000 : ℝ) ≤
+      (-(1724744807306295194459 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1770641251845221 / 2500000000000000 : ℝ) =
+            (1770641251845221 / 1250000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1770641251845221 / 1250000000000000 : ℝ) ≤
+            (1740991094193704805541 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7082042733292393 / 10000000000000000 : ℝ)) (q :=
+      (1770641251845221 / 2500000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_620_4 :
+    Uω (aρ 4) (bρ 4) (11549999389463 / 16000000000000 : ℝ) ≤
+      (-(1781024646579131442769 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2801316349543653 / 4000000000000000 : ℝ) ≤
+      (-(1781024646579131442769 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2801316349543653 / 4000000000000000 : ℝ) =
+            (2801316349543653 / 2000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2801316349543653 / 2000000000000000 : ℝ) ≤
+            (1684711254920868557231 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (700178864731389 / 1000000000000000 : ℝ)) (q :=
+      (2801316349543653 / 4000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_620_5 :
+    Uω (aρ 5) (bρ 5) (11549999389463 / 16000000000000 : ℝ) ≤
+      (-(233480682225324909683 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13765468611784181 / 20000000000000000 : ℝ) ≤
+      (-(233480682225324909683 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (13765468611784181 / 20000000000000000 : ℝ) =
+            (13765468611784181 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13765468611784181 / 10000000000000000 : ℝ) ≤
+            (199736305462175090317 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3439432715357403 / 5000000000000000 : ℝ)) (q :=
+      (13765468611784181 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_620_6 :
+    Uω (aρ 6) (bρ 6) (11549999389463 / 16000000000000 : ℝ) ≤
+      (-(124672999234403041657 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2684087350055289 / 4000000000000000 : ℝ) ≤
+      (-(124672999234403041657 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2684087350055289 / 4000000000000000 : ℝ) =
+            (2684087350055289 / 2000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2684087350055289 / 2000000000000000 : ℝ) ≤
+            (91935494609346958343 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (670107394434707 / 1000000000000000 : ℝ)) (q :=
+      (2684087350055289 / 4000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_620_7 :
+    Uω (aρ 7) (bρ 7) (11549999389463 / 16000000000000 : ℝ) ≤
+      (-(543165025054239187137 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (404729777930469 / 625000000000000 : ℝ) ≤
+      (-(543165025054239187137 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (404729777930469 / 625000000000000 : ℝ) =
+            (404729777930469 / 312500000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (404729777930469 / 312500000000000 : ℝ) ≤
+            (323268950320760812863 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6455651468605633 / 10000000000000000 : ℝ)) (q :=
+      (404729777930469 / 625000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_620_8 :
+    Uω (aρ 8) (bρ 8) (11549999389463 / 16000000000000 : ℝ) ≤
+      (-(2413388600116685197157 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12342573587683343 / 20000000000000000 : ℝ) ≤
+      (-(2413388600116685197157 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (12342573587683343 / 20000000000000000 : ℝ) =
+            (12342573587683343 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12342573587683343 / 10000000000000000 : ℝ) ≤
+            (1052347301383314802843 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (383150863365873 / 625000000000000 : ℝ)) (q :=
+      (12342573587683343 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_620_9 :
+    Uω (aρ 9) (bρ 9) (11549999389463 / 16000000000000 : ℝ) ≤
+      (-(1364828396129223884731 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11586039881026203 / 20000000000000000 : ℝ) ≤
+      (-(1364828396129223884731 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11586039881026203 / 20000000000000000 : ℝ) =
+            (11586039881026203 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11586039881026203 / 10000000000000000 : ℝ) ≤
+            (368039554620776115269 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1428711433889207 / 2500000000000000 : ℝ)) (q :=
+      (11586039881026203 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_620_10 :
+    Uω (aρ 10) (bρ 10) (11549999389463 / 16000000000000 : ℝ) ≤
+      (-(3135121683520498290713 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10683579500375607 / 20000000000000000 : ℝ) ≤
+      (-(3135121683520498290713 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10683579500375607 / 20000000000000000 : ℝ) =
+            (10683579500375607 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10683579500375607 / 10000000000000000 : ℝ) ≤
+            (330614217979501709287 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (325059024275077 / 625000000000000 : ℝ)) (q :=
+      (10683579500375607 / 20000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_620_11 :
+    Uω (aρ 11) (bρ 11) (11549999389463 / 16000000000000 : ℝ) ≤
+      (-(3645098135037624419509 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9647633441254317 / 20000000000000000 : ℝ) ≤
+      (-(3645098135037624419509 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9647633441254317 / 20000000000000000 : ℝ) =
+            (9647633441254317 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9647633441254317 / 5000000000000000 : ℝ) ≤
+            (3286373667962375580491 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2291737477742471 / 5000000000000000 : ℝ)) (q :=
+      (9647633441254317 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_620_12 :
+    Uω (aρ 12) (bρ 12) (11549999389463 / 16000000000000 : ℝ) ≤
+      (-(8557396661480946125929 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8499374791807127 / 20000000000000000 : ℝ) ≤
+      (-(8557396661480946125929 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8499374791807127 / 20000000000000000 : ℝ) =
+            (8499374791807127 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8499374791807127 / 5000000000000000 : ℝ) ≤
+            (5305546944519053874071 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (60295568792543 / 156250000000000 : ℝ)) (q :=
+      (8499374791807127 / 20000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_620_13 :
+    Uω (aρ 13) (bρ 13) (11549999389463 / 16000000000000 : ℝ) ≤
+      (-(506546171996047407941 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7261888573088961 / 20000000000000000 : ℝ) ≤
+      (-(506546171996047407941 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7261888573088961 / 20000000000000000 : ℝ) =
+            (7261888573088961 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7261888573088961 / 5000000000000000 : ℝ) ≤
+            (186601008303952592059 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1510405962799793 / 5000000000000000 : ℝ)) (q :=
+      (7261888573088961 / 20000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_620_14 :
+    Uω (aρ 14) (bρ 14) (11549999389463 / 16000000000000 : ℝ) ≤
+      (-(12145704453235629212067 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5936749893837629 / 20000000000000000 : ℝ) ≤
+      (-(12145704453235629212067 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5936749893837629 / 20000000000000000 : ℝ) =
+            (5936749893837629 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5936749893837629 / 5000000000000000 : ℝ) ≤
+            (1717239152764370787933 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1019800265696627 / 5000000000000000 : ℝ)) (q :=
+      (5936749893837629 / 20000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_620_15 :
+    Uω (aρ 15) (bρ 15) (11549999389463 / 16000000000000 : ℝ) ≤
+      (-(15418235905060781715109 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4279810283251159 / 20000000000000000 : ℝ) ≤
+      (-(15418235905060781715109 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4279810283251159 / 20000000000000000 : ℝ) =
+            (4279810283251159 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4279810283251159 / 2500000000000000 : ℝ) ≤
+            (5376179503939218284891 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (80279274777723 / 1250000000000000 : ℝ)) (q :=
+      (4279810283251159 / 20000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_620_16 :
+    Uω (aρ 16) (bρ 16) (11549999389463 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_620 :
+    Uρ (11549999389463 / 16000000000000 : ℝ) ≤
+      (-(71985901044622510087 / 125000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (11549999389463 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(133952532052508978969 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_620_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (11549999389463 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3382269313318846887479 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_620_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (11549999389463 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1724744807306295194459 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_620_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (11549999389463 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1781024646579131442769 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_620_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (11549999389463 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(233480682225324909683 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_620_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (11549999389463 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(124672999234403041657 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_620_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (11549999389463 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(543165025054239187137 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_620_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (11549999389463 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(2413388600116685197157 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_620_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (11549999389463 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(1364828396129223884731 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_620_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (11549999389463 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(3135121683520498290713 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_620_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (11549999389463 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(3645098135037624419509 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_620_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (11549999389463 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(8557396661480946125929 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_620_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (11549999389463 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) * (-(506546171996047407941 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_620_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (11549999389463 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(12145704453235629212067 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_620_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (11549999389463 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(15418235905060781715109 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_620_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (11549999389463 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_620_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_621_1 :
+    Uω (aρ 1) (bρ 1) (23130475497483 / 32000000000000 : ℝ) ≤
+      (-(667101948738974140143 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28655015229586461 / 40000000000000000 : ℝ) ≤
+      (-(667101948738974140143 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (28655015229586461 / 40000000000000000 : ℝ) =
+            (28655015229586461 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28655015229586461 / 20000000000000000 : ℝ) ≤
+            (719192411861025859857 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7163731239914793 / 10000000000000000 : ℝ)) (q :=
+      (28655015229586461 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_621_2 :
+    Uω (aρ 2) (bρ 2) (23130475497483 / 32000000000000 : ℝ) ≤
+      (-(3368920966598643556611 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28559435080868687 / 40000000000000000 : ℝ) ≤
+      (-(3368920966598643556611 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (28559435080868687 / 40000000000000000 : ℝ) =
+            (28559435080868687 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28559435080868687 / 20000000000000000 : ℝ) ≤
+            (3562550836401356443389 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3569855089032953 / 5000000000000000 : ℝ)) (q :=
+      (28559435080868687 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_621_3 :
+    Uω (aρ 3) (bρ 3) (23130475497483 / 32000000000000 : ℝ) ≤
+      (-(214753161810318558973 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28368358733376049 / 40000000000000000 : ℝ) ≤
+      (-(214753161810318558973 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (28368358733376049 / 40000000000000000 : ℝ) =
+            (28368358733376049 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28368358733376049 / 20000000000000000 : ℝ) ≤
+            (218463825877181441027 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7091568110669587 / 10000000000000000 : ℝ)) (q :=
+      (28368358733376049 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_621_4 :
+    Uω (aρ 4) (bρ 4) (23130475497483 / 32000000000000 : ℝ) ≤
+      (-(709691267490747128891 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28051267555963567 / 40000000000000000 : ℝ) ≤
+      (-(709691267490747128891 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (28051267555963567 / 40000000000000000 : ℝ) =
+            (28051267555963567 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28051267555963567 / 20000000000000000 : ℝ) ≤
+            (676603093109252871109 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3505658351514173 / 5000000000000000 : ℝ)) (q :=
+      (28051267555963567 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_621_5 :
+    Uω (aρ 5) (bρ 5) (23130475497483 / 32000000000000 : ℝ) ≤
+      (-(3721855232630060188191 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5513810903674307 / 8000000000000000 : ℝ) ≤
+      (-(3721855232630060188191 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5513810903674307 / 8000000000000000 : ℝ) =
+            (5513810903674307 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5513810903674307 / 4000000000000000 : ℝ) ≤
+            (3209616570369939811809 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (688840010356733 / 1000000000000000 : ℝ)) (q :=
+      (5513810903674307 / 8000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_621_6 :
+    Uω (aρ 6) (bρ 6) (23130475497483 / 32000000000000 : ℝ) ≤
+      (-(1987666742048291147841 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (26879021311355879 / 40000000000000000 : ℝ) ≤
+      (-(1987666742048291147841 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (26879021311355879 / 40000000000000000 : ℝ) =
+            (26879021311355879 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26879021311355879 / 20000000000000000 : ℝ) ≤
+            (1478069159451708852159 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3355311937599751 / 5000000000000000 : ℝ)) (q :=
+      (26879021311355879 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_621_7 :
+    Uω (aρ 7) (bρ 7) (23130475497483 / 32000000000000 : ℝ) ≤
+      (-(4330578213947306030187 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25940919681933677 / 40000000000000000 : ℝ) ≤
+      (-(4330578213947306030187 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (25940919681933677 / 40000000000000000 : ℝ) =
+            (25940919681933677 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25940919681933677 / 20000000000000000 : ℝ) ≤
+            (2600893589052693969813 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6465234441248401 / 10000000000000000 : ℝ)) (q :=
+      (25940919681933677 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_621_8 :
+    Uω (aρ 8) (bρ 8) (23130475497483 / 32000000000000 : ℝ) ≤
+      (-(4811253801470739519341 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24723496671703543 / 40000000000000000 : ℝ) ≤
+      (-(4811253801470739519341 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (24723496671703543 / 40000000000000000 : ℝ) =
+            (24723496671703543 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24723496671703543 / 20000000000000000 : ℝ) ≤
+            (2120218001529260480659 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3070032293736667 / 5000000000000000 : ℝ)) (q :=
+      (24723496671703543 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_621_9 :
+    Uω (aρ 9) (bρ 9) (23130475497483 / 32000000000000 : ℝ) ≤
+      (-(1360665628048339499749 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23210695900736581 / 40000000000000000 : ℝ) ≤
+      (-(1360665628048339499749 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23210695900736581 / 40000000000000000 : ℝ) =
+            (23210695900736581 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23210695900736581 / 20000000000000000 : ℝ) ≤
+            (372202322701660500251 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5724629830349853 / 10000000000000000 : ℝ)) (q :=
+      (23210695900736581 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_621_10 :
+    Uω (aρ 10) (bρ 10) (23130475497483 / 32000000000000 : ℝ) ≤
+      (-(1562987251951932787287 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21406284627007463 / 40000000000000000 : ℝ) ≤
+      (-(1562987251951932787287 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (21406284627007463 / 40000000000000000 : ℝ) =
+            (21406284627007463 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21406284627007463 / 20000000000000000 : ℝ) ≤
+            (169880698798067212713 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2605491613490147 / 5000000000000000 : ℝ)) (q :=
+      (21406284627007463 / 40000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_621_11 :
+    Uω (aρ 11) (bρ 11) (23130475497483 / 32000000000000 : ℝ) ≤
+      (-(726944114380871616587 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19335356040944007 / 40000000000000000 : ℝ) ≤
+      (-(726944114380871616587 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (19335356040944007 / 40000000000000000 : ℝ) =
+            (19335356040944007 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19335356040944007 / 10000000000000000 : ℝ) ≤
+            (659350246219128383413 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2296997780076783 / 5000000000000000 : ℝ)) (q :=
+      (19335356040944007 / 40000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_621_12 :
+    Uω (aρ 12) (bρ 12) (23130475497483 / 32000000000000 : ℝ) ≤
+      (-(8532752764931727160707 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17040692787205723 / 40000000000000000 : ℝ) ≤
+      (-(8532752764931727160707 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (17040692787205723 / 40000000000000000 : ℝ) =
+            (17040692787205723 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17040692787205723 / 10000000000000000 : ℝ) ≤
+            (5330190841068272839293 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (241897751873089 / 625000000000000 : ℝ)) (q :=
+      (17040692787205723 / 40000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_621_13 :
+    Uω (aρ 13) (bρ 13) (23130475497483 / 32000000000000 : ℝ) ≤
+      (-(5049732549018707484119 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14569538481916747 / 40000000000000000 : ℝ) ≤
+      (-(5049732549018707484119 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (14569538481916747 / 40000000000000000 : ℝ) =
+            (14569538481916747 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14569538481916747 / 10000000000000000 : ℝ) ≤
+            (1881739253981292515881 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1481527645957 / 4882812500000 : ℝ)) (q :=
+      (14569538481916747 / 40000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_621_14 :
+    Uω (aρ 14) (bρ 14) (23130475497483 / 32000000000000 : ℝ) ≤
+      (-(12099215801753549008383 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11928826590805561 / 40000000000000000 : ℝ) ≤
+      (-(12099215801753549008383 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (11928826590805561 / 40000000000000000 : ℝ) =
+            (11928826590805561 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11928826590805561 / 10000000000000000 : ℝ) ≤
+            (1763727804246450991617 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2057739958409343 / 10000000000000000 : ℝ)) (q :=
+      (11928826590805561 / 40000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_621_15 :
+    Uω (aρ 15) (bρ 15) (23130475497483 / 32000000000000 : ℝ) ≤
+      (-(1909462497718683193977 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1736499912346799 / 8000000000000000 : ℝ) ≤
+      (-(1909462497718683193977 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (1736499912346799 / 8000000000000000 : ℝ) =
+            (1736499912346799 / 1000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1736499912346799 / 1000000000000000 : ℝ) ≤
+            (689839428406316806023 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (8676871516107 / 125000000000000 : ℝ)) (q :=
+      (1736499912346799 / 8000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_621_16 :
+    Uω (aρ 16) (bρ 16) (23130475497483 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_621 :
+    Uρ (23130475497483 / 32000000000000 : ℝ) ≤
+      (-(1147521724675301463431 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (23130475497483 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(667101948738974140143 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_621_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (23130475497483 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3368920966598643556611 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_621_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (23130475497483 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(214753161810318558973 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_621_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (23130475497483 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(709691267490747128891 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_621_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (23130475497483 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(3721855232630060188191 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_621_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (23130475497483 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1987666742048291147841 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_621_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (23130475497483 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(4330578213947306030187 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_621_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (23130475497483 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(4811253801470739519341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_621_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (23130475497483 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(1360665628048339499749 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_621_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (23130475497483 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(1562987251951932787287 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_621_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (23130475497483 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(726944114380871616587 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_621_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (23130475497483 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(8532752764931727160707 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_621_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (23130475497483 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(5049732549018707484119 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_621_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (23130475497483 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(12099215801753549008383 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_621_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (23130475497483 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(1909462497718683193977 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_621_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (23130475497483 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_621_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_622_1 :
+    Uω (aρ 1) (bρ 1) (579023805401 / 800000000000 : ℝ) ≤
+      (-(664444772228277286747 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14346555623817321 / 20000000000000000 : ℝ) ≤
+      (-(664444772228277286747 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (14346555623817321 / 20000000000000000 : ℝ) =
+            (14346555623817321 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14346555623817321 / 10000000000000000 : ℝ) ≤
+            (721849588371722713253 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7173255274389821 / 10000000000000000 : ℝ)) (q :=
+      (14346555623817321 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_622_2 :
+    Uω (aρ 2) (bρ 2) (579023805401 / 800000000000 : ℝ) ≤
+      (-(1677795207352304648309 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14298765885431609 / 20000000000000000 : ℝ) ≤
+      (-(1677795207352304648309 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (14298765885431609 / 20000000000000000 : ℝ) =
+            (14298765885431609 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14298765885431609 / 10000000000000000 : ℝ) ≤
+            (1787940694147695351691 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7149234548514109 / 10000000000000000 : ℝ)) (q :=
+      (14298765885431609 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_622_3 :
+    Uω (aρ 3) (bρ 3) (579023805401 / 800000000000 : ℝ) ≤
+      (-(3422629602471597769299 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14203228714851061 / 20000000000000000 : ℝ) ≤
+      (-(3422629602471597769299 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (14203228714851061 / 20000000000000000 : ℝ) =
+            (14203228714851061 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14203228714851061 / 10000000000000000 : ℝ) ≤
+            (3508842200528402230701 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7101093484283561 / 10000000000000000 : ℝ)) (q :=
+      (14203228714851061 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_622_4 :
+    Uω (aρ 4) (bρ 4) (579023805401 / 800000000000 : ℝ) ≤
+      (-(17674409207002628977 / 50000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7022342898585249 / 10000000000000000 : ℝ) ≤
+      (-(17674409207002628977 / 50000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (7022342898585249 / 10000000000000000 : ℝ) =
+            (7022342898585249 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7022342898585249 / 5000000000000000 : ℝ) ≤
+            (16982949807997371023 / 50000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3510422373833999 / 5000000000000000 : ℝ)) (q :=
+      (7022342898585249 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_622_5 :
+    Uω (aρ 5) (bρ 5) (579023805401 / 800000000000 : ℝ) ≤
+      (-(23175241795223090519 / 62500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6901792938514239 / 10000000000000000 : ℝ) ≤
+      (-(23175241795223090519 / 62500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (6901792938514239 / 10000000000000000 : ℝ) =
+            (6901792938514239 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6901792938514239 / 5000000000000000 : ℝ) ≤
+            (20146456973526909481 / 62500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3448967373430489 / 5000000000000000 : ℝ)) (q :=
+      (6901792938514239 / 10000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_622_6 :
+    Uω (aρ 6) (bρ 6) (579023805401 / 800000000000 : ℝ) ≤
+      (-(1980575594952124207391 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13458584487407519 / 20000000000000000 : ℝ) ≤
+      (-(1980575594952124207391 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (13458584487407519 / 20000000000000000 : ℝ) =
+            (13458584487407519 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13458584487407519 / 10000000000000000 : ℝ) ≤
+            (1485160306547875792609 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6720173732380019 / 10000000000000000 : ℝ)) (q :=
+      (13458584487407519 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_622_7 :
+    Uω (aρ 7) (bρ 7) (579023805401 / 800000000000 : ℝ) ≤
+      (-(4315858062121709975551 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12989566614057469 / 20000000000000000 : ℝ) ≤
+      (-(4315858062121709975551 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (12989566614057469 / 20000000000000000 : ℝ) =
+            (12989566614057469 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12989566614057469 / 10000000000000000 : ℝ) ≤
+            (2615613740878290024449 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6474817239789969 / 10000000000000000 : ℝ)) (q :=
+      (12989566614057469 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_622_8 :
+    Uω (aρ 8) (bρ 8) (579023805401 / 800000000000 : ℝ) ≤
+      (-(119893869557067372297 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12380922688662013 / 20000000000000000 : ℝ) ≤
+      (-(119893869557067372297 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (12380922688662013 / 20000000000000000 : ℝ) =
+            (12380922688662013 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12380922688662013 / 10000000000000000 : ℝ) ≤
+            (53392925517932627703 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6149714965734513 / 10000000000000000 : ℝ)) (q :=
+      (12380922688662013 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_622_9 :
+    Uω (aρ 9) (bρ 9) (579023805401 / 800000000000 : ℝ) ≤
+      (-(5426039873039151492967 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11624655143871947 / 20000000000000000 : ℝ) ≤
+      (-(5426039873039151492967 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11624655143871947 / 20000000000000000 : ℝ) =
+            (11624655143871947 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11624655143871947 / 10000000000000000 : ℝ) ≤
+            (1505431929960848507033 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5734413049304447 / 10000000000000000 : ℝ)) (q :=
+      (11624655143871947 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_622_10 :
+    Uω (aρ 10) (bρ 10) (579023805401 / 800000000000 : ℝ) ≤
+      (-(6233689854961697792241 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (536135159873557 / 1000000000000000 : ℝ) ≤
+      (-(6233689854961697792241 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (536135159873557 / 1000000000000000 : ℝ) =
+            (536135159873557 / 500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (536135159873557 / 500000000000000 : ℝ) ≤
+            (697781948038302207759 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (65262751704983 / 125000000000000 : ℝ)) (q :=
+      (536135159873557 / 1000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10]) (by norm_num [bρ_10])
+      (by norm_num) (by norm_num [aρ_10, bρ_10]) (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_622_11 :
+    Uω (aρ 11) (bρ 11) (579023805401 / 800000000000 : ℝ) ≤
+      (-(906091685536126654847 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9687718261114239 / 20000000000000000 : ℝ) ≤
+      (-(906091685536126654847 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9687718261114239 / 20000000000000000 : ℝ) =
+            (9687718261114239 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9687718261114239 / 5000000000000000 : ℝ) ≤
+            (826776265213873345153 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4604511826246739 / 10000000000000000 : ℝ)) (q :=
+      (9687718261114239 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_622_12 :
+    Uω (aρ 12) (bρ 12) (579023805401 / 800000000000 : ℝ) ≤
+      (-(1063522702341544572123 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1708261520558867 / 4000000000000000 : ℝ) ≤
+      (-(1063522702341544572123 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1708261520558867 / 4000000000000000 : ℝ) =
+            (1708261520558867 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1708261520558867 / 1000000000000000 : ℝ) ≤
+            (669345248408455427877 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (776360252922367 / 2000000000000000 : ℝ)) (q :=
+      (1708261520558867 / 4000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_622_13 :
+    Uω (aρ 13) (bρ 13) (579023805401 / 800000000000 : ℝ) ≤
+      (-(10068144786604213098637 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7307621132830777 / 20000000000000000 : ℝ) ≤
+      (-(10068144786604213098637 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7307621132830777 / 20000000000000000 : ℝ) =
+            (7307621132830777 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7307621132830777 / 5000000000000000 : ℝ) ≤
+            (3794798819395786901363 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3047496536243277 / 10000000000000000 : ℝ)) (q :=
+      (7307621132830777 / 20000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_622_14 :
+    Uω (aρ 14) (bρ 14) (579023805401 / 800000000000 : ℝ) ≤
+      (-(753320867564564560021 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5991961883306781 / 20000000000000000 : ℝ) ≤
+      (-(753320867564564560021 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5991961883306781 / 20000000000000000 : ℝ) =
+            (5991961883306781 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5991961883306781 / 5000000000000000 : ℝ) ≤
+            (113113107810435439979 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2075764571764281 / 10000000000000000 : ℝ)) (q :=
+      (5991961883306781 / 20000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_622_15 :
+    Uω (aρ 15) (bρ 15) (579023805401 / 800000000000 : ℝ) ≤
+      (-(15143118237455441992627 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2199595035106191 / 10000000000000000 : ℝ) ≤
+      (-(15143118237455441992627 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2199595035106191 / 10000000000000000 : ℝ) =
+            (2199595035106191 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2199595035106191 / 1250000000000000 : ℝ) ≤
+            (5651297171544558007373 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (371283018042441 / 5000000000000000 : ℝ)) (q :=
+      (2199595035106191 / 10000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_622_16 :
+    Uω (aρ 16) (bρ 16) (579023805401 / 800000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_622 :
+    Uρ (579023805401 / 800000000000 : ℝ) ≤
+      (-(1429174612416460317899 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (579023805401 / 800000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(664444772228277286747 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_622_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (579023805401 / 800000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1677795207352304648309 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_622_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (579023805401 / 800000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3422629602471597769299 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_622_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (579023805401 / 800000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(17674409207002628977 / 50000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_622_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (579023805401 / 800000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(23175241795223090519 / 62500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_622_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (579023805401 / 800000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1980575594952124207391 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_622_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (579023805401 / 800000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(4315858062121709975551 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_622_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (579023805401 / 800000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(119893869557067372297 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_622_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (579023805401 / 800000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(5426039873039151492967 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_622_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (579023805401 / 800000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(6233689854961697792241 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_622_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (579023805401 / 800000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(906091685536126654847 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_622_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (579023805401 / 800000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(1063522702341544572123 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_622_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (579023805401 / 800000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(10068144786604213098637 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_622_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (579023805401 / 800000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) * (-(753320867564564560021 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_622_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (579023805401 / 800000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(15143118237455441992627 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_622_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (579023805401 / 800000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_622_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_623_1 :
+    Uω (aρ 1) (bρ 1) (23191428934597 / 32000000000000 : ℝ) ≤
+      (-(3308955606748218408403 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28731207265364987 / 40000000000000000 : ℝ) ≤
+      (-(3308955606748218408403 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (28731207265364987 / 40000000000000000 : ℝ) =
+            (28731207265364987 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28731207265364987 / 20000000000000000 : ℝ) ≤
+            (3622516196251781591597 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7182779308705931 / 10000000000000000 : ℝ)) (q :=
+      (28731207265364987 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_623_2 :
+    Uω (aρ 2) (bρ 2) (23191428934597 / 32000000000000 : ℝ) ≤
+      (-(835569402563246451051 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5727125691750179 / 8000000000000000 : ℝ) ≤
+      (-(835569402563246451051 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5727125691750179 / 8000000000000000 : ℝ) =
+            (5727125691750179 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5727125691750179 / 4000000000000000 : ℝ) ≤
+            (897298548186753548949 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1431751783581777 / 2000000000000000 : ℝ)) (q :=
+      (5727125691750179 / 8000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_623_3 :
+    Uω (aρ 3) (bρ 3) (23191428934597 / 32000000000000 : ℝ) ≤
+      (-(3409226606762139118281 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28444556118532007 / 40000000000000000 : ℝ) ≤
+      (-(3409226606762139118281 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (28444556118532007 / 40000000000000000 : ℝ) =
+            (28444556118532007 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28444556118532007 / 20000000000000000 : ℝ) ≤
+            (3522245196237860881719 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7110618854149441 / 10000000000000000 : ℝ)) (q :=
+      (28444556118532007 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_623_4 :
+    Uω (aρ 4) (bρ 4) (23191428934597 / 32000000000000 : ℝ) ≤
+      (-(3521325754907749846863 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28127475610658871 / 40000000000000000 : ℝ) ≤
+      (-(3521325754907749846863 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (28127475610658871 / 40000000000000000 : ℝ) =
+            (28127475610658871 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28127475610658871 / 20000000000000000 : ℝ) ≤
+            (3410146048092250153137 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7030372781277873 / 10000000000000000 : ℝ)) (q :=
+      (28127475610658871 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_623_5 :
+    Uω (aρ 5) (bρ 5) (23191428934597 / 32000000000000 : ℝ) ≤
+      (-(461780153311734872569 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5529057786173887 / 8000000000000000 : ℝ) ≤
+      (-(461780153311734872569 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5529057786173887 / 8000000000000000 : ℝ) =
+            (5529057786173887 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5529057786173887 / 4000000000000000 : ℝ) ≤
+            (404653822063265127431 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1381493872143631 / 2000000000000000 : ℝ)) (q :=
+      (5529057786173887 / 8000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_623_6 :
+    Uω (aρ 6) (bρ 6) (23191428934597 / 32000000000000 : ℝ) ≤
+      (-(3946989035406164734171 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (26955316491557633 / 40000000000000000 : ℝ) ≤
+      (-(3946989035406164734171 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (26955316491557633 / 40000000000000000 : ℝ) =
+            (26955316491557633 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26955316491557633 / 20000000000000000 : ℝ) ≤
+            (2984482767593835265829 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3364861758101127 / 5000000000000000 : ℝ)) (q :=
+      (26955316491557633 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_623_7 :
+    Uω (aρ 7) (bρ 7) (23191428934597 / 32000000000000 : ℝ) ≤
+      (-(4301159679979176426909 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (26017346427637547 / 40000000000000000 : ℝ) ≤
+      (-(4301159679979176426909 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (26017346427637547 / 40000000000000000 : ℝ) =
+            (26017346427637547 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26017346427637547 / 20000000000000000 : ℝ) ≤
+            (2630312123020823573091 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6484399865002211 / 10000000000000000 : ℝ)) (q :=
+      (26017346427637547 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_623_8 :
+    Uω (aρ 8) (bρ 8) (23191428934597 / 32000000000000 : ℝ) ≤
+      (-(4780280065221076428117 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24800193295944771 / 40000000000000000 : ℝ) ≤
+      (-(4780280065221076428117 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (24800193295944771 / 40000000000000000 : ℝ) =
+            (24800193295944771 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24800193295944771 / 20000000000000000 : ℝ) ≤
+            (2151191737778923571883 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6159364950495823 / 10000000000000000 : ℝ)) (q :=
+      (24800193295944771 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_623_9 :
+    Uω (aρ 9) (bρ 9) (23191428934597 / 32000000000000 : ℝ) ≤
+      (-(5409445567589630971793 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23287922932024447 / 40000000000000000 : ℝ) ≤
+      (-(5409445567589630971793 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23287922932024447 / 40000000000000000 : ℝ) =
+            (23287922932024447 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23287922932024447 / 20000000000000000 : ℝ) ≤
+            (1522026235410369028207 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5744195396895661 / 10000000000000000 : ℝ)) (q :=
+      (23287922932024447 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_623_10 :
+    Uω (aρ 10) (bρ 10) (23191428934597 / 32000000000000 : ℝ) ≤
+      (-(6215465766550052784111 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4296904865352947 / 8000000000000000 : ℝ) ≤
+      (-(6215465766550052784111 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (4296904865352947 / 8000000000000000 : ℝ) =
+            (4296904865352947 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4296904865352947 / 4000000000000000 : ℝ) ≤
+            (716006036449947215889 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1046211025552161 / 2000000000000000 : ℝ)) (q :=
+      (4296904865352947 / 8000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_623_11 :
+    Uω (aρ 11) (bρ 11) (23191428934597 / 32000000000000 : ℝ) ≤
+      (-(903509131940136337699 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3883101677135997 / 8000000000000000 : ℝ) ≤
+      (-(903509131940136337699 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3883101677135997 / 8000000000000000 : ℝ) =
+            (3883101677135997 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3883101677135997 / 2000000000000000 : ℝ) ≤
+            (829358818809863662301 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (461502378342343 / 1000000000000000 : ℝ)) (q :=
+      (3883101677135997 / 8000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_623_12 :
+    Uω (aρ 12) (bρ 12) (23191428934597 / 32000000000000 : ℝ) ≤
+      (-(848368272941469264369 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17124517021946789 / 40000000000000000 : ℝ) ≤
+      (-(848368272941469264369 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (17124517021946789 / 40000000000000000 : ℝ) =
+            (17124517021946789 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17124517021946789 / 10000000000000000 : ℝ) ≤
+            (537926087658530735631 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (486653524780229 / 1250000000000000 : ℝ)) (q :=
+      (17124517021946789 / 40000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_623_13 :
+    Uω (aρ 13) (bρ 13) (23191428934597 / 32000000000000 : ℝ) ≤
+      (-(5018480503871485516127 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14660889249224863 / 40000000000000000 : ℝ) ≤
+      (-(5018480503871485516127 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (14660889249224863 / 40000000000000000 : ℝ) =
+            (14660889249224863 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14660889249224863 / 10000000000000000 : ℝ) ≤
+            (1912991299128514483873 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3060796053475869 / 10000000000000000 : ℝ)) (q :=
+      (14660889249224863 / 40000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_623_14 :
+    Uω (aρ 14) (bρ 14) (23191428934597 / 32000000000000 : ℝ) ≤
+      (-(12007449152369101586589 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12038797245736077 / 40000000000000000 : ℝ) ≤
+      (-(12007449152369101586589 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (12038797245736077 / 40000000000000000 : ℝ) =
+            (12038797245736077 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12038797245736077 / 10000000000000000 : ℝ) ≤
+            (1855494453630898413411 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (523419334194119 / 2500000000000000 : ℝ)) (q :=
+      (12038797245736077 / 40000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_623_15 :
+    Uω (aρ 15) (bρ 15) (23191428934597 / 32000000000000 : ℝ) ≤
+      (-(15018676704649794032253 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8908552611326413 / 40000000000000000 : ℝ) ≤
+      (-(15018676704649794032253 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (8908552611326413 / 40000000000000000 : ℝ) =
+            (8908552611326413 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8908552611326413 / 5000000000000000 : ℝ) ≤
+            (5775738704350205967747 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (197032074246661 / 2500000000000000 : ℝ)) (q :=
+      (8908552611326413 / 40000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_623_16 :
+    Uω (aρ 16) (bρ 16) (23191428934597 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_623 :
+    Uρ (23191428934597 / 32000000000000 : ℝ) ≤
+      (-(5696085690247323872273 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (23191428934597 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3308955606748218408403 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_623_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (23191428934597 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(835569402563246451051 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_623_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (23191428934597 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3409226606762139118281 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_623_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (23191428934597 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3521325754907749846863 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_623_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (23191428934597 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(461780153311734872569 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_623_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (23191428934597 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3946989035406164734171 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_623_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (23191428934597 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(4301159679979176426909 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_623_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (23191428934597 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(4780280065221076428117 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_623_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (23191428934597 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(5409445567589630971793 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_623_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (23191428934597 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(6215465766550052784111 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_623_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (23191428934597 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(903509131940136337699 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_623_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (23191428934597 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(848368272941469264369 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_623_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (23191428934597 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(5018480503871485516127 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_623_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (23191428934597 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(12007449152369101586589 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_623_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (23191428934597 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(15018676704649794032253 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_623_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (23191428934597 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_623_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_624_1 :
+    Uω (aρ 1) (bρ 1) (11610952826577 / 16000000000000 : ℝ) ≤
+      (-(3295704933797769085771 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (719232582069469 / 1000000000000000 : ℝ) ≤
+      (-(3295704933797769085771 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (719232582069469 / 1000000000000000 : ℝ) =
+            (719232582069469 / 500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (719232582069469 / 500000000000000 : ℝ) ≤
+            (3635766869202230914229 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1438460668572751 / 2000000000000000 : ℝ)) (q :=
+      (719232582069469 / 1000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_624_2 :
+    Uω (aρ 2) (bρ 2) (11610952826577 / 16000000000000 : ℝ) ≤
+      (-(665796501209805653239 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (716843128613503 / 1000000000000000 : ℝ) ≤
+      (-(665796501209805653239 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (716843128613503 / 1000000000000000 : ℝ) =
+            (716843128613503 / 500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (716843128613503 / 500000000000000 : ℝ) ≤
+            (720497859390194346761 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1433656657250887 / 2000000000000000 : ℝ)) (q :=
+      (716843128613503 / 1000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_624_3 :
+    Uω (aρ 3) (bρ 3) (11610952826577 / 16000000000000 : ℝ) ≤
+      (-(3395841553661085159127 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7120663699973947 / 10000000000000000 : ℝ) ≤
+      (-(3395841553661085159127 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (7120663699973947 / 10000000000000000 : ℝ) =
+            (7120663699973947 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7120663699973947 / 5000000000000000 : ℝ) ≤
+            (3535630249338914840873 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7120144220282269 / 10000000000000000 : ℝ)) (q :=
+      (7120663699973947 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_624_4 :
+    Uω (aρ 4) (bρ 4) (11610952826577 / 16000000000000 : ℝ) ≤
+      (-(3507788028088219534127 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14082789802503379 / 20000000000000000 : ℝ) ≤
+      (-(3507788028088219534127 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (14082789802503379 / 20000000000000000 : ℝ) =
+            (14082789802503379 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14082789802503379 / 10000000000000000 : ℝ) ≤
+            (3423683774911780465873 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3519950401951377 / 5000000000000000 : ℝ)) (q :=
+      (14082789802503379 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_624_5 :
+    Uω (aρ 5) (bρ 5) (11610952826577 / 16000000000000 : ℝ) ≤
+      (-(3680462797695905282609 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6920851512263107 / 10000000000000000 : ℝ) ≤
+      (-(3680462797695905282609 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (6920851512263107 / 10000000000000000 : ℝ) =
+            (6920851512263107 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6920851512263107 / 5000000000000000 : ℝ) ≤
+            (3251009005304094717391 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6917003945260589 / 10000000000000000 : ℝ)) (q :=
+      (6920851512263107 / 10000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_624_6 :
+    Uω (aρ 6) (bρ 6) (11610952826577 / 16000000000000 : ℝ) ≤
+      (-(1966423481665302052459 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6748365965551843 / 10000000000000000 : ℝ) ≤
+      (-(1966423481665302052459 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (6748365965551843 / 10000000000000000 : ℝ) =
+            (6748365965551843 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6748365965551843 / 5000000000000000 : ℝ) ≤
+            (1499312419834697947541 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6739273226978061 / 10000000000000000 : ℝ)) (q :=
+      (6748365965551843 / 10000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_624_7 :
+    Uω (aρ 7) (bρ 7) (11610952826577 / 16000000000000 : ℝ) ≤
+      (-(2143241501416554488787 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2605555928203611 / 4000000000000000 : ℝ) ≤
+      (-(2143241501416554488787 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2605555928203611 / 4000000000000000 : ℝ) =
+            (2605555928203611 / 2000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2605555928203611 / 2000000000000000 : ℝ) ≤
+            (1322494400083445511213 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (649398231765243 / 1000000000000000 : ℝ)) (q :=
+      (2605555928203611 / 4000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_624_8 :
+    Uω (aρ 8) (bρ 8) (11610952826577 / 16000000000000 : ℝ) ≤
+      (-(4764829573210770707701 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12419270215629507 / 20000000000000000 : ℝ) ≤
+      (-(4764829573210770707701 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (12419270215629507 / 20000000000000000 : ℝ) =
+            (12419270215629507 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12419270215629507 / 10000000000000000 : ℝ) ≤
+            (2166642229789229292299 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3084507271801941 / 5000000000000000 : ℝ)) (q :=
+      (12419270215629507 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_624_9 :
+    Uω (aρ 9) (bρ 9) (11610952826577 / 16000000000000 : ℝ) ≤
+      (-(5392879496913676081697 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5831633460616669 / 10000000000000000 : ℝ) ≤
+      (-(5392879496913676081697 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5831633460616669 / 10000000000000000 : ℝ) =
+            (5831633460616669 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5831633460616669 / 5000000000000000 : ℝ) ≤
+            (1538592306086323918303 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5753976877567713 / 10000000000000000 : ℝ)) (q :=
+      (5831633460616669 / 10000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_624_10 :
+    Uω (aρ 10) (bρ 10) (11610952826577 / 16000000000000 : ℝ) ≤
+      (-(1549319150378315990693 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2690454805563697 / 5000000000000000 : ℝ) ≤
+      (-(1549319150378315990693 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2690454805563697 / 5000000000000000 : ℝ) =
+            (2690454805563697 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2690454805563697 / 2500000000000000 : ℝ) ≤
+            (183548800371684009307 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5241088212084163 / 10000000000000000 : ℝ)) (q :=
+      (2690454805563697 / 5000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_624_11 :
+    Uω (aρ 11) (bρ 11) (11610952826577 / 16000000000000 : ℝ) ≤
+      (-(7207459623385319839303 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9727785845026523 / 20000000000000000 : ℝ) ≤
+      (-(7207459623385319839303 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9727785845026523 / 20000000000000000 : ℝ) =
+            (9727785845026523 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9727785845026523 / 5000000000000000 : ℝ) ≤
+            (6655483982614680160697 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2312765730530449 / 5000000000000000 : ℝ)) (q :=
+      (9727785845026523 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_624_12 :
+    Uω (aρ 12) (bρ 12) (11610952826577 / 16000000000000 : ℝ) ≤
+      (-(1691851121799353991287 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8583199208577703 / 20000000000000000 : ℝ) ≤
+      (-(1691851121799353991287 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8583199208577703 / 20000000000000000 : ℝ) =
+            (8583199208577703 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8583199208577703 / 5000000000000000 : ℝ) ≤
+            (1080737599400646008713 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1952322460648539 / 5000000000000000 : ℝ)) (q :=
+      (8583199208577703 / 20000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_624_13 :
+    Uω (aρ 13) (bρ 13) (11610952826577 / 16000000000000 : ℝ) ≤
+      (-(2501478072666182089101 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1838310021227563 / 5000000000000000 : ℝ) ≤
+      (-(2501478072666182089101 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1838310021227563 / 5000000000000000 : ℝ) =
+            (1838310021227563 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1838310021227563 / 1250000000000000 : ℝ) ≤
+            (964257828833817910899 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3074067539224627 / 10000000000000000 : ℝ)) (q :=
+      (1838310021227563 / 5000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_624_14 :
+    Uω (aρ 14) (bρ 14) (11610952826577 / 16000000000000 : ℝ) ≤
+      (-(11962152448444704526753 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6046726360687117 / 20000000000000000 : ℝ) ≤
+      (-(11962152448444704526753 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6046726360687117 / 20000000000000000 : ℝ) =
+            (6046726360687117 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6046726360687117 / 5000000000000000 : ℝ) ≤
+            (1900791157555295473247 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (527870275011623 / 2500000000000000 : ℝ)) (q :=
+      (6046726360687117 / 20000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_624_15 :
+    Uω (aρ 15) (bρ 15) (11610952826577 / 16000000000000 : ℝ) ≤
+      (-(14901054336295950917693 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4506977895184479 / 20000000000000000 : ℝ) ≤
+      (-(14901054336295950917693 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4506977895184479 / 20000000000000000 : ℝ) =
+            (4506977895184479 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4506977895184479 / 2500000000000000 : ℝ) ≤
+            (5893361072704049082307 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (415652955979427 / 5000000000000000 : ℝ)) (q :=
+      (4506977895184479 / 20000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_624_16 :
+    Uω (aρ 16) (bρ 16) (11610952826577 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_624 :
+    Uρ (11610952826577 / 16000000000000 : ℝ) ≤
+      (-(2837864753785064472951 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (11610952826577 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3295704933797769085771 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_624_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (11610952826577 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(665796501209805653239 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_624_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (11610952826577 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3395841553661085159127 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_624_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (11610952826577 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3507788028088219534127 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_624_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (11610952826577 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(3680462797695905282609 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_624_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (11610952826577 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1966423481665302052459 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_624_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (11610952826577 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(2143241501416554488787 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_624_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (11610952826577 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(4764829573210770707701 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_624_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (11610952826577 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(5392879496913676081697 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_624_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (11610952826577 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(1549319150378315990693 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_624_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (11610952826577 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(7207459623385319839303 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_624_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (11610952826577 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(1691851121799353991287 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_624_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (11610952826577 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(2501478072666182089101 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_624_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (11610952826577 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(11962152448444704526753 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_624_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (11610952826577 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(14901054336295950917693 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_624_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (11610952826577 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_624_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_625_1 :
+    Uω (aρ 1) (bρ 1) (23252382371711 / 32000000000000 : ℝ) ≤
+      (-(3282471795757911769263 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5761479859975443 / 8000000000000000 : ℝ) ≤
+      (-(3282471795757911769263 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5761479859975443 / 8000000000000000 : ℝ) =
+            (5761479859975443 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5761479859975443 / 4000000000000000 : ℝ) ≤
+            (3649000007242088230737 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (90022842210799 / 125000000000000 : ℝ)) (q :=
+      (5761479859975443 / 8000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_625_2 :
+    Uω (aρ 2) (bρ 2) (23252382371711 / 32000000000000 : ℝ) ≤
+      (-(1657852527543002467351 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5742364365647849 / 8000000000000000 : ℝ) ≤
+      (-(1657852527543002467351 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5742364365647849 / 8000000000000000 : ℝ) =
+            (5742364365647849 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5742364365647849 / 4000000000000000 : ℝ) ≤
+            (1807883373956997532649 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1435561530710987 / 2000000000000000 : ℝ)) (q :=
+      (5742364365647849 / 8000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_625_3 :
+    Uω (aρ 3) (bρ 3) (23252382371711 / 32000000000000 : ℝ) ≤
+      (-(845618598796519162513 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28520753473823387 / 40000000000000000 : ℝ) ≤
+      (-(845618598796519162513 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (28520753473823387 / 40000000000000000 : ℝ) =
+            (28520753473823387 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28520753473823387 / 20000000000000000 : ℝ) ≤
+            (887249351953480837487 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3564834791348503 / 5000000000000000 : ℝ)) (q :=
+      (28520753473823387 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_625_4 :
+    Uω (aρ 4) (bρ 4) (23252382371711 / 32000000000000 : ℝ) ≤
+      (-(3494268611257336033207 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5640736715494749 / 8000000000000000 : ℝ) ≤
+      (-(3494268611257336033207 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5640736715494749 / 8000000000000000 : ℝ) =
+            (5640736715494749 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5640736715494749 / 4000000000000000 : ℝ) ≤
+            (3437203191742663966793 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1409885763117437 / 2000000000000000 : ℝ)) (q :=
+      (5640736715494749 / 8000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_625_5 :
+    Uω (aρ 5) (bρ 5) (23252382371711 / 32000000000000 : ℝ) ≤
+      (-(458337918543920478139 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27721523108848051 / 40000000000000000 : ℝ) ≤
+      (-(458337918543920478139 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (27721523108848051 / 40000000000000000 : ℝ) =
+            (27721523108848051 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27721523108848051 / 20000000000000000 : ℝ) ≤
+            (408096056831079521861 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3463269250304669 / 5000000000000000 : ℝ)) (q :=
+      (27721523108848051 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_625_6 :
+    Uω (aρ 6) (bρ 6) (23252382371711 / 32000000000000 : ℝ) ≤
+      (-(3918724916650439724693 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27031611087384427 / 40000000000000000 : ℝ) ≤
+      (-(3918724916650439724693 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (27031611087384427 / 40000000000000000 : ℝ) =
+            (27031611087384427 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27031611087384427 / 20000000000000000 : ℝ) ≤
+            (3012746886349560275307 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3374411432508763 / 5000000000000000 : ℝ)) (q :=
+      (27031611087384427 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_625_7 :
+    Uω (aρ 7) (bρ 7) (23252382371711 / 32000000000000 : ℝ) ≤
+      (-(4271827966286515829487 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5218754358567231 / 8000000000000000 : ℝ) ≤
+      (-(4271827966286515829487 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5218754358567231 / 8000000000000000 : ℝ) =
+            (5218754358567231 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5218754358567231 / 4000000000000000 : ℝ) ≤
+            (2659643836713484170513 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (650356459850339 / 1000000000000000 : ℝ)) (q :=
+      (5218754358567231 / 8000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_625_8 :
+    Uω (aρ 8) (bρ 8) (23252382371711 / 32000000000000 : ℝ) ≤
+      (-(2374701614773647720401 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4975377357387353 / 8000000000000000 : ℝ) ≤
+      (-(2374701614773647720401 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (4975377357387353 / 8000000000000000 : ℝ) =
+            (4975377357387353 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4975377357387353 / 4000000000000000 : ℝ) ≤
+            (1091034286726352279599 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1235732749378739 / 2000000000000000 : ℝ)) (q :=
+      (4975377357387353 / 8000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_625_9 :
+    Uω (aρ 9) (bρ 9) (23252382371711 / 32000000000000 : ℝ) ≤
+      (-(5376341562609795843151 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23365143027897899 / 40000000000000000 : ℝ) ≤
+      (-(5376341562609795843151 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23365143027897899 / 40000000000000000 : ℝ) =
+            (23365143027897899 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23365143027897899 / 20000000000000000 : ℝ) ≤
+            (1555130240390204156849 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2881878747867131 / 5000000000000000 : ℝ)) (q :=
+      (23365143027897899 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_625_10 :
+    Uω (aρ 10) (bρ 10) (23252382371711 / 32000000000000 : ℝ) ≤
+      (-(6179122219677369954149 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21562748770038979 / 40000000000000000 : ℝ) ≤
+      (-(6179122219677369954149 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (21562748770038979 / 40000000000000000 : ℝ) =
+            (21562748770038979 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21562748770038979 / 20000000000000000 : ℝ) ≤
+            (752349583322630045851 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2625559700149901 / 5000000000000000 : ℝ)) (q :=
+      (21562748770038979 / 40000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_625_11 :
+    Uω (aρ 11) (bρ 11) (23252382371711 / 32000000000000 : ℝ) ≤
+      (-(7186892955616715097121 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19495626493545693 / 40000000000000000 : ℝ) ≤
+      (-(7186892955616715097121 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (19495626493545693 / 40000000000000000 : ℝ) =
+            (19495626493545693 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19495626493545693 / 10000000000000000 : ℝ) ≤
+            (6676050650383284902879 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4636034888258159 / 10000000000000000 : ℝ)) (q :=
+      (19495626493545693 / 40000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_625_12 :
+    Uω (aρ 12) (bρ 12) (23252382371711 / 32000000000000 : ℝ) ≤
+      (-(8434899774897159942441 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17208259569819893 / 40000000000000000 : ℝ) ≤
+      (-(8434899774897159942441 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (17208259569819893 / 40000000000000000 : ℝ) =
+            (17208259569819893 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17208259569819893 / 10000000000000000 : ℝ) ≤
+            (5428043831102840057559 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3916051523080259 / 10000000000000000 : ℝ)) (q :=
+      (17208259569819893 / 40000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_625_13 :
+    Uω (aρ 13) (bρ 13) (23252382371711 / 32000000000000 : ℝ) ≤
+      (-(2493749297745844288991 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14752015750445453 / 40000000000000000 : ℝ) ≤
+      (-(2493749297745844288991 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (14752015750445453 / 40000000000000000 : ℝ) =
+            (14752015750445453 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14752015750445453 / 10000000000000000 : ℝ) ≤
+            (971986603754155711009 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3087311354988039 / 10000000000000000 : ℝ)) (q :=
+      (14752015750445453 / 40000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_625_14 :
+    Uω (aρ 14) (bρ 14) (23252382371711 / 32000000000000 : ℝ) ≤
+      (-(11917234953315698604143 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2429579132447089 / 8000000000000000 : ℝ) ≤
+      (-(11917234953315698604143 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2429579132447089 / 8000000000000000 : ℝ) =
+            (2429579132447089 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2429579132447089 / 2000000000000000 : ℝ) ≤
+            (1945708652684301395857 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (425835719185607 / 2000000000000000 : ℝ)) (q :=
+      (2429579132447089 / 8000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_625_15 :
+    Uω (aρ 15) (bρ 15) (23252382371711 / 32000000000000 : ℝ) ≤
+      (-(739462680758528667917 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9115297916670169 / 40000000000000000 : ℝ) ≤
+      (-(739462680758528667917 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (9115297916670169 / 40000000000000000 : ℝ) =
+            (9115297916670169 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9115297916670169 / 5000000000000000 : ℝ) ≤
+            (300258089691471332083 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (872453000560397 / 10000000000000000 : ℝ)) (q :=
+      (9115297916670169 / 40000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_625_16 :
+    Uω (aρ 16) (bρ 16) (23252382371711 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_625 :
+    Uρ (23252382371711 / 32000000000000 : ℝ) ≤
+      (-(1413899734414673350401 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (23252382371711 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3282471795757911769263 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_625_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (23252382371711 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1657852527543002467351 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_625_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (23252382371711 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(845618598796519162513 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_625_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (23252382371711 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3494268611257336033207 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_625_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (23252382371711 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(458337918543920478139 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_625_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (23252382371711 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3918724916650439724693 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_625_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (23252382371711 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(4271827966286515829487 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_625_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (23252382371711 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(2374701614773647720401 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_625_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (23252382371711 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(5376341562609795843151 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_625_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (23252382371711 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(6179122219677369954149 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_625_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (23252382371711 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(7186892955616715097121 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_625_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (23252382371711 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(8434899774897159942441 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_625_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (23252382371711 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(2493749297745844288991 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_625_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (23252382371711 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(11917234953315698604143 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_625_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (23252382371711 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(739462680758528667917 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_625_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (23252382371711 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_625_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_626_1 :
+    Uω (aρ 1) (bρ 1) (5820714772567 / 8000000000000 : ℝ) ≤
+      (-(3269256146281007372917 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7211373829165401 / 10000000000000000 : ℝ) ≤
+      (-(3269256146281007372917 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (7211373829165401 / 10000000000000000 : ℝ) =
+            (7211373829165401 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7211373829165401 / 5000000000000000 : ℝ) ≤
+            (3662215656718992627083 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1802837852676763 / 2500000000000000 : ℝ)) (q :=
+      (7211373829165401 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_626_2 :
+    Uω (aρ 2) (bρ 2) (5820714772567 / 8000000000000 : ℝ) ≤
+      (-(3302445210544198461269 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (449217476716509 / 625000000000000 : ℝ) ≤
+      (-(3302445210544198461269 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (449217476716509 / 625000000000000 : ℝ) =
+            (449217476716509 / 312500000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (449217476716509 / 312500000000000 : ℝ) ≤
+            (3629026592455801538731 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3593666009907269 / 5000000000000000 : ℝ)) (q :=
+      (449217476716509 / 625000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_626_3 :
+    Uω (aρ 3) (bρ 3) (5820714772567 / 8000000000000 : ℝ) ≤
+      (-(842281270886748966641 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7139713035086143 / 10000000000000000 : ℝ) ≤
+      (-(842281270886748966641 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (7139713035086143 / 10000000000000000 : ℝ) =
+            (7139713035086143 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7139713035086143 / 5000000000000000 : ℝ) ≤
+            (890586679863251033359 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (892399367676067 / 1250000000000000 : ℝ)) (q :=
+      (7139713035086143 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_626_4 :
+    Uω (aρ 4) (bρ 4) (5820714772567 / 8000000000000 : ℝ) ≤
+      (-(3480767454931998022547 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14120893764074217 / 20000000000000000 : ℝ) ≤
+      (-(3480767454931998022547 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (14120893764074217 / 20000000000000000 : ℝ) =
+            (14120893764074217 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14120893764074217 / 10000000000000000 : ℝ) ≤
+            (3450704348068001977453 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7058956816375467 / 10000000000000000 : ℝ)) (q :=
+      (14120893764074217 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_626_5 :
+    Uω (aρ 5) (bρ 5) (5820714772567 / 8000000000000 : ℝ) ≤
+      (-(3652962826186939902803 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13879820055248543 / 20000000000000000 : ℝ) ≤
+      (-(3652962826186939902803 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (13879820055248543 / 20000000000000000 : ℝ) =
+            (13879820055248543 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13879820055248543 / 10000000000000000 : ℝ) ≤
+            (3278508976813060097197 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6936073026884793 / 10000000000000000 : ℝ)) (q :=
+      (13879820055248543 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_626_6 :
+    Uω (aρ 6) (bρ 6) (5820714772567 / 8000000000000 : ℝ) ≤
+      (-(1952311419290875189301 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1353487908385273 / 2000000000000000 : ℝ) ≤
+      (-(1952311419290875189301 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1353487908385273 / 2000000000000000 : ℝ) =
+            (1353487908385273 / 1000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1353487908385273 / 1000000000000000 : ℝ) ≤
+            (1513424482209124810699 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (337918621531449 / 500000000000000 : ℝ)) (q :=
+      (1353487908385273 / 2000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_626_7 :
+    Uω (aρ 7) (bρ 7) (5820714772567 / 8000000000000 : ℝ) ≤
+      (-(4257194506230267187757 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3266497995194277 / 5000000000000000 : ℝ) ≤
+      (-(4257194506230267187757 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (3266497995194277 / 5000000000000000 : ℝ) =
+            (3266497995194277 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3266497995194277 / 2500000000000000 : ℝ) ≤
+            (2674277296769732812243 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3256573354156679 / 5000000000000000 : ℝ)) (q :=
+      (3266497995194277 / 5000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_626_8 :
+    Uω (aρ 8) (bρ 8) (5820714772567 / 8000000000000 : ℝ) ≤
+      (-(4734000957894396568091 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12457616183312499 / 20000000000000000 : ℝ) ≤
+      (-(4734000957894396568091 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (12457616183312499 / 20000000000000000 : ℝ) =
+            (12457616183312499 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12457616183312499 / 10000000000000000 : ℝ) ≤
+            (2197470845105603431909 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6188312562188749 / 10000000000000000 : ℝ)) (q :=
+      (12457616183312499 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_626_9 :
+    Uω (aρ 9) (bρ 9) (5820714772567 / 8000000000000 : ℝ) ≤
+      (-(2679915833401137352107 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11701875248542423 / 20000000000000000 : ℝ) ≤
+      (-(2679915833401137352107 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11701875248542423 / 20000000000000000 : ℝ) =
+            (11701875248542423 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11701875248542423 / 10000000000000000 : ℝ) ≤
+            (785820068098862647893 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5773537255778673 / 10000000000000000 : ℝ)) (q :=
+      (11701875248542423 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_626_10 :
+    Uω (aρ 10) (bρ 10) (5820714772567 / 8000000000000 : ℝ) ≤
+      (-(6161002481746330224137 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10800927662522133 / 20000000000000000 : ℝ) ≤
+      (-(6161002481746330224137 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10800927662522133 / 20000000000000000 : ℝ) =
+            (10800927662522133 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10800927662522133 / 10000000000000000 : ℝ) ≤
+            (770469321253669775863 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5261148703253383 / 10000000000000000 : ℝ)) (q :=
+      (10800927662522133 / 20000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_626_11 :
+    Uω (aρ 11) (bρ 11) (5820714772567 / 8000000000000 : ℝ) ≤
+      (-(7166372821784453306367 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (195356728538063 / 400000000000000 : ℝ) ≤
+      (-(7166372821784453306367 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (195356728538063 / 400000000000000 : ℝ) =
+            (195356728538063 / 100000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (195356728538063 / 100000000000000 : ℝ) ≤
+            (6696570784215546693633 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (23232670469197 / 50000000000000 : ℝ)) (q :=
+      (195356728538063 / 400000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11]) (by norm_num [bρ_11])
+      (by norm_num) (by norm_num [aρ_11, bρ_11]) (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_626_12 :
+    Uω (aρ 12) (bρ 12) (5820714772567 / 8000000000000 : ℝ) ≤
+      (-(4205307374925527529399 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8625050328156639 / 20000000000000000 : ℝ) ≤
+      (-(4205307374925527529399 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8625050328156639 / 20000000000000000 : ℝ) =
+            (8625050328156639 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8625050328156639 / 5000000000000000 : ℝ) ≤
+            (2726164428074472470601 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3927448091777889 / 10000000000000000 : ℝ)) (q :=
+      (8625050328156639 / 20000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_626_13 :
+    Uω (aρ 13) (bρ 13) (5820714772567 / 8000000000000 : ℝ) ≤
+      (-(497210714502570725607 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7398748350124519 / 20000000000000000 : ℝ) ≤
+      (-(497210714502570725607 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7398748350124519 / 20000000000000000 : ℝ) =
+            (7398748350124519 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7398748350124519 / 5000000000000000 : ℝ) ≤
+            (195936465797429274393 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3100527855340769 / 10000000000000000 : ℝ)) (q :=
+      (7398748350124519 / 20000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_626_14 :
+    Uω (aρ 14) (bρ 14) (5820714772567 / 8000000000000 : ℝ) ≤
+      (-(5936344091881768530959 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6101065662292823 / 20000000000000000 : ℝ) ≤
+      (-(5936344091881768530959 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6101065662292823 / 20000000000000000 : ℝ) =
+            (6101065662292823 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6101065662292823 / 5000000000000000 : ℝ) ≤
+            (995127711118231469041 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2146772452554073 / 10000000000000000 : ℝ)) (q :=
+      (6101065662292823 / 20000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_626_15 :
+    Uω (aρ 15) (bρ 15) (5820714772567 / 8000000000000 : ℝ) ≤
+      (-(14682499387665634183519 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2303282209888357 / 10000000000000000 : ℝ) ≤
+      (-(14682499387665634183519 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2303282209888357 / 10000000000000000 : ℝ) =
+            (2303282209888357 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2303282209888357 / 1250000000000000 : ℝ) ≤
+            (6111916021334365816481 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (227961121863241 / 2500000000000000 : ℝ)) (q :=
+      (2303282209888357 / 10000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_626_16 :
+    Uω (aρ 16) (bρ 16) (5820714772567 / 8000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_626 :
+    Uρ (5820714772567 / 8000000000000 : ℝ) ≤
+      (-(5635669807423770572593 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (5820714772567 / 8000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3269256146281007372917 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_626_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (5820714772567 / 8000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3302445210544198461269 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_626_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (5820714772567 / 8000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(842281270886748966641 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_626_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (5820714772567 / 8000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3480767454931998022547 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_626_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (5820714772567 / 8000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(3652962826186939902803 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_626_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (5820714772567 / 8000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1952311419290875189301 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_626_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (5820714772567 / 8000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(4257194506230267187757 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_626_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (5820714772567 / 8000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(4734000957894396568091 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_626_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (5820714772567 / 8000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(2679915833401137352107 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_626_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (5820714772567 / 8000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(6161002481746330224137 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_626_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (5820714772567 / 8000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(7166372821784453306367 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_626_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (5820714772567 / 8000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(4205307374925527529399 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_626_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (5820714772567 / 8000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) * (-(497210714502570725607 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_626_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (5820714772567 / 8000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(5936344091881768530959 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_626_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (5820714772567 / 8000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(14682499387665634183519 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_626_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (5820714772567 / 8000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_626_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_627_1 :
+    Uω (aρ 1) (bρ 1) (932533432353 / 1280000000000 : ℝ) ≤
+      (-(3256057939202933221281 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28883591333133169 / 40000000000000000 : ℝ) ≤
+      (-(3256057939202933221281 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (28883591333133169 / 40000000000000000 : ℝ) =
+            (28883591333133169 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28883591333133169 / 20000000000000000 : ℝ) ≤
+            (3675413863797066778719 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1805218861098443 / 2500000000000000 : ℝ)) (q :=
+      (28883591333133169 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_627_2 :
+    Uω (aρ 2) (bρ 2) (932533432353 / 1280000000000 : ℝ) ≤
+      (-(3289202925789912456111 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28788015189400381 / 40000000000000000 : ℝ) ≤
+      (-(3289202925789912456111 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (28788015189400381 / 40000000000000000 : ℝ) =
+            (28788015189400381 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28788015189400381 / 20000000000000000 : ℝ) ≤
+            (3642268877210087543889 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3598428192518689 / 5000000000000000 : ℝ)) (q :=
+      (28788015189400381 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_627_3 :
+    Uω (aρ 3) (bρ 3) (932533432353 / 1280000000000 : ℝ) ≤
+      (-(671158714228988945661 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28596950799488947 / 40000000000000000 : ℝ) ≤
+      (-(671158714228988945661 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (28596950799488947 / 40000000000000000 : ℝ) =
+            (28596950799488947 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28596950799488947 / 20000000000000000 : ℝ) ≤
+            (715135646371011054339 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7148720296431661 / 10000000000000000 : ℝ)) (q :=
+      (28596950799488947 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_627_4 :
+    Uω (aρ 4) (bρ 4) (932533432353 / 1280000000000 : ℝ) ≤
+      (-(3467284509829511449131 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28279891457118949 / 40000000000000000 : ℝ) ≤
+      (-(3467284509829511449131 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (28279891457118949 / 40000000000000000 : ℝ) =
+            (28279891457118949 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28279891457118949 / 20000000000000000 : ℝ) ≤
+            (3464187293170488550869 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3534242403155831 / 5000000000000000 : ℝ)) (q :=
+      (28279891457118949 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_627_5 :
+    Uω (aρ 5) (bρ 5) (932533432353 / 1280000000000 : ℝ) ≤
+      (-(1819620589572607906299 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27797757054238993 / 40000000000000000 : ℝ) ≤
+      (-(1819620589572607906299 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (27797757054238993 / 40000000000000000 : ℝ) =
+            (27797757054238993 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27797757054238993 / 20000000000000000 : ℝ) ≤
+            (1646115311927392093701 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1736401881051671 / 2500000000000000 : ℝ)) (q :=
+      (27797757054238993 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_627_6 :
+    Uω (aρ 6) (bρ 6) (932533432353 / 1280000000000 : ℝ) ≤
+      (-(243158792036402207847 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27107905103783649 / 40000000000000000 : ℝ) ≤
+      (-(243158792036402207847 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (27107905103783649 / 40000000000000000 : ℝ) =
+            (27107905103783649 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27107905103783649 / 20000000000000000 : ℝ) ≤
+            (190058195651097792153 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1691980481029753 / 2500000000000000 : ℝ)) (q :=
+      (27107905103783649 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_627_7 :
+    Uω (aρ 7) (bρ 7) (932533432353 / 1280000000000 : ℝ) ≤
+      (-(4242582558841379302559 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (26170195789697881 / 40000000000000000 : ℝ) ≤
+      (-(4242582558841379302559 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (26170195789697881 / 40000000000000000 : ℝ) =
+            (26170195789697881 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26170195789697881 / 20000000000000000 : ℝ) ≤
+            (2688889244158620697441 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (203835270244879 / 312500000000000 : ℝ)) (q :=
+      (26170195789697881 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_627_8 :
+    Uω (aρ 8) (bρ 8) (932533432353 / 1280000000000 : ℝ) ≤
+      (-(4718622682281668301957 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24953577173947831 / 40000000000000000 : ℝ) ≤
+      (-(4718622682281668301957 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (24953577173947831 / 40000000000000000 : ℝ) =
+            (24953577173947831 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24953577173947831 / 20000000000000000 : ℝ) ≤
+            (2212849120718331698043 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6197960991301103 / 10000000000000000 : ℝ)) (q :=
+      (24953577173947831 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_627_9 :
+    Uω (aρ 9) (bρ 9) (932533432353 / 1280000000000 : ℝ) ≤
+      (-(2671674856068678912033 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (937694250349367 / 1600000000000000 : ℝ) ≤
+      (-(2671674856068678912033 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (937694250349367 / 1600000000000000 : ℝ) =
+            (937694250349367 / 800000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (937694250349367 / 800000000000000 : ℝ) ≤
+            (794061045431321087967 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (231332646482171 / 400000000000000 : ℝ)) (q :=
+      (937694250349367 / 1600000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_627_10 :
+    Uω (aρ 10) (bρ 10) (932533432353 / 1280000000000 : ℝ) ≤
+      (-(1535729312323613446291 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4328191626209523 / 8000000000000000 : ℝ) ≤
+      (-(1535729312323613446291 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (4328191626209523 / 8000000000000000 : ℝ) =
+            (4328191626209523 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4328191626209523 / 4000000000000000 : ℝ) ≤
+            (197138638426386553709 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1054235226341199 / 2000000000000000 : ℝ)) (q :=
+      (4328191626209523 / 8000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_627_11 :
+    Uω (aρ 11) (bρ 11) (932533432353 / 1280000000000 : ℝ) ≤
+      (-(3572949496635842197997 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19575710827940307 / 40000000000000000 : ℝ) ≤
+      (-(3572949496635842197997 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (19575710827940307 / 40000000000000000 : ℝ) =
+            (19575710827940307 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19575710827940307 / 10000000000000000 : ℝ) ≤
+            (3358522306364157802003 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4657029106357341 / 10000000000000000 : ℝ)) (q :=
+      (19575710827940307 / 40000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_627_12 :
+    Uω (aρ 12) (bρ 12) (932533432353 / 1280000000000 : ℝ) ≤
+      (-(262075001932125642291 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3458384370162619 / 8000000000000000 : ℝ) ≤
+      (-(262075001932125642291 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3458384370162619 / 8000000000000000 : ℝ) =
+            (3458384370162619 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3458384370162619 / 2000000000000000 : ℝ) ≤
+            (171141985755374357709 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (787766942895747 / 2000000000000000 : ℝ)) (q :=
+      (3458384370162619 / 8000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_627_13 :
+    Uω (aρ 13) (bρ 13) (932533432353 / 1280000000000 : ℝ) ≤
+      (-(2478390548579272769413 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14842923714890069 / 40000000000000000 : ℝ) ≤
+      (-(2478390548579272769413 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (14842923714890069 / 40000000000000000 : ℝ) =
+            (14842923714890069 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14842923714890069 / 10000000000000000 : ℝ) ≤
+            (987345352920727230587 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1556858694056111 / 5000000000000000 : ℝ)) (q :=
+      (14842923714890069 / 40000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_627_14 :
+    Uω (aρ 14) (bρ 14) (932533432353 / 1280000000000 : ℝ) ≤
+      (-(11828503971903031566869 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12256164763485383 / 40000000000000000 : ℝ) ≤
+      (-(11828503971903031566869 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (12256164763485383 / 40000000000000000 : ℝ) =
+            (12256164763485383 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12256164763485383 / 10000000000000000 : ℝ) ≤
+            (2034439634096968433131 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2164265197454879 / 10000000000000000 : ℝ)) (q :=
+      (12256164763485383 / 40000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_627_15 :
+    Uω (aρ 15) (bρ 15) (932533432353 / 1280000000000 : ℝ) ≤
+      (-(1458017506958502499819 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9307885521012117 / 40000000000000000 : ℝ) ≤
+      (-(1458017506958502499819 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (9307885521012117 / 40000000000000000 : ℝ) =
+            (9307885521012117 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9307885521012117 / 5000000000000000 : ℝ) ≤
+            (621424033941497500181 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (474849426816623 / 5000000000000000 : ℝ)) (q :=
+      (9307885521012117 / 40000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_627_16 :
+    Uω (aρ 16) (bρ 16) (932533432353 / 1280000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_627 :
+    Uρ (932533432353 / 1280000000000 : ℝ) ≤
+      (-(5615922790488704981441 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (932533432353 / 1280000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3256057939202933221281 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_627_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (932533432353 / 1280000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3289202925789912456111 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_627_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (932533432353 / 1280000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(671158714228988945661 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_627_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (932533432353 / 1280000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3467284509829511449131 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_627_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (932533432353 / 1280000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1819620589572607906299 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_627_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (932533432353 / 1280000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(243158792036402207847 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_627_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (932533432353 / 1280000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(4242582558841379302559 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_627_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (932533432353 / 1280000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(4718622682281668301957 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_627_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (932533432353 / 1280000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(2671674856068678912033 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_627_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (932533432353 / 1280000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(1535729312323613446291 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_627_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (932533432353 / 1280000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(3572949496635842197997 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_627_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (932533432353 / 1280000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(262075001932125642291 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_627_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (932533432353 / 1280000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(2478390548579272769413 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_627_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (932533432353 / 1280000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(11828503971903031566869 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_627_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (932533432353 / 1280000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(1458017506958502499819 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_627_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (932533432353 / 1280000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_627_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/U52.lean b/LeanPool/Zeta5Irrational/Table/U52.lean
new file mode 100644
index 0000000000..75524aae81
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/U52.lean
@@ -0,0 +1,5284 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-! # Certified arcsine potential bounds (U52) -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma U_628_1 :
+    Uω (aρ 1) (bρ 1) (11671906263691 / 16000000000000 : ℝ) ≤
+      (-(3242877128542114788709 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3615210918661643 / 5000000000000000 : ℝ) ≤
+      (-(3242877128542114788709 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (3615210918661643 / 5000000000000000 : ℝ) =
+            (3615210918661643 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3615210918661643 / 2500000000000000 : ℝ) ≤
+            (3688594674457885211291 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7230399477924697 / 10000000000000000 : ℝ)) (q :=
+      (3615210918661643 / 5000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_628_2 :
+    Uω (aρ 2) (bρ 2) (11671906263691 / 16000000000000 : ℝ) ≤
+      (-(1637989077187245085489 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14413055933439441 / 20000000000000000 : ℝ) ≤
+      (-(1637989077187245085489 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (14413055933439441 / 20000000000000000 : ℝ) =
+            (14413055933439441 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14413055933439441 / 10000000000000000 : ℝ) ≤
+            (1827746824312754914511 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3603190374613783 / 5000000000000000 : ℝ)) (q :=
+      (14413055933439441 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_628_3 :
+    Uω (aρ 3) (bρ 3) (11671906263691 / 16000000000000 : ℝ) ≤
+      (-(3342479810571222740781 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14317524725642981 / 20000000000000000 : ℝ) ≤
+      (-(3342479810571222740781 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (14317524725642981 / 20000000000000000 : ℝ) =
+            (14317524725642981 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14317524725642981 / 10000000000000000 : ℝ) ≤
+            (3588991992428777259219 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3579122823890553 / 5000000000000000 : ℝ)) (q :=
+      (14317524725642981 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_628_4 :
+    Uω (aρ 4) (bρ 4) (11671906263691 / 16000000000000 : ℝ) ≤
+      (-(3453819726866507349207 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14158997682236471 / 20000000000000000 : ℝ) ≤
+      (-(3453819726866507349207 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (14158997682236471 / 20000000000000000 : ℝ) =
+            (14158997682236471 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14158997682236471 / 10000000000000000 : ℝ) ≤
+            (3477652076133492650793 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1769503196359899 / 2500000000000000 : ℝ)) (q :=
+      (14158997682236471 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_628_5 :
+    Uω (aρ 5) (bρ 5) (11671906263691 / 16000000000000 : ℝ) ≤
+      (-(3625538355383480504909 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6958968485077981 / 10000000000000000 : ℝ) ≤
+      (-(3625538355383480504909 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (6958968485077981 / 10000000000000000 : ℝ) =
+            (6958968485077981 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6958968485077981 / 5000000000000000 : ℝ) ≤
+            (3305933447616519495091 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6955141992694087 / 10000000000000000 : ℝ)) (q :=
+      (6958968485077981 / 10000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_628_6 :
+    Uω (aρ 6) (bρ 6) (11671906263691 / 16000000000000 : ℝ) ≤
+      (-(193823918117542028053 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2714605189622871 / 4000000000000000 : ℝ) ≤
+      (-(193823918117542028053 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2714605189622871 / 4000000000000000 : ℝ) =
+            (2714605189622871 / 2000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2714605189622871 / 2000000000000000 : ℝ) ≤
+            (152749672032457971947 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (42359195911203 / 62500000000000 : ℝ)) (q :=
+      (2714605189622871 / 4000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_628_7 :
+    Uω (aρ 7) (bρ 7) (11671906263691 / 16000000000000 : ℝ) ≤
+      (-(132124751893165640127 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6552101819691467 / 10000000000000000 : ℝ) ≤
+      (-(132124751893165640127 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (6552101819691467 / 10000000000000000 : ℝ) =
+            (6552101819691467 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6552101819691467 / 5000000000000000 : ℝ) ≤
+            (84483741950584359873 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6532310417821059 / 10000000000000000 : ℝ)) (q :=
+      (6552101819691467 / 10000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_628_8 :
+    Uω (aρ 8) (bρ 8) (11671906263691 / 16000000000000 : ℝ) ≤
+      (-(4703268327102192686097 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1561995075781669 / 2500000000000000 : ℝ) ≤
+      (-(4703268327102192686097 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1561995075781669 / 2500000000000000 : ℝ) =
+            (1561995075781669 / 1250000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1561995075781669 / 1250000000000000 : ℝ) ≤
+            (2228203475897807313903 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6207609036031477 / 10000000000000000 : ℝ)) (q :=
+      (1561995075781669 / 2500000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_628_9 :
+    Uω (aρ 9) (bρ 9) (11671906263691 / 16000000000000 : ℝ) ≤
+      (-(5326895601779464101117 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23480960321493 / 40000000000000 : ℝ) ≤
+      (-(5326895601779464101117 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23480960321493 / 40000000000000 : ℝ) = (23480960321493 / 20000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23480960321493 / 20000000000000 : ℝ) ≤
+            (1604576201220535898883 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (46344753751077 / 80000000000000 : ℝ)) (q :=
+      (23480960321493 / 40000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_628_10 :
+    Uω (aρ 10) (bρ 10) (11671906263691 / 16000000000000 : ℝ) ≤
+      (-(3062433192379457887933 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1084002860470187 / 2000000000000000 : ℝ) ≤
+      (-(3062433192379457887933 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1084002860470187 / 2000000000000000 : ℝ) =
+            (1084002860470187 / 1000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1084002860470187 / 1000000000000000 : ℝ) ≤
+            (403302709120542112067 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1056240339266999 / 2000000000000000 : ℝ)) (q :=
+      (1084002860470187 / 2000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_628_11 :
+    Uω (aρ 11) (bρ 11) (11671906263691 / 16000000000000 : ℝ) ≤
+      (-(7125471243255631514223 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4903935118129209 / 10000000000000000 : ℝ) ≤
+      (-(7125471243255631514223 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4903935118129209 / 10000000000000000 : ℝ) =
+            (4903935118129209 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4903935118129209 / 2500000000000000 : ℝ) ≤
+            (6737472362744368485777 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4667519954096543 / 10000000000000000 : ℝ)) (q :=
+      (4903935118129209 / 10000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_628_12 :
+    Uω (aρ 12) (bρ 12) (11671906263691 / 16000000000000 : ℝ) ≤
+      (-(8362255243951402653027 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8666861662668721 / 20000000000000000 : ℝ) ≤
+      (-(8362255243951402653027 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8666861662668721 / 20000000000000000 : ℝ) =
+            (8666861662668721 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8666861662668721 / 5000000000000000 : ℝ) ≤
+            (5500688362048597346973 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1975105738595923 / 5000000000000000 : ℝ)) (q :=
+      (8666861662668721 / 20000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_628_13 :
+    Uω (aρ 13) (bρ 13) (11671906263691 / 16000000000000 : ℝ) ≤
+      (-(9883039534702144509293 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1488829747688233 / 4000000000000000 : ℝ) ≤
+      (-(9883039534702144509293 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1488829747688233 / 4000000000000000 : ℝ) =
+            (1488829747688233 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1488829747688233 / 1000000000000000 : ℝ) ≤
+            (3979904071297855490707 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (312688029455929 / 1000000000000000 : ℝ)) (q :=
+      (1488829747688233 / 4000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_628_14 :
+    Uω (aρ 14) (bρ 14) (11671906263691 / 16000000000000 : ℝ) ≤
+      (-(589233722447172838279 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6155000421638059 / 20000000000000000 : ℝ) ≤
+      (-(589233722447172838279 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6155000421638059 / 20000000000000000 : ℝ) =
+            (6155000421638059 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6155000421638059 / 5000000000000000 : ℝ) ≤
+            (103913457852827161721 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (68176851962537 / 312500000000000 : ℝ)) (q :=
+      (6155000421638059 / 20000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_628_15 :
+    Uω (aρ 15) (bρ 15) (11671906263691 / 16000000000000 : ℝ) ≤
+      (-(7240890309318723000821 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2349980500237737 / 10000000000000000 : ℝ) ≤
+      (-(7240890309318723000821 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2349980500237737 / 10000000000000000 : ℝ) =
+            (2349980500237737 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2349980500237737 / 1250000000000000 : ℝ) ≤
+            (3156317395181276999179 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (986193119053599 / 10000000000000000 : ℝ)) (q :=
+      (2349980500237737 / 10000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_628_16 :
+    Uω (aρ 16) (bρ 16) (11671906263691 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_628 :
+    Uρ (11671906263691 / 16000000000000 : ℝ) ≤
+      (-(1399085531536063093099 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (11671906263691 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3242877128542114788709 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_628_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (11671906263691 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1637989077187245085489 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_628_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (11671906263691 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3342479810571222740781 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_628_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (11671906263691 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3453819726866507349207 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_628_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (11671906263691 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(3625538355383480504909 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_628_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (11671906263691 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(193823918117542028053 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_628_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (11671906263691 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(132124751893165640127 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_628_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (11671906263691 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(4703268327102192686097 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_628_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (11671906263691 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(5326895601779464101117 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_628_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (11671906263691 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(3062433192379457887933 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_628_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (11671906263691 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(7125471243255631514223 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_628_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (11671906263691 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(8362255243951402653027 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_628_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (11671906263691 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(9883039534702144509293 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_628_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (11671906263691 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) * (-(589233722447172838279 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_628_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (11671906263691 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(7240890309318723000821 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_628_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (11671906263691 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_628_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_629_1 :
+    Uω (aρ 1) (bρ 1) (23374289245939 / 32000000000000 : ℝ) ≤
+      (-(645942733699712492987 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28959783365142761 / 40000000000000000 : ℝ) ≤
+      (-(645942733699712492987 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (28959783365142761 / 40000000000000000 : ℝ) =
+            (28959783365142761 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28959783365142761 / 20000000000000000 : ℝ) ≤
+            (740351626900287507013 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7239923511300443 / 10000000000000000 : ℝ)) (q :=
+      (28959783365142761 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_629_2 :
+    Uω (aρ 2) (bρ 2) (23374289245939 / 32000000000000 : ℝ) ≤
+      (-(815692712508334048631 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28864208542300257 / 40000000000000000 : ℝ) ≤
+      (-(815692712508334048631 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (28864208542300257 / 40000000000000000 : ℝ) =
+            (28864208542300257 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28864208542300257 / 20000000000000000 : ℝ) ≤
+            (917175238241665951369 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7215905112389191 / 10000000000000000 : ℝ)) (q :=
+      (28864208542300257 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_629_3 :
+    Uω (aρ 3) (bρ 3) (23374289245939 / 32000000000000 : ℝ) ≤
+      (-(832295938651579155123 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28673148095764909 / 40000000000000000 : ℝ) ≤
+      (-(832295938651579155123 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (28673148095764909 / 40000000000000000 : ℝ) =
+            (28673148095764909 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28673148095764909 / 20000000000000000 : ℝ) ≤
+            (900572012098420844877 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7167770995471517 / 10000000000000000 : ℝ)) (q :=
+      (28673148095764909 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_629_4 :
+    Uω (aρ 4) (bρ 4) (23374289245939 / 32000000000000 : ℝ) ≤
+      (-(3440373057157870478887 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28356099250297589 / 40000000000000000 : ℝ) ≤
+      (-(3440373057157870478887 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (28356099250297589 / 40000000000000000 : ℝ) =
+            (28356099250297589 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28356099250297589 / 20000000000000000 : ℝ) ≤
+            (3491098745842129521113 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7087540753802857 / 10000000000000000 : ℝ)) (q :=
+      (28356099250297589 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_629_5 :
+    Uω (aρ 5) (bρ 5) (23374289245939 / 32000000000000 : ℝ) ≤
+      (-(722370860654510914959 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27873990768952717 / 40000000000000000 : ℝ) ≤
+      (-(722370860654510914959 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (27873990768952717 / 40000000000000000 : ℝ) =
+            (27873990768952717 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27873990768952717 / 20000000000000000 : ℝ) ≤
+            (663923499945489085041 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6964676432465421 / 10000000000000000 : ℝ)) (q :=
+      (27873990768952717 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_629_6 :
+    Uω (aρ 6) (bρ 6) (23374289245939 / 32000000000000 : ℝ) ≤
+      (-(965608962956098136493 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1087367941825877 / 1600000000000000 : ℝ) ≤
+      (-(965608962956098136493 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1087367941825877 / 1600000000000000 : ℝ) =
+            (1087367941825877 / 800000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1087367941825877 / 800000000000000 : ℝ) ≤
+            (767258987793901863507 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (271480827838101 / 400000000000000 : ℝ)) (q :=
+      (1087367941825877 / 1600000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_629_7 :
+    Uω (aρ 7) (bρ 7) (23374289245939 / 32000000000000 : ℝ) ≤
+      (-(105335573704855377237 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (26246618430248087 / 40000000000000000 : ℝ) ≤
+      (-(105335573704855377237 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (26246618430248087 / 40000000000000000 : ℝ) =
+            (26246618430248087 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26246618430248087 / 20000000000000000 : ℝ) ≤
+            (67951221370144622763 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3270946009506553 / 5000000000000000 : ℝ)) (q :=
+      (26246618430248087 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_629_8 :
+    Uω (aρ 8) (bρ 8) (23374289245939 / 32000000000000 : ℝ) ≤
+      (-(4687937817110196890423 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25030264485880559 / 40000000000000000 : ℝ) ≤
+      (-(4687937817110196890423 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (25030264485880559 / 40000000000000000 : ℝ) =
+            (25030264485880559 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25030264485880559 / 20000000000000000 : ℝ) ≤
+            (2243533985889803109577 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3108628349084671 / 5000000000000000 : ℝ)) (q :=
+      (25030264485880559 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_629_9 :
+    Uω (aρ 9) (bρ 9) (23374289245939 / 32000000000000 : ℝ) ≤
+      (-(5310469239407437719787 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4703912538789881 / 8000000000000000 : ℝ) ≤
+      (-(5310469239407437719787 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (4703912538789881 / 8000000000000000 : ℝ) =
+            (4703912538789881 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4703912538789881 / 4000000000000000 : ℝ) ≤
+            (1621002563592562280213 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1160574286112753 / 2000000000000000 : ℝ)) (q :=
+      (4703912538789881 / 8000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_629_10 :
+    Uω (aρ 10) (bρ 10) (23374289245939 / 32000000000000 : ℝ) ≤
+      (-(3053424875716178132791 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21719152581301551 / 40000000000000000 : ℝ) ≤
+      (-(3053424875716178132791 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (21719152581301551 / 40000000000000000 : ℝ) =
+            (21719152581301551 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21719152581301551 / 20000000000000000 : ℝ) ≤
+            (412311025783821867209 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2645612703867419 / 5000000000000000 : ℝ)) (q :=
+      (21719152581301551 / 40000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_629_11 :
+    Uω (aρ 11) (bρ 11) (23374289245939 / 32000000000000 : ℝ) ≤
+      (-(1421017869337593465473 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19655761843575217 / 40000000000000000 : ℝ) ≤
+      (-(1421017869337593465473 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (19655761843575217 / 40000000000000000 : ℝ) =
+            (19655761843575217 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19655761843575217 / 10000000000000000 : ℝ) ≤
+            (1351570851862406534527 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4678006665076671 / 10000000000000000 : ℝ)) (q :=
+      (19655761843575217 / 40000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_629_12 :
+    Uω (aρ 12) (bρ 12) (23374289245939 / 32000000000000 : ℝ) ≤
+      (-(833817983441931326163 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3475101049956067 / 8000000000000000 : ℝ) ≤
+      (-(833817983441931326163 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3475101049956067 / 8000000000000000 : ℝ) =
+            (3475101049956067 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3475101049956067 / 2000000000000000 : ℝ) ≤
+            (552476377158068673837 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (396157846486423 / 1000000000000000 : ℝ)) (q :=
+      (3475101049956067 / 8000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_629_13 :
+    Uω (aρ 13) (bρ 13) (23374289245939 / 32000000000000 : ℝ) ≤
+      (-(12315806207499133563 / 12500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2986723731185781 / 8000000000000000 : ℝ) ≤
+      (-(12315806207499133563 / 12500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2986723731185781 / 8000000000000000 : ℝ) =
+            (2986723731185781 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2986723731185781 / 2000000000000000 : ℝ) ≤
+            (5012873300000866437 / 12500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (628003381906703 / 2000000000000000 : ℝ)) (q :=
+      (2986723731185781 / 8000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_629_14 :
+    Uω (aρ 14) (bρ 14) (23374289245939 / 32000000000000 : ℝ) ≤
+      (-(469647681200431082067 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2472728849475393 / 8000000000000000 : ℝ) ≤
+      (-(469647681200431082067 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2472728849475393 / 8000000000000000 : ℝ) =
+            (2472728849475393 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2472728849475393 / 2000000000000000 : ℝ) ≤
+            (84870063039568917933 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (439791398060509 / 2000000000000000 : ℝ)) (q :=
+      (2472728849475393 / 8000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_629_15 :
+    Uω (aρ 15) (bρ 15) (23374289245939 / 32000000000000 : ℝ) ≤
+      (-(14386903830489276582459 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9489529854871201 / 40000000000000000 : ℝ) ≤
+      (-(14386903830489276582459 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (9489529854871201 / 40000000000000000 : ℝ) =
+            (9489529854871201 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9489529854871201 / 5000000000000000 : ℝ) ≤
+            (6407511578510723417541 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1021473071464663 / 10000000000000000 : ℝ)) (q :=
+      (9489529854871201 / 40000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_629_16 :
+    Uω (aρ 16) (bρ 16) (23374289245939 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_629 :
+    Uρ (23374289245939 / 32000000000000 : ℝ) ≤
+      (-(5576914744225668371273 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (23374289245939 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(645942733699712492987 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_629_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (23374289245939 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(815692712508334048631 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_629_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (23374289245939 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(832295938651579155123 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_629_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (23374289245939 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3440373057157870478887 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_629_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (23374289245939 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(722370860654510914959 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_629_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (23374289245939 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(965608962956098136493 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_629_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (23374289245939 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(105335573704855377237 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_629_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (23374289245939 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(4687937817110196890423 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_629_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (23374289245939 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(5310469239407437719787 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_629_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (23374289245939 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(3053424875716178132791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_629_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (23374289245939 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(1421017869337593465473 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_629_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (23374289245939 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(833817983441931326163 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_629_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (23374289245939 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) * (-(12315806207499133563 / 12500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_629_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (23374289245939 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) * (-(469647681200431082067 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_629_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (23374289245939 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(14386903830489276582459 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_629_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (23374289245939 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_629_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_630_1 :
+    Uω (aρ 1) (bρ 1) (1462797872781 / 2000000000000 : ℝ) ≤
+      (-(3216567513452919317099 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14498939690341621 / 20000000000000000 : ℝ) ≤
+      (-(3216567513452919317099 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (14498939690341621 / 20000000000000000 : ℝ) =
+            (14498939690341621 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14498939690341621 / 10000000000000000 : ℝ) ≤
+            (3714904289547080682901 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7249447544521621 / 10000000000000000 : ℝ)) (q :=
+      (14498939690341621 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_630_2 :
+    Uω (aρ 2) (bρ 2) (1462797872781 / 2000000000000 : ℝ) ≤
+      (-(1624790483342472992419 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14451152607836319 / 20000000000000000 : ℝ) ≤
+      (-(1624790483342472992419 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (14451152607836319 / 20000000000000000 : ℝ) =
+            (14451152607836319 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14451152607836319 / 10000000000000000 : ℝ) ≤
+            (1840945418157527007581 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7225429474526319 / 10000000000000000 : ℝ)) (q :=
+      (14451152607836319 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_630_3 :
+    Uω (aρ 3) (bρ 3) (1462797872781 / 2000000000000 : ℝ) ≤
+      (-(1657952678109447268993 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7177811683238731 / 10000000000000000 : ℝ) ≤
+      (-(1657952678109447268993 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (7177811683238731 / 10000000000000000 : ℝ) =
+            (7177811683238731 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7177811683238731 / 5000000000000000 : ℝ) ≤
+            (1807783223390552731007 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3588648169758731 / 5000000000000000 : ℝ)) (q :=
+      (7177811683238731 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_630_4 :
+    Uω (aρ 4) (bρ 4) (1462797872781 / 2000000000000 : ℝ) ≤
+      (-(3426944452015668801577 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7098550778669901 / 10000000000000000 : ℝ) ≤
+      (-(3426944452015668801577 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (7098550778669901 / 10000000000000000 : ℝ) =
+            (7098550778669901 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7098550778669901 / 5000000000000000 : ℝ) ≤
+            (3504527350984331198423 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3548534355722401 / 5000000000000000 : ℝ)) (q :=
+      (7098550778669901 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_630_5 :
+    Uω (aρ 5) (bρ 5) (1462797872781 / 2000000000000 : ℝ) ≤
+      (-(1799094485697806630207 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13956053770198459 / 20000000000000000 : ℝ) ≤
+      (-(1799094485697806630207 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (13956053770198459 / 20000000000000000 : ℝ) =
+            (13956053770198459 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13956053770198459 / 10000000000000000 : ℝ) ≤
+            (1666641415802193369793 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6974210843638459 / 10000000000000000 : ℝ)) (q :=
+      (13956053770198459 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_630_6 :
+    Uω (aρ 6) (bρ 6) (1462797872781 / 2000000000000 : ℝ) ≤
+      (-(38484130851782445057 / 100000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13611172526320581 / 20000000000000000 : ℝ) ≤
+      (-(38484130851782445057 / 100000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (13611172526320581 / 20000000000000000 : ℝ) =
+            (13611172526320581 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13611172526320581 / 10000000000000000 : ℝ) ≤
+            (30830587178217554943 / 100000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6796569974900581 / 10000000000000000 : ℝ)) (q :=
+      (13611172526320581 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_630_7 :
+    Uω (aρ 7) (bρ 7) (1462797872781 / 2000000000000 : ℝ) ≤
+      (-(524859394838169808527 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13142414622812853 / 20000000000000000 : ℝ) ≤
+      (-(524859394838169808527 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (13142414622812853 / 20000000000000000 : ℝ) =
+            (13142414622812853 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13142414622812853 / 10000000000000000 : ℝ) ≤
+            (341574580536830191473 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6551473452152853 / 10000000000000000 : ℝ)) (q :=
+      (13142414622812853 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_630_8 :
+    Uω (aρ 8) (bρ 8) (1462797872781 / 2000000000000 : ℝ) ≤
+      (-(4672631077418728873253 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1253430349881301 / 2000000000000000 : ℝ) ≤
+      (-(4672631077418728873253 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1253430349881301 / 2000000000000000 : ℝ) =
+            (1253430349881301 / 1000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1253430349881301 / 1000000000000000 : ℝ) ≤
+            (2258840725581271126747 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (622690397949301 / 1000000000000000 : ℝ)) (q :=
+      (1253430349881301 / 2000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_630_9 :
+    Uω (aρ 9) (bρ 9) (1462797872781 / 2000000000000 : ℝ) ≤
+      (-(529407052921083357707 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11779081692316477 / 20000000000000000 : ℝ) ≤
+      (-(529407052921083357707 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11779081692316477 / 20000000000000000 : ℝ) =
+            (11779081692316477 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11779081692316477 / 10000000000000000 : ℝ) ≤
+            (163740127378916642293 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5812647801356477 / 10000000000000000 : ℝ)) (q :=
+      (11779081692316477 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_630_10 :
+    Uω (aρ 10) (bρ 10) (1462797872781 / 2000000000000 : ℝ) ≤
+      (-(3044433606727779236829 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10879122133882899 / 20000000000000000 : ℝ) ≤
+      (-(3044433606727779236829 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10879122133882899 / 20000000000000000 : ℝ) =
+            (10879122133882899 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10879122133882899 / 10000000000000000 : ℝ) ≤
+            (421302294772220763171 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5301247276417899 / 10000000000000000 : ℝ)) (q :=
+      (10879122133882899 / 20000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_630_11 :
+    Uω (aρ 11) (bρ 11) (1462797872781 / 2000000000000 : ℝ) ≤
+      (-(7084753080275469726601 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1969577499663141 / 4000000000000000 : ℝ) ≤
+      (-(7084753080275469726601 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1969577499663141 / 4000000000000000 : ℝ) =
+            (1969577499663141 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1969577499663141 / 1000000000000000 : ℝ) ≤
+            (6778190525724530273399 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (937697853411141 / 2000000000000000 : ℝ)) (q :=
+      (1969577499663141 / 4000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_630_12 :
+    Uω (aρ 12) (bρ 12) (1462797872781 / 2000000000000 : ℝ) ≤
+      (-(1662834675285439792897 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2177158473993287 / 5000000000000000 : ℝ) ≤
+      (-(1662834675285439792897 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2177158473993287 / 5000000000000000 : ℝ) =
+            (2177158473993287 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2177158473993287 / 1250000000000000 : ℝ) ≤
+            (1109754045914560207103 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (993233940349537 / 2500000000000000 : ℝ)) (q :=
+      (2177158473993287 / 5000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_630_13 :
+    Uω (aρ 13) (bρ 13) (1462797872781 / 2000000000000 : ℝ) ≤
+      (-(9822377166288816452343 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7489443954622891 / 20000000000000000 : ℝ) ≤
+      (-(9822377166288816452343 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7489443954622891 / 20000000000000000 : ℝ) =
+            (7489443954622891 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7489443954622891 / 5000000000000000 : ℝ) ≤
+            (4040566439711183547657 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3153127561642891 / 10000000000000000 : ℝ)) (q :=
+      (7489443954622891 / 20000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_630_14 :
+    Uω (aρ 14) (bρ 14) (1462797872781 / 2000000000000 : ℝ) ≤
+      (-(5849024699973716311919 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (310427487186159 / 1000000000000000 : ℝ) ≤
+      (-(5849024699973716311919 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (310427487186159 / 1000000000000000 : ℝ) =
+            (310427487186159 / 250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (310427487186159 / 250000000000000 : ℝ) ≤
+            (1082447103026283688081 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (110808031789409 / 500000000000000 : ℝ)) (q :=
+      (310427487186159 / 1000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14]) (by norm_num [bρ_14])
+      (by norm_num) (by norm_num [aρ_14, bρ_14]) (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_630_15 :
+    Uω (aρ 15) (bρ 15) (1462797872781 / 2000000000000 : ℝ) ≤
+      (-(7147600066389320593217 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4788476294669653 / 20000000000000000 : ℝ) ≤
+      (-(7147600066389320593217 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (4788476294669653 / 20000000000000000 : ℝ) =
+            (4788476294669653 / 2500000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4788476294669653 / 2500000000000000 : ℝ) ≤
+            (3249607638110679406783 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1055660464149653 / 10000000000000000 : ℝ)) (q :=
+      (4788476294669653 / 20000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_630_16 :
+    Uω (aρ 16) (bρ 16) (1462797872781 / 2000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_630 :
+    Uρ (1462797872781 / 2000000000000 : ℝ) ≤
+      (-(5557629648906738505043 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1462797872781 / 2000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3216567513452919317099 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_630_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1462797872781 / 2000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1624790483342472992419 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_630_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1462797872781 / 2000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1657952678109447268993 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_630_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1462797872781 / 2000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3426944452015668801577 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_630_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1462797872781 / 2000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1799094485697806630207 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_630_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1462797872781 / 2000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) * (-(38484130851782445057 / 100000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_630_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1462797872781 / 2000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(524859394838169808527 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_630_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1462797872781 / 2000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(4672631077418728873253 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_630_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1462797872781 / 2000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(529407052921083357707 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_630_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1462797872781 / 2000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(3044433606727779236829 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_630_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1462797872781 / 2000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(7084753080275469726601 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_630_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1462797872781 / 2000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(1662834675285439792897 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_630_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1462797872781 / 2000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(9822377166288816452343 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_630_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1462797872781 / 2000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(5849024699973716311919 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_630_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1462797872781 / 2000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(7147600066389320593217 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_630_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1462797872781 / 2000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_630_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_631_1 :
+    Uω (aρ 1) (bρ 1) (23435242683053 / 32000000000000 : ℝ) ≤
+      (-(3203438617965511292341 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (29035975395915803 / 40000000000000000 : ℝ) ≤
+      (-(3203438617965511292341 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (29035975395915803 / 40000000000000000 : ℝ) =
+            (29035975395915803 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (29035975395915803 / 20000000000000000 : ℝ) ≤
+            (3728033185034488707659 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7258971577588839 / 10000000000000000 : ℝ)) (q :=
+      (29035975395915803 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_631_2 :
+    Uω (aρ 2) (bρ 2) (23435242683053 / 32000000000000 : ℝ) ≤
+      (-(3236408458429937679207 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28940401887004121 / 40000000000000000 : ℝ) ≤
+      (-(3236408458429937679207 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (28940401887004121 / 40000000000000000 : ℝ) =
+            (28940401887004121 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28940401887004121 / 20000000000000000 : ℝ) ≤
+            (3695063344570062320793 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3617476917821499 / 5000000000000000 : ℝ)) (q :=
+      (28940401887004121 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_631_3 :
+    Uω (aρ 3) (bρ 3) (23435242683053 / 32000000000000 : ℝ) ≤
+      (-(132105782742592253447 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28749345362884991 / 40000000000000000 : ℝ) ≤
+      (-(132105782742592253447 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (28749345362884991 / 40000000000000000 : ℝ) =
+            (28749345362884991 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28749345362884991 / 20000000000000000 : ℝ) ≤
+            (145153089377407746553 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7186821679933433 / 10000000000000000 : ℝ)) (q :=
+      (28749345362884991 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_631_4 :
+    Uω (aρ 4) (bρ 4) (23435242683053 / 32000000000000 : ℝ) ≤
+      (-(3413533862948097198867 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28432306957705233 / 40000000000000000 : ℝ) ≤
+      (-(3413533862948097198867 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (28432306957705233 / 40000000000000000 : ℝ) =
+            (28432306957705233 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28432306957705233 / 20000000000000000 : ℝ) ≤
+            (3517937940051902801133 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3553298329204277 / 5000000000000000 : ℝ)) (q :=
+      (28432306957705233 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_631_5 :
+    Uω (aρ 5) (bρ 5) (23435242683053 / 32000000000000 : ℝ) ≤
+      (-(3584542308547022505337 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27950224254878789 / 40000000000000000 : ℝ) ≤
+      (-(3584542308547022505337 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (27950224254878789 / 40000000000000000 : ℝ) =
+            (27950224254878789 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27950224254878789 / 20000000000000000 : ℝ) ≤
+            (3346929494452977494663 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1745936306582583 / 2500000000000000 : ℝ)) (q :=
+      (27950224254878789 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_631_6 :
+    Uω (aρ 6) (bρ 6) (23435242683053 / 32000000000000 : ℝ) ≤
+      (-(3834410006823929320503 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27260491417810903 / 40000000000000000 : ℝ) ≤
+      (-(3834410006823929320503 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (27260491417810903 / 40000000000000000 : ℝ) =
+            (27260491417810903 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27260491417810903 / 20000000000000000 : ℝ) ≤
+            (3097061796176070679497 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6806119182936389 / 10000000000000000 : ℝ)) (q :=
+      (27260491417810903 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_631_7 :
+    Uω (aρ 7) (bρ 7) (23435242683053 / 32000000000000 : ℝ) ≤
+      (-(836869725883868706071 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (26323039726371219 / 40000000000000000 : ℝ) ≤
+      (-(836869725883868706071 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (26323039726371219 / 40000000000000000 : ℝ) =
+            (26323039726371219 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26323039726371219 / 20000000000000000 : ℝ) ≤
+            (549424634716131293929 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6561054717976547 / 10000000000000000 : ℝ)) (q :=
+      (26323039726371219 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_631_8 :
+    Uω (aρ 8) (bρ 8) (23435242683053 / 32000000000000 : ℝ) ≤
+      (-(931469606699471203557 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25106948751277561 / 40000000000000000 : ℝ) ≤
+      (-(931469606699471203557 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (25106948751277561 / 40000000000000000 : ℝ) =
+            (25106948751277561 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25106948751277561 / 20000000000000000 : ℝ) ≤
+            (454824753900528796443 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3118275440884859 / 5000000000000000 : ℝ)) (q :=
+      (25106948751277561 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_631_9 :
+    Uω (aρ 9) (bρ 9) (23435242683053 / 32000000000000 : ℝ) ≤
+      (-(5277699375886241265793 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4719352480403037 / 8000000000000000 : ℝ) ≤
+      (-(5277699375886241265793 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (4719352480403037 / 8000000000000000 : ℝ) =
+            (4719352480403037 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4719352480403037 / 4000000000000000 : ℝ) ≤
+            (1653772427113758734207 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (582242333549853 / 1000000000000000 : ℝ)) (q :=
+      (4719352480403037 / 8000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_631_10 :
+    Uω (aρ 10) (bρ 10) (23435242683053 / 32000000000000 : ℝ) ≤
+      (-(758864829476275798249 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21797332289658697 / 40000000000000000 : ℝ) ≤
+      (-(758864829476275798249 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (21797332289658697 / 40000000000000000 : ℝ) =
+            (21797332289658697 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21797332289658697 / 20000000000000000 : ℝ) ≤
+            (107569145898724201751 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2655633656407643 / 5000000000000000 : ℝ)) (q :=
+      (21797332289658697 / 40000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_631_11 :
+    Uω (aρ 11) (bρ 11) (23435242683053 / 32000000000000 : ℝ) ≤
+      (-(7064462222460948975133 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19735779986684331 / 40000000000000000 : ℝ) ≤
+      (-(7064462222460948975133 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (19735779986684331 / 40000000000000000 : ℝ) =
+            (19735779986684331 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19735779986684331 / 10000000000000000 : ℝ) ≤
+            (6798481383539051024867 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4698967787533103 / 10000000000000000 : ℝ)) (q :=
+      (19735779986684331 / 40000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_631_12 :
+    Uω (aρ 12) (bρ 12) (23435242683053 / 32000000000000 : ℝ) ≤
+      (-(8290235418091904679413 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17459011117584283 / 40000000000000000 : ℝ) ≤
+      (-(8290235418091904679413 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (17459011117584283 / 40000000000000000 : ℝ) =
+            (17459011117584283 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17459011117584283 / 10000000000000000 : ℝ) ≤
+            (5572708187908095320587 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3984283449668079 / 10000000000000000 : ℝ)) (q :=
+      (17459011117584283 / 40000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_631_13 :
+    Uω (aρ 13) (bρ 13) (23435242683053 / 32000000000000 : ℝ) ≤
+      (-(9792234836373217187103 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15024105881875133 / 40000000000000000 : ℝ) ≤
+      (-(9792234836373217187103 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (15024105881875133 / 40000000000000000 : ℝ) =
+            (15024105881875133 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15024105881875133 / 10000000000000000 : ℝ) ≤
+            (4070708769626782812897 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (395776571676063 / 1250000000000000 : ℝ)) (q :=
+      (15024105881875133 / 40000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_631_14 :
+    Uω (aρ 14) (bρ 14) (23435242683053 / 32000000000000 : ℝ) ≤
+      (-(2913809875003440331781 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12470370911958191 / 40000000000000000 : ℝ) ≤
+      (-(2913809875003440331781 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (12470370911958191 / 40000000000000000 : ℝ) =
+            (12470370911958191 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12470370911958191 / 10000000000000000 : ℝ) ≤
+            (551926026496559668219 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2233272373495033 / 10000000000000000 : ℝ)) (q :=
+      (12470370911958191 / 40000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_631_15 :
+    Uω (aρ 15) (bρ 15) (23435242683053 / 32000000000000 : ℝ) ≤
+      (-(221974656099820521723 / 156250000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9662396035747569 / 40000000000000000 : ℝ) ≤
+      (-(221974656099820521723 / 156250000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (9662396035747569 / 40000000000000000 : ℝ) =
+            (9662396035747569 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9662396035747569 / 5000000000000000 : ℝ) ≤
+            (102938084665804478277 / 156250000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (544429106402361 / 5000000000000000 : ℝ)) (q :=
+      (9662396035747569 / 40000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_631_16 :
+    Uω (aρ 16) (bρ 16) (23435242683053 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_631 :
+    Uρ (23435242683053 / 32000000000000 : ℝ) ≤
+      (-(86538710522970460249 / 156250000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (23435242683053 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3203438617965511292341 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_631_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (23435242683053 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3236408458429937679207 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_631_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (23435242683053 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(132105782742592253447 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_631_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (23435242683053 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3413533862948097198867 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_631_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (23435242683053 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(3584542308547022505337 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_631_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (23435242683053 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3834410006823929320503 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_631_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (23435242683053 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(836869725883868706071 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_631_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (23435242683053 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(931469606699471203557 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_631_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (23435242683053 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(5277699375886241265793 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_631_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (23435242683053 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(758864829476275798249 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_631_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (23435242683053 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(7064462222460948975133 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_631_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (23435242683053 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(8290235418091904679413 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_631_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (23435242683053 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(9792234836373217187103 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_631_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (23435242683053 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(2913809875003440331781 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_631_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (23435242683053 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(221974656099820521723 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_631_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (23435242683053 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_631_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_632_1 :
+    Uω (aρ 1) (bρ 1) (2346571940161 / 3200000000000 : ℝ) ≤
+      (-(1595163468387701664069 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14537035705420829 / 20000000000000000 : ℝ) ≤
+      (-(1595163468387701664069 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (14537035705420829 / 20000000000000000 : ℝ) =
+            (14537035705420829 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14537035705420829 / 10000000000000000 : ℝ) ≤
+            (1870572433112298335931 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (454280975656419 / 625000000000000 : ℝ)) (q :=
+      (14537035705420829 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_632_2 :
+    Uω (aρ 2) (bρ 2) (2346571940161 / 3200000000000 : ℝ) ≤
+      (-(3223253279550098399567 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (905578079884461 / 1250000000000000 : ℝ) ≤
+      (-(3223253279550098399567 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (905578079884461 / 1250000000000000 : ℝ) =
+            (905578079884461 / 625000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (905578079884461 / 625000000000000 : ℝ) ≤
+            (3708218523449901600433 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7244478195743251 / 10000000000000000 : ℝ)) (q :=
+      (905578079884461 / 1250000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_632_3 :
+    Uω (aρ 3) (bρ 3) (2346571940161 / 3200000000000 : ℝ) ≤
+      (-(3289401344986094540583 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14393721992791967 / 20000000000000000 : ℝ) ≤
+      (-(3289401344986094540583 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (14393721992791967 / 20000000000000000 : ℝ) =
+            (14393721992791967 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14393721992791967 / 10000000000000000 : ℝ) ≤
+            (3642070458013905459417 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3598173508366921 / 5000000000000000 : ℝ)) (q :=
+      (14393721992791967 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_632_4 :
+    Uω (aρ 4) (bρ 4) (2346571940161 / 3200000000000 : ℝ) ≤
+      (-(680028248331685089271 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (889700336858133 / 1250000000000000 : ℝ) ≤
+      (-(680028248331685089271 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (889700336858133 / 1250000000000000 : ℝ) =
+            (889700336858133 / 625000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (889700336858133 / 625000000000000 : ℝ) ≤
+            (706266112268314910729 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7116124594737003 / 10000000000000000 : ℝ)) (q :=
+      (889700336858133 / 1250000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_632_5 :
+    Uω (aρ 5) (bρ 5) (2346571940161 / 3200000000000 : ℝ) ≤
+      (-(1785457131865592913579 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6997085228157827 / 10000000000000000 : ℝ) ≤
+      (-(1785457131865592913579 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (6997085228157827 / 10000000000000000 : ℝ) =
+            (6997085228157827 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6997085228157827 / 5000000000000000 : ℝ) ≤
+            (1680278769634407086421 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6993279580657529 / 10000000000000000 : ℝ)) (q :=
+      (6997085228157827 / 10000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_632_6 :
+    Uω (aρ 6) (bρ 6) (2346571940161 / 3200000000000 : ℝ) ≤
+      (-(1910213280704012408157 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3412329705219033 / 5000000000000000 : ℝ) ≤
+      (-(1910213280704012408157 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (3412329705219033 / 5000000000000000 : ℝ) =
+            (3412329705219033 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3412329705219033 / 2500000000000000 : ℝ) ≤
+            (1555522620795987591843 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6815668320358007 / 10000000000000000 : ℝ)) (q :=
+      (3412329705219033 / 5000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_632_7 :
+    Uω (aρ 7) (bρ 7) (2346571940161 / 3200000000000 : ℝ) ≤
+      (-(4169843297918505665339 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13180624936974249 / 20000000000000000 : ℝ) ≤
+      (-(4169843297918505665339 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (13180624936974249 / 20000000000000000 : ℝ) =
+            (13180624936974249 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13180624936974249 / 10000000000000000 : ℝ) ≤
+            (2761628505081494334661 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1642658954304031 / 2500000000000000 : ℝ)) (q :=
+      (13180624936974249 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_632_8 :
+    Uω (aρ 8) (bρ 8) (2346571940161 / 3200000000000 : ℝ) ≤
+      (-(2321044305584939681371 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (157158060939673 / 250000000000000 : ℝ) ≤
+      (-(2321044305584939681371 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (157158060939673 / 250000000000000 : ℝ) =
+            (157158060939673 / 125000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (157158060939673 / 125000000000000 : ℝ) ≤
+            (1144691595915060318629 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1249239481351143 / 2000000000000000 : ℝ)) (q :=
+      (157158060939673 / 250000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_632_9 :
+    Uω (aρ 9) (bρ 9) (2346571940161 / 3200000000000 : ℝ) ≤
+      (-(5261355684633632966319 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11817679877255061 / 20000000000000000 : ℝ) ≤
+      (-(5261355684633632966319 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11817679877255061 / 20000000000000000 : ℝ) =
+            (11817679877255061 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11817679877255061 / 10000000000000000 : ℝ) ≤
+            (1670116118366367033681 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (729024754649617 / 1250000000000000 : ℝ)) (q :=
+      (11817679877255061 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_632_10 :
+    Uω (aρ 10) (bρ 10) (2346571940161 / 3200000000000 : ℝ) ≤
+      (-(6053003884311726330449 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10918208333840763 / 20000000000000000 : ℝ) ≤
+      (-(6053003884311726330449 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10918208333840763 / 20000000000000000 : ℝ) =
+            (10918208333840763 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10918208333840763 / 10000000000000000 : ℝ) ≤
+            (878467918688273669551 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2660642763638819 / 5000000000000000 : ℝ)) (q :=
+      (10918208333840763 / 20000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_632_11 :
+    Uω (aρ 11) (bρ 11) (2346571940161 / 3200000000000 : ℝ) ≤
+      (-(7044216553404438458349 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1977577686822209 / 4000000000000000 : ℝ) ≤
+      (-(7044216553404438458349 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1977577686822209 / 4000000000000000 : ℝ) =
+            (1977577686822209 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1977577686822209 / 1000000000000000 : ℝ) ≤
+            (6818727052595561541651 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (117736056343823 / 250000000000000 : ℝ)) (q :=
+      (1977577686822209 / 4000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_632_12 :
+    Uω (aρ 12) (bρ 12) (2346571940161 / 3200000000000 : ℝ) ≤
+      (-(4133182756188624599249 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8750367695210443 / 20000000000000000 : ℝ) ≤
+      (-(4133182756188624599249 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8750367695210443 / 20000000000000000 : ℝ) =
+            (8750367695210443 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8750367695210443 / 5000000000000000 : ℝ) ≤
+            (2798289046811375400751 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1997810805768659 / 5000000000000000 : ℝ)) (q :=
+      (8750367695210443 / 20000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_632_13 :
+    Uω (aρ 13) (bρ 13) (2346571940161 / 3200000000000 : ℝ) ≤
+      (-(9762216699229738464429 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7534636603494341 / 20000000000000000 : ℝ) ≤
+      (-(9762216699229738464429 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7534636603494341 / 20000000000000000 : ℝ) =
+            (7534636603494341 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7534636603494341 / 5000000000000000 : ℝ) ≤
+            (4100726906770261535571 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (397409032677027 / 1250000000000000 : ℝ)) (q :=
+      (7534636603494341 / 20000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_632_14 :
+    Uω (aρ 14) (bρ 14) (2346571940161 / 3200000000000 : ℝ) ≤
+      (-(11612755515422022468479 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6261731357118699 / 20000000000000000 : ℝ) ≤
+      (-(11612755515422022468479 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6261731357118699 / 20000000000000000 : ℝ) =
+            (6261731357118699 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6261731357118699 / 5000000000000000 : ℝ) ≤
+            (2250188090577977531521 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1125147150042787 / 5000000000000000 : ℝ)) (q :=
+      (6261731357118699 / 20000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_632_15 :
+    Uω (aρ 15) (bρ 15) (2346571940161 / 3200000000000 : ℝ) ≤
+      (-(2824037625449027245121 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2436509005779369 / 10000000000000000 : ℝ) ≤
+      (-(2824037625449027245121 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2436509005779369 / 10000000000000000 : ℝ) =
+            (2436509005779369 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2436509005779369 / 1250000000000000 : ℝ) ≤
+            (1334845456350972754879 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1121154231940613 / 10000000000000000 : ℝ)) (q :=
+      (2436509005779369 / 10000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_632_16 :
+    Uω (aρ 16) (bρ 16) (2346571940161 / 3200000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_632 :
+    Uρ (2346571940161 / 3200000000000 : ℝ) ≤
+      (-(5519450151378080841463 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (2346571940161 / 3200000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1595163468387701664069 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_632_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (2346571940161 / 3200000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3223253279550098399567 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_632_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (2346571940161 / 3200000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3289401344986094540583 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_632_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (2346571940161 / 3200000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(680028248331685089271 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_632_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (2346571940161 / 3200000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1785457131865592913579 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_632_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (2346571940161 / 3200000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1910213280704012408157 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_632_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (2346571940161 / 3200000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(4169843297918505665339 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_632_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (2346571940161 / 3200000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(2321044305584939681371 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_632_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (2346571940161 / 3200000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(5261355684633632966319 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_632_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (2346571940161 / 3200000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(6053003884311726330449 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_632_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (2346571940161 / 3200000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(7044216553404438458349 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_632_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (2346571940161 / 3200000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(4133182756188624599249 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_632_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (2346571940161 / 3200000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(9762216699229738464429 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_632_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (2346571940161 / 3200000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(11612755515422022468479 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_632_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (2346571940161 / 3200000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(2824037625449027245121 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_632_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (2346571940161 / 3200000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_632_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_633_1 :
+    Uω (aρ 1) (bρ 1) (23496196120167 / 32000000000000 : ℝ) ≤
+      (-(1588616212399733649147 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (29112167425462007 / 40000000000000000 : ℝ) ≤
+      (-(1588616212399733649147 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (29112167425462007 / 40000000000000000 : ℝ) =
+            (29112167425462007 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (29112167425462007 / 20000000000000000 : ℝ) ≤
+            (1877119689100266350853 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (909752455407977 / 1250000000000000 : ℝ)) (q :=
+      (29112167425462007 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_633_2 :
+    Uω (aρ 2) (bρ 2) (23496196120167 / 32000000000000 : ℝ) ≤
+      (-(3210115384507427481169 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (29016595223576541 / 40000000000000000 : ℝ) ≤
+      (-(3210115384507427481169 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (29016595223576541 / 40000000000000000 : ℝ) =
+            (29016595223576541 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (29016595223576541 / 20000000000000000 : ℝ) ≤
+            (3721356418492572518831 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7254002554831083 / 10000000000000000 : ℝ)) (q :=
+      (29016595223576541 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_633_3 :
+    Uω (aρ 3) (bρ 3) (23496196120167 / 32000000000000 : ℝ) ≤
+      (-(131047025560400143257 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28825542601080433 / 40000000000000000 : ℝ) ≤
+      (-(131047025560400143257 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (28825542601080433 / 40000000000000000 : ℝ) =
+            (28825542601080433 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28825542601080433 / 20000000000000000 : ℝ) ≤
+            (146211846559599856743 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7205872349933029 / 10000000000000000 : ℝ)) (q :=
+      (28825542601080433 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_633_4 :
+    Uω (aρ 4) (bρ 4) (23496196120167 / 32000000000000 : ℝ) ≤
+      (-(846691635010987255199 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28508514580030001 / 40000000000000000 : ℝ) ≤
+      (-(846691635010987255199 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (28508514580030001 / 40000000000000000 : ℝ) =
+            (28508514580030001 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28508514580030001 / 20000000000000000 : ℝ) ≤
+            (886176315739012744801 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7125652520472813 / 10000000000000000 : ℝ)) (q :=
+      (28508514580030001 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_633_5 :
+    Uω (aρ 5) (bρ 5) (23496196120167 / 32000000000000 : ℝ) ≤
+      (-(3557304786161391909607 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28026457513886189 / 40000000000000000 : ℝ) ≤
+      (-(3557304786161391909607 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (28026457513886189 / 40000000000000000 : ℝ) =
+            (28026457513886189 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28026457513886189 / 20000000000000000 : ℝ) ≤
+            (3374167016838608090393 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7002813906735907 / 10000000000000000 : ℝ)) (q :=
+      (28026457513886189 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_633_6 :
+    Uω (aρ 6) (bρ 6) (23496196120167 / 32000000000000 : ℝ) ≤
+      (-(3806462693810826226399 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27336783725058021 / 40000000000000000 : ℝ) ≤
+      (-(3806462693810826226399 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (27336783725058021 / 40000000000000000 : ℝ) =
+            (27336783725058021 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27336783725058021 / 20000000000000000 : ℝ) ≤
+            (3125009109189173773601 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6825217387461823 / 10000000000000000 : ℝ)) (q :=
+      (27336783725058021 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_633_7 :
+    Uω (aρ 7) (bρ 7) (23496196120167 / 32000000000000 : ℝ) ≤
+      (-(2077679551030624965963 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (26399459689812871 / 40000000000000000 : ℝ) ≤
+      (-(2077679551030624965963 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (26399459689812871 / 40000000000000000 : ℝ) =
+            (26399459689812871 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26399459689812871 / 20000000000000000 : ℝ) ≤
+            (1388056350469375034037 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (411263546912453 / 625000000000000 : ℝ)) (q :=
+      (26399459689812871 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_633_8 :
+    Uω (aρ 8) (bρ 8) (23496196120167 / 32000000000000 : ℝ) ≤
+      (-(4626852736612063300721 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25183629998327079 / 40000000000000000 : ℝ) ≤
+      (-(4626852736612063300721 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (25183629998327079 / 40000000000000000 : ℝ) =
+            (25183629998327079 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25183629998327079 / 20000000000000000 : ℝ) ≤
+            (2304619066387936699279 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (95456597232 / 152587890625 : ℝ)) (q :=
+      (25183629998327079 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_633_9 :
+    Uω (aρ 9) (bρ 9) (23496196120167 / 32000000000000 : ℝ) ≤
+      (-(5245039361152753413503 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23673955450474761 / 40000000000000000 : ℝ) ≤
+      (-(5245039361152753413503 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23673955450474761 / 40000000000000000 : ℝ) =
+            (23673955450474761 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23673955450474761 / 20000000000000000 : ℝ) ≤
+            (1686432441847246586497 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5841971910630193 / 10000000000000000 : ℝ)) (q :=
+      (23673955450474761 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_633_10 :
+    Uω (aρ 10) (bρ 10) (23496196120167 / 32000000000000 : ℝ) ≤
+      (-(1508780706400550594719 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21875497422376211 / 40000000000000000 : ℝ) ≤
+      (-(1508780706400550594719 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (21875497422376211 / 40000000000000000 : ℝ) =
+            (21875497422376211 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21875497422376211 / 20000000000000000 : ℝ) ≤
+            (224087244349449405281 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2665650965037959 / 5000000000000000 : ℝ)) (q :=
+      (21875497422376211 / 40000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_633_11 :
+    Uω (aρ 11) (bρ 11) (23496196120167 / 32000000000000 : ℝ) ≤
+      (-(219500495467645514591 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19815765695228119 / 40000000000000000 : ℝ) ≤
+      (-(219500495467645514591 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (19815765695228119 / 40000000000000000 : ℝ) =
+            (19815765695228119 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19815765695228119 / 10000000000000000 : ℝ) ≤
+            (213716492219854485409 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (589989086588359 / 1250000000000000 : ℝ)) (q :=
+      (19815765695228119 / 40000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_633_12 :
+    Uω (aρ 12) (bρ 12) (23496196120167 / 32000000000000 : ℝ) ≤
+      (-(1030320402127628943133 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3508488154438577 / 8000000000000000 : ℝ) ≤
+      (-(1030320402127628943133 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3508488154438577 / 8000000000000000 : ℝ) =
+            (3508488154438577 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3508488154438577 / 2000000000000000 : ℝ) ≤
+            (702547548622371056867 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (801390065574851 / 2000000000000000 : ℝ)) (q :=
+      (3508488154438577 / 8000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_633_13 :
+    Uω (aρ 13) (bρ 13) (23496196120167 / 32000000000000 : ℝ) ≤
+      (-(304135046858737321201 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3022878101236307 / 8000000000000000 : ℝ) ≤
+      (-(304135046858737321201 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3022878101236307 / 8000000000000000 : ℝ) =
+            (3022878101236307 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3022878101236307 / 2000000000000000 : ℝ) ≤
+            (129081940828762678799 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (159615346823179 / 500000000000000 : ℝ)) (q :=
+      (3022878101236307 / 8000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_633_14 :
+    Uω (aρ 14) (bρ 14) (23496196120167 / 32000000000000 : ℝ) ≤
+      (-(11570590863640125216681 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12576378939996057 / 40000000000000000 : ℝ) ≤
+      (-(11570590863640125216681 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (12576378939996057 / 40000000000000000 : ℝ) =
+            (12576378939996057 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12576378939996057 / 10000000000000000 : ℝ) ≤
+            (2292352742359874783319 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2267228438415841 / 10000000000000000 : ℝ)) (q :=
+      (12576378939996057 / 40000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_633_15 :
+    Uω (aρ 15) (bρ 15) (23496196120167 / 32000000000000 : ℝ) ≤
+      (-(7018207705648636198711 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9828024152786267 / 40000000000000000 : ℝ) ≤
+      (-(7018207705648636198711 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (9828024152786267 / 40000000000000000 : ℝ) =
+            (9828024152786267 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9828024152786267 / 5000000000000000 : ℝ) ≤
+            (3378999998851363801289 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (576312161112973 / 5000000000000000 : ℝ)) (q :=
+      (9828024152786267 / 40000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_633_16 :
+    Uω (aρ 16) (bρ 16) (23496196120167 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_633 :
+    Uρ (23496196120167 / 32000000000000 : ℝ) ≤
+      (-(5500540668506838355503 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (23496196120167 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1588616212399733649147 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_633_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (23496196120167 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3210115384507427481169 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_633_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (23496196120167 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(131047025560400143257 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_633_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (23496196120167 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(846691635010987255199 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_633_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (23496196120167 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(3557304786161391909607 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_633_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (23496196120167 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3806462693810826226399 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_633_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (23496196120167 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(2077679551030624965963 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_633_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (23496196120167 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(4626852736612063300721 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_633_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (23496196120167 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(5245039361152753413503 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_633_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (23496196120167 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(1508780706400550594719 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_633_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (23496196120167 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) * (-(219500495467645514591 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_633_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (23496196120167 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(1030320402127628943133 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_633_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (23496196120167 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) * (-(304135046858737321201 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_633_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (23496196120167 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(11570590863640125216681 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_633_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (23496196120167 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(7018207705648636198711 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_633_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (23496196120167 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_633_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_634_1 :
+    Uω (aρ 1) (bρ 1) (5881668209681 / 8000000000000 : ℝ) ≤
+      (-(3164155037131447097627 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (583005268795561 / 800000000000000 : ℝ) ≤
+      (-(3164155037131447097627 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (583005268795561 / 800000000000000 : ℝ) =
+            (583005268795561 / 400000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (583005268795561 / 400000000000000 : ℝ) ≤
+            (3767316765868552902373 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (291501747034911 / 400000000000000 : ℝ)) (q :=
+      (583005268795561 / 800000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_634_2 :
+    Uω (aρ 2) (bρ 2) (5881668209681 / 8000000000000 : ℝ) ≤
+      (-(3196994727943193935909 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7263672972208363 / 10000000000000000 : ℝ) ≤
+      (-(3196994727943193935909 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (7263672972208363 / 10000000000000000 : ℝ) =
+            (7263672972208363 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7263672972208363 / 5000000000000000 : ℝ) ≤
+            (3734477075056806064091 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1815881728227619 / 2500000000000000 : ℝ)) (q :=
+      (7263672972208363 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_634_3 :
+    Uω (aρ 3) (bρ 3) (5881668209681 / 8000000000000 : ℝ) ≤
+      (-(163148370217400263421 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7215910302350753 / 10000000000000000 : ℝ) ≤
+      (-(163148370217400263421 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (7215910302350753 / 10000000000000000 : ℝ) =
+            (7215910302350753 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7215910302350753 / 5000000000000000 : ℝ) ≤
+            (183425219932599736579 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (901924709943157 / 1250000000000000 : ℝ)) (q :=
+      (7215910302350753 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_634_4 :
+    Uω (aρ 4) (bρ 4) (5881668209681 / 8000000000000 : ℝ) ≤
+      (-(3373409710194953931207 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14273309179749669 / 20000000000000000 : ℝ) ≤
+      (-(3373409710194953931207 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (14273309179749669 / 20000000000000000 : ℝ) =
+            (14273309179749669 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14273309179749669 / 10000000000000000 : ℝ) ≤
+            (3558062092805046068793 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7135180435658419 / 10000000000000000 : ℝ)) (q :=
+      (14273309179749669 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_634_5 :
+    Uω (aρ 5) (bρ 5) (5881668209681 / 8000000000000 : ℝ) ≤
+      (-(3543713825258676462001 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7016143514718471 / 10000000000000000 : ℝ) ≤
+      (-(3543713825258676462001 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (7016143514718471 / 10000000000000000 : ℝ) =
+            (7016143514718471 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7016143514718471 / 5000000000000000 : ℝ) ≤
+            (3387757977741323537999 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1753087051170173 / 2500000000000000 : ℝ)) (q :=
+      (7016143514718471 / 10000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_634_6 :
+    Uω (aρ 6) (bρ 6) (5881668209681 / 8000000000000 : ℝ) ≤
+      (-(3792518349145028688873 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13687464834158817 / 20000000000000000 : ℝ) ≤
+      (-(3792518349145028688873 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (13687464834158817 / 20000000000000000 : ℝ) =
+            (13687464834158817 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13687464834158817 / 10000000000000000 : ℝ) ≤
+            (3138953453854971311127 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6834766384542567 / 10000000000000000 : ℝ)) (q :=
+      (13687464834158817 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_634_7 :
+    Uω (aρ 7) (bρ 7) (5881668209681 / 8000000000000 : ℝ) ≤
+      (-(1035223994995105045411 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13218834587705587 / 20000000000000000 : ℝ) ≤
+      (-(1035223994995105045411 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (13218834587705587 / 20000000000000000 : ℝ) =
+            (13218834587705587 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13218834587705587 / 10000000000000000 : ℝ) ≤
+            (697643955754894954589 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6589797518849337 / 10000000000000000 : ℝ)) (q :=
+      (13218834587705587 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_634_8 :
+    Uω (aρ 8) (bρ 8) (5881668209681 / 8000000000000 : ℝ) ≤
+      (-(4611640336349390814731 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3152746187335603 / 5000000000000000 : ℝ) ≤
+      (-(4611640336349390814731 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (3152746187335603 / 5000000000000000 : ℝ) =
+            (3152746187335603 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3152746187335603 / 2500000000000000 : ℝ) ≤
+            (2319831466650609185269 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3132744665913081 / 5000000000000000 : ℝ)) (q :=
+      (3152746187335603 / 5000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_634_9 :
+    Uω (aρ 9) (bρ 9) (5881668209681 / 8000000000000 : ℝ) ≤
+      (-(5228750311639540664367 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5928137374552389 / 10000000000000000 : ℝ) ≤
+      (-(5228750311639540664367 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5928137374552389 / 10000000000000000 : ℝ) =
+            (5928137374552389 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5928137374552389 / 5000000000000000 : ℝ) ≤
+            (1702721491360459335633 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (365734059996783 / 625000000000000 : ℝ)) (q :=
+      (5928137374552389 / 10000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_634_10 :
+    Uω (aρ 10) (bρ 10) (5881668209681 / 8000000000000 : ℝ) ≤
+      (-(6017275327143369276991 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5478643643531723 / 10000000000000000 : ℝ) ≤
+      (-(6017275327143369276991 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5478643643531723 / 10000000000000000 : ℝ) =
+            (5478643643531723 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5478643643531723 / 5000000000000000 : ℝ) ≤
+            (914196475856630723009 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1335329132850549 / 2500000000000000 : ℝ)) (q :=
+      (5478643643531723 / 10000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_634_11 :
+    Uω (aρ 11) (bρ 11) (5881668209681 / 8000000000000 : ℝ) ≤
+      (-(437741244417544630727 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (620492078787101 / 1250000000000000 : ℝ) ≤
+      (-(437741244417544630727 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (620492078787101 / 1250000000000000 : ℝ) =
+            (620492078787101 / 312500000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (620492078787101 / 312500000000000 : ℝ) ≤
+            (428692730957455369273 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2365189565568683 / 5000000000000000 : ℝ)) (q :=
+      (620492078787101 / 1250000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11]) (by norm_num [bρ_11])
+      (by norm_num) (by norm_num [aρ_11, bρ_11]) (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_634_12 :
+    Uω (aρ 12) (bρ 12) (5881668209681 / 8000000000000 : ℝ) ≤
+      (-(4109414047231730727419 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8792063711339569 / 20000000000000000 : ℝ) ≤
+      (-(4109414047231730727419 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8792063711339569 / 20000000000000000 : ℝ) =
+            (8792063711339569 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8792063711339569 / 5000000000000000 : ℝ) ≤
+            (2822057755768269272581 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4018269678568319 / 10000000000000000 : ℝ)) (q :=
+      (8792063711339569 / 20000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_634_13 :
+    Uω (aρ 13) (bρ 13) (5881668209681 / 8000000000000 : ℝ) ≤
+      (-(1940509600574714462243 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1894932298719603 / 5000000000000000 : ℝ) ≤
+      (-(1940509600574714462243 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1894932298719603 / 5000000000000000 : ℝ) =
+            (1894932298719603 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1894932298719603 / 1250000000000000 : ℝ) ≤
+            (832079120625285537757 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1602658451851081 / 5000000000000000 : ℝ)) (q :=
+      (1894932298719603 / 5000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_634_14 :
+    Uω (aρ 14) (bρ 14) (5881668209681 / 8000000000000 : ℝ) ≤
+      (-(90068274870373059033 / 78125000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (157864043680091 / 500000000000000 : ℝ) ≤
+      (-(90068274870373059033 / 78125000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (157864043680091 / 500000000000000 : ℝ) =
+            (157864043680091 / 125000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (157864043680091 / 125000000000000 : ℝ) ≤
+            (18235972051501940967 / 78125000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (228407674107239 / 1000000000000000 : ℝ)) (q :=
+      (157864043680091 / 500000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14]) (by norm_num [bρ_14])
+      (by norm_num) (by norm_num [aρ_14, bρ_14]) (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_634_15 :
+    Uω (aρ 15) (bρ 15) (5881668209681 / 8000000000000 : ℝ) ≤
+      (-(13954872639935423773157 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2477123054752717 / 10000000000000000 : ℝ) ≤
+      (-(13954872639935423773157 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (2477123054752717 / 10000000000000000 : ℝ) =
+            (2477123054752717 / 1250000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2477123054752717 / 1250000000000000 : ℝ) ≤
+            (6839542769064576226843 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (18489599699831 / 156250000000000 : ℝ)) (q :=
+      (2477123054752717 / 10000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_634_16 :
+    Uω (aρ 16) (bρ 16) (5881668209681 / 8000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_634 :
+    Uρ (5881668209681 / 8000000000000 : ℝ) ≤
+      (-(2740871436653137694881 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (5881668209681 / 8000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3164155037131447097627 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_634_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (5881668209681 / 8000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3196994727943193935909 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_634_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (5881668209681 / 8000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(163148370217400263421 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_634_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (5881668209681 / 8000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3373409710194953931207 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_634_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (5881668209681 / 8000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(3543713825258676462001 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_634_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (5881668209681 / 8000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3792518349145028688873 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_634_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (5881668209681 / 8000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(1035223994995105045411 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_634_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (5881668209681 / 8000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(4611640336349390814731 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_634_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (5881668209681 / 8000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(5228750311639540664367 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_634_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (5881668209681 / 8000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(6017275327143369276991 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_634_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (5881668209681 / 8000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) * (-(437741244417544630727 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_634_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (5881668209681 / 8000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(4109414047231730727419 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_634_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (5881668209681 / 8000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(1940509600574714462243 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_634_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (5881668209681 / 8000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) * (-(90068274870373059033 / 78125000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_634_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (5881668209681 / 8000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(13954872639935423773157 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_634_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (5881668209681 / 8000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_634_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_635_1 :
+    Uω (aρ 1) (bρ 1) (23557149557281 / 32000000000000 : ℝ) ≤
+      (-(630218945808207475073 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (29188359453790979 / 40000000000000000 : ℝ) ≤
+      (-(630218945808207475073 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (29188359453790979 / 40000000000000000 : ℝ) =
+            (29188359453790979 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (29188359453790979 / 20000000000000000 : ℝ) ≤
+            (756075414791792524927 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7297067708330177 / 10000000000000000 : ℝ)) (q :=
+      (29188359453790979 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_635_2 :
+    Uω (aρ 2) (bρ 2) (23557149557281 / 32000000000000 : ℝ) ≤
+      (-(636778252935398947851 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (29092788552081409 / 40000000000000000 : ℝ) ≤
+      (-(636778252935398947851 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (29092788552081409 / 40000000000000000 : ℝ) =
+            (29092788552081409 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (29092788552081409 / 20000000000000000 : ℝ) ≤
+            (749516107664601052149 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (454565704374087 / 625000000000000 : ℝ)) (q :=
+      (29092788552081409 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_635_3 :
+    Uω (aρ 3) (bρ 3) (23557149557281 / 32000000000000 : ℝ) ≤
+      (-(649955318978965030579 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28901739810580043 / 40000000000000000 : ℝ) ≤
+      (-(649955318978965030579 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (28901739810580043 / 40000000000000000 : ℝ) =
+            (28901739810580043 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28901739810580043 / 20000000000000000 : ℝ) ≤
+            (736339041621034969421 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7224923005584709 / 10000000000000000 : ℝ)) (q :=
+      (28901739810580043 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_635_4 :
+    Uω (aρ 4) (bρ 4) (23557149557281 / 32000000000000 : ℝ) ≤
+      (-(672014140878737271299 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28584722117952679 / 40000000000000000 : ℝ) ≤
+      (-(672014140878737271299 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (28584722117952679 / 40000000000000000 : ℝ) =
+            (28584722117952679 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28584722117952679 / 20000000000000000 : ℝ) ≤
+            (714280219721262728701 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7144708340336027 / 10000000000000000 : ℝ)) (q :=
+      (28584722117952679 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_635_5 :
+    Uω (aρ 5) (bρ 5) (23557149557281 / 32000000000000 : ℝ) ≤
+      (-(882535332662672776061 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28102690547823587 / 40000000000000000 : ℝ) ≤
+      (-(882535332662672776061 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (28102690547823587 / 40000000000000000 : ℝ) =
+            (28102690547823587 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28102690547823587 / 20000000000000000 : ℝ) ≤
+            (850332618087327223939 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7021882474606481 / 10000000000000000 : ℝ)) (q :=
+      (28102690547823587 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_635_6 :
+    Uω (aρ 6) (bρ 6) (23557149557281 / 32000000000000 : ℝ) ≤
+      (-(1889296736377210212201 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27413075472117267 / 40000000000000000 : ℝ) ≤
+      (-(1889296736377210212201 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (27413075472117267 / 40000000000000000 : ℝ) =
+            (27413075472117267 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27413075472117267 / 20000000000000000 : ℝ) ≤
+            (1576439165122789787799 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6844315311893321 / 10000000000000000 : ℝ)) (q :=
+      (27413075472117267 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_635_7 :
+    Uω (aρ 7) (bρ 7) (23557149557281 / 32000000000000 : ℝ) ≤
+      (-(82529077401633511151 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5295175666436363 / 8000000000000000 : ℝ) ≤
+      (-(82529077401633511151 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5295175666436363 / 8000000000000000 : ℝ) =
+            (5295175666436363 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5295175666436363 / 4000000000000000 : ℝ) ≤
+            (56100358658366488849 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1319875624537119 / 2000000000000000 : ℝ)) (q :=
+      (5295175666436363 / 8000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_635_8 :
+    Uω (aρ 8) (bρ 8) (23557149557281 / 32000000000000 : ℝ) ≤
+      (-(1149112834313707893071 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25260308254868517 / 40000000000000000 : ℝ) ≤
+      (-(1149112834313707893071 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (25260308254868517 / 40000000000000000 : ℝ) =
+            (25260308254868517 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25260308254868517 / 20000000000000000 : ℝ) ≤
+            (583755116436292106929 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3137567367684473 / 5000000000000000 : ℝ)) (q :=
+      (25260308254868517 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_635_9 :
+    Uω (aρ 9) (bρ 9) (23557149557281 / 32000000000000 : ℝ) ≤
+      (-(2606244221391287807851 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23751141905958909 / 40000000000000000 : ℝ) ≤
+      (-(2606244221391287807851 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23751141905958909 / 40000000000000000 : ℝ) =
+            (23751141905958909 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23751141905958909 / 20000000000000000 : ℝ) ≤
+            (859491680108712192149 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2930758594637071 / 5000000000000000 : ℝ)) (q :=
+      (23751141905958909 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_635_10 :
+    Uω (aρ 10) (bρ 10) (23557149557281 / 32000000000000 : ℝ) ≤
+      (-(1499865314302420498799 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (21953648143161469 / 40000000000000000 : ℝ) ≤
+      (-(1499865314302420498799 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (21953648143161469 / 40000000000000000 : ℝ) =
+            (21953648143161469 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21953648143161469 / 20000000000000000 : ℝ) ≤
+            (233002636447579501201 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2675664670685211 / 5000000000000000 : ℝ)) (q :=
+      (21953648143161469 / 40000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_635_11 :
+    Uω (aρ 11) (bρ 11) (23557149557281 / 32000000000000 : ℝ) ≤
+      (-(6983748505754081904723 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3979143879818317 / 8000000000000000 : ℝ) ≤
+      (-(6983748505754081904723 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3979143879818317 / 8000000000000000 : ℝ) =
+            (3979143879818317 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3979143879818317 / 2000000000000000 : ℝ) ≤
+            (6879195100245918095277 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (925945624129 / 1953125000000 : ℝ)) (q :=
+      (3979143879818317 / 8000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_635_12 :
+    Uω (aρ 12) (bρ 12) (23557149557281 / 32000000000000 : ℝ) ≤
+      (-(1639031942355393550651 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17625795499731837 / 40000000000000000 : ℝ) ≤
+      (-(1639031942355393550651 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (17625795499731837 / 40000000000000000 : ℝ) =
+            (17625795499731837 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17625795499731837 / 10000000000000000 : ℝ) ≤
+            (1133556778844606449349 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2014789871272803 / 5000000000000000 : ℝ)) (q :=
+      (17625795499731837 / 40000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_635_13 :
+    Uω (aρ 13) (bρ 13) (23557149557281 / 32000000000000 : ℝ) ≤
+      (-(4836447497896644797437 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3040895491400307 / 8000000000000000 : ℝ) ≤
+      (-(4836447497896644797437 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3040895491400307 / 8000000000000000 : ℝ) =
+            (3040895491400307 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3040895491400307 / 2000000000000000 : ℝ) ≤
+            (2095024305103355202563 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (643660492555091 / 2000000000000000 : ℝ)) (q :=
+      (3040895491400307 / 8000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_635_14 :
+    Uω (aρ 14) (bρ 14) (23557149557281 / 32000000000000 : ℝ) ≤
+      (-(11487194324412495821493 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12681700148751069 / 40000000000000000 : ℝ) ≤
+      (-(11487194324412495821493 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (12681700148751069 / 40000000000000000 : ℝ) =
+            (12681700148751069 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12681700148751069 / 10000000000000000 : ℝ) ≤
+            (2375749281587504178507 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1150420546847611 / 5000000000000000 : ℝ)) (q :=
+      (12681700148751069 / 40000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_635_15 :
+    Uω (aρ 15) (bρ 15) (23557149557281 / 32000000000000 : ℝ) ≤
+      (-(346884892714684572593 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9987555644342411 / 40000000000000000 : ℝ) ≤
+      (-(346884892714684572593 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (9987555644342411 / 40000000000000000 : ℝ) =
+            (9987555644342411 / 5000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 15
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9987555644342411 / 5000000000000000 : ℝ) ≤
+            (172975492510315427407 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1213342118905893 / 10000000000000000 : ℝ)) (q :=
+      (9987555644342411 / 40000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_635_16 :
+    Uω (aρ 16) (bρ 16) (23557149557281 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_635 :
+    Uρ (23557149557281 / 32000000000000 : ℝ) ≤
+      (-(109261026582514005137 / 200000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (23557149557281 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(630218945808207475073 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_635_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (23557149557281 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(636778252935398947851 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_635_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (23557149557281 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(649955318978965030579 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_635_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (23557149557281 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(672014140878737271299 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_635_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (23557149557281 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(882535332662672776061 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_635_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (23557149557281 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1889296736377210212201 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_635_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (23557149557281 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) * (-(82529077401633511151 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_635_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (23557149557281 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(1149112834313707893071 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_635_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (23557149557281 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(2606244221391287807851 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_635_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (23557149557281 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(1499865314302420498799 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_635_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (23557149557281 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(6983748505754081904723 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_635_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (23557149557281 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(1639031942355393550651 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_635_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (23557149557281 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4836447497896644797437 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_635_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (23557149557281 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(11487194324412495821493 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_635_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (23557149557281 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(346884892714684572593 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_635_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (23557149557281 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_635_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_636_1 :
+    Uω (aρ 1) (bρ 1) (11793813137919 / 16000000000000 : ℝ) ≤
+      (-(627610291194593107191 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14613227733750989 / 20000000000000000 : ℝ) ≤
+      (-(627610291194593107191 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (14613227733750989 / 20000000000000000 : ℝ) =
+            (14613227733750989 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14613227733750989 / 10000000000000000 : ℝ) ≤
+            (758684069405406892809 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3653295870318307 / 5000000000000000 : ℝ)) (q :=
+      (14613227733750989 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_636_2 :
+    Uω (aρ 2) (bρ 2) (11793813137919 / 16000000000000 : ℝ) ≤
+      (-(634160989941164399659 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14565442606664147 / 20000000000000000 : ℝ) ≤
+      (-(634160989941164399659 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (14565442606664147 / 20000000000000000 : ℝ) =
+            (14565442606664147 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14565442606664147 / 10000000000000000 : ℝ) ≤
+            (752133370658835600341 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1820643906514943 / 2500000000000000 : ℝ)) (q :=
+      (14565442606664147 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_636_3 :
+    Uω (aρ 3) (bρ 3) (11793813137919 / 16000000000000 : ℝ) ≤
+      (-(1618301582363736978203 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14469919202319877 / 20000000000000000 : ℝ) ≤
+      (-(1618301582363736978203 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (14469919202319877 / 20000000000000000 : ℝ) =
+            (14469919202319877 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14469919202319877 / 10000000000000000 : ℝ) ≤
+            (1847434319136263021797 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3617224164032751 / 5000000000000000 : ℝ)) (q :=
+      (14469919202319877 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_636_4 :
+    Uω (aρ 4) (bρ 4) (11793813137919 / 16000000000000 : ℝ) ≤
+      (-(1673374737556661780137 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14311412927736997 / 20000000000000000 : ℝ) ≤
+      (-(1673374737556661780137 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (14311412927736997 / 20000000000000000 : ℝ) =
+            (14311412927736997 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14311412927736997 / 10000000000000000 : ℝ) ≤
+            (1792361163943338219863 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3577118117273811 / 5000000000000000 : ℝ)) (q :=
+      (14311412927736997 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_636_5 :
+    Uω (aρ 5) (bρ 5) (11793813137919 / 16000000000000 : ℝ) ≤
+      (-(3516587252170573535409 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3517600872620407 / 5000000000000000 : ℝ) ≤
+      (-(3516587252170573535409 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (3517600872620407 / 5000000000000000 : ℝ) =
+            (3517600872620407 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3517600872620407 / 2500000000000000 : ℝ) ≤
+            (3414884550829426464591 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7031416716627253 / 10000000000000000 : ℝ)) (q :=
+      (3517600872620407 / 5000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_636_6 :
+    Uω (aρ 6) (bρ 6) (11793813137919 / 16000000000000 : ℝ) ≤
+      (-(470586001276572675323 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13725610568519909 / 20000000000000000 : ℝ) ≤
+      (-(470586001276572675323 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (13725610568519909 / 20000000000000000 : ℝ) =
+            (13725610568519909 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13725610568519909 / 10000000000000000 : ℝ) ≤
+            (395847974098427324677 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3426932084902767 / 5000000000000000 : ℝ)) (q :=
+      (13725610568519909 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_636_7 :
+    Uω (aρ 7) (bρ 7) (11793813137919 / 16000000000000 : ℝ) ≤
+      (-(822406542208375767147 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13257043580777419 / 20000000000000000 : ℝ) ≤
+      (-(822406542208375767147 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (13257043580777419 / 20000000000000000 : ℝ) =
+            (13257043580777419 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13257043580777419 / 10000000000000000 : ℝ) ≤
+            (563887818391624232853 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1652239640705761 / 2500000000000000 : ℝ)) (q :=
+      (13257043580777419 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_636_8 :
+    Uω (aρ 8) (bρ 8) (11793813137919 / 16000000000000 : ℝ) ≤
+      (-(4581285666546628779379 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1581165391894029 / 2500000000000000 : ℝ) ≤
+      (-(4581285666546628779379 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1581165391894029 / 2500000000000000 : ℝ) =
+            (1581165391894029 / 1250000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1581165391894029 / 1250000000000000 : ℝ) ≤
+            (2350186136453371220621 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6284779768537857 / 10000000000000000 : ℝ)) (q :=
+      (1581165391894029 / 2500000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_636_9 :
+    Uω (aρ 9) (bρ 9) (11793813137919 / 16000000000000 : ℝ) ≤
+      (-(5196253661759570077241 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (594743317047791 / 1000000000000000 : ℝ) ≤
+      (-(5196253661759570077241 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (594743317047791 / 1000000000000000 : ℝ) =
+            (594743317047791 / 500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (594743317047791 / 500000000000000 : ℝ) ≤
+            (1735218141240429922759 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1174257720540289 / 2000000000000000 : ℝ)) (q :=
+      (594743317047791 / 1000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_636_10 :
+    Uω (aρ 10) (bρ 10) (11793813137919 / 16000000000000 : ℝ) ≤
+      (-(5981680484881463733891 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10996359074776561 / 20000000000000000 : ℝ) ≤
+      (-(5981680484881463733891 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10996359074776561 / 20000000000000000 : ℝ) =
+            (10996359074776561 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10996359074776561 / 10000000000000000 : ℝ) ≤
+            (949791318118536266109 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2680670185008593 / 5000000000000000 : ℝ)) (q :=
+      (10996359074776561 / 20000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_636_11 :
+    Uω (aρ 11) (bρ 11) (11793813137919 / 16000000000000 : ℝ) ≤
+      (-(278547257081232514053 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9967842190723267 / 20000000000000000 : ℝ) ≤
+      (-(278547257081232514053 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9967842190723267 / 20000000000000000 : ℝ) =
+            (9967842190723267 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9967842190723267 / 5000000000000000 : ℝ) ≤
+            (275970487158767485947 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1187825028042223 / 2500000000000000 : ℝ)) (q :=
+      (9967842190723267 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_636_12 :
+    Uω (aρ 12) (bρ 12) (11793813137919 / 16000000000000 : ℝ) ≤
+      (-(817155764059736202051 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1766744515930713 / 4000000000000000 : ℝ) ≤
+      (-(817155764059736202051 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1766744515930713 / 4000000000000000 : ℝ) =
+            (1766744515930713 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1766744515930713 / 1000000000000000 : ℝ) ≤
+            (569138596540263797949 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (404088059778419 / 1000000000000000 : ℝ)) (q :=
+      (1766744515930713 / 4000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_636_13 :
+    Uω (aρ 13) (bρ 13) (11793813137919 / 16000000000000 : ℝ) ≤
+      (-(9643361284767544950333 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7624724148226917 / 20000000000000000 : ℝ) ≤
+      (-(9643361284767544950333 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7624724148226917 / 20000000000000000 : ℝ) =
+            (7624724148226917 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7624724148226917 / 5000000000000000 : ℝ) ≤
+            (4219582321232455049667 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1615631953976271 / 5000000000000000 : ℝ)) (q :=
+      (7624724148226917 / 20000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_636_14 :
+    Uω (aρ 14) (bρ 14) (11793813137919 / 16000000000000 : ℝ) ≤
+      (-(457838013503125911517 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1273411254666331 / 4000000000000000 : ℝ) ≤
+      (-(457838013503125911517 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1273411254666331 / 4000000000000000 : ℝ) =
+            (1273411254666331 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1273411254666331 / 1000000000000000 : ℝ) ≤
+            (96679730736874088483 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (57938082952557 / 250000000000000 : ℝ)) (q :=
+      (1273411254666331 / 4000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_636_15 :
+    Uω (aρ 15) (bρ 15) (11793813137919 / 16000000000000 : ℝ) ≤
+      (-(6898919903209831485569 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2516329046367611 / 10000000000000000 : ℝ) ≤
+      (-(6898919903209831485569 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2516329046367611 / 10000000000000000 : ℝ) =
+            (2516329046367611 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2516329046367611 / 2500000000000000 : ℝ) ≤
+            (32551899790168514431 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1242698414920847 / 10000000000000000 : ℝ)) (q :=
+      (2516329046367611 / 10000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_636_16 :
+    Uω (aρ 16) (bρ 16) (11793813137919 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_636 :
+    Uρ (11793813137919 / 16000000000000 : ℝ) ≤
+      (-(5444461197856341892419 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (11793813137919 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(627610291194593107191 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_636_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (11793813137919 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(634160989941164399659 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_636_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (11793813137919 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1618301582363736978203 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_636_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (11793813137919 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1673374737556661780137 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_636_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (11793813137919 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(3516587252170573535409 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_636_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (11793813137919 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(470586001276572675323 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_636_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (11793813137919 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(822406542208375767147 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_636_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (11793813137919 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(4581285666546628779379 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_636_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (11793813137919 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(5196253661759570077241 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_636_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (11793813137919 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(5981680484881463733891 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_636_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (11793813137919 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) * (-(278547257081232514053 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_636_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (11793813137919 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(817155764059736202051 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_636_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (11793813137919 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(9643361284767544950333 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_636_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (11793813137919 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) * (-(457838013503125911517 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_636_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (11793813137919 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(6898919903209831485569 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_636_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (11793813137919 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_636_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_637_1 :
+    Uω (aρ 1) (bρ 1) (4723620598879 / 6400000000000 : ℝ) ≤
+      (-(195314073346629939723 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1170582059236489 / 1600000000000000 : ℝ) ≤
+      (-(195314073346629939723 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1170582059236489 / 1600000000000000 : ℝ) =
+            (1170582059236489 / 800000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1170582059236489 / 800000000000000 : ℝ) ≤
+            (237902914340870060277 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (292644630911707 / 400000000000000 : ℝ)) (q :=
+      (1170582059236489 / 1600000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_637_2 :
+    Uω (aρ 2) (bρ 2) (4723620598879 / 6400000000000 : ℝ) ≤
+      (-(25261885905625078749 / 80000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (29168981872581949 / 40000000000000000 : ℝ) ≤
+      (-(25261885905625078749 / 80000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (29168981872581949 / 40000000000000000 : ℝ) =
+            (29168981872581949 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (29168981872581949 / 20000000000000000 : ℝ) ≤
+            (30189888518374921251 / 80000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7292099981137537 / 10000000000000000 : ℝ)) (q :=
+      (29168981872581949 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_637_3 :
+    Uω (aρ 3) (bρ 3) (4723620598879 / 6400000000000 : ℝ) ≤
+      (-(3223447068104290737667 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28977936991610221 / 40000000000000000 : ℝ) ≤
+      (-(3223447068104290737667 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (28977936991610221 / 40000000000000000 : ℝ) =
+            (28977936991610221 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28977936991610221 / 20000000000000000 : ℝ) ≤
+            (3708024734895709262333 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7243973647001673 / 10000000000000000 : ℝ)) (q :=
+      (28977936991610221 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_637_4 :
+    Uω (aρ 4) (bρ 4) (4723620598879 / 6400000000000 : ℝ) ≤
+      (-(166672298750848071597 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28660929572146799 / 40000000000000000 : ℝ) ≤
+      (-(166672298750848071597 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (28660929572146799 / 40000000000000000 : ℝ) =
+            (28660929572146799 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28660929572146799 / 20000000000000000 : ℝ) ≤
+            (179901291399151928403 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3581882059167481 / 5000000000000000 : ℝ)) (q :=
+      (28660929572146799 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_637_5 :
+    Uω (aρ 5) (bρ 5) (4723620598879 / 6400000000000 : ℝ) ≤
+      (-(3503051539855834598479 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28178923358519609 / 40000000000000000 : ℝ) ≤
+      (-(3503051539855834598479 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (28178923358519609 / 40000000000000000 : ℝ) =
+            (28178923358519609 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28178923358519609 / 20000000000000000 : ℝ) ≤
+            (3428420263144165401521 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7040950930856367 / 10000000000000000 : ℝ)) (q :=
+      (28178923358519609 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_637_6 :
+    Uω (aρ 6) (bρ 6) (4723620598879 / 6400000000000 : ℝ) ≤
+      (-(750160381464319673849 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27489366663664929 / 40000000000000000 : ℝ) ≤
+      (-(750160381464319673849 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (27489366663664929 / 40000000000000000 : ℝ) =
+            (27489366663664929 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27489366663664929 / 20000000000000000 : ℝ) ≤
+            (636133979135680326151 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6863412958569027 / 10000000000000000 : ℝ)) (q :=
+      (27489366663664929 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_637_7 :
+    Uω (aρ 7) (bρ 7) (4723620598879 / 6400000000000 : ℝ) ≤
+      (-(1024408110451874314823 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5310459132990397 / 8000000000000000 : ℝ) ≤
+      (-(1024408110451874314823 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5310459132990397 / 8000000000000000 : ℝ) =
+            (5310459132990397 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5310459132990397 / 4000000000000000 : ℝ) ≤
+            (708459840298125685177 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1323707767994511 / 2000000000000000 : ℝ)) (q :=
+      (5310459132990397 / 8000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_637_8 :
+    Uω (aρ 8) (bρ 8) (4723620598879 / 6400000000000 : ℝ) ≤
+      (-(2283071625893052422273 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25336983548397839 / 40000000000000000 : ℝ) ≤
+      (-(2283071625893052422273 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (25336983548397839 / 40000000000000000 : ℝ) =
+            (25336983548397839 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25336983548397839 / 20000000000000000 : ℝ) ≤
+            (1182664275606947577727 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3147212216517741 / 5000000000000000 : ℝ)) (q :=
+      (25336983548397839 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_637_9 :
+    Uω (aρ 9) (bρ 9) (4723620598879 / 6400000000000 : ℝ) ≤
+      (-(647505734529234841183 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23828321834201471 / 40000000000000000 : ℝ) ≤
+      (-(647505734529234841183 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23828321834201471 / 40000000000000000 : ℝ) =
+            (23828321834201471 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23828321834201471 / 20000000000000000 : ℝ) ≤
+            (218928240845765158817 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2940529602148649 / 5000000000000000 : ℝ)) (q :=
+      (23828321834201471 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_637_10 :
+    Uω (aρ 10) (bρ 10) (4723620598879 / 6400000000000 : ℝ) ≤
+      (-(2981966440019059832977 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22031784613221831 / 40000000000000000 : ℝ) ≤
+      (-(2981966440019059832977 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (22031784613221831 / 40000000000000000 : ℝ) =
+            (22031784613221831 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22031784613221831 / 20000000000000000 : ℝ) ≤
+            (483769461480940167023 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2685674813651239 / 5000000000000000 : ℝ)) (q :=
+      (22031784613221831 / 40000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_637_11 :
+    Uω (aρ 11) (bρ 11) (4723620598879 / 6400000000000 : ℝ) ≤
+      (-(1388731692596799573783 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19975641520276431 / 40000000000000000 : ℝ) ≤
+      (-(1388731692596799573783 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (19975641520276431 / 40000000000000000 : ℝ) =
+            (19975641520276431 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 15
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19975641520276431 / 10000000000000000 : ℝ) ≤
+            (1383857028603200426217 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2380877353517389 / 5000000000000000 : ℝ)) (q :=
+      (19975641520276431 / 40000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_637_12 :
+    Uω (aρ 12) (bρ 12) (4723620598879 / 6400000000000 : ℝ) ≤
+      (-(4074010728528158471861 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17709076555495139 / 40000000000000000 : ℝ) ≤
+      (-(4074010728528158471861 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (17709076555495139 / 40000000000000000 : ℝ) =
+            (17709076555495139 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17709076555495139 / 10000000000000000 : ℝ) ≤
+            (2857461074471841528139 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1013043080332283 / 2500000000000000 : ℝ)) (q :=
+      (17709076555495139 / 40000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_637_13 :
+    Uω (aρ 13) (bρ 13) (4723620598879 / 6400000000000 : ℝ) ≤
+      (-(2403486424000805781369 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15294371486162191 / 40000000000000000 : ℝ) ≤
+      (-(2403486424000805781369 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (15294371486162191 / 40000000000000000 : ℝ) =
+            (15294371486162191 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15294371486162191 / 10000000000000000 : ℝ) ≤
+            (1062249477499194218631 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1622100764128829 / 5000000000000000 : ℝ)) (q :=
+      (15294371486162191 / 40000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_637_14 :
+    Uω (aρ 14) (bρ 14) (4723620598879 / 6400000000000 : ℝ) ≤
+      (-(11405001465921323028451 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12786364210016413 / 40000000000000000 : ℝ) ≤
+      (-(11405001465921323028451 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (12786364210016413 / 40000000000000000 : ℝ) =
+            (12786364210016413 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12786364210016413 / 10000000000000000 : ℝ) ≤
+            (2457942140078676971549 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2334125175229769 / 10000000000000000 : ℝ)) (q :=
+      (12786364210016413 / 40000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_637_15 :
+    Uω (aρ 15) (bρ 15) (4723620598879 / 6400000000000 : ℝ) ≤
+      (-(857629773117390833237 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10141864090335721 / 40000000000000000 : ℝ) ≤
+      (-(857629773117390833237 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (10141864090335721 / 40000000000000000 : ℝ) =
+            (10141864090335721 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10141864090335721 / 10000000000000000 : ℝ) ≤
+            (8804202257609166763 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1271448392804423 / 10000000000000000 : ℝ)) (q :=
+      (10141864090335721 / 40000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_637_16 :
+    Uω (aρ 16) (bρ 16) (4723620598879 / 6400000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_637 :
+    Uρ (4723620598879 / 6400000000000 : ℝ) ≤
+      (-(339123009167877599779 / 625000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (4723620598879 / 6400000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(195314073346629939723 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_637_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (4723620598879 / 6400000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) * (-(25261885905625078749 / 80000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_637_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (4723620598879 / 6400000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3223447068104290737667 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_637_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (4723620598879 / 6400000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(166672298750848071597 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_637_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (4723620598879 / 6400000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(3503051539855834598479 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_637_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (4723620598879 / 6400000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(750160381464319673849 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_637_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (4723620598879 / 6400000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(1024408110451874314823 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_637_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (4723620598879 / 6400000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(2283071625893052422273 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_637_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (4723620598879 / 6400000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(647505734529234841183 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_637_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (4723620598879 / 6400000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(2981966440019059832977 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_637_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (4723620598879 / 6400000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(1388731692596799573783 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_637_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (4723620598879 / 6400000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(4074010728528158471861 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_637_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (4723620598879 / 6400000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(2403486424000805781369 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_637_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (4723620598879 / 6400000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(11405001465921323028451 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_637_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (4723620598879 / 6400000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(857629773117390833237 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_637_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (4723620598879 / 6400000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_637_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_638_1 :
+    Uω (aρ 1) (bρ 1) (2956072464119 / 4000000000000 : ℝ) ≤
+      (-(622403167510487586771 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1831415468376431 / 2500000000000000 : ℝ) ≤
+      (-(622403167510487586771 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1831415468376431 / 2500000000000000 : ℝ) =
+            (1831415468376431 / 1250000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1831415468376431 / 1250000000000000 : ℝ) ≤
+            (763891193089512413229 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1831409951199737 / 2500000000000000 : ℝ)) (q :=
+      (1831415468376431 / 2500000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_638_2 :
+    Uω (aρ 2) (bρ 2) (2956072464119 / 4000000000000 : ℝ) ≤
+      (-(1572341792758970059237 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2920707852985017 / 4000000000000000 : ℝ) ≤
+      (-(1572341792758970059237 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2920707852985017 / 4000000000000000 : ℝ) =
+            (2920707852985017 / 2000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2920707852985017 / 2000000000000000 : ℝ) ≤
+            (1893394108741029940763 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1460324867044517 / 2000000000000000 : ℝ)) (q :=
+      (2920707852985017 / 4000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_638_3 :
+    Uω (aρ 3) (bρ 3) (2956072464119 / 4000000000000 : ℝ) ≤
+      (-(802577064865997302589 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2901603557151937 / 4000000000000000 : ℝ) ≤
+      (-(802577064865997302589 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2901603557151937 / 4000000000000000 : ℝ) =
+            (2901603557151937 / 2000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2901603557151937 / 2000000000000000 : ℝ) ≤
+            (930290885884002697411 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1450699792481437 / 2000000000000000 : ℝ)) (q :=
+      (2901603557151937 / 4000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_638_4 :
+    Uω (aρ 4) (bρ 4) (2956072464119 / 4000000000000 : ℝ) ≤
+      (-(20751000980978739367 / 62500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14349516634027087 / 20000000000000000 : ℝ) ≤
+      (-(20751000980978739367 / 62500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (14349516634027087 / 20000000000000000 : ℝ) =
+            (14349516634027087 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14349516634027087 / 10000000000000000 : ℝ) ≤
+            (22570697787771260633 / 62500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7173291991739587 / 10000000000000000 : ℝ)) (q :=
+      (14349516634027087 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_638_5 :
+    Uω (aρ 5) (bρ 5) (2956072464119 / 4000000000000 : ℝ) ≤
+      (-(872383535986811447349 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1410851984035907 / 2000000000000000 : ℝ) ≤
+      (-(872383535986811447349 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1410851984035907 / 2000000000000000 : ℝ) =
+            (1410851984035907 / 1000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1410851984035907 / 1000000000000000 : ℝ) ≤
+            (860484414763188552651 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (705048511740657 / 1000000000000000 : ℝ)) (q :=
+      (1410851984035907 / 2000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_638_6 :
+    Uω (aρ 6) (bρ 6) (2956072464119 / 4000000000000 : ℝ) ≤
+      (-(3736935110110778650981 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (430117375821391 / 625000000000000 : ℝ) ≤
+      (-(3736935110110778650981 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (430117375821391 / 625000000000000 : ℝ) =
+            (430117375821391 / 312500000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (430117375821391 / 312500000000000 : ℝ) ≤
+            (3194536692889221349019 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1718240419618003 / 2500000000000000 : ℝ)) (q :=
+      (430117375821391 / 625000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_638_7 :
+    Uω (aρ 7) (bρ 7) (2956072464119 / 4000000000000 : ℝ) ≤
+      (-(81665060031860360223 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (106362015375147 / 160000000000000 : ℝ) ≤
+      (-(81665060031860360223 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (106362015375147 / 160000000000000 : ℝ) = (106362015375147 / 80000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (106362015375147 / 80000000000000 : ℝ) ≤
+            (56964376028139639777 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (53024951638727 / 80000000000000 : ℝ)) (q :=
+      (106362015375147 / 160000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_638_8 :
+    Uω (aρ 8) (bρ 8) (2956072464119 / 4000000000000 : ℝ) ≤
+      (-(1137756005218871002723 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1585957505783303 / 2500000000000000 : ℝ) ≤
+      (-(1137756005218871002723 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1585957505783303 / 2500000000000000 : ℝ) =
+            (1585957505783303 / 1250000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1585957505783303 / 1250000000000000 : ℝ) ≤
+            (595111945531128997277 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1576017182638481 / 2500000000000000 : ℝ)) (q :=
+      (1585957505783303 / 2500000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_638_9 :
+    Uω (aρ 9) (bρ 9) (2956072464119 / 4000000000000 : ℝ) ≤
+      (-(1290966248587763081941 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11933454685453741 / 20000000000000000 : ℝ) ≤
+      (-(1290966248587763081941 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11933454685453741 / 20000000000000000 : ℝ) =
+            (11933454685453741 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11933454685453741 / 10000000000000000 : ℝ) ≤
+            (441901702162236918059 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5890828998101241 / 10000000000000000 : ℝ)) (q :=
+      (11933454685453741 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_638_10 :
+    Uω (aρ 10) (bρ 10) (2956072464119 / 4000000000000 : ℝ) ≤
+      (-(2973109156675716891163 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1379427972120991 / 2500000000000000 : ℝ) ≤
+      (-(2973109156675716891163 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1379427972120991 / 2500000000000000 : ℝ) =
+            (1379427972120991 / 1250000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1379427972120991 / 1250000000000000 : ℝ) ≤
+            (492626744824283108837 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1345339280777607 / 2500000000000000 : ℝ)) (q :=
+      (1379427972120991 / 2500000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_638_11 :
+    Uω (aρ 11) (bρ 11) (2956072464119 / 4000000000000 : ℝ) ≤
+      (-(6923679406295921670647 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10007795433565161 / 20000000000000000 : ℝ) ≤
+      (-(6923679406295921670647 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10007795433565161 / 20000000000000000 : ℝ) =
+            (10007795433565161 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 2
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10007795433565161 / 10000000000000000 : ℝ) ≤
+            (7792396704078329353 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4772205405912661 / 10000000000000000 : ℝ)) (q :=
+      (10007795433565161 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_638_12 :
+    Uω (aρ 12) (bρ 12) (2956072464119 / 4000000000000 : ℝ) ≤
+      (-(4062275370857562616801 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4437672460137349 / 10000000000000000 : ℝ) ≤
+      (-(4062275370857562616801 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4437672460137349 / 10000000000000000 : ℝ) =
+            (4437672460137349 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4437672460137349 / 2500000000000000 : ℝ) ≤
+            (2869196432142437383199 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2031727494653599 / 5000000000000000 : ℝ)) (q :=
+      (4437672460137349 / 10000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_638_13 :
+    Uω (aρ 13) (bρ 13) (2956072464119 / 4000000000000 : ℝ) ≤
+      (-(2396161768732546289149 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (958702974621041 / 2500000000000000 : ℝ) ≤
+      (-(2396161768732546289149 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (958702974621041 / 2500000000000000 : ℝ) =
+            (958702974621041 / 625000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (958702974621041 / 625000000000000 : ℝ) ≤
+            (1069574132767453710851 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (814278901898957 / 2500000000000000 : ℝ)) (q :=
+      (958702974621041 / 2500000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13]) (by norm_num [bρ_13])
+      (by norm_num) (by norm_num [aρ_13, bρ_13]) (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_638_14 :
+    Uω (aρ 14) (bρ 14) (2956072464119 / 4000000000000 : ℝ) ≤
+      (-(11364342135936268109407 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (80240365902861 / 250000000000000 : ℝ) ≤
+      (-(11364342135936268109407 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (80240365902861 / 250000000000000 : ℝ) = (80240365902861 / 62500000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (80240365902861 / 62500000000000 : ℝ) ≤
+            (2498601470063731890593 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (117532418395069 / 500000000000000 : ℝ)) (q :=
+      (80240365902861 / 250000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14]) (by norm_num [bρ_14])
+      (by norm_num) (by norm_num [aρ_14, bρ_14]) (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_638_15 :
+    Uω (aρ 15) (bρ 15) (2956072464119 / 4000000000000 : ℝ) ≤
+      (-(13647990650442621606171 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5108639918318627 / 20000000000000000 : ℝ) ≤
+      (-(13647990650442621606171 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5108639918318627 / 20000000000000000 : ℝ) =
+            (5108639918318627 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5108639918318627 / 5000000000000000 : ℝ) ≤
+            (214952955557378393829 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1299632291406127 / 10000000000000000 : ℝ)) (q :=
+      (5108639918318627 / 20000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_638_16 :
+    Uω (aρ 16) (bρ 16) (2956072464119 / 4000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_638 :
+    Uρ (2956072464119 / 4000000000000 : ℝ) ≤
+      (-(675946034224267949889 / 1250000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (2956072464119 / 4000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(622403167510487586771 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_638_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (2956072464119 / 4000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1572341792758970059237 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_638_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (2956072464119 / 4000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(802577064865997302589 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_638_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (2956072464119 / 4000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(20751000980978739367 / 62500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_638_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (2956072464119 / 4000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(872383535986811447349 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_638_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (2956072464119 / 4000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3736935110110778650981 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_638_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (2956072464119 / 4000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) * (-(81665060031860360223 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_638_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (2956072464119 / 4000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(1137756005218871002723 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_638_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (2956072464119 / 4000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(1290966248587763081941 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_638_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (2956072464119 / 4000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(2973109156675716891163 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_638_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (2956072464119 / 4000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(6923679406295921670647 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_638_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (2956072464119 / 4000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(4062275370857562616801 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_638_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (2956072464119 / 4000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(2396161768732546289149 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_638_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (2956072464119 / 4000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(11364342135936268109407 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_638_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (2956072464119 / 4000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(13647990650442621606171 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_638_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (2956072464119 / 4000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_638_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_639_1 :
+    Uω (aρ 1) (bρ 1) (23679056431509 / 32000000000000 : ℝ) ≤
+      (-(3099023403956418114027 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5868148701367031 / 8000000000000000 : ℝ) ≤
+      (-(3099023403956418114027 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5868148701367031 / 8000000000000000 : ℝ) =
+            (5868148701367031 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5868148701367031 / 4000000000000000 : ℝ) ≤
+            (3832448399043581885973 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1467032767331203 / 2000000000000000 : ℝ)) (q :=
+      (5868148701367031 / 8000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_639_2 :
+    Uω (aρ 2) (bρ 2) (23679056431509 / 32000000000000 : ℝ) ≤
+      (-(3131648447173872218449 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (29245175185140721 / 40000000000000000 : ℝ) ≤
+      (-(3131648447173872218449 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (29245175185140721 / 40000000000000000 : ℝ) =
+            (29245175185140721 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (29245175185140721 / 20000000000000000 : ℝ) ≤
+            (3799823355826127781551 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3655574344159399 / 5000000000000000 : ℝ)) (q :=
+      (29245175185140721 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_639_3 :
+    Uω (aρ 3) (bρ 3) (23679056431509 / 32000000000000 : ℝ) ≤
+      (-(3197186693424707257857 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (29054134144394987 / 40000000000000000 : ℝ) ≤
+      (-(3197186693424707257857 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (29054134144394987 / 40000000000000000 : ℝ) =
+            (29054134144394987 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (29054134144394987 / 20000000000000000 : ℝ) ≤
+            (3734285109575292742143 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7263024274295931 / 10000000000000000 : ℝ)) (q :=
+      (29054134144394987 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_639_4 :
+    Uω (aρ 4) (bρ 4) (23679056431509 / 32000000000000 : ℝ) ≤
+      (-(3306891973972135931103 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28737136943278751 / 40000000000000000 : ℝ) ≤
+      (-(3306891973972135931103 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (28737136943278751 / 40000000000000000 : ℝ) =
+            (28737136943278751 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28737136943278751 / 20000000000000000 : ℝ) ≤
+            (3624579829027864068897 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7182819854802813 / 10000000000000000 : ℝ)) (q :=
+      (28737136943278751 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_639_5 :
+    Uω (aρ 5) (bρ 5) (23679056431509 / 32000000000000 : ℝ) ≤
+      (-(54313047107620936833 / 156250000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28255155947783119 / 40000000000000000 : ℝ) ≤
+      (-(54313047107620936833 / 156250000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (28255155947783119 / 40000000000000000 : ℝ) =
+            (28255155947783119 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28255155947783119 / 20000000000000000 : ℝ) ≤
+            (53991199814254063167 / 156250000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7060019276389997 / 10000000000000000 : ℝ)) (q :=
+      (28255155947783119 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_639_6 :
+    Uω (aρ 6) (bρ 6) (23679056431509 / 32000000000000 : ℝ) ≤
+      (-(3723087564835383075901 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27565657304325319 / 40000000000000000 : ℝ) ≤
+      (-(3723087564835383075901 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (27565657304325319 / 40000000000000000 : ℝ) =
+            (27565657304325319 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27565657304325319 / 20000000000000000 : ℝ) ≤
+            (3208384238164616924099 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6882510329801097 / 10000000000000000 : ℝ)) (q :=
+      (27565657304325319 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_639_7 :
+    Uω (aρ 7) (bρ 7) (23679056431509 / 32000000000000 : ℝ) ≤
+      (-(2034447164939685717283 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (26628711699464443 / 40000000000000000 : ℝ) ≤
+      (-(2034447164939685717283 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (26628711699464443 / 40000000000000000 : ℝ) =
+            (26628711699464443 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26628711699464443 / 20000000000000000 : ℝ) ≤
+            (1431288736560314282717 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6637698908130659 / 10000000000000000 : ℝ)) (q :=
+      (26628711699464443 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_639_8 :
+    Uω (aρ 8) (bρ 8) (23679056431509 / 32000000000000 : ℝ) ≤
+      (-(2267963951027866300151 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25413655906072891 / 40000000000000000 : ℝ) ≤
+      (-(2267963951027866300151 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (25413655906072891 / 40000000000000000 : ℝ) =
+            (25413655906072891 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25413655906072891 / 20000000000000000 : ℝ) ≤
+            (1197771950472133699849 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6313712662774883 / 10000000000000000 : ℝ)) (q :=
+      (25413655906072891 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_639_9 :
+    Uω (aρ 9) (bρ 9) (23679056431509 / 32000000000000 : ℝ) ≤
+      (-(5147710924735413315221 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23905495300054583 / 40000000000000000 : ℝ) ≤
+      (-(5147710924735413315221 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23905495300054583 / 40000000000000000 : ℝ) =
+            (23905495300054583 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23905495300054583 / 20000000000000000 : ℝ) ≤
+            (1783760878264586684779 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5900597988125729 / 10000000000000000 : ℝ)) (q :=
+      (23905495300054583 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_639_10 :
+    Uω (aρ 10) (bρ 10) (23679056431509 / 32000000000000 : ℝ) ≤
+      (-(2964268328139464888913 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22109906991313217 / 40000000000000000 : ℝ) ≤
+      (-(2964268328139464888913 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (22109906991313217 / 40000000000000000 : ℝ) =
+            (22109906991313217 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22109906991313217 / 20000000000000000 : ℝ) ≤
+            (501467573360535111087 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2695681433625023 / 5000000000000000 : ℝ)) (q :=
+      (22109906991313217 / 40000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_639_11 :
+    Uω (aρ 11) (bρ 11) (23679056431509 / 32000000000000 : ℝ) ≤
+      (-(6903744043443189821783 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20055532473087563 / 40000000000000000 : ℝ) ≤
+      (-(6903744043443189821783 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (20055532473087563 / 40000000000000000 : ℝ) =
+            (20055532473087563 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20055532473087563 / 20000000000000000 : ℝ) ≤
+            (27727759556810178217 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4782652234342219 / 10000000000000000 : ℝ)) (q :=
+      (20055532473087563 / 40000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_639_12 :
+    Uω (aρ 12) (bρ 12) (23679056431509 / 32000000000000 : ℝ) ≤
+      (-(4050572539749861997181 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17792285164915683 / 40000000000000000 : ℝ) ≤
+      (-(4050572539749861997181 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (17792285164915683 / 40000000000000000 : ℝ) =
+            (17792285164915683 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17792285164915683 / 10000000000000000 : ℝ) ≤
+            (2880899263250138002819 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4074728676941279 / 10000000000000000 : ℝ)) (q :=
+      (17792285164915683 / 40000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_639_13 :
+    Uω (aρ 13) (bρ 13) (23679056431509 / 32000000000000 : ℝ) ≤
+      (-(1194433035719959725449 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15384077177592513 / 40000000000000000 : ℝ) ≤
+      (-(1194433035719959725449 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (15384077177592513 / 40000000000000000 : ℝ) =
+            (15384077177592513 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15384077177592513 / 10000000000000000 : ℝ) ≤
+            (538434915030040274551 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1635003212437347 / 5000000000000000 : ℝ)) (q :=
+      (15384077177592513 / 40000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_639_14 :
+    Uω (aρ 14) (bρ 14) (23679056431509 / 32000000000000 : ℝ) ≤
+      (-(11323966949471001526601 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12890398844629933 / 40000000000000000 : ℝ) ≤
+      (-(11323966949471001526601 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (12890398844629933 / 40000000000000000 : ℝ) =
+            (12890398844629933 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12890398844629933 / 10000000000000000 : ℝ) ≤
+            (2538976656528998473399 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (591773635859601 / 2500000000000000 : ℝ)) (q :=
+      (12890398844629933 / 40000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_639_15 :
+    Uω (aρ 15) (bρ 15) (23679056431509 / 32000000000000 : ℝ) ≤
+      (-(3393869927754534575737 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10291635546897611 / 40000000000000000 : ℝ) ≤
+      (-(3393869927754534575737 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (10291635546897611 / 40000000000000000 : ℝ) =
+            (10291635546897611 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10291635546897611 / 10000000000000000 : ℝ) ≤
+            (71865973745465424263 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1327286171987243 / 10000000000000000 : ℝ)) (q :=
+      (10291635546897611 / 40000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_639_16 :
+    Uω (aρ 16) (bρ 16) (23679056431509 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_639 :
+    Uρ (23679056431509 / 32000000000000 : ℝ) ≤
+      (-(2694629023050604486451 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (23679056431509 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3099023403956418114027 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_639_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (23679056431509 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3131648447173872218449 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_639_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (23679056431509 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3197186693424707257857 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_639_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (23679056431509 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3306891973972135931103 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_639_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (23679056431509 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(54313047107620936833 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_639_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (23679056431509 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3723087564835383075901 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_639_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (23679056431509 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(2034447164939685717283 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_639_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (23679056431509 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(2267963951027866300151 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_639_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (23679056431509 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(5147710924735413315221 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_639_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (23679056431509 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(2964268328139464888913 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_639_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (23679056431509 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(6903744043443189821783 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_639_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (23679056431509 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(4050572539749861997181 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_639_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (23679056431509 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(1194433035719959725449 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_639_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (23679056431509 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(11323966949471001526601 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_639_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (23679056431509 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(3393869927754534575737 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_639_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (23679056431509 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_639_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/U53.lean b/LeanPool/Zeta5Irrational/Table/U53.lean
new file mode 100644
index 0000000000..87d7869602
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/U53.lean
@@ -0,0 +1,5463 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-! # Certified arcsine potential bounds (U53) -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma U_640_1 :
+    Uω (aρ 1) (bρ 1) (11854766575033 / 16000000000000 : ℝ) ≤
+      (-(617209565778762565663 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14689419759675083 / 20000000000000000 : ℝ) ≤
+      (-(617209565778762565663 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (14689419759675083 / 20000000000000000 : ℝ) =
+            (14689419759675083 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14689419759675083 / 10000000000000000 : ℝ) ≤
+            (769084794821237434337 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3672343934182229 / 5000000000000000 : ℝ)) (q :=
+      (14689419759675083 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_640_2 :
+    Uω (aρ 2) (bρ 2) (11854766575033 / 16000000000000 : ℝ) ≤
+      (-(1559315139434144244523 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (732081795961533 / 1000000000000000 : ℝ) ≤
+      (-(1559315139434144244523 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (732081795961533 / 1000000000000000 : ℝ) =
+            (732081795961533 / 500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (732081795961533 / 500000000000000 : ℝ) ≤
+            (1906420762065855755477 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1464134608086007 / 2000000000000000 : ℝ)) (q :=
+      (732081795961533 / 1000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_640_3 :
+    Uω (aρ 3) (bρ 3) (11854766575033 / 16000000000000 : ℝ) ≤
+      (-(796020581195767598923 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14546116355132353 / 20000000000000000 : ℝ) ≤
+      (-(796020581195767598923 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (14546116355132353 / 20000000000000000 : ℝ) =
+            (14546116355132353 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14546116355132353 / 10000000000000000 : ℝ) ≤
+            (936847369554232401077 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (56816793614701 / 78125000000000 : ℝ)) (q :=
+      (14546116355132353 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_640_4 :
+    Uω (aρ 4) (bρ 4) (11854766575033 / 16000000000000 : ℝ) ≤
+      (-(3293641379290378129919 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3596905074737841 / 5000000000000000 : ℝ) ≤
+      (-(3293641379290378129919 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (3596905074737841 / 5000000000000000 : ℝ) =
+            (3596905074737841 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3596905074737841 / 2500000000000000 : ℝ) ≤
+            (3637830423709621870081 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7192347707565739 / 10000000000000000 : ℝ)) (q :=
+      (3596905074737841 / 5000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_640_5 :
+    Uω (aρ 5) (bρ 5) (11854766575033 / 16000000000000 : ℝ) ≤
+      (-(3462554103321312357713 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7073318039984403 / 10000000000000000 : ℝ) ≤
+      (-(3462554103321312357713 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (7073318039984403 / 10000000000000000 : ℝ) =
+            (7073318039984403 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7073318039984403 / 5000000000000000 : ℝ) ≤
+            (3468917699678687642287 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7069553407918181 / 10000000000000000 : ℝ)) (q :=
+      (7073318039984403 / 10000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_640_6 :
+    Uω (aρ 6) (bρ 6) (11854766575033 / 16000000000000 : ℝ) ≤
+      (-(1854629608987680224327 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6900950604875963 / 10000000000000000 : ℝ) ≤
+      (-(1854629608987680224327 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (6900950604875963 / 10000000000000000 : ℝ) =
+            (6900950604875963 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6900950604875963 / 5000000000000000 : ℝ) ≤
+            (1611106292512319775673 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6892058912841301 / 10000000000000000 : ℝ)) (q :=
+      (6900950604875963 / 10000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_640_7 :
+    Uω (aρ 7) (bρ 7) (11854766575033 / 16000000000000 : ℝ) ≤
+      (-(4054556366412365688859 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13333459616691127 / 20000000000000000 : ℝ) ≤
+      (-(4054556366412365688859 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (13333459616691127 / 20000000000000000 : ℝ) =
+            (13333459616691127 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13333459616691127 / 10000000000000000 : ℝ) ≤
+            (2876915436587634311141 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3323639350270251 / 5000000000000000 : ℝ)) (q :=
+      (13333459616691127 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_640_8 :
+    Uω (aρ 8) (bρ 8) (11854766575033 / 16000000000000 : ℝ) ≤
+      (-(141276713247013046609 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (318149887404509 / 500000000000000 : ℝ) ≤
+      (-(141276713247013046609 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (318149887404509 / 500000000000000 : ℝ) =
+            (318149887404509 / 250000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (318149887404509 / 250000000000000 : ℝ) ≤
+            (75331780596736953391 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1264671246273947 / 2000000000000000 : ℝ)) (q :=
+      (318149887404509 / 500000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_640_9 :
+    Uω (aρ 9) (bρ 9) (11854766575033 / 16000000000000 : ℝ) ≤
+      (-(160361986765208395623 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11972039814806983 / 20000000000000000 : ℝ) ≤
+      (-(160361986765208395623 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11972039814806983 / 20000000000000000 : ℝ) =
+            (11972039814806983 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11972039814806983 / 10000000000000000 : ℝ) ≤
+            (56246507078541604377 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2955183089178179 / 5000000000000000 : ℝ)) (q :=
+      (11972039814806983 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_640_10 :
+    Uω (aρ 10) (bρ 10) (11854766575033 / 16000000000000 : ℝ) ≤
+      (-(1477721945264326746369 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5537240736205787 / 10000000000000000 : ℝ) ≤
+      (-(1477721945264326746369 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5537240736205787 / 10000000000000000 : ℝ) =
+            (5537240736205787 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5537240736205787 / 5000000000000000 : ℝ) ≤
+            (255146005485673253631 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5401366869455949 / 10000000000000000 : ℝ)) (q :=
+      (5537240736205787 / 10000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_640_11 :
+    Uω (aρ 11) (bρ 11) (11854766575033 / 16000000000000 : ℝ) ≤
+      (-(430240760642288933031 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10047733194381673 / 20000000000000000 : ℝ) ≤
+      (-(430240760642288933031 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10047733194381673 / 20000000000000000 : ℝ) =
+            (10047733194381673 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10047733194381673 / 10000000000000000 : ℝ) ≤
+            (2976227045211066969 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (599136902203881 / 1250000000000000 : ℝ)) (q :=
+      (10047733194381673 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_640_12 :
+    Uω (aρ 12) (bρ 12) (11854766575033 / 16000000000000 : ℝ) ≤
+      (-(4038902029818474039807 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1114616417328769 / 2500000000000000 : ℝ) ≤
+      (-(4038902029818474039807 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1114616417328769 / 2500000000000000 : ℝ) =
+            (1114616417328769 / 625000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1114616417328769 / 625000000000000 : ℝ) ≤
+            (2892569773181525960193 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4085993458564527 / 10000000000000000 : ℝ)) (q :=
+      (1114616417328769 / 2500000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_640_13 :
+    Uω (aρ 13) (bρ 13) (11854766575033 / 16000000000000 : ℝ) ≤
+      (-(9526396211055741483633 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (192860759814833 / 500000000000000 : ℝ) ≤
+      (-(9526396211055741483633 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (192860759814833 / 500000000000000 : ℝ) =
+            (192860759814833 / 125000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (192860759814833 / 125000000000000 : ℝ) ≤
+            (4336547394944258516367 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (656574850824539 / 2000000000000000 : ℝ)) (q :=
+      (192860759814833 / 500000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13]) (by norm_num [bρ_13])
+      (by norm_num) (by norm_num [aρ_13, bρ_13]) (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_640_14 :
+    Uω (aρ 14) (bρ 14) (11854766575033 / 16000000000000 : ℝ) ≤
+      (-(11283870676060941852289 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3235547074773687 / 10000000000000000 : ℝ) ≤
+      (-(11283870676060941852289 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3235547074773687 / 10000000000000000 : ℝ) =
+            (3235547074773687 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3235547074773687 / 2500000000000000 : ℝ) ≤
+            (2579072929939058147711 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2383465296121749 / 10000000000000000 : ℝ)) (q :=
+      (3235547074773687 / 10000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_640_15 :
+    Uω (aρ 15) (bρ 15) (11854766575033 / 16000000000000 : ℝ) ≤
+      (-(3376112697703459964891 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5182498075348999 / 20000000000000000 : ℝ) ≤
+      (-(3376112697703459964891 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5182498075348999 / 20000000000000000 : ℝ) =
+            (5182498075348999 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5182498075348999 / 5000000000000000 : ℝ) ≤
+            (89623203796540035109 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (677221249669187 / 5000000000000000 : ℝ)) (q :=
+      (5182498075348999 / 20000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_640_16 :
+    Uω (aρ 16) (bρ 16) (11854766575033 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_640 :
+    Uρ (11854766575033 / 16000000000000 : ℝ) ≤
+      (-(1074206850063200334523 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (11854766575033 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(617209565778762565663 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_640_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (11854766575033 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1559315139434144244523 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_640_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (11854766575033 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(796020581195767598923 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_640_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (11854766575033 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3293641379290378129919 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_640_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (11854766575033 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(3462554103321312357713 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_640_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (11854766575033 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1854629608987680224327 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_640_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (11854766575033 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(4054556366412365688859 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_640_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (11854766575033 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(141276713247013046609 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_640_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (11854766575033 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(160361986765208395623 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_640_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (11854766575033 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(1477721945264326746369 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_640_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (11854766575033 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) * (-(430240760642288933031 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_640_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (11854766575033 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(4038902029818474039807 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_640_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (11854766575033 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(9526396211055741483633 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_640_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (11854766575033 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(11283870676060941852289 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_640_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (11854766575033 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(3376112697703459964891 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_640_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (11854766575033 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_640_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_641_1 :
+    Uω (aρ 1) (bρ 1) (23740009868623 / 32000000000000 : ℝ) ≤
+      (-(3073089068670948228763 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (29416935531569081 / 40000000000000000 : ℝ) ≤
+      (-(3073089068670948228763 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (29416935531569081 / 40000000000000000 : ℝ) =
+            (29416935531569081 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (29416935531569081 / 20000000000000000 : ℝ) ≤
+            (3858382734329051771237 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7354211899924853 / 10000000000000000 : ℝ)) (q :=
+      (29416935531569081 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_641_2 :
+    Uω (aρ 2) (bρ 2) (23740009868623 / 32000000000000 : ℝ) ≤
+      (-(3105629036471363676813 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5864273697963929 / 8000000000000000 : ℝ) ≤
+      (-(3105629036471363676813 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5864273697963929 / 8000000000000000 : ℝ) =
+            (5864273697963929 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5864273697963929 / 4000000000000000 : ℝ) ≤
+            (3825842766528636323187 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1466039478312027 / 2000000000000000 : ℝ)) (q :=
+      (5864273697963929 / 8000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_641_3 :
+    Uω (aρ 3) (bρ 3) (23740009868623 / 32000000000000 : ℝ) ≤
+      (-(1585497554256626544711 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (29130331269156021 / 40000000000000000 : ℝ) ≤
+      (-(1585497554256626544711 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (29130331269156021 / 40000000000000000 : ℝ) =
+            (29130331269156021 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (29130331269156021 / 20000000000000000 : ℝ) ≤
+            (1880238347243373455289 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7282074887578323 / 10000000000000000 : ℝ)) (q :=
+      (29130331269156021 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_641_4 :
+    Uω (aρ 4) (bρ 4) (23740009868623 / 32000000000000 : ℝ) ≤
+      (-(3280408326324041502759 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28813344232007867 / 40000000000000000 : ℝ) ≤
+      (-(3280408326324041502759 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (28813344232007867 / 40000000000000000 : ℝ) =
+            (28813344232007867 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28813344232007867 / 20000000000000000 : ℝ) ≤
+            (3651063476675958497241 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3600937775034623 / 5000000000000000 : ℝ)) (q :=
+      (28813344232007867 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_641_5 :
+    Uω (aρ 5) (bρ 5) (23740009868623 / 32000000000000 : ℝ) ≤
+      (-(1724545680045968091771 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28331388317403481 / 40000000000000000 : ℝ) ≤
+      (-(1724545680045968091771 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (28331388317403481 / 40000000000000000 : ℝ) =
+            (28331388317403481 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28331388317403481 / 20000000000000000 : ℝ) ≤
+            (1741190221454031908229 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7079087512102053 / 10000000000000000 : ℝ)) (q :=
+      (28331388317403481 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_641_6 :
+    Uω (aρ 6) (bρ 6) (23740009868623 / 32000000000000 : ℝ) ≤
+      (-(3695450016234095153087 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27641947398671501 / 40000000000000000 : ℝ) ≤
+      (-(3695450016234095153087 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (27641947398671501 / 40000000000000000 : ℝ) =
+            (27641947398671501 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27641947398671501 / 20000000000000000 : ℝ) ≤
+            (3236021786765904846913 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6901607427876063 / 10000000000000000 : ℝ)) (q :=
+      (27641947398671501 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_641_7 :
+    Uω (aρ 7) (bρ 7) (23740009868623 / 32000000000000 : ℝ) ≤
+      (-(808047810240227746469 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (26705126446929301 / 40000000000000000 : ℝ) ≤
+      (-(808047810240227746469 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (26705126446929301 / 40000000000000000 : ℝ) =
+            (26705126446929301 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26705126446929301 / 20000000000000000 : ℝ) ≤
+            (578246550359772253531 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6656858332764963 / 10000000000000000 : ℝ)) (q :=
+      (26705126446929301 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_641_8 :
+    Uω (aρ 8) (bρ 8) (23740009868623 / 32000000000000 : ℝ) ≤
+      (-(4505804715333577953633 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5098065070943721 / 8000000000000000 : ℝ) ≤
+      (-(4505804715333577953633 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5098065070943721 / 8000000000000000 : ℝ) =
+            (5098065070943721 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5098065070943721 / 4000000000000000 : ℝ) ≤
+            (2425667087666422046367 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1266599887599923 / 2000000000000000 : ℝ)) (q :=
+      (5098065070943721 / 8000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_641_9 :
+    Uω (aρ 9) (bρ 9) (23740009868623 / 32000000000000 : ℝ) ≤
+      (-(5115482859176546188077 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23982662367503569 / 40000000000000000 : ℝ) ≤
+      (-(5115482859176546188077 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23982662367503569 / 40000000000000000 : ℝ) =
+            (23982662367503569 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23982662367503569 / 20000000000000000 : ℝ) ≤
+            (1815988943823453811923 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5920133572752097 / 10000000000000000 : ℝ)) (q :=
+      (23982662367503569 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_641_10 :
+    Uω (aρ 10) (bρ 10) (23740009868623 / 32000000000000 : ℝ) ≤
+      (-(736658945086993470021 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22188015433787531 / 40000000000000000 : ℝ) ≤
+      (-(736658945086993470021 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (22188015433787531 / 40000000000000000 : ℝ) =
+            (22188015433787531 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22188015433787531 / 20000000000000000 : ℝ) ≤
+            (129775030288006529979 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2705684569694539 / 5000000000000000 : ℝ)) (q :=
+      (22188015433787531 / 40000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_641_11 :
+    Uω (aρ 11) (bρ 11) (23740009868623 / 32000000000000 : ℝ) ≤
+      (-(6864003581606836853937 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20135392664314169 / 40000000000000000 : ℝ) ≤
+      (-(6864003581606836853937 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (20135392664314169 / 40000000000000000 : ℝ) =
+            (20135392664314169 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20135392664314169 / 20000000000000000 : ℝ) ≤
+            (67468221393163146063 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4803534380857397 / 10000000000000000 : ℝ)) (q :=
+      (20135392664314169 / 40000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_641_12 :
+    Uω (aρ 12) (bρ 12) (23740009868623 / 32000000000000 : ℝ) ≤
+      (-(4027263637795985554169 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (715016900979913 / 1600000000000000 : ℝ) ≤
+      (-(4027263637795985554169 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (715016900979913 / 1600000000000000 : ℝ) =
+            (715016900979913 / 400000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (715016900979913 / 400000000000000 : ℝ) ≤
+            (2904208165204014445831 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (163889976305369 / 400000000000000 : ℝ)) (q :=
+      (715016900979913 / 1600000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12]) (by norm_num [bρ_12])
+      (by norm_num) (by norm_num [aρ_12, bρ_12]) (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_641_13 :
+    Uω (aρ 13) (bρ 13) (23740009868623 / 32000000000000 : ℝ) ≤
+      (-(9497441751319171954421 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15473598955246807 / 40000000000000000 : ℝ) ≤
+      (-(9497441751319171954421 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (15473598955246807 / 40000000000000000 : ℝ) =
+            (15473598955246807 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15473598955246807 / 10000000000000000 : ℝ) ≤
+            (4365501854680828045579 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (823929841150929 / 2500000000000000 : ℝ)) (q :=
+      (15473598955246807 / 40000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_641_14 :
+    Uω (aρ 14) (bρ 14) (23740009868623 / 32000000000000 : ℝ) ≤
+      (-(1124404824568891011653 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12993829994981327 / 40000000000000000 : ℝ) ≤
+      (-(1124404824568891011653 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (12993829994981327 / 40000000000000000 : ℝ) =
+            (12993829994981327 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12993829994981327 / 10000000000000000 : ℝ) ≤
+            (261889536031108988347 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (299970271189497 / 1250000000000000 : ℝ)) (q :=
+      (12993829994981327 / 40000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_641_15 :
+    Uω (aρ 15) (bρ 15) (23740009868623 / 32000000000000 : ℝ) ≤
+      (-(13434819948548071133851 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10437420347172571 / 40000000000000000 : ℝ) ≤
+      (-(13434819948548071133851 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (10437420347172571 / 40000000000000000 : ℝ) =
+            (10437420347172571 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10437420347172571 / 10000000000000000 : ℝ) ≤
+            (428123657451928866149 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (690565311513299 / 5000000000000000 : ℝ)) (q :=
+      (10437420347172571 / 40000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_641_16 :
+    Uω (aρ 16) (bρ 16) (23740009868623 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_641 :
+    Uρ (23740009868623 / 32000000000000 : ℝ) ≤
+      (-(2676446975214992581051 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (23740009868623 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3073089068670948228763 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_641_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (23740009868623 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3105629036471363676813 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_641_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (23740009868623 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1585497554256626544711 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_641_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (23740009868623 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3280408326324041502759 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_641_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (23740009868623 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1724545680045968091771 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_641_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (23740009868623 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3695450016234095153087 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_641_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (23740009868623 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(808047810240227746469 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_641_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (23740009868623 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(4505804715333577953633 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_641_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (23740009868623 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(5115482859176546188077 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_641_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (23740009868623 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(736658945086993470021 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_641_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (23740009868623 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(6864003581606836853937 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_641_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (23740009868623 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(4027263637795985554169 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_641_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (23740009868623 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(9497441751319171954421 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_641_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (23740009868623 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(1124404824568891011653 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_641_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (23740009868623 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(13434819948548071133851 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_641_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (23740009868623 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_641_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_642_1 :
+    Uω (aρ 1) (bρ 1) (1188524329359 / 1600000000000 : ℝ) ≤
+      (-(3060147079763797147191 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (589100630869861 / 800000000000000 : ℝ) ≤
+      (-(3060147079763797147191 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (589100630869861 / 800000000000000 : ℝ) =
+            (589100630869861 / 400000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (589100630869861 / 400000000000000 : ℝ) ≤
+            (3871324723236202852809 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (294549437253511 / 400000000000000 : ℝ)) (q :=
+      (589100630869861 / 800000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_642_2 :
+    Uω (aρ 2) (bρ 2) (1188524329359 / 1600000000000 : ℝ) ≤
+      (-(3092644676025192306147 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14679732569611667 / 20000000000000000 : ℝ) ≤
+      (-(3092644676025192306147 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (14679732569611667 / 20000000000000000 : ℝ) =
+            (14679732569611667 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14679732569611667 / 10000000000000000 : ℝ) ≤
+            (3838827126974807693853 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7339721741712917 / 10000000000000000 : ℝ)) (q :=
+      (14679732569611667 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_642_3 :
+    Uω (aρ 3) (bρ 3) (1188524329359 / 1600000000000 : ℝ) ≤
+      (-(3157924999766053070299 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14584214910548139 / 20000000000000000 : ℝ) ≤
+      (-(3157924999766053070299 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (14584214910548139 / 20000000000000000 : ℝ) =
+            (14584214910548139 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14584214910548139 / 10000000000000000 : ℝ) ≤
+            (3773546803233946929701 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7291600188999389 / 10000000000000000 : ℝ)) (q :=
+      (14584214910548139 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_642_4 :
+    Uω (aρ 4) (bρ 4) (1188524329359 / 1600000000000 : ℝ) ≤
+      (-(1633596384335385696813 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3606430980709437 / 5000000000000000 : ℝ) ≤
+      (-(1633596384335385696813 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (3606430980709437 / 5000000000000000 : ℝ) =
+            (3606430980709437 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3606430980709437 / 2500000000000000 : ℝ) ≤
+            (1832139517164614303187 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3605701691176999 / 5000000000000000 : ℝ)) (q :=
+      (3606430980709437 / 5000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_642_5 :
+    Uω (aρ 5) (bρ 5) (1188524329359 / 1600000000000 : ℝ) ≤
+      (-(1717823368121241994069 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7092376105100347 / 10000000000000000 : ℝ) ≤
+      (-(1717823368121241994069 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (7092376105100347 / 10000000000000000 : ℝ) =
+            (7092376105100347 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7092376105100347 / 5000000000000000 : ℝ) ≤
+            (1747912533378758005931 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (886077698631493 / 1250000000000000 : ℝ)) (q :=
+      (7092376105100347 / 10000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_642_6 :
+    Uω (aρ 6) (bρ 6) (1188524329359 / 1600000000000 : ℝ) ≤
+      (-(736331981307432108431 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2768009224239201 / 4000000000000000 : ℝ) ≤
+      (-(736331981307432108431 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2768009224239201 / 4000000000000000 : ℝ) =
+            (2768009224239201 / 2000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2768009224239201 / 2000000000000000 : ℝ) ≤
+            (649962379292567891569 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1382231175037451 / 2000000000000000 : ℝ)) (q :=
+      (2768009224239201 / 4000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_642_7 :
+    Uω (aρ 7) (bρ 7) (1188524329359 / 1600000000000 : ℝ) ≤
+      (-(805188464903322653011 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (267433333414867 / 400000000000000 : ℝ) ≤
+      (-(805188464903322653011 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (267433333414867 / 400000000000000 : ℝ) =
+            (267433333414867 / 200000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (267433333414867 / 200000000000000 : ℝ) ≤
+            (581105895696677346989 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (33332189027473 / 50000000000000 : ℝ)) (q :=
+      (267433333414867 / 400000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_642_8 :
+    Uω (aρ 8) (bρ 8) (1188524329359 / 1600000000000 : ℝ) ≤
+      (-(4490777505587620117577 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6382164749112121 / 10000000000000000 : ℝ) ≤
+      (-(4490777505587620117577 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (6382164749112121 / 10000000000000000 : ℝ) =
+            (6382164749112121 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6382164749112121 / 5000000000000000 : ℝ) ≤
+            (2440694297412379882423 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1585660571078873 / 2500000000000000 : ℝ)) (q :=
+      (6382164749112121 / 10000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_642_9 :
+    Uω (aρ 9) (bρ 9) (1188524329359 / 1600000000000 : ℝ) ≤
+      (-(2549704341422734584063 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6005310880397129 / 10000000000000000 : ℝ) ≤
+      (-(2549704341422734584063 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (6005310880397129 / 10000000000000000 : ℝ) =
+            (6005310880397129 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6005310880397129 / 5000000000000000 : ℝ) ≤
+            (916031560077265415937 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1482475043811377 / 2500000000000000 : ℝ)) (q :=
+      (6005310880397129 / 10000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_642_10 :
+    Uω (aρ 10) (bρ 10) (1188524329359 / 1600000000000 : ℝ) ≤
+      (-(5875687868970487449787 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2778383059672791 / 5000000000000000 : ℝ) ≤
+      (-(5875687868970487449787 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2778383059672791 / 5000000000000000 : ℝ) =
+            (2778383059672791 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2778383059672791 / 2500000000000000 : ℝ) ≤
+            (1055783934029512550213 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2710684843318707 / 5000000000000000 : ℝ)) (q :=
+      (2778383059672791 / 5000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_642_11 :
+    Uω (aρ 11) (bρ 11) (1188524329359 / 1600000000000 : ℝ) ≤
+      (-(3422099036893664444543 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5043827837360799 / 10000000000000000 : ℝ) ≤
+      (-(3422099036893664444543 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5043827837360799 / 10000000000000000 : ℝ) =
+            (5043827837360799 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5043827837360799 / 5000000000000000 : ℝ) ≤
+            (43636864606335555457 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (300873109304553 / 625000000000000 : ℝ)) (q :=
+      (5043827837360799 / 10000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_642_12 :
+    Uω (aρ 12) (bρ 12) (1188524329359 / 1600000000000 : ℝ) ≤
+      (-(31372321582058323511 / 39062500000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4479241212954563 / 10000000000000000 : ℝ) ≤
+      (-(31372321582058323511 / 39062500000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4479241212954563 / 10000000000000000 : ℝ) =
+            (4479241212954563 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4479241212954563 / 2500000000000000 : ℝ) ≤
+            (22779801878879176489 / 39062500000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (128390518648293 / 312500000000000 : ℝ)) (q :=
+      (4479241212954563 / 10000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_642_13 :
+    Uω (aρ 13) (bρ 13) (1188524329359 / 1600000000000 : ℝ) ≤
+      (-(4734299912291785360331 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7759146108497097 / 20000000000000000 : ℝ) ≤
+      (-(4734299912291785360331 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7759146108497097 / 20000000000000000 : ℝ) =
+            (7759146108497097 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7759146108497097 / 5000000000000000 : ℝ) ≤
+            (2197171890708214639669 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3308542020928347 / 10000000000000000 : ℝ)) (q :=
+      (7759146108497097 / 20000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_642_14 :
+    Uω (aρ 14) (bρ 14) (1188524329359 / 1600000000000 : ℝ) ≤
+      (-(2240898948388567398827 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (407666466324277 / 1250000000000000 : ℝ) ≤
+      (-(2240898948388567398827 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (407666466324277 / 1250000000000000 : ℝ) =
+            (407666466324277 / 312500000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (407666466324277 / 312500000000000 : ℝ) ≤
+            (531689772811432601173 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1207993329332341 / 5000000000000000 : ℝ)) (q :=
+      (407666466324277 / 1250000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14]) (by norm_num [bρ_14])
+      (by norm_num) (by norm_num [aρ_14, bρ_14]) (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_642_15 :
+    Uω (aρ 15) (bρ 15) (1188524329359 / 1600000000000 : ℝ) ≤
+      (-(13366510929828003237353 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5254480704081081 / 20000000000000000 : ℝ) ≤
+      (-(13366510929828003237353 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (5254480704081081 / 20000000000000000 : ℝ) =
+            (5254480704081081 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5254480704081081 / 5000000000000000 : ℝ) ≤
+            (496432676171996762647 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1407377178972331 / 10000000000000000 : ℝ)) (q :=
+      (5254480704081081 / 20000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_642_16 :
+    Uω (aρ 16) (bρ 16) (1188524329359 / 1600000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_642 :
+    Uρ (1188524329359 / 1600000000000 : ℝ) ≤
+      (-(666854306635595554443 / 1250000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1188524329359 / 1600000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3060147079763797147191 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_642_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1188524329359 / 1600000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3092644676025192306147 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_642_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1188524329359 / 1600000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3157924999766053070299 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_642_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1188524329359 / 1600000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1633596384335385696813 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_642_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1188524329359 / 1600000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1717823368121241994069 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_642_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1188524329359 / 1600000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(736331981307432108431 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_642_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1188524329359 / 1600000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(805188464903322653011 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_642_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1188524329359 / 1600000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(4490777505587620117577 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_642_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1188524329359 / 1600000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(2549704341422734584063 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_642_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1188524329359 / 1600000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(5875687868970487449787 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_642_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1188524329359 / 1600000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(3422099036893664444543 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_642_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1188524329359 / 1600000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(31372321582058323511 / 39062500000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_642_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1188524329359 / 1600000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4734299912291785360331 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_642_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1188524329359 / 1600000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(2240898948388567398827 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_642_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1188524329359 / 1600000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(13366510929828003237353 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_642_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1188524329359 / 1600000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_642_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_643_1 :
+    Uω (aρ 1) (bρ 1) (23800963305737 / 32000000000000 : ℝ) ≤
+      (-(1523610909408552015951 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5898625511024643 / 8000000000000000 : ℝ) ≤
+      (-(1523610909408552015951 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5898625511024643 / 8000000000000000 : ℝ) =
+            (5898625511024643 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5898625511024643 / 4000000000000000 : ℝ) ≤
+            (1942124992091447984049 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1474651992520759 / 2000000000000000 : ℝ)) (q :=
+      (5898625511024643 / 8000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_643_2 :
+    Uω (aρ 2) (bρ 2) (23800963305737 / 32000000000000 : ℝ) ≤
+      (-(1539838576871448098047 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (29397561786679987 / 40000000000000000 : ℝ) ≤
+      (-(1539838576871448098047 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (29397561786679987 / 40000000000000000 : ℝ) =
+            (29397561786679987 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (29397561786679987 / 20000000000000000 : ℝ) ≤
+            (1925897324628551901953 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7349246090892181 / 10000000000000000 : ℝ)) (q :=
+      (29397561786679987 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_643_3 :
+    Uω (aρ 3) (bρ 3) (23800963305737 / 32000000000000 : ℝ) ≤
+      (-(628974390773593105421 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5841305673222537 / 8000000000000000 : ℝ) ≤
+      (-(628974390773593105421 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5841305673222537 / 8000000000000000 : ℝ) =
+            (5841305673222537 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5841305673222537 / 4000000000000000 : ℝ) ≤
+            (757319969826406894579 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (730112548695853 / 1000000000000000 : ℝ)) (q :=
+      (5841305673222537 / 8000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_643_4 :
+    Uω (aρ 4) (bρ 4) (23800963305737 / 32000000000000 : ℝ) ≤
+      (-(813498665028040558633 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28889551438986517 / 40000000000000000 : ℝ) ≤
+      (-(813498665028040558633 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (28889551438986517 / 40000000000000000 : ℝ) =
+            (28889551438986517 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28889551438986517 / 20000000000000000 : ℝ) ≤
+            (919369285721959441367 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3610465602230223 / 5000000000000000 : ℝ)) (q :=
+      (28889551438986517 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_643_5 :
+    Uω (aρ 5) (bρ 5) (23800963305737 / 32000000000000 : ℝ) ≤
+      (-(1711110091506825365049 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5681524093830163 / 8000000000000000 : ℝ) ≤
+      (-(1711110091506825365049 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5681524093830163 / 8000000000000000 : ℝ) =
+            (5681524093830163 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5681524093830163 / 4000000000000000 : ℝ) ≤
+            (1754625809993174634951 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1419631127775519 / 2000000000000000 : ℝ)) (q :=
+      (5681524093830163 / 8000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_643_6 :
+    Uω (aρ 6) (bρ 6) (23800963305737 / 32000000000000 : ℝ) ≤
+      (-(3667888836031086007881 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27718236951226001 / 40000000000000000 : ℝ) ≤
+      (-(3667888836031086007881 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (27718236951226001 / 40000000000000000 : ℝ) =
+            (27718236951226001 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27718236951226001 / 20000000000000000 : ℝ) ≤
+            (3263582966968913992119 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1730176063763797 / 2500000000000000 : ℝ)) (q :=
+      (27718236951226001 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_643_7 :
+    Uω (aρ 7) (bρ 7) (23800963305737 / 32000000000000 : ℝ) ≤
+      (-(501458265861246567413 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (26781539918427609 / 40000000000000000 : ℝ) ≤
+      (-(501458265861246567413 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (26781539918427609 / 40000000000000000 : ℝ) =
+            (26781539918427609 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26781539918427609 / 20000000000000000 : ℝ) ≤
+            (364975709513753432587 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (834502139926999 / 1250000000000000 : ℝ)) (q :=
+      (26781539918427609 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_643_8 :
+    Uω (aρ 8) (bρ 8) (23800963305737 / 32000000000000 : ℝ) ≤
+      (-(4475773124241223698507 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25566991920832123 / 40000000000000000 : ℝ) ≤
+      (-(4475773124241223698507 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (25566991920832123 / 40000000000000000 : ℝ) =
+            (25566991920832123 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25566991920832123 / 20000000000000000 : ℝ) ≤
+            (2455698678758776301493 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6352284771958249 / 10000000000000000 : ℝ)) (q :=
+      (25566991920832123 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_643_9 :
+    Uω (aρ 9) (bρ 9) (23800963305737 / 32000000000000 : ℝ) ≤
+      (-(1270840239499814070163 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24059823099681563 / 40000000000000000 : ℝ) ≤
+      (-(1270840239499814070163 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (24059823099681563 / 40000000000000000 : ℝ) =
+            (24059823099681563 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24059823099681563 / 20000000000000000 : ℝ) ≤
+            (462027711250185929837 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5939665989742969 / 10000000000000000 : ℝ)) (q :=
+      (24059823099681563 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_643_10 :
+    Uω (aρ 10) (bρ 10) (23800963305737 / 32000000000000 : ℝ) ≤
+      (-(292906829020823116481 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (890644403785559 / 1600000000000000 : ℝ) ≤
+      (-(292906829020823116481 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (890644403785559 / 1600000000000000 : ℝ) =
+            (890644403785559 / 800000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (890644403785559 / 800000000000000 : ℝ) ≤
+            (53666761129176883519 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (217254740828667 / 400000000000000 : ℝ)) (q :=
+      (890644403785559 / 1600000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10]) (by norm_num [bρ_10])
+      (by norm_num) (by norm_num [aρ_10, bρ_10]) (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_643_11 :
+    Uω (aρ 11) (bρ 11) (23800963305737 / 32000000000000 : ℝ) ≤
+      (-(106631803823411271099 / 156250000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20215222493405561 / 40000000000000000 : ℝ) ≤
+      (-(106631803823411271099 / 156250000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (20215222493405561 / 40000000000000000 : ℝ) =
+            (20215222493405561 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20215222493405561 / 20000000000000000 : ℝ) ≤
+            (1672443098463728901 / 156250000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (603050168288121 / 1250000000000000 : ℝ)) (q :=
+      (20215222493405561 / 40000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_643_12 :
+    Uω (aρ 12) (bρ 12) (23800963305737 / 32000000000000 : ℝ) ≤
+      (-(1601632961928138856129 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17958489802713691 / 40000000000000000 : ℝ) ≤
+      (-(1601632961928138856129 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (17958489802713691 / 40000000000000000 : ℝ) =
+            (17958489802713691 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17958489802713691 / 10000000000000000 : ℝ) ≤
+            (1170955759271861143871 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4119735097644033 / 10000000000000000 : ℝ)) (q :=
+      (17958489802713691 / 40000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_643_13 :
+    Uω (aρ 13) (bρ 13) (23800963305737 / 32000000000000 : ℝ) ≤
+      (-(4719934683011539956371 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3112588218111867 / 8000000000000000 : ℝ) ≤
+      (-(4719934683011539956371 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3112588218111867 / 8000000000000000 : ℝ) =
+            (3112588218111867 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3112588218111867 / 2000000000000000 : ℝ) ≤
+            (2211537119988460043629 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (664268496632371 / 2000000000000000 : ℝ)) (q :=
+      (3112588218111867 / 8000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_643_14 :
+    Uω (aρ 14) (bρ 14) (23800963305737 / 32000000000000 : ℝ) ≤
+      (-(2791301348886200293647 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13096681978477213 / 40000000000000000 : ℝ) ≤
+      (-(2791301348886200293647 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (13096681978477213 / 40000000000000000 : ℝ) =
+            (13096681978477213 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13096681978477213 / 10000000000000000 : ℝ) ≤
+            (674434552613799706353 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1216070106082897 / 5000000000000000 : ℝ)) (q :=
+      (13096681978477213 / 40000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_643_15 :
+    Uω (aρ 15) (bρ 15) (23800963305737 / 32000000000000 : ℝ) ≤
+      (-(13299454215022681784837 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10579667853113777 / 40000000000000000 : ℝ) ≤
+      (-(13299454215022681784837 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (10579667853113777 / 40000000000000000 : ℝ) =
+            (10579667853113777 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10579667853113777 / 10000000000000000 : ℝ) ≤
+            (563489390977318215163 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (358301606724769 / 2500000000000000 : ℝ)) (q :=
+      (10579667853113777 / 40000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_643_16 :
+    Uω (aρ 16) (bρ 16) (23800963305737 / 32000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_643 :
+    Uρ (23800963305737 / 32000000000000 : ℝ) ≤
+      (-(5316853278250076301219 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (23800963305737 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1523610909408552015951 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_643_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (23800963305737 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1539838576871448098047 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_643_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (23800963305737 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(628974390773593105421 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_643_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (23800963305737 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(813498665028040558633 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_643_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (23800963305737 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1711110091506825365049 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_643_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (23800963305737 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3667888836031086007881 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_643_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (23800963305737 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(501458265861246567413 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_643_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (23800963305737 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(4475773124241223698507 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_643_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (23800963305737 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(1270840239499814070163 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_643_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (23800963305737 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) * (-(292906829020823116481 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_643_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (23800963305737 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) * (-(106631803823411271099 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_643_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (23800963305737 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(1601632961928138856129 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_643_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (23800963305737 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4719934683011539956371 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_643_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (23800963305737 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(2791301348886200293647 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_643_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (23800963305737 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(13299454215022681784837 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_643_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (23800963305737 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_643_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_644_1 :
+    Uω (aρ 1) (bρ 1) (11915720012147 / 16000000000000 : ℝ) ≤
+      (-(3034313242643511466231 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7382805891615179 / 10000000000000000 : ℝ) ≤
+      (-(3034313242643511466231 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (7382805891615179 / 10000000000000000 : ℝ) =
+            (7382805891615179 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7382805891615179 / 5000000000000000 : ℝ) ≤
+            (3897158560356488533769 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7382783993723483 / 10000000000000000 : ℝ)) (q :=
+      (7382805891615179 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_644_2 :
+    Uω (aρ 2) (bρ 2) (11915720012147 / 16000000000000 : ℝ) ≤
+      (-(766681606501934609241 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14717829216098583 / 20000000000000000 : ℝ) ≤
+      (-(766681606501934609241 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (14717829216098583 / 20000000000000000 : ℝ) =
+            (14717829216098583 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14717829216098583 / 10000000000000000 : ℝ) ≤
+            (966186344248065390759 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1839692609775427 / 2500000000000000 : ℝ)) (q :=
+      (14717829216098583 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_644_3 :
+    Uω (aρ 3) (bρ 3) (11915720012147 / 16000000000000 : ℝ) ≤
+      (-(3131835926320268809229 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14622313452116153 / 20000000000000000 : ℝ) ≤
+      (-(3131835926320268809229 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (14622313452116153 / 20000000000000000 : ℝ) =
+            (14622313452116153 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14622313452116153 / 10000000000000000 : ℝ) ≤
+            (3799635876679731190771 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3655325390734639 / 5000000000000000 : ℝ)) (q :=
+      (14622313452116153 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_644_4 :
+    Uω (aρ 4) (bρ 4) (11915720012147 / 16000000000000 : ℝ) ≤
+      (-(810203488653197127289 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14463827506010701 / 20000000000000000 : ℝ) ≤
+      (-(810203488653197127289 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (14463827506010701 / 20000000000000000 : ℝ) =
+            (14463827506010701 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14463827506010701 / 10000000000000000 : ℝ) ≤
+            (922664462096802872711 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3615229508214413 / 5000000000000000 : ℝ)) (q :=
+      (14463827506010701 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_644_5 :
+    Uω (aρ 5) (bρ 5) (11915720012147 / 16000000000000 : ℝ) ≤
+      (-(852202912960722596749 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14222868231935033 / 20000000000000000 : ℝ) ≤
+      (-(852202912960722596749 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (14222868231935033 / 20000000000000000 : ℝ) =
+            (14222868231935033 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14222868231935033 / 10000000000000000 : ℝ) ≤
+            (880665037789277403251 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3553844830844079 / 5000000000000000 : ℝ)) (q :=
+      (14222868231935033 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_644_6 :
+    Uω (aρ 6) (bρ 6) (11915720012147 / 16000000000000 : ℝ) ≤
+      (-(3654136752082123200103 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6939095381432753 / 10000000000000000 : ℝ) ≤
+      (-(3654136752082123200103 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (6939095381432753 / 10000000000000000 : ℝ) =
+            (6939095381432753 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6939095381432753 / 5000000000000000 : ℝ) ≤
+            (3277335050917876799897 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6930252567758631 / 10000000000000000 : ℝ)) (q :=
+      (6939095381432753 / 10000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_644_7 :
+    Uω (aρ 7) (bρ 7) (11915720012147 / 16000000000000 : ℝ) ≤
+      (-(3997410399111135964581 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13409873089558651 / 20000000000000000 : ℝ) ≤
+      (-(3997410399111135964581 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (13409873089558651 / 20000000000000000 : ℝ) =
+            (13409873089558651 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13409873089558651 / 10000000000000000 : ℝ) ≤
+            (2934061403888864035419 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (13057805225023 / 19531250000000 : ℝ)) (q :=
+      (13409873089558651 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_644_8 :
+    Uω (aρ 8) (bρ 8) (11915720012147 / 16000000000000 : ℝ) ≤
+      (-(178431660047890653203 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3200665516391409 / 5000000000000000 : ℝ) ≤
+      (-(178431660047890653203 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (3200665516391409 / 5000000000000000 : ℝ) =
+            (3200665516391409 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3200665516391409 / 2500000000000000 : ℝ) ≤
+            (98827212072109346797 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6361926902558761 / 10000000000000000 : ℝ)) (q :=
+      (3200665516391409 / 5000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_644_9 :
+    Uω (aρ 9) (bρ 9) (11915720012147 / 16000000000000 : ℝ) ≤
+      (-(633417449450731358873 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1204920055477177 / 2000000000000000 : ℝ) ≤
+      (-(633417449450731358873 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1204920055477177 / 2000000000000000 : ℝ) =
+            (1204920055477177 / 1000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1204920055477177 / 1000000000000000 : ℝ) ≤
+            (233016525924268641127 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1189886204024979 / 2000000000000000 : ℝ)) (q :=
+      (1204920055477177 / 2000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_644_10 :
+    Uω (aρ 10) (bρ 10) (11915720012147 / 16000000000000 : ℝ) ≤
+      (-(2920308785161509705411 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5576288076111443 / 10000000000000000 : ℝ) ≤
+      (-(2920308785161509705411 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5576288076111443 / 10000000000000000 : ℝ) =
+            (5576288076111443 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5576288076111443 / 5000000000000000 : ℝ) ≤
+            (545427116338490294589 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5441365651071011 / 10000000000000000 : ℝ)) (q :=
+      (5576288076111443 / 10000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_644_11 :
+    Uω (aρ 11) (bρ 11) (11915720012147 / 16000000000000 : ℝ) ≤
+      (-(3402357746865413340039 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5063781536253397 / 10000000000000000 : ℝ) ≤
+      (-(3402357746865413340039 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5063781536253397 / 10000000000000000 : ℝ) =
+            (5063781536253397 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5063781536253397 / 5000000000000000 : ℝ) ≤
+            (63378154634586659961 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4834829197559919 / 10000000000000000 : ℝ)) (q :=
+      (5063781536253397 / 10000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_644_12 :
+    Uω (aρ 12) (bρ 12) (11915720012147 / 16000000000000 : ℝ) ≤
+      (-(3992539167654871337849 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4499999379751123 / 10000000000000000 : ℝ) ≤
+      (-(3992539167654871337849 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4499999379751123 / 10000000000000000 : ℝ) =
+            (4499999379751123 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4499999379751123 / 2500000000000000 : ℝ) ≤
+            (2938932635345128662151 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4130964981240371 / 10000000000000000 : ℝ)) (q :=
+      (4499999379751123 / 10000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_644_13 :
+    Uω (aρ 13) (bρ 13) (11915720012147 / 16000000000000 : ℝ) ≤
+      (-(9411249327571361579049 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1950943260898971 / 5000000000000000 : ℝ) ≤
+      (-(9411249327571361579049 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1950943260898971 / 5000000000000000 : ℝ) =
+            (1950943260898971 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1950943260898971 / 1250000000000000 : ℝ) ≤
+            (4451694278428638420951 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3334121006929009 / 10000000000000000 : ℝ)) (q :=
+      (1950943260898971 / 5000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_644_14 :
+    Uω (aρ 14) (bρ 14) (11915720012147 / 16000000000000 : ℝ) ≤
+      (-(5563087789113514828389 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6573948985756573 / 20000000000000000 : ℝ) ≤
+      (-(5563087789113514828389 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6573948985756573 / 20000000000000000 : ℝ) =
+            (6573948985756573 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6573948985756573 / 5000000000000000 : ℝ) ≤
+            (1368384013886485171611 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1224112117067349 / 5000000000000000 : ℝ)) (q :=
+      (6573948985756573 / 20000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_644_15 :
+    Uω (aρ 15) (bρ 15) (11915720012147 / 16000000000000 : ℝ) ≤
+      (-(13233586213666000565321 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2662396004972763 / 10000000000000000 : ℝ) ≤
+      (-(13233586213666000565321 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2662396004972763 / 10000000000000000 : ℝ) =
+            (2662396004972763 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2662396004972763 / 2500000000000000 : ℝ) ≤
+            (629357392333999434679 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1458640535738651 / 10000000000000000 : ℝ)) (q :=
+      (2662396004972763 / 10000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_644_16 :
+    Uω (aρ 16) (bρ 16) (11915720012147 / 16000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_644 :
+    Uρ (11915720012147 / 16000000000000 : ℝ) ≤
+      (-(2649474067190841539383 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (11915720012147 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(3034313242643511466231 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_644_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (11915720012147 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(766681606501934609241 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_644_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (11915720012147 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3131835926320268809229 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_644_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (11915720012147 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(810203488653197127289 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_644_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (11915720012147 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(852202912960722596749 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_644_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (11915720012147 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3654136752082123200103 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_644_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (11915720012147 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(3997410399111135964581 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_644_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (11915720012147 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(178431660047890653203 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_644_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (11915720012147 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(633417449450731358873 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_644_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (11915720012147 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(2920308785161509705411 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_644_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (11915720012147 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(3402357746865413340039 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_644_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (11915720012147 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(3992539167654871337849 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_644_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (11915720012147 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(9411249327571361579049 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_644_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (11915720012147 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(5563087789113514828389 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_644_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (11915720012147 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(13233586213666000565321 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_644_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (11915720012147 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_644_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_645_1 :
+    Uω (aρ 1) (bρ 1) (746637295669 / 1000000000000 : ℝ) ≤
+      (-(1504272986350256580489 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7401853897065559 / 10000000000000000 : ℝ) ≤
+      (-(1504272986350256580489 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (7401853897065559 / 10000000000000000 : ℝ) =
+            (7401853897065559 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7401853897065559 / 5000000000000000 : ℝ) ≤
+            (1961462915149743419511 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3700916027763059 / 5000000000000000 : ℝ)) (q :=
+      (7401853897065559 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_645_2 :
+    Uω (aρ 2) (bρ 2) (746637295669 / 1000000000000 : ℝ) ≤
+      (-(3040875180557327393003 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1475592585872157 / 2000000000000000 : ℝ) ≤
+      (-(3040875180557327393003 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1475592585872157 / 2000000000000000 : ℝ) =
+            (1475592585872157 / 1000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1475592585872157 / 1000000000000000 : ℝ) ≤
+            (3890596622442672606997 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (737781913262657 / 1000000000000000 : ℝ)) (q :=
+      (1475592585872157 / 2000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_645_3 :
+    Uω (aρ 3) (bρ 3) (746637295669 / 1000000000000 : ℝ) ≤
+      (-(388226843643701977613 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1466041197994437 / 2000000000000000 : ℝ) ≤
+      (-(388226843643701977613 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1466041197994437 / 2000000000000000 : ℝ) =
+            (1466041197994437 / 1000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1466041197994437 / 1000000000000000 : ℝ) ≤
+            (478207131731298022387 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (732970136019937 / 1000000000000000 : ℝ)) (q :=
+      (1466041197994437 / 2000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_645_4 :
+    Uω (aρ 4) (bρ 4) (746637295669 / 1000000000000 : ℝ) ≤
+      (-(1607252284779575578089 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (580077241951651 / 800000000000000 : ℝ) ≤
+      (-(1607252284779575578089 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (580077241951651 / 800000000000000 : ℝ) =
+            (580077241951651 / 400000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (580077241951651 / 400000000000000 : ℝ) ≤
+            (1858483616720424421911 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (289980584404451 / 400000000000000 : ℝ)) (q :=
+      (580077241951651 / 800000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_645_5 :
+    Uω (aρ 5) (bρ 5) (746637295669 / 1000000000000 : ℝ) ≤
+      (-(3382048462402861285543 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (142609841460427 / 200000000000000 : ℝ) ≤
+      (-(3382048462402861285543 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (142609841460427 / 200000000000000 : ℝ) =
+            (142609841460427 / 100000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (142609841460427 / 100000000000000 : ℝ) ≤
+            (3549423340597138714457 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (71267576266977 / 100000000000000 : ℝ)) (q :=
+      (142609841460427 / 200000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_645_6 :
+    Uω (aρ 6) (bρ 6) (746637295669 / 1000000000000 : ℝ) ≤
+      (-(3626689334411872943689 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3479083784246109 / 5000000000000000 : ℝ) ≤
+      (-(3626689334411872943689 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (3479083784246109 / 5000000000000000 : ℝ) =
+            (3479083784246109 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3479083784246109 / 2500000000000000 : ℝ) ≤
+            (3304782468588127056311 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1737337248194859 / 2500000000000000 : ℝ)) (q :=
+      (3479083784246109 / 5000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_645_7 :
+    Uω (aρ 7) (bρ 7) (746637295669 / 1000000000000 : ℝ) ≤
+      (-(496120014692653319329 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (420252464955703 / 625000000000000 : ℝ) ≤
+      (-(496120014692653319329 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (420252464955703 / 625000000000000 : ℝ) =
+            (420252464955703 / 312500000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (420252464955703 / 312500000000000 : ℝ) ≤
+            (370313960682346680671 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (838094264392187 / 1250000000000000 : ℝ)) (q :=
+      (420252464955703 / 625000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_645_8 :
+    Uω (aρ 8) (bρ 8) (746637295669 / 1000000000000 : ℝ) ≤
+      (-(4430896251256838405569 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6420496605605981 / 10000000000000000 : ℝ) ≤
+      (-(4430896251256838405569 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (6420496605605981 / 10000000000000000 : ℝ) =
+            (6420496605605981 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6420496605605981 / 5000000000000000 : ℝ) ≤
+            (2500575551743161594431 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3190605049553481 / 5000000000000000 : ℝ)) (q :=
+      (6420496605605981 / 10000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_645_9 :
+    Uω (aρ 9) (bρ 9) (746637295669 / 1000000000000 : ℝ) ≤
+      (-(2517687802170217421033 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6043888113840151 / 10000000000000000 : ℝ) ≤
+      (-(2517687802170217421033 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (6043888113840151 / 10000000000000000 : ℝ) =
+            (6043888113840151 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6043888113840151 / 5000000000000000 : ℝ) ≤
+            (948048099329782578967 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2984479371967651 / 5000000000000000 : ℝ)) (q :=
+      (6043888113840151 / 10000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_645_10 :
+    Uω (aρ 10) (bρ 10) (746637295669 / 1000000000000 : ℝ) ≤
+      (-(2902837945202620132743 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2797903322069447 / 5000000000000000 : ℝ) ≤
+      (-(2902837945202620132743 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2797903322069447 / 5000000000000000 : ℝ) =
+            (2797903322069447 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2797903322069447 / 2500000000000000 : ℝ) ≤
+            (562897956297379867257 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1365338709506947 / 2500000000000000 : ℝ)) (q :=
+      (2797903322069447 / 5000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_645_11 :
+    Uω (aρ 11) (bρ 11) (746637295669 / 1000000000000 : ℝ) ≤
+      (-(1691350707796066750917 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1016745558211633 / 2000000000000000 : ℝ) ≤
+      (-(1691350707796066750917 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1016745558211633 / 2000000000000000 : ℝ) =
+            (1016745558211633 / 1000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1016745558211633 / 1000000000000000 : ℝ) ≤
+            (41517242953933249083 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (485567375807133 / 1000000000000000 : ℝ)) (q :=
+      (1016745558211633 / 2000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_645_12 :
+    Uω (aρ 12) (bρ 12) (746637295669 / 1000000000000 : ℝ) ≤
+      (-(3969546476480306766289 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9041480903424091 / 20000000000000000 : ℝ) ≤
+      (-(3969546476480306766289 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (9041480903424091 / 20000000000000000 : ℝ) =
+            (9041480903424091 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9041480903424091 / 5000000000000000 : ℝ) ≤
+            (2961925326519693233711 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4153399176064091 / 10000000000000000 : ℝ)) (q :=
+      (9041480903424091 / 20000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_645_13 :
+    Uω (aρ 13) (bρ 13) (746637295669 / 1000000000000 : ℝ) ≤
+      (-(935433640031612922849 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1569662645146701 / 4000000000000000 : ℝ) ≤
+      (-(935433640031612922849 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1569662645146701 / 4000000000000000 : ℝ) =
+            (1569662645146701 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1569662645146701 / 1000000000000000 : ℝ) ≤
+            (450860720568387077151 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (671922647993701 / 2000000000000000 : ℝ)) (q :=
+      (1569662645146701 / 4000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_645_14 :
+    Uω (aρ 14) (bρ 14) (746637295669 / 1000000000000 : ℝ) ≤
+      (-(1104887668831840048521 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3312480894648137 / 10000000000000000 : ℝ) ≤
+      (-(1104887668831840048521 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3312480894648137 / 10000000000000000 : ℝ) =
+            (3312480894648137 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3312480894648137 / 2500000000000000 : ℝ) ≤
+            (281406691768159951479 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1240094544288137 / 5000000000000000 : ℝ)) (q :=
+      (3312480894648137 / 10000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_645_15 :
+    Uω (aρ 15) (bρ 15) (746637295669 / 1000000000000 : ℝ) ≤
+      (-(6552593809832138435991 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (67420030040337 / 250000000000000 : ℝ) ≤
+      (-(6552593809832138435991 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (67420030040337 / 250000000000000 : ℝ) = (67420030040337 / 62500000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (67420030040337 / 62500000000000 : ℝ) ≤
+            (378877993167861564009 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (37710074498049 / 250000000000000 : ℝ)) (q :=
+      (67420030040337 / 250000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15]) (by norm_num [bρ_15])
+      (by norm_num) (by norm_num [aρ_15, bρ_15]) (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_645_16 :
+    Uω (aρ 16) (bρ 16) (746637295669 / 1000000000000 : ℝ) ≤
+      (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+  by rw [Uω_of_mem (by norm_num [aρ_16]) (by norm_num [bρ_16])]; exact Lmid_16
+
+lemma U_645 :
+    Uρ (746637295669 / 1000000000000 : ℝ) ≤
+      (-(5263357591767758389913 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (746637295669 / 1000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1504272986350256580489 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_645_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (746637295669 / 1000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3040875180557327393003 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_645_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (746637295669 / 1000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(388226843643701977613 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_645_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (746637295669 / 1000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1607252284779575578089 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_645_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (746637295669 / 1000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(3382048462402861285543 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_645_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (746637295669 / 1000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3626689334411872943689 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_645_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (746637295669 / 1000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(496120014692653319329 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_645_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (746637295669 / 1000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(4430896251256838405569 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_645_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (746637295669 / 1000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(2517687802170217421033 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_645_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (746637295669 / 1000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(2902837945202620132743 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_645_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (746637295669 / 1000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(1691350707796066750917 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_645_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (746637295669 / 1000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(3969546476480306766289 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_645_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (746637295669 / 1000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(935433640031612922849 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_645_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (746637295669 / 1000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(1104887668831840048521 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_645_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (746637295669 / 1000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(6552593809832138435991 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_645_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (746637295669 / 1000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(419641552645073499393 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_645_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_646_1 :
+    Uω (aρ 1) (bρ 1) (765809953469387 / 1024000000000000 : ℝ) ≤
+      (-(2992023359028118615229 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (949004006820246619 / 1280000000000000000 : ℝ) ≤
+      (-(2992023359028118615229 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (949004006820246619 / 1280000000000000000 : ℝ) =
+            (949004006820246619 / 640000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (949004006820246619 / 640000000000000000 : ℝ) ≤
+            (3939448443971881384771 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7414071997801871 / 10000000000000000 : ℝ)) (q :=
+      (949004006820246619 / 1280000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_646_2 :
+    Uω (aρ 2) (bρ 2) (765809953469387 / 1024000000000000 : ℝ) ≤
+      (-(3024298834984676636863 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (37837839552948499 / 51200000000000000 : ℝ) ≤
+      (-(3024298834984676636863 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (37837839552948499 / 51200000000000000 : ℝ) =
+            (37837839552948499 / 25600000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (37837839552948499 / 25600000000000000 : ℝ) ≤
+            (3907172968015323363137 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (295602379164641 / 400000000000000 : ℝ)) (q :=
+      (37837839552948499 / 51200000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_646_3 :
+    Uω (aρ 3) (bρ 3) (765809953469387 / 1024000000000000 : ℝ) ≤
+      (-(3089129675859665067647 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (187966635556047583 / 256000000000000000 : ℝ) ≤
+      (-(3089129675859665067647 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (187966635556047583 / 256000000000000000 : ℝ) =
+            (187966635556047583 / 128000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (187966635556047583 / 128000000000000000 : ℝ) ≤
+            (3842342127140334932353 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1468388582582347 / 2000000000000000 : ℝ)) (q :=
+      (187966635556047583 / 256000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_646_4 :
+    Uω (aρ 4) (bρ 4) (765809953469387 / 1024000000000000 : ℝ) ≤
+      (-(3197635102305245452937 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (929690603533680987 / 1280000000000000000 : ℝ) ≤
+      (-(3197635102305245452937 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (929690603533680987 / 1280000000000000000 : ℝ) =
+            (929690603533680987 / 640000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (929690603533680987 / 640000000000000000 : ℝ) ≤
+            (3733836700694754547063 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7261759371374283 / 10000000000000000 : ℝ)) (q :=
+      (929690603533680987 / 1280000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_646_5 :
+    Uω (aρ 5) (bρ 5) (765809953469387 / 1024000000000000 : ℝ) ≤
+      (-(3364888680715927873629 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (914270508277011163 / 1280000000000000000 : ℝ) ≤
+      (-(3364888680715927873629 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (914270508277011163 / 1280000000000000000 : ℝ) =
+            (914270508277011163 / 640000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (914270508277011163 / 640000000000000000 : ℝ) ≤
+            (3566583122284072126371 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7139010302323817 / 10000000000000000 : ℝ)) (q :=
+      (914270508277011163 / 1280000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_646_6 :
+    Uω (aρ 6) (bρ 6) (765809953469387 / 1024000000000000 : ℝ) ≤
+      (-(902272973128295129753 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (892214136760683483 / 1280000000000000000 : ℝ) ≤
+      (-(902272973128295129753 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (892214136760683483 / 1280000000000000000 : ℝ) =
+            (892214136760683483 / 640000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (892214136760683483 / 640000000000000000 : ℝ) ≤
+            (830594977621704870247 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6961619872521197 / 10000000000000000 : ℝ)) (q :=
+      (892214136760683483 / 1280000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_646_7 :
+    Uω (aρ 7) (bρ 7) (765809953469387 / 1024000000000000 : ℝ) ≤
+      (-(3950721346605848420679 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (172449649957237919 / 256000000000000000 : ℝ) ≤
+      (-(3950721346605848420679 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (172449649957237919 / 256000000000000000 : ℝ) =
+            (172449649957237919 / 128000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (172449649957237919 / 128000000000000000 : ℝ) ≤
+            (2980750456394151579321 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (671706426930473 / 1000000000000000 : ℝ)) (q :=
+      (172449649957237919 / 256000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_646_8 :
+    Uω (aρ 8) (bρ 8) (765809953469387 / 1024000000000000 : ℝ) ≤
+      (-(1102933434446520637007 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (823399895872270299 / 1280000000000000000 : ℝ) ≤
+      (-(1102933434446520637007 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (823399895872270299 / 1280000000000000000 : ℝ) =
+            (823399895872270299 / 640000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (823399895872270299 / 640000000000000000 : ℝ) ≤
+            (629934516303479362993 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6393600390739741 / 10000000000000000 : ℝ)) (q :=
+      (823399895872270299 / 1280000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8])
+      (by norm_num [bρ_8]) (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_646_9 :
+    Uω (aρ 9) (bρ 9) (765809953469387 / 1024000000000000 : ℝ) ≤
+      (-(5014891254244500333307 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (775204008300344347 / 1280000000000000000 : ℝ) ≤
+      (-(5014891254244500333307 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (775204008300344347 / 1280000000000000000 : ℝ) =
+            (775204008300344347 / 640000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (775204008300344347 / 640000000000000000 : ℝ) ≤
+            (1916580548755499666693 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2990752637894199 / 5000000000000000 : ℝ)) (q :=
+      (775204008300344347 / 1280000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9])
+      (by norm_num [bρ_9]) (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_646_10 :
+    Uω (aρ 10) (bρ 10) (765809953469387 / 1024000000000000 : ℝ) ≤
+      (-(2891645207438468629941 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28714737430711059 / 51200000000000000 : ℝ) ≤
+      (-(2891645207438468629941 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (28714737430711059 / 51200000000000000 : ℝ) =
+            (28714737430711059 / 25600000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28714737430711059 / 25600000000000000 : ℝ) ≤
+            (574090694061531370059 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (218967839538481 / 400000000000000 : ℝ)) (q :=
+      (28714737430711059 / 51200000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_646_11 :
+    Uω (aρ 11) (bρ 11) (765809953469387 / 1024000000000000 : ℝ) ≤
+      (-(6740230160113358478249 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (652357249560574811 / 1280000000000000000 : ℝ) ≤
+      (-(6740230160113358478249 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (652357249560574811 / 1280000000000000000 : ℝ) =
+            (652357249560574811 / 640000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (652357249560574811 / 640000000000000000 : ℝ) ≤
+            (191241642886641521751 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4869060330179499 / 10000000000000000 : ℝ)) (q :=
+      (652357249560574811 / 1280000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_646_12 :
+    Uω (aρ 12) (bρ 12) (765809953469387 / 1024000000000000 : ℝ) ≤
+      (-(3954837198590898932649 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (580359603050208539 / 1280000000000000000 : ℝ) ≤
+      (-(3954837198590898932649 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (580359603050208539 / 1280000000000000000 : ℝ) =
+            (580359603050208539 / 320000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (580359603050208539 / 320000000000000000 : ℝ) ≤
+            (2976634604409101067351 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2083898600070013 / 5000000000000000 : ℝ)) (q :=
+      (580359603050208539 / 1280000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_646_13 :
+    Uω (aρ 13) (bρ 13) (765809953469387 / 1024000000000000 : ℝ) ≤
+      (-(9317992342333403447889 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (100824180191920607 / 256000000000000000 : ℝ) ≤
+      (-(9317992342333403447889 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (100824180191920607 / 256000000000000000 : ℝ) =
+            (100824180191920607 / 64000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (100824180191920607 / 64000000000000000 : ℝ) ≤
+            (4544951263666596552111 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (675189844313863 / 2000000000000000 : ℝ)) (q :=
+      (100824180191920607 / 256000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_646_14 :
+    Uω (aρ 14) (bρ 14) (765809953469387 / 1024000000000000 : ℝ) ≤
+      (-(10999728173947622292111 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (85217313779522783 / 256000000000000000 : ℝ) ≤
+      (-(10999728173947622292111 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (85217313779522783 / 256000000000000000 : ℝ) =
+            (85217313779522783 / 64000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (85217313779522783 / 64000000000000000 : ℝ) ≤
+            (2863215432052377707889 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (500118013629147 / 2000000000000000 : ℝ)) (q :=
+      (85217313779522783 / 256000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_646_15 :
+    Uω (aρ 15) (bρ 15) (765809953469387 / 1024000000000000 : ℝ) ≤
+      (-(3256218652869368476347 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (347974045552011739 / 1280000000000000000 : ℝ) ≤
+      (-(3256218652869368476347 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (347974045552011739 / 1280000000000000000 : ℝ) =
+            (347974045552011739 / 320000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (347974045552011739 / 320000000000000000 : ℝ) ≤
+            (209517248630631523653 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1539655168285701 / 10000000000000000 : ℝ)) (q :=
+      (347974045552011739 / 1280000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_646_16 :
+    Uω (aρ 16) (bρ 16) (765809953469387 / 1024000000000000 : ℝ) ≤
+      (-(3195214162855050481337 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (259046679374740763 / 1280000000000000000 : ℝ) ≤
+      (-(3195214162855050481337 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (259046679374740763 / 1280000000000000000 : ℝ) =
+            (259046679374740763 / 160000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (259046679374740763 / 160000000000000000 : ℝ) ≤
+            (963668918944949518663 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (151268361637921 / 5000000000000000 : ℝ)) (q :=
+      (259046679374740763 / 1280000000000000000 : ℝ)) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [bρ_16]) (by norm_num) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [aρ_16, bρ_16]) hlg
+
+lemma U_646 :
+    Uρ (765809953469387 / 1024000000000000 : ℝ) ≤
+      (-(2617905894506130104443 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (765809953469387 / 1024000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2992023359028118615229 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_646_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) *
+        Uω (aρ 2) (bρ 2) (765809953469387 / 1024000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3024298834984676636863 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_646_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) *
+        Uω (aρ 3) (bρ 3) (765809953469387 / 1024000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3089129675859665067647 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_646_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (765809953469387 / 1024000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3197635102305245452937 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_646_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (765809953469387 / 1024000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(3364888680715927873629 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_646_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) *
+        Uω (aρ 6) (bρ 6) (765809953469387 / 1024000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(902272973128295129753 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_646_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) *
+        Uω (aρ 7) (bρ 7) (765809953469387 / 1024000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(3950721346605848420679 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_646_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) *
+        Uω (aρ 8) (bρ 8) (765809953469387 / 1024000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(1102933434446520637007 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_646_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (765809953469387 / 1024000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(5014891254244500333307 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_646_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) *
+        Uω (aρ 10) (bρ 10) (765809953469387 / 1024000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(2891645207438468629941 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_646_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) *
+        Uω (aρ 11) (bρ 11) (765809953469387 / 1024000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(6740230160113358478249 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_646_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) *
+        Uω (aρ 12) (bρ 12) (765809953469387 / 1024000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(3954837198590898932649 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_646_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) *
+        Uω (aρ 13) (bρ 13) (765809953469387 / 1024000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(9317992342333403447889 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_646_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) *
+        Uω (aρ 14) (bρ 14) (765809953469387 / 1024000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(10999728173947622292111 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_646_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (765809953469387 / 1024000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(3256218652869368476347 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_646_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) *
+        Uω (aρ 16) (bρ 16) (765809953469387 / 1024000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) *
+        (-(3195214162855050481337 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_646_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_647_1 :
+    Uω (aρ 1) (bρ 1) (383531658086859 / 512000000000000 : ℝ) ≤
+      (-(2975528000166417994777 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (475285357400443739 / 640000000000000000 : ℝ) ≤
+      (-(2975528000166417994777 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (475285357400443739 / 640000000000000000 : ℝ) =
+            (475285357400443739 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (475285357400443739 / 320000000000000000 : ℝ) ≤
+            (3955943802833582005223 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3713155969919951 / 5000000000000000 : ℝ)) (q :=
+      (475285357400443739 / 640000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_647_2 :
+    Uω (aρ 2) (bρ 2) (383531658086859 / 512000000000000 : ℝ) ≤
+      (-(3007749922533249645921 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (473756361294213403 / 640000000000000000 : ℝ) ≤
+      (-(3007749922533249645921 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (473756361294213403 / 640000000000000000 : ℝ) =
+            (473756361294213403 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (473756361294213403 / 320000000000000000 : ℝ) ≤
+            (3923721880466750354079 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1850574956007551 / 2500000000000000 : ℝ)) (q :=
+      (473756361294213403 / 640000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_647_3 :
+    Uω (aρ 3) (bρ 3) (383531658086859 / 512000000000000 : ℝ) ≤
+      (-(3072472399178550156159 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (470699994242783899 / 640000000000000000 : ℝ) ≤
+      (-(3072472399178550156159 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (470699994242783899 / 640000000000000000 : ℝ) =
+            (470699994242783899 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (470699994242783899 / 320000000000000000 : ℝ) ≤
+            (3858999403821449843841 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (919273057502879 / 1250000000000000 : ℝ)) (q :=
+      (470699994242783899 / 640000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_647_4 :
+    Uω (aρ 4) (bρ 4) (383531658086859 / 512000000000000 : ℝ) ≤
+      (-(3180794056339040181553 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (465628809445522491 / 640000000000000000 : ℝ) ≤
+      (-(3180794056339040181553 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (465628809445522491 / 640000000000000000 : ℝ) =
+            (465628809445522491 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (465628809445522491 / 320000000000000000 : ℝ) ≤
+            (3750677746660959818447 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7274004116173613 / 10000000000000000 : ℝ)) (q :=
+      (465628809445522491 / 640000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_647_5 :
+    Uω (aρ 5) (bρ 5) (383531658086859 / 512000000000000 : ℝ) ≤
+      (-(1673879162526766033253 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (457919014200195963 / 640000000000000000 : ℝ) ≤
+      (-(1673879162526766033253 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (457919014200195963 / 640000000000000000 : ℝ) =
+            (457919014200195963 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (457919014200195963 / 320000000000000000 : ℝ) ≤
+            (1791856738973233966747 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7151262934092159 / 10000000000000000 : ℝ)) (q :=
+      (457919014200195963 / 640000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_647_6 :
+    Uω (aρ 6) (bρ 6) (383531658086859 / 512000000000000 : ℝ) ≤
+      (-(1795762720630881838427 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (446891408889576027 / 640000000000000000 : ℝ) ≤
+      (-(1795762720630881838427 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (446891408889576027 / 640000000000000000 : ℝ) =
+            (446891408889576027 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (446891408889576027 / 320000000000000000 : ℝ) ≤
+            (1669973180869118161573 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3486945321637643 / 5000000000000000 : ℝ)) (q :=
+      (446891408889576027 / 640000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_647_7 :
+    Uω (aρ 7) (bρ 7) (383531658086859 / 512000000000000 : ℝ) ≤
+      (-(983128992591838766227 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (431909717466452347 / 640000000000000000 : ℝ) ≤
+      (-(983128992591838766227 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (431909717466452347 / 640000000000000000 : ℝ) =
+            (431909717466452347 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (431909717466452347 / 320000000000000000 : ℝ) ≤
+            (749738958158161233773 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6729374167062671 / 10000000000000000 : ℝ)) (q :=
+      (431909717466452347 / 640000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_647_8 :
+    Uω (aρ 8) (bρ 8) (383531658086859 / 512000000000000 : ℝ) ≤
+      (-(4392608322960627458259 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (412488094606284123 / 640000000000000000 : ℝ) ≤
+      (-(4392608322960627458259 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (412488094606284123 / 640000000000000000 : ℝ) =
+            (412488094606284123 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (412488094606284123 / 320000000000000000 : ℝ) ≤
+            (2538863480039372541741 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3202995052011207 / 5000000000000000 : ℝ)) (q :=
+      (412488094606284123 / 640000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8])
+      (by norm_num [bρ_8]) (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_647_9 :
+    Uω (aρ 9) (bρ 9) (383531658086859 / 512000000000000 : ℝ) ≤
+      (-(998889964806808706123 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (388395128435461531 / 640000000000000000 : ℝ) ≤
+      (-(998889964806808706123 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (388395128435461531 / 640000000000000000 : ℝ) =
+            (388395128435461531 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (388395128435461531 / 320000000000000000 : ℝ) ≤
+            (387404395793191293877 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (374628158721513 / 625000000000000 : ℝ)) (q :=
+      (388395128435461531 / 640000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9])
+      (by norm_num [bρ_9]) (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_647_10 :
+    Uω (aρ 10) (bρ 10) (383531658086859 / 512000000000000 : ℝ) ≤
+      (-(5760957383230505686231 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (359736722593189947 / 640000000000000000 : ℝ) ≤
+      (-(5760957383230505686231 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (359736722593189947 / 640000000000000000 : ℝ) =
+            (359736722593189947 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (359736722593189947 / 320000000000000000 : ℝ) ≤
+            (1170514419769494313769 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5487034390468221 / 10000000000000000 : ℝ)) (q :=
+      (359736722593189947 / 640000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_647_11 :
+    Uω (aρ 11) (bρ 11) (383531658086859 / 512000000000000 : ℝ) ≤
+      (-(3357563293086641684171 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (326998478334935771 / 640000000000000000 : ℝ) ≤
+      (-(3357563293086641684171 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (326998478334935771 / 640000000000000000 : ℝ) =
+            (326998478334935771 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (326998478334935771 / 320000000000000000 : ℝ) ≤
+            (108172608413358315829 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2441220441801389 / 5000000000000000 : ℝ)) (q :=
+      (326998478334935771 / 640000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_647_12 :
+    Uω (aρ 12) (bρ 12) (383531658086859 / 512000000000000 : ℝ) ≤
+      (-(1970089311762010817417 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (58206354853245951 / 128000000000000000 : ℝ) ≤
+      (-(1970089311762010817417 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (58206354853245951 / 128000000000000000 : ℝ) =
+            (58206354853245951 / 32000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (58206354853245951 / 32000000000000000 : ℝ) ≤
+            (1495646589737989182583 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (836436295628143 / 2000000000000000 : ℝ)) (q :=
+      (58206354853245951 / 128000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_647_13 :
+    Uω (aρ 13) (bρ 13) (383531658086859 / 512000000000000 : ℝ) ≤
+      (-(1856364708443179756327 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (252973773579265531 / 640000000000000000 : ℝ) ≤
+      (-(1856364708443179756327 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (252973773579265531 / 640000000000000000 : ℝ) =
+            (252973773579265531 / 160000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (252973773579265531 / 160000000000000000 : ℝ) ≤
+            (916224012756820243673 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3392250698268083 / 10000000000000000 : ℝ)) (q :=
+      (252973773579265531 / 640000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_647_14 :
+    Uω (aρ 14) (bρ 14) (383531658086859 / 512000000000000 : ℝ) ≤
+      (-(10950977982772224991237 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (42816882045091679 / 128000000000000000 : ℝ) ≤
+      (-(10950977982772224991237 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (42816882045091679 / 128000000000000000 : ℝ) =
+            (42816882045091679 / 32000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (42816882045091679 / 32000000000000000 : ℝ) ≤
+            (2911965623227775008763 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (252088537850661 / 1000000000000000 : ℝ)) (q :=
+      (42816882045091679 / 128000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_647_15 :
+    Uω (aρ 15) (bρ 15) (383531658086859 / 512000000000000 : ℝ) ≤
+      (-(80913513290171031217 / 62500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7014476874257139 / 25600000000000000 : ℝ) ≤
+      (-(80913513290171031217 / 62500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7014476874257139 / 25600000000000000 : ℝ) =
+            (7014476874257139 / 6400000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7014476874257139 / 6400000000000000 : ℝ) ≤
+            (5729884247328968783 / 62500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (62815235775577 / 400000000000000 : ℝ)) (q :=
+      (7014476874257139 / 25600000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_647_16 :
+    Uω (aρ 16) (bρ 16) (383531658086859 / 512000000000000 : ℝ) ≤
+      (-(7820519683689689857467 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5357451598432147 / 25600000000000000 : ℝ) ≤
+      (-(7820519683689689857467 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (5357451598432147 / 25600000000000000 : ℝ) =
+            (5357451598432147 / 3200000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5357451598432147 / 3200000000000000 : ℝ) ≤
+            (2576688020810310142533 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (8564030983223 / 200000000000000 : ℝ)) (q :=
+      (5357451598432147 / 25600000000000000 : ℝ)) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [bρ_16]) (by norm_num) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [aρ_16, bρ_16]) hlg
+
+lemma U_647 :
+    Uρ (383531658086859 / 512000000000000 : ℝ) ≤
+      (-(5211206336518075658923 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (383531658086859 / 512000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2975528000166417994777 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_647_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (383531658086859 / 512000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(3007749922533249645921 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_647_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (383531658086859 / 512000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3072472399178550156159 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_647_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (383531658086859 / 512000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3180794056339040181553 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_647_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (383531658086859 / 512000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1673879162526766033253 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_647_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (383531658086859 / 512000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1795762720630881838427 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_647_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (383531658086859 / 512000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(983128992591838766227 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_647_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (383531658086859 / 512000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(4392608322960627458259 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_647_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (383531658086859 / 512000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(998889964806808706123 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_647_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) *
+        Uω (aρ 10) (bρ 10) (383531658086859 / 512000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(5760957383230505686231 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_647_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) *
+        Uω (aρ 11) (bρ 11) (383531658086859 / 512000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(3357563293086641684171 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_647_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) *
+        Uω (aρ 12) (bρ 12) (383531658086859 / 512000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(1970089311762010817417 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_647_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) *
+        Uω (aρ 13) (bρ 13) (383531658086859 / 512000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(1856364708443179756327 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_647_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) *
+        Uω (aρ 14) (bρ 14) (383531658086859 / 512000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(10950977982772224991237 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_647_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (383531658086859 / 512000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) * (-(80913513290171031217 / 62500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_647_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) *
+        Uω (aρ 16) (bρ 16) (383531658086859 / 512000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) *
+        (-(7820519683689689857467 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_647_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_648_1 :
+    Uω (aρ 1) (bρ 1) (768316678878049 / 1024000000000000 : ℝ) ≤
+      (-(1479529903173407912513 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (190427484553277853 / 256000000000000000 : ℝ) ≤
+      (-(1479529903173407912513 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (190427484553277853 / 256000000000000000 : ℝ) =
+            (190427484553277853 / 128000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (190427484553277853 / 128000000000000000 : ℝ) ≤
+            (1986205998326592087487 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1487710376328277 / 2000000000000000 : ℝ)) (q :=
+      (190427484553277853 / 256000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_648_2 :
+    Uω (aρ 2) (bρ 2) (768316678878049 / 1024000000000000 : ℝ) ≤
+      (-(1495614176274166129169 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (949079456252822801 / 1280000000000000000 : ℝ) ≤
+      (-(1495614176274166129169 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (949079456252822801 / 1280000000000000000 : ℝ) =
+            (949079456252822801 / 640000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (949079456252822801 / 640000000000000000 : ℝ) ≤
+            (1970121725225833870831 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7414540167376909 / 10000000000000000 : ℝ)) (q :=
+      (949079456252822801 / 1280000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_648_3 :
+    Uω (aρ 3) (bρ 3) (768316678878049 / 1024000000000000 : ℝ) ≤
+      (-(1527921413315886781817 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (942966798834216401 / 1280000000000000000 : ℝ) ≤
+      (-(1527921413315886781817 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (942966798834216401 / 1280000000000000000 : ℝ) =
+            (942966798834216401 / 640000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (942966798834216401 / 640000000000000000 : ℝ) ≤
+            (1937814488184113218183 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (230200812548787 / 312500000000000 : ℝ)) (q :=
+      (942966798834216401 / 1280000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_648_4 :
+    Uω (aρ 4) (bρ 4) (768316678878049 / 1024000000000000 : ℝ) ≤
+      (-(1581990668008304209843 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (932824633200045713 / 1280000000000000000 : ℝ) ≤
+      (-(1581990668008304209843 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (932824633200045713 / 1280000000000000000 : ℝ) =
+            (932824633200045713 / 640000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (932824633200045713 / 640000000000000000 : ℝ) ≤
+            (1883745233491695790157 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7286248844592267 / 10000000000000000 : ℝ)) (q :=
+      (932824633200045713 / 1280000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_648_5 :
+    Uω (aρ 5) (bρ 5) (768316678878049 / 1024000000000000 : ℝ) ≤
+      (-(3330657294562922625271 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (917405545731278097 / 1280000000000000000 : ℝ) ≤
+      (-(3330657294562922625271 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (917405545731278097 / 1280000000000000000 : ℝ) =
+            (917405545731278097 / 640000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (917405545731278097 / 640000000000000000 : ℝ) ≤
+            (3600814508437077374729 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7163515522227773 / 10000000000000000 : ℝ)) (q :=
+      (917405545731278097 / 1280000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_648_6 :
+    Uω (aρ 6) (bρ 6) (768316678878049 / 1024000000000000 : ℝ) ≤
+      (-(714797974284375681511 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (895351491859164177 / 1280000000000000000 : ℝ) ≤
+      (-(714797974284375681511 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (895351491859164177 / 1280000000000000000 : ℝ) =
+            (895351491859164177 / 640000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (895351491859164177 / 640000000000000000 : ℝ) ≤
+            (671496386315624318489 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6986161305615993 / 10000000000000000 : ℝ)) (q :=
+      (895351491859164177 / 1280000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_648_7 :
+    Uω (aρ 7) (bρ 7) (768316678878049 / 1024000000000000 : ℝ) ≤
+      (-(489292983258718903463 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (865390603759317137 / 1280000000000000000 : ℝ) ≤
+      (-(489292983258718903463 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (865390603759317137 / 1280000000000000000 : ℝ) =
+            (865390603759317137 / 640000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (865390603759317137 / 640000000000000000 : ℝ) ≤
+            (377140992116281096537 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6741683809815883 / 10000000000000000 : ℝ)) (q :=
+      (865390603759317137 / 1280000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_648_8 :
+    Uω (aρ 8) (bρ 8) (768316678878049 / 1024000000000000 : ℝ) ≤
+      (-(4373519861695254145247 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (826552445752811537 / 1280000000000000000 : ℝ) ≤
+      (-(4373519861695254145247 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (826552445752811537 / 1280000000000000000 : ℝ) =
+            (826552445752811537 / 640000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (826552445752811537 / 640000000000000000 : ℝ) ≤
+            (2557951941304745854753 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6418379242304233 / 10000000000000000 : ℝ)) (q :=
+      (826552445752811537 / 1280000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8])
+      (by norm_num [bρ_8]) (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_648_9 :
+    Uω (aρ 9) (bρ 9) (768316678878049 / 1024000000000000 : ℝ) ≤
+      (-(2487025564965771604759 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (778376424791792401 / 1280000000000000000 : ℝ) ≤
+      (-(2487025564965771604759 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (778376424791792401 / 1280000000000000000 : ℝ) =
+            (778376424791792401 / 640000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (778376424791792401 / 640000000000000000 : ℝ) ≤
+            (978710336534228395241 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6006594543148309 / 10000000000000000 : ℝ)) (q :=
+      (778376424791792401 / 1280000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9])
+      (by norm_num [bρ_9]) (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_648_10 :
+    Uω (aρ 10) (bρ 10) (768316678878049 / 1024000000000000 : ℝ) ≤
+      (-(1434669134801156281813 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (721078279937399953 / 1280000000000000000 : ℝ) ≤
+      (-(1434669134801156281813 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (721078279937399953 / 1280000000000000000 : ℝ) =
+            (721078279937399953 / 640000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (721078279937399953 / 640000000000000000 : ℝ) ≤
+            (298198815948843718187 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5499870063293427 / 10000000000000000 : ℝ)) (q :=
+      (721078279937399953 / 1280000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_648_11 :
+    Uω (aρ 11) (bρ 11) (768316678878049 / 1024000000000000 : ℝ) ≤
+      (-(3345045850188347560643 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (655636281741624337 / 1280000000000000000 : ℝ) ≤
+      (-(3345045850188347560643 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (655636281741624337 / 1280000000000000000 : ℝ) =
+            (655636281741624337 / 640000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (655636281741624337 / 640000000000000000 : ℝ) ≤
+            (120690051311652439357 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4895815467689433 / 10000000000000000 : ℝ)) (q :=
+      (655636281741624337 / 1280000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_648_12 :
+    Uω (aρ 12) (bρ 12) (768316678878049 / 1024000000000000 : ℝ) ≤
+      (-(245348147203090920197 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (116753324662457309 / 256000000000000000 : ℝ) ≤
+      (-(245348147203090920197 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (116753324662457309 / 256000000000000000 : ℝ) =
+            (116753324662457309 / 64000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (116753324662457309 / 64000000000000000 : ℝ) ≤
+            (187868840484409079803 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (419655215141603 / 1000000000000000 : ℝ)) (q :=
+      (116753324662457309 / 256000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_648_13 :
+    Uω (aρ 13) (bρ 13) (768316678878049 / 1024000000000000 : ℝ) ≤
+      (-(9245827955184049930021 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (507772016727975633 / 1280000000000000000 : ℝ) ≤
+      (-(9245827955184049930021 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (507772016727975633 / 1280000000000000000 : ℝ) =
+            (507772016727975633 / 320000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (507772016727975633 / 320000000000000000 : ℝ) ≤
+            (4617115650815950069979 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (852129541282793 / 2500000000000000 : ℝ)) (q :=
+      (507772016727975633 / 1280000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_648_14 :
+    Uω (aρ 14) (bρ 14) (768316678878049 / 1024000000000000 : ℝ) ≤
+      (-(10902617715487759439317 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (86048894243380189 / 256000000000000000 : ℝ) ≤
+      (-(10902617715487759439317 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (86048894243380189 / 256000000000000000 : ℝ) =
+            (86048894243380189 / 64000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (86048894243380189 / 64000000000000000 : ℝ) ≤
+            (2960325890512240560683 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (254107755156563 / 1000000000000000 : ℝ)) (q :=
+      (86048894243380189 / 256000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_648_15 :
+    Uω (aρ 15) (bρ 15) (768316678878049 / 1024000000000000 : ℝ) ≤
+      (-(12868962730306120814047 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (353441888679318097 / 1280000000000000000 : ℝ) ≤
+      (-(12868962730306120814047 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (353441888679318097 / 1280000000000000000 : ℝ) =
+            (353441888679318097 / 320000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (353441888679318097 / 320000000000000000 : ℝ) ≤
+            (993980875693879185953 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (800305238415449 / 5000000000000000 : ℝ)) (q :=
+      (353441888679318097 / 1280000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_648_16 :
+    Uω (aρ 16) (bρ 16) (768316678878049 / 1024000000000000 : ℝ) ≤
+      (-(3076834627347482851323 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (274842454963529489 / 1280000000000000000 : ℝ) ≤
+      (-(3076834627347482851323 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (274842454963529489 / 1280000000000000000 : ℝ) =
+            (274842454963529489 / 160000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (274842454963529489 / 160000000000000000 : ℝ) ≤
+            (1082048454452517148677 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (524865976531701 / 10000000000000000 : ℝ)) (q :=
+      (274842454963529489 / 1280000000000000000 : ℝ)) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [bρ_16]) (by norm_num) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [aρ_16, bρ_16]) hlg
+
+lemma U_648 :
+    Uρ (768316678878049 / 1024000000000000 : ℝ) ≤
+      (-(1037435107378750881219 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (768316678878049 / 1024000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1479529903173407912513 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_648_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) *
+        Uω (aρ 2) (bρ 2) (768316678878049 / 1024000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1495614176274166129169 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_648_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) *
+        Uω (aρ 3) (bρ 3) (768316678878049 / 1024000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1527921413315886781817 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_648_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (768316678878049 / 1024000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1581990668008304209843 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_648_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (768316678878049 / 1024000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(3330657294562922625271 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_648_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) *
+        Uω (aρ 6) (bρ 6) (768316678878049 / 1024000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(714797974284375681511 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_648_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) *
+        Uω (aρ 7) (bρ 7) (768316678878049 / 1024000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(489292983258718903463 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_648_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) *
+        Uω (aρ 8) (bρ 8) (768316678878049 / 1024000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(4373519861695254145247 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_648_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (768316678878049 / 1024000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(2487025564965771604759 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_648_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) *
+        Uω (aρ 10) (bρ 10) (768316678878049 / 1024000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(1434669134801156281813 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_648_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) *
+        Uω (aρ 11) (bρ 11) (768316678878049 / 1024000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(3345045850188347560643 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_648_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) *
+        Uω (aρ 12) (bρ 12) (768316678878049 / 1024000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(245348147203090920197 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_648_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) *
+        Uω (aρ 13) (bρ 13) (768316678878049 / 1024000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(9245827955184049930021 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_648_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) *
+        Uω (aρ 14) (bρ 14) (768316678878049 / 1024000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(10902617715487759439317 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_648_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (768316678878049 / 1024000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(12868962730306120814047 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_648_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) *
+        Uω (aρ 16) (bρ 16) (768316678878049 / 1024000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) *
+        (-(3076834627347482851323 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_648_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_649_1 :
+    Uω (aρ 1) (bρ 1) (38478502079119 / 51200000000000 : ℝ) ≤
+      (-(1471309344121746862427 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (238426032679206651 / 320000000000000000 : ℝ) ≤
+      (-(1471309344121746862427 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (238426032679206651 / 320000000000000000 : ℝ) =
+            (238426032679206651 / 160000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (238426032679206651 / 160000000000000000 : ℝ) ≤
+            (1994426557378253137573 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3725395911603743 / 5000000000000000 : ℝ)) (q :=
+      (238426032679206651 / 320000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_649_2 :
+    Uω (aρ 2) (bρ 2) (38478502079119 / 51200000000000 : ℝ) ≤
+      (-(2974734034823848649063 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (237661547454349131 / 320000000000000000 : ℝ) ≤
+      (-(2974734034823848649063 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (237661547454349131 / 320000000000000000 : ℝ) =
+            (237661547454349131 / 160000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (237661547454349131 / 160000000000000000 : ℝ) ≤
+            (3956737768176151350937 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7426780509163891 / 10000000000000000 : ℝ)) (q :=
+      (237661547454349131 / 320000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_649_3 :
+    Uω (aρ 3) (bρ 3) (38478502079119 / 51200000000000 : ℝ) ≤
+      (-(3039240866205603229829 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (47226680441397951 / 64000000000000000 : ℝ) ≤
+      (-(3039240866205603229829 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (47226680441397951 / 64000000000000000 : ℝ) =
+            (47226680441397951 / 32000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (47226680441397951 / 32000000000000000 : ℝ) ≤
+            (3892230936794396770171 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (737866753755393 / 1000000000000000 : ℝ)) (q :=
+      (47226680441397951 / 64000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_649_4 :
+    Uω (aρ 4) (bρ 4) (38478502079119 / 51200000000000 : ℝ) ≤
+      (-(1573598423088097864679 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (233597911616489947 / 320000000000000000 : ℝ) ≤
+      (-(1573598423088097864679 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (233597911616489947 / 320000000000000000 : ℝ) =
+            (233597911616489947 / 160000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (233597911616489947 / 160000000000000000 : ℝ) ≤
+            (1892137478411902135321 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1824623389178173 / 2500000000000000 : ℝ)) (q :=
+      (233597911616489947 / 320000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_649_5 :
+    Uω (aρ 5) (bρ 5) (38478502079119 / 51200000000000 : ℝ) ≤
+      (-(82839637222737004547 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (229743265070993531 / 320000000000000000 : ℝ) ≤
+      (-(82839637222737004547 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (229743265070993531 / 320000000000000000 : ℝ) =
+            (229743265070993531 / 160000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (229743265070993531 / 160000000000000000 : ℝ) ≤
+            (90447157852262995453 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3587884033477083 / 5000000000000000 : ℝ)) (q :=
+      (229743265070993531 / 320000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_649_6 :
+    Uω (aρ 6) (bρ 6) (38478502079119 / 51200000000000 : ℝ) ≤
+      (-(3556485074335744005427 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (224230039759304091 / 320000000000000000 : ℝ) ≤
+      (-(3556485074335744005427 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (224230039759304091 / 320000000000000000 : ℝ) =
+            (224230039759304091 / 160000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (224230039759304091 / 160000000000000000 : ℝ) ≤
+            (3374986728664255994573 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (874803982514197 / 1250000000000000 : ℝ)) (q :=
+      (224230039759304091 / 320000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_649_7 :
+    Uω (aρ 7) (bρ 7) (38478502079119 / 51200000000000 : ℝ) ≤
+      (-(3896204911636402043661 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (43348087817733087 / 64000000000000000 : ℝ) ≤
+      (-(3896204911636402043661 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (43348087817733087 / 64000000000000000 : ℝ) =
+            (43348087817733087 / 32000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (43348087817733087 / 32000000000000000 : ℝ) ≤
+            (3035266891363597956339 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (337699659947933 / 500000000000000 : ℝ)) (q :=
+      (43348087817733087 / 64000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_649_8 :
+    Uω (aρ 8) (bρ 8) (38478502079119 / 51200000000000 : ℝ) ≤
+      (-(4354468209763952836931 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (207032166426422683 / 320000000000000000 : ℝ) ≤
+      (-(4354468209763952836931 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (207032166426422683 / 320000000000000000 : ℝ) =
+            (207032166426422683 / 160000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (207032166426422683 / 160000000000000000 : ℝ) ≤
+            (2577003593236047163069 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (803845976113561 / 1250000000000000 : ℝ)) (q :=
+      (207032166426422683 / 320000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8])
+      (by norm_num [bρ_8]) (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_649_9 :
+    Uω (aρ 9) (bρ 9) (38478502079119 / 51200000000000 : ℝ) ≤
+      (-(990738997872693485211 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (194990628141794779 / 320000000000000000 : ℝ) ≤
+      (-(990738997872693485211 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (194990628141794779 / 320000000000000000 : ℝ) =
+            (194990628141794779 / 160000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (194990628141794779 / 160000000000000000 : ℝ) ≤
+            (395555362727306514789 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1504784323619811 / 2500000000000000 : ℝ)) (q :=
+      (194990628141794779 / 320000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9])
+      (by norm_num [bρ_9]) (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_649_10 :
+    Uω (aρ 10) (bρ 10) (38478502079119 / 51200000000000 : ℝ) ≤
+      (-(142911190712137077017 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (180670735310228651 / 320000000000000000 : ℝ) ≤
+      (-(142911190712137077017 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (180670735310228651 / 320000000000000000 : ℝ) =
+            (180670735310228651 / 160000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (180670735310228651 / 160000000000000000 : ℝ) ≤
+            (30375604362862922983 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5512703026001361 / 10000000000000000 : ℝ)) (q :=
+      (180670735310228651 / 320000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_649_11 :
+    Uω (aρ 11) (bρ 11) (38478502079119 / 51200000000000 : ℝ) ≤
+      (-(1666281274396531617289 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (164318806974575771 / 320000000000000000 : ℝ) ≤
+      (-(1666281274396531617289 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (164318806974575771 / 320000000000000000 : ℝ) =
+            (164318806974575771 / 160000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (164318806974575771 / 160000000000000000 : ℝ) ≤
+            (66586676353468382711 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (613648016403507 / 1250000000000000 : ℝ)) (q :=
+      (164318806974575771 / 320000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_649_12 :
+    Uω (aρ 12) (bρ 12) (38478502079119 / 51200000000000 : ℝ) ≤
+      (-(1955506001388691126887 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (29273441815204607 / 64000000000000000 : ℝ) ≤
+      (-(1955506001388691126887 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (29273441815204607 / 64000000000000000 : ℝ) =
+            (29273441815204607 / 16000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (29273441815204607 / 16000000000000000 : ℝ) ≤
+            (1510229900111308873113 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (421090935925351 / 1000000000000000 : ℝ)) (q :=
+      (29273441815204607 / 64000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_649_13 :
+    Uω (aρ 13) (bρ 13) (38478502079119 / 51200000000000 : ℝ) ≤
+      (-(9210003576042587965751 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (127398585171176907 / 320000000000000000 : ℝ) ≤
+      (-(9210003576042587965751 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (127398585171176907 / 320000000000000000 : ℝ) =
+            (127398585171176907 / 80000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (127398585171176907 / 80000000000000000 : ℝ) ≤
+            (4652940029957412034249 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3424752106795627 / 10000000000000000 : ℝ)) (q :=
+      (127398585171176907 / 320000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_649_14 :
+    Uω (aρ 14) (bρ 14) (38478502079119 / 51200000000000 : ℝ) ≤
+      (-(2170927853472496869607 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (108078419329204123 / 320000000000000000 : ℝ) ≤
+      (-(2170927853472496869607 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (108078419329204123 / 320000000000000000 : ℝ) =
+            (108078419329204123 / 80000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (108078419329204123 / 80000000000000000 : ℝ) ≤
+            (601660867727503130393 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (160073064169833 / 625000000000000 : ℝ)) (q :=
+      (108078419329204123 / 320000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_649_15 :
+    Uω (aρ 15) (bρ 15) (38478502079119 / 51200000000000 : ℝ) ≤
+      (-(12793196732551274176083 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17806497336238479 / 64000000000000000 : ℝ) ≤
+      (-(12793196732551274176083 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (17806497336238479 / 64000000000000000 : ℝ) =
+            (17806497336238479 / 16000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17806497336238479 / 16000000000000000 : ℝ) ≤
+            (1069746873448725823917 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (326074302726319 / 2000000000000000 : ℝ)) (q :=
+      (17806497336238479 / 64000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_649_16 :
+    Uω (aρ 16) (bρ 16) (38478502079119 / 51200000000000 : ℝ) ≤
+      (-(758390116236913319073 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (70213511395988331 / 320000000000000000 : ℝ) ≤
+      (-(758390116236913319073 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (70213511395988331 / 320000000000000000 : ℝ) =
+            (70213511395988331 / 40000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (70213511395988331 / 40000000000000000 : ℝ) ≤
+            (281330654213086680927 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (606557209816341 / 10000000000000000 : ℝ)) (q :=
+      (70213511395988331 / 320000000000000000 : ℝ)) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [bρ_16]) (by norm_num) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [aρ_16, bρ_16]) hlg
+
+lemma U_649 :
+    Uρ (38478502079119 / 51200000000000 : ℝ) ≤
+      (-(161359119069458724063 / 312500000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (38478502079119 / 51200000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1471309344121746862427 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_649_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (38478502079119 / 51200000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(2974734034823848649063 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_649_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (38478502079119 / 51200000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(3039240866205603229829 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_649_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (38478502079119 / 51200000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1573598423088097864679 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_649_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (38478502079119 / 51200000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(82839637222737004547 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_649_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (38478502079119 / 51200000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3556485074335744005427 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_649_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (38478502079119 / 51200000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(3896204911636402043661 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_649_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (38478502079119 / 51200000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(4354468209763952836931 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_649_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (38478502079119 / 51200000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(990738997872693485211 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_649_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (38478502079119 / 51200000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) * (-(142911190712137077017 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_649_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (38478502079119 / 51200000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(1666281274396531617289 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_649_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (38478502079119 / 51200000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(1955506001388691126887 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_649_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (38478502079119 / 51200000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(9210003576042587965751 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_649_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (38478502079119 / 51200000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(2170927853472496869607 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_649_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (38478502079119 / 51200000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(12793196732551274176083 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_649_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (38478502079119 / 51200000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(758390116236913319073 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_649_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_650_1 :
+    Uω (aρ 1) (bρ 1) (770823404286711 / 1024000000000000 : ℝ) ≤
+      (-(1463102278485251628801 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (955270838652273607 / 1280000000000000000 : ℝ) ≤
+      (-(1463102278485251628801 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (955270838652273607 / 1280000000000000000 : ℝ) =
+            (955270838652273607 / 640000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (955270838652273607 / 640000000000000000 : ℝ) ≤
+            (2002633623014748371199 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7463031764539363 / 10000000000000000 : ℝ)) (q :=
+      (955270838652273607 / 1280000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_650_2 :
+    Uω (aρ 2) (bρ 2) (770823404286711 / 1024000000000000 : ℝ) ≤
+      (-(1479133439799703354753 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (952212923282640711 / 1280000000000000000 : ℝ) ≤
+      (-(1479133439799703354753 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (952212923282640711 / 1280000000000000000 : ℝ) =
+            (952212923282640711 / 640000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (952212923282640711 / 640000000000000000 : ℝ) ≤
+            (1986602461700296645247 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7439020849398849 / 10000000000000000 : ℝ)) (q :=
+      (952212923282640711 / 1280000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_650_3 :
+    Uω (aρ 3) (bρ 3) (770823404286711 / 1024000000000000 : ℝ) ≤
+      (-(1511333213172008917413 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (946100418468559111 / 1280000000000000000 : ℝ) ≤
+      (-(1511333213172008917413 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (946100418468559111 / 1280000000000000000 : ℝ) =
+            (946100418468559111 / 640000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (946100418468559111 / 640000000000000000 : ℝ) ≤
+            (1954402688327991082587 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (923863633503603 / 1250000000000000 : ℝ)) (q :=
+      (946100418468559111 / 1280000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_650_4 :
+    Uω (aρ 4) (bρ 4) (770823404286711 / 1024000000000000 : ℝ) ≤
+      (-(3130440492134986453193 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (187191731738805659 / 256000000000000000 : ℝ) ≤
+      (-(3130440492134986453193 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (187191731738805659 / 256000000000000000 : ℝ) =
+            (187191731738805659 / 128000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (187191731738805659 / 128000000000000000 : ℝ) ≤
+            (3801031310865013546807 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (365536912630839 / 500000000000000 : ℝ)) (q :=
+      (187191731738805659 / 256000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_650_5 :
+    Uω (aρ 5) (bρ 5) (770823404286711 / 1024000000000000 : ℝ) ≤
+      (-(1648271404136584240029 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (920540572072686919 / 1280000000000000000 : ℝ) ≤
+      (-(1648271404136584240029 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (920540572072686919 / 1280000000000000000 : ℝ) =
+            (920540572072686919 / 640000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (920540572072686919 / 640000000000000000 : ℝ) ≤
+            (1817464497363415759971 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7188020568493321 / 10000000000000000 : ℝ)) (q :=
+      (920540572072686919 / 1280000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_650_6 :
+    Uω (aρ 6) (bρ 6) (770823404286711 / 1024000000000000 : ℝ) ≤
+      (-(1769505470959731984053 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (179697763869909851 / 256000000000000000 : ℝ) ≤
+      (-(1769505470959731984053 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (179697763869909851 / 256000000000000000 : ℝ) =
+            (179697763869909851 / 128000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (179697763869909851 / 128000000000000000 : ℝ) ≤
+            (1696230430540268015947 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1402140461466859 / 2000000000000000 : ℝ)) (q :=
+      (179697763869909851 / 256000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_650_7 :
+    Uω (aρ 7) (bρ 7) (770823404286711 / 1024000000000000 : ℝ) ≤
+      (-(387809898566500223647 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (173706578561504411 / 256000000000000000 : ℝ) ≤
+      (-(387809898566500223647 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (173706578561504411 / 256000000000000000 : ℝ) =
+            (173706578561504411 / 128000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (173706578561504411 / 128000000000000000 : ℝ) ≤
+            (305337281733499776353 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (84578779198439 / 125000000000000 : ℝ)) (q :=
+      (173706578561504411 / 256000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_650_8 :
+    Uω (aρ 8) (bρ 8) (770823404286711 / 1024000000000000 : ℝ) ≤
+      (-(867090644758616515783 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (829704849282250631 / 1280000000000000000 : ℝ) ≤
+      (-(867090644758616515783 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (829704849282250631 / 1280000000000000000 : ℝ) =
+            (829704849282250631 / 640000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (829704849282250631 / 640000000000000000 : ℝ) ≤
+            (519203715841383484217 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3221577903566377 / 5000000000000000 : ℝ)) (q :=
+      (829704849282250631 / 1280000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8])
+      (by norm_num [bρ_8]) (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_650_9 :
+    Uω (aρ 9) (bρ 9) (770823404286711 / 1024000000000000 : ℝ) ≤
+      (-(4933381220949565653991 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (781548520697065799 / 1280000000000000000 : ℝ) ≤
+      (-(4933381220949565653991 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (781548520697065799 / 1280000000000000000 : ℝ) =
+            (781548520697065799 / 640000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (781548520697065799 / 640000000000000000 : ℝ) ≤
+            (1998090582050434346009 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6031678801349241 / 10000000000000000 : ℝ)) (q :=
+      (781548520697065799 / 1280000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9])
+      (by norm_num [bρ_9]) (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_650_10 :
+    Uω (aρ 10) (bρ 10) (770823404286711 / 1024000000000000 : ℝ) ≤
+      (-(2847135199343223354549 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (724287430305411079 / 1280000000000000000 : ℝ) ≤
+      (-(2847135199343223354549 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (724287430305411079 / 1280000000000000000 : ℝ) =
+            (724287430305411079 / 640000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (724287430305411079 / 640000000000000000 : ℝ) ≤
+            (618600702156776645451 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1381383324368659 / 2500000000000000 : ℝ)) (q :=
+      (724287430305411079 / 1280000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_650_11 :
+    Uω (aρ 11) (bρ 11) (770823404286711 / 1024000000000000 : ℝ) ≤
+      (-(1328045275292720239523 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (131782759645416603 / 256000000000000000 : ℝ) ≤
+      (-(1328045275292720239523 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (131782759645416603 / 256000000000000000 : ℝ) =
+            (131782759645416603 / 128000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (131782759645416603 / 128000000000000000 : ℝ) ≤
+            (58249085307279760477 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (61531836530697 / 125000000000000 : ℝ)) (q :=
+      (131782759645416603 / 256000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_650_12 :
+    Uω (aρ 12) (bρ 12) (770823404286711 / 1024000000000000 : ℝ) ≤
+      (-(7793006360136964195467 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (587170196292819527 / 1280000000000000000 : ℝ) ≤
+      (-(7793006360136964195467 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (587170196292819527 / 1280000000000000000 : ℝ) =
+            (587170196292819527 / 320000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (587170196292819527 / 320000000000000000 : ℝ) ≤
+            (6069937245863035804533 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4225253238917893 / 10000000000000000 : ℝ)) (q :=
+      (587170196292819527 / 1280000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_650_13 :
+    Uω (aρ 13) (bρ 13) (770823404286711 / 1024000000000000 : ℝ) ≤
+      (-(9174348438106759670099 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (511414549396843719 / 1280000000000000000 : ℝ) ≤
+      (-(9174348438106759670099 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (511414549396843719 / 1280000000000000000 : ℝ) =
+            (511414549396843719 / 320000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (511414549396843719 / 320000000000000000 : ℝ) ≤
+            (4688595167893240329901 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3440952997763271 / 10000000000000000 : ℝ)) (q :=
+      (511414549396843719 / 1280000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_650_14 :
+    Uω (aρ 14) (bρ 14) (770823404286711 / 1024000000000000 : ℝ) ≤
+      (-(10807034813892791887537 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (86875317848878747 / 256000000000000000 : ℝ) ≤
+      (-(10807034813892791887537 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (86875317848878747 / 256000000000000000 : ℝ) =
+            (86875317848878747 / 64000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (86875317848878747 / 64000000000000000 : ℝ) ≤
+            (3055908792107208112463 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (8066131735707 / 31250000000000 : ℝ)) (q :=
+      (86875317848878747 / 256000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_650_15 :
+    Uω (aρ 15) (bρ 15) (770823404286711 / 1024000000000000 : ℝ) ≤
+      (-(12718791257063904718547 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (358789631010213383 / 1280000000000000000 : ℝ) ≤
+      (-(12718791257063904718547 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (358789631010213383 / 1280000000000000000 : ℝ) =
+            (358789631010213383 / 320000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (358789631010213383 / 320000000000000000 : ℝ) ≤
+            (1144152348936095281453 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (414922302608043 / 2500000000000000 : ℝ)) (q :=
+      (358789631010213383 / 1280000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_650_16 :
+    Uω (aρ 16) (bρ 16) (770823404286711 / 1024000000000000 : ℝ) ≤
+      (-(936083213417966146663 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (57250953781302043 / 256000000000000000 : ℝ) ≤
+      (-(936083213417966146663 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (57250953781302043 / 256000000000000000 : ℝ) =
+            (57250953781302043 / 32000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (57250953781302043 / 32000000000000000 : ℝ) ≤
+            (363567749644533853337 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (67870364157181 / 1000000000000000 : ℝ)) (q :=
+      (57250953781302043 / 256000000000000000 : ℝ)) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [bρ_16]) (by norm_num) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [aρ_16, bρ_16]) hlg
+
+lemma U_650 :
+    Uρ (770823404286711 / 1024000000000000 : ℝ) ≤
+      (-(1028013062888184996239 / 2000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (770823404286711 / 1024000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1463102278485251628801 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_650_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) *
+        Uω (aρ 2) (bρ 2) (770823404286711 / 1024000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1479133439799703354753 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_650_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) *
+        Uω (aρ 3) (bρ 3) (770823404286711 / 1024000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1511333213172008917413 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_650_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (770823404286711 / 1024000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3130440492134986453193 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_650_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (770823404286711 / 1024000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1648271404136584240029 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_650_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) *
+        Uω (aρ 6) (bρ 6) (770823404286711 / 1024000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1769505470959731984053 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_650_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) *
+        Uω (aρ 7) (bρ 7) (770823404286711 / 1024000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(387809898566500223647 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_650_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) *
+        Uω (aρ 8) (bρ 8) (770823404286711 / 1024000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(867090644758616515783 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_650_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (770823404286711 / 1024000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(4933381220949565653991 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_650_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) *
+        Uω (aρ 10) (bρ 10) (770823404286711 / 1024000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(2847135199343223354549 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_650_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) *
+        Uω (aρ 11) (bρ 11) (770823404286711 / 1024000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(1328045275292720239523 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_650_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) *
+        Uω (aρ 12) (bρ 12) (770823404286711 / 1024000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(7793006360136964195467 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_650_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) *
+        Uω (aρ 13) (bρ 13) (770823404286711 / 1024000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(9174348438106759670099 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_650_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) *
+        Uω (aρ 14) (bρ 14) (770823404286711 / 1024000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(10807034813892791887537 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_650_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (770823404286711 / 1024000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(12718791257063904718547 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_650_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) *
+        Uω (aρ 16) (bρ 16) (770823404286711 / 1024000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(936083213417966146663 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_650_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_651_1 :
+    Uω (aρ 1) (bρ 1) (386038383495521 / 512000000000000 : ℝ) ≤
+      (-(1454908662039441301187 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (478418773286401937 / 640000000000000000 : ℝ) ≤
+      (-(1454908662039441301187 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (478418773286401937 / 640000000000000000 : ℝ) =
+            (478418773286401937 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (478418773286401937 / 320000000000000000 : ℝ) ≤
+            (2010827239460558698813 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3737635852819083 / 5000000000000000 : ℝ)) (q :=
+      (478418773286401937 / 640000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_651_2 :
+    Uω (aρ 2) (bρ 2) (386038383495521 / 512000000000000 : ℝ) ≤
+      (-(2941826797557366372611 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (476889828324522417 / 640000000000000000 : ℝ) ≤
+      (-(2941826797557366372611 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (476889828324522417 / 640000000000000000 : ℝ) =
+            (476889828324522417 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (476889828324522417 / 320000000000000000 : ℝ) ≤
+            (3989645005442633627389 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7451261188089431 / 10000000000000000 : ℝ)) (q :=
+      (476889828324522417 / 640000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_651_3 :
+    Uω (aρ 3) (bρ 3) (386038383495521 / 512000000000000 : ℝ) ≤
+      (-(150305970797283732639 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (473833613878884241 / 640000000000000000 : ℝ) ≤
+      (-(150305970797283732639 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (473833613878884241 / 640000000000000000 : ℝ) =
+            (473833613878884241 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (473833613878884241 / 320000000000000000 : ℝ) ≤
+            (196267619352716267361 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3701575296506619 / 5000000000000000 : ℝ)) (q :=
+      (473833613878884241 / 640000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_651_4 :
+    Uω (aρ 4) (bρ 4) (386038383495521 / 512000000000000 : ℝ) ≤
+      (-(3113712179685890930111 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (468762834944728689 / 640000000000000000 : ℝ) ≤
+      (-(3113712179685890930111 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (468762834944728689 / 640000000000000000 : ℝ) =
+            (468762834944728689 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (468762834944728689 / 320000000000000000 : ℝ) ≤
+            (3817759623314109069889 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7322982932385877 / 10000000000000000 : ℝ)) (q :=
+      (468762834944728689 / 640000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_651_5 :
+    Uω (aρ 5) (bρ 5) (386038383495521 / 512000000000000 : ℝ) ≤
+      (-(3279529153345017258651 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (461054040555763377 / 640000000000000000 : ℝ) ≤
+      (-(3279529153345017258651 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (461054040555763377 / 640000000000000000 : ℝ) =
+            (461054040555763377 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (461054040555763377 / 320000000000000000 : ℝ) ≤
+            (3651942649654982741349 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7200273027065711 / 10000000000000000 : ℝ)) (q :=
+      (461054040555763377 / 640000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_651_6 :
+    Uω (aρ 6) (bρ 6) (386038383495521 / 512000000000000 : ℝ) ≤
+      (-(880391841664744773029 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (450028736416075057 / 640000000000000000 : ℝ) ≤
+      (-(880391841664744773029 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (450028736416075057 / 640000000000000000 : ℝ) =
+            (450028736416075057 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (450028736416075057 / 320000000000000000 : ℝ) ≤
+            (852476109085255226971 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7022972647840451 / 10000000000000000 : ℝ)) (q :=
+      (450028736416075057 / 640000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_651_7 :
+    Uω (aρ 7) (bρ 7) (386038383495521 / 512000000000000 : ℝ) ≤
+      (-(772005193484518130031 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17402080264116729 / 25600000000000000 : ℝ) ≤
+      (-(772005193484518130031 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (17402080264116729 / 25600000000000000 : ℝ) =
+            (17402080264116729 / 12800000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17402080264116729 / 12800000000000000 : ℝ) ≤
+            (614289167115481869969 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (67786112219393 / 100000000000000 : ℝ)) (q :=
+      (17402080264116729 / 25600000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_651_8 :
+    Uω (aρ 8) (bρ 8) (386038383495521 / 512000000000000 : ℝ) ≤
+      (-(4316474761254591827999 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (83128099669190653 / 128000000000000000 : ℝ) ≤
+      (-(4316474761254591827999 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (83128099669190653 / 128000000000000000 : ℝ) =
+            (83128099669190653 / 64000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (83128099669190653 / 64000000000000000 : ℝ) ≤
+            (2614997041745408172001 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1291108648049829 / 2000000000000000 : ℝ)) (q :=
+      (83128099669190653 / 128000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8])
+      (by norm_num [bρ_8]) (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_651_9 :
+    Uω (aρ 9) (bρ 9) (386038383495521 / 512000000000000 : ℝ) ≤
+      (-(2456554822246143235107 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (391567224838617969 / 640000000000000000 : ℝ) ≤
+      (-(2456554822246143235107 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (391567224838617969 / 640000000000000000 : ℝ) =
+            (391567224838617969 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (391567224838617969 / 320000000000000000 : ℝ) ≤
+            (1009181079253856764893 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6044219071504917 / 10000000000000000 : ℝ)) (q :=
+      (391567224838617969 / 640000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9])
+      (by norm_num [bρ_9]) (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_651_10 :
+    Uω (aρ 10) (bρ 10) (386038383495521 / 512000000000000 : ℝ) ≤
+      (-(5672144599328013069273 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (362945874163962929 / 640000000000000000 : ℝ) ≤
+      (-(5672144599328013069273 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (362945874163962929 / 640000000000000000 : ℝ) =
+            (362945874163962929 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (362945874163962929 / 320000000000000000 : ℝ) ≤
+            (1259327203671986930727 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5538360896416947 / 10000000000000000 : ℝ)) (q :=
+      (362945874163962929 / 640000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_651_11 :
+    Uω (aρ 11) (bρ 11) (386038383495521 / 512000000000000 : ℝ) ≤
+      (-(66153951394211009971 / 100000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (330275997890179633 / 640000000000000000 : ℝ) ≤
+      (-(66153951394211009971 / 100000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (330275997890179633 / 640000000000000000 : ℝ) =
+            (330275997890179633 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (330275997890179633 / 320000000000000000 : ℝ) ≤
+            (3160766635788990029 / 100000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4935903889066219 / 10000000000000000 : ℝ)) (q :=
+      (330275997890179633 / 640000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_651_12 :
+    Uω (aρ 12) (bρ 12) (386038383495521 / 512000000000000 : ℝ) ≤
+      (-(7764087011892156874073 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (58887071193642237 / 128000000000000000 : ℝ) ≤
+      (-(7764087011892156874073 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (58887071193642237 / 128000000000000000 : ℝ) =
+            (58887071193642237 / 32000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (58887071193642237 / 32000000000000000 : ℝ) ≤
+            (6098856594107843125927 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (847916785137941 / 2000000000000000 : ℝ)) (q :=
+      (58887071193642237 / 128000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_651_13 :
+    Uω (aρ 13) (bρ 13) (386038383495521 / 512000000000000 : ℝ) ≤
+      (-(2284715153041468922039 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (51323267260200061 / 128000000000000000 : ℝ) ≤
+      (-(2284715153041468922039 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (51323267260200061 / 128000000000000000 : ℝ) =
+            (51323267260200061 / 32000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (51323267260200061 / 32000000000000000 : ℝ) ≤
+            (1181020748458531077961 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (691424260536873 / 2000000000000000 : ℝ)) (q :=
+      (51323267260200061 / 128000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_651_14 :
+    Uω (aρ 14) (bρ 14) (386038383495521 / 512000000000000 : ℝ) ≤
+      (-(10759796797346239902829 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (218216676070397777 / 640000000000000000 : ℝ) ≤
+      (-(10759796797346239902829 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (218216676070397777 / 640000000000000000 : ℝ) =
+            (218216676070397777 / 160000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (218216676070397777 / 160000000000000000 : ℝ) ≤
+            (3103146808653760097171 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (650264801380759 / 2500000000000000 : ℝ)) (q :=
+      (218216676070397777 / 640000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_651_15 :
+    Uω (aρ 15) (bρ 15) (386038383495521 / 512000000000000 : ℝ) ≤
+      (-(252913589034511244197 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (180711209722183089 / 640000000000000000 : ℝ) ≤
+      (-(252913589034511244197 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (180711209722183089 / 640000000000000000 : ℝ) =
+            (180711209722183089 / 160000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (180711209722183089 / 160000000000000000 : ℝ) ≤
+            (24345283085488755803 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1688586659556327 / 10000000000000000 : ℝ)) (q :=
+      (180711209722183089 / 640000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_651_16 :
+    Uω (aρ 16) (bρ 16) (386038383495521 / 512000000000000 : ℝ) ≤
+      (-(7402636371640588000721 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (145611323094618033 / 640000000000000000 : ℝ) ≤
+      (-(7402636371640588000721 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (145611323094618033 / 640000000000000000 : ℝ) =
+            (145611323094618033 / 80000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (145611323094618033 / 80000000000000000 : ℝ) ≤
+            (2994571332859411999279 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (744086853954919 / 10000000000000000 : ℝ)) (q :=
+      (145611323094618033 / 640000000000000000 : ℝ)) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [bρ_16]) (by norm_num) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [aρ_16, bρ_16]) hlg
+
+lemma U_651 :
+    Uρ (386038383495521 / 512000000000000 : ℝ) ≤
+      (-(511684876669522631533 / 1000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (386038383495521 / 512000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1454908662039441301187 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_651_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (386038383495521 / 512000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(2941826797557366372611 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_651_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (386038383495521 / 512000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(150305970797283732639 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_651_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (386038383495521 / 512000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3113712179685890930111 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_651_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (386038383495521 / 512000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(3279529153345017258651 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_651_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (386038383495521 / 512000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(880391841664744773029 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_651_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (386038383495521 / 512000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(772005193484518130031 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_651_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (386038383495521 / 512000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(4316474761254591827999 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_651_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (386038383495521 / 512000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(2456554822246143235107 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_651_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) *
+        Uω (aρ 10) (bρ 10) (386038383495521 / 512000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(5672144599328013069273 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_651_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) *
+        Uω (aρ 11) (bρ 11) (386038383495521 / 512000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) * (-(66153951394211009971 / 100000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_651_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) *
+        Uω (aρ 12) (bρ 12) (386038383495521 / 512000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(7764087011892156874073 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_651_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) *
+        Uω (aρ 13) (bρ 13) (386038383495521 / 512000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(2284715153041468922039 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_651_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) *
+        Uω (aρ 14) (bρ 14) (386038383495521 / 512000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(10759796797346239902829 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_651_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (386038383495521 / 512000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(252913589034511244197 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_651_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) *
+        Uω (aρ 16) (bρ 16) (386038383495521 / 512000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) *
+        (-(7402636371640588000721 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_651_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/U54.lean b/LeanPool/Zeta5Irrational/Table/U54.lean
new file mode 100644
index 0000000000..08c6370d38
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/U54.lean
@@ -0,0 +1,5593 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-! # Certified arcsine potential bounds (U54) -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma U_652_1 :
+    Uω (aρ 1) (bρ 1) (773330129695373 / 1024000000000000 : ℝ) ≤
+      (-(2893456901553794808471 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (191680850895698137 / 256000000000000000 : ℝ) ≤
+      (-(2893456901553794808471 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (191680850895698137 / 256000000000000000 : ℝ) =
+            (191680850895698137 / 128000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (191680850895698137 / 128000000000000000 : ℝ) ≤
+            (4038014901446205191529 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (93593895581313 / 125000000000000 : ℝ)) (q :=
+      (191680850895698137 / 256000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_652_2 :
+    Uω (aρ 2) (bρ 2) (773330129695373 / 1024000000000000 : ℝ) ≤
+      (-(1462706849909965999691 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (191069277983419033 / 256000000000000000 : ℝ) ≤
+      (-(1462706849909965999691 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (191069277983419033 / 256000000000000000 : ℝ) =
+            (191069277983419033 / 128000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (191069277983419033 / 128000000000000000 : ℝ) ≤
+            (2003029051590034000309 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1492700305048647 / 2000000000000000 : ℝ)) (q :=
+      (191069277983419033 / 256000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_652_3 :
+    Uω (aρ 3) (bρ 3) (773330129695373 / 1024000000000000 : ℝ) ≤
+      (-(1494799872180451229579 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (189846807339465497 / 256000000000000000 : ℝ) ≤
+      (-(1494799872180451229579 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (189846807339465497 / 256000000000000000 : ℝ) =
+            (189846807339465497 / 128000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (189846807339465497 / 128000000000000000 : ℝ) ≤
+            (1970936029319548770421 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1483078422506873 / 2000000000000000 : ℝ)) (q :=
+      (189846807339465497 / 256000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_652_4 :
+    Uω (aρ 4) (bρ 4) (773330129695373 / 1024000000000000 : ℝ) ≤
+      (-(3097011815094364335053 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (187818536011483673 / 256000000000000000 : ℝ) ≤
+      (-(3097011815094364335053 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (187818536011483673 / 256000000000000000 : ℝ) =
+            (187818536011483673 / 128000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (187818536011483673 / 128000000000000000 : ℝ) ≤
+            (3834459987905635664947 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1467045519220157 / 2000000000000000 : ℝ)) (q :=
+      (187818536011483673 / 256000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_652_5 :
+    Uω (aρ 5) (bρ 5) (773330129695373 / 1024000000000000 : ℝ) ≤
+      (-(3262544425323638530593 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (923675587414507837 / 1280000000000000000 : ℝ) ≤
+      (-(3262544425323638530593 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (923675587414507837 / 1280000000000000000 : ℝ) =
+            (923675587414507837 / 640000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (923675587414507837 / 640000000000000000 : ℝ) ≤
+            (3668927377676361469407 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1803131360722577 / 2500000000000000 : ℝ)) (q :=
+      (923675587414507837 / 1280000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_652_6 :
+    Uω (aρ 6) (bρ 6) (773330129695373 / 1024000000000000 : ℝ) ≤
+      (-(1752077120803032959657 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (901626119520754429 / 1280000000000000000 : ℝ) ≤
+      (-(1752077120803032959657 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (901626119520754429 / 1280000000000000000 : ℝ) =
+            (901626119520754429 / 640000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (901626119520754429 / 640000000000000000 : ℝ) ≤
+            (1713658780696967040343 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7035242882190411 / 10000000000000000 : ℝ)) (q :=
+      (901626119520754429 / 1280000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_652_7 :
+    Uω (aρ 7) (bρ 7) (773330129695373 / 1024000000000000 : ℝ) ≤
+      (-(3841985736840612427921 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (871675117636903997 / 1280000000000000000 : ℝ) ≤
+      (-(3841985736840612427921 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (871675117636903997 / 1280000000000000000 : ℝ) =
+            (871675117636903997 / 640000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (871675117636903997 / 640000000000000000 : ℝ) ≤
+            (3089486066159387572079 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (424432491157203 / 625000000000000 : ℝ)) (q :=
+      (871675117636903997 / 1280000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_652_8 :
+    Uω (aρ 8) (bρ 8) (773330129695373 / 1024000000000000 : ℝ) ≤
+      (-(4297532680459312403669 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (832857108142459837 / 1280000000000000000 : ℝ) ≤
+      (-(4297532680459312403669 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (832857108142459837 / 1280000000000000000 : ℝ) =
+            (832857108142459837 / 640000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (832857108142459837 / 640000000000000000 : ℝ) ≤
+            (2633939122540687596331 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3233965055752279 / 5000000000000000 : ℝ)) (q :=
+      (832857108142459837 / 1280000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8])
+      (by norm_num [bρ_8]) (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_652_9 :
+    Uω (aρ 9) (bρ 9) (773330129695373 / 1024000000000000 : ℝ) ≤
+      (-(2446440040483157550377 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (784720299999318333 / 1280000000000000000 : ℝ) ≤
+      (-(2446440040483157550377 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (784720299999318333 / 1280000000000000000 : ℝ) =
+            (784720299999318333 / 640000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (784720299999318333 / 640000000000000000 : ℝ) ≤
+            (1019295861016842449623 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1514189528156993 / 2500000000000000 : ℝ)) (q :=
+      (784720299999318333 / 1280000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9])
+      (by norm_num [bρ_9]) (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_652_10 :
+    Uω (aρ 10) (bρ 10) (773330129695373 / 1024000000000000 : ℝ) ≤
+      (-(5650069981818022359461 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (145499179298836569 / 256000000000000000 : ℝ) ≤
+      (-(5650069981818022359461 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (145499179298836569 / 256000000000000000 : ℝ) =
+            (145499179298836569 / 128000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (145499179298836569 / 128000000000000000 : ℝ) ≤
+            (1281401821181977640539 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (555118584135523 / 1000000000000000 : ℝ)) (q :=
+      (145499179298836569 / 256000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_652_11 :
+    Uω (aρ 11) (bρ 11) (773330129695373 / 1024000000000000 : ℝ) ≤
+      (-(6590630992571855497951 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (662189823576252669 / 1280000000000000000 : ℝ) ≤
+      (-(6590630992571855497951 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (662189823576252669 / 1280000000000000000 : ℝ) =
+            (662189823576252669 / 640000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (662189823576252669 / 640000000000000000 : ℝ) ≤
+            (340840810428144502049 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4949255078217571 / 10000000000000000 : ℝ)) (q :=
+      (662189823576252669 / 1280000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_652_12 :
+    Uω (aρ 12) (bρ 12) (773330129695373 / 1024000000000000 : ℝ) ≤
+      (-(7735265208020603162373 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (590570391768275581 / 1280000000000000000 : ℝ) ≤
+      (-(7735265208020603162373 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (590570391768275581 / 1280000000000000000 : ℝ) =
+            (590570391768275581 / 320000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (590570391768275581 / 320000000000000000 : ℝ) ≤
+            (6127678397979396837627 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4253901552902929 / 10000000000000000 : ℝ)) (q :=
+      (590570391768275581 / 1280000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_652_13 :
+    Uω (aρ 13) (bρ 13) (773330129695373 / 1024000000000000 : ℝ) ≤
+      (-(9103538205482609527087 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (515048739424809981 / 1280000000000000000 : ℝ) ≤
+      (-(9103538205482609527087 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (515048739424809981 / 1280000000000000000 : ℝ) =
+            (515048739424809981 / 320000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (515048739424809981 / 320000000000000000 : ℝ) ≤
+            (4759405400517390472913 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3473257476631279 / 10000000000000000 : ℝ)) (q :=
+      (515048739424809981 / 1280000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_652_14 :
+    Uω (aρ 14) (bρ 14) (773330129695373 / 1024000000000000 : ℝ) ≤
+      (-(10712917914125938837233 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (438484106052634813 / 1280000000000000000 : ℝ) ≤
+      (-(10712917914125938837233 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (438484106052634813 / 1280000000000000000 : ℝ) =
+            (438484106052634813 / 320000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (438484106052634813 / 320000000000000000 : ℝ) ≤
+            (3150025691874061162767 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1310431182618021 / 5000000000000000 : ℝ)) (q :=
+      (438484106052634813 / 1280000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_652_15 :
+    Uω (aρ 15) (bρ 15) (773330129695373 / 1024000000000000 : ℝ) ≤
+      (-(1571724976514098835237 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2912237359646401 / 10240000000000000 : ℝ) ≤
+      (-(1571724976514098835237 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2912237359646401 / 10240000000000000 : ℝ) =
+            (2912237359646401 / 2560000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2912237359646401 / 2560000000000000 : ℝ) ≤
+            (161142974235901164763 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (858542539319 / 5000000000000 : ℝ)) (q :=
+      (2912237359646401 / 10240000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_652_16 :
+    Uω (aρ 16) (bρ 16) (773330129695373 / 1024000000000000 : ℝ) ≤
+      (-(7323588502680710169883 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (295863339134328381 / 1280000000000000000 : ℝ) ≤
+      (-(7323588502680710169883 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (295863339134328381 / 1280000000000000000 : ℝ) =
+            (295863339134328381 / 160000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (295863339134328381 / 160000000000000000 : ℝ) ≤
+            (3073619201819289830117 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (100544726382813 / 1250000000000000 : ℝ)) (q :=
+      (295863339134328381 / 1280000000000000000 : ℝ)) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [bρ_16]) (by norm_num) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [aρ_16, bρ_16]) hlg
+
+lemma U_652 :
+    Uρ (773330129695373 / 1024000000000000 : ℝ) ≤
+      (-(636726646180617477787 / 1250000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (773330129695373 / 1024000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2893456901553794808471 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_652_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) *
+        Uω (aρ 2) (bρ 2) (773330129695373 / 1024000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1462706849909965999691 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_652_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) *
+        Uω (aρ 3) (bρ 3) (773330129695373 / 1024000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1494799872180451229579 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_652_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (773330129695373 / 1024000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3097011815094364335053 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_652_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (773330129695373 / 1024000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(3262544425323638530593 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_652_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) *
+        Uω (aρ 6) (bρ 6) (773330129695373 / 1024000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1752077120803032959657 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_652_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) *
+        Uω (aρ 7) (bρ 7) (773330129695373 / 1024000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(3841985736840612427921 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_652_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) *
+        Uω (aρ 8) (bρ 8) (773330129695373 / 1024000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(4297532680459312403669 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_652_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (773330129695373 / 1024000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(2446440040483157550377 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_652_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) *
+        Uω (aρ 10) (bρ 10) (773330129695373 / 1024000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(5650069981818022359461 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_652_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) *
+        Uω (aρ 11) (bρ 11) (773330129695373 / 1024000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(6590630992571855497951 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_652_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) *
+        Uω (aρ 12) (bρ 12) (773330129695373 / 1024000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(7735265208020603162373 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_652_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) *
+        Uω (aρ 13) (bρ 13) (773330129695373 / 1024000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(9103538205482609527087 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_652_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) *
+        Uω (aρ 14) (bρ 14) (773330129695373 / 1024000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(10712917914125938837233 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_652_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (773330129695373 / 1024000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(1571724976514098835237 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_652_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) *
+        Uω (aρ 16) (bρ 16) (773330129695373 / 1024000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) *
+        (-(7323588502680710169883 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_652_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_653_1 :
+    Uω (aρ 1) (bρ 1) (96822936549963 / 128000000000000 : ℝ) ≤
+      (-(2877123201811696091637 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (119996370296175819 / 160000000000000000 : ℝ) ≤
+      (-(2877123201811696091637 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (119996370296175819 / 160000000000000000 : ℝ) =
+            (119996370296175819 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (119996370296175819 / 80000000000000000 : ℝ) ≤
+            (4054348601188303908363 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3749875793570559 / 5000000000000000 : ℝ)) (q :=
+      (119996370296175819 / 160000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_653_2 :
+    Uω (aρ 2) (bρ 2) (96822936549963 / 128000000000000 : ℝ) ≤
+      (-(1454513748973133960733 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (119614140385909363 / 160000000000000000 : ℝ) ≤
+      (-(1454513748973133960733 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (119614140385909363 / 160000000000000000 : ℝ) =
+            (119614140385909363 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (119614140385909363 / 80000000000000000 : ℝ) ≤
+            (2011222152526866039267 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7475741860867811 / 10000000000000000 : ℝ)) (q :=
+      (119614140385909363 / 160000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_653_3 :
+    Uω (aρ 3) (bρ 3) (96822936549963 / 128000000000000 : ℝ) ≤
+      (-(185819207586795205137 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (118850105661120603 / 160000000000000000 : ℝ) ≤
+      (-(185819207586795205137 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (118850105661120603 / 160000000000000000 : ℝ) =
+            (118850105661120603 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (118850105661120603 / 80000000000000000 : ℝ) ≤
+            (247397780100704794863 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (464227101663701 / 625000000000000 : ℝ)) (q :=
+      (118850105661120603 / 160000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_653_4 :
+    Uω (aρ 4) (bρ 4) (96822936549963 / 128000000000000 : ℝ) ≤
+      (-(3080339305095248653731 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (117582461150381027 / 160000000000000000 : ℝ) ≤
+      (-(3080339305095248653731 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (117582461150381027 / 160000000000000000 : ℝ) =
+            (117582461150381027 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (117582461150381027 / 80000000000000000 : ℝ) ≤
+            (3851132497904751346269 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7347472243841769 / 10000000000000000 : ℝ)) (q :=
+      (117582461150381027 / 160000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_653_5 :
+    Uω (aρ 5) (bρ 5) (96822936549963 / 128000000000000 : ℝ) ≤
+      (-(64911770518235379883 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (115655386374443651 / 160000000000000000 : ℝ) ≤
+      (-(64911770518235379883 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (115655386374443651 / 160000000000000000 : ℝ) =
+            (115655386374443651 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (115655386374443651 / 80000000000000000 : ℝ) ≤
+            (73717665541764620117 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7224777816184597 / 10000000000000000 : ℝ)) (q :=
+      (115655386374443651 / 160000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_653_6 :
+    Uω (aρ 6) (bρ 6) (96822936549963 / 128000000000000 : ℝ) ≤
+      (-(174338573018717726637 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22579868986271223 / 32000000000000000 : ℝ) ≤
+      (-(174338573018717726637 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (22579868986271223 / 32000000000000000 : ℝ) =
+            (22579868986271223 / 16000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22579868986271223 / 16000000000000000 : ℝ) ≤
+            (172235017131282273363 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1409502602187731 / 2000000000000000 : ℝ)) (q :=
+      (22579868986271223 / 32000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_653_7 :
+    Uω (aρ 7) (bρ 7) (96822936549963 / 128000000000000 : ℝ) ≤
+      (-(1911989087255013801439 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (109155775773423779 / 160000000000000000 : ℝ) ≤
+      (-(1911989087255013801439 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (109155775773423779 / 160000000000000000 : ℝ) =
+            (109155775773423779 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (109155775773423779 / 80000000000000000 : ℝ) ≤
+            (1553746814244986198561 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6803228246957113 / 10000000000000000 : ℝ)) (q :=
+      (109155775773423779 / 160000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_653_8 :
+    Uω (aρ 8) (bρ 8) (96822936549963 / 128000000000000 : ℝ) ≤
+      (-(2139313420275157981527 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20860829596002847 / 32000000000000000 : ℝ) ≤
+      (-(2139313420275157981527 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (20860829596002847 / 32000000000000000 : ℝ) =
+            (20860829596002847 / 16000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20860829596002847 / 16000000000000000 : ℝ) ≤
+            (1326422481224842018473 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (162007910603023 / 250000000000000 : ℝ)) (q :=
+      (20860829596002847 / 32000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_653_9 :
+    Uω (aρ 9) (bρ 9) (96822936549963 / 128000000000000 : ℝ) ≤
+      (-(1218173088127056917877 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (786306072150831 / 1280000000000000 : ℝ) ≤
+      (-(1218173088127056917877 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (786306072150831 / 1280000000000000 : ℝ) =
+            (786306072150831 / 640000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (786306072150831 / 640000000000000 : ℝ) ≤
+            (514694862622943082123 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (48554367458687 / 80000000000000 : ℝ)) (q :=
+      (786306072150831 / 1280000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_653_10 :
+    Uω (aρ 10) (bρ 10) (96822936549963 / 128000000000000 : ℝ) ≤
+      (-(5628046299432133914873 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (91137484497341187 / 160000000000000000 : ℝ) ≤
+      (-(5628046299432133914873 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (91137484497341187 / 160000000000000000 : ℝ) =
+            (91137484497341187 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (91137484497341187 / 80000000000000000 : ℝ) ≤
+            (1303425503567866085127 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5564008150641789 / 10000000000000000 : ℝ)) (q :=
+      (91137484497341187 / 160000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_653_11 :
+    Uω (aρ 11) (bρ 11) (96822936549963 / 128000000000000 : ℝ) ≤
+      (-(6565933545682449290393 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (82978410574888283 / 160000000000000000 : ℝ) ≤
+      (-(6565933545682449290393 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (82978410574888283 / 160000000000000000 : ℝ) =
+            (82978410574888283 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (82978410574888283 / 80000000000000000 : ℝ) ≤
+            (365538257317550709607 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (310162533533761 / 625000000000000 : ℝ)) (q :=
+      (82978410574888283 / 160000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_653_12 :
+    Uω (aρ 12) (bρ 12) (96822936549963 / 128000000000000 : ℝ) ≤
+      (-(7706540205108390999763 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (74033655524941171 / 160000000000000000 : ℝ) ≤
+      (-(7706540205108390999763 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (74033655524941171 / 160000000000000000 : ℝ) =
+            (74033655524941171 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (74033655524941171 / 40000000000000000 : ℝ) ≤
+            (6156403400891609000237 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4268206251981787 / 10000000000000000 : ℝ)) (q :=
+      (74033655524941171 / 160000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_653_13 :
+    Uω (aρ 13) (bρ 13) (96822936549963 / 128000000000000 : ℝ) ≤
+      (-(2267094840206635581753 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (64607847299227699 / 160000000000000000 : ℝ) ≤
+      (-(2267094840206635581753 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (64607847299227699 / 160000000000000000 : ℝ) =
+            (64607847299227699 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (64607847299227699 / 40000000000000000 : ℝ) ≤
+            (1198641061293364418247 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3489361965362603 / 10000000000000000 : ℝ)) (q :=
+      (64607847299227699 / 160000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_653_14 :
+    Uω (aρ 14) (bρ 14) (96822936549963 / 128000000000000 : ℝ) ≤
+      (-(10666391102895760677611 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (55066123271789059 / 160000000000000000 : ℝ) ≤
+      (-(10666391102895760677611 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (55066123271789059 / 160000000000000000 : ℝ) =
+            (55066123271789059 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (55066123271789059 / 40000000000000000 : ℝ) ≤
+            (3196552503104239322389 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2640573746977773 / 10000000000000000 : ℝ)) (q :=
+      (55066123271789059 / 160000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_653_15 :
+    Uω (aρ 15) (bρ 15) (96822936549963 / 128000000000000 : ℝ) ≤
+      (-(12503095599865301573741 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (45826579202610331 / 160000000000000000 : ℝ) ≤
+      (-(12503095599865301573741 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (45826579202610331 / 160000000000000000 : ℝ) =
+            (45826579202610331 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (45826579202610331 / 40000000000000000 : ℝ) ≤
+            (1359848006134698426259 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (218150501968179 / 1250000000000000 : ℝ)) (q :=
+      (45826579202610331 / 160000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_653_16 :
+    Uω (aρ 16) (bρ 16) (96822936549963 / 128000000000000 : ℝ) ≤
+      (-(14500144393995872475567 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (37530704140993003 / 160000000000000000 : ℝ) ≤
+      (-(14500144393995872475567 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (37530704140993003 / 160000000000000000 : ℝ) =
+            (37530704140993003 / 20000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (37530704140993003 / 20000000000000000 : ℝ) ≤
+            (6294271015004127524433 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (430295642276633 / 5000000000000000 : ℝ)) (q :=
+      (37530704140993003 / 160000000000000000 : ℝ)) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [bρ_16]) (by norm_num) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [aρ_16, bρ_16]) hlg
+
+lemma U_653 :
+    Uρ (96822936549963 / 128000000000000 : ℝ) ≤
+      (-(2535469459548085995737 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (96822936549963 / 128000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2877123201811696091637 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_653_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (96822936549963 / 128000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1454513748973133960733 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_653_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (96822936549963 / 128000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(185819207586795205137 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_653_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (96822936549963 / 128000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3080339305095248653731 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_653_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (96822936549963 / 128000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(64911770518235379883 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_653_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (96822936549963 / 128000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(174338573018717726637 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_653_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (96822936549963 / 128000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(1911989087255013801439 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_653_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (96822936549963 / 128000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(2139313420275157981527 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_653_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (96822936549963 / 128000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(1218173088127056917877 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_653_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (96822936549963 / 128000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(5628046299432133914873 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_653_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (96822936549963 / 128000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(6565933545682449290393 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_653_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (96822936549963 / 128000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(7706540205108390999763 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_653_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (96822936549963 / 128000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(2267094840206635581753 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_653_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (96822936549963 / 128000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(10666391102895760677611 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_653_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (96822936549963 / 128000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(12503095599865301573741 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_653_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (96822936549963 / 128000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) *
+        (-(14500144393995872475567 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_653_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_654_1 :
+    Uω (aρ 1) (bρ 1) (155167371020807 / 204800000000000 : ℝ) ≤
+      (-(178801008606094785107 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (192307534049124759 / 256000000000000000 : ℝ) ≤
+      (-(178801008606094785107 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (192307534049124759 / 256000000000000000 : ℝ) =
+            (192307534049124759 / 128000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (192307534049124759 / 128000000000000000 : ℝ) ≤
+            (254415979081405214893 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (751199152754753 / 1000000000000000 : ℝ)) (q :=
+      (192307534049124759 / 256000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_654_2 :
+    Uω (aρ 2) (bρ 2) (155167371020807 / 204800000000000 : ℝ) ≤
+      (-(180791756495602557493 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (958479856160063987 / 1280000000000000000 : ℝ) ≤
+      (-(180791756495602557493 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (958479856160063987 / 1280000000000000000 : ℝ) =
+            (958479856160063987 / 640000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (958479856160063987 / 640000000000000000 : ℝ) ≤
+            (252425231191897442507 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3743991097485329 / 5000000000000000 : ℝ)) (q :=
+      (958479856160063987 / 1280000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_654_3 :
+    Uω (aρ 3) (bρ 3) (155167371020807 / 204800000000000 : ℝ) ≤
+      (-(2956642057273893298753 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7618941228275183 / 10240000000000000 : ℝ) ≤
+      (-(2956642057273893298753 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (7618941228275183 / 10240000000000000 : ℝ) =
+            (7618941228275183 / 5120000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7618941228275183 / 5120000000000000 : ℝ) ≤
+            (3974829745726106701247 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (59519001082357 / 80000000000000 : ℝ)) (q :=
+      (7618941228275183 / 10240000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_654_4 :
+    Uω (aρ 4) (bρ 4) (155167371020807 / 204800000000000 : ℝ) ≤
+      (-(612738911377929143541 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (942226697331449523 / 1280000000000000000 : ℝ) ≤
+      (-(612738911377929143541 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (942226697331449523 / 1280000000000000000 : ℝ) =
+            (942226697331449523 / 640000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (942226697331449523 / 640000000000000000 : ℝ) ≤
+            (773555449222070856459 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7359716875688557 / 10000000000000000 : ℝ)) (q :=
+      (942226697331449523 / 1280000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_654_5 :
+    Uω (aρ 5) (bρ 5) (155167371020807 / 204800000000000 : ℝ) ≤
+      (-(3228661357312839731263 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (926810591868475827 / 1280000000000000000 : ℝ) ≤
+      (-(3228661357312839731263 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (926810591868475827 / 1280000000000000000 : ℝ) =
+            (926810591868475827 / 640000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (926810591868475827 / 640000000000000000 : ℝ) ≤
+            (3702810445687160268737 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7237030147164593 / 10000000000000000 : ℝ)) (q :=
+      (926810591868475827 / 1280000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_654_6 :
+    Uω (aρ 6) (bρ 6) (155167371020807 / 204800000000000 : ℝ) ≤
+      (-(1734709458567699032261 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (904763392657673011 / 1280000000000000000 : ℝ) ≤
+      (-(1734709458567699032261 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (904763392657673011 / 1280000000000000000 : ℝ) =
+            (904763392657673011 / 640000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (904763392657673011 / 640000000000000000 : ℝ) ≤
+            (1731026442932300967739 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7059783034635799 / 10000000000000000 : ℝ)) (q :=
+      (904763392657673011 / 1280000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_654_7 :
+    Uω (aρ 7) (bρ 7) (155167371020807 / 204800000000000 : ℝ) ≤
+      (-(761200632335293118429 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (874817278943332787 / 1280000000000000000 : ℝ) ≤
+      (-(761200632335293118429 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (874817278943332787 / 1280000000000000000 : ℝ) =
+            (874817278943332787 / 640000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (874817278943332787 / 640000000000000000 : ℝ) ≤
+            (625093728264706881571 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6815536388609233 / 10000000000000000 : ℝ)) (q :=
+      (874817278943332787 / 1280000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_654_8 :
+    Uω (aρ 8) (bρ 8) (155167371020807 / 204800000000000 : ℝ) ≤
+      (-(425975710149634223111 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (33440368959579563 / 51200000000000000 : ℝ) ≤
+      (-(425975710149634223111 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (33440368959579563 / 51200000000000000 : ℝ) =
+            (33440368959579563 / 25600000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (33440368959579563 / 25600000000000000 : ℝ) ≤
+            (267171470150365776889 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (259708087251817 / 400000000000000 : ℝ)) (q :=
+      (33440368959579563 / 51200000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_654_9 :
+    Uω (aρ 9) (bρ 9) (155167371020807 / 204800000000000 : ℝ) ≤
+      (-(4852546282406263118157 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (787891766615224371 / 1280000000000000000 : ℝ) ≤
+      (-(4852546282406263118157 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (787891766615224371 / 1280000000000000000 : ℝ) =
+            (787891766615224371 / 640000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (787891766615224371 / 640000000000000000 : ℝ) ≤
+            (2078925520593736881843 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6081832538182539 / 10000000000000000 : ℝ)) (q :=
+      (787891766615224371 / 1280000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9])
+      (by norm_num [bρ_9]) (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_654_10 :
+    Uω (aρ 10) (bρ 10) (155167371020807 / 204800000000000 : ℝ) ≤
+      (-(2803036653647281027841 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (146140737589014167 / 256000000000000000 : ℝ) ≤
+      (-(2803036653647281027841 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (146140737589014167 / 256000000000000000 : ℝ) =
+            (146140737589014167 / 128000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (146140737589014167 / 128000000000000000 : ℝ) ≤
+            (662699247852718972159 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (557682784245639 / 1000000000000000 : ℝ)) (q :=
+      (146140737589014167 / 256000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_654_11 :
+    Uω (aρ 11) (bρ 11) (155167371020807 / 204800000000000 : ℝ) ≤
+      (-(817662801515715558399 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (665464381799973043 / 1280000000000000000 : ℝ) ≤
+      (-(817662801515715558399 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (665464381799973043 / 1280000000000000000 : ℝ) =
+            (665464381799973043 / 640000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (665464381799973043 / 640000000000000000 : ℝ) ≤
+            (48771173859284441601 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4975940310144237 / 10000000000000000 : ℝ)) (q :=
+      (665464381799973043 / 1280000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_654_12 :
+    Uω (aρ 12) (bρ 12) (155167371020807 / 204800000000000 : ℝ) ≤
+      (-(239934727155107788411 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23758691100856299 / 51200000000000000 : ℝ) ≤
+      (-(239934727155107788411 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (23758691100856299 / 51200000000000000 : ℝ) =
+            (23758691100856299 / 12800000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23758691100856299 / 12800000000000000 : ℝ) ≤
+            (193282260532392211589 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (85649963049533 / 200000000000000 : ℝ)) (q :=
+      (23758691100856299 / 51200000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_654_13 :
+    Uω (aρ 13) (bρ 13) (155167371020807 / 204800000000000 : ℝ) ≤
+      (-(2258345563885136111871 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (518674817457819507 / 1280000000000000000 : ℝ) ≤
+      (-(2258345563885136111871 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (518674817457819507 / 1280000000000000000 : ℝ) =
+            (518674817457819507 / 320000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (518674817457819507 / 320000000000000000 : ℝ) ≤
+            (1207390337614863888129 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (438179400697261 / 1250000000000000 : ℝ)) (q :=
+      (518674817457819507 / 1280000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_654_14 :
+    Uω (aρ 14) (bρ 14) (155167371020807 / 204800000000000 : ℝ) ≤
+      (-(5310104766705545786437 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (442568123075821491 / 1280000000000000000 : ℝ) ≤
+      (-(5310104766705545786437 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (442568123075821491 / 1280000000000000000 : ℝ) =
+            (442568123075821491 / 320000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (442568123075821491 / 320000000000000000 : ℝ) ≤
+            (1621367036294454213563 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2660195390904369 / 10000000000000000 : ℝ)) (q :=
+      (442568123075821491 / 1280000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_654_15 :
+    Uω (aρ 15) (bρ 15) (155167371020807 / 204800000000000 : ℝ) ≤
+      (-(12433514339998384626713 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (369172466004084083 / 1280000000000000000 : ℝ) ≤
+      (-(12433514339998384626713 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (369172466004084083 / 1280000000000000000 : ℝ) =
+            (369172466004084083 / 320000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (369172466004084083 / 320000000000000000 : ℝ) ≤
+            (1429429266001615373287 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (55405047705421 / 312500000000000 : ℝ)) (q :=
+      (369172466004084083 / 1280000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_654_16 :
+    Uω (aρ 16) (bρ 16) (155167371020807 / 204800000000000 : ℝ) ≤
+      (-(14362160113274664036023 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (304417265698092147 / 1280000000000000000 : ℝ) ≤
+      (-(14362160113274664036023 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (304417265698092147 / 1280000000000000000 : ℝ) =
+            (304417265698092147 / 160000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (304417265698092147 / 160000000000000000 : ℝ) ≤
+            (6432255295725335963977 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (228383293325587 / 2500000000000000 : ℝ)) (q :=
+      (304417265698092147 / 1280000000000000000 : ℝ)) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [bρ_16]) (by norm_num) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [aρ_16, bρ_16]) hlg
+
+lemma U_654 :
+    Uρ (155167371020807 / 204800000000000 : ℝ) ≤
+      (-(5048211830024685695573 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (155167371020807 / 204800000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(178801008606094785107 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_654_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (155167371020807 / 204800000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(180791756495602557493 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_654_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (155167371020807 / 204800000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(2956642057273893298753 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_654_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (155167371020807 / 204800000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(612738911377929143541 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_654_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (155167371020807 / 204800000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(3228661357312839731263 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_654_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (155167371020807 / 204800000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1734709458567699032261 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_654_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (155167371020807 / 204800000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(761200632335293118429 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_654_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (155167371020807 / 204800000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(425975710149634223111 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_654_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (155167371020807 / 204800000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(4852546282406263118157 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_654_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) *
+        Uω (aρ 10) (bρ 10) (155167371020807 / 204800000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(2803036653647281027841 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_654_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) *
+        Uω (aρ 11) (bρ 11) (155167371020807 / 204800000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(817662801515715558399 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_654_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) *
+        Uω (aρ 12) (bρ 12) (155167371020807 / 204800000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(239934727155107788411 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_654_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) *
+        Uω (aρ 13) (bρ 13) (155167371020807 / 204800000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(2258345563885136111871 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_654_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) *
+        Uω (aρ 14) (bρ 14) (155167371020807 / 204800000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(5310104766705545786437 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_654_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (155167371020807 / 204800000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(12433514339998384626713 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_654_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) *
+        Uω (aρ 16) (bρ 16) (155167371020807 / 204800000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) *
+        (-(14362160113274664036023 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_654_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_655_1 :
+    Uω (aρ 1) (bρ 1) (388545108904183 / 512000000000000 : ℝ) ≤
+      (-(44445869101279289721 / 156250000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (481552189053607047 / 640000000000000000 : ℝ) ≤
+      (-(44445869101279289721 / 156250000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (481552189053607047 / 640000000000000000 : ℝ) =
+            (481552189053607047 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (481552189053607047 / 320000000000000000 : ℝ) ≤
+            (63858377820595710279 / 156250000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1881057866931349 / 2500000000000000 : ℝ)) (q :=
+      (481552189053607047 / 640000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_655_2 :
+    Uω (aρ 2) (bρ 2) (388545108904183 / 512000000000000 : ℝ) ≤
+      (-(2876335430194582704191 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (480023294567969607 / 640000000000000000 : ℝ) ≤
+      (-(2876335430194582704191 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (480023294567969607 / 640000000000000000 : ℝ) =
+            (480023294567969607 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (480023294567969607 / 320000000000000000 : ℝ) ≤
+            (4055136372805417295809 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3750111263779613 / 5000000000000000 : ℝ)) (q :=
+      (480023294567969607 / 640000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_655_3 :
+    Uω (aρ 3) (bρ 3) (388545108904183 / 512000000000000 : ℝ) ≤
+      (-(2940203862703970074569 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (476967230717666407 / 640000000000000000 : ℝ) ≤
+      (-(2940203862703970074569 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (476967230717666407 / 640000000000000000 : ℝ) =
+            (476967230717666407 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (476967230717666407 / 320000000000000000 : ℝ) ≤
+            (3991267940296029925431 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7452116638587251 / 10000000000000000 : ℝ)) (q :=
+      (476967230717666407 / 640000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_655_4 :
+    Uω (aρ 4) (bρ 4) (388545108904183 / 512000000000000 : ℝ) ≤
+      (-(3047077478141810132203 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (471896852223845543 / 640000000000000000 : ℝ) ≤
+      (-(3047077478141810132203 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (471896852223845543 / 640000000000000000 : ℝ) =
+            (471896852223845543 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (471896852223845543 / 320000000000000000 : ℝ) ≤
+            (3884394324858189867797 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7371961491720349 / 10000000000000000 : ℝ)) (q :=
+      (471896852223845543 / 640000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_655_5 :
+    Uω (aρ 5) (bρ 5) (388545108904183 / 512000000000000 : ℝ) ≤
+      (-(80294070555689559269 / 250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (464189045023509479 / 640000000000000000 : ℝ) ≤
+      (-(80294070555689559269 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (464189045023509479 / 640000000000000000 : ℝ) =
+            (464189045023509479 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (464189045023509479 / 320000000000000000 : ℝ) ≤
+            (92992724519310440731 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7249282436044847 / 10000000000000000 : ℝ)) (q :=
+      (464189045023509479 / 640000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_655_6 :
+    Uω (aρ 6) (bρ 6) (388545108904183 / 512000000000000 : ℝ) ≤
+      (-(1726048253307382550747 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (453166009588110663 / 640000000000000000 : ℝ) ≤
+      (-(1726048253307382550747 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (453166009588110663 / 640000000000000000 : ℝ) =
+            (453166009588110663 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (453166009588110663 / 320000000000000000 : ℝ) ≤
+            (1739687648192617449253 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3536026476914317 / 5000000000000000 : ℝ)) (q :=
+      (453166009588110663 / 640000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_655_7 :
+    Uω (aρ 7) (bρ 7) (388545108904183 / 512000000000000 : ℝ) ≤
+      (-(3788060580235419980457 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (438194167995073671 / 640000000000000000 : ℝ) ≤
+      (-(3788060580235419980457 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (438194167995073671 / 640000000000000000 : ℝ) =
+            (438194167995073671 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (438194167995073671 / 320000000000000000 : ℝ) ≤
+            (3143411222764580019543 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1706961071201557 / 2500000000000000 : ℝ)) (q :=
+      (438194167995073671 / 640000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_655_8 :
+    Uω (aρ 8) (bρ 8) (388545108904183 / 512000000000000 : ℝ) ≤
+      (-(4240923324085287446963 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16751704575872767 / 25600000000000000 : ℝ) ≤
+      (-(4240923324085287446963 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (16751704575872767 / 25600000000000000 : ℝ) =
+            (16751704575872767 / 12800000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16751704575872767 / 12800000000000000 : ℝ) ≤
+            (2690548478914712553037 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (260203495448031 / 400000000000000 : ℝ)) (q :=
+      (16751704575872767 / 25600000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_655_9 :
+    Uω (aρ 9) (bρ 9) (388545108904183 / 512000000000000 : ℝ) ≤
+      (-(966488339018040144383 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (394738691935962311 / 640000000000000000 : ℝ) ≤
+      (-(966488339018040144383 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (394738691935962311 / 640000000000000000 : ℝ) =
+            (394738691935962311 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (394738691935962311 / 320000000000000000 : ℝ) ≤
+            (419806021581959855617 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3047183968829499 / 5000000000000000 : ℝ)) (q :=
+      (394738691935962311 / 640000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9])
+      (by norm_num [bρ_9]) (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_655_10 :
+    Uω (aρ 10) (bρ 10) (388545108904183 / 512000000000000 : ℝ) ≤
+      (-(5584150762359065444067 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14646146670916031 / 25600000000000000 : ℝ) ≤
+      (-(5584150762359065444067 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (14646146670916031 / 25600000000000000 : ℝ) =
+            (14646146670916031 / 12800000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14646146670916031 / 12800000000000000 : ℝ) ≤
+            (1347321040640934555933 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (223585797392333 / 400000000000000 : ℝ)) (q :=
+      (14646146670916031 / 25600000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_655_11 :
+    Uω (aρ 11) (bρ 11) (388545108904183 / 512000000000000 : ℝ) ≤
+      (-(6516737208834470122621 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (333550559048547303 / 640000000000000000 : ℝ) ≤
+      (-(6516737208834470122621 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (333550559048547303 / 640000000000000000 : ℝ) =
+            (333550559048547303 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (333550559048547303 / 320000000000000000 : ℝ) ≤
+            (414734594165529877379 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4989274444627279 / 10000000000000000 : ℝ)) (q :=
+      (333550559048547303 / 640000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_655_12 :
+    Uω (aρ 12) (bρ 12) (388545108904183 / 512000000000000 : ℝ) ≤
+      (-(7649377674455245816967 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (297832249953725863 / 640000000000000000 : ℝ) ≤
+      (-(7649377674455245816967 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (297832249953725863 / 640000000000000000 : ℝ) =
+            (297832249953725863 / 160000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (297832249953725863 / 160000000000000000 : ℝ) ≤
+            (6213565931544754183033 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4296777382099109 / 10000000000000000 : ℝ)) (q :=
+      (297832249953725863 / 640000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_655_13 :
+    Uω (aρ 13) (bρ 13) (388545108904183 / 512000000000000 : ℝ) ≤
+      (-(8998545100638613692049 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (52048488400157723 / 128000000000000000 : ℝ) ≤
+      (-(8998545100638613692049 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (52048488400157723 / 128000000000000000 : ℝ) =
+            (52048488400157723 / 32000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (52048488400157723 / 32000000000000000 : ℝ) ≤
+            (4864398505361386307951 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (176073881258241 / 500000000000000 : ℝ)) (q :=
+      (52048488400157723 / 128000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_655_14 :
+    Uω (aρ 14) (bρ 14) (388545108904183 / 512000000000000 : ℝ) ≤
+      (-(10574366596004491116633 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (44460164291706619 / 128000000000000000 : ℝ) ≤
+      (-(10574366596004491116633 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (44460164291706619 / 128000000000000000 : ℝ) =
+            (44460164291706619 / 32000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (44460164291706619 / 32000000000000000 : ℝ) ≤
+            (3288577009995508883367 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (535945853652867 / 2000000000000000 : ℝ)) (q :=
+      (44460164291706619 / 128000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_655_15 :
+    Uω (aρ 15) (bρ 15) (388545108904183 / 512000000000000 : ℝ) ≤
+      (-(247300147688838586911 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (185855118499611463 / 640000000000000000 : ℝ) ≤
+      (-(247300147688838586911 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (185855118499611463 / 640000000000000000 : ℝ) =
+            (185855118499611463 / 160000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (185855118499611463 / 160000000000000000 : ℝ) ≤
+            (29958724431161413089 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (900187164106517 / 5000000000000000 : ℝ)) (q :=
+      (185855118499611463 / 640000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_655_16 :
+    Uω (aρ 16) (bρ 16) (388545108904183 / 512000000000000 : ℝ) ≤
+      (-(889484798547236706907 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (154206482206035719 / 640000000000000000 : ℝ) ≤
+      (-(889484798547236706907 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (154206482206035719 / 640000000000000000 : ℝ) =
+            (154206482206035719 / 80000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (154206482206035719 / 80000000000000000 : ℝ) ≤
+            (410166164515263293093 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (120465761943599 / 1250000000000000 : ℝ)) (q :=
+      (154206482206035719 / 640000000000000000 : ℝ)) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [bρ_16]) (by norm_num) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [aρ_16, bρ_16]) hlg
+
+lemma U_655 :
+    Uρ (388545108904183 / 512000000000000 : ℝ) ≤
+      (-(5025621108762662119411 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (388545108904183 / 512000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(44445869101279289721 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_655_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (388545108904183 / 512000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(2876335430194582704191 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_655_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (388545108904183 / 512000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(2940203862703970074569 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_655_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (388545108904183 / 512000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3047077478141810132203 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_655_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (388545108904183 / 512000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(80294070555689559269 / 250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_655_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (388545108904183 / 512000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1726048253307382550747 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_655_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (388545108904183 / 512000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(3788060580235419980457 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_655_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (388545108904183 / 512000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(4240923324085287446963 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_655_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (388545108904183 / 512000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(966488339018040144383 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_655_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) *
+        Uω (aρ 10) (bρ 10) (388545108904183 / 512000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(5584150762359065444067 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_655_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) *
+        Uω (aρ 11) (bρ 11) (388545108904183 / 512000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(6516737208834470122621 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_655_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) *
+        Uω (aρ 12) (bρ 12) (388545108904183 / 512000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(7649377674455245816967 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_655_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) *
+        Uω (aρ 13) (bρ 13) (388545108904183 / 512000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(8998545100638613692049 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_655_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) *
+        Uω (aρ 14) (bρ 14) (388545108904183 / 512000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(10574366596004491116633 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_655_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (388545108904183 / 512000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(247300147688838586911 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_655_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) *
+        Uω (aρ 16) (bρ 16) (388545108904183 / 512000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(889484798547236706907 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_655_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_656_1 :
+    Uω (aρ 1) (bρ 1) (194899235804257 / 256000000000000 : ℝ) ≤
+      (-(703013473485130443647 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (241559448446699633 / 320000000000000000 : ℝ) ≤
+      (-(703013473485130443647 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (241559448446699633 / 320000000000000000 : ℝ) =
+            (241559448446699633 / 160000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (241559448446699633 / 160000000000000000 : ℝ) ≤
+            (1029854477264869556353 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3774355673699969 / 5000000000000000 : ℝ)) (q :=
+      (241559448446699633 / 320000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_656_2 :
+    Uω (aρ 2) (bρ 2) (194899235804257 / 256000000000000 : ℝ) ≤
+      (-(568749979081353075301 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (48159002739941997 / 64000000000000000 : ℝ) ≤
+      (-(568749979081353075301 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (48159002739941997 / 64000000000000000 : ℝ) =
+            (48159002739941997 / 32000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (48159002739941997 / 32000000000000000 : ℝ) ≤
+            (817544381518646924699 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1504940637644617 / 2000000000000000 : ℝ)) (q :=
+      (48159002739941997 / 64000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_656_3 :
+    Uω (aρ 3) (bρ 3) (194899235804257 / 256000000000000 : ℝ) ≤
+      (-(2907408327145662851251 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (239267019052619217 / 320000000000000000 : ℝ) ≤
+      (-(2907408327145662851251 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (239267019052619217 / 320000000000000000 : ℝ) =
+            (239267019052619217 / 160000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (239267019052619217 / 160000000000000000 : ℝ) ≤
+            (4024063475854337148749 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (934574953641239 / 1250000000000000 : ℝ)) (q :=
+      (239267019052619217 / 320000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_656_4 :
+    Uω (aρ 4) (bρ 4) (194899235804257 / 256000000000000 : ℝ) ≤
+      (-(3013925961973787554127 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (47346385783190109 / 64000000000000000 : ℝ) ≤
+      (-(3013925961973787554127 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (47346385783190109 / 64000000000000000 : ℝ) =
+            (47346385783190109 / 32000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (47346385783190109 / 32000000000000000 : ℝ) ≤
+            (3917545841026212445873 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (23113908364541 / 31250000000000 : ℝ)) (q :=
+      (47346385783190109 / 64000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_656_5 :
+    Uω (aρ 5) (bρ 5) (194899235804257 / 256000000000000 : ℝ) ≤
+      (-(1589025632933657282791 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (232878269593854273 / 320000000000000000 : ℝ) ≤
+      (-(1589025632933657282791 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (232878269593854273 / 320000000000000000 : ℝ) =
+            (232878269593854273 / 160000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (232878269593854273 / 160000000000000000 : ℝ) ≤
+            (1876710268566342717209 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7273786888357103 / 10000000000000000 : ℝ)) (q :=
+      (232878269593854273 / 320000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_656_6 :
+    Uω (aρ 6) (bρ 6) (194899235804257 / 256000000000000 : ℝ) ≤
+      (-(1708770832688818507607 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (227367313071814481 / 320000000000000000 : ℝ) ≤
+      (-(1708770832688818507607 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (227367313071814481 / 320000000000000000 : ℝ) =
+            (227367313071814481 / 160000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (227367313071814481 / 160000000000000000 : ℝ) ≤
+            (1756965068811181492393 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (443537030054351 / 625000000000000 : ℝ)) (q :=
+      (227367313071814481 / 320000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_656_7 :
+    Uω (aρ 7) (bρ 7) (194899235804257 / 256000000000000 : ℝ) ≤
+      (-(1876136121166841495429 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (43976520164748893 / 64000000000000000 : ℝ) ≤
+      (-(1876136121166841495429 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (43976520164748893 / 64000000000000000 : ℝ) =
+            (43976520164748893 / 32000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (43976520164748893 / 32000000000000000 : ℝ) ≤
+            (1589599780333158504571 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (171311483653131 / 250000000000000 : ℝ)) (q :=
+      (43976520164748893 / 64000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_656_8 :
+    Uω (aρ 8) (bρ 8) (194899235804257 / 256000000000000 : ℝ) ≤
+      (-(420336310140069993621 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (42036856658536173 / 64000000000000000 : ℝ) ≤
+      (-(420336310140069993621 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (42036856658536173 / 64000000000000000 : ℝ) =
+            (42036856658536173 / 32000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (42036856658536173 / 32000000000000000 : ℝ) ≤
+            (272810870159930006379 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1305971230354753 / 2000000000000000 : ℝ)) (q :=
+      (42036856658536173 / 64000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_656_9 :
+    Uω (aρ 9) (bρ 9) (194899235804257 / 256000000000000 : ℝ) ≤
+      (-(2396178136091364650849 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (198162097165019857 / 320000000000000000 : ℝ) ≤
+      (-(2396178136091364650849 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (198162097165019857 / 320000000000000000 : ℝ) =
+            (198162097165019857 / 160000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (198162097165019857 / 160000000000000000 : ℝ) ≤
+            (1069557765408635349151 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (764929393394369 / 1250000000000000 : ℝ)) (q :=
+      (198162097165019857 / 320000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9])
+      (by norm_num [bρ_9]) (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_656_10 :
+    Uω (aρ 10) (bρ 10) (194899235804257 / 256000000000000 : ℝ) ≤
+      (-(692557006368090820231 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (36775706510351901 / 64000000000000000 : ℝ) ≤
+      (-(692557006368090820231 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (36775706510351901 / 64000000000000000 : ℝ) =
+            (36775706510351901 / 32000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (36775706510351901 / 32000000000000000 : ℝ) ≤
+            (173876969006909179769 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (280763569616059 / 500000000000000 : ℝ)) (q :=
+      (36775706510351901 / 64000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_656_11 :
+    Uω (aρ 11) (bρ 11) (194899235804257 / 256000000000000 : ℝ) ≤
+      (-(6467803078306683504167 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (167593379872956993 / 320000000000000000 : ℝ) ≤
+      (-(6467803078306683504167 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (167593379872956993 / 320000000000000000 : ℝ) =
+            (167593379872956993 / 160000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (167593379872956993 / 160000000000000000 : ℝ) ≤
+            (463668724693316495833 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5015925976101023 / 10000000000000000 : ℝ)) (q :=
+      (167593379872956993 / 320000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_656_12 :
+    Uω (aρ 12) (bρ 12) (194899235804257 / 256000000000000 : ℝ) ≤
+      (-(237268551693682025271 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (29952827199544909 / 64000000000000000 : ℝ) ≤
+      (-(237268551693682025271 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (29952827199544909 / 64000000000000000 : ℝ) =
+            (29952827199544909 / 16000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (29952827199544909 / 16000000000000000 : ℝ) ≤
+            (195948435993817974729 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (865059666116799 / 2000000000000000 : ℝ)) (q :=
+      (29952827199544909 / 64000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_656_13 :
+    Uω (aρ 13) (bρ 13) (194899235804257 / 256000000000000 : ℝ) ≤
+      (-(4464671824597859984033 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (131024801584548321 / 320000000000000000 : ℝ) ≤
+      (-(4464671824597859984033 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (131024801584548321 / 320000000000000000 : ℝ) =
+            (131024801584548321 / 80000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (131024801584548321 / 80000000000000000 : ℝ) ≤
+            (2466799978402140015967 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3553471671355481 / 10000000000000000 : ℝ)) (q :=
+      (131024801584548321 / 320000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_656_14 :
+    Uω (aρ 14) (bρ 14) (194899235804257 / 256000000000000 : ℝ) ≤
+      (-(163807362855826435447 / 156250000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (112163078801764561 / 320000000000000000 : ℝ) ≤
+      (-(163807362855826435447 / 156250000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (112163078801764561 / 320000000000000000 : ℝ) =
+            (112163078801764561 / 80000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (112163078801764561 / 80000000000000000 : ℝ) ≤
+            (52801130987923564553 / 156250000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (169908830154781 / 625000000000000 : ℝ)) (q :=
+      (112163078801764561 / 320000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_656_15 :
+    Uω (aρ 15) (bρ 15) (194899235804257 / 256000000000000 : ℝ) ≤
+      (-(2446207738464103347739 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (94180874109960769 / 320000000000000000 : ℝ) ≤
+      (-(2446207738464103347739 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (94180874109960769 / 320000000000000000 : ℝ) =
+            (94180874109960769 / 80000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (94180874109960769 / 80000000000000000 : ℝ) ≤
+            (326380982735896652261 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1854226766653759 / 10000000000000000 : ℝ)) (q :=
+      (94180874109960769 / 320000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_656_16 :
+    Uω (aρ 16) (bρ 16) (194899235804257 / 256000000000000 : ℝ) ≤
+      (-(559580302932836820887 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (78993868669915009 / 320000000000000000 : ℝ) ≤
+      (-(559580302932836820887 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (78993868669915009 / 320000000000000000 : ℝ) =
+            (78993868669915009 / 40000000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (78993868669915009 / 40000000000000000 : ℝ) ≤
+            (272196313427163179113 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1057410578160899 / 10000000000000000 : ℝ)) (q :=
+      (78993868669915009 / 320000000000000000 : ℝ)) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [bρ_16]) (by norm_num) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [aρ_16, bρ_16]) hlg
+
+lemma U_656 :
+    Uρ (194899235804257 / 256000000000000 : ℝ) ≤
+      (-(498081621074527344563 / 1000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (194899235804257 / 256000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(703013473485130443647 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_656_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (194899235804257 / 256000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(568749979081353075301 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_656_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (194899235804257 / 256000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(2907408327145662851251 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_656_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (194899235804257 / 256000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(3013925961973787554127 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_656_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (194899235804257 / 256000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1589025632933657282791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_656_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (194899235804257 / 256000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1708770832688818507607 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_656_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (194899235804257 / 256000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(1876136121166841495429 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_656_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (194899235804257 / 256000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(420336310140069993621 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_656_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (194899235804257 / 256000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(2396178136091364650849 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_656_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) *
+        Uω (aρ 10) (bρ 10) (194899235804257 / 256000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(692557006368090820231 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_656_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) *
+        Uω (aρ 11) (bρ 11) (194899235804257 / 256000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(6467803078306683504167 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_656_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) *
+        Uω (aρ 12) (bρ 12) (194899235804257 / 256000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(237268551693682025271 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_656_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) *
+        Uω (aρ 13) (bρ 13) (194899235804257 / 256000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4464671824597859984033 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_656_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) *
+        Uω (aρ 14) (bρ 14) (194899235804257 / 256000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) * (-(163807362855826435447 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_656_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (194899235804257 / 256000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(2446207738464103347739 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_656_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) *
+        Uω (aρ 16) (bρ 16) (194899235804257 / 256000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(559580302932836820887 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_656_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_657_1 :
+    Uω (aρ 1) (bρ 1) (78210366862569 / 102400000000000 : ℝ) ≤
+      (-(1389838665380847924577 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (96937120940872441 / 128000000000000000 : ℝ) ≤
+      (-(1389838665380847924577 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (96937120940872441 / 128000000000000000 : ℝ) =
+            (96937120940872441 / 64000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (96937120940872441 / 64000000000000000 : ℝ) ≤
+            (2075897236119152075423 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1514638245234713 / 2000000000000000 : ℝ)) (q :=
+      (96937120940872441 / 128000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_657_2 :
+    Uω (aρ 2) (bρ 2) (78210366862569 / 102400000000000 : ℝ) ≤
+      (-(2811270201498031012067 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (483156760039864957 / 640000000000000000 : ℝ) ≤
+      (-(2811270201498031012067 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (483156760039864957 / 640000000000000000 : ℝ) =
+            (483156760039864957 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (483156760039864957 / 320000000000000000 : ℝ) ≤
+            (4120201601501968987933 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3774591921459013 / 5000000000000000 : ℝ)) (q :=
+      (483156760039864957 / 640000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_657_3 :
+    Uω (aρ 3) (bρ 3) (78210366862569 / 102400000000000 : ℝ) ≤
+      (-(57494400179207807219 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (480100844813911869 / 640000000000000000 : ℝ) ≤
+      (-(57494400179207807219 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (480100844813911869 / 640000000000000000 : ℝ) =
+            (480100844813911869 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (480100844813911869 / 320000000000000000 : ℝ) ≤
+            (81135035880792192781 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (234408831201781 / 312500000000000 : ℝ)) (q :=
+      (480100844813911869 / 640000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_657_4 :
+    Uω (aρ 4) (bρ 4) (78210366862569 / 102400000000000 : ℝ) ≤
+      (-(1490442013525983607791 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (475030861445632029 / 640000000000000000 : ℝ) ≤
+      (-(1490442013525983607791 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (475030861445632029 / 640000000000000000 : ℝ) =
+            (475030861445632029 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (475030861445632029 / 320000000000000000 : ℝ) ≤
+            (1975293887974016392209 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7420939799263247 / 10000000000000000 : ℝ)) (q :=
+      (475030861445632029 / 640000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_657_5 :
+    Uω (aρ 5) (bρ 5) (78210366862569 / 102400000000000 : ℝ) ≤
+      (-(3144453088255461251677 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (467324028044376893 / 640000000000000000 : ℝ) ≤
+      (-(3144453088255461251677 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (467324028044376893 / 640000000000000000 : ℝ) =
+            (467324028044376893 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (467324028044376893 / 320000000000000000 : ℝ) ≤
+            (3787018714744538748323 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (114035799606391 / 156250000000000 : ℝ)) (q :=
+      (467324028044376893 / 640000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_657_6 :
+    Uω (aρ 6) (bρ 6) (78210366862569 / 102400000000000 : ℝ) ≤
+      (-(3383106105401640094821 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (456303229529549373 / 640000000000000000 : ℝ) ≤
+      (-(3383106105401640094821 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (456303229529549373 / 640000000000000000 : ℝ) =
+            (456303229529549373 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (456303229529549373 / 320000000000000000 : ℝ) ≤
+            (3548365697598359905179 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (890141449545083 / 1250000000000000 : ℝ)) (q :=
+      (456303229529549373 / 640000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_657_7 :
+    Uω (aρ 7) (bρ 7) (78210366862569 / 102400000000000 : ℝ) ≤
+      (-(1858306114393186210517 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (441336204386488477 / 640000000000000000 : ℝ) ≤
+      (-(1858306114393186210517 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (441336204386488477 / 640000000000000000 : ℝ) =
+            (441336204386488477 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (441336204386488477 / 320000000000000000 : ℝ) ≤
+            (1607429787106813789483 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6877073441400011 / 10000000000000000 : ℝ)) (q :=
+      (441336204386488477 / 640000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_657_8 :
+    Uω (aρ 8) (bρ 8) (78210366862569 / 102400000000000 : ℝ) ≤
+      (-(2082972537469778711169 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (421944449282505021 / 640000000000000000 : ℝ) ≤
+      (-(2082972537469778711169 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (421944449282505021 / 640000000000000000 : ℝ) =
+            (421944449282505021 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (421944449282505021 / 320000000000000000 : ℝ) ≤
+            (1382763364030221288831 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (409663921608783 / 625000000000000 : ℝ)) (q :=
+      (421944449282505021 / 640000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8])
+      (by norm_num [bρ_8]) (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_657_9 :
+    Uω (aρ 9) (bρ 9) (78210366862569 / 102400000000000 : ℝ) ≤
+      (-(4752434701677612012609 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (397909545135901917 / 640000000000000000 : ℝ) ≤
+      (-(4752434701677612012609 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (397909545135901917 / 640000000000000000 : ℝ) =
+            (397909545135901917 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (397909545135901917 / 320000000000000000 : ℝ) ≤
+            (2179037101322387987391 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6144497619519181 / 10000000000000000 : ℝ)) (q :=
+      (397909545135901917 / 640000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9])
+      (by norm_num [bρ_9]) (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_657_10 :
+    Uω (aρ 10) (bρ 10) (78210366862569 / 102400000000000 : ℝ) ≤
+      (-(1374240064175814711503 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (369360137484945533 / 640000000000000000 : ℝ) ≤
+      (-(1374240064175814711503 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (369360137484945533 / 640000000000000000 : ℝ) =
+            (369360137484945533 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (369360137484945533 / 320000000000000000 : ℝ) ≤
+            (358627886574185288497 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2820443831960897 / 5000000000000000 : ℝ)) (q :=
+      (369360137484945533 / 640000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_657_11 :
+    Uω (aρ 11) (bρ 11) (78210366862569 / 102400000000000 : ℝ) ≤
+      (-(6419128159047950867803 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (67364451149068953 / 128000000000000000 : ℝ) ≤
+      (-(6419128159047950867803 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (67364451149068953 / 128000000000000000 : ℝ) =
+            (67364451149068953 / 64000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (67364451149068953 / 64000000000000000 : ℝ) ≤
+            (512343643952049132197 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (504255548576427 / 1000000000000000 : ℝ)) (q :=
+      (67364451149068953 / 128000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_657_12 :
+    Uω (aρ 12) (bρ 12) (78210366862569 / 102400000000000 : ℝ) ≤
+      (-(942022814831744407111 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (301222719783302717 / 640000000000000000 : ℝ) ≤
+      (-(942022814831744407111 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (301222719783302717 / 640000000000000000 : ℝ) =
+            (301222719783302717 / 160000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (301222719783302717 / 160000000000000000 : ℝ) ≤
+            (790845135918255592889 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (17006914389201 / 39062500000000 : ℝ)) (q :=
+      (301222719783302717 / 640000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_657_13 :
+    Uω (aρ 13) (bρ 13) (78210366862569 / 102400000000000 : ℝ) ≤
+      (-(4430380667806009793779 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (263852976883560349 / 640000000000000000 : ℝ) ≤
+      (-(4430380667806009793779 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (263852976883560349 / 640000000000000000 : ℝ) =
+            (263852976883560349 / 160000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (263852976883560349 / 160000000000000000 : ℝ) ≤
+            (2501091135193990206221 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3585347359613507 / 10000000000000000 : ℝ)) (q :=
+      (263852976883560349 / 640000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_657_14 :
+    Uω (aρ 14) (bρ 14) (78210366862569 / 102400000000000 : ℝ) ≤
+      (-(5197128077412783468093 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (45268193150524441 / 128000000000000000 : ℝ) ≤
+      (-(5197128077412783468093 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (45268193150524441 / 128000000000000000 : ℝ) =
+            (45268193150524441 / 32000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (45268193150524441 / 32000000000000000 : ℝ) ≤
+            (1734343725587216531907 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (275702429681669 / 1000000000000000 : ℝ)) (q :=
+      (45268193150524441 / 128000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_657_15 :
+    Uω (aρ 15) (bρ 15) (78210366862569 / 102400000000000 : ℝ) ≤
+      (-(12100863543607861278483 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (190829779040784189 / 640000000000000000 : ℝ) ≤
+      (-(12100863543607861278483 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (190829779040784189 / 640000000000000000 : ℝ) =
+            (190829779040784189 / 160000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (190829779040784189 / 160000000000000000 : ℝ) ≤
+            (1762080062392138721517 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (59589780921461 / 312500000000000 : ℝ)) (q :=
+      (190829779040784189 / 640000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_657_16 :
+    Uω (aρ 16) (bρ 16) (78210366862569 / 102400000000000 : ℝ) ≤
+      (-(6883547383324041154463 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (161540954567697789 / 640000000000000000 : ℝ) ≤
+      (-(6883547383324041154463 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (161540954567697789 / 640000000000000000 : ℝ) =
+            (161540954567697789 / 160000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (161540954567697789 / 160000000000000000 : ℝ) ≤
+            (47924419675958845537 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (571984438106401 / 5000000000000000 : ℝ)) (q :=
+      (161540954567697789 / 640000000000000000 : ℝ)) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [bρ_16]) (by norm_num) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [aρ_16, bρ_16]) hlg
+
+lemma U_657 :
+    Uρ (78210366862569 / 102400000000000 : ℝ) ≤
+      (-(4936472451356495359701 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (78210366862569 / 102400000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1389838665380847924577 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_657_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (78210366862569 / 102400000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(2811270201498031012067 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_657_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (78210366862569 / 102400000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) * (-(57494400179207807219 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_657_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (78210366862569 / 102400000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1490442013525983607791 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_657_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (78210366862569 / 102400000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(3144453088255461251677 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_657_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (78210366862569 / 102400000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3383106105401640094821 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_657_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (78210366862569 / 102400000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(1858306114393186210517 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_657_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (78210366862569 / 102400000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(2082972537469778711169 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_657_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (78210366862569 / 102400000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(4752434701677612012609 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_657_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (78210366862569 / 102400000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(1374240064175814711503 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_657_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (78210366862569 / 102400000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(6419128159047950867803 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_657_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (78210366862569 / 102400000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(942022814831744407111 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_657_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (78210366862569 / 102400000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4430380667806009793779 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_657_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (78210366862569 / 102400000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(5197128077412783468093 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_657_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (78210366862569 / 102400000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(12100863543607861278483 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_657_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (78210366862569 / 102400000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) *
+        (-(6883547383324041154463 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_657_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_658_1 :
+    Uω (aρ 1) (bρ 1) (49038149627147 / 64000000000000 : ℝ) ≤
+      (-(171712828384726161937 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (60781539060846823 / 80000000000000000 : ℝ) ≤
+      (-(171712828384726161937 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (60781539060846823 / 80000000000000000 : ℝ) =
+            (60781539060846823 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (60781539060846823 / 40000000000000000 : ℝ) ≤
+            (261504159302773838063 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7597671104054987 / 10000000000000000 : ℝ)) (q :=
+      (60781539060846823 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_658_2 :
+    Uω (aρ 2) (bρ 2) (49038149627147 / 64000000000000 : ℝ) ≤
+      (-(2778895663106141822649 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (60590436561394599 / 80000000000000000 : ℝ) ≤
+      (-(2778895663106141822649 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (60590436561394599 / 80000000000000000 : ℝ) =
+            (60590436561394599 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (60590436561394599 / 40000000000000000 : ℝ) ≤
+            (4152576139893858177351 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7573664491701931 / 10000000000000000 : ℝ)) (q :=
+      (60590436561394599 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_658_3 :
+    Uω (aρ 3) (bρ 3) (49038149627147 / 64000000000000 : ℝ) ≤
+      (-(2842138209295904021429 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (60208456356289091 / 80000000000000000 : ℝ) ≤
+      (-(2842138209295904021429 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (60208456356289091 / 80000000000000000 : ℝ) =
+            (60208456356289091 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (60208456356289091 / 40000000000000000 : ℝ) ≤
+            (4089333593704095978571 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3762782773387777 / 5000000000000000 : ℝ)) (q :=
+      (60208456356289091 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_658_4 :
+    Uω (aρ 4) (bρ 4) (49038149627147 / 64000000000000 : ℝ) ≤
+      (-(1473975475526614955667 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (59574732885589659 / 80000000000000000 : ℝ) ≤
+      (-(1473975475526614955667 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (59574732885589659 / 80000000000000000 : ℝ) =
+            (59574732885589659 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (59574732885589659 / 40000000000000000 : ℝ) ≤
+            (1991760425973385044333 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (116334825940089 / 156250000000000 : ℝ)) (q :=
+      (59574732885589659 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_658_5 :
+    Uω (aρ 5) (bρ 5) (49038149627147 / 64000000000000 : ℝ) ≤
+      (-(12443870114242395501 / 40000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11722287791169907 / 16000000000000000 : ℝ) ≤
+      (-(12443870114242395501 / 40000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11722287791169907 / 16000000000000000 : ℝ) =
+            (11722287791169907 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11722287791169907 / 8000000000000000 : ℝ) ≤
+            (15282017097757604499 / 40000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1464559059413133 / 2000000000000000 : ℝ)) (q :=
+      (11722287791169907 / 16000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_658_6 :
+    Uω (aρ 6) (bρ 6) (49038149627147 / 64000000000000 : ℝ) ≤
+      (-(3348789004045531172871 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (57233977485193759 / 80000000000000000 : ℝ) ≤
+      (-(3348789004045531172871 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (57233977485193759 / 80000000000000000 : ℝ) =
+            (57233977485193759 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (57233977485193759 / 40000000000000000 : ℝ) ≤
+            (3582682798954468827129 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7145670304541721 / 10000000000000000 : ℝ)) (q :=
+      (57233977485193759 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_658_7 :
+    Uω (aρ 7) (bρ 7) (49038149627147 / 64000000000000 : ℝ) ≤
+      (-(3681079617672882922721 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11072679413570503 / 16000000000000000 : ℝ) ≤
+      (-(3681079617672882922721 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11072679413570503 / 16000000000000000 : ℝ) =
+            (11072679413570503 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11072679413570503 / 8000000000000000 : ℝ) ≤
+            (3250392185327117077279 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (690168658096641 / 1000000000000000 : ℝ)) (q :=
+      (11072679413570503 / 16000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_658_8 :
+    Uω (aρ 8) (bρ 8) (49038149627147 / 64000000000000 : ℝ) ≤
+      (-(3225521999879289743 / 7812500000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (52940032909125299 / 80000000000000000 : ℝ) ≤
+      (-(3225521999879289743 / 7812500000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (52940032909125299 / 80000000000000000 : ℝ) =
+            (52940032909125299 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (52940032909125299 / 40000000000000000 : ℝ) ≤
+            (2189690346214460257 / 7812500000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3289693596312303 / 5000000000000000 : ℝ)) (q :=
+      (52940032909125299 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_658_9 :
+    Uω (aρ 9) (bρ 9) (49038149627147 / 64000000000000 : ℝ) ≤
+      (-(4712675619102600426311 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (49936843275116379 / 80000000000000000 : ℝ) ≤
+      (-(4712675619102600426311 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (49936843275116379 / 80000000000000000 : ℝ) =
+            (49936843275116379 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (49936843275116379 / 40000000000000000 : ℝ) ≤
+            (2218796183897399573689 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (771194426560297 / 1250000000000000 : ℝ)) (q :=
+      (49936843275116379 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_658_10 :
+    Uω (aρ 10) (bρ 10) (49038149627147 / 64000000000000 : ℝ) ≤
+      (-(2726830749623017945813 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (46370361042211667 / 80000000000000000 : ℝ) ≤
+      (-(2726830749623017945813 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (46370361042211667 / 80000000000000000 : ℝ) =
+            (46370361042211667 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (46370361042211667 / 40000000000000000 : ℝ) ≤
+            (738905151876982054187 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2833246943875599 / 5000000000000000 : ℝ)) (q :=
+      (46370361042211667 / 80000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_658_11 :
+    Uω (aρ 11) (bρ 11) (49038149627147 / 64000000000000 : ℝ) ≤
+      (-(6370709510127633117299 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (42307132269084127 / 80000000000000000 : ℝ) ≤
+      (-(6370709510127633117299 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (42307132269084127 / 80000000000000000 : ℝ) =
+            (42307132269084127 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (42307132269084127 / 40000000000000000 : ℝ) ≤
+            (560762292872366882701 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5069163320674313 / 10000000000000000 : ℝ)) (q :=
+      (42307132269084127 / 80000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_658_12 :
+    Uω (aρ 12) (bρ 12) (49038149627147 / 64000000000000 : ℝ) ≤
+      (-(7480138777450943837259 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7572890600044083 / 16000000000000000 : ℝ) ≤
+      (-(7480138777450943837259 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7572890600044083 / 16000000000000000 : ℝ) =
+            (7572890600044083 / 4000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7572890600044083 / 4000000000000000 : ℝ) ≤
+            (6382804828549056162741 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (876438720028677 / 2000000000000000 : ℝ)) (q :=
+      (7572890600044083 / 16000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_658_13 :
+    Uω (aρ 13) (bρ 13) (49038149627147 / 64000000000000 : ℝ) ≤
+      (-(8792784974843993706773 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (33206582909350039 / 80000000000000000 : ℝ) ≤
+      (-(8792784974843993706773 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (33206582909350039 / 80000000000000000 : ℝ) =
+            (33206582909350039 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (33206582909350039 / 20000000000000000 : ℝ) ≤
+            (5070158631156006293227 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3617107819020791 / 10000000000000000 : ℝ)) (q :=
+      (33206582909350039 / 80000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_658_14 :
+    Uω (aρ 14) (bρ 14) (49038149627147 / 64000000000000 : ℝ) ≤
+      (-(644129719662722860471 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28543210092723919 / 80000000000000000 : ℝ) ≤
+      (-(644129719662722860471 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (28543210092723919 / 80000000000000000 : ℝ) =
+            (28543210092723919 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28543210092723919 / 20000000000000000 : ℝ) ≤
+            (222304255712277139529 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2795191899909261 / 10000000000000000 : ℝ)) (q :=
+      (28543210092723919 / 80000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_658_15 :
+    Uω (aρ 15) (bρ 15) (49038149627147 / 64000000000000 : ℝ) ≤
+      (-(11974197185992771512583 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (24157790487552283 / 80000000000000000 : ℝ) ≤
+      (-(11974197185992771512583 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (24157790487552283 / 80000000000000000 : ℝ) =
+            (24157790487552283 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24157790487552283 / 20000000000000000 : ℝ) ≤
+            (1888746420007228487417 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (244801284503919 / 1250000000000000 : ℝ)) (q :=
+      (24157790487552283 / 80000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_658_16 :
+    Uω (aρ 16) (bρ 16) (49038149627147 / 64000000000000 : ℝ) ≤
+      (-(13560409783495217840437 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20614313318481031 / 80000000000000000 : ℝ) ≤
+      (-(13560409783495217840437 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (20614313318481031 / 80000000000000000 : ℝ) =
+            (20614313318481031 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20614313318481031 / 20000000000000000 : ℝ) ≤
+            (302533822504782159563 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1224912635273539 / 10000000000000000 : ℝ)) (q :=
+      (20614313318481031 / 80000000000000000 : ℝ)) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [bρ_16]) (by norm_num) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [aρ_16, bρ_16]) hlg
+
+lemma U_658 :
+    Uρ (49038149627147 / 64000000000000 : ℝ) ≤
+      (-(1223138141627158991151 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (49038149627147 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(171712828384726161937 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_658_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (49038149627147 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(2778895663106141822649 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_658_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (49038149627147 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(2842138209295904021429 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_658_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (49038149627147 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1473975475526614955667 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_658_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (49038149627147 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(12443870114242395501 / 40000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_658_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (49038149627147 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3348789004045531172871 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_658_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (49038149627147 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(3681079617672882922721 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_658_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (49038149627147 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) * (-(3225521999879289743 / 7812500000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_658_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (49038149627147 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(4712675619102600426311 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_658_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (49038149627147 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(2726830749623017945813 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_658_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (49038149627147 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(6370709510127633117299 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_658_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (49038149627147 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(7480138777450943837259 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_658_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (49038149627147 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(8792784974843993706773 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_658_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (49038149627147 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) * (-(644129719662722860471 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_658_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (49038149627147 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(11974197185992771512583 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_658_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (49038149627147 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) *
+        (-(13560409783495217840437 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_658_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_659_1 :
+    Uω (aρ 1) (bρ 1) (393558559721507 / 512000000000000 : ℝ) ≤
+      (-(2715236991883287655507 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (487819020240911443 / 640000000000000000 : ℝ) ≤
+      (-(2715236991883287655507 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (487819020240911443 / 640000000000000000 : ℝ) =
+            (487819020240911443 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (487819020240911443 / 320000000000000000 : ℝ) ≤
+            (4216234811116712344493 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7622150981052799 / 10000000000000000 : ℝ)) (q :=
+      (487819020240911443 / 640000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_659_2 :
+    Uω (aρ 2) (bρ 2) (393558559721507 / 512000000000000 : ℝ) ≤
+      (-(10986502406018490339 / 40000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (486290224755123731 / 640000000000000000 : ℝ) ≤
+      (-(10986502406018490339 / 40000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (486290224755123731 / 640000000000000000 : ℝ) =
+            (486290224755123731 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (486290224755123731 / 320000000000000000 : ℝ) ≤
+            (16739384805981509661 / 40000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7598145134631933 / 10000000000000000 : ℝ)) (q :=
+      (486290224755123731 / 640000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_659_3 :
+    Uω (aρ 3) (bρ 3) (393558559721507 / 512000000000000 : ℝ) ≤
+      (-(2809662236111423922899 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (483234456220980979 / 640000000000000000 : ℝ) ≤
+      (-(2809662236111423922899 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (483234456220980979 / 640000000000000000 : ℝ) =
+            (483234456220980979 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (483234456220980979 / 320000000000000000 : ℝ) ≤
+            (4121809566888576077101 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1887512118572493 / 2500000000000000 : ℝ)) (q :=
+      (483234456220980979 / 640000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_659_4 :
+    Uω (aρ 4) (bρ 4) (393558559721507 / 512000000000000 : ℝ) ≤
+      (-(1457563009387830351699 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (478164862768577587 / 640000000000000000 : ℝ) ≤
+      (-(1457563009387830351699 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (478164862768577587 / 640000000000000000 : ℝ) =
+            (478164862768577587 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (478164862768577587 / 320000000000000000 : ℝ) ≤
+            (2008172892112169648301 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3734958929983683 / 5000000000000000 : ℝ)) (q :=
+      (478164862768577587 / 640000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_659_5 :
+    Uω (aρ 5) (bρ 5) (393558559721507 / 512000000000000 : ℝ) ≤
+      (-(3077593833591589989139 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (470458990047537363 / 640000000000000000 : ℝ) ≤
+      (-(3077593833591589989139 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (470458990047537363 / 640000000000000000 : ℝ) =
+            (470458990047537363 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (470458990047537363 / 320000000000000000 : ℝ) ≤
+            (3853877969408410010861 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7347299256769859 / 10000000000000000 : ℝ)) (q :=
+      (470458990047537363 / 640000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_659_6 :
+    Uω (aρ 6) (bρ 6) (393558559721507 / 512000000000000 : ℝ) ≤
+      (-(1657294773584553884893 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (459440397333449523 / 640000000000000000 : ℝ) ≤
+      (-(1657294773584553884893 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (459440397333449523 / 640000000000000000 : ℝ) =
+            (459440397333449523 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (459440397333449523 / 320000000000000000 : ℝ) ≤
+            (1808441127915446115107 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3585104304797307 / 5000000000000000 : ℝ)) (q :=
+      (459440397333449523 / 640000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_659_7 :
+    Uω (aρ 7) (bρ 7) (393558559721507 / 512000000000000 : ℝ) ≤
+      (-(729134699404871384131 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (444478118442517427 / 640000000000000000 : ℝ) ≤
+      (-(729134699404871384131 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (444478118442517427 / 640000000000000000 : ℝ) =
+            (444478118442517427 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (444478118442517427 / 320000000000000000 : ℝ) ≤
+            (657159661195128615869 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3463149387506493 / 5000000000000000 : ℝ)) (q :=
+      (444478118442517427 / 640000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_659_8 :
+    Uω (aρ 8) (bρ 8) (393558559721507 / 512000000000000 : ℝ) ≤
+      (-(4091531283765524748763 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (425096009329768787 / 640000000000000000 : ℝ) ≤
+      (-(4091531283765524748763 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (425096009329768787 / 640000000000000000 : ℝ) =
+            (425096009329768787 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (425096009329768787 / 320000000000000000 : ℝ) ≤
+            (2839940519234475251237 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6604149516579591 / 10000000000000000 : ℝ)) (q :=
+      (425096009329768787 / 640000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8])
+      (by norm_num [bρ_8]) (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_659_9 :
+    Uω (aρ 9) (bρ 9) (393558559721507 / 512000000000000 : ℝ) ≤
+      (-(2336538838658455572537 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (401079799342275667 / 640000000000000000 : ℝ) ≤
+      (-(2336538838658455572537 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (401079799342275667 / 640000000000000000 : ℝ) =
+            (401079799342275667 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (401079799342275667 / 320000000000000000 : ℝ) ≤
+            (1129197062841544427463 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6194608582830431 / 10000000000000000 : ℝ)) (q :=
+      (401079799342275667 / 640000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9])
+      (by norm_num [bρ_9]) (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_659_10 :
+    Uω (aρ 10) (bρ 10) (393558559721507 / 512000000000000 : ℝ) ≤
+      (-(2705278962857846117447 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (372565322001013043 / 640000000000000000 : ℝ) ≤
+      (-(2705278962857846117447 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (372565322001013043 / 640000000000000000 : ℝ) =
+            (372565322001013043 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (372565322001013043 / 320000000000000000 : ℝ) ≤
+            (760456938642153882553 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2846045099705487 / 5000000000000000 : ℝ)) (q :=
+      (372565322001013043 / 640000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_659_11 :
+    Uω (aρ 11) (bρ 11) (393558559721507 / 512000000000000 : ℝ) ≤
+      (-(1580636060845473348809 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (340091177832826643 / 640000000000000000 : ℝ) ≤
+      (-(1580636060845473348809 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (340091177832826643 / 640000000000000000 : ℝ) =
+            (340091177832826643 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (340091177832826643 / 320000000000000000 : ℝ) ≤
+            (152231889904526651191 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5095749820360149 / 10000000000000000 : ℝ)) (q :=
+      (340091177832826643 / 640000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_659_12 :
+    Uω (aρ 12) (bρ 12) (393558559721507 / 512000000000000 : ℝ) ≤
+      (-(742445707098810187001 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (304607014493012307 / 640000000000000000 : ℝ) ≤
+      (-(742445707098810187001 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (304607014493012307 / 640000000000000000 : ℝ) =
+            (304607014493012307 / 160000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (304607014493012307 / 160000000000000000 : ℝ) ≤
+            (643848653501189812999 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4410569812675951 / 10000000000000000 : ℝ)) (q :=
+      (304607014493012307 / 640000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_659_13 :
+    Uω (aρ 13) (bρ 13) (393558559721507 / 512000000000000 : ℝ) ≤
+      (-(8725401843653626065141 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (267448758631353747 / 640000000000000000 : ℝ) ≤
+      (-(8725401843653626065141 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (267448758631353747 / 640000000000000000 : ℝ) =
+            (267448758631353747 / 160000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (267448758631353747 / 160000000000000000 : ℝ) ≤
+            (5137541762346373934859 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3648756058594121 / 10000000000000000 : ℝ)) (q :=
+      (267448758631353747 / 640000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_659_14 :
+    Uω (aρ 14) (bρ 14) (393558559721507 / 512000000000000 : ℝ) ≤
+      (-(5109543065045329487741 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (230340710504174291 / 640000000000000000 : ℝ) ≤
+      (-(5109543065045329487741 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (230340710504174291 / 640000000000000000 : ℝ) =
+            (230340710504174291 / 160000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (230340710504174291 / 160000000000000000 : ℝ) ≤
+            (1821928737954670512259 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2833056839664763 / 10000000000000000 : ℝ)) (q :=
+      (230340710504174291 / 640000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_659_15 :
+    Uω (aρ 15) (bρ 15) (393558559721507 / 512000000000000 : ℝ) ≤
+      (-(5925394563411376445939 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (195662113945982771 / 640000000000000000 : ℝ) ≤
+      (-(5925394563411376445939 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (195662113945982771 / 640000000000000000 : ℝ) =
+            (195662113945982771 / 160000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (195662113945982771 / 160000000000000000 : ℝ) ≤
+            (1006077239588623554061 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1004461987318139 / 5000000000000000 : ℝ)) (q :=
+      (195662113945982771 / 640000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_659_16 :
+    Uω (aρ 16) (bρ 16) (393558559721507 / 512000000000000 : ℝ) ≤
+      (-(6683298545853789424627 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (168141933781848243 / 640000000000000000 : ℝ) ≤
+      (-(6683298545853789424627 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (168141933781848243 / 640000000000000000 : ℝ) =
+            (168141933781848243 / 160000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (168141933781848243 / 160000000000000000 : ℝ) ≤
+            (248173257146210575373 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (650644998008537 / 5000000000000000 : ℝ)) (q :=
+      (168141933781848243 / 640000000000000000 : ℝ)) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [bρ_16]) (by norm_num) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [aρ_16, bρ_16]) hlg
+
+lemma U_659 :
+    Uρ (393558559721507 / 512000000000000 : ℝ) ≤
+      (-(2424513986067734309671 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (393558559721507 / 512000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2715236991883287655507 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_659_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (393558559721507 / 512000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) * (-(10986502406018490339 / 40000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_659_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (393558559721507 / 512000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(2809662236111423922899 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_659_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (393558559721507 / 512000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1457563009387830351699 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_659_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (393558559721507 / 512000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(3077593833591589989139 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_659_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (393558559721507 / 512000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1657294773584553884893 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_659_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (393558559721507 / 512000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(729134699404871384131 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_659_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (393558559721507 / 512000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(4091531283765524748763 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_659_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (393558559721507 / 512000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(2336538838658455572537 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_659_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) *
+        Uω (aρ 10) (bρ 10) (393558559721507 / 512000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(2705278962857846117447 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_659_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) *
+        Uω (aρ 11) (bρ 11) (393558559721507 / 512000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(1580636060845473348809 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_659_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) *
+        Uω (aρ 12) (bρ 12) (393558559721507 / 512000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(742445707098810187001 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_659_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) *
+        Uω (aρ 13) (bρ 13) (393558559721507 / 512000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(8725401843653626065141 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_659_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) *
+        Uω (aρ 14) (bρ 14) (393558559721507 / 512000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(5109543065045329487741 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_659_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (393558559721507 / 512000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(5925394563411376445939 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_659_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) *
+        Uω (aρ 16) (bρ 16) (393558559721507 / 512000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) *
+        (-(6683298545853789424627 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_659_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_660_1 :
+    Uω (aρ 1) (bρ 1) (197405961212919 / 256000000000000 : ℝ) ≤
+      (-(2683171878172451050851 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (244692863983522167 / 320000000000000000 : ℝ) ≤
+      (-(2683171878172451050851 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (244692863983522167 / 320000000000000000 : ℝ) =
+            (244692863983522167 / 160000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (244692863983522167 / 160000000000000000 : ℝ) ≤
+            (4248299924827548949149 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7646630857175487 / 10000000000000000 : ℝ)) (q :=
+      (244692863983522167 / 320000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_660_2 :
+    Uω (aρ 2) (bρ 2) (197405961212919 / 256000000000000 : ℝ) ≤
+      (-(1357229672258682030653 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (48785695683357067 / 64000000000000000 : ℝ) ≤
+      (-(1357229672258682030653 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (48785695683357067 / 64000000000000000 : ℝ) =
+            (48785695683357067 / 32000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (48785695683357067 / 32000000000000000 : ℝ) ≤
+            (2108506229241317969347 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1524525154352887 / 2000000000000000 : ℝ)) (q :=
+      (48785695683357067 / 64000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_660_3 :
+    Uω (aρ 3) (bρ 3) (197405961212919 / 256000000000000 : ℝ) ≤
+      (-(1388645702044695172863 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (242400630466186087 / 320000000000000000 : ℝ) ≤
+      (-(1388645702044695172863 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (242400630466186087 / 320000000000000000 : ℝ) =
+            (242400630466186087 / 160000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (242400630466186087 / 160000000000000000 : ℝ) ≤
+            (2077090199455304827137 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3787265690600991 / 5000000000000000 : ℝ)) (q :=
+      (242400630466186087 / 320000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_660_4 :
+    Uω (aρ 4) (bρ 4) (197405961212919 / 256000000000000 : ℝ) ≤
+      (-(720602130511302706029 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (239865930258189943 / 320000000000000000 : ℝ) ≤
+      (-(720602130511302706029 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (239865930258189943 / 320000000000000000 : ℝ) =
+            (239865930258189943 / 160000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (239865930258189943 / 160000000000000000 : ℝ) ≤
+            (1012265820238697293971 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7494406799267223 / 10000000000000000 : ℝ)) (q :=
+      (239865930258189943 / 320000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_660_5 :
+    Uω (aρ 5) (bρ 5) (197405961212919 / 256000000000000 : ℝ) ≤
+      (-(1522165628847763865193 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (47202646329847147 / 64000000000000000 : ℝ) ≤
+      (-(1522165628847763865193 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (47202646329847147 / 64000000000000000 : ℝ) =
+            (47202646329847147 / 32000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (47202646329847147 / 32000000000000000 : ℝ) ≤
+            (1943570272652236134807 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1474360611108517 / 2000000000000000 : ℝ)) (q :=
+      (47202646329847147 / 64000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_660_6 :
+    Uω (aρ 6) (bρ 6) (197405961212919 / 256000000000000 : ℝ) ≤
+      (-(20503168306351268131 / 62500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (230504481008619239 / 320000000000000000 : ℝ) ≤
+      (-(20503168306351268131 / 62500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (230504481008619239 / 320000000000000000 : ℝ) =
+            (230504481008619239 / 160000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (230504481008619239 / 160000000000000000 : ℝ) ≤
+            (22818530462398731869 / 62500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3597373257822027 / 5000000000000000 : ℝ)) (q :=
+      (230504481008619239 / 320000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_660_7 :
+    Uω (aρ 7) (bρ 7) (197405961212919 / 256000000000000 : ℝ) ≤
+      (-(1805196482340114028887 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (223024515203491959 / 320000000000000000 : ℝ) ≤
+      (-(1805196482340114028887 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (223024515203491959 / 320000000000000000 : ℝ) =
+            (223024515203491959 / 160000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (223024515203491959 / 160000000000000000 : ℝ) ≤
+            (1660539419159885971113 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6950910033583599 / 10000000000000000 : ℝ)) (q :=
+      (223024515203491959 / 320000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_660_8 :
+    Uω (aρ 8) (bρ 8) (197405961212919 / 256000000000000 : ℝ) ≤
+      (-(4054533386656685392731 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (42667168821410459 / 64000000000000000 : ℝ) ≤
+      (-(4054533386656685392731 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (42667168821410459 / 64000000000000000 : ℝ) =
+            (42667168821410459 / 32000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (42667168821410459 / 32000000000000000 : ℝ) ≤
+            (2876938416343314607269 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (165722743534903 / 250000000000000 : ℝ)) (q :=
+      (42667168821410459 / 64000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_660_9 :
+    Uω (aρ 9) (bρ 9) (197405961212919 / 256000000000000 : ℝ) ≤
+      (-(1158409886553347536791 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (201332353173736791 / 320000000000000000 : ℝ) ≤
+      (-(1158409886553347536791 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (201332353173736791 / 320000000000000000 : ℝ) =
+            (201332353173736791 / 160000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (201332353173736791 / 160000000000000000 : ℝ) ≤
+            (574458064196652463209 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6219657186423901 / 10000000000000000 : ℝ)) (q :=
+      (201332353173736791 / 320000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9])
+      (by norm_num [bρ_9]) (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_660_10 :
+    Uω (aρ 10) (bρ 10) (197405961212919 / 256000000000000 : ℝ) ≤
+      (-(5367647710234672246907 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (187083721367394247 / 320000000000000000 : ℝ) ≤
+      (-(5367647710234672246907 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (187083721367394247 / 320000000000000000 : ℝ) =
+            (187083721367394247 / 160000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (187083721367394247 / 160000000000000000 : ℝ) ≤
+            (1563824092765327753093 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1429419183005623 / 2500000000000000 : ℝ)) (q :=
+      (187083721367394247 / 320000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_660_11 :
+    Uω (aρ 11) (bρ 11) (197405961212919 / 256000000000000 : ℝ) ≤
+      (-(1568657380525846977687 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (170862312708270567 / 320000000000000000 : ℝ) ≤
+      (-(1568657380525846977687 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (170862312708270567 / 320000000000000000 : ℝ) =
+            (170862312708270567 / 160000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (170862312708270567 / 160000000000000000 : ℝ) ≤
+            (164210570224153022313 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2561157658516131 / 5000000000000000 : ℝ)) (q :=
+      (170862312708270567 / 320000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_660_12 :
+    Uω (aρ 12) (bρ 12) (197405961212919 / 256000000000000 : ℝ) ≤
+      (-(3684566083047103674919 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (153148460143557719 / 320000000000000000 : ℝ) ≤
+      (-(3684566083047103674919 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (153148460143557719 / 320000000000000000 : ℝ) =
+            (153148460143557719 / 80000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (153148460143557719 / 80000000000000000 : ℝ) ≤
+            (3246905719952896325081 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4438899628422709 / 10000000000000000 : ℝ)) (q :=
+      (153148460143557719 / 320000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_660_13 :
+    Uω (aρ 13) (bρ 13) (197405961212919 / 256000000000000 : ℝ) ≤
+      (-(4329299829043131844737 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (134620677797613799 / 320000000000000000 : ℝ) ≤
+      (-(4329299829043131844737 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (134620677797613799 / 320000000000000000 : ℝ) =
+            (134620677797613799 / 80000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (134620677797613799 / 80000000000000000 : ℝ) ≤
+            (2602171973956868155263 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1840147486698107 / 5000000000000000 : ℝ)) (q :=
+      (134620677797613799 / 320000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_660_14 :
+    Uω (aρ 14) (bρ 14) (197405961212919 / 256000000000000 : ℝ) ≤
+      (-(1013324731520483938837 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (116163219149155479 / 320000000000000000 : ℝ) ≤
+      (-(1013324731520483938837 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (116163219149155479 / 320000000000000000 : ℝ) =
+            (116163219149155479 / 80000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (116163219149155479 / 80000000000000000 : ℝ) ≤
+            (372969629079516061163 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2870631092912569 / 10000000000000000 : ℝ)) (q :=
+      (116163219149155479 / 320000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_660_15 :
+    Uω (aρ 15) (bρ 15) (197405961212919 / 256000000000000 : ℝ) ≤
+      (-(11730417611087330118851 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (99015780276553943 / 320000000000000000 : ℝ) ≤
+      (-(11730417611087330118851 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (99015780276553943 / 320000000000000000 : ℝ) =
+            (99015780276553943 / 80000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (99015780276553943 / 80000000000000000 : ℝ) ≤
+            (2132525994912669881149 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2058489440789973 / 10000000000000000 : ℝ)) (q :=
+      (99015780276553943 / 320000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_660_16 :
+    Uω (aρ 16) (bρ 16) (197405961212919 / 256000000000000 : ℝ) ≤
+      (-(659178816886656724683 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (85623806816932631 / 320000000000000000 : ℝ) ≤
+      (-(659178816886656724683 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (85623806816932631 / 320000000000000000 : ℝ) =
+            (85623806816932631 / 80000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (85623806816932631 / 80000000000000000 : ℝ) ≤
+            (33968363413343275317 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1373862751073641 / 10000000000000000 : ℝ)) (q :=
+      (85623806816932631 / 320000000000000000 : ℝ)) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [bρ_16]) (by norm_num) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [aρ_16, bρ_16]) hlg
+
+lemma U_660 :
+    Uρ (197405961212919 / 256000000000000 : ℝ) ≤
+      (-(2402937868237377408313 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (197405961212919 / 256000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2683171878172451050851 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_660_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (197405961212919 / 256000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1357229672258682030653 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_660_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (197405961212919 / 256000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1388645702044695172863 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_660_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (197405961212919 / 256000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(720602130511302706029 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_660_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (197405961212919 / 256000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1522165628847763865193 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_660_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (197405961212919 / 256000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) * (-(20503168306351268131 / 62500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_660_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (197405961212919 / 256000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(1805196482340114028887 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_660_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (197405961212919 / 256000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(4054533386656685392731 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_660_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (197405961212919 / 256000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(1158409886553347536791 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_660_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) *
+        Uω (aρ 10) (bρ 10) (197405961212919 / 256000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(5367647710234672246907 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_660_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) *
+        Uω (aρ 11) (bρ 11) (197405961212919 / 256000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(1568657380525846977687 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_660_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) *
+        Uω (aρ 12) (bρ 12) (197405961212919 / 256000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(3684566083047103674919 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_660_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) *
+        Uω (aρ 13) (bρ 13) (197405961212919 / 256000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4329299829043131844737 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_660_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) *
+        Uω (aρ 14) (bρ 14) (197405961212919 / 256000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(1013324731520483938837 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_660_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (197405961212919 / 256000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(11730417611087330118851 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_660_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) *
+        Uω (aρ 16) (bρ 16) (197405961212919 / 256000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(659178816886656724683 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_660_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_661_1 :
+    Uω (aρ 1) (bρ 1) (396065285130169 / 512000000000000 : ℝ) ≤
+      (-(662802313408731620701 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (490952435665441417 / 640000000000000000 : ℝ) ≤
+      (-(662802313408731620701 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (490952435665441417 / 640000000000000000 : ℝ) =
+            (490952435665441417 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (490952435665441417 / 320000000000000000 : ℝ) ≤
+            (1070065637341268379299 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7671110732431431 / 10000000000000000 : ℝ)) (q :=
+      (490952435665441417 / 640000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_661_2 :
+    Uω (aρ 2) (bρ 2) (396065285130169 / 512000000000000 : ℝ) ≤
+      (-(2682396226434606046731 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (489423688728279241 / 640000000000000000 : ℝ) ≤
+      (-(2682396226434606046731 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (489423688728279241 / 640000000000000000 : ℝ) =
+            (489423688728279241 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (489423688728279241 / 320000000000000000 : ℝ) ≤
+            (4249075576565393953269 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7647106403155113 / 10000000000000000 : ℝ)) (q :=
+      (489423688728279241 / 640000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_661_3 :
+    Uω (aρ 3) (bρ 3) (396065285130169 / 512000000000000 : ℝ) ≤
+      (-(54900500690973011711 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (486368064990858537 / 640000000000000000 : ℝ) ≤
+      (-(54900500690973011711 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (486368064990858537 / 640000000000000000 : ℝ) =
+            (486368064990858537 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (486368064990858537 / 320000000000000000 : ℝ) ≤
+            (83728935369026988289 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1899753566927679 / 2500000000000000 : ℝ)) (q :=
+      (486368064990858537 / 640000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_661_4 :
+    Uω (aρ 4) (bρ 4) (396065285130169 / 512000000000000 : ℝ) ≤
+      (-(2849797759623886644061 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (481298856347041321 / 640000000000000000 : ℝ) ≤
+      (-(2849797759623886644061 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (481298856347041321 / 640000000000000000 : ℝ) =
+            (481298856347041321 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (481298856347041321 / 320000000000000000 : ℝ) ≤
+            (4081674043376113355939 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1879723919664107 / 2500000000000000 : ℝ)) (q :=
+      (481298856347041321 / 640000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_661_5 :
+    Uω (aρ 5) (bρ 5) (396065285130169 / 512000000000000 : ℝ) ≤
+      (-(301117906265737937309 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (94718786290156437 / 128000000000000000 : ℝ) ≤
+      (-(301117906265737937309 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (94718786290156437 / 128000000000000000 : ℝ) =
+            (94718786290156437 / 64000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (94718786290156437 / 64000000000000000 : ℝ) ≤
+            (392029274034262062691 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (739630669498333 / 1000000000000000 : ℝ)) (q :=
+      (94718786290156437 / 128000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_661_6 :
+    Uω (aρ 6) (bρ 6) (396065285130169 / 512000000000000 : ℝ) ≤
+      (-(1623270176049848597967 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (462577514063111017 / 640000000000000000 : ℝ) ≤
+      (-(1623270176049848597967 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (462577514063111017 / 640000000000000000 : ℝ) =
+            (462577514063111017 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (462577514063111017 / 320000000000000000 : ℝ) ≤
+            (1842465725450151402033 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3609642013379303 / 5000000000000000 : ℝ)) (q :=
+      (462577514063111017 / 640000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_661_7 :
+    Uω (aρ 7) (bρ 7) (396065285130169 / 512000000000000 : ℝ) ≤
+      (-(3575237128147365079751 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (447619912753074921 / 640000000000000000 : ℝ) ≤
+      (-(3575237128147365079751 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (447619912753074921 / 640000000000000000 : ℝ) =
+            (447619912753074921 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (447619912753074921 / 320000000000000000 : ℝ) ≤
+            (3356234674852634920249 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3487760183289989 / 5000000000000000 : ℝ)) (q :=
+      (447619912753074921 / 640000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_661_8 :
+    Uω (aρ 8) (bρ 8) (396065285130169 / 512000000000000 : ℝ) ≤
+      (-(803534684119275944343 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (428247300675913961 / 640000000000000000 : ℝ) ≤
+      (-(803534684119275944343 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (428247300675913961 / 640000000000000000 : ℝ) =
+            (428247300675913961 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (428247300675913961 / 320000000000000000 : ℝ) ≤
+            (582759676480724055657 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3326833945254349 / 5000000000000000 : ℝ)) (q :=
+      (428247300675913961 / 640000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8])
+      (by norm_num [bρ_8]) (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_661_9 :
+    Uω (aρ 9) (bρ 9) (396065285130169 / 512000000000000 : ℝ) ≤
+      (-(459435991242715682401 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (404249468975058729 / 640000000000000000 : ℝ) ≤
+      (-(459435991242715682401 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (404249468975058729 / 640000000000000000 : ℝ) =
+            (404249468975058729 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (404249468975058729 / 320000000000000000 : ℝ) ≤
+            (233711189057284317599 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1561175319554243 / 2500000000000000 : ℝ)) (q :=
+      (404249468975058729 / 640000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9])
+      (by norm_num [bρ_9]) (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_661_10 :
+    Uω (aρ 10) (bρ 10) (396065285130169 / 512000000000000 : ℝ) ≤
+      (-(5324929053367865177511 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (375769254721378313 / 640000000000000000 : ℝ) ≤
+      (-(5324929053367865177511 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (375769254721378313 / 640000000000000000 : ℝ) =
+            (375769254721378313 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (375769254721378313 / 320000000000000000 : ℝ) ≤
+            (1606542749632134822489 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5743253616284459 / 10000000000000000 : ℝ)) (q :=
+      (375769254721378313 / 640000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_661_11 :
+    Uω (aρ 11) (bρ 11) (396065285130169 / 512000000000000 : ℝ) ≤
+      (-(15567406399431601151 / 25000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (343357411306843081 / 640000000000000000 : ℝ) ≤
+      (-(15567406399431601151 / 25000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (343357411306843081 / 640000000000000000 : ℝ) =
+            (343357411306843081 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (343357411306843081 / 320000000000000000 : ℝ) ≤
+            (1761273108068398849 / 25000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5148860135785233 / 10000000000000000 : ℝ)) (q :=
+      (343357411306843081 / 640000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_661_12 :
+    Uω (aρ 12) (bρ 12) (396065285130169 / 512000000000000 : ℝ) ≤
+      (-(3657079475934853258049 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (307985369630861833 / 640000000000000000 : ℝ) ≤
+      (-(3657079475934853258049 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (307985369630861833 / 640000000000000000 : ℝ) =
+            (307985369630861833 / 160000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (307985369630861833 / 160000000000000000 : ℝ) ≤
+            (3274392327065146741951 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4467183930095819 / 10000000000000000 : ℝ)) (q :=
+      (307985369630861833 / 640000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_661_13 :
+    Uω (aρ 13) (bρ 13) (396065285130169 / 512000000000000 : ℝ) ≤
+      (-(4296183276173925836353 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (271030543345022249 / 640000000000000000 : ℝ) ≤
+      (-(4296183276173925836353 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (271030543345022249 / 640000000000000000 : ℝ) =
+            (271030543345022249 / 160000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (271030543345022249 / 160000000000000000 : ℝ) ≤
+            (2635288526826074163647 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (927931837564583 / 2500000000000000 : ℝ)) (q :=
+      (271030543345022249 / 640000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_661_14 :
+    Uω (aρ 14) (bρ 14) (396065285130169 / 512000000000000 : ℝ) ≤
+      (-(5024260336321951904853 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (234303224706138217 / 640000000000000000 : ℝ) ≤
+      (-(5024260336321951904853 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (234303224706138217 / 640000000000000000 : ℝ) =
+            (234303224706138217 / 160000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (234303224706138217 / 160000000000000000 : ℝ) ≤
+            (1907211466678048095147 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1453962963919103 / 5000000000000000 : ℝ)) (q :=
+      (234303224706138217 / 640000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_661_15 :
+    Uω (aρ 15) (bρ 15) (396065285130169 / 512000000000000 : ℝ) ≤
+      (-(11612885194558307579307 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (200372804993721609 / 640000000000000000 : ℝ) ≤
+      (-(11612885194558307579307 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (200372804993721609 / 640000000000000000 : ℝ) =
+            (200372804993721609 / 160000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (200372804993721609 / 160000000000000000 : ℝ) ≤
+            (2250058411441692420693 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2107173589240187 / 10000000000000000 : ℝ)) (q :=
+      (200372804993721609 / 640000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_661_16 :
+    Uω (aρ 16) (bρ 16) (396065285130169 / 512000000000000 : ℝ) ≤
+      (-(13009776067628077066779 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (174249916260764873 / 640000000000000000 : ℝ) ≤
+      (-(13009776067628077066779 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (174249916260764873 / 640000000000000000 : ℝ) =
+            (174249916260764873 / 160000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (174249916260764873 / 160000000000000000 : ℝ) ≤
+            (853167538371922933221 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1443204967845289 / 10000000000000000 : ℝ)) (q :=
+      (174249916260764873 / 640000000000000000 : ℝ)) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [bρ_16]) (by norm_num) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [aρ_16, bρ_16]) hlg
+
+lemma U_661 :
+    Uρ (396065285130169 / 512000000000000 : ℝ) ≤
+      (-(4763076847769005740497 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (396065285130169 / 512000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(662802313408731620701 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_661_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (396065285130169 / 512000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(2682396226434606046731 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_661_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (396065285130169 / 512000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) * (-(54900500690973011711 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_661_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (396065285130169 / 512000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(2849797759623886644061 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_661_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (396065285130169 / 512000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(301117906265737937309 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_661_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (396065285130169 / 512000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1623270176049848597967 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_661_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (396065285130169 / 512000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(3575237128147365079751 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_661_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (396065285130169 / 512000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(803534684119275944343 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_661_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (396065285130169 / 512000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(459435991242715682401 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_661_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) *
+        Uω (aρ 10) (bρ 10) (396065285130169 / 512000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(5324929053367865177511 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_661_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) *
+        Uω (aρ 11) (bρ 11) (396065285130169 / 512000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) * (-(15567406399431601151 / 25000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_661_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) *
+        Uω (aρ 12) (bρ 12) (396065285130169 / 512000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(3657079475934853258049 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_661_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) *
+        Uω (aρ 13) (bρ 13) (396065285130169 / 512000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4296183276173925836353 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_661_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) *
+        Uω (aρ 14) (bρ 14) (396065285130169 / 512000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(5024260336321951904853 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_661_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (396065285130169 / 512000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(11612885194558307579307 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_661_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) *
+        Uω (aρ 16) (bρ 16) (396065285130169 / 512000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) *
+        (-(13009776067628077066779 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_661_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_662_1 :
+    Uω (aρ 1) (bρ 1) (794637295669 / 1024000000000 : ℝ) ≤
+      (-(2619348465185242405921 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (123129785834091833 / 160000000000000000 : ℝ) ≤
+      (-(2619348465185242405921 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (123129785834091833 / 160000000000000000 : ℝ) =
+            (123129785834091833 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (123129785834091833 / 80000000000000000 : ℝ) ≤
+            (4312123337814757594079 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7695590606828901 / 10000000000000000 : ℝ)) (q :=
+      (123129785834091833 / 160000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_662_2 :
+    Uω (aρ 2) (bρ 2) (794637295669 / 1024000000000 : ℝ) ≤
+      (-(530087117586050276873 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (122747605110252121 / 160000000000000000 : ℝ) ≤
+      (-(530087117586050276873 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (122747605110252121 / 160000000000000000 : ℝ) =
+            (122747605110252121 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (122747605110252121 / 80000000000000000 : ℝ) ≤
+            (856207243013949723127 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7671587028858937 / 10000000000000000 : ℝ)) (q :=
+      (122747605110252121 / 160000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_662_3 :
+    Uω (aρ 3) (bρ 3) (794637295669 / 1024000000000 : ℝ) ≤
+      (-(1356431227679515851249 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (121983717100682641 / 160000000000000000 : ℝ) ≤
+      (-(1356431227679515851249 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (121983717100682641 / 160000000000000000 : ℝ) =
+            (121983717100682641 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (121983717100682641 / 80000000000000000 : ℝ) ≤
+            (2109304673820484148751 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (476468570875797 / 625000000000000 : ℝ)) (q :=
+      (121983717100682641 / 160000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_662_4 :
+    Uω (aρ 4) (bρ 4) (794637295669 / 1024000000000 : ℝ) ≤
+      (-(56345860742388664727 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (120716462569808329 / 160000000000000000 : ℝ) ≤
+      (-(56345860742388664727 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (120716462569808329 / 160000000000000000 : ℝ) =
+            (120716462569808329 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (120716462569808329 / 80000000000000000 : ℝ) ≤
+            (82283575317611335273 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7543384498718463 / 10000000000000000 : ℝ)) (q :=
+      (120716462569808329 / 160000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_662_5 :
+    Uω (aρ 5) (bρ 5) (794637295669 / 1024000000000 : ℝ) ≤
+      (-(1489068258800647595981 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23758069727748837 / 32000000000000000 : ℝ) ≤
+      (-(1489068258800647595981 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23758069727748837 / 32000000000000000 : ℝ) =
+            (23758069727748837 / 16000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23758069727748837 / 16000000000000000 : ℝ) ≤
+            (1976667642699352404019 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1484162035334089 / 2000000000000000 : ℝ)) (q :=
+      (23758069727748837 / 32000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_662_6 :
+    Uω (aρ 6) (bρ 6) (794637295669 / 1024000000000 : ℝ) ≤
+      (-(401586128386062271131 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (116036513399873761 / 160000000000000000 : ℝ) ≤
+      (-(401586128386062271131 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (116036513399873761 / 160000000000000000 : ℝ) =
+            (116036513399873761 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (116036513399873761 / 80000000000000000 : ℝ) ≤
+            (464847846988937728869 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3621910573475821 / 5000000000000000 : ℝ)) (q :=
+      (116036513399873761 / 160000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_662_7 :
+    Uω (aρ 7) (bρ 7) (794637295669 / 1024000000000 : ℝ) ≤
+      (-(442525638057717244647 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (112297691448294529 / 160000000000000000 : ℝ) ≤
+      (-(442525638057717244647 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (112297691448294529 / 160000000000000000 : ℝ) =
+            (112297691448294529 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (112297691448294529 / 80000000000000000 : ℝ) ≤
+            (423908337317282755353 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3500064891882119 / 5000000000000000 : ℝ)) (q :=
+      (112297691448294529 / 160000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_662_8 :
+    Uω (aρ 8) (bρ 8) (794637295669 / 1024000000000 : ℝ) ≤
+      (-(39809503495964381451 / 100000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (107455711863479641 / 160000000000000000 : ℝ) ≤
+      (-(39809503495964381451 / 100000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (107455711863479641 / 160000000000000000 : ℝ) =
+            (107455711863479641 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (107455711863479641 / 80000000000000000 : ℝ) ≤
+            (29505214534035618549 / 100000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6678423987002377 / 10000000000000000 : ℝ)) (q :=
+      (107455711863479641 / 160000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8])
+      (by norm_num [bρ_8]) (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_662_9 :
+    Uω (aρ 9) (bρ 9) (794637295669 / 1024000000000 : ℝ) ≤
+      (-(455523747905055093527 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20291704447757669 / 32000000000000000 : ℝ) ≤
+      (-(455523747905055093527 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (20291704447757669 / 32000000000000000 : ℝ) =
+            (20291704447757669 / 16000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20291704447757669 / 16000000000000000 : ℝ) ≤
+            (237623432394944906473 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1253948182455193 / 2000000000000000 : ℝ)) (q :=
+      (20291704447757669 / 32000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_662_10 :
+    Uω (aρ 10) (bρ 10) (794637295669 / 1024000000000 : ℝ) ≤
+      (-(1056480036319776071903 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (94342690516851393 / 160000000000000000 : ℝ) ≤
+      (-(1056480036319776071903 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (94342690516851393 / 160000000000000000 : ℝ) =
+            (94342690516851393 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (94342690516851393 / 80000000000000000 : ℝ) ≤
+            (329814324280223928097 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2884410490264423 / 5000000000000000 : ℝ)) (q :=
+      (94342690516851393 / 160000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_662_11 :
+    Uω (aρ 11) (bρ 11) (794637295669 / 1024000000000 : ℝ) ≤
+      (-(1235908123765897962347 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (86247386421328329 / 160000000000000000 : ℝ) ≤
+      (-(1235908123765897962347 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (86247386421328329 / 160000000000000000 : ℝ) =
+            (86247386421328329 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (86247386421328329 / 80000000000000000 : ℝ) ≤
+            (150386236834102037653 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5175384594793463 / 10000000000000000 : ℝ)) (q :=
+      (86247386421328329 / 160000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_662_12 :
+    Uω (aρ 12) (bρ 12) (794637295669 / 1024000000000 : ℝ) ≤
+      (-(7259532435704126432831 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3096723900153329 / 6400000000000000 : ℝ) ≤
+      (-(7259532435704126432831 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3096723900153329 / 6400000000000000 : ℝ) =
+            (3096723900153329 / 1600000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3096723900153329 / 1600000000000000 : ℝ) ≤
+            (6603411170295873567169 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (179816943071663 / 400000000000000 : ℝ)) (q :=
+      (3096723900153329 / 6400000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_662_13 :
+    Uω (aρ 13) (bρ 13) (794637295669 / 1024000000000 : ℝ) ≤
+      (-(8526691058976463844233 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13640820388378613 / 32000000000000000 : ℝ) ≤
+      (-(8526691058976463844233 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (13640820388378613 / 32000000000000000 : ℝ) =
+            (13640820388378613 / 8000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13640820388378613 / 8000000000000000 : ℝ) ≤
+            (5336252547023536155767 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (748611174628811 / 2000000000000000 : ℝ)) (q :=
+      (13640820388378613 / 32000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_662_14 :
+    Uω (aρ 14) (bρ 14) (794637295669 / 1024000000000 : ℝ) ≤
+      (-(4982434958219247805323 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (59067852357270697 / 160000000000000000 : ℝ) ≤
+      (-(4982434958219247805323 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (59067852357270697 / 160000000000000000 : ℝ) =
+            (59067852357270697 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (59067852357270697 / 40000000000000000 : ℝ) ≤
+            (1949036844780752194677 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2944951960111259 / 10000000000000000 : ℝ)) (q :=
+      (59067852357270697 / 160000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_662_15 :
+    Uω (aρ 15) (bρ 15) (794637295669 / 1024000000000 : ℝ) ≤
+      (-(11498015157682552696681 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (50671939657254177 / 160000000000000000 : ℝ) ≤
+      (-(11498015157682552696681 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (50671939657254177 / 160000000000000000 : ℝ) =
+            (50671939657254177 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (50671939657254177 / 40000000000000000 : ℝ) ≤
+            (2364928448317447303319 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1077518075012097 / 5000000000000000 : ℝ)) (q :=
+      (50671939657254177 / 160000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_662_16 :
+    Uω (aρ 16) (bρ 16) (794637295669 / 1024000000000 : ℝ) ≤
+      (-(50171776057069747423 / 39062500000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (44290771968930889 / 160000000000000000 : ℝ) ≤
+      (-(50171776057069747423 / 39062500000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (44290771968930889 / 160000000000000000 : ℝ) =
+            (44290771968930889 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (44290771968930889 / 40000000000000000 : ℝ) ≤
+            (3980347403867752577 / 39062500000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1509761840493783 / 10000000000000000 : ℝ)) (q :=
+      (44290771968930889 / 160000000000000000 : ℝ)) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [bρ_16]) (by norm_num) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [aρ_16, bρ_16]) hlg
+
+lemma U_662 :
+    Uρ (794637295669 / 1024000000000 : ℝ) ≤
+      (-(590076893444211888841 / 1250000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (794637295669 / 1024000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2619348465185242405921 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_662_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (794637295669 / 1024000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(530087117586050276873 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_662_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (794637295669 / 1024000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1356431227679515851249 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_662_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (794637295669 / 1024000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(56345860742388664727 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_662_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (794637295669 / 1024000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1489068258800647595981 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_662_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (794637295669 / 1024000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(401586128386062271131 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_662_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (794637295669 / 1024000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(442525638057717244647 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_662_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (794637295669 / 1024000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) * (-(39809503495964381451 / 100000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_662_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (794637295669 / 1024000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(455523747905055093527 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_662_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (794637295669 / 1024000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(1056480036319776071903 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_662_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (794637295669 / 1024000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(1235908123765897962347 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_662_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (794637295669 / 1024000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(7259532435704126432831 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_662_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (794637295669 / 1024000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(8526691058976463844233 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_662_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (794637295669 / 1024000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(4982434958219247805323 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_662_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (794637295669 / 1024000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(11498015157682552696681 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_662_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (794637295669 / 1024000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(50171776057069747423 / 39062500000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_662_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_663_1 :
+    Uω (aρ 1) (bρ 1) (398572010538831 / 512000000000000 : ℝ) ≤
+      (-(1293794432980282284923 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (494085850980083391 / 640000000000000000 : ℝ) ≤
+      (-(1293794432980282284923 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (494085850980083391 / 640000000000000000 : ℝ) =
+            (494085850980083391 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (494085850980083391 / 320000000000000000 : ℝ) ≤
+            (2171941468519717715077 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7720070480376063 / 10000000000000000 : ℝ)) (q :=
+      (494085850980083391 / 640000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_663_2 :
+    Uω (aρ 2) (bρ 2) (398572010538831 / 512000000000000 : ℝ) ≤
+      (-(654644193995127234473 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (492557151973494943 / 640000000000000000 : ℝ) ≤
+      (-(654644193995127234473 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (492557151973494943 / 640000000000000000 : ℝ) =
+            (492557151973494943 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (492557151973494943 / 320000000000000000 : ℝ) ≤
+            (1078223756754872765527 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3848033824465087 / 5000000000000000 : ℝ)) (q :=
+      (492557151973494943 / 640000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_663_3 :
+    Uω (aρ 3) (bρ 3) (398572010538831 / 512000000000000 : ℝ) ≤
+      (-(2680803000857294788523 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (489501671174198047 / 640000000000000000 : ℝ) ≤
+      (-(2680803000857294788523 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (489501671174198047 / 640000000000000000 : ℝ) =
+            (489501671174198047 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (489501671174198047 / 320000000000000000 : ℝ) ≤
+            (4250668802142705211477 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3823989990151073 / 5000000000000000 : ℝ)) (q :=
+      (489501671174198047 / 640000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_663_4 :
+    Uω (aρ 4) (bρ 4) (398572010538831 / 512000000000000 : ℝ) ≤
+      (-(2784893666896482135699 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (484432842331385599 / 640000000000000000 : ℝ) ≤
+      (-(2784893666896482135699 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (484432842331385599 / 640000000000000000 : ℝ) =
+            (484432842331385599 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (484432842331385599 / 320000000000000000 : ℝ) ≤
+            (4146578136103517864301 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7567873260029257 / 10000000000000000 : ℝ)) (q :=
+      (484432842331385599 / 640000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_663_5 :
+    Uω (aρ 5) (bρ 5) (398572010538831 / 512000000000000 : ℝ) ≤
+      (-(368150362361694042147 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (476728852660897087 / 640000000000000000 : ℝ) ≤
+      (-(368150362361694042147 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (476728852660897087 / 640000000000000000 : ℝ) =
+            (476728852660897087 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (476728852660897087 / 320000000000000000 : ℝ) ≤
+            (498283613013305957853 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7445313502161491 / 10000000000000000 : ℝ)) (q :=
+      (476728852660897087 / 640000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_663_6 :
+    Uω (aρ 6) (bρ 6) (398572010538831 / 512000000000000 : ℝ) ≤
+      (-(1589476086348222504551 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (465714580753082303 / 640000000000000000 : ℝ) ≤
+      (-(1589476086348222504551 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (465714580753082303 / 640000000000000000 : ℝ) =
+            (465714580753082303 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (465714580753082303 / 320000000000000000 : ℝ) ≤
+            (1876259815151777495449 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7268357880182279 / 10000000000000000 : ℝ)) (q :=
+      (465714580753082303 / 640000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_663_7 :
+    Uω (aρ 7) (bρ 7) (398572010538831 / 512000000000000 : ℝ) ≤
+      (-(3505296020052566257047 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (450761589835292159 / 640000000000000000 : ℝ) ≤
+      (-(3505296020052566257047 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (450761589835292159 / 640000000000000000 : ℝ) =
+            (450761589835292159 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (450761589835292159 / 320000000000000000 : ℝ) ≤
+            (3426175782947433742953 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7024738294761337 / 10000000000000000 : ℝ)) (q :=
+      (450761589835292159 / 640000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_663_8 :
+    Uω (aρ 8) (bρ 8) (398572010538831 / 512000000000000 : ℝ) ≤
+      (-(493045393677593413659 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (86279665855173267 / 128000000000000000 : ℝ) ≤
+      (-(493045393677593413659 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (86279665855173267 / 128000000000000000 : ℝ) =
+            (86279665855173267 / 64000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (86279665855173267 / 64000000000000000 : ℝ) ≤
+            (373388581697406586341 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (83789725670241 / 125000000000000 : ℝ)) (q :=
+      (86279665855173267 / 128000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8])
+      (by norm_num [bρ_8]) (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_663_9 :
+    Uω (aρ 9) (bρ 9) (398572010538831 / 512000000000000 : ℝ) ≤
+      (-(903254193070843639721 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (407418567990045727 / 640000000000000000 : ℝ) ≤
+      (-(903254193070843639721 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (407418567990045727 / 640000000000000000 : ℝ) =
+            (407418567990045727 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (407418567990045727 / 320000000000000000 : ℝ) ≤
+            (483040167529156360279 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3147388070898693 / 5000000000000000 : ℝ)) (q :=
+      (407418567990045727 / 640000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9])
+      (by norm_num [bρ_9]) (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_663_10 :
+    Uω (aρ 10) (bρ 10) (398572010538831 / 512000000000000 : ℝ) ≤
+      (-(5240059346819541959201 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (378971968805497247 / 640000000000000000 : ℝ) ≤
+      (-(5240059346819541959201 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (378971968805497247 / 640000000000000000 : ℝ) =
+            (378971968805497247 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (378971968805497247 / 320000000000000000 : ℝ) ≤
+            (1691412456180458040799 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2897189475387623 / 5000000000000000 : ℝ)) (q :=
+      (378971968805497247 / 640000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_663_11 :
+    Uω (aρ 11) (bρ 11) (398572010538831 / 512000000000000 : ℝ) ≤
+      (-(6132361009483060921937 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (346621038518138783 / 640000000000000000 : ℝ) ≤
+      (-(6132361009483060921937 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (346621038518138783 / 640000000000000000 : ℝ) =
+            (346621038518138783 / 320000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (346621038518138783 / 320000000000000000 : ℝ) ≤
+            (799110793516939078063 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2600944502750147 / 5000000000000000 : ℝ)) (q :=
+      (346621038518138783 / 640000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_663_12 :
+    Uω (aρ 12) (bρ 12) (398572010538831 / 512000000000000 : ℝ) ≤
+      (-(1441049547891940481097 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (311358008202268671 / 640000000000000000 : ℝ) ≤
+      (-(1441049547891940481097 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (311358008202268671 / 640000000000000000 : ℝ) =
+            (311358008202268671 / 160000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (311358008202268671 / 160000000000000000 : ℝ) ≤
+            (1331539173308059518903 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4523619404814353 / 10000000000000000 : ℝ)) (q :=
+      (311358008202268671 / 640000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_663_13 :
+    Uω (aρ 13) (bρ 13) (398572010538831 / 512000000000000 : ℝ) ≤
+      (-(2115390522553072752363 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (274599031618965983 / 640000000000000000 : ℝ) ≤
+      (-(2115390522553072752363 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (274599031618965983 / 640000000000000000 : ℝ) =
+            (274599031618965983 / 160000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (274599031618965983 / 160000000000000000 : ℝ) ≤
+            (1350345378946927247637 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (471785391022643 / 1250000000000000 : ℝ)) (q :=
+      (274599031618965983 / 640000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_663_14 :
+    Uω (aρ 14) (bρ 14) (398572010538831 / 512000000000000 : ℝ) ≤
+      (-(9882260711854623304547 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (238231312906570783 / 640000000000000000 : ℝ) ≤
+      (-(9882260711854623304547 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (238231312906570783 / 640000000000000000 : ℝ) =
+            (238231312906570783 / 160000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (238231312906570783 / 160000000000000000 : ℝ) ≤
+            (3980682894145376695453 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1490859601731897 / 5000000000000000 : ℝ)) (q :=
+      (238231312906570783 / 640000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_663_15 :
+    Uω (aρ 15) (bρ 15) (398572010538831 / 512000000000000 : ℝ) ≤
+      (-(11385648572861481335083 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (204978135744948383 / 640000000000000000 : ℝ) ≤
+      (-(11385648572861481335083 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (204978135744948383 / 640000000000000000 : ℝ) =
+            (204978135744948383 / 160000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (204978135744948383 / 160000000000000000 : ℝ) ≤
+            (2477295033138518664917 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1101065347289047 / 5000000000000000 : ℝ)) (q :=
+      (204978135744948383 / 640000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_663_16 :
+    Uω (aρ 16) (bρ 16) (398572010538831 / 512000000000000 : ℝ) ≤
+      (-(3171300016281540266367 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (179998437324363327 / 640000000000000000 : ℝ) ≤
+      (-(3171300016281540266367 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (179998437324363327 / 640000000000000000 : ℝ) =
+            (179998437324363327 / 160000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (179998437324363327 / 160000000000000000 : ℝ) ≤
+            (294435885218459733633 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1573886770444811 / 10000000000000000 : ℝ)) (q :=
+      (179998437324363327 / 640000000000000000 : ℝ)) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [bρ_16]) (by norm_num) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [aρ_16, bρ_16]) hlg
+
+lemma U_663 :
+    Uρ (398572010538831 / 512000000000000 : ℝ) ≤
+      (-(4678476634791325595441 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (398572010538831 / 512000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1293794432980282284923 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_663_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (398572010538831 / 512000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(654644193995127234473 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_663_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (398572010538831 / 512000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(2680803000857294788523 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_663_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (398572010538831 / 512000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(2784893666896482135699 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_663_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (398572010538831 / 512000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(368150362361694042147 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_663_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (398572010538831 / 512000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1589476086348222504551 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_663_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (398572010538831 / 512000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(3505296020052566257047 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_663_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (398572010538831 / 512000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(493045393677593413659 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_663_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (398572010538831 / 512000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(903254193070843639721 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_663_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) *
+        Uω (aρ 10) (bρ 10) (398572010538831 / 512000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(5240059346819541959201 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_663_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) *
+        Uω (aρ 11) (bρ 11) (398572010538831 / 512000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(6132361009483060921937 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_663_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) *
+        Uω (aρ 12) (bρ 12) (398572010538831 / 512000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(1441049547891940481097 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_663_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) *
+        Uω (aρ 13) (bρ 13) (398572010538831 / 512000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(2115390522553072752363 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_663_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) *
+        Uω (aρ 14) (bρ 14) (398572010538831 / 512000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(9882260711854623304547 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_663_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (398572010538831 / 512000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(11385648572861481335083 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_663_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) *
+        Uω (aρ 16) (bρ 16) (398572010538831 / 512000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) *
+        (-(3171300016281540266367 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_663_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/U55.lean b/LeanPool/Zeta5Irrational/Table/U55.lean
new file mode 100644
index 0000000000..93ac5ee641
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/U55.lean
@@ -0,0 +1,5559 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-! # Certified arcsine potential bounds (U55) -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma U_664_1 :
+    Uω (aρ 1) (bρ 1) (199912686621581 / 256000000000000 : ℝ) ≤
+      (-(1277964907620946027673 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (247826279298423837 / 320000000000000000 : ℝ) ≤
+      (-(1277964907620946027673 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (247826279298423837 / 320000000000000000 : ℝ) =
+            (247826279298423837 / 160000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (247826279298423837 / 160000000000000000 : ℝ) ≤
+            (2187770993879053972327 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3872275176540491 / 5000000000000000 : ℝ)) (q :=
+      (247826279298423837 / 320000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_664_2 :
+    Uω (aρ 2) (bρ 2) (199912686621581 / 256000000000000 : ℝ) ≤
+      (-(103472765751352777257 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (49412388332745321 / 64000000000000000 : ℝ) ≤
+      (-(103472765751352777257 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (49412388332745321 / 64000000000000000 : ℝ) =
+            (49412388332745321 / 32000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (49412388332745321 / 32000000000000000 : ℝ) ≤
+            (173786106368647222743 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1544109652684481 / 2000000000000000 : ℝ)) (q :=
+      (49412388332745321 / 64000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_664_3 :
+    Uω (aρ 3) (bρ 3) (199912686621581 / 256000000000000 : ℝ) ≤
+      (-(2648846011764422714851 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (245534236655695789 / 320000000000000000 : ℝ) ≤
+      (-(2648846011764422714851 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (245534236655695789 / 320000000000000000 : ℝ) =
+            (245534236655695789 / 160000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (245534236655695789 / 160000000000000000 : ℝ) ≤
+            (4282625791235577285149 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7672462806770479 / 10000000000000000 : ℝ)) (q :=
+      (245534236655695789 / 320000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_664_4 :
+    Uω (aρ 4) (bρ 4) (199912686621581 / 256000000000000 : ℝ) ≤
+      (-(2752598967989138191723 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (242999916260845037 / 320000000000000000 : ℝ) ≤
+      (-(2752598967989138191723 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (242999916260845037 / 320000000000000000 : ℝ) =
+            (242999916260845037 / 160000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (242999916260845037 / 160000000000000000 : ℝ) ≤
+            (4178872835010861808277 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7592361963157307 / 10000000000000000 : ℝ)) (q :=
+      (242999916260845037 / 320000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_664_5 :
+    Uω (aρ 5) (bρ 5) (199912686621581 / 256000000000000 : ℝ) ≤
+      (-(291237749004709474387 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (47829630581773097 / 64000000000000000 : ℝ) ≤
+      (-(291237749004709474387 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (47829630581773097 / 64000000000000000 : ℝ) =
+            (47829630581773097 / 32000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (47829630581773097 / 32000000000000000 : ℝ) ≤
+            (401909431295290525613 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1493963334598717 / 2000000000000000 : ℝ)) (q :=
+      (47829630581773097 / 64000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_664_6 :
+    Uω (aρ 6) (bρ 6) (199912686621581 / 256000000000000 : ℝ) ≤
+      (-(3145329015573109930877 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (233641547824428733 / 320000000000000000 : ℝ) ≤
+      (-(3145329015573109930877 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (233641547824428733 / 320000000000000000 : ℝ) =
+            (233641547824428733 / 160000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (233641547824428733 / 160000000000000000 : ℝ) ≤
+            (3786142787426890069123 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (113951472349317 / 156250000000000 : ℝ)) (q :=
+      (233641547824428733 / 320000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_664_7 :
+    Uω (aρ 7) (bρ 7) (199912686621581 / 256000000000000 : ℝ) ≤
+      (-(3470509010608891617659 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (226166192591551821 / 320000000000000000 : ℝ) ≤
+      (-(3470509010608891617659 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (226166192591551821 / 320000000000000000 : ℝ) =
+            (226166192591551821 / 160000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (226166192591551821 / 160000000000000000 : ℝ) ≤
+            (3460962792391108382341 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7049345909061481 / 10000000000000000 : ℝ)) (q :=
+      (226166192591551821 / 320000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_664_8 :
+    Uω (aρ 8) (bρ 8) (199912686621581 / 256000000000000 : ℝ) ≤
+      (-(1953955403703193484059 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (216486873429367757 / 320000000000000000 : ℝ) ≤
+      (-(1953955403703193484059 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (216486873429367757 / 320000000000000000 : ℝ) =
+            (216486873429367757 / 160000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (216486873429367757 / 160000000000000000 : ℝ) ≤
+            (1511780497796806515941 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6727930112764977 / 10000000000000000 : ℝ)) (q :=
+      (216486873429367757 / 320000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8])
+      (by norm_num [bρ_8]) (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_664_9 :
+    Uω (aρ 9) (bρ 9) (199912686621581 / 256000000000000 : ℝ) ≤
+      (-(4477459106514180083733 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (204501453877376573 / 320000000000000000 : ℝ) ≤
+      (-(4477459106514180083733 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (204501453877376573 / 320000000000000000 : ℝ) =
+            (204501453877376573 / 160000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (204501453877376573 / 160000000000000000 : ℝ) ≤
+            (2454012696485819916267 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (789975877390691 / 1250000000000000 : ℝ)) (q :=
+      (204501453877376573 / 320000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9])
+      (by norm_num [bρ_9]) (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_664_10 :
+    Uω (aρ 10) (bρ 10) (199912686621581 / 256000000000000 : ℝ) ≤
+      (-(1299476206458049317223 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (38057287889597513 / 64000000000000000 : ℝ) ≤
+      (-(1299476206458049317223 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (38057287889597513 / 64000000000000000 : ℝ) =
+            (38057287889597513 / 32000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (38057287889597513 / 32000000000000000 : ℝ) ≤
+            (433391744291950682777 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1163985530156743 / 2000000000000000 : ℝ)) (q :=
+      (38057287889597513 / 64000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_664_11 :
+    Uω (aρ 11) (bρ 11) (199912686621581 / 256000000000000 : ℝ) ≤
+      (-(6085421088558797644903 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (174125949780980637 / 320000000000000000 : ℝ) ≤
+      (-(6085421088558797644903 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (174125949780980637 / 320000000000000000 : ℝ) =
+            (174125949780980637 / 160000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (174125949780980637 / 160000000000000000 : ℝ) ≤
+            (846050714441202355097 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2614186836400391 / 5000000000000000 : ℝ)) (q :=
+      (174125949780980637 / 320000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_664_12 :
+    Uω (aρ 12) (bρ 12) (199912686621581 / 256000000000000 : ℝ) ≤
+      (-(3575650047906401257211 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (156521125124646877 / 320000000000000000 : ℝ) ≤
+      (-(3575650047906401257211 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (156521125124646877 / 320000000000000000 : ℝ) =
+            (156521125124646877 / 80000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (156521125124646877 / 80000000000000000 : ℝ) ≤
+            (3355821755093598742789 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2275886114232461 / 5000000000000000 : ℝ)) (q :=
+      (156521125124646877 / 320000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_664_13 :
+    Uω (aρ 13) (bρ 13) (199912686621581 / 256000000000000 : ℝ) ≤
+      (-(4198484460239256737569 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27637849467573129 / 64000000000000000 : ℝ) ≤
+      (-(4198484460239256737569 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (27637849467573129 / 64000000000000000 : ℝ) =
+            (27637849467573129 / 16000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27637849467573129 / 16000000000000000 : ℝ) ≤
+            (2732987342760743262431 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (761082321677219 / 2000000000000000 : ℝ)) (q :=
+      (27637849467573129 / 64000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_664_14 :
+    Uω (aρ 14) (bρ 14) (199912686621581 / 256000000000000 : ℝ) ≤
+      (-(1960132106122831270009 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (120091618888945197 / 320000000000000000 : ℝ) ≤
+      (-(1960132106122831270009 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (120091618888945197 / 320000000000000000 : ℝ) =
+            (120091618888945197 / 80000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (120091618888945197 / 80000000000000000 : ℝ) ≤
+            (812456615077168729991 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3018237115373567 / 10000000000000000 : ℝ)) (q :=
+      (120091618888945197 / 320000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_664_15 :
+    Uω (aρ 15) (bρ 15) (199912686621581 / 256000000000000 : ℝ) ≤
+      (-(225512837703902044943 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (165796384488381 / 512000000000000 : ℝ) ≤
+      (-(225512837703902044943 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (165796384488381 / 512000000000000 : ℝ) =
+            (165796384488381 / 128000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (165796384488381 / 128000000000000 : ℝ) ≤
+            (51746034416097955057 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3597608770091 / 16000000000000 : ℝ)) (q :=
+      (165796384488381 / 512000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15]) (by norm_num [bρ_15])
+      (by norm_num) (by norm_num [aρ_15, bρ_15]) (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_664_16 :
+    Uω (aρ 16) (bρ 16) (199912686621581 / 256000000000000 : ℝ) ≤
+      (-(1253266360243979778637 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (91382558574823133 / 320000000000000000 : ℝ) ≤
+      (-(1253266360243979778637 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (91382558574823133 / 320000000000000000 : ℝ) =
+            (91382558574823133 / 80000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (91382558574823133 / 80000000000000000 : ℝ) ≤
+            (133028000356020221363 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (817932887332969 / 5000000000000000 : ℝ)) (q :=
+      (91382558574823133 / 320000000000000000 : ℝ)) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [bρ_16]) (by norm_num) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [aρ_16, bρ_16]) hlg
+
+lemma U_664 :
+    Uρ (199912686621581 / 256000000000000 : ℝ) ≤
+      (-(4636648990333002156281 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (199912686621581 / 256000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1277964907620946027673 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_664_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (199912686621581 / 256000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(103472765751352777257 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_664_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (199912686621581 / 256000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(2648846011764422714851 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_664_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (199912686621581 / 256000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(2752598967989138191723 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_664_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (199912686621581 / 256000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(291237749004709474387 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_664_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (199912686621581 / 256000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3145329015573109930877 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_664_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (199912686621581 / 256000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(3470509010608891617659 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_664_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (199912686621581 / 256000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(1953955403703193484059 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_664_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (199912686621581 / 256000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(4477459106514180083733 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_664_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) *
+        Uω (aρ 10) (bρ 10) (199912686621581 / 256000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(1299476206458049317223 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_664_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) *
+        Uω (aρ 11) (bρ 11) (199912686621581 / 256000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(6085421088558797644903 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_664_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) *
+        Uω (aρ 12) (bρ 12) (199912686621581 / 256000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(3575650047906401257211 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_664_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) *
+        Uω (aρ 13) (bρ 13) (199912686621581 / 256000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4198484460239256737569 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_664_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) *
+        Uω (aρ 14) (bρ 14) (199912686621581 / 256000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(1960132106122831270009 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_664_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (199912686621581 / 256000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(225512837703902044943 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_664_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) *
+        Uω (aρ 16) (bρ 16) (199912686621581 / 256000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) *
+        (-(1253266360243979778637 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_664_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_665_1 :
+    Uω (aρ 1) (bρ 1) (25145756165739 / 32000000000000 : ℝ) ≤
+      (-(249291082670160325427 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (31174123359408541 / 40000000000000000 : ℝ) ≤
+      (-(249291082670160325427 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (31174123359408541 / 40000000000000000 : ℝ) =
+            (31174123359408541 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (31174123359408541 / 20000000000000000 : ℝ) ≤
+            (443856097629839674573 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7793510095995833 / 10000000000000000 : ℝ)) (q :=
+      (31174123359408541 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_665_2 :
+    Uω (aρ 2) (bρ 2) (25145756165739 / 32000000000000 : ℝ) ≤
+      (-(252360486208296023661 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (31078584094158457 / 40000000000000000 : ℝ) ≤
+      (-(252360486208296023661 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (31078584094158457 / 40000000000000000 : ℝ) =
+            (31078584094158457 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (31078584094158457 / 20000000000000000 : ℝ) ≤
+            (440786694091703976339 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7769509475880791 / 10000000000000000 : ℝ)) (q :=
+      (31078584094158457 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_665_3 :
+    Uω (aρ 3) (bρ 3) (25145756165739 / 32000000000000 : ℝ) ≤
+      (-(2585236824123272191887 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (30887629731693159 / 40000000000000000 : ℝ) ≤
+      (-(2585236824123272191887 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (30887629731693159 / 40000000000000000 : ℝ) =
+            (30887629731693159 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (30887629731693159 / 20000000000000000 : ℝ) ≤
+            (4346234978876727808113 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3860714200499071 / 5000000000000000 : ℝ)) (q :=
+      (30887629731693159 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_665_4 :
+    Uω (aρ 4) (bρ 4) (25145756165739 / 32000000000000 : ℝ) ≤
+      (-(1344160446545207092199 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (30570862961772313 / 40000000000000000 : ℝ) ≤
+      (-(1344160446545207092199 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (30570862961772313 / 40000000000000000 : ℝ) =
+            (30570862961772313 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (30570862961772313 / 20000000000000000 : ℝ) ≤
+            (2121575454954792907801 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7641339197102719 / 10000000000000000 : ℝ)) (q :=
+      (30570862961772313 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_665_5 :
+    Uω (aρ 5) (bρ 5) (25145756165739 / 32000000000000 : ℝ) ≤
+      (-(355881058895171594713 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (30089449842355157 / 40000000000000000 : ℝ) ≤
+      (-(355881058895171594713 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (30089449842355157 / 40000000000000000 : ℝ) =
+            (30089449842355157 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (30089449842355157 / 20000000000000000 : ℝ) ≤
+            (510552916479828405287 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7518822556729141 / 10000000000000000 : ℝ)) (q :=
+      (30089449842355157 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_665_6 :
+    Uω (aρ 6) (bρ 6) (25145756165739 / 32000000000000 : ℝ) ≤
+      (-(3078420738767238145273 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1176050222896367 / 1600000000000000 : ℝ) ≤
+      (-(3078420738767238145273 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1176050222896367 / 1600000000000000 : ℝ) =
+            (1176050222896367 / 800000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1176050222896367 / 800000000000000 : ℝ) ≤
+            (3853051064232761854727 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (146839315937923 / 200000000000000 : ℝ)) (q :=
+      (1176050222896367 / 1600000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_665_7 :
+    Uω (aρ 7) (bρ 7) (25145756165739 / 32000000000000 : ℝ) ≤
+      (-(340129780488928500307 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5693423637368201 / 8000000000000000 : ℝ) ≤
+      (-(340129780488928500307 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5693423637368201 / 8000000000000000 : ℝ) =
+            (5693423637368201 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5693423637368201 / 4000000000000000 : ℝ) ≤
+            (353017399811071499693 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1419711696974413 / 2000000000000000 : ℝ)) (q :=
+      (5693423637368201 / 8000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_665_8 :
+    Uω (aρ 8) (bρ 8) (25145756165739 / 32000000000000 : ℝ) ≤
+      (-(3835406704026617134509 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (27257774507655987 / 40000000000000000 : ℝ) ≤
+      (-(3835406704026617134509 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (27257774507655987 / 40000000000000000 : ℝ) =
+            (27257774507655987 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27257774507655987 / 20000000000000000 : ℝ) ≤
+            (3096065098973382865491 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1694357074154889 / 2500000000000000 : ℝ)) (q :=
+      (27257774507655987 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_665_9 :
+    Uω (aρ 9) (bρ 9) (25145756165739 / 32000000000000 : ℝ) ≤
+      (-(4400294372036981046643 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25760698502538297 / 40000000000000000 : ℝ) ≤
+      (-(4400294372036981046643 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (25760698502538297 / 40000000000000000 : ℝ) =
+            (25760698502538297 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25760698502538297 / 20000000000000000 : ℝ) ≤
+            (2531177430963018953357 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6369855922420711 / 10000000000000000 : ℝ)) (q :=
+      (25760698502538297 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_665_10 :
+    Uω (aρ 10) (bρ 10) (25145756165739 / 32000000000000 : ℝ) ≤
+      (-(2557073977047964558419 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23985864038979301 / 40000000000000000 : ℝ) ≤
+      (-(2557073977047964558419 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23985864038979301 / 40000000000000000 : ℝ) =
+            (23985864038979301 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23985864038979301 / 20000000000000000 : ℝ) ≤
+            (908661924452035441581 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5870997724136213 / 10000000000000000 : ℝ)) (q :=
+      (23985864038979301 / 40000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_665_11 :
+    Uω (aρ 11) (bρ 11) (25145756165739 / 32000000000000 : ℝ) ≤
+      (-(5992249965690502151559 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4393897053689701 / 8000000000000000 : ℝ) ≤
+      (-(5992249965690502151559 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (4393897053689701 / 8000000000000000 : ℝ) =
+            (4393897053689701 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4393897053689701 / 4000000000000000 : ℝ) ≤
+            (939221837309497848441 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1056256993015163 / 2000000000000000 : ℝ)) (q :=
+      (4393897053689701 / 8000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_665_12 :
+    Uω (aρ 12) (bρ 12) (25145756165739 / 32000000000000 : ℝ) ≤
+      (-(1408879486035550407403 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19775419173586063 / 40000000000000000 : ℝ) ≤
+      (-(1408879486035550407403 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (19775419173586063 / 40000000000000000 : ℝ) =
+            (19775419173586063 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19775419173586063 / 10000000000000000 : ℝ) ≤
+            (1363709235164449592597 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2303976007164797 / 5000000000000000 : ℝ)) (q :=
+      (19775419173586063 / 40000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_665_13 :
+    Uω (aρ 13) (bρ 13) (25145756165739 / 32000000000000 : ℝ) ≤
+      (-(4134674394368595173987 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17495515216103551 / 40000000000000000 : ℝ) ≤
+      (-(4134674394368595173987 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (17495515216103551 / 40000000000000000 : ℝ) =
+            (17495515216103551 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17495515216103551 / 10000000000000000 : ℝ) ≤
+            (2796797408631404826013 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1933690888591669 / 5000000000000000 : ℝ)) (q :=
+      (17495515216103551 / 40000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_665_14 :
+    Uω (aρ 14) (bρ 14) (25145756165739 / 32000000000000 : ℝ) ≤
+      (-(9640365382064048121153 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15254017567212007 / 40000000000000000 : ℝ) ≤
+      (-(9640365382064048121153 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (15254017567212007 / 40000000000000000 : ℝ) =
+            (15254017567212007 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15254017567212007 / 10000000000000000 : ℝ) ≤
+            (4222578223935951878847 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1545280118891283 / 5000000000000000 : ℝ)) (q :=
+      (15254017567212007 / 40000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_665_15 :
+    Uω (aρ 15) (bρ 15) (25145756165739 / 32000000000000 : ℝ) ≤
+      (-(442487964087878460311 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13232283235222089 / 40000000000000000 : ℝ) ≤
+      (-(442487964087878460311 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (13232283235222089 / 40000000000000000 : ℝ) =
+            (13232283235222089 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13232283235222089 / 10000000000000000 : ℝ) ≤
+            (112029780152121539689 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2339266349202607 / 10000000000000000 : ℝ)) (q :=
+      (13232283235222089 / 40000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_665_16 :
+    Uω (aρ 16) (bρ 16) (25145756165739 / 32000000000000 : ℝ) ≤
+      (-(12243810141211648147373 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11757583533052329 / 40000000000000000 : ℝ) ≤
+      (-(12243810141211648147373 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (11757583533052329 / 40000000000000000 : ℝ) =
+            (11757583533052329 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11757583533052329 / 10000000000000000 : ℝ) ≤
+            (1619133464788351852627 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1754288149627727 / 10000000000000000 : ℝ)) (q :=
+      (11757583533052329 / 40000000000000000 : ℝ)) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [bρ_16]) (by norm_num) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [aρ_16, bρ_16]) hlg
+
+lemma U_665 :
+    Uρ (25145756165739 / 32000000000000 : ℝ) ≤
+      (-(4553883503740599041333 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (25145756165739 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(249291082670160325427 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_665_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (25145756165739 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(252360486208296023661 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_665_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (25145756165739 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(2585236824123272191887 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_665_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (25145756165739 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(1344160446545207092199 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_665_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (25145756165739 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(355881058895171594713 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_665_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (25145756165739 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(3078420738767238145273 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_665_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (25145756165739 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(340129780488928500307 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_665_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (25145756165739 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(3835406704026617134509 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_665_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (25145756165739 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(4400294372036981046643 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_665_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (25145756165739 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(2557073977047964558419 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_665_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (25145756165739 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(5992249965690502151559 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_665_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (25145756165739 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(1408879486035550407403 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_665_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (25145756165739 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4134674394368595173987 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_665_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (25145756165739 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(9640365382064048121153 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_665_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (25145756165739 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(442487964087878460311 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_665_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (25145756165739 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) *
+        (-(12243810141211648147373 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_665_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_666_1 :
+    Uω (aρ 1) (bρ 1) (202419412030243 / 256000000000000 : ℝ) ≤
+      (-(2430286493772126347773 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (250959694399717187 / 320000000000000000 : ℝ) ≤
+      (-(2430286493772126347773 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (250959694399717187 / 320000000000000000 : ℝ) =
+            (250959694399717187 / 160000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (250959694399717187 / 160000000000000000 : ℝ) ≤
+            (4501185309227873652227 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7842469835635957 / 10000000000000000 : ℝ)) (q :=
+      (250959694399717187 / 320000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_666_2 :
+    Uω (aρ 2) (bρ 2) (202419412030243 / 256000000000000 : ℝ) ≤
+      (-(49215753797925468473 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (250195403495771923 / 320000000000000000 : ℝ) ≤
+      (-(49215753797925468473 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (250195403495771923 / 320000000000000000 : ℝ) =
+            (250195403495771923 / 160000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (250195403495771923 / 160000000000000000 : ℝ) ≤
+            (89413682262074531527 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3909235333324689 / 5000000000000000 : ℝ)) (q :=
+      (250195403495771923 / 320000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_666_3 :
+    Uω (aρ 3) (bρ 3) (202419412030243 / 256000000000000 : ℝ) ≤
+      (-(1261014871413616783293 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (248667837818667059 / 320000000000000000 : ℝ) ≤
+      (-(1261014871413616783293 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (248667837818667059 / 320000000000000000 : ℝ) =
+            (248667837818667059 / 160000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (248667837818667059 / 160000000000000000 : ℝ) ≤
+            (2204721030086383216707 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1942598479545081 / 2500000000000000 : ℝ)) (q :=
+      (248667837818667059 / 320000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_666_4 :
+    Uω (aρ 4) (bρ 4) (202419412030243 / 256000000000000 : ℝ) ≤
+      (-(656113373712938037237 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (246133887510078227 / 320000000000000000 : ℝ) ≤
+      (-(656113373712938037237 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (246133887510078227 / 320000000000000000 : ℝ) =
+            (246133887510078227 / 160000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (246133887510078227 / 160000000000000000 : ℝ) ≤
+            (1076754577037061962763 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3845158102479261 / 5000000000000000 : ℝ)) (q :=
+      (246133887510078227 / 320000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_666_5 :
+    Uω (aρ 5) (bρ 5) (202419412030243 / 256000000000000 : ℝ) ≤
+      (-(1391071934225770335259 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (242283034957824227 / 320000000000000000 : ℝ) ≤
+      (-(1391071934225770335259 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (242283034957824227 / 320000000000000000 : ℝ) =
+            (242283034957824227 / 160000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (242283034957824227 / 160000000000000000 : ℝ) ≤
+            (2074663967274229664741 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7567827839777647 / 10000000000000000 : ℝ)) (q :=
+      (242283034957824227 / 320000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_666_6 :
+    Uω (aρ 6) (bρ 6) (202419412030243 / 256000000000000 : ℝ) ≤
+      (-(1505979082273865270421 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (236778517547688323 / 320000000000000000 : ℝ) ≤
+      (-(1505979082273865270421 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (236778517547688323 / 320000000000000000 : ℝ) =
+            (236778517547688323 / 160000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (236778517547688323 / 160000000000000000 : ℝ) ≤
+            (1959756819226134729579 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7391035876784153 / 10000000000000000 : ℝ)) (q :=
+      (236778517547688323 / 320000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_666_7 :
+    Uω (aρ 7) (bρ 7) (202419412030243 / 256000000000000 : ℝ) ≤
+      (-(208285297068072753121 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (45861528555831351 / 64000000000000000 : ℝ) ≤
+      (-(208285297068072753121 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (45861528555831351 / 64000000000000000 : ℝ) =
+            (45861528555831351 / 32000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (45861528555831351 / 32000000000000000 : ℝ) ≤
+            (224931690619427246879 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (714776758451093 / 1000000000000000 : ℝ)) (q :=
+      (45861528555831351 / 64000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_666_8 :
+    Uω (aρ 8) (bρ 8) (202419412030243 / 256000000000000 : ℝ) ≤
+      (-(37634301630337236271 / 100000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (219637394386552531 / 320000000000000000 : ℝ) ≤
+      (-(37634301630337236271 / 100000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (219637394386552531 / 320000000000000000 : ℝ) =
+            (219637394386552531 / 160000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (219637394386552531 / 160000000000000000 : ℝ) ≤
+            (31680416399662763729 / 100000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3413459355656583 / 5000000000000000 : ℝ)) (q :=
+      (219637394386552531 / 320000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8])
+      (by norm_num [bρ_8]) (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_666_9 :
+    Uω (aρ 9) (bρ 9) (202419412030243 / 256000000000000 : ℝ) ≤
+      (-(864746693025047178221 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (41533890668800807 / 64000000000000000 : ℝ) ≤
+      (-(864746693025047178221 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (41533890668800807 / 64000000000000000 : ℝ) =
+            (41533890668800807 / 32000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (41533890668800807 / 32000000000000000 : ℝ) ≤
+            (521547667574952821779 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1283977604902777 / 2000000000000000 : ℝ)) (q :=
+      (41533890668800807 / 64000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_666_10 :
+    Uω (aρ 10) (bρ 10) (202419412030243 / 256000000000000 : ℝ) ≤
+      (-(31444476630483468137 / 62500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (193486814774151011 / 320000000000000000 : ℝ) ≤
+      (-(31444476630483468137 / 62500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (193486814774151011 / 320000000000000000 : ℝ) =
+            (193486814774151011 / 160000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (193486814774151011 / 160000000000000000 : ℝ) ≤
+            (11877222138266531863 / 62500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5922032147393071 / 10000000000000000 : ℝ)) (q :=
+      (193486814774151011 / 320000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_666_11 :
+    Uω (aρ 11) (bρ 11) (202419412030243 / 256000000000000 : ℝ) ≤
+      (-(2950003497771861187207 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (177384606978711139 / 320000000000000000 : ℝ) ≤
+      (-(2950003497771861187207 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (177384606978711139 / 320000000000000000 : ℝ) =
+            (177384606978711139 / 160000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (177384606978711139 / 160000000000000000 : ℝ) ≤
+            (515732403728138812793 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5334120786383079 / 10000000000000000 : ℝ)) (q :=
+      (177384606978711139 / 320000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_666_12 :
+    Uω (aρ 12) (bρ 12) (202419412030243 / 256000000000000 : ℝ) ≤
+      (-(3469394193332567164563 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (159882977435271507 / 320000000000000000 : ℝ) ≤
+      (-(3469394193332567164563 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (159882977435271507 / 320000000000000000 : ℝ) =
+            (159882977435271507 / 80000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 15
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (159882977435271507 / 80000000000000000 : ℝ) ≤
+            (3462077609667432835437 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2331984518301551 / 5000000000000000 : ℝ)) (q :=
+      (159882977435271507 / 320000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_666_13 :
+    Uω (aρ 13) (bρ 13) (202419412030243 / 256000000000000 : ℝ) ≤
+      (-(8143751499644154206229 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (141733118963215363 / 320000000000000000 : ℝ) ≤
+      (-(8143751499644154206229 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (141733118963215363 / 320000000000000000 : ℝ) =
+            (141733118963215363 / 80000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (141733118963215363 / 80000000000000000 : ℝ) ≤
+            (5719192106355845793771 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3928984623694593 / 10000000000000000 : ℝ)) (q :=
+      (141733118963215363 / 320000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_666_14 :
+    Uω (aρ 14) (bρ 14) (202419412030243 / 256000000000000 : ℝ) ≤
+      (-(9483755677962245212443 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (123958325876728179 / 320000000000000000 : ℝ) ≤
+      (-(9483755677962245212443 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (123958325876728179 / 320000000000000000 : ℝ) =
+            (123958325876728179 / 80000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (123958325876728179 / 80000000000000000 : ℝ) ≤
+            (4379187928037754787557 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (98812104401067 / 312500000000000 : ℝ)) (q :=
+      (123958325876728179 / 320000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_666_15 :
+    Uω (aρ 15) (bρ 15) (202419412030243 / 256000000000000 : ℝ) ≤
+      (-(5428388530800015652507 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (108055316856504243 / 320000000000000000 : ℝ) ≤
+      (-(5428388530800015652507 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (108055316856504243 / 320000000000000000 : ℝ) =
+            (108055316856504243 / 80000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (108055316856504243 / 80000000000000000 : ℝ) ≤
+            (1503083272199984347493 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (606905638621287 / 2500000000000000 : ℝ)) (q :=
+      (108055316856504243 / 320000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_666_16 :
+    Uω (aρ 16) (bρ 16) (202419412030243 / 256000000000000 : ℝ) ≤
+      (-(748334846524065932333 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (96639275845260099 / 320000000000000000 : ℝ) ≤
+      (-(748334846524065932333 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (96639275845260099 / 320000000000000000 : ℝ) =
+            (96639275845260099 / 80000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (96639275845260099 / 80000000000000000 : ℝ) ≤
+            (118099128850934067667 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1866491642792389 / 10000000000000000 : ℝ)) (q :=
+      (96639275845260099 / 320000000000000000 : ℝ)) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [bρ_16]) (by norm_num) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [aρ_16, bρ_16]) hlg
+
+lemma U_666 :
+    Uρ (202419412030243 / 256000000000000 : ℝ) ≤
+      (-(4472242976673732246981 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (202419412030243 / 256000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2430286493772126347773 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_666_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (202419412030243 / 256000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) * (-(49215753797925468473 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_666_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (202419412030243 / 256000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1261014871413616783293 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_666_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (202419412030243 / 256000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(656113373712938037237 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_666_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (202419412030243 / 256000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1391071934225770335259 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_666_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (202419412030243 / 256000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1505979082273865270421 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_666_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (202419412030243 / 256000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(208285297068072753121 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_666_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (202419412030243 / 256000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) * (-(37634301630337236271 / 100000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_666_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (202419412030243 / 256000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(864746693025047178221 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_666_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) *
+        Uω (aρ 10) (bρ 10) (202419412030243 / 256000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) * (-(31444476630483468137 / 62500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_666_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) *
+        Uω (aρ 11) (bρ 11) (202419412030243 / 256000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(2950003497771861187207 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_666_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) *
+        Uω (aρ 12) (bρ 12) (202419412030243 / 256000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(3469394193332567164563 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_666_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) *
+        Uω (aρ 13) (bρ 13) (202419412030243 / 256000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(8143751499644154206229 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_666_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) *
+        Uω (aρ 14) (bρ 14) (202419412030243 / 256000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(9483755677962245212443 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_666_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (202419412030243 / 256000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(5428388530800015652507 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_666_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) *
+        Uω (aρ 16) (bρ 16) (202419412030243 / 256000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(748334846524065932333 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_666_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_667_1 :
+    Uω (aρ 1) (bρ 1) (101836387367287 / 128000000000000 : ℝ) ≤
+      (-(1184025952061820073691 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25252640187274563 / 32000000000000000 : ℝ) ≤
+      (-(1184025952061820073691 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (25252640187274563 / 32000000000000000 : ℝ) =
+            (25252640187274563 / 16000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25252640187274563 / 16000000000000000 : ℝ) ≤
+            (2281709949438179926309 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1578285914412461 / 2000000000000000 : ℝ)) (q :=
+      (25252640187274563 / 32000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_667_2 :
+    Uω (aρ 2) (bρ 2) (101836387367287 / 128000000000000 : ℝ) ≤
+      (-(149897666808250901199 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25176213389771851 / 32000000000000000 : ℝ) ≤
+      (-(149897666808250901199 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (25176213389771851 / 32000000000000000 : ℝ) =
+            (25176213389771851 / 16000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25176213389771851 / 16000000000000000 : ℝ) ≤
+            (283319320879249098801 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (786743183613311 / 1000000000000000 : ℝ)) (q :=
+      (25176213389771851 / 32000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_667_3 :
+    Uω (aρ 3) (bρ 3) (101836387367287 / 128000000000000 : ℝ) ≤
+      (-(2459219715338930544141 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (125117318287109767 / 160000000000000000 : ℝ) ≤
+      (-(2459219715338930544141 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (125117318287109767 / 160000000000000000 : ℝ) =
+            (125117318287109767 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (125117318287109767 / 80000000000000000 : ℝ) ≤
+            (4472252087661069455859 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (977419919970553 / 1250000000000000 : ℝ)) (q :=
+      (125117318287109767 / 160000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_667_4 :
+    Uω (aρ 4) (bρ 4) (101836387367287 / 128000000000000 : ℝ) ≤
+      (-(2560991557050394938723 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (123850433888610727 / 160000000000000000 : ℝ) ≤
+      (-(2560991557050394938723 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (123850433888610727 / 160000000000000000 : ℝ) =
+            (123850433888610727 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (123850433888610727 / 80000000000000000 : ℝ) ≤
+            (4370480245949605061277 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1934823247754261 / 2500000000000000 : ℝ)) (q :=
+      (123850433888610727 / 160000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_667_5 :
+    Uω (aρ 5) (bρ 5) (101836387367287 / 128000000000000 : ℝ) ≤
+      (-(1358829098960116310293 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (121925230875660191 / 160000000000000000 : ℝ) ≤
+      (-(1358829098960116310293 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (121925230875660191 / 160000000000000000 : ℝ) =
+            (121925230875660191 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (121925230875660191 / 80000000000000000 : ℝ) ≤
+            (2106906802539883689707 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7616832533733227 / 10000000000000000 : ℝ)) (q :=
+      (121925230875660191 / 160000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_667_6 :
+    Uω (aρ 6) (bρ 6) (101836387367287 / 128000000000000 : ℝ) ≤
+      (-(2945935381077970403423 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23834696720031283 / 32000000000000000 : ℝ) ≤
+      (-(2945935381077970403423 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23834696720031283 / 32000000000000000 : ℝ) =
+            (23834696720031283 / 16000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23834696720031283 / 16000000000000000 : ℝ) ≤
+            (3985536421922029596577 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1488020899887051 / 2000000000000000 : ℝ)) (q :=
+      (23834696720031283 / 32000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_667_7 :
+    Uω (aρ 7) (bρ 7) (101836387367287 / 128000000000000 : ℝ) ≤
+      (-(130572130440057185229 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (115439142792856383 / 160000000000000000 : ℝ) ≤
+      (-(130572130440057185229 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (115439142792856383 / 160000000000000000 : ℝ) =
+            (115439142792856383 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (115439142792856383 / 80000000000000000 : ℝ) ≤
+            (146686741679942814771 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7196973279282751 / 10000000000000000 : ℝ)) (q :=
+      (115439142792856383 / 160000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_667_8 :
+    Uω (aρ 8) (bρ 8) (101836387367287 / 128000000000000 : ℝ) ≤
+      (-(1845986741144000437987 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22121247108923667 / 32000000000000000 : ℝ) ≤
+      (-(1845986741144000437987 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (22121247108923667 / 32000000000000000 : ℝ) =
+            (22121247108923667 / 16000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22121247108923667 / 16000000000000000 : ℝ) ≤
+            (1619749160355999562013 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1375280304918599 / 2000000000000000 : ℝ)) (q :=
+      (22121247108923667 / 32000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_667_9 :
+    Uω (aρ 9) (bρ 9) (101836387367287 / 128000000000000 : ℝ) ≤
+      (-(2123883406626890648073 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20925305608484907 / 32000000000000000 : ℝ) ≤
+      (-(2123883406626890648073 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (20925305608484907 / 32000000000000000 : ℝ) =
+            (20925305608484907 / 16000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20925305608484907 / 16000000000000000 : ℝ) ≤
+            (1341852494873109351927 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (646990371517877 / 1000000000000000 : ℝ)) (q :=
+      (20925305608484907 / 32000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_667_10 :
+    Uω (aρ 10) (bρ 10) (101836387367287 / 128000000000000 : ℝ) ≤
+      (-(618599601174517764213 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19508616145571099 / 32000000000000000 : ℝ) ≤
+      (-(618599601174517764213 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (19508616145571099 / 32000000000000000 : ℝ) =
+            (19508616145571099 / 16000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19508616145571099 / 16000000000000000 : ℝ) ≤
+            (247834374200482235787 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1166607780147 / 1953125000000 : ℝ)) (q :=
+      (19508616145571099 / 32000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_667_11 :
+    Uω (aρ 11) (bρ 11) (101836387367287 / 128000000000000 : ℝ) ≤
+      (-(1452168150311807516473 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (89506079902867479 / 160000000000000000 : ℝ) ≤
+      (-(1452168150311807516473 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (89506079902867479 / 160000000000000000 : ℝ) =
+            (89506079902867479 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (89506079902867479 / 80000000000000000 : ℝ) ≤
+            (280699800438192483527 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2693441678717069 / 5000000000000000 : ℝ)) (q :=
+      (89506079902867479 / 160000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_667_12 :
+    Uω (aρ 12) (bρ 12) (101836387367287 / 128000000000000 : ℝ) ≤
+      (-(6834438478415439925663 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (80780044994235911 / 160000000000000000 : ℝ) ≤
+      (-(6834438478415439925663 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (80780044994235911 / 160000000000000000 : ℝ) =
+            (80780044994235911 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (80780044994235911 / 80000000000000000 : ℝ) ≤
+            (97033324584560074337 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (147494659079381 / 312500000000000 : ℝ)) (q :=
+      (80780044994235911 / 160000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_667_13 :
+    Uω (aρ 13) (bρ 13) (101836387367287 / 128000000000000 : ℝ) ≤
+      (-(802010272264151745109 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14349651124315259 / 32000000000000000 : ℝ) ≤
+      (-(802010272264151745109 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (14349651124315259 / 32000000000000000 : ℝ) =
+            (14349651124315259 / 8000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14349651124315259 / 8000000000000000 : ℝ) ≤
+            (584284088335848254891 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (199511858027637 / 500000000000000 : ℝ)) (q :=
+      (14349651124315259 / 32000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_667_14 :
+    Uω (aρ 14) (bρ 14) (101836387367287 / 128000000000000 : ℝ) ≤
+      (-(4665312136517286567177 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12587112524091627 / 32000000000000000 : ℝ) ≤
+      (-(4665312136517286567177 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (12587112524091627 / 32000000000000000 : ℝ) =
+            (12587112524091627 / 8000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12587112524091627 / 8000000000000000 : ℝ) ≤
+            (2266159666482713432823 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (323257782045797 / 1000000000000000 : ℝ)) (q :=
+      (12587112524091627 / 32000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_667_15 :
+    Uω (aρ 15) (bρ 15) (101836387367287 / 128000000000000 : ℝ) ≤
+      (-(10658612229026117417291 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (55108975179349567 / 160000000000000000 : ℝ) ≤
+      (-(10658612229026117417291 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (55108975179349567 / 160000000000000000 : ℝ) =
+            (55108975179349567 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (55108975179349567 / 40000000000000000 : ℝ) ≤
+            (3204331376973882582709 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2513827667734399 / 10000000000000000 : ℝ)) (q :=
+      (55108975179349567 / 160000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_667_16 :
+    Uω (aρ 16) (bρ 16) (101836387367287 / 128000000000000 : ℝ) ≤
+      (-(5859174174619187689721 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (49567678580104111 / 160000000000000000 : ℝ) ≤
+      (-(5859174174619187689721 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (49567678580104111 / 160000000000000000 : ℝ) =
+            (49567678580104111 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (49567678580104111 / 40000000000000000 : ℝ) ≤
+            (1072297628380812310279 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1973537244338717 / 10000000000000000 : ℝ)) (q :=
+      (49567678580104111 / 160000000000000000 : ℝ)) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [bρ_16]) (by norm_num) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [aρ_16, bρ_16]) hlg
+
+lemma U_667 :
+    Uρ (101836387367287 / 128000000000000 : ℝ) ≤
+      (-(1097916055907063284131 / 2500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (101836387367287 / 128000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1184025952061820073691 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_667_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (101836387367287 / 128000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(149897666808250901199 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_667_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (101836387367287 / 128000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(2459219715338930544141 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_667_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (101836387367287 / 128000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(2560991557050394938723 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_667_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (101836387367287 / 128000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1358829098960116310293 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_667_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (101836387367287 / 128000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(2945935381077970403423 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_667_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (101836387367287 / 128000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(130572130440057185229 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_667_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (101836387367287 / 128000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(1845986741144000437987 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_667_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (101836387367287 / 128000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(2123883406626890648073 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_667_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) *
+        Uω (aρ 10) (bρ 10) (101836387367287 / 128000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(618599601174517764213 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_667_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) *
+        Uω (aρ 11) (bρ 11) (101836387367287 / 128000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(1452168150311807516473 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_667_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) *
+        Uω (aρ 12) (bρ 12) (101836387367287 / 128000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(6834438478415439925663 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_667_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) *
+        Uω (aρ 13) (bρ 13) (101836387367287 / 128000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(802010272264151745109 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_667_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) *
+        Uω (aρ 14) (bρ 14) (101836387367287 / 128000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(4665312136517286567177 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_667_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (101836387367287 / 128000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(10658612229026117417291 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_667_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) *
+        Uω (aρ 16) (bρ 16) (101836387367287 / 128000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) *
+        (-(5859174174619187689721 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_667_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_668_1 :
+    Uω (aρ 1) (bρ 1) (40985227487781 / 51200000000000 : ℝ) ≤
+      (-(28827527957205322453 / 125000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (254093109295304793 / 320000000000000000 : ℝ) ≤
+      (-(28827527957205322453 / 125000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (254093109295304793 / 320000000000000000 : ℝ) =
+            (254093109295304793 / 160000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (254093109295304793 / 160000000000000000 : ℝ) ≤
+            (57815869580294677547 / 125000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7940389305334323 / 10000000000000000 : ℝ)) (q :=
+      (254093109295304793 / 320000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_668_2 :
+    Uω (aρ 2) (bρ 2) (40985227487781 / 51200000000000 : ℝ) ≤
+      (-(46726498663784903531 / 200000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (253328863965426249 / 320000000000000000 : ℝ) ≤
+      (-(46726498663784903531 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (253328863965426249 / 320000000000000000 : ℝ) =
+            (253328863965426249 / 160000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (253328863965426249 / 160000000000000000 : ℝ) ≤
+            (91902937396215096469 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3958196492363457 / 5000000000000000 : ℝ)) (q :=
+      (253328863965426249 / 320000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_668_3 :
+    Uω (aρ 3) (bρ 3) (40985227487781 / 51200000000000 : ℝ) ≤
+      (-(2396801783767159707439 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (50360286828556901 / 64000000000000000 : ℝ) ≤
+      (-(2396801783767159707439 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (50360286828556901 / 64000000000000000 : ℝ) =
+            (50360286828556901 / 32000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (50360286828556901 / 32000000000000000 : ℝ) ≤
+            (4534670019232840292561 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1573664945432361 / 2000000000000000 : ℝ)) (q :=
+      (50360286828556901 / 64000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_668_4 :
+    Uω (aρ 4) (bρ 4) (40985227487781 / 51200000000000 : ℝ) ≤
+      (-(312241245281524448941 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (249267844562557577 / 320000000000000000 : ℝ) ≤
+      (-(312241245281524448941 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (249267844562557577 / 320000000000000000 : ℝ) =
+            (249267844562557577 / 160000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (249267844562557577 / 160000000000000000 : ℝ) ≤
+            (554192730093475551059 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3894134779731311 / 5000000000000000 : ℝ)) (q :=
+      (249267844562557577 / 320000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_668_5 :
+    Uω (aρ 5) (bρ 5) (40985227487781 / 51200000000000 : ℝ) ≤
+      (-(663396520318592504421 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (49083575860018021 / 64000000000000000 : ℝ) ≤
+      (-(663396520318592504421 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (49083575860018021 / 64000000000000000 : ℝ) =
+            (49083575860018021 / 32000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (49083575860018021 / 32000000000000000 : ℝ) ≤
+            (1069471430431407495579 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1533167329978681 / 2000000000000000 : ℝ)) (q :=
+      (49083575860018021 / 64000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_668_6 :
+    Uω (aρ 6) (bρ 6) (40985227487781 / 51200000000000 : ℝ) ≤
+      (-(115213863744958042733 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (47983078799087621 / 64000000000000000 : ℝ) ≤
+      (-(115213863744958042733 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (47983078799087621 / 64000000000000000 : ℝ) =
+            (47983078799087621 / 32000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (47983078799087621 / 32000000000000000 : ℝ) ≤
+            (162045008375041957267 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1497834338698531 / 2000000000000000 : ℝ)) (q :=
+      (47983078799087621 / 64000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_668_7 :
+    Uω (aρ 7) (bρ 7) (40985227487781 / 51200000000000 : ℝ) ≤
+      (-(3196506870553017259521 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (232448875024204217 / 320000000000000000 : ℝ) ≤
+      (-(3196506870553017259521 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (232448875024204217 / 320000000000000000 : ℝ) =
+            (232448875024204217 / 160000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (232448875024204217 / 160000000000000000 : ℝ) ≤
+            (3734964932446982740479 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7246175638550537 / 10000000000000000 : ℝ)) (q :=
+      (232448875024204217 / 320000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_668_8 :
+    Uω (aρ 8) (bρ 8) (40985227487781 / 51200000000000 : ℝ) ≤
+      (-(3621029128780322054959 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (222787428776145417 / 320000000000000000 : ℝ) ≤
+      (-(3621029128780322054959 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (222787428776145417 / 320000000000000000 : ℝ) =
+            (222787428776145417 / 160000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (222787428776145417 / 160000000000000000 : ℝ) ≤
+            (3310442674219677945041 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3462938449693431 / 5000000000000000 : ℝ)) (q :=
+      (222787428776145417 / 320000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8])
+      (by norm_num [bρ_8]) (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_668_9 :
+    Uω (aρ 9) (bρ 9) (40985227487781 / 51200000000000 : ℝ) ≤
+      (-(2086192537048560607821 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (210836402285843529 / 320000000000000000 : ℝ) ≤
+      (-(2086192537048560607821 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (210836402285843529 / 320000000000000000 : ℝ) =
+            (210836402285843529 / 160000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (210836402285843529 / 160000000000000000 : ℝ) ≤
+            (1379543364451439392179 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3259951686051497 / 5000000000000000 : ℝ)) (q :=
+      (210836402285843529 / 320000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9])
+      (by norm_num [bρ_9]) (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_668_10 :
+    Uω (aρ 10) (bρ 10) (40985227487781 / 51200000000000 : ℝ) ≤
+      (-(4867177017714833062951 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (39336993294475941 / 64000000000000000 : ℝ) ≤
+      (-(4867177017714833062951 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (39336993294475941 / 64000000000000000 : ℝ) =
+            (39336993294475941 / 32000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (39336993294475941 / 32000000000000000 : ℝ) ≤
+            (2064294785285166937049 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1204799533451301 / 2000000000000000 : ℝ)) (q :=
+      (39336993294475941 / 64000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_668_11 :
+    Uω (aρ 11) (bρ 11) (40985227487781 / 51200000000000 : ℝ) ≤
+      (-(571822785259138887833 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (180638574733142217 / 320000000000000000 : ℝ) ≤
+      (-(571822785259138887833 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (180638574733142217 / 320000000000000000 : ℝ) =
+            (180638574733142217 / 160000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (180638574733142217 / 160000000000000000 : ℝ) ≤
+            (121324395040861112167 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2719787404879581 / 5000000000000000 : ℝ)) (q :=
+      (180638574733142217 / 320000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_668_12 :
+    Uω (aρ 12) (bρ 12) (40985227487781 / 51200000000000 : ℝ) ≤
+      (-(420707167491835697151 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6529391167203393 / 12800000000000000 : ℝ) ≤
+      (-(420707167491835697151 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (6529391167203393 / 12800000000000000 : ℝ) =
+            (6529391167203393 / 6400000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6529391167203393 / 6400000000000000 : ℝ) ≤
+            (12509820195664302849 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (191021507375123 / 400000000000000 : ℝ)) (q :=
+      (6529391167203393 / 12800000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_668_13 :
+    Uω (aρ 13) (bρ 13) (40985227487781 / 51200000000000 : ℝ) ≤
+      (-(1579666510826825980323 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (145254552806008569 / 320000000000000000 : ℝ) ≤
+      (-(1579666510826825980323 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (145254552806008569 / 320000000000000000 : ℝ) =
+            (145254552806008569 / 80000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (145254552806008569 / 80000000000000000 : ℝ) ≤
+            (1192922210373174019677 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4051155277593309 / 10000000000000000 : ℝ)) (q :=
+      (145254552806008569 / 320000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_668_14 :
+    Uω (aρ 14) (bρ 14) (40985227487781 / 51200000000000 : ℝ) ≤
+      (-(1836156577461968690781 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (127771397037108601 / 320000000000000000 : ℝ) ≤
+      (-(1836156577461968690781 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (127771397037108601 / 320000000000000000 : ℝ) =
+            (127771397037108601 / 80000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (127771397037108601 / 80000000000000000 : ℝ) ≤
+            (936432143738031309219 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3302385327082061 / 10000000000000000 : ℝ)) (q :=
+      (127771397037108601 / 320000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_668_15 :
+    Uω (aρ 15) (bρ 15) (40985227487781 / 51200000000000 : ℝ) ≤
+      (-(5233528120475439557461 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (112349592456445801 / 320000000000000000 : ℝ) ≤
+      (-(5233528120475439557461 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (112349592456445801 / 320000000000000000 : ℝ) =
+            (112349592456445801 / 80000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (112349592456445801 / 80000000000000000 : ℝ) ≤
+            (1697943682524560442539 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (649523954551409 / 2500000000000000 : ℝ)) (q :=
+      (112349592456445801 / 320000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_668_16 :
+    Uω (aρ 16) (bρ 16) (40985227487781 / 51200000000000 : ℝ) ≤
+      (-(11476547035735173474157 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (101561679294659737 / 320000000000000000 : ℝ) ≤
+      (-(11476547035735173474157 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (101561679294659737 / 320000000000000000 : ℝ) =
+            (101561679294659737 / 80000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (101561679294659737 / 80000000000000000 : ℝ) ≤
+            (2386396570264826525843 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2076222897104007 / 10000000000000000 : ℝ)) (q :=
+      (101561679294659737 / 320000000000000000 : ℝ)) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [bρ_16]) (by norm_num) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [aρ_16, bρ_16]) hlg
+
+lemma U_668 :
+    Uρ (40985227487781 / 51200000000000 : ℝ) ≤
+      (-(431209322955317224327 / 1000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (40985227487781 / 51200000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(28827527957205322453 / 125000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_668_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (40985227487781 / 51200000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) * (-(46726498663784903531 / 200000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_668_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (40985227487781 / 51200000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(2396801783767159707439 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_668_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (40985227487781 / 51200000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(312241245281524448941 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_668_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (40985227487781 / 51200000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(663396520318592504421 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_668_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (40985227487781 / 51200000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(115213863744958042733 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_668_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (40985227487781 / 51200000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(3196506870553017259521 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_668_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (40985227487781 / 51200000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(3621029128780322054959 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_668_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (40985227487781 / 51200000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(2086192537048560607821 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_668_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (40985227487781 / 51200000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(4867177017714833062951 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_668_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (40985227487781 / 51200000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(571822785259138887833 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_668_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (40985227487781 / 51200000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(420707167491835697151 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_668_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (40985227487781 / 51200000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(1579666510826825980323 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_668_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (40985227487781 / 51200000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(1836156577461968690781 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_668_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (40985227487781 / 51200000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(5233528120475439557461 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_668_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (40985227487781 / 51200000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) *
+        (-(11476547035735173474157 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_668_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_669_1 :
+    Uω (aρ 1) (bρ 1) (51544875035809 / 64000000000000 : ℝ) ≤
+      (-(2244732758859609584141 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (63914954167080633 / 80000000000000000 : ℝ) ≤
+      (-(2244732758859609584141 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (63914954167080633 / 80000000000000000 : ℝ) =
+            (63914954167080633 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (63914954167080633 / 40000000000000000 : ℝ) ≤
+            (4686739044140390415859 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3994674517755001 / 5000000000000000 : ℝ)) (q :=
+      (63914954167080633 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_669_2 :
+    Uω (aρ 2) (bρ 2) (51544875035809 / 64000000000000 : ℝ) ≤
+      (-(227466970668141009613 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (63723898426264653 / 80000000000000000 : ℝ) ≤
+      (-(227466970668141009613 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (63723898426264653 / 80000000000000000 : ℝ) =
+            (63723898426264653 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (63723898426264653 / 40000000000000000 : ℝ) ≤
+            (465680209631858990387 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7965354112816007 / 10000000000000000 : ℝ)) (q :=
+      (63723898426264653 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_669_3 :
+    Uω (aρ 3) (bρ 3) (51544875035809 / 64000000000000 : ℝ) ≤
+      (-(2334771082517334658739 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (63342057636596289 / 80000000000000000 : ℝ) ≤
+      (-(2334771082517334658739 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (63342057636596289 / 80000000000000000 : ℝ) =
+            (63342057636596289 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (63342057636596289 / 40000000000000000 : ℝ) ≤
+            (4596700720482665341261 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1979322505437229 / 2500000000000000 : ℝ)) (q :=
+      (63342057636596289 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_669_4 :
+    Uω (aρ 4) (bρ 4) (51544875035809 / 64000000000000 : ℝ) ≤
+      (-(304407961166478728341 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (62708704482840553 / 80000000000000000 : ℝ) ≤
+      (-(304407961166478728341 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (62708704482840553 / 80000000000000000 : ℝ) =
+            (62708704482840553 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (62708704482840553 / 40000000000000000 : ℝ) ≤
+            (562026014208521271659 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3918622957187491 / 5000000000000000 : ℝ)) (q :=
+      (62708704482840553 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_669_5 :
+    Uω (aρ 5) (bρ 5) (51544875035809 / 64000000000000 : ℝ) ≤
+      (-(647480560805871238739 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2469852877802989 / 3200000000000000 : ℝ) ≤
+      (-(647480560805871238739 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2469852877802989 / 3200000000000000 : ℝ) =
+            (2469852877802989 / 1600000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2469852877802989 / 1600000000000000 : ℝ) ≤
+            (1085387389944128761261 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (308593607970741 / 400000000000000 : ℝ)) (q :=
+      (2469852877802989 / 3200000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_669_6 :
+    Uω (aρ 6) (bρ 6) (51544875035809 / 64000000000000 : ℝ) ≤
+      (-(87974566296101048461 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12074189918970493 / 16000000000000000 : ℝ) ≤
+      (-(87974566296101048461 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (12074189918970493 / 16000000000000000 : ℝ) =
+            (12074189918970493 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12074189918970493 / 8000000000000000 : ℝ) ≤
+            (128633927547648951539 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (47113984292831 / 62500000000000 : ℝ)) (q :=
+      (12074189918970493 / 16000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_669_7 :
+    Uω (aρ 7) (bρ 7) (51544875035809 / 64000000000000 : ℝ) ≤
+      (-(391146157008232414821 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (58504853043606353 / 80000000000000000 : ℝ) ≤
+      (-(391146157008232414821 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (58504853043606353 / 80000000000000000 : ℝ) =
+            (58504853043606353 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (58504853043606353 / 40000000000000000 : ℝ) ≤
+            (475287818366767585179 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (911921841225179 / 1250000000000000 : ℝ)) (q :=
+      (58504853043606353 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_669_8 :
+    Uω (aρ 8) (bρ 8) (51544875035809 / 64000000000000 : ℝ) ≤
+      (-(355058973366974848083 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (56090567494944313 / 80000000000000000 : ℝ) ≤
+      (-(355058973366974848083 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (56090567494944313 / 80000000000000000 : ℝ) =
+            (56090567494944313 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (56090567494944313 / 40000000000000000 : ℝ) ≤
+            (338088206933025151917 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3487672496987961 / 5000000000000000 : ℝ)) (q :=
+      (56090567494944313 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_669_9 :
+    Uω (aρ 9) (bρ 9) (51544875035809 / 64000000000000 : ℝ) ≤
+      (-(1024394782019966389091 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (53104874451030497 / 80000000000000000 : ℝ) ≤
+      (-(1024394782019966389091 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (53104874451030497 / 80000000000000000 : ℝ) =
+            (53104874451030497 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (53104874451030497 / 40000000000000000 : ℝ) ≤
+            (708473168730033610909 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1642471840339367 / 2500000000000000 : ℝ)) (q :=
+      (53104874451030497 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_669_10 :
+    Uω (aρ 10) (bρ 10) (51544875035809 / 64000000000000 : ℝ) ≤
+      (-(11965611613780661111 / 25000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (49570810864276341 / 80000000000000000 : ℝ) ≤
+      (-(11965611613780661111 / 25000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (49570810864276341 / 80000000000000000 : ℝ) =
+            (49570810864276341 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (49570810864276341 / 40000000000000000 : ℝ) ≤
+            (5363067893719338889 / 25000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6074930498163929 / 10000000000000000 : ℝ)) (q :=
+      (49570810864276341 / 80000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_669_11 :
+    Uω (aρ 11) (bρ 11) (51544875035809 / 64000000000000 : ℝ) ≤
+      (-(5628654436701494422077 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (45565971127884609 / 80000000000000000 : ℝ) ≤
+      (-(5628654436701494422077 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (45565971127884609 / 80000000000000000 : ℝ) =
+            (45565971127884609 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (45565971127884609 / 40000000000000000 : ℝ) ≤
+            (1302817366298505577923 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1373049297567749 / 2500000000000000 : ℝ)) (q :=
+      (45565971127884609 / 80000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_669_12 :
+    Uω (aρ 12) (bρ 12) (51544875035809 / 64000000000000 : ℝ) ≤
+      (-(6629385355911510922651 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1649071288432777 / 3200000000000000 : ℝ) ≤
+      (-(6629385355911510922651 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1649071288432777 / 3200000000000000 : ℝ) =
+            (1649071288432777 / 1600000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1649071288432777 / 1600000000000000 : ℝ) ≤
+            (302086447088489077349 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (48311000576897 / 100000000000000 : ℝ)) (q :=
+      (1649071288432777 / 3200000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_669_13 :
+    Uω (aρ 13) (bρ 13) (51544875035809 / 64000000000000 : ℝ) ≤
+      (-(972296894257070220027 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (36751870368632741 / 80000000000000000 : ℝ) ≤
+      (-(972296894257070220027 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (36751870368632741 / 80000000000000000 : ℝ) =
+            (36751870368632741 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (36751870368632741 / 20000000000000000 : ℝ) ≤
+            (760571056492929779973 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4111753838738029 / 10000000000000000 : ℝ)) (q :=
+      (36751870368632741 / 80000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_669_14 :
+    Uω (aρ 14) (bρ 14) (51544875035809 / 64000000000000 : ℝ) ≤
+      (-(9034059768671761378607 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6482995971853097 / 16000000000000000 : ℝ) ≤
+      (-(9034059768671761378607 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6482995971853097 / 16000000000000000 : ℝ) =
+            (6482995971853097 / 4000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6482995971853097 / 4000000000000000 : ℝ) ≤
+            (4828883837328238621393 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (674291699288243 / 2000000000000000 : ℝ)) (q :=
+      (6482995971853097 / 16000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_669_15 :
+    Uω (aρ 15) (bρ 15) (51544875035809 / 64000000000000 : ℝ) ≤
+      (-(10281552864863602885801 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28613291501051061 / 80000000000000000 : ℝ) ≤
+      (-(10281552864863602885801 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (28613291501051061 / 80000000000000000 : ℝ) =
+            (28613291501051061 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28613291501051061 / 20000000000000000 : ℝ) ≤
+            (3581390741136397114199 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2680609684302609 / 10000000000000000 : ℝ)) (q :=
+      (28613291501051061 / 80000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_669_16 :
+    Uω (aρ 16) (bρ 16) (51544875035809 / 64000000000000 : ℝ) ≤
+      (-(5623107287608835128659 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25982030816194973 / 80000000000000000 : ℝ) ≤
+      (-(5623107287608835128659 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (25982030816194973 / 80000000000000000 : ℝ) =
+            (25982030816194973 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25982030816194973 / 20000000000000000 : ℝ) ≤
+            (1308364515391164871341 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2175166164598587 / 10000000000000000 : ℝ)) (q :=
+      (25982030816194973 / 80000000000000000 : ℝ)) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [bρ_16]) (by norm_num) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [aρ_16, bρ_16]) hlg
+
+lemma U_669 :
+    Uρ (51544875035809 / 64000000000000 : ℝ) ≤
+      (-(4233482968561500353719 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (51544875035809 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2244732758859609584141 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_669_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (51544875035809 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(227466970668141009613 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_669_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (51544875035809 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(2334771082517334658739 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_669_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (51544875035809 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(304407961166478728341 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_669_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (51544875035809 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(647480560805871238739 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_669_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (51544875035809 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) * (-(87974566296101048461 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_669_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (51544875035809 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(391146157008232414821 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_669_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (51544875035809 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(355058973366974848083 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_669_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (51544875035809 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(1024394782019966389091 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_669_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (51544875035809 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) * (-(11965611613780661111 / 25000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_669_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (51544875035809 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(5628654436701494422077 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_669_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (51544875035809 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(6629385355911510922651 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_669_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (51544875035809 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(972296894257070220027 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_669_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (51544875035809 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(9034059768671761378607 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_669_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (51544875035809 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(10281552864863602885801 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_669_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (51544875035809 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) *
+        (-(5623107287608835128659 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_669_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_670_1 :
+    Uω (aρ 1) (bρ 1) (104343112775949 / 128000000000000 : ℝ) ≤
+      (-(2122915875407152198241 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (129396615634666397 / 160000000000000000 : ℝ) ≤
+      (-(2122915875407152198241 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (129396615634666397 / 160000000000000000 : ℝ) =
+            (129396615634666397 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (129396615634666397 / 80000000000000000 : ℝ) ≤
+            (4808555927592847801759 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2021817121699321 / 2500000000000000 : ℝ)) (q :=
+      (129396615634666397 / 160000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_670_2 :
+    Uω (aρ 2) (bρ 2) (104343112775949 / 128000000000000 : ℝ) ≤
+      (-(2152488114323455304571 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (129014526112001253 / 160000000000000000 : ℝ) ≤
+      (-(2152488114323455304571 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (129014526112001253 / 160000000000000000 : ℝ) =
+            (129014526112001253 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (129014526112001253 / 80000000000000000 : ℝ) ≤
+            (4778983688676544695429 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (8063276308974141 / 10000000000000000 : ℝ)) (q :=
+      (129014526112001253 / 160000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_670_3 :
+    Uω (aρ 3) (bρ 3) (104343112775949 / 128000000000000 : ℝ) ≤
+      (-(2211852356495572801771 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (128250909972047621 / 160000000000000000 : ℝ) ≤
+      (-(2211852356495572801771 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (128250909972047621 / 160000000000000000 : ℝ) =
+            (128250909972047621 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (128250909972047621 / 80000000000000000 : ℝ) ≤
+            (4719619446504427198229 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (8015220397829937 / 10000000000000000 : ℝ)) (q :=
+      (128250909972047621 / 160000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_670_4 :
+    Uω (aρ 4) (bρ 4) (104343112775949 / 128000000000000 : ℝ) ≤
+      (-(23110974918272984411 / 100000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (126984377336210149 / 160000000000000000 : ℝ) ≤
+      (-(23110974918272984411 / 100000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (126984377336210149 / 160000000000000000 : ℝ) =
+            (126984377336210149 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (126984377336210149 / 80000000000000000 : ℝ) ≤
+            (46203743111727015589 / 100000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7935197999415253 / 10000000000000000 : ℝ)) (q :=
+      (126984377336210149 / 160000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_670_5 :
+    Uω (aρ 5) (bρ 5) (104343112775949 / 128000000000000 : ℝ) ≤
+      (-(2463798801145355917777 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (125060039108793597 / 160000000000000000 : ℝ) ≤
+      (-(2463798801145355917777 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (125060039108793597 / 160000000000000000 : ℝ) =
+            (125060039108793597 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (125060039108793597 / 80000000000000000 : ℝ) ≤
+            (4467673001854644082223 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (488302852520199 / 625000000000000 : ℝ)) (q :=
+      (125060039108793597 / 160000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_670_6 :
+    Uω (aρ 6) (bρ 6) (104343112775949 / 128000000000000 : ℝ) ≤
+      (-(671531988149197401087 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (122310270820928029 / 160000000000000000 : ℝ) ≤
+      (-(671531988149197401087 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (122310270820928029 / 160000000000000000 : ℝ) =
+            (122310270820928029 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (122310270820928029 / 80000000000000000 : ℝ) ≤
+            (1061335962600802598913 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1909091244869997 / 2500000000000000 : ℝ)) (q :=
+      (122310270820928029 / 160000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_670_7 :
+    Uω (aρ 7) (bρ 7) (104343112775949 / 128000000000000 : ℝ) ≤
+      (-(1497922846347504928719 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (118580166892592797 / 160000000000000000 : ℝ) ≤
+      (-(1497922846347504928719 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (118580166892592797 / 160000000000000000 : ℝ) =
+            (118580166892592797 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (118580166892592797 / 80000000000000000 : ℝ) ≤
+            (1967813055152495071281 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1848440842299521 / 2500000000000000 : ℝ)) (q :=
+      (118580166892592797 / 160000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_670_8 :
+    Uω (aρ 8) (bρ 8) (104343112775949 / 128000000000000 : ℝ) ≤
+      (-(426399641951322608969 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (113755806355612957 / 160000000000000000 : ℝ) ≤
+      (-(426399641951322608969 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (113755806355612957 / 160000000000000000 : ℝ) =
+            (113755806355612957 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (113755806355612957 / 80000000000000000 : ℝ) ≤
+            (440034333423677391031 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1768564988353901 / 2500000000000000 : ℝ)) (q :=
+      (113755806355612957 / 160000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8])
+      (by norm_num [bρ_8]) (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_670_9 :
+    Uω (aρ 9) (bρ 9) (104343112775949 / 128000000000000 : ℝ) ≤
+      (-(3949659208324676186013 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (107792479667179109 / 160000000000000000 : ℝ) ≤
+      (-(3949659208324676186013 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (107792479667179109 / 160000000000000000 : ℝ) =
+            (107792479667179109 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (107792479667179109 / 80000000000000000 : ℝ) ≤
+            (2981812594675323813987 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6669809745721373 / 10000000000000000 : ℝ)) (q :=
+      (107792479667179109 / 160000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9])
+      (by norm_num [bρ_9]) (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_670_10 :
+    Uω (aρ 10) (bρ 10) (104343112775949 / 128000000000000 : ℝ) ≤
+      (-(4626394831317498006823 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (100739132785923661 / 160000000000000000 : ℝ) ≤
+      (-(4626394831317498006823 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (100739132785923661 / 160000000000000000 : ℝ) =
+            (100739132785923661 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (100739132785923661 / 80000000000000000 : ℝ) ≤
+            (2305076971682501993177 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3088350209529721 / 5000000000000000 : ℝ)) (q :=
+      (100739132785923661 / 160000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_670_11 :
+    Uω (aρ 11) (bρ 11) (104343112775949 / 128000000000000 : ℝ) ≤
+      (-(2726025637081969480721 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (92755656630473309 / 160000000000000000 : ℝ) ≤
+      (-(2726025637081969480721 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (92755656630473309 / 160000000000000000 : ℝ) =
+            (92755656630473309 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (92755656630473309 / 80000000000000000 : ℝ) ≤
+            (739710264418030519279 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1399310631458287 / 2500000000000000 : ℝ)) (q :=
+      (92755656630473309 / 160000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_670_12 :
+    Uω (aρ 12) (bρ 12) (104343112775949 / 128000000000000 : ℝ) ≤
+      (-(1607247505027276502399 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (84122562513276669 / 160000000000000000 : ℝ) ≤
+      (-(1607247505027276502399 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (84122562513276669 / 160000000000000000 : ℝ) =
+            (84122562513276669 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (84122562513276669 / 80000000000000000 : ℝ) ≤
+            (125620445722723497601 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (617725732233571 / 1250000000000000 : ℝ)) (q :=
+      (84122562513276669 / 160000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_670_13 :
+    Uω (aρ 13) (bρ 13) (104343112775949 / 128000000000000 : ℝ) ≤
+      (-(7543653695001394939451 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (75249438756350077 / 160000000000000000 : ℝ) ≤
+      (-(7543653695001394939451 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (75249438756350077 / 160000000000000000 : ℝ) =
+            (75249438756350077 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (75249438756350077 / 40000000000000000 : ℝ) ≤
+            (6319289910998605060549 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (66125736403871 / 156250000000000 : ℝ)) (q :=
+      (75249438756350077 / 160000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_670_14 :
+    Uω (aρ 14) (bρ 14) (104343112775949 / 128000000000000 : ℝ) ≤
+      (-(8749352398163579758549 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2668089704939989 / 6400000000000000 : ℝ) ≤
+      (-(8749352398163579758549 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (2668089704939989 / 6400000000000000 : ℝ) =
+            (2668089704939989 / 1600000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2668089704939989 / 1600000000000000 : ℝ) ≤
+            (5113591207836420241451 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (70151497965729 / 200000000000000 : ℝ)) (q :=
+      (2668089704939989 / 6400000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_670_15 :
+    Uω (aρ 15) (bρ 15) (104343112775949 / 128000000000000 : ℝ) ≤
+      (-(2481709788466278231419 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (59292919606829613 / 160000000000000000 : ℝ) ≤
+      (-(2481709788466278231419 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (59292919606829613 / 160000000000000000 : ℝ) =
+            (59292919606829613 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (59292919606829613 / 40000000000000000 : ℝ) ≤
+            (984026113033721768581 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1420491399308843 / 5000000000000000 : ℝ)) (q :=
+      (59292919606829613 / 160000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_670_16 :
+    Uω (aρ 16) (bρ 16) (104343112775949 / 128000000000000 : ℝ) ≤
+      (-(1081467769326687780553 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (54255590911517429 / 160000000000000000 : ℝ) ≤
+      (-(1081467769326687780553 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (54255590911517429 / 160000000000000000 : ℝ) =
+            (54255590911517429 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (54255590911517429 / 40000000000000000 : ℝ) ≤
+            (304826591273312219447 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2363688363213663 / 10000000000000000 : ℝ)) (q :=
+      (54255590911517429 / 160000000000000000 : ℝ)) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [bρ_16]) (by norm_num) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [aρ_16, bρ_16]) hlg
+
+lemma U_670 :
+    Uρ (104343112775949 / 128000000000000 : ℝ) ≤
+      (-(32631863470159351417 / 80000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (104343112775949 / 128000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(2122915875407152198241 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_670_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (104343112775949 / 128000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(2152488114323455304571 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_670_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (104343112775949 / 128000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(2211852356495572801771 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_670_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (104343112775949 / 128000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(23110974918272984411 / 100000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_670_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (104343112775949 / 128000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(2463798801145355917777 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_670_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (104343112775949 / 128000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(671531988149197401087 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_670_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (104343112775949 / 128000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(1497922846347504928719 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_670_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (104343112775949 / 128000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(426399641951322608969 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_670_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (104343112775949 / 128000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(3949659208324676186013 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_670_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) *
+        Uω (aρ 10) (bρ 10) (104343112775949 / 128000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(4626394831317498006823 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_670_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) *
+        Uω (aρ 11) (bρ 11) (104343112775949 / 128000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(2726025637081969480721 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_670_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) *
+        Uω (aρ 12) (bρ 12) (104343112775949 / 128000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(1607247505027276502399 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_670_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) *
+        Uω (aρ 13) (bρ 13) (104343112775949 / 128000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(7543653695001394939451 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_670_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) *
+        Uω (aρ 14) (bρ 14) (104343112775949 / 128000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8749352398163579758549 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_670_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (104343112775949 / 128000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(2481709788466278231419 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_670_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) *
+        Uω (aρ 16) (bρ 16) (104343112775949 / 128000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) *
+        (-(1081467769326687780553 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_670_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_671_1 :
+    Uω (aρ 1) (bρ 1) (2639911887007 / 3200000000000 : ℝ) ≤
+      (-(500641273210134050889 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16370415355172451 / 20000000000000000 : ℝ) ≤
+      (-(500641273210134050889 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (16370415355172451 / 20000000000000000 : ℝ) =
+            (16370415355172451 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16370415355172451 / 10000000000000000 : ℝ) ≤
+            (1232226677539865949111 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (15986695168673 / 19531250000000 : ℝ)) (q :=
+      (16370415355172451 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_671_2 :
+    Uω (aρ 2) (bρ 2) (2639911887007 / 3200000000000 : ℝ) ≤
+      (-(1015890701030451598891 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8161328421903959 / 10000000000000000 : ℝ) ≤
+      (-(1015890701030451598891 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (8161328421903959 / 10000000000000000 : ℝ) =
+            (8161328421903959 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8161328421903959 / 5000000000000000 : ℝ) ≤
+            (2449845200469548401109 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (8161198427506043 / 10000000000000000 : ℝ)) (q :=
+      (8161328421903959 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_671_3 :
+    Uω (aρ 3) (bρ 3) (2639911887007 / 3200000000000 : ℝ) ≤
+      (-(1045213186822443644979 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8113606404156209 / 10000000000000000 : ℝ) ≤
+      (-(1045213186822443644979 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (8113606404156209 / 10000000000000000 : ℝ) =
+            (8113606404156209 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8113606404156209 / 5000000000000000 : ℝ) ≤
+            (2420522714677556355021 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (8113150498360543 / 10000000000000000 : ℝ)) (q :=
+      (8113606404156209 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_671_4 :
+    Uω (aρ 4) (bρ 4) (2639911887007 / 3200000000000 : ℝ) ≤
+      (-(2188454632258100397817 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8034458702845387 / 10000000000000000 : ℝ) ≤
+      (-(2188454632258100397817 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (8034458702845387 / 10000000000000000 : ℝ) =
+            (8034458702845387 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8034458702845387 / 5000000000000000 : ℝ) ≤
+            (4743017170741899602183 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (8033149276803899 / 10000000000000000 : ℝ)) (q :=
+      (8034458702845387 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_671_5 :
+    Uω (aρ 5) (bρ 5) (2639911887007 / 3200000000000 : ℝ) ≤
+      (-(2339247632052252958919 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15457448387817 / 19531250000000 : ℝ) ≤
+      (-(2339247632052252958919 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (15457448387817 / 19531250000000 : ℝ) = (15457448387817 / 9765625000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15457448387817 / 9765625000000 : ℝ) ≤
+            (4592224170947747041081 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7910848939572733 / 10000000000000000 : ℝ)) (q :=
+      (15457448387817 / 19531250000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_671_6 :
+    Uω (aρ 6) (bρ 6) (2639911887007 / 3200000000000 : ℝ) ≤
+      (-(127935879922268966083 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7742412510432781 / 10000000000000000 : ℝ) ≤
+      (-(127935879922268966083 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (7742412510432781 / 10000000000000000 : ℝ) =
+            (7742412510432781 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7742412510432781 / 5000000000000000 : ℝ) ≤
+            (218637710227731033917 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7734487186453687 / 10000000000000000 : ℝ)) (q :=
+      (7742412510432781 / 10000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_671_7 :
+    Uω (aρ 7) (bρ 7) (2639911887007 / 3200000000000 : ℝ) ≤
+      (-(143214223958901012311 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15018816155841109 / 20000000000000000 : ℝ) ≤
+      (-(143214223958901012311 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (15018816155841109 / 20000000000000000 : ℝ) =
+            (15018816155841109 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15018816155841109 / 10000000000000000 : ℝ) ≤
+            (203359366191098987689 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3746069851094617 / 5000000000000000 : ℝ)) (q :=
+      (15018816155841109 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_671_8 :
+    Uω (aρ 8) (bρ 8) (2639911887007 / 3200000000000 : ℝ) ≤
+      (-(3273739843396933459351 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1802035298080953 / 2500000000000000 : ℝ) ≤
+      (-(3273739843396933459351 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1802035298080953 / 2500000000000000 : ℝ) =
+            (1802035298080953 / 1250000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1802035298080953 / 1250000000000000 : ℝ) ≤
+            (3657731959603066540649 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7173147582335749 / 10000000000000000 : ℝ)) (q :=
+      (1802035298080953 / 2500000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_671_9 :
+    Uω (aρ 9) (bρ 9) (2639911887007 / 3200000000000 : ℝ) ≤
+      (-(3803938287939391401999 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13671842754950423 / 20000000000000000 : ℝ) ≤
+      (-(3803938287939391401999 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (13671842754950423 / 20000000000000000 : ℝ) =
+            (13671842754950423 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13671842754950423 / 10000000000000000 : ℝ) ≤
+            (3127533515060608598001 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1692418395249637 / 2500000000000000 : ℝ)) (q :=
+      (13671842754950423 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_671_10 :
+    Uω (aρ 10) (bρ 10) (2639911887007 / 3200000000000 : ℝ) ≤
+      (-(4469155812153975488917 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3197989499534529 / 5000000000000000 : ℝ) ≤
+      (-(4469155812153975488917 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (3197989499534529 / 5000000000000000 : ℝ) =
+            (3197989499534529 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3197989499534529 / 2500000000000000 : ℝ) ≤
+            (2462315990846024511083 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6278347857681241 / 10000000000000000 : ℝ)) (q :=
+      (3197989499534529 / 5000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_671_11 :
+    Uω (aρ 11) (bρ 11) (2639911887007 / 3200000000000 : ℝ) ≤
+      (-(5278727945726234280523 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11797167717509879 / 20000000000000000 : ℝ) ≤
+      (-(5278727945726234280523 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11797167717509879 / 20000000000000000 : ℝ) =
+            (11797167717509879 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11797167717509879 / 10000000000000000 : ℝ) ≤
+            (1652743857273765719477 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1425508550814501 / 2500000000000000 : ℝ)) (q :=
+      (11797167717509879 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_671_12 :
+    Uω (aρ 12) (bρ 12) (2639911887007 / 3200000000000 : ℝ) ≤
+      (-(1558256031449427925669 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10723417062856459 / 20000000000000000 : ℝ) ≤
+      (-(1558256031449427925669 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10723417062856459 / 20000000000000000 : ℝ) =
+            (10723417062856459 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10723417062856459 / 10000000000000000 : ℝ) ≤
+            (174611919300572074331 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (315748977830599 / 625000000000000 : ℝ)) (q :=
+      (10723417062856459 / 20000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_671_13 :
+    Uω (aρ 13) (bρ 13) (2639911887007 / 3200000000000 : ℝ) ≤
+      (-(7315481309970044122373 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (96232701758961 / 200000000000000 : ℝ) ≤
+      (-(7315481309970044122373 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (96232701758961 / 200000000000000 : ℝ) = (96232701758961 / 50000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (96232701758961 / 50000000000000 : ℝ) ≤
+            (6547462296029955877627 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (174048739996969 / 400000000000000 : ℝ)) (q :=
+      (96232701758961 / 200000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13]) (by norm_num [bρ_13])
+      (by norm_num) (by norm_num [aρ_13, bρ_13]) (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_671_14 :
+    Uω (aρ 14) (bρ 14) (2639911887007 / 3200000000000 : ℝ) ≤
+      (-(132427977383954035777 / 156250000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8569361431673801 / 20000000000000000 : ℝ) ≤
+      (-(132427977383954035777 / 156250000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (8569361431673801 / 20000000000000000 : ℝ) =
+            (8569361431673801 / 5000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8569361431673801 / 5000000000000000 : ℝ) ≤
+            (84180516459795964223 / 156250000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1820618520373463 / 5000000000000000 : ℝ)) (q :=
+      (8569361431673801 / 20000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_671_15 :
+    Uω (aρ 15) (bρ 15) (2639911887007 / 3200000000000 : ℝ) ≤
+      (-(4795647034182452320199 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1532905499401413 / 4000000000000000 : ℝ) ≤
+      (-(4795647034182452320199 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1532905499401413 / 4000000000000000 : ℝ) =
+            (1532905499401413 / 1000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1532905499401413 / 1000000000000000 : ℝ) ≤
+            (2135824768817547679801 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (299597638349519 / 1000000000000000 : ℝ)) (q :=
+      (1532905499401413 / 4000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_671_16 :
+    Uω (aρ 16) (bρ 16) (2639911887007 / 3200000000000 : ℝ) ≤
+      (-(10415420626330915702817 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1764550262759391 / 5000000000000000 : ℝ) ≤
+      (-(10415420626330915702817 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (1764550262759391 / 5000000000000000 : ℝ) =
+            (1764550262759391 / 1250000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1764550262759391 / 1250000000000000 : ℝ) ≤
+            (3447522979669084297183 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2542021589035689 / 10000000000000000 : ℝ)) (q :=
+      (1764550262759391 / 5000000000000000 : ℝ)) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [bρ_16]) (by norm_num) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [aρ_16, bρ_16]) hlg
+
+lemma U_671 :
+    Uρ (2639911887007 / 3200000000000 : ℝ) ≤
+      (-(122746261948983473009 / 312500000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (2639911887007 / 3200000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(500641273210134050889 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_671_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (2639911887007 / 3200000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1015890701030451598891 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_671_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (2639911887007 / 3200000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1045213186822443644979 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_671_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (2639911887007 / 3200000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(2188454632258100397817 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_671_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (2639911887007 / 3200000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(2339247632052252958919 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_671_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (2639911887007 / 3200000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(127935879922268966083 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_671_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (2639911887007 / 3200000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(143214223958901012311 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_671_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (2639911887007 / 3200000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(3273739843396933459351 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_671_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (2639911887007 / 3200000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(3803938287939391401999 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_671_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (2639911887007 / 3200000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(4469155812153975488917 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_671_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (2639911887007 / 3200000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(5278727945726234280523 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_671_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (2639911887007 / 3200000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(1558256031449427925669 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_671_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (2639911887007 / 3200000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(7315481309970044122373 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_671_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (2639911887007 / 3200000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) * (-(132427977383954035777 / 156250000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_671_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (2639911887007 / 3200000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(4795647034182452320199 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_671_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (2639911887007 / 3200000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) *
+        (-(10415420626330915702817 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_671_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_672_1 :
+    Uω (aρ 1) (bρ 1) (106849838184611 / 128000000000000 : ℝ) ≤
+      (-(235455692546499870873 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (132530029957627211 / 160000000000000000 : ℝ) ≤
+      (-(235455692546499870873 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (132530029957627211 / 160000000000000000 : ℝ) =
+            (132530029957627211 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (132530029957627211 / 80000000000000000 : ℝ) ≤
+            (630978282828500129127 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (8283107354615667 / 10000000000000000 : ℝ)) (q :=
+      (132530029957627211 / 160000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1])
+      (by norm_num [bρ_1]) (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_672_2 :
+    Uω (aρ 2) (bρ 2) (106849838184611 / 128000000000000 : ℝ) ≤
+      (-(191251438587919057173 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (132147982790004099 / 160000000000000000 : ℝ) ≤
+      (-(191251438587919057173 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (132147982790004099 / 160000000000000000 : ℝ) =
+            (132147982790004099 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (132147982790004099 / 80000000000000000 : ℝ) ≤
+            (501895741712080942827 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4129560235586389 / 5000000000000000 : ℝ)) (q :=
+      (132147982790004099 / 160000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2])
+      (by norm_num [bρ_2]) (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_672_3 :
+    Uω (aρ 3) (bρ 3) (106849838184611 / 128000000000000 : ℝ) ≤
+      (-(123153581774262376029 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (131384492835420723 / 160000000000000000 : ℝ) ≤
+      (-(123153581774262376029 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (131384492835420723 / 160000000000000000 : ℝ) =
+            (131384492835420723 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (131384492835420723 / 80000000000000000 : ℝ) ≤
+            (310063405913237623971 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (513192520824991 / 625000000000000 : ℝ)) (q :=
+      (131384492835420723 / 160000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3])
+      (by norm_num [bρ_3]) (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_672_4 :
+    Uω (aρ 4) (bρ 4) (106849838184611 / 128000000000000 : ℝ) ≤
+      (-(413459634607695491819 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (130118294927134203 / 160000000000000000 : ℝ) ≤
+      (-(413459634607695491819 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (130118294927134203 / 160000000000000000 : ℝ) =
+            (130118294927134203 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (130118294927134203 / 80000000000000000 : ℝ) ≤
+            (972834725992304508181 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (8131099775729041 / 10000000000000000 : ℝ)) (q :=
+      (130118294927134203 / 160000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4])
+      (by norm_num [bρ_4]) (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_672_5 :
+    Uω (aρ 5) (bρ 5) (106849838184611 / 128000000000000 : ℝ) ≤
+      (-(2216229985259282885979 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (128194778771775299 / 160000000000000000 : ℝ) ≤
+      (-(2216229985259282885979 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (128194778771775299 / 160000000000000000 : ℝ) =
+            (128194778771775299 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (128194778771775299 / 80000000000000000 : ℝ) ≤
+            (4715241817740717114021 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4004425087822089 / 5000000000000000 : ℝ)) (q :=
+      (128194778771775299 / 160000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5])
+      (by norm_num [bρ_5]) (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_672_6 :
+    Uω (aρ 6) (bρ 6) (106849838184611 / 128000000000000 : ℝ) ≤
+      (-(486582685306763027307 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (125446888816866827 / 160000000000000000 : ℝ) ≤
+      (-(486582685306763027307 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (125446888816866827 / 160000000000000000 : ℝ) =
+            (125446888816866827 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (125446888816866827 / 80000000000000000 : ℝ) ≤
+            (899711675293236972693 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7832604306420619 / 10000000000000000 : ℝ)) (q :=
+      (125446888816866827 / 160000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6])
+      (by norm_num [bρ_6]) (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_672_7 :
+    Uω (aρ 7) (bρ 7) (106849838184611 / 128000000000000 : ℝ) ≤
+      (-(341804927920539353553 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (121720796977537651 / 160000000000000000 : ℝ) ≤
+      (-(341804927920539353553 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (121720796977537651 / 160000000000000000 : ℝ) =
+            (121720796977537651 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (121720796977537651 / 80000000000000000 : ℝ) ≤
+            (524629047454460646447 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (948813025908059 / 1250000000000000 : ℝ)) (q :=
+      (121720796977537651 / 160000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7])
+      (by norm_num [bρ_7]) (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_672_8 :
+    Uω (aρ 8) (bρ 8) (106849838184611 / 128000000000000 : ℝ) ≤
+      (-(3138164350541981558903 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (116904502074220267 / 160000000000000000 : ℝ) ≤
+      (-(3138164350541981558903 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (116904502074220267 / 160000000000000000 : ℝ) =
+            (116904502074220267 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (116904502074220267 / 80000000000000000 : ℝ) ≤
+            (3793307452458018441097 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7272008995689799 / 10000000000000000 : ℝ)) (q :=
+      (116904502074220267 / 160000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8])
+      (by norm_num [bρ_8]) (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_672_9 :
+    Uω (aρ 9) (bρ 9) (106849838184611 / 128000000000000 : ℝ) ≤
+      (-(915087684084109970181 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (22191311289371623 / 32000000000000000 : ℝ) ≤
+      (-(915087684084109970181 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (22191311289371623 / 32000000000000000 : ℝ) =
+            (22191311289371623 / 16000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22191311289371623 / 16000000000000000 : ℝ) ≤
+            (817780266665890029819 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (686948142062953 / 1000000000000000 : ℝ)) (q :=
+      (22191311289371623 / 32000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_672_10 :
+    Uω (aρ 10) (bρ 10) (106849838184611 / 128000000000000 : ℝ) ≤
+      (-(2157220375553611875821 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (103931262160899803 / 160000000000000000 : ℝ) ≤
+      (-(2157220375553611875821 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (103931262160899803 / 160000000000000000 : ℝ) =
+            (103931262160899803 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (103931262160899803 / 80000000000000000 : ℝ) ≤
+            (1308515525946388124179 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6379878668379741 / 10000000000000000 : ℝ)) (q :=
+      (103931262160899803 / 160000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_672_11 :
+    Uω (aρ 11) (bρ 11) (106849838184611 / 128000000000000 : ℝ) ≤
+      (-(2554278773373393057629 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (95997107451389363 / 160000000000000000 : ℝ) ≤
+      (-(2554278773373393057629 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (95997107451389363 / 160000000000000000 : ℝ) =
+            (95997107451389363 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (95997107451389363 / 80000000000000000 : ℝ) ≤
+            (911457128126606942371 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (45363952780437 / 78125000000000 : ℝ)) (q :=
+      (95997107451389363 / 160000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_672_12 :
+    Uω (aρ 12) (bρ 12) (106849838184611 / 128000000000000 : ℝ) ≤
+      (-(188789903102745975883 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (87448156886536459 / 160000000000000000 : ℝ) ≤
+      (-(188789903102745975883 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (87448156886536459 / 160000000000000000 : ℝ) =
+            (87448156886536459 / 80000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (87448156886536459 / 80000000000000000 : ℝ) ≤
+            (27818590741004024117 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5161667231974323 / 10000000000000000 : ℝ)) (q :=
+      (87448156886536459 / 160000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_672_13 :
+    Uω (aρ 13) (bρ 13) (106849838184611 / 128000000000000 : ℝ) ≤
+      (-(1773362058072074792879 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15742925331419543 / 32000000000000000 : ℝ) ≤
+      (-(1773362058072074792879 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (15742925331419543 / 32000000000000000 : ℝ) =
+            (15742925331419543 / 8000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15742925331419543 / 8000000000000000 : ℝ) ≤
+            (1692373843427925207121 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (111733942372237 / 250000000000000 : ℝ)) (q :=
+      (15742925331419543 / 32000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_672_14 :
+    Uω (aρ 14) (bρ 14) (106849838184611 / 128000000000000 : ℝ) ≤
+      (-(8211227049645739158739 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (70389993110883323 / 160000000000000000 : ℝ) ≤
+      (-(8211227049645739158739 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (70389993110883323 / 160000000000000000 : ℝ) =
+            (70389993110883323 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (70389993110883323 / 40000000000000000 : ℝ) ≤
+            (5651716556354260841261 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3772705786657681 / 10000000000000000 : ℝ)) (q :=
+      (70389993110883323 / 160000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_672_15 :
+    Uω (aρ 15) (bρ 15) (106849838184611 / 128000000000000 : ℝ) ≤
+      (-(1854487086372971585059 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (63302844916412531 / 160000000000000000 : ℝ) ≤
+      (-(1854487086372971585059 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (63302844916412531 / 160000000000000000 : ℝ) =
+            (63302844916412531 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (63302844916412531 / 40000000000000000 : ℝ) ≤
+            (918101634827028414941 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (393298192470479 / 1250000000000000 : ℝ)) (q :=
+      (63302844916412531 / 160000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_672_16 :
+    Uω (aρ 16) (bρ 16) (106849838184611 / 128000000000000 : ℝ) ≤
+      (-(10042650893961810327003 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (58610198971289891 / 160000000000000000 : ℝ) ≤
+      (-(10042650893961810327003 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (58610198971289891 / 160000000000000000 : ℝ) =
+            (58610198971289891 / 40000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (58610198971289891 / 40000000000000000 : ℝ) ≤
+            (3820292712038189672997 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1356088224066751 / 5000000000000000 : ℝ)) (q :=
+      (58610198971289891 / 160000000000000000 : ℝ)) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [bρ_16]) (by norm_num) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [aρ_16, bρ_16]) hlg
+
+lemma U_672 :
+    Uρ (106849838184611 / 128000000000000 : ℝ) ≤
+      (-(3779941234837194175711 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (106849838184611 / 128000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(235455692546499870873 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_672_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (106849838184611 / 128000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(191251438587919057173 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_672_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (106849838184611 / 128000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(123153581774262376029 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_672_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (106849838184611 / 128000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(413459634607695491819 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_672_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (106849838184611 / 128000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(2216229985259282885979 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_672_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (106849838184611 / 128000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(486582685306763027307 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_672_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (106849838184611 / 128000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(341804927920539353553 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_672_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (106849838184611 / 128000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(3138164350541981558903 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_672_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (106849838184611 / 128000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(915087684084109970181 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_672_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) *
+        Uω (aρ 10) (bρ 10) (106849838184611 / 128000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(2157220375553611875821 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_672_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) *
+        Uω (aρ 11) (bρ 11) (106849838184611 / 128000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(2554278773373393057629 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_672_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) *
+        Uω (aρ 12) (bρ 12) (106849838184611 / 128000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(188789903102745975883 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_672_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) *
+        Uω (aρ 13) (bρ 13) (106849838184611 / 128000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(1773362058072074792879 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_672_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) *
+        Uω (aρ 14) (bρ 14) (106849838184611 / 128000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(8211227049645739158739 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_672_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) *
+        Uω (aρ 15) (bρ 15) (106849838184611 / 128000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(1854487086372971585059 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_672_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) *
+        Uω (aρ 16) (bρ 16) (106849838184611 / 128000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) *
+        (-(10042650893961810327003 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_672_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_673_1 :
+    Uω (aρ 1) (bρ 1) (54051600444471 / 64000000000000 : ℝ) ≤
+      (-(441530894174971766827 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (67048368493290031 / 80000000000000000 : ℝ) ≤
+      (-(441530894174971766827 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (67048368493290031 / 80000000000000000 : ℝ) =
+            (67048368493290031 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (67048368493290031 / 40000000000000000 : ℝ) ≤
+            (1291337056575028233173 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4190513385979457 / 5000000000000000 : ℝ)) (q :=
+      (67048368493290031 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_673_2 :
+    Uω (aρ 2) (bρ 2) (54051600444471 / 64000000000000 : ℝ) ≤
+      (-(358930625195043976947 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (66857355125838379 / 80000000000000000 : ℝ) ≤
+      (-(358930625195043976947 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (66857355125838379 / 80000000000000000 : ℝ) =
+            (66857355125838379 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (66857355125838379 / 40000000000000000 : ℝ) ≤
+            (1027363735404956023053 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (8357042442606001 / 10000000000000000 : ℝ)) (q :=
+      (66857355125838379 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_673_3 :
+    Uω (aρ 3) (bρ 3) (54051600444471 / 64000000000000 : ℝ) ≤
+      (-(1851910609720033326851 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (66475640576983063 / 80000000000000000 : ℝ) ≤
+      (-(1851910609720033326851 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (66475640576983063 / 80000000000000000 : ℝ) =
+            (66475640576983063 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (66475640576983063 / 40000000000000000 : ℝ) ≤
+            (5079561193279966673149 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2077252477935543 / 2500000000000000 : ℝ)) (q :=
+      (66475640576983063 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_673_4 :
+    Uω (aρ 4) (bρ 4) (54051600444471 / 64000000000000 : ℝ) ≤
+      (-(973796252202356616109 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (65842622301709847 / 80000000000000000 : ℝ) ≤
+      (-(973796252202356616109 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (65842622301709847 / 80000000000000000 : ℝ) =
+            (65842622301709847 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (65842622301709847 / 40000000000000000 : ℝ) ≤
+            (2491939649297643383891 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2057262380997217 / 2500000000000000 : ℝ)) (q :=
+      (65842622301709847 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_673_5 :
+    Uω (aρ 5) (bρ 5) (54051600444471 / 64000000000000 : ℝ) ≤
+      (-(1047354263203227591767 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12976212444371607 / 16000000000000000 : ℝ) ≤
+      (-(1047354263203227591767 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (12976212444371607 / 16000000000000000 : ℝ) =
+            (12976212444371607 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12976212444371607 / 8000000000000000 : ℝ) ≤
+            (2418381638296772408233 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1621369884672183 / 2000000000000000 : ℝ)) (q :=
+      (12976212444371607 / 16000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_673_6 :
+    Uω (aρ 6) (bρ 6) (54051600444471 / 64000000000000 : ℝ) ≤
+      (-(2308675365543395899577 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (63507569140601767 / 80000000000000000 : ℝ) ≤
+      (-(2308675365543395899577 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (63507569140601767 / 80000000000000000 : ℝ) =
+            (63507569140601767 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (63507569140601767 / 40000000000000000 : ℝ) ≤
+            (4622796437456604100423 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (247834891505839 / 312500000000000 : ℝ)) (q :=
+      (63507569140601767 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_673_7 :
+    Uω (aρ 7) (bρ 7) (54051600444471 / 64000000000000 : ℝ) ≤
+      (-(2606266133502291094023 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12329097371660311 / 16000000000000000 : ℝ) ≤
+      (-(2606266133502291094023 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (12329097371660311 / 16000000000000000 : ℝ) =
+            (12329097371660311 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12329097371660311 / 8000000000000000 : ℝ) ≤
+            (4325205669497708905977 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1537771467674359 / 2000000000000000 : ℝ)) (q :=
+      (12329097371660311 / 16000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_673_8 :
+    Uω (aρ 8) (bρ 8) (54051600444471 / 64000000000000 : ℝ) ≤
+      (-(375552419867593022027 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (59239271892758979 / 80000000000000000 : ℝ) ≤
+      (-(375552419867593022027 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (59239271892758979 / 80000000000000000 : ℝ) =
+            (59239271892758979 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (59239271892758979 / 40000000000000000 : ℝ) ≤
+            (490881555507406977973 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7370845248326151 / 10000000000000000 : ℝ)) (q :=
+      (59239271892758979 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_673_9 :
+    Uω (aρ 9) (bρ 9) (54051600444471 / 64000000000000 : ℝ) ≤
+      (-(3518833867664006306331 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (56268971067573579 / 80000000000000000 : ℝ) ≤
+      (-(3518833867664006306331 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (56268971067573579 / 80000000000000000 : ℝ) =
+            (56268971067573579 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (56268971067573579 / 40000000000000000 : ℝ) ≤
+            (3412637935335993693669 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6969235670389801 / 10000000000000000 : ℝ)) (q :=
+      (56268971067573579 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_673_10 :
+    Uω (aρ 10) (bρ 10) (54051600444471 / 64000000000000 : ℝ) ≤
+      (-(4162167202080293224863 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2110519423226207 / 3200000000000000 : ℝ) ≤
+      (-(4162167202080293224863 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2110519423226207 / 3200000000000000 : ℝ) =
+            (2110519423226207 / 1600000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2110519423226207 / 1600000000000000 : ℝ) ≤
+            (2769304600919706775137 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4050811457597 / 6250000000000 : ℝ)) (q :=
+      (2110519423226207 / 3200000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_673_11 :
+    Uω (aρ 11) (bρ 11) (54051600444471 / 64000000000000 : ℝ) ≤
+      (-(4941420751005724572059 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (48807527807521887 / 80000000000000000 : ℝ) ≤
+      (-(4941420751005724572059 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (48807527807521887 / 80000000000000000 : ℝ) =
+            (48807527807521887 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (48807527807521887 / 40000000000000000 : ℝ) ≤
+            (1990051051994275427941 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2955455257538439 / 5000000000000000 : ℝ)) (q :=
+      (48807527807521887 / 80000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_673_12 :
+    Uω (aρ 12) (bρ 12) (54051600444471 / 64000000000000 : ℝ) ≤
+      (-(1463388324566729199727 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (44552635240304839 / 80000000000000000 : ℝ) ≤
+      (-(1463388324566729199727 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (44552635240304839 / 80000000000000000 : ℝ) =
+            (44552635240304839 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (44552635240304839 / 40000000000000000 : ℝ) ≤
+            (269479626183270800273 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (329430466872351 / 625000000000000 : ℝ)) (q :=
+      (44552635240304839 / 80000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_673_13 :
+    Uω (aρ 13) (bρ 13) (54051600444471 / 64000000000000 : ℝ) ≤
+      (-(687718534739097609481 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (40217736294292543 / 80000000000000000 : ℝ) ≤
+      (-(687718534739097609481 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (40217736294292543 / 80000000000000000 : ℝ) =
+            (40217736294292543 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (40217736294292543 / 40000000000000000 : ℝ) ≤
+            (5428645560902390519 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2293272237524771 / 5000000000000000 : ℝ)) (q :=
+      (40217736294292543 / 80000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_673_14 :
+    Uω (aρ 14) (bρ 14) (54051600444471 / 64000000000000 : ℝ) ≤
+      (-(994505843548616338199 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (288837284746911 / 640000000000000 : ℝ) ≤
+      (-(994505843548616338199 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (288837284746911 / 640000000000000 : ℝ) =
+            (288837284746911 / 160000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (288837284746911 / 160000000000000 : ℝ) ≤
+            (738362107201383661801 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (31217622678899 / 80000000000000 : ℝ)) (q :=
+      (288837284746911 / 640000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14]) (by norm_num [bρ_14])
+      (by norm_num) (by norm_num [aρ_14, bρ_14]) (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_673_15 :
+    Uω (aρ 15) (bρ 15) (54051600444471 / 64000000000000 : ℝ) ≤
+      (-(8968279189791174831113 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (32628910325302671 / 80000000000000000 : ℝ) ≤
+      (-(8968279189791174831113 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (32628910325302671 / 80000000000000000 : ℝ) =
+            (32628910325302671 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (32628910325302671 / 20000000000000000 : ℝ) ≤
+            (4894664416208825168887 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1646419272631037 / 5000000000000000 : ℝ)) (q :=
+      (32628910325302671 / 80000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_673_16 :
+    Uω (aρ 16) (bρ 16) (54051600444471 / 64000000000000 : ℝ) ≤
+      (-(387685589835395029451 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6070097983094999 / 16000000000000000 : ℝ) ≤
+      (-(387685589835395029451 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (6070097983094999 / 16000000000000000 : ℝ) =
+            (6070097983094999 / 4000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6070097983094999 / 4000000000000000 : ℝ) ≤
+            (166832154404604970549 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (575121018863031 / 2000000000000000 : ℝ)) (q :=
+      (6070097983094999 / 16000000000000000 : ℝ)) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [bρ_16]) (by norm_num) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [aρ_16, bρ_16]) hlg
+
+lemma U_673 :
+    Uρ (54051600444471 / 64000000000000 : ℝ) ≤
+      (-(363496725301935166593 / 1000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (54051600444471 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(441530894174971766827 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_673_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (54051600444471 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(358930625195043976947 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_673_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (54051600444471 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1851910609720033326851 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_673_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (54051600444471 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(973796252202356616109 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_673_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (54051600444471 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1047354263203227591767 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_673_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (54051600444471 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(2308675365543395899577 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_673_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (54051600444471 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(2606266133502291094023 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_673_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (54051600444471 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(375552419867593022027 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_673_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (54051600444471 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(3518833867664006306331 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_673_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (54051600444471 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(4162167202080293224863 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_673_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (54051600444471 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(4941420751005724572059 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_673_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (54051600444471 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(1463388324566729199727 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_673_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (54051600444471 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(687718534739097609481 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_673_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (54051600444471 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(994505843548616338199 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_673_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (54051600444471 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(8968279189791174831113 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_673_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (54051600444471 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(387685589835395029451 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_673_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_674_1 :
+    Uω (aρ 1) (bρ 1) (27652481574401 / 32000000000000 : ℝ) ≤
+      (-(383785914618737645777 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (34307537698640223 / 40000000000000000 : ℝ) ≤
+      (-(383785914618737645777 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (34307537698640223 / 40000000000000000 : ℝ) =
+            (34307537698640223 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (34307537698640223 / 20000000000000000 : ℝ) ≤
+            (1349082036131262354223 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (8576865575404799 / 10000000000000000 : ℝ)) (q :=
+      (34307537698640223 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_674_2 :
+    Uω (aρ 2) (bρ 2) (27652481574401 / 32000000000000 : ℝ) ≤
+      (-(48844312258711844961 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (34212040880202477 / 40000000000000000 : ℝ) ≤
+      (-(48844312258711844961 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (34212040880202477 / 40000000000000000 : ℝ) =
+            (34212040880202477 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (34212040880202477 / 20000000000000000 : ℝ) ≤
+            (167764181585038155039 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4276443089347963 / 5000000000000000 : ℝ)) (q :=
+      (34212040880202477 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_674_3 :
+    Uω (aρ 3) (bρ 3) (27652481574401 / 32000000000000 : ℝ) ≤
+      (-(404738031473057972023 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (34021212980809573 / 40000000000000000 : ℝ) ≤
+      (-(404738031473057972023 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (34021212980809573 / 40000000000000000 : ℝ) =
+            (34021212980809573 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (34021212980809573 / 20000000000000000 : ℝ) ≤
+            (1328129919276942027977 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4252434167674737 / 5000000000000000 : ℝ)) (q :=
+      (34021212980809573 / 40000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_674_4 :
+    Uω (aρ 4) (bρ 4) (27652481574401 / 32000000000000 : ℝ) ≤
+      (-(342479473035474896289 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1348191267778473 / 1600000000000000 : ℝ) ≤
+      (-(342479473035474896289 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1348191267778473 / 1600000000000000 : ℝ) =
+            (1348191267778473 / 800000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1348191267778473 / 800000000000000 : ℝ) ≤
+            (1043814887564525103711 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (42124734366203 / 50000000000000 : ℝ)) (q :=
+      (1348191267778473 / 1600000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_674_5 :
+    Uω (aρ 5) (bρ 5) (27652481574401 / 32000000000000 : ℝ) ≤
+      (-(185601151129559812207 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (33224192580018133 / 40000000000000000 : ℝ) ≤
+      (-(185601151129559812207 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (33224192580018133 / 40000000000000000 : ℝ) =
+            (33224192580018133 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (33224192580018133 / 20000000000000000 : ℝ) ≤
+            (507546029170440187793 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4151421117676877 / 5000000000000000 : ℝ)) (q :=
+      (33224192580018133 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_674_6 :
+    Uω (aρ 6) (bρ 6) (27652481574401 / 32000000000000 : ℝ) ≤
+      (-(1032372319093198975119 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (32537881332450541 / 40000000000000000 : ℝ) ≤
+      (-(1032372319093198975119 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (32537881332450541 / 40000000000000000 : ℝ) =
+            (32537881332450541 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (32537881332450541 / 20000000000000000 : ℝ) ≤
+            (2433363582406801024881 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4063463493354979 / 5000000000000000 : ℝ)) (q :=
+      (32537881332450541 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_674_7 :
+    Uω (aρ 7) (bρ 7) (27652481574401 / 32000000000000 : ℝ) ≤
+      (-(1177382868026010592239 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (31607766959490703 / 40000000000000000 : ℝ) ≤
+      (-(1177382868026010592239 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (31607766959490703 / 40000000000000000 : ℝ) =
+            (31607766959490703 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (31607766959490703 / 20000000000000000 : ℝ) ≤
+            (2288353033473989407761 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7885531180990039 / 10000000000000000 : ℝ)) (q :=
+      (31607766959490703 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_674_8 :
+    Uω (aρ 8) (bρ 8) (27652481574401 / 32000000000000 : ℝ) ≤
+      (-(2742226011902361903109 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (30406513722986933 / 40000000000000000 : ℝ) ≤
+      (-(2742226011902361903109 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (30406513722986933 / 40000000000000000 : ℝ) =
+            (30406513722986933 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (30406513722986933 / 20000000000000000 : ℝ) ≤
+            (4189245791097638096891 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3784223107039077 / 5000000000000000 : ℝ)) (q :=
+      (30406513722986933 / 40000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_674_9 :
+    Uω (aρ 9) (bρ 9) (27652481574401 / 32000000000000 : ℝ) ≤
+      (-(129671004603477111227 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28924874733777511 / 40000000000000000 : ℝ) ≤
+      (-(129671004603477111227 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (28924874733777511 / 40000000000000000 : ℝ) =
+            (28924874733777511 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28924874733777511 / 20000000000000000 : ℝ) ≤
+            (147587867516522888773 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7168592347833443 / 10000000000000000 : ℝ)) (q :=
+      (28924874733777511 / 40000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_674_10 :
+    Uω (aρ 10) (bρ 10) (27652481574401 / 32000000000000 : ℝ) ≤
+      (-(483079380526028675361 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5435644174946641 / 8000000000000000 : ℝ) ≤
+      (-(483079380526028675361 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5435644174946641 / 8000000000000000 : ℝ) =
+            (5435644174946641 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5435644174946641 / 4000000000000000 : ℝ) ≤
+            (383354594848971324639 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (668382445180629 / 1000000000000000 : ℝ)) (q :=
+      (5435644174946641 / 8000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_674_11 :
+    Uω (aρ 11) (bρ 11) (27652481574401 / 32000000000000 : ℝ) ≤
+      (-(576975617282524960421 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (25211467739382093 / 40000000000000000 : ℝ) ≤
+      (-(576975617282524960421 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (25211467739382093 / 40000000000000000 : ℝ) =
+            (25211467739382093 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25211467739382093 / 20000000000000000 : ℝ) ≤
+            (289458358092475039579 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3059462255167867 / 5000000000000000 : ℝ)) (q :=
+      (25211467739382093 / 40000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_674_12 :
+    Uω (aρ 12) (bρ 12) (27652481574401 / 32000000000000 : ℝ) ≤
+      (-(2744733037009107092033 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (23102315404187439 / 40000000000000000 : ℝ) ≤
+      (-(2744733037009107092033 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23102315404187439 / 40000000000000000 : ℝ) =
+            (23102315404187439 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23102315404187439 / 20000000000000000 : ℝ) ≤
+            (721002864490892907967 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5488048439423407 / 10000000000000000 : ℝ)) (q :=
+      (23102315404187439 / 40000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_674_13 :
+    Uω (aρ 13) (bρ 13) (27652481574401 / 32000000000000 : ℝ) ≤
+      (-(3230332515999585509361 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20964131438699567 / 40000000000000000 : ℝ) ≤
+      (-(3230332515999585509361 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (20964131438699567 / 40000000000000000 : ℝ) =
+            (20964131438699567 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20964131438699567 / 20000000000000000 : ℝ) ≤
+            (235403385500414490639 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4818338198274471 / 10000000000000000 : ℝ)) (q :=
+      (20964131438699567 / 40000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_674_14 :
+    Uω (aρ 14) (bρ 14) (27652481574401 / 32000000000000 : ℝ) ≤
+      (-(298795737915287152287 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3790326625954983 / 8000000000000000 : ℝ) ≤
+      (-(298795737915287152287 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3790326625954983 / 8000000000000000 : ℝ) =
+            (3790326625954983 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3790326625954983 / 2000000000000000 : ℝ) ≤
+            (255722006324712847713 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (831203265771429 / 2000000000000000 : ℝ)) (q :=
+      (3790326625954983 / 8000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_674_15 :
+    Uω (aρ 15) (bρ 15) (27652481574401 / 32000000000000 : ℝ) ≤
+      (-(2099471820989259101859 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17272069640599751 / 40000000000000000 : ℝ) ≤
+      (-(2099471820989259101859 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (17272069640599751 / 40000000000000000 : ℝ) =
+            (17272069640599751 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17272069640599751 / 10000000000000000 : ℝ) ≤
+            (1366264080510740898141 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3575807861684563 / 10000000000000000 : ℝ)) (q :=
+      (17272069640599751 / 40000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_674_16 :
+    Uω (aρ 16) (bρ 16) (27652481574401 / 32000000000000 : ℝ) ≤
+      (-(9045797086010213353117 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16188477844103073 / 40000000000000000 : ℝ) ≤
+      (-(9045797086010213353117 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (16188477844103073 / 40000000000000000 : ℝ) =
+            (16188477844103073 / 10000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16188477844103073 / 10000000000000000 : ℝ) ≤
+            (4817146519989786646883 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (199148975934139 / 625000000000000 : ℝ)) (q :=
+      (16188477844103073 / 40000000000000000 : ℝ)) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [bρ_16]) (by norm_num) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [aρ_16, bρ_16]) hlg
+
+lemma U_674 :
+    Uρ (27652481574401 / 32000000000000 : ℝ) ≤
+      (-(104789093514343343243 / 312500000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (27652481574401 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(383785914618737645777 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_674_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (27652481574401 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) * (-(48844312258711844961 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_674_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (27652481574401 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(404738031473057972023 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_674_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (27652481574401 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(342479473035474896289 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_674_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (27652481574401 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(185601151129559812207 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_674_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (27652481574401 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1032372319093198975119 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_674_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (27652481574401 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(1177382868026010592239 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_674_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (27652481574401 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(2742226011902361903109 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_674_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (27652481574401 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(129671004603477111227 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_674_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (27652481574401 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(483079380526028675361 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_674_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (27652481574401 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(576975617282524960421 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_674_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (27652481574401 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(2744733037009107092033 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_674_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (27652481574401 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(3230332515999585509361 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_674_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (27652481574401 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) * (-(298795737915287152287 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_674_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (27652481574401 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(2099471820989259101859 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_674_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (27652481574401 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) *
+        (-(9045797086010213353117 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_674_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_675_1 :
+    Uω (aρ 1) (bρ 1) (56558325853133 / 64000000000000 : ℝ) ≤
+      (-(1309378706805599354279 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (70181782143953017 / 80000000000000000 : ℝ) ≤
+      (-(1309378706805599354279 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (70181782143953017 / 80000000000000000 : ℝ) =
+            (70181782143953017 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (70181782143953017 / 40000000000000000 : ℝ) ≤
+            (5622093096194400645721 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (8772704339521223 / 10000000000000000 : ℝ)) (q :=
+      (70181782143953017 / 80000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_675_2 :
+    Uω (aρ 2) (bρ 2) (56558325853133 / 64000000000000 : ℝ) ≤
+      (-(1336627245042598542631 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (69990807353880377 / 80000000000000000 : ℝ) ≤
+      (-(1336627245042598542631 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (69990807353880377 / 80000000000000000 : ℝ) =
+            (69990807353880377 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (69990807353880377 / 40000000000000000 : ℝ) ≤
+            (5594844557957401457369 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (8748729654513063 / 10000000000000000 : ℝ)) (q :=
+      (69990807353880377 / 80000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_675_3 :
+    Uω (aρ 3) (bρ 3) (56558325853133 / 64000000000000 : ℝ) ≤
+      (-(1391297819368064538101 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (69609207654522701 / 80000000000000000 : ℝ) ≤
+      (-(1391297819368064538101 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (69609207654522701 / 80000000000000000 : ℝ) =
+            (69609207654522701 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (69609207654522701 / 40000000000000000 : ℝ) ≤
+            (5540173983631935461899 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2175181459005911 / 2500000000000000 : ℝ)) (q :=
+      (69609207654522701 / 80000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_675_4 :
+    Uω (aρ 4) (bρ 4) (56558325853133 / 64000000000000 : ℝ) ≤
+      (-(741304296202974127503 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (68976493674625433 / 80000000000000000 : ℝ) ≤
+      (-(741304296202974127503 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (68976493674625433 / 80000000000000000 : ℝ) =
+            (68976493674625433 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (68976493674625433 / 40000000000000000 : ℝ) ≤
+            (2724431605297025872497 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (8620841522114327 / 10000000000000000 : ℝ)) (q :=
+      (68976493674625433 / 80000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_675_5 :
+    Uω (aρ 5) (bρ 5) (56558325853133 / 64000000000000 : ℝ) ≤
+      (-(1622883744341165361017 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13603135906815797 / 16000000000000000 : ℝ) ≤
+      (-(1622883744341165361017 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (13603135906815797 / 16000000000000000 : ℝ) =
+            (13603135906815797 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13603135906815797 / 8000000000000000 : ℝ) ≤
+            (5308588058658834638983 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1699765581262543 / 2000000000000000 : ℝ)) (q :=
+      (13603135906815797 / 16000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_675_6 :
+    Uω (aρ 6) (bρ 6) (56558325853133 / 64000000000000 : ℝ) ≤
+      (-(913316624783847168869 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (66643885998227629 / 80000000000000000 : ℝ) ≤
+      (-(913316624783847168869 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (66643885998227629 / 80000000000000000 : ℝ) =
+            (66643885998227629 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (66643885998227629 / 40000000000000000 : ℝ) ≤
+            (2552419276716152831131 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2080779974372469 / 2500000000000000 : ℝ)) (q :=
+      (66643885998227629 / 80000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_675_7 :
+    Uω (aρ 7) (bρ 7) (56558325853133 / 64000000000000 : ℝ) ≤
+      (-(527365195200048172033 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (64785418593637181 / 80000000000000000 : ℝ) ≤
+      (-(527365195200048172033 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (64785418593637181 / 80000000000000000 : ℝ) =
+            (64785418593637181 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (64785418593637181 / 40000000000000000 : ℝ) ≤
+            (1205502755549951827967 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1010270553387783 / 1250000000000000 : ℝ)) (q :=
+      (64785418593637181 / 80000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_675_8 :
+    Uω (aρ 8) (bρ 8) (56558325853133 / 64000000000000 : ℝ) ≤
+      (-(1243394595876810300753 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (62386425724127397 / 80000000000000000 : ℝ) ≤
+      (-(1243394595876810300753 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (62386425724127397 / 80000000000000000 : ℝ) =
+            (62386425724127397 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (62386425724127397 / 40000000000000000 : ℝ) ≤
+            (2222341305623189699247 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3882978930532409 / 5000000000000000 : ℝ)) (q :=
+      (62386425724127397 / 80000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_675_9 :
+    Uω (aρ 9) (bρ 9) (56558325853133 / 64000000000000 : ℝ) ≤
+      (-(2972313060724690363641 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (59429772916198877 / 80000000000000000 : ℝ) ≤
+      (-(2972313060724690363641 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (59429772916198877 / 80000000000000000 : ℝ) =
+            (59429772916198877 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (59429772916198877 / 40000000000000000 : ℝ) ≤
+            (3959158742275309636359 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (28780313622823 / 39062500000000 : ℝ)) (q :=
+      (59429772916198877 / 80000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_675_10 :
+    Uω (aρ 10) (bρ 10) (56558325853133 / 64000000000000 : ℝ) ≤
+      (-(893994214702231111179 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (55948350270039657 / 80000000000000000 : ℝ) ≤
+      (-(893994214702231111179 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (55948350270039657 / 80000000000000000 : ℝ) =
+            (55948350270039657 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (55948350270039657 / 40000000000000000 : ℝ) ≤
+            (838873736047768888821 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6885963659397883 / 10000000000000000 : ℝ)) (q :=
+      (55948350270039657 / 80000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_675_11 :
+    Uω (aρ 11) (bρ 11) (56558325853133 / 64000000000000 : ℝ) ≤
+      (-(215052782857109338981 / 500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (10407046827545529 / 16000000000000000 : ℝ) ≤
+      (-(215052782857109338981 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (10407046827545529 / 16000000000000000 : ℝ) =
+            (10407046827545529 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10407046827545529 / 8000000000000000 : ℝ) ≤
+            (131520807292890661019 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (79077015656561 / 125000000000000 : ℝ)) (q :=
+      (10407046827545529 / 16000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_675_12 :
+    Uω (aρ 12) (bρ 12) (56558325853133 / 64000000000000 : ℝ) ≤
+      (-(1027892171772619989907 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9570090251541869 / 16000000000000000 : ℝ) ≤
+      (-(1027892171772619989907 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (9570090251541869 / 16000000000000000 : ℝ) =
+            (9570090251541869 / 8000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9570090251541869 / 8000000000000000 : ℝ) ≤
+            (358402188827380010093 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1140733125841061 / 2000000000000000 : ℝ)) (q :=
+      (9570090251541869 / 16000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_675_13 :
+    Uω (aρ 13) (bρ 13) (56558325853133 / 64000000000000 : ℝ) ≤
+      (-(1212718825538876018277 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (43626607223548973 / 80000000000000000 : ℝ) ≤
+      (-(1212718825538876018277 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (43626607223548973 / 80000000000000000 : ℝ) =
+            (43626607223548973 / 40000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (43626607223548973 / 40000000000000000 : ℝ) ≤
+            (173575535061123981723 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1261771590565053 / 2500000000000000 : ℝ)) (q :=
+      (43626607223548973 / 80000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_675_14 :
+    Uω (aρ 14) (bρ 14) (56558325853133 / 64000000000000 : ℝ) ≤
+      (-(280489785711701105503 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7935641994942533 / 16000000000000000 : ℝ) ≤
+      (-(280489785711701105503 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (7935641994942533 / 16000000000000000 : ℝ) =
+            (7935641994942533 / 4000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7935641994942533 / 4000000000000000 : ℝ) ≤
+            (274027958528298894497 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (880782867019227 / 2000000000000000 : ℝ)) (q :=
+      (7935641994942533 / 16000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_675_15 :
+    Uω (aρ 15) (bρ 15) (56558325853133 / 64000000000000 : ℝ) ≤
+      (-(7870167065046628959241 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (36416061511973109 / 80000000000000000 : ℝ) ≤
+      (-(7870167065046628959241 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (36416061511973109 / 80000000000000000 : ℝ) =
+            (36416061511973109 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (36416061511973109 / 20000000000000000 : ℝ) ≤
+            (5992776540953371040759 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1923975248413123 / 5000000000000000 : ℝ)) (q :=
+      (36416061511973109 / 80000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_675_16 :
+    Uω (aρ 16) (bρ 16) (56558325853133 / 64000000000000 : ℝ) ≤
+      (-(4229065425846391883387 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (34336658657888021 / 80000000000000000 : ℝ) ≤
+      (-(4229065425846391883387 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (34336658657888021 / 80000000000000000 : ℝ) =
+            (34336658657888021 / 20000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (34336658657888021 / 20000000000000000 : ℝ) ≤
+            (2702406377153608116613 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1740235717407487 / 5000000000000000 : ℝ)) (q :=
+      (34336658657888021 / 80000000000000000 : ℝ)) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [bρ_16]) (by norm_num) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [aρ_16, bρ_16]) hlg
+
+lemma U_675 :
+    Uρ (56558325853133 / 64000000000000 : ℝ) ≤
+      (-(1540795269486682136973 / 5000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (56558325853133 / 64000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(1309378706805599354279 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_675_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (56558325853133 / 64000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(1336627245042598542631 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_675_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (56558325853133 / 64000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1391297819368064538101 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_675_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (56558325853133 / 64000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(741304296202974127503 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_675_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (56558325853133 / 64000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(1622883744341165361017 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_675_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (56558325853133 / 64000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(913316624783847168869 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_675_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (56558325853133 / 64000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(527365195200048172033 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_675_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (56558325853133 / 64000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(1243394595876810300753 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_675_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (56558325853133 / 64000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(2972313060724690363641 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_675_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (56558325853133 / 64000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(893994214702231111179 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_675_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (56558325853133 / 64000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) * (-(215052782857109338981 / 500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_675_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (56558325853133 / 64000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(1027892171772619989907 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_675_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (56558325853133 / 64000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(1212718825538876018277 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_675_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (56558325853133 / 64000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) * (-(280489785711701105503 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_675_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (56558325853133 / 64000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(7870167065046628959241 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_675_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (56558325853133 / 64000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) *
+        (-(4229065425846391883387 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_675_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/U56.lean b/LeanPool/Zeta5Irrational/Table/U56.lean
new file mode 100644
index 0000000000..20b1934c2c
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/U56.lean
@@ -0,0 +1,3379 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.UpperCertificate
+
+/-! # Certified arcsine potential bounds (U56) -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma U_676_1 :
+    Uω (aρ 1) (bρ 1) (7226461069683 / 8000000000000 : ℝ) ≤
+      (-(108859842802811722997 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (358742443718067 / 400000000000000 : ℝ) ≤
+      (-(108859842802811722997 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (358742443718067 / 400000000000000 : ℝ) =
+            (358742443718067 / 200000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (358742443718067 / 200000000000000 : ℝ) ≤
+            (584287337497188277003 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (44842715334423 / 50000000000000 : ℝ)) (q :=
+      (358742443718067 / 400000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_676_2 :
+    Uω (aρ 2) (bρ 2) (7226461069683 / 8000000000000 : ℝ) ≤
+      (-(223049730527027750149 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17889382993662361 / 20000000000000000 : ℝ) ≤
+      (-(223049730527027750149 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (17889382993662361 / 20000000000000000 : ℝ) =
+            (17889382993662361 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17889382993662361 / 10000000000000000 : ℝ) ≤
+            (1163244630072972249851 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (8944572887153611 / 10000000000000000 : ℝ)) (q :=
+      (17889382993662361 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_676_3 :
+    Uω (aρ 3) (bρ 3) (7226461069683 / 8000000000000 : ℝ) ≤
+      (-(1168711495788115444963 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3558799294975663 / 4000000000000000 : ℝ) ≤
+      (-(1168711495788115444963 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (3558799294975663 / 4000000000000000 : ℝ) =
+            (3558799294975663 / 2000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3558799294975663 / 2000000000000000 : ℝ) ≤
+            (5762760307211884555037 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1779316494943913 / 2000000000000000 : ℝ)) (q :=
+      (3558799294975663 / 4000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_676_4 :
+    Uω (aρ 4) (bρ 4) (7226461069683 / 8000000000000 : ℝ) ≤
+      (-(628991572116585160863 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8817926734849447 / 10000000000000000 : ℝ) ≤
+      (-(628991572116585160863 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (8817926734849447 / 10000000000000000 : ℝ) =
+            (8817926734849447 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8817926734849447 / 5000000000000000 : ℝ) ≤
+            (2836744329383414839137 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1102091706325643 / 1250000000000000 : ℝ)) (q :=
+      (8817926734849447 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_676_5 :
+    Uω (aρ 5) (bρ 5) (7226461069683 / 8000000000000 : ℝ) ≤
+      (-(55802843000971698371 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1087233551180307 / 1250000000000000 : ℝ) ≤
+      (-(55802843000971698371 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1087233551180307 / 1250000000000000 : ℝ) =
+            (1087233551180307 / 625000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1087233551180307 / 625000000000000 : ℝ) ≤
+            (221456029119028301629 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4347403459563081 / 5000000000000000 : ℝ)) (q :=
+      (1087233551180307 / 1250000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_676_6 :
+    Uω (aρ 6) (bρ 6) (7226461069683 / 8000000000000 : ℝ) ≤
+      (-(398517382703400588829 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1705298599747893 / 2000000000000000 : ℝ) ≤
+      (-(398517382703400588829 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1705298599747893 / 2000000000000000 : ℝ) =
+            (1705298599747893 / 1000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1705298599747893 / 1000000000000000 : ℝ) ≤
+            (1334350568046599411171 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (425964823643009 / 500000000000000 : ℝ)) (q :=
+      (1705298599747893 / 2000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_676_7 :
+    Uω (aρ 7) (bρ 7) (7226461069683 / 8000000000000 : ℝ) ≤
+      (-(1870052209910470026807 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16588788113255563 / 20000000000000000 : ℝ) ≤
+      (-(1870052209910470026807 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (16588788113255563 / 20000000000000000 : ℝ) =
+            (16588788113255563 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16588788113255563 / 10000000000000000 : ℝ) ≤
+            (5061419593089529973193 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (8278759969396813 / 10000000000000000 : ℝ)) (q :=
+      (16588788113255563 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_676_8 :
+    Uω (aρ 8) (bρ 8) (7226461069683 / 8000000000000 : ℝ) ≤
+      (-(1118883342221099544697 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3997468331946279 / 5000000000000000 : ℝ) ≤
+      (-(1118883342221099544697 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (3997468331946279 / 5000000000000000 : ℝ) =
+            (3997468331946279 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3997468331946279 / 2500000000000000 : ℝ) ≤
+            (2346852559278900455303 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3981693417633183 / 5000000000000000 : ℝ)) (q :=
+      (3997468331946279 / 5000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_676_9 :
+    Uω (aρ 9) (bρ 9) (7226461069683 / 8000000000000 : ℝ) ≤
+      (-(2710035841923158558309 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1906534407118849 / 2500000000000000 : ℝ) ≤
+      (-(2710035841923158558309 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1906534407118849 / 2500000000000000 : ℝ) =
+            (1906534407118849 / 1250000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1906534407118849 / 1250000000000000 : ℝ) ≤
+            (4221435961076841441691 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3783377196396021 / 5000000000000000 : ℝ)) (q :=
+      (1906534407118849 / 2500000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_676_10 :
+    Uω (aρ 10) (bρ 10) (7226461069683 / 8000000000000 : ℝ) ≤
+      (-(3295663745591862291993 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7192355444655383 / 10000000000000000 : ℝ) ≤
+      (-(3295663745591862291993 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (7192355444655383 / 10000000000000000 : ℝ) =
+            (7192355444655383 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7192355444655383 / 5000000000000000 : ℝ) ≤
+            (3635808057408137708007 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (885968632330877 / 1250000000000000 : ℝ)) (q :=
+      (7192355444655383 / 10000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_676_11 :
+    Uω (aρ 11) (bρ 11) (7226461069683 / 8000000000000 : ℝ) ≤
+      (-(999108980076215164303 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6705589958471729 / 10000000000000000 : ℝ) ≤
+      (-(999108980076215164303 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (6705589958471729 / 10000000000000000 : ℝ) =
+            (6705589958471729 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6705589958471729 / 5000000000000000 : ℝ) ≤
+            (733758970673784835697 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1633173678121177 / 2500000000000000 : ℝ)) (q :=
+      (6705589958471729 / 10000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_676_12 :
+    Uω (aρ 12) (bρ 12) (7226461069683 / 8000000000000 : ℝ) ≤
+      (-(4802404152764222826933 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (618634644546397 / 1000000000000000 : ℝ) ≤
+      (-(4802404152764222826933 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (618634644546397 / 1000000000000000 : ℝ) =
+            (618634644546397 / 500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (618634644546397 / 500000000000000 : ℝ) ≤
+            (2129067650235777173067 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (591790778315419 / 1000000000000000 : ℝ)) (q :=
+      (618634644546397 / 1000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12]) (by norm_num [bρ_12])
+      (by norm_num) (by norm_num [aρ_12, bρ_12]) (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_676_13 :
+    Uω (aρ 13) (bρ 13) (7226461069683 / 8000000000000 : ℝ) ≤
+      (-(2842013848850894396079 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (11328588692629349 / 20000000000000000 : ℝ) ≤
+      (-(2842013848850894396079 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (11328588692629349 / 20000000000000000 : ℝ) =
+            (11328588692629349 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11328588692629349 / 10000000000000000 : ℝ) ≤
+            (623722052649105603921 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5273185326450599 / 10000000000000000 : ℝ)) (q :=
+      (11328588692629349 / 20000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_676_14 :
+    Uω (aρ 14) (bρ 14) (7226461069683 / 8000000000000 : ℝ) ≤
+      (-(6579422346950211503277 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1294789970322837 / 2500000000000000 : ℝ) ≤
+      (-(6579422346950211503277 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1294789970322837 / 2500000000000000 : ℝ) =
+            (1294789970322837 / 1250000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1294789970322837 / 1250000000000000 : ℝ) ≤
+            (352049456049788496723 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2323421840724473 / 5000000000000000 : ℝ)) (q :=
+      (1294789970322837 / 2500000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_676_15 :
+    Uω (aρ 15) (bρ 15) (7226461069683 / 8000000000000 : ℝ) ≤
+      (-(7377973537692901902479 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4781659866852911 / 10000000000000000 : ℝ) ≤
+      (-(7377973537692901902479 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4781659866852911 / 10000000000000000 : ℝ) =
+            (4781659866852911 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4781659866852911 / 2500000000000000 : ℝ) ≤
+            (6484970068307098097521 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (8030111191381 / 19531250000000 : ℝ)) (q :=
+      (4781659866852911 / 10000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_676_16 :
+    Uω (aρ 16) (bρ 16) (7226461069683 / 8000000000000 : ℝ) ≤
+      (-(7917178800593945107141 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4530658134130151 / 10000000000000000 : ℝ) ≤
+      (-(7917178800593945107141 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (4530658134130151 / 10000000000000000 : ℝ) =
+            (4530658134130151 / 2500000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4530658134130151 / 2500000000000000 : ℝ) ≤
+            (5945764805406054892859 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (117555784876611 / 312500000000000 : ℝ)) (q :=
+      (4530658134130151 / 10000000000000000 : ℝ)) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [bρ_16]) (by norm_num) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [aρ_16, bρ_16]) hlg
+
+lemma U_676 :
+    Uρ (7226461069683 / 8000000000000 : ℝ) ≤
+      (-(352383937996564167861 / 1250000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (7226461069683 / 8000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(108859842802811722997 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_676_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (7226461069683 / 8000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(223049730527027750149 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_676_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (7226461069683 / 8000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(1168711495788115444963 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_676_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (7226461069683 / 8000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) *
+        (-(628991572116585160863 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_676_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (7226461069683 / 8000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(55802843000971698371 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_676_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (7226461069683 / 8000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(398517382703400588829 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_676_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (7226461069683 / 8000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(1870052209910470026807 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_676_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (7226461069683 / 8000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(1118883342221099544697 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_676_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (7226461069683 / 8000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(2710035841923158558309 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_676_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (7226461069683 / 8000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(3295663745591862291993 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_676_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (7226461069683 / 8000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(999108980076215164303 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_676_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (7226461069683 / 8000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(4802404152764222826933 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_676_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (7226461069683 / 8000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(2842013848850894396079 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_676_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (7226461069683 / 8000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(6579422346950211503277 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_676_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (7226461069683 / 8000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(7377973537692901902479 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_676_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (7226461069683 / 8000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) *
+        (-(7917178800593945107141 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_676_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_677_1 :
+    Uω (aρ 1) (bρ 1) (30159206983063 / 32000000000000 : ℝ) ≤
+      (-(82643010106364090371 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (37440950769403939 / 40000000000000000 : ℝ) ≤
+      (-(82643010106364090371 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (37440950769403939 / 40000000000000000 : ℝ) =
+            (37440950769403939 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (37440950769403939 / 20000000000000000 : ℝ) ≤
+            (783790965268635909629 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4680110210289891 / 5000000000000000 : ℝ)) (q :=
+      (37440950769403939 / 40000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_677_2 :
+    Uω (aρ 2) (bρ 2) (30159206983063 / 32000000000000 : ℝ) ≤
+      (-(85834148424330625231 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (37345489271458807 / 40000000000000000 : ℝ) ≤
+      (-(85834148424330625231 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (37345489271458807 / 40000000000000000 : ℝ) =
+            (37345489271458807 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (37345489271458807 / 20000000000000000 : ℝ) ≤
+            (780599826950669374769 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (291758083878663 / 312500000000000 : ℝ)) (q :=
+      (37345489271458807 / 40000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_677_3 :
+    Uω (aρ 3) (bρ 3) (30159206983063 / 32000000000000 : ℝ) ≤
+      (-(368936939244914093781 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7430953291820317 / 8000000000000000 : ℝ) ≤
+      (-(368936939244914093781 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (7430953291820317 / 8000000000000000 : ℝ) =
+            (7430953291820317 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7430953291820317 / 4000000000000000 : ℝ) ≤
+            (3096798962255085906219 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1857658676857721 / 2000000000000000 : ℝ)) (q :=
+      (7430953291820317 / 8000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_677_4 :
+    Uω (aρ 4) (bρ 4) (30159206983063 / 32000000000000 : ℝ) ≤
+      (-(82332884107747569223 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (36838613308049463 / 40000000000000000 : ℝ) ≤
+      (-(82332884107747569223 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (36838613308049463 / 40000000000000000 : ℝ) =
+            (36838613308049463 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (36838613308049463 / 20000000000000000 : ℝ) ≤
+            (610814296192252430777 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1151063873728443 / 1250000000000000 : ℝ)) (q :=
+      (36838613308049463 / 40000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_677_5 :
+    Uω (aρ 5) (bρ 5) (30159206983063 / 32000000000000 : ℝ) ≤
+      (-(477229023155082316277 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (36358704876728341 / 40000000000000000 : ℝ) ≤
+      (-(477229023155082316277 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (36358704876728341 / 40000000000000000 : ℝ) =
+            (36358704876728341 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (36358704876728341 / 20000000000000000 : ℝ) ≤
+            (2988506878344917683723 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (9086746693501983 / 10000000000000000 : ℝ)) (q :=
+      (36358704876728341 / 40000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_677_6 :
+    Uω (aρ 6) (bρ 6) (30159206983063 / 32000000000000 : ℝ) ≤
+      (-(1144589866729602112133 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (35673940604441891 / 40000000000000000 : ℝ) ≤
+      (-(1144589866729602112133 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (35673940604441891 / 40000000000000000 : ℝ) =
+            (35673940604441891 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (35673940604441891 / 20000000000000000 : ℝ) ≤
+            (5786881936270397887867 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4455802466249379 / 5000000000000000 : ℝ)) (q :=
+      (35673940604441891 / 40000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_677_7 :
+    Uω (aρ 7) (bρ 7) (30159206983063 / 32000000000000 : ℝ) ≤
+      (-(1407832126127480045027 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (34747104376406801 / 40000000000000000 : ℝ) ≤
+      (-(1407832126127480045027 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (34747104376406801 / 40000000000000000 : ℝ) =
+            (34747104376406801 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (34747104376406801 / 20000000000000000 : ℝ) ≤
+            (5523639676872519954973 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (8671848199241213 / 10000000000000000 : ℝ)) (q :=
+      (34747104376406801 / 40000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_677_8 :
+    Uω (aρ 8) (bρ 8) (30159206983063 / 32000000000000 : ℝ) ≤
+      (-(1757720990443800197271 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (6710473013911293 / 8000000000000000 : ℝ) ≤
+      (-(1757720990443800197271 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (6710473013911293 / 8000000000000000 : ℝ) =
+            (6710473013911293 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6710473013911293 / 4000000000000000 : ℝ) ≤
+            (5173750812556199802729 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1671604039431209 / 2000000000000000 : ℝ)) (q :=
+      (6710473013911293 / 8000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_677_9 :
+    Uω (aρ 9) (bρ 9) (30159206983063 / 32000000000000 : ℝ) ≤
+      (-(2205550341869910017897 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1283317593561117 / 1600000000000000 : ℝ) ≤
+      (-(2205550341869910017897 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1283317593561117 / 1600000000000000 : ℝ) =
+            (1283317593561117 / 800000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1283317593561117 / 800000000000000 : ℝ) ≤
+            (4725921461130089982103 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (318570928410071 / 400000000000000 : ℝ)) (q :=
+      (1283317593561117 / 1600000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_677_10 :
+    Uω (aρ 10) (bρ 10) (30159206983063 / 32000000000000 : ℝ) ≤
+      (-(2758186921991172196263 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (30358020869620537 / 40000000000000000 : ℝ) ≤
+      (-(2758186921991172196263 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (30358020869620537 / 40000000000000000 : ℝ) =
+            (30358020869620537 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (30358020869620537 / 20000000000000000 : ℝ) ≤
+            (4173284881008827803737 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7490372759043081 / 10000000000000000 : ℝ)) (q :=
+      (30358020869620537 / 40000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_677_11 :
+    Uω (aρ 11) (bρ 11) (30159206983063 / 32000000000000 : ℝ) ≤
+      (-(426875502031516809391 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (28428127279949587 / 40000000000000000 : ℝ) ≤
+      (-(426875502031516809391 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (28428127279949587 / 40000000000000000 : ℝ) =
+            (28428127279949587 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28428127279949587 / 20000000000000000 : ℝ) ≤
+            (439558473343483190609 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3471951295206303 / 5000000000000000 : ℝ)) (q :=
+      (28428127279949587 / 40000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_677_12 :
+    Uω (aρ 12) (bρ 12) (30159206983063 / 32000000000000 : ℝ) ≤
+      (-(4163275610179741765353 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (26378568806350523 / 40000000000000000 : ℝ) ≤
+      (-(4163275610179741765353 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (26378568806350523 / 40000000000000000 : ℝ) =
+            (26378568806350523 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26378568806350523 / 20000000000000000 : ℝ) ≤
+            (2768196192820258234647 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3171411725149037 / 5000000000000000 : ℝ)) (q :=
+      (26378568806350523 / 40000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_677_13 :
+    Uω (aρ 13) (bρ 13) (30159206983063 / 32000000000000 : ℝ) ≤
+      (-(4971061494768781944211 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4866307285834539 / 8000000000000000 : ℝ) ≤
+      (-(4971061494768781944211 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (4866307285834539 / 8000000000000000 : ℝ) =
+            (4866307285834539 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4866307285834539 / 4000000000000000 : ℝ) ≤
+            (1960410308231218055789 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (35741806270651 / 62500000000000 : ℝ)) (q :=
+      (4866307285834539 / 8000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_677_14 :
+    Uω (aρ 14) (bρ 14) (30159206983063 / 32000000000000 : ℝ) ≤
+      (-(2888495107361552193757 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4489505310941811 / 8000000000000000 : ℝ) ≤
+      (-(2888495107361552193757 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (4489505310941811 / 8000000000000000 : ℝ) =
+            (4489505310941811 / 4000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4489505310941811 / 4000000000000000 : ℝ) ≤
+            (577240794138447806243 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (256030567555867 / 500000000000000 : ℝ)) (q :=
+      (4489505310941811 / 8000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_677_15 :
+    Uω (aρ 15) (bρ 15) (30159206983063 / 32000000000000 : ℝ) ≤
+      (-(810002854756473131841 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (20923588733206163 / 40000000000000000 : ℝ) ≤
+      (-(810002854756473131841 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (20923588733206163 / 40000000000000000 : ℝ) =
+            (20923588733206163 / 20000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20923588733206163 / 20000000000000000 : ℝ) ≤
+            (56431120618526868159 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2309107858890447 / 5000000000000000 : ℝ)) (q :=
+      (20923588733206163 / 40000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_677_16 :
+    Uω (aρ 16) (bρ 16) (30159206983063 / 32000000000000 : ℝ) ≤
+      (-(434015474906195459093 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3994892943791167 / 8000000000000000 : ℝ) ≤
+      (-(434015474906195459093 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (3994892943791167 / 8000000000000000 : ℝ) =
+            (3994892943791167 / 2000000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 15
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3994892943791167 / 2000000000000000 : ℝ) ≤
+            (432418500468804540907 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (429602536216573 / 1000000000000000 : ℝ)) (q :=
+      (3994892943791167 / 8000000000000000 : ℝ)) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [bρ_16]) (by norm_num) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [aρ_16, bρ_16]) hlg
+
+lemma U_677 :
+    Uρ (30159206983063 / 32000000000000 : ℝ) ≤
+      (-(2318547818050359873459 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (30159206983063 / 32000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(82643010106364090371 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_677_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (30159206983063 / 32000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(85834148424330625231 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_677_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (30159206983063 / 32000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (-(368936939244914093781 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_677_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (30159206983063 / 32000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(82332884107747569223 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_677_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (30159206983063 / 32000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (-(477229023155082316277 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_677_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (30159206983063 / 32000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(1144589866729602112133 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_677_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (30159206983063 / 32000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(1407832126127480045027 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_677_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (30159206983063 / 32000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(1757720990443800197271 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_677_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (30159206983063 / 32000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(2205550341869910017897 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_677_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (30159206983063 / 32000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(2758186921991172196263 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_677_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (30159206983063 / 32000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(426875502031516809391 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_677_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (30159206983063 / 32000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(4163275610179741765353 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_677_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (30159206983063 / 32000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4971061494768781944211 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_677_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (30159206983063 / 32000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(2888495107361552193757 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_677_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (30159206983063 / 32000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(810002854756473131841 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_677_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (30159206983063 / 32000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (-(434015474906195459093 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_677_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_678_1 :
+    Uω (aρ 1) (bρ 1) (15706284843697 / 16000000000000 : ℝ) ≤
+      (-(31401869492385685679 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (4875957115579501 / 5000000000000000 : ℝ) ≤
+      (-(31401869492385685679 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (4875957115579501 / 5000000000000000 : ℝ) =
+            (4875957115579501 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4875957115579501 / 2500000000000000 : ℝ) ≤
+            (835032105882614314321 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (9751897653092379 / 10000000000000000 : ℝ)) (q :=
+      (4875957115579501 / 5000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_678_2 :
+    Uω (aρ 2) (bρ 2) (15706284843697 / 16000000000000 : ℝ) ≤
+      (-(275713462460201346549 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9728052738201763 / 10000000000000000 : ℝ) ≤
+      (-(275713462460201346549 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (9728052738201763 / 10000000000000000 : ℝ) =
+            (9728052738201763 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9728052738201763 / 5000000000000000 : ℝ) ≤
+            (6655758340539798653451 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (9727943679687901 / 10000000000000000 : ℝ)) (q :=
+      (9728052738201763 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_678_3 :
+    Uω (aρ 3) (bρ 3) (15706284843697 / 16000000000000 : ℝ) ≤
+      (-(20302229308895077003 / 625000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3872153429302717 / 4000000000000000 : ℝ) ≤
+      (-(20302229308895077003 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (3872153429302717 / 4000000000000000 : ℝ) =
+            (3872153429302717 / 2000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3872153429302717 / 2000000000000000 : ℝ) ≤
+            (412914758378604922997 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (242000036403699 / 250000000000000 : ℝ)) (q :=
+      (3872153429302717 / 4000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_678_4 :
+    Uω (aρ 4) (bρ 4) (15706284843697 / 16000000000000 : ℝ) ≤
+      (-(81357388672044063251 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9601375777845239 / 10000000000000000 : ℝ) ≤
+      (-(81357388672044063251 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (9601375777845239 / 10000000000000000 : ℝ) =
+            (9601375777845239 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9601375777845239 / 5000000000000000 : ℝ) ≤
+            (1304936971927955936749 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (9600280046389853 / 10000000000000000 : ℝ)) (q :=
+      (9601375777845239 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_678_5 :
+    Uω (aρ 5) (bρ 5) (15706284843697 / 16000000000000 : ℝ) ≤
+      (-(532453962168235848837 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (592592069951071 / 625000000000000 : ℝ) ≤
+      (-(532453962168235848837 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (592592069951071 / 625000000000000 : ℝ) =
+            (592592069951071 / 312500000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (592592069951071 / 312500000000000 : ℝ) ≤
+            (6399017840831764151163 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (9478664648468647 / 10000000000000000 : ℝ)) (q :=
+      (592592069951071 / 625000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_678_6 :
+    Uω (aρ 6) (bρ 6) (15706284843697 / 16000000000000 : ℝ) ≤
+      (-(357237988318402216991 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3724180260044697 / 4000000000000000 : ℝ) ≤
+      (-(357237988318402216991 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (3724180260044697 / 4000000000000000 : ℝ) =
+            (3724180260044697 / 2000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3724180260044697 / 2000000000000000 : ℝ) ≤
+            (3108497913181597783009 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (465193004269893 / 500000000000000 : ℝ)) (q :=
+      (3724180260044697 / 4000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_678_7 :
+    Uω (aρ 7) (bρ 7) (15706284843697 / 16000000000000 : ℝ) ≤
+      (-(966103549437018389189 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1815819399022757 / 2000000000000000 : ℝ) ≤
+      (-(966103549437018389189 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1815819399022757 / 2000000000000000 : ℝ) =
+            (1815819399022757 / 1000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1815819399022757 / 1000000000000000 : ℝ) ≤
+            (5965368253562981610811 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1812962831232389 / 2000000000000000 : ℝ)) (q :=
+      (1815819399022757 / 2000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_678_8 :
+    Uω (aρ 8) (bρ 8) (15706284843697 / 16000000000000 : ℝ) ≤
+      (-(1299819252231914866819 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17562226044293351 / 20000000000000000 : ℝ) ≤
+      (-(1299819252231914866819 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (17562226044293351 / 20000000000000000 : ℝ) =
+            (17562226044293351 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17562226044293351 / 10000000000000000 : ℝ) ≤
+            (5631652550768085133181 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4376193930783863 / 5000000000000000 : ℝ)) (q :=
+      (17562226044293351 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_678_9 :
+    Uω (aρ 9) (bρ 9) (15706284843697 / 16000000000000 : ℝ) ≤
+      (-(172562545086624586799 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16830113098059219 / 20000000000000000 : ℝ) ≤
+      (-(172562545086624586799 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (16830113098059219 / 20000000000000000 : ℝ) =
+            (16830113098059219 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16830113098059219 / 10000000000000000 : ℝ) ≤
+            (520584635213375413201 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4180620271846797 / 5000000000000000 : ℝ)) (q :=
+      (16830113098059219 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_678_10 :
+    Uω (aρ 10) (bρ 10) (15706284843697 / 16000000000000 : ℝ) ≤
+      (-(2248820499400769661627 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7986104091424179 / 10000000000000000 : ℝ) ≤
+      (-(2248820499400769661627 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (7986104091424179 / 10000000000000000 : ℝ) =
+            (7986104091424179 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7986104091424179 / 5000000000000000 : ℝ) ≤
+            (4682651303599230338373 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7891894661977733 / 10000000000000000 : ℝ)) (q :=
+      (7986104091424179 / 10000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_678_11 :
+    Uω (aρ 11) (bρ 11) (15706284843697 / 16000000000000 : ℝ) ≤
+      (-(286694897106712835459 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7507407468771171 / 10000000000000000 : ℝ) ≤
+      (-(286694897106712835459 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (7507407468771171 / 10000000000000000 : ℝ) =
+            (7507407468771171 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7507407468771171 / 5000000000000000 : ℝ) ≤
+            (406452283193287164541 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7352978042876717 / 10000000000000000 : ℝ)) (q :=
+      (7507407468771171 / 10000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_678_12 :
+    Uω (aρ 12) (bρ 12) (15706284843697 / 16000000000000 : ℝ) ≤
+      (-(44567725956253433693 / 125000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3500466007121139 / 5000000000000000 : ℝ) ≤
+      (-(44567725956253433693 / 125000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (3500466007121139 / 5000000000000000 : ℝ) =
+            (3500466007121139 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3500466007121139 / 2500000000000000 : ℝ) ≤
+            (42075671581246566307 / 125000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6763727230503931 / 10000000000000000 : ℝ)) (q :=
+      (3500466007121139 / 5000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_678_13 :
+    Uω (aρ 13) (bρ 13) (15706284843697 / 16000000000000 : ℝ) ≤
+      (-(4311189061635353508999 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (12995632859396249 / 20000000000000000 : ℝ) ≤
+      (-(4311189061635353508999 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (12995632859396249 / 20000000000000000 : ℝ) =
+            (12995632859396249 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12995632859396249 / 10000000000000000 : ℝ) ≤
+            (2620282741364646491001 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (96201215672041 / 156250000000000 : ℝ)) (q :=
+      (12995632859396249 / 20000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_678_14 :
+    Uω (aρ 14) (bρ 14) (15706284843697 / 16000000000000 : ℝ) ≤
+      (-(504471450642891062021 / 1000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3019123210339151 / 5000000000000000 : ℝ) ≤
+      (-(504471450642891062021 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (3019123210339151 / 5000000000000000 : ℝ) =
+            (3019123210339151 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3019123210339151 / 2500000000000000 : ℝ) ≤
+            (188675729657108937979 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5581665070015979 / 10000000000000000 : ℝ)) (q :=
+      (3019123210339151 / 5000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_678_15 :
+    Uω (aρ 15) (bρ 15) (15706284843697 / 16000000000000 : ℝ) ≤
+      (-(1134783711466407468651 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (5670023356259313 / 10000000000000000 : ℝ) ≤
+      (-(1134783711466407468651 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (5670023356259313 / 10000000000000000 : ℝ) =
+            (5670023356259313 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5670023356259313 / 5000000000000000 : ℝ) ≤
+            (251510649133592531349 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (5104792218593001 / 10000000000000000 : ℝ)) (q :=
+      (5670023356259313 / 10000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_678_16 :
+    Uω (aρ 16) (bρ 16) (15706284843697 / 16000000000000 : ℝ) ≤
+      (-(1520695928810576621543 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (435429698376879 / 800000000000000 : ℝ) ≤
+      (-(1520695928810576621543 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (435429698376879 / 800000000000000 : ℝ) =
+            (435429698376879 / 400000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (435429698376879 / 400000000000000 : ℝ) ≤
+            (212172021939423378457 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (96057192340127 / 200000000000000 : ℝ)) (q :=
+      (435429698376879 / 800000000000000 : ℝ)) (by norm_num [aρ_16, bρ_16]) (by norm_num [bρ_16])
+      (by norm_num) (by norm_num [aρ_16, bρ_16]) (by norm_num [aρ_16, bρ_16]) hlg
+
+lemma U_678 :
+    Uρ (15706284843697 / 16000000000000 : ℝ) ≤
+      (-(92343274023702849147 / 500000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (15706284843697 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (-(31401869492385685679 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_678_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (15706284843697 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) *
+        (-(275713462460201346549 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_678_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (15706284843697 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) * (-(20302229308895077003 / 625000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_678_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (15706284843697 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (-(81357388672044063251 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_678_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (15706284843697 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) *
+        (-(532453962168235848837 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_678_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (15706284843697 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) *
+        (-(357237988318402216991 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_678_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (15706284843697 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(966103549437018389189 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_678_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (15706284843697 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (-(1299819252231914866819 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_678_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (15706284843697 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(172562545086624586799 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_678_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (15706284843697 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(2248820499400769661627 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_678_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (15706284843697 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(286694897106712835459 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_678_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (15706284843697 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(44567725956253433693 / 125000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_678_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (15706284843697 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(4311189061635353508999 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_678_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (15706284843697 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(504471450642891062021 / 1000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_678_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (15706284843697 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(1134783711466407468651 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_678_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (15706284843697 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) *
+        (-(1520695928810576621543 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_678_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_679_1 :
+    Uω (aρ 1) (bρ 1) (4239911887007 / 4000000000000 : ℝ) ≤
+      (32589570407983029189 / 625000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (1316908394254119 / 1250000000000000 : ℝ) ≤
+      (32589570407983029189 / 625000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (2633812952158351 / 2500000000000000 : ℝ)) (q :=
+      (1316908394254119 / 1250000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_679_2 :
+    Uω (aρ 2) (bρ 2) (4239911887007 / 4000000000000 : ℝ) ≤
+      (99752968003727494617 / 2000000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (164240821186893 / 156250000000000 : ℝ) ≤
+      (99752968003727494617 / 2000000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (2627827906249951 / 2500000000000000 : ℝ)) (q :=
+      (164240821186893 / 156250000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_679_3 :
+    Uω (aρ 3) (bρ 3) (4239911887007 / 4000000000000 : ℝ) ≤
+      (226665678130179312083 / 5000000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (20927527742097363 / 20000000000000000 : ℝ) ≤
+      (226665678130179312083 / 5000000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (10463410361524863 / 10000000000000000 : ℝ)) (q :=
+      (20927527742097363 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_679_4 :
+    Uω (aρ 4) (bρ 4) (4239911887007 / 4000000000000 : ℝ) ≤
+      (75518162225807609173 / 2000000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (20769620260968647 / 20000000000000000 : ℝ) ≤
+      (75518162225807609173 / 2000000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (10383797061461147 / 10000000000000000 : ℝ)) (q :=
+      (20769620260968647 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_679_5 :
+    Uω (aρ 5) (bρ 5) (4239911887007 / 4000000000000 : ℝ) ≤
+      (261587752722163667687 / 10000000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (4106015675897693 / 4000000000000000 : ℝ) ≤
+      (261587752722163667687 / 10000000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (2052489019863193 / 2000000000000000 : ℝ)) (q :=
+      (4106015675897693 / 4000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_679_6 :
+    Uω (aρ 6) (bρ 6) (4239911887007 / 4000000000000 : ℝ) ≤
+      (23468035726243644037 / 2500000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (20188628247545629 / 20000000000000000 : ℝ) ≤
+      (23468035726243644037 / 2500000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (10088235342513129 / 10000000000000000 : ℝ)) (q :=
+      (20188628247545629 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_679_7 :
+    Uω (aρ 7) (bρ 7) (4239911887007 / 4000000000000 : ℝ) ≤
+      (-(17169293930414594967 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19727169305494247 / 20000000000000000 : ℝ) ≤
+      (-(17169293930414594967 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (19727169305494247 / 20000000000000000 : ℝ) =
+            (19727169305494247 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19727169305494247 / 10000000000000000 : ℝ) ≤
+            (849264681444585405033 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (9850437781221747 / 10000000000000000 : ℝ)) (q :=
+      (19727169305494247 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_679_8 :
+    Uω (aρ 8) (bρ 8) (4239911887007 / 4000000000000 : ℝ) ≤
+      (-(13838695655646965701 / 312500000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19133647745154483 / 20000000000000000 : ℝ) ≤
+      (-(13838695655646965701 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (19133647745154483 / 20000000000000000 : ℝ) =
+            (19133647745154483 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19133647745154483 / 10000000000000000 : ℝ) ≤
+            (202769798188103034299 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (9540457872221983 / 10000000000000000 : ℝ)) (q :=
+      (19133647745154483 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_679_9 :
+    Uω (aρ 9) (bρ 9) (4239911887007 / 4000000000000 : ℝ) ≤
+      (-(4152692092358634187 / 50000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9203015949090823 / 10000000000000000 : ℝ) ≤
+      (-(4152692092358634187 / 50000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (9203015949090823 / 10000000000000000 : ℝ) =
+            (9203015949090823 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9203015949090823 / 5000000000000000 : ℝ) ≤
+            (30504666922641365813 / 50000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4576903826804573 / 5000000000000000 : ℝ)) (q :=
+      (9203015949090823 / 10000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_679_10 :
+    Uω (aρ 10) (bρ 10) (4239911887007 / 4000000000000 : ℝ) ≤
+      (-(325854264266228926319 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8777954317313903 / 10000000000000000 : ℝ) ≤
+      (-(325854264266228926319 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (8777954317313903 / 10000000000000000 : ℝ) =
+            (8777954317313903 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8777954317313903 / 5000000000000000 : ℝ) ≤
+            (1407013686483771073681 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4346121711775153 / 5000000000000000 : ℝ)) (q :=
+      (8777954317313903 / 10000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_679_11 :
+    Uω (aρ 11) (bρ 11) (4239911887007 / 4000000000000 : ℝ) ≤
+      (-(928275361804200092781 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (8305600284736179 / 10000000000000000 : ℝ) ≤
+      (-(928275361804200092781 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (8305600284736179 / 10000000000000000 : ℝ) =
+            (8305600284736179 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8305600284736179 / 5000000000000000 : ℝ) ≤
+            (2537460539695799907219 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4083005992299929 / 5000000000000000 : ℝ)) (q :=
+      (8305600284736179 / 10000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_679_12 :
+    Uω (aρ 12) (bρ 12) (4239911887007 / 4000000000000 : ℝ) ≤
+      (-(98932252879129752837 / 400000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3123529863022419 / 4000000000000000 : ℝ) ≤
+      (-(98932252879129752837 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (3123529863022419 / 4000000000000000 : ℝ) =
+            (3123529863022419 / 2000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3123529863022419 / 2000000000000000 : ℝ) ≤
+            (178326619240870247163 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1519232165384919 / 2000000000000000 : ℝ)) (q :=
+      (3123529863022419 / 4000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_679_13 :
+    Uω (aρ 13) (bρ 13) (4239911887007 / 4000000000000 : ℝ) ≤
+      (-(1560256889330704127261 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3659719606775781 / 5000000000000000 : ℝ) ≤
+      (-(1560256889330704127261 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (3659719606775781 / 5000000000000000 : ℝ) =
+            (3659719606775781 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3659719606775781 / 2500000000000000 : ℝ) ≤
+            (1905479012169295872739 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (219274115015957 / 312500000000000 : ℝ)) (q :=
+      (3659719606775781 / 5000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_679_14 :
+    Uω (aρ 14) (bρ 14) (4239911887007 / 4000000000000 : ℝ) ≤
+      (-(748717128402485919173 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (13754605439581289 / 20000000000000000 : ℝ) ≤
+      (-(748717128402485919173 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (13754605439581289 / 20000000000000000 : ℝ) =
+            (13754605439581289 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13754605439581289 / 10000000000000000 : ℝ) ≤
+            (637577232197514080827 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (6476425978033789 / 10000000000000000 : ℝ)) (q :=
+      (13754605439581289 / 20000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_679_15 :
+    Uω (aρ 15) (bρ 15) (4239911887007 / 4000000000000 : ℝ) ≤
+      (-(2132701068385280181461 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2611054459411399 / 4000000000000000 : ℝ) ≤
+      (-(2132701068385280181461 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2611054459411399 / 4000000000000000 : ℝ) =
+            (2611054459411399 / 2000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2611054459411399 / 2000000000000000 : ℝ) ≤
+            (1333034833114719818539 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1207333222584899 / 2000000000000000 : ℝ)) (q :=
+      (2611054459411399 / 4000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_679_16 :
+    Uω (aρ 16) (bρ 16) (4239911887007 / 4000000000000 : ℝ) ≤
+      (-(2298623924628737086939 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (3157287040140563 / 5000000000000000 : ℝ) ≤
+      (-(2298623924628737086939 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (3157287040140563 / 5000000000000000 : ℝ) =
+            (3157287040140563 / 2500000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3157287040140563 / 2500000000000000 : ℝ) ≤
+            (1167111976871262913061 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (720364203492469 / 1250000000000000 : ℝ)) (q :=
+      (3157287040140563 / 5000000000000000 : ℝ)) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [bρ_16]) (by norm_num) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [aρ_16, bρ_16]) hlg
+
+lemma U_679 :
+    Uρ (4239911887007 / 4000000000000 : ℝ) ≤
+      (-(976182167040415310791 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (4239911887007 / 4000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (32589570407983029189 / 625000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_679_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (4239911887007 / 4000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) * (99752968003727494617 / 2000000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_679_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (4239911887007 / 4000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) * (226665678130179312083 / 5000000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_679_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (4239911887007 / 4000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (75518162225807609173 / 2000000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_679_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (4239911887007 / 4000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (261587752722163667687 / 10000000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_679_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (4239911887007 / 4000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) * (23468035726243644037 / 2500000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_679_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (4239911887007 / 4000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (-(17169293930414594967 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_679_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (4239911887007 / 4000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) * (-(13838695655646965701 / 312500000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_679_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (4239911887007 / 4000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(4152692092358634187 / 50000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_679_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (4239911887007 / 4000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(325854264266228926319 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_679_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (4239911887007 / 4000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(928275361804200092781 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_679_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (4239911887007 / 4000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (-(98932252879129752837 / 400000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_679_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (4239911887007 / 4000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(1560256889330704127261 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_679_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (4239911887007 / 4000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(748717128402485919173 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_679_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (4239911887007 / 4000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(2132701068385280181461 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_679_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (4239911887007 / 4000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) *
+        (-(2298623924628737086939 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_679_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_680_1 :
+    Uω (aρ 1) (bρ 1) (18213010252359 / 16000000000000 : ℝ) ≤
+      (1238640559424707941031 / 10000000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (22637239812565641 / 20000000000000000 : ℝ) ≤
+      (1238640559424707941031 / 10000000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (5659302811463133 / 5000000000000000 : ℝ)) (q :=
+      (22637239812565641 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_680_2 :
+    Uω (aρ 2) (bρ 2) (18213010252359 / 16000000000000 : ℝ) ≤
+      (121754804071227033031 / 1000000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (5647385623161349 / 5000000000000000 : ℝ) ≤
+      (121754804071227033031 / 1000000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (11294677315516021 / 10000000000000000 : ℝ)) (q :=
+      (5647385623161349 / 5000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_680_3 :
+    Uω (aρ 3) (bρ 3) (18213010252359 / 16000000000000 : ℝ) ≤
+      (117528797438582498587 / 1000000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (22494280367204911 / 20000000000000000 : ℝ) ≤
+      (117528797438582498587 / 1000000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (175731426506649 / 156250000000000 : ℝ)) (q :=
+      (22494280367204911 / 20000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_680_4 :
+    Uω (aρ 4) (bρ 4) (18213010252359 / 16000000000000 : ℝ) ≤
+      (17263797762555920119 / 156250000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (22336465769744637 / 20000000000000000 : ℝ) ≤
+      (17263797762555920119 / 156250000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (5583645440015131 / 5000000000000000 : ℝ)) (q :=
+      (22336465769744637 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_680_5 :
+    Uω (aρ 5) (bρ 5) (18213010252359 / 16000000000000 : ℝ) ≤
+      (49858176718355945399 / 500000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (22097149691943109 / 20000000000000000 : ℝ) ≤
+      (49858176718355945399 / 500000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (5523082360781867 / 5000000000000000 : ℝ)) (q :=
+      (22097149691943109 / 20000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_680_6 :
+    Uω (aρ 6) (bρ 6) (18213010252359 / 16000000000000 : ℝ) ≤
+      (841668503982988491161 / 10000000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (21756207605261641 / 20000000000000000 : ℝ) ≤
+      (841668503982988491161 / 10000000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (5436231505011133 / 5000000000000000 : ℝ)) (q :=
+      (21756207605261641 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_680_7 :
+    Uω (aρ 7) (bρ 7) (18213010252359 / 16000000000000 : ℝ) ≤
+      (125556082777140847159 / 2000000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (10647904735856001 / 10000000000000000 : ℝ) ≤
+      (125556082777140847159 / 2000000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (10635726257232627 / 10000000000000000 : ℝ)) (q :=
+      (10647904735856001 / 10000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_680_8 :
+    Uω (aρ 8) (bρ 8) (18213010252359 / 16000000000000 : ℝ) ≤
+      (34611624805998184127 / 1000000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (10352175778881961 / 10000000000000000 : ℝ) ≤
+      (34611624805998184127 / 1000000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (10327809994624547 / 10000000000000000 : ℝ)) (q :=
+      (10352175778881961 / 10000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_680_9 :
+    Uω (aρ 9) (bρ 9) (18213010252359 / 16000000000000 : ℝ) ≤
+      (-(9759515284691867981 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19980490485952811 / 20000000000000000 : ℝ) ≤
+      (-(9759515284691867981 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (19980490485952811 / 20000000000000000 : ℝ) =
+            (19980490485952811 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 15
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19980490485952811 / 10000000000000000 : ℝ) ≤
+            (6921712287715308132019 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (2486228637793359 / 2500000000000000 : ℝ)) (q :=
+      (19980490485952811 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_680_10 :
+    Uω (aρ 10) (bρ 10) (18213010252359 / 16000000000000 : ℝ) ≤
+      (-(441205039158688405329 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (9568386474729171 / 10000000000000000 : ℝ) ≤
+      (-(441205039158688405329 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (9568386474729171 / 10000000000000000 : ℝ) =
+            (9568386474729171 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9568386474729171 / 5000000000000000 : ℝ) ≤
+            (6490266763841311594671 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (9489756048173967 / 10000000000000000 : ℝ)) (q :=
+      (9568386474729171 / 10000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_680_11 :
+    Uω (aρ 11) (bρ 11) (18213010252359 / 16000000000000 : ℝ) ≤
+      (-(235459762249886509823 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (18202307438725301 / 20000000000000000 : ℝ) ≤
+      (-(235459762249886509823 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (18202307438725301 / 20000000000000000 : ℝ) =
+            (18202307438725301 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18202307438725301 / 10000000000000000 : ℝ) ≤
+            (1497408188500113490177 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (4486883581822963 / 5000000000000000 : ℝ)) (q :=
+      (18202307438725301 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_680_12 :
+    Uω (aρ 12) (bρ 12) (18213010252359 / 16000000000000 : ℝ) ≤
+      (-(186784175247454332741 / 1250000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (2153002525543727 / 2500000000000000 : ℝ) ≤
+      (-(186784175247454332741 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (2153002525543727 / 2500000000000000 : ℝ) =
+            (2153002525543727 / 1250000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2153002525543727 / 1250000000000000 : ℝ) ≤
+            (679649800127545667259 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (8419180025955441 / 10000000000000000 : ℝ)) (q :=
+      (2153002525543727 / 2500000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_680_13 :
+    Uω (aρ 13) (bρ 13) (18213010252359 / 16000000000000 : ℝ) ≤
+      (-(2066466640027391588759 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16266139392844263 / 20000000000000000 : ℝ) ≤
+      (-(2066466640027391588759 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (16266139392844263 / 20000000000000000 : ℝ) =
+            (16266139392844263 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16266139392844263 / 10000000000000000 : ℝ) ≤
+            (4865005162972608411241 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (982585119505611 / 1250000000000000 : ℝ)) (q :=
+      (16266139392844263 / 20000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_680_14 :
+    Uω (aρ 14) (bρ 14) (18213010252359 / 16000000000000 : ℝ) ≤
+      (-(521780271597462009859 / 2000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (15407311007233661 / 20000000000000000 : ℝ) ≤
+      (-(521780271597462009859 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (15407311007233661 / 20000000000000000 : ℝ) =
+            (15407311007233661 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15407311007233661 / 10000000000000000 : ℝ) ≤
+            (864514089002537990141 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (3672889927739643 / 5000000000000000 : ℝ)) (q :=
+      (15407311007233661 / 20000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_680_15 :
+    Uω (aρ 15) (bρ 15) (18213010252359 / 16000000000000 : ℝ) ≤
+      (-(3055850814904569992253 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (46043263943697 / 62500000000000 : ℝ) ≤
+      (-(3055850814904569992253 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (46043263943697 / 62500000000000 : ℝ) = (46043263943697 / 31250000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (46043263943697 / 31250000000000 : ℝ) ≤
+            (3875620988095430007747 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1386377317528733 / 2000000000000000 : ℝ)) (q :=
+      (46043263943697 / 62500000000000 : ℝ)) (by norm_num [aρ_15, bρ_15]) (by norm_num [bρ_15])
+      (by norm_num) (by norm_num [aρ_15, bρ_15]) (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_680_16 :
+    Uω (aρ 16) (bρ 16) (18213010252359 / 16000000000000 : ℝ) ≤
+      (-(834040341612878975103 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (14326574032209963 / 20000000000000000 : ℝ) ≤
+      (-(834040341612878975103 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (14326574032209963 / 20000000000000000 : ℝ) =
+            (14326574032209963 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14326574032209963 / 10000000000000000 : ℝ) ≤
+            (898827609137121024897 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1669246952345147 / 2500000000000000 : ℝ)) (q :=
+      (14326574032209963 / 20000000000000000 : ℝ)) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [bρ_16]) (by norm_num) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [aρ_16, bρ_16]) hlg
+
+lemma U_680 :
+    Uρ (18213010252359 / 16000000000000 : ℝ) ≤
+      (-(185691588642492027487 / 10000000000000000000000 : ℝ)) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (18213010252359 / 16000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (1238640559424707941031 / 10000000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_680_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (18213010252359 / 16000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) * (121754804071227033031 / 1000000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_680_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (18213010252359 / 16000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) * (117528797438582498587 / 1000000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_680_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (18213010252359 / 16000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (17263797762555920119 / 156250000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_680_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (18213010252359 / 16000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (49858176718355945399 / 500000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_680_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (18213010252359 / 16000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) * (841668503982988491161 / 10000000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_680_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (18213010252359 / 16000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) * (125556082777140847159 / 2000000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_680_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (18213010252359 / 16000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) * (34611624805998184127 / 1000000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_680_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (18213010252359 / 16000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (-(9759515284691867981 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_680_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (18213010252359 / 16000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) *
+        (-(441205039158688405329 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_680_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (18213010252359 / 16000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) *
+        (-(235459762249886509823 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_680_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (18213010252359 / 16000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(186784175247454332741 / 1250000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_680_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (18213010252359 / 16000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(2066466640027391588759 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_680_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (18213010252359 / 16000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(521780271597462009859 / 2000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_680_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (18213010252359 / 16000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(3055850814904569992253 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_680_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (18213010252359 / 16000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) *
+        (-(834040341612878975103 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_680_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_681_1 :
+    Uω (aρ 1) (bρ 1) (1946637295669 / 1600000000000 : ℝ) ≤
+      (381566729181073695513 / 2000000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (2420394504208369 / 2000000000000000 : ℝ) ≤
+      (381566729181073695513 / 2000000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (4727327797749 / 3906250000000 : ℝ)) (q :=
+      (2420394504208369 / 2000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_681_2 :
+    Uω (aρ 2) (bρ 2) (1946637295669 / 1600000000000 : ℝ) ≤
+      (944056028915092092057 / 5000000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (603906451432549 / 500000000000000 : ℝ) ≤
+      (944056028915092092057 / 5000000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (1207804118996571 / 1000000000000000 : ℝ)) (q :=
+      (603906451432549 / 500000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_681_3 :
+    Uω (aρ 3) (bρ 3) (1946637295669 / 1600000000000 : ℝ) ≤
+      (1848611035360319879371 / 10000000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (4812205315787301 / 4000000000000000 : ℝ) ≤
+      (1848611035360319879371 / 10000000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (2406041163590051 / 2000000000000000 : ℝ)) (q :=
+      (4812205315787301 / 4000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_681_4 :
+    Uω (aρ 4) (bρ 4) (1946637295669 / 1600000000000 : ℝ) ≤
+      (44570985859756852759 / 250000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (23903292644316659 / 20000000000000000 : ℝ) ≤
+      (44570985859756852759 / 250000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (11950766064395409 / 10000000000000000 : ℝ)) (q :=
+      (23903292644316659 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_681_5 :
+    Uω (aρ 5) (bρ 5) (1946637295669 / 1600000000000 : ℝ) ≤
+      (210287390033879084137 / 1250000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (739505383293149 / 625000000000000 : ℝ) ≤
+      (210287390033879084137 / 1250000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (5914917802397259 / 5000000000000000 : ℝ)) (q :=
+      (739505383293149 / 625000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_681_6 :
+    Uω (aρ 6) (bρ 6) (1946637295669 / 1600000000000 : ℝ) ≤
+      (38434103813816542803 / 250000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (2915458645235873 / 2500000000000000 : ℝ) ≤
+      (38434103813816542803 / 250000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (5828286438220367 / 5000000000000000 : ℝ)) (q :=
+      (2915458645235873 / 2500000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_681_7 :
+    Uω (aρ 7) (bρ 7) (1946637295669 / 1600000000000 : ℝ) ≤
+      (1338393787175108167323 / 10000000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (22864183623441699 / 20000000000000000 : ℝ) ≤
+      (1338393787175108167323 / 10000000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (11420748718755449 / 10000000000000000 : ℝ)) (q :=
+      (22864183623441699 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_681_8 :
+    Uω (aρ 8) (bρ 8) (1946637295669 / 1600000000000 : ℝ) ≤
+      (538549030281882261609 / 5000000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (22274490047633971 / 20000000000000000 : ℝ) ≤
+      (538549030281882261609 / 5000000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (11114596794287721 / 10000000000000000 : ℝ)) (q :=
+      (22274490047633971 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_681_9 :
+    Uω (aρ 9) (bρ 9) (1946637295669 / 1600000000000 : ℝ) ≤
+      (748204042513207311507 / 10000000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (10776905848559021 / 10000000000000000 : ℝ) ≤
+      (748204042513207311507 / 10000000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (670930254508237 / 625000000000000 : ℝ)) (q :=
+      (10776905848559021 / 10000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_681_10 :
+    Uω (aρ 10) (bρ 10) (1946637295669 / 1600000000000 : ℝ) ≤
+      (351480511177384359847 / 10000000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (20715460896111329 / 20000000000000000 : ℝ) ≤
+      (351480511177384359847 / 10000000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (10285092304620079 / 10000000000000000 : ℝ)) (q :=
+      (20715460896111329 / 20000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_681_11 :
+    Uω (aρ 11) (bρ 11) (1946637295669 / 1600000000000 : ℝ) ≤
+      (-(26459032811501839539 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19789443919795677 / 20000000000000000 : ℝ) ≤
+      (-(26459032811501839539 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (19789443919795677 / 20000000000000000 : ℝ) =
+            (19789443919795677 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19789443919795677 / 10000000000000000 : ℝ) ≤
+            (1706408917938498160461 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (9777551954509427 / 10000000000000000 : ℝ)) (q :=
+      (19789443919795677 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_681_12 :
+    Uω (aρ 12) (bρ 12) (1946637295669 / 1600000000000 : ℝ) ≤
+      (-(151566259945222584619 / 2500000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (18823493977979713 / 20000000000000000 : ℝ) ≤
+      (-(151566259945222584619 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (18823493977979713 / 20000000000000000 : ℝ) =
+            (18823493977979713 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18823493977979713 / 10000000000000000 : ℝ) ≤
+            (1581301690804777415381 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (9235302109378463 / 10000000000000000 : ℝ)) (q :=
+      (18823493977979713 / 20000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_681_13 :
+    Uω (aρ 13) (bρ 13) (1946637295669 / 1600000000000 : ℝ) ≤
+      (-(1119336165660419824783 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1788207217931651 / 2000000000000000 : ℝ) ≤
+      (-(1119336165660419824783 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1788207217931651 / 2000000000000000 : ℝ) =
+            (1788207217931651 / 1000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1788207217931651 / 1000000000000000 : ℝ) ≤
+            (5812135637339580175217 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (434663102615513 / 500000000000000 : ℝ)) (q :=
+      (1788207217931651 / 2000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_681_14 :
+    Uω (aρ 14) (bρ 14) (1946637295669 / 1600000000000 : ℝ) ≤
+      (-(800052579600481050621 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (17042696554316489 / 20000000000000000 : ℝ) ≤
+      (-(800052579600481050621 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (17042696554316489 / 20000000000000000 : ℝ) =
+            (17042696554316489 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17042696554316489 / 10000000000000000 : ℝ) ≤
+            (2665683321899518949379 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (8197813712355239 / 10000000000000000 : ℝ)) (q :=
+      (17042696554316489 / 20000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_681_15 :
+    Uω (aρ 15) (bρ 15) (1946637295669 / 1600000000000 : ℝ) ≤
+      (-(1991591901709806684277 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (16388388784320793 / 20000000000000000 : ℝ) ≤
+      (-(1991591901709806684277 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (16388388784320793 / 20000000000000000 : ℝ) =
+            (16388388784320793 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16388388784320793 / 10000000000000000 : ℝ) ≤
+            (4939879901290193315723 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (7803079219774543 / 10000000000000000 : ℝ)) (q :=
+      (16388388784320793 / 20000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_681_16 :
+    Uω (aρ 16) (bρ 16) (1946637295669 / 1600000000000 : ℝ) ≤
+      (-(1117370294237785446319 / 5000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (7997356374547131 / 10000000000000000 : ℝ) ≤
+      (-(1117370294237785446319 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (7997356374547131 / 10000000000000000 : ℝ) =
+            (7997356374547131 / 5000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7997356374547131 / 5000000000000000 : ℝ) ≤
+            (2348365607262214553681 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (1890443709014503 / 2500000000000000 : ℝ)) (q :=
+      (7997356374547131 / 10000000000000000 : ℝ)) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [bρ_16]) (by norm_num) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [aρ_16, bρ_16]) hlg
+
+lemma U_681 :
+    Uρ (1946637295669 / 1600000000000 : ℝ) ≤
+      (539206832017378636869 / 10000000000000000000000 : ℝ) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (1946637295669 / 1600000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (381566729181073695513 / 2000000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_681_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (1946637295669 / 1600000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) * (944056028915092092057 / 5000000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_681_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (1946637295669 / 1600000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (1848611035360319879371 / 10000000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_681_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (1946637295669 / 1600000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (44570985859756852759 / 250000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_681_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (1946637295669 / 1600000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (210287390033879084137 / 1250000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_681_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (1946637295669 / 1600000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) * (38434103813816542803 / 250000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_681_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (1946637295669 / 1600000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (1338393787175108167323 / 10000000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_681_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (1946637295669 / 1600000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) * (538549030281882261609 / 5000000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_681_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (1946637295669 / 1600000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (748204042513207311507 / 10000000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_681_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (1946637295669 / 1600000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) * (351480511177384359847 / 10000000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_681_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (1946637295669 / 1600000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) * (-(26459032811501839539 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_681_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (1946637295669 / 1600000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) *
+        (-(151566259945222584619 / 2500000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_681_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (1946637295669 / 1600000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) *
+        (-(1119336165660419824783 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_681_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (1946637295669 / 1600000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) *
+        (-(800052579600481050621 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_681_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (1946637295669 / 1600000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(1991591901709806684277 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_681_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (1946637295669 / 1600000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) *
+        (-(1117370294237785446319 / 5000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_681_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_682_1 :
+    Uω (aρ 1) (bρ 1) (2746637295669 / 2000000000000 : ℝ) ≤
+      (1562609017187395006033 / 5000000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (2733735486528567 / 2000000000000000 : ℝ) ≤
+      (1562609017187395006033 / 5000000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (1366866560502567 / 1000000000000000 : ℝ)) (q :=
+      (2733735486528567 / 2000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_682_2 :
+    Uω (aρ 2) (bρ 2) (2746637295669 / 2000000000000 : ℝ) ≤
+      (621553036055481657661 / 2000000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (13644842494784013 / 10000000000000000 : ℝ) ≤
+      (621553036055481657661 / 2000000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (6822382370909013 / 5000000000000000 : ℝ)) (q :=
+      (13644842494784013 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_682_3 :
+    Uω (aρ 3) (bρ 3) (2746637295669 / 2000000000000 : ℝ) ≤
+      (768206569321752291863 / 2500000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (5438900839273781 / 4000000000000000 : ℝ) ≤
+      (768206569321752291863 / 2500000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (2719396010993781 / 2000000000000000 : ℝ)) (q :=
+      (5438900839273781 / 4000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_682_4 :
+    Uω (aρ 4) (bρ 4) (2746637295669 / 2000000000000 : ℝ) ≤
+      (3014704537217327050923 / 10000000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (1351845172554389 / 1000000000000000 : ℝ) ≤
+      (3014704537217327050923 / 10000000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (675883674537639 / 500000000000000 : ℝ)) (q :=
+      (1351845172554389 / 1000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_682_5 :
+    Uω (aρ 5) (bρ 5) (2746637295669 / 2000000000000 : ℝ) ≤
+      (585197835390417916143 / 2000000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (1339905270349957 / 1000000000000000 : ℝ) ≤
+      (585197835390417916143 / 2000000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (669853268299957 / 500000000000000 : ℝ)) (q :=
+      (1339905270349957 / 1000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_682_6 :
+    Uω (aρ 6) (bρ 6) (2746637295669 / 2000000000000 : ℝ) ≤
+      (1399192462686000616281 / 5000000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (26458322685085169 / 20000000000000000 : ℝ) ≤
+      (1399192462686000616281 / 5000000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (13224523019225169 / 10000000000000000 : ℝ)) (q :=
+      (26458322685085169 / 20000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_682_7 :
+    Uω (aρ 7) (bρ 7) (2746637295669 / 2000000000000 : ℝ) ≤
+      (40996380119763539301 / 156250000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (13000163388913701 / 10000000000000000 : ℝ) ≤
+      (40996380119763539301 / 156250000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (6495094246363701 / 5000000000000000 : ℝ)) (q :=
+      (13000163388913701 / 10000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_682_8 :
+    Uω (aρ 8) (bρ 8) (2746637295669 / 2000000000000 : ℝ) ≤
+      (2395479286330499915887 / 10000000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (25413491697247327 / 20000000000000000 : ℝ) ≤
+      (2395479286330499915887 / 10000000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (12686895063487327 / 10000000000000000 : ℝ)) (q :=
+      (25413491697247327 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_682_9 :
+    Uω (aρ 9) (bρ 9) (2746637295669 / 2000000000000 : ℝ) ≤
+      (2109866511084314583863 / 10000000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (24697917409205917 / 20000000000000000 : ℝ) ≤
+      (2109866511084314583863 / 10000000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (12312286403805917 / 10000000000000000 : ℝ)) (q :=
+      (24697917409205917 / 20000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_682_10 :
+    Uω (aρ 10) (bρ 10) (2746637295669 / 2000000000000 : ℝ) ≤
+      (1768087379828123108613 / 10000000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (11934028186673021 / 10000000000000000 : ℝ) ≤
+      (1768087379828123108613 / 10000000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (5935492200720521 / 5000000000000000 : ℝ)) (q :=
+      (11934028186673021 / 10000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_682_11 :
+    Uω (aρ 11) (bρ 11) (2746637295669 / 2000000000000 : ℝ) ≤
+      (1378108348866220666213 / 10000000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (11477584138870181 / 10000000000000000 : ℝ) ≤
+      (1378108348866220666213 / 10000000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (5688286466020181 / 5000000000000000 : ℝ)) (q :=
+      (11477584138870181 / 10000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_682_12 :
+    Uω (aρ 12) (bρ 12) (2746637295669 / 2000000000000 : ℝ) ≤
+      (956720482436799679737 / 10000000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (22007962546276019 / 20000000000000000 : ℝ) ≤
+      (956720482436799679737 / 10000000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (10853067297261019 / 10000000000000000 : ℝ)) (q :=
+      (22007962546276019 / 20000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_682_13 :
+    Uω (aρ 13) (bρ 13) (2746637295669 / 2000000000000 : ℝ) ≤
+      (531080877900712560969 / 10000000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (21090872444982791 / 20000000000000000 : ℝ) ≤
+      (531080877900712560969 / 10000000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (10335358937562791 / 10000000000000000 : ℝ)) (q :=
+      (21090872444982791 / 20000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_682_14 :
+    Uω (aρ 14) (bρ 14) (2746637295669 / 2000000000000 : ℝ) ≤
+      (34680649393365873701 / 2500000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (20279378520574633 / 20000000000000000 : ℝ) ≤
+      (34680649393365873701 / 2500000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (9867792298199633 / 10000000000000000 : ℝ)) (q :=
+      (20279378520574633 / 20000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_682_15 :
+    Uω (aρ 15) (bρ 15) (2746637295669 / 2000000000000 : ℝ) ≤
+      (-(175689130303678076623 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (1965169040399109 / 2000000000000000 : ℝ) ≤
+      (-(175689130303678076623 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (1965169040399109 / 2000000000000000 : ℝ) =
+            (1965169040399109 / 1000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1965169040399109 / 1000000000000000 : ℝ) ≤
+            (6755782672696321923377 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (949967745903109 / 1000000000000000 : ℝ)) (q :=
+      (1965169040399109 / 2000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_682_16 :
+    Uω (aρ 16) (bρ 16) (2746637295669 / 2000000000000 : ℝ) ≤
+      (-(368491277579048029599 / 10000000000000000000000 : ℝ)) := by
+  have hlg :
+    Real.log (19276430760921297 / 20000000000000000 : ℝ) ≤
+      (-(368491277579048029599 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (19276430760921297 / 20000000000000000 : ℝ) =
+            (19276430760921297 / 10000000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19276430760921297 / 10000000000000000 : ℝ) ≤
+            (6562980525420951970401 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  exact
+    Uω_le_right_certificate (s := (9276789467471297 / 10000000000000000 : ℝ)) (q :=
+      (19276430760921297 / 20000000000000000 : ℝ)) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [bρ_16]) (by norm_num) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [aρ_16, bρ_16]) hlg
+
+lemma U_682 :
+    Uρ (2746637295669 / 2000000000000 : ℝ) ≤
+      (1832301135489243993659 / 10000000000000000000000 : ℝ) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (2746637295669 / 2000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (1562609017187395006033 / 5000000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_682_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (2746637295669 / 2000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) * (621553036055481657661 / 2000000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_682_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (2746637295669 / 2000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) * (768206569321752291863 / 2500000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_682_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (2746637295669 / 2000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (3014704537217327050923 / 10000000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_682_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (2746637295669 / 2000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (585197835390417916143 / 2000000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_682_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (2746637295669 / 2000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) * (1399192462686000616281 / 5000000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_682_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (2746637295669 / 2000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) * (40996380119763539301 / 156250000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_682_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (2746637295669 / 2000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (2395479286330499915887 / 10000000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_682_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (2746637295669 / 2000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (2109866511084314583863 / 10000000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_682_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (2746637295669 / 2000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) * (1768087379828123108613 / 10000000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_682_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (2746637295669 / 2000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) * (1378108348866220666213 / 10000000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_682_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (2746637295669 / 2000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (956720482436799679737 / 10000000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_682_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (2746637295669 / 2000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) * (531080877900712560969 / 10000000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_682_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (2746637295669 / 2000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) * (34680649393365873701 / 2500000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_682_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (2746637295669 / 2000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) *
+        (-(175689130303678076623 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_682_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (2746637295669 / 2000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) *
+        (-(368491277579048029599 / 10000000000000000000000 : ℝ)) :=
+    mul_le_mul_of_nonneg_left U_682_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_683_1 :
+    Uω (aρ 1) (bρ 1) (6746637295669 / 4000000000000 : ℝ) ≤
+      (648647470165495546441 / 1250000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (33604172798388249 / 20000000000000000 : ℝ) ≤
+      (648647470165495546441 / 1250000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (16802076777300749 / 10000000000000000 : ℝ)) (q :=
+      (33604172798388249 / 20000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_683_2 :
+    Uω (aρ 2) (bρ 2) (6746637295669 / 4000000000000 : ℝ) ≤
+      (646873916084388180027 / 1250000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (16778263776351059 / 10000000000000000 : ℝ) ≤
+      (646873916084388180027 / 1250000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (8389100272062309 / 5000000000000000 : ℝ)) (q :=
+      (16778263776351059 / 10000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_683_3 :
+    Uω (aρ 3) (bρ 3) (6746637295669 / 4000000000000 : ℝ) ≤
+      (5146608484439031249377 / 10000000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (16730709809385323 / 10000000000000000 : ℝ) ≤
+      (5146608484439031249377 / 10000000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (8365244358271573 / 5000000000000000 : ℝ)) (q :=
+      (16730709809385323 / 10000000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_683_4 :
+    Uω (aρ 4) (bρ 4) (6746637295669 / 4000000000000 : ℝ) ≤
+      (637431915769893316069 / 1250000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (16652004933609187 / 10000000000000000 : ℝ) ≤
+      (637431915769893316069 / 1250000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (8325686573027937 / 5000000000000000 : ℝ)) (q :=
+      (16652004933609187 / 10000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_683_5 :
+    Uω (aρ 5) (bρ 5) (6746637295669 / 4000000000000 : ℝ) ≤
+      (15711355649568528379 / 31250000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (16532836162270769 / 10000000000000000 : ℝ) ≤
+      (15711355649568528379 / 31250000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (8265612761357019 / 5000000000000000 : ℝ)) (q :=
+      (16532836162270769 / 10000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_683_6 :
+    Uω (aρ 6) (bρ 6) (6746637295669 / 4000000000000 : ℝ) ≤
+      (1231163739354149495203 / 2500000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (2045432067349599 / 1250000000000000 : ℝ) ≤
+      (1231163739354149495203 / 2500000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (4089926662726521 / 2500000000000000 : ℝ)) (q :=
+      (2045432067349599 / 1250000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_683_7 :
+    Uω (aρ 7) (bρ 7) (6746637295669 / 4000000000000 : ℝ) ≤
+      (4784372414919632677529 / 10000000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (16135508405435549 / 10000000000000000 : ℝ) ≤
+      (4784372414919632677529 / 10000000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (8063735882471799 / 5000000000000000 : ℝ)) (q :=
+      (16135508405435549 / 10000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_683_8 :
+    Uω (aρ 8) (bρ 8) (6746637295669 / 4000000000000 : ℝ) ≤
+      (1150527614874777660297 / 2500000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (15844083326734951 / 10000000000000000 : ℝ) ≤
+      (1150527614874777660297 / 2500000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (7914081629441201 / 5000000000000000 : ℝ)) (q :=
+      (15844083326734951 / 10000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_683_9 :
+    Uω (aρ 9) (bρ 9) (6746637295669 / 4000000000000 : ℝ) ≤
+      (4375967434278497798067 / 10000000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (15489801450345909 / 10000000000000000 : ℝ) ≤
+      (4375967434278497798067 / 10000000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (7730282567232159 / 5000000000000000 : ℝ)) (q :=
+      (15489801450345909 / 10000000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_683_10 :
+    Uω (aρ 10) (bρ 10) (6746637295669 / 4000000000000 : ℝ) ≤
+      (4108233282580037516371 / 10000000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (6032235606512157 / 4000000000000000 : ℝ) ≤
+      (4108233282580037516371 / 10000000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (3006139859965657 / 2000000000000000 : ℝ)) (q :=
+      (6032235606512157 / 4000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_683_11 :
+    Uω (aρ 11) (bρ 11) (6746637295669 / 4000000000000 : ℝ) ≤
+      (761357075335845670987 / 2000000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (29265542826264307 / 20000000000000000 : ℝ) ≤
+      (761357075335845670987 / 2000000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (14553540719736807 / 10000000000000000 : ℝ)) (q :=
+      (29265542826264307 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_683_12 :
+    Uω (aρ 12) (bρ 12) (6746637295669 / 4000000000000 : ℝ) ≤
+      (139448290184602758597 / 400000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (28342232281010363 / 20000000000000000 : ℝ) ≤
+      (139448290184602758597 / 400000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (14053930271167863 / 10000000000000000 : ℝ)) (q :=
+      (28342232281010363 / 20000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_683_13 :
+    Uω (aρ 13) (bρ 13) (6746637295669 / 4000000000000 : ℝ) ≤
+      (3168189274869361871549 / 10000000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (6863769907032257 / 5000000000000000 : ℝ) ≤
+      (3168189274869361871549 / 10000000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (1695769919985191 / 1250000000000000 : ℝ)) (q :=
+      (6863769907032257 / 5000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_683_14 :
+    Uω (aρ 14) (bρ 14) (6746637295669 / 4000000000000 : ℝ) ≤
+      (288054402101994101473 / 1000000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (13338298652980011 / 10000000000000000 : ℝ) ≤
+      (288054402101994101473 / 1000000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (6565802161378761 / 5000000000000000 : ℝ)) (q :=
+      (13338298652980011 / 10000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_683_15 :
+    Uω (aρ 15) (bρ 15) (6746637295669 / 4000000000000 : ℝ) ≤
+      (1327044432574458074269 / 5000000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (3259910100253591 / 2500000000000000 : ℝ) ≤
+      (1327044432574458074269 / 5000000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (3198465274060307 / 2500000000000000 : ℝ)) (q :=
+      (3259910100253591 / 2500000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_683_16 :
+    Uω (aρ 16) (bρ 16) (6746637295669 / 4000000000000 : ℝ) ≤
+      (2517089256916817378273 / 10000000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (803888497956787 / 625000000000000 : ℝ) ≤
+      (2517089256916817378273 / 10000000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (3147845970084921 / 2500000000000000 : ℝ)) (q :=
+      (803888497956787 / 625000000000000 : ℝ)) (by norm_num [aρ_16, bρ_16]) (by norm_num [bρ_16])
+      (by norm_num) (by norm_num [aρ_16, bρ_16]) (by norm_num [aρ_16, bρ_16]) hlg
+
+lemma U_683 :
+    Uρ (6746637295669 / 4000000000000 : ℝ) ≤
+      (991641173695049392639 / 2500000000000000000000 : ℝ) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (6746637295669 / 4000000000000 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (648647470165495546441 / 1250000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_683_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (6746637295669 / 4000000000000 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) * (646873916084388180027 / 1250000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_683_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (6746637295669 / 4000000000000 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) *
+        (5146608484439031249377 / 10000000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_683_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (6746637295669 / 4000000000000 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (637431915769893316069 / 1250000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_683_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (6746637295669 / 4000000000000 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (15711355649568528379 / 31250000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_683_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (6746637295669 / 4000000000000 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) * (1231163739354149495203 / 2500000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_683_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (6746637295669 / 4000000000000 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) *
+        (4784372414919632677529 / 10000000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_683_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (6746637295669 / 4000000000000 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) * (1150527614874777660297 / 2500000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_683_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (6746637295669 / 4000000000000 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (4375967434278497798067 / 10000000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_683_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (6746637295669 / 4000000000000 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) * (4108233282580037516371 / 10000000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_683_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (6746637295669 / 4000000000000 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) * (761357075335845670987 / 2000000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_683_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (6746637295669 / 4000000000000 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (139448290184602758597 / 400000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_683_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (6746637295669 / 4000000000000 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) * (3168189274869361871549 / 10000000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_683_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (6746637295669 / 4000000000000 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) * (288054402101994101473 / 1000000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_683_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (6746637295669 / 4000000000000 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) * (1327044432574458074269 / 5000000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_683_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (6746637295669 / 4000000000000 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (2517089256916817378273 / 10000000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_683_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+lemma U_684_1 : Uω (aρ 1) (bρ 1) (2 : ℝ) ≤ (431197938622519678389 / 625000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (19935494672365351 / 10000000000000000 : ℝ) ≤
+      (431197938622519678389 / 625000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 14
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (9967743281407851 / 5000000000000000 : ℝ)) (q :=
+      (19935494672365351 / 10000000000000000 : ℝ)) (by norm_num [aρ_1, bρ_1]) (by norm_num [bρ_1])
+      (by norm_num) (by norm_num [aρ_1, bρ_1]) (by norm_num [aρ_1, bρ_1]) hlg
+
+lemma U_684_2 :
+    Uω (aρ 2) (bρ 2) (2 : ℝ) ≤ (1721803564318625397281 / 2500000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (39823360975531723 / 20000000000000000 : ℝ) ≤
+      (1721803564318625397281 / 2500000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 14
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (19911627206126723 / 10000000000000000 : ℝ)) (q :=
+      (39823360975531723 / 20000000000000000 : ℝ)) (by norm_num [aρ_2, bρ_2]) (by norm_num [bρ_2])
+      (by norm_num) (by norm_num [aρ_2, bρ_2]) (by norm_num [aρ_2, bρ_2]) hlg
+
+lemma U_684_3 :
+    Uω (aρ 3) (bρ 3) (2 : ℝ) ≤ (3431657896680287743107 / 5000000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (993207572309059 / 500000000000000 : ℝ) ≤
+      (3431657896680287743107 / 5000000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 14
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (124149782683171 / 62500000000000 : ℝ)) (q :=
+      (993207572309059 / 500000000000000 : ℝ)) (by norm_num [aρ_3, bρ_3]) (by norm_num [bρ_3])
+      (by norm_num) (by norm_num [aρ_3, bρ_3]) (by norm_num [aρ_3, bρ_3]) hlg
+
+lemma U_684_4 :
+    Uω (aρ 4) (bρ 4) (2 : ℝ) ≤ (6823648472170288821807 / 10000000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (39571023506073419 / 20000000000000000 : ℝ) ≤
+      (6823648472170288821807 / 10000000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 14
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (19784980024083419 / 10000000000000000 : ℝ)) (q :=
+      (39571023506073419 / 20000000000000000 : ℝ)) (by norm_num [aρ_4, bρ_4]) (by norm_num [bρ_4])
+      (by norm_num) (by norm_num [aρ_4, bρ_4]) (by norm_num [aρ_4, bρ_4]) hlg
+
+lemma U_684_5 :
+    Uω (aρ 5) (bρ 5) (2 : ℝ) ≤ (6763315653946060878467 / 10000000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (7866599825072169 / 4000000000000000 : ℝ) ≤
+      (6763315653946060878467 / 10000000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 14
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (3933029112541169 / 2000000000000000 : ℝ)) (q :=
+      (7866599825072169 / 4000000000000000 : ℝ)) (by norm_num [aρ_5, bρ_5]) (by norm_num [bρ_5])
+      (by norm_num) (by norm_num [aρ_5, bρ_5]) (by norm_num [aρ_5, bρ_5]) hlg
+
+lemma U_684_6 : Uω (aρ 6) (bρ 6) (2 : ℝ) ≤ (333849709093393011791 / 500000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (1949746605406931 / 1000000000000000 : ℝ) ≤
+      (333849709093393011791 / 500000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 14
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (974715946031181 / 500000000000000 : ℝ)) (q :=
+      (1949746605406931 / 1000000000000000 : ℝ)) (by norm_num [aρ_6, bρ_6]) (by norm_num [bρ_6])
+      (by norm_num) (by norm_num [aρ_6, bρ_6]) (by norm_num [aρ_6, bρ_6]) hlg
+
+lemma U_684_7 :
+    Uω (aρ 7) (bρ 7) (2 : ℝ) ≤ (3279879678491493456121 / 5000000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (38540444995663327 / 20000000000000000 : ℝ) ≤
+      (3279879678491493456121 / 5000000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 14
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (19263493188908327 / 10000000000000000 : ℝ)) (q :=
+      (38540444995663327 / 20000000000000000 : ℝ)) (by norm_num [aρ_7, bρ_7]) (by norm_num [bρ_7])
+      (by norm_num) (by norm_num [aρ_7, bρ_7]) (by norm_num [aρ_7, bρ_7]) hlg
+
+lemma U_684_8 :
+    Uω (aρ 8) (bρ 8) (2 : ℝ) ≤ (6408070500101273547501 / 10000000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (37960241040023251 / 20000000000000000 : ℝ) ≤
+      (6408070500101273547501 / 10000000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 14
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (18966830884608251 / 10000000000000000 : ℝ)) (q :=
+      (37960241040023251 / 20000000000000000 : ℝ)) (by norm_num [aρ_8, bρ_8]) (by norm_num [bρ_8])
+      (by norm_num) (by norm_num [aρ_8, bρ_8]) (by norm_num [aρ_8, bρ_8]) hlg
+
+lemma U_684_9 :
+    Uω (aρ 9) (bρ 9) (2 : ℝ) ≤ (3110439556119279087301 / 5000000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (298050139723413 / 160000000000000 : ℝ) ≤
+      (3110439556119279087301 / 5000000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 13
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (148830583506973 / 80000000000000 : ℝ)) (q :=
+      (298050139723413 / 160000000000000 : ℝ)) (by norm_num [aρ_9, bρ_9]) (by norm_num [bρ_9])
+      (by norm_num) (by norm_num [aρ_9, bρ_9]) (by norm_num [aρ_9, bρ_9]) hlg
+
+lemma U_684_10 :
+    Uω (aρ 10) (bρ 10) (2 : ℝ) ≤ (6000773764786687711391 / 10000000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (3644519589963523 / 2000000000000000 : ℝ) ≤
+      (6000773764786687711391 / 10000000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 13
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (1818131040607523 / 1000000000000000 : ℝ)) (q :=
+      (3644519589963523 / 2000000000000000 : ℝ)) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [bρ_10]) (by norm_num) (by norm_num [aρ_10, bρ_10])
+      (by norm_num [aρ_10, bρ_10]) hlg
+
+lemma U_684_11 :
+    Uω (aρ 11) (bρ 11) (2 : ℝ) ≤ (5755005827102536483859 / 10000000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (35560407008524539 / 20000000000000000 : ℝ) ≤
+      (5755005827102536483859 / 10000000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 13
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (17714998141169539 / 10000000000000000 : ℝ)) (q :=
+      (35560407008524539 / 20000000000000000 : ℝ)) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [bρ_11]) (by norm_num) (by norm_num [aρ_11, bρ_11])
+      (by norm_num [aρ_11, bρ_11]) hlg
+
+lemma U_684_12 :
+    Uω (aρ 12) (bρ 12) (2 : ℝ) ≤ (1099230232572658765387 / 2000000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (34651720907416137 / 20000000000000000 : ℝ) ≤
+      (1099230232572658765387 / 2000000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (17230012136746137 / 10000000000000000 : ℝ)) (q :=
+      (34651720907416137 / 20000000000000000 : ℝ)) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [bρ_12]) (by norm_num) (by norm_num [aρ_12, bρ_12])
+      (by norm_num [aρ_12, bρ_12]) hlg
+
+lemma U_684_13 :
+    Uω (aρ 13) (bρ 13) (2 : ℝ) ≤ (2621030225880535406049 / 5000000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (8445586165416359 / 5000000000000000 : ℝ) ≤
+      (2621030225880535406049 / 5000000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (4190004408147609 / 2500000000000000 : ℝ)) (q :=
+      (8445586165416359 / 5000000000000000 : ℝ)) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [bρ_13]) (by norm_num) (by norm_num [aρ_13, bρ_13])
+      (by norm_num [aρ_13, bρ_13]) hlg
+
+lemma U_684_14 :
+    Uω (aρ 14) (bρ 14) (2 : ℝ) ≤ (250733810854960598567 / 500000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (16511427463010601 / 10000000000000000 : ℝ) ≤
+      (250733810854960598567 / 500000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (8172227590995601 / 5000000000000000 : ℝ)) (q :=
+      (16511427463010601 / 10000000000000000 : ℝ)) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [bρ_14]) (by norm_num) (by norm_num [aρ_14, bρ_14])
+      (by norm_num [aρ_14, bρ_14]) hlg
+
+lemma U_684_15 :
+    Uω (aρ 15) (bρ 15) (2 : ℝ) ≤ (2418686312017971573191 / 5000000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (16221253963311517 / 10000000000000000 : ℝ) ≤
+      (2418686312017971573191 / 5000000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (8011840730004017 / 5000000000000000 : ℝ)) (q :=
+      (16221253963311517 / 10000000000000000 : ℝ)) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [bρ_15]) (by norm_num) (by norm_num [aρ_15, bρ_15])
+      (by norm_num [aρ_15, bρ_15]) hlg
+
+lemma U_684_16 :
+    Uω (aρ 16) (bρ 16) (2 : ℝ) ≤ (4730867613563951102639 / 10000000000000000000000 : ℝ) := by
+  have hlg :
+    Real.log (1604940623761263 / 1000000000000000 : ℝ) ≤
+      (4730867613563951102639 / 10000000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  exact
+    Uω_le_right_certificate (s := (791617883006013 / 500000000000000 : ℝ)) (q :=
+      (1604940623761263 / 1000000000000000 : ℝ)) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [bρ_16]) (by norm_num) (by norm_num [aρ_16, bρ_16])
+      (by norm_num [aρ_16, bρ_16]) hlg
+
+lemma U_684 : Uρ (2 : ℝ) ≤ (1138744120442109294651 / 2000000000000000000000 : ℝ) := by
+  rw [Uρ_expand, cρ_1, cρ_2, cρ_3, cρ_4, cρ_5, cρ_6, cρ_7, cρ_8, cρ_9, cρ_10, cρ_11, cρ_12, cρ_13,
+    cρ_14, cρ_15, cρ_16]
+  have hm1 :
+    (525779809 / 50000000000 : ℝ) * Uω (aρ 1) (bρ 1) (2 : ℝ) ≤
+      (525779809 / 50000000000 : ℝ) * (431197938622519678389 / 625000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_684_1 (by norm_num)
+  have hm2 :
+    (29471737793 / 1000000000000 : ℝ) * Uω (aρ 2) (bρ 2) (2 : ℝ) ≤
+      (29471737793 / 1000000000000 : ℝ) * (1721803564318625397281 / 2500000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_684_2 (by norm_num)
+  have hm3 :
+    (42934365099 / 1000000000000 : ℝ) * Uω (aρ 3) (bρ 3) (2 : ℝ) ≤
+      (42934365099 / 1000000000000 : ℝ) * (3431657896680287743107 / 5000000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_684_3 (by norm_num)
+  have hm4 :
+    (29102115983 / 500000000000 : ℝ) * Uω (aρ 4) (bρ 4) (2 : ℝ) ≤
+      (29102115983 / 500000000000 : ℝ) * (6823648472170288821807 / 10000000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_684_4 (by norm_num)
+  have hm5 :
+    (6903762131 / 100000000000 : ℝ) * Uω (aρ 5) (bρ 5) (2 : ℝ) ≤
+      (6903762131 / 100000000000 : ℝ) * (6763315653946060878467 / 10000000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_684_5 (by norm_num)
+  have hm6 :
+    (78873099189 / 1000000000000 : ℝ) * Uω (aρ 6) (bρ 6) (2 : ℝ) ≤
+      (78873099189 / 1000000000000 : ℝ) * (333849709093393011791 / 500000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_684_6 (by norm_num)
+  have hm7 :
+    (84856120711 / 1000000000000 : ℝ) * Uω (aρ 7) (bρ 7) (2 : ℝ) ≤
+      (84856120711 / 1000000000000 : ℝ) * (3279879678491493456121 / 5000000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_684_7 (by norm_num)
+  have hm8 :
+    (88396082127 / 1000000000000 : ℝ) * Uω (aρ 8) (bρ 8) (2 : ℝ) ≤
+      (88396082127 / 1000000000000 : ℝ) *
+        (6408070500101273547501 / 10000000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_684_8 (by norm_num)
+  have hm9 :
+    (706427057 / 8000000000 : ℝ) * Uω (aρ 9) (bρ 9) (2 : ℝ) ≤
+      (706427057 / 8000000000 : ℝ) * (3110439556119279087301 / 5000000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_684_9 (by norm_num)
+  have hm10 :
+    (17094464251 / 200000000000 : ℝ) * Uω (aρ 10) (bρ 10) (2 : ℝ) ≤
+      (17094464251 / 200000000000 : ℝ) * (6000773764786687711391 / 10000000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_684_10 (by norm_num)
+  have hm11 :
+    (39449592119 / 500000000000 : ℝ) * Uω (aρ 11) (bρ 11) (2 : ℝ) ≤
+      (39449592119 / 500000000000 : ℝ) * (5755005827102536483859 / 10000000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_684_11 (by norm_num)
+  have hm12 :
+    (35176735959 / 500000000000 : ℝ) * Uω (aρ 12) (bρ 12) (2 : ℝ) ≤
+      (35176735959 / 500000000000 : ℝ) * (1099230232572658765387 / 2000000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_684_12 (by norm_num)
+  have hm13 :
+    (11767795323 / 200000000000 : ℝ) * Uω (aρ 13) (bρ 13) (2 : ℝ) ≤
+      (11767795323 / 200000000000 : ℝ) * (2621030225880535406049 / 5000000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_684_13 (by norm_num)
+  have hm14 :
+    (22210660553 / 500000000000 : ℝ) * Uω (aρ 14) (bρ 14) (2 : ℝ) ≤
+      (22210660553 / 500000000000 : ℝ) * (250733810854960598567 / 500000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_684_14 (by norm_num)
+  have hm15 :
+    (30462865791 / 1000000000000 : ℝ) * Uω (aρ 15) (bρ 15) (2 : ℝ) ≤
+      (30462865791 / 1000000000000 : ℝ) * (2418686312017971573191 / 5000000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_684_15 (by norm_num)
+  have hm16 :
+    (5959622577 / 1000000000000 : ℝ) * Uω (aρ 16) (bρ 16) (2 : ℝ) ≤
+      (5959622577 / 1000000000000 : ℝ) * (4730867613563951102639 / 10000000000000000000000 : ℝ) :=
+    mul_le_mul_of_nonneg_left U_684_16 (by norm_num)
+  linarith only [hm1, hm2, hm3, hm4, hm5, hm6, hm7, hm8, hm9, hm10, hm11, hm12, hm13, hm14, hm15,
+    hm16]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/UpperCertificate.lean b/LeanPool/Zeta5Irrational/Table/UpperCertificate.lean
new file mode 100644
index 0000000000..eeed4a6faf
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/UpperCertificate.lean
@@ -0,0 +1,39 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Table.Common
+public import LeanPool.Zeta5Irrational.Certificates
+public import Mathlib.Analysis.Complex.ExponentialBounds
+
+/-! # Reusable upper certificates for arcsine potentials -/
+
+@[expose] public section
+
+namespace Zeta5Irrational
+
+/-- An upper certificate for an arcsine potential to the left of its support. -/
+lemma Uω_le_left_certificate {a b t s q U : ℝ} (hab : a < b) (ht : t ≤ a) (hs : 0 ≤ s)
+    (hsq : (t - a) * (t - b) ≤ s ^ 2) (hq : ((a + b) / 2 - t + s) / 2 ≤ q)
+    (hlog : Real.log q ≤ U) : Uω a b t ≤ U := by
+  rw [Uω_of_le hab ht]
+  have hroot := sqrt_le_of_sq_le hs hsq
+  have hroot0 := Real.sqrt_nonneg ((t - a) * (t - b))
+  apply (Real.log_le_log (by linarith) ?_).trans hlog
+  linarith
+
+/-- An upper certificate for an arcsine potential to the right of its support. -/
+lemma Uω_le_right_certificate {a b t s q U : ℝ} (hab : a < b) (ht : b ≤ t) (hs : 0 ≤ s)
+    (hsq : (t - a) * (t - b) ≤ s ^ 2) (hq : (t - (a + b) / 2 + s) / 2 ≤ q)
+    (hlog : Real.log q ≤ U) : Uω a b t ≤ U := by
+  rw [Uω_of_ge hab ht]
+  have hroot := sqrt_le_of_sq_le hs hsq
+  have hroot0 := Real.sqrt_nonneg ((t - a) * (t - b))
+  apply (Real.log_le_log (by linarith) ?_).trans hlog
+  linarith
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/V00.lean b/LeanPool/Zeta5Irrational/Table/V00.lean
new file mode 100644
index 0000000000..8dc82aafc6
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/V00.lean
@@ -0,0 +1,1380 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Energy
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # V00: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_1 :
+    ((8967502915859873 / 250000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(25870887852728765963031 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((14973028876951 / 2500000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (11978279880015751031 / 2000000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (796870528491162609301 / 10000000000000000000000 : ℝ))))) ≤
+      Vfield (7174131 / 200000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (7174131 / 200000000000 : ℝ)) = Real.log (200007174131 / 200000000000 : ℝ) := by
+    norm_num
+  have e2 :
+    Real.log ((7174131 / 200000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1132174131 / 200000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (8967502915859873 / 250000000000000000000 : ℝ) ≤ Real.log (200007174131 / 200000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 2
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1132174131 / 200000000000 : ℝ) ≤
+      (-(25870887852728765963031 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show (1132174131 / 200000000000 : ℝ) = (1132174131 / 781250000 : ℝ) / 2 ^ 8 by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1132174131 / 781250000 : ℝ) ≤
+            (1854999359271234036969 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (59892115507801 / 10000000000000000 : ℝ) ≤ Real.sqrt (7174131 / 200000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi : Real.sqrt (7174131 / 200000000000 : ℝ) ≤ (14973028876951 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (7174131 / 200000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau3 :
+    Real.arctan (14973028876951 / 2500000000000000 : ℝ) ≤
+      (11978279880015751031 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 2
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (11978279880015751031 / 2000000000000000000000 : ℝ)) ≤
+      Real.arctan (2500000000000000 / 14973028876951 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 14973028876951 : ℝ) =
+              (14973028876951 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau3)
+  have hz1 :
+    (2500000000000000 / 14973028876951 : ℝ) ≤ 1 / Real.sqrt (7174131 / 200000000000 : ℝ) := by
+    rw [show (2500000000000000 / 14973028876951 : ℝ) = 1 / (14973028876951 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hal4 :
+    (796870528491162609301 / 10000000000000000000000 : ℝ) ≤
+      Real.arctan (59892115507801 / 750000000000000 : ℝ) :=
+    arctan_ge_of_series (by norm_num) (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat2 :
+    Real.arctan (750000000000000 / 59892115507801 : ℝ) ≤
+      (Real.pi / 2 - (796870528491162609301 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (750000000000000 / 59892115507801 : ℝ) =
+              (59892115507801 / 750000000000000 : ℝ)
+            by
+        norm_num];
+        exact hal4)
+  have hz2 :
+    (3 / 40) / Real.sqrt (7174131 / 200000000000 : ℝ) ≤ (750000000000000 / 59892115507801 : ℝ) := by
+    rw [show
+        (750000000000000 / 59892115507801 : ℝ) =
+          (3 / 40) / (59892115507801 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (11978279880015751031 / 2000000000000000000000 : ℝ)) -
+        6 * (Real.pi / 2 - (796870528491162609301 / 10000000000000000000000 : ℝ))) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (7174131 / 200000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (7174131 / 200000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((14973028876951 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (11978279880015751031 / 2000000000000000000000 : ℝ)) -
+          6 * (Real.pi / 2 - (796870528491162609301 / 10000000000000000000000 : ℝ)))) ≤
+      Real.sqrt (7174131 / 200000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (7174131 / 200000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (7174131 / 200000000000 : ℝ))) := by
+    have h1 :=
+      mul_le_mul_of_nonpos_right hs_hi
+        (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20] :
+          (Real.pi + (Real.pi / 2 - (11978279880015751031 / 2000000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (796870528491162609301 / 10000000000000000000000 : ℝ))) ≤
+            0)
+    have h2 := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg (7174131 / 200000000000 : ℝ))
+    linarith only [h1, h2]
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_2 :
+    ((67255668762557231 / 1250000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(25855071403081196359633 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((4584535081561 / 625000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (73351245745236760791 / 10000000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (974933462669317096681 / 10000000000000000000000 : ℝ))))) ≤
+      Vfield (21522393 / 400000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (21522393 / 400000000000 : ℝ)) = Real.log (400021522393 / 400000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((21522393 / 400000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (2271522393 / 400000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (67255668762557231 / 1250000000000000000000 : ℝ) ≤
+      Real.log (400021522393 / 400000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 2
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (2271522393 / 400000000000 : ℝ) ≤
+      (-(25855071403081196359633 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show (2271522393 / 400000000000 : ℝ) = (2271522393 / 1562500000 : ℝ) / 2 ^ 8 by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2271522393 / 1562500000 : ℝ) ≤
+            (1870815808918803640367 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (73352561304973 / 10000000000000000 : ℝ) ≤ Real.sqrt (21522393 / 400000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi : Real.sqrt (21522393 / 400000000000 : ℝ) ≤ (4584535081561 / 625000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (21522393 / 400000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau5 :
+    Real.arctan (4584535081561 / 625000000000000 : ℝ) ≤
+      (73351245745236760791 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 2
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (73351245745236760791 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (625000000000000 / 4584535081561 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (625000000000000 / 4584535081561 : ℝ) =
+              (4584535081561 / 625000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau5)
+  have hz1 :
+    (625000000000000 / 4584535081561 : ℝ) ≤ 1 / Real.sqrt (21522393 / 400000000000 : ℝ) := by
+    rw [show (625000000000000 / 4584535081561 : ℝ) = 1 / (4584535081561 / 625000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hal6 :
+    (974933462669317096681 / 10000000000000000000000 : ℝ) ≤
+      Real.arctan (73352561304973 / 750000000000000 : ℝ) :=
+    arctan_ge_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat2 :
+    Real.arctan (750000000000000 / 73352561304973 : ℝ) ≤
+      (Real.pi / 2 - (974933462669317096681 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (750000000000000 / 73352561304973 : ℝ) =
+              (73352561304973 / 750000000000000 : ℝ)
+            by
+        norm_num];
+        exact hal6)
+  have hz2 :
+    (3 / 40) / Real.sqrt (21522393 / 400000000000 : ℝ) ≤ (750000000000000 / 73352561304973 : ℝ) :=
+  by
+    rw [show
+        (750000000000000 / 73352561304973 : ℝ) =
+          (3 / 40) / (73352561304973 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (73351245745236760791 / 10000000000000000000000 : ℝ)) -
+        6 * (Real.pi / 2 - (974933462669317096681 / 10000000000000000000000 : ℝ))) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (21522393 / 400000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (21522393 / 400000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((4584535081561 / 625000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (73351245745236760791 / 10000000000000000000000 : ℝ)) -
+          6 * (Real.pi / 2 - (974933462669317096681 / 10000000000000000000000 : ℝ)))) ≤
+      Real.sqrt (21522393 / 400000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (21522393 / 400000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (21522393 / 400000000000 : ℝ))) := by
+    have h1 :=
+      mul_le_mul_of_nonpos_right hs_hi
+        (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20] :
+          (Real.pi + (Real.pi / 2 - (73351245745236760791 / 10000000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (974933462669317096681 / 10000000000000000000000 : ℝ))) ≤
+            0)
+    have h2 := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg (21522393 / 400000000000 : ℝ))
+    linarith only [h1, h2]
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_3 :
+    ((358693683576464417 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(25839304827724134032909 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((5293765126897 / 625000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (16939643323085595789 / 2000000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (35142868108192574273 / 312500000000000000000 : ℝ))))) ≤
+      Vfield (7174131 / 100000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (7174131 / 100000000000 : ℝ)) = Real.log (100007174131 / 100000000000 : ℝ) := by
+    norm_num
+  have e2 :
+    Real.log ((7174131 / 100000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (569674131 / 100000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (358693683576464417 / 5000000000000000000000 : ℝ) ≤
+      Real.log (100007174131 / 100000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 2
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (569674131 / 100000000000 : ℝ) ≤
+      (-(25839304827724134032909 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show (569674131 / 100000000000 : ℝ) = (569674131 / 390625000 : ℝ) / 2 ^ 8 by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (569674131 / 390625000 : ℝ) ≤
+            (1886582384275865967091 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (84700242030349 / 10000000000000000 : ℝ) ≤ Real.sqrt (7174131 / 100000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi : Real.sqrt (7174131 / 100000000000 : ℝ) ≤ (5293765126897 / 625000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (7174131 / 100000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau7 :
+    Real.arctan (5293765126897 / 625000000000000 : ℝ) ≤
+      (16939643323085595789 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 2
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (16939643323085595789 / 2000000000000000000000 : ℝ)) ≤
+      Real.arctan (625000000000000 / 5293765126897 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (625000000000000 / 5293765126897 : ℝ) =
+              (5293765126897 / 625000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau7)
+  have hz1 : (625000000000000 / 5293765126897 : ℝ) ≤ 1 / Real.sqrt (7174131 / 100000000000 : ℝ) :=
+  by
+    rw [show (625000000000000 / 5293765126897 : ℝ) = 1 / (5293765126897 / 625000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hal8 :
+    (35142868108192574273 / 312500000000000000000 : ℝ) ≤
+      Real.arctan (84700242030349 / 750000000000000 : ℝ) :=
+    arctan_ge_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat2 :
+    Real.arctan (750000000000000 / 84700242030349 : ℝ) ≤
+      (Real.pi / 2 - (35142868108192574273 / 312500000000000000000 : ℝ)) :=
+    arctan_le_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (750000000000000 / 84700242030349 : ℝ) =
+              (84700242030349 / 750000000000000 : ℝ)
+            by
+        norm_num];
+        exact hal8)
+  have hz2 :
+    (3 / 40) / Real.sqrt (7174131 / 100000000000 : ℝ) ≤ (750000000000000 / 84700242030349 : ℝ) := by
+    rw [show
+        (750000000000000 / 84700242030349 : ℝ) =
+          (3 / 40) / (84700242030349 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (16939643323085595789 / 2000000000000000000000 : ℝ)) -
+        6 * (Real.pi / 2 - (35142868108192574273 / 312500000000000000000 : ℝ))) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (7174131 / 100000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (7174131 / 100000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((5293765126897 / 625000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (16939643323085595789 / 2000000000000000000000 : ℝ)) -
+          6 * (Real.pi / 2 - (35142868108192574273 / 312500000000000000000 : ℝ)))) ≤
+      Real.sqrt (7174131 / 100000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (7174131 / 100000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (7174131 / 100000000000 : ℝ))) := by
+    have h1 :=
+      mul_le_mul_of_nonpos_right hs_hi
+        (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20] :
+          (Real.pi + (Real.pi / 2 - (16939643323085595789 / 2000000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (35142868108192574273 / 312500000000000000000 : ℝ))) ≤
+            0)
+    have h2 := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg (7174131 / 100000000000 : ℝ))
+    linarith only [h1, h2]
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_4 :
+    ((37063408769787157 / 500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(51674418216178446788859 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((86098527861981 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (17219280094892480421 / 2000000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (1142976954465331252621 / 10000000000000000000000 : ℝ))))) ≤
+      Vfield (14825913 / 200000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (14825913 / 200000000000 : ℝ)) = Real.log (200014825913 / 200000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((14825913 / 200000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1139825913 / 200000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (37063408769787157 / 500000000000000000000 : ℝ) ≤
+      Real.log (200014825913 / 200000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 2
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1139825913 / 200000000000 : ℝ) ≤
+      (-(51674418216178446788859 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show (1139825913 / 200000000000 : ℝ) = (1139825913 / 781250000 : ℝ) / 2 ^ 8 by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1139825913 / 781250000 : ℝ) ≤
+            (3777356207821553211141 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (43049263930989 / 5000000000000000 : ℝ) ≤ Real.sqrt (14825913 / 200000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (14825913 / 200000000000 : ℝ) ≤ (86098527861981 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (14825913 / 200000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau9 :
+    Real.arctan (86098527861981 / 10000000000000000 : ℝ) ≤
+      (17219280094892480421 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 2
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (17219280094892480421 / 2000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 86098527861981 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 86098527861981 : ℝ) =
+              (86098527861981 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau9)
+  have hz1 :
+    (10000000000000000 / 86098527861981 : ℝ) ≤ 1 / Real.sqrt (14825913 / 200000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 86098527861981 : ℝ) = 1 / (86098527861981 / 10000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hal10 :
+    (1142976954465331252621 / 10000000000000000000000 : ℝ) ≤
+      Real.arctan (14349754643663 / 125000000000000 : ℝ) :=
+    arctan_ge_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat2 :
+    Real.arctan (125000000000000 / 14349754643663 : ℝ) ≤
+      (Real.pi / 2 - (1142976954465331252621 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (125000000000000 / 14349754643663 : ℝ) =
+              (14349754643663 / 125000000000000 : ℝ)
+            by
+        norm_num];
+        exact hal10)
+  have hz2 :
+    (3 / 40) / Real.sqrt (14825913 / 200000000000 : ℝ) ≤ (125000000000000 / 14349754643663 : ℝ) :=
+  by
+    rw [show
+        (125000000000000 / 14349754643663 : ℝ) =
+          (3 / 40) / (43049263930989 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (17219280094892480421 / 2000000000000000000000 : ℝ)) -
+        6 * (Real.pi / 2 - (1142976954465331252621 / 10000000000000000000000 : ℝ))) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (14825913 / 200000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (14825913 / 200000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((86098527861981 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (17219280094892480421 / 2000000000000000000000 : ℝ)) -
+          6 * (Real.pi / 2 - (1142976954465331252621 / 10000000000000000000000 : ℝ)))) ≤
+      Real.sqrt (14825913 / 200000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (14825913 / 200000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (14825913 / 200000000000 : ℝ))) := by
+    have h1 :=
+      mul_le_mul_of_nonpos_right hs_hi
+        (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20] :
+          (Real.pi + (Real.pi / 2 - (17219280094892480421 / 2000000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (1142976954465331252621 / 10000000000000000000000 : ℝ))) ≤
+            0)
+    have h2 := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg (14825913 / 200000000000 : ℝ))
+    linarith only [h1, h2]
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_5 :
+    ((196661542144736661 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(51666458509222761164599 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((88694820029133 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (44346247166374961649 / 5000000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (117713038222569168397 / 1000000000000000000000 : ℝ))))) ≤
+      Vfield (78667711 / 1000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (78667711 / 1000000000000 : ℝ)) =
+      Real.log (1000078667711 / 1000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((78667711 / 1000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (5703667711 / 1000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (196661542144736661 / 2500000000000000000000 : ℝ) ≤
+      Real.log (1000078667711 / 1000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 2
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (5703667711 / 1000000000000 : ℝ) ≤
+      (-(51666458509222761164599 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show (5703667711 / 1000000000000 : ℝ) = (5703667711 / 3906250000 : ℝ) / 2 ^ 8 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5703667711 / 3906250000 : ℝ) ≤
+            (3785315914777238835401 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (8869482002913 / 1000000000000000 : ℝ) ≤ Real.sqrt (78667711 / 1000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (78667711 / 1000000000000 : ℝ) ≤ (88694820029133 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (78667711 / 1000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau11 :
+    Real.arctan (88694820029133 / 10000000000000000 : ℝ) ≤
+      (44346247166374961649 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 2
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (44346247166374961649 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 88694820029133 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 88694820029133 : ℝ) =
+              (88694820029133 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau11)
+  have hz1 :
+    (10000000000000000 / 88694820029133 : ℝ) ≤ 1 / Real.sqrt (78667711 / 1000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 88694820029133 : ℝ) = 1 / (88694820029133 / 10000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hal12 :
+    (117713038222569168397 / 1000000000000000000000 : ℝ) ≤
+      Real.arctan (2956494000971 / 25000000000000 : ℝ) :=
+    arctan_ge_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat2 :
+    Real.arctan (25000000000000 / 2956494000971 : ℝ) ≤
+      (Real.pi / 2 - (117713038222569168397 / 1000000000000000000000 : ℝ)) :=
+    arctan_le_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (25000000000000 / 2956494000971 : ℝ) = (2956494000971 / 25000000000000 : ℝ)
+            by
+        norm_num];
+        exact hal12)
+  have hz2 :
+    (3 / 40) / Real.sqrt (78667711 / 1000000000000 : ℝ) ≤ (25000000000000 / 2956494000971 : ℝ) := by
+    rw [show
+        (25000000000000 / 2956494000971 : ℝ) = (3 / 40) / (8869482002913 / 1000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (44346247166374961649 / 5000000000000000000000 : ℝ)) -
+        6 * (Real.pi / 2 - (117713038222569168397 / 1000000000000000000000 : ℝ))) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (78667711 / 1000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (78667711 / 1000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((88694820029133 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (44346247166374961649 / 5000000000000000000000 : ℝ)) -
+          6 * (Real.pi / 2 - (117713038222569168397 / 1000000000000000000000 : ℝ)))) ≤
+      Real.sqrt (78667711 / 1000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (78667711 / 1000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (78667711 / 1000000000000 : ℝ))) := by
+    have h1 :=
+      mul_le_mul_of_nonpos_right hs_hi
+        (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20] :
+          (Real.pi + (Real.pi / 2 - (44346247166374961649 / 5000000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (117713038222569168397 / 1000000000000000000000 : ℝ))) ≤
+            0)
+    have h2 := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg (78667711 / 1000000000000 : ℝ))
+    linarith only [h1, h2]
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_6 :
+    ((85811957048696007 / 1000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(51653934194779139713727 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((92636730836101 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (92634081079116824847 / 10000000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (614466029692359913717 / 5000000000000000000000 : ℝ))))) ≤
+      Vfield (85815639 / 1000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (85815639 / 1000000000000 : ℝ)) =
+      Real.log (1000085815639 / 1000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((85815639 / 1000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (5710815639 / 1000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (85811957048696007 / 1000000000000000000000 : ℝ) ≤
+      Real.log (1000085815639 / 1000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 2
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (5710815639 / 1000000000000 : ℝ) ≤
+      (-(51653934194779139713727 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show (5710815639 / 1000000000000 : ℝ) = (5710815639 / 3906250000 : ℝ) / 2 ^ 8 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5710815639 / 3906250000 : ℝ) ≤
+            (3797840229220860286273 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (46318365418049 / 5000000000000000 : ℝ) ≤ Real.sqrt (85815639 / 1000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (85815639 / 1000000000000 : ℝ) ≤ (92636730836101 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (85815639 / 1000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau13 :
+    Real.arctan (92636730836101 / 10000000000000000 : ℝ) ≤
+      (92634081079116824847 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 2
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (92634081079116824847 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 92636730836101 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 92636730836101 : ℝ) =
+              (92636730836101 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau13)
+  have hz1 :
+    (10000000000000000 / 92636730836101 : ℝ) ≤ 1 / Real.sqrt (85815639 / 1000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 92636730836101 : ℝ) = 1 / (92636730836101 / 10000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hal14 :
+    (614466029692359913717 / 5000000000000000000000 : ℝ) ≤
+      Real.arctan (46318365418049 / 375000000000000 : ℝ) :=
+    arctan_ge_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat2 :
+    Real.arctan (375000000000000 / 46318365418049 : ℝ) ≤
+      (Real.pi / 2 - (614466029692359913717 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (375000000000000 / 46318365418049 : ℝ) =
+              (46318365418049 / 375000000000000 : ℝ)
+            by
+        norm_num];
+        exact hal14)
+  have hz2 :
+    (3 / 40) / Real.sqrt (85815639 / 1000000000000 : ℝ) ≤
+      (375000000000000 / 46318365418049 : ℝ) := by
+    rw [show
+        (375000000000000 / 46318365418049 : ℝ) =
+          (3 / 40) / (46318365418049 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (92634081079116824847 / 10000000000000000000000 : ℝ)) -
+        6 * (Real.pi / 2 - (614466029692359913717 / 5000000000000000000000 : ℝ))) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (85815639 / 1000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (85815639 / 1000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((92636730836101 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (92634081079116824847 / 10000000000000000000000 : ℝ)) -
+          6 * (Real.pi / 2 - (614466029692359913717 / 5000000000000000000000 : ℝ)))) ≤
+      Real.sqrt (85815639 / 1000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (85815639 / 1000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (85815639 / 1000000000000 : ℝ))) := by
+    have h1 :=
+      mul_le_mul_of_nonpos_right hs_hi
+        (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20] :
+          (Real.pi + (Real.pi / 2 - (92634081079116824847 / 10000000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (614466029692359913717 / 5000000000000000000000 : ℝ))) ≤
+            0)
+    have h2 := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg (85815639 / 1000000000000 : ℝ))
+    linarith only [h1, h2]
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_7 :
+    ((30107723030942933 / 312500000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(25817752990537051829671 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((19631541457563 / 2000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (49077277496392064609 / 5000000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (1301372761809135071741 / 10000000000000000000000 : ℝ))))) ≤
+      Vfield (19269871 / 200000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (19269871 / 200000000000 : ℝ)) = Real.log (200019269871 / 200000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((19269871 / 200000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1144269871 / 200000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (30107723030942933 / 312500000000000000000 : ℝ) ≤
+      Real.log (200019269871 / 200000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 2
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1144269871 / 200000000000 : ℝ) ≤
+      (-(25817752990537051829671 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show (1144269871 / 200000000000 : ℝ) = (1144269871 / 781250000 : ℝ) / 2 ^ 8 by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1144269871 / 781250000 : ℝ) ≤
+            (1908134221462948170329 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (24539426821953 / 2500000000000000 : ℝ) ≤ Real.sqrt (19269871 / 200000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (19269871 / 200000000000 : ℝ) ≤ (19631541457563 / 2000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (19269871 / 200000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau15 :
+    Real.arctan (19631541457563 / 2000000000000000 : ℝ) ≤
+      (49077277496392064609 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 2
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (49077277496392064609 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (2000000000000000 / 19631541457563 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2000000000000000 / 19631541457563 : ℝ) =
+              (19631541457563 / 2000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau15)
+  have hz1 :
+    (2000000000000000 / 19631541457563 : ℝ) ≤ 1 / Real.sqrt (19269871 / 200000000000 : ℝ) := by
+    rw [show (2000000000000000 / 19631541457563 : ℝ) = 1 / (19631541457563 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hal16 :
+    (1301372761809135071741 / 10000000000000000000000 : ℝ) ≤
+      Real.arctan (8179808940651 / 62500000000000 : ℝ) :=
+    arctan_ge_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat2 :
+    Real.arctan (62500000000000 / 8179808940651 : ℝ) ≤
+      (Real.pi / 2 - (1301372761809135071741 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (62500000000000 / 8179808940651 : ℝ) = (8179808940651 / 62500000000000 : ℝ)
+            by
+        norm_num];
+        exact hal16)
+  have hz2 :
+    (3 / 40) / Real.sqrt (19269871 / 200000000000 : ℝ) ≤ (62500000000000 / 8179808940651 : ℝ) := by
+    rw [show
+        (62500000000000 / 8179808940651 : ℝ) = (3 / 40) / (24539426821953 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (49077277496392064609 / 5000000000000000000000 : ℝ)) -
+        6 * (Real.pi / 2 - (1301372761809135071741 / 10000000000000000000000 : ℝ))) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (19269871 / 200000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (19269871 / 200000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((19631541457563 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (49077277496392064609 / 5000000000000000000000 : ℝ)) -
+          6 * (Real.pi / 2 - (1301372761809135071741 / 10000000000000000000000 : ℝ)))) ≤
+      Real.sqrt (19269871 / 200000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (19269871 / 200000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (19269871 / 200000000000 : ℝ))) := by
+    have h1 :=
+      mul_le_mul_of_nonpos_right hs_hi
+        (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20] :
+          (Real.pi + (Real.pi / 2 - (49077277496392064609 / 5000000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (1301372761809135071741 / 10000000000000000000000 : ℝ))) ≤
+            0)
+    have h2 := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg (19269871 / 200000000000 : ℝ))
+    linarith only [h1, h2]
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_8 :
+    ((278789739678365227 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(51609021536891161835761 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((105604031173059 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (52800052853726802719 / 5000000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (1398857469718843080699 / 10000000000000000000000 : ℝ))))) ≤
+      Vfield (55761057 / 500000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (55761057 / 500000000000 : ℝ)) = Real.log (500055761057 / 500000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((55761057 / 500000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (2868261057 / 500000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (278789739678365227 / 2500000000000000000000 : ℝ) ≤
+      Real.log (500055761057 / 500000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 2
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (2868261057 / 500000000000 : ℝ) ≤
+      (-(51609021536891161835761 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show (2868261057 / 500000000000 : ℝ) = (2868261057 / 1953125000 : ℝ) / 2 ^ 8 by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2868261057 / 1953125000 : ℝ) ≤
+            (3842752887108838164239 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo : (1650062987079 / 156250000000000 : ℝ) ≤ Real.sqrt (55761057 / 500000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (55761057 / 500000000000 : ℝ) ≤ (105604031173059 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (55761057 / 500000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau17 :
+    Real.arctan (105604031173059 / 10000000000000000 : ℝ) ≤
+      (52800052853726802719 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 2
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (52800052853726802719 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 105604031173059 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 105604031173059 : ℝ) =
+              (105604031173059 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau17)
+  have hz1 :
+    (10000000000000000 / 105604031173059 : ℝ) ≤ 1 / Real.sqrt (55761057 / 500000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 105604031173059 : ℝ) = 1 / (105604031173059 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hal18 :
+    (1398857469718843080699 / 10000000000000000000000 : ℝ) ≤
+      Real.arctan (550020995693 / 3906250000000 : ℝ) :=
+    arctan_ge_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat2 :
+    Real.arctan (3906250000000 / 550020995693 : ℝ) ≤
+      (Real.pi / 2 - (1398857469718843080699 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_inv (by norm_num)
+      (by
+        rw [show (1 : ℝ) / (3906250000000 / 550020995693 : ℝ) = (550020995693 / 3906250000000 : ℝ)
+            by
+        norm_num];
+        exact hal18)
+  have hz2 :
+    (3 / 40) / Real.sqrt (55761057 / 500000000000 : ℝ) ≤ (3906250000000 / 550020995693 : ℝ) := by
+    rw [show (3906250000000 / 550020995693 : ℝ) = (3 / 40) / (1650062987079 / 156250000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (52800052853726802719 / 5000000000000000000000 : ℝ)) -
+        6 * (Real.pi / 2 - (1398857469718843080699 / 10000000000000000000000 : ℝ))) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (55761057 / 500000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (55761057 / 500000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((105604031173059 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (52800052853726802719 / 5000000000000000000000 : ℝ)) -
+          6 * (Real.pi / 2 - (1398857469718843080699 / 10000000000000000000000 : ℝ)))) ≤
+      Real.sqrt (55761057 / 500000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (55761057 / 500000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (55761057 / 500000000000 : ℝ))) := by
+    have h1 :=
+      mul_le_mul_of_nonpos_right hs_hi
+        (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20] :
+          (Real.pi + (Real.pi / 2 - (52800052853726802719 / 5000000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (1398857469718843080699 / 10000000000000000000000 : ℝ))) ≤
+            0)
+    have h2 := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg (55761057 / 500000000000 : ℝ))
+    linarith only [h1, h2]
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_9 :
+    ((1333382420614834419 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(51571047996245974723013 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((57738014340641 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (115470896292829511109 / 10000000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (763841962364687947279 / 5000000000000000000000 : ℝ))))) ≤
+      Vfield (33336783 / 250000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (33336783 / 250000000000 : ℝ)) = Real.log (250033336783 / 250000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((33336783 / 250000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1439586783 / 250000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1333382420614834419 / 10000000000000000000000 : ℝ) ≤
+      Real.log (250033336783 / 250000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 2
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1439586783 / 250000000000 : ℝ) ≤
+      (-(51571047996245974723013 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show (1439586783 / 250000000000 : ℝ) = (1439586783 / 976562500 : ℝ) / 2 ^ 8 by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1439586783 / 976562500 : ℝ) ≤
+            (3880726427754025276987 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (115476028681279 / 10000000000000000 : ℝ) ≤ Real.sqrt (33336783 / 250000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (33336783 / 250000000000 : ℝ) ≤ (57738014340641 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (33336783 / 250000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau19 :
+    Real.arctan (57738014340641 / 5000000000000000 : ℝ) ≤
+      (115470896292829511109 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 2
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (115470896292829511109 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (5000000000000000 / 57738014340641 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 57738014340641 : ℝ) =
+              (57738014340641 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau19)
+  have hz1 :
+    (5000000000000000 / 57738014340641 : ℝ) ≤ 1 / Real.sqrt (33336783 / 250000000000 : ℝ) := by
+    rw [show (5000000000000000 / 57738014340641 : ℝ) = 1 / (57738014340641 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hal20 :
+    (763841962364687947279 / 5000000000000000000000 : ℝ) ≤
+      Real.arctan (115476028681279 / 750000000000000 : ℝ) :=
+    arctan_ge_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat2 :
+    Real.arctan (750000000000000 / 115476028681279 : ℝ) ≤
+      (Real.pi / 2 - (763841962364687947279 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (750000000000000 / 115476028681279 : ℝ) =
+              (115476028681279 / 750000000000000 : ℝ)
+            by
+        norm_num];
+        exact hal20)
+  have hz2 :
+    (3 / 40) / Real.sqrt (33336783 / 250000000000 : ℝ) ≤
+      (750000000000000 / 115476028681279 : ℝ) := by
+    rw [show
+        (750000000000000 / 115476028681279 : ℝ) =
+          (3 / 40) / (115476028681279 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (115470896292829511109 / 10000000000000000000000 : ℝ)) -
+        6 * (Real.pi / 2 - (763841962364687947279 / 5000000000000000000000 : ℝ))) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (33336783 / 250000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (33336783 / 250000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((57738014340641 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (115470896292829511109 / 10000000000000000000000 : ℝ)) -
+          6 * (Real.pi / 2 - (763841962364687947279 / 5000000000000000000000 : ℝ)))) ≤
+      Real.sqrt (33336783 / 250000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (33336783 / 250000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (33336783 / 250000000000 : ℝ))) := by
+    have h1 :=
+      mul_le_mul_of_nonpos_right hs_hi
+        (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20] :
+          (Real.pi + (Real.pi / 2 - (115470896292829511109 / 10000000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (763841962364687947279 / 5000000000000000000000 : ℝ))) ≤
+            0)
+    have h2 := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg (33336783 / 250000000000 : ℝ))
+    linarith only [h1, h2]
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_10 :
+    ((66033623550755633 / 400000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(51516061139574022265869 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((128490344384317 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (25696654786369851871 / 2000000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (1696732463001094446147 / 10000000000000000000000 : ℝ))))) ≤
+      Vfield (82548843 / 500000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (82548843 / 500000000000 : ℝ)) = Real.log (500082548843 / 500000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((82548843 / 500000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (2895048843 / 500000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (66033623550755633 / 400000000000000000000 : ℝ) ≤
+      Real.log (500082548843 / 500000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 2
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (2895048843 / 500000000000 : ℝ) ≤
+      (-(51516061139574022265869 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show (2895048843 / 500000000000 : ℝ) = (2895048843 / 1953125000 : ℝ) / 2 ^ 8 by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2895048843 / 1953125000 : ℝ) ≤
+            (3935713284425977734131 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (64245172192157 / 5000000000000000 : ℝ) ≤ Real.sqrt (82548843 / 500000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (82548843 / 500000000000 : ℝ) ≤ (128490344384317 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (82548843 / 500000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau21 :
+    Real.arctan (128490344384317 / 10000000000000000 : ℝ) ≤
+      (25696654786369851871 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 2
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (25696654786369851871 / 2000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 128490344384317 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 128490344384317 : ℝ) =
+              (128490344384317 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau21)
+  have hz1 :
+    (10000000000000000 / 128490344384317 : ℝ) ≤ 1 / Real.sqrt (82548843 / 500000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 128490344384317 : ℝ) = 1 / (128490344384317 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hal22 :
+    (1696732463001094446147 / 10000000000000000000000 : ℝ) ≤
+      Real.arctan (64245172192157 / 375000000000000 : ℝ) :=
+    arctan_ge_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat2 :
+    Real.arctan (375000000000000 / 64245172192157 : ℝ) ≤
+      (Real.pi / 2 - (1696732463001094446147 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (375000000000000 / 64245172192157 : ℝ) =
+              (64245172192157 / 375000000000000 : ℝ)
+            by
+        norm_num];
+        exact hal22)
+  have hz2 :
+    (3 / 40) / Real.sqrt (82548843 / 500000000000 : ℝ) ≤ (375000000000000 / 64245172192157 : ℝ) :=
+  by
+    rw [show
+        (375000000000000 / 64245172192157 : ℝ) =
+          (3 / 40) / (64245172192157 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (25696654786369851871 / 2000000000000000000000 : ℝ)) -
+        6 * (Real.pi / 2 - (1696732463001094446147 / 10000000000000000000000 : ℝ))) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (82548843 / 500000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (82548843 / 500000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((128490344384317 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (25696654786369851871 / 2000000000000000000000 : ℝ)) -
+          6 * (Real.pi / 2 - (1696732463001094446147 / 10000000000000000000000 : ℝ)))) ≤
+      Real.sqrt (82548843 / 500000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (82548843 / 500000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (82548843 / 500000000000 : ℝ))) := by
+    have h1 :=
+      mul_le_mul_of_nonpos_right hs_hi
+        (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20] :
+          (Real.pi + (Real.pi / 2 - (25696654786369851871 / 2000000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (1696732463001094446147 / 10000000000000000000000 : ℝ))) ≤
+            0)
+    have h2 := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg (82548843 / 500000000000 : ℝ))
+    linarith only [h1, h2]
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_11 :
+    ((1060918377258494863 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(51435029869943759185693 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((18209123228481 / 1250000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (145662682903283762149 / 10000000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (1918420011923721109387 / 10000000000000000000000 : ℝ))))) ≤
+      Vfield (53051547 / 250000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (53051547 / 250000000000 : ℝ)) = Real.log (250053051547 / 250000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((53051547 / 250000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1459301547 / 250000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1060918377258494863 / 5000000000000000000000 : ℝ) ≤
+      Real.log (250053051547 / 250000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 2
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1459301547 / 250000000000 : ℝ) ≤
+      (-(51435029869943759185693 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show (1459301547 / 250000000000 : ℝ) = (1459301547 / 976562500 : ℝ) / 2 ^ 8 by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1459301547 / 976562500 : ℝ) ≤
+            (4016744554056240814307 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (29134597165569 / 2000000000000000 : ℝ) ≤ Real.sqrt (53051547 / 250000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (53051547 / 250000000000 : ℝ) ≤ (18209123228481 / 1250000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (53051547 / 250000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau23 :
+    Real.arctan (18209123228481 / 1250000000000000 : ℝ) ≤
+      (145662682903283762149 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 2
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (145662682903283762149 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (1250000000000000 / 18209123228481 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1250000000000000 / 18209123228481 : ℝ) =
+              (18209123228481 / 1250000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau23)
+  have hz1 :
+    (1250000000000000 / 18209123228481 : ℝ) ≤ 1 / Real.sqrt (53051547 / 250000000000 : ℝ) := by
+    rw [show (1250000000000000 / 18209123228481 : ℝ) = 1 / (18209123228481 / 1250000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hal24 :
+    (1918420011923721109387 / 10000000000000000000000 : ℝ) ≤
+      Real.arctan (9711532388523 / 50000000000000 : ℝ) :=
+    arctan_ge_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat2 :
+    Real.arctan (50000000000000 / 9711532388523 : ℝ) ≤
+      (Real.pi / 2 - (1918420011923721109387 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (50000000000000 / 9711532388523 : ℝ) = (9711532388523 / 50000000000000 : ℝ)
+            by
+        norm_num];
+        exact hal24)
+  have hz2 :
+    (3 / 40) / Real.sqrt (53051547 / 250000000000 : ℝ) ≤ (50000000000000 / 9711532388523 : ℝ) := by
+    rw [show
+        (50000000000000 / 9711532388523 : ℝ) = (3 / 40) / (29134597165569 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (145662682903283762149 / 10000000000000000000000 : ℝ)) -
+        6 * (Real.pi / 2 - (1918420011923721109387 / 10000000000000000000000 : ℝ))) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (53051547 / 250000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (53051547 / 250000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((18209123228481 / 1250000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (145662682903283762149 / 10000000000000000000000 : ℝ)) -
+          6 * (Real.pi / 2 - (1918420011923721109387 / 10000000000000000000000 : ℝ)))) ≤
+      Real.sqrt (53051547 / 250000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (53051547 / 250000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (53051547 / 250000000000 : ℝ))) := by
+    have h1 :=
+      mul_le_mul_of_nonpos_right hs_hi
+        (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20] :
+          (Real.pi + (Real.pi / 2 - (145662682903283762149 / 10000000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (1918420011923721109387 / 10000000000000000000000 : ℝ))) ≤
+            0)
+    have h2 := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg (53051547 / 250000000000 : ℝ))
+    linarith only [h1, h2]
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_12 :
+    ((2480279280302765263 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(12843449268270218651961 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((9843674394551 / 625000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (39371442317394135319 / 2500000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (2069906494283519593871 / 10000000000000000000000 : ℝ))))) ≤
+      Vfield (496117379 / 2000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (496117379 / 2000000000000 : ℝ)) =
+      Real.log (2000496117379 / 2000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((496117379 / 2000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (11746117379 / 2000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2480279280302765263 / 10000000000000000000000 : ℝ) ≤
+      Real.log (2000496117379 / 2000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 2
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (11746117379 / 2000000000000 : ℝ) ≤
+      (-(12843449268270218651961 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show (11746117379 / 2000000000000 : ℝ) = (11746117379 / 7812500000 : ℝ) / 2 ^ 8 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11746117379 / 7812500000 : ℝ) ≤
+            (1019494337729781348039 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (157498790312813 / 10000000000000000 : ℝ) ≤ Real.sqrt (496117379 / 2000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (496117379 / 2000000000000 : ℝ) ≤ (9843674394551 / 625000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (496117379 / 2000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau25 :
+    Real.arctan (9843674394551 / 625000000000000 : ℝ) ≤
+      (39371442317394135319 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 2
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (39371442317394135319 / 2500000000000000000000 : ℝ)) ≤
+      Real.arctan (625000000000000 / 9843674394551 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (625000000000000 / 9843674394551 : ℝ) =
+              (9843674394551 / 625000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau25)
+  have hz1 :
+    (625000000000000 / 9843674394551 : ℝ) ≤ 1 / Real.sqrt (496117379 / 2000000000000 : ℝ) := by
+    rw [show (625000000000000 / 9843674394551 : ℝ) = 1 / (9843674394551 / 625000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hal26 :
+    (2069906494283519593871 / 10000000000000000000000 : ℝ) ≤
+      Real.arctan (157498790312813 / 750000000000000 : ℝ) :=
+    arctan_ge_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat2 :
+    Real.arctan (750000000000000 / 157498790312813 : ℝ) ≤
+      (Real.pi / 2 - (2069906494283519593871 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (750000000000000 / 157498790312813 : ℝ) =
+              (157498790312813 / 750000000000000 : ℝ)
+            by
+        norm_num];
+        exact hal26)
+  have hz2 :
+    (3 / 40) / Real.sqrt (496117379 / 2000000000000 : ℝ) ≤
+      (750000000000000 / 157498790312813 : ℝ) := by
+    rw [show
+        (750000000000000 / 157498790312813 : ℝ) =
+          (3 / 40) / (157498790312813 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (39371442317394135319 / 2500000000000000000000 : ℝ)) -
+        6 * (Real.pi / 2 - (2069906494283519593871 / 10000000000000000000000 : ℝ))) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (496117379 / 2000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (496117379 / 2000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((9843674394551 / 625000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (39371442317394135319 / 2500000000000000000000 : ℝ)) -
+          6 * (Real.pi / 2 - (2069906494283519593871 / 10000000000000000000000 : ℝ)))) ≤
+      Real.sqrt (496117379 / 2000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (496117379 / 2000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (496117379 / 2000000000000 : ℝ))) := by
+    have h1 :=
+      mul_le_mul_of_nonpos_right hs_hi
+        (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20] :
+          (Real.pi + (Real.pi / 2 - (39371442317394135319 / 2500000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (2069906494283519593871 / 10000000000000000000000 : ℝ))) ≤
+            0)
+    have h2 := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg (496117379 / 2000000000000 : ℝ))
+    linarith only [h1, h2]
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/V01.lean b/LeanPool/Zeta5Irrational/Table/V01.lean
new file mode 100644
index 0000000000..7d2b95c047
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/V01.lean
@@ -0,0 +1,1411 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Energy
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # V01: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_13 :
+    ((2838708958444624399 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(51312936940985871685561 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((84248322090117 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (168480700868393777449 / 10000000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (138120555504676214793 / 625000000000000000000 : ℝ))))) ≤
+      Vfield (283911191 / 1000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (283911191 / 1000000000000 : ℝ)) =
+      Real.log (1000283911191 / 1000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((283911191 / 1000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (5908911191 / 1000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2838708958444624399 / 10000000000000000000000 : ℝ) ≤
+      Real.log (1000283911191 / 1000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 2
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (5908911191 / 1000000000000 : ℝ) ≤
+      (-(51312936940985871685561 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show (5908911191 / 1000000000000 : ℝ) = (5908911191 / 3906250000 : ℝ) / 2 ^ 8 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5908911191 / 3906250000 : ℝ) ≤
+            (4138837483014128314439 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (168496644180231 / 10000000000000000 : ℝ) ≤ Real.sqrt (283911191 / 1000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (283911191 / 1000000000000 : ℝ) ≤ (84248322090117 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (283911191 / 1000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau27 :
+    Real.arctan (84248322090117 / 5000000000000000 : ℝ) ≤
+      (168480700868393777449 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 2
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (168480700868393777449 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (5000000000000000 / 84248322090117 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 84248322090117 : ℝ) =
+              (84248322090117 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau27)
+  have hz1 :
+    (5000000000000000 / 84248322090117 : ℝ) ≤ 1 / Real.sqrt (283911191 / 1000000000000 : ℝ) := by
+    rw [show (5000000000000000 / 84248322090117 : ℝ) = 1 / (84248322090117 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hal28 :
+    (138120555504676214793 / 625000000000000000000 : ℝ) ≤
+      Real.arctan (56165548060077 / 250000000000000 : ℝ) :=
+    arctan_ge_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat2 :
+    Real.arctan (250000000000000 / 56165548060077 : ℝ) ≤
+      (Real.pi / 2 - (138120555504676214793 / 625000000000000000000 : ℝ)) :=
+    arctan_le_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (250000000000000 / 56165548060077 : ℝ) =
+              (56165548060077 / 250000000000000 : ℝ)
+            by
+        norm_num];
+        exact hal28)
+  have hz2 :
+    (3 / 40) / Real.sqrt (283911191 / 1000000000000 : ℝ) ≤
+      (250000000000000 / 56165548060077 : ℝ) := by
+    rw [show
+        (250000000000000 / 56165548060077 : ℝ) =
+          (3 / 40) / (168496644180231 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (168480700868393777449 / 10000000000000000000000 : ℝ)) -
+        6 * (Real.pi / 2 - (138120555504676214793 / 625000000000000000000 : ℝ))) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (283911191 / 1000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (283911191 / 1000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((84248322090117 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (168480700868393777449 / 10000000000000000000000 : ℝ)) -
+          6 * (Real.pi / 2 - (138120555504676214793 / 625000000000000000000 : ℝ)))) ≤
+      Real.sqrt (283911191 / 1000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (283911191 / 1000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (283911191 / 1000000000000 : ℝ))) := by
+    have h1 :=
+      mul_le_mul_of_nonpos_right hs_hi
+        (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20] :
+          (Real.pi + (Real.pi / 2 - (168480700868393777449 / 10000000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (138120555504676214793 / 625000000000000000000 : ℝ))) ≤
+            0)
+    have h2 := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg (283911191 / 1000000000000 : ℝ))
+    linarith only [h1, h2]
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_14 :
+    ((3400600750179924673 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(51218264559151743459197 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((36884571408653 / 2000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (46100488200992503859 / 2500000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (37673974722398950663 / 156250000000000000000 : ℝ))))) ≤
+      Vfield (170058951 / 500000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (170058951 / 500000000000 : ℝ)) = Real.log (500170058951 / 500000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((170058951 / 500000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (2982558951 / 500000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (3400600750179924673 / 10000000000000000000000 : ℝ) ≤
+      Real.log (500170058951 / 500000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 2
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (2982558951 / 500000000000 : ℝ) ≤
+      (-(51218264559151743459197 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show (2982558951 / 500000000000 : ℝ) = (2982558951 / 1953125000 : ℝ) / 2 ^ 8 by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2982558951 / 1953125000 : ℝ) ≤
+            (4233509864848256540803 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (92211428521631 / 5000000000000000 : ℝ) ≤ Real.sqrt (170058951 / 500000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (170058951 / 500000000000 : ℝ) ≤ (36884571408653 / 2000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (170058951 / 500000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau29 :
+    Real.arctan (36884571408653 / 2000000000000000 : ℝ) ≤
+      (46100488200992503859 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 2
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (46100488200992503859 / 2500000000000000000000 : ℝ)) ≤
+      Real.arctan (2000000000000000 / 36884571408653 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2000000000000000 / 36884571408653 : ℝ) =
+              (36884571408653 / 2000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau29)
+  have hz1 :
+    (2000000000000000 / 36884571408653 : ℝ) ≤ 1 / Real.sqrt (170058951 / 500000000000 : ℝ) := by
+    rw [show (2000000000000000 / 36884571408653 : ℝ) = 1 / (36884571408653 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hal30 :
+    (37673974722398950663 / 156250000000000000000 : ℝ) ≤
+      Real.arctan (92211428521631 / 375000000000000 : ℝ) :=
+    arctan_ge_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat2 :
+    Real.arctan (375000000000000 / 92211428521631 : ℝ) ≤
+      (Real.pi / 2 - (37673974722398950663 / 156250000000000000000 : ℝ)) :=
+    arctan_le_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (375000000000000 / 92211428521631 : ℝ) =
+              (92211428521631 / 375000000000000 : ℝ)
+            by
+        norm_num];
+        exact hal30)
+  have hz2 :
+    (3 / 40) / Real.sqrt (170058951 / 500000000000 : ℝ) ≤
+      (375000000000000 / 92211428521631 : ℝ) := by
+    rw [show
+        (375000000000000 / 92211428521631 : ℝ) =
+          (3 / 40) / (92211428521631 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (46100488200992503859 / 2500000000000000000000 : ℝ)) -
+        6 * (Real.pi / 2 - (37673974722398950663 / 156250000000000000000 : ℝ))) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (170058951 / 500000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (170058951 / 500000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((36884571408653 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (46100488200992503859 / 2500000000000000000000 : ℝ)) -
+          6 * (Real.pi / 2 - (37673974722398950663 / 156250000000000000000 : ℝ)))) ≤
+      Real.sqrt (170058951 / 500000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (170058951 / 500000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (170058951 / 500000000000 : ℝ))) := by
+    have h1 :=
+      mul_le_mul_of_nonpos_right hs_hi
+        (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20] :
+          (Real.pi + (Real.pi / 2 - (46100488200992503859 / 2500000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (37673974722398950663 / 156250000000000000000 : ℝ))) ≤
+            0)
+    have h2 := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg (170058951 / 500000000000 : ℝ))
+    linarith only [h1, h2]
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_15 :
+    ((396246097145057287 / 1000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(25562240032020096608843 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((24884879099817 / 1250000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (99526369538609389173 / 5000000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (324319511825085218099 / 1250000000000000000000 : ℝ))))) ≤
+      Vfield (396324613 / 1000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (396324613 / 1000000000000 : ℝ)) =
+      Real.log (1000396324613 / 1000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((396324613 / 1000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (6021324613 / 1000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (396246097145057287 / 1000000000000000000000 : ℝ) ≤
+      Real.log (1000396324613 / 1000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 2
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (6021324613 / 1000000000000 : ℝ) ≤
+      (-(25562240032020096608843 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show (6021324613 / 1000000000000 : ℝ) = (6021324613 / 3906250000 : ℝ) / 2 ^ 8 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6021324613 / 3906250000 : ℝ) ≤
+            (2163647179979903391157 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (199079032798533 / 10000000000000000 : ℝ) ≤ Real.sqrt (396324613 / 1000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (396324613 / 1000000000000 : ℝ) ≤ (24884879099817 / 1250000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (396324613 / 1000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau31 :
+    Real.arctan (24884879099817 / 1250000000000000 : ℝ) ≤
+      (99526369538609389173 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 2
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (99526369538609389173 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (1250000000000000 / 24884879099817 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1250000000000000 / 24884879099817 : ℝ) =
+              (24884879099817 / 1250000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau31)
+  have hz1 :
+    (1250000000000000 / 24884879099817 : ℝ) ≤ 1 / Real.sqrt (396324613 / 1000000000000 : ℝ) := by
+    rw [show (1250000000000000 / 24884879099817 : ℝ) = 1 / (24884879099817 / 1250000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hal32 :
+    (324319511825085218099 / 1250000000000000000000 : ℝ) ≤
+      Real.arctan (66359677599511 / 250000000000000 : ℝ) :=
+    arctan_ge_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat2 :
+    Real.arctan (250000000000000 / 66359677599511 : ℝ) ≤
+      (Real.pi / 2 - (324319511825085218099 / 1250000000000000000000 : ℝ)) :=
+    arctan_le_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (250000000000000 / 66359677599511 : ℝ) =
+              (66359677599511 / 250000000000000 : ℝ)
+            by
+        norm_num];
+        exact hal32)
+  have hz2 :
+    (3 / 40) / Real.sqrt (396324613 / 1000000000000 : ℝ) ≤
+      (250000000000000 / 66359677599511 : ℝ) := by
+    rw [show
+        (250000000000000 / 66359677599511 : ℝ) =
+          (3 / 40) / (199079032798533 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (99526369538609389173 / 5000000000000000000000 : ℝ)) -
+        6 * (Real.pi / 2 - (324319511825085218099 / 1250000000000000000000 : ℝ))) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (396324613 / 1000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (396324613 / 1000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((24884879099817 / 1250000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (99526369538609389173 / 5000000000000000000000 : ℝ)) -
+          6 * (Real.pi / 2 - (324319511825085218099 / 1250000000000000000000 : ℝ)))) ≤
+      Real.sqrt (396324613 / 1000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (396324613 / 1000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (396324613 / 1000000000000 : ℝ))) := by
+    have h1 :=
+      mul_le_mul_of_nonpos_right hs_hi
+        (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20] :
+          (Real.pi + (Real.pi / 2 - (99526369538609389173 / 5000000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (324319511825085218099 / 1250000000000000000000 : ℝ))) ≤
+            0)
+    have h2 := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg (396324613 / 1000000000000 : ℝ))
+    linarith only [h1, h2]
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_16 :
+    ((2435712931404190679 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(25487292403074346048679 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((110369975480201 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (27588013595377272983 / 1250000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (143118806689254327513 / 500000000000000000000 : ℝ))))) ≤
+      Vfield (974522519 / 2000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (974522519 / 2000000000000 : ℝ)) =
+      Real.log (2000974522519 / 2000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((974522519 / 2000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (12224522519 / 2000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2435712931404190679 / 5000000000000000000000 : ℝ) ≤
+      Real.log (2000974522519 / 2000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 2
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (12224522519 / 2000000000000 : ℝ) ≤
+      (-(25487292403074346048679 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show (12224522519 / 2000000000000 : ℝ) = (12224522519 / 7812500000 : ℝ) / 2 ^ 8 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12224522519 / 7812500000 : ℝ) ≤
+            (2238594808925653951321 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (220739950960399 / 10000000000000000 : ℝ) ≤ Real.sqrt (974522519 / 2000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (974522519 / 2000000000000 : ℝ) ≤ (110369975480201 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (974522519 / 2000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau33 :
+    Real.arctan (110369975480201 / 5000000000000000 : ℝ) ≤
+      (27588013595377272983 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 2
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (27588013595377272983 / 1250000000000000000000 : ℝ)) ≤
+      Real.arctan (5000000000000000 / 110369975480201 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 110369975480201 : ℝ) =
+              (110369975480201 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau33)
+  have hz1 :
+    (5000000000000000 / 110369975480201 : ℝ) ≤ 1 / Real.sqrt (974522519 / 2000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 110369975480201 : ℝ) = 1 / (110369975480201 / 5000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hal34 :
+    (143118806689254327513 / 500000000000000000000 : ℝ) ≤
+      Real.arctan (220739950960399 / 750000000000000 : ℝ) :=
+    arctan_ge_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat2 :
+    Real.arctan (750000000000000 / 220739950960399 : ℝ) ≤
+      (Real.pi / 2 - (143118806689254327513 / 500000000000000000000 : ℝ)) :=
+    arctan_le_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (750000000000000 / 220739950960399 : ℝ) =
+              (220739950960399 / 750000000000000 : ℝ)
+            by
+        norm_num];
+        exact hal34)
+  have hz2 :
+    (3 / 40) / Real.sqrt (974522519 / 2000000000000 : ℝ) ≤
+      (750000000000000 / 220739950960399 : ℝ) := by
+    rw [show
+        (750000000000000 / 220739950960399 : ℝ) =
+          (3 / 40) / (220739950960399 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (27588013595377272983 / 1250000000000000000000 : ℝ)) -
+        6 * (Real.pi / 2 - (143118806689254327513 / 500000000000000000000 : ℝ))) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (974522519 / 2000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (974522519 / 2000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((110369975480201 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (27588013595377272983 / 1250000000000000000000 : ℝ)) -
+          6 * (Real.pi / 2 - (143118806689254327513 / 500000000000000000000 : ℝ)))) ≤
+      Real.sqrt (974522519 / 2000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (974522519 / 2000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (974522519 / 2000000000000 : ℝ))) := by
+    have h1 :=
+      mul_le_mul_of_nonpos_right hs_hi
+        (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20] :
+          (Real.pi + (Real.pi / 2 - (27588013595377272983 / 1250000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (143118806689254327513 / 500000000000000000000 : ℝ))) ≤
+            0)
+    have h2 := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg (974522519 / 2000000000000 : ℝ))
+    linarith only [h1, h2]
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_17 :
+    ((2890154069979049949 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(25413451632679054566369 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((30057182637849 / 1250000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (120205566586796683229 / 5000000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (3102561376671980779519 / 10000000000000000000000 : ℝ))))) ≤
+      Vfield (289098953 / 500000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (289098953 / 500000000000 : ℝ)) = Real.log (500289098953 / 500000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((289098953 / 500000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (3101598953 / 500000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2890154069979049949 / 5000000000000000000000 : ℝ) ≤
+      Real.log (500289098953 / 500000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 2
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (3101598953 / 500000000000 : ℝ) ≤
+      (-(25413451632679054566369 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show (3101598953 / 500000000000 : ℝ) = (3101598953 / 1953125000 : ℝ) / 2 ^ 8 by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3101598953 / 1953125000 : ℝ) ≤
+            (2312435579320945433631 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (240457461102789 / 10000000000000000 : ℝ) ≤ Real.sqrt (289098953 / 500000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (289098953 / 500000000000 : ℝ) ≤ (30057182637849 / 1250000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (289098953 / 500000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau35 :
+    Real.arctan (30057182637849 / 1250000000000000 : ℝ) ≤
+      (120205566586796683229 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 2
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (120205566586796683229 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (1250000000000000 / 30057182637849 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1250000000000000 / 30057182637849 : ℝ) =
+              (30057182637849 / 1250000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau35)
+  have hz1 :
+    (1250000000000000 / 30057182637849 : ℝ) ≤ 1 / Real.sqrt (289098953 / 500000000000 : ℝ) := by
+    rw [show (1250000000000000 / 30057182637849 : ℝ) = 1 / (30057182637849 / 1250000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hal36 :
+    (3102561376671980779519 / 10000000000000000000000 : ℝ) ≤
+      Real.arctan (80152487034263 / 250000000000000 : ℝ) :=
+    arctan_ge_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat2 :
+    Real.arctan (250000000000000 / 80152487034263 : ℝ) ≤
+      (Real.pi / 2 - (3102561376671980779519 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (250000000000000 / 80152487034263 : ℝ) =
+              (80152487034263 / 250000000000000 : ℝ)
+            by
+        norm_num];
+        exact hal36)
+  have hz2 :
+    (3 / 40) / Real.sqrt (289098953 / 500000000000 : ℝ) ≤
+      (250000000000000 / 80152487034263 : ℝ) := by
+    rw [show
+        (250000000000000 / 80152487034263 : ℝ) =
+          (3 / 40) / (240457461102789 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (120205566586796683229 / 5000000000000000000000 : ℝ)) -
+        6 * (Real.pi / 2 - (3102561376671980779519 / 10000000000000000000000 : ℝ))) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (289098953 / 500000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (289098953 / 500000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((30057182637849 / 1250000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (120205566586796683229 / 5000000000000000000000 : ℝ)) -
+          6 * (Real.pi / 2 - (3102561376671980779519 / 10000000000000000000000 : ℝ)))) ≤
+      Real.sqrt (289098953 / 500000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (289098953 / 500000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (289098953 / 500000000000 : ℝ))) := by
+    have h1 :=
+      mul_le_mul_of_nonpos_right hs_hi
+        (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20] :
+          (Real.pi + (Real.pi / 2 - (120205566586796683229 / 5000000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (3102561376671980779519 / 10000000000000000000000 : ℝ))) ≤
+            0)
+    have h2 := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg (289098953 / 500000000000 : ℝ))
+    linarith only [h1, h2]
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_18 :
+    ((7296953385895796397 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(50585194098875661653047 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((270178021126813 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (54022462008273651983 / 2000000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (691531386986268494069 / 2000000000000000000000 : ℝ))))) ≤
+      Vfield (729961631 / 1000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (729961631 / 1000000000000 : ℝ)) =
+      Real.log (1000729961631 / 1000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((729961631 / 1000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (6354961631 / 1000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (7296953385895796397 / 10000000000000000000000 : ℝ) ≤
+      Real.log (1000729961631 / 1000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 2
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (6354961631 / 1000000000000 : ℝ) ≤
+      (-(50585194098875661653047 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show (6354961631 / 1000000000000 : ℝ) = (6354961631 / 3906250000 : ℝ) / 2 ^ 8 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6354961631 / 3906250000 : ℝ) ≤
+            (4866580325124338346953 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (27017802112681 / 1000000000000000 : ℝ) ≤ Real.sqrt (729961631 / 1000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (729961631 / 1000000000000 : ℝ) ≤ (270178021126813 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (729961631 / 1000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau37 :
+    Real.arctan (270178021126813 / 10000000000000000 : ℝ) ≤
+      (54022462008273651983 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 2
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (54022462008273651983 / 2000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 270178021126813 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 270178021126813 : ℝ) =
+              (270178021126813 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau37)
+  have hz1 :
+    (10000000000000000 / 270178021126813 : ℝ) ≤ 1 / Real.sqrt (729961631 / 1000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 270178021126813 : ℝ) = 1 / (270178021126813 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hal38 :
+    (691531386986268494069 / 2000000000000000000000 : ℝ) ≤
+      Real.arctan (27017802112681 / 75000000000000 : ℝ) :=
+    arctan_ge_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat2 :
+    Real.arctan (75000000000000 / 27017802112681 : ℝ) ≤
+      (Real.pi / 2 - (691531386986268494069 / 2000000000000000000000 : ℝ)) :=
+    arctan_le_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (75000000000000 / 27017802112681 : ℝ) =
+              (27017802112681 / 75000000000000 : ℝ)
+            by
+        norm_num];
+        exact hal38)
+  have hz2 :
+    (3 / 40) / Real.sqrt (729961631 / 1000000000000 : ℝ) ≤
+      (75000000000000 / 27017802112681 : ℝ) := by
+    rw [show
+        (75000000000000 / 27017802112681 : ℝ) = (3 / 40) / (27017802112681 / 1000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (54022462008273651983 / 2000000000000000000000 : ℝ)) -
+        6 * (Real.pi / 2 - (691531386986268494069 / 2000000000000000000000 : ℝ))) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (729961631 / 1000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (729961631 / 1000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((270178021126813 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (54022462008273651983 / 2000000000000000000000 : ℝ)) -
+          6 * (Real.pi / 2 - (691531386986268494069 / 2000000000000000000000 : ℝ)))) ≤
+      Real.sqrt (729961631 / 1000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (729961631 / 1000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (729961631 / 1000000000000 : ℝ))) := by
+    have h1 :=
+      mul_le_mul_of_nonpos_right hs_hi
+        (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20] :
+          (Real.pi + (Real.pi / 2 - (54022462008273651983 / 2000000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (691531386986268494069 / 2000000000000000000000 : ℝ))) ≤
+            0)
+    have h2 := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg (729961631 / 1000000000000 : ℝ))
+    linarith only [h1, h2]
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_19 :
+    ((1611037951921033759 / 2000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(50466495663078367309339 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((283873826461689 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (283797610683746791347 / 10000000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (3618342583696765969299 / 10000000000000000000000 : ℝ))))) ≤
+      Vfield (1611686987 / 2000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1611686987 / 2000000000000 : ℝ)) =
+      Real.log (2001611686987 / 2000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1611686987 / 2000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (12861686987 / 2000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1611037951921033759 / 2000000000000000000000 : ℝ) ≤
+      Real.log (2001611686987 / 2000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 2
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (12861686987 / 2000000000000 : ℝ) ≤
+      (-(50466495663078367309339 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show (12861686987 / 2000000000000 : ℝ) = (12861686987 / 7812500000 : ℝ) / 2 ^ 8 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12861686987 / 7812500000 : ℝ) ≤
+            (4985278760921632690661 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (141936913230843 / 5000000000000000 : ℝ) ≤ Real.sqrt (1611686987 / 2000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1611686987 / 2000000000000 : ℝ) ≤ (283873826461689 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1611686987 / 2000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau39 :
+    Real.arctan (283873826461689 / 10000000000000000 : ℝ) ≤
+      (283797610683746791347 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (283797610683746791347 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 283873826461689 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 283873826461689 : ℝ) =
+              (283873826461689 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau39)
+  have hz1 :
+    (10000000000000000 / 283873826461689 : ℝ) ≤ 1 / Real.sqrt (1611686987 / 2000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 283873826461689 : ℝ) = 1 / (283873826461689 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hal40 :
+    (3618342583696765969299 / 10000000000000000000000 : ℝ) ≤
+      Real.arctan (47312304410281 / 125000000000000 : ℝ) :=
+    arctan_ge_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat2 :
+    Real.arctan (125000000000000 / 47312304410281 : ℝ) ≤
+      (Real.pi / 2 - (3618342583696765969299 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (125000000000000 / 47312304410281 : ℝ) =
+              (47312304410281 / 125000000000000 : ℝ)
+            by
+        norm_num];
+        exact hal40)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1611686987 / 2000000000000 : ℝ) ≤
+      (125000000000000 / 47312304410281 : ℝ) := by
+    rw [show
+        (125000000000000 / 47312304410281 : ℝ) =
+          (3 / 40) / (141936913230843 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (283797610683746791347 / 10000000000000000000000 : ℝ)) -
+        6 * (Real.pi / 2 - (3618342583696765969299 / 10000000000000000000000 : ℝ))) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1611686987 / 2000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1611686987 / 2000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((283873826461689 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (283797610683746791347 / 10000000000000000000000 : ℝ)) -
+          6 * (Real.pi / 2 - (3618342583696765969299 / 10000000000000000000000 : ℝ)))) ≤
+      Real.sqrt (1611686987 / 2000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1611686987 / 2000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1611686987 / 2000000000000 : ℝ))) := by
+    have h1 :=
+      mul_le_mul_of_nonpos_right hs_hi
+        (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20] :
+          (Real.pi + (Real.pi / 2 - (283797610683746791347 / 10000000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (3618342583696765969299 / 10000000000000000000000 : ℝ))) ≤
+            0)
+    have h2 := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg (1611686987 / 2000000000000 : ℝ))
+    linarith only [h1, h2]
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_20 :
+    ((35253474581734419 / 40000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(6293648705936709382501 / 1250000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((148469302887837 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (74212844787754867117 / 2500000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (3769825878752123006391 / 10000000000000000000000 : ℝ))))) ≤
+      Vfield (220431339 / 250000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (220431339 / 250000000000 : ℝ)) = Real.log (250220431339 / 250000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((220431339 / 250000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1626681339 / 250000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (35253474581734419 / 40000000000000000000 : ℝ) ≤ Real.log (250220431339 / 250000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 2
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1626681339 / 250000000000 : ℝ) ≤
+      (-(6293648705936709382501 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show (1626681339 / 250000000000 : ℝ) = (1626681339 / 976562500 : ℝ) / 2 ^ 8 by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1626681339 / 976562500 : ℝ) ≤
+            (637823097063290617499 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (296938605775671 / 10000000000000000 : ℝ) ≤ Real.sqrt (220431339 / 250000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (220431339 / 250000000000 : ℝ) ≤ (148469302887837 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (220431339 / 250000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau41 :
+    Real.arctan (148469302887837 / 5000000000000000 : ℝ) ≤
+      (74212844787754867117 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (74212844787754867117 / 2500000000000000000000 : ℝ)) ≤
+      Real.arctan (5000000000000000 / 148469302887837 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 148469302887837 : ℝ) =
+              (148469302887837 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau41)
+  have hz1 :
+    (5000000000000000 / 148469302887837 : ℝ) ≤ 1 / Real.sqrt (220431339 / 250000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 148469302887837 : ℝ) = 1 / (148469302887837 / 5000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hal42 :
+    (3769825878752123006391 / 10000000000000000000000 : ℝ) ≤
+      Real.arctan (98979535258557 / 250000000000000 : ℝ) :=
+    arctan_ge_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat2 :
+    Real.arctan (250000000000000 / 98979535258557 : ℝ) ≤
+      (Real.pi / 2 - (3769825878752123006391 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (250000000000000 / 98979535258557 : ℝ) =
+              (98979535258557 / 250000000000000 : ℝ)
+            by
+        norm_num];
+        exact hal42)
+  have hz2 :
+    (3 / 40) / Real.sqrt (220431339 / 250000000000 : ℝ) ≤
+      (250000000000000 / 98979535258557 : ℝ) := by
+    rw [show
+        (250000000000000 / 98979535258557 : ℝ) =
+          (3 / 40) / (296938605775671 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (74212844787754867117 / 2500000000000000000000 : ℝ)) -
+        6 * (Real.pi / 2 - (3769825878752123006391 / 10000000000000000000000 : ℝ))) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (220431339 / 250000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (220431339 / 250000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((148469302887837 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (74212844787754867117 / 2500000000000000000000 : ℝ)) -
+          6 * (Real.pi / 2 - (3769825878752123006391 / 10000000000000000000000 : ℝ)))) ≤
+      Real.sqrt (220431339 / 250000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (220431339 / 250000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (220431339 / 250000000000 : ℝ))) := by
+    have h1 :=
+      mul_le_mul_of_nonpos_right hs_hi
+        (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20] :
+          (Real.pi + (Real.pi / 2 - (74212844787754867117 / 2500000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (3769825878752123006391 / 10000000000000000000000 : ℝ))) ≤
+            0)
+    have h2 := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg (220431339 / 250000000000 : ℝ))
+    linarith only [h1, h2]
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_21 :
+    ((2528385230620046493 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(50151154605132541364319 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3180983626569 / 100000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (79497784202311396663 / 2500000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (2005672468682107829937 / 5000000000000000000000 : ℝ))))) ≤
+      Vfield (4047462733 / 4000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (4047462733 / 4000000000000 : ℝ)) =
+      Real.log (4004047462733 / 4000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((4047462733 / 4000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (26547462733 / 4000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2528385230620046493 / 2500000000000000000000 : ℝ) ≤
+      Real.log (4004047462733 / 4000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 2
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (26547462733 / 4000000000000 : ℝ) ≤
+      (-(50151154605132541364319 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show (26547462733 / 4000000000000 : ℝ) = (26547462733 / 15625000000 : ℝ) / 2 ^ 8 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26547462733 / 15625000000 : ℝ) ≤
+            (5300619818867458635681 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (318098362656897 / 10000000000000000 : ℝ) ≤ Real.sqrt (4047462733 / 4000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (4047462733 / 4000000000000 : ℝ) ≤ (3180983626569 / 100000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (4047462733 / 4000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau43 :
+    Real.arctan (3180983626569 / 100000000000000 : ℝ) ≤
+      (79497784202311396663 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (79497784202311396663 / 2500000000000000000000 : ℝ)) ≤
+      Real.arctan (100000000000000 / 3180983626569 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (100000000000000 / 3180983626569 : ℝ) =
+              (3180983626569 / 100000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau43)
+  have hz1 :
+    (100000000000000 / 3180983626569 : ℝ) ≤ 1 / Real.sqrt (4047462733 / 4000000000000 : ℝ) := by
+    rw [show (100000000000000 / 3180983626569 : ℝ) = 1 / (3180983626569 / 100000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hal44 :
+    (2005672468682107829937 / 5000000000000000000000 : ℝ) ≤
+      Real.arctan (106032787552299 / 250000000000000 : ℝ) :=
+    arctan_ge_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat2 :
+    Real.arctan (250000000000000 / 106032787552299 : ℝ) ≤
+      (Real.pi / 2 - (2005672468682107829937 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (250000000000000 / 106032787552299 : ℝ) =
+              (106032787552299 / 250000000000000 : ℝ)
+            by
+        norm_num];
+        exact hal44)
+  have hz2 :
+    (3 / 40) / Real.sqrt (4047462733 / 4000000000000 : ℝ) ≤
+      (250000000000000 / 106032787552299 : ℝ) := by
+    rw [show
+        (250000000000000 / 106032787552299 : ℝ) =
+          (3 / 40) / (318098362656897 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (79497784202311396663 / 2500000000000000000000 : ℝ)) -
+        6 * (Real.pi / 2 - (2005672468682107829937 / 5000000000000000000000 : ℝ))) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (4047462733 / 4000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (4047462733 / 4000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3180983626569 / 100000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (79497784202311396663 / 2500000000000000000000 : ℝ)) -
+          6 * (Real.pi / 2 - (2005672468682107829937 / 5000000000000000000000 : ℝ)))) ≤
+      Real.sqrt (4047462733 / 4000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (4047462733 / 4000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (4047462733 / 4000000000000 : ℝ))) := by
+    have h1 :=
+      mul_le_mul_of_nonpos_right hs_hi
+        (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20] :
+          (Real.pi + (Real.pi / 2 - (79497784202311396663 / 2500000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (2005672468682107829937 / 5000000000000000000000 : ℝ))) ≤
+            0)
+    have h2 := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg (4047462733 / 4000000000000 : ℝ))
+    linarith only [h1, h2]
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_22 :
+    ((5706772088353681693 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(24978482657227015538013 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((67587158854327 / 2000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (337807240776056924649 / 10000000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (4233370302187970192887 / 10000000000000000000000 : ℝ))))) ≤
+      Vfield (2284012021 / 2000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (2284012021 / 2000000000000 : ℝ)) =
+      Real.log (2002284012021 / 2000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((2284012021 / 2000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (13534012021 / 2000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (5706772088353681693 / 5000000000000000000000 : ℝ) ≤
+      Real.log (2002284012021 / 2000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 2
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (13534012021 / 2000000000000 : ℝ) ≤
+      (-(24978482657227015538013 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show (13534012021 / 2000000000000 : ℝ) = (13534012021 / 7812500000 : ℝ) / 2 ^ 8 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13534012021 / 7812500000 : ℝ) ≤
+            (2747404554772984461987 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (21120987141977 / 625000000000000 : ℝ) ≤ Real.sqrt (2284012021 / 2000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (2284012021 / 2000000000000 : ℝ) ≤ (67587158854327 / 2000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (2284012021 / 2000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau45 :
+    Real.arctan (67587158854327 / 2000000000000000 : ℝ) ≤
+      (337807240776056924649 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (337807240776056924649 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (2000000000000000 / 67587158854327 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2000000000000000 / 67587158854327 : ℝ) =
+              (67587158854327 / 2000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau45)
+  have hz1 :
+    (2000000000000000 / 67587158854327 : ℝ) ≤ 1 / Real.sqrt (2284012021 / 2000000000000 : ℝ) := by
+    rw [show (2000000000000000 / 67587158854327 : ℝ) = 1 / (67587158854327 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hal46 :
+    (4233370302187970192887 / 10000000000000000000000 : ℝ) ≤
+      Real.arctan (21120987141977 / 46875000000000 : ℝ) :=
+    arctan_ge_of_series (by norm_num) (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat2 :
+    Real.arctan (46875000000000 / 21120987141977 : ℝ) ≤
+      (Real.pi / 2 - (4233370302187970192887 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (46875000000000 / 21120987141977 : ℝ) =
+              (21120987141977 / 46875000000000 : ℝ)
+            by
+        norm_num];
+        exact hal46)
+  have hz2 :
+    (3 / 40) / Real.sqrt (2284012021 / 2000000000000 : ℝ) ≤
+      (46875000000000 / 21120987141977 : ℝ) := by
+    rw [show
+        (46875000000000 / 21120987141977 : ℝ) = (3 / 40) / (21120987141977 / 625000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (337807240776056924649 / 10000000000000000000000 : ℝ)) -
+        6 * (Real.pi / 2 - (4233370302187970192887 / 10000000000000000000000 : ℝ))) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (2284012021 / 2000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (2284012021 / 2000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((67587158854327 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (337807240776056924649 / 10000000000000000000000 : ℝ)) -
+          6 * (Real.pi / 2 - (4233370302187970192887 / 10000000000000000000000 : ℝ)))) ≤
+      Real.sqrt (2284012021 / 2000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (2284012021 / 2000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (2284012021 / 2000000000000 : ℝ))) := by
+    have h1 :=
+      mul_le_mul_of_nonpos_right hs_hi
+        (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20] :
+          (Real.pi + (Real.pi / 2 - (337807240776056924649 / 10000000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (4233370302187970192887 / 10000000000000000000000 : ℝ))) ≤
+            0)
+    have h2 := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg (2284012021 / 2000000000000 : ℝ))
+    linarith only [h1, h2]
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_23 :
+    ((6356689226027724381 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(12441618812709760022989 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((44583950618293 / 1250000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (178260237063611828863 / 5000000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (221953446107719388427 / 500000000000000000000 : ℝ))))) ≤
+      Vfield (5088585351 / 4000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (5088585351 / 4000000000000 : ℝ)) =
+      Real.log (4005088585351 / 4000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((5088585351 / 4000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (27588585351 / 4000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (6356689226027724381 / 5000000000000000000000 : ℝ) ≤
+      Real.log (4005088585351 / 4000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 2
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (27588585351 / 4000000000000 : ℝ) ≤
+      (-(12441618812709760022989 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show (27588585351 / 4000000000000 : ℝ) = (27588585351 / 15625000000 : ℝ) / 2 ^ 8 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27588585351 / 15625000000 : ℝ) ≤
+            (1421324793290239977011 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (356671604946341 / 10000000000000000 : ℝ) ≤ Real.sqrt (5088585351 / 4000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (5088585351 / 4000000000000 : ℝ) ≤ (44583950618293 / 1250000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (5088585351 / 4000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau47 :
+    Real.arctan (44583950618293 / 1250000000000000 : ℝ) ≤
+      (178260237063611828863 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (178260237063611828863 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (1250000000000000 / 44583950618293 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1250000000000000 / 44583950618293 : ℝ) =
+              (44583950618293 / 1250000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau47)
+  have hz1 :
+    (1250000000000000 / 44583950618293 : ℝ) ≤ 1 / Real.sqrt (5088585351 / 4000000000000 : ℝ) := by
+    rw [show (1250000000000000 / 44583950618293 : ℝ) = 1 / (44583950618293 / 1250000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hal48 :
+    (221953446107719388427 / 500000000000000000000 : ℝ) ≤
+      Real.arctan (356671604946341 / 750000000000000 : ℝ) :=
+    arctan_ge_of_series (by norm_num) (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat2 :
+    Real.arctan (750000000000000 / 356671604946341 : ℝ) ≤
+      (Real.pi / 2 - (221953446107719388427 / 500000000000000000000 : ℝ)) :=
+    arctan_le_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (750000000000000 / 356671604946341 : ℝ) =
+              (356671604946341 / 750000000000000 : ℝ)
+            by
+        norm_num];
+        exact hal48)
+  have hz2 :
+    (3 / 40) / Real.sqrt (5088585351 / 4000000000000 : ℝ) ≤
+      (750000000000000 / 356671604946341 : ℝ) := by
+    rw [show
+        (750000000000000 / 356671604946341 : ℝ) =
+          (3 / 40) / (356671604946341 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (178260237063611828863 / 5000000000000000000000 : ℝ)) -
+        6 * (Real.pi / 2 - (221953446107719388427 / 500000000000000000000 : ℝ))) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (5088585351 / 4000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (5088585351 / 4000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((44583950618293 / 1250000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (178260237063611828863 / 5000000000000000000000 : ℝ)) -
+          6 * (Real.pi / 2 - (221953446107719388427 / 500000000000000000000 : ℝ)))) ≤
+      Real.sqrt (5088585351 / 4000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (5088585351 / 4000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (5088585351 / 4000000000000 : ℝ))) := by
+    have h1 :=
+      mul_le_mul_of_nonpos_right hs_hi
+        (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20] :
+          (Real.pi + (Real.pi / 2 - (178260237063611828863 / 5000000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (221953446107719388427 / 500000000000000000000 : ℝ))) ≤
+            0)
+    have h2 := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg (5088585351 / 4000000000000 : ℝ))
+    linarith only [h1, h2]
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_24 :
+    ((350326094811190417 / 250000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(49579546107330803619953 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((374471182469361 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (187148145473045818201 / 5000000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (4630833778944860274891 / 10000000000000000000000 : ℝ))))) ≤
+      Vfield (280457333 / 200000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (280457333 / 200000000000 : ℝ)) = Real.log (200280457333 / 200000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((280457333 / 200000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1405457333 / 200000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (350326094811190417 / 250000000000000000000 : ℝ) ≤
+      Real.log (200280457333 / 200000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 2
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1405457333 / 200000000000 : ℝ) ≤
+      (-(49579546107330803619953 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show (1405457333 / 200000000000 : ℝ) = (1405457333 / 781250000 : ℝ) / 2 ^ 8 by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1405457333 / 781250000 : ℝ) ≤
+            (5872228316669196380047 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (187235591234679 / 5000000000000000 : ℝ) ≤ Real.sqrt (280457333 / 200000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (280457333 / 200000000000 : ℝ) ≤ (374471182469361 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (280457333 / 200000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau49 :
+    Real.arctan (374471182469361 / 10000000000000000 : ℝ) ≤
+      (187148145473045818201 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (187148145473045818201 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 374471182469361 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 374471182469361 : ℝ) =
+              (374471182469361 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau49)
+  have hz1 :
+    (10000000000000000 / 374471182469361 : ℝ) ≤ 1 / Real.sqrt (280457333 / 200000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 374471182469361 : ℝ) = 1 / (374471182469361 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hal50 :
+    (4630833778944860274891 / 10000000000000000000000 : ℝ) ≤
+      Real.arctan (62411863744893 / 125000000000000 : ℝ) :=
+    arctan_ge_of_series (by norm_num) (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat2 :
+    Real.arctan (125000000000000 / 62411863744893 : ℝ) ≤
+      (Real.pi / 2 - (4630833778944860274891 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (125000000000000 / 62411863744893 : ℝ) =
+              (62411863744893 / 125000000000000 : ℝ)
+            by
+        norm_num];
+        exact hal50)
+  have hz2 :
+    (3 / 40) / Real.sqrt (280457333 / 200000000000 : ℝ) ≤
+      (125000000000000 / 62411863744893 : ℝ) := by
+    rw [show
+        (125000000000000 / 62411863744893 : ℝ) =
+          (3 / 40) / (187235591234679 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (187148145473045818201 / 5000000000000000000000 : ℝ)) -
+        6 * (Real.pi / 2 - (4630833778944860274891 / 10000000000000000000000 : ℝ))) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (280457333 / 200000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (280457333 / 200000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((374471182469361 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (187148145473045818201 / 5000000000000000000000 : ℝ)) -
+          6 * (Real.pi / 2 - (4630833778944860274891 / 10000000000000000000000 : ℝ)))) ≤
+      Real.sqrt (280457333 / 200000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (280457333 / 200000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (280457333 / 200000000000 : ℝ))) := by
+    have h1 :=
+      mul_le_mul_of_nonpos_right hs_hi
+        (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20] :
+          (Real.pi + (Real.pi / 2 - (187148145473045818201 / 5000000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (4630833778944860274891 / 10000000000000000000000 : ℝ))) ≤
+            0)
+    have h2 := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg (280457333 / 200000000000 : ℝ))
+    linarith only [h1, h2]
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/V02.lean b/LeanPool/Zeta5Irrational/Table/V02.lean
new file mode 100644
index 0000000000..a608a3b1c8
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/V02.lean
@@ -0,0 +1,1510 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Potential
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # V02: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_25 :
+    ((1017781512086277763 / 625000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(24630475597680706138947 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((50463121818531 / 1250000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (403485872391495007731 / 10000000000000000000000 : ℝ)) -
+              6 *
+                (Real.pi / 2 - (2 * (2468977145022571269173 / 10000000000000000000000 : ℝ)))))) ≤
+      Vfield (6519108259 / 4000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (6519108259 / 4000000000000 : ℝ)) =
+      Real.log (4006519108259 / 4000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((6519108259 / 4000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (29019108259 / 4000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1017781512086277763 / 625000000000000000000 : ℝ) ≤
+      Real.log (4006519108259 / 4000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 2
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (29019108259 / 4000000000000 : ℝ) ≤
+      (-(24630475597680706138947 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show (29019108259 / 4000000000000 : ℝ) = (29019108259 / 15625000000 : ℝ) / 2 ^ 8 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (29019108259 / 15625000000 : ℝ) ≤
+            (3095411614319293861053 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (80740994909649 / 2000000000000000 : ℝ) ≤ Real.sqrt (6519108259 / 4000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (6519108259 / 4000000000000 : ℝ) ≤ (50463121818531 / 1250000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (6519108259 / 4000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau51 :
+    Real.arctan (50463121818531 / 1250000000000000 : ℝ) ≤
+      (403485872391495007731 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (403485872391495007731 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (1250000000000000 / 50463121818531 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1250000000000000 / 50463121818531 : ℝ) =
+              (50463121818531 / 1250000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau51)
+  have hz1 :
+    (1250000000000000 / 50463121818531 : ℝ) ≤ 1 / Real.sqrt (6519108259 / 4000000000000 : ℝ) := by
+    rw [show (1250000000000000 / 50463121818531 : ℝ) = 1 / (50463121818531 / 1250000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hal53 :
+    (2468977145022571269173 / 10000000000000000000000 : ℝ) ≤
+      Real.arctan (5382732993976600 / 21356663880050529 : ℝ) :=
+    arctan_ge_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hal52 :
+    (2 * (2468977145022571269173 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (26913664969883 / 50000000000000 : ℝ) :=
+    arctan_ge_half (by norm_num)
+      (sqrt_le_of_sq_le (by norm_num) (by norm_num) :
+        Real.sqrt (1 + (26913664969883 / 50000000000000 : ℝ) ^ 2) ≤
+          (11356663880050529 / 10000000000000000 : ℝ))
+      (by
+        rw [show
+            (26913664969883 / 50000000000000 : ℝ) /
+                (1 + (11356663880050529 / 10000000000000000 : ℝ)) =
+              (5382732993976600 / 21356663880050529 : ℝ)
+            by
+        norm_num];
+        exact hal53)
+  have hat2 :
+    Real.arctan (50000000000000 / 26913664969883 : ℝ) ≤
+      (Real.pi / 2 - (2 * (2468977145022571269173 / 10000000000000000000000 : ℝ))) :=
+    arctan_le_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (50000000000000 / 26913664969883 : ℝ) =
+              (26913664969883 / 50000000000000 : ℝ)
+            by
+        norm_num];
+        exact hal52)
+  have hz2 :
+    (3 / 40) / Real.sqrt (6519108259 / 4000000000000 : ℝ) ≤
+      (50000000000000 / 26913664969883 : ℝ) := by
+    rw [show
+        (50000000000000 / 26913664969883 : ℝ) = (3 / 40) / (80740994909649 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (403485872391495007731 / 10000000000000000000000 : ℝ)) -
+        6 * (Real.pi / 2 - (2 * (2468977145022571269173 / 10000000000000000000000 : ℝ)))) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (6519108259 / 4000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (6519108259 / 4000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((50463121818531 / 1250000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (403485872391495007731 / 10000000000000000000000 : ℝ)) -
+          6 * (Real.pi / 2 - (2 * (2468977145022571269173 / 10000000000000000000000 : ℝ))))) ≤
+      Real.sqrt (6519108259 / 4000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (6519108259 / 4000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (6519108259 / 4000000000000 : ℝ))) := by
+    have h1 :=
+      mul_le_mul_of_nonpos_right hs_hi
+        (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20] :
+          (Real.pi + (Real.pi / 2 - (403485872391495007731 / 10000000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (2 * (2468977145022571269173 / 10000000000000000000000 : ℝ)))) ≤
+            0)
+    have h2 := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg (6519108259 / 4000000000000 : ℝ))
+    linarith only [h1, h2]
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_26 :
+    ((9277724379122655699 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(48952193938876747273721 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((430960260871001 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (53836719370999942317 / 1250000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (2 * (130385975621936090201 / 500000000000000000000 : ℝ)))))) ≤
+      Vfield (3714534929 / 2000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (3714534929 / 2000000000000 : ℝ)) =
+      Real.log (2003714534929 / 2000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((3714534929 / 2000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (14964534929 / 2000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (9277724379122655699 / 5000000000000000000000 : ℝ) ≤
+      Real.log (2003714534929 / 2000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 2
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (14964534929 / 2000000000000 : ℝ) ≤
+      (-(48952193938876747273721 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show (14964534929 / 2000000000000 : ℝ) = (14964534929 / 7812500000 : ℝ) / 2 ^ 8 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14964534929 / 7812500000 : ℝ) ≤
+            (6499580485123252726279 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (215480130435499 / 5000000000000000 : ℝ) ≤ Real.sqrt (3714534929 / 2000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (3714534929 / 2000000000000 : ℝ) ≤ (430960260871001 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (3714534929 / 2000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau54 :
+    Real.arctan (430960260871001 / 10000000000000000 : ℝ) ≤
+      (53836719370999942317 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (53836719370999942317 / 1250000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 430960260871001 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 430960260871001 : ℝ) =
+              (430960260871001 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau54)
+  have hz1 :
+    (10000000000000000 / 430960260871001 : ℝ) ≤ 1 / Real.sqrt (3714534929 / 2000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 430960260871001 : ℝ) = 1 / (430960260871001 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hal56 :
+    (130385975621936090201 / 500000000000000000000 : ℝ) ≤
+      Real.arctan (17238410434839920 / 64600040380323231 : ℝ) :=
+    arctan_ge_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hal55 :
+    (2 * (130385975621936090201 / 500000000000000000000 : ℝ)) ≤
+      Real.arctan (215480130435499 / 375000000000000 : ℝ) :=
+    arctan_ge_half (by norm_num)
+      (sqrt_le_of_sq_le (by norm_num) (by norm_num) :
+        Real.sqrt (1 + (215480130435499 / 375000000000000 : ℝ) ^ 2) ≤
+          (11533346793441077 / 10000000000000000 : ℝ))
+      (by
+        rw [show
+            (215480130435499 / 375000000000000 : ℝ) /
+                (1 + (11533346793441077 / 10000000000000000 : ℝ)) =
+              (17238410434839920 / 64600040380323231 : ℝ)
+            by
+        norm_num];
+        exact hal56)
+  have hat2 :
+    Real.arctan (375000000000000 / 215480130435499 : ℝ) ≤
+      (Real.pi / 2 - (2 * (130385975621936090201 / 500000000000000000000 : ℝ))) :=
+    arctan_le_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (375000000000000 / 215480130435499 : ℝ) =
+              (215480130435499 / 375000000000000 : ℝ)
+            by
+        norm_num];
+        exact hal55)
+  have hz2 :
+    (3 / 40) / Real.sqrt (3714534929 / 2000000000000 : ℝ) ≤
+      (375000000000000 / 215480130435499 : ℝ) := by
+    rw [show
+        (375000000000000 / 215480130435499 : ℝ) =
+          (3 / 40) / (215480130435499 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (53836719370999942317 / 1250000000000000000000 : ℝ)) -
+        6 * (Real.pi / 2 - (2 * (130385975621936090201 / 500000000000000000000 : ℝ)))) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (3714534929 / 2000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (3714534929 / 2000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((430960260871001 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (53836719370999942317 / 1250000000000000000000 : ℝ)) -
+          6 * (Real.pi / 2 - (2 * (130385975621936090201 / 500000000000000000000 : ℝ))))) ≤
+      Real.sqrt (3714534929 / 2000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (3714534929 / 2000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (3714534929 / 2000000000000 : ℝ))) := by
+    have h1 :=
+      mul_le_mul_of_nonpos_right hs_hi
+        (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20] :
+          (Real.pi + (Real.pi / 2 - (53836719370999942317 / 1250000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (2 * (130385975621936090201 / 500000000000000000000 : ℝ)))) ≤
+            0)
+    have h2 := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg (3714534929 / 2000000000000 : ℝ))
+    linarith only [h1, h2]
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_27 :
+    ((10412938860637891291 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(6081585619044427772321 / 1250000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((456591487464453 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (57034323691496256243 / 1250000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (2 * (683570867995966803463 / 2500000000000000000000 : ℝ)))))) ≤
+      Vfield (8339031457 / 4000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (8339031457 / 4000000000000 : ℝ)) =
+      Real.log (4008339031457 / 4000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((8339031457 / 4000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (30839031457 / 4000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (10412938860637891291 / 5000000000000000000000 : ℝ) ≤
+      Real.log (4008339031457 / 4000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 2
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (30839031457 / 4000000000000 : ℝ) ≤
+      (-(6081585619044427772321 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 8
+        (show (30839031457 / 4000000000000 : ℝ) = (30839031457 / 15625000000 : ℝ) / 2 ^ 8 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (30839031457 / 15625000000 : ℝ) ≤
+            (849886183955572227679 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (9131829749289 / 200000000000000 : ℝ) ≤ Real.sqrt (8339031457 / 4000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (8339031457 / 4000000000000 : ℝ) ≤ (456591487464453 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (8339031457 / 4000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau57 :
+    Real.arctan (456591487464453 / 10000000000000000 : ℝ) ≤
+      (57034323691496256243 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (57034323691496256243 / 1250000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 456591487464453 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 456591487464453 : ℝ) =
+              (456591487464453 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau57)
+  have hz1 :
+    (10000000000000000 / 456591487464453 : ℝ) ≤ 1 / Real.sqrt (8339031457 / 4000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 456591487464453 : ℝ) = 1 / (456591487464453 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hal59 :
+    (683570867995966803463 / 2500000000000000000000 : ℝ) ≤
+      Real.arctan (1521971624881500 / 5426840895392921 : ℝ) :=
+    arctan_ge_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hal58 :
+    (2 * (683570867995966803463 / 2500000000000000000000 : ℝ)) ≤
+      Real.arctan (3043943249763 / 5000000000000 : ℝ) :=
+    arctan_ge_half (by norm_num)
+      (sqrt_le_of_sq_le (by norm_num) (by norm_num) :
+        Real.sqrt (1 + (3043943249763 / 5000000000000 : ℝ) ^ 2) ≤
+          (2926840895392921 / 2500000000000000 : ℝ))
+      (by
+        rw [show
+            (3043943249763 / 5000000000000 : ℝ) /
+                (1 + (2926840895392921 / 2500000000000000 : ℝ)) =
+              (1521971624881500 / 5426840895392921 : ℝ)
+            by
+        norm_num];
+        exact hal59)
+  have hat2 :
+    Real.arctan (5000000000000 / 3043943249763 : ℝ) ≤
+      (Real.pi / 2 - (2 * (683570867995966803463 / 2500000000000000000000 : ℝ))) :=
+    arctan_le_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000 / 3043943249763 : ℝ) = (3043943249763 / 5000000000000 : ℝ) by
+            norm_num];
+        exact hal58)
+  have hz2 :
+    (3 / 40) / Real.sqrt (8339031457 / 4000000000000 : ℝ) ≤ (5000000000000 / 3043943249763 : ℝ) :=
+  by
+    rw [show
+        (5000000000000 / 3043943249763 : ℝ) = (3 / 40) / (9131829749289 / 200000000000000 : ℝ) by
+        norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (57034323691496256243 / 1250000000000000000000 : ℝ)) -
+        6 * (Real.pi / 2 - (2 * (683570867995966803463 / 2500000000000000000000 : ℝ)))) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (8339031457 / 4000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (8339031457 / 4000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((456591487464453 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (57034323691496256243 / 1250000000000000000000 : ℝ)) -
+          6 * (Real.pi / 2 - (2 * (683570867995966803463 / 2500000000000000000000 : ℝ))))) ≤
+      Real.sqrt (8339031457 / 4000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (8339031457 / 4000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (8339031457 / 4000000000000 : ℝ))) := by
+    have h1 :=
+      mul_le_mul_of_nonpos_right hs_hi
+        (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20] :
+          (Real.pi + (Real.pi / 2 - (57034323691496256243 / 1250000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (2 * (683570867995966803463 / 2500000000000000000000 : ℝ)))) ≤
+            0)
+    have h2 := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg (8339031457 / 4000000000000 : ℝ))
+    linarith only [h1, h2]
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_28 :
+    ((23095791316545841351 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(48361886282182873980137 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((480858426566491 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (480488318545920830527 / 10000000000000000000000 : ℝ)) -
+              6 *
+                (Real.pi / 2 - (2 * (2850623752102888537797 / 10000000000000000000000 : ℝ)))))) ≤
+      Vfield (289031033 / 125000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (289031033 / 125000000000 : ℝ)) = Real.log (125289031033 / 125000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((289031033 / 125000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (992156033 / 125000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (23095791316545841351 / 10000000000000000000000 : ℝ) ≤
+      Real.log (125289031033 / 125000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 2
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (992156033 / 125000000000 : ℝ) ≤
+      (-(48361886282182873980137 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show (992156033 / 125000000000 : ℝ) = (992156033 / 976562500 : ℝ) / 2 ^ 7 by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (992156033 / 976562500 : ℝ) ≤
+            (158416338817126019863 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (60107303320811 / 1250000000000000 : ℝ) ≤ Real.sqrt (289031033 / 125000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (289031033 / 125000000000 : ℝ) ≤ (480858426566491 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (289031033 / 125000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau60 :
+    Real.arctan (480858426566491 / 10000000000000000 : ℝ) ≤
+      (480488318545920830527 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (480488318545920830527 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 480858426566491 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 480858426566491 : ℝ) =
+              (480858426566491 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau60)
+  have hz1 :
+    (10000000000000000 / 480858426566491 : ℝ) ≤ 1 / Real.sqrt (289031033 / 125000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 480858426566491 : ℝ) = 1 / (480858426566491 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hal62 :
+    (2850623752102888537797 / 10000000000000000000000 : ℝ) ≤
+      Real.arctan (19234337062659520 / 65636494247330193 : ℝ) :=
+    arctan_ge_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hal61 :
+    (2 * (2850623752102888537797 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (60107303320811 / 93750000000000 : ℝ) :=
+    arctan_ge_half (by norm_num)
+      (sqrt_le_of_sq_le (by norm_num) (by norm_num) :
+        Real.sqrt (1 + (60107303320811 / 93750000000000 : ℝ) ^ 2) ≤
+          (11878831415776731 / 10000000000000000 : ℝ))
+      (by
+        rw [show
+            (60107303320811 / 93750000000000 : ℝ) /
+                (1 + (11878831415776731 / 10000000000000000 : ℝ)) =
+              (19234337062659520 / 65636494247330193 : ℝ)
+            by
+        norm_num];
+        exact hal62)
+  have hat2 :
+    Real.arctan (93750000000000 / 60107303320811 : ℝ) ≤
+      (Real.pi / 2 - (2 * (2850623752102888537797 / 10000000000000000000000 : ℝ))) :=
+    arctan_le_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (93750000000000 / 60107303320811 : ℝ) =
+              (60107303320811 / 93750000000000 : ℝ)
+            by
+        norm_num];
+        exact hal61)
+  have hz2 :
+    (3 / 40) / Real.sqrt (289031033 / 125000000000 : ℝ) ≤ (93750000000000 / 60107303320811 : ℝ) :=
+  by
+    rw [show
+        (93750000000000 / 60107303320811 : ℝ) = (3 / 40) / (60107303320811 / 1250000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (480488318545920830527 / 10000000000000000000000 : ℝ)) -
+        6 * (Real.pi / 2 - (2 * (2850623752102888537797 / 10000000000000000000000 : ℝ)))) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (289031033 / 125000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (289031033 / 125000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((480858426566491 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (480488318545920830527 / 10000000000000000000000 : ℝ)) -
+          6 * (Real.pi / 2 - (2 * (2850623752102888537797 / 10000000000000000000000 : ℝ))))) ≤
+      Real.sqrt (289031033 / 125000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (289031033 / 125000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (289031033 / 125000000000 : ℝ))) := by
+    have h1 :=
+      mul_le_mul_of_nonpos_right hs_hi
+        (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20] :
+          (Real.pi + (Real.pi / 2 - (480488318545920830527 / 10000000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (2 * (2850623752102888537797 / 10000000000000000000000 : ℝ)))) ≤
+            0)
+    have h2 := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg (289031033 / 125000000000 : ℝ))
+    linarith only [h1, h2]
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_29 :
+    ((1552336842063143213 / 500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(23702373925704740231503 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((278814372611959 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (557051839319540677711 / 10000000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (2 * (639331329091312161727 / 2000000000000000000000 : ℝ)))))) ≤
+      Vfield (124379927 / 40000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (124379927 / 40000000000 : ℝ)) = Real.log (40124379927 / 40000000000 : ℝ) := by
+    norm_num
+  have e2 :
+    Real.log ((124379927 / 40000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (349379927 / 40000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1552336842063143213 / 500000000000000000000 : ℝ) ≤
+      Real.log (40124379927 / 40000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (349379927 / 40000000000 : ℝ) ≤
+      (-(23702373925704740231503 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show (349379927 / 40000000000 : ℝ) = (349379927 / 312500000 : ℝ) / 2 ^ 7 by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (349379927 / 312500000 : ℝ) ≤
+            (557777384795259768497 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (111525749044783 / 2000000000000000 : ℝ) ≤ Real.sqrt (124379927 / 40000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (124379927 / 40000000000 : ℝ) ≤ (278814372611959 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (124379927 / 40000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau63 :
+    Real.arctan (278814372611959 / 5000000000000000 : ℝ) ≤
+      (557051839319540677711 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (557051839319540677711 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (5000000000000000 / 278814372611959 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 278814372611959 : ℝ) =
+              (278814372611959 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau63)
+  have hz1 :
+    (5000000000000000 / 278814372611959 : ℝ) ≤ 1 / Real.sqrt (124379927 / 40000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 278814372611959 : ℝ) = 1 / (278814372611959 / 5000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hal65 :
+    (639331329091312161727 / 2000000000000000000000 : ℝ) ≤
+      Real.arctan (22305149808956600 / 67383414878793471 : ℝ) :=
+    arctan_ge_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hal64 :
+    (2 * (639331329091312161727 / 2000000000000000000000 : ℝ)) ≤
+      Real.arctan (111525749044783 / 150000000000000 : ℝ) :=
+    arctan_ge_half (by norm_num)
+      (sqrt_le_of_sq_le (by norm_num) (by norm_num) :
+        Real.sqrt (1 + (111525749044783 / 150000000000000 : ℝ) ^ 2) ≤
+          (12461138292931157 / 10000000000000000 : ℝ))
+      (by
+        rw [show
+            (111525749044783 / 150000000000000 : ℝ) /
+                (1 + (12461138292931157 / 10000000000000000 : ℝ)) =
+              (22305149808956600 / 67383414878793471 : ℝ)
+            by
+        norm_num];
+        exact hal65)
+  have hat2 :
+    Real.arctan (150000000000000 / 111525749044783 : ℝ) ≤
+      (Real.pi / 2 - (2 * (639331329091312161727 / 2000000000000000000000 : ℝ))) :=
+    arctan_le_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (150000000000000 / 111525749044783 : ℝ) =
+              (111525749044783 / 150000000000000 : ℝ)
+            by
+        norm_num];
+        exact hal64)
+  have hz2 :
+    (3 / 40) / Real.sqrt (124379927 / 40000000000 : ℝ) ≤
+      (150000000000000 / 111525749044783 : ℝ) := by
+    rw [show
+        (150000000000000 / 111525749044783 : ℝ) =
+          (3 / 40) / (111525749044783 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (557051839319540677711 / 10000000000000000000000 : ℝ)) -
+        6 * (Real.pi / 2 - (2 * (639331329091312161727 / 2000000000000000000000 : ℝ)))) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (124379927 / 40000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (124379927 / 40000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((278814372611959 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (557051839319540677711 / 10000000000000000000000 : ℝ)) -
+          6 * (Real.pi / 2 - (2 * (639331329091312161727 / 2000000000000000000000 : ℝ))))) ≤
+      Real.sqrt (124379927 / 40000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (124379927 / 40000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (124379927 / 40000000000 : ℝ))) := by
+    have h1 :=
+      mul_le_mul_of_nonpos_right hs_hi
+        (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20] :
+          (Real.pi + (Real.pi / 2 - (557051839319540677711 / 10000000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (2 * (639331329091312161727 / 2000000000000000000000 : ℝ)))) ≤
+            0)
+    have h2 := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg (124379927 / 40000000000 : ℝ))
+    linarith only [h1, h2]
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_30 :
+    ((35019840201976078429 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(46958475676102112366331 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((74036763782639 / 1250000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (591602950926025152249 / 10000000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (2 * (1671111291423061004301 / 5000000000000000000000 : ℝ)))))) ≤
+      Vfield (7016246261 / 2000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (7016246261 / 2000000000000 : ℝ)) =
+      Real.log (2007016246261 / 2000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((7016246261 / 2000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (18266246261 / 2000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (35019840201976078429 / 10000000000000000000000 : ℝ) ≤
+      Real.log (2007016246261 / 2000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (18266246261 / 2000000000000 : ℝ) ≤
+      (-(46958475676102112366331 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show (18266246261 / 2000000000000 : ℝ) = (18266246261 / 15625000000 : ℝ) / 2 ^ 7 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18266246261 / 15625000000 : ℝ) ≤
+            (1561826944897887633669 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (592294110261109 / 10000000000000000 : ℝ) ≤ Real.sqrt (7016246261 / 2000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (7016246261 / 2000000000000 : ℝ) ≤ (74036763782639 / 1250000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (7016246261 / 2000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau66 :
+    Real.arctan (74036763782639 / 1250000000000000 : ℝ) ≤
+      (591602950926025152249 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (591602950926025152249 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (1250000000000000 / 74036763782639 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1250000000000000 / 74036763782639 : ℝ) =
+              (74036763782639 / 1250000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau66)
+  have hz1 :
+    (1250000000000000 / 74036763782639 : ℝ) ≤ 1 / Real.sqrt (7016246261 / 2000000000000 : ℝ) := by
+    rw [show (1250000000000000 / 74036763782639 : ℝ) = 1 / (74036763782639 / 1250000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hal68 :
+    (1671111291423061004301 / 5000000000000000000000 : ℝ) ≤
+      Real.arctan (11845882205222180 / 34113474964537443 : ℝ) :=
+    arctan_ge_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hal67 :
+    (2 * (1671111291423061004301 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (592294110261109 / 750000000000000 : ℝ) :=
+    arctan_ge_half (by norm_num)
+      (sqrt_le_of_sq_le (by norm_num) (by norm_num) :
+        Real.sqrt (1 + (592294110261109 / 750000000000000 : ℝ) ^ 2) ≤
+          (6371158321512481 / 5000000000000000 : ℝ))
+      (by
+        rw [show
+            (592294110261109 / 750000000000000 : ℝ) /
+                (1 + (6371158321512481 / 5000000000000000 : ℝ)) =
+              (11845882205222180 / 34113474964537443 : ℝ)
+            by
+        norm_num];
+        exact hal68)
+  have hat2 :
+    Real.arctan (750000000000000 / 592294110261109 : ℝ) ≤
+      (Real.pi / 2 - (2 * (1671111291423061004301 / 5000000000000000000000 : ℝ))) :=
+    arctan_le_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (750000000000000 / 592294110261109 : ℝ) =
+              (592294110261109 / 750000000000000 : ℝ)
+            by
+        norm_num];
+        exact hal67)
+  have hz2 :
+    (3 / 40) / Real.sqrt (7016246261 / 2000000000000 : ℝ) ≤
+      (750000000000000 / 592294110261109 : ℝ) := by
+    rw [show
+        (750000000000000 / 592294110261109 : ℝ) =
+          (3 / 40) / (592294110261109 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (591602950926025152249 / 10000000000000000000000 : ℝ)) -
+        6 * (Real.pi / 2 - (2 * (1671111291423061004301 / 5000000000000000000000 : ℝ)))) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (7016246261 / 2000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (7016246261 / 2000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((74036763782639 / 1250000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (591602950926025152249 / 10000000000000000000000 : ℝ)) -
+          6 * (Real.pi / 2 - (2 * (1671111291423061004301 / 5000000000000000000000 : ℝ))))) ≤
+      Real.sqrt (7016246261 / 2000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (7016246261 / 2000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (7016246261 / 2000000000000 : ℝ))) := by
+    have h1 :=
+      mul_le_mul_of_nonpos_right hs_hi
+        (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20] :
+          (Real.pi + (Real.pi / 2 - (591602950926025152249 / 10000000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (2 * (1671111291423061004301 / 5000000000000000000000 : ℝ)))) ≤
+            0)
+    have h2 := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg (7016246261 / 2000000000000 : ℝ))
+    linarith only [h1, h2]
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_31 :
+    ((487392070432052579 / 125000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(9306254293033936914081 / 2000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((625039845609863 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (624227790518306883007 / 10000000000000000000000 : ℝ)) -
+              6 *
+                (Real.pi / 2 - (2 * (3473848146500016081923 / 10000000000000000000000 : ℝ)))))) ≤
+      Vfield (1953374043 / 500000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1953374043 / 500000000000 : ℝ)) = Real.log (501953374043 / 500000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1953374043 / 500000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (4765874043 / 500000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (487392070432052579 / 125000000000000000000 : ℝ) ≤
+      Real.log (501953374043 / 500000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (4765874043 / 500000000000 : ℝ) ≤
+      (-(9306254293033936914081 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show (4765874043 / 500000000000 : ℝ) = (4765874043 / 3906250000 : ℝ) / 2 ^ 7 by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4765874043 / 3906250000 : ℝ) ≤
+            (397806231166063085919 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (31251992280493 / 500000000000000 : ℝ) ≤ Real.sqrt (1953374043 / 500000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1953374043 / 500000000000 : ℝ) ≤ (625039845609863 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1953374043 / 500000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau69 :
+    Real.arctan (625039845609863 / 10000000000000000 : ℝ) ≤
+      (624227790518306883007 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (624227790518306883007 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 625039845609863 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 625039845609863 : ℝ) =
+              (625039845609863 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau69)
+  have hz1 :
+    (10000000000000000 / 625039845609863 : ℝ) ≤ 1 / Real.sqrt (1953374043 / 500000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 625039845609863 : ℝ) = 1 / (625039845609863 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hal71 :
+    (3473848146500016081923 / 10000000000000000000000 : ℝ) ≤
+      Real.arctan (25001593824394400 / 69052268740240899 : ℝ) :=
+    arctan_ge_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hal70 :
+    (2 * (3473848146500016081923 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (31251992280493 / 37500000000000 : ℝ) :=
+    arctan_ge_half (by norm_num)
+      (sqrt_le_of_sq_le (by norm_num) (by norm_num) :
+        Real.sqrt (1 + (31251992280493 / 37500000000000 : ℝ) ^ 2) ≤
+          (13017422913413633 / 10000000000000000 : ℝ))
+      (by
+        rw [show
+            (31251992280493 / 37500000000000 : ℝ) /
+                (1 + (13017422913413633 / 10000000000000000 : ℝ)) =
+              (25001593824394400 / 69052268740240899 : ℝ)
+            by
+        norm_num];
+        exact hal71)
+  have hat2 :
+    Real.arctan (37500000000000 / 31251992280493 : ℝ) ≤
+      (Real.pi / 2 - (2 * (3473848146500016081923 / 10000000000000000000000 : ℝ))) :=
+    arctan_le_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (37500000000000 / 31251992280493 : ℝ) =
+              (31251992280493 / 37500000000000 : ℝ)
+            by
+        norm_num];
+        exact hal70)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1953374043 / 500000000000 : ℝ) ≤
+      (37500000000000 / 31251992280493 : ℝ) := by
+    rw [show
+        (37500000000000 / 31251992280493 : ℝ) = (3 / 40) / (31251992280493 / 500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (624227790518306883007 / 10000000000000000000000 : ℝ)) -
+        6 * (Real.pi / 2 - (2 * (3473848146500016081923 / 10000000000000000000000 : ℝ)))) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1953374043 / 500000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1953374043 / 500000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((625039845609863 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (624227790518306883007 / 10000000000000000000000 : ℝ)) -
+          6 * (Real.pi / 2 - (2 * (3473848146500016081923 / 10000000000000000000000 : ℝ))))) ≤
+      Real.sqrt (1953374043 / 500000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1953374043 / 500000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1953374043 / 500000000000 : ℝ))) := by
+    have h1 :=
+      mul_le_mul_of_nonpos_right hs_hi
+        (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20] :
+          (Real.pi + (Real.pi / 2 - (624227790518306883007 / 10000000000000000000000 : ℝ)) -
+              6 * (Real.pi / 2 - (2 * (3473848146500016081923 / 10000000000000000000000 : ℝ)))) ≤
+            0)
+    have h2 := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg (1953374043 / 500000000000 : ℝ))
+    linarith only [h1, h2]
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_32 :
+    ((236122014583635193 / 40000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(11153647148939723420669 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((384724177171127 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (38396760876845215791 / 500000000000000000000 : ℝ)) -
+              6 * (2 * (772599247056886688553 / 2000000000000000000000 : ℝ))))) ≤
+      Vfield qm := by
+  rw [Vfield_eq (by norm_num)]
+  rw [show qm = (59205077 / 10000000000 : ℝ) by norm_num]
+  have e1 :
+    Real.log (1 + (59205077 / 10000000000 : ℝ)) = Real.log (10059205077 / 10000000000 : ℝ) := by
+    norm_num
+  have e2 :
+    Real.log ((59205077 / 10000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (115455077 / 10000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (236122014583635193 / 40000000000000000000 : ℝ) ≤ Real.log (10059205077 / 10000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (115455077 / 10000000000 : ℝ) ≤
+      (-(11153647148939723420669 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show (115455077 / 10000000000 : ℝ) = (115455077 / 78125000 : ℝ) / 2 ^ 7 by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (115455077 / 78125000 : ℝ) ≤
+            (976428506310276579331 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (769448354342251 / 10000000000000000 : ℝ) ≤ Real.sqrt (59205077 / 10000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (59205077 / 10000000000 : ℝ) ≤ (384724177171127 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (59205077 / 10000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau72 :
+    Real.arctan (384724177171127 / 5000000000000000 : ℝ) ≤
+      (38396760876845215791 / 500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (38396760876845215791 / 500000000000000000000 : ℝ)) ≤
+      Real.arctan (5000000000000000 / 384724177171127 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 384724177171127 : ℝ) =
+              (384724177171127 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau72)
+  have hz1 :
+    (5000000000000000 / 384724177171127 : ℝ) ≤ 1 / Real.sqrt (59205077 / 10000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 384724177171127 : ℝ) = 1 / (384724177171127 / 5000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hau73 :
+    Real.arctan (625000000000000000000000000000 / 1536623830262694597801594657771 : ℝ) ≤
+      (772599247056886688553 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat2 :
+    Real.arctan (750000000000000 / 769448354342251 : ℝ) ≤
+      (2 * (772599247056886688553 / 2000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (3491138280786563 / 2500000000000000 : ℝ) ≤
+          Real.sqrt (1 + (750000000000000 / 769448354342251 : ℝ) ^ 2))
+      (by
+        rw [show
+            (750000000000000 / 769448354342251 : ℝ) /
+                (1 + (3491138280786563 / 2500000000000000 : ℝ)) =
+              (625000000000000000000000000000 / 1536623830262694597801594657771 : ℝ)
+            by
+        norm_num];
+        exact hau73)
+  have hz2 :
+    (3 / 40) / Real.sqrt (59205077 / 10000000000 : ℝ) ≤ (750000000000000 / 769448354342251 : ℝ) :=
+  by
+    rw [show
+        (750000000000000 / 769448354342251 : ℝ) =
+          (3 / 40) / (769448354342251 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (38396760876845215791 / 500000000000000000000 : ℝ)) -
+        6 * (2 * (772599247056886688553 / 2000000000000000000000 : ℝ))) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (59205077 / 10000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (59205077 / 10000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((384724177171127 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (38396760876845215791 / 500000000000000000000 : ℝ)) -
+          6 * (2 * (772599247056886688553 / 2000000000000000000000 : ℝ)))) ≤
+      Real.sqrt (59205077 / 10000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (59205077 / 10000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (59205077 / 10000000000 : ℝ))) := by
+    have h1 :=
+      mul_le_mul_of_nonpos_right hs_hi
+        (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20] :
+          (Real.pi + (Real.pi / 2 - (38396760876845215791 / 500000000000000000000 : ℝ)) -
+              6 * (2 * (772599247056886688553 / 2000000000000000000000 : ℝ))) ≤
+            0)
+    have h2 := mul_le_mul_of_nonneg_left hP (Real.sqrt_nonneg (59205077 / 10000000000 : ℝ))
+    linarith only [h1, h2]
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_33 :
+    ((29515252817068737431 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(5576823552816418281887 / 1250000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((384724183669287 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (6143481843653882843 / 80000000000000000000 : ℝ)) -
+              6 * (2 * (1931498096536080841753 / 5000000000000000000000 : ℝ))))) ≤
+      Vfield qp := by
+  rw [Vfield_eq (by norm_num)]
+  rw [show qp = (59205079 / 10000000000 : ℝ) by norm_num]
+  have e1 :
+    Real.log (1 + (59205079 / 10000000000 : ℝ)) = Real.log (10059205079 / 10000000000 : ℝ) := by
+    norm_num
+  have e2 :
+    Real.log ((59205079 / 10000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (115455079 / 10000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (29515252817068737431 / 5000000000000000000000 : ℝ) ≤
+      Real.log (10059205079 / 10000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (115455079 / 10000000000 : ℝ) ≤
+      (-(5576823552816418281887 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show (115455079 / 10000000000 : ℝ) = (115455079 / 78125000 : ℝ) / 2 ^ 7 by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (115455079 / 78125000 : ℝ) ≤
+            (488214274808581718113 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (384724183669287 / 5000000000000000 : ℝ) ≤ Real.sqrt (59205079 / 10000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (59205079 / 10000000000 : ℝ) ≤ (769448367338577 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (59205079 / 10000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau74 :
+    Real.arctan (769448367338577 / 10000000000000000 : ℝ) ≤
+      (6143481843653882843 / 80000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (6143481843653882843 / 80000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 769448367338577 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 769448367338577 : ℝ) =
+              (769448367338577 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau74)
+  have hz1 :
+    (10000000000000000 / 769448367338577 : ℝ) ≤ 1 / Real.sqrt (59205079 / 10000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 769448367338577 : ℝ) = 1 / (769448367338577 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hau75 :
+    Real.arctan (625000000000000000000000000000 / 1536623848848506770019191008793 : ℝ) ≤
+      (1931498096536080841753 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat2 :
+    Real.arctan (125000000000000 / 128241394556429 : ℝ) ≤
+      (2 * (1931498096536080841753 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6982276504115517 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (125000000000000 / 128241394556429 : ℝ) ^ 2))
+      (by
+        rw [show
+            (125000000000000 / 128241394556429 : ℝ) /
+                (1 + (6982276504115517 / 5000000000000000 : ℝ)) =
+              (625000000000000000000000000000 / 1536623848848506770019191008793 : ℝ)
+            by
+        norm_num];
+        exact hau75)
+  have hz2 :
+    (3 / 40) / Real.sqrt (59205079 / 10000000000 : ℝ) ≤ (125000000000000 / 128241394556429 : ℝ) :=
+  by
+    rw [show
+        (125000000000000 / 128241394556429 : ℝ) =
+          (3 / 40) / (384724183669287 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (6143481843653882843 / 80000000000000000000 : ℝ)) -
+        6 * (2 * (1931498096536080841753 / 5000000000000000000000 : ℝ))) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (59205079 / 10000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (59205079 / 10000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((384724183669287 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (6143481843653882843 / 80000000000000000000 : ℝ)) -
+          6 * (2 * (1931498096536080841753 / 5000000000000000000000 : ℝ)))) ≤
+      Real.sqrt (59205079 / 10000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (59205079 / 10000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (59205079 / 10000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_34 :
+    ((22381255075260993563 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1056380614920881581967 / 250000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((237074560146697 / 2500000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (945470894474292487187 / 10000000000000000000000 : ℝ)) -
+              6 * (2 * (3345807944831543455677 / 10000000000000000000000 : ℝ))))) ≤
+      Vfield (8992695531 / 1000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (8992695531 / 1000000000000 : ℝ)) =
+      Real.log (1008992695531 / 1000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((8992695531 / 1000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (14617695531 / 1000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (22381255075260993563 / 2500000000000000000000 : ℝ) ≤
+      Real.log (1008992695531 / 1000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (14617695531 / 1000000000000 : ℝ) ≤
+      (-(1056380614920881581967 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show (14617695531 / 1000000000000 : ℝ) = (14617695531 / 7812500000 : ℝ) / 2 ^ 7 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14617695531 / 7812500000 : ℝ) ≤
+            (156626950604118418033 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (237074560146697 / 2500000000000000 : ℝ) ≤ Real.sqrt (8992695531 / 1000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (8992695531 / 1000000000000 : ℝ) ≤ (948298240586791 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (8992695531 / 1000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau76 :
+    Real.arctan (948298240586791 / 10000000000000000 : ℝ) ≤
+      (945470894474292487187 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (945470894474292487187 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 948298240586791 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 948298240586791 : ℝ) =
+              (948298240586791 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau76)
+  have hz1 :
+    (10000000000000000 / 948298240586791 : ℝ) ≤ 1 / Real.sqrt (8992695531 / 1000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 948298240586791 : ℝ) = 1 / (948298240586791 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hau77 :
+    Real.arctan (1875000000000000000000000000000 / 5393337156221760652486041622069 : ℝ) ≤
+      (3345807944831543455677 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat2 :
+    Real.arctan (187500000000000 / 237074560146697 : ℝ) ≤
+      (2 * (3345807944831543455677 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12749539861571277 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (187500000000000 / 237074560146697 : ℝ) ^ 2))
+      (by
+        rw [show
+            (187500000000000 / 237074560146697 : ℝ) /
+                (1 + (12749539861571277 / 10000000000000000 : ℝ)) =
+              (1875000000000000000000000000000 / 5393337156221760652486041622069 : ℝ)
+            by
+        norm_num];
+        exact hau77)
+  have hz2 :
+    (3 / 40) / Real.sqrt (8992695531 / 1000000000000 : ℝ) ≤
+      (187500000000000 / 237074560146697 : ℝ) := by
+    rw [show
+        (187500000000000 / 237074560146697 : ℝ) =
+          (3 / 40) / (237074560146697 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (945470894474292487187 / 10000000000000000000000 : ℝ)) -
+        6 * (2 * (3345807944831543455677 / 10000000000000000000000 : ℝ))) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (8992695531 / 1000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (8992695531 / 1000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((237074560146697 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (945470894474292487187 / 10000000000000000000000 : ℝ)) -
+          6 * (2 * (3345807944831543455677 / 10000000000000000000000 : ℝ)))) ≤
+      Real.sqrt (8992695531 / 1000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (8992695531 / 1000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (8992695531 / 1000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_35 :
+    ((9738658275375887661 / 1000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(834531719438274254441 / 200000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((989253927032891 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (986045720575090130693 / 10000000000000000000000 : ℝ)) -
+              6 * (2 * (1621737244125958055291 / 5000000000000000000000 : ℝ))))) ≤
+      Vfield (19572466643 / 2000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (19572466643 / 2000000000000 : ℝ)) =
+      Real.log (2019572466643 / 2000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((19572466643 / 2000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (30822466643 / 2000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (9738658275375887661 / 1000000000000000000000 : ℝ) ≤
+      Real.log (2019572466643 / 2000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (30822466643 / 2000000000000 : ℝ) ≤
+      (-(834531719438274254441 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 7
+        (show (30822466643 / 2000000000000 : ℝ) = (30822466643 / 15625000000 : ℝ) / 2 ^ 7 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (30822466643 / 15625000000 : ℝ) ≤
+            (135874332981725745559 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (989253927032891 / 10000000000000000 : ℝ) ≤ Real.sqrt (19572466643 / 2000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (19572466643 / 2000000000000 : ℝ) ≤ (494626963516447 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (19572466643 / 2000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau78 :
+    Real.arctan (494626963516447 / 5000000000000000 : ℝ) ≤
+      (986045720575090130693 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (986045720575090130693 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (5000000000000000 / 494626963516447 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 494626963516447 : ℝ) =
+              (494626963516447 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau78)
+  have hz1 :
+    (5000000000000000 / 494626963516447 : ℝ) ≤ 1 / Real.sqrt (19572466643 / 2000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 494626963516447 : ℝ) = 1 / (494626963516447 / 5000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hau79 :
+    Real.arctan (7500000000000000000000000000000 / 22306738125470534291120193719987 : ℝ) ≤
+      (1621737244125958055291 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat2 :
+    Real.arctan (750000000000000 / 989253927032891 : ℝ) ≤
+      (2 * (1621737244125958055291 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12549051882337257 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (750000000000000 / 989253927032891 : ℝ) ^ 2))
+      (by
+        rw [show
+            (750000000000000 / 989253927032891 : ℝ) /
+                (1 + (12549051882337257 / 10000000000000000 : ℝ)) =
+              (7500000000000000000000000000000 / 22306738125470534291120193719987 : ℝ)
+            by
+        norm_num];
+        exact hau79)
+  have hz2 :
+    (3 / 40) / Real.sqrt (19572466643 / 2000000000000 : ℝ) ≤
+      (750000000000000 / 989253927032891 : ℝ) := by
+    rw [show
+        (750000000000000 / 989253927032891 : ℝ) =
+          (3 / 40) / (989253927032891 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (986045720575090130693 / 10000000000000000000000 : ℝ)) -
+        6 * (2 * (1621737244125958055291 / 5000000000000000000000 : ℝ))) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (19572466643 / 2000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (19572466643 / 2000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((989253927032891 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (986045720575090130693 / 10000000000000000000000 : ℝ)) -
+          6 * (2 * (1621737244125958055291 / 5000000000000000000000 : ℝ)))) ≤
+      Real.sqrt (19572466643 / 2000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (19572466643 / 2000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (19572466643 / 2000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_36 :
+    ((101315047537995983751 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(20736194980936662302051 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1009108627291927 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1005704152080836199331 / 10000000000000000000000 : ℝ)) -
+              6 * (2 * (3195771674208240842333 / 10000000000000000000000 : ℝ))))) ≤
+      Vfield (40732008867 / 4000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (40732008867 / 4000000000000 : ℝ)) =
+      Real.log (4040732008867 / 4000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((40732008867 / 4000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (63232008867 / 4000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (101315047537995983751 / 10000000000000000000000 : ℝ) ≤
+      Real.log (4040732008867 / 4000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (63232008867 / 4000000000000 : ℝ) ≤
+      (-(20736194980936662302051 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show (63232008867 / 4000000000000 : ℝ) = (63232008867 / 62500000000 : ℝ) / 2 ^ 6 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (63232008867 / 62500000000 : ℝ) ≤
+            (58220428063337697949 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1009108627291927 / 10000000000000000 : ℝ) ≤ Real.sqrt (40732008867 / 4000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (40732008867 / 4000000000000 : ℝ) ≤ (100910862729193 / 1000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (40732008867 / 4000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau80 :
+    Real.arctan (100910862729193 / 1000000000000000 : ℝ) ≤
+      (1005704152080836199331 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1005704152080836199331 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (1000000000000000 / 100910862729193 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1000000000000000 / 100910862729193 : ℝ) =
+              (100910862729193 / 1000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau80)
+  have hz1 :
+    (1000000000000000 / 100910862729193 : ℝ) ≤ 1 / Real.sqrt (40732008867 / 4000000000000 : ℝ) := by
+    rw [show
+        (1000000000000000 / 100910862729193 : ℝ) = 1 / (100910862729193 / 1000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hau81 :
+    Real.arctan (2500000000000000000000000000000 / 7554691356957750317036587677079 : ℝ) ≤
+      (3195771674208240842333 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat2 :
+    Real.arctan (750000000000000 / 1009108627291927 : ℝ) ≤
+      (2 * (3195771674208240842333 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12459498866534531 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (750000000000000 / 1009108627291927 : ℝ) ^ 2))
+      (by
+        rw [show
+            (750000000000000 / 1009108627291927 : ℝ) /
+                (1 + (12459498866534531 / 10000000000000000 : ℝ)) =
+              (2500000000000000000000000000000 / 7554691356957750317036587677079 : ℝ)
+            by
+        norm_num];
+        exact hau81)
+  have hz2 :
+    (3 / 40) / Real.sqrt (40732008867 / 4000000000000 : ℝ) ≤
+      (750000000000000 / 1009108627291927 : ℝ) := by
+    rw [show
+        (750000000000000 / 1009108627291927 : ℝ) =
+          (3 / 40) / (1009108627291927 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1005704152080836199331 / 10000000000000000000000 : ℝ)) -
+        6 * (2 * (3195771674208240842333 / 10000000000000000000000 : ℝ))) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (40732008867 / 4000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (40732008867 / 4000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1009108627291927 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1005704152080836199331 / 10000000000000000000000 : ℝ)) -
+          6 * (2 * (3195771674208240842333 / 10000000000000000000000 : ℝ)))) ≤
+      Real.sqrt (40732008867 / 4000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (40732008867 / 4000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (40732008867 / 4000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/V03.lean b/LeanPool/Zeta5Irrational/Table/V03.lean
new file mode 100644
index 0000000000..2f8a162225
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/V03.lean
@@ -0,0 +1,1396 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Energy
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # V03: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_37 :
+    ((21048393928938516659 / 2000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(41224495654581728135701 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((20571602866087 / 200000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1024975616002557514417 / 10000000000000000000000 : ℝ)) -
+              6 * (2 * (630029876112921474829 / 2000000000000000000000 : ℝ))))) ≤
+      Vfield (1322471389 / 125000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1322471389 / 125000000000 : ℝ)) = Real.log (126322471389 / 125000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1322471389 / 125000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (2025596389 / 125000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (21048393928938516659 / 2000000000000000000000 : ℝ) ≤
+      Real.log (126322471389 / 125000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (2025596389 / 125000000000 : ℝ) ≤
+      (-(41224495654581728135701 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show (2025596389 / 125000000000 : ℝ) = (2025596389 / 1953125000 : ℝ) / 2 ^ 6 by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2025596389 / 1953125000 : ℝ) ≤
+            (364335163418271864299 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (20571602866087 / 200000000000000 : ℝ) ≤ Real.sqrt (1322471389 / 125000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1322471389 / 125000000000 : ℝ) ≤ (1028580143304353 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1322471389 / 125000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau82 :
+    Real.arctan (1028580143304353 / 10000000000000000 : ℝ) ≤
+      (1024975616002557514417 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1024975616002557514417 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 1028580143304353 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 1028580143304353 : ℝ) =
+              (1028580143304353 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau82)
+  have hz1 :
+    (10000000000000000 / 1028580143304353 : ℝ) ≤ 1 / Real.sqrt (1322471389 / 125000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 1028580143304353 : ℝ) =
+          1 / (1028580143304353 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hau83 :
+    Real.arctan (75000000000000000000000000000 / 230155976262274147498241660923 : ℝ) ≤
+      (630029876112921474829 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat2 :
+    Real.arctan (15000000000000 / 20571602866087 : ℝ) ≤
+      (2 * (630029876112921474829 / 2000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6188042942521229 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (15000000000000 / 20571602866087 : ℝ) ^ 2))
+      (by
+        rw [show
+            (15000000000000 / 20571602866087 : ℝ) /
+                (1 + (6188042942521229 / 5000000000000000 : ℝ)) =
+              (75000000000000000000000000000 / 230155976262274147498241660923 : ℝ)
+            by
+        norm_num];
+        exact hau83)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1322471389 / 125000000000 : ℝ) ≤
+      (15000000000000 / 20571602866087 : ℝ) := by
+    rw [show
+        (15000000000000 / 20571602866087 : ℝ) = (3 / 40) / (20571602866087 / 200000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1024975616002557514417 / 10000000000000000000000 : ℝ)) -
+        6 * (2 * (630029876112921474829 / 2000000000000000000000 : ℝ))) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1322471389 / 125000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1322471389 / 125000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((20571602866087 / 200000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1024975616002557514417 / 10000000000000000000000 : ℝ)) -
+          6 * (2 * (630029876112921474829 / 2000000000000000000000 : ℝ)))) ≤
+      Real.sqrt (1322471389 / 125000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1322471389 / 125000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1322471389 / 125000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_38 :
+    ((113091190668515940261 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(40746414146378761469201 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1066457167564641 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1062441486169975621307 / 10000000000000000000000 : ℝ)) -
+              6 * (2 * (3064563126433209565083 / 10000000000000000000000 : ℝ))))) ≤
+      Vfield (4549323561 / 400000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (4549323561 / 400000000000 : ℝ)) = Real.log (404549323561 / 400000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((4549323561 / 400000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (6799323561 / 400000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (113091190668515940261 / 10000000000000000000000 : ℝ) ≤
+      Real.log (404549323561 / 400000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (6799323561 / 400000000000 : ℝ) ≤
+      (-(40746414146378761469201 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show (6799323561 / 400000000000 : ℝ) = (6799323561 / 6250000000 : ℝ) / 2 ^ 6 by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6799323561 / 6250000000 : ℝ) ≤
+            (842416671621238530799 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1066457167564641 / 10000000000000000 : ℝ) ≤ Real.sqrt (4549323561 / 400000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (4549323561 / 400000000000 : ℝ) ≤ (266614291891161 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (4549323561 / 400000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau84 :
+    Real.arctan (266614291891161 / 2500000000000000 : ℝ) ≤
+      (1062441486169975621307 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1062441486169975621307 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (2500000000000000 / 266614291891161 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 266614291891161 : ℝ) =
+              (266614291891161 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau84)
+  have hz1 :
+    (2500000000000000 / 266614291891161 : ℝ) ≤ 1 / Real.sqrt (4549323561 / 400000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 266614291891161 : ℝ) = 1 / (266614291891161 / 2500000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hau85 :
+    Real.arctan (2500000000000000000000000000000 / 7900775987878507578486688127791 : ℝ) ≤
+      (3064563126433209565083 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat2 :
+    Real.arctan (250000000000000 / 355485722521547 : ℝ) ≤
+      (2 * (3064563126433209565083 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12225297634561453 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (250000000000000 / 355485722521547 : ℝ) ^ 2))
+      (by
+        rw [show
+            (250000000000000 / 355485722521547 : ℝ) /
+                (1 + (12225297634561453 / 10000000000000000 : ℝ)) =
+              (2500000000000000000000000000000 / 7900775987878507578486688127791 : ℝ)
+            by
+        norm_num];
+        exact hau85)
+  have hz2 :
+    (3 / 40) / Real.sqrt (4549323561 / 400000000000 : ℝ) ≤
+      (250000000000000 / 355485722521547 : ℝ) := by
+    rw [show
+        (250000000000000 / 355485722521547 : ℝ) =
+          (3 / 40) / (1066457167564641 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1062441486169975621307 / 10000000000000000000000 : ℝ)) -
+        6 * (2 * (3064563126433209565083 / 10000000000000000000000 : ℝ))) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (4549323561 / 400000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (4549323561 / 400000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1066457167564641 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1062441486169975621307 / 10000000000000000000000 : ℝ)) -
+          6 * (2 * (3064563126433209565083 / 10000000000000000000000 : ℝ)))) ≤
+      Real.sqrt (4549323561 / 400000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (4549323561 / 400000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (4549323561 / 400000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_39 :
+    ((120934255497089002089 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(8058029952033732590147 / 2000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((551517150526617 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (549296596867263340437 / 5000000000000000000000 : ℝ)) -
+              6 * (2 * (1492843499897578576763 / 5000000000000000000000 : ℝ))))) ≤
+      Vfield (12166846693 / 1000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (12166846693 / 1000000000000 : ℝ)) =
+      Real.log (1012166846693 / 1000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((12166846693 / 1000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (17791846693 / 1000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (120934255497089002089 / 10000000000000000000000 : ℝ) ≤
+      Real.log (1012166846693 / 1000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (17791846693 / 1000000000000 : ℝ) ≤
+      (-(8058029952033732590147 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show (17791846693 / 1000000000000 : ℝ) = (17791846693 / 15625000000 : ℝ) / 2 ^ 6 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17791846693 / 15625000000 : ℝ) ≤
+            (259736211566267409853 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (551517150526617 / 5000000000000000 : ℝ) ≤ Real.sqrt (12166846693 / 1000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (12166846693 / 1000000000000 : ℝ) ≤ (1103034301053237 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (12166846693 / 1000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau86 :
+    Real.arctan (1103034301053237 / 10000000000000000 : ℝ) ≤
+      (549296596867263340437 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (549296596867263340437 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 1103034301053237 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 1103034301053237 : ℝ) =
+              (1103034301053237 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau86)
+  have hz1 :
+    (10000000000000000 / 1103034301053237 : ℝ) ≤
+      1 / Real.sqrt (12166846693 / 1000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 1103034301053237 : ℝ) =
+          1 / (1103034301053237 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hau87 :
+    Real.arctan (50000000000000000000000000000 / 162459674291281643418659719961 : ℝ) ≤
+      (1492843499897578576763 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat2 :
+    Real.arctan (125000000000000 / 183839050175539 : ℝ) ≤
+      (2 * (1492843499897578576763 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (483706014234499 / 400000000000000 : ℝ) ≤
+          Real.sqrt (1 + (125000000000000 / 183839050175539 : ℝ) ^ 2))
+      (by
+        rw [show
+            (125000000000000 / 183839050175539 : ℝ) /
+                (1 + (483706014234499 / 400000000000000 : ℝ)) =
+              (50000000000000000000000000000 / 162459674291281643418659719961 : ℝ)
+            by
+        norm_num];
+        exact hau87)
+  have hz2 :
+    (3 / 40) / Real.sqrt (12166846693 / 1000000000000 : ℝ) ≤
+      (125000000000000 / 183839050175539 : ℝ) := by
+    rw [show
+        (125000000000000 / 183839050175539 : ℝ) =
+          (3 / 40) / (551517150526617 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (549296596867263340437 / 5000000000000000000000 : ℝ)) -
+        6 * (2 * (1492843499897578576763 / 5000000000000000000000 : ℝ))) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (12166846693 / 1000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (12166846693 / 1000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((551517150526617 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (549296596867263340437 / 5000000000000000000000 : ℝ)) -
+          6 * (2 * (1492843499897578576763 / 5000000000000000000000 : ℝ)))) ≤
+      Real.sqrt (12166846693 / 1000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (12166846693 / 1000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (12166846693 / 1000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_40 :
+    ((152245145501698111849 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(38648533017005889889951 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((309646954762597 / 2500000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (154038970346850627713 / 1250000000000000000000 : ℝ)) -
+              6 * (2 * (2722372641804015738093 / 10000000000000000000000 : ℝ))))) ≤
+      Vfield (3068199571 / 200000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (3068199571 / 200000000000 : ℝ)) = Real.log (203068199571 / 200000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((3068199571 / 200000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (4193199571 / 200000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (152245145501698111849 / 10000000000000000000000 : ℝ) ≤
+      Real.log (203068199571 / 200000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (4193199571 / 200000000000 : ℝ) ≤
+      (-(38648533017005889889951 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show (4193199571 / 200000000000 : ℝ) = (4193199571 / 3125000000 : ℝ) / 2 ^ 6 by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4193199571 / 3125000000 : ℝ) ≤
+            (2940297800994110110049 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (309646954762597 / 2500000000000000 : ℝ) ≤ Real.sqrt (3068199571 / 200000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (3068199571 / 200000000000 : ℝ) ≤ (1238587819050391 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (3068199571 / 200000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau88 :
+    Real.arctan (1238587819050391 / 10000000000000000 : ℝ) ≤
+      (154038970346850627713 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (154038970346850627713 / 1250000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 1238587819050391 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 1238587819050391 : ℝ) =
+              (1238587819050391 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau88)
+  have hz1 :
+    (10000000000000000 / 1238587819050391 : ℝ) ≤ 1 / Real.sqrt (3068199571 / 200000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 1238587819050391 : ℝ) =
+          1 / (1238587819050391 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hau89 :
+    Real.arctan (1875000000000000000000000000000 / 6716379582333321103149502846301 : ℝ) ≤
+      (2722372641804015738093 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat2 :
+    Real.arctan (187500000000000 / 309646954762597 : ℝ) ≤
+      (2 * (2722372641804015738093 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (11690442870599833 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (187500000000000 / 309646954762597 : ℝ) ^ 2))
+      (by
+        rw [show
+            (187500000000000 / 309646954762597 : ℝ) /
+                (1 + (11690442870599833 / 10000000000000000 : ℝ)) =
+              (1875000000000000000000000000000 / 6716379582333321103149502846301 : ℝ)
+            by
+        norm_num];
+        exact hau89)
+  have hz2 :
+    (3 / 40) / Real.sqrt (3068199571 / 200000000000 : ℝ) ≤
+      (187500000000000 / 309646954762597 : ℝ) := by
+    rw [show
+        (187500000000000 / 309646954762597 : ℝ) =
+          (3 / 40) / (309646954762597 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (154038970346850627713 / 1250000000000000000000 : ℝ)) -
+        6 * (2 * (2722372641804015738093 / 10000000000000000000000 : ℝ))) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (3068199571 / 200000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (3068199571 / 200000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((309646954762597 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (154038970346850627713 / 1250000000000000000000 : ℝ)) -
+          6 * (2 * (2722372641804015738093 / 10000000000000000000000 : ℝ)))) ≤
+      Real.sqrt (3068199571 / 200000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (3068199571 / 200000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (3068199571 / 200000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_41 :
+    ((158638233080768093329 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(38343528723300591384029 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((126452844113181 / 1000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (251570460987800448443 / 2000000000000000000000 : ℝ)) -
+              6 * (2 * (1338338273905248023557 / 5000000000000000000000 : ℝ))))) ≤
+      Vfield (255845148549 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (255845148549 / 16000000000000 : ℝ)) =
+      Real.log (16255845148549 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((255845148549 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (345845148549 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (158638233080768093329 / 10000000000000000000000 : ℝ) ≤
+      Real.log (16255845148549 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (345845148549 / 16000000000000 : ℝ) ≤
+      (-(38343528723300591384029 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show (345845148549 / 16000000000000 : ℝ) = (345845148549 / 250000000000 : ℝ) / 2 ^ 6 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (345845148549 / 250000000000 : ℝ) ≤
+            (3245302094699408615971 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (126452844113181 / 1000000000000000 : ℝ) ≤ Real.sqrt (255845148549 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (255845148549 / 16000000000000 : ℝ) ≤ (1264528441131813 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (255845148549 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau90 :
+    Real.arctan (1264528441131813 / 10000000000000000 : ℝ) ≤
+      (251570460987800448443 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (251570460987800448443 / 2000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 1264528441131813 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 1264528441131813 : ℝ) =
+              (1264528441131813 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau90)
+  have hz1 :
+    (10000000000000000 / 1264528441131813 : ℝ) ≤
+      1 / Real.sqrt (255845148549 / 16000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 1264528441131813 : ℝ) =
+          1 / (1264528441131813 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hau91 :
+    Real.arctan (25000000000000000000000000000 / 91158115069490436044377004569 : ℝ) ≤
+      (1338338273905248023557 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat2 :
+    Real.arctan (25000000000000 / 42150948037727 : ℝ) ≤
+      (2 * (1338338273905248023557 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (1162658713818247 / 1000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (25000000000000 / 42150948037727 : ℝ) ^ 2))
+      (by
+        rw [show
+            (25000000000000 / 42150948037727 : ℝ) /
+                (1 + (1162658713818247 / 1000000000000000 : ℝ)) =
+              (25000000000000000000000000000 / 91158115069490436044377004569 : ℝ)
+            by
+        norm_num];
+        exact hau91)
+  have hz2 :
+    (3 / 40) / Real.sqrt (255845148549 / 16000000000000 : ℝ) ≤
+      (25000000000000 / 42150948037727 : ℝ) := by
+    rw [show
+        (25000000000000 / 42150948037727 : ℝ) =
+          (3 / 40) / (126452844113181 / 1000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (251570460987800448443 / 2000000000000000000000 : ℝ)) -
+        6 * (2 * (1338338273905248023557 / 5000000000000000000000 : ℝ))) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (255845148549 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (255845148549 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((126452844113181 / 1000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (251570460987800448443 / 2000000000000000000000 : ℝ)) -
+          6 * (2 * (1338338273905248023557 / 5000000000000000000000 : ℝ)))) ≤
+      Real.sqrt (255845148549 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (255845148549 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (255845148549 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_42 :
+    ((165027236114105152063 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(4755944063788979246611 / 1250000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1289947507212017 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (160357918015002146829 / 1250000000000000000000 : ℝ)) -
+              6 * (2 * (2633224767274595554807 / 10000000000000000000000 : ℝ))))) ≤
+      Vfield (133117165709 / 8000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (133117165709 / 8000000000000 : ℝ)) =
+      Real.log (8133117165709 / 8000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((133117165709 / 8000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (178117165709 / 8000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (165027236114105152063 / 10000000000000000000000 : ℝ) ≤
+      Real.log (8133117165709 / 8000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (178117165709 / 8000000000000 : ℝ) ≤
+      (-(4755944063788979246611 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show (178117165709 / 8000000000000 : ℝ) = (178117165709 / 125000000000 : ℝ) / 2 ^ 6 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (178117165709 / 125000000000 : ℝ) ≤
+            (442659788461020753389 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1289947507212017 / 10000000000000000 : ℝ) ≤ Real.sqrt (133117165709 / 8000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (133117165709 / 8000000000000 : ℝ) ≤ (64497375360601 / 500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (133117165709 / 8000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau92 :
+    Real.arctan (64497375360601 / 500000000000000 : ℝ) ≤
+      (160357918015002146829 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (160357918015002146829 / 1250000000000000000000 : ℝ)) ≤
+      Real.arctan (500000000000000 / 64497375360601 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (500000000000000 / 64497375360601 : ℝ) =
+              (64497375360601 / 500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau92)
+  have hz1 :
+    (500000000000000 / 64497375360601 : ℝ) ≤ 1 / Real.sqrt (133117165709 / 8000000000000 : ℝ) := by
+    rw [show (500000000000000 / 64497375360601 : ℝ) = 1 / (64497375360601 / 500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hau93 :
+    Real.arctan (7500000000000000000000000000000 / 27820817409063237865736181214393 : ℝ) ≤
+      (2633224767274595554807 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat2 :
+    Real.arctan (750000000000000 / 1289947507212017 : ℝ) ≤
+      (2 * (2633224767274595554807 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (11567402745862729 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (750000000000000 / 1289947507212017 : ℝ) ^ 2))
+      (by
+        rw [show
+            (750000000000000 / 1289947507212017 : ℝ) /
+                (1 + (11567402745862729 / 10000000000000000 : ℝ)) =
+              (7500000000000000000000000000000 / 27820817409063237865736181214393 : ℝ)
+            by
+        norm_num];
+        exact hau93)
+  have hz2 :
+    (3 / 40) / Real.sqrt (133117165709 / 8000000000000 : ℝ) ≤
+      (750000000000000 / 1289947507212017 : ℝ) := by
+    rw [show
+        (750000000000000 / 1289947507212017 : ℝ) =
+          (3 / 40) / (1289947507212017 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (160357918015002146829 / 1250000000000000000000 : ℝ)) -
+        6 * (2 * (2633224767274595554807 / 10000000000000000000000 : ℝ))) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (133117165709 / 8000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (133117165709 / 8000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1289947507212017 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (160357918015002146829 / 1250000000000000000000 : ℝ)) -
+          6 * (2 * (2633224767274595554807 / 10000000000000000000000 : ℝ)))) ≤
+      Real.sqrt (133117165709 / 8000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (133117165709 / 8000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (133117165709 / 8000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_43 :
+    ((168220207556483795939 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(37902785946772965577681 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((130247102379597 / 1000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (647589960594501769007 / 5000000000000000000000 : ℝ)) -
+              6 * (2 * (1306141833947405153479 / 5000000000000000000000 : ℝ))))) ≤
+      Vfield (108571569141 / 6400000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (108571569141 / 6400000000000 : ℝ)) =
+      Real.log (6508571569141 / 6400000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((108571569141 / 6400000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (144571569141 / 6400000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (168220207556483795939 / 10000000000000000000000 : ℝ) ≤
+      Real.log (6508571569141 / 6400000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (144571569141 / 6400000000000 : ℝ) ≤
+      (-(37902785946772965577681 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show (144571569141 / 6400000000000 : ℝ) = (144571569141 / 100000000000 : ℝ) / 2 ^ 6 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (144571569141 / 100000000000 : ℝ) ≤
+            (3686044871227034422319 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (130247102379597 / 1000000000000000 : ℝ) ≤ Real.sqrt (108571569141 / 6400000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (108571569141 / 6400000000000 : ℝ) ≤ (1302471023795973 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (108571569141 / 6400000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau94 :
+    Real.arctan (1302471023795973 / 10000000000000000 : ℝ) ≤
+      (647589960594501769007 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (647589960594501769007 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 1302471023795973 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 1302471023795973 : ℝ) =
+              (1302471023795973 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau94)
+  have hz1 :
+    (10000000000000000 / 1302471023795973 : ℝ) ≤
+      1 / Real.sqrt (108571569141 / 6400000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 1302471023795973 : ℝ) =
+          1 / (1302471023795973 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hau95 :
+    Real.arctan (1953125000000000000000000000 / 7305846349559108574878267487 : ℝ) ≤
+      (1306141833947405153479 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat2 :
+    Real.arctan (25000000000000 / 43415700793199 : ℝ) ≤
+      (2 * (1306141833947405153479 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (90151596164113 / 78125000000000 : ℝ) ≤
+          Real.sqrt (1 + (25000000000000 / 43415700793199 : ℝ) ^ 2))
+      (by
+        rw [show
+            (25000000000000 / 43415700793199 : ℝ) / (1 + (90151596164113 / 78125000000000 : ℝ)) =
+              (1953125000000000000000000000 / 7305846349559108574878267487 : ℝ)
+            by
+        norm_num];
+        exact hau95)
+  have hz2 :
+    (3 / 40) / Real.sqrt (108571569141 / 6400000000000 : ℝ) ≤
+      (25000000000000 / 43415700793199 : ℝ) := by
+    rw [show
+        (25000000000000 / 43415700793199 : ℝ) =
+          (3 / 40) / (130247102379597 / 1000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (647589960594501769007 / 5000000000000000000000 : ℝ)) -
+        6 * (2 * (1306141833947405153479 / 5000000000000000000000 : ℝ))) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (108571569141 / 6400000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (108571569141 / 6400000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((130247102379597 / 1000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (647589960594501769007 / 5000000000000000000000 : ℝ)) -
+          6 * (2 * (1306141833947405153479 / 5000000000000000000000 : ℝ)))) ≤
+      Real.sqrt (108571569141 / 6400000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (108571569141 / 6400000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (108571569141 / 6400000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_44 :
+    ((42853039954403033183 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(7552017049555776737193 / 2000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1314875265678743 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (653687650225280262137 / 5000000000000000000000 : ℝ)) -
+              6 * (2 * (1295919005291486724873 / 5000000000000000000000 : ℝ))))) ≤
+      Vfield (276623514287 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (276623514287 / 16000000000000 : ℝ)) =
+      Real.log (16276623514287 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((276623514287 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (366623514287 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (42853039954403033183 / 2500000000000000000000 : ℝ) ≤
+      Real.log (16276623514287 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (366623514287 / 16000000000000 : ℝ) ≤
+      (-(7552017049555776737193 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show (366623514287 / 16000000000000 : ℝ) = (366623514287 / 250000000000 : ℝ) / 2 ^ 6 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (366623514287 / 250000000000 : ℝ) ≤
+            (765749114044223262807 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1314875265678743 / 10000000000000000 : ℝ) ≤ Real.sqrt (276623514287 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (276623514287 / 16000000000000 : ℝ) ≤ (657437632839373 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (276623514287 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau96 :
+    Real.arctan (657437632839373 / 5000000000000000 : ℝ) ≤
+      (653687650225280262137 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (653687650225280262137 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (5000000000000000 / 657437632839373 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 657437632839373 : ℝ) =
+              (657437632839373 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau96)
+  have hz1 :
+    (5000000000000000 / 657437632839373 : ℝ) ≤
+      1 / Real.sqrt (276623514287 / 16000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 657437632839373 : ℝ) = 1 / (657437632839373 / 5000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hau97 :
+    Real.arctan (7500000000000000000000000000000 / 28286113603444361580753162742349 : ℝ) ≤
+      (1295919005291486724873 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat2 :
+    Real.arctan (750000000000000 / 1314875265678743 : ℝ) ≤
+      (2 * (1295919005291486724873 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (11512393108134843 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (750000000000000 / 1314875265678743 : ℝ) ^ 2))
+      (by
+        rw [show
+            (750000000000000 / 1314875265678743 : ℝ) /
+                (1 + (11512393108134843 / 10000000000000000 : ℝ)) =
+              (7500000000000000000000000000000 / 28286113603444361580753162742349 : ℝ)
+            by
+        norm_num];
+        exact hau97)
+  have hz2 :
+    (3 / 40) / Real.sqrt (276623514287 / 16000000000000 : ℝ) ≤
+      (750000000000000 / 1314875265678743 : ℝ) := by
+    rw [show
+        (750000000000000 / 1314875265678743 : ℝ) =
+          (3 / 40) / (1314875265678743 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (653687650225280262137 / 5000000000000000000000 : ℝ)) -
+        6 * (2 * (1295919005291486724873 / 5000000000000000000000 : ℝ))) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (276623514287 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (276623514287 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1314875265678743 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (653687650225280262137 / 5000000000000000000000 : ℝ)) -
+          6 * (2 * (1295919005291486724873 / 5000000000000000000000 : ℝ)))) ≤
+      Real.sqrt (276623514287 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (276623514287 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (276623514287 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_45 :
+    ((174603093547929167631 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(7523878456169585870279 / 2000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1327163577242599 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1319452846275249601221 / 10000000000000000000000 : ℝ)) -
+              6 * (2 * (514373683695854022843 / 2000000000000000000000 : ℝ))))) ≤
+      Vfield (563636211443 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (563636211443 / 32000000000000 : ℝ)) =
+      Real.log (32563636211443 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((563636211443 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (743636211443 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (174603093547929167631 / 10000000000000000000000 : ℝ) ≤
+      Real.log (32563636211443 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (743636211443 / 32000000000000 : ℝ) ≤
+      (-(7523878456169585870279 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show (743636211443 / 32000000000000 : ℝ) = (743636211443 / 500000000000 : ℝ) / 2 ^ 6 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (743636211443 / 500000000000 : ℝ) ≤
+            (793887707430414129721 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1327163577242599 / 10000000000000000 : ℝ) ≤ Real.sqrt (563636211443 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (563636211443 / 32000000000000 : ℝ) ≤ (663581788621301 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (563636211443 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau98 :
+    Real.arctan (663581788621301 / 5000000000000000 : ℝ) ≤
+      (1319452846275249601221 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1319452846275249601221 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (5000000000000000 / 663581788621301 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 663581788621301 : ℝ) =
+              (663581788621301 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau98)
+  have hz1 :
+    (5000000000000000 / 663581788621301 : ℝ) ≤
+      1 / Real.sqrt (563636211443 / 32000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 663581788621301 : ℝ) = 1 / (663581788621301 / 5000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hau99 :
+    Real.arctan (156250000000000000000000000000 / 594080378247320218277056913069 : ℝ) ≤
+      (514373683695854022843 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat2 :
+    Real.arctan (750000000000000 / 1327163577242599 : ℝ) ≤
+      (2 * (514373683695854022843 / 2000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (717894851322593 / 625000000000000 : ℝ) ≤
+          Real.sqrt (1 + (750000000000000 / 1327163577242599 : ℝ) ^ 2))
+      (by
+        rw [show
+            (750000000000000 / 1327163577242599 : ℝ) /
+                (1 + (717894851322593 / 625000000000000 : ℝ)) =
+              (156250000000000000000000000000 / 594080378247320218277056913069 : ℝ)
+            by
+        norm_num];
+        exact hau99)
+  have hz2 :
+    (3 / 40) / Real.sqrt (563636211443 / 32000000000000 : ℝ) ≤
+      (750000000000000 / 1327163577242599 : ℝ) := by
+    rw [show
+        (750000000000000 / 1327163577242599 : ℝ) =
+          (3 / 40) / (1327163577242599 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1319452846275249601221 / 10000000000000000000000 : ℝ)) -
+        6 * (2 * (514373683695854022843 / 2000000000000000000000 : ℝ))) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (563636211443 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (563636211443 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1327163577242599 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1319452846275249601221 / 10000000000000000000000 : ℝ)) -
+          6 * (2 * (514373683695854022843 / 2000000000000000000000 : ℝ)))) ≤
+      Real.sqrt (563636211443 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (563636211443 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (563636211443 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_46 :
+    ((177793009397219830411 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(37480651333224899688083 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1339339149440871 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1331415769064528188357 / 10000000000000000000000 : ℝ)) -
+              6 * (2 * (510471313435554669691 / 2000000000000000000000 : ℝ))))) ≤
+      Vfield (71753174289 / 4000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (71753174289 / 4000000000000 : ℝ)) =
+      Real.log (4071753174289 / 4000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((71753174289 / 4000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (94253174289 / 4000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (177793009397219830411 / 10000000000000000000000 : ℝ) ≤
+      Real.log (4071753174289 / 4000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (94253174289 / 4000000000000 : ℝ) ≤
+      (-(37480651333224899688083 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show (94253174289 / 4000000000000 : ℝ) = (94253174289 / 62500000000 : ℝ) / 2 ^ 6 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (94253174289 / 62500000000 : ℝ) ≤
+            (4108179484775100311917 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1339339149440871 / 10000000000000000 : ℝ) ≤ Real.sqrt (71753174289 / 4000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (71753174289 / 4000000000000 : ℝ) ≤ (669669574720437 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (71753174289 / 4000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau100 :
+    Real.arctan (669669574720437 / 5000000000000000 : ℝ) ≤
+      (1331415769064528188357 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1331415769064528188357 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (5000000000000000 / 669669574720437 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 669669574720437 : ℝ) =
+              (669669574720437 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau100)
+  have hz1 :
+    (5000000000000000 / 669669574720437 : ℝ) ≤ 1 / Real.sqrt (71753174289 / 4000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 669669574720437 : ℝ) = 1 / (669669574720437 / 5000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hau101 :
+    Real.arctan (312500000000000000000000000000 / 1197655475685803382537060230697 : ℝ) ≤
+      (510471313435554669691 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat2 :
+    Real.arctan (250000000000000 / 446446383146957 : ℝ) ≤
+      (2 * (510471313435554669691 / 2000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (1432641232847821 / 1250000000000000 : ℝ) ≤
+          Real.sqrt (1 + (250000000000000 / 446446383146957 : ℝ) ^ 2))
+      (by
+        rw [show
+            (250000000000000 / 446446383146957 : ℝ) /
+                (1 + (1432641232847821 / 1250000000000000 : ℝ)) =
+              (312500000000000000000000000000 / 1197655475685803382537060230697 : ℝ)
+            by
+        norm_num];
+        exact hau101)
+  have hz2 :
+    (3 / 40) / Real.sqrt (71753174289 / 4000000000000 : ℝ) ≤
+      (250000000000000 / 446446383146957 : ℝ) := by
+    rw [show
+        (250000000000000 / 446446383146957 : ℝ) =
+          (3 / 40) / (1339339149440871 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1331415769064528188357 / 10000000000000000000000 : ℝ)) -
+        6 * (2 * (510471313435554669691 / 2000000000000000000000 : ℝ))) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (71753174289 / 4000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (71753174289 / 4000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1339339149440871 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1331415769064528188357 / 10000000000000000000000 : ℝ)) -
+          6 * (2 * (510471313435554669691 / 2000000000000000000000 : ℝ)))) ≤
+      Real.sqrt (71753174289 / 4000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (71753174289 / 4000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (71753174289 / 4000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_47 :
+    ((180981908014677709171 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(3734380897942405989187 / 1000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1351405029475109 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (335816783738066689991 / 2500000000000000000000 : ℝ)) -
+              6 * (2 * (2533285111816478099343 / 10000000000000000000000 : ℝ))))) ≤
+      Vfield (584414577181 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (584414577181 / 32000000000000 : ℝ)) =
+      Real.log (32584414577181 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((584414577181 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (764414577181 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (180981908014677709171 / 10000000000000000000000 : ℝ) ≤
+      Real.log (32584414577181 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (764414577181 / 32000000000000 : ℝ) ≤
+      (-(3734380897942405989187 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show (764414577181 / 32000000000000 : ℝ) = (764414577181 / 500000000000 : ℝ) / 2 ^ 6 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (764414577181 / 500000000000 : ℝ) ≤
+            (424502183857594010813 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1351405029475109 / 10000000000000000 : ℝ) ≤ Real.sqrt (584414577181 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (584414577181 / 32000000000000 : ℝ) ≤ (168925628684389 / 1250000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (584414577181 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau102 :
+    Real.arctan (168925628684389 / 1250000000000000 : ℝ) ≤
+      (335816783738066689991 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (335816783738066689991 / 2500000000000000000000 : ℝ)) ≤
+      Real.arctan (1250000000000000 / 168925628684389 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1250000000000000 / 168925628684389 : ℝ) =
+              (168925628684389 / 1250000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau102)
+  have hz1 :
+    (1250000000000000 / 168925628684389 : ℝ) ≤
+      1 / Real.sqrt (584414577181 / 32000000000000 : ℝ) := by
+    rw [show
+        (1250000000000000 / 168925628684389 : ℝ) = 1 / (168925628684389 / 1250000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hau103 :
+    Real.arctan (2500000000000000000000000000000 / 9656593059943326973085658954717 : ℝ) ≤
+      (2533285111816478099343 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat2 :
+    Real.arctan (750000000000000 / 1351405029475109 : ℝ) ≤
+      (2 * (2533285111816478099343 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (11436785085136139 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (750000000000000 / 1351405029475109 : ℝ) ^ 2))
+      (by
+        rw [show
+            (750000000000000 / 1351405029475109 : ℝ) /
+                (1 + (11436785085136139 / 10000000000000000 : ℝ)) =
+              (2500000000000000000000000000000 / 9656593059943326973085658954717 : ℝ)
+            by
+        norm_num];
+        exact hau103)
+  have hz2 :
+    (3 / 40) / Real.sqrt (584414577181 / 32000000000000 : ℝ) ≤
+      (750000000000000 / 1351405029475109 : ℝ) := by
+    rw [show
+        (750000000000000 / 1351405029475109 : ℝ) =
+          (3 / 40) / (1351405029475109 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (335816783738066689991 / 2500000000000000000000 : ℝ)) -
+        6 * (2 * (2533285111816478099343 / 10000000000000000000000 : ℝ))) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (584414577181 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (584414577181 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1351405029475109 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (335816783738066689991 / 2500000000000000000000 : ℝ)) -
+          6 * (2 * (2533285111816478099343 / 10000000000000000000000 : ℝ)))) ≤
+      Real.sqrt (584414577181 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (584414577181 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (584414577181 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_48 :
+    ((184169790048865831767 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(18604406978856690119019 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1363364129701323 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (677504937366921683649 / 5000000000000000000000 : ℝ)) -
+              6 * (2 * (1257318810145620304883 / 5000000000000000000000 : ℝ))))) ≤
+      Vfield (11896075201 / 640000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (11896075201 / 640000000000 : ℝ)) =
+      Real.log (651896075201 / 640000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((11896075201 / 640000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (15496075201 / 640000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (184169790048865831767 / 10000000000000000000000 : ℝ) ≤
+      Real.log (651896075201 / 640000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (15496075201 / 640000000000 : ℝ) ≤
+      (-(18604406978856690119019 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show (15496075201 / 640000000000 : ℝ) = (15496075201 / 10000000000 : ℝ) / 2 ^ 6 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15496075201 / 10000000000 : ℝ) ≤
+            (2190008430143309880981 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1363364129701323 / 10000000000000000 : ℝ) ≤ Real.sqrt (11896075201 / 640000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (11896075201 / 640000000000 : ℝ) ≤ (681682064850663 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (11896075201 / 640000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau104 :
+    Real.arctan (681682064850663 / 5000000000000000 : ℝ) ≤
+      (677504937366921683649 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (677504937366921683649 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (5000000000000000 / 681682064850663 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 681682064850663 : ℝ) =
+              (681682064850663 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau104)
+  have hz1 :
+    (5000000000000000 / 681682064850663 : ℝ) ≤ 1 / Real.sqrt (11896075201 / 640000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 681682064850663 : ℝ) = 1 / (681682064850663 / 5000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hau105 :
+    Real.arctan (250000000000000000000000000000 / 973134845774848592644041727587 : ℝ) ≤
+      (1257318810145620304883 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat2 :
+    Real.arctan (250000000000000 / 454454709900441 : ℝ) ≤
+      (2 * (1257318810145620304883 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (1141324150844507 / 1000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (250000000000000 / 454454709900441 : ℝ) ^ 2))
+      (by
+        rw [show
+            (250000000000000 / 454454709900441 : ℝ) /
+                (1 + (1141324150844507 / 1000000000000000 : ℝ)) =
+              (250000000000000000000000000000 / 973134845774848592644041727587 : ℝ)
+            by
+        norm_num];
+        exact hau105)
+  have hz2 :
+    (3 / 40) / Real.sqrt (11896075201 / 640000000000 : ℝ) ≤
+      (250000000000000 / 454454709900441 : ℝ) := by
+    rw [show
+        (250000000000000 / 454454709900441 : ℝ) =
+          (3 / 40) / (1363364129701323 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (677504937366921683649 / 5000000000000000000000 : ℝ)) -
+        6 * (2 * (1257318810145620304883 / 5000000000000000000000 : ℝ))) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (11896075201 / 640000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (11896075201 / 640000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1363364129701323 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (677504937366921683649 / 5000000000000000000000 : ℝ)) -
+          6 * (2 * (1257318810145620304883 / 5000000000000000000000 : ℝ)))) ≤
+      Real.sqrt (11896075201 / 640000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (11896075201 / 640000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (11896075201 / 640000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/V04.lean b/LeanPool/Zeta5Irrational/Table/V04.lean
new file mode 100644
index 0000000000..051631c3ee
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/V04.lean
@@ -0,0 +1,1295 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Energy
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # V04: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_49 :
+    ((9367832807386358049 / 500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(37075617054825365057033 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1375219235839097 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (683323396048354980263 / 5000000000000000000000 : ℝ)) -
+              6 * (2 * (62409962799765955357 / 250000000000000000000 : ℝ))))) ≤
+      Vfield (605192942919 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (605192942919 / 32000000000000 : ℝ)) =
+      Real.log (32605192942919 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((605192942919 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (785192942919 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (9367832807386358049 / 500000000000000000000 : ℝ) ≤
+      Real.log (32605192942919 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (785192942919 / 32000000000000 : ℝ) ≤
+      (-(37075617054825365057033 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show (785192942919 / 32000000000000 : ℝ) = (785192942919 / 500000000000 : ℝ) / 2 ^ 6 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (785192942919 / 500000000000 : ℝ) ≤
+            (4513213763174634942967 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1375219235839097 / 10000000000000000 : ℝ) ≤ Real.sqrt (605192942919 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (605192942919 / 32000000000000 : ℝ) ≤ (13752192358391 / 100000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (605192942919 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau106 :
+    Real.arctan (13752192358391 / 100000000000000 : ℝ) ≤
+      (683323396048354980263 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (683323396048354980263 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (100000000000000 / 13752192358391 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (100000000000000 / 13752192358391 : ℝ) =
+              (13752192358391 / 100000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau106)
+  have hz1 :
+    (100000000000000 / 13752192358391 : ℝ) ≤ 1 / Real.sqrt (605192942919 / 32000000000000 : ℝ) := by
+    rw [show (100000000000000 / 13752192358391 : ℝ) = 1 / (13752192358391 / 100000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hau107 :
+    Real.arctan (1500000000000000000000000000000 / 5883314432594533368393189293789 : ℝ) ≤
+      (62409962799765955357 / 250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat2 :
+    Real.arctan (750000000000000 / 1375219235839097 : ℝ) ≤
+      (2 * (62409962799765955357 / 250000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (2278092015637637 / 2000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (750000000000000 / 1375219235839097 : ℝ) ^ 2))
+      (by
+        rw [show
+            (750000000000000 / 1375219235839097 : ℝ) /
+                (1 + (2278092015637637 / 2000000000000000 : ℝ)) =
+              (1500000000000000000000000000000 / 5883314432594533368393189293789 : ℝ)
+            by
+        norm_num];
+        exact hau107)
+  have hz2 :
+    (3 / 40) / Real.sqrt (605192942919 / 32000000000000 : ℝ) ≤
+      (750000000000000 / 1375219235839097 : ℝ) := by
+    rw [show
+        (750000000000000 / 1375219235839097 : ℝ) =
+          (3 / 40) / (1375219235839097 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (683323396048354980263 / 5000000000000000000000 : ℝ)) -
+        6 * (2 * (62409962799765955357 / 250000000000000000000 : ℝ))) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (605192942919 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (605192942919 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1375219235839097 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (683323396048354980263 / 5000000000000000000000 : ℝ)) -
+          6 * (2 * (62409962799765955357 / 250000000000000000000 : ℝ)))) ≤
+      Real.sqrt (605192942919 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (605192942919 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (605192942919 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_50 :
+    ((19054250695858538469 / 1000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(7388834199649120704961 / 2000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((173371626818641 / 1250000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1378180571218092944099 / 10000000000000000000000 : ℝ)) -
+              6 * (2 * (1239276500758368893079 / 5000000000000000000000 : ℝ))))) ≤
+      Vfield (153895531447 / 8000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (153895531447 / 8000000000000 : ℝ)) =
+      Real.log (8153895531447 / 8000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((153895531447 / 8000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (198895531447 / 8000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (19054250695858538469 / 1000000000000000000000 : ℝ) ≤
+      Real.log (8153895531447 / 8000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (198895531447 / 8000000000000 : ℝ) ≤
+      (-(7388834199649120704961 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show (198895531447 / 8000000000000 : ℝ) = (198895531447 / 125000000000 : ℝ) / 2 ^ 6 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (198895531447 / 125000000000 : ℝ) ≤
+            (928931963950879295039 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (173371626818641 / 1250000000000000 : ℝ) ≤ Real.sqrt (153895531447 / 8000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (153895531447 / 8000000000000 : ℝ) ≤ (1386973014549131 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (153895531447 / 8000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau108 :
+    Real.arctan (1386973014549131 / 10000000000000000 : ℝ) ≤
+      (1378180571218092944099 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1378180571218092944099 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 1386973014549131 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 1386973014549131 : ℝ) =
+              (1386973014549131 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau108)
+  have hz1 :
+    (10000000000000000 / 1386973014549131 : ℝ) ≤
+      1 / Real.sqrt (153895531447 / 8000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 1386973014549131 : ℝ) =
+          1 / (1386973014549131 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hau109 :
+    Real.arctan (312500000000000000000000000000 / 1234891668870499026098149566991 : ℝ) ≤
+      (1239276500758368893079 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat2 :
+    Real.arctan (93750000000000 / 173371626818641 : ℝ) ≤
+      (2 * (1239276500758368893079 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (11368404245793053 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (93750000000000 / 173371626818641 : ℝ) ^ 2))
+      (by
+        rw [show
+            (93750000000000 / 173371626818641 : ℝ) /
+                (1 + (11368404245793053 / 10000000000000000 : ℝ)) =
+              (312500000000000000000000000000 / 1234891668870499026098149566991 : ℝ)
+            by
+        norm_num];
+        exact hau109)
+  have hz2 :
+    (3 / 40) / Real.sqrt (153895531447 / 8000000000000 : ℝ) ≤
+      (93750000000000 / 173371626818641 : ℝ) := by
+    rw [show
+        (93750000000000 / 173371626818641 : ℝ) =
+          (3 / 40) / (173371626818641 / 1250000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1378180571218092944099 / 10000000000000000000000 : ℝ)) -
+        6 * (2 * (1239276500758368893079 / 5000000000000000000000 : ℝ))) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (153895531447 / 8000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (153895531447 / 8000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((173371626818641 / 1250000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1378180571218092944099 / 10000000000000000000000 : ℝ)) -
+          6 * (2 * (1239276500758368893079 / 5000000000000000000000 : ℝ)))) ≤
+      Real.sqrt (153895531447 / 8000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (153895531447 / 8000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (153895531447 / 8000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_51 :
+    ((19691116530249562487 / 1000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(36686351439827391625567 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1410186702539329 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (140094889551191353099 / 1000000000000000000000 : ℝ)) -
+              6 * (2 * (1221993650998643614353 / 5000000000000000000000 : ℝ))))) ≤
+      Vfield (318180245763 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (318180245763 / 16000000000000 : ℝ)) =
+      Real.log (16318180245763 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((318180245763 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (408180245763 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (19691116530249562487 / 1000000000000000000000 : ℝ) ≤
+      Real.log (16318180245763 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (408180245763 / 16000000000000 : ℝ) ≤
+      (-(36686351439827391625567 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show (408180245763 / 16000000000000 : ℝ) = (408180245763 / 250000000000 : ℝ) / 2 ^ 6 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (408180245763 / 250000000000 : ℝ) ≤
+            (4902479378172608374433 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1410186702539329 / 10000000000000000 : ℝ) ≤ Real.sqrt (318180245763 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (318180245763 / 16000000000000 : ℝ) ≤ (352546675634833 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (318180245763 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau110 :
+    Real.arctan (352546675634833 / 2500000000000000 : ℝ) ≤
+      (140094889551191353099 / 1000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (140094889551191353099 / 1000000000000000000000 : ℝ)) ≤
+      Real.arctan (2500000000000000 / 352546675634833 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 352546675634833 : ℝ) =
+              (352546675634833 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau110)
+  have hz1 :
+    (2500000000000000 / 352546675634833 : ℝ) ≤
+      1 / Real.sqrt (318180245763 / 16000000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 352546675634833 : ℝ) = 1 / (352546675634833 / 2500000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hau111 :
+    Real.arctan (7500000000000000000000000000000 / 30074113379675286162750332138029 : ℝ) ≤
+      (1221993650998643614353 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat2 :
+    Real.arctan (750000000000000 / 1410186702539329 : ℝ) ≤
+      (2 * (1221993650998643614353 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (11326334538200301 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (750000000000000 / 1410186702539329 : ℝ) ^ 2))
+      (by
+        rw [show
+            (750000000000000 / 1410186702539329 : ℝ) /
+                (1 + (11326334538200301 / 10000000000000000 : ℝ)) =
+              (7500000000000000000000000000000 / 30074113379675286162750332138029 : ℝ)
+            by
+        norm_num];
+        exact hau111)
+  have hz2 :
+    (3 / 40) / Real.sqrt (318180245763 / 16000000000000 : ℝ) ≤
+      (750000000000000 / 1410186702539329 : ℝ) := by
+    rw [show
+        (750000000000000 / 1410186702539329 : ℝ) =
+          (3 / 40) / (1410186702539329 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (140094889551191353099 / 1000000000000000000000 : ℝ)) -
+        6 * (2 * (1221993650998643614353 / 5000000000000000000000 : ℝ))) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (318180245763 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (318180245763 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1410186702539329 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (140094889551191353099 / 1000000000000000000000 : ℝ)) -
+          6 * (2 * (1221993650998643614353 / 5000000000000000000000 : ℝ)))) ≤
+      Real.sqrt (318180245763 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (318180245763 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (318180245763 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_52 :
+    ((203275770246830504773 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(36435012247301481743849 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1433024399286347 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1423334185029087098537 / 10000000000000000000000 : ℝ)) -
+              6 * (2 * (150677267173378803647 / 625000000000000000000 : ℝ))))) ≤
+      Vfield (41071178579 / 2000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (41071178579 / 2000000000000 : ℝ)) =
+      Real.log (2041071178579 / 2000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((41071178579 / 2000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (52321178579 / 2000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (203275770246830504773 / 10000000000000000000000 : ℝ) ≤
+      Real.log (2041071178579 / 2000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (52321178579 / 2000000000000 : ℝ) ≤
+      (-(36435012247301481743849 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show (52321178579 / 2000000000000 : ℝ) = (52321178579 / 31250000000 : ℝ) / 2 ^ 6 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (52321178579 / 31250000000 : ℝ) ≤
+            (5153818570698518256151 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1433024399286347 / 10000000000000000 : ℝ) ≤ Real.sqrt (41071178579 / 2000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (41071178579 / 2000000000000 : ℝ) ≤ (28660487985727 / 200000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (41071178579 / 2000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau112 :
+    Real.arctan (28660487985727 / 200000000000000 : ℝ) ≤
+      (1423334185029087098537 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1423334185029087098537 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (200000000000000 / 28660487985727 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (200000000000000 / 28660487985727 : ℝ) =
+              (28660487985727 / 200000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau112)
+  have hz1 :
+    (200000000000000 / 28660487985727 : ℝ) ≤ 1 / Real.sqrt (41071178579 / 2000000000000 : ℝ) := by
+    rw [show (200000000000000 / 28660487985727 : ℝ) = 1 / (28660487985727 / 200000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hau113 :
+    Real.arctan (312500000000000000000000000000 / 1271019977433822553363219150469 : ℝ) ≤
+      (150677267173378803647 / 625000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat2 :
+    Real.arctan (750000000000000 / 1433024399286347 : ℝ) ≤
+      (2 * (150677267173378803647 / 625000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (1410847878235981 / 1250000000000000 : ℝ) ≤
+          Real.sqrt (1 + (750000000000000 / 1433024399286347 : ℝ) ^ 2))
+      (by
+        rw [show
+            (750000000000000 / 1433024399286347 : ℝ) /
+                (1 + (1410847878235981 / 1250000000000000 : ℝ)) =
+              (312500000000000000000000000000 / 1271019977433822553363219150469 : ℝ)
+            by
+        norm_num];
+        exact hau113)
+  have hz2 :
+    (3 / 40) / Real.sqrt (41071178579 / 2000000000000 : ℝ) ≤
+      (750000000000000 / 1433024399286347 : ℝ) := by
+    rw [show
+        (750000000000000 / 1433024399286347 : ℝ) =
+          (3 / 40) / (1433024399286347 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1423334185029087098537 / 10000000000000000000000 : ℝ)) -
+        6 * (2 * (150677267173378803647 / 625000000000000000000 : ℝ))) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (41071178579 / 2000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (41071178579 / 2000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1433024399286347 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1423334185029087098537 / 10000000000000000000000 : ℝ)) -
+          6 * (2 * (150677267173378803647 / 625000000000000000000 : ℝ)))) ≤
+      Real.sqrt (41071178579 / 2000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (41071178579 / 2000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (41071178579 / 2000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_53 :
+    ((107996420276222433859 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(35950526579446718972183 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1477641267294771 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (146702560506701594711 / 1000000000000000000000 : ℝ)) -
+              6 * (2 * (2348409106350313720017 / 10000000000000000000000 : ℝ))))) ≤
+      Vfield (34934779437 / 1600000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (34934779437 / 1600000000000 : ℝ)) =
+      Real.log (1634934779437 / 1600000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((34934779437 / 1600000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (43934779437 / 1600000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (107996420276222433859 / 5000000000000000000000 : ℝ) ≤
+      Real.log (1634934779437 / 1600000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (43934779437 / 1600000000000 : ℝ) ≤
+      (-(35950526579446718972183 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show (43934779437 / 1600000000000 : ℝ) = (43934779437 / 25000000000 : ℝ) / 2 ^ 6 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (43934779437 / 25000000000 : ℝ) ≤
+            (5638304238553281027817 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1477641267294771 / 10000000000000000 : ℝ) ≤ Real.sqrt (34934779437 / 1600000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (34934779437 / 1600000000000 : ℝ) ≤ (738820633647387 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (34934779437 / 1600000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau114 :
+    Real.arctan (738820633647387 / 5000000000000000 : ℝ) ≤
+      (146702560506701594711 / 1000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (146702560506701594711 / 1000000000000000000000 : ℝ)) ≤
+      Real.arctan (5000000000000000 / 738820633647387 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 738820633647387 : ℝ) =
+              (738820633647387 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau114)
+  have hz1 :
+    (5000000000000000 / 738820633647387 : ℝ) ≤ 1 / Real.sqrt (34934779437 / 1600000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 738820633647387 : ℝ) = 1 / (738820633647387 / 5000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hau115 :
+    Real.arctan (1250000000000000000000000000000 / 5224540267429134358032754116807 : ℝ) ≤
+      (2348409106350313720017 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat2 :
+    Real.arctan (250000000000000 / 492547089098257 : ℝ) ≤
+      (2 * (2348409106350313720017 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (5607189410040151 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (250000000000000 / 492547089098257 : ℝ) ^ 2))
+      (by
+        rw [show
+            (250000000000000 / 492547089098257 : ℝ) /
+                (1 + (5607189410040151 / 5000000000000000 : ℝ)) =
+              (1250000000000000000000000000000 / 5224540267429134358032754116807 : ℝ)
+            by
+        norm_num];
+        exact hau115)
+  have hz2 :
+    (3 / 40) / Real.sqrt (34934779437 / 1600000000000 : ℝ) ≤
+      (250000000000000 / 492547089098257 : ℝ) := by
+    rw [show
+        (250000000000000 / 492547089098257 : ℝ) =
+          (3 / 40) / (1477641267294771 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (146702560506701594711 / 1000000000000000000000 : ℝ)) -
+        6 * (2 * (2348409106350313720017 / 10000000000000000000000 : ℝ))) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (34934779437 / 1600000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (34934779437 / 1600000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1477641267294771 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (146702560506701594711 / 1000000000000000000000 : ℝ)) -
+          6 * (2 * (2348409106350313720017 / 10000000000000000000000 : ℝ)))) ≤
+      Real.sqrt (34934779437 / 1600000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (34934779437 / 1600000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (34934779437 / 1600000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_54 :
+    ((114346879504294972067 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(35488432866774594506247 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1520949867903277 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1509382020844949857689 / 10000000000000000000000 : ℝ)) -
+              6 * (4581227649370826993687 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (92531540027 / 4000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (92531540027 / 4000000000000 : ℝ)) =
+      Real.log (4092531540027 / 4000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((92531540027 / 4000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (115031540027 / 4000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (114346879504294972067 / 5000000000000000000000 : ℝ) ≤
+      Real.log (4092531540027 / 4000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (115031540027 / 4000000000000 : ℝ) ≤
+      (-(35488432866774594506247 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 6
+        (show (115031540027 / 4000000000000 : ℝ) = (115031540027 / 62500000000 : ℝ) / 2 ^ 6 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (115031540027 / 62500000000 : ℝ) ≤
+            (6100397951225405493753 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1520949867903277 / 10000000000000000 : ℝ) ≤ Real.sqrt (92531540027 / 4000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (92531540027 / 4000000000000 : ℝ) ≤ (19011873348791 / 125000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (92531540027 / 4000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau116 :
+    Real.arctan (19011873348791 / 125000000000000 : ℝ) ≤
+      (1509382020844949857689 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1509382020844949857689 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (125000000000000 / 19011873348791 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (125000000000000 / 19011873348791 : ℝ) =
+              (19011873348791 / 125000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau116)
+  have hz1 :
+    (125000000000000 / 19011873348791 : ℝ) ≤ 1 / Real.sqrt (92531540027 / 4000000000000 : ℝ) := by
+    rw [show (125000000000000 / 19011873348791 : ℝ) = 1 / (19011873348791 / 125000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 1520949867903277 : ℝ) ≤
+      (4581227649370826993687 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (92531540027 / 4000000000000 : ℝ) ≤
+      (750000000000000 / 1520949867903277 : ℝ) := by
+    rw [show
+        (750000000000000 / 1520949867903277 : ℝ) =
+          (3 / 40) / (1520949867903277 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1509382020844949857689 / 10000000000000000000000 : ℝ)) -
+        6 * (4581227649370826993687 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (92531540027 / 4000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (92531540027 / 4000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1520949867903277 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1509382020844949857689 / 10000000000000000000000 : ℝ)) -
+          6 * (4581227649370826993687 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (92531540027 / 4000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (92531540027 / 4000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (92531540027 / 4000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_55 :
+    ((127023652061630830299 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(34623757699024104622901 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((40101574722821 / 250000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1590513943220726835857 / 10000000000000000000000 : ℝ)) -
+              6 * (218681344812415938491 / 500000000000000000000 : ℝ)))) ≤
+      Vfield (6432545181 / 250000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (6432545181 / 250000000000 : ℝ)) = Real.log (256432545181 / 250000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((6432545181 / 250000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (7838795181 / 250000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (127023652061630830299 / 5000000000000000000000 : ℝ) ≤
+      Real.log (256432545181 / 250000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (7838795181 / 250000000000 : ℝ) ≤
+      (-(34623757699024104622901 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show (7838795181 / 250000000000 : ℝ) = (7838795181 / 7812500000 : ℝ) / 2 ^ 5 by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7838795181 / 7812500000 : ℝ) ≤
+            (33601315975895377099 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (40101574722821 / 250000000000000 : ℝ) ≤ Real.sqrt (6432545181 / 250000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (6432545181 / 250000000000 : ℝ) ≤ (1604062988912843 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (6432545181 / 250000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau117 :
+    Real.arctan (1604062988912843 / 10000000000000000 : ℝ) ≤
+      (1590513943220726835857 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1590513943220726835857 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 1604062988912843 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 1604062988912843 : ℝ) =
+              (1604062988912843 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau117)
+  have hz1 :
+    (10000000000000000 / 1604062988912843 : ℝ) ≤ 1 / Real.sqrt (6432545181 / 250000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 1604062988912843 : ℝ) =
+          1 / (1604062988912843 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (18750000000000 / 40101574722821 : ℝ) ≤
+      (218681344812415938491 / 500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (6432545181 / 250000000000 : ℝ) ≤
+      (18750000000000 / 40101574722821 : ℝ) := by
+    rw [show
+        (18750000000000 / 40101574722821 : ℝ) = (3 / 40) / (40101574722821 / 250000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1590513943220726835857 / 10000000000000000000000 : ℝ)) -
+        6 * (218681344812415938491 / 500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (6432545181 / 250000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (6432545181 / 250000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((40101574722821 / 250000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1590513943220726835857 / 10000000000000000000000 : ℝ)) -
+          6 * (218681344812415938491 / 500000000000000000000 : ℝ))) ≤
+      Real.sqrt (6432545181 / 250000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (6432545181 / 250000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (6432545181 / 250000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_56 :
+    ((32987613906975619173 / 1250000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(34306346441541073575049 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1635279580656621 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (324186506259879103831 / 2000000000000000000000 : ℝ)) -
+              6 * (1075033352401525906013 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (213931144553 / 8000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (213931144553 / 8000000000000 : ℝ)) =
+      Real.log (8213931144553 / 8000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((213931144553 / 8000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (258931144553 / 8000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (32987613906975619173 / 1250000000000000000000 : ℝ) ≤
+      Real.log (8213931144553 / 8000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (258931144553 / 8000000000000 : ℝ) ≤
+      (-(34306346441541073575049 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show (258931144553 / 8000000000000 : ℝ) = (258931144553 / 250000000000 : ℝ) / 2 ^ 5 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (258931144553 / 250000000000 : ℝ) ≤
+            (351012573458926424951 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1635279580656621 / 10000000000000000 : ℝ) ≤ Real.sqrt (213931144553 / 8000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (213931144553 / 8000000000000 : ℝ) ≤ (102204973791039 / 625000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (213931144553 / 8000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau118 :
+    Real.arctan (102204973791039 / 625000000000000 : ℝ) ≤
+      (324186506259879103831 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (324186506259879103831 / 2000000000000000000000 : ℝ)) ≤
+      Real.arctan (625000000000000 / 102204973791039 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (625000000000000 / 102204973791039 : ℝ) =
+              (102204973791039 / 625000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau118)
+  have hz1 :
+    (625000000000000 / 102204973791039 : ℝ) ≤ 1 / Real.sqrt (213931144553 / 8000000000000 : ℝ) := by
+    rw [show (625000000000000 / 102204973791039 : ℝ) = 1 / (102204973791039 / 625000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 545093193552207 : ℝ) ≤
+      (1075033352401525906013 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (213931144553 / 8000000000000 : ℝ) ≤
+      (250000000000000 / 545093193552207 : ℝ) := by
+    rw [show
+        (250000000000000 / 545093193552207 : ℝ) =
+          (3 / 40) / (1635279580656621 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (324186506259879103831 / 2000000000000000000000 : ℝ)) -
+        6 * (1075033352401525906013 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (213931144553 / 8000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (213931144553 / 8000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1635279580656621 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (324186506259879103831 / 2000000000000000000000 : ℝ)) -
+          6 * (1075033352401525906013 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (213931144553 / 8000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (213931144553 / 8000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (213931144553 / 8000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_57 :
+    ((13441203810162759651 / 500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(17075670345849735943963 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1650666509071031 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (817957512773989300277 / 5000000000000000000000 : ℝ)) -
+              6 * (2132376865167301685317 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (435951987867 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (435951987867 / 16000000000000 : ℝ)) =
+      Real.log (16435951987867 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((435951987867 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (525951987867 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (13441203810162759651 / 500000000000000000000 : ℝ) ≤
+      Real.log (16435951987867 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (525951987867 / 16000000000000 : ℝ) ≤
+      (-(17075670345849735943963 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show (525951987867 / 16000000000000 : ℝ) = (525951987867 / 500000000000 : ℝ) / 2 ^ 5 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (525951987867 / 500000000000 : ℝ) ≤
+            (253009161650264056037 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1650666509071031 / 10000000000000000 : ℝ) ≤ Real.sqrt (435951987867 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (435951987867 / 16000000000000 : ℝ) ≤ (825333254535517 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (435951987867 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau119 :
+    Real.arctan (825333254535517 / 5000000000000000 : ℝ) ≤
+      (817957512773989300277 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (817957512773989300277 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (5000000000000000 / 825333254535517 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 825333254535517 : ℝ) =
+              (825333254535517 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau119)
+  have hz1 :
+    (5000000000000000 / 825333254535517 : ℝ) ≤
+      1 / Real.sqrt (435951987867 / 16000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 825333254535517 : ℝ) = 1 / (825333254535517 / 5000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 1650666509071031 : ℝ) ≤
+      (2132376865167301685317 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (435951987867 / 16000000000000 : ℝ) ≤
+      (750000000000000 / 1650666509071031 : ℝ) := by
+    rw [show
+        (750000000000000 / 1650666509071031 : ℝ) =
+          (3 / 40) / (1650666509071031 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (817957512773989300277 / 5000000000000000000000 : ℝ)) -
+        6 * (2132376865167301685317 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (435951987867 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (435951987867 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1650666509071031 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (817957512773989300277 / 5000000000000000000000 : ℝ)) -
+          6 * (2132376865167301685317 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (435951987867 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (435951987867 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (435951987867 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_58 :
+    ((34218102323501764669 / 1250000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(33998700991624565113719 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((83295566229917 / 500000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (66030073509107242779 / 400000000000000000000 : ℝ)) -
+              6 * (846047177419245798601 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (111010421657 / 4000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (111010421657 / 4000000000000 : ℝ)) =
+      Real.log (4111010421657 / 4000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((111010421657 / 4000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (133510421657 / 4000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (34218102323501764669 / 1250000000000000000000 : ℝ) ≤
+      Real.log (4111010421657 / 4000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (133510421657 / 4000000000000 : ℝ) ≤
+      (-(33998700991624565113719 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show (133510421657 / 4000000000000 : ℝ) = (133510421657 / 125000000000 : ℝ) / 2 ^ 5 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (133510421657 / 125000000000 : ℝ) ≤
+            (658658023375434886281 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (83295566229917 / 500000000000000 : ℝ) ≤ Real.sqrt (111010421657 / 4000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (111010421657 / 4000000000000 : ℝ) ≤ (1665911324598343 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (111010421657 / 4000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau120 :
+    Real.arctan (1665911324598343 / 10000000000000000 : ℝ) ≤
+      (66030073509107242779 / 400000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (66030073509107242779 / 400000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 1665911324598343 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 1665911324598343 : ℝ) =
+              (1665911324598343 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau120)
+  have hz1 :
+    (10000000000000000 / 1665911324598343 : ℝ) ≤
+      1 / Real.sqrt (111010421657 / 4000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 1665911324598343 : ℝ) =
+          1 / (1665911324598343 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (37500000000000 / 83295566229917 : ℝ) ≤
+      (846047177419245798601 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (111010421657 / 4000000000000 : ℝ) ≤
+      (37500000000000 / 83295566229917 : ℝ) := by
+    rw [show
+        (37500000000000 / 83295566229917 : ℝ) = (3 / 40) / (83295566229917 / 500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (66030073509107242779 / 400000000000000000000 : ℝ)) -
+        6 * (846047177419245798601 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (111010421657 / 4000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (111010421657 / 4000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((83295566229917 / 500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (66030073509107242779 / 400000000000000000000 : ℝ)) -
+          6 * (846047177419245798601 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (111010421657 / 4000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (111010421657 / 4000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (111010421657 / 4000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_59 :
+    ((27866314079307059799 / 1000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(33848356193919409894199 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((42025447340579 / 250000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (208180859025354406709 / 1250000000000000000000 : ℝ)) -
+              6 * (4196545340860832545349 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (452131385389 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (452131385389 / 16000000000000 : ℝ)) =
+      Real.log (16452131385389 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((452131385389 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (542131385389 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (27866314079307059799 / 1000000000000000000000 : ℝ) ≤
+      Real.log (16452131385389 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (542131385389 / 16000000000000 : ℝ) ≤
+      (-(33848356193919409894199 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show (542131385389 / 16000000000000 : ℝ) = (542131385389 / 500000000000 : ℝ) / 2 ^ 5 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (542131385389 / 500000000000 : ℝ) ≤
+            (809002821080590105801 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (42025447340579 / 250000000000000 : ℝ) ≤ Real.sqrt (452131385389 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (452131385389 / 16000000000000 : ℝ) ≤ (1681017893623163 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (452131385389 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau121 :
+    Real.arctan (1681017893623163 / 10000000000000000 : ℝ) ≤
+      (208180859025354406709 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (208180859025354406709 / 1250000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 1681017893623163 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 1681017893623163 : ℝ) =
+              (1681017893623163 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau121)
+  have hz1 :
+    (10000000000000000 / 1681017893623163 : ℝ) ≤
+      1 / Real.sqrt (452131385389 / 16000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 1681017893623163 : ℝ) =
+          1 / (1681017893623163 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (18750000000000 / 42025447340579 : ℝ) ≤
+      (4196545340860832545349 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (452131385389 / 16000000000000 : ℝ) ≤
+      (18750000000000 / 42025447340579 : ℝ) := by
+    rw [show
+        (18750000000000 / 42025447340579 : ℝ) = (3 / 40) / (42025447340579 / 250000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (208180859025354406709 / 1250000000000000000000 : ℝ)) -
+        6 * (4196545340860832545349 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (452131385389 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (452131385389 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((42025447340579 / 250000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (208180859025354406709 / 1250000000000000000000 : ℝ)) -
+          6 * (4196545340860832545349 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (452131385389 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (452131385389 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (452131385389 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_60 :
+    ((4392521797998957757 / 156250000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(33774023019545920101553 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((844260248280909 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (418185598816008147827 / 2500000000000000000000 : ℝ)) -
+              6 * (1045000009396149589471 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (912352469539 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (912352469539 / 32000000000000 : ℝ)) =
+      Real.log (32912352469539 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((912352469539 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1092352469539 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (4392521797998957757 / 156250000000000000000 : ℝ) ≤
+      Real.log (32912352469539 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1092352469539 / 32000000000000 : ℝ) ≤
+      (-(33774023019545920101553 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show (1092352469539 / 32000000000000 : ℝ) = (1092352469539 / 1000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1092352469539 / 1000000000000 : ℝ) ≤
+            (883335995454079898447 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (844260248280909 / 5000000000000000 : ℝ) ≤ Real.sqrt (912352469539 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (912352469539 / 32000000000000 : ℝ) ≤ (1688520496561821 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (912352469539 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau122 :
+    Real.arctan (1688520496561821 / 10000000000000000 : ℝ) ≤
+      (418185598816008147827 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (418185598816008147827 / 2500000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 1688520496561821 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 1688520496561821 : ℝ) =
+              (1688520496561821 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau122)
+  have hz1 :
+    (10000000000000000 / 1688520496561821 : ℝ) ≤
+      1 / Real.sqrt (912352469539 / 32000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 1688520496561821 : ℝ) =
+          1 / (1688520496561821 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (125000000000000 / 281420082760303 : ℝ) ≤
+      (1045000009396149589471 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (912352469539 / 32000000000000 : ℝ) ≤
+      (125000000000000 / 281420082760303 : ℝ) := by
+    rw [show
+        (125000000000000 / 281420082760303 : ℝ) =
+          (3 / 40) / (844260248280909 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (418185598816008147827 / 2500000000000000000000 : ℝ)) -
+        6 * (1045000009396149589471 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (912352469539 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (912352469539 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((844260248280909 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (418185598816008147827 / 2500000000000000000000 : ℝ)) -
+          6 * (1045000009396149589471 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (912352469539 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (912352469539 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (912352469539 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/V05.lean b/LeanPool/Zeta5Irrational/Table/V05.lean
new file mode 100644
index 0000000000..54e002838c
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/V05.lean
@@ -0,0 +1,1224 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Energy
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # V05: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_61 :
+    ((283579045197899142671 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(33700238312826287098913 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1695989910328919 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1680003860151625494353 / 10000000000000000000000 : ℝ)) -
+              6 * (4163649465840225933501 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (9204421683 / 320000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (9204421683 / 320000000000 : ℝ)) = Real.log (329204421683 / 320000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((9204421683 / 320000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (11004421683 / 320000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (283579045197899142671 / 10000000000000000000000 : ℝ) ≤
+      Real.log (329204421683 / 320000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (11004421683 / 320000000000 : ℝ) ≤
+      (-(33700238312826287098913 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show (11004421683 / 320000000000 : ℝ) = (11004421683 / 10000000000 : ℝ) / 2 ^ 5 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11004421683 / 10000000000 : ℝ) ≤
+            (957120702173712901087 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1695989910328919 / 10000000000000000 : ℝ) ≤ Real.sqrt (9204421683 / 320000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (9204421683 / 320000000000 : ℝ) ≤ (847994955164461 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (9204421683 / 320000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau123 :
+    Real.arctan (847994955164461 / 5000000000000000 : ℝ) ≤
+      (1680003860151625494353 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1680003860151625494353 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (5000000000000000 / 847994955164461 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 847994955164461 : ℝ) =
+              (847994955164461 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau123)
+  have hz1 :
+    (5000000000000000 / 847994955164461 : ℝ) ≤ 1 / Real.sqrt (9204421683 / 320000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 847994955164461 : ℝ) = 1 / (847994955164461 / 5000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 1695989910328919 : ℝ) ≤
+      (4163649465840225933501 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (9204421683 / 320000000000 : ℝ) ≤
+      (750000000000000 / 1695989910328919 : ℝ) := by
+    rw [show
+        (750000000000000 / 1695989910328919 : ℝ) =
+          (3 / 40) / (1695989910328919 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1680003860151625494353 / 10000000000000000000000 : ℝ)) -
+        6 * (4163649465840225933501 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (9204421683 / 320000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (9204421683 / 320000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1695989910328919 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1680003860151625494353 / 10000000000000000000000 : ℝ)) -
+          6 * (4163649465840225933501 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (9204421683 / 320000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (9204421683 / 320000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (9204421683 / 320000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_62 :
+    ((28603609146785446397 / 1000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(6725398807852979702343 / 2000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((851713285760769 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (42180792698504378977 / 250000000000000000000 : ℝ)) -
+              6 * (2073744912138374589189 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (928531867061 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (928531867061 / 32000000000000 : ℝ)) =
+      Real.log (32928531867061 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((928531867061 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1108531867061 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (28603609146785446397 / 1000000000000000000000 : ℝ) ≤
+      Real.log (32928531867061 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1108531867061 / 32000000000000 : ℝ) ≤
+      (-(6725398807852979702343 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show (1108531867061 / 32000000000000 : ℝ) = (1108531867061 / 1000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1108531867061 / 1000000000000 : ℝ) ≤
+            (206072995147020297657 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (851713285760769 / 5000000000000000 : ℝ) ≤ Real.sqrt (928531867061 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (928531867061 / 32000000000000 : ℝ) ≤ (1703426571521541 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (928531867061 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau124 :
+    Real.arctan (1703426571521541 / 10000000000000000 : ℝ) ≤
+      (42180792698504378977 / 250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (42180792698504378977 / 250000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 1703426571521541 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 1703426571521541 : ℝ) =
+              (1703426571521541 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau124)
+  have hz1 :
+    (10000000000000000 / 1703426571521541 : ℝ) ≤
+      1 / Real.sqrt (928531867061 / 32000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 1703426571521541 : ℝ) =
+          1 / (1703426571521541 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (125000000000000 / 283904428586923 : ℝ) ≤
+      (2073744912138374589189 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (928531867061 / 32000000000000 : ℝ) ≤
+      (125000000000000 / 283904428586923 : ℝ) := by
+    rw [show
+        (125000000000000 / 283904428586923 : ℝ) =
+          (3 / 40) / (851713285760769 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (42180792698504378977 / 250000000000000000000 : ℝ)) -
+        6 * (2073744912138374589189 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (928531867061 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (928531867061 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((851713285760769 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (42180792698504378977 / 250000000000000000000 : ℝ)) -
+          6 * (2073744912138374589189 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (928531867061 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (928531867061 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (928531867061 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_63 :
+    ((2307940273427734419 / 80000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(33554282339629042240773 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1710830907247629 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (847213185079392045819 / 5000000000000000000000 : ℝ)) -
+              6 * (2065758707467686423337 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (468310782911 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (468310782911 / 16000000000000 : ℝ)) =
+      Real.log (16468310782911 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((468310782911 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (558310782911 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2307940273427734419 / 80000000000000000000 : ℝ) ≤
+      Real.log (16468310782911 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (558310782911 / 16000000000000 : ℝ) ≤
+      (-(33554282339629042240773 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show (558310782911 / 16000000000000 : ℝ) = (558310782911 / 500000000000 : ℝ) / 2 ^ 5 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (558310782911 / 500000000000 : ℝ) ≤
+            (1103076675370957759227 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1710830907247629 / 10000000000000000 : ℝ) ≤ Real.sqrt (468310782911 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (468310782911 / 16000000000000 : ℝ) ≤ (106926931702977 / 625000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (468310782911 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau125 :
+    Real.arctan (106926931702977 / 625000000000000 : ℝ) ≤
+      (847213185079392045819 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (847213185079392045819 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (625000000000000 / 106926931702977 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (625000000000000 / 106926931702977 : ℝ) =
+              (106926931702977 / 625000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau125)
+  have hz1 :
+    (625000000000000 / 106926931702977 : ℝ) ≤ 1 / Real.sqrt (468310782911 / 16000000000000 : ℝ) :=
+  by
+    rw [show (625000000000000 / 106926931702977 : ℝ) = 1 / (106926931702977 / 625000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 570276969082543 : ℝ) ≤
+      (2065758707467686423337 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (468310782911 / 16000000000000 : ℝ) ≤
+      (250000000000000 / 570276969082543 : ℝ) := by
+    rw [show
+        (250000000000000 / 570276969082543 : ℝ) =
+          (3 / 40) / (1710830907247629 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (847213185079392045819 / 5000000000000000000000 : ℝ)) -
+        6 * (2065758707467686423337 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (468310782911 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (468310782911 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1710830907247629 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (847213185079392045819 / 5000000000000000000000 : ℝ)) -
+          6 * (2065758707467686423337 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (468310782911 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (468310782911 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (468310782911 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_64 :
+    ((290948373626185129557 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(418526194061101859357 / 125000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((429550833853069 / 2500000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1701588269078549890509 / 10000000000000000000000 : ℝ)) -
+              6 * (2057864319825470333217 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (944711264583 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (944711264583 / 32000000000000 : ℝ)) =
+      Real.log (32944711264583 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((944711264583 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1124711264583 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (290948373626185129557 / 10000000000000000000000 : ℝ) ≤
+      Real.log (32944711264583 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1124711264583 / 32000000000000 : ℝ) ≤
+      (-(418526194061101859357 / 125000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show (1124711264583 / 32000000000000 : ℝ) = (1124711264583 / 1000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1124711264583 / 1000000000000 : ℝ) ≤
+            (14690793626398140643 / 125000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (429550833853069 / 2500000000000000 : ℝ) ≤ Real.sqrt (944711264583 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (944711264583 / 32000000000000 : ℝ) ≤ (1718203335412279 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (944711264583 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau126 :
+    Real.arctan (1718203335412279 / 10000000000000000 : ℝ) ≤
+      (1701588269078549890509 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1701588269078549890509 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 1718203335412279 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 1718203335412279 : ℝ) =
+              (1718203335412279 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau126)
+  have hz1 :
+    (10000000000000000 / 1718203335412279 : ℝ) ≤
+      1 / Real.sqrt (944711264583 / 32000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 1718203335412279 : ℝ) =
+          1 / (1718203335412279 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (187500000000000 / 429550833853069 : ℝ) ≤
+      (2057864319825470333217 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (944711264583 / 32000000000000 : ℝ) ≤
+      (187500000000000 / 429550833853069 : ℝ) := by
+    rw [show
+        (187500000000000 / 429550833853069 : ℝ) =
+          (3 / 40) / (429550833853069 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1701588269078549890509 / 10000000000000000000000 : ℝ)) -
+        6 * (2057864319825470333217 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (944711264583 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (944711264583 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((429550833853069 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1701588269078549890509 / 10000000000000000000000 : ℝ)) -
+          6 * (2057864319825470333217 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (944711264583 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (944711264583 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (944711264583 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_65 :
+    ((293403610107240061481 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(3341042607133438815809 / 1000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1725544264992931 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1708717817988970253053 / 10000000000000000000000 : ℝ)) -
+              6 * (4100119996605776193839 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (59550060209 / 2000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (59550060209 / 2000000000000 : ℝ)) =
+      Real.log (2059550060209 / 2000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((59550060209 / 2000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (70800060209 / 2000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (293403610107240061481 / 10000000000000000000000 : ℝ) ≤
+      Real.log (2059550060209 / 2000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (70800060209 / 2000000000000 : ℝ) ≤
+      (-(3341042607133438815809 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show (70800060209 / 2000000000000 : ℝ) = (70800060209 / 62500000000 : ℝ) / 2 ^ 5 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (70800060209 / 62500000000 : ℝ) ≤
+            (124693294366561184191 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1725544264992931 / 10000000000000000 : ℝ) ≤ Real.sqrt (59550060209 / 2000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (59550060209 / 2000000000000 : ℝ) ≤ (862772132496467 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (59550060209 / 2000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau127 :
+    Real.arctan (862772132496467 / 5000000000000000 : ℝ) ≤
+      (1708717817988970253053 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1708717817988970253053 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (5000000000000000 / 862772132496467 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 862772132496467 : ℝ) =
+              (862772132496467 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau127)
+  have hz1 :
+    (5000000000000000 / 862772132496467 : ℝ) ≤ 1 / Real.sqrt (59550060209 / 2000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 862772132496467 : ℝ) = 1 / (862772132496467 / 5000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 1725544264992931 : ℝ) ≤
+      (4100119996605776193839 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (59550060209 / 2000000000000 : ℝ) ≤
+      (750000000000000 / 1725544264992931 : ℝ) := by
+    rw [show
+        (750000000000000 / 1725544264992931 : ℝ) =
+          (3 / 40) / (1725544264992931 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1708717817988970253053 / 10000000000000000000000 : ℝ)) -
+        6 * (4100119996605776193839 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (59550060209 / 2000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (59550060209 / 2000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1725544264992931 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1708717817988970253053 / 10000000000000000000000 : ℝ)) -
+          6 * (4100119996605776193839 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (59550060209 / 2000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (59550060209 / 2000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (59550060209 / 2000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_66 :
+    ((147929121958822036403 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(104185208174615246663 / 31250000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((433213524076041 / 2500000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (428953855365956309143 / 2500000000000000000000 : ℝ)) -
+              6 * (816937615405656711109 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (192178132421 / 6400000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (192178132421 / 6400000000000 : ℝ)) =
+      Real.log (6592178132421 / 6400000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((192178132421 / 6400000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (228178132421 / 6400000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (147929121958822036403 / 5000000000000000000000 : ℝ) ≤
+      Real.log (6592178132421 / 6400000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (228178132421 / 6400000000000 : ℝ) ≤
+      (-(104185208174615246663 / 31250000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show (228178132421 / 6400000000000 : ℝ) = (228178132421 / 200000000000 : ℝ) / 2 ^ 5 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (228178132421 / 200000000000 : ℝ) ≤
+            (4119038747259753337 / 31250000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (433213524076041 / 2500000000000000 : ℝ) ≤ Real.sqrt (192178132421 / 6400000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (192178132421 / 6400000000000 : ℝ) ≤ (1732854096304167 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (192178132421 / 6400000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau128 :
+    Real.arctan (1732854096304167 / 10000000000000000 : ℝ) ≤
+      (428953855365956309143 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (428953855365956309143 / 2500000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 1732854096304167 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 1732854096304167 : ℝ) =
+              (1732854096304167 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau128)
+  have hz1 :
+    (10000000000000000 / 1732854096304167 : ℝ) ≤
+      1 / Real.sqrt (192178132421 / 6400000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 1732854096304167 : ℝ) =
+          1 / (1732854096304167 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (62500000000000 / 144404508025347 : ℝ) ≤
+      (816937615405656711109 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (192178132421 / 6400000000000 : ℝ) ≤
+      (62500000000000 / 144404508025347 : ℝ) := by
+    rw [show
+        (62500000000000 / 144404508025347 : ℝ) =
+          (3 / 40) / (433213524076041 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (428953855365956309143 / 2500000000000000000000 : ℝ)) -
+        6 * (816937615405656711109 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (192178132421 / 6400000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (192178132421 / 6400000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((433213524076041 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (428953855365956309143 / 2500000000000000000000 : ℝ)) -
+          6 * (816937615405656711109 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (192178132421 / 6400000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (192178132421 / 6400000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (192178132421 / 6400000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_67 :
+    ((9322258604787240969 / 312500000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(33268609951502147843163 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1740133221252397 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (861440737808505226089 / 5000000000000000000000 : ℝ)) -
+              6 * (813885912405834617621 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (484490180433 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (484490180433 / 16000000000000 : ℝ)) =
+      Real.log (16484490180433 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((484490180433 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (574490180433 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (9322258604787240969 / 312500000000000000000 : ℝ) ≤
+      Real.log (16484490180433 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (574490180433 / 16000000000000 : ℝ) ≤
+      (-(33268609951502147843163 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show (574490180433 / 16000000000000 : ℝ) = (574490180433 / 500000000000 : ℝ) / 2 ^ 5 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (574490180433 / 500000000000 : ℝ) ≤
+            (1388749063497852156837 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1740133221252397 / 10000000000000000 : ℝ) ≤ Real.sqrt (484490180433 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (484490180433 / 16000000000000 : ℝ) ≤ (4350333053131 / 25000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (484490180433 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau129 :
+    Real.arctan (4350333053131 / 25000000000000 : ℝ) ≤
+      (861440737808505226089 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (861440737808505226089 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (25000000000000 / 4350333053131 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (25000000000000 / 4350333053131 : ℝ) = (4350333053131 / 25000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau129)
+  have hz1 :
+    (25000000000000 / 4350333053131 : ℝ) ≤ 1 / Real.sqrt (484490180433 / 16000000000000 : ℝ) := by
+    rw [show (25000000000000 / 4350333053131 : ℝ) = 1 / (4350333053131 / 25000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 1740133221252397 : ℝ) ≤
+      (813885912405834617621 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (484490180433 / 16000000000000 : ℝ) ≤
+      (750000000000000 / 1740133221252397 : ℝ) := by
+    rw [show
+        (750000000000000 / 1740133221252397 : ℝ) =
+          (3 / 40) / (1740133221252397 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (861440737808505226089 / 5000000000000000000000 : ℝ)) -
+        6 * (813885912405834617621 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (484490180433 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (484490180433 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1740133221252397 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (861440737808505226089 / 5000000000000000000000 : ℝ)) -
+          6 * (813885912405834617621 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (484490180433 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (484490180433 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (484490180433 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_68 :
+    ((75191426177364952563 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(33198449022893842451719 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1747382023581097 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1729916368348796967283 / 10000000000000000000000 : ℝ)) -
+              6 * (4054341219568550892639 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (977070059627 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (977070059627 / 32000000000000 : ℝ)) =
+      Real.log (32977070059627 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((977070059627 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1157070059627 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (75191426177364952563 / 2500000000000000000000 : ℝ) ≤
+      Real.log (32977070059627 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1157070059627 / 32000000000000 : ℝ) ≤
+      (-(33198449022893842451719 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show (1157070059627 / 32000000000000 : ℝ) = (1157070059627 / 1000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1157070059627 / 1000000000000 : ℝ) ≤
+            (1458909992106157548281 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1747382023581097 / 10000000000000000 : ℝ) ≤ Real.sqrt (977070059627 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (977070059627 / 32000000000000 : ℝ) ≤ (17473820235811 / 100000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (977070059627 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau130 :
+    Real.arctan (17473820235811 / 100000000000000 : ℝ) ≤
+      (1729916368348796967283 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1729916368348796967283 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (100000000000000 / 17473820235811 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (100000000000000 / 17473820235811 : ℝ) =
+              (17473820235811 / 100000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau130)
+  have hz1 :
+    (100000000000000 / 17473820235811 : ℝ) ≤ 1 / Real.sqrt (977070059627 / 32000000000000 : ℝ) := by
+    rw [show (100000000000000 / 17473820235811 : ℝ) = 1 / (17473820235811 / 100000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 1747382023581097 : ℝ) ≤
+      (4054341219568550892639 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (977070059627 / 32000000000000 : ℝ) ≤
+      (750000000000000 / 1747382023581097 : ℝ) := by
+    rw [show
+        (750000000000000 / 1747382023581097 : ℝ) =
+          (3 / 40) / (1747382023581097 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1729916368348796967283 / 10000000000000000000000 : ℝ)) -
+        6 * (4054341219568550892639 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (977070059627 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (977070059627 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1747382023581097 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1729916368348796967283 / 10000000000000000000000 : ℝ)) -
+          6 * (4054341219568550892639 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (977070059627 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (977070059627 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (977070059627 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_69 :
+    ((303218532281807704997 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(6625755384441007730879 / 2000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((43865021977671 / 250000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1736920479582931750383 / 10000000000000000000000 : ℝ)) -
+              6 * (201970995077376040719 / 500000000000000000000 : ℝ)))) ≤
+      Vfield (246289939597 / 8000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (246289939597 / 8000000000000 : ℝ)) =
+      Real.log (8246289939597 / 8000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((246289939597 / 8000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (291289939597 / 8000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (303218532281807704997 / 10000000000000000000000 : ℝ) ≤
+      Real.log (8246289939597 / 8000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (291289939597 / 8000000000000 : ℝ) ≤
+      (-(6625755384441007730879 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show (291289939597 / 8000000000000 : ℝ) = (291289939597 / 250000000000 : ℝ) / 2 ^ 5 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (291289939597 / 250000000000 : ℝ) ≤
+            (305716418558992269121 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (43865021977671 / 250000000000000 : ℝ) ≤ Real.sqrt (246289939597 / 8000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (246289939597 / 8000000000000 : ℝ) ≤ (1754600879106843 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (246289939597 / 8000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau131 :
+    Real.arctan (1754600879106843 / 10000000000000000 : ℝ) ≤
+      (1736920479582931750383 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1736920479582931750383 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 1754600879106843 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 1754600879106843 : ℝ) =
+              (1754600879106843 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau131)
+  have hz1 :
+    (10000000000000000 / 1754600879106843 : ℝ) ≤
+      1 / Real.sqrt (246289939597 / 8000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 1754600879106843 : ℝ) =
+          1 / (1754600879106843 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (6250000000000 / 14621673992557 : ℝ) ≤
+      (201970995077376040719 / 500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (246289939597 / 8000000000000 : ℝ) ≤
+      (6250000000000 / 14621673992557 : ℝ) := by
+    rw [show
+        (6250000000000 / 14621673992557 : ℝ) = (3 / 40) / (43865021977671 / 250000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1736920479582931750383 / 10000000000000000000000 : ℝ)) -
+        6 * (201970995077376040719 / 500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (246289939597 / 8000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (246289939597 / 8000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((43865021977671 / 250000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1736920479582931750383 / 10000000000000000000000 : ℝ)) -
+          6 * (201970995077376040719 / 500000000000000000000 : ℝ))) ≤
+      Real.sqrt (246289939597 / 8000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (246289939597 / 8000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (246289939597 / 8000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_70 :
+    ((154061191627547000041 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(32990872286197016689579 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((442237553684297 / 2500000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1750837838732508012519 / 10000000000000000000000 : ℝ)) -
+              6 * (501258268688367222693 / 1250000000000000000000 : ℝ)))) ≤
+      Vfield (100133915591 / 3200000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (100133915591 / 3200000000000 : ℝ)) =
+      Real.log (3300133915591 / 3200000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((100133915591 / 3200000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (118133915591 / 3200000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (154061191627547000041 / 5000000000000000000000 : ℝ) ≤
+      Real.log (3300133915591 / 3200000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (118133915591 / 3200000000000 : ℝ) ≤
+      (-(32990872286197016689579 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show (118133915591 / 3200000000000 : ℝ) = (118133915591 / 100000000000 : ℝ) / 2 ^ 5 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (118133915591 / 100000000000 : ℝ) ≤
+            (1666486728802983310421 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (442237553684297 / 2500000000000000 : ℝ) ≤ Real.sqrt (100133915591 / 3200000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (100133915591 / 3200000000000 : ℝ) ≤ (1768950214737191 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (100133915591 / 3200000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau132 :
+    Real.arctan (1768950214737191 / 10000000000000000 : ℝ) ≤
+      (1750837838732508012519 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1750837838732508012519 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 1768950214737191 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 1768950214737191 : ℝ) =
+              (1768950214737191 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau132)
+  have hz1 :
+    (10000000000000000 / 1768950214737191 : ℝ) ≤
+      1 / Real.sqrt (100133915591 / 3200000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 1768950214737191 : ℝ) =
+          1 / (1768950214737191 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (187500000000000 / 442237553684297 : ℝ) ≤
+      (501258268688367222693 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (100133915591 / 3200000000000 : ℝ) ≤
+      (187500000000000 / 442237553684297 : ℝ) := by
+    rw [show
+        (187500000000000 / 442237553684297 : ℝ) =
+          (3 / 40) / (442237553684297 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1750837838732508012519 / 10000000000000000000000 : ℝ)) -
+        6 * (501258268688367222693 / 1250000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (100133915591 / 3200000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (100133915591 / 3200000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((442237553684297 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1750837838732508012519 / 10000000000000000000000 : ℝ)) -
+          6 * (501258268688367222693 / 1250000000000000000000 : ℝ))) ≤
+      Real.sqrt (100133915591 / 3200000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (100133915591 / 3200000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (100133915591 / 3200000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_71 :
+    ((156511915315791689151 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(16427421789228452963099 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1783184084573153 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (220579555174712374839 / 1250000000000000000000 : ℝ)) -
+              6 * (3981344696696963220873 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (127189819179 / 4000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (127189819179 / 4000000000000 : ℝ)) =
+      Real.log (4127189819179 / 4000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((127189819179 / 4000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (149689819179 / 4000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (156511915315791689151 / 5000000000000000000000 : ℝ) ≤
+      Real.log (4127189819179 / 4000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (149689819179 / 4000000000000 : ℝ) ≤
+      (-(16427421789228452963099 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show (149689819179 / 4000000000000 : ℝ) = (149689819179 / 125000000000 : ℝ) / 2 ^ 5 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (149689819179 / 125000000000 : ℝ) ≤
+            (901257718271547036901 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1783184084573153 / 10000000000000000 : ℝ) ≤ Real.sqrt (127189819179 / 4000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (127189819179 / 4000000000000 : ℝ) ≤ (445796021143289 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (127189819179 / 4000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau133 :
+    Real.arctan (445796021143289 / 2500000000000000 : ℝ) ≤
+      (220579555174712374839 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (220579555174712374839 / 1250000000000000000000 : ℝ)) ≤
+      Real.arctan (2500000000000000 / 445796021143289 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 445796021143289 : ℝ) =
+              (445796021143289 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau133)
+  have hz1 :
+    (2500000000000000 / 445796021143289 : ℝ) ≤ 1 / Real.sqrt (127189819179 / 4000000000000 : ℝ) :=
+  by
+    rw [show
+        (2500000000000000 / 445796021143289 : ℝ) = 1 / (445796021143289 / 2500000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 1783184084573153 : ℝ) ≤
+      (3981344696696963220873 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (127189819179 / 4000000000000 : ℝ) ≤
+      (750000000000000 / 1783184084573153 : ℝ) := by
+    rw [show
+        (750000000000000 / 1783184084573153 : ℝ) =
+          (3 / 40) / (1783184084573153 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (220579555174712374839 / 1250000000000000000000 : ℝ)) -
+        6 * (3981344696696963220873 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (127189819179 / 4000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (127189819179 / 4000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1783184084573153 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (220579555174712374839 / 1250000000000000000000 : ℝ)) -
+          6 * (3981344696696963220873 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (127189819179 / 4000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (127189819179 / 4000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (127189819179 / 4000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_72 :
+    ((317922876766342253191 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(32720640446352325306753 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1797305231932307 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (355663812413146259823 / 2000000000000000000000 : ℝ)) -
+              6 * (1976616554840196050209 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (516848975477 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (516848975477 / 16000000000000 : ℝ)) =
+      Real.log (16516848975477 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((516848975477 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (606848975477 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (317922876766342253191 / 10000000000000000000000 : ℝ) ≤
+      Real.log (16516848975477 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (606848975477 / 16000000000000 : ℝ) ≤
+      (-(32720640446352325306753 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show (606848975477 / 16000000000000 : ℝ) = (606848975477 / 500000000000 : ℝ) / 2 ^ 5 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (606848975477 / 500000000000 : ℝ) ≤
+            (1936718568647674693247 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1797305231932307 / 10000000000000000 : ℝ) ≤ Real.sqrt (516848975477 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (516848975477 / 16000000000000 : ℝ) ≤ (179730523193231 / 1000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (516848975477 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau134 :
+    Real.arctan (179730523193231 / 1000000000000000 : ℝ) ≤
+      (355663812413146259823 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (355663812413146259823 / 2000000000000000000000 : ℝ)) ≤
+      Real.arctan (1000000000000000 / 179730523193231 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1000000000000000 / 179730523193231 : ℝ) =
+              (179730523193231 / 1000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau134)
+  have hz1 :
+    (1000000000000000 / 179730523193231 : ℝ) ≤
+      1 / Real.sqrt (516848975477 / 16000000000000 : ℝ) := by
+    rw [show
+        (1000000000000000 / 179730523193231 : ℝ) = 1 / (179730523193231 / 1000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 1797305231932307 : ℝ) ≤
+      (1976616554840196050209 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (516848975477 / 16000000000000 : ℝ) ≤
+      (750000000000000 / 1797305231932307 : ℝ) := by
+    rw [show
+        (750000000000000 / 1797305231932307 : ℝ) =
+          (3 / 40) / (1797305231932307 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (355663812413146259823 / 2000000000000000000000 : ℝ)) -
+        6 * (1976616554840196050209 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (516848975477 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (516848975477 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1797305231932307 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (355663812413146259823 / 2000000000000000000000 : ℝ)) -
+          6 * (1976616554840196050209 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (516848975477 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (516848975477 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (516848975477 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/V06.lean b/LeanPool/Zeta5Irrational/Table/V06.lean
new file mode 100644
index 0000000000..d2649f26ed
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/V06.lean
@@ -0,0 +1,1226 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Energy
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # V06: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_73 :
+    ((20176220250686091201 / 625000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2036763408608682511547 / 625000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((905658146596647 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (895944183790336958567 / 5000000000000000000000 : ℝ)) -
+              6 * (785142011276042781277 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (262469337119 / 8000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (262469337119 / 8000000000000 : ℝ)) =
+      Real.log (8262469337119 / 8000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((262469337119 / 8000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (307469337119 / 8000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (20176220250686091201 / 625000000000000000000 : ℝ) ≤
+      Real.log (8262469337119 / 8000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (307469337119 / 8000000000000 : ℝ) ≤
+      (-(2036763408608682511547 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show (307469337119 / 8000000000000 : ℝ) = (307469337119 / 250000000000 : ℝ) / 2 ^ 5 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (307469337119 / 250000000000 : ℝ) ≤
+            (129321529828817488453 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (905658146596647 / 5000000000000000 : ℝ) ≤ Real.sqrt (262469337119 / 8000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (262469337119 / 8000000000000 : ℝ) ≤ (1811316293193297 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (262469337119 / 8000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau135 :
+    Real.arctan (1811316293193297 / 10000000000000000 : ℝ) ≤
+      (895944183790336958567 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (895944183790336958567 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 1811316293193297 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 1811316293193297 : ℝ) =
+              (1811316293193297 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau135)
+  have hz1 :
+    (10000000000000000 / 1811316293193297 : ℝ) ≤
+      1 / Real.sqrt (262469337119 / 8000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 1811316293193297 : ℝ) =
+          1 / (1811316293193297 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (125000000000000 / 301886048865549 : ℝ) ≤
+      (785142011276042781277 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (262469337119 / 8000000000000 : ℝ) ≤
+      (125000000000000 / 301886048865549 : ℝ) := by
+    rw [show
+        (125000000000000 / 301886048865549 : ℝ) =
+          (3 / 40) / (905658146596647 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (895944183790336958567 / 5000000000000000000000 : ℝ)) -
+        6 * (785142011276042781277 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (262469337119 / 8000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (262469337119 / 8000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((905658146596647 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (895944183790336958567 / 5000000000000000000000 : ℝ)) -
+          6 * (785142011276042781277 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (262469337119 / 8000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (262469337119 / 8000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (262469337119 / 8000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_74 :
+    ((332605631219046927529 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(32328510364068521412621 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1839018202329709 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1818697196560381043083 / 10000000000000000000000 : ℝ)) -
+              6 * (3872349322266150066581 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (6763975897 / 200000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (6763975897 / 200000000000 : ℝ)) = Real.log (206763975897 / 200000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((6763975897 / 200000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (7888975897 / 200000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (332605631219046927529 / 10000000000000000000000 : ℝ) ≤
+      Real.log (206763975897 / 200000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (7888975897 / 200000000000 : ℝ) ≤
+      (-(32328510364068521412621 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show (7888975897 / 200000000000 : ℝ) = (7888975897 / 6250000000 : ℝ) / 2 ^ 5 by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7888975897 / 6250000000 : ℝ) ≤
+            (2328848650931478587379 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1839018202329709 / 10000000000000000 : ℝ) ≤ Real.sqrt (6763975897 / 200000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (6763975897 / 200000000000 : ℝ) ≤ (114938637645607 / 625000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (6763975897 / 200000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau136 :
+    Real.arctan (114938637645607 / 625000000000000 : ℝ) ≤
+      (1818697196560381043083 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1818697196560381043083 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (625000000000000 / 114938637645607 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (625000000000000 / 114938637645607 : ℝ) =
+              (114938637645607 / 625000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau136)
+  have hz1 :
+    (625000000000000 / 114938637645607 : ℝ) ≤ 1 / Real.sqrt (6763975897 / 200000000000 : ℝ) := by
+    rw [show (625000000000000 / 114938637645607 : ℝ) = 1 / (114938637645607 / 625000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 1839018202329709 : ℝ) ≤
+      (3872349322266150066581 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (6763975897 / 200000000000 : ℝ) ≤
+      (750000000000000 / 1839018202329709 : ℝ) := by
+    rw [show
+        (750000000000000 / 1839018202329709 : ℝ) =
+          (3 / 40) / (1839018202329709 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1818697196560381043083 / 10000000000000000000000 : ℝ)) -
+        6 * (3872349322266150066581 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (6763975897 / 200000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (6763975897 / 200000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1839018202329709 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1818697196560381043083 / 10000000000000000000000 : ℝ)) -
+          6 * (3872349322266150066581 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (6763975897 / 200000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (6763975897 / 200000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (6763975897 / 200000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_75 :
+    ((68476434199942366179 / 2000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(16037690220153861181957 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((466577243270909 / 2500000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1845082320061846925631 / 10000000000000000000000 : ℝ)) -
+              6 * (764222275881632837763 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (278648734641 / 8000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (278648734641 / 8000000000000 : ℝ)) =
+      Real.log (8278648734641 / 8000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((278648734641 / 8000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (323648734641 / 8000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (68476434199942366179 / 2000000000000000000000 : ℝ) ≤
+      Real.log (8278648734641 / 8000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (323648734641 / 8000000000000 : ℝ) ≤
+      (-(16037690220153861181957 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show (323648734641 / 8000000000000 : ℝ) = (323648734641 / 250000000000 : ℝ) / 2 ^ 5 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (323648734641 / 250000000000 : ℝ) ≤
+            (1290989287346138818043 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (466577243270909 / 2500000000000000 : ℝ) ≤ Real.sqrt (278648734641 / 8000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (278648734641 / 8000000000000 : ℝ) ≤ (1866308973083639 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (278648734641 / 8000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau137 :
+    Real.arctan (1866308973083639 / 10000000000000000 : ℝ) ≤
+      (1845082320061846925631 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1845082320061846925631 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 1866308973083639 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 1866308973083639 : ℝ) =
+              (1866308973083639 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau137)
+  have hz1 :
+    (10000000000000000 / 1866308973083639 : ℝ) ≤
+      1 / Real.sqrt (278648734641 / 8000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 1866308973083639 : ℝ) =
+          1 / (1866308973083639 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (187500000000000 / 466577243270909 : ℝ) ≤
+      (764222275881632837763 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (278648734641 / 8000000000000 : ℝ) ≤
+      (187500000000000 / 466577243270909 : ℝ) := by
+    rw [show
+        (187500000000000 / 466577243270909 : ℝ) =
+          (3 / 40) / (466577243270909 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1845082320061846925631 / 10000000000000000000000 : ℝ)) -
+        6 * (764222275881632837763 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (278648734641 / 8000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (278648734641 / 8000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((466577243270909 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1845082320061846925631 / 10000000000000000000000 : ℝ)) -
+          6 * (764222275881632837763 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (278648734641 / 8000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (278648734641 / 8000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (278648734641 / 8000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_76 :
+    ((352149162041970693603 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(7957125030406035210083 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((75728255413947 / 400000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (935530869456240818211 / 5000000000000000000000 : ℝ)) -
+              6 * (3771858836909198659297 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (143369216701 / 4000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (143369216701 / 4000000000000 : ℝ)) =
+      Real.log (4143369216701 / 4000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((143369216701 / 4000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (165869216701 / 4000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (352149162041970693603 / 10000000000000000000000 : ℝ) ≤
+      Real.log (4143369216701 / 4000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (165869216701 / 4000000000000 : ℝ) ≤
+      (-(7957125030406035210083 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show (165869216701 / 4000000000000 : ℝ) = (165869216701 / 125000000000 : ℝ) / 2 ^ 5 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (165869216701 / 125000000000 : ℝ) ≤
+            (707214723343964789917 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (75728255413947 / 400000000000000 : ℝ) ≤ Real.sqrt (143369216701 / 4000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (143369216701 / 4000000000000 : ℝ) ≤ (946603192674339 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (143369216701 / 4000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau138 :
+    Real.arctan (946603192674339 / 5000000000000000 : ℝ) ≤
+      (935530869456240818211 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (935530869456240818211 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (5000000000000000 / 946603192674339 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 946603192674339 : ℝ) =
+              (946603192674339 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau138)
+  have hz1 :
+    (5000000000000000 / 946603192674339 : ℝ) ≤ 1 / Real.sqrt (143369216701 / 4000000000000 : ℝ) :=
+  by
+    rw [show
+        (5000000000000000 / 946603192674339 : ℝ) = 1 / (946603192674339 / 5000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (10000000000000 / 25242751804649 : ℝ) ≤
+      (3771858836909198659297 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (143369216701 / 4000000000000 : ℝ) ≤
+      (10000000000000 / 25242751804649 : ℝ) := by
+    rw [show
+        (10000000000000 / 25242751804649 : ℝ) = (3 / 40) / (75728255413947 / 400000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (935530869456240818211 / 5000000000000000000000 : ℝ)) -
+        6 * (3771858836909198659297 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (143369216701 / 4000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (143369216701 / 4000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((75728255413947 / 400000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (935530869456240818211 / 5000000000000000000000 : ℝ)) -
+          6 * (3771858836909198659297 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (143369216701 / 4000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (143369216701 / 4000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (143369216701 / 4000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_77 :
+    ((371654572393941879647 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(313523048550250673131 / 100000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1945886144292619 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (48046732795733132627 / 250000000000000000000 : ℝ)) -
+              6 * (183940991600097138117 / 500000000000000000000 : ℝ)))) ≤
+      Vfield (75729457731 / 2000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (75729457731 / 2000000000000 : ℝ)) =
+      Real.log (2075729457731 / 2000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((75729457731 / 2000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (86979457731 / 2000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (371654572393941879647 / 10000000000000000000000 : ℝ) ≤
+      Real.log (2075729457731 / 2000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (86979457731 / 2000000000000 : ℝ) ≤
+      (-(313523048550250673131 / 100000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show (86979457731 / 2000000000000 : ℝ) = (86979457731 / 62500000000 : ℝ) / 2 ^ 5 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (86979457731 / 62500000000 : ℝ) ≤
+            (33050541599749326869 / 100000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1945886144292619 / 10000000000000000 : ℝ) ≤ Real.sqrt (75729457731 / 2000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (75729457731 / 2000000000000 : ℝ) ≤ (972943072146311 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (75729457731 / 2000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau139 :
+    Real.arctan (972943072146311 / 5000000000000000 : ℝ) ≤
+      (48046732795733132627 / 250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (48046732795733132627 / 250000000000000000000 : ℝ)) ≤
+      Real.arctan (5000000000000000 / 972943072146311 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 972943072146311 : ℝ) =
+              (972943072146311 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau139)
+  have hz1 :
+    (5000000000000000 / 972943072146311 : ℝ) ≤ 1 / Real.sqrt (75729457731 / 2000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 972943072146311 : ℝ) = 1 / (972943072146311 / 5000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 1945886144292619 : ℝ) ≤
+      (183940991600097138117 / 500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (75729457731 / 2000000000000 : ℝ) ≤
+      (750000000000000 / 1945886144292619 : ℝ) := by
+    rw [show
+        (750000000000000 / 1945886144292619 : ℝ) =
+          (3 / 40) / (1945886144292619 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (48046732795733132627 / 250000000000000000000 : ℝ)) -
+        6 * (183940991600097138117 / 500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (75729457731 / 2000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (75729457731 / 2000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1945886144292619 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (48046732795733132627 / 250000000000000000000 : ℝ)) -
+          6 * (183940991600097138117 / 500000000000000000000 : ℝ))) ≤
+      Real.sqrt (75729457731 / 2000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (75729457731 / 2000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (75729457731 / 2000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_78 :
+    ((51318953063325241837 / 1250000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(15231489342023646879373 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((409436579929053 / 2000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2019282332713757840697 / 10000000000000000000000 : ℝ)) -
+              6 * (877929728397766997881 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (20954789123 / 500000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (20954789123 / 500000000000 : ℝ)) =
+      Real.log (520954789123 / 500000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((20954789123 / 500000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (23767289123 / 500000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (51318953063325241837 / 1250000000000000000000 : ℝ) ≤
+      Real.log (520954789123 / 500000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (23767289123 / 500000000000 : ℝ) ≤
+      (-(15231489342023646879373 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show (23767289123 / 500000000000 : ℝ) = (23767289123 / 15625000000 : ℝ) / 2 ^ 5 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23767289123 / 15625000000 : ℝ) ≤
+            (2097190165476353120627 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (409436579929053 / 2000000000000000 : ℝ) ≤ Real.sqrt (20954789123 / 500000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (20954789123 / 500000000000 : ℝ) ≤ (511795724911317 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (20954789123 / 500000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau140 :
+    Real.arctan (511795724911317 / 2500000000000000 : ℝ) ≤
+      (2019282332713757840697 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (2019282332713757840697 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (2500000000000000 / 511795724911317 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 511795724911317 : ℝ) =
+              (511795724911317 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau140)
+  have hz1 :
+    (2500000000000000 / 511795724911317 : ℝ) ≤ 1 / Real.sqrt (20954789123 / 500000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 511795724911317 : ℝ) = 1 / (511795724911317 / 2500000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (50000000000000 / 136478859976351 : ℝ) ≤
+      (877929728397766997881 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (20954789123 / 500000000000 : ℝ) ≤
+      (50000000000000 / 136478859976351 : ℝ) := by
+    rw [show
+        (50000000000000 / 136478859976351 : ℝ) =
+          (3 / 40) / (409436579929053 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2019282332713757840697 / 10000000000000000000000 : ℝ)) -
+        6 * (877929728397766997881 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (20954789123 / 500000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (20954789123 / 500000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((409436579929053 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2019282332713757840697 / 10000000000000000000000 : ℝ)) -
+          6 * (877929728397766997881 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (20954789123 / 500000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (20954789123 / 500000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (20954789123 / 500000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_79 :
+    ((26556429962361706593 / 625000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1884565064399020768501 / 625000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((16276631329523 / 78125000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2054026228998431747243 / 10000000000000000000000 : ℝ)) -
+              6 * (431930043648711482693 / 1250000000000000000000 : ℝ)))) ≤
+      Vfield (694494763253 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (694494763253 / 16000000000000 : ℝ)) =
+      Real.log (16694494763253 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((694494763253 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (784494763253 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (26556429962361706593 / 625000000000000000000 : ℝ) ≤
+      Real.log (16694494763253 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (784494763253 / 16000000000000 : ℝ) ≤
+      (-(1884565064399020768501 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show (784494763253 / 16000000000000 : ℝ) = (784494763253 / 500000000000 : ℝ) / 2 ^ 5 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (784494763253 / 500000000000 : ℝ) ≤
+            (281519874038479231499 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (16276631329523 / 78125000000000 : ℝ) ≤ Real.sqrt (694494763253 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (694494763253 / 16000000000000 : ℝ) ≤ (2083408810178947 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (694494763253 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau141 :
+    Real.arctan (2083408810178947 / 10000000000000000 : ℝ) ≤
+      (2054026228998431747243 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (2054026228998431747243 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 2083408810178947 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 2083408810178947 : ℝ) =
+              (2083408810178947 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau141)
+  have hz1 :
+    (10000000000000000 / 2083408810178947 : ℝ) ≤
+      1 / Real.sqrt (694494763253 / 16000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 2083408810178947 : ℝ) =
+          1 / (2083408810178947 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (5859375000000 / 16276631329523 : ℝ) ≤
+      (431930043648711482693 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (694494763253 / 16000000000000 : ℝ) ≤
+      (5859375000000 / 16276631329523 : ℝ) := by
+    rw [show
+        (5859375000000 / 16276631329523 : ℝ) = (3 / 40) / (16276631329523 / 78125000000000 : ℝ) by
+        norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2054026228998431747243 / 10000000000000000000000 : ℝ)) -
+        6 * (431930043648711482693 / 1250000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (694494763253 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (694494763253 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((16276631329523 / 78125000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2054026228998431747243 / 10000000000000000000000 : ℝ)) -
+          6 * (431930043648711482693 / 1250000000000000000000 : ℝ))) ≤
+      Real.sqrt (694494763253 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (694494763253 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (694494763253 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_80 :
+    ((54008848847639878331 / 1250000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(30001601622668503149493 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2101287579841671 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (258894388269868848451 / 1250000000000000000000 : ℝ)) -
+              6 * (1714149204325831240059 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (1412931037823 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1412931037823 / 32000000000000 : ℝ)) =
+      Real.log (33412931037823 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1412931037823 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1592931037823 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (54008848847639878331 / 1250000000000000000000 : ℝ) ≤
+      Real.log (33412931037823 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1592931037823 / 32000000000000 : ℝ) ≤
+      (-(30001601622668503149493 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show (1592931037823 / 32000000000000 : ℝ) = (1592931037823 / 1000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1592931037823 / 1000000000000 : ℝ) ≤
+            (4655757392331496850507 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2101287579841671 / 10000000000000000 : ℝ) ≤ Real.sqrt (1412931037823 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1412931037823 / 32000000000000 : ℝ) ≤ (1050643789920837 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1412931037823 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau142 :
+    Real.arctan (1050643789920837 / 5000000000000000 : ℝ) ≤
+      (258894388269868848451 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (258894388269868848451 / 1250000000000000000000 : ℝ)) ≤
+      Real.arctan (5000000000000000 / 1050643789920837 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 1050643789920837 : ℝ) =
+              (1050643789920837 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau142)
+  have hz1 :
+    (5000000000000000 / 1050643789920837 : ℝ) ≤
+      1 / Real.sqrt (1412931037823 / 32000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 1050643789920837 : ℝ) = 1 / (1050643789920837 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 700429193280557 : ℝ) ≤
+      (1714149204325831240059 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1412931037823 / 32000000000000 : ℝ) ≤
+      (250000000000000 / 700429193280557 : ℝ) := by
+    rw [show
+        (250000000000000 / 700429193280557 : ℝ) =
+          (3 / 40) / (2101287579841671 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (258894388269868848451 / 1250000000000000000000 : ℝ)) -
+        6 * (1714149204325831240059 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1412931037823 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1412931037823 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2101287579841671 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (258894388269868848451 / 1250000000000000000000 : ℝ)) -
+          6 * (1714149204325831240059 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (1412931037823 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1412931037823 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1412931037823 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_81 :
+    ((219616783974515609901 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1492621071671766733419 / 500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((105950775316447 / 500000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1044063650657751722727 / 5000000000000000000000 : ℝ)) -
+              6 * (850446742255768075683 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (71843627457 / 1600000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (71843627457 / 1600000000000 : ℝ)) =
+      Real.log (1671843627457 / 1600000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((71843627457 / 1600000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (80843627457 / 1600000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (219616783974515609901 / 5000000000000000000000 : ℝ) ≤
+      Real.log (1671843627457 / 1600000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (80843627457 / 1600000000000 : ℝ) ≤
+      (-(1492621071671766733419 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show (80843627457 / 1600000000000 : ℝ) = (80843627457 / 50000000000 : ℝ) / 2 ^ 5 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (80843627457 / 50000000000 : ℝ) ≤
+            (240246879078233266581 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (105950775316447 / 500000000000000 : ℝ) ≤ Real.sqrt (71843627457 / 1600000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (71843627457 / 1600000000000 : ℝ) ≤ (2119015506328943 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (71843627457 / 1600000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau143 :
+    Real.arctan (2119015506328943 / 10000000000000000 : ℝ) ≤
+      (1044063650657751722727 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1044063650657751722727 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 2119015506328943 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 2119015506328943 : ℝ) =
+              (2119015506328943 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau143)
+  have hz1 :
+    (10000000000000000 / 2119015506328943 : ℝ) ≤
+      1 / Real.sqrt (71843627457 / 1600000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 2119015506328943 : ℝ) =
+          1 / (2119015506328943 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (37500000000000 / 105950775316447 : ℝ) ≤
+      (850446742255768075683 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (71843627457 / 1600000000000 : ℝ) ≤
+      (37500000000000 / 105950775316447 : ℝ) := by
+    rw [show
+        (37500000000000 / 105950775316447 : ℝ) =
+          (3 / 40) / (105950775316447 / 500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1044063650657751722727 / 5000000000000000000000 : ℝ)) -
+        6 * (850446742255768075683 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (71843627457 / 1600000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (71843627457 / 1600000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((105950775316447 / 500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1044063650657751722727 / 5000000000000000000000 : ℝ)) -
+          6 * (850446742255768075683 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (71843627457 / 1600000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (71843627457 / 1600000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (71843627457 / 1600000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_82 :
+    ((442813033499629176129 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(14889328837725568225533 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1063912041417817 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1048277920962015937779 / 5000000000000000000000 : ℝ)) -
+              6 * (3388760114959635706763 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (2897686609597 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (2897686609597 / 64000000000000 : ℝ)) =
+      Real.log (66897686609597 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((2897686609597 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (3257686609597 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (442813033499629176129 / 10000000000000000000000 : ℝ) ≤
+      Real.log (66897686609597 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (3257686609597 / 64000000000000 : ℝ) ≤
+      (-(14889328837725568225533 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show (3257686609597 / 64000000000000 : ℝ) = (3257686609597 / 2000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3257686609597 / 2000000000000 : ℝ) ≤
+            (2439350669774431774467 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1063912041417817 / 5000000000000000 : ℝ) ≤ Real.sqrt (2897686609597 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (2897686609597 / 64000000000000 : ℝ) ≤ (2127824082835637 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (2897686609597 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau144 :
+    Real.arctan (2127824082835637 / 10000000000000000 : ℝ) ≤
+      (1048277920962015937779 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1048277920962015937779 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 2127824082835637 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 2127824082835637 : ℝ) =
+              (2127824082835637 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau144)
+  have hz1 :
+    (10000000000000000 / 2127824082835637 : ℝ) ≤
+      1 / Real.sqrt (2897686609597 / 64000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 2127824082835637 : ℝ) =
+          1 / (2127824082835637 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 1063912041417817 : ℝ) ≤
+      (3388760114959635706763 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (2897686609597 / 64000000000000 : ℝ) ≤
+      (375000000000000 / 1063912041417817 : ℝ) := by
+    rw [show
+        (375000000000000 / 1063912041417817 : ℝ) =
+          (3 / 40) / (1063912041417817 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1048277920962015937779 / 5000000000000000000000 : ℝ)) -
+        6 * (3388760114959635706763 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (2897686609597 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (2897686609597 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1063912041417817 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1048277920962015937779 / 5000000000000000000000 : ℝ)) -
+          6 * (3388760114959635706763 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (2897686609597 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (2897686609597 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (2897686609597 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_83 :
+    ((223195609125654108827 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2970543404491423273459 / 1000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1068298172202887 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1052473315449602418319 / 5000000000000000000000 : ℝ)) -
+              6 * (1687940980739126299293 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (1460814060457 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1460814060457 / 32000000000000 : ℝ)) =
+      Real.log (33460814060457 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1460814060457 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1640814060457 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (223195609125654108827 / 5000000000000000000000 : ℝ) ≤
+      Real.log (33460814060457 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1640814060457 / 32000000000000 : ℝ) ≤
+      (-(2970543404491423273459 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show (1640814060457 / 32000000000000 : ℝ) = (1640814060457 / 1000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1640814060457 / 1000000000000 : ℝ) ≤
+            (495192497008576726541 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1068298172202887 / 5000000000000000 : ℝ) ≤ Real.sqrt (1460814060457 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1460814060457 / 32000000000000 : ℝ) ≤ (2136596344405777 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1460814060457 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau145 :
+    Real.arctan (2136596344405777 / 10000000000000000 : ℝ) ≤
+      (1052473315449602418319 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1052473315449602418319 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 2136596344405777 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 2136596344405777 : ℝ) =
+              (2136596344405777 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau145)
+  have hz1 :
+    (10000000000000000 / 2136596344405777 : ℝ) ≤
+      1 / Real.sqrt (1460814060457 / 32000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 2136596344405777 : ℝ) =
+          1 / (2136596344405777 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 1068298172202887 : ℝ) ≤
+      (1687940980739126299293 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1460814060457 / 32000000000000 : ℝ) ≤
+      (375000000000000 / 1068298172202887 : ℝ) := by
+    rw [show
+        (375000000000000 / 1068298172202887 : ℝ) =
+          (3 / 40) / (1068298172202887 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1052473315449602418319 / 5000000000000000000000 : ℝ)) -
+        6 * (1687940980739126299293 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1460814060457 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1460814060457 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1068298172202887 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1052473315449602418319 / 5000000000000000000000 : ℝ)) -
+          6 * (1687940980739126299293 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (1460814060457 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1460814060457 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1460814060457 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_84 :
+    ((449968123120327541017 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(29632742689235428438627 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((536333184128633 / 2500000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2113300121198763880449 / 10000000000000000000000 : ℝ)) -
+              6 * (420393712183787165013 / 1250000000000000000000 : ℝ)))) ≤
+      Vfield (2945569632231 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (2945569632231 / 64000000000000 : ℝ)) =
+      Real.log (66945569632231 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((2945569632231 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (3305569632231 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (449968123120327541017 / 10000000000000000000000 : ℝ) ≤
+      Real.log (66945569632231 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (3305569632231 / 64000000000000 : ℝ) ≤
+      (-(29632742689235428438627 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show (3305569632231 / 64000000000000 : ℝ) = (3305569632231 / 2000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3305569632231 / 2000000000000 : ℝ) ≤
+            (5024616325764571561373 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (536333184128633 / 2500000000000000 : ℝ) ≤ Real.sqrt (2945569632231 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (2945569632231 / 64000000000000 : ℝ) ≤ (429066547302907 / 2000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (2945569632231 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau146 :
+    Real.arctan (429066547302907 / 2000000000000000 : ℝ) ≤
+      (2113300121198763880449 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (2113300121198763880449 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (2000000000000000 / 429066547302907 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2000000000000000 / 429066547302907 : ℝ) =
+              (429066547302907 / 2000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau146)
+  have hz1 :
+    (2000000000000000 / 429066547302907 : ℝ) ≤
+      1 / Real.sqrt (2945569632231 / 64000000000000 : ℝ) := by
+    rw [show
+        (2000000000000000 / 429066547302907 : ℝ) = 1 / (429066547302907 / 2000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (187500000000000 / 536333184128633 : ℝ) ≤
+      (420393712183787165013 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (2945569632231 / 64000000000000 : ℝ) ≤
+      (187500000000000 / 536333184128633 : ℝ) := by
+    rw [show
+        (187500000000000 / 536333184128633 : ℝ) =
+          (3 / 40) / (536333184128633 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2113300121198763880449 / 10000000000000000000000 : ℝ)) -
+        6 * (420393712183787165013 / 1250000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (2945569632231 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (2945569632231 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((536333184128633 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2113300121198763880449 / 10000000000000000000000 : ℝ)) -
+          6 * (420393712183787165013 / 1250000000000000000000 : ℝ))) ≤
+      Real.sqrt (2945569632231 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (2945569632231 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (2945569632231 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/V07.lean b/LeanPool/Zeta5Irrational/Table/V07.lean
new file mode 100644
index 0000000000..8b9a20d8ae
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/V07.lean
@@ -0,0 +1,1230 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Energy
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # V07: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_85 :
+    ((226771874510981741379 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(14780287962918279265411 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((430806739120629 / 2000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1060808378329828977447 / 5000000000000000000000 : ℝ)) -
+              6 * (41882007321997290791 / 125000000000000000000 : ℝ)))) ≤
+      Vfield (742377785887 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (742377785887 / 16000000000000 : ℝ)) =
+      Real.log (16742377785887 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((742377785887 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (832377785887 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (226771874510981741379 / 5000000000000000000000 : ℝ) ≤
+      Real.log (16742377785887 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (832377785887 / 16000000000000 : ℝ) ≤
+      (-(14780287962918279265411 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show (832377785887 / 16000000000000 : ℝ) = (832377785887 / 500000000000 : ℝ) / 2 ^ 5 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (832377785887 / 500000000000 : ℝ) ≤
+            (2548391544581720734589 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (430806739120629 / 2000000000000000 : ℝ) ≤ Real.sqrt (742377785887 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (742377785887 / 16000000000000 : ℝ) ≤ (538508423900787 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (742377785887 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau147 :
+    Real.arctan (538508423900787 / 2500000000000000 : ℝ) ≤
+      (1060808378329828977447 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1060808378329828977447 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (2500000000000000 / 538508423900787 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 538508423900787 : ℝ) =
+              (538508423900787 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau147)
+  have hz1 :
+    (2500000000000000 / 538508423900787 : ℝ) ≤
+      1 / Real.sqrt (742377785887 / 16000000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 538508423900787 : ℝ) = 1 / (538508423900787 / 2500000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (50000000000000 / 143602246373543 : ℝ) ≤
+      (41882007321997290791 / 125000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (742377785887 / 16000000000000 : ℝ) ≤
+      (50000000000000 / 143602246373543 : ℝ) := by
+    rw [show
+        (50000000000000 / 143602246373543 : ℝ) =
+          (3 / 40) / (430806739120629 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1060808378329828977447 / 5000000000000000000000 : ℝ)) -
+        6 * (41882007321997290791 / 125000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (742377785887 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (742377785887 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((430806739120629 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1060808378329828977447 / 5000000000000000000000 : ℝ)) -
+          6 * (41882007321997290791 / 125000000000000000000 : ℝ))) ≤
+      Real.sqrt (742377785887 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (742377785887 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (742377785887 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_86 :
+    ((457118096870510924543 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(29488926237277907956139 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2162699649333341 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2129896972254185019601 / 10000000000000000000000 : ℝ)) -
+              6 * (834527990154732413477 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (598690530973 / 12800000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (598690530973 / 12800000000000 : ℝ)) =
+      Real.log (13398690530973 / 12800000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((598690530973 / 12800000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (670690530973 / 12800000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (457118096870510924543 / 10000000000000000000000 : ℝ) ≤
+      Real.log (13398690530973 / 12800000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (670690530973 / 12800000000000 : ℝ) ≤
+      (-(29488926237277907956139 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show (670690530973 / 12800000000000 : ℝ) = (670690530973 / 400000000000 : ℝ) / 2 ^ 5 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (670690530973 / 400000000000 : ℝ) ≤
+            (5168432777722092043861 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2162699649333341 / 10000000000000000 : ℝ) ≤ Real.sqrt (598690530973 / 12800000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (598690530973 / 12800000000000 : ℝ) ≤ (67584364041667 / 312500000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (598690530973 / 12800000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau148 :
+    Real.arctan (67584364041667 / 312500000000000 : ℝ) ≤
+      (2129896972254185019601 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (2129896972254185019601 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (312500000000000 / 67584364041667 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (312500000000000 / 67584364041667 : ℝ) =
+              (67584364041667 / 312500000000000 : ℝ)
+            by
+        norm_num];
+        exact hau148)
+  have hz1 :
+    (312500000000000 / 67584364041667 : ℝ) ≤ 1 / Real.sqrt (598690530973 / 12800000000000 : ℝ) := by
+    rw [show (312500000000000 / 67584364041667 : ℝ) = 1 / (67584364041667 / 312500000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 2162699649333341 : ℝ) ≤
+      (834527990154732413477 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (598690530973 / 12800000000000 : ℝ) ≤
+      (750000000000000 / 2162699649333341 : ℝ) := by
+    rw [show
+        (750000000000000 / 2162699649333341 : ℝ) =
+          (3 / 40) / (2162699649333341 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2129896972254185019601 / 10000000000000000000000 : ℝ)) -
+        6 * (834527990154732413477 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (598690530973 / 12800000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (598690530973 / 12800000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2162699649333341 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2129896972254185019601 / 10000000000000000000000 : ℝ)) -
+          6 * (834527990154732413477 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (598690530973 / 12800000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (598690530973 / 12800000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (598690530973 / 12800000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_87 :
+    ((460691167579284695889 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(29417786266558953428469 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2171331016832617 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2138141194336938362321 / 10000000000000000000000 : ℝ)) -
+              6 * (3325801225385149696601 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (1508697083091 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1508697083091 / 32000000000000 : ℝ)) =
+      Real.log (33508697083091 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1508697083091 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1688697083091 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (460691167579284695889 / 10000000000000000000000 : ℝ) ≤
+      Real.log (33508697083091 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1688697083091 / 32000000000000 : ℝ) ≤
+      (-(29417786266558953428469 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show (1688697083091 / 32000000000000 : ℝ) = (1688697083091 / 1000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1688697083091 / 1000000000000 : ℝ) ≤
+            (5239572748441046571531 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2171331016832617 / 10000000000000000 : ℝ) ≤ Real.sqrt (1508697083091 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1508697083091 / 32000000000000 : ℝ) ≤ (108566550841631 / 500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1508697083091 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau149 :
+    Real.arctan (108566550841631 / 500000000000000 : ℝ) ≤
+      (2138141194336938362321 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (2138141194336938362321 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (500000000000000 / 108566550841631 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (500000000000000 / 108566550841631 : ℝ) =
+              (108566550841631 / 500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau149)
+  have hz1 :
+    (500000000000000 / 108566550841631 : ℝ) ≤
+      1 / Real.sqrt (1508697083091 / 32000000000000 : ℝ) := by
+    rw [show (500000000000000 / 108566550841631 : ℝ) = 1 / (108566550841631 / 500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 2171331016832617 : ℝ) ≤
+      (3325801225385149696601 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1508697083091 / 32000000000000 : ℝ) ≤
+      (750000000000000 / 2171331016832617 : ℝ) := by
+    rw [show
+        (750000000000000 / 2171331016832617 : ℝ) =
+          (3 / 40) / (2171331016832617 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2138141194336938362321 / 10000000000000000000000 : ℝ)) -
+        6 * (3325801225385149696601 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1508697083091 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1508697083091 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2171331016832617 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2138141194336938362321 / 10000000000000000000000 : ℝ)) -
+          6 * (3325801225385149696601 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (1508697083091 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1508697083091 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1508697083091 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_88 :
+    ((464262962060620974379 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(29347148812585080077493 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((435985641786157 / 2000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (429269968176594904411 / 2000000000000000000000 : ℝ)) -
+              6 * (1656812925087286155121 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (3041335677499 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (3041335677499 / 64000000000000 : ℝ)) =
+      Real.log (67041335677499 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((3041335677499 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (3401335677499 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (464262962060620974379 / 10000000000000000000000 : ℝ) ≤
+      Real.log (67041335677499 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (3401335677499 / 64000000000000 : ℝ) ≤
+      (-(29347148812585080077493 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show (3401335677499 / 64000000000000 : ℝ) = (3401335677499 / 2000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3401335677499 / 2000000000000 : ℝ) ≤
+            (5310210202414919922507 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (435985641786157 / 2000000000000000 : ℝ) ≤ Real.sqrt (3041335677499 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (3041335677499 / 64000000000000 : ℝ) ≤ (544982052232697 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (3041335677499 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau150 :
+    Real.arctan (544982052232697 / 2500000000000000 : ℝ) ≤
+      (429269968176594904411 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (429269968176594904411 / 2000000000000000000000 : ℝ)) ≤
+      Real.arctan (2500000000000000 / 544982052232697 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 544982052232697 : ℝ) =
+              (544982052232697 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau150)
+  have hz1 :
+    (2500000000000000 / 544982052232697 : ℝ) ≤
+      1 / Real.sqrt (3041335677499 / 64000000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 544982052232697 : ℝ) = 1 / (544982052232697 / 2500000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (150000000000000 / 435985641786157 : ℝ) ≤
+      (1656812925087286155121 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (3041335677499 / 64000000000000 : ℝ) ≤
+      (150000000000000 / 435985641786157 : ℝ) := by
+    rw [show
+        (150000000000000 / 435985641786157 : ℝ) =
+          (3 / 40) / (435985641786157 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (429269968176594904411 / 2000000000000000000000 : ℝ)) -
+        6 * (1656812925087286155121 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (3041335677499 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (3041335677499 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((435985641786157 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (429269968176594904411 / 2000000000000000000000 : ℝ)) -
+          6 * (1656812925087286155121 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (3041335677499 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (3041335677499 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (3041335677499 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_89 :
+    ((233916740612939341691 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(3659625853224161050433 / 1250000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((437698325677629 / 2000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2154523321717564278481 / 10000000000000000000000 : ℝ)) -
+              6 * (1320633347875011933 / 4000000000000000000 : ℝ)))) ≤
+      Vfield (191579824301 / 4000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (191579824301 / 4000000000000 : ℝ)) =
+      Real.log (4191579824301 / 4000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((191579824301 / 4000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (214079824301 / 4000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (233916740612939341691 / 5000000000000000000000 : ℝ) ≤
+      Real.log (4191579824301 / 4000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (214079824301 / 4000000000000 : ℝ) ≤
+      (-(3659625853224161050433 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show (214079824301 / 4000000000000 : ℝ) = (214079824301 / 125000000000 : ℝ) / 2 ^ 5 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (214079824301 / 125000000000 : ℝ) ≤
+            (672544023650838949567 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (437698325677629 / 2000000000000000 : ℝ) ≤ Real.sqrt (191579824301 / 4000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (191579824301 / 4000000000000 : ℝ) ≤ (547122907097037 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (191579824301 / 4000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau151 :
+    Real.arctan (547122907097037 / 2500000000000000 : ℝ) ≤
+      (2154523321717564278481 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (2154523321717564278481 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (2500000000000000 / 547122907097037 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 547122907097037 : ℝ) =
+              (547122907097037 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau151)
+  have hz1 :
+    (2500000000000000 / 547122907097037 : ℝ) ≤ 1 / Real.sqrt (191579824301 / 4000000000000 : ℝ) :=
+  by
+    rw [show
+        (2500000000000000 / 547122907097037 : ℝ) = 1 / (547122907097037 / 2500000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (50000000000000 / 145899441892543 : ℝ) ≤
+      (1320633347875011933 / 4000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (191579824301 / 4000000000000 : ℝ) ≤
+      (50000000000000 / 145899441892543 : ℝ) := by
+    rw [show
+        (50000000000000 / 145899441892543 : ℝ) =
+          (3 / 40) / (437698325677629 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2154523321717564278481 / 10000000000000000000000 : ℝ)) -
+        6 * (1320633347875011933 / 4000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (191579824301 / 4000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (191579824301 / 4000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((437698325677629 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2154523321717564278481 / 10000000000000000000000 : ℝ)) -
+          6 * (1320633347875011933 / 4000000000000000000 : ℝ))) ≤
+      Real.sqrt (191579824301 / 4000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (191579824301 / 4000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (191579824301 / 4000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_90 :
+    ((47140272598544088727 / 1000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(29207353403930246574293 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((109851083505783 / 500000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (432532407747580506091 / 2000000000000000000000 : ℝ)) -
+              6 * (1644835690550822215319 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (3089218700133 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (3089218700133 / 64000000000000 : ℝ)) =
+      Real.log (67089218700133 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((3089218700133 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (3449218700133 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (47140272598544088727 / 1000000000000000000000 : ℝ) ≤
+      Real.log (67089218700133 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (3449218700133 / 64000000000000 : ℝ) ≤
+      (-(29207353403930246574293 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show (3449218700133 / 64000000000000 : ℝ) = (3449218700133 / 2000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3449218700133 / 2000000000000 : ℝ) ≤
+            (5450005611069753425707 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (109851083505783 / 500000000000000 : ℝ) ≤ Real.sqrt (3089218700133 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (3089218700133 / 64000000000000 : ℝ) ≤ (2197021670115663 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (3089218700133 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau152 :
+    Real.arctan (2197021670115663 / 10000000000000000 : ℝ) ≤
+      (432532407747580506091 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (432532407747580506091 / 2000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 2197021670115663 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 2197021670115663 : ℝ) =
+              (2197021670115663 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau152)
+  have hz1 :
+    (10000000000000000 / 2197021670115663 : ℝ) ≤
+      1 / Real.sqrt (3089218700133 / 64000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 2197021670115663 : ℝ) =
+          1 / (2197021670115663 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (12500000000000 / 36617027835261 : ℝ) ≤
+      (1644835690550822215319 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (3089218700133 / 64000000000000 : ℝ) ≤
+      (12500000000000 / 36617027835261 : ℝ) := by
+    rw [show
+        (12500000000000 / 36617027835261 : ℝ) = (3 / 40) / (109851083505783 / 500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (432532407747580506091 / 2000000000000000000000 : ℝ)) -
+        6 * (1644835690550822215319 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (3089218700133 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (3089218700133 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((109851083505783 / 500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (432532407747580506091 / 2000000000000000000000 : ℝ)) -
+          6 * (1644835690550822215319 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (3089218700133 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (3089218700133 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (3089218700133 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_91 :
+    ((118742674312179046039 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(29138181787976365168227 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((220551872138747 / 1000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2170766386127115701763 / 10000000000000000000000 : ℝ)) -
+              6 * (655577508409682210587 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (62263204229 / 1280000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (62263204229 / 1280000000000 : ℝ)) =
+      Real.log (1342263204229 / 1280000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((62263204229 / 1280000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (69463204229 / 1280000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (118742674312179046039 / 2500000000000000000000 : ℝ) ≤
+      Real.log (1342263204229 / 1280000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (69463204229 / 1280000000000 : ℝ) ≤
+      (-(29138181787976365168227 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show (69463204229 / 1280000000000 : ℝ) = (69463204229 / 40000000000 : ℝ) / 2 ^ 5 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (69463204229 / 40000000000 : ℝ) ≤
+            (5519177227023634831773 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (220551872138747 / 1000000000000000 : ℝ) ≤ Real.sqrt (62263204229 / 1280000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (62263204229 / 1280000000000 : ℝ) ≤ (2205518721387473 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (62263204229 / 1280000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau153 :
+    Real.arctan (2205518721387473 / 10000000000000000 : ℝ) ≤
+      (2170766386127115701763 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (2170766386127115701763 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 2205518721387473 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 2205518721387473 : ℝ) =
+              (2205518721387473 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau153)
+  have hz1 :
+    (10000000000000000 / 2205518721387473 : ℝ) ≤
+      1 / Real.sqrt (62263204229 / 1280000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 2205518721387473 : ℝ) =
+          1 / (2205518721387473 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (75000000000000 / 220551872138747 : ℝ) ≤
+      (655577508409682210587 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (62263204229 / 1280000000000 : ℝ) ≤
+      (75000000000000 / 220551872138747 : ℝ) := by
+    rw [show
+        (75000000000000 / 220551872138747 : ℝ) =
+          (3 / 40) / (220551872138747 / 1000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2170766386127115701763 / 10000000000000000000000 : ℝ)) -
+        6 * (655577508409682210587 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (62263204229 / 1280000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (62263204229 / 1280000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((220551872138747 / 1000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2170766386127115701763 / 10000000000000000000000 : ℝ)) -
+          6 * (655577508409682210587 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (62263204229 / 1280000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (62263204229 / 1280000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (62263204229 / 1280000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_92 :
+    ((478537395924140096139 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1816842834888117217761 / 625000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2213983162046051 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (544709187640218728401 / 2500000000000000000000 : ℝ)) -
+              6 * (25517418505229754007 / 78125000000000000000 : ℝ)))) ≤
+      Vfield (3137101722767 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (3137101722767 / 64000000000000 : ℝ)) =
+      Real.log (67137101722767 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((3137101722767 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (3497101722767 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (478537395924140096139 / 10000000000000000000000 : ℝ) ≤
+      Real.log (67137101722767 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (3497101722767 / 64000000000000 : ℝ) ≤
+      (-(1816842834888117217761 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show (3497101722767 / 64000000000000 : ℝ) = (3497101722767 / 2000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3497101722767 / 2000000000000 : ℝ) ≤
+            (349242103549382782239 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2213983162046051 / 10000000000000000 : ℝ) ≤ Real.sqrt (3137101722767 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (3137101722767 / 64000000000000 : ℝ) ≤ (1106991581023027 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (3137101722767 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau154 :
+    Real.arctan (1106991581023027 / 5000000000000000 : ℝ) ≤
+      (544709187640218728401 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (544709187640218728401 / 2500000000000000000000 : ℝ)) ≤
+      Real.arctan (5000000000000000 / 1106991581023027 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 1106991581023027 : ℝ) =
+              (1106991581023027 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau154)
+  have hz1 :
+    (5000000000000000 / 1106991581023027 : ℝ) ≤
+      1 / Real.sqrt (3137101722767 / 64000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 1106991581023027 : ℝ) = 1 / (1106991581023027 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 2213983162046051 : ℝ) ≤
+      (25517418505229754007 / 78125000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (3137101722767 / 64000000000000 : ℝ) ≤
+      (750000000000000 / 2213983162046051 : ℝ) := by
+    rw [show
+        (750000000000000 / 2213983162046051 : ℝ) =
+          (3 / 40) / (2213983162046051 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (544709187640218728401 / 2500000000000000000000 : ℝ)) -
+        6 * (25517418505229754007 / 78125000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (3137101722767 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (3137101722767 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2213983162046051 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (544709187640218728401 / 2500000000000000000000 : ℝ)) -
+          6 * (25517418505229754007 / 78125000000000000000 : ℝ))) ≤
+      Real.sqrt (3137101722767 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (3137101722767 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (3137101722767 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_93 :
+    ((241051411459588228539 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(14500628815202590003913 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1111207682350181 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2186873511406957655783 / 10000000000000000000000 : ℝ)) -
+              6 * (3254695233748423021389 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (790260808521 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (790260808521 / 16000000000000 : ℝ)) =
+      Real.log (16790260808521 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((790260808521 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (880260808521 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (241051411459588228539 / 5000000000000000000000 : ℝ) ≤
+      Real.log (16790260808521 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (880260808521 / 16000000000000 : ℝ) ≤
+      (-(14500628815202590003913 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show (880260808521 / 16000000000000 : ℝ) = (880260808521 / 500000000000 : ℝ) / 2 ^ 5 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (880260808521 / 500000000000 : ℝ) ≤
+            (2828050692297409996087 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1111207682350181 / 5000000000000000 : ℝ) ≤ Real.sqrt (790260808521 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (790260808521 / 16000000000000 : ℝ) ≤ (444483072940073 / 2000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (790260808521 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau155 :
+    Real.arctan (444483072940073 / 2000000000000000 : ℝ) ≤
+      (2186873511406957655783 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (2186873511406957655783 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (2000000000000000 / 444483072940073 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2000000000000000 / 444483072940073 : ℝ) =
+              (444483072940073 / 2000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau155)
+  have hz1 :
+    (2000000000000000 / 444483072940073 : ℝ) ≤
+      1 / Real.sqrt (790260808521 / 16000000000000 : ℝ) := by
+    rw [show
+        (2000000000000000 / 444483072940073 : ℝ) = 1 / (444483072940073 / 2000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 1111207682350181 : ℝ) ≤
+      (3254695233748423021389 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (790260808521 / 16000000000000 : ℝ) ≤
+      (375000000000000 / 1111207682350181 : ℝ) := by
+    rw [show
+        (375000000000000 / 1111207682350181 : ℝ) =
+          (3 / 40) / (1111207682350181 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2186873511406957655783 / 10000000000000000000000 : ℝ)) -
+        6 * (3254695233748423021389 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (790260808521 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (790260808521 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1111207682350181 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2186873511406957655783 / 10000000000000000000000 : ℝ)) -
+          6 * (3254695233748423021389 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (790260808521 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (790260808521 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (790260808521 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_94 :
+    ((7588546549067481217 / 156250000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(361668653152000194621 / 125000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2230815694917233 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2194877040918014085383 / 10000000000000000000000 : ℝ)) -
+              6 * (1621641182458017941551 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (3184984745401 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (3184984745401 / 64000000000000 : ℝ)) =
+      Real.log (67184984745401 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((3184984745401 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (3544984745401 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (7588546549067481217 / 156250000000000000000 : ℝ) ≤
+      Real.log (67184984745401 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (3544984745401 / 64000000000000 : ℝ) ≤
+      (-(361668653152000194621 / 125000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show (3544984745401 / 64000000000000 : ℝ) = (3544984745401 / 2000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3544984745401 / 2000000000000 : ℝ) ≤
+            (71548334535499805379 / 125000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2230815694917233 / 10000000000000000 : ℝ) ≤ Real.sqrt (3184984745401 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (3184984745401 / 64000000000000 : ℝ) ≤ (557703923729309 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (3184984745401 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau156 :
+    Real.arctan (557703923729309 / 2500000000000000 : ℝ) ≤
+      (2194877040918014085383 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (2194877040918014085383 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (2500000000000000 / 557703923729309 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 557703923729309 : ℝ) =
+              (557703923729309 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau156)
+  have hz1 :
+    (2500000000000000 / 557703923729309 : ℝ) ≤
+      1 / Real.sqrt (3184984745401 / 64000000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 557703923729309 : ℝ) = 1 / (557703923729309 / 2500000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 2230815694917233 : ℝ) ≤
+      (1621641182458017941551 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (3184984745401 / 64000000000000 : ℝ) ≤
+      (750000000000000 / 2230815694917233 : ℝ) := by
+    rw [show
+        (750000000000000 / 2230815694917233 : ℝ) =
+          (3 / 40) / (2230815694917233 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2194877040918014085383 / 10000000000000000000000 : ℝ)) -
+        6 * (1621641182458017941551 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (3184984745401 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (3184984745401 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2230815694917233 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2194877040918014085383 / 10000000000000000000000 : ℝ)) -
+          6 * (1621641182458017941551 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (3184984745401 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (3184984745401 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (3184984745401 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_95 :
+    ((489229865493091729373 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(14433091499673113264557 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((22391845114063 / 100000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1101423852208921727903 / 5000000000000000000000 : ℝ)) -
+              6 * (807997210730837352441 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (1604463128359 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1604463128359 / 32000000000000 : ℝ)) =
+      Real.log (33604463128359 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1604463128359 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1784463128359 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (489229865493091729373 / 10000000000000000000000 : ℝ) ≤
+      Real.log (33604463128359 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1784463128359 / 32000000000000 : ℝ) ≤
+      (-(14433091499673113264557 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show (1784463128359 / 32000000000000 : ℝ) = (1784463128359 / 1000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1784463128359 / 1000000000000 : ℝ) ≤
+            (2895588007826886735443 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (22391845114063 / 100000000000000 : ℝ) ≤ Real.sqrt (1604463128359 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1604463128359 / 32000000000000 : ℝ) ≤ (2239184511406303 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1604463128359 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau157 :
+    Real.arctan (2239184511406303 / 10000000000000000 : ℝ) ≤
+      (1101423852208921727903 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1101423852208921727903 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 2239184511406303 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 2239184511406303 : ℝ) =
+              (2239184511406303 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau157)
+  have hz1 :
+    (10000000000000000 / 2239184511406303 : ℝ) ≤
+      1 / Real.sqrt (1604463128359 / 32000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 2239184511406303 : ℝ) =
+          1 / (2239184511406303 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (7500000000000 / 22391845114063 : ℝ) ≤
+      (807997210730837352441 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1604463128359 / 32000000000000 : ℝ) ≤
+      (7500000000000 / 22391845114063 : ℝ) := by
+    rw [show
+        (7500000000000 / 22391845114063 : ℝ) = (3 / 40) / (22391845114063 / 100000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1101423852208921727903 / 5000000000000000000000 : ℝ)) -
+        6 * (807997210730837352441 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1604463128359 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1604463128359 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((22391845114063 / 100000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1101423852208921727903 / 5000000000000000000000 : ℝ)) -
+          6 * (807997210730837352441 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (1604463128359 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1604463128359 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1604463128359 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_96 :
+    ((123197870720513080647 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(14399661886339595032899 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2247522166198741 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2210785860481439136199 / 10000000000000000000000 : ℝ)) -
+              6 * (3220812599981731487411 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (646573553607 / 12800000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (646573553607 / 12800000000000 : ℝ)) =
+      Real.log (13446573553607 / 12800000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((646573553607 / 12800000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (718573553607 / 12800000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (123197870720513080647 / 2500000000000000000000 : ℝ) ≤
+      Real.log (13446573553607 / 12800000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (718573553607 / 12800000000000 : ℝ) ≤
+      (-(14399661886339595032899 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show (718573553607 / 12800000000000 : ℝ) = (718573553607 / 400000000000 : ℝ) / 2 ^ 5 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (718573553607 / 400000000000 : ℝ) ≤
+            (2929017621160404967101 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2247522166198741 / 10000000000000000 : ℝ) ≤ Real.sqrt (646573553607 / 12800000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (646573553607 / 12800000000000 : ℝ) ≤ (280940270774843 / 1250000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (646573553607 / 12800000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau158 :
+    Real.arctan (280940270774843 / 1250000000000000 : ℝ) ≤
+      (2210785860481439136199 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (2210785860481439136199 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (1250000000000000 / 280940270774843 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1250000000000000 / 280940270774843 : ℝ) =
+              (280940270774843 / 1250000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau158)
+  have hz1 :
+    (1250000000000000 / 280940270774843 : ℝ) ≤
+      1 / Real.sqrt (646573553607 / 12800000000000 : ℝ) := by
+    rw [show
+        (1250000000000000 / 280940270774843 : ℝ) = 1 / (280940270774843 / 1250000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 2247522166198741 : ℝ) ≤
+      (3220812599981731487411 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (646573553607 / 12800000000000 : ℝ) ≤
+      (750000000000000 / 2247522166198741 : ℝ) := by
+    rw [show
+        (750000000000000 / 2247522166198741 : ℝ) =
+          (3 / 40) / (2247522166198741 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2210785860481439136199 / 10000000000000000000000 : ℝ)) -
+        6 * (3220812599981731487411 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (646573553607 / 12800000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (646573553607 / 12800000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2247522166198741 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2210785860481439136199 / 10000000000000000000000 : ℝ)) -
+          6 * (3220812599981731487411 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (646573553607 / 12800000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (646573553607 / 12800000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (646573553607 / 12800000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/V08.lean b/LeanPool/Zeta5Irrational/Table/V08.lean
new file mode 100644
index 0000000000..e09f79668b
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/V08.lean
@@ -0,0 +1,1229 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Energy
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # V08: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_97 :
+    ((15510994756587233961 / 312500000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(14366454297200582094051 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2255829004820067 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1109345930554525236403 / 5000000000000000000000 : ℝ)) -
+              6 * (1604875809082805327399 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (407101159919 / 8000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (407101159919 / 8000000000000 : ℝ)) =
+      Real.log (8407101159919 / 8000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((407101159919 / 8000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (452101159919 / 8000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (15510994756587233961 / 312500000000000000000 : ℝ) ≤
+      Real.log (8407101159919 / 8000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (452101159919 / 8000000000000 : ℝ) ≤
+      (-(14366454297200582094051 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show (452101159919 / 8000000000000 : ℝ) = (452101159919 / 250000000000 : ℝ) / 2 ^ 5 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (452101159919 / 250000000000 : ℝ) ≤
+            (2962225210299417905949 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2255829004820067 / 10000000000000000 : ℝ) ≤ Real.sqrt (407101159919 / 8000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (407101159919 / 8000000000000 : ℝ) ≤ (225582900482007 / 1000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (407101159919 / 8000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau159 :
+    Real.arctan (225582900482007 / 1000000000000000 : ℝ) ≤
+      (1109345930554525236403 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1109345930554525236403 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (1000000000000000 / 225582900482007 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1000000000000000 / 225582900482007 : ℝ) =
+              (225582900482007 / 1000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau159)
+  have hz1 :
+    (1000000000000000 / 225582900482007 : ℝ) ≤ 1 / Real.sqrt (407101159919 / 8000000000000 : ℝ) :=
+  by
+    rw [show
+        (1000000000000000 / 225582900482007 : ℝ) = 1 / (225582900482007 / 1000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 751943001606689 : ℝ) ≤
+      (1604875809082805327399 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (407101159919 / 8000000000000 : ℝ) ≤
+      (250000000000000 / 751943001606689 : ℝ) := by
+    rw [show
+        (250000000000000 / 751943001606689 : ℝ) =
+          (3 / 40) / (2255829004820067 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1109345930554525236403 / 5000000000000000000000 : ℝ)) -
+        6 * (1604875809082805327399 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (407101159919 / 8000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (407101159919 / 8000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2255829004820067 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1109345930554525236403 / 5000000000000000000000 : ℝ)) -
+          6 * (1604875809082805327399 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (407101159919 / 8000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (407101159919 / 8000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (407101159919 / 8000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_98 :
+    ((503468730297146605923 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(28601387060477260572887 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1136175792060049 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (89376350887521459017 / 400000000000000000000 : ℝ)) -
+              6 * (318796760635867034687 / 1000000000000000000000 : ℝ)))) ≤
+      Vfield (1652346150993 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1652346150993 / 32000000000000 : ℝ)) =
+      Real.log (33652346150993 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1652346150993 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1832346150993 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (503468730297146605923 / 10000000000000000000000 : ℝ) ≤
+      Real.log (33652346150993 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1832346150993 / 32000000000000 : ℝ) ≤
+      (-(28601387060477260572887 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show (1832346150993 / 32000000000000 : ℝ) = (1832346150993 / 1000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1832346150993 / 1000000000000 : ℝ) ≤
+            (6055971954522739427113 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1136175792060049 / 5000000000000000 : ℝ) ≤ Real.sqrt (1652346150993 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1652346150993 / 32000000000000 : ℝ) ≤ (2272351584120101 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1652346150993 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau160 :
+    Real.arctan (2272351584120101 / 10000000000000000 : ℝ) ≤
+      (89376350887521459017 / 400000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (89376350887521459017 / 400000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 2272351584120101 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 2272351584120101 : ℝ) =
+              (2272351584120101 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau160)
+  have hz1 :
+    (10000000000000000 / 2272351584120101 : ℝ) ≤
+      1 / Real.sqrt (1652346150993 / 32000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 2272351584120101 : ℝ) =
+          1 / (2272351584120101 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 1136175792060049 : ℝ) ≤
+      (318796760635867034687 / 1000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1652346150993 / 32000000000000 : ℝ) ≤
+      (375000000000000 / 1136175792060049 : ℝ) := by
+    rw [show
+        (375000000000000 / 1136175792060049 : ℝ) =
+          (3 / 40) / (1136175792060049 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (89376350887521459017 / 400000000000000000000 : ℝ)) -
+        6 * (318796760635867034687 / 1000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1652346150993 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1652346150993 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1136175792060049 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (89376350887521459017 / 400000000000000000000 : ℝ)) -
+          6 * (318796760635867034687 / 1000000000000000000000 : ℝ))) ≤
+      Real.sqrt (1652346150993 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1652346150993 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1652346150993 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_99 :
+    ((510580566961623545383 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(28471572886770229474981 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((457750977922221 / 2000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2250001128422295373419 / 10000000000000000000000 : ℝ)) -
+              6 * (1583310802186801713659 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (167628766231 / 3200000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (167628766231 / 3200000000000 : ℝ)) =
+      Real.log (3367628766231 / 3200000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((167628766231 / 3200000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (185628766231 / 3200000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (510580566961623545383 / 10000000000000000000000 : ℝ) ≤
+      Real.log (3367628766231 / 3200000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (185628766231 / 3200000000000 : ℝ) ≤
+      (-(28471572886770229474981 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show (185628766231 / 3200000000000 : ℝ) = (185628766231 / 100000000000 : ℝ) / 2 ^ 5 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (185628766231 / 100000000000 : ℝ) ≤
+            (6185786128229770525019 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (457750977922221 / 2000000000000000 : ℝ) ≤ Real.sqrt (167628766231 / 3200000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (167628766231 / 3200000000000 : ℝ) ≤ (572188722402777 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (167628766231 / 3200000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau161 :
+    Real.arctan (572188722402777 / 2500000000000000 : ℝ) ≤
+      (2250001128422295373419 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (2250001128422295373419 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (2500000000000000 / 572188722402777 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 572188722402777 : ℝ) =
+              (572188722402777 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau161)
+  have hz1 :
+    (2500000000000000 / 572188722402777 : ℝ) ≤ 1 / Real.sqrt (167628766231 / 3200000000000 : ℝ) :=
+  by
+    rw [show
+        (2500000000000000 / 572188722402777 : ℝ) = 1 / (572188722402777 / 2500000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (50000000000000 / 152583659307407 : ℝ) ≤
+      (1583310802186801713659 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (167628766231 / 3200000000000 : ℝ) ≤
+      (50000000000000 / 152583659307407 : ℝ) := by
+    rw [show
+        (50000000000000 / 152583659307407 : ℝ) =
+          (3 / 40) / (457750977922221 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2250001128422295373419 / 10000000000000000000000 : ℝ)) -
+        6 * (1583310802186801713659 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (167628766231 / 3200000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (167628766231 / 3200000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((457750977922221 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2250001128422295373419 / 10000000000000000000000 : ℝ)) -
+          6 * (1583310802186801713659 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (167628766231 / 3200000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (167628766231 / 3200000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (167628766231 / 3200000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_100 :
+    ((6471091867478608207 / 125000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(28343422312220233724777 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((72032545864051 / 312500000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2265471525602191722833 / 10000000000000000000000 : ℝ)) -
+              6 * (1572849557147011448877 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (1700229173627 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1700229173627 / 32000000000000 : ℝ)) =
+      Real.log (33700229173627 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1700229173627 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1880229173627 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (6471091867478608207 / 125000000000000000000 : ℝ) ≤
+      Real.log (33700229173627 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1880229173627 / 32000000000000 : ℝ) ≤
+      (-(28343422312220233724777 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show (1880229173627 / 32000000000000 : ℝ) = (1880229173627 / 1000000000000 : ℝ) / 2 ^ 5
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1880229173627 / 1000000000000 : ℝ) ≤
+            (6313936702779766275223 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (72032545864051 / 312500000000000 : ℝ) ≤ Real.sqrt (1700229173627 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1700229173627 / 32000000000000 : ℝ) ≤ (461008293529927 / 2000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1700229173627 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau162 :
+    Real.arctan (461008293529927 / 2000000000000000 : ℝ) ≤
+      (2265471525602191722833 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (2265471525602191722833 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (2000000000000000 / 461008293529927 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2000000000000000 / 461008293529927 : ℝ) =
+              (461008293529927 / 2000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau162)
+  have hz1 :
+    (2000000000000000 / 461008293529927 : ℝ) ≤
+      1 / Real.sqrt (1700229173627 / 32000000000000 : ℝ) := by
+    rw [show
+        (2000000000000000 / 461008293529927 : ℝ) = 1 / (461008293529927 / 2000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (23437500000000 / 72032545864051 : ℝ) ≤
+      (1572849557147011448877 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1700229173627 / 32000000000000 : ℝ) ≤
+      (23437500000000 / 72032545864051 : ℝ) := by
+    rw [show
+        (23437500000000 / 72032545864051 : ℝ) = (3 / 40) / (72032545864051 / 312500000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2265471525602191722833 / 10000000000000000000000 : ℝ)) -
+        6 * (1572849557147011448877 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1700229173627 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1700229173627 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((72032545864051 / 312500000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2265471525602191722833 / 10000000000000000000000 : ℝ)) -
+          6 * (1572849557147011448877 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (1700229173627 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1700229173627 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1700229173627 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_101 :
+    ((524789084785909236581 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(28216893236949584838917 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1160606887629269 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2280822469190898733897 / 10000000000000000000000 : ℝ)) -
+              6 * (3125186301868497031861 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (107760667809 / 2000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (107760667809 / 2000000000000 : ℝ)) =
+      Real.log (2107760667809 / 2000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((107760667809 / 2000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (119010667809 / 2000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (524789084785909236581 / 10000000000000000000000 : ℝ) ≤
+      Real.log (2107760667809 / 2000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (119010667809 / 2000000000000 : ℝ) ≤
+      (-(28216893236949584838917 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show (119010667809 / 2000000000000 : ℝ) = (119010667809 / 62500000000 : ℝ) / 2 ^ 5 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (119010667809 / 62500000000 : ℝ) ≤
+            (6440465778050415161083 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1160606887629269 / 5000000000000000 : ℝ) ≤ Real.sqrt (107760667809 / 2000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (107760667809 / 2000000000000 : ℝ) ≤ (2321213775258541 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (107760667809 / 2000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau163 :
+    Real.arctan (2321213775258541 / 10000000000000000 : ℝ) ≤
+      (2280822469190898733897 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (2280822469190898733897 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 2321213775258541 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 2321213775258541 : ℝ) =
+              (2321213775258541 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau163)
+  have hz1 :
+    (10000000000000000 / 2321213775258541 : ℝ) ≤
+      1 / Real.sqrt (107760667809 / 2000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 2321213775258541 : ℝ) =
+          1 / (2321213775258541 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 1160606887629269 : ℝ) ≤
+      (3125186301868497031861 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (107760667809 / 2000000000000 : ℝ) ≤
+      (375000000000000 / 1160606887629269 : ℝ) := by
+    rw [show
+        (375000000000000 / 1160606887629269 : ℝ) =
+          (3 / 40) / (1160606887629269 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2280822469190898733897 / 10000000000000000000000 : ℝ)) -
+        6 * (3125186301868497031861 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (107760667809 / 2000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (107760667809 / 2000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1160606887629269 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2280822469190898733897 / 10000000000000000000000 : ℝ)) -
+          6 * (3125186301868497031861 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (107760667809 / 2000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (107760667809 / 2000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (107760667809 / 2000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_102 :
+    ((134744360763168781831 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(27968538997595321615371 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2353224986307353 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (23111755917091873053 / 100000000000000000000 : ℝ)) -
+              6 * (192833591403646283279 / 625000000000000000000 : ℝ)))) ≤
+      Vfield (886026853789 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (886026853789 / 16000000000000 : ℝ)) =
+      Real.log (16886026853789 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((886026853789 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (976026853789 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (134744360763168781831 / 2500000000000000000000 : ℝ) ≤
+      Real.log (16886026853789 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (976026853789 / 16000000000000 : ℝ) ≤
+      (-(27968538997595321615371 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show (976026853789 / 16000000000000 : ℝ) = (976026853789 / 500000000000 : ℝ) / 2 ^ 5 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (976026853789 / 500000000000 : ℝ) ≤
+            (6688820017404678384629 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2353224986307353 / 10000000000000000 : ℝ) ≤ Real.sqrt (886026853789 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (886026853789 / 16000000000000 : ℝ) ≤ (588306246576839 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (886026853789 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau164 :
+    Real.arctan (588306246576839 / 2500000000000000 : ℝ) ≤
+      (23111755917091873053 / 100000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (23111755917091873053 / 100000000000000000000 : ℝ)) ≤
+      Real.arctan (2500000000000000 / 588306246576839 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 588306246576839 : ℝ) =
+              (588306246576839 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau164)
+  have hz1 :
+    (2500000000000000 / 588306246576839 : ℝ) ≤
+      1 / Real.sqrt (886026853789 / 16000000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 588306246576839 : ℝ) = 1 / (588306246576839 / 2500000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 2353224986307353 : ℝ) ≤
+      (192833591403646283279 / 625000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (886026853789 / 16000000000000 : ℝ) ≤
+      (750000000000000 / 2353224986307353 : ℝ) := by
+    rw [show
+        (750000000000000 / 2353224986307353 : ℝ) =
+          (3 / 40) / (2353224986307353 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (23111755917091873053 / 100000000000000000000 : ℝ)) -
+        6 * (192833591403646283279 / 625000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (886026853789 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (886026853789 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2353224986307353 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (23111755917091873053 / 100000000000000000000 : ℝ)) -
+          6 * (192833591403646283279 / 625000000000000000000 : ℝ))) ≤
+      Real.sqrt (886026853789 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (886026853789 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (886026853789 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_103 :
+    ((138286424721773515033 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(13863101781672242510999 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1192403275103739 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2341078892322328510047 / 10000000000000000000000 : ℝ)) -
+              6 * (3046976284942680492891 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (454984182553 / 8000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (454984182553 / 8000000000000 : ℝ)) =
+      Real.log (8454984182553 / 8000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((454984182553 / 8000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (499984182553 / 8000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (138286424721773515033 / 2500000000000000000000 : ℝ) ≤
+      Real.log (8454984182553 / 8000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (499984182553 / 8000000000000 : ℝ) ≤
+      (-(13863101781672242510999 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 5
+        (show (499984182553 / 8000000000000 : ℝ) = (499984182553 / 250000000000 : ℝ) / 2 ^ 5 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 15
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (499984182553 / 250000000000 : ℝ) ≤
+            (3465577725827757489001 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1192403275103739 / 5000000000000000 : ℝ) ≤ Real.sqrt (454984182553 / 8000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (454984182553 / 8000000000000 : ℝ) ≤ (2384806550207481 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (454984182553 / 8000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau165 :
+    Real.arctan (2384806550207481 / 10000000000000000 : ℝ) ≤
+      (2341078892322328510047 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (2341078892322328510047 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 2384806550207481 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 2384806550207481 : ℝ) =
+              (2384806550207481 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau165)
+  have hz1 :
+    (10000000000000000 / 2384806550207481 : ℝ) ≤
+      1 / Real.sqrt (454984182553 / 8000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 2384806550207481 : ℝ) =
+          1 / (2384806550207481 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (125000000000000 / 397467758367913 : ℝ) ≤
+      (3046976284942680492891 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (454984182553 / 8000000000000 : ℝ) ≤
+      (125000000000000 / 397467758367913 : ℝ) := by
+    rw [show
+        (125000000000000 / 397467758367913 : ℝ) =
+          (3 / 40) / (1192403275103739 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2341078892322328510047 / 10000000000000000000000 : ℝ)) -
+        6 * (3046976284942680492891 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (454984182553 / 8000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (454984182553 / 8000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1192403275103739 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2341078892322328510047 / 10000000000000000000000 : ℝ)) -
+          6 * (3046976284942680492891 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (454984182553 / 8000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (454984182553 / 8000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (454984182553 / 8000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_104 :
+    ((72677766318577952713 / 1250000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(218067756075849292299 / 80000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2446747059541503 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1199801878895283653787 / 5000000000000000000000 : ℝ)) -
+              6 * (594872574128161186261 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (47892569387 / 800000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (47892569387 / 800000000000 : ℝ)) =
+      Real.log (847892569387 / 800000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((47892569387 / 800000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (52392569387 / 800000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (72677766318577952713 / 1250000000000000000000 : ℝ) ≤
+      Real.log (847892569387 / 800000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (52392569387 / 800000000000 : ℝ) ≤
+      (-(218067756075849292299 / 80000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (52392569387 / 800000000000 : ℝ) = (52392569387 / 50000000000 : ℝ) / 2 ^ 4 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (52392569387 / 50000000000 : ℝ) ≤
+            (3739341620150707701 / 80000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2446747059541503 / 10000000000000000 : ℝ) ≤ Real.sqrt (47892569387 / 800000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (47892569387 / 800000000000 : ℝ) ≤ (1223373529770753 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (47892569387 / 800000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau166 :
+    Real.arctan (1223373529770753 / 5000000000000000 : ℝ) ≤
+      (1199801878895283653787 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1199801878895283653787 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (5000000000000000 / 1223373529770753 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 1223373529770753 : ℝ) =
+              (1223373529770753 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau166)
+  have hz1 :
+    (5000000000000000 / 1223373529770753 : ℝ) ≤ 1 / Real.sqrt (47892569387 / 800000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 1223373529770753 : ℝ) = 1 / (1223373529770753 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 815582353180501 : ℝ) ≤
+      (594872574128161186261 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (47892569387 / 800000000000 : ℝ) ≤
+      (250000000000000 / 815582353180501 : ℝ) := by
+    rw [show
+        (250000000000000 / 815582353180501 : ℝ) =
+          (3 / 40) / (2446747059541503 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1199801878895283653787 / 5000000000000000000000 : ℝ)) -
+        6 * (594872574128161186261 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (47892569387 / 800000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (47892569387 / 800000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2446747059541503 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1199801878895283653787 / 5000000000000000000000 : ℝ)) -
+          6 * (594872574128161186261 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (47892569387 / 800000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (47892569387 / 800000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (47892569387 / 800000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_105 :
+    ((609618831947161312673 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(26811638883410333602363 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1253578883121989 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1228261108513296682027 / 5000000000000000000000 : ℝ)) -
+              6 * (290670848125288106191 / 1000000000000000000000 : ℝ)))) ≤
+      Vfield (502867205187 / 8000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (502867205187 / 8000000000000 : ℝ)) =
+      Real.log (8502867205187 / 8000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((502867205187 / 8000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (547867205187 / 8000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (609618831947161312673 / 10000000000000000000000 : ℝ) ≤
+      Real.log (8502867205187 / 8000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (547867205187 / 8000000000000 : ℝ) ≤
+      (-(26811638883410333602363 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (547867205187 / 8000000000000 : ℝ) = (547867205187 / 500000000000 : ℝ) / 2 ^ 4 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (547867205187 / 500000000000 : ℝ) ≤
+            (914248328589666397637 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1253578883121989 / 5000000000000000 : ℝ) ≤ Real.sqrt (502867205187 / 8000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (502867205187 / 8000000000000 : ℝ) ≤ (2507157766243981 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (502867205187 / 8000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau167 :
+    Real.arctan (2507157766243981 / 10000000000000000 : ℝ) ≤
+      (1228261108513296682027 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1228261108513296682027 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 2507157766243981 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 2507157766243981 : ℝ) =
+              (2507157766243981 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau167)
+  have hz1 :
+    (10000000000000000 / 2507157766243981 : ℝ) ≤
+      1 / Real.sqrt (502867205187 / 8000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 2507157766243981 : ℝ) =
+          1 / (2507157766243981 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 1253578883121989 : ℝ) ≤
+      (290670848125288106191 / 1000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (502867205187 / 8000000000000 : ℝ) ≤
+      (375000000000000 / 1253578883121989 : ℝ) := by
+    rw [show
+        (375000000000000 / 1253578883121989 : ℝ) =
+          (3 / 40) / (1253578883121989 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1228261108513296682027 / 5000000000000000000000 : ℝ)) -
+        6 * (290670848125288106191 / 1000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (502867205187 / 8000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (502867205187 / 8000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1253578883121989 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1228261108513296682027 / 5000000000000000000000 : ℝ)) -
+          6 * (290670848125288106191 / 1000000000000000000000 : ℝ))) ≤
+      Real.sqrt (502867205187 / 8000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (502867205187 / 8000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (502867205187 / 8000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_106 :
+    ((63773625144515066253 / 1000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(26383922958767512167317 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2566146713712993 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2511944817458580581319 / 10000000000000000000000 : ℝ)) -
+              6 * (2843472865197113350371 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (65851089563 / 1000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (65851089563 / 1000000000000 : ℝ)) =
+      Real.log (1065851089563 / 1000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((65851089563 / 1000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (71476089563 / 1000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (63773625144515066253 / 1000000000000000000000 : ℝ) ≤
+      Real.log (1065851089563 / 1000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (71476089563 / 1000000000000 : ℝ) ≤
+      (-(26383922958767512167317 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (71476089563 / 1000000000000 : ℝ) = (71476089563 / 62500000000 : ℝ) / 2 ^ 4 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (71476089563 / 62500000000 : ℝ) ≤
+            (1341964253232487832683 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2566146713712993 / 10000000000000000 : ℝ) ≤ Real.sqrt (65851089563 / 1000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (65851089563 / 1000000000000 : ℝ) ≤ (641536678428249 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (65851089563 / 1000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau168 :
+    Real.arctan (641536678428249 / 2500000000000000 : ℝ) ≤
+      (2511944817458580581319 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (2511944817458580581319 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (2500000000000000 / 641536678428249 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 641536678428249 : ℝ) =
+              (641536678428249 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau168)
+  have hz1 :
+    (2500000000000000 / 641536678428249 : ℝ) ≤ 1 / Real.sqrt (65851089563 / 1000000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 641536678428249 : ℝ) = 1 / (641536678428249 / 2500000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 855382237904331 : ℝ) ≤
+      (2843472865197113350371 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (65851089563 / 1000000000000 : ℝ) ≤
+      (250000000000000 / 855382237904331 : ℝ) := by
+    rw [show
+        (250000000000000 / 855382237904331 : ℝ) =
+          (3 / 40) / (2566146713712993 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2511944817458580581319 / 10000000000000000000000 : ℝ)) -
+        6 * (2843472865197113350371 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (65851089563 / 1000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (65851089563 / 1000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2566146713712993 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2511944817458580581319 / 10000000000000000000000 : ℝ)) -
+          6 * (2843472865197113350371 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (65851089563 / 1000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (65851089563 / 1000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (65851089563 / 1000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_107 :
+    ((161897864139303837243 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(13118980103704165315979 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((646635646287919 / 2500000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2531071162844353185537 / 10000000000000000000000 : ℝ)) -
+              6 * (2822227043582772569641 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (2140864814337 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (2140864814337 / 32000000000000 : ℝ)) =
+      Real.log (34140864814337 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((2140864814337 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (2320864814337 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (161897864139303837243 / 2500000000000000000000 : ℝ) ≤
+      Real.log (34140864814337 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (2320864814337 / 32000000000000 : ℝ) ≤
+      (-(13118980103704165315979 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (2320864814337 / 32000000000000 : ℝ) = (2320864814337 / 2000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2320864814337 / 2000000000000 : ℝ) ≤
+            (743963502295834684021 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (646635646287919 / 2500000000000000 : ℝ) ≤ Real.sqrt (2140864814337 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (2140864814337 / 32000000000000 : ℝ) ≤ (2586542585151679 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (2140864814337 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau169 :
+    Real.arctan (2586542585151679 / 10000000000000000 : ℝ) ≤
+      (2531071162844353185537 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (2531071162844353185537 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 2586542585151679 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 2586542585151679 : ℝ) =
+              (2586542585151679 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau169)
+  have hz1 :
+    (10000000000000000 / 2586542585151679 : ℝ) ≤
+      1 / Real.sqrt (2140864814337 / 32000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 2586542585151679 : ℝ) =
+          1 / (2586542585151679 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (187500000000000 / 646635646287919 : ℝ) ≤
+      (2822227043582772569641 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (2140864814337 / 32000000000000 : ℝ) ≤
+      (187500000000000 / 646635646287919 : ℝ) := by
+    rw [show
+        (187500000000000 / 646635646287919 : ℝ) =
+          (3 / 40) / (646635646287919 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2531071162844353185537 / 10000000000000000000000 : ℝ)) -
+        6 * (2822227043582772569641 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (2140864814337 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (2140864814337 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((646635646287919 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2531071162844353185537 / 10000000000000000000000 : ℝ)) -
+          6 * (2822227043582772569641 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (2140864814337 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (2140864814337 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (2140864814337 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_108 :
+    ((26297478348966672627 / 400000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(26094097354576342594779 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2606778880784913 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2550029181381709669393 / 10000000000000000000000 : ℝ)) -
+              6 * (1400725358029980269551 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (1087247381329 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1087247381329 / 16000000000000 : ℝ)) =
+      Real.log (17087247381329 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1087247381329 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1177247381329 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (26297478348966672627 / 400000000000000000000 : ℝ) ≤
+      Real.log (17087247381329 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1177247381329 / 16000000000000 : ℝ) ≤
+      (-(26094097354576342594779 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (1177247381329 / 16000000000000 : ℝ) = (1177247381329 / 1000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1177247381329 / 1000000000000 : ℝ) ≤
+            (1631789857423657405221 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2606778880784913 / 10000000000000000 : ℝ) ≤ Real.sqrt (1087247381329 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1087247381329 / 16000000000000 : ℝ) ≤ (651694720196229 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1087247381329 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau170 :
+    Real.arctan (651694720196229 / 2500000000000000 : ℝ) ≤
+      (2550029181381709669393 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (2550029181381709669393 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (2500000000000000 / 651694720196229 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 651694720196229 : ℝ) =
+              (651694720196229 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau170)
+  have hz1 :
+    (2500000000000000 / 651694720196229 : ℝ) ≤
+      1 / Real.sqrt (1087247381329 / 16000000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 651694720196229 : ℝ) = 1 / (651694720196229 / 2500000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 868926293594971 : ℝ) ≤
+      (1400725358029980269551 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1087247381329 / 16000000000000 : ℝ) ≤
+      (250000000000000 / 868926293594971 : ℝ) := by
+    rw [show
+        (250000000000000 / 868926293594971 : ℝ) =
+          (3 / 40) / (2606778880784913 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2550029181381709669393 / 10000000000000000000000 : ℝ)) -
+        6 * (1400725358029980269551 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1087247381329 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1087247381329 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2606778880784913 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2550029181381709669393 / 10000000000000000000000 : ℝ)) -
+          6 * (1400725358029980269551 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (1087247381329 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1087247381329 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1087247381329 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/V09.lean b/LeanPool/Zeta5Irrational/Table/V09.lean
new file mode 100644
index 0000000000..be9cfa2025
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/V09.lean
@@ -0,0 +1,1235 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Energy
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # V09: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_109 :
+    ((83409097129161147991 / 1250000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(3244034354456449727219 / 1250000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2626859288543901 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1284411327890970380713 / 5000000000000000000000 : ℝ)) -
+              6 * (695281708072396394957 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (2208124710979 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (2208124710979 / 32000000000000 : ℝ)) =
+      Real.log (34208124710979 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((2208124710979 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (2388124710979 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (83409097129161147991 / 1250000000000000000000 : ℝ) ≤
+      Real.log (34208124710979 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (2388124710979 / 32000000000000 : ℝ) ≤
+      (-(3244034354456449727219 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (2388124710979 / 32000000000000 : ℝ) = (2388124710979 / 2000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2388124710979 / 2000000000000 : ℝ) ≤
+            (221701547043550272781 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2626859288543901 / 10000000000000000 : ℝ) ≤ Real.sqrt (2208124710979 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (2208124710979 / 32000000000000 : ℝ) ≤ (82089352766997 / 312500000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (2208124710979 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau171 :
+    Real.arctan (82089352766997 / 312500000000000 : ℝ) ≤
+      (1284411327890970380713 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1284411327890970380713 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (312500000000000 / 82089352766997 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (312500000000000 / 82089352766997 : ℝ) =
+              (82089352766997 / 312500000000000 : ℝ)
+            by
+        norm_num];
+        exact hau171)
+  have hz1 :
+    (312500000000000 / 82089352766997 : ℝ) ≤ 1 / Real.sqrt (2208124710979 / 32000000000000 : ℝ) :=
+  by
+    rw [show (312500000000000 / 82089352766997 : ℝ) = 1 / (82089352766997 / 312500000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 875619762847967 : ℝ) ≤
+      (695281708072396394957 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (2208124710979 / 32000000000000 : ℝ) ≤
+      (250000000000000 / 875619762847967 : ℝ) := by
+    rw [show
+        (250000000000000 / 875619762847967 : ℝ) =
+          (3 / 40) / (2626859288543901 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1284411327890970380713 / 5000000000000000000000 : ℝ)) -
+        6 * (695281708072396394957 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (2208124710979 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (2208124710979 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2626859288543901 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1284411327890970380713 / 5000000000000000000000 : ℝ)) -
+          6 * (695281708072396394957 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (2208124710979 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (2208124710979 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (2208124710979 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_110 :
+    ((16804676517263733869 / 250000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(6470527693651944065921 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((659210537122859 / 2500000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1289079416002930597313 / 5000000000000000000000 : ℝ)) -
+              6 * (554225892774949028251 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (4449879370279 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (4449879370279 / 64000000000000 : ℝ)) =
+      Real.log (68449879370279 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((4449879370279 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (4809879370279 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (16804676517263733869 / 250000000000000000000 : ℝ) ≤
+      Real.log (68449879370279 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (4809879370279 / 64000000000000 : ℝ) ≤
+      (-(6470527693651944065921 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (4809879370279 / 64000000000000 : ℝ) = (4809879370279 / 4000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4809879370279 / 4000000000000 : ℝ) ≤
+            (460944109348055934079 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (659210537122859 / 2500000000000000 : ℝ) ≤ Real.sqrt (4449879370279 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (4449879370279 / 64000000000000 : ℝ) ≤ (2636842148491439 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (4449879370279 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau172 :
+    Real.arctan (2636842148491439 / 10000000000000000 : ℝ) ≤
+      (1289079416002930597313 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1289079416002930597313 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 2636842148491439 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 2636842148491439 : ℝ) =
+              (2636842148491439 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau172)
+  have hz1 :
+    (10000000000000000 / 2636842148491439 : ℝ) ≤
+      1 / Real.sqrt (4449879370279 / 64000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 2636842148491439 : ℝ) =
+          1 / (2636842148491439 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (187500000000000 / 659210537122859 : ℝ) ≤
+      (554225892774949028251 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (4449879370279 / 64000000000000 : ℝ) ≤
+      (187500000000000 / 659210537122859 : ℝ) := by
+    rw [show
+        (187500000000000 / 659210537122859 : ℝ) =
+          (3 / 40) / (659210537122859 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1289079416002930597313 / 5000000000000000000000 : ℝ)) -
+        6 * (554225892774949028251 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (4449879370279 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (4449879370279 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((659210537122859 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1289079416002930597313 / 5000000000000000000000 : ℝ)) -
+          6 * (554225892774949028251 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (4449879370279 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (4449879370279 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (4449879370279 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_111 :
+    ((33854946525780009319 / 500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1613277224061336718483 / 625000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((66169683911481 / 250000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2587455226799105179707 / 10000000000000000000000 : ℝ)) -
+              6 * (2761239196892071779283 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (22417546593 / 320000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (22417546593 / 320000000000 : ℝ)) =
+      Real.log (342417546593 / 320000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((22417546593 / 320000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (24217546593 / 320000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (33854946525780009319 / 500000000000000000000 : ℝ) ≤
+      Real.log (342417546593 / 320000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (24217546593 / 320000000000 : ℝ) ≤
+      (-(1613277224061336718483 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (24217546593 / 320000000000 : ℝ) = (24217546593 / 20000000000 : ℝ) / 2 ^ 4 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24217546593 / 20000000000 : ℝ) ≤
+            (119590726688663281517 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (66169683911481 / 250000000000000 : ℝ) ≤ Real.sqrt (22417546593 / 320000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (22417546593 / 320000000000 : ℝ) ≤ (2646787356459243 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (22417546593 / 320000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau173 :
+    Real.arctan (2646787356459243 / 10000000000000000 : ℝ) ≤
+      (2587455226799105179707 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (2587455226799105179707 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 2646787356459243 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 2646787356459243 : ℝ) =
+              (2646787356459243 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau173)
+  have hz1 :
+    (10000000000000000 / 2646787356459243 : ℝ) ≤ 1 / Real.sqrt (22417546593 / 320000000000 : ℝ) :=
+  by
+    rw [show
+        (10000000000000000 / 2646787356459243 : ℝ) =
+          1 / (2646787356459243 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (6250000000000 / 22056561303827 : ℝ) ≤
+      (2761239196892071779283 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (22417546593 / 320000000000 : ℝ) ≤
+      (6250000000000 / 22056561303827 : ℝ) := by
+    rw [show
+        (6250000000000 / 22056561303827 : ℝ) = (3 / 40) / (66169683911481 / 250000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2587455226799105179707 / 10000000000000000000000 : ℝ)) -
+        6 * (2761239196892071779283 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (22417546593 / 320000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (22417546593 / 320000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((66169683911481 / 250000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2587455226799105179707 / 10000000000000000000000 : ℝ)) -
+          6 * (2761239196892071779283 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (22417546593 / 320000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (22417546593 / 320000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (22417546593 / 320000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_112 :
+    ((170502097219640879853 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(25743242501413712938097 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((531339067058467 / 2000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (40573629285751086729 / 156250000000000000000 : ℝ)) -
+              6 * (343931766448437447047 / 1250000000000000000000 : ℝ)))) ≤
+      Vfield (4517139266921 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (4517139266921 / 64000000000000 : ℝ)) =
+      Real.log (68517139266921 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((4517139266921 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (4877139266921 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (170502097219640879853 / 2500000000000000000000 : ℝ) ≤
+      Real.log (68517139266921 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (4877139266921 / 64000000000000 : ℝ) ≤
+      (-(25743242501413712938097 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (4877139266921 / 64000000000000 : ℝ) = (4877139266921 / 4000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4877139266921 / 4000000000000 : ℝ) ≤
+            (1982644710586287061903 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (531339067058467 / 2000000000000000 : ℝ) ≤ Real.sqrt (4517139266921 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (4517139266921 / 64000000000000 : ℝ) ≤ (1328347667646169 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (4517139266921 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau174 :
+    Real.arctan (1328347667646169 / 5000000000000000 : ℝ) ≤
+      (40573629285751086729 / 156250000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (40573629285751086729 / 156250000000000000000 : ℝ)) ≤
+      Real.arctan (5000000000000000 / 1328347667646169 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 1328347667646169 : ℝ) =
+              (1328347667646169 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau174)
+  have hz1 :
+    (5000000000000000 / 1328347667646169 : ℝ) ≤
+      1 / Real.sqrt (4517139266921 / 64000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 1328347667646169 : ℝ) = 1 / (1328347667646169 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (150000000000000 / 531339067058467 : ℝ) ≤
+      (343931766448437447047 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (4517139266921 / 64000000000000 : ℝ) ≤
+      (150000000000000 / 531339067058467 : ℝ) := by
+    rw [show
+        (150000000000000 / 531339067058467 : ℝ) =
+          (3 / 40) / (531339067058467 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (40573629285751086729 / 156250000000000000000 : ℝ)) -
+        6 * (343931766448437447047 / 1250000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (4517139266921 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (4517139266921 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((531339067058467 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (40573629285751086729 / 156250000000000000000 : ℝ)) -
+          6 * (343931766448437447047 / 1250000000000000000000 : ℝ))) ≤
+      Real.sqrt (4517139266921 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (4517139266921 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (4517139266921 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_113 :
+    ((343457719073036051437 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(12837262449008737335481 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1333283249990003 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2605930400624028400873 / 10000000000000000000000 : ℝ)) -
+              6 * (1370886207538278677507 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (2275384607621 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (2275384607621 / 32000000000000 : ℝ)) =
+      Real.log (34275384607621 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((2275384607621 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (2455384607621 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (343457719073036051437 / 5000000000000000000000 : ℝ) ≤
+      Real.log (34275384607621 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (2455384607621 / 32000000000000 : ℝ) ≤
+      (-(12837262449008737335481 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (2455384607621 / 32000000000000 : ℝ) = (2455384607621 / 2000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2455384607621 / 2000000000000 : ℝ) ≤
+            (1025681156991262664519 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1333283249990003 / 5000000000000000 : ℝ) ≤ Real.sqrt (2275384607621 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (2275384607621 / 32000000000000 : ℝ) ≤ (2666566499980009 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (2275384607621 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau175 :
+    Real.arctan (2666566499980009 / 10000000000000000 : ℝ) ≤
+      (2605930400624028400873 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (2605930400624028400873 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 2666566499980009 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 2666566499980009 : ℝ) =
+              (2666566499980009 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau175)
+  have hz1 :
+    (10000000000000000 / 2666566499980009 : ℝ) ≤
+      1 / Real.sqrt (2275384607621 / 32000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 2666566499980009 : ℝ) =
+          1 / (2666566499980009 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 1333283249990003 : ℝ) ≤
+      (1370886207538278677507 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (2275384607621 / 32000000000000 : ℝ) ≤
+      (375000000000000 / 1333283249990003 : ℝ) := by
+    rw [show
+        (375000000000000 / 1333283249990003 : ℝ) =
+          (3 / 40) / (1333283249990003 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2605930400624028400873 / 10000000000000000000000 : ℝ)) -
+        6 * (1370886207538278677507 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (2275384607621 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (2275384607621 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1333283249990003 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2605930400624028400873 / 10000000000000000000000 : ℝ)) -
+          6 * (1370886207538278677507 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (2275384607621 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (2275384607621 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (2275384607621 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_114 :
+    ((345910040340638219639 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(3200784535571160288559 / 1250000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((334550157232327 / 1250000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1307555012094042014081 / 5000000000000000000000 : ℝ)) -
+              6 * (170762014991703940461 / 625000000000000000000 : ℝ)))) ≤
+      Vfield (4584399163563 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (4584399163563 / 64000000000000 : ℝ)) =
+      Real.log (68584399163563 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((4584399163563 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (4944399163563 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (345910040340638219639 / 5000000000000000000000 : ℝ) ≤
+      Real.log (68584399163563 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (4944399163563 / 64000000000000 : ℝ) ≤
+      (-(3200784535571160288559 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (4944399163563 / 64000000000000 : ℝ) = (4944399163563 / 4000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4944399163563 / 4000000000000 : ℝ) ≤
+            (264951365928839711441 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (334550157232327 / 1250000000000000 : ℝ) ≤ Real.sqrt (4584399163563 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (4584399163563 / 64000000000000 : ℝ) ≤ (2676401257858619 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (4584399163563 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau176 :
+    Real.arctan (2676401257858619 / 10000000000000000 : ℝ) ≤
+      (1307555012094042014081 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1307555012094042014081 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 2676401257858619 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 2676401257858619 : ℝ) =
+              (2676401257858619 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau176)
+  have hz1 :
+    (10000000000000000 / 2676401257858619 : ℝ) ≤
+      1 / Real.sqrt (4584399163563 / 64000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 2676401257858619 : ℝ) =
+          1 / (2676401257858619 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (93750000000000 / 334550157232327 : ℝ) ≤
+      (170762014991703940461 / 625000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (4584399163563 / 64000000000000 : ℝ) ≤
+      (93750000000000 / 334550157232327 : ℝ) := by
+    rw [show
+        (93750000000000 / 334550157232327 : ℝ) =
+          (3 / 40) / (334550157232327 / 1250000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1307555012094042014081 / 5000000000000000000000 : ℝ)) -
+        6 * (170762014991703940461 / 625000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (4584399163563 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (4584399163563 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((334550157232327 / 1250000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1307555012094042014081 / 5000000000000000000000 : ℝ)) -
+          6 * (170762014991703940461 / 625000000000000000000 : ℝ))) ≤
+      Real.sqrt (4584399163563 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (4584399163563 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (4584399163563 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_115 :
+    ((13934446376877032259 / 200000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(25538490302831131762179 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2686200008807747 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (524850311157877829367 / 2000000000000000000000 : ℝ)) -
+              6 * (2722711842439540008903 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (1154507277971 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1154507277971 / 16000000000000 : ℝ)) =
+      Real.log (17154507277971 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1154507277971 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1244507277971 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (13934446376877032259 / 200000000000000000000 : ℝ) ≤
+      Real.log (17154507277971 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1244507277971 / 16000000000000 : ℝ) ≤
+      (-(25538490302831131762179 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (1244507277971 / 16000000000000 : ℝ) = (1244507277971 / 1000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1244507277971 / 1000000000000 : ℝ) ≤
+            (2187396909168868237821 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2686200008807747 / 10000000000000000 : ℝ) ≤ Real.sqrt (1154507277971 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1154507277971 / 16000000000000 : ℝ) ≤ (10744800035231 / 40000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1154507277971 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau177 :
+    Real.arctan (10744800035231 / 40000000000000 : ℝ) ≤
+      (524850311157877829367 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (524850311157877829367 / 2000000000000000000000 : ℝ)) ≤
+      Real.arctan (40000000000000 / 10744800035231 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (40000000000000 / 10744800035231 : ℝ) =
+              (10744800035231 / 40000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau177)
+  have hz1 :
+    (40000000000000 / 10744800035231 : ℝ) ≤ 1 / Real.sqrt (1154507277971 / 16000000000000 : ℝ) := by
+    rw [show (40000000000000 / 10744800035231 : ℝ) = 1 / (10744800035231 / 40000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 2686200008807747 : ℝ) ≤
+      (2722711842439540008903 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1154507277971 / 16000000000000 : ℝ) ≤
+      (750000000000000 / 2686200008807747 : ℝ) := by
+    rw [show
+        (750000000000000 / 2686200008807747 : ℝ) =
+          (3 / 40) / (2686200008807747 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (524850311157877829367 / 2000000000000000000000 : ℝ)) -
+        6 * (2722711842439540008903 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1154507277971 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1154507277971 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2686200008807747 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (524850311157877829367 / 2000000000000000000000 : ℝ)) -
+          6 * (2722711842439540008903 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (1154507277971 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1154507277971 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1154507277971 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_116 :
+    ((701622154990004101623 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(6367790180748935590831 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2695963145439919 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2633355398856895131241 / 10000000000000000000000 : ℝ)) -
+              6 * (1356664750939314845041 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (930331812041 / 12800000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (930331812041 / 12800000000000 : ℝ)) =
+      Real.log (13730331812041 / 12800000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((930331812041 / 12800000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1002331812041 / 12800000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (701622154990004101623 / 10000000000000000000000 : ℝ) ≤
+      Real.log (13730331812041 / 12800000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1002331812041 / 12800000000000 : ℝ) ≤
+      (-(6367790180748935590831 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (1002331812041 / 12800000000000 : ℝ) = (1002331812041 / 800000000000 : ℝ) / 2 ^ 4 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1002331812041 / 800000000000 : ℝ) ≤
+            (563681622251064409169 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2695963145439919 / 10000000000000000 : ℝ) ≤ Real.sqrt (930331812041 / 12800000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (930331812041 / 12800000000000 : ℝ) ≤ (1347981572719961 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (930331812041 / 12800000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau178 :
+    Real.arctan (1347981572719961 / 5000000000000000 : ℝ) ≤
+      (2633355398856895131241 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (2633355398856895131241 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (5000000000000000 / 1347981572719961 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 1347981572719961 : ℝ) =
+              (1347981572719961 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau178)
+  have hz1 :
+    (5000000000000000 / 1347981572719961 : ℝ) ≤
+      1 / Real.sqrt (930331812041 / 12800000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 1347981572719961 : ℝ) = 1 / (1347981572719961 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 2695963145439919 : ℝ) ≤
+      (1356664750939314845041 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (930331812041 / 12800000000000 : ℝ) ≤
+      (750000000000000 / 2695963145439919 : ℝ) := by
+    rw [show
+        (750000000000000 / 2695963145439919 : ℝ) =
+          (3 / 40) / (2695963145439919 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2633355398856895131241 / 10000000000000000000000 : ℝ)) -
+        6 * (1356664750939314845041 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (930331812041 / 12800000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (930331812041 / 12800000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2695963145439919 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2633355398856895131241 / 10000000000000000000000 : ℝ)) -
+          6 * (1356664750939314845041 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (930331812041 / 12800000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (930331812041 / 12800000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (930331812041 / 12800000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_117 :
+    ((706519591472478571351 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(5080856288050152583271 / 2000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((541138210656829 / 2000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (21139375596999086211 / 80000000000000000000 : ℝ)) -
+              6 * (338005442320010944411 / 1250000000000000000000 : ℝ)))) ≤
+      Vfield (2342644504263 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (2342644504263 / 32000000000000 : ℝ)) =
+      Real.log (34342644504263 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((2342644504263 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (2522644504263 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (706519591472478571351 / 10000000000000000000000 : ℝ) ≤
+      Real.log (34342644504263 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (2522644504263 / 32000000000000 : ℝ) ≤
+      (-(5080856288050152583271 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (2522644504263 / 32000000000000 : ℝ) = (2522644504263 / 2000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2522644504263 / 2000000000000 : ℝ) ≤
+            (464321154349847416729 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (541138210656829 / 2000000000000000 : ℝ) ≤ Real.sqrt (2342644504263 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (2342644504263 / 32000000000000 : ℝ) ≤ (676422763321037 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (2342644504263 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau179 :
+    Real.arctan (676422763321037 / 2500000000000000 : ℝ) ≤
+      (21139375596999086211 / 80000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (21139375596999086211 / 80000000000000000000 : ℝ)) ≤
+      Real.arctan (2500000000000000 / 676422763321037 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 676422763321037 : ℝ) =
+              (676422763321037 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau179)
+  have hz1 :
+    (2500000000000000 / 676422763321037 : ℝ) ≤
+      1 / Real.sqrt (2342644504263 / 32000000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 676422763321037 : ℝ) = 1 / (676422763321037 / 2500000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (150000000000000 / 541138210656829 : ℝ) ≤
+      (338005442320010944411 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (2342644504263 / 32000000000000 : ℝ) ≤
+      (150000000000000 / 541138210656829 : ℝ) := by
+    rw [show
+        (150000000000000 / 541138210656829 : ℝ) =
+          (3 / 40) / (541138210656829 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (21139375596999086211 / 80000000000000000000 : ℝ)) -
+        6 * (338005442320010944411 / 1250000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (2342644504263 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (2342644504263 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((541138210656829 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (21139375596999086211 / 80000000000000000000 : ℝ)) -
+          6 * (338005442320010944411 / 1250000000000000000000 : ℝ))) ≤
+      Real.sqrt (2342644504263 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (2342644504263 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (2342644504263 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_118 :
+    ((354483705287062168277 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(3171376098236488809029 / 1250000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((677635478748883 / 2500000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1323470681148397034747 / 5000000000000000000000 : ℝ)) -
+              6 * (1349718092230963117081 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (9404207965373 / 128000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (9404207965373 / 128000000000000 : ℝ)) =
+      Real.log (137404207965373 / 128000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((9404207965373 / 128000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (10124207965373 / 128000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (354483705287062168277 / 5000000000000000000000 : ℝ) ≤
+      Real.log (137404207965373 / 128000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (10124207965373 / 128000000000000 : ℝ) ≤
+      (-(3171376098236488809029 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (10124207965373 / 128000000000000 : ℝ) = (10124207965373 / 8000000000000 : ℝ) / 2 ^ 4 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10124207965373 / 8000000000000 : ℝ) ≤
+            (294359803263511190971 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (677635478748883 / 2500000000000000 : ℝ) ≤ Real.sqrt (9404207965373 / 128000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (9404207965373 / 128000000000000 : ℝ) ≤ (542108382999107 / 2000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (9404207965373 / 128000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau180 :
+    Real.arctan (542108382999107 / 2000000000000000 : ℝ) ≤
+      (1323470681148397034747 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1323470681148397034747 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (2000000000000000 / 542108382999107 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2000000000000000 / 542108382999107 : ℝ) =
+              (542108382999107 / 2000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau180)
+  have hz1 :
+    (2000000000000000 / 542108382999107 : ℝ) ≤
+      1 / Real.sqrt (9404207965373 / 128000000000000 : ℝ) := by
+    rw [show
+        (2000000000000000 / 542108382999107 : ℝ) = 1 / (542108382999107 / 2000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (187500000000000 / 677635478748883 : ℝ) ≤
+      (1349718092230963117081 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (9404207965373 / 128000000000000 : ℝ) ≤
+      (187500000000000 / 677635478748883 : ℝ) := by
+    rw [show
+        (187500000000000 / 677635478748883 : ℝ) =
+          (3 / 40) / (677635478748883 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1323470681148397034747 / 5000000000000000000000 : ℝ)) -
+        6 * (1349718092230963117081 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (9404207965373 / 128000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (9404207965373 / 128000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((677635478748883 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1323470681148397034747 / 5000000000000000000000 : ℝ)) -
+          6 * (1349718092230963117081 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (9404207965373 / 128000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (9404207965373 / 128000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (9404207965373 / 128000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_119 :
+    ((711414630640564654789 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(12668923235728554122461 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((678846027740893 / 2500000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (662862899328136929311 / 2500000000000000000000 : ℝ)) -
+              6 * (2694852312884068981851 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (4718918956847 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (4718918956847 / 64000000000000 : ℝ)) =
+      Real.log (68718918956847 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((4718918956847 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (5078918956847 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (711414630640564654789 / 10000000000000000000000 : ℝ) ≤
+      Real.log (68718918956847 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (5078918956847 / 64000000000000 : ℝ) ≤
+      (-(12668923235728554122461 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (5078918956847 / 64000000000000 : ℝ) = (5078918956847 / 4000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5078918956847 / 4000000000000 : ℝ) ≤
+            (1194020370271445877539 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (678846027740893 / 2500000000000000 : ℝ) ≤ Real.sqrt (4718918956847 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (4718918956847 / 64000000000000 : ℝ) ≤ (108615364438543 / 400000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (4718918956847 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau181 :
+    Real.arctan (108615364438543 / 400000000000000 : ℝ) ≤
+      (662862899328136929311 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (662862899328136929311 / 2500000000000000000000 : ℝ)) ≤
+      Real.arctan (400000000000000 / 108615364438543 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (400000000000000 / 108615364438543 : ℝ) =
+              (108615364438543 / 400000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau181)
+  have hz1 :
+    (400000000000000 / 108615364438543 : ℝ) ≤
+      1 / Real.sqrt (4718918956847 / 64000000000000 : ℝ) := by
+    rw [show (400000000000000 / 108615364438543 : ℝ) = 1 / (108615364438543 / 400000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (187500000000000 / 678846027740893 : ℝ) ≤
+      (2694852312884068981851 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (4718918956847 / 64000000000000 : ℝ) ≤
+      (187500000000000 / 678846027740893 : ℝ) := by
+    rw [show
+        (187500000000000 / 678846027740893 : ℝ) =
+          (3 / 40) / (678846027740893 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (662862899328136929311 / 2500000000000000000000 : ℝ)) -
+        6 * (2694852312884068981851 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (4718918956847 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (4718918956847 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((678846027740893 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (662862899328136929311 / 2500000000000000000000 : ℝ)) -
+          6 * (2694852312884068981851 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (4718918956847 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (4718918956847 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (4718918956847 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_120 :
+    ((713861251964921990511 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1581549610471153343741 / 625000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2720217687465327 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (663988175564238770757 / 2500000000000000000000 : ℝ)) -
+              6 * (2690291724930460355717 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (1894293572403 / 25600000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1894293572403 / 25600000000000 : ℝ)) =
+      Real.log (27494293572403 / 25600000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1894293572403 / 25600000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (2038293572403 / 25600000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (713861251964921990511 / 10000000000000000000000 : ℝ) ≤
+      Real.log (27494293572403 / 25600000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (2038293572403 / 25600000000000 : ℝ) ≤
+      (-(1581549610471153343741 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (2038293572403 / 25600000000000 : ℝ) = (2038293572403 / 1600000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2038293572403 / 1600000000000 : ℝ) ≤
+            (151318340278846656259 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2720217687465327 / 10000000000000000 : ℝ) ≤ Real.sqrt (1894293572403 / 25600000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1894293572403 / 25600000000000 : ℝ) ≤ (272021768746533 / 1000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1894293572403 / 25600000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau182 :
+    Real.arctan (272021768746533 / 1000000000000000 : ℝ) ≤
+      (663988175564238770757 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (663988175564238770757 / 2500000000000000000000 : ℝ)) ≤
+      Real.arctan (1000000000000000 / 272021768746533 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1000000000000000 / 272021768746533 : ℝ) =
+              (272021768746533 / 1000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau182)
+  have hz1 :
+    (1000000000000000 / 272021768746533 : ℝ) ≤
+      1 / Real.sqrt (1894293572403 / 25600000000000 : ℝ) := by
+    rw [show
+        (1000000000000000 / 272021768746533 : ℝ) = 1 / (272021768746533 / 1000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 906739229155109 : ℝ) ≤
+      (2690291724930460355717 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1894293572403 / 25600000000000 : ℝ) ≤
+      (250000000000000 / 906739229155109 : ℝ) := by
+    rw [show
+        (250000000000000 / 906739229155109 : ℝ) =
+          (3 / 40) / (2720217687465327 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (663988175564238770757 / 2500000000000000000000 : ℝ)) -
+        6 * (2690291724930460355717 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1894293572403 / 25600000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1894293572403 / 25600000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2720217687465327 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (663988175564238770757 / 2500000000000000000000 : ℝ)) -
+          6 * (2690291724930460355717 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (1894293572403 / 25600000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1894293572403 / 25600000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1894293572403 / 25600000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/V10.lean b/LeanPool/Zeta5Irrational/Table/V10.lean
new file mode 100644
index 0000000000..a49640cb61
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/V10.lean
@@ -0,0 +1,1240 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Energy
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # V10: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_121 :
+    ((358153637420051856447 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(505436999038738140509 / 200000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((681260672591859 / 2500000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2660444724297807303401 / 10000000000000000000000 : ℝ)) -
+              6 * (2685754224056738553489 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (297034306573 / 4000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (297034306573 / 4000000000000 : ℝ)) =
+      Real.log (4297034306573 / 4000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((297034306573 / 4000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (319534306573 / 4000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (358153637420051856447 / 5000000000000000000000 : ℝ) ≤
+      Real.log (4297034306573 / 4000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (319534306573 / 4000000000000 : ℝ) ≤
+      (-(505436999038738140509 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (319534306573 / 4000000000000 : ℝ) = (319534306573 / 250000000000 : ℝ) / 2 ^ 4 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (319534306573 / 250000000000 : ℝ) ≤
+            (49080745201261859491 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (681260672591859 / 2500000000000000 : ℝ) ≤ Real.sqrt (297034306573 / 4000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (297034306573 / 4000000000000 : ℝ) ≤ (2725042690367439 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (297034306573 / 4000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau183 :
+    Real.arctan (2725042690367439 / 10000000000000000 : ℝ) ≤
+      (2660444724297807303401 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (2660444724297807303401 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 2725042690367439 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 2725042690367439 : ℝ) =
+              (2725042690367439 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau183)
+  have hz1 :
+    (10000000000000000 / 2725042690367439 : ℝ) ≤
+      1 / Real.sqrt (297034306573 / 4000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 2725042690367439 : ℝ) =
+          1 / (2725042690367439 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (62500000000000 / 227086890863953 : ℝ) ≤
+      (2685754224056738553489 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (297034306573 / 4000000000000 : ℝ) ≤
+      (62500000000000 / 227086890863953 : ℝ) := by
+    rw [show
+        (62500000000000 / 227086890863953 : ℝ) =
+          (3 / 40) / (681260672591859 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2660444724297807303401 / 10000000000000000000000 : ℝ)) -
+        6 * (2685754224056738553489 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (297034306573 / 4000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (297034306573 / 4000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((681260672591859 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2660444724297807303401 / 10000000000000000000000 : ℝ)) -
+          6 * (2685754224056738553489 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (297034306573 / 4000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (297034306573 / 4000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (297034306573 / 4000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_122 :
+    ((143750539911760461247 / 2000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(25239014309567609724181 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2729859165131193 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1332463855095506303277 / 5000000000000000000000 : ℝ)) -
+              6 * (2681239616034587774699 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (9538727758657 / 128000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (9538727758657 / 128000000000000 : ℝ)) =
+      Real.log (137538727758657 / 128000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((9538727758657 / 128000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (10258727758657 / 128000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (143750539911760461247 / 2000000000000000000000 : ℝ) ≤
+      Real.log (137538727758657 / 128000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (10258727758657 / 128000000000000 : ℝ) ≤
+      (-(25239014309567609724181 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (10258727758657 / 128000000000000 : ℝ) = (10258727758657 / 8000000000000 : ℝ) / 2 ^ 4 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10258727758657 / 8000000000000 : ℝ) ≤
+            (2486872902432390275819 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2729859165131193 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (9538727758657 / 128000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (9538727758657 / 128000000000000 : ℝ) ≤ (682464791282799 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (9538727758657 / 128000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau184 :
+    Real.arctan (682464791282799 / 2500000000000000 : ℝ) ≤
+      (1332463855095506303277 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1332463855095506303277 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (2500000000000000 / 682464791282799 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 682464791282799 : ℝ) =
+              (682464791282799 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau184)
+  have hz1 :
+    (2500000000000000 / 682464791282799 : ℝ) ≤
+      1 / Real.sqrt (9538727758657 / 128000000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 682464791282799 : ℝ) = 1 / (682464791282799 / 2500000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 909953055043731 : ℝ) ≤
+      (2681239616034587774699 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (9538727758657 / 128000000000000 : ℝ) ≤
+      (250000000000000 / 909953055043731 : ℝ) := by
+    rw [show
+        (250000000000000 / 909953055043731 : ℝ) =
+          (3 / 40) / (2729859165131193 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1332463855095506303277 / 5000000000000000000000 : ℝ)) -
+        6 * (2681239616034587774699 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (9538727758657 / 128000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (9538727758657 / 128000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2729859165131193 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1332463855095506303277 / 5000000000000000000000 : ℝ)) -
+          6 * (2681239616034587774699 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (9538727758657 / 128000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (9538727758657 / 128000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (9538727758657 / 128000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_123 :
+    ((180299381603373895023 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(25206286132366748618349 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((546933431363509 / 2000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2669401706285649911771 / 10000000000000000000000 : ℝ)) -
+              6 * (2676747708916748668133 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (4786178853489 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (4786178853489 / 64000000000000 : ℝ)) =
+      Real.log (68786178853489 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((4786178853489 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (5146178853489 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (180299381603373895023 / 2500000000000000000000 : ℝ) ≤
+      Real.log (68786178853489 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (5146178853489 / 64000000000000 : ℝ) ≤
+      (-(25206286132366748618349 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (5146178853489 / 64000000000000 : ℝ) = (5146178853489 / 4000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5146178853489 / 4000000000000 : ℝ) ≤
+            (2519601079633251381651 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (546933431363509 / 2000000000000000 : ℝ) ≤ Real.sqrt (4786178853489 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (4786178853489 / 64000000000000 : ℝ) ≤ (683666789204387 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (4786178853489 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau185 :
+    Real.arctan (683666789204387 / 2500000000000000 : ℝ) ≤
+      (2669401706285649911771 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (2669401706285649911771 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (2500000000000000 / 683666789204387 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 683666789204387 : ℝ) =
+              (683666789204387 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau185)
+  have hz1 :
+    (2500000000000000 / 683666789204387 : ℝ) ≤
+      1 / Real.sqrt (4786178853489 / 64000000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 683666789204387 : ℝ) = 1 / (683666789204387 / 2500000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (150000000000000 / 546933431363509 : ℝ) ≤
+      (2676747708916748668133 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (4786178853489 / 64000000000000 : ℝ) ≤
+      (150000000000000 / 546933431363509 : ℝ) := by
+    rw [show
+        (150000000000000 / 546933431363509 : ℝ) =
+          (3 / 40) / (546933431363509 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2669401706285649911771 / 10000000000000000000000 : ℝ)) -
+        6 * (2676747708916748668133 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (4786178853489 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (4786178853489 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((546933431363509 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2669401706285649911771 / 10000000000000000000000 : ℝ)) -
+          6 * (2676747708916748668133 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (4786178853489 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (4786178853489 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (4786178853489 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_124 :
+    ((180910438924111719793 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(6293416179799985485973 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2739466710092009 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2673866758528940898129 / 10000000000000000000000 : ℝ)) -
+              6 * (1336139156501337456721 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (9605987655299 / 128000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (9605987655299 / 128000000000000 : ℝ)) =
+      Real.log (137605987655299 / 128000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((9605987655299 / 128000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (10325987655299 / 128000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (180910438924111719793 / 2500000000000000000000 : ℝ) ≤
+      Real.log (137605987655299 / 128000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (10325987655299 / 128000000000000 : ℝ) ≤
+      (-(6293416179799985485973 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (10325987655299 / 128000000000000 : ℝ) = (10325987655299 / 8000000000000 : ℝ) / 2 ^ 4 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10325987655299 / 8000000000000 : ℝ) ≤
+            (638055623200014514027 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2739466710092009 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (9605987655299 / 128000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (9605987655299 / 128000000000000 : ℝ) ≤ (684866677523003 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (9605987655299 / 128000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau186 :
+    Real.arctan (684866677523003 / 2500000000000000 : ℝ) ≤
+      (2673866758528940898129 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (2673866758528940898129 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (2500000000000000 / 684866677523003 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 684866677523003 : ℝ) =
+              (684866677523003 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau186)
+  have hz1 :
+    (2500000000000000 / 684866677523003 : ℝ) ≤
+      1 / Real.sqrt (9605987655299 / 128000000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 684866677523003 : ℝ) = 1 / (684866677523003 / 2500000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 2739466710092009 : ℝ) ≤
+      (1336139156501337456721 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (9605987655299 / 128000000000000 : ℝ) ≤
+      (750000000000000 / 2739466710092009 : ℝ) := by
+    rw [show
+        (750000000000000 / 2739466710092009 : ℝ) =
+          (3 / 40) / (2739466710092009 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2673866758528940898129 / 10000000000000000000000 : ℝ)) -
+        6 * (1336139156501337456721 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (9605987655299 / 128000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (9605987655299 / 128000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2739466710092009 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2673866758528940898129 / 10000000000000000000000 : ℝ)) -
+          6 * (1336139156501337456721 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (9605987655299 / 128000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (9605987655299 / 128000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (9605987655299 / 128000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_125 :
+    ((726085387699705292959 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(6285287343943028423973 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((548851573845903 / 2000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2678322912471147172359 / 10000000000000000000000 : ℝ)) -
+              6 * (1333915620402408418013 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (481980880181 / 6400000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (481980880181 / 6400000000000 : ℝ)) =
+      Real.log (6881980880181 / 6400000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((481980880181 / 6400000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (517980880181 / 6400000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (726085387699705292959 / 10000000000000000000000 : ℝ) ≤
+      Real.log (6881980880181 / 6400000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (517980880181 / 6400000000000 : ℝ) ≤
+      (-(6285287343943028423973 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (517980880181 / 6400000000000 : ℝ) = (517980880181 / 400000000000 : ℝ) / 2 ^ 4 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (517980880181 / 400000000000 : ℝ) ≤
+            (646184459056971576027 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (548851573845903 / 2000000000000000 : ℝ) ≤ Real.sqrt (481980880181 / 6400000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (481980880181 / 6400000000000 : ℝ) ≤ (1372128934614759 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (481980880181 / 6400000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau187 :
+    Real.arctan (1372128934614759 / 5000000000000000 : ℝ) ≤
+      (2678322912471147172359 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (2678322912471147172359 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (5000000000000000 / 1372128934614759 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 1372128934614759 : ℝ) =
+              (1372128934614759 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau187)
+  have hz1 :
+    (5000000000000000 / 1372128934614759 : ℝ) ≤
+      1 / Real.sqrt (481980880181 / 6400000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 1372128934614759 : ℝ) = 1 / (1372128934614759 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (50000000000000 / 182950524615301 : ℝ) ≤
+      (1333915620402408418013 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (481980880181 / 6400000000000 : ℝ) ≤
+      (50000000000000 / 182950524615301 : ℝ) := by
+    rw [show
+        (50000000000000 / 182950524615301 : ℝ) =
+          (3 / 40) / (548851573845903 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2678322912471147172359 / 10000000000000000000000 : ℝ)) -
+        6 * (1333915620402408418013 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (481980880181 / 6400000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (481980880181 / 6400000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((548851573845903 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2678322912471147172359 / 10000000000000000000000 : ℝ)) -
+          6 * (1333915620402408418013 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (481980880181 / 6400000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (481980880181 / 6400000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (481980880181 / 6400000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_126 :
+    ((72852842271510586509 / 1000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(12554369707269414330891 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2749040678119169 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2682770213270387066649 / 10000000000000000000000 : ℝ)) -
+              6 * (1331703153507762705967 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (9673247551941 / 128000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (9673247551941 / 128000000000000 : ℝ)) =
+      Real.log (137673247551941 / 128000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((9673247551941 / 128000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (10393247551941 / 128000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (72852842271510586509 / 1000000000000000000000 : ℝ) ≤
+      Real.log (137673247551941 / 128000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (10393247551941 / 128000000000000 : ℝ) ≤
+      (-(12554369707269414330891 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show
+          (10393247551941 / 128000000000000 : ℝ) = (10393247551941 / 8000000000000 : ℝ) / 2 ^ 4 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10393247551941 / 8000000000000 : ℝ) ≤
+            (1308573898730585669109 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2749040678119169 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (9673247551941 / 128000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (9673247551941 / 128000000000000 : ℝ) ≤ (687260169529793 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (9673247551941 / 128000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau188 :
+    Real.arctan (687260169529793 / 2500000000000000 : ℝ) ≤
+      (2682770213270387066649 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (2682770213270387066649 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (2500000000000000 / 687260169529793 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 687260169529793 : ℝ) =
+              (687260169529793 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau188)
+  have hz1 :
+    (2500000000000000 / 687260169529793 : ℝ) ≤
+      1 / Real.sqrt (9673247551941 / 128000000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 687260169529793 : ℝ) = 1 / (687260169529793 / 2500000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 2749040678119169 : ℝ) ≤
+      (1331703153507762705967 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (9673247551941 / 128000000000000 : ℝ) ≤
+      (750000000000000 / 2749040678119169 : ℝ) := by
+    rw [show
+        (750000000000000 / 2749040678119169 : ℝ) =
+          (3 / 40) / (2749040678119169 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2682770213270387066649 / 10000000000000000000000 : ℝ)) -
+        6 * (1331703153507762705967 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (9673247551941 / 128000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (9673247551941 / 128000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2749040678119169 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2682770213270387066649 / 10000000000000000000000 : ℝ)) -
+          6 * (1331703153507762705967 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (9673247551941 / 128000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (9673247551941 / 128000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (9673247551941 / 128000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_127 :
+    ((730970861034269802467 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(25076434154619059633243 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((172113448766809 / 625000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (335901088212172452663 / 1250000000000000000000 : ℝ)) -
+              6 * (10636013313898229031 / 40000000000000000000 : ℝ)))) ≤
+      Vfield (4853438750131 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (4853438750131 / 64000000000000 : ℝ)) =
+      Real.log (68853438750131 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((4853438750131 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (5213438750131 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (730970861034269802467 / 10000000000000000000000 : ℝ) ≤
+      Real.log (68853438750131 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (5213438750131 / 64000000000000 : ℝ) ≤
+      (-(25076434154619059633243 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (5213438750131 / 64000000000000 : ℝ) = (5213438750131 / 4000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5213438750131 / 4000000000000 : ℝ) ≤
+            (2649453057380940366757 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (172113448766809 / 625000000000000 : ℝ) ≤ Real.sqrt (4853438750131 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (4853438750131 / 64000000000000 : ℝ) ≤ (2753815180268947 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (4853438750131 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau189 :
+    Real.arctan (2753815180268947 / 10000000000000000 : ℝ) ≤
+      (335901088212172452663 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (335901088212172452663 / 1250000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 2753815180268947 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 2753815180268947 : ℝ) =
+              (2753815180268947 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau189)
+  have hz1 :
+    (10000000000000000 / 2753815180268947 : ℝ) ≤
+      1 / Real.sqrt (4853438750131 / 64000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 2753815180268947 : ℝ) =
+          1 / (2753815180268947 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (46875000000000 / 172113448766809 : ℝ) ≤
+      (10636013313898229031 / 40000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (4853438750131 / 64000000000000 : ℝ) ≤
+      (46875000000000 / 172113448766809 : ℝ) := by
+    rw [show
+        (46875000000000 / 172113448766809 : ℝ) =
+          (3 / 40) / (172113448766809 / 625000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (335901088212172452663 / 1250000000000000000000 : ℝ)) -
+        6 * (10636013313898229031 / 40000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (4853438750131 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (4853438750131 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((172113448766809 / 625000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (335901088212172452663 / 1250000000000000000000 : ℝ)) -
+          6 * (10636013313898229031 / 40000000000000000000 : ℝ))) ≤
+      Real.sqrt (4853438750131 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (4853438750131 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (4853438750131 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_128 :
+    ((735853948749304669843 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(5002427009599881711513 / 2000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2763339436502733 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (5265741098844656167 / 19531250000000000000 : ℝ)) -
+              6 * (2650262515042783265311 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (1221767174613 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1221767174613 / 16000000000000 : ℝ)) =
+      Real.log (17221767174613 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1221767174613 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1311767174613 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (735853948749304669843 / 10000000000000000000000 : ℝ) ≤
+      Real.log (17221767174613 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1311767174613 / 16000000000000 : ℝ) ≤
+      (-(5002427009599881711513 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (1311767174613 / 16000000000000 : ℝ) = (1311767174613 / 1000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1311767174613 / 1000000000000 : ℝ) ≤
+            (542750432800118288487 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2763339436502733 / 10000000000000000 : ℝ) ≤ Real.sqrt (1221767174613 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1221767174613 / 16000000000000 : ℝ) ≤ (172708714781421 / 625000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1221767174613 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau190 :
+    Real.arctan (172708714781421 / 625000000000000 : ℝ) ≤
+      (5265741098844656167 / 19531250000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (5265741098844656167 / 19531250000000000000 : ℝ)) ≤
+      Real.arctan (625000000000000 / 172708714781421 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (625000000000000 / 172708714781421 : ℝ) =
+              (172708714781421 / 625000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau190)
+  have hz1 :
+    (625000000000000 / 172708714781421 : ℝ) ≤
+      1 / Real.sqrt (1221767174613 / 16000000000000 : ℝ) := by
+    rw [show (625000000000000 / 172708714781421 : ℝ) = 1 / (172708714781421 / 625000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 921113145500911 : ℝ) ≤
+      (2650262515042783265311 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1221767174613 / 16000000000000 : ℝ) ≤
+      (250000000000000 / 921113145500911 : ℝ) := by
+    rw [show
+        (250000000000000 / 921113145500911 : ℝ) =
+          (3 / 40) / (2763339436502733 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (5265741098844656167 / 19531250000000000000 : ℝ)) -
+        6 * (2650262515042783265311 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1221767174613 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1221767174613 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2763339436502733 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (5265741098844656167 / 19531250000000000000 : ℝ)) -
+          6 * (2650262515042783265311 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (1221767174613 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1221767174613 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1221767174613 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_129 :
+    ((740734653173510746083 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(6237061684726463436083 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1386415489272859 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (676218868449462891921 / 2500000000000000000000 : ℝ)) -
+              6 * (52832147539552359909 / 200000000000000000000 : ℝ)))) ≤
+      Vfield (4920698646773 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (4920698646773 / 64000000000000 : ℝ)) =
+      Real.log (68920698646773 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((4920698646773 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (5280698646773 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (740734653173510746083 / 10000000000000000000000 : ℝ) ≤
+      Real.log (68920698646773 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (5280698646773 / 64000000000000 : ℝ) ≤
+      (-(6237061684726463436083 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (5280698646773 / 64000000000000 : ℝ) = (5280698646773 / 4000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5280698646773 / 4000000000000 : ℝ) ≤
+            (694410118273536563917 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1386415489272859 / 5000000000000000 : ℝ) ≤ Real.sqrt (4920698646773 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (4920698646773 / 64000000000000 : ℝ) ≤ (2772830978545721 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (4920698646773 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau191 :
+    Real.arctan (2772830978545721 / 10000000000000000 : ℝ) ≤
+      (676218868449462891921 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (676218868449462891921 / 2500000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 2772830978545721 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 2772830978545721 : ℝ) =
+              (2772830978545721 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau191)
+  have hz1 :
+    (10000000000000000 / 2772830978545721 : ℝ) ≤
+      1 / Real.sqrt (4920698646773 / 64000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 2772830978545721 : ℝ) =
+          1 / (2772830978545721 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 1386415489272859 : ℝ) ≤
+      (52832147539552359909 / 200000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (4920698646773 / 64000000000000 : ℝ) ≤
+      (375000000000000 / 1386415489272859 : ℝ) := by
+    rw [show
+        (375000000000000 / 1386415489272859 : ℝ) =
+          (3 / 40) / (1386415489272859 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (676218868449462891921 / 2500000000000000000000 : ℝ)) -
+        6 * (52832147539552359909 / 200000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (4920698646773 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (4920698646773 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1386415489272859 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (676218868449462891921 / 2500000000000000000000 : ℝ)) -
+          6 * (52832147539552359909 / 200000000000000000000 : ℝ))) ≤
+      Real.sqrt (4920698646773 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (4920698646773 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (4920698646773 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_130 :
+    ((37280648831609041759 / 500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(6221191002898389833279 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2782290141202813 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (542731428790419567221 / 2000000000000000000000 : ℝ)) -
+              6 * (658259130780697986243 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (2477164297547 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (2477164297547 / 32000000000000 : ℝ)) =
+      Real.log (34477164297547 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((2477164297547 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (2657164297547 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (37280648831609041759 / 500000000000000000000 : ℝ) ≤
+      Real.log (34477164297547 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (2657164297547 / 32000000000000 : ℝ) ≤
+      (-(6221191002898389833279 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (2657164297547 / 32000000000000 : ℝ) = (2657164297547 / 2000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2657164297547 / 2000000000000 : ℝ) ≤
+            (710280800101610166721 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2782290141202813 / 10000000000000000 : ℝ) ≤ Real.sqrt (2477164297547 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (2477164297547 / 32000000000000 : ℝ) ≤ (21736641728147 / 78125000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (2477164297547 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau192 :
+    Real.arctan (21736641728147 / 78125000000000 : ℝ) ≤
+      (542731428790419567221 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (542731428790419567221 / 2000000000000000000000 : ℝ)) ≤
+      Real.arctan (78125000000000 / 21736641728147 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (78125000000000 / 21736641728147 : ℝ) =
+              (21736641728147 / 78125000000000 : ℝ)
+            by
+        norm_num];
+        exact hau192)
+  have hz1 :
+    (78125000000000 / 21736641728147 : ℝ) ≤ 1 / Real.sqrt (2477164297547 / 32000000000000 : ℝ) := by
+    rw [show (78125000000000 / 21736641728147 : ℝ) = 1 / (21736641728147 / 78125000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 2782290141202813 : ℝ) ≤
+      (658259130780697986243 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (2477164297547 / 32000000000000 : ℝ) ≤
+      (750000000000000 / 2782290141202813 : ℝ) := by
+    rw [show
+        (750000000000000 / 2782290141202813 : ℝ) =
+          (3 / 40) / (2782290141202813 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (542731428790419567221 / 2000000000000000000000 : ℝ)) -
+        6 * (658259130780697986243 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (2477164297547 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (2477164297547 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2782290141202813 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (542731428790419567221 / 2000000000000000000000 : ℝ)) -
+          6 * (658259130780697986243 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (2477164297547 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (2477164297547 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (2477164297547 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_131 :
+    ((750488921447206341103 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(24821681749024979579647 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1395858626803403 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2722404791990247254829 / 10000000000000000000000 : ℝ)) -
+              6 * (2624548593756629872647 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (997591708683 / 12800000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (997591708683 / 12800000000000 : ℝ)) =
+      Real.log (13797591708683 / 12800000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((997591708683 / 12800000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1069591708683 / 12800000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (750488921447206341103 / 10000000000000000000000 : ℝ) ≤
+      Real.log (13797591708683 / 12800000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1069591708683 / 12800000000000 : ℝ) ≤
+      (-(24821681749024979579647 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (1069591708683 / 12800000000000 : ℝ) = (1069591708683 / 800000000000 : ℝ) / 2 ^ 4 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1069591708683 / 800000000000 : ℝ) ≤
+            (2904205462975020420353 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1395858626803403 / 5000000000000000 : ℝ) ≤ Real.sqrt (997591708683 / 12800000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (997591708683 / 12800000000000 : ℝ) ≤ (2791717253606809 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (997591708683 / 12800000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau193 :
+    Real.arctan (2791717253606809 / 10000000000000000 : ℝ) ≤
+      (2722404791990247254829 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (2722404791990247254829 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 2791717253606809 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 2791717253606809 : ℝ) =
+              (2791717253606809 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau193)
+  have hz1 :
+    (10000000000000000 / 2791717253606809 : ℝ) ≤
+      1 / Real.sqrt (997591708683 / 12800000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 2791717253606809 : ℝ) =
+          1 / (2791717253606809 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 1395858626803403 : ℝ) ≤
+      (2624548593756629872647 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (997591708683 / 12800000000000 : ℝ) ≤
+      (375000000000000 / 1395858626803403 : ℝ) := by
+    rw [show
+        (375000000000000 / 1395858626803403 : ℝ) =
+          (3 / 40) / (1395858626803403 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2722404791990247254829 / 10000000000000000000000 : ℝ)) -
+        6 * (2624548593756629872647 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (997591708683 / 12800000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (997591708683 / 12800000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1395858626803403 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2722404791990247254829 / 10000000000000000000000 : ℝ)) -
+          6 * (2624548593756629872647 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (997591708683 / 12800000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (997591708683 / 12800000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (997591708683 / 12800000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_132 :
+    ((377681244968541949289 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(24758994930394767520483 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1400556319675997 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2731118751199019664549 / 10000000000000000000000 : ℝ)) -
+              6 * (2616142259684255857913 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (627698561467 / 8000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (627698561467 / 8000000000000 : ℝ)) =
+      Real.log (8627698561467 / 8000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((627698561467 / 8000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (672698561467 / 8000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (377681244968541949289 / 5000000000000000000000 : ℝ) ≤
+      Real.log (8627698561467 / 8000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (672698561467 / 8000000000000 : ℝ) ≤
+      (-(24758994930394767520483 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (672698561467 / 8000000000000 : ℝ) = (672698561467 / 500000000000 : ℝ) / 2 ^ 4 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (672698561467 / 500000000000 : ℝ) ≤
+            (2966892281605232479517 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1400556319675997 / 5000000000000000 : ℝ) ≤ Real.sqrt (627698561467 / 8000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (627698561467 / 8000000000000 : ℝ) ≤ (2801112639351997 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (627698561467 / 8000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau194 :
+    Real.arctan (2801112639351997 / 10000000000000000 : ℝ) ≤
+      (2731118751199019664549 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (2731118751199019664549 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 2801112639351997 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 2801112639351997 : ℝ) =
+              (2801112639351997 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau194)
+  have hz1 :
+    (10000000000000000 / 2801112639351997 : ℝ) ≤
+      1 / Real.sqrt (627698561467 / 8000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 2801112639351997 : ℝ) =
+          1 / (2801112639351997 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 1400556319675997 : ℝ) ≤
+      (2616142259684255857913 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (627698561467 / 8000000000000 : ℝ) ≤
+      (375000000000000 / 1400556319675997 : ℝ) := by
+    rw [show
+        (375000000000000 / 1400556319675997 : ℝ) =
+          (3 / 40) / (1400556319675997 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2731118751199019664549 / 10000000000000000000000 : ℝ)) -
+        6 * (2616142259684255857913 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (627698561467 / 8000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (627698561467 / 8000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1400556319675997 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2731118751199019664549 / 10000000000000000000000 : ℝ)) -
+          6 * (2616142259684255857913 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (627698561467 / 8000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (627698561467 / 8000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (627698561467 / 8000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/V11.lean b/LeanPool/Zeta5Irrational/Table/V11.lean
new file mode 100644
index 0000000000..77c27faea0
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/V11.lean
@@ -0,0 +1,1233 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Energy
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # V11: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_133 :
+    ((190058421104230496931 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(6174174657182944863893 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2810476616623781 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2739799349363924849661 / 10000000000000000000000 : ℝ)) -
+              6 * (2607816221361664600063 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (5055218440057 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (5055218440057 / 64000000000000 : ℝ)) =
+      Real.log (69055218440057 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((5055218440057 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (5415218440057 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (190058421104230496931 / 2500000000000000000000 : ℝ) ≤
+      Real.log (69055218440057 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (5415218440057 / 64000000000000 : ℝ) ≤
+      (-(6174174657182944863893 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (5415218440057 / 64000000000000 : ℝ) = (5415218440057 / 4000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5415218440057 / 4000000000000 : ℝ) ≤
+            (757297145817055136107 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2810476616623781 / 10000000000000000 : ℝ) ≤ Real.sqrt (5055218440057 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (5055218440057 / 64000000000000 : ℝ) ≤ (351309577077973 / 1250000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (5055218440057 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau195 :
+    Real.arctan (351309577077973 / 1250000000000000 : ℝ) ≤
+      (2739799349363924849661 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (2739799349363924849661 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (1250000000000000 / 351309577077973 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1250000000000000 / 351309577077973 : ℝ) =
+              (351309577077973 / 1250000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau195)
+  have hz1 :
+    (1250000000000000 / 351309577077973 : ℝ) ≤
+      1 / Real.sqrt (5055218440057 / 64000000000000 : ℝ) := by
+    rw [show
+        (1250000000000000 / 351309577077973 : ℝ) = 1 / (351309577077973 / 1250000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 2810476616623781 : ℝ) ≤
+      (2607816221361664600063 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (5055218440057 / 64000000000000 : ℝ) ≤
+      (750000000000000 / 2810476616623781 : ℝ) := by
+    rw [show
+        (750000000000000 / 2810476616623781 : ℝ) =
+          (3 / 40) / (2810476616623781 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2739799349363924849661 / 10000000000000000000000 : ℝ)) -
+        6 * (2607816221361664600063 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (5055218440057 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (5055218440057 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2810476616623781 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2739799349363924849661 / 10000000000000000000000 : ℝ)) -
+          6 * (2607816221361664600063 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (5055218440057 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (5055218440057 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (5055218440057 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_134 :
+    ((382551253599223766289 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1231739400428765560097 / 500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((140990474916221 / 500000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (687111727224121272759 / 2500000000000000000000 : ℝ)) -
+              6 * (1299784604025157638269 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (2544424194189 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (2544424194189 / 32000000000000 : ℝ)) =
+      Real.log (34544424194189 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((2544424194189 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (2724424194189 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (382551253599223766289 / 5000000000000000000000 : ℝ) ≤
+      Real.log (34544424194189 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (2724424194189 / 32000000000000 : ℝ) ≤
+      (-(1231739400428765560097 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (2724424194189 / 32000000000000 : ℝ) = (2724424194189 / 2000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2724424194189 / 2000000000000 : ℝ) ≤
+            (154554960171234439903 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (140990474916221 / 500000000000000 : ℝ) ≤ Real.sqrt (2544424194189 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (2544424194189 / 32000000000000 : ℝ) ≤ (2819809498324423 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (2544424194189 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau196 :
+    Real.arctan (2819809498324423 / 10000000000000000 : ℝ) ≤
+      (687111727224121272759 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (687111727224121272759 / 2500000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 2819809498324423 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 2819809498324423 : ℝ) =
+              (2819809498324423 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau196)
+  have hz1 :
+    (10000000000000000 / 2819809498324423 : ℝ) ≤
+      1 / Real.sqrt (2544424194189 / 32000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 2819809498324423 : ℝ) =
+          1 / (2819809498324423 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (37500000000000 / 140990474916221 : ℝ) ≤
+      (1299784604025157638269 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (2544424194189 / 32000000000000 : ℝ) ≤
+      (37500000000000 / 140990474916221 : ℝ) := by
+    rw [show
+        (37500000000000 / 140990474916221 : ℝ) =
+          (3 / 40) / (140990474916221 / 500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (687111727224121272759 / 2500000000000000000000 : ℝ)) -
+        6 * (1299784604025157638269 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (2544424194189 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (2544424194189 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((140990474916221 / 500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (687111727224121272759 / 2500000000000000000000 : ℝ)) -
+          6 * (1299784604025157638269 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (2544424194189 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (2544424194189 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (2544424194189 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_135 :
+    ((30798758423600499343 / 400000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(4914651664744565192507 / 2000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2829111592195009 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (689265436739419728229 / 2500000000000000000000 : ℝ)) -
+              6 * (1295699988500512258437 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (5122478336699 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (5122478336699 / 64000000000000 : ℝ)) =
+      Real.log (69122478336699 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((5122478336699 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (5482478336699 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (30798758423600499343 / 400000000000000000000 : ℝ) ≤
+      Real.log (69122478336699 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (5482478336699 / 64000000000000 : ℝ) ≤
+      (-(4914651664744565192507 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (5482478336699 / 64000000000000 : ℝ) = (5482478336699 / 4000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5482478336699 / 4000000000000 : ℝ) ≤
+            (630525777655434807493 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2829111592195009 / 10000000000000000 : ℝ) ≤ Real.sqrt (5122478336699 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (5122478336699 / 64000000000000 : ℝ) ≤ (707277898048753 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (5122478336699 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau197 :
+    Real.arctan (707277898048753 / 2500000000000000 : ℝ) ≤
+      (689265436739419728229 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (689265436739419728229 / 2500000000000000000000 : ℝ)) ≤
+      Real.arctan (2500000000000000 / 707277898048753 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 707277898048753 : ℝ) =
+              (707277898048753 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau197)
+  have hz1 :
+    (2500000000000000 / 707277898048753 : ℝ) ≤
+      1 / Real.sqrt (5122478336699 / 64000000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 707277898048753 : ℝ) = 1 / (707277898048753 / 2500000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 2829111592195009 : ℝ) ≤
+      (1295699988500512258437 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (5122478336699 / 64000000000000 : ℝ) ≤
+      (750000000000000 / 2829111592195009 : ℝ) := by
+    rw [show
+        (750000000000000 / 2829111592195009 : ℝ) =
+          (3 / 40) / (2829111592195009 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (689265436739419728229 / 2500000000000000000000 : ℝ)) -
+        6 * (1295699988500512258437 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (5122478336699 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (5122478336699 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2829111592195009 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (689265436739419728229 / 2500000000000000000000 : ℝ)) -
+          6 * (1295699988500512258437 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (5122478336699 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (5122478336699 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (5122478336699 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_136 :
+    ((774833046896600384011 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(24512104915046550125607 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((567676640186779 / 2000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (691411043894439901673 / 2500000000000000000000 : ℝ)) -
+              6 * (645826828166487246769 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (257805414251 / 3200000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (257805414251 / 3200000000000 : ℝ)) =
+      Real.log (3457805414251 / 3200000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((257805414251 / 3200000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (275805414251 / 3200000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (774833046896600384011 / 10000000000000000000000 : ℝ) ≤
+      Real.log (3457805414251 / 3200000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (275805414251 / 3200000000000 : ℝ) ≤
+      (-(24512104915046550125607 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (275805414251 / 3200000000000 : ℝ) = (275805414251 / 200000000000 : ℝ) / 2 ^ 4 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (275805414251 / 200000000000 : ℝ) ≤
+            (3213782296953449874393 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (567676640186779 / 2000000000000000 : ℝ) ≤ Real.sqrt (257805414251 / 3200000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (257805414251 / 3200000000000 : ℝ) ≤ (1419191600466949 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (257805414251 / 3200000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau198 :
+    Real.arctan (1419191600466949 / 5000000000000000 : ℝ) ≤
+      (691411043894439901673 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (691411043894439901673 / 2500000000000000000000 : ℝ)) ≤
+      Real.arctan (5000000000000000 / 1419191600466949 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 1419191600466949 : ℝ) =
+              (1419191600466949 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau198)
+  have hz1 :
+    (5000000000000000 / 1419191600466949 : ℝ) ≤
+      1 / Real.sqrt (257805414251 / 3200000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 1419191600466949 : ℝ) = 1 / (1419191600466949 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (150000000000000 / 567676640186779 : ℝ) ≤
+      (645826828166487246769 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (257805414251 / 3200000000000 : ℝ) ≤
+      (150000000000000 / 567676640186779 : ℝ) := by
+    rw [show
+        (150000000000000 / 567676640186779 : ℝ) =
+          (3 / 40) / (567676640186779 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (691411043894439901673 / 2500000000000000000000 : ℝ)) -
+        6 * (645826828166487246769 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (257805414251 / 3200000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (257805414251 / 3200000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((567676640186779 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (691411043894439901673 / 2500000000000000000000 : ℝ)) -
+          6 * (645826828166487246769 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (257805414251 / 3200000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (257805414251 / 3200000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (257805414251 / 3200000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_137 :
+    ((196138531864494114447 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(3048863589055876755923 / 1250000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2856836149282431 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2782713027656475583597 / 10000000000000000000000 : ℝ)) -
+              6 * (2567346953413312793429 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (2611684090831 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (2611684090831 / 32000000000000 : ℝ)) =
+      Real.log (34611684090831 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((2611684090831 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (2791684090831 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (196138531864494114447 / 2500000000000000000000 : ℝ) ≤
+      Real.log (34611684090831 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (2791684090831 / 32000000000000 : ℝ) ≤
+      (-(3048863589055876755923 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (2791684090831 / 32000000000000 : ℝ) = (2791684090831 / 2000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2791684090831 / 2000000000000 : ℝ) ≤
+            (416872312444123244077 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2856836149282431 / 10000000000000000 : ℝ) ≤ Real.sqrt (2611684090831 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (2611684090831 / 32000000000000 : ℝ) ≤ (1428418074641217 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (2611684090831 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau199 :
+    Real.arctan (1428418074641217 / 5000000000000000 : ℝ) ≤
+      (2782713027656475583597 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (2782713027656475583597 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (5000000000000000 / 1428418074641217 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 1428418074641217 : ℝ) =
+              (1428418074641217 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau199)
+  have hz1 :
+    (5000000000000000 / 1428418074641217 : ℝ) ≤
+      1 / Real.sqrt (2611684090831 / 32000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 1428418074641217 : ℝ) = 1 / (1428418074641217 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 952278716427477 : ℝ) ≤
+      (2567346953413312793429 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (2611684090831 / 32000000000000 : ℝ) ≤
+      (250000000000000 / 952278716427477 : ℝ) := by
+    rw [show
+        (250000000000000 / 952278716427477 : ℝ) =
+          (3 / 40) / (2856836149282431 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2782713027656475583597 / 10000000000000000000000 : ℝ)) -
+        6 * (2567346953413312793429 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (2611684090831 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (2611684090831 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2856836149282431 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2782713027656475583597 / 10000000000000000000000 : ℝ)) -
+          6 * (2567346953413312793429 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (2611684090831 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (2611684090831 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (2611684090831 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_138 :
+    ((198566441813831119773 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(3033895473797801538299 / 1250000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1437585334193243 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (279965586309664145921 / 1000000000000000000000 : ℝ)) -
+              6 * (2551678921654980299883 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (165332127447 / 2000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (165332127447 / 2000000000000 : ℝ)) =
+      Real.log (2165332127447 / 2000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((165332127447 / 2000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (176582127447 / 2000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (198566441813831119773 / 2500000000000000000000 : ℝ) ≤
+      Real.log (2165332127447 / 2000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (176582127447 / 2000000000000 : ℝ) ≤
+      (-(3033895473797801538299 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (176582127447 / 2000000000000 : ℝ) = (176582127447 / 125000000000 : ℝ) / 2 ^ 4 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (176582127447 / 125000000000 : ℝ) ≤
+            (431840427702198461701 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1437585334193243 / 5000000000000000 : ℝ) ≤ Real.sqrt (165332127447 / 2000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (165332127447 / 2000000000000 : ℝ) ≤ (2875170668386489 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (165332127447 / 2000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau200 :
+    Real.arctan (2875170668386489 / 10000000000000000 : ℝ) ≤
+      (279965586309664145921 / 1000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (279965586309664145921 / 1000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 2875170668386489 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 2875170668386489 : ℝ) =
+              (2875170668386489 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau200)
+  have hz1 :
+    (10000000000000000 / 2875170668386489 : ℝ) ≤
+      1 / Real.sqrt (165332127447 / 2000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 2875170668386489 : ℝ) =
+          1 / (2875170668386489 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 1437585334193243 : ℝ) ≤
+      (2551678921654980299883 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (165332127447 / 2000000000000 : ℝ) ≤
+      (375000000000000 / 1437585334193243 : ℝ) := by
+    rw [show
+        (375000000000000 / 1437585334193243 : ℝ) =
+          (3 / 40) / (1437585334193243 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (279965586309664145921 / 1000000000000000000000 : ℝ)) -
+        6 * (2551678921654980299883 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (165332127447 / 2000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (165332127447 / 2000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1437585334193243 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (279965586309664145921 / 1000000000000000000000 : ℝ)) -
+          6 * (2551678921654980299883 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (165332127447 / 2000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (165332127447 / 2000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (165332127447 / 2000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_139 :
+    ((803967984607916239937 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1207641790130783507417 / 500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2893389009596379 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2816475004873911548537 / 10000000000000000000000 : ℝ)) -
+              6 * (1268147198934660719937 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (2678943987473 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (2678943987473 / 32000000000000 : ℝ)) =
+      Real.log (34678943987473 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((2678943987473 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (2858943987473 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (803967984607916239937 / 10000000000000000000000 : ℝ) ≤
+      Real.log (34678943987473 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (2858943987473 / 32000000000000 : ℝ) ≤
+      (-(1207641790130783507417 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (2858943987473 / 32000000000000 : ℝ) = (2858943987473 / 2000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2858943987473 / 2000000000000 : ℝ) ≤
+            (178652570469216492583 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2893389009596379 / 10000000000000000 : ℝ) ≤ Real.sqrt (2678943987473 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (2678943987473 / 32000000000000 : ℝ) ≤ (1446694504798191 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (2678943987473 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau201 :
+    Real.arctan (1446694504798191 / 5000000000000000 : ℝ) ≤
+      (2816475004873911548537 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (2816475004873911548537 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (5000000000000000 / 1446694504798191 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 1446694504798191 : ℝ) =
+              (1446694504798191 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau201)
+  have hz1 :
+    (5000000000000000 / 1446694504798191 : ℝ) ≤
+      1 / Real.sqrt (2678943987473 / 32000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 1446694504798191 : ℝ) = 1 / (1446694504798191 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 964463003198793 : ℝ) ≤
+      (1268147198934660719937 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (2678943987473 / 32000000000000 : ℝ) ≤
+      (250000000000000 / 964463003198793 : ℝ) := by
+    rw [show
+        (250000000000000 / 964463003198793 : ℝ) =
+          (3 / 40) / (2893389009596379 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2816475004873911548537 / 10000000000000000000000 : ℝ)) -
+        6 * (1268147198934660719937 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (2678943987473 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (2678943987473 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2893389009596379 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2816475004873911548537 / 10000000000000000000000 : ℝ)) -
+          6 * (1268147198934660719937 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (2678943987473 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (2678943987473 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (2678943987473 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_140 :
+    ((406830398890876967997 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(24035891607929812287671 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2911493353823127 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1416586352121228549643 / 5000000000000000000000 : ℝ)) -
+              6 * (1260592465377202839901 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (1356286967897 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1356286967897 / 16000000000000 : ℝ)) =
+      Real.log (17356286967897 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1356286967897 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1446286967897 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (406830398890876967997 / 5000000000000000000000 : ℝ) ≤
+      Real.log (17356286967897 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1446286967897 / 16000000000000 : ℝ) ≤
+      (-(24035891607929812287671 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (1446286967897 / 16000000000000 : ℝ) = (1446286967897 / 1000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1446286967897 / 1000000000000 : ℝ) ≤
+            (3689995604070187712329 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2911493353823127 / 10000000000000000 : ℝ) ≤ Real.sqrt (1356286967897 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1356286967897 / 16000000000000 : ℝ) ≤ (291149335382313 / 1000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1356286967897 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau202 :
+    Real.arctan (291149335382313 / 1000000000000000 : ℝ) ≤
+      (1416586352121228549643 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1416586352121228549643 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (1000000000000000 / 291149335382313 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1000000000000000 / 291149335382313 : ℝ) =
+              (291149335382313 / 1000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau202)
+  have hz1 :
+    (1000000000000000 / 291149335382313 : ℝ) ≤
+      1 / Real.sqrt (1356286967897 / 16000000000000 : ℝ) := by
+    rw [show
+        (1000000000000000 / 291149335382313 : ℝ) = 1 / (291149335382313 / 1000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 970497784607709 : ℝ) ≤
+      (1260592465377202839901 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1356286967897 / 16000000000000 : ℝ) ≤
+      (250000000000000 / 970497784607709 : ℝ) := by
+    rw [show
+        (250000000000000 / 970497784607709 : ℝ) =
+          (3 / 40) / (2911493353823127 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1416586352121228549643 / 5000000000000000000000 : ℝ)) -
+        6 * (1260592465377202839901 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1356286967897 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1356286967897 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2911493353823127 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1416586352121228549643 / 5000000000000000000000 : ℝ)) -
+          6 * (1260592465377202839901 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (1356286967897 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1356286967897 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1356286967897 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_141 :
+    ((52063642774504058401 / 625000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2380602772690295968647 / 1000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2947368440891381 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1433106220253817106089 / 5000000000000000000000 : ℝ)) -
+              6 * (1245879543219645552791 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (694958458109 / 8000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (694958458109 / 8000000000000 : ℝ)) =
+      Real.log (8694958458109 / 8000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((694958458109 / 8000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (739958458109 / 8000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (52063642774504058401 / 625000000000000000000 : ℝ) ≤
+      Real.log (8694958458109 / 8000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (739958458109 / 8000000000000 : ℝ) ≤
+      (-(2380602772690295968647 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (739958458109 / 8000000000000 : ℝ) = (739958458109 / 500000000000 : ℝ) / 2 ^ 4 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (739958458109 / 500000000000 : ℝ) ≤
+            (391985948509704031353 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2947368440891381 / 10000000000000000 : ℝ) ≤ Real.sqrt (694958458109 / 8000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (694958458109 / 8000000000000 : ℝ) ≤ (368421055111423 / 1250000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (694958458109 / 8000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau203 :
+    Real.arctan (368421055111423 / 1250000000000000 : ℝ) ≤
+      (1433106220253817106089 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1433106220253817106089 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (1250000000000000 / 368421055111423 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1250000000000000 / 368421055111423 : ℝ) =
+              (368421055111423 / 1250000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau203)
+  have hz1 :
+    (1250000000000000 / 368421055111423 : ℝ) ≤ 1 / Real.sqrt (694958458109 / 8000000000000 : ℝ) :=
+  by
+    rw [show
+        (1250000000000000 / 368421055111423 : ℝ) = 1 / (368421055111423 / 1250000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 2947368440891381 : ℝ) ≤
+      (1245879543219645552791 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (694958458109 / 8000000000000 : ℝ) ≤
+      (750000000000000 / 2947368440891381 : ℝ) := by
+    rw [show
+        (750000000000000 / 2947368440891381 : ℝ) =
+          (3 / 40) / (2947368440891381 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1433106220253817106089 / 5000000000000000000000 : ℝ)) -
+        6 * (1245879543219645552791 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (694958458109 / 8000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (694958458109 / 8000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2947368440891381 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1433106220253817106089 / 5000000000000000000000 : ℝ)) -
+          6 * (1245879543219645552791 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (694958458109 / 8000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (694958458109 / 8000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (694958458109 / 8000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_142 :
+    ((852338372156698860517 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(23581329078528711081391 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1491406039897313 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2898791760613854838001 / 10000000000000000000000 : ℝ)) -
+              6 * (2463340430743041968497 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (1423546864539 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1423546864539 / 16000000000000 : ℝ)) =
+      Real.log (17423546864539 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1423546864539 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1513546864539 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (852338372156698860517 / 10000000000000000000000 : ℝ) ≤
+      Real.log (17423546864539 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1513546864539 / 16000000000000 : ℝ) ≤
+      (-(23581329078528711081391 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (1513546864539 / 16000000000000 : ℝ) = (1513546864539 / 1000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1513546864539 / 1000000000000 : ℝ) ≤
+            (4144558133471288918609 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1491406039897313 / 5000000000000000 : ℝ) ≤ Real.sqrt (1423546864539 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1423546864539 / 16000000000000 : ℝ) ≤ (2982812079794629 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1423546864539 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau204 :
+    Real.arctan (2982812079794629 / 10000000000000000 : ℝ) ≤
+      (2898791760613854838001 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (2898791760613854838001 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 2982812079794629 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 2982812079794629 : ℝ) =
+              (2982812079794629 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau204)
+  have hz1 :
+    (10000000000000000 / 2982812079794629 : ℝ) ≤
+      1 / Real.sqrt (1423546864539 / 16000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 2982812079794629 : ℝ) =
+          1 / (2982812079794629 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 1491406039897313 : ℝ) ≤
+      (2463340430743041968497 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1423546864539 / 16000000000000 : ℝ) ≤
+      (375000000000000 / 1491406039897313 : ℝ) := by
+    rw [show
+        (375000000000000 / 1491406039897313 : ℝ) =
+          (3 / 40) / (1491406039897313 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2898791760613854838001 / 10000000000000000000000 : ℝ)) -
+        6 * (2463340430743041968497 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1423546864539 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1423546864539 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1491406039897313 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2898791760613854838001 / 10000000000000000000000 : ℝ)) -
+          6 * (2463340430743041968497 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (1423546864539 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1423546864539 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1423546864539 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_143 :
+    ((43581060265344278991 / 500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(23361568621129862900791 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3017839472267369 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1465463193295560484297 / 5000000000000000000000 : ℝ)) -
+              6 * (487174553803326026091 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (72858840643 / 800000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (72858840643 / 800000000000 : ℝ)) =
+      Real.log (872858840643 / 800000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((72858840643 / 800000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (77358840643 / 800000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (43581060265344278991 / 500000000000000000000 : ℝ) ≤
+      Real.log (872858840643 / 800000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (77358840643 / 800000000000 : ℝ) ≤
+      (-(23361568621129862900791 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (77358840643 / 800000000000 : ℝ) = (77358840643 / 50000000000 : ℝ) / 2 ^ 4 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (77358840643 / 50000000000 : ℝ) ≤
+            (4364318590870137099209 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3017839472267369 / 10000000000000000 : ℝ) ≤ Real.sqrt (72858840643 / 800000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (72858840643 / 800000000000 : ℝ) ≤ (754459868066843 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (72858840643 / 800000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau205 :
+    Real.arctan (754459868066843 / 2500000000000000 : ℝ) ≤
+      (1465463193295560484297 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1465463193295560484297 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (2500000000000000 / 754459868066843 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 754459868066843 : ℝ) =
+              (754459868066843 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau205)
+  have hz1 :
+    (2500000000000000 / 754459868066843 : ℝ) ≤ 1 / Real.sqrt (72858840643 / 800000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 754459868066843 : ℝ) = 1 / (754459868066843 / 2500000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 3017839472267369 : ℝ) ≤
+      (487174553803326026091 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (72858840643 / 800000000000 : ℝ) ≤
+      (750000000000000 / 3017839472267369 : ℝ) := by
+    rw [show
+        (750000000000000 / 3017839472267369 : ℝ) =
+          (3 / 40) / (3017839472267369 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1465463193295560484297 / 5000000000000000000000 : ℝ)) -
+        6 * (487174553803326026091 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (72858840643 / 800000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (72858840643 / 800000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3017839472267369 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1465463193295560484297 / 5000000000000000000000 : ℝ)) -
+          6 * (487174553803326026091 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (72858840643 / 800000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (72858840643 / 800000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (72858840643 / 800000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_144 :
+    ((910075680531480208983 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(22936026120516716875781 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1543351016002799 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (18712000414536656493 / 62500000000000000000 : ℝ)) -
+              6 * (595896677737549471971 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (762218354751 / 8000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (762218354751 / 8000000000000 : ℝ)) =
+      Real.log (8762218354751 / 8000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((762218354751 / 8000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (807218354751 / 8000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (910075680531480208983 / 10000000000000000000000 : ℝ) ≤
+      Real.log (8762218354751 / 8000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (807218354751 / 8000000000000 : ℝ) ≤
+      (-(22936026120516716875781 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (807218354751 / 8000000000000 : ℝ) = (807218354751 / 500000000000 : ℝ) / 2 ^ 4 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (807218354751 / 500000000000 : ℝ) ≤
+            (4789861091483283124219 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1543351016002799 / 5000000000000000 : ℝ) ≤ Real.sqrt (762218354751 / 8000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (762218354751 / 8000000000000 : ℝ) ≤ (3086702032005601 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (762218354751 / 8000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau206 :
+    Real.arctan (3086702032005601 / 10000000000000000 : ℝ) ≤
+      (18712000414536656493 / 62500000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (18712000414536656493 / 62500000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 3086702032005601 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 3086702032005601 : ℝ) =
+              (3086702032005601 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau206)
+  have hz1 :
+    (10000000000000000 / 3086702032005601 : ℝ) ≤
+      1 / Real.sqrt (762218354751 / 8000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 3086702032005601 : ℝ) =
+          1 / (3086702032005601 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 1543351016002799 : ℝ) ≤
+      (595896677737549471971 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (762218354751 / 8000000000000 : ℝ) ≤
+      (375000000000000 / 1543351016002799 : ℝ) := by
+    rw [show
+        (375000000000000 / 1543351016002799 : ℝ) =
+          (3 / 40) / (1543351016002799 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (18712000414536656493 / 62500000000000000000 : ℝ)) -
+        6 * (595896677737549471971 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (762218354751 / 8000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (762218354751 / 8000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1543351016002799 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (18712000414536656493 / 62500000000000000000 : ℝ)) -
+          6 * (595896677737549471971 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (762218354751 / 8000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (762218354751 / 8000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (762218354751 / 8000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/V12.lean b/LeanPool/Zeta5Irrational/Table/V12.lean
new file mode 100644
index 0000000000..27af5c5c10
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/V12.lean
@@ -0,0 +1,1228 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Energy
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # V12: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_145 :
+    ((474191423687705405539 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(11263927763828573808227 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3154061475050857 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (95478208413835360219 / 312500000000000000000 : ℝ)) -
+              6 * (1167265072719666656229 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (24870259471 / 250000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (24870259471 / 250000000000 : ℝ)) =
+      Real.log (274870259471 / 250000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((24870259471 / 250000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (26276509471 / 250000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (474191423687705405539 / 5000000000000000000000 : ℝ) ≤
+      Real.log (274870259471 / 250000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (26276509471 / 250000000000 : ℝ) ≤
+      (-(11263927763828573808227 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (26276509471 / 250000000000 : ℝ) = (26276509471 / 15625000000 : ℝ) / 2 ^ 4 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (26276509471 / 15625000000 : ℝ) ≤
+            (2599015842171426191773 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3154061475050857 / 10000000000000000 : ℝ) ≤ Real.sqrt (24870259471 / 250000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (24870259471 / 250000000000 : ℝ) ≤ (157703073752543 / 500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (24870259471 / 250000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau207 :
+    Real.arctan (157703073752543 / 500000000000000 : ℝ) ≤
+      (95478208413835360219 / 312500000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (95478208413835360219 / 312500000000000000000 : ℝ)) ≤
+      Real.arctan (500000000000000 / 157703073752543 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (500000000000000 / 157703073752543 : ℝ) =
+              (157703073752543 / 500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau207)
+  have hz1 :
+    (500000000000000 / 157703073752543 : ℝ) ≤ 1 / Real.sqrt (24870259471 / 250000000000 : ℝ) := by
+    rw [show (500000000000000 / 157703073752543 : ℝ) = 1 / (157703073752543 / 500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 3154061475050857 : ℝ) ≤
+      (1167265072719666656229 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (24870259471 / 250000000000 : ℝ) ≤
+      (750000000000000 / 3154061475050857 : ℝ) := by
+    rw [show
+        (750000000000000 / 3154061475050857 : ℝ) =
+          (3 / 40) / (3154061475050857 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (95478208413835360219 / 312500000000000000000 : ℝ)) -
+        6 * (1167265072719666656229 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (24870259471 / 250000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (24870259471 / 250000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3154061475050857 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (95478208413835360219 / 312500000000000000000 : ℝ)) -
+          6 * (1167265072719666656229 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (24870259471 / 250000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (24870259471 / 250000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (24870259471 / 250000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_146 :
+    ((120140089155551917327 / 1250000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(22395404883448735089487 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((198512470516247 / 625000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (3075424865340954955163 / 10000000000000000000000 : ℝ)) -
+              6 * (2318837467048294312633 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (1614118950931 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1614118950931 / 16000000000000 : ℝ)) =
+      Real.log (17614118950931 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1614118950931 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1704118950931 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (120140089155551917327 / 1250000000000000000000 : ℝ) ≤
+      Real.log (17614118950931 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1704118950931 / 16000000000000 : ℝ) ≤
+      (-(22395404883448735089487 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (1704118950931 / 16000000000000 : ℝ) = (1704118950931 / 1000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1704118950931 / 1000000000000 : ℝ) ≤
+            (5330482328551264910513 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (198512470516247 / 625000000000000 : ℝ) ≤ Real.sqrt (1614118950931 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1614118950931 / 16000000000000 : ℝ) ≤ (635239905651991 / 2000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1614118950931 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau208 :
+    Real.arctan (635239905651991 / 2000000000000000 : ℝ) ≤
+      (3075424865340954955163 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (3075424865340954955163 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (2000000000000000 / 635239905651991 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2000000000000000 / 635239905651991 : ℝ) =
+              (635239905651991 / 2000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau208)
+  have hz1 :
+    (2000000000000000 / 635239905651991 : ℝ) ≤
+      1 / Real.sqrt (1614118950931 / 16000000000000 : ℝ) := by
+    rw [show
+        (2000000000000000 / 635239905651991 : ℝ) = 1 / (635239905651991 / 2000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (46875000000000 / 198512470516247 : ℝ) ≤
+      (2318837467048294312633 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1614118950931 / 16000000000000 : ℝ) ≤
+      (46875000000000 / 198512470516247 : ℝ) := by
+    rw [show
+        (46875000000000 / 198512470516247 : ℝ) =
+          (3 / 40) / (198512470516247 / 625000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (3075424865340954955163 / 10000000000000000000000 : ℝ)) -
+        6 * (2318837467048294312633 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1614118950931 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1614118950931 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((198512470516247 / 625000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (3075424865340954955163 / 10000000000000000000000 : ℝ)) -
+          6 * (2318837467048294312633 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (1614118950931 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1614118950931 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1614118950931 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_147 :
+    ((194768474885970078339 / 2000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(22264685648903831350143 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((799546086001203 / 2500000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1547691177866349120623 / 5000000000000000000000 : ℝ)) -
+              6 * (575864282886168339093 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (818270647859 / 8000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (818270647859 / 8000000000000 : ℝ)) =
+      Real.log (8818270647859 / 8000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((818270647859 / 8000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (863270647859 / 8000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (194768474885970078339 / 2000000000000000000000 : ℝ) ≤
+      Real.log (8818270647859 / 8000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (863270647859 / 8000000000000 : ℝ) ≤
+      (-(22264685648903831350143 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (863270647859 / 8000000000000 : ℝ) = (863270647859 / 500000000000 : ℝ) / 2 ^ 4 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (863270647859 / 500000000000 : ℝ) ≤
+            (5461201563096168649857 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (799546086001203 / 2500000000000000 : ℝ) ≤ Real.sqrt (818270647859 / 8000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (818270647859 / 8000000000000 : ℝ) ≤ (639636868800963 / 2000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (818270647859 / 8000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau209 :
+    Real.arctan (639636868800963 / 2000000000000000 : ℝ) ≤
+      (1547691177866349120623 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1547691177866349120623 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (2000000000000000 / 639636868800963 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2000000000000000 / 639636868800963 : ℝ) =
+              (639636868800963 / 2000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau209)
+  have hz1 :
+    (2000000000000000 / 639636868800963 : ℝ) ≤ 1 / Real.sqrt (818270647859 / 8000000000000 : ℝ) :=
+  by
+    rw [show
+        (2000000000000000 / 639636868800963 : ℝ) = 1 / (639636868800963 / 2000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (62500000000000 / 266515362000401 : ℝ) ≤
+      (575864282886168339093 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (818270647859 / 8000000000000 : ℝ) ≤
+      (62500000000000 / 266515362000401 : ℝ) := by
+    rw [show
+        (62500000000000 / 266515362000401 : ℝ) =
+          (3 / 40) / (799546086001203 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1547691177866349120623 / 5000000000000000000000 : ℝ)) -
+        6 * (575864282886168339093 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (818270647859 / 8000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (818270647859 / 8000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((799546086001203 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1547691177866349120623 / 5000000000000000000000 : ℝ)) -
+          6 * (575864282886168339093 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (818270647859 / 8000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (818270647859 / 8000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (818270647859 / 8000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_148 :
+    ((98654787210944095483 / 1000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(22135653141131857303987 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3220019060992689 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (124607136184504540319 / 400000000000000000000 : ℝ)) -
+              6 * (457675782124995330019 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (331792728101 / 3200000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (331792728101 / 3200000000000 : ℝ)) =
+      Real.log (3531792728101 / 3200000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((331792728101 / 3200000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (349792728101 / 3200000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (98654787210944095483 / 1000000000000000000000 : ℝ) ≤
+      Real.log (3531792728101 / 3200000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (349792728101 / 3200000000000 : ℝ) ≤
+      (-(22135653141131857303987 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (349792728101 / 3200000000000 : ℝ) = (349792728101 / 200000000000 : ℝ) / 2 ^ 4 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (349792728101 / 200000000000 : ℝ) ≤
+            (5590234070868142696013 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3220019060992689 / 10000000000000000 : ℝ) ≤ Real.sqrt (331792728101 / 3200000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (331792728101 / 3200000000000 : ℝ) ≤ (805004765248173 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (331792728101 / 3200000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau210 :
+    Real.arctan (805004765248173 / 2500000000000000 : ℝ) ≤
+      (124607136184504540319 / 400000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (124607136184504540319 / 400000000000000000000 : ℝ)) ≤
+      Real.arctan (2500000000000000 / 805004765248173 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 805004765248173 : ℝ) =
+              (805004765248173 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau210)
+  have hz1 :
+    (2500000000000000 / 805004765248173 : ℝ) ≤ 1 / Real.sqrt (331792728101 / 3200000000000 : ℝ) :=
+  by
+    rw [show
+        (2500000000000000 / 805004765248173 : ℝ) = 1 / (805004765248173 / 2500000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 1073339686997563 : ℝ) ≤
+      (457675782124995330019 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (331792728101 / 3200000000000 : ℝ) ≤
+      (250000000000000 / 1073339686997563 : ℝ) := by
+    rw [show
+        (250000000000000 / 1073339686997563 : ℝ) =
+          (3 / 40) / (3220019060992689 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (124607136184504540319 / 400000000000000000000 : ℝ)) -
+        6 * (457675782124995330019 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (331792728101 / 3200000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (331792728101 / 3200000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3220019060992689 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (124607136184504540319 / 400000000000000000000 : ℝ)) -
+          6 * (457675782124995330019 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (331792728101 / 3200000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (331792728101 / 3200000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (331792728101 / 3200000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_149 :
+    ((496447286229849446731 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(4414351183143760254287 / 2000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3230881084257917 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (781254219709386649659 / 2500000000000000000000 : ℝ)) -
+              6 * (1140475012063116264297 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (3340349625797 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (3340349625797 / 32000000000000 : ℝ)) =
+      Real.log (35340349625797 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((3340349625797 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (3520349625797 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (496447286229849446731 / 5000000000000000000000 : ℝ) ≤
+      Real.log (35340349625797 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (3520349625797 / 32000000000000 : ℝ) ≤
+      (-(4414351183143760254287 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (3520349625797 / 32000000000000 : ℝ) = (3520349625797 / 2000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3520349625797 / 2000000000000 : ℝ) ≤
+            (1130826259256239745713 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3230881084257917 / 10000000000000000 : ℝ) ≤ Real.sqrt (3340349625797 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (3340349625797 / 32000000000000 : ℝ) ≤ (5048251694153 / 15625000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (3340349625797 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau211 :
+    Real.arctan (5048251694153 / 15625000000000 : ℝ) ≤
+      (781254219709386649659 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (781254219709386649659 / 2500000000000000000000 : ℝ)) ≤
+      Real.arctan (15625000000000 / 5048251694153 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (15625000000000 / 5048251694153 : ℝ) = (5048251694153 / 15625000000000 : ℝ)
+            by
+        norm_num];
+        exact hau211)
+  have hz1 :
+    (15625000000000 / 5048251694153 : ℝ) ≤ 1 / Real.sqrt (3340349625797 / 32000000000000 : ℝ) := by
+    rw [show (15625000000000 / 5048251694153 : ℝ) = 1 / (5048251694153 / 15625000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 3230881084257917 : ℝ) ≤
+      (1140475012063116264297 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (3340349625797 / 32000000000000 : ℝ) ≤
+      (750000000000000 / 3230881084257917 : ℝ) := by
+    rw [show
+        (750000000000000 / 3230881084257917 : ℝ) =
+          (3 / 40) / (3230881084257917 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (781254219709386649659 / 2500000000000000000000 : ℝ)) -
+        6 * (1140475012063116264297 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (3340349625797 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (3340349625797 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3230881084257917 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (781254219709386649659 / 2500000000000000000000 : ℝ)) -
+          6 * (1140475012063116264297 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (3340349625797 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (3340349625797 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (3340349625797 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_150 :
+    ((499618623652078033613 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(22008264384942687831393 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((648341342444703 / 2000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (313481616803270820263 / 1000000000000000000000 : ℝ)) -
+              6 * (2273593038966975577993 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (420346496323 / 4000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (420346496323 / 4000000000000 : ℝ)) =
+      Real.log (4420346496323 / 4000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((420346496323 / 4000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (442846496323 / 4000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (499618623652078033613 / 5000000000000000000000 : ℝ) ≤
+      Real.log (4420346496323 / 4000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (442846496323 / 4000000000000 : ℝ) ≤
+      (-(22008264384942687831393 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (442846496323 / 4000000000000 : ℝ) = (442846496323 / 250000000000 : ℝ) / 2 ^ 4 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (442846496323 / 250000000000 : ℝ) ≤
+            (5717622827057312168607 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (648341342444703 / 2000000000000000 : ℝ) ≤ Real.sqrt (420346496323 / 4000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (420346496323 / 4000000000000 : ℝ) ≤ (1620853356111759 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (420346496323 / 4000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau212 :
+    Real.arctan (1620853356111759 / 5000000000000000 : ℝ) ≤
+      (313481616803270820263 / 1000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (313481616803270820263 / 1000000000000000000000 : ℝ)) ≤
+      Real.arctan (5000000000000000 / 1620853356111759 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 1620853356111759 : ℝ) =
+              (1620853356111759 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau212)
+  have hz1 :
+    (5000000000000000 / 1620853356111759 : ℝ) ≤
+      1 / Real.sqrt (420346496323 / 4000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 1620853356111759 : ℝ) = 1 / (1620853356111759 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (50000000000000 / 216113780814901 : ℝ) ≤
+      (2273593038966975577993 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (420346496323 / 4000000000000 : ℝ) ≤
+      (50000000000000 / 216113780814901 : ℝ) := by
+    rw [show
+        (50000000000000 / 216113780814901 : ℝ) =
+          (3 / 40) / (648341342444703 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (313481616803270820263 / 1000000000000000000000 : ℝ)) -
+        6 * (2273593038966975577993 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (420346496323 / 4000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (420346496323 / 4000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((648341342444703 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (313481616803270820263 / 1000000000000000000000 : ℝ)) -
+          6 * (2273593038966975577993 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (420346496323 / 4000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (420346496323 / 4000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (420346496323 / 4000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_151 :
+    ((201115180349214172211 / 2000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2194517342963678964519 / 1000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1626248154152247 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (3144576650626855419069 / 10000000000000000000000 : ℝ)) -
+              6 * (18130454419663968323 / 80000000000000000000 : ℝ)))) ≤
+      Vfield (3385194315371 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (3385194315371 / 32000000000000 : ℝ)) =
+      Real.log (35385194315371 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((3385194315371 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (3565194315371 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (201115180349214172211 / 2000000000000000000000 : ℝ) ≤
+      Real.log (35385194315371 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (3565194315371 / 32000000000000 : ℝ) ≤
+      (-(2194517342963678964519 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (3565194315371 / 32000000000000 : ℝ) = (3565194315371 / 2000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3565194315371 / 2000000000000 : ℝ) ≤
+            (578071378236321035481 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1626248154152247 / 5000000000000000 : ℝ) ≤ Real.sqrt (3385194315371 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (3385194315371 / 32000000000000 : ℝ) ≤ (3252496308304497 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (3385194315371 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau213 :
+    Real.arctan (3252496308304497 / 10000000000000000 : ℝ) ≤
+      (3144576650626855419069 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (3144576650626855419069 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 3252496308304497 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 3252496308304497 : ℝ) =
+              (3252496308304497 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau213)
+  have hz1 :
+    (10000000000000000 / 3252496308304497 : ℝ) ≤
+      1 / Real.sqrt (3385194315371 / 32000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 3252496308304497 : ℝ) =
+          1 / (3252496308304497 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (125000000000000 / 542082718050749 : ℝ) ≤
+      (18130454419663968323 / 80000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (3385194315371 / 32000000000000 : ℝ) ≤
+      (125000000000000 / 542082718050749 : ℝ) := by
+    rw [show
+        (125000000000000 / 542082718050749 : ℝ) =
+          (3 / 40) / (1626248154152247 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (3144576650626855419069 / 10000000000000000000000 : ℝ)) -
+        6 * (18130454419663968323 / 80000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (3385194315371 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (3385194315371 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1626248154152247 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (3144576650626855419069 / 10000000000000000000000 : ℝ)) -
+          6 * (18130454419663968323 / 80000000000000000000 : ℝ))) ≤
+      Real.sqrt (3385194315371 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (3385194315371 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (3385194315371 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_152 :
+    ((1011910540879013487081 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(21882478026919803602261 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1631625114953933 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (630859739779612780437 / 2000000000000000000000 : ℝ)) -
+              6 * (2259090187622654074271 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (1703808330079 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1703808330079 / 16000000000000 : ℝ)) =
+      Real.log (17703808330079 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1703808330079 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1793808330079 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1011910540879013487081 / 10000000000000000000000 : ℝ) ≤
+      Real.log (17703808330079 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1793808330079 / 16000000000000 : ℝ) ≤
+      (-(21882478026919803602261 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (1793808330079 / 16000000000000 : ℝ) = (1793808330079 / 1000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1793808330079 / 1000000000000 : ℝ) ≤
+            (5843409185080196397739 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1631625114953933 / 5000000000000000 : ℝ) ≤ Real.sqrt (1703808330079 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1703808330079 / 16000000000000 : ℝ) ≤ (3263250229907869 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1703808330079 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau214 :
+    Real.arctan (3263250229907869 / 10000000000000000 : ℝ) ≤
+      (630859739779612780437 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (630859739779612780437 / 2000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 3263250229907869 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 3263250229907869 : ℝ) =
+              (3263250229907869 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau214)
+  have hz1 :
+    (10000000000000000 / 3263250229907869 : ℝ) ≤
+      1 / Real.sqrt (1703808330079 / 16000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 3263250229907869 : ℝ) =
+          1 / (3263250229907869 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 1631625114953933 : ℝ) ≤
+      (2259090187622654074271 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1703808330079 / 16000000000000 : ℝ) ≤
+      (375000000000000 / 1631625114953933 : ℝ) := by
+    rw [show
+        (375000000000000 / 1631625114953933 : ℝ) =
+          (3 / 40) / (1631625114953933 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (630859739779612780437 / 2000000000000000000000 : ℝ)) -
+        6 * (2259090187622654074271 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1703808330079 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1703808330079 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1631625114953933 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (630859739779612780437 / 2000000000000000000000 : ℝ)) -
+          6 * (2259090187622654074271 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (1703808330079 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1703808330079 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1703808330079 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_153 :
+    ((2545602924467127949 / 25000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(21820173247796202238733 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1636984414285389 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (3163982679114103670613 / 10000000000000000000000 : ℝ)) -
+              6 * (225194209246394264897 / 1000000000000000000000 : ℝ)))) ≤
+      Vfield (686007800989 / 6400000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (686007800989 / 6400000000000 : ℝ)) =
+      Real.log (7086007800989 / 6400000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((686007800989 / 6400000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (722007800989 / 6400000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2545602924467127949 / 25000000000000000000 : ℝ) ≤
+      Real.log (7086007800989 / 6400000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (722007800989 / 6400000000000 : ℝ) ≤
+      (-(21820173247796202238733 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (722007800989 / 6400000000000 : ℝ) = (722007800989 / 400000000000 : ℝ) / 2 ^ 4 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (722007800989 / 400000000000 : ℝ) ≤
+            (5905713964203797761267 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1636984414285389 / 5000000000000000 : ℝ) ≤ Real.sqrt (686007800989 / 6400000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (686007800989 / 6400000000000 : ℝ) ≤ (3273968828570781 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (686007800989 / 6400000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau215 :
+    Real.arctan (3273968828570781 / 10000000000000000 : ℝ) ≤
+      (3163982679114103670613 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (3163982679114103670613 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 3273968828570781 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 3273968828570781 : ℝ) =
+              (3273968828570781 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau215)
+  have hz1 :
+    (10000000000000000 / 3273968828570781 : ℝ) ≤
+      1 / Real.sqrt (686007800989 / 6400000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 3273968828570781 : ℝ) =
+          1 / (3273968828570781 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (125000000000000 / 545661471428463 : ℝ) ≤
+      (225194209246394264897 / 1000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (686007800989 / 6400000000000 : ℝ) ≤
+      (125000000000000 / 545661471428463 : ℝ) := by
+    rw [show
+        (125000000000000 / 545661471428463 : ℝ) =
+          (3 / 40) / (1636984414285389 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (3163982679114103670613 / 10000000000000000000000 : ℝ)) -
+        6 * (225194209246394264897 / 1000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (686007800989 / 6400000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (686007800989 / 6400000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1636984414285389 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (3163982679114103670613 / 10000000000000000000000 : ℝ)) -
+          6 * (225194209246394264897 / 1000000000000000000000 : ℝ))) ≤
+      Real.sqrt (686007800989 / 6400000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (686007800989 / 6400000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (686007800989 / 6400000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_154 :
+    ((1024567793543830700313 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(21758254254830878525977 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((656930490018921 / 2000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (3173628951668722226459 / 10000000000000000000000 : ℝ)) -
+              6 * (89794457570276818707 / 400000000000000000000 : ℝ)))) ≤
+      Vfield (863115337433 / 8000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (863115337433 / 8000000000000 : ℝ)) =
+      Real.log (8863115337433 / 8000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((863115337433 / 8000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (908115337433 / 8000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1024567793543830700313 / 10000000000000000000000 : ℝ) ≤
+      Real.log (8863115337433 / 8000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (908115337433 / 8000000000000 : ℝ) ≤
+      (-(21758254254830878525977 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (908115337433 / 8000000000000 : ℝ) = (908115337433 / 500000000000 : ℝ) / 2 ^ 4 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (908115337433 / 500000000000 : ℝ) ≤
+            (5967632957169121474023 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (656930490018921 / 2000000000000000 : ℝ) ≤ Real.sqrt (863115337433 / 8000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (863115337433 / 8000000000000 : ℝ) ≤ (205290778130913 / 625000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (863115337433 / 8000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau216 :
+    Real.arctan (205290778130913 / 625000000000000 : ℝ) ≤
+      (3173628951668722226459 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (3173628951668722226459 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (625000000000000 / 205290778130913 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (625000000000000 / 205290778130913 : ℝ) =
+              (205290778130913 / 625000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau216)
+  have hz1 :
+    (625000000000000 / 205290778130913 : ℝ) ≤ 1 / Real.sqrt (863115337433 / 8000000000000 : ℝ) := by
+    rw [show (625000000000000 / 205290778130913 : ℝ) = 1 / (205290778130913 / 625000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (50000000000000 / 218976830006307 : ℝ) ≤
+      (89794457570276818707 / 400000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (863115337433 / 8000000000000 : ℝ) ≤
+      (50000000000000 / 218976830006307 : ℝ) := by
+    rw [show
+        (50000000000000 / 218976830006307 : ℝ) =
+          (3 / 40) / (656930490018921 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (3173628951668722226459 / 10000000000000000000000 : ℝ)) -
+        6 * (89794457570276818707 / 400000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (863115337433 / 8000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (863115337433 / 8000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((656930490018921 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (3173628951668722226459 / 10000000000000000000000 : ℝ)) -
+          6 * (89794457570276818707 / 400000000000000000000 : ℝ))) ≤
+      Real.sqrt (863115337433 / 8000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (863115337433 / 8000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (863115337433 / 8000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_155 :
+    ((20554592101476308083 / 200000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(21727437940939080223441 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1644990625478013 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1589219029311252440703 / 5000000000000000000000 : ℝ)) -
+              6 * (448269214612905134469 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (6927345044251 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (6927345044251 / 64000000000000 : ℝ)) =
+      Real.log (70927345044251 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((6927345044251 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (7287345044251 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (20554592101476308083 / 200000000000000000000 : ℝ) ≤
+      Real.log (70927345044251 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (7287345044251 / 64000000000000 : ℝ) ≤
+      (-(21727437940939080223441 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (7287345044251 / 64000000000000 : ℝ) = (7287345044251 / 4000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7287345044251 / 4000000000000 : ℝ) ≤
+            (5998449271060919776559 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1644990625478013 / 5000000000000000 : ℝ) ≤ Real.sqrt (6927345044251 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (6927345044251 / 64000000000000 : ℝ) ≤ (3289981250956029 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (6927345044251 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau217 :
+    Real.arctan (3289981250956029 / 10000000000000000 : ℝ) ≤
+      (1589219029311252440703 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1589219029311252440703 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 3289981250956029 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 3289981250956029 : ℝ) =
+              (3289981250956029 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau217)
+  have hz1 :
+    (10000000000000000 / 3289981250956029 : ℝ) ≤
+      1 / Real.sqrt (6927345044251 / 64000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 3289981250956029 : ℝ) =
+          1 / (3289981250956029 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (125000000000000 / 548330208492671 : ℝ) ≤
+      (448269214612905134469 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (6927345044251 / 64000000000000 : ℝ) ≤
+      (125000000000000 / 548330208492671 : ℝ) := by
+    rw [show
+        (125000000000000 / 548330208492671 : ℝ) =
+          (3 / 40) / (1644990625478013 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1589219029311252440703 / 5000000000000000000000 : ℝ)) -
+        6 * (448269214612905134469 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (6927345044251 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (6927345044251 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1644990625478013 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1589219029311252440703 / 5000000000000000000000 : ℝ)) -
+          6 * (448269214612905134469 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (6927345044251 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (6927345044251 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (6927345044251 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_156 :
+    ((1030890417214564711111 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(10848358149947670282237 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1647650717337557 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (795809467803487892921 / 2500000000000000000000 : ℝ)) -
+              6 * (279730896733188049123 / 1250000000000000000000 : ℝ)))) ≤
+      Vfield (3474883694519 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (3474883694519 / 32000000000000 : ℝ)) =
+      Real.log (35474883694519 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((3474883694519 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (3654883694519 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1030890417214564711111 / 10000000000000000000000 : ℝ) ≤
+      Real.log (35474883694519 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (3654883694519 / 32000000000000 : ℝ) ≤
+      (-(10848358149947670282237 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (3654883694519 / 32000000000000 : ℝ) = (3654883694519 / 2000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3654883694519 / 2000000000000 : ℝ) ≤
+            (3014585456052329717763 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1647650717337557 / 5000000000000000 : ℝ) ≤ Real.sqrt (3474883694519 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (3474883694519 / 32000000000000 : ℝ) ≤ (3295301434675117 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (3474883694519 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau218 :
+    Real.arctan (3295301434675117 / 10000000000000000 : ℝ) ≤
+      (795809467803487892921 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (795809467803487892921 / 2500000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 3295301434675117 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 3295301434675117 : ℝ) =
+              (3295301434675117 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau218)
+  have hz1 :
+    (10000000000000000 / 3295301434675117 : ℝ) ≤
+      1 / Real.sqrt (3474883694519 / 32000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 3295301434675117 : ℝ) =
+          1 / (3295301434675117 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 1647650717337557 : ℝ) ≤
+      (279730896733188049123 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (3474883694519 / 32000000000000 : ℝ) ≤
+      (375000000000000 / 1647650717337557 : ℝ) := by
+    rw [show
+        (375000000000000 / 1647650717337557 : ℝ) =
+          (3 / 40) / (1647650717337557 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (795809467803487892921 / 2500000000000000000000 : ℝ)) -
+        6 * (279730896733188049123 / 1250000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (3474883694519 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (3474883694519 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1647650717337557 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (795809467803487892921 / 2500000000000000000000 : ℝ)) -
+          6 * (279730896733188049123 / 1250000000000000000000 : ℝ))) ≤
+      Real.sqrt (3474883694519 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (3474883694519 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (3474883694519 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/V13.lean b/LeanPool/Zeta5Irrational/Table/V13.lean
new file mode 100644
index 0000000000..5f2f61df15
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/V13.lean
@@ -0,0 +1,1240 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Energy
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # V13: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_157 :
+    ((206810046119531065579 / 2000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2166608875177930686979 / 1000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3300613042921201 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (3188028432898504270633 / 10000000000000000000000 : ℝ)) -
+              6 * (2234364613470351213961 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (278887589353 / 2560000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (278887589353 / 2560000000000 : ℝ)) =
+      Real.log (2838887589353 / 2560000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((278887589353 / 2560000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (293287589353 / 2560000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (206810046119531065579 / 2000000000000000000000 : ℝ) ≤
+      Real.log (2838887589353 / 2560000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (293287589353 / 2560000000000 : ℝ) ≤
+      (-(2166608875177930686979 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (293287589353 / 2560000000000 : ℝ) = (293287589353 / 160000000000 : ℝ) / 2 ^ 4 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (293287589353 / 160000000000 : ℝ) ≤
+            (605979846022069313021 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3300613042921201 / 10000000000000000 : ℝ) ≤ Real.sqrt (278887589353 / 2560000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (278887589353 / 2560000000000 : ℝ) ≤ (825153260730301 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (278887589353 / 2560000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau219 :
+    Real.arctan (825153260730301 / 2500000000000000 : ℝ) ≤
+      (3188028432898504270633 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (3188028432898504270633 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (2500000000000000 / 825153260730301 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 825153260730301 : ℝ) =
+              (825153260730301 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau219)
+  have hz1 :
+    (2500000000000000 / 825153260730301 : ℝ) ≤ 1 / Real.sqrt (278887589353 / 2560000000000 : ℝ) :=
+  by
+    rw [show
+        (2500000000000000 / 825153260730301 : ℝ) = 1 / (825153260730301 / 2500000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 3300613042921201 : ℝ) ≤
+      (2234364613470351213961 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (278887589353 / 2560000000000 : ℝ) ≤
+      (750000000000000 / 3300613042921201 : ℝ) := by
+    rw [show
+        (750000000000000 / 3300613042921201 : ℝ) =
+          (3 / 40) / (3300613042921201 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (3188028432898504270633 / 10000000000000000000000 : ℝ)) -
+        6 * (2234364613470351213961 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (278887589353 / 2560000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (278887589353 / 2560000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3300613042921201 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (3188028432898504270633 / 10000000000000000000000 : ℝ)) -
+          6 * (2234364613470351213961 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (278887589353 / 2560000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (278887589353 / 2560000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (278887589353 / 2560000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_158 :
+    ((1037209045854065450943 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(21635554721982830516497 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((330591611702887 / 1000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (798202446697152099411 / 2500000000000000000000 : ℝ)) -
+              6 * (2230898265082615487957 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (1748653019653 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1748653019653 / 16000000000000 : ℝ)) =
+      Real.log (17748653019653 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1748653019653 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1838653019653 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1037209045854065450943 / 10000000000000000000000 : ℝ) ≤
+      Real.log (17748653019653 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1838653019653 / 16000000000000 : ℝ) ≤
+      (-(21635554721982830516497 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (1838653019653 / 16000000000000 : ℝ) = (1838653019653 / 1000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1838653019653 / 1000000000000 : ℝ) ≤
+            (6090332490017169483503 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (330591611702887 / 1000000000000000 : ℝ) ≤ Real.sqrt (1748653019653 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1748653019653 / 16000000000000 : ℝ) ≤ (3305916117028873 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1748653019653 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau220 :
+    Real.arctan (3305916117028873 / 10000000000000000 : ℝ) ≤
+      (798202446697152099411 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (798202446697152099411 / 2500000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 3305916117028873 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 3305916117028873 : ℝ) =
+              (3305916117028873 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau220)
+  have hz1 :
+    (10000000000000000 / 3305916117028873 : ℝ) ≤
+      1 / Real.sqrt (1748653019653 / 16000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 3305916117028873 : ℝ) =
+          1 / (3305916117028873 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (75000000000000 / 330591611702887 : ℝ) ≤
+      (2230898265082615487957 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1748653019653 / 16000000000000 : ℝ) ≤
+      (75000000000000 / 330591611702887 : ℝ) := by
+    rw [show
+        (75000000000000 / 330591611702887 : ℝ) =
+          (3 / 40) / (330591611702887 / 1000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (798202446697152099411 / 2500000000000000000000 : ℝ)) -
+        6 * (2230898265082615487957 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1748653019653 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1748653019653 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((330591611702887 / 1000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (798202446697152099411 / 2500000000000000000000 : ℝ)) -
+          6 * (2230898265082615487957 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (1748653019653 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1748653019653 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1748653019653 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_159 :
+    ((520183431807087761219 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(5401278410286320760879 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3311210698001703 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (3197581975657524269899 / 10000000000000000000000 : ℝ)) -
+              6 * (445489600655899282081 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (7017034423399 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (7017034423399 / 64000000000000 : ℝ)) =
+      Real.log (71017034423399 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((7017034423399 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (7377034423399 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (520183431807087761219 / 5000000000000000000000 : ℝ) ≤
+      Real.log (71017034423399 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (7377034423399 / 64000000000000 : ℝ) ≤
+      (-(5401278410286320760879 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (7377034423399 / 64000000000000 : ℝ) = (7377034423399 / 4000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7377034423399 / 4000000000000 : ℝ) ≤
+            (1530193392713679239121 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3311210698001703 / 10000000000000000 : ℝ) ≤ Real.sqrt (7017034423399 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (7017034423399 / 64000000000000 : ℝ) ≤ (1655605349000853 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (7017034423399 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau221 :
+    Real.arctan (1655605349000853 / 5000000000000000 : ℝ) ≤
+      (3197581975657524269899 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (3197581975657524269899 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (5000000000000000 / 1655605349000853 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 1655605349000853 : ℝ) =
+              (1655605349000853 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau221)
+  have hz1 :
+    (5000000000000000 / 1655605349000853 : ℝ) ≤
+      1 / Real.sqrt (7017034423399 / 64000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 1655605349000853 : ℝ) = 1 / (1655605349000853 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 1103736899333901 : ℝ) ≤
+      (445489600655899282081 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (7017034423399 / 64000000000000 : ℝ) ≤
+      (250000000000000 / 1103736899333901 : ℝ) := by
+    rw [show
+        (250000000000000 / 1103736899333901 : ℝ) =
+          (3 / 40) / (3311210698001703 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (3197581975657524269899 / 10000000000000000000000 : ℝ)) -
+        6 * (445489600655899282081 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (7017034423399 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (7017034423399 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3311210698001703 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (3197581975657524269899 / 10000000000000000000000 : ℝ)) -
+          6 * (445489600655899282081 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (7017034423399 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (7017034423399 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (7017034423399 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_160 :
+    ((1043523684507768985099 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(5393691236272463449183 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3316496826515987 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (3202345041943094693627 / 10000000000000000000000 : ℝ)) -
+              6 * (139000856499547490079 / 625000000000000000000 : ℝ)))) ≤
+      Vfield (3519728384093 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (3519728384093 / 32000000000000 : ℝ)) =
+      Real.log (35519728384093 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((3519728384093 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (3699728384093 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1043523684507768985099 / 10000000000000000000000 : ℝ) ≤
+      Real.log (35519728384093 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (3699728384093 / 32000000000000 : ℝ) ≤
+      (-(5393691236272463449183 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (3699728384093 / 32000000000000 : ℝ) = (3699728384093 / 2000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3699728384093 / 2000000000000 : ℝ) ≤
+            (1537780566727536550817 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3316496826515987 / 10000000000000000 : ℝ) ≤ Real.sqrt (3519728384093 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (3519728384093 / 32000000000000 : ℝ) ≤ (331649682651599 / 1000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (3519728384093 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau222 :
+    Real.arctan (331649682651599 / 1000000000000000 : ℝ) ≤
+      (3202345041943094693627 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (3202345041943094693627 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (1000000000000000 / 331649682651599 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1000000000000000 / 331649682651599 : ℝ) =
+              (331649682651599 / 1000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau222)
+  have hz1 :
+    (1000000000000000 / 331649682651599 : ℝ) ≤
+      1 / Real.sqrt (3519728384093 / 32000000000000 : ℝ) := by
+    rw [show
+        (1000000000000000 / 331649682651599 : ℝ) = 1 / (331649682651599 / 1000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 3316496826515987 : ℝ) ≤
+      (139000856499547490079 / 625000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (3519728384093 / 32000000000000 : ℝ) ≤
+      (750000000000000 / 3316496826515987 : ℝ) := by
+    rw [show
+        (750000000000000 / 3316496826515987 : ℝ) =
+          (3 / 40) / (3316496826515987 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (3202345041943094693627 / 10000000000000000000000 : ℝ)) -
+        6 * (139000856499547490079 / 625000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (3519728384093 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (3519728384093 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3316496826515987 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (3202345041943094693627 / 10000000000000000000000 : ℝ)) -
+          6 * (139000856499547490079 / 625000000000000000000 : ℝ))) ≤
+      Real.sqrt (3519728384093 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (3519728384093 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (3519728384093 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_161 :
+    ((1046679509164033035797 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(21544508074760811294807 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1660887271462177 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (801774756937861789931 / 2500000000000000000000 : ℝ)) -
+              6 * (2220595244489986577137 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (7061879112973 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (7061879112973 / 64000000000000 : ℝ)) =
+      Real.log (71061879112973 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((7061879112973 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (7421879112973 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1046679509164033035797 / 10000000000000000000000 : ℝ) ≤
+      Real.log (71061879112973 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (7421879112973 / 64000000000000 : ℝ) ≤
+      (-(21544508074760811294807 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (7421879112973 / 64000000000000 : ℝ) = (7421879112973 / 4000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7421879112973 / 4000000000000 : ℝ) ≤
+            (6181379137239188705193 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1660887271462177 / 5000000000000000 : ℝ) ≤ Real.sqrt (7061879112973 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (7061879112973 / 64000000000000 : ℝ) ≤ (3321774542924357 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (7061879112973 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau223 :
+    Real.arctan (3321774542924357 / 10000000000000000 : ℝ) ≤
+      (801774756937861789931 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (801774756937861789931 / 2500000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 3321774542924357 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 3321774542924357 : ℝ) =
+              (3321774542924357 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau223)
+  have hz1 :
+    (10000000000000000 / 3321774542924357 : ℝ) ≤
+      1 / Real.sqrt (7061879112973 / 64000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 3321774542924357 : ℝ) =
+          1 / (3321774542924357 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 1660887271462177 : ℝ) ≤
+      (2220595244489986577137 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (7061879112973 / 64000000000000 : ℝ) ≤
+      (375000000000000 / 1660887271462177 : ℝ) := by
+    rw [show
+        (375000000000000 / 1660887271462177 : ℝ) =
+          (3 / 40) / (1660887271462177 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (801774756937861789931 / 2500000000000000000000 : ℝ)) -
+        6 * (2220595244489986577137 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (7061879112973 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (7061879112973 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1660887271462177 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (801774756937861789931 / 2500000000000000000000 : ℝ)) -
+          6 * (2220595244489986577137 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (7061879112973 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (7061879112973 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (7061879112973 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_162 :
+    ((41993373528462375119 / 400000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(21514342476161710872833 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1663521943629689 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1605921987430325739573 / 5000000000000000000000 : ℝ)) -
+              6 * (138574531459758201139 / 625000000000000000000 : ℝ)))) ≤
+      Vfield (44276884111 / 400000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (44276884111 / 400000000000 : ℝ)) =
+      Real.log (444276884111 / 400000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((44276884111 / 400000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (46526884111 / 400000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (41993373528462375119 / 400000000000000000000 : ℝ) ≤
+      Real.log (444276884111 / 400000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (46526884111 / 400000000000 : ℝ) ≤
+      (-(21514342476161710872833 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (46526884111 / 400000000000 : ℝ) = (46526884111 / 25000000000 : ℝ) / 2 ^ 4 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (46526884111 / 25000000000 : ℝ) ≤
+            (6211544735838289127167 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1663521943629689 / 5000000000000000 : ℝ) ≤ Real.sqrt (44276884111 / 400000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (44276884111 / 400000000000 : ℝ) ≤ (3327043887259381 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (44276884111 / 400000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau224 :
+    Real.arctan (3327043887259381 / 10000000000000000 : ℝ) ≤
+      (1605921987430325739573 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1605921987430325739573 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 3327043887259381 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 3327043887259381 : ℝ) =
+              (3327043887259381 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau224)
+  have hz1 :
+    (10000000000000000 / 3327043887259381 : ℝ) ≤ 1 / Real.sqrt (44276884111 / 400000000000 : ℝ) :=
+  by
+    rw [show
+        (10000000000000000 / 3327043887259381 : ℝ) =
+          1 / (3327043887259381 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 1663521943629689 : ℝ) ≤
+      (138574531459758201139 / 625000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (44276884111 / 400000000000 : ℝ) ≤
+      (375000000000000 / 1663521943629689 : ℝ) := by
+    rw [show
+        (375000000000000 / 1663521943629689 : ℝ) =
+          (3 / 40) / (1663521943629689 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1605921987430325739573 / 5000000000000000000000 : ℝ)) -
+        6 * (138574531459758201139 / 625000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (44276884111 / 400000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (44276884111 / 400000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1663521943629689 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1605921987430325739573 / 5000000000000000000000 : ℝ)) -
+          6 * (138574531459758201139 / 625000000000000000000 : ℝ))) ≤
+      Real.sqrt (44276884111 / 400000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (44276884111 / 400000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (44276884111 / 400000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_163 :
+    ((526494086139172486441 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(21484267600294531528243 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((666460979847423 / 2000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (804144981181080084799 / 2500000000000000000000 : ℝ)) -
+              6 * (2213805360475395042061 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (7106723802547 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (7106723802547 / 64000000000000 : ℝ)) =
+      Real.log (71106723802547 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((7106723802547 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (7466723802547 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (526494086139172486441 / 5000000000000000000000 : ℝ) ≤
+      Real.log (71106723802547 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (7466723802547 / 64000000000000 : ℝ) ≤
+      (-(21484267600294531528243 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (7466723802547 / 64000000000000 : ℝ) = (7466723802547 / 4000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7466723802547 / 4000000000000 : ℝ) ≤
+            (6241619611705468471757 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (666460979847423 / 2000000000000000 : ℝ) ≤ Real.sqrt (7106723802547 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (7106723802547 / 64000000000000 : ℝ) ≤ (1666152449618559 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (7106723802547 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau225 :
+    Real.arctan (1666152449618559 / 5000000000000000 : ℝ) ≤
+      (804144981181080084799 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (804144981181080084799 / 2500000000000000000000 : ℝ)) ≤
+      Real.arctan (5000000000000000 / 1666152449618559 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 1666152449618559 : ℝ) =
+              (1666152449618559 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau225)
+  have hz1 :
+    (5000000000000000 / 1666152449618559 : ℝ) ≤
+      1 / Real.sqrt (7106723802547 / 64000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 1666152449618559 : ℝ) = 1 / (1666152449618559 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (50000000000000 / 222153659949141 : ℝ) ≤
+      (2213805360475395042061 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (7106723802547 / 64000000000000 : ℝ) ≤
+      (50000000000000 / 222153659949141 : ℝ) := by
+    rw [show
+        (50000000000000 / 222153659949141 : ℝ) =
+          (3 / 40) / (666460979847423 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (804144981181080084799 / 2500000000000000000000 : ℝ)) -
+        6 * (2213805360475395042061 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (7106723802547 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (7106723802547 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((666460979847423 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (804144981181080084799 / 2500000000000000000000 : ℝ)) -
+          6 * (2213805360475395042061 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (7106723802547 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (7106723802547 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (7106723802547 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_164 :
+    ((132017626498974098703 / 1250000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(21454282903099725251803 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3337557618260599 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1610653459237582293741 / 5000000000000000000000 : ℝ)) -
+              6 * (442086739402679778117 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (3564573073667 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (3564573073667 / 32000000000000 : ℝ)) =
+      Real.log (35564573073667 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((3564573073667 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (3744573073667 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (132017626498974098703 / 1250000000000000000000 : ℝ) ≤
+      Real.log (35564573073667 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (3744573073667 / 32000000000000 : ℝ) ≤
+      (-(21454282903099725251803 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (3744573073667 / 32000000000000 : ℝ) = (3744573073667 / 2000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3744573073667 / 2000000000000 : ℝ) ≤
+            (6271604308900274748197 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3337557618260599 / 10000000000000000 : ℝ) ≤ Real.sqrt (3564573073667 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (3564573073667 / 32000000000000 : ℝ) ≤ (1668778809130301 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (3564573073667 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau226 :
+    Real.arctan (1668778809130301 / 5000000000000000 : ℝ) ≤
+      (1610653459237582293741 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1610653459237582293741 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (5000000000000000 / 1668778809130301 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 1668778809130301 : ℝ) =
+              (1668778809130301 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau226)
+  have hz1 :
+    (5000000000000000 / 1668778809130301 : ℝ) ≤
+      1 / Real.sqrt (3564573073667 / 32000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 1668778809130301 : ℝ) = 1 / (1668778809130301 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 3337557618260599 : ℝ) ≤
+      (442086739402679778117 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (3564573073667 / 32000000000000 : ℝ) ≤
+      (750000000000000 / 3337557618260599 : ℝ) := by
+    rw [show
+        (750000000000000 / 3337557618260599 : ℝ) =
+          (3 / 40) / (3337557618260599 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1610653459237582293741 / 5000000000000000000000 : ℝ)) -
+        6 * (442086739402679778117 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (3564573073667 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (3564573073667 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3337557618260599 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1610653459237582293741 / 5000000000000000000000 : ℝ)) -
+          6 * (442086739402679778117 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (3564573073667 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (3564573073667 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (3564573073667 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_165 :
+    ((1059292857978712554033 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2142438784539716247631 / 1000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((668560416684657 / 2000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (806506249232123808033 / 2500000000000000000000 : ℝ)) -
+              6 * (2207077395399656451427 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (7151568492121 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (7151568492121 / 64000000000000 : ℝ)) =
+      Real.log (71151568492121 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((7151568492121 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (7511568492121 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1059292857978712554033 / 10000000000000000000000 : ℝ) ≤
+      Real.log (71151568492121 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (7511568492121 / 64000000000000 : ℝ) ≤
+      (-(2142438784539716247631 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (7511568492121 / 64000000000000 : ℝ) = (7511568492121 / 4000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7511568492121 / 4000000000000 : ℝ) ≤
+            (630149936660283752369 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (668560416684657 / 2000000000000000 : ℝ) ≤ Real.sqrt (7151568492121 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (7151568492121 / 64000000000000 : ℝ) ≤ (417850260427911 / 1250000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (7151568492121 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau227 :
+    Real.arctan (417850260427911 / 1250000000000000 : ℝ) ≤
+      (806506249232123808033 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (806506249232123808033 / 2500000000000000000000 : ℝ)) ≤
+      Real.arctan (1250000000000000 / 417850260427911 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1250000000000000 / 417850260427911 : ℝ) =
+              (417850260427911 / 1250000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau227)
+  have hz1 :
+    (1250000000000000 / 417850260427911 : ℝ) ≤
+      1 / Real.sqrt (7151568492121 / 64000000000000 : ℝ) := by
+    rw [show
+        (1250000000000000 / 417850260427911 : ℝ) = 1 / (417850260427911 / 1250000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (50000000000000 / 222853472228219 : ℝ) ≤
+      (2207077395399656451427 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (7151568492121 / 64000000000000 : ℝ) ≤
+      (50000000000000 / 222853472228219 : ℝ) := by
+    rw [show
+        (50000000000000 / 222853472228219 : ℝ) =
+          (3 / 40) / (668560416684657 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (806506249232123808033 / 2500000000000000000000 : ℝ)) -
+        6 * (2207077395399656451427 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (7151568492121 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (7151568492121 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((668560416684657 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (806506249232123808033 / 2500000000000000000000 : ℝ)) -
+          6 * (2207077395399656451427 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (7151568492121 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (7151568492121 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (7151568492121 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_166 :
+    ((531221855432660748179 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(10697290946413978844261 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1674019166756219 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (807683550146417840233 / 2500000000000000000000 : ℝ)) -
+              6 * (2203736339310343190637 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (1793497709227 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1793497709227 / 16000000000000 : ℝ)) =
+      Real.log (17793497709227 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1793497709227 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1883497709227 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (531221855432660748179 / 5000000000000000000000 : ℝ) ≤
+      Real.log (17793497709227 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1883497709227 / 16000000000000 : ℝ) ≤
+      (-(10697290946413978844261 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (1883497709227 / 16000000000000 : ℝ) = (1883497709227 / 1000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1883497709227 / 1000000000000 : ℝ) ≤
+            (3165652659586021155739 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1674019166756219 / 5000000000000000 : ℝ) ≤ Real.sqrt (1793497709227 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1793497709227 / 16000000000000 : ℝ) ≤ (3348038333512441 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1793497709227 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau228 :
+    Real.arctan (3348038333512441 / 10000000000000000 : ℝ) ≤
+      (807683550146417840233 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (807683550146417840233 / 2500000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 3348038333512441 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 3348038333512441 : ℝ) =
+              (3348038333512441 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau228)
+  have hz1 :
+    (10000000000000000 / 3348038333512441 : ℝ) ≤
+      1 / Real.sqrt (1793497709227 / 16000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 3348038333512441 : ℝ) =
+          1 / (3348038333512441 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 1674019166756219 : ℝ) ≤
+      (2203736339310343190637 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1793497709227 / 16000000000000 : ℝ) ≤
+      (375000000000000 / 1674019166756219 : ℝ) := by
+    rw [show
+        (375000000000000 / 1674019166756219 : ℝ) =
+          (3 / 40) / (1674019166756219 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (807683550146417840233 / 2500000000000000000000 : ℝ)) -
+        6 * (2203736339310343190637 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1793497709227 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1793497709227 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1674019166756219 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (807683550146417840233 / 2500000000000000000000 : ℝ)) -
+          6 * (2203736339310343190637 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (1793497709227 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1793497709227 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1793497709227 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_167 :
+    ((266398392819311274443 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(21364864515797159093447 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1676633203506243 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (647086913927502052769 / 2000000000000000000000 : ℝ)) -
+              6 * (2200410413651344220241 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (1439282636339 / 12800000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1439282636339 / 12800000000000 : ℝ)) =
+      Real.log (14239282636339 / 12800000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1439282636339 / 12800000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1511282636339 / 12800000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (266398392819311274443 / 2500000000000000000000 : ℝ) ≤
+      Real.log (14239282636339 / 12800000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1511282636339 / 12800000000000 : ℝ) ≤
+      (-(21364864515797159093447 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (1511282636339 / 12800000000000 : ℝ) = (1511282636339 / 800000000000 : ℝ) / 2 ^ 4 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1511282636339 / 800000000000 : ℝ) ≤
+            (6361022696202840906553 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1676633203506243 / 5000000000000000 : ℝ) ≤ Real.sqrt (1439282636339 / 12800000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1439282636339 / 12800000000000 : ℝ) ≤ (3353266407012489 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1439282636339 / 12800000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau229 :
+    Real.arctan (3353266407012489 / 10000000000000000 : ℝ) ≤
+      (647086913927502052769 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (647086913927502052769 / 2000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 3353266407012489 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 3353266407012489 : ℝ) =
+              (3353266407012489 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau229)
+  have hz1 :
+    (10000000000000000 / 3353266407012489 : ℝ) ≤
+      1 / Real.sqrt (1439282636339 / 12800000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 3353266407012489 : ℝ) =
+          1 / (3353266407012489 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (125000000000000 / 558877734502081 : ℝ) ≤
+      (2200410413651344220241 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1439282636339 / 12800000000000 : ℝ) ≤
+      (125000000000000 / 558877734502081 : ℝ) := by
+    rw [show
+        (125000000000000 / 558877734502081 : ℝ) =
+          (3 / 40) / (1676633203506243 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (647086913927502052769 / 2000000000000000000000 : ℝ)) -
+        6 * (2200410413651344220241 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1439282636339 / 12800000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1439282636339 / 12800000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1676633203506243 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (647086913927502052769 / 2000000000000000000000 : ℝ)) -
+          6 * (2200410413651344220241 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (1439282636339 / 12800000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1439282636339 / 12800000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1439282636339 / 12800000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_168 :
+    ((16699100622492466183 / 156250000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(21335235189417287986779 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3358486342108319 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1620063071983821825819 / 5000000000000000000000 : ℝ)) -
+              6 * (137318719033849222699 / 625000000000000000000 : ℝ)))) ≤
+      Vfield (3609417763241 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (3609417763241 / 32000000000000 : ℝ)) =
+      Real.log (35609417763241 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((3609417763241 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (3789417763241 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (16699100622492466183 / 156250000000000000000 : ℝ) ≤
+      Real.log (35609417763241 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (3789417763241 / 32000000000000 : ℝ) ≤
+      (-(21335235189417287986779 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (3789417763241 / 32000000000000 : ℝ) = (3789417763241 / 2000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3789417763241 / 2000000000000 : ℝ) ≤
+            (6390652022582712013221 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3358486342108319 / 10000000000000000 : ℝ) ≤ Real.sqrt (3609417763241 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (3609417763241 / 32000000000000 : ℝ) ≤ (1679243171054161 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (3609417763241 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau230 :
+    Real.arctan (1679243171054161 / 5000000000000000 : ℝ) ≤
+      (1620063071983821825819 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1620063071983821825819 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (5000000000000000 / 1679243171054161 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 1679243171054161 : ℝ) =
+              (1679243171054161 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau230)
+  have hz1 :
+    (5000000000000000 / 1679243171054161 : ℝ) ≤
+      1 / Real.sqrt (3609417763241 / 32000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 1679243171054161 : ℝ) = 1 / (1679243171054161 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 3358486342108319 : ℝ) ≤
+      (137318719033849222699 / 625000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (3609417763241 / 32000000000000 : ℝ) ≤
+      (750000000000000 / 3358486342108319 : ℝ) := by
+    rw [show
+        (750000000000000 / 3358486342108319 : ℝ) =
+          (3 / 40) / (3358486342108319 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1620063071983821825819 / 5000000000000000000000 : ℝ)) -
+        6 * (137318719033849222699 / 625000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (3609417763241 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (3609417763241 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3358486342108319 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1620063071983821825819 / 5000000000000000000000 : ℝ)) -
+          6 * (137318719033849222699 / 625000000000000000000 : ℝ))) ≤
+      Real.sqrt (3609417763241 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (3609417763241 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (3609417763241 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/V14.lean b/LeanPool/Zeta5Irrational/Table/V14.lean
new file mode 100644
index 0000000000..50ea4ee335
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/V14.lean
@@ -0,0 +1,1231 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Energy
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # V14: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_169 :
+    ((1071890317176583928039 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(21305693393452711451859 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3363698176688539 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1622404481577912283161 / 5000000000000000000000 : ℝ)) -
+              6 * (548450874824163752413 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (7241257871269 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (7241257871269 / 64000000000000 : ℝ)) =
+      Real.log (71241257871269 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((7241257871269 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (7601257871269 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1071890317176583928039 / 10000000000000000000000 : ℝ) ≤
+      Real.log (71241257871269 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (7601257871269 / 64000000000000 : ℝ) ≤
+      (-(21305693393452711451859 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (7601257871269 / 64000000000000 : ℝ) = (7601257871269 / 4000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7601257871269 / 4000000000000 : ℝ) ≤
+            (6420193818547288548141 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3363698176688539 / 10000000000000000 : ℝ) ≤ Real.sqrt (7241257871269 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (7241257871269 / 64000000000000 : ℝ) ≤ (1681849088344271 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (7241257871269 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau231 :
+    Real.arctan (1681849088344271 / 5000000000000000 : ℝ) ≤
+      (1622404481577912283161 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1622404481577912283161 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (5000000000000000 / 1681849088344271 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 1681849088344271 : ℝ) =
+              (1681849088344271 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau231)
+  have hz1 :
+    (5000000000000000 / 1681849088344271 : ℝ) ≤
+      1 / Real.sqrt (7241257871269 / 64000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 1681849088344271 : ℝ) = 1 / (1681849088344271 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 3363698176688539 : ℝ) ≤
+      (548450874824163752413 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (7241257871269 / 64000000000000 : ℝ) ≤
+      (750000000000000 / 3363698176688539 : ℝ) := by
+    rw [show
+        (750000000000000 / 3363698176688539 : ℝ) =
+          (3 / 40) / (3363698176688539 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1622404481577912283161 / 5000000000000000000000 : ℝ)) -
+        6 * (548450874824163752413 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (7241257871269 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (7241257871269 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3363698176688539 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1622404481577912283161 / 5000000000000000000000 : ℝ)) -
+          6 * (548450874824163752413 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (7241257871269 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (7241257871269 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (7241257871269 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_170 :
+    ((1075037203912298076033 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(10638119306132417311027 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1684450974174337 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (812370766620299188099 / 2500000000000000000000 : ℝ)) -
+              6 * (273815285801581864431 / 1250000000000000000000 : ℝ)))) ≤
+      Vfield (907960027007 / 8000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (907960027007 / 8000000000000 : ℝ)) =
+      Real.log (8907960027007 / 8000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((907960027007 / 8000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (952960027007 / 8000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1075037203912298076033 / 10000000000000000000000 : ℝ) ≤
+      Real.log (8907960027007 / 8000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (952960027007 / 8000000000000 : ℝ) ≤
+      (-(10638119306132417311027 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (952960027007 / 8000000000000 : ℝ) = (952960027007 / 500000000000 : ℝ) / 2 ^ 4 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (952960027007 / 500000000000 : ℝ) ≤
+            (3224824299867582688973 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1684450974174337 / 5000000000000000 : ℝ) ≤ Real.sqrt (907960027007 / 8000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (907960027007 / 8000000000000 : ℝ) ≤ (3368901948348677 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (907960027007 / 8000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau232 :
+    Real.arctan (3368901948348677 / 10000000000000000 : ℝ) ≤
+      (812370766620299188099 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (812370766620299188099 / 2500000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 3368901948348677 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 3368901948348677 : ℝ) =
+              (3368901948348677 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau232)
+  have hz1 :
+    (10000000000000000 / 3368901948348677 : ℝ) ≤
+      1 / Real.sqrt (907960027007 / 8000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 3368901948348677 : ℝ) =
+          1 / (3368901948348677 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 1684450974174337 : ℝ) ≤
+      (273815285801581864431 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (907960027007 / 8000000000000 : ℝ) ≤
+      (375000000000000 / 1684450974174337 : ℝ) := by
+    rw [show
+        (375000000000000 / 1684450974174337 : ℝ) =
+          (3 / 40) / (1684450974174337 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (812370766620299188099 / 2500000000000000000000 : ℝ)) -
+        6 * (273815285801581864431 / 1250000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (907960027007 / 8000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (907960027007 / 8000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1684450974174337 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (812370766620299188099 / 2500000000000000000000 : ℝ)) -
+          6 * (273815285801581864431 / 1250000000000000000000 : ℝ))) ≤
+      Real.sqrt (907960027007 / 8000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (907960027007 / 8000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (907960027007 / 8000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_171 :
+    ((539091550334963103107 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(5311717583689524422171 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((105440552949823 / 312500000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (406768561615688575177 / 1250000000000000000000 : ℝ)) -
+              6 * (1093627877775185649047 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (7286102560843 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (7286102560843 / 64000000000000 : ℝ)) =
+      Real.log (71286102560843 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((7286102560843 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (7646102560843 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (539091550334963103107 / 5000000000000000000000 : ℝ) ≤
+      Real.log (71286102560843 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (7646102560843 / 64000000000000 : ℝ) ≤
+      (-(5311717583689524422171 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (7646102560843 / 64000000000000 : ℝ) = (7646102560843 / 4000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7646102560843 / 4000000000000 : ℝ) ≤
+            (1619754219310475577829 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (105440552949823 / 312500000000000 : ℝ) ≤ Real.sqrt (7286102560843 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (7286102560843 / 64000000000000 : ℝ) ≤ (3374097694394339 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (7286102560843 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau233 :
+    Real.arctan (3374097694394339 / 10000000000000000 : ℝ) ≤
+      (406768561615688575177 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (406768561615688575177 / 1250000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 3374097694394339 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 3374097694394339 : ℝ) =
+              (3374097694394339 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau233)
+  have hz1 :
+    (10000000000000000 / 3374097694394339 : ℝ) ≤
+      1 / Real.sqrt (7286102560843 / 64000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 3374097694394339 : ℝ) =
+          1 / (3374097694394339 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (23437500000000 / 105440552949823 : ℝ) ≤
+      (1093627877775185649047 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (7286102560843 / 64000000000000 : ℝ) ≤
+      (23437500000000 / 105440552949823 : ℝ) := by
+    rw [show
+        (23437500000000 / 105440552949823 : ℝ) =
+          (3 / 40) / (105440552949823 / 312500000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (406768561615688575177 / 1250000000000000000000 : ℝ)) -
+        6 * (1093627877775185649047 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (7286102560843 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (7286102560843 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((105440552949823 / 312500000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (406768561615688575177 / 1250000000000000000000 : ℝ)) -
+          6 * (1093627877775185649047 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (7286102560843 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (7286102560843 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (7286102560843 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_172 :
+    ((1081328008072146210999 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(21217588054326763577179 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3379285451844349 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1629402640588147556519 / 5000000000000000000000 : ℝ)) -
+              6 * (2184003797519659522247 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (730852490563 / 6400000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (730852490563 / 6400000000000 : ℝ)) =
+      Real.log (7130852490563 / 6400000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((730852490563 / 6400000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (766852490563 / 6400000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1081328008072146210999 / 10000000000000000000000 : ℝ) ≤
+      Real.log (7130852490563 / 6400000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (766852490563 / 6400000000000 : ℝ) ≤
+      (-(21217588054326763577179 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (766852490563 / 6400000000000 : ℝ) = (766852490563 / 400000000000 : ℝ) / 2 ^ 4 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (766852490563 / 400000000000 : ℝ) ≤
+            (6508299157673236422821 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3379285451844349 / 10000000000000000 : ℝ) ≤ Real.sqrt (730852490563 / 6400000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (730852490563 / 6400000000000 : ℝ) ≤ (13200333796267 / 39062500000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (730852490563 / 6400000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau234 :
+    Real.arctan (13200333796267 / 39062500000000 : ℝ) ≤
+      (1629402640588147556519 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1629402640588147556519 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (39062500000000 / 13200333796267 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (39062500000000 / 13200333796267 : ℝ) =
+              (13200333796267 / 39062500000000 : ℝ)
+            by
+        norm_num];
+        exact hau234)
+  have hz1 :
+    (39062500000000 / 13200333796267 : ℝ) ≤ 1 / Real.sqrt (730852490563 / 6400000000000 : ℝ) := by
+    rw [show (39062500000000 / 13200333796267 : ℝ) = 1 / (13200333796267 / 39062500000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 3379285451844349 : ℝ) ≤
+      (2184003797519659522247 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (730852490563 / 6400000000000 : ℝ) ≤
+      (750000000000000 / 3379285451844349 : ℝ) := by
+    rw [show
+        (750000000000000 / 3379285451844349 : ℝ) =
+          (3 / 40) / (3379285451844349 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1629402640588147556519 / 5000000000000000000000 : ℝ)) -
+        6 * (2184003797519659522247 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (730852490563 / 6400000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (730852490563 / 6400000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3379285451844349 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1629402640588147556519 / 5000000000000000000000 : ℝ)) -
+          6 * (2184003797519659522247 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (730852490563 / 6400000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (730852490563 / 6400000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (730852490563 / 6400000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_173 :
+    ((271117981685262172049 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1059419563440124122021 / 500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3384465257433817 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (3263453469630001012291 / 10000000000000000000000 : ℝ)) -
+              6 * (2180766304264109440283 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (7330947250417 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (7330947250417 / 64000000000000 : ℝ)) =
+      Real.log (71330947250417 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((7330947250417 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (7690947250417 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (271117981685262172049 / 2500000000000000000000 : ℝ) ≤
+      Real.log (71330947250417 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (7690947250417 / 64000000000000 : ℝ) ≤
+      (-(1059419563440124122021 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (7690947250417 / 64000000000000 : ℝ) = (7690947250417 / 4000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7690947250417 / 4000000000000 : ℝ) ≤
+            (326874797159875877979 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3384465257433817 / 10000000000000000 : ℝ) ≤ Real.sqrt (7330947250417 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (7330947250417 / 64000000000000 : ℝ) ≤ (169223262871691 / 500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (7330947250417 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau235 :
+    Real.arctan (169223262871691 / 500000000000000 : ℝ) ≤
+      (3263453469630001012291 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (3263453469630001012291 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (500000000000000 / 169223262871691 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (500000000000000 / 169223262871691 : ℝ) =
+              (169223262871691 / 500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau235)
+  have hz1 :
+    (500000000000000 / 169223262871691 : ℝ) ≤
+      1 / Real.sqrt (7330947250417 / 64000000000000 : ℝ) := by
+    rw [show (500000000000000 / 169223262871691 : ℝ) = 1 / (169223262871691 / 500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 3384465257433817 : ℝ) ≤
+      (2180766304264109440283 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (7330947250417 / 64000000000000 : ℝ) ≤
+      (750000000000000 / 3384465257433817 : ℝ) := by
+    rw [show
+        (750000000000000 / 3384465257433817 : ℝ) =
+          (3 / 40) / (3384465257433817 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (3263453469630001012291 / 10000000000000000000000 : ℝ)) -
+        6 * (2180766304264109440283 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (7330947250417 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (7330947250417 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3384465257433817 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (3263453469630001012291 / 10000000000000000000000 : ℝ)) -
+          6 * (2180766304264109440283 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (7330947250417 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (7330947250417 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (7330947250417 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_174 :
+    ((217522971459627535867 / 2000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(5289819870100654850123 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1694818573808583 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (3268093096395075635043 / 10000000000000000000000 : ℝ)) -
+              6 * (2177543168845947543807 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (1838342398801 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1838342398801 / 16000000000000 : ℝ)) =
+      Real.log (17838342398801 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1838342398801 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1928342398801 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (217522971459627535867 / 2000000000000000000000 : ℝ) ≤
+      Real.log (17838342398801 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1928342398801 / 16000000000000 : ℝ) ≤
+      (-(5289819870100654850123 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (1928342398801 / 16000000000000 : ℝ) = (1928342398801 / 1000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1928342398801 / 1000000000000 : ℝ) ≤
+            (1641651932899345149877 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1694818573808583 / 5000000000000000 : ℝ) ≤ Real.sqrt (1838342398801 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1838342398801 / 16000000000000 : ℝ) ≤ (3389637147617169 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1838342398801 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau236 :
+    Real.arctan (3389637147617169 / 10000000000000000 : ℝ) ≤
+      (3268093096395075635043 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (3268093096395075635043 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 3389637147617169 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 3389637147617169 : ℝ) =
+              (3389637147617169 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau236)
+  have hz1 :
+    (10000000000000000 / 3389637147617169 : ℝ) ≤
+      1 / Real.sqrt (1838342398801 / 16000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 3389637147617169 : ℝ) =
+          1 / (3389637147617169 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (125000000000000 / 564939524602861 : ℝ) ≤
+      (2177543168845947543807 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1838342398801 / 16000000000000 : ℝ) ≤
+      (125000000000000 / 564939524602861 : ℝ) := by
+    rw [show
+        (125000000000000 / 564939524602861 : ℝ) =
+          (3 / 40) / (1694818573808583 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (3268093096395075635043 / 10000000000000000000000 : ℝ)) -
+        6 * (2177543168845947543807 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1838342398801 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1838342398801 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1694818573808583 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (3268093096395075635043 / 10000000000000000000000 : ℝ)) -
+          6 * (2177543168845947543807 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (1838342398801 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1838342398801 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1838342398801 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_175 :
+    ((1093897756559963010019 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2637663615683673257443 / 1250000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((339995732619773 / 1000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1638673407935689605673 / 5000000000000000000000 : ℝ)) -
+              6 * (2171139549275033092439 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (3699107142389 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (3699107142389 / 32000000000000 : ℝ)) =
+      Real.log (35699107142389 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((3699107142389 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (3879107142389 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1093897756559963010019 / 10000000000000000000000 : ℝ) ≤
+      Real.log (35699107142389 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (3879107142389 / 32000000000000 : ℝ) ≤
+      (-(2637663615683673257443 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (3879107142389 / 32000000000000 : ℝ) = (3879107142389 / 2000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3879107142389 / 2000000000000 : ℝ) ≤
+            (828072285816326742557 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (339995732619773 / 1000000000000000 : ℝ) ≤ Real.sqrt (3699107142389 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (3699107142389 / 32000000000000 : ℝ) ≤ (3399957326197733 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (3699107142389 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau237 :
+    Real.arctan (3399957326197733 / 10000000000000000 : ℝ) ≤
+      (1638673407935689605673 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1638673407935689605673 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 3399957326197733 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 3399957326197733 : ℝ) =
+              (3399957326197733 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau237)
+  have hz1 :
+    (10000000000000000 / 3399957326197733 : ℝ) ≤
+      1 / Real.sqrt (3699107142389 / 32000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 3399957326197733 : ℝ) =
+          1 / (3399957326197733 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (75000000000000 / 339995732619773 : ℝ) ≤
+      (2171139549275033092439 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (3699107142389 / 32000000000000 : ℝ) ≤
+      (75000000000000 / 339995732619773 : ℝ) := by
+    rw [show
+        (75000000000000 / 339995732619773 : ℝ) =
+          (3 / 40) / (339995732619773 / 1000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1638673407935689605673 / 5000000000000000000000 : ℝ)) -
+        6 * (2171139549275033092439 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (3699107142389 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (3699107142389 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((339995732619773 / 1000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1638673407935689605673 / 5000000000000000000000 : ℝ)) -
+          6 * (2171139549275033092439 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (3699107142389 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (3699107142389 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (3699107142389 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_176 :
+    ((4400706843271805573 / 40000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(10521836246531778932779 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((852561568430141 / 2500000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1643283369413787703769 / 5000000000000000000000 : ℝ)) -
+              6 * (135299506569951335849 / 625000000000000000000 : ℝ)))) ≤
+      Vfield (465191185897 / 4000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (465191185897 / 4000000000000 : ℝ)) =
+      Real.log (4465191185897 / 4000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((465191185897 / 4000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (487691185897 / 4000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (4400706843271805573 / 40000000000000000000 : ℝ) ≤
+      Real.log (4465191185897 / 4000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (487691185897 / 4000000000000 : ℝ) ≤
+      (-(10521836246531778932779 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (487691185897 / 4000000000000 : ℝ) = (487691185897 / 250000000000 : ℝ) / 2 ^ 4 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (487691185897 / 250000000000 : ℝ) ≤
+            (3341107359468221067221 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (852561568430141 / 2500000000000000 : ℝ) ≤ Real.sqrt (465191185897 / 4000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (465191185897 / 4000000000000 : ℝ) ≤ (3410246273720567 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (465191185897 / 4000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau238 :
+    Real.arctan (3410246273720567 / 10000000000000000 : ℝ) ≤
+      (1643283369413787703769 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1643283369413787703769 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 3410246273720567 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 3410246273720567 : ℝ) =
+              (3410246273720567 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau238)
+  have hz1 :
+    (10000000000000000 / 3410246273720567 : ℝ) ≤
+      1 / Real.sqrt (465191185897 / 4000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 3410246273720567 : ℝ) =
+          1 / (3410246273720567 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (187500000000000 / 852561568430141 : ℝ) ≤
+      (135299506569951335849 / 625000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (465191185897 / 4000000000000 : ℝ) ≤
+      (187500000000000 / 852561568430141 : ℝ) := by
+    rw [show
+        (187500000000000 / 852561568430141 : ℝ) =
+          (3 / 40) / (852561568430141 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1643283369413787703769 / 5000000000000000000000 : ℝ)) -
+        6 * (135299506569951335849 / 625000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (465191185897 / 4000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (465191185897 / 4000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((852561568430141 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1643283369413787703769 / 5000000000000000000000 : ℝ)) -
+          6 * (135299506569951335849 / 625000000000000000000 : ℝ))) ≤
+      Real.sqrt (465191185897 / 4000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (465191185897 / 4000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (465191185897 / 4000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_177 :
+    ((1106451725023094177469 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(10493183176854784838949 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3420504272016681 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (3295753160068787053413 / 10000000000000000000000 : ℝ)) -
+              6 * (431700003932662026033 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (3743951831963 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (3743951831963 / 32000000000000 : ℝ)) =
+      Real.log (35743951831963 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((3743951831963 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (3923951831963 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1106451725023094177469 / 10000000000000000000000 : ℝ) ≤
+      Real.log (35743951831963 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (3923951831963 / 32000000000000 : ℝ) ≤
+      (-(10493183176854784838949 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (3923951831963 / 32000000000000 : ℝ) = (3923951831963 / 2000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3923951831963 / 2000000000000 : ℝ) ≤
+            (3369760429145215161051 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3420504272016681 / 10000000000000000 : ℝ) ≤ Real.sqrt (3743951831963 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (3743951831963 / 32000000000000 : ℝ) ≤ (855126068004171 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (3743951831963 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau239 :
+    Real.arctan (855126068004171 / 2500000000000000 : ℝ) ≤
+      (3295753160068787053413 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (3295753160068787053413 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (2500000000000000 / 855126068004171 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 855126068004171 : ℝ) =
+              (855126068004171 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau239)
+  have hz1 :
+    (2500000000000000 / 855126068004171 : ℝ) ≤
+      1 / Real.sqrt (3743951831963 / 32000000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 855126068004171 : ℝ) = 1 / (855126068004171 / 2500000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 1140168090672227 : ℝ) ≤
+      (431700003932662026033 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (3743951831963 / 32000000000000 : ℝ) ≤
+      (250000000000000 / 1140168090672227 : ℝ) := by
+    rw [show
+        (250000000000000 / 1140168090672227 : ℝ) =
+          (3 / 40) / (3420504272016681 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (3295753160068787053413 / 10000000000000000000000 : ℝ)) -
+        6 * (431700003932662026033 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (3743951831963 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (3743951831963 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3420504272016681 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (3295753160068787053413 / 10000000000000000000000 : ℝ)) -
+          6 * (431700003932662026033 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (3743951831963 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (3743951831963 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (3743951831963 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_178 :
+    ((222544560823413658539 / 2000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2092938674339300144207 / 1000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((686146319740731 / 2000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (66098127401491722431 / 200000000000000000000 : ℝ)) -
+              6 * (1076131246361408251279 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (15065496707 / 128000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (15065496707 / 128000000000 : ℝ)) =
+      Real.log (143065496707 / 128000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((15065496707 / 128000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (15785496707 / 128000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (222544560823413658539 / 2000000000000000000000 : ℝ) ≤
+      Real.log (143065496707 / 128000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (15785496707 / 128000000000 : ℝ) ≤
+      (-(2092938674339300144207 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (15785496707 / 128000000000 : ℝ) = (15785496707 / 8000000000 : ℝ) / 2 ^ 4 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15785496707 / 8000000000 : ℝ) ≤
+            (679650046860699855793 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (686146319740731 / 2000000000000000 : ℝ) ≤ Real.sqrt (15065496707 / 128000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (15065496707 / 128000000000 : ℝ) ≤ (1715365799351829 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (15065496707 / 128000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau240 :
+    Real.arctan (1715365799351829 / 5000000000000000 : ℝ) ≤
+      (66098127401491722431 / 200000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (66098127401491722431 / 200000000000000000000 : ℝ)) ≤
+      Real.arctan (5000000000000000 / 1715365799351829 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 1715365799351829 : ℝ) =
+              (1715365799351829 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau240)
+  have hz1 :
+    (5000000000000000 / 1715365799351829 : ℝ) ≤ 1 / Real.sqrt (15065496707 / 128000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 1715365799351829 : ℝ) = 1 / (1715365799351829 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (50000000000000 / 228715439913577 : ℝ) ≤
+      (1076131246361408251279 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (15065496707 / 128000000000 : ℝ) ≤
+      (50000000000000 / 228715439913577 : ℝ) := by
+    rw [show
+        (50000000000000 / 228715439913577 : ℝ) =
+          (3 / 40) / (686146319740731 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (66098127401491722431 / 200000000000000000000 : ℝ)) -
+        6 * (1076131246361408251279 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (15065496707 / 128000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (15065496707 / 128000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((686146319740731 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (66098127401491722431 / 200000000000000000000 : ℝ)) -
+          6 * (1076131246361408251279 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (15065496707 / 128000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (15065496707 / 128000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (15065496707 / 128000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_179 :
+    ((1118989953032259600469 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(20872729962077034974551 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3440928527273287 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (3314026655088241252493 / 10000000000000000000000 : ℝ)) -
+              6 * (429215748043235457857 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (3788796521537 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (3788796521537 / 32000000000000 : ℝ)) =
+      Real.log (35788796521537 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((3788796521537 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (3968796521537 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1118989953032259600469 / 10000000000000000000000 : ℝ) ≤
+      Real.log (35788796521537 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (3968796521537 / 32000000000000 : ℝ) ≤
+      (-(20872729962077034974551 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (3968796521537 / 32000000000000 : ℝ) = (3968796521537 / 2000000000000 : ℝ) / 2 ^ 4
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3968796521537 / 2000000000000 : ℝ) ≤
+            (6853157249922965025449 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3440928527273287 / 10000000000000000 : ℝ) ≤ Real.sqrt (3788796521537 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (3788796521537 / 32000000000000 : ℝ) ≤ (344092852727329 / 1000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (3788796521537 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau241 :
+    Real.arctan (344092852727329 / 1000000000000000 : ℝ) ≤
+      (3314026655088241252493 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (3314026655088241252493 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (1000000000000000 / 344092852727329 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1000000000000000 / 344092852727329 : ℝ) =
+              (344092852727329 / 1000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau241)
+  have hz1 :
+    (1000000000000000 / 344092852727329 : ℝ) ≤
+      1 / Real.sqrt (3788796521537 / 32000000000000 : ℝ) := by
+    rw [show
+        (1000000000000000 / 344092852727329 : ℝ) = 1 / (344092852727329 / 1000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 3440928527273287 : ℝ) ≤
+      (429215748043235457857 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (3788796521537 / 32000000000000 : ℝ) ≤
+      (750000000000000 / 3440928527273287 : ℝ) := by
+    rw [show
+        (750000000000000 / 3440928527273287 : ℝ) =
+          (3 / 40) / (3440928527273287 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (3314026655088241252493 / 10000000000000000000000 : ℝ)) -
+        6 * (429215748043235457857 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (3788796521537 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (3788796521537 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3440928527273287 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (3314026655088241252493 / 10000000000000000000000 : ℝ)) -
+          6 * (429215748043235457857 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (3788796521537 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (3788796521537 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (3788796521537 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_180 :
+    ((45010127067671446827 / 400000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(20816392372260701008297 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3451095327176937 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (332311429720360928943 / 1000000000000000000000 : ℝ)) -
+              6 * (2139947993750407451587 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (952804716581 / 8000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (952804716581 / 8000000000000 : ℝ)) =
+      Real.log (8952804716581 / 8000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((952804716581 / 8000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (997804716581 / 8000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (45010127067671446827 / 400000000000000000000 : ℝ) ≤
+      Real.log (8952804716581 / 8000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (997804716581 / 8000000000000 : ℝ) ≤
+      (-(20816392372260701008297 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 4
+        (show (997804716581 / 8000000000000 : ℝ) = (997804716581 / 500000000000 : ℝ) / 2 ^ 4 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 15
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (997804716581 / 500000000000 : ℝ) ≤
+            (6909494839739298991703 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3451095327176937 / 10000000000000000 : ℝ) ≤ Real.sqrt (952804716581 / 8000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (952804716581 / 8000000000000 : ℝ) ≤ (172554766358847 / 500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (952804716581 / 8000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau242 :
+    Real.arctan (172554766358847 / 500000000000000 : ℝ) ≤
+      (332311429720360928943 / 1000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (332311429720360928943 / 1000000000000000000000 : ℝ)) ≤
+      Real.arctan (500000000000000 / 172554766358847 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (500000000000000 / 172554766358847 : ℝ) =
+              (172554766358847 / 500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau242)
+  have hz1 :
+    (500000000000000 / 172554766358847 : ℝ) ≤ 1 / Real.sqrt (952804716581 / 8000000000000 : ℝ) := by
+    rw [show (500000000000000 / 172554766358847 : ℝ) = 1 / (172554766358847 / 500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 1150365109058979 : ℝ) ≤
+      (2139947993750407451587 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (952804716581 / 8000000000000 : ℝ) ≤
+      (250000000000000 / 1150365109058979 : ℝ) := by
+    rw [show
+        (250000000000000 / 1150365109058979 : ℝ) =
+          (3 / 40) / (3451095327176937 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (332311429720360928943 / 1000000000000000000000 : ℝ)) -
+        6 * (2139947993750407451587 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (952804716581 / 8000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (952804716581 / 8000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3451095327176937 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (332311429720360928943 / 1000000000000000000000 : ℝ)) -
+          6 * (2139947993750407451587 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (952804716581 / 8000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (952804716581 / 8000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (952804716581 / 8000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/V15.lean b/LeanPool/Zeta5Irrational/Table/V15.lean
new file mode 100644
index 0000000000..fd73ffb46f
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/V15.lean
@@ -0,0 +1,1230 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Energy
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # V15: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_181 :
+    ((565756240004760742753 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(20760370400176964455387 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1730616131954301 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (3332169574449736592719 / 10000000000000000000000 : ℝ)) -
+              6 * (213386950021971650621 / 1000000000000000000000 : ℝ)))) ≤
+      Vfield (3833641211111 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (3833641211111 / 32000000000000 : ℝ)) =
+      Real.log (35833641211111 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((3833641211111 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (4013641211111 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (565756240004760742753 / 5000000000000000000000 : ℝ) ≤
+      Real.log (35833641211111 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (4013641211111 / 32000000000000 : ℝ) ≤
+      (-(20760370400176964455387 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (4013641211111 / 32000000000000 : ℝ) = (4013641211111 / 4000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4013641211111 / 4000000000000 : ℝ) ≤
+            (34045008823035544613 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1730616131954301 / 5000000000000000 : ℝ) ≤ Real.sqrt (3833641211111 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (3833641211111 / 32000000000000 : ℝ) ≤ (692246452781721 / 2000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (3833641211111 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau243 :
+    Real.arctan (692246452781721 / 2000000000000000 : ℝ) ≤
+      (3332169574449736592719 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (3332169574449736592719 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (2000000000000000 / 692246452781721 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2000000000000000 / 692246452781721 : ℝ) =
+              (692246452781721 / 2000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau243)
+  have hz1 :
+    (2000000000000000 / 692246452781721 : ℝ) ≤
+      1 / Real.sqrt (3833641211111 / 32000000000000 : ℝ) := by
+    rw [show
+        (2000000000000000 / 692246452781721 : ℝ) = 1 / (692246452781721 / 2000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (125000000000000 / 576872043984767 : ℝ) ≤
+      (213386950021971650621 / 1000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (3833641211111 / 32000000000000 : ℝ) ≤
+      (125000000000000 / 576872043984767 : ℝ) := by
+    rw [show
+        (125000000000000 / 576872043984767 : ℝ) =
+          (3 / 40) / (1730616131954301 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (3332169574449736592719 / 10000000000000000000000 : ℝ)) -
+        6 * (213386950021971650621 / 1000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (3833641211111 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (3833641211111 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1730616131954301 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (3332169574449736592719 / 10000000000000000000000 : ℝ)) -
+          6 * (213386950021971650621 / 1000000000000000000000 : ℝ))) ≤
+      Real.sqrt (3833641211111 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (3833641211111 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (3833641211111 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_182 :
+    ((227553573578023514173 / 2000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(4140932104806516067421 / 2000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3471339599085811 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (668238552174649568123 / 2000000000000000000000 : ℝ)) -
+              6 * (265980315177075916949 / 1250000000000000000000 : ℝ)))) ≤
+      Vfield (1928031777949 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1928031777949 / 16000000000000 : ℝ)) =
+      Real.log (17928031777949 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1928031777949 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (2018031777949 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (227553573578023514173 / 2000000000000000000000 : ℝ) ≤
+      Real.log (17928031777949 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (2018031777949 / 16000000000000 : ℝ) ≤
+      (-(4140932104806516067421 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (2018031777949 / 16000000000000 : ℝ) = (2018031777949 / 2000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2018031777949 / 2000000000000 : ℝ) ≤
+            (17950976993483932579 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3471339599085811 / 10000000000000000 : ℝ) ≤ Real.sqrt (1928031777949 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1928031777949 / 16000000000000 : ℝ) ≤ (1735669799542907 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1928031777949 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau244 :
+    Real.arctan (1735669799542907 / 5000000000000000 : ℝ) ≤
+      (668238552174649568123 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (668238552174649568123 / 2000000000000000000000 : ℝ)) ≤
+      Real.arctan (5000000000000000 / 1735669799542907 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 1735669799542907 : ℝ) =
+              (1735669799542907 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau244)
+  have hz1 :
+    (5000000000000000 / 1735669799542907 : ℝ) ≤
+      1 / Real.sqrt (1928031777949 / 16000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 1735669799542907 : ℝ) = 1 / (1735669799542907 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 3471339599085811 : ℝ) ≤
+      (265980315177075916949 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1928031777949 / 16000000000000 : ℝ) ≤
+      (750000000000000 / 3471339599085811 : ℝ) := by
+    rw [show
+        (750000000000000 / 3471339599085811 : ℝ) =
+          (3 / 40) / (3471339599085811 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (668238552174649568123 / 2000000000000000000000 : ℝ)) -
+        6 * (265980315177075916949 / 1250000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1928031777949 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1928031777949 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3471339599085811 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (668238552174649568123 / 2000000000000000000000 : ℝ)) -
+          6 * (265980315177075916949 / 1250000000000000000000 : ℝ))) ≤
+      Real.sqrt (1928031777949 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1928031777949 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1928031777949 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_183 :
+    ((115026691691253115783 / 1000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(10297081645966272621029 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((139658659693411 / 400000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (3359143938006206998433 / 10000000000000000000000 : ℝ)) -
+              6 * (1057970113702477507409 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (121903382671 / 1000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (121903382671 / 1000000000000 : ℝ)) =
+      Real.log (1121903382671 / 1000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((121903382671 / 1000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (127528382671 / 1000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (115026691691253115783 / 1000000000000000000000 : ℝ) ≤
+      Real.log (1121903382671 / 1000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (127528382671 / 1000000000000 : ℝ) ≤
+      (-(10297081645966272621029 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (127528382671 / 1000000000000 : ℝ) = (127528382671 / 125000000000 : ℝ) / 2 ^ 3 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (127528382671 / 125000000000 : ℝ) ≤
+            (100126058533727378971 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (139658659693411 / 400000000000000 : ℝ) ≤ Real.sqrt (121903382671 / 1000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (121903382671 / 1000000000000 : ℝ) ≤ (1745733246167639 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (121903382671 / 1000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau245 :
+    Real.arctan (1745733246167639 / 5000000000000000 : ℝ) ≤
+      (3359143938006206998433 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (3359143938006206998433 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (5000000000000000 / 1745733246167639 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 1745733246167639 : ℝ) =
+              (1745733246167639 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau245)
+  have hz1 :
+    (5000000000000000 / 1745733246167639 : ℝ) ≤
+      1 / Real.sqrt (121903382671 / 1000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 1745733246167639 : ℝ) = 1 / (1745733246167639 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (30000000000000 / 139658659693411 : ℝ) ≤
+      (1057970113702477507409 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (121903382671 / 1000000000000 : ℝ) ≤
+      (30000000000000 / 139658659693411 : ℝ) := by
+    rw [show
+        (30000000000000 / 139658659693411 : ℝ) =
+          (3 / 40) / (139658659693411 / 400000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (3359143938006206998433 / 10000000000000000000000 : ℝ)) -
+        6 * (1057970113702477507409 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (121903382671 / 1000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (121903382671 / 1000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((139658659693411 / 400000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (3359143938006206998433 / 10000000000000000000000 : ℝ)) -
+          6 * (1057970113702477507409 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (121903382671 / 1000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (121903382671 / 1000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (121903382671 / 1000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_184 :
+    ((72671897675793028739 / 625000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(10242436845908227152729 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((877869506319801 / 2500000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1688484972162490177399 / 5000000000000000000000 : ℝ)) -
+              6 * (263029436248314576797 / 1250000000000000000000 : ℝ)))) ≤
+      Vfield (1972876467523 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1972876467523 / 16000000000000 : ℝ)) =
+      Real.log (17972876467523 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1972876467523 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (2062876467523 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (72671897675793028739 / 625000000000000000000 : ℝ) ≤
+      Real.log (17972876467523 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (2062876467523 / 16000000000000 : ℝ) ≤
+      (-(10242436845908227152729 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (2062876467523 / 16000000000000 : ℝ) = (2062876467523 / 2000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2062876467523 / 2000000000000 : ℝ) ≤
+            (154770858591772847271 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (877869506319801 / 2500000000000000 : ℝ) ≤ Real.sqrt (1972876467523 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1972876467523 / 16000000000000 : ℝ) ≤ (3511478025279207 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1972876467523 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau246 :
+    Real.arctan (3511478025279207 / 10000000000000000 : ℝ) ≤
+      (1688484972162490177399 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1688484972162490177399 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 3511478025279207 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 3511478025279207 : ℝ) =
+              (3511478025279207 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau246)
+  have hz1 :
+    (10000000000000000 / 3511478025279207 : ℝ) ≤
+      1 / Real.sqrt (1972876467523 / 16000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 3511478025279207 : ℝ) =
+          1 / (3511478025279207 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (62500000000000 / 292623168773267 : ℝ) ≤
+      (263029436248314576797 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1972876467523 / 16000000000000 : ℝ) ≤
+      (62500000000000 / 292623168773267 : ℝ) := by
+    rw [show
+        (62500000000000 / 292623168773267 : ℝ) =
+          (3 / 40) / (877869506319801 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1688484972162490177399 / 5000000000000000000000 : ℝ)) -
+        6 * (263029436248314576797 / 1250000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1972876467523 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1972876467523 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((877869506319801 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1688484972162490177399 / 5000000000000000000000 : ℝ)) -
+          6 * (263029436248314576797 / 1250000000000000000000 : ℝ))) ≤
+      Real.sqrt (1972876467523 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1972876467523 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1972876467523 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_185 :
+    ((293804561124544004693 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(101883828061061805467 / 50000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3531376159082673 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (424334104594847274703 / 1250000000000000000000 : ℝ)) -
+              6 * (1046361451427109468381 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (199529881231 / 1600000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (199529881231 / 1600000000000 : ℝ)) =
+      Real.log (1799529881231 / 1600000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((199529881231 / 1600000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (208529881231 / 1600000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (293804561124544004693 / 2500000000000000000000 : ℝ) ≤
+      Real.log (1799529881231 / 1600000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (208529881231 / 1600000000000 : ℝ) ≤
+      (-(101883828061061805467 / 50000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (208529881231 / 1600000000000 : ℝ) = (208529881231 / 200000000000 : ℝ) / 2 ^ 3 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (208529881231 / 200000000000 : ℝ) ≤
+            (2088248983938194533 / 50000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3531376159082673 / 10000000000000000 : ℝ) ≤ Real.sqrt (199529881231 / 1600000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (199529881231 / 1600000000000 : ℝ) ≤ (882844039770669 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (199529881231 / 1600000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau247 :
+    Real.arctan (882844039770669 / 2500000000000000 : ℝ) ≤
+      (424334104594847274703 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (424334104594847274703 / 1250000000000000000000 : ℝ)) ≤
+      Real.arctan (2500000000000000 / 882844039770669 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 882844039770669 : ℝ) =
+              (882844039770669 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau247)
+  have hz1 :
+    (2500000000000000 / 882844039770669 : ℝ) ≤ 1 / Real.sqrt (199529881231 / 1600000000000 : ℝ) :=
+  by
+    rw [show
+        (2500000000000000 / 882844039770669 : ℝ) = 1 / (882844039770669 / 2500000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 1177125386360891 : ℝ) ≤
+      (1046361451427109468381 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (199529881231 / 1600000000000 : ℝ) ≤
+      (250000000000000 / 1177125386360891 : ℝ) := by
+    rw [show
+        (250000000000000 / 1177125386360891 : ℝ) =
+          (3 / 40) / (3531376159082673 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (424334104594847274703 / 1250000000000000000000 : ℝ)) -
+        6 * (1046361451427109468381 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (199529881231 / 1600000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (199529881231 / 1600000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3531376159082673 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (424334104594847274703 / 1250000000000000000000 : ℝ)) -
+          6 * (1046361451427109468381 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (199529881231 / 1600000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (199529881231 / 1600000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (199529881231 / 1600000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_186 :
+    ((74229412545702061359 / 625000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(158357920152138572839 / 78125000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1775581399982569 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (106632956741550105427 / 312500000000000000000 : ℝ)) -
+              6 * (2081397264622508967207 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (2017721157097 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (2017721157097 / 16000000000000 : ℝ)) =
+      Real.log (18017721157097 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((2017721157097 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (2107721157097 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (74229412545702061359 / 625000000000000000000 : ℝ) ≤
+      Real.log (18017721157097 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (2107721157097 / 16000000000000 : ℝ) ≤
+      (-(158357920152138572839 / 78125000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (2107721157097 / 16000000000000 : ℝ) = (2107721157097 / 2000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2107721157097 / 2000000000000 : ℝ) ≤
+            (4098450230673927161 / 78125000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1775581399982569 / 5000000000000000 : ℝ) ≤ Real.sqrt (2017721157097 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (2017721157097 / 16000000000000 : ℝ) ≤ (3551162799965141 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (2017721157097 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau248 :
+    Real.arctan (3551162799965141 / 10000000000000000 : ℝ) ≤
+      (106632956741550105427 / 312500000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (106632956741550105427 / 312500000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 3551162799965141 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 3551162799965141 : ℝ) =
+              (3551162799965141 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau248)
+  have hz1 :
+    (10000000000000000 / 3551162799965141 : ℝ) ≤
+      1 / Real.sqrt (2017721157097 / 16000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 3551162799965141 : ℝ) =
+          1 / (3551162799965141 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 1775581399982569 : ℝ) ≤
+      (2081397264622508967207 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (2017721157097 / 16000000000000 : ℝ) ≤
+      (375000000000000 / 1775581399982569 : ℝ) := by
+    rw [show
+        (375000000000000 / 1775581399982569 : ℝ) =
+          (3 / 40) / (1775581399982569 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (106632956741550105427 / 312500000000000000000 : ℝ)) -
+        6 * (2081397264622508967207 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (2017721157097 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (2017721157097 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1775581399982569 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (106632956741550105427 / 312500000000000000000 : ℝ)) -
+          6 * (2081397264622508967207 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (2017721157097 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (2017721157097 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (2017721157097 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_187 :
+    ((1200107470129474167961 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(5040998430578654879721 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((446354975166469 / 1250000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (342971722732483462099 / 1000000000000000000000 : ℝ)) -
+              6 * (82810142757838897227 / 400000000000000000000 : ℝ)))) ≤
+      Vfield (510035875471 / 4000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (510035875471 / 4000000000000 : ℝ)) =
+      Real.log (4510035875471 / 4000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((510035875471 / 4000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (532535875471 / 4000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1200107470129474167961 / 10000000000000000000000 : ℝ) ≤
+      Real.log (4510035875471 / 4000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (532535875471 / 4000000000000 : ℝ) ≤
+      (-(5040998430578654879721 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (532535875471 / 4000000000000 : ℝ) = (532535875471 / 500000000000 : ℝ) / 2 ^ 3 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (532535875471 / 500000000000 : ℝ) ≤
+            (157605421671345120279 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (446354975166469 / 1250000000000000 : ℝ) ≤ Real.sqrt (510035875471 / 4000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (510035875471 / 4000000000000 : ℝ) ≤ (714167960266351 / 2000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (510035875471 / 4000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau249 :
+    Real.arctan (714167960266351 / 2000000000000000 : ℝ) ≤
+      (342971722732483462099 / 1000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (342971722732483462099 / 1000000000000000000000 : ℝ)) ≤
+      Real.arctan (2000000000000000 / 714167960266351 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2000000000000000 / 714167960266351 : ℝ) =
+              (714167960266351 / 2000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau249)
+  have hz1 :
+    (2000000000000000 / 714167960266351 : ℝ) ≤ 1 / Real.sqrt (510035875471 / 4000000000000 : ℝ) :=
+  by
+    rw [show
+        (2000000000000000 / 714167960266351 : ℝ) = 1 / (714167960266351 / 2000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (93750000000000 / 446354975166469 : ℝ) ≤
+      (82810142757838897227 / 400000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (510035875471 / 4000000000000 : ℝ) ≤
+      (93750000000000 / 446354975166469 : ℝ) := by
+    rw [show
+        (93750000000000 / 446354975166469 : ℝ) =
+          (3 / 40) / (446354975166469 / 1250000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (342971722732483462099 / 1000000000000000000000 : ℝ)) -
+        6 * (82810142757838897227 / 400000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (510035875471 / 4000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (510035875471 / 4000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((446354975166469 / 1250000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (342971722732483462099 / 1000000000000000000000 : ℝ)) -
+          6 * (82810142757838897227 / 400000000000000000000 : ℝ))) ≤
+      Real.sqrt (510035875471 / 4000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (510035875471 / 4000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (510035875471 / 4000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_188 :
+    ((612467451086576923913 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(9977827430743743884387 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((360987204712473 / 1000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1732146236679985576439 / 5000000000000000000000 : ℝ)) -
+              6 * (128030806236496645127 / 625000000000000000000 : ℝ)))) ≤
+      Vfield (1042494095729 / 8000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1042494095729 / 8000000000000 : ℝ)) =
+      Real.log (9042494095729 / 8000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1042494095729 / 8000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1087494095729 / 8000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (612467451086576923913 / 5000000000000000000000 : ℝ) ≤
+      Real.log (9042494095729 / 8000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1087494095729 / 8000000000000 : ℝ) ≤
+      (-(9977827430743743884387 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (1087494095729 / 8000000000000 : ℝ) = (1087494095729 / 1000000000000 : ℝ) / 2 ^ 3 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1087494095729 / 1000000000000 : ℝ) ≤
+            (419380273756256115613 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (360987204712473 / 1000000000000000 : ℝ) ≤ Real.sqrt (1042494095729 / 8000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1042494095729 / 8000000000000 : ℝ) ≤ (3609872047124733 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1042494095729 / 8000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau250 :
+    Real.arctan (3609872047124733 / 10000000000000000 : ℝ) ≤
+      (1732146236679985576439 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1732146236679985576439 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 3609872047124733 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 3609872047124733 : ℝ) =
+              (3609872047124733 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau250)
+  have hz1 :
+    (10000000000000000 / 3609872047124733 : ℝ) ≤
+      1 / Real.sqrt (1042494095729 / 8000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 3609872047124733 : ℝ) =
+          1 / (3609872047124733 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (25000000000000 / 120329068237491 : ℝ) ≤
+      (128030806236496645127 / 625000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1042494095729 / 8000000000000 : ℝ) ≤
+      (25000000000000 / 120329068237491 : ℝ) := by
+    rw [show
+        (25000000000000 / 120329068237491 : ℝ) =
+          (3 / 40) / (360987204712473 / 1000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1732146236679985576439 / 5000000000000000000000 : ℝ)) -
+        6 * (128030806236496645127 / 625000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1042494095729 / 8000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1042494095729 / 8000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((360987204712473 / 1000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1732146236679985576439 / 5000000000000000000000 : ℝ)) -
+          6 * (128030806236496645127 / 625000000000000000000 : ℝ))) ≤
+      Real.sqrt (1042494095729 / 8000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1042494095729 / 8000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1042494095729 / 8000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_189 :
+    ((24994016934096835669 / 200000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(19751568072237638037033 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((145939469679453 / 400000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1749206566456722486477 / 5000000000000000000000 : ℝ)) -
+              6 * (1013702202475916540091 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (266229110129 / 2000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (266229110129 / 2000000000000 : ℝ)) =
+      Real.log (2266229110129 / 2000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((266229110129 / 2000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (277479110129 / 2000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (24994016934096835669 / 200000000000000000000 : ℝ) ≤
+      Real.log (2266229110129 / 2000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (277479110129 / 2000000000000 : ℝ) ≤
+      (-(19751568072237638037033 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (277479110129 / 2000000000000 : ℝ) = (277479110129 / 250000000000 : ℝ) / 2 ^ 3 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (277479110129 / 250000000000 : ℝ) ≤
+            (1042847336762361962967 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (145939469679453 / 400000000000000 : ℝ) ≤ Real.sqrt (266229110129 / 2000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (266229110129 / 2000000000000 : ℝ) ≤ (456060842748291 / 1250000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (266229110129 / 2000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau251 :
+    Real.arctan (456060842748291 / 1250000000000000 : ℝ) ≤
+      (1749206566456722486477 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1749206566456722486477 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (1250000000000000 / 456060842748291 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1250000000000000 / 456060842748291 : ℝ) =
+              (456060842748291 / 1250000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau251)
+  have hz1 :
+    (1250000000000000 / 456060842748291 : ℝ) ≤ 1 / Real.sqrt (266229110129 / 2000000000000 : ℝ) :=
+  by
+    rw [show
+        (1250000000000000 / 456060842748291 : ℝ) = 1 / (456060842748291 / 1250000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (10000000000000 / 48646489893151 : ℝ) ≤
+      (1013702202475916540091 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (266229110129 / 2000000000000 : ℝ) ≤
+      (10000000000000 / 48646489893151 : ℝ) := by
+    rw [show
+        (10000000000000 / 48646489893151 : ℝ) = (3 / 40) / (145939469679453 / 400000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1749206566456722486477 / 5000000000000000000000 : ℝ)) -
+        6 * (1013702202475916540091 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (266229110129 / 2000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (266229110129 / 2000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((145939469679453 / 400000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1749206566456722486477 / 5000000000000000000000 : ℝ)) -
+          6 * (1013702202475916540091 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (266229110129 / 2000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (266229110129 / 2000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (266229110129 / 2000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_190 :
+    ((259809897242961142701 / 2000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1935548032090052720183 / 1000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((465564410925817 / 1250000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (3565345242591427109191 / 10000000000000000000000 : ℝ)) -
+              6 * (397422127297066665873 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (110976113009 / 800000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (110976113009 / 800000000000 : ℝ)) =
+      Real.log (910976113009 / 800000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((110976113009 / 800000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (115476113009 / 800000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (259809897242961142701 / 2000000000000000000000 : ℝ) ≤
+      Real.log (910976113009 / 800000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (115476113009 / 800000000000 : ℝ) ≤
+      (-(1935548032090052720183 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (115476113009 / 800000000000 : ℝ) = (115476113009 / 100000000000 : ℝ) / 2 ^ 3 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (115476113009 / 100000000000 : ℝ) ≤
+            (143893508809947279817 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (465564410925817 / 1250000000000000 : ℝ) ≤ Real.sqrt (110976113009 / 800000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (110976113009 / 800000000000 : ℝ) ≤ (3724515287406539 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (110976113009 / 800000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau252 :
+    Real.arctan (3724515287406539 / 10000000000000000 : ℝ) ≤
+      (3565345242591427109191 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (3565345242591427109191 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 3724515287406539 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 3724515287406539 : ℝ) =
+              (3724515287406539 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau252)
+  have hz1 :
+    (10000000000000000 / 3724515287406539 : ℝ) ≤
+      1 / Real.sqrt (110976113009 / 800000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 3724515287406539 : ℝ) =
+          1 / (3724515287406539 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (93750000000000 / 465564410925817 : ℝ) ≤
+      (397422127297066665873 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (110976113009 / 800000000000 : ℝ) ≤
+      (93750000000000 / 465564410925817 : ℝ) := by
+    rw [show
+        (93750000000000 / 465564410925817 : ℝ) =
+          (3 / 40) / (465564410925817 / 1250000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (3565345242591427109191 / 10000000000000000000000 : ℝ)) -
+        6 * (397422127297066665873 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (110976113009 / 800000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (110976113009 / 800000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((465564410925817 / 1250000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (3565345242591427109191 / 10000000000000000000000 : ℝ)) -
+          6 * (397422127297066665873 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (110976113009 / 800000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (110976113009 / 800000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (110976113009 / 800000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_191 :
+    ((13481557922957465779 / 100000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(4743621304204845889853 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3799022604012773 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (3630616003248135794793 / 10000000000000000000000 : ℝ)) -
+              6 * (1949127934440516860467 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (72162863729 / 500000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (72162863729 / 500000000000 : ℝ)) =
+      Real.log (572162863729 / 500000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((72162863729 / 500000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (74975363729 / 500000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (13481557922957465779 / 100000000000000000000 : ℝ) ≤
+      Real.log (572162863729 / 500000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (74975363729 / 500000000000 : ℝ) ≤
+      (-(4743621304204845889853 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (74975363729 / 500000000000 : ℝ) = (74975363729 / 62500000000 : ℝ) / 2 ^ 3 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (74975363729 / 62500000000 : ℝ) ≤
+            (454982548045154110147 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3799022604012773 / 10000000000000000 : ℝ) ≤ Real.sqrt (72162863729 / 500000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (72162863729 / 500000000000 : ℝ) ≤ (474877825501597 / 1250000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (72162863729 / 500000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau253 :
+    Real.arctan (474877825501597 / 1250000000000000 : ℝ) ≤
+      (3630616003248135794793 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (3630616003248135794793 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (1250000000000000 / 474877825501597 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1250000000000000 / 474877825501597 : ℝ) =
+              (474877825501597 / 1250000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau253)
+  have hz1 :
+    (1250000000000000 / 474877825501597 : ℝ) ≤ 1 / Real.sqrt (72162863729 / 500000000000 : ℝ) := by
+    rw [show
+        (1250000000000000 / 474877825501597 : ℝ) = 1 / (474877825501597 / 1250000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 3799022604012773 : ℝ) ≤
+      (1949127934440516860467 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (72162863729 / 500000000000 : ℝ) ≤
+      (750000000000000 / 3799022604012773 : ℝ) := by
+    rw [show
+        (750000000000000 / 3799022604012773 : ℝ) =
+          (3 / 40) / (3799022604012773 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (3630616003248135794793 / 10000000000000000000000 : ℝ)) -
+        6 * (1949127934440516860467 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (72162863729 / 500000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (72162863729 / 500000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3799022604012773 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (3630616003248135794793 / 10000000000000000000000 : ℝ)) -
+          6 * (1949127934440516860467 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (72162863729 / 500000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (72162863729 / 500000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (72162863729 / 500000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_192 :
+    ((1363649648793105799231 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(18856849239285061292259 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1911152162703291 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1825472812071868163633 / 5000000000000000000000 : ℝ)) -
+              6 * (968775768368206191403 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (4675203313927 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (4675203313927 / 32000000000000 : ℝ)) =
+      Real.log (36675203313927 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((4675203313927 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (4855203313927 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1363649648793105799231 / 10000000000000000000000 : ℝ) ≤
+      Real.log (36675203313927 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (4855203313927 / 32000000000000 : ℝ) ≤
+      (-(18856849239285061292259 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (4855203313927 / 32000000000000 : ℝ) = (4855203313927 / 4000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4855203313927 / 4000000000000 : ℝ) ≤
+            (1937566169714938707741 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1911152162703291 / 5000000000000000 : ℝ) ≤ Real.sqrt (4675203313927 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (4675203313927 / 32000000000000 : ℝ) ≤ (764460865081317 / 2000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (4675203313927 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau254 :
+    Real.arctan (764460865081317 / 2000000000000000 : ℝ) ≤
+      (1825472812071868163633 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1825472812071868163633 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (2000000000000000 / 764460865081317 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2000000000000000 / 764460865081317 : ℝ) =
+              (764460865081317 / 2000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau254)
+  have hz1 :
+    (2000000000000000 / 764460865081317 : ℝ) ≤
+      1 / Real.sqrt (4675203313927 / 32000000000000 : ℝ) := by
+    rw [show
+        (2000000000000000 / 764460865081317 : ℝ) = 1 / (764460865081317 / 2000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (125000000000000 / 637050720901097 : ℝ) ≤
+      (968775768368206191403 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (4675203313927 / 32000000000000 : ℝ) ≤
+      (125000000000000 / 637050720901097 : ℝ) := by
+    rw [show
+        (125000000000000 / 637050720901097 : ℝ) =
+          (3 / 40) / (1911152162703291 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1825472812071868163633 / 5000000000000000000000 : ℝ)) -
+        6 * (968775768368206191403 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (4675203313927 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (4675203313927 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1911152162703291 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1825472812071868163633 / 5000000000000000000000 : ℝ)) -
+          6 * (968775768368206191403 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (4675203313927 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (4675203313927 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (4675203313927 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/V16.lean b/LeanPool/Zeta5Irrational/Table/V16.lean
new file mode 100644
index 0000000000..6c9360206c
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/V16.lean
@@ -0,0 +1,1244 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Energy
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # V16: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_193 :
+    ((1379119536463717589619 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1874058101002048808097 / 1000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1922722546692823 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (3671120904872284070399 / 10000000000000000000000 : ℝ)) -
+              6 * (1926179010437621127729 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (2365991674599 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (2365991674599 / 16000000000000 : ℝ)) =
+      Real.log (18365991674599 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((2365991674599 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (2455991674599 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1379119536463717589619 / 10000000000000000000000 : ℝ) ≤
+      Real.log (18365991674599 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (2455991674599 / 16000000000000 : ℝ) ≤
+      (-(1874058101002048808097 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (2455991674599 / 16000000000000 : ℝ) = (2455991674599 / 2000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2455991674599 / 2000000000000 : ℝ) ≤
+            (205383439897951191903 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1922722546692823 / 5000000000000000 : ℝ) ≤ Real.sqrt (2365991674599 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (2365991674599 / 16000000000000 : ℝ) ≤ (3845445093385649 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (2365991674599 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau255 :
+    Real.arctan (3845445093385649 / 10000000000000000 : ℝ) ≤
+      (3671120904872284070399 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (3671120904872284070399 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 3845445093385649 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 3845445093385649 : ℝ) =
+              (3845445093385649 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau255)
+  have hz1 :
+    (10000000000000000 / 3845445093385649 : ℝ) ≤
+      1 / Real.sqrt (2365991674599 / 16000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 3845445093385649 : ℝ) =
+          1 / (3845445093385649 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 1922722546692823 : ℝ) ≤
+      (1926179010437621127729 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (2365991674599 / 16000000000000 : ℝ) ≤
+      (375000000000000 / 1922722546692823 : ℝ) := by
+    rw [show
+        (375000000000000 / 1922722546692823 : ℝ) =
+          (3 / 40) / (1922722546692823 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (3671120904872284070399 / 10000000000000000000000 : ℝ)) -
+        6 * (1926179010437621127729 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (2365991674599 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (2365991674599 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1922722546692823 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (3671120904872284070399 / 10000000000000000000000 : ℝ)) -
+          6 * (1926179010437621127729 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (2365991674599 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (2365991674599 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (2365991674599 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_194 :
+    ((1394565529352076953773 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(18625649088734036599843 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((120888982420927 / 312500000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (922786131046177653937 / 2500000000000000000000 : ℝ)) -
+              6 * (1915004440272120860647 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (4788763384469 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (4788763384469 / 32000000000000 : ℝ)) =
+      Real.log (36788763384469 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((4788763384469 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (4968763384469 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1394565529352076953773 / 10000000000000000000000 : ℝ) ≤
+      Real.log (36788763384469 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (4968763384469 / 32000000000000 : ℝ) ≤
+      (-(18625649088734036599843 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (4968763384469 / 32000000000000 : ℝ) = (4968763384469 / 4000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4968763384469 / 4000000000000 : ℝ) ≤
+            (2168766320265963400157 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (120888982420927 / 312500000000000 : ℝ) ≤ Real.sqrt (4788763384469 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (4788763384469 / 32000000000000 : ℝ) ≤ (3868447437469667 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (4788763384469 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau256 :
+    Real.arctan (3868447437469667 / 10000000000000000 : ℝ) ≤
+      (922786131046177653937 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (922786131046177653937 / 2500000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 3868447437469667 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 3868447437469667 : ℝ) =
+              (3868447437469667 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau256)
+  have hz1 :
+    (10000000000000000 / 3868447437469667 : ℝ) ≤
+      1 / Real.sqrt (4788763384469 / 32000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 3868447437469667 : ℝ) =
+          1 / (3868447437469667 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (23437500000000 / 120888982420927 : ℝ) ≤
+      (1915004440272120860647 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (4788763384469 / 32000000000000 : ℝ) ≤
+      (23437500000000 / 120888982420927 : ℝ) := by
+    rw [show
+        (23437500000000 / 120888982420927 : ℝ) =
+          (3 / 40) / (120888982420927 / 312500000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (922786131046177653937 / 2500000000000000000000 : ℝ)) -
+        6 * (1915004440272120860647 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (4788763384469 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (4788763384469 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((120888982420927 / 312500000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (922786131046177653937 / 2500000000000000000000 : ℝ)) -
+          6 * (1915004440272120860647 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (4788763384469 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (4788763384469 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (4788763384469 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_195 :
+    ((175284948537260638459 / 1250000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(4642168678222045719597 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3879897470497971 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1850550138337685246867 / 5000000000000000000000 : ℝ)) -
+              6 * (1909489605510279217527 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (9634306804209 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (9634306804209 / 64000000000000 : ℝ)) =
+      Real.log (73634306804209 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((9634306804209 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (9994306804209 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (175284948537260638459 / 1250000000000000000000 : ℝ) ≤
+      Real.log (73634306804209 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (9994306804209 / 64000000000000 : ℝ) ≤
+      (-(4642168678222045719597 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (9994306804209 / 64000000000000 : ℝ) = (9994306804209 / 8000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9994306804209 / 8000000000000 : ℝ) ≤
+            (556435174027954280403 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3879897470497971 / 10000000000000000 : ℝ) ≤ Real.sqrt (9634306804209 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (9634306804209 / 64000000000000 : ℝ) ≤ (1939948735248987 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (9634306804209 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau257 :
+    Real.arctan (1939948735248987 / 5000000000000000 : ℝ) ≤
+      (1850550138337685246867 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1850550138337685246867 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (5000000000000000 / 1939948735248987 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 1939948735248987 : ℝ) =
+              (1939948735248987 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau257)
+  have hz1 :
+    (5000000000000000 / 1939948735248987 : ℝ) ≤
+      1 / Real.sqrt (9634306804209 / 64000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 1939948735248987 : ℝ) = 1 / (1939948735248987 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 1293299156832657 : ℝ) ≤
+      (1909489605510279217527 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (9634306804209 / 64000000000000 : ℝ) ≤
+      (250000000000000 / 1293299156832657 : ℝ) := by
+    rw [show
+        (250000000000000 / 1293299156832657 : ℝ) =
+          (3 / 40) / (3879897470497971 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1850550138337685246867 / 5000000000000000000000 : ℝ)) -
+        6 * (1909489605510279217527 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (9634306804209 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (9634306804209 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3879897470497971 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1850550138337685246867 / 5000000000000000000000 : ℝ)) -
+          6 * (1909489605510279217527 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (9634306804209 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (9634306804209 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (9634306804209 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_196 :
+    ((1409987701160100614743 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(4628005776724920317491 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3891313812414451 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (742203816691221276071 / 2000000000000000000000 : ℝ)) -
+              6 * (1904022148523259736843 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (242277170987 / 1600000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (242277170987 / 1600000000000 : ℝ)) =
+      Real.log (1842277170987 / 1600000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((242277170987 / 1600000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (251277170987 / 1600000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1409987701160100614743 / 10000000000000000000000 : ℝ) ≤
+      Real.log (1842277170987 / 1600000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (251277170987 / 1600000000000 : ℝ) ≤
+      (-(4628005776724920317491 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (251277170987 / 1600000000000 : ℝ) = (251277170987 / 200000000000 : ℝ) / 2 ^ 3 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (251277170987 / 200000000000 : ℝ) ≤
+            (570598075525079682509 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3891313812414451 / 10000000000000000 : ℝ) ≤ Real.sqrt (242277170987 / 1600000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (242277170987 / 1600000000000 : ℝ) ≤ (1945656906207227 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (242277170987 / 1600000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau258 :
+    Real.arctan (1945656906207227 / 5000000000000000 : ℝ) ≤
+      (742203816691221276071 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (742203816691221276071 / 2000000000000000000000 : ℝ)) ≤
+      Real.arctan (5000000000000000 / 1945656906207227 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 1945656906207227 : ℝ) =
+              (1945656906207227 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau258)
+  have hz1 :
+    (5000000000000000 / 1945656906207227 : ℝ) ≤
+      1 / Real.sqrt (242277170987 / 1600000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 1945656906207227 : ℝ) = 1 / (1945656906207227 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 3891313812414451 : ℝ) ≤
+      (1904022148523259736843 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (242277170987 / 1600000000000 : ℝ) ≤
+      (750000000000000 / 3891313812414451 : ℝ) := by
+    rw [show
+        (750000000000000 / 3891313812414451 : ℝ) =
+          (3 / 40) / (3891313812414451 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (742203816691221276071 / 2000000000000000000000 : ℝ)) -
+        6 * (1904022148523259736843 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (242277170987 / 1600000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (242277170987 / 1600000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3891313812414451 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (742203816691221276071 / 2000000000000000000000 : ℝ)) -
+          6 * (1904022148523259736843 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (242277170987 / 1600000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (242277170987 / 1600000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (242277170987 / 1600000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_197 :
+    ((283537975419536278287 / 2000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(18455690634264303292773 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3902696758883327 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (744180251624464067841 / 2000000000000000000000 : ℝ)) -
+              6 * (1898601394724880011151 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (9747866874751 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (9747866874751 / 64000000000000 : ℝ)) =
+      Real.log (73747866874751 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((9747866874751 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (10107866874751 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (283537975419536278287 / 2000000000000000000000 : ℝ) ≤
+      Real.log (73747866874751 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (10107866874751 / 64000000000000 : ℝ) ≤
+      (-(18455690634264303292773 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (10107866874751 / 64000000000000 : ℝ) = (10107866874751 / 8000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10107866874751 / 8000000000000 : ℝ) ≤
+            (2338724774735696707227 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3902696758883327 / 10000000000000000 : ℝ) ≤ Real.sqrt (9747866874751 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (9747866874751 / 64000000000000 : ℝ) ≤ (390269675888333 / 1000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (9747866874751 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau259 :
+    Real.arctan (390269675888333 / 1000000000000000 : ℝ) ≤
+      (744180251624464067841 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (744180251624464067841 / 2000000000000000000000 : ℝ)) ≤
+      Real.arctan (1000000000000000 / 390269675888333 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1000000000000000 / 390269675888333 : ℝ) =
+              (390269675888333 / 1000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau259)
+  have hz1 :
+    (1000000000000000 / 390269675888333 : ℝ) ≤
+      1 / Real.sqrt (9747866874751 / 64000000000000 : ℝ) := by
+    rw [show
+        (1000000000000000 / 390269675888333 : ℝ) = 1 / (390269675888333 / 1000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 3902696758883327 : ℝ) ≤
+      (1898601394724880011151 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (9747866874751 / 64000000000000 : ℝ) ≤
+      (750000000000000 / 3902696758883327 : ℝ) := by
+    rw [show
+        (750000000000000 / 3902696758883327 : ℝ) =
+          (3 / 40) / (3902696758883327 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (744180251624464067841 / 2000000000000000000000 : ℝ)) -
+        6 * (1898601394724880011151 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (9747866874751 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (9747866874751 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3902696758883327 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (744180251624464067841 / 2000000000000000000000 : ℝ)) -
+          6 * (1898601394724880011151 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (9747866874751 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (9747866874751 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (9747866874751 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_198 :
+    ((1425386125249237081831 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(73598694878360117537 / 40000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((782809320253901 / 2000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (3730747109815750035747 / 10000000000000000000000 : ℝ)) -
+              6 * (1893226682901490889443 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (4902323455011 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (4902323455011 / 32000000000000 : ℝ)) =
+      Real.log (36902323455011 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((4902323455011 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (5082323455011 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1425386125249237081831 / 10000000000000000000000 : ℝ) ≤
+      Real.log (36902323455011 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (5082323455011 / 32000000000000 : ℝ) ≤
+      (-(73598694878360117537 / 40000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (5082323455011 / 32000000000000 : ℝ) = (5082323455011 / 4000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5082323455011 / 4000000000000 : ℝ) ≤
+            (9578966757639882463 / 40000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (782809320253901 / 2000000000000000 : ℝ) ≤ Real.sqrt (4902323455011 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (4902323455011 / 32000000000000 : ℝ) ≤ (978511650317377 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (4902323455011 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau260 :
+    Real.arctan (978511650317377 / 2500000000000000 : ℝ) ≤
+      (3730747109815750035747 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (3730747109815750035747 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (2500000000000000 / 978511650317377 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 978511650317377 : ℝ) =
+              (978511650317377 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau260)
+  have hz1 :
+    (2500000000000000 / 978511650317377 : ℝ) ≤
+      1 / Real.sqrt (4902323455011 / 32000000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 978511650317377 : ℝ) = 1 / (978511650317377 / 2500000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (150000000000000 / 782809320253901 : ℝ) ≤
+      (1893226682901490889443 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (4902323455011 / 32000000000000 : ℝ) ≤
+      (150000000000000 / 782809320253901 : ℝ) := by
+    rw [show
+        (150000000000000 / 782809320253901 : ℝ) =
+          (3 / 40) / (782809320253901 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (3730747109815750035747 / 10000000000000000000000 : ℝ)) -
+        6 * (1893226682901490889443 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (4902323455011 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (4902323455011 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((782809320253901 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (3730747109815750035747 / 10000000000000000000000 : ℝ)) -
+          6 * (1893226682901490889443 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (4902323455011 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (4902323455011 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (4902323455011 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_199 :
+    ((716538227366047056267 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(18343968847235335596581 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3925363626725593 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (935139235828401014473 / 2500000000000000000000 : ℝ)) -
+              6 * (47197434121825702827 / 250000000000000000000 : ℝ)))) ≤
+      Vfield (9861426945293 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (9861426945293 / 64000000000000 : ℝ)) =
+      Real.log (73861426945293 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((9861426945293 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (10221426945293 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (716538227366047056267 / 5000000000000000000000 : ℝ) ≤
+      Real.log (73861426945293 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (10221426945293 / 64000000000000 : ℝ) ≤
+      (-(18343968847235335596581 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (10221426945293 / 64000000000000 : ℝ) = (10221426945293 / 8000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10221426945293 / 8000000000000 : ℝ) ≤
+            (2450446561764664403419 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3925363626725593 / 10000000000000000 : ℝ) ≤ Real.sqrt (9861426945293 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (9861426945293 / 64000000000000 : ℝ) ≤ (981340906681399 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (9861426945293 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau261 :
+    Real.arctan (981340906681399 / 2500000000000000 : ℝ) ≤
+      (935139235828401014473 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (935139235828401014473 / 2500000000000000000000 : ℝ)) ≤
+      Real.arctan (2500000000000000 / 981340906681399 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 981340906681399 : ℝ) =
+              (981340906681399 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau261)
+  have hz1 :
+    (2500000000000000 / 981340906681399 : ℝ) ≤
+      1 / Real.sqrt (9861426945293 / 64000000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 981340906681399 : ℝ) = 1 / (981340906681399 / 2500000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 3925363626725593 : ℝ) ≤
+      (47197434121825702827 / 250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (9861426945293 / 64000000000000 : ℝ) ≤
+      (750000000000000 / 3925363626725593 : ℝ) := by
+    rw [show
+        (750000000000000 / 3925363626725593 : ℝ) =
+          (3 / 40) / (3925363626725593 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (935139235828401014473 / 2500000000000000000000 : ℝ)) -
+        6 * (47197434121825702827 / 250000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (9861426945293 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (9861426945293 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3925363626725593 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (935139235828401014473 / 2500000000000000000000 : ℝ)) -
+          6 * (47197434121825702827 / 250000000000000000000 : ℝ))) ≤
+      Real.sqrt (9861426945293 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (9861426945293 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (9861426945293 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_200 :
+    ((360190218660640124399 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(914428627999259877791 / 500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3936648118276669 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (468791382389424016227 / 1250000000000000000000 : ℝ)) -
+              6 * (117663300322781655899 / 625000000000000000000 : ℝ)))) ≤
+      Vfield (2479551745141 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (2479551745141 / 16000000000000 : ℝ)) =
+      Real.log (18479551745141 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((2479551745141 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (2569551745141 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (360190218660640124399 / 2500000000000000000000 : ℝ) ≤
+      Real.log (18479551745141 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (2569551745141 / 16000000000000 : ℝ) ≤
+      (-(914428627999259877791 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (2569551745141 / 16000000000000 : ℝ) = (2569551745141 / 2000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2569551745141 / 2000000000000 : ℝ) ≤
+            (125292142450740122209 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3936648118276669 / 10000000000000000 : ℝ) ≤ Real.sqrt (2479551745141 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (2479551745141 / 16000000000000 : ℝ) ≤ (61510126848073 / 156250000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (2479551745141 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau262 :
+    Real.arctan (61510126848073 / 156250000000000 : ℝ) ≤
+      (468791382389424016227 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (468791382389424016227 / 1250000000000000000000 : ℝ)) ≤
+      Real.arctan (156250000000000 / 61510126848073 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (156250000000000 / 61510126848073 : ℝ) =
+              (61510126848073 / 156250000000000 : ℝ)
+            by
+        norm_num];
+        exact hau262)
+  have hz1 :
+    (156250000000000 / 61510126848073 : ℝ) ≤ 1 / Real.sqrt (2479551745141 / 16000000000000 : ℝ) :=
+  by
+    rw [show (156250000000000 / 61510126848073 : ℝ) = 1 / (61510126848073 / 156250000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 3936648118276669 : ℝ) ≤
+      (117663300322781655899 / 625000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (2479551745141 / 16000000000000 : ℝ) ≤
+      (750000000000000 / 3936648118276669 : ℝ) := by
+    rw [show
+        (750000000000000 / 3936648118276669 : ℝ) =
+          (3 / 40) / (3936648118276669 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (468791382389424016227 / 1250000000000000000000 : ℝ)) -
+        6 * (117663300322781655899 / 625000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (2479551745141 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (2479551745141 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3936648118276669 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (468791382389424016227 / 1250000000000000000000 : ℝ)) -
+          6 * (117663300322781655899 / 625000000000000000000 : ℝ))) ≤
+      Real.sqrt (2479551745141 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (2479551745141 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (2479551745141 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_201 :
+    ((722300435672506061083 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(18260989071053057397457 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((394227825117491 / 1000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (3755204815637543892277 / 10000000000000000000000 : ℝ)) -
+              6 * (1879987114489769237383 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (19893193996399 / 128000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (19893193996399 / 128000000000000 : ℝ)) =
+      Real.log (147893193996399 / 128000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((19893193996399 / 128000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (20613193996399 / 128000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (722300435672506061083 / 5000000000000000000000 : ℝ) ≤
+      Real.log (147893193996399 / 128000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (20613193996399 / 128000000000000 : ℝ) ≤
+      (-(18260989071053057397457 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (20613193996399 / 128000000000000 : ℝ) = (20613193996399 / 16000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20613193996399 / 16000000000000 : ℝ) ≤
+            (2533426337946942602543 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (394227825117491 / 1000000000000000 : ℝ) ≤ Real.sqrt (19893193996399 / 128000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (19893193996399 / 128000000000000 : ℝ) ≤
+      (3942278251174913 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (19893193996399 / 128000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau263 :
+    Real.arctan (3942278251174913 / 10000000000000000 : ℝ) ≤
+      (3755204815637543892277 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (3755204815637543892277 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 3942278251174913 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 3942278251174913 : ℝ) =
+              (3942278251174913 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau263)
+  have hz1 :
+    (10000000000000000 / 3942278251174913 : ℝ) ≤
+      1 / Real.sqrt (19893193996399 / 128000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 3942278251174913 : ℝ) =
+          1 / (3942278251174913 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (75000000000000 / 394227825117491 : ℝ) ≤
+      (1879987114489769237383 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (19893193996399 / 128000000000000 : ℝ) ≤
+      (75000000000000 / 394227825117491 : ℝ) := by
+    rw [show
+        (75000000000000 / 394227825117491 : ℝ) =
+          (3 / 40) / (394227825117491 / 1000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (3755204815637543892277 / 10000000000000000000000 : ℝ)) -
+        6 * (1879987114489769237383 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (19893193996399 / 128000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (19893193996399 / 128000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((394227825117491 / 1000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (3755204815637543892277 / 10000000000000000000000 : ℝ)) -
+          6 * (1879987114489769237383 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (19893193996399 / 128000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (19893193996399 / 128000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (19893193996399 / 128000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_202 :
+    ((1448439394055990394649 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(18233481457763565163467 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1973950177451433 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (3760069753527545238699 / 10000000000000000000000 : ℝ)) -
+              6 * (187737238068757978517 / 1000000000000000000000 : ℝ)))) ≤
+      Vfield (1994997403167 / 12800000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1994997403167 / 12800000000000 : ℝ)) =
+      Real.log (14794997403167 / 12800000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1994997403167 / 12800000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (2066997403167 / 12800000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1448439394055990394649 / 10000000000000000000000 : ℝ) ≤
+      Real.log (14794997403167 / 12800000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (2066997403167 / 12800000000000 : ℝ) ≤
+      (-(18233481457763565163467 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (2066997403167 / 12800000000000 : ℝ) = (2066997403167 / 1600000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2066997403167 / 1600000000000 : ℝ) ≤
+            (2560933951236434836533 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1973950177451433 / 5000000000000000 : ℝ) ≤ Real.sqrt (1994997403167 / 12800000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1994997403167 / 12800000000000 : ℝ) ≤ (3947900354902869 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1994997403167 / 12800000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau264 :
+    Real.arctan (3947900354902869 / 10000000000000000 : ℝ) ≤
+      (3760069753527545238699 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (3760069753527545238699 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 3947900354902869 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 3947900354902869 : ℝ) =
+              (3947900354902869 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau264)
+  have hz1 :
+    (10000000000000000 / 3947900354902869 : ℝ) ≤
+      1 / Real.sqrt (1994997403167 / 12800000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 3947900354902869 : ℝ) =
+          1 / (3947900354902869 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (125000000000000 / 657983392483811 : ℝ) ≤
+      (187737238068757978517 / 1000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1994997403167 / 12800000000000 : ℝ) ≤
+      (125000000000000 / 657983392483811 : ℝ) := by
+    rw [show
+        (125000000000000 / 657983392483811 : ℝ) =
+          (3 / 40) / (1973950177451433 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (3760069753527545238699 / 10000000000000000000000 : ℝ)) -
+        6 * (187737238068757978517 / 1000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1994997403167 / 12800000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1994997403167 / 12800000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1973950177451433 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (3760069753527545238699 / 10000000000000000000000 : ℝ)) -
+          6 * (187737238068757978517 / 1000000000000000000000 : ℝ))) ≤
+      Real.sqrt (1994997403167 / 12800000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1994997403167 / 12800000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1994997403167 / 12800000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_203 :
+    ((1452276443906651095289 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(18206049303829734797543 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1976757231857117 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1882462954591030446627 / 5000000000000000000000 : ℝ)) -
+              6 * (937384263878327770559 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (20006754066941 / 128000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (20006754066941 / 128000000000000 : ℝ)) =
+      Real.log (148006754066941 / 128000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((20006754066941 / 128000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (20726754066941 / 128000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1452276443906651095289 / 10000000000000000000000 : ℝ) ≤
+      Real.log (148006754066941 / 128000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (20726754066941 / 128000000000000 : ℝ) ≤
+      (-(18206049303829734797543 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (20726754066941 / 128000000000000 : ℝ) = (20726754066941 / 16000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20726754066941 / 16000000000000 : ℝ) ≤
+            (2588366105170265202457 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1976757231857117 / 5000000000000000 : ℝ) ≤
+      Real.sqrt (20006754066941 / 128000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (20006754066941 / 128000000000000 : ℝ) ≤
+      (3953514463714237 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (20006754066941 / 128000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau265 :
+    Real.arctan (3953514463714237 / 10000000000000000 : ℝ) ≤
+      (1882462954591030446627 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1882462954591030446627 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 3953514463714237 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 3953514463714237 : ℝ) =
+              (3953514463714237 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau265)
+  have hz1 :
+    (10000000000000000 / 3953514463714237 : ℝ) ≤
+      1 / Real.sqrt (20006754066941 / 128000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 3953514463714237 : ℝ) =
+          1 / (3953514463714237 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 1976757231857117 : ℝ) ≤
+      (937384263878327770559 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (20006754066941 / 128000000000000 : ℝ) ≤
+      (375000000000000 / 1976757231857117 : ℝ) := by
+    rw [show
+        (375000000000000 / 1976757231857117 : ℝ) =
+          (3 / 40) / (1976757231857117 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1882462954591030446627 / 5000000000000000000000 : ℝ)) -
+        6 * (937384263878327770559 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (20006754066941 / 128000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (20006754066941 / 128000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1976757231857117 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1882462954591030446627 / 5000000000000000000000 : ℝ)) -
+          6 * (937384263878327770559 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (20006754066941 / 128000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (20006754066941 / 128000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (20006754066941 / 128000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_204 :
+    ((145611202202684841337 / 1000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(18178692196381109013093 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3959120611619847 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (3769773318745880151089 / 10000000000000000000000 : ℝ)) -
+              6 * (1872175480431801522837 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (5015883525553 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (5015883525553 / 32000000000000 : ℝ)) =
+      Real.log (37015883525553 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((5015883525553 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (5195883525553 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (145611202202684841337 / 1000000000000000000000 : ℝ) ≤
+      Real.log (37015883525553 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (5195883525553 / 32000000000000 : ℝ) ≤
+      (-(18178692196381109013093 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (5195883525553 / 32000000000000 : ℝ) = (5195883525553 / 4000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5195883525553 / 4000000000000 : ℝ) ≤
+            (2615723212618890986907 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3959120611619847 / 10000000000000000 : ℝ) ≤ Real.sqrt (5015883525553 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (5015883525553 / 32000000000000 : ℝ) ≤ (79182412232397 / 200000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (5015883525553 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau266 :
+    Real.arctan (79182412232397 / 200000000000000 : ℝ) ≤
+      (3769773318745880151089 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (3769773318745880151089 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (200000000000000 / 79182412232397 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (200000000000000 / 79182412232397 : ℝ) =
+              (79182412232397 / 200000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau266)
+  have hz1 :
+    (200000000000000 / 79182412232397 : ℝ) ≤ 1 / Real.sqrt (5015883525553 / 32000000000000 : ℝ) :=
+  by
+    rw [show (200000000000000 / 79182412232397 : ℝ) = 1 / (79182412232397 / 200000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 1319706870539949 : ℝ) ≤
+      (1872175480431801522837 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (5015883525553 / 32000000000000 : ℝ) ≤
+      (250000000000000 / 1319706870539949 : ℝ) := by
+    rw [show
+        (250000000000000 / 1319706870539949 : ℝ) =
+          (3 / 40) / (3959120611619847 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (3769773318745880151089 / 10000000000000000000000 : ℝ)) -
+        6 * (1872175480431801522837 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (5015883525553 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (5015883525553 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3959120611619847 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (3769773318745880151089 / 10000000000000000000000 : ℝ)) -
+          6 * (1872175480431801522837 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (5015883525553 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (5015883525553 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (5015883525553 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/V17.lean b/LeanPool/Zeta5Irrational/Table/V17.lean
new file mode 100644
index 0000000000..e12cbbce01
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/V17.lean
@@ -0,0 +1,1250 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Energy
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # V17: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_205 :
+    ((729973064772568471697 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(9075704862963237331003 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3964718832390083 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (3774612018114385332253 / 10000000000000000000000 : ℝ)) -
+              6 * (1869593164174766211853 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (20120314137483 / 128000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (20120314137483 / 128000000000000 : ℝ)) =
+      Real.log (148120314137483 / 128000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((20120314137483 / 128000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (20840314137483 / 128000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (729973064772568471697 / 5000000000000000000000 : ℝ) ≤
+      Real.log (148120314137483 / 128000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (20840314137483 / 128000000000000 : ℝ) ≤
+      (-(9075704862963237331003 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (20840314137483 / 128000000000000 : ℝ) = (20840314137483 / 16000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20840314137483 / 16000000000000 : ℝ) ≤
+            (1321502841536762668997 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3964718832390083 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (20120314137483 / 128000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (20120314137483 / 128000000000000 : ℝ) ≤
+      (1982359416195043 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (20120314137483 / 128000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau267 :
+    Real.arctan (1982359416195043 / 5000000000000000 : ℝ) ≤
+      (3774612018114385332253 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (3774612018114385332253 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (5000000000000000 / 1982359416195043 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 1982359416195043 : ℝ) =
+              (1982359416195043 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau267)
+  have hz1 :
+    (5000000000000000 / 1982359416195043 : ℝ) ≤
+      1 / Real.sqrt (20120314137483 / 128000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 1982359416195043 : ℝ) = 1 / (1982359416195043 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 3964718832390083 : ℝ) ≤
+      (1869593164174766211853 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (20120314137483 / 128000000000000 : ℝ) ≤
+      (750000000000000 / 3964718832390083 : ℝ) := by
+    rw [show
+        (750000000000000 / 3964718832390083 : ℝ) =
+          (3 / 40) / (3964718832390083 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (3774612018114385332253 / 10000000000000000000000 : ℝ)) -
+        6 * (1869593164174766211853 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (20120314137483 / 128000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (20120314137483 / 128000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3964718832390083 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (3774612018114385332253 / 10000000000000000000000 : ℝ)) -
+          6 * (1869593164174766211853 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (20120314137483 / 128000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (20120314137483 / 128000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (20120314137483 / 128000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_206 :
+    ((1463778767588773677433 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(18124201486317085329591 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((248144322472327 / 625000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (3779442042935979217611 / 10000000000000000000000 : ℝ)) -
+              6 * (373404301033048224383 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (10088547086377 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (10088547086377 / 64000000000000 : ℝ)) =
+      Real.log (74088547086377 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((10088547086377 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (10448547086377 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1463778767588773677433 / 10000000000000000000000 : ℝ) ≤
+      Real.log (74088547086377 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (10448547086377 / 64000000000000 : ℝ) ≤
+      (-(18124201486317085329591 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (10448547086377 / 64000000000000 : ℝ) = (10448547086377 / 8000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10448547086377 / 8000000000000 : ℝ) ≤
+            (2670213922682914670409 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (248144322472327 / 625000000000000 : ℝ) ≤ Real.sqrt (10088547086377 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (10088547086377 / 64000000000000 : ℝ) ≤ (794061831911447 / 2000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (10088547086377 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau268 :
+    Real.arctan (794061831911447 / 2000000000000000 : ℝ) ≤
+      (3779442042935979217611 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (3779442042935979217611 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (2000000000000000 / 794061831911447 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2000000000000000 / 794061831911447 : ℝ) =
+              (794061831911447 / 2000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau268)
+  have hz1 :
+    (2000000000000000 / 794061831911447 : ℝ) ≤
+      1 / Real.sqrt (10088547086377 / 64000000000000 : ℝ) := by
+    rw [show
+        (2000000000000000 / 794061831911447 : ℝ) = 1 / (794061831911447 / 2000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (46875000000000 / 248144322472327 : ℝ) ≤
+      (373404301033048224383 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (10088547086377 / 64000000000000 : ℝ) ≤
+      (46875000000000 / 248144322472327 : ℝ) := by
+    rw [show
+        (46875000000000 / 248144322472327 : ℝ) =
+          (3 / 40) / (248144322472327 / 625000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (3779442042935979217611 / 10000000000000000000000 : ℝ)) -
+        6 * (373404301033048224383 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (10088547086377 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (10088547086377 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((248144322472327 / 625000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (3779442042935979217611 / 10000000000000000000000 : ℝ)) -
+          6 * (373404301033048224383 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (10088547086377 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (10088547086377 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (10088547086377 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_207 :
+    ((1467609937283719993659 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(226213338433879786221 / 125000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3975891626417843 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1892131714307245280953 / 5000000000000000000000 : ℝ)) -
+              6 * (233057553786498956527 / 1250000000000000000000 : ℝ)))) ≤
+      Vfield (809354968321 / 5120000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (809354968321 / 5120000000000 : ℝ)) =
+      Real.log (5929354968321 / 5120000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((809354968321 / 5120000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (838154968321 / 5120000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1467609937283719993659 / 10000000000000000000000 : ℝ) ≤
+      Real.log (5929354968321 / 5120000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (838154968321 / 5120000000000 : ℝ) ≤
+      (-(226213338433879786221 / 125000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (838154968321 / 5120000000000 : ℝ) = (838154968321 / 640000000000 : ℝ) / 2 ^ 3 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (838154968321 / 640000000000 : ℝ) ≤
+            (33716854178620213779 / 125000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3975891626417843 / 10000000000000000 : ℝ) ≤ Real.sqrt (809354968321 / 5120000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (809354968321 / 5120000000000 : ℝ) ≤ (1987945813208923 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (809354968321 / 5120000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau269 :
+    Real.arctan (1987945813208923 / 5000000000000000 : ℝ) ≤
+      (1892131714307245280953 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1892131714307245280953 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (5000000000000000 / 1987945813208923 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 1987945813208923 : ℝ) =
+              (1987945813208923 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau269)
+  have hz1 :
+    (5000000000000000 / 1987945813208923 : ℝ) ≤
+      1 / Real.sqrt (809354968321 / 5120000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 1987945813208923 : ℝ) = 1 / (1987945813208923 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 3975891626417843 : ℝ) ≤
+      (233057553786498956527 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (809354968321 / 5120000000000 : ℝ) ≤
+      (750000000000000 / 3975891626417843 : ℝ) := by
+    rw [show
+        (750000000000000 / 3975891626417843 : ℝ) =
+          (3 / 40) / (3975891626417843 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1892131714307245280953 / 5000000000000000000000 : ℝ)) -
+        6 * (233057553786498956527 / 1250000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (809354968321 / 5120000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (809354968321 / 5120000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3975891626417843 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1892131714307245280953 / 5000000000000000000000 : ℝ)) -
+          6 * (233057553786498956527 / 1250000000000000000000 : ℝ))) ≤
+      Real.sqrt (809354968321 / 5120000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (809354968321 / 5120000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (809354968321 / 5120000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_208 :
+    ((367859909938660910269 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2258750761441776246791 / 1250000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1990733133017519 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (378907621031154650991 / 1000000000000000000000 : ℝ)) -
+              6 * (1861909867144123045899 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (634082945103 / 4000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (634082945103 / 4000000000000 : ℝ)) =
+      Real.log (4634082945103 / 4000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((634082945103 / 4000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (656582945103 / 4000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (367859909938660910269 / 2500000000000000000000 : ℝ) ≤
+      Real.log (4634082945103 / 4000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (656582945103 / 4000000000000 : ℝ) ≤
+      (-(2258750761441776246791 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (656582945103 / 4000000000000 : ℝ) = (656582945103 / 500000000000 : ℝ) / 2 ^ 3 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (656582945103 / 500000000000 : ℝ) ≤
+            (340551164683223753209 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1990733133017519 / 5000000000000000 : ℝ) ≤ Real.sqrt (634082945103 / 4000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (634082945103 / 4000000000000 : ℝ) ≤ (3981466266035041 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (634082945103 / 4000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau270 :
+    Real.arctan (3981466266035041 / 10000000000000000 : ℝ) ≤
+      (378907621031154650991 / 1000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (378907621031154650991 / 1000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 3981466266035041 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 3981466266035041 : ℝ) =
+              (3981466266035041 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau270)
+  have hz1 :
+    (10000000000000000 / 3981466266035041 : ℝ) ≤
+      1 / Real.sqrt (634082945103 / 4000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 3981466266035041 : ℝ) =
+          1 / (3981466266035041 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 1990733133017519 : ℝ) ≤
+      (1861909867144123045899 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (634082945103 / 4000000000000 : ℝ) ≤
+      (375000000000000 / 1990733133017519 : ℝ) := by
+    rw [show
+        (375000000000000 / 1990733133017519 : ℝ) =
+          (3 / 40) / (1990733133017519 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (378907621031154650991 / 1000000000000000000000 : ℝ)) -
+        6 * (1861909867144123045899 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (634082945103 / 4000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (634082945103 / 4000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1990733133017519 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (378907621031154650991 / 1000000000000000000000 : ℝ)) -
+          6 * (1861909867144123045899 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (634082945103 / 4000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (634082945103 / 4000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (634082945103 / 4000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_209 :
+    ((1475267876124920720441 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(18043018140451505241589 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((797406622248159 / 2000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (3793880422948913718569 / 10000000000000000000000 : ℝ)) -
+              6 * (1859369744002478350221 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (20347434278567 / 128000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (20347434278567 / 128000000000000 : ℝ)) =
+      Real.log (148347434278567 / 128000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((20347434278567 / 128000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (21067434278567 / 128000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1475267876124920720441 / 10000000000000000000000 : ℝ) ≤
+      Real.log (148347434278567 / 128000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (21067434278567 / 128000000000000 : ℝ) ≤
+      (-(18043018140451505241589 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (21067434278567 / 128000000000000 : ℝ) = (21067434278567 / 16000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21067434278567 / 16000000000000 : ℝ) ≤
+            (2751397268548494758411 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (797406622248159 / 2000000000000000 : ℝ) ≤ Real.sqrt (20347434278567 / 128000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (20347434278567 / 128000000000000 : ℝ) ≤
+      (1993516555620399 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (20347434278567 / 128000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau271 :
+    Real.arctan (1993516555620399 / 5000000000000000 : ℝ) ≤
+      (3793880422948913718569 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (3793880422948913718569 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (5000000000000000 / 1993516555620399 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 1993516555620399 : ℝ) =
+              (1993516555620399 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau271)
+  have hz1 :
+    (5000000000000000 / 1993516555620399 : ℝ) ≤
+      1 / Real.sqrt (20347434278567 / 128000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 1993516555620399 : ℝ) = 1 / (1993516555620399 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (50000000000000 / 265802207416053 : ℝ) ≤
+      (1859369744002478350221 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (20347434278567 / 128000000000000 : ℝ) ≤
+      (50000000000000 / 265802207416053 : ℝ) := by
+    rw [show
+        (50000000000000 / 265802207416053 : ℝ) =
+          (3 / 40) / (797406622248159 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (3793880422948913718569 / 10000000000000000000000 : ℝ)) -
+        6 * (1859369744002478350221 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (20347434278567 / 128000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (20347434278567 / 128000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((797406622248159 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (3793880422948913718569 / 10000000000000000000000 : ℝ)) -
+          6 * (1859369744002478350221 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (20347434278567 / 128000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (20347434278567 / 128000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (20347434278567 / 128000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_210 :
+    ((1479094647516637661407 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(9008051414162736961301 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1996296097319103 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (474834512651351679093 / 1250000000000000000000 : ℝ)) -
+              6 * (928419994915581758669 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (10202107156919 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (10202107156919 / 64000000000000 : ℝ)) =
+      Real.log (74202107156919 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((10202107156919 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (10562107156919 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1479094647516637661407 / 10000000000000000000000 : ℝ) ≤
+      Real.log (74202107156919 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (10562107156919 / 64000000000000 : ℝ) ≤
+      (-(9008051414162736961301 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (10562107156919 / 64000000000000 : ℝ) = (10562107156919 / 8000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10562107156919 / 8000000000000 : ℝ) ≤
+            (1389156290337263038699 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1996296097319103 / 5000000000000000 : ℝ) ≤ Real.sqrt (10202107156919 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (10202107156919 / 64000000000000 : ℝ) ≤
+      (3992592194638209 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (10202107156919 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau272 :
+    Real.arctan (3992592194638209 / 10000000000000000 : ℝ) ≤
+      (474834512651351679093 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (474834512651351679093 / 1250000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 3992592194638209 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 3992592194638209 : ℝ) =
+              (3992592194638209 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau272)
+  have hz1 :
+    (10000000000000000 / 3992592194638209 : ℝ) ≤
+      1 / Real.sqrt (10202107156919 / 64000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 3992592194638209 : ℝ) =
+          1 / (3992592194638209 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (125000000000000 / 665432032439701 : ℝ) ≤
+      (928419994915581758669 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (10202107156919 / 64000000000000 : ℝ) ≤
+      (125000000000000 / 665432032439701 : ℝ) := by
+    rw [show
+        (125000000000000 / 665432032439701 : ℝ) =
+          (3 / 40) / (1996296097319103 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (474834512651351679093 / 1250000000000000000000 : ℝ)) -
+        6 * (928419994915581758669 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (10202107156919 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (10202107156919 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1996296097319103 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (474834512651351679093 / 1250000000000000000000 : ℝ)) -
+          6 * (928419994915581758669 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (10202107156919 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (10202107156919 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (10202107156919 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_211 :
+    ((11863359640404745567 / 80000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1798925976518522571577 / 1000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3998143548603701 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1901731639773101406791 / 5000000000000000000000 : ℝ)) -
+              6 * (927160267134600840667 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (20460994349109 / 128000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (20460994349109 / 128000000000000 : ℝ)) =
+      Real.log (148460994349109 / 128000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((20460994349109 / 128000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (21180994349109 / 128000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (11863359640404745567 / 80000000000000000000 : ℝ) ≤
+      Real.log (148460994349109 / 128000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (21180994349109 / 128000000000000 : ℝ) ≤
+      (-(1798925976518522571577 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (21180994349109 / 128000000000000 : ℝ) = (21180994349109 / 16000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21180994349109 / 16000000000000 : ℝ) ≤
+            (280515564381477428423 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3998143548603701 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (20460994349109 / 128000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (20460994349109 / 128000000000000 : ℝ) ≤ (499767943575463 / 1250000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (20460994349109 / 128000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau273 :
+    Real.arctan (499767943575463 / 1250000000000000 : ℝ) ≤
+      (1901731639773101406791 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1901731639773101406791 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (1250000000000000 / 499767943575463 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1250000000000000 / 499767943575463 : ℝ) =
+              (499767943575463 / 1250000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau273)
+  have hz1 :
+    (1250000000000000 / 499767943575463 : ℝ) ≤
+      1 / Real.sqrt (20460994349109 / 128000000000000 : ℝ) := by
+    rw [show
+        (1250000000000000 / 499767943575463 : ℝ) = 1 / (499767943575463 / 1250000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 3998143548603701 : ℝ) ≤
+      (927160267134600840667 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (20460994349109 / 128000000000000 : ℝ) ≤
+      (750000000000000 / 3998143548603701 : ℝ) := by
+    rw [show
+        (750000000000000 / 3998143548603701 : ℝ) =
+          (3 / 40) / (3998143548603701 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1901731639773101406791 / 5000000000000000000000 : ℝ)) -
+        6 * (927160267134600840667 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (20460994349109 / 128000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (20460994349109 / 128000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3998143548603701 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1901731639773101406791 / 5000000000000000000000 : ℝ)) -
+          6 * (927160267134600840667 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (20460994349109 / 128000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (20460994349109 / 128000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (20460994349109 / 128000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_212 :
+    ((185842974980787540947 / 1250000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2245311070523984085901 / 1250000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1000921801322313 / 2500000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1904120996085517332403 / 5000000000000000000000 : ℝ)) -
+              6 * (185181130762230820097 / 1000000000000000000000 : ℝ)))) ≤
+      Vfield (1025888719219 / 6400000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1025888719219 / 6400000000000 : ℝ)) =
+      Real.log (7425888719219 / 6400000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1025888719219 / 6400000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1061888719219 / 6400000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (185842974980787540947 / 1250000000000000000000 : ℝ) ≤
+      Real.log (7425888719219 / 6400000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1061888719219 / 6400000000000 : ℝ) ≤
+      (-(2245311070523984085901 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (1061888719219 / 6400000000000 : ℝ) = (1061888719219 / 800000000000 : ℝ) / 2 ^ 3 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1061888719219 / 800000000000 : ℝ) ≤
+            (353990855601015914099 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1000921801322313 / 2500000000000000 : ℝ) ≤ Real.sqrt (1025888719219 / 6400000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1025888719219 / 6400000000000 : ℝ) ≤ (800737441057851 / 2000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1025888719219 / 6400000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau274 :
+    Real.arctan (800737441057851 / 2000000000000000 : ℝ) ≤
+      (1904120996085517332403 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1904120996085517332403 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (2000000000000000 / 800737441057851 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2000000000000000 / 800737441057851 : ℝ) =
+              (800737441057851 / 2000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau274)
+  have hz1 :
+    (2000000000000000 / 800737441057851 : ℝ) ≤
+      1 / Real.sqrt (1025888719219 / 6400000000000 : ℝ) := by
+    rw [show
+        (2000000000000000 / 800737441057851 : ℝ) = 1 / (800737441057851 / 2000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (62500000000000 / 333640600440771 : ℝ) ≤
+      (185181130762230820097 / 1000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1025888719219 / 6400000000000 : ℝ) ≤
+      (62500000000000 / 333640600440771 : ℝ) := by
+    rw [show
+        (62500000000000 / 333640600440771 : ℝ) =
+          (3 / 40) / (1000921801322313 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1904120996085517332403 / 5000000000000000000000 : ℝ)) -
+        6 * (185181130762230820097 / 1000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1025888719219 / 6400000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1025888719219 / 6400000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1000921801322313 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1904120996085517332403 / 5000000000000000000000 : ℝ)) -
+          6 * (185181130762230820097 / 1000000000000000000000 : ℝ))) ≤
+      Real.sqrt (1025888719219 / 6400000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1025888719219 / 6400000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1025888719219 / 6400000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_213 :
+    ((372641545755497074473 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(896789442080253987649 / 500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((801844639324909 / 2000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (3813012273070485265727 / 10000000000000000000000 : ℝ)) -
+              6 * (1849312240854789952491 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (20574554419651 / 128000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (20574554419651 / 128000000000000 : ℝ)) =
+      Real.log (148574554419651 / 128000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((20574554419651 / 128000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (21294554419651 / 128000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (372641545755497074473 / 2500000000000000000000 : ℝ) ≤
+      Real.log (148574554419651 / 128000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (21294554419651 / 128000000000000 : ℝ) ≤
+      (-(896789442080253987649 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (21294554419651 / 128000000000000 : ℝ) = (21294554419651 / 16000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21294554419651 / 16000000000000 : ℝ) ≤
+            (142931328369746012351 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (801844639324909 / 2000000000000000 : ℝ) ≤ Real.sqrt (20574554419651 / 128000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (20574554419651 / 128000000000000 : ℝ) ≤
+      (1002305799156137 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (20574554419651 / 128000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau275 :
+    Real.arctan (1002305799156137 / 2500000000000000 : ℝ) ≤
+      (3813012273070485265727 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (3813012273070485265727 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (2500000000000000 / 1002305799156137 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 1002305799156137 : ℝ) =
+              (1002305799156137 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau275)
+  have hz1 :
+    (2500000000000000 / 1002305799156137 : ℝ) ≤
+      1 / Real.sqrt (20574554419651 / 128000000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 1002305799156137 : ℝ) = 1 / (1002305799156137 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (150000000000000 / 801844639324909 : ℝ) ≤
+      (1849312240854789952491 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (20574554419651 / 128000000000000 : ℝ) ≤
+      (150000000000000 / 801844639324909 : ℝ) := by
+    rw [show
+        (150000000000000 / 801844639324909 : ℝ) =
+          (3 / 40) / (801844639324909 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (3813012273070485265727 / 10000000000000000000000 : ℝ)) -
+        6 * (1849312240854789952491 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (20574554419651 / 128000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (20574554419651 / 128000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((801844639324909 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (3813012273070485265727 / 10000000000000000000000 : ℝ)) -
+          6 * (1849312240854789952491 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (20574554419651 / 128000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (20574554419651 / 128000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (20574554419651 / 128000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_214 :
+    ((747193552847302273959 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(17909160216750060519991 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((4014751554319121 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (763554831200230249377 / 2000000000000000000000 : ℝ)) -
+              6 * (369364653116313286371 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (10315667227461 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (10315667227461 / 64000000000000 : ℝ)) =
+      Real.log (74315667227461 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((10315667227461 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (10675667227461 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (747193552847302273959 / 5000000000000000000000 : ℝ) ≤
+      Real.log (74315667227461 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (10675667227461 / 64000000000000 : ℝ) ≤
+      (-(17909160216750060519991 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (10675667227461 / 64000000000000 : ℝ) = (10675667227461 / 8000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10675667227461 / 8000000000000 : ℝ) ≤
+            (2885255192249939480009 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (4014751554319121 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (10315667227461 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (10315667227461 / 64000000000000 : ℝ) ≤ (1003687888579781 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (10315667227461 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau276 :
+    Real.arctan (1003687888579781 / 2500000000000000 : ℝ) ≤
+      (763554831200230249377 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (763554831200230249377 / 2000000000000000000000 : ℝ)) ≤
+      Real.arctan (2500000000000000 / 1003687888579781 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 1003687888579781 : ℝ) =
+              (1003687888579781 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau276)
+  have hz1 :
+    (2500000000000000 / 1003687888579781 : ℝ) ≤
+      1 / Real.sqrt (10315667227461 / 64000000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 1003687888579781 : ℝ) = 1 / (1003687888579781 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 4014751554319121 : ℝ) ≤
+      (369364653116313286371 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (10315667227461 / 64000000000000 : ℝ) ≤
+      (750000000000000 / 4014751554319121 : ℝ) := by
+    rw [show
+        (750000000000000 / 4014751554319121 : ℝ) =
+          (3 / 40) / (4014751554319121 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (763554831200230249377 / 2000000000000000000000 : ℝ)) -
+        6 * (369364653116313286371 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (10315667227461 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (10315667227461 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((4014751554319121 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (763554831200230249377 / 2000000000000000000000 : ℝ)) -
+          6 * (369364653116313286371 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (10315667227461 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (10315667227461 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (10315667227461 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_215 :
+    ((1498206568979816676777 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(17882602311985011388837 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((4020272309864503 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (29863497456978361057 / 78125000000000000000 : ℝ)) -
+              6 * (461086078515075774277 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (20688114490193 / 128000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (20688114490193 / 128000000000000 : ℝ)) =
+      Real.log (148688114490193 / 128000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((20688114490193 / 128000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (21408114490193 / 128000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1498206568979816676777 / 10000000000000000000000 : ℝ) ≤
+      Real.log (148688114490193 / 128000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (21408114490193 / 128000000000000 : ℝ) ≤
+      (-(17882602311985011388837 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (21408114490193 / 128000000000000 : ℝ) = (21408114490193 / 16000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21408114490193 / 16000000000000 : ℝ) ≤
+            (2911813097014988611163 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (4020272309864503 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (20688114490193 / 128000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (20688114490193 / 128000000000000 : ℝ) ≤
+      (2010136154932253 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (20688114490193 / 128000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau277 :
+    Real.arctan (2010136154932253 / 5000000000000000 : ℝ) ≤
+      (29863497456978361057 / 78125000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (29863497456978361057 / 78125000000000000000 : ℝ)) ≤
+      Real.arctan (5000000000000000 / 2010136154932253 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 2010136154932253 : ℝ) =
+              (2010136154932253 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau277)
+  have hz1 :
+    (5000000000000000 / 2010136154932253 : ℝ) ≤
+      1 / Real.sqrt (20688114490193 / 128000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 2010136154932253 : ℝ) = 1 / (2010136154932253 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 4020272309864503 : ℝ) ≤
+      (461086078515075774277 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (20688114490193 / 128000000000000 : ℝ) ≤
+      (750000000000000 / 4020272309864503 : ℝ) := by
+    rw [show
+        (750000000000000 / 4020272309864503 : ℝ) =
+          (3 / 40) / (4020272309864503 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (29863497456978361057 / 78125000000000000000 : ℝ)) -
+        6 * (461086078515075774277 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (20688114490193 / 128000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (20688114490193 / 128000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((4020272309864503 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (29863497456978361057 / 78125000000000000000 : ℝ)) -
+          6 * (461086078515075774277 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (20688114490193 / 128000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (20688114490193 / 128000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (20688114490193 / 128000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_216 :
+    ((300404914798402879239 / 2000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(17856114752668976953491 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2012892747268141 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (382727286185266192809 / 1000000000000000000000 : ℝ)) -
+              6 * (460468829795916814311 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (2593111815683 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (2593111815683 / 16000000000000 : ℝ)) =
+      Real.log (18593111815683 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((2593111815683 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (2683111815683 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (300404914798402879239 / 2000000000000000000000 : ℝ) ≤
+      Real.log (18593111815683 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (2683111815683 / 16000000000000 : ℝ) ≤
+      (-(17856114752668976953491 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (2683111815683 / 16000000000000 : ℝ) = (2683111815683 / 2000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2683111815683 / 2000000000000 : ℝ) ≤
+            (2938300656331023046509 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2012892747268141 / 5000000000000000 : ℝ) ≤ Real.sqrt (2593111815683 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (2593111815683 / 16000000000000 : ℝ) ≤ (805157098907257 / 2000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (2593111815683 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau278 :
+    Real.arctan (805157098907257 / 2000000000000000 : ℝ) ≤
+      (382727286185266192809 / 1000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (382727286185266192809 / 1000000000000000000000 : ℝ)) ≤
+      Real.arctan (2000000000000000 / 805157098907257 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2000000000000000 / 805157098907257 : ℝ) =
+              (805157098907257 / 2000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau278)
+  have hz1 :
+    (2000000000000000 / 805157098907257 : ℝ) ≤
+      1 / Real.sqrt (2593111815683 / 16000000000000 : ℝ) := by
+    rw [show
+        (2000000000000000 / 805157098907257 : ℝ) = 1 / (805157098907257 / 2000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 2012892747268141 : ℝ) ≤
+      (460468829795916814311 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (2593111815683 / 16000000000000 : ℝ) ≤
+      (375000000000000 / 2012892747268141 : ℝ) := by
+    rw [show
+        (375000000000000 / 2012892747268141 : ℝ) =
+          (3 / 40) / (2012892747268141 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (382727286185266192809 / 1000000000000000000000 : ℝ)) -
+        6 * (460468829795916814311 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (2593111815683 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (2593111815683 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2012892747268141 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (382727286185266192809 / 1000000000000000000000 : ℝ)) -
+          6 * (460468829795916814311 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (2593111815683 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (2593111815683 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (2593111815683 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/V18.lean b/LeanPool/Zeta5Irrational/Table/V18.lean
new file mode 100644
index 0000000000..cb5d342781
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/V18.lean
@@ -0,0 +1,1252 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Energy
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # V18: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_217 :
+    ((301168224368862296271 / 2000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(17829697167130139861507 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((4031291139396187 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (3832009751163251614927 / 10000000000000000000000 : ℝ)) -
+              6 * (919708107235846697917 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (4160334912147 / 25600000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (4160334912147 / 25600000000000 : ℝ)) =
+      Real.log (29760334912147 / 25600000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((4160334912147 / 25600000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (4304334912147 / 25600000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (301168224368862296271 / 2000000000000000000000 : ℝ) ≤
+      Real.log (29760334912147 / 25600000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (4304334912147 / 25600000000000 : ℝ) ≤
+      (-(17829697167130139861507 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (4304334912147 / 25600000000000 : ℝ) = (4304334912147 / 3200000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4304334912147 / 3200000000000 : ℝ) ≤
+            (2964718241869860138493 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (4031291139396187 / 10000000000000000 : ℝ) ≤ Real.sqrt (4160334912147 / 25600000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (4160334912147 / 25600000000000 : ℝ) ≤ (403129113939619 / 1000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (4160334912147 / 25600000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau279 :
+    Real.arctan (403129113939619 / 1000000000000000 : ℝ) ≤
+      (3832009751163251614927 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (3832009751163251614927 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (1000000000000000 / 403129113939619 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1000000000000000 / 403129113939619 : ℝ) =
+              (403129113939619 / 1000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau279)
+  have hz1 :
+    (1000000000000000 / 403129113939619 : ℝ) ≤
+      1 / Real.sqrt (4160334912147 / 25600000000000 : ℝ) := by
+    rw [show
+        (1000000000000000 / 403129113939619 : ℝ) = 1 / (403129113939619 / 1000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 4031291139396187 : ℝ) ≤
+      (919708107235846697917 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (4160334912147 / 25600000000000 : ℝ) ≤
+      (750000000000000 / 4031291139396187 : ℝ) := by
+    rw [show
+        (750000000000000 / 4031291139396187 : ℝ) =
+          (3 / 40) / (4031291139396187 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (3832009751163251614927 / 10000000000000000000000 : ℝ)) -
+        6 * (919708107235846697917 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (4160334912147 / 25600000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (4160334912147 / 25600000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((4031291139396187 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (3832009751163251614927 / 10000000000000000000000 : ℝ)) -
+          6 * (919708107235846697917 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (4160334912147 / 25600000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (4160334912147 / 25600000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (4160334912147 / 25600000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_218 :
+    ((11794189169129279047 / 78125000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(4450837296658632106129 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2018394637647063 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1918369187644382154269 / 5000000000000000000000 : ℝ)) -
+              6 * (1836966934064261994179 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (10429227298003 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (10429227298003 / 64000000000000 : ℝ)) =
+      Real.log (74429227298003 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((10429227298003 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (10789227298003 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (11794189169129279047 / 78125000000000000000 : ℝ) ≤
+      Real.log (74429227298003 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (10789227298003 / 64000000000000 : ℝ) ≤
+      (-(4450837296658632106129 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (10789227298003 / 64000000000000 : ℝ) = (10789227298003 / 8000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10789227298003 / 8000000000000 : ℝ) ≤
+            (747766555591367893871 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2018394637647063 / 5000000000000000 : ℝ) ≤ Real.sqrt (10429227298003 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (10429227298003 / 64000000000000 : ℝ) ≤
+      (4036789275294129 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (10429227298003 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau280 :
+    Real.arctan (4036789275294129 / 10000000000000000 : ℝ) ≤
+      (1918369187644382154269 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1918369187644382154269 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 4036789275294129 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 4036789275294129 : ℝ) =
+              (4036789275294129 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau280)
+  have hz1 :
+    (10000000000000000 / 4036789275294129 : ℝ) ≤
+      1 / Real.sqrt (10429227298003 / 64000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 4036789275294129 : ℝ) =
+          1 / (4036789275294129 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (125000000000000 / 672798212549021 : ℝ) ≤
+      (1836966934064261994179 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (10429227298003 / 64000000000000 : ℝ) ≤
+      (125000000000000 / 672798212549021 : ℝ) := by
+    rw [show
+        (125000000000000 / 672798212549021 : ℝ) =
+          (3 / 40) / (2018394637647063 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1918369187644382154269 / 5000000000000000000000 : ℝ)) -
+        6 * (1836966934064261994179 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (10429227298003 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (10429227298003 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2018394637647063 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1918369187644382154269 / 5000000000000000000000 : ℝ)) -
+          6 * (1836966934064261994179 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (10429227298003 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (10429227298003 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (10429227298003 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_219 :
+    ((189183731314411355741 / 1250000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1777707044535513541937 / 1000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1010569983217551 / 2500000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (480182345859374426031 / 1250000000000000000000 : ℝ)) -
+              6 * (229315926589210924701 / 1250000000000000000000 : ℝ)))) ≤
+      Vfield (20915234631277 / 128000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (20915234631277 / 128000000000000 : ℝ)) =
+      Real.log (148915234631277 / 128000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((20915234631277 / 128000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (21635234631277 / 128000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (189183731314411355741 / 1250000000000000000000 : ℝ) ≤
+      Real.log (148915234631277 / 128000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (21635234631277 / 128000000000000 : ℝ) ≤
+      (-(1777707044535513541937 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (21635234631277 / 128000000000000 : ℝ) = (21635234631277 / 16000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21635234631277 / 16000000000000 : ℝ) ≤
+            (301734496364486458063 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1010569983217551 / 2500000000000000 : ℝ) ≤
+      Real.sqrt (20915234631277 / 128000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (20915234631277 / 128000000000000 : ℝ) ≤
+      (4042279932870207 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (20915234631277 / 128000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau281 :
+    Real.arctan (4042279932870207 / 10000000000000000 : ℝ) ≤
+      (480182345859374426031 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (480182345859374426031 / 1250000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 4042279932870207 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 4042279932870207 : ℝ) =
+              (4042279932870207 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau281)
+  have hz1 :
+    (10000000000000000 / 4042279932870207 : ℝ) ≤
+      1 / Real.sqrt (20915234631277 / 128000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 4042279932870207 : ℝ) =
+          1 / (4042279932870207 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (62500000000000 / 336856661072517 : ℝ) ≤
+      (229315926589210924701 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (20915234631277 / 128000000000000 : ℝ) ≤
+      (62500000000000 / 336856661072517 : ℝ) := by
+    rw [show
+        (62500000000000 / 336856661072517 : ℝ) =
+          (3 / 40) / (1010569983217551 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (480182345859374426031 / 1250000000000000000000 : ℝ)) -
+        6 * (229315926589210924701 / 1250000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (20915234631277 / 128000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (20915234631277 / 128000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1010569983217551 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (480182345859374426031 / 1250000000000000000000 : ℝ)) -
+          6 * (229315926589210924701 / 1250000000000000000000 : ℝ))) ≤
+      Real.sqrt (20915234631277 / 128000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (20915234631277 / 128000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (20915234631277 / 128000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_220 :
+    ((1517282033553838498547 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(17750860580341441455871 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((809552628511343 / 2000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1923085479175907476307 / 5000000000000000000000 : ℝ)) -
+              6 * (458024396444353584973 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (5243003666637 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (5243003666637 / 32000000000000 : ℝ)) =
+      Real.log (37243003666637 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((5243003666637 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (5423003666637 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1517282033553838498547 / 10000000000000000000000 : ℝ) ≤
+      Real.log (37243003666637 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (5423003666637 / 32000000000000 : ℝ) ≤
+      (-(17750860580341441455871 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (5423003666637 / 32000000000000 : ℝ) = (5423003666637 / 4000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5423003666637 / 4000000000000 : ℝ) ≤
+            (3043554828658558544129 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (809552628511343 / 2000000000000000 : ℝ) ≤ Real.sqrt (5243003666637 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (5243003666637 / 32000000000000 : ℝ) ≤ (2023881571278359 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (5243003666637 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau282 :
+    Real.arctan (2023881571278359 / 5000000000000000 : ℝ) ≤
+      (1923085479175907476307 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1923085479175907476307 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (5000000000000000 / 2023881571278359 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 2023881571278359 : ℝ) =
+              (2023881571278359 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau282)
+  have hz1 :
+    (5000000000000000 / 2023881571278359 : ℝ) ≤
+      1 / Real.sqrt (5243003666637 / 32000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 2023881571278359 : ℝ) = 1 / (2023881571278359 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (150000000000000 / 809552628511343 : ℝ) ≤
+      (458024396444353584973 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (5243003666637 / 32000000000000 : ℝ) ≤
+      (150000000000000 / 809552628511343 : ℝ) := by
+    rw [show
+        (150000000000000 / 809552628511343 : ℝ) =
+          (3 / 40) / (809552628511343 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1923085479175907476307 / 5000000000000000000000 : ℝ)) -
+        6 * (458024396444353584973 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (5243003666637 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (5243003666637 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((809552628511343 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1923085479175907476307 / 5000000000000000000000 : ℝ)) -
+          6 * (458024396444353584973 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (5243003666637 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (5243003666637 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (5243003666637 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_221 :
+    ((1521092763872220139037 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(17724719231489336947553 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((4053238934580107 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (3850874981935184802889 / 10000000000000000000000 : ℝ)) -
+              6 * (182967738921082186107 / 1000000000000000000000 : ℝ)))) ≤
+      Vfield (21028794701819 / 128000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (21028794701819 / 128000000000000 : ℝ)) =
+      Real.log (149028794701819 / 128000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((21028794701819 / 128000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (21748794701819 / 128000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1521092763872220139037 / 10000000000000000000000 : ℝ) ≤
+      Real.log (149028794701819 / 128000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (21748794701819 / 128000000000000 : ℝ) ≤
+      (-(17724719231489336947553 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (21748794701819 / 128000000000000 : ℝ) = (21748794701819 / 16000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21748794701819 / 16000000000000 : ℝ) ≤
+            (3069696177510663052447 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (4053238934580107 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (21028794701819 / 128000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (21028794701819 / 128000000000000 : ℝ) ≤ (405323893458011 / 1000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (21028794701819 / 128000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau283 :
+    Real.arctan (405323893458011 / 1000000000000000 : ℝ) ≤
+      (3850874981935184802889 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (3850874981935184802889 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (1000000000000000 / 405323893458011 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1000000000000000 / 405323893458011 : ℝ) =
+              (405323893458011 / 1000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau283)
+  have hz1 :
+    (1000000000000000 / 405323893458011 : ℝ) ≤
+      1 / Real.sqrt (21028794701819 / 128000000000000 : ℝ) := by
+    rw [show
+        (1000000000000000 / 405323893458011 : ℝ) = 1 / (405323893458011 / 1000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 4053238934580107 : ℝ) ≤
+      (182967738921082186107 / 1000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (21028794701819 / 128000000000000 : ℝ) ≤
+      (750000000000000 / 4053238934580107 : ℝ) := by
+    rw [show
+        (750000000000000 / 4053238934580107 : ℝ) =
+          (3 / 40) / (4053238934580107 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (3850874981935184802889 / 10000000000000000000000 : ℝ)) -
+        6 * (182967738921082186107 / 1000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (21028794701819 / 128000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (21028794701819 / 128000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((4053238934580107 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (3850874981935184802889 / 10000000000000000000000 : ℝ)) -
+          6 * (182967738921082186107 / 1000000000000000000000 : ℝ))) ≤
+      Real.sqrt (21028794701819 / 128000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (21028794701819 / 128000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (21028794701819 / 128000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_222 :
+    ((1524902042577198992607 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2212330755188929494453 / 1250000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((405870733896293 / 1000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (3855570869629157385863 / 10000000000000000000000 : ℝ)) -
+              6 * (228408344945016066593 / 1250000000000000000000 : ℝ)))) ≤
+      Vfield (2108557473709 / 12800000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (2108557473709 / 12800000000000 : ℝ)) =
+      Real.log (14908557473709 / 12800000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((2108557473709 / 12800000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (2180557473709 / 12800000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1524902042577198992607 / 10000000000000000000000 : ℝ) ≤
+      Real.log (14908557473709 / 12800000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (2180557473709 / 12800000000000 : ℝ) ≤
+      (-(2212330755188929494453 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (2180557473709 / 12800000000000 : ℝ) = (2180557473709 / 1600000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2180557473709 / 1600000000000 : ℝ) ≤
+            (386971170936070505547 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (405870733896293 / 1000000000000000 : ℝ) ≤ Real.sqrt (2108557473709 / 12800000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (2108557473709 / 12800000000000 : ℝ) ≤ (4058707338962933 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (2108557473709 / 12800000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau284 :
+    Real.arctan (4058707338962933 / 10000000000000000 : ℝ) ≤
+      (3855570869629157385863 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (3855570869629157385863 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 4058707338962933 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 4058707338962933 : ℝ) =
+              (4058707338962933 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau284)
+  have hz1 :
+    (10000000000000000 / 4058707338962933 : ℝ) ≤
+      1 / Real.sqrt (2108557473709 / 12800000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 4058707338962933 : ℝ) =
+          1 / (4058707338962933 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (75000000000000 / 405870733896293 : ℝ) ≤
+      (228408344945016066593 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (2108557473709 / 12800000000000 : ℝ) ≤
+      (75000000000000 / 405870733896293 : ℝ) := by
+    rw [show
+        (75000000000000 / 405870733896293 : ℝ) =
+          (3 / 40) / (405870733896293 / 1000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (3855570869629157385863 / 10000000000000000000000 : ℝ)) -
+        6 * (228408344945016066593 / 1250000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (2108557473709 / 12800000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (2108557473709 / 12800000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((405870733896293 / 1000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (3855570869629157385863 / 10000000000000000000000 : ℝ)) -
+          6 * (228408344945016066593 / 1250000000000000000000 : ℝ))) ≤
+      Real.sqrt (2108557473709 / 12800000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (2108557473709 / 12800000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (2108557473709 / 12800000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_223 :
+    ((1528709870774273975149 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(17672640655907776151439 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((812833677105151 / 2000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (3860258653227845737129 / 10000000000000000000000 : ℝ)) -
+              6 * (912432816977701343561 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (21142354772361 / 128000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (21142354772361 / 128000000000000 : ℝ)) =
+      Real.log (149142354772361 / 128000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((21142354772361 / 128000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (21862354772361 / 128000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1528709870774273975149 / 10000000000000000000000 : ℝ) ≤
+      Real.log (149142354772361 / 128000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (21862354772361 / 128000000000000 : ℝ) ≤
+      (-(17672640655907776151439 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show
+          (21862354772361 / 128000000000000 : ℝ) = (21862354772361 / 16000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21862354772361 / 16000000000000 : ℝ) ≤
+            (3121774753092223848561 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (812833677105151 / 2000000000000000 : ℝ) ≤ Real.sqrt (21142354772361 / 128000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (21142354772361 / 128000000000000 : ℝ) ≤
+      (2032084192762879 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (21142354772361 / 128000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau285 :
+    Real.arctan (2032084192762879 / 5000000000000000 : ℝ) ≤
+      (3860258653227845737129 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (3860258653227845737129 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (5000000000000000 / 2032084192762879 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 2032084192762879 : ℝ) =
+              (2032084192762879 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau285)
+  have hz1 :
+    (5000000000000000 / 2032084192762879 : ℝ) ≤
+      1 / Real.sqrt (21142354772361 / 128000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 2032084192762879 : ℝ) = 1 / (2032084192762879 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (150000000000000 / 812833677105151 : ℝ) ≤
+      (912432816977701343561 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (21142354772361 / 128000000000000 : ℝ) ≤
+      (150000000000000 / 812833677105151 : ℝ) := by
+    rw [show
+        (150000000000000 / 812833677105151 : ℝ) =
+          (3 / 40) / (812833677105151 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (3860258653227845737129 / 10000000000000000000000 : ℝ)) -
+        6 * (912432816977701343561 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (21142354772361 / 128000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (21142354772361 / 128000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((812833677105151 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (3860258653227845737129 / 10000000000000000000000 : ℝ)) -
+          6 * (912432816977701343561 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (21142354772361 / 128000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (21142354772361 / 128000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (21142354772361 / 128000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_224 :
+    ((76625812478384080911 / 500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(8823351361468449374261 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1017405525972267 / 2500000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (241558647769835521489 / 625000000000000000000 : ℝ)) -
+              6 * (1822473950103674257329 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (1324945925477 / 8000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1324945925477 / 8000000000000 : ℝ)) =
+      Real.log (9324945925477 / 8000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1324945925477 / 8000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1369945925477 / 8000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (76625812478384080911 / 500000000000000000000 : ℝ) ≤
+      Real.log (9324945925477 / 8000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1369945925477 / 8000000000000 : ℝ) ≤
+      (-(8823351361468449374261 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (1369945925477 / 8000000000000 : ℝ) = (1369945925477 / 1000000000000 : ℝ) / 2 ^ 3 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1369945925477 / 1000000000000 : ℝ) ≤
+            (1573856343031550625739 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1017405525972267 / 2500000000000000 : ℝ) ≤ Real.sqrt (1324945925477 / 8000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1324945925477 / 8000000000000 : ℝ) ≤ (4069622103889071 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1324945925477 / 8000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau286 :
+    Real.arctan (4069622103889071 / 10000000000000000 : ℝ) ≤
+      (241558647769835521489 / 625000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (241558647769835521489 / 625000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 4069622103889071 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 4069622103889071 : ℝ) =
+              (4069622103889071 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau286)
+  have hz1 :
+    (10000000000000000 / 4069622103889071 : ℝ) ≤
+      1 / Real.sqrt (1324945925477 / 8000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 4069622103889071 : ℝ) =
+          1 / (4069622103889071 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (62500000000000 / 339135175324089 : ℝ) ≤
+      (1822473950103674257329 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1324945925477 / 8000000000000 : ℝ) ≤
+      (62500000000000 / 339135175324089 : ℝ) := by
+    rw [show
+        (62500000000000 / 339135175324089 : ℝ) =
+          (3 / 40) / (1017405525972267 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (241558647769835521489 / 625000000000000000000 : ℝ)) -
+        6 * (1822473950103674257329 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1324945925477 / 8000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1324945925477 / 8000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1017405525972267 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (241558647769835521489 / 625000000000000000000 : ℝ)) -
+          6 * (1822473950103674257329 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (1324945925477 / 8000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1324945925477 / 8000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1324945925477 / 8000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_225 :
+    ((1540124663354140614713 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(17595027821549266170449 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((408050767350993 / 1000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (3874273694284965512503 / 10000000000000000000000 : ℝ)) -
+              6 * (1817718661331479209709 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (10656347439087 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (10656347439087 / 64000000000000 : ℝ)) =
+      Real.log (74656347439087 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((10656347439087 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (11016347439087 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1540124663354140614713 / 10000000000000000000000 : ℝ) ≤
+      Real.log (74656347439087 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (11016347439087 / 64000000000000 : ℝ) ≤
+      (-(17595027821549266170449 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (11016347439087 / 64000000000000 : ℝ) = (11016347439087 / 8000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11016347439087 / 8000000000000 : ℝ) ≤
+            (3199387587450733829551 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (408050767350993 / 1000000000000000 : ℝ) ≤ Real.sqrt (10656347439087 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (10656347439087 / 64000000000000 : ℝ) ≤
+      (4080507673509933 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (10656347439087 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau287 :
+    Real.arctan (4080507673509933 / 10000000000000000 : ℝ) ≤
+      (3874273694284965512503 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (3874273694284965512503 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 4080507673509933 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 4080507673509933 : ℝ) =
+              (4080507673509933 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau287)
+  have hz1 :
+    (10000000000000000 / 4080507673509933 : ℝ) ≤
+      1 / Real.sqrt (10656347439087 / 64000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 4080507673509933 : ℝ) =
+          1 / (4080507673509933 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (25000000000000 / 136016922450331 : ℝ) ≤
+      (1817718661331479209709 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (10656347439087 / 64000000000000 : ℝ) ≤
+      (25000000000000 / 136016922450331 : ℝ) := by
+    rw [show
+        (25000000000000 / 136016922450331 : ℝ) =
+          (3 / 40) / (408050767350993 / 1000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (3874273694284965512503 / 10000000000000000000000 : ℝ)) -
+        6 * (1817718661331479209709 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (10656347439087 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (10656347439087 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((408050767350993 / 1000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (3874273694284965512503 / 10000000000000000000000 : ℝ)) -
+          6 * (1817718661331479209709 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (10656347439087 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (10656347439087 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (10656347439087 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_226 :
+    ((773863646372646733217 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(17543618577511353923213 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((255710267553653 / 625000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (970894277033406227889 / 2500000000000000000000 : ℝ)) -
+              6 * (453250101545841468879 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (5356563737179 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (5356563737179 / 32000000000000 : ℝ)) =
+      Real.log (37356563737179 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((5356563737179 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (5536563737179 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (773863646372646733217 / 5000000000000000000000 : ℝ) ≤
+      Real.log (37356563737179 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (5536563737179 / 32000000000000 : ℝ) ≤
+      (-(17543618577511353923213 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (5536563737179 / 32000000000000 : ℝ) = (5536563737179 / 4000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5536563737179 / 4000000000000 : ℝ) ≤
+            (3250796831488646076787 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (255710267553653 / 625000000000000 : ℝ) ≤ Real.sqrt (5356563737179 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (5356563737179 / 32000000000000 : ℝ) ≤ (4091364280858451 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (5356563737179 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau288 :
+    Real.arctan (4091364280858451 / 10000000000000000 : ℝ) ≤
+      (970894277033406227889 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (970894277033406227889 / 2500000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 4091364280858451 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 4091364280858451 : ℝ) =
+              (4091364280858451 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau288)
+  have hz1 :
+    (10000000000000000 / 4091364280858451 : ℝ) ≤
+      1 / Real.sqrt (5356563737179 / 32000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 4091364280858451 : ℝ) =
+          1 / (4091364280858451 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (46875000000000 / 255710267553653 : ℝ) ≤
+      (453250101545841468879 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (5356563737179 / 32000000000000 : ℝ) ≤
+      (46875000000000 / 255710267553653 : ℝ) := by
+    rw [show
+        (46875000000000 / 255710267553653 : ℝ) =
+          (3 / 40) / (255710267553653 / 625000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (970894277033406227889 / 2500000000000000000000 : ℝ)) -
+        6 * (453250101545841468879 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (5356563737179 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (5356563737179 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((255710267553653 / 625000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (970894277033406227889 / 2500000000000000000000 : ℝ)) -
+          6 * (453250101545841468879 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (5356563737179 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (5356563737179 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (5356563737179 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_227 :
+    ((1555324146529782093153 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(3498494454666579830101 / 2000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((256387009742743 / 625000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (973212212814486599171 / 2500000000000000000000 : ℝ)) -
+              6 * (1808318706405921828007 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (10769907509629 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (10769907509629 / 64000000000000 : ℝ)) =
+      Real.log (74769907509629 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((10769907509629 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (11129907509629 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1555324146529782093153 / 10000000000000000000000 : ℝ) ≤
+      Real.log (74769907509629 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (11129907509629 / 64000000000000 : ℝ) ≤
+      (-(3498494454666579830101 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (11129907509629 / 64000000000000 : ℝ) = (11129907509629 / 8000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11129907509629 / 8000000000000 : ℝ) ≤
+            (660388627133420169899 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (256387009742743 / 625000000000000 : ℝ) ≤ Real.sqrt (10769907509629 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (10769907509629 / 64000000000000 : ℝ) ≤
+      (4102192155883891 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (10769907509629 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau289 :
+    Real.arctan (4102192155883891 / 10000000000000000 : ℝ) ≤
+      (973212212814486599171 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (973212212814486599171 / 2500000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 4102192155883891 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 4102192155883891 : ℝ) =
+              (4102192155883891 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau289)
+  have hz1 :
+    (10000000000000000 / 4102192155883891 : ℝ) ≤
+      1 / Real.sqrt (10769907509629 / 64000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 4102192155883891 : ℝ) =
+          1 / (4102192155883891 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (46875000000000 / 256387009742743 : ℝ) ≤
+      (1808318706405921828007 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (10769907509629 / 64000000000000 : ℝ) ≤
+      (46875000000000 / 256387009742743 : ℝ) := by
+    rw [show
+        (46875000000000 / 256387009742743 : ℝ) =
+          (3 / 40) / (256387009742743 / 625000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (973212212814486599171 / 2500000000000000000000 : ℝ)) -
+        6 * (1808318706405921828007 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (10769907509629 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (10769907509629 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((256387009742743 / 625000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (973212212814486599171 / 2500000000000000000000 : ℝ)) -
+          6 * (1808318706405921828007 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (10769907509629 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (10769907509629 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (10769907509629 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_228 :
+    ((1562915233476233808219 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(17441586233008606041231 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2056495762754341 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (3902089165903819241319 / 10000000000000000000000 : ℝ)) -
+              6 * (1803673092348793482377 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (108266875449 / 640000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (108266875449 / 640000000000 : ℝ)) =
+      Real.log (748266875449 / 640000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((108266875449 / 640000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (111866875449 / 640000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1562915233476233808219 / 10000000000000000000000 : ℝ) ≤
+      Real.log (748266875449 / 640000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (111866875449 / 640000000000 : ℝ) ≤
+      (-(17441586233008606041231 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (111866875449 / 640000000000 : ℝ) = (111866875449 / 80000000000 : ℝ) / 2 ^ 3 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (111866875449 / 80000000000 : ℝ) ≤
+            (3352829175991393958769 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2056495762754341 / 5000000000000000 : ℝ) ≤ Real.sqrt (108266875449 / 640000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (108266875449 / 640000000000 : ℝ) ≤ (822598305101737 / 2000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (108266875449 / 640000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau290 :
+    Real.arctan (822598305101737 / 2000000000000000 : ℝ) ≤
+      (3902089165903819241319 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (3902089165903819241319 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (2000000000000000 / 822598305101737 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2000000000000000 / 822598305101737 : ℝ) =
+              (822598305101737 / 2000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau290)
+  have hz1 :
+    (2000000000000000 / 822598305101737 : ℝ) ≤ 1 / Real.sqrt (108266875449 / 640000000000 : ℝ) := by
+    rw [show
+        (2000000000000000 / 822598305101737 : ℝ) = 1 / (822598305101737 / 2000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 2056495762754341 : ℝ) ≤
+      (1803673092348793482377 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (108266875449 / 640000000000 : ℝ) ≤
+      (375000000000000 / 2056495762754341 : ℝ) := by
+    rw [show
+        (375000000000000 / 2056495762754341 : ℝ) =
+          (3 / 40) / (2056495762754341 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (3902089165903819241319 / 10000000000000000000000 : ℝ)) -
+        6 * (1803673092348793482377 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (108266875449 / 640000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (108266875449 / 640000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2056495762754341 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (3902089165903819241319 / 10000000000000000000000 : ℝ)) -
+          6 * (1803673092348793482377 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (108266875449 / 640000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (108266875449 / 640000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (108266875449 / 640000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/V19.lean b/LeanPool/Zeta5Irrational/Table/V19.lean
new file mode 100644
index 0000000000..8ebb101544
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/V19.lean
@@ -0,0 +1,1245 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Energy
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # V19: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_229 :
+    ((196312570291665255689 / 1250000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(8695478910589005680281 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((4123762613683913 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1955649145612707344259 / 5000000000000000000000 : ℝ)) -
+              6 * (224882887845839136873 / 1250000000000000000000 : ℝ)))) ≤
+      Vfield (10883467580171 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (10883467580171 / 64000000000000 : ℝ)) =
+      Real.log (74883467580171 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((10883467580171 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (11243467580171 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (196312570291665255689 / 1250000000000000000000 : ℝ) ≤
+      Real.log (74883467580171 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (11243467580171 / 64000000000000 : ℝ) ≤
+      (-(8695478910589005680281 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (11243467580171 / 64000000000000 : ℝ) = (11243467580171 / 8000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11243467580171 / 8000000000000 : ℝ) ≤
+            (1701728793910994319719 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (4123762613683913 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (10883467580171 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (10883467580171 / 64000000000000 : ℝ) ≤ (1030940653420979 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (10883467580171 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau291 :
+    Real.arctan (1030940653420979 / 2500000000000000 : ℝ) ≤
+      (1955649145612707344259 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1955649145612707344259 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (2500000000000000 / 1030940653420979 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 1030940653420979 : ℝ) =
+              (1030940653420979 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau291)
+  have hz1 :
+    (2500000000000000 / 1030940653420979 : ℝ) ≤
+      1 / Real.sqrt (10883467580171 / 64000000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 1030940653420979 : ℝ) = 1 / (1030940653420979 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 4123762613683913 : ℝ) ≤
+      (224882887845839136873 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (10883467580171 / 64000000000000 : ℝ) ≤
+      (750000000000000 / 4123762613683913 : ℝ) := by
+    rw [show
+        (750000000000000 / 4123762613683913 : ℝ) =
+          (3 / 40) / (4123762613683913 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1955649145612707344259 / 5000000000000000000000 : ℝ)) -
+        6 * (224882887845839136873 / 1250000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (10883467580171 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (10883467580171 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((4123762613683913 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1955649145612707344259 / 5000000000000000000000 : ℝ)) -
+          6 * (224882887845839136873 / 1250000000000000000000 : ℝ))) ≤
+      Real.sqrt (10883467580171 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (10883467580171 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (10883467580171 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_230 :
+    ((197260017728728357033 / 1250000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(17340584442306510414111 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2067252820721749 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (39204764633408501653 / 100000000000000000000 : ℝ)) -
+              6 * (112155517789191659431 / 625000000000000000000 : ℝ)))) ≤
+      Vfield (5470123807721 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (5470123807721 / 32000000000000 : ℝ)) =
+      Real.log (37470123807721 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((5470123807721 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (5650123807721 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (197260017728728357033 / 1250000000000000000000 : ℝ) ≤
+      Real.log (37470123807721 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (5650123807721 / 32000000000000 : ℝ) ≤
+      (-(17340584442306510414111 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (5650123807721 / 32000000000000 : ℝ) = (5650123807721 / 4000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5650123807721 / 4000000000000 : ℝ) ≤
+            (3453830966693489585889 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2067252820721749 / 5000000000000000 : ℝ) ≤ Real.sqrt (5470123807721 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (5470123807721 / 32000000000000 : ℝ) ≤ (4134505641443501 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (5470123807721 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau292 :
+    Real.arctan (4134505641443501 / 10000000000000000 : ℝ) ≤
+      (39204764633408501653 / 100000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (39204764633408501653 / 100000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 4134505641443501 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 4134505641443501 : ℝ) =
+              (4134505641443501 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau292)
+  have hz1 :
+    (10000000000000000 / 4134505641443501 : ℝ) ≤
+      1 / Real.sqrt (5470123807721 / 32000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 4134505641443501 : ℝ) =
+          1 / (4134505641443501 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 2067252820721749 : ℝ) ≤
+      (112155517789191659431 / 625000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (5470123807721 / 32000000000000 : ℝ) ≤
+      (375000000000000 / 2067252820721749 : ℝ) := by
+    rw [show
+        (375000000000000 / 2067252820721749 : ℝ) =
+          (3 / 40) / (2067252820721749 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (39204764633408501653 / 100000000000000000000 : ℝ)) -
+        6 * (112155517789191659431 / 625000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (5470123807721 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (5470123807721 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2067252820721749 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (39204764633408501653 / 100000000000000000000 : ℝ)) -
+          6 * (112155517789191659431 / 625000000000000000000 : ℝ))) ≤
+      Real.sqrt (5470123807721 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (5470123807721 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (5470123807721 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_231 :
+    ((792826990337347588387 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(8645231769943453544543 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2072610413478559 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (785924783077314604701 / 2000000000000000000000 : ℝ)) -
+              6 * (447487048230703640181 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (10997027650713 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (10997027650713 / 64000000000000 : ℝ)) =
+      Real.log (74997027650713 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((10997027650713 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (11357027650713 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (792826990337347588387 / 5000000000000000000000 : ℝ) ≤
+      Real.log (74997027650713 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (11357027650713 / 64000000000000 : ℝ) ≤
+      (-(8645231769943453544543 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (11357027650713 / 64000000000000 : ℝ) = (11357027650713 / 8000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11357027650713 / 8000000000000 : ℝ) ≤
+            (1751975934556546455457 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2072610413478559 / 5000000000000000 : ℝ) ≤ Real.sqrt (10997027650713 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (10997027650713 / 64000000000000 : ℝ) ≤
+      (4145220826957121 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (10997027650713 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau293 :
+    Real.arctan (4145220826957121 / 10000000000000000 : ℝ) ≤
+      (785924783077314604701 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (785924783077314604701 / 2000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 4145220826957121 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 4145220826957121 : ℝ) =
+              (4145220826957121 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau293)
+  have hz1 :
+    (10000000000000000 / 4145220826957121 : ℝ) ≤
+      1 / Real.sqrt (10997027650713 / 64000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 4145220826957121 : ℝ) =
+          1 / (4145220826957121 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 2072610413478559 : ℝ) ≤
+      (447487048230703640181 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (10997027650713 / 64000000000000 : ℝ) ≤
+      (375000000000000 / 2072610413478559 : ℝ) := by
+    rw [show
+        (375000000000000 / 2072610413478559 : ℝ) =
+          (3 / 40) / (2072610413478559 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (785924783077314604701 / 2000000000000000000000 : ℝ)) -
+        6 * (447487048230703640181 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (10997027650713 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (10997027650713 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2072610413478559 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (785924783077314604701 / 2000000000000000000000 : ℝ)) -
+          6 * (447487048230703640181 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (10997027650713 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (10997027650713 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (10997027650713 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_232 :
+    ((796611043778550434037 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(4310148148915218457899 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((259744274098869 / 625000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1969370438785238650359 / 5000000000000000000000 : ℝ)) -
+              6 * (892721195245301638437 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (345431490187 / 2000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (345431490187 / 2000000000000 : ℝ)) =
+      Real.log (2345431490187 / 2000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((345431490187 / 2000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (356681490187 / 2000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (796611043778550434037 / 5000000000000000000000 : ℝ) ≤
+      Real.log (2345431490187 / 2000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (356681490187 / 2000000000000 : ℝ) ≤
+      (-(4310148148915218457899 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (356681490187 / 2000000000000 : ℝ) = (356681490187 / 250000000000 : ℝ) / 2 ^ 3 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (356681490187 / 250000000000 : ℝ) ≤
+            (888455703334781542101 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (259744274098869 / 625000000000000 : ℝ) ≤ Real.sqrt (345431490187 / 2000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (345431490187 / 2000000000000 : ℝ) ≤ (4155908385581907 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (345431490187 / 2000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau294 :
+    Real.arctan (4155908385581907 / 10000000000000000 : ℝ) ≤
+      (1969370438785238650359 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1969370438785238650359 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 4155908385581907 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 4155908385581907 : ℝ) =
+              (4155908385581907 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau294)
+  have hz1 :
+    (10000000000000000 / 4155908385581907 : ℝ) ≤
+      1 / Real.sqrt (345431490187 / 2000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 4155908385581907 : ℝ) =
+          1 / (4155908385581907 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (15625000000000 / 86581424699623 : ℝ) ≤
+      (892721195245301638437 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (345431490187 / 2000000000000 : ℝ) ≤
+      (15625000000000 / 86581424699623 : ℝ) := by
+    rw [show
+        (15625000000000 / 86581424699623 : ℝ) = (3 / 40) / (259744274098869 / 625000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1969370438785238650359 / 5000000000000000000000 : ℝ)) -
+        6 * (892721195245301638437 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (345431490187 / 2000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (345431490187 / 2000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((259744274098869 / 625000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1969370438785238650359 / 5000000000000000000000 : ℝ)) -
+          6 * (892721195245301638437 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (345431490187 / 2000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (345431490187 / 2000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (345431490187 / 2000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_233 :
+    ((804170570046356540339 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(17141590695463957545947 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((4177201469832627 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (3956884238851918576391 / 10000000000000000000000 : ℝ)) -
+              6 * (177653194295087677753 / 1000000000000000000000 : ℝ)))) ≤
+      Vfield (5583683878263 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (5583683878263 / 32000000000000 : ℝ)) =
+      Real.log (37583683878263 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((5583683878263 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (5763683878263 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (804170570046356540339 / 5000000000000000000000 : ℝ) ≤
+      Real.log (37583683878263 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (5763683878263 / 32000000000000 : ℝ) ≤
+      (-(17141590695463957545947 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (5763683878263 / 32000000000000 : ℝ) = (5763683878263 / 4000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5763683878263 / 4000000000000 : ℝ) ≤
+            (3652824713536042454053 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (4177201469832627 / 10000000000000000 : ℝ) ≤ Real.sqrt (5583683878263 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (5583683878263 / 32000000000000 : ℝ) ≤ (417720146983263 / 1000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (5583683878263 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau295 :
+    Real.arctan (417720146983263 / 1000000000000000 : ℝ) ≤
+      (3956884238851918576391 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (3956884238851918576391 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (1000000000000000 / 417720146983263 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1000000000000000 / 417720146983263 : ℝ) =
+              (417720146983263 / 1000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau295)
+  have hz1 :
+    (1000000000000000 / 417720146983263 : ℝ) ≤
+      1 / Real.sqrt (5583683878263 / 32000000000000 : ℝ) := by
+    rw [show
+        (1000000000000000 / 417720146983263 : ℝ) = 1 / (417720146983263 / 1000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 1392400489944209 : ℝ) ≤
+      (177653194295087677753 / 1000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (5583683878263 / 32000000000000 : ℝ) ≤
+      (250000000000000 / 1392400489944209 : ℝ) := by
+    rw [show
+        (250000000000000 / 1392400489944209 : ℝ) =
+          (3 / 40) / (4177201469832627 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (3956884238851918576391 / 10000000000000000000000 : ℝ)) -
+        6 * (177653194295087677753 / 1000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (5583683878263 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (5583683878263 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((4177201469832627 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (3956884238851918576391 / 10000000000000000000000 : ℝ)) -
+          6 * (177653194295087677753 / 1000000000000000000000 : ℝ))) ≤
+      Real.sqrt (5583683878263 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (5583683878263 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (5583683878263 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_234 :
+    ((1623437368556860115687 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(17043559332240260533931 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2099193281346059 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (3974908332220430602867 / 10000000000000000000000 : ℝ)) -
+              6 * (1767753594973346520647 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (2820231956767 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (2820231956767 / 16000000000000 : ℝ)) =
+      Real.log (18820231956767 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((2820231956767 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (2910231956767 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1623437368556860115687 / 10000000000000000000000 : ℝ) ≤
+      Real.log (18820231956767 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (2910231956767 / 16000000000000 : ℝ) ≤
+      (-(17043559332240260533931 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (2910231956767 / 16000000000000 : ℝ) = (2910231956767 / 2000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2910231956767 / 2000000000000 : ℝ) ≤
+            (3750856076759739466069 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2099193281346059 / 5000000000000000 : ℝ) ≤ Real.sqrt (2820231956767 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (2820231956767 / 16000000000000 : ℝ) ≤ (4198386562692121 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (2820231956767 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau296 :
+    Real.arctan (4198386562692121 / 10000000000000000 : ℝ) ≤
+      (3974908332220430602867 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (3974908332220430602867 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 4198386562692121 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 4198386562692121 : ℝ) =
+              (4198386562692121 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau296)
+  have hz1 :
+    (10000000000000000 / 4198386562692121 : ℝ) ≤
+      1 / Real.sqrt (2820231956767 / 16000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 4198386562692121 : ℝ) =
+          1 / (4198386562692121 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 2099193281346059 : ℝ) ≤
+      (1767753594973346520647 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (2820231956767 / 16000000000000 : ℝ) ≤
+      (375000000000000 / 2099193281346059 : ℝ) := by
+    rw [show
+        (375000000000000 / 2099193281346059 : ℝ) =
+          (3 / 40) / (2099193281346059 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (3974908332220430602867 / 10000000000000000000000 : ℝ)) -
+        6 * (1767753594973346520647 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (2820231956767 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (2820231956767 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2099193281346059 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (3974908332220430602867 / 10000000000000000000000 : ℝ)) -
+          6 * (1767753594973346520647 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (2820231956767 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (2820231956767 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (2820231956767 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_235 :
+    ((8192554208785925893 / 50000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(8473239830903755215511 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2109732645385169 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1996407449528079098729 / 5000000000000000000000 : ℝ)) -
+              6 * (879552057096567979033 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (1139448789761 / 6400000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1139448789761 / 6400000000000 : ℝ)) =
+      Real.log (7539448789761 / 6400000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1139448789761 / 6400000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1175448789761 / 6400000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (8192554208785925893 / 50000000000000000000 : ℝ) ≤
+      Real.log (7539448789761 / 6400000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1175448789761 / 6400000000000 : ℝ) ≤
+      (-(8473239830903755215511 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (1175448789761 / 6400000000000 : ℝ) = (1175448789761 / 800000000000 : ℝ) / 2 ^ 3 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1175448789761 / 800000000000 : ℝ) ≤
+            (1923967873596244784489 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2109732645385169 / 5000000000000000 : ℝ) ≤ Real.sqrt (1139448789761 / 6400000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1139448789761 / 6400000000000 : ℝ) ≤ (4219465290770341 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1139448789761 / 6400000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau297 :
+    Real.arctan (4219465290770341 / 10000000000000000 : ℝ) ≤
+      (1996407449528079098729 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1996407449528079098729 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 4219465290770341 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 4219465290770341 : ℝ) =
+              (4219465290770341 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau297)
+  have hz1 :
+    (10000000000000000 / 4219465290770341 : ℝ) ≤
+      1 / Real.sqrt (1139448789761 / 6400000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 4219465290770341 : ℝ) =
+          1 / (4219465290770341 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 2109732645385169 : ℝ) ≤
+      (879552057096567979033 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1139448789761 / 6400000000000 : ℝ) ≤
+      (375000000000000 / 2109732645385169 : ℝ) := by
+    rw [show
+        (375000000000000 / 2109732645385169 : ℝ) =
+          (3 / 40) / (2109732645385169 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1996407449528079098729 / 5000000000000000000000 : ℝ)) -
+        6 * (879552057096567979033 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1139448789761 / 6400000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1139448789761 / 6400000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2109732645385169 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1996407449528079098729 / 5000000000000000000000 : ℝ)) -
+          6 * (879552057096567979033 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (1139448789761 / 6400000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1139448789761 / 6400000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1139448789761 / 6400000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_236 :
+    ((25836900440478381773 / 156250000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2106291672941955151379 / 1250000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((4240439240248289 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (4010605638583281302999 / 10000000000000000000000 : ℝ)) -
+              6 * (54705636809765470191 / 312500000000000000000 : ℝ)))) ≤
+      Vfield (1438505996019 / 8000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1438505996019 / 8000000000000 : ℝ)) =
+      Real.log (9438505996019 / 8000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1438505996019 / 8000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1483505996019 / 8000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (25836900440478381773 / 156250000000000000000 : ℝ) ≤
+      Real.log (9438505996019 / 8000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1483505996019 / 8000000000000 : ℝ) ≤
+      (-(2106291672941955151379 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (1483505996019 / 8000000000000 : ℝ) = (1483505996019 / 1000000000000 : ℝ) / 2 ^ 3 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1483505996019 / 1000000000000 : ℝ) ≤
+            (493010253183044848621 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (4240439240248289 / 10000000000000000 : ℝ) ≤ Real.sqrt (1438505996019 / 8000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1438505996019 / 8000000000000 : ℝ) ≤ (1060109810062073 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1438505996019 / 8000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau298 :
+    Real.arctan (1060109810062073 / 2500000000000000 : ℝ) ≤
+      (4010605638583281302999 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (4010605638583281302999 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (2500000000000000 / 1060109810062073 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 1060109810062073 : ℝ) =
+              (1060109810062073 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau298)
+  have hz1 :
+    (2500000000000000 / 1060109810062073 : ℝ) ≤
+      1 / Real.sqrt (1438505996019 / 8000000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 1060109810062073 : ℝ) = 1 / (1060109810062073 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 4240439240248289 : ℝ) ≤
+      (54705636809765470191 / 312500000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1438505996019 / 8000000000000 : ℝ) ≤
+      (750000000000000 / 4240439240248289 : ℝ) := by
+    rw [show
+        (750000000000000 / 4240439240248289 : ℝ) =
+          (3 / 40) / (4240439240248289 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (4010605638583281302999 / 10000000000000000000000 : ℝ)) -
+        6 * (54705636809765470191 / 312500000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1438505996019 / 8000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1438505996019 / 8000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((4240439240248289 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (4010605638583281302999 / 10000000000000000000000 : ℝ)) -
+          6 * (54705636809765470191 / 312500000000000000000 : ℝ))) ≤
+      Real.sqrt (1438505996019 / 8000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1438505996019 / 8000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1438505996019 / 8000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_237 :
+    ((1683595413202401175063 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(8330385197728652168027 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2141039477139623 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (4045846230365333599963 / 10000000000000000000000 : ℝ)) -
+              6 * (86694908410619904871 / 500000000000000000000 : ℝ)))) ≤
+      Vfield (2933792027309 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (2933792027309 / 16000000000000 : ℝ)) =
+      Real.log (18933792027309 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((2933792027309 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (3023792027309 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1683595413202401175063 / 10000000000000000000000 : ℝ) ≤
+      Real.log (18933792027309 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (3023792027309 / 16000000000000 : ℝ) ≤
+      (-(8330385197728652168027 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (3023792027309 / 16000000000000 : ℝ) = (3023792027309 / 2000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3023792027309 / 2000000000000 : ℝ) ≤
+            (2066822506771347831973 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2141039477139623 / 5000000000000000 : ℝ) ≤ Real.sqrt (2933792027309 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (2933792027309 / 16000000000000 : ℝ) ≤ (4282078954279249 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (2933792027309 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau299 :
+    Real.arctan (4282078954279249 / 10000000000000000 : ℝ) ≤
+      (4045846230365333599963 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (4045846230365333599963 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 4282078954279249 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 4282078954279249 : ℝ) =
+              (4282078954279249 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau299)
+  have hz1 :
+    (10000000000000000 / 4282078954279249 : ℝ) ≤
+      1 / Real.sqrt (2933792027309 / 16000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 4282078954279249 : ℝ) =
+          1 / (4282078954279249 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 2141039477139623 : ℝ) ≤
+      (86694908410619904871 / 500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (2933792027309 / 16000000000000 : ℝ) ≤
+      (375000000000000 / 2141039477139623 : ℝ) := by
+    rw [show
+        (375000000000000 / 2141039477139623 : ℝ) =
+          (3 / 40) / (2141039477139623 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (4045846230365333599963 / 10000000000000000000000 : ℝ)) -
+        6 * (86694908410619904871 / 500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (2933792027309 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (2933792027309 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2141039477139623 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (4045846230365333599963 / 10000000000000000000000 : ℝ)) -
+          6 * (86694908410619904871 / 500000000000000000000 : ℝ))) ≤
+      Real.sqrt (2933792027309 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (2933792027309 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (2933792027309 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_238 :
+    ((1713539265424779612481 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2059341758896377698357 / 1250000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2161658818542197 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2040321455583755349747 / 5000000000000000000000 : ℝ)) -
+              6 * (343536800730602077419 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (149528603129 / 800000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (149528603129 / 800000000000 : ℝ)) =
+      Real.log (949528603129 / 800000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((149528603129 / 800000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (154028603129 / 800000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1713539265424779612481 / 10000000000000000000000 : ℝ) ≤
+      Real.log (949528603129 / 800000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (154028603129 / 800000000000 : ℝ) ≤
+      (-(2059341758896377698357 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (154028603129 / 800000000000 : ℝ) = (154028603129 / 100000000000 : ℝ) / 2 ^ 3 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (154028603129 / 100000000000 : ℝ) ≤
+            (539960167228622301643 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2161658818542197 / 5000000000000000 : ℝ) ≤ Real.sqrt (149528603129 / 800000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (149528603129 / 800000000000 : ℝ) ≤ (4323317637084397 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (149528603129 / 800000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau300 :
+    Real.arctan (4323317637084397 / 10000000000000000 : ℝ) ≤
+      (2040321455583755349747 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (2040321455583755349747 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 4323317637084397 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 4323317637084397 : ℝ) =
+              (4323317637084397 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau300)
+  have hz1 :
+    (10000000000000000 / 4323317637084397 : ℝ) ≤
+      1 / Real.sqrt (149528603129 / 800000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 4323317637084397 : ℝ) =
+          1 / (4323317637084397 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 2161658818542197 : ℝ) ≤
+      (343536800730602077419 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (149528603129 / 800000000000 : ℝ) ≤
+      (375000000000000 / 2161658818542197 : ℝ) := by
+    rw [show
+        (375000000000000 / 2161658818542197 : ℝ) =
+          (3 / 40) / (2161658818542197 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2040321455583755349747 / 5000000000000000000000 : ℝ)) -
+        6 * (343536800730602077419 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (149528603129 / 800000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (149528603129 / 800000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2161658818542197 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2040321455583755349747 / 5000000000000000000000 : ℝ)) -
+          6 * (343536800730602077419 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (149528603129 / 800000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (149528603129 / 800000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (149528603129 / 800000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_239 :
+    ((871696860918730643327 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(16292095582403450641143 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2182083328585823 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (4115007895609233887117 / 10000000000000000000000 : ℝ)) -
+              6 * (1701916397919497671761 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (3047352097851 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (3047352097851 / 16000000000000 : ℝ)) =
+      Real.log (19047352097851 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((3047352097851 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (3137352097851 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (871696860918730643327 / 5000000000000000000000 : ℝ) ≤
+      Real.log (19047352097851 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (3137352097851 / 16000000000000 : ℝ) ≤
+      (-(16292095582403450641143 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (3137352097851 / 16000000000000 : ℝ) = (3137352097851 / 2000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3137352097851 / 2000000000000 : ℝ) ≤
+            (4502319826596549358857 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2182083328585823 / 5000000000000000 : ℝ) ≤ Real.sqrt (3047352097851 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (3047352097851 / 16000000000000 : ℝ) ≤ (4364166657171649 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (3047352097851 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau301 :
+    Real.arctan (4364166657171649 / 10000000000000000 : ℝ) ≤
+      (4115007895609233887117 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (4115007895609233887117 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 4364166657171649 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 4364166657171649 : ℝ) =
+              (4364166657171649 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau301)
+  have hz1 :
+    (10000000000000000 / 4364166657171649 : ℝ) ≤
+      1 / Real.sqrt (3047352097851 / 16000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 4364166657171649 : ℝ) =
+          1 / (4364166657171649 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 2182083328585823 : ℝ) ≤
+      (1701916397919497671761 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (3047352097851 / 16000000000000 : ℝ) ≤
+      (375000000000000 / 2182083328585823 : ℝ) := by
+    rw [show
+        (375000000000000 / 2182083328585823 : ℝ) =
+          (3 / 40) / (2182083328585823 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (4115007895609233887117 / 10000000000000000000000 : ℝ)) -
+        6 * (1701916397919497671761 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (3047352097851 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (3047352097851 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2182083328585823 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (4115007895609233887117 / 10000000000000000000000 : ℝ)) -
+          6 * (1701916397919497671761 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (3047352097851 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (3047352097851 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (3047352097851 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_240 :
+    ((443289828656272310679 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(4028183258387475971649 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((4404636855861387 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2074476423521889182987 / 5000000000000000000000 : ℝ)) -
+              6 * (421643805205669706173 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (1552066066561 / 8000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1552066066561 / 8000000000000 : ℝ)) =
+      Real.log (9552066066561 / 8000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1552066066561 / 8000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1597066066561 / 8000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (443289828656272310679 / 2500000000000000000000 : ℝ) ≤
+      Real.log (9552066066561 / 8000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1597066066561 / 8000000000000 : ℝ) ≤
+      (-(4028183258387475971649 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (1597066066561 / 8000000000000 : ℝ) = (1597066066561 / 1000000000000 : ℝ) / 2 ^ 3 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1597066066561 / 1000000000000 : ℝ) ≤
+            (1170420593862524028351 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (4404636855861387 / 10000000000000000 : ℝ) ≤ Real.sqrt (1552066066561 / 8000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1552066066561 / 8000000000000 : ℝ) ≤ (440463685586139 / 1000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1552066066561 / 8000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau302 :
+    Real.arctan (440463685586139 / 1000000000000000 : ℝ) ≤
+      (2074476423521889182987 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (2074476423521889182987 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (1000000000000000 / 440463685586139 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1000000000000000 / 440463685586139 : ℝ) =
+              (440463685586139 / 1000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau302)
+  have hz1 :
+    (1000000000000000 / 440463685586139 : ℝ) ≤
+      1 / Real.sqrt (1552066066561 / 8000000000000 : ℝ) := by
+    rw [show
+        (1000000000000000 / 440463685586139 : ℝ) = 1 / (440463685586139 / 1000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 1468212285287129 : ℝ) ≤
+      (421643805205669706173 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1552066066561 / 8000000000000 : ℝ) ≤
+      (250000000000000 / 1468212285287129 : ℝ) := by
+    rw [show
+        (250000000000000 / 1468212285287129 : ℝ) =
+          (3 / 40) / (4404636855861387 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2074476423521889182987 / 5000000000000000000000 : ℝ)) -
+        6 * (421643805205669706173 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1552066066561 / 8000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1552066066561 / 8000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((4404636855861387 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2074476423521889182987 / 5000000000000000000000 : ℝ)) -
+          6 * (421643805205669706173 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (1552066066561 / 8000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1552066066561 / 8000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1552066066561 / 8000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/V20.lean b/LeanPool/Zeta5Irrational/Table/V20.lean
new file mode 100644
index 0000000000..40cdf442e2
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/V20.lean
@@ -0,0 +1,1235 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Energy
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # V20: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_241 :
+    ((916213007214556584881 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(15763379946783387163977 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((280280107331961 / 625000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (526953344135841385621 / 1250000000000000000000 : ℝ)) -
+              6 * (828548887056965813683 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (201105762729 / 1000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (201105762729 / 1000000000000 : ℝ)) =
+      Real.log (1201105762729 / 1000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((201105762729 / 1000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (206730762729 / 1000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (916213007214556584881 / 5000000000000000000000 : ℝ) ≤
+      Real.log (1201105762729 / 1000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (206730762729 / 1000000000000 : ℝ) ≤
+      (-(15763379946783387163977 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (206730762729 / 1000000000000 : ℝ) = (206730762729 / 125000000000 : ℝ) / 2 ^ 3 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (206730762729 / 125000000000 : ℝ) ≤
+            (5031035462216612836023 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (280280107331961 / 625000000000000 : ℝ) ≤ Real.sqrt (201105762729 / 1000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (201105762729 / 1000000000000 : ℝ) ≤ (4484481717311379 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (201105762729 / 1000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau303 :
+    Real.arctan (4484481717311379 / 10000000000000000 : ℝ) ≤
+      (526953344135841385621 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (526953344135841385621 / 1250000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 4484481717311379 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 4484481717311379 : ℝ) =
+              (4484481717311379 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau303)
+  have hz1 :
+    (10000000000000000 / 4484481717311379 : ℝ) ≤
+      1 / Real.sqrt (201105762729 / 1000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 4484481717311379 : ℝ) =
+          1 / (4484481717311379 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (15625000000000 / 93426702443987 : ℝ) ≤
+      (828548887056965813683 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (201105762729 / 1000000000000 : ℝ) ≤
+      (15625000000000 / 93426702443987 : ℝ) := by
+    rw [show
+        (15625000000000 / 93426702443987 : ℝ) = (3 / 40) / (280280107331961 / 625000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (526953344135841385621 / 1250000000000000000000 : ℝ)) -
+        6 * (828548887056965813683 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (201105762729 / 1000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (201105762729 / 1000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((280280107331961 / 625000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (526953344135841385621 / 1250000000000000000000 : ℝ)) -
+          6 * (828548887056965813683 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (201105762729 / 1000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (201105762729 / 1000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (201105762729 / 1000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_242 :
+    ((925082402975123894543 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(978797156642601188271 / 625000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((4508195537539797 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (132354768328260144823 / 312500000000000000000 : ℝ)) -
+              6 * (1648538651467957972391 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (3251812320751 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (3251812320751 / 16000000000000 : ℝ)) =
+      Real.log (19251812320751 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((3251812320751 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (3341812320751 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (925082402975123894543 / 5000000000000000000000 : ℝ) ≤
+      Real.log (19251812320751 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (3341812320751 / 16000000000000 : ℝ) ≤
+      (-(978797156642601188271 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (3341812320751 / 16000000000000 : ℝ) = (3341812320751 / 2000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3341812320751 / 2000000000000 : ℝ) ≤
+            (320853806419898811729 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (4508195537539797 / 10000000000000000 : ℝ) ≤ Real.sqrt (3251812320751 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (3251812320751 / 16000000000000 : ℝ) ≤ (22540977687699 / 50000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (3251812320751 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau304 :
+    Real.arctan (22540977687699 / 50000000000000 : ℝ) ≤
+      (132354768328260144823 / 312500000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (132354768328260144823 / 312500000000000000000 : ℝ)) ≤
+      Real.arctan (50000000000000 / 22540977687699 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (50000000000000 / 22540977687699 : ℝ) =
+              (22540977687699 / 50000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau304)
+  have hz1 :
+    (50000000000000 / 22540977687699 : ℝ) ≤ 1 / Real.sqrt (3251812320751 / 16000000000000 : ℝ) := by
+    rw [show (50000000000000 / 22540977687699 : ℝ) = 1 / (22540977687699 / 50000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 1502731845846599 : ℝ) ≤
+      (1648538651467957972391 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (3251812320751 / 16000000000000 : ℝ) ≤
+      (250000000000000 / 1502731845846599 : ℝ) := by
+    rw [show
+        (250000000000000 / 1502731845846599 : ℝ) =
+          (3 / 40) / (4508195537539797 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (132354768328260144823 / 312500000000000000000 : ℝ)) -
+        6 * (1648538651467957972391 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (3251812320751 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (3251812320751 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((4508195537539797 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (132354768328260144823 / 312500000000000000000 : ℝ)) -
+          6 * (1648538651467957972391 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (3251812320751 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (3251812320751 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (3251812320751 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_243 :
+    ((233484023338699112559 / 1250000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(15559171574152312277573 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((906357054068373 / 2000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2127470226874638958149 / 5000000000000000000000 : ℝ)) -
+              6 * (205013851103837126691 / 1250000000000000000000 : ℝ)))) ≤
+      Vfield (1642966218919 / 8000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1642966218919 / 8000000000000 : ℝ)) =
+      Real.log (9642966218919 / 8000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1642966218919 / 8000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1687966218919 / 8000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (233484023338699112559 / 1250000000000000000000 : ℝ) ≤
+      Real.log (9642966218919 / 8000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1687966218919 / 8000000000000 : ℝ) ≤
+      (-(15559171574152312277573 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (1687966218919 / 8000000000000 : ℝ) = (1687966218919 / 1000000000000 : ℝ) / 2 ^ 3 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1687966218919 / 1000000000000 : ℝ) ≤
+            (5235243834847687722427 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (906357054068373 / 2000000000000000 : ℝ) ≤ Real.sqrt (1642966218919 / 8000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1642966218919 / 8000000000000 : ℝ) ≤ (1132946317585467 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1642966218919 / 8000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau305 :
+    Real.arctan (1132946317585467 / 2500000000000000 : ℝ) ≤
+      (2127470226874638958149 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (2127470226874638958149 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (2500000000000000 / 1132946317585467 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 1132946317585467 : ℝ) =
+              (1132946317585467 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau305)
+  have hz1 :
+    (2500000000000000 / 1132946317585467 : ℝ) ≤
+      1 / Real.sqrt (1642966218919 / 8000000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 1132946317585467 : ℝ) = 1 / (1132946317585467 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (50000000000000 / 302119018022791 : ℝ) ≤
+      (205013851103837126691 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1642966218919 / 8000000000000 : ℝ) ≤
+      (50000000000000 / 302119018022791 : ℝ) := by
+    rw [show
+        (50000000000000 / 302119018022791 : ℝ) =
+          (3 / 40) / (906357054068373 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2127470226874638958149 / 5000000000000000000000 : ℝ)) -
+        6 * (205013851103837126691 / 1250000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1642966218919 / 8000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1642966218919 / 8000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((906357054068373 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2127470226874638958149 / 5000000000000000000000 : ℝ)) -
+          6 * (205013851103837126691 / 1250000000000000000000 : ℝ))) ≤
+      Real.sqrt (1642966218919 / 8000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1642966218919 / 8000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1642966218919 / 8000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_244 :
+    ((942774133875520337323 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(15458610182818172849901 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((569406605438411 / 1250000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (427439244048937257737 / 1000000000000000000000 : ℝ)) -
+              6 * (326362184796567935759 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (132802102197 / 640000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (132802102197 / 640000000000 : ℝ)) =
+      Real.log (772802102197 / 640000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((132802102197 / 640000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (136402102197 / 640000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (942774133875520337323 / 5000000000000000000000 : ℝ) ≤
+      Real.log (772802102197 / 640000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (136402102197 / 640000000000 : ℝ) ≤
+      (-(15458610182818172849901 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (136402102197 / 640000000000 : ℝ) = (136402102197 / 80000000000 : ℝ) / 2 ^ 3 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (136402102197 / 80000000000 : ℝ) ≤
+            (5335805226181827150099 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (569406605438411 / 1250000000000000 : ℝ) ≤ Real.sqrt (132802102197 / 640000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (132802102197 / 640000000000 : ℝ) ≤ (4555252843507291 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (132802102197 / 640000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau306 :
+    Real.arctan (4555252843507291 / 10000000000000000 : ℝ) ≤
+      (427439244048937257737 / 1000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (427439244048937257737 / 1000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 4555252843507291 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 4555252843507291 : ℝ) =
+              (4555252843507291 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau306)
+  have hz1 :
+    (10000000000000000 / 4555252843507291 : ℝ) ≤
+      1 / Real.sqrt (132802102197 / 640000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 4555252843507291 : ℝ) =
+          1 / (4555252843507291 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (93750000000000 / 569406605438411 : ℝ) ≤
+      (326362184796567935759 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (132802102197 / 640000000000 : ℝ) ≤
+      (93750000000000 / 569406605438411 : ℝ) := by
+    rw [show
+        (93750000000000 / 569406605438411 : ℝ) =
+          (3 / 40) / (569406605438411 / 1250000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (427439244048937257737 / 1000000000000000000000 : ℝ)) -
+        6 * (326362184796567935759 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (132802102197 / 640000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (132802102197 / 640000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((569406605438411 / 1250000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (427439244048937257737 / 1000000000000000000000 : ℝ)) -
+          6 * (326362184796567935759 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (132802102197 / 640000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (132802102197 / 640000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (132802102197 / 640000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_245 :
+    ((473593651354481801167 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(15408706184512598421353 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((4566941409102827 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1071017029048578874257 / 2500000000000000000000 : ℝ)) -
+              6 * (325541591908918348673 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (6674225226937 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (6674225226937 / 32000000000000 : ℝ)) =
+      Real.log (38674225226937 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((6674225226937 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (6854225226937 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (473593651354481801167 / 2500000000000000000000 : ℝ) ≤
+      Real.log (38674225226937 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (6854225226937 / 32000000000000 : ℝ) ≤
+      (-(15408706184512598421353 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (6854225226937 / 32000000000000 : ℝ) = (6854225226937 / 4000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6854225226937 / 4000000000000 : ℝ) ≤
+            (5385709224487401578647 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (4566941409102827 / 10000000000000000 : ℝ) ≤ Real.sqrt (6674225226937 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (6674225226937 / 32000000000000 : ℝ) ≤ (456694140910283 / 1000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (6674225226937 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau307 :
+    Real.arctan (456694140910283 / 1000000000000000 : ℝ) ≤
+      (1071017029048578874257 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1071017029048578874257 / 2500000000000000000000 : ℝ)) ≤
+      Real.arctan (1000000000000000 / 456694140910283 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1000000000000000 / 456694140910283 : ℝ) =
+              (456694140910283 / 1000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau307)
+  have hz1 :
+    (1000000000000000 / 456694140910283 : ℝ) ≤
+      1 / Real.sqrt (6674225226937 / 32000000000000 : ℝ) := by
+    rw [show
+        (1000000000000000 / 456694140910283 : ℝ) = 1 / (456694140910283 / 1000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 4566941409102827 : ℝ) ≤
+      (325541591908918348673 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (6674225226937 / 32000000000000 : ℝ) ≤
+      (750000000000000 / 4566941409102827 : ℝ) := by
+    rw [show
+        (750000000000000 / 4566941409102827 : ℝ) =
+          (3 / 40) / (4566941409102827 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1071017029048578874257 / 2500000000000000000000 : ℝ)) -
+        6 * (325541591908918348673 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (6674225226937 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (6674225226937 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((4566941409102827 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1071017029048578874257 / 2500000000000000000000 : ℝ)) -
+          6 * (325541591908918348673 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (6674225226937 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (6674225226937 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (6674225226937 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_246 :
+    ((380638631906135117007 / 2000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(3071809998195709492653 / 2000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2289300067710379 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (268356911279549160589 / 625000000000000000000 : ℝ)) -
+              6 * (1623635791236049692747 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (838543168003 / 4000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (838543168003 / 4000000000000 : ℝ)) =
+      Real.log (4838543168003 / 4000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((838543168003 / 4000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (861043168003 / 4000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (380638631906135117007 / 2000000000000000000000 : ℝ) ≤
+      Real.log (4838543168003 / 4000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (861043168003 / 4000000000000 : ℝ) ≤
+      (-(3071809998195709492653 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (861043168003 / 4000000000000 : ℝ) = (861043168003 / 500000000000 : ℝ) / 2 ^ 3 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (861043168003 / 500000000000 : ℝ) ≤
+            (1087073083604290507347 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2289300067710379 / 5000000000000000 : ℝ) ≤ Real.sqrt (838543168003 / 4000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (838543168003 / 4000000000000 : ℝ) ≤ (4578600135420761 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (838543168003 / 4000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau308 :
+    Real.arctan (4578600135420761 / 10000000000000000 : ℝ) ≤
+      (268356911279549160589 / 625000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (268356911279549160589 / 625000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 4578600135420761 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 4578600135420761 : ℝ) =
+              (4578600135420761 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau308)
+  have hz1 :
+    (10000000000000000 / 4578600135420761 : ℝ) ≤
+      1 / Real.sqrt (838543168003 / 4000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 4578600135420761 : ℝ) =
+          1 / (4578600135420761 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 2289300067710379 : ℝ) ≤
+      (1623635791236049692747 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (838543168003 / 4000000000000 : ℝ) ≤
+      (375000000000000 / 2289300067710379 : ℝ) := by
+    rw [show
+        (375000000000000 / 2289300067710379 : ℝ) =
+          (3 / 40) / (2289300067710379 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (268356911279549160589 / 625000000000000000000 : ℝ)) -
+        6 * (1623635791236049692747 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (838543168003 / 4000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (838543168003 / 4000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2289300067710379 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (268356911279549160589 / 625000000000000000000 : ℝ)) -
+          6 * (1623635791236049692747 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (838543168003 / 4000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (838543168003 / 4000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (838543168003 / 4000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_247 :
+    ((956001971902565436247 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(3061927830671543104789 / 2000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1147557312456873 / 2500000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1075830019932161256039 / 2500000000000000000000 : ℝ)) -
+              6 * (1619594035700487665621 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (6742465461111 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (6742465461111 / 32000000000000 : ℝ)) =
+      Real.log (38742465461111 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((6742465461111 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (6922465461111 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (956001971902565436247 / 5000000000000000000000 : ℝ) ≤
+      Real.log (38742465461111 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (6922465461111 / 32000000000000 : ℝ) ≤
+      (-(3061927830671543104789 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (6922465461111 / 32000000000000 : ℝ) = (6922465461111 / 4000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6922465461111 / 4000000000000 : ℝ) ≤
+            (1096955251128456895211 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1147557312456873 / 2500000000000000 : ℝ) ≤ Real.sqrt (6742465461111 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (6742465461111 / 32000000000000 : ℝ) ≤ (918045849965499 / 2000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (6742465461111 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau309 :
+    Real.arctan (918045849965499 / 2000000000000000 : ℝ) ≤
+      (1075830019932161256039 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1075830019932161256039 / 2500000000000000000000 : ℝ)) ≤
+      Real.arctan (2000000000000000 / 918045849965499 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2000000000000000 / 918045849965499 : ℝ) =
+              (918045849965499 / 2000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau309)
+  have hz1 :
+    (2000000000000000 / 918045849965499 : ℝ) ≤
+      1 / Real.sqrt (6742465461111 / 32000000000000 : ℝ) := by
+    rw [show
+        (2000000000000000 / 918045849965499 : ℝ) = 1 / (918045849965499 / 2000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (62500000000000 / 382519104152291 : ℝ) ≤
+      (1619594035700487665621 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (6742465461111 / 32000000000000 : ℝ) ≤
+      (62500000000000 / 382519104152291 : ℝ) := by
+    rw [show
+        (62500000000000 / 382519104152291 : ℝ) =
+          (3 / 40) / (1147557312456873 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1075830019932161256039 / 2500000000000000000000 : ℝ)) -
+        6 * (1619594035700487665621 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (6742465461111 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (6742465461111 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1147557312456873 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1075830019932161256039 / 2500000000000000000000 : ℝ)) -
+          6 * (1619594035700487665621 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (6742465461111 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (6742465461111 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (6742465461111 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_248 :
+    ((30012608936264309779 / 156250000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(15260471258913580618157 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((4601828976816581 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (215644842867158688389 / 500000000000000000000 : ℝ)) -
+              6 * (64623292649177857249 / 400000000000000000000 : ℝ)))) ≤
+      Vfield (3388292789099 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (3388292789099 / 16000000000000 : ℝ)) =
+      Real.log (19388292789099 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((3388292789099 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (3478292789099 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (30012608936264309779 / 156250000000000000000 : ℝ) ≤
+      Real.log (19388292789099 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (3478292789099 / 16000000000000 : ℝ) ≤
+      (-(15260471258913580618157 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (3478292789099 / 16000000000000 : ℝ) = (3478292789099 / 2000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3478292789099 / 2000000000000 : ℝ) ≤
+            (5533944150086419381843 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (4601828976816581 / 10000000000000000 : ℝ) ≤ Real.sqrt (3388292789099 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (3388292789099 / 16000000000000 : ℝ) ≤ (575228622102073 / 1250000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (3388292789099 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau310 :
+    Real.arctan (575228622102073 / 1250000000000000 : ℝ) ≤
+      (215644842867158688389 / 500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (215644842867158688389 / 500000000000000000000 : ℝ)) ≤
+      Real.arctan (1250000000000000 / 575228622102073 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1250000000000000 / 575228622102073 : ℝ) =
+              (575228622102073 / 1250000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau310)
+  have hz1 :
+    (1250000000000000 / 575228622102073 : ℝ) ≤
+      1 / Real.sqrt (3388292789099 / 16000000000000 : ℝ) := by
+    rw [show
+        (1250000000000000 / 575228622102073 : ℝ) = 1 / (575228622102073 / 1250000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 4601828976816581 : ℝ) ≤
+      (64623292649177857249 / 400000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (3388292789099 / 16000000000000 : ℝ) ≤
+      (750000000000000 / 4601828976816581 : ℝ) := by
+    rw [show
+        (750000000000000 / 4601828976816581 : ℝ) =
+          (3 / 40) / (4601828976816581 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (215644842867158688389 / 500000000000000000000 : ℝ)) -
+        6 * (64623292649177857249 / 400000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (3388292789099 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (3388292789099 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((4601828976816581 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (215644842867158688389 / 500000000000000000000 : ℝ)) -
+          6 * (64623292649177857249 / 400000000000000000000 : ℝ))) ≤
+      Real.sqrt (3388292789099 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (3388292789099 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (3388292789099 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_249 :
+    ((1929602257521558340209 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1901442991290557573503 / 1250000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1153349884514821 / 2500000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (86448823074548479773 / 200000000000000000000 : ℝ)) -
+              6 * (805800131307608396503 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (1362141139057 / 6400000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1362141139057 / 6400000000000 : ℝ)) =
+      Real.log (7762141139057 / 6400000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1362141139057 / 6400000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1398141139057 / 6400000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1929602257521558340209 / 10000000000000000000000 : ℝ) ≤
+      Real.log (7762141139057 / 6400000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1398141139057 / 6400000000000 : ℝ) ≤
+      (-(1901442991290557573503 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (1398141139057 / 6400000000000 : ℝ) = (1398141139057 / 800000000000 : ℝ) / 2 ^ 3 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1398141139057 / 800000000000 : ℝ) ≤
+            (697858934834442426497 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1153349884514821 / 2500000000000000 : ℝ) ≤ Real.sqrt (1362141139057 / 6400000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1362141139057 / 6400000000000 : ℝ) ≤ (4613399538059287 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1362141139057 / 6400000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau311 :
+    Real.arctan (4613399538059287 / 10000000000000000 : ℝ) ≤
+      (86448823074548479773 / 200000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (86448823074548479773 / 200000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 4613399538059287 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 4613399538059287 : ℝ) =
+              (4613399538059287 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau311)
+  have hz1 :
+    (10000000000000000 / 4613399538059287 : ℝ) ≤
+      1 / Real.sqrt (1362141139057 / 6400000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 4613399538059287 : ℝ) =
+          1 / (4613399538059287 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (187500000000000 / 1153349884514821 : ℝ) ≤
+      (805800131307608396503 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1362141139057 / 6400000000000 : ℝ) ≤
+      (187500000000000 / 1153349884514821 : ℝ) := by
+    rw [show
+        (187500000000000 / 1153349884514821 : ℝ) =
+          (3 / 40) / (1153349884514821 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (86448823074548479773 / 200000000000000000000 : ℝ)) -
+        6 * (805800131307608396503 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1362141139057 / 6400000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1362141139057 / 6400000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1153349884514821 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (86448823074548479773 / 200000000000000000000 : ℝ)) -
+          6 * (805800131307608396503 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (1362141139057 / 6400000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1362141139057 / 6400000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1362141139057 / 6400000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_250 :
+    ((484597453553654580303 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(3790713706248445209689 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((924988230490799 / 2000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (4331953206373391881623 / 10000000000000000000000 : ℝ)) -
+              6 * (1607647511007304005851 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (1711206453093 / 8000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1711206453093 / 8000000000000 : ℝ)) =
+      Real.log (9711206453093 / 8000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1711206453093 / 8000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1756206453093 / 8000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (484597453553654580303 / 2500000000000000000000 : ℝ) ≤
+      Real.log (9711206453093 / 8000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1756206453093 / 8000000000000 : ℝ) ≤
+      (-(3790713706248445209689 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (1756206453093 / 8000000000000 : ℝ) = (1756206453093 / 1000000000000 : ℝ) / 2 ^ 3 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1756206453093 / 1000000000000 : ℝ) ≤
+            (1407890146001554790311 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (924988230490799 / 2000000000000000 : ℝ) ≤ Real.sqrt (1711206453093 / 8000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1711206453093 / 8000000000000 : ℝ) ≤ (2312470576226999 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1711206453093 / 8000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau312 :
+    Real.arctan (2312470576226999 / 5000000000000000 : ℝ) ≤
+      (4331953206373391881623 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (4331953206373391881623 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (5000000000000000 / 2312470576226999 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 2312470576226999 : ℝ) =
+              (2312470576226999 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau312)
+  have hz1 :
+    (5000000000000000 / 2312470576226999 : ℝ) ≤
+      1 / Real.sqrt (1711206453093 / 8000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 2312470576226999 : ℝ) = 1 / (2312470576226999 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (150000000000000 / 924988230490799 : ℝ) ≤
+      (1607647511007304005851 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1711206453093 / 8000000000000 : ℝ) ≤
+      (150000000000000 / 924988230490799 : ℝ) := by
+    rw [show
+        (150000000000000 / 924988230490799 : ℝ) =
+          (3 / 40) / (924988230490799 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (4331953206373391881623 / 10000000000000000000000 : ℝ)) -
+        6 * (1607647511007304005851 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1711206453093 / 8000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1711206453093 / 8000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((924988230490799 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (4331953206373391881623 / 10000000000000000000000 : ℝ)) -
+          6 * (1607647511007304005851 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (1711206453093 / 8000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1711206453093 / 8000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1711206453093 / 8000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_251 :
+    ((971390349231681716199 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(15138598883318111695953 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((4630701172242809 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (4336697214724926254481 / 10000000000000000000000 : ℝ)) -
+              6 * (321136402291257269187 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (13723771741831 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (13723771741831 / 64000000000000 : ℝ)) =
+      Real.log (77723771741831 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((13723771741831 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (14083771741831 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (971390349231681716199 / 5000000000000000000000 : ℝ) ≤
+      Real.log (77723771741831 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (14083771741831 / 64000000000000 : ℝ) ≤
+      (-(15138598883318111695953 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (14083771741831 / 64000000000000 : ℝ) = (14083771741831 / 8000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14083771741831 / 8000000000000 : ℝ) ≤
+            (5655816525681888304047 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (4630701172242809 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (13723771741831 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (13723771741831 / 64000000000000 : ℝ) ≤ (1157675293060703 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (13723771741831 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau313 :
+    Real.arctan (1157675293060703 / 2500000000000000 : ℝ) ≤
+      (4336697214724926254481 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (4336697214724926254481 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (2500000000000000 / 1157675293060703 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 1157675293060703 : ℝ) =
+              (1157675293060703 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau313)
+  have hz1 :
+    (2500000000000000 / 1157675293060703 : ℝ) ≤
+      1 / Real.sqrt (13723771741831 / 64000000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 1157675293060703 : ℝ) = 1 / (1157675293060703 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 4630701172242809 : ℝ) ≤
+      (321136402291257269187 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (13723771741831 / 64000000000000 : ℝ) ≤
+      (750000000000000 / 4630701172242809 : ℝ) := by
+    rw [show
+        (750000000000000 / 4630701172242809 : ℝ) =
+          (3 / 40) / (4630701172242809 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (4336697214724926254481 / 10000000000000000000000 : ℝ)) -
+        6 * (321136402291257269187 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (13723771741831 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (13723771741831 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((4630701172242809 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (4336697214724926254481 / 10000000000000000000000 : ℝ)) -
+          6 * (321136402291257269187 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (13723771741831 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (13723771741831 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (13723771741831 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_252 :
+    ((486792413892953500607 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(944650102148565756543 / 625000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((4636454036174559 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (4341433249904055055669 / 10000000000000000000000 : ℝ)) -
+              6 * (400930925943187151183 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (6878945929459 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (6878945929459 / 32000000000000 : ℝ)) =
+      Real.log (38878945929459 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((6878945929459 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (7058945929459 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (486792413892953500607 / 2500000000000000000000 : ℝ) ≤
+      Real.log (38878945929459 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (7058945929459 / 32000000000000 : ℝ) ≤
+      (-(944650102148565756543 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (7058945929459 / 32000000000000 : ℝ) = (7058945929459 / 4000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7058945929459 / 4000000000000 : ℝ) ≤
+            (355000860913934243457 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (4636454036174559 / 10000000000000000 : ℝ) ≤ Real.sqrt (6878945929459 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (6878945929459 / 32000000000000 : ℝ) ≤ (2318227018087281 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (6878945929459 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau314 :
+    Real.arctan (2318227018087281 / 5000000000000000 : ℝ) ≤
+      (4341433249904055055669 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (4341433249904055055669 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (5000000000000000 / 2318227018087281 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 2318227018087281 : ℝ) =
+              (2318227018087281 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau314)
+  have hz1 :
+    (5000000000000000 / 2318227018087281 : ℝ) ≤
+      1 / Real.sqrt (6878945929459 / 32000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 2318227018087281 : ℝ) = 1 / (2318227018087281 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 1545484678724853 : ℝ) ≤
+      (400930925943187151183 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (6878945929459 / 32000000000000 : ℝ) ≤
+      (250000000000000 / 1545484678724853 : ℝ) := by
+    rw [show
+        (250000000000000 / 1545484678724853 : ℝ) =
+          (3 / 40) / (4636454036174559 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (4341433249904055055669 / 10000000000000000000000 : ℝ)) -
+        6 * (400930925943187151183 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (6878945929459 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (6878945929459 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((4636454036174559 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (4341433249904055055669 / 10000000000000000000000 : ℝ)) -
+          6 * (400930925943187151183 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (6878945929459 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (6878945929459 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (6878945929459 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/V21.lean b/LeanPool/Zeta5Irrational/Table/V21.lean
new file mode 100644
index 0000000000..136f0c5617
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/V21.lean
@@ -0,0 +1,1241 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Energy
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # V21: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_253 :
+    ((243944585903856897783 / 1250000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(3018052558963085022733 / 2000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((580274971356627 / 1250000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (4346161340798394868929 / 10000000000000000000000 : ℝ)) -
+              6 * (100110784012609814337 / 625000000000000000000 : ℝ)))) ≤
+      Vfield (2758402395201 / 12800000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (2758402395201 / 12800000000000 : ℝ)) =
+      Real.log (15558402395201 / 12800000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((2758402395201 / 12800000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (2830402395201 / 12800000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (243944585903856897783 / 1250000000000000000000 : ℝ) ≤
+      Real.log (15558402395201 / 12800000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (2830402395201 / 12800000000000 : ℝ) ≤
+      (-(3018052558963085022733 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (2830402395201 / 12800000000000 : ℝ) = (2830402395201 / 1600000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2830402395201 / 1600000000000 : ℝ) ≤
+            (1140830522836914977267 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (580274971356627 / 1250000000000000 : ℝ) ≤ Real.sqrt (2758402395201 / 12800000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (2758402395201 / 12800000000000 : ℝ) ≤ (4642199770853019 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (2758402395201 / 12800000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau315 :
+    Real.arctan (4642199770853019 / 10000000000000000 : ℝ) ≤
+      (4346161340798394868929 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (4346161340798394868929 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 4642199770853019 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 4642199770853019 : ℝ) =
+              (4642199770853019 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau315)
+  have hz1 :
+    (10000000000000000 / 4642199770853019 : ℝ) ≤
+      1 / Real.sqrt (2758402395201 / 12800000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 4642199770853019 : ℝ) =
+          1 / (4642199770853019 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (31250000000000 / 193424990452209 : ℝ) ≤
+      (100110784012609814337 / 625000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (2758402395201 / 12800000000000 : ℝ) ≤
+      (31250000000000 / 193424990452209 : ℝ) := by
+    rw [show
+        (31250000000000 / 193424990452209 : ℝ) =
+          (3 / 40) / (580274971356627 / 1250000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (4346161340798394868929 / 10000000000000000000000 : ℝ)) -
+        6 * (100110784012609814337 / 625000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (2758402395201 / 12800000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (2758402395201 / 12800000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((580274971356627 / 1250000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (4346161340798394868929 / 10000000000000000000000 : ℝ)) -
+          6 * (100110784012609814337 / 625000000000000000000 : ℝ))) ≤
+      Real.sqrt (2758402395201 / 12800000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (2758402395201 / 12800000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (2758402395201 / 12800000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_254 :
+    ((488985448782286927153 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(15066182083325074857749 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2323969201358757 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (4350881516122305889661 / 10000000000000000000000 : ℝ)) -
+              6 * (1599828489360179180629 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (3456533023273 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (3456533023273 / 16000000000000 : ℝ)) =
+      Real.log (19456533023273 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((3456533023273 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (3546533023273 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (488985448782286927153 / 2500000000000000000000 : ℝ) ≤
+      Real.log (19456533023273 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (3546533023273 / 16000000000000 : ℝ) ≤
+      (-(15066182083325074857749 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (3546533023273 / 16000000000000 : ℝ) = (3546533023273 / 2000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3546533023273 / 2000000000000 : ℝ) ≤
+            (5728233325674925142251 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2323969201358757 / 5000000000000000 : ℝ) ≤ Real.sqrt (3456533023273 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (3456533023273 / 16000000000000 : ℝ) ≤ (4647938402717517 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (3456533023273 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau316 :
+    Real.arctan (4647938402717517 / 10000000000000000 : ℝ) ≤
+      (4350881516122305889661 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (4350881516122305889661 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 4647938402717517 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 4647938402717517 : ℝ) =
+              (4647938402717517 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau316)
+  have hz1 :
+    (10000000000000000 / 4647938402717517 : ℝ) ≤
+      1 / Real.sqrt (3456533023273 / 16000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 4647938402717517 : ℝ) =
+          1 / (4647938402717517 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (125000000000000 / 774656400452919 : ℝ) ≤
+      (1599828489360179180629 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (3456533023273 / 16000000000000 : ℝ) ≤
+      (125000000000000 / 774656400452919 : ℝ) := by
+    rw [show
+        (125000000000000 / 774656400452919 : ℝ) =
+          (3 / 40) / (2323969201358757 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (4350881516122305889661 / 10000000000000000000000 : ℝ)) -
+        6 * (1599828489360179180629 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (3456533023273 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (3456533023273 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2323969201358757 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (4350881516122305889661 / 10000000000000000000000 : ℝ)) -
+          6 * (1599828489360179180629 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (3456533023273 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (3456533023273 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (3456533023273 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_255 :
+    ((1960324980953131803927 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1504215922062519674107 / 1000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((290854372377773 / 625000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (544449225552295190347 / 1250000000000000000000 : ℝ)) -
+              6 * (1597891496232632013767 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (13860252210179 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (13860252210179 / 64000000000000 : ℝ)) =
+      Real.log (77860252210179 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((13860252210179 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (14220252210179 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1960324980953131803927 / 10000000000000000000000 : ℝ) ≤
+      Real.log (77860252210179 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (14220252210179 / 64000000000000 : ℝ) ≤
+      (-(1504215922062519674107 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (14220252210179 / 64000000000000 : ℝ) = (14220252210179 / 8000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14220252210179 / 8000000000000 : ℝ) ≤
+            (575225618837480325893 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (290854372377773 / 625000000000000 : ℝ) ≤ Real.sqrt (13860252210179 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (13860252210179 / 64000000000000 : ℝ) ≤
+      (4653669958044371 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (13860252210179 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau317 :
+    Real.arctan (4653669958044371 / 10000000000000000 : ℝ) ≤
+      (544449225552295190347 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (544449225552295190347 / 1250000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 4653669958044371 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 4653669958044371 : ℝ) =
+              (4653669958044371 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau317)
+  have hz1 :
+    (10000000000000000 / 4653669958044371 : ℝ) ≤
+      1 / Real.sqrt (13860252210179 / 64000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 4653669958044371 : ℝ) =
+          1 / (4653669958044371 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (46875000000000 / 290854372377773 : ℝ) ≤
+      (1597891496232632013767 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (13860252210179 / 64000000000000 : ℝ) ≤
+      (46875000000000 / 290854372377773 : ℝ) := by
+    rw [show
+        (46875000000000 / 290854372377773 : ℝ) =
+          (3 / 40) / (290854372377773 / 625000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (544449225552295190347 / 1250000000000000000000 : ℝ)) -
+        6 * (1597891496232632013767 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (13860252210179 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (13860252210179 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((290854372377773 / 625000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (544449225552295190347 / 1250000000000000000000 : ℝ)) -
+          6 * (1597891496232632013767 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (13860252210179 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (13860252210179 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (13860252210179 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_256 :
+    ((982353123193515533941 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(7509096964721451375789 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2329697231474141 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (872059646811761308523 / 2000000000000000000000 : ℝ)) -
+              6 * (1595961522167481885227 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (6947186163633 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (6947186163633 / 32000000000000 : ℝ)) =
+      Real.log (38947186163633 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((6947186163633 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (7127186163633 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (982353123193515533941 / 5000000000000000000000 : ℝ) ≤
+      Real.log (38947186163633 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (7127186163633 / 32000000000000 : ℝ) ≤
+      (-(7509096964721451375789 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (7127186163633 / 32000000000000 : ℝ) = (7127186163633 / 4000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7127186163633 / 4000000000000 : ℝ) ≤
+            (2888110739778548624211 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2329697231474141 / 5000000000000000 : ℝ) ≤ Real.sqrt (6947186163633 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (6947186163633 / 32000000000000 : ℝ) ≤ (931878892589657 / 2000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (6947186163633 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau318 :
+    Real.arctan (931878892589657 / 2000000000000000 : ℝ) ≤
+      (872059646811761308523 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (872059646811761308523 / 2000000000000000000000 : ℝ)) ≤
+      Real.arctan (2000000000000000 / 931878892589657 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2000000000000000 / 931878892589657 : ℝ) =
+              (931878892589657 / 2000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau318)
+  have hz1 :
+    (2000000000000000 / 931878892589657 : ℝ) ≤
+      1 / Real.sqrt (6947186163633 / 32000000000000 : ℝ) := by
+    rw [show
+        (2000000000000000 / 931878892589657 : ℝ) = 1 / (931878892589657 / 2000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 2329697231474141 : ℝ) ≤
+      (1595961522167481885227 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (6947186163633 / 32000000000000 : ℝ) ≤
+      (375000000000000 / 2329697231474141 : ℝ) := by
+    rw [show
+        (375000000000000 / 2329697231474141 : ℝ) =
+          (3 / 40) / (2329697231474141 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (872059646811761308523 / 2000000000000000000000 : ℝ)) -
+        6 * (1595961522167481885227 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (6947186163633 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (6947186163633 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2329697231474141 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (872059646811761308523 / 2000000000000000000000 : ℝ)) -
+          6 * (1595961522167481885227 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (6947186163633 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (6947186163633 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (6947186163633 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_257 :
+    ((246135699139107045039 / 1250000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(14994285934494019374537 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((4665111943383733 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (17459979332987969381 / 40000000000000000000 : ℝ)) -
+              6 * (398509631218225111797 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (13928492444353 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (13928492444353 / 64000000000000 : ℝ)) =
+      Real.log (77928492444353 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((13928492444353 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (14288492444353 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (246135699139107045039 / 1250000000000000000000 : ℝ) ≤
+      Real.log (77928492444353 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (14288492444353 / 64000000000000 : ℝ) ≤
+      (-(14994285934494019374537 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (14288492444353 / 64000000000000 : ℝ) = (14288492444353 / 8000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14288492444353 / 8000000000000 : ℝ) ≤
+            (5800129474505980625463 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (4665111943383733 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (13928492444353 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (13928492444353 / 64000000000000 : ℝ) ≤ (583138992922967 / 1250000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (13928492444353 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau319 :
+    Real.arctan (583138992922967 / 1250000000000000 : ℝ) ≤
+      (17459979332987969381 / 40000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (17459979332987969381 / 40000000000000000000 : ℝ)) ≤
+      Real.arctan (1250000000000000 / 583138992922967 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1250000000000000 / 583138992922967 : ℝ) =
+              (583138992922967 / 1250000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau319)
+  have hz1 :
+    (1250000000000000 / 583138992922967 : ℝ) ≤
+      1 / Real.sqrt (13928492444353 / 64000000000000 : ℝ) := by
+    rw [show
+        (1250000000000000 / 583138992922967 : ℝ) = 1 / (583138992922967 / 1250000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 4665111943383733 : ℝ) ≤
+      (398509631218225111797 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (13928492444353 / 64000000000000 : ℝ) ≤
+      (750000000000000 / 4665111943383733 : ℝ) := by
+    rw [show
+        (750000000000000 / 4665111943383733 : ℝ) =
+          (3 / 40) / (4665111943383733 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (17459979332987969381 / 40000000000000000000 : ℝ)) -
+        6 * (398509631218225111797 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (13928492444353 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (13928492444353 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((4665111943383733 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (17459979332987969381 / 40000000000000000000 : ℝ)) -
+          6 * (398509631218225111797 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (13928492444353 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (13928492444353 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (13928492444353 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_258 :
+    ((1973463022810409675657 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(14970434962464114446783 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((934164485029269 / 2000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2184841815009400424567 / 5000000000000000000000 : ℝ)) -
+              6 * (796061231206485465923 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (87266328509 / 400000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (87266328509 / 400000000000 : ℝ)) =
+      Real.log (487266328509 / 400000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((87266328509 / 400000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (89516328509 / 400000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1973463022810409675657 / 10000000000000000000000 : ℝ) ≤
+      Real.log (487266328509 / 400000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (89516328509 / 400000000000 : ℝ) ≤
+      (-(14970434962464114446783 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (89516328509 / 400000000000 : ℝ) = (89516328509 / 50000000000 : ℝ) / 2 ^ 3 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (89516328509 / 50000000000 : ℝ) ≤
+            (5823980446535885553217 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (934164485029269 / 2000000000000000 : ℝ) ≤ Real.sqrt (87266328509 / 400000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (87266328509 / 400000000000 : ℝ) ≤ (1167705606286587 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (87266328509 / 400000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau320 :
+    Real.arctan (1167705606286587 / 2500000000000000 : ℝ) ≤
+      (2184841815009400424567 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (2184841815009400424567 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (2500000000000000 / 1167705606286587 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 1167705606286587 : ℝ) =
+              (1167705606286587 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau320)
+  have hz1 :
+    (2500000000000000 / 1167705606286587 : ℝ) ≤ 1 / Real.sqrt (87266328509 / 400000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 1167705606286587 : ℝ) = 1 / (1167705606286587 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (50000000000000 / 311388161676423 : ℝ) ≤
+      (796061231206485465923 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (87266328509 / 400000000000 : ℝ) ≤
+      (50000000000000 / 311388161676423 : ℝ) := by
+    rw [show
+        (50000000000000 / 311388161676423 : ℝ) =
+          (3 / 40) / (934164485029269 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2184841815009400424567 / 5000000000000000000000 : ℝ)) -
+        6 * (796061231206485465923 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (87266328509 / 400000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (87266328509 / 400000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((934164485029269 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2184841815009400424567 / 5000000000000000000000 : ℝ)) -
+          6 * (796061231206485465923 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (87266328509 / 400000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (87266328509 / 400000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (87266328509 / 400000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_259 :
+    ((395567707431457601787 / 2000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1868330092748718717143 / 1250000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((18706103735497 / 40000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (109359116306101365259 / 250000000000000000000 : ℝ)) -
+              6 * (1590213293203845113049 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (13996732678527 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (13996732678527 / 64000000000000 : ℝ)) =
+      Real.log (77996732678527 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((13996732678527 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (14356732678527 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (395567707431457601787 / 2000000000000000000000 : ℝ) ≤
+      Real.log (77996732678527 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (14356732678527 / 64000000000000 : ℝ) ≤
+      (-(1868330092748718717143 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (14356732678527 / 64000000000000 : ℝ) = (14356732678527 / 8000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14356732678527 / 8000000000000 : ℝ) ≤
+            (730971833376281282857 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (18706103735497 / 40000000000000 : ℝ) ≤ Real.sqrt (13996732678527 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (13996732678527 / 64000000000000 : ℝ) ≤
+      (4676525933874253 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (13996732678527 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau321 :
+    Real.arctan (4676525933874253 / 10000000000000000 : ℝ) ≤
+      (109359116306101365259 / 250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (109359116306101365259 / 250000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 4676525933874253 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 4676525933874253 : ℝ) =
+              (4676525933874253 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau321)
+  have hz1 :
+    (10000000000000000 / 4676525933874253 : ℝ) ≤
+      1 / Real.sqrt (13996732678527 / 64000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 4676525933874253 : ℝ) =
+          1 / (4676525933874253 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (3000000000000 / 18706103735497 : ℝ) ≤
+      (1590213293203845113049 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (13996732678527 / 64000000000000 : ℝ) ≤
+      (3000000000000 / 18706103735497 : ℝ) := by
+    rw [show
+        (3000000000000 / 18706103735497 : ℝ) = (3 / 40) / (18706103735497 / 40000000000000 : ℝ) by
+        norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (109359116306101365259 / 250000000000000000000 : ℝ)) -
+        6 * (1590213293203845113049 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (13996732678527 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (13996732678527 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((18706103735497 / 40000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (109359116306101365259 / 250000000000000000000 : ℝ)) -
+          6 * (1590213293203845113049 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (13996732678527 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (13996732678527 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (13996732678527 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_260 :
+    ((1982212137828887213253 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(7461451501819978019017 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((187288899801977 / 400000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2189518963813952082979 / 5000000000000000000000 : ℝ)) -
+              6 * (1588310976009953894083 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (7015426397807 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (7015426397807 / 32000000000000 : ℝ)) =
+      Real.log (39015426397807 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((7015426397807 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (7195426397807 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1982212137828887213253 / 10000000000000000000000 : ℝ) ≤
+      Real.log (39015426397807 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (7195426397807 / 32000000000000 : ℝ) ≤
+      (-(7461451501819978019017 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (7195426397807 / 32000000000000 : ℝ) = (7195426397807 / 4000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7195426397807 / 4000000000000 : ℝ) ≤
+            (2935756202680021980983 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (187288899801977 / 400000000000000 : ℝ) ≤ Real.sqrt (7015426397807 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (7015426397807 / 32000000000000 : ℝ) ≤ (1170555623762357 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (7015426397807 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau322 :
+    Real.arctan (1170555623762357 / 2500000000000000 : ℝ) ≤
+      (2189518963813952082979 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (2189518963813952082979 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (2500000000000000 / 1170555623762357 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 1170555623762357 : ℝ) =
+              (1170555623762357 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau322)
+  have hz1 :
+    (2500000000000000 / 1170555623762357 : ℝ) ≤
+      1 / Real.sqrt (7015426397807 / 32000000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 1170555623762357 : ℝ) = 1 / (1170555623762357 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (30000000000000 / 187288899801977 : ℝ) ≤
+      (1588310976009953894083 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (7015426397807 / 32000000000000 : ℝ) ≤
+      (30000000000000 / 187288899801977 : ℝ) := by
+    rw [show
+        (30000000000000 / 187288899801977 : ℝ) =
+          (3 / 40) / (187288899801977 / 400000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2189518963813952082979 / 5000000000000000000000 : ℝ)) -
+        6 * (1588310976009953894083 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (7015426397807 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (7015426397807 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((187288899801977 / 400000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2189518963813952082979 / 5000000000000000000000 : ℝ)) -
+          6 * (1588310976009953894083 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (7015426397807 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (7015426397807 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (7015426397807 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_261 :
+    ((198658382649840584889 / 1000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(7449610739948963803307 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2343956066999513 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (219185174185610494233 / 500000000000000000000 : ℝ)) -
+              6 * (1586415469940263764489 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (14064972912701 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (14064972912701 / 64000000000000 : ℝ)) =
+      Real.log (78064972912701 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((14064972912701 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (14424972912701 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (198658382649840584889 / 1000000000000000000000 : ℝ) ≤
+      Real.log (78064972912701 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (14424972912701 / 64000000000000 : ℝ) ≤
+      (-(7449610739948963803307 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (14424972912701 / 64000000000000 : ℝ) = (14424972912701 / 8000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14424972912701 / 8000000000000 : ℝ) ≤
+            (2947596964551036196693 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2343956066999513 / 5000000000000000 : ℝ) ≤ Real.sqrt (14064972912701 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (14064972912701 / 64000000000000 : ℝ) ≤
+      (4687912133999029 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (14064972912701 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau323 :
+    Real.arctan (4687912133999029 / 10000000000000000 : ℝ) ≤
+      (219185174185610494233 / 500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (219185174185610494233 / 500000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 4687912133999029 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 4687912133999029 : ℝ) =
+              (4687912133999029 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau323)
+  have hz1 :
+    (10000000000000000 / 4687912133999029 : ℝ) ≤
+      1 / Real.sqrt (14064972912701 / 64000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 4687912133999029 : ℝ) =
+          1 / (4687912133999029 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 2343956066999513 : ℝ) ≤
+      (1586415469940263764489 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (14064972912701 / 64000000000000 : ℝ) ≤
+      (375000000000000 / 2343956066999513 : ℝ) := by
+    rw [show
+        (375000000000000 / 2343956066999513 : ℝ) =
+          (3 / 40) / (2343956066999513 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (219185174185610494233 / 500000000000000000000 : ℝ)) -
+        6 * (1586415469940263764489 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (14064972912701 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (14064972912701 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2343956066999513 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (219185174185610494233 / 500000000000000000000 : ℝ)) -
+          6 * (1586415469940263764489 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (14064972912701 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (14064972912701 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (14064972912701 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_262 :
+    ((248869200604606127997 / 1250000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(14875595905142932846117 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2346797437948349 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (4388361347876901126497 / 10000000000000000000000 : ℝ)) -
+              6 * (396131683611146364051 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (3524773257447 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (3524773257447 / 16000000000000 : ℝ)) =
+      Real.log (19524773257447 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((3524773257447 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (3614773257447 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (248869200604606127997 / 1250000000000000000000 : ℝ) ≤
+      Real.log (19524773257447 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (3614773257447 / 16000000000000 : ℝ) ≤
+      (-(14875595905142932846117 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (3614773257447 / 16000000000000 : ℝ) = (3614773257447 / 2000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3614773257447 / 2000000000000 : ℝ) ≤
+            (5918819503857067153883 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2346797437948349 / 5000000000000000 : ℝ) ≤ Real.sqrt (3524773257447 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (3524773257447 / 16000000000000 : ℝ) ≤ (4693594875896701 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (3524773257447 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau324 :
+    Real.arctan (4693594875896701 / 10000000000000000 : ℝ) ≤
+      (4388361347876901126497 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (4388361347876901126497 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 4693594875896701 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 4693594875896701 : ℝ) =
+              (4693594875896701 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau324)
+  have hz1 :
+    (10000000000000000 / 4693594875896701 : ℝ) ≤
+      1 / Real.sqrt (3524773257447 / 16000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 4693594875896701 : ℝ) =
+          1 / (4693594875896701 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 2346797437948349 : ℝ) ≤
+      (396131683611146364051 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (3524773257447 / 16000000000000 : ℝ) ≤
+      (375000000000000 / 2346797437948349 : ℝ) := by
+    rw [show
+        (375000000000000 / 2346797437948349 : ℝ) =
+          (3 / 40) / (2346797437948349 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (4388361347876901126497 / 10000000000000000000000 : ℝ)) -
+        6 * (396131683611146364051 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (3524773257447 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (3524773257447 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2346797437948349 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (4388361347876901126497 / 10000000000000000000000 : ℝ)) -
+          6 * (396131683611146364051 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (3524773257447 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (3524773257447 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (3524773257447 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_263 :
+    ((1995321474513032226771 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(14852026015632438169299 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1174817686440969 / 2500000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1098252886835331265209 / 2500000000000000000000 : ℝ)) -
+              6 * (1582644729309929085451 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (4522628207 / 20480000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (4522628207 / 20480000000 : ℝ)) = Real.log (25002628207 / 20480000000 : ℝ) := by
+    norm_num
+  have e2 :
+    Real.log ((4522628207 / 20480000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (4637828207 / 20480000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1995321474513032226771 / 10000000000000000000000 : ℝ) ≤
+      Real.log (25002628207 / 20480000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (4637828207 / 20480000000 : ℝ) ≤
+      (-(14852026015632438169299 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (4637828207 / 20480000000 : ℝ) = (4637828207 / 2560000000 : ℝ) / 2 ^ 3 by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4637828207 / 2560000000 : ℝ) ≤
+            (5942389393367561830701 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1174817686440969 / 2500000000000000 : ℝ) ≤ Real.sqrt (4522628207 / 20480000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (4522628207 / 20480000000 : ℝ) ≤ (4699270745763879 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (4522628207 / 20480000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau325 :
+    Real.arctan (4699270745763879 / 10000000000000000 : ℝ) ≤
+      (1098252886835331265209 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1098252886835331265209 / 2500000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 4699270745763879 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 4699270745763879 : ℝ) =
+              (4699270745763879 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau325)
+  have hz1 :
+    (10000000000000000 / 4699270745763879 : ℝ) ≤ 1 / Real.sqrt (4522628207 / 20480000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 4699270745763879 : ℝ) =
+          1 / (4699270745763879 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (62500000000000 / 391605895480323 : ℝ) ≤
+      (1582644729309929085451 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (4522628207 / 20480000000 : ℝ) ≤
+      (62500000000000 / 391605895480323 : ℝ) := by
+    rw [show
+        (62500000000000 / 391605895480323 : ℝ) =
+          (3 / 40) / (1174817686440969 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1098252886835331265209 / 2500000000000000000000 : ℝ)) -
+        6 * (1582644729309929085451 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (4522628207 / 20480000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (4522628207 / 20480000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1174817686440969 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1098252886835331265209 / 2500000000000000000000 : ℝ)) -
+          6 * (1582644729309929085451 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (4522628207 / 20480000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (4522628207 / 20480000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (4522628207 / 20480000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_264 :
+    ((999843718596792574793 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(14828511549484442036283 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((4704939768471071 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (4397654109165580255539 / 10000000000000000000000 : ℝ)) -
+              6 * (158076941465690711107 / 1000000000000000000000 : ℝ)))) ≤
+      Vfield (7083666631981 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (7083666631981 / 32000000000000 : ℝ)) =
+      Real.log (39083666631981 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((7083666631981 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (7263666631981 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (999843718596792574793 / 5000000000000000000000 : ℝ) ≤
+      Real.log (39083666631981 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (7263666631981 / 32000000000000 : ℝ) ≤
+      (-(14828511549484442036283 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (7263666631981 / 32000000000000 : ℝ) = (7263666631981 / 4000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7263666631981 / 4000000000000 : ℝ) ≤
+            (5965903859515557963717 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (4704939768471071 / 10000000000000000 : ℝ) ≤ Real.sqrt (7083666631981 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (7083666631981 / 32000000000000 : ℝ) ≤ (2352469884235537 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (7083666631981 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau326 :
+    Real.arctan (2352469884235537 / 5000000000000000 : ℝ) ≤
+      (4397654109165580255539 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (4397654109165580255539 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (5000000000000000 / 2352469884235537 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 2352469884235537 : ℝ) =
+              (2352469884235537 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau326)
+  have hz1 :
+    (5000000000000000 / 2352469884235537 : ℝ) ≤
+      1 / Real.sqrt (7083666631981 / 32000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 2352469884235537 : ℝ) = 1 / (2352469884235537 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 4704939768471071 : ℝ) ≤
+      (158076941465690711107 / 1000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (7083666631981 / 32000000000000 : ℝ) ≤
+      (750000000000000 / 4704939768471071 : ℝ) := by
+    rw [show
+        (750000000000000 / 4704939768471071 : ℝ) =
+          (3 / 40) / (4704939768471071 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (4397654109165580255539 / 10000000000000000000000 : ℝ)) -
+        6 * (158076941465690711107 / 1000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (7083666631981 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (7083666631981 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((4704939768471071 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (4397654109165580255539 / 10000000000000000000000 : ℝ)) -
+          6 * (158076941465690711107 / 1000000000000000000000 : ℝ))) ≤
+      Real.sqrt (7083666631981 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (7083666631981 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (7083666631981 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/V22.lean b/LeanPool/Zeta5Irrational/Table/V22.lean
new file mode 100644
index 0000000000..53f9b3bd2c
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/V22.lean
@@ -0,0 +1,1245 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Energy
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # V22: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_265 :
+    ((501012873635738876091 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(7402526123330008104227 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((4710601968739139 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2201144530125916720343 / 5000000000000000000000 : ℝ)) -
+              6 * (789450375468092547361 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (14201453381049 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (14201453381049 / 64000000000000 : ℝ)) =
+      Real.log (78201453381049 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((14201453381049 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (14561453381049 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (501012873635738876091 / 2500000000000000000000 : ℝ) ≤
+      Real.log (78201453381049 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (14561453381049 / 64000000000000 : ℝ) ≤
+      (-(7402526123330008104227 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (14561453381049 / 64000000000000 : ℝ) = (14561453381049 / 8000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14561453381049 / 8000000000000 : ℝ) ≤
+            (2994681581169991895773 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (4710601968739139 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (14201453381049 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (14201453381049 / 64000000000000 : ℝ) ≤ (2355300984369571 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (14201453381049 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau327 :
+    Real.arctan (2355300984369571 / 5000000000000000 : ℝ) ≤
+      (2201144530125916720343 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (2201144530125916720343 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (5000000000000000 / 2355300984369571 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 2355300984369571 : ℝ) =
+              (2355300984369571 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau327)
+  have hz1 :
+    (5000000000000000 / 2355300984369571 : ℝ) ≤
+      1 / Real.sqrt (14201453381049 / 64000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 2355300984369571 : ℝ) = 1 / (2355300984369571 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 4710601968739139 : ℝ) ≤
+      (789450375468092547361 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (14201453381049 / 64000000000000 : ℝ) ≤
+      (750000000000000 / 4710601968739139 : ℝ) := by
+    rw [show
+        (750000000000000 / 4710601968739139 : ℝ) =
+          (3 / 40) / (4710601968739139 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2201144530125916720343 / 5000000000000000000000 : ℝ)) -
+        6 * (789450375468092547361 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (14201453381049 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (14201453381049 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((4710601968739139 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2201144530125916720343 / 5000000000000000000000 : ℝ)) -
+          6 * (789450375468092547361 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (14201453381049 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (14201453381049 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (14201453381049 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_266 :
+    ((1004206824111706414963 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(14781647848946051308547 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((4716257371140541 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (4406916427345624348069 / 10000000000000000000000 : ℝ)) -
+              6 * (39425967473124423959 / 250000000000000000000 : ℝ)))) ≤
+      Vfield (1779446687267 / 8000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1779446687267 / 8000000000000 : ℝ)) =
+      Real.log (9779446687267 / 8000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1779446687267 / 8000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1824446687267 / 8000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1004206824111706414963 / 5000000000000000000000 : ℝ) ≤
+      Real.log (9779446687267 / 8000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1824446687267 / 8000000000000 : ℝ) ≤
+      (-(14781647848946051308547 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (1824446687267 / 8000000000000 : ℝ) = (1824446687267 / 1000000000000 : ℝ) / 2 ^ 3 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1824446687267 / 1000000000000 : ℝ) ≤
+            (6012767560053948691453 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (4716257371140541 / 10000000000000000 : ℝ) ≤ Real.sqrt (1779446687267 / 8000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1779446687267 / 8000000000000 : ℝ) ≤ (73691521424071 / 156250000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1779446687267 / 8000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau328 :
+    Real.arctan (73691521424071 / 156250000000000 : ℝ) ≤
+      (4406916427345624348069 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (4406916427345624348069 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (156250000000000 / 73691521424071 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (156250000000000 / 73691521424071 : ℝ) =
+              (73691521424071 / 156250000000000 : ℝ)
+            by
+        norm_num];
+        exact hau328)
+  have hz1 :
+    (156250000000000 / 73691521424071 : ℝ) ≤ 1 / Real.sqrt (1779446687267 / 8000000000000 : ℝ) := by
+    rw [show (156250000000000 / 73691521424071 : ℝ) = 1 / (73691521424071 / 156250000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 4716257371140541 : ℝ) ≤
+      (39425967473124423959 / 250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1779446687267 / 8000000000000 : ℝ) ≤
+      (750000000000000 / 4716257371140541 : ℝ) := by
+    rw [show
+        (750000000000000 / 4716257371140541 : ℝ) =
+          (3 / 40) / (4716257371140541 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (4406916427345624348069 / 10000000000000000000000 : ℝ)) -
+        6 * (39425967473124423959 / 250000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1779446687267 / 8000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1779446687267 / 8000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((4716257371140541 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (4406916427345624348069 / 10000000000000000000000 : ℝ)) -
+          6 * (39425967473124423959 / 250000000000000000000 : ℝ))) ≤
+      Real.sqrt (1779446687267 / 8000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1779446687267 / 8000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1779446687267 / 8000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_267 :
+    ((125798368743440768183 / 625000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(7379149049969101912307 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((4721906000100587 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2205768118518581484771 / 5000000000000000000000 : ℝ)) -
+              6 * (1575183219723586415793 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (14269693615223 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (14269693615223 / 64000000000000 : ℝ)) =
+      Real.log (78269693615223 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((14269693615223 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (14629693615223 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (125798368743440768183 / 625000000000000000000 : ℝ) ≤
+      Real.log (78269693615223 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (14629693615223 / 64000000000000 : ℝ) ≤
+      (-(7379149049969101912307 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (14629693615223 / 64000000000000 : ℝ) = (14629693615223 / 8000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14629693615223 / 8000000000000 : ℝ) ≤
+            (3018058654530898087693 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (4721906000100587 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (14269693615223 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (14269693615223 / 64000000000000 : ℝ) ≤ (472190600010059 / 1000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (14269693615223 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau329 :
+    Real.arctan (472190600010059 / 1000000000000000 : ℝ) ≤
+      (2205768118518581484771 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (2205768118518581484771 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (1000000000000000 / 472190600010059 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1000000000000000 / 472190600010059 : ℝ) =
+              (472190600010059 / 1000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau329)
+  have hz1 :
+    (1000000000000000 / 472190600010059 : ℝ) ≤
+      1 / Real.sqrt (14269693615223 / 64000000000000 : ℝ) := by
+    rw [show
+        (1000000000000000 / 472190600010059 : ℝ) = 1 / (472190600010059 / 1000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 4721906000100587 : ℝ) ≤
+      (1575183219723586415793 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (14269693615223 / 64000000000000 : ℝ) ≤
+      (750000000000000 / 4721906000100587 : ℝ) := by
+    rw [show
+        (750000000000000 / 4721906000100587 : ℝ) =
+          (3 / 40) / (4721906000100587 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2205768118518581484771 / 5000000000000000000000 : ℝ)) -
+        6 * (1575183219723586415793 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (14269693615223 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (14269693615223 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((4721906000100587 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2205768118518581484771 / 5000000000000000000000 : ℝ)) -
+          6 * (1575183219723586415793 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (14269693615223 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (14269693615223 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (14269693615223 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_268 :
+    ((2017132251215798468539 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(92093767156400260889 / 62500000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((945509575979733 / 2000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2208074257881297602699 / 5000000000000000000000 : ℝ)) -
+              6 * (314666854950398720743 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (1430381373231 / 6400000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1430381373231 / 6400000000000 : ℝ)) =
+      Real.log (7830381373231 / 6400000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1430381373231 / 6400000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1466381373231 / 6400000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2017132251215798468539 / 10000000000000000000000 : ℝ) ≤
+      Real.log (7830381373231 / 6400000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1466381373231 / 6400000000000 : ℝ) ≤
+      (-(92093767156400260889 / 62500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (1466381373231 / 6400000000000 : ℝ) = (1466381373231 / 800000000000 : ℝ) / 2 ^ 3 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1466381373231 / 800000000000 : ℝ) ≤
+            (37871329149849739111 / 62500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (945509575979733 / 2000000000000000 : ℝ) ≤ Real.sqrt (1430381373231 / 6400000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1430381373231 / 6400000000000 : ℝ) ≤ (1181886969974667 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1430381373231 / 6400000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau330 :
+    Real.arctan (1181886969974667 / 2500000000000000 : ℝ) ≤
+      (2208074257881297602699 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (2208074257881297602699 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (2500000000000000 / 1181886969974667 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 1181886969974667 : ℝ) =
+              (1181886969974667 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau330)
+  have hz1 :
+    (2500000000000000 / 1181886969974667 : ℝ) ≤
+      1 / Real.sqrt (1430381373231 / 6400000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 1181886969974667 : ℝ) = 1 / (1181886969974667 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (50000000000000 / 315169858659911 : ℝ) ≤
+      (314666854950398720743 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1430381373231 / 6400000000000 : ℝ) ≤
+      (50000000000000 / 315169858659911 : ℝ) := by
+    rw [show
+        (50000000000000 / 315169858659911 : ℝ) =
+          (3 / 40) / (945509575979733 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2208074257881297602699 / 5000000000000000000000 : ℝ)) -
+        6 * (314666854950398720743 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1430381373231 / 6400000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1430381373231 / 6400000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((945509575979733 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2208074257881297602699 / 5000000000000000000000 : ℝ)) -
+          6 * (314666854950398720743 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (1430381373231 / 6400000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1430381373231 / 6400000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1430381373231 / 6400000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_269 :
+    ((1010744351920704571427 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(3677940382841596508797 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((236659151733473 / 500000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2210376644902636965323 / 5000000000000000000000 : ℝ)) -
+              6 * (785745912873242323847 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (14337933849397 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (14337933849397 / 64000000000000 : ℝ)) =
+      Real.log (78337933849397 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((14337933849397 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (14697933849397 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1010744351920704571427 / 5000000000000000000000 : ℝ) ≤
+      Real.log (78337933849397 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (14697933849397 / 64000000000000 : ℝ) ≤
+      (-(3677940382841596508797 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (14697933849397 / 64000000000000 : ℝ) = (14697933849397 / 8000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14697933849397 / 8000000000000 : ℝ) ≤
+            (1520663469408403491203 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (236659151733473 / 500000000000000 : ℝ) ≤ Real.sqrt (14337933849397 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (14337933849397 / 64000000000000 : ℝ) ≤
+      (4733183034669463 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (14337933849397 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau331 :
+    Real.arctan (4733183034669463 / 10000000000000000 : ℝ) ≤
+      (2210376644902636965323 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (2210376644902636965323 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 4733183034669463 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 4733183034669463 : ℝ) =
+              (4733183034669463 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau331)
+  have hz1 :
+    (10000000000000000 / 4733183034669463 : ℝ) ≤
+      1 / Real.sqrt (14337933849397 / 64000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 4733183034669463 : ℝ) =
+          1 / (4733183034669463 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (37500000000000 / 236659151733473 : ℝ) ≤
+      (785745912873242323847 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (14337933849397 / 64000000000000 : ℝ) ≤
+      (37500000000000 / 236659151733473 : ℝ) := by
+    rw [show
+        (37500000000000 / 236659151733473 : ℝ) =
+          (3 / 40) / (236659151733473 / 500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2210376644902636965323 / 5000000000000000000000 : ℝ)) -
+        6 * (785745912873242323847 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (14337933849397 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (14337933849397 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((236659151733473 / 500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2210376644902636965323 / 5000000000000000000000 : ℝ)) -
+          6 * (785745912873242323847 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (14337933849397 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (14337933849397 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (14337933849397 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_270 :
+    ((506460814856369766769 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(7344287103943422644799 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2369405744202079 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (4425350585297006953787 / 10000000000000000000000 : ℝ)) -
+              6 * (9810348967227032483 / 62500000000000000000 : ℝ)))) ≤
+      Vfield (3593013491621 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (3593013491621 / 16000000000000 : ℝ)) =
+      Real.log (19593013491621 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((3593013491621 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (3683013491621 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (506460814856369766769 / 2500000000000000000000 : ℝ) ≤
+      Real.log (19593013491621 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (3683013491621 / 16000000000000 : ℝ) ≤
+      (-(7344287103943422644799 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (3683013491621 / 16000000000000 : ℝ) = (3683013491621 / 2000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3683013491621 / 2000000000000 : ℝ) ≤
+            (3052920600556577355201 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2369405744202079 / 5000000000000000 : ℝ) ≤ Real.sqrt (3593013491621 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (3593013491621 / 16000000000000 : ℝ) ≤ (4738811488404161 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (3593013491621 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau332 :
+    Real.arctan (4738811488404161 / 10000000000000000 : ℝ) ≤
+      (4425350585297006953787 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (4425350585297006953787 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 4738811488404161 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 4738811488404161 : ℝ) =
+              (4738811488404161 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau332)
+  have hz1 :
+    (10000000000000000 / 4738811488404161 : ℝ) ≤
+      1 / Real.sqrt (3593013491621 / 16000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 4738811488404161 : ℝ) =
+          1 / (4738811488404161 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 2369405744202079 : ℝ) ≤
+      (9810348967227032483 / 62500000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (3593013491621 / 16000000000000 : ℝ) ≤
+      (375000000000000 / 2369405744202079 : ℝ) := by
+    rw [show
+        (375000000000000 / 2369405744202079 : ℝ) =
+          (3 / 40) / (2369405744202079 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (4425350585297006953787 / 10000000000000000000000 : ℝ)) -
+        6 * (9810348967227032483 / 62500000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (3593013491621 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (3593013491621 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2369405744202079 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (4425350585297006953787 / 10000000000000000000000 : ℝ)) -
+          6 * (9810348967227032483 / 62500000000000000000 : ℝ))) ≤
+      Real.sqrt (3593013491621 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (3593013491621 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (3593013491621 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_271 :
+    ((2030195919619443733491 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(14665440525249540781007 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((4744433264951639 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (4429940428219294879569 / 10000000000000000000000 : ℝ)) -
+              6 * (1567826264140476018089 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (14406174083571 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (14406174083571 / 64000000000000 : ℝ)) =
+      Real.log (78406174083571 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((14406174083571 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (14766174083571 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2030195919619443733491 / 10000000000000000000000 : ℝ) ≤
+      Real.log (78406174083571 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (14766174083571 / 64000000000000 : ℝ) ≤
+      (-(14665440525249540781007 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (14766174083571 / 64000000000000 : ℝ) = (14766174083571 / 8000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14766174083571 / 8000000000000 : ℝ) ≤
+            (6128974883750459218993 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (4744433264951639 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (14406174083571 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (14406174083571 / 64000000000000 : ℝ) ≤ (2372216632475821 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (14406174083571 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau333 :
+    Real.arctan (2372216632475821 / 5000000000000000 : ℝ) ≤
+      (4429940428219294879569 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (4429940428219294879569 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (5000000000000000 / 2372216632475821 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 2372216632475821 : ℝ) =
+              (2372216632475821 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau333)
+  have hz1 :
+    (5000000000000000 / 2372216632475821 : ℝ) ≤
+      1 / Real.sqrt (14406174083571 / 64000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 2372216632475821 : ℝ) = 1 / (2372216632475821 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 4744433264951639 : ℝ) ≤
+      (1567826264140476018089 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (14406174083571 / 64000000000000 : ℝ) ≤
+      (750000000000000 / 4744433264951639 : ℝ) := by
+    rw [show
+        (750000000000000 / 4744433264951639 : ℝ) =
+          (3 / 40) / (4744433264951639 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (4429940428219294879569 / 10000000000000000000000 : ℝ)) -
+        6 * (1567826264140476018089 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (14406174083571 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (14406174083571 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((4744433264951639 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (4429940428219294879569 / 10000000000000000000000 : ℝ)) -
+          6 * (1567826264140476018089 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (14406174083571 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (14406174083571 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (14406174083571 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_272 :
+    ((1017273343036291565619 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(58569440943380077467 / 40000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2375024194009827 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (4434522844404553221673 / 10000000000000000000000 : ℝ)) -
+              6 * (1566003076564351681379 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (7220147100329 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (7220147100329 / 32000000000000 : ℝ)) =
+      Real.log (39220147100329 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((7220147100329 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (7400147100329 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1017273343036291565619 / 5000000000000000000000 : ℝ) ≤
+      Real.log (39220147100329 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (7400147100329 / 32000000000000 : ℝ) ≤
+      (-(58569440943380077467 / 40000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (7400147100329 / 32000000000000 : ℝ) = (7400147100329 / 4000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7400147100329 / 4000000000000 : ℝ) ≤
+            (24608220692619922533 / 40000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2375024194009827 / 5000000000000000 : ℝ) ≤ Real.sqrt (7220147100329 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (7220147100329 / 32000000000000 : ℝ) ≤ (4750048388019657 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (7220147100329 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau334 :
+    Real.arctan (4750048388019657 / 10000000000000000 : ℝ) ≤
+      (4434522844404553221673 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (4434522844404553221673 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 4750048388019657 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 4750048388019657 : ℝ) =
+              (4750048388019657 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau334)
+  have hz1 :
+    (10000000000000000 / 4750048388019657 : ℝ) ≤
+      1 / Real.sqrt (7220147100329 / 32000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 4750048388019657 : ℝ) =
+          1 / (4750048388019657 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (125000000000000 / 791674731336609 : ℝ) ≤
+      (1566003076564351681379 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (7220147100329 / 32000000000000 : ℝ) ≤
+      (125000000000000 / 791674731336609 : ℝ) := by
+    rw [show
+        (125000000000000 / 791674731336609 : ℝ) =
+          (3 / 40) / (2375024194009827 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (4434522844404553221673 / 10000000000000000000000 : ℝ)) -
+        6 * (1566003076564351681379 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (7220147100329 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (7220147100329 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2375024194009827 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (4434522844404553221673 / 10000000000000000000000 : ℝ)) -
+          6 * (1566003076564351681379 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (7220147100329 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (7220147100329 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (7220147100329 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_273 :
+    ((407779112086405099183 / 2000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(14619333093774351598831 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((475565688117599 / 1000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (4439097859537322069829 / 10000000000000000000000 : ℝ)) -
+              6 * (1564186234996620458847 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (2894882863549 / 12800000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (2894882863549 / 12800000000000 : ℝ)) =
+      Real.log (15694882863549 / 12800000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((2894882863549 / 12800000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (2966882863549 / 12800000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (407779112086405099183 / 2000000000000000000000 : ℝ) ≤
+      Real.log (15694882863549 / 12800000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (2966882863549 / 12800000000000 : ℝ) ≤
+      (-(14619333093774351598831 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (2966882863549 / 12800000000000 : ℝ) = (2966882863549 / 1600000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2966882863549 / 1600000000000 : ℝ) ≤
+            (6175082315225648401169 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (475565688117599 / 1000000000000000 : ℝ) ≤ Real.sqrt (2894882863549 / 12800000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (2894882863549 / 12800000000000 : ℝ) ≤ (4755656881175993 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (2894882863549 / 12800000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau335 :
+    Real.arctan (4755656881175993 / 10000000000000000 : ℝ) ≤
+      (4439097859537322069829 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (4439097859537322069829 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 4755656881175993 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 4755656881175993 : ℝ) =
+              (4755656881175993 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau335)
+  have hz1 :
+    (10000000000000000 / 4755656881175993 : ℝ) ≤
+      1 / Real.sqrt (2894882863549 / 12800000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 4755656881175993 : ℝ) =
+          1 / (4755656881175993 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (75000000000000 / 475565688117599 : ℝ) ≤
+      (1564186234996620458847 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (2894882863549 / 12800000000000 : ℝ) ≤
+      (75000000000000 / 475565688117599 : ℝ) := by
+    rw [show
+        (75000000000000 / 475565688117599 : ℝ) =
+          (3 / 40) / (475565688117599 / 1000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (4439097859537322069829 / 10000000000000000000000 : ℝ)) -
+        6 * (1564186234996620458847 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (2894882863549 / 12800000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (2894882863549 / 12800000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((475565688117599 / 1000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (4439097859537322069829 / 10000000000000000000000 : ℝ)) -
+          6 * (1564186234996620458847 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (2894882863549 / 12800000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (2894882863549 / 12800000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (2894882863549 / 12800000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_274 :
+    ((2043242544342751051039 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(7298179427416706252041 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((4761258767849631 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (4443665499155469865247 / 10000000000000000000000 : ℝ)) -
+              6 * (390593925676510568093 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (906783402177 / 4000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (906783402177 / 4000000000000 : ℝ)) =
+      Real.log (4906783402177 / 4000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((906783402177 / 4000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (929283402177 / 4000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2043242544342751051039 / 10000000000000000000000 : ℝ) ≤
+      Real.log (4906783402177 / 4000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (929283402177 / 4000000000000 : ℝ) ≤
+      (-(7298179427416706252041 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (929283402177 / 4000000000000 : ℝ) = (929283402177 / 500000000000 : ℝ) / 2 ^ 3 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (929283402177 / 500000000000 : ℝ) ≤
+            (3099028277083293747959 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (4761258767849631 / 10000000000000000 : ℝ) ≤ Real.sqrt (906783402177 / 4000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (906783402177 / 4000000000000 : ℝ) ≤ (2380629383924817 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (906783402177 / 4000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau336 :
+    Real.arctan (2380629383924817 / 5000000000000000 : ℝ) ≤
+      (4443665499155469865247 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (4443665499155469865247 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (5000000000000000 / 2380629383924817 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 2380629383924817 : ℝ) =
+              (2380629383924817 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau336)
+  have hz1 :
+    (5000000000000000 / 2380629383924817 : ℝ) ≤
+      1 / Real.sqrt (906783402177 / 4000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 2380629383924817 : ℝ) = 1 / (2380629383924817 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 1587086255949877 : ℝ) ≤
+      (390593925676510568093 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (906783402177 / 4000000000000 : ℝ) ≤
+      (250000000000000 / 1587086255949877 : ℝ) := by
+    rw [show
+        (250000000000000 / 1587086255949877 : ℝ) =
+          (3 / 40) / (4761258767849631 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (4443665499155469865247 / 10000000000000000000000 : ℝ)) -
+        6 * (390593925676510568093 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (906783402177 / 4000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (906783402177 / 4000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((4761258767849631 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (4443665499155469865247 / 10000000000000000000000 : ℝ)) -
+          6 * (390593925676510568093 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (906783402177 / 4000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (906783402177 / 4000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (906783402177 / 4000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_275 :
+    ((1023793819723797870683 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(3643359319124335629843 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((4766854071331891 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (88964515773027434811 / 200000000000000000000 : ℝ)) -
+              6 * (390142860814587977897 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (14542654551919 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (14542654551919 / 64000000000000 : ℝ)) =
+      Real.log (78542654551919 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((14542654551919 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (14902654551919 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1023793819723797870683 / 5000000000000000000000 : ℝ) ≤
+      Real.log (78542654551919 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (14902654551919 / 64000000000000 : ℝ) ≤
+      (-(3643359319124335629843 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (14902654551919 / 64000000000000 : ℝ) = (14902654551919 / 8000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (14902654551919 / 8000000000000 : ℝ) ≤
+            (1555244533125664370157 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (4766854071331891 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (14542654551919 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (14542654551919 / 64000000000000 : ℝ) ≤ (2383427035665947 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (14542654551919 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau337 :
+    Real.arctan (2383427035665947 / 5000000000000000 : ℝ) ≤
+      (88964515773027434811 / 200000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (88964515773027434811 / 200000000000000000000 : ℝ)) ≤
+      Real.arctan (5000000000000000 / 2383427035665947 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 2383427035665947 : ℝ) =
+              (2383427035665947 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau337)
+  have hz1 :
+    (5000000000000000 / 2383427035665947 : ℝ) ≤
+      1 / Real.sqrt (14542654551919 / 64000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 2383427035665947 : ℝ) = 1 / (2383427035665947 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 4766854071331891 : ℝ) ≤
+      (390142860814587977897 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (14542654551919 / 64000000000000 : ℝ) ≤
+      (750000000000000 / 4766854071331891 : ℝ) := by
+    rw [show
+        (750000000000000 / 4766854071331891 : ℝ) =
+          (3 / 40) / (4766854071331891 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (88964515773027434811 / 200000000000000000000 : ℝ)) -
+        6 * (390142860814587977897 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (14542654551919 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (14542654551919 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((4766854071331891 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (88964515773027434811 / 200000000000000000000 : ℝ)) -
+          6 * (390142860814587977897 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (14542654551919 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (14542654551919 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (14542654551919 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_276 :
+    ((2051930847387254958117 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(14550568117905186109883 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((95448856295551 / 200000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2226389376636543872639 / 5000000000000000000000 : ℝ)) -
+              6 * (1558773420513176924499 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (7288387334503 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (7288387334503 / 32000000000000 : ℝ)) =
+      Real.log (39288387334503 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((7288387334503 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (7468387334503 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2051930847387254958117 / 10000000000000000000000 : ℝ) ≤
+      Real.log (39288387334503 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (7468387334503 / 32000000000000 : ℝ) ≤
+      (-(14550568117905186109883 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (7468387334503 / 32000000000000 : ℝ) = (7468387334503 / 4000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7468387334503 / 4000000000000 : ℝ) ≤
+            (6243847291094813890117 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (95448856295551 / 200000000000000 : ℝ) ≤ Real.sqrt (7288387334503 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (7288387334503 / 32000000000000 : ℝ) ≤ (4772442814777553 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (7288387334503 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau338 :
+    Real.arctan (4772442814777553 / 10000000000000000 : ℝ) ≤
+      (2226389376636543872639 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (2226389376636543872639 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 4772442814777553 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 4772442814777553 : ℝ) =
+              (4772442814777553 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau338)
+  have hz1 :
+    (10000000000000000 / 4772442814777553 : ℝ) ≤
+      1 / Real.sqrt (7288387334503 / 32000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 4772442814777553 : ℝ) =
+          1 / (4772442814777553 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (15000000000000 / 95448856295551 : ℝ) ≤
+      (1558773420513176924499 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (7288387334503 / 32000000000000 : ℝ) ≤
+      (15000000000000 / 95448856295551 : ℝ) := by
+    rw [show
+        (15000000000000 / 95448856295551 : ℝ) = (3 / 40) / (95448856295551 / 200000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2226389376636543872639 / 5000000000000000000000 : ℝ)) -
+        6 * (1558773420513176924499 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (7288387334503 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (7288387334503 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((95448856295551 / 200000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2226389376636543872639 / 5000000000000000000000 : ℝ)) -
+          6 * (1558773420513176924499 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (7288387334503 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (7288387334503 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (7288387334503 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/V23.lean b/LeanPool/Zeta5Irrational/Table/V23.lean
new file mode 100644
index 0000000000..e2031baf14
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/V23.lean
@@ -0,0 +1,1234 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Energy
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # V23: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_277 :
+    ((64394112760097439463 / 312500000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(226640407886521376383 / 156250000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2391800356751099 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (4461862808171570115379 / 10000000000000000000000 : ℝ)) -
+              6 * (1555195942020142671139 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (732250745159 / 3200000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (732250745159 / 3200000000000 : ℝ)) =
+      Real.log (3932250745159 / 3200000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((732250745159 / 3200000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (750250745159 / 3200000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (64394112760097439463 / 312500000000000000000 : ℝ) ≤
+      Real.log (3932250745159 / 3200000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (750250745159 / 3200000000000 : ℝ) ≤
+      (-(226640407886521376383 / 156250000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (750250745159 / 3200000000000 : ℝ) = (750250745159 / 400000000000 : ℝ) / 2 ^ 3 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (750250745159 / 400000000000 : ℝ) ≤
+            (98272332879103623617 / 156250000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2391800356751099 / 5000000000000000 : ℝ) ≤ Real.sqrt (732250745159 / 3200000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (732250745159 / 3200000000000 : ℝ) ≤ (4783600713502201 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (732250745159 / 3200000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau339 :
+    Real.arctan (4783600713502201 / 10000000000000000 : ℝ) ≤
+      (4461862808171570115379 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (4461862808171570115379 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 4783600713502201 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 4783600713502201 : ℝ) =
+              (4783600713502201 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau339)
+  have hz1 :
+    (10000000000000000 / 4783600713502201 : ℝ) ≤
+      1 / Real.sqrt (732250745159 / 3200000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 4783600713502201 : ℝ) =
+          1 / (4783600713502201 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 2391800356751099 : ℝ) ≤
+      (1555195942020142671139 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (732250745159 / 3200000000000 : ℝ) ≤
+      (375000000000000 / 2391800356751099 : ℝ) := by
+    rw [show
+        (375000000000000 / 2391800356751099 : ℝ) =
+          (3 / 40) / (2391800356751099 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (4461862808171570115379 / 10000000000000000000000 : ℝ)) -
+        6 * (1555195942020142671139 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (732250745159 / 3200000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (732250745159 / 3200000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2391800356751099 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (4461862808171570115379 / 10000000000000000000000 : ℝ)) -
+          6 * (1555195942020142671139 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (732250745159 / 3200000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (732250745159 / 3200000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (732250745159 / 3200000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_278 :
+    ((2069284840233233473751 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2891922184229524885251 / 2000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((29967079041083 / 62500000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2235458931496437735461 / 5000000000000000000000 : ℝ)) -
+              6 * (310328596773444075099 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (7356627568677 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (7356627568677 / 32000000000000 : ℝ)) =
+      Real.log (39356627568677 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((7356627568677 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (7536627568677 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2069284840233233473751 / 10000000000000000000000 : ℝ) ≤
+      Real.log (39356627568677 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (7536627568677 / 32000000000000 : ℝ) ≤
+      (-(2891922184229524885251 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (7536627568677 / 32000000000000 : ℝ) = (7536627568677 / 4000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7536627568677 / 4000000000000 : ℝ) ≤
+            (1266960897570475114749 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (29967079041083 / 62500000000000 : ℝ) ≤ Real.sqrt (7356627568677 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (7356627568677 / 32000000000000 : ℝ) ≤ (4794732646573283 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (7356627568677 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau340 :
+    Real.arctan (4794732646573283 / 10000000000000000 : ℝ) ≤
+      (2235458931496437735461 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (2235458931496437735461 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 4794732646573283 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 4794732646573283 : ℝ) =
+              (4794732646573283 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau340)
+  have hz1 :
+    (10000000000000000 / 4794732646573283 : ℝ) ≤
+      1 / Real.sqrt (7356627568677 / 32000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 4794732646573283 : ℝ) =
+          1 / (4794732646573283 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (4687500000000 / 29967079041083 : ℝ) ≤
+      (310328596773444075099 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (7356627568677 / 32000000000000 : ℝ) ≤
+      (4687500000000 / 29967079041083 : ℝ) := by
+    rw [show
+        (4687500000000 / 29967079041083 : ℝ) = (3 / 40) / (29967079041083 / 62500000000000 : ℝ) by
+        norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2235458931496437735461 / 5000000000000000000000 : ℝ)) -
+        6 * (310328596773444075099 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (7356627568677 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (7356627568677 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((29967079041083 / 62500000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2235458931496437735461 / 5000000000000000000000 : ℝ)) -
+          6 * (310328596773444075099 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (7356627568677 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (7356627568677 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (7356627568677 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_279 :
+    ((2077950556166482502539 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(7207220349310972146853 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((600729849303283 / 1250000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2239972057321476567427 / 5000000000000000000000 : ℝ)) -
+              6 * (774057133603146563687 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (1847686921441 / 8000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1847686921441 / 8000000000000 : ℝ)) =
+      Real.log (9847686921441 / 8000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1847686921441 / 8000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1892686921441 / 8000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2077950556166482502539 / 10000000000000000000000 : ℝ) ≤
+      Real.log (9847686921441 / 8000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1892686921441 / 8000000000000 : ℝ) ≤
+      (-(7207220349310972146853 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (1892686921441 / 8000000000000 : ℝ) = (1892686921441 / 1000000000000 : ℝ) / 2 ^ 3 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1892686921441 / 1000000000000 : ℝ) ≤
+            (3189987355189027853147 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (600729849303283 / 1250000000000000 : ℝ) ≤ Real.sqrt (1847686921441 / 8000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1847686921441 / 8000000000000 : ℝ) ≤ (4805838794426267 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1847686921441 / 8000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau341 :
+    Real.arctan (4805838794426267 / 10000000000000000 : ℝ) ≤
+      (2239972057321476567427 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (2239972057321476567427 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 4805838794426267 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 4805838794426267 : ℝ) =
+              (4805838794426267 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau341)
+  have hz1 :
+    (10000000000000000 / 4805838794426267 : ℝ) ≤
+      1 / Real.sqrt (1847686921441 / 8000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 4805838794426267 : ℝ) =
+          1 / (4805838794426267 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (93750000000000 / 600729849303283 : ℝ) ≤
+      (774057133603146563687 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1847686921441 / 8000000000000 : ℝ) ≤
+      (93750000000000 / 600729849303283 : ℝ) := by
+    rw [show
+        (93750000000000 / 600729849303283 : ℝ) =
+          (3 / 40) / (600729849303283 / 1250000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2239972057321476567427 / 5000000000000000000000 : ℝ)) -
+        6 * (774057133603146563687 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1847686921441 / 8000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1847686921441 / 8000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((600729849303283 / 1250000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2239972057321476567427 / 5000000000000000000000 : ℝ)) -
+          6 * (774057133603146563687 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (1847686921441 / 8000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1847686921441 / 8000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1847686921441 / 8000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_280 :
+    ((2086608769137852456389 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(14369473593852945214773 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((4816919335416503 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (280558859864199691039 / 625000000000000000000 : ℝ)) -
+              6 * (308921903521780670773 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (7424867802851 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (7424867802851 / 32000000000000 : ℝ)) =
+      Real.log (39424867802851 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((7424867802851 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (7604867802851 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2086608769137852456389 / 10000000000000000000000 : ℝ) ≤
+      Real.log (39424867802851 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (7604867802851 / 32000000000000 : ℝ) ≤
+      (-(14369473593852945214773 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (7604867802851 / 32000000000000 : ℝ) = (7604867802851 / 4000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7604867802851 / 4000000000000 : ℝ) ≤
+            (6424941815147054785227 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (4816919335416503 / 10000000000000000 : ℝ) ≤ Real.sqrt (7424867802851 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (7424867802851 / 32000000000000 : ℝ) ≤ (2408459667708253 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (7424867802851 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau342 :
+    Real.arctan (2408459667708253 / 5000000000000000 : ℝ) ≤
+      (280558859864199691039 / 625000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (280558859864199691039 / 625000000000000000000 : ℝ)) ≤
+      Real.arctan (5000000000000000 / 2408459667708253 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 2408459667708253 : ℝ) =
+              (2408459667708253 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau342)
+  have hz1 :
+    (5000000000000000 / 2408459667708253 : ℝ) ≤
+      1 / Real.sqrt (7424867802851 / 32000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 2408459667708253 : ℝ) = 1 / (2408459667708253 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 4816919335416503 : ℝ) ≤
+      (308921903521780670773 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (7424867802851 / 32000000000000 : ℝ) ≤
+      (750000000000000 / 4816919335416503 : ℝ) := by
+    rw [show
+        (750000000000000 / 4816919335416503 : ℝ) =
+          (3 / 40) / (4816919335416503 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (280558859864199691039 / 625000000000000000000 : ℝ)) -
+        6 * (308921903521780670773 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (7424867802851 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (7424867802851 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((4816919335416503 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (280558859864199691039 / 625000000000000000000 : ℝ)) -
+          6 * (308921903521780670773 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (7424867802851 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (7424867802851 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (7424867802851 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_281 :
+    ((2095259492128553921179 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(14324707788288511762533 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((4827974445852653 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (4497910985085163099573 / 10000000000000000000000 : ℝ)) -
+              6 * (1541128464976589575851 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (3729493959969 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (3729493959969 / 16000000000000 : ℝ)) =
+      Real.log (19729493959969 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((3729493959969 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (3819493959969 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2095259492128553921179 / 10000000000000000000000 : ℝ) ≤
+      Real.log (19729493959969 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (3819493959969 / 16000000000000 : ℝ) ≤
+      (-(14324707788288511762533 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (3819493959969 / 16000000000000 : ℝ) = (3819493959969 / 2000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3819493959969 / 2000000000000 : ℝ) ≤
+            (6469707620711488237467 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (4827974445852653 / 10000000000000000 : ℝ) ≤ Real.sqrt (3729493959969 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (3729493959969 / 16000000000000 : ℝ) ≤ (301748402865791 / 625000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (3729493959969 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau343 :
+    Real.arctan (301748402865791 / 625000000000000 : ℝ) ≤
+      (4497910985085163099573 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (4497910985085163099573 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (625000000000000 / 301748402865791 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (625000000000000 / 301748402865791 : ℝ) =
+              (301748402865791 / 625000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau343)
+  have hz1 :
+    (625000000000000 / 301748402865791 : ℝ) ≤
+      1 / Real.sqrt (3729493959969 / 16000000000000 : ℝ) := by
+    rw [show (625000000000000 / 301748402865791 : ℝ) = 1 / (301748402865791 / 625000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 4827974445852653 : ℝ) ≤
+      (1541128464976589575851 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (3729493959969 / 16000000000000 : ℝ) ≤
+      (750000000000000 / 4827974445852653 : ℝ) := by
+    rw [show
+        (750000000000000 / 4827974445852653 : ℝ) =
+          (3 / 40) / (4827974445852653 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (4497910985085163099573 / 10000000000000000000000 : ℝ)) -
+        6 * (1541128464976589575851 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (3729493959969 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (3729493959969 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((4827974445852653 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (4497910985085163099573 / 10000000000000000000000 : ℝ)) -
+          6 * (1541128464976589575851 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (3729493959969 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (3729493959969 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (3729493959969 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_282 :
+    ((2103902738086137537693 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(14280141487690492393971 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((4839004300029409 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1126712996706153351403 / 2500000000000000000000 : ℝ)) -
+              6 * (1537670843453434960069 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (299724321481 / 1280000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (299724321481 / 1280000000000 : ℝ)) =
+      Real.log (1579724321481 / 1280000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((299724321481 / 1280000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (306924321481 / 1280000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2103902738086137537693 / 10000000000000000000000 : ℝ) ≤
+      Real.log (1579724321481 / 1280000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (306924321481 / 1280000000000 : ℝ) ≤
+      (-(14280141487690492393971 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (306924321481 / 1280000000000 : ℝ) = (306924321481 / 160000000000 : ℝ) / 2 ^ 3 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (306924321481 / 160000000000 : ℝ) ≤
+            (6514273921309507606029 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (4839004300029409 / 10000000000000000 : ℝ) ≤ Real.sqrt (299724321481 / 1280000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (299724321481 / 1280000000000 : ℝ) ≤ (1209751075007353 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (299724321481 / 1280000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau344 :
+    Real.arctan (1209751075007353 / 2500000000000000 : ℝ) ≤
+      (1126712996706153351403 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1126712996706153351403 / 2500000000000000000000 : ℝ)) ≤
+      Real.arctan (2500000000000000 / 1209751075007353 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 1209751075007353 : ℝ) =
+              (1209751075007353 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau344)
+  have hz1 :
+    (2500000000000000 / 1209751075007353 : ℝ) ≤
+      1 / Real.sqrt (299724321481 / 1280000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 1209751075007353 : ℝ) = 1 / (1209751075007353 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 4839004300029409 : ℝ) ≤
+      (1537670843453434960069 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (299724321481 / 1280000000000 : ℝ) ≤
+      (750000000000000 / 4839004300029409 : ℝ) := by
+    rw [show
+        (750000000000000 / 4839004300029409 : ℝ) =
+          (3 / 40) / (4839004300029409 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1126712996706153351403 / 2500000000000000000000 : ℝ)) -
+        6 * (1537670843453434960069 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (299724321481 / 1280000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (299724321481 / 1280000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((4839004300029409 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1126712996706153351403 / 2500000000000000000000 : ℝ)) -
+          6 * (1537670843453434960069 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (299724321481 / 1280000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (299724321481 / 1280000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (299724321481 / 1280000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_283 :
+    ((264067314990576284203 / 1250000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(7117886460851593151109 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2425004535129779 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (4515764951354820672259 / 10000000000000000000000 : ℝ)) -
+              6 * (383559097835192323907 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (29403234977 / 125000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (29403234977 / 125000000000 : ℝ)) =
+      Real.log (154403234977 / 125000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((29403234977 / 125000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (30106359977 / 125000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (264067314990576284203 / 1250000000000000000000 : ℝ) ≤
+      Real.log (154403234977 / 125000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (30106359977 / 125000000000 : ℝ) ≤
+      (-(7117886460851593151109 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (30106359977 / 125000000000 : ℝ) = (30106359977 / 15625000000 : ℝ) / 2 ^ 3 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (30106359977 / 15625000000 : ℝ) ≤
+            (3279321243648406848891 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2425004535129779 / 5000000000000000 : ℝ) ≤ Real.sqrt (29403234977 / 125000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (29403234977 / 125000000000 : ℝ) ≤ (4850009070259561 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (29403234977 / 125000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau345 :
+    Real.arctan (4850009070259561 / 10000000000000000 : ℝ) ≤
+      (4515764951354820672259 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (4515764951354820672259 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 4850009070259561 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 4850009070259561 : ℝ) =
+              (4850009070259561 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau345)
+  have hz1 :
+    (10000000000000000 / 4850009070259561 : ℝ) ≤ 1 / Real.sqrt (29403234977 / 125000000000 : ℝ) :=
+  by
+    rw [show
+        (10000000000000000 / 4850009070259561 : ℝ) =
+          1 / (4850009070259561 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 2425004535129779 : ℝ) ≤
+      (383559097835192323907 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (29403234977 / 125000000000 : ℝ) ≤
+      (375000000000000 / 2425004535129779 : ℝ) := by
+    rw [show
+        (375000000000000 / 2425004535129779 : ℝ) =
+          (3 / 40) / (2425004535129779 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (4515764951354820672259 / 10000000000000000000000 : ℝ)) -
+        6 * (383559097835192323907 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (29403234977 / 125000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (29403234977 / 125000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2425004535129779 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (4515764951354820672259 / 10000000000000000000000 : ℝ)) -
+          6 * (383559097835192323907 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (29403234977 / 125000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (29403234977 / 125000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (29403234977 / 125000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_284 :
+    ((2121166850524551193967 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1419160034343136087147 / 1000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((972197785381079 / 2000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2262325032459621594699 / 5000000000000000000000 : ℝ)) -
+              6 * (61232994040558072229 / 400000000000000000000 : ℝ)))) ≤
+      Vfield (7561348271199 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (7561348271199 / 32000000000000 : ℝ)) =
+      Real.log (39561348271199 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((7561348271199 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (7741348271199 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2121166850524551193967 / 10000000000000000000000 : ℝ) ≤
+      Real.log (39561348271199 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (7741348271199 / 32000000000000 : ℝ) ≤
+      (-(1419160034343136087147 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (7741348271199 / 32000000000000 : ℝ) = (7741348271199 / 4000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7741348271199 / 4000000000000 : ℝ) ≤
+            (660281506556863912853 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (972197785381079 / 2000000000000000 : ℝ) ≤ Real.sqrt (7561348271199 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (7561348271199 / 32000000000000 : ℝ) ≤ (2430494463452699 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (7561348271199 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau346 :
+    Real.arctan (2430494463452699 / 5000000000000000 : ℝ) ≤
+      (2262325032459621594699 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (2262325032459621594699 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (5000000000000000 / 2430494463452699 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 2430494463452699 : ℝ) =
+              (2430494463452699 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau346)
+  have hz1 :
+    (5000000000000000 / 2430494463452699 : ℝ) ≤
+      1 / Real.sqrt (7561348271199 / 32000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 2430494463452699 : ℝ) = 1 / (2430494463452699 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (150000000000000 / 972197785381079 : ℝ) ≤
+      (61232994040558072229 / 400000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (7561348271199 / 32000000000000 : ℝ) ≤
+      (150000000000000 / 972197785381079 : ℝ) := by
+    rw [show
+        (150000000000000 / 972197785381079 : ℝ) =
+          (3 / 40) / (972197785381079 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2262325032459621594699 / 5000000000000000000000 : ℝ)) -
+        6 * (61232994040558072229 / 400000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (7561348271199 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (7561348271199 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((972197785381079 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2262325032459621594699 / 5000000000000000000000 : ℝ)) -
+          6 * (61232994040558072229 / 400000000000000000000 : ℝ))) ≤
+      Real.sqrt (7561348271199 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (7561348271199 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (7561348271199 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_285 :
+    ((266223467841653341547 / 1250000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(14147622029027548248203 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2435972019204743 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (1133376877931879777319 / 2500000000000000000000 : ℝ)) -
+              6 * (95464748052574244179 / 625000000000000000000 : ℝ)))) ≤
+      Vfield (3797734194143 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (3797734194143 / 16000000000000 : ℝ)) =
+      Real.log (19797734194143 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((3797734194143 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (3887734194143 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (266223467841653341547 / 1250000000000000000000 : ℝ) ≤
+      Real.log (19797734194143 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (3887734194143 / 16000000000000 : ℝ) ≤
+      (-(14147622029027548248203 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (3887734194143 / 16000000000000 : ℝ) = (3887734194143 / 2000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3887734194143 / 2000000000000 : ℝ) ≤
+            (6646793379972451751797 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2435972019204743 / 5000000000000000 : ℝ) ≤ Real.sqrt (3797734194143 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (3797734194143 / 16000000000000 : ℝ) ≤ (4871944038409489 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (3797734194143 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau347 :
+    Real.arctan (4871944038409489 / 10000000000000000 : ℝ) ≤
+      (1133376877931879777319 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (1133376877931879777319 / 2500000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 4871944038409489 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 4871944038409489 : ℝ) =
+              (4871944038409489 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau347)
+  have hz1 :
+    (10000000000000000 / 4871944038409489 : ℝ) ≤
+      1 / Real.sqrt (3797734194143 / 16000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 4871944038409489 : ℝ) =
+          1 / (4871944038409489 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 2435972019204743 : ℝ) ≤
+      (95464748052574244179 / 625000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (3797734194143 / 16000000000000 : ℝ) ≤
+      (375000000000000 / 2435972019204743 : ℝ) := by
+    rw [show
+        (375000000000000 / 2435972019204743 : ℝ) =
+          (3 / 40) / (2435972019204743 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (1133376877931879777319 / 2500000000000000000000 : ℝ)) -
+        6 * (95464748052574244179 / 625000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (3797734194143 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (3797734194143 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2435972019204743 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (1133376877931879777319 / 2500000000000000000000 : ℝ)) -
+          6 * (95464748052574244179 / 625000000000000000000 : ℝ))) ≤
+      Real.sqrt (3797734194143 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (3797734194143 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (3797734194143 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_286 :
+    ((2147007263199972397 / 10000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1757530176157463034543 / 1250000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((4893780690344377 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (568892516491583260051 / 1250000000000000000000 : ℝ)) -
+              6 * (760362591960730908383 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (383185431123 / 1600000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (383185431123 / 1600000000000 : ℝ)) =
+      Real.log (1983185431123 / 1600000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((383185431123 / 1600000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (392185431123 / 1600000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2147007263199972397 / 10000000000000000000 : ℝ) ≤
+      Real.log (1983185431123 / 1600000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (392185431123 / 1600000000000 : ℝ) ≤
+      (-(1757530176157463034543 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (392185431123 / 1600000000000 : ℝ) = (392185431123 / 200000000000 : ℝ) / 2 ^ 3 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (392185431123 / 200000000000 : ℝ) ≤
+            (841771749967536965457 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (4893780690344377 / 10000000000000000 : ℝ) ≤ Real.sqrt (383185431123 / 1600000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (383185431123 / 1600000000000 : ℝ) ≤ (244689034517219 / 500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (383185431123 / 1600000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau348 :
+    Real.arctan (244689034517219 / 500000000000000 : ℝ) ≤
+      (568892516491583260051 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (568892516491583260051 / 1250000000000000000000 : ℝ)) ≤
+      Real.arctan (500000000000000 / 244689034517219 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (500000000000000 / 244689034517219 : ℝ) =
+              (244689034517219 / 500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau348)
+  have hz1 :
+    (500000000000000 / 244689034517219 : ℝ) ≤ 1 / Real.sqrt (383185431123 / 1600000000000 : ℝ) := by
+    rw [show (500000000000000 / 244689034517219 : ℝ) = 1 / (244689034517219 / 500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 4893780690344377 : ℝ) ≤
+      (760362591960730908383 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (383185431123 / 1600000000000 : ℝ) ≤
+      (750000000000000 / 4893780690344377 : ℝ) := by
+    rw [show
+        (750000000000000 / 4893780690344377 : ℝ) =
+          (3 / 40) / (4893780690344377 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (568892516491583260051 / 1250000000000000000000 : ℝ)) -
+        6 * (760362591960730908383 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (383185431123 / 1600000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (383185431123 / 1600000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((4893780690344377 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (568892516491583260051 / 1250000000000000000000 : ℝ)) -
+          6 * (760362591960730908383 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (383185431123 / 1600000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (383185431123 / 1600000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (383185431123 / 1600000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_287 :
+    ((541049295860268807619 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(13973617717453968005883 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((4915520336340929 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (228433212307367662667 / 500000000000000000000 : ℝ)) -
+              6 * (75705104221060272141 / 500000000000000000000 : ℝ)))) ≤
+      Vfield (3865974428317 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (3865974428317 / 16000000000000 : ℝ)) =
+      Real.log (19865974428317 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((3865974428317 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (3955974428317 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (541049295860268807619 / 2500000000000000000000 : ℝ) ≤
+      Real.log (19865974428317 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (3955974428317 / 16000000000000 : ℝ) ≤
+      (-(13973617717453968005883 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (3955974428317 / 16000000000000 : ℝ) = (3955974428317 / 2000000000000 : ℝ) / 2 ^ 3
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3955974428317 / 2000000000000 : ℝ) ≤
+            (6820797691546031994117 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (4915520336340929 / 10000000000000000 : ℝ) ≤ Real.sqrt (3865974428317 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (3865974428317 / 16000000000000 : ℝ) ≤ (1228880084085233 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (3865974428317 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau349 :
+    Real.arctan (1228880084085233 / 2500000000000000 : ℝ) ≤
+      (228433212307367662667 / 500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (228433212307367662667 / 500000000000000000000 : ℝ)) ≤
+      Real.arctan (2500000000000000 / 1228880084085233 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 1228880084085233 : ℝ) =
+              (1228880084085233 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau349)
+  have hz1 :
+    (2500000000000000 / 1228880084085233 : ℝ) ≤
+      1 / Real.sqrt (3865974428317 / 16000000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 1228880084085233 : ℝ) = 1 / (1228880084085233 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 4915520336340929 : ℝ) ≤
+      (75705104221060272141 / 500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (3865974428317 / 16000000000000 : ℝ) ≤
+      (750000000000000 / 4915520336340929 : ℝ) := by
+    rw [show
+        (750000000000000 / 4915520336340929 : ℝ) =
+          (3 / 40) / (4915520336340929 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (228433212307367662667 / 500000000000000000000 : ℝ)) -
+        6 * (75705104221060272141 / 500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (3865974428317 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (3865974428317 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((4915520336340929 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (228433212307367662667 / 500000000000000000000 : ℝ)) -
+          6 * (75705104221060272141 / 500000000000000000000 : ℝ))) ≤
+      Real.sqrt (3865974428317 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (3865974428317 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (3865974428317 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_288 :
+    ((545339401261763887393 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(3471934488122335850373 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((4937164257828069 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (36688650063609893199 / 80000000000000000000 : ℝ)) -
+              6 * (12060518217518261529 / 80000000000000000000 : ℝ)))) ≤
+      Vfield (975023636351 / 4000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (975023636351 / 4000000000000 : ℝ)) =
+      Real.log (4975023636351 / 4000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((975023636351 / 4000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (997523636351 / 4000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (545339401261763887393 / 2500000000000000000000 : ℝ) ≤
+      Real.log (4975023636351 / 4000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (997523636351 / 4000000000000 : ℝ) ≤
+      (-(3471934488122335850373 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 3
+        (show (997523636351 / 4000000000000 : ℝ) = (997523636351 / 500000000000 : ℝ) / 2 ^ 3 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 15
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (997523636351 / 500000000000 : ℝ) ≤
+            (1726669364127664149627 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (4937164257828069 / 10000000000000000 : ℝ) ≤ Real.sqrt (975023636351 / 4000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (975023636351 / 4000000000000 : ℝ) ≤ (617145532228509 / 1250000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (975023636351 / 4000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau350 :
+    Real.arctan (617145532228509 / 1250000000000000 : ℝ) ≤
+      (36688650063609893199 / 80000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (36688650063609893199 / 80000000000000000000 : ℝ)) ≤
+      Real.arctan (1250000000000000 / 617145532228509 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1250000000000000 / 617145532228509 : ℝ) =
+              (617145532228509 / 1250000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau350)
+  have hz1 :
+    (1250000000000000 / 617145532228509 : ℝ) ≤ 1 / Real.sqrt (975023636351 / 4000000000000 : ℝ) :=
+  by
+    rw [show
+        (1250000000000000 / 617145532228509 : ℝ) = 1 / (617145532228509 / 1250000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 1645721419276023 : ℝ) ≤
+      (12060518217518261529 / 80000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (975023636351 / 4000000000000 : ℝ) ≤
+      (250000000000000 / 1645721419276023 : ℝ) := by
+    rw [show
+        (250000000000000 / 1645721419276023 : ℝ) =
+          (3 / 40) / (4937164257828069 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (36688650063609893199 / 80000000000000000000 : ℝ)) -
+        6 * (12060518217518261529 / 80000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (975023636351 / 4000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (975023636351 / 4000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((4937164257828069 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (36688650063609893199 / 80000000000000000000 : ℝ)) -
+          6 * (12060518217518261529 / 80000000000000000000 : ℝ))) ≤
+      Real.sqrt (975023636351 / 4000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (975023636351 / 4000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (975023636351 / 4000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/V24.lean b/LeanPool/Zeta5Irrational/Table/V24.lean
new file mode 100644
index 0000000000..0f46be5c85
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/V24.lean
@@ -0,0 +1,1398 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Energy
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # V24: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_289 :
+    ((2198488629086328392477 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1725323680994778098129 / 1250000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((4958713708268379 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (184135701649771270463 / 400000000000000000000 : ℝ)) -
+              6 * (750555712906224550967 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (3934214662491 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (3934214662491 / 16000000000000 : ℝ)) =
+      Real.log (19934214662491 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((3934214662491 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (4024214662491 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2198488629086328392477 / 10000000000000000000000 : ℝ) ≤
+      Real.log (19934214662491 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (4024214662491 / 16000000000000 : ℝ) ≤
+      (-(1725323680994778098129 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (4024214662491 / 16000000000000 : ℝ) = (4024214662491 / 4000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4024214662491 / 4000000000000 : ℝ) ≤
+            (7544269755221901871 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (4958713708268379 / 10000000000000000 : ℝ) ≤ Real.sqrt (3934214662491 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (3934214662491 / 16000000000000 : ℝ) ≤ (2479356854134191 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (3934214662491 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau351 :
+    Real.arctan (2479356854134191 / 5000000000000000 : ℝ) ≤
+      (184135701649771270463 / 400000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (184135701649771270463 / 400000000000000000000 : ℝ)) ≤
+      Real.arctan (5000000000000000 / 2479356854134191 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 2479356854134191 : ℝ) =
+              (2479356854134191 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau351)
+  have hz1 :
+    (5000000000000000 / 2479356854134191 : ℝ) ≤
+      1 / Real.sqrt (3934214662491 / 16000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 2479356854134191 : ℝ) = 1 / (2479356854134191 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 1652904569422793 : ℝ) ≤
+      (750555712906224550967 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (3934214662491 / 16000000000000 : ℝ) ≤
+      (250000000000000 / 1652904569422793 : ℝ) := by
+    rw [show
+        (250000000000000 / 1652904569422793 : ℝ) =
+          (3 / 40) / (4958713708268379 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (184135701649771270463 / 400000000000000000000 : ℝ)) -
+        6 * (750555712906224550967 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (3934214662491 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (3934214662491 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((4958713708268379 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (184135701649771270463 / 400000000000000000000 : ℝ)) -
+          6 * (750555712906224550967 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (3934214662491 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (3934214662491 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (3934214662491 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_290 :
+    ((1107795178054387541519 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(13718159850552214083469 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((622521239250649 / 1250000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2310299720564564919953 / 5000000000000000000000 : ℝ)) -
+              6 * (1494740248442681908947 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (1984167389789 / 8000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1984167389789 / 8000000000000 : ℝ)) =
+      Real.log (9984167389789 / 8000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1984167389789 / 8000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (2029167389789 / 8000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1107795178054387541519 / 5000000000000000000000 : ℝ) ≤
+      Real.log (9984167389789 / 8000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (2029167389789 / 8000000000000 : ℝ) ≤
+      (-(13718159850552214083469 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (2029167389789 / 8000000000000 : ℝ) = (2029167389789 / 2000000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2029167389789 / 2000000000000 : ℝ) ≤
+            (144783755447785916531 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (622521239250649 / 1250000000000000 : ℝ) ≤ Real.sqrt (1984167389789 / 8000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1984167389789 / 8000000000000 : ℝ) ≤ (996033982801039 / 2000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1984167389789 / 8000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau352 :
+    Real.arctan (996033982801039 / 2000000000000000 : ℝ) ≤
+      (2310299720564564919953 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (Real.pi / 2 - (2310299720564564919953 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (2000000000000000 / 996033982801039 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2000000000000000 / 996033982801039 : ℝ) =
+              (996033982801039 / 2000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau352)
+  have hz1 :
+    (2000000000000000 / 996033982801039 : ℝ) ≤
+      1 / Real.sqrt (1984167389789 / 8000000000000 : ℝ) := by
+    rw [show
+        (2000000000000000 / 996033982801039 : ℝ) = 1 / (996033982801039 / 2000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (93750000000000 / 622521239250649 : ℝ) ≤
+      (1494740248442681908947 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1984167389789 / 8000000000000 : ℝ) ≤
+      (93750000000000 / 622521239250649 : ℝ) := by
+    rw [show
+        (93750000000000 / 622521239250649 : ℝ) =
+          (3 / 40) / (622521239250649 / 1250000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2310299720564564919953 / 5000000000000000000000 : ℝ)) -
+        6 * (1494740248442681908947 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1984167389789 / 8000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1984167389789 / 8000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((622521239250649 / 1250000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2310299720564564919953 / 5000000000000000000000 : ℝ)) -
+          6 * (1494740248442681908947 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (1984167389789 / 8000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1984167389789 / 8000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1984167389789 / 8000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_291 :
+    ((281213289841407876621 / 1250000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(3387852383232238012393 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((502280736600061 / 1000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (581838166519744333303 / 2500000000000000000000 : ℝ))) -
+              6 * (1482237548872014842613 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (504571876719 / 2000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (504571876719 / 2000000000000 : ℝ)) =
+      Real.log (2504571876719 / 2000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((504571876719 / 2000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (515821876719 / 2000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (281213289841407876621 / 1250000000000000000000 : ℝ) ≤
+      Real.log (2504571876719 / 2000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (515821876719 / 2000000000000 : ℝ) ≤
+      (-(3387852383232238012393 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (515821876719 / 2000000000000 : ℝ) = (515821876719 / 500000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (515821876719 / 500000000000 : ℝ) ≤
+            (77883518267761987607 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (502280736600061 / 1000000000000000 : ℝ) ≤ Real.sqrt (504571876719 / 2000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (504571876719 / 2000000000000 : ℝ) ≤ (5022807366000613 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (504571876719 / 2000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau354 :
+    Real.arctan (5022807366000613 / 21190558245054175 : ℝ) ≤
+      (581838166519744333303 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau353 :
+    Real.arctan (5022807366000613 / 10000000000000000 : ℝ) ≤
+      (2 * (581838166519744333303 / 2500000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (447622329802167 / 400000000000000 : ℝ) ≤
+          Real.sqrt (1 + (5022807366000613 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (5022807366000613 / 10000000000000000 : ℝ) /
+                (1 + (447622329802167 / 400000000000000 : ℝ)) =
+              (5022807366000613 / 21190558245054175 : ℝ)
+            by
+        norm_num];
+        exact hau354)
+  have hat1 :
+    (Real.pi / 2 - (2 * (581838166519744333303 / 2500000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 5022807366000613 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 5022807366000613 : ℝ) =
+              (5022807366000613 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau353)
+  have hz1 :
+    (10000000000000000 / 5022807366000613 : ℝ) ≤
+      1 / Real.sqrt (504571876719 / 2000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 5022807366000613 : ℝ) =
+          1 / (5022807366000613 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (75000000000000 / 502280736600061 : ℝ) ≤
+      (1482237548872014842613 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (504571876719 / 2000000000000 : ℝ) ≤
+      (75000000000000 / 502280736600061 : ℝ) := by
+    rw [show
+        (75000000000000 / 502280736600061 : ℝ) =
+          (3 / 40) / (502280736600061 / 1000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (581838166519744333303 / 2500000000000000000000 : ℝ))) -
+        6 * (1482237548872014842613 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (504571876719 / 2000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (504571876719 / 2000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((502280736600061 / 1000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (581838166519744333303 / 2500000000000000000000 : ℝ))) -
+          6 * (1482237548872014842613 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (504571876719 / 2000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (504571876719 / 2000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (504571876719 / 2000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_292 :
+    ((1141853143538352302107 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(13387394237893872067869 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((31656786952219 / 62500000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2344204573684607949139 / 10000000000000000000000 : ℝ))) -
+              6 * (36751085971136076113 / 250000000000000000000 : ℝ)))) ≤
+      Vfield (2052407623963 / 8000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (2052407623963 / 8000000000000 : ℝ)) =
+      Real.log (10052407623963 / 8000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((2052407623963 / 8000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (2097407623963 / 8000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1141853143538352302107 / 5000000000000000000000 : ℝ) ≤
+      Real.log (10052407623963 / 8000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (2097407623963 / 8000000000000 : ℝ) ≤
+      (-(13387394237893872067869 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (2097407623963 / 8000000000000 : ℝ) = (2097407623963 / 2000000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2097407623963 / 2000000000000 : ℝ) ≤
+            (475549368106127932131 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (31656786952219 / 62500000000000 : ℝ) ≤ Real.sqrt (2052407623963 / 8000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (2052407623963 / 8000000000000 : ℝ) ≤ (5065085912355043 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (2052407623963 / 8000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau356 :
+    Real.arctan (5065085912355043 / 21209598355852786 : ℝ) ≤
+      (2344204573684607949139 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau355 :
+    Real.arctan (5065085912355043 / 10000000000000000 : ℝ) ≤
+      (2 * (2344204573684607949139 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (5604799177926393 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (5065085912355043 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (5065085912355043 / 10000000000000000 : ℝ) /
+                (1 + (5604799177926393 / 5000000000000000 : ℝ)) =
+              (5065085912355043 / 21209598355852786 : ℝ)
+            by
+        norm_num];
+        exact hau356)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2344204573684607949139 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 5065085912355043 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 5065085912355043 : ℝ) =
+              (5065085912355043 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau355)
+  have hz1 :
+    (10000000000000000 / 5065085912355043 : ℝ) ≤
+      1 / Real.sqrt (2052407623963 / 8000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 5065085912355043 : ℝ) =
+          1 / (5065085912355043 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (4687500000000 / 31656786952219 : ℝ) ≤
+      (36751085971136076113 / 250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (2052407623963 / 8000000000000 : ℝ) ≤
+      (4687500000000 / 31656786952219 : ℝ) := by
+    rw [show
+        (4687500000000 / 31656786952219 : ℝ) = (3 / 40) / (31656786952219 / 62500000000000 : ℝ) by
+        norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2344204573684607949139 / 10000000000000000000000 : ℝ))) -
+        6 * (36751085971136076113 / 250000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (2052407623963 / 8000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (2052407623963 / 8000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((31656786952219 / 62500000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2344204573684607949139 / 10000000000000000000000 : ℝ))) -
+          6 * (36751085971136076113 / 250000000000000000000 : ℝ))) ≤
+      Real.sqrt (2052407623963 / 8000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (2052407623963 / 8000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (2052407623963 / 8000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_293 :
+    ((1158795523617137301057 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(13226025692197263579119 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2553507233352051 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2 * (147553771669833188447 / 625000000000000000000 : ℝ))) -
+              6 * (72907271447543795003 / 500000000000000000000 : ℝ)))) ≤
+      Vfield (41730554821 / 160000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (41730554821 / 160000000000 : ℝ)) =
+      Real.log (201730554821 / 160000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((41730554821 / 160000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (42630554821 / 160000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1158795523617137301057 / 5000000000000000000000 : ℝ) ≤
+      Real.log (201730554821 / 160000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (42630554821 / 160000000000 : ℝ) ≤
+      (-(13226025692197263579119 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (42630554821 / 160000000000 : ℝ) = (42630554821 / 40000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (42630554821 / 40000000000 : ℝ) ≤
+            (636917913802736420881 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2553507233352051 / 5000000000000000 : ℝ) ≤ Real.sqrt (41730554821 / 160000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (41730554821 / 160000000000 : ℝ) ≤ (1021402893340821 / 2000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (41730554821 / 160000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau358 :
+    Real.arctan (5107014466704105 / 21228606180783302 : ℝ) ≤
+      (147553771669833188447 / 625000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau357 :
+    Real.arctan (1021402893340821 / 2000000000000000 : ℝ) ≤
+      (2 * (147553771669833188447 / 625000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (5614303090391651 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1021402893340821 / 2000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1021402893340821 / 2000000000000000 : ℝ) /
+                (1 + (5614303090391651 / 5000000000000000 : ℝ)) =
+              (5107014466704105 / 21228606180783302 : ℝ)
+            by
+        norm_num];
+        exact hau358)
+  have hat1 :
+    (Real.pi / 2 - (2 * (147553771669833188447 / 625000000000000000000 : ℝ))) ≤
+      Real.arctan (2000000000000000 / 1021402893340821 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2000000000000000 / 1021402893340821 : ℝ) =
+              (1021402893340821 / 2000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau357)
+  have hz1 :
+    (2000000000000000 / 1021402893340821 : ℝ) ≤ 1 / Real.sqrt (41730554821 / 160000000000 : ℝ) := by
+    rw [show
+        (2000000000000000 / 1021402893340821 : ℝ) = 1 / (1021402893340821 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (125000000000000 / 851169077784017 : ℝ) ≤
+      (72907271447543795003 / 500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (41730554821 / 160000000000 : ℝ) ≤
+      (125000000000000 / 851169077784017 : ℝ) := by
+    rw [show
+        (125000000000000 / 851169077784017 : ℝ) =
+          (3 / 40) / (2553507233352051 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (147553771669833188447 / 625000000000000000000 : ℝ))) -
+        6 * (72907271447543795003 / 500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (41730554821 / 160000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (41730554821 / 160000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2553507233352051 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (147553771669833188447 / 625000000000000000000 : ℝ))) -
+          6 * (72907271447543795003 / 500000000000000000000 : ℝ))) ≤
+      Real.sqrt (41730554821 / 160000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (41730554821 / 160000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (41730554821 / 160000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_294 :
+    ((2385018047629768990871 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(20173275815075893487 / 15625000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2594927729740271 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1196801073083827964937 / 5000000000000000000000 : ℝ))) -
+              6 * (1435191184916654243353 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (269345996903 / 1000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (269345996903 / 1000000000000 : ℝ)) =
+      Real.log (1269345996903 / 1000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((269345996903 / 1000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (274970996903 / 1000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2385018047629768990871 / 10000000000000000000000 : ℝ) ≤
+      Real.log (1269345996903 / 1000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (274970996903 / 1000000000000 : ℝ) ≤
+      (-(20173275815075893487 / 15625000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (274970996903 / 1000000000000 : ℝ) = (274970996903 / 250000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (274970996903 / 250000000000 : ℝ) ≤
+            (1487573569299106513 / 15625000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2594927729740271 / 5000000000000000 : ℝ) ≤ Real.sqrt (269345996903 / 1000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (269345996903 / 1000000000000 : ℝ) ≤ (1037971091896109 / 2000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (269345996903 / 1000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau360 :
+    Real.arctan (5189855459480545 / 21266525626398761 : ℝ) ≤
+      (1196801073083827964937 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau359 :
+    Real.arctan (1037971091896109 / 2000000000000000 : ℝ) ≤
+      (2 * (1196801073083827964937 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (11266525626398761 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1037971091896109 / 2000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1037971091896109 / 2000000000000000 : ℝ) /
+                (1 + (11266525626398761 / 10000000000000000 : ℝ)) =
+              (5189855459480545 / 21266525626398761 : ℝ)
+            by
+        norm_num];
+        exact hau360)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1196801073083827964937 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (2000000000000000 / 1037971091896109 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2000000000000000 / 1037971091896109 : ℝ) =
+              (1037971091896109 / 2000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau359)
+  have hz1 :
+    (2000000000000000 / 1037971091896109 : ℝ) ≤
+      1 / Real.sqrt (269345996903 / 1000000000000 : ℝ) := by
+    rw [show
+        (2000000000000000 / 1037971091896109 : ℝ) = 1 / (1037971091896109 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 2594927729740271 : ℝ) ≤
+      (1435191184916654243353 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (269345996903 / 1000000000000 : ℝ) ≤
+      (375000000000000 / 2594927729740271 : ℝ) := by
+    rw [show
+        (375000000000000 / 2594927729740271 : ℝ) =
+          (3 / 40) / (2594927729740271 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1196801073083827964937 / 5000000000000000000000 : ℝ))) -
+        6 * (1435191184916654243353 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (269345996903 / 1000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (269345996903 / 1000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2594927729740271 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1196801073083827964937 / 5000000000000000000000 : ℝ))) -
+          6 * (1435191184916654243353 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (269345996903 / 1000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (269345996903 / 1000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (269345996903 / 1000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_295 :
+    ((300517427895070866779 / 1250000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(12822930291729512624071 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((5213209021966759 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2 * (96111696549200328527 / 400000000000000000000 : ℝ))) -
+              6 * (714424657120258015799 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (8696815458149 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (8696815458149 / 32000000000000 : ℝ)) =
+      Real.log (40696815458149 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((8696815458149 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (8876815458149 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (300517427895070866779 / 1250000000000000000000 : ℝ) ≤
+      Real.log (40696815458149 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (8876815458149 / 32000000000000 : ℝ) ≤
+      (-(12822930291729512624071 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (8876815458149 / 32000000000000 : ℝ) = (8876815458149 / 8000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8876815458149 / 8000000000000 : ℝ) ≤
+            (1040013314270487375929 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (5213209021966759 / 10000000000000000 : ℝ) ≤ Real.sqrt (8696815458149 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (8696815458149 / 32000000000000 : ℝ) ≤ (2606604510983381 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (8696815458149 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau362 :
+    Real.arctan (2606604510983381 / 10638651175297059 : ℝ) ≤
+      (96111696549200328527 / 400000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau361 :
+    Real.arctan (2606604510983381 / 5000000000000000 : ℝ) ≤
+      (2 * (96111696549200328527 / 400000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (5638651175297059 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (2606604510983381 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (2606604510983381 / 5000000000000000 : ℝ) /
+                (1 + (5638651175297059 / 5000000000000000 : ℝ)) =
+              (2606604510983381 / 10638651175297059 : ℝ)
+            by
+        norm_num];
+        exact hau362)
+  have hat1 :
+    (Real.pi / 2 - (2 * (96111696549200328527 / 400000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 2606604510983381 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 2606604510983381 : ℝ) =
+              (2606604510983381 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau361)
+  have hz1 :
+    (5000000000000000 / 2606604510983381 : ℝ) ≤
+      1 / Real.sqrt (8696815458149 / 32000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 2606604510983381 : ℝ) = 1 / (2606604510983381 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 5213209021966759 : ℝ) ≤
+      (714424657120258015799 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (8696815458149 / 32000000000000 : ℝ) ≤
+      (750000000000000 / 5213209021966759 : ℝ) := by
+    rw [show
+        (750000000000000 / 5213209021966759 : ℝ) =
+          (3 / 40) / (5213209021966759 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (96111696549200328527 / 400000000000000000000 : ℝ))) -
+        6 * (714424657120258015799 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (8696815458149 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (8696815458149 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((5213209021966759 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (96111696549200328527 / 400000000000000000000 : ℝ))) -
+          6 * (714424657120258015799 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (8696815458149 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (8696815458149 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (8696815458149 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_296 :
+    ((1211612152879776726763 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(12735731124909343586647 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2618229216623863 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2411924233564044653293 / 10000000000000000000000 : ℝ))) -
+              6 * (56903631257368425313 / 400000000000000000000 : ℝ)))) ≤
+      Vfield (4387279507701 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (4387279507701 / 16000000000000 : ℝ)) =
+      Real.log (20387279507701 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((4387279507701 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (4477279507701 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1211612152879776726763 / 5000000000000000000000 : ℝ) ≤
+      Real.log (20387279507701 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (4477279507701 / 16000000000000 : ℝ) ≤
+      (-(12735731124909343586647 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (4477279507701 / 16000000000000 : ℝ) = (4477279507701 / 4000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4477279507701 / 4000000000000 : ℝ) ≤
+            (1127212481090656413353 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2618229216623863 / 5000000000000000 : ℝ) ≤ Real.sqrt (4387279507701 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (4387279507701 / 16000000000000 : ℝ) ≤ (5236458433247729 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (4387279507701 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau364 :
+    Real.arctan (5236458433247729 / 21288068786250872 : ℝ) ≤
+      (2411924233564044653293 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau363 :
+    Real.arctan (5236458433247729 / 10000000000000000 : ℝ) ≤
+      (2 * (2411924233564044653293 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (1411008598281359 / 1250000000000000 : ℝ) ≤
+          Real.sqrt (1 + (5236458433247729 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (5236458433247729 / 10000000000000000 : ℝ) /
+                (1 + (1411008598281359 / 1250000000000000 : ℝ)) =
+              (5236458433247729 / 21288068786250872 : ℝ)
+            by
+        norm_num];
+        exact hau364)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2411924233564044653293 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 5236458433247729 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 5236458433247729 : ℝ) =
+              (5236458433247729 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau363)
+  have hz1 :
+    (10000000000000000 / 5236458433247729 : ℝ) ≤
+      1 / Real.sqrt (4387279507701 / 16000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 5236458433247729 : ℝ) =
+          1 / (5236458433247729 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 2618229216623863 : ℝ) ≤
+      (56903631257368425313 / 400000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (4387279507701 / 16000000000000 : ℝ) ≤
+      (375000000000000 / 2618229216623863 : ℝ) := by
+    rw [show
+        (375000000000000 / 2618229216623863 : ℝ) =
+          (3 / 40) / (2618229216623863 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2411924233564044653293 / 10000000000000000000000 : ℝ))) -
+        6 * (56903631257368425313 / 400000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (4387279507701 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (4387279507701 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2618229216623863 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2411924233564044653293 / 10000000000000000000000 : ℝ))) -
+          6 * (56903631257368425313 / 400000000000000000000 : ℝ))) ≤
+      Real.sqrt (4387279507701 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (4387279507701 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (4387279507701 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_297 :
+    ((152642052153384038571 / 625000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2529857151842140007031 / 2000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((5259605074484857 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1210499184897321206457 / 5000000000000000000000 : ℝ))) -
+              6 * (708206888526691726481 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (1770460514531 / 6400000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1770460514531 / 6400000000000 : ℝ)) =
+      Real.log (8170460514531 / 6400000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1770460514531 / 6400000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1806460514531 / 6400000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (152642052153384038571 / 625000000000000000000 : ℝ) ≤
+      Real.log (8170460514531 / 6400000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1806460514531 / 6400000000000 : ℝ) ≤
+      (-(2529857151842140007031 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (1806460514531 / 6400000000000 : ℝ) = (1806460514531 / 1600000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1806460514531 / 1600000000000 : ℝ) ≤
+            (242731569357859992969 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (5259605074484857 / 10000000000000000 : ℝ) ≤ Real.sqrt (1770460514531 / 6400000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1770460514531 / 6400000000000 : ℝ) ≤ (262980253724243 / 500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1770460514531 / 6400000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau366 :
+    Real.arctan (5259605074484860 / 21298824962780283 : ℝ) ≤
+      (1210499184897321206457 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau365 :
+    Real.arctan (262980253724243 / 500000000000000 : ℝ) ≤
+      (2 * (1210499184897321206457 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (11298824962780283 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (262980253724243 / 500000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (262980253724243 / 500000000000000 : ℝ) /
+                (1 + (11298824962780283 / 10000000000000000 : ℝ)) =
+              (5259605074484860 / 21298824962780283 : ℝ)
+            by
+        norm_num];
+        exact hau366)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1210499184897321206457 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (500000000000000 / 262980253724243 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (500000000000000 / 262980253724243 : ℝ) =
+              (262980253724243 / 500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau365)
+  have hz1 :
+    (500000000000000 / 262980253724243 : ℝ) ≤ 1 / Real.sqrt (1770460514531 / 6400000000000 : ℝ) :=
+  by
+    rw [show (500000000000000 / 262980253724243 : ℝ) = 1 / (262980253724243 / 500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 5259605074484857 : ℝ) ≤
+      (708206888526691726481 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1770460514531 / 6400000000000 : ℝ) ≤
+      (750000000000000 / 5259605074484857 : ℝ) := by
+    rw [show
+        (750000000000000 / 5259605074484857 : ℝ) =
+          (3 / 40) / (5259605074484857 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1210499184897321206457 / 5000000000000000000000 : ℝ))) -
+        6 * (708206888526691726481 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1770460514531 / 6400000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1770460514531 / 6400000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((5259605074484857 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1210499184897321206457 / 5000000000000000000000 : ℝ))) -
+          6 * (708206888526691726481 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (1770460514531 / 6400000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1770460514531 / 6400000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1770460514531 / 6400000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_298 :
+    ((1225891754658183301957 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(6303170850486857560717 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((658892534956321 / 1250000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (303189255113144357391 / 1250000000000000000000 : ℝ))) -
+              6 * (353338824384936034991 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (17782348702563 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (17782348702563 / 64000000000000 : ℝ)) =
+      Real.log (81782348702563 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((17782348702563 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (18142348702563 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1225891754658183301957 / 5000000000000000000000 : ℝ) ≤
+      Real.log (81782348702563 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (18142348702563 / 64000000000000 : ℝ) ≤
+      (-(6303170850486857560717 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (18142348702563 / 64000000000000 : ℝ) = (18142348702563 / 16000000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18142348702563 / 16000000000000 : ℝ) ≤
+            (628300952513142439283 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (658892534956321 / 1250000000000000 : ℝ) ≤ Real.sqrt (17782348702563 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (17782348702563 / 64000000000000 : ℝ) ≤
+      (5271140279650571 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (17782348702563 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau368 :
+    Real.arctan (5271140279650571 / 21304199213024984 : ℝ) ≤
+      (303189255113144357391 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau367 :
+    Real.arctan (5271140279650571 / 10000000000000000 : ℝ) ≤
+      (2 * (303189255113144357391 / 1250000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (1413024901628123 / 1250000000000000 : ℝ) ≤
+          Real.sqrt (1 + (5271140279650571 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (5271140279650571 / 10000000000000000 : ℝ) /
+                (1 + (1413024901628123 / 1250000000000000 : ℝ)) =
+              (5271140279650571 / 21304199213024984 : ℝ)
+            by
+        norm_num];
+        exact hau368)
+  have hat1 :
+    (Real.pi / 2 - (2 * (303189255113144357391 / 1250000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 5271140279650571 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 5271140279650571 : ℝ) =
+              (5271140279650571 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau367)
+  have hz1 :
+    (10000000000000000 / 5271140279650571 : ℝ) ≤
+      1 / Real.sqrt (17782348702563 / 64000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 5271140279650571 : ℝ) =
+          1 / (5271140279650571 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (93750000000000 / 658892534956321 : ℝ) ≤
+      (353338824384936034991 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (17782348702563 / 64000000000000 : ℝ) ≤
+      (93750000000000 / 658892534956321 : ℝ) := by
+    rw [show
+        (93750000000000 / 658892534956321 : ℝ) =
+          (3 / 40) / (658892534956321 / 1250000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (303189255113144357391 / 1250000000000000000000 : ℝ))) -
+        6 * (353338824384936034991 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (17782348702563 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (17782348702563 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((658892534956321 / 1250000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (303189255113144357391 / 1250000000000000000000 : ℝ))) -
+          6 * (353338824384936034991 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (17782348702563 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (17782348702563 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (17782348702563 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_299 :
+    ((1230642573739392974039 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(6281790636822936316249 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2641325148290271 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2430015570348441889311 / 10000000000000000000000 : ℝ))) -
+              6 * (282063309236787619361 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (2232511532477 / 8000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (2232511532477 / 8000000000000 : ℝ)) =
+      Real.log (10232511532477 / 8000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((2232511532477 / 8000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (2277511532477 / 8000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1230642573739392974039 / 5000000000000000000000 : ℝ) ≤
+      Real.log (10232511532477 / 8000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (2277511532477 / 8000000000000 : ℝ) ≤
+      (-(6281790636822936316249 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (2277511532477 / 8000000000000 : ℝ) = (2277511532477 / 2000000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2277511532477 / 2000000000000 : ℝ) ≤
+            (649681166177063683751 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2641325148290271 / 5000000000000000 : ℝ) ≤ Real.sqrt (2232511532477 / 8000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (2232511532477 / 8000000000000 : ℝ) ≤ (1056530059316109 / 2000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (2232511532477 / 8000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau370 :
+    Real.arctan (5282650296580545 / 21309570909453748 : ℝ) ≤
+      (2430015570348441889311 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau369 :
+    Real.arctan (1056530059316109 / 2000000000000000 : ℝ) ≤
+      (2 * (2430015570348441889311 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (2827392727363437 / 2500000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1056530059316109 / 2000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1056530059316109 / 2000000000000000 : ℝ) /
+                (1 + (2827392727363437 / 2500000000000000 : ℝ)) =
+              (5282650296580545 / 21309570909453748 : ℝ)
+            by
+        norm_num];
+        exact hau370)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2430015570348441889311 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (2000000000000000 / 1056530059316109 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2000000000000000 / 1056530059316109 : ℝ) =
+              (1056530059316109 / 2000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau369)
+  have hz1 :
+    (2000000000000000 / 1056530059316109 : ℝ) ≤
+      1 / Real.sqrt (2232511532477 / 8000000000000 : ℝ) := by
+    rw [show
+        (2000000000000000 / 1056530059316109 : ℝ) = 1 / (1056530059316109 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (125000000000000 / 880441716096757 : ℝ) ≤
+      (282063309236787619361 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (2232511532477 / 8000000000000 : ℝ) ≤
+      (125000000000000 / 880441716096757 : ℝ) := by
+    rw [show
+        (125000000000000 / 880441716096757 : ℝ) =
+          (3 / 40) / (2641325148290271 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2430015570348441889311 / 10000000000000000000000 : ℝ))) -
+        6 * (282063309236787619361 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (2232511532477 / 8000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (2232511532477 / 8000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2641325148290271 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2430015570348441889311 / 10000000000000000000000 : ℝ))) -
+          6 * (282063309236787619361 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (2232511532477 / 8000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (2232511532477 / 8000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (2232511532477 / 8000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_300 :
+    ((2470777766097794190011 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(6260501456740909970931 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((5294135289560543 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2434503049163477909489 / 10000000000000000000000 : ℝ))) -
+              6 * (1407297311798374862231 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (17937835817069 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (17937835817069 / 64000000000000 : ℝ)) =
+      Real.log (81937835817069 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((17937835817069 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (18297835817069 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2470777766097794190011 / 10000000000000000000000 : ℝ) ≤
+      Real.log (81937835817069 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (18297835817069 / 64000000000000 : ℝ) ≤
+      (-(6260501456740909970931 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (18297835817069 / 64000000000000 : ℝ) = (18297835817069 / 16000000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18297835817069 / 16000000000000 : ℝ) ≤
+            (670970346259090029069 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (5294135289560543 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (17937835817069 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (17937835817069 / 64000000000000 : ℝ) ≤ (2647067644780273 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (17937835817069 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau372 :
+    Real.arctan (5294135289560546 / 21314940055703799 : ℝ) ≤
+      (2434503049163477909489 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau371 :
+    Real.arctan (2647067644780273 / 5000000000000000 : ℝ) ≤
+      (2 * (2434503049163477909489 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (11314940055703799 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (2647067644780273 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (2647067644780273 / 5000000000000000 : ℝ) /
+                (1 + (11314940055703799 / 10000000000000000 : ℝ)) =
+              (5294135289560546 / 21314940055703799 : ℝ)
+            by
+        norm_num];
+        exact hau372)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2434503049163477909489 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 2647067644780273 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 2647067644780273 : ℝ) =
+              (2647067644780273 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau371)
+  have hz1 :
+    (5000000000000000 / 2647067644780273 : ℝ) ≤
+      1 / Real.sqrt (17937835817069 / 64000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 2647067644780273 : ℝ) = 1 / (2647067644780273 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 5294135289560543 : ℝ) ≤
+      (1407297311798374862231 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (17937835817069 / 64000000000000 : ℝ) ≤
+      (750000000000000 / 5294135289560543 : ℝ) := by
+    rw [show
+        (750000000000000 / 5294135289560543 : ℝ) =
+          (3 / 40) / (5294135289560543 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2434503049163477909489 / 10000000000000000000000 : ℝ))) -
+        6 * (1407297311798374862231 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (17937835817069 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (17937835817069 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((5294135289560543 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2434503049163477909489 / 10000000000000000000000 : ℝ))) -
+          6 * (1407297311798374862231 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (17937835817069 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (17937835817069 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (17937835817069 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/V25.lean b/LeanPool/Zeta5Irrational/Table/V25.lean
new file mode 100644
index 0000000000..231a02e67f
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/V25.lean
@@ -0,0 +1,1439 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Energy
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # V25: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_301 :
+    ((1240130691140485734679 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(12478605076626120675669 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((5305595421098193 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2 * (152436035464757284741 / 625000000000000000000 : ℝ))) -
+              6 * (175537173293389352119 / 1250000000000000000000 : ℝ)))) ≤
+      Vfield (9007789687161 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (9007789687161 / 32000000000000 : ℝ)) =
+      Real.log (41007789687161 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((9007789687161 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (9187789687161 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1240130691140485734679 / 5000000000000000000000 : ℝ) ≤
+      Real.log (41007789687161 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (9187789687161 / 32000000000000 : ℝ) ≤
+      (-(12478605076626120675669 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (9187789687161 / 32000000000000 : ℝ) = (9187789687161 / 8000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9187789687161 / 8000000000000 : ℝ) ≤
+            (1384338529373879324331 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (5305595421098193 / 10000000000000000 : ℝ) ≤ Real.sqrt (9007789687161 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (9007789687161 / 32000000000000 : ℝ) ≤ (1326398855274549 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (9007789687161 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau374 :
+    Real.arctan (1326398855274549 / 5330076663850934 : ℝ) ≤
+      (152436035464757284741 / 625000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau373 :
+    Real.arctan (1326398855274549 / 2500000000000000 : ℝ) ≤
+      (2 * (152436035464757284741 / 625000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (1415038331925467 / 1250000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1326398855274549 / 2500000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1326398855274549 / 2500000000000000 : ℝ) /
+                (1 + (1415038331925467 / 1250000000000000 : ℝ)) =
+              (1326398855274549 / 5330076663850934 : ℝ)
+            by
+        norm_num];
+        exact hau374)
+  have hat1 :
+    (Real.pi / 2 - (2 * (152436035464757284741 / 625000000000000000000 : ℝ))) ≤
+      Real.arctan (2500000000000000 / 1326398855274549 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 1326398855274549 : ℝ) =
+              (1326398855274549 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau373)
+  have hz1 :
+    (2500000000000000 / 1326398855274549 : ℝ) ≤
+      1 / Real.sqrt (9007789687161 / 32000000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 1326398855274549 : ℝ) = 1 / (1326398855274549 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 1768531807032731 : ℝ) ≤
+      (175537173293389352119 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (9007789687161 / 32000000000000 : ℝ) ≤
+      (250000000000000 / 1768531807032731 : ℝ) := by
+    rw [show
+        (250000000000000 / 1768531807032731 : ℝ) =
+          (3 / 40) / (5305595421098193 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (152436035464757284741 / 625000000000000000000 : ℝ))) -
+        6 * (175537173293389352119 / 1250000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (9007789687161 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (9007789687161 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((5305595421098193 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (152436035464757284741 / 625000000000000000000 : ℝ))) -
+          6 * (175537173293389352119 / 1250000000000000000000 : ℝ))) ≤
+      Real.sqrt (9007789687161 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (9007789687161 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (9007789687161 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_302 :
+    ((1244868006543635756541 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(12436386238777361036033 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((5317030851949791 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2443436214313161845841 / 10000000000000000000000 : ℝ))) -
+              6 * (10947785663169174833 / 78125000000000000000 : ℝ)))) ≤
+      Vfield (723732917263 / 2560000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (723732917263 / 2560000000000 : ℝ)) =
+      Real.log (3283732917263 / 2560000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((723732917263 / 2560000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (738132917263 / 2560000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1244868006543635756541 / 5000000000000000000000 : ℝ) ≤
+      Real.log (3283732917263 / 2560000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (738132917263 / 2560000000000 : ℝ) ≤
+      (-(12436386238777361036033 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (738132917263 / 2560000000000 : ℝ) = (738132917263 / 640000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (738132917263 / 640000000000 : ℝ) ≤
+            (1426557367222638963967 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (5317030851949791 / 10000000000000000 : ℝ) ≤ Real.sqrt (723732917263 / 2560000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (723732917263 / 2560000000000 : ℝ) ≤ (2658515425974897 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (723732917263 / 2560000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau376 :
+    Real.arctan (2658515425974897 / 10662835356086779 : ℝ) ≤
+      (2443436214313161845841 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau375 :
+    Real.arctan (2658515425974897 / 5000000000000000 : ℝ) ≤
+      (2 * (2443436214313161845841 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (5662835356086779 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (2658515425974897 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (2658515425974897 / 5000000000000000 : ℝ) /
+                (1 + (5662835356086779 / 5000000000000000 : ℝ)) =
+              (2658515425974897 / 10662835356086779 : ℝ)
+            by
+        norm_num];
+        exact hau376)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2443436214313161845841 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 2658515425974897 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 2658515425974897 : ℝ) =
+              (2658515425974897 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau375)
+  have hz1 :
+    (5000000000000000 / 2658515425974897 : ℝ) ≤
+      1 / Real.sqrt (723732917263 / 2560000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 2658515425974897 : ℝ) = 1 / (2658515425974897 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 1772343617316597 : ℝ) ≤
+      (10947785663169174833 / 78125000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (723732917263 / 2560000000000 : ℝ) ≤
+      (250000000000000 / 1772343617316597 : ℝ) := by
+    rw [show
+        (250000000000000 / 1772343617316597 : ℝ) =
+          (3 / 40) / (5317030851949791 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2443436214313161845841 / 10000000000000000000000 : ℝ))) -
+        6 * (10947785663169174833 / 78125000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (723732917263 / 2560000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (723732917263 / 2560000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((5317030851949791 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2443436214313161845841 / 10000000000000000000000 : ℝ))) -
+          6 * (10947785663169174833 / 78125000000000000000 : ℝ))) ≤
+      Real.sqrt (723732917263 / 2560000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (723732917263 / 2560000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (723732917263 / 2560000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_303 :
+    ((2499201675527205748003 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(6197172447429658443921 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((532844174114663 / 1000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2447882078016176642779 / 10000000000000000000000 : ℝ))) -
+              6 * (349588661375526816723 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (4542766622207 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (4542766622207 / 16000000000000 : ℝ)) =
+      Real.log (20542766622207 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((4542766622207 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (4632766622207 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2499201675527205748003 / 10000000000000000000000 : ℝ) ≤
+      Real.log (20542766622207 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (4632766622207 / 16000000000000 : ℝ) ≤
+      (-(6197172447429658443921 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (4632766622207 / 16000000000000 : ℝ) = (4632766622207 / 4000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4632766622207 / 4000000000000 : ℝ) ≤
+            (734299355570341556079 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (532844174114663 / 1000000000000000 : ℝ) ≤ Real.sqrt (4542766622207 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (4542766622207 / 16000000000000 : ℝ) ≤ (5328441741146633 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (4542766622207 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau378 :
+    Real.arctan (5328441741146633 / 21331032229624701 : ℝ) ≤
+      (2447882078016176642779 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau377 :
+    Real.arctan (5328441741146633 / 10000000000000000 : ℝ) ≤
+      (2 * (2447882078016176642779 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (11331032229624701 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (5328441741146633 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (5328441741146633 / 10000000000000000 : ℝ) /
+                (1 + (11331032229624701 / 10000000000000000 : ℝ)) =
+              (5328441741146633 / 21331032229624701 : ℝ)
+            by
+        norm_num];
+        exact hau378)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2447882078016176642779 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 5328441741146633 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 5328441741146633 : ℝ) =
+              (5328441741146633 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau377)
+  have hz1 :
+    (10000000000000000 / 5328441741146633 : ℝ) ≤
+      1 / Real.sqrt (4542766622207 / 16000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 5328441741146633 : ℝ) =
+          1 / (5328441741146633 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (75000000000000 / 532844174114663 : ℝ) ≤
+      (349588661375526816723 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (4542766622207 / 16000000000000 : ℝ) ≤
+      (75000000000000 / 532844174114663 : ℝ) := by
+    rw [show
+        (75000000000000 / 532844174114663 : ℝ) =
+          (3 / 40) / (532844174114663 / 1000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2447882078016176642779 / 10000000000000000000000 : ℝ))) -
+        6 * (349588661375526816723 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (4542766622207 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (4542766622207 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((532844174114663 / 1000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2447882078016176642779 / 10000000000000000000000 : ℝ))) -
+          6 * (349588661375526816723 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (4542766622207 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (4542766622207 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (4542766622207 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_304 :
+    ((62598278722809241977 / 250000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(6186695158982756371369 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((666767253982059 / 1250000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1225049934248501779659 / 5000000000000000000000 : ℝ))) -
+              6 * (698440355903575839597 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (36419876534909 / 128000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (36419876534909 / 128000000000000 : ℝ)) =
+      Real.log (164419876534909 / 128000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((36419876534909 / 128000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (37139876534909 / 128000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (62598278722809241977 / 250000000000000000000 : ℝ) ≤
+      Real.log (164419876534909 / 128000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (37139876534909 / 128000000000000 : ℝ) ≤
+      (-(6186695158982756371369 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (37139876534909 / 128000000000000 : ℝ) = (37139876534909 / 32000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (37139876534909 / 32000000000000 : ℝ) ≤
+            (744776644017243628631 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (666767253982059 / 1250000000000000 : ℝ) ≤ Real.sqrt (36419876534909 / 128000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (36419876534909 / 128000000000000 : ℝ) ≤ (213365521274259 / 400000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (36419876534909 / 128000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau380 :
+    Real.arctan (1066827606371295 / 4266742407446401 : ℝ) ≤
+      (1225049934248501779659 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau379 :
+    Real.arctan (213365521274259 / 400000000000000 : ℝ) ≤
+      (2 * (1225049934248501779659 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (2266742407446401 / 2000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (213365521274259 / 400000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (213365521274259 / 400000000000000 : ℝ) /
+                (1 + (2266742407446401 / 2000000000000000 : ℝ)) =
+              (1066827606371295 / 4266742407446401 : ℝ)
+            by
+        norm_num];
+        exact hau380)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1225049934248501779659 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (400000000000000 / 213365521274259 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (400000000000000 / 213365521274259 : ℝ) =
+              (213365521274259 / 400000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau379)
+  have hz1 :
+    (400000000000000 / 213365521274259 : ℝ) ≤
+      1 / Real.sqrt (36419876534909 / 128000000000000 : ℝ) := by
+    rw [show (400000000000000 / 213365521274259 : ℝ) = 1 / (213365521274259 / 400000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (31250000000000 / 222255751327353 : ℝ) ≤
+      (698440355903575839597 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (36419876534909 / 128000000000000 : ℝ) ≤
+      (31250000000000 / 222255751327353 : ℝ) := by
+    rw [show
+        (31250000000000 / 222255751327353 : ℝ) =
+          (3 / 40) / (666767253982059 / 1250000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1225049934248501779659 / 5000000000000000000000 : ℝ))) -
+        6 * (698440355903575839597 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (36419876534909 / 128000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (36419876534909 / 128000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((666767253982059 / 1250000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1225049934248501779659 / 5000000000000000000000 : ℝ))) -
+          6 * (698440355903575839597 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (36419876534909 / 128000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (36419876534909 / 128000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (36419876534909 / 128000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_305 :
+    ((1254329193281513271393 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2470495911739800849337 / 2000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((5339828246020797 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (245231424585502598539 / 1000000000000000000000 : ℝ))) -
+              6 * (348852857314937662517 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (18248810046081 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (18248810046081 / 64000000000000 : ℝ)) =
+      Real.log (82248810046081 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((18248810046081 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (18608810046081 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1254329193281513271393 / 5000000000000000000000 : ℝ) ≤
+      Real.log (82248810046081 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (18608810046081 / 64000000000000 : ℝ) ≤
+      (-(2470495911739800849337 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (18608810046081 / 64000000000000 : ℝ) = (18608810046081 / 16000000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18608810046081 / 16000000000000 : ℝ) ≤
+            (302092809460199150663 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (5339828246020797 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (18248810046081 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (18248810046081 / 64000000000000 : ℝ) ≤ (3337392653763 / 6250000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (18248810046081 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau382 :
+    Real.arctan (762832606574400 / 3048055887337151 : ℝ) ≤
+      (245231424585502598539 / 1000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau381 :
+    Real.arctan (3337392653763 / 6250000000000 : ℝ) ≤
+      (2 * (245231424585502598539 / 1000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (11336391211360057 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (3337392653763 / 6250000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (3337392653763 / 6250000000000 : ℝ) /
+                (1 + (11336391211360057 / 10000000000000000 : ℝ)) =
+              (762832606574400 / 3048055887337151 : ℝ)
+            by
+        norm_num];
+        exact hau382)
+  have hat1 :
+    (Real.pi / 2 - (2 * (245231424585502598539 / 1000000000000000000000 : ℝ))) ≤
+      Real.arctan (6250000000000 / 3337392653763 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (6250000000000 / 3337392653763 : ℝ) = (3337392653763 / 6250000000000 : ℝ) by
+            norm_num];
+        exact hau381)
+  have hz1 :
+    (6250000000000 / 3337392653763 : ℝ) ≤ 1 / Real.sqrt (18248810046081 / 64000000000000 : ℝ) := by
+    rw [show (6250000000000 / 3337392653763 : ℝ) = 1 / (3337392653763 / 6250000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 1779942748673599 : ℝ) ≤
+      (348852857314937662517 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (18248810046081 / 64000000000000 : ℝ) ≤
+      (250000000000000 / 1779942748673599 : ℝ) := by
+    rw [show
+        (250000000000000 / 1779942748673599 : ℝ) =
+          (3 / 40) / (5339828246020797 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (245231424585502598539 / 1000000000000000000000 : ℝ))) -
+        6 * (348852857314937662517 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (18248810046081 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (18248810046081 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((5339828246020797 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (245231424585502598539 / 1000000000000000000000 : ℝ))) -
+          6 * (348852857314937662517 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (18248810046081 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (18248810046081 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (18248810046081 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_306 :
+    ((2513383390591947328477 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(6165806217095040688203 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((167047262595149 / 312500000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2454525220863704290673 / 10000000000000000000000 : ℝ))) -
+              6 * (1393946773448474205463 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (7315072729883 / 25600000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (7315072729883 / 25600000000000 : ℝ)) =
+      Real.log (32915072729883 / 25600000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((7315072729883 / 25600000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (7459072729883 / 25600000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2513383390591947328477 / 10000000000000000000000 : ℝ) ≤
+      Real.log (32915072729883 / 25600000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (7459072729883 / 25600000000000 : ℝ) ≤
+      (-(6165806217095040688203 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (7459072729883 / 25600000000000 : ℝ) = (7459072729883 / 6400000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7459072729883 / 6400000000000 : ℝ) ≤
+            (765665585904959311797 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (167047262595149 / 312500000000000 : ℝ) ≤ Real.sqrt (7315072729883 / 25600000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (7315072729883 / 25600000000000 : ℝ) ≤ (5345512403044771 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (7315072729883 / 25600000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau384 :
+    Real.arctan (5345512403044771 / 21339069752457891 : ℝ) ≤
+      (2454525220863704290673 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau383 :
+    Real.arctan (5345512403044771 / 10000000000000000 : ℝ) ≤
+      (2 * (2454525220863704290673 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (11339069752457891 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (5345512403044771 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (5345512403044771 / 10000000000000000 : ℝ) /
+                (1 + (11339069752457891 / 10000000000000000 : ℝ)) =
+              (5345512403044771 / 21339069752457891 : ℝ)
+            by
+        norm_num];
+        exact hau384)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2454525220863704290673 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 5345512403044771 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 5345512403044771 : ℝ) =
+              (5345512403044771 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau383)
+  have hz1 :
+    (10000000000000000 / 5345512403044771 : ℝ) ≤
+      1 / Real.sqrt (7315072729883 / 25600000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 5345512403044771 : ℝ) =
+          1 / (5345512403044771 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (23437500000000 / 167047262595149 : ℝ) ≤
+      (1393946773448474205463 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (7315072729883 / 25600000000000 : ℝ) ≤
+      (23437500000000 / 167047262595149 : ℝ) := by
+    rw [show
+        (23437500000000 / 167047262595149 : ℝ) =
+          (3 / 40) / (167047262595149 / 312500000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2454525220863704290673 / 10000000000000000000000 : ℝ))) -
+        6 * (1393946773448474205463 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (7315072729883 / 25600000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (7315072729883 / 25600000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((167047262595149 / 312500000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2454525220863704290673 / 10000000000000000000000 : ℝ))) -
+          6 * (1393946773448474205463 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (7315072729883 / 25600000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (7315072729883 / 25600000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (7315072729883 / 25600000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_307 :
+    ((314763270388613698039 / 1250000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(6155394381355720490859 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1337797630557621 / 2500000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1228366402120582497673 / 5000000000000000000000 : ℝ))) -
+              6 * (278497344028178964147 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (9163276801667 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (9163276801667 / 32000000000000 : ℝ)) =
+      Real.log (41163276801667 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((9163276801667 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (9343276801667 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (314763270388613698039 / 1250000000000000000000 : ℝ) ≤
+      Real.log (41163276801667 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (9343276801667 / 32000000000000 : ℝ) ≤
+      (-(6155394381355720490859 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (9343276801667 / 32000000000000 : ℝ) = (9343276801667 / 8000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9343276801667 / 8000000000000 : ℝ) ≤
+            (776077421644279509141 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1337797630557621 / 2500000000000000 : ℝ) ≤ Real.sqrt (9163276801667 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (9163276801667 / 32000000000000 : ℝ) ≤ (5351190522230487 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (9163276801667 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau386 :
+    Real.arctan (5351190522230487 / 21341747660974007 : ℝ) ≤
+      (1228366402120582497673 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau385 :
+    Real.arctan (5351190522230487 / 10000000000000000 : ℝ) ≤
+      (2 * (1228366402120582497673 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (11341747660974007 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (5351190522230487 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (5351190522230487 / 10000000000000000 : ℝ) /
+                (1 + (11341747660974007 / 10000000000000000 : ℝ)) =
+              (5351190522230487 / 21341747660974007 : ℝ)
+            by
+        norm_num];
+        exact hau386)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1228366402120582497673 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 5351190522230487 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 5351190522230487 : ℝ) =
+              (5351190522230487 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau385)
+  have hz1 :
+    (10000000000000000 / 5351190522230487 : ℝ) ≤
+      1 / Real.sqrt (9163276801667 / 32000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 5351190522230487 : ℝ) =
+          1 / (5351190522230487 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (62500000000000 / 445932543519207 : ℝ) ≤
+      (278497344028178964147 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (9163276801667 / 32000000000000 : ℝ) ≤
+      (62500000000000 / 445932543519207 : ℝ) := by
+    rw [show
+        (62500000000000 / 445932543519207 : ℝ) =
+          (3 / 40) / (1337797630557621 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1228366402120582497673 / 5000000000000000000000 : ℝ))) -
+        6 * (278497344028178964147 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (9163276801667 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (9163276801667 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1337797630557621 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1228366402120582497673 / 5000000000000000000000 : ℝ))) -
+          6 * (278497344028178964147 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (9163276801667 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (9163276801667 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (9163276801667 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_308 :
+    ((2522826706220703069523 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(12290008363668690903519 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1071372524555549 / 2000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2458937006650601874589 / 10000000000000000000000 : ℝ))) -
+              6 * (347757811320475804929 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (36730850763921 / 128000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (36730850763921 / 128000000000000 : ℝ)) =
+      Real.log (164730850763921 / 128000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((36730850763921 / 128000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (37450850763921 / 128000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2522826706220703069523 / 10000000000000000000000 : ℝ) ≤
+      Real.log (164730850763921 / 128000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (37450850763921 / 128000000000000 : ℝ) ≤
+      (-(12290008363668690903519 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (37450850763921 / 128000000000000 : ℝ) = (37450850763921 / 32000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (37450850763921 / 32000000000000 : ℝ) ≤
+            (1572935242331309096481 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1071372524555549 / 2000000000000000 : ℝ) ≤
+      Real.sqrt (36730850763921 / 128000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (36730850763921 / 128000000000000 : ℝ) ≤
+      (1339215655694437 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (36730850763921 / 128000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau388 :
+    Real.arctan (2678431311388874 / 10672212468678189 : ℝ) ≤
+      (2458937006650601874589 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau387 :
+    Real.arctan (1339215655694437 / 2500000000000000 : ℝ) ≤
+      (2 * (2458937006650601874589 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (5672212468678189 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1339215655694437 / 2500000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1339215655694437 / 2500000000000000 : ℝ) /
+                (1 + (5672212468678189 / 5000000000000000 : ℝ)) =
+              (2678431311388874 / 10672212468678189 : ℝ)
+            by
+        norm_num];
+        exact hau388)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2458937006650601874589 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (2500000000000000 / 1339215655694437 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 1339215655694437 : ℝ) =
+              (1339215655694437 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau387)
+  have hz1 :
+    (2500000000000000 / 1339215655694437 : ℝ) ≤
+      1 / Real.sqrt (36730850763921 / 128000000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 1339215655694437 : ℝ) = 1 / (1339215655694437 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (150000000000000 / 1071372524555549 : ℝ) ≤
+      (347757811320475804929 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (36730850763921 / 128000000000000 : ℝ) ≤
+      (150000000000000 / 1071372524555549 : ℝ) := by
+    rw [show
+        (150000000000000 / 1071372524555549 : ℝ) =
+          (3 / 40) / (1071372524555549 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2458937006650601874589 / 10000000000000000000000 : ℝ))) -
+        6 * (347757811320475804929 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (36730850763921 / 128000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (36730850763921 / 128000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1071372524555549 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2458937006650601874589 / 10000000000000000000000 : ℝ))) -
+          6 * (347757811320475804929 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (36730850763921 / 128000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (36730850763921 / 128000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (36730850763921 / 128000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_309 :
+    ((2527545022031135393343 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(12269271057590952551449 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((5362528723784813 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (492227567740135145223 / 2000000000000000000000 : ℝ))) -
+              6 * (1389580324992042662019 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (18404297160587 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (18404297160587 / 64000000000000 : ℝ)) =
+      Real.log (82404297160587 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((18404297160587 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (18764297160587 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2527545022031135393343 / 10000000000000000000000 : ℝ) ≤
+      Real.log (82404297160587 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (18764297160587 / 64000000000000 : ℝ) ≤
+      (-(12269271057590952551449 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (18764297160587 / 64000000000000 : ℝ) = (18764297160587 / 16000000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18764297160587 / 16000000000000 : ℝ) ≤
+            (1593672548409047448551 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (5362528723784813 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (18404297160587 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (18404297160587 / 64000000000000 : ℝ) ≤ (335158045236551 / 625000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (18404297160587 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau390 :
+    Real.arctan (335158045236551 / 1334193848878278 : ℝ) ≤
+      (492227567740135145223 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau389 :
+    Real.arctan (335158045236551 / 625000000000000 : ℝ) ≤
+      (2 * (492227567740135145223 / 2000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (354596924439139 / 312500000000000 : ℝ) ≤
+          Real.sqrt (1 + (335158045236551 / 625000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (335158045236551 / 625000000000000 : ℝ) /
+                (1 + (354596924439139 / 312500000000000 : ℝ)) =
+              (335158045236551 / 1334193848878278 : ℝ)
+            by
+        norm_num];
+        exact hau390)
+  have hat1 :
+    (Real.pi / 2 - (2 * (492227567740135145223 / 2000000000000000000000 : ℝ))) ≤
+      Real.arctan (625000000000000 / 335158045236551 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (625000000000000 / 335158045236551 : ℝ) =
+              (335158045236551 / 625000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau389)
+  have hz1 :
+    (625000000000000 / 335158045236551 : ℝ) ≤
+      1 / Real.sqrt (18404297160587 / 64000000000000 : ℝ) := by
+    rw [show (625000000000000 / 335158045236551 : ℝ) = 1 / (335158045236551 / 625000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 5362528723784813 : ℝ) ≤
+      (1389580324992042662019 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (18404297160587 / 64000000000000 : ℝ) ≤
+      (750000000000000 / 5362528723784813 : ℝ) := by
+    rw [show
+        (750000000000000 / 5362528723784813 : ℝ) =
+          (3 / 40) / (5362528723784813 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (492227567740135145223 / 2000000000000000000000 : ℝ))) -
+        6 * (1389580324992042662019 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (18404297160587 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (18404297160587 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((5362528723784813 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (492227567740135145223 / 2000000000000000000000 : ℝ))) -
+          6 * (1389580324992042662019 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (18404297160587 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (18404297160587 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (18404297160587 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_310 :
+    ((50645222252820752813 / 200000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(122485766661215607187 / 100000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1073637768849831 / 2000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2 * (24633353109459053127 / 100000000000000000000 : ℝ))) -
+              6 * (277626787113173012921 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (36886337878427 / 128000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (36886337878427 / 128000000000000 : ℝ)) =
+      Real.log (164886337878427 / 128000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((36886337878427 / 128000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (37606337878427 / 128000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (50645222252820752813 / 200000000000000000000 : ℝ) ≤
+      Real.log (164886337878427 / 128000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (37606337878427 / 128000000000000 : ℝ) ≤
+      (-(122485766661215607187 / 100000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (37606337878427 / 128000000000000 : ℝ) = (37606337878427 / 32000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (37606337878427 / 32000000000000 : ℝ) ≤
+            (16143669398784392813 / 100000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1073637768849831 / 2000000000000000 : ℝ) ≤
+      Real.sqrt (36886337878427 / 128000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (36886337878427 / 128000000000000 : ℝ) ≤
+      (2684094422124579 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (36886337878427 / 128000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau392 :
+    Real.arctan (5368188844249158 / 21349777595509133 : ℝ) ≤
+      (24633353109459053127 / 100000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau391 :
+    Real.arctan (2684094422124579 / 5000000000000000 : ℝ) ≤
+      (2 * (24633353109459053127 / 100000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (11349777595509133 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (2684094422124579 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (2684094422124579 / 5000000000000000 : ℝ) /
+                (1 + (11349777595509133 / 10000000000000000 : ℝ)) =
+              (5368188844249158 / 21349777595509133 : ℝ)
+            by
+        norm_num];
+        exact hau392)
+  have hat1 :
+    (Real.pi / 2 - (2 * (24633353109459053127 / 100000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 2684094422124579 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 2684094422124579 : ℝ) =
+              (2684094422124579 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau391)
+  have hz1 :
+    (5000000000000000 / 2684094422124579 : ℝ) ≤
+      1 / Real.sqrt (36886337878427 / 128000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 2684094422124579 : ℝ) = 1 / (2684094422124579 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (50000000000000 / 357879256283277 : ℝ) ≤
+      (277626787113173012921 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (36886337878427 / 128000000000000 : ℝ) ≤
+      (50000000000000 / 357879256283277 : ℝ) := by
+    rw [show
+        (50000000000000 / 357879256283277 : ℝ) =
+          (3 / 40) / (1073637768849831 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (24633353109459053127 / 100000000000000000000 : ℝ))) -
+        6 * (277626787113173012921 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (36886337878427 / 128000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (36886337878427 / 128000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1073637768849831 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (24633353109459053127 / 100000000000000000000 : ℝ))) -
+          6 * (277626787113173012921 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (36886337878427 / 128000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (36886337878427 / 128000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (36886337878427 / 128000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_311 :
+    ((2536974980148269984337 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(12227925012008859438829 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2686921501534097 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (308191179235882659497 / 1250000000000000000000 : ℝ))) -
+              6 * (693346026735151028923 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (231025508973 / 800000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (231025508973 / 800000000000 : ℝ)) =
+      Real.log (1031025508973 / 800000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((231025508973 / 800000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (235525508973 / 800000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2536974980148269984337 / 10000000000000000000000 : ℝ) ≤
+      Real.log (1031025508973 / 800000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (235525508973 / 800000000000 : ℝ) ≤
+      (-(12227925012008859438829 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (235525508973 / 800000000000 : ℝ) = (235525508973 / 200000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (235525508973 / 200000000000 : ℝ) ≤
+            (1635018593991140561171 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2686921501534097 / 5000000000000000 : ℝ) ≤ Real.sqrt (231025508973 / 800000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (231025508973 / 800000000000 : ℝ) ≤ (5373843003068197 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (231025508973 / 800000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau394 :
+    Real.arctan (5373843003068197 / 21352452978172822 : ℝ) ≤
+      (308191179235882659497 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau393 :
+    Real.arctan (5373843003068197 / 10000000000000000 : ℝ) ≤
+      (2 * (308191179235882659497 / 1250000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (5676226489086411 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (5373843003068197 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (5373843003068197 / 10000000000000000 : ℝ) /
+                (1 + (5676226489086411 / 5000000000000000 : ℝ)) =
+              (5373843003068197 / 21352452978172822 : ℝ)
+            by
+        norm_num];
+        exact hau394)
+  have hat1 :
+    (Real.pi / 2 - (2 * (308191179235882659497 / 1250000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 5373843003068197 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 5373843003068197 : ℝ) =
+              (5373843003068197 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau393)
+  have hz1 :
+    (10000000000000000 / 5373843003068197 : ℝ) ≤
+      1 / Real.sqrt (231025508973 / 800000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 5373843003068197 : ℝ) =
+          1 / (5373843003068197 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 2686921501534097 : ℝ) ≤
+      (693346026735151028923 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (231025508973 / 800000000000 : ℝ) ≤
+      (375000000000000 / 2686921501534097 : ℝ) := by
+    rw [show
+        (375000000000000 / 2686921501534097 : ℝ) =
+          (3 / 40) / (2686921501534097 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (308191179235882659497 / 1250000000000000000000 : ℝ))) -
+        6 * (693346026735151028923 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (231025508973 / 800000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (231025508973 / 800000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2686921501534097 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (308191179235882659497 / 1250000000000000000000 : ℝ))) -
+          6 * (693346026735151028923 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (231025508973 / 800000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (231025508973 / 800000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (231025508973 / 800000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_312 :
+    ((2541686626647727284469 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(3051828979774273160183 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((5379491219040039 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1233860108985775068049 / 5000000000000000000000 : ℝ))) -
+              6 * (277050931068611634373 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (37041824992933 / 128000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (37041824992933 / 128000000000000 : ℝ)) =
+      Real.log (165041824992933 / 128000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((37041824992933 / 128000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (37761824992933 / 128000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2541686626647727284469 / 10000000000000000000000 : ℝ) ≤
+      Real.log (165041824992933 / 128000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (37761824992933 / 128000000000000 : ℝ) ≤
+      (-(3051828979774273160183 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (37761824992933 / 128000000000000 : ℝ) = (37761824992933 / 32000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (37761824992933 / 32000000000000 : ℝ) ≤
+            (413906921725726839817 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (5379491219040039 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (37041824992933 / 128000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (37041824992933 / 128000000000000 : ℝ) ≤
+      (2689745609520021 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (37041824992933 / 128000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau396 :
+    Real.arctan (5379491219040042 / 21355127730489379 : ℝ) ≤
+      (1233860108985775068049 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau395 :
+    Real.arctan (2689745609520021 / 5000000000000000 : ℝ) ≤
+      (2 * (1233860108985775068049 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (11355127730489379 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (2689745609520021 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (2689745609520021 / 5000000000000000 : ℝ) /
+                (1 + (11355127730489379 / 10000000000000000 : ℝ)) =
+              (5379491219040042 / 21355127730489379 : ℝ)
+            by
+        norm_num];
+        exact hau396)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1233860108985775068049 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 2689745609520021 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 2689745609520021 : ℝ) =
+              (2689745609520021 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau395)
+  have hz1 :
+    (5000000000000000 / 2689745609520021 : ℝ) ≤
+      1 / Real.sqrt (37041824992933 / 128000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 2689745609520021 : ℝ) = 1 / (2689745609520021 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 1793163739680013 : ℝ) ≤
+      (277050931068611634373 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (37041824992933 / 128000000000000 : ℝ) ≤
+      (250000000000000 / 1793163739680013 : ℝ) := by
+    rw [show
+        (250000000000000 / 1793163739680013 : ℝ) =
+          (3 / 40) / (5379491219040039 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1233860108985775068049 / 5000000000000000000000 : ℝ))) -
+        6 * (277050931068611634373 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (37041824992933 / 128000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (37041824992933 / 128000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((5379491219040039 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1233860108985775068049 / 5000000000000000000000 : ℝ))) -
+          6 * (277050931068611634373 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (37041824992933 / 128000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (37041824992933 / 128000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (37041824992933 / 128000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/V26.lean b/LeanPool/Zeta5Irrational/Table/V26.lean
new file mode 100644
index 0000000000..b0640a5cd1
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/V26.lean
@@ -0,0 +1,1450 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Energy
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # V26: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_313 :
+    ((1273198027115672337129 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2437349842463477713527 / 2000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((269256675543211 / 500000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2 * (154369229599612503803 / 625000000000000000000 : ℝ))) -
+              6 * (1383821717991019276453 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (18559784275093 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (18559784275093 / 64000000000000 : ℝ)) =
+      Real.log (82559784275093 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((18559784275093 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (18919784275093 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1273198027115672337129 / 5000000000000000000000 : ℝ) ≤
+      Real.log (82559784275093 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (18919784275093 / 64000000000000 : ℝ) ≤
+      (-(2437349842463477713527 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (18919784275093 / 64000000000000 : ℝ) = (18919784275093 / 16000000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18919784275093 / 16000000000000 : ℝ) ≤
+            (335238878736522286473 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (269256675543211 / 500000000000000 : ℝ) ≤ Real.sqrt (18559784275093 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (18559784275093 / 64000000000000 : ℝ) ≤
+      (5385133510864223 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (18559784275093 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau398 :
+    Real.arctan (5385133510864223 / 21357801852904144 : ℝ) ≤
+      (154369229599612503803 / 625000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau397 :
+    Real.arctan (5385133510864223 / 10000000000000000 : ℝ) ≤
+      (2 * (154369229599612503803 / 625000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (709862615806509 / 625000000000000 : ℝ) ≤
+          Real.sqrt (1 + (5385133510864223 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (5385133510864223 / 10000000000000000 : ℝ) /
+                (1 + (709862615806509 / 625000000000000 : ℝ)) =
+              (5385133510864223 / 21357801852904144 : ℝ)
+            by
+        norm_num];
+        exact hau398)
+  have hat1 :
+    (Real.pi / 2 - (2 * (154369229599612503803 / 625000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 5385133510864223 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 5385133510864223 : ℝ) =
+              (5385133510864223 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau397)
+  have hz1 :
+    (10000000000000000 / 5385133510864223 : ℝ) ≤
+      1 / Real.sqrt (18559784275093 / 64000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 5385133510864223 : ℝ) =
+          1 / (5385133510864223 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (37500000000000 / 269256675543211 : ℝ) ≤
+      (1383821717991019276453 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (18559784275093 / 64000000000000 : ℝ) ≤
+      (37500000000000 / 269256675543211 : ℝ) := by
+    rw [show
+        (37500000000000 / 269256675543211 : ℝ) =
+          (3 / 40) / (269256675543211 / 500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (154369229599612503803 / 625000000000000000000 : ℝ))) -
+        6 * (1383821717991019276453 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (18559784275093 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (18559784275093 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((269256675543211 / 500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (154369229599612503803 / 625000000000000000000 : ℝ))) -
+          6 * (1383821717991019276453 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (18559784275093 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (18559784275093 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (18559784275093 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_314 :
+    ((31888790812351289161 / 125000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(6083112358839418315309 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2695384948571201 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2472091811095637514179 / 10000000000000000000000 : ℝ))) -
+              6 * (691196609194341337837 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (37197312107439 / 128000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (37197312107439 / 128000000000000 : ℝ)) =
+      Real.log (165197312107439 / 128000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((37197312107439 / 128000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (37917312107439 / 128000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (31888790812351289161 / 125000000000000000000 : ℝ) ≤
+      Real.log (165197312107439 / 128000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (37917312107439 / 128000000000000 : ℝ) ≤
+      (-(6083112358839418315309 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (37917312107439 / 128000000000000 : ℝ) = (37917312107439 / 32000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (37917312107439 / 32000000000000 : ℝ) ≤
+            (848359444160581684691 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2695384948571201 / 5000000000000000 : ℝ) ≤
+      Real.sqrt (37197312107439 / 128000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (37197312107439 / 128000000000000 : ℝ) ≤
+      (1078153979428481 / 2000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (37197312107439 / 128000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau400 :
+    Real.arctan (5390769897142405 / 21360475345861928 : ℝ) ≤
+      (2472091811095637514179 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau399 :
+    Real.arctan (1078153979428481 / 2000000000000000 : ℝ) ≤
+      (2 * (2472091811095637514179 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (1420059418232741 / 1250000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1078153979428481 / 2000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1078153979428481 / 2000000000000000 : ℝ) /
+                (1 + (1420059418232741 / 1250000000000000 : ℝ)) =
+              (5390769897142405 / 21360475345861928 : ℝ)
+            by
+        norm_num];
+        exact hau400)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2472091811095637514179 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (2000000000000000 / 1078153979428481 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2000000000000000 / 1078153979428481 : ℝ) =
+              (1078153979428481 / 2000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau399)
+  have hz1 :
+    (2000000000000000 / 1078153979428481 : ℝ) ≤
+      1 / Real.sqrt (37197312107439 / 128000000000000 : ℝ) := by
+    rw [show
+        (2000000000000000 / 1078153979428481 : ℝ) = 1 / (1078153979428481 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 2695384948571201 : ℝ) ≤
+      (691196609194341337837 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (37197312107439 / 128000000000000 : ℝ) ≤
+      (375000000000000 / 2695384948571201 : ℝ) := by
+    rw [show
+        (375000000000000 / 2695384948571201 : ℝ) =
+          (3 / 40) / (2695384948571201 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2472091811095637514179 / 10000000000000000000000 : ℝ))) -
+        6 * (691196609194341337837 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (37197312107439 / 128000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (37197312107439 / 128000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2695384948571201 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2472091811095637514179 / 10000000000000000000000 : ℝ))) -
+          6 * (691196609194341337837 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (37197312107439 / 128000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (37197312107439 / 128000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (37197312107439 / 128000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_315 :
+    ((511161652200807009039 / 2000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(12145742262259655687553 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((5396400396379109 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (247427264076666588767 / 1000000000000000000000 : ℝ))) -
+              6 * (1380969133676600751081 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (9318763916173 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (9318763916173 / 32000000000000 : ℝ)) =
+      Real.log (41318763916173 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((9318763916173 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (9498763916173 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (511161652200807009039 / 2000000000000000000000 : ℝ) ≤
+      Real.log (41318763916173 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (9498763916173 / 32000000000000 : ℝ) ≤
+      (-(12145742262259655687553 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (9498763916173 / 32000000000000 : ℝ) = (9498763916173 / 8000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9498763916173 / 8000000000000 : ℝ) ≤
+            (1717201343740344312447 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (5396400396379109 / 10000000000000000 : ℝ) ≤ Real.sqrt (9318763916173 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (9318763916173 / 32000000000000 : ℝ) ≤ (674550049547389 / 1250000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (9318763916173 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau402 :
+    Real.arctan (5396400396379112 / 21363148209807025 : ℝ) ≤
+      (247427264076666588767 / 1000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau401 :
+    Real.arctan (674550049547389 / 1250000000000000 : ℝ) ≤
+      (2 * (247427264076666588767 / 1000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (454525928392281 / 400000000000000 : ℝ) ≤
+          Real.sqrt (1 + (674550049547389 / 1250000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (674550049547389 / 1250000000000000 : ℝ) /
+                (1 + (454525928392281 / 400000000000000 : ℝ)) =
+              (5396400396379112 / 21363148209807025 : ℝ)
+            by
+        norm_num];
+        exact hau402)
+  have hat1 :
+    (Real.pi / 2 - (2 * (247427264076666588767 / 1000000000000000000000 : ℝ))) ≤
+      Real.arctan (1250000000000000 / 674550049547389 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1250000000000000 / 674550049547389 : ℝ) =
+              (674550049547389 / 1250000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau401)
+  have hz1 :
+    (1250000000000000 / 674550049547389 : ℝ) ≤
+      1 / Real.sqrt (9318763916173 / 32000000000000 : ℝ) := by
+    rw [show
+        (1250000000000000 / 674550049547389 : ℝ) = 1 / (674550049547389 / 1250000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 5396400396379109 : ℝ) ≤
+      (1380969133676600751081 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (9318763916173 / 32000000000000 : ℝ) ≤
+      (750000000000000 / 5396400396379109 : ℝ) := by
+    rw [show
+        (750000000000000 / 5396400396379109 : ℝ) =
+          (3 / 40) / (5396400396379109 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (247427264076666588767 / 1000000000000000000000 : ℝ))) -
+        6 * (1380969133676600751081 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (9318763916173 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (9318763916173 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((5396400396379109 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (247427264076666588767 / 1000000000000000000000 : ℝ))) -
+          6 * (1380969133676600751081 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (9318763916173 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (9318763916173 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (9318763916173 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_316 :
+    ((1280255522181114874191 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(47364459664837705379 / 39062500000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((5402025026982429 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (247645017284463756857 / 1000000000000000000000 : ℝ))) -
+              6 * (689774720579922654177 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (7470559844389 / 25600000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (7470559844389 / 25600000000000 : ℝ)) =
+      Real.log (33070559844389 / 25600000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((7470559844389 / 25600000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (7614559844389 / 25600000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1280255522181114874191 / 5000000000000000000000 : ℝ) ≤
+      Real.log (33070559844389 / 25600000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (7614559844389 / 25600000000000 : ℝ) ≤
+      (-(47364459664837705379 / 39062500000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (7614559844389 / 25600000000000 : ℝ) = (7614559844389 / 6400000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7614559844389 / 6400000000000 : ℝ) ≤
+            (6787663796099794621 / 39062500000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (5402025026982429 / 10000000000000000 : ℝ) ≤ Real.sqrt (7470559844389 / 25600000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (7470559844389 / 25600000000000 : ℝ) ≤ (168813282093201 / 312500000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (7470559844389 / 25600000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau404 :
+    Real.arctan (1800675008994144 / 7121940148394401 : ℝ) ≤
+      (247645017284463756857 / 1000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau403 :
+    Real.arctan (168813282093201 / 312500000000000 : ℝ) ≤
+      (2 * (247645017284463756857 / 1000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (11365820445183203 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (168813282093201 / 312500000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (168813282093201 / 312500000000000 : ℝ) /
+                (1 + (11365820445183203 / 10000000000000000 : ℝ)) =
+              (1800675008994144 / 7121940148394401 : ℝ)
+            by
+        norm_num];
+        exact hau404)
+  have hat1 :
+    (Real.pi / 2 - (2 * (247645017284463756857 / 1000000000000000000000 : ℝ))) ≤
+      Real.arctan (312500000000000 / 168813282093201 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (312500000000000 / 168813282093201 : ℝ) =
+              (168813282093201 / 312500000000000 : ℝ)
+            by
+        norm_num];
+        exact hau403)
+  have hz1 :
+    (312500000000000 / 168813282093201 : ℝ) ≤
+      1 / Real.sqrt (7470559844389 / 25600000000000 : ℝ) := by
+    rw [show (312500000000000 / 168813282093201 : ℝ) = 1 / (168813282093201 / 312500000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 1800675008994143 : ℝ) ≤
+      (689774720579922654177 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (7470559844389 / 25600000000000 : ℝ) ≤
+      (250000000000000 / 1800675008994143 : ℝ) := by
+    rw [show
+        (250000000000000 / 1800675008994143 : ℝ) =
+          (3 / 40) / (5402025026982429 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (247645017284463756857 / 1000000000000000000000 : ℝ))) -
+        6 * (689774720579922654177 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (7470559844389 / 25600000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (7470559844389 / 25600000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((5402025026982429 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (247645017284463756857 / 1000000000000000000000 : ℝ))) -
+          6 * (689774720579922654177 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (7470559844389 / 25600000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (7470559844389 / 25600000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (7470559844389 / 25600000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_317 :
+    ((1282605808571419532773 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1210490278268556981051 / 1000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1081528761452943 / 2000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (619656104378955515141 / 2500000000000000000000 : ℝ))) -
+              6 * (344533529576622148267 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (18715271389599 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (18715271389599 / 64000000000000 : ℝ)) =
+      Real.log (82715271389599 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((18715271389599 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (19075271389599 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1282605808571419532773 / 5000000000000000000000 : ℝ) ≤
+      Real.log (82715271389599 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (19075271389599 / 64000000000000 : ℝ) ≤
+      (-(1210490278268556981051 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (19075271389599 / 64000000000000 : ℝ) = (19075271389599 / 16000000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19075271389599 / 16000000000000 : ℝ) ≤
+            (175804082331443018949 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1081528761452943 / 2000000000000000 : ℝ) ≤ Real.sqrt (18715271389599 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (18715271389599 / 64000000000000 : ℝ) ≤ (2703821903632359 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (18715271389599 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau406 :
+    Real.arctan (901273967877453 / 3561415342072285 : ℝ) ≤
+      (619656104378955515141 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau405 :
+    Real.arctan (2703821903632359 / 5000000000000000 : ℝ) ≤
+      (2 * (619656104378955515141 / 2500000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (1136849205243371 / 1000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (2703821903632359 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (2703821903632359 / 5000000000000000 : ℝ) /
+                (1 + (1136849205243371 / 1000000000000000 : ℝ)) =
+              (901273967877453 / 3561415342072285 : ℝ)
+            by
+        norm_num];
+        exact hau406)
+  have hat1 :
+    (Real.pi / 2 - (2 * (619656104378955515141 / 2500000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 2703821903632359 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 2703821903632359 : ℝ) =
+              (2703821903632359 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau405)
+  have hz1 :
+    (5000000000000000 / 2703821903632359 : ℝ) ≤
+      1 / Real.sqrt (18715271389599 / 64000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 2703821903632359 : ℝ) = 1 / (2703821903632359 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (50000000000000 / 360509587150981 : ℝ) ≤
+      (344533529576622148267 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (18715271389599 / 64000000000000 : ℝ) ≤
+      (50000000000000 / 360509587150981 : ℝ) := by
+    rw [show
+        (50000000000000 / 360509587150981 : ℝ) =
+          (3 / 40) / (1081528761452943 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (619656104378955515141 / 2500000000000000000000 : ℝ))) -
+        6 * (344533529576622148267 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (18715271389599 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (18715271389599 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1081528761452943 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (619656104378955515141 / 2500000000000000000000 : ℝ))) -
+          6 * (344533529576622148267 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (18715271389599 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (18715271389599 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (18715271389599 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_318 :
+    ((1284954990711541413163 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(6042272708977260666909 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2706628377721641 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2480795384915372937519 / 10000000000000000000000 : ℝ))) -
+              6 * (43022598210815649167 / 312500000000000000000 : ℝ)))) ≤
+      Vfield (37508286336451 / 128000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (37508286336451 / 128000000000000 : ℝ)) =
+      Real.log (165508286336451 / 128000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((37508286336451 / 128000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (38228286336451 / 128000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1284954990711541413163 / 5000000000000000000000 : ℝ) ≤
+      Real.log (165508286336451 / 128000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (38228286336451 / 128000000000000 : ℝ) ≤
+      (-(6042272708977260666909 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (38228286336451 / 128000000000000 : ℝ) = (38228286336451 / 32000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (38228286336451 / 32000000000000 : ℝ) ≤
+            (889199094022739333091 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2706628377721641 / 5000000000000000 : ℝ) ≤
+      Real.sqrt (37508286336451 / 128000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (37508286336451 / 128000000000000 : ℝ) ≤
+      (1082651351088657 / 2000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (37508286336451 / 128000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau408 :
+    Real.arctan (360883783696219 / 1424744202133418 : ℝ) ≤
+      (2480795384915372937519 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau407 :
+    Real.arctan (1082651351088657 / 2000000000000000 : ℝ) ≤
+      (2 * (2480795384915372937519 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (1137116303200127 / 1000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1082651351088657 / 2000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1082651351088657 / 2000000000000000 : ℝ) /
+                (1 + (1137116303200127 / 1000000000000000 : ℝ)) =
+              (360883783696219 / 1424744202133418 : ℝ)
+            by
+        norm_num];
+        exact hau408)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2480795384915372937519 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (2000000000000000 / 1082651351088657 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2000000000000000 / 1082651351088657 : ℝ) =
+              (1082651351088657 / 2000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau407)
+  have hz1 :
+    (2000000000000000 / 1082651351088657 : ℝ) ≤
+      1 / Real.sqrt (37508286336451 / 128000000000000 : ℝ) := by
+    rw [show
+        (2000000000000000 / 1082651351088657 : ℝ) = 1 / (1082651351088657 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (125000000000000 / 902209459240547 : ℝ) ≤
+      (43022598210815649167 / 312500000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (37508286336451 / 128000000000000 : ℝ) ≤
+      (125000000000000 / 902209459240547 : ℝ) := by
+    rw [show
+        (125000000000000 / 902209459240547 : ℝ) =
+          (3 / 40) / (2706628377721641 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2480795384915372937519 / 10000000000000000000000 : ℝ))) -
+        6 * (43022598210815649167 / 312500000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (37508286336451 / 128000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (37508286336451 / 128000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2706628377721641 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2480795384915372937519 / 10000000000000000000000 : ℝ))) -
+          6 * (43022598210815649167 / 312500000000000000000 : ℝ))) ≤
+      Real.sqrt (37508286336451 / 128000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (37508286336451 / 128000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (37508286336451 / 128000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_319 :
+    ((20114110463103549801 / 78125000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(12064229411273515284231 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((5418863889641097 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (496592617025537929121 / 2000000000000000000000 : ℝ))) -
+              6 * (275063298453652888357 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (4698253736713 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (4698253736713 / 16000000000000 : ℝ)) =
+      Real.log (20698253736713 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((4698253736713 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (4788253736713 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (20114110463103549801 / 78125000000000000000 : ℝ) ≤
+      Real.log (20698253736713 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (4788253736713 / 16000000000000 : ℝ) ≤
+      (-(12064229411273515284231 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (4788253736713 / 16000000000000 : ℝ) = (4788253736713 / 4000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4788253736713 / 4000000000000 : ℝ) ≤
+            (1798714194726484715769 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (5418863889641097 / 10000000000000000 : ℝ) ≤ Real.sqrt (4698253736713 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (4698253736713 / 16000000000000 : ℝ) ≤ (54188638896411 / 100000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (4698253736713 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau410 :
+    Real.arctan (5418863889641100 / 21373833384328093 : ℝ) ≤
+      (496592617025537929121 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau409 :
+    Real.arctan (54188638896411 / 100000000000000 : ℝ) ≤
+      (2 * (496592617025537929121 / 2000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (11373833384328093 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (54188638896411 / 100000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (54188638896411 / 100000000000000 : ℝ) /
+                (1 + (11373833384328093 / 10000000000000000 : ℝ)) =
+              (5418863889641100 / 21373833384328093 : ℝ)
+            by
+        norm_num];
+        exact hau410)
+  have hat1 :
+    (Real.pi / 2 - (2 * (496592617025537929121 / 2000000000000000000000 : ℝ))) ≤
+      Real.arctan (100000000000000 / 54188638896411 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (100000000000000 / 54188638896411 : ℝ) =
+              (54188638896411 / 100000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau409)
+  have hz1 :
+    (100000000000000 / 54188638896411 : ℝ) ≤ 1 / Real.sqrt (4698253736713 / 16000000000000 : ℝ) :=
+  by
+    rw [show (100000000000000 / 54188638896411 : ℝ) = 1 / (54188638896411 / 100000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 1806287963213699 : ℝ) ≤
+      (275063298453652888357 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (4698253736713 / 16000000000000 : ℝ) ≤
+      (250000000000000 / 1806287963213699 : ℝ) := by
+    rw [show
+        (250000000000000 / 1806287963213699 : ℝ) =
+          (3 / 40) / (5418863889641097 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (496592617025537929121 / 2000000000000000000000 : ℝ))) -
+        6 * (275063298453652888357 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (4698253736713 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (4698253736713 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((5418863889641097 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (496592617025537929121 / 2000000000000000000000 : ℝ))) -
+          6 * (275063298453652888357 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (4698253736713 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (4698253736713 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (4698253736713 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_320 :
+    ((2579300092776726062851 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(3010988648734265671267 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((5424465227887459 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (310640941023347092163 / 1250000000000000000000 : ℝ))) -
+              6 * (1373914144821106679677 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (37663773450957 / 128000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (37663773450957 / 128000000000000 : ℝ)) =
+      Real.log (165663773450957 / 128000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((37663773450957 / 128000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (38383773450957 / 128000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2579300092776726062851 / 10000000000000000000000 : ℝ) ≤
+      Real.log (165663773450957 / 128000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (38383773450957 / 128000000000000 : ℝ) ≤
+      (-(3010988648734265671267 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (38383773450957 / 128000000000000 : ℝ) = (38383773450957 / 32000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (38383773450957 / 32000000000000 : ℝ) ≤
+            (454747252765734328733 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (5424465227887459 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (37663773450957 / 128000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (37663773450957 / 128000000000000 : ℝ) ≤
+      (2712232613943731 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (37663773450957 / 128000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau412 :
+    Real.arctan (2712232613943731 / 10688251554927932 : ℝ) ≤
+      (310640941023347092163 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau411 :
+    Real.arctan (2712232613943731 / 5000000000000000 : ℝ) ≤
+      (2 * (310640941023347092163 / 1250000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (1422062888731983 / 1250000000000000 : ℝ) ≤
+          Real.sqrt (1 + (2712232613943731 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (2712232613943731 / 5000000000000000 : ℝ) /
+                (1 + (1422062888731983 / 1250000000000000 : ℝ)) =
+              (2712232613943731 / 10688251554927932 : ℝ)
+            by
+        norm_num];
+        exact hau412)
+  have hat1 :
+    (Real.pi / 2 - (2 * (310640941023347092163 / 1250000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 2712232613943731 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 2712232613943731 : ℝ) =
+              (2712232613943731 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau411)
+  have hz1 :
+    (5000000000000000 / 2712232613943731 : ℝ) ≤
+      1 / Real.sqrt (37663773450957 / 128000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 2712232613943731 : ℝ) = 1 / (2712232613943731 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 5424465227887459 : ℝ) ≤
+      (1373914144821106679677 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (37663773450957 / 128000000000000 : ℝ) ≤
+      (750000000000000 / 5424465227887459 : ℝ) := by
+    rw [show
+        (750000000000000 / 5424465227887459 : ℝ) =
+          (3 / 40) / (5424465227887459 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (310640941023347092163 / 1250000000000000000000 : ℝ))) -
+        6 * (1373914144821106679677 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (37663773450957 / 128000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (37663773450957 / 128000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((5424465227887459 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (310640941023347092163 / 1250000000000000000000 : ℝ))) -
+          6 * (1373914144821106679677 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (37663773450957 / 128000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (37663773450957 / 128000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (37663773450957 / 128000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_321 :
+    ((645997960997488683677 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(12023720802257671034363 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((5430060788118679 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (621822181019150037887 / 2500000000000000000000 : ℝ))) -
+              6 * (1372516078509845972423 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (3774151700821 / 12800000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (3774151700821 / 12800000000000 : ℝ)) =
+      Real.log (16574151700821 / 12800000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((3774151700821 / 12800000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (3846151700821 / 12800000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (645997960997488683677 / 2500000000000000000000 : ℝ) ≤
+      Real.log (16574151700821 / 12800000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (3846151700821 / 12800000000000 : ℝ) ≤
+      (-(12023720802257671034363 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (3846151700821 / 12800000000000 : ℝ) = (3846151700821 / 3200000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3846151700821 / 3200000000000 : ℝ) ≤
+            (1839222803742328965637 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (5430060788118679 / 10000000000000000 : ℝ) ≤ Real.sqrt (3774151700821 / 12800000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (3774151700821 / 12800000000000 : ℝ) ≤ (2715030394059341 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (3774151700821 / 12800000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau414 :
+    Real.arctan (2715030394059341 / 10689586104512877 : ℝ) ≤
+      (621822181019150037887 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau413 :
+    Real.arctan (2715030394059341 / 5000000000000000 : ℝ) ≤
+      (2 * (621822181019150037887 / 2500000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (5689586104512877 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (2715030394059341 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (2715030394059341 / 5000000000000000 : ℝ) /
+                (1 + (5689586104512877 / 5000000000000000 : ℝ)) =
+              (2715030394059341 / 10689586104512877 : ℝ)
+            by
+        norm_num];
+        exact hau414)
+  have hat1 :
+    (Real.pi / 2 - (2 * (621822181019150037887 / 2500000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 2715030394059341 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 2715030394059341 : ℝ) =
+              (2715030394059341 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau413)
+  have hz1 :
+    (5000000000000000 / 2715030394059341 : ℝ) ≤
+      1 / Real.sqrt (3774151700821 / 12800000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 2715030394059341 : ℝ) = 1 / (2715030394059341 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 5430060788118679 : ℝ) ≤
+      (1372516078509845972423 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (3774151700821 / 12800000000000 : ℝ) ≤
+      (750000000000000 / 5430060788118679 : ℝ) := by
+    rw [show
+        (750000000000000 / 5430060788118679 : ℝ) =
+          (3 / 40) / (5430060788118679 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (621822181019150037887 / 2500000000000000000000 : ℝ))) -
+        6 * (1372516078509845972423 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (3774151700821 / 12800000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (3774151700821 / 12800000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((5430060788118679 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (621822181019150037887 / 2500000000000000000000 : ℝ))) -
+          6 * (1372516078509845972423 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (3774151700821 / 12800000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (3774151700821 / 12800000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (3774151700821 / 12800000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_322 :
+    ((647170348745621798299 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(120035278675576217591 / 100000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2717825294089373 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (248944668273143873453 / 1000000000000000000000 : ℝ))) -
+              6 * (1371122271595357286801 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (37819260565463 / 128000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (37819260565463 / 128000000000000 : ℝ)) =
+      Real.log (165819260565463 / 128000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((37819260565463 / 128000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (38539260565463 / 128000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (647170348745621798299 / 2500000000000000000000 : ℝ) ≤
+      Real.log (165819260565463 / 128000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (38539260565463 / 128000000000000 : ℝ) ≤
+      (-(120035278675576217591 / 100000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (38539260565463 / 128000000000000 : ℝ) = (38539260565463 / 32000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (38539260565463 / 32000000000000 : ℝ) ≤
+            (18594157384423782409 / 100000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2717825294089373 / 5000000000000000 : ℝ) ≤
+      Real.sqrt (37819260565463 / 128000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (37819260565463 / 128000000000000 : ℝ) ≤
+      (5435650588178749 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (37819260565463 / 128000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau416 :
+    Real.arctan (5435650588178749 / 21381840682278414 : ℝ) ≤
+      (248944668273143873453 / 1000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau415 :
+    Real.arctan (5435650588178749 / 10000000000000000 : ℝ) ≤
+      (2 * (248944668273143873453 / 1000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (5690920341139207 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (5435650588178749 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (5435650588178749 / 10000000000000000 : ℝ) /
+                (1 + (5690920341139207 / 5000000000000000 : ℝ)) =
+              (5435650588178749 / 21381840682278414 : ℝ)
+            by
+        norm_num];
+        exact hau416)
+  have hat1 :
+    (Real.pi / 2 - (2 * (248944668273143873453 / 1000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 5435650588178749 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 5435650588178749 : ℝ) =
+              (5435650588178749 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau415)
+  have hz1 :
+    (10000000000000000 / 5435650588178749 : ℝ) ≤
+      1 / Real.sqrt (37819260565463 / 128000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 5435650588178749 : ℝ) =
+          1 / (5435650588178749 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 2717825294089373 : ℝ) ≤
+      (1371122271595357286801 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (37819260565463 / 128000000000000 : ℝ) ≤
+      (375000000000000 / 2717825294089373 : ℝ) := by
+    rw [show
+        (375000000000000 / 2717825294089373 : ℝ) =
+          (3 / 40) / (2717825294089373 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (248944668273143873453 / 1000000000000000000000 : ℝ))) -
+        6 * (1371122271595357286801 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (37819260565463 / 128000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (37819260565463 / 128000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2717825294089373 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (248944668273143873453 / 1000000000000000000000 : ℝ))) -
+          6 * (1371122271595357286801 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (37819260565463 / 128000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (37819260565463 / 128000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (37819260565463 / 128000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_323 :
+    ((518673749563393131451 / 2000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(5991687813080415260473 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1360308661454999 / 2500000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2491601414036235246563 / 10000000000000000000000 : ℝ))) -
+              6 * (684866351246375430211 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (9474251030679 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (9474251030679 / 32000000000000 : ℝ)) =
+      Real.log (41474251030679 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((9474251030679 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (9654251030679 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (518673749563393131451 / 2000000000000000000000 : ℝ) ≤
+      Real.log (41474251030679 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (9654251030679 / 32000000000000 : ℝ) ≤
+      (-(5991687813080415260473 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (9654251030679 / 32000000000000 : ℝ) = (9654251030679 / 8000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9654251030679 / 8000000000000 : ℝ) ≤
+            (939783989919584739527 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1360308661454999 / 2500000000000000 : ℝ) ≤ Real.sqrt (9474251030679 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (9474251030679 / 32000000000000 : ℝ) ≤ (5441234645819999 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (9474251030679 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau418 :
+    Real.arctan (418556511216923 / 1644962194619537 : ℝ) ≤
+      (2491601414036235246563 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau417 :
+    Real.arctan (5441234645819999 / 10000000000000000 : ℝ) ≤
+      (2 * (2491601414036235246563 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (11384508530053981 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (5441234645819999 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (5441234645819999 / 10000000000000000 : ℝ) /
+                (1 + (11384508530053981 / 10000000000000000 : ℝ)) =
+              (418556511216923 / 1644962194619537 : ℝ)
+            by
+        norm_num];
+        exact hau418)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2491601414036235246563 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 5441234645819999 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 5441234645819999 : ℝ) =
+              (5441234645819999 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau417)
+  have hz1 :
+    (10000000000000000 / 5441234645819999 : ℝ) ≤
+      1 / Real.sqrt (9474251030679 / 32000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 5441234645819999 : ℝ) =
+          1 / (5441234645819999 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (187500000000000 / 1360308661454999 : ℝ) ≤
+      (684866351246375430211 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (9474251030679 / 32000000000000 : ℝ) ≤
+      (187500000000000 / 1360308661454999 : ℝ) := by
+    rw [show
+        (187500000000000 / 1360308661454999 : ℝ) =
+          (3 / 40) / (1360308661454999 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2491601414036235246563 / 10000000000000000000000 : ℝ))) -
+        6 * (684866351246375430211 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (9474251030679 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (9474251030679 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1360308661454999 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2491601414036235246563 / 10000000000000000000000 : ℝ))) -
+          6 * (684866351246375430211 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (9474251030679 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (9474251030679 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (9474251030679 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_324 :
+    ((1299026952276566602519 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(11963263914384789373721 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1361703244675941 / 2500000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2493752927826940413923 / 10000000000000000000000 : ℝ))) -
+              6 * (1368347349769969504417 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (37974747679969 / 128000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (37974747679969 / 128000000000000 : ℝ)) =
+      Real.log (165974747679969 / 128000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((37974747679969 / 128000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (38694747679969 / 128000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1299026952276566602519 / 5000000000000000000000 : ℝ) ≤
+      Real.log (165974747679969 / 128000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (38694747679969 / 128000000000000 : ℝ) ≤
+      (-(11963263914384789373721 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (38694747679969 / 128000000000000 : ℝ) = (38694747679969 / 32000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (38694747679969 / 32000000000000 : ℝ) ≤
+            (1899679691615210626279 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1361703244675941 / 2500000000000000 : ℝ) ≤
+      Real.sqrt (37974747679969 / 128000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (37974747679969 / 128000000000000 : ℝ) ≤
+      (5446812978703767 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (37974747679969 / 128000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau420 :
+    Real.arctan (1815604326234589 / 7129058584264025 : ℝ) ≤
+      (2493752927826940413923 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau419 :
+    Real.arctan (5446812978703767 / 10000000000000000 : ℝ) ≤
+      (2 * (2493752927826940413923 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (455487030111683 / 400000000000000 : ℝ) ≤
+          Real.sqrt (1 + (5446812978703767 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (5446812978703767 / 10000000000000000 : ℝ) /
+                (1 + (455487030111683 / 400000000000000 : ℝ)) =
+              (1815604326234589 / 7129058584264025 : ℝ)
+            by
+        norm_num];
+        exact hau420)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2493752927826940413923 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 5446812978703767 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 5446812978703767 : ℝ) =
+              (5446812978703767 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau419)
+  have hz1 :
+    (10000000000000000 / 5446812978703767 : ℝ) ≤
+      1 / Real.sqrt (37974747679969 / 128000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 5446812978703767 : ℝ) =
+          1 / (5446812978703767 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (62500000000000 / 453901081558647 : ℝ) ≤
+      (1368347349769969504417 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (37974747679969 / 128000000000000 : ℝ) ≤
+      (62500000000000 / 453901081558647 : ℝ) := by
+    rw [show
+        (62500000000000 / 453901081558647 : ℝ) =
+          (3 / 40) / (1361703244675941 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2493752927826940413923 / 10000000000000000000000 : ℝ))) -
+        6 * (1368347349769969504417 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (37974747679969 / 128000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (37974747679969 / 128000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1361703244675941 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2493752927826940413923 / 10000000000000000000000 : ℝ))) -
+          6 * (1368347349769969504417 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (37974747679969 / 128000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (37974747679969 / 128000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (37974747679969 / 128000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/V27.lean b/LeanPool/Zeta5Irrational/Table/V27.lean
new file mode 100644
index 0000000000..95edaa31c0
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/V27.lean
@@ -0,0 +1,1448 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Energy
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # V27: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_325 :
+    ((2602736867247839204543 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(5971596284766294893433 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2726192802200519 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2495901233890856705717 / 10000000000000000000000 : ℝ))) -
+              6 * (683483096073199743303 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (19026245618611 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (19026245618611 / 64000000000000 : ℝ)) =
+      Real.log (83026245618611 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((19026245618611 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (19386245618611 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2602736867247839204543 / 10000000000000000000000 : ℝ) ≤
+      Real.log (83026245618611 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (19386245618611 / 64000000000000 : ℝ) ≤
+      (-(5971596284766294893433 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (19386245618611 / 64000000000000 : ℝ) = (19386245618611 / 16000000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19386245618611 / 16000000000000 : ℝ) ≤
+            (959875518233705106567 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2726192802200519 / 5000000000000000 : ℝ) ≤ Real.sqrt (19026245618611 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (19026245618611 / 64000000000000 : ℝ) ≤
+      (5452385604401041 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (19026245618611 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau422 :
+    Real.arctan (5452385604401041 / 21389842350931802 : ℝ) ≤
+      (2495901233890856705717 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau421 :
+    Real.arctan (5452385604401041 / 10000000000000000 : ℝ) ≤
+      (2 * (2495901233890856705717 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (5694921175465901 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (5452385604401041 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (5452385604401041 / 10000000000000000 : ℝ) /
+                (1 + (5694921175465901 / 5000000000000000 : ℝ)) =
+              (5452385604401041 / 21389842350931802 : ℝ)
+            by
+        norm_num];
+        exact hau422)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2495901233890856705717 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 5452385604401041 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 5452385604401041 : ℝ) =
+              (5452385604401041 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau421)
+  have hz1 :
+    (10000000000000000 / 5452385604401041 : ℝ) ≤
+      1 / Real.sqrt (19026245618611 / 64000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 5452385604401041 : ℝ) =
+          1 / (5452385604401041 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 2726192802200519 : ℝ) ≤
+      (683483096073199743303 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (19026245618611 / 64000000000000 : ℝ) ≤
+      (375000000000000 / 2726192802200519 : ℝ) := by
+    rw [show
+        (375000000000000 / 2726192802200519 : ℝ) =
+          (3 / 40) / (2726192802200519 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2495901233890856705717 / 10000000000000000000000 : ℝ))) -
+        6 * (683483096073199743303 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (19026245618611 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (19026245618611 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2726192802200519 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2495901233890856705717 / 10000000000000000000000 : ℝ))) -
+          6 * (683483096073199743303 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (19026245618611 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (19026245618611 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (19026245618611 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_326 :
+    ((651854409488761182881 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(11923161429885025563023 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1364488135098277 / 2500000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2498046341966979991561 / 10000000000000000000000 : ℝ))) -
+              6 * (1365589208491496565637 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (1525209391779 / 5120000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1525209391779 / 5120000000000 : ℝ)) =
+      Real.log (6645209391779 / 5120000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1525209391779 / 5120000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1554009391779 / 5120000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (651854409488761182881 / 2500000000000000000000 : ℝ) ≤
+      Real.log (6645209391779 / 5120000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1554009391779 / 5120000000000 : ℝ) ≤
+      (-(11923161429885025563023 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (1554009391779 / 5120000000000 : ℝ) = (1554009391779 / 1280000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1554009391779 / 1280000000000 : ℝ) ≤
+            (1939782176114974436977 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1364488135098277 / 2500000000000000 : ℝ) ≤ Real.sqrt (1525209391779 / 5120000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1525209391779 / 5120000000000 : ℝ) ≤ (5457952540393111 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1525209391779 / 5120000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau424 :
+    Real.arctan (5457952540393111 / 21392508324911752 : ℝ) ≤
+      (2498046341966979991561 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau423 :
+    Real.arctan (5457952540393111 / 10000000000000000 : ℝ) ≤
+      (2 * (2498046341966979991561 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (1424063540613969 / 1250000000000000 : ℝ) ≤
+          Real.sqrt (1 + (5457952540393111 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (5457952540393111 / 10000000000000000 : ℝ) /
+                (1 + (1424063540613969 / 1250000000000000 : ℝ)) =
+              (5457952540393111 / 21392508324911752 : ℝ)
+            by
+        norm_num];
+        exact hau424)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2498046341966979991561 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 5457952540393111 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 5457952540393111 : ℝ) =
+              (5457952540393111 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau423)
+  have hz1 :
+    (10000000000000000 / 5457952540393111 : ℝ) ≤
+      1 / Real.sqrt (1525209391779 / 5120000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 5457952540393111 : ℝ) =
+          1 / (5457952540393111 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (187500000000000 / 1364488135098277 : ℝ) ≤
+      (1365589208491496565637 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1525209391779 / 5120000000000 : ℝ) ≤
+      (187500000000000 / 1364488135098277 : ℝ) := by
+    rw [show
+        (187500000000000 / 1364488135098277 : ℝ) =
+          (3 / 40) / (1364488135098277 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2498046341966979991561 / 10000000000000000000000 : ℝ))) -
+        6 * (1365589208491496565637 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1525209391779 / 5120000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1525209391779 / 5120000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1364488135098277 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2498046341966979991561 / 10000000000000000000000 : ℝ))) -
+          6 * (1365589208491496565637 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (1525209391779 / 5120000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1525209391779 / 5120000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1525209391779 / 5120000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_327 :
+    ((2612096218725827974741 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(5951585167346387345697 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((27317569020361 / 50000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2500188261746333242647 / 10000000000000000000000 : ℝ))) -
+              6 * (341054094455857365227 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (2387998646983 / 8000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (2387998646983 / 8000000000000 : ℝ)) =
+      Real.log (10387998646983 / 8000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((2387998646983 / 8000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (2432998646983 / 8000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2612096218725827974741 / 10000000000000000000000 : ℝ) ≤
+      Real.log (10387998646983 / 8000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (2432998646983 / 8000000000000 : ℝ) ≤
+      (-(5951585167346387345697 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (2432998646983 / 8000000000000 : ℝ) = (2432998646983 / 2000000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2432998646983 / 2000000000000 : ℝ) ≤
+            (979886635653612654303 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (27317569020361 / 50000000000000 : ℝ) ≤ Real.sqrt (2387998646983 / 8000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (2387998646983 / 8000000000000 : ℝ) ≤ (5463513804072203 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (2387998646983 / 8000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau426 :
+    Real.arctan (5463513804072203 / 21395173675170005 : ℝ) ≤
+      (2500188261746333242647 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau425 :
+    Real.arctan (5463513804072203 / 10000000000000000 : ℝ) ≤
+      (2 * (2500188261746333242647 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (2279034735034001 / 2000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (5463513804072203 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (5463513804072203 / 10000000000000000 : ℝ) /
+                (1 + (2279034735034001 / 2000000000000000 : ℝ)) =
+              (5463513804072203 / 21395173675170005 : ℝ)
+            by
+        norm_num];
+        exact hau426)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2500188261746333242647 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 5463513804072203 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 5463513804072203 : ℝ) =
+              (5463513804072203 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau425)
+  have hz1 :
+    (10000000000000000 / 5463513804072203 : ℝ) ≤
+      1 / Real.sqrt (2387998646983 / 8000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 5463513804072203 : ℝ) =
+          1 / (5463513804072203 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (3750000000000 / 27317569020361 : ℝ) ≤
+      (341054094455857365227 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (2387998646983 / 8000000000000 : ℝ) ≤
+      (3750000000000 / 27317569020361 : ℝ) := by
+    rw [show
+        (3750000000000 / 27317569020361 : ℝ) = (3 / 40) / (27317569020361 / 50000000000000 : ℝ) by
+        norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2500188261746333242647 / 10000000000000000000000 : ℝ))) -
+        6 * (341054094455857365227 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (2387998646983 / 8000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (2387998646983 / 8000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((27317569020361 / 50000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2500188261746333242647 / 10000000000000000000000 : ℝ))) -
+          6 * (341054094455857365227 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (2387998646983 / 8000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (2387998646983 / 8000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (2387998646983 / 8000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_328 :
+    ((2616772611608389628563 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(11883219124168659191989 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1367267353185529 / 2500000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2502327002872303514111 / 10000000000000000000000 : ℝ))) -
+              6 * (681423839653867688303 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (38285721908981 / 128000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (38285721908981 / 128000000000000 : ℝ)) =
+      Real.log (166285721908981 / 128000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((38285721908981 / 128000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (39005721908981 / 128000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2616772611608389628563 / 10000000000000000000000 : ℝ) ≤
+      Real.log (166285721908981 / 128000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (39005721908981 / 128000000000000 : ℝ) ≤
+      (-(11883219124168659191989 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (39005721908981 / 128000000000000 : ℝ) = (39005721908981 / 32000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (39005721908981 / 32000000000000 : ℝ) ≤
+            (1979724481831340808011 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1367267353185529 / 2500000000000000 : ℝ) ≤
+      Real.sqrt (38285721908981 / 128000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (38285721908981 / 128000000000000 : ℝ) ≤
+      (5469069412742119 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (38285721908981 / 128000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau428 :
+    Real.arctan (5469069412742119 / 21397838402144127 : ℝ) ≤
+      (2502327002872303514111 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau427 :
+    Real.arctan (5469069412742119 / 10000000000000000 : ℝ) ≤
+      (2 * (2502327002872303514111 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (11397838402144127 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (5469069412742119 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (5469069412742119 / 10000000000000000 : ℝ) /
+                (1 + (11397838402144127 / 10000000000000000 : ℝ)) =
+              (5469069412742119 / 21397838402144127 : ℝ)
+            by
+        norm_num];
+        exact hau428)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2502327002872303514111 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 5469069412742119 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 5469069412742119 : ℝ) =
+              (5469069412742119 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau427)
+  have hz1 :
+    (10000000000000000 / 5469069412742119 : ℝ) ≤
+      1 / Real.sqrt (38285721908981 / 128000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 5469069412742119 : ℝ) =
+          1 / (5469069412742119 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (187500000000000 / 1367267353185529 : ℝ) ≤
+      (681423839653867688303 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (38285721908981 / 128000000000000 : ℝ) ≤
+      (187500000000000 / 1367267353185529 : ℝ) := by
+    rw [show
+        (187500000000000 / 1367267353185529 : ℝ) =
+          (3 / 40) / (1367267353185529 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2502327002872303514111 / 10000000000000000000000 : ℝ))) -
+        6 * (681423839653867688303 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (38285721908981 / 128000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (38285721908981 / 128000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1367267353185529 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2502327002872303514111 / 10000000000000000000000 : ℝ))) -
+          6 * (681423839653867688303 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (38285721908981 / 128000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (38285721908981 / 128000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (38285721908981 / 128000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_329 :
+    ((262144681864805827297 / 1000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(11863307639479982018991 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1094923876723771 / 2000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1252231287470484669311 / 5000000000000000000000 : ℝ))) -
+              6 * (340370773063998395589 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (19181732733117 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (19181732733117 / 64000000000000 : ℝ)) =
+      Real.log (83181732733117 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((19181732733117 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (19541732733117 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (262144681864805827297 / 1000000000000000000000 : ℝ) ≤
+      Real.log (83181732733117 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (19541732733117 / 64000000000000 : ℝ) ≤
+      (-(11863307639479982018991 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (19541732733117 / 64000000000000 : ℝ) = (19541732733117 / 16000000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19541732733117 / 16000000000000 : ℝ) ≤
+            (1999635966520017981009 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1094923876723771 / 2000000000000000 : ℝ) ≤ Real.sqrt (19181732733117 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (19181732733117 / 64000000000000 : ℝ) ≤ (2737309691809429 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (19181732733117 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau430 :
+    Real.arctan (5474619383618858 / 21400502506271173 : ℝ) ≤
+      (1252231287470484669311 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau429 :
+    Real.arctan (2737309691809429 / 5000000000000000 : ℝ) ≤
+      (2 * (1252231287470484669311 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (11400502506271173 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (2737309691809429 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (2737309691809429 / 5000000000000000 : ℝ) /
+                (1 + (11400502506271173 / 10000000000000000 : ℝ)) =
+              (5474619383618858 / 21400502506271173 : ℝ)
+            by
+        norm_num];
+        exact hau430)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1252231287470484669311 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 2737309691809429 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 2737309691809429 : ℝ) =
+              (2737309691809429 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau429)
+  have hz1 :
+    (5000000000000000 / 2737309691809429 : ℝ) ≤
+      1 / Real.sqrt (19181732733117 / 64000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 2737309691809429 : ℝ) = 1 / (2737309691809429 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (150000000000000 / 1094923876723771 : ℝ) ≤
+      (340370773063998395589 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (19181732733117 / 64000000000000 : ℝ) ≤
+      (150000000000000 / 1094923876723771 : ℝ) := by
+    rw [show
+        (150000000000000 / 1094923876723771 : ℝ) =
+          (3 / 40) / (1094923876723771 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1252231287470484669311 / 5000000000000000000000 : ℝ))) -
+        6 * (340370773063998395589 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (19181732733117 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (19181732733117 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1094923876723771 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1252231287470484669311 / 5000000000000000000000 : ℝ))) -
+          6 * (340370773063998395589 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (19181732733117 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (19181732733117 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (19181732733117 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_330 :
+    ((525223768377459148197 / 2000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(11843435722740940104459 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1370040933457811 / 2500000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1253297493750716190239 / 5000000000000000000000 : ℝ))) -
+              6 * (272024519224901943929 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (38441209023487 / 128000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (38441209023487 / 128000000000000 : ℝ)) =
+      Real.log (166441209023487 / 128000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((38441209023487 / 128000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (39161209023487 / 128000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (525223768377459148197 / 2000000000000000000000 : ℝ) ≤
+      Real.log (166441209023487 / 128000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (39161209023487 / 128000000000000 : ℝ) ≤
+      (-(11843435722740940104459 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (39161209023487 / 128000000000000 : ℝ) = (39161209023487 / 32000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (39161209023487 / 32000000000000 : ℝ) ≤
+            (2019507883259059895541 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1370040933457811 / 2500000000000000 : ℝ) ≤
+      Real.sqrt (38441209023487 / 128000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (38441209023487 / 128000000000000 : ℝ) ≤
+      (5480163733831247 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (38441209023487 / 128000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau432 :
+    Real.arctan (5480163733831247 / 21403165987987687 : ℝ) ≤
+      (1253297493750716190239 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau431 :
+    Real.arctan (5480163733831247 / 10000000000000000 : ℝ) ≤
+      (2 * (1253297493750716190239 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (11403165987987687 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (5480163733831247 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (5480163733831247 / 10000000000000000 : ℝ) /
+                (1 + (11403165987987687 / 10000000000000000 : ℝ)) =
+              (5480163733831247 / 21403165987987687 : ℝ)
+            by
+        norm_num];
+        exact hau432)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1253297493750716190239 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 5480163733831247 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 5480163733831247 : ℝ) =
+              (5480163733831247 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau431)
+  have hz1 :
+    (10000000000000000 / 5480163733831247 : ℝ) ≤
+      1 / Real.sqrt (38441209023487 / 128000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 5480163733831247 : ℝ) =
+          1 / (5480163733831247 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (187500000000000 / 1370040933457811 : ℝ) ≤
+      (272024519224901943929 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (38441209023487 / 128000000000000 : ℝ) ≤
+      (187500000000000 / 1370040933457811 : ℝ) := by
+    rw [show
+        (187500000000000 / 1370040933457811 : ℝ) =
+          (3 / 40) / (1370040933457811 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1253297493750716190239 / 5000000000000000000000 : ℝ))) -
+        6 * (272024519224901943929 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (38441209023487 / 128000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (38441209023487 / 128000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1370040933457811 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1253297493750716190239 / 5000000000000000000000 : ℝ))) -
+          6 * (272024519224901943929 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (38441209023487 / 128000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (38441209023487 / 128000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (38441209023487 / 128000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_331 :
+    ((2630788683365702473583 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(11823603217005112636329 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((5485702480421547 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2508724250056138755739 / 10000000000000000000000 : ℝ))) -
+              6 * (135876617051301854297 / 1000000000000000000000 : ℝ)))) ≤
+      Vfield (1925947629037 / 6400000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1925947629037 / 6400000000000 : ℝ)) =
+      Real.log (8325947629037 / 6400000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1925947629037 / 6400000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1961947629037 / 6400000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2630788683365702473583 / 10000000000000000000000 : ℝ) ≤
+      Real.log (8325947629037 / 6400000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1961947629037 / 6400000000000 : ℝ) ≤
+      (-(11823603217005112636329 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (1961947629037 / 6400000000000 : ℝ) = (1961947629037 / 1600000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1961947629037 / 1600000000000 : ℝ) ≤
+            (2039340388994887363671 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (5485702480421547 / 10000000000000000 : ℝ) ≤ Real.sqrt (1925947629037 / 6400000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1925947629037 / 6400000000000 : ℝ) ≤ (109714049608431 / 200000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1925947629037 / 6400000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau434 :
+    Real.arctan (914283746736925 / 3567638141288284 : ℝ) ≤
+      (2508724250056138755739 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau433 :
+    Real.arctan (109714049608431 / 200000000000000 : ℝ) ≤
+      (2 * (2508724250056138755739 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (1425728605966213 / 1250000000000000 : ℝ) ≤
+          Real.sqrt (1 + (109714049608431 / 200000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (109714049608431 / 200000000000000 : ℝ) /
+                (1 + (1425728605966213 / 1250000000000000 : ℝ)) =
+              (914283746736925 / 3567638141288284 : ℝ)
+            by
+        norm_num];
+        exact hau434)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2508724250056138755739 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (200000000000000 / 109714049608431 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (200000000000000 / 109714049608431 : ℝ) =
+              (109714049608431 / 200000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau433)
+  have hz1 :
+    (200000000000000 / 109714049608431 : ℝ) ≤ 1 / Real.sqrt (1925947629037 / 6400000000000 : ℝ) :=
+  by
+    rw [show (200000000000000 / 109714049608431 : ℝ) = 1 / (109714049608431 / 200000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 1828567493473849 : ℝ) ≤
+      (135876617051301854297 / 1000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1925947629037 / 6400000000000 : ℝ) ≤
+      (250000000000000 / 1828567493473849 : ℝ) := by
+    rw [show
+        (250000000000000 / 1828567493473849 : ℝ) =
+          (3 / 40) / (5485702480421547 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2508724250056138755739 / 10000000000000000000000 : ℝ))) -
+        6 * (135876617051301854297 / 1000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1925947629037 / 6400000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1925947629037 / 6400000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((5485702480421547 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2508724250056138755739 / 10000000000000000000000 : ℝ))) -
+          6 * (135876617051301854297 / 1000000000000000000000 : ℝ))) ≤
+      Real.sqrt (1925947629037 / 6400000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1925947629037 / 6400000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1925947629037 / 6400000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_332 :
+    ((66003045729603764453 / 250000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(736503488463111140191 / 625000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1099352646095163 / 2000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (314121670365840661703 / 1250000000000000000000 : ℝ))) -
+              6 * (678032724979199444369 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (19337219847623 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (19337219847623 / 64000000000000 : ℝ)) =
+      Real.log (83337219847623 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((19337219847623 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (19697219847623 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (66003045729603764453 / 250000000000000000000 : ℝ) ≤
+      Real.log (83337219847623 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (19697219847623 / 64000000000000 : ℝ) ≤
+      (-(736503488463111140191 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (19697219847623 / 64000000000000 : ℝ) = (19697219847623 / 16000000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19697219847623 / 16000000000000 : ℝ) ≤
+            (129930486911888859809 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1099352646095163 / 2000000000000000 : ℝ) ≤ Real.sqrt (19337219847623 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (19337219847623 / 64000000000000 : ℝ) ≤ (2748381615237909 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (19337219847623 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau436 :
+    Real.arctan (916127205079303 / 3568525450505307 : ℝ) ≤
+      (314121670365840661703 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau435 :
+    Real.arctan (2748381615237909 / 5000000000000000 : ℝ) ≤
+      (2 * (314121670365840661703 / 1250000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (5705576351515921 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (2748381615237909 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (2748381615237909 / 5000000000000000 : ℝ) /
+                (1 + (5705576351515921 / 5000000000000000 : ℝ)) =
+              (916127205079303 / 3568525450505307 : ℝ)
+            by
+        norm_num];
+        exact hau436)
+  have hat1 :
+    (Real.pi / 2 - (2 * (314121670365840661703 / 1250000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 2748381615237909 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 2748381615237909 : ℝ) =
+              (2748381615237909 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau435)
+  have hz1 :
+    (5000000000000000 / 2748381615237909 : ℝ) ≤
+      1 / Real.sqrt (19337219847623 / 64000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 2748381615237909 : ℝ) = 1 / (2748381615237909 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (50000000000000 / 366450882031721 : ℝ) ≤
+      (678032724979199444369 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (19337219847623 / 64000000000000 : ℝ) ≤
+      (50000000000000 / 366450882031721 : ℝ) := by
+    rw [show
+        (50000000000000 / 366450882031721 : ℝ) =
+          (3 / 40) / (1099352646095163 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (314121670365840661703 / 1250000000000000000000 : ℝ))) -
+        6 * (678032724979199444369 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (19337219847623 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (19337219847623 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1099352646095163 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (314121670365840661703 / 1250000000000000000000 : ℝ))) -
+          6 * (678032724979199444369 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (19337219847623 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (19337219847623 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (19337219847623 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_333 :
+    ((2649446272363180973399 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2348932839525893305633 / 2000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((5507801768411671 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1258604994325691642003 / 5000000000000000000000 : ℝ))) -
+              6 * (270676154042019708727 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (4853740851219 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (4853740851219 / 16000000000000 : ℝ)) =
+      Real.log (20853740851219 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((4853740851219 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (4943740851219 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2649446272363180973399 / 10000000000000000000000 : ℝ) ≤
+      Real.log (20853740851219 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (4943740851219 / 16000000000000 : ℝ) ≤
+      (-(2348932839525893305633 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (4943740851219 / 16000000000000 : ℝ) = (4943740851219 / 4000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4943740851219 / 4000000000000 : ℝ) ≤
+            (423655881674106694367 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (5507801768411671 / 10000000000000000 : ℝ) ≤ Real.sqrt (4853740851219 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (4853740851219 / 16000000000000 : ℝ) ≤ (2753900884205837 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (4853740851219 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau438 :
+    Real.arctan (5507801768411674 / 21416474075655703 : ℝ) ≤
+      (1258604994325691642003 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau437 :
+    Real.arctan (2753900884205837 / 5000000000000000 : ℝ) ≤
+      (2 * (1258604994325691642003 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (11416474075655703 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (2753900884205837 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (2753900884205837 / 5000000000000000 : ℝ) /
+                (1 + (11416474075655703 / 10000000000000000 : ℝ)) =
+              (5507801768411674 / 21416474075655703 : ℝ)
+            by
+        norm_num];
+        exact hau438)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1258604994325691642003 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 2753900884205837 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 2753900884205837 : ℝ) =
+              (2753900884205837 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau437)
+  have hz1 :
+    (5000000000000000 / 2753900884205837 : ℝ) ≤
+      1 / Real.sqrt (4853740851219 / 16000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 2753900884205837 : ℝ) = 1 / (2753900884205837 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 5507801768411671 : ℝ) ≤
+      (270676154042019708727 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (4853740851219 / 16000000000000 : ℝ) ≤
+      (750000000000000 / 5507801768411671 : ℝ) := by
+    rw [show
+        (750000000000000 / 5507801768411671 : ℝ) =
+          (3 / 40) / (5507801768411671 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1258604994325691642003 / 5000000000000000000000 : ℝ))) -
+        6 * (270676154042019708727 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (4853740851219 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (4853740851219 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((5507801768411671 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1258604994325691642003 / 5000000000000000000000 : ℝ))) -
+          6 * (270676154042019708727 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (4853740851219 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (4853740851219 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (4853740851219 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_334 :
+    ((166172626819820649641 / 625000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2926356785290338176897 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((5518818227512711 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2521434201602609004749 / 10000000000000000000000 : ℝ))) -
+              6 * (1350711973100143615783 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (19492706962129 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (19492706962129 / 64000000000000 : ℝ)) =
+      Real.log (83492706962129 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((19492706962129 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (19852706962129 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (166172626819820649641 / 625000000000000000000 : ℝ) ≤
+      Real.log (83492706962129 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (19852706962129 / 64000000000000 : ℝ) ≤
+      (-(2926356785290338176897 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (19852706962129 / 64000000000000 : ℝ) = (19852706962129 / 16000000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19852706962129 / 16000000000000 : ℝ) ≤
+            (539379116209661823103 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (5518818227512711 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (19492706962129 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (19492706962129 / 64000000000000 : ℝ) ≤ (2759409113756357 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (19492706962129 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau440 :
+    Real.arctan (2759409113756357 / 10710896484535649 : ℝ) ≤
+      (2521434201602609004749 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau439 :
+    Real.arctan (2759409113756357 / 5000000000000000 : ℝ) ≤
+      (2 * (2521434201602609004749 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (5710896484535649 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (2759409113756357 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (2759409113756357 / 5000000000000000 : ℝ) /
+                (1 + (5710896484535649 / 5000000000000000 : ℝ)) =
+              (2759409113756357 / 10710896484535649 : ℝ)
+            by
+        norm_num];
+        exact hau440)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2521434201602609004749 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 2759409113756357 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 2759409113756357 : ℝ) =
+              (2759409113756357 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau439)
+  have hz1 :
+    (5000000000000000 / 2759409113756357 : ℝ) ≤
+      1 / Real.sqrt (19492706962129 / 64000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 2759409113756357 : ℝ) = 1 / (2759409113756357 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 5518818227512711 : ℝ) ≤
+      (1350711973100143615783 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (19492706962129 / 64000000000000 : ℝ) ≤
+      (750000000000000 / 5518818227512711 : ℝ) := by
+    rw [show
+        (750000000000000 / 5518818227512711 : ℝ) =
+          (3 / 40) / (5518818227512711 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2521434201602609004749 / 10000000000000000000000 : ℝ))) -
+        6 * (1350711973100143615783 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (19492706962129 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (19492706962129 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((5518818227512711 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2521434201602609004749 / 10000000000000000000000 : ℝ))) -
+          6 * (1350711973100143615783 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (19492706962129 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (19492706962129 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (19492706962129 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_335 :
+    ((533613823123012619069 / 2000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(729146464864790530219 / 625000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((345613296233431 / 625000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1262823037718319180193 / 5000000000000000000000 : ℝ))) -
+              6 * (337014725658869163649 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (9785225259691 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (9785225259691 / 32000000000000 : ℝ)) =
+      Real.log (41785225259691 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((9785225259691 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (9965225259691 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (533613823123012619069 / 2000000000000000000000 : ℝ) ≤
+      Real.log (41785225259691 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (9965225259691 / 32000000000000 : ℝ) ≤
+      (-(729146464864790530219 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (9965225259691 / 32000000000000 : ℝ) = (9965225259691 / 8000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9965225259691 / 8000000000000 : ℝ) ≤
+            (137287510510209469781 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (345613296233431 / 625000000000000 : ℝ) ≤ Real.sqrt (9785225259691 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (9785225259691 / 32000000000000 : ℝ) ≤ (5529812739734899 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (9785225259691 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau442 :
+    Real.arctan (5529812739734899 / 21427109386740566 : ℝ) ≤
+      (1262823037718319180193 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau441 :
+    Real.arctan (5529812739734899 / 10000000000000000 : ℝ) ≤
+      (2 * (1262823037718319180193 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (5713554693370283 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (5529812739734899 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (5529812739734899 / 10000000000000000 : ℝ) /
+                (1 + (5713554693370283 / 5000000000000000 : ℝ)) =
+              (5529812739734899 / 21427109386740566 : ℝ)
+            by
+        norm_num];
+        exact hau442)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1262823037718319180193 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 5529812739734899 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 5529812739734899 : ℝ) =
+              (5529812739734899 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau441)
+  have hz1 :
+    (10000000000000000 / 5529812739734899 : ℝ) ≤
+      1 / Real.sqrt (9785225259691 / 32000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 5529812739734899 : ℝ) =
+          1 / (5529812739734899 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (46875000000000 / 345613296233431 : ℝ) ≤
+      (337014725658869163649 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (9785225259691 / 32000000000000 : ℝ) ≤
+      (46875000000000 / 345613296233431 : ℝ) := by
+    rw [show
+        (46875000000000 / 345613296233431 : ℝ) =
+          (3 / 40) / (345613296233431 / 625000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1262823037718319180193 / 5000000000000000000000 : ℝ))) -
+        6 * (337014725658869163649 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (9785225259691 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (9785225259691 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((345613296233431 / 625000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1262823037718319180193 / 5000000000000000000000 : ℝ))) -
+          6 * (337014725658869163649 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (9785225259691 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (9785225259691 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (9785225259691 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_336 :
+    ((1338683773990469618213 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2906852973399345608809 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((554078543572499 / 1000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1264922841551584207157 / 5000000000000000000000 : ℝ))) -
+              6 * (672710702479827681713 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (3929638815327 / 12800000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (3929638815327 / 12800000000000 : ℝ)) =
+      Real.log (16729638815327 / 12800000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((3929638815327 / 12800000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (4001638815327 / 12800000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1338683773990469618213 / 5000000000000000000000 : ℝ) ≤
+      Real.log (16729638815327 / 12800000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (4001638815327 / 12800000000000 : ℝ) ≤
+      (-(2906852973399345608809 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (4001638815327 / 12800000000000 : ℝ) = (4001638815327 / 3200000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4001638815327 / 3200000000000 : ℝ) ≤
+            (558882928100654391191 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (554078543572499 / 1000000000000000 : ℝ) ≤ Real.sqrt (3929638815327 / 12800000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (3929638815327 / 12800000000000 : ℝ) ≤ (5540785435724993 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (3929638815327 / 12800000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau444 :
+    Real.arctan (5540785435724993 / 21432423332117393 : ℝ) ≤
+      (1264922841551584207157 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau443 :
+    Real.arctan (5540785435724993 / 10000000000000000 : ℝ) ≤
+      (2 * (1264922841551584207157 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (11432423332117393 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (5540785435724993 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (5540785435724993 / 10000000000000000 : ℝ) /
+                (1 + (11432423332117393 / 10000000000000000 : ℝ)) =
+              (5540785435724993 / 21432423332117393 : ℝ)
+            by
+        norm_num];
+        exact hau444)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1264922841551584207157 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 5540785435724993 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 5540785435724993 : ℝ) =
+              (5540785435724993 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau443)
+  have hz1 :
+    (10000000000000000 / 5540785435724993 : ℝ) ≤
+      1 / Real.sqrt (3929638815327 / 12800000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 5540785435724993 : ℝ) =
+          1 / (5540785435724993 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (75000000000000 / 554078543572499 : ℝ) ≤
+      (672710702479827681713 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (3929638815327 / 12800000000000 : ℝ) ≤
+      (75000000000000 / 554078543572499 : ℝ) := by
+    rw [show
+        (75000000000000 / 554078543572499 : ℝ) =
+          (3 / 40) / (554078543572499 / 1000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1264922841551584207157 / 5000000000000000000000 : ℝ))) -
+        6 * (672710702479827681713 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (3929638815327 / 12800000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (3929638815327 / 12800000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((554078543572499 / 1000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1264922841551584207157 / 5000000000000000000000 : ℝ))) -
+          6 * (672710702479827681713 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (3929638815327 / 12800000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (3929638815327 / 12800000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (3929638815327 / 12800000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/V28.lean b/LeanPool/Zeta5Irrational/Table/V28.lean
new file mode 100644
index 0000000000..1ed7fb262f
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/V28.lean
@@ -0,0 +1,1428 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Energy
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # V28: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_337 :
+    ((537331468458756497103 / 2000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(5794315664138754364461 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1387934111209669 / 2500000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1267016548427454690201 / 5000000000000000000000 : ℝ))) -
+              6 * (1342799328315371089453 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (616435551059 / 2000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (616435551059 / 2000000000000 : ℝ)) =
+      Real.log (2616435551059 / 2000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((616435551059 / 2000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (627685551059 / 2000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (537331468458756497103 / 2000000000000000000000 : ℝ) ≤
+      Real.log (2616435551059 / 2000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (627685551059 / 2000000000000 : ℝ) ≤
+      (-(5794315664138754364461 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (627685551059 / 2000000000000 : ℝ) = (627685551059 / 500000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (627685551059 / 500000000000 : ℝ) ≤
+            (1137156138861245635539 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1387934111209669 / 2500000000000000 : ℝ) ≤ Real.sqrt (616435551059 / 2000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (616435551059 / 2000000000000 : ℝ) ≤ (5551736444838679 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (616435551059 / 2000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau446 :
+    Real.arctan (5551736444838679 / 21437734808647645 : ℝ) ≤
+      (1267016548427454690201 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau445 :
+    Real.arctan (5551736444838679 / 10000000000000000 : ℝ) ≤
+      (2 * (1267016548427454690201 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (2287546961729529 / 2000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (5551736444838679 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (5551736444838679 / 10000000000000000 : ℝ) /
+                (1 + (2287546961729529 / 2000000000000000 : ℝ)) =
+              (5551736444838679 / 21437734808647645 : ℝ)
+            by
+        norm_num];
+        exact hau446)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1267016548427454690201 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 5551736444838679 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 5551736444838679 : ℝ) =
+              (5551736444838679 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau445)
+  have hz1 :
+    (10000000000000000 / 5551736444838679 : ℝ) ≤
+      1 / Real.sqrt (616435551059 / 2000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 5551736444838679 : ℝ) =
+          1 / (5551736444838679 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (187500000000000 / 1387934111209669 : ℝ) ≤
+      (1342799328315371089453 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (616435551059 / 2000000000000 : ℝ) ≤
+      (187500000000000 / 1387934111209669 : ℝ) := by
+    rw [show
+        (187500000000000 / 1387934111209669 : ℝ) =
+          (3 / 40) / (1387934111209669 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1267016548427454690201 / 5000000000000000000000 : ℝ))) -
+        6 * (1342799328315371089453 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (616435551059 / 2000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (616435551059 / 2000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1387934111209669 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1267016548427454690201 / 5000000000000000000000 : ℝ))) -
+          6 * (1342799328315371089453 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (616435551059 / 2000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (616435551059 / 2000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (616435551059 / 2000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_338 :
+    ((1347969257293923422131 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(11550000575388243838883 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1112533179031669 / 2000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1269104194128484300137 / 5000000000000000000000 : ℝ))) -
+              6 * (670096261503884641029 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (19803681191141 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (19803681191141 / 64000000000000 : ℝ)) =
+      Real.log (83803681191141 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((19803681191141 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (20163681191141 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1347969257293923422131 / 5000000000000000000000 : ℝ) ≤
+      Real.log (83803681191141 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (20163681191141 / 64000000000000 : ℝ) ≤
+      (-(11550000575388243838883 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (20163681191141 / 64000000000000 : ℝ) = (20163681191141 / 16000000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20163681191141 / 16000000000000 : ℝ) ≤
+            (2312943030611756161117 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1112533179031669 / 2000000000000000 : ℝ) ≤ Real.sqrt (19803681191141 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (19803681191141 / 64000000000000 : ℝ) ≤ (1390666473789587 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (19803681191141 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau448 :
+    Real.arctan (5562665895158348 / 21443043819769187 : ℝ) ≤
+      (1269104194128484300137 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau447 :
+    Real.arctan (1390666473789587 / 2500000000000000 : ℝ) ≤
+      (2 * (1269104194128484300137 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (11443043819769187 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1390666473789587 / 2500000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1390666473789587 / 2500000000000000 : ℝ) /
+                (1 + (11443043819769187 / 10000000000000000 : ℝ)) =
+              (5562665895158348 / 21443043819769187 : ℝ)
+            by
+        norm_num];
+        exact hau448)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1269104194128484300137 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (2500000000000000 / 1390666473789587 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 1390666473789587 : ℝ) =
+              (1390666473789587 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau447)
+  have hz1 :
+    (2500000000000000 / 1390666473789587 : ℝ) ≤
+      1 / Real.sqrt (19803681191141 / 64000000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 1390666473789587 : ℝ) = 1 / (1390666473789587 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (150000000000000 / 1112533179031669 : ℝ) ≤
+      (670096261503884641029 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (19803681191141 / 64000000000000 : ℝ) ≤
+      (150000000000000 / 1112533179031669 : ℝ) := by
+    rw [show
+        (150000000000000 / 1112533179031669 : ℝ) =
+          (3 / 40) / (1112533179031669 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1269104194128484300137 / 5000000000000000000000 : ℝ))) -
+        6 * (670096261503884641029 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (19803681191141 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (19803681191141 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1112533179031669 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1269104194128484300137 / 5000000000000000000000 : ℝ))) -
+          6 * (670096261503884641029 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (19803681191141 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (19803681191141 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (19803681191141 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_339 :
+    ((1352605540426391349941 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(11511518481907533559663 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((5573573913510577 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2 * (79449113381127256143 / 312500000000000000000 : ℝ))) -
+              6 * (167200105171068611039 / 1250000000000000000000 : ℝ)))) ≤
+      Vfield (9940712374197 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (9940712374197 / 32000000000000 : ℝ)) =
+      Real.log (41940712374197 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((9940712374197 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (10120712374197 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1352605540426391349941 / 5000000000000000000000 : ℝ) ≤
+      Real.log (41940712374197 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (10120712374197 / 32000000000000 : ℝ) ≤
+      (-(11511518481907533559663 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (10120712374197 / 32000000000000 : ℝ) = (10120712374197 / 8000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10120712374197 / 8000000000000 : ℝ) ≤
+            (2351425124092466440337 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (5573573913510577 / 10000000000000000 : ℝ) ≤ Real.sqrt (9940712374197 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (9940712374197 / 32000000000000 : ℝ) ≤ (278678695675529 / 500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (9940712374197 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau450 :
+    Real.arctan (1393393478377645 / 5362087592227979 : ℝ) ≤
+      (79449113381127256143 / 312500000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau449 :
+    Real.arctan (278678695675529 / 500000000000000 : ℝ) ≤
+      (2 * (79449113381127256143 / 312500000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (2862087592227979 / 2500000000000000 : ℝ) ≤
+          Real.sqrt (1 + (278678695675529 / 500000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (278678695675529 / 500000000000000 : ℝ) /
+                (1 + (2862087592227979 / 2500000000000000 : ℝ)) =
+              (1393393478377645 / 5362087592227979 : ℝ)
+            by
+        norm_num];
+        exact hau450)
+  have hat1 :
+    (Real.pi / 2 - (2 * (79449113381127256143 / 312500000000000000000 : ℝ))) ≤
+      Real.arctan (500000000000000 / 278678695675529 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (500000000000000 / 278678695675529 : ℝ) =
+              (278678695675529 / 500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau449)
+  have hz1 :
+    (500000000000000 / 278678695675529 : ℝ) ≤
+      1 / Real.sqrt (9940712374197 / 32000000000000 : ℝ) := by
+    rw [show (500000000000000 / 278678695675529 : ℝ) = 1 / (278678695675529 / 500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 5573573913510577 : ℝ) ≤
+      (167200105171068611039 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (9940712374197 / 32000000000000 : ℝ) ≤
+      (750000000000000 / 5573573913510577 : ℝ) := by
+    rw [show
+        (750000000000000 / 5573573913510577 : ℝ) =
+          (3 / 40) / (5573573913510577 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (79449113381127256143 / 312500000000000000000 : ℝ))) -
+        6 * (167200105171068611039 / 1250000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (9940712374197 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (9940712374197 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((5573573913510577 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (79449113381127256143 / 312500000000000000000 : ℝ))) -
+          6 * (167200105171068611039 / 1250000000000000000000 : ℝ))) ≤
+      Real.sqrt (9940712374197 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (9940712374197 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (9940712374197 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_340 :
+    ((680932614757960827541 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(11434995727182164588973 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((349707884715169 / 625000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (510132446778923655351 / 2000000000000000000000 : ℝ))) -
+              6 * (1332462268344726654813 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (200369118629 / 640000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (200369118629 / 640000000000 : ℝ)) =
+      Real.log (840369118629 / 640000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((200369118629 / 640000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (203969118629 / 640000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (680932614757960827541 / 2500000000000000000000 : ℝ) ≤
+      Real.log (840369118629 / 640000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (203969118629 / 640000000000 : ℝ) ≤
+      (-(11434995727182164588973 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (203969118629 / 640000000000 : ℝ) = (203969118629 / 160000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (203969118629 / 160000000000 : ℝ) ≤
+            (2427947878817835411027 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (349707884715169 / 625000000000000 : ℝ) ≤ Real.sqrt (200369118629 / 640000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (200369118629 / 640000000000 : ℝ) ≤ (5595326155442707 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (200369118629 / 640000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau452 :
+    Real.arctan (5595326155442707 / 21458956094940814 : ℝ) ≤
+      (510132446778923655351 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau451 :
+    Real.arctan (5595326155442707 / 10000000000000000 : ℝ) ≤
+      (2 * (510132446778923655351 / 2000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (5729478047470407 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (5595326155442707 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (5595326155442707 / 10000000000000000 : ℝ) /
+                (1 + (5729478047470407 / 5000000000000000 : ℝ)) =
+              (5595326155442707 / 21458956094940814 : ℝ)
+            by
+        norm_num];
+        exact hau452)
+  have hat1 :
+    (Real.pi / 2 - (2 * (510132446778923655351 / 2000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 5595326155442707 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 5595326155442707 : ℝ) =
+              (5595326155442707 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau451)
+  have hz1 :
+    (10000000000000000 / 5595326155442707 : ℝ) ≤
+      1 / Real.sqrt (200369118629 / 640000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 5595326155442707 : ℝ) =
+          1 / (5595326155442707 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (46875000000000 / 349707884715169 : ℝ) ≤
+      (1332462268344726654813 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (200369118629 / 640000000000 : ℝ) ≤
+      (46875000000000 / 349707884715169 : ℝ) := by
+    rw [show
+        (46875000000000 / 349707884715169 : ℝ) =
+          (3 / 40) / (349707884715169 / 625000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (510132446778923655351 / 2000000000000000000000 : ℝ))) -
+        6 * (1332462268344726654813 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (200369118629 / 640000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (200369118629 / 640000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((349707884715169 / 625000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (510132446778923655351 / 2000000000000000000000 : ℝ))) -
+          6 * (1332462268344726654813 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (200369118629 / 640000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (200369118629 / 640000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (200369118629 / 640000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_341 :
+    ((548443120772471356537 / 2000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2271810820302100756551 / 2000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((5616994160776461 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2558905467817166652031 / 10000000000000000000000 : ℝ))) -
+              6 * (265476493422338033503 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (10096199488703 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (10096199488703 / 32000000000000 : ℝ)) =
+      Real.log (42096199488703 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((10096199488703 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (10276199488703 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (548443120772471356537 / 2000000000000000000000 : ℝ) ≤
+      Real.log (42096199488703 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (10276199488703 / 32000000000000 : ℝ) ≤
+      (-(2271810820302100756551 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (10276199488703 / 32000000000000 : ℝ) = (10276199488703 / 8000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10276199488703 / 8000000000000 : ℝ) ≤
+            (500777900897899243449 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (5616994160776461 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (10096199488703 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (10096199488703 / 32000000000000 : ℝ) ≤ (351062135048529 / 625000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (10096199488703 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau454 :
+    Real.arctan (1872331386925488 / 7156517338005019 : ℝ) ≤
+      (2558905467817166652031 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau453 :
+    Real.arctan (351062135048529 / 625000000000000 : ℝ) ≤
+      (2 * (2558905467817166652031 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (11469552014015057 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (351062135048529 / 625000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (351062135048529 / 625000000000000 : ℝ) /
+                (1 + (11469552014015057 / 10000000000000000 : ℝ)) =
+              (1872331386925488 / 7156517338005019 : ℝ)
+            by
+        norm_num];
+        exact hau454)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2558905467817166652031 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (625000000000000 / 351062135048529 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (625000000000000 / 351062135048529 : ℝ) =
+              (351062135048529 / 625000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau453)
+  have hz1 :
+    (625000000000000 / 351062135048529 : ℝ) ≤
+      1 / Real.sqrt (10096199488703 / 32000000000000 : ℝ) := by
+    rw [show (625000000000000 / 351062135048529 : ℝ) = 1 / (351062135048529 / 625000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 1872331386925487 : ℝ) ≤
+      (265476493422338033503 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (10096199488703 / 32000000000000 : ℝ) ≤
+      (250000000000000 / 1872331386925487 : ℝ) := by
+    rw [show
+        (250000000000000 / 1872331386925487 : ℝ) =
+          (3 / 40) / (5616994160776461 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2558905467817166652031 / 10000000000000000000000 : ℝ))) -
+        6 * (265476493422338033503 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (10096199488703 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (10096199488703 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((5616994160776461 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2558905467817166652031 / 10000000000000000000000 : ℝ))) -
+          6 * (265476493422338033503 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (10096199488703 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (10096199488703 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (10096199488703 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_342 :
+    ((1380333320836278337613 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(5641842422477859503957 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((5638578900628463 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2 * (40110966774065083747 / 156250000000000000000 : ℝ))) -
+              6 * (1322360325788739087959 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (2543485761489 / 8000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (2543485761489 / 8000000000000 : ℝ)) =
+      Real.log (10543485761489 / 8000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((2543485761489 / 8000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (2588485761489 / 8000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1380333320836278337613 / 5000000000000000000000 : ℝ) ≤
+      Real.log (10543485761489 / 8000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (2588485761489 / 8000000000000 : ℝ) ≤
+      (-(5641842422477859503957 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (2588485761489 / 8000000000000 : ℝ) = (2588485761489 / 2000000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2588485761489 / 2000000000000 : ℝ) ≤
+            (1289629380522140496043 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (5638578900628463 / 10000000000000000 : ℝ) ≤ Real.sqrt (2543485761489 / 8000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (2543485761489 / 8000000000000 : ℝ) ≤ (2819289450314233 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (2543485761489 / 8000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau456 :
+    Real.arctan (5638578900628466 / 21480138153289467 : ℝ) ≤
+      (40110966774065083747 / 156250000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau455 :
+    Real.arctan (2819289450314233 / 5000000000000000 : ℝ) ≤
+      (2 * (40110966774065083747 / 156250000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (11480138153289467 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (2819289450314233 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (2819289450314233 / 5000000000000000 : ℝ) /
+                (1 + (11480138153289467 / 10000000000000000 : ℝ)) =
+              (5638578900628466 / 21480138153289467 : ℝ)
+            by
+        norm_num];
+        exact hau456)
+  have hat1 :
+    (Real.pi / 2 - (2 * (40110966774065083747 / 156250000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 2819289450314233 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 2819289450314233 : ℝ) =
+              (2819289450314233 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau455)
+  have hz1 :
+    (5000000000000000 / 2819289450314233 : ℝ) ≤
+      1 / Real.sqrt (2543485761489 / 8000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 2819289450314233 : ℝ) = 1 / (2819289450314233 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 5638578900628463 : ℝ) ≤
+      (1322360325788739087959 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (2543485761489 / 8000000000000 : ℝ) ≤
+      (750000000000000 / 5638578900628463 : ℝ) := by
+    rw [show
+        (750000000000000 / 5638578900628463 : ℝ) =
+          (3 / 40) / (5638578900628463 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (40110966774065083747 / 156250000000000000000 : ℝ))) -
+        6 * (1322360325788739087959 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (2543485761489 / 8000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (2543485761489 / 8000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((5638578900628463 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (40110966774065083747 / 156250000000000000000 : ℝ))) -
+          6 * (1322360325788739087959 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (2543485761489 / 8000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (2543485761489 / 8000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (2543485761489 / 8000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_343 :
+    ((2779083698092697689807 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2241775878834215454377 / 2000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1415020331899601 / 2500000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1287625992309098770811 / 5000000000000000000000 : ℝ))) -
+              6 * (1317394761724191804781 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (10251686603209 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (10251686603209 / 32000000000000 : ℝ)) =
+      Real.log (42251686603209 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((10251686603209 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (10431686603209 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2779083698092697689807 / 10000000000000000000000 : ℝ) ≤
+      Real.log (42251686603209 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (10431686603209 / 32000000000000 : ℝ) ≤
+      (-(2241775878834215454377 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (10431686603209 / 32000000000000 : ℝ) = (10431686603209 / 8000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10431686603209 / 8000000000000 : ℝ) ≤
+            (530812842365784545623 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1415020331899601 / 2500000000000000 : ℝ) ≤ Real.sqrt (10251686603209 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (10251686603209 / 32000000000000 : ℝ) ≤
+      (5660081327598407 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (10251686603209 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau458 :
+    Real.arctan (5660081327598407 / 21490714539793777 : ℝ) ≤
+      (1287625992309098770811 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau457 :
+    Real.arctan (5660081327598407 / 10000000000000000 : ℝ) ≤
+      (2 * (1287625992309098770811 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (11490714539793777 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (5660081327598407 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (5660081327598407 / 10000000000000000 : ℝ) /
+                (1 + (11490714539793777 / 10000000000000000 : ℝ)) =
+              (5660081327598407 / 21490714539793777 : ℝ)
+            by
+        norm_num];
+        exact hau458)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1287625992309098770811 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 5660081327598407 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 5660081327598407 : ℝ) =
+              (5660081327598407 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau457)
+  have hz1 :
+    (10000000000000000 / 5660081327598407 : ℝ) ≤
+      1 / Real.sqrt (10251686603209 / 32000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 5660081327598407 : ℝ) =
+          1 / (5660081327598407 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (187500000000000 / 1415020331899601 : ℝ) ≤
+      (1317394761724191804781 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (10251686603209 / 32000000000000 : ℝ) ≤
+      (187500000000000 / 1415020331899601 : ℝ) := by
+    rw [show
+        (187500000000000 / 1415020331899601 : ℝ) =
+          (3 / 40) / (1415020331899601 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1287625992309098770811 / 5000000000000000000000 : ℝ))) -
+        6 * (1317394761724191804781 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (10251686603209 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (10251686603209 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1415020331899601 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1287625992309098770811 / 5000000000000000000000 : ℝ))) -
+          6 * (1317394761724191804781 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (10251686603209 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (10251686603209 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (10251686603209 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_344 :
+    ((1398733449030094056213 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2783657344140277014661 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2840751188129811 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (322919540605409180593 / 1250000000000000000000 : ℝ))) -
+              6 * (131248472051481762243 / 1000000000000000000000 : ℝ)))) ≤
+      Vfield (5164715080231 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (5164715080231 / 16000000000000 : ℝ)) =
+      Real.log (21164715080231 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((5164715080231 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (5254715080231 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1398733449030094056213 / 5000000000000000000000 : ℝ) ≤
+      Real.log (21164715080231 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (5254715080231 / 16000000000000 : ℝ) ≤
+      (-(2783657344140277014661 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (5254715080231 / 16000000000000 : ℝ) = (5254715080231 / 4000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5254715080231 / 4000000000000 : ℝ) ≤
+            (682078557359722985339 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2840751188129811 / 5000000000000000 : ℝ) ≤ Real.sqrt (5164715080231 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (5164715080231 / 16000000000000 : ℝ) ≤ (45452019010077 / 80000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (5164715080231 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau460 :
+    Real.arctan (5681502376259625 / 21501281200433443 : ℝ) ≤
+      (322919540605409180593 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau459 :
+    Real.arctan (45452019010077 / 80000000000000 : ℝ) ≤
+      (2 * (322919540605409180593 / 1250000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (11501281200433443 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (45452019010077 / 80000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (45452019010077 / 80000000000000 : ℝ) /
+                (1 + (11501281200433443 / 10000000000000000 : ℝ)) =
+              (5681502376259625 / 21501281200433443 : ℝ)
+            by
+        norm_num];
+        exact hau460)
+  have hat1 :
+    (Real.pi / 2 - (2 * (322919540605409180593 / 1250000000000000000000 : ℝ))) ≤
+      Real.arctan (80000000000000 / 45452019010077 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (80000000000000 / 45452019010077 : ℝ) =
+              (45452019010077 / 80000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau459)
+  have hz1 :
+    (80000000000000 / 45452019010077 : ℝ) ≤ 1 / Real.sqrt (5164715080231 / 16000000000000 : ℝ) := by
+    rw [show (80000000000000 / 45452019010077 : ℝ) = 1 / (45452019010077 / 80000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (125000000000000 / 946917062709937 : ℝ) ≤
+      (131248472051481762243 / 1000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (5164715080231 / 16000000000000 : ℝ) ≤
+      (125000000000000 / 946917062709937 : ℝ) := by
+    rw [show
+        (125000000000000 / 946917062709937 : ℝ) =
+          (3 / 40) / (2840751188129811 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (322919540605409180593 / 1250000000000000000000 : ℝ))) -
+        6 * (131248472051481762243 / 1000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (5164715080231 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (5164715080231 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2840751188129811 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (322919540605409180593 / 1250000000000000000000 : ℝ))) -
+          6 * (131248472051481762243 / 1000000000000000000000 : ℝ))) ≤
+      Real.sqrt (5164715080231 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (5164715080231 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (5164715080231 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_345 :
+    ((2834132224953083779733 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(5493881535351685480757 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1431025997411363 / 2500000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2 * (40616089696646176937 / 156250000000000000000 : ℝ))) -
+              6 * (1302827124684973012057 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (1310614659371 / 4000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1310614659371 / 4000000000000 : ℝ)) =
+      Real.log (5310614659371 / 4000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1310614659371 / 4000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1333114659371 / 4000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2834132224953083779733 / 10000000000000000000000 : ℝ) ≤
+      Real.log (5310614659371 / 4000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1333114659371 / 4000000000000 : ℝ) ≤
+      (-(5493881535351685480757 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (1333114659371 / 4000000000000 : ℝ) = (1333114659371 / 1000000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1333114659371 / 1000000000000 : ℝ) ≤
+            (1437590267648314519243 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1431025997411363 / 2500000000000000 : ℝ) ≤ Real.sqrt (1310614659371 / 4000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1310614659371 / 4000000000000 : ℝ) ≤ (1144820797929091 / 2000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1310614659371 / 4000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau462 :
+    Real.arctan (5724103989645455 / 21522385451124041 : ℝ) ≤
+      (40616089696646176937 / 156250000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau461 :
+    Real.arctan (1144820797929091 / 2000000000000000 : ℝ) ≤
+      (2 * (40616089696646176937 / 156250000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (11522385451124041 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1144820797929091 / 2000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1144820797929091 / 2000000000000000 : ℝ) /
+                (1 + (11522385451124041 / 10000000000000000 : ℝ)) =
+              (5724103989645455 / 21522385451124041 : ℝ)
+            by
+        norm_num];
+        exact hau462)
+  have hat1 :
+    (Real.pi / 2 - (2 * (40616089696646176937 / 156250000000000000000 : ℝ))) ≤
+      Real.arctan (2000000000000000 / 1144820797929091 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2000000000000000 / 1144820797929091 : ℝ) =
+              (1144820797929091 / 2000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau461)
+  have hz1 :
+    (2000000000000000 / 1144820797929091 : ℝ) ≤
+      1 / Real.sqrt (1310614659371 / 4000000000000 : ℝ) := by
+    rw [show
+        (2000000000000000 / 1144820797929091 : ℝ) = 1 / (1144820797929091 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (187500000000000 / 1431025997411363 : ℝ) ≤
+      (1302827124684973012057 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1310614659371 / 4000000000000 : ℝ) ≤
+      (187500000000000 / 1431025997411363 : ℝ) := by
+    rw [show
+        (187500000000000 / 1431025997411363 : ℝ) =
+          (3 / 40) / (1431025997411363 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (40616089696646176937 / 156250000000000000000 : ℝ))) -
+        6 * (1302827124684973012057 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1310614659371 / 4000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1310614659371 / 4000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1431025997411363 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (40616089696646176937 / 156250000000000000000 : ℝ))) -
+          6 * (1302827124684973012057 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (1310614659371 / 4000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1310614659371 / 4000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1310614659371 / 4000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_346 :
+    ((2870663608185664775589 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2710755638254187861199 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((5766390874464393 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2615326125171193657161 / 10000000000000000000000 : ℝ))) -
+              6 * (1293379633233145650797 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (5320202194737 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (5320202194737 / 16000000000000 : ℝ)) =
+      Real.log (21320202194737 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((5320202194737 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (5410202194737 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2870663608185664775589 / 10000000000000000000000 : ℝ) ≤
+      Real.log (21320202194737 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (5410202194737 / 16000000000000 : ℝ) ≤
+      (-(2710755638254187861199 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (5410202194737 / 16000000000000 : ℝ) = (5410202194737 / 4000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5410202194737 / 4000000000000 : ℝ) ≤
+            (754980263245812138801 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (5766390874464393 / 10000000000000000 : ℝ) ≤ Real.sqrt (5320202194737 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (5320202194737 / 16000000000000 : ℝ) ≤ (1441597718616099 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (5320202194737 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau464 :
+    Real.arctan (5766390874464396 / 21543451118149469 : ℝ) ≤
+      (2615326125171193657161 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau463 :
+    Real.arctan (1441597718616099 / 2500000000000000 : ℝ) ≤
+      (2 * (2615326125171193657161 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (11543451118149469 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1441597718616099 / 2500000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1441597718616099 / 2500000000000000 : ℝ) /
+                (1 + (11543451118149469 / 10000000000000000 : ℝ)) =
+              (5766390874464396 / 21543451118149469 : ℝ)
+            by
+        norm_num];
+        exact hau464)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2615326125171193657161 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (2500000000000000 / 1441597718616099 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 1441597718616099 : ℝ) =
+              (1441597718616099 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau463)
+  have hz1 :
+    (2500000000000000 / 1441597718616099 : ℝ) ≤
+      1 / Real.sqrt (5320202194737 / 16000000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 1441597718616099 : ℝ) = 1 / (1441597718616099 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 1922130291488131 : ℝ) ≤
+      (1293379633233145650797 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (5320202194737 / 16000000000000 : ℝ) ≤
+      (250000000000000 / 1922130291488131 : ℝ) := by
+    rw [show
+        (250000000000000 / 1922130291488131 : ℝ) =
+          (3 / 40) / (5766390874464393 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2615326125171193657161 / 10000000000000000000000 : ℝ))) -
+        6 * (1293379633233145650797 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (5320202194737 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (5320202194737 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((5766390874464393 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2615326125171193657161 / 10000000000000000000000 : ℝ))) -
+          6 * (1293379633233145650797 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (5320202194737 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (5320202194737 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (5320202194737 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_347 :
+    ((726765505706868317119 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(5350173580933170941977 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((580836990470971 / 1000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2631049342012675989687 / 10000000000000000000000 : ℝ))) -
+              6 * (128413473666420098043 / 1000000000000000000000 : ℝ)))) ≤
+      Vfield (539794575199 / 1600000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (539794575199 / 1600000000000 : ℝ)) =
+      Real.log (2139794575199 / 1600000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((539794575199 / 1600000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (548794575199 / 1600000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (726765505706868317119 / 2500000000000000000000 : ℝ) ≤
+      Real.log (2139794575199 / 1600000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (548794575199 / 1600000000000 : ℝ) ≤
+      (-(5350173580933170941977 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (548794575199 / 1600000000000 : ℝ) = (548794575199 / 400000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (548794575199 / 400000000000 : ℝ) ≤
+            (1581298222066829058023 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (580836990470971 / 1000000000000000 : ℝ) ≤ Real.sqrt (539794575199 / 1600000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (539794575199 / 1600000000000 : ℝ) ≤ (5808369904709713 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (539794575199 / 1600000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau466 :
+    Real.arctan (5808369904709713 / 21564478412359871 : ℝ) ≤
+      (2631049342012675989687 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau465 :
+    Real.arctan (5808369904709713 / 10000000000000000 : ℝ) ≤
+      (2 * (2631049342012675989687 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (11564478412359871 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (5808369904709713 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (5808369904709713 / 10000000000000000 : ℝ) /
+                (1 + (11564478412359871 / 10000000000000000 : ℝ)) =
+              (5808369904709713 / 21564478412359871 : ℝ)
+            by
+        norm_num];
+        exact hau466)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2631049342012675989687 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 5808369904709713 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 5808369904709713 : ℝ) =
+              (5808369904709713 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau465)
+  have hz1 :
+    (10000000000000000 / 5808369904709713 : ℝ) ≤
+      1 / Real.sqrt (539794575199 / 1600000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 5808369904709713 : ℝ) =
+          1 / (5808369904709713 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (75000000000000 / 580836990470971 : ℝ) ≤
+      (128413473666420098043 / 1000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (539794575199 / 1600000000000 : ℝ) ≤
+      (75000000000000 / 580836990470971 : ℝ) := by
+    rw [show
+        (75000000000000 / 580836990470971 : ℝ) =
+          (3 / 40) / (580836990470971 / 1000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2631049342012675989687 / 10000000000000000000000 : ℝ))) -
+        6 * (128413473666420098043 / 1000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (539794575199 / 1600000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (539794575199 / 1600000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((580836990470971 / 1000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2631049342012675989687 / 10000000000000000000000 : ℝ))) -
+          6 * (128413473666420098043 / 1000000000000000000000 : ℝ))) ≤
+      Real.sqrt (539794575199 / 1600000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (539794575199 / 1600000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (539794575199 / 1600000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_348 :
+    ((1471664216669667297137 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(10559678795699479359667 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((5850047707734419 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2646603120442483009779 / 10000000000000000000000 : ℝ))) -
+              6 * (637542647980929447969 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (5475689309243 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (5475689309243 / 16000000000000 : ℝ)) =
+      Real.log (21475689309243 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((5475689309243 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (5565689309243 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1471664216669667297137 / 5000000000000000000000 : ℝ) ≤
+      Real.log (21475689309243 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (5565689309243 / 16000000000000 : ℝ) ≤
+      (-(10559678795699479359667 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (5565689309243 / 16000000000000 : ℝ) = (5565689309243 / 4000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5565689309243 / 4000000000000 : ℝ) ≤
+            (3303264810300520640333 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (5850047707734419 / 10000000000000000 : ℝ) ≤ Real.sqrt (5475689309243 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (5475689309243 / 16000000000000 : ℝ) ≤ (2925023853867211 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (5475689309243 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau468 :
+    Real.arctan (5850047707734422 / 21585467542691953 : ℝ) ≤
+      (2646603120442483009779 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau467 :
+    Real.arctan (2925023853867211 / 5000000000000000 : ℝ) ≤
+      (2 * (2646603120442483009779 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (11585467542691953 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (2925023853867211 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (2925023853867211 / 5000000000000000 : ℝ) /
+                (1 + (11585467542691953 / 10000000000000000 : ℝ)) =
+              (5850047707734422 / 21585467542691953 : ℝ)
+            by
+        norm_num];
+        exact hau468)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2646603120442483009779 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 2925023853867211 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 2925023853867211 : ℝ) =
+              (2925023853867211 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau467)
+  have hz1 :
+    (5000000000000000 / 2925023853867211 : ℝ) ≤
+      1 / Real.sqrt (5475689309243 / 16000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 2925023853867211 : ℝ) = 1 / (2925023853867211 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 5850047707734419 : ℝ) ≤
+      (637542647980929447969 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (5475689309243 / 16000000000000 : ℝ) ≤
+      (750000000000000 / 5850047707734419 : ℝ) := by
+    rw [show
+        (750000000000000 / 5850047707734419 : ℝ) =
+          (3 / 40) / (5850047707734419 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2646603120442483009779 / 10000000000000000000000 : ℝ))) -
+        6 * (637542647980929447969 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (5475689309243 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (5475689309243 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((5850047707734419 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2646603120442483009779 / 10000000000000000000000 : ℝ))) -
+          6 * (637542647980929447969 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (5475689309243 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (5475689309243 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (5475689309243 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/V29.lean b/LeanPool/Zeta5Irrational/Table/V29.lean
new file mode 100644
index 0000000000..972ce0cf57
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/V29.lean
@@ -0,0 +1,1436 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Energy
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # V29: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_349 :
+    ((372432974215837231393 / 1250000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(10420961770985507744611 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2945715338232803 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2661991062193999150973 / 10000000000000000000000 : ℝ))) -
+              6 * (1266224519414994741439 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (86772388539 / 250000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (86772388539 / 250000000000 : ℝ)) =
+      Real.log (336772388539 / 250000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((86772388539 / 250000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (88178638539 / 250000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (372432974215837231393 / 1250000000000000000000 : ℝ) ≤
+      Real.log (336772388539 / 250000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (88178638539 / 250000000000 : ℝ) ≤
+      (-(10420961770985507744611 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (88178638539 / 250000000000 : ℝ) = (88178638539 / 62500000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (88178638539 / 62500000000 : ℝ) ≤
+            (3441981835014492255389 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2945715338232803 / 5000000000000000 : ℝ) ≤ Real.sqrt (86772388539 / 250000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (86772388539 / 250000000000 : ℝ) ≤ (5891430676465609 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (86772388539 / 250000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau470 :
+    Real.arctan (5891430676465609 / 21606418716193208 : ℝ) ≤
+      (2661991062193999150973 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau469 :
+    Real.arctan (5891430676465609 / 10000000000000000 : ℝ) ≤
+      (2 * (2661991062193999150973 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (1450802339524151 / 1250000000000000 : ℝ) ≤
+          Real.sqrt (1 + (5891430676465609 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (5891430676465609 / 10000000000000000 : ℝ) /
+                (1 + (1450802339524151 / 1250000000000000 : ℝ)) =
+              (5891430676465609 / 21606418716193208 : ℝ)
+            by
+        norm_num];
+        exact hau470)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2661991062193999150973 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 5891430676465609 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 5891430676465609 : ℝ) =
+              (5891430676465609 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau469)
+  have hz1 :
+    (10000000000000000 / 5891430676465609 : ℝ) ≤ 1 / Real.sqrt (86772388539 / 250000000000 : ℝ) :=
+  by
+    rw [show
+        (10000000000000000 / 5891430676465609 : ℝ) =
+          1 / (5891430676465609 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 2945715338232803 : ℝ) ≤
+      (1266224519414994741439 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (86772388539 / 250000000000 : ℝ) ≤
+      (375000000000000 / 2945715338232803 : ℝ) := by
+    rw [show
+        (375000000000000 / 2945715338232803 : ℝ) =
+          (3 / 40) / (2945715338232803 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2661991062193999150973 / 10000000000000000000000 : ℝ))) -
+        6 * (1266224519414994741439 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (86772388539 / 250000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (86772388539 / 250000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2945715338232803 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2661991062193999150973 / 10000000000000000000000 : ℝ))) -
+          6 * (1266224519414994741439 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (86772388539 / 250000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (86772388539 / 250000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (86772388539 / 250000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_350 :
+    ((119954631949475568979 / 400000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(10347063308137192662121 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2956796049364893 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1335104378906465501497 / 5000000000000000000000 : ℝ))) -
+              6 * (630764785401862489949 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (11190582883251 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (11190582883251 / 32000000000000 : ℝ)) =
+      Real.log (43190582883251 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((11190582883251 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (11370582883251 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (119954631949475568979 / 400000000000000000000 : ℝ) ≤
+      Real.log (43190582883251 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (11370582883251 / 32000000000000 : ℝ) ≤
+      (-(10347063308137192662121 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (11370582883251 / 32000000000000 : ℝ) = (11370582883251 / 8000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11370582883251 / 8000000000000 : ℝ) ≤
+            (3515880297862807337879 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2956796049364893 / 5000000000000000 : ℝ) ≤ Real.sqrt (11190582883251 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (11190582883251 / 32000000000000 : ℝ) ≤
+      (5913592098729789 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (11190582883251 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau472 :
+    Real.arctan (5913592098729789 / 21617683569032140 : ℝ) ≤
+      (1335104378906465501497 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau471 :
+    Real.arctan (5913592098729789 / 10000000000000000 : ℝ) ≤
+      (2 * (1335104378906465501497 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (580884178451607 / 500000000000000 : ℝ) ≤
+          Real.sqrt (1 + (5913592098729789 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (5913592098729789 / 10000000000000000 : ℝ) /
+                (1 + (580884178451607 / 500000000000000 : ℝ)) =
+              (5913592098729789 / 21617683569032140 : ℝ)
+            by
+        norm_num];
+        exact hau472)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1335104378906465501497 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 5913592098729789 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 5913592098729789 : ℝ) =
+              (5913592098729789 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau471)
+  have hz1 :
+    (10000000000000000 / 5913592098729789 : ℝ) ≤
+      1 / Real.sqrt (11190582883251 / 32000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 5913592098729789 : ℝ) =
+          1 / (5913592098729789 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (125000000000000 / 985598683121631 : ℝ) ≤
+      (630764785401862489949 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (11190582883251 / 32000000000000 : ℝ) ≤
+      (125000000000000 / 985598683121631 : ℝ) := by
+    rw [show
+        (125000000000000 / 985598683121631 : ℝ) =
+          (3 / 40) / (2956796049364893 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1335104378906465501497 / 5000000000000000000000 : ℝ))) -
+        6 * (630764785401862489949 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (11190582883251 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (11190582883251 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2956796049364893 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1335104378906465501497 / 5000000000000000000000 : ℝ))) -
+          6 * (630764785401862489949 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (11190582883251 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (11190582883251 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (11190582883251 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_351 :
+    ((3018230232850539159649 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2568426735005533212383 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((5935670779677621 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (334797489789192552847 / 1250000000000000000000 : ℝ))) -
+              6 * (628443231648490909653 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (1127430003351 / 3200000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1127430003351 / 3200000000000 : ℝ)) =
+      Real.log (4327430003351 / 3200000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1127430003351 / 3200000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1145430003351 / 3200000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (3018230232850539159649 / 10000000000000000000000 : ℝ) ≤
+      Real.log (4327430003351 / 3200000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1145430003351 / 3200000000000 : ℝ) ≤
+      (-(2568426735005533212383 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (1145430003351 / 3200000000000 : ℝ) = (1145430003351 / 800000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1145430003351 / 800000000000 : ℝ) ≤
+            (897309166494466787617 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (5935670779677621 / 10000000000000000 : ℝ) ≤ Real.sqrt (1127430003351 / 3200000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1127430003351 / 3200000000000 : ℝ) ≤ (741958847459703 / 1250000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1127430003351 / 3200000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau474 :
+    Real.arctan (5935670779677624 / 21628937509709077 : ℝ) ≤
+      (334797489789192552847 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau473 :
+    Real.arctan (741958847459703 / 1250000000000000 : ℝ) ≤
+      (2 * (334797489789192552847 / 1250000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (11628937509709077 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (741958847459703 / 1250000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (741958847459703 / 1250000000000000 : ℝ) /
+                (1 + (11628937509709077 / 10000000000000000 : ℝ)) =
+              (5935670779677624 / 21628937509709077 : ℝ)
+            by
+        norm_num];
+        exact hau474)
+  have hat1 :
+    (Real.pi / 2 - (2 * (334797489789192552847 / 1250000000000000000000 : ℝ))) ≤
+      Real.arctan (1250000000000000 / 741958847459703 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1250000000000000 / 741958847459703 : ℝ) =
+              (741958847459703 / 1250000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau473)
+  have hz1 :
+    (1250000000000000 / 741958847459703 : ℝ) ≤
+      1 / Real.sqrt (1127430003351 / 3200000000000 : ℝ) := by
+    rw [show
+        (1250000000000000 / 741958847459703 : ℝ) = 1 / (741958847459703 / 1250000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 1978556926559207 : ℝ) ≤
+      (628443231648490909653 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1127430003351 / 3200000000000 : ℝ) ≤
+      (250000000000000 / 1978556926559207 : ℝ) := by
+    rw [show
+        (250000000000000 / 1978556926559207 : ℝ) =
+          (3 / 40) / (5935670779677621 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (334797489789192552847 / 1250000000000000000000 : ℝ))) -
+        6 * (628443231648490909653 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1127430003351 / 3200000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1127430003351 / 3200000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((5935670779677621 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (334797489789192552847 / 1250000000000000000000 : ℝ))) -
+          6 * (628443231648490909653 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (1127430003351 / 3200000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1127430003351 / 3200000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1127430003351 / 3200000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_352 :
+    ((759389310323588584061 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(10200884771266944986441 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((5957667639208997 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2686505063364219207891 / 10000000000000000000000 : ℝ))) -
+              6 * (1252294249803677508751 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (11358017183769 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (11358017183769 / 32000000000000 : ℝ)) =
+      Real.log (43358017183769 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((11358017183769 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (11538017183769 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (759389310323588584061 / 2500000000000000000000 : ℝ) ≤
+      Real.log (43358017183769 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (11538017183769 / 32000000000000 : ℝ) ≤
+      (-(10200884771266944986441 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (11538017183769 / 32000000000000 : ℝ) = (11538017183769 / 8000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11538017183769 / 8000000000000 : ℝ) ≤
+            (3662058834733055013559 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (5957667639208997 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (11358017183769 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (11358017183769 / 32000000000000 : ℝ) ≤ (5957667639209 / 10000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (11358017183769 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau476 :
+    Real.arctan (2978833819604500 / 10820090284937127 : ℝ) ≤
+      (2686505063364219207891 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau475 :
+    Real.arctan (5957667639209 / 10000000000000 : ℝ) ≤
+      (2 * (2686505063364219207891 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (5820090284937127 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (5957667639209 / 10000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (5957667639209 / 10000000000000 : ℝ) /
+                (1 + (5820090284937127 / 5000000000000000 : ℝ)) =
+              (2978833819604500 / 10820090284937127 : ℝ)
+            by
+        norm_num];
+        exact hau476)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2686505063364219207891 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000 / 5957667639209 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000 / 5957667639209 : ℝ) = (5957667639209 / 10000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau475)
+  have hz1 :
+    (10000000000000 / 5957667639209 : ℝ) ≤ 1 / Real.sqrt (11358017183769 / 32000000000000 : ℝ) := by
+    rw [show (10000000000000 / 5957667639209 : ℝ) = 1 / (5957667639209 / 10000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 5957667639208997 : ℝ) ≤
+      (1252294249803677508751 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (11358017183769 / 32000000000000 : ℝ) ≤
+      (750000000000000 / 5957667639208997 : ℝ) := by
+    rw [show
+        (750000000000000 / 5957667639208997 : ℝ) =
+          (3 / 40) / (5957667639208997 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2686505063364219207891 / 10000000000000000000000 : ℝ))) -
+        6 * (1252294249803677508751 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (11358017183769 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (11358017183769 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((5957667639208997 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2686505063364219207891 / 10000000000000000000000 : ℝ))) -
+          6 * (1252294249803677508751 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (11358017183769 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (11358017183769 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (11358017183769 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_353 :
+    ((2437765404834750633 / 8000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(5082335795652477889393 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((596863566877371 / 1000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1345275269978941688457 / 5000000000000000000000 : ℝ))) -
+              6 * (1250016938943352110893 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (22799751517797 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (22799751517797 / 64000000000000 : ℝ)) =
+      Real.log (86799751517797 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((22799751517797 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (23159751517797 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2437765404834750633 / 8000000000000000000 : ℝ) ≤
+      Real.log (86799751517797 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (23159751517797 / 64000000000000 : ℝ) ≤
+      (-(5082335795652477889393 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (23159751517797 / 64000000000000 : ℝ) = (23159751517797 / 16000000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23159751517797 / 16000000000000 : ℝ) ≤
+            (1849136007347522110607 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (596863566877371 / 1000000000000000 : ℝ) ≤ Real.sqrt (22799751517797 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (22799751517797 / 64000000000000 : ℝ) ≤
+      (5968635668773713 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (22799751517797 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau478 :
+    Real.arctan (5968635668773713 / 21645798029613848 : ℝ) ≤
+      (1345275269978941688457 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau477 :
+    Real.arctan (5968635668773713 / 10000000000000000 : ℝ) ≤
+      (2 * (1345275269978941688457 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (1455724753701731 / 1250000000000000 : ℝ) ≤
+          Real.sqrt (1 + (5968635668773713 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (5968635668773713 / 10000000000000000 : ℝ) /
+                (1 + (1455724753701731 / 1250000000000000 : ℝ)) =
+              (5968635668773713 / 21645798029613848 : ℝ)
+            by
+        norm_num];
+        exact hau478)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1345275269978941688457 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 5968635668773713 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 5968635668773713 : ℝ) =
+              (5968635668773713 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau477)
+  have hz1 :
+    (10000000000000000 / 5968635668773713 : ℝ) ≤
+      1 / Real.sqrt (22799751517797 / 64000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 5968635668773713 : ℝ) =
+          1 / (5968635668773713 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (25000000000000 / 198954522292457 : ℝ) ≤
+      (1250016938943352110893 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (22799751517797 / 64000000000000 : ℝ) ≤
+      (25000000000000 / 198954522292457 : ℝ) := by
+    rw [show
+        (25000000000000 / 198954522292457 : ℝ) =
+          (3 / 40) / (596863566877371 / 1000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1345275269978941688457 / 5000000000000000000000 : ℝ))) -
+        6 * (1250016938943352110893 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (22799751517797 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (22799751517797 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((596863566877371 / 1000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1345275269978941688457 / 5000000000000000000000 : ℝ))) -
+          6 * (1250016938943352110893 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (22799751517797 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (22799751517797 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (22799751517797 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_354 :
+    ((3056846968454616536319 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(10128589077740964565741 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((5979583580303689 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1347292351694098581751 / 5000000000000000000000 : ℝ))) -
+              6 * (623876003640546214071 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (2860433583507 / 8000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (2860433583507 / 8000000000000 : ℝ)) =
+      Real.log (10860433583507 / 8000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((2860433583507 / 8000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (2905433583507 / 8000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (3056846968454616536319 / 10000000000000000000000 : ℝ) ≤
+      Real.log (10860433583507 / 8000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (2905433583507 / 8000000000000 : ℝ) ≤
+      (-(10128589077740964565741 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (2905433583507 / 8000000000000 : ℝ) = (2905433583507 / 2000000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2905433583507 / 2000000000000 : ℝ) ≤
+            (3734354528259035434259 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (5979583580303689 / 10000000000000000 : ℝ) ≤ Real.sqrt (2860433583507 / 8000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (2860433583507 / 8000000000000 : ℝ) ≤ (1494895895075923 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (2860433583507 / 8000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau480 :
+    Real.arctan (1494895895075923 / 5412853195256301 : ℝ) ≤
+      (1347292351694098581751 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau479 :
+    Real.arctan (1494895895075923 / 2500000000000000 : ℝ) ≤
+      (2 * (1347292351694098581751 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (2912853195256301 / 2500000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1494895895075923 / 2500000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1494895895075923 / 2500000000000000 : ℝ) /
+                (1 + (2912853195256301 / 2500000000000000 : ℝ)) =
+              (1494895895075923 / 5412853195256301 : ℝ)
+            by
+        norm_num];
+        exact hau480)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1347292351694098581751 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (2500000000000000 / 1494895895075923 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 1494895895075923 : ℝ) =
+              (1494895895075923 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau479)
+  have hz1 :
+    (2500000000000000 / 1494895895075923 : ℝ) ≤
+      1 / Real.sqrt (2860433583507 / 8000000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 1494895895075923 : ℝ) = 1 / (1494895895075923 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 5979583580303689 : ℝ) ≤
+      (623876003640546214071 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (2860433583507 / 8000000000000 : ℝ) ≤
+      (750000000000000 / 5979583580303689 : ℝ) := by
+    rw [show
+        (750000000000000 / 5979583580303689 : ℝ) =
+          (3 / 40) / (5979583580303689 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1347292351694098581751 / 5000000000000000000000 : ℝ))) -
+        6 * (623876003640546214071 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (2860433583507 / 8000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (2860433583507 / 8000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((5979583580303689 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1347292351694098581751 / 5000000000000000000000 : ℝ))) -
+          6 * (623876003640546214071 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (2860433583507 / 8000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (2860433583507 / 8000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (2860433583507 / 8000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_355 :
+    ((766619474111480287379 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(5046318145504354096527 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((5990511484098597 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1349303808025649067077 / 5000000000000000000000 : ℝ))) -
+              6 * (1245499343066240103687 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (4593437163663 / 12800000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (4593437163663 / 12800000000000 : ℝ)) =
+      Real.log (17393437163663 / 12800000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((4593437163663 / 12800000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (4665437163663 / 12800000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (766619474111480287379 / 2500000000000000000000 : ℝ) ≤
+      Real.log (17393437163663 / 12800000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (4665437163663 / 12800000000000 : ℝ) ≤
+      (-(5046318145504354096527 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (4665437163663 / 12800000000000 : ℝ) = (4665437163663 / 3200000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4665437163663 / 3200000000000 : ℝ) ≤
+            (1885153657495645903473 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (5990511484098597 / 10000000000000000 : ℝ) ≤ Real.sqrt (4593437163663 / 12800000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (4593437163663 / 12800000000000 : ℝ) ≤ (29952557420493 / 50000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (4593437163663 / 12800000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau482 :
+    Real.arctan (5990511484098600 / 21657024828021821 : ℝ) ≤
+      (1349303808025649067077 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau481 :
+    Real.arctan (29952557420493 / 50000000000000 : ℝ) ≤
+      (2 * (1349303808025649067077 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (11657024828021821 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (29952557420493 / 50000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (29952557420493 / 50000000000000 : ℝ) /
+                (1 + (11657024828021821 / 10000000000000000 : ℝ)) =
+              (5990511484098600 / 21657024828021821 : ℝ)
+            by
+        norm_num];
+        exact hau482)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1349303808025649067077 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (50000000000000 / 29952557420493 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (50000000000000 / 29952557420493 : ℝ) =
+              (29952557420493 / 50000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau481)
+  have hz1 :
+    (50000000000000 / 29952557420493 : ℝ) ≤ 1 / Real.sqrt (4593437163663 / 12800000000000 : ℝ) := by
+    rw [show (50000000000000 / 29952557420493 : ℝ) = 1 / (29952557420493 / 50000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 1996837161366199 : ℝ) ≤
+      (1245499343066240103687 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (4593437163663 / 12800000000000 : ℝ) ≤
+      (250000000000000 / 1996837161366199 : ℝ) := by
+    rw [show
+        (250000000000000 / 1996837161366199 : ℝ) =
+          (3 / 40) / (5990511484098597 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1349303808025649067077 / 5000000000000000000000 : ℝ))) -
+        6 * (1245499343066240103687 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (4593437163663 / 12800000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (4593437163663 / 12800000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((5990511484098597 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1349303808025649067077 / 5000000000000000000000 : ℝ))) -
+          6 * (1245499343066240103687 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (4593437163663 / 12800000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (4593437163663 / 12800000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (4593437163663 / 12800000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_356 :
+    ((3076099557883663825427 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2514203075409914914533 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((6001419489453879 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2702619339793887572677 / 10000000000000000000000 : ℝ))) -
+              6 * (310814708988853740969 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (11525451484287 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (11525451484287 / 32000000000000 : ℝ)) =
+      Real.log (43525451484287 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((11525451484287 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (11705451484287 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (3076099557883663825427 / 10000000000000000000000 : ℝ) ≤
+      Real.log (43525451484287 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (11705451484287 / 32000000000000 : ℝ) ≤
+      (-(2514203075409914914533 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (11705451484287 / 32000000000000 : ℝ) = (11705451484287 / 8000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11705451484287 / 8000000000000 : ℝ) ≤
+            (951532826090085085467 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (6001419489453879 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (11525451484287 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (11525451484287 / 32000000000000 : ℝ) ≤ (3000709744726941 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (11525451484287 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau484 :
+    Real.arctan (3000709744726941 / 10831317087253892 : ℝ) ≤
+      (2702619339793887572677 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau483 :
+    Real.arctan (3000709744726941 / 5000000000000000 : ℝ) ≤
+      (2 * (2702619339793887572677 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (1457829271813473 / 1250000000000000 : ℝ) ≤
+          Real.sqrt (1 + (3000709744726941 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (3000709744726941 / 5000000000000000 : ℝ) /
+                (1 + (1457829271813473 / 1250000000000000 : ℝ)) =
+              (3000709744726941 / 10831317087253892 : ℝ)
+            by
+        norm_num];
+        exact hau484)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2702619339793887572677 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 3000709744726941 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 3000709744726941 : ℝ) =
+              (3000709744726941 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau483)
+  have hz1 :
+    (5000000000000000 / 3000709744726941 : ℝ) ≤
+      1 / Real.sqrt (11525451484287 / 32000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 3000709744726941 : ℝ) = 1 / (3000709744726941 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 2000473163151293 : ℝ) ≤
+      (310814708988853740969 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (11525451484287 / 32000000000000 : ℝ) ≤
+      (250000000000000 / 2000473163151293 : ℝ) := by
+    rw [show
+        (250000000000000 / 2000473163151293 : ℝ) =
+          (3 / 40) / (6001419489453879 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2702619339793887572677 / 10000000000000000000000 : ℝ))) -
+        6 * (310814708988853740969 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (11525451484287 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (11525451484287 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((6001419489453879 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2702619339793887572677 / 10000000000000000000000 : ℝ))) -
+          6 * (310814708988853740969 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (11525451484287 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (11525451484287 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (11525451484287 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_357 :
+    ((3085711970582634738283 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1252639523764855944051 / 1250000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3006153852336759 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2 * (8458187299750094947 / 31250000000000000000 : ℝ))) -
+              6 * (620515188494905216727 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (23134620118833 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (23134620118833 / 64000000000000 : ℝ)) =
+      Real.log (87134620118833 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((23134620118833 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (23494620118833 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (3085711970582634738283 / 10000000000000000000000 : ℝ) ≤
+      Real.log (87134620118833 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (23494620118833 / 64000000000000 : ℝ) ≤
+      (-(1252639523764855944051 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (23494620118833 / 64000000000000 : ℝ) = (23494620118833 / 16000000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23494620118833 / 16000000000000 : ℝ) ≤
+            (480228426985144055949 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3006153852336759 / 5000000000000000 : ℝ) ≤ Real.sqrt (23134620118833 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (23134620118833 / 64000000000000 : ℝ) ≤
+      (6012307704673521 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (23134620118833 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau486 :
+    Real.arctan (6012307704673521 / 21668240824377793 : ℝ) ≤
+      (8458187299750094947 / 31250000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau485 :
+    Real.arctan (6012307704673521 / 10000000000000000 : ℝ) ≤
+      (2 * (8458187299750094947 / 31250000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (11668240824377793 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (6012307704673521 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (6012307704673521 / 10000000000000000 : ℝ) /
+                (1 + (11668240824377793 / 10000000000000000 : ℝ)) =
+              (6012307704673521 / 21668240824377793 : ℝ)
+            by
+        norm_num];
+        exact hau486)
+  have hat1 :
+    (Real.pi / 2 - (2 * (8458187299750094947 / 31250000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 6012307704673521 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 6012307704673521 : ℝ) =
+              (6012307704673521 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau485)
+  have hz1 :
+    (10000000000000000 / 6012307704673521 : ℝ) ≤
+      1 / Real.sqrt (23134620118833 / 64000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 6012307704673521 : ℝ) =
+          1 / (6012307704673521 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (125000000000000 / 1002051284112253 : ℝ) ≤
+      (620515188494905216727 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (23134620118833 / 64000000000000 : ℝ) ≤
+      (125000000000000 / 1002051284112253 : ℝ) := by
+    rw [show
+        (125000000000000 / 1002051284112253 : ℝ) =
+          (3 / 40) / (3006153852336759 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (8458187299750094947 / 31250000000000000000 : ℝ))) -
+        6 * (620515188494905216727 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (23134620118833 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (23134620118833 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3006153852336759 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (8458187299750094947 / 31250000000000000000 : ℝ))) -
+          6 * (620515188494905216727 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (23134620118833 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (23134620118833 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (23134620118833 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_358 :
+    ((773828788076575112559 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(4992773523371619840171 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((6023176237082577 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1355304732598921570641 / 5000000000000000000000 : ℝ))) -
+              6 * (1238813858572936387099 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (5804584317273 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (5804584317273 / 16000000000000 : ℝ)) =
+      Real.log (21804584317273 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((5804584317273 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (5894584317273 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (773828788076575112559 / 2500000000000000000000 : ℝ) ≤
+      Real.log (21804584317273 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (5894584317273 / 16000000000000 : ℝ) ≤
+      (-(4992773523371619840171 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (5894584317273 / 16000000000000 : ℝ) = (5894584317273 / 4000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5894584317273 / 4000000000000 : ℝ) ≤
+            (1938698279628380159829 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (6023176237082577 / 10000000000000000 : ℝ) ≤ Real.sqrt (5804584317273 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (5804584317273 / 16000000000000 : ℝ) ≤ (301158811854129 / 500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (5804584317273 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau488 :
+    Real.arctan (1003862706180430 / 3612307463586199 : ℝ) ≤
+      (1355304732598921570641 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau487 :
+    Real.arctan (301158811854129 / 500000000000000 : ℝ) ≤
+      (2 * (1355304732598921570641 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (5836922390758597 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (301158811854129 / 500000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (301158811854129 / 500000000000000 : ℝ) /
+                (1 + (5836922390758597 / 5000000000000000 : ℝ)) =
+              (1003862706180430 / 3612307463586199 : ℝ)
+            by
+        norm_num];
+        exact hau488)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1355304732598921570641 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (500000000000000 / 301158811854129 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (500000000000000 / 301158811854129 : ℝ) =
+              (301158811854129 / 500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau487)
+  have hz1 :
+    (500000000000000 / 301158811854129 : ℝ) ≤
+      1 / Real.sqrt (5804584317273 / 16000000000000 : ℝ) := by
+    rw [show (500000000000000 / 301158811854129 : ℝ) = 1 / (301158811854129 / 500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 2007725412360859 : ℝ) ≤
+      (1238813858572936387099 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (5804584317273 / 16000000000000 : ℝ) ≤
+      (250000000000000 / 2007725412360859 : ℝ) := by
+    rw [show
+        (250000000000000 / 2007725412360859 : ℝ) =
+          (3 / 40) / (6023176237082577 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1355304732598921570641 / 5000000000000000000000 : ℝ))) -
+        6 * (1238813858572936387099 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (5804584317273 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (5804584317273 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((6023176237082577 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1355304732598921570641 / 5000000000000000000000 : ℝ))) -
+          6 * (1238813858572936387099 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (5804584317273 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (5804584317273 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (5804584317273 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_359 :
+    ((620022657417898369221 / 2000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(498390490323817900053 / 500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((602860315550759 / 1000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (542520019722747216779 / 2000000000000000000000 : ℝ))) -
+              6 * (1237710043813499311807 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (46520391688443 / 128000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (46520391688443 / 128000000000000 : ℝ)) =
+      Real.log (174520391688443 / 128000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((46520391688443 / 128000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (47240391688443 / 128000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (620022657417898369221 / 2000000000000000000000 : ℝ) ≤
+      Real.log (174520391688443 / 128000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (47240391688443 / 128000000000000 : ℝ) ≤
+      (-(498390490323817900053 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (47240391688443 / 128000000000000 : ℝ) = (47240391688443 / 32000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (47240391688443 / 32000000000000 : ℝ) ≤
+            (194756689976182099947 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (602860315550759 / 1000000000000000 : ℝ) ≤ Real.sqrt (46520391688443 / 128000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (46520391688443 / 128000000000000 : ℝ) ≤
+      (6028603155507593 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (46520391688443 / 128000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau490 :
+    Real.arctan (6028603155507593 / 21676645751524539 : ℝ) ≤
+      (542520019722747216779 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau489 :
+    Real.arctan (6028603155507593 / 10000000000000000 : ℝ) ≤
+      (2 * (542520019722747216779 / 2000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (11676645751524539 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (6028603155507593 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (6028603155507593 / 10000000000000000 : ℝ) /
+                (1 + (11676645751524539 / 10000000000000000 : ℝ)) =
+              (6028603155507593 / 21676645751524539 : ℝ)
+            by
+        norm_num];
+        exact hau490)
+  have hat1 :
+    (Real.pi / 2 - (2 * (542520019722747216779 / 2000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 6028603155507593 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 6028603155507593 : ℝ) =
+              (6028603155507593 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau489)
+  have hz1 :
+    (10000000000000000 / 6028603155507593 : ℝ) ≤
+      1 / Real.sqrt (46520391688443 / 128000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 6028603155507593 : ℝ) =
+          1 / (6028603155507593 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (75000000000000 / 602860315550759 : ℝ) ≤
+      (1237710043813499311807 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (46520391688443 / 128000000000000 : ℝ) ≤
+      (75000000000000 / 602860315550759 : ℝ) := by
+    rw [show
+        (75000000000000 / 602860315550759 : ℝ) =
+          (3 / 40) / (602860315550759 / 1000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (542520019722747216779 / 2000000000000000000000 : ℝ))) -
+        6 * (1237710043813499311807 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (46520391688443 / 128000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (46520391688443 / 128000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((602860315550759 / 1000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (542520019722747216779 / 2000000000000000000000 : ℝ))) -
+          6 * (1237710043813499311807 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (46520391688443 / 128000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (46520391688443 / 128000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (46520391688443 / 128000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_360 :
+    ((3104909120767000870261 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(4975051985741318175863 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1206805038607909 / 2000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2714587987866077907253 / 10000000000000000000000 : ℝ))) -
+              6 * (1236609174448796264479 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (23302054419351 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (23302054419351 / 64000000000000 : ℝ)) =
+      Real.log (87302054419351 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((23302054419351 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (23662054419351 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (3104909120767000870261 / 10000000000000000000000 : ℝ) ≤
+      Real.log (87302054419351 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (23662054419351 / 64000000000000 : ℝ) ≤
+      (-(4975051985741318175863 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (23662054419351 / 64000000000000 : ℝ) = (23662054419351 / 16000000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23662054419351 / 16000000000000 : ℝ) ≤
+            (1956419817258681824137 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1206805038607909 / 2000000000000000 : ℝ) ≤ Real.sqrt (23302054419351 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (23302054419351 / 64000000000000 : ℝ) ≤ (1508506298259887 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (23302054419351 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau492 :
+    Real.arctan (1508506298259887 / 5419861512450504 : ℝ) ≤
+      (2714587987866077907253 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau491 :
+    Real.arctan (1508506298259887 / 2500000000000000 : ℝ) ≤
+      (2 * (2714587987866077907253 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (364982689056313 / 312500000000000 : ℝ) ≤
+          Real.sqrt (1 + (1508506298259887 / 2500000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1508506298259887 / 2500000000000000 : ℝ) /
+                (1 + (364982689056313 / 312500000000000 : ℝ)) =
+              (1508506298259887 / 5419861512450504 : ℝ)
+            by
+        norm_num];
+        exact hau492)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2714587987866077907253 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (2500000000000000 / 1508506298259887 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 1508506298259887 : ℝ) =
+              (1508506298259887 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau491)
+  have hz1 :
+    (2500000000000000 / 1508506298259887 : ℝ) ≤
+      1 / Real.sqrt (23302054419351 / 64000000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 1508506298259887 : ℝ) = 1 / (1508506298259887 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (150000000000000 / 1206805038607909 : ℝ) ≤
+      (1236609174448796264479 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (23302054419351 / 64000000000000 : ℝ) ≤
+      (150000000000000 / 1206805038607909 : ℝ) := by
+    rw [show
+        (150000000000000 / 1206805038607909 : ℝ) =
+          (3 / 40) / (1206805038607909 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2714587987866077907253 / 10000000000000000000000 : ℝ))) -
+        6 * (1236609174448796264479 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (23302054419351 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (23302054419351 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1206805038607909 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2714587987866077907253 / 10000000000000000000000 : ℝ))) -
+          6 * (1236609174448796264479 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (23302054419351 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (23302054419351 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (23302054419351 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/V30.lean b/LeanPool/Zeta5Irrational/Table/V30.lean
new file mode 100644
index 0000000000..e49ab6c62c
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/V30.lean
@@ -0,0 +1,1444 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Energy
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # V30: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_361 :
+    ((621940531108982726093 / 2000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(993242943074726082441 / 1000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((6039442362824217 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2716573140403061887061 / 10000000000000000000000 : ℝ))) -
+              6 * (617755618701384853011 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (46687825988961 / 128000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (46687825988961 / 128000000000000 : ℝ)) =
+      Real.log (174687825988961 / 128000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((46687825988961 / 128000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (47407825988961 / 128000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (621940531108982726093 / 2000000000000000000000 : ℝ) ≤
+      Real.log (174687825988961 / 128000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (47407825988961 / 128000000000000 : ℝ) ≤
+      (-(993242943074726082441 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (47407825988961 / 128000000000000 : ℝ) = (47407825988961 / 32000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (47407825988961 / 32000000000000 : ℝ) ≤
+            (393051417525273917559 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (6039442362824217 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (46687825988961 / 128000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (46687825988961 / 128000000000000 : ℝ) ≤ (301972118141211 / 500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (46687825988961 / 128000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau494 :
+    Real.arctan (6039442362824220 / 21682245676832677 : ℝ) ≤
+      (2716573140403061887061 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau493 :
+    Real.arctan (301972118141211 / 500000000000000 : ℝ) ≤
+      (2 * (2716573140403061887061 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (11682245676832677 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (301972118141211 / 500000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (301972118141211 / 500000000000000 : ℝ) /
+                (1 + (11682245676832677 / 10000000000000000 : ℝ)) =
+              (6039442362824220 / 21682245676832677 : ℝ)
+            by
+        norm_num];
+        exact hau494)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2716573140403061887061 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (500000000000000 / 301972118141211 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (500000000000000 / 301972118141211 : ℝ) =
+              (301972118141211 / 500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau493)
+  have hz1 :
+    (500000000000000 / 301972118141211 : ℝ) ≤
+      1 / Real.sqrt (46687825988961 / 128000000000000 : ℝ) := by
+    rw [show (500000000000000 / 301972118141211 : ℝ) = 1 / (301972118141211 / 500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 2013147454274739 : ℝ) ≤
+      (617755618701384853011 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (46687825988961 / 128000000000000 : ℝ) ≤
+      (250000000000000 / 2013147454274739 : ℝ) := by
+    rw [show
+        (250000000000000 / 2013147454274739 : ℝ) =
+          (3 / 40) / (6039442362824217 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2716573140403061887061 / 10000000000000000000000 : ℝ))) -
+        6 * (617755618701384853011 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (46687825988961 / 128000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (46687825988961 / 128000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((6039442362824217 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2716573140403061887061 / 10000000000000000000000 : ℝ))) -
+          6 * (617755618701384853011 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (46687825988961 / 128000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (46687825988961 / 128000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (46687825988961 / 128000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_362 :
+    ((1557246946813072634581 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2478696518460755041279 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((6044854677948479 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2 * (42477430681884392329 / 156250000000000000000 : ℝ))) -
+              6 * (617208109840245433709 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (2338577156961 / 6400000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (2338577156961 / 6400000000000 : ℝ)) =
+      Real.log (8738577156961 / 6400000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((2338577156961 / 6400000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (2374577156961 / 6400000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1557246946813072634581 / 5000000000000000000000 : ℝ) ≤
+      Real.log (8738577156961 / 6400000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (2374577156961 / 6400000000000 : ℝ) ≤
+      (-(2478696518460755041279 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (2374577156961 / 6400000000000 : ℝ) = (2374577156961 / 1600000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2374577156961 / 1600000000000 : ℝ) ≤
+            (987039383039244958721 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (6044854677948479 / 10000000000000000 : ℝ) ≤ Real.sqrt (2338577156961 / 6400000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (2338577156961 / 6400000000000 : ℝ) ≤ (3022427338974241 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (2338577156961 / 6400000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau496 :
+    Real.arctan (3022427338974241 / 10842522316549497 : ℝ) ≤
+      (42477430681884392329 / 156250000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau495 :
+    Real.arctan (3022427338974241 / 5000000000000000 : ℝ) ≤
+      (2 * (42477430681884392329 / 156250000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (5842522316549497 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (3022427338974241 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (3022427338974241 / 5000000000000000 : ℝ) /
+                (1 + (5842522316549497 / 5000000000000000 : ℝ)) =
+              (3022427338974241 / 10842522316549497 : ℝ)
+            by
+        norm_num];
+        exact hau496)
+  have hat1 :
+    (Real.pi / 2 - (2 * (42477430681884392329 / 156250000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 3022427338974241 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 3022427338974241 : ℝ) =
+              (3022427338974241 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau495)
+  have hz1 :
+    (5000000000000000 / 3022427338974241 : ℝ) ≤
+      1 / Real.sqrt (2338577156961 / 6400000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 3022427338974241 : ℝ) = 1 / (3022427338974241 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 6044854677948479 : ℝ) ≤
+      (617208109840245433709 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (2338577156961 / 6400000000000 : ℝ) ≤
+      (750000000000000 / 6044854677948479 : ℝ) := by
+    rw [show
+        (750000000000000 / 6044854677948479 : ℝ) =
+          (3 / 40) / (6044854677948479 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (42477430681884392329 / 156250000000000000000 : ℝ))) -
+        6 * (617208109840245433709 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (2338577156961 / 6400000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (2338577156961 / 6400000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((6044854677948479 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (42477430681884392329 / 156250000000000000000 : ℝ))) -
+          6 * (617208109840245433709 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (2338577156961 / 6400000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (2338577156961 / 6400000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (2338577156961 / 6400000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_363 :
+    ((623856567442089207543 / 2000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(154643340483221346247 / 156250000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((6050262151440667 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (544107052992504643051 / 2000000000000000000000 : ℝ))) -
+              6 * (9635344596621206931 / 78125000000000000000 : ℝ)))) ≤
+      Vfield (46855260289479 / 128000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (46855260289479 / 128000000000000 : ℝ)) =
+      Real.log (174855260289479 / 128000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((46855260289479 / 128000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (47575260289479 / 128000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (623856567442089207543 / 2000000000000000000000 : ℝ) ≤
+      Real.log (174855260289479 / 128000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (47575260289479 / 128000000000000 : ℝ) ≤
+      (-(154643340483221346247 / 156250000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (47575260289479 / 128000000000000 : ℝ) = (47575260289479 / 32000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (47575260289479 / 32000000000000 : ℝ) ≤
+            (61965153360528653753 / 156250000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (6050262151440667 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (46855260289479 / 128000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (46855260289479 / 128000000000000 : ℝ) ≤ (605026215144067 / 1000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (46855260289479 / 128000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau498 :
+    Real.arctan (3025131075720335 / 10843921459541432 : ℝ) ≤
+      (544107052992504643051 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau497 :
+    Real.arctan (605026215144067 / 1000000000000000 : ℝ) ≤
+      (2 * (544107052992504643051 / 2000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (730490182442679 / 625000000000000 : ℝ) ≤
+          Real.sqrt (1 + (605026215144067 / 1000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (605026215144067 / 1000000000000000 : ℝ) /
+                (1 + (730490182442679 / 625000000000000 : ℝ)) =
+              (3025131075720335 / 10843921459541432 : ℝ)
+            by
+        norm_num];
+        exact hau498)
+  have hat1 :
+    (Real.pi / 2 - (2 * (544107052992504643051 / 2000000000000000000000 : ℝ))) ≤
+      Real.arctan (1000000000000000 / 605026215144067 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1000000000000000 / 605026215144067 : ℝ) =
+              (605026215144067 / 1000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau497)
+  have hz1 :
+    (1000000000000000 / 605026215144067 : ℝ) ≤
+      1 / Real.sqrt (46855260289479 / 128000000000000 : ℝ) := by
+    rw [show
+        (1000000000000000 / 605026215144067 : ℝ) = 1 / (605026215144067 / 1000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 6050262151440667 : ℝ) ≤
+      (9635344596621206931 / 78125000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (46855260289479 / 128000000000000 : ℝ) ≤
+      (750000000000000 / 6050262151440667 : ℝ) := by
+    rw [show
+        (750000000000000 / 6050262151440667 : ℝ) =
+          (3 / 40) / (6050262151440667 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (544107052992504643051 / 2000000000000000000000 : ℝ))) -
+        6 * (9635344596621206931 / 78125000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (46855260289479 / 128000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (46855260289479 / 128000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((6050262151440667 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (544107052992504643051 / 2000000000000000000000 : ℝ))) -
+          6 * (9635344596621206931 / 78125000000000000000 : ℝ))) ≤
+      Real.sqrt (46855260289479 / 128000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (46855260289479 / 128000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (46855260289479 / 128000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_364 :
+    ((1562034744247203678047 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(9879592472732310410423 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((6055664796270951 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (544502450344153676819 / 2000000000000000000000 : ℝ))) -
+              6 * (1232234890629238011593 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (23469488719869 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (23469488719869 / 64000000000000 : ℝ)) =
+      Real.log (87469488719869 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((23469488719869 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (23829488719869 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1562034744247203678047 / 5000000000000000000000 : ℝ) ≤
+      Real.log (87469488719869 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (23829488719869 / 64000000000000 : ℝ) ≤
+      (-(9879592472732310410423 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (23829488719869 / 64000000000000 : ℝ) = (23829488719869 / 16000000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23829488719869 / 16000000000000 : ℝ) ≤
+            (3983351133267689589577 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (6055664796270951 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (23469488719869 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (23469488719869 / 64000000000000 : ℝ) ≤ (3027832398135477 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (23469488719869 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau500 :
+    Real.arctan (1009277466045159 / 3615106755877601 : ℝ) ≤
+      (544502450344153676819 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau499 :
+    Real.arctan (3027832398135477 / 5000000000000000 : ℝ) ≤
+      (2 * (544502450344153676819 / 2000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (5845320267632803 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (3027832398135477 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (3027832398135477 / 5000000000000000 : ℝ) /
+                (1 + (5845320267632803 / 5000000000000000 : ℝ)) =
+              (1009277466045159 / 3615106755877601 : ℝ)
+            by
+        norm_num];
+        exact hau500)
+  have hat1 :
+    (Real.pi / 2 - (2 * (544502450344153676819 / 2000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 3027832398135477 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 3027832398135477 : ℝ) =
+              (3027832398135477 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau499)
+  have hz1 :
+    (5000000000000000 / 3027832398135477 : ℝ) ≤
+      1 / Real.sqrt (23469488719869 / 64000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 3027832398135477 : ℝ) = 1 / (3027832398135477 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 2018554932090317 : ℝ) ≤
+      (1232234890629238011593 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (23469488719869 / 64000000000000 : ℝ) ≤
+      (250000000000000 / 2018554932090317 : ℝ) := by
+    rw [show
+        (250000000000000 / 2018554932090317 : ℝ) =
+          (3 / 40) / (6055664796270951 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (544502450344153676819 / 2000000000000000000000 : ℝ))) -
+        6 * (1232234890629238011593 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (23469488719869 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (23469488719869 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((6055664796270951 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (544502450344153676819 / 2000000000000000000000 : ℝ))) -
+          6 * (1232234890629238011593 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (23469488719869 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (23469488719869 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (23469488719869 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_365 :
+    ((3128853849671467858077 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(9862042010572357117663 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((6061062625351691 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2724486531235580131221 / 10000000000000000000000 : ℝ))) -
+              6 * (1231148553710269436801 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (47022694589997 / 128000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (47022694589997 / 128000000000000 : ℝ)) =
+      Real.log (175022694589997 / 128000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((47022694589997 / 128000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (47742694589997 / 128000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (3128853849671467858077 / 10000000000000000000000 : ℝ) ≤
+      Real.log (175022694589997 / 128000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (47742694589997 / 128000000000000 : ℝ) ≤
+      (-(9862042010572357117663 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (47742694589997 / 128000000000000 : ℝ) = (47742694589997 / 32000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (47742694589997 / 32000000000000 : ℝ) ≤
+            (4000901595427642882337 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (6061062625351691 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (47022694589997 / 128000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (47022694589997 / 128000000000000 : ℝ) ≤
+      (3030531312675847 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (47022694589997 / 128000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau502 :
+    Real.arctan (3030531312675847 / 10846718741063981 : ℝ) ≤
+      (2724486531235580131221 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau501 :
+    Real.arctan (3030531312675847 / 5000000000000000 : ℝ) ≤
+      (2 * (2724486531235580131221 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (5846718741063981 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (3030531312675847 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (3030531312675847 / 5000000000000000 : ℝ) /
+                (1 + (5846718741063981 / 5000000000000000 : ℝ)) =
+              (3030531312675847 / 10846718741063981 : ℝ)
+            by
+        norm_num];
+        exact hau502)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2724486531235580131221 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 3030531312675847 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 3030531312675847 : ℝ) =
+              (3030531312675847 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau501)
+  have hz1 :
+    (5000000000000000 / 3030531312675847 : ℝ) ≤
+      1 / Real.sqrt (47022694589997 / 128000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 3030531312675847 : ℝ) = 1 / (3030531312675847 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 6061062625351691 : ℝ) ≤
+      (1231148553710269436801 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (47022694589997 / 128000000000000 : ℝ) ≤
+      (750000000000000 / 6061062625351691 : ℝ) := by
+    rw [show
+        (750000000000000 / 6061062625351691 : ℝ) =
+          (3 / 40) / (6061062625351691 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2724486531235580131221 / 10000000000000000000000 : ℝ))) -
+        6 * (1231148553710269436801 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (47022694589997 / 128000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (47022694589997 / 128000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((6061062625351691 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2724486531235580131221 / 10000000000000000000000 : ℝ))) -
+          6 * (1231148553710269436801 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (47022694589997 / 128000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (47022694589997 / 128000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (47022694589997 / 128000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_366 :
+    ((313363592293191942199 / 1000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(9844522296328471492081 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3033227825768903 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2726458110795700910111 / 10000000000000000000000 : ℝ))) -
+              6 * (615032542466899038457 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (1472075366883 / 4000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1472075366883 / 4000000000000 : ℝ)) =
+      Real.log (5472075366883 / 4000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1472075366883 / 4000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1494575366883 / 4000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (313363592293191942199 / 1000000000000000000000 : ℝ) ≤
+      Real.log (5472075366883 / 4000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1494575366883 / 4000000000000 : ℝ) ≤
+      (-(9844522296328471492081 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (1494575366883 / 4000000000000 : ℝ) = (1494575366883 / 1000000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1494575366883 / 1000000000000 : ℝ) ≤
+            (4018421309671528507919 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3033227825768903 / 5000000000000000 : ℝ) ≤ Real.sqrt (1472075366883 / 4000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1472075366883 / 4000000000000 : ℝ) ≤ (6066455651537809 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1472075366883 / 4000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau504 :
+    Real.arctan (6066455651537809 / 21696233760150102 : ℝ) ≤
+      (2726458110795700910111 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau503 :
+    Real.arctan (6066455651537809 / 10000000000000000 : ℝ) ≤
+      (2 * (2726458110795700910111 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (5848116880075051 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (6066455651537809 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (6066455651537809 / 10000000000000000 : ℝ) /
+                (1 + (5848116880075051 / 5000000000000000 : ℝ)) =
+              (6066455651537809 / 21696233760150102 : ℝ)
+            by
+        norm_num];
+        exact hau504)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2726458110795700910111 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 6066455651537809 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 6066455651537809 : ℝ) =
+              (6066455651537809 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau503)
+  have hz1 :
+    (10000000000000000 / 6066455651537809 : ℝ) ≤
+      1 / Real.sqrt (1472075366883 / 4000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 6066455651537809 : ℝ) =
+          1 / (6066455651537809 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 3033227825768903 : ℝ) ≤
+      (615032542466899038457 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1472075366883 / 4000000000000 : ℝ) ≤
+      (375000000000000 / 3033227825768903 : ℝ) := by
+    rw [show
+        (375000000000000 / 3033227825768903 : ℝ) =
+          (3 / 40) / (3033227825768903 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2726458110795700910111 / 10000000000000000000000 : ℝ))) -
+        6 * (615032542466899038457 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1472075366883 / 4000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1472075366883 / 4000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3033227825768903 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2726458110795700910111 / 10000000000000000000000 : ℝ))) -
+          6 * (615032542466899038457 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (1472075366883 / 4000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1472075366883 / 4000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1472075366883 / 4000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_367 :
+    ((627683142092582637407 / 2000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(491351661122504170979 / 500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((6071843887627121 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (545685399531711867479 / 2000000000000000000000 : ℝ))) -
+              6 * (19202882370327738227 / 156250000000000000000 : ℝ)))) ≤
+      Vfield (9438025778103 / 25600000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (9438025778103 / 25600000000000 : ℝ)) =
+      Real.log (35038025778103 / 25600000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((9438025778103 / 25600000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (9582025778103 / 25600000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (627683142092582637407 / 2000000000000000000000 : ℝ) ≤
+      Real.log (35038025778103 / 25600000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (9582025778103 / 25600000000000 : ℝ) ≤
+      (-(491351661122504170979 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (9582025778103 / 25600000000000 : ℝ) = (9582025778103 / 6400000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9582025778103 / 6400000000000 : ℝ) ≤
+            (201795519177495829021 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (6071843887627121 / 10000000000000000 : ℝ) ≤ Real.sqrt (9438025778103 / 25600000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (9438025778103 / 25600000000000 : ℝ) ≤ (1517960971906781 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (9438025778103 / 25600000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau506 :
+    Real.arctan (3035921943813562 / 10849514684905809 : ℝ) ≤
+      (545685399531711867479 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau505 :
+    Real.arctan (1517960971906781 / 2500000000000000 : ℝ) ≤
+      (2 * (545685399531711867479 / 2000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (5849514684905809 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1517960971906781 / 2500000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1517960971906781 / 2500000000000000 : ℝ) /
+                (1 + (5849514684905809 / 5000000000000000 : ℝ)) =
+              (3035921943813562 / 10849514684905809 : ℝ)
+            by
+        norm_num];
+        exact hau506)
+  have hat1 :
+    (Real.pi / 2 - (2 * (545685399531711867479 / 2000000000000000000000 : ℝ))) ≤
+      Real.arctan (2500000000000000 / 1517960971906781 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 1517960971906781 : ℝ) =
+              (1517960971906781 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau505)
+  have hz1 :
+    (2500000000000000 / 1517960971906781 : ℝ) ≤
+      1 / Real.sqrt (9438025778103 / 25600000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 1517960971906781 : ℝ) = 1 / (1517960971906781 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 6071843887627121 : ℝ) ≤
+      (19202882370327738227 / 156250000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (9438025778103 / 25600000000000 : ℝ) ≤
+      (750000000000000 / 6071843887627121 : ℝ) := by
+    rw [show
+        (750000000000000 / 6071843887627121 : ℝ) =
+          (3 / 40) / (6071843887627121 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (545685399531711867479 / 2000000000000000000000 : ℝ))) -
+        6 * (19202882370327738227 / 156250000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (9438025778103 / 25600000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (9438025778103 / 25600000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((6071843887627121 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (545685399531711867479 / 2000000000000000000000 : ℝ))) -
+          6 * (19202882370327738227 / 156250000000000000000 : ℝ))) ≤
+      Real.sqrt (9438025778103 / 25600000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (9438025778103 / 25600000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (9438025778103 / 25600000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_368 :
+    ((12572772857793862221 / 40000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(4904787340974963006027 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((6077227346360729 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1365196599525230821389 / 5000000000000000000000 : ℝ))) -
+              6 * (613953350745148672973 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (23636923020387 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (23636923020387 / 64000000000000 : ℝ)) =
+      Real.log (87636923020387 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((23636923020387 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (23996923020387 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (12572772857793862221 / 40000000000000000000 : ℝ) ≤
+      Real.log (87636923020387 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (23996923020387 / 64000000000000 : ℝ) ≤
+      (-(4904787340974963006027 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (23996923020387 / 64000000000000 : ℝ) = (23996923020387 / 16000000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23996923020387 / 16000000000000 : ℝ) ≤
+            (2026684462025036993973 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (6077227346360729 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (23636923020387 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (23636923020387 / 64000000000000 : ℝ) ≤ (1519306836590183 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (23636923020387 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau508 :
+    Real.arctan (3038613673180366 / 10850912155795767 : ℝ) ≤
+      (1365196599525230821389 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau507 :
+    Real.arctan (1519306836590183 / 2500000000000000 : ℝ) ≤
+      (2 * (1365196599525230821389 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (5850912155795767 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1519306836590183 / 2500000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1519306836590183 / 2500000000000000 : ℝ) /
+                (1 + (5850912155795767 / 5000000000000000 : ℝ)) =
+              (3038613673180366 / 10850912155795767 : ℝ)
+            by
+        norm_num];
+        exact hau508)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1365196599525230821389 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (2500000000000000 / 1519306836590183 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 1519306836590183 : ℝ) =
+              (1519306836590183 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau507)
+  have hz1 :
+    (2500000000000000 / 1519306836590183 : ℝ) ≤
+      1 / Real.sqrt (23636923020387 / 64000000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 1519306836590183 : ℝ) = 1 / (1519306836590183 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 6077227346360729 : ℝ) ≤
+      (613953350745148672973 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (23636923020387 / 64000000000000 : ℝ) ≤
+      (750000000000000 / 6077227346360729 : ℝ) := by
+    rw [show
+        (750000000000000 / 6077227346360729 : ℝ) =
+          (3 / 40) / (6077227346360729 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1365196599525230821389 / 5000000000000000000000 : ℝ))) -
+        6 * (613953350745148672973 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (23636923020387 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (23636923020387 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((6077227346360729 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1365196599525230821389 / 5000000000000000000000 : ℝ))) -
+          6 * (613953350745148672973 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (23636923020387 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (23636923020387 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (23636923020387 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_369 :
+    ((49187006829210377799 / 156250000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1958429313680021735671 / 2000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((6082606040423341 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2732356722166779862839 / 10000000000000000000000 : ℝ))) -
+              6 * (61341588092849836077 / 500000000000000000000 : ℝ)))) ≤
+      Vfield (47357563191033 / 128000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (47357563191033 / 128000000000000 : ℝ)) =
+      Real.log (175357563191033 / 128000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((47357563191033 / 128000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (48077563191033 / 128000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (49187006829210377799 / 156250000000000000000 : ℝ) ≤
+      Real.log (175357563191033 / 128000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (48077563191033 / 128000000000000 : ℝ) ≤
+      (-(1958429313680021735671 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (48077563191033 / 128000000000000 : ℝ) = (48077563191033 / 32000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (48077563191033 / 32000000000000 : ℝ) ≤
+            (814159407519978264329 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (6082606040423341 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (47357563191033 / 128000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (47357563191033 / 128000000000000 : ℝ) ≤ (380162877526459 / 625000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (47357563191033 / 128000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau510 :
+    Real.arctan (3041303020211672 / 10852309292984149 : ℝ) ≤
+      (2732356722166779862839 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau509 :
+    Real.arctan (380162877526459 / 625000000000000 : ℝ) ≤
+      (2 * (2732356722166779862839 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (5852309292984149 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (380162877526459 / 625000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (380162877526459 / 625000000000000 : ℝ) /
+                (1 + (5852309292984149 / 5000000000000000 : ℝ)) =
+              (3041303020211672 / 10852309292984149 : ℝ)
+            by
+        norm_num];
+        exact hau510)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2732356722166779862839 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (625000000000000 / 380162877526459 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (625000000000000 / 380162877526459 : ℝ) =
+              (380162877526459 / 625000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau509)
+  have hz1 :
+    (625000000000000 / 380162877526459 : ℝ) ≤
+      1 / Real.sqrt (47357563191033 / 128000000000000 : ℝ) := by
+    rw [show (625000000000000 / 380162877526459 : ℝ) = 1 / (380162877526459 / 625000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 6082606040423341 : ℝ) ≤
+      (61341588092849836077 / 500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (47357563191033 / 128000000000000 : ℝ) ≤
+      (750000000000000 / 6082606040423341 : ℝ) := by
+    rw [show
+        (750000000000000 / 6082606040423341 : ℝ) =
+          (3 / 40) / (6082606040423341 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2732356722166779862839 / 10000000000000000000000 : ℝ))) -
+        6 * (61341588092849836077 / 500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (47357563191033 / 128000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (47357563191033 / 128000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((6082606040423341 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2732356722166779862839 / 10000000000000000000000 : ℝ))) -
+          6 * (61341588092849836077 / 500000000000000000000 : ℝ))) ≤
+      Real.sqrt (47357563191033 / 128000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (47357563191033 / 128000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (47357563191033 / 128000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_370 :
+    ((3152741380503673934997 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(9774748775928224355261 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((6087979982443631 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (683579393543033958129 / 2500000000000000000000 : ℝ))) -
+              6 * (1225759640432438821581 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (11860320085323 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (11860320085323 / 32000000000000 : ℝ)) =
+      Real.log (43860320085323 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((11860320085323 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (12040320085323 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (3152741380503673934997 / 10000000000000000000000 : ℝ) ≤
+      Real.log (43860320085323 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (12040320085323 / 32000000000000 : ℝ) ≤
+      (-(9774748775928224355261 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (12040320085323 / 32000000000000 : ℝ) = (12040320085323 / 8000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12040320085323 / 8000000000000 : ℝ) ≤
+            (4088194830071775644739 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (6087979982443631 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (11860320085323 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (11860320085323 / 32000000000000 : ℝ) ≤ (3043989991221817 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (11860320085323 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau512 :
+    Real.arctan (6087979982443634 / 21707412193419789 : ℝ) ≤
+      (683579393543033958129 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau511 :
+    Real.arctan (3043989991221817 / 5000000000000000 : ℝ) ≤
+      (2 * (683579393543033958129 / 2500000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (11707412193419789 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (3043989991221817 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (3043989991221817 / 5000000000000000 : ℝ) /
+                (1 + (11707412193419789 / 10000000000000000 : ℝ)) =
+              (6087979982443634 / 21707412193419789 : ℝ)
+            by
+        norm_num];
+        exact hau512)
+  have hat1 :
+    (Real.pi / 2 - (2 * (683579393543033958129 / 2500000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 3043989991221817 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 3043989991221817 : ℝ) =
+              (3043989991221817 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau511)
+  have hz1 :
+    (5000000000000000 / 3043989991221817 : ℝ) ≤
+      1 / Real.sqrt (11860320085323 / 32000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 3043989991221817 : ℝ) = 1 / (3043989991221817 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 6087979982443631 : ℝ) ≤
+      (1225759640432438821581 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (11860320085323 / 32000000000000 : ℝ) ≤
+      (750000000000000 / 6087979982443631 : ℝ) := by
+    rw [show
+        (750000000000000 / 6087979982443631 : ℝ) =
+          (3 / 40) / (6087979982443631 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (683579393543033958129 / 2500000000000000000000 : ℝ))) -
+        6 * (1225759640432438821581 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (11860320085323 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (11860320085323 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((6087979982443631 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (683579393543033958129 / 2500000000000000000000 : ℝ))) -
+          6 * (1225759640432438821581 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (11860320085323 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (11860320085323 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (11860320085323 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_371 :
+    ((394689005865717860253 / 1250000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(4878690599606745271231 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((6093349184994587 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1368137881100294212459 / 5000000000000000000000 : ℝ))) -
+              6 * (244938064984704678467 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (47524997491551 / 128000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (47524997491551 / 128000000000000 : ℝ)) =
+      Real.log (175524997491551 / 128000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((47524997491551 / 128000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (48244997491551 / 128000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (394689005865717860253 / 1250000000000000000000 : ℝ) ≤
+      Real.log (175524997491551 / 128000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (48244997491551 / 128000000000000 : ℝ) ≤
+      (-(4878690599606745271231 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (48244997491551 / 128000000000000 : ℝ) = (48244997491551 / 32000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (48244997491551 / 32000000000000 : ℝ) ≤
+            (2052781203393254728769 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (6093349184994587 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (47524997491551 / 128000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (47524997491551 / 128000000000000 : ℝ) ≤ (609334918499459 / 1000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (47524997491551 / 128000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau514 :
+    Real.arctan (3046674592497295 / 10855102567211658 : ℝ) ≤
+      (1368137881100294212459 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau513 :
+    Real.arctan (609334918499459 / 1000000000000000 : ℝ) ≤
+      (2 * (1368137881100294212459 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (2927551283605829 / 2500000000000000 : ℝ) ≤
+          Real.sqrt (1 + (609334918499459 / 1000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (609334918499459 / 1000000000000000 : ℝ) /
+                (1 + (2927551283605829 / 2500000000000000 : ℝ)) =
+              (3046674592497295 / 10855102567211658 : ℝ)
+            by
+        norm_num];
+        exact hau514)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1368137881100294212459 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (1000000000000000 / 609334918499459 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1000000000000000 / 609334918499459 : ℝ) =
+              (609334918499459 / 1000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau513)
+  have hz1 :
+    (1000000000000000 / 609334918499459 : ℝ) ≤
+      1 / Real.sqrt (47524997491551 / 128000000000000 : ℝ) := by
+    rw [show
+        (1000000000000000 / 609334918499459 : ℝ) = 1 / (609334918499459 / 1000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 6093349184994587 : ℝ) ≤
+      (244938064984704678467 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (47524997491551 / 128000000000000 : ℝ) ≤
+      (750000000000000 / 6093349184994587 : ℝ) := by
+    rw [show
+        (750000000000000 / 6093349184994587 : ℝ) =
+          (3 / 40) / (6093349184994587 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1368137881100294212459 / 5000000000000000000000 : ℝ))) -
+        6 * (244938064984704678467 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (47524997491551 / 128000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (47524997491551 / 128000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((6093349184994587 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1368137881100294212459 / 5000000000000000000000 : ℝ))) -
+          6 * (244938064984704678467 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (47524997491551 / 128000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (47524997491551 / 128000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (47524997491551 / 128000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_372 :
+    ((1581140219253604103593 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(974004373348292378993 / 1000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((6098713660593851 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2738231293355815583263 / 10000000000000000000000 : ℝ))) -
+              6 * (1223623803112093052309 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (4760871464181 / 12800000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (4760871464181 / 12800000000000 : ℝ)) =
+      Real.log (17560871464181 / 12800000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((4760871464181 / 12800000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (4832871464181 / 12800000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1581140219253604103593 / 5000000000000000000000 : ℝ) ≤
+      Real.log (17560871464181 / 12800000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (4832871464181 / 12800000000000 : ℝ) ≤
+      (-(974004373348292378993 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (4832871464181 / 12800000000000 : ℝ) = (4832871464181 / 3200000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4832871464181 / 3200000000000 : ℝ) ≤
+            (412289987251707621007 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (6098713660593851 / 10000000000000000 : ℝ) ≤ Real.sqrt (4760871464181 / 12800000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (4760871464181 / 12800000000000 : ℝ) ≤ (3049356830296927 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (4760871464181 / 12800000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau516 :
+    Real.arctan (6098713660593854 / 21712997409455619 : ℝ) ≤
+      (2738231293355815583263 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau515 :
+    Real.arctan (3049356830296927 / 5000000000000000 : ℝ) ≤
+      (2 * (2738231293355815583263 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (11712997409455619 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (3049356830296927 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (3049356830296927 / 5000000000000000 : ℝ) /
+                (1 + (11712997409455619 / 10000000000000000 : ℝ)) =
+              (6098713660593854 / 21712997409455619 : ℝ)
+            by
+        norm_num];
+        exact hau516)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2738231293355815583263 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 3049356830296927 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 3049356830296927 : ℝ) =
+              (3049356830296927 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau515)
+  have hz1 :
+    (5000000000000000 / 3049356830296927 : ℝ) ≤
+      1 / Real.sqrt (4760871464181 / 12800000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 3049356830296927 : ℝ) = 1 / (3049356830296927 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 6098713660593851 : ℝ) ≤
+      (1223623803112093052309 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (4760871464181 / 12800000000000 : ℝ) ≤
+      (750000000000000 / 6098713660593851 : ℝ) := by
+    rw [show
+        (750000000000000 / 6098713660593851 : ℝ) =
+          (3 / 40) / (6098713660593851 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2738231293355815583263 / 10000000000000000000000 : ℝ))) -
+        6 * (1223623803112093052309 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (4760871464181 / 12800000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (4760871464181 / 12800000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((6098713660593851 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2738231293355815583263 / 10000000000000000000000 : ℝ))) -
+          6 * (1223623803112093052309 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (4760871464181 / 12800000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (4760871464181 / 12800000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (4760871464181 / 12800000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/V31.lean b/LeanPool/Zeta5Irrational/Table/V31.lean
new file mode 100644
index 0000000000..586b1241be
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/V31.lean
@@ -0,0 +1,1447 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Energy
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # V31: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_373 :
+    ((3167046557416502155937 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(9722736274507547290279 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3052036710852029 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2740184174711295780057 / 10000000000000000000000 : ℝ))) -
+              6 * (1222560062854347182191 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (47692431792069 / 128000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (47692431792069 / 128000000000000 : ℝ)) =
+      Real.log (175692431792069 / 128000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((47692431792069 / 128000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (48412431792069 / 128000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (3167046557416502155937 / 10000000000000000000000 : ℝ) ≤
+      Real.log (175692431792069 / 128000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (48412431792069 / 128000000000000 : ℝ) ≤
+      (-(9722736274507547290279 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (48412431792069 / 128000000000000 : ℝ) = (48412431792069 / 32000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (48412431792069 / 32000000000000 : ℝ) ≤
+            (4140207331492452709721 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3052036710852029 / 5000000000000000 : ℝ) ≤
+      Real.sqrt (47692431792069 / 128000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (47692431792069 / 128000000000000 : ℝ) ≤
+      (6104073421704061 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (47692431792069 / 128000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau518 :
+    Real.arctan (164974957343353 / 586913216729537 : ℝ) ≤
+      (2740184174711295780057 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau517 :
+    Real.arctan (6104073421704061 / 10000000000000000 : ℝ) ≤
+      (2 * (2740184174711295780057 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (11715789018992869 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (6104073421704061 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (6104073421704061 / 10000000000000000 : ℝ) /
+                (1 + (11715789018992869 / 10000000000000000 : ℝ)) =
+              (164974957343353 / 586913216729537 : ℝ)
+            by
+        norm_num];
+        exact hau518)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2740184174711295780057 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 6104073421704061 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 6104073421704061 : ℝ) =
+              (6104073421704061 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau517)
+  have hz1 :
+    (10000000000000000 / 6104073421704061 : ℝ) ≤
+      1 / Real.sqrt (47692431792069 / 128000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 6104073421704061 : ℝ) =
+          1 / (6104073421704061 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 3052036710852029 : ℝ) ≤
+      (1222560062854347182191 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (47692431792069 / 128000000000000 : ℝ) ≤
+      (375000000000000 / 3052036710852029 : ℝ) := by
+    rw [show
+        (375000000000000 / 3052036710852029 : ℝ) =
+          (3 / 40) / (3052036710852029 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2740184174711295780057 / 10000000000000000000000 : ℝ))) -
+        6 * (1222560062854347182191 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (47692431792069 / 128000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (47692431792069 / 128000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3052036710852029 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2740184174711295780057 / 10000000000000000000000 : ℝ))) -
+          6 * (1222560062854347182191 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (47692431792069 / 128000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (47692431792069 / 128000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (47692431792069 / 128000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_374 :
+    ((3171810405818957948811 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(4852729359299315616467 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((305471424036659 / 500000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2742134413310491105127 / 10000000000000000000000 : ℝ))) -
+              6 * (610749546040129944137 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (5972018617791 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (5972018617791 / 16000000000000 : ℝ)) =
+      Real.log (21972018617791 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((5972018617791 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (6062018617791 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (3171810405818957948811 / 10000000000000000000000 : ℝ) ≤
+      Real.log (21972018617791 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (6062018617791 / 16000000000000 : ℝ) ≤
+      (-(4852729359299315616467 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (6062018617791 / 16000000000000 : ℝ) = (6062018617791 / 4000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6062018617791 / 4000000000000 : ℝ) ≤
+            (2078742443700684383533 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (305471424036659 / 500000000000000 : ℝ) ≤ Real.sqrt (5972018617791 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (5972018617791 / 16000000000000 : ℝ) ≤ (6109428480733183 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (5972018617791 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau520 :
+    Real.arctan (6109428480733183 / 21718579963510670 : ℝ) ≤
+      (2742134413310491105127 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau519 :
+    Real.arctan (6109428480733183 / 10000000000000000 : ℝ) ≤
+      (2 * (2742134413310491105127 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (1171857996351067 / 1000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (6109428480733183 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (6109428480733183 / 10000000000000000 : ℝ) /
+                (1 + (1171857996351067 / 1000000000000000 : ℝ)) =
+              (6109428480733183 / 21718579963510670 : ℝ)
+            by
+        norm_num];
+        exact hau520)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2742134413310491105127 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 6109428480733183 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 6109428480733183 : ℝ) =
+              (6109428480733183 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau519)
+  have hz1 :
+    (10000000000000000 / 6109428480733183 : ℝ) ≤
+      1 / Real.sqrt (5972018617791 / 16000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 6109428480733183 : ℝ) =
+          1 / (6109428480733183 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (37500000000000 / 305471424036659 : ℝ) ≤
+      (610749546040129944137 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (5972018617791 / 16000000000000 : ℝ) ≤
+      (37500000000000 / 305471424036659 : ℝ) := by
+    rw [show
+        (37500000000000 / 305471424036659 : ℝ) =
+          (3 / 40) / (305471424036659 / 500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2742134413310491105127 / 10000000000000000000000 : ℝ))) -
+        6 * (610749546040129944137 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (5972018617791 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (5972018617791 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((305471424036659 / 500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2742134413310491105127 / 10000000000000000000000 : ℝ))) -
+          6 * (610749546040129944137 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (5972018617791 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (5972018617791 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (5972018617791 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_375 :
+    ((1588285992938407841959 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(9688210962603965581069 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3057389425017427 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1372041008083511690423 / 5000000000000000000000 : ℝ))) -
+              6 * (305110219698251537417 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (47859866092587 / 128000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (47859866092587 / 128000000000000 : ℝ)) =
+      Real.log (175859866092587 / 128000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((47859866092587 / 128000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (48579866092587 / 128000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1588285992938407841959 / 5000000000000000000000 : ℝ) ≤
+      Real.log (175859866092587 / 128000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (48579866092587 / 128000000000000 : ℝ) ≤
+      (-(9688210962603965581069 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (48579866092587 / 128000000000000 : ℝ) = (48579866092587 / 32000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (48579866092587 / 32000000000000 : ℝ) ≤
+            (4174732643396034418931 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3057389425017427 / 5000000000000000 : ℝ) ≤
+      Real.sqrt (47859866092587 / 128000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (47859866092587 / 128000000000000 : ℝ) ≤
+      (6114778850034857 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (47859866092587 / 128000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau522 :
+    Real.arctan (6114778850034857 / 21721370243484060 : ℝ) ≤
+      (1372041008083511690423 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau521 :
+    Real.arctan (6114778850034857 / 10000000000000000 : ℝ) ≤
+      (2 * (1372041008083511690423 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (586068512174203 / 500000000000000 : ℝ) ≤
+          Real.sqrt (1 + (6114778850034857 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (6114778850034857 / 10000000000000000 : ℝ) /
+                (1 + (586068512174203 / 500000000000000 : ℝ)) =
+              (6114778850034857 / 21721370243484060 : ℝ)
+            by
+        norm_num];
+        exact hau522)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1372041008083511690423 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 6114778850034857 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 6114778850034857 : ℝ) =
+              (6114778850034857 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau521)
+  have hz1 :
+    (10000000000000000 / 6114778850034857 : ℝ) ≤
+      1 / Real.sqrt (47859866092587 / 128000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 6114778850034857 : ℝ) =
+          1 / (6114778850034857 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 3057389425017427 : ℝ) ≤
+      (305110219698251537417 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (47859866092587 / 128000000000000 : ℝ) ≤
+      (375000000000000 / 3057389425017427 : ℝ) := by
+    rw [show
+        (375000000000000 / 3057389425017427 : ℝ) =
+          (3 / 40) / (3057389425017427 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1372041008083511690423 / 5000000000000000000000 : ℝ))) -
+        6 * (305110219698251537417 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (47859866092587 / 128000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (47859866092587 / 128000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3057389425017427 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1372041008083511690423 / 5000000000000000000000 : ℝ))) -
+          6 * (305110219698251537417 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (47859866092587 / 128000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (47859866092587 / 128000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (47859866092587 / 128000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_376 :
+    ((3181331299749228225409 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(4835496451952082468179 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((76501556773859 / 125000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (686506747566213541199 / 2500000000000000000000 : ℝ))) -
+              6 * (243877082213678186227 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (23971791621423 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (23971791621423 / 64000000000000 : ℝ)) =
+      Real.log (87971791621423 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((23971791621423 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (24331791621423 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (3181331299749228225409 / 10000000000000000000000 : ℝ) ≤
+      Real.log (87971791621423 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (24331791621423 / 64000000000000 : ℝ) ≤
+      (-(4835496451952082468179 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (24331791621423 / 64000000000000 : ℝ) = (24331791621423 / 16000000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24331791621423 / 16000000000000 : ℝ) ≤
+            (2095975351047917531821 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (76501556773859 / 125000000000000 : ℝ) ≤ Real.sqrt (23971791621423 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (23971791621423 / 64000000000000 : ℝ) ≤
+      (6120124541908723 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (23971791621423 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau524 :
+    Real.arctan (6120124541908723 / 21724159859387513 : ℝ) ≤
+      (686506747566213541199 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau523 :
+    Real.arctan (6120124541908723 / 10000000000000000 : ℝ) ≤
+      (2 * (686506747566213541199 / 2500000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (11724159859387513 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (6120124541908723 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (6120124541908723 / 10000000000000000 : ℝ) /
+                (1 + (11724159859387513 / 10000000000000000 : ℝ)) =
+              (6120124541908723 / 21724159859387513 : ℝ)
+            by
+        norm_num];
+        exact hau524)
+  have hat1 :
+    (Real.pi / 2 - (2 * (686506747566213541199 / 2500000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 6120124541908723 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 6120124541908723 : ℝ) =
+              (6120124541908723 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau523)
+  have hz1 :
+    (10000000000000000 / 6120124541908723 : ℝ) ≤
+      1 / Real.sqrt (23971791621423 / 64000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 6120124541908723 : ℝ) =
+          1 / (6120124541908723 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (9375000000000 / 76501556773859 : ℝ) ≤
+      (243877082213678186227 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (23971791621423 / 64000000000000 : ℝ) ≤
+      (9375000000000 / 76501556773859 : ℝ) := by
+    rw [show
+        (9375000000000 / 76501556773859 : ℝ) = (3 / 40) / (76501556773859 / 125000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (686506747566213541199 / 2500000000000000000000 : ℝ))) -
+        6 * (243877082213678186227 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (23971791621423 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (23971791621423 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((76501556773859 / 125000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (686506747566213541199 / 2500000000000000000000 : ℝ))) -
+          6 * (243877082213678186227 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (23971791621423 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (23971791621423 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (23971791621423 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_377 :
+    ((19913052184951669237 / 62500000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(4826902220204502580303 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3062732784300373 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2 * (171748083909903721723 / 625000000000000000000 : ℝ))) -
+              6 * (15229158463178572059 / 125000000000000000000 : ℝ)))) ≤
+      Vfield (9605460078621 / 25600000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (9605460078621 / 25600000000000 : ℝ)) =
+      Real.log (35205460078621 / 25600000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((9605460078621 / 25600000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (9749460078621 / 25600000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (19913052184951669237 / 62500000000000000000 : ℝ) ≤
+      Real.log (35205460078621 / 25600000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (9749460078621 / 25600000000000 : ℝ) ≤
+      (-(4826902220204502580303 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (9749460078621 / 25600000000000 : ℝ) = (9749460078621 / 6400000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9749460078621 / 6400000000000 : ℝ) ≤
+            (2104569582795497419697 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3062732784300373 / 5000000000000000 : ℝ) ≤ Real.sqrt (9605460078621 / 25600000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (9605460078621 / 25600000000000 : ℝ) ≤ (6125465568600749 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (9605460078621 / 25600000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau526 :
+    Real.arctan (6125465568600749 / 21726948811694936 : ℝ) ≤
+      (171748083909903721723 / 625000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau525 :
+    Real.arctan (6125465568600749 / 10000000000000000 : ℝ) ≤
+      (2 * (171748083909903721723 / 625000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (1465868601461867 / 1250000000000000 : ℝ) ≤
+          Real.sqrt (1 + (6125465568600749 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (6125465568600749 / 10000000000000000 : ℝ) /
+                (1 + (1465868601461867 / 1250000000000000 : ℝ)) =
+              (6125465568600749 / 21726948811694936 : ℝ)
+            by
+        norm_num];
+        exact hau526)
+  have hat1 :
+    (Real.pi / 2 - (2 * (171748083909903721723 / 625000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 6125465568600749 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 6125465568600749 : ℝ) =
+              (6125465568600749 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau525)
+  have hz1 :
+    (10000000000000000 / 6125465568600749 : ℝ) ≤
+      1 / Real.sqrt (9605460078621 / 25600000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 6125465568600749 : ℝ) =
+          1 / (6125465568600749 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 3062732784300373 : ℝ) ≤
+      (15229158463178572059 / 125000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (9605460078621 / 25600000000000 : ℝ) ≤
+      (375000000000000 / 3062732784300373 : ℝ) := by
+    rw [show
+        (375000000000000 / 3062732784300373 : ℝ) =
+          (3 / 40) / (3062732784300373 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (171748083909903721723 / 625000000000000000000 : ℝ))) -
+        6 * (15229158463178572059 / 125000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (9605460078621 / 25600000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (9605460078621 / 25600000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3062732784300373 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (171748083909903721723 / 625000000000000000000 : ℝ))) -
+          6 * (15229158463178572059 / 125000000000000000000 : ℝ))) ≤
+      Real.sqrt (9605460078621 / 25600000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (9605460078621 / 25600000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (9605460078621 / 25600000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_378 :
+    ((1595421568779464123529 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(9636645470553791427333 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((153270048557589 / 250000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2 * (54998181599460134389 / 200000000000000000000 : ℝ))) -
+              6 * (608641332485034393921 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (12027754385841 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (12027754385841 / 32000000000000 : ℝ)) =
+      Real.log (44027754385841 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((12027754385841 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (12207754385841 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1595421568779464123529 / 5000000000000000000000 : ℝ) ≤
+      Real.log (44027754385841 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (12207754385841 / 32000000000000 : ℝ) ≤
+      (-(9636645470553791427333 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (12207754385841 / 32000000000000 : ℝ) = (12207754385841 / 8000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12207754385841 / 8000000000000 : ℝ) ≤
+            (4226298135446208572667 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (153270048557589 / 250000000000000 : ℝ) ≤ Real.sqrt (12027754385841 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (12027754385841 / 32000000000000 : ℝ) ≤
+      (6130801942303563 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (12027754385841 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau528 :
+    Real.arctan (6130801942303563 / 21729737100879674 : ℝ) ≤
+      (54998181599460134389 / 200000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau527 :
+    Real.arctan (6130801942303563 / 10000000000000000 : ℝ) ≤
+      (2 * (54998181599460134389 / 200000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (5864868550439837 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (6130801942303563 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (6130801942303563 / 10000000000000000 : ℝ) /
+                (1 + (5864868550439837 / 5000000000000000 : ℝ)) =
+              (6130801942303563 / 21729737100879674 : ℝ)
+            by
+        norm_num];
+        exact hau528)
+  have hat1 :
+    (Real.pi / 2 - (2 * (54998181599460134389 / 200000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 6130801942303563 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 6130801942303563 : ℝ) =
+              (6130801942303563 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau527)
+  have hz1 :
+    (10000000000000000 / 6130801942303563 : ℝ) ≤
+      1 / Real.sqrt (12027754385841 / 32000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 6130801942303563 : ℝ) =
+          1 / (6130801942303563 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (6250000000000 / 51090016185863 : ℝ) ≤
+      (608641332485034393921 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (12027754385841 / 32000000000000 : ℝ) ≤
+      (6250000000000 / 51090016185863 : ℝ) := by
+    rw [show
+        (6250000000000 / 51090016185863 : ℝ) = (3 / 40) / (153270048557589 / 250000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (54998181599460134389 / 200000000000000000000 : ℝ))) -
+        6 * (608641332485034393921 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (12027754385841 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (12027754385841 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((153270048557589 / 250000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (54998181599460134389 / 200000000000000000000 : ℝ))) -
+          6 * (608641332485034393921 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (12027754385841 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (12027754385841 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (12027754385841 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_379 :
+    ((798898916449784521491 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(9619515893295757380253 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1534033418789193 / 2500000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2751846209404526497207 / 10000000000000000000000 : ℝ))) -
+              6 * (1216235363106071541963 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (48194734693623 / 128000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (48194734693623 / 128000000000000 : ℝ)) =
+      Real.log (176194734693623 / 128000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((48194734693623 / 128000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (48914734693623 / 128000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (798898916449784521491 / 2500000000000000000000 : ℝ) ≤
+      Real.log (176194734693623 / 128000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (48914734693623 / 128000000000000 : ℝ) ≤
+      (-(9619515893295757380253 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (48914734693623 / 128000000000000 : ℝ) = (48914734693623 / 32000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (48914734693623 / 32000000000000 : ℝ) ≤
+            (4243427712704242619747 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1534033418789193 / 2500000000000000 : ℝ) ≤
+      Real.sqrt (48194734693623 / 128000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (48194734693623 / 128000000000000 : ℝ) ≤ (245445347006271 / 400000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (48194734693623 / 128000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau530 :
+    Real.arctan (6136133675156775 / 21732524727414511 : ℝ) ≤
+      (2751846209404526497207 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau529 :
+    Real.arctan (245445347006271 / 400000000000000 : ℝ) ≤
+      (2 * (2751846209404526497207 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (11732524727414511 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (245445347006271 / 400000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (245445347006271 / 400000000000000 : ℝ) /
+                (1 + (11732524727414511 / 10000000000000000 : ℝ)) =
+              (6136133675156775 / 21732524727414511 : ℝ)
+            by
+        norm_num];
+        exact hau530)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2751846209404526497207 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (400000000000000 / 245445347006271 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (400000000000000 / 245445347006271 : ℝ) =
+              (245445347006271 / 400000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau529)
+  have hz1 :
+    (400000000000000 / 245445347006271 : ℝ) ≤
+      1 / Real.sqrt (48194734693623 / 128000000000000 : ℝ) := by
+    rw [show (400000000000000 / 245445347006271 : ℝ) = 1 / (245445347006271 / 400000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (62500000000000 / 511344472929731 : ℝ) ≤
+      (1216235363106071541963 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (48194734693623 / 128000000000000 : ℝ) ≤
+      (62500000000000 / 511344472929731 : ℝ) := by
+    rw [show
+        (62500000000000 / 511344472929731 : ℝ) =
+          (3 / 40) / (1534033418789193 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2751846209404526497207 / 10000000000000000000000 : ℝ))) -
+        6 * (1216235363106071541963 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (48194734693623 / 128000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (48194734693623 / 128000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1534033418789193 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2751846209404526497207 / 10000000000000000000000 : ℝ))) -
+          6 * (1216235363106071541963 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (48194734693623 / 128000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (48194734693623 / 128000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (48194734693623 / 128000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_380 :
+    ((1600172968229879563989 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(9602415608110495079503 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1228292155849459 / 2000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2753780737720085654817 / 10000000000000000000000 : ℝ))) -
+              6 * (303797689955757674163 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (24139225921941 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (24139225921941 / 64000000000000 : ℝ)) =
+      Real.log (88139225921941 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((24139225921941 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (24499225921941 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1600172968229879563989 / 5000000000000000000000 : ℝ) ≤
+      Real.log (88139225921941 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (24499225921941 / 64000000000000 : ℝ) ≤
+      (-(9602415608110495079503 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (24499225921941 / 64000000000000 : ℝ) = (24499225921941 / 16000000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24499225921941 / 16000000000000 : ℝ) ≤
+            (4260527997889504920497 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1228292155849459 / 2000000000000000 : ℝ) ≤ Real.sqrt (24139225921941 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (24139225921941 / 64000000000000 : ℝ) ≤ (3070730389623649 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (24139225921941 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau532 :
+    Real.arctan (6141460779247298 / 21735311691771667 : ℝ) ≤
+      (2753780737720085654817 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau531 :
+    Real.arctan (3070730389623649 / 5000000000000000 : ℝ) ≤
+      (2 * (2753780737720085654817 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (11735311691771667 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (3070730389623649 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (3070730389623649 / 5000000000000000 : ℝ) /
+                (1 + (11735311691771667 / 10000000000000000 : ℝ)) =
+              (6141460779247298 / 21735311691771667 : ℝ)
+            by
+        norm_num];
+        exact hau532)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2753780737720085654817 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 3070730389623649 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 3070730389623649 : ℝ) =
+              (3070730389623649 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau531)
+  have hz1 :
+    (5000000000000000 / 3070730389623649 : ℝ) ≤
+      1 / Real.sqrt (24139225921941 / 64000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 3070730389623649 : ℝ) = 1 / (3070730389623649 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (150000000000000 / 1228292155849459 : ℝ) ≤
+      (303797689955757674163 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (24139225921941 / 64000000000000 : ℝ) ≤
+      (150000000000000 / 1228292155849459 : ℝ) := by
+    rw [show
+        (150000000000000 / 1228292155849459 : ℝ) =
+          (3 / 40) / (1228292155849459 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2753780737720085654817 / 10000000000000000000000 : ℝ))) -
+        6 * (303797689955757674163 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (24139225921941 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (24139225921941 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1228292155849459 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2753780737720085654817 / 10000000000000000000000 : ℝ))) -
+          6 * (303797689955757674163 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (24139225921941 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (24139225921941 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (24139225921941 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_381 :
+    ((1602546975842297952737 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(9585344514988415420279 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((6146783266609669 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (275571267175795931953 / 1000000000000000000000 : ℝ))) -
+              6 * (75884302721971491671 / 625000000000000000000 : ℝ)))) ≤
+      Vfield (48362168994141 / 128000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (48362168994141 / 128000000000000 : ℝ)) =
+      Real.log (176362168994141 / 128000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((48362168994141 / 128000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (49082168994141 / 128000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1602546975842297952737 / 5000000000000000000000 : ℝ) ≤
+      Real.log (176362168994141 / 128000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (49082168994141 / 128000000000000 : ℝ) ≤
+      (-(9585344514988415420279 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (49082168994141 / 128000000000000 : ℝ) = (49082168994141 / 32000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (49082168994141 / 32000000000000 : ℝ) ≤
+            (4277599091011584579721 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (6146783266609669 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (48362168994141 / 128000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (48362168994141 / 128000000000000 : ℝ) ≤ (768347908326209 / 1250000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (48362168994141 / 128000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau534 :
+    Real.arctan (6146783266609672 / 21738097994422803 : ℝ) ≤
+      (275571267175795931953 / 1000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau533 :
+    Real.arctan (768347908326209 / 1250000000000000 : ℝ) ≤
+      (2 * (275571267175795931953 / 1000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (11738097994422803 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (768347908326209 / 1250000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (768347908326209 / 1250000000000000 : ℝ) /
+                (1 + (11738097994422803 / 10000000000000000 : ℝ)) =
+              (6146783266609672 / 21738097994422803 : ℝ)
+            by
+        norm_num];
+        exact hau534)
+  have hat1 :
+    (Real.pi / 2 - (2 * (275571267175795931953 / 1000000000000000000000 : ℝ))) ≤
+      Real.arctan (1250000000000000 / 768347908326209 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1250000000000000 / 768347908326209 : ℝ) =
+              (768347908326209 / 1250000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau533)
+  have hz1 :
+    (1250000000000000 / 768347908326209 : ℝ) ≤
+      1 / Real.sqrt (48362168994141 / 128000000000000 : ℝ) := by
+    rw [show
+        (1250000000000000 / 768347908326209 : ℝ) = 1 / (768347908326209 / 1250000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 6146783266609669 : ℝ) ≤
+      (75884302721971491671 / 625000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (48362168994141 / 128000000000000 : ℝ) ≤
+      (750000000000000 / 6146783266609669 : ℝ) := by
+    rw [show
+        (750000000000000 / 6146783266609669 : ℝ) =
+          (3 / 40) / (6146783266609669 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (275571267175795931953 / 1000000000000000000000 : ℝ))) -
+        6 * (75884302721971491671 / 625000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (48362168994141 / 128000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (48362168994141 / 128000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((6146783266609669 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (275571267175795931953 / 1000000000000000000000 : ℝ))) -
+          6 * (75884302721971491671 / 625000000000000000000 : ℝ))) ≤
+      Real.sqrt (48362168994141 / 128000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (48362168994141 / 128000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (48362168994141 / 128000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_382 :
+    ((320983971361440075487 / 1000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1196037814303904839033 / 1250000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((49216809193811 / 80000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (551528403665559915819 / 2000000000000000000000 : ℝ))) -
+              6 * (1213109602791532020873 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (121114715361 / 320000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (121114715361 / 320000000000 : ℝ)) =
+      Real.log (441114715361 / 320000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((121114715361 / 320000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (122914715361 / 320000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (320983971361440075487 / 1000000000000000000000 : ℝ) ≤
+      Real.log (441114715361 / 320000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (122914715361 / 320000000000 : ℝ) ≤
+      (-(1196037814303904839033 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (122914715361 / 320000000000 : ℝ) = (122914715361 / 80000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (122914715361 / 80000000000 : ℝ) ≤
+            (536830136446095160967 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (49216809193811 / 80000000000000 : ℝ) ≤ Real.sqrt (121114715361 / 320000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (121114715361 / 320000000000 : ℝ) ≤ (3076050574613189 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (121114715361 / 320000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau536 :
+    Real.arctan (3076050574613189 / 10870441817919510 : ℝ) ≤
+      (551528403665559915819 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau535 :
+    Real.arctan (3076050574613189 / 5000000000000000 : ℝ) ≤
+      (2 * (551528403665559915819 / 2000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (587044181791951 / 500000000000000 : ℝ) ≤
+          Real.sqrt (1 + (3076050574613189 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (3076050574613189 / 5000000000000000 : ℝ) /
+                (1 + (587044181791951 / 500000000000000 : ℝ)) =
+              (3076050574613189 / 10870441817919510 : ℝ)
+            by
+        norm_num];
+        exact hau536)
+  have hat1 :
+    (Real.pi / 2 - (2 * (551528403665559915819 / 2000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 3076050574613189 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 3076050574613189 : ℝ) =
+              (3076050574613189 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau535)
+  have hz1 :
+    (5000000000000000 / 3076050574613189 : ℝ) ≤ 1 / Real.sqrt (121114715361 / 320000000000 : ℝ) :=
+  by
+    rw [show
+        (5000000000000000 / 3076050574613189 : ℝ) = 1 / (3076050574613189 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (6000000000000 / 49216809193811 : ℝ) ≤
+      (1213109602791532020873 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (121114715361 / 320000000000 : ℝ) ≤
+      (6000000000000 / 49216809193811 : ℝ) := by
+    rw [show
+        (6000000000000 / 49216809193811 : ℝ) = (3 / 40) / (49216809193811 / 80000000000000 : ℝ) by
+        norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (551528403665559915819 / 2000000000000000000000 : ℝ))) -
+        6 * (1213109602791532020873 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (121114715361 / 320000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (121114715361 / 320000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((49216809193811 / 80000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (551528403665559915819 / 2000000000000000000000 : ℝ))) -
+          6 * (1213109602791532020873 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (121114715361 / 320000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (121114715361 / 320000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (121114715361 / 320000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_383 :
+    ((3214583224386879607881 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1193911188431064389001 / 1250000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1539353609757037 / 2500000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2 * (86236524506587603067 / 312500000000000000000 : ℝ))) -
+              6 * (60603651305585332871 / 500000000000000000000 : ℝ)))) ≤
+      Vfield (48529603294659 / 128000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (48529603294659 / 128000000000000 : ℝ)) =
+      Real.log (176529603294659 / 128000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((48529603294659 / 128000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (49249603294659 / 128000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (3214583224386879607881 / 10000000000000000000000 : ℝ) ≤
+      Real.log (176529603294659 / 128000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (49249603294659 / 128000000000000 : ℝ) ≤
+      (-(1193911188431064389001 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (49249603294659 / 128000000000000 : ℝ) = (49249603294659 / 32000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (49249603294659 / 32000000000000 : ℝ) ≤
+            (538956762318935610999 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1539353609757037 / 2500000000000000 : ℝ) ≤
+      Real.sqrt (48529603294659 / 128000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (48529603294659 / 128000000000000 : ℝ) ≤
+      (6157414439028151 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (48529603294659 / 128000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau538 :
+    Real.arctan (6157414439028151 / 21743668616490859 : ℝ) ≤
+      (86236524506587603067 / 312500000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau537 :
+    Real.arctan (6157414439028151 / 10000000000000000 : ℝ) ≤
+      (2 * (86236524506587603067 / 312500000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (11743668616490859 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (6157414439028151 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (6157414439028151 / 10000000000000000 : ℝ) /
+                (1 + (11743668616490859 / 10000000000000000 : ℝ)) =
+              (6157414439028151 / 21743668616490859 : ℝ)
+            by
+        norm_num];
+        exact hau538)
+  have hat1 :
+    (Real.pi / 2 - (2 * (86236524506587603067 / 312500000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 6157414439028151 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 6157414439028151 : ℝ) =
+              (6157414439028151 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau537)
+  have hz1 :
+    (10000000000000000 / 6157414439028151 : ℝ) ≤
+      1 / Real.sqrt (48529603294659 / 128000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 6157414439028151 : ℝ) =
+          1 / (6157414439028151 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (187500000000000 / 1539353609757037 : ℝ) ≤
+      (60603651305585332871 / 500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (48529603294659 / 128000000000000 : ℝ) ≤
+      (187500000000000 / 1539353609757037 : ℝ) := by
+    rw [show
+        (187500000000000 / 1539353609757037 : ℝ) =
+          (3 / 40) / (1539353609757037 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (86236524506587603067 / 312500000000000000000 : ℝ))) -
+        6 * (60603651305585332871 / 500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (48529603294659 / 128000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (48529603294659 / 128000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1539353609757037 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (86236524506587603067 / 312500000000000000000 : ℝ))) -
+          6 * (60603651305585332871 / 500000000000000000000 : ℝ))) ≤
+      Real.sqrt (48529603294659 / 128000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (48529603294659 / 128000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (48529603294659 / 128000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_384 :
+    ((643864897227339553809 / 2000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(9534305395554174604017 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1232544629578859 / 2000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (276149297615988133653 / 1000000000000000000000 : ℝ))) -
+              6 * (242207820429808106391 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (24306660222459 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (24306660222459 / 64000000000000 : ℝ)) =
+      Real.log (88306660222459 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((24306660222459 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (24666660222459 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (643864897227339553809 / 2000000000000000000000 : ℝ) ≤
+      Real.log (88306660222459 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (24666660222459 / 64000000000000 : ℝ) ≤
+      (-(9534305395554174604017 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (24666660222459 / 64000000000000 : ℝ) = (24666660222459 / 16000000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24666660222459 / 16000000000000 : ℝ) ≤
+            (4328638210445825395983 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1232544629578859 / 2000000000000000 : ℝ) ≤ Real.sqrt (24306660222459 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (24306660222459 / 64000000000000 : ℝ) ≤ (3081361573947149 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (24306660222459 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau540 :
+    Real.arctan (143319142974286 / 505731463647635 : ℝ) ≤
+      (276149297615988133653 / 1000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau539 :
+    Real.arctan (3081361573947149 / 5000000000000000 : ℝ) ≤
+      (2 * (276149297615988133653 / 1000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (2349290587369661 / 2000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (3081361573947149 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (3081361573947149 / 5000000000000000 : ℝ) /
+                (1 + (2349290587369661 / 2000000000000000 : ℝ)) =
+              (143319142974286 / 505731463647635 : ℝ)
+            by
+        norm_num];
+        exact hau540)
+  have hat1 :
+    (Real.pi / 2 - (2 * (276149297615988133653 / 1000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 3081361573947149 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 3081361573947149 : ℝ) =
+              (3081361573947149 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau539)
+  have hz1 :
+    (5000000000000000 / 3081361573947149 : ℝ) ≤
+      1 / Real.sqrt (24306660222459 / 64000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 3081361573947149 : ℝ) = 1 / (3081361573947149 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (150000000000000 / 1232544629578859 : ℝ) ≤
+      (242207820429808106391 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (24306660222459 / 64000000000000 : ℝ) ≤
+      (150000000000000 / 1232544629578859 : ℝ) := by
+    rw [show
+        (150000000000000 / 1232544629578859 : ℝ) =
+          (3 / 40) / (1232544629578859 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (276149297615988133653 / 1000000000000000000000 : ℝ))) -
+        6 * (242207820429808106391 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (24306660222459 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (24306660222459 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1232544629578859 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (276149297615988133653 / 1000000000000000000000 : ℝ))) -
+          6 * (242207820429808106391 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (24306660222459 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (24306660222459 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (24306660222459 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/V32.lean b/LeanPool/Zeta5Irrational/Table/V32.lean
new file mode 100644
index 0000000000..479e51ab17
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/V32.lean
@@ -0,0 +1,1444 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Energy
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # V32: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_385 :
+    ((403007937624435709943 / 1250000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(9517350080763106229959 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((6168027287652999 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2763414600899822286143 / 10000000000000000000000 : ℝ))) -
+              6 * (605003909804122796891 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (48697037595177 / 128000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (48697037595177 / 128000000000000 : ℝ)) =
+      Real.log (176697037595177 / 128000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((48697037595177 / 128000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (49417037595177 / 128000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (403007937624435709943 / 1250000000000000000000 : ℝ) ≤
+      Real.log (176697037595177 / 128000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (49417037595177 / 128000000000000 : ℝ) ≤
+      (-(9517350080763106229959 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (49417037595177 / 128000000000000 : ℝ) = (49417037595177 / 32000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (49417037595177 / 32000000000000 : ℝ) ≤
+            (4345593525236893770041 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (6168027287652999 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (48697037595177 / 128000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (48697037595177 / 128000000000000 : ℝ) ≤
+      (3084013643826501 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (48697037595177 / 128000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau542 :
+    Real.arctan (6168027287653002 / 21749236597380785 : ℝ) ≤
+      (2763414600899822286143 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau541 :
+    Real.arctan (3084013643826501 / 5000000000000000 : ℝ) ≤
+      (2 * (2763414600899822286143 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (2349847319476157 / 2000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (3084013643826501 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (3084013643826501 / 5000000000000000 : ℝ) /
+                (1 + (2349847319476157 / 2000000000000000 : ℝ)) =
+              (6168027287653002 / 21749236597380785 : ℝ)
+            by
+        norm_num];
+        exact hau542)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2763414600899822286143 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 3084013643826501 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 3084013643826501 : ℝ) =
+              (3084013643826501 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau541)
+  have hz1 :
+    (5000000000000000 / 3084013643826501 : ℝ) ≤
+      1 / Real.sqrt (48697037595177 / 128000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 3084013643826501 : ℝ) = 1 / (3084013643826501 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 2056009095884333 : ℝ) ≤
+      (605003909804122796891 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (48697037595177 / 128000000000000 : ℝ) ≤
+      (250000000000000 / 2056009095884333 : ℝ) := by
+    rw [show
+        (250000000000000 / 2056009095884333 : ℝ) =
+          (3 / 40) / (6168027287652999 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2763414600899822286143 / 10000000000000000000000 : ℝ))) -
+        6 * (605003909804122796891 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (48697037595177 / 128000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (48697037595177 / 128000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((6168027287652999 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2763414600899822286143 / 10000000000000000000000 : ℝ))) -
+          6 * (605003909804122796891 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (48697037595177 / 128000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (48697037595177 / 128000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (48697037595177 / 128000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_386 :
+    ((3228800271091844667397 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(4750211732793883134469 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((617332687008163 / 1000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2765333665127458345457 / 10000000000000000000000 : ℝ))) -
+              6 * (1208979167261253981733 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (12195188686359 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (12195188686359 / 32000000000000 : ℝ)) =
+      Real.log (44195188686359 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((12195188686359 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (12375188686359 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (3228800271091844667397 / 10000000000000000000000 : ℝ) ≤
+      Real.log (44195188686359 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (12375188686359 / 32000000000000 : ℝ) ≤
+      (-(4750211732793883134469 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (12375188686359 / 32000000000000 : ℝ) = (12375188686359 / 8000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12375188686359 / 8000000000000 : ℝ) ≤
+            (2181260070206116865531 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (617332687008163 / 1000000000000000 : ℝ) ≤ Real.sqrt (12195188686359 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (12195188686359 / 32000000000000 : ℝ) ≤
+      (6173326870081633 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (12195188686359 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau544 :
+    Real.arctan (6173326870081633 / 21752019598557171 : ℝ) ≤
+      (2765333665127458345457 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau543 :
+    Real.arctan (6173326870081633 / 10000000000000000 : ℝ) ≤
+      (2 * (2765333665127458345457 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (11752019598557171 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (6173326870081633 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (6173326870081633 / 10000000000000000 : ℝ) /
+                (1 + (11752019598557171 / 10000000000000000 : ℝ)) =
+              (6173326870081633 / 21752019598557171 : ℝ)
+            by
+        norm_num];
+        exact hau544)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2765333665127458345457 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 6173326870081633 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 6173326870081633 : ℝ) =
+              (6173326870081633 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau543)
+  have hz1 :
+    (10000000000000000 / 6173326870081633 : ℝ) ≤
+      1 / Real.sqrt (12195188686359 / 32000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 6173326870081633 : ℝ) =
+          1 / (6173326870081633 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (75000000000000 / 617332687008163 : ℝ) ≤
+      (1208979167261253981733 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (12195188686359 / 32000000000000 : ℝ) ≤
+      (75000000000000 / 617332687008163 : ℝ) := by
+    rw [show
+        (75000000000000 / 617332687008163 : ℝ) =
+          (3 / 40) / (617332687008163 / 1000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2765333665127458345457 / 10000000000000000000000 : ℝ))) -
+        6 * (1208979167261253981733 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (12195188686359 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (12195188686359 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((617332687008163 / 1000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2765333665127458345457 / 10000000000000000000000 : ℝ))) -
+          6 * (1208979167261253981733 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (12195188686359 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (12195188686359 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (12195188686359 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_387 :
+    ((202095924909459542751 / 625000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(4741762726517407558013 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((6178621906907049 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1383625087755914228753 / 5000000000000000000000 : ℝ))) -
+              6 * (301988283486676190467 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (9772894379139 / 25600000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (9772894379139 / 25600000000000 : ℝ)) =
+      Real.log (35372894379139 / 25600000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((9772894379139 / 25600000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (9916894379139 / 25600000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (202095924909459542751 / 625000000000000000000 : ℝ) ≤
+      Real.log (35372894379139 / 25600000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (9916894379139 / 25600000000000 : ℝ) ≤
+      (-(4741762726517407558013 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (9916894379139 / 25600000000000 : ℝ) = (9916894379139 / 6400000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9916894379139 / 6400000000000 : ℝ) ≤
+            (2189709076482592441987 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (6178621906907049 / 10000000000000000 : ℝ) ≤ Real.sqrt (9772894379139 / 25600000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (9772894379139 / 25600000000000 : ℝ) ≤ (1544655476726763 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (9772894379139 / 25600000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau546 :
+    Real.arctan (2059540635635684 / 7251600646948593 : ℝ) ≤
+      (1383625087755914228753 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau545 :
+    Real.arctan (1544655476726763 / 2500000000000000 : ℝ) ≤
+      (2 * (1383625087755914228753 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (11754801940845779 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1544655476726763 / 2500000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1544655476726763 / 2500000000000000 : ℝ) /
+                (1 + (11754801940845779 / 10000000000000000 : ℝ)) =
+              (2059540635635684 / 7251600646948593 : ℝ)
+            by
+        norm_num];
+        exact hau546)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1383625087755914228753 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (2500000000000000 / 1544655476726763 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 1544655476726763 : ℝ) =
+              (1544655476726763 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau545)
+  have hz1 :
+    (2500000000000000 / 1544655476726763 : ℝ) ≤
+      1 / Real.sqrt (9772894379139 / 25600000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 1544655476726763 : ℝ) = 1 / (1544655476726763 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 2059540635635683 : ℝ) ≤
+      (301988283486676190467 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (9772894379139 / 25600000000000 : ℝ) ≤
+      (250000000000000 / 2059540635635683 : ℝ) := by
+    rw [show
+        (250000000000000 / 2059540635635683 : ℝ) =
+          (3 / 40) / (6178621906907049 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1383625087755914228753 / 5000000000000000000000 : ℝ))) -
+        6 * (301988283486676190467 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (9772894379139 / 25600000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (9772894379139 / 25600000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((6178621906907049 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1383625087755914228753 / 5000000000000000000000 : ℝ))) -
+          6 * (301988283486676190467 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (9772894379139 / 25600000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (9772894379139 / 25600000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (9772894379139 / 25600000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_388 :
+    ((323826708549657002721 / 1000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(9466655946601782328347 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((6183912409805911 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2 * (173072758668396268933 / 625000000000000000000 : ℝ))) -
+              6 * (1206929708569435187719 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (24474094522977 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (24474094522977 / 64000000000000 : ℝ)) =
+      Real.log (88474094522977 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((24474094522977 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (24834094522977 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (323826708549657002721 / 1000000000000000000000 : ℝ) ≤
+      Real.log (88474094522977 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (24834094522977 / 64000000000000 : ℝ) ≤
+      (-(9466655946601782328347 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (24834094522977 / 64000000000000 : ℝ) = (24834094522977 / 16000000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (24834094522977 / 16000000000000 : ℝ) ≤
+            (4396287659398217671653 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (6183912409805911 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (24474094522977 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (24474094522977 / 64000000000000 : ℝ) ≤ (3091956204902957 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (24474094522977 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau548 :
+    Real.arctan (3091956204902957 / 10878791812357186 : ℝ) ≤
+      (173072758668396268933 / 625000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau547 :
+    Real.arctan (3091956204902957 / 5000000000000000 : ℝ) ≤
+      (2 * (173072758668396268933 / 625000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (2939395906178593 / 2500000000000000 : ℝ) ≤
+          Real.sqrt (1 + (3091956204902957 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (3091956204902957 / 5000000000000000 : ℝ) /
+                (1 + (2939395906178593 / 2500000000000000 : ℝ)) =
+              (3091956204902957 / 10878791812357186 : ℝ)
+            by
+        norm_num];
+        exact hau548)
+  have hat1 :
+    (Real.pi / 2 - (2 * (173072758668396268933 / 625000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 3091956204902957 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 3091956204902957 : ℝ) =
+              (3091956204902957 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau547)
+  have hz1 :
+    (5000000000000000 / 3091956204902957 : ℝ) ≤
+      1 / Real.sqrt (24474094522977 / 64000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 3091956204902957 : ℝ) = 1 / (3091956204902957 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 6183912409805911 : ℝ) ≤
+      (1206929708569435187719 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (24474094522977 / 64000000000000 : ℝ) ≤
+      (750000000000000 / 6183912409805911 : ℝ) := by
+    rw [show
+        (750000000000000 / 6183912409805911 : ℝ) =
+          (3 / 40) / (6183912409805911 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (173072758668396268933 / 625000000000000000000 : ℝ))) -
+        6 * (1206929708569435187719 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (24474094522977 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (24474094522977 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((6183912409805911 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (173072758668396268933 / 625000000000000000000 : ℝ))) -
+          6 * (1206929708569435187719 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (24474094522977 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (24474094522977 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (24474094522977 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_389 :
+    ((405965618289914982473 / 1250000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(9433002068520124417443 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1548619965070337 / 2500000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (554596889976445321921 / 2000000000000000000000 : ℝ))) -
+              6 * (1204890637574057060631 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (6139452918309 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (6139452918309 / 16000000000000 : ℝ)) =
+      Real.log (22139452918309 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((6139452918309 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (6229452918309 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (405965618289914982473 / 1250000000000000000000 : ℝ) ≤
+      Real.log (22139452918309 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (6229452918309 / 16000000000000 : ℝ) ≤
+      (-(9433002068520124417443 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (6229452918309 / 16000000000000 : ℝ) = (6229452918309 / 4000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6229452918309 / 4000000000000 : ℝ) ≤
+            (4429941537479875582557 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1548619965070337 / 2500000000000000 : ℝ) ≤ Real.sqrt (6139452918309 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (6139452918309 / 16000000000000 : ℝ) ≤ (6194479860281351 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (6139452918309 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau550 :
+    Real.arctan (6194479860281351 / 21763145019059794 : ℝ) ≤
+      (554596889976445321921 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau549 :
+    Real.arctan (6194479860281351 / 10000000000000000 : ℝ) ≤
+      (2 * (554596889976445321921 / 2000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (5881572509529897 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (6194479860281351 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (6194479860281351 / 10000000000000000 : ℝ) /
+                (1 + (5881572509529897 / 5000000000000000 : ℝ)) =
+              (6194479860281351 / 21763145019059794 : ℝ)
+            by
+        norm_num];
+        exact hau550)
+  have hat1 :
+    (Real.pi / 2 - (2 * (554596889976445321921 / 2000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 6194479860281351 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 6194479860281351 : ℝ) =
+              (6194479860281351 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau549)
+  have hz1 :
+    (10000000000000000 / 6194479860281351 : ℝ) ≤
+      1 / Real.sqrt (6139452918309 / 16000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 6194479860281351 : ℝ) =
+          1 / (6194479860281351 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (187500000000000 / 1548619965070337 : ℝ) ≤
+      (1204890637574057060631 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (6139452918309 / 16000000000000 : ℝ) ≤
+      (187500000000000 / 1548619965070337 : ℝ) := by
+    rw [show
+        (187500000000000 / 1548619965070337 : ℝ) =
+          (3 / 40) / (1548619965070337 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (554596889976445321921 / 2000000000000000000000 : ℝ))) -
+        6 * (1204890637574057060631 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (6139452918309 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (6139452918309 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1548619965070337 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (554596889976445321921 / 2000000000000000000000 : ℝ))) -
+          6 * (1204890637574057060631 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (6139452918309 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (6139452918309 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (6139452918309 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_390 :
+    ((203573366905027506113 / 625000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(4699730534507726419707 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3102514656964207 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2776794651275838889401 / 10000000000000000000000 : ℝ))) -
+              6 * (240572373363749729447 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (4928305764699 / 12800000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (4928305764699 / 12800000000000 : ℝ)) =
+      Real.log (17728305764699 / 12800000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((4928305764699 / 12800000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (5000305764699 / 12800000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (203573366905027506113 / 625000000000000000000 : ℝ) ≤
+      Real.log (17728305764699 / 12800000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (5000305764699 / 12800000000000 : ℝ) ≤
+      (-(4699730534507726419707 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (5000305764699 / 12800000000000 : ℝ) = (5000305764699 / 3200000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5000305764699 / 3200000000000 : ℝ) ≤
+            (2231741268492273580293 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3102514656964207 / 5000000000000000 : ℝ) ≤ Real.sqrt (4928305764699 / 12800000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (4928305764699 / 12800000000000 : ℝ) ≤ (6205029313928417 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (4928305764699 / 12800000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau552 :
+    Real.arctan (6205029313928417 / 21768703785324488 : ℝ) ≤
+      (2776794651275838889401 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau551 :
+    Real.arctan (6205029313928417 / 10000000000000000 : ℝ) ≤
+      (2 * (2776794651275838889401 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (1471087973165561 / 1250000000000000 : ℝ) ≤
+          Real.sqrt (1 + (6205029313928417 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (6205029313928417 / 10000000000000000 : ℝ) /
+                (1 + (1471087973165561 / 1250000000000000 : ℝ)) =
+              (6205029313928417 / 21768703785324488 : ℝ)
+            by
+        norm_num];
+        exact hau552)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2776794651275838889401 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 6205029313928417 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 6205029313928417 : ℝ) =
+              (6205029313928417 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau551)
+  have hz1 :
+    (10000000000000000 / 6205029313928417 : ℝ) ≤
+      1 / Real.sqrt (4928305764699 / 12800000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 6205029313928417 : ℝ) =
+          1 / (6205029313928417 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 3102514656964207 : ℝ) ≤
+      (240572373363749729447 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (4928305764699 / 12800000000000 : ℝ) ≤
+      (375000000000000 / 3102514656964207 : ℝ) := by
+    rw [show
+        (375000000000000 / 3102514656964207 : ℝ) =
+          (3 / 40) / (3102514656964207 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2776794651275838889401 / 10000000000000000000000 : ℝ))) -
+        6 * (240572373363749729447 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (4928305764699 / 12800000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (4928305764699 / 12800000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3102514656964207 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2776794651275838889401 / 10000000000000000000000 : ℝ))) -
+          6 * (240572373363749729447 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (4928305764699 / 12800000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (4928305764699 / 12800000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (4928305764699 / 12800000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_391 :
+    ((816653468713089555349 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2341508048351393276649 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3107780431191633 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2 * (86893587344624066073 / 312500000000000000000 : ℝ))) -
+              6 * (1200843309874540748747 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (12362622986877 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (12362622986877 / 32000000000000 : ℝ)) =
+      Real.log (44362622986877 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((12362622986877 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (12542622986877 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (816653468713089555349 / 2500000000000000000000 : ℝ) ≤
+      Real.log (44362622986877 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (12542622986877 / 32000000000000 : ℝ) ≤
+      (-(2341508048351393276649 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (12542622986877 / 32000000000000 : ℝ) = (12542622986877 / 8000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12542622986877 / 8000000000000 : ℝ) ≤
+            (1124227853148606723351 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3107780431191633 / 5000000000000000 : ℝ) ≤ Real.sqrt (12362622986877 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (12362622986877 / 32000000000000 : ℝ) ≤
+      (6215560862383269 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (12362622986877 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau554 :
+    Real.arctan (6215560862383269 / 21774259927230697 : ℝ) ≤
+      (86893587344624066073 / 312500000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau553 :
+    Real.arctan (6215560862383269 / 10000000000000000 : ℝ) ≤
+      (2 * (86893587344624066073 / 312500000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (11774259927230697 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (6215560862383269 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (6215560862383269 / 10000000000000000 : ℝ) /
+                (1 + (11774259927230697 / 10000000000000000 : ℝ)) =
+              (6215560862383269 / 21774259927230697 : ℝ)
+            by
+        norm_num];
+        exact hau554)
+  have hat1 :
+    (Real.pi / 2 - (2 * (86893587344624066073 / 312500000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 6215560862383269 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 6215560862383269 : ℝ) =
+              (6215560862383269 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau553)
+  have hz1 :
+    (10000000000000000 / 6215560862383269 : ℝ) ≤
+      1 / Real.sqrt (12362622986877 / 32000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 6215560862383269 : ℝ) =
+          1 / (6215560862383269 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (125000000000000 / 1035926810397211 : ℝ) ≤
+      (1200843309874540748747 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (12362622986877 / 32000000000000 : ℝ) ≤
+      (125000000000000 / 1035926810397211 : ℝ) := by
+    rw [show
+        (125000000000000 / 1035926810397211 : ℝ) =
+          (3 / 40) / (3107780431191633 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (86893587344624066073 / 312500000000000000000 : ℝ))) -
+        6 * (1200843309874540748747 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (12362622986877 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (12362622986877 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3107780431191633 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (86893587344624066073 / 312500000000000000000 : ℝ))) -
+          6 * (1200843309874540748747 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (12362622986877 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (12362622986877 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (12362622986877 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_392 :
+    ((3276044976259764027657 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(9332714694551548750447 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((6226074596507039 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2 * (21753007288003625813 / 78125000000000000000 : ℝ))) -
+              6 * (299708720331099562257 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (24808963124013 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (24808963124013 / 64000000000000 : ℝ)) =
+      Real.log (88808963124013 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((24808963124013 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (25168963124013 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (3276044976259764027657 / 10000000000000000000000 : ℝ) ≤
+      Real.log (88808963124013 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (25168963124013 / 64000000000000 : ℝ) ≤
+      (-(9332714694551548750447 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (25168963124013 / 64000000000000 : ℝ) = (25168963124013 / 16000000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25168963124013 / 16000000000000 : ℝ) ≤
+            (4530228911448451249553 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (6226074596507039 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (24808963124013 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (24808963124013 / 64000000000000 : ℝ) ≤ (3113037298253521 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (24808963124013 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau556 :
+    Real.arctan (3113037298253521 / 10889906724245943 : ℝ) ≤
+      (21753007288003625813 / 78125000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau555 :
+    Real.arctan (3113037298253521 / 5000000000000000 : ℝ) ≤
+      (2 * (21753007288003625813 / 78125000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (5889906724245943 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (3113037298253521 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (3113037298253521 / 5000000000000000 : ℝ) /
+                (1 + (5889906724245943 / 5000000000000000 : ℝ)) =
+              (3113037298253521 / 10889906724245943 : ℝ)
+            by
+        norm_num];
+        exact hau556)
+  have hat1 :
+    (Real.pi / 2 - (2 * (21753007288003625813 / 78125000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 3113037298253521 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 3113037298253521 : ℝ) =
+              (3113037298253521 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau555)
+  have hz1 :
+    (5000000000000000 / 3113037298253521 : ℝ) ≤
+      1 / Real.sqrt (24808963124013 / 64000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 3113037298253521 : ℝ) = 1 / (3113037298253521 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 6226074596507039 : ℝ) ≤
+      (299708720331099562257 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (24808963124013 / 64000000000000 : ℝ) ≤
+      (750000000000000 / 6226074596507039 : ℝ) := by
+    rw [show
+        (750000000000000 / 6226074596507039 : ℝ) =
+          (3 / 40) / (6226074596507039 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (21753007288003625813 / 78125000000000000000 : ℝ))) -
+        6 * (299708720331099562257 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (24808963124013 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (24808963124013 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((6226074596507039 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (21753007288003625813 / 78125000000000000000 : ℝ))) -
+          6 * (299708720331099562257 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (24808963124013 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (24808963124013 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (24808963124013 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_393 :
+    ((657093438295957909727 / 2000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2324876958189376444893 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((6236570606394991 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2788165116089121008927 / 10000000000000000000000 : ℝ))) -
+              6 * (1196836496748037631457 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (777896258571 / 2000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (777896258571 / 2000000000000 : ℝ)) =
+      Real.log (2777896258571 / 2000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((777896258571 / 2000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (789146258571 / 2000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (657093438295957909727 / 2000000000000000000000 : ℝ) ≤
+      Real.log (2777896258571 / 2000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (789146258571 / 2000000000000 : ℝ) ≤
+      (-(2324876958189376444893 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (789146258571 / 2000000000000 : ℝ) = (789146258571 / 500000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (789146258571 / 500000000000 : ℝ) ≤
+            (1140858943310623555107 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (6236570606394991 / 10000000000000000 : ℝ) ≤ Real.sqrt (777896258571 / 2000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (777896258571 / 2000000000000 : ℝ) ≤ (3118285303197497 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (777896258571 / 2000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau558 :
+    Real.arctan (6236570606394994 / 21785364352812771 : ℝ) ≤
+      (2788165116089121008927 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau557 :
+    Real.arctan (3118285303197497 / 5000000000000000 : ℝ) ≤
+      (2 * (2788165116089121008927 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (11785364352812771 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (3118285303197497 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (3118285303197497 / 5000000000000000 : ℝ) /
+                (1 + (11785364352812771 / 10000000000000000 : ℝ)) =
+              (6236570606394994 / 21785364352812771 : ℝ)
+            by
+        norm_num];
+        exact hau558)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2788165116089121008927 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 3118285303197497 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 3118285303197497 : ℝ) =
+              (3118285303197497 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau557)
+  have hz1 :
+    (5000000000000000 / 3118285303197497 : ℝ) ≤
+      1 / Real.sqrt (777896258571 / 2000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 3118285303197497 : ℝ) = 1 / (3118285303197497 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 6236570606394991 : ℝ) ≤
+      (1196836496748037631457 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (777896258571 / 2000000000000 : ℝ) ≤
+      (750000000000000 / 6236570606394991 : ℝ) := by
+    rw [show
+        (750000000000000 / 6236570606394991 : ℝ) =
+          (3 / 40) / (6236570606394991 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2788165116089121008927 / 10000000000000000000000 : ℝ))) -
+        6 * (1196836496748037631457 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (777896258571 / 2000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (777896258571 / 2000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((6236570606394991 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2788165116089121008927 / 10000000000000000000000 : ℝ))) -
+          6 * (1196836496748037631457 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (777896258571 / 2000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (777896258571 / 2000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (777896258571 / 2000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_394 :
+    ((1647440268621094061217 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(9266410875672095229157 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((195220280668297 / 312500000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2791935395588520664673 / 10000000000000000000000 : ℝ))) -
+              6 * (1194848072707023067937 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (24976397424531 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (24976397424531 / 64000000000000 : ℝ)) =
+      Real.log (88976397424531 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((24976397424531 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (25336397424531 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1647440268621094061217 / 5000000000000000000000 : ℝ) ≤
+      Real.log (88976397424531 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (25336397424531 / 64000000000000 : ℝ) ≤
+      (-(9266410875672095229157 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (25336397424531 / 64000000000000 : ℝ) = (25336397424531 / 16000000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25336397424531 / 16000000000000 : ℝ) ≤
+            (4596532730327904770843 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (195220280668297 / 312500000000000 : ℝ) ≤ Real.sqrt (24976397424531 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (24976397424531 / 64000000000000 : ℝ) ≤
+      (6247048981385507 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (24976397424531 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau560 :
+    Real.arctan (6247048981385507 / 21790912643889347 : ℝ) ≤
+      (2791935395588520664673 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau559 :
+    Real.arctan (6247048981385507 / 10000000000000000 : ℝ) ≤
+      (2 * (2791935395588520664673 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (11790912643889347 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (6247048981385507 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (6247048981385507 / 10000000000000000 : ℝ) /
+                (1 + (11790912643889347 / 10000000000000000 : ℝ)) =
+              (6247048981385507 / 21790912643889347 : ℝ)
+            by
+        norm_num];
+        exact hau560)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2791935395588520664673 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 6247048981385507 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 6247048981385507 : ℝ) =
+              (6247048981385507 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau559)
+  have hz1 :
+    (10000000000000000 / 6247048981385507 : ℝ) ≤
+      1 / Real.sqrt (24976397424531 / 64000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 6247048981385507 : ℝ) =
+          1 / (6247048981385507 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (23437500000000 / 195220280668297 : ℝ) ≤
+      (1194848072707023067937 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (24976397424531 / 64000000000000 : ℝ) ≤
+      (23437500000000 / 195220280668297 : ℝ) := by
+    rw [show
+        (23437500000000 / 195220280668297 : ℝ) =
+          (3 / 40) / (195220280668297 / 312500000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2791935395588520664673 / 10000000000000000000000 : ℝ))) -
+        6 * (1194848072707023067937 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (24976397424531 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (24976397424531 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((195220280668297 / 312500000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2791935395588520664673 / 10000000000000000000000 : ℝ))) -
+          6 * (1194848072707023067937 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (24976397424531 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (24976397424531 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (24976397424531 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_395 :
+    ((660857006045902524463 / 2000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(923342309819158098361 / 1000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((6257509810068967 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2 * (139784791091839437819 / 500000000000000000000 : ℝ))) -
+              6 * (596434763365065326249 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (2506011457479 / 6400000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (2506011457479 / 6400000000000 : ℝ)) =
+      Real.log (8906011457479 / 6400000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((2506011457479 / 6400000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (2542011457479 / 6400000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (660857006045902524463 / 2000000000000000000000 : ℝ) ≤
+      Real.log (8906011457479 / 6400000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (2542011457479 / 6400000000000 : ℝ) ≤
+      (-(923342309819158098361 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (2542011457479 / 6400000000000 : ℝ) = (2542011457479 / 1600000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2542011457479 / 1600000000000 : ℝ) ≤
+            (462952050780841901639 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (6257509810068967 / 10000000000000000 : ℝ) ≤ Real.sqrt (2506011457479 / 6400000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (2506011457479 / 6400000000000 : ℝ) ≤ (625750981006897 / 1000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (2506011457479 / 6400000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau562 :
+    Real.arctan (6257509810068970 / 21796458325408917 : ℝ) ≤
+      (139784791091839437819 / 500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau561 :
+    Real.arctan (625750981006897 / 1000000000000000 : ℝ) ≤
+      (2 * (139784791091839437819 / 500000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (11796458325408917 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (625750981006897 / 1000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (625750981006897 / 1000000000000000 : ℝ) /
+                (1 + (11796458325408917 / 10000000000000000 : ℝ)) =
+              (6257509810068970 / 21796458325408917 : ℝ)
+            by
+        norm_num];
+        exact hau562)
+  have hat1 :
+    (Real.pi / 2 - (2 * (139784791091839437819 / 500000000000000000000 : ℝ))) ≤
+      Real.arctan (1000000000000000 / 625750981006897 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1000000000000000 / 625750981006897 : ℝ) =
+              (625750981006897 / 1000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau561)
+  have hz1 :
+    (1000000000000000 / 625750981006897 : ℝ) ≤
+      1 / Real.sqrt (2506011457479 / 6400000000000 : ℝ) := by
+    rw [show
+        (1000000000000000 / 625750981006897 : ℝ) = 1 / (625750981006897 / 1000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 6257509810068967 : ℝ) ≤
+      (596434763365065326249 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (2506011457479 / 6400000000000 : ℝ) ≤
+      (750000000000000 / 6257509810068967 : ℝ) := by
+    rw [show
+        (750000000000000 / 6257509810068967 : ℝ) =
+          (3 / 40) / (6257509810068967 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (139784791091839437819 / 500000000000000000000 : ℝ))) -
+        6 * (596434763365065326249 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (2506011457479 / 6400000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (2506011457479 / 6400000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((6257509810068967 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (139784791091839437819 / 500000000000000000000 : ℝ))) -
+          6 * (596434763365065326249 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (2506011457479 / 6400000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (2506011457479 / 6400000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (2506011457479 / 6400000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_396 :
+    ((828420171769323212123 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(4600271891182260404739 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((6267953180296503 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (349930805612534943393 / 1250000000000000000000 : ℝ))) -
+              6 * (1190900777298982651693 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (25143831725049 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (25143831725049 / 64000000000000 : ℝ)) =
+      Real.log (89143831725049 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((25143831725049 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (25503831725049 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (828420171769323212123 / 2500000000000000000000 : ℝ) ≤
+      Real.log (89143831725049 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (25503831725049 / 64000000000000 : ℝ) ≤
+      (-(4600271891182260404739 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show
+          (25503831725049 / 64000000000000 : ℝ) = (25503831725049 / 16000000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25503831725049 / 16000000000000 : ℝ) ≤
+            (2331199911817739595261 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (6267953180296503 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (25143831725049 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (25143831725049 / 64000000000000 : ℝ) ≤ (3133976590148253 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (25143831725049 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau564 :
+    Real.arctan (3133976590148253 / 10901000700525061 : ℝ) ≤
+      (349930805612534943393 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau563 :
+    Real.arctan (3133976590148253 / 5000000000000000 : ℝ) ≤
+      (2 * (349930805612534943393 / 1250000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (5901000700525061 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (3133976590148253 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (3133976590148253 / 5000000000000000 : ℝ) /
+                (1 + (5901000700525061 / 5000000000000000 : ℝ)) =
+              (3133976590148253 / 10901000700525061 : ℝ)
+            by
+        norm_num];
+        exact hau564)
+  have hat1 :
+    (Real.pi / 2 - (2 * (349930805612534943393 / 1250000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 3133976590148253 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 3133976590148253 : ℝ) =
+              (3133976590148253 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau563)
+  have hz1 :
+    (5000000000000000 / 3133976590148253 : ℝ) ≤
+      1 / Real.sqrt (25143831725049 / 64000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 3133976590148253 : ℝ) = 1 / (3133976590148253 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 2089317726765501 : ℝ) ≤
+      (1190900777298982651693 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (25143831725049 / 64000000000000 : ℝ) ≤
+      (250000000000000 / 2089317726765501 : ℝ) := by
+    rw [show
+        (250000000000000 / 2089317726765501 : ℝ) =
+          (3 / 40) / (6267953180296503 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (349930805612534943393 / 1250000000000000000000 : ℝ))) -
+        6 * (1190900777298982651693 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (25143831725049 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (25143831725049 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((6267953180296503 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (349930805612534943393 / 1250000000000000000000 : ℝ))) -
+          6 * (1190900777298982651693 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (25143831725049 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (25143831725049 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (25143831725049 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/V33.lean b/LeanPool/Zeta5Irrational/Table/V33.lean
new file mode 100644
index 0000000000..90d2c22584
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/V33.lean
@@ -0,0 +1,1432 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Energy
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # V33: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_397 :
+    ((830766881093553021839 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(8952902555955087179 / 9765625000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1255675835837717 / 2000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (700796828610549806021 / 2500000000000000000000 : ℝ))) -
+              6 * (118894174383393800607 / 1000000000000000000000 : ℝ)))) ≤
+      Vfield (6306887218827 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (6306887218827 / 16000000000000 : ℝ)) =
+      Real.log (22306887218827 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((6306887218827 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (6396887218827 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (830766881093553021839 / 2500000000000000000000 : ℝ) ≤
+      Real.log (22306887218827 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (6396887218827 / 16000000000000 : ℝ) ≤
+      (-(8952902555955087179 / 9765625000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (6396887218827 / 16000000000000 : ℝ) = (6396887218827 / 4000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6396887218827 / 4000000000000 : ℝ) ≤
+            (4585128309279287821 / 9765625000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1255675835837717 / 2000000000000000 : ℝ) ≤ Real.sqrt (6306887218827 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (6306887218827 / 16000000000000 : ℝ) ≤ (1569594794797147 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (6306887218827 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau566 :
+    Real.arctan (6278379179188588 / 21807541874482967 : ℝ) ≤
+      (700796828610549806021 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau565 :
+    Real.arctan (1569594794797147 / 2500000000000000 : ℝ) ≤
+      (2 * (700796828610549806021 / 2500000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (11807541874482967 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1569594794797147 / 2500000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1569594794797147 / 2500000000000000 : ℝ) /
+                (1 + (11807541874482967 / 10000000000000000 : ℝ)) =
+              (6278379179188588 / 21807541874482967 : ℝ)
+            by
+        norm_num];
+        exact hau566)
+  have hat1 :
+    (Real.pi / 2 - (2 * (700796828610549806021 / 2500000000000000000000 : ℝ))) ≤
+      Real.arctan (2500000000000000 / 1569594794797147 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 1569594794797147 : ℝ) =
+              (1569594794797147 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau565)
+  have hz1 :
+    (2500000000000000 / 1569594794797147 : ℝ) ≤
+      1 / Real.sqrt (6306887218827 / 16000000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 1569594794797147 : ℝ) = 1 / (1569594794797147 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (150000000000000 / 1255675835837717 : ℝ) ≤
+      (118894174383393800607 / 1000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (6306887218827 / 16000000000000 : ℝ) ≤
+      (150000000000000 / 1255675835837717 : ℝ) := by
+    rw [show
+        (150000000000000 / 1255675835837717 : ℝ) =
+          (3 / 40) / (1255675835837717 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (700796828610549806021 / 2500000000000000000000 : ℝ))) -
+        6 * (118894174383393800607 / 1000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (6306887218827 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (6306887218827 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1255675835837717 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (700796828610549806021 / 2500000000000000000000 : ℝ))) -
+          6 * (118894174383393800607 / 1000000000000000000000 : ℝ))) ≤
+      Real.sqrt (6306887218827 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (6306887218827 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (6306887218827 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_398 :
+    ((3341814806437021744403 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(4551274765307920417809 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1574794851961441 / 2500000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1405319994812823534829 / 5000000000000000000000 : ℝ))) -
+              6 * (237010501418881799479 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (12697491587913 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (12697491587913 / 32000000000000 : ℝ)) =
+      Real.log (44697491587913 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((12697491587913 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (12877491587913 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (3341814806437021744403 / 10000000000000000000000 : ℝ) ≤
+      Real.log (44697491587913 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (12877491587913 / 32000000000000 : ℝ) ≤
+      (-(4551274765307920417809 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (12877491587913 / 32000000000000 : ℝ) = (12877491587913 / 8000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12877491587913 / 8000000000000 : ℝ) ≤
+            (2380197037692079582191 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1574794851961441 / 2500000000000000 : ℝ) ≤ Real.sqrt (12697491587913 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (12697491587913 / 32000000000000 : ℝ) ≤
+      (6299179407845767 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (12697491587913 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau568 :
+    Real.arctan (6299179407845767 / 21818615029360594 : ℝ) ≤
+      (1405319994812823534829 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau567 :
+    Real.arctan (6299179407845767 / 10000000000000000 : ℝ) ≤
+      (2 * (1405319994812823534829 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (5909307514680297 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (6299179407845767 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (6299179407845767 / 10000000000000000 : ℝ) /
+                (1 + (5909307514680297 / 5000000000000000 : ℝ)) =
+              (6299179407845767 / 21818615029360594 : ℝ)
+            by
+        norm_num];
+        exact hau568)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1405319994812823534829 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 6299179407845767 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 6299179407845767 : ℝ) =
+              (6299179407845767 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau567)
+  have hz1 :
+    (10000000000000000 / 6299179407845767 : ℝ) ≤
+      1 / Real.sqrt (12697491587913 / 32000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 6299179407845767 : ℝ) =
+          1 / (6299179407845767 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (187500000000000 / 1574794851961441 : ℝ) ≤
+      (237010501418881799479 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (12697491587913 / 32000000000000 : ℝ) ≤
+      (187500000000000 / 1574794851961441 : ℝ) := by
+    rw [show
+        (187500000000000 / 1574794851961441 : ℝ) =
+          (3 / 40) / (1574794851961441 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1405319994812823534829 / 5000000000000000000000 : ℝ))) -
+        6 * (237010501418881799479 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (12697491587913 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (12697491587913 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1574794851961441 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1405319994812823534829 / 5000000000000000000000 : ℝ))) -
+          6 * (237010501418881799479 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (12697491587913 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (12697491587913 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (12697491587913 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_399 :
+    ((840131752049283713763 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2259437372176322009597 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1263982235742061 / 2000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2818054236801107742807 / 10000000000000000000000 : ℝ))) -
+              6 * (590600595064456682237 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (3195302184543 / 8000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (3195302184543 / 8000000000000 : ℝ)) =
+      Real.log (11195302184543 / 8000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((3195302184543 / 8000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (3240302184543 / 8000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (840131752049283713763 / 2500000000000000000000 : ℝ) ≤
+      Real.log (11195302184543 / 8000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (3240302184543 / 8000000000000 : ℝ) ≤
+      (-(2259437372176322009597 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (3240302184543 / 8000000000000 : ℝ) = (3240302184543 / 2000000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3240302184543 / 2000000000000 : ℝ) ≤
+            (1206298529323677990403 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1263982235742061 / 2000000000000000 : ℝ) ≤ Real.sqrt (3195302184543 / 8000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (3195302184543 / 8000000000000 : ℝ) ≤ (1579977794677577 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (3195302184543 / 8000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau570 :
+    Real.arctan (6319911178710308 / 21829677819230221 : ℝ) ≤
+      (2818054236801107742807 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau569 :
+    Real.arctan (1579977794677577 / 2500000000000000 : ℝ) ≤
+      (2 * (2818054236801107742807 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (11829677819230221 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1579977794677577 / 2500000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1579977794677577 / 2500000000000000 : ℝ) /
+                (1 + (11829677819230221 / 10000000000000000 : ℝ)) =
+              (6319911178710308 / 21829677819230221 : ℝ)
+            by
+        norm_num];
+        exact hau570)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2818054236801107742807 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (2500000000000000 / 1579977794677577 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 1579977794677577 : ℝ) =
+              (1579977794677577 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau569)
+  have hz1 :
+    (2500000000000000 / 1579977794677577 : ℝ) ≤
+      1 / Real.sqrt (3195302184543 / 8000000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 1579977794677577 : ℝ) = 1 / (1579977794677577 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (150000000000000 / 1263982235742061 : ℝ) ≤
+      (590600595064456682237 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (3195302184543 / 8000000000000 : ℝ) ≤
+      (150000000000000 / 1263982235742061 : ℝ) := by
+    rw [show
+        (150000000000000 / 1263982235742061 : ℝ) =
+          (3 / 40) / (1263982235742061 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2818054236801107742807 / 10000000000000000000000 : ℝ))) -
+        6 * (590600595064456682237 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (3195302184543 / 8000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (3195302184543 / 8000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1263982235742061 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2818054236801107742807 / 10000000000000000000000 : ℝ))) -
+          6 * (590600595064456682237 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (3195302184543 / 8000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (3195302184543 / 8000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (3195302184543 / 8000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_400 :
+    ((3379204260695362054447 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(8973366649314295692837 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((99071486926427 / 156250000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2825430439216674767041 / 10000000000000000000000 : ℝ))) -
+              6 * (1177387180713437419171 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (12864925888431 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (12864925888431 / 32000000000000 : ℝ)) =
+      Real.log (44864925888431 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((12864925888431 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (13044925888431 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (3379204260695362054447 / 10000000000000000000000 : ℝ) ≤
+      Real.log (44864925888431 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (13044925888431 / 32000000000000 : ℝ) ≤
+      (-(8973366649314295692837 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (13044925888431 / 32000000000000 : ℝ) = (13044925888431 / 8000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13044925888431 / 8000000000000 : ℝ) ≤
+            (4889576956685704307163 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (99071486926427 / 156250000000000 : ℝ) ≤ Real.sqrt (12864925888431 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (12864925888431 / 32000000000000 : ℝ) ≤
+      (6340575163291331 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (12864925888431 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau572 :
+    Real.arctan (6340575163291331 / 21840730273143918 : ℝ) ≤
+      (2825430439216674767041 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau571 :
+    Real.arctan (6340575163291331 / 10000000000000000 : ℝ) ≤
+      (2 * (2825430439216674767041 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (5920365136571959 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (6340575163291331 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (6340575163291331 / 10000000000000000 : ℝ) /
+                (1 + (5920365136571959 / 5000000000000000 : ℝ)) =
+              (6340575163291331 / 21840730273143918 : ℝ)
+            by
+        norm_num];
+        exact hau572)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2825430439216674767041 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 6340575163291331 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 6340575163291331 : ℝ) =
+              (6340575163291331 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau571)
+  have hz1 :
+    (10000000000000000 / 6340575163291331 : ℝ) ≤
+      1 / Real.sqrt (12864925888431 / 32000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 6340575163291331 : ℝ) =
+          1 / (6340575163291331 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (11718750000000 / 99071486926427 : ℝ) ≤
+      (1177387180713437419171 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (12864925888431 / 32000000000000 : ℝ) ≤
+      (11718750000000 / 99071486926427 : ℝ) := by
+    rw [show
+        (11718750000000 / 99071486926427 : ℝ) = (3 / 40) / (99071486926427 / 156250000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2825430439216674767041 / 10000000000000000000000 : ℝ))) -
+        6 * (1177387180713437419171 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (12864925888431 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (12864925888431 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((99071486926427 / 156250000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2825430439216674767041 / 10000000000000000000000 : ℝ))) -
+          6 * (1177387180713437419171 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (12864925888431 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (12864925888431 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (12864925888431 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_401 :
+    ((3397846694239637457303 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(8909395674635741552251 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1272234404438211 / 2000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2832768974090960128569 / 10000000000000000000000 : ℝ))) -
+              6 * (293402470093434932507 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (1294864303869 / 3200000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1294864303869 / 3200000000000 : ℝ)) =
+      Real.log (4494864303869 / 3200000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1294864303869 / 3200000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1312864303869 / 3200000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (3397846694239637457303 / 10000000000000000000000 : ℝ) ≤
+      Real.log (4494864303869 / 3200000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1312864303869 / 3200000000000 : ℝ) ≤
+      (-(8909395674635741552251 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (1312864303869 / 3200000000000 : ℝ) = (1312864303869 / 800000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1312864303869 / 800000000000 : ℝ) ≤
+            (4953547931364258447749 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1272234404438211 / 2000000000000000 : ℝ) ≤ Real.sqrt (1294864303869 / 3200000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1294864303869 / 3200000000000 : ℝ) ≤ (3180586011095529 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1294864303869 / 3200000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau574 :
+    Real.arctan (3180586011095529 / 10925886210009146 : ℝ) ≤
+      (2832768974090960128569 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau573 :
+    Real.arctan (3180586011095529 / 5000000000000000 : ℝ) ≤
+      (2 * (2832768974090960128569 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (2962943105004573 / 2500000000000000 : ℝ) ≤
+          Real.sqrt (1 + (3180586011095529 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (3180586011095529 / 5000000000000000 : ℝ) /
+                (1 + (2962943105004573 / 2500000000000000 : ℝ)) =
+              (3180586011095529 / 10925886210009146 : ℝ)
+            by
+        norm_num];
+        exact hau574)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2832768974090960128569 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 3180586011095529 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 3180586011095529 : ℝ) =
+              (3180586011095529 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau573)
+  have hz1 :
+    (5000000000000000 / 3180586011095529 : ℝ) ≤
+      1 / Real.sqrt (1294864303869 / 3200000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 3180586011095529 : ℝ) = 1 / (3180586011095529 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (50000000000000 / 424078134812737 : ℝ) ≤
+      (293402470093434932507 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1294864303869 / 3200000000000 : ℝ) ≤
+      (50000000000000 / 424078134812737 : ℝ) := by
+    rw [show
+        (50000000000000 / 424078134812737 : ℝ) =
+          (3 / 40) / (1272234404438211 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2832768974090960128569 / 10000000000000000000000 : ℝ))) -
+        6 * (293402470093434932507 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1294864303869 / 3200000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1294864303869 / 3200000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1272234404438211 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2832768974090960128569 / 10000000000000000000000 : ℝ))) -
+          6 * (293402470093434932507 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (1294864303869 / 3200000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1294864303869 / 3200000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1294864303869 / 3200000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_402 :
+    ((3416454438410473535733 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(8845831328651810423357 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((255268096214049 / 400000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2 * (14200351063728030601 / 50000000000000000000 : ℝ))) -
+              6 * (584934351995421838111 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (13032360188949 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (13032360188949 / 32000000000000 : ℝ)) =
+      Real.log (45032360188949 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((13032360188949 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (13212360188949 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (3416454438410473535733 / 10000000000000000000000 : ℝ) ≤
+      Real.log (45032360188949 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (13212360188949 / 32000000000000 : ℝ) ≤
+      (-(8845831328651810423357 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (13212360188949 / 32000000000000 : ℝ) = (13212360188949 / 8000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (13212360188949 / 8000000000000 : ℝ) ≤
+            (5017112277348189576643 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (255268096214049 / 400000000000000 : ℝ) ≤ Real.sqrt (13032360188949 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (13032360188949 / 32000000000000 : ℝ) ≤ (1595425601337807 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (13032360188949 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau576 :
+    Real.arctan (6381702405351228 / 21862804288635365 : ℝ) ≤
+      (14200351063728030601 / 50000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau575 :
+    Real.arctan (1595425601337807 / 2500000000000000 : ℝ) ≤
+      (2 * (14200351063728030601 / 50000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (2372560857727073 / 2000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1595425601337807 / 2500000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1595425601337807 / 2500000000000000 : ℝ) /
+                (1 + (2372560857727073 / 2000000000000000 : ℝ)) =
+              (6381702405351228 / 21862804288635365 : ℝ)
+            by
+        norm_num];
+        exact hau576)
+  have hat1 :
+    (Real.pi / 2 - (2 * (14200351063728030601 / 50000000000000000000 : ℝ))) ≤
+      Real.arctan (2500000000000000 / 1595425601337807 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 1595425601337807 : ℝ) =
+              (1595425601337807 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau575)
+  have hz1 :
+    (2500000000000000 / 1595425601337807 : ℝ) ≤
+      1 / Real.sqrt (13032360188949 / 32000000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 1595425601337807 : ℝ) = 1 / (1595425601337807 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (10000000000000 / 85089365404683 : ℝ) ≤
+      (584934351995421838111 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (13032360188949 / 32000000000000 : ℝ) ≤
+      (10000000000000 / 85089365404683 : ℝ) := by
+    rw [show
+        (10000000000000 / 85089365404683 : ℝ) = (3 / 40) / (255268096214049 / 400000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (14200351063728030601 / 50000000000000000000 : ℝ))) -
+        6 * (584934351995421838111 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (13032360188949 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (13032360188949 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((255268096214049 / 400000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (14200351063728030601 / 50000000000000000000 : ℝ))) -
+          6 * (584934351995421838111 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (13032360188949 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (13032360188949 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (13032360188949 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_403 :
+    ((1717513811033182669897 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(4391334237281119106909 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3201083476146201 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (355916815091755312081 / 1250000000000000000000 : ℝ))) -
+              6 * (583081539710138134777 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (1639509667401 / 4000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1639509667401 / 4000000000000 : ℝ)) =
+      Real.log (5639509667401 / 4000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1639509667401 / 4000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1662009667401 / 4000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1717513811033182669897 / 5000000000000000000000 : ℝ) ≤
+      Real.log (5639509667401 / 4000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1662009667401 / 4000000000000 : ℝ) ≤
+      (-(4391334237281119106909 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (1662009667401 / 4000000000000 : ℝ) = (1662009667401 / 1000000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1662009667401 / 1000000000000 : ℝ) ≤
+            (2540137565718880893091 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3201083476146201 / 5000000000000000 : ℝ) ≤ Real.sqrt (1639509667401 / 4000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1639509667401 / 4000000000000 : ℝ) ≤ (1280433390458481 / 2000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1639509667401 / 4000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau578 :
+    Real.arctan (914595278898915 / 3124832272520494 : ℝ) ≤
+      (355916815091755312081 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau577 :
+    Real.arctan (1280433390458481 / 2000000000000000 : ℝ) ≤
+      (2 * (355916815091755312081 / 1250000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (5936912953821729 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1280433390458481 / 2000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1280433390458481 / 2000000000000000 : ℝ) /
+                (1 + (5936912953821729 / 5000000000000000 : ℝ)) =
+              (914595278898915 / 3124832272520494 : ℝ)
+            by
+        norm_num];
+        exact hau578)
+  have hat1 :
+    (Real.pi / 2 - (2 * (355916815091755312081 / 1250000000000000000000 : ℝ))) ≤
+      Real.arctan (2000000000000000 / 1280433390458481 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2000000000000000 / 1280433390458481 : ℝ) =
+              (1280433390458481 / 2000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau577)
+  have hz1 :
+    (2000000000000000 / 1280433390458481 : ℝ) ≤
+      1 / Real.sqrt (1639509667401 / 4000000000000 : ℝ) := by
+    rw [show
+        (2000000000000000 / 1280433390458481 : ℝ) = 1 / (1280433390458481 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (125000000000000 / 1067027825382067 : ℝ) ≤
+      (583081539710138134777 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1639509667401 / 4000000000000 : ℝ) ≤
+      (125000000000000 / 1067027825382067 : ℝ) := by
+    rw [show
+        (125000000000000 / 1067027825382067 : ℝ) =
+          (3 / 40) / (3201083476146201 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (355916815091755312081 / 1250000000000000000000 : ℝ))) -
+        6 * (583081539710138134777 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1639509667401 / 4000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1639509667401 / 4000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3201083476146201 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (355916815091755312081 / 1250000000000000000000 : ℝ))) -
+          6 * (583081539710138134777 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (1639509667401 / 4000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1639509667401 / 4000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1639509667401 / 4000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_404 :
+    ((868017704922321109341 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(541095448505268199687 / 625000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((805362630610271 / 1250000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (715438444708057385753 / 2500000000000000000000 : ℝ))) -
+              6 * (115885625981794946927 / 1000000000000000000000 : ℝ)))) ≤
+      Vfield (6641755819863 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (6641755819863 / 16000000000000 : ℝ)) =
+      Real.log (22641755819863 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((6641755819863 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (6731755819863 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (868017704922321109341 / 2500000000000000000000 : ℝ) ≤
+      Real.log (22641755819863 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (6731755819863 / 16000000000000 : ℝ) ≤
+      (-(541095448505268199687 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (6731755819863 / 16000000000000 : ℝ) = (6731755819863 / 4000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6731755819863 / 4000000000000 : ℝ) ≤
+            (325338526869731800313 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (805362630610271 / 1250000000000000 : ℝ) ≤ Real.sqrt (6641755819863 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (6641755819863 / 16000000000000 : ℝ) ≤ (6442901044882171 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (6641755819863 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau580 :
+    Real.arctan (6442901044882171 / 21895838510762650 : ℝ) ≤
+      (715438444708057385753 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau579 :
+    Real.arctan (6442901044882171 / 10000000000000000 : ℝ) ≤
+      (2 * (715438444708057385753 / 2500000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (237916770215253 / 200000000000000 : ℝ) ≤
+          Real.sqrt (1 + (6442901044882171 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (6442901044882171 / 10000000000000000 : ℝ) /
+                (1 + (237916770215253 / 200000000000000 : ℝ)) =
+              (6442901044882171 / 21895838510762650 : ℝ)
+            by
+        norm_num];
+        exact hau580)
+  have hat1 :
+    (Real.pi / 2 - (2 * (715438444708057385753 / 2500000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 6442901044882171 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 6442901044882171 : ℝ) =
+              (6442901044882171 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau579)
+  have hz1 :
+    (10000000000000000 / 6442901044882171 : ℝ) ≤
+      1 / Real.sqrt (6641755819863 / 16000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 6442901044882171 : ℝ) =
+          1 / (6442901044882171 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (93750000000000 / 805362630610271 : ℝ) ≤
+      (115885625981794946927 / 1000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (6641755819863 / 16000000000000 : ℝ) ≤
+      (93750000000000 / 805362630610271 : ℝ) := by
+    rw [show
+        (93750000000000 / 805362630610271 : ℝ) =
+          (3 / 40) / (805362630610271 / 1250000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (715438444708057385753 / 2500000000000000000000 : ℝ))) -
+        6 * (115885625981794946927 / 1000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (6641755819863 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (6641755819863 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((805362630610271 / 1250000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (715438444708057385753 / 2500000000000000000000 : ℝ))) -
+          6 * (115885625981794946927 / 1000000000000000000000 : ℝ))) ≤
+      Real.sqrt (6641755819863 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (6641755819863 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (6641755819863 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_405 :
+    ((175448865186911473401 / 500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(8533932576410381810459 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((6483379216370309 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2876029562117140236423 / 10000000000000000000000 : ℝ))) -
+              6 * (115168509024532050863 / 1000000000000000000000 : ℝ)))) ≤
+      Vfield (3362736485061 / 8000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (3362736485061 / 8000000000000 : ℝ)) =
+      Real.log (11362736485061 / 8000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((3362736485061 / 8000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (3407736485061 / 8000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (175448865186911473401 / 500000000000000000000 : ℝ) ≤
+      Real.log (11362736485061 / 8000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (3407736485061 / 8000000000000 : ℝ) ≤
+      (-(8533932576410381810459 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (3407736485061 / 8000000000000 : ℝ) = (3407736485061 / 2000000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3407736485061 / 2000000000000 : ℝ) ≤
+            (5329011029589618189541 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (6483379216370309 / 10000000000000000 : ℝ) ≤ Real.sqrt (3362736485061 / 8000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (3362736485061 / 8000000000000 : ℝ) ≤ (810422402046289 / 1250000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (3362736485061 / 8000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau582 :
+    Real.arctan (1620844804092578 / 5479452613980277 : ℝ) ≤
+      (2876029562117140236423 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau581 :
+    Real.arctan (810422402046289 / 1250000000000000 : ℝ) ≤
+      (2 * (2876029562117140236423 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (2979452613980277 / 2500000000000000 : ℝ) ≤
+          Real.sqrt (1 + (810422402046289 / 1250000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (810422402046289 / 1250000000000000 : ℝ) /
+                (1 + (2979452613980277 / 2500000000000000 : ℝ)) =
+              (1620844804092578 / 5479452613980277 : ℝ)
+            by
+        norm_num];
+        exact hau582)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2876029562117140236423 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (1250000000000000 / 810422402046289 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1250000000000000 / 810422402046289 : ℝ) =
+              (810422402046289 / 1250000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau581)
+  have hz1 :
+    (1250000000000000 / 810422402046289 : ℝ) ≤
+      1 / Real.sqrt (3362736485061 / 8000000000000 : ℝ) := by
+    rw [show
+        (1250000000000000 / 810422402046289 : ℝ) = 1 / (810422402046289 / 1250000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 6483379216370309 : ℝ) ≤
+      (115168509024532050863 / 1000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (3362736485061 / 8000000000000 : ℝ) ≤
+      (750000000000000 / 6483379216370309 : ℝ) := by
+    rw [show
+        (750000000000000 / 6483379216370309 : ℝ) =
+          (3 / 40) / (6483379216370309 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2876029562117140236423 / 10000000000000000000000 : ℝ))) -
+        6 * (115168509024532050863 / 1000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (3362736485061 / 8000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (3362736485061 / 8000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((6483379216370309 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2876029562117140236423 / 10000000000000000000000 : ℝ))) -
+          6 * (115168509024532050863 / 1000000000000000000000 : ℝ))) ≤
+      Real.sqrt (3362736485061 / 8000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (3362736485061 / 8000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (3362736485061 / 8000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_406 :
+    ((886437019907079153121 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(8411846908633533086469 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((652360623063511 / 1000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1445082299841854199793 / 5000000000000000000000 : ℝ))) -
+              6 * (1144645424719571608529 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (6809190120381 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (6809190120381 / 16000000000000 : ℝ)) =
+      Real.log (22809190120381 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((6809190120381 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (6899190120381 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (886437019907079153121 / 2500000000000000000000 : ℝ) ≤
+      Real.log (22809190120381 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (6899190120381 / 16000000000000 : ℝ) ≤
+      (-(8411846908633533086469 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (6899190120381 / 16000000000000 : ℝ) = (6899190120381 / 4000000000000 : ℝ) / 2 ^ 2
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (6899190120381 / 4000000000000 : ℝ) ≤
+            (5451096697366466913531 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (652360623063511 / 1000000000000000 : ℝ) ≤ Real.sqrt (6809190120381 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (6809190120381 / 16000000000000 : ℝ) ≤ (6523606230635113 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (6809190120381 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau584 :
+    Real.arctan (6523606230635113 / 21939741967579585 : ℝ) ≤
+      (1445082299841854199793 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau583 :
+    Real.arctan (6523606230635113 / 10000000000000000 : ℝ) ≤
+      (2 * (1445082299841854199793 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (2387948393515917 / 2000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (6523606230635113 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (6523606230635113 / 10000000000000000 : ℝ) /
+                (1 + (2387948393515917 / 2000000000000000 : ℝ)) =
+              (6523606230635113 / 21939741967579585 : ℝ)
+            by
+        norm_num];
+        exact hau584)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1445082299841854199793 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 6523606230635113 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 6523606230635113 : ℝ) =
+              (6523606230635113 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau583)
+  have hz1 :
+    (10000000000000000 / 6523606230635113 : ℝ) ≤
+      1 / Real.sqrt (6809190120381 / 16000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 6523606230635113 : ℝ) =
+          1 / (6523606230635113 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (75000000000000 / 652360623063511 : ℝ) ≤
+      (1144645424719571608529 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (6809190120381 / 16000000000000 : ℝ) ≤
+      (75000000000000 / 652360623063511 : ℝ) := by
+    rw [show
+        (75000000000000 / 652360623063511 : ℝ) =
+          (3 / 40) / (652360623063511 / 1000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1445082299841854199793 / 5000000000000000000000 : ℝ))) -
+        6 * (1144645424719571608529 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (6809190120381 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (6809190120381 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((652360623063511 / 1000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1445082299841854199793 / 5000000000000000000000 : ℝ))) -
+          6 * (1144645424719571608529 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (6809190120381 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (6809190120381 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (6809190120381 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_407 :
+    ((3582384141724289505683 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(8291233772453230333 / 10000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3281793352783657 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (290416153952159878763 / 1000000000000000000000 : ℝ))) -
+              6 * (568866646261417588901 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (86161340883 / 200000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (86161340883 / 200000000000 : ℝ)) =
+      Real.log (286161340883 / 200000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((86161340883 / 200000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (87286340883 / 200000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (3582384141724289505683 / 10000000000000000000000 : ℝ) ≤
+      Real.log (286161340883 / 200000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (87286340883 / 200000000000 : ℝ) ≤
+      (-(8291233772453230333 / 10000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (87286340883 / 200000000000 : ℝ) = (87286340883 / 50000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (87286340883 / 50000000000 : ℝ) ≤
+            (5571709833546769667 / 10000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3281793352783657 / 5000000000000000 : ℝ) ≤ Real.sqrt (86161340883 / 200000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (86161340883 / 200000000000 : ℝ) ≤ (6563586705567317 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (86161340883 / 200000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau586 :
+    Real.arctan (6563586705567317 / 21961633268141102 : ℝ) ≤
+      (290416153952159878763 / 1000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau585 :
+    Real.arctan (6563586705567317 / 10000000000000000 : ℝ) ≤
+      (2 * (290416153952159878763 / 1000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (5980816634070551 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (6563586705567317 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (6563586705567317 / 10000000000000000 : ℝ) /
+                (1 + (5980816634070551 / 5000000000000000 : ℝ)) =
+              (6563586705567317 / 21961633268141102 : ℝ)
+            by
+        norm_num];
+        exact hau586)
+  have hat1 :
+    (Real.pi / 2 - (2 * (290416153952159878763 / 1000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 6563586705567317 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 6563586705567317 : ℝ) =
+              (6563586705567317 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau585)
+  have hz1 :
+    (10000000000000000 / 6563586705567317 : ℝ) ≤ 1 / Real.sqrt (86161340883 / 200000000000 : ℝ) :=
+  by
+    rw [show
+        (10000000000000000 / 6563586705567317 : ℝ) =
+          1 / (6563586705567317 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 3281793352783657 : ℝ) ≤
+      (568866646261417588901 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (86161340883 / 200000000000 : ℝ) ≤
+      (375000000000000 / 3281793352783657 : ℝ) := by
+    rw [show
+        (375000000000000 / 3281793352783657 : ℝ) =
+          (3 / 40) / (3281793352783657 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (290416153952159878763 / 1000000000000000000000 : ℝ))) -
+        6 * (568866646261417588901 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (86161340883 / 200000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (86161340883 / 200000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3281793352783657 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (290416153952159878763 / 1000000000000000000000 : ℝ))) -
+          6 * (568866646261417588901 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (86161340883 / 200000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (86161340883 / 200000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (86161340883 / 200000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_408 :
+    ((3600878263421158486701 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(8230729569040064947449 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1316746451171219 / 2000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2911194962262022984371 / 10000000000000000000000 : ℝ))) -
+              6 * (1134281773410074366533 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (54181913021 / 125000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (54181913021 / 125000000000 : ℝ)) =
+      Real.log (179181913021 / 125000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((54181913021 / 125000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (54885038021 / 125000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (3600878263421158486701 / 10000000000000000000000 : ℝ) ≤
+      Real.log (179181913021 / 125000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (54885038021 / 125000000000 : ℝ) ≤
+      (-(8230729569040064947449 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (54885038021 / 125000000000 : ℝ) = (54885038021 / 31250000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (54885038021 / 31250000000 : ℝ) ≤
+            (5632214036959935052551 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1316746451171219 / 2000000000000000 : ℝ) ≤ Real.sqrt (54181913021 / 125000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (54181913021 / 125000000000 : ℝ) ≤ (3291866127928049 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (54181913021 / 125000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau588 :
+    Real.arctan (940533179408014 / 3138957054124321 : ℝ) ≤
+      (2911194962262022984371 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau587 :
+    Real.arctan (3291866127928049 / 5000000000000000 : ℝ) ≤
+      (2 * (2911194962262022984371 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (11972699378870247 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (3291866127928049 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (3291866127928049 / 5000000000000000 : ℝ) /
+                (1 + (11972699378870247 / 10000000000000000 : ℝ)) =
+              (940533179408014 / 3138957054124321 : ℝ)
+            by
+        norm_num];
+        exact hau588)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2911194962262022984371 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 3291866127928049 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 3291866127928049 : ℝ) =
+              (3291866127928049 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau587)
+  have hz1 :
+    (5000000000000000 / 3291866127928049 : ℝ) ≤ 1 / Real.sqrt (54181913021 / 125000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 3291866127928049 : ℝ) = 1 / (3291866127928049 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (150000000000000 / 1316746451171219 : ℝ) ≤
+      (1134281773410074366533 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (54181913021 / 125000000000 : ℝ) ≤
+      (150000000000000 / 1316746451171219 : ℝ) := by
+    rw [show
+        (150000000000000 / 1316746451171219 : ℝ) =
+          (3 / 40) / (1316746451171219 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2911194962262022984371 / 10000000000000000000000 : ℝ))) -
+        6 * (1134281773410074366533 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (54181913021 / 125000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (54181913021 / 125000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1316746451171219 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2911194962262022984371 / 10000000000000000000000 : ℝ))) -
+          6 * (1134281773410074366533 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (54181913021 / 125000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (54181913021 / 125000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (54181913021 / 125000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/V34.lean b/LeanPool/Zeta5Irrational/Table/V34.lean
new file mode 100644
index 0000000000..6e18b124a7
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/V34.lean
@@ -0,0 +1,1416 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Energy
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # V34: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_409 :
+    ((1809669122496870992029 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(408529462049720624371 / 500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1320763270114671 / 2000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2918193984890556559269 / 10000000000000000000000 : ℝ))) -
+              6 * (1130861477778600441087 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (436103903921 / 1000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (436103903921 / 1000000000000 : ℝ)) =
+      Real.log (1436103903921 / 1000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((436103903921 / 1000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (441728903921 / 1000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1809669122496870992029 / 5000000000000000000000 : ℝ) ≤
+      Real.log (1436103903921 / 1000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (441728903921 / 1000000000000 : ℝ) ≤
+      (-(408529462049720624371 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (441728903921 / 1000000000000 : ℝ) = (441728903921 / 250000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (441728903921 / 250000000000 : ℝ) ≤
+            (284617718250279375629 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1320763270114671 / 2000000000000000 : ℝ) ≤ Real.sqrt (436103903921 / 1000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (436103903921 / 1000000000000 : ℝ) ≤ (3301908175286679 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (436103903921 / 1000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau590 :
+    Real.arctan (6603816350573358 / 21983755270869811 : ℝ) ≤
+      (2918193984890556559269 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau589 :
+    Real.arctan (3301908175286679 / 5000000000000000 : ℝ) ≤
+      (2 * (2918193984890556559269 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (11983755270869811 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (3301908175286679 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (3301908175286679 / 5000000000000000 : ℝ) /
+                (1 + (11983755270869811 / 10000000000000000 : ℝ)) =
+              (6603816350573358 / 21983755270869811 : ℝ)
+            by
+        norm_num];
+        exact hau590)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2918193984890556559269 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 3301908175286679 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 3301908175286679 : ℝ) =
+              (3301908175286679 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau589)
+  have hz1 :
+    (5000000000000000 / 3301908175286679 : ℝ) ≤
+      1 / Real.sqrt (436103903921 / 1000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 3301908175286679 : ℝ) = 1 / (3301908175286679 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (50000000000000 / 440254423371557 : ℝ) ≤
+      (1130861477778600441087 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (436103903921 / 1000000000000 : ℝ) ≤
+      (50000000000000 / 440254423371557 : ℝ) := by
+    rw [show
+        (50000000000000 / 440254423371557 : ℝ) =
+          (3 / 40) / (1320763270114671 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2918193984890556559269 / 10000000000000000000000 : ℝ))) -
+        6 * (1130861477778600441087 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (436103903921 / 1000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (436103903921 / 1000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1320763270114671 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2918193984890556559269 / 10000000000000000000000 : ℝ))) -
+          6 * (1130861477778600441087 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (436103903921 / 1000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (436103903921 / 1000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (436103903921 / 1000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_410 :
+    ((28420032908242764527 / 78125000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(8110808437737774119867 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((103497492949001 / 156250000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1462579464314678080479 / 5000000000000000000000 : ℝ))) -
+              6 * (563735968840652122499 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (219376251837 / 500000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (219376251837 / 500000000000 : ℝ)) =
+      Real.log (719376251837 / 500000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((219376251837 / 500000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (222188751837 / 500000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (28420032908242764527 / 78125000000000000000 : ℝ) ≤
+      Real.log (719376251837 / 500000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (222188751837 / 500000000000 : ℝ) ≤
+      (-(8110808437737774119867 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (222188751837 / 500000000000 : ℝ) = (222188751837 / 125000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (222188751837 / 125000000000 : ℝ) ≤
+            (5752135168262225880133 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (103497492949001 / 156250000000000 : ℝ) ≤ Real.sqrt (219376251837 / 500000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (219376251837 / 500000000000 : ℝ) ≤ (6623839548736067 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (219376251837 / 500000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau592 :
+    Real.arctan (6623839548736067 / 21994800972396332 : ℝ) ≤
+      (1462579464314678080479 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau591 :
+    Real.arctan (6623839548736067 / 10000000000000000 : ℝ) ≤
+      (2 * (1462579464314678080479 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (2998700243099083 / 2500000000000000 : ℝ) ≤
+          Real.sqrt (1 + (6623839548736067 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (6623839548736067 / 10000000000000000 : ℝ) /
+                (1 + (2998700243099083 / 2500000000000000 : ℝ)) =
+              (6623839548736067 / 21994800972396332 : ℝ)
+            by
+        norm_num];
+        exact hau592)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1462579464314678080479 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 6623839548736067 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 6623839548736067 : ℝ) =
+              (6623839548736067 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau591)
+  have hz1 :
+    (10000000000000000 / 6623839548736067 : ℝ) ≤
+      1 / Real.sqrt (219376251837 / 500000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 6623839548736067 : ℝ) =
+          1 / (6623839548736067 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (11718750000000 / 103497492949001 : ℝ) ≤
+      (563735968840652122499 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (219376251837 / 500000000000 : ℝ) ≤
+      (11718750000000 / 103497492949001 : ℝ) := by
+    rw [show
+        (11718750000000 / 103497492949001 : ℝ) =
+          (3 / 40) / (103497492949001 / 156250000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1462579464314678080479 / 5000000000000000000000 : ℝ))) -
+        6 * (563735968840652122499 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (219376251837 / 500000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (219376251837 / 500000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((103497492949001 / 156250000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1462579464314678080479 / 5000000000000000000000 : ℝ))) -
+          6 * (563735968840652122499 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (219376251837 / 500000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (219376251837 / 500000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (219376251837 / 500000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_411 :
+    ((91174111991140909019 / 250000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(8081051519608223970771 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((6633828483993989 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (292862872001000197323 / 1000000000000000000000 : ℝ))) -
+              6 * (1125788557400433642409 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (880153607101 / 2000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (880153607101 / 2000000000000 : ℝ)) =
+      Real.log (2880153607101 / 2000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((880153607101 / 2000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (891403607101 / 2000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (91174111991140909019 / 250000000000000000000 : ℝ) ≤
+      Real.log (2880153607101 / 2000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (891403607101 / 2000000000000 : ℝ) ≤
+      (-(8081051519608223970771 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (891403607101 / 2000000000000 : ℝ) = (891403607101 / 500000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (891403607101 / 500000000000 : ℝ) ≤
+            (5781892086391776029229 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (6633828483993989 / 10000000000000000 : ℝ) ≤ Real.sqrt (880153607101 / 2000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (880153607101 / 2000000000000 : ℝ) ≤ (829228560499249 / 1250000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (880153607101 / 2000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau594 :
+    Real.arctan (3316914241996996 / 11000160005263401 : ℝ) ≤
+      (292862872001000197323 / 1000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau593 :
+    Real.arctan (829228560499249 / 1250000000000000 : ℝ) ≤
+      (2 * (292862872001000197323 / 1000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6000160005263401 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (829228560499249 / 1250000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (829228560499249 / 1250000000000000 : ℝ) /
+                (1 + (6000160005263401 / 5000000000000000 : ℝ)) =
+              (3316914241996996 / 11000160005263401 : ℝ)
+            by
+        norm_num];
+        exact hau594)
+  have hat1 :
+    (Real.pi / 2 - (2 * (292862872001000197323 / 1000000000000000000000 : ℝ))) ≤
+      Real.arctan (1250000000000000 / 829228560499249 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1250000000000000 / 829228560499249 : ℝ) =
+              (829228560499249 / 1250000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau593)
+  have hz1 :
+    (1250000000000000 / 829228560499249 : ℝ) ≤ 1 / Real.sqrt (880153607101 / 2000000000000 : ℝ) :=
+  by
+    rw [show
+        (1250000000000000 / 829228560499249 : ℝ) = 1 / (829228560499249 / 1250000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 6633828483993989 : ℝ) ≤
+      (1125788557400433642409 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (880153607101 / 2000000000000 : ℝ) ≤
+      (750000000000000 / 6633828483993989 : ℝ) := by
+    rw [show
+        (750000000000000 / 6633828483993989 : ℝ) =
+          (3 / 40) / (6633828483993989 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (292862872001000197323 / 1000000000000000000000 : ℝ))) -
+        6 * (1125788557400433642409 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (880153607101 / 2000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (880153607101 / 2000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((6633828483993989 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (292862872001000197323 / 1000000000000000000000 : ℝ))) -
+          6 * (1125788557400433642409 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (880153607101 / 2000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (880153607101 / 2000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (880153607101 / 2000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_412 :
+    ((3656156290323998442747 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(503211430390812632561 / 625000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((6643802400937281 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (586418022000673760631 / 2000000000000000000000 : ℝ))) -
+              6 * (224822538986222456981 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (441401103427 / 1000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (441401103427 / 1000000000000 : ℝ)) =
+      Real.log (1441401103427 / 1000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((441401103427 / 1000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (447026103427 / 1000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (3656156290323998442747 / 10000000000000000000000 : ℝ) ≤
+      Real.log (1441401103427 / 1000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (447026103427 / 1000000000000 : ℝ) ≤
+      (-(503211430390812632561 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (447026103427 / 1000000000000 : ℝ) = (447026103427 / 250000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (447026103427 / 250000000000 : ℝ) ≤
+            (363222544984187367439 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (6643802400937281 / 10000000000000000 : ℝ) ≤ Real.sqrt (441401103427 / 1000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (441401103427 / 1000000000000 : ℝ) ≤ (1660950600234321 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (441401103427 / 1000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau596 :
+    Real.arctan (3321901200468642 / 11002918255788179 : ℝ) ≤
+      (586418022000673760631 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau595 :
+    Real.arctan (1660950600234321 / 2500000000000000 : ℝ) ≤
+      (2 * (586418022000673760631 / 2000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6002918255788179 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1660950600234321 / 2500000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1660950600234321 / 2500000000000000 : ℝ) /
+                (1 + (6002918255788179 / 5000000000000000 : ℝ)) =
+              (3321901200468642 / 11002918255788179 : ℝ)
+            by
+        norm_num];
+        exact hau596)
+  have hat1 :
+    (Real.pi / 2 - (2 * (586418022000673760631 / 2000000000000000000000 : ℝ))) ≤
+      Real.arctan (2500000000000000 / 1660950600234321 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 1660950600234321 : ℝ) =
+              (1660950600234321 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau595)
+  have hz1 :
+    (2500000000000000 / 1660950600234321 : ℝ) ≤
+      1 / Real.sqrt (441401103427 / 1000000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 1660950600234321 : ℝ) = 1 / (1660950600234321 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 2214600800312427 : ℝ) ≤
+      (224822538986222456981 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (441401103427 / 1000000000000 : ℝ) ≤
+      (250000000000000 / 2214600800312427 : ℝ) := by
+    rw [show
+        (250000000000000 / 2214600800312427 : ℝ) =
+          (3 / 40) / (6643802400937281 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (586418022000673760631 / 2000000000000000000000 : ℝ))) -
+        6 * (224822538986222456981 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (441401103427 / 1000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (441401103427 / 1000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((6643802400937281 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (586418022000673760631 / 2000000000000000000000 : ℝ))) -
+          6 * (224822538986222456981 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (441401103427 / 1000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (441401103427 / 1000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (441401103427 / 1000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_413 :
+    ((733067931964476973299 / 2000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(8021802015363241262467 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((6653761367102819 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2935543137456839238447 / 10000000000000000000000 : ℝ))) -
+              6 * (1122444294482276282733 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (885450806607 / 2000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (885450806607 / 2000000000000 : ℝ)) =
+      Real.log (2885450806607 / 2000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((885450806607 / 2000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (896700806607 / 2000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (733067931964476973299 / 2000000000000000000000 : ℝ) ≤
+      Real.log (2885450806607 / 2000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (896700806607 / 2000000000000 : ℝ) ≤
+      (-(8021802015363241262467 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (896700806607 / 2000000000000 : ℝ) = (896700806607 / 500000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (896700806607 / 500000000000 : ℝ) ≤
+            (5841141590636758737533 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (6653761367102819 / 10000000000000000 : ℝ) ≤ Real.sqrt (885450806607 / 2000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (885450806607 / 2000000000000 : ℝ) ≤ (3326880683551411 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (885450806607 / 2000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau598 :
+    Real.arctan (6653761367102822 / 22011350479040647 : ℝ) ≤
+      (2935543137456839238447 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau597 :
+    Real.arctan (3326880683551411 / 5000000000000000 : ℝ) ≤
+      (2 * (2935543137456839238447 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12011350479040647 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (3326880683551411 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (3326880683551411 / 5000000000000000 : ℝ) /
+                (1 + (12011350479040647 / 10000000000000000 : ℝ)) =
+              (6653761367102822 / 22011350479040647 : ℝ)
+            by
+        norm_num];
+        exact hau598)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2935543137456839238447 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 3326880683551411 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 3326880683551411 : ℝ) =
+              (3326880683551411 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau597)
+  have hz1 :
+    (5000000000000000 / 3326880683551411 : ℝ) ≤
+      1 / Real.sqrt (885450806607 / 2000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 3326880683551411 : ℝ) = 1 / (3326880683551411 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 6653761367102819 : ℝ) ≤
+      (1122444294482276282733 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (885450806607 / 2000000000000 : ℝ) ≤
+      (750000000000000 / 6653761367102819 : ℝ) := by
+    rw [show
+        (750000000000000 / 6653761367102819 : ℝ) =
+          (3 / 40) / (6653761367102819 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2935543137456839238447 / 10000000000000000000000 : ℝ))) -
+        6 * (1122444294482276282733 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (885450806607 / 2000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (885450806607 / 2000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((6653761367102819 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2935543137456839238447 / 10000000000000000000000 : ℝ))) -
+          6 * (1122444294482276282733 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (885450806607 / 2000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (885450806607 / 2000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (885450806607 / 2000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_414 :
+    ((146980584145210727889 / 400000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(7992308389251518931253 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((6663705449522809 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (587797568187209344421 / 2000000000000000000000 : ℝ))) -
+              6 * (560391650420402333093 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (22202485159 / 50000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (22202485159 / 50000000000 : ℝ)) = Real.log (72202485159 / 50000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((22202485159 / 50000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (22483735159 / 50000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (146980584145210727889 / 400000000000000000000 : ℝ) ≤
+      Real.log (72202485159 / 50000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (22483735159 / 50000000000 : ℝ) ≤
+      (-(7992308389251518931253 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (22483735159 / 50000000000 : ℝ) = (22483735159 / 12500000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22483735159 / 12500000000 : ℝ) ≤
+            (5870635216748481068747 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (6663705449522809 / 10000000000000000 : ℝ) ≤ Real.sqrt (22202485159 / 50000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (22202485159 / 50000000000 : ℝ) ≤ (1665926362380703 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (22202485159 / 50000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau600 :
+    Real.arctan (3331852724761406 / 11008430958203647 : ℝ) ≤
+      (587797568187209344421 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau599 :
+    Real.arctan (1665926362380703 / 2500000000000000 : ℝ) ≤
+      (2 * (587797568187209344421 / 2000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6008430958203647 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1665926362380703 / 2500000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1665926362380703 / 2500000000000000 : ℝ) /
+                (1 + (6008430958203647 / 5000000000000000 : ℝ)) =
+              (3331852724761406 / 11008430958203647 : ℝ)
+            by
+        norm_num];
+        exact hau600)
+  have hat1 :
+    (Real.pi / 2 - (2 * (587797568187209344421 / 2000000000000000000000 : ℝ))) ≤
+      Real.arctan (2500000000000000 / 1665926362380703 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 1665926362380703 : ℝ) =
+              (1665926362380703 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau599)
+  have hz1 :
+    (2500000000000000 / 1665926362380703 : ℝ) ≤ 1 / Real.sqrt (22202485159 / 50000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 1665926362380703 : ℝ) = 1 / (1665926362380703 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 6663705449522809 : ℝ) ≤
+      (560391650420402333093 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (22202485159 / 50000000000 : ℝ) ≤
+      (750000000000000 / 6663705449522809 : ℝ) := by
+    rw [show
+        (750000000000000 / 6663705449522809 : ℝ) =
+          (3 / 40) / (6663705449522809 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (587797568187209344421 / 2000000000000000000000 : ℝ))) -
+        6 * (560391650420402333093 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (22202485159 / 50000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (22202485159 / 50000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((6663705449522809 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (587797568187209344421 / 2000000000000000000000 : ℝ))) -
+          6 * (560391650420402333093 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (22202485159 / 50000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (22202485159 / 50000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (22202485159 / 50000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_415 :
+    ((1841840568597262780059 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1592580298959499282641 / 2000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3336817357365023 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2942424258727729199681 / 10000000000000000000000 : ℝ))) -
+              6 * (559564829681922406319 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (890748006113 / 2000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (890748006113 / 2000000000000 : ℝ)) =
+      Real.log (2890748006113 / 2000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((890748006113 / 2000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (901998006113 / 2000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1841840568597262780059 / 5000000000000000000000 : ℝ) ≤
+      Real.log (2890748006113 / 2000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (901998006113 / 2000000000000 : ℝ) ≤
+      (-(1592580298959499282641 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (901998006113 / 2000000000000 : ℝ) = (901998006113 / 500000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (901998006113 / 500000000000 : ℝ) ≤
+            (1180008422240500717359 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3336817357365023 / 5000000000000000 : ℝ) ≤ Real.sqrt (890748006113 / 2000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (890748006113 / 2000000000000 : ℝ) ≤ (6673634714730049 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (890748006113 / 2000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau602 :
+    Real.arctan (6673634714730049 / 22022370827155931 : ℝ) ≤
+      (2942424258727729199681 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau601 :
+    Real.arctan (6673634714730049 / 10000000000000000 : ℝ) ≤
+      (2 * (2942424258727729199681 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12022370827155931 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (6673634714730049 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (6673634714730049 / 10000000000000000 : ℝ) /
+                (1 + (12022370827155931 / 10000000000000000 : ℝ)) =
+              (6673634714730049 / 22022370827155931 : ℝ)
+            by
+        norm_num];
+        exact hau602)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2942424258727729199681 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 6673634714730049 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 6673634714730049 : ℝ) =
+              (6673634714730049 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau601)
+  have hz1 :
+    (10000000000000000 / 6673634714730049 : ℝ) ≤
+      1 / Real.sqrt (890748006113 / 2000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 6673634714730049 : ℝ) =
+          1 / (6673634714730049 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 3336817357365023 : ℝ) ≤
+      (559564829681922406319 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (890748006113 / 2000000000000 : ℝ) ≤
+      (375000000000000 / 3336817357365023 : ℝ) := by
+    rw [show
+        (375000000000000 / 3336817357365023 : ℝ) =
+          (3 / 40) / (3336817357365023 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2942424258727729199681 / 10000000000000000000000 : ℝ))) -
+        6 * (559564829681922406319 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (890748006113 / 2000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (890748006113 / 2000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3336817357365023 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2942424258727729199681 / 10000000000000000000000 : ℝ))) -
+          6 * (559564829681922406319 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (890748006113 / 2000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (890748006113 / 2000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (890748006113 / 2000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_416 :
+    ((923209818979901109979 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1983395205848588651167 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((6683549228763111 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2945852428842548212577 / 10000000000000000000000 : ℝ))) -
+              6 * (558741657985630589877 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (446698302933 / 1000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (446698302933 / 1000000000000 : ℝ)) =
+      Real.log (1446698302933 / 1000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((446698302933 / 1000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (452323302933 / 1000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (923209818979901109979 / 2500000000000000000000 : ℝ) ≤
+      Real.log (1446698302933 / 1000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (452323302933 / 1000000000000 : ℝ) ≤
+      (-(1983395205848588651167 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (452323302933 / 1000000000000 : ℝ) = (452323302933 / 250000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (452323302933 / 250000000000 : ℝ) ≤
+            (1482340695651411348833 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (6683549228763111 / 10000000000000000 : ℝ) ≤ Real.sqrt (446698302933 / 1000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (446698302933 / 1000000000000 : ℝ) ≤ (3341774614381557 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (446698302933 / 1000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau604 :
+    Real.arctan (6683549228763114 / 22027877214758221 : ℝ) ≤
+      (2945852428842548212577 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau603 :
+    Real.arctan (3341774614381557 / 5000000000000000 : ℝ) ≤
+      (2 * (2945852428842548212577 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12027877214758221 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (3341774614381557 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (3341774614381557 / 5000000000000000 : ℝ) /
+                (1 + (12027877214758221 / 10000000000000000 : ℝ)) =
+              (6683549228763114 / 22027877214758221 : ℝ)
+            by
+        norm_num];
+        exact hau604)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2945852428842548212577 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 3341774614381557 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 3341774614381557 : ℝ) =
+              (3341774614381557 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau603)
+  have hz1 :
+    (5000000000000000 / 3341774614381557 : ℝ) ≤
+      1 / Real.sqrt (446698302933 / 1000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 3341774614381557 : ℝ) = 1 / (3341774614381557 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 2227849742921037 : ℝ) ≤
+      (558741657985630589877 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (446698302933 / 1000000000000 : ℝ) ≤
+      (250000000000000 / 2227849742921037 : ℝ) := by
+    rw [show
+        (250000000000000 / 2227849742921037 : ℝ) =
+          (3 / 40) / (6683549228763111 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2945852428842548212577 / 10000000000000000000000 : ℝ))) -
+        6 * (558741657985630589877 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (446698302933 / 1000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (446698302933 / 1000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((6683549228763111 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2945852428842548212577 / 10000000000000000000000 : ℝ))) -
+          6 * (558741657985630589877 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (446698302933 / 1000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (446698302933 / 1000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (446698302933 / 1000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_417 :
+    ((3701989035167639417921 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(3952172935448006424207 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1338689811434299 / 2000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2949272389017869224799 / 10000000000000000000000 : ℝ))) -
+              6 * (1115844217138206506899 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (896045205619 / 2000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (896045205619 / 2000000000000 : ℝ)) =
+      Real.log (2896045205619 / 2000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((896045205619 / 2000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (907295205619 / 2000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (3701989035167639417921 / 10000000000000000000000 : ℝ) ≤
+      Real.log (2896045205619 / 2000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (907295205619 / 2000000000000 : ℝ) ≤
+      (-(3952172935448006424207 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (907295205619 / 2000000000000 : ℝ) = (907295205619 / 500000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (907295205619 / 500000000000 : ℝ) ≤
+            (2979298867551993575793 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1338689811434299 / 2000000000000000 : ℝ) ≤ Real.sqrt (896045205619 / 2000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (896045205619 / 2000000000000 : ℝ) ≤ (3346724528585749 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (896045205619 / 2000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau606 :
+    Real.arctan (6693449057171498 / 22033381082677885 : ℝ) ≤
+      (2949272389017869224799 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau605 :
+    Real.arctan (3346724528585749 / 5000000000000000 : ℝ) ≤
+      (2 * (2949272389017869224799 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (2406676216535577 / 2000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (3346724528585749 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (3346724528585749 / 5000000000000000 : ℝ) /
+                (1 + (2406676216535577 / 2000000000000000 : ℝ)) =
+              (6693449057171498 / 22033381082677885 : ℝ)
+            by
+        norm_num];
+        exact hau606)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2949272389017869224799 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 3346724528585749 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 3346724528585749 : ℝ) =
+              (3346724528585749 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau605)
+  have hz1 :
+    (5000000000000000 / 3346724528585749 : ℝ) ≤
+      1 / Real.sqrt (896045205619 / 2000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 3346724528585749 : ℝ) = 1 / (3346724528585749 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (150000000000000 / 1338689811434299 : ℝ) ≤
+      (1115844217138206506899 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (896045205619 / 2000000000000 : ℝ) ≤
+      (150000000000000 / 1338689811434299 : ℝ) := by
+    rw [show
+        (150000000000000 / 1338689811434299 : ℝ) =
+          (3 / 40) / (1338689811434299 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2949272389017869224799 / 10000000000000000000000 : ℝ))) -
+        6 * (1115844217138206506899 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (896045205619 / 2000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (896045205619 / 2000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1338689811434299 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2949272389017869224799 / 10000000000000000000000 : ℝ))) -
+          6 * (1115844217138206506899 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (896045205619 / 2000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (896045205619 / 2000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (896045205619 / 2000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_418 :
+    ((926640194319229491321 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1972440095724352398719 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((6698393484618157 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2950979302097091334933 / 10000000000000000000000 : ℝ))) -
+              6 * (278756841963755830087 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (1794739010991 / 4000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1794739010991 / 4000000000000 : ℝ)) =
+      Real.log (5794739010991 / 4000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1794739010991 / 4000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1817239010991 / 4000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (926640194319229491321 / 2500000000000000000000 : ℝ) ≤
+      Real.log (5794739010991 / 4000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1817239010991 / 4000000000000 : ℝ) ≤
+      (-(1972440095724352398719 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (1817239010991 / 4000000000000 : ℝ) = (1817239010991 / 1000000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1817239010991 / 1000000000000 : ℝ) ≤
+            (1493295805775647601281 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (6698393484618157 / 10000000000000000 : ℝ) ≤ Real.sqrt (1794739010991 / 4000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1794739010991 / 4000000000000 : ℝ) ≤ (83729918557727 / 125000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1794739010991 / 4000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau608 :
+    Real.arctan (3349196742309080 / 11018066036418489 : ℝ) ≤
+      (2950979302097091334933 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau607 :
+    Real.arctan (83729918557727 / 125000000000000 : ℝ) ≤
+      (2 * (2950979302097091334933 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6018066036418489 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (83729918557727 / 125000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (83729918557727 / 125000000000000 : ℝ) /
+                (1 + (6018066036418489 / 5000000000000000 : ℝ)) =
+              (3349196742309080 / 11018066036418489 : ℝ)
+            by
+        norm_num];
+        exact hau608)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2950979302097091334933 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (125000000000000 / 83729918557727 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (125000000000000 / 83729918557727 : ℝ) =
+              (83729918557727 / 125000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau607)
+  have hz1 :
+    (125000000000000 / 83729918557727 : ℝ) ≤ 1 / Real.sqrt (1794739010991 / 4000000000000 : ℝ) := by
+    rw [show (125000000000000 / 83729918557727 : ℝ) = 1 / (83729918557727 / 125000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 6698393484618157 : ℝ) ≤
+      (278756841963755830087 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1794739010991 / 4000000000000 : ℝ) ≤
+      (750000000000000 / 6698393484618157 : ℝ) := by
+    rw [show
+        (750000000000000 / 6698393484618157 : ℝ) =
+          (3 / 40) / (6698393484618157 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2950979302097091334933 / 10000000000000000000000 : ℝ))) -
+        6 * (278756841963755830087 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1794739010991 / 4000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1794739010991 / 4000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((6698393484618157 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2950979302097091334933 / 10000000000000000000000 : ℝ))) -
+          6 * (278756841963755830087 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (1794739010991 / 4000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1794739010991 / 4000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1794739010991 / 4000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_419 :
+    ((3711130430258663998133 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1968799034391279262693 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((6703334265020653 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (738171044180126359817 / 2500000000000000000000 : ℝ))) -
+              6 * (222842461977562569229 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (224673451343 / 500000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (224673451343 / 500000000000 : ℝ)) =
+      Real.log (724673451343 / 500000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((224673451343 / 500000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (227485951343 / 500000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (3711130430258663998133 / 10000000000000000000000 : ℝ) ≤
+      Real.log (724673451343 / 500000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (227485951343 / 500000000000 : ℝ) ≤
+      (-(1968799034391279262693 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (227485951343 / 500000000000 : ℝ) = (227485951343 / 125000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (227485951343 / 125000000000 : ℝ) ≤
+            (1496936867108720737307 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (6703334265020653 / 10000000000000000 : ℝ) ≤ Real.sqrt (224673451343 / 500000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (224673451343 / 500000000000 : ℝ) ≤ (418958391563791 / 625000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (224673451343 / 500000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau610 :
+    Real.arctan (6703334265020656 / 22038882434370725 : ℝ) ≤
+      (738171044180126359817 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau609 :
+    Real.arctan (418958391563791 / 625000000000000 : ℝ) ≤
+      (2 * (738171044180126359817 / 2500000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (481555297374829 / 400000000000000 : ℝ) ≤
+          Real.sqrt (1 + (418958391563791 / 625000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (418958391563791 / 625000000000000 : ℝ) /
+                (1 + (481555297374829 / 400000000000000 : ℝ)) =
+              (6703334265020656 / 22038882434370725 : ℝ)
+            by
+        norm_num];
+        exact hau610)
+  have hat1 :
+    (Real.pi / 2 - (2 * (738171044180126359817 / 2500000000000000000000 : ℝ))) ≤
+      Real.arctan (625000000000000 / 418958391563791 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (625000000000000 / 418958391563791 : ℝ) =
+              (418958391563791 / 625000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau609)
+  have hz1 :
+    (625000000000000 / 418958391563791 : ℝ) ≤ 1 / Real.sqrt (224673451343 / 500000000000 : ℝ) := by
+    rw [show (625000000000000 / 418958391563791 : ℝ) = 1 / (418958391563791 / 625000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 6703334265020653 : ℝ) ≤
+      (222842461977562569229 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (224673451343 / 500000000000 : ℝ) ≤
+      (750000000000000 / 6703334265020653 : ℝ) := by
+    rw [show
+        (750000000000000 / 6703334265020653 : ℝ) =
+          (3 / 40) / (6703334265020653 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (738171044180126359817 / 2500000000000000000000 : ℝ))) -
+        6 * (222842461977562569229 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (224673451343 / 500000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (224673451343 / 500000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((6703334265020653 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (738171044180126359817 / 2500000000000000000000 : ℝ))) -
+          6 * (222842461977562569229 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (224673451343 / 500000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (224673451343 / 500000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (224673451343 / 500000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_420 :
+    ((1857848998010661530569 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(786065307311242032783 / 1000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((6708271406437353 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2 * (5908774035059268563 / 20000000000000000000 : ℝ))) -
+              6 * (278349759174727094639 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (1800036210497 / 4000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1800036210497 / 4000000000000 : ℝ)) =
+      Real.log (5800036210497 / 4000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1800036210497 / 4000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1822536210497 / 4000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1857848998010661530569 / 5000000000000000000000 : ℝ) ≤
+      Real.log (5800036210497 / 4000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1822536210497 / 4000000000000 : ℝ) ≤
+      (-(786065307311242032783 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (1822536210497 / 4000000000000 : ℝ) = (1822536210497 / 1000000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1822536210497 / 1000000000000 : ℝ) ≤
+            (600229053288757967217 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (6708271406437353 / 10000000000000000 : ℝ) ≤ Real.sqrt (1800036210497 / 4000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1800036210497 / 4000000000000 : ℝ) ≤ (1677067851609339 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1800036210497 / 4000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau612 :
+    Real.arctan (2236090468812452 / 7347210722569955 : ℝ) ≤
+      (5908774035059268563 / 20000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau611 :
+    Real.arctan (1677067851609339 / 2500000000000000 : ℝ) ≤
+      (2 * (5908774035059268563 / 20000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (2408326433541973 / 2000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1677067851609339 / 2500000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1677067851609339 / 2500000000000000 : ℝ) /
+                (1 + (2408326433541973 / 2000000000000000 : ℝ)) =
+              (2236090468812452 / 7347210722569955 : ℝ)
+            by
+        norm_num];
+        exact hau612)
+  have hat1 :
+    (Real.pi / 2 - (2 * (5908774035059268563 / 20000000000000000000 : ℝ))) ≤
+      Real.arctan (2500000000000000 / 1677067851609339 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 1677067851609339 : ℝ) =
+              (1677067851609339 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau611)
+  have hz1 :
+    (2500000000000000 / 1677067851609339 : ℝ) ≤
+      1 / Real.sqrt (1800036210497 / 4000000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 1677067851609339 : ℝ) = 1 / (1677067851609339 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 2236090468812451 : ℝ) ≤
+      (278349759174727094639 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1800036210497 / 4000000000000 : ℝ) ≤
+      (250000000000000 / 2236090468812451 : ℝ) := by
+    rw [show
+        (250000000000000 / 2236090468812451 : ℝ) =
+          (3 / 40) / (6708271406437353 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (5908774035059268563 / 20000000000000000000 : ℝ))) -
+        6 * (278349759174727094639 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1800036210497 / 4000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1800036210497 / 4000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((6708271406437353 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (5908774035059268563 / 20000000000000000000 : ℝ))) -
+          6 * (278349759174727094639 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (1800036210497 / 4000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1800036210497 / 4000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1800036210497 / 4000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/V35.lean b/LeanPool/Zeta5Irrational/Table/V35.lean
new file mode 100644
index 0000000000..4fa073abe7
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/V35.lean
@@ -0,0 +1,1421 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Energy
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # V35: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_421 :
+    ((1860131738235363403791 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1569226225604356732877 / 2000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3356602458448497 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1478043914574713354633 / 5000000000000000000000 : ℝ))) -
+              6 * (222517508356799869161 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (7210739241 / 16000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (7210739241 / 16000000000 : ℝ)) = Real.log (23210739241 / 16000000000 : ℝ) := by
+    norm_num
+  have e2 :
+    Real.log ((7210739241 / 16000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (7300739241 / 16000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1860131738235363403791 / 5000000000000000000000 : ℝ) ≤
+      Real.log (23210739241 / 16000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (7300739241 / 16000000000 : ℝ) ≤
+      (-(1569226225604356732877 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (7300739241 / 16000000000 : ℝ) = (7300739241 / 4000000000 : ℝ) / 2 ^ 2 by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7300739241 / 4000000000 : ℝ) ≤
+            (1203362495595643267123 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3356602458448497 / 5000000000000000 : ℝ) ≤ Real.sqrt (7210739241 / 16000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (7210739241 / 16000000000 : ℝ) ≤ (6713204916896997 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (7210739241 / 16000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau614 :
+    Real.arctan (6713204916896997 / 22044381273284651 : ℝ) ≤
+      (1478043914574713354633 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau613 :
+    Real.arctan (6713204916896997 / 10000000000000000 : ℝ) ≤
+      (2 * (1478043914574713354633 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12044381273284651 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (6713204916896997 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (6713204916896997 / 10000000000000000 : ℝ) /
+                (1 + (12044381273284651 / 10000000000000000 : ℝ)) =
+              (6713204916896997 / 22044381273284651 : ℝ)
+            by
+        norm_num];
+        exact hau614)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1478043914574713354633 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 6713204916896997 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 6713204916896997 : ℝ) =
+              (6713204916896997 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau613)
+  have hz1 :
+    (10000000000000000 / 6713204916896997 : ℝ) ≤ 1 / Real.sqrt (7210739241 / 16000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 6713204916896997 : ℝ) =
+          1 / (6713204916896997 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (125000000000000 / 1118867486149499 : ℝ) ≤
+      (222517508356799869161 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (7210739241 / 16000000000 : ℝ) ≤
+      (125000000000000 / 1118867486149499 : ℝ) := by
+    rw [show
+        (125000000000000 / 1118867486149499 : ℝ) =
+          (3 / 40) / (3356602458448497 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1478043914574713354633 / 5000000000000000000000 : ℝ))) -
+        6 * (222517508356799869161 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (7210739241 / 16000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (7210739241 / 16000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3356602458448497 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1478043914574713354633 / 5000000000000000000000 : ℝ))) -
+          6 * (222517508356799869161 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (7210739241 / 16000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (7210739241 / 16000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (7210739241 / 16000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_422 :
+    ((116400839797190555333 / 312500000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1957907560260822272463 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1679533701099709 / 2500000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2957786616188343050519 / 10000000000000000000000 : ℝ))) -
+              6 * (555888909335956105031 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (1805333410003 / 4000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1805333410003 / 4000000000000 : ℝ)) =
+      Real.log (5805333410003 / 4000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1805333410003 / 4000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1827833410003 / 4000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (116400839797190555333 / 312500000000000000000 : ℝ) ≤
+      Real.log (5805333410003 / 4000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1827833410003 / 4000000000000 : ℝ) ≤
+      (-(1957907560260822272463 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (1827833410003 / 4000000000000 : ℝ) = (1827833410003 / 1000000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1827833410003 / 1000000000000 : ℝ) ≤
+            (1507828341239177727537 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1679533701099709 / 2500000000000000 : ℝ) ≤ Real.sqrt (1805333410003 / 4000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1805333410003 / 4000000000000 : ℝ) ≤ (6718134804398839 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1805333410003 / 4000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau616 :
+    Real.arctan (6718134804398839 / 22047129751524842 : ℝ) ≤
+      (2957786616188343050519 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau615 :
+    Real.arctan (6718134804398839 / 10000000000000000 : ℝ) ≤
+      (2 * (2957786616188343050519 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6023564875762421 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (6718134804398839 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (6718134804398839 / 10000000000000000 : ℝ) /
+                (1 + (6023564875762421 / 5000000000000000 : ℝ)) =
+              (6718134804398839 / 22047129751524842 : ℝ)
+            by
+        norm_num];
+        exact hau616)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2957786616188343050519 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 6718134804398839 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 6718134804398839 : ℝ) =
+              (6718134804398839 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau615)
+  have hz1 :
+    (10000000000000000 / 6718134804398839 : ℝ) ≤
+      1 / Real.sqrt (1805333410003 / 4000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 6718134804398839 : ℝ) =
+          1 / (6718134804398839 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (187500000000000 / 1679533701099709 : ℝ) ≤
+      (555888909335956105031 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1805333410003 / 4000000000000 : ℝ) ≤
+      (187500000000000 / 1679533701099709 : ℝ) := by
+    rw [show
+        (187500000000000 / 1679533701099709 : ℝ) =
+          (3 / 40) / (1679533701099709 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2957786616188343050519 / 10000000000000000000000 : ℝ))) -
+        6 * (555888909335956105031 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1805333410003 / 4000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1805333410003 / 4000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1679533701099709 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2957786616188343050519 / 10000000000000000000000 : ℝ))) -
+          6 * (555888909335956105031 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (1805333410003 / 4000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1805333410003 / 4000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1805333410003 / 4000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_423 :
+    ((3729388189040054123869 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(7817150351193086393177 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3361530538456403 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2959483383238434001347 / 10000000000000000000000 : ℝ))) -
+              6 * (1110969860924394921403 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (451995502439 / 1000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (451995502439 / 1000000000000 : ℝ)) =
+      Real.log (1451995502439 / 1000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((451995502439 / 1000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (457620502439 / 1000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (3729388189040054123869 / 10000000000000000000000 : ℝ) ≤
+      Real.log (1451995502439 / 1000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (457620502439 / 1000000000000 : ℝ) ≤
+      (-(7817150351193086393177 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (457620502439 / 1000000000000 : ℝ) = (457620502439 / 250000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (457620502439 / 250000000000 : ℝ) ≤
+            (6045793254806913606823 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3361530538456403 / 5000000000000000 : ℝ) ≤ Real.sqrt (451995502439 / 1000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (451995502439 / 1000000000000 : ℝ) ≤ (6723061076912809 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (451995502439 / 1000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau618 :
+    Real.arctan (6723061076912809 / 22049877602859706 : ℝ) ≤
+      (2959483383238434001347 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau617 :
+    Real.arctan (6723061076912809 / 10000000000000000 : ℝ) ≤
+      (2 * (2959483383238434001347 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6024938801429853 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (6723061076912809 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (6723061076912809 / 10000000000000000 : ℝ) /
+                (1 + (6024938801429853 / 5000000000000000 : ℝ)) =
+              (6723061076912809 / 22049877602859706 : ℝ)
+            by
+        norm_num];
+        exact hau618)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2959483383238434001347 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 6723061076912809 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 6723061076912809 : ℝ) =
+              (6723061076912809 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau617)
+  have hz1 :
+    (10000000000000000 / 6723061076912809 : ℝ) ≤
+      1 / Real.sqrt (451995502439 / 1000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 6723061076912809 : ℝ) =
+          1 / (6723061076912809 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 3361530538456403 : ℝ) ≤
+      (1110969860924394921403 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (451995502439 / 1000000000000 : ℝ) ≤
+      (375000000000000 / 3361530538456403 : ℝ) := by
+    rw [show
+        (375000000000000 / 3361530538456403 : ℝ) =
+          (3 / 40) / (3361530538456403 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2959483383238434001347 / 10000000000000000000000 : ℝ))) -
+        6 * (1110969860924394921403 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (451995502439 / 1000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (451995502439 / 1000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3361530538456403 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2959483383238434001347 / 10000000000000000000000 : ℝ))) -
+          6 * (1110969860924394921403 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (451995502439 / 1000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (451995502439 / 1000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (451995502439 / 1000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_424 :
+    ((3733947424958614430597 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(975336424718981787153 / 1250000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((6727983742379657 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2961178134875425243097 / 10000000000000000000000 : ℝ))) -
+              6 * (1110163662135900473617 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (1810630609509 / 4000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1810630609509 / 4000000000000 : ℝ)) =
+      Real.log (5810630609509 / 4000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1810630609509 / 4000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1833130609509 / 4000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (3733947424958614430597 / 10000000000000000000000 : ℝ) ≤
+      Real.log (5810630609509 / 4000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1833130609509 / 4000000000000 : ℝ) ≤
+      (-(975336424718981787153 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (1833130609509 / 4000000000000 : ℝ) = (1833130609509 / 1000000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1833130609509 / 1000000000000 : ℝ) ≤
+            (757531526031018212847 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (6727983742379657 / 10000000000000000 : ℝ) ≤ Real.sqrt (1810630609509 / 4000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1810630609509 / 4000000000000 : ℝ) ≤ (336399187118983 / 500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1810630609509 / 4000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau620 :
+    Real.arctan (3363991871189830 / 11026312413859013 : ℝ) ≤
+      (2961178134875425243097 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau619 :
+    Real.arctan (336399187118983 / 500000000000000 : ℝ) ≤
+      (2 * (2961178134875425243097 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6026312413859013 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (336399187118983 / 500000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (336399187118983 / 500000000000000 : ℝ) /
+                (1 + (6026312413859013 / 5000000000000000 : ℝ)) =
+              (3363991871189830 / 11026312413859013 : ℝ)
+            by
+        norm_num];
+        exact hau620)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2961178134875425243097 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (500000000000000 / 336399187118983 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (500000000000000 / 336399187118983 : ℝ) =
+              (336399187118983 / 500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau619)
+  have hz1 :
+    (500000000000000 / 336399187118983 : ℝ) ≤ 1 / Real.sqrt (1810630609509 / 4000000000000 : ℝ) :=
+  by
+    rw [show (500000000000000 / 336399187118983 : ℝ) = 1 / (336399187118983 / 500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 6727983742379657 : ℝ) ≤
+      (1110163662135900473617 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1810630609509 / 4000000000000 : ℝ) ≤
+      (750000000000000 / 6727983742379657 : ℝ) := by
+    rw [show
+        (750000000000000 / 6727983742379657 : ℝ) =
+          (3 / 40) / (6727983742379657 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2961178134875425243097 / 10000000000000000000000 : ℝ))) -
+        6 * (1110163662135900473617 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1810630609509 / 4000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1810630609509 / 4000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((6727983742379657 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2961178134875425243097 / 10000000000000000000000 : ℝ))) -
+          6 * (1110163662135900473617 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (1810630609509 / 4000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1810630609509 / 4000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1810630609509 / 4000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_425 :
+    ((3738504583161202382823 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1557650664052654548403 / 2000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((841612851088889 / 1250000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (592574175131759472259 / 2000000000000000000000 : ℝ))) -
+              6 * (1109359215933374475979 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (906639604631 / 2000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (906639604631 / 2000000000000 : ℝ)) =
+      Real.log (2906639604631 / 2000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((906639604631 / 2000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (917889604631 / 2000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (3738504583161202382823 / 10000000000000000000000 : ℝ) ≤
+      Real.log (2906639604631 / 2000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (917889604631 / 2000000000000 : ℝ) ≤
+      (-(1557650664052654548403 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (917889604631 / 2000000000000 : ℝ) = (917889604631 / 500000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (917889604631 / 500000000000 : ℝ) ≤
+            (1214938057147345451597 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (841612851088889 / 1250000000000000 : ℝ) ≤ Real.sqrt (906639604631 / 2000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (906639604631 / 2000000000000 : ℝ) ≤ (1346580561742223 / 2000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (906639604631 / 2000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau622 :
+    Real.arctan (6732902808711115 / 22055371426528093 : ℝ) ≤
+      (592574175131759472259 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau621 :
+    Real.arctan (1346580561742223 / 2000000000000000 : ℝ) ≤
+      (2 * (592574175131759472259 / 2000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12055371426528093 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1346580561742223 / 2000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1346580561742223 / 2000000000000000 : ℝ) /
+                (1 + (12055371426528093 / 10000000000000000 : ℝ)) =
+              (6732902808711115 / 22055371426528093 : ℝ)
+            by
+        norm_num];
+        exact hau622)
+  have hat1 :
+    (Real.pi / 2 - (2 * (592574175131759472259 / 2000000000000000000000 : ℝ))) ≤
+      Real.arctan (2000000000000000 / 1346580561742223 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2000000000000000 / 1346580561742223 : ℝ) =
+              (1346580561742223 / 2000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau621)
+  have hz1 :
+    (2000000000000000 / 1346580561742223 : ℝ) ≤
+      1 / Real.sqrt (906639604631 / 2000000000000 : ℝ) := by
+    rw [show
+        (2000000000000000 / 1346580561742223 : ℝ) = 1 / (1346580561742223 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (93750000000000 / 841612851088889 : ℝ) ≤
+      (1109359215933374475979 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (906639604631 / 2000000000000 : ℝ) ≤
+      (93750000000000 / 841612851088889 : ℝ) := by
+    rw [show
+        (93750000000000 / 841612851088889 : ℝ) =
+          (3 / 40) / (841612851088889 / 1250000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (592574175131759472259 / 2000000000000000000000 : ℝ))) -
+        6 * (1109359215933374475979 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (906639604631 / 2000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (906639604631 / 2000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((841612851088889 / 1250000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (592574175131759472259 / 2000000000000000000000 : ℝ))) -
+          6 * (1109359215933374475979 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (906639604631 / 2000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (906639604631 / 2000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (906639604631 / 2000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_426 :
+    ((93576491638516288227 / 250000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(7773836058532506180713 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1684454570947503 / 2500000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1482280805065932959781 / 5000000000000000000000 : ℝ))) -
+              6 * (1108556515976043602471 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (363185561803 / 800000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (363185561803 / 800000000000 : ℝ)) =
+      Real.log (1163185561803 / 800000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((363185561803 / 800000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (367685561803 / 800000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (93576491638516288227 / 250000000000000000000 : ℝ) ≤
+      Real.log (1163185561803 / 800000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (367685561803 / 800000000000 : ℝ) ≤
+      (-(7773836058532506180713 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (367685561803 / 800000000000 : ℝ) = (367685561803 / 200000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (367685561803 / 200000000000 : ℝ) ≤
+            (6089107547467493819287 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1684454570947503 / 2500000000000000 : ℝ) ≤ Real.sqrt (363185561803 / 800000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (363185561803 / 800000000000 : ℝ) ≤ (1347563656758003 / 2000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (363185561803 / 800000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau624 :
+    Real.arctan (1347563656758003 / 4411623479943542 : ℝ) ≤
+      (1482280805065932959781 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau623 :
+    Real.arctan (1347563656758003 / 2000000000000000 : ℝ) ≤
+      (2 * (1482280805065932959781 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (1205811739971771 / 1000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1347563656758003 / 2000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1347563656758003 / 2000000000000000 : ℝ) /
+                (1 + (1205811739971771 / 1000000000000000 : ℝ)) =
+              (1347563656758003 / 4411623479943542 : ℝ)
+            by
+        norm_num];
+        exact hau624)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1482280805065932959781 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (2000000000000000 / 1347563656758003 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2000000000000000 / 1347563656758003 : ℝ) =
+              (1347563656758003 / 2000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau623)
+  have hz1 :
+    (2000000000000000 / 1347563656758003 : ℝ) ≤ 1 / Real.sqrt (363185561803 / 800000000000 : ℝ) :=
+  by
+    rw [show
+        (2000000000000000 / 1347563656758003 : ℝ) = 1 / (1347563656758003 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (62500000000000 / 561484856982501 : ℝ) ≤
+      (1108556515976043602471 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (363185561803 / 800000000000 : ℝ) ≤
+      (62500000000000 / 561484856982501 : ℝ) := by
+    rw [show
+        (62500000000000 / 561484856982501 : ℝ) =
+          (3 / 40) / (1684454570947503 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1482280805065932959781 / 5000000000000000000000 : ℝ))) -
+        6 * (1108556515976043602471 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (363185561803 / 800000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (363185561803 / 800000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1684454570947503 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1482280805065932959781 / 5000000000000000000000 : ℝ))) -
+          6 * (1108556515976043602471 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (363185561803 / 800000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (363185561803 / 800000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (363185561803 / 800000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_427 :
+    ((1873806336993604995811 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(3879719776312348896627 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3371365087735233 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2 * (92695323213183203139 / 312500000000000000000 : ℝ))) -
+              6 * (1107755555955205237929 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (28415256387 / 62500000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (28415256387 / 62500000000 : ℝ)) = Real.log (90915256387 / 62500000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((28415256387 / 62500000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (28766818887 / 62500000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1873806336993604995811 / 5000000000000000000000 : ℝ) ≤
+      Real.log (90915256387 / 62500000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (28766818887 / 62500000000 : ℝ) ≤
+      (-(3879719776312348896627 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (28766818887 / 62500000000 : ℝ) = (28766818887 / 15625000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (28766818887 / 15625000000 : ℝ) ≤
+            (3051752026687651103373 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3371365087735233 / 5000000000000000 : ℝ) ≤ Real.sqrt (28415256387 / 62500000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (28415256387 / 62500000000 : ℝ) ≤ (6742730175470469 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (28415256387 / 62500000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau626 :
+    Real.arctan (6742730175470469 / 22060862747714194 : ℝ) ≤
+      (92695323213183203139 / 312500000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau625 :
+    Real.arctan (6742730175470469 / 10000000000000000 : ℝ) ≤
+      (2 * (92695323213183203139 / 312500000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6030431373857097 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (6742730175470469 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (6742730175470469 / 10000000000000000 : ℝ) /
+                (1 + (6030431373857097 / 5000000000000000 : ℝ)) =
+              (6742730175470469 / 22060862747714194 : ℝ)
+            by
+        norm_num];
+        exact hau626)
+  have hat1 :
+    (Real.pi / 2 - (2 * (92695323213183203139 / 312500000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 6742730175470469 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 6742730175470469 : ℝ) =
+              (6742730175470469 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau625)
+  have hz1 :
+    (10000000000000000 / 6742730175470469 : ℝ) ≤ 1 / Real.sqrt (28415256387 / 62500000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 6742730175470469 : ℝ) =
+          1 / (6742730175470469 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (125000000000000 / 1123788362578411 : ℝ) ≤
+      (1107755555955205237929 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (28415256387 / 62500000000 : ℝ) ≤
+      (125000000000000 / 1123788362578411 : ℝ) := by
+    rw [show
+        (125000000000000 / 1123788362578411 : ℝ) =
+          (3 / 40) / (3371365087735233 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (92695323213183203139 / 312500000000000000000 : ℝ))) -
+        6 * (1107755555955205237929 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (28415256387 / 62500000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (28415256387 / 62500000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3371365087735233 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (92695323213183203139 / 312500000000000000000 : ℝ))) -
+          6 * (1107755555955205237929 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (28415256387 / 62500000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (28415256387 / 62500000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (28415256387 / 62500000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_428 :
+    ((938040902597136293219 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(968132967857934315427 / 1250000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1686909622894499 / 2500000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (593587415648002736951 / 2000000000000000000000 : ℝ))) -
+              6 * (2213912659188038901 / 20000000000000000000 : ℝ)))) ≤
+      Vfield (1821225008521 / 4000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1821225008521 / 4000000000000 : ℝ)) =
+      Real.log (5821225008521 / 4000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1821225008521 / 4000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1843725008521 / 4000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (938040902597136293219 / 2500000000000000000000 : ℝ) ≤
+      Real.log (5821225008521 / 4000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1843725008521 / 4000000000000 : ℝ) ≤
+      (-(968132967857934315427 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (1843725008521 / 4000000000000 : ℝ) = (1843725008521 / 1000000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1843725008521 / 1000000000000 : ℝ) ≤
+            (764734982892065684573 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1686909622894499 / 2500000000000000 : ℝ) ≤ Real.sqrt (1821225008521 / 4000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1821225008521 / 4000000000000 : ℝ) ≤ (6747638491577999 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1821225008521 / 4000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau628 :
+    Real.arctan (6747638491577999 / 22063607470944377 : ℝ) ≤
+      (593587415648002736951 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau627 :
+    Real.arctan (6747638491577999 / 10000000000000000 : ℝ) ≤
+      (2 * (593587415648002736951 / 2000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12063607470944377 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (6747638491577999 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (6747638491577999 / 10000000000000000 : ℝ) /
+                (1 + (12063607470944377 / 10000000000000000 : ℝ)) =
+              (6747638491577999 / 22063607470944377 : ℝ)
+            by
+        norm_num];
+        exact hau628)
+  have hat1 :
+    (Real.pi / 2 - (2 * (593587415648002736951 / 2000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 6747638491577999 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 6747638491577999 : ℝ) =
+              (6747638491577999 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau627)
+  have hz1 :
+    (10000000000000000 / 6747638491577999 : ℝ) ≤
+      1 / Real.sqrt (1821225008521 / 4000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 6747638491577999 : ℝ) =
+          1 / (6747638491577999 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (187500000000000 / 1686909622894499 : ℝ) ≤
+      (2213912659188038901 / 20000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1821225008521 / 4000000000000 : ℝ) ≤
+      (187500000000000 / 1686909622894499 : ℝ) := by
+    rw [show
+        (187500000000000 / 1686909622894499 : ℝ) =
+          (3 / 40) / (1686909622894499 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (593587415648002736951 / 2000000000000000000000 : ℝ))) -
+        6 * (2213912659188038901 / 20000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1821225008521 / 4000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1821225008521 / 4000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1686909622894499 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (593587415648002736951 / 2000000000000000000000 : ℝ))) -
+          6 * (2213912659188038901 / 20000000000000000000 : ℝ))) ≤
+      Real.sqrt (1821225008521 / 4000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1821225008521 / 4000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1821225008521 / 4000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_429 :
+    ((3756712476629748451179 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(386535428491473142307 / 500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1350508647981937 / 2000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1484810910440810834293 / 5000000000000000000000 : ℝ))) -
+              6 * (276539707661825492393 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (911936804137 / 2000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (911936804137 / 2000000000000 : ℝ)) =
+      Real.log (2911936804137 / 2000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((911936804137 / 2000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (923186804137 / 2000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (3756712476629748451179 / 10000000000000000000000 : ℝ) ≤
+      Real.log (2911936804137 / 2000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (923186804137 / 2000000000000 : ℝ) ≤
+      (-(386535428491473142307 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (923186804137 / 2000000000000 : ℝ) = (923186804137 / 500000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (923186804137 / 500000000000 : ℝ) ≤
+            (306611751808526857693 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1350508647981937 / 2000000000000000 : ℝ) ≤ Real.sqrt (911936804137 / 2000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (911936804137 / 2000000000000 : ℝ) ≤ (844067904988711 / 1250000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (911936804137 / 2000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau630 :
+    Real.arctan (6752543239909688 / 22066351569834603 : ℝ) ≤
+      (1484810910440810834293 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau629 :
+    Real.arctan (844067904988711 / 1250000000000000 : ℝ) ≤
+      (2 * (1484810910440810834293 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12066351569834603 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (844067904988711 / 1250000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (844067904988711 / 1250000000000000 : ℝ) /
+                (1 + (12066351569834603 / 10000000000000000 : ℝ)) =
+              (6752543239909688 / 22066351569834603 : ℝ)
+            by
+        norm_num];
+        exact hau630)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1484810910440810834293 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (1250000000000000 / 844067904988711 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1250000000000000 / 844067904988711 : ℝ) =
+              (844067904988711 / 1250000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau629)
+  have hz1 :
+    (1250000000000000 / 844067904988711 : ℝ) ≤ 1 / Real.sqrt (911936804137 / 2000000000000 : ℝ) :=
+  by
+    rw [show
+        (1250000000000000 / 844067904988711 : ℝ) = 1 / (844067904988711 / 1250000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (150000000000000 / 1350508647981937 : ℝ) ≤
+      (276539707661825492393 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (911936804137 / 2000000000000 : ℝ) ≤
+      (150000000000000 / 1350508647981937 : ℝ) := by
+    rw [show
+        (150000000000000 / 1350508647981937 : ℝ) =
+          (3 / 40) / (1350508647981937 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1484810910440810834293 / 5000000000000000000000 : ℝ))) -
+        6 * (276539707661825492393 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (911936804137 / 2000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (911936804137 / 2000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1350508647981937 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1484810910440810834293 / 5000000000000000000000 : ℝ))) -
+          6 * (276539707661825492393 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (911936804137 / 2000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (911936804137 / 2000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (911936804137 / 2000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_430 :
+    ((3761259274593339865841 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1929093493589703793197 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3378722214117157 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2971304575226139195549 / 10000000000000000000000 : ℝ))) -
+              6 * (1105363052901320432927 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (1826522208027 / 4000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1826522208027 / 4000000000000 : ℝ)) =
+      Real.log (5826522208027 / 4000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1826522208027 / 4000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1849022208027 / 4000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (3761259274593339865841 / 10000000000000000000000 : ℝ) ≤
+      Real.log (5826522208027 / 4000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1849022208027 / 4000000000000 : ℝ) ≤
+      (-(1929093493589703793197 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (1849022208027 / 4000000000000 : ℝ) = (1849022208027 / 1000000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1849022208027 / 1000000000000 : ℝ) ≤
+            (1536642407910296206803 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3378722214117157 / 5000000000000000 : ℝ) ≤ Real.sqrt (1826522208027 / 4000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1826522208027 / 4000000000000 : ℝ) ≤ (6757444428234317 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1826522208027 / 4000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau632 :
+    Real.arctan (6757444428234317 / 22069095044810733 : ℝ) ≤
+      (2971304575226139195549 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau631 :
+    Real.arctan (6757444428234317 / 10000000000000000 : ℝ) ≤
+      (2 * (2971304575226139195549 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12069095044810733 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (6757444428234317 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (6757444428234317 / 10000000000000000 : ℝ) /
+                (1 + (12069095044810733 / 10000000000000000 : ℝ)) =
+              (6757444428234317 / 22069095044810733 : ℝ)
+            by
+        norm_num];
+        exact hau632)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2971304575226139195549 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 6757444428234317 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 6757444428234317 : ℝ) =
+              (6757444428234317 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau631)
+  have hz1 :
+    (10000000000000000 / 6757444428234317 : ℝ) ≤
+      1 / Real.sqrt (1826522208027 / 4000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 6757444428234317 : ℝ) =
+          1 / (6757444428234317 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 3378722214117157 : ℝ) ≤
+      (1105363052901320432927 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1826522208027 / 4000000000000 : ℝ) ≤
+      (375000000000000 / 3378722214117157 : ℝ) := by
+    rw [show
+        (375000000000000 / 3378722214117157 : ℝ) =
+          (3 / 40) / (3378722214117157 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2971304575226139195549 / 10000000000000000000000 : ℝ))) -
+        6 * (1105363052901320432927 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1826522208027 / 4000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1826522208027 / 4000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3378722214117157 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2971304575226139195549 / 10000000000000000000000 : ℝ))) -
+          6 * (1105363052901320432927 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (1826522208027 / 4000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1826522208027 / 4000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1826522208027 / 4000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_431 :
+    ((1882902003079636395783 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1925514974385436700747 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((6762342064292517 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2972985345737252723841 / 10000000000000000000000 : ℝ))) -
+              6 * (1104568990173589174241 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (91458540389 / 200000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (91458540389 / 200000000000 : ℝ)) =
+      Real.log (291458540389 / 200000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((91458540389 / 200000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (92583540389 / 200000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1882902003079636395783 / 5000000000000000000000 : ℝ) ≤
+      Real.log (291458540389 / 200000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (92583540389 / 200000000000 : ℝ) ≤
+      (-(1925514974385436700747 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (92583540389 / 200000000000 : ℝ) = (92583540389 / 50000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (92583540389 / 50000000000 : ℝ) ≤
+            (1540220927114563299253 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (6762342064292517 / 10000000000000000 : ℝ) ≤ Real.sqrt (91458540389 / 200000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (91458540389 / 200000000000 : ℝ) ≤ (169058551607313 / 250000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (91458540389 / 200000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau634 :
+    Real.arctan (6762342064292520 / 22071837896298143 : ℝ) ≤
+      (2972985345737252723841 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau633 :
+    Real.arctan (169058551607313 / 250000000000000 : ℝ) ≤
+      (2 * (2972985345737252723841 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12071837896298143 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (169058551607313 / 250000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (169058551607313 / 250000000000000 : ℝ) /
+                (1 + (12071837896298143 / 10000000000000000 : ℝ)) =
+              (6762342064292520 / 22071837896298143 : ℝ)
+            by
+        norm_num];
+        exact hau634)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2972985345737252723841 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (250000000000000 / 169058551607313 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (250000000000000 / 169058551607313 : ℝ) =
+              (169058551607313 / 250000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau633)
+  have hz1 :
+    (250000000000000 / 169058551607313 : ℝ) ≤ 1 / Real.sqrt (91458540389 / 200000000000 : ℝ) := by
+    rw [show (250000000000000 / 169058551607313 : ℝ) = 1 / (169058551607313 / 250000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 2254114021430839 : ℝ) ≤
+      (1104568990173589174241 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (91458540389 / 200000000000 : ℝ) ≤
+      (250000000000000 / 2254114021430839 : ℝ) := by
+    rw [show
+        (250000000000000 / 2254114021430839 : ℝ) =
+          (3 / 40) / (6762342064292517 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2972985345737252723841 / 10000000000000000000000 : ℝ))) -
+        6 * (1104568990173589174241 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (91458540389 / 200000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (91458540389 / 200000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((6762342064292517 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2972985345737252723841 / 10000000000000000000000 : ℝ))) -
+          6 * (1104568990173589174241 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (91458540389 / 200000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (91458540389 / 200000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (91458540389 / 200000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_432 :
+    ((1885173336602469301127 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(768776628072108333261 / 1000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((6767236155796913 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2974664136862955190747 / 10000000000000000000000 : ℝ))) -
+              6 * (551888318156334600501 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (1831819407533 / 4000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1831819407533 / 4000000000000 : ℝ)) =
+      Real.log (5831819407533 / 4000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1831819407533 / 4000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1854319407533 / 4000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1885173336602469301127 / 5000000000000000000000 : ℝ) ≤
+      Real.log (5831819407533 / 4000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1854319407533 / 4000000000000 : ℝ) ≤
+      (-(768776628072108333261 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (1854319407533 / 4000000000000 : ℝ) = (1854319407533 / 1000000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1854319407533 / 1000000000000 : ℝ) ≤
+            (617517732527891666739 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (6767236155796913 / 10000000000000000 : ℝ) ≤ Real.sqrt (1831819407533 / 4000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1831819407533 / 4000000000000 : ℝ) ≤ (1691809038949229 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1831819407533 / 4000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau636 :
+    Real.arctan (1691809038949229 / 5518645031180432 : ℝ) ≤
+      (2974664136862955190747 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau635 :
+    Real.arctan (1691809038949229 / 2500000000000000 : ℝ) ≤
+      (2 * (2974664136862955190747 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (188665314448777 / 156250000000000 : ℝ) ≤
+          Real.sqrt (1 + (1691809038949229 / 2500000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1691809038949229 / 2500000000000000 : ℝ) /
+                (1 + (188665314448777 / 156250000000000 : ℝ)) =
+              (1691809038949229 / 5518645031180432 : ℝ)
+            by
+        norm_num];
+        exact hau636)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2974664136862955190747 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (2500000000000000 / 1691809038949229 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 1691809038949229 : ℝ) =
+              (1691809038949229 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau635)
+  have hz1 :
+    (2500000000000000 / 1691809038949229 : ℝ) ≤
+      1 / Real.sqrt (1831819407533 / 4000000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 1691809038949229 : ℝ) = 1 / (1691809038949229 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 6767236155796913 : ℝ) ≤
+      (551888318156334600501 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1831819407533 / 4000000000000 : ℝ) ≤
+      (750000000000000 / 6767236155796913 : ℝ) := by
+    rw [show
+        (750000000000000 / 6767236155796913 : ℝ) =
+          (3 / 40) / (6767236155796913 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2974664136862955190747 / 10000000000000000000000 : ℝ))) -
+        6 * (551888318156334600501 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1831819407533 / 4000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1831819407533 / 4000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((6767236155796913 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2974664136862955190747 / 10000000000000000000000 : ℝ))) -
+          6 * (551888318156334600501 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (1831819407533 / 4000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1831819407533 / 4000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1831819407533 / 4000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/V36.lean b/LeanPool/Zeta5Irrational/Table/V36.lean
new file mode 100644
index 0000000000..d305c5dbd1
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/V36.lean
@@ -0,0 +1,1425 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Energy
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # V36: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_433 :
+    ((3774887277605171324213 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(479593316593176151771 / 625000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3386063355216127 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2976340953035626311821 / 10000000000000000000000 : ℝ))) -
+              6 * (11029859851979670127 / 100000000000000000000 : ℝ)))) ≤
+      Vfield (917234003643 / 2000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (917234003643 / 2000000000000 : ℝ)) =
+      Real.log (2917234003643 / 2000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((917234003643 / 2000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (928484003643 / 2000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (3774887277605171324213 / 10000000000000000000000 : ℝ) ≤
+      Real.log (2917234003643 / 2000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (928484003643 / 2000000000000 : ℝ) ≤
+      (-(479593316593176151771 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (928484003643 / 2000000000000 : ℝ) = (928484003643 / 500000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (928484003643 / 500000000000 : ℝ) ≤
+            (386840658781823848229 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3386063355216127 / 5000000000000000 : ℝ) ≤ Real.sqrt (917234003643 / 2000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (917234003643 / 2000000000000 : ℝ) ≤ (6772126710432257 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (917234003643 / 2000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau638 :
+    Real.arctan (6772126710432257 / 22077321730505898 : ℝ) ≤
+      (2976340953035626311821 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau637 :
+    Real.arctan (6772126710432257 / 10000000000000000 : ℝ) ≤
+      (2 * (2976340953035626311821 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6038660865252949 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (6772126710432257 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (6772126710432257 / 10000000000000000 : ℝ) /
+                (1 + (6038660865252949 / 5000000000000000 : ℝ)) =
+              (6772126710432257 / 22077321730505898 : ℝ)
+            by
+        norm_num];
+        exact hau638)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2976340953035626311821 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 6772126710432257 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 6772126710432257 : ℝ) =
+              (6772126710432257 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau637)
+  have hz1 :
+    (10000000000000000 / 6772126710432257 : ℝ) ≤
+      1 / Real.sqrt (917234003643 / 2000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 6772126710432257 : ℝ) =
+          1 / (6772126710432257 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 3386063355216127 : ℝ) ≤
+      (11029859851979670127 / 100000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (917234003643 / 2000000000000 : ℝ) ≤
+      (375000000000000 / 3386063355216127 : ℝ) := by
+    rw [show
+        (375000000000000 / 3386063355216127 : ℝ) =
+          (3 / 40) / (3386063355216127 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2976340953035626311821 / 10000000000000000000000 : ℝ))) -
+        6 * (11029859851979670127 / 100000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (917234003643 / 2000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (917234003643 / 2000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3386063355216127 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2976340953035626311821 / 10000000000000000000000 : ℝ))) -
+          6 * (11029859851979670127 / 100000000000000000000 : ℝ))) ≤
+      Real.sqrt (917234003643 / 2000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (917234003643 / 2000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (917234003643 / 2000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_434 :
+    ((377942582123225227881 / 1000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(7659240193694681880157 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((6777013735855563 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2978015798672107485851 / 10000000000000000000000 : ℝ))) -
+              6 * (1102197030739536467523 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (1837116607039 / 4000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1837116607039 / 4000000000000 : ℝ)) =
+      Real.log (5837116607039 / 4000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1837116607039 / 4000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1859616607039 / 4000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (377942582123225227881 / 1000000000000000000000 : ℝ) ≤
+      Real.log (5837116607039 / 4000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1859616607039 / 4000000000000 : ℝ) ≤
+      (-(7659240193694681880157 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (1859616607039 / 4000000000000 : ℝ) = (1859616607039 / 1000000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1859616607039 / 1000000000000 : ℝ) ≤
+            (6203703412305318119843 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (6777013735855563 / 10000000000000000 : ℝ) ≤ Real.sqrt (1837116607039 / 4000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1837116607039 / 4000000000000 : ℝ) ≤ (3388506867927783 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1837116607039 / 4000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau640 :
+    Real.arctan (6777013735855566 / 22080062714074583 : ℝ) ≤
+      (2978015798672107485851 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau639 :
+    Real.arctan (3388506867927783 / 5000000000000000 : ℝ) ≤
+      (2 * (2978015798672107485851 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12080062714074583 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (3388506867927783 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (3388506867927783 / 5000000000000000 : ℝ) /
+                (1 + (12080062714074583 / 10000000000000000 : ℝ)) =
+              (6777013735855566 / 22080062714074583 : ℝ)
+            by
+        norm_num];
+        exact hau640)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2978015798672107485851 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 3388506867927783 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 3388506867927783 : ℝ) =
+              (3388506867927783 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau639)
+  have hz1 :
+    (5000000000000000 / 3388506867927783 : ℝ) ≤
+      1 / Real.sqrt (1837116607039 / 4000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 3388506867927783 : ℝ) = 1 / (3388506867927783 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 2259004578618521 : ℝ) ≤
+      (1102197030739536467523 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1837116607039 / 4000000000000 : ℝ) ≤
+      (250000000000000 / 2259004578618521 : ℝ) := by
+    rw [show
+        (250000000000000 / 2259004578618521 : ℝ) =
+          (3 / 40) / (6777013735855563 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2978015798672107485851 / 10000000000000000000000 : ℝ))) -
+        6 * (1102197030739536467523 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1837116607039 / 4000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1837116607039 / 4000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((6777013735855563 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2978015798672107485851 / 10000000000000000000000 : ℝ))) -
+          6 * (1102197030739536467523 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (1837116607039 / 4000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1837116607039 / 4000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1837116607039 / 4000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_435 :
+    ((18919811529779573573 / 50000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1529001521484943568809 / 2000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((6781897239696277 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2979688678173779422881 / 10000000000000000000000 : ℝ))) -
+              6 * (34419055214933789157 / 312500000000000000000 : ℝ)))) ≤
+      Vfield (229970650849 / 500000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (229970650849 / 500000000000 : ℝ)) =
+      Real.log (729970650849 / 500000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((229970650849 / 500000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (232783150849 / 500000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (18919811529779573573 / 50000000000000000000 : ℝ) ≤
+      Real.log (729970650849 / 500000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (232783150849 / 500000000000 : ℝ) ≤
+      (-(1529001521484943568809 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (232783150849 / 500000000000 : ℝ) = (232783150849 / 125000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (232783150849 / 125000000000 : ℝ) ≤
+            (1243587199715056431191 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (6781897239696277 / 10000000000000000 : ℝ) ≤ Real.sqrt (229970650849 / 500000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (229970650849 / 500000000000 : ℝ) ≤ (169547430992407 / 250000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (229970650849 / 500000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau642 :
+    Real.arctan (65210550381695 / 212334644960108 : ℝ) ≤
+      (2979688678173779422881 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau641 :
+    Real.arctan (169547430992407 / 250000000000000 : ℝ) ≤
+      (2 * (2979688678173779422881 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (377587596120351 / 312500000000000 : ℝ) ≤
+          Real.sqrt (1 + (169547430992407 / 250000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (169547430992407 / 250000000000000 : ℝ) /
+                (1 + (377587596120351 / 312500000000000 : ℝ)) =
+              (65210550381695 / 212334644960108 : ℝ)
+            by
+        norm_num];
+        exact hau642)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2979688678173779422881 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (250000000000000 / 169547430992407 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (250000000000000 / 169547430992407 : ℝ) =
+              (169547430992407 / 250000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau641)
+  have hz1 :
+    (250000000000000 / 169547430992407 : ℝ) ≤ 1 / Real.sqrt (229970650849 / 500000000000 : ℝ) := by
+    rw [show (250000000000000 / 169547430992407 : ℝ) = 1 / (169547430992407 / 250000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 6781897239696277 : ℝ) ≤
+      (34419055214933789157 / 312500000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (229970650849 / 500000000000 : ℝ) ≤
+      (750000000000000 / 6781897239696277 : ℝ) := by
+    rw [show
+        (750000000000000 / 6781897239696277 : ℝ) =
+          (3 / 40) / (6781897239696277 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2979688678173779422881 / 10000000000000000000000 : ℝ))) -
+        6 * (34419055214933789157 / 312500000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (229970650849 / 500000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (229970650849 / 500000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((6781897239696277 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2979688678173779422881 / 10000000000000000000000 : ℝ))) -
+          6 * (34419055214933789157 / 312500000000000000000 : ℝ))) ≤
+      Real.sqrt (229970650849 / 500000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (229970650849 / 500000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (229970650849 / 500000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_436 :
+    ((3788496733643348428943 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1907698812254968296973 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((6786777229556381 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2981359595926635812237 / 10000000000000000000000 : ℝ))) -
+              6 * (44024967503350408363 / 400000000000000000000 : ℝ)))) ≤
+      Vfield (368482761309 / 800000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (368482761309 / 800000000000 : ℝ)) =
+      Real.log (1168482761309 / 800000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((368482761309 / 800000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (372982761309 / 800000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (3788496733643348428943 / 10000000000000000000000 : ℝ) ≤
+      Real.log (1168482761309 / 800000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (372982761309 / 800000000000 : ℝ) ≤
+      (-(1907698812254968296973 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (372982761309 / 800000000000 : ℝ) = (372982761309 / 200000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (372982761309 / 200000000000 : ℝ) ≤
+            (1558037089245031703027 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (6786777229556381 / 10000000000000000 : ℝ) ≤ Real.sqrt (368482761309 / 800000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (368482761309 / 800000000000 : ℝ) ≤ (212086788423637 / 312500000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (368482761309 / 800000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau644 :
+    Real.arctan (6786777229556384 / 22085542816258813 : ℝ) ≤
+      (2981359595926635812237 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau643 :
+    Real.arctan (212086788423637 / 312500000000000 : ℝ) ≤
+      (2 * (2981359595926635812237 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12085542816258813 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (212086788423637 / 312500000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (212086788423637 / 312500000000000 : ℝ) /
+                (1 + (12085542816258813 / 10000000000000000 : ℝ)) =
+              (6786777229556384 / 22085542816258813 : ℝ)
+            by
+        norm_num];
+        exact hau644)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2981359595926635812237 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (312500000000000 / 212086788423637 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (312500000000000 / 212086788423637 : ℝ) =
+              (212086788423637 / 312500000000000 : ℝ)
+            by
+        norm_num];
+        exact hau643)
+  have hz1 :
+    (312500000000000 / 212086788423637 : ℝ) ≤ 1 / Real.sqrt (368482761309 / 800000000000 : ℝ) := by
+    rw [show (312500000000000 / 212086788423637 : ℝ) = 1 / (212086788423637 / 312500000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 6786777229556381 : ℝ) ≤
+      (44024967503350408363 / 400000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (368482761309 / 800000000000 : ℝ) ≤
+      (750000000000000 / 6786777229556381 : ℝ) := by
+    rw [show
+        (750000000000000 / 6786777229556381 : ℝ) =
+          (3 / 40) / (6786777229556381 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2981359595926635812237 / 10000000000000000000000 : ℝ))) -
+        6 * (44024967503350408363 / 400000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (368482761309 / 800000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (368482761309 / 800000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((6786777229556381 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2981359595926635812237 / 10000000000000000000000 : ℝ))) -
+          6 * (44024967503350408363 / 400000000000000000000 : ℝ))) ≤
+      Real.sqrt (368482761309 / 800000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (368482761309 / 800000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (368482761309 / 800000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_437 :
+    ((3793029106159204379151 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(7616603061064595854687 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((6791653713010549 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2983028556301359654301 / 10000000000000000000000 : ℝ))) -
+              6 * (549920143428996453391 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (922531203149 / 2000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (922531203149 / 2000000000000 : ℝ)) =
+      Real.log (2922531203149 / 2000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((922531203149 / 2000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (933781203149 / 2000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (3793029106159204379151 / 10000000000000000000000 : ℝ) ≤
+      Real.log (2922531203149 / 2000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (933781203149 / 2000000000000 : ℝ) ≤
+      (-(7616603061064595854687 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (933781203149 / 2000000000000 : ℝ) = (933781203149 / 500000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (933781203149 / 500000000000 : ℝ) ≤
+            (6246340544935404145313 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (6791653713010549 / 10000000000000000 : ℝ) ≤ Real.sqrt (922531203149 / 2000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (922531203149 / 2000000000000 : ℝ) ≤ (848956714126319 / 1250000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (922531203149 / 2000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau646 :
+    Real.arctan (3395826856505276 / 11044140967859907 : ℝ) ≤
+      (2983028556301359654301 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau645 :
+    Real.arctan (848956714126319 / 1250000000000000 : ℝ) ≤
+      (2 * (2983028556301359654301 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6044140967859907 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (848956714126319 / 1250000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (848956714126319 / 1250000000000000 : ℝ) /
+                (1 + (6044140967859907 / 5000000000000000 : ℝ)) =
+              (3395826856505276 / 11044140967859907 : ℝ)
+            by
+        norm_num];
+        exact hau646)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2983028556301359654301 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (1250000000000000 / 848956714126319 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1250000000000000 / 848956714126319 : ℝ) =
+              (848956714126319 / 1250000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau645)
+  have hz1 :
+    (1250000000000000 / 848956714126319 : ℝ) ≤ 1 / Real.sqrt (922531203149 / 2000000000000 : ℝ) :=
+  by
+    rw [show
+        (1250000000000000 / 848956714126319 : ℝ) = 1 / (848956714126319 / 1250000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 6791653713010549 : ℝ) ≤
+      (549920143428996453391 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (922531203149 / 2000000000000 : ℝ) ≤
+      (750000000000000 / 6791653713010549 : ℝ) := by
+    rw [show
+        (750000000000000 / 6791653713010549 : ℝ) =
+          (3 / 40) / (6791653713010549 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2983028556301359654301 / 10000000000000000000000 : ℝ))) -
+        6 * (549920143428996453391 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (922531203149 / 2000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (922531203149 / 2000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((6791653713010549 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2983028556301359654301 / 10000000000000000000000 : ℝ))) -
+          6 * (549920143428996453391 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (922531203149 / 2000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (922531203149 / 2000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (922531203149 / 2000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_438 :
+    ((1898779712682799641597 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(7602430986387443157753 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1699131674401571 / 2500000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2984695563653398434181 / 10000000000000000000000 : ℝ))) -
+              6 * (549529029365633383129 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (1847711006051 / 4000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1847711006051 / 4000000000000 : ℝ)) =
+      Real.log (5847711006051 / 4000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1847711006051 / 4000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1870211006051 / 4000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1898779712682799641597 / 5000000000000000000000 : ℝ) ≤
+      Real.log (5847711006051 / 4000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1870211006051 / 4000000000000 : ℝ) ≤
+      (-(7602430986387443157753 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (1870211006051 / 4000000000000 : ℝ) = (1870211006051 / 1000000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1870211006051 / 1000000000000 : ℝ) ≤
+            (6260512619612556842247 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1699131674401571 / 2500000000000000 : ℝ) ≤ Real.sqrt (1847711006051 / 4000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1847711006051 / 4000000000000 : ℝ) ≤ (6796526697606287 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1847711006051 / 4000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau648 :
+    Real.arctan (6796526697606287 / 22091020434656249 : ℝ) ≤
+      (2984695563653398434181 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau647 :
+    Real.arctan (6796526697606287 / 10000000000000000 : ℝ) ≤
+      (2 * (2984695563653398434181 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12091020434656249 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (6796526697606287 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (6796526697606287 / 10000000000000000 : ℝ) /
+                (1 + (12091020434656249 / 10000000000000000 : ℝ)) =
+              (6796526697606287 / 22091020434656249 : ℝ)
+            by
+        norm_num];
+        exact hau648)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2984695563653398434181 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 6796526697606287 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 6796526697606287 : ℝ) =
+              (6796526697606287 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau647)
+  have hz1 :
+    (10000000000000000 / 6796526697606287 : ℝ) ≤
+      1 / Real.sqrt (1847711006051 / 4000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 6796526697606287 : ℝ) =
+          1 / (6796526697606287 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (187500000000000 / 1699131674401571 : ℝ) ≤
+      (549529029365633383129 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1847711006051 / 4000000000000 : ℝ) ≤
+      (187500000000000 / 1699131674401571 : ℝ) := by
+    rw [show
+        (187500000000000 / 1699131674401571 : ℝ) =
+          (3 / 40) / (1699131674401571 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2984695563653398434181 / 10000000000000000000000 : ℝ))) -
+        6 * (549529029365633383129 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1847711006051 / 4000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1847711006051 / 4000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1699131674401571 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2984695563653398434181 / 10000000000000000000000 : ℝ))) -
+          6 * (549529029365633383129 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (1847711006051 / 4000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1847711006051 / 4000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1847711006051 / 4000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_439 :
+    ((3802087693122120209983 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(758827896805969992371 / 1000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((425087261929003 / 625000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2986360622323037088307 / 10000000000000000000000 : ℝ))) -
+              6 * (1098277497263947141799 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (462589901451 / 1000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (462589901451 / 1000000000000 : ℝ)) =
+      Real.log (1462589901451 / 1000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((462589901451 / 1000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (468214901451 / 1000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (3802087693122120209983 / 10000000000000000000000 : ℝ) ≤
+      Real.log (1462589901451 / 1000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (468214901451 / 1000000000000 : ℝ) ≤
+      (-(758827896805969992371 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (468214901451 / 1000000000000 : ℝ) = (468214901451 / 250000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (468214901451 / 250000000000 : ℝ) ≤
+            (627466463794030007629 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (425087261929003 / 625000000000000 : ℝ) ≤ Real.sqrt (462589901451 / 1000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (462589901451 / 1000000000000 : ℝ) ≤ (6801396190864051 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (462589901451 / 1000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau650 :
+    Real.arctan (6801396190864051 / 22093758313489648 : ℝ) ≤
+      (2986360622323037088307 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau649 :
+    Real.arctan (6801396190864051 / 10000000000000000 : ℝ) ≤
+      (2 * (2986360622323037088307 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (755859894593103 / 625000000000000 : ℝ) ≤
+          Real.sqrt (1 + (6801396190864051 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (6801396190864051 / 10000000000000000 : ℝ) /
+                (1 + (755859894593103 / 625000000000000 : ℝ)) =
+              (6801396190864051 / 22093758313489648 : ℝ)
+            by
+        norm_num];
+        exact hau650)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2986360622323037088307 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 6801396190864051 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 6801396190864051 : ℝ) =
+              (6801396190864051 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau649)
+  have hz1 :
+    (10000000000000000 / 6801396190864051 : ℝ) ≤
+      1 / Real.sqrt (462589901451 / 1000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 6801396190864051 : ℝ) =
+          1 / (6801396190864051 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (46875000000000 / 425087261929003 : ℝ) ≤
+      (1098277497263947141799 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (462589901451 / 1000000000000 : ℝ) ≤
+      (46875000000000 / 425087261929003 : ℝ) := by
+    rw [show
+        (46875000000000 / 425087261929003 : ℝ) =
+          (3 / 40) / (425087261929003 / 625000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2986360622323037088307 / 10000000000000000000000 : ℝ))) -
+        6 * (1098277497263947141799 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (462589901451 / 1000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (462589901451 / 1000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((425087261929003 / 625000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2986360622323037088307 / 10000000000000000000000 : ℝ))) -
+          6 * (1098277497263947141799 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (462589901451 / 1000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (462589901451 / 1000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (462589901451 / 1000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_440 :
+    ((3806613911285829159271 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(946768368674250799957 / 1250000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3403131100138701 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (597604747327094440931 / 2000000000000000000000 : ℝ))) -
+              6 * (274374649136471787029 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (1853008205557 / 4000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1853008205557 / 4000000000000 : ℝ)) =
+      Real.log (5853008205557 / 4000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1853008205557 / 4000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1875508205557 / 4000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (3806613911285829159271 / 10000000000000000000000 : ℝ) ≤
+      Real.log (5853008205557 / 4000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1875508205557 / 4000000000000 : ℝ) ≤
+      (-(946768368674250799957 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (1875508205557 / 4000000000000 : ℝ) = (1875508205557 / 1000000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1875508205557 / 1000000000000 : ℝ) ≤
+            (786099582075749200043 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3403131100138701 / 5000000000000000 : ℝ) ≤ Real.sqrt (1853008205557 / 4000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1853008205557 / 4000000000000 : ℝ) ≤ (1361252440055481 / 2000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1853008205557 / 4000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau652 :
+    Real.arctan (6806262200277405 / 22096495572641069 : ℝ) ≤
+      (597604747327094440931 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau651 :
+    Real.arctan (1361252440055481 / 2000000000000000 : ℝ) ≤
+      (2 * (597604747327094440931 / 2000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12096495572641069 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1361252440055481 / 2000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1361252440055481 / 2000000000000000 : ℝ) /
+                (1 + (12096495572641069 / 10000000000000000 : ℝ)) =
+              (6806262200277405 / 22096495572641069 : ℝ)
+            by
+        norm_num];
+        exact hau652)
+  have hat1 :
+    (Real.pi / 2 - (2 * (597604747327094440931 / 2000000000000000000000 : ℝ))) ≤
+      Real.arctan (2000000000000000 / 1361252440055481 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2000000000000000 / 1361252440055481 : ℝ) =
+              (1361252440055481 / 2000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau651)
+  have hz1 :
+    (2000000000000000 / 1361252440055481 : ℝ) ≤
+      1 / Real.sqrt (1853008205557 / 4000000000000 : ℝ) := by
+    rw [show
+        (2000000000000000 / 1361252440055481 : ℝ) = 1 / (1361252440055481 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (125000000000000 / 1134377033379567 : ℝ) ≤
+      (274374649136471787029 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1853008205557 / 4000000000000 : ℝ) ≤
+      (125000000000000 / 1134377033379567 : ℝ) := by
+    rw [show
+        (125000000000000 / 1134377033379567 : ℝ) =
+          (3 / 40) / (3403131100138701 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (597604747327094440931 / 2000000000000000000000 : ℝ))) -
+        6 * (274374649136471787029 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1853008205557 / 4000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1853008205557 / 4000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3403131100138701 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (597604747327094440931 / 2000000000000000000000 : ℝ))) -
+          6 * (274374649136471787029 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (1853008205557 / 4000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1853008205557 / 4000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1853008205557 / 4000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_441 :
+    ((1905569040855633815593 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(75600348739429958419 / 100000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((6811124733313139 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (298968491090088576719 / 1000000000000000000000 : ℝ))) -
+              6 * (68545084418514996731 / 625000000000000000000 : ℝ)))) ≤
+      Vfield (185565680531 / 400000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (185565680531 / 400000000000 : ℝ)) =
+      Real.log (585565680531 / 400000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((185565680531 / 400000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (187815680531 / 400000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1905569040855633815593 / 5000000000000000000000 : ℝ) ≤
+      Real.log (585565680531 / 400000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (187815680531 / 400000000000 : ℝ) ≤
+      (-(75600348739429958419 / 100000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (187815680531 / 400000000000 : ℝ) = (187815680531 / 100000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (187815680531 / 100000000000 : ℝ) ≤
+            (63029087320570041581 / 100000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (6811124733313139 / 10000000000000000 : ℝ) ≤ Real.sqrt (185565680531 / 400000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (185565680531 / 400000000000 : ℝ) ≤ (3405562366656571 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (185565680531 / 400000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau654 :
+    Real.arctan (3405562366656571 / 11049616106265545 : ℝ) ≤
+      (298968491090088576719 / 1000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau653 :
+    Real.arctan (3405562366656571 / 5000000000000000 : ℝ) ≤
+      (2 * (298968491090088576719 / 1000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (1209923221253109 / 1000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (3405562366656571 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (3405562366656571 / 5000000000000000 : ℝ) /
+                (1 + (1209923221253109 / 1000000000000000 : ℝ)) =
+              (3405562366656571 / 11049616106265545 : ℝ)
+            by
+        norm_num];
+        exact hau654)
+  have hat1 :
+    (Real.pi / 2 - (2 * (298968491090088576719 / 1000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 3405562366656571 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 3405562366656571 : ℝ) =
+              (3405562366656571 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau653)
+  have hz1 :
+    (5000000000000000 / 3405562366656571 : ℝ) ≤ 1 / Real.sqrt (185565680531 / 400000000000 : ℝ) :=
+  by
+    rw [show
+        (5000000000000000 / 3405562366656571 : ℝ) = 1 / (3405562366656571 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 6811124733313139 : ℝ) ≤
+      (68545084418514996731 / 625000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (185565680531 / 400000000000 : ℝ) ≤
+      (750000000000000 / 6811124733313139 : ℝ) := by
+    rw [show
+        (750000000000000 / 6811124733313139 : ℝ) =
+          (3 / 40) / (6811124733313139 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (298968491090088576719 / 1000000000000000000000 : ℝ))) -
+        6 * (68545084418514996731 / 625000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (185565680531 / 400000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (185565680531 / 400000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((6811124733313139 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (298968491090088576719 / 1000000000000000000000 : ℝ))) -
+          6 * (68545084418514996731 / 625000000000000000000 : ℝ))) ≤
+      Real.sqrt (185565680531 / 400000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (185565680531 / 400000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (185565680531 / 400000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_442 :
+    ((23847876289065382409 / 62500000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(7545942685497941771449 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3407991898705709 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2991344149414517301659 / 10000000000000000000000 : ℝ))) -
+              6 * (1095945753863272201041 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (1858305405063 / 4000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1858305405063 / 4000000000000 : ℝ)) =
+      Real.log (5858305405063 / 4000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1858305405063 / 4000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1880805405063 / 4000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (23847876289065382409 / 62500000000000000000 : ℝ) ≤
+      Real.log (5858305405063 / 4000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1880805405063 / 4000000000000 : ℝ) ≤
+      (-(7545942685497941771449 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (1880805405063 / 4000000000000 : ℝ) = (1880805405063 / 1000000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1880805405063 / 1000000000000 : ℝ) ≤
+            (6317000920502058228551 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3407991898705709 / 5000000000000000 : ℝ) ≤ Real.sqrt (1858305405063 / 4000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1858305405063 / 4000000000000 : ℝ) ≤ (6815983797411421 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1858305405063 / 4000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau656 :
+    Real.arctan (6815983797411421 / 22101968233579816 : ℝ) ≤
+      (2991344149414517301659 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau655 :
+    Real.arctan (6815983797411421 / 10000000000000000 : ℝ) ≤
+      (2 * (2991344149414517301659 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (1512746029197477 / 1250000000000000 : ℝ) ≤
+          Real.sqrt (1 + (6815983797411421 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (6815983797411421 / 10000000000000000 : ℝ) /
+                (1 + (1512746029197477 / 1250000000000000 : ℝ)) =
+              (6815983797411421 / 22101968233579816 : ℝ)
+            by
+        norm_num];
+        exact hau656)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2991344149414517301659 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 6815983797411421 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 6815983797411421 : ℝ) =
+              (6815983797411421 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau655)
+  have hz1 :
+    (10000000000000000 / 6815983797411421 : ℝ) ≤
+      1 / Real.sqrt (1858305405063 / 4000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 6815983797411421 : ℝ) =
+          1 / (6815983797411421 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 3407991898705709 : ℝ) ≤
+      (1095945753863272201041 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1858305405063 / 4000000000000 : ℝ) ≤
+      (375000000000000 / 3407991898705709 : ℝ) := by
+    rw [show
+        (375000000000000 / 3407991898705709 : ℝ) =
+          (3 / 40) / (3407991898705709 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2991344149414517301659 / 10000000000000000000000 : ℝ))) -
+        6 * (1095945753863272201041 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1858305405063 / 4000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1858305405063 / 4000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3407991898705709 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2991344149414517301659 / 10000000000000000000000 : ℝ))) -
+          6 * (1095945753863272201041 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (1858305405063 / 4000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1858305405063 / 4000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1858305405063 / 4000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_443 :
+    ((382018028675292399023 / 1000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(3765935164043707208489 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3410419699992949 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (598600291291347412667 / 2000000000000000000000 : ℝ))) -
+              6 * (136896475028022353711 / 1250000000000000000000 : ℝ)))) ≤
+      Vfield (116309625301 / 250000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (116309625301 / 250000000000 : ℝ)) =
+      Real.log (366309625301 / 250000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((116309625301 / 250000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (117715875301 / 250000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (382018028675292399023 / 1000000000000000000000 : ℝ) ≤
+      Real.log (366309625301 / 250000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (117715875301 / 250000000000 : ℝ) ≤
+      (-(3765935164043707208489 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (117715875301 / 250000000000 : ℝ) = (117715875301 / 62500000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (117715875301 / 62500000000 : ℝ) ≤
+            (3165536638956292791511 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3410419699992949 / 5000000000000000 : ℝ) ≤ Real.sqrt (116309625301 / 250000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (116309625301 / 250000000000 : ℝ) ≤ (6820839399985901 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (116309625301 / 250000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau658 :
+    Real.arctan (6820839399985901 / 22104703636206877 : ℝ) ≤
+      (598600291291347412667 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau657 :
+    Real.arctan (6820839399985901 / 10000000000000000 : ℝ) ≤
+      (2 * (598600291291347412667 / 2000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12104703636206877 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (6820839399985901 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (6820839399985901 / 10000000000000000 : ℝ) /
+                (1 + (12104703636206877 / 10000000000000000 : ℝ)) =
+              (6820839399985901 / 22104703636206877 : ℝ)
+            by
+        norm_num];
+        exact hau658)
+  have hat1 :
+    (Real.pi / 2 - (2 * (598600291291347412667 / 2000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 6820839399985901 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 6820839399985901 : ℝ) =
+              (6820839399985901 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau657)
+  have hz1 :
+    (10000000000000000 / 6820839399985901 : ℝ) ≤
+      1 / Real.sqrt (116309625301 / 250000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 6820839399985901 : ℝ) =
+          1 / (6820839399985901 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 3410419699992949 : ℝ) ≤
+      (136896475028022353711 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (116309625301 / 250000000000 : ℝ) ≤
+      (375000000000000 / 3410419699992949 : ℝ) := by
+    rw [show
+        (375000000000000 / 3410419699992949 : ℝ) =
+          (3 / 40) / (3410419699992949 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (598600291291347412667 / 2000000000000000000000 : ℝ))) -
+        6 * (136896475028022353711 / 1250000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (116309625301 / 250000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (116309625301 / 250000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3410419699992949 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (598600291291347412667 / 2000000000000000000000 : ℝ))) -
+          6 * (136896475028022353711 / 1250000000000000000000 : ℝ))) ≤
+      Real.sqrt (116309625301 / 250000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (116309625301 / 250000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (116309625301 / 250000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_444 :
+    ((3824698325065663360863 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1879454436493986812577 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1365138309684773 / 2000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2994656836293116231503 / 10000000000000000000000 : ℝ))) -
+              6 * (68399967749056262157 / 625000000000000000000 : ℝ)))) ≤
+      Vfield (1863602604569 / 4000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1863602604569 / 4000000000000 : ℝ)) =
+      Real.log (5863602604569 / 4000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1863602604569 / 4000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1886102604569 / 4000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (3824698325065663360863 / 10000000000000000000000 : ℝ) ≤
+      Real.log (5863602604569 / 4000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1886102604569 / 4000000000000 : ℝ) ≤
+      (-(1879454436493986812577 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (1886102604569 / 4000000000000 : ℝ) = (1886102604569 / 1000000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1886102604569 / 1000000000000 : ℝ) ≤
+            (1586281465006013187423 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1365138309684773 / 2000000000000000 : ℝ) ≤ Real.sqrt (1863602604569 / 4000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1863602604569 / 4000000000000 : ℝ) ≤ (1706422887105967 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1863602604569 / 4000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau660 :
+    Real.arctan (3412845774211934 / 11053719210415713 : ℝ) ≤
+      (2994656836293116231503 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau659 :
+    Real.arctan (1706422887105967 / 2500000000000000 : ℝ) ≤
+      (2 * (2994656836293116231503 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6053719210415713 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1706422887105967 / 2500000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1706422887105967 / 2500000000000000 : ℝ) /
+                (1 + (6053719210415713 / 5000000000000000 : ℝ)) =
+              (3412845774211934 / 11053719210415713 : ℝ)
+            by
+        norm_num];
+        exact hau660)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2994656836293116231503 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (2500000000000000 / 1706422887105967 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 1706422887105967 : ℝ) =
+              (1706422887105967 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau659)
+  have hz1 :
+    (2500000000000000 / 1706422887105967 : ℝ) ≤
+      1 / Real.sqrt (1863602604569 / 4000000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 1706422887105967 : ℝ) = 1 / (1706422887105967 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (150000000000000 / 1365138309684773 : ℝ) ≤
+      (68399967749056262157 / 625000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1863602604569 / 4000000000000 : ℝ) ≤
+      (150000000000000 / 1365138309684773 : ℝ) := by
+    rw [show
+        (150000000000000 / 1365138309684773 : ℝ) =
+          (3 / 40) / (1365138309684773 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2994656836293116231503 / 10000000000000000000000 : ℝ))) -
+        6 * (68399967749056262157 / 625000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1863602604569 / 4000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1863602604569 / 4000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1365138309684773 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2994656836293116231503 / 10000000000000000000000 : ℝ))) -
+          6 * (68399967749056262157 / 625000000000000000000 : ℝ))) ≤
+      Real.sqrt (1863602604569 / 4000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1863602604569 / 4000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1863602604569 / 4000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/V37.lean b/LeanPool/Zeta5Irrational/Table/V37.lean
new file mode 100644
index 0000000000..6460a76ad9
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/V37.lean
@@ -0,0 +1,1422 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Energy
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # V37: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_445 :
+    ((765842864606636857627 / 2000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(234493277614459757331 / 312500000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((6830540250086371 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (2996310293174500557039 / 10000000000000000000000 : ℝ))) -
+              6 * (546814399689969324883 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (933125602161 / 2000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (933125602161 / 2000000000000 : ℝ)) =
+      Real.log (2933125602161 / 2000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((933125602161 / 2000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (944375602161 / 2000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (765842864606636857627 / 2000000000000000000000 : ℝ) ≤
+      Real.log (2933125602161 / 2000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (944375602161 / 2000000000000 : ℝ) ≤
+      (-(234493277614459757331 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (944375602161 / 2000000000000 : ℝ) = (944375602161 / 500000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (944375602161 / 500000000000 : ℝ) ≤
+            (198723710073040242669 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (6830540250086371 / 10000000000000000 : ℝ) ≤ Real.sqrt (933125602161 / 2000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (933125602161 / 2000000000000 : ℝ) ≤ (3415270125043187 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (933125602161 / 2000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau662 :
+    Real.arctan (3415270125043187 / 11055086293936074 : ℝ) ≤
+      (2996310293174500557039 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau661 :
+    Real.arctan (3415270125043187 / 5000000000000000 : ℝ) ≤
+      (2 * (2996310293174500557039 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (3027543146968037 / 2500000000000000 : ℝ) ≤
+          Real.sqrt (1 + (3415270125043187 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (3415270125043187 / 5000000000000000 : ℝ) /
+                (1 + (3027543146968037 / 2500000000000000 : ℝ)) =
+              (3415270125043187 / 11055086293936074 : ℝ)
+            by
+        norm_num];
+        exact hau662)
+  have hat1 :
+    (Real.pi / 2 - (2 * (2996310293174500557039 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 3415270125043187 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 3415270125043187 : ℝ) =
+              (3415270125043187 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau661)
+  have hz1 :
+    (5000000000000000 / 3415270125043187 : ℝ) ≤
+      1 / Real.sqrt (933125602161 / 2000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 3415270125043187 : ℝ) = 1 / (3415270125043187 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 6830540250086371 : ℝ) ≤
+      (546814399689969324883 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (933125602161 / 2000000000000 : ℝ) ≤
+      (750000000000000 / 6830540250086371 : ℝ) := by
+    rw [show
+        (750000000000000 / 6830540250086371 : ℝ) =
+          (3 / 40) / (6830540250086371 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (2996310293174500557039 / 10000000000000000000000 : ℝ))) -
+        6 * (546814399689969324883 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (933125602161 / 2000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (933125602161 / 2000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((6830540250086371 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (2996310293174500557039 / 10000000000000000000000 : ℝ))) -
+          6 * (546814399689969324883 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (933125602161 / 2000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (933125602161 / 2000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (933125602161 / 2000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_446 :
+    ((3838240205298106251957 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(747577809759293719037 / 1000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1710056835599697 / 2500000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (599922291000491826411 / 2000000000000000000000 : ℝ))) -
+              6 * (218418460430541568817 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (467887100957 / 1000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (467887100957 / 1000000000000 : ℝ)) =
+      Real.log (1467887100957 / 1000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((467887100957 / 1000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (473512100957 / 1000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (3838240205298106251957 / 10000000000000000000000 : ℝ) ≤
+      Real.log (1467887100957 / 1000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (473512100957 / 1000000000000 : ℝ) ≤
+      (-(747577809759293719037 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (473512100957 / 1000000000000 : ℝ) = (473512100957 / 250000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (473512100957 / 250000000000 : ℝ) ≤
+            (638716550840706280963 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1710056835599697 / 2500000000000000 : ℝ) ≤ Real.sqrt (467887100957 / 1000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (467887100957 / 1000000000000 : ℝ) ≤ (6840227342398791 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (467887100957 / 1000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau664 :
+    Real.arctan (207279616436327 / 670170880935590 : ℝ) ≤
+      (599922291000491826411 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau663 :
+    Real.arctan (6840227342398791 / 10000000000000000 : ℝ) ≤
+      (2 * (599922291000491826411 / 2000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (1211563907087447 / 1000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (6840227342398791 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (6840227342398791 / 10000000000000000 : ℝ) /
+                (1 + (1211563907087447 / 1000000000000000 : ℝ)) =
+              (207279616436327 / 670170880935590 : ℝ)
+            by
+        norm_num];
+        exact hau664)
+  have hat1 :
+    (Real.pi / 2 - (2 * (599922291000491826411 / 2000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 6840227342398791 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 6840227342398791 : ℝ) =
+              (6840227342398791 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau663)
+  have hz1 :
+    (10000000000000000 / 6840227342398791 : ℝ) ≤
+      1 / Real.sqrt (467887100957 / 1000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 6840227342398791 : ℝ) =
+          1 / (6840227342398791 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (62500000000000 / 570018945199899 : ℝ) ≤
+      (218418460430541568817 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (467887100957 / 1000000000000 : ℝ) ≤
+      (62500000000000 / 570018945199899 : ℝ) := by
+    rw [show
+        (62500000000000 / 570018945199899 : ℝ) =
+          (3 / 40) / (1710056835599697 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (599922291000491826411 / 2000000000000000000000 : ℝ))) -
+        6 * (218418460430541568817 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (467887100957 / 1000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (467887100957 / 1000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1710056835599697 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (599922291000491826411 / 2000000000000000000000 : ℝ))) -
+          6 * (218418460430541568817 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (467887100957 / 1000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (467887100957 / 1000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (467887100957 / 1000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_447 :
+    ((240453621765865922073 / 625000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(7447849530514487331947 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1369980147058343 / 2000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (750726243913883062749 / 2500000000000000000000 : ℝ))) -
+              6 * (218112452596585238241 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (938422801667 / 2000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (938422801667 / 2000000000000 : ℝ)) =
+      Real.log (2938422801667 / 2000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((938422801667 / 2000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (949672801667 / 2000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (240453621765865922073 / 625000000000000000000 : ℝ) ≤
+      Real.log (2938422801667 / 2000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (949672801667 / 2000000000000 : ℝ) ≤
+      (-(7447849530514487331947 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (949672801667 / 2000000000000 : ℝ) = (949672801667 / 500000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (949672801667 / 500000000000 : ℝ) ≤
+            (6415094075485512668053 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1369980147058343 / 2000000000000000 : ℝ) ≤ Real.sqrt (938422801667 / 2000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (938422801667 / 2000000000000 : ℝ) ≤ (3424950367645859 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (938422801667 / 2000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau666 :
+    Real.arctan (3424950367645859 / 11060551544276931 : ℝ) ≤
+      (750726243913883062749 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau665 :
+    Real.arctan (3424950367645859 / 5000000000000000 : ℝ) ≤
+      (2 * (750726243913883062749 / 2500000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6060551544276931 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (3424950367645859 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (3424950367645859 / 5000000000000000 : ℝ) /
+                (1 + (6060551544276931 / 5000000000000000 : ℝ)) =
+              (3424950367645859 / 11060551544276931 : ℝ)
+            by
+        norm_num];
+        exact hau666)
+  have hat1 :
+    (Real.pi / 2 - (2 * (750726243913883062749 / 2500000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 3424950367645859 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 3424950367645859 : ℝ) =
+              (3424950367645859 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau665)
+  have hz1 :
+    (5000000000000000 / 3424950367645859 : ℝ) ≤
+      1 / Real.sqrt (938422801667 / 2000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 3424950367645859 : ℝ) = 1 / (3424950367645859 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (150000000000000 / 1369980147058343 : ℝ) ≤
+      (218112452596585238241 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (938422801667 / 2000000000000 : ℝ) ≤
+      (150000000000000 / 1369980147058343 : ℝ) := by
+    rw [show
+        (150000000000000 / 1369980147058343 : ℝ) =
+          (3 / 40) / (1369980147058343 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (750726243913883062749 / 2500000000000000000000 : ℝ))) -
+        6 * (218112452596585238241 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (938422801667 / 2000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (938422801667 / 2000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1369980147058343 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (750726243913883062749 / 2500000000000000000000 : ℝ))) -
+          6 * (218112452596585238241 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (938422801667 / 2000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (938422801667 / 2000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (938422801667 / 2000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_448 :
+    ((3856267566566845575453 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1854999686683688517279 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3429780243361081 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1503095444308757277833 / 5000000000000000000000 : ℝ))) -
+              6 * (217807727351321952453 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (47053570071 / 100000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (47053570071 / 100000000000 : ℝ)) =
+      Real.log (147053570071 / 100000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((47053570071 / 100000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (47616070071 / 100000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (3856267566566845575453 / 10000000000000000000000 : ℝ) ≤
+      Real.log (147053570071 / 100000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (47616070071 / 100000000000 : ℝ) ≤
+      (-(1854999686683688517279 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (47616070071 / 100000000000 : ℝ) = (47616070071 / 25000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (47616070071 / 25000000000 : ℝ) ≤
+            (1610736214816311482721 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3429780243361081 / 5000000000000000 : ℝ) ≤ Real.sqrt (47053570071 / 100000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (47053570071 / 100000000000 : ℝ) ≤ (1371912097344433 / 2000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (47053570071 / 100000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau668 :
+    Real.arctan (1371912097344433 / 4425312928848564 : ℝ) ≤
+      (1503095444308757277833 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau667 :
+    Real.arctan (1371912097344433 / 2000000000000000 : ℝ) ≤
+      (2 * (1503095444308757277833 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (606328232212141 / 500000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1371912097344433 / 2000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1371912097344433 / 2000000000000000 : ℝ) /
+                (1 + (606328232212141 / 500000000000000 : ℝ)) =
+              (1371912097344433 / 4425312928848564 : ℝ)
+            by
+        norm_num];
+        exact hau668)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1503095444308757277833 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (2000000000000000 / 1371912097344433 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2000000000000000 / 1371912097344433 : ℝ) =
+              (1371912097344433 / 2000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau667)
+  have hz1 :
+    (2000000000000000 / 1371912097344433 : ℝ) ≤ 1 / Real.sqrt (47053570071 / 100000000000 : ℝ) := by
+    rw [show
+        (2000000000000000 / 1371912097344433 : ℝ) = 1 / (1371912097344433 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 3429780243361081 : ℝ) ≤
+      (217807727351321952453 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (47053570071 / 100000000000 : ℝ) ≤
+      (375000000000000 / 3429780243361081 : ℝ) := by
+    rw [show
+        (375000000000000 / 3429780243361081 : ℝ) =
+          (3 / 40) / (3429780243361081 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1503095444308757277833 / 5000000000000000000000 : ℝ))) -
+        6 * (217807727351321952453 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (47053570071 / 100000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (47053570071 / 100000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3429780243361081 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1503095444308757277833 / 5000000000000000000000 : ℝ))) -
+          6 * (217807727351321952453 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (47053570071 / 100000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (47053570071 / 100000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (47053570071 / 100000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_449 :
+    ((3865269074863888535449 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(7392225314191339633631 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1717301663559907 / 2500000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3009469227143283066133 / 10000000000000000000000 : ℝ))) -
+              6 * (1087521378799764588307 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (943720001173 / 2000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (943720001173 / 2000000000000 : ℝ)) =
+      Real.log (2943720001173 / 2000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((943720001173 / 2000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (954970001173 / 2000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (3865269074863888535449 / 10000000000000000000000 : ℝ) ≤
+      Real.log (2943720001173 / 2000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (954970001173 / 2000000000000 : ℝ) ≤
+      (-(7392225314191339633631 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (954970001173 / 2000000000000 : ℝ) = (954970001173 / 500000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (954970001173 / 500000000000 : ℝ) ≤
+            (6470718291808660366369 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1717301663559907 / 2500000000000000 : ℝ) ≤ Real.sqrt (943720001173 / 2000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (943720001173 / 2000000000000 : ℝ) ≤ (6869206654239631 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (943720001173 / 2000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau670 :
+    Real.arctan (528400511864587 / 1702463364712795 : ℝ) ≤
+      (3009469227143283066133 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau669 :
+    Real.arctan (6869206654239631 / 10000000000000000 : ℝ) ≤
+      (2 * (3009469227143283066133 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (2426404748253267 / 2000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (6869206654239631 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (6869206654239631 / 10000000000000000 : ℝ) /
+                (1 + (2426404748253267 / 2000000000000000 : ℝ)) =
+              (528400511864587 / 1702463364712795 : ℝ)
+            by
+        norm_num];
+        exact hau670)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3009469227143283066133 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 6869206654239631 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 6869206654239631 : ℝ) =
+              (6869206654239631 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau669)
+  have hz1 :
+    (10000000000000000 / 6869206654239631 : ℝ) ≤
+      1 / Real.sqrt (943720001173 / 2000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 6869206654239631 : ℝ) =
+          1 / (6869206654239631 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (187500000000000 / 1717301663559907 : ℝ) ≤
+      (1087521378799764588307 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (943720001173 / 2000000000000 : ℝ) ≤
+      (187500000000000 / 1717301663559907 : ℝ) := by
+    rw [show
+        (187500000000000 / 1717301663559907 : ℝ) =
+          (3 / 40) / (1717301663559907 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3009469227143283066133 / 10000000000000000000000 : ℝ))) -
+        6 * (1087521378799764588307 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (943720001173 / 2000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (943720001173 / 2000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1717301663559907 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3009469227143283066133 / 10000000000000000000000 : ℝ))) -
+          6 * (1087521378799764588307 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (943720001173 / 2000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (943720001173 / 2000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (943720001173 / 2000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_450 :
+    ((3874262487732329959907 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(7364528804411839307839 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3439419647495053 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (753185006065245064909 / 2500000000000000000000 : ℝ))) -
+              6 * (1086010444872883962773 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (473184300463 / 1000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (473184300463 / 1000000000000 : ℝ)) =
+      Real.log (1473184300463 / 1000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((473184300463 / 1000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (478809300463 / 1000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (3874262487732329959907 / 10000000000000000000000 : ℝ) ≤
+      Real.log (1473184300463 / 1000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (478809300463 / 1000000000000 : ℝ) ≤
+      (-(7364528804411839307839 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (478809300463 / 1000000000000 : ℝ) = (478809300463 / 250000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (478809300463 / 250000000000 : ℝ) ≤
+            (6498414801588160692161 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3439419647495053 / 5000000000000000 : ℝ) ≤ Real.sqrt (473184300463 / 1000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (473184300463 / 1000000000000 : ℝ) ≤ (6878839294990109 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (473184300463 / 1000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau672 :
+    Real.arctan (6878839294990109 / 22137480382941922 : ℝ) ≤
+      (753185006065245064909 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau671 :
+    Real.arctan (6878839294990109 / 10000000000000000 : ℝ) ≤
+      (2 * (753185006065245064909 / 2500000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6068740191470961 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (6878839294990109 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (6878839294990109 / 10000000000000000 : ℝ) /
+                (1 + (6068740191470961 / 5000000000000000 : ℝ)) =
+              (6878839294990109 / 22137480382941922 : ℝ)
+            by
+        norm_num];
+        exact hau672)
+  have hat1 :
+    (Real.pi / 2 - (2 * (753185006065245064909 / 2500000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 6878839294990109 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 6878839294990109 : ℝ) =
+              (6878839294990109 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau671)
+  have hz1 :
+    (10000000000000000 / 6878839294990109 : ℝ) ≤
+      1 / Real.sqrt (473184300463 / 1000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 6878839294990109 : ℝ) =
+          1 / (6878839294990109 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 3439419647495053 : ℝ) ≤
+      (1086010444872883962773 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (473184300463 / 1000000000000 : ℝ) ≤
+      (375000000000000 / 3439419647495053 : ℝ) := by
+    rw [show
+        (375000000000000 / 3439419647495053 : ℝ) =
+          (3 / 40) / (3439419647495053 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (753185006065245064909 / 2500000000000000000000 : ℝ))) -
+        6 * (1086010444872883962773 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (473184300463 / 1000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (473184300463 / 1000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3439419647495053 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (753185006065245064909 / 2500000000000000000000 : ℝ))) -
+          6 * (1086010444872883962773 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (473184300463 / 1000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (473184300463 / 1000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (473184300463 / 1000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_451 :
+    ((776649563944038904551 / 2000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(3668454396237089531377 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((215264327053751 / 312500000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (754000828193542139837 / 2500000000000000000000 : ℝ))) -
+              6 * (54225289558276524857 / 500000000000000000000 : ℝ)))) ≤
+      Vfield (949017200679 / 2000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (949017200679 / 2000000000000 : ℝ)) =
+      Real.log (2949017200679 / 2000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((949017200679 / 2000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (960267200679 / 2000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (776649563944038904551 / 2000000000000000000000 : ℝ) ≤
+      Real.log (2949017200679 / 2000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (960267200679 / 2000000000000 : ℝ) ≤
+      (-(3668454396237089531377 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (960267200679 / 2000000000000 : ℝ) = (960267200679 / 500000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (960267200679 / 500000000000 : ℝ) ≤
+            (3263017406762910468623 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (215264327053751 / 312500000000000 : ℝ) ≤ Real.sqrt (949017200679 / 2000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (949017200679 / 2000000000000 : ℝ) ≤ (1377691693144007 / 2000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (949017200679 / 2000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau674 :
+    Real.arctan (6888458465720035 / 22142934572579644 : ℝ) ≤
+      (754000828193542139837 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau673 :
+    Real.arctan (1377691693144007 / 2000000000000000 : ℝ) ≤
+      (2 * (754000828193542139837 / 2500000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (3035733643144911 / 2500000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1377691693144007 / 2000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1377691693144007 / 2000000000000000 : ℝ) /
+                (1 + (3035733643144911 / 2500000000000000 : ℝ)) =
+              (6888458465720035 / 22142934572579644 : ℝ)
+            by
+        norm_num];
+        exact hau674)
+  have hat1 :
+    (Real.pi / 2 - (2 * (754000828193542139837 / 2500000000000000000000 : ℝ))) ≤
+      Real.arctan (2000000000000000 / 1377691693144007 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2000000000000000 / 1377691693144007 : ℝ) =
+              (1377691693144007 / 2000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau673)
+  have hz1 :
+    (2000000000000000 / 1377691693144007 : ℝ) ≤
+      1 / Real.sqrt (949017200679 / 2000000000000 : ℝ) := by
+    rw [show
+        (2000000000000000 / 1377691693144007 : ℝ) = 1 / (1377691693144007 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (23437500000000 / 215264327053751 : ℝ) ≤
+      (54225289558276524857 / 500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (949017200679 / 2000000000000 : ℝ) ≤
+      (23437500000000 / 215264327053751 : ℝ) := by
+    rw [show
+        (23437500000000 / 215264327053751 : ℝ) =
+          (3 / 40) / (215264327053751 / 312500000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (754000828193542139837 / 2500000000000000000000 : ℝ))) -
+        6 * (54225289558276524857 / 500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (949017200679 / 2000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (949017200679 / 2000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((215264327053751 / 312500000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (754000828193542139837 / 2500000000000000000000 : ℝ))) -
+          6 * (54225289558276524857 / 500000000000000000000 : ℝ))) ≤
+      Real.sqrt (949017200679 / 2000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (949017200679 / 2000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (949017200679 / 2000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_452 :
+    ((7784450170672652891 / 20000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(7309364856967499426647 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((862258027847523 / 1250000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3019259125263958673499 / 10000000000000000000000 : ℝ))) -
+              6 * (270751843572750147929 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (59479112527 / 125000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (59479112527 / 125000000000 : ℝ)) =
+      Real.log (184479112527 / 125000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((59479112527 / 125000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (60182237527 / 125000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (7784450170672652891 / 20000000000000000000 : ℝ) ≤
+      Real.log (184479112527 / 125000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (60182237527 / 125000000000 : ℝ) ≤
+      (-(7309364856967499426647 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (60182237527 / 125000000000 : ℝ) = (60182237527 / 31250000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (60182237527 / 31250000000 : ℝ) ≤
+            (6553578749032500573353 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (862258027847523 / 1250000000000000 : ℝ) ≤ Real.sqrt (59479112527 / 125000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (59479112527 / 125000000000 : ℝ) ≤ (6898064222780187 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (59479112527 / 125000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau676 :
+    Real.arctan (6898064222780187 / 22148386313482132 : ℝ) ≤
+      (3019259125263958673499 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau675 :
+    Real.arctan (6898064222780187 / 10000000000000000 : ℝ) ≤
+      (2 * (3019259125263958673499 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (3037096578370533 / 2500000000000000 : ℝ) ≤
+          Real.sqrt (1 + (6898064222780187 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (6898064222780187 / 10000000000000000 : ℝ) /
+                (1 + (3037096578370533 / 2500000000000000 : ℝ)) =
+              (6898064222780187 / 22148386313482132 : ℝ)
+            by
+        norm_num];
+        exact hau676)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3019259125263958673499 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 6898064222780187 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 6898064222780187 : ℝ) =
+              (6898064222780187 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau675)
+  have hz1 :
+    (10000000000000000 / 6898064222780187 : ℝ) ≤ 1 / Real.sqrt (59479112527 / 125000000000 : ℝ) :=
+  by
+    rw [show
+        (10000000000000000 / 6898064222780187 : ℝ) =
+          1 / (6898064222780187 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (31250000000000 / 287419342615841 : ℝ) ≤
+      (270751843572750147929 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (59479112527 / 125000000000 : ℝ) ≤
+      (31250000000000 / 287419342615841 : ℝ) := by
+    rw [show
+        (31250000000000 / 287419342615841 : ℝ) =
+          (3 / 40) / (862258027847523 / 1250000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3019259125263958673499 / 10000000000000000000000 : ℝ))) -
+        6 * (270751843572750147929 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (59479112527 / 125000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (59479112527 / 125000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((862258027847523 / 1250000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3019259125263958673499 / 10000000000000000000000 : ℝ))) -
+          6 * (270751843572750147929 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (59479112527 / 125000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (59479112527 / 125000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (59479112527 / 125000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_453 :
+    ((780238859810106012883 / 2000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(7281896579953576581699 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1726914155532383 / 2500000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (604501498818222252083 / 2000000000000000000000 : ℝ))) -
+              6 * (216303030256213961533 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (190862880037 / 400000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (190862880037 / 400000000000 : ℝ)) =
+      Real.log (590862880037 / 400000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((190862880037 / 400000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (193112880037 / 400000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (780238859810106012883 / 2000000000000000000000 : ℝ) ≤
+      Real.log (590862880037 / 400000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (193112880037 / 400000000000 : ℝ) ≤
+      (-(7281896579953576581699 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (193112880037 / 400000000000 : ℝ) = (193112880037 / 100000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (193112880037 / 100000000000 : ℝ) ≤
+            (6581047026046423418301 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1726914155532383 / 2500000000000000 : ℝ) ≤ Real.sqrt (190862880037 / 400000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (190862880037 / 400000000000 : ℝ) ≤ (1381531324425907 / 2000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (190862880037 / 400000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau678 :
+    Real.arctan (6907656622129535 / 22153835608944609 : ℝ) ≤
+      (604501498818222252083 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau677 :
+    Real.arctan (1381531324425907 / 2000000000000000 : ℝ) ≤
+      (2 * (604501498818222252083 / 2000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12153835608944609 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1381531324425907 / 2000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1381531324425907 / 2000000000000000 : ℝ) /
+                (1 + (12153835608944609 / 10000000000000000 : ℝ)) =
+              (6907656622129535 / 22153835608944609 : ℝ)
+            by
+        norm_num];
+        exact hau678)
+  have hat1 :
+    (Real.pi / 2 - (2 * (604501498818222252083 / 2000000000000000000000 : ℝ))) ≤
+      Real.arctan (2000000000000000 / 1381531324425907 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2000000000000000 / 1381531324425907 : ℝ) =
+              (1381531324425907 / 2000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau677)
+  have hz1 :
+    (2000000000000000 / 1381531324425907 : ℝ) ≤ 1 / Real.sqrt (190862880037 / 400000000000 : ℝ) :=
+  by
+    rw [show
+        (2000000000000000 / 1381531324425907 : ℝ) = 1 / (1381531324425907 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (187500000000000 / 1726914155532383 : ℝ) ≤
+      (216303030256213961533 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (190862880037 / 400000000000 : ℝ) ≤
+      (187500000000000 / 1726914155532383 : ℝ) := by
+    rw [show
+        (187500000000000 / 1726914155532383 : ℝ) =
+          (3 / 40) / (1726914155532383 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (604501498818222252083 / 2000000000000000000000 : ℝ))) -
+        6 * (216303030256213961533 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (190862880037 / 400000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (190862880037 / 400000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1726914155532383 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (604501498818222252083 / 2000000000000000000000 : ℝ))) -
+          6 * (216303030256213961533 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (190862880037 / 400000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (190862880037 / 400000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (190862880037 / 400000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_454 :
+    ((782031095058741953577 / 2000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1813625886732192964727 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((6917235719339047 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3025748451398115421169 / 10000000000000000000000 : ℝ))) -
+              6 * (33750908736900517841 / 312500000000000000000 : ℝ)))) ≤
+      Vfield (478481499969 / 1000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (478481499969 / 1000000000000 : ℝ)) =
+      Real.log (1478481499969 / 1000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((478481499969 / 1000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (484106499969 / 1000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (782031095058741953577 / 2000000000000000000000 : ℝ) ≤
+      Real.log (1478481499969 / 1000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (484106499969 / 1000000000000 : ℝ) ≤
+      (-(1813625886732192964727 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (484106499969 / 1000000000000 : ℝ) = (484106499969 / 250000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (484106499969 / 250000000000 : ℝ) ≤
+            (1652110014767807035273 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (6917235719339047 / 10000000000000000 : ℝ) ≤ Real.sqrt (478481499969 / 1000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (478481499969 / 1000000000000 : ℝ) ≤ (138344714386781 / 200000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (478481499969 / 1000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau680 :
+    Real.arctan (988176531334150 / 3165611780322131 : ℝ) ≤
+      (3025748451398115421169 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau679 :
+    Real.arctan (138344714386781 / 200000000000000 : ℝ) ≤
+      (2 * (3025748451398115421169 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12159282462254917 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (138344714386781 / 200000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (138344714386781 / 200000000000000 : ℝ) /
+                (1 + (12159282462254917 / 10000000000000000 : ℝ)) =
+              (988176531334150 / 3165611780322131 : ℝ)
+            by
+        norm_num];
+        exact hau680)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3025748451398115421169 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (200000000000000 / 138344714386781 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (200000000000000 / 138344714386781 : ℝ) =
+              (138344714386781 / 200000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau679)
+  have hz1 :
+    (200000000000000 / 138344714386781 : ℝ) ≤ 1 / Real.sqrt (478481499969 / 1000000000000 : ℝ) := by
+    rw [show (200000000000000 / 138344714386781 : ℝ) = 1 / (138344714386781 / 200000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 6917235719339047 : ℝ) ≤
+      (33750908736900517841 / 312500000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (478481499969 / 1000000000000 : ℝ) ≤
+      (750000000000000 / 6917235719339047 : ℝ) := by
+    rw [show
+        (750000000000000 / 6917235719339047 : ℝ) =
+          (3 / 40) / (6917235719339047 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3025748451398115421169 / 10000000000000000000000 : ℝ))) -
+        6 * (33750908736900517841 / 312500000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (478481499969 / 1000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (478481499969 / 1000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((6917235719339047 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3025748451398115421169 / 10000000000000000000000 : ℝ))) -
+          6 * (33750908736900517841 / 312500000000000000000 : ℝ))) ≤
+      Real.sqrt (478481499969 / 1000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (478481499969 / 1000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (478481499969 / 1000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_455 :
+    ((3919108628458009307323 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(451699084174156310643 / 625000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((6926801569595449 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (757245507277809449937 / 2500000000000000000000 : ℝ))) -
+              6 * (269637279260881504219 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (959611599691 / 2000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (959611599691 / 2000000000000 : ℝ)) =
+      Real.log (2959611599691 / 2000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((959611599691 / 2000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (970861599691 / 2000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (3919108628458009307323 / 10000000000000000000000 : ℝ) ≤
+      Real.log (2959611599691 / 2000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (970861599691 / 2000000000000 : ℝ) ≤
+      (-(451699084174156310643 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (970861599691 / 2000000000000 : ℝ) = (970861599691 / 500000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (970861599691 / 500000000000 : ℝ) ≤
+            (414734891200843689357 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (6926801569595449 / 10000000000000000 : ℝ) ≤ Real.sqrt (959611599691 / 2000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (959611599691 / 2000000000000 : ℝ) ≤ (1731700392398863 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (959611599691 / 2000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau682 :
+    Real.arctan (1731700392398863 / 5541181719173383 : ℝ) ≤
+      (757245507277809449937 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau681 :
+    Real.arctan (1731700392398863 / 2500000000000000 : ℝ) ≤
+      (2 * (757245507277809449937 / 2500000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (3041181719173383 / 2500000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1731700392398863 / 2500000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1731700392398863 / 2500000000000000 : ℝ) /
+                (1 + (3041181719173383 / 2500000000000000 : ℝ)) =
+              (1731700392398863 / 5541181719173383 : ℝ)
+            by
+        norm_num];
+        exact hau682)
+  have hat1 :
+    (Real.pi / 2 - (2 * (757245507277809449937 / 2500000000000000000000 : ℝ))) ≤
+      Real.arctan (2500000000000000 / 1731700392398863 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 1731700392398863 : ℝ) =
+              (1731700392398863 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau681)
+  have hz1 :
+    (2500000000000000 / 1731700392398863 : ℝ) ≤
+      1 / Real.sqrt (959611599691 / 2000000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 1731700392398863 : ℝ) = 1 / (1731700392398863 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 6926801569595449 : ℝ) ≤
+      (269637279260881504219 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (959611599691 / 2000000000000 : ℝ) ≤
+      (750000000000000 / 6926801569595449 : ℝ) := by
+    rw [show
+        (750000000000000 / 6926801569595449 : ℝ) =
+          (3 / 40) / (6926801569595449 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (757245507277809449937 / 2500000000000000000000 : ℝ))) -
+        6 * (269637279260881504219 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (959611599691 / 2000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (959611599691 / 2000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((6926801569595449 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (757245507277809449937 / 2500000000000000000000 : ℝ))) -
+          6 * (269637279260881504219 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (959611599691 / 2000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (959611599691 / 2000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (959611599691 / 2000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_456 :
+    ((245503360806059405313 / 625000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(7199941571780214482551 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1734088556926231 / 2500000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (606441651788510147911 / 2000000000000000000000 : ℝ))) -
+              6 * (538537610962834018837 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (240565049861 / 500000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (240565049861 / 500000000000 : ℝ)) =
+      Real.log (740565049861 / 500000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((240565049861 / 500000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (243377549861 / 500000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (245503360806059405313 / 625000000000000000000 : ℝ) ≤
+      Real.log (740565049861 / 500000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (243377549861 / 500000000000 : ℝ) ≤
+      (-(7199941571780214482551 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (243377549861 / 500000000000 : ℝ) = (243377549861 / 125000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (243377549861 / 125000000000 : ℝ) ≤
+            (6663002034219785517449 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1734088556926231 / 2500000000000000 : ℝ) ≤ Real.sqrt (240565049861 / 500000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (240565049861 / 500000000000 : ℝ) ≤ (6936354227704927 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (240565049861 / 500000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau684 :
+    Real.arctan (6936354227704927 / 22170168855533598 : ℝ) ≤
+      (606441651788510147911 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau683 :
+    Real.arctan (6936354227704927 / 10000000000000000 : ℝ) ≤
+      (2 * (606441651788510147911 / 2000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6085084427766799 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (6936354227704927 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (6936354227704927 / 10000000000000000 : ℝ) /
+                (1 + (6085084427766799 / 5000000000000000 : ℝ)) =
+              (6936354227704927 / 22170168855533598 : ℝ)
+            by
+        norm_num];
+        exact hau684)
+  have hat1 :
+    (Real.pi / 2 - (2 * (606441651788510147911 / 2000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 6936354227704927 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 6936354227704927 : ℝ) =
+              (6936354227704927 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau683)
+  have hz1 :
+    (10000000000000000 / 6936354227704927 : ℝ) ≤
+      1 / Real.sqrt (240565049861 / 500000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 6936354227704927 : ℝ) =
+          1 / (6936354227704927 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (187500000000000 / 1734088556926231 : ℝ) ≤
+      (538537610962834018837 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (240565049861 / 500000000000 : ℝ) ≤
+      (187500000000000 / 1734088556926231 : ℝ) := by
+    rw [show
+        (187500000000000 / 1734088556926231 : ℝ) =
+          (3 / 40) / (1734088556926231 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (606441651788510147911 / 2000000000000000000000 : ℝ))) -
+        6 * (538537610962834018837 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (240565049861 / 500000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (240565049861 / 500000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1734088556926231 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (606441651788510147911 / 2000000000000000000000 : ℝ))) -
+          6 * (538537610962834018837 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (240565049861 / 500000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (240565049861 / 500000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (240565049861 / 500000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/V38.lean b/LeanPool/Zeta5Irrational/Table/V38.lean
new file mode 100644
index 0000000000..8cd135a850
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/V38.lean
@@ -0,0 +1,1419 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Energy
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # V38: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_457 :
+    ((1972960046410281505363 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(7145675682770966266981 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((6955420184827081 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3038638800749049350961 / 10000000000000000000000 : ℝ))) -
+              6 * (16783522952507147781 / 156250000000000000000 : ℝ)))) ≤
+      Vfield (19351147979 / 40000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (19351147979 / 40000000000 : ℝ)) = Real.log (59351147979 / 40000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((19351147979 / 40000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (19576147979 / 40000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1972960046410281505363 / 5000000000000000000000 : ℝ) ≤
+      Real.log (59351147979 / 40000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (19576147979 / 40000000000 : ℝ) ≤
+      (-(7145675682770966266981 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (19576147979 / 40000000000 : ℝ) = (19576147979 / 10000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19576147979 / 10000000000 : ℝ) ≤
+            (6717267923229033733019 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (6955420184827081 / 10000000000000000 : ℝ) ≤ Real.sqrt (19351147979 / 40000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (19351147979 / 40000000000 : ℝ) ≤ (1738855046206771 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (19351147979 / 40000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau686 :
+    Real.arctan (3477710092413542 / 11090522759737049 : ℝ) ≤
+      (3038638800749049350961 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau685 :
+    Real.arctan (1738855046206771 / 2500000000000000 : ℝ) ≤
+      (2 * (3038638800749049350961 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6090522759737049 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1738855046206771 / 2500000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1738855046206771 / 2500000000000000 : ℝ) /
+                (1 + (6090522759737049 / 5000000000000000 : ℝ)) =
+              (3477710092413542 / 11090522759737049 : ℝ)
+            by
+        norm_num];
+        exact hau686)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3038638800749049350961 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (2500000000000000 / 1738855046206771 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 1738855046206771 : ℝ) =
+              (1738855046206771 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau685)
+  have hz1 :
+    (2500000000000000 / 1738855046206771 : ℝ) ≤ 1 / Real.sqrt (19351147979 / 40000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 1738855046206771 : ℝ) = 1 / (1738855046206771 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 6955420184827081 : ℝ) ≤
+      (16783522952507147781 / 156250000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (19351147979 / 40000000000 : ℝ) ≤
+      (750000000000000 / 6955420184827081 : ℝ) := by
+    rw [show
+        (750000000000000 / 6955420184827081 : ℝ) =
+          (3 / 40) / (6955420184827081 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3038638800749049350961 / 10000000000000000000000 : ℝ))) -
+        6 * (16783522952507147781 / 156250000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (19351147979 / 40000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (19351147979 / 40000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((6955420184827081 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3038638800749049350961 / 10000000000000000000000 : ℝ))) -
+          6 * (16783522952507147781 / 156250000000000000000 : ℝ))) ≤
+      Real.sqrt (19351147979 / 40000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (19351147979 / 40000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (19351147979 / 40000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_458 :
+    ((3963754549125513984239 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(709170268375400157447 / 1000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((6974434021682331 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (304504032630575797579 / 1000000000000000000000 : ℝ))) -
+              6 * (21424789892291219471 / 200000000000000000000 : ℝ)))) ≤
+      Vfield (121606824807 / 250000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (121606824807 / 250000000000 : ℝ)) =
+      Real.log (371606824807 / 250000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((121606824807 / 250000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (123013074807 / 250000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (3963754549125513984239 / 10000000000000000000000 : ℝ) ≤
+      Real.log (371606824807 / 250000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (123013074807 / 250000000000 : ℝ) ≤
+      (-(709170268375400157447 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (123013074807 / 250000000000 : ℝ) = (123013074807 / 62500000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (123013074807 / 62500000000 : ℝ) ≤
+            (677124092224599842553 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (6974434021682331 / 10000000000000000 : ℝ) ≤ Real.sqrt (121606824807 / 250000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (121606824807 / 250000000000 : ℝ) ≤ (3487217010841167 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (121606824807 / 250000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau688 :
+    Real.arctan (112490871317457 / 357934072259930 : ℝ) ≤
+      (304504032630575797579 / 1000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau687 :
+    Real.arctan (3487217010841167 / 5000000000000000 : ℝ) ≤
+      (2 * (304504032630575797579 / 1000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (609595624005783 / 500000000000000 : ℝ) ≤
+          Real.sqrt (1 + (3487217010841167 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (3487217010841167 / 5000000000000000 : ℝ) /
+                (1 + (609595624005783 / 500000000000000 : ℝ)) =
+              (112490871317457 / 357934072259930 : ℝ)
+            by
+        norm_num];
+        exact hau688)
+  have hat1 :
+    (Real.pi / 2 - (2 * (304504032630575797579 / 1000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 3487217010841167 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 3487217010841167 : ℝ) =
+              (3487217010841167 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau687)
+  have hz1 :
+    (5000000000000000 / 3487217010841167 : ℝ) ≤ 1 / Real.sqrt (121606824807 / 250000000000 : ℝ) :=
+  by
+    rw [show
+        (5000000000000000 / 3487217010841167 : ℝ) = 1 / (3487217010841167 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 2324811340560777 : ℝ) ≤
+      (21424789892291219471 / 200000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (121606824807 / 250000000000 : ℝ) ≤
+      (250000000000000 / 2324811340560777 : ℝ) := by
+    rw [show
+        (250000000000000 / 2324811340560777 : ℝ) =
+          (3 / 40) / (6974434021682331 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (304504032630575797579 / 1000000000000000000000 : ℝ))) -
+        6 * (21424789892291219471 / 200000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (121606824807 / 250000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (121606824807 / 250000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((6974434021682331 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (304504032630575797579 / 1000000000000000000000 : ℝ))) -
+          6 * (21424789892291219471 / 200000000000000000000 : ℝ))) ≤
+      Real.sqrt (121606824807 / 250000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (121606824807 / 250000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (121606824807 / 250000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_459 :
+    ((995389313815897849753 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(703801943005654114471 / 1000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3496698081694357 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3051413081799342518727 / 10000000000000000000000 : ℝ))) -
+              6 * (534178489479691317771 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (489075898981 / 1000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (489075898981 / 1000000000000 : ℝ)) =
+      Real.log (1489075898981 / 1000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((489075898981 / 1000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (494700898981 / 1000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (995389313815897849753 / 2500000000000000000000 : ℝ) ≤
+      Real.log (1489075898981 / 1000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (494700898981 / 1000000000000 : ℝ) ≤
+      (-(703801943005654114471 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (494700898981 / 1000000000000 : ℝ) = (494700898981 / 250000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (494700898981 / 250000000000 : ℝ) ≤
+            (682492417594345885529 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3496698081694357 / 5000000000000000 : ℝ) ≤ Real.sqrt (489075898981 / 1000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (489075898981 / 1000000000000 : ℝ) ≤ (6993396163388717 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (489075898981 / 1000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau690 :
+    Real.arctan (6993396163388717 / 22202769763381590 : ℝ) ≤
+      (3051413081799342518727 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau689 :
+    Real.arctan (6993396163388717 / 10000000000000000 : ℝ) ≤
+      (2 * (3051413081799342518727 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (1220276976338159 / 1000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (6993396163388717 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (6993396163388717 / 10000000000000000 : ℝ) /
+                (1 + (1220276976338159 / 1000000000000000 : ℝ)) =
+              (6993396163388717 / 22202769763381590 : ℝ)
+            by
+        norm_num];
+        exact hau690)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3051413081799342518727 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 6993396163388717 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 6993396163388717 : ℝ) =
+              (6993396163388717 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau689)
+  have hz1 :
+    (10000000000000000 / 6993396163388717 : ℝ) ≤
+      1 / Real.sqrt (489075898981 / 1000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 6993396163388717 : ℝ) =
+          1 / (6993396163388717 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 3496698081694357 : ℝ) ≤
+      (534178489479691317771 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (489075898981 / 1000000000000 : ℝ) ≤
+      (375000000000000 / 3496698081694357 : ℝ) := by
+    rw [show
+        (375000000000000 / 3496698081694357 : ℝ) =
+          (3 / 40) / (3496698081694357 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3051413081799342518727 / 10000000000000000000000 : ℝ))) -
+        6 * (534178489479691317771 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (489075898981 / 1000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (489075898981 / 1000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3496698081694357 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3051413081799342518727 / 10000000000000000000000 : ℝ))) -
+          6 * (534178489479691317771 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (489075898981 / 1000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (489075898981 / 1000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (489075898981 / 1000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_460 :
+    ((499916040510216863701 / 1250000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(873077853422577693929 / 1250000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1402461405863277 / 2000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3057757310175657760821 / 10000000000000000000000 : ℝ))) -
+              6 * (213099521612080179629 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (245862249367 / 500000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (245862249367 / 500000000000 : ℝ)) =
+      Real.log (745862249367 / 500000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((245862249367 / 500000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (248674749367 / 500000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (499916040510216863701 / 1250000000000000000000 : ℝ) ≤
+      Real.log (745862249367 / 500000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (248674749367 / 500000000000 : ℝ) ≤
+      (-(873077853422577693929 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (248674749367 / 500000000000 : ℝ) = (248674749367 / 125000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (248674749367 / 125000000000 : ℝ) ≤
+            (859790097327422306071 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1402461405863277 / 2000000000000000 : ℝ) ≤ Real.sqrt (245862249367 / 500000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (245862249367 / 500000000000 : ℝ) ≤ (1753076757329097 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (245862249367 / 500000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau692 :
+    Real.arctan (1753076757329097 / 5553404348769992 : ℝ) ≤
+      (3057757310175657760821 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau691 :
+    Real.arctan (1753076757329097 / 2500000000000000 : ℝ) ≤
+      (2 * (3057757310175657760821 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (381675543596249 / 312500000000000 : ℝ) ≤
+          Real.sqrt (1 + (1753076757329097 / 2500000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1753076757329097 / 2500000000000000 : ℝ) /
+                (1 + (381675543596249 / 312500000000000 : ℝ)) =
+              (1753076757329097 / 5553404348769992 : ℝ)
+            by
+        norm_num];
+        exact hau692)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3057757310175657760821 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (2500000000000000 / 1753076757329097 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 1753076757329097 : ℝ) =
+              (1753076757329097 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau691)
+  have hz1 :
+    (2500000000000000 / 1753076757329097 : ℝ) ≤ 1 / Real.sqrt (245862249367 / 500000000000 : ℝ) :=
+  by
+    rw [show
+        (2500000000000000 / 1753076757329097 : ℝ) = 1 / (1753076757329097 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (50000000000000 / 467487135287759 : ℝ) ≤
+      (213099521612080179629 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (245862249367 / 500000000000 : ℝ) ≤
+      (50000000000000 / 467487135287759 : ℝ) := by
+    rw [show
+        (50000000000000 / 467487135287759 : ℝ) =
+          (3 / 40) / (1402461405863277 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3057757310175657760821 / 10000000000000000000000 : ℝ))) -
+        6 * (213099521612080179629 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (245862249367 / 500000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (245862249367 / 500000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1402461405863277 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3057757310175657760821 / 10000000000000000000000 : ℝ))) -
+          6 * (213099521612080179629 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (245862249367 / 500000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (245862249367 / 500000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (245862249367 / 500000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_461 :
+    ((4017067867826327729629 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(6931509830732862114179 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((7031167033195839 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1532036625599493404347 / 5000000000000000000000 : ℝ))) -
+              6 * (132832634229193100987 / 1250000000000000000000 : ℝ)))) ≤
+      Vfield (494373098487 / 1000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (494373098487 / 1000000000000 : ℝ)) =
+      Real.log (1494373098487 / 1000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((494373098487 / 1000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (499998098487 / 1000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (4017067867826327729629 / 10000000000000000000000 : ℝ) ≤
+      Real.log (1494373098487 / 1000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (499998098487 / 1000000000000 : ℝ) ≤
+      (-(6931509830732862114179 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 2
+        (show (499998098487 / 1000000000000 : ℝ) = (499998098487 / 250000000000 : ℝ) / 2 ^ 2 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 15
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (499998098487 / 250000000000 : ℝ) ≤
+            (6931433775267137885821 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (7031167033195839 / 10000000000000000 : ℝ) ≤ Real.sqrt (494373098487 / 1000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (494373098487 / 1000000000000 : ℝ) ≤ (3515583516597921 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (494373098487 / 1000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau694 :
+    Real.arctan (1171861172199307 / 3704075900150728 : ℝ) ≤
+      (1532036625599493404347 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau693 :
+    Real.arctan (3515583516597921 / 5000000000000000 : ℝ) ≤
+      (2 * (1532036625599493404347 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (764028462556523 / 625000000000000 : ℝ) ≤
+          Real.sqrt (1 + (3515583516597921 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (3515583516597921 / 5000000000000000 : ℝ) /
+                (1 + (764028462556523 / 625000000000000 : ℝ)) =
+              (1171861172199307 / 3704075900150728 : ℝ)
+            by
+        norm_num];
+        exact hau694)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1532036625599493404347 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 3515583516597921 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 3515583516597921 : ℝ) =
+              (3515583516597921 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau693)
+  have hz1 :
+    (5000000000000000 / 3515583516597921 : ℝ) ≤
+      1 / Real.sqrt (494373098487 / 1000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 3515583516597921 : ℝ) = 1 / (3515583516597921 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 2343722344398613 : ℝ) ≤
+      (132832634229193100987 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (494373098487 / 1000000000000 : ℝ) ≤
+      (250000000000000 / 2343722344398613 : ℝ) := by
+    rw [show
+        (250000000000000 / 2343722344398613 : ℝ) =
+          (3 / 40) / (7031167033195839 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1532036625599493404347 / 5000000000000000000000 : ℝ))) -
+        6 * (132832634229193100987 / 1250000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (494373098487 / 1000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (494373098487 / 1000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((7031167033195839 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1532036625599493404347 / 5000000000000000000000 : ℝ))) -
+          6 * (132832634229193100987 / 1250000000000000000000 : ℝ))) ≤
+      Real.sqrt (494373098487 / 1000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (494373098487 / 1000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (494373098487 / 1000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_462 :
+    ((4034775998147450528869 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1719669361495489087719 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((44062353645147 / 62500000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (383795142688735162467 / 1250000000000000000000 : ℝ))) -
+              6 * (1059847073905755610613 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (1553192807 / 3125000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1553192807 / 3125000000 : ℝ)) = Real.log (4678192807 / 3125000000 : ℝ) := by
+    norm_num
+  have e2 :
+    Real.log ((1553192807 / 3125000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (392692733 / 781250000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (4034775998147450528869 / 10000000000000000000000 : ℝ) ≤
+      Real.log (4678192807 / 3125000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (392692733 / 781250000 : ℝ) ≤
+      (-(1719669361495489087719 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (392692733 / 781250000 : ℝ) = (392692733 / 390625000 : ℝ) / 2 ^ 1 by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (392692733 / 390625000 : ℝ) ≤
+            (13198589254510912281 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo : (44062353645147 / 62500000000000 : ℝ) ≤ Real.sqrt (1553192807 / 3125000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1553192807 / 3125000000 : ℝ) ≤ (7049976583223523 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1553192807 / 3125000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau696 :
+    Real.arctan (7049976583223523 / 22235283806434568 : ℝ) ≤
+      (383795142688735162467 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau695 :
+    Real.arctan (7049976583223523 / 10000000000000000 : ℝ) ≤
+      (2 * (383795142688735162467 / 1250000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (1529410475804321 / 1250000000000000 : ℝ) ≤
+          Real.sqrt (1 + (7049976583223523 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (7049976583223523 / 10000000000000000 : ℝ) /
+                (1 + (1529410475804321 / 1250000000000000 : ℝ)) =
+              (7049976583223523 / 22235283806434568 : ℝ)
+            by
+        norm_num];
+        exact hau696)
+  have hat1 :
+    (Real.pi / 2 - (2 * (383795142688735162467 / 1250000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 7049976583223523 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 7049976583223523 : ℝ) =
+              (7049976583223523 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau695)
+  have hz1 :
+    (10000000000000000 / 7049976583223523 : ℝ) ≤ 1 / Real.sqrt (1553192807 / 3125000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 7049976583223523 : ℝ) =
+          1 / (7049976583223523 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (1562500000000 / 14687451215049 : ℝ) ≤
+      (1059847073905755610613 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1553192807 / 3125000000 : ℝ) ≤ (1562500000000 / 14687451215049 : ℝ) := by
+    rw [show
+        (1562500000000 / 14687451215049 : ℝ) = (3 / 40) / (44062353645147 / 62500000000000 : ℝ) by
+        norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (383795142688735162467 / 1250000000000000000000 : ℝ))) -
+        6 * (1059847073905755610613 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1553192807 / 3125000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1553192807 / 3125000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((44062353645147 / 62500000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (383795142688735162467 / 1250000000000000000000 : ℝ))) -
+          6 * (1059847073905755610613 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (1553192807 / 3125000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1553192807 / 3125000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1553192807 / 3125000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_463 :
+    ((4052452826103097732711 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(6826122718446278959801 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((70687360821649 / 100000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (769155303670411559609 / 2500000000000000000000 : ℝ))) -
+              6 * (105705531147958899277 / 1000000000000000000000 : ℝ)))) ≤
+      Vfield (499670297993 / 1000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (499670297993 / 1000000000000 : ℝ)) =
+      Real.log (1499670297993 / 1000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((499670297993 / 1000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (505295297993 / 1000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (4052452826103097732711 / 10000000000000000000000 : ℝ) ≤
+      Real.log (1499670297993 / 1000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (505295297993 / 1000000000000 : ℝ) ≤
+      (-(6826122718446278959801 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (505295297993 / 1000000000000 : ℝ) = (505295297993 / 500000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 3
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (505295297993 / 500000000000 : ℝ) ≤
+            (105349084553721040199 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (70687360821649 / 100000000000000 : ℝ) ≤ Real.sqrt (499670297993 / 1000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (499670297993 / 1000000000000 : ℝ) ≤ (7068736082164903 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (499670297993 / 1000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau698 :
+    Real.arctan (7068736082164903 / 22246102637137253 : ℝ) ≤
+      (769155303670411559609 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau697 :
+    Real.arctan (7068736082164903 / 10000000000000000 : ℝ) ≤
+      (2 * (769155303670411559609 / 2500000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12246102637137253 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (7068736082164903 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (7068736082164903 / 10000000000000000 : ℝ) /
+                (1 + (12246102637137253 / 10000000000000000 : ℝ)) =
+              (7068736082164903 / 22246102637137253 : ℝ)
+            by
+        norm_num];
+        exact hau698)
+  have hat1 :
+    (Real.pi / 2 - (2 * (769155303670411559609 / 2500000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 7068736082164903 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 7068736082164903 : ℝ) =
+              (7068736082164903 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau697)
+  have hz1 :
+    (10000000000000000 / 7068736082164903 : ℝ) ≤
+      1 / Real.sqrt (499670297993 / 1000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 7068736082164903 : ℝ) =
+          1 / (7068736082164903 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (7500000000000 / 70687360821649 : ℝ) ≤
+      (105705531147958899277 / 1000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (499670297993 / 1000000000000 : ℝ) ≤
+      (7500000000000 / 70687360821649 : ℝ) := by
+    rw [show
+        (7500000000000 / 70687360821649 : ℝ) = (3 / 40) / (70687360821649 / 100000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (769155303670411559609 / 2500000000000000000000 : ℝ))) -
+        6 * (105705531147958899277 / 1000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (499670297993 / 1000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (499670297993 / 1000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((70687360821649 / 100000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (769155303670411559609 / 2500000000000000000000 : ℝ))) -
+          6 * (105705531147958899277 / 1000000000000000000000 : ℝ))) ≤
+      Real.sqrt (499670297993 / 1000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (499670297993 / 1000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (499670297993 / 1000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_464 :
+    ((4087713016214549625359 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(3360917337612814841707 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3553053255648583 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1544529414447257797889 / 5000000000000000000000 : ℝ))) -
+              6 * (210307467807043208833 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (504967497499 / 1000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (504967497499 / 1000000000000 : ℝ)) =
+      Real.log (1504967497499 / 1000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((504967497499 / 1000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (510592497499 / 1000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (4087713016214549625359 / 10000000000000000000000 : ℝ) ≤
+      Real.log (1504967497499 / 1000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (510592497499 / 1000000000000 : ℝ) ≤
+      (-(3360917337612814841707 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (510592497499 / 1000000000000 : ℝ) = (510592497499 / 500000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (510592497499 / 500000000000 : ℝ) ≤
+            (104818563887185158293 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3553053255648583 / 5000000000000000 : ℝ) ≤ Real.sqrt (504967497499 / 1000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (504967497499 / 1000000000000 : ℝ) ≤ (7106106511297169 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (504967497499 / 1000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau700 :
+    Real.arctan (7106106511297169 / 22267711675365540 : ℝ) ≤
+      (1544529414447257797889 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau699 :
+    Real.arctan (7106106511297169 / 10000000000000000 : ℝ) ≤
+      (2 * (1544529414447257797889 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (613385583768277 / 500000000000000 : ℝ) ≤
+          Real.sqrt (1 + (7106106511297169 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (7106106511297169 / 10000000000000000 : ℝ) /
+                (1 + (613385583768277 / 500000000000000 : ℝ)) =
+              (7106106511297169 / 22267711675365540 : ℝ)
+            by
+        norm_num];
+        exact hau700)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1544529414447257797889 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 7106106511297169 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 7106106511297169 : ℝ) =
+              (7106106511297169 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau699)
+  have hz1 :
+    (10000000000000000 / 7106106511297169 : ℝ) ≤
+      1 / Real.sqrt (504967497499 / 1000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 7106106511297169 : ℝ) =
+          1 / (7106106511297169 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 3553053255648583 : ℝ) ≤
+      (210307467807043208833 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (504967497499 / 1000000000000 : ℝ) ≤
+      (375000000000000 / 3553053255648583 : ℝ) := by
+    rw [show
+        (375000000000000 / 3553053255648583 : ℝ) =
+          (3 / 40) / (3553053255648583 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1544529414447257797889 / 5000000000000000000000 : ℝ))) -
+        6 * (210307467807043208833 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (504967497499 / 1000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (504967497499 / 1000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3553053255648583 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1544529414447257797889 / 5000000000000000000000 : ℝ))) -
+          6 * (210307467807043208833 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (504967497499 / 1000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (504967497499 / 1000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (504967497499 / 1000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_465 :
+    ((4122849314940802649517 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(6618623015859148835301 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1785820359465433 / 2500000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1550693951571935343901 / 5000000000000000000000 : ℝ))) -
+              6 * (209220977745917649767 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (102052939401 / 200000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (102052939401 / 200000000000 : ℝ)) =
+      Real.log (302052939401 / 200000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((102052939401 / 200000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (103177939401 / 200000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (4122849314940802649517 / 10000000000000000000000 : ℝ) ≤
+      Real.log (302052939401 / 200000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (103177939401 / 200000000000 : ℝ) ≤
+      (-(6618623015859148835301 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (103177939401 / 200000000000 : ℝ) = (103177939401 / 100000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (103177939401 / 100000000000 : ℝ) ≤
+            (312848787140851164699 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1785820359465433 / 2500000000000000 : ℝ) ≤ Real.sqrt (102052939401 / 200000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (102052939401 / 200000000000 : ℝ) ≤ (1428656287572347 / 2000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (102052939401 / 200000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau702 :
+    Real.arctan (7143281437861735 / 22289282717087274 : ℝ) ≤
+      (1550693951571935343901 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau701 :
+    Real.arctan (1428656287572347 / 2000000000000000 : ℝ) ≤
+      (2 * (1550693951571935343901 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6144641358543637 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1428656287572347 / 2000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1428656287572347 / 2000000000000000 : ℝ) /
+                (1 + (6144641358543637 / 5000000000000000 : ℝ)) =
+              (7143281437861735 / 22289282717087274 : ℝ)
+            by
+        norm_num];
+        exact hau702)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1550693951571935343901 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (2000000000000000 / 1428656287572347 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2000000000000000 / 1428656287572347 : ℝ) =
+              (1428656287572347 / 2000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau701)
+  have hz1 :
+    (2000000000000000 / 1428656287572347 : ℝ) ≤ 1 / Real.sqrt (102052939401 / 200000000000 : ℝ) :=
+  by
+    rw [show
+        (2000000000000000 / 1428656287572347 : ℝ) = 1 / (1428656287572347 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (187500000000000 / 1785820359465433 : ℝ) ≤
+      (209220977745917649767 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (102052939401 / 200000000000 : ℝ) ≤
+      (187500000000000 / 1785820359465433 : ℝ) := by
+    rw [show
+        (187500000000000 / 1785820359465433 : ℝ) =
+          (3 / 40) / (1785820359465433 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1550693951571935343901 / 5000000000000000000000 : ℝ))) -
+        6 * (209220977745917649767 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (102052939401 / 200000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (102052939401 / 200000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1785820359465433 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1550693951571935343901 / 5000000000000000000000 : ℝ))) -
+          6 * (209220977745917649767 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (102052939401 / 200000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (102052939401 / 200000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (102052939401 / 200000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_466 :
+    ((4157862589852264614611 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1629116436967895470151 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1436052779686039 / 2000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3113610201061039069011 / 10000000000000000000000 : ℝ))) -
+              6 * (520377886983577812947 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (515561896511 / 1000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (515561896511 / 1000000000000 : ℝ)) =
+      Real.log (1515561896511 / 1000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((515561896511 / 1000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (521186896511 / 1000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (4157862589852264614611 / 10000000000000000000000 : ℝ) ≤
+      Real.log (1515561896511 / 1000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (521186896511 / 1000000000000 : ℝ) ≤
+      (-(1629116436967895470151 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (521186896511 / 1000000000000 : ℝ) = (521186896511 / 500000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (521186896511 / 500000000000 : ℝ) ≤
+            (103751513782104529849 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1436052779686039 / 2000000000000000 : ℝ) ≤ Real.sqrt (515561896511 / 1000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (515561896511 / 1000000000000 : ℝ) ≤ (3590131949215099 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (515561896511 / 1000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau704 :
+    Real.arctan (3590131949215099 / 11155407981017586 : ℝ) ≤
+      (3113610201061039069011 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau703 :
+    Real.arctan (3590131949215099 / 5000000000000000 : ℝ) ≤
+      (2 * (3113610201061039069011 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (3077703990508793 / 2500000000000000 : ℝ) ≤
+          Real.sqrt (1 + (3590131949215099 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (3590131949215099 / 5000000000000000 : ℝ) /
+                (1 + (3077703990508793 / 2500000000000000 : ℝ)) =
+              (3590131949215099 / 11155407981017586 : ℝ)
+            by
+        norm_num];
+        exact hau704)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3113610201061039069011 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 3590131949215099 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 3590131949215099 : ℝ) =
+              (3590131949215099 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau703)
+  have hz1 :
+    (5000000000000000 / 3590131949215099 : ℝ) ≤
+      1 / Real.sqrt (515561896511 / 1000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 3590131949215099 : ℝ) = 1 / (3590131949215099 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (150000000000000 / 1436052779686039 : ℝ) ≤
+      (520377886983577812947 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (515561896511 / 1000000000000 : ℝ) ≤
+      (150000000000000 / 1436052779686039 : ℝ) := by
+    rw [show
+        (150000000000000 / 1436052779686039 : ℝ) =
+          (3 / 40) / (1436052779686039 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3113610201061039069011 / 10000000000000000000000 : ℝ))) -
+        6 * (520377886983577812947 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (515561896511 / 1000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (515561896511 / 1000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1436052779686039 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3113610201061039069011 / 10000000000000000000000 : ℝ))) -
+          6 * (520377886983577812947 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (515561896511 / 1000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (515561896511 / 1000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (515561896511 / 1000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_467 :
+    ((834864249188940764473 / 2000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(6468680488511488354407 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((899703389990447 / 1250000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (623866760360916799687 / 2000000000000000000000 : ℝ))) -
+              6 * (1038263128909289262889 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (16577867570387 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (16577867570387 / 32000000000000 : ℝ)) =
+      Real.log (48577867570387 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((16577867570387 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (16757867570387 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (834864249188940764473 / 2000000000000000000000 : ℝ) ≤
+      Real.log (48577867570387 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (16757867570387 / 32000000000000 : ℝ) ≤
+      (-(6468680488511488354407 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (16757867570387 / 32000000000000 : ℝ) = (16757867570387 / 16000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16757867570387 / 16000000000000 : ℝ) ≤
+            (462791314488511645593 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (899703389990447 / 1250000000000000 : ℝ) ≤ Real.sqrt (16577867570387 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (16577867570387 / 32000000000000 : ℝ) ≤
+      (7197627119923579 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (16577867570387 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau706 :
+    Real.arctan (7197627119923579 / 22320951106041260 : ℝ) ≤
+      (623866760360916799687 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau705 :
+    Real.arctan (7197627119923579 / 10000000000000000 : ℝ) ≤
+      (2 * (623866760360916799687 / 2000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (616047555302063 / 500000000000000 : ℝ) ≤
+          Real.sqrt (1 + (7197627119923579 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (7197627119923579 / 10000000000000000 : ℝ) /
+                (1 + (616047555302063 / 500000000000000 : ℝ)) =
+              (7197627119923579 / 22320951106041260 : ℝ)
+            by
+        norm_num];
+        exact hau706)
+  have hat1 :
+    (Real.pi / 2 - (2 * (623866760360916799687 / 2000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 7197627119923579 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 7197627119923579 : ℝ) =
+              (7197627119923579 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau705)
+  have hz1 :
+    (10000000000000000 / 7197627119923579 : ℝ) ≤
+      1 / Real.sqrt (16577867570387 / 32000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 7197627119923579 : ℝ) =
+          1 / (7197627119923579 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (93750000000000 / 899703389990447 : ℝ) ≤
+      (1038263128909289262889 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (16577867570387 / 32000000000000 : ℝ) ≤
+      (93750000000000 / 899703389990447 : ℝ) := by
+    rw [show
+        (93750000000000 / 899703389990447 : ℝ) =
+          (3 / 40) / (899703389990447 / 1250000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (623866760360916799687 / 2000000000000000000000 : ℝ))) -
+        6 * (1038263128909289262889 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (16577867570387 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (16577867570387 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((899703389990447 / 1250000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (623866760360916799687 / 2000000000000000000000 : ℝ))) -
+          6 * (1038263128909289262889 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (16577867570387 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (16577867570387 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (16577867570387 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_468 :
+    ((1047688214451545209707 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(3210561243363384980221 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((7214948555867791 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3125034246207891380201 / 10000000000000000000000 : ℝ))) -
+              6 * (103578830857744472109 / 1000000000000000000000 : ℝ)))) ≤
+      Vfield (8328877226211 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (8328877226211 / 16000000000000 : ℝ)) =
+      Real.log (24328877226211 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((8328877226211 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (8418877226211 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1047688214451545209707 / 2500000000000000000000 : ℝ) ≤
+      Real.log (24328877226211 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (8418877226211 / 16000000000000 : ℝ) ≤
+      (-(3210561243363384980221 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (8418877226211 / 16000000000000 : ℝ) = (8418877226211 / 8000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8418877226211 / 8000000000000 : ℝ) ≤
+            (255174658136615019779 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (7214948555867791 / 10000000000000000 : ℝ) ≤ Real.sqrt (8328877226211 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (8328877226211 / 16000000000000 : ℝ) ≤ (3607474277933897 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (8328877226211 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau708 :
+    Real.arctan (3607474277933897 / 11165538959892694 : ℝ) ≤
+      (3125034246207891380201 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau707 :
+    Real.arctan (3607474277933897 / 5000000000000000 : ℝ) ≤
+      (2 * (3125034246207891380201 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (3082769479946347 / 2500000000000000 : ℝ) ≤
+          Real.sqrt (1 + (3607474277933897 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (3607474277933897 / 5000000000000000 : ℝ) /
+                (1 + (3082769479946347 / 2500000000000000 : ℝ)) =
+              (3607474277933897 / 11165538959892694 : ℝ)
+            by
+        norm_num];
+        exact hau708)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3125034246207891380201 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 3607474277933897 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 3607474277933897 : ℝ) =
+              (3607474277933897 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau707)
+  have hz1 :
+    (5000000000000000 / 3607474277933897 : ℝ) ≤
+      1 / Real.sqrt (8328877226211 / 16000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 3607474277933897 : ℝ) = 1 / (3607474277933897 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 7214948555867791 : ℝ) ≤
+      (103578830857744472109 / 1000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (8328877226211 / 16000000000000 : ℝ) ≤
+      (750000000000000 / 7214948555867791 : ℝ) := by
+    rw [show
+        (750000000000000 / 7214948555867791 : ℝ) =
+          (3 / 40) / (7214948555867791 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3125034246207891380201 / 10000000000000000000000 : ℝ))) -
+        6 * (103578830857744472109 / 1000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (8328877226211 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (8328877226211 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((7214948555867791 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3125034246207891380201 / 10000000000000000000000 : ℝ))) -
+          6 * (103578830857744472109 / 1000000000000000000000 : ℝ))) ≤
+      Real.sqrt (8328877226211 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (8328877226211 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (8328877226211 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/V39.lean b/LeanPool/Zeta5Irrational/Table/V39.lean
new file mode 100644
index 0000000000..36265ddc57
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/V39.lean
@@ -0,0 +1,1453 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Energy
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # V39: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_469 :
+    ((841431502833399730677 / 2000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(6373789591157519530799 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((904028563312041 / 1250000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1565355854444793571363 / 5000000000000000000000 : ℝ))) -
+              6 * (258332775384579108939 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (16737641334457 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (16737641334457 / 32000000000000 : ℝ)) =
+      Real.log (48737641334457 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((16737641334457 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (16917641334457 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (841431502833399730677 / 2000000000000000000000 : ℝ) ≤
+      Real.log (48737641334457 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (16917641334457 / 32000000000000 : ℝ) ≤
+      (-(6373789591157519530799 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (16917641334457 / 32000000000000 : ℝ) = (16917641334457 / 16000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (16917641334457 / 16000000000000 : ℝ) ≤
+            (557682211842480469201 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (904028563312041 / 1250000000000000 : ℝ) ≤ Real.sqrt (16737641334457 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (16737641334457 / 32000000000000 : ℝ) ≤
+      (7232228506496331 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (16737641334457 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau710 :
+    Real.arctan (556325269730487 / 1718553571059557 : ℝ) ≤
+      (1565355854444793571363 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau709 :
+    Real.arctan (7232228506496331 / 10000000000000000 : ℝ) ≤
+      (2 * (1565355854444793571363 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12341196423774241 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (7232228506496331 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (7232228506496331 / 10000000000000000 : ℝ) /
+                (1 + (12341196423774241 / 10000000000000000 : ℝ)) =
+              (556325269730487 / 1718553571059557 : ℝ)
+            by
+        norm_num];
+        exact hau710)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1565355854444793571363 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 7232228506496331 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 7232228506496331 : ℝ) =
+              (7232228506496331 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau709)
+  have hz1 :
+    (10000000000000000 / 7232228506496331 : ℝ) ≤
+      1 / Real.sqrt (16737641334457 / 32000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 7232228506496331 : ℝ) =
+          1 / (7232228506496331 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (31250000000000 / 301342854437347 : ℝ) ≤
+      (258332775384579108939 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (16737641334457 / 32000000000000 : ℝ) ≤
+      (31250000000000 / 301342854437347 : ℝ) := by
+    rw [show
+        (31250000000000 / 301342854437347 : ℝ) =
+          (3 / 40) / (904028563312041 / 1250000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1565355854444793571363 / 5000000000000000000000 : ℝ))) -
+        6 * (258332775384579108939 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (16737641334457 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (16737641334457 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((904028563312041 / 1250000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1565355854444793571363 / 5000000000000000000000 : ℝ))) -
+          6 * (258332775384579108939 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (16737641334457 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (16737641334457 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (16737641334457 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_470 :
+    ((1053837440410898379117 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(6350206894233353416501 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((7240853017660129 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3133541876096710658859 / 10000000000000000000000 : ℝ))) -
+              6 * (1032109037888209828613 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (33555169550949 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (33555169550949 / 64000000000000 : ℝ)) =
+      Real.log (97555169550949 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((33555169550949 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (33915169550949 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1053837440410898379117 / 2500000000000000000000 : ℝ) ≤
+      Real.log (97555169550949 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (33915169550949 / 64000000000000 : ℝ) ≤
+      (-(6350206894233353416501 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (33915169550949 / 64000000000000 : ℝ) = (33915169550949 / 32000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (33915169550949 / 32000000000000 : ℝ) ≤
+            (581264908766646583499 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (7240853017660129 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (33555169550949 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (33555169550949 / 64000000000000 : ℝ) ≤ (1810213254415033 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (33555169550949 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau712 :
+    Real.arctan (7240853017660132 / 22346252565995797 : ℝ) ≤
+      (3133541876096710658859 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau711 :
+    Real.arctan (1810213254415033 / 2500000000000000 : ℝ) ≤
+      (2 * (3133541876096710658859 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12346252565995797 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1810213254415033 / 2500000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1810213254415033 / 2500000000000000 : ℝ) /
+                (1 + (12346252565995797 / 10000000000000000 : ℝ)) =
+              (7240853017660132 / 22346252565995797 : ℝ)
+            by
+        norm_num];
+        exact hau712)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3133541876096710658859 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (2500000000000000 / 1810213254415033 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 1810213254415033 : ℝ) =
+              (1810213254415033 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau711)
+  have hz1 :
+    (2500000000000000 / 1810213254415033 : ℝ) ≤
+      1 / Real.sqrt (33555169550949 / 64000000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 1810213254415033 : ℝ) = 1 / (1810213254415033 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 7240853017660129 : ℝ) ≤
+      (1032109037888209828613 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (33555169550949 / 64000000000000 : ℝ) ≤
+      (750000000000000 / 7240853017660129 : ℝ) := by
+    rw [show
+        (750000000000000 / 7240853017660129 : ℝ) =
+          (3 / 40) / (7240853017660129 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3133541876096710658859 / 10000000000000000000000 : ℝ))) -
+        6 * (1032109037888209828613 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (33555169550949 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (33555169550949 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((7240853017660129 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3133541876096710658859 / 10000000000000000000000 : ℝ))) -
+          6 * (1032109037888209828613 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (33555169550949 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (33555169550949 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (33555169550949 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_471 :
+    ((422353530332149810381 / 1000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(6326679680849111885173 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3624733634232227 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (784091590609151552069 / 2500000000000000000000 : ℝ))) -
+              6 * (1030891299853228078583 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (4204382054123 / 8000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (4204382054123 / 8000000000000 : ℝ)) =
+      Real.log (12204382054123 / 8000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((4204382054123 / 8000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (4249382054123 / 8000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (422353530332149810381 / 1000000000000000000000 : ℝ) ≤
+      Real.log (12204382054123 / 8000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (4249382054123 / 8000000000000 : ℝ) ≤
+      (-(6326679680849111885173 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (4249382054123 / 8000000000000 : ℝ) = (4249382054123 / 4000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4249382054123 / 4000000000000 : ℝ) ≤
+            (604792122150888114827 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3624733634232227 / 5000000000000000 : ℝ) ≤ Real.sqrt (4204382054123 / 8000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (4204382054123 / 8000000000000 : ℝ) ≤ (7249467268464457 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (4204382054123 / 8000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau714 :
+    Real.arctan (7249467268464457 / 22351306638430506 : ℝ) ≤
+      (784091590609151552069 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau713 :
+    Real.arctan (7249467268464457 / 10000000000000000 : ℝ) ≤
+      (2 * (784091590609151552069 / 2500000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6175653319215253 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (7249467268464457 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (7249467268464457 / 10000000000000000 : ℝ) /
+                (1 + (6175653319215253 / 5000000000000000 : ℝ)) =
+              (7249467268464457 / 22351306638430506 : ℝ)
+            by
+        norm_num];
+        exact hau714)
+  have hat1 :
+    (Real.pi / 2 - (2 * (784091590609151552069 / 2500000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 7249467268464457 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 7249467268464457 : ℝ) =
+              (7249467268464457 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau713)
+  have hz1 :
+    (10000000000000000 / 7249467268464457 : ℝ) ≤
+      1 / Real.sqrt (4204382054123 / 8000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 7249467268464457 : ℝ) =
+          1 / (7249467268464457 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 3624733634232227 : ℝ) ≤
+      (1030891299853228078583 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (4204382054123 / 8000000000000 : ℝ) ≤
+      (375000000000000 / 3624733634232227 : ℝ) := by
+    rw [show
+        (375000000000000 / 3624733634232227 : ℝ) =
+          (3 / 40) / (3624733634232227 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (784091590609151552069 / 2500000000000000000000 : ℝ))) -
+        6 * (1030891299853228078583 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (4204382054123 / 8000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (4204382054123 / 8000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3624733634232227 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (784091590609151552069 / 2500000000000000000000 : ℝ))) -
+          6 * (1030891299853228078583 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (4204382054123 / 8000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (4204382054123 / 8000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (4204382054123 / 8000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_472 :
+    ((846342830033970747269 / 2000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(393950480658920792153 / 625000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3629035647720933 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3139185189170562755643 / 10000000000000000000000 : ℝ))) -
+              6 * (1029677861974945545847 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (33714943315019 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (33714943315019 / 64000000000000 : ℝ)) =
+      Real.log (97714943315019 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((33714943315019 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (34074943315019 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (846342830033970747269 / 2000000000000000000000 : ℝ) ≤
+      Real.log (97714943315019 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (34074943315019 / 64000000000000 : ℝ) ≤
+      (-(393950480658920792153 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (34074943315019 / 64000000000000 : ℝ) = (34074943315019 / 32000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (34074943315019 / 32000000000000 : ℝ) ≤
+            (39266507028579207847 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3629035647720933 / 5000000000000000 : ℝ) ≤ Real.sqrt (33714943315019 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (33714943315019 / 64000000000000 : ℝ) ≤
+      (7258071295441869 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (33714943315019 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau716 :
+    Real.arctan (7258071295441869 / 22356358643618158 : ℝ) ≤
+      (3139185189170562755643 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau715 :
+    Real.arctan (7258071295441869 / 10000000000000000 : ℝ) ≤
+      (2 * (3139185189170562755643 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6178179321809079 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (7258071295441869 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (7258071295441869 / 10000000000000000 : ℝ) /
+                (1 + (6178179321809079 / 5000000000000000 : ℝ)) =
+              (7258071295441869 / 22356358643618158 : ℝ)
+            by
+        norm_num];
+        exact hau716)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3139185189170562755643 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 7258071295441869 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 7258071295441869 : ℝ) =
+              (7258071295441869 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau715)
+  have hz1 :
+    (10000000000000000 / 7258071295441869 : ℝ) ≤
+      1 / Real.sqrt (33714943315019 / 64000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 7258071295441869 : ℝ) =
+          1 / (7258071295441869 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (125000000000000 / 1209678549240311 : ℝ) ≤
+      (1029677861974945545847 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (33714943315019 / 64000000000000 : ℝ) ≤
+      (125000000000000 / 1209678549240311 : ℝ) := by
+    rw [show
+        (125000000000000 / 1209678549240311 : ℝ) =
+          (3 / 40) / (3629035647720933 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3139185189170562755643 / 10000000000000000000000 : ℝ))) -
+        6 * (1029677861974945545847 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (33714943315019 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (33714943315019 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3629035647720933 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3139185189170562755643 / 10000000000000000000000 : ℝ))) -
+          6 * (1029677861974945545847 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (33714943315019 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (33714943315019 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (33714943315019 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_473 :
+    ((33919090505047273979 / 80000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(6279790664681930096523 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((7266665134908643 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1570999188718471372011 / 5000000000000000000000 : ℝ))) -
+              6 * (102846869900421664989 / 1000000000000000000000 : ℝ)))) ≤
+      Vfield (16897415098527 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (16897415098527 / 32000000000000 : ℝ)) =
+      Real.log (48897415098527 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((16897415098527 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (17077415098527 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (33919090505047273979 / 80000000000000000000 : ℝ) ≤
+      Real.log (48897415098527 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (17077415098527 / 32000000000000 : ℝ) ≤
+      (-(6279790664681930096523 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (17077415098527 / 32000000000000 : ℝ) = (17077415098527 / 16000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17077415098527 / 16000000000000 : ℝ) ≤
+            (651681138318069903477 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (7266665134908643 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (16897415098527 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (16897415098527 / 32000000000000 : ℝ) ≤ (3633332567454323 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (16897415098527 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau718 :
+    Real.arctan (3633332567454323 / 11180704292046677 : ℝ) ≤
+      (1570999188718471372011 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau717 :
+    Real.arctan (3633332567454323 / 5000000000000000 : ℝ) ≤
+      (2 * (1570999188718471372011 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6180704292046677 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (3633332567454323 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (3633332567454323 / 5000000000000000 : ℝ) /
+                (1 + (6180704292046677 / 5000000000000000 : ℝ)) =
+              (3633332567454323 / 11180704292046677 : ℝ)
+            by
+        norm_num];
+        exact hau718)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1570999188718471372011 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 3633332567454323 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 3633332567454323 : ℝ) =
+              (3633332567454323 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau717)
+  have hz1 :
+    (5000000000000000 / 3633332567454323 : ℝ) ≤
+      1 / Real.sqrt (16897415098527 / 32000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 3633332567454323 : ℝ) = 1 / (3633332567454323 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 7266665134908643 : ℝ) ≤
+      (102846869900421664989 / 1000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (16897415098527 / 32000000000000 : ℝ) ≤
+      (750000000000000 / 7266665134908643 : ℝ) := by
+    rw [show
+        (750000000000000 / 7266665134908643 : ℝ) =
+          (3 / 40) / (7266665134908643 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1570999188718471372011 / 5000000000000000000000 : ℝ))) -
+        6 * (102846869900421664989 / 1000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (16897415098527 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (16897415098527 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((7266665134908643 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1570999188718471372011 / 5000000000000000000000 : ℝ))) -
+          6 * (102846869900421664989 / 1000000000000000000000 : ℝ))) ≤
+      Real.sqrt (16897415098527 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (16897415098527 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (16897415098527 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_474 :
+    ((531006475390013602341 / 1250000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(782053543305886984727 / 1250000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((291009952918663 / 400000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3144805948252172401271 / 10000000000000000000000 : ℝ))) -
+              6 * (1027263785898967908693 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (33874717079089 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (33874717079089 / 64000000000000 : ℝ)) =
+      Real.log (97874717079089 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((33874717079089 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (34234717079089 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (531006475390013602341 / 1250000000000000000000 : ℝ) ≤
+      Real.log (97874717079089 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (34234717079089 / 64000000000000 : ℝ) ≤
+      (-(782053543305886984727 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (34234717079089 / 64000000000000 : ℝ) = (34234717079089 / 32000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (34234717079089 / 32000000000000 : ℝ) ≤
+            (84380432069113015273 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (291009952918663 / 400000000000000 : ℝ) ≤ Real.sqrt (33874717079089 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (33874717079089 / 64000000000000 : ℝ) ≤ (3637624411483289 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (33874717079089 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau720 :
+    Real.arctan (3637624411483289 / 11183228231192759 : ℝ) ≤
+      (3144805948252172401271 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau719 :
+    Real.arctan (3637624411483289 / 5000000000000000 : ℝ) ≤
+      (2 * (3144805948252172401271 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6183228231192759 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (3637624411483289 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (3637624411483289 / 5000000000000000 : ℝ) /
+                (1 + (6183228231192759 / 5000000000000000 : ℝ)) =
+              (3637624411483289 / 11183228231192759 : ℝ)
+            by
+        norm_num];
+        exact hau720)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3144805948252172401271 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 3637624411483289 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 3637624411483289 : ℝ) =
+              (3637624411483289 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau719)
+  have hz1 :
+    (5000000000000000 / 3637624411483289 : ℝ) ≤
+      1 / Real.sqrt (33874717079089 / 64000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 3637624411483289 : ℝ) = 1 / (3637624411483289 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (30000000000000 / 291009952918663 : ℝ) ≤
+      (1027263785898967908693 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (33874717079089 / 64000000000000 : ℝ) ≤
+      (30000000000000 / 291009952918663 : ℝ) := by
+    rw [show
+        (30000000000000 / 291009952918663 : ℝ) =
+          (3 / 40) / (291009952918663 / 400000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3144805948252172401271 / 10000000000000000000000 : ℝ))) -
+        6 * (1027263785898967908693 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (33874717079089 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (33874717079089 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((291009952918663 / 400000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3144805948252172401271 / 10000000000000000000000 : ℝ))) -
+          6 * (1027263785898967908693 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (33874717079089 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (33874717079089 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (33874717079089 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_475 :
+    ((425621063102617945859 / 1000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1246624096162879854307 / 2000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((910477799438091 / 1250000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (629521584502343043317 / 2000000000000000000000 : ℝ))) -
+              6 * (513031548911010165729 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (8488650990281 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (8488650990281 / 16000000000000 : ℝ)) =
+      Real.log (24488650990281 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((8488650990281 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (8578650990281 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (425621063102617945859 / 1000000000000000000000 : ℝ) ≤
+      Real.log (24488650990281 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (8578650990281 / 16000000000000 : ℝ) ≤
+      (-(1246624096162879854307 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (8578650990281 / 16000000000000 : ℝ) = (8578650990281 / 8000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8578650990281 / 8000000000000 : ℝ) ≤
+            (139670264437120145693 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (910477799438091 / 1250000000000000 : ℝ) ≤ Real.sqrt (8488650990281 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (8488650990281 / 16000000000000 : ℝ) ≤ (7283822395504731 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (8488650990281 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau722 :
+    Real.arctan (7283822395504731 / 22371502281018916 : ℝ) ≤
+      (629521584502343043317 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau721 :
+    Real.arctan (7283822395504731 / 10000000000000000 : ℝ) ≤
+      (2 * (629521584502343043317 / 2000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (3092875570254729 / 2500000000000000 : ℝ) ≤
+          Real.sqrt (1 + (7283822395504731 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (7283822395504731 / 10000000000000000 : ℝ) /
+                (1 + (3092875570254729 / 2500000000000000 : ℝ)) =
+              (7283822395504731 / 22371502281018916 : ℝ)
+            by
+        norm_num];
+        exact hau722)
+  have hat1 :
+    (Real.pi / 2 - (2 * (629521584502343043317 / 2000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 7283822395504731 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 7283822395504731 : ℝ) =
+              (7283822395504731 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau721)
+  have hz1 :
+    (10000000000000000 / 7283822395504731 : ℝ) ≤
+      1 / Real.sqrt (8488650990281 / 16000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 7283822395504731 : ℝ) =
+          1 / (7283822395504731 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (31250000000000 / 303492599812697 : ℝ) ≤
+      (513031548911010165729 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (8488650990281 / 16000000000000 : ℝ) ≤
+      (31250000000000 / 303492599812697 : ℝ) := by
+    rw [show
+        (31250000000000 / 303492599812697 : ℝ) =
+          (3 / 40) / (910477799438091 / 1250000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (629521584502343043317 / 2000000000000000000000 : ℝ))) -
+        6 * (513031548911010165729 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (8488650990281 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (8488650990281 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((910477799438091 / 1250000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (629521584502343043317 / 2000000000000000000000 : ℝ))) -
+          6 * (513031548911010165729 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (8488650990281 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (8488650990281 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (8488650990281 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_476 :
+    ((4264362807711218122773 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(776233351817385457859 / 1250000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3646192944100597 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2 * (157520216049551920661 / 500000000000000000000 : ℝ))) -
+              6 * (512433305069469542579 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (34034490843159 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (34034490843159 / 64000000000000 : ℝ)) =
+      Real.log (98034490843159 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((34034490843159 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (34394490843159 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (4264362807711218122773 / 10000000000000000000000 : ℝ) ≤
+      Real.log (98034490843159 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (34394490843159 / 64000000000000 : ℝ) ≤
+      (-(776233351817385457859 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (34394490843159 / 64000000000000 : ℝ) = (34394490843159 / 32000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (34394490843159 / 32000000000000 : ℝ) ≤
+            (90200623557614542141 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3646192944100597 / 5000000000000000 : ℝ) ≤ Real.sqrt (34034490843159 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (34034490843159 / 64000000000000 : ℝ) ≤
+      (7292385888201197 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (34034490843159 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau724 :
+    Real.arctan (7292385888201197 / 22376546042512666 : ℝ) ≤
+      (157520216049551920661 / 500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau723 :
+    Real.arctan (7292385888201197 / 10000000000000000 : ℝ) ≤
+      (2 * (157520216049551920661 / 500000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6188273021256333 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (7292385888201197 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (7292385888201197 / 10000000000000000 : ℝ) /
+                (1 + (6188273021256333 / 5000000000000000 : ℝ)) =
+              (7292385888201197 / 22376546042512666 : ℝ)
+            by
+        norm_num];
+        exact hau724)
+  have hat1 :
+    (Real.pi / 2 - (2 * (157520216049551920661 / 500000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 7292385888201197 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 7292385888201197 : ℝ) =
+              (7292385888201197 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau723)
+  have hz1 :
+    (10000000000000000 / 7292385888201197 : ℝ) ≤
+      1 / Real.sqrt (34034490843159 / 64000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 7292385888201197 : ℝ) =
+          1 / (7292385888201197 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 3646192944100597 : ℝ) ≤
+      (512433305069469542579 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (34034490843159 / 64000000000000 : ℝ) ≤
+      (375000000000000 / 3646192944100597 : ℝ) := by
+    rw [show
+        (375000000000000 / 3646192944100597 : ℝ) =
+          (3 / 40) / (3646192944100597 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (157520216049551920661 / 500000000000000000000 : ℝ))) -
+        6 * (512433305069469542579 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (34034490843159 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (34034490843159 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3646192944100597 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (157520216049551920661 / 500000000000000000000 : ℝ))) -
+          6 * (512433305069469542579 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (34034490843159 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (34034490843159 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (34034490843159 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_477 :
+    ((2136254172005389240579 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(6186667096138956931663 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((7300939336524829 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3153195164346571015303 / 10000000000000000000000 : ℝ))) -
+              6 * (1023674298415909778301 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (17057188862597 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (17057188862597 / 32000000000000 : ℝ)) =
+      Real.log (49057188862597 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((17057188862597 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (17237188862597 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2136254172005389240579 / 5000000000000000000000 : ℝ) ≤
+      Real.log (49057188862597 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (17237188862597 / 32000000000000 : ℝ) ≤
+      (-(6186667096138956931663 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (17237188862597 / 32000000000000 : ℝ) = (17237188862597 / 16000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17237188862597 / 16000000000000 : ℝ) ≤
+            (744804706861043068337 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (7300939336524829 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (17057188862597 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (17057188862597 / 32000000000000 : ℝ) ≤ (228154354266401 / 312500000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (17057188862597 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau726 :
+    Real.arctan (1825234834131208 / 5595396937345189 : ℝ) ≤
+      (3153195164346571015303 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau725 :
+    Real.arctan (228154354266401 / 312500000000000 : ℝ) ≤
+      (2 * (3153195164346571015303 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (3095396937345189 / 2500000000000000 : ℝ) ≤
+          Real.sqrt (1 + (228154354266401 / 312500000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (228154354266401 / 312500000000000 : ℝ) /
+                (1 + (3095396937345189 / 2500000000000000 : ℝ)) =
+              (1825234834131208 / 5595396937345189 : ℝ)
+            by
+        norm_num];
+        exact hau726)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3153195164346571015303 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (312500000000000 / 228154354266401 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (312500000000000 / 228154354266401 : ℝ) =
+              (228154354266401 / 312500000000000 : ℝ)
+            by
+        norm_num];
+        exact hau725)
+  have hz1 :
+    (312500000000000 / 228154354266401 : ℝ) ≤
+      1 / Real.sqrt (17057188862597 / 32000000000000 : ℝ) := by
+    rw [show (312500000000000 / 228154354266401 : ℝ) = 1 / (228154354266401 / 312500000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 7300939336524829 : ℝ) ≤
+      (1023674298415909778301 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (17057188862597 / 32000000000000 : ℝ) ≤
+      (750000000000000 / 7300939336524829 : ℝ) := by
+    rw [show
+        (750000000000000 / 7300939336524829 : ℝ) =
+          (3 / 40) / (7300939336524829 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3153195164346571015303 / 10000000000000000000000 : ℝ))) -
+        6 * (1023674298415909778301 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (17057188862597 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (17057188862597 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((7300939336524829 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3153195164346571015303 / 10000000000000000000000 : ℝ))) -
+          6 * (1023674298415909778301 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (17057188862597 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (17057188862597 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (17057188862597 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_478 :
+    ((4280647250733957331043 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(770440134484759110209 / 1250000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((11421066837089 / 15625000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3155980473116647307711 / 10000000000000000000000 : ℝ))) -
+              6 * (127810767302205454203 / 1250000000000000000000 : ℝ)))) ≤
+      Vfield (34194264607229 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (34194264607229 / 64000000000000 : ℝ)) =
+      Real.log (98194264607229 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((34194264607229 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (34554264607229 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (4280647250733957331043 / 10000000000000000000000 : ℝ) ≤
+      Real.log (98194264607229 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (34554264607229 / 64000000000000 : ℝ) ≤
+      (-(770440134484759110209 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (34554264607229 / 64000000000000 : ℝ) = (34554264607229 / 32000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (34554264607229 / 32000000000000 : ℝ) ≤
+            (95993840890240889791 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (11421066837089 / 15625000000000 : ℝ) ≤ Real.sqrt (34194264607229 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (34194264607229 / 64000000000000 : ℝ) ≤
+      (7309482775736963 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (34194264607229 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau728 :
+    Real.arctan (7309482775736963 / 22386627404132059 : ℝ) ≤
+      (3155980473116647307711 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau727 :
+    Real.arctan (7309482775736963 / 10000000000000000 : ℝ) ≤
+      (2 * (3155980473116647307711 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12386627404132059 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (7309482775736963 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (7309482775736963 / 10000000000000000 : ℝ) /
+                (1 + (12386627404132059 / 10000000000000000 : ℝ)) =
+              (7309482775736963 / 22386627404132059 : ℝ)
+            by
+        norm_num];
+        exact hau728)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3155980473116647307711 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 7309482775736963 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 7309482775736963 : ℝ) =
+              (7309482775736963 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau727)
+  have hz1 :
+    (10000000000000000 / 7309482775736963 : ℝ) ≤
+      1 / Real.sqrt (34194264607229 / 64000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 7309482775736963 : ℝ) =
+          1 / (7309482775736963 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (1171875000000 / 11421066837089 : ℝ) ≤
+      (127810767302205454203 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (34194264607229 / 64000000000000 : ℝ) ≤
+      (1171875000000 / 11421066837089 : ℝ) := by
+    rw [show
+        (1171875000000 / 11421066837089 : ℝ) = (3 / 40) / (11421066837089 / 15625000000000 : ℝ) by
+        norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3155980473116647307711 / 10000000000000000000000 : ℝ))) -
+        6 * (127810767302205454203 / 1250000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (34194264607229 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (34194264607229 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((11421066837089 / 15625000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3155980473116647307711 / 10000000000000000000000 : ℝ))) -
+          6 * (127810767302205454203 / 1250000000000000000000 : ℝ))) ≤
+      Real.sqrt (34194264607229 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (34194264607229 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (34194264607229 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_479 :
+    ((4284714221375033324851 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(6151968124980879444211 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((7313750752889049 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (631474211683661753521 / 2000000000000000000000 : ℝ))) -
+              6 * (1021893607795543436653 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (68468416096493 / 128000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (68468416096493 / 128000000000000 : ℝ)) =
+      Real.log (196468416096493 / 128000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((68468416096493 / 128000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (69188416096493 / 128000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (4284714221375033324851 / 10000000000000000000000 : ℝ) ≤
+      Real.log (196468416096493 / 128000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (69188416096493 / 128000000000000 : ℝ) ≤
+      (-(6151968124980879444211 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (69188416096493 / 128000000000000 : ℝ) = (69188416096493 / 64000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (69188416096493 / 64000000000000 : ℝ) ≤
+            (779503678019120555789 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (7313750752889049 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (68468416096493 / 128000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (68468416096493 / 128000000000000 : ℝ) ≤
+      (1828437688222263 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (68468416096493 / 128000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau730 :
+    Real.arctan (7313750752889052 / 22389146462746541 : ℝ) ≤
+      (631474211683661753521 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau729 :
+    Real.arctan (1828437688222263 / 2500000000000000 : ℝ) ≤
+      (2 * (631474211683661753521 / 2000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12389146462746541 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1828437688222263 / 2500000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1828437688222263 / 2500000000000000 : ℝ) /
+                (1 + (12389146462746541 / 10000000000000000 : ℝ)) =
+              (7313750752889052 / 22389146462746541 : ℝ)
+            by
+        norm_num];
+        exact hau730)
+  have hat1 :
+    (Real.pi / 2 - (2 * (631474211683661753521 / 2000000000000000000000 : ℝ))) ≤
+      Real.arctan (2500000000000000 / 1828437688222263 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 1828437688222263 : ℝ) =
+              (1828437688222263 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau729)
+  have hz1 :
+    (2500000000000000 / 1828437688222263 : ℝ) ≤
+      1 / Real.sqrt (68468416096493 / 128000000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 1828437688222263 : ℝ) = 1 / (1828437688222263 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 2437916917629683 : ℝ) ≤
+      (1021893607795543436653 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (68468416096493 / 128000000000000 : ℝ) ≤
+      (250000000000000 / 2437916917629683 : ℝ) := by
+    rw [show
+        (250000000000000 / 2437916917629683 : ℝ) =
+          (3 / 40) / (7313750752889049 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (631474211683661753521 / 2000000000000000000000 : ℝ))) -
+        6 * (1021893607795543436653 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (68468416096493 / 128000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (68468416096493 / 128000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((7313750752889049 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (631474211683661753521 / 2000000000000000000000 : ℝ))) -
+          6 * (1021893607795543436653 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (68468416096493 / 128000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (68468416096493 / 128000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (68468416096493 / 128000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_480 :
+    ((4288779538663480659127 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(383776781609412568073 / 625000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3659008120446543 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1579380133861223043083 / 5000000000000000000000 : ℝ))) -
+              6 * (127662763263163460067 / 1250000000000000000000 : ℝ)))) ≤
+      Vfield (2142134468079 / 4000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (2142134468079 / 4000000000000 : ℝ)) =
+      Real.log (6142134468079 / 4000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((2142134468079 / 4000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (2164634468079 / 4000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (4288779538663480659127 / 10000000000000000000000 : ℝ) ≤
+      Real.log (6142134468079 / 4000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (2164634468079 / 4000000000000 : ℝ) ≤
+      (-(383776781609412568073 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (2164634468079 / 4000000000000 : ℝ) = (2164634468079 / 2000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2164634468079 / 2000000000000 : ℝ) ≤
+            (49440206078087431927 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3659008120446543 / 5000000000000000 : ℝ) ≤ Real.sqrt (2142134468079 / 4000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (2142134468079 / 4000000000000 : ℝ) ≤ (7318016240893089 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (2142134468079 / 4000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau732 :
+    Real.arctan (7318016240893089 / 22391665009270343 : ℝ) ≤
+      (1579380133861223043083 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau731 :
+    Real.arctan (7318016240893089 / 10000000000000000 : ℝ) ≤
+      (2 * (1579380133861223043083 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12391665009270343 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (7318016240893089 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (7318016240893089 / 10000000000000000 : ℝ) /
+                (1 + (12391665009270343 / 10000000000000000 : ℝ)) =
+              (7318016240893089 / 22391665009270343 : ℝ)
+            by
+        norm_num];
+        exact hau732)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1579380133861223043083 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 7318016240893089 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 7318016240893089 : ℝ) =
+              (7318016240893089 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau731)
+  have hz1 :
+    (10000000000000000 / 7318016240893089 : ℝ) ≤
+      1 / Real.sqrt (2142134468079 / 4000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 7318016240893089 : ℝ) =
+          1 / (7318016240893089 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (125000000000000 / 1219669373482181 : ℝ) ≤
+      (127662763263163460067 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (2142134468079 / 4000000000000 : ℝ) ≤
+      (125000000000000 / 1219669373482181 : ℝ) := by
+    rw [show
+        (125000000000000 / 1219669373482181 : ℝ) =
+          (3 / 40) / (3659008120446543 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1579380133861223043083 / 5000000000000000000000 : ℝ))) -
+        6 * (127662763263163460067 / 1250000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (2142134468079 / 4000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (2142134468079 / 4000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3659008120446543 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1579380133861223043083 / 5000000000000000000000 : ℝ))) -
+          6 * (127662763263163460067 / 1250000000000000000000 : ℝ))) ≤
+      Real.sqrt (2142134468079 / 4000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (2142134468079 / 4000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (2142134468079 / 4000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/V40.lean b/LeanPool/Zeta5Irrational/Table/V40.lean
new file mode 100644
index 0000000000..d4b56d0244
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/V40.lean
@@ -0,0 +1,1452 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Energy
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # V40: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_481 :
+    ((536605400492879213153 / 1250000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(306445109372711187137 / 500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((7322279244099123 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3160148103563553089317 / 10000000000000000000000 : ℝ))) -
+              6 * (10207116303724433279 / 100000000000000000000 : ℝ)))) ≤
+      Vfield (68628189860563 / 128000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (68628189860563 / 128000000000000 : ℝ)) =
+      Real.log (196628189860563 / 128000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((68628189860563 / 128000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (69348189860563 / 128000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (536605400492879213153 / 1250000000000000000000 : ℝ) ≤
+      Real.log (196628189860563 / 128000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (69348189860563 / 128000000000000 : ℝ) ≤
+      (-(306445109372711187137 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (69348189860563 / 128000000000000 : ℝ) = (69348189860563 / 64000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (69348189860563 / 64000000000000 : ℝ) ≤
+            (40128480777288812863 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (7322279244099123 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (68628189860563 / 128000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (68628189860563 / 128000000000000 : ℝ) ≤
+      (3661139622049563 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (68628189860563 / 128000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau734 :
+    Real.arctan (2440759748033042 / 7464727681338547 : ℝ) ≤
+      (3160148103563553089317 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau733 :
+    Real.arctan (3661139622049563 / 5000000000000000 : ℝ) ≤
+      (2 * (3160148103563553089317 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12394183044015641 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (3661139622049563 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (3661139622049563 / 5000000000000000 : ℝ) /
+                (1 + (12394183044015641 / 10000000000000000 : ℝ)) =
+              (2440759748033042 / 7464727681338547 : ℝ)
+            by
+        norm_num];
+        exact hau734)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3160148103563553089317 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 3661139622049563 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 3661139622049563 : ℝ) =
+              (3661139622049563 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau733)
+  have hz1 :
+    (5000000000000000 / 3661139622049563 : ℝ) ≤
+      1 / Real.sqrt (68628189860563 / 128000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 3661139622049563 : ℝ) = 1 / (3661139622049563 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 2440759748033041 : ℝ) ≤
+      (10207116303724433279 / 100000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (68628189860563 / 128000000000000 : ℝ) ≤
+      (250000000000000 / 2440759748033041 : ℝ) := by
+    rw [show
+        (250000000000000 / 2440759748033041 : ℝ) =
+          (3 / 40) / (7322279244099123 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3160148103563553089317 / 10000000000000000000000 : ℝ))) -
+        6 * (10207116303724433279 / 100000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (68628189860563 / 128000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (68628189860563 / 128000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((7322279244099123 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3160148103563553089317 / 10000000000000000000000 : ℝ))) -
+          6 * (10207116303724433279 / 100000000000000000000 : ℝ))) ≤
+      Real.sqrt (68628189860563 / 128000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (68628189860563 / 128000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (68628189860563 / 128000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_482 :
+    ((537113152319473669229 / 1250000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(122347782789297651323 / 200000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1465307953368911 / 2000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3161534568468916080673 / 10000000000000000000000 : ℝ))) -
+              6 * (1020122177634482114133 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (34354038371299 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (34354038371299 / 64000000000000 : ℝ)) =
+      Real.log (98354038371299 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((34354038371299 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (34714038371299 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (537113152319473669229 / 1250000000000000000000 : ℝ) ≤
+      Real.log (98354038371299 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (34714038371299 / 64000000000000 : ℝ) ≤
+      (-(122347782789297651323 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (34714038371299 / 64000000000000 : ℝ) = (34714038371299 / 32000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (34714038371299 / 32000000000000 : ℝ) ≤
+            (16281653270702348677 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1465307953368911 / 2000000000000000 : ℝ) ≤ Real.sqrt (34354038371299 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (34354038371299 / 64000000000000 : ℝ) ≤ (3663269883422279 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (34354038371299 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau736 :
+    Real.arctan (3663269883422279 / 11198350283647147 : ℝ) ≤
+      (3161534568468916080673 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau735 :
+    Real.arctan (3663269883422279 / 5000000000000000 : ℝ) ≤
+      (2 * (3161534568468916080673 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6198350283647147 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (3663269883422279 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (3663269883422279 / 5000000000000000 : ℝ) /
+                (1 + (6198350283647147 / 5000000000000000 : ℝ)) =
+              (3663269883422279 / 11198350283647147 : ℝ)
+            by
+        norm_num];
+        exact hau736)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3161534568468916080673 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 3663269883422279 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 3663269883422279 : ℝ) =
+              (3663269883422279 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau735)
+  have hz1 :
+    (5000000000000000 / 3663269883422279 : ℝ) ≤
+      1 / Real.sqrt (34354038371299 / 64000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 3663269883422279 : ℝ) = 1 / (3663269883422279 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (150000000000000 / 1465307953368911 : ℝ) ≤
+      (1020122177634482114133 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (34354038371299 / 64000000000000 : ℝ) ≤
+      (150000000000000 / 1465307953368911 : ℝ) := by
+    rw [show
+        (150000000000000 / 1465307953368911 : ℝ) =
+          (3 / 40) / (1465307953368911 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3161534568468916080673 / 10000000000000000000000 : ℝ))) -
+        6 * (1020122177634482114133 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (34354038371299 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (34354038371299 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1465307953368911 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3161534568468916080673 / 10000000000000000000000 : ℝ))) -
+          6 * (1020122177634482114133 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (34354038371299 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (34354038371299 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (34354038371299 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_483 :
+    ((4300965583842209674487 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(6105889331261373677087 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((293231912538167 / 400000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1581459832479321819787 / 5000000000000000000000 : ℝ))) -
+              6 * (1019533744940917633747 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (68787963624633 / 128000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (68787963624633 / 128000000000000 : ℝ)) =
+      Real.log (196787963624633 / 128000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((68787963624633 / 128000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (69507963624633 / 128000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (4300965583842209674487 / 10000000000000000000000 : ℝ) ≤
+      Real.log (196787963624633 / 128000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (69507963624633 / 128000000000000 : ℝ) ≤
+      (-(6105889331261373677087 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (69507963624633 / 128000000000000 : ℝ) = (69507963624633 / 64000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (69507963624633 / 64000000000000 : ℝ) ≤
+            (825582471738626322913 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (293231912538167 / 400000000000000 : ℝ) ≤ Real.sqrt (68787963624633 / 128000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (68787963624633 / 128000000000000 : ℝ) ≤
+      (3665398906727089 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (68787963624633 / 128000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau738 :
+    Real.arctan (7330797813454178 / 22399217579417845 : ℝ) ≤
+      (1581459832479321819787 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau737 :
+    Real.arctan (3665398906727089 / 5000000000000000 : ℝ) ≤
+      (2 * (1581459832479321819787 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (2479843515883569 / 2000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (3665398906727089 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (3665398906727089 / 5000000000000000 : ℝ) /
+                (1 + (2479843515883569 / 2000000000000000 : ℝ)) =
+              (7330797813454178 / 22399217579417845 : ℝ)
+            by
+        norm_num];
+        exact hau738)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1581459832479321819787 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 3665398906727089 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 3665398906727089 : ℝ) =
+              (3665398906727089 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau737)
+  have hz1 :
+    (5000000000000000 / 3665398906727089 : ℝ) ≤
+      1 / Real.sqrt (68787963624633 / 128000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 3665398906727089 : ℝ) = 1 / (3665398906727089 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (30000000000000 / 293231912538167 : ℝ) ≤
+      (1019533744940917633747 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (68787963624633 / 128000000000000 : ℝ) ≤
+      (30000000000000 / 293231912538167 : ℝ) := by
+    rw [show
+        (30000000000000 / 293231912538167 : ℝ) =
+          (3 / 40) / (293231912538167 / 400000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1581459832479321819787 / 5000000000000000000000 : ℝ))) -
+        6 * (1019533744940917633747 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (68787963624633 / 128000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (68787963624633 / 128000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((293231912538167 / 400000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1581459832479321819787 / 5000000000000000000000 : ℝ))) -
+          6 * (1019533744940917633747 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (68787963624633 / 128000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (68787963624633 / 128000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (68787963624633 / 128000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_484 :
+    ((2152512150570562284839 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(6094402732427668675489 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((7335053388240221 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (791075848886423354773 / 2500000000000000000000 : ℝ))) -
+              6 * (203789265870628778459 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (17216962626667 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (17216962626667 / 32000000000000 : ℝ)) =
+      Real.log (49216962626667 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((17216962626667 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (17396962626667 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2152512150570562284839 / 5000000000000000000000 : ℝ) ≤
+      Real.log (49216962626667 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (17396962626667 / 32000000000000 : ℝ) ≤
+      (-(6094402732427668675489 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (17396962626667 / 32000000000000 : ℝ) = (17396962626667 / 16000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17396962626667 / 16000000000000 : ℝ) ≤
+            (837069070572331324511 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (7335053388240221 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (17216962626667 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (17216962626667 / 32000000000000 : ℝ) ≤ (229220418382507 / 312500000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (17216962626667 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau740 :
+    Real.arctan (458440836765014 / 1400108380043595 : ℝ) ≤
+      (791075848886423354773 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau739 :
+    Real.arctan (229220418382507 / 312500000000000 : ℝ) ≤
+      (2 * (791075848886423354773 / 2500000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (155021676008719 / 125000000000000 : ℝ) ≤
+          Real.sqrt (1 + (229220418382507 / 312500000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (229220418382507 / 312500000000000 : ℝ) /
+                (1 + (155021676008719 / 125000000000000 : ℝ)) =
+              (458440836765014 / 1400108380043595 : ℝ)
+            by
+        norm_num];
+        exact hau740)
+  have hat1 :
+    (Real.pi / 2 - (2 * (791075848886423354773 / 2500000000000000000000 : ℝ))) ≤
+      Real.arctan (312500000000000 / 229220418382507 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (312500000000000 / 229220418382507 : ℝ) =
+              (229220418382507 / 312500000000000 : ℝ)
+            by
+        norm_num];
+        exact hau739)
+  have hz1 :
+    (312500000000000 / 229220418382507 : ℝ) ≤
+      1 / Real.sqrt (17216962626667 / 32000000000000 : ℝ) := by
+    rw [show (312500000000000 / 229220418382507 : ℝ) = 1 / (229220418382507 / 312500000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 7335053388240221 : ℝ) ≤
+      (203789265870628778459 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (17216962626667 / 32000000000000 : ℝ) ≤
+      (750000000000000 / 7335053388240221 : ℝ) := by
+    rw [show
+        (750000000000000 / 7335053388240221 : ℝ) =
+          (3 / 40) / (7335053388240221 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (791075848886423354773 / 2500000000000000000000 : ℝ))) -
+        6 * (203789265870628778459 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (17216962626667 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (17216962626667 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((7335053388240221 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (791075848886423354773 / 2500000000000000000000 : ℝ))) -
+          6 * (203789265870628778459 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (17216962626667 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (17216962626667 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (17216962626667 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_485 :
+    ((4309081371789734423577 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(3041464656326215977173 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((229353327984451 / 312500000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3165685762735902141383 / 10000000000000000000000 : ℝ))) -
+              6 * (1018359927944392969983 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (68947737388703 / 128000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (68947737388703 / 128000000000000 : ℝ)) =
+      Real.log (196947737388703 / 128000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((68947737388703 / 128000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (69667737388703 / 128000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (4309081371789734423577 / 10000000000000000000000 : ℝ) ≤
+      Real.log (196947737388703 / 128000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (69667737388703 / 128000000000000 : ℝ) ≤
+      (-(3041464656326215977173 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (69667737388703 / 128000000000000 : ℝ) = (69667737388703 / 64000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (69667737388703 / 64000000000000 : ℝ) ≤
+            (424271245173784022827 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (229353327984451 / 312500000000000 : ℝ) ≤ Real.sqrt (68947737388703 / 128000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (68947737388703 / 128000000000000 : ℝ) ≤
+      (1467861299100487 / 2000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (68947737388703 / 128000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau742 :
+    Real.arctan (1467861299100487 / 4480850014288846 : ℝ) ≤
+      (3165685762735902141383 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau741 :
+    Real.arctan (1467861299100487 / 2000000000000000 : ℝ) ≤
+      (2 * (3165685762735902141383 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (1240425007144423 / 1000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1467861299100487 / 2000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1467861299100487 / 2000000000000000 : ℝ) /
+                (1 + (1240425007144423 / 1000000000000000 : ℝ)) =
+              (1467861299100487 / 4480850014288846 : ℝ)
+            by
+        norm_num];
+        exact hau742)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3165685762735902141383 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (2000000000000000 / 1467861299100487 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2000000000000000 / 1467861299100487 : ℝ) =
+              (1467861299100487 / 2000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau741)
+  have hz1 :
+    (2000000000000000 / 1467861299100487 : ℝ) ≤
+      1 / Real.sqrt (68947737388703 / 128000000000000 : ℝ) := by
+    rw [show
+        (2000000000000000 / 1467861299100487 : ℝ) = 1 / (1467861299100487 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (23437500000000 / 229353327984451 : ℝ) ≤
+      (1018359927944392969983 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (68947737388703 / 128000000000000 : ℝ) ≤
+      (23437500000000 / 229353327984451 : ℝ) := by
+    rw [show
+        (23437500000000 / 229353327984451 : ℝ) =
+          (3 / 40) / (229353327984451 / 312500000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3165685762735902141383 / 10000000000000000000000 : ℝ))) -
+        6 * (1018359927944392969983 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (68947737388703 / 128000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (68947737388703 / 128000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((229353327984451 / 312500000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3165685762735902141383 / 10000000000000000000000 : ℝ))) -
+          6 * (1018359927944392969983 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (68947737388703 / 128000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (68947737388703 / 128000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (68947737388703 / 128000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_486 :
+    ((4313136797123612745393 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(3035734520864270388191 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3671778569764047 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (633413353805602304743 / 2000000000000000000000 : ℝ))) -
+              6 * (6361090861247965123 / 62500000000000000000 : ℝ)))) ≤
+      Vfield (34513812135369 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (34513812135369 / 64000000000000 : ℝ)) =
+      Real.log (98513812135369 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((34513812135369 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (34873812135369 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (4313136797123612745393 / 10000000000000000000000 : ℝ) ≤
+      Real.log (98513812135369 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (34873812135369 / 64000000000000 : ℝ) ≤
+      (-(3035734520864270388191 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (34873812135369 / 64000000000000 : ℝ) = (34873812135369 / 32000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (34873812135369 / 32000000000000 : ℝ) ≤
+            (430001380635729611809 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3671778569764047 / 5000000000000000 : ℝ) ≤ Real.sqrt (34513812135369 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (34513812135369 / 64000000000000 : ℝ) ≤
+      (7343557139528097 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (34513812135369 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau744 :
+    Real.arctan (2447852379842699 / 7468921850656191 : ℝ) ≤
+      (633413353805602304743 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau743 :
+    Real.arctan (7343557139528097 / 10000000000000000 : ℝ) ≤
+      (2 * (633413353805602304743 / 2000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12406765551968573 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (7343557139528097 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (7343557139528097 / 10000000000000000 : ℝ) /
+                (1 + (12406765551968573 / 10000000000000000 : ℝ)) =
+              (2447852379842699 / 7468921850656191 : ℝ)
+            by
+        norm_num];
+        exact hau744)
+  have hat1 :
+    (Real.pi / 2 - (2 * (633413353805602304743 / 2000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 7343557139528097 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 7343557139528097 : ℝ) =
+              (7343557139528097 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau743)
+  have hz1 :
+    (10000000000000000 / 7343557139528097 : ℝ) ≤
+      1 / Real.sqrt (34513812135369 / 64000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 7343557139528097 : ℝ) =
+          1 / (7343557139528097 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (125000000000000 / 1223926189921349 : ℝ) ≤
+      (6361090861247965123 / 62500000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (34513812135369 / 64000000000000 : ℝ) ≤
+      (125000000000000 / 1223926189921349 : ℝ) := by
+    rw [show
+        (125000000000000 / 1223926189921349 : ℝ) =
+          (3 / 40) / (3671778569764047 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (633413353805602304743 / 2000000000000000000000 : ℝ))) -
+        6 * (6361090861247965123 / 62500000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (34513812135369 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (34513812135369 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3671778569764047 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (633413353805602304743 / 2000000000000000000000 : ℝ))) -
+          6 * (6361090861247965123 / 62500000000000000000 : ℝ))) ≤
+      Real.sqrt (34513812135369 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (34513812135369 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (34513812135369 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_487 :
+    ((1079297644619177201843 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(6060021889552608097611 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((7347805324592091 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2 * (63368928338273947477 / 200000000000000000000 : ℝ))) -
+              6 * (127148769501964290459 / 1250000000000000000000 : ℝ)))) ≤
+      Vfield (69107511152773 / 128000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (69107511152773 / 128000000000000 : ℝ)) =
+      Real.log (197107511152773 / 128000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((69107511152773 / 128000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (69827511152773 / 128000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1079297644619177201843 / 2500000000000000000000 : ℝ) ≤
+      Real.log (197107511152773 / 128000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (69827511152773 / 128000000000000 : ℝ) ≤
+      (-(6060021889552608097611 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (69827511152773 / 128000000000000 : ℝ) = (69827511152773 / 64000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (69827511152773 / 64000000000000 : ℝ) ≤
+            (871449913447391902389 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (7347805324592091 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (69107511152773 / 128000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (69107511152773 / 128000000000000 : ℝ) ≤
+      (3673902662296047 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (69107511152773 / 128000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau746 :
+    Real.arctan (2449268441530698 / 7469760174193609 : ℝ) ≤
+      (63368928338273947477 / 200000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau745 :
+    Real.arctan (3673902662296047 / 5000000000000000 : ℝ) ≤
+      (2 * (63368928338273947477 / 200000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12409280522580827 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (3673902662296047 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (3673902662296047 / 5000000000000000 : ℝ) /
+                (1 + (12409280522580827 / 10000000000000000 : ℝ)) =
+              (2449268441530698 / 7469760174193609 : ℝ)
+            by
+        norm_num];
+        exact hau746)
+  have hat1 :
+    (Real.pi / 2 - (2 * (63368928338273947477 / 200000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 3673902662296047 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 3673902662296047 : ℝ) =
+              (3673902662296047 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau745)
+  have hz1 :
+    (5000000000000000 / 3673902662296047 : ℝ) ≤
+      1 / Real.sqrt (69107511152773 / 128000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 3673902662296047 : ℝ) = 1 / (3673902662296047 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 2449268441530697 : ℝ) ≤
+      (127148769501964290459 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (69107511152773 / 128000000000000 : ℝ) ≤
+      (250000000000000 / 2449268441530697 : ℝ) := by
+    rw [show
+        (250000000000000 / 2449268441530697 : ℝ) =
+          (3 / 40) / (7347805324592091 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (63368928338273947477 / 200000000000000000000 : ℝ))) -
+        6 * (127148769501964290459 / 1250000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (69107511152773 / 128000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (69107511152773 / 128000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((7347805324592091 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (63368928338273947477 / 200000000000000000000 : ℝ))) -
+          6 * (127148769501964290459 / 1250000000000000000000 : ℝ))) ≤
+      Real.sqrt (69107511152773 / 128000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (69107511152773 / 128000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (69107511152773 / 128000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_488 :
+    ((4321242717181350277451 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1209717565224901623787 / 2000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((7352051054956959 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2 * (79245617721939937011 / 250000000000000000000 : ℝ))) -
+              6 * (1016606779700894286959 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (8648424754351 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (8648424754351 / 16000000000000 : ℝ)) =
+      Real.log (24648424754351 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((8648424754351 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (8738424754351 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (4321242717181350277451 / 10000000000000000000000 : ℝ) ≤
+      Real.log (24648424754351 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (8738424754351 / 16000000000000 : ℝ) ≤
+      (-(1209717565224901623787 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (8738424754351 / 16000000000000 : ℝ) = (8738424754351 / 8000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8738424754351 / 8000000000000 : ℝ) ≤
+            (176576795375098376213 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (7352051054956959 / 10000000000000000 : ℝ) ≤ Real.sqrt (8648424754351 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (8648424754351 / 16000000000000 : ℝ) ≤ (3676025527478481 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (8648424754351 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau748 :
+    Real.arctan (7352051054956962 / 22411794983590961 : ℝ) ≤
+      (79245617721939937011 / 250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau747 :
+    Real.arctan (3676025527478481 / 5000000000000000 : ℝ) ≤
+      (2 * (79245617721939937011 / 250000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12411794983590961 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (3676025527478481 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (3676025527478481 / 5000000000000000 : ℝ) /
+                (1 + (12411794983590961 / 10000000000000000 : ℝ)) =
+              (7352051054956962 / 22411794983590961 : ℝ)
+            by
+        norm_num];
+        exact hau748)
+  have hat1 :
+    (Real.pi / 2 - (2 * (79245617721939937011 / 250000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 3676025527478481 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 3676025527478481 : ℝ) =
+              (3676025527478481 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau747)
+  have hz1 :
+    (5000000000000000 / 3676025527478481 : ℝ) ≤
+      1 / Real.sqrt (8648424754351 / 16000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 3676025527478481 : ℝ) = 1 / (3676025527478481 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 2450683684985653 : ℝ) ≤
+      (1016606779700894286959 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (8648424754351 / 16000000000000 : ℝ) ≤
+      (250000000000000 / 2450683684985653 : ℝ) := by
+    rw [show
+        (250000000000000 / 2450683684985653 : ℝ) =
+          (3 / 40) / (7352051054956959 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (79245617721939937011 / 250000000000000000000 : ℝ))) -
+        6 * (1016606779700894286959 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (8648424754351 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (8648424754351 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((7352051054956959 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (79245617721939937011 / 250000000000000000000 : ℝ))) -
+          6 * (1016606779700894286959 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (8648424754351 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (8648424754351 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (8648424754351 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_489 :
+    ((270330825910515365411 / 625000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(6037166821546904547073 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((7356294334872931 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1585600823698669241149 / 5000000000000000000000 : ℝ))) -
+              6 * (508012202987596007037 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (69267284916843 / 128000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (69267284916843 / 128000000000000 : ℝ)) =
+      Real.log (197267284916843 / 128000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((69267284916843 / 128000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (69987284916843 / 128000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (270330825910515365411 / 625000000000000000000 : ℝ) ≤
+      Real.log (197267284916843 / 128000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (69987284916843 / 128000000000000 : ℝ) ≤
+      (-(6037166821546904547073 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (69987284916843 / 128000000000000 : ℝ) = (69987284916843 / 64000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (69987284916843 / 64000000000000 : ℝ) ≤
+            (894304981453095452927 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (7356294334872931 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (69267284916843 / 128000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (69267284916843 / 128000000000000 : ℝ) ≤
+      (3678147167436467 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (69267284916843 / 128000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau750 :
+    Real.arctan (3678147167436467 / 11207154467654313 : ℝ) ≤
+      (1585600823698669241149 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau749 :
+    Real.arctan (3678147167436467 / 5000000000000000 : ℝ) ≤
+      (2 * (1585600823698669241149 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6207154467654313 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (3678147167436467 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (3678147167436467 / 5000000000000000 : ℝ) /
+                (1 + (6207154467654313 / 5000000000000000 : ℝ)) =
+              (3678147167436467 / 11207154467654313 : ℝ)
+            by
+        norm_num];
+        exact hau750)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1585600823698669241149 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 3678147167436467 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 3678147167436467 : ℝ) =
+              (3678147167436467 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau749)
+  have hz1 :
+    (5000000000000000 / 3678147167436467 : ℝ) ≤
+      1 / Real.sqrt (69267284916843 / 128000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 3678147167436467 : ℝ) = 1 / (3678147167436467 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 7356294334872931 : ℝ) ≤
+      (508012202987596007037 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (69267284916843 / 128000000000000 : ℝ) ≤
+      (750000000000000 / 7356294334872931 : ℝ) := by
+    rw [show
+        (750000000000000 / 7356294334872931 : ℝ) =
+          (3 / 40) / (7356294334872931 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1585600823698669241149 / 5000000000000000000000 : ℝ))) -
+        6 * (508012202987596007037 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (69267284916843 / 128000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (69267284916843 / 128000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((7356294334872931 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1585600823698669241149 / 5000000000000000000000 : ℝ))) -
+          6 * (508012202987596007037 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (69267284916843 / 128000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (69267284916843 / 128000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (69267284916843 / 128000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_490 :
+    ((432934207196648785071 / 1000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(6025758846024781546297 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((736053516857799 / 1000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3172577234943563944481 / 10000000000000000000000 : ℝ))) -
+              6 * (203088606394024225601 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (34673585899439 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (34673585899439 / 64000000000000 : ℝ)) =
+      Real.log (98673585899439 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((34673585899439 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (35033585899439 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (432934207196648785071 / 1000000000000000000000 : ℝ) ≤
+      Real.log (98673585899439 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (35033585899439 / 64000000000000 : ℝ) ≤
+      (-(6025758846024781546297 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (35033585899439 / 64000000000000 : ℝ) = (35033585899439 / 32000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (35033585899439 / 32000000000000 : ℝ) ≤
+            (905712956975218453703 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (736053516857799 / 1000000000000000 : ℝ) ≤ Real.sqrt (34673585899439 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (34673585899439 / 64000000000000 : ℝ) ≤
+      (7360535168577993 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (34673585899439 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau752 :
+    Real.arctan (2453511722859331 / 7472274126014387 : ℝ) ≤
+      (3172577234943563944481 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau751 :
+    Real.arctan (7360535168577993 / 10000000000000000 : ℝ) ≤
+      (2 * (3172577234943563944481 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12416822378043161 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (7360535168577993 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (7360535168577993 / 10000000000000000 : ℝ) /
+                (1 + (12416822378043161 / 10000000000000000 : ℝ)) =
+              (2453511722859331 / 7472274126014387 : ℝ)
+            by
+        norm_num];
+        exact hau752)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3172577234943563944481 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 7360535168577993 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 7360535168577993 : ℝ) =
+              (7360535168577993 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau751)
+  have hz1 :
+    (10000000000000000 / 7360535168577993 : ℝ) ≤
+      1 / Real.sqrt (34673585899439 / 64000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 7360535168577993 : ℝ) =
+          1 / (7360535168577993 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (25000000000000 / 245351172285933 : ℝ) ≤
+      (203088606394024225601 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (34673585899439 / 64000000000000 : ℝ) ≤
+      (25000000000000 / 245351172285933 : ℝ) := by
+    rw [show
+        (25000000000000 / 245351172285933 : ℝ) =
+          (3 / 40) / (736053516857799 / 1000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3172577234943563944481 / 10000000000000000000000 : ℝ))) -
+        6 * (203088606394024225601 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (34673585899439 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (34673585899439 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((736053516857799 / 1000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3172577234943563944481 / 10000000000000000000000 : ℝ))) -
+          6 * (203088606394024225601 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (34673585899439 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (34673585899439 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (34673585899439 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_491 :
+    ((4333389290703554887527 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1202872773972995472511 / 2000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((7364773560297917 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (634790294795992208869 / 2000000000000000000000 : ℝ))) -
+              6 * (1014862654828672016827 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (69427058680913 / 128000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (69427058680913 / 128000000000000 : ℝ)) =
+      Real.log (197427058680913 / 128000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((69427058680913 / 128000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (70147058680913 / 128000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (4333389290703554887527 / 10000000000000000000000 : ℝ) ≤
+      Real.log (197427058680913 / 128000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (70147058680913 / 128000000000000 : ℝ) ≤
+      (-(1202872773972995472511 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (70147058680913 / 128000000000000 : ℝ) = (70147058680913 / 64000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (70147058680913 / 64000000000000 : ℝ) ≤
+            (183421586627004527489 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (7364773560297917 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (69427058680913 / 128000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (69427058680913 / 128000000000000 : ℝ) ≤ (23014917375931 / 31250000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (69427058680913 / 128000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau754 :
+    Real.arctan (3682386780148960 / 11209667656051797 : ℝ) ≤
+      (634790294795992208869 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau753 :
+    Real.arctan (23014917375931 / 31250000000000 : ℝ) ≤
+      (2 * (634790294795992208869 / 2000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6209667656051797 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (23014917375931 / 31250000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (23014917375931 / 31250000000000 : ℝ) /
+                (1 + (6209667656051797 / 5000000000000000 : ℝ)) =
+              (3682386780148960 / 11209667656051797 : ℝ)
+            by
+        norm_num];
+        exact hau754)
+  have hat1 :
+    (Real.pi / 2 - (2 * (634790294795992208869 / 2000000000000000000000 : ℝ))) ≤
+      Real.arctan (31250000000000 / 23014917375931 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (31250000000000 / 23014917375931 : ℝ) =
+              (23014917375931 / 31250000000000 : ℝ)
+            by
+        norm_num];
+        exact hau753)
+  have hz1 :
+    (31250000000000 / 23014917375931 : ℝ) ≤
+      1 / Real.sqrt (69427058680913 / 128000000000000 : ℝ) := by
+    rw [show (31250000000000 / 23014917375931 : ℝ) = 1 / (23014917375931 / 31250000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 7364773560297917 : ℝ) ≤
+      (1014862654828672016827 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (69427058680913 / 128000000000000 : ℝ) ≤
+      (750000000000000 / 7364773560297917 : ℝ) := by
+    rw [show
+        (750000000000000 / 7364773560297917 : ℝ) =
+          (3 / 40) / (7364773560297917 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (634790294795992208869 / 2000000000000000000000 : ℝ))) -
+        6 * (1014862654828672016827 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (69427058680913 / 128000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (69427058680913 / 128000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((7364773560297917 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (634790294795992208869 / 2000000000000000000000 : ℝ))) -
+          6 * (1014862654828672016827 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (69427058680913 / 128000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (69427058680913 / 128000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (69427058680913 / 128000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_492 :
+    ((542179359013164303477 / 1250000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1200596372695144120349 / 2000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3684504757123171 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1587662183481644612061 / 5000000000000000000000 : ℝ))) -
+              6 * (25357081792631315167 / 250000000000000000000 : ℝ)))) ≤
+      Vfield (17376736390737 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (17376736390737 / 32000000000000 : ℝ)) =
+      Real.log (49376736390737 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((17376736390737 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (17556736390737 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (542179359013164303477 / 1250000000000000000000 : ℝ) ≤
+      Real.log (49376736390737 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (17556736390737 / 32000000000000 : ℝ) ≤
+      (-(1200596372695144120349 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (17556736390737 / 32000000000000 : ℝ) = (17556736390737 / 16000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17556736390737 / 16000000000000 : ℝ) ≤
+            (185697987904855879651 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3684504757123171 / 5000000000000000 : ℝ) ≤ Real.sqrt (17376736390737 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (17376736390737 / 32000000000000 : ℝ) ≤ (1473801902849269 / 2000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (17376736390737 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau756 :
+    Real.arctan (7369009514246345 / 22421847737798637 : ℝ) ≤
+      (1587662183481644612061 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau755 :
+    Real.arctan (1473801902849269 / 2000000000000000 : ℝ) ≤
+      (2 * (1587662183481644612061 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12421847737798637 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1473801902849269 / 2000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1473801902849269 / 2000000000000000 : ℝ) /
+                (1 + (12421847737798637 / 10000000000000000 : ℝ)) =
+              (7369009514246345 / 22421847737798637 : ℝ)
+            by
+        norm_num];
+        exact hau756)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1587662183481644612061 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (2000000000000000 / 1473801902849269 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2000000000000000 / 1473801902849269 : ℝ) =
+              (1473801902849269 / 2000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau755)
+  have hz1 :
+    (2000000000000000 / 1473801902849269 : ℝ) ≤
+      1 / Real.sqrt (17376736390737 / 32000000000000 : ℝ) := by
+    rw [show
+        (2000000000000000 / 1473801902849269 : ℝ) = 1 / (1473801902849269 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 3684504757123171 : ℝ) ≤
+      (25357081792631315167 / 250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (17376736390737 / 32000000000000 : ℝ) ≤
+      (375000000000000 / 3684504757123171 : ℝ) := by
+    rw [show
+        (375000000000000 / 3684504757123171 : ℝ) =
+          (3 / 40) / (3684504757123171 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1587662183481644612061 / 5000000000000000000000 : ℝ))) -
+        6 * (25357081792631315167 / 250000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (17376736390737 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (17376736390737 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3684504757123171 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1587662183481644612061 / 5000000000000000000000 : ℝ))) -
+          6 * (25357081792631315167 / 250000000000000000000 : ℝ))) ≤
+      Real.sqrt (17376736390737 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (17376736390737 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (17376736390737 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/V41.lean b/LeanPool/Zeta5Irrational/Table/V41.lean
new file mode 100644
index 0000000000..6f39f2a9ed
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/V41.lean
@@ -0,0 +1,1445 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Energy
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # V41: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_493 :
+    ((868295763499205084321 / 2000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(5991612797366169144041 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1843310758656197 / 2500000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2 * (198543494771462783499 / 625000000000000000000 : ℝ))) -
+              6 * (50685243988281521377 / 500000000000000000000 : ℝ)))) ≤
+      Vfield (69586832444983 / 128000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (69586832444983 / 128000000000000 : ℝ)) =
+      Real.log (197586832444983 / 128000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((69586832444983 / 128000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (70306832444983 / 128000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (868295763499205084321 / 2000000000000000000000 : ℝ) ≤
+      Real.log (197586832444983 / 128000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (70306832444983 / 128000000000000 : ℝ) ≤
+      (-(5991612797366169144041 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (70306832444983 / 128000000000000 : ℝ) = (70306832444983 / 64000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (70306832444983 / 64000000000000 : ℝ) ≤
+            (939859005633830855959 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1843310758656197 / 2500000000000000 : ℝ) ≤
+      Real.sqrt (69586832444983 / 128000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (69586832444983 / 128000000000000 : ℝ) ≤
+      (7373243034624791 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (69586832444983 / 128000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau758 :
+    Real.arctan (7373243034624791 / 22424359655436692 : ℝ) ≤
+      (198543494771462783499 / 625000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau757 :
+    Real.arctan (7373243034624791 / 10000000000000000 : ℝ) ≤
+      (2 * (198543494771462783499 / 625000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (3106089913859173 / 2500000000000000 : ℝ) ≤
+          Real.sqrt (1 + (7373243034624791 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (7373243034624791 / 10000000000000000 : ℝ) /
+                (1 + (3106089913859173 / 2500000000000000 : ℝ)) =
+              (7373243034624791 / 22424359655436692 : ℝ)
+            by
+        norm_num];
+        exact hau758)
+  have hat1 :
+    (Real.pi / 2 - (2 * (198543494771462783499 / 625000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 7373243034624791 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 7373243034624791 : ℝ) =
+              (7373243034624791 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau757)
+  have hz1 :
+    (10000000000000000 / 7373243034624791 : ℝ) ≤
+      1 / Real.sqrt (69586832444983 / 128000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 7373243034624791 : ℝ) =
+          1 / (7373243034624791 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (187500000000000 / 1843310758656197 : ℝ) ≤
+      (50685243988281521377 / 500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (69586832444983 / 128000000000000 : ℝ) ≤
+      (187500000000000 / 1843310758656197 : ℝ) := by
+    rw [show
+        (187500000000000 / 1843310758656197 : ℝ) =
+          (3 / 40) / (1843310758656197 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (198543494771462783499 / 625000000000000000000 : ℝ))) -
+        6 * (50685243988281521377 / 500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (69586832444983 / 128000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (69586832444983 / 128000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1843310758656197 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (198543494771462783499 / 625000000000000000000 : ℝ))) -
+          6 * (50685243988281521377 / 500000000000000000000 : ℝ))) ≤
+      Real.sqrt (69586832444983 / 128000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (69586832444983 / 128000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (69586832444983 / 128000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_494 :
+    ((2172760564099170449509 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2990128321072975838143 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((295098965024909 / 400000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3178066124563289035779 / 10000000000000000000000 : ℝ))) -
+              6 * (506563738093436821663 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (34833359663509 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (34833359663509 / 64000000000000 : ℝ)) =
+      Real.log (98833359663509 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((34833359663509 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (35193359663509 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2172760564099170449509 / 5000000000000000000000 : ℝ) ≤
+      Real.log (98833359663509 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (35193359663509 / 64000000000000 : ℝ) ≤
+      (-(2990128321072975838143 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (35193359663509 / 64000000000000 : ℝ) = (35193359663509 / 32000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (35193359663509 / 32000000000000 : ℝ) ≤
+            (475607580427024161857 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (295098965024909 / 400000000000000 : ℝ) ≤ Real.sqrt (34833359663509 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (34833359663509 / 64000000000000 : ℝ) ≤ (922184265702841 / 1250000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (34833359663509 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau760 :
+    Real.arctan (7377474125622728 / 22426871065325849 : ℝ) ≤
+      (3178066124563289035779 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau759 :
+    Real.arctan (922184265702841 / 1250000000000000 : ℝ) ≤
+      (2 * (3178066124563289035779 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12426871065325849 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (922184265702841 / 1250000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (922184265702841 / 1250000000000000 : ℝ) /
+                (1 + (12426871065325849 / 10000000000000000 : ℝ)) =
+              (7377474125622728 / 22426871065325849 : ℝ)
+            by
+        norm_num];
+        exact hau760)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3178066124563289035779 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (1250000000000000 / 922184265702841 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1250000000000000 / 922184265702841 : ℝ) =
+              (922184265702841 / 1250000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau759)
+  have hz1 :
+    (1250000000000000 / 922184265702841 : ℝ) ≤
+      1 / Real.sqrt (34833359663509 / 64000000000000 : ℝ) := by
+    rw [show
+        (1250000000000000 / 922184265702841 : ℝ) = 1 / (922184265702841 / 1250000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (30000000000000 / 295098965024909 : ℝ) ≤
+      (506563738093436821663 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (34833359663509 / 64000000000000 : ℝ) ≤
+      (30000000000000 / 295098965024909 : ℝ) := by
+    rw [show
+        (30000000000000 / 295098965024909 : ℝ) =
+          (3 / 40) / (295098965024909 / 400000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3178066124563289035779 / 10000000000000000000000 : ℝ))) -
+        6 * (506563738093436821663 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (34833359663509 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (34833359663509 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((295098965024909 / 400000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3178066124563289035779 / 10000000000000000000000 : ℝ))) -
+          6 * (506563738093436821663 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (34833359663509 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (34833359663509 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (34833359663509 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_495 :
+    ((2174780902766655282939 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(298445668426235591233 / 500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((7381702791417617 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (397429374257384632401 / 1250000000000000000000 : ℝ))) -
+              6 * (1012551058157293370951 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (69746606209053 / 128000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (69746606209053 / 128000000000000 : ℝ)) =
+      Real.log (197746606209053 / 128000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((69746606209053 / 128000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (70466606209053 / 128000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2174780902766655282939 / 5000000000000000000000 : ℝ) ≤
+      Real.log (197746606209053 / 128000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (70466606209053 / 128000000000000 : ℝ) ≤
+      (-(298445668426235591233 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (70466606209053 / 128000000000000 : ℝ) = (70466606209053 / 64000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (70466606209053 / 64000000000000 : ℝ) ≤
+            (48127921723764408767 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (7381702791417617 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (69746606209053 / 128000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (69746606209053 / 128000000000000 : ℝ) ≤ (369085139570881 / 500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (69746606209053 / 128000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau762 :
+    Real.arctan (3690851395708810 / 11214690983886943 : ℝ) ≤
+      (397429374257384632401 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau761 :
+    Real.arctan (369085139570881 / 500000000000000 : ℝ) ≤
+      (2 * (397429374257384632401 / 1250000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6214690983886943 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (369085139570881 / 500000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (369085139570881 / 500000000000000 : ℝ) /
+                (1 + (6214690983886943 / 5000000000000000 : ℝ)) =
+              (3690851395708810 / 11214690983886943 : ℝ)
+            by
+        norm_num];
+        exact hau762)
+  have hat1 :
+    (Real.pi / 2 - (2 * (397429374257384632401 / 1250000000000000000000 : ℝ))) ≤
+      Real.arctan (500000000000000 / 369085139570881 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (500000000000000 / 369085139570881 : ℝ) =
+              (369085139570881 / 500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau761)
+  have hz1 :
+    (500000000000000 / 369085139570881 : ℝ) ≤
+      1 / Real.sqrt (69746606209053 / 128000000000000 : ℝ) := by
+    rw [show (500000000000000 / 369085139570881 : ℝ) = 1 / (369085139570881 / 500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 7381702791417617 : ℝ) ≤
+      (1012551058157293370951 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (69746606209053 / 128000000000000 : ℝ) ≤
+      (750000000000000 / 7381702791417617 : ℝ) := by
+    rw [show
+        (750000000000000 / 7381702791417617 : ℝ) =
+          (3 / 40) / (7381702791417617 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (397429374257384632401 / 1250000000000000000000 : ℝ))) -
+        6 * (1012551058157293370951 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (69746606209053 / 128000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (69746606209053 / 128000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((7381702791417617 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (397429374257384632401 / 1250000000000000000000 : ℝ))) -
+          6 * (1012551058157293370951 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (69746606209053 / 128000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (69746606209053 / 128000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (69746606209053 / 128000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_496 :
+    ((4353600850820383394063 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(5957582947311654869927 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((7385929036174967 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2 * (159040126363004019747 / 500000000000000000000 : ℝ))) -
+              6 * (25299390571909664953 / 250000000000000000000 : ℝ)))) ≤
+      Vfield (4364155818193 / 8000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (4364155818193 / 8000000000000 : ℝ)) =
+      Real.log (12364155818193 / 8000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((4364155818193 / 8000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (4409155818193 / 8000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (4353600850820383394063 / 10000000000000000000000 : ℝ) ≤
+      Real.log (12364155818193 / 8000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (4409155818193 / 8000000000000 : ℝ) ≤
+      (-(5957582947311654869927 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (4409155818193 / 8000000000000 : ℝ) = (4409155818193 / 4000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4409155818193 / 4000000000000 : ℝ) ≤
+            (973888855688345130073 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (7385929036174967 / 10000000000000000 : ℝ) ≤ Real.sqrt (4364155818193 / 8000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (4364155818193 / 8000000000000 : ℝ) ≤ (738592903617497 / 1000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (4364155818193 / 8000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau764 :
+    Real.arctan (7385929036174970 / 22431892363088271 : ℝ) ≤
+      (159040126363004019747 / 500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau763 :
+    Real.arctan (738592903617497 / 1000000000000000 : ℝ) ≤
+      (2 * (159040126363004019747 / 500000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12431892363088271 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (738592903617497 / 1000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (738592903617497 / 1000000000000000 : ℝ) /
+                (1 + (12431892363088271 / 10000000000000000 : ℝ)) =
+              (7385929036174970 / 22431892363088271 : ℝ)
+            by
+        norm_num];
+        exact hau764)
+  have hat1 :
+    (Real.pi / 2 - (2 * (159040126363004019747 / 500000000000000000000 : ℝ))) ≤
+      Real.arctan (1000000000000000 / 738592903617497 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1000000000000000 / 738592903617497 : ℝ) =
+              (738592903617497 / 1000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau763)
+  have hz1 :
+    (1000000000000000 / 738592903617497 : ℝ) ≤
+      1 / Real.sqrt (4364155818193 / 8000000000000 : ℝ) := by
+    rw [show
+        (1000000000000000 / 738592903617497 : ℝ) = 1 / (738592903617497 / 1000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 7385929036174967 : ℝ) ≤
+      (25299390571909664953 / 250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (4364155818193 / 8000000000000 : ℝ) ≤
+      (750000000000000 / 7385929036174967 : ℝ) := by
+    rw [show
+        (750000000000000 / 7385929036174967 : ℝ) =
+          (3 / 40) / (7385929036174967 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (159040126363004019747 / 500000000000000000000 : ℝ))) -
+        6 * (25299390571909664953 / 250000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (4364155818193 / 8000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (4364155818193 / 8000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((7385929036174967 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (159040126363004019747 / 500000000000000000000 : ℝ))) -
+          6 * (25299390571909664953 / 250000000000000000000 : ℝ))) ≤
+      Real.sqrt (4364155818193 / 8000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (4364155818193 / 8000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (4364155818193 / 8000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_497 :
+    ((4357638265377410206641 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2973132674707548513867 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((739015286404837 / 1000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1591084363294408453171 / 5000000000000000000000 : ℝ))) -
+              6 * (1011401167554778336271 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (69906379973123 / 128000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (69906379973123 / 128000000000000 : ℝ)) =
+      Real.log (197906379973123 / 128000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((69906379973123 / 128000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (70626379973123 / 128000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (4357638265377410206641 / 10000000000000000000000 : ℝ) ≤
+      Real.log (197906379973123 / 128000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (70626379973123 / 128000000000000 : ℝ) ≤
+      (-(2973132674707548513867 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (70626379973123 / 128000000000000 : ℝ) = (70626379973123 / 64000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (70626379973123 / 64000000000000 : ℝ) ≤
+            (492603226792451486133 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (739015286404837 / 1000000000000000 : ℝ) ≤ Real.sqrt (69906379973123 / 128000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (69906379973123 / 128000000000000 : ℝ) ≤
+      (7390152864048373 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (69906379973123 / 128000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau766 :
+    Real.arctan (7390152864048373 / 22434402251576162 : ℝ) ≤
+      (1591084363294408453171 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau765 :
+    Real.arctan (7390152864048373 / 10000000000000000 : ℝ) ≤
+      (2 * (1591084363294408453171 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6217201125788081 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (7390152864048373 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (7390152864048373 / 10000000000000000 : ℝ) /
+                (1 + (6217201125788081 / 5000000000000000 : ℝ)) =
+              (7390152864048373 / 22434402251576162 : ℝ)
+            by
+        norm_num];
+        exact hau766)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1591084363294408453171 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 7390152864048373 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 7390152864048373 : ℝ) =
+              (7390152864048373 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau765)
+  have hz1 :
+    (10000000000000000 / 7390152864048373 : ℝ) ≤
+      1 / Real.sqrt (69906379973123 / 128000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 7390152864048373 : ℝ) =
+          1 / (7390152864048373 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (75000000000000 / 739015286404837 : ℝ) ≤
+      (1011401167554778336271 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (69906379973123 / 128000000000000 : ℝ) ≤
+      (75000000000000 / 739015286404837 : ℝ) := by
+    rw [show
+        (75000000000000 / 739015286404837 : ℝ) =
+          (3 / 40) / (739015286404837 / 1000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1591084363294408453171 / 5000000000000000000000 : ℝ))) -
+        6 * (1011401167554778336271 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (69906379973123 / 128000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (69906379973123 / 128000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((739015286404837 / 1000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1591084363294408453171 / 5000000000000000000000 : ℝ))) -
+          6 * (1011401167554778336271 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (69906379973123 / 128000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (69906379973123 / 128000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (69906379973123 / 128000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_498 :
+    ((4361674050520646257781 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(5934960545842017276557 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1478874855835911 / 2000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3183533594461034959257 / 10000000000000000000000 : ℝ))) -
+              6 * (252706922353541398699 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (34993133427579 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (34993133427579 / 64000000000000 : ℝ)) =
+      Real.log (98993133427579 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((34993133427579 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (35353133427579 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (4361674050520646257781 / 10000000000000000000000 : ℝ) ≤
+      Real.log (98993133427579 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (35353133427579 / 64000000000000 : ℝ) ≤
+      (-(5934960545842017276557 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (35353133427579 / 64000000000000 : ℝ) = (35353133427579 / 32000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (35353133427579 / 32000000000000 : ℝ) ≤
+            (996511257157982723443 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1478874855835911 / 2000000000000000 : ℝ) ≤ Real.sqrt (34993133427579 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (34993133427579 / 64000000000000 : ℝ) ≤ (3697187139589779 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (34993133427579 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau768 :
+    Real.arctan (7394374279179558 / 22436911633544405 : ℝ) ≤
+      (3183533594461034959257 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau767 :
+    Real.arctan (3697187139589779 / 5000000000000000 : ℝ) ≤
+      (2 * (3183533594461034959257 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (2487382326708881 / 2000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (3697187139589779 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (3697187139589779 / 5000000000000000 : ℝ) /
+                (1 + (2487382326708881 / 2000000000000000 : ℝ)) =
+              (7394374279179558 / 22436911633544405 : ℝ)
+            by
+        norm_num];
+        exact hau768)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3183533594461034959257 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 3697187139589779 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 3697187139589779 : ℝ) =
+              (3697187139589779 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau767)
+  have hz1 :
+    (5000000000000000 / 3697187139589779 : ℝ) ≤
+      1 / Real.sqrt (34993133427579 / 64000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 3697187139589779 : ℝ) = 1 / (3697187139589779 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (50000000000000 / 492958285278637 : ℝ) ≤
+      (252706922353541398699 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (34993133427579 / 64000000000000 : ℝ) ≤
+      (50000000000000 / 492958285278637 : ℝ) := by
+    rw [show
+        (50000000000000 / 492958285278637 : ℝ) =
+          (3 / 40) / (1478874855835911 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3183533594461034959257 / 10000000000000000000000 : ℝ))) -
+        6 * (252706922353541398699 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (34993133427579 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (34993133427579 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1478874855835911 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3183533594461034959257 / 10000000000000000000000 : ℝ))) -
+          6 * (252706922353541398699 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (34993133427579 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (34993133427579 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (34993133427579 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_499 :
+    ((2182854103782376903733 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(5923668507697611716043 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3699296642849219 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2 * (15924485666428705901 / 50000000000000000000 : ℝ))) -
+              6 * (1010255185687260178931 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (70066153737193 / 128000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (70066153737193 / 128000000000000 : ℝ)) =
+      Real.log (198066153737193 / 128000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((70066153737193 / 128000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (70786153737193 / 128000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2182854103782376903733 / 5000000000000000000000 : ℝ) ≤
+      Real.log (198066153737193 / 128000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (70786153737193 / 128000000000000 : ℝ) ≤
+      (-(5923668507697611716043 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (70786153737193 / 128000000000000 : ℝ) = (70786153737193 / 64000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (70786153737193 / 64000000000000 : ℝ) ≤
+            (1007803295302388283957 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3699296642849219 / 5000000000000000 : ℝ) ≤
+      Real.sqrt (70066153737193 / 128000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (70066153737193 / 128000000000000 : ℝ) ≤
+      (7398593285698441 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (70066153737193 / 128000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau770 :
+    Real.arctan (7398593285698441 / 22439420509299540 : ℝ) ≤
+      (15924485666428705901 / 50000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau769 :
+    Real.arctan (7398593285698441 / 10000000000000000 : ℝ) ≤
+      (2 * (15924485666428705901 / 50000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (621971025464977 / 500000000000000 : ℝ) ≤
+          Real.sqrt (1 + (7398593285698441 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (7398593285698441 / 10000000000000000 : ℝ) /
+                (1 + (621971025464977 / 500000000000000 : ℝ)) =
+              (7398593285698441 / 22439420509299540 : ℝ)
+            by
+        norm_num];
+        exact hau770)
+  have hat1 :
+    (Real.pi / 2 - (2 * (15924485666428705901 / 50000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 7398593285698441 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 7398593285698441 : ℝ) =
+              (7398593285698441 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau769)
+  have hz1 :
+    (10000000000000000 / 7398593285698441 : ℝ) ≤
+      1 / Real.sqrt (70066153737193 / 128000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 7398593285698441 : ℝ) =
+          1 / (7398593285698441 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 3699296642849219 : ℝ) ≤
+      (1010255185687260178931 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (70066153737193 / 128000000000000 : ℝ) ≤
+      (375000000000000 / 3699296642849219 : ℝ) := by
+    rw [show
+        (375000000000000 / 3699296642849219 : ℝ) =
+          (3 / 40) / (3699296642849219 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (15924485666428705901 / 50000000000000000000 : ℝ))) -
+        6 * (1010255185687260178931 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (70066153737193 / 128000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (70066153737193 / 128000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3699296642849219 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (15924485666428705901 / 50000000000000000000 : ℝ))) -
+          6 * (1010255185687260178931 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (70066153737193 / 128000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (70066153737193 / 128000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (70066153737193 / 128000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_500 :
+    ((4369740737822804691011 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(739048650773106305367 / 1250000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1480561977544633 / 2000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2 * (159312967273261300701 / 500000000000000000000 : ℝ))) -
+              6 * (1009683653617733165789 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (17536510154807 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (17536510154807 / 32000000000000 : ℝ)) =
+      Real.log (49536510154807 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((17536510154807 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (17716510154807 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (4369740737822804691011 / 10000000000000000000000 : ℝ) ≤
+      Real.log (49536510154807 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 10
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (17716510154807 / 32000000000000 : ℝ) ≤
+      (-(739048650773106305367 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (17716510154807 / 32000000000000 : ℝ) = (17716510154807 / 16000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17716510154807 / 16000000000000 : ℝ) ≤
+            (127385324601893694633 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1480561977544633 / 2000000000000000 : ℝ) ≤ Real.sqrt (17536510154807 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (17536510154807 / 32000000000000 : ℝ) ≤ (231337808991349 / 312500000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (17536510154807 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau772 :
+    Real.arctan (7402809887723168 / 22441928879147793 : ℝ) ≤
+      (159312967273261300701 / 500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau771 :
+    Real.arctan (231337808991349 / 312500000000000 : ℝ) ≤
+      (2 * (159312967273261300701 / 500000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12441928879147793 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (231337808991349 / 312500000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (231337808991349 / 312500000000000 : ℝ) /
+                (1 + (12441928879147793 / 10000000000000000 : ℝ)) =
+              (7402809887723168 / 22441928879147793 : ℝ)
+            by
+        norm_num];
+        exact hau772)
+  have hat1 :
+    (Real.pi / 2 - (2 * (159312967273261300701 / 500000000000000000000 : ℝ))) ≤
+      Real.arctan (312500000000000 / 231337808991349 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (312500000000000 / 231337808991349 : ℝ) =
+              (231337808991349 / 312500000000000 : ℝ)
+            by
+        norm_num];
+        exact hau771)
+  have hz1 :
+    (312500000000000 / 231337808991349 : ℝ) ≤
+      1 / Real.sqrt (17536510154807 / 32000000000000 : ℝ) := by
+    rw [show (312500000000000 / 231337808991349 : ℝ) = 1 / (231337808991349 / 312500000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (150000000000000 / 1480561977544633 : ℝ) ≤
+      (1009683653617733165789 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (17536510154807 / 32000000000000 : ℝ) ≤
+      (150000000000000 / 1480561977544633 : ℝ) := by
+    rw [show
+        (150000000000000 / 1480561977544633 : ℝ) =
+          (3 / 40) / (1480561977544633 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (159312967273261300701 / 500000000000000000000 : ℝ))) -
+        6 * (1009683653617733165789 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (17536510154807 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (17536510154807 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1480561977544633 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (159312967273261300701 / 500000000000000000000 : ℝ))) -
+          6 * (1009683653617733165789 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (17536510154807 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (17536510154807 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (17536510154807 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_501 :
+    ((4377800923225096597531 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(5889868698352369758921 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((7411235894704171 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1594489899732132286837 / 5000000000000000000000 : ℝ))) -
+              6 * (1008543493479960270289 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (35152907191649 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (35152907191649 / 64000000000000 : ℝ)) =
+      Real.log (99152907191649 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((35152907191649 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (35512907191649 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (4377800923225096597531 / 10000000000000000000000 : ℝ) ≤
+      Real.log (99152907191649 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (35512907191649 / 64000000000000 : ℝ) ≤
+      (-(5889868698352369758921 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (35512907191649 / 64000000000000 : ℝ) = (35512907191649 / 32000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (35512907191649 / 32000000000000 : ℝ) ≤
+            (1041603104647630241079 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (7411235894704171 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (35152907191649 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (35152907191649 / 64000000000000 : ℝ) ≤ (3705617947352087 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (35152907191649 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau774 :
+    Real.arctan (7411235894704174 / 22446944102347031 : ℝ) ≤
+      (1594489899732132286837 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau773 :
+    Real.arctan (3705617947352087 / 5000000000000000 : ℝ) ≤
+      (2 * (1594489899732132286837 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12446944102347031 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (3705617947352087 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (3705617947352087 / 5000000000000000 : ℝ) /
+                (1 + (12446944102347031 / 10000000000000000 : ℝ)) =
+              (7411235894704174 / 22446944102347031 : ℝ)
+            by
+        norm_num];
+        exact hau774)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1594489899732132286837 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 3705617947352087 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 3705617947352087 : ℝ) =
+              (3705617947352087 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau773)
+  have hz1 :
+    (5000000000000000 / 3705617947352087 : ℝ) ≤
+      1 / Real.sqrt (35152907191649 / 64000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 3705617947352087 : ℝ) = 1 / (3705617947352087 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 7411235894704171 : ℝ) ≤
+      (1008543493479960270289 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (35152907191649 / 64000000000000 : ℝ) ≤
+      (750000000000000 / 7411235894704171 : ℝ) := by
+    rw [show
+        (750000000000000 / 7411235894704171 : ℝ) =
+          (3 / 40) / (7411235894704171 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1594489899732132286837 / 5000000000000000000000 : ℝ))) -
+        6 * (1008543493479960270289 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (35152907191649 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (35152907191649 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((7411235894704171 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1594489899732132286837 / 5000000000000000000000 : ℝ))) -
+          6 * (1008543493479960270289 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (35152907191649 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (35152907191649 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (35152907191649 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_502 :
+    ((4385854617200366767653 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(5867398793907133653737 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((7419652332834149 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3191694975543130921949 / 10000000000000000000000 : ℝ))) -
+              6 * (503703593583643605337 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (8808198518421 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (8808198518421 / 16000000000000 : ℝ)) =
+      Real.log (24808198518421 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((8808198518421 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (8898198518421 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (4385854617200366767653 / 10000000000000000000000 : ℝ) ≤
+      Real.log (24808198518421 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (8898198518421 / 16000000000000 : ℝ) ≤
+      (-(5867398793907133653737 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (8898198518421 / 16000000000000 : ℝ) = (8898198518421 / 8000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8898198518421 / 8000000000000 : ℝ) ≤
+            (1064073009092866346263 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (7419652332834149 / 10000000000000000 : ℝ) ≤ Real.sqrt (8808198518421 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (8808198518421 / 16000000000000 : ℝ) ≤ (927456541604269 / 1250000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (8808198518421 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau776 :
+    Real.arctan (3709826166417076 / 11225978652792893 : ℝ) ≤
+      (3191694975543130921949 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau775 :
+    Real.arctan (927456541604269 / 1250000000000000 : ℝ) ≤
+      (2 * (3191694975543130921949 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6225978652792893 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (927456541604269 / 1250000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (927456541604269 / 1250000000000000 : ℝ) /
+                (1 + (6225978652792893 / 5000000000000000 : ℝ)) =
+              (3709826166417076 / 11225978652792893 : ℝ)
+            by
+        norm_num];
+        exact hau776)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3191694975543130921949 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (1250000000000000 / 927456541604269 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1250000000000000 / 927456541604269 : ℝ) =
+              (927456541604269 / 1250000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau775)
+  have hz1 :
+    (1250000000000000 / 927456541604269 : ℝ) ≤
+      1 / Real.sqrt (8808198518421 / 16000000000000 : ℝ) := by
+    rw [show
+        (1250000000000000 / 927456541604269 : ℝ) = 1 / (927456541604269 / 1250000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 7419652332834149 : ℝ) ≤
+      (503703593583643605337 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (8808198518421 / 16000000000000 : ℝ) ≤
+      (750000000000000 / 7419652332834149 : ℝ) := by
+    rw [show
+        (750000000000000 / 7419652332834149 : ℝ) =
+          (3 / 40) / (7419652332834149 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3191694975543130921949 / 10000000000000000000000 : ℝ))) -
+        6 * (503703593583643605337 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (8808198518421 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (8808198518421 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((7419652332834149 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3191694975543130921949 / 10000000000000000000000 : ℝ))) -
+          6 * (503703593583643605337 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (8808198518421 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (8808198518421 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (8808198518421 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_503 :
+    ((2196950915098102340867 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(292248963297407018761 / 500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((7428059234639349 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3194404892680992092831 / 10000000000000000000000 : ℝ))) -
+              6 * (251568678254492200593 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (35312680955719 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (35312680955719 / 64000000000000 : ℝ)) =
+      Real.log (99312680955719 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((35312680955719 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (35672680955719 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2196950915098102340867 / 5000000000000000000000 : ℝ) ≤
+      Real.log (99312680955719 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (35672680955719 / 64000000000000 : ℝ) ≤
+      (-(292248963297407018761 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (35672680955719 / 64000000000000 : ℝ) = (35672680955719 / 32000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (35672680955719 / 32000000000000 : ℝ) ≤
+            (54324626852592981239 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (7428059234639349 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (35312680955719 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (35312680955719 / 64000000000000 : ℝ) ≤ (928507404329919 / 1250000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (35312680955719 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau778 :
+    Real.arctan (1238009872439892 / 3742828081883801 : ℝ) ≤
+      (3194404892680992092831 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau777 :
+    Real.arctan (928507404329919 / 1250000000000000 : ℝ) ≤
+      (2 * (3194404892680992092831 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6228484245651403 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (928507404329919 / 1250000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (928507404329919 / 1250000000000000 : ℝ) /
+                (1 + (6228484245651403 / 5000000000000000 : ℝ)) =
+              (1238009872439892 / 3742828081883801 : ℝ)
+            by
+        norm_num];
+        exact hau778)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3194404892680992092831 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (1250000000000000 / 928507404329919 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1250000000000000 / 928507404329919 : ℝ) =
+              (928507404329919 / 1250000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau777)
+  have hz1 :
+    (1250000000000000 / 928507404329919 : ℝ) ≤
+      1 / Real.sqrt (35312680955719 / 64000000000000 : ℝ) := by
+    rw [show
+        (1250000000000000 / 928507404329919 : ℝ) = 1 / (928507404329919 / 1250000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 2476019744879783 : ℝ) ≤
+      (251568678254492200593 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (35312680955719 / 64000000000000 : ℝ) ≤
+      (250000000000000 / 2476019744879783 : ℝ) := by
+    rw [show
+        (250000000000000 / 2476019744879783 : ℝ) =
+          (3 / 40) / (7428059234639349 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3194404892680992092831 / 10000000000000000000000 : ℝ))) -
+        6 * (251568678254492200593 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (35312680955719 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (35312680955719 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((7428059234639349 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3194404892680992092831 / 10000000000000000000000 : ℝ))) -
+          6 * (251568678254492200593 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (35312680955719 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (35312680955719 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (35312680955719 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_504 :
+    ((4401942572634988737343 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2911304944548540144719 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((743645663246217 / 1000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3197109569752017389649 / 10000000000000000000000 : ℝ))) -
+              6 * (1005146049540338767613 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (17696283918877 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (17696283918877 / 32000000000000 : ℝ)) =
+      Real.log (49696283918877 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((17696283918877 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (17876283918877 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (4401942572634988737343 / 10000000000000000000000 : ℝ) ≤
+      Real.log (49696283918877 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (17876283918877 / 32000000000000 : ℝ) ≤
+      (-(2911304944548540144719 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (17876283918877 / 32000000000000 : ℝ) = (17876283918877 / 16000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (17876283918877 / 16000000000000 : ℝ) ≤
+            (554430956951459855281 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (743645663246217 / 1000000000000000 : ℝ) ≤ Real.sqrt (17696283918877 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (17696283918877 / 32000000000000 : ℝ) ≤
+      (7436456632462173 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (17696283918877 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau780 :
+    Real.arctan (2478818877487391 / 7487325887310646 : ℝ) ≤
+      (3197109569752017389649 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau779 :
+    Real.arctan (7436456632462173 / 10000000000000000 : ℝ) ≤
+      (2 * (3197109569752017389649 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6230988830965969 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (7436456632462173 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (7436456632462173 / 10000000000000000 : ℝ) /
+                (1 + (6230988830965969 / 5000000000000000 : ℝ)) =
+              (2478818877487391 / 7487325887310646 : ℝ)
+            by
+        norm_num];
+        exact hau780)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3197109569752017389649 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 7436456632462173 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 7436456632462173 : ℝ) =
+              (7436456632462173 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau779)
+  have hz1 :
+    (10000000000000000 / 7436456632462173 : ℝ) ≤
+      1 / Real.sqrt (17696283918877 / 32000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 7436456632462173 : ℝ) =
+          1 / (7436456632462173 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (25000000000000 / 247881887748739 : ℝ) ≤
+      (1005146049540338767613 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (17696283918877 / 32000000000000 : ℝ) ≤
+      (25000000000000 / 247881887748739 : ℝ) := by
+    rw [show
+        (25000000000000 / 247881887748739 : ℝ) =
+          (3 / 40) / (743645663246217 / 1000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3197109569752017389649 / 10000000000000000000000 : ℝ))) -
+        6 * (1005146049540338767613 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (17696283918877 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (17696283918877 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((743645663246217 / 1000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3197109569752017389649 / 10000000000000000000000 : ℝ))) -
+          6 * (1005146049540338767613 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (17696283918877 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (17696283918877 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (17696283918877 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/V42.lean b/LeanPool/Zeta5Irrational/Table/V42.lean
new file mode 100644
index 0000000000..96ea9aec6c
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/V42.lean
@@ -0,0 +1,1440 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Energy
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # V42: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_505 :
+    ((4409976854913976428969 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(5800290439484741597991 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((7444844558462607 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3199809025526181901943 / 10000000000000000000000 : ℝ))) -
+              6 * (1004021175411097253659 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (35472454719789 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (35472454719789 / 64000000000000 : ℝ)) =
+      Real.log (99472454719789 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((35472454719789 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (35832454719789 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (4409976854913976428969 / 10000000000000000000000 : ℝ) ≤
+      Real.log (99472454719789 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (35832454719789 / 64000000000000 : ℝ) ≤
+      (-(5800290439484741597991 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (35832454719789 / 64000000000000 : ℝ) = (35832454719789 / 32000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (35832454719789 / 32000000000000 : ℝ) ≤
+            (1131181363515258402009 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (7444844558462607 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (35472454719789 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (35472454719789 / 64000000000000 : ℝ) ≤ (744484455846261 / 1000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (35472454719789 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau782 :
+    Real.arctan (3722422279231305 / 11233492409951068 : ℝ) ≤
+      (3199809025526181901943 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau781 :
+    Real.arctan (744484455846261 / 1000000000000000 : ℝ) ≤
+      (2 * (3199809025526181901943 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (1558373102487767 / 1250000000000000 : ℝ) ≤
+          Real.sqrt (1 + (744484455846261 / 1000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (744484455846261 / 1000000000000000 : ℝ) /
+                (1 + (1558373102487767 / 1250000000000000 : ℝ)) =
+              (3722422279231305 / 11233492409951068 : ℝ)
+            by
+        norm_num];
+        exact hau782)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3199809025526181901943 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (1000000000000000 / 744484455846261 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1000000000000000 / 744484455846261 : ℝ) =
+              (744484455846261 / 1000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau781)
+  have hz1 :
+    (1000000000000000 / 744484455846261 : ℝ) ≤
+      1 / Real.sqrt (35472454719789 / 64000000000000 : ℝ) := by
+    rw [show
+        (1000000000000000 / 744484455846261 : ℝ) = 1 / (744484455846261 / 1000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 2481614852820869 : ℝ) ≤
+      (1004021175411097253659 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (35472454719789 / 64000000000000 : ℝ) ≤
+      (250000000000000 / 2481614852820869 : ℝ) := by
+    rw [show
+        (250000000000000 / 2481614852820869 : ℝ) =
+          (3 / 40) / (7444844558462607 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3199809025526181901943 / 10000000000000000000000 : ℝ))) -
+        6 * (1004021175411097253659 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (35472454719789 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (35472454719789 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((7444844558462607 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3199809025526181901943 / 10000000000000000000000 : ℝ))) -
+          6 * (1004021175411097253659 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (35472454719789 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (35472454719789 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (35472454719789 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_506 :
+    ((883600937481077002919 / 2000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(5778020694737567426223 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((931652880577461 / 1250000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3202503278670062354709 / 10000000000000000000000 : ℝ))) -
+              6 * (250725017368404689187 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (1111010675057 / 2000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1111010675057 / 2000000000000 : ℝ)) =
+      Real.log (3111010675057 / 2000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1111010675057 / 2000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1122260675057 / 2000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (883600937481077002919 / 2000000000000000000000 : ℝ) ≤
+      Real.log (3111010675057 / 2000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1122260675057 / 2000000000000 : ℝ) ≤
+      (-(5778020694737567426223 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (1122260675057 / 2000000000000 : ℝ) = (1122260675057 / 1000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1122260675057 / 1000000000000 : ℝ) ≤
+            (1153451108262432573777 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (931652880577461 / 1250000000000000 : ℝ) ≤ Real.sqrt (1111010675057 / 2000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1111010675057 / 2000000000000 : ℝ) ≤ (7453223044619691 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1111010675057 / 2000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau784 :
+    Real.arctan (2484407681539897 / 7490663322545827 : ℝ) ≤
+      (3202503278670062354709 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau783 :
+    Real.arctan (7453223044619691 / 10000000000000000 : ℝ) ≤
+      (2 * (3202503278670062354709 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12471989967637481 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (7453223044619691 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (7453223044619691 / 10000000000000000 : ℝ) /
+                (1 + (12471989967637481 / 10000000000000000 : ℝ)) =
+              (2484407681539897 / 7490663322545827 : ℝ)
+            by
+        norm_num];
+        exact hau784)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3202503278670062354709 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 7453223044619691 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 7453223044619691 : ℝ) =
+              (7453223044619691 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau783)
+  have hz1 :
+    (10000000000000000 / 7453223044619691 : ℝ) ≤
+      1 / Real.sqrt (1111010675057 / 2000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 7453223044619691 : ℝ) =
+          1 / (7453223044619691 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (31250000000000 / 310550960192487 : ℝ) ≤
+      (250725017368404689187 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1111010675057 / 2000000000000 : ℝ) ≤
+      (31250000000000 / 310550960192487 : ℝ) := by
+    rw [show
+        (31250000000000 / 310550960192487 : ℝ) =
+          (3 / 40) / (931652880577461 / 1250000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3202503278670062354709 / 10000000000000000000000 : ℝ))) -
+        6 * (250725017368404689187 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1111010675057 / 2000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1111010675057 / 2000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((931652880577461 / 1250000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3202503278670062354709 / 10000000000000000000000 : ℝ))) -
+          6 * (250725017368404689187 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (1111010675057 / 2000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1111010675057 / 2000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1111010675057 / 2000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_507 :
+    ((885205216091294371751 / 2000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(719475054245545283031 / 1250000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((74615921227329 / 100000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3205192347747627695933 / 10000000000000000000000 : ℝ))) -
+              6 * (500891355368140157209 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (35632228483859 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (35632228483859 / 64000000000000 : ℝ)) =
+      Real.log (99632228483859 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((35632228483859 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (35992228483859 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (885205216091294371751 / 2000000000000000000000 : ℝ) ≤
+      Real.log (99632228483859 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (35992228483859 / 64000000000000 : ℝ) ≤
+      (-(719475054245545283031 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (35992228483859 / 64000000000000 : ℝ) = (35992228483859 / 32000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (35992228483859 / 32000000000000 : ℝ) ≤
+            (146958921129454716969 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (74615921227329 / 100000000000000 : ℝ) ≤ Real.sqrt (35632228483859 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (35632228483859 / 64000000000000 : ℝ) ≤
+      (7461592122732903 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (35632228483859 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau786 :
+    Real.arctan (7461592122732903 / 22476993107557192 : ℝ) ≤
+      (3205192347747627695933 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau785 :
+    Real.arctan (7461592122732903 / 10000000000000000 : ℝ) ≤
+      (2 * (3205192347747627695933 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (1559624138444649 / 1250000000000000 : ℝ) ≤
+          Real.sqrt (1 + (7461592122732903 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (7461592122732903 / 10000000000000000 : ℝ) /
+                (1 + (1559624138444649 / 1250000000000000 : ℝ)) =
+              (7461592122732903 / 22476993107557192 : ℝ)
+            by
+        norm_num];
+        exact hau786)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3205192347747627695933 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 7461592122732903 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 7461592122732903 : ℝ) =
+              (7461592122732903 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau785)
+  have hz1 :
+    (10000000000000000 / 7461592122732903 : ℝ) ≤
+      1 / Real.sqrt (35632228483859 / 64000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 7461592122732903 : ℝ) =
+          1 / (7461592122732903 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (2500000000000 / 24871973742443 : ℝ) ≤
+      (500891355368140157209 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (35632228483859 / 64000000000000 : ℝ) ≤
+      (2500000000000 / 24871973742443 : ℝ) := by
+    rw [show
+        (2500000000000 / 24871973742443 : ℝ) = (3 / 40) / (74615921227329 / 100000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3205192347747627695933 / 10000000000000000000000 : ℝ))) -
+        6 * (500891355368140157209 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (35632228483859 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (35632228483859 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((74615921227329 / 100000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3205192347747627695933 / 10000000000000000000000 : ℝ))) -
+          6 * (500891355368140157209 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (35632228483859 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (35632228483859 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (35632228483859 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_508 :
+    ((2217020522194807226801 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(5733629437743145774003 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((933743978052949 / 1250000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1603938125610509220697 / 5000000000000000000000 : ℝ))) -
+              6 * (125083634796351411801 / 1250000000000000000000 : ℝ)))) ≤
+      Vfield (17856057682947 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (17856057682947 / 32000000000000 : ℝ)) =
+      Real.log (49856057682947 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((17856057682947 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (18036057682947 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2217020522194807226801 / 5000000000000000000000 : ℝ) ≤
+      Real.log (49856057682947 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (18036057682947 / 32000000000000 : ℝ) ≤
+      (-(5733629437743145774003 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (18036057682947 / 32000000000000 : ℝ) = (18036057682947 / 16000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18036057682947 / 16000000000000 : ℝ) ≤
+            (1197842365256854225997 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (933743978052949 / 1250000000000000 : ℝ) ≤ Real.sqrt (17856057682947 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (17856057682947 / 32000000000000 : ℝ) ≤ (1493990364884719 / 2000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (17856057682947 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau788 :
+    Real.arctan (7469951824423595 / 22481994242075637 : ℝ) ≤
+      (1603938125610509220697 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau787 :
+    Real.arctan (1493990364884719 / 2000000000000000 : ℝ) ≤
+      (2 * (1603938125610509220697 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12481994242075637 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1493990364884719 / 2000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1493990364884719 / 2000000000000000 : ℝ) /
+                (1 + (12481994242075637 / 10000000000000000 : ℝ)) =
+              (7469951824423595 / 22481994242075637 : ℝ)
+            by
+        norm_num];
+        exact hau788)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1603938125610509220697 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (2000000000000000 / 1493990364884719 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2000000000000000 / 1493990364884719 : ℝ) =
+              (1493990364884719 / 2000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau787)
+  have hz1 :
+    (2000000000000000 / 1493990364884719 : ℝ) ≤
+      1 / Real.sqrt (17856057682947 / 32000000000000 : ℝ) := by
+    rw [show
+        (2000000000000000 / 1493990364884719 : ℝ) = 1 / (1493990364884719 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (93750000000000 / 933743978052949 : ℝ) ≤
+      (125083634796351411801 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (17856057682947 / 32000000000000 : ℝ) ≤
+      (93750000000000 / 933743978052949 : ℝ) := by
+    rw [show
+        (93750000000000 / 933743978052949 : ℝ) =
+          (3 / 40) / (933743978052949 / 1250000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1603938125610509220697 / 5000000000000000000000 : ℝ))) -
+        6 * (125083634796351411801 / 1250000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (17856057682947 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (17856057682947 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((933743978052949 / 1250000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1603938125610509220697 / 5000000000000000000000 : ℝ))) -
+          6 * (125083634796351411801 / 1250000000000000000000 : ℝ))) ≤
+      Real.sqrt (17856057682947 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (17856057682947 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (17856057682947 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_509 :
+    ((111051239737559752989 / 250000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(5711507488108152006311 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((7478302181136373 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2 * (128422200298052838079 / 400000000000000000000 : ℝ))) -
+              6 * (999559151710662100997 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (35792002247929 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (35792002247929 / 64000000000000 : ℝ)) =
+      Real.log (99792002247929 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((35792002247929 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (36152002247929 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (111051239737559752989 / 250000000000000000000 : ℝ) ≤
+      Real.log (99792002247929 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (36152002247929 / 64000000000000 : ℝ) ≤
+      (-(5711507488108152006311 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (36152002247929 / 64000000000000 : ℝ) = (36152002247929 / 32000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (36152002247929 / 32000000000000 : ℝ) ≤
+            (1219964314891847993689 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (7478302181136373 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (35792002247929 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (35792002247929 / 64000000000000 : ℝ) ≤ (934787772642047 / 1250000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (35792002247929 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau790 :
+    Real.arctan (934787772642047 / 2810874171700294 : ℝ) ≤
+      (128422200298052838079 / 400000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau789 :
+    Real.arctan (934787772642047 / 1250000000000000 : ℝ) ≤
+      (2 * (128422200298052838079 / 400000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (780437085850147 / 625000000000000 : ℝ) ≤
+          Real.sqrt (1 + (934787772642047 / 1250000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (934787772642047 / 1250000000000000 : ℝ) /
+                (1 + (780437085850147 / 625000000000000 : ℝ)) =
+              (934787772642047 / 2810874171700294 : ℝ)
+            by
+        norm_num];
+        exact hau790)
+  have hat1 :
+    (Real.pi / 2 - (2 * (128422200298052838079 / 400000000000000000000 : ℝ))) ≤
+      Real.arctan (1250000000000000 / 934787772642047 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1250000000000000 / 934787772642047 : ℝ) =
+              (934787772642047 / 1250000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau789)
+  have hz1 :
+    (1250000000000000 / 934787772642047 : ℝ) ≤
+      1 / Real.sqrt (35792002247929 / 64000000000000 : ℝ) := by
+    rw [show
+        (1250000000000000 / 934787772642047 : ℝ) = 1 / (934787772642047 / 1250000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 7478302181136373 : ℝ) ≤
+      (999559151710662100997 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (35792002247929 / 64000000000000 : ℝ) ≤
+      (750000000000000 / 7478302181136373 : ℝ) := by
+    rw [show
+        (750000000000000 / 7478302181136373 : ℝ) =
+          (3 / 40) / (7478302181136373 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (128422200298052838079 / 400000000000000000000 : ℝ))) -
+        6 * (999559151710662100997 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (35792002247929 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (35792002247929 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((7478302181136373 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (128422200298052838079 / 400000000000000000000 : ℝ))) -
+          6 * (999559151710662100997 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (35792002247929 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (35792002247929 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (35792002247929 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_510 :
+    ((4450051726067655387087 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(5689434368536972102147 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((7486643224140491 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (803307158674833361741 / 2500000000000000000000 : ℝ))) -
+              6 * (499226455124696702811 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (8967972282491 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (8967972282491 / 16000000000000 : ℝ)) =
+      Real.log (24967972282491 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((8967972282491 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (9057972282491 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (4450051726067655387087 / 10000000000000000000000 : ℝ) ≤
+      Real.log (24967972282491 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (9057972282491 / 16000000000000 : ℝ) ≤
+      (-(5689434368536972102147 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (9057972282491 / 16000000000000 : ℝ) = (9057972282491 / 8000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9057972282491 / 8000000000000 : ℝ) ≤
+            (1242037434463027897853 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (7486643224140491 / 10000000000000000 : ℝ) ≤ Real.sqrt (8967972282491 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (8967972282491 / 16000000000000 : ℝ) ≤ (3743321612070247 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (8967972282491 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau792 :
+    Real.arctan (3743321612070247 / 11245995252271025 : ℝ) ≤
+      (803307158674833361741 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau791 :
+    Real.arctan (3743321612070247 / 5000000000000000 : ℝ) ≤
+      (2 * (803307158674833361741 / 2500000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (249839810090841 / 200000000000000 : ℝ) ≤
+          Real.sqrt (1 + (3743321612070247 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (3743321612070247 / 5000000000000000 : ℝ) /
+                (1 + (249839810090841 / 200000000000000 : ℝ)) =
+              (3743321612070247 / 11245995252271025 : ℝ)
+            by
+        norm_num];
+        exact hau792)
+  have hat1 :
+    (Real.pi / 2 - (2 * (803307158674833361741 / 2500000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 3743321612070247 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 3743321612070247 : ℝ) =
+              (3743321612070247 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau791)
+  have hz1 :
+    (5000000000000000 / 3743321612070247 : ℝ) ≤
+      1 / Real.sqrt (8967972282491 / 16000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 3743321612070247 : ℝ) = 1 / (3743321612070247 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 7486643224140491 : ℝ) ≤
+      (499226455124696702811 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (8967972282491 / 16000000000000 : ℝ) ≤
+      (750000000000000 / 7486643224140491 : ℝ) := by
+    rw [show
+        (750000000000000 / 7486643224140491 : ℝ) =
+          (3 / 40) / (7486643224140491 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (803307158674833361741 / 2500000000000000000000 : ℝ))) -
+        6 * (499226455124696702811 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (8967972282491 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (8967972282491 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((7486643224140491 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (803307158674833361741 / 2500000000000000000000 : ℝ))) -
+          6 * (499226455124696702811 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (8967972282491 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (8967972282491 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (8967972282491 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_511 :
+    ((2229023732166812530537 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(5667409863937838582023 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((7494974984531197 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2 * (100496785972697601687 / 312500000000000000000 : ℝ))) -
+              6 * (997350333639085100309 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (35951776011999 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (35951776011999 / 64000000000000 : ℝ)) =
+      Real.log (99951776011999 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((35951776011999 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (36311776011999 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2229023732166812530537 / 5000000000000000000000 : ℝ) ≤
+      Real.log (99951776011999 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (36311776011999 / 64000000000000 : ℝ) ≤
+      (-(5667409863937838582023 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (36311776011999 / 64000000000000 : ℝ) = (36311776011999 / 32000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (36311776011999 / 32000000000000 : ℝ) ≤
+            (1264061939062161417977 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (7494974984531197 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (35951776011999 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (35951776011999 / 64000000000000 : ℝ) ≤ (1171089841333 / 1562500000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (35951776011999 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau794 :
+    Real.arctan (1873743746132800 / 5624246409323659 : ℝ) ≤
+      (100496785972697601687 / 312500000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau793 :
+    Real.arctan (1171089841333 / 1562500000000 : ℝ) ≤
+      (2 * (100496785972697601687 / 312500000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (3124246409323659 / 2500000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1171089841333 / 1562500000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1171089841333 / 1562500000000 : ℝ) /
+                (1 + (3124246409323659 / 2500000000000000 : ℝ)) =
+              (1873743746132800 / 5624246409323659 : ℝ)
+            by
+        norm_num];
+        exact hau794)
+  have hat1 :
+    (Real.pi / 2 - (2 * (100496785972697601687 / 312500000000000000000 : ℝ))) ≤
+      Real.arctan (1562500000000 / 1171089841333 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1562500000000 / 1171089841333 : ℝ) = (1171089841333 / 1562500000000 : ℝ) by
+            norm_num];
+        exact hau793)
+  have hz1 :
+    (1562500000000 / 1171089841333 : ℝ) ≤ 1 / Real.sqrt (35951776011999 / 64000000000000 : ℝ) := by
+    rw [show (1562500000000 / 1171089841333 : ℝ) = 1 / (1171089841333 / 1562500000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 7494974984531197 : ℝ) ≤
+      (997350333639085100309 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (35951776011999 / 64000000000000 : ℝ) ≤
+      (750000000000000 / 7494974984531197 : ℝ) := by
+    rw [show
+        (750000000000000 / 7494974984531197 : ℝ) =
+          (3 / 40) / (7494974984531197 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (100496785972697601687 / 312500000000000000000 : ℝ))) -
+        6 * (997350333639085100309 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (35951776011999 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (35951776011999 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((7494974984531197 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (100496785972697601687 / 312500000000000000000 : ℝ))) -
+          6 * (997350333639085100309 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (35951776011999 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (35951776011999 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (35951776011999 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_512 :
+    ((4466036814523950969417 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(5645433760637049356529 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((75032974932311 / 100000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (643712114958955676583 / 2000000000000000000000 : ℝ))) -
+              6 * (498125700844381961051 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (18015831447017 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (18015831447017 / 32000000000000 : ℝ)) =
+      Real.log (50015831447017 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((18015831447017 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (18195831447017 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (4466036814523950969417 / 10000000000000000000000 : ℝ) ≤
+      Real.log (50015831447017 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (18195831447017 / 32000000000000 : ℝ) ≤
+      (-(5645433760637049356529 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (18195831447017 / 32000000000000 : ℝ) = (18195831447017 / 16000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18195831447017 / 16000000000000 : ℝ) ≤
+            (1286038042362950643471 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (75032974932311 / 100000000000000 : ℝ) ≤ Real.sqrt (18015831447017 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (18015831447017 / 32000000000000 : ℝ) ≤
+      (7503297493231103 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (18015831447017 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau796 :
+    Real.arctan (7503297493231103 / 22501978774255223 : ℝ) ≤
+      (643712114958955676583 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau795 :
+    Real.arctan (7503297493231103 / 10000000000000000 : ℝ) ≤
+      (2 * (643712114958955676583 / 2000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12501978774255223 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (7503297493231103 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (7503297493231103 / 10000000000000000 : ℝ) /
+                (1 + (12501978774255223 / 10000000000000000 : ℝ)) =
+              (7503297493231103 / 22501978774255223 : ℝ)
+            by
+        norm_num];
+        exact hau796)
+  have hat1 :
+    (Real.pi / 2 - (2 * (643712114958955676583 / 2000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 7503297493231103 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 7503297493231103 : ℝ) =
+              (7503297493231103 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau795)
+  have hz1 :
+    (10000000000000000 / 7503297493231103 : ℝ) ≤
+      1 / Real.sqrt (18015831447017 / 32000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 7503297493231103 : ℝ) =
+          1 / (7503297493231103 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (7500000000000 / 75032974932311 : ℝ) ≤
+      (498125700844381961051 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (18015831447017 / 32000000000000 : ℝ) ≤
+      (7500000000000 / 75032974932311 : ℝ) := by
+    rw [show
+        (7500000000000 / 75032974932311 : ℝ) = (3 / 40) / (75032974932311 / 100000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (643712114958955676583 / 2000000000000000000000 : ℝ))) -
+        6 * (498125700844381961051 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (18015831447017 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (18015831447017 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((75032974932311 / 100000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (643712114958955676583 / 2000000000000000000000 : ℝ))) -
+          6 * (498125700844381961051 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (18015831447017 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (18015831447017 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (18015831447017 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_513 :
+    ((4474019786837800397257 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1124701169273305923953 / 2000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((469475673811969 / 625000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2 * (161060946183457558827 / 500000000000000000000 : ℝ))) -
+              6 * (124394511795356336081 / 1250000000000000000000 : ℝ)))) ≤
+      Vfield (36111549776069 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (36111549776069 / 64000000000000 : ℝ)) =
+      Real.log (100111549776069 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((36111549776069 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (36471549776069 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (4474019786837800397257 / 10000000000000000000000 : ℝ) ≤
+      Real.log (100111549776069 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (36471549776069 / 64000000000000 : ℝ) ≤
+      (-(1124701169273305923953 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (36471549776069 / 64000000000000 : ℝ) = (36471549776069 / 32000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (36471549776069 / 32000000000000 : ℝ) ≤
+            (261593191326694076047 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (469475673811969 / 625000000000000 : ℝ) ≤ Real.sqrt (36111549776069 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (36111549776069 / 64000000000000 : ℝ) ≤
+      (7511610780991507 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (36111549776069 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau798 :
+    Real.arctan (7511610780991507 / 22506969917814139 : ℝ) ≤
+      (161060946183457558827 / 500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau797 :
+    Real.arctan (7511610780991507 / 10000000000000000 : ℝ) ≤
+      (2 * (161060946183457558827 / 500000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12506969917814139 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (7511610780991507 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (7511610780991507 / 10000000000000000 : ℝ) /
+                (1 + (12506969917814139 / 10000000000000000 : ℝ)) =
+              (7511610780991507 / 22506969917814139 : ℝ)
+            by
+        norm_num];
+        exact hau798)
+  have hat1 :
+    (Real.pi / 2 - (2 * (161060946183457558827 / 500000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 7511610780991507 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 7511610780991507 : ℝ) =
+              (7511610780991507 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau797)
+  have hz1 :
+    (10000000000000000 / 7511610780991507 : ℝ) ≤
+      1 / Real.sqrt (36111549776069 / 64000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 7511610780991507 : ℝ) =
+          1 / (7511610780991507 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (46875000000000 / 469475673811969 : ℝ) ≤
+      (124394511795356336081 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (36111549776069 / 64000000000000 : ℝ) ≤
+      (46875000000000 / 469475673811969 : ℝ) := by
+    rw [show
+        (46875000000000 / 469475673811969 : ℝ) =
+          (3 / 40) / (469475673811969 / 625000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (161060946183457558827 / 500000000000000000000 : ℝ))) -
+        6 * (124394511795356336081 / 1250000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (36111549776069 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (36111549776069 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((469475673811969 / 625000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (161060946183457558827 / 500000000000000000000 : ℝ))) -
+          6 * (124394511795356336081 / 1250000000000000000000 : ℝ))) ≤
+      Real.sqrt (36111549776069 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (36111549776069 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (36111549776069 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_514 :
+    ((2240998195724967104201 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(5601625910251529815539 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((939989359799217 / 1250000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3223872215616594728717 / 10000000000000000000000 : ℝ))) -
+              6 * (497032195889812852133 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (4523929582263 / 8000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (4523929582263 / 8000000000000 : ℝ)) =
+      Real.log (12523929582263 / 8000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((4523929582263 / 8000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (4568929582263 / 8000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2240998195724967104201 / 5000000000000000000000 : ℝ) ≤
+      Real.log (12523929582263 / 8000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (4568929582263 / 8000000000000 : ℝ) ≤
+      (-(5601625910251529815539 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (4568929582263 / 8000000000000 : ℝ) = (4568929582263 / 4000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4568929582263 / 4000000000000 : ℝ) ≤
+            (1329845892748470184461 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (939989359799217 / 1250000000000000 : ℝ) ≤ Real.sqrt (4523929582263 / 8000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (4523929582263 / 8000000000000 : ℝ) ≤ (7519914878393739 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (4523929582263 / 8000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau800 :
+    Real.arctan (2506638292797913 / 7503986356785649 : ℝ) ≤
+      (3223872215616594728717 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau799 :
+    Real.arctan (7519914878393739 / 10000000000000000 : ℝ) ≤
+      (2 * (3223872215616594728717 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12511959070356947 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (7519914878393739 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (7519914878393739 / 10000000000000000 : ℝ) /
+                (1 + (12511959070356947 / 10000000000000000 : ℝ)) =
+              (2506638292797913 / 7503986356785649 : ℝ)
+            by
+        norm_num];
+        exact hau800)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3223872215616594728717 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 7519914878393739 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 7519914878393739 : ℝ) =
+              (7519914878393739 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau799)
+  have hz1 :
+    (10000000000000000 / 7519914878393739 : ℝ) ≤
+      1 / Real.sqrt (4523929582263 / 8000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 7519914878393739 : ℝ) =
+          1 / (7519914878393739 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (31250000000000 / 313329786599739 : ℝ) ≤
+      (497032195889812852133 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (4523929582263 / 8000000000000 : ℝ) ≤
+      (31250000000000 / 313329786599739 : ℝ) := by
+    rw [show
+        (31250000000000 / 313329786599739 : ℝ) =
+          (3 / 40) / (939989359799217 / 1250000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3223872215616594728717 / 10000000000000000000000 : ℝ))) -
+        6 * (497032195889812852133 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (4523929582263 / 8000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (4523929582263 / 8000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((939989359799217 / 1250000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3223872215616594728717 / 10000000000000000000000 : ℝ))) -
+          6 * (497032195889812852133 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (4523929582263 / 8000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (4523929582263 / 8000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (4523929582263 / 8000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_515 :
+    ((2248965269073266439973 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(5558009135882844102449 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((7536495623606959 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3229163699717176018831 / 10000000000000000000000 : ℝ))) -
+              6 * (30996616314830743811 / 312500000000000000000 : ℝ)))) ≤
+      Vfield (18175605211087 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (18175605211087 / 32000000000000 : ℝ)) =
+      Real.log (50175605211087 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((18175605211087 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (18355605211087 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2248965269073266439973 / 5000000000000000000000 : ℝ) ≤
+      Real.log (50175605211087 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (18355605211087 / 32000000000000 : ℝ) ≤
+      (-(5558009135882844102449 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (18355605211087 / 32000000000000 : ℝ) = (18355605211087 / 16000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18355605211087 / 16000000000000 : ℝ) ≤
+            (1373462667117155897551 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (7536495623606959 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (18175605211087 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (18175605211087 / 32000000000000 : ℝ) ≤ (3768247811803481 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (18175605211087 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau802 :
+    Real.arctan (7536495623606962 / 22521931411912735 : ℝ) ≤
+      (3229163699717176018831 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau801 :
+    Real.arctan (3768247811803481 / 5000000000000000 : ℝ) ≤
+      (2 * (3229163699717176018831 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (2504386282382547 / 2000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (3768247811803481 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (3768247811803481 / 5000000000000000 : ℝ) /
+                (1 + (2504386282382547 / 2000000000000000 : ℝ)) =
+              (7536495623606962 / 22521931411912735 : ℝ)
+            by
+        norm_num];
+        exact hau802)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3229163699717176018831 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 3768247811803481 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 3768247811803481 : ℝ) =
+              (3768247811803481 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau801)
+  have hz1 :
+    (5000000000000000 / 3768247811803481 : ℝ) ≤
+      1 / Real.sqrt (18175605211087 / 32000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 3768247811803481 : ℝ) = 1 / (3768247811803481 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 7536495623606959 : ℝ) ≤
+      (30996616314830743811 / 312500000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (18175605211087 / 32000000000000 : ℝ) ≤
+      (750000000000000 / 7536495623606959 : ℝ) := by
+    rw [show
+        (750000000000000 / 7536495623606959 : ℝ) =
+          (3 / 40) / (7536495623606959 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3229163699717176018831 / 10000000000000000000000 : ℝ))) -
+        6 * (30996616314830743811 / 312500000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (18175605211087 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (18175605211087 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((7536495623606959 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3229163699717176018831 / 10000000000000000000000 : ℝ))) -
+          6 * (30996616314830743811 / 312500000000000000000 : ℝ))) ≤
+      Real.sqrt (18175605211087 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (18175605211087 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (18175605211087 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_516 :
+    ((4513839335526654517601 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(5514581777940426081753 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((7553039970171363 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1617217584057645801541 / 5000000000000000000000 : ℝ))) -
+              6 * (989733236539927218279 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (9127746046561 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (9127746046561 / 16000000000000 : ℝ)) =
+      Real.log (25127746046561 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((9127746046561 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (9217746046561 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (4513839335526654517601 / 10000000000000000000000 : ℝ) ≤
+      Real.log (25127746046561 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (9217746046561 / 16000000000000 : ℝ) ≤
+      (-(5514581777940426081753 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (9217746046561 / 16000000000000 : ℝ) = (9217746046561 / 8000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9217746046561 / 8000000000000 : ℝ) ≤
+            (1416890025059573918247 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (7553039970171363 / 10000000000000000 : ℝ) ≤ Real.sqrt (9127746046561 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (9127746046561 / 16000000000000 : ℝ) ≤ (3776519985085683 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (9127746046561 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau804 :
+    Real.arctan (3776519985085683 / 11265947908956119 : ℝ) ≤
+      (1617217584057645801541 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau803 :
+    Real.arctan (3776519985085683 / 5000000000000000 : ℝ) ≤
+      (2 * (1617217584057645801541 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6265947908956119 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (3776519985085683 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (3776519985085683 / 5000000000000000 : ℝ) /
+                (1 + (6265947908956119 / 5000000000000000 : ℝ)) =
+              (3776519985085683 / 11265947908956119 : ℝ)
+            by
+        norm_num];
+        exact hau804)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1617217584057645801541 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 3776519985085683 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 3776519985085683 : ℝ) =
+              (3776519985085683 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau803)
+  have hz1 :
+    (5000000000000000 / 3776519985085683 : ℝ) ≤
+      1 / Real.sqrt (9127746046561 / 16000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 3776519985085683 : ℝ) = 1 / (3776519985085683 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 2517679990057121 : ℝ) ≤
+      (989733236539927218279 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (9127746046561 / 16000000000000 : ℝ) ≤
+      (250000000000000 / 2517679990057121 : ℝ) := by
+    rw [show
+        (250000000000000 / 2517679990057121 : ℝ) =
+          (3 / 40) / (7553039970171363 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1617217584057645801541 / 5000000000000000000000 : ℝ))) -
+        6 * (989733236539927218279 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (9127746046561 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (9127746046561 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((7553039970171363 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1617217584057645801541 / 5000000000000000000000 : ℝ))) -
+          6 * (989733236539927218279 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (9127746046561 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (9127746046561 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (9127746046561 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/V43.lean b/LeanPool/Zeta5Irrational/Table/V43.lean
new file mode 100644
index 0000000000..b21d09223b
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/V43.lean
@@ -0,0 +1,1440 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Energy
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # V43: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_517 :
+    ((452972286411765166689 / 1000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(5471342198361810627931 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((7569548156750547 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1619843380159431640001 / 5000000000000000000000 : ℝ))) -
+              6 * (246897195377139917649 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (18335378975157 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (18335378975157 / 32000000000000 : ℝ)) =
+      Real.log (50335378975157 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((18335378975157 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (18515378975157 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (452972286411765166689 / 1000000000000000000000 : ℝ) ≤
+      Real.log (50335378975157 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (18515378975157 / 32000000000000 : ℝ) ≤
+      (-(5471342198361810627931 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (18515378975157 / 32000000000000 : ℝ) = (18515378975157 / 16000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18515378975157 / 16000000000000 : ℝ) ≤
+            (1460129604638189372069 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (7569548156750547 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (18335378975157 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (18335378975157 / 32000000000000 : ℝ) ≤ (151390963135011 / 200000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (18335378975157 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau806 :
+    Real.arctan (2523182718916850 / 7513950769089891 : ℝ) ≤
+      (1619843380159431640001 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau805 :
+    Real.arctan (151390963135011 / 200000000000000 : ℝ) ≤
+      (2 * (1619843380159431640001 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12541852307269673 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (151390963135011 / 200000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (151390963135011 / 200000000000000 : ℝ) /
+                (1 + (12541852307269673 / 10000000000000000 : ℝ)) =
+              (2523182718916850 / 7513950769089891 : ℝ)
+            by
+        norm_num];
+        exact hau806)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1619843380159431640001 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (200000000000000 / 151390963135011 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (200000000000000 / 151390963135011 : ℝ) =
+              (151390963135011 / 200000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau805)
+  have hz1 :
+    (200000000000000 / 151390963135011 : ℝ) ≤
+      1 / Real.sqrt (18335378975157 / 32000000000000 : ℝ) := by
+    rw [show (200000000000000 / 151390963135011 : ℝ) = 1 / (151390963135011 / 200000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 2523182718916849 : ℝ) ≤
+      (246897195377139917649 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (18335378975157 / 32000000000000 : ℝ) ≤
+      (250000000000000 / 2523182718916849 : ℝ) := by
+    rw [show
+        (250000000000000 / 2523182718916849 : ℝ) =
+          (3 / 40) / (7569548156750547 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1619843380159431640001 / 5000000000000000000000 : ℝ))) -
+        6 * (246897195377139917649 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (18335378975157 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (18335378975157 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((7569548156750547 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1619843380159431640001 / 5000000000000000000000 : ℝ))) -
+          6 * (246897195377139917649 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (18335378975157 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (18335378975157 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (18335378975157 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_518 :
+    ((4545581204063767592191 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(5428288780241887976119 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3793010209705643 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3244918614347308492101 / 10000000000000000000000 : ℝ))) -
+              6 * (492729102817023540833 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (2301908232149 / 4000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (2301908232149 / 4000000000000 : ℝ)) =
+      Real.log (6301908232149 / 4000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((2301908232149 / 4000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (2324408232149 / 4000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (4545581204063767592191 / 10000000000000000000000 : ℝ) ≤
+      Real.log (6301908232149 / 4000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (2324408232149 / 4000000000000 : ℝ) ≤
+      (-(5428288780241887976119 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (2324408232149 / 4000000000000 : ℝ) = (2324408232149 / 2000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2324408232149 / 2000000000000 : ℝ) ≤
+            (1503183022758112023881 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3793010209705643 / 5000000000000000 : ℝ) ≤ Real.sqrt (2301908232149 / 4000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (2301908232149 / 4000000000000 : ℝ) ≤ (7586020419411289 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (2301908232149 / 4000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau808 :
+    Real.arctan (7586020419411289 / 22551800898824240 : ℝ) ≤
+      (3244918614347308492101 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau807 :
+    Real.arctan (7586020419411289 / 10000000000000000 : ℝ) ≤
+      (2 * (3244918614347308492101 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (156897511235303 / 125000000000000 : ℝ) ≤
+          Real.sqrt (1 + (7586020419411289 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (7586020419411289 / 10000000000000000 : ℝ) /
+                (1 + (156897511235303 / 125000000000000 : ℝ)) =
+              (7586020419411289 / 22551800898824240 : ℝ)
+            by
+        norm_num];
+        exact hau808)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3244918614347308492101 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 7586020419411289 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 7586020419411289 : ℝ) =
+              (7586020419411289 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau807)
+  have hz1 :
+    (10000000000000000 / 7586020419411289 : ℝ) ≤
+      1 / Real.sqrt (2301908232149 / 4000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 7586020419411289 : ℝ) =
+          1 / (7586020419411289 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 3793010209705643 : ℝ) ≤
+      (492729102817023540833 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (2301908232149 / 4000000000000 : ℝ) ≤
+      (375000000000000 / 3793010209705643 : ℝ) := by
+    rw [show
+        (375000000000000 / 3793010209705643 : ℝ) =
+          (3 / 40) / (3793010209705643 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3244918614347308492101 / 10000000000000000000000 : ℝ))) -
+        6 * (492729102817023540833 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (2301908232149 / 4000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (2301908232149 / 4000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3793010209705643 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3244918614347308492101 / 10000000000000000000000 : ℝ))) -
+          6 * (492729102817023540833 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (2301908232149 / 4000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (2301908232149 / 4000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (2301908232149 / 4000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_519 :
+    ((114035360878214065547 / 250000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1346354981867526213689 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1900614247915729 / 2500000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3250130866753457043071 / 10000000000000000000000 : ℝ))) -
+              6 * (245835339961437818473 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (18495152739227 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (18495152739227 / 32000000000000 : ℝ)) =
+      Real.log (50495152739227 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((18495152739227 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (18675152739227 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (114035360878214065547 / 250000000000000000000 : ℝ) ≤
+      Real.log (50495152739227 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (18675152739227 / 32000000000000 : ℝ) ≤
+      (-(1346354981867526213689 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (18675152739227 / 32000000000000 : ℝ) = (18675152739227 / 16000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (18675152739227 / 16000000000000 : ℝ) ≤
+            (386512968882473786311 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1900614247915729 / 2500000000000000 : ℝ) ≤ Real.sqrt (18495152739227 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (18495152739227 / 32000000000000 : ℝ) ≤
+      (7602456991662919 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (18495152739227 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau810 :
+    Real.arctan (7602456991662919 / 22561741611340537 : ℝ) ≤
+      (3250130866753457043071 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau809 :
+    Real.arctan (7602456991662919 / 10000000000000000 : ℝ) ≤
+      (2 * (3250130866753457043071 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12561741611340537 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (7602456991662919 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (7602456991662919 / 10000000000000000 : ℝ) /
+                (1 + (12561741611340537 / 10000000000000000 : ℝ)) =
+              (7602456991662919 / 22561741611340537 : ℝ)
+            by
+        norm_num];
+        exact hau810)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3250130866753457043071 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 7602456991662919 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 7602456991662919 : ℝ) =
+              (7602456991662919 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau809)
+  have hz1 :
+    (10000000000000000 / 7602456991662919 : ℝ) ≤
+      1 / Real.sqrt (18495152739227 / 32000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 7602456991662919 : ℝ) =
+          1 / (7602456991662919 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (187500000000000 / 1900614247915729 : ℝ) ≤
+      (245835339961437818473 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (18495152739227 / 32000000000000 : ℝ) ≤
+      (187500000000000 / 1900614247915729 : ℝ) := by
+    rw [show
+        (187500000000000 / 1900614247915729 : ℝ) =
+          (3 / 40) / (1900614247915729 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3250130866753457043071 / 10000000000000000000000 : ℝ))) -
+        6 * (245835339961437818473 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (18495152739227 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (18495152739227 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1900614247915729 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3250130866753457043071 / 10000000000000000000000 : ℝ))) -
+          6 * (245835339961437818473 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (18495152739227 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (18495152739227 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (18495152739227 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_520 :
+    ((228861131834866065769 / 500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(5342734064375408869163 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((7618858104495957 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1627661826322526439887 / 5000000000000000000000 : ℝ))) -
+              6 * (245309524326254742311 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (9287519810631 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (9287519810631 / 16000000000000 : ℝ)) =
+      Real.log (25287519810631 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((9287519810631 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (9377519810631 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (228861131834866065769 / 500000000000000000000 : ℝ) ≤
+      Real.log (25287519810631 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (9377519810631 / 16000000000000 : ℝ) ≤
+      (-(5342734064375408869163 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (9377519810631 / 16000000000000 : ℝ) = (9377519810631 / 8000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9377519810631 / 8000000000000 : ℝ) ≤
+            (1588737738624591130837 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (7618858104495957 / 10000000000000000 : ℝ) ≤ Real.sqrt (9287519810631 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (9287519810631 / 16000000000000 : ℝ) ≤ (190471452612399 / 250000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (9287519810631 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau812 :
+    Real.arctan (7618858104495960 / 22571674463508977 : ℝ) ≤
+      (1627661826322526439887 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau811 :
+    Real.arctan (190471452612399 / 250000000000000 : ℝ) ≤
+      (2 * (1627661826322526439887 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12571674463508977 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (190471452612399 / 250000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (190471452612399 / 250000000000000 : ℝ) /
+                (1 + (12571674463508977 / 10000000000000000 : ℝ)) =
+              (7618858104495960 / 22571674463508977 : ℝ)
+            by
+        norm_num];
+        exact hau812)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1627661826322526439887 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (250000000000000 / 190471452612399 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (250000000000000 / 190471452612399 : ℝ) =
+              (190471452612399 / 250000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau811)
+  have hz1 :
+    (250000000000000 / 190471452612399 : ℝ) ≤
+      1 / Real.sqrt (9287519810631 / 16000000000000 : ℝ) := by
+    rw [show (250000000000000 / 190471452612399 : ℝ) = 1 / (190471452612399 / 250000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 2539619368165319 : ℝ) ≤
+      (245309524326254742311 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (9287519810631 / 16000000000000 : ℝ) ≤
+      (250000000000000 / 2539619368165319 : ℝ) := by
+    rw [show
+        (250000000000000 / 2539619368165319 : ℝ) =
+          (3 / 40) / (7618858104495957 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1627661826322526439887 / 5000000000000000000000 : ℝ))) -
+        6 * (245309524326254742311 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (9287519810631 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (9287519810631 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((7618858104495957 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1627661826322526439887 / 5000000000000000000000 : ℝ))) -
+          6 * (245309524326254742311 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (9287519810631 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (9287519810631 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (9287519810631 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_521 :
+    ((4608764267010799293243 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(131447627616321705781 / 250000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3825777431750391 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (816412839554078022283 / 2500000000000000000000 : ℝ))) -
+              6 * (61066984094740587393 / 625000000000000000000 : ℝ)))) ≤
+      Vfield (4683703346333 / 8000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (4683703346333 / 8000000000000 : ℝ)) =
+      Real.log (12683703346333 / 8000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((4683703346333 / 8000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (4728703346333 / 8000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (4608764267010799293243 / 10000000000000000000000 : ℝ) ≤
+      Real.log (12683703346333 / 8000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (4728703346333 / 8000000000000 : ℝ) ≤
+      (-(131447627616321705781 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (4728703346333 / 8000000000000 : ℝ) = (4728703346333 / 4000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4728703346333 / 4000000000000 : ℝ) ≤
+            (41839167458678294219 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3825777431750391 / 5000000000000000 : ℝ) ≤ Real.sqrt (4683703346333 / 8000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (4683703346333 / 8000000000000 : ℝ) ≤ (1530310972700157 / 2000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (4683703346333 / 8000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau814 :
+    Real.arctan (510103657566719 / 1506101110746363 : ℝ) ≤
+      (816412839554078022283 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau813 :
+    Real.arctan (1530310972700157 / 2000000000000000 : ℝ) ≤
+      (2 * (816412839554078022283 / 2500000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (2518303332239089 / 2000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1530310972700157 / 2000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1530310972700157 / 2000000000000000 : ℝ) /
+                (1 + (2518303332239089 / 2000000000000000 : ℝ)) =
+              (510103657566719 / 1506101110746363 : ℝ)
+            by
+        norm_num];
+        exact hau814)
+  have hat1 :
+    (Real.pi / 2 - (2 * (816412839554078022283 / 2500000000000000000000 : ℝ))) ≤
+      Real.arctan (2000000000000000 / 1530310972700157 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2000000000000000 / 1530310972700157 : ℝ) =
+              (1530310972700157 / 2000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau813)
+  have hz1 :
+    (2000000000000000 / 1530310972700157 : ℝ) ≤
+      1 / Real.sqrt (4683703346333 / 8000000000000 : ℝ) := by
+    rw [show
+        (2000000000000000 / 1530310972700157 : ℝ) = 1 / (1530310972700157 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (125000000000000 / 1275259143916797 : ℝ) ≤
+      (61066984094740587393 / 625000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (4683703346333 / 8000000000000 : ℝ) ≤
+      (125000000000000 / 1275259143916797 : ℝ) := by
+    rw [show
+        (125000000000000 / 1275259143916797 : ℝ) =
+          (3 / 40) / (3825777431750391 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (816412839554078022283 / 2500000000000000000000 : ℝ))) -
+        6 * (61066984094740587393 / 625000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (4683703346333 / 8000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (4683703346333 / 8000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3825777431750391 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (816412839554078022283 / 2500000000000000000000 : ℝ))) -
+          6 * (61066984094740587393 / 625000000000000000000 : ℝ))) ≤
+      Real.sqrt (4683703346333 / 8000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (4683703346333 / 8000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (4683703346333 / 8000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_522 :
+    ((290012920163202756309 / 625000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(323361855717053871977 / 625000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((7684112495394717 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3275902783813129556593 / 10000000000000000000000 : ℝ))) -
+              6 * (486479009297971681703 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (9447293574701 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (9447293574701 / 16000000000000 : ℝ)) =
+      Real.log (25447293574701 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((9447293574701 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (9537293574701 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (290012920163202756309 / 625000000000000000000 : ℝ) ≤
+      Real.log (25447293574701 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (9537293574701 / 16000000000000 : ℝ) ≤
+      (-(323361855717053871977 / 625000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (9537293574701 / 16000000000000 : ℝ) = (9537293574701 / 8000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9537293574701 / 8000000000000 : ℝ) ≤
+            (109855131970446128023 / 625000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (7684112495394717 / 10000000000000000 : ℝ) ≤ Real.sqrt (9447293574701 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (9447293574701 / 16000000000000 : ℝ) ≤ (48025703096217 / 62500000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (9447293574701 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau816 :
+    Real.arctan (452006617376160 / 1330078096466981 : ℝ) ≤
+      (3275902783813129556593 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau815 :
+    Real.arctan (48025703096217 / 62500000000000 : ℝ) ≤
+      (2 * (3275902783813129556593 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12611327639938677 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (48025703096217 / 62500000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (48025703096217 / 62500000000000 : ℝ) /
+                (1 + (12611327639938677 / 10000000000000000 : ℝ)) =
+              (452006617376160 / 1330078096466981 : ℝ)
+            by
+        norm_num];
+        exact hau816)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3275902783813129556593 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (62500000000000 / 48025703096217 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (62500000000000 / 48025703096217 : ℝ) =
+              (48025703096217 / 62500000000000 : ℝ)
+            by
+        norm_num];
+        exact hau815)
+  have hz1 :
+    (62500000000000 / 48025703096217 : ℝ) ≤ 1 / Real.sqrt (9447293574701 / 16000000000000 : ℝ) := by
+    rw [show (62500000000000 / 48025703096217 : ℝ) = 1 / (48025703096217 / 62500000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 2561370831798239 : ℝ) ≤
+      (486479009297971681703 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (9447293574701 / 16000000000000 : ℝ) ≤
+      (250000000000000 / 2561370831798239 : ℝ) := by
+    rw [show
+        (250000000000000 / 2561370831798239 : ℝ) =
+          (3 / 40) / (7684112495394717 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3275902783813129556593 / 10000000000000000000000 : ℝ))) -
+        6 * (486479009297971681703 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (9447293574701 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (9447293574701 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((7684112495394717 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3275902783813129556593 / 10000000000000000000000 : ℝ))) -
+          6 * (486479009297971681703 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (9447293574701 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (9447293574701 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (9447293574701 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_523 :
+    ((467155062520418752421 / 1000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(19884280940523666413 / 39062500000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((308661310447811 / 400000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (410759870046985860257 / 1250000000000000000000 : ℝ))) -
+              6 * (121111977243712197883 / 1250000000000000000000 : ℝ)))) ≤
+      Vfield (297724389273 / 500000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (297724389273 / 500000000000 : ℝ)) =
+      Real.log (797724389273 / 500000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((297724389273 / 500000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (300536889273 / 500000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (467155062520418752421 / 1000000000000000000000 : ℝ) ≤
+      Real.log (797724389273 / 500000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (300536889273 / 500000000000 : ℝ) ≤
+      (-(19884280940523666413 / 39062500000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (300536889273 / 500000000000 : ℝ) = (300536889273 / 250000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (300536889273 / 250000000000 : ℝ) ≤
+            (7191780789945083587 / 39062500000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (308661310447811 / 400000000000000 : ℝ) ≤ Real.sqrt (297724389273 / 500000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (297724389273 / 500000000000 : ℝ) ≤ (3858266380597639 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (297724389273 / 500000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau818 :
+    Real.arctan (3858266380597639 / 11315553773316319 : ℝ) ≤
+      (410759870046985860257 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau817 :
+    Real.arctan (3858266380597639 / 5000000000000000 : ℝ) ≤
+      (2 * (410759870046985860257 / 1250000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6315553773316319 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (3858266380597639 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (3858266380597639 / 5000000000000000 : ℝ) /
+                (1 + (6315553773316319 / 5000000000000000 : ℝ)) =
+              (3858266380597639 / 11315553773316319 : ℝ)
+            by
+        norm_num];
+        exact hau818)
+  have hat1 :
+    (Real.pi / 2 - (2 * (410759870046985860257 / 1250000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 3858266380597639 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 3858266380597639 : ℝ) =
+              (3858266380597639 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau817)
+  have hz1 :
+    (5000000000000000 / 3858266380597639 : ℝ) ≤ 1 / Real.sqrt (297724389273 / 500000000000 : ℝ) :=
+  by
+    rw [show
+        (5000000000000000 / 3858266380597639 : ℝ) = 1 / (3858266380597639 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (30000000000000 / 308661310447811 : ℝ) ≤
+      (121111977243712197883 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (297724389273 / 500000000000 : ℝ) ≤
+      (30000000000000 / 308661310447811 : ℝ) := by
+    rw [show
+        (30000000000000 / 308661310447811 : ℝ) =
+          (3 / 40) / (308661310447811 / 400000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (410759870046985860257 / 1250000000000000000000 : ℝ))) -
+        6 * (121111977243712197883 / 1250000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (297724389273 / 500000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (297724389273 / 500000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((308661310447811 / 400000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (410759870046985860257 / 1250000000000000000000 : ℝ))) -
+          6 * (121111977243712197883 / 1250000000000000000000 : ℝ))) ≤
+      Real.sqrt (297724389273 / 500000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (297724389273 / 500000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (297724389273 / 500000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_524 :
+    ((4685015939116695900419 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(5054674252401897476843 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3865224920526233 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1645218759627469874719 / 5000000000000000000000 : ℝ))) -
+              6 * (193432477556515180871 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (19123153518409 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (19123153518409 / 32000000000000 : ℝ)) =
+      Real.log (51123153518409 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((19123153518409 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (19303153518409 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (4685015939116695900419 / 10000000000000000000000 : ℝ) ≤
+      Real.log (51123153518409 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (19303153518409 / 32000000000000 : ℝ) ≤
+      (-(5054674252401897476843 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (19303153518409 / 32000000000000 : ℝ) = (19303153518409 / 16000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19303153518409 / 16000000000000 : ℝ) ≤
+            (1876797550598102523157 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3865224920526233 / 5000000000000000 : ℝ) ≤ Real.sqrt (19123153518409 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (19123153518409 / 32000000000000 : ℝ) ≤
+      (7730449841052469 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (19123153518409 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau820 :
+    Real.arctan (7730449841052469 / 22639614501440624 : ℝ) ≤
+      (1645218759627469874719 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau819 :
+    Real.arctan (7730449841052469 / 10000000000000000 : ℝ) ≤
+      (2 * (1645218759627469874719 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (789975906340039 / 625000000000000 : ℝ) ≤
+          Real.sqrt (1 + (7730449841052469 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (7730449841052469 / 10000000000000000 : ℝ) /
+                (1 + (789975906340039 / 625000000000000 : ℝ)) =
+              (7730449841052469 / 22639614501440624 : ℝ)
+            by
+        norm_num];
+        exact hau820)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1645218759627469874719 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 7730449841052469 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 7730449841052469 : ℝ) =
+              (7730449841052469 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau819)
+  have hz1 :
+    (10000000000000000 / 7730449841052469 : ℝ) ≤
+      1 / Real.sqrt (19123153518409 / 32000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 7730449841052469 : ℝ) =
+          1 / (7730449841052469 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 3865224920526233 : ℝ) ≤
+      (193432477556515180871 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (19123153518409 / 32000000000000 : ℝ) ≤
+      (375000000000000 / 3865224920526233 : ℝ) := by
+    rw [show
+        (375000000000000 / 3865224920526233 : ℝ) =
+          (3 / 40) / (3865224920526233 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1645218759627469874719 / 5000000000000000000000 : ℝ))) -
+        6 * (193432477556515180871 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (19123153518409 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (19123153518409 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3865224920526233 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1645218759627469874719 / 5000000000000000000000 : ℝ))) -
+          6 * (193432477556515180871 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (19123153518409 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (19123153518409 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (19123153518409 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_525 :
+    ((4698463145940359342123 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(5019099591638264099907 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((7744341911063601 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3294782395019543481039 / 10000000000000000000000 : ℝ))) -
+              6 * (120679778530763911613 / 1250000000000000000000 : ℝ)))) ≤
+      Vfield (9595973061673 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (9595973061673 / 16000000000000 : ℝ)) =
+      Real.log (25595973061673 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((9595973061673 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (9685973061673 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (4698463145940359342123 / 10000000000000000000000 : ℝ) ≤
+      Real.log (25595973061673 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (9685973061673 / 16000000000000 : ℝ) ≤
+      (-(5019099591638264099907 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (9685973061673 / 16000000000000 : ℝ) = (9685973061673 / 8000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9685973061673 / 8000000000000 : ℝ) ≤
+            (1912372211361735900093 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (7744341911063601 / 10000000000000000 : ℝ) ≤ Real.sqrt (9595973061673 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (9595973061673 / 16000000000000 : ℝ) ≤ (1936085477765901 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (9595973061673 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau822 :
+    Real.arctan (7744341911063604 / 22648115734584983 : ℝ) ≤
+      (3294782395019543481039 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau821 :
+    Real.arctan (1936085477765901 / 2500000000000000 : ℝ) ≤
+      (2 * (3294782395019543481039 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12648115734584983 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1936085477765901 / 2500000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1936085477765901 / 2500000000000000 : ℝ) /
+                (1 + (12648115734584983 / 10000000000000000 : ℝ)) =
+              (7744341911063604 / 22648115734584983 : ℝ)
+            by
+        norm_num];
+        exact hau822)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3294782395019543481039 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (2500000000000000 / 1936085477765901 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 1936085477765901 : ℝ) =
+              (1936085477765901 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau821)
+  have hz1 :
+    (2500000000000000 / 1936085477765901 : ℝ) ≤
+      1 / Real.sqrt (9595973061673 / 16000000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 1936085477765901 : ℝ) = 1 / (1936085477765901 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 2581447303687867 : ℝ) ≤
+      (120679778530763911613 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (9595973061673 / 16000000000000 : ℝ) ≤
+      (250000000000000 / 2581447303687867 : ℝ) := by
+    rw [show
+        (250000000000000 / 2581447303687867 : ℝ) =
+          (3 / 40) / (7744341911063601 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3294782395019543481039 / 10000000000000000000000 : ℝ))) -
+        6 * (120679778530763911613 / 1250000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (9595973061673 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (9595973061673 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((7744341911063601 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3294782395019543481039 / 10000000000000000000000 : ℝ))) -
+          6 * (120679778530763911613 / 1250000000000000000000 : ℝ))) ≤
+      Real.sqrt (9595973061673 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (9595973061673 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (9595973061673 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_526 :
+    ((2355946147153874507959 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(155739094938564335577 / 312500000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3879104552789353 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3299113666383077039093 / 10000000000000000000000 : ℝ))) -
+              6 * (481861628499394707899 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (19260738728283 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (19260738728283 / 32000000000000 : ℝ)) =
+      Real.log (51260738728283 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((19260738728283 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (19440738728283 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2355946147153874507959 / 5000000000000000000000 : ℝ) ≤
+      Real.log (51260738728283 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (19440738728283 / 32000000000000 : ℝ) ≤
+      (-(155739094938564335577 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (19440738728283 / 32000000000000 : ℝ) = (19440738728283 / 16000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19440738728283 / 16000000000000 : ℝ) ≤
+            (60869398905185664423 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3879104552789353 / 5000000000000000 : ℝ) ≤ Real.sqrt (19260738728283 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (19260738728283 / 32000000000000 : ℝ) ≤
+      (7758209105578709 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (19260738728283 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau824 :
+    Real.arctan (7758209105578709 / 22656611257595153 : ℝ) ≤
+      (3299113666383077039093 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau823 :
+    Real.arctan (7758209105578709 / 10000000000000000 : ℝ) ≤
+      (2 * (3299113666383077039093 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12656611257595153 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (7758209105578709 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (7758209105578709 / 10000000000000000 : ℝ) /
+                (1 + (12656611257595153 / 10000000000000000 : ℝ)) =
+              (7758209105578709 / 22656611257595153 : ℝ)
+            by
+        norm_num];
+        exact hau824)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3299113666383077039093 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 7758209105578709 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 7758209105578709 : ℝ) =
+              (7758209105578709 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau823)
+  have hz1 :
+    (10000000000000000 / 7758209105578709 : ℝ) ≤
+      1 / Real.sqrt (19260738728283 / 32000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 7758209105578709 : ℝ) =
+          1 / (7758209105578709 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 3879104552789353 : ℝ) ≤
+      (481861628499394707899 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (19260738728283 / 32000000000000 : ℝ) ≤
+      (375000000000000 / 3879104552789353 : ℝ) := by
+    rw [show
+        (375000000000000 / 3879104552789353 : ℝ) =
+          (3 / 40) / (3879104552789353 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3299113666383077039093 / 10000000000000000000000 : ℝ))) -
+        6 * (481861628499394707899 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (19260738728283 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (19260738728283 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3879104552789353 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3299113666383077039093 / 10000000000000000000000 : ℝ))) -
+          6 * (481861628499394707899 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (19260738728283 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (19260738728283 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (19260738728283 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_527 :
+    ((589825013964311084409 / 1250000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(4965973772639253023391 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((776513341618149 / 1000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1650637112411661779237 / 5000000000000000000000 : ℝ))) -
+              6 * (481434595765630853183 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (38590270061503 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (38590270061503 / 64000000000000 : ℝ)) =
+      Real.log (102590270061503 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((38590270061503 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (38950270061503 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (589825013964311084409 / 1250000000000000000000 : ℝ) ≤
+      Real.log (102590270061503 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (38950270061503 / 64000000000000 : ℝ) ≤
+      (-(4965973772639253023391 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (38950270061503 / 64000000000000 : ℝ) = (38950270061503 / 32000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (38950270061503 / 32000000000000 : ℝ) ≤
+            (1965498030360746976609 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (776513341618149 / 1000000000000000 : ℝ) ≤ Real.sqrt (38590270061503 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (38590270061503 / 64000000000000 : ℝ) ≤
+      (7765133416181493 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (38590270061503 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau826 :
+    Real.arctan (7765133416181493 / 22660856881392287 : ℝ) ≤
+      (1650637112411661779237 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau825 :
+    Real.arctan (7765133416181493 / 10000000000000000 : ℝ) ≤
+      (2 * (1650637112411661779237 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12660856881392287 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (7765133416181493 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (7765133416181493 / 10000000000000000 : ℝ) /
+                (1 + (12660856881392287 / 10000000000000000 : ℝ)) =
+              (7765133416181493 / 22660856881392287 : ℝ)
+            by
+        norm_num];
+        exact hau826)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1650637112411661779237 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 7765133416181493 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 7765133416181493 : ℝ) =
+              (7765133416181493 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau825)
+  have hz1 :
+    (10000000000000000 / 7765133416181493 : ℝ) ≤
+      1 / Real.sqrt (38590270061503 / 64000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 7765133416181493 : ℝ) =
+          1 / (7765133416181493 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (25000000000000 / 258837780539383 : ℝ) ≤
+      (481434595765630853183 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (38590270061503 / 64000000000000 : ℝ) ≤
+      (25000000000000 / 258837780539383 : ℝ) := by
+    rw [show
+        (25000000000000 / 258837780539383 : ℝ) =
+          (3 / 40) / (776513341618149 / 1000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1650637112411661779237 / 5000000000000000000000 : ℝ))) -
+        6 * (481434595765630853183 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (38590270061503 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (38590270061503 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((776513341618149 / 1000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1650637112411661779237 / 5000000000000000000000 : ℝ))) -
+          6 * (481434595765630853183 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (38590270061503 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (38590270061503 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (38590270061503 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_528 :
+    ((4725303432655770558119 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2474163850341135916233 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3886025778874623 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2 * (20646446321052335661 / 62500000000000000000 : ℝ))) -
+              6 * (48100869635968825743 / 500000000000000000000 : ℝ)))) ≤
+      Vfield (966476566661 / 1600000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (966476566661 / 1600000000000 : ℝ)) =
+      Real.log (2566476566661 / 1600000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((966476566661 / 1600000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (975476566661 / 1600000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (4725303432655770558119 / 10000000000000000000000 : ℝ) ≤
+      Real.log (2566476566661 / 1600000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (975476566661 / 1600000000000 : ℝ) ≤
+      (-(2474163850341135916233 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (975476566661 / 1600000000000 : ℝ) = (975476566661 / 800000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (975476566661 / 800000000000 : ℝ) ≤
+            (991572051158864083767 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3886025778874623 / 5000000000000000 : ℝ) ≤ Real.sqrt (966476566661 / 1600000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (966476566661 / 1600000000000 : ℝ) ≤ (7772051557749249 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (966476566661 / 1600000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau828 :
+    Real.arctan (2590683852583083 / 7555033693987300 : ℝ) ≤
+      (20646446321052335661 / 62500000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau827 :
+    Real.arctan (7772051557749249 / 10000000000000000 : ℝ) ≤
+      (2 * (20646446321052335661 / 62500000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (126651010819619 / 100000000000000 : ℝ) ≤
+          Real.sqrt (1 + (7772051557749249 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (7772051557749249 / 10000000000000000 : ℝ) /
+                (1 + (126651010819619 / 100000000000000 : ℝ)) =
+              (2590683852583083 / 7555033693987300 : ℝ)
+            by
+        norm_num];
+        exact hau828)
+  have hat1 :
+    (Real.pi / 2 - (2 * (20646446321052335661 / 62500000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 7772051557749249 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 7772051557749249 : ℝ) =
+              (7772051557749249 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau827)
+  have hz1 :
+    (10000000000000000 / 7772051557749249 : ℝ) ≤
+      1 / Real.sqrt (966476566661 / 1600000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 7772051557749249 : ℝ) =
+          1 / (7772051557749249 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (125000000000000 / 1295341926291541 : ℝ) ≤
+      (48100869635968825743 / 500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (966476566661 / 1600000000000 : ℝ) ≤
+      (125000000000000 / 1295341926291541 : ℝ) := by
+    rw [show
+        (125000000000000 / 1295341926291541 : ℝ) =
+          (3 / 40) / (3886025778874623 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (20646446321052335661 / 62500000000000000000 : ℝ))) -
+        6 * (48100869635968825743 / 500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (966476566661 / 1600000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (966476566661 / 1600000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3886025778874623 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (20646446321052335661 / 62500000000000000000 : ℝ))) -
+          6 * (48100869635968825743 / 500000000000000000000 : ℝ))) ≤
+      Real.sqrt (966476566661 / 1600000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (966476566661 / 1600000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (966476566661 / 1600000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/V44.lean b/LeanPool/Zeta5Irrational/Table/V44.lean
new file mode 100644
index 0000000000..4aebdc9009
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/V44.lean
@@ -0,0 +1,1446 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Energy
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # V44: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_529 :
+    ((4732002263155807105589 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(986142542453729802313 / 2000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3889481773370539 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (661117047132910503789 / 2000000000000000000000 : ℝ))) -
+              6 * (192233570110948262549 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (38727855271377 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (38727855271377 / 64000000000000 : ℝ)) =
+      Real.log (102727855271377 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((38727855271377 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (39087855271377 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (4732002263155807105589 / 10000000000000000000000 : ℝ) ≤
+      Real.log (102727855271377 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (39087855271377 / 64000000000000 : ℝ) ≤
+      (-(986142542453729802313 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (39087855271377 / 64000000000000 : ℝ) = (39087855271377 / 32000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (39087855271377 / 32000000000000 : ℝ) ≤
+            (400151818146270197687 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3889481773370539 / 5000000000000000 : ℝ) ≤ Real.sqrt (38727855271377 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (38727855271377 / 64000000000000 : ℝ) ≤
+      (7778963546741081 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (38727855271377 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau830 :
+    Real.arctan (7778963546741081 / 22669343860734325 : ℝ) ≤
+      (661117047132910503789 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau829 :
+    Real.arctan (7778963546741081 / 10000000000000000 : ℝ) ≤
+      (2 * (661117047132910503789 / 2000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (506773754429373 / 400000000000000 : ℝ) ≤
+          Real.sqrt (1 + (7778963546741081 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (7778963546741081 / 10000000000000000 : ℝ) /
+                (1 + (506773754429373 / 400000000000000 : ℝ)) =
+              (7778963546741081 / 22669343860734325 : ℝ)
+            by
+        norm_num];
+        exact hau830)
+  have hat1 :
+    (Real.pi / 2 - (2 * (661117047132910503789 / 2000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 7778963546741081 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 7778963546741081 : ℝ) =
+              (7778963546741081 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau829)
+  have hz1 :
+    (10000000000000000 / 7778963546741081 : ℝ) ≤
+      1 / Real.sqrt (38727855271377 / 64000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 7778963546741081 : ℝ) =
+          1 / (7778963546741081 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 3889481773370539 : ℝ) ≤
+      (192233570110948262549 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (38727855271377 / 64000000000000 : ℝ) ≤
+      (375000000000000 / 3889481773370539 : ℝ) := by
+    rw [show
+        (375000000000000 / 3889481773370539 : ℝ) =
+          (3 / 40) / (3889481773370539 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (661117047132910503789 / 2000000000000000000000 : ℝ))) -
+        6 * (192233570110948262549 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (38727855271377 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (38727855271377 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3889481773370539 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (661117047132910503789 / 2000000000000000000000 : ℝ))) -
+          6 * (192233570110948262549 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (38727855271377 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (38727855271377 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (38727855271377 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_530 :
+    ((4738696609226712308587 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(4913128698083634611777 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3892934699771517 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3307735707316062547853 / 10000000000000000000000 : ℝ))) -
+              6 * (960320555090723355467 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (19398323938157 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (19398323938157 / 32000000000000 : ℝ)) =
+      Real.log (51398323938157 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((19398323938157 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (19578323938157 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (4738696609226712308587 / 10000000000000000000000 : ℝ) ≤
+      Real.log (51398323938157 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (19578323938157 / 32000000000000 : ℝ) ≤
+      (-(4913128698083634611777 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (19578323938157 / 32000000000000 : ℝ) = (19578323938157 / 16000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19578323938157 / 16000000000000 : ℝ) ≤
+            (2018343104916365388223 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3892934699771517 / 5000000000000000 : ℝ) ≤ Real.sqrt (19398323938157 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (19398323938157 / 32000000000000 : ℝ) ≤
+      (7785869399543037 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (19398323938157 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau832 :
+    Real.arctan (7785869399543037 / 22673585219137504 : ℝ) ≤
+      (3307735707316062547853 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau831 :
+    Real.arctan (7785869399543037 / 10000000000000000 : ℝ) ≤
+      (2 * (3307735707316062547853 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (396049538098047 / 312500000000000 : ℝ) ≤
+          Real.sqrt (1 + (7785869399543037 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (7785869399543037 / 10000000000000000 : ℝ) /
+                (1 + (396049538098047 / 312500000000000 : ℝ)) =
+              (7785869399543037 / 22673585219137504 : ℝ)
+            by
+        norm_num];
+        exact hau832)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3307735707316062547853 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 7785869399543037 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 7785869399543037 : ℝ) =
+              (7785869399543037 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau831)
+  have hz1 :
+    (10000000000000000 / 7785869399543037 : ℝ) ≤
+      1 / Real.sqrt (19398323938157 / 32000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 7785869399543037 : ℝ) =
+          1 / (7785869399543037 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (125000000000000 / 1297644899923839 : ℝ) ≤
+      (960320555090723355467 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (19398323938157 / 32000000000000 : ℝ) ≤
+      (125000000000000 / 1297644899923839 : ℝ) := by
+    rw [show
+        (125000000000000 / 1297644899923839 : ℝ) =
+          (3 / 40) / (3892934699771517 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3307735707316062547853 / 10000000000000000000000 : ℝ))) -
+        6 * (960320555090723355467 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (19398323938157 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (19398323938157 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3892934699771517 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3307735707316062547853 / 10000000000000000000000 : ℝ))) -
+          6 * (960320555090723355467 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (19398323938157 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (19398323938157 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (19398323938157 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_531 :
+    ((474538647686853408261 / 1000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(4895575549388124547303 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((974096141558569 / 1250000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3309882835885235928563 / 10000000000000000000000 : ℝ))) -
+              6 * (479737748220980712743 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (38865440481251 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (38865440481251 / 64000000000000 : ℝ)) =
+      Real.log (102865440481251 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((38865440481251 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (39225440481251 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (474538647686853408261 / 1000000000000000000000 : ℝ) ≤
+      Real.log (102865440481251 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (39225440481251 / 64000000000000 : ℝ) ≤
+      (-(4895575549388124547303 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (39225440481251 / 64000000000000 : ℝ) = (39225440481251 / 32000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (39225440481251 / 32000000000000 : ℝ) ≤
+            (2035896253611875452697 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (974096141558569 / 1250000000000000 : ℝ) ≤ Real.sqrt (38865440481251 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (38865440481251 / 64000000000000 : ℝ) ≤ (1558553826493711 / 2000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (38865440481251 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau834 :
+    Real.arctan (1558553826493711 / 4535565031719397 : ℝ) ≤
+      (3309882835885235928563 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau833 :
+    Real.arctan (1558553826493711 / 2000000000000000 : ℝ) ≤
+      (2 * (3309882835885235928563 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (2535565031719397 / 2000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1558553826493711 / 2000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1558553826493711 / 2000000000000000 : ℝ) /
+                (1 + (2535565031719397 / 2000000000000000 : ℝ)) =
+              (1558553826493711 / 4535565031719397 : ℝ)
+            by
+        norm_num];
+        exact hau834)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3309882835885235928563 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (2000000000000000 / 1558553826493711 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2000000000000000 / 1558553826493711 : ℝ) =
+              (1558553826493711 / 2000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau833)
+  have hz1 :
+    (2000000000000000 / 1558553826493711 : ℝ) ≤
+      1 / Real.sqrt (38865440481251 / 64000000000000 : ℝ) := by
+    rw [show
+        (2000000000000000 / 1558553826493711 : ℝ) = 1 / (1558553826493711 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (93750000000000 / 974096141558569 : ℝ) ≤
+      (479737748220980712743 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (38865440481251 / 64000000000000 : ℝ) ≤
+      (93750000000000 / 974096141558569 : ℝ) := by
+    rw [show
+        (93750000000000 / 974096141558569 : ℝ) =
+          (3 / 40) / (974096141558569 / 1250000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3309882835885235928563 / 10000000000000000000000 : ℝ))) -
+        6 * (479737748220980712743 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (38865440481251 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (38865440481251 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((974096141558569 / 1250000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3309882835885235928563 / 10000000000000000000000 : ℝ))) -
+          6 * (479737748220980712743 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (38865440481251 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (38865440481251 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (38865440481251 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_532 :
+    ((4752071872069286534347 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2439026579007312957233 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1949915690439729 / 2500000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3312026630892787282743 / 10000000000000000000000 : ℝ))) -
+              6 * (479316332391940315167 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (9733558271547 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (9733558271547 / 16000000000000 : ℝ)) =
+      Real.log (25733558271547 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((9733558271547 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (9823558271547 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (4752071872069286534347 / 10000000000000000000000 : ℝ) ≤
+      Real.log (25733558271547 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (9823558271547 / 16000000000000 : ℝ) ≤
+      (-(2439026579007312957233 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (9823558271547 / 16000000000000 : ℝ) = (9823558271547 / 8000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9823558271547 / 8000000000000 : ℝ) ≤
+            (1026709322492687042767 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1949915690439729 / 2500000000000000 : ℝ) ≤ Real.sqrt (9733558271547 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (9733558271547 / 16000000000000 : ℝ) ≤ (7799662761758919 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (9733558271547 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau836 :
+    Real.arctan (7799662761758919 / 22682063680535938 : ℝ) ≤
+      (3312026630892787282743 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau835 :
+    Real.arctan (7799662761758919 / 10000000000000000 : ℝ) ≤
+      (2 * (3312026630892787282743 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6341031840267969 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (7799662761758919 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (7799662761758919 / 10000000000000000 : ℝ) /
+                (1 + (6341031840267969 / 5000000000000000 : ℝ)) =
+              (7799662761758919 / 22682063680535938 : ℝ)
+            by
+        norm_num];
+        exact hau836)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3312026630892787282743 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 7799662761758919 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 7799662761758919 : ℝ) =
+              (7799662761758919 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau835)
+  have hz1 :
+    (10000000000000000 / 7799662761758919 : ℝ) ≤
+      1 / Real.sqrt (9733558271547 / 16000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 7799662761758919 : ℝ) =
+          1 / (7799662761758919 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (62500000000000 / 649971896813243 : ℝ) ≤
+      (479316332391940315167 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (9733558271547 / 16000000000000 : ℝ) ≤
+      (62500000000000 / 649971896813243 : ℝ) := by
+    rw [show
+        (62500000000000 / 649971896813243 : ℝ) =
+          (3 / 40) / (1949915690439729 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3312026630892787282743 / 10000000000000000000000 : ℝ))) -
+        6 * (479316332391940315167 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (9733558271547 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (9733558271547 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1949915690439729 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3312026630892787282743 / 10000000000000000000000 : ℝ))) -
+          6 * (479316332391940315167 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (9733558271547 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (9733558271547 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (9733558271547 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_533 :
+    ((594844100100622765061 / 1250000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(972112283272651519513 / 2000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3903275151791851 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3314167101818064698889 / 10000000000000000000000 : ℝ))) -
+              6 * (239448012588053166339 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (312024205529 / 512000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (312024205529 / 512000000000 : ℝ)) =
+      Real.log (824024205529 / 512000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((312024205529 / 512000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (314904205529 / 512000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (594844100100622765061 / 1250000000000000000000 : ℝ) ≤
+      Real.log (824024205529 / 512000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (314904205529 / 512000000000 : ℝ) ≤
+      (-(972112283272651519513 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (314904205529 / 512000000000 : ℝ) = (314904205529 / 256000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (314904205529 / 256000000000 : ℝ) ≤
+            (414182077327348480487 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3903275151791851 / 5000000000000000 : ℝ) ≤ Real.sqrt (312024205529 / 512000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (312024205529 / 512000000000 : ℝ) ≤ (1561310060716741 / 2000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (312024205529 / 512000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau838 :
+    Real.arctan (7806550303583705 / 22686300786375152 : ℝ) ≤
+      (3314167101818064698889 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau837 :
+    Real.arctan (1561310060716741 / 2000000000000000 : ℝ) ≤
+      (2 * (3314167101818064698889 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (792893799148447 / 625000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1561310060716741 / 2000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1561310060716741 / 2000000000000000 : ℝ) /
+                (1 + (792893799148447 / 625000000000000 : ℝ)) =
+              (7806550303583705 / 22686300786375152 : ℝ)
+            by
+        norm_num];
+        exact hau838)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3314167101818064698889 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (2000000000000000 / 1561310060716741 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2000000000000000 / 1561310060716741 : ℝ) =
+              (1561310060716741 / 2000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau837)
+  have hz1 :
+    (2000000000000000 / 1561310060716741 : ℝ) ≤ 1 / Real.sqrt (312024205529 / 512000000000 : ℝ) :=
+  by
+    rw [show
+        (2000000000000000 / 1561310060716741 : ℝ) = 1 / (1561310060716741 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 3903275151791851 : ℝ) ≤
+      (239448012588053166339 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (312024205529 / 512000000000 : ℝ) ≤
+      (375000000000000 / 3903275151791851 : ℝ) := by
+    rw [show
+        (375000000000000 / 3903275151791851 : ℝ) =
+          (3 / 40) / (3903275151791851 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3314167101818064698889 / 10000000000000000000000 : ℝ))) -
+        6 * (239448012588053166339 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (312024205529 / 512000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (312024205529 / 512000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3903275151791851 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3314167101818064698889 / 10000000000000000000000 : ℝ))) -
+          6 * (239448012588053166339 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (312024205529 / 512000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (312024205529 / 512000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (312024205529 / 512000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_534 :
+    ((953085853807932739837 / 2000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(4843100217397785791881 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1562686354808243 / 2000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (829076064524823682427 / 2500000000000000000000 : ℝ))) -
+              6 * (956953643442521360779 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (19535909148031 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (19535909148031 / 32000000000000 : ℝ)) =
+      Real.log (51535909148031 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((19535909148031 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (19715909148031 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (953085853807932739837 / 2000000000000000000000 : ℝ) ≤
+      Real.log (51535909148031 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (19715909148031 / 32000000000000 : ℝ) ≤
+      (-(4843100217397785791881 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (19715909148031 / 32000000000000 : ℝ) = (19715909148031 / 16000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19715909148031 / 16000000000000 : ℝ) ≤
+            (2088371585602214208119 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1562686354808243 / 2000000000000000 : ℝ) ≤ Real.sqrt (19535909148031 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (19535909148031 / 32000000000000 : ℝ) ≤ (3906715887020609 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (19535909148031 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau840 :
+    Real.arctan (3906715887020609 / 11345268238766523 : ℝ) ≤
+      (829076064524823682427 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau839 :
+    Real.arctan (3906715887020609 / 5000000000000000 : ℝ) ≤
+      (2 * (829076064524823682427 / 2500000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6345268238766523 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (3906715887020609 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (3906715887020609 / 5000000000000000 : ℝ) /
+                (1 + (6345268238766523 / 5000000000000000 : ℝ)) =
+              (3906715887020609 / 11345268238766523 : ℝ)
+            by
+        norm_num];
+        exact hau840)
+  have hat1 :
+    (Real.pi / 2 - (2 * (829076064524823682427 / 2500000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 3906715887020609 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 3906715887020609 : ℝ) =
+              (3906715887020609 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau839)
+  have hz1 :
+    (5000000000000000 / 3906715887020609 : ℝ) ≤
+      1 / Real.sqrt (19535909148031 / 32000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 3906715887020609 : ℝ) = 1 / (3906715887020609 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (150000000000000 / 1562686354808243 : ℝ) ≤
+      (956953643442521360779 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (19535909148031 / 32000000000000 : ℝ) ≤
+      (150000000000000 / 1562686354808243 : ℝ) := by
+    rw [show
+        (150000000000000 / 1562686354808243 : ℝ) =
+          (3 / 40) / (1562686354808243 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (829076064524823682427 / 2500000000000000000000 : ℝ))) -
+        6 * (956953643442521360779 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (19535909148031 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (19535909148031 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1562686354808243 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (829076064524823682427 / 2500000000000000000000 : ℝ))) -
+          6 * (956953643442521360779 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (19535909148031 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (19535909148031 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (19535909148031 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_535 :
+    ((4772101282725436474587 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(4825669454641694078661 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3910153594579467 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (829609527283457739137 / 2500000000000000000000 : ℝ))) -
+              6 * (478058717204865695451 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (39140610900999 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (39140610900999 / 64000000000000 : ℝ)) =
+      Real.log (103140610900999 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((39140610900999 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (39500610900999 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (4772101282725436474587 / 10000000000000000000000 : ℝ) ≤
+      Real.log (103140610900999 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (39500610900999 / 64000000000000 : ℝ) ≤
+      (-(4825669454641694078661 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (39500610900999 / 64000000000000 : ℝ) = (39500610900999 / 32000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (39500610900999 / 32000000000000 : ℝ) ≤
+            (2105802348358305921339 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3910153594579467 / 5000000000000000 : ℝ) ≤ Real.sqrt (39140610900999 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (39140610900999 / 64000000000000 : ℝ) ≤
+      (7820307189158937 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (39140610900999 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau842 :
+    Real.arctan (2606769063052979 / 7564923585141891 : ℝ) ≤
+      (829609527283457739137 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau841 :
+    Real.arctan (7820307189158937 / 10000000000000000 : ℝ) ≤
+      (2 * (829609527283457739137 / 2500000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12694770755425673 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (7820307189158937 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (7820307189158937 / 10000000000000000 : ℝ) /
+                (1 + (12694770755425673 / 10000000000000000 : ℝ)) =
+              (2606769063052979 / 7564923585141891 : ℝ)
+            by
+        norm_num];
+        exact hau842)
+  have hat1 :
+    (Real.pi / 2 - (2 * (829609527283457739137 / 2500000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 7820307189158937 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 7820307189158937 : ℝ) =
+              (7820307189158937 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau841)
+  have hz1 :
+    (10000000000000000 / 7820307189158937 : ℝ) ≤
+      1 / Real.sqrt (39140610900999 / 64000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 7820307189158937 : ℝ) =
+          1 / (7820307189158937 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (125000000000000 / 1303384531526489 : ℝ) ≤
+      (478058717204865695451 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (39140610900999 / 64000000000000 : ℝ) ≤
+      (125000000000000 / 1303384531526489 : ℝ) := by
+    rw [show
+        (125000000000000 / 1303384531526489 : ℝ) =
+          (3 / 40) / (3910153594579467 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (829609527283457739137 / 2500000000000000000000 : ℝ))) -
+        6 * (478058717204865695451 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (39140610900999 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (39140610900999 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3910153594579467 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (829609527283457739137 / 2500000000000000000000 : ℝ))) -
+          6 * (478058717204865695451 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (39140610900999 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (39140610900999 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (39140610900999 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_536 :
+    ((4778768847802499834877 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(4808269022174287691333 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1565435312978789 / 2000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3320568664278397156717 / 10000000000000000000000 : ℝ))) -
+              6 * (238820853416915635233 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (2450587719121 / 4000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (2450587719121 / 4000000000000 : ℝ)) =
+      Real.log (6450587719121 / 4000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((2450587719121 / 4000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (2473087719121 / 4000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (4778768847802499834877 / 10000000000000000000000 : ℝ) ≤
+      Real.log (6450587719121 / 4000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (2473087719121 / 4000000000000 : ℝ) ≤
+      (-(4808269022174287691333 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (2473087719121 / 4000000000000 : ℝ) = (2473087719121 / 2000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2473087719121 / 2000000000000 : ℝ) ≤
+            (2123202780825712308667 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1565435312978789 / 2000000000000000 : ℝ) ≤ Real.sqrt (2450587719121 / 4000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (2450587719121 / 4000000000000 : ℝ) ≤ (1956794141223487 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (2450587719121 / 4000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau844 :
+    Real.arctan (7827176564893948 / 22699003621466725 : ℝ) ≤
+      (3320568664278397156717 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau843 :
+    Real.arctan (1956794141223487 / 2500000000000000 : ℝ) ≤
+      (2 * (3320568664278397156717 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (507960144858669 / 400000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1956794141223487 / 2500000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1956794141223487 / 2500000000000000 : ℝ) /
+                (1 + (507960144858669 / 400000000000000 : ℝ)) =
+              (7827176564893948 / 22699003621466725 : ℝ)
+            by
+        norm_num];
+        exact hau844)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3320568664278397156717 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (2500000000000000 / 1956794141223487 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 1956794141223487 : ℝ) =
+              (1956794141223487 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau843)
+  have hz1 :
+    (2500000000000000 / 1956794141223487 : ℝ) ≤
+      1 / Real.sqrt (2450587719121 / 4000000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 1956794141223487 : ℝ) = 1 / (1956794141223487 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (150000000000000 / 1565435312978789 : ℝ) ≤
+      (238820853416915635233 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (2450587719121 / 4000000000000 : ℝ) ≤
+      (150000000000000 / 1565435312978789 : ℝ) := by
+    rw [show
+        (150000000000000 / 1565435312978789 : ℝ) =
+          (3 / 40) / (1565435312978789 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3320568664278397156717 / 10000000000000000000000 : ℝ))) -
+        6 * (238820853416915635233 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (2450587719121 / 4000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (2450587719121 / 4000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1565435312978789 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3320568664278397156717 / 10000000000000000000000 : ℝ))) -
+          6 * (238820853416915635233 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (2450587719121 / 4000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (2450587719121 / 4000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (2450587719121 / 4000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_537 :
+    ((4785431970199179084243 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2395449407313415808313 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((7834039917133371 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (664539186569865807171 / 2000000000000000000000 : ℝ))) -
+              6 * (954451571688568381881 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (39278196110873 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (39278196110873 / 64000000000000 : ℝ)) =
+      Real.log (103278196110873 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((39278196110873 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (39638196110873 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (4785431970199179084243 / 10000000000000000000000 : ℝ) ≤
+      Real.log (103278196110873 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (39638196110873 / 64000000000000 : ℝ) ≤
+      (-(2395449407313415808313 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (39638196110873 / 64000000000000 : ℝ) = (39638196110873 / 32000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (39638196110873 / 32000000000000 : ℝ) ≤
+            (1070286494186584191687 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (7834039917133371 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (39278196110873 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (39278196110873 / 64000000000000 : ℝ) ≤ (3917019958566687 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (39278196110873 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau846 :
+    Real.arctan (3917019958566687 / 11351617538533768 : ℝ) ≤
+      (664539186569865807171 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau845 :
+    Real.arctan (3917019958566687 / 5000000000000000 : ℝ) ≤
+      (2 * (664539186569865807171 / 2000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (793952192316721 / 625000000000000 : ℝ) ≤
+          Real.sqrt (1 + (3917019958566687 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (3917019958566687 / 5000000000000000 : ℝ) /
+                (1 + (793952192316721 / 625000000000000 : ℝ)) =
+              (3917019958566687 / 11351617538533768 : ℝ)
+            by
+        norm_num];
+        exact hau846)
+  have hat1 :
+    (Real.pi / 2 - (2 * (664539186569865807171 / 2000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 3917019958566687 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 3917019958566687 : ℝ) =
+              (3917019958566687 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau845)
+  have hz1 :
+    (5000000000000000 / 3917019958566687 : ℝ) ≤
+      1 / Real.sqrt (39278196110873 / 64000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 3917019958566687 : ℝ) = 1 / (3917019958566687 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 2611346639044457 : ℝ) ≤
+      (954451571688568381881 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (39278196110873 / 64000000000000 : ℝ) ≤
+      (250000000000000 / 2611346639044457 : ℝ) := by
+    rw [show
+        (250000000000000 / 2611346639044457 : ℝ) =
+          (3 / 40) / (7834039917133371 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (664539186569865807171 / 2000000000000000000000 : ℝ))) -
+        6 * (954451571688568381881 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (39278196110873 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (39278196110873 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((7834039917133371 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (664539186569865807171 / 2000000000000000000000 : ℝ))) -
+          6 * (954451571688568381881 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (39278196110873 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (39278196110873 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (39278196110873 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_538 :
+    ((2396045327915978534597 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2386779363589361089281 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3920448630847403 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3324819924122815790111 / 10000000000000000000000 : ℝ))) -
+              6 * (953621899002678461567 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (3934698871581 / 6400000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (3934698871581 / 6400000000000 : ℝ)) =
+      Real.log (10334698871581 / 6400000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((3934698871581 / 6400000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (3970698871581 / 6400000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2396045327915978534597 / 5000000000000000000000 : ℝ) ≤
+      Real.log (10334698871581 / 6400000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (3970698871581 / 6400000000000 : ℝ) ≤
+      (-(2386779363589361089281 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (3970698871581 / 6400000000000 : ℝ) = (3970698871581 / 3200000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (3970698871581 / 3200000000000 : ℝ) ≤
+            (1078956537910638910719 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3920448630847403 / 5000000000000000 : ℝ) ≤ Real.sqrt (3934698871581 / 6400000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (3934698871581 / 6400000000000 : ℝ) ≤ (7840897261694809 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (3934698871581 / 6400000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau848 :
+    Real.arctan (7840897261694809 / 22707465123637094 : ℝ) ≤
+      (3324819924122815790111 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau847 :
+    Real.arctan (7840897261694809 / 10000000000000000 : ℝ) ≤
+      (2 * (3324819924122815790111 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6353732561818547 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (7840897261694809 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (7840897261694809 / 10000000000000000 : ℝ) /
+                (1 + (6353732561818547 / 5000000000000000 : ℝ)) =
+              (7840897261694809 / 22707465123637094 : ℝ)
+            by
+        norm_num];
+        exact hau848)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3324819924122815790111 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 7840897261694809 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 7840897261694809 : ℝ) =
+              (7840897261694809 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau847)
+  have hz1 :
+    (10000000000000000 / 7840897261694809 : ℝ) ≤
+      1 / Real.sqrt (3934698871581 / 6400000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 7840897261694809 : ℝ) =
+          1 / (7840897261694809 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 3920448630847403 : ℝ) ≤
+      (953621899002678461567 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (3934698871581 / 6400000000000 : ℝ) ≤
+      (375000000000000 / 3920448630847403 : ℝ) := by
+    rw [show
+        (375000000000000 / 3920448630847403 : ℝ) =
+          (3 / 40) / (3920448630847403 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3324819924122815790111 / 10000000000000000000000 : ℝ))) -
+        6 * (953621899002678461567 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (3934698871581 / 6400000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (3934698871581 / 6400000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3920448630847403 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3324819924122815790111 / 10000000000000000000000 : ℝ))) -
+          6 * (953621899002678461567 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (3934698871581 / 6400000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (3934698871581 / 6400000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (3934698871581 / 6400000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_539 :
+    ((2399372455302752849833 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1189062163888422939389 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((7847748614326733 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3326940647335136480411 / 10000000000000000000000 : ℝ))) -
+              6 * (119099298274718298781 / 1250000000000000000000 : ℝ)))) ≤
+      Vfield (39415781320747 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (39415781320747 / 64000000000000 : ℝ)) =
+      Real.log (103415781320747 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((39415781320747 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (39775781320747 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2399372455302752849833 / 5000000000000000000000 : ℝ) ≤
+      Real.log (103415781320747 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (39775781320747 / 64000000000000 : ℝ) ≤
+      (-(1189062163888422939389 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (39775781320747 / 64000000000000 : ℝ) = (39775781320747 / 32000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (39775781320747 / 32000000000000 : ℝ) ≤
+            (543805786861577060611 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (7847748614326733 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (39415781320747 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (39415781320747 / 64000000000000 : ℝ) ≤ (490484288395421 / 625000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (39415781320747 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau850 :
+    Real.arctan (7847748614326736 / 22711693762582041 : ℝ) ≤
+      (3326940647335136480411 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau849 :
+    Real.arctan (490484288395421 / 625000000000000 : ℝ) ≤
+      (2 * (3326940647335136480411 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12711693762582041 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (490484288395421 / 625000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (490484288395421 / 625000000000000 : ℝ) /
+                (1 + (12711693762582041 / 10000000000000000 : ℝ)) =
+              (7847748614326736 / 22711693762582041 : ℝ)
+            by
+        norm_num];
+        exact hau850)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3326940647335136480411 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (625000000000000 / 490484288395421 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (625000000000000 / 490484288395421 : ℝ) =
+              (490484288395421 / 625000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau849)
+  have hz1 :
+    (625000000000000 / 490484288395421 : ℝ) ≤
+      1 / Real.sqrt (39415781320747 / 64000000000000 : ℝ) := by
+    rw [show (625000000000000 / 490484288395421 : ℝ) = 1 / (490484288395421 / 625000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 7847748614326733 : ℝ) ≤
+      (119099298274718298781 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (39415781320747 / 64000000000000 : ℝ) ≤
+      (750000000000000 / 7847748614326733 : ℝ) := by
+    rw [show
+        (750000000000000 / 7847748614326733 : ℝ) =
+          (3 / 40) / (7847748614326733 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3326940647335136480411 / 10000000000000000000000 : ℝ))) -
+        6 * (119099298274718298781 / 1250000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (39415781320747 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (39415781320747 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((7847748614326733 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3326940647335136480411 / 10000000000000000000000 : ℝ))) -
+          6 * (119099298274718298781 / 1250000000000000000000 : ℝ))) ≤
+      Real.sqrt (39415781320747 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (39415781320747 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (39415781320747 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_540 :
+    ((4805394740412717365303 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(4738968496016046301109 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1570918798141791 / 2000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2 * (83226452792072501471 / 250000000000000000000 : ℝ))) -
+              6 * (475984511959300131801 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (9871143481421 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (9871143481421 / 16000000000000 : ℝ)) =
+      Real.log (25871143481421 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((9871143481421 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (9961143481421 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (4805394740412717365303 / 10000000000000000000000 : ℝ) ≤
+      Real.log (25871143481421 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (9961143481421 / 16000000000000 : ℝ) ≤
+      (-(4738968496016046301109 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (9961143481421 / 16000000000000 : ℝ) = (9961143481421 / 8000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (9961143481421 / 8000000000000 : ℝ) ≤
+            (2192503306983953698891 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1570918798141791 / 2000000000000000 : ℝ) ≤ Real.sqrt (9871143481421 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (9871143481421 / 16000000000000 : ℝ) ≤ (3927296995354479 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (9871143481421 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau852 :
+    Real.arctan (3927296995354479 / 11357960497653340 : ℝ) ≤
+      (83226452792072501471 / 250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau851 :
+    Real.arctan (3927296995354479 / 5000000000000000 : ℝ) ≤
+      (2 * (83226452792072501471 / 250000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (317898024882667 / 250000000000000 : ℝ) ≤
+          Real.sqrt (1 + (3927296995354479 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (3927296995354479 / 5000000000000000 : ℝ) /
+                (1 + (317898024882667 / 250000000000000 : ℝ)) =
+              (3927296995354479 / 11357960497653340 : ℝ)
+            by
+        norm_num];
+        exact hau852)
+  have hat1 :
+    (Real.pi / 2 - (2 * (83226452792072501471 / 250000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 3927296995354479 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 3927296995354479 : ℝ) =
+              (3927296995354479 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau851)
+  have hz1 :
+    (5000000000000000 / 3927296995354479 : ℝ) ≤
+      1 / Real.sqrt (9871143481421 / 16000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 3927296995354479 : ℝ) = 1 / (3927296995354479 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (50000000000000 / 523639599380597 : ℝ) ≤
+      (475984511959300131801 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (9871143481421 / 16000000000000 : ℝ) ≤
+      (50000000000000 / 523639599380597 : ℝ) := by
+    rw [show
+        (50000000000000 / 523639599380597 : ℝ) =
+          (3 / 40) / (1570918798141791 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (83226452792072501471 / 250000000000000000000 : ℝ))) -
+        6 * (475984511959300131801 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (9871143481421 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (9871143481421 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1570918798141791 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (83226452792072501471 / 250000000000000000000 : ℝ))) -
+          6 * (475984511959300131801 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (9871143481421 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (9871143481421 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (9871143481421 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/V45.lean b/LeanPool/Zeta5Irrational/Table/V45.lean
new file mode 100644
index 0000000000..3ec6a7a05e
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/V45.lean
@@ -0,0 +1,1443 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Energy
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # V45: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_541 :
+    ((1203010037783684061843 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(4721718145366935286647 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1965358351613251 / 2500000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1665586163161638182387 / 5000000000000000000000 : ℝ))) -
+              6 * (951145802866699611319 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (39553366530621 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (39553366530621 / 64000000000000 : ℝ)) =
+      Real.log (103553366530621 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((39553366530621 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (39913366530621 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1203010037783684061843 / 2500000000000000000000 : ℝ) ≤
+      Real.log (103553366530621 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (39913366530621 / 64000000000000 : ℝ) ≤
+      (-(4721718145366935286647 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (39913366530621 / 64000000000000 : ℝ) = (39913366530621 / 32000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (39913366530621 / 32000000000000 : ℝ) ≤
+            (2209753657633064713353 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1965358351613251 / 2500000000000000 : ℝ) ≤ Real.sqrt (39553366530621 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (39553366530621 / 64000000000000 : ℝ) ≤
+      (7861433406453007 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (39553366530621 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau854 :
+    Real.arctan (7861433406453007 / 22720146823212981 : ℝ) ≤
+      (1665586163161638182387 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau853 :
+    Real.arctan (7861433406453007 / 10000000000000000 : ℝ) ≤
+      (2 * (1665586163161638182387 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12720146823212981 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (7861433406453007 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (7861433406453007 / 10000000000000000 : ℝ) /
+                (1 + (12720146823212981 / 10000000000000000 : ℝ)) =
+              (7861433406453007 / 22720146823212981 : ℝ)
+            by
+        norm_num];
+        exact hau854)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1665586163161638182387 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 7861433406453007 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 7861433406453007 : ℝ) =
+              (7861433406453007 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau853)
+  have hz1 :
+    (10000000000000000 / 7861433406453007 : ℝ) ≤
+      1 / Real.sqrt (39553366530621 / 64000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 7861433406453007 : ℝ) =
+          1 / (7861433406453007 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (187500000000000 / 1965358351613251 : ℝ) ≤
+      (951145802866699611319 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (39553366530621 / 64000000000000 : ℝ) ≤
+      (187500000000000 / 1965358351613251 : ℝ) := by
+    rw [show
+        (187500000000000 / 1965358351613251 : ℝ) =
+          (3 / 40) / (1965358351613251 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1665586163161638182387 / 5000000000000000000000 : ℝ))) -
+        6 * (951145802866699611319 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (39553366530621 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (39553366530621 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1965358351613251 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1665586163161638182387 / 5000000000000000000000 : ℝ))) -
+          6 * (951145802866699611319 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (39553366530621 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (39553366530621 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (39553366530621 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_542 :
+    ((481868114864098952669 / 1000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(117612437523516345031 / 250000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1573653375420513 / 2000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3333283300374232651811 / 10000000000000000000000 : ℝ))) -
+              6 * (118790589224961794759 / 1250000000000000000000 : ℝ)))) ≤
+      Vfield (19811079567779 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (19811079567779 / 32000000000000 : ℝ)) =
+      Real.log (51811079567779 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((19811079567779 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (19991079567779 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (481868114864098952669 / 1000000000000000000000 : ℝ) ≤
+      Real.log (51811079567779 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (19991079567779 / 32000000000000 : ℝ) ≤
+      (-(117612437523516345031 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (19991079567779 / 32000000000000 : ℝ) = (19991079567779 / 16000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (19991079567779 / 16000000000000 : ℝ) ≤
+            (55674357551483654969 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1573653375420513 / 2000000000000000 : ℝ) ≤ Real.sqrt (19811079567779 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (19811079567779 / 32000000000000 : ℝ) ≤ (983533359637821 / 1250000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (19811079567779 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau856 :
+    Real.arctan (3934133438551284 / 11362185623850293 : ℝ) ≤
+      (3333283300374232651811 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau855 :
+    Real.arctan (983533359637821 / 1250000000000000 : ℝ) ≤
+      (2 * (3333283300374232651811 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6362185623850293 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (983533359637821 / 1250000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (983533359637821 / 1250000000000000 : ℝ) /
+                (1 + (6362185623850293 / 5000000000000000 : ℝ)) =
+              (3934133438551284 / 11362185623850293 : ℝ)
+            by
+        norm_num];
+        exact hau856)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3333283300374232651811 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (1250000000000000 / 983533359637821 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1250000000000000 / 983533359637821 : ℝ) =
+              (983533359637821 / 1250000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau855)
+  have hz1 :
+    (1250000000000000 / 983533359637821 : ℝ) ≤
+      1 / Real.sqrt (19811079567779 / 32000000000000 : ℝ) := by
+    rw [show
+        (1250000000000000 / 983533359637821 : ℝ) = 1 / (983533359637821 / 1250000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (50000000000000 / 524551125140171 : ℝ) ≤
+      (118790589224961794759 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (19811079567779 / 32000000000000 : ℝ) ≤
+      (50000000000000 / 524551125140171 : ℝ) := by
+    rw [show
+        (50000000000000 / 524551125140171 : ℝ) =
+          (3 / 40) / (1573653375420513 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3333283300374232651811 / 10000000000000000000000 : ℝ))) -
+        6 * (118790589224961794759 / 1250000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (19811079567779 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (19811079567779 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1573653375420513 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3333283300374232651811 / 10000000000000000000000 : ℝ))) -
+          6 * (118790589224961794759 / 1250000000000000000000 : ℝ))) ≤
+      Real.sqrt (19811079567779 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (19811079567779 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (19811079567779 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_543 :
+    ((603164717348652311 / 1250000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(4687306460600976406899 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((7875094418133881 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2 * (66707820858295245557 / 200000000000000000000 : ℝ))) -
+              6 * (237376436882747463663 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (7938190348099 / 12800000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (7938190348099 / 12800000000000 : ℝ)) =
+      Real.log (20738190348099 / 12800000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((7938190348099 / 12800000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (8010190348099 / 12800000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (603164717348652311 / 1250000000000000000 : ℝ) ≤
+      Real.log (20738190348099 / 12800000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (8010190348099 / 12800000000000 : ℝ) ≤
+      (-(4687306460600976406899 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (8010190348099 / 12800000000000 : ℝ) = (8010190348099 / 6400000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8010190348099 / 6400000000000 : ℝ) ≤
+            (2244165342399023593101 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (7875094418133881 / 10000000000000000 : ℝ) ≤ Real.sqrt (7938190348099 / 12800000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (7938190348099 / 12800000000000 : ℝ) ≤ (1968773604533471 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (7938190348099 / 12800000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau858 :
+    Real.arctan (7875094418133884 / 22728594270166813 : ℝ) ≤
+      (66707820858295245557 / 200000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau857 :
+    Real.arctan (1968773604533471 / 2500000000000000 : ℝ) ≤
+      (2 * (66707820858295245557 / 200000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12728594270166813 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1968773604533471 / 2500000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1968773604533471 / 2500000000000000 : ℝ) /
+                (1 + (12728594270166813 / 10000000000000000 : ℝ)) =
+              (7875094418133884 / 22728594270166813 : ℝ)
+            by
+        norm_num];
+        exact hau858)
+  have hat1 :
+    (Real.pi / 2 - (2 * (66707820858295245557 / 200000000000000000000 : ℝ))) ≤
+      Real.arctan (2500000000000000 / 1968773604533471 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 1968773604533471 : ℝ) =
+              (1968773604533471 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau857)
+  have hz1 :
+    (2500000000000000 / 1968773604533471 : ℝ) ≤
+      1 / Real.sqrt (7938190348099 / 12800000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 1968773604533471 : ℝ) = 1 / (1968773604533471 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 7875094418133881 : ℝ) ≤
+      (237376436882747463663 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (7938190348099 / 12800000000000 : ℝ) ≤
+      (750000000000000 / 7875094418133881 : ℝ) := by
+    rw [show
+        (750000000000000 / 7875094418133881 : ℝ) =
+          (3 / 40) / (7875094418133881 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (66707820858295245557 / 200000000000000000000 : ℝ))) -
+        6 * (237376436882747463663 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (7938190348099 / 12800000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (7938190348099 / 12800000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((7875094418133881 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (66707820858295245557 / 200000000000000000000 : ℝ))) -
+          6 * (237376436882747463663 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (7938190348099 / 12800000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (7938190348099 / 12800000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (7938190348099 / 12800000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_544 :
+    ((603993740928188690151 / 1250000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(4670144922737522464169 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((985239505619521 / 1250000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3337495562985116148511 / 10000000000000000000000 : ℝ))) -
+              6 * (948688894929314443929 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (4969968043179 / 8000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (4969968043179 / 8000000000000 : ℝ)) =
+      Real.log (12969968043179 / 8000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((4969968043179 / 8000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (5014968043179 / 8000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (603993740928188690151 / 1250000000000000000000 : ℝ) ≤
+      Real.log (12969968043179 / 8000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (5014968043179 / 8000000000000 : ℝ) ≤
+      (-(4670144922737522464169 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (5014968043179 / 8000000000000 : ℝ) = (5014968043179 / 4000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5014968043179 / 4000000000000 : ℝ) ≤
+            (2261326880262477535831 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (985239505619521 / 1250000000000000 : ℝ) ≤ Real.sqrt (4969968043179 / 8000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (4969968043179 / 8000000000000 : ℝ) ≤ (7881916044956171 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (4969968043179 / 8000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau860 :
+    Real.arctan (7881916044956171 / 22732815892006665 : ℝ) ≤
+      (3337495562985116148511 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau859 :
+    Real.arctan (7881916044956171 / 10000000000000000 : ℝ) ≤
+      (2 * (3337495562985116148511 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (2546563178401333 / 2000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (7881916044956171 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (7881916044956171 / 10000000000000000 : ℝ) /
+                (1 + (2546563178401333 / 2000000000000000 : ℝ)) =
+              (7881916044956171 / 22732815892006665 : ℝ)
+            by
+        norm_num];
+        exact hau860)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3337495562985116148511 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 7881916044956171 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 7881916044956171 : ℝ) =
+              (7881916044956171 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau859)
+  have hz1 :
+    (10000000000000000 / 7881916044956171 : ℝ) ≤
+      1 / Real.sqrt (4969968043179 / 8000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 7881916044956171 : ℝ) =
+          1 / (7881916044956171 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (93750000000000 / 985239505619521 : ℝ) ≤
+      (948688894929314443929 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (4969968043179 / 8000000000000 : ℝ) ≤
+      (93750000000000 / 985239505619521 : ℝ) := by
+    rw [show
+        (93750000000000 / 985239505619521 : ℝ) =
+          (3 / 40) / (985239505619521 / 1250000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3337495562985116148511 / 10000000000000000000000 : ℝ))) -
+        6 * (948688894929314443929 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (4969968043179 / 8000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (4969968043179 / 8000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((985239505619521 / 1250000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3337495562985116148511 / 10000000000000000000000 : ℝ))) -
+          6 * (948688894929314443929 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (4969968043179 / 8000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (4969968043179 / 8000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (4969968043179 / 8000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_545 :
+    ((38708621763074600159 / 80000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(4653012786262152590651 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1577746354582403 / 2000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3339596869587028725083 / 10000000000000000000000 : ℝ))) -
+              6 * (947874146918292524199 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (39828536950369 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (39828536950369 / 64000000000000 : ℝ)) =
+      Real.log (103828536950369 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((39828536950369 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (40188536950369 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (38708621763074600159 / 80000000000000000000 : ℝ) ≤
+      Real.log (103828536950369 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (40188536950369 / 64000000000000 : ℝ) ≤
+      (-(4653012786262152590651 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (40188536950369 / 64000000000000 : ℝ) = (40188536950369 / 32000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (40188536950369 / 32000000000000 : ℝ) ≤
+            (2278459016737847409349 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1577746354582403 / 2000000000000000 : ℝ) ≤ Real.sqrt (39828536950369 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (39828536950369 / 64000000000000 : ℝ) ≤ (3944365886456009 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (39828536950369 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau862 :
+    Real.arctan (3944365886456009 / 11368518057306416 : ℝ) ≤
+      (3339596869587028725083 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau861 :
+    Real.arctan (3944365886456009 / 5000000000000000 : ℝ) ≤
+      (2 * (3339596869587028725083 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (398032378581651 / 312500000000000 : ℝ) ≤
+          Real.sqrt (1 + (3944365886456009 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (3944365886456009 / 5000000000000000 : ℝ) /
+                (1 + (398032378581651 / 312500000000000 : ℝ)) =
+              (3944365886456009 / 11368518057306416 : ℝ)
+            by
+        norm_num];
+        exact hau862)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3339596869587028725083 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 3944365886456009 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 3944365886456009 : ℝ) =
+              (3944365886456009 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau861)
+  have hz1 :
+    (5000000000000000 / 3944365886456009 : ℝ) ≤
+      1 / Real.sqrt (39828536950369 / 64000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 3944365886456009 : ℝ) = 1 / (3944365886456009 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (150000000000000 / 1577746354582403 : ℝ) ≤
+      (947874146918292524199 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (39828536950369 / 64000000000000 : ℝ) ≤
+      (150000000000000 / 1577746354582403 : ℝ) := by
+    rw [show
+        (150000000000000 / 1577746354582403 : ℝ) =
+          (3 / 40) / (1577746354582403 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3339596869587028725083 / 10000000000000000000000 : ℝ))) -
+        6 * (947874146918292524199 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (39828536950369 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (39828536950369 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1577746354582403 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3339596869587028725083 / 10000000000000000000000 : ℝ))) -
+          6 * (947874146918292524199 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (39828536950369 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (39828536950369 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (39828536950369 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_546 :
+    ((4845201123488534177387 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2317954975302697975151 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((7895541617277791 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1670847485841972874563 / 5000000000000000000000 : ℝ))) -
+              6 * (947061494476020427973 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (19948664777653 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (19948664777653 / 32000000000000 : ℝ)) =
+      Real.log (51948664777653 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((19948664777653 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (20128664777653 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (4845201123488534177387 / 10000000000000000000000 : ℝ) ≤
+      Real.log (51948664777653 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (20128664777653 / 32000000000000 : ℝ) ≤
+      (-(2317954975302697975151 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (20128664777653 / 32000000000000 : ℝ) = (20128664777653 / 16000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20128664777653 / 16000000000000 : ℝ) ≤
+            (1147780926197302024849 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (7895541617277791 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (19948664777653 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (19948664777653 / 32000000000000 : ℝ) ≤ (3947770808638897 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (19948664777653 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau864 :
+    Real.arctan (7895541617277794 / 22741254939375697 : ℝ) ≤
+      (1670847485841972874563 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau863 :
+    Real.arctan (3947770808638897 / 5000000000000000 : ℝ) ≤
+      (2 * (1670847485841972874563 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12741254939375697 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (3947770808638897 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (3947770808638897 / 5000000000000000 : ℝ) /
+                (1 + (12741254939375697 / 10000000000000000 : ℝ)) =
+              (7895541617277794 / 22741254939375697 : ℝ)
+            by
+        norm_num];
+        exact hau864)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1670847485841972874563 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 3947770808638897 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 3947770808638897 : ℝ) =
+              (3947770808638897 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau863)
+  have hz1 :
+    (5000000000000000 / 3947770808638897 : ℝ) ≤
+      1 / Real.sqrt (19948664777653 / 32000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 3947770808638897 : ℝ) = 1 / (3947770808638897 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 7895541617277791 : ℝ) ≤
+      (947061494476020427973 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (19948664777653 / 32000000000000 : ℝ) ≤
+      (750000000000000 / 7895541617277791 : ℝ) := by
+    rw [show
+        (750000000000000 / 7895541617277791 : ℝ) =
+          (3 / 40) / (7895541617277791 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1670847485841972874563 / 5000000000000000000000 : ℝ))) -
+        6 * (947061494476020427973 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (19948664777653 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (19948664777653 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((7895541617277791 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1670847485841972874563 / 5000000000000000000000 : ℝ))) -
+          6 * (947061494476020427973 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (19948664777653 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (19948664777653 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (19948664777653 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_547 :
+    ((4851820142549443681207 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(4618836315712907967727 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1580469118652809 / 2000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1671894939100624160823 / 5000000000000000000000 : ℝ))) -
+              6 * (946250928634646755839 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (39966122160243 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (39966122160243 / 64000000000000 : ℝ)) =
+      Real.log (103966122160243 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((39966122160243 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (40326122160243 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (4851820142549443681207 / 10000000000000000000000 : ℝ) ≤
+      Real.log (103966122160243 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (40326122160243 / 64000000000000 : ℝ) ≤
+      (-(4618836315712907967727 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (40326122160243 / 64000000000000 : ℝ) = (40326122160243 / 32000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (40326122160243 / 32000000000000 : ℝ) ≤
+            (2312635487287092032273 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1580469118652809 / 2000000000000000 : ℝ) ≤ Real.sqrt (39966122160243 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (39966122160243 / 64000000000000 : ℝ) ≤ (493896599579003 / 625000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (39966122160243 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau866 :
+    Real.arctan (7902345593264048 / 22745472367683345 : ℝ) ≤
+      (1671894939100624160823 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau865 :
+    Real.arctan (493896599579003 / 625000000000000 : ℝ) ≤
+      (2 * (1671894939100624160823 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (2549094473536669 / 2000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (493896599579003 / 625000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (493896599579003 / 625000000000000 : ℝ) /
+                (1 + (2549094473536669 / 2000000000000000 : ℝ)) =
+              (7902345593264048 / 22745472367683345 : ℝ)
+            by
+        norm_num];
+        exact hau866)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1671894939100624160823 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (625000000000000 / 493896599579003 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (625000000000000 / 493896599579003 : ℝ) =
+              (493896599579003 / 625000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau865)
+  have hz1 :
+    (625000000000000 / 493896599579003 : ℝ) ≤
+      1 / Real.sqrt (39966122160243 / 64000000000000 : ℝ) := by
+    rw [show (625000000000000 / 493896599579003 : ℝ) = 1 / (493896599579003 / 625000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (150000000000000 / 1580469118652809 : ℝ) ≤
+      (946250928634646755839 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (39966122160243 / 64000000000000 : ℝ) ≤
+      (150000000000000 / 1580469118652809 : ℝ) := by
+    rw [show
+        (150000000000000 / 1580469118652809 : ℝ) =
+          (3 / 40) / (1580469118652809 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1671894939100624160823 / 5000000000000000000000 : ℝ))) -
+        6 * (946250928634646755839 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (39966122160243 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (39966122160243 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1580469118652809 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1671894939100624160823 / 5000000000000000000000 : ℝ))) -
+          6 * (946250928634646755839 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (39966122160243 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (39966122160243 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (39966122160243 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_548 :
+    ((2429217391683414152803 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(4601791782041958375991 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1581828743203179 / 2000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3345881598026473956461 / 10000000000000000000000 : ℝ))) -
+              6 * (236360610119989398587 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (2001745738259 / 3200000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (2001745738259 / 3200000000000 : ℝ)) =
+      Real.log (5201745738259 / 3200000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((2001745738259 / 3200000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (2019745738259 / 3200000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2429217391683414152803 / 5000000000000000000000 : ℝ) ≤
+      Real.log (5201745738259 / 3200000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (2019745738259 / 3200000000000 : ℝ) ≤
+      (-(4601791782041958375991 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (2019745738259 / 3200000000000 : ℝ) = (2019745738259 / 1600000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2019745738259 / 1600000000000 : ℝ) ≤
+            (2329680020958041624009 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1581828743203179 / 2000000000000000 : ℝ) ≤ Real.sqrt (2001745738259 / 3200000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (2001745738259 / 3200000000000 : ℝ) ≤ (3954571858007949 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (2001745738259 / 3200000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau868 :
+    Real.arctan (2636381238671966 / 7583229466973853 : ℝ) ≤
+      (3345881598026473956461 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau867 :
+    Real.arctan (3954571858007949 / 5000000000000000 : ℝ) ≤
+      (2 * (3345881598026473956461 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12749688400921559 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (3954571858007949 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (3954571858007949 / 5000000000000000 : ℝ) /
+                (1 + (12749688400921559 / 10000000000000000 : ℝ)) =
+              (2636381238671966 / 7583229466973853 : ℝ)
+            by
+        norm_num];
+        exact hau868)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3345881598026473956461 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 3954571858007949 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 3954571858007949 : ℝ) =
+              (3954571858007949 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau867)
+  have hz1 :
+    (5000000000000000 / 3954571858007949 : ℝ) ≤
+      1 / Real.sqrt (2001745738259 / 3200000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 3954571858007949 : ℝ) = 1 / (3954571858007949 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (50000000000000 / 527276247734393 : ℝ) ≤
+      (236360610119989398587 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (2001745738259 / 3200000000000 : ℝ) ≤
+      (50000000000000 / 527276247734393 : ℝ) := by
+    rw [show
+        (50000000000000 / 527276247734393 : ℝ) =
+          (3 / 40) / (1581828743203179 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3345881598026473956461 / 10000000000000000000000 : ℝ))) -
+        6 * (236360610119989398587 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (2001745738259 / 3200000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (2001745738259 / 3200000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1581828743203179 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3345881598026473956461 / 10000000000000000000000 : ℝ))) -
+          6 * (236360610119989398587 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (2001745738259 / 3200000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (2001745738259 / 3200000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (2001745738259 / 3200000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_549 :
+    ((2432522525864480701643 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(4584776250557949002001 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((7915936000613433 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (418496267501192554361 / 1250000000000000000000 : ℝ))) -
+              6 * (472318010575481681801 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (40103707370117 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (40103707370117 / 64000000000000 : ℝ)) =
+      Real.log (104103707370117 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((40103707370117 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (40463707370117 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2432522525864480701643 / 5000000000000000000000 : ℝ) ≤
+      Real.log (104103707370117 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (40463707370117 / 64000000000000 : ℝ) ≤
+      (-(4584776250557949002001 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (40463707370117 / 64000000000000 : ℝ) = (40463707370117 / 32000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (40463707370117 / 32000000000000 : ℝ) ≤
+            (2346695552442050997999 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (7915936000613433 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (40103707370117 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (40103707370117 / 64000000000000 : ℝ) ≤ (1978984000153359 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (40103707370117 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau870 :
+    Real.arctan (3957968000306718 / 11376951520236919 : ℝ) ≤
+      (418496267501192554361 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau869 :
+    Real.arctan (1978984000153359 / 2500000000000000 : ℝ) ≤
+      (2 * (418496267501192554361 / 1250000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6376951520236919 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1978984000153359 / 2500000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1978984000153359 / 2500000000000000 : ℝ) /
+                (1 + (6376951520236919 / 5000000000000000 : ℝ)) =
+              (3957968000306718 / 11376951520236919 : ℝ)
+            by
+        norm_num];
+        exact hau870)
+  have hat1 :
+    (Real.pi / 2 - (2 * (418496267501192554361 / 1250000000000000000000 : ℝ))) ≤
+      Real.arctan (2500000000000000 / 1978984000153359 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 1978984000153359 : ℝ) =
+              (1978984000153359 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau869)
+  have hz1 :
+    (2500000000000000 / 1978984000153359 : ℝ) ≤
+      1 / Real.sqrt (40103707370117 / 64000000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 1978984000153359 : ℝ) = 1 / (1978984000153359 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 2638645333537811 : ℝ) ≤
+      (472318010575481681801 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (40103707370117 / 64000000000000 : ℝ) ≤
+      (250000000000000 / 2638645333537811 : ℝ) := by
+    rw [show
+        (250000000000000 / 2638645333537811 : ℝ) =
+          (3 / 40) / (7915936000613433 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (418496267501192554361 / 1250000000000000000000 : ℝ))) -
+        6 * (472318010575481681801 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (40103707370117 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (40103707370117 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((7915936000613433 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (418496267501192554361 / 1250000000000000000000 : ℝ))) -
+          6 * (472318010575481681801 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (40103707370117 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (40103707370117 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (40103707370117 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_550 :
+    ((4871650953412645296287 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(4567789622730961196307 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((7922722462072103 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3350055512962961865401 / 10000000000000000000000 : ℝ))) -
+              6 * (943831661839492373971 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (20086249987527 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (20086249987527 / 32000000000000 : ℝ)) =
+      Real.log (52086249987527 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((20086249987527 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (20266249987527 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (4871650953412645296287 / 10000000000000000000000 : ℝ) ≤
+      Real.log (52086249987527 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (20266249987527 / 32000000000000 : ℝ) ≤
+      (-(4567789622730961196307 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (20266249987527 / 32000000000000 : ℝ) = (20266249987527 / 16000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20266249987527 / 16000000000000 : ℝ) ≤
+            (2363682180269038803693 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (7922722462072103 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (20086249987527 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (20086249987527 / 32000000000000 : ℝ) ≤ (3961361231036053 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (20086249987527 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau872 :
+    Real.arctan (7922722462072106 / 22758116287721391 : ℝ) ≤
+      (3350055512962961865401 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau871 :
+    Real.arctan (3961361231036053 / 5000000000000000 : ℝ) ≤
+      (2 * (3350055512962961865401 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12758116287721391 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (3961361231036053 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (3961361231036053 / 5000000000000000 : ℝ) /
+                (1 + (12758116287721391 / 10000000000000000 : ℝ)) =
+              (7922722462072106 / 22758116287721391 : ℝ)
+            by
+        norm_num];
+        exact hau872)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3350055512962961865401 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 3961361231036053 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 3961361231036053 : ℝ) =
+              (3961361231036053 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau871)
+  have hz1 :
+    (5000000000000000 / 3961361231036053 : ℝ) ≤
+      1 / Real.sqrt (20086249987527 / 32000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 3961361231036053 : ℝ) = 1 / (3961361231036053 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 7922722462072103 : ℝ) ≤
+      (943831661839492373971 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (20086249987527 / 32000000000000 : ℝ) ≤
+      (750000000000000 / 7922722462072103 : ℝ) := by
+    rw [show
+        (750000000000000 / 7922722462072103 : ℝ) =
+          (3 / 40) / (7922722462072103 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3350055512962961865401 / 10000000000000000000000 : ℝ))) -
+        6 * (943831661839492373971 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (20086249987527 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (20086249987527 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((7922722462072103 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3350055512962961865401 / 10000000000000000000000 : ℝ))) -
+          6 * (943831661839492373971 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (20086249987527 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (20086249987527 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (20086249987527 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_551 :
+    ((609781561772905195823 / 1250000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1137707950133083130891 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((7929503115343099 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3352137725662069253959 / 10000000000000000000000 : ℝ))) -
+              6 * (471514676894892731199 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (40241292579991 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (40241292579991 / 64000000000000 : ℝ)) =
+      Real.log (104241292579991 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((40241292579991 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (40601292579991 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (609781561772905195823 / 1250000000000000000000 : ℝ) ≤
+      Real.log (104241292579991 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (40601292579991 / 64000000000000 : ℝ) ≤
+      (-(1137707950133083130891 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (40601292579991 / 64000000000000 : ℝ) = (40601292579991 / 32000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (40601292579991 / 32000000000000 : ℝ) ≤
+            (595160000616916869109 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (7929503115343099 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (40241292579991 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (40241292579991 / 64000000000000 : ℝ) ≤ (3964751557671551 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (40241292579991 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau874 :
+    Real.arctan (3964751557671551 / 11381164072021576 : ℝ) ≤
+      (3352137725662069253959 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau873 :
+    Real.arctan (3964751557671551 / 5000000000000000 : ℝ) ≤
+      (2 * (3352137725662069253959 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (797645509002697 / 625000000000000 : ℝ) ≤
+          Real.sqrt (1 + (3964751557671551 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (3964751557671551 / 5000000000000000 : ℝ) /
+                (1 + (797645509002697 / 625000000000000 : ℝ)) =
+              (3964751557671551 / 11381164072021576 : ℝ)
+            by
+        norm_num];
+        exact hau874)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3352137725662069253959 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 3964751557671551 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 3964751557671551 : ℝ) =
+              (3964751557671551 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau873)
+  have hz1 :
+    (5000000000000000 / 3964751557671551 : ℝ) ≤
+      1 / Real.sqrt (40241292579991 / 64000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 3964751557671551 : ℝ) = 1 / (3964751557671551 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 7929503115343099 : ℝ) ≤
+      (471514676894892731199 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (40241292579991 / 64000000000000 : ℝ) ≤
+      (750000000000000 / 7929503115343099 : ℝ) := by
+    rw [show
+        (750000000000000 / 7929503115343099 : ℝ) =
+          (3 / 40) / (7929503115343099 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3352137725662069253959 / 10000000000000000000000 : ℝ))) -
+        6 * (471514676894892731199 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (40241292579991 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (40241292579991 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((7929503115343099 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3352137725662069253959 / 10000000000000000000000 : ℝ))) -
+          6 * (471514676894892731199 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (40241292579991 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (40241292579991 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (40241292579991 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_552 :
+    ((2442424839897350623381 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(4533902686431262429811 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((7936277975313741 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3354216786845224709769 / 10000000000000000000000 : ℝ))) -
+              6 * (942229088298096658871 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (1259690162029 / 2000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1259690162029 / 2000000000000 : ℝ)) =
+      Real.log (3259690162029 / 2000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1259690162029 / 2000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1270940162029 / 2000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2442424839897350623381 / 5000000000000000000000 : ℝ) ≤
+      Real.log (3259690162029 / 2000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1270940162029 / 2000000000000 : ℝ) ≤
+      (-(4533902686431262429811 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (1270940162029 / 2000000000000 : ℝ) = (1270940162029 / 1000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1270940162029 / 1000000000000 : ℝ) ≤
+            (2397569116568737570189 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (7936277975313741 / 10000000000000000 : ℝ) ≤ Real.sqrt (1259690162029 / 2000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1259690162029 / 2000000000000 : ℝ) ≤ (496017373457109 / 625000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1259690162029 / 2000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau876 :
+    Real.arctan (7936277975313744 / 22766538610815775 : ℝ) ≤
+      (3354216786845224709769 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau875 :
+    Real.arctan (496017373457109 / 625000000000000 : ℝ) ≤
+      (2 * (3354216786845224709769 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (510661544432631 / 400000000000000 : ℝ) ≤
+          Real.sqrt (1 + (496017373457109 / 625000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (496017373457109 / 625000000000000 : ℝ) /
+                (1 + (510661544432631 / 400000000000000 : ℝ)) =
+              (7936277975313744 / 22766538610815775 : ℝ)
+            by
+        norm_num];
+        exact hau876)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3354216786845224709769 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (625000000000000 / 496017373457109 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (625000000000000 / 496017373457109 : ℝ) =
+              (496017373457109 / 625000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau875)
+  have hz1 :
+    (625000000000000 / 496017373457109 : ℝ) ≤ 1 / Real.sqrt (1259690162029 / 2000000000000 : ℝ) :=
+  by
+    rw [show (625000000000000 / 496017373457109 : ℝ) = 1 / (496017373457109 / 625000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 2645425991771247 : ℝ) ≤
+      (942229088298096658871 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1259690162029 / 2000000000000 : ℝ) ≤
+      (250000000000000 / 2645425991771247 : ℝ) := by
+    rw [show
+        (250000000000000 / 2645425991771247 : ℝ) =
+          (3 / 40) / (7936277975313741 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3354216786845224709769 / 10000000000000000000000 : ℝ))) -
+        6 * (942229088298096658871 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1259690162029 / 2000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1259690162029 / 2000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((7936277975313741 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3354216786845224709769 / 10000000000000000000000 : ℝ))) -
+          6 * (942229088298096658871 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (1259690162029 / 2000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1259690162029 / 2000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1259690162029 / 2000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/V46.lean b/LeanPool/Zeta5Irrational/Table/V46.lean
new file mode 100644
index 0000000000..fe9c667ead
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/V46.lean
@@ -0,0 +1,1441 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Energy
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # V46: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_553 :
+    ((2445721257994797455589 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(4517002183391446592761 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3971523528403931 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3356292705214036771549 / 10000000000000000000000 : ℝ))) -
+              6 * (941430856712295929961 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (8075775557973 / 12800000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (8075775557973 / 12800000000000 : ℝ)) =
+      Real.log (20875775557973 / 12800000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((8075775557973 / 12800000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (8147775557973 / 12800000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2445721257994797455589 / 5000000000000000000000 : ℝ) ≤
+      Real.log (20875775557973 / 12800000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (8147775557973 / 12800000000000 : ℝ) ≤
+      (-(4517002183391446592761 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (8147775557973 / 12800000000000 : ℝ) = (8147775557973 / 6400000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8147775557973 / 6400000000000 : ℝ) ≤
+            (2414469619608553407239 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3971523528403931 / 5000000000000000 : ℝ) ≤ Real.sqrt (8075775557973 / 12800000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (8075775557973 / 12800000000000 : ℝ) ≤ (1588609411361573 / 2000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (8075775557973 / 12800000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau878 :
+    Real.arctan (7943047056807865 / 22770747689413649 : ℝ) ≤
+      (3356292705214036771549 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau877 :
+    Real.arctan (1588609411361573 / 2000000000000000 : ℝ) ≤
+      (2 * (3356292705214036771549 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12770747689413649 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1588609411361573 / 2000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1588609411361573 / 2000000000000000 : ℝ) /
+                (1 + (12770747689413649 / 10000000000000000 : ℝ)) =
+              (7943047056807865 / 22770747689413649 : ℝ)
+            by
+        norm_num];
+        exact hau878)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3356292705214036771549 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (2000000000000000 / 1588609411361573 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2000000000000000 / 1588609411361573 : ℝ) =
+              (1588609411361573 / 2000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau877)
+  have hz1 :
+    (2000000000000000 / 1588609411361573 : ℝ) ≤
+      1 / Real.sqrt (8075775557973 / 12800000000000 : ℝ) := by
+    rw [show
+        (2000000000000000 / 1588609411361573 : ℝ) = 1 / (1588609411361573 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 3971523528403931 : ℝ) ≤
+      (941430856712295929961 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (8075775557973 / 12800000000000 : ℝ) ≤
+      (375000000000000 / 3971523528403931 : ℝ) := by
+    rw [show
+        (375000000000000 / 3971523528403931 : ℝ) =
+          (3 / 40) / (3971523528403931 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3356292705214036771549 / 10000000000000000000000 : ℝ))) -
+        6 * (941430856712295929961 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (8075775557973 / 12800000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (8075775557973 / 12800000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3971523528403931 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3356292705214036771549 / 10000000000000000000000 : ℝ))) -
+          6 * (941430856712295929961 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (8075775557973 / 12800000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (8075775557973 / 12800000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (8075775557973 / 12800000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_554 :
+    ((2449015504249571333993 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(4500130194867739664721 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((7949810374586183 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (671673097886714578427 / 2000000000000000000000 : ℝ))) -
+              6 * (37625386017259067461 / 400000000000000000000 : ℝ)))) ≤
+      Vfield (20223835197401 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (20223835197401 / 32000000000000 : ℝ)) =
+      Real.log (52223835197401 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((20223835197401 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (20403835197401 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2449015504249571333993 / 5000000000000000000000 : ℝ) ≤
+      Real.log (52223835197401 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (20403835197401 / 32000000000000 : ℝ) ≤
+      (-(4500130194867739664721 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (20403835197401 / 32000000000000 : ℝ) = (20403835197401 / 16000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20403835197401 / 16000000000000 : ℝ) ≤
+            (2431341608132260335279 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (7949810374586183 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (20223835197401 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (20223835197401 / 32000000000000 : ℝ) ≤ (3974905187293093 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (20223835197401 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau880 :
+    Real.arctan (3974905187293093 / 11387477690604448 : ℝ) ≤
+      (671673097886714578427 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau879 :
+    Real.arctan (3974905187293093 / 5000000000000000 : ℝ) ≤
+      (2 * (671673097886714578427 / 2000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (199608677831389 / 156250000000000 : ℝ) ≤
+          Real.sqrt (1 + (3974905187293093 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (3974905187293093 / 5000000000000000 : ℝ) /
+                (1 + (199608677831389 / 156250000000000 : ℝ)) =
+              (3974905187293093 / 11387477690604448 : ℝ)
+            by
+        norm_num];
+        exact hau880)
+  have hat1 :
+    (Real.pi / 2 - (2 * (671673097886714578427 / 2000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 3974905187293093 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 3974905187293093 : ℝ) =
+              (3974905187293093 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau879)
+  have hz1 :
+    (5000000000000000 / 3974905187293093 : ℝ) ≤
+      1 / Real.sqrt (20223835197401 / 32000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 3974905187293093 : ℝ) = 1 / (3974905187293093 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 7949810374586183 : ℝ) ≤
+      (37625386017259067461 / 400000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (20223835197401 / 32000000000000 : ℝ) ≤
+      (750000000000000 / 7949810374586183 : ℝ) := by
+    rw [show
+        (750000000000000 / 7949810374586183 : ℝ) =
+          (3 / 40) / (7949810374586183 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (671673097886714578427 / 2000000000000000000000 : ℝ))) -
+        6 * (37625386017259067461 / 400000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (20223835197401 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (20223835197401 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((7949810374586183 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (671673097886714578427 / 2000000000000000000000 : ℝ))) -
+          6 * (37625386017259067461 / 400000000000000000000 : ℝ))) ≤
+      Real.sqrt (20223835197401 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (20223835197401 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (20223835197401 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_555 :
+    ((4904615163043244052437 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(4483286624802846121209 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((7956567943346689 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3360435148132570495601 / 10000000000000000000000 : ℝ))) -
+              6 * (187968092181113295817 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (40516462999739 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (40516462999739 / 64000000000000 : ℝ)) =
+      Real.log (104516462999739 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((40516462999739 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (40876462999739 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (4904615163043244052437 / 10000000000000000000000 : ℝ) ≤
+      Real.log (104516462999739 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (40876462999739 / 64000000000000 : ℝ) ≤
+      (-(4483286624802846121209 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (40876462999739 / 64000000000000 : ℝ) = (40876462999739 / 32000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (40876462999739 / 32000000000000 : ℝ) ≤
+            (2448185178197153878791 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (7956567943346689 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (40516462999739 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (40516462999739 / 64000000000000 : ℝ) ≤ (1989141985836673 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (40516462999739 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau882 :
+    Real.arctan (7956567943346692 / 22779161687571379 : ℝ) ≤
+      (3360435148132570495601 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau881 :
+    Real.arctan (1989141985836673 / 2500000000000000 : ℝ) ≤
+      (2 * (3360435148132570495601 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12779161687571379 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1989141985836673 / 2500000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1989141985836673 / 2500000000000000 : ℝ) /
+                (1 + (12779161687571379 / 10000000000000000 : ℝ)) =
+              (7956567943346692 / 22779161687571379 : ℝ)
+            by
+        norm_num];
+        exact hau882)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3360435148132570495601 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (2500000000000000 / 1989141985836673 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 1989141985836673 : ℝ) =
+              (1989141985836673 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau881)
+  have hz1 :
+    (2500000000000000 / 1989141985836673 : ℝ) ≤
+      1 / Real.sqrt (40516462999739 / 64000000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 1989141985836673 : ℝ) = 1 / (1989141985836673 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 7956567943346689 : ℝ) ≤
+      (187968092181113295817 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (40516462999739 / 64000000000000 : ℝ) ≤
+      (750000000000000 / 7956567943346689 : ℝ) := by
+    rw [show
+        (750000000000000 / 7956567943346689 : ℝ) =
+          (3 / 40) / (7956567943346689 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3360435148132570495601 / 10000000000000000000000 : ℝ))) -
+        6 * (187968092181113295817 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (40516462999739 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (40516462999739 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((7956567943346689 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3360435148132570495601 / 10000000000000000000000 : ℝ))) -
+          6 * (187968092181113295817 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (40516462999739 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (40516462999739 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (40516462999739 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_556 :
+    ((2455597492665253910909 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2233235688812019465853 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1592663955545001 / 2000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3362501689903647580097 / 10000000000000000000000 : ℝ))) -
+              6 * (234762069908735241679 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (10146313901169 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (10146313901169 / 16000000000000 : ℝ)) =
+      Real.log (26146313901169 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((10146313901169 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (10236313901169 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2455597492665253910909 / 5000000000000000000000 : ℝ) ≤
+      Real.log (26146313901169 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (10236313901169 / 16000000000000 : ℝ) ≤
+      (-(2233235688812019465853 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (10236313901169 / 16000000000000 : ℝ) = (10236313901169 / 8000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10236313901169 / 8000000000000 : ℝ) ≤
+            (1232500212687980534147 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1592663955545001 / 2000000000000000 : ℝ) ≤ Real.sqrt (10146313901169 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (10146313901169 / 16000000000000 : ℝ) ≤ (497707486107813 / 625000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (10146313901169 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau884 :
+    Real.arctan (3981659888862504 / 11391683304934355 : ℝ) ≤
+      (3362501689903647580097 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau883 :
+    Real.arctan (497707486107813 / 625000000000000 : ℝ) ≤
+      (2 * (3362501689903647580097 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (1278336660986871 / 1000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (497707486107813 / 625000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (497707486107813 / 625000000000000 : ℝ) /
+                (1 + (1278336660986871 / 1000000000000000 : ℝ)) =
+              (3981659888862504 / 11391683304934355 : ℝ)
+            by
+        norm_num];
+        exact hau884)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3362501689903647580097 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (625000000000000 / 497707486107813 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (625000000000000 / 497707486107813 : ℝ) =
+              (497707486107813 / 625000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau883)
+  have hz1 :
+    (625000000000000 / 497707486107813 : ℝ) ≤
+      1 / Real.sqrt (10146313901169 / 16000000000000 : ℝ) := by
+    rw [show (625000000000000 / 497707486107813 : ℝ) = 1 / (497707486107813 / 625000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (50000000000000 / 530887985181667 : ℝ) ≤
+      (234762069908735241679 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (10146313901169 / 16000000000000 : ℝ) ≤
+      (50000000000000 / 530887985181667 : ℝ) := by
+    rw [show
+        (50000000000000 / 530887985181667 : ℝ) =
+          (3 / 40) / (1592663955545001 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3362501689903647580097 / 10000000000000000000000 : ℝ))) -
+        6 * (234762069908735241679 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (10146313901169 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (10146313901169 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1592663955545001 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3362501689903647580097 / 10000000000000000000000 : ℝ))) -
+          6 * (234762069908735241679 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (10146313901169 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (10146313901169 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (10146313901169 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_557 :
+    ((4917770481058281652449 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(4449684358239905772299 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((7970065892294761 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (672913024660701932149 / 2000000000000000000000 : ℝ))) -
+              6 * (117282262271255301799 / 1250000000000000000000 : ℝ)))) ≤
+      Vfield (40654048209613 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (40654048209613 / 64000000000000 : ℝ)) =
+      Real.log (104654048209613 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((40654048209613 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (41014048209613 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (4917770481058281652449 / 10000000000000000000000 : ℝ) ≤
+      Real.log (104654048209613 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (41014048209613 / 64000000000000 : ℝ) ≤
+      (-(4449684358239905772299 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (41014048209613 / 64000000000000 : ℝ) = (41014048209613 / 32000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (41014048209613 / 32000000000000 : ℝ) ≤
+            (2481787444760094227701 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (7970065892294761 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (40654048209613 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (40654048209613 / 64000000000000 : ℝ) ≤ (1992516473073691 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (40654048209613 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau886 :
+    Real.arctan (3985032946147382 / 11393785074733125 : ℝ) ≤
+      (672913024660701932149 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau885 :
+    Real.arctan (1992516473073691 / 2500000000000000 : ℝ) ≤
+      (2 * (672913024660701932149 / 2000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (10230056119573 / 8000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1992516473073691 / 2500000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1992516473073691 / 2500000000000000 : ℝ) /
+                (1 + (10230056119573 / 8000000000000 : ℝ)) =
+              (3985032946147382 / 11393785074733125 : ℝ)
+            by
+        norm_num];
+        exact hau886)
+  have hat1 :
+    (Real.pi / 2 - (2 * (672913024660701932149 / 2000000000000000000000 : ℝ))) ≤
+      Real.arctan (2500000000000000 / 1992516473073691 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 1992516473073691 : ℝ) =
+              (1992516473073691 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau885)
+  have hz1 :
+    (2500000000000000 / 1992516473073691 : ℝ) ≤
+      1 / Real.sqrt (40654048209613 / 64000000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 1992516473073691 : ℝ) = 1 / (1992516473073691 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 7970065892294761 : ℝ) ≤
+      (117282262271255301799 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (40654048209613 / 64000000000000 : ℝ) ≤
+      (750000000000000 / 7970065892294761 : ℝ) := by
+    rw [show
+        (750000000000000 / 7970065892294761 : ℝ) =
+          (3 / 40) / (7970065892294761 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (672913024660701932149 / 2000000000000000000000 : ℝ))) -
+        6 * (117282262271255301799 / 1250000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (40654048209613 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (40654048209613 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((7970065892294761 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (672913024660701932149 / 2000000000000000000000 : ℝ))) -
+          6 * (117282262271255301799 / 1250000000000000000000 : ℝ))) ≤
+      Real.sqrt (40654048209613 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (40654048209613 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (40654048209613 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_558 :
+    ((4924341655912681739097 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(4432925472037122503297 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((3988403150783981 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1683312728426578497401 / 5000000000000000000000 : ℝ))) -
+              6 * (937469908111000802269 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (814456816291 / 1280000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (814456816291 / 1280000000000 : ℝ)) =
+      Real.log (2094456816291 / 1280000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((814456816291 / 1280000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (821656816291 / 1280000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (4924341655912681739097 / 10000000000000000000000 : ℝ) ≤
+      Real.log (2094456816291 / 1280000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (821656816291 / 1280000000000 : ℝ) ≤
+      (-(4432925472037122503297 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (821656816291 / 1280000000000 : ℝ) = (821656816291 / 640000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (821656816291 / 640000000000 : ℝ) ≤
+            (2498546330962877496703 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (3988403150783981 / 5000000000000000 : ℝ) ≤ Real.sqrt (814456816291 / 1280000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (814456816291 / 1280000000000 : ℝ) ≤ (1595361260313593 / 2000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (814456816291 / 1280000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau888 :
+    Real.arctan (7976806301567965 / 22791772307727118 : ℝ) ≤
+      (1683312728426578497401 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau887 :
+    Real.arctan (1595361260313593 / 2000000000000000 : ℝ) ≤
+      (2 * (1683312728426578497401 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6395886153863559 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1595361260313593 / 2000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1595361260313593 / 2000000000000000 : ℝ) /
+                (1 + (6395886153863559 / 5000000000000000 : ℝ)) =
+              (7976806301567965 / 22791772307727118 : ℝ)
+            by
+        norm_num];
+        exact hau888)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1683312728426578497401 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (2000000000000000 / 1595361260313593 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2000000000000000 / 1595361260313593 : ℝ) =
+              (1595361260313593 / 2000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau887)
+  have hz1 :
+    (2000000000000000 / 1595361260313593 : ℝ) ≤
+      1 / Real.sqrt (814456816291 / 1280000000000 : ℝ) := by
+    rw [show
+        (2000000000000000 / 1595361260313593 : ℝ) = 1 / (1595361260313593 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 3988403150783981 : ℝ) ≤
+      (937469908111000802269 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (814456816291 / 1280000000000 : ℝ) ≤
+      (375000000000000 / 3988403150783981 : ℝ) := by
+    rw [show
+        (375000000000000 / 3988403150783981 : ℝ) =
+          (3 / 40) / (3988403150783981 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1683312728426578497401 / 5000000000000000000000 : ℝ))) -
+        6 * (937469908111000802269 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (814456816291 / 1280000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (814456816291 / 1280000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((3988403150783981 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1683312728426578497401 / 5000000000000000000000 : ℝ))) -
+          6 * (937469908111000802269 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (814456816291 / 1280000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (814456816291 / 1280000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (814456816291 / 1280000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_559 :
+    ((2465454257784311148601 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2208097312438626819047 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((7983541019995351 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1684341349519044723379 / 5000000000000000000000 : ℝ))) -
+              6 * (936683701107258921899 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (40791633419487 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (40791633419487 / 64000000000000 : ℝ)) =
+      Real.log (104791633419487 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((40791633419487 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (41151633419487 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2465454257784311148601 / 5000000000000000000000 : ℝ) ≤
+      Real.log (104791633419487 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (41151633419487 / 64000000000000 : ℝ) ≤
+      (-(2208097312438626819047 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (41151633419487 / 64000000000000 : ℝ) = (41151633419487 / 32000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (41151633419487 / 32000000000000 : ℝ) ≤
+            (1257638589061373180953 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (7983541019995351 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (40791633419487 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (40791633419487 / 64000000000000 : ℝ) ≤ (3991770509997677 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (40791633419487 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau890 :
+    Real.arctan (3991770509997677 / 11397986543006097 : ℝ) ≤
+      (1684341349519044723379 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau889 :
+    Real.arctan (3991770509997677 / 5000000000000000 : ℝ) ≤
+      (2 * (1684341349519044723379 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6397986543006097 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (3991770509997677 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (3991770509997677 / 5000000000000000 : ℝ) /
+                (1 + (6397986543006097 / 5000000000000000 : ℝ)) =
+              (3991770509997677 / 11397986543006097 : ℝ)
+            by
+        norm_num];
+        exact hau890)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1684341349519044723379 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 3991770509997677 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 3991770509997677 : ℝ) =
+              (3991770509997677 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau889)
+  have hz1 :
+    (5000000000000000 / 3991770509997677 : ℝ) ≤
+      1 / Real.sqrt (40791633419487 / 64000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 3991770509997677 : ℝ) = 1 / (3991770509997677 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 7983541019995351 : ℝ) ≤
+      (936683701107258921899 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (40791633419487 / 64000000000000 : ℝ) ≤
+      (750000000000000 / 7983541019995351 : ℝ) := by
+    rw [show
+        (750000000000000 / 7983541019995351 : ℝ) =
+          (3 / 40) / (7983541019995351 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1684341349519044723379 / 5000000000000000000000 : ℝ))) -
+        6 * (936683701107258921899 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (40791633419487 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (40791633419487 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((7983541019995351 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1684341349519044723379 / 5000000000000000000000 : ℝ))) -
+          6 * (936683701107258921899 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (40791633419487 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (40791633419487 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (40791633419487 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_560 :
+    ((4937471065689844968309 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(549936465386697441599 / 1250000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1997567515491693 / 2500000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (842684214577127660523 / 2500000000000000000000 : ℝ))) -
+              6 * (467949734428600301737 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (5107553253053 / 8000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (5107553253053 / 8000000000000 : ℝ)) =
+      Real.log (13107553253053 / 8000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((5107553253053 / 8000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (5152553253053 / 8000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (4937471065689844968309 / 10000000000000000000000 : ℝ) ≤
+      Real.log (13107553253053 / 8000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (5152553253053 / 8000000000000 : ℝ) ≤
+      (-(549936465386697441599 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (5152553253053 / 8000000000000 : ℝ) = (5152553253053 / 4000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5152553253053 / 4000000000000 : ℝ) ≤
+            (316497509988302558401 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1997567515491693 / 2500000000000000 : ℝ) ≤ Real.sqrt (5107553253053 / 8000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (5107553253053 / 8000000000000 : ℝ) ≤ (319610802478671 / 400000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (5107553253053 / 8000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau892 :
+    Real.arctan (2663423353988925 / 7600057495226708 : ℝ) ≤
+      (842684214577127660523 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau891 :
+    Real.arctan (319610802478671 / 400000000000000 : ℝ) ≤
+      (2 * (842684214577127660523 / 2500000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (3200043121420031 / 2500000000000000 : ℝ) ≤
+          Real.sqrt (1 + (319610802478671 / 400000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (319610802478671 / 400000000000000 : ℝ) /
+                (1 + (3200043121420031 / 2500000000000000 : ℝ)) =
+              (2663423353988925 / 7600057495226708 : ℝ)
+            by
+        norm_num];
+        exact hau892)
+  have hat1 :
+    (Real.pi / 2 - (2 * (842684214577127660523 / 2500000000000000000000 : ℝ))) ≤
+      Real.arctan (400000000000000 / 319610802478671 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (400000000000000 / 319610802478671 : ℝ) =
+              (319610802478671 / 400000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau891)
+  have hz1 :
+    (400000000000000 / 319610802478671 : ℝ) ≤ 1 / Real.sqrt (5107553253053 / 8000000000000 : ℝ) :=
+  by
+    rw [show (400000000000000 / 319610802478671 : ℝ) = 1 / (319610802478671 / 400000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (62500000000000 / 665855838497231 : ℝ) ≤
+      (467949734428600301737 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (5107553253053 / 8000000000000 : ℝ) ≤
+      (62500000000000 / 665855838497231 : ℝ) := by
+    rw [show
+        (62500000000000 / 665855838497231 : ℝ) =
+          (3 / 40) / (1997567515491693 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (842684214577127660523 / 2500000000000000000000 : ℝ))) -
+        6 * (467949734428600301737 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (5107553253053 / 8000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (5107553253053 / 8000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1997567515491693 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (842684214577127660523 / 2500000000000000000000 : ℝ))) -
+          6 * (467949734428600301737 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (5107553253053 / 8000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (5107553253053 / 8000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (5107553253053 / 8000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_561 :
+    ((4950583259927416104161 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2183084691663832143643 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((8003711173798727 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (843708990432839064989 / 2500000000000000000000 : ℝ))) -
+              6 * (467168447827085537807 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (20499005617149 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (20499005617149 / 32000000000000 : ℝ)) =
+      Real.log (52499005617149 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((20499005617149 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (20679005617149 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (4950583259927416104161 / 10000000000000000000000 : ℝ) ≤
+      Real.log (52499005617149 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (20679005617149 / 32000000000000 : ℝ) ≤
+      (-(2183084691663832143643 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (20679005617149 / 32000000000000 : ℝ) = (20679005617149 / 16000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20679005617149 / 16000000000000 : ℝ) ≤
+            (1282651209836167856357 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (8003711173798727 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (20499005617149 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (20499005617149 / 32000000000000 : ℝ) ≤ (800371117379873 / 1000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (20499005617149 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau894 :
+    Real.arctan (8003711173798730 / 22808567154588003 : ℝ) ≤
+      (843708990432839064989 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau893 :
+    Real.arctan (800371117379873 / 1000000000000000 : ℝ) ≤
+      (2 * (843708990432839064989 / 2500000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12808567154588003 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (800371117379873 / 1000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (800371117379873 / 1000000000000000 : ℝ) /
+                (1 + (12808567154588003 / 10000000000000000 : ℝ)) =
+              (8003711173798730 / 22808567154588003 : ℝ)
+            by
+        norm_num];
+        exact hau894)
+  have hat1 :
+    (Real.pi / 2 - (2 * (843708990432839064989 / 2500000000000000000000 : ℝ))) ≤
+      Real.arctan (1000000000000000 / 800371117379873 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1000000000000000 / 800371117379873 : ℝ) =
+              (800371117379873 / 1000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau893)
+  have hz1 :
+    (1000000000000000 / 800371117379873 : ℝ) ≤
+      1 / Real.sqrt (20499005617149 / 32000000000000 : ℝ) := by
+    rw [show
+        (1000000000000000 / 800371117379873 : ℝ) = 1 / (800371117379873 / 1000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 8003711173798727 : ℝ) ≤
+      (467168447827085537807 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (20499005617149 / 32000000000000 : ℝ) ≤
+      (750000000000000 / 8003711173798727 : ℝ) := by
+    rw [show
+        (750000000000000 / 8003711173798727 : ℝ) =
+          (3 / 40) / (8003711173798727 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (843708990432839064989 / 2500000000000000000000 : ℝ))) -
+        6 * (467168447827085537807 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (20499005617149 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (20499005617149 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((8003711173798727 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (843708990432839064989 / 2500000000000000000000 : ℝ))) -
+          6 * (467168447827085537807 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (20499005617149 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (20499005617149 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (20499005617149 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_562 :
+    ((2481839141856494072461 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2166478856360513774443 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1603425950195627 / 2000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (844730708506255858439 / 2500000000000000000000 : ℝ))) -
+              6 * (466391061526950188599 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (10283899111043 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (10283899111043 / 16000000000000 : ℝ)) =
+      Real.log (26283899111043 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((10283899111043 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (10373899111043 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2481839141856494072461 / 5000000000000000000000 : ℝ) ≤
+      Real.log (26283899111043 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (10373899111043 / 16000000000000 : ℝ) ≤
+      (-(2166478856360513774443 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (10373899111043 / 16000000000000 : ℝ) = (10373899111043 / 8000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10373899111043 / 8000000000000 : ℝ) ≤
+            (1299257045139486225557 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1603425950195627 / 2000000000000000 : ℝ) ≤ Real.sqrt (10283899111043 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (10283899111043 / 16000000000000 : ℝ) ≤ (4008564875489069 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (10283899111043 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau896 :
+    Real.arctan (8017129750978138 / 22816956325275465 : ℝ) ≤
+      (844730708506255858439 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau895 :
+    Real.arctan (4008564875489069 / 5000000000000000 : ℝ) ≤
+      (2 * (844730708506255858439 / 2500000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (2563391265055093 / 2000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (4008564875489069 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (4008564875489069 / 5000000000000000 : ℝ) /
+                (1 + (2563391265055093 / 2000000000000000 : ℝ)) =
+              (8017129750978138 / 22816956325275465 : ℝ)
+            by
+        norm_num];
+        exact hau896)
+  have hat1 :
+    (Real.pi / 2 - (2 * (844730708506255858439 / 2500000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 4008564875489069 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 4008564875489069 : ℝ) =
+              (4008564875489069 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau895)
+  have hz1 :
+    (5000000000000000 / 4008564875489069 : ℝ) ≤
+      1 / Real.sqrt (10283899111043 / 16000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 4008564875489069 : ℝ) = 1 / (4008564875489069 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (150000000000000 / 1603425950195627 : ℝ) ≤
+      (466391061526950188599 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (10283899111043 / 16000000000000 : ℝ) ≤
+      (150000000000000 / 1603425950195627 : ℝ) := by
+    rw [show
+        (150000000000000 / 1603425950195627 : ℝ) =
+          (3 / 40) / (1603425950195627 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (844730708506255858439 / 2500000000000000000000 : ℝ))) -
+        6 * (466391061526950188599 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (10283899111043 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (10283899111043 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1603425950195627 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (844730708506255858439 / 2500000000000000000000 : ℝ))) -
+          6 * (466391061526950188599 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (10283899111043 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (10283899111043 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (10283899111043 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_563 :
+    ((4976756181957258726983 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1074963994651026888953 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((4015262953233787 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (845749385385216735619 / 2500000000000000000000 : ℝ))) -
+              6 * (931235086368221112029 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (20636590827023 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (20636590827023 / 32000000000000 : ℝ)) =
+      Real.log (52636590827023 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((20636590827023 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (20816590827023 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (4976756181957258726983 / 10000000000000000000000 : ℝ) ≤
+      Real.log (52636590827023 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (20816590827023 / 32000000000000 : ℝ) ≤
+      (-(1074963994651026888953 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (20816590827023 / 32000000000000 : ℝ) = (20816590827023 / 16000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20816590827023 / 16000000000000 : ℝ) ≤
+            (657903956098973111047 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (4015262953233787 / 5000000000000000 : ℝ) ≤ Real.sqrt (20636590827023 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (20636590827023 / 32000000000000 : ℝ) ≤
+      (8030525906467577 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (20636590827023 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau898 :
+    Real.arctan (8030525906467577 / 22825340008531815 : ℝ) ≤
+      (845749385385216735619 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau897 :
+    Real.arctan (8030525906467577 / 10000000000000000 : ℝ) ≤
+      (2 * (845749385385216735619 / 2500000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (2565068001706363 / 2000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (8030525906467577 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (8030525906467577 / 10000000000000000 : ℝ) /
+                (1 + (2565068001706363 / 2000000000000000 : ℝ)) =
+              (8030525906467577 / 22825340008531815 : ℝ)
+            by
+        norm_num];
+        exact hau898)
+  have hat1 :
+    (Real.pi / 2 - (2 * (845749385385216735619 / 2500000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 8030525906467577 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 8030525906467577 : ℝ) =
+              (8030525906467577 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau897)
+  have hz1 :
+    (10000000000000000 / 8030525906467577 : ℝ) ≤
+      1 / Real.sqrt (20636590827023 / 32000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 8030525906467577 : ℝ) =
+          1 / (8030525906467577 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 4015262953233787 : ℝ) ≤
+      (931235086368221112029 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (20636590827023 / 32000000000000 : ℝ) ≤
+      (375000000000000 / 4015262953233787 : ℝ) := by
+    rw [show
+        (375000000000000 / 4015262953233787 : ℝ) =
+          (3 / 40) / (4015262953233787 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (845749385385216735619 / 2500000000000000000000 : ℝ))) -
+        6 * (931235086368221112029 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (20636590827023 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (20636590827023 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((4015262953233787 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (845749385385216735619 / 2500000000000000000000 : ℝ))) -
+          6 * (931235086368221112029 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (20636590827023 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (20636590827023 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (20636590827023 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_564 :
+    ((498981699939495428459 / 1000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(853372691111829526357 / 2000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2010974938072249 / 2500000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3387060150084523494817 / 10000000000000000000000 : ℝ))) -
+              6 * (929695721657504509173 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (517634585799 / 800000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (517634585799 / 800000000000 : ℝ)) =
+      Real.log (1317634585799 / 800000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((517634585799 / 800000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (522134585799 / 800000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (498981699939495428459 / 1000000000000000000000 : ℝ) ≤
+      Real.log (1317634585799 / 800000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (522134585799 / 800000000000 : ℝ) ≤
+      (-(853372691111829526357 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (522134585799 / 800000000000 : ℝ) = (522134585799 / 400000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (522134585799 / 400000000000 : ℝ) ≤
+            (532921669488170473643 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2010974938072249 / 2500000000000000 : ℝ) ≤ Real.sqrt (517634585799 / 800000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (517634585799 / 800000000000 : ℝ) ≤ (8043899752288999 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (517634585799 / 800000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau900 :
+    Real.arctan (8043899752288999 / 22833718215111121 : ℝ) ≤
+      (3387060150084523494817 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau899 :
+    Real.arctan (8043899752288999 / 10000000000000000 : ℝ) ≤
+      (2 * (3387060150084523494817 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12833718215111121 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (8043899752288999 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (8043899752288999 / 10000000000000000 : ℝ) /
+                (1 + (12833718215111121 / 10000000000000000 : ℝ)) =
+              (8043899752288999 / 22833718215111121 : ℝ)
+            by
+        norm_num];
+        exact hau900)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3387060150084523494817 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 8043899752288999 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 8043899752288999 : ℝ) =
+              (8043899752288999 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau899)
+  have hz1 :
+    (10000000000000000 / 8043899752288999 : ℝ) ≤
+      1 / Real.sqrt (517634585799 / 800000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 8043899752288999 : ℝ) =
+          1 / (8043899752288999 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (187500000000000 / 2010974938072249 : ℝ) ≤
+      (929695721657504509173 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (517634585799 / 800000000000 : ℝ) ≤
+      (187500000000000 / 2010974938072249 : ℝ) := by
+    rw [show
+        (187500000000000 / 2010974938072249 : ℝ) =
+          (3 / 40) / (2010974938072249 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3387060150084523494817 / 10000000000000000000000 : ℝ))) -
+        6 * (929695721657504509173 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (517634585799 / 800000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (517634585799 / 800000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2010974938072249 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3387060150084523494817 / 10000000000000000000000 : ℝ))) -
+          6 * (929695721657504509173 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (517634585799 / 800000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (517634585799 / 800000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (517634585799 / 800000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/V47.lean b/LeanPool/Zeta5Irrational/Table/V47.lean
new file mode 100644
index 0000000000..1aa20d0c73
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/V47.lean
@@ -0,0 +1,1434 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Energy
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # V47: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_565 :
+    ((156339399393304630697 / 312500000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1058494856331202102067 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2014312849883663 / 2500000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3391110724922073870779 / 10000000000000000000000 : ℝ))) -
+              6 * (464081982859780515481 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (20774176036897 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (20774176036897 / 32000000000000 : ℝ)) =
+      Real.log (52774176036897 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((20774176036897 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (20954176036897 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (156339399393304630697 / 312500000000000000000 : ℝ) ≤
+      Real.log (52774176036897 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (20954176036897 / 32000000000000 : ℝ) ≤
+      (-(1058494856331202102067 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (20954176036897 / 32000000000000 : ℝ) = (20954176036897 / 16000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (20954176036897 / 16000000000000 : ℝ) ≤
+            (674373094418797897933 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2014312849883663 / 2500000000000000 : ℝ) ≤ Real.sqrt (20774176036897 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (20774176036897 / 32000000000000 : ℝ) ≤ (1611450279906931 / 2000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (20774176036897 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau902 :
+    Real.arctan (8057251399534655 / 22842090955732369 : ℝ) ≤
+      (3391110724922073870779 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau901 :
+    Real.arctan (1611450279906931 / 2000000000000000 : ℝ) ≤
+      (2 * (3391110724922073870779 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12842090955732369 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1611450279906931 / 2000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1611450279906931 / 2000000000000000 : ℝ) /
+                (1 + (12842090955732369 / 10000000000000000 : ℝ)) =
+              (8057251399534655 / 22842090955732369 : ℝ)
+            by
+        norm_num];
+        exact hau902)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3391110724922073870779 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (2000000000000000 / 1611450279906931 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2000000000000000 / 1611450279906931 : ℝ) =
+              (1611450279906931 / 2000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau901)
+  have hz1 :
+    (2000000000000000 / 1611450279906931 : ℝ) ≤
+      1 / Real.sqrt (20774176036897 / 32000000000000 : ℝ) := by
+    rw [show
+        (2000000000000000 / 1611450279906931 : ℝ) = 1 / (1611450279906931 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (187500000000000 / 2014312849883663 : ℝ) ≤
+      (464081982859780515481 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (20774176036897 / 32000000000000 : ℝ) ≤
+      (187500000000000 / 2014312849883663 : ℝ) := by
+    rw [show
+        (187500000000000 / 2014312849883663 : ℝ) =
+          (3 / 40) / (2014312849883663 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3391110724922073870779 / 10000000000000000000000 : ℝ))) -
+        6 * (464081982859780515481 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (20774176036897 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (20774176036897 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2014312849883663 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3391110724922073870779 / 10000000000000000000000 : ℝ))) -
+          6 * (464081982859780515481 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (20774176036897 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (20774176036897 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (20774176036897 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_566 :
+    ((2507943784957586433653 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(4201203176702343280537 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((63051413737327 / 78125000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2 * (169757466539305760399 / 500000000000000000000 : ℝ))) -
+              6 * (926639756078741289001 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (10421484320917 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (10421484320917 / 16000000000000 : ℝ)) =
+      Real.log (26421484320917 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((10421484320917 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (10511484320917 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2507943784957586433653 / 5000000000000000000000 : ℝ) ≤
+      Real.log (26421484320917 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (10511484320917 / 16000000000000 : ℝ) ≤
+      (-(4201203176702343280537 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (10511484320917 / 16000000000000 : ℝ) = (10511484320917 / 8000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10511484320917 / 8000000000000 : ℝ) ≤
+            (2730268626297656719463 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (63051413737327 / 78125000000000 : ℝ) ≤ Real.sqrt (10421484320917 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (10421484320917 / 16000000000000 : ℝ) ≤
+      (8070580958377859 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (10421484320917 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau904 :
+    Real.arctan (8070580958377859 / 22850458241079625 : ℝ) ≤
+      (169757466539305760399 / 500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau903 :
+    Real.arctan (8070580958377859 / 10000000000000000 : ℝ) ≤
+      (2 * (169757466539305760399 / 500000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (102803665928637 / 80000000000000 : ℝ) ≤
+          Real.sqrt (1 + (8070580958377859 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (8070580958377859 / 10000000000000000 : ℝ) /
+                (1 + (102803665928637 / 80000000000000 : ℝ)) =
+              (8070580958377859 / 22850458241079625 : ℝ)
+            by
+        norm_num];
+        exact hau904)
+  have hat1 :
+    (Real.pi / 2 - (2 * (169757466539305760399 / 500000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 8070580958377859 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 8070580958377859 : ℝ) =
+              (8070580958377859 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau903)
+  have hz1 :
+    (10000000000000000 / 8070580958377859 : ℝ) ≤
+      1 / Real.sqrt (10421484320917 / 16000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 8070580958377859 : ℝ) =
+          1 / (8070580958377859 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (5859375000000 / 63051413737327 : ℝ) ≤
+      (926639756078741289001 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (10421484320917 / 16000000000000 : ℝ) ≤
+      (5859375000000 / 63051413737327 : ℝ) := by
+    rw [show
+        (5859375000000 / 63051413737327 : ℝ) = (3 / 40) / (63051413737327 / 78125000000000 : ℝ) by
+        norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (169757466539305760399 / 500000000000000000000 : ℝ))) -
+        6 * (926639756078741289001 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (10421484320917 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (10421484320917 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((63051413737327 / 78125000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (169757466539305760399 / 500000000000000000000 : ℝ))) -
+          6 * (926639756078741289001 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (10421484320917 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (10421484320917 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (10421484320917 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_567 :
+    ((314306088224720380513 / 625000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(4168534005463306455681 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((8083888538083597 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3399176031881748881431 / 10000000000000000000000 : ℝ))) -
+              6 * (925123030975232317823 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (20911761246771 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (20911761246771 / 32000000000000 : ℝ)) =
+      Real.log (52911761246771 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((20911761246771 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (21091761246771 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (314306088224720380513 / 625000000000000000000 : ℝ) ≤
+      Real.log (52911761246771 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (21091761246771 / 32000000000000 : ℝ) ≤
+      (-(4168534005463306455681 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (21091761246771 / 32000000000000 : ℝ) = (21091761246771 / 16000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21091761246771 / 16000000000000 : ℝ) ≤
+            (2762937797536693544319 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (8083888538083597 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (20911761246771 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (20911761246771 / 32000000000000 : ℝ) ≤ (20209721345209 / 25000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (20911761246771 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau906 :
+    Real.arctan (8083888538083600 / 22858820081802193 : ℝ) ≤
+      (3399176031881748881431 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau905 :
+    Real.arctan (20209721345209 / 25000000000000 : ℝ) ≤
+      (2 * (3399176031881748881431 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12858820081802193 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (20209721345209 / 25000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (20209721345209 / 25000000000000 : ℝ) /
+                (1 + (12858820081802193 / 10000000000000000 : ℝ)) =
+              (8083888538083600 / 22858820081802193 : ℝ)
+            by
+        norm_num];
+        exact hau906)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3399176031881748881431 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (25000000000000 / 20209721345209 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (25000000000000 / 20209721345209 : ℝ) =
+              (20209721345209 / 25000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau905)
+  have hz1 :
+    (25000000000000 / 20209721345209 : ℝ) ≤ 1 / Real.sqrt (20911761246771 / 32000000000000 : ℝ) :=
+  by
+    rw [show (25000000000000 / 20209721345209 : ℝ) = 1 / (20209721345209 / 25000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 8083888538083597 : ℝ) ≤
+      (925123030975232317823 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (20911761246771 / 32000000000000 : ℝ) ≤
+      (750000000000000 / 8083888538083597 : ℝ) := by
+    rw [show
+        (750000000000000 / 8083888538083597 : ℝ) =
+          (3 / 40) / (8083888538083597 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3399176031881748881431 / 10000000000000000000000 : ℝ))) -
+        6 * (925123030975232317823 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (20911761246771 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (20911761246771 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((8083888538083597 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3399176031881748881431 / 10000000000000000000000 : ℝ))) -
+          6 * (925123030975232317823 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (20911761246771 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (20911761246771 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (20911761246771 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_568 :
+    ((1008378069933354245329 / 2000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2067985607129381824541 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((4048587123509493 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3403190891892478111163 / 10000000000000000000000 : ℝ))) -
+              6 * (923613729354546866129 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (5245138462927 / 8000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (5245138462927 / 8000000000000 : ℝ)) =
+      Real.log (13245138462927 / 8000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((5245138462927 / 8000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (5290138462927 / 8000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1008378069933354245329 / 2000000000000000000000 : ℝ) ≤
+      Real.log (13245138462927 / 8000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 11
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (5290138462927 / 8000000000000 : ℝ) ≤
+      (-(2067985607129381824541 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (5290138462927 / 8000000000000 : ℝ) = (5290138462927 / 4000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5290138462927 / 4000000000000 : ℝ) ≤
+            (1397750294370618175459 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (4048587123509493 / 5000000000000000 : ℝ) ≤ Real.sqrt (5245138462927 / 8000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (5245138462927 / 8000000000000 : ℝ) ≤ (8097174247018989 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (5245138462927 / 8000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau908 :
+    Real.arctan (8097174247018989 / 22867176488514778 : ℝ) ≤
+      (3403190891892478111163 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau907 :
+    Real.arctan (8097174247018989 / 10000000000000000 : ℝ) ≤
+      (2 * (3403190891892478111163 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6433588244257389 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (8097174247018989 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (8097174247018989 / 10000000000000000 : ℝ) /
+                (1 + (6433588244257389 / 5000000000000000 : ℝ)) =
+              (8097174247018989 / 22867176488514778 : ℝ)
+            by
+        norm_num];
+        exact hau908)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3403190891892478111163 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 8097174247018989 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 8097174247018989 : ℝ) =
+              (8097174247018989 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau907)
+  have hz1 :
+    (10000000000000000 / 8097174247018989 : ℝ) ≤
+      1 / Real.sqrt (5245138462927 / 8000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 8097174247018989 : ℝ) =
+          1 / (8097174247018989 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (125000000000000 / 1349529041169831 : ℝ) ≤
+      (923613729354546866129 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (5245138462927 / 8000000000000 : ℝ) ≤
+      (125000000000000 / 1349529041169831 : ℝ) := by
+    rw [show
+        (125000000000000 / 1349529041169831 : ℝ) =
+          (3 / 40) / (4048587123509493 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3403190891892478111163 / 10000000000000000000000 : ℝ))) -
+        6 * (923613729354546866129 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (5245138462927 / 8000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (5245138462927 / 8000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((4048587123509493 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3403190891892478111163 / 10000000000000000000000 : ℝ))) -
+          6 * (923613729354546866129 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (5245138462927 / 8000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (5245138462927 / 8000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (5245138462927 / 8000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_569 :
+    ((2533912845142745232631 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1017790504105132442423 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((8123680481619383 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1705592670410026563051 / 5000000000000000000000 : ℝ))) -
+              6 * (920617155808572349771 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (10559069530791 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (10559069530791 / 16000000000000 : ℝ)) =
+      Real.log (26559069530791 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((10559069530791 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (10649069530791 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2533912845142745232631 / 5000000000000000000000 : ℝ) ≤
+      Real.log (26559069530791 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (10649069530791 / 16000000000000 : ℝ) ≤
+      (-(1017790504105132442423 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (10649069530791 / 16000000000000 : ℝ) = (10649069530791 / 8000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10649069530791 / 8000000000000 : ℝ) ≤
+            (715077446644867557577 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (8123680481619383 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (10559069530791 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (10559069530791 / 16000000000000 : ℝ) ≤ (4061840240809693 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (10559069530791 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau910 :
+    Real.arctan (8123680481619386 / 22883873042196735 : ℝ) ≤
+      (1705592670410026563051 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau909 :
+    Real.arctan (4061840240809693 / 5000000000000000 : ℝ) ≤
+      (2 * (1705592670410026563051 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (2576774608439347 / 2000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (4061840240809693 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (4061840240809693 / 5000000000000000 : ℝ) /
+                (1 + (2576774608439347 / 2000000000000000 : ℝ)) =
+              (8123680481619386 / 22883873042196735 : ℝ)
+            by
+        norm_num];
+        exact hau910)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1705592670410026563051 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 4061840240809693 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 4061840240809693 : ℝ) =
+              (4061840240809693 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau909)
+  have hz1 :
+    (5000000000000000 / 4061840240809693 : ℝ) ≤
+      1 / Real.sqrt (10559069530791 / 16000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 4061840240809693 : ℝ) = 1 / (4061840240809693 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 8123680481619383 : ℝ) ≤
+      (920617155808572349771 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (10559069530791 / 16000000000000 : ℝ) ≤
+      (750000000000000 / 8123680481619383 : ℝ) := by
+    rw [show
+        (750000000000000 / 8123680481619383 : ℝ) =
+          (3 / 40) / (8123680481619383 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1705592670410026563051 / 5000000000000000000000 : ℝ))) -
+        6 * (920617155808572349771 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (10559069530791 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (10559069530791 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((8123680481619383 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1705592670410026563051 / 5000000000000000000000 : ℝ))) -
+          6 * (920617155808572349771 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (10559069530791 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (10559069530791 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (10559069530791 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_570 :
+    ((1018738788135635374369 / 2000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(801354027725657851151 / 2000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((8150100511545853 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1709566588411900250929 / 5000000000000000000000 : ℝ))) -
+              6 * (9176495607237304067 / 100000000000000000000 : ℝ)))) ≤
+      Vfield (664241383483 / 1000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (664241383483 / 1000000000000 : ℝ)) =
+      Real.log (1664241383483 / 1000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((664241383483 / 1000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (669866383483 / 1000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1018738788135635374369 / 2000000000000000000000 : ℝ) ≤
+      Real.log (1664241383483 / 1000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (669866383483 / 1000000000000 : ℝ) ≤
+      (-(801354027725657851151 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (669866383483 / 1000000000000 : ℝ) = (669866383483 / 500000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (669866383483 / 500000000000 : ℝ) ≤
+            (584940332874342148849 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (8150100511545853 / 10000000000000000 : ℝ) ≤ Real.sqrt (664241383483 / 1000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (664241383483 / 1000000000000 : ℝ) ≤ (15918165061613 / 19531250000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (664241383483 / 1000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau912 :
+    Real.arctan (8150100511545856 / 22900547986357013 : ℝ) ≤
+      (1709566588411900250929 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau911 :
+    Real.arctan (15918165061613 / 19531250000000 : ℝ) ≤
+      (2 * (1709566588411900250929 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12900547986357013 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (15918165061613 / 19531250000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (15918165061613 / 19531250000000 : ℝ) /
+                (1 + (12900547986357013 / 10000000000000000 : ℝ)) =
+              (8150100511545856 / 22900547986357013 : ℝ)
+            by
+        norm_num];
+        exact hau912)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1709566588411900250929 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (19531250000000 / 15918165061613 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (19531250000000 / 15918165061613 : ℝ) =
+              (15918165061613 / 19531250000000 : ℝ)
+            by
+        norm_num];
+        exact hau911)
+  have hz1 :
+    (19531250000000 / 15918165061613 : ℝ) ≤ 1 / Real.sqrt (664241383483 / 1000000000000 : ℝ) := by
+    rw [show (19531250000000 / 15918165061613 : ℝ) = 1 / (15918165061613 / 19531250000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 8150100511545853 : ℝ) ≤
+      (9176495607237304067 / 100000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (664241383483 / 1000000000000 : ℝ) ≤
+      (750000000000000 / 8150100511545853 : ℝ) := by
+    rw [show
+        (750000000000000 / 8150100511545853 : ℝ) =
+          (3 / 40) / (8150100511545853 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1709566588411900250929 / 5000000000000000000000 : ℝ))) -
+        6 * (9176495607237304067 / 100000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (664241383483 / 1000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (664241383483 / 1000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((8150100511545853 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1709566588411900250929 / 5000000000000000000000 : ℝ))) -
+          6 * (9176495607237304067 / 100000000000000000000 : ℝ))) ≤
+      Real.sqrt (664241383483 / 1000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (664241383483 / 1000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (664241383483 / 1000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_571 :
+    ((510343822688833403141 / 1000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(796515710353325413713 / 2000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((8160048151107357 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1711060181048130356787 / 5000000000000000000000 : ℝ))) -
+              6 * (183307428647100260073 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (4261528693017 / 6400000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (4261528693017 / 6400000000000 : ℝ)) =
+      Real.log (10661528693017 / 6400000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((4261528693017 / 6400000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (4297528693017 / 6400000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (510343822688833403141 / 1000000000000000000000 : ℝ) ≤
+      Real.log (10661528693017 / 6400000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (4297528693017 / 6400000000000 : ℝ) ≤
+      (-(796515710353325413713 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (4297528693017 / 6400000000000 : ℝ) = (4297528693017 / 3200000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4297528693017 / 3200000000000 : ℝ) ≤
+            (589778650246674586287 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (8160048151107357 / 10000000000000000 : ℝ) ≤ Real.sqrt (4261528693017 / 6400000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (4261528693017 / 6400000000000 : ℝ) ≤ (51000300944421 / 62500000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (4261528693017 / 6400000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau914 :
+    Real.arctan (32000188827872 / 89830724899411 : ℝ) ≤
+      (1711060181048130356787 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau913 :
+    Real.arctan (51000300944421 / 62500000000000 : ℝ) ≤
+      (2 * (1711060181048130356787 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (2581366969869961 / 2000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (51000300944421 / 62500000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (51000300944421 / 62500000000000 : ℝ) /
+                (1 + (2581366969869961 / 2000000000000000 : ℝ)) =
+              (32000188827872 / 89830724899411 : ℝ)
+            by
+        norm_num];
+        exact hau914)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1711060181048130356787 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (62500000000000 / 51000300944421 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (62500000000000 / 51000300944421 : ℝ) =
+              (51000300944421 / 62500000000000 : ℝ)
+            by
+        norm_num];
+        exact hau913)
+  have hz1 :
+    (62500000000000 / 51000300944421 : ℝ) ≤ 1 / Real.sqrt (4261528693017 / 6400000000000 : ℝ) := by
+    rw [show (62500000000000 / 51000300944421 : ℝ) = 1 / (51000300944421 / 62500000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 2720016050369119 : ℝ) ≤
+      (183307428647100260073 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (4261528693017 / 6400000000000 : ℝ) ≤
+      (250000000000000 / 2720016050369119 : ℝ) := by
+    rw [show
+        (250000000000000 / 2720016050369119 : ℝ) =
+          (3 / 40) / (8160048151107357 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1711060181048130356787 / 5000000000000000000000 : ℝ))) -
+        6 * (183307428647100260073 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (4261528693017 / 6400000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (4261528693017 / 6400000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((8160048151107357 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1711060181048130356787 / 5000000000000000000000 : ℝ))) -
+          6 * (183307428647100260073 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (4261528693017 / 6400000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (4261528693017 / 6400000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (4261528693017 / 6400000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_572 :
+    ((5113173027229668871761 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(791689069397068086157 / 2000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((8169983678593319 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (428137625724666504333 / 1250000000000000000000 : ℝ))) -
+              6 * (915428761561533214527 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (10679781329357 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (10679781329357 / 16000000000000 : ℝ)) =
+      Real.log (26679781329357 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((10679781329357 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (10769781329357 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (5113173027229668871761 / 10000000000000000000000 : ℝ) ≤
+      Real.log (26679781329357 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (10769781329357 / 16000000000000 : ℝ) ≤
+      (-(791689069397068086157 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (10769781329357 / 16000000000000 : ℝ) = (10769781329357 / 8000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10769781329357 / 8000000000000 : ℝ) ≤
+            (594605291202931913843 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (8169983678593319 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (10679781329357 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (10679781329357 / 16000000000000 : ℝ) ≤ (4084991839296661 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (10679781329357 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau916 :
+    Real.arctan (4084991839296661 / 11456559325764792 : ℝ) ≤
+      (428137625724666504333 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau915 :
+    Real.arctan (4084991839296661 / 5000000000000000 : ℝ) ≤
+      (2 * (428137625724666504333 / 1250000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (807069915720599 / 625000000000000 : ℝ) ≤
+          Real.sqrt (1 + (4084991839296661 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (4084991839296661 / 5000000000000000 : ℝ) /
+                (1 + (807069915720599 / 625000000000000 : ℝ)) =
+              (4084991839296661 / 11456559325764792 : ℝ)
+            by
+        norm_num];
+        exact hau916)
+  have hat1 :
+    (Real.pi / 2 - (2 * (428137625724666504333 / 1250000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 4084991839296661 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 4084991839296661 : ℝ) =
+              (4084991839296661 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau915)
+  have hz1 :
+    (5000000000000000 / 4084991839296661 : ℝ) ≤
+      1 / Real.sqrt (10679781329357 / 16000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 4084991839296661 : ℝ) = 1 / (4084991839296661 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 8169983678593319 : ℝ) ≤
+      (915428761561533214527 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (10679781329357 / 16000000000000 : ℝ) ≤
+      (750000000000000 / 8169983678593319 : ℝ) := by
+    rw [show
+        (750000000000000 / 8169983678593319 : ℝ) =
+          (3 / 40) / (8169983678593319 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (428137625724666504333 / 1250000000000000000000 : ℝ))) -
+        6 * (915428761561533214527 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (10679781329357 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (10679781329357 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((8169983678593319 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (428137625724666504333 / 1250000000000000000000 : ℝ))) -
+          6 * (915428761561533214527 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (10679781329357 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (10679781329357 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (10679781329357 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_573 :
+    ((5122898360152830778087 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1967185121586613830879 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((8179907138138663 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3428075133878351375991 / 10000000000000000000000 : ℝ))) -
+              6 * (3571579653740407077 / 39062500000000000000 : ℝ)))) ≤
+      Vfield (21411481852343 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (21411481852343 / 32000000000000 : ℝ)) =
+      Real.log (53411481852343 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((21411481852343 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (21591481852343 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (5122898360152830778087 / 10000000000000000000000 : ℝ) ≤
+      Real.log (53411481852343 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (21591481852343 / 32000000000000 : ℝ) ≤
+      (-(1967185121586613830879 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (21591481852343 / 32000000000000 : ℝ) = (21591481852343 / 16000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21591481852343 / 16000000000000 : ℝ) ≤
+            (1498550779913386169121 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (8179907138138663 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (21411481852343 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (21411481852343 / 32000000000000 : ℝ) ≤ (4089953569069333 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (21411481852343 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau918 :
+    Real.arctan (584279081295619 / 1637099956954468 : ℝ) ≤
+      (3428075133878351375991 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau917 :
+    Real.arctan (4089953569069333 / 5000000000000000 : ℝ) ≤
+      (2 * (3428075133878351375991 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (1614924924670319 / 1250000000000000 : ℝ) ≤
+          Real.sqrt (1 + (4089953569069333 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (4089953569069333 / 5000000000000000 : ℝ) /
+                (1 + (1614924924670319 / 1250000000000000 : ℝ)) =
+              (584279081295619 / 1637099956954468 : ℝ)
+            by
+        norm_num];
+        exact hau918)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3428075133878351375991 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 4089953569069333 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 4089953569069333 : ℝ) =
+              (4089953569069333 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau917)
+  have hz1 :
+    (5000000000000000 / 4089953569069333 : ℝ) ≤
+      1 / Real.sqrt (21411481852343 / 32000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 4089953569069333 : ℝ) = 1 / (4089953569069333 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 8179907138138663 : ℝ) ≤
+      (3571579653740407077 / 39062500000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (21411481852343 / 32000000000000 : ℝ) ≤
+      (750000000000000 / 8179907138138663 : ℝ) := by
+    rw [show
+        (750000000000000 / 8179907138138663 : ℝ) =
+          (3 / 40) / (8179907138138663 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3428075133878351375991 / 10000000000000000000000 : ℝ))) -
+        6 * (3571579653740407077 / 39062500000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (21411481852343 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (21411481852343 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((8179907138138663 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3428075133878351375991 / 10000000000000000000000 : ℝ))) -
+          6 * (3571579653740407077 / 39062500000000000000 : ℝ))) ≤
+      Real.sqrt (21411481852343 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (21411481852343 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (21411481852343 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_574 :
+    ((320788390253418113291 / 625000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1955176480622274021633 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2047454643402731 / 2500000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2 * (85776069303373838337 / 250000000000000000000 : ℝ))) -
+              6 * (182644801696868735909 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (5365850261493 / 8000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (5365850261493 / 8000000000000 : ℝ)) =
+      Real.log (13365850261493 / 8000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((5365850261493 / 8000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (5410850261493 / 8000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (320788390253418113291 / 625000000000000000000 : ℝ) ≤
+      Real.log (13365850261493 / 8000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (5410850261493 / 8000000000000 : ℝ) ≤
+      (-(1955176480622274021633 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (5410850261493 / 8000000000000 : ℝ) = (5410850261493 / 4000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5410850261493 / 4000000000000 : ℝ) ≤
+            (1510559420877725978367 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2047454643402731 / 2500000000000000 : ℝ) ≤ Real.sqrt (5365850261493 / 8000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (5365850261493 / 8000000000000 : ℝ) ≤ (8189818573610927 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (5365850261493 / 8000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau920 :
+    Real.arctan (8189818573610927 / 22925677091304056 : ℝ) ≤
+      (85776069303373838337 / 250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau919 :
+    Real.arctan (8189818573610927 / 10000000000000000 : ℝ) ≤
+      (2 * (85776069303373838337 / 250000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (1615709636413007 / 1250000000000000 : ℝ) ≤
+          Real.sqrt (1 + (8189818573610927 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (8189818573610927 / 10000000000000000 : ℝ) /
+                (1 + (1615709636413007 / 1250000000000000 : ℝ)) =
+              (8189818573610927 / 22925677091304056 : ℝ)
+            by
+        norm_num];
+        exact hau920)
+  have hat1 :
+    (Real.pi / 2 - (2 * (85776069303373838337 / 250000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 8189818573610927 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 8189818573610927 : ℝ) =
+              (8189818573610927 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau919)
+  have hz1 :
+    (10000000000000000 / 8189818573610927 : ℝ) ≤
+      1 / Real.sqrt (5365850261493 / 8000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 8189818573610927 : ℝ) =
+          1 / (8189818573610927 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (187500000000000 / 2047454643402731 : ℝ) ≤
+      (182644801696868735909 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (5365850261493 / 8000000000000 : ℝ) ≤
+      (187500000000000 / 2047454643402731 : ℝ) := by
+    rw [show
+        (187500000000000 / 2047454643402731 : ℝ) =
+          (3 / 40) / (2047454643402731 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (85776069303373838337 / 250000000000000000000 : ℝ))) -
+        6 * (182644801696868735909 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (5365850261493 / 8000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (5365850261493 / 8000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2047454643402731 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (85776069303373838337 / 250000000000000000000 : ℝ))) -
+          6 * (182644801696868735909 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (5365850261493 / 8000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (5365850261493 / 8000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (5365850261493 / 8000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_575 :
+    ((5142320697278545517473 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(3886393224119610109169 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((4099859014306257 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (686800789241670516879 / 2000000000000000000000 : ℝ))) -
+              6 * (114015948625701972549 / 1250000000000000000000 : ℝ)))) ≤
+      Vfield (21515320239601 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (21515320239601 / 32000000000000 : ℝ)) =
+      Real.log (53515320239601 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((21515320239601 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (21695320239601 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (5142320697278545517473 / 10000000000000000000000 : ℝ) ≤
+      Real.log (53515320239601 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (21695320239601 / 32000000000000 : ℝ) ≤
+      (-(3886393224119610109169 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (21695320239601 / 32000000000000 : ℝ) = (21695320239601 / 16000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21695320239601 / 16000000000000 : ℝ) ≤
+            (3045078578880389890831 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (4099859014306257 / 5000000000000000 : ℝ) ≤ Real.sqrt (21515320239601 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (21515320239601 / 32000000000000 : ℝ) ≤
+      (8199718028612517 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (21515320239601 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau922 :
+    Real.arctan (8199718028612517 / 22931951737798634 : ℝ) ≤
+      (686800789241670516879 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau921 :
+    Real.arctan (8199718028612517 / 10000000000000000 : ℝ) ≤
+      (2 * (686800789241670516879 / 2000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6465975868899317 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (8199718028612517 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (8199718028612517 / 10000000000000000 : ℝ) /
+                (1 + (6465975868899317 / 5000000000000000 : ℝ)) =
+              (8199718028612517 / 22931951737798634 : ℝ)
+            by
+        norm_num];
+        exact hau922)
+  have hat1 :
+    (Real.pi / 2 - (2 * (686800789241670516879 / 2000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 8199718028612517 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 8199718028612517 : ℝ) =
+              (8199718028612517 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau921)
+  have hz1 :
+    (10000000000000000 / 8199718028612517 : ℝ) ≤
+      1 / Real.sqrt (21515320239601 / 32000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 8199718028612517 : ℝ) =
+          1 / (8199718028612517 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (125000000000000 / 1366619671435419 : ℝ) ≤
+      (114015948625701972549 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (21515320239601 / 32000000000000 : ℝ) ≤
+      (125000000000000 / 1366619671435419 : ℝ) := by
+    rw [show
+        (125000000000000 / 1366619671435419 : ℝ) =
+          (3 / 40) / (4099859014306257 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (686800789241670516879 / 2000000000000000000000 : ℝ))) -
+        6 * (114015948625701972549 / 1250000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (21515320239601 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (21515320239601 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((4099859014306257 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (686800789241670516879 / 2000000000000000000000 : ℝ))) -
+          6 * (114015948625701972549 / 1250000000000000000000 : ℝ))) ≤
+      Real.sqrt (21515320239601 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (21515320239601 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (21515320239601 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_576 :
+    ((5147170393103906195523 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1937217424407472507173 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((8204663276990617 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1717741058573659101171 / 5000000000000000000000 : ℝ))) -
+              6 * (911580858114015239713 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (43082559672831 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (43082559672831 / 64000000000000 : ℝ)) =
+      Real.log (107082559672831 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((43082559672831 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (43442559672831 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (5147170393103906195523 / 10000000000000000000000 : ℝ) ≤
+      Real.log (107082559672831 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (43442559672831 / 64000000000000 : ℝ) ≤
+      (-(1937217424407472507173 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (43442559672831 / 64000000000000 : ℝ) = (43442559672831 / 32000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (43442559672831 / 32000000000000 : ℝ) ≤
+            (1528518477092527492827 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (8204663276990617 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (43082559672831 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (43082559672831 / 64000000000000 : ℝ) ≤ (410233163849531 / 500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (43082559672831 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau924 :
+    Real.arctan (1640932655398124 / 4587017583927859 : ℝ) ≤
+      (1717741058573659101171 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau923 :
+    Real.arctan (410233163849531 / 500000000000000 : ℝ) ≤
+      (2 * (1717741058573659101171 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (2587017583927859 / 2000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (410233163849531 / 500000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (410233163849531 / 500000000000000 : ℝ) /
+                (1 + (2587017583927859 / 2000000000000000 : ℝ)) =
+              (1640932655398124 / 4587017583927859 : ℝ)
+            by
+        norm_num];
+        exact hau924)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1717741058573659101171 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (500000000000000 / 410233163849531 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (500000000000000 / 410233163849531 : ℝ) =
+              (410233163849531 / 500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau923)
+  have hz1 :
+    (500000000000000 / 410233163849531 : ℝ) ≤
+      1 / Real.sqrt (43082559672831 / 64000000000000 : ℝ) := by
+    rw [show (500000000000000 / 410233163849531 : ℝ) = 1 / (410233163849531 / 500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 8204663276990617 : ℝ) ≤
+      (911580858114015239713 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (43082559672831 / 64000000000000 : ℝ) ≤
+      (750000000000000 / 8204663276990617 : ℝ) := by
+    rw [show
+        (750000000000000 / 8204663276990617 : ℝ) =
+          (3 / 40) / (8204663276990617 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1717741058573659101171 / 5000000000000000000000 : ℝ))) -
+        6 * (911580858114015239713 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (43082559672831 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (43082559672831 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((8204663276990617 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1717741058573659101171 / 5000000000000000000000 : ℝ))) -
+          6 * (911580858114015239713 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (43082559672831 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (43082559672831 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (43082559672831 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/V48.lean b/LeanPool/Zeta5Irrational/Table/V48.lean
new file mode 100644
index 0000000000..544413654c
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/V48.lean
@@ -0,0 +1,1442 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Energy
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # V48: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_577 :
+    ((5152017738114334704557 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(3862490756705591713337 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((8209605546482957 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1718479340793304782101 / 5000000000000000000000 : ℝ))) -
+              6 * (911035109185732816891 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (2156723943323 / 3200000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (2156723943323 / 3200000000000 : ℝ)) =
+      Real.log (5356723943323 / 3200000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((2156723943323 / 3200000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (2174723943323 / 3200000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (5152017738114334704557 / 10000000000000000000000 : ℝ) ≤
+      Real.log (5356723943323 / 3200000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (2174723943323 / 3200000000000 : ℝ) ≤
+      (-(3862490756705591713337 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (2174723943323 / 3200000000000 : ℝ) = (2174723943323 / 1600000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2174723943323 / 1600000000000 : ℝ) ≤
+            (3068981046294408286663 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (8209605546482957 / 10000000000000000 : ℝ) ≤ Real.sqrt (2156723943323 / 3200000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (2156723943323 / 3200000000000 : ℝ) ≤ (102620069331037 / 125000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (2156723943323 / 3200000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau926 :
+    Real.arctan (2052401386620740 / 5734555835320011 : ℝ) ≤
+      (1718479340793304782101 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau925 :
+    Real.arctan (102620069331037 / 125000000000000 : ℝ) ≤
+      (2 * (1718479340793304782101 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (3234555835320011 / 2500000000000000 : ℝ) ≤
+          Real.sqrt (1 + (102620069331037 / 125000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (102620069331037 / 125000000000000 : ℝ) /
+                (1 + (3234555835320011 / 2500000000000000 : ℝ)) =
+              (2052401386620740 / 5734555835320011 : ℝ)
+            by
+        norm_num];
+        exact hau926)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1718479340793304782101 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (125000000000000 / 102620069331037 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (125000000000000 / 102620069331037 : ℝ) =
+              (102620069331037 / 125000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau925)
+  have hz1 :
+    (125000000000000 / 102620069331037 : ℝ) ≤ 1 / Real.sqrt (2156723943323 / 3200000000000 : ℝ) :=
+  by
+    rw [show (125000000000000 / 102620069331037 : ℝ) = 1 / (102620069331037 / 125000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 8209605546482957 : ℝ) ≤
+      (911035109185732816891 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (2156723943323 / 3200000000000 : ℝ) ≤
+      (750000000000000 / 8209605546482957 : ℝ) := by
+    rw [show
+        (750000000000000 / 8209605546482957 : ℝ) =
+          (3 / 40) / (8209605546482957 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1718479340793304782101 / 5000000000000000000000 : ℝ))) -
+        6 * (911035109185732816891 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (2156723943323 / 3200000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (2156723943323 / 3200000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((8209605546482957 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1718479340793304782101 / 5000000000000000000000 : ℝ))) -
+          6 * (911035109185732816891 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (2156723943323 / 3200000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (2156723943323 / 3200000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (2156723943323 / 3200000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_578 :
+    ((1031372546917553828993 / 2000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(3850560913712286605501 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((410727242123313 / 500000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3438433642688509428973 / 10000000000000000000000 : ℝ))) -
+              6 * (9104903392848186487 / 100000000000000000000 : ℝ)))) ≤
+      Vfield (43186398060089 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (43186398060089 / 64000000000000 : ℝ)) =
+      Real.log (107186398060089 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((43186398060089 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (43546398060089 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1031372546917553828993 / 2000000000000000000000 : ℝ) ≤
+      Real.log (107186398060089 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (43546398060089 / 64000000000000 : ℝ) ≤
+      (-(3850560913712286605501 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (43546398060089 / 64000000000000 : ℝ) = (43546398060089 / 32000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (43546398060089 / 32000000000000 : ℝ) ≤
+            (3080910889287713394499 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (410727242123313 / 500000000000000 : ℝ) ≤ Real.sqrt (43186398060089 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (43186398060089 / 64000000000000 : ℝ) ≤
+      (8214544842466263 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (43186398060089 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau928 :
+    Real.arctan (2738181614155421 / 7647119334424474 : ℝ) ≤
+      (3438433642688509428973 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau927 :
+    Real.arctan (8214544842466263 / 10000000000000000 : ℝ) ≤
+      (2 * (3438433642688509428973 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6470679001636711 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (8214544842466263 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (8214544842466263 / 10000000000000000 : ℝ) /
+                (1 + (6470679001636711 / 5000000000000000 : ℝ)) =
+              (2738181614155421 / 7647119334424474 : ℝ)
+            by
+        norm_num];
+        exact hau928)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3438433642688509428973 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 8214544842466263 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 8214544842466263 : ℝ) =
+              (8214544842466263 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau927)
+  have hz1 :
+    (10000000000000000 / 8214544842466263 : ℝ) ≤
+      1 / Real.sqrt (43186398060089 / 64000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 8214544842466263 : ℝ) =
+          1 / (8214544842466263 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (12500000000000 / 136909080707771 : ℝ) ≤
+      (9104903392848186487 / 100000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (43186398060089 / 64000000000000 : ℝ) ≤
+      (12500000000000 / 136909080707771 : ℝ) := by
+    rw [show
+        (12500000000000 / 136909080707771 : ℝ) =
+          (3 / 40) / (410727242123313 / 500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3438433642688509428973 / 10000000000000000000000 : ℝ))) -
+        6 * (9104903392848186487 / 100000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (43186398060089 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (43186398060089 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((410727242123313 / 500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3438433642688509428973 / 10000000000000000000000 : ℝ))) -
+          6 * (9104903392848186487 / 100000000000000000000 : ℝ))) ≤
+      Real.sqrt (43186398060089 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (43186398060089 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (43186398060089 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_579 :
+    ((5161705384798838239711 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(959661321469397220031 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((8219481170301101 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1719953501802942922099 / 5000000000000000000000 : ℝ))) -
+              6 * (227486636371899300713 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (21619158626859 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (21619158626859 / 32000000000000 : ℝ)) =
+      Real.log (53619158626859 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((21619158626859 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (21799158626859 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (5161705384798838239711 / 10000000000000000000000 : ℝ) ≤
+      Real.log (53619158626859 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (21799158626859 / 32000000000000 : ℝ) ≤
+      (-(959661321469397220031 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (21799158626859 / 32000000000000 : ℝ) = (21799158626859 / 16000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21799158626859 / 16000000000000 : ℝ) ≤
+            (773206629280602779969 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (8219481170301101 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (21619158626859 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (21619158626859 / 32000000000000 : ℝ) ≤ (513717573143819 / 625000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (21619158626859 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau930 :
+    Real.arctan (8219481170301104 / 22944491906171303 : ℝ) ≤
+      (1719953501802942922099 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau929 :
+    Real.arctan (513717573143819 / 625000000000000 : ℝ) ≤
+      (2 * (1719953501802942922099 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12944491906171303 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (513717573143819 / 625000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (513717573143819 / 625000000000000 : ℝ) /
+                (1 + (12944491906171303 / 10000000000000000 : ℝ)) =
+              (8219481170301104 / 22944491906171303 : ℝ)
+            by
+        norm_num];
+        exact hau930)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1719953501802942922099 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (625000000000000 / 513717573143819 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (625000000000000 / 513717573143819 : ℝ) =
+              (513717573143819 / 625000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau929)
+  have hz1 :
+    (625000000000000 / 513717573143819 : ℝ) ≤
+      1 / Real.sqrt (21619158626859 / 32000000000000 : ℝ) := by
+    rw [show (625000000000000 / 513717573143819 : ℝ) = 1 / (513717573143819 / 625000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 8219481170301101 : ℝ) ≤
+      (227486636371899300713 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (21619158626859 / 32000000000000 : ℝ) ≤
+      (750000000000000 / 8219481170301101 : ℝ) := by
+    rw [show
+        (750000000000000 / 8219481170301101 : ℝ) =
+          (3 / 40) / (8219481170301101 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1719953501802942922099 / 5000000000000000000000 : ℝ))) -
+        6 * (227486636371899300713 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (21619158626859 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (21619158626859 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((8219481170301101 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1719953501802942922099 / 5000000000000000000000 : ℝ))) -
+          6 * (227486636371899300713 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (21619158626859 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (21619158626859 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (21619158626859 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_580 :
+    ((5166545691018867741703 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1913371919682650284589 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((8224414535331963 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (860344691870556942057 / 2500000000000000000000 : ℝ))) -
+              6 * (454701862441301176601 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (43290236447347 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (43290236447347 / 64000000000000 : ℝ)) =
+      Real.log (107290236447347 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((43290236447347 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (43650236447347 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (5166545691018867741703 / 10000000000000000000000 : ℝ) ≤
+      Real.log (107290236447347 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (43650236447347 / 64000000000000 : ℝ) ≤
+      (-(1913371919682650284589 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (43650236447347 / 64000000000000 : ℝ) = (43650236447347 / 32000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (43650236447347 / 32000000000000 : ℝ) ≤
+            (1552363981817349715411 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (8224414535331963 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (43290236447347 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (43290236447347 / 64000000000000 : ℝ) ≤ (4112207267665983 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (43290236447347 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau932 :
+    Real.arctan (8224414535331966 / 22947625050524891 : ℝ) ≤
+      (860344691870556942057 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau931 :
+    Real.arctan (4112207267665983 / 5000000000000000 : ℝ) ≤
+      (2 * (860344691870556942057 / 2500000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12947625050524891 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (4112207267665983 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (4112207267665983 / 5000000000000000 : ℝ) /
+                (1 + (12947625050524891 / 10000000000000000 : ℝ)) =
+              (8224414535331966 / 22947625050524891 : ℝ)
+            by
+        norm_num];
+        exact hau932)
+  have hat1 :
+    (Real.pi / 2 - (2 * (860344691870556942057 / 2500000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 4112207267665983 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 4112207267665983 : ℝ) =
+              (4112207267665983 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau931)
+  have hz1 :
+    (5000000000000000 / 4112207267665983 : ℝ) ≤
+      1 / Real.sqrt (43290236447347 / 64000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 4112207267665983 : ℝ) = 1 / (4112207267665983 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 2741471511777321 : ℝ) ≤
+      (454701862441301176601 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (43290236447347 / 64000000000000 : ℝ) ≤
+      (250000000000000 / 2741471511777321 : ℝ) := by
+    rw [show
+        (250000000000000 / 2741471511777321 : ℝ) =
+          (3 / 40) / (8224414535331963 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (860344691870556942057 / 2500000000000000000000 : ℝ))) -
+        6 * (454701862441301176601 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (43290236447347 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (43290236447347 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((8224414535331963 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (860344691870556942057 / 2500000000000000000000 : ℝ))) -
+          6 * (454701862441301176601 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (43290236447347 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (43290236447347 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (43290236447347 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_581 :
+    ((2585691827757943412823 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1907428270229945023543 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((514334058930457 / 625000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (860712234362921461299 / 2500000000000000000000 : ℝ))) -
+              6 * (908861874570511082349 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (2708884727561 / 4000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (2708884727561 / 4000000000000 : ℝ)) =
+      Real.log (6708884727561 / 4000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((2708884727561 / 4000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (2731384727561 / 4000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2585691827757943412823 / 5000000000000000000000 : ℝ) ≤
+      Real.log (6708884727561 / 4000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (2731384727561 / 4000000000000 : ℝ) ≤
+      (-(1907428270229945023543 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (2731384727561 / 4000000000000 : ℝ) = (2731384727561 / 2000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2731384727561 / 2000000000000 : ℝ) ≤
+            (1558307631270054976457 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (514334058930457 / 625000000000000 : ℝ) ≤ Real.sqrt (2708884727561 / 4000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (2708884727561 / 4000000000000 : ℝ) ≤ (1645868988577463 / 2000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (2708884727561 / 4000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau934 :
+    Real.arctan (8229344942887315 / 22950757436884724 : ℝ) ≤
+      (860712234362921461299 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau933 :
+    Real.arctan (1645868988577463 / 2000000000000000 : ℝ) ≤
+      (2 * (860712234362921461299 / 2500000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (3237689359221181 / 2500000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1645868988577463 / 2000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1645868988577463 / 2000000000000000 : ℝ) /
+                (1 + (3237689359221181 / 2500000000000000 : ℝ)) =
+              (8229344942887315 / 22950757436884724 : ℝ)
+            by
+        norm_num];
+        exact hau934)
+  have hat1 :
+    (Real.pi / 2 - (2 * (860712234362921461299 / 2500000000000000000000 : ℝ))) ≤
+      Real.arctan (2000000000000000 / 1645868988577463 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2000000000000000 / 1645868988577463 : ℝ) =
+              (1645868988577463 / 2000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau933)
+  have hz1 :
+    (2000000000000000 / 1645868988577463 : ℝ) ≤
+      1 / Real.sqrt (2708884727561 / 4000000000000 : ℝ) := by
+    rw [show
+        (2000000000000000 / 1645868988577463 : ℝ) = 1 / (1645868988577463 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (46875000000000 / 514334058930457 : ℝ) ≤
+      (908861874570511082349 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (2708884727561 / 4000000000000 : ℝ) ≤
+      (46875000000000 / 514334058930457 : ℝ) := by
+    rw [show
+        (46875000000000 / 514334058930457 : ℝ) =
+          (3 / 40) / (514334058930457 / 625000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (860712234362921461299 / 2500000000000000000000 : ℝ))) -
+        6 * (908861874570511082349 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (2708884727561 / 4000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (2708884727561 / 4000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((514334058930457 / 625000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (860712234362921461299 / 2500000000000000000000 : ℝ))) -
+          6 * (908861874570511082349 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (2708884727561 / 4000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (2708884727561 / 4000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (2708884727561 / 4000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_582 :
+    ((2588109640277317232227 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(3802983355565918952933 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((8234272398279661 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (430539689579888660491 / 1250000000000000000000 : ℝ))) -
+              6 * (908320991664078785569 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (8678814966921 / 12800000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (8678814966921 / 12800000000000 : ℝ)) =
+      Real.log (21478814966921 / 12800000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((8678814966921 / 12800000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (8750814966921 / 12800000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2588109640277317232227 / 5000000000000000000000 : ℝ) ≤
+      Real.log (21478814966921 / 12800000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (8750814966921 / 12800000000000 : ℝ) ≤
+      (-(3802983355565918952933 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (8750814966921 / 12800000000000 : ℝ) = (8750814966921 / 6400000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8750814966921 / 6400000000000 : ℝ) ≤
+            (3128488447434081047067 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (8234272398279661 / 10000000000000000 : ℝ) ≤ Real.sqrt (8678814966921 / 12800000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (8678814966921 / 12800000000000 : ℝ) ≤ (514642024892479 / 625000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (8678814966921 / 12800000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau936 :
+    Real.arctan (8234272398279664 / 22953889065800675 : ℝ) ≤
+      (430539689579888660491 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau935 :
+    Real.arctan (514642024892479 / 625000000000000 : ℝ) ≤
+      (2 * (430539689579888660491 / 1250000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (518155562632027 / 400000000000000 : ℝ) ≤
+          Real.sqrt (1 + (514642024892479 / 625000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (514642024892479 / 625000000000000 : ℝ) /
+                (1 + (518155562632027 / 400000000000000 : ℝ)) =
+              (8234272398279664 / 22953889065800675 : ℝ)
+            by
+        norm_num];
+        exact hau936)
+  have hat1 :
+    (Real.pi / 2 - (2 * (430539689579888660491 / 1250000000000000000000 : ℝ))) ≤
+      Real.arctan (625000000000000 / 514642024892479 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (625000000000000 / 514642024892479 : ℝ) =
+              (514642024892479 / 625000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau935)
+  have hz1 :
+    (625000000000000 / 514642024892479 : ℝ) ≤
+      1 / Real.sqrt (8678814966921 / 12800000000000 : ℝ) := by
+    rw [show (625000000000000 / 514642024892479 : ℝ) = 1 / (514642024892479 / 625000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 8234272398279661 : ℝ) ≤
+      (908320991664078785569 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (8678814966921 / 12800000000000 : ℝ) ≤
+      (750000000000000 / 8234272398279661 : ℝ) := by
+    rw [show
+        (750000000000000 / 8234272398279661 : ℝ) =
+          (3 / 40) / (8234272398279661 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (430539689579888660491 / 1250000000000000000000 : ℝ))) -
+        6 * (908320991664078785569 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (8678814966921 / 12800000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (8678814966921 / 12800000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((8234272398279661 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (430539689579888660491 / 1250000000000000000000 : ℝ))) -
+          6 * (908320991664078785569 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (8678814966921 / 12800000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (8678814966921 / 12800000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (8678814966921 / 12800000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_583 :
+    ((2590526284198282895123 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(473890531400934063847 / 1250000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((4119598453402819 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (430723063520010527707 / 1250000000000000000000 : ℝ))) -
+              6 * (181556214657614903937 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (21722997014117 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (21722997014117 / 32000000000000 : ℝ)) =
+      Real.log (53722997014117 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((21722997014117 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (21902997014117 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2590526284198282895123 / 5000000000000000000000 : ℝ) ≤
+      Real.log (53722997014117 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (21902997014117 / 32000000000000 : ℝ) ≤
+      (-(473890531400934063847 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (21902997014117 / 32000000000000 : ℝ) = (21902997014117 / 16000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (21902997014117 / 16000000000000 : ℝ) ≤
+            (392543443974065936153 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (4119598453402819 / 5000000000000000 : ℝ) ≤ Real.sqrt (21722997014117 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (21722997014117 / 32000000000000 : ℝ) ≤
+      (8239196906805641 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (21722997014117 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau938 :
+    Real.arctan (8239196906805641 / 22957019937821953 : ℝ) ≤
+      (430723063520010527707 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau937 :
+    Real.arctan (8239196906805641 / 10000000000000000 : ℝ) ≤
+      (2 * (430723063520010527707 / 1250000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12957019937821953 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (8239196906805641 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (8239196906805641 / 10000000000000000 : ℝ) /
+                (1 + (12957019937821953 / 10000000000000000 : ℝ)) =
+              (8239196906805641 / 22957019937821953 : ℝ)
+            by
+        norm_num];
+        exact hau938)
+  have hat1 :
+    (Real.pi / 2 - (2 * (430723063520010527707 / 1250000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 8239196906805641 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 8239196906805641 : ℝ) =
+              (8239196906805641 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau937)
+  have hz1 :
+    (10000000000000000 / 8239196906805641 : ℝ) ≤
+      1 / Real.sqrt (21722997014117 / 32000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 8239196906805641 : ℝ) =
+          1 / (8239196906805641 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 4119598453402819 : ℝ) ≤
+      (181556214657614903937 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (21722997014117 / 32000000000000 : ℝ) ≤
+      (375000000000000 / 4119598453402819 : ℝ) := by
+    rw [show
+        (375000000000000 / 4119598453402819 : ℝ) =
+          (3 / 40) / (4119598453402819 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (430723063520010527707 / 1250000000000000000000 : ℝ))) -
+        6 * (181556214657614903937 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (21722997014117 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (21722997014117 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((4119598453402819 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (430723063520010527707 / 1250000000000000000000 : ℝ))) -
+          6 * (181556214657614903937 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (21722997014117 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (21722997014117 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (21722997014117 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_584 :
+    ((2592941760649929219989 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(755855838805518644779 / 2000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((8244118473746051 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3447249915120971146511 / 10000000000000000000000 : ℝ))) -
+              6 * (181448423315843392787 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (43497913221863 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (43497913221863 / 64000000000000 : ℝ)) =
+      Real.log (107497913221863 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((43497913221863 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (43857913221863 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2592941760649929219989 / 5000000000000000000000 : ℝ) ≤
+      Real.log (107497913221863 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (43857913221863 / 64000000000000 : ℝ) ≤
+      (-(755855838805518644779 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (43857913221863 / 64000000000000 : ℝ) = (43857913221863 / 32000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (43857913221863 / 32000000000000 : ℝ) ≤
+            (630438521794481355221 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (8244118473746051 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (43497913221863 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (43497913221863 / 64000000000000 : ℝ) ≤ (4122059236873027 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (43497913221863 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau940 :
+    Real.arctan (4122059236873027 / 11480075026748551 : ℝ) ≤
+      (3447249915120971146511 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau939 :
+    Real.arctan (4122059236873027 / 5000000000000000 : ℝ) ≤
+      (2 * (3447249915120971146511 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6480075026748551 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (4122059236873027 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (4122059236873027 / 5000000000000000 : ℝ) /
+                (1 + (6480075026748551 / 5000000000000000 : ℝ)) =
+              (4122059236873027 / 11480075026748551 : ℝ)
+            by
+        norm_num];
+        exact hau940)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3447249915120971146511 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 4122059236873027 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 4122059236873027 : ℝ) =
+              (4122059236873027 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau939)
+  have hz1 :
+    (5000000000000000 / 4122059236873027 : ℝ) ≤
+      1 / Real.sqrt (43497913221863 / 64000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 4122059236873027 : ℝ) = 1 / (4122059236873027 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 8244118473746051 : ℝ) ≤
+      (181448423315843392787 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (43497913221863 / 64000000000000 : ℝ) ≤
+      (750000000000000 / 8244118473746051 : ℝ) := by
+    rw [show
+        (750000000000000 / 8244118473746051 : ℝ) =
+          (3 / 40) / (8244118473746051 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3447249915120971146511 / 10000000000000000000000 : ℝ))) -
+        6 * (181448423315843392787 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (43497913221863 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (43497913221863 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((8244118473746051 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3447249915120971146511 / 10000000000000000000000 : ℝ))) -
+          6 * (181448423315843392787 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (43497913221863 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (43497913221863 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (43497913221863 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_585 :
+    ((5190712141519418885757 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(75348963015754358381 / 200000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((8249037104365953 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2 * (215544608788683865353 / 625000000000000000000 : ℝ))) -
+              6 * (113338014835763844939 / 1250000000000000000000 : ℝ)))) ≤
+      Vfield (10887458103873 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (10887458103873 / 16000000000000 : ℝ)) =
+      Real.log (26887458103873 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((10887458103873 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (10977458103873 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (5190712141519418885757 / 10000000000000000000000 : ℝ) ≤
+      Real.log (26887458103873 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (10977458103873 / 16000000000000 : ℝ) ≤
+      (-(75348963015754358381 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (10977458103873 / 16000000000000 : ℝ) = (10977458103873 / 8000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (10977458103873 / 8000000000000 : ℝ) ≤
+            (63280473044245641619 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (8249037104365953 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (10887458103873 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (10887458103873 / 16000000000000 : ℝ) ≤ (2062259276091489 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (10887458103873 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau942 :
+    Real.arctan (8249037104365956 / 22963279413374003 : ℝ) ≤
+      (215544608788683865353 / 625000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau941 :
+    Real.arctan (2062259276091489 / 2500000000000000 : ℝ) ≤
+      (2 * (215544608788683865353 / 625000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12963279413374003 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (2062259276091489 / 2500000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (2062259276091489 / 2500000000000000 : ℝ) /
+                (1 + (12963279413374003 / 10000000000000000 : ℝ)) =
+              (8249037104365956 / 22963279413374003 : ℝ)
+            by
+        norm_num];
+        exact hau942)
+  have hat1 :
+    (Real.pi / 2 - (2 * (215544608788683865353 / 625000000000000000000 : ℝ))) ≤
+      Real.arctan (2500000000000000 / 2062259276091489 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 2062259276091489 : ℝ) =
+              (2062259276091489 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau941)
+  have hz1 :
+    (2500000000000000 / 2062259276091489 : ℝ) ≤
+      1 / Real.sqrt (10887458103873 / 16000000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 2062259276091489 : ℝ) = 1 / (2062259276091489 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 2749679034788651 : ℝ) ≤
+      (113338014835763844939 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (10887458103873 / 16000000000000 : ℝ) ≤
+      (250000000000000 / 2749679034788651 : ℝ) := by
+    rw [show
+        (250000000000000 / 2749679034788651 : ℝ) =
+          (3 / 40) / (8249037104365953 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (215544608788683865353 / 625000000000000000000 : ℝ))) -
+        6 * (113338014835763844939 / 1250000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (10887458103873 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (10887458103873 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((8249037104365953 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (215544608788683865353 / 625000000000000000000 : ℝ))) -
+          6 * (113338014835763844939 / 1250000000000000000000 : ℝ))) ≤
+      Real.sqrt (10887458103873 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (10887458103873 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (10887458103873 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_586 :
+    ((41564307450455109999 / 80000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(3755631088367118116969 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1031744100489339 / 1250000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1725087993871008903277 / 5000000000000000000000 : ℝ))) -
+              6 * (1132708845961478349 / 12500000000000000000 : ℝ)))) ≤
+      Vfield (43601751609121 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (43601751609121 / 64000000000000 : ℝ)) =
+      Real.log (107601751609121 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((43601751609121 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (43961751609121 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (41564307450455109999 / 80000000000000000000 : ℝ) ≤
+      Real.log (107601751609121 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (43961751609121 / 64000000000000 : ℝ) ≤
+      (-(3755631088367118116969 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (43961751609121 / 64000000000000 : ℝ) = (43961751609121 / 32000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (43961751609121 / 32000000000000 : ℝ) ≤
+            (3175840714632881883031 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1031744100489339 / 1250000000000000 : ℝ) ≤ Real.sqrt (43601751609121 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (43601751609121 / 64000000000000 : ℝ) ≤ (1650790560782943 / 2000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (43601751609121 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau944 :
+    Real.arctan (2751317601304905 / 7655469339333293 : ℝ) ≤
+      (1725087993871008903277 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau943 :
+    Real.arctan (1650790560782943 / 2000000000000000 : ℝ) ≤
+      (2 * (1725087993871008903277 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12966408017999879 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1650790560782943 / 2000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1650790560782943 / 2000000000000000 : ℝ) /
+                (1 + (12966408017999879 / 10000000000000000 : ℝ)) =
+              (2751317601304905 / 7655469339333293 : ℝ)
+            by
+        norm_num];
+        exact hau944)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1725087993871008903277 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (2000000000000000 / 1650790560782943 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2000000000000000 / 1650790560782943 : ℝ) =
+              (1650790560782943 / 2000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau943)
+  have hz1 :
+    (2000000000000000 / 1650790560782943 : ℝ) ≤
+      1 / Real.sqrt (43601751609121 / 64000000000000 : ℝ) := by
+    rw [show
+        (2000000000000000 / 1650790560782943 : ℝ) = 1 / (1650790560782943 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (31250000000000 / 343914700163113 : ℝ) ≤
+      (1132708845961478349 / 12500000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (43601751609121 / 64000000000000 : ℝ) ≤
+      (31250000000000 / 343914700163113 : ℝ) := by
+    rw [show
+        (31250000000000 / 343914700163113 : ℝ) =
+          (3 / 40) / (1031744100489339 / 1250000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1725087993871008903277 / 5000000000000000000000 : ℝ))) -
+        6 * (1132708845961478349 / 12500000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (43601751609121 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (43601751609121 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1031744100489339 / 1250000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1725087993871008903277 / 5000000000000000000000 : ℝ))) -
+          6 * (1132708845961478349 / 12500000000000000000 : ℝ))) ≤
+      Real.sqrt (43601751609121 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (43601751609121 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (43601751609121 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_587 :
+    ((2600181196455325552307 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(58497312090036620399 / 156250000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((8258865577626073 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2 * (53931822805767294657 / 156250000000000000000 : ℝ))) -
+              6 * (226407747000154761683 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (174614683211 / 256000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (174614683211 / 256000000000 : ℝ)) =
+      Real.log (430614683211 / 256000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((174614683211 / 256000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (176054683211 / 256000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2600181196455325552307 / 5000000000000000000000 : ℝ) ≤
+      Real.log (430614683211 / 256000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (176054683211 / 256000000000 : ℝ) ≤
+      (-(58497312090036620399 / 156250000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (176054683211 / 256000000000 : ℝ) = (176054683211 / 128000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (176054683211 / 128000000000 : ℝ) ≤
+            (49806934831838379601 / 156250000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (8258865577626073 / 10000000000000000 : ℝ) ≤ Real.sqrt (174614683211 / 256000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (174614683211 / 256000000000 : ℝ) ≤ (2064716394406519 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (174614683211 / 256000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau946 :
+    Real.arctan (4129432788813038 / 11484767933960645 : ℝ) ≤
+      (53931822805767294657 / 156250000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau945 :
+    Real.arctan (2064716394406519 / 2500000000000000 : ℝ) ≤
+      (2 * (53931822805767294657 / 156250000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (1296953586792129 / 1000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (2064716394406519 / 2500000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (2064716394406519 / 2500000000000000 : ℝ) /
+                (1 + (1296953586792129 / 1000000000000000 : ℝ)) =
+              (4129432788813038 / 11484767933960645 : ℝ)
+            by
+        norm_num];
+        exact hau946)
+  have hat1 :
+    (Real.pi / 2 - (2 * (53931822805767294657 / 156250000000000000000 : ℝ))) ≤
+      Real.arctan (2500000000000000 / 2064716394406519 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 2064716394406519 : ℝ) =
+              (2064716394406519 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau945)
+  have hz1 :
+    (2500000000000000 / 2064716394406519 : ℝ) ≤ 1 / Real.sqrt (174614683211 / 256000000000 : ℝ) :=
+  by
+    rw [show
+        (2500000000000000 / 2064716394406519 : ℝ) = 1 / (2064716394406519 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 8258865577626073 : ℝ) ≤
+      (226407747000154761683 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (174614683211 / 256000000000 : ℝ) ≤
+      (750000000000000 / 8258865577626073 : ℝ) := by
+    rw [show
+        (750000000000000 / 8258865577626073 : ℝ) =
+          (3 / 40) / (8258865577626073 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (53931822805767294657 / 156250000000000000000 : ℝ))) -
+        6 * (226407747000154761683 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (174614683211 / 256000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (174614683211 / 256000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((8258865577626073 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (53931822805767294657 / 156250000000000000000 : ℝ))) -
+          6 * (226407747000154761683 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (174614683211 / 256000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (174614683211 / 256000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (174614683211 / 256000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_588 :
+    ((2602592014287918378433 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1866019387043334946147 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((516485964419889 / 625000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2 * (215818484948127477083 / 625000000000000000000 : ℝ))) -
+              6 * (452547924782151544459 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (43705589996379 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (43705589996379 / 64000000000000 : ℝ)) =
+      Real.log (107705589996379 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((43705589996379 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (44065589996379 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2602592014287918378433 / 5000000000000000000000 : ℝ) ≤
+      Real.log (107705589996379 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (44065589996379 / 64000000000000 : ℝ) ≤
+      (-(1866019387043334946147 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (44065589996379 / 64000000000000 : ℝ) = (44065589996379 / 32000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (44065589996379 / 32000000000000 : ℝ) ≤
+            (1599716514456665053853 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (516485964419889 / 625000000000000 : ℝ) ≤ Real.sqrt (43705589996379 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (43705589996379 / 64000000000000 : ℝ) ≤
+      (8263775430718227 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (43705589996379 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau948 :
+    Real.arctan (1180539347245461 / 3281808994812020 : ℝ) ≤
+      (215818484948127477083 / 625000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau947 :
+    Real.arctan (8263775430718227 / 10000000000000000 : ℝ) ≤
+      (2 * (215818484948127477083 / 625000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (648633148184207 / 500000000000000 : ℝ) ≤
+          Real.sqrt (1 + (8263775430718227 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (8263775430718227 / 10000000000000000 : ℝ) /
+                (1 + (648633148184207 / 500000000000000 : ℝ)) =
+              (1180539347245461 / 3281808994812020 : ℝ)
+            by
+        norm_num];
+        exact hau948)
+  have hat1 :
+    (Real.pi / 2 - (2 * (215818484948127477083 / 625000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 8263775430718227 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 8263775430718227 : ℝ) =
+              (8263775430718227 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau947)
+  have hz1 :
+    (10000000000000000 / 8263775430718227 : ℝ) ≤
+      1 / Real.sqrt (43705589996379 / 64000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 8263775430718227 : ℝ) =
+          1 / (8263775430718227 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (15625000000000 / 172161988139963 : ℝ) ≤
+      (452547924782151544459 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (43705589996379 / 64000000000000 : ℝ) ≤
+      (15625000000000 / 172161988139963 : ℝ) := by
+    rw [show
+        (15625000000000 / 172161988139963 : ℝ) =
+          (3 / 40) / (516485964419889 / 625000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (215818484948127477083 / 625000000000000000000 : ℝ))) -
+        6 * (452547924782151544459 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (43705589996379 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (43705589996379 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((516485964419889 / 625000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (215818484948127477083 / 625000000000000000000 : ℝ))) -
+          6 * (452547924782151544459 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (43705589996379 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (43705589996379 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (43705589996379 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/V49.lean b/LeanPool/Zeta5Irrational/Table/V49.lean
new file mode 100644
index 0000000000..4f09bc7e02
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/V49.lean
@@ -0,0 +1,1442 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Energy
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # V49: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_589 :
+    ((5210003340544330517601 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(3720263456569547413113 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((8268682368393859 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3454553289605606480941 / 10000000000000000000000 : ℝ))) -
+              6 * (452280829327876491589 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (5469688648751 / 8000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (5469688648751 / 8000000000000 : ℝ)) =
+      Real.log (13469688648751 / 8000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((5469688648751 / 8000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (5514688648751 / 8000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (5210003340544330517601 / 10000000000000000000000 : ℝ) ≤
+      Real.log (13469688648751 / 8000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (5514688648751 / 8000000000000 : ℝ) ≤
+      (-(3720263456569547413113 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (5514688648751 / 8000000000000 : ℝ) = (5514688648751 / 4000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5514688648751 / 4000000000000 : ℝ) ≤
+            (3211208346430452586887 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (8268682368393859 / 10000000000000000 : ℝ) ≤ Real.sqrt (5469688648751 / 8000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (5469688648751 / 8000000000000 : ℝ) ≤ (4134341184196931 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (5469688648751 / 8000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau950 :
+    Real.arctan (4134341184196931 / 11487894652916836 : ℝ) ≤
+      (3454553289605606480941 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau949 :
+    Real.arctan (4134341184196931 / 5000000000000000 : ℝ) ≤
+      (2 * (3454553289605606480941 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (1621973663229209 / 1250000000000000 : ℝ) ≤
+          Real.sqrt (1 + (4134341184196931 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (4134341184196931 / 5000000000000000 : ℝ) /
+                (1 + (1621973663229209 / 1250000000000000 : ℝ)) =
+              (4134341184196931 / 11487894652916836 : ℝ)
+            by
+        norm_num];
+        exact hau950)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3454553289605606480941 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 4134341184196931 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 4134341184196931 : ℝ) =
+              (4134341184196931 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau949)
+  have hz1 :
+    (5000000000000000 / 4134341184196931 : ℝ) ≤
+      1 / Real.sqrt (5469688648751 / 8000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 4134341184196931 : ℝ) = 1 / (4134341184196931 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 8268682368393859 : ℝ) ≤
+      (452280829327876491589 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (5469688648751 / 8000000000000 : ℝ) ≤
+      (750000000000000 / 8268682368393859 : ℝ) := by
+    rw [show
+        (750000000000000 / 8268682368393859 : ℝ) =
+          (3 / 40) / (8268682368393859 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3454553289605606480941 / 10000000000000000000000 : ℝ))) -
+        6 * (452280829327876491589 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (5469688648751 / 8000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (5469688648751 / 8000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((8268682368393859 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3454553289605606480941 / 10000000000000000000000 : ℝ))) -
+          6 * (452280829327876491589 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (5469688648751 / 8000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (5469688648751 / 8000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (5469688648751 / 8000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_590 :
+    ((1303705082763694364061 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(370850198855605597403 / 1000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((4136793197920123 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2 * (138240370157103788799 / 400000000000000000000 : ℝ))) -
+              6 * (452014206241029862917 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (43809428383637 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (43809428383637 / 64000000000000 : ℝ)) =
+      Real.log (107809428383637 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((43809428383637 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (44169428383637 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1303705082763694364061 / 2500000000000000000000 : ℝ) ≤
+      Real.log (107809428383637 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (44169428383637 / 64000000000000 : ℝ) ≤
+      (-(370850198855605597403 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (44169428383637 / 64000000000000 : ℝ) = (44169428383637 / 32000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (44169428383637 / 32000000000000 : ℝ) ≤
+            (322296981444394402597 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (4136793197920123 / 5000000000000000 : ℝ) ≤ Real.sqrt (43809428383637 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (43809428383637 / 64000000000000 : ℝ) ≤
+      (8273586395840249 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (43809428383637 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau952 :
+    Real.arctan (285296082615181 / 792376375686706 : ℝ) ≤
+      (138240370157103788799 / 400000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau951 :
+    Real.arctan (8273586395840249 / 10000000000000000 : ℝ) ≤
+      (2 * (138240370157103788799 / 400000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6489457447457237 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (8273586395840249 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (8273586395840249 / 10000000000000000 : ℝ) /
+                (1 + (6489457447457237 / 5000000000000000 : ℝ)) =
+              (285296082615181 / 792376375686706 : ℝ)
+            by
+        norm_num];
+        exact hau952)
+  have hat1 :
+    (Real.pi / 2 - (2 * (138240370157103788799 / 400000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 8273586395840249 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 8273586395840249 : ℝ) =
+              (8273586395840249 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau951)
+  have hz1 :
+    (10000000000000000 / 8273586395840249 : ℝ) ≤
+      1 / Real.sqrt (43809428383637 / 64000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 8273586395840249 : ℝ) =
+          1 / (8273586395840249 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 4136793197920123 : ℝ) ≤
+      (452014206241029862917 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (43809428383637 / 64000000000000 : ℝ) ≤
+      (375000000000000 / 4136793197920123 : ℝ) := by
+    rw [show
+        (375000000000000 / 4136793197920123 : ℝ) =
+          (3 / 40) / (4136793197920123 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (138240370157103788799 / 400000000000000000000 : ℝ))) -
+        6 * (452014206241029862917 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (43809428383637 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (43809428383637 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((4136793197920123 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (138240370157103788799 / 400000000000000000000 : ℝ))) -
+          6 * (452014206241029862917 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (43809428383637 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (43809428383637 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (43809428383637 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_591 :
+    ((521963500234258913999 / 1000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(924188584376590225871 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((8278487518229287 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2 * (86436591379470589641 / 250000000000000000000 : ℝ))) -
+              6 * (903496108261826234217 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (21930673788633 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (21930673788633 / 32000000000000 : ℝ)) =
+      Real.log (53930673788633 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((21930673788633 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (22110673788633 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (521963500234258913999 / 1000000000000000000000 : ℝ) ≤
+      Real.log (53930673788633 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (22110673788633 / 32000000000000 : ℝ) ≤
+      (-(924188584376590225871 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (22110673788633 / 32000000000000 : ℝ) = (22110673788633 / 16000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22110673788633 / 16000000000000 : ℝ) ≤
+            (808679366373409774129 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (8278487518229287 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (21930673788633 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (21930673788633 / 32000000000000 : ℝ) ≤ (827848751822929 / 1000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (21930673788633 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau954 :
+    Real.arctan (8278487518229290 / 22982039731470479 : ℝ) ≤
+      (86436591379470589641 / 250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau953 :
+    Real.arctan (827848751822929 / 1000000000000000 : ℝ) ≤
+      (2 * (86436591379470589641 / 250000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12982039731470479 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (827848751822929 / 1000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (827848751822929 / 1000000000000000 : ℝ) /
+                (1 + (12982039731470479 / 10000000000000000 : ℝ)) =
+              (8278487518229290 / 22982039731470479 : ℝ)
+            by
+        norm_num];
+        exact hau954)
+  have hat1 :
+    (Real.pi / 2 - (2 * (86436591379470589641 / 250000000000000000000 : ℝ))) ≤
+      Real.arctan (1000000000000000 / 827848751822929 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1000000000000000 / 827848751822929 : ℝ) =
+              (827848751822929 / 1000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau953)
+  have hz1 :
+    (1000000000000000 / 827848751822929 : ℝ) ≤
+      1 / Real.sqrt (21930673788633 / 32000000000000 : ℝ) := by
+    rw [show
+        (1000000000000000 / 827848751822929 : ℝ) = 1 / (827848751822929 / 1000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 8278487518229287 : ℝ) ≤
+      (903496108261826234217 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (21930673788633 / 32000000000000 : ℝ) ≤
+      (750000000000000 / 8278487518229287 : ℝ) := by
+    rw [show
+        (750000000000000 / 8278487518229287 : ℝ) =
+          (3 / 40) / (8278487518229287 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (86436591379470589641 / 250000000000000000000 : ℝ))) -
+        6 * (903496108261826234217 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (21930673788633 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (21930673788633 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((8278487518229287 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (86436591379470589641 / 250000000000000000000 : ℝ))) -
+          6 * (903496108261826234217 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (21930673788633 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (21930673788633 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (21930673788633 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_592 :
+    ((5224447356639949858109 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(737004094199034598467 / 2000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1656677148143517 / 2000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (345891649639318181503 / 1000000000000000000000 : ℝ))) -
+              6 * (180592948645021209329 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (8782653354179 / 12800000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (8782653354179 / 12800000000000 : ℝ)) =
+      Real.log (21582653354179 / 12800000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((8782653354179 / 12800000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (8854653354179 / 12800000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (5224447356639949858109 / 10000000000000000000000 : ℝ) ≤
+      Real.log (21582653354179 / 12800000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (8854653354179 / 12800000000000 : ℝ) ≤
+      (-(737004094199034598467 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (8854653354179 / 12800000000000 : ℝ) = (8854653354179 / 6400000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8854653354179 / 6400000000000 : ℝ) ≤
+            (649290266400965401533 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1656677148143517 / 2000000000000000 : ℝ) ≤ Real.sqrt (8782653354179 / 12800000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (8782653354179 / 12800000000000 : ℝ) ≤ (2070846435179397 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (8782653354179 / 12800000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau956 :
+    Real.arctan (8283385740717588 / 22985163816044965 : ℝ) ≤
+      (345891649639318181503 / 1000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau955 :
+    Real.arctan (2070846435179397 / 2500000000000000 : ℝ) ≤
+      (2 * (345891649639318181503 / 1000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (2597032763208993 / 2000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (2070846435179397 / 2500000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (2070846435179397 / 2500000000000000 : ℝ) /
+                (1 + (2597032763208993 / 2000000000000000 : ℝ)) =
+              (8283385740717588 / 22985163816044965 : ℝ)
+            by
+        norm_num];
+        exact hau956)
+  have hat1 :
+    (Real.pi / 2 - (2 * (345891649639318181503 / 1000000000000000000000 : ℝ))) ≤
+      Real.arctan (2500000000000000 / 2070846435179397 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 2070846435179397 : ℝ) =
+              (2070846435179397 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau955)
+  have hz1 :
+    (2500000000000000 / 2070846435179397 : ℝ) ≤
+      1 / Real.sqrt (8782653354179 / 12800000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 2070846435179397 : ℝ) = 1 / (2070846435179397 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (50000000000000 / 552225716047839 : ℝ) ≤
+      (180592948645021209329 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (8782653354179 / 12800000000000 : ℝ) ≤
+      (50000000000000 / 552225716047839 : ℝ) := by
+    rw [show
+        (50000000000000 / 552225716047839 : ℝ) =
+          (3 / 40) / (1656677148143517 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (345891649639318181503 / 1000000000000000000000 : ℝ))) -
+        6 * (180592948645021209329 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (8782653354179 / 12800000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (8782653354179 / 12800000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1656677148143517 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (345891649639318181503 / 1000000000000000000000 : ℝ))) -
+          6 * (180592948645021209329 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (8782653354179 / 12800000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (8782653354179 / 12800000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (8782653354179 / 12800000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_593 :
+    ((2614628698087911415639 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(14693201426844845997 / 40000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1036035133555813 / 1250000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3460367780595662617999 / 10000000000000000000000 : ℝ))) -
+              6 * (180486862922668628021 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (10991296491131 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (10991296491131 / 16000000000000 : ℝ)) =
+      Real.log (26991296491131 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((10991296491131 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (11081296491131 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2614628698087911415639 / 5000000000000000000000 : ℝ) ≤
+      Real.log (26991296491131 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (11081296491131 / 16000000000000 : ℝ) ≤
+      (-(14693201426844845997 / 40000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (11081296491131 / 16000000000000 : ℝ) = (11081296491131 / 8000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11081296491131 / 8000000000000 : ℝ) ≤
+            (13032685785155154003 / 40000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1036035133555813 / 1250000000000000 : ℝ) ≤ Real.sqrt (10991296491131 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (10991296491131 / 16000000000000 : ℝ) ≤
+      (8288281068446507 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (10991296491131 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau958 :
+    Real.arctan (8288281068446507 / 22988287149180554 : ℝ) ≤
+      (3460367780595662617999 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau957 :
+    Real.arctan (8288281068446507 / 10000000000000000 : ℝ) ≤
+      (2 * (3460367780595662617999 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6494143574590277 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (8288281068446507 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (8288281068446507 / 10000000000000000 : ℝ) /
+                (1 + (6494143574590277 / 5000000000000000 : ℝ)) =
+              (8288281068446507 / 22988287149180554 : ℝ)
+            by
+        norm_num];
+        exact hau958)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3460367780595662617999 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 8288281068446507 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 8288281068446507 : ℝ) =
+              (8288281068446507 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau957)
+  have hz1 :
+    (10000000000000000 / 8288281068446507 : ℝ) ≤
+      1 / Real.sqrt (10991296491131 / 16000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 8288281068446507 : ℝ) =
+          1 / (8288281068446507 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (93750000000000 / 1036035133555813 : ℝ) ≤
+      (180486862922668628021 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (10991296491131 / 16000000000000 : ℝ) ≤
+      (93750000000000 / 1036035133555813 : ℝ) := by
+    rw [show
+        (93750000000000 / 1036035133555813 : ℝ) =
+          (3 / 40) / (1036035133555813 / 1250000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3460367780595662617999 / 10000000000000000000000 : ℝ))) -
+        6 * (180486862922668628021 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (10991296491131 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (10991296491131 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1036035133555813 / 1250000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3460367780595662617999 / 10000000000000000000000 : ℝ))) -
+          6 * (180486862922668628021 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (10991296491131 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (10991296491131 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (10991296491131 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_594 :
+    ((654258140396994550749 / 1250000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(3661593962456670299233 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1658634701308447 / 2000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2 * (216363594425149969639 / 625000000000000000000 : ℝ))) -
+              6 * (901904819679311450819 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (44017105158153 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (44017105158153 / 64000000000000 : ℝ)) =
+      Real.log (108017105158153 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((44017105158153 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (44377105158153 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (654258140396994550749 / 1250000000000000000000 : ℝ) ≤
+      Real.log (108017105158153 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (44377105158153 / 64000000000000 : ℝ) ≤
+      (-(3661593962456670299233 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (44377105158153 / 64000000000000 : ℝ) = (44377105158153 / 32000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (44377105158153 / 32000000000000 : ℝ) ≤
+            (3269877840543329700767 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1658634701308447 / 2000000000000000 : ℝ) ≤ Real.sqrt (44017105158153 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (44017105158153 / 64000000000000 : ℝ) ≤ (4146586753271119 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (44017105158153 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau960 :
+    Real.arctan (8293173506542238 / 22991409731419221 : ℝ) ≤
+      (216363594425149969639 / 625000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau959 :
+    Real.arctan (4146586753271119 / 5000000000000000 : ℝ) ≤
+      (2 * (216363594425149969639 / 625000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12991409731419221 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (4146586753271119 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (4146586753271119 / 5000000000000000 : ℝ) /
+                (1 + (12991409731419221 / 10000000000000000 : ℝ)) =
+              (8293173506542238 / 22991409731419221 : ℝ)
+            by
+        norm_num];
+        exact hau960)
+  have hat1 :
+    (Real.pi / 2 - (2 * (216363594425149969639 / 625000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 4146586753271119 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 4146586753271119 : ℝ) =
+              (4146586753271119 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau959)
+  have hz1 :
+    (5000000000000000 / 4146586753271119 : ℝ) ≤
+      1 / Real.sqrt (44017105158153 / 64000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 4146586753271119 : ℝ) = 1 / (4146586753271119 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (150000000000000 / 1658634701308447 : ℝ) ≤
+      (901904819679311450819 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (44017105158153 / 64000000000000 : ℝ) ≤
+      (150000000000000 / 1658634701308447 : ℝ) := by
+    rw [show
+        (150000000000000 / 1658634701308447 : ℝ) =
+          (3 / 40) / (1658634701308447 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (216363594425149969639 / 625000000000000000000 : ℝ))) -
+        6 * (901904819679311450819 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (44017105158153 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (44017105158153 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1658634701308447 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (216363594425149969639 / 625000000000000000000 : ℝ))) -
+          6 * (901904819679311450819 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (44017105158153 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (44017105158153 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (44017105158153 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_595 :
+    ((1047774107972578046817 / 2000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(729980251229337430647 / 2000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((4149031530057929 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3463265690020702793097 / 10000000000000000000000 : ℝ))) -
+              6 * (901376255687055180941 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (22034512175891 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (22034512175891 / 32000000000000 : ℝ)) =
+      Real.log (54034512175891 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((22034512175891 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (22214512175891 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1047774107972578046817 / 2000000000000000000000 : ℝ) ≤
+      Real.log (54034512175891 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (22214512175891 / 32000000000000 : ℝ) ≤
+      (-(729980251229337430647 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (22214512175891 / 32000000000000 : ℝ) = (22214512175891 / 16000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22214512175891 / 16000000000000 : ℝ) ≤
+            (656314109370662569353 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (4149031530057929 / 5000000000000000 : ℝ) ≤ Real.sqrt (22034512175891 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (22034512175891 / 32000000000000 : ℝ) ≤
+      (8298063060115861 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (22034512175891 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau962 :
+    Real.arctan (8298063060115861 / 22994531563302287 : ℝ) ≤
+      (3463265690020702793097 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau961 :
+    Real.arctan (8298063060115861 / 10000000000000000 : ℝ) ≤
+      (2 * (3463265690020702793097 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (12994531563302287 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (8298063060115861 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (8298063060115861 / 10000000000000000 : ℝ) /
+                (1 + (12994531563302287 / 10000000000000000 : ℝ)) =
+              (8298063060115861 / 22994531563302287 : ℝ)
+            by
+        norm_num];
+        exact hau962)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3463265690020702793097 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 8298063060115861 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 8298063060115861 : ℝ) =
+              (8298063060115861 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau961)
+  have hz1 :
+    (10000000000000000 / 8298063060115861 : ℝ) ≤
+      1 / Real.sqrt (22034512175891 / 32000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 8298063060115861 : ℝ) =
+          1 / (8298063060115861 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 4149031530057929 : ℝ) ≤
+      (901376255687055180941 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (22034512175891 / 32000000000000 : ℝ) ≤
+      (375000000000000 / 4149031530057929 : ℝ) := by
+    rw [show
+        (375000000000000 / 4149031530057929 : ℝ) =
+          (3 / 40) / (4149031530057929 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3463265690020702793097 / 10000000000000000000000 : ℝ))) -
+        6 * (901376255687055180941 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (22034512175891 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (22034512175891 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((4149031530057929 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3463265690020702793097 / 10000000000000000000000 : ℝ))) -
+          6 * (901376255687055180941 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (22034512175891 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (22034512175891 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (22034512175891 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_596 :
+    ((2621836824227980718709 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(727644441161763215397 / 2000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((41514748671317 / 50000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1732356160624546634617 / 5000000000000000000000 : ℝ))) -
+              6 * (900848619911829885433 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (44120943545411 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (44120943545411 / 64000000000000 : ℝ)) =
+      Real.log (108120943545411 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((44120943545411 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (44480943545411 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2621836824227980718709 / 5000000000000000000000 : ℝ) ≤
+      Real.log (108120943545411 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (44480943545411 / 64000000000000 : ℝ) ≤
+      (-(727644441161763215397 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (44480943545411 / 64000000000000 : ℝ) = (44480943545411 / 32000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (44480943545411 / 32000000000000 : ℝ) ≤
+            (658649919438236784603 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (41514748671317 / 50000000000000 : ℝ) ≤ Real.sqrt (44120943545411 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (44120943545411 / 64000000000000 : ℝ) ≤
+      (8302949734263403 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (44120943545411 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau964 :
+    Real.arctan (8302949734263403 / 22997652645370420 : ℝ) ≤
+      (1732356160624546634617 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau963 :
+    Real.arctan (8302949734263403 / 10000000000000000 : ℝ) ≤
+      (2 * (1732356160624546634617 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (649882632268521 / 500000000000000 : ℝ) ≤
+          Real.sqrt (1 + (8302949734263403 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (8302949734263403 / 10000000000000000 : ℝ) /
+                (1 + (649882632268521 / 500000000000000 : ℝ)) =
+              (8302949734263403 / 22997652645370420 : ℝ)
+            by
+        norm_num];
+        exact hau964)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1732356160624546634617 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 8302949734263403 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 8302949734263403 : ℝ) =
+              (8302949734263403 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau963)
+  have hz1 :
+    (10000000000000000 / 8302949734263403 : ℝ) ≤
+      1 / Real.sqrt (44120943545411 / 64000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 8302949734263403 : ℝ) =
+          1 / (8302949734263403 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (3750000000000 / 41514748671317 : ℝ) ≤
+      (900848619911829885433 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (44120943545411 / 64000000000000 : ℝ) ≤
+      (3750000000000 / 41514748671317 : ℝ) := by
+    rw [show
+        (3750000000000 / 41514748671317 : ℝ) = (3 / 40) / (41514748671317 / 50000000000000 : ℝ) by
+        norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1732356160624546634617 / 5000000000000000000000 : ℝ))) -
+        6 * (900848619911829885433 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (44120943545411 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (44120943545411 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((41514748671317 / 50000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1732356160624546634617 / 5000000000000000000000 : ℝ))) -
+          6 * (900848619911829885433 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (44120943545411 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (44120943545411 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (44120943545411 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_597 :
+    ((5248474451171310757761 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(453319597447812848417 / 1250000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((519239595879119 / 625000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2 * (69323148149546782071 / 200000000000000000000 : ℝ))) -
+              6 * (225080477410010679099 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (276080392119 / 400000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (276080392119 / 400000000000 : ℝ)) =
+      Real.log (676080392119 / 400000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((276080392119 / 400000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (278330392119 / 400000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (5248474451171310757761 / 10000000000000000000000 : ℝ) ≤
+      Real.log (676080392119 / 400000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (278330392119 / 400000000000 : ℝ) ≤
+      (-(453319597447812848417 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (278330392119 / 400000000000 : ℝ) = (278330392119 / 200000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (278330392119 / 200000000000 : ℝ) ≤
+            (413114377927187151583 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (519239595879119 / 625000000000000 : ℝ) ≤ Real.sqrt (276080392119 / 400000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (276080392119 / 400000000000 : ℝ) ≤ (8307833534065907 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (276080392119 / 400000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau966 :
+    Real.arctan (488696090239171 / 1352986645774332 : ℝ) ≤
+      (69323148149546782071 / 200000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau965 :
+    Real.arctan (8307833534065907 / 10000000000000000 : ℝ) ≤
+      (2 * (69323148149546782071 / 200000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (3250193244540911 / 2500000000000000 : ℝ) ≤
+          Real.sqrt (1 + (8307833534065907 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (8307833534065907 / 10000000000000000 : ℝ) /
+                (1 + (3250193244540911 / 2500000000000000 : ℝ)) =
+              (488696090239171 / 1352986645774332 : ℝ)
+            by
+        norm_num];
+        exact hau966)
+  have hat1 :
+    (Real.pi / 2 - (2 * (69323148149546782071 / 200000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 8307833534065907 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 8307833534065907 : ℝ) =
+              (8307833534065907 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau965)
+  have hz1 :
+    (10000000000000000 / 8307833534065907 : ℝ) ≤
+      1 / Real.sqrt (276080392119 / 400000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 8307833534065907 : ℝ) =
+          1 / (8307833534065907 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (46875000000000 / 519239595879119 : ℝ) ≤
+      (225080477410010679099 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (276080392119 / 400000000000 : ℝ) ≤
+      (46875000000000 / 519239595879119 : ℝ) := by
+    rw [show
+        (46875000000000 / 519239595879119 : ℝ) =
+          (3 / 40) / (519239595879119 / 625000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (69323148149546782071 / 200000000000000000000 : ℝ))) -
+        6 * (225080477410010679099 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (276080392119 / 400000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (276080392119 / 400000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((519239595879119 / 625000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (69323148149546782071 / 200000000000000000000 : ℝ))) -
+          6 * (225080477410010679099 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (276080392119 / 400000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (276080392119 / 400000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (276080392119 / 400000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_598 :
+    ((656659118777736086491 / 1250000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(3614904945718563204627 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((8312714464589487 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3467600951686490417953 / 10000000000000000000000 : ℝ))) -
+              6 * (899796122169194200699 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (44224781932669 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (44224781932669 / 64000000000000 : ℝ)) =
+      Real.log (108224781932669 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((44224781932669 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (44584781932669 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (656659118777736086491 / 1250000000000000000000 : ℝ) ≤
+      Real.log (108224781932669 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (44584781932669 / 64000000000000 : ℝ) ≤
+      (-(3614904945718563204627 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (44584781932669 / 64000000000000 : ℝ) = (44584781932669 / 32000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (44584781932669 / 32000000000000 : ℝ) ≤
+            (3316566857281436795373 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (8312714464589487 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (44224781932669 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (44224781932669 / 64000000000000 : ℝ) ≤ (831271446458949 / 1000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (44224781932669 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau968 :
+    Real.arctan (2770904821529830 / 7667964187407111 : ℝ) ≤
+      (3467600951686490417953 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau967 :
+    Real.arctan (831271446458949 / 1000000000000000 : ℝ) ≤
+      (2 * (3467600951686490417953 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (13003892562221333 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (831271446458949 / 1000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (831271446458949 / 1000000000000000 : ℝ) /
+                (1 + (13003892562221333 / 10000000000000000 : ℝ)) =
+              (2770904821529830 / 7667964187407111 : ℝ)
+            by
+        norm_num];
+        exact hau968)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3467600951686490417953 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (1000000000000000 / 831271446458949 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1000000000000000 / 831271446458949 : ℝ) =
+              (831271446458949 / 1000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau967)
+  have hz1 :
+    (1000000000000000 / 831271446458949 : ℝ) ≤
+      1 / Real.sqrt (44224781932669 / 64000000000000 : ℝ) := by
+    rw [show
+        (1000000000000000 / 831271446458949 : ℝ) = 1 / (831271446458949 / 1000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 2770904821529829 : ℝ) ≤
+      (899796122169194200699 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (44224781932669 / 64000000000000 : ℝ) ≤
+      (250000000000000 / 2770904821529829 : ℝ) := by
+    rw [show
+        (250000000000000 / 2770904821529829 : ℝ) =
+          (3 / 40) / (8312714464589487 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3467600951686490417953 / 10000000000000000000000 : ℝ))) -
+        6 * (899796122169194200699 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (44224781932669 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (44224781932669 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((8312714464589487 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3467600951686490417953 / 10000000000000000000000 : ℝ))) -
+          6 * (899796122169194200699 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (44224781932669 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (44224781932669 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (44224781932669 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_599 :
+    ((5258069147817461612193 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(3603266672578664989709 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((41587962654427 / 50000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3469042956848913514917 / 10000000000000000000000 : ℝ))) -
+              6 * (899271254807820140911 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (22138350563149 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (22138350563149 / 32000000000000 : ℝ)) =
+      Real.log (54138350563149 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((22138350563149 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (22318350563149 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (5258069147817461612193 / 10000000000000000000000 : ℝ) ≤
+      Real.log (54138350563149 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (22318350563149 / 32000000000000 : ℝ) ≤
+      (-(3603266672578664989709 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (22318350563149 / 32000000000000 : ℝ) = (22318350563149 / 16000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22318350563149 / 16000000000000 : ℝ) ≤
+            (3328205130421335010291 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (41587962654427 / 50000000000000 : ℝ) ≤ Real.sqrt (22138350563149 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (22138350563149 / 32000000000000 : ℝ) ≤
+      (8317592530885403 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (22138350563149 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau970 :
+    Real.arctan (8317592530885403 / 23007011398082213 : ℝ) ≤
+      (3469042956848913514917 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau969 :
+    Real.arctan (8317592530885403 / 10000000000000000 : ℝ) ≤
+      (2 * (3469042956848913514917 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (13007011398082213 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (8317592530885403 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (8317592530885403 / 10000000000000000 : ℝ) /
+                (1 + (13007011398082213 / 10000000000000000 : ℝ)) =
+              (8317592530885403 / 23007011398082213 : ℝ)
+            by
+        norm_num];
+        exact hau970)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3469042956848913514917 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 8317592530885403 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 8317592530885403 : ℝ) =
+              (8317592530885403 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau969)
+  have hz1 :
+    (10000000000000000 / 8317592530885403 : ℝ) ≤
+      1 / Real.sqrt (22138350563149 / 32000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 8317592530885403 : ℝ) =
+          1 / (8317592530885403 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (3750000000000 / 41587962654427 : ℝ) ≤
+      (899271254807820140911 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (22138350563149 / 32000000000000 : ℝ) ≤
+      (3750000000000 / 41587962654427 : ℝ) := by
+    rw [show
+        (3750000000000 / 41587962654427 : ℝ) = (3 / 40) / (41587962654427 / 50000000000000 : ℝ) by
+        norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3469042956848913514917 / 10000000000000000000000 : ℝ))) -
+        6 * (899271254807820140911 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (22138350563149 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (22138350563149 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((41587962654427 / 50000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3469042956848913514917 / 10000000000000000000000 : ℝ))) -
+          6 * (899271254807820140911 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (22138350563149 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (22138350563149 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (22138350563149 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_600 :
+    ((131571576154115446801 / 250000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(897910482158703023617 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2080616934497523 / 2500000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3470483425928326162743 / 10000000000000000000000 : ℝ))) -
+              6 * (898747304875433424139 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (44328620319927 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (44328620319927 / 64000000000000 : ℝ)) =
+      Real.log (108328620319927 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((44328620319927 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (44688620319927 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (131571576154115446801 / 250000000000000000000 : ℝ) ≤
+      Real.log (108328620319927 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (44688620319927 / 64000000000000 : ℝ) ≤
+      (-(897910482158703023617 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (44688620319927 / 64000000000000 : ℝ) = (44688620319927 / 32000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (44688620319927 / 32000000000000 : ℝ) ≤
+            (834957468591296976383 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2080616934497523 / 2500000000000000 : ℝ) ≤ Real.sqrt (44328620319927 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (44328620319927 / 64000000000000 : ℝ) ≤ (1664493547598019 / 2000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (44328620319927 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau972 :
+    Real.arctan (2774155912663365 / 7670043162094789 : ℝ) ≤
+      (3470483425928326162743 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau971 :
+    Real.arctan (1664493547598019 / 2000000000000000 : ℝ) ≤
+      (2 * (3470483425928326162743 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (13010129486284367 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1664493547598019 / 2000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1664493547598019 / 2000000000000000 : ℝ) /
+                (1 + (13010129486284367 / 10000000000000000 : ℝ)) =
+              (2774155912663365 / 7670043162094789 : ℝ)
+            by
+        norm_num];
+        exact hau972)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3470483425928326162743 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (2000000000000000 / 1664493547598019 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2000000000000000 / 1664493547598019 : ℝ) =
+              (1664493547598019 / 2000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau971)
+  have hz1 :
+    (2000000000000000 / 1664493547598019 : ℝ) ≤
+      1 / Real.sqrt (44328620319927 / 64000000000000 : ℝ) := by
+    rw [show
+        (2000000000000000 / 1664493547598019 : ℝ) = 1 / (1664493547598019 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (62500000000000 / 693538978165841 : ℝ) ≤
+      (898747304875433424139 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (44328620319927 / 64000000000000 : ℝ) ≤
+      (62500000000000 / 693538978165841 : ℝ) := by
+    rw [show
+        (62500000000000 / 693538978165841 : ℝ) =
+          (3 / 40) / (2080616934497523 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3470483425928326162743 / 10000000000000000000000 : ℝ))) -
+        6 * (898747304875433424139 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (44328620319927 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (44328620319927 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2080616934497523 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3470483425928326162743 / 10000000000000000000000 : ℝ))) -
+          6 * (898747304875433424139 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (44328620319927 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (44328620319927 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (44328620319927 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/V50.lean b/LeanPool/Zeta5Irrational/Table/V50.lean
new file mode 100644
index 0000000000..c4115afea9
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/V50.lean
@@ -0,0 +1,1445 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Energy
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # V50: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_601 :
+    ((5267654647466773897281 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(895007670617208076217 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((4163670045462633 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2 * (138876894475193245607 / 400000000000000000000 : ℝ))) -
+              6 * (898224269702467027377 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (11095134878389 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (11095134878389 / 16000000000000 : ℝ)) =
+      Real.log (27095134878389 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((11095134878389 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (11185134878389 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (5267654647466773897281 / 10000000000000000000000 : ℝ) ≤
+      Real.log (27095134878389 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (11185134878389 / 16000000000000 : ℝ) ≤
+      (-(895007670617208076217 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (11185134878389 / 16000000000000 : ℝ) = (11185134878389 / 8000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11185134878389 / 8000000000000 : ℝ) ≤
+            (837860280132791923783 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (4163670045462633 / 5000000000000000 : ℝ) ≤ Real.sqrt (11095134878389 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (11095134878389 / 16000000000000 : ℝ) ≤
+      (8327340090925269 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (11095134878389 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau974 :
+    Real.arctan (2775780030308423 / 7671082275788410 : ℝ) ≤
+      (138876894475193245607 / 400000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau973 :
+    Real.arctan (8327340090925269 / 10000000000000000 : ℝ) ≤
+      (2 * (138876894475193245607 / 400000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (1301324682736523 / 1000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (8327340090925269 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (8327340090925269 / 10000000000000000 : ℝ) /
+                (1 + (1301324682736523 / 1000000000000000 : ℝ)) =
+              (2775780030308423 / 7671082275788410 : ℝ)
+            by
+        norm_num];
+        exact hau974)
+  have hat1 :
+    (Real.pi / 2 - (2 * (138876894475193245607 / 400000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 8327340090925269 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 8327340090925269 : ℝ) =
+              (8327340090925269 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau973)
+  have hz1 :
+    (10000000000000000 / 8327340090925269 : ℝ) ≤
+      1 / Real.sqrt (11095134878389 / 16000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 8327340090925269 : ℝ) =
+          1 / (8327340090925269 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (125000000000000 / 1387890015154211 : ℝ) ≤
+      (898224269702467027377 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (11095134878389 / 16000000000000 : ℝ) ≤
+      (125000000000000 / 1387890015154211 : ℝ) := by
+    rw [show
+        (125000000000000 / 1387890015154211 : ℝ) =
+          (3 / 40) / (4163670045462633 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (138876894475193245607 / 400000000000000000000 : ℝ))) -
+        6 * (898224269702467027377 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (11095134878389 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (11095134878389 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((4163670045462633 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (138876894475193245607 / 400000000000000000000 : ℝ))) -
+          6 * (898224269702467027377 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (11095134878389 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (11095134878389 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (11095134878389 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_602 :
+    ((5272443953924180262591 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1784216451385933877829 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2083052398674487 / 2500000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1736679883824976458027 / 5000000000000000000000 : ℝ))) -
+              6 * (897702146630215959721 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (8886491741437 / 12800000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (8886491741437 / 12800000000000 : ℝ)) =
+      Real.log (21686491741437 / 12800000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((8886491741437 / 12800000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (8958491741437 / 12800000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (5272443953924180262591 / 10000000000000000000000 : ℝ) ≤
+      Real.log (21686491741437 / 12800000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (8958491741437 / 12800000000000 : ℝ) ≤
+      (-(1784216451385933877829 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (8958491741437 / 12800000000000 : ℝ) = (8958491741437 / 6400000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (8958491741437 / 6400000000000 : ℝ) ≤
+            (1681519450114066122171 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2083052398674487 / 2500000000000000 : ℝ) ≤ Real.sqrt (8886491741437 / 12800000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (8886491741437 / 12800000000000 : ℝ) ≤ (8332209594697951 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (8886491741437 / 12800000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau976 :
+    Real.arctan (490129976158703 / 1353903730697741 : ℝ) ≤
+      (1736679883824976458027 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau975 :
+    Real.arctan (8332209594697951 / 10000000000000000 : ℝ) ≤
+      (2 * (1736679883824976458027 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (13016363421861597 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (8332209594697951 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (8332209594697951 / 10000000000000000 : ℝ) /
+                (1 + (13016363421861597 / 10000000000000000 : ℝ)) =
+              (490129976158703 / 1353903730697741 : ℝ)
+            by
+        norm_num];
+        exact hau976)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1736679883824976458027 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 8332209594697951 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 8332209594697951 : ℝ) =
+              (8332209594697951 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau975)
+  have hz1 :
+    (10000000000000000 / 8332209594697951 : ℝ) ≤
+      1 / Real.sqrt (8886491741437 / 12800000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 8332209594697951 : ℝ) =
+          1 / (8332209594697951 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (187500000000000 / 2083052398674487 : ℝ) ≤
+      (897702146630215959721 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (8886491741437 / 12800000000000 : ℝ) ≤
+      (187500000000000 / 2083052398674487 : ℝ) := by
+    rw [show
+        (187500000000000 / 2083052398674487 : ℝ) =
+          (3 / 40) / (2083052398674487 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1736679883824976458027 / 5000000000000000000000 : ℝ))) -
+        6 * (897702146630215959721 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (8886491741437 / 12800000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (8886491741437 / 12800000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2083052398674487 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1736679883824976458027 / 5000000000000000000000 : ℝ))) -
+          6 * (897702146630215959721 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (8886491741437 / 12800000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (8886491741437 / 12800000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (8886491741437 / 12800000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_603 :
+    ((5277230967733927753499 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(111151517448245887379 / 312500000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1042134531787567 / 1250000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (434349455772083549517 / 1250000000000000000000 : ℝ))) -
+              6 * (897180933010781682679 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (22242188950407 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (22242188950407 / 32000000000000 : ℝ)) =
+      Real.log (54242188950407 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((22242188950407 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (22422188950407 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (5277230967733927753499 / 10000000000000000000000 : ℝ) ≤
+      Real.log (54242188950407 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (22422188950407 / 32000000000000 : ℝ) ≤
+      (-(111151517448245887379 / 312500000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (22422188950407 / 32000000000000 : ℝ) = (22422188950407 / 16000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22422188950407 / 16000000000000 : ℝ) ≤
+            (105456976395504112621 / 312500000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1042134531787567 / 1250000000000000 : ℝ) ≤ Real.sqrt (22242188950407 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (22242188950407 / 32000000000000 : ℝ) ≤
+      (8337076254300539 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (22242188950407 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau978 :
+    Real.arctan (8337076254300539 / 23019479270309618 : ℝ) ≤
+      (434349455772083549517 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau977 :
+    Real.arctan (8337076254300539 / 10000000000000000 : ℝ) ≤
+      (2 * (434349455772083549517 / 1250000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6509739635154809 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (8337076254300539 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (8337076254300539 / 10000000000000000 : ℝ) /
+                (1 + (6509739635154809 / 5000000000000000 : ℝ)) =
+              (8337076254300539 / 23019479270309618 : ℝ)
+            by
+        norm_num];
+        exact hau978)
+  have hat1 :
+    (Real.pi / 2 - (2 * (434349455772083549517 / 1250000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 8337076254300539 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 8337076254300539 : ℝ) =
+              (8337076254300539 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau977)
+  have hz1 :
+    (10000000000000000 / 8337076254300539 : ℝ) ≤
+      1 / Real.sqrt (22242188950407 / 32000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 8337076254300539 : ℝ) =
+          1 / (8337076254300539 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (93750000000000 / 1042134531787567 : ℝ) ≤
+      (897180933010781682679 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (22242188950407 / 32000000000000 : ℝ) ≤
+      (93750000000000 / 1042134531787567 : ℝ) := by
+    rw [show
+        (93750000000000 / 1042134531787567 : ℝ) =
+          (3 / 40) / (1042134531787567 / 1250000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (434349455772083549517 / 1250000000000000000000 : ℝ))) -
+        6 * (897180933010781682679 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (22242188950407 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (22242188950407 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1042134531787567 / 1250000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (434349455772083549517 / 1250000000000000000000 : ℝ))) -
+          6 * (897180933010781682679 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (22242188950407 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (22242188950407 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (22242188950407 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_604 :
+    ((5286798126183046578949 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(3533720051035585403877 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((8346801060892279 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3477662833209266870603 / 10000000000000000000000 : ℝ))) -
+              6 * (448070611796229685211 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (5573527036009 / 8000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (5573527036009 / 8000000000000 : ℝ)) =
+      Real.log (13573527036009 / 8000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((5573527036009 / 8000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (5618527036009 / 8000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (5286798126183046578949 / 10000000000000000000000 : ℝ) ≤
+      Real.log (13573527036009 / 8000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (5618527036009 / 8000000000000 : ℝ) ≤
+      (-(3533720051035585403877 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (5618527036009 / 8000000000000 : ℝ) = (5618527036009 / 4000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5618527036009 / 4000000000000 : ℝ) ≤
+            (3397751751964414596123 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (8346801060892279 / 10000000000000000 : ℝ) ≤ Real.sqrt (5573527036009 / 8000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (5573527036009 / 8000000000000 : ℝ) ≤ (4173400530446141 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (5573527036009 / 8000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau980 :
+    Real.arctan (4173400530446141 / 11512854365601009 : ℝ) ≤
+      (3477662833209266870603 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau979 :
+    Real.arctan (4173400530446141 / 5000000000000000 : ℝ) ≤
+      (2 * (3477662833209266870603 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6512854365601009 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (4173400530446141 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (4173400530446141 / 5000000000000000 : ℝ) /
+                (1 + (6512854365601009 / 5000000000000000 : ℝ)) =
+              (4173400530446141 / 11512854365601009 : ℝ)
+            by
+        norm_num];
+        exact hau980)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3477662833209266870603 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 4173400530446141 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 4173400530446141 : ℝ) =
+              (4173400530446141 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau979)
+  have hz1 :
+    (5000000000000000 / 4173400530446141 : ℝ) ≤
+      1 / Real.sqrt (5573527036009 / 8000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 4173400530446141 : ℝ) = 1 / (4173400530446141 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 8346801060892279 : ℝ) ≤
+      (448070611796229685211 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (5573527036009 / 8000000000000 : ℝ) ≤
+      (750000000000000 / 8346801060892279 : ℝ) := by
+    rw [show
+        (750000000000000 / 8346801060892279 : ℝ) =
+          (3 / 40) / (8346801060892279 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3477662833209266870603 / 10000000000000000000000 : ℝ))) -
+        6 * (448070611796229685211 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (5573527036009 / 8000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (5573527036009 / 8000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((8346801060892279 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3477662833209266870603 / 10000000000000000000000 : ℝ))) -
+          6 * (448070611796229685211 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (5573527036009 / 8000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (5573527036009 / 8000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (5573527036009 / 8000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_605 :
+    ((5296356140327890318239 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(3510644913100674866389 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1044564318793689 / 1250000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3480523946311819555931 / 10000000000000000000000 : ℝ))) -
+              6 * (111888140059786550771 / 1250000000000000000000 : ℝ)))) ≤
+      Vfield (4469205467533 / 6400000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (4469205467533 / 6400000000000 : ℝ)) =
+      Real.log (10869205467533 / 6400000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((4469205467533 / 6400000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (4505205467533 / 6400000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (5296356140327890318239 / 10000000000000000000000 : ℝ) ≤
+      Real.log (10869205467533 / 6400000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (4505205467533 / 6400000000000 : ℝ) ≤
+      (-(3510644913100674866389 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (4505205467533 / 6400000000000 : ℝ) = (4505205467533 / 3200000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4505205467533 / 3200000000000 : ℝ) ≤
+            (3420826889899325133611 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1044564318793689 / 1250000000000000 : ℝ) ≤ Real.sqrt (4469205467533 / 6400000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (4469205467533 / 6400000000000 : ℝ) ≤ (1671302910069903 / 2000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (4469205467533 / 6400000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau982 :
+    Real.arctan (2785504850116505 / 7677311738106276 : ℝ) ≤
+      (3480523946311819555931 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau981 :
+    Real.arctan (1671302910069903 / 2000000000000000 : ℝ) ≤
+      (2 * (3480523946311819555931 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (3257983803579707 / 2500000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1671302910069903 / 2000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1671302910069903 / 2000000000000000 : ℝ) /
+                (1 + (3257983803579707 / 2500000000000000 : ℝ)) =
+              (2785504850116505 / 7677311738106276 : ℝ)
+            by
+        norm_num];
+        exact hau982)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3480523946311819555931 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (2000000000000000 / 1671302910069903 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2000000000000000 / 1671302910069903 : ℝ) =
+              (1671302910069903 / 2000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau981)
+  have hz1 :
+    (2000000000000000 / 1671302910069903 : ℝ) ≤
+      1 / Real.sqrt (4469205467533 / 6400000000000 : ℝ) := by
+    rw [show
+        (2000000000000000 / 1671302910069903 : ℝ) = 1 / (1671302910069903 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (31250000000000 / 348188106264563 : ℝ) ≤
+      (111888140059786550771 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (4469205467533 / 6400000000000 : ℝ) ≤
+      (31250000000000 / 348188106264563 : ℝ) := by
+    rw [show
+        (31250000000000 / 348188106264563 : ℝ) =
+          (3 / 40) / (1044564318793689 / 1250000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3480523946311819555931 / 10000000000000000000000 : ℝ))) -
+        6 * (111888140059786550771 / 1250000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (4469205467533 / 6400000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (4469205467533 / 6400000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1044564318793689 / 1250000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3480523946311819555931 / 10000000000000000000000 : ℝ))) -
+          6 * (111888140059786550771 / 1250000000000000000000 : ℝ))) ≤
+      Real.sqrt (4469205467533 / 6400000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (4469205467533 / 6400000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (4469205467533 / 6400000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_606 :
+    ((5305905027632047828519 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(3487622898804823612337 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((4183108381045539 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (870844752170985371119 / 2500000000000000000000 : ℝ))) -
+              6 * (894072602868394696077 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (11198973265647 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (11198973265647 / 16000000000000 : ℝ)) =
+      Real.log (27198973265647 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((11198973265647 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (11288973265647 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (5305905027632047828519 / 10000000000000000000000 : ℝ) ≤
+      Real.log (27198973265647 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (11288973265647 / 16000000000000 : ℝ) ≤
+      (-(3487622898804823612337 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (11288973265647 / 16000000000000 : ℝ) = (11288973265647 / 8000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11288973265647 / 8000000000000 : ℝ) ≤
+            (3443848904195176387663 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (4183108381045539 / 5000000000000000 : ℝ) ≤ Real.sqrt (11198973265647 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (11198973265647 / 16000000000000 : ℝ) ≤
+      (8366216762091081 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (11198973265647 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau984 :
+    Real.arctan (8366216762091081 / 23038158723926233 : ℝ) ≤
+      (870844752170985371119 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau983 :
+    Real.arctan (8366216762091081 / 10000000000000000 : ℝ) ≤
+      (2 * (870844752170985371119 / 2500000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (13038158723926233 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (8366216762091081 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (8366216762091081 / 10000000000000000 : ℝ) /
+                (1 + (13038158723926233 / 10000000000000000 : ℝ)) =
+              (8366216762091081 / 23038158723926233 : ℝ)
+            by
+        norm_num];
+        exact hau984)
+  have hat1 :
+    (Real.pi / 2 - (2 * (870844752170985371119 / 2500000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 8366216762091081 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 8366216762091081 : ℝ) =
+              (8366216762091081 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau983)
+  have hz1 :
+    (10000000000000000 / 8366216762091081 : ℝ) ≤
+      1 / Real.sqrt (11198973265647 / 16000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 8366216762091081 : ℝ) =
+          1 / (8366216762091081 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (125000000000000 / 1394369460348513 : ℝ) ≤
+      (894072602868394696077 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (11198973265647 / 16000000000000 : ℝ) ≤
+      (125000000000000 / 1394369460348513 : ℝ) := by
+    rw [show
+        (125000000000000 / 1394369460348513 : ℝ) =
+          (3 / 40) / (4183108381045539 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (870844752170985371119 / 2500000000000000000000 : ℝ))) -
+        6 * (894072602868394696077 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (11198973265647 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (11198973265647 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((4183108381045539 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (870844752170985371119 / 2500000000000000000000 : ℝ))) -
+          6 * (894072602868394696077 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (11198973265647 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (11198973265647 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (11198973265647 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_607 :
+    ((332215300344320520497 / 625000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(866163441026753018953 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((6543677918209 / 7812500000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3486228043391734482853 / 10000000000000000000000 : ℝ))) -
+              6 * (446521825065223545333 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (22449865724923 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (22449865724923 / 32000000000000 : ℝ)) =
+      Real.log (54449865724923 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((22449865724923 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (22629865724923 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (332215300344320520497 / 625000000000000000000 : ℝ) ≤
+      Real.log (54449865724923 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (22629865724923 / 32000000000000 : ℝ) ≤
+      (-(866163441026753018953 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (22629865724923 / 32000000000000 : ℝ) = (22629865724923 / 16000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22629865724923 / 16000000000000 : ℝ) ≤
+            (866704509723246981047 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (6543677918209 / 7812500000000 : ℝ) ≤ Real.sqrt (22449865724923 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (22449865724923 / 32000000000000 : ℝ) ≤
+      (8375907735307523 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (22449865724923 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau986 :
+    Real.arctan (8375907735307523 / 23044379264280242 : ℝ) ≤
+      (3486228043391734482853 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau985 :
+    Real.arctan (8375907735307523 / 10000000000000000 : ℝ) ≤
+      (2 * (3486228043391734482853 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6522189632140121 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (8375907735307523 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (8375907735307523 / 10000000000000000 : ℝ) /
+                (1 + (6522189632140121 / 5000000000000000 : ℝ)) =
+              (8375907735307523 / 23044379264280242 : ℝ)
+            by
+        norm_num];
+        exact hau986)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3486228043391734482853 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 8375907735307523 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 8375907735307523 : ℝ) =
+              (8375907735307523 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau985)
+  have hz1 :
+    (10000000000000000 / 8375907735307523 : ℝ) ≤
+      1 / Real.sqrt (22449865724923 / 32000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 8375907735307523 : ℝ) =
+          1 / (8375907735307523 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (585937500000 / 6543677918209 : ℝ) ≤
+      (446521825065223545333 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (22449865724923 / 32000000000000 : ℝ) ≤
+      (585937500000 / 6543677918209 : ℝ) := by
+    rw [show (585937500000 / 6543677918209 : ℝ) = (3 / 40) / (6543677918209 / 7812500000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3486228043391734482853 / 10000000000000000000000 : ℝ))) -
+        6 * (446521825065223545333 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (22449865724923 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (22449865724923 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((6543677918209 / 7812500000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3486228043391734482853 / 10000000000000000000000 : ℝ))) -
+          6 * (446521825065223545333 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (22449865724923 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (22449865724923 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (22449865724923 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_608 :
+    ((1064995098264590386373 / 2000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(3441737266643992525549 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((8385587508962921 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3489071073368835962193 / 10000000000000000000000 : ℝ))) -
+              6 * (892018241797966445617 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (2812723114819 / 4000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (2812723114819 / 4000000000000 : ℝ)) =
+      Real.log (6812723114819 / 4000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((2812723114819 / 4000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (2835223114819 / 4000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1064995098264590386373 / 2000000000000000000000 : ℝ) ≤
+      Real.log (6812723114819 / 4000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (2835223114819 / 4000000000000 : ℝ) ≤
+      (-(3441737266643992525549 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (2835223114819 / 4000000000000 : ℝ) = (2835223114819 / 2000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2835223114819 / 2000000000000 : ℝ) ≤
+            (3489734536356007474451 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (8385587508962921 / 10000000000000000 : ℝ) ≤ Real.sqrt (2812723114819 / 4000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (2812723114819 / 4000000000000 : ℝ) ≤ (2096396877240731 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (2812723114819 / 4000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau988 :
+    Real.arctan (4192793754481462 / 11525298419813361 : ℝ) ≤
+      (3489071073368835962193 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau987 :
+    Real.arctan (2096396877240731 / 2500000000000000 : ℝ) ≤
+      (2 * (3489071073368835962193 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6525298419813361 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (2096396877240731 / 2500000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (2096396877240731 / 2500000000000000 : ℝ) /
+                (1 + (6525298419813361 / 5000000000000000 : ℝ)) =
+              (4192793754481462 / 11525298419813361 : ℝ)
+            by
+        norm_num];
+        exact hau988)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3489071073368835962193 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (2500000000000000 / 2096396877240731 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 2096396877240731 : ℝ) =
+              (2096396877240731 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau987)
+  have hz1 :
+    (2500000000000000 / 2096396877240731 : ℝ) ≤
+      1 / Real.sqrt (2812723114819 / 4000000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 2096396877240731 : ℝ) = 1 / (2096396877240731 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 8385587508962921 : ℝ) ≤
+      (892018241797966445617 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (2812723114819 / 4000000000000 : ℝ) ≤
+      (750000000000000 / 8385587508962921 : ℝ) := by
+    rw [show
+        (750000000000000 / 8385587508962921 : ℝ) =
+          (3 / 40) / (8385587508962921 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3489071073368835962193 / 10000000000000000000000 : ℝ))) -
+        6 * (892018241797966445617 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (2812723114819 / 4000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (2812723114819 / 4000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((8385587508962921 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3489071073368835962193 / 10000000000000000000000 : ℝ))) -
+          6 * (892018241797966445617 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (2812723114819 / 4000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (2812723114819 / 4000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (2812723114819 / 4000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_609 :
+    ((5334497102387739281797 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(3418873165714944740207 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((4197628060898369 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3491908121417462665577 / 10000000000000000000000 : ℝ))) -
+              6 * (890996357568595104887 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (22553704112181 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (22553704112181 / 32000000000000 : ℝ)) =
+      Real.log (54553704112181 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((22553704112181 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (22733704112181 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (5334497102387739281797 / 10000000000000000000000 : ℝ) ≤
+      Real.log (54553704112181 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (22733704112181 / 32000000000000 : ℝ) ≤
+      (-(3418873165714944740207 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (22733704112181 / 32000000000000 : ℝ) = (22733704112181 / 16000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22733704112181 / 16000000000000 : ℝ) ≤
+            (3512598637285055259793 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (4197628060898369 / 5000000000000000 : ℝ) ≤ Real.sqrt (22553704112181 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (22553704112181 / 32000000000000 : ℝ) ≤
+      (8395256121796741 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (22553704112181 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau990 :
+    Real.arctan (8395256121796741 / 23056811454201428 : ℝ) ≤
+      (3491908121417462665577 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau989 :
+    Real.arctan (8395256121796741 / 10000000000000000 : ℝ) ≤
+      (2 * (3491908121417462665577 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (3264202863550357 / 2500000000000000 : ℝ) ≤
+          Real.sqrt (1 + (8395256121796741 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (8395256121796741 / 10000000000000000 : ℝ) /
+                (1 + (3264202863550357 / 2500000000000000 : ℝ)) =
+              (8395256121796741 / 23056811454201428 : ℝ)
+            by
+        norm_num];
+        exact hau990)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3491908121417462665577 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 8395256121796741 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 8395256121796741 : ℝ) =
+              (8395256121796741 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau989)
+  have hz1 :
+    (10000000000000000 / 8395256121796741 : ℝ) ≤
+      1 / Real.sqrt (22553704112181 / 32000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 8395256121796741 : ℝ) =
+          1 / (8395256121796741 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 4197628060898369 : ℝ) ≤
+      (890996357568595104887 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (22553704112181 / 32000000000000 : ℝ) ≤
+      (375000000000000 / 4197628060898369 : ℝ) := by
+    rw [show
+        (375000000000000 / 4197628060898369 : ℝ) =
+          (3 / 40) / (4197628060898369 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3491908121417462665577 / 10000000000000000000000 : ℝ))) -
+        6 * (890996357568595104887 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (22553704112181 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (22553704112181 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((4197628060898369 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3491908121417462665577 / 10000000000000000000000 : ℝ))) -
+          6 * (890996357568595104887 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (22553704112181 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (22553704112181 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (22553704112181 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_610 :
+    ((2672004827984150136307 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(339606122226630670097 / 1000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((8404913612325603 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3494739210209437387999 / 10000000000000000000000 : ℝ))) -
+              6 * (889977977302413373633 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (2260562330581 / 3200000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (2260562330581 / 3200000000000 : ℝ)) =
+      Real.log (5460562330581 / 3200000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((2260562330581 / 3200000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (2278562330581 / 3200000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2672004827984150136307 / 5000000000000000000000 : ℝ) ≤
+      Real.log (5460562330581 / 3200000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (2278562330581 / 3200000000000 : ℝ) ≤
+      (-(339606122226630670097 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (2278562330581 / 3200000000000 : ℝ) = (2278562330581 / 1600000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2278562330581 / 1600000000000 : ℝ) ≤
+            (353541058073369329903 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (8404913612325603 / 10000000000000000 : ℝ) ≤ Real.sqrt (2260562330581 / 3200000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (2260562330581 / 3200000000000 : ℝ) ≤ (4202456806162803 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (2260562330581 / 3200000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau992 :
+    Real.arctan (2801637870775202 / 7687674370743347 : ℝ) ≤
+      (3494739210209437387999 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau991 :
+    Real.arctan (4202456806162803 / 5000000000000000 : ℝ) ≤
+      (2 * (3494739210209437387999 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (13063023112230041 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (4202456806162803 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (4202456806162803 / 5000000000000000 : ℝ) /
+                (1 + (13063023112230041 / 10000000000000000 : ℝ)) =
+              (2801637870775202 / 7687674370743347 : ℝ)
+            by
+        norm_num];
+        exact hau992)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3494739210209437387999 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 4202456806162803 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 4202456806162803 : ℝ) =
+              (4202456806162803 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau991)
+  have hz1 :
+    (5000000000000000 / 4202456806162803 : ℝ) ≤
+      1 / Real.sqrt (2260562330581 / 3200000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 4202456806162803 : ℝ) = 1 / (4202456806162803 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 2801637870775201 : ℝ) ≤
+      (889977977302413373633 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (2260562330581 / 3200000000000 : ℝ) ≤
+      (250000000000000 / 2801637870775201 : ℝ) := by
+    rw [show
+        (250000000000000 / 2801637870775201 : ℝ) =
+          (3 / 40) / (8404913612325603 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3494739210209437387999 / 10000000000000000000000 : ℝ))) -
+        6 * (889977977302413373633 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (2260562330581 / 3200000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (2260562330581 / 3200000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((8404913612325603 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3494739210209437387999 / 10000000000000000000000 : ℝ))) -
+          6 * (889977977302413373633 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (2260562330581 / 3200000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (2260562330581 / 3200000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (2260562330581 / 3200000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_611 :
+    ((1070702633856044376569 / 2000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(674660239775355582283 / 2000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2103640004711281 / 2500000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1748782181143606580643 / 5000000000000000000000 : ℝ))) -
+              6 * (888963081020271195629 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (22657542499439 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (22657542499439 / 32000000000000 : ℝ)) =
+      Real.log (54657542499439 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((22657542499439 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (22837542499439 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1070702633856044376569 / 2000000000000000000000 : ℝ) ≤
+      Real.log (54657542499439 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (22837542499439 / 32000000000000 : ℝ) ≤
+      (-(674660239775355582283 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (22837542499439 / 32000000000000 : ℝ) = (22837542499439 / 16000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22837542499439 / 16000000000000 : ℝ) ≤
+            (711634120824644417717 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2103640004711281 / 2500000000000000 : ℝ) ≤ Real.sqrt (22657542499439 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (22657542499439 / 32000000000000 : ℝ) ≤
+      (8414560018845127 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (22657542499439 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau994 :
+    Real.arctan (8414560018845127 / 23069231817928201 : ℝ) ≤
+      (1748782181143606580643 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau993 :
+    Real.arctan (8414560018845127 / 10000000000000000 : ℝ) ≤
+      (2 * (1748782181143606580643 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (13069231817928201 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (8414560018845127 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (8414560018845127 / 10000000000000000 : ℝ) /
+                (1 + (13069231817928201 / 10000000000000000 : ℝ)) =
+              (8414560018845127 / 23069231817928201 : ℝ)
+            by
+        norm_num];
+        exact hau994)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1748782181143606580643 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 8414560018845127 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 8414560018845127 : ℝ) =
+              (8414560018845127 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau993)
+  have hz1 :
+    (10000000000000000 / 8414560018845127 : ℝ) ≤
+      1 / Real.sqrt (22657542499439 / 32000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 8414560018845127 : ℝ) =
+          1 / (8414560018845127 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (187500000000000 / 2103640004711281 : ℝ) ≤
+      (888963081020271195629 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (22657542499439 / 32000000000000 : ℝ) ≤
+      (187500000000000 / 2103640004711281 : ℝ) := by
+    rw [show
+        (187500000000000 / 2103640004711281 : ℝ) =
+          (3 / 40) / (2103640004711281 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1748782181143606580643 / 5000000000000000000000 : ℝ))) -
+        6 * (888963081020271195629 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (22657542499439 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (22657542499439 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2103640004711281 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1748782181143606580643 / 5000000000000000000000 : ℝ))) -
+          6 * (888963081020271195629 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (22657542499439 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (22657542499439 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (22657542499439 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_612 :
+    ((1340751914872513778333 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1675296429871246496299 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1684839075886329 / 2000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3500383600064879102301 / 10000000000000000000000 : ℝ))) -
+              6 * (887951648902143076857 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (5677365423267 / 8000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (5677365423267 / 8000000000000 : ℝ)) =
+      Real.log (13677365423267 / 8000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((5677365423267 / 8000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (5722365423267 / 8000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1340751914872513778333 / 2500000000000000000000 : ℝ) ≤
+      Real.log (13677365423267 / 8000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (5722365423267 / 8000000000000 : ℝ) ≤
+      (-(1675296429871246496299 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (5722365423267 / 8000000000000 : ℝ) = (5722365423267 / 4000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5722365423267 / 4000000000000 : ℝ) ≤
+            (1790439471628753503701 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1684839075886329 / 2000000000000000 : ℝ) ≤ Real.sqrt (5677365423267 / 8000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (5677365423267 / 8000000000000 : ℝ) ≤ (263256105607239 / 312500000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (5677365423267 / 8000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau996 :
+    Real.arctan (263256105607239 / 721107424234423 : ℝ) ≤
+      (3500383600064879102301 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau995 :
+    Real.arctan (263256105607239 / 312500000000000 : ℝ) ≤
+      (2 * (3500383600064879102301 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (408607424234423 / 312500000000000 : ℝ) ≤
+          Real.sqrt (1 + (263256105607239 / 312500000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (263256105607239 / 312500000000000 : ℝ) /
+                (1 + (408607424234423 / 312500000000000 : ℝ)) =
+              (263256105607239 / 721107424234423 : ℝ)
+            by
+        norm_num];
+        exact hau996)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3500383600064879102301 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (312500000000000 / 263256105607239 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (312500000000000 / 263256105607239 : ℝ) =
+              (263256105607239 / 312500000000000 : ℝ)
+            by
+        norm_num];
+        exact hau995)
+  have hz1 :
+    (312500000000000 / 263256105607239 : ℝ) ≤ 1 / Real.sqrt (5677365423267 / 8000000000000 : ℝ) :=
+  by
+    rw [show (312500000000000 / 263256105607239 : ℝ) = 1 / (263256105607239 / 312500000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (50000000000000 / 561613025295443 : ℝ) ≤
+      (887951648902143076857 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (5677365423267 / 8000000000000 : ℝ) ≤
+      (50000000000000 / 561613025295443 : ℝ) := by
+    rw [show
+        (50000000000000 / 561613025295443 : ℝ) =
+          (3 / 40) / (1684839075886329 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3500383600064879102301 / 10000000000000000000000 : ℝ))) -
+        6 * (887951648902143076857 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (5677365423267 / 8000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (5677365423267 / 8000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1684839075886329 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3500383600064879102301 / 10000000000000000000000 : ℝ))) -
+          6 * (887951648902143076857 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (5677365423267 / 8000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (5677365423267 / 8000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (5677365423267 / 8000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/V51.lean b/LeanPool/Zeta5Irrational/Table/V51.lean
new file mode 100644
index 0000000000..a29484a3a2
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/V51.lean
@@ -0,0 +1,1435 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Energy
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # V51: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_613 :
+    ((5372493143715501039287 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(332793597066236323861 / 1000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1054227466493001 / 1250000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1751598472914581166259 / 5000000000000000000000 : ℝ))) -
+              6 * (886943661285501435459 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (22761380886697 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (22761380886697 / 32000000000000 : ℝ)) =
+      Real.log (54761380886697 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((22761380886697 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (22941380886697 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (5372493143715501039287 / 10000000000000000000000 : ℝ) ≤
+      Real.log (54761380886697 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (22941380886697 / 32000000000000 : ℝ) ≤
+      (-(332793597066236323861 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (22941380886697 / 32000000000000 : ℝ) = (22941380886697 / 16000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (22941380886697 / 16000000000000 : ℝ) ≤
+            (360353583233763676139 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1054227466493001 / 1250000000000000 : ℝ) ≤ Real.sqrt (22761380886697 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (22761380886697 / 32000000000000 : ℝ) ≤
+      (8433819731944011 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (22761380886697 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau998 :
+    Real.arctan (2811273243981337 / 7693880129715234 : ℝ) ≤
+      (1751598472914581166259 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau997 :
+    Real.arctan (8433819731944011 / 10000000000000000 : ℝ) ≤
+      (2 * (1751598472914581166259 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6540820194572851 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (8433819731944011 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (8433819731944011 / 10000000000000000 : ℝ) /
+                (1 + (6540820194572851 / 5000000000000000 : ℝ)) =
+              (2811273243981337 / 7693880129715234 : ℝ)
+            by
+        norm_num];
+        exact hau998)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1751598472914581166259 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 8433819731944011 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 8433819731944011 : ℝ) =
+              (8433819731944011 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau997)
+  have hz1 :
+    (10000000000000000 / 8433819731944011 : ℝ) ≤
+      1 / Real.sqrt (22761380886697 / 32000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 8433819731944011 : ℝ) =
+          1 / (8433819731944011 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (31250000000000 / 351409155497667 : ℝ) ≤
+      (886943661285501435459 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (22761380886697 / 32000000000000 : ℝ) ≤
+      (31250000000000 / 351409155497667 : ℝ) := by
+    rw [show
+        (31250000000000 / 351409155497667 : ℝ) =
+          (3 / 40) / (1054227466493001 / 1250000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1751598472914581166259 / 5000000000000000000000 : ℝ))) -
+        6 * (886943661285501435459 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (22761380886697 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (22761380886697 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1054227466493001 / 1250000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1751598472914581166259 / 5000000000000000000000 : ℝ))) -
+          6 * (886943661285501435459 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (22761380886697 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (22761380886697 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (22761380886697 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_614 :
+    ((5381969639025595980783 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1652665149511791927237 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1688686622805057 / 2000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (876501105435104276007 / 2500000000000000000000 : ℝ))) -
+              6 * (885939098663711472987 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (11406650040163 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (11406650040163 / 16000000000000 : ℝ)) =
+      Real.log (27406650040163 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((11406650040163 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (11496650040163 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (5381969639025595980783 / 10000000000000000000000 : ℝ) ≤
+      Real.log (27406650040163 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (11496650040163 / 16000000000000 : ℝ) ≤
+      (-(1652665149511791927237 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (11496650040163 / 16000000000000 : ℝ) = (11496650040163 / 8000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11496650040163 / 8000000000000 : ℝ) ≤
+            (1813070751988208072763 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1688686622805057 / 2000000000000000 : ℝ) ≤ Real.sqrt (11406650040163 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (11406650040163 / 16000000000000 : ℝ) ≤ (1055429139253161 / 1250000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (11406650040163 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1000 :
+    Real.arctan (4221716557012644 / 11543920131523205 : ℝ) ≤
+      (876501105435104276007 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau999 :
+    Real.arctan (1055429139253161 / 1250000000000000 : ℝ) ≤
+      (2 * (876501105435104276007 / 2500000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (1308784026304641 / 1000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1055429139253161 / 1250000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1055429139253161 / 1250000000000000 : ℝ) /
+                (1 + (1308784026304641 / 1000000000000000 : ℝ)) =
+              (4221716557012644 / 11543920131523205 : ℝ)
+            by
+        norm_num];
+        exact hau1000)
+  have hat1 :
+    (Real.pi / 2 - (2 * (876501105435104276007 / 2500000000000000000000 : ℝ))) ≤
+      Real.arctan (1250000000000000 / 1055429139253161 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1250000000000000 / 1055429139253161 : ℝ) =
+              (1055429139253161 / 1250000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau999)
+  have hz1 :
+    (1250000000000000 / 1055429139253161 : ℝ) ≤
+      1 / Real.sqrt (11406650040163 / 16000000000000 : ℝ) := by
+    rw [show
+        (1250000000000000 / 1055429139253161 : ℝ) = 1 / (1055429139253161 / 1250000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (50000000000000 / 562895540935019 : ℝ) ≤
+      (885939098663711472987 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (11406650040163 / 16000000000000 : ℝ) ≤
+      (50000000000000 / 562895540935019 : ℝ) := by
+    rw [show
+        (50000000000000 / 562895540935019 : ℝ) =
+          (3 / 40) / (1688686622805057 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (876501105435104276007 / 2500000000000000000000 : ℝ))) -
+        6 * (885939098663711472987 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (11406650040163 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (11406650040163 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1688686622805057 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (876501105435104276007 / 2500000000000000000000 : ℝ))) -
+          6 * (885939098663711472987 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (11406650040163 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (11406650040163 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (11406650040163 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_615 :
+    ((5400895730933659307041 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(815067921368615468109 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((846262711639831 / 1000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1755800926009630248063 / 5000000000000000000000 : ℝ))) -
+              6 * (88394017114811751033 / 1000000000000000000000 : ℝ)))) ≤
+      Vfield (89520072139 / 125000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (89520072139 / 125000000000 : ℝ)) =
+      Real.log (214520072139 / 125000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((89520072139 / 125000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (90223197139 / 125000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (5400895730933659307041 / 10000000000000000000000 : ℝ) ≤
+      Real.log (214520072139 / 125000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (90223197139 / 125000000000 : ℝ) ≤
+      (-(815067921368615468109 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (90223197139 / 125000000000 : ℝ) = (90223197139 / 62500000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 9
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (90223197139 / 62500000000 : ℝ) ≤
+            (917800029381384531891 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (846262711639831 / 1000000000000000 : ℝ) ≤ Real.sqrt (89520072139 / 125000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (89520072139 / 125000000000 : ℝ) ≤ (8462627116398313 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (89520072139 / 125000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau1002 :
+    Real.arctan (8462627116398313 / 23100231208310790 : ℝ) ≤
+      (1755800926009630248063 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1001 :
+    Real.arctan (8462627116398313 / 10000000000000000 : ℝ) ≤
+      (2 * (1755800926009630248063 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (1310023120831079 / 1000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (8462627116398313 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (8462627116398313 / 10000000000000000 : ℝ) /
+                (1 + (1310023120831079 / 1000000000000000 : ℝ)) =
+              (8462627116398313 / 23100231208310790 : ℝ)
+            by
+        norm_num];
+        exact hau1002)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1755800926009630248063 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 8462627116398313 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 8462627116398313 : ℝ) =
+              (8462627116398313 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1001)
+  have hz1 :
+    (10000000000000000 / 8462627116398313 : ℝ) ≤ 1 / Real.sqrt (89520072139 / 125000000000 : ℝ) :=
+  by
+    rw [show
+        (10000000000000000 / 8462627116398313 : ℝ) =
+          1 / (8462627116398313 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (75000000000000 / 846262711639831 : ℝ) ≤
+      (88394017114811751033 / 1000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (89520072139 / 125000000000 : ℝ) ≤
+      (75000000000000 / 846262711639831 : ℝ) := by
+    rw [show
+        (75000000000000 / 846262711639831 : ℝ) =
+          (3 / 40) / (846262711639831 / 1000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1755800926009630248063 / 5000000000000000000000 : ℝ))) -
+        6 * (88394017114811751033 / 1000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (89520072139 / 125000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (89520072139 / 125000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((846262711639831 / 1000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1755800926009630248063 / 5000000000000000000000 : ℝ))) -
+          6 * (88394017114811751033 / 1000000000000000000000 : ℝ))) ≤
+      Real.sqrt (89520072139 / 125000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (89520072139 / 125000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (89520072139 / 125000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_616 :
+    ((216479549629272587089 / 400000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(808479101909136909281 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((8473873801407549 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3514876734834164438587 / 10000000000000000000000 : ℝ))) -
+              6 * (17655461571844476817 / 200000000000000000000 : ℝ)))) ≤
+      Vfield (11489045952349 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (11489045952349 / 16000000000000 : ℝ)) =
+      Real.log (27489045952349 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((11489045952349 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (11579045952349 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (216479549629272587089 / 400000000000000000000 : ℝ) ≤
+      Real.log (27489045952349 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (11579045952349 / 16000000000000 : ℝ) ≤
+      (-(808479101909136909281 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (11579045952349 / 16000000000000 : ℝ) = (11579045952349 / 8000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11579045952349 / 8000000000000 : ℝ) ≤
+            (924388848840863090719 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (8473873801407549 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (11489045952349 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (11489045952349 / 16000000000000 : ℝ) ≤ (132404278146993 / 156250000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (11489045952349 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1004 :
+    Real.arctan (8473873801407552 / 23107499273399989 : ℝ) ≤
+      (3514876734834164438587 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1003 :
+    Real.arctan (132404278146993 / 156250000000000 : ℝ) ≤
+      (2 * (3514876734834164438587 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (13107499273399989 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (132404278146993 / 156250000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (132404278146993 / 156250000000000 : ℝ) /
+                (1 + (13107499273399989 / 10000000000000000 : ℝ)) =
+              (8473873801407552 / 23107499273399989 : ℝ)
+            by
+        norm_num];
+        exact hau1004)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3514876734834164438587 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (156250000000000 / 132404278146993 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (156250000000000 / 132404278146993 : ℝ) =
+              (132404278146993 / 156250000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1003)
+  have hz1 :
+    (156250000000000 / 132404278146993 : ℝ) ≤
+      1 / Real.sqrt (11489045952349 / 16000000000000 : ℝ) := by
+    rw [show (156250000000000 / 132404278146993 : ℝ) = 1 / (132404278146993 / 156250000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 2824624600469183 : ℝ) ≤
+      (17655461571844476817 / 200000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (11489045952349 / 16000000000000 : ℝ) ≤
+      (250000000000000 / 2824624600469183 : ℝ) := by
+    rw [show
+        (250000000000000 / 2824624600469183 : ℝ) =
+          (3 / 40) / (8473873801407549 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3514876734834164438587 / 10000000000000000000000 : ℝ))) -
+        6 * (17655461571844476817 / 200000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (11489045952349 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (11489045952349 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((8473873801407549 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3514876734834164438587 / 10000000000000000000000 : ℝ))) -
+          6 * (17655461571844476817 / 200000000000000000000 : ℝ))) ≤
+      Real.sqrt (11489045952349 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (11489045952349 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (11489045952349 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_617 :
+    ((2708765317241712306349 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(3220764770559494080803 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((8479491550067837 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3516511187288392257217 / 10000000000000000000000 : ℝ))) -
+              6 * (882191263222842597187 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (4601713724651 / 6400000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (4601713724651 / 6400000000000 : ℝ)) =
+      Real.log (11001713724651 / 6400000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((4601713724651 / 6400000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (4637713724651 / 6400000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2708765317241712306349 / 5000000000000000000000 : ℝ) ≤
+      Real.log (11001713724651 / 6400000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (4637713724651 / 6400000000000 : ℝ) ≤
+      (-(3220764770559494080803 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (4637713724651 / 6400000000000 : ℝ) = (4637713724651 / 3200000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4637713724651 / 3200000000000 : ℝ) ≤
+            (3710707032440505919197 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (8479491550067837 / 10000000000000000 : ℝ) ≤ Real.sqrt (4601713724651 / 6400000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (4601713724651 / 6400000000000 : ℝ) ≤ (13249205546981 / 15625000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (4601713724651 / 6400000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1006 :
+    Real.arctan (8479491550067840 / 23111131795069099 : ℝ) ≤
+      (3516511187288392257217 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1005 :
+    Real.arctan (13249205546981 / 15625000000000 : ℝ) ≤
+      (2 * (3516511187288392257217 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (13111131795069099 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (13249205546981 / 15625000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (13249205546981 / 15625000000000 : ℝ) /
+                (1 + (13111131795069099 / 10000000000000000 : ℝ)) =
+              (8479491550067840 / 23111131795069099 : ℝ)
+            by
+        norm_num];
+        exact hau1006)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3516511187288392257217 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (15625000000000 / 13249205546981 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (15625000000000 / 13249205546981 : ℝ) =
+              (13249205546981 / 15625000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1005)
+  have hz1 :
+    (15625000000000 / 13249205546981 : ℝ) ≤ 1 / Real.sqrt (4601713724651 / 6400000000000 : ℝ) := by
+    rw [show (15625000000000 / 13249205546981 : ℝ) = 1 / (13249205546981 / 15625000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 8479491550067837 : ℝ) ≤
+      (882191263222842597187 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (4601713724651 / 6400000000000 : ℝ) ≤
+      (750000000000000 / 8479491550067837 : ℝ) := by
+    rw [show
+        (750000000000000 / 8479491550067837 : ℝ) =
+          (3 / 40) / (8479491550067837 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3516511187288392257217 / 10000000000000000000000 : ℝ))) -
+        6 * (882191263222842597187 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (4601713724651 / 6400000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (4601713724651 / 6400000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((8479491550067837 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3516511187288392257217 / 10000000000000000000000 : ℝ))) -
+          6 * (882191263222842597187 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (4601713724651 / 6400000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (4601713724651 / 6400000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (4601713724651 / 6400000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_618 :
+    ((2711534729338718410971 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1603815203661389908143 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((212127639484407 / 250000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (439767956615355164713 / 1250000000000000000000 : ℝ))) -
+              6 * (881610596728614691281 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (5759761335453 / 8000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (5759761335453 / 8000000000000 : ℝ)) =
+      Real.log (13759761335453 / 8000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((5759761335453 / 8000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (5804761335453 / 8000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2711534729338718410971 / 5000000000000000000000 : ℝ) ≤
+      Real.log (13759761335453 / 8000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (5804761335453 / 8000000000000 : ℝ) ≤
+      (-(1603815203661389908143 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (5804761335453 / 8000000000000 : ℝ) = (5804761335453 / 4000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5804761335453 / 4000000000000 : ℝ) ≤
+            (1861920697838610091857 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (212127639484407 / 250000000000000 : ℝ) ≤ Real.sqrt (5759761335453 / 8000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (5759761335453 / 8000000000000 : ℝ) ≤ (8485105579376283 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (5759761335453 / 8000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1008 :
+    Real.arctan (8485105579376283 / 23114763310603912 : ℝ) ≤
+      (439767956615355164713 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1007 :
+    Real.arctan (8485105579376283 / 10000000000000000 : ℝ) ≤
+      (2 * (439767956615355164713 / 1250000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (1639345413825489 / 1250000000000000 : ℝ) ≤
+          Real.sqrt (1 + (8485105579376283 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (8485105579376283 / 10000000000000000 : ℝ) /
+                (1 + (1639345413825489 / 1250000000000000 : ℝ)) =
+              (8485105579376283 / 23114763310603912 : ℝ)
+            by
+        norm_num];
+        exact hau1008)
+  have hat1 :
+    (Real.pi / 2 - (2 * (439767956615355164713 / 1250000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 8485105579376283 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 8485105579376283 : ℝ) =
+              (8485105579376283 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1007)
+  have hz1 :
+    (10000000000000000 / 8485105579376283 : ℝ) ≤
+      1 / Real.sqrt (5759761335453 / 8000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 8485105579376283 : ℝ) =
+          1 / (8485105579376283 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (6250000000000 / 70709213161469 : ℝ) ≤
+      (881610596728614691281 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (5759761335453 / 8000000000000 : ℝ) ≤
+      (6250000000000 / 70709213161469 : ℝ) := by
+    rw [show
+        (6250000000000 / 70709213161469 : ℝ) = (3 / 40) / (212127639484407 / 250000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (439767956615355164713 / 1250000000000000000000 : ℝ))) -
+        6 * (881610596728614691281 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (5759761335453 / 8000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (5759761335453 / 8000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((212127639484407 / 250000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (439767956615355164713 / 1250000000000000000000 : ℝ))) -
+          6 * (881610596728614691281 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (5759761335453 / 8000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (5759761335453 / 8000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (5759761335453 / 8000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_619 :
+    ((1085721043342463421133 / 2000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(3194513272609733674267 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((4245357948355271 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3519774136081336267697 / 10000000000000000000000 : ℝ))) -
+              6 * (22025776883336704059 / 250000000000000000000 : ℝ)))) ≤
+      Vfield (23069522060369 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (23069522060369 / 32000000000000 : ℝ)) =
+      Real.log (55069522060369 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((23069522060369 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (23249522060369 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1085721043342463421133 / 2000000000000000000000 : ℝ) ≤
+      Real.log (55069522060369 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (23249522060369 / 32000000000000 : ℝ) ≤
+      (-(3194513272609733674267 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23249522060369 / 32000000000000 : ℝ) = (23249522060369 / 16000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23249522060369 / 16000000000000 : ℝ) ≤
+            (3736958530390266325733 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (4245357948355271 / 5000000000000000 : ℝ) ≤ Real.sqrt (23069522060369 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (23069522060369 / 32000000000000 : ℝ) ≤ (1698143179342109 / 2000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (23069522060369 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1010 :
+    Real.arctan (8490715896710545 / 23118393820840001 : ℝ) ≤
+      (3519774136081336267697 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1009 :
+    Real.arctan (1698143179342109 / 2000000000000000 : ℝ) ≤
+      (2 * (3519774136081336267697 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (13118393820840001 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1698143179342109 / 2000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1698143179342109 / 2000000000000000 : ℝ) /
+                (1 + (13118393820840001 / 10000000000000000 : ℝ)) =
+              (8490715896710545 / 23118393820840001 : ℝ)
+            by
+        norm_num];
+        exact hau1010)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3519774136081336267697 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (2000000000000000 / 1698143179342109 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2000000000000000 / 1698143179342109 : ℝ) =
+              (1698143179342109 / 2000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1009)
+  have hz1 :
+    (2000000000000000 / 1698143179342109 : ℝ) ≤
+      1 / Real.sqrt (23069522060369 / 32000000000000 : ℝ) := by
+    rw [show
+        (2000000000000000 / 1698143179342109 : ℝ) = 1 / (1698143179342109 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 4245357948355271 : ℝ) ≤
+      (22025776883336704059 / 250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (23069522060369 / 32000000000000 : ℝ) ≤
+      (375000000000000 / 4245357948355271 : ℝ) := by
+    rw [show
+        (375000000000000 / 4245357948355271 : ℝ) =
+          (3 / 40) / (4245357948355271 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3519774136081336267697 / 10000000000000000000000 : ℝ))) -
+        6 * (22025776883336704059 / 250000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (23069522060369 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (23069522060369 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((4245357948355271 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3519774136081336267697 / 10000000000000000000000 : ℝ))) -
+          6 * (22025776883336704059 / 250000000000000000000 : ℝ))) ≤
+      Real.sqrt (23069522060369 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (23069522060369 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (23069522060369 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_620 :
+    ((2717068955990445230723 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(3181413321281778418051 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((8496322509423929 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2 * (176070132054670847363 / 500000000000000000000 : ℝ))) -
+              6 * (440226347639341872671 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (11549999389463 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (11549999389463 / 16000000000000 : ℝ)) =
+      Real.log (27549999389463 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((11549999389463 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (11639999389463 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2717068955990445230723 / 5000000000000000000000 : ℝ) ≤
+      Real.log (27549999389463 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (11639999389463 / 16000000000000 : ℝ) ≤
+      (-(3181413321281778418051 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (11639999389463 / 16000000000000 : ℝ) = (11639999389463 / 8000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11639999389463 / 8000000000000 : ℝ) ≤
+            (3750058481718221581949 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (8496322509423929 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (11549999389463 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (11549999389463 / 16000000000000 : ℝ) ≤ (2124080627355983 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (11549999389463 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1012 :
+    Real.arctan (4248161254711966 / 11561011663305891 : ℝ) ≤
+      (176070132054670847363 / 500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1011 :
+    Real.arctan (2124080627355983 / 2500000000000000 : ℝ) ≤
+      (2 * (176070132054670847363 / 500000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6561011663305891 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (2124080627355983 / 2500000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (2124080627355983 / 2500000000000000 : ℝ) /
+                (1 + (6561011663305891 / 5000000000000000 : ℝ)) =
+              (4248161254711966 / 11561011663305891 : ℝ)
+            by
+        norm_num];
+        exact hau1012)
+  have hat1 :
+    (Real.pi / 2 - (2 * (176070132054670847363 / 500000000000000000000 : ℝ))) ≤
+      Real.arctan (2500000000000000 / 2124080627355983 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 2124080627355983 : ℝ) =
+              (2124080627355983 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1011)
+  have hz1 :
+    (2500000000000000 / 2124080627355983 : ℝ) ≤
+      1 / Real.sqrt (11549999389463 / 16000000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 2124080627355983 : ℝ) = 1 / (2124080627355983 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 8496322509423929 : ℝ) ≤
+      (440226347639341872671 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (11549999389463 / 16000000000000 : ℝ) ≤
+      (750000000000000 / 8496322509423929 : ℝ) := by
+    rw [show
+        (750000000000000 / 8496322509423929 : ℝ) =
+          (3 / 40) / (8496322509423929 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (176070132054670847363 / 500000000000000000000 : ℝ))) -
+        6 * (440226347639341872671 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (11549999389463 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (11549999389463 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((8496322509423929 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (176070132054670847363 / 500000000000000000000 : ℝ))) -
+          6 * (440226347639341872671 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (11549999389463 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (11549999389463 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (11549999389463 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_621 :
+    ((543966754787035356037 / 1000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(3168330508377498755171 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((8501925424845501 / 10000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2 * (220189323267150120757 / 625000000000000000000 : ℝ))) -
+              6 * (439937726411396261549 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (23130475497483 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (23130475497483 / 32000000000000 : ℝ)) =
+      Real.log (55130475497483 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((23130475497483 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (23310475497483 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (543966754787035356037 / 1000000000000000000000 : ℝ) ≤
+      Real.log (55130475497483 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (23310475497483 / 32000000000000 : ℝ) ≤
+      (-(3168330508377498755171 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23310475497483 / 32000000000000 : ℝ) = (23310475497483 / 16000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23310475497483 / 16000000000000 : ℝ) ≤
+            (3763141294622501244829 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (8501925424845501 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (23130475497483 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (23130475497483 / 32000000000000 : ℝ) ≤ (132842584763211 / 156250000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (23130475497483 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1014 :
+    Real.arctan (1062740678105688 / 2890706478594065 : ℝ) ≤
+      (220189323267150120757 / 625000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1013 :
+    Real.arctan (132842584763211 / 156250000000000 : ℝ) ≤
+      (2 * (220189323267150120757 / 625000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (328141295718813 / 250000000000000 : ℝ) ≤
+          Real.sqrt (1 + (132842584763211 / 156250000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (132842584763211 / 156250000000000 : ℝ) /
+                (1 + (328141295718813 / 250000000000000 : ℝ)) =
+              (1062740678105688 / 2890706478594065 : ℝ)
+            by
+        norm_num];
+        exact hau1014)
+  have hat1 :
+    (Real.pi / 2 - (2 * (220189323267150120757 / 625000000000000000000 : ℝ))) ≤
+      Real.arctan (156250000000000 / 132842584763211 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (156250000000000 / 132842584763211 : ℝ) =
+              (132842584763211 / 156250000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1013)
+  have hz1 :
+    (156250000000000 / 132842584763211 : ℝ) ≤
+      1 / Real.sqrt (23130475497483 / 32000000000000 : ℝ) := by
+    rw [show (156250000000000 / 132842584763211 : ℝ) = 1 / (132842584763211 / 156250000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 2833975141615167 : ℝ) ≤
+      (439937726411396261549 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (23130475497483 / 32000000000000 : ℝ) ≤
+      (250000000000000 / 2833975141615167 : ℝ) := by
+    rw [show
+        (250000000000000 / 2833975141615167 : ℝ) =
+          (3 / 40) / (8501925424845501 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (220189323267150120757 / 625000000000000000000 : ℝ))) -
+        6 * (439937726411396261549 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (23130475497483 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (23130475497483 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((8501925424845501 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (220189323267150120757 / 625000000000000000000 : ℝ))) -
+          6 * (439937726411396261549 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (23130475497483 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (23130475497483 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (23130475497483 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_622 :
+    ((5445194127762287189187 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(126210591564468617503 / 400000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1063440581285023 / 1250000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3524653733925453359381 / 10000000000000000000000 : ℝ))) -
+              6 * (219824836060368561831 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (579023805401 / 800000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (579023805401 / 800000000000 : ℝ)) =
+      Real.log (1379023805401 / 800000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((579023805401 / 800000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (583523805401 / 800000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (5445194127762287189187 / 10000000000000000000000 : ℝ) ≤
+      Real.log (1379023805401 / 800000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (583523805401 / 800000000000 : ℝ) ≤
+      (-(126210591564468617503 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (583523805401 / 800000000000 : ℝ) = (583523805401 / 400000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (583523805401 / 400000000000 : ℝ) ≤
+            (151048280555531382497 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1063440581285023 / 1250000000000000 : ℝ) ≤ Real.sqrt (579023805401 / 800000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (579023805401 / 800000000000 : ℝ) ≤ (8507524650280187 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (579023805401 / 800000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau1016 :
+    Real.arctan (8507524650280187 / 23129279328094325 : ℝ) ≤
+      (3524653733925453359381 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1015 :
+    Real.arctan (8507524650280187 / 10000000000000000 : ℝ) ≤
+      (2 * (3524653733925453359381 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (525171173123773 / 400000000000000 : ℝ) ≤
+          Real.sqrt (1 + (8507524650280187 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (8507524650280187 / 10000000000000000 : ℝ) /
+                (1 + (525171173123773 / 400000000000000 : ℝ)) =
+              (8507524650280187 / 23129279328094325 : ℝ)
+            by
+        norm_num];
+        exact hau1016)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3524653733925453359381 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 8507524650280187 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 8507524650280187 : ℝ) =
+              (8507524650280187 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1015)
+  have hz1 :
+    (10000000000000000 / 8507524650280187 : ℝ) ≤
+      1 / Real.sqrt (579023805401 / 800000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 8507524650280187 : ℝ) =
+          1 / (8507524650280187 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (93750000000000 / 1063440581285023 : ℝ) ≤
+      (219824836060368561831 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (579023805401 / 800000000000 : ℝ) ≤
+      (93750000000000 / 1063440581285023 : ℝ) := by
+    rw [show
+        (93750000000000 / 1063440581285023 : ℝ) =
+          (3 / 40) / (1063440581285023 / 1250000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3524653733925453359381 / 10000000000000000000000 : ℝ))) -
+        6 * (219824836060368561831 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (579023805401 / 800000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (579023805401 / 800000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1063440581285023 / 1250000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3524653733925453359381 / 10000000000000000000000 : ℝ))) -
+          6 * (219824836060368561831 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (579023805401 / 800000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (579023805401 / 800000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (579023805401 / 800000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_623 :
+    ((85167463359885447783 / 156250000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(314221611887456540269 / 1000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((8513120193008883 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3526276330333639916337 / 10000000000000000000000 : ℝ))) -
+              6 * (87872436582745622019 / 1000000000000000000000 : ℝ)))) ≤
+      Vfield (23191428934597 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (23191428934597 / 32000000000000 : ℝ)) =
+      Real.log (55191428934597 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((23191428934597 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (23371428934597 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (85167463359885447783 / 156250000000000000000 : ℝ) ≤
+      Real.log (55191428934597 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (23371428934597 / 32000000000000 : ℝ) ≤
+      (-(314221611887456540269 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23371428934597 / 32000000000000 : ℝ) = (23371428934597 / 16000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23371428934597 / 16000000000000 : ℝ) ≤
+            (378925568412543459731 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (8513120193008883 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (23191428934597 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (23191428934597 / 32000000000000 : ℝ) ≤ (4256560096504443 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (23191428934597 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1018 :
+    Real.arctan (8513120193008886 / 23132905825468163 : ℝ) ≤
+      (3526276330333639916337 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1017 :
+    Real.arctan (4256560096504443 / 5000000000000000 : ℝ) ≤
+      (2 * (3526276330333639916337 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (13132905825468163 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (4256560096504443 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (4256560096504443 / 5000000000000000 : ℝ) /
+                (1 + (13132905825468163 / 10000000000000000 : ℝ)) =
+              (8513120193008886 / 23132905825468163 : ℝ)
+            by
+        norm_num];
+        exact hau1018)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3526276330333639916337 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 4256560096504443 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 4256560096504443 : ℝ) =
+              (4256560096504443 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1017)
+  have hz1 :
+    (5000000000000000 / 4256560096504443 : ℝ) ≤
+      1 / Real.sqrt (23191428934597 / 32000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 4256560096504443 : ℝ) = 1 / (4256560096504443 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 2837706731002961 : ℝ) ≤
+      (87872436582745622019 / 1000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (23191428934597 / 32000000000000 : ℝ) ≤
+      (250000000000000 / 2837706731002961 : ℝ) := by
+    rw [show
+        (250000000000000 / 2837706731002961 : ℝ) =
+          (3 / 40) / (8513120193008883 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3526276330333639916337 / 10000000000000000000000 : ℝ))) -
+        6 * (87872436582745622019 / 1000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (23191428934597 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (23191428934597 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((8513120193008883 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3526276330333639916337 / 10000000000000000000000 : ℝ))) -
+          6 * (87872436582745622019 / 1000000000000000000000 : ℝ))) ≤
+      Real.sqrt (23191428934597 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (23191428934597 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (23191428934597 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_624 :
+    ((5456238133051884174373 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1564592226615293954067 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((8518712060288587 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1763948482885999733877 / 5000000000000000000000 : ℝ))) -
+              6 * (878150513890413655597 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (11610952826577 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (11610952826577 / 16000000000000 : ℝ)) =
+      Real.log (27610952826577 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((11610952826577 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (11700952826577 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (5456238133051884174373 / 10000000000000000000000 : ℝ) ≤
+      Real.log (27610952826577 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (11700952826577 / 16000000000000 : ℝ) ≤
+      (-(1564592226615293954067 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (11700952826577 / 16000000000000 : ℝ) = (11700952826577 / 8000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11700952826577 / 8000000000000 : ℝ) ≤
+            (1901143674884706045933 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (8518712060288587 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (11610952826577 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (11610952826577 / 16000000000000 : ℝ) ≤ (851871206028859 / 1000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (11610952826577 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1020 :
+    Real.arctan (8518712060288590 / 23136531321703847 : ℝ) ≤
+      (1763948482885999733877 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1019 :
+    Real.arctan (851871206028859 / 1000000000000000 : ℝ) ≤
+      (2 * (1763948482885999733877 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (13136531321703847 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (851871206028859 / 1000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (851871206028859 / 1000000000000000 : ℝ) /
+                (1 + (13136531321703847 / 10000000000000000 : ℝ)) =
+              (8518712060288590 / 23136531321703847 : ℝ)
+            by
+        norm_num];
+        exact hau1020)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1763948482885999733877 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (1000000000000000 / 851871206028859 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1000000000000000 / 851871206028859 : ℝ) =
+              (851871206028859 / 1000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1019)
+  have hz1 :
+    (1000000000000000 / 851871206028859 : ℝ) ≤
+      1 / Real.sqrt (11610952826577 / 16000000000000 : ℝ) := by
+    rw [show
+        (1000000000000000 / 851871206028859 : ℝ) = 1 / (851871206028859 / 1000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 8518712060288587 : ℝ) ≤
+      (878150513890413655597 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (11610952826577 / 16000000000000 : ℝ) ≤
+      (750000000000000 / 8518712060288587 : ℝ) := by
+    rw [show
+        (750000000000000 / 8518712060288587 : ℝ) =
+          (3 / 40) / (8518712060288587 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1763948482885999733877 / 5000000000000000000000 : ℝ))) -
+        6 * (878150513890413655597 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (11610952826577 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (11610952826577 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((8518712060288587 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1763948482885999733877 / 5000000000000000000000 : ℝ))) -
+          6 * (878150513890413655597 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (11610952826577 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (11610952826577 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (11610952826577 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/V52.lean b/LeanPool/Zeta5Irrational/Table/V52.lean
new file mode 100644
index 0000000000..f0def0ecbf
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/V52.lean
@@ -0,0 +1,1443 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Energy
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # V52: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_625 :
+    ((2730877782592370590811 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(3116169747917816613287 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1704860051870497 / 2000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3529515644499602674989 / 10000000000000000000000 : ℝ))) -
+              6 * (877577784756869561493 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (23252382371711 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (23252382371711 / 32000000000000 : ℝ)) =
+      Real.log (55252382371711 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((23252382371711 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (23432382371711 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2730877782592370590811 / 5000000000000000000000 : ℝ) ≤
+      Real.log (55252382371711 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (23432382371711 / 32000000000000 : ℝ) ≤
+      (-(3116169747917816613287 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23432382371711 / 32000000000000 : ℝ) = (23432382371711 / 16000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23432382371711 / 16000000000000 : ℝ) ≤
+            (3815302055082183386713 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1704860051870497 / 2000000000000000 : ℝ) ≤ Real.sqrt (23252382371711 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (23252382371711 / 32000000000000 : ℝ) ≤ (1065537532419061 / 1250000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (23252382371711 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1022 :
+    Real.arctan (4262150129676244 / 11570077908815025 : ℝ) ≤
+      (3529515644499602674989 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1021 :
+    Real.arctan (1065537532419061 / 1250000000000000 : ℝ) ≤
+      (2 * (3529515644499602674989 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (262803116352601 / 200000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1065537532419061 / 1250000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1065537532419061 / 1250000000000000 : ℝ) /
+                (1 + (262803116352601 / 200000000000000 : ℝ)) =
+              (4262150129676244 / 11570077908815025 : ℝ)
+            by
+        norm_num];
+        exact hau1022)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3529515644499602674989 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (1250000000000000 / 1065537532419061 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1250000000000000 / 1065537532419061 : ℝ) =
+              (1065537532419061 / 1250000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1021)
+  have hz1 :
+    (1250000000000000 / 1065537532419061 : ℝ) ≤
+      1 / Real.sqrt (23252382371711 / 32000000000000 : ℝ) := by
+    rw [show
+        (1250000000000000 / 1065537532419061 : ℝ) = 1 / (1065537532419061 / 1250000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (150000000000000 / 1704860051870497 : ℝ) ≤
+      (877577784756869561493 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (23252382371711 / 32000000000000 : ℝ) ≤
+      (150000000000000 / 1704860051870497 : ℝ) := by
+    rw [show
+        (150000000000000 / 1704860051870497 : ℝ) =
+          (3 / 40) / (1704860051870497 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3529515644499602674989 / 10000000000000000000000 : ℝ))) -
+        6 * (877577784756869561493 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (23252382371711 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (23252382371711 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1704860051870497 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3529515644499602674989 / 10000000000000000000000 : ℝ))) -
+          6 * (877577784756869561493 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (23252382371711 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (23252382371711 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (23252382371711 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_626 :
+    ((546726995479048066533 / 1000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(77579298971171939033 / 250000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((8529884797410073 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1765566185380807726961 / 5000000000000000000000 : ℝ))) -
+              6 * (219251543692524048579 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (5820714772567 / 8000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (5820714772567 / 8000000000000 : ℝ)) =
+      Real.log (13820714772567 / 8000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((5820714772567 / 8000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (5865714772567 / 8000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (546726995479048066533 / 1000000000000000000000 : ℝ) ≤
+      Real.log (13820714772567 / 8000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (5865714772567 / 8000000000000 : ℝ) ≤
+      (-(77579298971171939033 / 250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (5865714772567 / 8000000000000 : ℝ) = (5865714772567 / 4000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5865714772567 / 4000000000000 : ℝ) ≤
+            (95707496103828060967 / 250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (8529884797410073 / 10000000000000000 : ℝ) ≤ Real.sqrt (5820714772567 / 8000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (5820714772567 / 8000000000000 : ℝ) ≤ (2132471199352519 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (5820714772567 / 8000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1024 :
+    Real.arctan (8529884797410076 / 23143779314074301 : ℝ) ≤
+      (1765566185380807726961 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1023 :
+    Real.arctan (2132471199352519 / 2500000000000000 : ℝ) ≤
+      (2 * (1765566185380807726961 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (13143779314074301 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (2132471199352519 / 2500000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (2132471199352519 / 2500000000000000 : ℝ) /
+                (1 + (13143779314074301 / 10000000000000000 : ℝ)) =
+              (8529884797410076 / 23143779314074301 : ℝ)
+            by
+        norm_num];
+        exact hau1024)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1765566185380807726961 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (2500000000000000 / 2132471199352519 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 2132471199352519 : ℝ) =
+              (2132471199352519 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1023)
+  have hz1 :
+    (2500000000000000 / 2132471199352519 : ℝ) ≤
+      1 / Real.sqrt (5820714772567 / 8000000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 2132471199352519 : ℝ) = 1 / (2132471199352519 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 8529884797410073 : ℝ) ≤
+      (219251543692524048579 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (5820714772567 / 8000000000000 : ℝ) ≤
+      (750000000000000 / 8529884797410073 : ℝ) := by
+    rw [show
+        (750000000000000 / 8529884797410073 : ℝ) =
+          (3 / 40) / (8529884797410073 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1765566185380807726961 / 5000000000000000000000 : ℝ))) -
+        6 * (219251543692524048579 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (5820714772567 / 8000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (5820714772567 / 8000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((8529884797410073 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1765566185380807726961 / 5000000000000000000000 : ℝ))) -
+          6 * (219251543692524048579 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (5820714772567 / 8000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (5820714772567 / 8000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (5820714772567 / 8000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_627 :
+    ((5472781305222789424267 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(3090191042100093015749 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((8535465681647259 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3532747148789359859367 / 10000000000000000000000 : ℝ))) -
+              6 * (175287136058003471421 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (932533432353 / 1280000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (932533432353 / 1280000000000 : ℝ)) =
+      Real.log (2212533432353 / 1280000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((932533432353 / 1280000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (939733432353 / 1280000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (5472781305222789424267 / 10000000000000000000000 : ℝ) ≤
+      Real.log (2212533432353 / 1280000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (939733432353 / 1280000000000 : ℝ) ≤
+      (-(3090191042100093015749 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (939733432353 / 1280000000000 : ℝ) = (939733432353 / 640000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (939733432353 / 640000000000 : ℝ) ≤
+            (3841280760899906984251 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (8535465681647259 / 10000000000000000 : ℝ) ≤ Real.sqrt (932533432353 / 1280000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (932533432353 / 1280000000000 : ℝ) ≤ (4267732840823631 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (932533432353 / 1280000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau1026 :
+    Real.arctan (251043108283743 / 680805935643029 : ℝ) ≤
+      (3532747148789359859367 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1025 :
+    Real.arctan (4267732840823631 / 5000000000000000 : ℝ) ≤
+      (2 * (3532747148789359859367 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6573700905931493 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (4267732840823631 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (4267732840823631 / 5000000000000000 : ℝ) /
+                (1 + (6573700905931493 / 5000000000000000 : ℝ)) =
+              (251043108283743 / 680805935643029 : ℝ)
+            by
+        norm_num];
+        exact hau1026)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3532747148789359859367 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 4267732840823631 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 4267732840823631 : ℝ) =
+              (4267732840823631 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1025)
+  have hz1 :
+    (5000000000000000 / 4267732840823631 : ℝ) ≤
+      1 / Real.sqrt (932533432353 / 1280000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 4267732840823631 : ℝ) = 1 / (4267732840823631 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 2845155227215753 : ℝ) ≤
+      (175287136058003471421 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (932533432353 / 1280000000000 : ℝ) ≤
+      (250000000000000 / 2845155227215753 : ℝ) := by
+    rw [show
+        (250000000000000 / 2845155227215753 : ℝ) =
+          (3 / 40) / (8535465681647259 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3532747148789359859367 / 10000000000000000000000 : ℝ))) -
+        6 * (175287136058003471421 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (932533432353 / 1280000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (932533432353 / 1280000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((8535465681647259 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3532747148789359859367 / 10000000000000000000000 : ℝ))) -
+          6 * (175287136058003471421 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (932533432353 / 1280000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (932533432353 / 1280000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (932533432353 / 1280000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_628 :
+    ((5478289619829812308189 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(769306738482647776489 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2135260729806619 / 2500000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (441794997850047114757 / 1250000000000000000000 : ℝ))) -
+              6 * (218966574423277628579 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (11671906263691 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (11671906263691 / 16000000000000 : ℝ)) =
+      Real.log (27671906263691 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((11671906263691 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (11761906263691 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (5478289619829812308189 / 10000000000000000000000 : ℝ) ≤
+      Real.log (27671906263691 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (11761906263691 / 16000000000000 : ℝ) ≤
+      (-(769306738482647776489 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (11761906263691 / 16000000000000 : ℝ) = (11761906263691 / 8000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11761906263691 / 8000000000000 : ℝ) ≤
+            (963561212267352223511 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2135260729806619 / 2500000000000000 : ℝ) ≤ Real.sqrt (11671906263691 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (11671906263691 / 16000000000000 : ℝ) ≤
+      (8541042919226479 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (11671906263691 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1028 :
+    Real.arctan (8541042919226479 / 23151023311821356 : ℝ) ≤
+      (441794997850047114757 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1027 :
+    Real.arctan (8541042919226479 / 10000000000000000 : ℝ) ≤
+      (2 * (441794997850047114757 / 1250000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (3287755827955339 / 2500000000000000 : ℝ) ≤
+          Real.sqrt (1 + (8541042919226479 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (8541042919226479 / 10000000000000000 : ℝ) /
+                (1 + (3287755827955339 / 2500000000000000 : ℝ)) =
+              (8541042919226479 / 23151023311821356 : ℝ)
+            by
+        norm_num];
+        exact hau1028)
+  have hat1 :
+    (Real.pi / 2 - (2 * (441794997850047114757 / 1250000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 8541042919226479 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 8541042919226479 : ℝ) =
+              (8541042919226479 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1027)
+  have hz1 :
+    (10000000000000000 / 8541042919226479 : ℝ) ≤
+      1 / Real.sqrt (11671906263691 / 16000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 8541042919226479 : ℝ) =
+          1 / (8541042919226479 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (187500000000000 / 2135260729806619 : ℝ) ≤
+      (218966574423277628579 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (11671906263691 / 16000000000000 : ℝ) ≤
+      (187500000000000 / 2135260729806619 : ℝ) := by
+    rw [show
+        (187500000000000 / 2135260729806619 : ℝ) =
+          (3 / 40) / (2135260729806619 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (441794997850047114757 / 1250000000000000000000 : ℝ))) -
+        6 * (218966574423277628579 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (11671906263691 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (11671906263691 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2135260729806619 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (441794997850047114757 / 1250000000000000000000 : ℝ))) -
+          6 * (218966574423277628579 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (11671906263691 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (11671906263691 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (11671906263691 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_629 :
+    ((5483794901954164421831 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(38303495634517765457 / 125000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2136654129321697 / 2500000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3535970876998488086731 / 10000000000000000000000 : ℝ))) -
+              6 * (875298023372310254261 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (23374289245939 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (23374289245939 / 32000000000000 : ℝ)) =
+      Real.log (55374289245939 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((23374289245939 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (23554289245939 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (5483794901954164421831 / 10000000000000000000000 : ℝ) ≤
+      Real.log (55374289245939 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (23554289245939 / 32000000000000 : ℝ) ≤
+      (-(38303495634517765457 / 125000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23554289245939 / 32000000000000 : ℝ) = (23554289245939 / 16000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23554289245939 / 16000000000000 : ℝ) ≤
+            (48339901902982234543 / 125000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2136654129321697 / 2500000000000000 : ℝ) ≤ Real.sqrt (23374289245939 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (23374289245939 / 32000000000000 : ℝ) ≤
+      (8546616517286791 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (23374289245939 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1030 :
+    Real.arctan (8546616517286791 / 23154643814773525 : ℝ) ≤
+      (3535970876998488086731 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1029 :
+    Real.arctan (8546616517286791 / 10000000000000000 : ℝ) ≤
+      (2 * (3535970876998488086731 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (526185752590941 / 400000000000000 : ℝ) ≤
+          Real.sqrt (1 + (8546616517286791 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (8546616517286791 / 10000000000000000 : ℝ) /
+                (1 + (526185752590941 / 400000000000000 : ℝ)) =
+              (8546616517286791 / 23154643814773525 : ℝ)
+            by
+        norm_num];
+        exact hau1030)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3535970876998488086731 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 8546616517286791 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 8546616517286791 : ℝ) =
+              (8546616517286791 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1029)
+  have hz1 :
+    (10000000000000000 / 8546616517286791 : ℝ) ≤
+      1 / Real.sqrt (23374289245939 / 32000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 8546616517286791 : ℝ) =
+          1 / (8546616517286791 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (187500000000000 / 2136654129321697 : ℝ) ≤
+      (875298023372310254261 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (23374289245939 / 32000000000000 : ℝ) ≤
+      (187500000000000 / 2136654129321697 : ℝ) := by
+    rw [show
+        (187500000000000 / 2136654129321697 : ℝ) =
+          (3 / 40) / (2136654129321697 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3535970876998488086731 / 10000000000000000000000 : ℝ))) -
+        6 * (875298023372310254261 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (23374289245939 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (23374289245939 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2136654129321697 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3535970876998488086731 / 10000000000000000000000 : ℝ))) -
+          6 * (875298023372310254261 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (23374289245939 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (23374289245939 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (23374289245939 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_630 :
+    ((5489297154932943295337 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(122053963567387008637 / 400000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1710437296588799 / 2000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (884394958893463844413 / 2500000000000000000000 : ℝ))) -
+              6 * (174946170747382495469 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (1462797872781 / 2000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1462797872781 / 2000000000000 : ℝ)) =
+      Real.log (3462797872781 / 2000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1462797872781 / 2000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1474047872781 / 2000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (5489297154932943295337 / 10000000000000000000000 : ℝ) ≤
+      Real.log (3462797872781 / 2000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1474047872781 / 2000000000000 : ℝ) ≤
+      (-(122053963567387008637 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (1474047872781 / 2000000000000 : ℝ) = (1474047872781 / 1000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1474047872781 / 1000000000000 : ℝ) ≤
+            (155204908552612991363 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1710437296588799 / 2000000000000000 : ℝ) ≤ Real.sqrt (1462797872781 / 2000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1462797872781 / 2000000000000 : ℝ) ≤ (4276093241471999 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1462797872781 / 2000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1032 :
+    Real.arctan (4276093241471999 / 11579131660771237 : ℝ) ≤
+      (884394958893463844413 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1031 :
+    Real.arctan (4276093241471999 / 5000000000000000 : ℝ) ≤
+      (2 * (884394958893463844413 / 2500000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6579131660771237 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (4276093241471999 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (4276093241471999 / 5000000000000000 : ℝ) /
+                (1 + (6579131660771237 / 5000000000000000 : ℝ)) =
+              (4276093241471999 / 11579131660771237 : ℝ)
+            by
+        norm_num];
+        exact hau1032)
+  have hat1 :
+    (Real.pi / 2 - (2 * (884394958893463844413 / 2500000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 4276093241471999 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 4276093241471999 : ℝ) =
+              (4276093241471999 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1031)
+  have hz1 :
+    (5000000000000000 / 4276093241471999 : ℝ) ≤
+      1 / Real.sqrt (1462797872781 / 2000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 4276093241471999 : ℝ) = 1 / (4276093241471999 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (150000000000000 / 1710437296588799 : ℝ) ≤
+      (174946170747382495469 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1462797872781 / 2000000000000 : ℝ) ≤
+      (150000000000000 / 1710437296588799 : ℝ) := by
+    rw [show
+        (150000000000000 / 1710437296588799 : ℝ) =
+          (3 / 40) / (1710437296588799 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (884394958893463844413 / 2500000000000000000000 : ℝ))) -
+        6 * (174946170747382495469 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1462797872781 / 2000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1462797872781 / 2000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1710437296588799 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (884394958893463844413 / 2500000000000000000000 : ℝ))) -
+          6 * (174946170747382495469 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (1462797872781 / 2000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1462797872781 / 2000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1462797872781 / 2000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_631 :
+    ((1098959276419548204243 / 2000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(3038435225960614059479 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((4278876411645369 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (884796715675760448579 / 2500000000000000000000 : ℝ))) -
+              6 * (437082392606239601383 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (23435242683053 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (23435242683053 / 32000000000000 : ℝ)) =
+      Real.log (55435242683053 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((23435242683053 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (23615242683053 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1098959276419548204243 / 2000000000000000000000 : ℝ) ≤
+      Real.log (55435242683053 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (23615242683053 / 32000000000000 : ℝ) ≤
+      (-(3038435225960614059479 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23615242683053 / 32000000000000 : ℝ) = (23615242683053 / 16000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23615242683053 / 16000000000000 : ℝ) ≤
+            (3893036577039385940521 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (4278876411645369 / 5000000000000000 : ℝ) ≤ Real.sqrt (23435242683053 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (23435242683053 / 32000000000000 : ℝ) ≤
+      (8557752823290741 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (23435242683053 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1034 :
+    Real.arctan (8557752823290741 / 23161881832950052 : ℝ) ≤
+      (884796715675760448579 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1033 :
+    Real.arctan (8557752823290741 / 10000000000000000 : ℝ) ≤
+      (2 * (884796715675760448579 / 2500000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (3290470458237513 / 2500000000000000 : ℝ) ≤
+          Real.sqrt (1 + (8557752823290741 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (8557752823290741 / 10000000000000000 : ℝ) /
+                (1 + (3290470458237513 / 2500000000000000 : ℝ)) =
+              (8557752823290741 / 23161881832950052 : ℝ)
+            by
+        norm_num];
+        exact hau1034)
+  have hat1 :
+    (Real.pi / 2 - (2 * (884796715675760448579 / 2500000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 8557752823290741 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 8557752823290741 : ℝ) =
+              (8557752823290741 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1033)
+  have hz1 :
+    (10000000000000000 / 8557752823290741 : ℝ) ≤
+      1 / Real.sqrt (23435242683053 / 32000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 8557752823290741 : ℝ) =
+          1 / (8557752823290741 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (125000000000000 / 1426292137215123 : ℝ) ≤
+      (437082392606239601383 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (23435242683053 / 32000000000000 : ℝ) ≤
+      (125000000000000 / 1426292137215123 : ℝ) := by
+    rw [show
+        (125000000000000 / 1426292137215123 : ℝ) =
+          (3 / 40) / (4278876411645369 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (884796715675760448579 / 2500000000000000000000 : ℝ))) -
+        6 * (437082392606239601383 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (23435242683053 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (23435242683053 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((4278876411645369 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (884796715675760448579 / 2500000000000000000000 : ℝ))) -
+          6 * (437082392606239601383 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (23435242683053 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (23435242683053 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (23435242683053 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_632 :
+    ((5500292586774656357943 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(184664185669970729 / 610351562500000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((8563315545396609 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (708158392509814711217 / 2000000000000000000000 : ℝ))) -
+              6 * (436799907120371740757 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (2346571940161 / 3200000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (2346571940161 / 3200000000000 : ℝ)) =
+      Real.log (5546571940161 / 3200000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((2346571940161 / 3200000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (2364571940161 / 3200000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (5500292586774656357943 / 10000000000000000000000 : ℝ) ≤
+      Real.log (5546571940161 / 3200000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (2364571940161 / 3200000000000 : ℝ) ≤
+      (-(184664185669970729 / 610351562500000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (2364571940161 / 3200000000000 : ℝ) = (2364571940161 / 1600000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2364571940161 / 1600000000000 : ℝ) ≤
+            (953597115474413959 / 2441406250000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (8563315545396609 / 10000000000000000 : ℝ) ≤ Real.sqrt (2346571940161 / 3200000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (2346571940161 / 3200000000000 : ℝ) ≤ (2140828886349153 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (2346571940161 / 3200000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1036 :
+    Real.arctan (2854438515132204 / 7721833116605659 : ℝ) ≤
+      (708158392509814711217 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1035 :
+    Real.arctan (2140828886349153 / 2500000000000000 : ℝ) ≤
+      (2 * (708158392509814711217 / 2000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (13165499349816977 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (2140828886349153 / 2500000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (2140828886349153 / 2500000000000000 : ℝ) /
+                (1 + (13165499349816977 / 10000000000000000 : ℝ)) =
+              (2854438515132204 / 7721833116605659 : ℝ)
+            by
+        norm_num];
+        exact hau1036)
+  have hat1 :
+    (Real.pi / 2 - (2 * (708158392509814711217 / 2000000000000000000000 : ℝ))) ≤
+      Real.arctan (2500000000000000 / 2140828886349153 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 2140828886349153 : ℝ) =
+              (2140828886349153 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1035)
+  have hz1 :
+    (2500000000000000 / 2140828886349153 : ℝ) ≤
+      1 / Real.sqrt (2346571940161 / 3200000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 2140828886349153 : ℝ) = 1 / (2140828886349153 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 2854438515132203 : ℝ) ≤
+      (436799907120371740757 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (2346571940161 / 3200000000000 : ℝ) ≤
+      (250000000000000 / 2854438515132203 : ℝ) := by
+    rw [show
+        (250000000000000 / 2854438515132203 : ℝ) =
+          (3 / 40) / (8563315545396609 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (708158392509814711217 / 2000000000000000000000 : ℝ))) -
+        6 * (436799907120371740757 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (2346571940161 / 3200000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (2346571940161 / 3200000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((8563315545396609 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (708158392509814711217 / 2000000000000000000000 : ℝ))) -
+          6 * (436799907120371740757 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (2346571940161 / 3200000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (2346571940161 / 3200000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (2346571940161 / 3200000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_633 :
+    ((5505785772284306800337 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(3012657422447236628907 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((8568874656308251 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3542395139261497851501 / 10000000000000000000000 : ℝ))) -
+              6 * (109129492159939492009 / 1250000000000000000000 : ℝ)))) ≤
+      Vfield (23496196120167 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (23496196120167 / 32000000000000 : ℝ)) =
+      Real.log (55496196120167 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((23496196120167 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (23676196120167 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (5505785772284306800337 / 10000000000000000000000 : ℝ) ≤
+      Real.log (55496196120167 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (23676196120167 / 32000000000000 : ℝ) ≤
+      (-(3012657422447236628907 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23676196120167 / 32000000000000 : ℝ) = (23676196120167 / 16000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23676196120167 / 16000000000000 : ℝ) ≤
+            (3918814380552763371093 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (8568874656308251 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (23496196120167 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (23496196120167 / 32000000000000 : ℝ) ≤ (4284437328154127 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (23496196120167 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1038 :
+    Real.arctan (8568874656308254 / 23169115872962841 : ℝ) ≤
+      (3542395139261497851501 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1037 :
+    Real.arctan (4284437328154127 / 5000000000000000 : ℝ) ≤
+      (2 * (3542395139261497851501 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (13169115872962841 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (4284437328154127 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (4284437328154127 / 5000000000000000 : ℝ) /
+                (1 + (13169115872962841 / 10000000000000000 : ℝ)) =
+              (8568874656308254 / 23169115872962841 : ℝ)
+            by
+        norm_num];
+        exact hau1038)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3542395139261497851501 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 4284437328154127 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 4284437328154127 : ℝ) =
+              (4284437328154127 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1037)
+  have hz1 :
+    (5000000000000000 / 4284437328154127 : ℝ) ≤
+      1 / Real.sqrt (23496196120167 / 32000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 4284437328154127 : ℝ) = 1 / (4284437328154127 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 8568874656308251 : ℝ) ≤
+      (109129492159939492009 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (23496196120167 / 32000000000000 : ℝ) ≤
+      (750000000000000 / 8568874656308251 : ℝ) := by
+    rw [show
+        (750000000000000 / 8568874656308251 : ℝ) =
+          (3 / 40) / (8568874656308251 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3542395139261497851501 / 10000000000000000000000 : ℝ))) -
+        6 * (109129492159939492009 / 1250000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (23496196120167 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (23496196120167 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((8568874656308251 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3542395139261497851501 / 10000000000000000000000 : ℝ))) -
+          6 * (109129492159939492009 / 1250000000000000000000 : ℝ))) ≤
+      Real.sqrt (23496196120167 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (23496196120167 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (23496196120167 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_634 :
+    ((5511275941941840616771 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2999793396511508222773 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1714886032609893 / 2000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (885999099244110855617 / 2500000000000000000000 : ℝ))) -
+              6 * (872473150802591192013 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (5881668209681 / 8000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (5881668209681 / 8000000000000 : ℝ)) =
+      Real.log (13881668209681 / 8000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((5881668209681 / 8000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (5926668209681 / 8000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (5511275941941840616771 / 10000000000000000000000 : ℝ) ≤
+      Real.log (13881668209681 / 8000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (5926668209681 / 8000000000000 : ℝ) ≤
+      (-(2999793396511508222773 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (5926668209681 / 8000000000000 : ℝ) = (5926668209681 / 4000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (5926668209681 / 4000000000000 : ℝ) ≤
+            (3931678406488491777227 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1714886032609893 / 2000000000000000 : ℝ) ≤ Real.sqrt (5881668209681 / 8000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (5881668209681 / 8000000000000 : ℝ) ≤ (2143607540762367 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (5881668209681 / 8000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1040 :
+    Real.arctan (8574430163049468 / 23172731403206113 : ℝ) ≤
+      (885999099244110855617 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1039 :
+    Real.arctan (2143607540762367 / 2500000000000000 : ℝ) ≤
+      (2 * (885999099244110855617 / 2500000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (13172731403206113 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (2143607540762367 / 2500000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (2143607540762367 / 2500000000000000 : ℝ) /
+                (1 + (13172731403206113 / 10000000000000000 : ℝ)) =
+              (8574430163049468 / 23172731403206113 : ℝ)
+            by
+        norm_num];
+        exact hau1040)
+  have hat1 :
+    (Real.pi / 2 - (2 * (885999099244110855617 / 2500000000000000000000 : ℝ))) ≤
+      Real.arctan (2500000000000000 / 2143607540762367 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 2143607540762367 : ℝ) =
+              (2143607540762367 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1039)
+  have hz1 :
+    (2500000000000000 / 2143607540762367 : ℝ) ≤
+      1 / Real.sqrt (5881668209681 / 8000000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 2143607540762367 : ℝ) = 1 / (2143607540762367 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (50000000000000 / 571628677536631 : ℝ) ≤
+      (872473150802591192013 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (5881668209681 / 8000000000000 : ℝ) ≤
+      (50000000000000 / 571628677536631 : ℝ) := by
+    rw [show
+        (50000000000000 / 571628677536631 : ℝ) =
+          (3 / 40) / (1714886032609893 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (885999099244110855617 / 2500000000000000000000 : ℝ))) -
+        6 * (872473150802591192013 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (5881668209681 / 8000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (5881668209681 / 8000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1714886032609893 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (885999099244110855617 / 2500000000000000000000 : ℝ))) -
+          6 * (872473150802591192013 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (5881668209681 / 8000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (5881668209681 / 8000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (5881668209681 / 8000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_635 :
+    ((1379190774764237213347 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2986945897633933050071 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((33515554971177 / 39062500000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1772797869908340061831 / 5000000000000000000000 : ℝ))) -
+              6 * (174382290259931083323 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (23557149557281 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (23557149557281 / 32000000000000 : ℝ)) =
+      Real.log (55557149557281 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((23557149557281 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (23737149557281 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1379190774764237213347 / 2500000000000000000000 : ℝ) ≤
+      Real.log (55557149557281 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (23737149557281 / 32000000000000 : ℝ) ≤
+      (-(2986945897633933050071 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23737149557281 / 32000000000000 : ℝ) = (23737149557281 / 16000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23737149557281 / 16000000000000 : ℝ) ≤
+            (3944525905366066949929 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (33515554971177 / 39062500000000 : ℝ) ≤ Real.sqrt (23557149557281 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (23557149557281 / 32000000000000 : ℝ) ≤ (1715996414524263 / 2000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (23557149557281 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1042 :
+    Real.arctan (1715996414524263 / 4635269188272827 : ℝ) ≤
+      (1772797869908340061831 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1041 :
+    Real.arctan (1715996414524263 / 2000000000000000 : ℝ) ≤
+      (2 * (1772797869908340061831 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (2635269188272827 / 2000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1715996414524263 / 2000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1715996414524263 / 2000000000000000 : ℝ) /
+                (1 + (2635269188272827 / 2000000000000000 : ℝ)) =
+              (1715996414524263 / 4635269188272827 : ℝ)
+            by
+        norm_num];
+        exact hau1042)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1772797869908340061831 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (2000000000000000 / 1715996414524263 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2000000000000000 / 1715996414524263 : ℝ) =
+              (1715996414524263 / 2000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1041)
+  have hz1 :
+    (2000000000000000 / 1715996414524263 : ℝ) ≤
+      1 / Real.sqrt (23557149557281 / 32000000000000 : ℝ) := by
+    rw [show
+        (2000000000000000 / 1715996414524263 : ℝ) = 1 / (1715996414524263 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (976562500000 / 11171851657059 : ℝ) ≤
+      (174382290259931083323 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (23557149557281 / 32000000000000 : ℝ) ≤
+      (976562500000 / 11171851657059 : ℝ) := by
+    rw [show
+        (976562500000 / 11171851657059 : ℝ) = (3 / 40) / (33515554971177 / 39062500000000 : ℝ) by
+        norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1772797869908340061831 / 5000000000000000000000 : ℝ))) -
+        6 * (174382290259931083323 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (23557149557281 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (23557149557281 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((33515554971177 / 39062500000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1772797869908340061831 / 5000000000000000000000 : ℝ))) -
+          6 * (174382290259931083323 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (23557149557281 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (23557149557281 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (23557149557281 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_636 :
+    ((5522247246933877305379 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2974114883402715778061 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1073191299000277 / 1250000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3547193171891676928569 / 10000000000000000000000 : ℝ))) -
+              6 * (435675417638097174757 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (11793813137919 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (11793813137919 / 16000000000000 : ℝ)) =
+      Real.log (27793813137919 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((11793813137919 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (11883813137919 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (5522247246933877305379 / 10000000000000000000000 : ℝ) ≤
+      Real.log (27793813137919 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (11883813137919 / 16000000000000 : ℝ) ≤
+      (-(2974114883402715778061 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (11883813137919 / 16000000000000 : ℝ) = (11883813137919 / 8000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11883813137919 / 8000000000000 : ℝ) ≤
+            (3957356919597284221939 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1073191299000277 / 1250000000000000 : ℝ) ≤ Real.sqrt (11793813137919 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (11793813137919 / 16000000000000 : ℝ) ≤
+      (8585530392002219 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (11793813137919 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1044 :
+    Real.arctan (8585530392002219 / 23179959488253131 : ℝ) ≤
+      (3547193171891676928569 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1043 :
+    Real.arctan (8585530392002219 / 10000000000000000 : ℝ) ≤
+      (2 * (3547193171891676928569 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (13179959488253131 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (8585530392002219 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (8585530392002219 / 10000000000000000 : ℝ) /
+                (1 + (13179959488253131 / 10000000000000000 : ℝ)) =
+              (8585530392002219 / 23179959488253131 : ℝ)
+            by
+        norm_num];
+        exact hau1044)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3547193171891676928569 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 8585530392002219 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 8585530392002219 : ℝ) =
+              (8585530392002219 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1043)
+  have hz1 :
+    (10000000000000000 / 8585530392002219 : ℝ) ≤
+      1 / Real.sqrt (11793813137919 / 16000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 8585530392002219 : ℝ) =
+          1 / (8585530392002219 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (93750000000000 / 1073191299000277 : ℝ) ≤
+      (435675417638097174757 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (11793813137919 / 16000000000000 : ℝ) ≤
+      (93750000000000 / 1073191299000277 : ℝ) := by
+    rw [show
+        (93750000000000 / 1073191299000277 : ℝ) =
+          (3 / 40) / (1073191299000277 / 1250000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3547193171891676928569 / 10000000000000000000000 : ℝ))) -
+        6 * (435675417638097174757 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (11793813137919 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (11793813137919 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1073191299000277 / 1250000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3547193171891676928569 / 10000000000000000000000 : ℝ))) -
+          6 * (435675417638097174757 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (11793813137919 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (11793813137919 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (11793813137919 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/V53.lean b/LeanPool/Zeta5Irrational/Table/V53.lean
new file mode 100644
index 0000000000..677793076f
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/V53.lean
@@ -0,0 +1,1453 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Energy
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # V53: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_637 :
+    ((1381932097217859613867 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(370162538946138480059 / 1250000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2147768782037017 / 2500000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (443598587162207701421 / 1250000000000000000000 : ℝ))) -
+              6 * (217697824813350455459 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (4723620598879 / 6400000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (4723620598879 / 6400000000000 : ℝ)) =
+      Real.log (11123620598879 / 6400000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((4723620598879 / 6400000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (4759620598879 / 6400000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1381932097217859613867 / 2500000000000000000000 : ℝ) ≤
+      Real.log (11123620598879 / 6400000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (4759620598879 / 6400000000000 : ℝ) ≤
+      (-(370162538946138480059 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (4759620598879 / 6400000000000 : ℝ) = (4759620598879 / 3200000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (4759620598879 / 3200000000000 : ℝ) ≤
+            (496271436428861519941 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2147768782037017 / 2500000000000000 : ℝ) ≤ Real.sqrt (4723620598879 / 6400000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (4723620598879 / 6400000000000 : ℝ) ≤ (8591075128148071 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (4723620598879 / 6400000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1046 :
+    Real.arctan (8591075128148071 / 23183572044688206 : ℝ) ≤
+      (443598587162207701421 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1045 :
+    Real.arctan (8591075128148071 / 10000000000000000 : ℝ) ≤
+      (2 * (443598587162207701421 / 1250000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6591786022344103 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (8591075128148071 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (8591075128148071 / 10000000000000000 : ℝ) /
+                (1 + (6591786022344103 / 5000000000000000 : ℝ)) =
+              (8591075128148071 / 23183572044688206 : ℝ)
+            by
+        norm_num];
+        exact hau1046)
+  have hat1 :
+    (Real.pi / 2 - (2 * (443598587162207701421 / 1250000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 8591075128148071 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 8591075128148071 : ℝ) =
+              (8591075128148071 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1045)
+  have hz1 :
+    (10000000000000000 / 8591075128148071 : ℝ) ≤
+      1 / Real.sqrt (4723620598879 / 6400000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 8591075128148071 : ℝ) =
+          1 / (8591075128148071 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (187500000000000 / 2147768782037017 : ℝ) ≤
+      (217697824813350455459 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (4723620598879 / 6400000000000 : ℝ) ≤
+      (187500000000000 / 2147768782037017 : ℝ) := by
+    rw [show
+        (187500000000000 / 2147768782037017 : ℝ) =
+          (3 / 40) / (2147768782037017 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (443598587162207701421 / 1250000000000000000000 : ℝ))) -
+        6 * (217697824813350455459 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (4723620598879 / 6400000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (4723620598879 / 6400000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2147768782037017 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (443598587162207701421 / 1250000000000000000000 : ℝ))) -
+          6 * (217697824813350455459 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (4723620598879 / 6400000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (4723620598879 / 6400000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (4723620598879 / 6400000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_638 :
+    ((691650816020377922461 / 1250000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(147425107002328637843 / 500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((8596616287992327 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3550382320117678762279 / 10000000000000000000000 : ℝ))) -
+              6 * (870232839768089163139 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (2956072464119 / 4000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (2956072464119 / 4000000000000 : ℝ)) =
+      Real.log (6956072464119 / 4000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((2956072464119 / 4000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (2978572464119 / 4000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (691650816020377922461 / 1250000000000000000000 : ℝ) ≤
+      Real.log (6956072464119 / 4000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (2978572464119 / 4000000000000 : ℝ) ≤
+      (-(147425107002328637843 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (2978572464119 / 4000000000000 : ℝ) = (2978572464119 / 2000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2978572464119 / 2000000000000 : ℝ) ≤
+            (199148483147671362157 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (8596616287992327 / 10000000000000000 : ℝ) ≤ Real.sqrt (2956072464119 / 4000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (2956072464119 / 4000000000000 : ℝ) ≤ (859661628799233 / 1000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (2956072464119 / 4000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1048 :
+    Real.arctan (4298308143996165 / 11593591805741674 : ℝ) ≤
+      (3550382320117678762279 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1047 :
+    Real.arctan (859661628799233 / 1000000000000000 : ℝ) ≤
+      (2 * (3550382320117678762279 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (3296795902870837 / 2500000000000000 : ℝ) ≤
+          Real.sqrt (1 + (859661628799233 / 1000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (859661628799233 / 1000000000000000 : ℝ) /
+                (1 + (3296795902870837 / 2500000000000000 : ℝ)) =
+              (4298308143996165 / 11593591805741674 : ℝ)
+            by
+        norm_num];
+        exact hau1048)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3550382320117678762279 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (1000000000000000 / 859661628799233 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1000000000000000 / 859661628799233 : ℝ) =
+              (859661628799233 / 1000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1047)
+  have hz1 :
+    (1000000000000000 / 859661628799233 : ℝ) ≤
+      1 / Real.sqrt (2956072464119 / 4000000000000 : ℝ) := by
+    rw [show
+        (1000000000000000 / 859661628799233 : ℝ) = 1 / (859661628799233 / 1000000000000000 : ℝ) by
+        norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 2865538762664109 : ℝ) ≤
+      (870232839768089163139 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (2956072464119 / 4000000000000 : ℝ) ≤
+      (250000000000000 / 2865538762664109 : ℝ) := by
+    rw [show
+        (250000000000000 / 2865538762664109 : ℝ) =
+          (3 / 40) / (8596616287992327 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3550382320117678762279 / 10000000000000000000000 : ℝ))) -
+        6 * (870232839768089163139 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (2956072464119 / 4000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (2956072464119 / 4000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((8596616287992327 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3550382320117678762279 / 10000000000000000000000 : ℝ))) -
+          6 * (870232839768089163139 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (2956072464119 / 4000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (2956072464119 / 4000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (2956072464119 / 4000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_639 :
+    ((1384670417024153405141 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(36696504086374459829 / 125000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((8602153878446119 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3551974044421646939057 / 10000000000000000000000 : ℝ))) -
+              6 * (21741886334314884711 / 250000000000000000000 : ℝ)))) ≤
+      Vfield (23679056431509 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (23679056431509 / 32000000000000 : ℝ)) =
+      Real.log (55679056431509 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((23679056431509 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (23859056431509 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1384670417024153405141 / 2500000000000000000000 : ℝ) ≤
+      Real.log (55679056431509 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (23859056431509 / 32000000000000 : ℝ) ≤
+      (-(36696504086374459829 / 125000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23859056431509 / 32000000000000 : ℝ) = (23859056431509 / 16000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23859056431509 / 16000000000000 : ℝ) ≤
+            (49946893451125540171 / 125000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (8602153878446119 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (23679056431509 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (23679056431509 / 32000000000000 : ℝ) ≤ (4301076939223061 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (23679056431509 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1050 :
+    Real.arctan (4301076939223061 / 11595397094725715 : ℝ) ≤
+      (3551974044421646939057 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1049 :
+    Real.arctan (4301076939223061 / 5000000000000000 : ℝ) ≤
+      (2 * (3551974044421646939057 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (1319079418945143 / 1000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (4301076939223061 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (4301076939223061 / 5000000000000000 : ℝ) /
+                (1 + (1319079418945143 / 1000000000000000 : ℝ)) =
+              (4301076939223061 / 11595397094725715 : ℝ)
+            by
+        norm_num];
+        exact hau1050)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3551974044421646939057 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 4301076939223061 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 4301076939223061 : ℝ) =
+              (4301076939223061 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1049)
+  have hz1 :
+    (5000000000000000 / 4301076939223061 : ℝ) ≤
+      1 / Real.sqrt (23679056431509 / 32000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 4301076939223061 : ℝ) = 1 / (4301076939223061 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 8602153878446119 : ℝ) ≤
+      (21741886334314884711 / 250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (23679056431509 / 32000000000000 : ℝ) ≤
+      (750000000000000 / 8602153878446119 : ℝ) := by
+    rw [show
+        (750000000000000 / 8602153878446119 : ℝ) =
+          (3 / 40) / (8602153878446119 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3551974044421646939057 / 10000000000000000000000 : ℝ))) -
+        6 * (21741886334314884711 / 250000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (23679056431509 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (23679056431509 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((8602153878446119 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3551974044421646939057 / 10000000000000000000000 : ℝ))) -
+          6 * (21741886334314884711 / 250000000000000000000 : ℝ))) ≤
+      Real.sqrt (23679056431509 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (23679056431509 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (23679056431509 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_640 :
+    ((1108830762390958605563 / 2000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2922954830394664874721 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((8607687906398339 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3553563874266416429189 / 10000000000000000000000 : ℝ))) -
+              6 * (434559568317348828441 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (11854766575033 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (11854766575033 / 16000000000000 : ℝ)) =
+      Real.log (27854766575033 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((11854766575033 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (11944766575033 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1108830762390958605563 / 2000000000000000000000 : ℝ) ≤
+      Real.log (27854766575033 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (11944766575033 / 16000000000000 : ℝ) ≤
+      (-(2922954830394664874721 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (11944766575033 / 16000000000000 : ℝ) = (11944766575033 / 8000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (11944766575033 / 8000000000000 : ℝ) ≤
+            (4008516972605335125279 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (8607687906398339 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (11854766575033 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (11854766575033 / 16000000000000 : ℝ) ≤ (4303843953199171 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (11854766575033 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1052 :
+    Real.arctan (8607687906398342 / 23194403779404215 : ℝ) ≤
+      (3553563874266416429189 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1051 :
+    Real.arctan (4303843953199171 / 5000000000000000 : ℝ) ≤
+      (2 * (3553563874266416429189 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (2638880755880843 / 2000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (4303843953199171 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (4303843953199171 / 5000000000000000 : ℝ) /
+                (1 + (2638880755880843 / 2000000000000000 : ℝ)) =
+              (8607687906398342 / 23194403779404215 : ℝ)
+            by
+        norm_num];
+        exact hau1052)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3553563874266416429189 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 4303843953199171 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 4303843953199171 : ℝ) =
+              (4303843953199171 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1051)
+  have hz1 :
+    (5000000000000000 / 4303843953199171 : ℝ) ≤
+      1 / Real.sqrt (11854766575033 / 16000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 4303843953199171 : ℝ) = 1 / (4303843953199171 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 8607687906398339 : ℝ) ≤
+      (434559568317348828441 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (11854766575033 / 16000000000000 : ℝ) ≤
+      (750000000000000 / 8607687906398339 : ℝ) := by
+    rw [show
+        (750000000000000 / 8607687906398339 : ℝ) =
+          (3 / 40) / (8607687906398339 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3553563874266416429189 / 10000000000000000000000 : ℝ))) -
+        6 * (434559568317348828441 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (11854766575033 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (11854766575033 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((8607687906398339 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3553563874266416429189 / 10000000000000000000000 : ℝ))) -
+          6 * (434559568317348828441 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (11854766575033 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (11854766575033 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (11854766575033 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_641 :
+    ((2774811481507379783337 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2910205608895841854953 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((4306609189357877 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1777575906847914134599 / 5000000000000000000000 : ℝ))) -
+              6 * (868563886137522201071 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (23740009868623 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (23740009868623 / 32000000000000 : ℝ)) =
+      Real.log (55740009868623 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((23740009868623 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (23920009868623 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2774811481507379783337 / 5000000000000000000000 : ℝ) ≤
+      Real.log (55740009868623 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (23920009868623 / 32000000000000 : ℝ) ≤
+      (-(2910205608895841854953 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23920009868623 / 32000000000000 : ℝ) = (23920009868623 / 16000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23920009868623 / 16000000000000 : ℝ) ≤
+            (4021266194104158145047 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (4306609189357877 / 5000000000000000 : ℝ) ≤ Real.sqrt (23740009868623 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (23740009868623 / 32000000000000 : ℝ) ≤
+      (8613218378715757 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (23740009868623 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1054 :
+    Real.arctan (8613218378715757 / 23198012382152354 : ℝ) ≤
+      (1777575906847914134599 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1053 :
+    Real.arctan (8613218378715757 / 10000000000000000 : ℝ) ≤
+      (2 * (1777575906847914134599 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6599006191076177 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (8613218378715757 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (8613218378715757 / 10000000000000000 : ℝ) /
+                (1 + (6599006191076177 / 5000000000000000 : ℝ)) =
+              (8613218378715757 / 23198012382152354 : ℝ)
+            by
+        norm_num];
+        exact hau1054)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1777575906847914134599 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 8613218378715757 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 8613218378715757 : ℝ) =
+              (8613218378715757 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1053)
+  have hz1 :
+    (10000000000000000 / 8613218378715757 : ℝ) ≤
+      1 / Real.sqrt (23740009868623 / 32000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 8613218378715757 : ℝ) =
+          1 / (8613218378715757 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 4306609189357877 : ℝ) ≤
+      (868563886137522201071 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (23740009868623 / 32000000000000 : ℝ) ≤
+      (375000000000000 / 4306609189357877 : ℝ) := by
+    rw [show
+        (375000000000000 / 4306609189357877 : ℝ) =
+          (3 / 40) / (4306609189357877 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1777575906847914134599 / 5000000000000000000000 : ℝ))) -
+        6 * (868563886137522201071 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (23740009868623 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (23740009868623 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((4306609189357877 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1777575906847914134599 / 5000000000000000000000 : ℝ))) -
+          6 * (868563886137522201071 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (23740009868623 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (23740009868623 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (23740009868623 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_642 :
+    ((5555089124548337093933 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2897472620967558860071 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((4309372651121549 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (355673786674076914369 / 1000000000000000000000 : ℝ))) -
+              6 * (217002424619864156391 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (1188524329359 / 1600000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1188524329359 / 1600000000000 : ℝ)) =
+      Real.log (2788524329359 / 1600000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1188524329359 / 1600000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1197524329359 / 1600000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (5555089124548337093933 / 10000000000000000000000 : ℝ) ≤
+      Real.log (2788524329359 / 1600000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (1197524329359 / 1600000000000 : ℝ) ≤
+      (-(2897472620967558860071 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (1197524329359 / 1600000000000 : ℝ) = (1197524329359 / 800000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (1197524329359 / 800000000000 : ℝ) ≤
+            (4033999182032441139929 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (4309372651121549 / 5000000000000000 : ℝ) ≤ Real.sqrt (1188524329359 / 1600000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1188524329359 / 1600000000000 : ℝ) ≤ (8618745302243101 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1188524329359 / 1600000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1056 :
+    Real.arctan (8618745302243101 / 23201619998505391 : ℝ) ≤
+      (355673786674076914369 / 1000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1055 :
+    Real.arctan (8618745302243101 / 10000000000000000 : ℝ) ≤
+      (2 * (355673786674076914369 / 1000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (13201619998505391 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (8618745302243101 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (8618745302243101 / 10000000000000000 : ℝ) /
+                (1 + (13201619998505391 / 10000000000000000 : ℝ)) =
+              (8618745302243101 / 23201619998505391 : ℝ)
+            by
+        norm_num];
+        exact hau1056)
+  have hat1 :
+    (Real.pi / 2 - (2 * (355673786674076914369 / 1000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 8618745302243101 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 8618745302243101 : ℝ) =
+              (8618745302243101 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1055)
+  have hz1 :
+    (10000000000000000 / 8618745302243101 : ℝ) ≤
+      1 / Real.sqrt (1188524329359 / 1600000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 8618745302243101 : ℝ) =
+          1 / (8618745302243101 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 4309372651121549 : ℝ) ≤
+      (217002424619864156391 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1188524329359 / 1600000000000 : ℝ) ≤
+      (375000000000000 / 4309372651121549 : ℝ) := by
+    rw [show
+        (375000000000000 / 4309372651121549 : ℝ) =
+          (3 / 40) / (4309372651121549 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (355673786674076914369 / 1000000000000000000000 : ℝ))) -
+        6 * (217002424619864156391 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1188524329359 / 1600000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1188524329359 / 1600000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((4309372651121549 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (355673786674076914369 / 1000000000000000000000 : ℝ))) -
+          6 * (217002424619864156391 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (1188524329359 / 1600000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1188524329359 / 1600000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1188524329359 / 1600000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_643 :
+    ((222422091992879484073 / 400000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2884755825322004909539 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2156067170950793 / 2500000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3558322037419228638997 / 10000000000000000000000 : ℝ))) -
+              6 * (216864142568515565517 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (23800963305737 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (23800963305737 / 32000000000000 : ℝ)) =
+      Real.log (55800963305737 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((23800963305737 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (23980963305737 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (222422091992879484073 / 400000000000000000000 : ℝ) ≤
+      Real.log (55800963305737 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (23980963305737 / 32000000000000 : ℝ) ≤
+      (-(2884755825322004909539 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (23980963305737 / 32000000000000 : ℝ) = (23980963305737 / 16000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (23980963305737 / 16000000000000 : ℝ) ≤
+            (4046715977677995090461 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2156067170950793 / 2500000000000000 : ℝ) ≤ Real.sqrt (23800963305737 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (23800963305737 / 32000000000000 : ℝ) ≤ (344970747352127 / 400000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (23800963305737 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1058 :
+    Real.arctan (8624268683803175 / 23205226629271764 : ℝ) ≤
+      (3558322037419228638997 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1057 :
+    Real.arctan (344970747352127 / 400000000000000 : ℝ) ≤
+      (2 * (3558322037419228638997 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (3301306657317941 / 2500000000000000 : ℝ) ≤
+          Real.sqrt (1 + (344970747352127 / 400000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (344970747352127 / 400000000000000 : ℝ) /
+                (1 + (3301306657317941 / 2500000000000000 : ℝ)) =
+              (8624268683803175 / 23205226629271764 : ℝ)
+            by
+        norm_num];
+        exact hau1058)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3558322037419228638997 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (400000000000000 / 344970747352127 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (400000000000000 / 344970747352127 : ℝ) =
+              (344970747352127 / 400000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1057)
+  have hz1 :
+    (400000000000000 / 344970747352127 : ℝ) ≤
+      1 / Real.sqrt (23800963305737 / 32000000000000 : ℝ) := by
+    rw [show (400000000000000 / 344970747352127 : ℝ) = 1 / (344970747352127 / 400000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (187500000000000 / 2156067170950793 : ℝ) ≤
+      (216864142568515565517 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (23800963305737 / 32000000000000 : ℝ) ≤
+      (187500000000000 / 2156067170950793 : ℝ) := by
+    rw [show
+        (187500000000000 / 2156067170950793 : ℝ) =
+          (3 / 40) / (2156067170950793 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3558322037419228638997 / 10000000000000000000000 : ℝ))) -
+        6 * (216864142568515565517 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (23800963305737 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (23800963305737 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2156067170950793 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3558322037419228638997 / 10000000000000000000000 : ℝ))) -
+          6 * (216864142568515565517 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (23800963305737 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (23800963305737 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (23800963305737 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_644 :
+    ((69575156151210255373 / 125000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(718013795207170907293 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((539361783137309 / 625000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (889976082434089063109 / 2500000000000000000000 : ℝ))) -
+              6 * (866904498149987081409 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (11915720012147 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (11915720012147 / 16000000000000 : ℝ)) =
+      Real.log (27915720012147 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((11915720012147 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (12005720012147 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (69575156151210255373 / 125000000000000000000 : ℝ) ≤
+      Real.log (27915720012147 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (12005720012147 / 16000000000000 : ℝ) ≤
+      (-(718013795207170907293 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (12005720012147 / 16000000000000 : ℝ) = (12005720012147 / 8000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (12005720012147 / 8000000000000 : ℝ) ≤
+            (1014854155542829092707 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (539361783137309 / 625000000000000 : ℝ) ≤ Real.sqrt (11915720012147 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (11915720012147 / 16000000000000 : ℝ) ≤
+      (8629788530196947 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (11915720012147 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1060 :
+    Real.arctan (8629788530196947 / 23208832275258807 : ℝ) ≤
+      (889976082434089063109 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1059 :
+    Real.arctan (8629788530196947 / 10000000000000000 : ℝ) ≤
+      (2 * (889976082434089063109 / 2500000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (13208832275258807 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (8629788530196947 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (8629788530196947 / 10000000000000000 : ℝ) /
+                (1 + (13208832275258807 / 10000000000000000 : ℝ)) =
+              (8629788530196947 / 23208832275258807 : ℝ)
+            by
+        norm_num];
+        exact hau1060)
+  have hat1 :
+    (Real.pi / 2 - (2 * (889976082434089063109 / 2500000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 8629788530196947 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 8629788530196947 : ℝ) =
+              (8629788530196947 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1059)
+  have hz1 :
+    (10000000000000000 / 8629788530196947 : ℝ) ≤
+      1 / Real.sqrt (11915720012147 / 16000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 8629788530196947 : ℝ) =
+          1 / (8629788530196947 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (46875000000000 / 539361783137309 : ℝ) ≤
+      (866904498149987081409 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (11915720012147 / 16000000000000 : ℝ) ≤
+      (46875000000000 / 539361783137309 : ℝ) := by
+    rw [show
+        (46875000000000 / 539361783137309 : ℝ) =
+          (3 / 40) / (539361783137309 / 625000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (889976082434089063109 / 2500000000000000000000 : ℝ))) -
+        6 * (866904498149987081409 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (11915720012147 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (11915720012147 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((539361783137309 / 625000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (889976082434089063109 / 2500000000000000000000 : ℝ))) -
+          6 * (866904498149987081409 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (11915720012147 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (11915720012147 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (11915720012147 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_645 :
+    ((5576923940667797187807 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2846702181558542557229 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((8640817644580863 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3563063295243344731693 / 10000000000000000000000 : ℝ))) -
+              6 * (865803508735303275437 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (746637295669 / 1000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (746637295669 / 1000000000000 : ℝ)) =
+      Real.log (1746637295669 / 1000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((746637295669 / 1000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (752262295669 / 1000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (5576923940667797187807 / 10000000000000000000000 : ℝ) ≤
+      Real.log (1746637295669 / 1000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (752262295669 / 1000000000000 : ℝ) ≤
+      (-(2846702181558542557229 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (752262295669 / 1000000000000 : ℝ) = (752262295669 / 500000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (752262295669 / 500000000000 : ℝ) ≤
+            (4084769621441457442771 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (8640817644580863 / 10000000000000000 : ℝ) ≤ Real.sqrt (746637295669 / 1000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (746637295669 / 1000000000000 : ℝ) ≤ (4320408822290433 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (746637295669 / 1000000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau1062 :
+    Real.arctan (4320408822290433 / 11608020308059366 : ℝ) ≤
+      (3563063295243344731693 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1061 :
+    Real.arctan (4320408822290433 / 5000000000000000 : ℝ) ≤
+      (2 * (3563063295243344731693 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (3304010154029683 / 2500000000000000 : ℝ) ≤
+          Real.sqrt (1 + (4320408822290433 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (4320408822290433 / 5000000000000000 : ℝ) /
+                (1 + (3304010154029683 / 2500000000000000 : ℝ)) =
+              (4320408822290433 / 11608020308059366 : ℝ)
+            by
+        norm_num];
+        exact hau1062)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3563063295243344731693 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 4320408822290433 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 4320408822290433 : ℝ) =
+              (4320408822290433 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1061)
+  have hz1 :
+    (5000000000000000 / 4320408822290433 : ℝ) ≤
+      1 / Real.sqrt (746637295669 / 1000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 4320408822290433 : ℝ) = 1 / (4320408822290433 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 2880272548193621 : ℝ) ≤
+      (865803508735303275437 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (746637295669 / 1000000000000 : ℝ) ≤
+      (250000000000000 / 2880272548193621 : ℝ) := by
+    rw [show
+        (250000000000000 / 2880272548193621 : ℝ) =
+          (3 / 40) / (8640817644580863 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3563063295243344731693 / 10000000000000000000000 : ℝ))) -
+        6 * (865803508735303275437 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (746637295669 / 1000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (746637295669 / 1000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((8640817644580863 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3563063295243344731693 / 10000000000000000000000 : ℝ))) -
+          6 * (865803508735303275437 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (746637295669 / 1000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (746637295669 / 1000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (746637295669 / 1000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_646 :
+    ((1395982290881571788087 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(283044465619030769971 / 1000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((8647897332212889 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (356508924778024978169 / 1000000000000000000000 : ℝ))) -
+              6 * (865098242589412292353 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (765809953469387 / 1024000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (765809953469387 / 1024000000000000 : ℝ)) =
+      Real.log (1789809953469387 / 1024000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((765809953469387 / 1024000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (771569953469387 / 1024000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1395982290881571788087 / 2500000000000000000000 : ℝ) ≤
+      Real.log (1789809953469387 / 1024000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (771569953469387 / 1024000000000000 : ℝ) ≤
+      (-(283044465619030769971 / 1000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (771569953469387 / 1024000000000000 : ℝ) =
+            (771569953469387 / 512000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (771569953469387 / 512000000000000 : ℝ) ≤
+            (410102714680969230029 / 1000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (8647897332212889 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (765809953469387 / 1024000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (765809953469387 / 1024000000000000 : ℝ) ≤
+      (2161974333053223 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (765809953469387 / 1024000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1064 :
+    Real.arctan (665222871708684 / 1786205422491955 : ℝ) ≤
+      (356508924778024978169 / 1000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1063 :
+    Real.arctan (2161974333053223 / 2500000000000000 : ℝ) ≤
+      (2 * (356508924778024978169 / 1000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (2644134098479083 / 2000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (2161974333053223 / 2500000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (2161974333053223 / 2500000000000000 : ℝ) /
+                (1 + (2644134098479083 / 2000000000000000 : ℝ)) =
+              (665222871708684 / 1786205422491955 : ℝ)
+            by
+        norm_num];
+        exact hau1064)
+  have hat1 :
+    (Real.pi / 2 - (2 * (356508924778024978169 / 1000000000000000000000 : ℝ))) ≤
+      Real.arctan (2500000000000000 / 2161974333053223 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 2161974333053223 : ℝ) =
+              (2161974333053223 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1063)
+  have hz1 :
+    (2500000000000000 / 2161974333053223 : ℝ) ≤
+      1 / Real.sqrt (765809953469387 / 1024000000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 2161974333053223 : ℝ) = 1 / (2161974333053223 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 2882632444070963 : ℝ) ≤
+      (865098242589412292353 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (765809953469387 / 1024000000000000 : ℝ) ≤
+      (250000000000000 / 2882632444070963 : ℝ) := by
+    rw [show
+        (250000000000000 / 2882632444070963 : ℝ) =
+          (3 / 40) / (8647897332212889 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (356508924778024978169 / 1000000000000000000000 : ℝ))) -
+        6 * (865098242589412292353 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (765809953469387 / 1024000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (765809953469387 / 1024000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((8647897332212889 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (356508924778024978169 / 1000000000000000000000 : ℝ))) -
+          6 * (865098242589412292353 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (765809953469387 / 1024000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (765809953469387 / 1024000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (765809953469387 / 1024000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_647 :
+    ((2795464741252561992059 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1407106759320460963231 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((4327485614363427 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (356711212603540415007 / 1000000000000000000000 : ℝ))) -
+              6 * (432197348567961835751 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (383531658086859 / 512000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (383531658086859 / 512000000000000 : ℝ)) =
+      Real.log (895531658086859 / 512000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((383531658086859 / 512000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (386411658086859 / 512000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2795464741252561992059 / 5000000000000000000000 : ℝ) ≤
+      Real.log (895531658086859 / 512000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (386411658086859 / 512000000000000 : ℝ) ≤
+      (-(1407106759320460963231 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (386411658086859 / 512000000000000 : ℝ) =
+            (386411658086859 / 256000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (386411658086859 / 256000000000000 : ℝ) ≤
+            (2058629142179539036769 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (4327485614363427 / 5000000000000000 : ℝ) ≤
+      Real.sqrt (383531658086859 / 512000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (383531658086859 / 512000000000000 : ℝ) ≤
+      (8654971228726857 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (383531658086859 / 512000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1066 :
+    Real.arctan (8654971228726857 / 23225298747857821 : ℝ) ≤
+      (356711212603540415007 / 1000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1065 :
+    Real.arctan (8654971228726857 / 10000000000000000 : ℝ) ≤
+      (2 * (356711212603540415007 / 1000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (13225298747857821 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (8654971228726857 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (8654971228726857 / 10000000000000000 : ℝ) /
+                (1 + (13225298747857821 / 10000000000000000 : ℝ)) =
+              (8654971228726857 / 23225298747857821 : ℝ)
+            by
+        norm_num];
+        exact hau1066)
+  have hat1 :
+    (Real.pi / 2 - (2 * (356711212603540415007 / 1000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 8654971228726857 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 8654971228726857 : ℝ) =
+              (8654971228726857 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1065)
+  have hz1 :
+    (10000000000000000 / 8654971228726857 : ℝ) ≤
+      1 / Real.sqrt (383531658086859 / 512000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 8654971228726857 : ℝ) =
+          1 / (8654971228726857 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (125000000000000 / 1442495204787809 : ℝ) ≤
+      (432197348567961835751 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (383531658086859 / 512000000000000 : ℝ) ≤
+      (125000000000000 / 1442495204787809 : ℝ) := by
+    rw [show
+        (125000000000000 / 1442495204787809 : ℝ) =
+          (3 / 40) / (4327485614363427 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (356711212603540415007 / 1000000000000000000000 : ℝ))) -
+        6 * (432197348567961835751 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (383531658086859 / 512000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (383531658086859 / 512000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((4327485614363427 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (356711212603540415007 / 1000000000000000000000 : ℝ))) -
+          6 * (432197348567961835751 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (383531658086859 / 512000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (383531658086859 / 512000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (383531658086859 / 512000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_648 :
+    ((5597924904465249726207 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1399004341694148145729 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1082754918538849 / 1250000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1784565969182207295463 / 5000000000000000000000 : ℝ))) -
+              6 * (431846432694659931273 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (768316678878049 / 1024000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (768316678878049 / 1024000000000000 : ℝ)) =
+      Real.log (1792316678878049 / 1024000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((768316678878049 / 1024000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (774076678878049 / 1024000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (5597924904465249726207 / 10000000000000000000000 : ℝ) ≤
+      Real.log (1792316678878049 / 1024000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (774076678878049 / 1024000000000000 : ℝ) ≤
+      (-(1399004341694148145729 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (774076678878049 / 1024000000000000 : ℝ) =
+            (774076678878049 / 512000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (774076678878049 / 512000000000000 : ℝ) ≤
+            (2066731559805851854271 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1082754918538849 / 1250000000000000 : ℝ) ≤
+      Real.sqrt (768316678878049 / 1024000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (768316678878049 / 1024000000000000 : ℝ) ≤
+      (1732407869662159 / 2000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (768316678878049 / 1024000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1068 :
+    Real.arctan (1732407869662159 / 4645985076841398 : ℝ) ≤
+      (1784565969182207295463 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1067 :
+    Real.arctan (1732407869662159 / 2000000000000000 : ℝ) ≤
+      (2 * (1784565969182207295463 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (1322992538420699 / 1000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1732407869662159 / 2000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1732407869662159 / 2000000000000000 : ℝ) /
+                (1 + (1322992538420699 / 1000000000000000 : ℝ)) =
+              (1732407869662159 / 4645985076841398 : ℝ)
+            by
+        norm_num];
+        exact hau1068)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1784565969182207295463 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (2000000000000000 / 1732407869662159 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2000000000000000 / 1732407869662159 : ℝ) =
+              (1732407869662159 / 2000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1067)
+  have hz1 :
+    (2000000000000000 / 1732407869662159 : ℝ) ≤
+      1 / Real.sqrt (768316678878049 / 1024000000000000 : ℝ) := by
+    rw [show
+        (2000000000000000 / 1732407869662159 : ℝ) = 1 / (1732407869662159 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (93750000000000 / 1082754918538849 : ℝ) ≤
+      (431846432694659931273 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (768316678878049 / 1024000000000000 : ℝ) ≤
+      (93750000000000 / 1082754918538849 : ℝ) := by
+    rw [show
+        (93750000000000 / 1082754918538849 : ℝ) =
+          (3 / 40) / (1082754918538849 / 1250000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1784565969182207295463 / 5000000000000000000000 : ℝ))) -
+        6 * (431846432694659931273 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (768316678878049 / 1024000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (768316678878049 / 1024000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1082754918538849 / 1250000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1784565969182207295463 / 5000000000000000000000 : ℝ))) -
+          6 * (431846432694659931273 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (768316678878049 / 1024000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (768316678878049 / 1024000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (768316678878049 / 1024000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/V54.lean b/LeanPool/Zeta5Irrational/Table/V54.lean
new file mode 100644
index 0000000000..048353cc50
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/V54.lean
@@ -0,0 +1,1483 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Energy
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # V54: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_649 :
+    ((5604915436253217929901 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(695457516331357721269 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((4334550852547449 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3571148693088848187633 / 10000000000000000000000 : ℝ))) -
+              6 * (862992740403722005993 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (38478502079119 / 51200000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (38478502079119 / 51200000000000 : ℝ)) =
+      Real.log (89678502079119 / 51200000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((38478502079119 / 51200000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (38766502079119 / 51200000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (5604915436253217929901 / 10000000000000000000000 : ℝ) ≤
+      Real.log (89678502079119 / 51200000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (38766502079119 / 51200000000000 : ℝ) ≤
+      (-(695457516331357721269 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (38766502079119 / 51200000000000 : ℝ) = (38766502079119 / 25600000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (38766502079119 / 25600000000000 : ℝ) ≤
+            (1037410434418642278731 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (4334550852547449 / 5000000000000000 : ℝ) ≤ Real.sqrt (38478502079119 / 51200000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (38478502079119 / 51200000000000 : ℝ) ≤
+      (8669101705094901 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (38478502079119 / 51200000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1070 :
+    Real.arctan (8669101705094901 / 23234550403140988 : ℝ) ≤
+      (3571148693088848187633 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1069 :
+    Real.arctan (8669101705094901 / 10000000000000000 : ℝ) ≤
+      (2 * (3571148693088848187633 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (3308637600785247 / 2500000000000000 : ℝ) ≤
+          Real.sqrt (1 + (8669101705094901 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (8669101705094901 / 10000000000000000 : ℝ) /
+                (1 + (3308637600785247 / 2500000000000000 : ℝ)) =
+              (8669101705094901 / 23234550403140988 : ℝ)
+            by
+        norm_num];
+        exact hau1070)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3571148693088848187633 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 8669101705094901 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 8669101705094901 : ℝ) =
+              (8669101705094901 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1069)
+  have hz1 :
+    (10000000000000000 / 8669101705094901 : ℝ) ≤
+      1 / Real.sqrt (38478502079119 / 51200000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 8669101705094901 : ℝ) =
+          1 / (8669101705094901 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (125000000000000 / 1444850284182483 : ℝ) ≤
+      (862992740403722005993 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (38478502079119 / 51200000000000 : ℝ) ≤
+      (125000000000000 / 1444850284182483 : ℝ) := by
+    rw [show
+        (125000000000000 / 1444850284182483 : ℝ) =
+          (3 / 40) / (4334550852547449 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3571148693088848187633 / 10000000000000000000000 : ℝ))) -
+        6 * (862992740403722005993 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (38478502079119 / 51200000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (38478502079119 / 51200000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((4334550852547449 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3571148693088848187633 / 10000000000000000000000 : ℝ))) -
+          6 * (862992740403722005993 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (38478502079119 / 51200000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (38478502079119 / 51200000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (38478502079119 / 51200000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_650 :
+    ((5611901084701233859877 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2765677579757732943883 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((4338079156575929 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3573162398496421435577 / 10000000000000000000000 : ℝ))) -
+              6 * (215573578818150291501 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (770823404286711 / 1024000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (770823404286711 / 1024000000000000 : ℝ)) =
+      Real.log (1794823404286711 / 1024000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((770823404286711 / 1024000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (776583404286711 / 1024000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (5611901084701233859877 / 10000000000000000000000 : ℝ) ≤
+      Real.log (1794823404286711 / 1024000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (776583404286711 / 1024000000000000 : ℝ) ≤
+      (-(2765677579757732943883 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (776583404286711 / 1024000000000000 : ℝ) =
+            (776583404286711 / 512000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (776583404286711 / 512000000000000 : ℝ) ≤
+            (4165794223242267056117 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (4338079156575929 / 5000000000000000 : ℝ) ≤
+      Real.sqrt (770823404286711 / 1024000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (770823404286711 / 1024000000000000 : ℝ) ≤
+      (8676158313151861 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (770823404286711 / 1024000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1072 :
+    Real.arctan (8676158313151861 / 23239173806354916 : ℝ) ≤
+      (3573162398496421435577 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1071 :
+    Real.arctan (8676158313151861 / 10000000000000000 : ℝ) ≤
+      (2 * (3573162398496421435577 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (3309793451588729 / 2500000000000000 : ℝ) ≤
+          Real.sqrt (1 + (8676158313151861 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (8676158313151861 / 10000000000000000 : ℝ) /
+                (1 + (3309793451588729 / 2500000000000000 : ℝ)) =
+              (8676158313151861 / 23239173806354916 : ℝ)
+            by
+        norm_num];
+        exact hau1072)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3573162398496421435577 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 8676158313151861 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 8676158313151861 : ℝ) =
+              (8676158313151861 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1071)
+  have hz1 :
+    (10000000000000000 / 8676158313151861 : ℝ) ≤
+      1 / Real.sqrt (770823404286711 / 1024000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 8676158313151861 : ℝ) =
+          1 / (8676158313151861 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 4338079156575929 : ℝ) ≤
+      (215573578818150291501 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (770823404286711 / 1024000000000000 : ℝ) ≤
+      (375000000000000 / 4338079156575929 : ℝ) := by
+    rw [show
+        (375000000000000 / 4338079156575929 : ℝ) =
+          (3 / 40) / (4338079156575929 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3573162398496421435577 / 10000000000000000000000 : ℝ))) -
+        6 * (215573578818150291501 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (770823404286711 / 1024000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (770823404286711 / 1024000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((4338079156575929 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3573162398496421435577 / 10000000000000000000000 : ℝ))) -
+          6 * (215573578818150291501 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (770823404286711 / 1024000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (770823404286711 / 1024000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (770823404286711 / 1024000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_651 :
+    ((112377637132543891181 / 200000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2749551142400356585123 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((4341604593248587 / 5000000000000000 : ℝ) *
+            (Real.pi + (Real.pi / 2 - (2 * (178758653142059377991 / 500000000000000000000 : ℝ))) -
+              6 * (861597583128492385527 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (386038383495521 / 512000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (386038383495521 / 512000000000000 : ℝ)) =
+      Real.log (898038383495521 / 512000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((386038383495521 / 512000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (388918383495521 / 512000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (112377637132543891181 / 200000000000000000000 : ℝ) ≤
+      Real.log (898038383495521 / 512000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (388918383495521 / 512000000000000 : ℝ) ≤
+      (-(2749551142400356585123 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (388918383495521 / 512000000000000 : ℝ) =
+            (388918383495521 / 256000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (388918383495521 / 256000000000000 : ℝ) ≤
+            (4181920660599643414877 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (4341604593248587 / 5000000000000000 : ℝ) ≤
+      Real.sqrt (386038383495521 / 512000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (386038383495521 / 512000000000000 : ℝ) ≤
+      (8683209186497177 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (386038383495521 / 512000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1074 :
+    Real.arctan (8683209186497177 / 23243795595540915 : ℝ) ≤
+      (178758653142059377991 / 500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1073 :
+    Real.arctan (8683209186497177 / 10000000000000000 : ℝ) ≤
+      (2 * (178758653142059377991 / 500000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (2648759119108183 / 2000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (8683209186497177 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (8683209186497177 / 10000000000000000 : ℝ) /
+                (1 + (2648759119108183 / 2000000000000000 : ℝ)) =
+              (8683209186497177 / 23243795595540915 : ℝ)
+            by
+        norm_num];
+        exact hau1074)
+  have hat1 :
+    (Real.pi / 2 - (2 * (178758653142059377991 / 500000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 8683209186497177 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 8683209186497177 : ℝ) =
+              (8683209186497177 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1073)
+  have hz1 :
+    (10000000000000000 / 8683209186497177 : ℝ) ≤
+      1 / Real.sqrt (386038383495521 / 512000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 8683209186497177 : ℝ) =
+          1 / (8683209186497177 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 4341604593248587 : ℝ) ≤
+      (861597583128492385527 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (386038383495521 / 512000000000000 : ℝ) ≤
+      (375000000000000 / 4341604593248587 : ℝ) := by
+    rw [show
+        (375000000000000 / 4341604593248587 : ℝ) =
+          (3 / 40) / (4341604593248587 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (178758653142059377991 / 500000000000000000000 : ℝ))) -
+        6 * (861597583128492385527 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (386038383495521 / 512000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (386038383495521 / 512000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((4341604593248587 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (178758653142059377991 / 500000000000000000000 : ℝ))) -
+          6 * (861597583128492385527 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (386038383495521 / 512000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (386038383495521 / 512000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (386038383495521 / 512000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_652 :
+    ((2812928879417364386841 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(136672533468778197849 / 500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2172563584772373 / 2500000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (894295173585930991351 / 2500000000000000000000 : ℝ))) -
+              6 * (215225634285677686871 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (773330129695373 / 1024000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (773330129695373 / 1024000000000000 : ℝ)) =
+      Real.log (1797330129695373 / 1024000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((773330129695373 / 1024000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (779090129695373 / 1024000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2812928879417364386841 / 5000000000000000000000 : ℝ) ≤
+      Real.log (1797330129695373 / 1024000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (779090129695373 / 1024000000000000 : ℝ) ≤
+      (-(136672533468778197849 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (779090129695373 / 1024000000000000 : ℝ) =
+            (779090129695373 / 512000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (779090129695373 / 512000000000000 : ℝ) ≤
+            (209901056681221802151 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2172563584772373 / 2500000000000000 : ℝ) ≤
+      Real.sqrt (773330129695373 / 1024000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (773330129695373 / 1024000000000000 : ℝ) ≤
+      (1738050867817899 / 2000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (773330129695373 / 1024000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1076 :
+    Real.arctan (1738050867817899 / 4649683154477635 : ℝ) ≤
+      (894295173585930991351 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1075 :
+    Real.arctan (1738050867817899 / 2000000000000000 : ℝ) ≤
+      (2 * (894295173585930991351 / 2500000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (529936630895527 / 400000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1738050867817899 / 2000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1738050867817899 / 2000000000000000 : ℝ) /
+                (1 + (529936630895527 / 400000000000000 : ℝ)) =
+              (1738050867817899 / 4649683154477635 : ℝ)
+            by
+        norm_num];
+        exact hau1076)
+  have hat1 :
+    (Real.pi / 2 - (2 * (894295173585930991351 / 2500000000000000000000 : ℝ))) ≤
+      Real.arctan (2000000000000000 / 1738050867817899 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2000000000000000 / 1738050867817899 : ℝ) =
+              (1738050867817899 / 2000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1075)
+  have hz1 :
+    (2000000000000000 / 1738050867817899 : ℝ) ≤
+      1 / Real.sqrt (773330129695373 / 1024000000000000 : ℝ) := by
+    rw [show
+        (2000000000000000 / 1738050867817899 : ℝ) = 1 / (1738050867817899 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (62500000000000 / 724187861590791 : ℝ) ≤
+      (215225634285677686871 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (773330129695373 / 1024000000000000 : ℝ) ≤
+      (62500000000000 / 724187861590791 : ℝ) := by
+    rw [show
+        (62500000000000 / 724187861590791 : ℝ) =
+          (3 / 40) / (2172563584772373 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (894295173585930991351 / 2500000000000000000000 : ℝ))) -
+        6 * (215225634285677686871 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (773330129695373 / 1024000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (773330129695373 / 1024000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2172563584772373 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (894295173585930991351 / 2500000000000000000000 : ℝ))) -
+          6 * (215225634285677686871 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (773330129695373 / 1024000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (773330129695373 / 1024000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (773330129695373 / 1024000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_653 :
+    ((5632828798113236739333 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(543475215442021519619 / 2000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((8697293784830921 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3579185301191316278443 / 10000000000000000000000 : ℝ))) -
+              6 * (21505229263126765113 / 250000000000000000000 : ℝ)))) ≤
+      Vfield (96822936549963 / 128000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (96822936549963 / 128000000000000 : ℝ)) =
+      Real.log (224822936549963 / 128000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((96822936549963 / 128000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (97542936549963 / 128000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (5632828798113236739333 / 10000000000000000000000 : ℝ) ≤
+      Real.log (224822936549963 / 128000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (97542936549963 / 128000000000000 : ℝ) ≤
+      (-(543475215442021519619 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (97542936549963 / 128000000000000 : ℝ) = (97542936549963 / 64000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (97542936549963 / 64000000000000 : ℝ) ≤
+            (842819145157978480381 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (8697293784830921 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (96822936549963 / 128000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (96822936549963 / 128000000000000 : ℝ) ≤
+      (2174323446207731 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (96822936549963 / 128000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1078 :
+    Real.arctan (8697293784830924 / 23253034338582941 : ℝ) ≤
+      (3579185301191316278443 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1077 :
+    Real.arctan (2174323446207731 / 2500000000000000 : ℝ) ≤
+      (2 * (3579185301191316278443 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (13253034338582941 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (2174323446207731 / 2500000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (2174323446207731 / 2500000000000000 : ℝ) /
+                (1 + (13253034338582941 / 10000000000000000 : ℝ)) =
+              (8697293784830924 / 23253034338582941 : ℝ)
+            by
+        norm_num];
+        exact hau1078)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3579185301191316278443 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (2500000000000000 / 2174323446207731 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 2174323446207731 : ℝ) =
+              (2174323446207731 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1077)
+  have hz1 :
+    (2500000000000000 / 2174323446207731 : ℝ) ≤
+      1 / Real.sqrt (96822936549963 / 128000000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 2174323446207731 : ℝ) = 1 / (2174323446207731 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 8697293784830921 : ℝ) ≤
+      (21505229263126765113 / 250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (96822936549963 / 128000000000000 : ℝ) ≤
+      (750000000000000 / 8697293784830921 : ℝ) := by
+    rw [show
+        (750000000000000 / 8697293784830921 : ℝ) =
+          (3 / 40) / (8697293784830921 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3579185301191316278443 / 10000000000000000000000 : ℝ))) -
+        6 * (21505229263126765113 / 250000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (96822936549963 / 128000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (96822936549963 / 128000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((8697293784830921 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3579185301191316278443 / 10000000000000000000000 : ℝ))) -
+          6 * (21505229263126765113 / 250000000000000000000 : ℝ))) ≤
+      Real.sqrt (96822936549963 / 128000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (96822936549963 / 128000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (96822936549963 / 128000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_654 :
+    ((5639794981237929828231 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1350663641416316901041 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2176081884391839 / 2500000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1790593445769071451627 / 5000000000000000000000 : ℝ))) -
+              6 * (85951747652360666837 / 1000000000000000000000 : ℝ)))) ≤
+      Vfield (155167371020807 / 204800000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (155167371020807 / 204800000000000 : ℝ)) =
+      Real.log (359967371020807 / 204800000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((155167371020807 / 204800000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (156319371020807 / 204800000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (5639794981237929828231 / 10000000000000000000000 : ℝ) ≤
+      Real.log (359967371020807 / 204800000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (156319371020807 / 204800000000000 : ℝ) ≤
+      (-(1350663641416316901041 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (156319371020807 / 204800000000000 : ℝ) =
+            (156319371020807 / 102400000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (156319371020807 / 102400000000000 : ℝ) ≤
+            (2115072260083683098959 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2176081884391839 / 2500000000000000 : ℝ) ≤
+      Real.sqrt (155167371020807 / 204800000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (155167371020807 / 204800000000000 : ℝ) ≤
+      (8704327537567359 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (155167371020807 / 204800000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1080 :
+    Real.arctan (8704327537567359 / 23257651295808523 : ℝ) ≤
+      (1790593445769071451627 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1079 :
+    Real.arctan (8704327537567359 / 10000000000000000 : ℝ) ≤
+      (2 * (1790593445769071451627 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (13257651295808523 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (8704327537567359 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (8704327537567359 / 10000000000000000 : ℝ) /
+                (1 + (13257651295808523 / 10000000000000000 : ℝ)) =
+              (8704327537567359 / 23257651295808523 : ℝ)
+            by
+        norm_num];
+        exact hau1080)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1790593445769071451627 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 8704327537567359 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 8704327537567359 : ℝ) =
+              (8704327537567359 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1079)
+  have hz1 :
+    (10000000000000000 / 8704327537567359 : ℝ) ≤
+      1 / Real.sqrt (155167371020807 / 204800000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 8704327537567359 : ℝ) =
+          1 / (8704327537567359 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (62500000000000 / 725360628130613 : ℝ) ≤
+      (85951747652360666837 / 1000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (155167371020807 / 204800000000000 : ℝ) ≤
+      (62500000000000 / 725360628130613 : ℝ) := by
+    rw [show
+        (62500000000000 / 725360628130613 : ℝ) =
+          (3 / 40) / (2176081884391839 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1790593445769071451627 / 5000000000000000000000 : ℝ))) -
+        6 * (85951747652360666837 / 1000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (155167371020807 / 204800000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (155167371020807 / 204800000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2176081884391839 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1790593445769071451627 / 5000000000000000000000 : ℝ))) -
+          6 * (85951747652360666837 / 1000000000000000000000 : ℝ))) ≤
+      Real.sqrt (155167371020807 / 204800000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (155167371020807 / 204800000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (155167371020807 / 204800000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_655 :
+    ((1411689078742467514661 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2685304203571106785121 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((4355677805544397 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1791592736752728819669 / 5000000000000000000000 : ℝ))) -
+              6 * (107353431053037252639 / 1250000000000000000000 : ℝ)))) ≤
+      Vfield (388545108904183 / 512000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (388545108904183 / 512000000000000 : ℝ)) =
+      Real.log (900545108904183 / 512000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((388545108904183 / 512000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (391425108904183 / 512000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1411689078742467514661 / 2500000000000000000000 : ℝ) ≤
+      Real.log (900545108904183 / 512000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (391425108904183 / 512000000000000 : ℝ) ≤
+      (-(2685304203571106785121 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (391425108904183 / 512000000000000 : ℝ) =
+            (391425108904183 / 256000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (391425108904183 / 256000000000000 : ℝ) ≤
+            (4246167599428893214879 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (4355677805544397 / 5000000000000000 : ℝ) ≤
+      Real.sqrt (388545108904183 / 512000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (388545108904183 / 512000000000000 : ℝ) ≤
+      (8711355611088797 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (388545108904183 / 512000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1082 :
+    Real.arctan (8711355611088797 / 23262266645745298 : ℝ) ≤
+      (1791592736752728819669 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1081 :
+    Real.arctan (8711355611088797 / 10000000000000000 : ℝ) ≤
+      (2 * (1791592736752728819669 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6631133322872649 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (8711355611088797 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (8711355611088797 / 10000000000000000 : ℝ) /
+                (1 + (6631133322872649 / 5000000000000000 : ℝ)) =
+              (8711355611088797 / 23262266645745298 : ℝ)
+            by
+        norm_num];
+        exact hau1082)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1791592736752728819669 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 8711355611088797 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 8711355611088797 : ℝ) =
+              (8711355611088797 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1081)
+  have hz1 :
+    (10000000000000000 / 8711355611088797 : ℝ) ≤
+      1 / Real.sqrt (388545108904183 / 512000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 8711355611088797 : ℝ) =
+          1 / (8711355611088797 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 4355677805544397 : ℝ) ≤
+      (107353431053037252639 / 1250000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (388545108904183 / 512000000000000 : ℝ) ≤
+      (375000000000000 / 4355677805544397 : ℝ) := by
+    rw [show
+        (375000000000000 / 4355677805544397 : ℝ) =
+          (3 / 40) / (4355677805544397 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1791592736752728819669 / 5000000000000000000000 : ℝ))) -
+        6 * (107353431053037252639 / 1250000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (388545108904183 / 512000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (388545108904183 / 512000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((4355677805544397 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1791592736752728819669 / 5000000000000000000000 : ℝ))) -
+          6 * (107353431053037252639 / 1250000000000000000000 : ℝ))) ≤
+      Real.sqrt (388545108904183 / 512000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (388545108904183 / 512000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (388545108904183 / 512000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_656 :
+    ((5660664461229229342117 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1326667430844514298871 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((872539477536907 / 1000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1793586822311514017451 / 5000000000000000000000 : ℝ))) -
+              6 * (428726181632072282809 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (194899235804257 / 256000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (194899235804257 / 256000000000000 : ℝ)) =
+      Real.log (450899235804257 / 256000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((194899235804257 / 256000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (196339235804257 / 256000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (5660664461229229342117 / 10000000000000000000000 : ℝ) ≤
+      Real.log (450899235804257 / 256000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (196339235804257 / 256000000000000 : ℝ) ≤
+      (-(1326667430844514298871 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (196339235804257 / 256000000000000 : ℝ) =
+            (196339235804257 / 128000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (196339235804257 / 128000000000000 : ℝ) ≤
+            (2139068470655485701129 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (872539477536907 / 1000000000000000 : ℝ) ≤
+      Real.sqrt (194899235804257 / 256000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (194899235804257 / 256000000000000 : ℝ) ≤
+      (8725394775369073 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (194899235804257 / 256000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1084 :
+    Real.arctan (8725394775369073 / 23271492530459333 : ℝ) ≤
+      (1793586822311514017451 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1083 :
+    Real.arctan (8725394775369073 / 10000000000000000 : ℝ) ≤
+      (2 * (1793586822311514017451 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (13271492530459333 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (8725394775369073 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (8725394775369073 / 10000000000000000 : ℝ) /
+                (1 + (13271492530459333 / 10000000000000000 : ℝ)) =
+              (8725394775369073 / 23271492530459333 : ℝ)
+            by
+        norm_num];
+        exact hau1084)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1793586822311514017451 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 8725394775369073 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 8725394775369073 : ℝ) =
+              (8725394775369073 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1083)
+  have hz1 :
+    (10000000000000000 / 8725394775369073 : ℝ) ≤
+      1 / Real.sqrt (194899235804257 / 256000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 8725394775369073 : ℝ) =
+          1 / (8725394775369073 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (75000000000000 / 872539477536907 : ℝ) ≤
+      (428726181632072282809 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (194899235804257 / 256000000000000 : ℝ) ≤
+      (75000000000000 / 872539477536907 : ℝ) := by
+    rw [show
+        (75000000000000 / 872539477536907 : ℝ) =
+          (3 / 40) / (872539477536907 / 1000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1793586822311514017451 / 5000000000000000000000 : ℝ))) -
+        6 * (428726181632072282809 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (194899235804257 / 256000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (194899235804257 / 256000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((872539477536907 / 1000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1793586822311514017451 / 5000000000000000000000 : ℝ))) -
+          6 * (428726181632072282809 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (194899235804257 / 256000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (194899235804257 / 256000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (194899235804257 / 256000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_657 :
+    ((2837276645349158187007 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(524293479615471020593 / 2000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1092426423360713 / 1250000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (897787469715603738619 / 2500000000000000000000 : ℝ))) -
+              6 * (85608386208098783843 / 1000000000000000000000 : ℝ)))) ≤
+      Vfield (78210366862569 / 102400000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (78210366862569 / 102400000000000 : ℝ)) =
+      Real.log (180610366862569 / 102400000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((78210366862569 / 102400000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (78786366862569 / 102400000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2837276645349158187007 / 5000000000000000000000 : ℝ) ≤
+      Real.log (180610366862569 / 102400000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (78786366862569 / 102400000000000 : ℝ) ≤
+      (-(524293479615471020593 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (78786366862569 / 102400000000000 : ℝ) = (78786366862569 / 51200000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (78786366862569 / 51200000000000 : ℝ) ≤
+            (862000880984528979407 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1092426423360713 / 1250000000000000 : ℝ) ≤
+      Real.sqrt (78210366862569 / 102400000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (78210366862569 / 102400000000000 : ℝ) ≤
+      (8739411386885707 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (78210366862569 / 102400000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1086 :
+    Real.arctan (794491944262337 / 2116428364191795 : ℝ) ≤
+      (897787469715603738619 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1085 :
+    Real.arctan (8739411386885707 / 10000000000000000 : ℝ) ≤
+      (2 * (897787469715603738619 / 2500000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (2656142401221949 / 2000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (8739411386885707 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (8739411386885707 / 10000000000000000 : ℝ) /
+                (1 + (2656142401221949 / 2000000000000000 : ℝ)) =
+              (794491944262337 / 2116428364191795 : ℝ)
+            by
+        norm_num];
+        exact hau1086)
+  have hat1 :
+    (Real.pi / 2 - (2 * (897787469715603738619 / 2500000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 8739411386885707 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 8739411386885707 : ℝ) =
+              (8739411386885707 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1085)
+  have hz1 :
+    (10000000000000000 / 8739411386885707 : ℝ) ≤
+      1 / Real.sqrt (78210366862569 / 102400000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 8739411386885707 : ℝ) =
+          1 / (8739411386885707 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (93750000000000 / 1092426423360713 : ℝ) ≤
+      (85608386208098783843 / 1000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (78210366862569 / 102400000000000 : ℝ) ≤
+      (93750000000000 / 1092426423360713 : ℝ) := by
+    rw [show
+        (93750000000000 / 1092426423360713 : ℝ) =
+          (3 / 40) / (1092426423360713 / 1250000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (897787469715603738619 / 2500000000000000000000 : ℝ))) -
+        6 * (85608386208098783843 / 1000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (78210366862569 / 102400000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (78210366862569 / 102400000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1092426423360713 / 1250000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (897787469715603738619 / 2500000000000000000000 : ℝ))) -
+          6 * (85608386208098783843 / 1000000000000000000000 : ℝ))) ≤
+      Real.sqrt (78210366862569 / 102400000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (78210366862569 / 102400000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (78210366862569 / 102400000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_658 :
+    ((711052857120031112197 / 1250000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1294850582738612798973 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2188351388494561 / 2500000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (898778560006669417019 / 2500000000000000000000 : ℝ))) -
+              6 * (17094437850020888933 / 200000000000000000000 : ℝ)))) ≤
+      Vfield (49038149627147 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (49038149627147 / 64000000000000 : ℝ)) =
+      Real.log (113038149627147 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((49038149627147 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (49398149627147 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (711052857120031112197 / 1250000000000000000000 : ℝ) ≤
+      Real.log (113038149627147 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (49398149627147 / 64000000000000 : ℝ) ≤
+      (-(1294850582738612798973 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (49398149627147 / 64000000000000 : ℝ) = (49398149627147 / 32000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (49398149627147 / 32000000000000 : ℝ) ≤
+            (2170885318761387201027 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2188351388494561 / 2500000000000000 : ℝ) ≤ Real.sqrt (49038149627147 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (49038149627147 / 64000000000000 : ℝ) ≤
+      (8753405553978247 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (49038149627147 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1088 :
+    Real.arctan (8753405553978247 / 23289925086034804 : ℝ) ≤
+      (898778560006669417019 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1087 :
+    Real.arctan (8753405553978247 / 10000000000000000 : ℝ) ≤
+      (2 * (898778560006669417019 / 2500000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (3322481271508701 / 2500000000000000 : ℝ) ≤
+          Real.sqrt (1 + (8753405553978247 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (8753405553978247 / 10000000000000000 : ℝ) /
+                (1 + (3322481271508701 / 2500000000000000 : ℝ)) =
+              (8753405553978247 / 23289925086034804 : ℝ)
+            by
+        norm_num];
+        exact hau1088)
+  have hat1 :
+    (Real.pi / 2 - (2 * (898778560006669417019 / 2500000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 8753405553978247 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 8753405553978247 : ℝ) =
+              (8753405553978247 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1087)
+  have hz1 :
+    (10000000000000000 / 8753405553978247 : ℝ) ≤
+      1 / Real.sqrt (49038149627147 / 64000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 8753405553978247 : ℝ) =
+          1 / (8753405553978247 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (187500000000000 / 2188351388494561 : ℝ) ≤
+      (17094437850020888933 / 200000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (49038149627147 / 64000000000000 : ℝ) ≤
+      (187500000000000 / 2188351388494561 : ℝ) := by
+    rw [show
+        (187500000000000 / 2188351388494561 : ℝ) =
+          (3 / 40) / (2188351388494561 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (898778560006669417019 / 2500000000000000000000 : ℝ))) -
+        6 * (17094437850020888933 / 200000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (49038149627147 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (49038149627147 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2188351388494561 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (898778560006669417019 / 2500000000000000000000 : ℝ))) -
+          6 * (17094437850020888933 / 200000000000000000000 : ℝ))) ≤
+      Real.sqrt (49038149627147 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (49038149627147 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (49038149627147 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_659 :
+    ((1425568303343877683293 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(319754440347319404593 / 1250000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((4383688692060797 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3599066791409163219773 / 10000000000000000000000 : ℝ))) -
+              6 * (853366402731951753291 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (393558559721507 / 512000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (393558559721507 / 512000000000000 : ℝ)) =
+      Real.log (905558559721507 / 512000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((393558559721507 / 512000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (396438559721507 / 512000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1425568303343877683293 / 2500000000000000000000 : ℝ) ≤
+      Real.log (905558559721507 / 512000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 13
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (396438559721507 / 512000000000000 : ℝ) ≤
+      (-(319754440347319404593 / 1250000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (396438559721507 / 512000000000000 : ℝ) =
+            (396438559721507 / 256000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 10
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (396438559721507 / 256000000000000 : ℝ) ≤
+            (546679535027680595407 / 1250000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (4383688692060797 / 5000000000000000 : ℝ) ≤
+      Real.sqrt (393558559721507 / 512000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (393558559721507 / 512000000000000 : ℝ) ≤
+      (8767377384121597 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (393558559721507 / 512000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1090 :
+    Real.arctan (8767377384121597 / 23299131783526578 : ℝ) ≤
+      (3599066791409163219773 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1089 :
+    Real.arctan (8767377384121597 / 10000000000000000 : ℝ) ≤
+      (2 * (3599066791409163219773 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6649565891763289 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (8767377384121597 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (8767377384121597 / 10000000000000000 : ℝ) /
+                (1 + (6649565891763289 / 5000000000000000 : ℝ)) =
+              (8767377384121597 / 23299131783526578 : ℝ)
+            by
+        norm_num];
+        exact hau1090)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3599066791409163219773 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 8767377384121597 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 8767377384121597 : ℝ) =
+              (8767377384121597 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1089)
+  have hz1 :
+    (10000000000000000 / 8767377384121597 : ℝ) ≤
+      1 / Real.sqrt (393558559721507 / 512000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 8767377384121597 : ℝ) =
+          1 / (8767377384121597 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 4383688692060797 : ℝ) ≤
+      (853366402731951753291 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 4
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (393558559721507 / 512000000000000 : ℝ) ≤
+      (375000000000000 / 4383688692060797 : ℝ) := by
+    rw [show
+        (375000000000000 / 4383688692060797 : ℝ) =
+          (3 / 40) / (4383688692060797 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3599066791409163219773 / 10000000000000000000000 : ℝ))) -
+        6 * (853366402731951753291 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (393558559721507 / 512000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (393558559721507 / 512000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((4383688692060797 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3599066791409163219773 / 10000000000000000000000 : ℝ))) -
+          6 * (853366402731951753291 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (393558559721507 / 512000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (393558559721507 / 512000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (393558559721507 / 512000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_660 :
+    ((5716104413083145056471 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2526469834942398681093 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((4390663491967827 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (720601519159810012143 / 2000000000000000000000 : ℝ))) -
+              6 * (6656385480894487883 / 78125000000000000000 : ℝ)))) ≤
+      Vfield (197405961212919 / 256000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (197405961212919 / 256000000000000 : ℝ)) =
+      Real.log (453405961212919 / 256000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((197405961212919 / 256000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (198845961212919 / 256000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (5716104413083145056471 / 10000000000000000000000 : ℝ) ≤
+      Real.log (453405961212919 / 256000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 13
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (198845961212919 / 256000000000000 : ℝ) ≤
+      (-(2526469834942398681093 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (198845961212919 / 256000000000000 : ℝ) =
+            (198845961212919 / 128000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (198845961212919 / 128000000000000 : ℝ) ≤
+            (4405001968057601318907 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (4390663491967827 / 5000000000000000 : ℝ) ≤
+      Real.sqrt (197405961212919 / 256000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (197405961212919 / 256000000000000 : ℝ) ≤
+      (8781326983935657 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (197405961212919 / 256000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1092 :
+    Real.arctan (8781326983935657 / 23308332111831161 : ℝ) ≤
+      (720601519159810012143 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1091 :
+    Real.arctan (8781326983935657 / 10000000000000000 : ℝ) ≤
+      (2 * (720601519159810012143 / 2000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (13308332111831161 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (8781326983935657 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (8781326983935657 / 10000000000000000 : ℝ) /
+                (1 + (13308332111831161 / 10000000000000000 : ℝ)) =
+              (8781326983935657 / 23308332111831161 : ℝ)
+            by
+        norm_num];
+        exact hau1092)
+  have hat1 :
+    (Real.pi / 2 - (2 * (720601519159810012143 / 2000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 8781326983935657 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 8781326983935657 : ℝ) =
+              (8781326983935657 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1091)
+  have hz1 :
+    (10000000000000000 / 8781326983935657 : ℝ) ≤
+      1 / Real.sqrt (197405961212919 / 256000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 8781326983935657 : ℝ) =
+          1 / (8781326983935657 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (125000000000000 / 1463554497322609 : ℝ) ≤
+      (6656385480894487883 / 78125000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (197405961212919 / 256000000000000 : ℝ) ≤
+      (125000000000000 / 1463554497322609 : ℝ) := by
+    rw [show
+        (125000000000000 / 1463554497322609 : ℝ) =
+          (3 / 40) / (4390663491967827 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (720601519159810012143 / 2000000000000000000000 : ℝ))) -
+        6 * (6656385480894487883 / 78125000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (197405961212919 / 256000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (197405961212919 / 256000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((4390663491967827 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (720601519159810012143 / 2000000000000000000000 : ℝ))) -
+          6 * (6656385480894487883 / 78125000000000000000 : ℝ))) ≤
+      Real.sqrt (197405961212919 / 256000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (197405961212919 / 256000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (197405961212919 / 256000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/V55.lean b/LeanPool/Zeta5Irrational/Table/V55.lean
new file mode 100644
index 0000000000..a948a96436
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/V55.lean
@@ -0,0 +1,1463 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Energy
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # V55: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_661 :
+    ((1145983301800406914357 / 2000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(499000694584906999959 / 2000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1759050891838961 / 2000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3606936715486809115069 / 10000000000000000000000 : ℝ))) -
+              6 * (170134931662895184751 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (396065285130169 / 512000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (396065285130169 / 512000000000000 : ℝ)) =
+      Real.log (908065285130169 / 512000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((396065285130169 / 512000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (398945285130169 / 512000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1145983301800406914357 / 2000000000000000000000 : ℝ) ≤
+      Real.log (908065285130169 / 512000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 13
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (398945285130169 / 512000000000000 : ℝ) ≤
+      (-(499000694584906999959 / 2000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (398945285130169 / 512000000000000 : ℝ) =
+            (398945285130169 / 256000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (398945285130169 / 256000000000000 : ℝ) ≤
+            (887293666015093000041 / 2000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1759050891838961 / 2000000000000000 : ℝ) ≤
+      Real.sqrt (396065285130169 / 512000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (396065285130169 / 512000000000000 : ℝ) ≤
+      (1099406807399351 / 1250000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (396065285130169 / 512000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1094 :
+    Real.arctan (2198813614798702 / 5829381521037223 : ℝ) ≤
+      (3606936715486809115069 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1093 :
+    Real.arctan (1099406807399351 / 1250000000000000 : ℝ) ≤
+      (2 * (3606936715486809115069 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (3329381521037223 / 2500000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1099406807399351 / 1250000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1099406807399351 / 1250000000000000 : ℝ) /
+                (1 + (3329381521037223 / 2500000000000000 : ℝ)) =
+              (2198813614798702 / 5829381521037223 : ℝ)
+            by
+        norm_num];
+        exact hau1094)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3606936715486809115069 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (1250000000000000 / 1099406807399351 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1250000000000000 / 1099406807399351 : ℝ) =
+              (1099406807399351 / 1250000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1093)
+  have hz1 :
+    (1250000000000000 / 1099406807399351 : ℝ) ≤
+      1 / Real.sqrt (396065285130169 / 512000000000000 : ℝ) := by
+    rw [show
+        (1250000000000000 / 1099406807399351 : ℝ) = 1 / (1099406807399351 / 1250000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (150000000000000 / 1759050891838961 : ℝ) ≤
+      (170134931662895184751 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (396065285130169 / 512000000000000 : ℝ) ≤
+      (150000000000000 / 1759050891838961 : ℝ) := by
+    rw [show
+        (150000000000000 / 1759050891838961 : ℝ) =
+          (3 / 40) / (1759050891838961 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3606936715486809115069 / 10000000000000000000000 : ℝ))) -
+        6 * (170134931662895184751 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (396065285130169 / 512000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (396065285130169 / 512000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1759050891838961 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3606936715486809115069 / 10000000000000000000000 : ℝ))) -
+          6 * (170134931662895184751 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (396065285130169 / 512000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (396065285130169 / 512000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (396065285130169 / 512000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_662 :
+    ((1148741910766415787971 / 2000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2463635813600251484823 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((8809159914837269 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3610854212269592993969 / 10000000000000000000000 : ℝ))) -
+              6 * (169867660582945952537 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (794637295669 / 1024000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (794637295669 / 1024000000000 : ℝ)) =
+      Real.log (1818637295669 / 1024000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((794637295669 / 1024000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (800397295669 / 1024000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1148741910766415787971 / 2000000000000000000000 : ℝ) ≤
+      Real.log (1818637295669 / 1024000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 13
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (800397295669 / 1024000000000 : ℝ) ≤
+      (-(2463635813600251484823 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (800397295669 / 1024000000000 : ℝ) = (800397295669 / 512000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (800397295669 / 512000000000 : ℝ) ≤
+            (4467835989399748515177 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (8809159914837269 / 10000000000000000 : ℝ) ≤ Real.sqrt (794637295669 / 1024000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (794637295669 / 1024000000000 : ℝ) ≤ (1101144989354659 / 1250000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (794637295669 / 1024000000000 : ℝ) := Real.sqrt_pos.mpr (by norm_num)
+  have hau1096 :
+    Real.arctan (2202289978709318 / 5831678428408643 : ℝ) ≤
+      (3610854212269592993969 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1095 :
+    Real.arctan (1101144989354659 / 1250000000000000 : ℝ) ≤
+      (2 * (3610854212269592993969 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (3331678428408643 / 2500000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1101144989354659 / 1250000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1101144989354659 / 1250000000000000 : ℝ) /
+                (1 + (3331678428408643 / 2500000000000000 : ℝ)) =
+              (2202289978709318 / 5831678428408643 : ℝ)
+            by
+        norm_num];
+        exact hau1096)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3610854212269592993969 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (1250000000000000 / 1101144989354659 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (1250000000000000 / 1101144989354659 : ℝ) =
+              (1101144989354659 / 1250000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1095)
+  have hz1 :
+    (1250000000000000 / 1101144989354659 : ℝ) ≤
+      1 / Real.sqrt (794637295669 / 1024000000000 : ℝ) := by
+    rw [show
+        (1250000000000000 / 1101144989354659 : ℝ) = 1 / (1101144989354659 / 1250000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 8809159914837269 : ℝ) ≤
+      (169867660582945952537 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (794637295669 / 1024000000000 : ℝ) ≤
+      (750000000000000 / 8809159914837269 : ℝ) := by
+    rw [show
+        (750000000000000 / 8809159914837269 : ℝ) =
+          (3 / 40) / (8809159914837269 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3610854212269592993969 / 10000000000000000000000 : ℝ))) -
+        6 * (169867660582945952537 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (794637295669 / 1024000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (794637295669 / 1024000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((8809159914837269 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3610854212269592993969 / 10000000000000000000000 : ℝ))) -
+          6 * (169867660582945952537 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (794637295669 / 1024000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (794637295669 / 1024000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (794637295669 / 1024000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_663 :
+    ((5757483600055410461667 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1216183119845151522901 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2205760863743583 / 2500000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3614760147456547547451 / 10000000000000000000000 : ℝ))) -
+              6 * (42400411290363432093 / 500000000000000000000 : ℝ)))) ≤
+      Vfield (398572010538831 / 512000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (398572010538831 / 512000000000000 : ℝ)) =
+      Real.log (910572010538831 / 512000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((398572010538831 / 512000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (401452010538831 / 512000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (5757483600055410461667 / 10000000000000000000000 : ℝ) ≤
+      Real.log (910572010538831 / 512000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 13
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (401452010538831 / 512000000000000 : ℝ) ≤
+      (-(1216183119845151522901 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (401452010538831 / 512000000000000 : ℝ) =
+            (401452010538831 / 256000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (401452010538831 / 256000000000000 : ℝ) ≤
+            (2249552781654848477099 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2205760863743583 / 2500000000000000 : ℝ) ≤
+      Real.sqrt (398572010538831 / 512000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (398572010538831 / 512000000000000 : ℝ) ≤
+      (1764608690994867 / 2000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (398572010538831 / 512000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1098 :
+    Real.arctan (8823043454974335 / 23335895013397691 : ℝ) ≤
+      (3614760147456547547451 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1097 :
+    Real.arctan (1764608690994867 / 2000000000000000 : ℝ) ≤
+      (2 * (3614760147456547547451 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (13335895013397691 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1764608690994867 / 2000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1764608690994867 / 2000000000000000 : ℝ) /
+                (1 + (13335895013397691 / 10000000000000000 : ℝ)) =
+              (8823043454974335 / 23335895013397691 : ℝ)
+            by
+        norm_num];
+        exact hau1098)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3614760147456547547451 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (2000000000000000 / 1764608690994867 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2000000000000000 / 1764608690994867 : ℝ) =
+              (1764608690994867 / 2000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1097)
+  have hz1 :
+    (2000000000000000 / 1764608690994867 : ℝ) ≤
+      1 / Real.sqrt (398572010538831 / 512000000000000 : ℝ) := by
+    rw [show
+        (2000000000000000 / 1764608690994867 : ℝ) = 1 / (1764608690994867 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (62500000000000 / 735253621247861 : ℝ) ≤
+      (42400411290363432093 / 500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (398572010538831 / 512000000000000 : ℝ) ≤
+      (62500000000000 / 735253621247861 : ℝ) := by
+    rw [show
+        (62500000000000 / 735253621247861 : ℝ) =
+          (3 / 40) / (2205760863743583 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3614760147456547547451 / 10000000000000000000000 : ℝ))) -
+        6 * (42400411290363432093 / 500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (398572010538831 / 512000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (398572010538831 / 512000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2205760863743583 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3614760147456547547451 / 10000000000000000000000 : ℝ))) -
+          6 * (42400411290363432093 / 500000000000000000000 : ℝ))) ≤
+      Real.sqrt (398572010538831 / 512000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (398572010538831 / 512000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (398572010538831 / 512000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_664 :
+    ((5771238699937592243929 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(120059706984401290531 / 500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2209226295724861 / 2500000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1809327290937027190591 / 5000000000000000000000 : ℝ))) -
+              6 * (846684377985567652479 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (199912686621581 / 256000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (199912686621581 / 256000000000000 : ℝ)) =
+      Real.log (455912686621581 / 256000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((199912686621581 / 256000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (201352686621581 / 256000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (5771238699937592243929 / 10000000000000000000000 : ℝ) ≤
+      Real.log (455912686621581 / 256000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 13
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (201352686621581 / 256000000000000 : ℝ) ≤
+      (-(120059706984401290531 / 500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (201352686621581 / 256000000000000 : ℝ) =
+            (201352686621581 / 128000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (201352686621581 / 128000000000000 : ℝ) ≤
+            (226513883165598709469 / 500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2209226295724861 / 2500000000000000 : ℝ) ≤
+      Real.sqrt (199912686621581 / 256000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (199912686621581 / 256000000000000 : ℝ) ≤
+      (8836905182899447 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (199912686621581 / 256000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1100 :
+    Real.arctan (8836905182899447 / 23345069996502644 : ℝ) ≤
+      (1809327290937027190591 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1099 :
+    Real.arctan (8836905182899447 / 10000000000000000 : ℝ) ≤
+      (2 * (1809327290937027190591 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (3336267499125661 / 2500000000000000 : ℝ) ≤
+          Real.sqrt (1 + (8836905182899447 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (8836905182899447 / 10000000000000000 : ℝ) /
+                (1 + (3336267499125661 / 2500000000000000 : ℝ)) =
+              (8836905182899447 / 23345069996502644 : ℝ)
+            by
+        norm_num];
+        exact hau1100)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1809327290937027190591 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 8836905182899447 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 8836905182899447 : ℝ) =
+              (8836905182899447 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1099)
+  have hz1 :
+    (10000000000000000 / 8836905182899447 : ℝ) ≤
+      1 / Real.sqrt (199912686621581 / 256000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 8836905182899447 : ℝ) =
+          1 / (8836905182899447 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (187500000000000 / 2209226295724861 : ℝ) ≤
+      (846684377985567652479 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (199912686621581 / 256000000000000 : ℝ) ≤
+      (187500000000000 / 2209226295724861 : ℝ) := by
+    rw [show
+        (187500000000000 / 2209226295724861 : ℝ) =
+          (3 / 40) / (2209226295724861 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1809327290937027190591 / 5000000000000000000000 : ℝ))) -
+        6 * (846684377985567652479 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (199912686621581 / 256000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (199912686621581 / 256000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2209226295724861 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1809327290937027190591 / 5000000000000000000000 : ℝ))) -
+          6 * (846684377985567652479 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (199912686621581 / 256000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (199912686621581 / 256000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (199912686621581 / 256000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_665 :
+    ((5798692268665044501883 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(46782798876266465479 / 200000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((886456361125207 / 1000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3626409189323357591577 / 10000000000000000000000 : ℝ))) -
+              6 * (844055176835286775381 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (25145756165739 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (25145756165739 / 32000000000000 : ℝ)) =
+      Real.log (57145756165739 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((25145756165739 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (25325756165739 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (5798692268665044501883 / 10000000000000000000000 : ℝ) ≤
+      Real.log (57145756165739 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 13
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (25325756165739 / 32000000000000 : ℝ) ≤
+      (-(46782798876266465479 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (25325756165739 / 32000000000000 : ℝ) = (25325756165739 / 16000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (25325756165739 / 16000000000000 : ℝ) ≤
+            (91846637183733534521 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (886456361125207 / 1000000000000000 : ℝ) ≤ Real.sqrt (25145756165739 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (25145756165739 / 32000000000000 : ℝ) ≤
+      (8864563611252073 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (25145756165739 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1102 :
+    Real.arctan (8864563611252073 / 23363401064771436 : ℝ) ≤
+      (3626409189323357591577 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1101 :
+    Real.arctan (8864563611252073 / 10000000000000000 : ℝ) ≤
+      (2 * (3626409189323357591577 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (3340850266192859 / 2500000000000000 : ℝ) ≤
+          Real.sqrt (1 + (8864563611252073 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (8864563611252073 / 10000000000000000 : ℝ) /
+                (1 + (3340850266192859 / 2500000000000000 : ℝ)) =
+              (8864563611252073 / 23363401064771436 : ℝ)
+            by
+        norm_num];
+        exact hau1102)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3626409189323357591577 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 8864563611252073 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 8864563611252073 : ℝ) =
+              (8864563611252073 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1101)
+  have hz1 :
+    (10000000000000000 / 8864563611252073 : ℝ) ≤
+      1 / Real.sqrt (25145756165739 / 32000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 8864563611252073 : ℝ) =
+          1 / (8864563611252073 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (75000000000000 / 886456361125207 : ℝ) ≤
+      (844055176835286775381 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (25145756165739 / 32000000000000 : ℝ) ≤
+      (75000000000000 / 886456361125207 : ℝ) := by
+    rw [show
+        (75000000000000 / 886456361125207 : ℝ) =
+          (3 / 40) / (886456361125207 / 1000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3626409189323357591577 / 10000000000000000000000 : ℝ))) -
+        6 * (844055176835286775381 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (25145756165739 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (25145756165739 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((886456361125207 / 1000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3626409189323357591577 / 10000000000000000000000 : ℝ))) -
+          6 * (844055176835286775381 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (25145756165739 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (25145756165739 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (25145756165739 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_666 :
+    ((728258834231571345467 / 1250000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2277468446675428642341 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2223034002555877 / 2500000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (363411851176786735763 / 1000000000000000000000 : ℝ))) -
+              6 * (420725158977982490263 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (202419412030243 / 256000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (202419412030243 / 256000000000000 : ℝ)) =
+      Real.log (458419412030243 / 256000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((202419412030243 / 256000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (203859412030243 / 256000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (728258834231571345467 / 1250000000000000000000 : ℝ) ≤
+      Real.log (458419412030243 / 256000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 13
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (203859412030243 / 256000000000000 : ℝ) ≤
+      (-(2277468446675428642341 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (203859412030243 / 256000000000000 : ℝ) =
+            (203859412030243 / 128000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (203859412030243 / 128000000000000 : ℝ) ≤
+            (4654003356324571357659 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2223034002555877 / 2500000000000000 : ℝ) ≤
+      Real.sqrt (202419412030243 / 256000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (202419412030243 / 256000000000000 : ℝ) ≤
+      (8892136010223511 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (202419412030243 / 256000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1104 :
+    Real.arctan (8892136010223511 / 23381707022062382 : ℝ) ≤
+      (363411851176786735763 / 1000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1103 :
+    Real.arctan (8892136010223511 / 10000000000000000 : ℝ) ≤
+      (2 * (363411851176786735763 / 1000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6690853511031191 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (8892136010223511 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (8892136010223511 / 10000000000000000 : ℝ) /
+                (1 + (6690853511031191 / 5000000000000000 : ℝ)) =
+              (8892136010223511 / 23381707022062382 : ℝ)
+            by
+        norm_num];
+        exact hau1104)
+  have hat1 :
+    (Real.pi / 2 - (2 * (363411851176786735763 / 1000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 8892136010223511 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 8892136010223511 : ℝ) =
+              (8892136010223511 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1103)
+  have hz1 :
+    (10000000000000000 / 8892136010223511 : ℝ) ≤
+      1 / Real.sqrt (202419412030243 / 256000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 8892136010223511 : ℝ) =
+          1 / (8892136010223511 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (187500000000000 / 2223034002555877 : ℝ) ≤
+      (420725158977982490263 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (202419412030243 / 256000000000000 : ℝ) ≤
+      (187500000000000 / 2223034002555877 : ℝ) := by
+    rw [show
+        (187500000000000 / 2223034002555877 : ℝ) =
+          (3 / 40) / (2223034002555877 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (363411851176786735763 / 1000000000000000000000 : ℝ))) -
+        6 * (420725158977982490263 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (202419412030243 / 256000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (202419412030243 / 256000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2223034002555877 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (363411851176786735763 / 1000000000000000000000 : ℝ))) -
+          6 * (420725158977982490263 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (202419412030243 / 256000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (202419412030243 / 256000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (202419412030243 / 256000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_667 :
+    ((5853374325948511987951 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(44323499137129365541 / 200000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1114952897202199 / 1250000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1820891509478601631143 / 5000000000000000000000 : ℝ))) -
+              6 * (167773885606187475007 / 2000000000000000000000 : ℝ)))) ≤
+      Vfield (101836387367287 / 128000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (101836387367287 / 128000000000000 : ℝ)) =
+      Real.log (229836387367287 / 128000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((101836387367287 / 128000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (102556387367287 / 128000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (5853374325948511987951 / 10000000000000000000000 : ℝ) ≤
+      Real.log (229836387367287 / 128000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 13
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (102556387367287 / 128000000000000 : ℝ) ≤
+      (-(44323499137129365541 / 200000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (102556387367287 / 128000000000000 : ℝ) = (102556387367287 / 64000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (102556387367287 / 64000000000000 : ℝ) ≤
+            (94305936922870634459 / 200000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1114952897202199 / 1250000000000000 : ℝ) ≤
+      Real.sqrt (101836387367287 / 128000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (101836387367287 / 128000000000000 : ℝ) ≤
+      (1783924635523519 / 2000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (101836387367287 / 128000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1106 :
+    Real.arctan (1783924635523519 / 4679997594257823 : ℝ) ≤
+      (1820891509478601631143 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1105 :
+    Real.arctan (1783924635523519 / 2000000000000000 : ℝ) ≤
+      (2 * (1820891509478601631143 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (2679997594257823 / 2000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1783924635523519 / 2000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1783924635523519 / 2000000000000000 : ℝ) /
+                (1 + (2679997594257823 / 2000000000000000 : ℝ)) =
+              (1783924635523519 / 4679997594257823 : ℝ)
+            by
+        norm_num];
+        exact hau1106)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1820891509478601631143 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (2000000000000000 / 1783924635523519 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2000000000000000 / 1783924635523519 : ℝ) =
+              (1783924635523519 / 2000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1105)
+  have hz1 :
+    (2000000000000000 / 1783924635523519 : ℝ) ≤
+      1 / Real.sqrt (101836387367287 / 128000000000000 : ℝ) := by
+    rw [show
+        (2000000000000000 / 1783924635523519 : ℝ) = 1 / (1783924635523519 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (93750000000000 / 1114952897202199 : ℝ) ≤
+      (167773885606187475007 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (101836387367287 / 128000000000000 : ℝ) ≤
+      (93750000000000 / 1114952897202199 : ℝ) := by
+    rw [show
+        (93750000000000 / 1114952897202199 : ℝ) =
+          (3 / 40) / (1114952897202199 / 1250000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1820891509478601631143 / 5000000000000000000000 : ℝ))) -
+        6 * (167773885606187475007 / 2000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (101836387367287 / 128000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (101836387367287 / 128000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1114952897202199 / 1250000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1820891509478601631143 / 5000000000000000000000 : ℝ))) -
+          6 * (167773885606187475007 / 2000000000000000000000 : ℝ))) ≤
+      Real.sqrt (101836387367287 / 128000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (101836387367287 / 128000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (101836387367287 / 128000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_668 :
+    ((2940301816024167914831 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(86210194747194041833 / 400000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((4473512949491491 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (729880634679149746393 / 2000000000000000000000 : ℝ))) -
+              6 * (418156070855025838797 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (40985227487781 / 51200000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (40985227487781 / 51200000000000 : ℝ)) =
+      Real.log (92185227487781 / 51200000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((40985227487781 / 51200000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (41273227487781 / 51200000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (2940301816024167914831 / 5000000000000000000000 : ℝ) ≤
+      Real.log (92185227487781 / 51200000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 13
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (41273227487781 / 51200000000000 : ℝ) ≤
+      (-(86210194747194041833 / 400000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (41273227487781 / 51200000000000 : ℝ) = (41273227487781 / 25600000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (41273227487781 / 25600000000000 : ℝ) ≤
+            (191048677372805958167 / 400000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (4473512949491491 / 5000000000000000 : ℝ) ≤ Real.sqrt (40985227487781 / 51200000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (40985227487781 / 51200000000000 : ℝ) ≤ (1789405179796597 / 2000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (40985227487781 / 51200000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1108 :
+    Real.arctan (8947025898982985 / 23418244014664223 : ℝ) ≤
+      (729880634679149746393 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1107 :
+    Real.arctan (1789405179796597 / 2000000000000000 : ℝ) ≤
+      (2 * (729880634679149746393 / 2000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (13418244014664223 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1789405179796597 / 2000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1789405179796597 / 2000000000000000 : ℝ) /
+                (1 + (13418244014664223 / 10000000000000000 : ℝ)) =
+              (8947025898982985 / 23418244014664223 : ℝ)
+            by
+        norm_num];
+        exact hau1108)
+  have hat1 :
+    (Real.pi / 2 - (2 * (729880634679149746393 / 2000000000000000000000 : ℝ))) ≤
+      Real.arctan (2000000000000000 / 1789405179796597 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2000000000000000 / 1789405179796597 : ℝ) =
+              (1789405179796597 / 2000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1107)
+  have hz1 :
+    (2000000000000000 / 1789405179796597 : ℝ) ≤
+      1 / Real.sqrt (40985227487781 / 51200000000000 : ℝ) := by
+    rw [show
+        (2000000000000000 / 1789405179796597 : ℝ) = 1 / (1789405179796597 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 4473512949491491 : ℝ) ≤
+      (418156070855025838797 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (40985227487781 / 51200000000000 : ℝ) ≤
+      (375000000000000 / 4473512949491491 : ℝ) := by
+    rw [show
+        (375000000000000 / 4473512949491491 : ℝ) =
+          (3 / 40) / (4473512949491491 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (729880634679149746393 / 2000000000000000000000 : ℝ))) -
+        6 * (418156070855025838797 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (40985227487781 / 51200000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (40985227487781 / 51200000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((4473512949491491 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (729880634679149746393 / 2000000000000000000000 : ℝ))) -
+          6 * (418156070855025838797 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (40985227487781 / 51200000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (40985227487781 / 51200000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (40985227487781 / 51200000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_669 :
+    ((1476939748982764032159 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(2094703660133547240031 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((8974344947875111 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3656979430493994796031 / 10000000000000000000000 : ℝ))) -
+              6 * (833778101392414941191 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (51544875035809 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (51544875035809 / 64000000000000 : ℝ)) =
+      Real.log (115544875035809 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((51544875035809 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (51904875035809 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1476939748982764032159 / 2500000000000000000000 : ℝ) ≤
+      Real.log (115544875035809 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 13
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (51904875035809 / 64000000000000 : ℝ) ≤
+      (-(2094703660133547240031 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (51904875035809 / 64000000000000 : ℝ) = (51904875035809 / 32000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (51904875035809 / 32000000000000 : ℝ) ≤
+            (4836768142866452759969 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (8974344947875111 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (51544875035809 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (51544875035809 / 64000000000000 : ℝ) ≤ (4487172473937557 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (51544875035809 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1110 :
+    Real.arctan (8974344947875114 / 23436475253705919 : ℝ) ≤
+      (3656979430493994796031 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1109 :
+    Real.arctan (4487172473937557 / 5000000000000000 : ℝ) ≤
+      (2 * (3656979430493994796031 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (13436475253705919 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (4487172473937557 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (4487172473937557 / 5000000000000000 : ℝ) /
+                (1 + (13436475253705919 / 10000000000000000 : ℝ)) =
+              (8974344947875114 / 23436475253705919 : ℝ)
+            by
+        norm_num];
+        exact hau1110)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3656979430493994796031 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 4487172473937557 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 4487172473937557 : ℝ) =
+              (4487172473937557 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1109)
+  have hz1 :
+    (5000000000000000 / 4487172473937557 : ℝ) ≤
+      1 / Real.sqrt (51544875035809 / 64000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 4487172473937557 : ℝ) = 1 / (4487172473937557 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 8974344947875111 : ℝ) ≤
+      (833778101392414941191 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (51544875035809 / 64000000000000 : ℝ) ≤
+      (750000000000000 / 8974344947875111 : ℝ) := by
+    rw [show
+        (750000000000000 / 8974344947875111 : ℝ) =
+          (3 / 40) / (8974344947875111 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3656979430493994796031 / 10000000000000000000000 : ℝ))) -
+        6 * (833778101392414941191 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (51544875035809 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (51544875035809 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((8974344947875111 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3656979430493994796031 / 10000000000000000000000 : ℝ))) -
+          6 * (833778101392414941191 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (51544875035809 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (51544875035809 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (51544875035809 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_670 :
+    ((5961849495794045218043 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1974690200181473962903 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((2257183766004251 / 2500000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3672002039722087261413 / 10000000000000000000000 : ℝ))) -
+              6 * (828778365856198504881 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (104343112775949 / 128000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (104343112775949 / 128000000000000 : ℝ)) =
+      Real.log (232343112775949 / 128000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((104343112775949 / 128000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (105063112775949 / 128000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (5961849495794045218043 / 10000000000000000000000 : ℝ) ≤
+      Real.log (232343112775949 / 128000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 13
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (105063112775949 / 128000000000000 : ℝ) ≤
+      (-(1974690200181473962903 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (105063112775949 / 128000000000000 : ℝ) = (105063112775949 / 64000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 11
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (105063112775949 / 64000000000000 : ℝ) ≤
+            (4956781602818526037097 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (2257183766004251 / 2500000000000000 : ℝ) ≤
+      Real.sqrt (104343112775949 / 128000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (104343112775949 / 128000000000000 : ℝ) ≤
+      (9028735064017007 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (104343112775949 / 128000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1112 :
+    Real.arctan (9028735064017007 / 23472863721429463 : ℝ) ≤
+      (3672002039722087261413 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1111 :
+    Real.arctan (9028735064017007 / 10000000000000000 : ℝ) ≤
+      (2 * (3672002039722087261413 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (13472863721429463 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (9028735064017007 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (9028735064017007 / 10000000000000000 : ℝ) /
+                (1 + (13472863721429463 / 10000000000000000 : ℝ)) =
+              (9028735064017007 / 23472863721429463 : ℝ)
+            by
+        norm_num];
+        exact hau1112)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3672002039722087261413 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 9028735064017007 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 9028735064017007 : ℝ) =
+              (9028735064017007 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1111)
+  have hz1 :
+    (10000000000000000 / 9028735064017007 : ℝ) ≤
+      1 / Real.sqrt (104343112775949 / 128000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 9028735064017007 : ℝ) =
+          1 / (9028735064017007 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (187500000000000 / 2257183766004251 : ℝ) ≤
+      (828778365856198504881 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (104343112775949 / 128000000000000 : ℝ) ≤
+      (187500000000000 / 2257183766004251 : ℝ) := by
+    rw [show
+        (187500000000000 / 2257183766004251 : ℝ) =
+          (3 / 40) / (2257183766004251 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3672002039722087261413 / 10000000000000000000000 : ℝ))) -
+        6 * (828778365856198504881 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (104343112775949 / 128000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (104343112775949 / 128000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((2257183766004251 / 2500000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3672002039722087261413 / 10000000000000000000000 : ℝ))) -
+          6 * (828778365856198504881 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (104343112775949 / 128000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (104343112775949 / 128000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (104343112775949 / 128000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_671 :
+    ((3007824495396823922311 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1856099999143922723371 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1135349935516639 / 1250000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (737370870709823805733 / 2000000000000000000000 : ℝ))) -
+              6 * (82386750778903374503 / 1000000000000000000000 : ℝ)))) ≤
+      Vfield (2639911887007 / 3200000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (2639911887007 / 3200000000000 : ℝ)) =
+      Real.log (5839911887007 / 3200000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((2639911887007 / 3200000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (2657911887007 / 3200000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (3007824495396823922311 / 5000000000000000000000 : ℝ) ≤
+      Real.log (5839911887007 / 3200000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 13
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (2657911887007 / 3200000000000 : ℝ) ≤
+      (-(1856099999143922723371 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (2657911887007 / 3200000000000 : ℝ) = (2657911887007 / 1600000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (2657911887007 / 1600000000000 : ℝ) ≤
+            (5075371803856077276629 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1135349935516639 / 1250000000000000 : ℝ) ≤ Real.sqrt (2639911887007 / 3200000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (2639911887007 / 3200000000000 : ℝ) ≤ (1816559896826623 / 2000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (2639911887007 / 3200000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1114 :
+    Real.arctan (139735376678971 / 361679294968710 : ℝ) ≤
+      (737370870709823805733 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1113 :
+    Real.arctan (1816559896826623 / 2000000000000000 : ℝ) ≤
+      (2 * (737370870709823805733 / 2000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (270183083459323 / 200000000000000 : ℝ) ≤
+          Real.sqrt (1 + (1816559896826623 / 2000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (1816559896826623 / 2000000000000000 : ℝ) /
+                (1 + (270183083459323 / 200000000000000 : ℝ)) =
+              (139735376678971 / 361679294968710 : ℝ)
+            by
+        norm_num];
+        exact hau1114)
+  have hat1 :
+    (Real.pi / 2 - (2 * (737370870709823805733 / 2000000000000000000000 : ℝ))) ≤
+      Real.arctan (2000000000000000 / 1816559896826623 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2000000000000000 / 1816559896826623 : ℝ) =
+              (1816559896826623 / 2000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1113)
+  have hz1 :
+    (2000000000000000 / 1816559896826623 : ℝ) ≤
+      1 / Real.sqrt (2639911887007 / 3200000000000 : ℝ) := by
+    rw [show
+        (2000000000000000 / 1816559896826623 : ℝ) = 1 / (1816559896826623 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (93750000000000 / 1135349935516639 : ℝ) ≤
+      (82386750778903374503 / 1000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (2639911887007 / 3200000000000 : ℝ) ≤
+      (93750000000000 / 1135349935516639 : ℝ) := by
+    rw [show
+        (93750000000000 / 1135349935516639 : ℝ) =
+          (3 / 40) / (1135349935516639 / 1250000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (737370870709823805733 / 2000000000000000000000 : ℝ))) -
+        6 * (82386750778903374503 / 1000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (2639911887007 / 3200000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (2639911887007 / 3200000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1135349935516639 / 1250000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (737370870709823805733 / 2000000000000000000000 : ℝ))) -
+          6 * (82386750778903374503 / 1000000000000000000000 : ℝ))) ≤
+      Real.sqrt (2639911887007 / 3200000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (2639911887007 / 3200000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (2639911887007 / 3200000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_672 :
+    ((1517290148843070986257 / 2500000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(434724923741976688371 / 2500000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((9136543990028577 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (925384943603547055683 / 2500000000000000000000 : ℝ))) -
+              6 * (409521462399309001603 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (106849838184611 / 128000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (106849838184611 / 128000000000000 : ℝ)) =
+      Real.log (234849838184611 / 128000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((106849838184611 / 128000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (107569838184611 / 128000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1517290148843070986257 / 2500000000000000000000 : ℝ) ≤
+      Real.log (234849838184611 / 128000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 13
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (107569838184611 / 128000000000000 : ℝ) ≤
+      (-(434724923741976688371 / 2500000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (107569838184611 / 128000000000000 : ℝ) = (107569838184611 / 64000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (107569838184611 / 64000000000000 : ℝ) ≤
+            (1298143027008023311629 / 2500000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (9136543990028577 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (106849838184611 / 128000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (106849838184611 / 128000000000000 : ℝ) ≤ (456827199501429 / 500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (106849838184611 / 128000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1116 :
+    Real.arctan (761378665835715 / 1962112283010469 : ℝ) ≤
+      (925384943603547055683 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1115 :
+    Real.arctan (456827199501429 / 500000000000000 : ℝ) ≤
+      (2 * (925384943603547055683 / 2500000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (3386336849031407 / 2500000000000000 : ℝ) ≤
+          Real.sqrt (1 + (456827199501429 / 500000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (456827199501429 / 500000000000000 : ℝ) /
+                (1 + (3386336849031407 / 2500000000000000 : ℝ)) =
+              (761378665835715 / 1962112283010469 : ℝ)
+            by
+        norm_num];
+        exact hau1116)
+  have hat1 :
+    (Real.pi / 2 - (2 * (925384943603547055683 / 2500000000000000000000 : ℝ))) ≤
+      Real.arctan (500000000000000 / 456827199501429 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (500000000000000 / 456827199501429 : ℝ) =
+              (456827199501429 / 500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1115)
+  have hz1 :
+    (500000000000000 / 456827199501429 : ℝ) ≤
+      1 / Real.sqrt (106849838184611 / 128000000000000 : ℝ) := by
+    rw [show (500000000000000 / 456827199501429 : ℝ) = 1 / (456827199501429 / 500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 3045514663342859 : ℝ) ≤
+      (409521462399309001603 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (106849838184611 / 128000000000000 : ℝ) ≤
+      (250000000000000 / 3045514663342859 : ℝ) := by
+    rw [show
+        (250000000000000 / 3045514663342859 : ℝ) =
+          (3 / 40) / (9136543990028577 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (925384943603547055683 / 2500000000000000000000 : ℝ))) -
+        6 * (409521462399309001603 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (106849838184611 / 128000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (106849838184611 / 128000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((9136543990028577 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (925384943603547055683 / 2500000000000000000000 : ℝ))) -
+          6 * (409521462399309001603 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (106849838184611 / 128000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (106849838184611 / 128000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (106849838184611 / 128000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Table/V56.lean b/LeanPool/Zeta5Irrational/Table/V56.lean
new file mode 100644
index 0000000000..3d6a859f7e
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Table/V56.lean
@@ -0,0 +1,1266 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Energy
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.PotentialV
+import LeanPool.Zeta5Irrational.TableCheck
+import Mathlib.Analysis.Complex.ExponentialBounds
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # V56: certified bounds for the zeta(5) proof -/
+
+@[expose] public section
+
+open Finset
+
+namespace Zeta5Irrational
+
+lemma V_673 :
+    ((6122387374240496023273 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1623057085064450056693 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1837994838888139 / 2000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1858030802236809204597 / 5000000000000000000000 : ℝ))) -
+              6 * (407151059969191178087 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (54051600444471 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (54051600444471 / 64000000000000 : ℝ)) =
+      Real.log (118051600444471 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((54051600444471 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (54411600444471 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (6122387374240496023273 / 10000000000000000000000 : ℝ) ≤
+      Real.log (118051600444471 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 13
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (54411600444471 / 64000000000000 : ℝ) ≤
+      (-(1623057085064450056693 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (54411600444471 / 64000000000000 : ℝ) = (54411600444471 / 32000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (54411600444471 / 32000000000000 : ℝ) ≤
+            (5308414717935549943307 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1837994838888139 / 2000000000000000 : ℝ) ≤ Real.sqrt (54051600444471 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (54051600444471 / 64000000000000 : ℝ) ≤ (4594987097220349 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (54051600444471 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1118 :
+    Real.arctan (4594987097220349 / 11790722084110163 : ℝ) ≤
+      (1858030802236809204597 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1117 :
+    Real.arctan (4594987097220349 / 5000000000000000 : ℝ) ≤
+      (2 * (1858030802236809204597 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6790722084110163 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (4594987097220349 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (4594987097220349 / 5000000000000000 : ℝ) /
+                (1 + (6790722084110163 / 5000000000000000 : ℝ)) =
+              (4594987097220349 / 11790722084110163 : ℝ)
+            by
+        norm_num];
+        exact hau1118)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1858030802236809204597 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 4594987097220349 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 4594987097220349 : ℝ) =
+              (4594987097220349 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1117)
+  have hz1 :
+    (5000000000000000 / 4594987097220349 : ℝ) ≤
+      1 / Real.sqrt (54051600444471 / 64000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 4594987097220349 : ℝ) = 1 / (4594987097220349 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (150000000000000 / 1837994838888139 : ℝ) ≤
+      (407151059969191178087 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (54051600444471 / 64000000000000 : ℝ) ≤
+      (150000000000000 / 1837994838888139 : ℝ) := by
+    rw [show
+        (150000000000000 / 1837994838888139 : ℝ) =
+          (3 / 40) / (1837994838888139 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1858030802236809204597 / 5000000000000000000000 : ℝ))) -
+        6 * (407151059969191178087 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (54051600444471 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (54051600444471 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1837994838888139 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1858030802236809204597 / 5000000000000000000000 : ℝ))) -
+          6 * (407151059969191178087 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (54051600444471 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (54051600444471 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (54051600444471 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_674 :
+    ((389249904457505137419 / 625000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(1395321619366648151237 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((9295913345121237 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (1872313611514181767439 / 5000000000000000000000 : ℝ))) -
+              6 * (805062351806162380989 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (27652481574401 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (27652481574401 / 32000000000000 : ℝ)) =
+      Real.log (59652481574401 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((27652481574401 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (27832481574401 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (389249904457505137419 / 625000000000000000000 : ℝ) ≤
+      Real.log (59652481574401 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 13
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (27832481574401 / 32000000000000 : ℝ) ≤
+      (-(1395321619366648151237 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (27832481574401 / 32000000000000 : ℝ) = (27832481574401 / 16000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 12
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (27832481574401 / 16000000000000 : ℝ) ≤
+            (5536150183633351848763 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (9295913345121237 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (27652481574401 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (27652481574401 / 32000000000000 : ℝ) ≤ (232397833628031 / 250000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (27652481574401 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1120 :
+    Real.arctan (4647956672560620 / 11826675708571543 : ℝ) ≤
+      (1872313611514181767439 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1119 :
+    Real.arctan (232397833628031 / 250000000000000 : ℝ) ≤
+      (2 * (1872313611514181767439 / 5000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (6826675708571543 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (232397833628031 / 250000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (232397833628031 / 250000000000000 : ℝ) /
+                (1 + (6826675708571543 / 5000000000000000 : ℝ)) =
+              (4647956672560620 / 11826675708571543 : ℝ)
+            by
+        norm_num];
+        exact hau1120)
+  have hat1 :
+    (Real.pi / 2 - (2 * (1872313611514181767439 / 5000000000000000000000 : ℝ))) ≤
+      Real.arctan (250000000000000 / 232397833628031 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (250000000000000 / 232397833628031 : ℝ) =
+              (232397833628031 / 250000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1119)
+  have hz1 :
+    (250000000000000 / 232397833628031 : ℝ) ≤
+      1 / Real.sqrt (27652481574401 / 32000000000000 : ℝ) := by
+    rw [show (250000000000000 / 232397833628031 : ℝ) = 1 / (232397833628031 / 250000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 3098637781707079 : ℝ) ≤
+      (805062351806162380989 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (27652481574401 / 32000000000000 : ℝ) ≤
+      (250000000000000 / 3098637781707079 : ℝ) := by
+    rw [show
+        (250000000000000 / 3098637781707079 : ℝ) =
+          (3 / 40) / (9295913345121237 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (1872313611514181767439 / 5000000000000000000000 : ℝ))) -
+        6 * (805062351806162380989 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (27652481574401 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (27652481574401 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((9295913345121237 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (1872313611514181767439 / 5000000000000000000000 : ℝ))) -
+          6 * (805062351806162380989 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (27652481574401 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (27652481574401 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (27652481574401 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_675 :
+    ((6332505844415552817503 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(586328612943406901771 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((1880131741613021 / 2000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3772575766913640194333 / 10000000000000000000000 : ℝ))) -
+              6 * (796130138632003110727 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (56558325853133 / 64000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (56558325853133 / 64000000000000 : ℝ)) =
+      Real.log (120558325853133 / 64000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((56558325853133 / 64000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (56918325853133 / 64000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (6332505844415552817503 / 10000000000000000000000 : ℝ) ≤
+      Real.log (120558325853133 / 64000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 14
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (56918325853133 / 64000000000000 : ℝ) ≤
+      (-(586328612943406901771 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (56918325853133 / 64000000000000 : ℝ) = (56918325853133 / 32000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (56918325853133 / 32000000000000 : ℝ) ≤
+            (2879407288556593098229 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (1880131741613021 / 2000000000000000 : ℝ) ≤ Real.sqrt (56558325853133 / 64000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (56558325853133 / 64000000000000 : ℝ) ≤ (2350164677016277 / 2500000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (56558325853133 / 64000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1122 :
+    Real.arctan (9400658708065108 / 23724881935576725 : ℝ) ≤
+      (3772575766913640194333 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1121 :
+    Real.arctan (2350164677016277 / 2500000000000000 : ℝ) ≤
+      (2 * (3772575766913640194333 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (548995277423069 / 400000000000000 : ℝ) ≤
+          Real.sqrt (1 + (2350164677016277 / 2500000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (2350164677016277 / 2500000000000000 : ℝ) /
+                (1 + (548995277423069 / 400000000000000 : ℝ)) =
+              (9400658708065108 / 23724881935576725 : ℝ)
+            by
+        norm_num];
+        exact hau1122)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3772575766913640194333 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (2500000000000000 / 2350164677016277 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (2500000000000000 / 2350164677016277 : ℝ) =
+              (2350164677016277 / 2500000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1121)
+  have hz1 :
+    (2500000000000000 / 2350164677016277 : ℝ) ≤
+      1 / Real.sqrt (56558325853133 / 64000000000000 : ℝ) := by
+    rw [show
+        (2500000000000000 / 2350164677016277 : ℝ) = 1 / (2350164677016277 / 2500000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (150000000000000 / 1880131741613021 : ℝ) ≤
+      (796130138632003110727 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (56558325853133 / 64000000000000 : ℝ) ≤
+      (150000000000000 / 1880131741613021 : ℝ) := by
+    rw [show
+        (150000000000000 / 1880131741613021 : ℝ) =
+          (3 / 40) / (1880131741613021 / 2000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3772575766913640194333 / 10000000000000000000000 : ℝ))) -
+        6 * (796130138632003110727 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (56558325853133 / 64000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (56558325853133 / 64000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((1880131741613021 / 2000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3772575766913640194333 / 10000000000000000000000 : ℝ))) -
+          6 * (796130138632003110727 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (56558325853133 / 64000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (56558325853133 / 64000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (56558325853133 / 64000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_676 :
+    ((1287186464959551226363 / 2000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(954842976206395349259 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((9504249753191331 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (379993042391329462331 / 1000000000000000000000 : ℝ))) -
+              6 * (196872197076237575761 / 2500000000000000000000 : ℝ)))) ≤
+      Vfield (7226461069683 / 8000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (7226461069683 / 8000000000000 : ℝ)) =
+      Real.log (15226461069683 / 8000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((7226461069683 / 8000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (7271461069683 / 8000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1287186464959551226363 / 2000000000000000000000 : ℝ) ≤
+      Real.log (15226461069683 / 8000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 14
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (7271461069683 / 8000000000000 : ℝ) ≤
+      (-(954842976206395349259 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (7271461069683 / 8000000000000 : ℝ) = (7271461069683 / 4000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 13
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (7271461069683 / 4000000000000 : ℝ) ≤
+            (5976628826793604650741 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (9504249753191331 / 10000000000000000 : ℝ) ≤ Real.sqrt (7226461069683 / 8000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (7226461069683 / 8000000000000 : ℝ) ≤ (4752124876595667 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (7226461069683 / 8000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1124 :
+    Real.arctan (9504249753191334 / 23796041583404911 : ℝ) ≤
+      (379993042391329462331 / 1000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1123 :
+    Real.arctan (4752124876595667 / 5000000000000000 : ℝ) ≤
+      (2 * (379993042391329462331 / 1000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (13796041583404911 / 10000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (4752124876595667 / 5000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (4752124876595667 / 5000000000000000 : ℝ) /
+                (1 + (13796041583404911 / 10000000000000000 : ℝ)) =
+              (9504249753191334 / 23796041583404911 : ℝ)
+            by
+        norm_num];
+        exact hau1124)
+  have hat1 :
+    (Real.pi / 2 - (2 * (379993042391329462331 / 1000000000000000000000 : ℝ))) ≤
+      Real.arctan (5000000000000000 / 4752124876595667 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (5000000000000000 / 4752124876595667 : ℝ) =
+              (4752124876595667 / 5000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1123)
+  have hz1 :
+    (5000000000000000 / 4752124876595667 : ℝ) ≤
+      1 / Real.sqrt (7226461069683 / 8000000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 4752124876595667 : ℝ) = 1 / (4752124876595667 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (250000000000000 / 3168083251063777 : ℝ) ≤
+      (196872197076237575761 / 2500000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (7226461069683 / 8000000000000 : ℝ) ≤
+      (250000000000000 / 3168083251063777 : ℝ) := by
+    rw [show
+        (250000000000000 / 3168083251063777 : ℝ) =
+          (3 / 40) / (9504249753191331 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (379993042391329462331 / 1000000000000000000000 : ℝ))) -
+        6 * (196872197076237575761 / 2500000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (7226461069683 / 8000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (7226461069683 / 8000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((9504249753191331 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (379993042391329462331 / 1000000000000000000000 : ℝ))) -
+          6 * (196872197076237575761 / 2500000000000000000000 : ℝ))) ≤
+      Real.sqrt (7226461069683 / 8000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (7226461069683 / 8000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (7226461069683 / 8000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_677 :
+    ((6639630455055536452119 / 10000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(532950666319523470387 / 10000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((9708116285978029 / 10000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (770588915703197956467 / 2000000000000000000000 : ℝ))) -
+              6 * (385508999245285394263 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (30159206983063 / 32000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (30159206983063 / 32000000000000 : ℝ)) =
+      Real.log (62159206983063 / 32000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((30159206983063 / 32000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (30339206983063 / 32000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (6639630455055536452119 / 10000000000000000000000 : ℝ) ≤
+      Real.log (62159206983063 / 32000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 14
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (30339206983063 / 32000000000000 : ℝ) ≤
+      (-(532950666319523470387 / 10000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show
+          (30339206983063 / 32000000000000 : ℝ) = (30339206983063 / 16000000000000 : ℝ) / 2 ^ 1 by
+          norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (30339206983063 / 16000000000000 : ℝ) ≤
+            (6398521136680476529613 / 10000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (9708116285978029 / 10000000000000000 : ℝ) ≤
+      Real.sqrt (30159206983063 / 32000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (30159206983063 / 32000000000000 : ℝ) ≤ (606757267873627 / 625000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (30159206983063 / 32000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1126 :
+    Real.arctan (4854058142989016 / 11968635480172425 : ℝ) ≤
+      (770588915703197956467 / 2000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1125 :
+    Real.arctan (606757267873627 / 625000000000000 : ℝ) ≤
+      (2 * (770588915703197956467 / 2000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (278745419206897 / 200000000000000 : ℝ) ≤
+          Real.sqrt (1 + (606757267873627 / 625000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (606757267873627 / 625000000000000 : ℝ) /
+                (1 + (278745419206897 / 200000000000000 : ℝ)) =
+              (4854058142989016 / 11968635480172425 : ℝ)
+            by
+        norm_num];
+        exact hau1126)
+  have hat1 :
+    (Real.pi / 2 - (2 * (770588915703197956467 / 2000000000000000000000 : ℝ))) ≤
+      Real.arctan (625000000000000 / 606757267873627 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (625000000000000 / 606757267873627 : ℝ) =
+              (606757267873627 / 625000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1125)
+  have hz1 :
+    (625000000000000 / 606757267873627 : ℝ) ≤
+      1 / Real.sqrt (30159206983063 / 32000000000000 : ℝ) := by
+    rw [show (625000000000000 / 606757267873627 : ℝ) = 1 / (606757267873627 / 625000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 9708116285978029 : ℝ) ≤
+      (385508999245285394263 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (30159206983063 / 32000000000000 : ℝ) ≤
+      (750000000000000 / 9708116285978029 : ℝ) := by
+    rw [show
+        (750000000000000 / 9708116285978029 : ℝ) =
+          (3 / 40) / (9708116285978029 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (770588915703197956467 / 2000000000000000000000 : ℝ))) -
+        6 * (385508999245285394263 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (30159206983063 / 32000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (30159206983063 / 32000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((9708116285978029 / 10000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (770588915703197956467 / 2000000000000000000000 : ℝ))) -
+          6 * (385508999245285394263 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (30159206983063 / 32000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (30159206983063 / 32000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (30159206983063 / 32000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_678 :
+    ((3419630995230294538113 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (-(64069730353516965177 / 5000000000000000000000 : ℝ)) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((4953894434510747 / 5000000000000000 : ℝ) *
+            (Real.pi +
+                (Real.pi / 2 - (2 * (3903831421400330573611 / 10000000000000000000000 : ℝ))) -
+              6 * (377769635349620863439 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (15706284843697 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (15706284843697 / 16000000000000 : ℝ)) =
+      Real.log (31706284843697 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((15706284843697 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (15796284843697 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (3419630995230294538113 / 5000000000000000000000 : ℝ) ≤
+      Real.log (31706284843697 / 16000000000000 : ℝ) :=
+    log_ge_of_ge_one (by norm_num) 14
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hU2 :
+    Real.log (15796284843697 / 16000000000000 : ℝ) ≤
+      (-(64069730353516965177 / 5000000000000000000000 : ℝ)) := by
+    have h :=
+      log_le_of_split_neg 1
+        (show (15796284843697 / 16000000000000 : ℝ) = (15796284843697 / 8000000000000 : ℝ) / 2 ^ 1
+          by norm_num)
+        (by norm_num)
+        (log_le_of_ge_one (by norm_num) 14
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          Real.log (15796284843697 / 8000000000000 : ℝ) ≤
+            (3401666171146483034823 / 5000000000000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hs_lo :
+    (4953894434510747 / 5000000000000000 : ℝ) ≤ Real.sqrt (15706284843697 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (15706284843697 / 16000000000000 : ℝ) ≤
+      (9907788869021497 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (15706284843697 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hau1128 :
+    Real.arctan (9907788869021497 / 24077083514460878 : ℝ) ≤
+      (3903831421400330573611 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hau1127 :
+    Real.arctan (9907788869021497 / 10000000000000000 : ℝ) ≤
+      (2 * (3903831421400330573611 / 10000000000000000000000 : ℝ)) :=
+    arctan_le_half (by norm_num) (by norm_num)
+      (le_sqrt_of_sq_le (by norm_num) (by norm_num) :
+        (7038541757230439 / 5000000000000000 : ℝ) ≤
+          Real.sqrt (1 + (9907788869021497 / 10000000000000000 : ℝ) ^ 2))
+      (by
+        rw [show
+            (9907788869021497 / 10000000000000000 : ℝ) /
+                (1 + (7038541757230439 / 5000000000000000 : ℝ)) =
+              (9907788869021497 / 24077083514460878 : ℝ)
+            by
+        norm_num];
+        exact hau1128)
+  have hat1 :
+    (Real.pi / 2 - (2 * (3903831421400330573611 / 10000000000000000000000 : ℝ))) ≤
+      Real.arctan (10000000000000000 / 9907788869021497 : ℝ) :=
+    arctan_ge_inv (by norm_num)
+      (by
+        rw [show
+            (1 : ℝ) / (10000000000000000 / 9907788869021497 : ℝ) =
+              (9907788869021497 / 10000000000000000 : ℝ)
+            by
+        norm_num];
+        exact hau1127)
+  have hz1 :
+    (10000000000000000 / 9907788869021497 : ℝ) ≤
+      1 / Real.sqrt (15706284843697 / 16000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 9907788869021497 : ℝ) =
+          1 / (9907788869021497 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 4953894434510747 : ℝ) ≤
+      (377769635349620863439 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (15706284843697 / 16000000000000 : ℝ) ≤
+      (375000000000000 / 4953894434510747 : ℝ) := by
+    rw [show
+        (375000000000000 / 4953894434510747 : ℝ) =
+          (3 / 40) / (4953894434510747 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (Real.pi / 2 - (2 * (3903831421400330573611 / 10000000000000000000000 : ℝ))) -
+        6 * (377769635349620863439 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (15706284843697 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (15706284843697 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((4953894434510747 / 5000000000000000 : ℝ) *
+        (Real.pi + (Real.pi / 2 - (2 * (3903831421400330573611 / 10000000000000000000000 : ℝ))) -
+          6 * (377769635349620863439 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (15706284843697 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (15706284843697 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (15706284843697 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_679 :
+    ((3613476445799885952621 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (158852024015426556119 / 2500000000000000000000 : ℝ) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((5147761580900357 / 5000000000000000 : ℝ) *
+            (Real.pi + (2 * (481773868711436053381 / 1250000000000000000000 : ℝ)) -
+              6 * (727187456097735281609 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (4239911887007 / 4000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (4239911887007 / 4000000000000 : ℝ)) =
+      Real.log (8239911887007 / 4000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((4239911887007 / 4000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (4262411887007 / 4000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (3613476445799885952621 / 5000000000000000000000 : ℝ) ≤
+      Real.log (8239911887007 / 4000000000000 : ℝ) := by
+    have h :=
+      log_ge_of_split_pos 1
+        (show (8239911887007 / 4000000000000 : ℝ) = 2 ^ 1 * (8239911887007 / 8000000000000 : ℝ) by
+          norm_num)
+        (by norm_num)
+        (log_ge_of_ge_one (by norm_num) 4
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          (147740544299885952621 / 5000000000000000000000 : ℝ) ≤
+            Real.log (8239911887007 / 8000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hU2 :
+    Real.log (4262411887007 / 4000000000000 : ℝ) ≤
+      (158852024015426556119 / 2500000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 5
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hs_lo :
+    (5147761580900357 / 5000000000000000 : ℝ) ≤ Real.sqrt (4239911887007 / 4000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (4239911887007 / 4000000000000 : ℝ) ≤ (10295523161800717 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (4239911887007 / 4000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hal1129 :
+    (481773868711436053381 / 1250000000000000000000 : ℝ) ≤
+      Real.arctan (10000000000000000000000000000000 / 24648146516963879478122821351257 : ℝ) :=
+    arctan_ge_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (2 * (481773868711436053381 / 1250000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 10295523161800717 : ℝ) :=
+    arctan_ge_half (by norm_num)
+      (sqrt_le_of_sq_le (by norm_num) (by norm_num) :
+        Real.sqrt (1 + (10000000000000000 / 10295523161800717 : ℝ) ^ 2) ≤
+          (1394064500618621 / 1000000000000000 : ℝ))
+      (by
+        rw [show
+            (10000000000000000 / 10295523161800717 : ℝ) /
+                (1 + (1394064500618621 / 1000000000000000 : ℝ)) =
+              (10000000000000000000000000000000 / 24648146516963879478122821351257 : ℝ)
+            by
+        norm_num];
+        exact hal1129)
+  have hz1 :
+    (10000000000000000 / 10295523161800717 : ℝ) ≤
+      1 / Real.sqrt (4239911887007 / 4000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 10295523161800717 : ℝ) =
+          1 / (10295523161800717 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 5147761580900357 : ℝ) ≤
+      (727187456097735281609 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (4239911887007 / 4000000000000 : ℝ) ≤
+      (375000000000000 / 5147761580900357 : ℝ) := by
+    rw [show
+        (375000000000000 / 5147761580900357 : ℝ) =
+          (3 / 40) / (5147761580900357 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (2 * (481773868711436053381 / 1250000000000000000000 : ℝ)) -
+        6 * (727187456097735281609 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (4239911887007 / 4000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (4239911887007 / 4000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((5147761580900357 / 5000000000000000 : ℝ) *
+        (Real.pi + (2 * (481773868711436053381 / 1250000000000000000000 : ℝ)) -
+          6 * (727187456097735281609 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (4239911887007 / 4000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (4239911887007 / 4000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (4239911887007 / 4000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_680 :
+    ((1520034531928750963607 / 2000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (1344768187442564119861 / 10000000000000000000000 : ℝ) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((5334587942785359 / 5000000000000000 : ℝ) *
+            (Real.pi + (2 * (3765169602758050855633 / 10000000000000000000000 : ℝ)) -
+              6 * (21931411260193838793 / 312500000000000000000 : ℝ)))) ≤
+      Vfield (18213010252359 / 16000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (18213010252359 / 16000000000000 : ℝ)) =
+      Real.log (34213010252359 / 16000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((18213010252359 / 16000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (18303010252359 / 16000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (1520034531928750963607 / 2000000000000000000000 : ℝ) ≤
+      Real.log (34213010252359 / 16000000000000 : ℝ) := by
+    have h :=
+      log_ge_of_split_pos 1
+        (show
+          (34213010252359 / 16000000000000 : ℝ) = 2 ^ 1 * (34213010252359 / 32000000000000 : ℝ) by
+          norm_num)
+        (by norm_num)
+        (log_ge_of_ge_one (by norm_num) 5
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          (133740171328750963607 / 2000000000000000000000 : ℝ) ≤
+            Real.log (34213010252359 / 32000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hU2 :
+    Real.log (18303010252359 / 16000000000000 : ℝ) ≤
+      (1344768187442564119861 / 10000000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 6
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hs_lo :
+    (5334587942785359 / 5000000000000000 : ℝ) ≤ Real.sqrt (18213010252359 / 16000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (18213010252359 / 16000000000000 : ℝ) ≤
+      (10669175885570721 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (18213010252359 / 16000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hal1130 :
+    (3765169602758050855633 / 10000000000000000000000 : ℝ) ≤
+      Real.arctan (100000000000000000000000000000000 / 252921480211266056994548940029691 : ℝ) :=
+    arctan_ge_of_series (by norm_num) (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (2 * (3765169602758050855633 / 10000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 10669175885570721 : ℝ) :=
+    arctan_ge_half (by norm_num)
+      (sqrt_le_of_sq_le (by norm_num) (by norm_num) :
+        Real.sqrt (1 + (10000000000000000 / 10669175885570721 : ℝ) ^ 2) ≤
+          (13705812231788571 / 10000000000000000 : ℝ))
+      (by
+        rw [show
+            (10000000000000000 / 10669175885570721 : ℝ) /
+                (1 + (13705812231788571 / 10000000000000000 : ℝ)) =
+              (100000000000000000000000000000000 / 252921480211266056994548940029691 : ℝ)
+            by
+        norm_num];
+        exact hal1130)
+  have hz1 :
+    (10000000000000000 / 10669175885570721 : ℝ) ≤
+      1 / Real.sqrt (18213010252359 / 16000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 10669175885570721 : ℝ) =
+          1 / (10669175885570721 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (125000000000000 / 1778195980928453 : ℝ) ≤
+      (21931411260193838793 / 312500000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (18213010252359 / 16000000000000 : ℝ) ≤
+      (125000000000000 / 1778195980928453 : ℝ) := by
+    rw [show
+        (125000000000000 / 1778195980928453 : ℝ) =
+          (3 / 40) / (5334587942785359 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (2 * (3765169602758050855633 / 10000000000000000000000 : ℝ)) -
+        6 * (21931411260193838793 / 312500000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (18213010252359 / 16000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (18213010252359 / 16000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((5334587942785359 / 5000000000000000 : ℝ) *
+        (Real.pi + (2 * (3765169602758050855633 / 10000000000000000000000 : ℝ)) -
+          6 * (21931411260193838793 / 312500000000000000000 : ℝ))) ≤
+      Real.sqrt (18213010252359 / 16000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (18213010252359 / 16000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (18213010252359 / 16000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_681 :
+    ((3979981422045415289587 / 5000000000000000000000 : ℝ) -
+              6 * (3 / 40) * (1003562467551241527391 / 5000000000000000000000 : ℝ) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((11030178193452383 / 10000000000000000 : ℝ) *
+            (Real.pi + (2 * (1841128949775744109199 / 5000000000000000000000 : ℝ)) -
+              6 * (339453879744246898597 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (1946637295669 / 1600000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (1946637295669 / 1600000000000 : ℝ)) =
+      Real.log (3546637295669 / 1600000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((1946637295669 / 1600000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (1955637295669 / 1600000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (3979981422045415289587 / 5000000000000000000000 : ℝ) ≤
+      Real.log (3546637295669 / 1600000000000 : ℝ) := by
+    have h :=
+      log_ge_of_split_pos 1
+        (show (3546637295669 / 1600000000000 : ℝ) = 2 ^ 1 * (3546637295669 / 3200000000000 : ℝ) by
+          norm_num)
+        (by norm_num)
+        (log_ge_of_ge_one (by norm_num) 6
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          (514245520545415289587 / 5000000000000000000000 : ℝ) ≤
+            Real.log (3546637295669 / 3200000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hU2 :
+    Real.log (1955637295669 / 1600000000000 : ℝ) ≤
+      (1003562467551241527391 / 5000000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hs_lo :
+    (11030178193452383 / 10000000000000000 : ℝ) ≤ Real.sqrt (1946637295669 / 1600000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (1946637295669 / 1600000000000 : ℝ) ≤ (5515089096726193 / 5000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (1946637295669 / 1600000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hal1131 :
+    (1841128949775744109199 / 5000000000000000000000 : ℝ) ≤
+      Real.arctan (50000000000000000000000000000000 / 129592954174045864388334571323681 : ℝ) :=
+    arctan_ge_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (2 * (1841128949775744109199 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (5000000000000000 / 5515089096726193 : ℝ) :=
+    arctan_ge_half (by norm_num)
+      (sqrt_le_of_sq_le (by norm_num) (by norm_num) :
+        Real.sqrt (1 + (5000000000000000 / 5515089096726193 : ℝ) ^ 2) ≤
+          (13497889499368817 / 10000000000000000 : ℝ))
+      (by
+        rw [show
+            (5000000000000000 / 5515089096726193 : ℝ) /
+                (1 + (13497889499368817 / 10000000000000000 : ℝ)) =
+              (50000000000000000000000000000000 / 129592954174045864388334571323681 : ℝ)
+            by
+        norm_num];
+        exact hal1131)
+  have hz1 :
+    (5000000000000000 / 5515089096726193 : ℝ) ≤
+      1 / Real.sqrt (1946637295669 / 1600000000000 : ℝ) := by
+    rw [show
+        (5000000000000000 / 5515089096726193 : ℝ) = 1 / (5515089096726193 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 11030178193452383 : ℝ) ≤
+      (339453879744246898597 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (1946637295669 / 1600000000000 : ℝ) ≤
+      (750000000000000 / 11030178193452383 : ℝ) := by
+    rw [show
+        (750000000000000 / 11030178193452383 : ℝ) =
+          (3 / 40) / (11030178193452383 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (2 * (1841128949775744109199 / 5000000000000000000000 : ℝ)) -
+        6 * (339453879744246898597 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (1946637295669 / 1600000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (1946637295669 / 1600000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((11030178193452383 / 10000000000000000 : ℝ) *
+        (Real.pi + (2 * (1841128949775744109199 / 5000000000000000000000 : ℝ)) -
+          6 * (339453879744246898597 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (1946637295669 / 1600000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (1946637295669 / 1600000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (1946637295669 / 1600000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_682 :
+    ((216072312191625710833 / 250000000000000000000 : ℝ) -
+              6 * (3 / 40) * (3213177334616844766239 / 10000000000000000000000 : ℝ) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((5859433948417073 / 5000000000000000 : ℝ) *
+            (Real.pi + (2 * (1766052720082905933429 / 5000000000000000000000 : ℝ)) -
+              6 * (639121915477979050969 / 10000000000000000000000 : ℝ)))) ≤
+      Vfield (2746637295669 / 2000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (2746637295669 / 2000000000000 : ℝ)) =
+      Real.log (4746637295669 / 2000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((2746637295669 / 2000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (2757887295669 / 2000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (216072312191625710833 / 250000000000000000000 : ℝ) ≤
+      Real.log (4746637295669 / 2000000000000 : ℝ) := by
+    have h :=
+      log_ge_of_split_pos 1
+        (show (4746637295669 / 2000000000000 : ℝ) = 2 ^ 1 * (4746637295669 / 4000000000000 : ℝ) by
+          norm_num)
+        (by norm_num)
+        (log_ge_of_ge_one (by norm_num) 7
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          (42785517116625710833 / 250000000000000000000 : ℝ) ≤
+            Real.log (4746637295669 / 4000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hU2 :
+    Real.log (2757887295669 / 2000000000000 : ℝ) ≤
+      (3213177334616844766239 / 10000000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 9
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hs_lo :
+    (5859433948417073 / 5000000000000000 : ℝ) ≤ Real.sqrt (2746637295669 / 2000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (2746637295669 / 2000000000000 : ℝ) ≤ (11718867896834149 / 10000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (2746637295669 / 2000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hal1132 :
+    (1766052720082905933429 / 5000000000000000000000 : ℝ) ≤
+      Real.arctan (12500000000000000000000000000000 / 33905558639632100095102679201933 : ℝ) :=
+    arctan_ge_of_series (by norm_num) (by norm_num) 8
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (2 * (1766052720082905933429 / 5000000000000000000000 : ℝ)) ≤
+      Real.arctan (10000000000000000 / 11718867896834149 : ℝ) :=
+    arctan_ge_half (by norm_num)
+      (sqrt_le_of_sq_le (by norm_num) (by norm_num) :
+        Real.sqrt (1 + (10000000000000000 / 11718867896834149 : ℝ) ^ 2) ≤
+          (1643245229668617 / 1250000000000000 : ℝ))
+      (by
+        rw [show
+            (10000000000000000 / 11718867896834149 : ℝ) /
+                (1 + (1643245229668617 / 1250000000000000 : ℝ)) =
+              (12500000000000000000000000000000 / 33905558639632100095102679201933 : ℝ)
+            by
+        norm_num];
+        exact hal1132)
+  have hz1 :
+    (10000000000000000 / 11718867896834149 : ℝ) ≤
+      1 / Real.sqrt (2746637295669 / 2000000000000 : ℝ) := by
+    rw [show
+        (10000000000000000 / 11718867896834149 : ℝ) =
+          1 / (11718867896834149 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (375000000000000 / 5859433948417073 : ℝ) ≤
+      (639121915477979050969 / 10000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (2746637295669 / 2000000000000 : ℝ) ≤
+      (375000000000000 / 5859433948417073 : ℝ) := by
+    rw [show
+        (375000000000000 / 5859433948417073 : ℝ) =
+          (3 / 40) / (5859433948417073 / 5000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (2 * (1766052720082905933429 / 5000000000000000000000 : ℝ)) -
+        6 * (639121915477979050969 / 10000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (2746637295669 / 2000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (2746637295669 / 2000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((5859433948417073 / 5000000000000000 : ℝ) *
+        (Real.pi + (2 * (1766052720082905933429 / 5000000000000000000000 : ℝ)) -
+          6 * (639121915477979050969 / 10000000000000000000000 : ℝ))) ≤
+      Real.sqrt (2746637295669 / 2000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (2746637295669 / 2000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (2746637295669 / 2000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+lemma V_683 :
+    ((247074633638366187549 / 250000000000000000000 : ℝ) -
+              6 * (3 / 40) * (1052158574414174657667 / 2000000000000000000000 : ℝ) -
+            2 +
+          12 * (3 / 40) +
+        2 *
+          ((12987144889918067 / 10000000000000000 : ℝ) *
+            (Real.pi + (2 * (102527157563947062571 / 312500000000000000000 : ℝ)) -
+              6 * (288426718122331681813 / 5000000000000000000000 : ℝ)))) ≤
+      Vfield (6746637295669 / 4000000000000 : ℝ) := by
+  rw [Vfield_eq (by norm_num)]
+  have e1 :
+    Real.log (1 + (6746637295669 / 4000000000000 : ℝ)) =
+      Real.log (10746637295669 / 4000000000000 : ℝ) :=
+    by norm_num
+  have e2 :
+    Real.log ((6746637295669 / 4000000000000 : ℝ) + (3 / 40) ^ 2) =
+      Real.log (6769137295669 / 4000000000000 : ℝ) :=
+    by norm_num
+  have hL1 :
+    (247074633638366187549 / 250000000000000000000 : ℝ) ≤
+      Real.log (10746637295669 / 4000000000000 : ℝ) := by
+    have h :=
+      log_ge_of_split_pos 1
+        (show (10746637295669 / 4000000000000 : ℝ) = 2 ^ 1 * (10746637295669 / 8000000000000 : ℝ)
+          by norm_num)
+        (by norm_num)
+        (log_ge_of_ge_one (by norm_num) 8
+            (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num) :
+          (73787838563366187549 / 250000000000000000000 : ℝ) ≤
+            Real.log (10746637295669 / 8000000000000 : ℝ))
+    linarith only [h, Real.log_two_gt_d9]
+  have hU2 :
+    Real.log (6769137295669 / 4000000000000 : ℝ) ≤
+      (1052158574414174657667 / 2000000000000000000000 : ℝ) :=
+    log_le_of_ge_one (by norm_num) 12
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hs_lo :
+    (12987144889918067 / 10000000000000000 : ℝ) ≤ Real.sqrt (6746637295669 / 4000000000000 : ℝ) :=
+    le_sqrt_of_sq_le (by norm_num) (by norm_num)
+  have hs_hi :
+    Real.sqrt (6746637295669 / 4000000000000 : ℝ) ≤ (1298714488991807 / 1000000000000000 : ℝ) :=
+    sqrt_le_of_sq_le (by norm_num) (by norm_num)
+  have hspos : 0 < Real.sqrt (6746637295669 / 4000000000000 : ℝ) :=
+    Real.sqrt_pos.mpr (by norm_num)
+  have hal1133 :
+    (102527157563947062571 / 312500000000000000000 : ℝ) ≤
+      Real.arctan (5000000000000000000000000000000 / 14689088481056534722184183966923 : ℝ) :=
+    arctan_ge_of_series (by norm_num) (by norm_num) 7
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hat1 :
+    (2 * (102527157563947062571 / 312500000000000000000 : ℝ)) ≤
+      Real.arctan (1000000000000000 / 1298714488991807 : ℝ) :=
+    arctan_ge_half (by norm_num)
+      (sqrt_le_of_sq_le (by norm_num) (by norm_num) :
+        Real.sqrt (1 + (1000000000000000 / 1298714488991807 : ℝ) ^ 2) ≤
+          (6310483255222389 / 5000000000000000 : ℝ))
+      (by
+        rw [show
+            (1000000000000000 / 1298714488991807 : ℝ) /
+                (1 + (6310483255222389 / 5000000000000000 : ℝ)) =
+              (5000000000000000000000000000000 / 14689088481056534722184183966923 : ℝ)
+            by
+        norm_num];
+        exact hal1133)
+  have hz1 :
+    (1000000000000000 / 1298714488991807 : ℝ) ≤
+      1 / Real.sqrt (6746637295669 / 4000000000000 : ℝ) := by
+    rw [show
+        (1000000000000000 / 1298714488991807 : ℝ) = 1 / (1298714488991807 / 1000000000000000 : ℝ)
+        by
+    norm_num];
+    exact one_div_le_one_div_of_le hspos hs_hi
+  have hA1 := arctan_ge_of_le hz1 hat1
+  have hat2 :
+    Real.arctan (750000000000000 / 12987144889918067 : ℝ) ≤
+      (288426718122331681813 / 5000000000000000000000 : ℝ) :=
+    arctan_le_of_series (by norm_num) (by norm_num) 3
+      (by simp only [Finset.sum_range_succ, Finset.sum_range_zero]; norm_num)
+  have hz2 :
+    (3 / 40) / Real.sqrt (6746637295669 / 4000000000000 : ℝ) ≤
+      (750000000000000 / 12987144889918067 : ℝ) := by
+    rw [show
+        (750000000000000 / 12987144889918067 : ℝ) =
+          (3 / 40) / (12987144889918067 / 10000000000000000 : ℝ)
+        by
+    norm_num];
+    exact div_le_div_of_nonneg_left (by norm_num) (by norm_num) hs_lo
+  have hA2 := arctan_le_of_ge hz2 hat2
+  have hP :
+    (Real.pi + (2 * (102527157563947062571 / 312500000000000000000 : ℝ)) -
+        6 * (288426718122331681813 / 5000000000000000000000 : ℝ)) ≤
+      (Real.pi + Real.arctan (1 / Real.sqrt (6746637295669 / 4000000000000 : ℝ)) -
+        6 * Real.arctan ((3 / 40) / Real.sqrt (6746637295669 / 4000000000000 : ℝ))) :=
+    by linarith only [hA1, hA2]
+  have hprod :
+    ((12987144889918067 / 10000000000000000 : ℝ) *
+        (Real.pi + (2 * (102527157563947062571 / 312500000000000000000 : ℝ)) -
+          6 * (288426718122331681813 / 5000000000000000000000 : ℝ))) ≤
+      Real.sqrt (6746637295669 / 4000000000000 : ℝ) *
+        (Real.pi + Real.arctan (1 / Real.sqrt (6746637295669 / 4000000000000 : ℝ)) -
+          6 * Real.arctan ((3 / 40) / Real.sqrt (6746637295669 / 4000000000000 : ℝ))) :=
+    mul_le_mul hs_lo hP (by linarith only [Real.pi_gt_d20, Real.pi_lt_d20]) (Real.sqrt_nonneg _)
+  linarith only [e1, e2, hL1, hU2, hprod]
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/TableCheck.lean b/LeanPool/Zeta5Irrational/TableCheck.lean
new file mode 100644
index 0000000000..b9be58f6b5
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/TableCheck.lean
@@ -0,0 +1,338 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import LeanPool.Zeta5Irrational.Potential
+import LeanPool.Zeta5Irrational.Certificates
+import LeanPool.Zeta5Irrational.EnergyConst
+import Mathlib.Tactic.ReduceModChar
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+import Mathlib.Tactic.Positivity.Basic
+import Mathlib.Tactic.Ring.Basic
+
+/-! # Reduction of the potential inequality (6.2) to a finite check (Table 2)
+
+For each interval `[l, r]` of the partition of `(0, 2]`, the inequality `2Uρ(t) - V(t) ≤ M₀` on
+`[l, r]` follows from monotonicity of `Uρ` and `V` and one numerical inequality at the endpoints.
+The tail `t ≥ 2` is handled by the elementary bound (6.8).
+-/
+
+@[expose] public section
+
+open Filter Topology Set Finset
+
+namespace Zeta5Irrational
+
+lemma aρ_one_lt_qm : aρ 1 < qm := by norm_num [aρ, table1]
+
+lemma qp_lt_bρ_one : qp < bρ 1 := by norm_num [bρ, table1]
+
+lemma aρ_one_pos : 0 < aρ 1 := by norm_num [aρ, table1]
+
+/-- Intervals to the left of `a₁`: `Uρ` nonincreasing, `V` nonincreasing. -/
+lemma check_left {l r U V : ℝ} (hlr : l ≤ r) (hr : r ≤ aρ 1) (hU : Uρ l ≤ U) (hV : V ≤ Vfield r)
+    (hnum : 2 * U - V ≤ M0) : ∀ t, l ≤ t → t ≤ r → 0 < t → 2 * Uρ t - Vfield t ≤ M0 := by
+  intro t ht1 ht2 ht0
+  have h1 : Uρ t ≤ Uρ l :=
+    Uρ_antitoneOn (show l ∈ Iic (aρ 1) from hlr.trans hr) (show t ∈ Iic (aρ 1) from ht2.trans hr)
+      ht1
+  have h2 : Vfield r ≤ Vfield t :=
+    Vfield_antitoneOn ⟨ht0, ht2.trans (hr.trans aρ_one_lt_qm.le)⟩
+      ⟨ht0.trans_le ht2, hr.trans aρ_one_lt_qm.le⟩ ht2
+  linarith
+
+/-- Intervals to the right of `b₁`: `Uρ` nondecreasing, `V` nondecreasing. -/
+lemma check_right {l r U V : ℝ} (hl : bρ 1 ≤ l) (hlr : l ≤ r) (hU : Uρ r ≤ U) (hV : V ≤ Vfield l)
+    (hnum : 2 * U - V ≤ M0) : ∀ t, l ≤ t → t ≤ r → 0 < t → 2 * Uρ t - Vfield t ≤ M0 := by
+  intro t ht1 ht2 _
+  have h1 : Uρ t ≤ Uρ r :=
+    Uρ_monotoneOn (show t ∈ Ici (bρ 1) from hl.trans ht1) (show r ∈ Ici (bρ 1) from hl.trans hlr)
+      ht2
+  have h2 : Vfield l ≤ Vfield t :=
+    Vfield_monotoneOn (show l ∈ Ici qp from qp_lt_bρ_one.le.trans hl)
+      (show t ∈ Ici qp from qp_lt_bρ_one.le.trans (hl.trans ht1)) ht1
+  linarith
+
+/-- The constant value of `Uρ` on `[a₁, b₁]`. -/
+noncomputable def Uρmid : ℝ :=
+  ∑ j ∈ Icc 1 16, cρ j * Real.log ((bρ j - aρ j) / 4)
+
+/-- The interval `[a₁, q₋]`. -/
+lemma check_mid_left {U V : ℝ} (hU : Uρmid ≤ U) (hV : V ≤ Vfield qm) (hnum : 2 * U - V ≤ M0) :
+    ∀ t, aρ 1 ≤ t → t ≤ qm → 0 < t → 2 * Uρ t - Vfield t ≤ M0 := by
+  intro t ht1 ht2 _
+  have ht : t ∈ Icc (aρ 1) qm := ⟨ht1, ht2⟩
+  have hqmb : qm ≤ bρ 1 := by norm_num [bρ, table1]
+  rw [Uρ_of_mem ht.1 (ht.2.trans hqmb)]
+  have h2 : Vfield qm ≤ Vfield t :=
+    Vfield_antitoneOn ⟨aρ_one_pos.trans_le ht.1, ht.2⟩ ⟨by norm_num, le_rfl⟩ ht.2
+  change 2 * Uρmid - Vfield t ≤ M0
+  linarith
+
+/-- The interval `[q₊, b₁]`. -/
+lemma check_mid_right {U V : ℝ} (hU : Uρmid ≤ U) (hV : V ≤ Vfield qp) (hnum : 2 * U - V ≤ M0) :
+    ∀ t, qp ≤ t → t ≤ bρ 1 → 0 < t → 2 * Uρ t - Vfield t ≤ M0 := by
+  intro t ht1 ht2 _
+  have ht : t ∈ Icc qp (bρ 1) := ⟨ht1, ht2⟩
+  rw [Uρ_of_mem ((aρ_one_lt_qm.le.trans (by norm_num)).trans ht.1) ht.2]
+  have h2 : Vfield qp ≤ Vfield t :=
+    Vfield_monotoneOn (Set.mem_Ici.mpr le_rfl) (show t ∈ Ici qp from ht.1) ht.1
+  change 2 * Uρmid - Vfield t ≤ M0
+  linarith
+
+/-- The mixed lower bound for `V` on `[q₋, q₊]` (A.6):
+`V(t) ≥ log(1+q₋) - 6α log(q₊ + α²) - 2 + 12α + 2√q₊ · P(√q₋)`. -/
+lemma Vfield_ge_on_bracket {t : ℝ} (h1 : qm ≤ t) (h2 : t ≤ qp) :
+    Real.log (1 + qm) - 6 * (3 / 40) * Real.log (qp + (3 / 40) ^ 2) - 2 + 12 * (3 / 40) +
+        2 * Real.sqrt qp * Pfun (Real.sqrt qm) ≤
+      Vfield t := by
+  have ht : 0 < t := by
+    have : (0 : ℝ) < qm := by norm_num
+    linarith
+  rw [Vfield_eq_Φ ht]
+  unfold Φ
+  have hs1 : Real.sqrt qm ≤ Real.sqrt t := Real.sqrt_le_sqrt h1
+  have hs2 : Real.sqrt t ≤ Real.sqrt qp := Real.sqrt_le_sqrt h2
+  have hst : 0 < Real.sqrt t := Real.sqrt_pos.mpr ht
+  have hsq : Real.sqrt t ^ 2 = t := Real.sq_sqrt ht.le
+  have hP : Pfun (Real.sqrt qm) ≤ Pfun (Real.sqrt t) :=
+    Pfun_monotoneOn ⟨sqrt_qm_pos, sqrt_qm_le_y0⟩ ⟨hst, hs2.trans sqrt_qp_le_y0⟩ hs1
+  have hPneg : Pfun (Real.sqrt qm) < 0 := Pfun_sqrt_qm_neg
+  have hA : Real.log (1 + qm) ≤ Real.log (1 + Real.sqrt t ^ 2) := by
+    rw [hsq];
+    exact Real.log_le_log (by norm_num) (by linarith)
+  have hB : Real.log (Real.sqrt t ^ 2 + (3 / 40) ^ 2) ≤ Real.log (qp + (3 / 40) ^ 2) := by
+    rw [hsq]; exact Real.log_le_log (by positivity) (add_le_add h2 le_rfl)
+  have hC1 : Real.sqrt qp * Pfun (Real.sqrt qm) ≤ Real.sqrt t * Pfun (Real.sqrt qm) :=
+    mul_le_mul_of_nonpos_right hs2 hPneg.le
+  have hC2 : Real.sqrt t * Pfun (Real.sqrt qm) ≤ Real.sqrt t * Pfun (Real.sqrt t) :=
+    mul_le_mul_of_nonneg_left hP hst.le
+  have hB' :
+    -(6 * (3 / 40)) * Real.log (qp + (3 / 40) ^ 2) ≤
+      -(6 * (3 / 40)) * Real.log (Real.sqrt t ^ 2 + (3 / 40) ^ 2) :=
+    mul_le_mul_of_nonpos_left hB (by norm_num)
+  have key :=
+    add_le_add (add_le_add hA hB')
+      (mul_le_mul_of_nonneg_left (hC1.trans hC2) (by norm_num : (0 : ℝ) ≤ 2))
+  have e1 :
+    Real.log (1 + qm) - 6 * (3 / 40) * Real.log (qp + (3 / 40) ^ 2) - 2 + 12 * (3 / 40) +
+        2 * Real.sqrt qp * Pfun (Real.sqrt qm) =
+      (Real.log (1 + qm) + -(6 * (3 / 40)) * Real.log (qp + (3 / 40) ^ 2) +
+          2 * (Real.sqrt qp * Pfun (Real.sqrt qm))) +
+        (-2 + 12 * (3 / 40)) :=
+    by ring
+  have e2 :
+    Real.log (1 + Real.sqrt t ^ 2) - 6 * (3 / 40) * Real.log (Real.sqrt t ^ 2 + (3 / 40) ^ 2) -
+            2 +
+          12 * (3 / 40) +
+        2 * Real.sqrt t * Pfun (Real.sqrt t) =
+      (Real.log (1 + Real.sqrt t ^ 2) +
+            -(6 * (3 / 40)) * Real.log (Real.sqrt t ^ 2 + (3 / 40) ^ 2) +
+          2 * (Real.sqrt t * Pfun (Real.sqrt t))) +
+        (-2 + 12 * (3 / 40)) :=
+    by ring
+  rw [e1, e2]
+  exact add_le_add key le_rfl
+
+/-- The interval `[q₋, q₊]`. -/
+lemma check_mid {U V : ℝ} (hU : Uρmid ≤ U)
+    (hV :
+      V ≤
+        Real.log (1 + qm) - 6 * (3 / 40) * Real.log (qp + (3 / 40) ^ 2) - 2 + 12 * (3 / 40) +
+          2 * Real.sqrt qp * Pfun (Real.sqrt qm))
+    (hnum : 2 * U - V ≤ M0) : ∀ t, qm ≤ t → t ≤ qp → 0 < t → 2 * Uρ t - Vfield t ≤ M0 := by
+  intro t ht1 ht2 _
+  have ht : t ∈ Icc qm qp := ⟨ht1, ht2⟩
+  rw [Uρ_of_mem (aρ_one_lt_qm.le.trans ht.1) (ht.2.trans qp_lt_bρ_one.le)]
+  have := Vfield_ge_on_bracket ht.1 ht.2
+  change 2 * Uρmid - Vfield t ≤ M0
+  linarith
+
+/-! ### Wrappers for certified bounds -/
+
+/-- Upper bound for `log r`, `r ≥ 1`, from a numerical bound on the truncated series. -/
+lemma log_le_of_ge_one {r U : ℝ} (hr : 1 ≤ r) (m : ℕ)
+    (hU :
+      2 * ∑ k ∈ range m, (1 / (2 * (k : ℝ) + 1)) * ((r - 1) / (r + 1)) ^ (2 * k + 1) +
+          2 * ((r - 1) / (r + 1)) ^ (2 * m + 1) / ((2 * m + 1) * (1 - ((r - 1) / (r + 1)) ^ 2)) ≤
+        U) :
+    Real.log r ≤ U :=
+  (log_le_artanh hr m).trans hU
+
+/-- Lower bound for `log r`, `r ≥ 1`. -/
+lemma log_ge_of_ge_one {r L : ℝ} (hr : 1 ≤ r) (m : ℕ)
+    (hL : L ≤ 2 * ∑ k ∈ range m, (1 / (2 * (k : ℝ) + 1)) * ((r - 1) / (r + 1)) ^ (2 * k + 1)) :
+    L ≤ Real.log r :=
+  hL.trans (log_ge_artanh hr m)
+
+/-- Upper bound for `log w`, `0 < w ≤ 1`, via `log w = - log w⁻¹`. -/
+lemma log_le_of_le_one {w L : ℝ} (hw : 0 < w) (hw1 : w ≤ 1) (m : ℕ)
+    (hL :
+      L ≤ 2 * ∑ k ∈ range m, (1 / (2 * (k : ℝ) + 1)) * ((w⁻¹ - 1) / (w⁻¹ + 1)) ^ (2 * k + 1)) :
+    Real.log w ≤ -L := by
+  have h1 : 1 ≤ w⁻¹ := one_le_inv_iff₀.mpr ⟨hw, hw1⟩
+  have := log_ge_of_ge_one h1 m hL
+  rw [Real.log_inv] at this
+  linarith
+
+/-- Lower bound for `log w`, `0 < w ≤ 1`. -/
+lemma log_ge_of_le_one {w U : ℝ} (hw : 0 < w) (hw1 : w ≤ 1) (m : ℕ)
+    (hU :
+      2 * ∑ k ∈ range m, (1 / (2 * (k : ℝ) + 1)) * ((w⁻¹ - 1) / (w⁻¹ + 1)) ^ (2 * k + 1) +
+          2 * ((w⁻¹ - 1) / (w⁻¹ + 1)) ^ (2 * m + 1) /
+            ((2 * m + 1) * (1 - ((w⁻¹ - 1) / (w⁻¹ + 1)) ^ 2)) ≤
+        U) :
+    -U ≤ Real.log w := by
+  have h1 : 1 ≤ w⁻¹ := one_le_inv_iff₀.mpr ⟨hw, hw1⟩
+  have := log_le_of_ge_one h1 m hU
+  rw [Real.log_inv] at this
+  linarith
+
+/-- `arctan z ≥ L` from a truncated alternating series, `0 ≤ z < 1`. -/
+lemma arctan_ge_of_series {z L : ℝ} (hz0 : 0 ≤ z) (hz1 : z < 1) (k : ℕ)
+    (hL : L ≤ ∑ i ∈ range (2 * k), (-1 : ℝ) ^ i * (z ^ (2 * i + 1) / (2 * (i : ℝ) + 1))) :
+    L ≤ Real.arctan z :=
+  hL.trans (arctan_ge_partial hz0 hz1 k)
+
+/-- `arctan z ≤ U` from a truncated alternating series, `0 ≤ z < 1`. -/
+lemma arctan_le_of_series {z U : ℝ} (hz0 : 0 ≤ z) (hz1 : z < 1) (k : ℕ)
+    (hU : ∑ i ∈ range (2 * k + 1), (-1 : ℝ) ^ i * (z ^ (2 * i + 1) / (2 * (i : ℝ) + 1)) ≤ U) :
+    Real.arctan z ≤ U :=
+  (arctan_le_partial hz0 hz1 k).trans hU
+
+/--
+Half-angle, lower bound: `2 L ≤ arctan z` if `L ≤ arctan (z / (1 + hi))` with `√(1+z²) ≤ hi`. -/
+lemma arctan_ge_half {z hi L : ℝ} (hz : 0 ≤ z) (hhi : Real.sqrt (1 + z ^ 2) ≤ hi)
+    (hL : L ≤ Real.arctan (z / (1 + hi))) : 2 * L ≤ Real.arctan z := by
+  rw [arctan_eq_two_mul_arctan hz]
+  have h : z / (1 + hi) ≤ z / (1 + Real.sqrt (1 + z ^ 2)) :=
+    div_le_div_of_nonneg_left hz (by positivity) (by linarith)
+  linarith [Real.arctan_strictMono.monotone h]
+
+/--
+Half-angle, upper bound: `arctan z ≤ 2 U` if `arctan (z / (1 + lo)) ≤ U` with `0 ≤ lo ≤ √(1+z²)`. -/
+lemma arctan_le_half {z lo U : ℝ} (hz : 0 ≤ z) (hlo0 : 0 ≤ lo) (hlo : lo ≤ Real.sqrt (1 + z ^ 2))
+    (hU : Real.arctan (z / (1 + lo)) ≤ U) : Real.arctan z ≤ 2 * U := by
+  rw [arctan_eq_two_mul_arctan hz]
+  have h : z / (1 + Real.sqrt (1 + z ^ 2)) ≤ z / (1 + lo) :=
+    div_le_div_of_nonneg_left hz (by linarith) (by linarith)
+  linarith [Real.arctan_strictMono.monotone h]
+
+/-- Inversion, lower bound: `π/2 - U ≤ arctan z` if `arctan (1/z) ≤ U`, `z > 0`. -/
+lemma arctan_ge_inv {z U : ℝ} (hz : 0 < z) (hU : Real.arctan (1 / z) ≤ U) :
+    Real.pi / 2 - U ≤ Real.arctan z := by
+  rw [one_div, Real.arctan_inv_of_pos hz] at hU
+  linarith
+
+/-- Inversion, upper bound: `arctan z ≤ π/2 - L` if `L ≤ arctan (1/z)`, `z > 0`. -/
+lemma arctan_le_inv {z L : ℝ} (hz : 0 < z) (hL : L ≤ Real.arctan (1 / z)) :
+    Real.arctan z ≤ Real.pi / 2 - L := by
+  rw [one_div, Real.arctan_inv_of_pos hz] at hL
+  linarith
+
+/-- Monotonicity transfer for arctan with a rational enclosure of the argument. -/
+lemma arctan_ge_of_le {x z L : ℝ} (hzx : z ≤ x) (hL : L ≤ Real.arctan z) : L ≤ Real.arctan x :=
+  hL.trans (Real.arctan_strictMono.monotone hzx)
+
+lemma arctan_le_of_ge {x z U : ℝ} (hxz : x ≤ z) (hU : Real.arctan z ≤ U) : Real.arctan x ≤ U :=
+  (Real.arctan_strictMono.monotone hxz).trans hU
+
+/-- The expanded form of `Uρ`. -/
+lemma Uρ_expand (x : ℝ) :
+    Uρ x =
+      cρ 1 * Uω (aρ 1) (bρ 1) x + cρ 2 * Uω (aρ 2) (bρ 2) x + cρ 3 * Uω (aρ 3) (bρ 3) x +
+                                cρ 4 * Uω (aρ 4) (bρ 4) x +
+                              cρ 5 * Uω (aρ 5) (bρ 5) x +
+                            cρ 6 * Uω (aρ 6) (bρ 6) x +
+                          cρ 7 * Uω (aρ 7) (bρ 7) x +
+                        cρ 8 * Uω (aρ 8) (bρ 8) x +
+                      cρ 9 * Uω (aρ 9) (bρ 9) x +
+                    cρ 10 * Uω (aρ 10) (bρ 10) x +
+                  cρ 11 * Uω (aρ 11) (bρ 11) x +
+                cρ 12 * Uω (aρ 12) (bρ 12) x +
+              cρ 13 * Uω (aρ 13) (bρ 13) x +
+            cρ 14 * Uω (aρ 14) (bρ 14) x +
+          cρ 15 * Uω (aρ 15) (bρ 15) x +
+        cρ 16 * Uω (aρ 16) (bρ 16) x := by
+  unfold Uρ
+  simp only [Finset.sum_Icc_succ_top (show 1 ≤ 16 by norm_num),
+    Finset.sum_Icc_succ_top (show 1 ≤ 15 by norm_num),
+    Finset.sum_Icc_succ_top (show 1 ≤ 14 by norm_num),
+    Finset.sum_Icc_succ_top (show 1 ≤ 13 by norm_num),
+    Finset.sum_Icc_succ_top (show 1 ≤ 12 by norm_num),
+    Finset.sum_Icc_succ_top (show 1 ≤ 11 by norm_num),
+    Finset.sum_Icc_succ_top (show 1 ≤ 10 by norm_num),
+    Finset.sum_Icc_succ_top (show 1 ≤ 9 by norm_num),
+    Finset.sum_Icc_succ_top (show 1 ≤ 8 by norm_num),
+    Finset.sum_Icc_succ_top (show 1 ≤ 7 by norm_num),
+    Finset.sum_Icc_succ_top (show 1 ≤ 6 by norm_num),
+    Finset.sum_Icc_succ_top (show 1 ≤ 5 by norm_num),
+    Finset.sum_Icc_succ_top (show 1 ≤ 4 by norm_num),
+    Finset.sum_Icc_succ_top (show 1 ≤ 3 by norm_num),
+    Finset.sum_Icc_succ_top (show 1 ≤ 2 by norm_num), Finset.Icc_self, Finset.sum_singleton]
+
+/-- The same expansion of `Uρmid`. -/
+lemma Uρmid_expand :
+    Uρmid =
+      cρ 1 * Real.log ((bρ 1 - aρ 1) / 4) + cρ 2 * Real.log ((bρ 2 - aρ 2) / 4) +
+                                  cρ 3 * Real.log ((bρ 3 - aρ 3) / 4) +
+                                cρ 4 * Real.log ((bρ 4 - aρ 4) / 4) +
+                              cρ 5 * Real.log ((bρ 5 - aρ 5) / 4) +
+                            cρ 6 * Real.log ((bρ 6 - aρ 6) / 4) +
+                          cρ 7 * Real.log ((bρ 7 - aρ 7) / 4) +
+                        cρ 8 * Real.log ((bρ 8 - aρ 8) / 4) +
+                      cρ 9 * Real.log ((bρ 9 - aρ 9) / 4) +
+                    cρ 10 * Real.log ((bρ 10 - aρ 10) / 4) +
+                  cρ 11 * Real.log ((bρ 11 - aρ 11) / 4) +
+                cρ 12 * Real.log ((bρ 12 - aρ 12) / 4) +
+              cρ 13 * Real.log ((bρ 13 - aρ 13) / 4) +
+            cρ 14 * Real.log ((bρ 14 - aρ 14) / 4) +
+          cρ 15 * Real.log ((bρ 15 - aρ 15) / 4) +
+        cρ 16 * Real.log ((bρ 16 - aρ 16) / 4) := by
+  unfold Uρmid
+  simp only [Finset.sum_Icc_succ_top (show 1 ≤ 16 by norm_num),
+    Finset.sum_Icc_succ_top (show 1 ≤ 15 by norm_num),
+    Finset.sum_Icc_succ_top (show 1 ≤ 14 by norm_num),
+    Finset.sum_Icc_succ_top (show 1 ≤ 13 by norm_num),
+    Finset.sum_Icc_succ_top (show 1 ≤ 12 by norm_num),
+    Finset.sum_Icc_succ_top (show 1 ≤ 11 by norm_num),
+    Finset.sum_Icc_succ_top (show 1 ≤ 10 by norm_num),
+    Finset.sum_Icc_succ_top (show 1 ≤ 9 by norm_num),
+    Finset.sum_Icc_succ_top (show 1 ≤ 8 by norm_num),
+    Finset.sum_Icc_succ_top (show 1 ≤ 7 by norm_num),
+    Finset.sum_Icc_succ_top (show 1 ≤ 6 by norm_num),
+    Finset.sum_Icc_succ_top (show 1 ≤ 5 by norm_num),
+    Finset.sum_Icc_succ_top (show 1 ≤ 4 by norm_num),
+    Finset.sum_Icc_succ_top (show 1 ≤ 3 by norm_num),
+    Finset.sum_Icc_succ_top (show 1 ≤ 2 by norm_num), Finset.Icc_self, Finset.sum_singleton]
+
+/-! ### Binary range reduction for logarithms -/
+
+lemma log_le_of_split_pos {r r' U : ℝ} (k : ℕ) (hr : r = 2 ^ k * r') (hr' : 0 < r')
+    (hU : Real.log r' ≤ U) : Real.log r ≤ k * Real.log 2 + U := by
+  rw [hr, Real.log_mul (by positivity) hr'.ne', Real.log_pow]
+  linarith
+
+lemma log_ge_of_split_pos {r r' L : ℝ} (k : ℕ) (hr : r = 2 ^ k * r') (hr' : 0 < r')
+    (hL : L ≤ Real.log r') : k * Real.log 2 + L ≤ Real.log r := by
+  rw [hr, Real.log_mul (by positivity) hr'.ne', Real.log_pow]
+  linarith
+
+lemma log_le_of_split_neg {r r' U : ℝ} (k : ℕ) (hr : r = r' / 2 ^ k) (hr' : 0 < r')
+    (hU : Real.log r' ≤ U) : Real.log r ≤ U - k * Real.log 2 := by
+  rw [hr, Real.log_div hr'.ne' (by positivity), Real.log_pow]
+  linarith
+
+lemma log_ge_of_split_neg {r r' L : ℝ} (k : ℕ) (hr : r = r' / 2 ^ k) (hr' : 0 < r')
+    (hL : L ≤ Real.log r') : L - k * Real.log 2 ≤ Real.log r := by
+  rw [hr, Real.log_div hr'.ne' (by positivity), Real.log_pow]
+  linarith
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Weight.lean b/LeanPool/Zeta5Irrational/Weight.lean
new file mode 100644
index 0000000000..5e7f807d2e
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Weight.lean
@@ -0,0 +1,172 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import Mathlib.MeasureTheory.Measure.Haar.OfBasis
+import Mathlib.Analysis.Real.Pi.Bounds
+import Mathlib.Tactic.Polynomial.Basic
+import Mathlib.Tactic.ReduceModChar
+import Mathlib.Tactic.FieldSimp
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+import Mathlib.Tactic.Positivity.Basic
+import Mathlib.Tactic.Ring.Basic
+import Mathlib.Tactic.Ring.RingNF
+
+/-! # The weight `w` of (2.10)
+
+`w(y) = (2π)⁴ y⁵ / 12 · ∑_{ℓ ≥ 1} ℓ⁴ e^{-2πℓy}`, its positivity, the bound
+`w(y) ≤ (32 y + 11) e^{-π y}` for `y > 0` (a crude form of (6.11)), and measurability on `(0, ∞)`.
+-/
+
+@[expose] public section
+
+open MeasureTheory Set Filter Topology
+
+namespace Zeta5Irrational
+
+/-- The series `∑_{ℓ ≥ 1} ℓ⁴ e^{-2πℓy}`. -/
+noncomputable def wS (y : ℝ) : ℝ :=
+  ∑' ℓ : ℕ, ((ℓ : ℝ) + 1) ^ 4 * Real.exp (-(2 * Real.pi * ((ℓ : ℝ) + 1) * y))
+
+/-- The weight `w(y) = (2π)⁴ y⁵ / 12 · ∑_{ℓ ≥ 1} ℓ⁴ e^{-2πℓy}` of (2.10). -/
+noncomputable def w (y : ℝ) : ℝ :=
+  (2 * Real.pi) ^ 4 * y ^ 5 / 12 * wS y
+
+lemma summable_w_term {y : ℝ} (hy : 0 < y) :
+    Summable fun ℓ : ℕ => ((ℓ : ℝ) + 1) ^ 4 * Real.exp (-(2 * Real.pi * ((ℓ : ℝ) + 1) * y)) := by
+  have h := Real.summable_pow_mul_exp_neg_nat_mul 4 (r := 2 * Real.pi * y) (by positivity)
+  have h2 := (summable_nat_add_iff 1).mpr h
+  refine h2.congr fun ℓ => ?_
+  push_cast
+  ring_nf
+
+lemma wS_pos {y : ℝ} (hy : 0 < y) : 0 < wS y :=
+  (summable_w_term hy).tsum_pos (fun ℓ => by positivity) 0 (by positivity)
+
+lemma w_pos {y : ℝ} (hy : 0 < y) : 0 < w y := by unfold w; have := wS_pos hy; positivity
+
+/-- `1 / (1 - e^{-x}) ≤ 1 + 1 / x` for `x > 0`. -/
+lemma one_div_one_sub_exp_neg_le {x : ℝ} (hx : 0 < x) : 1 / (1 - Real.exp (-x)) ≤ 1 + 1 / x := by
+  have h1 : x + 1 ≤ Real.exp x := Real.add_one_le_exp x
+  have h2 : Real.exp (-x) = 1 / Real.exp x := by rw [Real.exp_neg, one_div]
+  have h3 : 0 < Real.exp x := Real.exp_pos x
+  have h4 : Real.exp (-x) < 1 := by rw [h2, div_lt_one h3]; linarith
+  rw [div_le_iff₀ (by linarith), h2]
+  have hid : (1 + 1 / x) * (1 - 1 / Real.exp x) = 1 + (Real.exp x - x - 1) / (x * Real.exp x) := by
+    field_simp
+    ring
+  rw [hid]
+  have : 0 ≤ (Real.exp x - x - 1) / (x * Real.exp x) := div_nonneg (by linarith) (by positivity)
+  linarith
+
+/-- The bound `wS y ≤ 24 (1 + 1/(πy)) e^{-πy} / (πy)⁴`. -/
+lemma wS_le {y : ℝ} (hy : 0 < y) :
+    wS y ≤ 24 / (Real.pi * y) ^ 4 * (1 + 1 / (Real.pi * y)) * Real.exp (-(Real.pi * y)) := by
+  have hpy : 0 < Real.pi * y := by positivity
+  set q : ℝ := Real.exp (-(Real.pi * y)) with hq
+  have hq0 : 0 ≤ q := (Real.exp_pos _).le
+  have hq1 : q < 1 := by
+    rw [hq, ← Real.exp_zero];
+    exact
+      Real.exp_lt_exp.mpr
+        (by linarith)
+          -- termwise bound
+  have hterm :
+    ∀ ℓ : ℕ,
+      ((ℓ : ℝ) + 1) ^ 4 * Real.exp (-(2 * Real.pi * ((ℓ : ℝ) + 1) * y)) ≤
+        24 / (Real.pi * y) ^ 4 * (q * q ^ ℓ) := by
+    intro ℓ
+    have hl : (0 : ℝ) ≤ (ℓ : ℝ) + 1 := by positivity
+    have hpow :=
+      Real.pow_div_factorial_le_exp (x := Real.pi * y * ((ℓ : ℝ) + 1)) (by positivity) 4
+    have hexp :
+      Real.exp (-(2 * Real.pi * ((ℓ : ℝ) + 1) * y)) =
+        Real.exp (-(Real.pi * y * ((ℓ : ℝ) + 1))) * (q * q ^ ℓ) :=
+      by rw [hq, ← Real.exp_nat_mul, ← Real.exp_add, ← Real.exp_add]; congr 1; ring
+    rw [hexp]
+    have hE :
+      Real.exp (-(Real.pi * y * ((ℓ : ℝ) + 1))) = 1 / Real.exp (Real.pi * y * ((ℓ : ℝ) + 1)) := by
+      rw [Real.exp_neg, one_div]
+    rw [hE]
+    have hEpos : 0 < Real.exp (Real.pi * y * ((ℓ : ℝ) + 1)) := Real.exp_pos _
+    have hqq : 0 ≤ q * q ^ ℓ := by positivity
+    have key :
+      ((ℓ : ℝ) + 1) ^ 4 * (1 / Real.exp (Real.pi * y * ((ℓ : ℝ) + 1))) ≤ 24 / (Real.pi * y) ^ 4 :=
+  by
+      rw [mul_one_div, div_le_div_iff₀ hEpos (by positivity)]
+      have :
+        (Real.pi * y * ((ℓ : ℝ) + 1)) ^ 4 / (4 : ℕ).factorial =
+          (Real.pi * y) ^ 4 * ((ℓ : ℝ) + 1) ^ 4 / 24 :=
+        by rw [Nat.factorial]; norm_num; ring
+      rw [this] at hpow
+      nlinarith [pow_pos hpy 4]
+    calc
+      ((ℓ : ℝ) + 1) ^ 4 * (1 / Real.exp (Real.pi * y * ((ℓ : ℝ) + 1)) * (q * q ^ ℓ)) =
+          (((ℓ : ℝ) + 1) ^ 4 * (1 / Real.exp (Real.pi * y * ((ℓ : ℝ) + 1)))) * (q * q ^ ℓ) :=
+        by ring
+      _ ≤ 24 / (Real.pi * y) ^ 4 * (q * q ^ ℓ) := mul_le_mul_of_nonneg_right key hqq
+  have hgeom : Summable fun ℓ : ℕ => 24 / (Real.pi * y) ^ 4 * (q * q ^ ℓ) :=
+    ((summable_geometric_of_lt_one hq0 hq1).mul_left q).mul_left _
+  calc
+    wS y ≤ ∑' ℓ : ℕ, 24 / (Real.pi * y) ^ 4 * (q * q ^ ℓ) :=
+      (summable_w_term hy).tsum_le_tsum hterm hgeom
+    _ = 24 / (Real.pi * y) ^ 4 * (q * (1 / (1 - q))) := by
+      rw [tsum_mul_left, tsum_mul_left, tsum_geometric_of_lt_one hq0 hq1, one_div]
+    _ ≤ 24 / (Real.pi * y) ^ 4 * (1 + 1 / (Real.pi * y)) * q := by
+      have h := one_div_one_sub_exp_neg_le hpy
+      rw [← hq] at h
+      have h24 : 0 ≤ 24 / (Real.pi * y) ^ 4 := by positivity
+      calc
+        24 / (Real.pi * y) ^ 4 * (q * (1 / (1 - q))) ≤
+            24 / (Real.pi * y) ^ 4 * (q * (1 + 1 / (Real.pi * y))) :=
+          by gcongr
+        _ = 24 / (Real.pi * y) ^ 4 * (1 + 1 / (Real.pi * y)) * q := by ring
+
+/-- The bound `w(y) ≤ (32 y + 11) e^{-πy}` for `y > 0`. -/
+lemma w_le {y : ℝ} (hy : 0 < y) : w y ≤ (32 * y + 11) * Real.exp (-(Real.pi * y)) := by
+  unfold w
+  have h := wS_le hy
+  have hpi := Real.pi_gt_three
+  have hpy : 0 < Real.pi * y := by positivity
+  have hc : 0 ≤ (2 * Real.pi) ^ 4 * y ^ 5 / 12 := by positivity
+  calc
+    (2 * Real.pi) ^ 4 * y ^ 5 / 12 * wS y ≤
+        (2 * Real.pi) ^ 4 * y ^ 5 / 12 *
+          (24 / (Real.pi * y) ^ 4 * (1 + 1 / (Real.pi * y)) * Real.exp (-(Real.pi * y))) :=
+      mul_le_mul_of_nonneg_left h hc
+    _ = (32 * y + 32 / Real.pi) * Real.exp (-(Real.pi * y)) := by
+      field_simp
+      ring
+    _ ≤ (32 * y + 11) * Real.exp (-(Real.pi * y)) := by
+      have : 32 / Real.pi ≤ 11 := by rw [div_le_iff₀ (by positivity)]; linarith
+      have he : 0 ≤ Real.exp (-(Real.pi * y)) := (Real.exp_pos _).le
+      nlinarith
+
+/-- `wS` is a.e.-measurable on `(0, ∞)`, as the pointwise limit of continuous partial sums. -/
+lemma aemeasurable_wS : AEMeasurable wS (volume.restrict (Ioi 0)) := by
+  refine
+    aemeasurable_of_tendsto_metrizable_ae' (f := fun n y =>
+      ∑ ℓ ∈ Finset.range n, ((ℓ : ℝ) + 1) ^ 4 * Real.exp (-(2 * Real.pi * ((ℓ : ℝ) + 1) * y))) ?_
+      ?_
+  · intro n
+    refine (Continuous.measurable ?_).aemeasurable
+    exact
+      continuous_finsetSum _ fun ℓ _ =>
+        continuous_const.mul (Real.continuous_exp.comp (by fun_prop))
+  · refine (ae_restrict_iff' measurableSet_Ioi).mpr (Filter.Eventually.of_forall fun y hy => ?_)
+    exact (summable_w_term hy).hasSum.tendsto_sum_nat
+
+lemma aemeasurable_w : AEMeasurable w (volume.restrict (Ioi 0)) := by
+  unfold w
+  exact
+    (aemeasurable_const.mul (measurable_id.pow_const 5).aemeasurable |>.div_const 12).mul
+      aemeasurable_wS
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/ZeroMass.lean b/LeanPool/Zeta5Irrational/ZeroMass.lean
new file mode 100644
index 0000000000..242ed4fe1e
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/ZeroMass.lean
@@ -0,0 +1,396 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+import Mathlib.MeasureTheory.Measure.Lebesgue.Complex
+
+public import Mathlib.Analysis.CStarAlgebra.Classes
+public import Mathlib.LinearAlgebra.Complex.FiniteDimensional
+public import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
+import LeanPool.Zeta5Irrational.Gaussian
+import Mathlib.Tactic.Polynomial.Basic
+import Mathlib.Tactic.ReduceModChar
+import Mathlib.Tactic.FieldSimp
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.Positivity.Basic
+import Mathlib.Tactic.Ring.Basic
+
+import Mathlib.MeasureTheory.Integral.Prod
+import Mathlib.MeasureTheory.Integral.DominatedConvergence
+
+/-! # Lemma 6.2 (zero-mass logarithmic energy) for the truncated kernel
+
+We work with explicit "atoms": a finite family of continuous curves `γ k : ℝ → ℂ`
+(parametrised over `[0, 2π]`) with real weights `s k`.  The double integral of a kernel `f`
+against the atoms `k, l` is `pairInt γ f k l`, and the energy is
+`energy s γ f = ∑ k l, s k * s l * pairInt γ f k l`.
+
+For the truncated kernel `Ltr a b r = (1/2) ∫_a^b (e^{-s} - e^{-s r²})/s ds` and zero total mass
+`∑ s k = 0`, the energy is nonpositive (`energy_Ltr_nonpos`): this is the Gaussian
+positivity argument of the paper.
+-/
+
+@[expose] public section
+
+open MeasureTheory Set Real intervalIntegral
+
+namespace Zeta5Irrational
+
+/-- The truncated logarithmic kernel `L_{a,b}(r)`. -/
+noncomputable def Ltr (a b r : ℝ) : ℝ :=
+  (1 / 2) * ∫ s in a..b, (Real.exp (-s) - Real.exp (-s * r ^ 2)) / s
+
+variable {ι : Type*}
+
+/-- Double integral of a kernel against two atoms. -/
+noncomputable def pairInt (γ : ι → ℝ → ℂ) (f : ℂ → ℂ → ℝ) (k l : ι) : ℝ :=
+  ∫ θ in (0 : ℝ)..2 * π, ∫ φ in (0 : ℝ)..2 * π, f (γ k θ) (γ l φ)
+
+/-- The energy of the signed combination of atoms. -/
+noncomputable def energy [Fintype ι] (s : ι → ℝ) (γ : ι → ℝ → ℂ) (f : ℂ → ℂ → ℝ) : ℝ :=
+  ∑ k, ∑ l, s k * s l * pairInt γ f k l
+
+/-- Gaussian pair integral. -/
+noncomputable def Gk (γ : ι → ℝ → ℂ) (t : ℝ) (k l : ι) : ℝ :=
+  ∫ θ in (0 : ℝ)..2 * π, ∫ φ in (0 : ℝ)..2 * π, Real.exp (-t * ‖γ k θ - γ l φ‖ ^ 2)
+
+/-- Gaussian smoothing of one atom. -/
+noncomputable def gk (γ : ι → ℝ → ℂ) (t : ℝ) (k : ι) (u : ℂ) : ℝ :=
+  ∫ θ in (0 : ℝ)..2 * π, Real.exp (-(2 * t) * ‖γ k θ - u‖ ^ 2)
+
+lemma sq_sub_comm' (a b : ℝ) : (a - b) ^ 2 = (b - a) ^ 2 := by ring
+
+/-- Integrability of a Gaussian on `ℂ`. -/
+lemma integrable_gaussC {c : ℝ} (hc : 0 < c) (z : ℂ) :
+    Integrable (fun u : ℂ => Real.exp (-c * ‖z - u‖ ^ 2)) := by
+  have e :
+    (fun u : ℂ => Real.exp (-c * ‖z - u‖ ^ 2)) =
+      (fun p : ℝ × ℝ => Real.exp (-c * (p.1 - z.re) ^ 2) * Real.exp (-c * (p.2 - z.im) ^ 2)) ∘
+        Complex.measurableEquivRealProd := by
+    funext u
+    simp only [Function.comp_apply, Complex.measurableEquivRealProd_apply, normSq_eq,
+      Complex.sub_re, Complex.sub_im, ← Real.exp_add]
+    congr 1; ring
+  rw [e,
+    Complex.volume_preserving_equiv_real_prod.integrable_comp_emb
+      Complex.measurableEquivRealProd.measurableEmbedding,
+    Measure.volume_eq_prod]
+  exact (integrable_gaussian_sub hc z.re).mul_prod (integrable_gaussian_sub hc z.im)
+
+/-- `‖γ - u‖² ≥ ‖u‖²/2 - ‖γ‖²`, in exponential form. -/
+lemma exp_gauss_le {t R : ℝ} (ht : 0 ≤ t) {γ u : ℂ} (hγ : ‖γ‖ ≤ R) :
+    Real.exp (-(2 * t) * ‖γ - u‖ ^ 2) ≤ Real.exp (2 * t * R ^ 2) * Real.exp (-t * ‖u‖ ^ 2) := by
+  rw [← Real.exp_add]
+  apply Real.exp_le_exp.mpr
+  have h1 : ‖u‖ ≤ ‖γ - u‖ + ‖γ‖ := by
+    calc
+      ‖u‖ = ‖γ - (γ - u)‖ := by rw [sub_sub_cancel]
+      _ ≤ ‖γ‖ + ‖γ - u‖ := (norm_sub_le _ _)
+      _ = _ := add_comm _ _
+  have h2 : ‖u‖ ^ 2 ≤ 2 * ‖γ - u‖ ^ 2 + 2 * ‖γ‖ ^ 2 := by
+    have := mul_self_le_mul_self (norm_nonneg u) h1
+    nlinarith [sq_nonneg (‖γ - u‖ - ‖γ‖)]
+  have h3 : ‖γ‖ ^ 2 ≤ R ^ 2 := by nlinarith [norm_nonneg γ]
+  nlinarith
+
+/-- Fubini between an interval integral and an integral over `ℂ`, for a continuous integrand
+with a Gaussian bound in the complex variable. -/
+lemma swap_interval_complex {F : ℝ → ℂ → ℝ} (hF : Continuous (Function.uncurry F)) {c C : ℝ}
+    (hc : 0 < c) {a b : ℝ} (hab : a ≤ b)
+    (hbound : ∀ θ ∈ Icc a b, ∀ u, |F θ u| ≤ C * Real.exp (-c * ‖u‖ ^ 2)) :
+    ∫ θ in a..b, ∫ u : ℂ, F θ u = ∫ u : ℂ, ∫ θ in a..b, F θ u := by
+  rw [intervalIntegral.integral_of_le hab]
+  simp_rw [intervalIntegral.integral_of_le hab]
+  apply integral_integral_swap
+  have hg :
+    Integrable (fun p : ℝ × ℂ => C * Real.exp (-c * ‖(0 : ℂ) - p.2‖ ^ 2))
+      ((volume.restrict (Ioc a b)).prod volume) :=
+    (integrable_const C).mul_prod (integrable_gaussC hc 0)
+  refine hg.mono' hF.aestronglyMeasurable ?_
+  rw [Measure.restrict_prod_eq_prod_univ]
+  filter_upwards [ae_restrict_mem (measurableSet_Ioc.prod MeasurableSet.univ)] with p hp
+  simp only [Real.norm_eq_abs, zero_sub, norm_neg]
+  exact hbound p.1 (Ioc_subset_Icc_self hp.1) p.2
+
+lemma gk_nonneg (γ : ι → ℝ → ℂ) (t : ℝ) (k : ι) (u : ℂ) : 0 ≤ gk γ t k u :=
+  intervalIntegral.integral_nonneg (by positivity) fun _ _ => (Real.exp_pos _).le
+
+lemma continuous_gk {γ : ι → ℝ → ℂ} (hγ : ∀ k, Continuous (γ k)) (t : ℝ) (k : ι) :
+    Continuous (gk γ t k) := by
+  unfold gk
+  apply continuous_parametric_intervalIntegral_of_continuous'
+  fun_prop
+
+lemma gk_le_two_pi {γ : ι → ℝ → ℂ} (hγ : ∀ k, Continuous (γ k)) {t : ℝ} (ht : 0 ≤ t) (k : ι)
+    (u : ℂ) : gk γ t k u ≤ 2 * π := by
+  unfold gk
+  have :=
+    intervalIntegral.integral_mono_on (μ := volume) (a := 0) (b := 2 * π) (f := fun θ =>
+      Real.exp (-(2 * t) * ‖γ k θ - u‖ ^ 2)) (g := fun _ => (1 : ℝ)) (by positivity)
+      (by apply Continuous.intervalIntegrable; fun_prop) intervalIntegrable_const
+      (fun θ _ => by
+        apply Real.exp_le_one_iff.mpr
+        have : 0 ≤ ‖γ k θ - u‖ ^ 2 := by positivity
+        nlinarith)
+  simpa using this
+
+lemma gk_le {γ : ι → ℝ → ℂ} (hγ : ∀ k, Continuous (γ k)) {R : ℝ} (hR : ∀ k θ, ‖γ k θ‖ ≤ R) {t : ℝ}
+    (ht : 0 ≤ t) (k : ι) (u : ℂ) :
+    gk γ t k u ≤ 2 * π * Real.exp (2 * t * R ^ 2) * Real.exp (-t * ‖u‖ ^ 2) := by
+  unfold gk
+  have :=
+    intervalIntegral.integral_mono_on (μ := volume) (a := 0) (b := 2 * π) (f := fun θ =>
+      Real.exp (-(2 * t) * ‖γ k θ - u‖ ^ 2)) (g := fun _ =>
+      Real.exp (2 * t * R ^ 2) * Real.exp (-t * ‖u‖ ^ 2)) (by positivity)
+      (by apply Continuous.intervalIntegrable; fun_prop) intervalIntegrable_const
+      (fun θ _ => exp_gauss_le ht (hR k θ))
+  rw [intervalIntegral.integral_const] at this
+  simp only [sub_zero, smul_eq_mul] at this
+  linarith
+
+/-- The Gaussian pair integral as an integral over `ℂ` of a product of smoothed atoms. -/
+lemma Gk_eq {γ : ι → ℝ → ℂ} (hγ : ∀ k, Continuous (γ k)) {R : ℝ} (hR : ∀ k θ, ‖γ k θ‖ ≤ R) {t : ℝ}
+    (ht : 0 < t) (k l : ι) : Gk γ t k l = 4 * t / π * ∫ u : ℂ, gk γ t k u * gk γ t l u := by
+  have hconv :
+    ∀ z w : ℂ,
+      Real.exp (-t * ‖z - w‖ ^ 2) =
+        4 * t / π *
+          ∫ u : ℂ, Real.exp (-(2 * t) * ‖z - u‖ ^ 2) * Real.exp (-(2 * t) * ‖w - u‖ ^ 2) := by
+    intro z w
+    rw [gaussian_conv ht]
+    field_simp
+  unfold Gk
+  simp_rw [hconv]
+  simp_rw [intervalIntegral.integral_const_mul]
+  congr 1
+    -- inner swap
+  have hinner :
+    ∀ θ,
+      ∫ φ in (0 : ℝ)..2 * π,
+          ∫ u : ℂ, Real.exp (-(2 * t) * ‖γ k θ - u‖ ^ 2) * Real.exp (-(2 * t) * ‖γ l φ - u‖ ^ 2) =
+        ∫ u : ℂ, Real.exp (-(2 * t) * ‖γ k θ - u‖ ^ 2) * gk γ t l u := by
+    intro θ
+    rw [swap_interval_complex (c := t) (C := Real.exp (2 * t * R ^ 2)) (by fun_prop) ht
+        (by positivity)]
+    · unfold gk
+      congr 1; funext u
+      rw [intervalIntegral.integral_const_mul]
+    · intro φ _ u
+      rw [abs_of_nonneg (by positivity)]
+      calc
+        Real.exp (-(2 * t) * ‖γ k θ - u‖ ^ 2) * Real.exp (-(2 * t) * ‖γ l φ - u‖ ^ 2) ≤
+            1 * (Real.exp (2 * t * R ^ 2) * Real.exp (-t * ‖u‖ ^ 2)) := by
+          apply mul_le_mul _ (exp_gauss_le ht.le (hR l φ)) (Real.exp_pos _).le zero_le_one
+          apply Real.exp_le_one_iff.mpr
+          have : 0 ≤ ‖γ k θ - u‖ ^ 2 := by positivity
+          nlinarith
+        _ = _ := by ring
+  simp_rw [hinner]
+  rw [swap_interval_complex (c := t) (C := Real.exp (2 * t * R ^ 2) * (2 * π))
+      (by
+        have := continuous_gk hγ t l
+        fun_prop)
+      ht (by positivity)]
+  · unfold gk
+    congr 1; funext u
+    rw [intervalIntegral.integral_mul_const]
+  · intro θ _ u
+    rw [abs_of_nonneg (mul_nonneg (Real.exp_pos _).le (gk_nonneg γ t l u))]
+    calc
+      Real.exp (-(2 * t) * ‖γ k θ - u‖ ^ 2) * gk γ t l u ≤
+          (Real.exp (2 * t * R ^ 2) * Real.exp (-t * ‖u‖ ^ 2)) * (2 * π) :=
+        mul_le_mul (exp_gauss_le ht.le (hR k θ)) (gk_le_two_pi hγ ht.le l u) (gk_nonneg γ t l u)
+          (by positivity)
+      _ = _ := by ring
+
+/-- The Gaussian energy is nonnegative. -/
+lemma gaussian_energy_nonneg [Fintype ι] {s : ι → ℝ} {γ : ι → ℝ → ℂ} (hγ : ∀ k, Continuous (γ k))
+    {R : ℝ} (hR : ∀ k θ, ‖γ k θ‖ ≤ R) {t : ℝ} (ht : 0 < t) :
+    0 ≤ ∑ k, ∑ l, s k * s l * Gk γ t k l := by
+  simp_rw [Gk_eq hγ hR ht]
+  have hint : ∀ k l : ι, Integrable (fun u : ℂ => gk γ t k u * gk γ t l u) := by
+    intro k l
+    have hg :
+      Integrable
+        (fun u : ℂ =>
+          (2 * π * Real.exp (2 * t * R ^ 2) * (2 * π)) * Real.exp (-t * ‖(0 : ℂ) - u‖ ^ 2)) :=
+      (integrable_gaussC ht 0).const_mul _
+    refine hg.mono' ((continuous_gk hγ t k).mul (continuous_gk hγ t l)).aestronglyMeasurable ?_
+    filter_upwards with u
+    rw [Real.norm_eq_abs, abs_of_nonneg (mul_nonneg (gk_nonneg _ _ _ _) (gk_nonneg _ _ _ _))]
+    simp only [zero_sub, norm_neg]
+    calc
+      gk γ t k u * gk γ t l u ≤
+          (2 * π * Real.exp (2 * t * R ^ 2) * Real.exp (-t * ‖u‖ ^ 2)) * (2 * π) :=
+        mul_le_mul (gk_le hγ hR ht.le k u) (gk_le_two_pi hγ ht.le l u) (gk_nonneg _ _ _ _)
+          (by positivity)
+      _ = _ := by ring
+  have e :
+    ∑ k, ∑ l, s k * s l * (4 * t / π * ∫ u : ℂ, gk γ t k u * gk γ t l u) =
+      4 * t / π * ∫ u : ℂ, (∑ k, s k * gk γ t k u) ^ 2 := by
+    have :
+      ∀ u : ℂ, (∑ k, s k * gk γ t k u) ^ 2 = ∑ k, ∑ l, s k * s l * (gk γ t k u * gk γ t l u) := by
+      intro u
+      rw [sq, Finset.sum_mul_sum]
+      refine Finset.sum_congr rfl fun k _ => Finset.sum_congr rfl fun l _ => by ring
+    simp_rw [this]
+    rw [integral_finsetSum _
+        (fun k _ => integrable_finsetSum _ (fun l _ => (hint k l).const_mul _))]
+    simp_rw [integral_finsetSum _ (fun l _ => (hint _ l).const_mul _),
+      MeasureTheory.integral_const_mul, Finset.mul_sum]
+    refine Finset.sum_congr rfl fun k _ => Finset.sum_congr rfl fun l _ => by ring
+  rw [e]
+  exact mul_nonneg (by positivity) (integral_nonneg fun u => sq_nonneg _)
+
+/-! ### The truncated kernel -/
+
+lemma continuous_inv_max {a : ℝ} (ha : 0 < a) : Continuous fun s : ℝ => (max s a)⁻¹ :=
+  (continuous_id.max continuous_const).inv₀ fun _s => (lt_max_of_lt_right ha).ne'
+
+lemma Ltr_eq_max {a b : ℝ} (_ha : 0 < a) (hab : a ≤ b) (r : ℝ) :
+    Ltr a b r = (1 / 2) * ∫ s in a..b, (Real.exp (-s) - Real.exp (-s * r ^ 2)) * (max s a)⁻¹ := by
+  unfold Ltr
+  congr 1
+  apply intervalIntegral.integral_congr
+  intro s hs
+  rw [uIcc_of_le hab] at hs
+  simp only [max_eq_left hs.1, div_eq_mul_inv]
+
+/-- Continuity of the Gaussian pair integral in the parameter. -/
+lemma continuous_Gk {γ : ι → ℝ → ℂ} (hγ : ∀ k, Continuous (γ k)) (k l : ι) :
+    Continuous fun t => Gk γ t k l := by
+  unfold Gk
+  apply continuous_parametric_intervalIntegral_of_continuous'
+  apply continuous_parametric_intervalIntegral_of_continuous'
+  fun_prop
+
+/-- The truncated-kernel pair integral in terms of Gaussian pair integrals. -/
+lemma pairInt_Ltr {γ : ι → ℝ → ℂ} (hγ : ∀ k, Continuous (γ k)) {a b : ℝ} (ha : 0 < a)
+    (hab : a ≤ b) (k l : ι) :
+    pairInt γ (fun z w => Ltr a b ‖z - w‖) k l =
+      (1 / 2) * ∫ s in a..b, (Real.exp (-s) * (2 * π) ^ 2 - Gk γ s k l) / s := by
+  unfold pairInt
+  simp_rw [Ltr_eq_max ha hab, intervalIntegral.integral_const_mul]
+  congr 1
+  have hinv := continuous_inv_max ha
+  have hH :
+    ∀ θ,
+      Continuous
+        (Function.uncurry fun φ s =>
+          (Real.exp (-s) - Real.exp (-s * ‖γ k θ - γ l φ‖ ^ 2)) * (max s a)⁻¹) := by
+    intro θ
+    have h1 :
+      Continuous fun p : ℝ × ℝ => Real.exp (-p.2) - Real.exp (-p.2 * ‖γ k θ - γ l p.1‖ ^ 2) := by
+      fun_prop
+    exact h1.mul (hinv.comp continuous_snd)
+  have h1 :
+    ∀ θ,
+      (∫ φ in (0 : ℝ)..2 * π,
+          ∫ s in a..b, (Real.exp (-s) - Real.exp (-s * ‖γ k θ - γ l φ‖ ^ 2)) * (max s a)⁻¹) =
+        ∫ s in a..b,
+          ∫ φ in (0 : ℝ)..2 * π,
+            (Real.exp (-s) - Real.exp (-s * ‖γ k θ - γ l φ‖ ^ 2)) * (max s a)⁻¹ := by
+    intro θ
+    exact
+      intervalIntegral_swap_of_continuous (f := fun φ s =>
+        (Real.exp (-s) - Real.exp (-s * ‖γ k θ - γ l φ‖ ^ 2)) * (max s a)⁻¹) (hH θ)
+        (by positivity) hab
+  simp_rw [h1]
+  have hH2 :
+    Continuous
+      (Function.uncurry fun (p : ℝ × ℝ) φ =>
+        (Real.exp (-p.2) - Real.exp (-p.2 * ‖γ k p.1 - γ l φ‖ ^ 2)) * (max p.2 a)⁻¹) := by
+    have h1 :
+      Continuous fun q : (ℝ × ℝ) × ℝ =>
+        Real.exp (-q.1.2) - Real.exp (-q.1.2 * ‖γ k q.1.1 - γ l q.2‖ ^ 2) :=
+      by fun_prop
+    exact h1.mul (hinv.comp (continuous_snd.comp continuous_fst))
+  rw [intervalIntegral_swap_of_continuous (f := fun θ s =>
+      ∫ φ in (0 : ℝ)..2 * π, (Real.exp (-s) - Real.exp (-s * ‖γ k θ - γ l φ‖ ^ 2)) * (max s a)⁻¹)
+      (by exact continuous_parametric_intervalIntegral_of_continuous' hH2 0 (2 * π))
+      (by positivity) hab]
+  apply intervalIntegral.integral_congr
+  intro s hs
+  rw [uIcc_of_le hab] at hs
+  simp only
+  simp_rw [intervalIntegral.integral_mul_const]
+  rw [max_eq_left hs.1, div_eq_mul_inv]
+  congr 1
+  unfold Gk
+  have hi :
+    ∀ θ, IntervalIntegrable (fun φ => Real.exp (-s * ‖γ k θ - γ l φ‖ ^ 2)) volume 0 (2 * π) :=
+    fun θ => by apply Continuous.intervalIntegrable; fun_prop
+  simp_rw [intervalIntegral.integral_sub intervalIntegrable_const (hi _),
+    intervalIntegral.integral_const]
+  rw [intervalIntegral.integral_sub intervalIntegrable_const
+      (by
+        apply Continuous.intervalIntegrable
+        apply continuous_parametric_intervalIntegral_of_continuous'
+        fun_prop),
+    intervalIntegral.integral_const]
+  simp only [sub_zero, smul_eq_mul]
+  ring
+
+/-- **Lemma 6.2 for the truncated kernel**: the energy of a zero-mass combination of atoms
+with respect to `L_{a,b}` is nonpositive. -/
+theorem energy_Ltr_nonpos [Fintype ι] {s : ι → ℝ} {γ : ι → ℝ → ℂ} (hγ : ∀ k, Continuous (γ k))
+    {R : ℝ} (hR : ∀ k θ, ‖γ k θ‖ ≤ R) (hmass : ∑ k, s k = 0) {a b : ℝ} (ha : 0 < a)
+    (hab : a ≤ b) : energy s γ (fun z w => Ltr a b ‖z - w‖) ≤ 0 := by
+  unfold energy
+  simp_rw [pairInt_Ltr hγ ha hab]
+  have hc :
+    ∀ k l : ι,
+      ContinuousOn (fun t => (Real.exp (-t) * (2 * π) ^ 2 - Gk γ t k l) / t) (uIcc a b) := by
+    intro k l
+    rw [uIcc_of_le hab]
+    apply
+      ContinuousOn.div (Continuous.continuousOn (by have := continuous_Gk hγ k l; fun_prop))
+        continuousOn_id
+    intro t ht; exact (ha.trans_le ht.1).ne'
+  have e :
+    ∑ k,
+        ∑ l, s k * s l * ((1 / 2) * ∫ t in a..b, (Real.exp (-t) * (2 * π) ^ 2 - Gk γ t k l) / t) =
+      (1 / 2) *
+        ∫ t in a..b, ∑ k, ∑ l, s k * s l * ((Real.exp (-t) * (2 * π) ^ 2 - Gk γ t k l) / t) := by
+    rw [intervalIntegral.integral_finsetSum (f := fun k t =>
+        ∑ l, s k * s l * ((Real.exp (-t) * (2 * π) ^ 2 - Gk γ t k l) / t))
+        (fun k _ =>
+          (continuousOn_finsetSum _ fun l _ =>
+              continuousOn_const.mul (hc k l)).intervalIntegrable),
+      Finset.mul_sum]
+    refine Finset.sum_congr rfl fun k _ => ?_
+    rw [intervalIntegral.integral_finsetSum (f := fun l t =>
+        s k * s l * ((Real.exp (-t) * (2 * π) ^ 2 - Gk γ t k l) / t))
+        (fun l _ => (continuousOn_const.mul (hc k l)).intervalIntegrable),
+      Finset.mul_sum]
+    refine Finset.sum_congr rfl fun l _ => ?_
+    rw [intervalIntegral.integral_const_mul]; ring
+  rw [e]
+  have hpt :
+    ∀ t ∈ Icc a b,
+      0 ≤ -(∑ k, ∑ l, s k * s l * ((Real.exp (-t) * (2 * π) ^ 2 - Gk γ t k l) / t)) := by
+    intro t ht
+    have ht0 : 0 < t := ha.trans_le ht.1
+    have hJ := gaussian_energy_nonneg (s := s) hγ hR ht0
+    have hA : ∑ k, ∑ l, s k * s l * (Real.exp (-t) * (2 * π) ^ 2) = 0 := by
+      simp_rw [← Finset.sum_mul, ← Finset.mul_sum, ← Finset.sum_mul, hmass, zero_mul]
+    have :
+      ∑ k, ∑ l, s k * s l * ((Real.exp (-t) * (2 * π) ^ 2 - Gk γ t k l) / t) =
+        (∑ k, ∑ l, s k * s l * (Real.exp (-t) * (2 * π) ^ 2) - ∑ k, ∑ l, s k * s l * Gk γ t k l) /
+          t := by
+      rw [← Finset.sum_sub_distrib, Finset.sum_div]
+      refine Finset.sum_congr rfl fun k _ => ?_
+      rw [← Finset.sum_sub_distrib, Finset.sum_div]
+      refine Finset.sum_congr rfl fun l _ => ?_
+      ring
+    rw [this, hA, zero_sub, neg_nonneg]
+    exact div_nonpos_of_nonpos_of_nonneg (neg_nonpos.mpr hJ) ht0.le
+  have := intervalIntegral.integral_nonneg (μ := volume) hab hpt
+  rw [intervalIntegral.integral_neg] at this
+  linarith
+
+end Zeta5Irrational
diff --git a/LeanPool/Zeta5Irrational/Zeta5.lean b/LeanPool/Zeta5Irrational/Zeta5.lean
new file mode 100644
index 0000000000..f205de76ed
--- /dev/null
+++ b/LeanPool/Zeta5Irrational/Zeta5.lean
@@ -0,0 +1,33 @@
+/-
+Copyright (c) 2026 Moritz Firsching. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Moritz Firsching
+-/
+
+module
+
+public import Mathlib.NumberTheory.LSeries.RiemannZeta
+import Mathlib.Tactic.NormNum.Ineq
+import Mathlib.Tactic.NormNum.Inv
+import Mathlib.Tactic.NormNum.Pow
+
+/-! # `ζ(5)` as a real number
+-/
+
+@[expose] public section
+
+namespace Zeta5Irrational
+
+/-- `ζ(5)` as a real number. -/
+noncomputable def zeta5 : ℝ :=
+  ∑' n : ℕ, 1 / (n : ℝ) ^ 5
+
+/-- Mathlib's `riemannZeta 5` is the real number `zeta5`. -/
+theorem riemannZeta_five : riemannZeta 5 = (zeta5 : ℂ) := by
+  have h := zeta_nat_eq_tsum_of_gt_one (k := 5) (by norm_num)
+  rw [Nat.cast_ofNat] at h
+  rw [h, zeta5, Complex.ofReal_tsum]
+  push_cast
+  rfl
+
+end Zeta5Irrational
diff --git a/LeanPool/projects.yml b/LeanPool/projects.yml
index b284342cec..bdc7ab8604 100644
--- a/LeanPool/projects.yml
+++ b/LeanPool/projects.yml
@@ -1,4 +1,65 @@
 projects:
+  - slug: zeta-five-irrational
+    title: Irrationality of the Riemann zeta function at five
+    summary: >-
+      Proves that Mathlib's Riemann zeta function at five is an irrational
+      real number, following Aabir Fauzan's September 2026 preprint.
+      Constructs integer polynomials of degree 37n with positive values
+      at zeta(5) that decay exponentially in n squared. Includes a general
+      integer-polynomial irrationality criterion, Hankel determinant and
+      moment-integral identities, logarithmic-energy estimates, prime-adic
+      integrality bounds, and kernel-checked rational certificates. The
+      formalization uses a different normalizer and a positive decay
+      constant in place of the paper's specific constant 139/5. Reuses the
+      prime number theorem infrastructure already in the MooreBound project.
+    branch: number theory
+    entry_module: LeanPool.Zeta5Irrational
+    authors:
+      - Moritz Firsching
+    source:
+      title: Zeta5 formalization of Aabir Fauzan's zeta(5) is irrational
+      url: https://github.com/mo271/Zeta5/tree/f19a1960609f7d38e7b63fd2acb05e6f60a7b741
+      github_repo: mo271/Zeta5
+      commit: f19a1960609f7d38e7b63fd2acb05e6f60a7b741
+    license: Apache-2.0
+    status: verified
+    provenance: mix
+    main_declarations:
+      - Zeta5Irrational.irrational_five
+    main_results:
+      - declaration: Zeta5Irrational.irrational_five
+        informal: >-
+          There exists an irrational real number x whose complex coercion
+          equals Mathlib's Riemann zeta function evaluated at five.
+        source_ref: >-
+          Fauzan, zeta(5) is irrational, Theorem 1.1, September 17, 2026,
+          https://doi.org/10.5281/zenodo.22826419; upstream Apery/Main.lean.
+          The completed formalization was released on September 23, 2026.
+      - declaration: Zeta5Irrational.main_estimate
+        informal: >-
+          There is a positive real c such that, for all sufficiently large
+          n, an integer polynomial Q of degree 37n is a positive rational
+          multiple of the constructed determinant polynomial and satisfies
+          0 < Q(zeta(5)) < exp(-c*n^2).
+        source_ref: >-
+          Upstream Apery/MainEstimate.lean; variant of the paper's Theorem 2.1
+          with c = -800*(Aeff + U) instead of 139/5.
+      - declaration: Zeta5Irrational.irrational_of_eventually_exists_int_poly
+        informal: >-
+          A real xi is irrational if, eventually, integer polynomials of
+          degree at most d(n) have positive values at xi bounded by epsilon(n),
+          and b^d(n)*epsilon(n) tends to zero for every positive natural b.
+        source_ref: Upstream Apery/Criterion.lean; the paper's Section 1.1.
+    tags:
+      - irrationality
+      - zeta-function
+      - hankel-determinants
+      - potential-theory
+      - certified-computation
+    msc:
+      - 11J72
+      - 11M06
+      - 31A15
   - slug: block-spectral-sensitivity
     title: Block sensitivity can exceed spectral sensitivity squared
     summary: >-